Atlas › Modular forms › Weight 2 › Level 664 › 664.2.a

Newspace 664.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(664))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
664.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 4q^{5} - 5q^{7} + O(q^{8})$	+	664a1 rank 2
664.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + q^{7} + O(q^{8})$	−	664c1 rank 1
664.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - q^{7} + O(q^{8})$	−	664b1 rank 1
664.2.a.d	4	degree 4	$4q + 2q^{3} + q^{5} + 5q^{7} + O(q^{8})$	−
664.2.a.e	6	degree 6	$6q - 3q^{3} - 4q^{5} - 5q^{7} + O(q^{8})$	+
664.2.a.f	8	degree 8	$8q + 2q^{3} + 7q^{5} + q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
664.2.a.a	$x + 15$
664.2.a.b	$x + 1$
664.2.a.c	$x + 5$
664.2.a.d	$x^{4} - 5x^{3} - 56x^{2} + 255x + 289$
664.2.a.e	$x^{6} + 15x^{5} - 42x^{4} - 1199x^{3} - 1148x^{2} + 21660x + 38569$
664.2.a.f	$x^{8} - 23x^{7} - 15x^{6} + 3687x^{5} - 24444x^{4} - 9157x^{3} + 327774x^{2} - 142349x - 1236370$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(664))$	21
Old forms	60
Cusp forms $S_{2}(\Gamma_0(664))$	81
Eisenstein series $E_{2}(\Gamma_0(664))$	7
Modular forms $M_{2}(\Gamma_0(664))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(664)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(166))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(332))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 83, 166, 332, 664]: [0, 1, 2, 3, 8, 23, 45, 88] (certified at ell = 2147483323)
T = [(3, 1), (5, 3)]; h of degree 21 = 1 x 21 (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(664, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 664:2 1000

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