Atlas › Modular forms › Weight 2 › Level 407 › 407.2.a

Newspace 407.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(407))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
407.2.a.a	4	degree 4	$4q - q^{2} + q^{4} + q^{5} - 4q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
407.2.a.b	4	degree 4	$4q + q^{2} - 4q^{3} + q^{4} - 5q^{5} - 6q^{6} - q^{7} + O(q^{8})$	− −	$-1$
407.2.a.c	11	degree 11	$11q + 2q^{2} + 14q^{4} + q^{5} + 4q^{6} + 9q^{7} + O(q^{8})$	+ −	$+1$
407.2.a.d	12	degree 12	$12q + q^{2} + 8q^{3} + 13q^{4} + 5q^{5} - 10q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

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 $T$
407.2.a.a	$x^{4} + x^{3} - 4x^{2} - 2x + 3$
407.2.a.b	$x^{4} + 3x^{3} - 2x^{2} - 4x - 1$
407.2.a.c	$x^{11} - 2x^{10} - 44x^{9} + 66x^{8} + 625x^{7} - 565x^{6} - 3803x^{5} + 1333x^{4} + 9382x^{3} + 271x^{2} - 5223x + 1409$
407.2.a.d	$x^{12} - 9x^{11} + 4x^{10} + 164x^{9} - 357x^{8} - 770x^{7} + 2626x^{6} + 120x^{5} - 5037x^{4} + 3217x^{3} + 204x^{2} - 134x - 11$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(407))$	31
Old forms	6
Cusp forms $S_{2}(\Gamma_0(407))$	37
Eisenstein series $E_{2}(\Gamma_0(407))$	3
Modular forms $M_{2}(\Gamma_0(407))$	40

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(407)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.188 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, 11, 37, 407]: [0, 2, 3, 40] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 31 = 1 x 31 (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(407, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 407:2 1000

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