Atlas › Modular forms › Weight 2 › Level 399 › 399.2.a

Newspace 399.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(399))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
399.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	399b1 rank 1
399.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} + 4q^{5} - q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	399c1 rank 0
399.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$	399a1 rank 1
399.2.a.d	3	degree 3	$3q + q^{2} - 3q^{3} + q^{4} + 4q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ + −	$+1$
399.2.a.e	3	degree 3	$3q + q^{2} + 3q^{3} + 9q^{4} + q^{6} - 3q^{7} + O(q^{8})$	− + +	$+1$
399.2.a.f	5	degree 5	$5q + q^{2} + 5q^{3} + 7q^{4} - 2q^{5} + q^{6} + 5q^{7} + O(q^{8})$	− − −	$+1$
399.2.a.g	5	degree 5	$5q + 3q^{2} - 5q^{3} + 7q^{4} + 4q^{5} - 3q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
399.2.a.a	$x + 47$
399.2.a.b	$x - 29$
399.2.a.c	$x + 73$
399.2.a.d	$x^{3} - 127x^{2} + 1993x - 2129$
399.2.a.e	$x^{3} - 143x^{2} + 6333x - 81441$
399.2.a.f	$x^{5} - 23x^{4} - 7244x^{3} + 80226x^{2} + 12451365x + 79229023$
399.2.a.g	$x^{5} - 121x^{4} + 4136x^{3} - 45454x^{2} + 6853x + 1325417$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(399))$	19
Old forms	30
Cusp forms $S_{2}(\Gamma_0(399))$	49
Eisenstein series $E_{2}(\Gamma_0(399))$	7
Modular forms $M_{2}(\Gamma_0(399))$	56

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(399)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(133))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 7, 19, 21, 57, 133, 399]: [0, 1, 1, 2, 4, 8, 14, 56] (certified at ell = 2147483647)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 19 = 1 x 19 (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(399, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 399:2 1000

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