Atlas › Modular forms › Weight 2 › Level 797 › 797.2.a › 797.2.a.d

Newform orbit 797.2.a.d

Trace form

The 38 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$38q + 9q^{2} + 13q^{3} + 53q^{4} - 2q^{5} + 3q^{6} + 19q^{7} + 15q^{8} + 49q^{9} + 12q^{10} + 14q^{11} + 26q^{12} + 11q^{13} + 8q^{14} + q^{15} + 75q^{16} + 7q^{17} + 19q^{18} + 39q^{19} - q^{20} - 2q^{21} + 22q^{22} + 24q^{23} - 25q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 38. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{38} - 22x^{37} + 93x^{36} + 1345x^{35} - 12975x^{34} - 11965x^{33} + 526883x^{32} - 1088106x^{31} - 10383263x^{30} + 42631083x^{29} + 102744619x^{28} - 757826731x^{27} - 270869333x^{26} + 8067832110x^{25} - 5337021730x^{24} - 56130872192x^{23} + 74267878253x^{22} + 265421818683x^{21} - 491647251729x^{20} - 868104027906x^{19} + 2049223274196x^{18} + 1972575595598x^{17} - 5770439440696x^{16} - 3103745021446x^{15} + 11198670300861x^{14} + 3375232255865x^{13} - 14894442034658x^{12} - 2591363981983x^{11} + 13199857347391x^{10} + 1505528107177x^{9} - 7357131544947x^{8} - 689967135631x^{7} + 2316473842503x^{6} + 205023072547x^{5} - 338868266135x^{4} - 23824511954x^{3} + 18590952329x^{2} + 900207304x - 137292782$

Atkin–Lehner signs

$p$	$w_p$
797	$-1$

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
$\operatorname{tr} a_p$	9	13	−2	19	14	11	7	39	24	3	15	27	−7	150	−15	−12	7	3	97	−22	50	32	44	−31	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.369 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, 797]: [0, 67] (certified at ell = 2147483029)
T = [(2, 1), (3, 1)]; h of degree 66 = 1 x 66 (phi(ord eps) x dim S_k^new); 7 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(797, 2)[3]   # 797.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 797:2 1000