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

Newform orbit 34.10.a.b

Trace form

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

$3q - 48q^{2} + 84q^{3} + 768q^{4} - 1304q^{5} - 1344q^{6} + 690q^{7} - 12288q^{8} - 1117q^{9} + 20864q^{10} + 45364q^{11} + 21504q^{12} + 74358q^{13} - 11040q^{14} + 155236q^{15} + 196608q^{16} - 250563q^{17} + 17872q^{18} - 708560q^{19} - 333824q^{20} - 2762432q^{21} - 725824q^{22} - 4543638q^{23} - 344064q^{24} + O(q^{25})$

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

$x^{3} + 1220x^{2} - 159790x - 84290348$

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 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
$\operatorname{tr} a_p$	−48	84	−1304	690	45364	74358	−250563	−708560	−4543638	−7825752	−3857870	−2504764	−6074582	44493412	−21786656	−26880170	−108222348	−78395508	70665056	−134285430	119495414	−217624358	−1035085876	247104742	2730005298

$\operatorname{tr} a_n$ 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)[1]   # 34.10.a.b

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

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