Atlas › Modular forms › Weight 2 › Level 705 › 705.2.a › 705.2.a.k

Newform orbit 705.2.a.k

Trace form

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

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

Coefficient field

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

$x^{4} - 132x^{3} + 6234x^{2} - 124780x + 886361$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	$-1$
47	$+1$

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	−4	4	8	4	−4	4	−8	16	4	0	−4	−8	24	−4	12	0	−28	−8	16	−24	−4	24	−8	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.622 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, 47, 141, 235, 705]: [0, 1, 1, 4, 5, 18, 26, 100] (certified at ell = 2147483137)
T = [(2, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 9 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(705, 2)[10]   # 705.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 705:2 1000