Embedded newform 605.2.k.b.56.13

• Label: 605.2.k.b.56.13
• 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			56.13
Character	\(\chi\)	\(=\)	605.56
Dual form			605.2.k.b.551.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.105289 + 0.0309156i) q^{2} -0.0677895 q^{3} +(-1.67238 - 1.07477i) q^{4} +(0.654861 - 0.755750i) q^{5} +(-0.00713748 - 0.00209575i) q^{6} +(-0.157614 + 0.345127i) q^{7} +(-0.286576 - 0.330727i) q^{8} -2.99540 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 2 q^{2} - 24 q^{4} + 22 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 212 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.105289	+	0.0309156i	0.0744505	+	0.0218606i	0.318746	−	0.947840i	\(-0.396738\pi\)
							−0.244295	+	0.969701i	\(0.578557\pi\)
\(3\)	−0.0677895			−0.0391383			−0.0195691	−	0.999809i	\(-0.506229\pi\)
							−0.0195691	+	0.999809i	\(0.506229\pi\)
\(5\)	0.654861	−	0.755750i	0.292863	−	0.337981i
							\(7\)	−0.157614	+	0.345127i	−0.0595727	+	0.130446i	−0.937077	−	0.349123i	\(-0.886480\pi\)
0.877504	+	0.479569i	\(0.159207\pi\)
\(11\)	−0.632938	+	3.25567i	−0.190838	+	0.981622i
							\(13\)	−4.16323	−	2.67555i	−1.15467	−	0.742064i	−0.184109	−	0.982906i	\(-0.558940\pi\)
−0.970564	+	0.240842i	\(0.922576\pi\)
\(17\)	0.991337	+	6.89490i	0.240434	+	1.67226i	0.649967	+	0.759963i	\(0.274783\pi\)
							−0.409532	+	0.912296i	\(0.634308\pi\)
\(19\)	0.193765	−	1.34766i	0.0444527	−	0.309175i	−0.955449	−	0.295156i	\(-0.904628\pi\)
							0.999902	−	0.0140190i	\(-0.00446254\pi\)
\(23\)	0.595360	+	1.30366i	0.124141	+	0.271831i	0.961491	−	0.274836i	\(-0.0886236\pi\)
							−0.837350	+	0.546667i	\(0.815896\pi\)
\(29\)	−0.411768	+	2.86391i	−0.0764634	+	0.531814i	0.915204	+	0.402991i	\(0.132029\pi\)
							−0.991667	+	0.128824i	\(0.958880\pi\)
\(31\)	−2.63030	+	1.69039i	−0.472415	+	0.303603i	−0.755100	−	0.655610i	\(-0.772412\pi\)
							0.282684	+	0.959213i	\(0.408775\pi\)
\(37\)	−3.33073	+	2.14053i	−0.547568	+	0.351901i	−0.784992	−	0.619505i	\(-0.787333\pi\)
							0.237424	+	0.971406i	\(0.423697\pi\)
\(41\)	−8.43883	−	2.47786i	−1.31792	−	0.386977i	−0.454180	−	0.890910i	\(-0.650068\pi\)
							−0.863743	+	0.503933i	\(0.831886\pi\)
\(43\)	−4.66646	−	5.38538i	−0.711628	−	0.821262i	0.278646	−	0.960394i	\(-0.410114\pi\)
							−0.990274	+	0.139132i	\(0.955569\pi\)
\(47\)	−2.93087	+	0.860582i	−0.427512	+	0.125529i	−0.488408	−	0.872615i	\(-0.662422\pi\)
							0.0608965	+	0.998144i	\(0.480604\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.k.b.56.13	✓	220
121.67	even	11	inner	605.2.k.b.551.13	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.k.b.56.13	✓	220	1.1	even	1	trivial
605.2.k.b.551.13	yes	220	121.67	even	11	inner