Atlas › Modular forms › Weight 2 › Level 396 › 396.2.a

Newspace 396.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(396))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
396.2.a.a	1	$\mathbb{Q}$	$q - 2q^{5} - 2q^{7} + O(q^{8})$	+	396b1 rank 1
396.2.a.b	1	$\mathbb{Q}$	$q - 2q^{5} + 2q^{7} + O(q^{8})$	−	396a1 rank 0
396.2.a.c	1	$\mathbb{Q}$	$q + 3q^{5} + 2q^{7} + O(q^{8})$	−	396c1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
396.2.a.a	$x + 8$
396.2.a.b	$x - 4$
396.2.a.c	$x - 9$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(396))$	3
Old forms	58
Cusp forms $S_{2}(\Gamma_0(396))$	61
Eisenstein series $E_{2}(\Gamma_0(396))$	23
Modular forms $M_{2}(\Gamma_0(396))$	84

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(396)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 9}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(66))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(99))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(132))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(198))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.26 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, 4, 6, 9, 11, 12, 18, 22, 33, 36, 44, 66, 99, 132, 198, 396]: [0, 1, 1, 2, 3, 2, 2, 5, 5, 5, 6, 9, 9, 16, 14, 30, 40, 78] (certified at ell = 2147482951)
T = [(5, 1), (7, 3)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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(396, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 396:2 1000

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