Atlas › Modular forms › Weight 2 › Level 505 › 505.2.a

Newspace 505.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(505))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
505.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{5} + O(q^{8})$	+ +	$-1$	505a1 rank 1
505.2.a.b	6	degree 6	$6q - 3q^{2} - q^{3} + 5q^{4} - 6q^{5} - 8q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
505.2.a.c	8	degree 8	$8q - 5q^{2} - 11q^{3} + 7q^{4} + 8q^{5} + 4q^{6} - 12q^{7} + O(q^{8})$	− −	$-1$
505.2.a.d	9	degree 9	$9q + 2q^{2} + q^{3} + 10q^{4} - 9q^{5} + 4q^{6} + O(q^{8})$	+ −	$+1$
505.2.a.e	9	degree 9	$9q + 2q^{2} + 7q^{3} + 6q^{4} + 9q^{5} + 4q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
505.2.a.a	$x - 1$
505.2.a.b	$x^{6} + 4x^{5} - 2x^{4} - 17x^{3} - 5x^{2} + 16x + 7$
505.2.a.c	$x^{8} + 16x^{7} + 95x^{6} + 247x^{5} + 211x^{4} - 187x^{3} - 338x^{2} + 8x + 113$
505.2.a.d	$x^{9} - 3x^{8} - 31x^{7} + 68x^{6} + 314x^{5} - 408x^{4} - 1063x^{3} + 318x^{2} + 157x + 13$
505.2.a.e	$x^{9} - 9x^{8} + 7x^{7} + 128x^{6} - 306x^{5} - 216x^{4} + 865x^{3} + 196x^{2} - 723x - 253$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(505))$	33
Old forms	16
Cusp forms $S_{2}(\Gamma_0(505))$	49
Eisenstein series $E_{2}(\Gamma_0(505))$	3
Modular forms $M_{2}(\Gamma_0(505))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(505)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(101))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.199 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, 101, 505]: [0, 1, 9, 52] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 4 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(505, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 505:2 1000

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