Atlas › Modular forms › Weight 2 › Level 449 › 449.2.a › 449.2.a.b

Newform orbit 449.2.a.b

Trace form

The 23 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$23q + 3q^{3} + 30q^{4} + 3q^{5} + 6q^{6} + 26q^{7} - 3q^{8} + 22q^{9} + 11q^{10} + 6q^{11} - 5q^{12} + 10q^{13} - 2q^{14} + 2q^{15} + 32q^{16} - 8q^{18} + 41q^{19} + q^{20} + 5q^{21} - 9q^{22} + 2q^{23} + 14q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 23. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{23} - 3x^{22} - 85x^{21} + 266x^{20} + 2860x^{19} - 9102x^{18} - 50365x^{17} + 158186x^{16} + 527071x^{15} - 1580022x^{14} - 3483483x^{13} + 9616835x^{12} + 14803993x^{11} - 36439399x^{10} - 39689209x^{9} + 85376416x^{8} + 62919746x^{7} - 118886914x^{6} - 50446161x^{5} + 88782974x^{4} + 11218709x^{3} - 26594058x^{2} + 4210915x + 22925$

Atkin–Lehner signs

$p$	$w_p$
449	$-1$

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	3	3	26	6	10	0	41	2	−7	73	8	0	22	23	−14	−3	38	1	−4	27	80	14	−34	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.188 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, 449]: [0, 38] (certified at ell = 2147479937)
T = [(2, 1), (3, 1)]; h of degree 37 = 1 x 37 (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(449, 2)[1]   # 449.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 449:2 1000