Atlas › Modular forms › Weight 2 › Level 494 › 494.2.a

Newspace 494.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(494))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
494.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$	494a1 rank 1
494.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+ + −	$+1$	494b1 rank 0
494.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} + q^{4} - 3q^{5} - 3q^{6} + 3q^{7} + O(q^{8})$	+ + −	$+1$	494c1 rank 0
494.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$	494d1 rank 1
494.2.a.e	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} - 6q^{7} + O(q^{8})$	+ + −	$+1$
494.2.a.f	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} + q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
494.2.a.g	3	degree 3	$3q + 3q^{2} + 5q^{3} + 3q^{4} + 5q^{5} + 5q^{6} - 2q^{7} + O(q^{8})$	− + +	$+1$
494.2.a.h	4	degree 4	$4q + 4q^{2} + 2q^{3} + 4q^{4} + 2q^{5} + 2q^{6} + 7q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
494.2.a.a	$x + 3$
494.2.a.b	$x - 54$
494.2.a.c	$x - 9$
494.2.a.d	$x + 47$
494.2.a.e	$x^{3} + 45x^{2} - 669x - 8081$
494.2.a.f	$x^{3} - 5x^{2} - 93x - 151$
494.2.a.g	$x^{3} - 3x^{2} - 2097x + 9099$
494.2.a.h	$x^{4} - 50x^{3} + 704x^{2} - 3062x + 263$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(494))$	17
Old forms	50
Cusp forms $S_{2}(\Gamma_0(494))$	67
Eisenstein series $E_{2}(\Gamma_0(494))$	7
Modular forms $M_{2}(\Gamma_0(494))$	74

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(494)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(247))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 19, 26, 38, 247, 494]: [0, 1, 1, 2, 5, 7, 24, 74] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 17 = 1 x 17 (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(494, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 494:2 1000

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