Atlas › Modular forms › Weight 4 › Level 119 › 119.4.a

Newspace 119.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(119))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
119.4.a.a	1	$\mathbb{Q}$	$q - q^{2} - 6q^{3} - 7q^{4} - 20q^{5} + 6q^{6} - 7q^{7} + O(q^{8})$	+ +	$+1$
119.4.a.b	3	degree 3	$3q - q^{2} + 5q^{3} + 5q^{4} - 19q^{5} - 22q^{6} - 21q^{7} + O(q^{8})$	+ −	$-1$
119.4.a.c	4	degree 4	$4q - 2q^{2} - 7q^{3} - 2q^{4} - 9q^{5} + 16q^{6} + 28q^{7} + O(q^{8})$	− +	$-1$
119.4.a.d	7	degree 7	$7q + 4q^{2} + 5q^{3} + 52q^{4} + 35q^{5} + 51q^{6} - 49q^{7} + O(q^{8})$	+ +	$+1$
119.4.a.e	9	degree 9	$9q + 2q^{2} + 11q^{3} + 38q^{4} - 3q^{5} + 9q^{6} + 63q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
119.4.a.a	$x + 7$
119.4.a.b	$x^{3} - 4x^{2} - 10x - 3$
119.4.a.c	$x^{4} + 9x^{3} - 30x^{2} - 103x + 127$
119.4.a.d	$x^{7} - 9x^{6} - 223x^{5} + 1816x^{4} + 13472x^{3} - 105827x^{2} - 188571x + 1542267$
119.4.a.e	$x^{9} - 13x^{8} - 138x^{7} + 1941x^{6} + 6333x^{5} - 88583x^{4} - 148807x^{3} + 1123722x^{2} + 2670975x + 1377081$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(119))$	24
Old forms	10
Cusp forms $S_{4}(\Gamma_0(119))$	34
Eisenstein series $E_{4}(\Gamma_0(119))$	4
Modular forms $M_{4}(\Gamma_0(119))$	38

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 7, 17, 119]: [1, 3, 6, 38] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (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(119, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 119:4 1000

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