Atlas › Modular forms › Weight 2 › Level 936 › 936.2.a

Newspace 936.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(936))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
936.2.a.a	1	$\mathbb{Q}$	$q - 4q^{5} + O(q^{8})$	+	936d1 rank 0
936.2.a.b	1	$\mathbb{Q}$	$q - 2q^{5} + O(q^{8})$	−	936e1 rank 1
936.2.a.c	1	$\mathbb{Q}$	$q - 2q^{5} + 2q^{7} + O(q^{8})$	+	936a1 rank 1
936.2.a.d	1	$\mathbb{Q}$	$q - 4q^{7} + O(q^{8})$	+	936h1 rank 1
936.2.a.e	1	$\mathbb{Q}$	$q + O(q^{8})$	+	936g1 rank 1
936.2.a.f	1	$\mathbb{Q}$	$q + q^{5} + 5q^{7} + O(q^{8})$	+	936b1 rank 0
936.2.a.g	1	$\mathbb{Q}$	$q + 2q^{5} + 2q^{7} + O(q^{8})$	+	936f1 rank 0
936.2.a.h	1	$\mathbb{Q}$	$q + 2q^{5} + 4q^{7} + O(q^{8})$	−	936i1 rank 0
936.2.a.i	1	$\mathbb{Q}$	$q + 4q^{5} - 4q^{7} + O(q^{8})$	+	936c1 rank 0
936.2.a.j	2	$\mathbb{Q}(\sqrt{17})$	$2q - 3q^{5} - q^{7} + O(q^{8})$	−
936.2.a.k	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{7} + O(q^{8})$	−
936.2.a.l	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
936.2.a.a	$x - 310$
936.2.a.b	$x + 234$
936.2.a.c	$x + 50$
936.2.a.d	$x + 242$
936.2.a.e	$x + 134$
936.2.a.f	$x + 95$
936.2.a.g	$x - 66$
936.2.a.h	$x - 30$
936.2.a.i	$x - 106$
936.2.a.j	$x^{2} - 567x + 80334$
936.2.a.k	$x^{2} + 204x - 21348$
936.2.a.l	$x^{2} - 156x + 4284$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(936))$	15
Old forms	138
Cusp forms $S_{2}(\Gamma_0(936))$	153
Eisenstein series $E_{2}(\Gamma_0(936))$	31
Modular forms $M_{2}(\Gamma_0(936))$	184

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.145 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, 3, 4, 6, 8, 9, 12, 13, 18, 24, 26, 36, 39, 52, 72, 78, 104, 117, 156, 234, 312, 468, 936]: [0, 1, 1, 2, 3, 3, 2, 5, 1, 5, 8, 5, 9, 6, 10, 16, 18, 18, 16, 34, 46, 64, 90, 176] (certified at ell = 2147483629)
T = [(5, 1), (7, 2), (11, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 5 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(936, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 936:2 1000

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