Atlas › Modular forms › Weight 2 › Level 335 › 335.2.a

Newspace 335.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(335))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
335.2.a.a	1	$\mathbb{Q}$	$q - 2q^{4} + q^{5} - 2q^{7} + O(q^{8})$	− −	$-1$	335a1 rank 1
335.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{5} - 4q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
335.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} - 2q^{5} + 5q^{6} + O(q^{8})$	+ −	$+1$
335.2.a.d	7	degree 7	$7q + 2q^{2} + 4q^{3} + 14q^{4} - 7q^{5} - 9q^{6} + 10q^{7} + O(q^{8})$	+ −	$+1$
335.2.a.e	11	degree 11	$11q + 14q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
335.2.a.a	$x$
335.2.a.b	$x^{2} - 8$
335.2.a.c	$x^{2} - x - 61$
335.2.a.d	$x^{7} - 14x^{6} - 33x^{5} + 1038x^{4} - 1707x^{3} - 14725x^{2} + 19029x + 62508$
335.2.a.e	$x^{11} - 273x^{9} + 239x^{8} + 25323x^{7} - 47964x^{6} - 950391x^{5} + 2548553x^{4} + 12225995x^{3} - 35463805x^{2} - 41689188x + 109010584$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(335))$	23
Old forms	10
Cusp forms $S_{2}(\Gamma_0(335))$	33
Eisenstein series $E_{2}(\Gamma_0(335))$	3
Modular forms $M_{2}(\Gamma_0(335))$	36

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(335)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(67))^{\oplus 2}$

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, 5, 67, 335]: [0, 1, 6, 36] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 3 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(335, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 335:2 1000

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