Atlas › Modular forms › Weight 2 › Level 306 › 306.2.a

Newspace 306.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(306))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
306.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 4q^{7} + O(q^{8})$	+ −	306b1 rank 1
306.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{5} + O(q^{8})$	+ +	306c1 rank 0
306.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{7} + O(q^{8})$	− −	306a1 rank 0
306.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 4q^{5} - 2q^{7} + O(q^{8})$	− −	306d1 rank 0
306.2.a.e	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{2} + 2q^{4} + 4q^{7} + O(q^{8})$	+ −
306.2.a.f	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{2} + 2q^{4} + 4q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
306.2.a.a	$x + 78$
306.2.a.b	$x - 174$
306.2.a.c	$x + 228$
306.2.a.d	$x + 88$
306.2.a.e	$x^{2} - 108x + 2190$
306.2.a.f	$x^{2} + 276x + 5790$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(306))$	8
Old forms	39
Cusp forms $S_{2}(\Gamma_0(306))$	47
Eisenstein series $E_{2}(\Gamma_0(306))$	15
Modular forms $M_{2}(\Gamma_0(306))$	62

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(306)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(102))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(153))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.415 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, 9, 17, 18, 34, 51, 102, 153, 306]: [0, 1, 1, 3, 2, 2, 5, 6, 8, 22, 20, 58] (certified at ell = 2147483137)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; 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(306, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 306:2 1000

← 305.2.a · 307.2.a →