Atlas › Modular forms › Weight 6 › Level 102 › 102.6.a

Newspace 102.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(102))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
102.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} - 9q^{3} + 16q^{4} + 81q^{5} + 36q^{6} + 92q^{7} + O(q^{8})$	+ + −	$+1$
102.6.a.b	1	$\mathbb{Q}$	$q - 4q^{2} + 9q^{3} + 16q^{4} + 9q^{5} - 36q^{6} - 88q^{7} + O(q^{8})$	+ − −	$-1$
102.6.a.c	1	$\mathbb{Q}$	$q + 4q^{2} - 9q^{3} + 16q^{4} - 9q^{5} - 36q^{6} - 52q^{7} + O(q^{8})$	− + +	$+1$
102.6.a.d	1	$\mathbb{Q}$	$q + 4q^{2} - 9q^{3} + 16q^{4} + 25q^{5} - 36q^{6} - 188q^{7} + O(q^{8})$	− + −	$-1$
102.6.a.e	1	$\mathbb{Q}$	$q + 4q^{2} + 9q^{3} + 16q^{4} - 81q^{5} + 36q^{6} - 88q^{7} + O(q^{8})$	− − +	$-1$
102.6.a.f	2	$\mathbb{Q}(\sqrt{769})$	$2q - 8q^{2} - 18q^{3} + 32q^{4} - 53q^{5} + 72q^{6} + 130q^{7} + O(q^{8})$	+ + +	$-1$
102.6.a.g	2	$\mathbb{Q}(\sqrt{20209})$	$2q - 8q^{2} + 18q^{3} + 32q^{4} - 25q^{5} - 72q^{6} + 146q^{7} + O(q^{8})$	+ − +	$+1$
102.6.a.h	3	degree 3	$3q + 12q^{2} + 27q^{3} + 48q^{4} + 53q^{5} + 108q^{6} + 168q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
102.6.a.a	$x - 173$
102.6.a.b	$x + 79$
102.6.a.c	$x + 61$
102.6.a.d	$x + 163$
102.6.a.e	$x + 169$
102.6.a.f	$x^{2} - 77x - 7938$
102.6.a.g	$x^{2} - 121x - 41810$
102.6.a.h	$x^{3} - 221x^{2} - 3760x + 156572$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(102))$	12
Old forms	74
Cusp forms $S_{6}(\Gamma_0(102))$	86
Eisenstein series $E_{6}(\Gamma_0(102))$	8
Modular forms $M_{6}(\Gamma_0(102))$	94

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(102)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.327 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, 3, 6, 17, 34, 51, 102]: [1, 2, 3, 7, 8, 24, 32, 94] (certified at ell = 2147483489)
T = [(5, 1), (7, 1)]; h of degree 12 = 1 x 12 (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(102, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 102:6 1000

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