Atlas › Modular forms › Weight 2 › Level 974 › 974.2.a

Newspace 974.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(974))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
974.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ −	$+1$	974b1 rank 0
974.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 3q^{5} - 4q^{7} + O(q^{8})$	+ −	$+1$	974a1 rank 0
974.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 4q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	+ −	$+1$	974c1 rank 0
974.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} + q^{4} - 2q^{5} - 3q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$	974d1 rank 0
974.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} - 3q^{5} - 3q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$	974h1 rank 1
974.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} + q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	974g1 rank 1
974.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$	974f1 rank 1
974.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$	974e1 rank 1
974.2.a.i	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
974.2.a.j	2	$\mathbb{Q}(\sqrt{41})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} - 6q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
974.2.a.k	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} - q^{5} + q^{6} + O(q^{8})$	+ +	$-1$
974.2.a.l	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} + q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
974.2.a.m	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} + 9q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$
974.2.a.n	6	degree 6	$6q - 6q^{2} + q^{3} + 6q^{4} - 2q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$
974.2.a.o	14	degree 14	$14q + 14q^{2} + 8q^{3} + 14q^{4} + 8q^{5} + 8q^{6} + 12q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ 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 $T$
974.2.a.a	$x - 6$
974.2.a.b	$x - 77$
974.2.a.c	$x - 14$
974.2.a.d	$x + 2$
974.2.a.e	$x + 31$
974.2.a.f	$x + 20$
974.2.a.g	$x + 64$
974.2.a.h	$x + 81$
974.2.a.i	$x^{2} + 38x - 5039$
974.2.a.j	$x^{2} + 111x + 2988$
974.2.a.k	$x^{3} + 91x^{2} - 4241x - 133981$
974.2.a.l	$x^{3} + 10x^{2} - 64x - 92$
974.2.a.m	$x^{3} - 63x^{2} + 431x + 8197$
974.2.a.n	$x^{6} + x^{5} - 15932x^{4} + 409152x^{3} + 51065868x^{2} - 1806043072x - 437451356$
974.2.a.o	$x^{14} - 299x^{13} + 16751x^{12} + 2529025x^{11} - 260247567x^{10} - 4462452253x^{9} + 1025099734669x^{8} - 5946737561013x^{7} - 1607407026395366x^{6} + 20184081654374324x^{5} + 1009847895367996148x^{4} - 13538671823270778144x^{3} - 193553422337216703564x^{2} + 1866382488193653570464x - 2847451559208374067904$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(974))$	41
Old forms	80
Cusp forms $S_{2}(\Gamma_0(974))$	121
Eisenstein series $E_{2}(\Gamma_0(974))$	3
Modular forms $M_{2}(\Gamma_0(974))$	124

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(974)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(487))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.706 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, 487, 974]: [0, 1, 41, 124] (certified at ell = 2147481883)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 41 = 1 x 41 (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(974, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 974:2 1000

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