Atlas › Modular forms › Weight 2 › Level 823 › 823.2.a › 823.2.a.b

Newform orbit 823.2.a.b

Trace form

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

$38q + 9q^{2} + 10q^{3} + 39q^{4} + 18q^{5} + 6q^{7} + 24q^{8} + 50q^{9} + 9q^{10} + 10q^{11} + 14q^{12} + 17q^{13} + 9q^{14} - q^{15} + 33q^{16} + 83q^{17} + 6q^{18} + 4q^{19} + 37q^{20} + 2q^{21} + 5q^{22} + 9q^{23} - 3q^{24} + O(q^{25})$

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

$x^{38} - 19x^{37} + 41x^{36} + 1390x^{35} - 8912x^{34} - 30949x^{33} + 407587x^{32} - 135711x^{31} - 9087069x^{30} + 19016778x^{29} + 113636427x^{28} - 406427070x^{27} - 788005998x^{26} + 4665199394x^{25} + 2136696137x^{24} - 33691210160x^{23} + 10487985648x^{22} + 161984638602x^{21} - 126576356826x^{20} - 532298664369x^{19} + 580823894114x^{18} + 1216831594989x^{17} - 1563657612732x^{16} - 1981772588733x^{15} + 2676213480631x^{14} + 2389862061309x^{13} - 2956336874840x^{12} - 2210612594575x^{11} + 2047224039389x^{10} + 1548965393739x^{9} - 791212616934x^{8} - 748922897305x^{7} + 96414556484x^{6} + 206282621073x^{5} + 34293826137x^{4} - 19501951112x^{3} - 9067517561x^{2} - 1423291205x - 79855915$

Atkin–Lehner signs

$p$	$w_p$
823	$-1$

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	10	18	6	10	17	83	4	9	33	−1	12	81	−5	19	54	11	21	−8	13	51	−19	39	64	37

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 823]: [0, 69] (certified at ell = 2147477467)
T = [(2, 1), (3, 1)]; h of degree 68 = 1 x 68 (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(823, 2)[1]   # 823.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 823:2 1000