Atlas › Modular forms › Weight 2 › Level 416 › 416.2.a

Newspace 416.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(416))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
416.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + q^{5} - 3q^{7} + O(q^{8})$	−	416b1 rank 1
416.2.a.b	1	$\mathbb{Q}$	$q + q^{3} + q^{5} + 3q^{7} + O(q^{8})$	−	416a1 rank 0
416.2.a.c	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} - 3q^{5} - 3q^{7} + O(q^{8})$	+
416.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 6q^{5} + O(q^{8})$	+
416.2.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} - 3q^{5} + 3q^{7} + O(q^{8})$	+
416.2.a.f	4	degree 4	$4q - 2q^{5} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
416.2.a.a	$x - 2$
416.2.a.b	$x - 4$
416.2.a.c	$x^{2} + 10x + 8$
416.2.a.d	$x^{2} - 18x + 76$
416.2.a.e	$x^{2} + 8x - 52$
416.2.a.f	$x^{4} + 6x^{3} - 184x^{2} - 456x + 7088$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(416))$	12
Old forms	37
Cusp forms $S_{2}(\Gamma_0(416))$	49
Eisenstein series $E_{2}(\Gamma_0(416))$	15
Modular forms $M_{2}(\Gamma_0(416))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(416)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(104))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(208))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 4, 8, 13, 16, 26, 32, 52, 104, 208, 416]: [0, 1, 2, 3, 1, 4, 5, 6, 10, 18, 32, 60] (certified at ell = 2147483497)
T = [(3, 1), (5, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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(416, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 416:2 1000

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