Atlas › Modular forms › Weight 2 › Level 101 › 101.2.a › 101.2.a.b

Newform orbit 101.2.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 4q^{3} + 12q^{4} - 3q^{5} - 6q^{6} + 2q^{7} - 6q^{8} + 9q^{9} - 4q^{10} + 8q^{11} - 4q^{12} - q^{13} - 10q^{14} + 2q^{15} + 22q^{16} - 7q^{17} - 24q^{18} + 19q^{19} - 28q^{20} - 8q^{21} - 10q^{22} - 7q^{23} - 42q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} - 4x^{6} - 14x^{5} + 88x^{4} - 141x^{3} + 90x^{2} - 23x + 2$

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
$\operatorname{tr} a_p$	0	4	−3	2	8	−1	−7	19	−7	−2	7	0	−2	32	−13	−8	16	−6	34	9	−2	15	−22	−22	−28

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 101]: [0, 9] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 1 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, 2)[1]   # 101.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 101:2 1000