Atlas › Modular forms › Weight 2 › Level 849 › 849.2.a › 849.2.a.c

Newform orbit 849.2.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{10} + 2x^{9} - 28x^{8} - 37x^{7} + 208x^{6} + 117x^{5} - 470x^{4} - 109x^{3} + 298x^{2} + 52x - 40$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
283	$+1$

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 odd 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
$\operatorname{tr} a_p$	−1	−10	−1	−6	−15	5	−4	−23	−19	−10	−24	22	−20	−16	−25	−10	−27	3	10	−44	0	−28	−11	−39	15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 3, 283, 849]: [0, 1, 24, 96] (certified at ell = 2147480197)
T = [(2, 1), (5, 1)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 5 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(849, 2)[2]   # 849.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 849:2 1000