Embedded newform 80.6.j.a.43.12

• Label: 80.6.j.a.43.12
• Level: $80$
• Weight: $6$
• Character: 80.43
• Analytic conductor: $12.831$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8307055850\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			43.12
Character	\(\chi\)	\(=\)	80.43
Dual form			80.6.j.a.67.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.58344 + 3.31543i) q^{2} -14.5939i q^{3} +(10.0158 - 30.3922i) q^{4} +(52.9491 - 17.9275i) q^{5} +(48.3851 + 66.8903i) q^{6} +(47.3215 - 47.3215i) q^{7} +(54.8563 + 172.507i) q^{8} +30.0178 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} - 8748 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.58344	+	3.31543i	−0.810245	+	0.586091i
							\(3\)		−	14.5939i		−	0.936200i	−0.883676	−	0.468100i	\(-0.844939\pi\)
0.883676	−	0.468100i	\(-0.155061\pi\)
\(5\)	52.9491	−	17.9275i	0.947182	−	0.320698i
							\(7\)	47.3215	−	47.3215i	0.365018	−	0.365018i	−0.500639	−	0.865656i	\(-0.666902\pi\)
0.865656	+	0.500639i	\(0.166902\pi\)
\(11\)	28.1268	−	28.1268i	0.0700871	−	0.0700871i	−0.671194	−	0.741281i	\(-0.734218\pi\)
							0.741281	+	0.671194i	\(0.234218\pi\)
\(13\)	−55.8021			−0.0915783			−0.0457891	−	0.998951i	\(-0.514580\pi\)
							−0.0457891	+	0.998951i	\(0.514580\pi\)
\(17\)	−92.8082	+	92.8082i	−0.0778868	+	0.0778868i	−0.744977	−	0.667090i	\(-0.767540\pi\)
							0.667090	+	0.744977i	\(0.267540\pi\)
\(19\)	879.583	−	879.583i	0.558975	−	0.558975i	−0.370040	−	0.929016i	\(-0.620656\pi\)
							0.929016	+	0.370040i	\(0.120656\pi\)
\(23\)	−128.064	−	128.064i	−0.0504786	−	0.0504786i	0.681417	−	0.731895i	\(-0.261364\pi\)
							−0.731895	+	0.681417i	\(0.761364\pi\)
\(29\)	3399.85	+	3399.85i	0.750696	+	0.750696i	0.974609	−	0.223913i	\(-0.0718831\pi\)
							−0.223913	+	0.974609i	\(0.571883\pi\)
\(31\)		−	5968.27i		−	1.11543i	−0.830031	−	0.557717i	\(-0.811677\pi\)
							0.830031	−	0.557717i	\(-0.188323\pi\)
\(37\)	−8959.01			−1.07586			−0.537930	−	0.842990i	\(-0.680793\pi\)
							−0.537930	+	0.842990i	\(0.680793\pi\)
\(41\)		−	16429.3i		−	1.52636i	−0.646183	−	0.763182i	\(-0.723636\pi\)
							0.646183	−	0.763182i	\(-0.276364\pi\)
\(43\)	−15387.3			−1.26909			−0.634543	−	0.772888i	\(-0.718811\pi\)
							−0.634543	+	0.772888i	\(0.718811\pi\)
\(47\)	−11254.2	−	11254.2i	−0.743137	−	0.743137i	0.230043	−	0.973180i	\(-0.426113\pi\)
							−0.973180	+	0.230043i	\(0.926113\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.j.a.43.12	✓	116
5.2	odd	4		80.6.s.a.27.18	yes	116
16.3	odd	4		80.6.s.a.3.18	yes	116
80.67	even	4	inner	80.6.j.a.67.12	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.12	✓	116	1.1	even	1	trivial
80.6.j.a.67.12	yes	116	80.67	even	4	inner
80.6.s.a.3.18	yes	116	16.3	odd	4
80.6.s.a.27.18	yes	116	5.2	odd	4