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

Newform orbit 673.2.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} - 3x^{3} - 17x^{2} + 35x + 49$

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
$\operatorname{tr} a_p$	−1	4	−4	4	−4	1	0	2	11	26	−3	−16	31	5	2	3	2	−11	23	−15	11	−2	−2	14	13

$\operatorname{tr} a_n$ 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)[1]   # 673.2.a.b

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

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