Atlas › Modular forms › Weight 2 › Level 298 › 298.2.a › 298.2.a.d

Newform orbit 298.2.a.d

Trace form

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

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

Coefficient field

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

$x^{3} + 2x^{2} - 29x - 83$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
149	$+1$

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 odd 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	−5	1	−4	−5	0	−9	−2	−19	−1	−4	7	−2	−15	−2	−1	−23	5	4	15	13	16	−15	−6	−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 0.273 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, 149, 298]: [0, 1, 13, 39] (certified at ell = 2147481629)
T = [(3, 1), (5, 3)]; h of degree 12 = 1 x 12 (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(298, 2)[3]   # 298.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 298:2 1000