Atlas › Modular forms › Weight 2 › Level 414 › 414.2.a

Newspace 414.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(414))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
414.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} + O(q^{8})$	+ −	414c1 rank 1
414.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 4q^{5} - 4q^{7} + O(q^{8})$	− +	414d1 rank 1
414.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{7} + O(q^{8})$	− −	414a1 rank 0
414.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} - 2q^{7} + O(q^{8})$	− −	414b1 rank 0
414.2.a.e	2	$\mathbb{Q}(\sqrt{7})$	$2q - 2q^{2} + 2q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	+ −
414.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} + O(q^{8})$	+ +
414.2.a.g	2	$\mathbb{Q}(\sqrt{7})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{29} + 8T_{31}$ 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 $T$
414.2.a.a	$x + 118$
414.2.a.b	$x + 42$
414.2.a.c	$x - 34$
414.2.a.d	$x - 30$
414.2.a.e	$x^{2} - 188x + 7828$
414.2.a.f	$x^{2} - 80x - 820$
414.2.a.g	$x^{2} - 28x - 1596$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(414))$	10
Old forms	55
Cusp forms $S_{2}(\Gamma_0(414))$	65
Eisenstein series $E_{2}(\Gamma_0(414))$	15
Modular forms $M_{2}(\Gamma_0(414))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(414)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(46))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(69))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(138))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(207))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.461 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, 23, 46, 69, 138, 207, 414]: [0, 1, 1, 3, 2, 5, 3, 8, 10, 28, 26, 76] (certified at ell = 2147483647)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (29, 7), (31, 8)]; h of degree 10 = 1 x 10 (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(414, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 414:2 1000

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