Atlas › Modular forms › Weight 4 › Level 37 › 37.4.a › 37.4.a.b

Newform orbit 37.4.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 4q^{2} + 13q^{3} + 18q^{4} + 11q^{5} + 9q^{6} + 24q^{7} + 30q^{8} + 46q^{9} - 75q^{10} + 61q^{11} - 65q^{12} - 37q^{13} - 128q^{14} - 116q^{15} - 182q^{16} + 130q^{17} - 159q^{18} - 22q^{19} - 59q^{20} - 44q^{21} - 95q^{22} + 73q^{23} - 105q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 17x^{4} + 16x^{3} + 955x^{2} - 4905x + 6008$

Atkin–Lehner signs

$p$	$w_p$
37	$+1$

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	13	11	24	61	−37	130	−22	73	271	363	−185	381	−408	276	156	100	−1711	787	1578	−313	569	2422	−2466	−2406

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.092 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, 37]: [1, 11] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 9 = 1 x 9 (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(37, 4)[1]   # 37.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 37:4 1000