Atlas › Modular forms › Weight 2 › Level 835 › 835.2.a › 835.2.a.d

Newform orbit 835.2.a.d

Trace form

The 22 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$22q + 2q^{2} + 3q^{3} + 30q^{4} + 22q^{5} + 9q^{6} - q^{7} + 6q^{8} + 41q^{9} + 2q^{10} + 10q^{11} - 3q^{12} + 8q^{13} + 8q^{14} + 3q^{15} + 38q^{16} + 17q^{17} - 15q^{18} + 9q^{19} + 30q^{20} + 14q^{21} - 24q^{22} + 12q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 22. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{22} - 5x^{21} - 87x^{20} + 470x^{19} + 2794x^{18} - 17064x^{17} - 40308x^{16} + 309899x^{15} + 213551x^{14} - 3000568x^{13} + 728032x^{12} + 15067444x^{11} - 12509763x^{10} - 34050310x^{9} + 45184448x^{8} + 16787179x^{7} - 48831906x^{6} + 22371376x^{5} - 216026x^{4} - 1780827x^{3} + 212378x^{2} + 20776x + 392$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
167	$+1$

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
$\operatorname{tr} a_p$	2	3	22	−1	10	8	17	9	12	26	6	−14	27	−19	22	44	36	30	−19	0	−35	−6	2	57	−26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.247 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, 5, 167, 835]: [0, 1, 15, 86] (certified at ell = 2147483237)
T = [(2, 1), (3, 1)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 6 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(835, 2)[3]   # 835.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 835:2 1000