Atlas › Modular forms › Weight 2 › Level 438 › 438.2.a

Newspace 438.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(438))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
438.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} - 2q^{7} + O(q^{8})$	+ + +	$-1$	438c1 rank 1
438.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 4q^{5} - q^{6} + O(q^{8})$	+ − −	$-1$	438g1 rank 1
438.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - 4q^{7} + O(q^{8})$	+ − −	$-1$	438d1 rank 1
438.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ − +	$+1$	438e1 rank 0
438.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− + −	$-1$	438f1 rank 1
438.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} - 2q^{7} + O(q^{8})$	− − −	$+1$	438b1 rank 0
438.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$	438a1 rank 0
438.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} + O(q^{8})$	+ + −	$+1$
438.2.a.i	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ 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 $T$
438.2.a.a	$x + 152$
438.2.a.b	$x + 142$
438.2.a.c	$x - 26$
438.2.a.d	$x - 148$
438.2.a.e	$x - 14$
438.2.a.f	$x + 318$
438.2.a.g	$x - 50$
438.2.a.h	$x^{2} - 500x + 62468$
438.2.a.i	$x^{2} - 84x - 656$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(438))$	11
Old forms	60
Cusp forms $S_{2}(\Gamma_0(438))$	71
Eisenstein series $E_{2}(\Gamma_0(438))$	7
Modular forms $M_{2}(\Gamma_0(438))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(438)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(146))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(219))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.691 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, 6, 73, 146, 219, 438]: [0, 1, 1, 3, 6, 20, 26, 78] (certified at ell = 2147482873)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; 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(438, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 438:2 1000

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