Atlas › Modular forms › Weight 2 › Level 853 › 853.2.a › 853.2.a.b

Newform orbit 853.2.a.b

Trace form

The 37 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$37q + 13q^{2} + 18q^{3} + 39q^{4} + 14q^{5} + q^{6} + 9q^{7} + 33q^{8} + 41q^{9} - q^{10} + 44q^{11} + 23q^{12} + 3q^{13} + 8q^{14} + 23q^{15} + 39q^{16} + 14q^{17} + 16q^{18} + 13q^{19} + 24q^{20} + 23q^{22} + 51q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 37. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{37} - 31x^{36} + 347x^{35} - 1023x^{34} - 11731x^{33} + 110290x^{32} - 142113x^{31} - 2256717x^{30} + 10382395x^{29} + 9146433x^{28} - 169755525x^{27} + 251989151x^{26} + 1242206362x^{25} - 4192906312x^{24} - 3018739386x^{23} + 30063608697x^{22} - 17926967106x^{21} - 119985065878x^{20} + 178275701137x^{19} + 262411415246x^{18} - 703067543078x^{17} - 197649999419x^{16} + 1593531989477x^{15} - 472613092140x^{14} - 2185805172089x^{13} + 1528928872933x^{12} + 1724158046259x^{11} - 1952020730939x^{10} - 600163280781x^{9} + 1312492474311x^{8} - 93926017444x^{7} - 442452034977x^{6} + 136091778203x^{5} + 51533196302x^{4} - 29230179920x^{3} + 3188105710x^{2} + 37856504x - 2384633$

Atkin–Lehner signs

$p$	$w_p$
853	$-1$

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
$\operatorname{tr} a_p$	13	18	14	9	44	3	14	13	51	22	7	27	16	31	45	52	38	−20	56	43	−17	4	71	21	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 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, 853]: [0, 71] (certified at ell = 2147481337)
T = [(2, 1), (3, 1)]; h of degree 70 = 1 x 70 (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(853, 2)[1]   # 853.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 853:2 1000