Atlas › Modular forms › Weight 4 › Level 194 › 194.4.a › 194.4.a.d

Newform orbit 194.4.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 18q^{2} + 10q^{3} + 36q^{4} + 29q^{5} + 20q^{6} + 19q^{7} + 72q^{8} + 153q^{9} + 58q^{10} + 81q^{11} + 40q^{12} + 112q^{13} + 38q^{14} + 151q^{15} + 144q^{16} + 197q^{17} + 306q^{18} + 163q^{19} + 116q^{20} + 266q^{21} + 162q^{22} + 136q^{23} + 80q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{9} - 39x^{8} - 288x^{7} + 26819x^{6} - 226468x^{5} - 3175186x^{4} + 57843464x^{3} - 267690036x^{2} + 134631904x + 1021518704$

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
$\operatorname{tr} a_p$	18	10	29	19	81	112	197	163	136	335	−230	162	265	149	−257	−150	−175	−917	−405	−1759	−143	710	−80	−1234	873

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.133 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 engine's checks

dim M^+ at levels [1, 2, 97, 194]: [1, 2, 26, 76] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 5 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, 4)[3]   # 194.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 194:4 1000