Atlas › Modular forms › Weight 2 › Level 698 › 698.2.a › 698.2.a.d

Newform orbit 698.2.a.d

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{12} - 16x^{11} - 304x^{10} + 6312x^{9} + 3851x^{8} - 613196x^{7} + 2878012x^{6} + 10416513x^{5} - 103601405x^{4} + 140131587x^{3} + 634570800x^{2} - 2029316532x + 1576255080$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
349	$+1$

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
$\operatorname{tr} a_p$	12	1	5	9	−6	14	3	5	11	11	−5	17	11	29	−9	2	−29	17	−2	−17	8	3	−36	−14	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 0.237 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, 349, 698]: [0, 1, 29, 89] (certified at ell = 2147481901)
T = [(3, 1), (5, 3)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 5 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(698, 2)[3]   # 698.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 698:2 1000