Atlas › Modular forms › Weight 2 › Level 498 › 498.2.a

Newspace 498.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(498))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
498.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ − −	$-1$	498b1 rank 1
498.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$	498a1 rank 0
498.2.a.c	2	$\mathbb{Q}(\sqrt{21})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 3q^{5} - 2q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$
498.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 5q^{5} - 2q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$
498.2.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} + 3q^{5} - 2q^{6} + O(q^{8})$	− + +	$+1$
498.2.a.f	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
498.2.a.g	4	degree 4	$4q + 4q^{2} + 4q^{3} + 4q^{4} + 2q^{5} + 4q^{6} + q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
498.2.a.a	$x + 76$
498.2.a.b	$x + 4$
498.2.a.c	$x^{2} - 76x + 1360$
498.2.a.d	$x^{2} + 164x + 6644$
498.2.a.e	$x^{2} - 90x - 32$
498.2.a.f	$x^{3} + 126x^{2} + 3264x - 19744$
498.2.a.g	$x^{4} - 58x^{3} - 3396x^{2} + 104648x + 2700352$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(498))$	15
Old forms	66
Cusp forms $S_{2}(\Gamma_0(498))$	81
Eisenstein series $E_{2}(\Gamma_0(498))$	7
Modular forms $M_{2}(\Gamma_0(498))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(498)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(166))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(249))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.385 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, 3, 6, 83, 166, 249, 498]: [0, 1, 1, 3, 8, 23, 30, 88] (certified at ell = 2147483323)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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(498, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 498:2 1000

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