Embedded newform 605.2.k.b.56.6

• Label: 605.2.k.b.56.6
• 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			56.6
Character	\(\chi\)	\(=\)	605.56
Dual form			605.2.k.b.551.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.47955 - 0.434434i) q^{2} +1.25158 q^{3} +(0.317819 + 0.204250i) q^{4} +(0.654861 - 0.755750i) q^{5} +(-1.85177 - 0.543727i) q^{6} +(1.94824 - 4.26605i) q^{7} +(1.63811 + 1.89048i) q^{8} -1.43356 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 2 q^{2} - 24 q^{4} + 22 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 212 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.47955	−	0.434434i	−1.04620	−	0.307191i	−0.286918	−	0.957955i	\(-0.592631\pi\)
							−0.759280	+	0.650764i	\(0.774449\pi\)
\(3\)	1.25158			0.722598			0.361299	−	0.932450i	\(-0.382333\pi\)
							0.361299	+	0.932450i	\(0.382333\pi\)
\(5\)	0.654861	−	0.755750i	0.292863	−	0.337981i
							\(7\)	1.94824	−	4.26605i	0.736365	−	1.61241i	−0.0530784	−	0.998590i	\(-0.516903\pi\)
0.789443	−	0.613824i	\(-0.210369\pi\)
\(11\)	−1.74884	−	2.81808i	−0.527295	−	0.849682i
							\(13\)	−2.37858	−	1.52862i	−0.659699	−	0.423963i	0.167500	−	0.985872i	\(-0.446431\pi\)
−0.827199	+	0.561909i	\(0.810067\pi\)
\(17\)	0.251085	+	1.74633i	0.0608969	+	0.423548i	0.997350	+	0.0727543i	\(0.0231789\pi\)
							−0.936453	+	0.350793i	\(0.885912\pi\)
\(19\)	−0.159731	+	1.11095i	−0.0366447	+	0.254870i	−0.999906	−	0.0136956i	\(-0.995640\pi\)
							0.963261	+	0.268565i	\(0.0865495\pi\)
\(23\)	1.38530	+	3.03337i	0.288854	+	0.632502i	0.997314	−	0.0732491i	\(-0.0233368\pi\)
							−0.708460	+	0.705751i	\(0.750610\pi\)
\(29\)	0.283045	−	1.96862i	0.0525601	−	0.365564i	−0.946519	−	0.322649i	\(-0.895427\pi\)
							0.999079	−	0.0429145i	\(-0.0136643\pi\)
\(31\)	−0.612785	+	0.393813i	−0.110059	+	0.0707309i	−0.594513	−	0.804086i	\(-0.702655\pi\)
							0.484453	+	0.874817i	\(0.339019\pi\)
\(37\)	−0.263050	+	0.169052i	−0.0432452	+	0.0277920i	−0.562085	−	0.827080i	\(-0.690001\pi\)
							0.518840	+	0.854872i	\(0.326364\pi\)
\(41\)	6.56473	+	1.92758i	1.02524	+	0.301037i	0.750773	−	0.660560i	\(-0.229681\pi\)
							0.274465	+	0.961597i	\(0.411499\pi\)
\(43\)	−3.46315	−	3.99669i	−0.528125	−	0.609489i	0.427521	−	0.904005i	\(-0.359387\pi\)
							−0.955646	+	0.294516i	\(0.904841\pi\)
\(47\)	−5.20164	+	1.52734i	−0.758737	+	0.222785i	−0.638144	−	0.769917i	\(-0.720298\pi\)
							−0.120593	+	0.992702i	\(0.538480\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.k.b.56.6	✓	220
121.67	even	11	inner	605.2.k.b.551.6	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.k.b.56.6	✓	220	1.1	even	1	trivial
605.2.k.b.551.6	yes	220	121.67	even	11	inner