Atlas › Modular forms › Weight 2 › Level 294 › 294.2.a

Newspace 294.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(294))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
294.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 3q^{5} + q^{6} + O(q^{8})$	+ +	294e1 rank 0
294.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 4q^{5} + q^{6} + O(q^{8})$	+ +	294f1 rank 0
294.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 4q^{5} - q^{6} + O(q^{8})$	+ −	294g1 rank 1
294.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} + O(q^{8})$	+ −	294d1 rank 0
294.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + O(q^{8})$	− +	294a1 rank 0
294.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + O(q^{8})$	− −	294b1 rank 0
294.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + O(q^{8})$	− −	294c1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
294.2.a.a	$x - 140$
294.2.a.b	$x$
294.2.a.c	$x + 8$
294.2.a.d	$x + 78$
294.2.a.e	$x + 86$
294.2.a.f	$x + 40$
294.2.a.g	$x + 52$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(294))$	7
Old forms	34
Cusp forms $S_{2}(\Gamma_0(294))$	41
Eisenstein series $E_{2}(\Gamma_0(294))$	31
Modular forms $M_{2}(\Gamma_0(294))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(294)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(42))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(98))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(147))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.473 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, 14, 21, 42, 49, 98, 147, 294]: [0, 1, 1, 3, 1, 4, 4, 12, 5, 16, 20, 60] (certified at ell = 2147483647)
T = [(5, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 7 = 1 x 7 (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(294, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 294:2 1000

← 293.2.a · 295.2.a →