Atlas › Modular forms › Weight 2 › Level 296 › 296.2.a

Newspace 296.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(296))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
296.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + q^{7} + O(q^{8})$	+	296a1 rank 1
296.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 3q^{7} + O(q^{8})$	−	296b1 rank 1
296.2.a.c	3	degree 3	$3q + 2q^{3} - q^{5} + 7q^{7} + O(q^{8})$	+
296.2.a.d	4	degree 4	$4q + 2q^{3} + 5q^{5} - q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
296.2.a.a	$x + 3$
296.2.a.b	$x + 1$
296.2.a.c	$x^{3} - x^{2} - 21x + 37$
296.2.a.d	$x^{4} - 7x^{3} + x^{2} + 39x - 2$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(296))$	9
Old forms	26
Cusp forms $S_{2}(\Gamma_0(296))$	35
Eisenstein series $E_{2}(\Gamma_0(296))$	7
Modular forms $M_{2}(\Gamma_0(296))$	42

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(296)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(74))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(148))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 8, 37, 74, 148, 296]: [0, 1, 2, 3, 3, 11, 22, 42] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 1 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(296, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 296:2 1000

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