Atlas › Modular forms › Weight 2 › Level 981 › 981.2.a

Newspace 981.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(981))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
981.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - 3q^{5} + 2q^{7} + O(q^{8})$	−	981b1 rank 1
981.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} - 2q^{7} + O(q^{8})$	−	981a1 rank 1
981.2.a.c	3	degree 3	$3q + 2q^{2} + 6q^{5} - q^{7} + O(q^{8})$	+
981.2.a.d	3	degree 3	$3q + 3q^{2} + 5q^{4} + 3q^{5} - 2q^{7} + O(q^{8})$	+
981.2.a.e	4	degree 4	$4q + q^{2} + 3q^{4} - q^{5} - 3q^{7} + O(q^{8})$	−
981.2.a.f	6	degree 6	$6q - 4q^{2} + 8q^{4} - 5q^{5} - 2q^{7} + O(q^{8})$	−
981.2.a.g	6	degree 6	$6q - 8q^{7} + O(q^{8})$	+
981.2.a.h	9	degree 9	$9q - 3q^{2} + 13q^{4} - q^{5} + 6q^{7} + O(q^{8})$	+
981.2.a.i	12	degree 12	$12q + 18q^{4} + 8q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
981.2.a.a	$x + 10$
981.2.a.b	$x - 4$
981.2.a.c	$x^{3} - 20x^{2} + 68x + 328$
981.2.a.d	$x^{3} - 12x^{2} + 44x - 46$
981.2.a.e	$x^{4} + 2x^{3} - 20x^{2} - 32x + 48$
981.2.a.f	$x^{6} + 19x^{5} - 41x^{4} - 2351x^{3} - 12785x^{2} - 18179x + 5848$
981.2.a.g	$x^{6} - 75x^{4} + 1727x^{2} - 12544$
981.2.a.h	$x^{9} + 6x^{8} - 284x^{7} - 1448x^{6} + 26254x^{5} + 109802x^{4} - 789947x^{3} - 2630896x^{2} + 1184976x + 2803226$
981.2.a.i	$x^{12} - 396x^{10} + 58010x^{8} - 3911281x^{6} + 122480620x^{4} - 1477509444x^{2} + 1280258892$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(981))$	45
Old forms	62
Cusp forms $S_{2}(\Gamma_0(981))$	107
Eisenstein series $E_{2}(\Gamma_0(981))$	7
Modular forms $M_{2}(\Gamma_0(981))$	114

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(981)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(109))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(327))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.305 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, 9, 109, 327, 981]: [0, 1, 2, 9, 38, 112] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 45 = 1 x 45 (phi(ord eps) x dim S_k^new); 7 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(981, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 981:2 1000

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