Atlas › Modular forms › Weight 2 › Level 645 › 645.2.a › 645.2.a.h

Newform orbit 645.2.a.h

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 2q^{2} + 3q^{3} + 2q^{4} - 3q^{5} - 2q^{6} - 2q^{7} - 6q^{8} + 3q^{9} + 2q^{10} - 11q^{11} + 2q^{12} - 5q^{13} - 10q^{14} - 3q^{15} + 4q^{16} - 7q^{17} - 2q^{18} - 8q^{19} - 2q^{20} - 2q^{21} + 14q^{22} + 7q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{11} + 7T_{23} + 8T_{29}$ , whose minimal polynomial is

$x^{3} + 186x^{2} + 8756x + 25562$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	$+1$
43	$-1$

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
$\operatorname{tr} a_p$	−2	3	−3	−2	−11	−5	−7	−8	7	−10	−11	−6	−7	3	−4	−1	−20	18	−21	−30	16	4	−5	−20	−9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.659 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, 3, 5, 15, 43, 129, 215, 645]: [0, 1, 1, 4, 4, 16, 24, 92] (certified at ell = 2147483353)
T = [(2, 1), (7, 2), (13, 3), (17, 4), (19, 5), (11, 6), (23, 7), (29, 8)]; h of degree 27 = 1 x 27 (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(645, 2)[7]   # 645.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 645:2 1000