Atlas › Modular forms › Weight 10 › Level 33 › 33.10.a › 33.10.a.d

Newform orbit 33.10.a.d

Trace form

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

$4q + 19q^{2} - 324q^{3} + 1937q^{4} + 1138q^{5} - 1539q^{6} + 8122q^{7} - 7149q^{8} + 26244q^{9} - 70802q^{10} + 58564q^{11} - 156897q^{12} + 230998q^{13} - 175760q^{14} - 92178q^{15} + 1786769q^{16} - 116q^{17} + 124659q^{18} + 1179324q^{19} + 4708262q^{20} - 657882q^{21} + 278179q^{22} + 317378q^{23} + 579069q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} - 1157x^{3} - 4118808x^{2} + 3761411332x - 650348382608$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
11	$-1$

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
$\operatorname{tr} a_p$	19	−324	1138	8122	58564	230998	−116	1179324	317378	252516	−7812776	13990056	−68389448	6207552	−30703042	41285850	−132805180	276571338	210077288	270008586	314963656	1553310662	500616048	−1406681304	2126336996

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.327 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, 3, 11, 33]: [1, 4, 10, 38] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 6 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(33, 10)[3]   # 33.10.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 33:10 1000