Embedded newform 189.2.v.b.85.1

• Label: 189.2.v.b.85.1
• 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			85.1
Character	\(\chi\)	\(=\)	189.85
Dual form			189.2.v.b.169.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.47328 - 0.900200i) q^{2} +(-0.969270 + 1.43545i) q^{3} +(3.77466 + 3.16732i) q^{4} +(-0.697772 + 3.95726i) q^{5} +(3.68947 - 2.67773i) q^{6} +(-0.766044 + 0.642788i) q^{7} +(-3.85257 - 6.67284i) q^{8} +(-1.12103 - 2.78268i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 3 q^{3} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.47328	−	0.900200i	−1.74887	−	0.636538i	−0.749206	−	0.662337i	\(-0.769565\pi\)
							−0.999667	+	0.0257998i	\(0.991787\pi\)
\(3\)	−0.969270	+	1.43545i	−0.559608	+	0.828757i
							\(5\)	−0.697772	+	3.95726i	−0.312053	+	1.76974i	0.276234	+	0.961090i	\(0.410914\pi\)
−0.588287	+	0.808652i	\(0.700198\pi\)
\(7\)	−0.766044	+	0.642788i	−0.289538	+	0.242951i
							\(11\)	−0.0973664	−	0.552192i	−0.0293571	−	0.166492i	0.966605	−	0.256273i	\(-0.0824945\pi\)
−0.995962	+	0.0897806i	\(0.971383\pi\)
\(13\)	−1.05869	+	0.385333i	−0.293629	+	0.106872i	−0.484634	−	0.874717i	\(-0.661047\pi\)
							0.191006	+	0.981589i	\(0.438825\pi\)
\(17\)	−0.215939	+	0.374018i	−0.0523730	+	0.0907127i	−0.891023	−	0.453957i	\(-0.850012\pi\)
							0.838650	+	0.544670i	\(0.183345\pi\)
\(19\)	1.21001	+	2.09580i	0.277596	+	0.480810i	0.970787	−	0.239944i	\(-0.0771291\pi\)
							−0.693191	+	0.720754i	\(0.743796\pi\)
\(23\)	−4.41042	−	3.70078i	−0.919636	−	0.771666i	0.0542920	−	0.998525i	\(-0.482710\pi\)
							−0.973928	+	0.226859i	\(0.927154\pi\)
\(29\)	5.19875	+	1.89219i	0.965383	+	0.351371i	0.776141	−	0.630560i	\(-0.217175\pi\)
							0.189242	+	0.981930i	\(0.439397\pi\)
\(31\)	−5.02464	−	4.21618i	−0.902452	−	0.757248i	0.0682157	−	0.997671i	\(-0.478269\pi\)
							−0.970668	+	0.240423i	\(0.922714\pi\)
\(37\)	−4.67073	+	8.08995i	−0.767864	+	1.32998i	0.170855	+	0.985296i	\(0.445347\pi\)
							−0.938719	+	0.344683i	\(0.887986\pi\)
\(41\)	5.27652	−	1.92050i	0.824054	−	0.299931i	0.104638	−	0.994510i	\(-0.466632\pi\)
							0.719416	+	0.694579i	\(0.244409\pi\)
\(43\)	−0.957373	−	5.42953i	−0.145998	−	0.827996i	−0.966560	−	0.256439i	\(-0.917451\pi\)
							0.820563	−	0.571557i	\(-0.193660\pi\)
\(47\)	−2.21505	+	1.85865i	−0.323098	+	0.271111i	−0.789881	−	0.613261i	\(-0.789857\pi\)
							0.466783	+	0.884372i	\(0.345413\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.b.85.1	✓	54
3.2	odd	2		567.2.v.a.64.9		54
27.7	even	9	inner	189.2.v.b.169.1	yes	54
27.13	even	9		5103.2.a.g.1.27		27
27.14	odd	18		5103.2.a.h.1.1		27
27.20	odd	18		567.2.v.a.505.9		54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.b.85.1	✓	54	1.1	even	1	trivial
189.2.v.b.169.1	yes	54	27.7	even	9	inner
567.2.v.a.64.9		54	3.2	odd	2
567.2.v.a.505.9		54	27.20	odd	18
5103.2.a.g.1.27		27	27.13	even	9
5103.2.a.h.1.1		27	27.14	odd	18