Atlas › Modular forms › Weight 2 › Level 948 › 948.2.a

Newspace 948.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(948))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
948.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + O(q^{8})$	+ +	948a1 rank 0
948.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + 4q^{5} - 3q^{7} + O(q^{8})$	+ +	948b1 rank 0
948.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - q^{7} + O(q^{8})$	− −	948c1 rank 0
948.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} - q^{5} - 2q^{7} + O(q^{8})$	+ −
948.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} - 3q^{5} - 4q^{7} + O(q^{8})$	− +
948.2.a.f	3	degree 3	$3q - 3q^{3} - 3q^{5} + q^{7} + O(q^{8})$	+ +
948.2.a.g	4	degree 4	$4q + 4q^{3} + q^{5} + 5q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
948.2.a.a	$x - 78$
948.2.a.b	$x + 20$
948.2.a.c	$x - 20$
948.2.a.d	$x^{2} + 67x + 1061$
948.2.a.e	$x^{2} + 87x - 9$
948.2.a.f	$x^{3} - 25x^{2} - 213x - 260$
948.2.a.g	$x^{4} - 123x^{3} + 4811x^{2} - 59918x + 17128$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(948))$	14
Old forms	141
Cusp forms $S_{2}(\Gamma_0(948))$	155
Eisenstein series $E_{2}(\Gamma_0(948))$	11
Modular forms $M_{2}(\Gamma_0(948))$	166

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(948)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(158))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(237))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(316))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(474))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.615 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, 4, 6, 12, 79, 158, 237, 316, 474, 948]: [0, 1, 1, 2, 3, 5, 7, 22, 28, 43, 84, 166] (certified at ell = 2147483053)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(948, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 948:2 1000

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