Atlas › Modular forms › Weight 4 › Level 175 › 175.4.a

Newspace 175.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(175))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
175.4.a.a	1	$\mathbb{Q}$	$q - q^{2} + 8q^{3} - 7q^{4} - 8q^{6} - 7q^{7} + O(q^{8})$	+
175.4.a.b	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - 7q^{4} + 2q^{6} + 7q^{7} + O(q^{8})$	−
175.4.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 8q^{2} - 2q^{3} + 20q^{4} - 8q^{6} + 14q^{7} + O(q^{8})$	−
175.4.a.d	2	$\mathbb{Q}(\sqrt{41})$	$2q - q^{2} + 5q^{3} + 5q^{4} - 23q^{6} - 14q^{7} + O(q^{8})$	+
175.4.a.e	2	$\mathbb{Q}(\sqrt{41})$	$2q + q^{2} - 5q^{3} + 5q^{4} - 23q^{6} + 14q^{7} + O(q^{8})$	−
175.4.a.f	3	degree 3	$3q + 3q^{2} - 2q^{3} + 13q^{4} + 24q^{6} - 21q^{7} + O(q^{8})$	+
175.4.a.g	4	degree 4	$4q - 4q^{2} + 3q^{3} + 36q^{4} + q^{6} - 28q^{7} + O(q^{8})$	+
175.4.a.h	4	degree 4	$4q + 4q^{2} - 3q^{3} + 36q^{4} + q^{6} + 28q^{7} + O(q^{8})$	−
175.4.a.i	5	degree 5	$5q - 4q^{2} - 10q^{3} + 18q^{4} + 6q^{6} - 35q^{7} + O(q^{8})$	+
175.4.a.j	5	degree 5	$5q + 4q^{2} + 10q^{3} + 18q^{4} + 6q^{6} + 35q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
175.4.a.a	$x - 23$
175.4.a.b	$x - 7$
175.4.a.c	$x^{2} + 14x - 193$
175.4.a.d	$x^{2} - 14x + 8$
175.4.a.e	$x^{2} + 14x + 8$
175.4.a.f	$x^{3} + 3x^{2} - 815x - 4809$
175.4.a.g	$x^{4} - 5x^{3} - 785x^{2} + 2872x + 121728$
175.4.a.h	$x^{4} + 5x^{3} - 785x^{2} - 2872x + 121728$
175.4.a.i	$x^{5} + 34x^{4} - 180x^{3} - 9718x^{2} + 2211x + 629392$
175.4.a.j	$x^{5} - 34x^{4} - 180x^{3} + 9718x^{2} + 2211x - 629392$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(175))$	29
Old forms	25
Cusp forms $S_{4}(\Gamma_0(175))$	54
Eisenstein series $E_{4}(\Gamma_0(175))$	12
Modular forms $M_{4}(\Gamma_0(175))$	66

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(175)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(25))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 5, 7, 25, 35, 175]: [1, 3, 3, 9, 14, 62] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 6 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(175, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 175:4 1000

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