Atlas › Modular forms › Weight 2 › Level 194 › 194.2.a › 194.2.a.c

Newform orbit 194.2.a.c

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{4} + 4x^{3} - 24x^{2} - 24x + 92$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
97	$+1$

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
$\operatorname{tr} a_p$	4	2	−2	2	−2	4	−8	0	−10	6	8	14	0	10	−28	−4	−20	12	16	12	10	8	−12	−18	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.088 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, 2, 97, 194]: [0, 1, 8, 26] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 9 = 1 x 9 (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(194, 2)[2]   # 194.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 194:2 1000