Atlas › Modular forms › Weight 6 › Level 109 › 109.6.a

Newspace 109.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(109))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
109.6.a.a	21	degree 21	$21q - 8q^{2} - 64q^{3} + 288q^{4} - 218q^{5} - 155q^{6} - 293q^{7} + O(q^{8})$	+	$-1$
109.6.a.b	24	degree 24	$24q + 12q^{2} + 44q^{3} + 432q^{4} + 182q^{5} + 349q^{6} + 99q^{7} + O(q^{8})$	−	$+1$

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 $T$
109.6.a.a	$x^{21} + 72x^{20} - 891x^{19} - 160958x^{18} - 1008373x^{17} + 144727188x^{16} + 1735713959x^{15} - 68291246184x^{14} - 942608162186x^{13} + 18806478611202x^{12} + 241628105742557x^{11} - 3222368801373630x^{10} - 29779318337915178x^{9} + 342877395517225252x^{8} + 1503249438271286683x^{7} - 18975206093165015550x^{6} - 14103577644763518096x^{5} + 412196823511775437220x^{4} - 595701134743005678980x^{3} - 674029858758527773472x^{2} + 214359555563686045520x - 13347600053314858208$
109.6.a.b	$x^{24} - 56x^{23} - 3511x^{22} + 236226x^{21} + 4328953x^{20} - 403005404x^{19} - 2044086111x^{18} + 372889654714x^{17} - 157251095636x^{16} - 210548840782972x^{15} + 582880923891653x^{14} + 76575860864381464x^{13} - 265632375138438416x^{12} - 18246472991531084388x^{11} + 58220224555327234611x^{10} + 2809862917492167539472x^{9} - 6576189382691140382800x^{8} - 266541616181812758583064x^{7} + 331703699452029390126896x^{6} + 14139224350257811346716160x^{5} - 3139065750762623138621184x^{4} - 350611611715677291444124416x^{3} - 102170448036277400403903744x^{2} + 3164003897449722928352100864x + 1398147460711239957746162688$

Orbit

Characteristic polynomial of

T

109.6.a.a

x^{21} + 72x^{20} - 891x^{19} - 160958x^{18} - 1008373x^{17} + 144727188x^{16} + 1735713959x^{15} - 68291246184x^{14} - 942608162186x^{13} + 18806478611202x^{12} + 241628105742557x^{11} - 3222368801373630x^{10} - 29779318337915178x^{9} + 342877395517225252x^{8} + 1503249438271286683x^{7} - 18975206093165015550x^{6} - 14103577644763518096x^{5} + 412196823511775437220x^{4} - 595701134743005678980x^{3} - 674029858758527773472x^{2} + 214359555563686045520x - 13347600053314858208

109.6.a.b

x^{24} - 56x^{23} - 3511x^{22} + 236226x^{21} + 4328953x^{20} - 403005404x^{19} - 2044086111x^{18} + 372889654714x^{17} - 157251095636x^{16} - 210548840782972x^{15} + 582880923891653x^{14} + 76575860864381464x^{13} - 265632375138438416x^{12} - 18246472991531084388x^{11} + 58220224555327234611x^{10} + 2809862917492167539472x^{9} - 6576189382691140382800x^{8} - 266541616181812758583064x^{7} + 331703699452029390126896x^{6} + 14139224350257811346716160x^{5} - 3139065750762623138621184x^{4} - 350611611715677291444124416x^{3} - 102170448036277400403903744x^{2} + 3164003897449722928352100864x + 1398147460711239957746162688

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(109))$	45
Old forms	0
Cusp forms $S_{6}(\Gamma_0(109))$	45
Eisenstein series $E_{6}(\Gamma_0(109))$	2
Modular forms $M_{6}(\Gamma_0(109))$	47

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.278 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, 109]: [1, 47] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (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(109, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 109:6 1000

← 108.6.a · 110.6.a →