Atlas › Modular forms › Weight 2 › Level 352 › 352.2.a

Newspace 352.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(352))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
352.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} + q^{5} + O(q^{8})$	−	352f1 rank 1
352.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 3q^{5} + 4q^{7} + O(q^{8})$	+	352d1 rank 1
352.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + q^{5} - 4q^{7} + O(q^{8})$	+	352c1 rank 1
352.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} - 4q^{7} + O(q^{8})$	−	352b1 rank 1
352.2.a.e	1	$\mathbb{Q}$	$q + q^{3} + q^{5} + 4q^{7} + O(q^{8})$	−	352a1 rank 0
352.2.a.f	1	$\mathbb{Q}$	$q + 3q^{3} + q^{5} + O(q^{8})$	+	352e1 rank 0
352.2.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} + 3q^{5} + O(q^{8})$	+
352.2.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} + 3q^{5} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
352.2.a.a	$x + 134$
352.2.a.b	$x + 82$
352.2.a.c	$x + 38$
352.2.a.d	$x - 38$
352.2.a.e	$x - 90$
352.2.a.f	$x + 14$
352.2.a.g	$x^{2} - 84x - 11564$
352.2.a.h	$x^{2} - 116x + 3092$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(352))$	10
Old forms	31
Cusp forms $S_{2}(\Gamma_0(352))$	41
Eisenstein series $E_{2}(\Gamma_0(352))$	15
Modular forms $M_{2}(\Gamma_0(352))$	56

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(352)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(88))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(176))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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, 4, 8, 11, 16, 22, 32, 44, 88, 176, 352]: [0, 1, 2, 3, 2, 4, 5, 6, 9, 16, 28, 52] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 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(352, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 352:2 1000

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