Atlas › Modular forms › Weight 6 › Level 101 › 101.6.a › 101.6.a.b

Newform orbit 101.6.a.b

Trace form

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

$24q + 7q^{2} + 44q^{3} + 479q^{4} + 21q^{5} + 22q^{6} + 390q^{7} + 441q^{8} + 2538q^{9} + 1034q^{10} + 1320q^{11} + 1012q^{12} + 1351q^{13} + 2091q^{14} + 2410q^{15} + 10719q^{16} + 375q^{17} + 3475q^{18} + 7219q^{19} - 1356q^{20} + 2288q^{21} + 2942q^{22} + 6469q^{23} + 5106q^{24} + O(q^{25})$

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

$x^{24} - 51x^{23} - 3530x^{22} + 201726x^{21} + 5046920x^{20} - 335145650x^{19} - 3896928639x^{18} + 309878762341x^{17} + 1904326733059x^{16} - 178197998113737x^{15} - 702428934652193x^{14} + 67070469509876795x^{13} + 230682716329425697x^{12} - 16771081577643325261x^{11} - 65685029738146957430x^{10} + 2735016961389269047202x^{9} + 13622388853429951480167x^{8} - 272564043296847623233655x^{7} - 1760938505600754822379967x^{6} + 14053694791704046900902685x^{5} + 121410130611924085795427892x^{4} - 194782992383171263760754386x^{3} - 3269265409770992893681401888x^{2} - 4858033344535080234112603832x + 6923908099239411794747316768$

Atkin–Lehner signs

$p$	$w_p$
101	$-1$

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
$\operatorname{tr} a_p$	7	44	21	390	1320	1351	375	7219	6469	24770	17715	20574	4784	41216	10807	−30248	−2256	109926	172334	−23603	−56136	267803	−27398	19248	−151552

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 101]: [1, 43] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (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(101, 6)[1]   # 101.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 101:6 1000