Atlas › Modular forms › Weight 2 › Level 733 › 733.2.a › 733.2.a.c

Newform orbit 733.2.a.c

Trace form

The 25 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$25q - 11q^{2} - 18q^{3} + 25q^{4} - 6q^{5} + 4q^{6} - 11q^{7} - 42q^{8} + 19q^{9} - 5q^{10} - 37q^{11} - 20q^{12} - 4q^{13} - 10q^{14} - 19q^{15} + 25q^{16} - 9q^{17} - 26q^{18} - 18q^{19} - 26q^{20} - 12q^{21} + 10q^{22} - 50q^{23} + 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 25. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{25} + 29x^{24} + 332x^{23} + 1696x^{22} + 1121x^{21} - 29473x^{20} - 124924x^{19} - 18993x^{18} + 1088921x^{17} + 2189629x^{16} - 2830171x^{15} - 13391105x^{14} - 2899213x^{13} + 36943251x^{12} + 29754047x^{11} - 55263229x^{10} - 64865183x^{9} + 47808627x^{8} + 68789791x^{7} - 25525722x^{6} - 38499528x^{5} + 9511586x^{4} + 10874628x^{3} - 2529219x^{2} - 1223880x + 330790$

Atkin–Lehner signs

$p$	$w_p$
733	$+1$

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
$\operatorname{tr} a_p$	−11	−18	−6	−11	−37	−4	−9	−18	−50	−7	−16	−10	−6	−26	−66	−34	−42	10	−62	−51	−7	−18	−101	−2	−18

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.247 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, 733]: [0, 61] (certified at ell = 2147479381)
T = [(2, 1), (3, 1)]; h of degree 60 = 1 x 60 (phi(ord eps) x dim S_k^new); 6 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(733, 2)[2]   # 733.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 733:2 1000