Atlas › Modular forms › Weight 2 › Level 982 › 982.2.a

Newspace 982.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(982))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
982.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$	982a1 rank 1
982.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
982.2.a.c	5	degree 5	$5q + 5q^{2} - 3q^{3} + 5q^{4} - 7q^{5} - 3q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
982.2.a.d	9	degree 9	$9q - 9q^{2} + 9q^{4} + 11q^{5} + 5q^{7} + O(q^{8})$	+ −	$+1$
982.2.a.e	10	degree 10	$10q - 10q^{2} - 2q^{3} + 10q^{4} - 10q^{5} + 2q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
982.2.a.f	13	degree 13	$13q + 13q^{2} + 5q^{3} + 13q^{4} + 8q^{5} + 5q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
982.2.a.a	$x + 5$
982.2.a.b	$x^{2} + 9x - 41$
982.2.a.c	$x^{5} + 24x^{4} + 204x^{3} + 720x^{2} + 832x - 224$
982.2.a.d	$x^{9} - 33x^{8} + 281x^{7} + 1126x^{6} - 23924x^{5} + 56088x^{4} + 342048x^{3} - 1454304x^{2} + 691328x + 1689088$
982.2.a.e	$x^{10} + 32x^{9} + 202x^{8} - 2630x^{7} - 30753x^{6} + 13569x^{5} + 1095594x^{4} + 2677588x^{3} - 9149496x^{2} - 43823680x - 46375776$
982.2.a.f	$x^{13} - 29x^{12} + 40x^{11} + 6068x^{10} - 52339x^{9} - 221050x^{8} + 3927827x^{7} - 3962892x^{6} - 98157492x^{5} + 294246376x^{4} + 641063408x^{3} - 3543824032x^{2} + 3568702976x + 315917824$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(982))$	40
Old forms	82
Cusp forms $S_{2}(\Gamma_0(982))$	122
Eisenstein series $E_{2}(\Gamma_0(982))$	3
Modular forms $M_{2}(\Gamma_0(982))$	125

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(982)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(491))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.304 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, 491, 982]: [0, 1, 42, 125] (certified at ell = 2147474491)
T = [(3, 1), (5, 3)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 6 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(982, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 982:2 1000

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