Atlas › Modular forms › Weight 2 › Level 350 › 350.2.a

Newspace 350.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(350))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
350.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} + O(q^{8})$	+ +	350c1 rank 1
350.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{7} + O(q^{8})$	+ −	350a1 rank 0
350.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} + q^{4} - 3q^{6} + q^{7} + O(q^{8})$	+ −	350e1 rank 0
350.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} - 3q^{6} - q^{7} + O(q^{8})$	− +	350f1 rank 1
350.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + O(q^{8})$	− −	350b1 rank 0
350.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} - q^{7} + O(q^{8})$	− +	350d1 rank 0
350.2.a.g	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{2} + 2q^{4} - 2q^{7} + O(q^{8})$	+ +
350.2.a.h	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{2} + 2q^{4} + 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{11} + 3T_{19} + 4T_{23} + 5T_{29} + 6T_{13} + 7T_{17} + 8T_{31}$ 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 $T$
350.2.a.a	$x + 111$
350.2.a.b	$x - 124$
350.2.a.c	$x + 35$
350.2.a.d	$x + 127$
350.2.a.e	$x + 43$
350.2.a.f	$x + 72$
350.2.a.g	$x^{2} - 144x - 960$
350.2.a.h	$x^{2} - 72x + 1080$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(350))$	10
Old forms	39
Cusp forms $S_{2}(\Gamma_0(350))$	49
Eisenstein series $E_{2}(\Gamma_0(350))$	23
Modular forms $M_{2}(\Gamma_0(350))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(350)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(70))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(175))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.44 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, 7, 10, 14, 25, 35, 50, 70, 175, 350]: [0, 1, 1, 1, 3, 4, 3, 6, 9, 16, 22, 64] (certified at ell = 2147482921)
T = [(3, 1), (11, 2), (19, 3), (23, 4), (29, 5), (13, 6), (17, 7), (31, 8)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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(350, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 350:2 1000

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