Atlas › Modular forms › Weight 10 › Level 7 › 7.10.a › 7.10.a.b

Newform orbit 7.10.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 21q^{2} + 84q^{3} + 1557q^{4} + 1554q^{5} + 4914q^{6} + 7203q^{7} + 14055q^{8} - 26001q^{9} - 97860q^{10} - 3444q^{11} - 106386q^{12} - 19782q^{13} + 50421q^{14} + 200304q^{15} + 482961q^{16} + 1016694q^{17} - 273267q^{18} + 222852q^{19} - 1922088q^{20} + 201684q^{21} - 2847048q^{22} + 1885632q^{23} - 1449630q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 105x^{2} - 17472x + 845676$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$

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
$\operatorname{tr} a_p$	21	84	1554	7203	−3444	−19782	1016694	222852	1885632	4081818	2869440	1395618	−14420658	−61631172	−10368960	67502610	−42590100	191746842	−255175788	296514504	344213310	−960412656	−1100517180	506816478	−647498250

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.273 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, 7]: [1, 7] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 5 = 1 x 5 (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(7, 10)[1]   # 7.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 7:10 1000