Atlas › Modular forms › Weight 2 › Level 710 › 710.2.a

Newspace 710.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(710))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
710.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	710a1 rank 1
710.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	710b1 rank 1
710.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$	710c1 rank 1
710.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$	710d1 rank 0
710.2.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} - q^{3} + 2q^{4} - 2q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
710.2.a.f	2	$\mathbb{Q}(\sqrt{10})$	$2q + 2q^{2} + 4q^{3} + 2q^{4} + 2q^{5} + 4q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$
710.2.a.g	3	degree 3	$3q + 3q^{2} + 3q^{4} + 3q^{5} - 2q^{7} + O(q^{8})$	− − −	$+1$
710.2.a.h	4	degree 4	$4q - 4q^{2} + 3q^{3} + 4q^{4} - 4q^{5} - 3q^{6} + 7q^{7} + O(q^{8})$	+ + −	$+1$
710.2.a.i	5	degree 5	$5q - 5q^{2} - q^{3} + 5q^{4} + 5q^{5} + q^{6} - q^{7} + O(q^{8})$	+ − +	$+1$
710.2.a.j	5	degree 5	$5q + 5q^{2} + 3q^{3} + 5q^{4} - 5q^{5} + 3q^{6} + 3q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
710.2.a.a	$x + 15$
710.2.a.b	$x + 21$
710.2.a.c	$x + 61$
710.2.a.d	$x - 11$
710.2.a.e	$x^{2} + 25x + 50$
710.2.a.f	$x^{2} + 40x + 360$
710.2.a.g	$x^{3} - 60x^{2} + 592x - 436$
710.2.a.h	$x^{4} - 81x^{3} + 2036x^{2} - 15976x + 37308$
710.2.a.i	$x^{5} - 51x^{4} - 402x^{3} + 66920x^{2} - 1356380x + 8522388$
710.2.a.j	$x^{5} - 77x^{4} + 206x^{3} + 46256x^{2} + 393820x + 749292$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(710))$	25
Old forms	80
Cusp forms $S_{2}(\Gamma_0(710))$	105
Eisenstein series $E_{2}(\Gamma_0(710))$	7
Modular forms $M_{2}(\Gamma_0(710))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(710)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(142))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(355))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.346 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, 5, 10, 71, 142, 355, 710]: [0, 1, 1, 3, 7, 20, 38, 112] (certified at ell = 2147482681)
T = [(3, 1), (7, 3), (11, 5), (13, 7)]; h of degree 25 = 1 x 25 (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(710, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 710:2 1000

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