Atlas › Modular forms › Weight 2 › Level 663 › 663.2.a

Newspace 663.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(663))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
663.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + O(q^{8})$	+ − −	$-1$	663b1 rank 1
663.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$	663c1 rank 1
663.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - 4q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$	663a1 rank 0
663.2.a.d	3	degree 3	$3q - 3q^{2} + 3q^{3} + 5q^{4} - 6q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	− − +	$-1$
663.2.a.e	3	degree 3	$3q - q^{2} - 3q^{3} + q^{4} + q^{6} - 4q^{7} + O(q^{8})$	+ + +	$-1$
663.2.a.f	5	degree 5	$5q + q^{2} - 5q^{3} + 7q^{4} + 2q^{5} - q^{6} + 8q^{7} + O(q^{8})$	+ − +	$+1$
663.2.a.g	5	degree 5	$5q + 3q^{2} + 5q^{3} + 7q^{4} + 3q^{6} + 2q^{7} + O(q^{8})$	− + +	$+1$
663.2.a.h	6	degree 6	$6q + 2q^{2} - 6q^{3} + 10q^{4} - 2q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$
663.2.a.i	6	degree 6	$6q + 4q^{2} + 6q^{3} + 6q^{4} + 4q^{5} + 4q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{19}$ 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 $T$
663.2.a.a	$x + 15$
663.2.a.b	$x + 25$
663.2.a.c	$x - 77$
663.2.a.d	$x^{3} + 123x^{2} + 4781x + 56549$
663.2.a.e	$x^{3} + 43x^{2} - 1997x - 8245$
663.2.a.f	$x^{5} - 19x^{4} - 1824x^{3} + 19986x^{2} + 384619x + 1033255$
663.2.a.g	$x^{5} - 11x^{4} - 12924x^{3} + 259870x^{2} + 40468057x - 1078482851$
663.2.a.h	$x^{6} + 72x^{5} - 7173x^{4} - 266200x^{3} - 2869951x^{2} - 9630810x + 2509025$
663.2.a.i	$x^{6} - 46x^{5} - 3533x^{4} - 2592x^{3} + 997023x^{2} - 1010264x - 43523461$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(663))$	31
Old forms	50
Cusp forms $S_{2}(\Gamma_0(663))$	81
Eisenstein series $E_{2}(\Gamma_0(663))$	7
Modular forms $M_{2}(\Gamma_0(663))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(663)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(221))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.425 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, 3, 13, 17, 39, 51, 221, 663]: [0, 1, 1, 2, 6, 8, 22, 88] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (7, 5), (11, 7), (19, 9)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 8 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(663, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 663:2 1000

← 662.2.a · 664.2.a →