Atlas › Modular forms › Weight 4 › Level 139 › 139.4.a

Newspace 139.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(139))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
139.4.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 4q^{3} + 2q^{4} + 10q^{5} - 28q^{6} - 34q^{7} + O(q^{8})$	−	$-1$
139.4.a.b	12	degree 12	$12q - 9q^{2} - 7q^{3} + 39q^{4} - 67q^{5} - 6q^{6} + O(q^{8})$	−	$-1$
139.4.a.c	20	degree 20	$20q + 7q^{2} + 7q^{3} + 101q^{4} + 63q^{5} + 14q^{6} + 8q^{7} + O(q^{8})$	+	$+1$

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 $T$
139.4.a.a	$x^{2} + 2x - 1$
139.4.a.b	$x^{12} + 16x^{11} - 112x^{10} - 2049x^{9} + 5544x^{8} + 75971x^{7} - 201041x^{6} - 696191x^{5} + 1621237x^{4} + 2580938x^{3} - 4359735x^{2} - 3560985x + 3054264$
139.4.a.c	$x^{20} - 14x^{19} - 463x^{18} + 6723x^{17} + 83878x^{16} - 1267130x^{15} - 8089567x^{14} + 126191976x^{13} + 465680358x^{12} - 7386765757x^{11} - 16712112798x^{10} + 263765245077x^{9} + 376971094319x^{8} - 5712463072437x^{7} - 5379648783268x^{6} + 71505565592670x^{5} + 52172618137780x^{4} - 463523850587436x^{3} - 378489393801248x^{2} + 1223336764923608x + 1345839519759872$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(139))$	34
Old forms	0
Cusp forms $S_{4}(\Gamma_0(139))$	34
Eisenstein series $E_{4}(\Gamma_0(139))$	2
Modular forms $M_{4}(\Gamma_0(139))$	36

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.112 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, 139]: [1, 36] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (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(139, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 139:4 1000

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