Atlas › Modular forms › Weight 4 › Level 213 › 213.4.a

Newspace 213.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(213))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
213.4.a.a	1	$\mathbb{Q}$	$q + 5q^{2} - 3q^{3} + 17q^{4} + 13q^{5} - 15q^{6} - 7q^{7} + O(q^{8})$	+ +	$+1$
213.4.a.b	8	degree 8	$8q - 9q^{2} + 24q^{3} + 27q^{4} - 32q^{5} - 27q^{6} - 56q^{7} + O(q^{8})$	− +	$-1$
213.4.a.c	8	degree 8	$8q - 5q^{2} - 24q^{3} + 31q^{4} + 12q^{5} + 15q^{6} - 42q^{7} + O(q^{8})$	+ −	$-1$
213.4.a.d	9	degree 9	$9q - 2q^{2} - 27q^{3} + 38q^{4} - 21q^{5} + 6q^{6} + 49q^{7} + O(q^{8})$	+ +	$+1$
213.4.a.e	10	degree 10	$10q + 7q^{2} + 30q^{3} + 51q^{4} + 28q^{5} + 21q^{6} + 28q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
213.4.a.a	$x - 18$
213.4.a.b	$x^{8} + 41x^{7} + 200x^{6} - 8983x^{5} - 99961x^{4} - 71770x^{3} + 841252x^{2} - 454032x - 173664$
213.4.a.c	$x^{8} - 7x^{7} - 362x^{6} + 1387x^{5} + 31607x^{4} + 5614x^{3} - 593984x^{2} - 750232x + 1087600$
213.4.a.d	$x^{9} + 23x^{8} - 544x^{7} - 14929x^{6} + 69593x^{5} + 3140288x^{4} + 3619924x^{3} - 223224072x^{2} - 665535744x + 2939693472$
213.4.a.e	$x^{10} - 35x^{9} - 244x^{8} + 17421x^{7} - 67173x^{6} - 2374754x^{5} + 19226536x^{4} + 65335464x^{3} - 1081168848x^{2} + 3177330496x - 1847260288$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(213))$	36
Old forms	34
Cusp forms $S_{4}(\Gamma_0(213))$	70
Eisenstein series $E_{4}(\Gamma_0(213))$	4
Modular forms $M_{4}(\Gamma_0(213))$	74

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(213)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.141 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, 3, 71, 213]: [1, 2, 19, 74] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 36 = 1 x 36 (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(213, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 213:4 1000

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