Atlas › Modular forms › Weight 4 › Level 174 › 174.4.a

Newspace 174.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(174))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
174.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} - 14q^{5} + 6q^{6} - 21q^{7} + O(q^{8})$	+ + +	$+1$
174.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} - 8q^{5} + 6q^{6} + 19q^{7} + O(q^{8})$	+ + −	$-1$
174.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} + 21q^{5} + 6q^{6} + 19q^{7} + O(q^{8})$	+ + +	$+1$
174.4.a.d	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{6} - 17q^{7} + O(q^{8})$	+ − +	$-1$
174.4.a.e	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} - 10q^{5} - 6q^{6} + 7q^{7} + O(q^{8})$	− + +	$-1$
174.4.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 4q^{4} - 18q^{5} + 6q^{6} - 29q^{7} + O(q^{8})$	− − −	$-1$
174.4.a.g	2	$\mathbb{Q}(\sqrt{457})$	$2q + 4q^{2} - 6q^{3} + 8q^{4} - 5q^{5} - 12q^{6} + 14q^{7} + O(q^{8})$	− + −	$+1$
174.4.a.h	3	degree 3	$3q - 6q^{2} + 9q^{3} + 12q^{4} - 5q^{5} - 18q^{6} + 18q^{7} + O(q^{8})$	+ − −	$+1$
174.4.a.i	3	degree 3	$3q + 6q^{2} + 9q^{3} + 12q^{4} + 7q^{5} + 18q^{6} + 34q^{7} + O(q^{8})$	− − +	$+1$

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 $T$
174.4.a.a	$x + 35$
174.4.a.b	$x - 11$
174.4.a.c	$x - 40$
174.4.a.d	$x + 17$
174.4.a.e	$x + 3$
174.4.a.f	$x + 47$
174.4.a.g	$x^{2} - 9x - 94$
174.4.a.h	$x^{3} - 13x^{2} - 1760x + 31356$
174.4.a.i	$x^{3} - 41x^{2} + 270x + 2400$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(174))$	14
Old forms	72
Cusp forms $S_{4}(\Gamma_0(174))$	86
Eisenstein series $E_{4}(\Gamma_0(174))$	8
Modular forms $M_{4}(\Gamma_0(174))$	94

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(174)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(87))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.158 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, 29, 58, 87, 174]: [1, 2, 2, 5, 9, 25, 32, 94] (certified at ell = 2147483549)
T = [(5, 1), (7, 1)]; h of degree 14 = 1 x 14 (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(174, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 174:4 1000

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