Atlas › Modular forms › Weight 2 › Level 701 › 701.2.a › 701.2.a.c

Newform orbit 701.2.a.c

Trace form

The 36 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$36q - q^{2} + 14q^{3} + 49q^{4} + q^{5} + 3q^{6} + 8q^{7} + 46q^{9} + 6q^{10} + 10q^{11} + 24q^{12} + 20q^{13} + 7q^{14} + 8q^{15} + 79q^{16} - 6q^{17} - 2q^{18} + 68q^{19} - 2q^{20} + 5q^{21} + 13q^{22} - 5q^{23} + 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 36. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{36} - 13x^{35} - 56x^{34} + 1365x^{33} - 800x^{32} - 59933x^{31} + 143346x^{30} + 1450278x^{29} - 5178655x^{28} - 21281758x^{27} + 100781212x^{26} + 193839778x^{25} - 1239509292x^{24} - 1040133571x^{23} + 10258758276x^{22} + 2344652103x^{21} - 58949677654x^{20} + 7665549249x^{19} + 239177929956x^{18} - 79819505234x^{17} - 690955231086x^{16} + 297485829245x^{15} + 1424099151387x^{14} - 635102799522x^{13} - 2085403211329x^{12} + 823884763097x^{11} + 2142931310486x^{10} - 622123656482x^{9} - 1503226788483x^{8} + 221456669403x^{7} + 678178898782x^{6} + 9196195342x^{5} - 172068088339x^{4} - 29754574558x^{3} + 16691610612x^{2} + 5629059399x + 419410201$

Atkin–Lehner signs

$p$	$w_p$
701	$-1$

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
$\operatorname{tr} a_p$	−1	14	1	8	10	20	−6	68	−5	−1	64	12	0	54	−6	−24	80	43	37	31	6	44	2	−6	−16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 701]: [0, 59] (certified at ell = 2147479601)
T = [(2, 1), (3, 1)]; h of degree 58 = 1 x 58 (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(701, 2)[2]   # 701.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 701:2 1000