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

Newform orbit 673.2.a.c

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q + 3q^{2} + 9q^{3} + 31q^{4} + 20q^{5} - 4q^{6} + 7q^{7} + 12q^{8} + 17q^{9} + 28q^{11} + 16q^{12} + 6q^{13} + 9q^{14} + 4q^{15} + 29q^{16} + 30q^{17} - 19q^{18} + 7q^{19} + 44q^{20} - 23q^{21} - 6q^{22} + 46q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} - 12x^{23} - 8x^{22} + 610x^{21} - 1596x^{20} - 9935x^{19} + 45490x^{18} + 49439x^{17} - 489840x^{16} + 200145x^{15} + 2472679x^{14} - 2818419x^{13} - 5827910x^{12} + 9799744x^{11} + 5497673x^{10} - 13784295x^{9} - 959315x^{8} + 7931136x^{7} - 940885x^{6} - 1522842x^{5} + 242098x^{4} + 53321x^{3} + 380x^{2} - 160x - 3$

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 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
$\operatorname{tr} a_p$	3	9	20	7	28	6	30	7	46	−2	15	14	−44	4	102	20	9	−12	−27	72	−33	15	49	13	−40

$\operatorname{tr} a_n$ 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)[2]   # 673.2.a.c

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

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