Atlas › Modular forms › Weight 2 › Level 522 › 522.2.a

Newspace 522.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(522))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
522.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 3q^{5} - q^{7} + O(q^{8})$	+ −	522c1 rank 0
522.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} - 2q^{7} + O(q^{8})$	+ −	522f1 rank 1
522.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + q^{7} + O(q^{8})$	+ −	522e1 rank 1
522.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + q^{7} + O(q^{8})$	+ +	522d1 rank 0
522.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+ −	522b1 rank 0
522.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 3q^{5} - 5q^{7} + O(q^{8})$	+ +	522a1 rank 1
522.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{5} - 5q^{7} + O(q^{8})$	− −	522i1 rank 1
522.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{5} - 3q^{7} + O(q^{8})$	− +	522j1 rank 1
522.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} + O(q^{8})$	− −	522k1 rank 0
522.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	− +	522g1 rank 0
522.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 3q^{5} - 2q^{7} + O(q^{8})$	− −	522l1 rank 0
522.2.a.l	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 3q^{5} - q^{7} + O(q^{8})$	− +	522h1 rank 0
522.2.a.m	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 3q^{5} + 5q^{7} + O(q^{8})$	− −	522m1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
522.2.a.a	$x - 212$
522.2.a.b	$x + 317$
522.2.a.c	$x + 152$
522.2.a.d	$x - 286$
522.2.a.e	$x - 112$
522.2.a.f	$x + 102$
522.2.a.g	$x - 14$
522.2.a.h	$x + 92$
522.2.a.i	$x + 114$
522.2.a.j	$x - 32$
522.2.a.k	$x + 97$
522.2.a.l	$x + 28$
522.2.a.m	$x + 34$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(522))$	13
Old forms	70
Cusp forms $S_{2}(\Gamma_0(522))$	83
Eisenstein series $E_{2}(\Gamma_0(522))$	15
Modular forms $M_{2}(\Gamma_0(522))$	98

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(522)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(87))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(174))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(261))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.74 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, 2, 3, 6, 9, 18, 29, 58, 87, 174, 261, 522]: [0, 1, 1, 3, 2, 5, 3, 9, 12, 34, 32, 94] (certified at ell = 2147483353)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 13 = 1 x 13 (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(522, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 522:2 1000

← 521.2.a · 523.2.a →