Atlas › Modular forms › Weight 2 › Level 309 › 309.2.a › 309.2.a.b

Newform orbit 309.2.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + q^{2} - 3q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + 3q^{8} + 3q^{9} + 7q^{10} + 8q^{11} - q^{12} - 3q^{13} + 8q^{14} - q^{15} - 3q^{16} + 4q^{17} + q^{18} + 12q^{19} + 5q^{20} + 2q^{21} - 2q^{22} + 7q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} - 2x^{2} - 12x + 8$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
103	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	1	−3	1	−2	8	−3	4	12	7	0	7	−6	2	−1	12	−8	15	−19	−3	8	−12	2	17	24	3

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 3, 103, 309]: [0, 1, 9, 36] (certified at ell = 2147482501)
T = [(2, 1), (5, 1)]; h of degree 17 = 1 x 17 (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(309, 2)[1]   # 309.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 309:2 1000