Atlas › Modular forms › Weight 6 › Level 78 › 78.6.a

Newspace 78.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(78))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
78.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} - 9q^{3} + 16q^{4} + 4q^{5} + 36q^{6} - 8q^{7} + O(q^{8})$	+ + +	$-1$
78.6.a.b	1	$\mathbb{Q}$	$q - 4q^{2} + 9q^{3} + 16q^{4} + 24q^{5} - 36q^{6} - 238q^{7} + O(q^{8})$	+ − −	$-1$
78.6.a.c	1	$\mathbb{Q}$	$q - 4q^{2} + 9q^{3} + 16q^{4} + 76q^{5} - 36q^{6} + 100q^{7} + O(q^{8})$	+ − +	$+1$
78.6.a.d	1	$\mathbb{Q}$	$q + 4q^{2} - 9q^{3} + 16q^{4} + 4q^{5} - 36q^{6} - 146q^{7} + O(q^{8})$	− + −	$-1$
78.6.a.e	1	$\mathbb{Q}$	$q + 4q^{2} - 9q^{3} + 16q^{4} + 56q^{5} - 36q^{6} - 68q^{7} + O(q^{8})$	− + +	$+1$
78.6.a.f	1	$\mathbb{Q}$	$q + 4q^{2} + 9q^{3} + 16q^{4} - 88q^{5} + 36q^{6} - 176q^{7} + O(q^{8})$	− − +	$-1$
78.6.a.g	2	$\mathbb{Q}(\sqrt{433})$	$2q - 8q^{2} - 18q^{3} + 32q^{4} + 2q^{5} + 72q^{6} - 150q^{7} + O(q^{8})$	+ + −	$+1$
78.6.a.h	2	$\mathbb{Q}(\sqrt{3241})$	$2q + 8q^{2} + 18q^{3} + 32q^{4} + 10q^{5} + 72q^{6} + 138q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
78.6.a.a	$x + 20$
78.6.a.b	$x + 690$
78.6.a.c	$x - 376$
78.6.a.d	$x + 434$
78.6.a.e	$x + 148$
78.6.a.f	$x + 616$
78.6.a.g	$x^{2} + 448x + 48444$
78.6.a.h	$x^{2} - 424x + 31980$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(78))$	10
Old forms	56
Cusp forms $S_{6}(\Gamma_0(78))$	66
Eisenstein series $E_{6}(\Gamma_0(78))$	8
Modular forms $M_{6}(\Gamma_0(78))$	74

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(78)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.301 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, 2, 3, 6, 13, 26, 39, 78]: [1, 2, 3, 7, 7, 19, 26, 74] (certified at ell = 2147483629)
T = [(5, 1), (7, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 4 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(78, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 78:6 1000

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