Atlas › Modular forms › Weight 2 › Level 325 › 325.2.a

Newspace 325.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(325))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
325.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + O(q^{8})$	−	325e1 rank 0
325.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + 4q^{7} + O(q^{8})$	+	325b1 rank 1
325.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - 4q^{7} + O(q^{8})$	−	325a1 rank 1
325.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - q^{4} + 2q^{6} + 4q^{7} + O(q^{8})$	−	325c1 rank 0
325.2.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	+	325d1 rank 0
325.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 8q^{6} - 2q^{7} + O(q^{8})$	+
325.2.a.g	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{4} - 6q^{6} - 4q^{7} + O(q^{8})$	+
325.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 8q^{6} + 2q^{7} + O(q^{8})$	−
325.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 4q^{6} - 4q^{7} + O(q^{8})$	−
325.2.a.j	3	degree 3	$3q - 3q^{2} - 4q^{3} + 5q^{4} + 2q^{6} - 2q^{7} + O(q^{8})$	−
325.2.a.k	3	degree 3	$3q + 3q^{2} + 4q^{3} + 5q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
325.2.a.a	$x + 5$
325.2.a.b	$x + 3$
325.2.a.c	$x - 3$
325.2.a.d	$x - 7$
325.2.a.e	$x - 5$
325.2.a.f	$x^{2} + 2x - 49$
325.2.a.g	$x^{2} + 6x - 3$
325.2.a.h	$x^{2} - 2x - 49$
325.2.a.i	$x^{2} - 2x - 31$
325.2.a.j	$x^{3} + 15x^{2} + 47x + 37$
325.2.a.k	$x^{3} - 15x^{2} + 47x - 37$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(325))$	19
Old forms	10
Cusp forms $S_{2}(\Gamma_0(325))$	29
Eisenstein series $E_{2}(\Gamma_0(325))$	11
Modular forms $M_{2}(\Gamma_0(325))$	40

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(325)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.375 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, 5, 13, 25, 65, 325]: [0, 1, 1, 3, 8, 36] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 3 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(325, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 325:2 1000

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