Atlas › Modular forms › Weight 2 › Level 348 › 348.2.a

Newspace 348.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(348))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
348.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + q^{7} + O(q^{8})$	+ +	348c1 rank 0
348.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 3q^{7} + O(q^{8})$	+ −	348a1 rank 1
348.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} - 3q^{7} + O(q^{8})$	− +	348d1 rank 1
348.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + q^{7} + O(q^{8})$	− −	348b1 rank 0

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 $T$
348.2.a.a	$x - 42$
348.2.a.b	$x + 36$
348.2.a.c	$x + 84$
348.2.a.d	$x + 38$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(348))$	4
Old forms	51
Cusp forms $S_{2}(\Gamma_0(348))$	55
Eisenstein series $E_{2}(\Gamma_0(348))$	11
Modular forms $M_{2}(\Gamma_0(348))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(348)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(87))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(116))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(174))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.26 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, 29, 58, 87, 116, 174, 348]: [0, 1, 1, 2, 3, 5, 3, 9, 12, 18, 34, 66] (certified at ell = 2147483549)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 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(348, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 348:2 1000

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