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

Newform orbit 733.2.a.d

Trace form

The 32 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$32q + 12q^{2} + 17q^{3} + 38q^{4} + 8q^{5} + 5q^{6} + 3q^{7} + 39q^{8} + 43q^{9} - q^{10} + 37q^{11} + 21q^{12} - 4q^{13} - 2q^{14} + 21q^{15} + 42q^{16} + 6q^{17} + 24q^{18} + 5q^{19} + 8q^{20} - 2q^{21} + 8q^{22} + 38q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 32. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{32} - 29x^{31} + 295x^{30} - 619x^{29} - 10852x^{28} + 81274x^{27} - 34934x^{26} - 1739281x^{25} + 5897992x^{24} + 10720447x^{23} - 95387053x^{22} + 73143224x^{21} + 669346645x^{20} - 1428198234x^{19} - 2129700606x^{18} + 8932410838x^{17} + 793675370x^{16} - 29883366484x^{15} + 15831434640x^{14} + 58476042103x^{13} - 52219493333x^{12} - 67406584268x^{11} + 76243003734x^{10} + 44253601406x^{9} - 55668066925x^{8} - 15317471191x^{7} + 18447187180x^{6} + 2317998937x^{5} - 1913456802x^{4} - 72678275x^{3} + 39614568x^{2} + 2404080x + 21600$

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 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
$\operatorname{tr} a_p$	12	17	8	3	37	−4	6	5	38	19	−7	13	1	42	57	51	65	−41	68	31	−7	−17	82	14	−9

$\operatorname{tr} a_n$ 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)[3]   # 733.2.a.d

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

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