Atlas › Modular forms › Weight 2 › Level 433 › 433.2.a › 433.2.a.d

Newform orbit 433.2.a.d

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + 7q^{2} + 6q^{3} + 27q^{4} + 7q^{5} - q^{6} + 7q^{7} + 33q^{8} + 14q^{9} - 7q^{10} + 12q^{11} - 2q^{12} - 3q^{13} - q^{14} - 16q^{15} + 29q^{16} + 17q^{17} + 12q^{18} + 9q^{19} - 4q^{20} - 17q^{21} - 18q^{22} + 47q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 13x^{15} + 25x^{14} + 328x^{13} - 1499x^{12} - 1295x^{11} + 16641x^{10} - 11749x^{9} - 71809x^{8} + 96032x^{7} + 128699x^{6} - 227616x^{5} - 82858x^{4} + 185074x^{3} + 29413x^{2} - 46005x - 11233$

Atkin–Lehner signs

$p$	$w_p$
433	$-1$

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
$\operatorname{tr} a_p$	7	6	7	7	12	−3	17	9	47	10	17	−7	−5	12	44	21	8	10	−15	35	−28	15	49	−26	−45

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.271 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, 433]: [0, 36] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 35 = 1 x 35 (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(433, 2)[3]   # 433.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 433:2 1000