Atlas › Modular forms › Weight 4 › Level 102 › 102.4.a

Newspace 102.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(102))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
102.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} - 3q^{5} + 6q^{6} + 20q^{7} + O(q^{8})$	+ + −	$-1$
102.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} - 5q^{5} - 6q^{6} - 32q^{7} + O(q^{8})$	+ − +	$-1$
102.4.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} - 12q^{5} - 6q^{6} - 22q^{7} + O(q^{8})$	− + +	$-1$
102.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} + 5q^{5} - 6q^{6} + 12q^{7} + O(q^{8})$	− + −	$+1$
102.4.a.e	2	$\mathbb{Q}(\sqrt{15})$	$2q - 4q^{2} + 6q^{3} + 8q^{4} + 12q^{5} - 12q^{6} + 16q^{7} + O(q^{8})$	+ − −	$+1$
102.4.a.f	2	$\mathbb{Q}(\sqrt{393})$	$2q + 4q^{2} + 6q^{3} + 8q^{4} + 3q^{5} + 12q^{6} + 22q^{7} + O(q^{8})$	− − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
102.4.a.a	$x + 625$
102.4.a.b	$x + 449$
102.4.a.c	$x + 304$
102.4.a.d	$x - 359$
102.4.a.e	$x^{2} - 472x + 54736$
102.4.a.f	$x^{2} - 343x - 90944$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(102))$	8
Old forms	42
Cusp forms $S_{4}(\Gamma_0(102))$	50
Eisenstein series $E_{4}(\Gamma_0(102))$	8
Modular forms $M_{4}(\Gamma_0(102))$	58

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 2, 5, 6, 16, 20, 58] (certified at ell = 2147483489)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 102:4 1000

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