Atlas › Modular forms › Weight 2 › Level 673 › 673.2.a › 673.2.a.d

Newform orbit 673.2.a.d

Trace form

The 25 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$25q - 7q^{2} - 15q^{3} + 21q^{4} - 16q^{5} - 6q^{6} - 21q^{7} - 18q^{8} + 18q^{9} - 7q^{10} - 14q^{11} - 31q^{12} - 9q^{13} - 12q^{14} - 12q^{15} + 13q^{16} - 32q^{17} - 5q^{18} - 3q^{19} - 31q^{20} + 5q^{21} - 11q^{22} - 71q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 25. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{25} + 22x^{24} + 166x^{23} + 252x^{22} - 3053x^{21} - 14680x^{20} + 4956x^{19} + 167307x^{18} + 230007x^{17} - 813713x^{16} - 2130938x^{15} + 1492036x^{14} + 8440018x^{13} + 1691295x^{12} - 16903366x^{11} - 11399312x^{10} + 15948574x^{9} + 16676927x^{8} - 4882401x^{7} - 8782975x^{6} - 255401x^{5} + 1813831x^{4} + 263557x^{3} - 126396x^{2} - 21752x + 1273$

Atkin–Lehner signs

$p$	$w_p$
673	$+1$

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
$\operatorname{tr} a_p$	−7	−15	−16	−21	−14	−9	−32	−3	−71	−8	−8	−10	−15	−11	−88	−41	−35	5	−14	−41	−14	−11	−41	−17	−11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.226 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, 673]: [0, 56] (certified at ell = 2147480161)
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(673, 2)[3]   # 673.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 673:2 1000