Atlas › Modular forms › Weight 2 › Level 357 › 357.2.a

Newspace 357.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(357))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
357.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - 3q^{5} - 2q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	357d1 rank 1
357.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + q^{5} - q^{7} + O(q^{8})$	+ + −	$+1$	357a1 rank 0
357.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + q^{5} + q^{7} + O(q^{8})$	+ − −	$-1$	357b1 rank 1
357.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + 2q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	357c1 rank 0
357.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{3} + 4q^{4} - 4q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
357.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	+ + +	$-1$
357.2.a.g	3	degree 3	$3q + q^{2} + 3q^{3} + 3q^{4} + 2q^{5} + q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$
357.2.a.h	4	degree 4	$4q + 2q^{2} - 4q^{3} + 6q^{4} - 2q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{19}$ 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 $T$
357.2.a.a	$x + 82$
357.2.a.b	$x - 66$
357.2.a.c	$x + 102$
357.2.a.d	$x - 26$
357.2.a.e	$x^{2} + 128x + 3904$
357.2.a.f	$x^{2} + 128x + 3448$
357.2.a.g	$x^{3} + 13x^{2} - 2844x - 3544$
357.2.a.h	$x^{4} - 110x^{3} + 3649x^{2} - 21736x - 485392$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(357))$	15
Old forms	30
Cusp forms $S_{2}(\Gamma_0(357))$	45
Eisenstein series $E_{2}(\Gamma_0(357))$	7
Modular forms $M_{2}(\Gamma_0(357))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(357)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(119))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.331 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, 3, 7, 17, 21, 51, 119, 357]: [0, 1, 1, 2, 4, 8, 14, 52] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (19, 9)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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(357, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 357:2 1000

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