Atlas › Modular forms › Weight 6 › Level 97 › 97.6.a › 97.6.a.b

Newform orbit 97.6.a.b

Trace form

The 21 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$21q + 19q^{2} + 36q^{3} + 367q^{4} + 161q^{5} + 71q^{6} + 431q^{7} + 804q^{8} + 1835q^{9} + 634q^{10} + 1081q^{11} + 651q^{12} + 1276q^{13} + 2335q^{14} + 4189q^{15} + 6879q^{16} + 3667q^{17} + 5949q^{18} - 31q^{19} + 5591q^{20} + 4134q^{21} + 3736q^{22} + 12830q^{23} + 2407q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 21. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{21} - 55x^{20} - 2547x^{19} + 168860x^{18} + 2504403x^{17} - 215916290x^{16} - 1259627675x^{15} + 151709284357x^{14} + 399225634775x^{13} - 64340187943430x^{12} - 119898480991918x^{11} + 16816624245301463x^{10} + 38856972476800042x^{9} - 2609970037080355949x^{8} - 8633908487208140123x^{7} + 213249827629529255929x^{6} + 928608538993680509432x^{5} - 6617738129308778421454x^{4} - 32082235446548653441595x^{3} + 14884766154031266322350x^{2} + 29190294826633520852028x - 2173801845288971483640$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	19	36	161	431	1081	1276	3667	−31	12830	14771	20980	3344	36099	31005	65511	17482	27413	−48051	−66911	140823	−44987	19136	117898	3016	197589

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.261 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, 97]: [1, 42] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 8 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(97, 6)[1]   # 97.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 97:6 1000