Atlas › Modular forms › Weight 2 › Level 605 › 605.2.a

Newspace 605.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(605))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
605.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} - q^{4} + q^{5} + 3q^{6} - 3q^{7} + O(q^{8})$	−	605c1 rank 1
605.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} + O(q^{8})$	−	605b1 rank 1
605.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} - q^{4} + q^{5} - 3q^{6} + 3q^{7} + O(q^{8})$	−	605a1 rank 1
605.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 2q^{5} + 8q^{6} + 4q^{7} + O(q^{8})$	+
605.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{4} - 2q^{5} + O(q^{8})$	+
605.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q + 4q^{3} + 2q^{4} + 2q^{5} + O(q^{8})$	−
605.2.a.g	3	degree 3	$3q - q^{2} + q^{3} + 9q^{4} - 3q^{5} - 5q^{6} + q^{7} + O(q^{8})$	+
605.2.a.h	3	degree 3	$3q + q^{2} + q^{3} + 9q^{4} - 3q^{5} + 5q^{6} - q^{7} + O(q^{8})$	+
605.2.a.i	4	degree 4	$4q - 3q^{2} - 2q^{3} + 7q^{4} + 4q^{5} + q^{6} - 11q^{7} + O(q^{8})$	−
605.2.a.j	4	degree 4	$4q - q^{2} - q^{4} - 4q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+
605.2.a.k	4	degree 4	$4q + q^{2} - q^{4} - 4q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+
605.2.a.l	4	degree 4	$4q + 3q^{2} - 2q^{3} + 7q^{4} + 4q^{5} - q^{6} + 11q^{7} + O(q^{8})$	−
605.2.a.m	6	degree 6	$6q + 6q^{3} + 6q^{4} + 6q^{5} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{13}$ 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 $T$
605.2.a.a	$x - 3$
605.2.a.b	$x + 15$
605.2.a.c	$x + 21$
605.2.a.d	$x^{2} - 74x + 487$
605.2.a.e	$x^{2} + 6x - 291$
605.2.a.f	$x^{2} - 12x - 327$
605.2.a.g	$x^{3} + 35x^{2} - 1751x + 14577$
605.2.a.h	$x^{3} - 41x^{2} - 875x + 4569$
605.2.a.i	$x^{4} + 113x^{3} + 4551x^{2} + 75335x + 405595$
605.2.a.j	$x^{4} + 23x^{3} - 755x^{2} - 8289x + 121039$
605.2.a.k	$x^{4} - 23x^{3} - 1355x^{2} + 30609x - 51131$
605.2.a.l	$x^{4} - 101x^{3} + 3579x^{2} - 53555x + 287905$
605.2.a.m	$x^{6} - 18x^{5} - 4653x^{4} + 60984x^{3} + 4854651x^{2} - 56854782x - 858809811$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(605))$	37
Old forms	18
Cusp forms $S_{2}(\Gamma_0(605))$	55
Eisenstein series $E_{2}(\Gamma_0(605))$	23
Modular forms $M_{2}(\Gamma_0(605))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(605)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(121))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.268 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, 5, 11, 55, 121, 605]: [0, 1, 2, 8, 12, 68] (certified at ell = 2147482921)
T = [(2, 1), (3, 3), (7, 5), (13, 7)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 9 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(605, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 605:2 1000

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