Atlas › Modular forms › Weight 2 › Level 481 › 481.2.a

Newspace 481.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(481))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
481.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 2q^{5} + 2q^{7} + O(q^{8})$	+ +	$-1$	481a1 rank 1
481.2.a.b	7	degree 7	$7q - 5q^{2} + q^{3} + 7q^{4} - 4q^{5} - 6q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
481.2.a.c	7	degree 7	$7q - q^{2} - 7q^{3} + 3q^{4} - 2q^{5} - 4q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$
481.2.a.d	11	degree 11	$11q + 3q^{2} + q^{3} + 15q^{4} + 4q^{5} + 4q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
481.2.a.e	11	degree 11	$11q + 3q^{2} + 5q^{3} + 11q^{4} + 2q^{5} - 6q^{6} + O(q^{8})$	− +	$+1$

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 $T$
481.2.a.a	$x - 1$
481.2.a.b	$x^{7} + 2x^{6} - 94x^{5} - 231x^{4} + 2447x^{3} + 5741x^{2} - 19679x - 38651$
481.2.a.c	$x^{7} + 22x^{6} + 142x^{5} + 65x^{4} - 1759x^{3} - 2643x^{2} + 2403x + 2577$
481.2.a.d	$x^{11} - 6x^{10} - 224x^{9} + 1293x^{8} + 18094x^{7} - 97185x^{6} - 641889x^{5} + 3001738x^{4} + 9815003x^{3} - 35247667x^{2} - 47630405x + 107849999$
481.2.a.e	$x^{11} - 18x^{10} - 48x^{9} + 2061x^{8} - 2032x^{7} - 81065x^{6} + 138457x^{5} + 1192066x^{4} - 2037683x^{3} - 4796859x^{2} + 4307477x + 2145467$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(481))$	37
Old forms	4
Cusp forms $S_{2}(\Gamma_0(481))$	41
Eisenstein series $E_{2}(\Gamma_0(481))$	3
Modular forms $M_{2}(\Gamma_0(481))$	44

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(481)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.199 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, 13, 37, 481]: [0, 1, 3, 44] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 37 = 1 x 37 (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(481, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 481:2 1000

← 480.2.a · 482.2.a →