Atlas › Modular forms › Weight 4 › Level 247 › 247.4.a › 247.4.a.e

Newform orbit 247.4.a.e

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + 11q^{2} + 20q^{3} + 53q^{4} + 26q^{5} + 28q^{6} - 9q^{7} + 87q^{8} + 184q^{9} + 42q^{10} + 27q^{11} + 194q^{12} + 208q^{13} + 128q^{14} - 71q^{15} + 297q^{16} + 419q^{17} + 320q^{18} + 304q^{19} + 301q^{20} + 53q^{21} + 10q^{22} - 88q^{23} + 291q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{16} - 381x^{15} - 1350902x^{14} + 609568057x^{13} + 659371258801x^{12} - 346881738888904x^{11} - 141213161393034812x^{10} + 90819620311054790555x^{9} + 11965943912922076289611x^{8} - 11608003124128288310508962x^{7} + 15880909578969616236605021x^{6} + 687462299400174944413608196317x^{5} - 48007670325942859443861552038806x^{4} - 14663090652520479849746101506280584x^{3} + 1516567140281532248486518319064457920x^{2} - 4245152422339531144367575939655071744x - 19000049925303081321666855546697805824$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
19	$-1$

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
$\operatorname{tr} a_p$	11	20	26	−9	27	208	419	304	−88	867	−210	546	872	586	283	2135	1295	1367	−265	−806	−240	−1556	1495	1525	4068

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.254 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 engine's checks

dim M^+ at levels [1, 13, 19, 247]: [1, 5, 6, 72] (certified at ell = 2147483629)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 54 = 1 x 54 (phi(ord eps) x dim S_k^new); 18 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(247, 4)[4]   # 247.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 247:4 1000