Atlas › Modular forms › Weight 6 › Level 65 › 65.6.a › 65.6.a.c

Newform orbit 65.6.a.c

Trace form

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

$4q - 9q^{2} - 4q^{3} + 93q^{4} - 100q^{5} - 74q^{6} - 136q^{7} - 447q^{8} + 644q^{9} + 225q^{10} - 516q^{11} - 710q^{12} - 676q^{13} + 1604q^{14} + 100q^{15} - 719q^{16} + 344q^{17} - 6905q^{18} - 5012q^{19} - 2325q^{20} - 4448q^{21} - 9318q^{22} + 708q^{23} - 2094q^{24} + O(q^{25})$

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

$x^{4} + 13x^{3} - 760x^{2} + 1728x + 50544$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
13	$+1$

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 odd 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
$\operatorname{tr} a_p$	−9	−4	−100	−136	−516	−676	344	−5012	708	5416	−10124	1448	21576	−25740	−51096	39448	37764	95016	−50592	−89308	−108312	−25464	20584	−148328	−77816

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 5, 13, 65]: [1, 3, 7, 36] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 4 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(65, 6)[2]   # 65.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 65:6 1000