Atlas › Modular forms › Weight 2 › Level 326 › 326.2.a

Newspace 326.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(326))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
326.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - 3q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ −	$+1$	326c1 rank 0
326.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} - q^{7} + O(q^{8})$	+ +	$-1$	326a1 rank 1
326.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$	326b1 rank 1
326.2.a.d	5	degree 5	$5q - 5q^{2} + 3q^{3} + 5q^{4} + 9q^{5} - 3q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
326.2.a.e	6	degree 6	$6q + 6q^{2} + 5q^{3} + 6q^{4} + 5q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$

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 $T$
326.2.a.a	$x + 5$
326.2.a.b	$x + 1$
326.2.a.c	$x + 3$
326.2.a.d	$x^{5} - 12x^{4} + 45x^{3} - 50x^{2} - 14x + 5$
326.2.a.e	$x^{6} - 5x^{5} - 7x^{4} + 61x^{3} - 64x^{2} - 3x + 9$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(326))$	14
Old forms	26
Cusp forms $S_{2}(\Gamma_0(326))$	40
Eisenstein series $E_{2}(\Gamma_0(326))$	3
Modular forms $M_{2}(\Gamma_0(326))$	43

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 163, 326]: [0, 1, 14, 43] (certified at ell = 2147483179)
T = [(3, 1), (5, 1)]; h of degree 14 = 1 x 14 (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(326, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 326:2 1000

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