Atlas › Modular forms › Weight 4 › Level 153 › 153.4.a › 153.4.a.i

Newform orbit 153.4.a.i

Trace form

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

$4q + 4q^{2} + 26q^{4} + 22q^{5} - 24q^{7} + 102q^{8} - 2q^{10} + 50q^{11} + 26q^{13} + 80q^{14} + 138q^{16} - 68q^{17} + 34q^{19} + 312q^{20} - 254q^{22} + 382q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} - 26x^{3} - 8x^{2} + 4254x - 25857$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 153$ )
17	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	0	22	−24	50	26	−68	34	382	540	−356	−404	−114	570	496	92	−48	−1036	812	1044	−1212	488	1708	−8	−76

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.124 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, 3, 9, 17, 51, 153]: [1, 2, 4, 6, 20, 56] (certified at ell = 2147483137)
T = [(2, 1), (5, 1)]; h of degree 20 = 1 x 20 (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(153, 4)[8]   # 153.4.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 153:4 1000