Atlas › Modular forms › Weight 10 › Level 34 › 34.10.a › 34.10.a.c

Newform orbit 34.10.a.c

Trace form

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

$3q - 48q^{2} + 84q^{3} + 768q^{4} + 1314q^{5} - 1344q^{6} + 6912q^{7} - 12288q^{8} + 27783q^{9} - 21024q^{10} - 71460q^{11} + 21504q^{12} - 248982q^{13} - 110592q^{14} + 231192q^{15} + 196608q^{16} + 250563q^{17} - 444528q^{18} + 248268q^{19} + 336384q^{20} + 3954336q^{21} + 1143360q^{22} + 2198256q^{23} - 344064q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 1398x^{2} - 2315220x + 2180455704$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
17	$-1$

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
$\operatorname{tr} a_p$	−48	84	1314	6912	−71460	−248982	250563	248268	2198256	6311754	10188312	−5090430	23455710	−34059372	−15060912	−44390238	27709380	−66961206	−283780116	186084432	281360574	19593336	613032300	383569998	−1104920730

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.329 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, 2, 17, 34]: [1, 3, 14, 42] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (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(34, 10)[2]   # 34.10.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 34:10 1000