Embedded newform 109.2.f.a.38.6

• Label: 109.2.f.a.38.6
• Level: $109$
• Weight: $2$
• Character: 109.38
• Analytic conductor: $0.870$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	109.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.870369382032\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			38.6
Character	\(\chi\)	\(=\)	109.38
Dual form			109.2.f.a.66.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.809610 - 1.40229i) q^{2} +(0.309749 - 1.75667i) q^{3} +(-0.310937 - 0.538559i) q^{4} +(-1.08704 + 0.395650i) q^{5} +(-2.21258 - 1.85658i) q^{6} +(-1.51205 + 0.550341i) q^{7} +2.23149 q^{8} +(-0.170882 - 0.0621961i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 6 q^{3} - 12 q^{4} - 6 q^{5} + 12 q^{6} + 3 q^{7} - 12 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/109\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.809610	−	1.40229i	0.572481	−	0.991566i	−0.423830	−	0.905742i	\(-0.639314\pi\)
							0.996310	−	0.0858238i	\(-0.0273522\pi\)
\(3\)	0.309749	−	1.75667i	0.178834	−	1.01422i	−0.754791	−	0.655965i	\(-0.772262\pi\)
							0.933625	−	0.358252i	\(-0.116627\pi\)
\(5\)	−1.08704	+	0.395650i	−0.486138	+	0.176940i	−0.573449	−	0.819241i	\(-0.694395\pi\)
							0.0873106	+	0.996181i	\(0.472173\pi\)
\(7\)	−1.51205	+	0.550341i	−0.571501	+	0.208009i	−0.611574	−	0.791187i	\(-0.709463\pi\)
							0.0400728	+	0.999197i	\(0.487241\pi\)
\(11\)	0.407473	+	2.31089i	0.122858	+	0.696760i	0.982557	+	0.185960i	\(0.0595396\pi\)
							−0.859700	+	0.510800i	\(0.829349\pi\)
\(13\)	−3.73026	+	1.35770i	−1.03459	+	0.376560i	−0.802826	−	0.596213i	\(-0.796671\pi\)
							−0.231763	+	0.972772i	\(0.574449\pi\)
\(17\)	1.10125	−	1.90743i	0.267093	−	0.462619i	−0.701017	−	0.713145i	\(-0.747270\pi\)
							0.968110	+	0.250526i	\(0.0806036\pi\)
\(19\)	1.54489	−	2.67583i	0.354422	−	0.613878i	−0.632597	−	0.774482i	\(-0.718011\pi\)
							0.987019	+	0.160604i	\(0.0513442\pi\)
\(23\)	1.57691	+	2.73129i	0.328809	+	0.569514i	0.982276	−	0.187441i	\(-0.0600193\pi\)
							−0.653467	+	0.756955i	\(0.726686\pi\)
\(29\)	−1.15502	+	6.55044i	−0.214482	+	1.21639i	0.667321	+	0.744770i	\(0.267441\pi\)
							−0.881803	+	0.471617i	\(0.843670\pi\)
\(31\)	−1.12920	−	0.410997i	−0.202811	−	0.0738172i	0.238617	−	0.971114i	\(-0.423306\pi\)
							−0.441428	+	0.897297i	\(0.645528\pi\)
\(37\)	−8.73913	−	3.18078i	−1.43670	−	0.522918i	−0.497859	−	0.867258i	\(-0.665880\pi\)
							−0.938845	+	0.344340i	\(0.888103\pi\)
\(41\)	−5.68979			−0.888596			−0.444298	−	0.895879i	\(-0.646547\pi\)
							−0.444298	+	0.895879i	\(0.646547\pi\)
\(43\)	2.07947	−	3.60175i	0.317117	−	0.549262i	−0.662768	−	0.748824i	\(-0.730619\pi\)
							0.979885	+	0.199562i	\(0.0639520\pi\)
\(47\)	2.76114	−	2.31687i	0.402754	−	0.337950i	−0.418803	−	0.908077i	\(-0.637550\pi\)
							0.821557	+	0.570127i	\(0.193106\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.2.f.a.38.6	✓	42
3.2	odd	2		981.2.w.a.910.2		42
109.66	even	9	inner	109.2.f.a.66.6	yes	42
327.284	odd	18		981.2.w.a.829.2		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.2.f.a.38.6	✓	42	1.1	even	1	trivial
109.2.f.a.66.6	yes	42	109.66	even	9	inner
981.2.w.a.829.2		42	327.284	odd	18
981.2.w.a.910.2		42	3.2	odd	2