Embedded newform 16.22.e.a.5.6

• Label: 16.22.e.a.5.6
• Level: $16$
• Weight: $22$
• Character: 16.5
• Analytic conductor: $44.716$
• Analytic rank: $0$
• Dimension: $82$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(44.7163750859\)
Analytic rank:	\(0\)
Dimension:	\(82\)
Relative dimension:	\(41\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	16.5
Dual form			16.22.e.a.13.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1391.35 - 401.620i) q^{2} +(12781.5 - 12781.5i) q^{3} +(1.77455e6 + 1.11759e6i) q^{4} +(5.32911e6 + 5.32911e6i) q^{5} +(-2.29168e7 + 1.26502e7i) q^{6} -4.26451e8i q^{7} +(-2.02018e9 - 2.26765e9i) q^{8} +1.01336e10i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 2 q^{2} - 2 q^{3} + 1604328 q^{4} - 2 q^{5} - 142760912 q^{6} - 5437750796 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1391.35	−	401.620i	−0.960774	−	0.277332i
							\(3\)	12781.5	−	12781.5i	0.124970	−	0.124970i	−0.641855	−	0.766826i	\(-0.721835\pi\)
0.766826	+	0.641855i	\(0.221835\pi\)
\(5\)	5.32911e6	+	5.32911e6i	0.244045	+	0.244045i	0.818521	−	0.574476i	\(-0.194794\pi\)
							−0.574476	+	0.818521i	\(0.694794\pi\)
\(7\)		−	4.26451e8i		−	0.570611i	−0.958437	−	0.285305i	\(-0.907905\pi\)
							0.958437	−	0.285305i	\(-0.0920950\pi\)
\(11\)	1.10471e11	+	1.10471e11i	1.28417	+	1.28417i	0.938271	+	0.345901i	\(0.112427\pi\)
							0.345901	+	0.938271i	\(0.387573\pi\)
\(13\)	4.36552e10	−	4.36552e10i	0.0878275	−	0.0878275i	−0.661828	−	0.749656i	\(-0.730219\pi\)
							0.749656	+	0.661828i	\(0.230219\pi\)
\(17\)	−8.80589e12			−1.05940			−0.529700	−	0.848185i	\(-0.677695\pi\)
							−0.529700	+	0.848185i	\(0.677695\pi\)
\(19\)	6.57989e12	−	6.57989e12i	0.246210	−	0.246210i	−0.573203	−	0.819413i	\(-0.694299\pi\)
							0.819413	+	0.573203i	\(0.194299\pi\)
\(23\)			5.08091e13i			0.255740i	0.991791	+	0.127870i	\(0.0408141\pi\)
							−0.991791	+	0.127870i	\(0.959186\pi\)
\(29\)	1.44772e15	−	1.44772e15i	0.639005	−	0.639005i	−0.311305	−	0.950310i	\(-0.600766\pi\)
							0.950310	+	0.311305i	\(0.100766\pi\)
\(31\)	−7.69478e15			−1.68616			−0.843079	−	0.537790i	\(-0.819259\pi\)
							−0.843079	+	0.537790i	\(0.819259\pi\)
\(37\)	−1.33778e16	−	1.33778e16i	−0.457368	−	0.457368i	0.440423	−	0.897790i	\(-0.354829\pi\)
							−0.897790	+	0.440423i	\(0.854829\pi\)
\(41\)			1.32174e17i			1.53786i	0.639336	+	0.768928i	\(0.279209\pi\)
							−0.639336	+	0.768928i	\(0.720791\pi\)
\(43\)	1.72068e17	+	1.72068e17i	1.21417	+	1.21417i	0.969641	+	0.244531i	\(0.0786340\pi\)
							0.244531	+	0.969641i	\(0.421366\pi\)
\(47\)	−4.97106e17			−1.37855			−0.689274	−	0.724501i	\(-0.742070\pi\)
							−0.689274	+	0.724501i	\(0.742070\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.22.e.a.5.6	✓	82
4.3	odd	2		64.22.e.a.49.19		82
16.3	odd	4		64.22.e.a.17.19		82
16.13	even	4	inner	16.22.e.a.13.6	yes	82

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.22.e.a.5.6	✓	82	1.1	even	1	trivial
16.22.e.a.13.6	yes	82	16.13	even	4	inner
64.22.e.a.17.19		82	16.3	odd	4
64.22.e.a.49.19		82	4.3	odd	2