Atlas › Modular forms › Weight 2 › Level 949 › 949.2.a › 949.2.a.d

Newform orbit 949.2.a.d

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q - q^{2} + 3q^{3} + 17q^{4} + 4q^{5} + 8q^{6} - 3q^{7} - 9q^{8} + 19q^{9} + 8q^{10} + 23q^{11} + 3q^{12} + 16q^{13} + 6q^{14} - 17q^{15} + 3q^{16} + 14q^{17} + 10q^{18} + 10q^{19} + 21q^{20} - 28q^{21} - 23q^{22} + 30q^{23} + 35q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{16} - 8x^{15} - 318x^{14} + 2197x^{13} + 40434x^{12} - 225803x^{11} - 2680125x^{10} + 10924384x^{9} + 100144198x^{8} - 252344787x^{7} - 2083327704x^{6} + 2322460722x^{5} + 21731067424x^{4} - 1640010450x^{3} - 87358716685x^{2} - 23209973263x + 102132288983$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
73	$+1$

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
$\operatorname{tr} a_p$	−1	3	4	−3	23	16	14	10	30	20	−5	3	11	28	33	25	59	−17	7	−1	−16	21	2	9	20

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.291 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, 13, 73, 949]: [0, 1, 6, 86] (certified at ell = 2147482873)
T = [(2, 1), (3, 3)]; h of degree 73 = 1 x 73 (phi(ord eps) x dim S_k^new); 10 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(949, 2)[3]   # 949.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 949:2 1000