Atlas › Modular forms › Weight 2 › Level 155 › 155.2.a

Newspace 155.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(155))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
155.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	155a1 rank 1
155.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} - q^{4} - q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$	155b1 rank 0
155.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - q^{5} + O(q^{8})$	+ +	$-1$	155c1 rank 1
155.2.a.d	4	degree 4	$4q - q^{2} - q^{3} + 9q^{4} - 4q^{5} + 8q^{6} + O(q^{8})$	+ −	$+1$
155.2.a.e	4	degree 4	$4q + q^{2} + q^{3} + 5q^{4} + 4q^{5} - 4q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$

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 $T$
155.2.a.a	$x + 3$
155.2.a.b	$x - 1$
155.2.a.c	$x + 1$
155.2.a.d	$x^{4} + 2x^{3} - 24x^{2} - 32x + 101$
155.2.a.e	$x^{4} - 2x^{3} - 6x^{2} + 12x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(155))$	11
Old forms	4
Cusp forms $S_{2}(\Gamma_0(155))$	15
Eisenstein series $E_{2}(\Gamma_0(155))$	3
Modular forms $M_{2}(\Gamma_0(155))$	18

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.178 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]: [0, 1, 3, 18] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 155:2 1000

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