Embedded newform 198.6.e.b.133.4

• Label: 198.6.e.b.133.4
• Level: $198$
• Weight: $6$
• Character: 198.133
• Analytic conductor: $31.756$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	198.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(31.7559963230\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			133.4
Character	\(\chi\)	\(=\)	198.133
Dual form			198.6.e.b.67.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00000 - 3.46410i) q^{2} +(-7.29840 - 13.7744i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(-6.92025 - 11.9862i) q^{5} +(-62.3126 - 2.26633i) q^{6} +(62.0895 - 107.542i) q^{7} -64.0000 q^{8} +(-136.467 + 201.062i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{2} + 21 q^{3} - 192 q^{4} - 7 q^{5} + 168 q^{6} + 24 q^{7} - 1536 q^{8} - 543 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/198\mathbb{Z}\right)^\times\).

\(n\)	\(145\)	\(155\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00000	−	3.46410i	0.353553	−	0.612372i
							\(3\)	−7.29840	−	13.7744i	−0.468193	−	0.883626i
\(5\)	−6.92025	−	11.9862i	−0.123793	−	0.214416i								0.797467	−	0.603362i	\(-0.206173\pi\)
							−0.921261	+	0.388946i	\(0.872839\pi\)
\(7\)	62.0895	−	107.542i	0.478931	−	0.829533i	−0.520777	−	0.853693i	\(-0.674358\pi\)
							0.999708	+	0.0241601i	\(0.00769114\pi\)
\(11\)	−60.5000	+	104.789i	−0.150756	+	0.261116i
							\(13\)	−315.556	−	546.559i	−0.517867	−	0.896972i	−0.999785	−	0.0207551i	\(-0.993393\pi\)
0.481918	−	0.876216i	\(-0.339940\pi\)
\(17\)	−1195.48			−1.00327			−0.501635	−	0.865079i	\(-0.667268\pi\)
							−0.501635	+	0.865079i	\(0.667268\pi\)
\(19\)	−1529.80			−0.972192			−0.486096	−	0.873905i	\(-0.661579\pi\)
							−0.486096	+	0.873905i	\(0.661579\pi\)
\(23\)	1193.61	+	2067.40i	0.470483	+	0.814900i	0.999430	−	0.0337547i	\(-0.0107465\pi\)
							−0.528947	+	0.848655i	\(0.677413\pi\)
\(29\)	−1128.03	+	1953.80i	−0.249072	+	0.431406i	−0.963269	−	0.268540i	\(-0.913459\pi\)
							0.714196	+	0.699945i	\(0.246792\pi\)
\(31\)	949.020	+	1643.75i	0.177366	+	0.307207i	0.940978	−	0.338469i	\(-0.109909\pi\)
							−0.763611	+	0.645676i	\(0.776576\pi\)
\(37\)	−839.673			−0.100834			−0.0504169	−	0.998728i	\(-0.516055\pi\)
							−0.0504169	+	0.998728i	\(0.516055\pi\)
\(41\)	2676.86	+	4636.45i	0.248694	+	0.430751i	0.963164	−	0.268916i	\(-0.0866653\pi\)
							−0.714470	+	0.699666i	\(0.753332\pi\)
\(43\)	−4427.23	+	7668.19i	−0.365141	+	0.632443i	−0.988799	−	0.149254i	\(-0.952313\pi\)
							0.623658	+	0.781698i	\(0.285646\pi\)
\(47\)	4362.84	−	7556.65i	0.288087	−	0.498982i	−0.685266	−	0.728293i	\(-0.740314\pi\)
							0.973353	+	0.229311i	\(0.0736473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	198.6.e.b.133.4	yes	24
3.2	odd	2		594.6.e.a.397.8		24
9.4	even	3	inner	198.6.e.b.67.4	✓	24
9.5	odd	6		594.6.e.a.199.8		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.6.e.b.67.4	✓	24	9.4	even	3	inner
198.6.e.b.133.4	yes	24	1.1	even	1	trivial
594.6.e.a.199.8		24	9.5	odd	6
594.6.e.a.397.8		24	3.2	odd	2