Embedded newform 605.2.k.a.56.16

• Label: 605.2.k.a.56.16
• Level: $605$
• Weight: $2$
• Character: 605.56
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.k (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			56.16
Character	\(\chi\)	\(=\)	605.56
Dual form			605.2.k.a.551.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.05521 + 0.309837i) q^{2} -3.06769 q^{3} +(-0.665042 - 0.427397i) q^{4} +(-0.654861 + 0.755750i) q^{5} +(-3.23705 - 0.950483i) q^{6} +(0.923710 - 2.02264i) q^{7} +(-2.00971 - 2.31933i) q^{8} +6.41070 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 4 q^{2} - 4 q^{3} - 24 q^{4} - 22 q^{5} - 12 q^{6} - 8 q^{7} - 12 q^{8} + 212 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.05521	+	0.309837i	0.746145	+	0.219088i	0.632636	−	0.774449i	\(-0.281973\pi\)
							0.113509	+	0.993537i	\(0.463791\pi\)
\(3\)	−3.06769			−1.77113			−0.885565	−	0.464516i	\(-0.846228\pi\)
							−0.885565	+	0.464516i	\(0.846228\pi\)
\(5\)	−0.654861	+	0.755750i	−0.292863	+	0.337981i
							\(7\)	0.923710	−	2.02264i	0.349130	−	0.764487i	−0.650856	−	0.759201i	\(-0.725590\pi\)
0.999986	−	0.00528645i	\(-0.00168274\pi\)
\(11\)	2.67155	+	1.96540i	0.805503	+	0.592591i
							\(13\)	−4.93617	−	3.17229i	−1.36905	−	0.879834i	−0.370254	−	0.928931i	\(-0.620729\pi\)
−0.998794	+	0.0490969i	\(0.984366\pi\)
\(17\)	0.885588	+	6.15940i	0.214787	+	1.49387i	0.756879	+	0.653555i	\(0.226723\pi\)
							−0.542092	+	0.840319i	\(0.682368\pi\)
\(19\)	−0.880405	+	6.12335i	−0.201979	+	1.40479i	0.596424	+	0.802670i	\(0.296588\pi\)
							−0.798403	+	0.602124i	\(0.794321\pi\)
\(23\)	2.21157	+	4.84266i	0.461144	+	1.00976i	0.987225	+	0.159330i	\(0.0509332\pi\)
							−0.526082	+	0.850434i	\(0.676339\pi\)
\(29\)	−0.109902	+	0.764385i	−0.0204083	+	0.141943i	−0.997478	−	0.0709772i	\(-0.977388\pi\)
							0.977070	+	0.212920i	\(0.0682973\pi\)
\(31\)	−4.51143	+	2.89932i	−0.810277	+	0.520734i	−0.878955	−	0.476905i	\(-0.841758\pi\)
							0.0686773	+	0.997639i	\(0.478122\pi\)
\(37\)	7.27152	−	4.67312i	1.19543	−	0.768256i	0.217270	−	0.976111i	\(-0.430285\pi\)
							0.978160	+	0.207855i	\(0.0666483\pi\)
\(41\)	11.4988	+	3.37635i	1.79581	+	0.527297i	0.997216	−	0.0745733i	\(-0.0237595\pi\)
							0.798594	+	0.601870i	\(0.205578\pi\)
\(43\)	−1.71107	−	1.97468i	−0.260936	−	0.301136i	0.610131	−	0.792301i	\(-0.291117\pi\)
							−0.871067	+	0.491164i	\(0.836571\pi\)
\(47\)	−0.304794	+	0.0894956i	−0.0444588	+	0.0130543i	−0.303886	−	0.952708i	\(-0.598284\pi\)
							0.259428	+	0.965763i	\(0.416466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.k.a.56.16	✓	220
121.67	even	11	inner	605.2.k.a.551.16	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.k.a.56.16	✓	220	1.1	even	1	trivial
605.2.k.a.551.16	yes	220	121.67	even	11	inner