Atlas › Modular forms › Weight 2 › Level 597 › 597.2.a

Newspace 597.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(597))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
597.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + O(q^{8})$	+ −	$+1$
597.2.a.b	3	degree 3	$3q - 2q^{2} + 3q^{3} - q^{5} - 2q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$
597.2.a.c	3	degree 3	$3q - 3q^{3} + 3q^{5} + 3q^{7} + O(q^{8})$	+ −	$+1$
597.2.a.d	11	degree 11	$11q - 5q^{2} - 11q^{3} + 11q^{4} - 5q^{5} + 5q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
597.2.a.e	14	degree 14	$14q + 2q^{2} + 14q^{3} + 20q^{4} + q^{5} + 2q^{6} + 11q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
597.2.a.a	$x^{2} - 2x - 1$
597.2.a.b	$x^{3} + 5x^{2} - x - 13$
597.2.a.c	$x^{3} - 9x^{2} - 57x + 361$
597.2.a.d	$x^{11} + 20x^{10} - 148x^{9} - 4621x^{8} - 2300x^{7} + 303189x^{6} + 467312x^{5} - 8113765x^{4} - 6927012x^{3} + 90936676x^{2} - 12798101x - 233649995$
597.2.a.e	$x^{14} - 5x^{13} - 451x^{12} + 2986x^{11} + 70521x^{10} - 560724x^{9} - 4592932x^{8} + 43302864x^{7} + 124417808x^{6} - 1544551121x^{5} - 713062996x^{4} + 24248009298x^{3} - 18955958278x^{2} - 120099054178x + 151519161847$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(597))$	33
Old forms	32
Cusp forms $S_{2}(\Gamma_0(597))$	65
Eisenstein series $E_{2}(\Gamma_0(597))$	3
Modular forms $M_{2}(\Gamma_0(597))$	68

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(597)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(199))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.221 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, 3, 199, 597]: [0, 1, 17, 68] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 33 = 1 x 33 (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(597, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 597:2 1000

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