Atlas › Modular forms › Weight 12 › Level 7 › 7.12.a

Newspace 7.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(7))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
7.12.a.a	2	$\mathbb{Q}(\sqrt{3369})$	$2q - 54q^{2} + 120q^{3} + 4100q^{4} - 13500q^{5} - 43668q^{6} + 33614q^{7} + O(q^{8})$	−	$-1$
7.12.a.b	3	degree 3	$3q + 77q^{2} - 140q^{3} + 5493q^{4} + 5026q^{5} + 64274q^{6} - 50421q^{7} + O(q^{8})$	+	$+1$

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 $T$
7.12.a.a	$x^{2} - 66x - 83136$
7.12.a.b	$x^{3} + 63x^{2} - 663088x - 106812564$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(7))$	5
Old forms	2
Cusp forms $S_{12}(\Gamma_0(7))$	7
Eisenstein series $E_{12}(\Gamma_0(7))$	2
Modular forms $M_{12}(\Gamma_0(7))$	9

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(7)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.311 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, 7]: [2, 9] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 2 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(7, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 7:12 1000

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