Atlas › Modular forms › Weight 4 › Level 138 › 138.4.a

Newspace 138.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(138))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
138.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} - 10q^{5} + 6q^{6} + 32q^{7} + O(q^{8})$	+ + −	$-1$
138.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} + 2q^{5} - 6q^{6} - 32q^{7} + O(q^{8})$	+ − +	$-1$
138.4.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} - 2q^{5} - 6q^{6} - 34q^{7} + O(q^{8})$	− + +	$-1$
138.4.a.d	2	$\mathbb{Q}(\sqrt{277})$	$2q - 4q^{2} + 6q^{3} + 8q^{4} + 2q^{5} - 12q^{6} + 28q^{7} + O(q^{8})$	+ − −	$+1$
138.4.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{2} - 6q^{3} + 8q^{4} + 8q^{5} - 12q^{6} + 12q^{7} + O(q^{8})$	− + −	$+1$
138.4.a.f	3	degree 3	$3q + 6q^{2} + 9q^{3} + 12q^{4} + 20q^{5} + 18q^{6} + 10q^{7} + O(q^{8})$	− − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
138.4.a.a	$x - 86$
138.4.a.b	$x + 94$
138.4.a.c	$x + 104$
138.4.a.d	$x^{2} - 86x + 1572$
138.4.a.e	$x^{2} - 44x - 28$
138.4.a.f	$x^{3} - 50x^{2} - 2084x + 61992$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(138))$	10
Old forms	58
Cusp forms $S_{4}(\Gamma_0(138))$	68
Eisenstein series $E_{4}(\Gamma_0(138))$	8
Modular forms $M_{4}(\Gamma_0(138))$	76

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(138)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(46))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(69))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.14 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, 23, 46, 69, 138]: [1, 2, 2, 5, 7, 20, 26, 76] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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(138, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 138:4 1000

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