Atlas › Modular forms › Weight 2 › Level 683 › 683.2.a › 683.2.a.c

Newform orbit 683.2.a.c

Trace form

The 31 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$31q + 2q^{2} + 3q^{3} + 30q^{4} + 18q^{5} + 7q^{6} + 20q^{7} + 34q^{9} - q^{10} - 4q^{11} - 7q^{12} + 62q^{13} - 8q^{14} + 9q^{15} + 36q^{16} + 27q^{17} + 12q^{18} + 16q^{19} + 22q^{20} + 10q^{21} - q^{22} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 31. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{31} - 5x^{30} - 104x^{29} + 532x^{28} + 4575x^{27} - 23909x^{26} - 113096x^{25} + 600227x^{24} + 1769011x^{23} - 9392808x^{22} - 18755020x^{21} + 96720526x^{20} + 141116851x^{19} - 672467224x^{18} - 775278914x^{17} + 3173240002x^{16} + 3130892456x^{15} - 10008752748x^{14} - 9089042740x^{13} + 20226436632x^{12} + 17892980899x^{11} - 23969518582x^{10} - 21456104663x^{9} + 13749264263x^{8} + 12831504056x^{7} - 2232679256x^{6} - 2478721820x^{5} - 811126x^{4} + 64562587x^{3} + 793916x^{2} - 115362x - 1931$

Atkin–Lehner signs

$p$	$w_p$
683	$-1$

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
$\operatorname{tr} a_p$	2	3	18	20	−4	62	27	16	0	21	2	59	−2	37	−15	8	−3	74	−2	1	107	−3	−20	6	51

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.23 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, 683]: [0, 58] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (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(683, 2)[2]   # 683.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 683:2 1000