Atlas › Modular forms › Weight 2 › Level 359 › 359.2.a

Newspace 359.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(359))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
359.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} - q^{7} + O(q^{8})$	+	$-1$	359b1 rank 1
359.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} + q^{5} - 2q^{6} + q^{7} + O(q^{8})$	+	$-1$	359a1 rank 1
359.2.a.c	4	degree 4	$4q - 2q^{2} - q^{3} + 2q^{4} - 6q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+	$-1$
359.2.a.d	24	degree 24	$24q + q^{2} + 5q^{3} + 31q^{4} + 6q^{5} + 7q^{6} + 5q^{7} + O(q^{8})$	−	$+1$

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 $T$
359.2.a.a	$x + 1$
359.2.a.b	$x + 5$
359.2.a.c	$x^{4} + 5x^{3} - 36x^{2} - 44x - 8$
359.2.a.d	$x^{24} - 16x^{23} - 432x^{22} + 7916x^{21} + 71773x^{20} - 1647787x^{19} - 5314242x^{18} + 189413891x^{17} + 79776321x^{16} - 13299677197x^{15} + 15341122200x^{14} + 597599376360x^{13} - 1254390106026x^{12} - 17542145001610x^{11} + 46730428439423x^{10} + 337385732003421x^{9} - 997367055335226x^{8} - 4179745081758668x^{7} + 12579163914229316x^{6} + 31772047024681079x^{5} - 89283636557071556x^{4} - 132128550962174365x^{3} + 305461123745837596x^{2} + 218978553474786084x - 315336482962270584$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(359))$	30
Old forms	0
Cusp forms $S_{2}(\Gamma_0(359))$	30
Eisenstein series $E_{2}(\Gamma_0(359))$	1
Modular forms $M_{2}(\Gamma_0(359))$	31

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.184 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, 359]: [0, 31] (certified at ell = 2147476963)
T = [(2, 1), (3, 3)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 4 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(359, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 359:2 1000

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