Atlas › Modular forms › Weight 4 › Level 195 › 195.4.a

Newspace 195.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(195))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
195.4.a.a	1	$\mathbb{Q}$	$q - 5q^{2} + 3q^{3} + 17q^{4} - 5q^{5} - 15q^{6} + 8q^{7} + O(q^{8})$	− + +	$-1$
195.4.a.b	1	$\mathbb{Q}$	$q - 3q^{2} - 3q^{3} + q^{4} - 5q^{5} + 9q^{6} + 2q^{7} + O(q^{8})$	+ + −	$-1$
195.4.a.c	1	$\mathbb{Q}$	$q - 3q^{2} + 3q^{3} + q^{4} + 5q^{5} - 9q^{6} - 16q^{7} + O(q^{8})$	− − −	$-1$
195.4.a.d	1	$\mathbb{Q}$	$q + 4q^{2} + 3q^{3} + 8q^{4} + 5q^{5} + 12q^{6} + 18q^{7} + O(q^{8})$	− − +	$+1$
195.4.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + 6q^{3} - 7q^{4} - 10q^{5} + 3q^{6} - 20q^{7} + O(q^{8})$	− + +	$-1$
195.4.a.f	3	degree 3	$3q - 2q^{2} - 9q^{3} + 10q^{4} + 15q^{5} + 6q^{6} - 30q^{7} + O(q^{8})$	+ − +	$-1$
195.4.a.g	3	degree 3	$3q + q^{2} + 9q^{3} + 25q^{4} + 15q^{5} + 3q^{6} - 12q^{7} + O(q^{8})$	− − +	$+1$
195.4.a.h	4	degree 4	$4q - 3q^{2} - 12q^{3} + 33q^{4} - 20q^{5} + 9q^{6} - 4q^{7} + O(q^{8})$	+ + +	$+1$
195.4.a.i	4	degree 4	$4q + 12q^{3} + 18q^{4} - 20q^{5} - 6q^{7} + O(q^{8})$	− + −	$+1$
195.4.a.j	4	degree 4	$4q + 2q^{2} - 12q^{3} + 18q^{4} + 20q^{5} - 6q^{6} + 4q^{7} + O(q^{8})$	+ − −	$+1$

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 $T$
195.4.a.a	$x - 3$
195.4.a.b	$x + 1$
195.4.a.c	$x + 19$
195.4.a.d	$x - 22$
195.4.a.e	$x^{2} + 19x - 118$
195.4.a.f	$x^{3} + 32x^{2} - 267x - 8766$
195.4.a.g	$x^{3} + 11x^{2} - 315x - 81$
195.4.a.h	$x^{4} + 7x^{3} - 1177x^{2} - 21x + 165690$
195.4.a.i	$x^{4} + 6x^{3} - 1206x^{2} + 9566x + 80713$
195.4.a.j	$x^{4} - 6x^{3} - 950x^{2} + 12690x + 12073$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(195))$	24
Old forms	56
Cusp forms $S_{4}(\Gamma_0(195))$	80
Eisenstein series $E_{4}(\Gamma_0(195))$	8
Modular forms $M_{4}(\Gamma_0(195))$	88

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(195)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.161 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, 13, 15, 39, 65, 195]: [1, 2, 3, 5, 8, 16, 24, 88] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 5 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(195, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 195:4 1000

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