Atlas › Modular forms › Weight 2 › Level 305 › 305.2.a

Newspace 305.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(305))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
305.2.a.a	3	degree 3	$3q - 3q^{5} - 6q^{6} - 6q^{7} + O(q^{8})$	+ +	$-1$
305.2.a.b	4	degree 4	$4q - 3q^{2} - 6q^{3} + 3q^{4} + 4q^{5} + 4q^{6} - 10q^{7} + O(q^{8})$	− −	$-1$
305.2.a.c	7	degree 7	$7q - 2q^{2} + 12q^{4} - 7q^{5} + 12q^{7} + O(q^{8})$	+ −	$+1$
305.2.a.d	7	degree 7	$7q + 2q^{2} + 6q^{3} + 8q^{4} + 7q^{5} - 2q^{6} + 8q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
305.2.a.a	$x^{3} - 12x - 8$
305.2.a.b	$x^{4} + 21x^{3} + 122x^{2} + 96x - 568$
305.2.a.c	$x^{7} + 2x^{6} - 146x^{5} - 148x^{4} + 6089x^{3} - 2766x^{2} - 69688x + 91272$
305.2.a.d	$x^{7} - 20x^{6} + 78x^{5} + 596x^{4} - 3671x^{3} - 3816x^{2} + 31292x + 14584$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(305))$	21
Old forms	8
Cusp forms $S_{2}(\Gamma_0(305))$	29
Eisenstein series $E_{2}(\Gamma_0(305))$	3
Modular forms $M_{2}(\Gamma_0(305))$	32

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(305)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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, 5, 61, 305]: [0, 1, 5, 32] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 3 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(305, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 305:2 1000

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