Atlas › Modular forms › Weight 2 › Level 497 › 497.2.a

Newspace 497.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(497))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
497.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} + q^{7} + O(q^{8})$	− −	$-1$	497a1 rank 1
497.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} - 2q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$
497.2.a.c	8	degree 8	$8q + 5q^{3} + 8q^{4} + 2q^{5} + 5q^{6} - 8q^{7} + O(q^{8})$	+ −	$+1$
497.2.a.d	9	degree 9	$9q - 2q^{2} - 7q^{3} + 4q^{4} - 4q^{5} - q^{6} - 9q^{7} + O(q^{8})$	+ +	$-1$
497.2.a.e	15	degree 15	$15q + 2q^{2} + 5q^{3} + 22q^{4} - 5q^{6} + 15q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
497.2.a.a	$x + 2$
497.2.a.b	$x^{2} + 8x - 2$
497.2.a.c	$x^{8} - 15x^{7} - 36x^{6} + 1146x^{5} - 1716x^{4} - 20126x^{3} + 51004x^{2} + 12718x - 25898$
497.2.a.d	$x^{9} + 23x^{8} + 108x^{7} - 871x^{6} - 7234x^{5} + 786x^{4} + 97130x^{3} + 166112x^{2} - 39944x - 153392$
497.2.a.e	$x^{15} - 17x^{14} - 177x^{13} + 4102x^{12} + 7958x^{11} - 389197x^{10} + 242772x^{9} + 18586482x^{8} - 30197934x^{7} - 473497196x^{6} + 899502652x^{5} + 6173331812x^{4} - 10504170430x^{3} - 32820781824x^{2} + 37049517288x + 2201548464$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(497))$	35
Old forms	12
Cusp forms $S_{2}(\Gamma_0(497))$	47
Eisenstein series $E_{2}(\Gamma_0(497))$	3
Modular forms $M_{2}(\Gamma_0(497))$	50

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(497)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.201 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, 7, 71, 497]: [0, 1, 7, 50] (certified at ell = 2147482681)
T = [(2, 1), (3, 3)]; h of degree 35 = 1 x 35 (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(497, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 497:2 1000

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