Atlas › Modular forms › Weight 2 › Level 595 › 595.2.a

Newspace 595.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(595))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
595.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 4q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	595a1 rank 0
595.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{3} + 2q^{4} - q^{5} + 4q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	595b1 rank 0
595.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{3} + 2q^{4} + q^{5} + 4q^{6} + q^{7} + O(q^{8})$	− − −	$+1$	595c1 rank 0
595.2.a.d	3	degree 3	$3q - 4q^{2} - 2q^{3} + 4q^{4} + 3q^{5} + 5q^{6} + 3q^{7} + O(q^{8})$	− − +	$-1$
595.2.a.e	3	degree 3	$3q - 2q^{2} - 4q^{3} + 3q^{5} + 5q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$
595.2.a.f	3	degree 3	$3q - 2q^{2} - 2q^{3} + 4q^{4} - 3q^{5} - 3q^{6} + 3q^{7} + O(q^{8})$	+ − −	$-1$
595.2.a.g	3	degree 3	$3q - 3q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
595.2.a.h	3	degree 3	$3q + 2q^{2} - 3q^{5} + 7q^{6} - 3q^{7} + O(q^{8})$	+ + −	$+1$
595.2.a.i	4	degree 4	$4q + q^{2} - 2q^{3} + 3q^{4} + 4q^{5} - 5q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$
595.2.a.j	4	degree 4	$4q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} + 7q^{6} - 4q^{7} + O(q^{8})$	− + +	$+1$
595.2.a.k	5	degree 5	$5q + 2q^{2} - 2q^{3} + 10q^{4} - 5q^{5} - 5q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{11} + 7T_{13} + 9T_{19}$ 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 $T$
595.2.a.a	$x - 61$
595.2.a.b	$x + 9$
595.2.a.c	$x - 9$
595.2.a.d	$x^{3} + 117x^{2} + 3912x + 35687$
595.2.a.e	$x^{3} + 45x^{2} - 2496x - 109843$
595.2.a.f	$x^{3} + 65x^{2} - 1638x - 44603$
595.2.a.g	$x^{3} + 141x^{2} + 4494x - 33877$
595.2.a.h	$x^{3} + 57x^{2} - 310x - 2647$
595.2.a.i	$x^{4} + 4x^{3} - 545x^{2} + 5877x - 18281$
595.2.a.j	$x^{4} - 54x^{3} - 7287x^{2} + 382267x - 4297849$
595.2.a.k	$x^{5} - 5x^{4} - 9071x^{3} - 73528x^{2} + 17180204x + 309986711$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(595))$	31
Old forms	38
Cusp forms $S_{2}(\Gamma_0(595))$	69
Eisenstein series $E_{2}(\Gamma_0(595))$	7
Modular forms $M_{2}(\Gamma_0(595))$	76

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(595)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(85))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(119))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 7, 17, 35, 85, 119, 595]: [0, 1, 1, 2, 6, 10, 14, 76] (certified at ell = 2147483137)
T = [(2, 1), (3, 3), (11, 5), (13, 7), (19, 9)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 8 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(595, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 595:2 1000

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