Atlas › Modular forms › Weight 2 › Level 975 › 975.2.a

Newspace 975.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(975))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
975.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} + q^{7} + O(q^{8})$	+ −	975d1 rank 0
975.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} + 3q^{7} + O(q^{8})$	+ +	975b1 rank 1
975.2.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - 2q^{6} - 3q^{7} + O(q^{8})$	− −	975i1 rank 1
975.2.a.d	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{7} + O(q^{8})$	+ +	975e1 rank 0
975.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{6} - q^{7} + O(q^{8})$	− +	975k1 rank 1
975.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{6} + 4q^{7} + O(q^{8})$	− +	975g1 rank 0
975.2.a.g	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + q^{7} + O(q^{8})$	+ −	975f1 rank 1
975.2.a.h	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{7} + O(q^{8})$	− +	975j1 rank 1
975.2.a.i	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} + O(q^{8})$	+ +	975a1 rank 1
975.2.a.j	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} + q^{7} + O(q^{8})$	+ −	975c1 rank 0
975.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} + q^{6} - 3q^{7} + O(q^{8})$	− −	975h1 rank 1
975.2.a.l	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + O(q^{8})$	+ −
975.2.a.m	3	degree 3	$3q - 3q^{2} - 3q^{3} + 5q^{4} + 3q^{6} + q^{7} + O(q^{8})$	+ −
975.2.a.n	3	degree 3	$3q - q^{2} - 3q^{3} + 5q^{4} + q^{6} - 5q^{7} + O(q^{8})$	+ +
975.2.a.o	3	degree 3	$3q + 3q^{3} + 8q^{4} - q^{7} + O(q^{8})$	− +
975.2.a.p	3	degree 3	$3q + q^{2} + 3q^{3} + 5q^{4} + q^{6} + 5q^{7} + O(q^{8})$	− −
975.2.a.q	3	degree 3	$3q + 3q^{2} + 3q^{3} + 5q^{4} + 3q^{6} - q^{7} + O(q^{8})$	− +
975.2.a.r	5	degree 5	$5q - 5q^{3} + 6q^{4} - 5q^{7} + O(q^{8})$	+ +
975.2.a.s	5	degree 5	$5q + 5q^{3} + 6q^{4} + 5q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
975.2.a.a	$x - 25$
975.2.a.b	$x - 39$
975.2.a.c	$x + 61$
975.2.a.d	$x - 13$
975.2.a.e	$x + 7$
975.2.a.f	$x + 27$
975.2.a.g	$x + 87$
975.2.a.h	$x + 135$
975.2.a.i	$x + 141$
975.2.a.j	$x - 55$
975.2.a.k	$x + 37$
975.2.a.l	$x^{2} + 14x - 401$
975.2.a.m	$x^{3} + 101x^{2} - 1933x - 146137$
975.2.a.n	$x^{3} + 71x^{2} - 4461x - 311427$
975.2.a.o	$x^{3} - 251x^{2} + 16299x - 69841$
975.2.a.p	$x^{3} - 55x^{2} - 2021x + 68667$
975.2.a.q	$x^{3} - 53x^{2} - 11757x + 769145$
975.2.a.r	$x^{5} - 259x^{4} + 13480x^{3} + 355284x^{2} - 19408271x - 338011087$
975.2.a.s	$x^{5} - 155x^{4} + 1320x^{3} + 734892x^{2} - 32070391x + 230909257$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(975))$	38
Old forms	91
Cusp forms $S_{2}(\Gamma_0(975))$	129
Eisenstein series $E_{2}(\Gamma_0(975))$	23
Modular forms $M_{2}(\Gamma_0(975))$	152

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(975)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(195))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(325))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.111 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, 5, 13, 15, 25, 39, 65, 75, 195, 325, 975]: [0, 1, 1, 1, 4, 3, 6, 8, 12, 32, 36, 144] (certified at ell = 2147482921)
T = [(2, 1), (7, 2), (11, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 38 = 1 x 38 (phi(ord eps) x dim S_k^new); 11 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(975, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 975:2 1000

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