Atlas › Modular forms › Weight 2 › Level 973 › 973.2.a › 973.2.a.h

Newform orbit 973.2.a.h

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 7q^{2} + 7q^{3} + 15q^{4} - q^{5} + 9q^{6} - 15q^{7} + 18q^{8} + 6q^{9} - 2q^{10} + 13q^{11} + 21q^{12} - 2q^{13} - 7q^{14} + 4q^{15} + 39q^{16} - 4q^{17} + 33q^{18} + 12q^{19} - 6q^{20} - 7q^{21} + 24q^{23} + 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{15} - 28x^{14} + 113x^{13} + 2917x^{12} - 21138x^{11} - 127116x^{10} + 1091983x^{9} + 3268088x^{8} - 25396112x^{7} - 56685489x^{6} + 251890354x^{5} + 535177186x^{4} - 540002487x^{3} - 736784243x^{2} + 442946622x + 25418565$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
139	$-1$

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
$\operatorname{tr} a_p$	7	7	−1	−15	13	−2	−4	12	24	15	13	16	17	15	33	28	14	−18	−2	34	14	14	71	−23	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.39 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, 7, 139, 973]: [0, 1, 12, 94] (certified at ell = 2147483137)
T = [(2, 1), (3, 3)]; h of degree 69 = 1 x 69 (phi(ord eps) x dim S_k^new); 9 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(973, 2)[7]   # 973.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 973:2 1000