Atlas › Modular forms › Weight 6 › Level 107 › 107.6.a › 107.6.a.b

Newform orbit 107.6.a.b

Trace form

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

$25q + 4q^{2} + 55q^{3} + 464q^{4} + 149q^{5} + 201q^{6} + 380q^{7} - 21q^{8} + 2652q^{9} + 1432q^{10} + 346q^{11} + 2412q^{12} + 3556q^{13} + 319q^{14} + 59q^{15} + 8196q^{16} + 3453q^{17} - 1284q^{18} + 4208q^{19} + 6073q^{20} + 3815q^{21} + 6557q^{22} + 3904q^{23} + 6035q^{24} + O(q^{25})$

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

$x^{25} - 59x^{24} - 3456x^{23} + 253294x^{22} + 3910668x^{21} - 439394578x^{20} - 998299544x^{19} + 412473702956x^{18} - 1443467661995x^{17} - 235449846170355x^{16} + 1485342471365875x^{15} + 86431766860733725x^{14} - 655230250411378709x^{13} - 20885274038919762155x^{12} + 165143502115432244665x^{11} + 3318353481847233911713x^{10} - 25165090638779293339161x^{9} - 337703236958638973773121x^{8} + 2291280065261137537822260x^{7} + 20702626239337816415925520x^{6} - 116209500871394322636424684x^{5} - 674740192221856454077542620x^{4} + 2728605558791848301144165200x^{3} + 8988208753262057014903547488x^{2} - 14392492587796328071214353504x - 31643015214665313867411339904$

Atkin–Lehner signs

$p$	$w_p$
107	$-1$

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
$\operatorname{tr} a_p$	4	55	149	380	346	3556	3453	4208	3904	26478	24906	61897	18704	22155	33308	42422	−11907	112045	37148	−111079	65266	53895	−187151	−99478	308322

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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 engine's checks

dim M^+ at levels [1, 107]: [1, 46] (certified at ell = 2147482621)
T = [(2, 1), (3, 1)]; h of degree 44 = 1 x 44 (phi(ord eps) x dim S_k^new); 8 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(107, 6)[1]   # 107.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 107:6 1000