Atlas › Modular forms › Weight 2 › Level 233 › 233.2.a › 233.2.a.c

Newform orbit 233.2.a.c

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} - 8x^{10} - 6x^{9} + 195x^{8} - 347x^{7} - 1223x^{6} + 4406x^{5} - 1331x^{4} - 12042x^{3} + 20420x^{2} - 13020x + 2978$

Atkin–Lehner signs

$p$	$w_p$
233	$-1$

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	10	−1	15	−5	4	7	8	8	−9	24	14	−8	15	8	−5	5	−2	−2	3	23	36	6	−19	33

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.168 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, 233]: [0, 20] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 2 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(233, 2)[2]   # 233.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 233:2 1000