Atlas › Modular forms › Weight 2 › Level 230 › 230.2.a › 230.2.a.d

Newform orbit 230.2.a.d

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{3} - 10x^{2} - 123x + 1236$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$+1$
23	$+1$

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
$\operatorname{tr} a_p$	3	1	−3	3	3	−1	−7	3	−3	−4	−5	−2	1	24	−14	−18	14	1	8	11	−8	−4	8	18	−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 0.099 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, 2, 5, 10, 23, 46, 115, 230]: [0, 1, 1, 3, 3, 8, 14, 40] (certified at ell = 2147483053)
T = [(3, 1), (7, 3)]; h of degree 9 = 1 x 9 (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(230, 2)[3]   # 230.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 230:2 1000