Atlas › Modular forms › Weight 2 › Level 418 › 418.2.a

Newspace 418.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(418))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
418.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$	418b1 rank 1
418.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	− − −	$+1$	418a1 rank 0
418.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} + q^{4} - 2q^{5} + 3q^{6} + q^{7} + O(q^{8})$	− − −	$+1$	418c1 rank 0
418.2.a.d	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} - 3q^{3} + 2q^{4} - q^{5} + 3q^{6} - q^{7} + O(q^{8})$	+ − −	$-1$
418.2.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} + q^{3} + 2q^{4} + 4q^{5} - q^{6} + 3q^{7} + O(q^{8})$	+ − +	$+1$
418.2.a.f	2	$\mathbb{Q}(\sqrt{21})$	$2q + 2q^{2} - q^{3} + 2q^{4} + 3q^{5} - q^{6} - 5q^{7} + O(q^{8})$	− − −	$+1$
418.2.a.g	3	degree 3	$3q - 3q^{2} + 3q^{4} - 3q^{5} - 6q^{7} + O(q^{8})$	+ + +	$-1$
418.2.a.h	3	degree 3	$3q + 3q^{2} + q^{3} + 3q^{4} + 5q^{5} + q^{6} + q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
418.2.a.a	$x + 7$
418.2.a.b	$x - 6$
418.2.a.c	$x + 3$
418.2.a.d	$x^{2} + 6x - 4$
418.2.a.e	$x^{2} - 13x + 38$
418.2.a.f	$x^{2} - 8x - 68$
418.2.a.g	$x^{3} + 9x^{2} - 60x - 516$
418.2.a.h	$x^{3} - 16x^{2} + 64x - 56$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(418))$	15
Old forms	42
Cusp forms $S_{2}(\Gamma_0(418))$	57
Eisenstein series $E_{2}(\Gamma_0(418))$	7
Modular forms $M_{2}(\Gamma_0(418))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(418)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(209))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.318 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, 11, 19, 22, 38, 209, 418]: [0, 1, 2, 2, 5, 7, 22, 64] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 15 = 1 x 15 (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(418, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 418:2 1000

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