Atlas › Modular forms › Weight 2 › Level 491 › 491.2.a › 491.2.a.c

Newform orbit 491.2.a.c

Trace form

The 29 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$29q + 5q^{3} + 40q^{4} + 12q^{5} + 3q^{7} - 3q^{8} + 40q^{9} + 17q^{10} + 4q^{12} + 35q^{13} + 5q^{14} + 3q^{15} + 58q^{16} + 3q^{17} - 9q^{18} + 24q^{19} + 17q^{20} + 31q^{21} - 7q^{22} - 5q^{23} - 17q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 29. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{29} - 15x^{28} - 508x^{27} + 8470x^{26} + 105900x^{25} - 2068203x^{24} - 11395058x^{23} + 288092335x^{22} + 607199795x^{21} - 25447159059x^{20} - 3549960431x^{19} + 1499369059275x^{18} - 1671699297919x^{17} - 60372858562804x^{16} + 118547509096850x^{15} + 1674103404634478x^{14} - 4280470940484677x^{13} - 31813217870552647x^{12} + 94465133527623865x^{11} + 407769720150392060x^{10} - 1314788566114377001x^{9} - 3436528966123750655x^{8} + 11289582404991676004x^{7} + 18473951319027690155x^{6} - 55694246841114426726x^{5} - 62015635093403660324x^{4} + 134806387664166876684x^{3} + 125685460115986926358x^{2} - 107914741621532953472x - 97617544757250161560$

Atkin–Lehner signs

$p$	$w_p$
491	$-1$

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
$\operatorname{tr} a_p$	0	5	12	3	0	35	3	24	−5	12	36	40	16	12	10	24	−3	100	5	−18	2	11	−35	11	27

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.201 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, 491]: [0, 42] (certified at ell = 2147474491)
T = [(2, 1), (3, 3)]; h of degree 41 = 1 x 41 (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(491, 2)[2]   # 491.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 491:2 1000