Atlas › Modular forms › Weight 2 › Level 915 › 915.2.a

Newspace 915.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(915))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
915.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− − +	$-1$	915d1 rank 1
915.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + O(q^{8})$	+ − −	$-1$	915b1 rank 1
915.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} + q^{5} - 2q^{6} + 3q^{7} + O(q^{8})$	+ − +	$+1$	915a1 rank 0
915.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} - q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− + +	$+1$	915c1 rank 0
915.2.a.e	3	degree 3	$3q - 2q^{2} - 3q^{3} + 4q^{4} + 3q^{5} + 2q^{6} - 5q^{7} + O(q^{8})$	+ − +	$+1$
915.2.a.f	4	degree 4	$4q + q^{2} + 4q^{3} + 3q^{4} - 4q^{5} + q^{6} - q^{7} + O(q^{8})$	− + +	$+1$
915.2.a.g	5	degree 5	$5q - 3q^{2} + 5q^{3} + 3q^{4} - 5q^{5} - 3q^{6} - 4q^{7} + O(q^{8})$	− + −	$-1$
915.2.a.h	5	degree 5	$5q + q^{2} - 5q^{3} + 3q^{4} - 5q^{5} - q^{6} - 6q^{7} + O(q^{8})$	+ + +	$-1$
915.2.a.i	5	degree 5	$5q + q^{2} - 5q^{3} + 5q^{4} - 5q^{5} - q^{6} + 6q^{7} + O(q^{8})$	+ + −	$+1$
915.2.a.j	5	degree 5	$5q + q^{2} - 5q^{3} + 11q^{4} + 5q^{5} - q^{6} + 6q^{7} + O(q^{8})$	+ − +	$+1$
915.2.a.k	8	degree 8	$8q + 2q^{2} + 8q^{3} + 10q^{4} + 8q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{7}$ 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 $T$
915.2.a.a	$x + 3$
915.2.a.b	$x - 1$
915.2.a.c	$x - 5$
915.2.a.d	$x - 3$
915.2.a.e	$x^{3} + 7x^{2} + 9x - 1$
915.2.a.f	$x^{4} - 24x^{2} + 2x + 137$
915.2.a.g	$x^{5} + 7x^{4} + 4x^{3} - 26x^{2} - 13x + 29$
915.2.a.h	$x^{5} + 5x^{4} - 8x^{3} - 54x^{2} - 35x + 9$
915.2.a.i	$x^{5} - 7x^{4} + 10x^{3} + 20x^{2} - 51x + 23$
915.2.a.j	$x^{5} - 7x^{4} - 32x^{3} + 276x^{2} - 475x + 235$
915.2.a.k	$x^{8} - 4x^{7} - 38x^{6} + 152x^{5} + 320x^{4} - 1034x^{3} - 1384x^{2} + 1258x + 1369$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(915))$	39
Old forms	82
Cusp forms $S_{2}(\Gamma_0(915))$	121
Eisenstein series $E_{2}(\Gamma_0(915))$	7
Modular forms $M_{2}(\Gamma_0(915))$	128

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(915)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(183))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(305))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.423 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, 61, 183, 305, 915]: [0, 1, 1, 4, 5, 22, 32, 128] (certified at ell = 2147482921)
T = [(2, 1), (7, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 5 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(915, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 915:2 1000

← 914.2.a · 916.2.a →