Atlas › Modular forms › Weight 2 › Level 907 › 907.2.a

Newspace 907.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(907))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
907.2.a.a	35	degree 35	$35q - 11q^{2} - 13q^{3} + 33q^{4} - 32q^{5} - 8q^{6} - 9q^{7} + O(q^{8})$	+	$-1$
907.2.a.b	40	degree 40	$40q + 9q^{2} + 9q^{3} + 41q^{4} + 32q^{5} + 8q^{6} + 7q^{7} + O(q^{8})$	−	$+1$

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 $T$
907.2.a.a	$x^{35} + 24x^{34} + 179x^{33} - 134x^{32} - 8203x^{31} - 28049x^{30} + 112787x^{29} + 827652x^{28} - 65266x^{27} - 10819454x^{26} - 14916130x^{25} + 78685703x^{24} + 192230325x^{23} - 330606103x^{22} - 1253585393x^{21} + 735285896x^{20} + 5033610931x^{19} - 426025111x^{18} - 13287254443x^{17} - 1716065843x^{16} + 23722929293x^{15} + 3997397786x^{14} - 28599392441x^{13} - 2766315067x^{12} + 22270776836x^{11} - 474862348x^{10} - 10190568684x^{9} + 1160328828x^{8} + 2530922795x^{7} - 411769952x^{6} - 303457548x^{5} + 49285123x^{4} + 13467097x^{3} - 1124818x^{2} - 161172x + 4792$
907.2.a.b	$x^{40} - 18x^{39} + 12x^{38} + 1612x^{37} - 7798x^{36} - 53489x^{35} + 452550x^{34} + 613879x^{33} - 12784203x^{32} + 8867260x^{31} + 213748620x^{30} - 409588525x^{29} - 2254714071x^{28} + 6676332596x^{27} + 15112581324x^{26} - 64496631229x^{25} - 60385119142x^{24} + 411638293761x^{23} + 99176513746x^{22} - 1808372829236x^{21} + 287955436780x^{20} + 5556032573757x^{19} - 2167330400564x^{18} - 11946076943105x^{17} + 6076557526133x^{16} + 17727993474801x^{15} - 9694031691242x^{14} - 17571556804232x^{13} + 9321497496307x^{12} + 10929268310246x^{11} - 5361732747689x^{10} - 3813362781411x^{9} + 1779012935785x^{8} + 608533323533x^{7} - 299200012603x^{6} - 24967191090x^{5} + 17231424608x^{4} - 22937076x^{3} - 308878180x^{2} + 9586136x - 58328$

Orbit

Characteristic polynomial of

T

907.2.a.a

x^{35} + 24x^{34} + 179x^{33} - 134x^{32} - 8203x^{31} - 28049x^{30} + 112787x^{29} + 827652x^{28} - 65266x^{27} - 10819454x^{26} - 14916130x^{25} + 78685703x^{24} + 192230325x^{23} - 330606103x^{22} - 1253585393x^{21} + 735285896x^{20} + 5033610931x^{19} - 426025111x^{18} - 13287254443x^{17} - 1716065843x^{16} + 23722929293x^{15} + 3997397786x^{14} - 28599392441x^{13} - 2766315067x^{12} + 22270776836x^{11} - 474862348x^{10} - 10190568684x^{9} + 1160328828x^{8} + 2530922795x^{7} - 411769952x^{6} - 303457548x^{5} + 49285123x^{4} + 13467097x^{3} - 1124818x^{2} - 161172x + 4792

907.2.a.b

x^{40} - 18x^{39} + 12x^{38} + 1612x^{37} - 7798x^{36} - 53489x^{35} + 452550x^{34} + 613879x^{33} - 12784203x^{32} + 8867260x^{31} + 213748620x^{30} - 409588525x^{29} - 2254714071x^{28} + 6676332596x^{27} + 15112581324x^{26} - 64496631229x^{25} - 60385119142x^{24} + 411638293761x^{23} + 99176513746x^{22} - 1808372829236x^{21} + 287955436780x^{20} + 5556032573757x^{19} - 2167330400564x^{18} - 11946076943105x^{17} + 6076557526133x^{16} + 17727993474801x^{15} - 9694031691242x^{14} - 17571556804232x^{13} + 9321497496307x^{12} + 10929268310246x^{11} - 5361732747689x^{10} - 3813362781411x^{9} + 1779012935785x^{8} + 608533323533x^{7} - 299200012603x^{6} - 24967191090x^{5} + 17231424608x^{4} - 22937076x^{3} - 308878180x^{2} + 9586136x - 58328

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(907))$	75
Old forms	0
Cusp forms $S_{2}(\Gamma_0(907))$	75
Eisenstein series $E_{2}(\Gamma_0(907))$	1
Modular forms $M_{2}(\Gamma_0(907))$	76

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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, 907]: [0, 76] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 75 = 1 x 75 (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(907, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 907:2 1000

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