Atlas › Modular forms › Weight 2 › Level 665 › 665.2.a

Newspace 665.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(665))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
665.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	665d1 rank 1
665.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 2q^{4} + q^{5} - 6q^{6} + q^{7} + O(q^{8})$	− − −	$+1$	665e1 rank 0
665.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	665a1 rank 1
665.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	665c1 rank 1
665.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} - q^{7} + O(q^{8})$	− + −	$-1$	665b1 rank 1
665.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} - q^{3} - 2q^{4} + 2q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
665.2.a.g	3	degree 3	$3q + q^{2} + q^{3} + q^{4} + 3q^{5} + 7q^{6} - 3q^{7} + O(q^{8})$	− + +	$+1$
665.2.a.h	4	degree 4	$4q + q^{2} + 2q^{3} + 3q^{4} - 4q^{5} + 3q^{6} - 4q^{7} + O(q^{8})$	+ + −	$+1$
665.2.a.i	6	degree 6	$6q - q^{2} - 4q^{3} + 5q^{4} - 6q^{5} - 3q^{6} - 6q^{7} + O(q^{8})$	+ + +	$-1$
665.2.a.j	7	degree 7	$7q + 3q^{2} + q^{3} + 13q^{4} - 7q^{5} - q^{6} + 7q^{7} + O(q^{8})$	+ − +	$+1$
665.2.a.k	8	degree 8	$8q + 2q^{2} - 3q^{3} + 12q^{4} + 8q^{5} + 9q^{6} + 8q^{7} + O(q^{8})$	− − −	$+1$

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 $T$
665.2.a.a	$x + 5$
665.2.a.b	$x - 7$
665.2.a.c	$x + 4$
665.2.a.d	$x + 2$
665.2.a.e	$x - 1$
665.2.a.f	$x^{2} + 5x - 32$
665.2.a.g	$x^{3} - 4x^{2} - 48x + 64$
665.2.a.h	$x^{4} - 7x^{3} - 62x^{2} + 268x + 1304$
665.2.a.i	$x^{6} + 13x^{5} - 32x^{4} - 654x^{3} + 555x^{2} + 8527x - 9986$
665.2.a.j	$x^{7} - 6x^{6} - 132x^{5} + 386x^{4} + 5321x^{3} - 2668x^{2} - 66560x - 74784$
665.2.a.k	$x^{8} + 7x^{7} - 156x^{6} - 846x^{5} + 7245x^{4} + 29663x^{3} - 118970x^{2} - 270868x + 674808$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(665))$	35
Old forms	42
Cusp forms $S_{2}(\Gamma_0(665))$	77
Eisenstein series $E_{2}(\Gamma_0(665))$	7
Modular forms $M_{2}(\Gamma_0(665))$	84

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(665)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(95))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(133))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.257 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, 7, 19, 35, 95, 133, 665]: [0, 1, 1, 2, 6, 12, 14, 84] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 5 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(665, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 665:2 1000

← 664.2.a · 666.2.a →