Atlas › Modular forms › Weight 2 › Level 390 › 390.2.a

Newspace 390.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(390))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
390.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + O(q^{8})$	+ + + +	$-1$	390a1 rank 1
390.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ + − +	$+1$	390f1 rank 0
390.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ − + +	$+1$	390g1 rank 0
390.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − − −	$+1$	390d1 rank 0
390.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− + + +	$+1$	390e1 rank 0
390.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + O(q^{8})$	− + − −	$+1$	390b1 rank 0
390.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− − + −	$+1$	390c1 rank 0
390.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + O(q^{8})$	− − − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{11} + 3T_{19} + 4T_{23} + 5T_{29} + 6T_{17} + 7T_{31} + 8T_{37}$ 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 $T$
390.2.a.a	$x + 150$
390.2.a.b	$x - 148$
390.2.a.c	$x - 6$
390.2.a.d	$x - 56$
390.2.a.e	$x - 28$
390.2.a.f	$x + 90$
390.2.a.g	$x + 28$
390.2.a.h	$x^{2} + 124x + 316$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(390))$	9
Old forms	68
Cusp forms $S_{2}(\Gamma_0(390))$	77
Eisenstein series $E_{2}(\Gamma_0(390))$	15
Modular forms $M_{2}(\Gamma_0(390))$	92

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(390)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(78))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(130))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(195))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.444 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, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, 195, 390]: [0, 1, 1, 1, 3, 3, 1, 4, 5, 10, 6, 8, 18, 24, 32, 92] (certified at ell = 2147483629)
T = [(7, 1), (11, 2), (19, 3), (23, 4), (29, 5), (17, 6), (31, 7), (37, 8)]; h of degree 9 = 1 x 9 (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(390, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 390:2 1000

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