Atlas › Modular forms › Weight 2 › Level 938 › 938.2.a

Newspace 938.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(938))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
938.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 2q^{5} + 2q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	938b1 rank 1
938.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	938a1 rank 1
938.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	938c1 rank 1
938.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 3q^{5} + q^{6} + q^{7} + O(q^{8})$	− − −	$+1$	938d1 rank 0
938.2.a.e	3	degree 3	$3q - 3q^{2} + q^{3} + 3q^{4} - 3q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ + −	$+1$
938.2.a.f	4	degree 4	$4q + 4q^{2} - 4q^{3} + 4q^{4} - 6q^{5} - 4q^{6} - 4q^{7} + O(q^{8})$	− + −	$-1$
938.2.a.g	4	degree 4	$4q + 4q^{2} + q^{3} + 4q^{4} + 5q^{5} + q^{6} - 4q^{7} + O(q^{8})$	− + +	$+1$
938.2.a.h	5	degree 5	$5q - 5q^{2} + 4q^{3} + 5q^{4} - 2q^{5} - 4q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$
938.2.a.i	6	degree 6	$6q - 6q^{2} - 4q^{3} + 6q^{4} + 4q^{6} - 6q^{7} + O(q^{8})$	+ + +	$-1$
938.2.a.j	7	degree 7	$7q + 7q^{2} - q^{3} + 7q^{4} + q^{5} - q^{6} + 7q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
938.2.a.a	$x + 20$
938.2.a.b	$x + 59$
938.2.a.c	$x + 45$
938.2.a.d	$x - 69$
938.2.a.e	$x^{3} - 41x^{2} - 2832x + 110426$
938.2.a.f	$x^{4} + 194x^{3} + 11471x^{2} + 216973x + 1077119$
938.2.a.g	$x^{4} - 147x^{3} + 5550x^{2} - 8392x - 255784$
938.2.a.h	$x^{5} - 38x^{4} - 1271x^{3} + 42353x^{2} + 77609x - 4129288$
938.2.a.i	$x^{6} + 104x^{5} - 9209x^{4} - 900507x^{3} + 7235191x^{2} + 893680144x - 8307379436$
938.2.a.j	$x^{7} + 7x^{6} - 7150x^{5} - 64019x^{4} + 11212626x^{3} + 220710390x^{2} - 51845712x - 12409118104$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(938))$	33
Old forms	100
Cusp forms $S_{2}(\Gamma_0(938))$	133
Eisenstein series $E_{2}(\Gamma_0(938))$	7
Modular forms $M_{2}(\Gamma_0(938))$	140

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(938)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(67))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(134))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(469))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.51 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, 7, 14, 67, 134, 469, 938]: [0, 1, 1, 4, 6, 19, 46, 140] (certified at ell = 2147483647)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 9 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(938, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 938:2 1000

← 937.2.a · 939.2.a →