Atlas › Modular forms › Weight 2 › Level 614 › 614.2.a

Newspace 614.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(614))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
614.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - q^{7} + O(q^{8})$	− −	$-1$	614b1 rank 1
614.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} - 3q^{7} + O(q^{8})$	− −	$-1$	614a1 rank 1
614.2.a.c	3	degree 3	$3q - 3q^{2} + 3q^{4} - 2q^{5} - 3q^{7} + O(q^{8})$	+ +	$-1$
614.2.a.d	4	degree 4	$4q - 4q^{2} - q^{3} + 4q^{4} + 3q^{5} + q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
614.2.a.e	6	degree 6	$6q - 6q^{2} + 6q^{4} - 3q^{7} + O(q^{8})$	+ −	$+1$
614.2.a.f	11	degree 11	$11q + 11q^{2} + 3q^{3} + 11q^{4} - 3q^{5} + 3q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

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 $T$
614.2.a.a	$x + 2$
614.2.a.b	$x + 6$
614.2.a.c	$x^{3} + 6x^{2} - 16x - 100$
614.2.a.d	$x^{4} - 8x^{3} - 173x^{2} + 1078x + 199$
614.2.a.e	$x^{6} - 162x^{4} + 104x^{3} + 7413x^{2} - 3480x - 105396$
614.2.a.f	$x^{11} + 6x^{10} - 363x^{9} - 756x^{8} + 46026x^{7} - 63164x^{6} - 1969831x^{5} + 7653068x^{4} + 5623463x^{3} - 43799550x^{2} + 2829008x + 39270748$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(614))$	26
Old forms	50
Cusp forms $S_{2}(\Gamma_0(614))$	76
Eisenstein series $E_{2}(\Gamma_0(614))$	3
Modular forms $M_{2}(\Gamma_0(614))$	79

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(614)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(307))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.318 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, 307, 614]: [0, 1, 26, 79] (certified at ell = 2147478013)
T = [(3, 1), (5, 3)]; h of degree 26 = 1 x 26 (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(614, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 614:2 1000

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