Atlas › Modular forms › Weight 2 › Level 510 › 510.2.a

Newspace 510.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(510))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
510.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ + + −	$+1$	510a1 rank 0
510.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ − − −	$+1$	510b1 rank 0
510.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− + + −	$-1$	510d1 rank 1
510.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− + + +	$+1$	510c1 rank 0
510.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + O(q^{8})$	− + − −	$+1$	510e1 rank 0
510.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + O(q^{8})$	− − + −	$+1$	510f1 rank 0
510.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− − − +	$+1$	510g1 rank 0
510.2.a.h	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + O(q^{8})$	+ + − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{19} + 5T_{23} + 6T_{29} + 7T_{31} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53} + 13T_{59} + 14T_{61} + 15T_{67} + 16T_{71}$ 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 $T$
510.2.a.a	$x - 434$
510.2.a.b	$x - 122$
510.2.a.c	$x - 80$
510.2.a.d	$x + 242$
510.2.a.e	$x + 124$
510.2.a.f	$x + 100$
510.2.a.g	$x + 182$
510.2.a.h	$x^{2} - 200x - 30344$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(510))$	9
Old forms	92
Cusp forms $S_{2}(\Gamma_0(510))$	101
Eisenstein series $E_{2}(\Gamma_0(510))$	15
Modular forms $M_{2}(\Gamma_0(510))$	116

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.426 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, 3, 5, 6, 10, 15, 17, 30, 34, 51, 85, 102, 170, 255, 510]: [0, 1, 1, 1, 3, 3, 4, 2, 10, 6, 8, 10, 22, 30, 40, 116] (certified at ell = 2147483489)
T = [(7, 1), (11, 2), (13, 3), (19, 4), (23, 5), (29, 6), (31, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12), (59, 13), (61, 14), (67, 15), (71, 16)]; h of degree 9 = 1 x 9 (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(510, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 510:2 1000

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