Atlas › Modular forms › Weight 2 › Level 980 › 980.2.a

Newspace 980.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(980))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
980.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - q^{5} + O(q^{8})$	+	980b1 rank 0
980.2.a.b	1	$\mathbb{Q}$	$q - 3q^{3} + q^{5} + O(q^{8})$	−	980i1 rank 0
980.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + O(q^{8})$	+	980d1 rank 1
980.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + O(q^{8})$	−	980e1 rank 0
980.2.a.e	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + O(q^{8})$	−	980f1 rank 0
980.2.a.f	1	$\mathbb{Q}$	$q + q^{3} - q^{5} + O(q^{8})$	+	980c1 rank 1
980.2.a.g	1	$\mathbb{Q}$	$q + q^{3} - q^{5} + O(q^{8})$	+	980a1 rank 0
980.2.a.h	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} + O(q^{8})$	−	980g1 rank 0
980.2.a.i	1	$\mathbb{Q}$	$q + 3q^{3} + q^{5} + O(q^{8})$	−	980h1 rank 0
980.2.a.j	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} + 2q^{5} + O(q^{8})$	−
980.2.a.k	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} - 2q^{5} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{11} + 3T_{19} + 4T_{23} + 5T_{29} + 6T_{13} + 7T_{17} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47} + 13T_{53} + 14T_{59} + 15T_{61} + 16T_{67}$ 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 $T$
980.2.a.a	$x - 192$
980.2.a.b	$x - 661$
980.2.a.c	$x + 299$
980.2.a.d	$x - 377$
980.2.a.e	$x - 244$
980.2.a.f	$x - 113$
980.2.a.g	$x - 176$
980.2.a.h	$x + 194$
980.2.a.i	$x + 68$
980.2.a.j	$x^{2} + 794x + 132521$
980.2.a.k	$x^{2} - 358x + 12041$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(980))$	13
Old forms	132
Cusp forms $S_{2}(\Gamma_0(980))$	145
Eisenstein series $E_{2}(\Gamma_0(980))$	47
Modular forms $M_{2}(\Gamma_0(980))$	192

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.602 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, 4, 5, 7, 10, 14, 20, 28, 35, 49, 70, 98, 140, 196, 245, 490, 980]: [0, 1, 2, 1, 1, 3, 4, 6, 7, 6, 5, 16, 16, 30, 31, 30, 88, 174] (certified at ell = 2147483353)
T = [(3, 1), (11, 2), (19, 3), (23, 4), (29, 5), (13, 6), (17, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12), (53, 13), (59, 14), (61, 15), (67, 16)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 5 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(980, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 980:2 1000

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