Atlas › Modular forms › Weight 2 › Level 695 › 695.2.a › 695.2.a.c

Newform orbit 695.2.a.c

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{17} - 11x^{16} - 5x^{15} + 431x^{14} - 978x^{13} - 5180x^{12} + 18527x^{11} + 23533x^{10} - 136027x^{9} - 10577x^{8} + 477607x^{7} - 243557x^{6} - 764724x^{5} + 723392x^{4} + 321244x^{3} - 617544x^{2} + 255097x - 34223$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
139	$+1$

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
$\operatorname{tr} a_p$	5	6	17	7	−3	21	15	11	16	−19	12	32	2	11	−5	46	11	0	12	−37	56	−5	−25	20	66

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.226 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, 5, 139, 695]: [0, 1, 12, 72] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 47 = 1 x 47 (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(695, 2)[2]   # 695.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 695:2 1000