Atlas › Modular forms › Weight 4 › Level 145 › 145.4.a › 145.4.a.c

Newform orbit 145.4.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - q^{2} - q^{3} + 21q^{4} - 30q^{5} - 29q^{6} + 3q^{7} - 99q^{8} - 15q^{9} + 5q^{10} - 148q^{11} + 3q^{12} + 13q^{13} - 153q^{14} + 5q^{15} + 85q^{16} - 53q^{17} - 168q^{18} - 188q^{19} - 105q^{20} - 220q^{21} + 340q^{22} - 235q^{23} - 445q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 2x^{5} - 77x^{4} + 133x^{3} + 324x^{2} + 183x + 32$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
29	$-1$

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
$\operatorname{tr} a_p$	−1	−1	−30	3	−148	13	−53	−188	−235	174	−347	−538	−574	421	−170	51	−39	247	−558	−2540	731	−2297	−540	−2698	409

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.116 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, 5, 29, 145]: [1, 3, 9, 48] (certified at ell = 2147483549)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 4 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(145, 4)[2]   # 145.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 145:4 1000