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

Newform orbit 433.2.a.c

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q - 10q^{2} - 8q^{3} + 12q^{4} - 5q^{5} + q^{6} - 15q^{7} - 24q^{8} + 5q^{9} - 3q^{10} - 14q^{11} - 12q^{12} - 9q^{13} + 5q^{14} - 20q^{15} + 14q^{16} - 25q^{17} - 9q^{18} - q^{19} - q^{21} + 4q^{22} - 45q^{23} + 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} + 18x^{14} + 115x^{13} + 237x^{12} - 532x^{11} - 2987x^{10} - 2184x^{9} + 8027x^{8} + 13096x^{7} - 3756x^{6} - 16393x^{5} - 3891x^{4} + 5893x^{3} + 1865x^{2} - 476x + 23$

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 odd 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
$\operatorname{tr} a_p$	−10	−8	−5	−15	−14	−9	−25	−1	−45	−10	−3	−9	−11	−26	−48	−37	−12	14	5	−27	2	−5	−33	20	13

$\operatorname{tr} a_n$ 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)[2]   # 433.2.a.c

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

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