Embedded newform 109.2.f.a.16.5

• Label: 109.2.f.a.16.5
• Level: $109$
• Weight: $2$
• Character: 109.16
• 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			16.5
Character	\(\chi\)	\(=\)	109.16
Dual form			109.2.f.a.75.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.503663 + 0.872370i) q^{2} +(-0.352094 + 0.295442i) q^{3} +(0.492647 - 0.853289i) q^{4} +(0.517671 + 2.93586i) q^{5} +(-0.435072 - 0.158353i) q^{6} +(-0.307807 - 1.74566i) q^{7} +3.00717 q^{8} +(-0.484260 + 2.74638i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 6 q^{3} - 12 q^{4} - 6 q^{5} + 12 q^{6} + 3 q^{7} - 12 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.503663	+	0.872370i	0.356144	+	0.616859i	0.987313	−	0.158786i	\(-0.0507580\pi\)
							−0.631169	+	0.775645i	\(0.717425\pi\)
\(3\)	−0.352094	+	0.295442i	−0.203282	+	0.170573i	−0.738745	−	0.673985i	\(-0.764581\pi\)
							0.535464	+	0.844558i	\(0.320137\pi\)
\(5\)	0.517671	+	2.93586i	0.231509	+	1.31296i	0.849841	+	0.527038i	\(0.176698\pi\)
							−0.618332	+	0.785917i	\(0.712191\pi\)
\(7\)	−0.307807	−	1.74566i	−0.116340	−	0.659798i	−0.986078	−	0.166284i	\(-0.946823\pi\)
							0.869738	−	0.493514i	\(-0.164288\pi\)
\(11\)	−1.58443	−	1.32949i	−0.477722	−	0.400857i	0.371880	−	0.928281i	\(-0.378713\pi\)
							−0.849602	+	0.527424i	\(0.823158\pi\)
\(13\)	−0.513248	−	2.91078i	−0.142349	−	0.807304i	−0.969457	−	0.245260i	\(-0.921127\pi\)
							0.827108	−	0.562043i	\(-0.189985\pi\)
\(17\)	−0.859368	−	1.48847i	−0.208427	−	0.361007i	0.742792	−	0.669522i	\(-0.233501\pi\)
							−0.951219	+	0.308516i	\(0.900168\pi\)
\(19\)	−2.50803	−	4.34403i	−0.575381	−	0.996590i	−0.996000	−	0.0893522i	\(-0.971520\pi\)
							0.420619	−	0.907237i	\(-0.361813\pi\)
\(23\)	−1.38141	+	2.39267i	−0.288043	+	0.498905i	−0.973343	−	0.229356i	\(-0.926338\pi\)
							0.685299	+	0.728261i	\(0.259671\pi\)
\(29\)	3.16031	−	2.65182i	0.586856	−	0.492430i	−0.300335	−	0.953834i	\(-0.597098\pi\)
							0.887190	+	0.461403i	\(0.152654\pi\)
\(31\)	0.604915	−	3.43064i	0.108646	−	0.616161i	−0.881055	−	0.473013i	\(-0.843166\pi\)
							0.989701	−	0.143148i	\(-0.0457226\pi\)
\(37\)	−1.89359	+	10.7391i	−0.311304	+	1.76550i	0.280929	+	0.959729i	\(0.409358\pi\)
							−0.592233	+	0.805767i	\(0.701754\pi\)
\(41\)	−8.90422			−1.39061			−0.695303	−	0.718717i	\(-0.744730\pi\)
							−0.695303	+	0.718717i	\(0.744730\pi\)
\(43\)	3.82458	+	6.62436i	0.583243	+	1.01021i	0.995092	+	0.0989541i	\(0.0315497\pi\)
							−0.411849	+	0.911252i	\(0.635117\pi\)
\(47\)	−2.40908	+	0.876833i	−0.351400	+	0.127899i	−0.511688	−	0.859171i	\(-0.670980\pi\)
							0.160288	+	0.987070i	\(0.448758\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.2.f.a.16.5	✓	42
3.2	odd	2		981.2.w.a.343.3		42
109.75	even	9	inner	109.2.f.a.75.5	yes	42
327.293	odd	18		981.2.w.a.838.3		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.2.f.a.16.5	✓	42	1.1	even	1	trivial
109.2.f.a.75.5	yes	42	109.75	even	9	inner
981.2.w.a.343.3		42	3.2	odd	2
981.2.w.a.838.3		42	327.293	odd	18