Embedded newform 189.2.v.a.85.4

• Label: 189.2.v.a.85.4
• Level: $189$
• Weight: $2$
• Character: 189.85
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.v (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(9\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			85.4
Character	\(\chi\)	\(=\)	189.85
Dual form			189.2.v.a.169.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08295 - 0.394160i) q^{2} +(0.106738 + 1.72876i) q^{3} +(-0.514680 - 0.431868i) q^{4} +(-0.132560 + 0.751786i) q^{5} +(0.565816 - 1.91422i) q^{6} +(0.766044 - 0.642788i) q^{7} +(1.53959 + 2.66665i) q^{8} +(-2.97721 + 0.369049i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 3 q^{3} - 3 q^{5} - 27 q^{8} - 9 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08295	−	0.394160i	−0.765758	−	0.278713i	−0.0705370	−	0.997509i	\(-0.522471\pi\)
							−0.695221	+	0.718796i	\(0.744693\pi\)
\(3\)	0.106738	+	1.72876i	0.0616253	+	0.998099i
							\(5\)	−0.132560	+	0.751786i	−0.0592827	+	0.336209i	−0.999996	−	0.00299518i	\(-0.999047\pi\)
0.940713	+	0.339204i	\(0.110158\pi\)
\(7\)	0.766044	−	0.642788i	0.289538	−	0.242951i
							\(11\)	0.836546	+	4.74429i	0.252228	+	1.43046i	0.803089	+	0.595859i	\(0.203188\pi\)
−0.550861	+	0.834597i	\(0.685701\pi\)
\(13\)	−4.57237	+	1.66421i	−1.26815	+	0.461568i	−0.886495	−	0.462738i	\(-0.846867\pi\)
							−0.381653	+	0.924306i	\(0.624645\pi\)
\(17\)	−2.56297	+	4.43920i	−0.621612	+	1.07666i	0.367574	+	0.929994i	\(0.380189\pi\)
							−0.989186	+	0.146669i	\(0.953145\pi\)
\(19\)	−0.229399	−	0.397331i	−0.0526278	−	0.0911540i	0.838511	−	0.544884i	\(-0.183426\pi\)
							−0.891139	+	0.453730i	\(0.850093\pi\)
\(23\)	−2.40975	−	2.02202i	−0.502468	−	0.421620i	0.356002	−	0.934485i	\(-0.384140\pi\)
							−0.858469	+	0.512865i	\(0.828584\pi\)
\(29\)	6.15200	+	2.23914i	1.14240	+	0.415799i	0.842779	−	0.538260i	\(-0.180918\pi\)
							0.299619	+	0.954059i	\(0.403141\pi\)
\(31\)	3.76234	+	3.15698i	0.675736	+	0.567010i	0.914757	−	0.404004i	\(-0.132382\pi\)
							−0.239021	+	0.971014i	\(0.576826\pi\)
\(37\)	4.24314	−	7.34934i	0.697568	−	1.20822i	−0.271739	−	0.962371i	\(-0.587599\pi\)
							0.969307	−	0.245853i	\(-0.0790680\pi\)
\(41\)	−3.12785	+	1.13844i	−0.488488	+	0.177795i	−0.574509	−	0.818498i	\(-0.694807\pi\)
							0.0860209	+	0.996293i	\(0.472585\pi\)
\(43\)	−0.456012	−	2.58617i	−0.0695412	−	0.394388i	−0.999634	−	0.0270625i	\(-0.991385\pi\)
							0.930093	−	0.367325i	\(-0.119726\pi\)
\(47\)	−3.93971	+	3.30581i	−0.574665	+	0.482201i	−0.883190	−	0.469015i	\(-0.844609\pi\)
							0.308525	+	0.951216i	\(0.400165\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.a.85.4	✓	54
3.2	odd	2		567.2.v.b.64.6		54
27.7	even	9	inner	189.2.v.a.169.4	yes	54
27.13	even	9		5103.2.a.i.1.17		27
27.14	odd	18		5103.2.a.f.1.11		27
27.20	odd	18		567.2.v.b.505.6		54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.a.85.4	✓	54	1.1	even	1	trivial
189.2.v.a.169.4	yes	54	27.7	even	9	inner
567.2.v.b.64.6		54	3.2	odd	2
567.2.v.b.505.6		54	27.20	odd	18
5103.2.a.f.1.11		27	27.14	odd	18
5103.2.a.i.1.17		27	27.13	even	9