Atlas › Modular forms › Weight 6 › Level 75 › 75.6.a

Newspace 75.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(75))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
75.6.a.a	1	$\mathbb{Q}$	$q - 7q^{2} - 9q^{3} + 17q^{4} + 63q^{6} - 12q^{7} + O(q^{8})$	+
75.6.a.b	1	$\mathbb{Q}$	$q - 4q^{2} - 9q^{3} - 16q^{4} + 36q^{6} + 225q^{7} + O(q^{8})$	+
75.6.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + 9q^{3} - 28q^{4} + 18q^{6} + 132q^{7} + O(q^{8})$	−
75.6.a.d	1	$\mathbb{Q}$	$q + 4q^{2} + 9q^{3} - 16q^{4} + 36q^{6} - 225q^{7} + O(q^{8})$	−
75.6.a.e	1	$\mathbb{Q}$	$q + 6q^{2} - 9q^{3} + 4q^{4} - 54q^{6} + 40q^{7} + O(q^{8})$	+
75.6.a.f	2	$\mathbb{Q}(\sqrt{89})$	$2q - 9q^{2} + 18q^{3} + 21q^{4} - 81q^{6} - 108q^{7} + O(q^{8})$	−
75.6.a.g	2	$\mathbb{Q}(\sqrt{31})$	$2q - 6q^{2} - 18q^{3} + 16q^{4} + 54q^{6} - 102q^{7} + O(q^{8})$	+
75.6.a.h	2	$\mathbb{Q}(\sqrt{409})$	$2q + q^{2} + 18q^{3} + 141q^{4} + 9q^{6} + 112q^{7} + O(q^{8})$	−
75.6.a.i	2	$\mathbb{Q}(\sqrt{31})$	$2q + 6q^{2} + 18q^{3} + 16q^{4} + 54q^{6} + 102q^{7} + O(q^{8})$	−
75.6.a.j	2	$\mathbb{Q}(\sqrt{89})$	$2q + 9q^{2} - 18q^{3} + 21q^{4} - 81q^{6} + 108q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{7}$ 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 $T$
75.6.a.a	$x + 19$
75.6.a.b	$x - 221$
75.6.a.c	$x - 134$
75.6.a.d	$x + 221$
75.6.a.e	$x - 46$
75.6.a.f	$x^{2} + 117x - 10484$
75.6.a.g	$x^{2} + 108x - 16459$
75.6.a.h	$x^{2} - 113x - 19814$
75.6.a.i	$x^{2} - 108x - 16459$
75.6.a.j	$x^{2} - 117x - 10484$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(75))$	15
Old forms	29
Cusp forms $S_{6}(\Gamma_0(75))$	44
Eisenstein series $E_{6}(\Gamma_0(75))$	12
Modular forms $M_{6}(\Gamma_0(75))$	56

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(75)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(25))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.272 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, 3, 5, 15, 25, 75]: [1, 3, 3, 12, 13, 52] (certified at ell = 2147482921)
T = [(2, 1), (7, 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(75, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 75:6 1000

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