Atlas › Modular forms › Weight 6 › Level 104 › 104.6.a

Newspace 104.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(104))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
104.6.a.a	1	$\mathbb{Q}$	$q - 23q^{3} - 9q^{5} + 165q^{7} + O(q^{8})$	+
104.6.a.b	2	$\mathbb{Q}(\sqrt{337})$	$2q + 23q^{3} - 47q^{5} - 95q^{7} + O(q^{8})$	+
104.6.a.c	3	degree 3	$3q - 20q^{3} + 56q^{5} - 76q^{7} + O(q^{8})$	−
104.6.a.d	4	degree 4	$4q - 11q^{3} + 31q^{5} + 39q^{7} + O(q^{8})$	+
104.6.a.e	5	degree 5	$5q + 9q^{3} + 19q^{5} - 109q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
104.6.a.a	$x + 32$
104.6.a.b	$x^{2} + 24x - 1204$
104.6.a.c	$x^{3} - 36x^{2} - 1532x + 21120$
104.6.a.d	$x^{4} - 20x^{3} - 10116x^{2} + 82512x + 24758784$
104.6.a.e	$x^{5} - 28x^{4} - 14656x^{3} + 590480x^{2} + 48582640x - 2287509120$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(104))$	15
Old forms	51
Cusp forms $S_{6}(\Gamma_0(104))$	66
Eisenstein series $E_{6}(\Gamma_0(104))$	8
Modular forms $M_{6}(\Gamma_0(104))$	74

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(104)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.285 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, 4, 7, 7, 19, 38, 74] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 104:6 1000

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