Atlas › Modular forms › Weight 2 › Level 608 › 608.2.a

Newspace 608.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(608))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
608.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} + q^{7} + O(q^{8})$	−	608f1 rank 1
608.2.a.b	1	$\mathbb{Q}$	$q - q^{5} - q^{7} + O(q^{8})$	−	608d1 rank 1
608.2.a.c	1	$\mathbb{Q}$	$q - q^{5} + q^{7} + O(q^{8})$	+	608a1 rank 1
608.2.a.d	1	$\mathbb{Q}$	$q + 3q^{5} - 5q^{7} + O(q^{8})$	−	608e1 rank 1
608.2.a.e	1	$\mathbb{Q}$	$q + 3q^{5} + 5q^{7} + O(q^{8})$	+	608b1 rank 0
608.2.a.f	1	$\mathbb{Q}$	$q + 3q^{3} - q^{7} + O(q^{8})$	+	608c1 rank 0
608.2.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} - q^{5} - 6q^{7} + O(q^{8})$	+
608.2.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} - q^{5} + 6q^{7} + O(q^{8})$	−
608.2.a.i	4	degree 4	$4q - 2q^{3} + q^{5} + q^{7} + O(q^{8})$	+
608.2.a.j	4	degree 4	$4q + 2q^{3} + q^{5} - q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{23} + 8T_{29}$ 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 $T$
608.2.a.a	$x - 8$
608.2.a.b	$x + 87$
608.2.a.c	$x - 7$
608.2.a.d	$x + 67$
608.2.a.e	$x - 3$
608.2.a.f	$x - 66$
608.2.a.g	$x^{2} + 125x + 2372$
608.2.a.h	$x^{2} + 13x - 6422$
608.2.a.i	$x^{4} - 191x^{3} + 8500x^{2} - 7744x - 1764664$
608.2.a.j	$x^{4} - 133x^{3} + 552x^{2} + 290256x - 3763384$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(608))$	18
Old forms	55
Cusp forms $S_{2}(\Gamma_0(608))$	73
Eisenstein series $E_{2}(\Gamma_0(608))$	15
Modular forms $M_{2}(\Gamma_0(608))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(608)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(76))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(152))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(304))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.397 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, 16, 19, 32, 38, 76, 152, 304, 608]: [0, 1, 2, 3, 4, 2, 6, 7, 13, 24, 44, 84] (certified at ell = 2147482873)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (23, 7), (29, 8)]; h of degree 18 = 1 x 18 (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(608, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 608:2 1000

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