Embedded newform 108.4.l.a.59.16

• Label: 108.4.l.a.59.16
• Level: $108$
• Weight: $4$
• Character: 108.59
• Analytic conductor: $6.372$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.37220628062\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(52\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			59.16
Character	\(\chi\)	\(=\)	108.59
Dual form			108.4.l.a.11.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.76579 + 2.20953i) q^{2} +(4.72349 + 2.16532i) q^{3} +(-1.76401 - 7.80309i) q^{4} +(-1.80584 - 4.96151i) q^{5} +(-13.1250 + 6.61318i) q^{6} +(9.40209 - 1.65784i) q^{7} +(20.3560 + 9.88097i) q^{8} +(17.6228 + 20.4558i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.76579	+	2.20953i	−0.624299	+	0.781185i
							\(3\)	4.72349	+	2.16532i	0.909037	+	0.416716i
\(5\)	−1.80584	−	4.96151i	−0.161520	−	0.443771i								0.832361	−	0.554234i	\(-0.186989\pi\)
							−0.993880	+	0.110463i	\(0.964767\pi\)
\(7\)	9.40209	−	1.65784i	0.507665	−	0.0895150i	0.0860515	−	0.996291i	\(-0.472575\pi\)
							0.421613	+	0.906776i	\(0.361464\pi\)
\(11\)	53.5277	+	19.4825i	1.46720	+	0.534017i	0.947339	−	0.320234i	\(-0.103761\pi\)
							0.519861	+	0.854251i	\(0.325984\pi\)
\(13\)	−18.5620	−	15.5753i	−0.396012	−	0.332294i	0.422937	−	0.906159i	\(-0.360999\pi\)
							−0.818950	+	0.573865i	\(0.805443\pi\)
\(17\)	35.7325	+	20.6302i	0.509788	+	0.294326i	0.732747	−	0.680502i	\(-0.238238\pi\)
							−0.222958	+	0.974828i	\(0.571571\pi\)
\(19\)	21.9438	−	12.6693i	0.264961	−	0.152975i	−0.361635	−	0.932320i	\(-0.617781\pi\)
							0.626595	+	0.779345i	\(0.284448\pi\)
\(23\)	−26.1866	+	148.511i	−0.237403	+	1.34638i	0.600090	+	0.799933i	\(0.295131\pi\)
							−0.837493	+	0.546448i	\(0.815980\pi\)
\(29\)	40.0603	+	47.7420i	0.256518	+	0.305706i	0.878899	−	0.477009i	\(-0.158279\pi\)
							−0.622381	+	0.782715i	\(0.713835\pi\)
\(31\)	−258.526	−	45.5852i	−1.49783	−	0.264108i	−0.636152	−	0.771564i	\(-0.719475\pi\)
							−0.861678	+	0.507456i	\(0.830586\pi\)
\(37\)	−14.6109	+	25.3069i	−0.0649196	+	0.112444i	−0.896658	−	0.442723i	\(-0.854012\pi\)
							0.831739	+	0.555167i	\(0.187346\pi\)
\(41\)	261.539	−	311.690i	0.996233	−	1.18726i	0.0139416	−	0.999903i	\(-0.495562\pi\)
							0.982291	−	0.187361i	\(-0.0599934\pi\)
\(43\)	59.7544	−	164.174i	0.211918	−	0.582239i	−0.787502	−	0.616313i	\(-0.788626\pi\)
							0.999419	+	0.0340736i	\(0.0108480\pi\)
\(47\)	1.92189	+	10.8996i	0.00596460	+	0.0338270i	0.987645	−	0.156709i	\(-0.0500885\pi\)
							−0.981680	+	0.190536i	\(0.938977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.59.16	yes	312
4.3	odd	2	inner	108.4.l.a.59.46	yes	312
27.11	odd	18	inner	108.4.l.a.11.46	yes	312
108.11	even	18	inner	108.4.l.a.11.16	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.16	✓	312	108.11	even	18	inner
108.4.l.a.11.46	yes	312	27.11	odd	18	inner
108.4.l.a.59.16	yes	312	1.1	even	1	trivial
108.4.l.a.59.46	yes	312	4.3	odd	2	inner