Atlas › Modular forms › Weight 2 › Level 408 › 408.2.a

Newspace 408.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(408))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
408.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + 3q^{5} + O(q^{8})$	+ +	408c1 rank 0
408.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} - 4q^{7} + O(q^{8})$	− −	408d1 rank 1
408.2.a.c	1	$\mathbb{Q}$	$q + q^{3} + 2q^{7} + O(q^{8})$	− +	408a1 rank 0
408.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	− −	408b1 rank 0
408.2.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{3} - q^{5} - 2q^{7} + O(q^{8})$	+ +
408.2.a.f	2	$\mathbb{Q}(\sqrt{57})$	$2q + 2q^{3} - q^{5} + 8q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{19} + 11T_{23}$ 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 $T$
408.2.a.a	$x - 105$
408.2.a.b	$x + 97$
408.2.a.c	$x - 78$
408.2.a.d	$x - 44$
408.2.a.e	$x^{2} + 221x + 11866$
408.2.a.f	$x^{2} + 45x - 13188$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(408))$	8
Old forms	57
Cusp forms $S_{2}(\Gamma_0(408))$	65
Eisenstein series $E_{2}(\Gamma_0(408))$	15
Modular forms $M_{2}(\Gamma_0(408))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(408)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(68))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(102))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(136))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(204))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.457 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, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, 408]: [0, 1, 1, 2, 3, 3, 5, 2, 8, 6, 8, 12, 22, 22, 42, 80] (certified at ell = 2147483489)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (19, 9), (23, 11)]; h of degree 8 = 1 x 8 (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(408, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 408:2 1000

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