Atlas › Modular forms › Weight 2 › Level 669 › 669.2.a › 669.2.a.h

Newform orbit 669.2.a.h

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{7} - 11x^{6} - 89x^{5} + 1044x^{4} + 480x^{3} - 12032x^{2} - 1792x + 12160$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
223	$-1$

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	−7	3	−2	15	−2	5	20	9	9	10	−16	16	−5	21	5	26	12	−3	21	−23	13	26	20	−9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.335 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, 223, 669]: [0, 1, 19, 76] (certified at ell = 2147482591)
T = [(2, 1), (5, 3)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 6 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(669, 2)[7]   # 669.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 669:2 1000