Atlas › Modular forms › Weight 2 › Level 978 › 978.2.a › 978.2.a.o

Newform orbit 978.2.a.o

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ , whose minimal polynomial is

$x^{4} - 202x^{3} + 4905x^{2} + 669144x - 13906252$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$+1$
163	$+1$

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
$\operatorname{tr} a_p$	4	−4	−3	−1	−4	16	3	11	−2	11	1	2	2	12	15	6	−14	11	−1	−19	2	13	0	−5	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.028 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, 3, 6, 163, 326, 489, 978]: [0, 1, 1, 3, 14, 43, 56, 168] (certified at ell = 2147483179)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 8 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(978, 2)[14]   # 978.2.a.o

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

cargo run --release -p sagebrush-web --example atlas -- 978:2 1000