Atlas › Modular forms › Weight 4 › Level 238 › 238.4.a

Newspace 238.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(238))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
238.4.a.a	2	$\mathbb{Q}(\sqrt{93})$	$2q - 4q^{2} - 5q^{3} + 8q^{4} + 9q^{5} + 10q^{6} - 14q^{7} + O(q^{8})$	+ + −	$-1$
238.4.a.b	2	$\mathbb{Q}(\sqrt{69})$	$2q + 4q^{2} - 11q^{3} + 8q^{4} - 7q^{5} - 22q^{6} + 14q^{7} + O(q^{8})$	− − −	$-1$
238.4.a.c	3	degree 3	$3q - 6q^{2} - 5q^{3} + 12q^{4} - 21q^{5} + 10q^{6} + 21q^{7} + O(q^{8})$	+ − +	$-1$
238.4.a.d	3	degree 3	$3q - 6q^{2} + q^{3} + 12q^{4} - 5q^{5} - 2q^{6} - 21q^{7} + O(q^{8})$	+ + +	$+1$
238.4.a.e	3	degree 3	$3q + 6q^{2} - 5q^{3} + 12q^{4} - 15q^{5} - 10q^{6} - 21q^{7} + O(q^{8})$	− + +	$-1$
238.4.a.f	3	degree 3	$3q + 6q^{2} + 7q^{3} + 12q^{4} + 19q^{5} + 14q^{6} + 21q^{7} + O(q^{8})$	− − +	$+1$
238.4.a.g	4	degree 4	$4q - 8q^{2} + q^{3} + 16q^{4} + 33q^{5} - 2q^{6} + 28q^{7} + O(q^{8})$	+ − −	$+1$
238.4.a.h	4	degree 4	$4q + 8q^{2} + q^{3} + 16q^{4} + 19q^{5} + 2q^{6} - 28q^{7} + O(q^{8})$	− + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
238.4.a.a	$x^{2} - 22x - 251$
238.4.a.b	$x^{2} + 32x + 187$
238.4.a.c	$x^{3} + 68x^{2} + 469x - 22062$
238.4.a.d	$x^{3} + 14x^{2} - 2689x - 66482$
238.4.a.e	$x^{3} + 50x^{2} - 369x - 9286$
238.4.a.f	$x^{3} - 64x^{2} + 629x + 4726$
238.4.a.g	$x^{4} - 100x^{3} + 2679x^{2} - 8068x - 130484$
238.4.a.h	$x^{4} - 58x^{3} - 2335x^{2} + 174148x - 1905948$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(238))$	24
Old forms	80
Cusp forms $S_{4}(\Gamma_0(238))$	104
Eisenstein series $E_{4}(\Gamma_0(238))$	8
Modular forms $M_{4}(\Gamma_0(238))$	112

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(238)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(119))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 17, 34, 119, 238]: [1, 2, 3, 8, 6, 16, 38, 112] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 24 = 1 x 24 (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(238, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 238:4 1000

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