Atlas › Modular forms › Weight 4 › Level 121 › 121.4.a › 121.4.a.f

Newform orbit 121.4.a.f

Trace form

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

$4q - 4q^{2} + 6q^{3} + 14q^{4} - 11q^{5} - 43q^{6} - 25q^{7} - 66q^{8} - 62q^{9} - 20q^{10} + 95q^{12} - 25q^{13} + 132q^{14} + 2q^{15} + 34q^{16} - 232q^{17} - 49q^{18} - 154q^{19} - 254q^{20} - 167q^{21} - 6q^{23} - 99q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{4} - 14x^{3} - 3x^{2} + 364x + 491$

Atkin–Lehner signs

$p$	$w_p$
11	— ( $11^2 \mid 121$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	−4	6	−11	−25	0	−25	−232	−154	−6	−363	37	93	−152	325	−869	811	178	105	43	629	270	−977	−1686	1891	−1772

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.109 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, 121]: [1, 4, 34] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 4 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(121, 4)[5]   # 121.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 121:4 1000