Atlas › Modular forms › Weight 6 › Level 103 › 103.6.a › 103.6.a.b

Newform orbit 103.6.a.b

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q + 19q^{2} + 16q^{3} + 431q^{4} + 153q^{5} + 79q^{6} + 186q^{7} + 804q^{8} + 2802q^{9} + 398q^{10} + 695q^{11} + 375q^{12} + 1115q^{13} + 2815q^{14} + 2587q^{15} + 9691q^{16} + 7057q^{17} + 4692q^{18} + 1433q^{19} + 6429q^{20} + 1894q^{21} + 1065q^{22} + 8504q^{23} + 6930q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} - 35x^{23} - 4383x^{22} + 148980x^{21} + 8311278x^{20} - 267518025x^{19} - 9078671965x^{18} + 266355207017x^{17} + 6399371770322x^{16} - 162162357188462x^{15} - 3075492210238726x^{14} + 62635875528303158x^{13} + 1030175978246886509x^{12} - 15337392073989523707x^{11} - 239317938640135857571x^{10} + 2277313446476760809086x^{9} + 37302206430043010007360x^{8} - 177309460718736986968869x^{7} - 3630455695043346187527617x^{6} + 3104017908306489069798077x^{5} + 189704115896341021105174872x^{4} + 413534408898629635603783436x^{3} - 3455787368767587163837300080x^{2} - 16587599604975377746852804080x - 19605100753764986764003901280$

Atkin–Lehner signs

$p$	$w_p$
103	$-1$

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	16	153	186	695	1115	7057	1433	8504	31546	10648	24138	20580	26926	12017	77727	−20453	−22235	8964	−13809	63415	−81261	276927	92014	−265950

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.273 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, 103]: [1, 45] (certified at ell = 2147482501)
T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 8 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(103, 6)[1]   # 103.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 103:6 1000