Atlas › Modular forms › Weight 2 › Level 978 › 978.2.a

Newspace 978.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(978))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
978.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 4q^{5} + q^{6} + 5q^{7} + O(q^{8})$	+ + −	$+1$	978d1 rank 0
978.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 3q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ + −	$+1$	978c1 rank 0
978.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} + 3q^{7} + O(q^{8})$	+ + −	$+1$	978a1 rank 0
978.2.a.d	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$	978b1 rank 0
978.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 3q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	978e1 rank 1
978.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	978f1 rank 1
978.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$	978g1 rank 1
978.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + O(q^{8})$	− − −	$+1$	978h1 rank 0
978.2.a.i	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 2q^{6} - 3q^{7} + O(q^{8})$	+ − −	$-1$
978.2.a.j	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} + 3q^{5} - 2q^{6} + q^{7} + O(q^{8})$	− + +	$+1$
978.2.a.k	2	$\mathbb{Q}(\sqrt{13})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 5q^{5} + 2q^{6} + 5q^{7} + O(q^{8})$	− − −	$+1$
978.2.a.l	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} + 2q^{5} + 3q^{6} - 6q^{7} + O(q^{8})$	+ + +	$-1$
978.2.a.m	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} - q^{7} + O(q^{8})$	+ − +	$+1$
978.2.a.n	3	degree 3	$3q + 3q^{2} + 3q^{3} + 3q^{4} + q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	− − −	$+1$
978.2.a.o	4	degree 4	$4q + 4q^{2} - 4q^{3} + 4q^{4} - 3q^{5} - 4q^{6} - q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
978.2.a.a	$x + 40$
978.2.a.b	$x + 62$
978.2.a.c	$x - 28$
978.2.a.d	$x - 8$
978.2.a.e	$x + 102$
978.2.a.f	$x + 38$
978.2.a.g	$x + 90$
978.2.a.h	$x$
978.2.a.i	$x^{2} + 74x - 251$
978.2.a.j	$x^{2} - 66x + 936$
978.2.a.k	$x^{2} + 186x + 6777$
978.2.a.l	$x^{3} + 48x^{2} - 1995x + 15562$
978.2.a.m	$x^{3} - 198x^{2} + 11492x - 178648$
978.2.a.n	$x^{3} - 278x^{2} + 24232x - 672832$
978.2.a.o	$x^{4} - 202x^{3} + 4905x^{2} + 669144x - 13906252$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(978))$	27
Old forms	134
Cusp forms $S_{2}(\Gamma_0(978))$	161
Eisenstein series $E_{2}(\Gamma_0(978))$	7
Modular forms $M_{2}(\Gamma_0(978))$	168

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(978)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(163))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(326))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(489))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.028 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, 3, 6, 163, 326, 489, 978]: [0, 1, 1, 3, 14, 43, 56, 168] (certified at ell = 2147483179)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 8 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(978, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 978:2 1000

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