Atlas › Modular forms › Weight 2 › Level 940 › 940.2.a

Newspace 940.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(940))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
940.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - q^{5} + 2q^{7} + O(q^{8})$	+ +	940a1 rank 0
940.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + q^{7} + O(q^{8})$	− +	940d1 rank 1
940.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - q^{7} + O(q^{8})$	+ −	940c1 rank 1
940.2.a.d	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} + 2q^{7} + O(q^{8})$	− −	940e1 rank 0
940.2.a.e	1	$\mathbb{Q}$	$q + 3q^{3} - q^{5} - 3q^{7} + O(q^{8})$	+ +	940b1 rank 0
940.2.a.f	2	$\mathbb{Q}(\sqrt{13})$	$2q - 3q^{3} - 2q^{5} - q^{7} + O(q^{8})$	+ −
940.2.a.g	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{3} + 2q^{5} - 5q^{7} + O(q^{8})$	− +
940.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{3} - 2q^{5} + 3q^{7} + O(q^{8})$	+ +
940.2.a.i	3	degree 3	$3q + 2q^{3} + 3q^{5} - 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{17} + 9T_{19} + 11T_{23}$ 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 $T$
940.2.a.a	$x - 13$
940.2.a.b	$x + 78$
940.2.a.c	$x + 38$
940.2.a.d	$x + 41$
940.2.a.e	$x - 58$
940.2.a.f	$x^{2} + 119x + 2991$
940.2.a.g	$x^{2} + 179x + 7461$
940.2.a.h	$x^{2} - 213x + 7561$
940.2.a.i	$x^{3} - 207x^{2} + 11877x - 207398$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(940))$	14
Old forms	125
Cusp forms $S_{2}(\Gamma_0(940))$	139
Eisenstein series $E_{2}(\Gamma_0(940))$	11
Modular forms $M_{2}(\Gamma_0(940))$	150

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(940)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(94))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(188))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(235))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(470))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.642 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, 5, 10, 20, 47, 94, 188, 235, 470, 940]: [0, 1, 2, 1, 3, 6, 5, 14, 27, 26, 76, 150] (certified at ell = 2147483137)
T = [(3, 1), (7, 3), (11, 5), (17, 7), (19, 9), (23, 11)]; h of degree 14 = 1 x 14 (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(940, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 940:2 1000

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