Atlas › Modular forms › Weight 2 › Level 235 › 235.2.a

Newspace 235.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(235))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
235.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	+ −	$+1$	235b1 rank 0
235.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + q^{7} + O(q^{8})$	− −	$-1$	235a1 rank 1
235.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{3} + 2q^{4} - q^{5} + 4q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$	235c1 rank 0
235.2.a.d	5	degree 5	$5q - 4q^{2} - 5q^{3} + 6q^{4} - 5q^{5} + q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
235.2.a.e	7	degree 7	$7q + q^{2} + q^{3} + 7q^{4} + 7q^{5} + q^{6} - 3q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
235.2.a.a	$x - 86$
235.2.a.b	$x + 94$
235.2.a.c	$x - 29$
235.2.a.d	$x^{5} + 227x^{4} + 18995x^{3} + 724177x^{2} + 12663129x + 81429688$
235.2.a.e	$x^{7} - 131x^{6} + 4906x^{5} + 12310x^{4} - 3898854x^{3} + 57876471x^{2} + 29352837x - 3289336256$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(235))$	15
Old forms	8
Cusp forms $S_{2}(\Gamma_0(235))$	23
Eisenstein series $E_{2}(\Gamma_0(235))$	3
Modular forms $M_{2}(\Gamma_0(235))$	26

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(235)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 47, 235]: [0, 1, 5, 26] (certified at ell = 2147483137)
T = [(2, 1), (3, 3), (11, 5), (13, 7), (17, 9)]; h of degree 15 = 1 x 15 (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(235, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 235:2 1000

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