Embedded newform 605.2.k.b.56.20

• Label: 605.2.k.b.56.20
• 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			56.20
Character	\(\chi\)	\(=\)	605.56
Dual form			605.2.k.b.551.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.11159 + 0.620020i) q^{2} -3.15222 q^{3} +(2.39189 + 1.53718i) q^{4} +(0.654861 - 0.755750i) q^{5} +(-6.65621 - 1.95444i) q^{6} +(-0.373094 + 0.816963i) q^{7} +(1.21527 + 1.40250i) q^{8} +6.93651 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 2 q^{2} - 24 q^{4} + 22 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 212 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.11159	+	0.620020i	1.49312	+	0.438420i	0.923536	−	0.383512i	\(-0.125286\pi\)
							0.569586	+	0.821932i	\(0.307104\pi\)
\(3\)	−3.15222			−1.81994			−0.909968	−	0.414677i	\(-0.863894\pi\)
							−0.909968	+	0.414677i	\(0.863894\pi\)
\(5\)	0.654861	−	0.755750i	0.292863	−	0.337981i
							\(7\)	−0.373094	+	0.816963i	−0.141016	+	0.308783i	−0.966942	−	0.254996i	\(-0.917926\pi\)
0.825926	+	0.563779i	\(0.190653\pi\)
\(11\)	−2.01924	+	2.63110i	−0.608823	+	0.793306i
							\(13\)	6.00504	+	3.85921i	1.66550	+	1.07035i	0.909428	+	0.415861i	\(0.136520\pi\)
0.756070	+	0.654490i	\(0.227117\pi\)
\(17\)	0.684687	+	4.76210i	0.166061	+	1.15498i	0.886928	+	0.461907i	\(0.152835\pi\)
							−0.720867	+	0.693073i	\(0.756256\pi\)
\(19\)	−0.550740	+	3.83048i	−0.126349	+	0.878773i	0.823779	+	0.566911i	\(0.191862\pi\)
							−0.950127	+	0.311862i	\(0.899047\pi\)
\(23\)	1.93208	+	4.23068i	0.402868	+	0.882157i	0.996971	+	0.0777685i	\(0.0247795\pi\)
							−0.594104	+	0.804388i	\(0.702493\pi\)
\(29\)	1.23923	−	8.61903i	0.230119	−	1.60051i	−0.467466	−	0.884011i	\(-0.654833\pi\)
							0.697585	−	0.716502i	\(-0.254258\pi\)
\(31\)	−2.13172	+	1.36997i	−0.382868	+	0.246054i	−0.717891	−	0.696156i	\(-0.754892\pi\)
							0.335023	+	0.942210i	\(0.391256\pi\)
\(37\)	−3.42149	+	2.19886i	−0.562489	+	0.361490i	−0.790777	−	0.612104i	\(-0.790323\pi\)
							0.228288	+	0.973594i	\(0.426687\pi\)
\(41\)	−2.76386	−	0.811542i	−0.431642	−	0.126742i	0.0586920	−	0.998276i	\(-0.481307\pi\)
							−0.490334	+	0.871535i	\(0.663125\pi\)
\(43\)	4.66868	+	5.38794i	0.711966	+	0.821653i	0.990317	−	0.138827i	\(-0.0443330\pi\)
							−0.278350	+	0.960480i	\(0.589788\pi\)
\(47\)	8.54975	−	2.51043i	1.24711	−	0.366184i	0.409427	−	0.912343i	\(-0.365729\pi\)
							0.837682	+	0.546159i	\(0.183910\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.k.b.56.20	✓	220
121.67	even	11	inner	605.2.k.b.551.20	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.k.b.56.20	✓	220	1.1	even	1	trivial
605.2.k.b.551.20	yes	220	121.67	even	11	inner