Atlas › Modular forms › Weight 2 › Level 598 › 598.2.a

Newspace 598.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(598))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
598.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} + q^{4} - 3q^{5} + 3q^{6} + 3q^{7} + O(q^{8})$	+ − −	$-1$	598b1 rank 1
598.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + O(q^{8})$	+ − −	$-1$	598a1 rank 1
598.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	598d1 rank 1
598.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} + 3q^{7} + O(q^{8})$	− + +	$+1$	598c1 rank 0
598.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+ + −	$+1$
598.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ − +	$+1$
598.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 6q^{7} + O(q^{8})$	− − +	$-1$
598.2.a.h	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{2} + 4q^{3} + 2q^{4} + 4q^{6} + O(q^{8})$	− + +	$+1$
598.2.a.i	4	degree 4	$4q - 4q^{2} + 4q^{4} - 4q^{5} - 2q^{7} + O(q^{8})$	+ + +	$-1$
598.2.a.j	5	degree 5	$5q + 5q^{2} + 2q^{3} + 5q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{17} + 4T_{19} + 5T_{29} + 6T_{7} + 7T_{11} + 8T_{31}$ 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 $T$
598.2.a.a	$x + 40$
598.2.a.b	$x + 10$
598.2.a.c	$x + 122$
598.2.a.d	$x - 82$
598.2.a.e	$x^{2} - 138x + 4398$
598.2.a.f	$x^{2} + 70x + 1072$
598.2.a.g	$x^{2} + 124x + 3826$
598.2.a.h	$x^{2} + 4x - 2$
598.2.a.i	$x^{4} + 108x^{3} - 2184x^{2} - 560520x - 15408972$
598.2.a.j	$x^{5} - 110x^{4} - 814x^{3} + 28328x^{2} + 98928x - 1726312$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(598))$	21
Old forms	60
Cusp forms $S_{2}(\Gamma_0(598))$	81
Eisenstein series $E_{2}(\Gamma_0(598))$	7
Modular forms $M_{2}(\Gamma_0(598))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(598)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(46))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(299))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.4 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, 13, 23, 26, 46, 299, 598]: [0, 1, 1, 3, 5, 8, 30, 88] (certified at ell = 2147483053)
T = [(3, 1), (5, 2), (17, 3), (19, 4), (29, 5), (7, 6), (11, 7), (31, 8)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 6 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(598, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 598:2 1000

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