Atlas › Modular forms › Weight 2 › Level 435 › 435.2.a › 435.2.a.f

Newform orbit 435.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{21})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} - x - 5$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	$-1$
29	$-1$

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
$\operatorname{tr} a_p$	−1	2	2	2	10	0	−6	−2	−8	2	8	−8	0	−10	12	10	−6	−2	10	−10	8	6	−14	12	14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.321 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, 3, 5, 15, 29, 87, 145, 435]: [0, 1, 1, 4, 3, 12, 16, 64] (certified at ell = 2147483549)
T = [(2, 1), (7, 1)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 3 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(435, 2)[5]   # 435.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 435:2 1000