Atlas › Modular forms › Weight 4 › Level 104 › 104.4.a

Newspace 104.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(104))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
104.4.a.a	1	$\mathbb{Q}$	$q + q^{3} - 7q^{5} - 21q^{7} + O(q^{8})$	−
104.4.a.b	1	$\mathbb{Q}$	$q + 5q^{3} + 19q^{5} - 3q^{7} + O(q^{8})$	+
104.4.a.c	2	$\mathbb{Q}(\sqrt{73})$	$2q - 3q^{3} - 3q^{5} - 25q^{7} + O(q^{8})$	+
104.4.a.d	2	$\mathbb{Q}(\sqrt{321})$	$2q - q^{3} - 11q^{5} + q^{7} + O(q^{8})$	+
104.4.a.e	3	degree 3	$3q - 8q^{5} + 36q^{7} + O(q^{8})$	−

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 $T$
104.4.a.a	$x + 20$
104.4.a.b	$x - 62$
104.4.a.c	$x^{2} + 12x - 1132$
104.4.a.d	$x^{2} + 34x - 32$
104.4.a.e	$x^{3} + 24x^{2} - 3068x + 45488$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(104))$	9
Old forms	29
Cusp forms $S_{4}(\Gamma_0(104))$	38
Eisenstein series $E_{4}(\Gamma_0(104))$	8
Modular forms $M_{4}(\Gamma_0(104))$	46

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(104)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.115 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, 8, 13, 26, 52, 104]: [1, 2, 3, 5, 5, 13, 24, 46] (certified at ell = 2147483629)
T = [(3, 1), (5, 3)]; h of degree 9 = 1 x 9 (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(104, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 104:4 1000

← 103.4.a · 105.4.a →