Atlas › Modular forms › Weight 6 › Level 79 › 79.6.a

Newspace 79.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(79))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
79.6.a.a	14	degree 14	$14q - 8q^{2} - 37q^{3} + 192q^{4} - 201q^{5} - 159q^{6} - 157q^{7} + O(q^{8})$	+	$-1$
79.6.a.b	19	degree 19	$19q + 8q^{2} + 17q^{3} + 352q^{4} + 199q^{5} + 201q^{6} + 39q^{7} + O(q^{8})$	−	$+1$

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 $T$
79.6.a.a	$x^{14} + 45x^{13} - 956x^{12} - 55232x^{11} + 320017x^{10} + 24302749x^{9} - 69043350x^{8} - 4743555482x^{7} + 13321554304x^{6} + 411504328944x^{5} - 1521265929888x^{4} - 12781417798336x^{3} + 63423242231488x^{2} - 5920808092928x - 188859596097536$
79.6.a.b	$x^{19} - 25x^{18} - 3796x^{17} + 87572x^{16} + 5911257x^{15} - 118699364x^{14} - 5086486233x^{13} + 81870294760x^{12} + 2692793282610x^{11} - 30831667632887x^{10} - 904085700208615x^{9} + 6058750661275016x^{8} + 187607222495222040x^{7} - 454950677654686656x^{6} - 21960253942720551712x^{5} - 21276571156911119904x^{4} + 1168489366510480801856x^{3} + 3730518629110530777472x^{2} - 15428410480253436636672x - 60090647579363652243456$

Orbit

Characteristic polynomial of

T

79.6.a.a

x^{14} + 45x^{13} - 956x^{12} - 55232x^{11} + 320017x^{10} + 24302749x^{9} - 69043350x^{8} - 4743555482x^{7} + 13321554304x^{6} + 411504328944x^{5} - 1521265929888x^{4} - 12781417798336x^{3} + 63423242231488x^{2} - 5920808092928x - 188859596097536

79.6.a.b

x^{19} - 25x^{18} - 3796x^{17} + 87572x^{16} + 5911257x^{15} - 118699364x^{14} - 5086486233x^{13} + 81870294760x^{12} + 2692793282610x^{11} - 30831667632887x^{10} - 904085700208615x^{9} + 6058750661275016x^{8} + 187607222495222040x^{7} - 454950677654686656x^{6} - 21960253942720551712x^{5} - 21276571156911119904x^{4} + 1168489366510480801856x^{3} + 3730518629110530777472x^{2} - 15428410480253436636672x - 60090647579363652243456

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(79))$	33
Old forms	0
Cusp forms $S_{6}(\Gamma_0(79))$	33
Eisenstein series $E_{6}(\Gamma_0(79))$	2
Modular forms $M_{6}(\Gamma_0(79))$	35

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.246 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, 79]: [1, 35] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 6 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(79, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 79:6 1000

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