Atlas › Modular forms › Weight 4 › Level 11 › 11.4.a › 11.4.a.a

Newform orbit 11.4.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 2q^{2} - 2q^{3} - 8q^{4} + 2q^{5} - 26q^{6} + 20q^{7} - 12q^{8} + 44q^{9} + 50q^{10} - 22q^{11} - 40q^{12} + 80q^{13} - 4q^{14} - 194q^{15} - 8q^{16} - 124q^{17} + 92q^{18} + 72q^{19} + 88q^{20} + 76q^{21} - 22q^{22} - 98q^{23} + 252q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} - 27$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	2	−2	2	20	−22	80	−124	72	−98	144	−34	54	536	−60	−272	−492	634	840	754	−678	−400	316	468	−1842	2194

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.113 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, 11]: [1, 4] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 2 = 1 x 2 (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(11, 4)[0]   # 11.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 11:4 1000