Atlas › Modular forms › Weight 2 › Level 957 › 957.2.a

Newspace 957.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(957))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
957.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$	957a1 rank 0
957.2.a.b	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} - 6q^{7} + O(q^{8})$	− − +	$-1$
957.2.a.c	3	degree 3	$3q - q^{2} + 3q^{3} - q^{4} - 4q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
957.2.a.d	3	degree 3	$3q + q^{2} - 3q^{3} + 3q^{4} + 2q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$
957.2.a.e	3	degree 3	$3q + q^{2} + 3q^{3} + 7q^{4} - 4q^{5} + q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$
957.2.a.f	4	degree 4	$4q + 2q^{2} - 4q^{3} + 6q^{4} + 4q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$
957.2.a.g	6	degree 6	$6q + 3q^{2} + 6q^{3} + 5q^{4} + 4q^{5} + 3q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$
957.2.a.h	7	degree 7	$7q - 6q^{2} - 7q^{3} + 10q^{4} - 2q^{5} + 6q^{6} - 8q^{7} + O(q^{8})$	+ − −	$-1$
957.2.a.i	7	degree 7	$7q - 2q^{2} - 7q^{3} + 6q^{4} - 6q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+ + +	$-1$
957.2.a.j	10	degree 10	$10q + q^{2} + 10q^{3} + 15q^{4} + 2q^{5} + q^{6} + 10q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{17}$ 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 $T$
957.2.a.a	$x - 53$
957.2.a.b	$x^{3} + 177x^{2} + 8847x + 127927$
957.2.a.c	$x^{3} - x^{2} - 1605x - 13187$
957.2.a.d	$x^{3} + 95x^{2} + 287x - 83383$
957.2.a.e	$x^{3} - 37x^{2} - 1469x + 12457$
957.2.a.f	$x^{4} - 86x^{3} + 1708x^{2} + 12934x - 437341$
957.2.a.g	$x^{6} - 91x^{5} + 201x^{4} + 125709x^{3} - 1729131x^{2} - 6922528x - 5886301$
957.2.a.h	$x^{7} + 150x^{6} + 3098x^{5} - 360922x^{4} - 12501530x^{3} + 130699329x^{2} + 7431188263x + 47507385779$
957.2.a.i	$x^{7} + 50x^{6} - 8368x^{5} - 576326x^{4} + 9047596x^{3} + 1533962121x^{2} + 41420361247x + 355326445871$
957.2.a.j	$x^{10} - 35x^{9} - 9967x^{8} + 163585x^{7} + 37982716x^{6} + 37117919x^{5} - 60387659068x^{4} - 888674569773x^{3} + 26296411247571x^{2} + 669396299263152x + 3381182440119579$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(957))$	47
Old forms	70
Cusp forms $S_{2}(\Gamma_0(957))$	117
Eisenstein series $E_{2}(\Gamma_0(957))$	7
Modular forms $M_{2}(\Gamma_0(957))$	124

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(957)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(87))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(319))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.492 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, 11, 29, 33, 87, 319, 957]: [0, 1, 2, 3, 6, 12, 32, 124] (certified at ell = 2147482681)
T = [(2, 1), (5, 3), (7, 5), (13, 7), (17, 9)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 12 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(957, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 957:2 1000

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