Atlas › Modular forms › Weight 2 › Level 991 › 991.2.a

Newspace 991.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(991))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
991.2.a.a	33	degree 33	$33q - 5q^{2} - 6q^{3} + 25q^{4} - 24q^{5} - 18q^{6} - 9q^{7} + O(q^{8})$	+	$-1$
991.2.a.b	49	degree 49	$49q + 4q^{2} + 4q^{3} + 52q^{4} + 26q^{5} + 12q^{6} + 3q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
991.2.a.a	$x^{33} + 11x^{32} - 19x^{31} - 575x^{30} - 617x^{29} + 13030x^{28} + 26635x^{27} - 171254x^{26} - 431965x^{25} + 1478677x^{24} + 4024780x^{23} - 9035164x^{22} - 24024686x^{21} + 40979360x^{20} + 95361911x^{19} - 140820851x^{18} - 251587355x^{17} + 362269549x^{16} + 424348718x^{15} - 668217428x^{14} - 408179442x^{13} + 824221042x^{12} + 138808147x^{11} - 614290178x^{10} + 95530256x^{9} + 234053309x^{8} - 95077656x^{7} - 28851200x^{6} + 22719248x^{5} - 2769846x^{4} - 445600x^{3} + 63061x^{2} + 2431x - 169$
991.2.a.b	$x^{49} - 8x^{48} - 167x^{47} + 1482x^{46} + 11997x^{45} - 122937x^{44} - 477998x^{43} + 6056504x^{42} + 11043597x^{41} - 198270366x^{40} - 121924200x^{39} + 4575077640x^{38} - 756739998x^{37} - 77072586900x^{36} + 53938951963x^{35} + 969424731636x^{34} - 1044527079995x^{33} - 9242450464891x^{32} + 12428086356136x^{31} + 67504574903519x^{30} - 102934477744534x^{29} - 380714040736493x^{28} + 619450781800602x^{27} + 1668300728077565x^{26} - 2754374825166251x^{25} - 5705922388317379x^{24} + 9082863609046682x^{23} + 15258349253671926x^{22} - 22085144978989170x^{21} - 31785816214736454x^{20} + 38997867429696570x^{19} + 50930163611967636x^{18} - 48674159708753594x^{17} - 61201797026987042x^{16} + 41075950918389835x^{15} + 53024221054160421x^{14} - 21734025378112711x^{13} - 31417383233256855x^{12} + 6149170960198181x^{11} + 11938152478108160x^{10} - 408954290650682x^{9} - 2673825936081040x^{8} - 217073115631265x^{7} + 306781729923469x^{6} + 49257009369793x^{5} - 13024272957857x^{4} - 2739782193853x^{3} + 63502712325x^{2} + 24302281308x + 137189455$

Orbit

Characteristic polynomial of

T

991.2.a.a

x^{33} + 11x^{32} - 19x^{31} - 575x^{30} - 617x^{29} + 13030x^{28} + 26635x^{27} - 171254x^{26} - 431965x^{25} + 1478677x^{24} + 4024780x^{23} - 9035164x^{22} - 24024686x^{21} + 40979360x^{20} + 95361911x^{19} - 140820851x^{18} - 251587355x^{17} + 362269549x^{16} + 424348718x^{15} - 668217428x^{14} - 408179442x^{13} + 824221042x^{12} + 138808147x^{11} - 614290178x^{10} + 95530256x^{9} + 234053309x^{8} - 95077656x^{7} - 28851200x^{6} + 22719248x^{5} - 2769846x^{4} - 445600x^{3} + 63061x^{2} + 2431x - 169

991.2.a.b

x^{49} - 8x^{48} - 167x^{47} + 1482x^{46} + 11997x^{45} - 122937x^{44} - 477998x^{43} + 6056504x^{42} + 11043597x^{41} - 198270366x^{40} - 121924200x^{39} + 4575077640x^{38} - 756739998x^{37} - 77072586900x^{36} + 53938951963x^{35} + 969424731636x^{34} - 1044527079995x^{33} - 9242450464891x^{32} + 12428086356136x^{31} + 67504574903519x^{30} - 102934477744534x^{29} - 380714040736493x^{28} + 619450781800602x^{27} + 1668300728077565x^{26} - 2754374825166251x^{25} - 5705922388317379x^{24} + 9082863609046682x^{23} + 15258349253671926x^{22} - 22085144978989170x^{21} - 31785816214736454x^{20} + 38997867429696570x^{19} + 50930163611967636x^{18} - 48674159708753594x^{17} - 61201797026987042x^{16} + 41075950918389835x^{15} + 53024221054160421x^{14} - 21734025378112711x^{13} - 31417383233256855x^{12} + 6149170960198181x^{11} + 11938152478108160x^{10} - 408954290650682x^{9} - 2673825936081040x^{8} - 217073115631265x^{7} + 306781729923469x^{6} + 49257009369793x^{5} - 13024272957857x^{4} - 2739782193853x^{3} + 63502712325x^{2} + 24302281308x + 137189455

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(991))$	82
Old forms	0
Cusp forms $S_{2}(\Gamma_0(991))$	82
Eisenstein series $E_{2}(\Gamma_0(991))$	1
Modular forms $M_{2}(\Gamma_0(991))$	83

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.303 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 991]: [0, 83] (certified at ell = 2147476321)
T = [(2, 1), (3, 1)]; h of degree 82 = 1 x 82 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(991, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 991:2 1000

← 990.2.a · 992.2.a →