Atlas › Modular forms › Weight 2 › Level 330 › 330.2.a

Newspace 330.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(330))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
330.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + O(q^{8})$	+ + + −	$+1$	330a1 rank 0
330.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ + − −	$-1$	330e1 rank 1
330.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 4q^{7} + O(q^{8})$	− + + +	$+1$	330d1 rank 0
330.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + O(q^{8})$	− + − −	$+1$	330c1 rank 0
330.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + O(q^{8})$	− − − +	$+1$	330b1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{13} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{29} + 7T_{31} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53}$ 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 $T$
330.2.a.a	$x - 216$
330.2.a.b	$x + 92$
330.2.a.c	$x + 324$
330.2.a.d	$x + 160$
330.2.a.e	$x - 16$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(330))$	5
Old forms	60
Cusp forms $S_{2}(\Gamma_0(330))$	65
Eisenstein series $E_{2}(\Gamma_0(330))$	15
Modular forms $M_{2}(\Gamma_0(330))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(330)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(66))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(110))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(165))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.548 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, 11, 15, 22, 30, 33, 55, 66, 110, 165, 330]: [0, 1, 1, 1, 3, 3, 2, 4, 5, 10, 6, 8, 16, 22, 28, 80] (certified at ell = 2147482921)
T = [(7, 1), (13, 2), (17, 3), (19, 4), (23, 5), (29, 6), (31, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 2 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(330, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 330:2 1000

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