Atlas › Modular forms › Weight 2 › Level 697 › 697.2.a

Newspace 697.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(697))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
697.2.a.a	3	degree 3	$3q - 3q^{2} - 2q^{3} - 3q^{4} + 2q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
697.2.a.b	3	degree 3	$3q + q^{2} + 2q^{3} + 5q^{4} - 2q^{5} - 4q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
697.2.a.c	7	degree 7	$7q - 3q^{2} - 2q^{3} + 9q^{4} - 5q^{5} + 2q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
697.2.a.d	10	degree 10	$10q + 4q^{2} + 2q^{3} + 6q^{4} + 11q^{5} + 10q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$
697.2.a.e	15	degree 15	$15q - 5q^{2} - 8q^{3} + 17q^{4} - 9q^{5} - 8q^{6} - 19q^{7} + O(q^{8})$	+ +	$-1$
697.2.a.f	15	degree 15	$15q + 5q^{2} + 4q^{3} + 17q^{4} + 7q^{5} + 6q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
697.2.a.a	$x^{3} + 5x^{2} + 3x - 7$
697.2.a.b	$x^{3} - 3x^{2} - x + 5$
697.2.a.c	$x^{7} + 5x^{6} - 15x^{5} - 73x^{4} + 21x^{3} + 178x^{2} - 7x - 109$
697.2.a.d	$x^{10} - 6x^{9} - 27x^{8} + 174x^{7} + 228x^{6} - 1491x^{5} - 799x^{4} + 3853x^{3} + 1616x^{2} - 2348x - 439$
697.2.a.e	$x^{15} + 13x^{14} + 39x^{13} - 125x^{12} - 689x^{11} + 228x^{10} + 3791x^{9} + 515x^{8} - 8439x^{7} - 905x^{6} + 5673x^{5} + 985x^{4} - 997x^{3} - 140x^{2} + 41x + 5$
697.2.a.f	$x^{15} - 9x^{14} - 21x^{13} + 389x^{12} - 447x^{11} - 4990x^{10} + 12021x^{9} + 18405x^{8} - 66683x^{7} - 10249x^{6} + 113303x^{5} - 11393x^{4} - 54763x^{3} + 9996x^{2} + 6149x - 833$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(697))$	53
Old forms	8
Cusp forms $S_{2}(\Gamma_0(697))$	61
Eisenstein series $E_{2}(\Gamma_0(697))$	3
Modular forms $M_{2}(\Gamma_0(697))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(697)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.225 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, 17, 41, 697]: [0, 2, 4, 64] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 53 = 1 x 53 (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(697, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 697:2 1000

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