Atlas › Modular forms › Weight 2 › Level 385 › 385.2.a

Newspace 385.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(385))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
385.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} - q^{4} + q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	385b1 rank 1
385.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} - q^{7} + O(q^{8})$	− + −	$-1$	385a1 rank 1
385.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{4} - 2q^{5} + 6q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
385.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 2q^{5} + 4q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$
385.2.a.e	3	degree 3	$3q - 3q^{2} - 4q^{3} + 5q^{4} - 3q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	+ − −	$-1$
385.2.a.f	3	degree 3	$3q - 3q^{2} - 2q^{3} + 5q^{4} - 3q^{5} + 4q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
385.2.a.g	3	degree 3	$3q + q^{2} + q^{4} + 3q^{5} - 2q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$
385.2.a.h	4	degree 4	$4q + 2q^{2} + 2q^{3} + 8q^{4} + 4q^{5} + 4q^{6} - 4q^{7} + O(q^{8})$	− + +	$+1$

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 $T$
385.2.a.a	$x + 7$
385.2.a.b	$x + 1$
385.2.a.c	$x^{2} - 6x - 39$
385.2.a.d	$x^{2} - 2x - 31$
385.2.a.e	$x^{3} + 15x^{2} + 47x + 37$
385.2.a.f	$x^{3} + 9x^{2} - 13x - 17$
385.2.a.g	$x^{3} - x^{2} - 33x + 85$
385.2.a.h	$x^{4} - 8x^{3} - 78x^{2} + 596x + 241$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(385))$	19
Old forms	26
Cusp forms $S_{2}(\Gamma_0(385))$	45
Eisenstein series $E_{2}(\Gamma_0(385))$	7
Modular forms $M_{2}(\Gamma_0(385))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(385)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(77))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.31 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, 11, 35, 55, 77, 385]: [0, 1, 1, 2, 6, 8, 10, 52] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 19 = 1 x 19 (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(385, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 385:2 1000

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