Atlas › Modular forms › Weight 2 › Level 611 › 611.2.a

Newspace 611.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(611))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
611.2.a.a	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} + 6q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$	611a1 rank 0
611.2.a.b	5	degree 5	$5q - 2q^{3} - 5q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
611.2.a.c	9	degree 9	$9q - 3q^{2} - 2q^{3} + 3q^{4} - 7q^{5} - 3q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
611.2.a.d	14	degree 14	$14q + 18q^{4} + 11q^{5} + 9q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$
611.2.a.e	18	degree 18	$18q - 2q^{2} + q^{3} + 26q^{4} + 9q^{5} - 7q^{6} - q^{7} + O(q^{8})$	− +	$+1$

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 $T$
611.2.a.a	$x - 5$
611.2.a.b	$x^{5} + 2x^{4} - 7x^{3} + x^{2} + 3x - 1$
611.2.a.c	$x^{9} + 5x^{8} - 11x^{7} - 72x^{6} + 13x^{5} + 294x^{4} + 132x^{3} - 355x^{2} - 281x - 19$
611.2.a.d	$x^{14} - 63x^{12} + 21x^{11} + 1384x^{10} - 750x^{9} - 13121x^{8} + 8097x^{7} + 55951x^{6} - 31058x^{5} - 110487x^{4} + 43687x^{3} + 98867x^{2} - 19716x - 32837$
611.2.a.e	$x^{18} + x^{17} - 61x^{16} - 46x^{15} + 1428x^{14} + 1100x^{13} - 16872x^{12} - 15329x^{11} + 107312x^{10} + 119767x^{9} - 351072x^{8} - 490649x^{7} + 474825x^{6} + 926635x^{5} - 29144x^{4} - 627202x^{3} - 287327x^{2} - 3340x + 11647$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(611))$	47
Old forms	8
Cusp forms $S_{2}(\Gamma_0(611))$	55
Eisenstein series $E_{2}(\Gamma_0(611))$	3
Modular forms $M_{2}(\Gamma_0(611))$	58

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(611)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.214 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, 47, 611]: [0, 1, 5, 58] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 47 = 1 x 47 (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(611, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 611:2 1000

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