Embedded newform 605.2.k.a.56.13

• Label: 605.2.k.a.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.a.551.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.228264 + 0.0670242i) q^{2} -2.04300 q^{3} +(-1.63490 - 1.05068i) q^{4} +(-0.654861 + 0.755750i) q^{5} +(-0.466343 - 0.136931i) q^{6} +(-0.615760 + 1.34833i) q^{7} +(-0.614349 - 0.708996i) q^{8} +1.17386 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 4 q^{2} - 4 q^{3} - 24 q^{4} - 22 q^{5} - 12 q^{6} - 8 q^{7} - 12 q^{8} + 212 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.228264	+	0.0670242i	0.161407	+	0.0473933i	0.361438	−	0.932396i	\(-0.382286\pi\)
							−0.200031	+	0.979790i	\(0.564104\pi\)
\(3\)	−2.04300			−1.17953			−0.589764	−	0.807575i	\(-0.700779\pi\)
							−0.589764	+	0.807575i	\(0.700779\pi\)
\(5\)	−0.654861	+	0.755750i	−0.292863	+	0.337981i
							\(7\)	−0.615760	+	1.34833i	−0.232735	+	0.509619i	−0.989582	−	0.143974i	\(-0.954012\pi\)
0.756846	+	0.653593i	\(0.226739\pi\)
\(11\)	−0.266688	+	3.30589i	−0.0804095	+	0.996762i
							\(13\)	0.917698	+	0.589769i	0.254524	+	0.163572i	0.661681	−	0.749786i	\(-0.269843\pi\)
−0.407157	+	0.913358i	\(0.633480\pi\)
\(17\)	−0.729418	−	5.07322i	−0.176910	−	1.23044i	−0.863860	−	0.503731i	\(-0.831960\pi\)
							0.686950	−	0.726704i	\(-0.258949\pi\)
\(19\)	0.912467	−	6.34634i	0.209334	−	1.45595i	−0.566003	−	0.824403i	\(-0.691511\pi\)
							0.775337	−	0.631548i	\(-0.217580\pi\)
\(23\)	1.11958	+	2.45153i	0.233448	+	0.511180i	0.989710	−	0.143089i	\(-0.0457036\pi\)
							−0.756262	+	0.654269i	\(0.772976\pi\)
\(29\)	1.29596	−	9.01363i	0.240655	−	1.67379i	−0.408212	−	0.912887i	\(-0.633847\pi\)
							0.648866	−	0.760902i	\(-0.275243\pi\)
\(31\)	5.75936	−	3.70132i	1.03441	−	0.664777i	0.0908138	−	0.995868i	\(-0.471053\pi\)
							0.943599	+	0.331091i	\(0.107417\pi\)
\(37\)	−3.16157	+	2.03182i	−0.519758	+	0.334028i	−0.774076	−	0.633092i	\(-0.781785\pi\)
							0.254318	+	0.967121i	\(0.418149\pi\)
\(41\)	4.95790	+	1.45577i	0.774293	+	0.227353i	0.644927	−	0.764244i	\(-0.276888\pi\)
							0.129366	+	0.991597i	\(0.458706\pi\)
\(43\)	6.54395	+	7.55212i	0.997943	+	1.15169i	0.988421	+	0.151733i	\(0.0484855\pi\)
							0.00952175	+	0.999955i	\(0.496969\pi\)
\(47\)	−5.15348	+	1.51320i	−0.751713	+	0.220723i	−0.635074	−	0.772452i	\(-0.719030\pi\)
							−0.116639	+	0.993174i	\(0.537212\pi\)

(See \(a_n\) instead)

Twists

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

    

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