Atlas › Modular forms › Weight 2 › Level 406 › 406.2.a

Newspace 406.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(406))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
406.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{7} + O(q^{8})$	+ + +	$-1$	406a1 rank 1
406.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 3q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	406b1 rank 1
406.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + 2q^{5} - 2q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	406d1 rank 0
406.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 3q^{5} - q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	406c1 rank 1
406.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} + 4q^{3} + 2q^{4} + 2q^{5} + 4q^{6} - 2q^{7} + O(q^{8})$	− + +	$+1$
406.2.a.f	3	degree 3	$3q - 3q^{2} + q^{3} + 3q^{4} + 5q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ + −	$+1$
406.2.a.g	4	degree 4	$4q + 4q^{2} + q^{3} + 4q^{4} - q^{5} + q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
406.2.a.a	$x + 86$
406.2.a.b	$x + 139$
406.2.a.c	$x - 68$
406.2.a.d	$x + 101$
406.2.a.e	$x^{2} - 30x - 87498$
406.2.a.f	$x^{3} - 261x^{2} - 3982x + 3341642$
406.2.a.g	$x^{4} + 265x^{3} + 2012x^{2} - 2357392x - 95661652$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(406))$	13
Old forms	44
Cusp forms $S_{2}(\Gamma_0(406))$	57
Eisenstein series $E_{2}(\Gamma_0(406))$	7
Modular forms $M_{2}(\Gamma_0(406))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(406)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(203))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.43 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, 7, 14, 29, 58, 203, 406]: [0, 1, 1, 4, 3, 9, 22, 64] (certified at ell = 2147483353)
T = [(3, 1), (5, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 5 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(406, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 406:2 1000

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