Atlas › Modular forms › Weight 2 › Level 445 › 445.2.a › 445.2.a.g

Newform orbit 445.2.a.g

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{8} - 19x^{7} - 11x^{6} + 1875x^{5} - 5942x^{4} - 47629x^{3} + 202749x^{2} + 57461x - 271357$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
89	$-1$

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	6	−8	−6	14	−7	17	17	−1	10	1	−11	15	−5	12	−1	26	13	−25	28	−17	−7	44	8	1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.289 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, 5, 89, 445]: [0, 1, 8, 46] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 4 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(445, 2)[6]   # 445.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 445:2 1000