Atlas › Modular forms › Weight 2 › Level 615 › 615.2.a

Newspace 615.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(615))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
615.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} + O(q^{8})$	+ + +	$-1$	615a1 rank 1
615.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{5} + O(q^{8})$	− + −	$-1$	615b1 rank 1
615.2.a.c	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} - 2q^{3} + 5q^{4} - 2q^{5} + q^{6} + O(q^{8})$	+ + +	$-1$
615.2.a.d	4	degree 4	$4q + 3q^{2} - 4q^{3} + 3q^{4} - 4q^{5} - 3q^{6} + O(q^{8})$	+ + −	$+1$
615.2.a.e	5	degree 5	$5q + q^{2} - 5q^{3} + 9q^{4} + 5q^{5} - q^{6} + O(q^{8})$	+ − +	$+1$
615.2.a.f	6	degree 6	$6q + q^{2} + 6q^{3} + 9q^{4} - 6q^{5} + q^{6} + 8q^{7} + O(q^{8})$	− + +	$+1$
615.2.a.g	8	degree 8	$8q + 2q^{2} + 8q^{3} + 14q^{4} + 8q^{5} + 2q^{6} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7} + 5T_{17} + 7T_{19} + 9T_{23}$ 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 $T$
615.2.a.a	$x + 59$
615.2.a.b	$x + 57$
615.2.a.c	$x^{2} + 18x + 13$
615.2.a.d	$x^{4} - 171x^{3} + 8514x^{2} - 104814x - 862893$
615.2.a.e	$x^{5} + 10x^{4} - 2555x^{3} - 37356x^{2} + 485365x - 1124961$
615.2.a.f	$x^{6} - 69x^{5} - 11073x^{4} + 1440957x^{3} - 61388631x^{2} + 1110654234x - 7096340803$
615.2.a.g	$x^{8} - 135x^{7} + 764x^{6} + 449344x^{5} - 10611252x^{4} - 327133905x^{3} + 8944645390x^{2} + 64301282472x - 1830946225063$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(615))$	27
Old forms	54
Cusp forms $S_{2}(\Gamma_0(615))$	81
Eisenstein series $E_{2}(\Gamma_0(615))$	7
Modular forms $M_{2}(\Gamma_0(615))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(615)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(123))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(205))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.337 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, 5, 15, 41, 123, 205, 615]: [0, 1, 1, 4, 4, 16, 22, 88] (certified at ell = 2147482921)
T = [(2, 1), (7, 3), (17, 5), (19, 7), (23, 9)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 7 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(615, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 615:2 1000

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