Embedded newform 605.2.s.a.16.14

• Label: 605.2.s.a.16.14
• Level: $605$
• Weight: $2$
• Character: 605.16
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $880$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.s (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(880\)
Relative dimension:	\(22\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			16.14
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.a.416.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.710733 + 0.0815490i) q^{2} +(2.35793 + 1.71314i) q^{3} +(-1.44953 - 0.337074i) q^{4} +(0.564443 - 0.825472i) q^{5} +(1.53616 + 1.40987i) q^{6} +(2.60666 + 2.68219i) q^{7} +(-2.35033 - 0.838595i) q^{8} +(1.69796 + 5.22577i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 2 q^{2} + 5 q^{3} + 24 q^{4} - 22 q^{5} + 7 q^{6} + q^{7} - 4 q^{8} - 217 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.710733	+	0.0815490i	0.502564	+	0.0576639i	0.361567	−	0.932346i	\(-0.382242\pi\)
							0.140997	+	0.990010i	\(0.454969\pi\)
\(3\)	2.35793	+	1.71314i	1.36135	+	0.989081i	0.998358	+	0.0572909i	\(0.0182463\pi\)
							0.362996	+	0.931791i	\(0.381754\pi\)
\(5\)	0.564443	−	0.825472i	0.252427	−	0.369162i
							\(7\)	2.60666	+	2.68219i	0.985223	+	1.01377i	0.999904	+	0.0138784i	\(0.00441778\pi\)
−0.0146805	+	0.999892i	\(0.504673\pi\)
\(11\)	−2.81356	+	1.75610i	−0.848320	+	0.529483i
							\(13\)	−0.988534	+	1.63930i	−0.274170	+	0.454660i	−0.962709	−	0.270540i	\(-0.912798\pi\)
0.688539	+	0.725199i	\(0.258252\pi\)
\(17\)	2.12095	−	1.73429i	0.514406	−	0.420628i	−0.339850	−	0.940480i	\(-0.610376\pi\)
							0.854257	+	0.519852i	\(0.174013\pi\)
\(19\)	1.20181	+	0.0687219i	0.275714	+	0.0157659i	0.194399	−	0.980923i	\(-0.437724\pi\)
							0.0813150	+	0.996688i	\(0.474088\pi\)
\(23\)	0.447899	−	3.11521i	0.0933934	−	0.649566i	−0.888323	−	0.459218i	\(-0.848130\pi\)
							0.981717	−	0.190347i	\(-0.0609614\pi\)
\(29\)	2.28272	−	5.86313i	0.423891	−	1.08876i	−0.544377	−	0.838841i	\(-0.683234\pi\)
							0.968268	−	0.249915i	\(-0.0804026\pi\)
\(31\)	1.07834	+	0.455712i	0.193676	+	0.0818483i	0.483951	−	0.875095i	\(-0.339201\pi\)
							−0.290275	+	0.956943i	\(0.593747\pi\)
\(37\)	−0.387852	+	4.51568i	−0.0637624	+	0.742372i	0.892726	+	0.450601i	\(0.148790\pi\)
							−0.956488	+	0.291772i	\(0.905755\pi\)
\(41\)	9.44419	−	5.33338i	1.47493	−	0.832934i	0.476017	−	0.879436i	\(-0.342080\pi\)
							0.998918	+	0.0465022i	\(0.0148075\pi\)
\(43\)	−3.54367	+	1.04052i	−0.540405	+	0.158677i	−0.540533	−	0.841323i	\(-0.681777\pi\)
							0.000128218		1.00000i	\(0.499959\pi\)
\(47\)	1.27973	−	6.31571i	0.186668	−	0.921241i	−0.771762	−	0.635912i	\(-0.780624\pi\)
							0.958429	−	0.285329i	\(-0.0921030\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.a.16.14	✓	880
121.53	even	55	inner	605.2.s.a.416.14	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.a.16.14	✓	880	1.1	even	1	trivial
605.2.s.a.416.14	yes	880	121.53	even	55	inner