Atlas › Modular forms › Weight 2 › Level 979 › 979.2.a › 979.2.a.f

Newform orbit 979.2.a.f

Trace form

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

$25q + 4q^{2} + 3q^{3} + 34q^{4} + 18q^{5} - 4q^{6} - q^{7} + 6q^{8} + 46q^{9} - 4q^{10} + 25q^{11} - 3q^{12} - 4q^{13} + 15q^{14} + 13q^{15} + 52q^{16} + 13q^{17} - 14q^{18} - 9q^{19} + 35q^{20} + 17q^{21} + 4q^{22} + 19q^{23} - 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 25. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{25} - 96x^{24} - 3650x^{23} + 612853x^{22} - 2682063x^{21} - 1518174652x^{20} + 31369043824x^{19} + 1821595660710x^{18} - 61028065922216x^{17} - 1001301034966767x^{16} + 56179648921397855x^{15} + 64872139126608092x^{14} - 27043021839487685071x^{13} + 183346173173820720068x^{12} + 6484030403964784645333x^{11} - 77509090625587413468778x^{10} - 666298234383381832141318x^{9} + 11151151992525459471878902x^{8} + 34677183605420208839310899x^{7} - 754671862983684640529038385x^{6} - 1236160138629896194450729554x^{5} + 24722288192343297019292096576x^{4} + 34452802742977666287814160875x^{3} - 336454860327398427183540087260x^{2} - 460433485897048970178233814371x + 936958585176353020813869299473$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
89	$+1$

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	3	18	−1	25	−4	13	−9	19	58	12	16	12	3	25	49	7	7	13	26	−21	−9	18	−25	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.366 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, 11, 89, 979]: [0, 2, 8, 92] (certified at ell = 2147482921)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 73 = 1 x 73 (phi(ord eps) x dim S_k^new); 16 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(979, 2)[5]   # 979.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 979:2 1000