Atlas › Modular forms › Weight 10 › Level 35 › 35.10.a › 35.10.a.d

Newform orbit 35.10.a.d

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 2q^{2} + 140q^{3} + 832q^{4} - 3125q^{5} - 144q^{6} + 12005q^{7} - 14136q^{8} + 67797q^{9} - 1250q^{10} + 10312q^{11} + 63388q^{12} + 158638q^{13} + 4802q^{14} - 87500q^{15} - 526696q^{16} + 31614q^{17} + 1816986q^{18} + 1655376q^{19} - 520000q^{20} + 336140q^{21} + 3659464q^{22} + 796104q^{23} + 6504576q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} - 142x^{4} - 74576x^{3} + 6322840x^{2} + 1239115411x + 3845372466$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
7	$-1$

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
$\operatorname{tr} a_p$	2	140	−3125	12005	10312	158638	31614	1655376	796104	−3353726	2678120	50994846	6330194	6149468	−6897780	−70886738	−152335168	257015698	133467828	522788960	159370858	464174900	2207636832	−1106708326	1956142254

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.324 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, 7, 35]: [1, 5, 7, 38] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 18 = 1 x 18 (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(35, 10)[3]   # 35.10.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 35:10 1000