Atlas › Modular forms › Weight 2 › Level 644 › 644.2.a

Newspace 644.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(644))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
644.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + q^{7} + O(q^{8})$	− +	644b1 rank 1
644.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - q^{7} + O(q^{8})$	+ −	644a1 rank 1
644.2.a.c	5	degree 5	$5q + q^{3} + 4q^{5} + 5q^{7} + O(q^{8})$	− −
644.2.a.d	5	degree 5	$5q + 3q^{3} + 2q^{5} - 5q^{7} + O(q^{8})$	+ +

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
644.2.a.a	$x + 1$
644.2.a.b	$x + 5$
644.2.a.c	$x^{5} - 13x^{4} - 90x^{3} + 1936x^{2} - 7696x + 4248$
644.2.a.d	$x^{5} - 9x^{4} - 182x^{3} + 1216x^{2} + 6088x - 25544$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(644))$	12
Old forms	79
Cusp forms $S_{2}(\Gamma_0(644))$	91
Eisenstein series $E_{2}(\Gamma_0(644))$	11
Modular forms $M_{2}(\Gamma_0(644))$	102

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(644)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(46))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(92))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(161))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(322))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 4, 7, 14, 23, 28, 46, 92, 161, 322, 644]: [0, 1, 2, 1, 4, 3, 7, 8, 15, 18, 52, 102] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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(644, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 644:2 1000

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