Atlas › Modular forms › Weight 2 › Level 506 › 506.2.a

Newspace 506.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(506))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
506.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$	506a1 rank 1
506.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 3q^{5} + 2q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$	506c1 rank 0
506.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 3q^{5} + 3q^{7} + O(q^{8})$	+ + −	$+1$	506b1 rank 0
506.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + q^{7} + O(q^{8})$	+ − −	$-1$	506d1 rank 1
506.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} - q^{7} + O(q^{8})$	− − +	$-1$	506f1 rank 1
506.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{5} - 3q^{7} + O(q^{8})$	− + −	$-1$	506e1 rank 1
506.2.a.g	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 7q^{5} - 3q^{6} + O(q^{8})$	+ − +	$+1$
506.2.a.h	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + O(q^{8})$	+ + −	$+1$
506.2.a.i	4	degree 4	$4q + 4q^{2} - q^{3} + 4q^{4} + 2q^{5} - q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$
506.2.a.j	5	degree 5	$5q + 5q^{2} + q^{3} + 5q^{4} + 4q^{5} + q^{6} + q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{29} + 8T_{31}$ 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 $T$
506.2.a.a	$x + 75$
506.2.a.b	$x - 73$
506.2.a.c	$x - 155$
506.2.a.d	$x + 85$
506.2.a.e	$x + 121$
506.2.a.f	$x + 55$
506.2.a.g	$x^{3} - 43x^{2} - 4957x + 132295$
506.2.a.h	$x^{3} - 21x^{2} - 13281x + 692477$
506.2.a.i	$x^{4} - 74x^{3} - 4152x^{2} + 168978x + 12143$
506.2.a.j	$x^{5} - 162x^{4} + 4128x^{3} + 65910x^{2} - 664761x - 3736972$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(506))$	21
Old forms	48
Cusp forms $S_{2}(\Gamma_0(506))$	69
Eisenstein series $E_{2}(\Gamma_0(506))$	7
Modular forms $M_{2}(\Gamma_0(506))$	76

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(506)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(46))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(253))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.456 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, 11, 22, 23, 46, 253, 506]: [0, 1, 2, 5, 3, 8, 26, 76] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (29, 7), (31, 8)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 6 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(506, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 506:2 1000

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