Atlas › Modular forms › Weight 2 › Level 715 › 715.2.a

Newspace 715.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(715))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
715.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} + q^{5} + O(q^{8})$	− + −	$-1$	715b1 rank 1
715.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{4} + q^{5} + 2q^{7} + O(q^{8})$	− + −	$-1$	715a1 rank 1
715.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{5} - 4q^{6} - 4q^{7} + O(q^{8})$	− − +	$-1$
715.2.a.d	3	degree 3	$3q + 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} + 6q^{7} + O(q^{8})$	+ − +	$+1$
715.2.a.e	3	degree 3	$3q + 2q^{2} + 2q^{4} - 3q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+ + −	$+1$
715.2.a.f	6	degree 6	$6q - 5q^{2} - 2q^{3} + 7q^{4} - 6q^{5} + 3q^{6} - 8q^{7} + O(q^{8})$	+ − −	$-1$
715.2.a.g	6	degree 6	$6q - q^{2} - 4q^{3} + 7q^{4} - 6q^{5} - 3q^{6} + 2q^{7} + O(q^{8})$	+ + +	$-1$
715.2.a.h	8	degree 8	$8q + 2q^{2} + 8q^{4} + 8q^{5} + 5q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$
715.2.a.i	9	degree 9	$9q + q^{2} + 2q^{3} + 15q^{4} + 9q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
715.2.a.a	$x + 2$
715.2.a.b	$x + 6$
715.2.a.c	$x^{2} - 8$
715.2.a.d	$x^{3} - 6x^{2} - 28x - 4$
715.2.a.e	$x^{3} - 2x^{2} - 44x + 20$
715.2.a.f	$x^{6} + 11x^{5} - 57x^{4} - 537x^{3} + 1801x^{2} + 4328x - 13436$
715.2.a.g	$x^{6} + 13x^{5} - 15x^{4} - 485x^{3} - 301x^{2} + 488x - 112$
715.2.a.h	$x^{8} - 2x^{7} - 178x^{6} + 534x^{5} + 7660x^{4} - 20899x^{3} - 106026x^{2} + 192704x + 409460$
715.2.a.i	$x^{9} - 7x^{8} - 214x^{7} + 1346x^{6} + 15398x^{5} - 79227x^{4} - 430187x^{3} + 1317154x^{2} + 4527436x + 169276$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(715))$	39
Old forms	42
Cusp forms $S_{2}(\Gamma_0(715))$	81
Eisenstein series $E_{2}(\Gamma_0(715))$	7
Modular forms $M_{2}(\Gamma_0(715))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(715)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(143))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.365 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, 11, 13, 55, 65, 143, 715]: [0, 1, 2, 1, 8, 8, 16, 88] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 39 = 1 x 39 (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(715, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 715:2 1000

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