Embedded newform 16.22.e.a.5.8

• Label: 16.22.e.a.5.8
• Level: $16$
• Weight: $22$
• Character: 16.5
• Analytic conductor: $44.716$
• Analytic rank: $0$
• Dimension: $82$
• CM: no
• 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			5.8
Character	\(\chi\)	\(=\)	16.5
Dual form			16.22.e.a.13.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1186.35 + 830.497i) q^{2} +(117950. - 117950. i) q^{3} +(717703. - 1.97052e6i) q^{4} +(2.07522e7 + 2.07522e7i) q^{5} +(-4.19730e7 + 2.37888e8i) q^{6} -1.29970e9i q^{7} +(7.85064e8 + 2.93378e9i) q^{8} -1.73642e10i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 2 q^{2} - 2 q^{3} + 1604328 q^{4} - 2 q^{5} - 142760912 q^{6} - 5437750796 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1186.35	+	830.497i	−0.819215	+	0.573486i
							\(3\)	117950.	−	117950.i	1.15326	−	1.15326i	0.167360	−	0.985896i	\(-0.446476\pi\)
0.985896	−	0.167360i	\(-0.0535244\pi\)
\(5\)	2.07522e7	+	2.07522e7i	0.950338	+	0.950338i	0.998824	−	0.0484857i	\(-0.0154395\pi\)
							−0.0484857	+	0.998824i	\(0.515440\pi\)
\(7\)		−	1.29970e9i		−	1.73906i	−0.493883	−	0.869528i	\(-0.664423\pi\)
							0.493883	−	0.869528i	\(-0.335577\pi\)
\(11\)	5.54435e10	+	5.54435e10i	0.644507	+	0.644507i	0.951660	−	0.307153i	\(-0.0993764\pi\)
							−0.307153	+	0.951660i	\(0.599376\pi\)
\(13\)	5.16123e11	−	5.16123e11i	1.03836	−	1.03836i	0.0391257	−	0.999234i	\(-0.487543\pi\)
							0.999234	−	0.0391257i	\(-0.0124573\pi\)
\(17\)	9.53487e12			1.14710			0.573550	−	0.819171i	\(-0.305566\pi\)
							0.573550	+	0.819171i	\(0.305566\pi\)
\(19\)	−9.83239e12	+	9.83239e12i	−0.367914	+	0.367914i	−0.866716	−	0.498802i	\(-0.833773\pi\)
							0.498802	+	0.866716i	\(0.333773\pi\)
\(23\)		−	2.30915e12i		−	0.0116228i	−0.999983	−	0.00581140i	\(-0.998150\pi\)
							0.999983	−	0.00581140i	\(-0.00184984\pi\)
\(29\)	−6.33852e14	+	6.33852e14i	−0.279775	+	0.279775i	−0.833019	−	0.553244i	\(-0.813390\pi\)
							0.553244	+	0.833019i	\(0.313390\pi\)
\(31\)	−5.23093e15			−1.14625			−0.573127	−	0.819466i	\(-0.694270\pi\)
							−0.573127	+	0.819466i	\(0.694270\pi\)
\(37\)	3.18380e16	+	3.18380e16i	1.08850	+	1.08850i	0.995683	+	0.0928149i	\(0.0295865\pi\)
							0.0928149	+	0.995683i	\(0.470414\pi\)
\(41\)			2.66193e15i			0.0309717i	0.999880	+	0.0154859i	\(0.00492950\pi\)
							−0.999880	+	0.0154859i	\(0.995071\pi\)
\(43\)	−2.83749e15	−	2.83749e15i	−0.0200224	−	0.0200224i	0.697025	−	0.717047i	\(-0.254507\pi\)
							−0.717047	+	0.697025i	\(0.754507\pi\)
\(47\)	−1.54686e17			−0.428968			−0.214484	−	0.976728i	\(-0.568807\pi\)
							−0.214484	+	0.976728i	\(0.568807\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.22.e.a.5.8	✓	82
4.3	odd	2		64.22.e.a.49.4		82
16.3	odd	4		64.22.e.a.17.4		82
16.13	even	4	inner	16.22.e.a.13.8	yes	82

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.22.e.a.5.8	✓	82	1.1	even	1	trivial
16.22.e.a.13.8	yes	82	16.13	even	4	inner
64.22.e.a.17.4		82	16.3	odd	4
64.22.e.a.49.4		82	4.3	odd	2