Atlas › Modular forms › Weight 2 › Level 378 › 378.2.a

Newspace 378.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(378))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
378.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 4q^{5} - q^{7} + O(q^{8})$	+ +	378h1 rank 0
378.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 3q^{5} + q^{7} + O(q^{8})$	+ −	378f1 rank 1
378.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} - q^{7} + O(q^{8})$	+ +	378d1 rank 1
378.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{7} + O(q^{8})$	+ −	378b1 rank 0
378.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{7} + O(q^{8})$	− −	378a1 rank 0
378.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{5} - q^{7} + O(q^{8})$	− +	378c1 rank 0
378.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 3q^{5} + q^{7} + O(q^{8})$	− −	378e1 rank 0
378.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 4q^{5} - q^{7} + O(q^{8})$	− +	378g1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
378.2.a.a	$x - 90$
378.2.a.b	$x + 119$
378.2.a.c	$x + 39$
378.2.a.d	$x - 22$
378.2.a.e	$x - 64$
378.2.a.f	$x - 21$
378.2.a.g	$x + 47$
378.2.a.h	$x + 24$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(378))$	8
Old forms	53
Cusp forms $S_{2}(\Gamma_0(378))$	61
Eisenstein series $E_{2}(\Gamma_0(378))$	23
Modular forms $M_{2}(\Gamma_0(378))$	84

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(378)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(42))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(54))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(63))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(126))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(189))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.42 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, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, 378]: [0, 1, 1, 3, 1, 2, 4, 5, 4, 4, 12, 11, 10, 28, 26, 76] (certified at ell = 2147483647)
T = [(5, 1), (11, 3), (13, 5), (17, 7), (19, 9)]; h of degree 8 = 1 x 8 (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(378, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 378:2 1000

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