Atlas › Modular forms › Weight 4 › Level 155 › 155.4.a

Newspace 155.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(155))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
155.4.a.a	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - 7q^{4} - 5q^{5} + 2q^{6} + 16q^{7} + O(q^{8})$	+ −	$-1$
155.4.a.b	4	degree 4	$4q + 2q^{2} - 4q^{3} + 6q^{4} - 20q^{5} - 18q^{6} - 24q^{7} + O(q^{8})$	+ −	$-1$
155.4.a.c	6	degree 6	$6q - 3q^{2} - 12q^{3} + 11q^{4} + 30q^{5} - 36q^{6} - 34q^{7} + O(q^{8})$	− +	$-1$
155.4.a.d	9	degree 9	$9q - q^{2} + 4q^{3} + 39q^{4} - 45q^{5} + 20q^{6} + 20q^{7} + O(q^{8})$	+ +	$+1$
155.4.a.e	10	degree 10	$10q + 9q^{2} + 6q^{3} + 51q^{4} + 50q^{5} + 50q^{7} + O(q^{8})$	− −	$+1$

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 $T$
155.4.a.a	$x - 3$
155.4.a.b	$x^{4} + 2x^{3} - 76x^{2} + 74x + 507$
155.4.a.c	$x^{6} + 15x^{5} + 20x^{4} - 252x^{3} - 969x^{2} - 1219x - 484$
155.4.a.d	$x^{9} - 3x^{8} - 280x^{7} + 592x^{6} + 25246x^{5} - 22774x^{4} - 847524x^{3} - 243820x^{2} + 9335981x + 12048357$
155.4.a.e	$x^{10} - 15x^{9} - 138x^{8} + 2394x^{7} + 6896x^{6} - 138888x^{5} - 157858x^{4} + 3451578x^{3} + 1886739x^{2} - 30779253x - 13200704$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(155))$	30
Old forms	16
Cusp forms $S_{4}(\Gamma_0(155))$	46
Eisenstein series $E_{4}(\Gamma_0(155))$	4
Modular forms $M_{4}(\Gamma_0(155))$	50

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(155)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.121 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, 5, 31, 155]: [1, 3, 9, 50] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 30 = 1 x 30 (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(155, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 155:4 1000

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