Atlas › Modular forms › Weight 2 › Level 368 › 368.2.a

Newspace 368.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(368))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
368.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} + 2q^{7} + O(q^{8})$	+	368g1 rank 1
368.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{7} + O(q^{8})$	−	368e1 rank 1
368.2.a.c	1	$\mathbb{Q}$	$q - 4q^{7} + O(q^{8})$	+	368a1 rank 1
368.2.a.d	1	$\mathbb{Q}$	$q + 4q^{5} + 4q^{7} + O(q^{8})$	+	368b1 rank 0
368.2.a.e	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} - 2q^{7} + O(q^{8})$	+	368d1 rank 1
368.2.a.f	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} + 4q^{7} + O(q^{8})$	−	368c1 rank 0
368.2.a.g	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{5} + 4q^{7} + O(q^{8})$	+	368f1 rank 0
368.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{5} - 2q^{7} + O(q^{8})$	+
368.2.a.i	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} + 4q^{5} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{7} + 7T_{19} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43} + 13T_{47} + 14T_{53} + 15T_{59} + 16T_{61}$ 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 $T$
368.2.a.a	$x + 328$
368.2.a.b	$x - 258$
368.2.a.c	$x + 148$
368.2.a.d	$x + 420$
368.2.a.e	$x + 44$
368.2.a.f	$x + 196$
368.2.a.g	$x + 102$
368.2.a.h	$x^{2} + 30x - 68220$
368.2.a.i	$x^{2} - 308x - 832$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(368))$	11
Old forms	32
Cusp forms $S_{2}(\Gamma_0(368))$	43
Eisenstein series $E_{2}(\Gamma_0(368))$	11
Modular forms $M_{2}(\Gamma_0(368))$	54

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(368)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(46))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(92))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(184))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.736 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, 16, 23, 46, 92, 184, 368]: [0, 1, 2, 3, 4, 3, 8, 15, 28, 52] (certified at ell = 2147483053)
T = [(3, 1), (5, 2), (11, 3), (13, 4), (17, 5), (7, 6), (19, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12), (47, 13), (53, 14), (59, 15), (61, 16)]; h of degree 11 = 1 x 11 (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(368, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 368:2 1000

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