Embedded newform 605.2.k.b.56.8

• Label: 605.2.k.b.56.8
• 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			56.8
Character	\(\chi\)	\(=\)	605.56
Dual form			605.2.k.b.551.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.960803 - 0.282117i) q^{2} -1.66611 q^{3} +(-0.838954 - 0.539163i) q^{4} +(0.654861 - 0.755750i) q^{5} +(1.60080 + 0.470037i) q^{6} +(-0.0635449 + 0.139144i) q^{7} +(1.96547 + 2.26828i) q^{8} -0.224089 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 2 q^{2} - 24 q^{4} + 22 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 212 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.960803	−	0.282117i	−0.679390	−	0.199487i	−0.0762100	−	0.997092i	\(-0.524282\pi\)
							−0.603180	+	0.797605i	\(0.706100\pi\)
\(3\)	−1.66611			−0.961927			−0.480964	−	0.876741i	\(-0.659713\pi\)
							−0.480964	+	0.876741i	\(0.659713\pi\)
\(5\)	0.654861	−	0.755750i	0.292863	−	0.337981i
							\(7\)	−0.0635449	+	0.139144i	−0.0240177	+	0.0525915i	−0.921261	−	0.388946i	\(-0.872839\pi\)
0.897243	+	0.441537i	\(0.145567\pi\)
\(11\)	−3.17831	+	0.947823i	−0.958295	+	0.285779i
							\(13\)	−0.501527	−	0.322312i	−0.139098	−	0.0893932i	0.469241	−	0.883070i	\(-0.344528\pi\)
−0.608339	+	0.793677i	\(0.708164\pi\)
\(17\)	−0.517167	−	3.59697i	−0.125431	−	0.872394i	−0.951242	−	0.308446i	\(-0.900191\pi\)
							0.825810	−	0.563948i	\(-0.190718\pi\)
\(19\)	−0.544111	+	3.78437i	−0.124828	+	0.868195i	0.827140	+	0.561996i	\(0.189966\pi\)
							−0.951968	+	0.306199i	\(0.900943\pi\)
\(23\)	1.38503	+	3.03280i	0.288800	+	0.632383i	0.997309	−	0.0733190i	\(-0.0233591\pi\)
							−0.708509	+	0.705702i	\(0.750632\pi\)
\(29\)	−0.325354	+	2.26288i	−0.0604167	+	0.420207i	0.937057	+	0.349175i	\(0.113538\pi\)
							−0.997474	+	0.0710317i	\(0.977371\pi\)
\(31\)	−0.804901	+	0.517278i	−0.144564	+	0.0929059i	−0.610924	−	0.791689i	\(-0.709202\pi\)
							0.466359	+	0.884595i	\(0.345565\pi\)
\(37\)	6.91625	−	4.44480i	1.13702	−	0.730721i	0.170010	−	0.985442i	\(-0.445620\pi\)
							0.967015	+	0.254721i	\(0.0819837\pi\)
\(41\)	6.22291	+	1.82721i	0.971855	+	0.285362i	0.728858	−	0.684665i	\(-0.240051\pi\)
							0.242997	+	0.970027i	\(0.421870\pi\)
\(43\)	1.38759	+	1.60137i	0.211606	+	0.244206i	0.851623	−	0.524154i	\(-0.175618\pi\)
							−0.640018	+	0.768360i	\(0.721073\pi\)
\(47\)	10.4989	−	3.08275i	1.53142	−	0.449665i	0.595934	−	0.803034i	\(-0.296782\pi\)
							0.935484	+	0.353369i	\(0.114964\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.k.b.56.8	✓	220
121.67	even	11	inner	605.2.k.b.551.8	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.k.b.56.8	✓	220	1.1	even	1	trivial
605.2.k.b.551.8	yes	220	121.67	even	11	inner