Atlas › Modular forms › Weight 2 › Level 355 › 355.2.a

Newspace 355.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(355))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
355.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{4} + q^{5} - q^{7} + O(q^{8})$	− +	$+1$	355a1 rank 0
355.2.a.b	4	degree 4	$4q - 4q^{2} - 3q^{3} + 4q^{4} + 4q^{5} - q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$
355.2.a.c	4	degree 4	$4q - 2q^{2} - q^{3} - 4q^{5} - q^{6} + q^{7} + O(q^{8})$	+ +	$-1$
355.2.a.d	6	degree 6	$6q + 3q^{2} + 3q^{3} + 9q^{4} + 6q^{5} - 5q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$
355.2.a.e	8	degree 8	$8q + 4q^{2} - q^{3} + 10q^{4} - 8q^{5} + 7q^{6} - 5q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
355.2.a.a	$x + 2$
355.2.a.b	$x^{4} + 7x^{3} + 14x^{2} + 4x - 7$
355.2.a.c	$x^{4} + 3x^{3} - 2x^{2} - 2x + 1$
355.2.a.d	$x^{6} - 6x^{5} + x^{4} + 40x^{3} - 51x^{2} + 11x + 2$
355.2.a.e	$x^{8} - 3x^{7} - 35x^{6} + 85x^{5} + 395x^{4} - 696x^{3} - 1459x^{2} + 1580x + 1436$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(355))$	23
Old forms	12
Cusp forms $S_{2}(\Gamma_0(355))$	35
Eisenstein series $E_{2}(\Gamma_0(355))$	3
Modular forms $M_{2}(\Gamma_0(355))$	38

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(355)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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, 5, 71, 355]: [0, 1, 7, 38] (certified at ell = 2147482681)
T = [(2, 1), (3, 1)]; h of degree 23 = 1 x 23 (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(355, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 355:2 1000

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