Atlas › Modular forms › Weight 6 › Level 74 › 74.6.a

Newspace 74.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(74))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
74.6.a.a	1	$\mathbb{Q}$	$q + 4q^{2} - 15q^{3} + 16q^{4} + 4q^{5} - 60q^{6} + 15q^{7} + O(q^{8})$	− −	$-1$
74.6.a.b	1	$\mathbb{Q}$	$q + 4q^{2} + 7q^{3} + 16q^{4} - 84q^{5} + 28q^{6} - 139q^{7} + O(q^{8})$	− −	$-1$
74.6.a.c	3	degree 3	$3q - 12q^{2} - 8q^{3} + 48q^{4} - 42q^{5} + 32q^{6} + 84q^{7} + O(q^{8})$	+ +	$-1$
74.6.a.d	5	degree 5	$5q - 20q^{2} + 19q^{3} + 80q^{4} - 67q^{5} - 76q^{6} + 182q^{7} + O(q^{8})$	+ −	$+1$
74.6.a.e	5	degree 5	$5q + 20q^{2} + 19q^{3} + 80q^{4} + 95q^{5} + 76q^{6} + 170q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
74.6.a.a	$x + 11$
74.6.a.b	$x + 77$
74.6.a.c	$x^{3} + 50x^{2} + 383x - 1226$
74.6.a.d	$x^{5} + 48x^{4} - 17272x^{3} - 936130x^{2} + 57264615x + 3143577250$
74.6.a.e	$x^{5} - 114x^{4} - 2018x^{3} + 431800x^{2} - 6144255x - 22198214$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(74))$	15
Old forms	30
Cusp forms $S_{6}(\Gamma_0(74))$	45
Eisenstein series $E_{6}(\Gamma_0(74))$	4
Modular forms $M_{6}(\Gamma_0(74))$	49

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(74)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 37, 74]: [1, 2, 17, 49] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(74, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 74:6 1000

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