Atlas › Modular forms › Weight 2 › Level 490 › 490.2.a

Newspace 490.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(490))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
490.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} + O(q^{8})$	+ −	490c1 rank 0
490.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + O(q^{8})$	+ −	490d1 rank 1
490.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + O(q^{8})$	+ +	490a1 rank 1
490.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - q^{5} - 2q^{6} + O(q^{8})$	+ +	490b1 rank 0
490.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} + q^{5} - 3q^{6} + O(q^{8})$	− −	490k1 rank 0
490.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + O(q^{8})$	− +	490g1 rank 1
490.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + O(q^{8})$	− +	490e1 rank 0
490.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{5} + O(q^{8})$	− −	490h1 rank 0
490.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + q^{5} + 2q^{6} + O(q^{8})$	− −	490j1 rank 0
490.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + q^{5} + 2q^{6} + O(q^{8})$	− −	490i1 rank 0
490.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} + q^{4} - q^{5} + 3q^{6} + O(q^{8})$	− +	490f1 rank 0
490.2.a.l	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} + 4q^{6} + O(q^{8})$	+ +
490.2.a.m	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 4q^{3} + 2q^{4} + 2q^{5} - 4q^{6} + O(q^{8})$	+ −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{11} + 3T_{19} + 4T_{23} + 5T_{29} + 6T_{13} + 7T_{17} + 8T_{31}$ 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 $T$
490.2.a.a	$x - 83$
490.2.a.b	$x + 86$
490.2.a.c	$x - 8$
490.2.a.d	$x - 61$
490.2.a.e	$x - 128$
490.2.a.f	$x + 138$
490.2.a.g	$x - 23$
490.2.a.h	$x + 4$
490.2.a.i	$x - 30$
490.2.a.j	$x + 47$
490.2.a.k	$x + 22$
490.2.a.l	$x^{2} + 92x + 194$
490.2.a.m	$x^{2} - 4x - 3694$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(490))$	15
Old forms	54
Cusp forms $S_{2}(\Gamma_0(490))$	69
Eisenstein series $E_{2}(\Gamma_0(490))$	31
Modular forms $M_{2}(\Gamma_0(490))$	100

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(490)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(70))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(98))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(245))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.449 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, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490]: [0, 1, 1, 1, 3, 4, 6, 5, 16, 16, 30, 88] (certified at ell = 2147483353)
T = [(3, 1), (11, 2), (19, 3), (23, 4), (29, 5), (13, 6), (17, 7), (31, 8)]; h of degree 15 = 1 x 15 (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(490, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 490:2 1000

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