Atlas › Modular forms › Weight 2 › Level 193 › 193.2.a › 193.2.a.c

Newform orbit 193.2.a.c

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 7x^{7} + x^{6} + 80x^{5} - 117x^{4} - 214x^{3} + 457x^{2} - 30x - 169$

Atkin–Lehner signs

$p$	$w_p$
193	$-1$

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
$\operatorname{tr} a_p$	2	5	8	5	9	−4	7	0	23	−2	7	−13	−3	−4	40	8	0	−6	3	25	−9	−9	44	24	−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 0.166 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, 193]: [0, 16] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 2 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(193, 2)[2]   # 193.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 193:2 1000