Atlas › Modular forms › Weight 2 › Level 851 › 851.2.a › 851.2.a.h

Newform orbit 851.2.a.h

Trace form

The 21 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$21q + 2q^{2} + 2q^{3} + 36q^{4} + 4q^{5} + 8q^{6} + 7q^{7} + 6q^{8} + 35q^{9} + 5q^{10} - 8q^{11} + 13q^{12} + 12q^{13} - 2q^{14} - 4q^{15} + 42q^{16} + 2q^{17} + 22q^{18} + 15q^{19} - 10q^{20} + 30q^{21} - 11q^{22} - 21q^{23} + 44q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 21. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{21} - 5x^{20} - 20115x^{19} + 142492x^{18} + 147075845x^{17} - 1766753130x^{16} - 517190339564x^{15} + 8970056264495x^{14} + 939057029273209x^{13} - 20832961327570560x^{12} - 871835533180274433x^{11} + 23183463352905247982x^{10} + 403284968956375852896x^{9} - 13068303960213411632928x^{8} - 79669721863614419235717x^{7} + 3715153658917693003220147x^{6} + 1249228437129639146488684x^{5} - 490393151169770673315813352x^{4} + 1333751053220262583455651451x^{3} + 22610787600350211405475807086x^{2} - 90416196041291670360131952525x - 98003514715360978794045061239$

Atkin–Lehner signs

$p$	$w_p$
23	$+1$
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
$\operatorname{tr} a_p$	2	2	4	7	−8	12	2	15	−21	1	15	21	27	−2	−9	10	0	64	9	9	66	15	−47	16	−11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.332 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 23, 37, 851]: [0, 3, 3, 78] (certified at ell = 2147483053)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 67 = 1 x 67 (phi(ord eps) x dim S_k^new); 16 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(851, 2)[7]   # 851.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 851:2 1000