Atlas › Modular forms › Weight 4 › Level 182 › 182.4.a

Newspace 182.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(182))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
182.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 7q^{3} + 4q^{4} - 14q^{6} + 7q^{7} + O(q^{8})$	+ − −	$+1$
182.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - 8q^{3} + 4q^{4} + 3q^{5} - 16q^{6} + 7q^{7} + O(q^{8})$	− − −	$-1$
182.4.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - 2q^{3} + 4q^{4} - 5q^{5} - 4q^{6} - 7q^{7} + O(q^{8})$	− + +	$-1$
182.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 5q^{3} + 4q^{4} + 16q^{5} + 10q^{6} + 7q^{7} + O(q^{8})$	− − +	$+1$
182.4.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 4q^{2} - 10q^{3} + 8q^{4} - 8q^{5} + 20q^{6} + 14q^{7} + O(q^{8})$	+ − +	$-1$
182.4.a.f	2	$\mathbb{Q}(\sqrt{1169})$	$2q - 4q^{2} - 8q^{3} + 8q^{4} + 5q^{5} + 16q^{6} + 14q^{7} + O(q^{8})$	+ − −	$+1$
182.4.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{2} - 4q^{3} + 8q^{4} + 6q^{5} + 8q^{6} - 14q^{7} + O(q^{8})$	+ + −	$-1$
182.4.a.h	2	$\mathbb{Q}(\sqrt{43})$	$2q + 4q^{2} + 2q^{3} + 8q^{4} + 4q^{5} + 4q^{6} + 14q^{7} + O(q^{8})$	− − +	$+1$
182.4.a.i	3	degree 3	$3q - 6q^{2} - q^{3} + 12q^{4} + 13q^{5} + 2q^{6} - 21q^{7} + O(q^{8})$	+ + +	$+1$
182.4.a.j	3	degree 3	$3q + 6q^{2} + 7q^{3} + 12q^{4} - 2q^{5} + 14q^{6} - 21q^{7} + O(q^{8})$	− + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{17}$ 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 $T$
182.4.a.a	$x - 370$
182.4.a.b	$x + 941$
182.4.a.c	$x + 15$
182.4.a.d	$x + 330$
182.4.a.e	$x^{2} - 322x - 84671$
182.4.a.f	$x^{2} + 621x - 116640$
182.4.a.g	$x^{2} + 378x + 22273$
182.4.a.h	$x^{2} - 1234x + 336657$
182.4.a.i	$x^{3} + 63x^{2} - 134952x + 16352896$
182.4.a.j	$x^{3} - 422x^{2} - 975975x + 471777624$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(182))$	18
Old forms	62
Cusp forms $S_{4}(\Gamma_0(182))$	80
Eisenstein series $E_{4}(\Gamma_0(182))$	8
Modular forms $M_{4}(\Gamma_0(182))$	88

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(182)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(91))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.223 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, 7, 13, 14, 26, 91, 182]: [1, 2, 3, 5, 8, 13, 30, 88] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (11, 5), (17, 7)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 7 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(182, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 182:4 1000

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