Atlas › Modular forms › Weight 4 › Level 209 › 209.4.a

Newspace 209.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(209))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
209.4.a.a	8	degree 8	$8q - q^{3} + 22q^{4} - 12q^{5} + q^{6} - 59q^{7} + O(q^{8})$	+ −	$-1$
209.4.a.b	10	degree 10	$10q - 6q^{2} - 9q^{3} + 14q^{4} - 10q^{5} + q^{6} - 53q^{7} + O(q^{8})$	− +	$-1$
209.4.a.c	13	degree 13	$13q - 2q^{2} + 11q^{3} + 82q^{4} + 8q^{5} + 13q^{6} + 39q^{7} + O(q^{8})$	+ +	$+1$
209.4.a.d	15	degree 15	$15q + 4q^{2} + 3q^{3} + 74q^{4} + 10q^{5} + 61q^{6} + 73q^{7} + O(q^{8})$	− −	$+1$

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 $T$
209.4.a.a	$x^{8} + x^{7} - 142x^{6} + 202x^{5} + 4030x^{4} - 19056x^{3} + 34001x^{2} - 26701x + 7508$
209.4.a.b	$x^{10} + 15x^{9} - 103x^{8} - 1913x^{7} + 3172x^{6} + 78946x^{5} - 3953x^{4} - 1075395x^{3} - 1027633x^{2} + 1568551x + 1360688$
209.4.a.c	$x^{13} - 9x^{12} - 319x^{11} + 2439x^{10} + 36869x^{9} - 216377x^{8} - 2090825x^{7} + 8555183x^{6} + 61792009x^{5} - 159968289x^{4} - 892800811x^{3} + 1339569583x^{2} + 4930561818x - 3821607640$
209.4.a.d	$x^{15} - 7x^{14} - 416x^{13} + 3000x^{12} + 59420x^{11} - 407384x^{10} - 3832582x^{9} + 22709284x^{8} + 120679850x^{7} - 519056428x^{6} - 1902876560x^{5} + 4574274300x^{4} + 11882387433x^{3} - 15794071257x^{2} - 22356999886x + 11217989776$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(209))$	46
Old forms	12
Cusp forms $S_{4}(\Gamma_0(209))$	58
Eisenstein series $E_{4}(\Gamma_0(209))$	4
Modular forms $M_{4}(\Gamma_0(209))$	62

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(209)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 11, 19, 209]: [1, 4, 6, 62] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 46 = 1 x 46 (phi(ord eps) x dim S_k^new); 7 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(209, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 209:4 1000

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