Atlas › Modular forms › Weight 2 › Level 739 › 739.2.a › 739.2.a.d

Newform orbit 739.2.a.d

Trace form

The 34 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$34q + 3q^{2} + 4q^{3} + 35q^{4} + 32q^{5} + 3q^{6} - 2q^{7} + 12q^{8} + 50q^{9} + 3q^{10} + 13q^{11} + q^{12} + 10q^{13} + 17q^{14} + 10q^{15} + 41q^{16} + 20q^{17} + 4q^{19} + 50q^{20} + 45q^{21} - 11q^{22} + q^{23} - 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 34. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{34} - 7x^{33} - 106x^{32} + 820x^{31} + 4612x^{30} - 41108x^{29} - 106483x^{28} + 1167935x^{27} + 1397104x^{26} - 21055091x^{25} - 9953917x^{24} + 255767100x^{23} + 24857991x^{22} - 2166101506x^{21} + 148387195x^{20} + 12999469457x^{19} - 1246179806x^{18} - 55299670634x^{17} + 1723497956x^{16} + 163927890558x^{15} + 14452993144x^{14} - 325938779986x^{13} - 74353567770x^{12} + 407658412705x^{11} + 146335227514x^{10} - 292283074525x^{9} - 135004074058x^{8} + 105940838841x^{7} + 55294410648x^{6} - 17989766611x^{5} - 9499265341x^{4} + 1428149124x^{3} + 566489124x^{2} - 51332032x + 1047632$

Atkin–Lehner signs

$p$	$w_p$
739	$-1$

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
$\operatorname{tr} a_p$	3	4	32	−2	13	10	20	4	1	55	18	12	74	−28	43	47	82	29	2	25	−8	−2	18	98	−22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 739]: [0, 62] (certified at ell = 2147483323)
T = [(2, 1), (3, 1)]; h of degree 61 = 1 x 61 (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(739, 2)[3]   # 739.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 739:2 1000