Atlas › Modular forms › Weight 2 › Level 395 › 395.2.a

Newspace 395.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(395))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
395.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$	395c1 rank 0
395.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} - 4q^{7} + O(q^{8})$	− +	$+1$	395a1 rank 0
395.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} - q^{4} + q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$	395b1 rank 0
395.2.a.d	3	degree 3	$3q - 2q^{2} - 2q^{3} + 3q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
395.2.a.e	3	degree 3	$3q - 3q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
395.2.a.f	3	degree 3	$3q + 6q^{2} + q^{3} + 6q^{4} + 3q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$
395.2.a.g	4	degree 4	$4q + q^{2} + 2q^{3} + 7q^{4} + 4q^{5} - 9q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$
395.2.a.h	11	degree 11	$11q - 2q^{3} + 20q^{4} - 11q^{5} + 3q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
395.2.a.a	$x + 3$
395.2.a.b	$x + 1$
395.2.a.c	$x - 1$
395.2.a.d	$x^{3} + 4x^{2} + 3x - 1$
395.2.a.e	$x^{3} - 3x - 1$
395.2.a.f	$x^{3} - 7x^{2} + 11x + 1$
395.2.a.g	$x^{4} - 3x^{3} - 5x^{2} + 16x - 1$
395.2.a.h	$x^{11} + 2x^{10} - 49x^{9} - 93x^{8} + 778x^{7} + 1450x^{6} - 4092x^{5} - 8096x^{4} + 1985x^{3} + 5404x^{2} + 185x - 523$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(395))$	27
Old forms	12
Cusp forms $S_{2}(\Gamma_0(395))$	39
Eisenstein series $E_{2}(\Gamma_0(395))$	3
Modular forms $M_{2}(\Gamma_0(395))$	42

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(395)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.194 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, 79, 395]: [0, 1, 7, 42] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 27 = 1 x 27 (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(395, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 395:2 1000

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