Atlas › Modular forms › Weight 2 › Level 616 › 616.2.a

Newspace 616.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(616))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
616.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + 2q^{5} + q^{7} + O(q^{8})$	− +	616c1 rank 0
616.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + q^{7} + O(q^{8})$	− −	616d1 rank 1
616.2.a.c	1	$\mathbb{Q}$	$q - 2q^{5} + q^{7} + O(q^{8})$	− +	616e1 rank 1
616.2.a.d	1	$\mathbb{Q}$	$q - q^{7} + O(q^{8})$	+ +	616a1 rank 1
616.2.a.e	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{5} + q^{7} + O(q^{8})$	− +	616b1 rank 0
616.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} - 3q^{5} - 2q^{7} + O(q^{8})$	+ −
616.2.a.g	3	degree 3	$3q + q^{3} + q^{5} + 3q^{7} + O(q^{8})$	− −
616.2.a.h	4	degree 4	$4q + q^{3} + 5q^{5} - 4q^{7} + O(q^{8})$	+ −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{13} + 7T_{17} + 9T_{19} + 11T_{23}$ 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 $T$
616.2.a.a	$x - 32$
616.2.a.b	$x + 113$
616.2.a.c	$x + 134$
616.2.a.d	$x + 4$
616.2.a.e	$x - 28$
616.2.a.f	$x^{2} + 143x + 5006$
616.2.a.g	$x^{3} - 95x^{2} - 21104x + 1134896$
616.2.a.h	$x^{4} - 127x^{3} + 1384x^{2} + 55272x - 789872$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(616))$	14
Old forms	75
Cusp forms $S_{2}(\Gamma_0(616))$	89
Eisenstein series $E_{2}(\Gamma_0(616))$	15
Modular forms $M_{2}(\Gamma_0(616))$	104

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(616)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(77))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(88))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(154))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(308))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.392 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, 7, 8, 11, 14, 22, 28, 44, 56, 77, 88, 154, 308, 616]: [0, 1, 2, 1, 3, 2, 4, 5, 7, 9, 12, 10, 16, 28, 54, 104] (certified at ell = 2147482951)
T = [(3, 1), (5, 3), (13, 5), (17, 7), (19, 9), (23, 11)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(616, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 616:2 1000

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