Embedded newform 605.2.s.b.16.9

• Label: 605.2.s.b.16.9
• 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			16.9
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.b.416.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.457329 - 0.0524736i) q^{2} +(-2.01066 - 1.46083i) q^{3} +(-1.74163 - 0.404998i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(0.842877 + 0.773585i) q^{6} +(1.60972 + 1.65636i) q^{7} +(1.64236 + 0.585993i) q^{8} +(0.981676 + 3.02129i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.457329	−	0.0524736i	−0.323380	−	0.0371044i	−0.0492185	−	0.998788i	\(-0.515673\pi\)
							−0.274162	+	0.961684i	\(0.588400\pi\)
\(3\)	−2.01066	−	1.46083i	−1.16085	−	0.843410i	−0.170969	−	0.985276i	\(-0.554690\pi\)
							−0.989886	+	0.141866i	\(0.954690\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	1.60972	+	1.65636i	0.608416	+	0.626045i	0.949417	−	0.314018i	\(-0.101675\pi\)
−0.341001	+	0.940063i	\(0.610766\pi\)
\(11\)	2.92655	−	1.56055i	0.882387	−	0.470524i
							\(13\)	−0.0161511	+	0.0267836i	−0.00447951	+	0.00742844i	−0.858581	−	0.512678i	\(-0.828653\pi\)
0.854101	+	0.520107i	\(0.174108\pi\)
\(17\)	3.34208	−	2.73280i	0.810573	−	0.662802i	−0.134045	−	0.990975i	\(-0.542797\pi\)
							0.944618	+	0.328173i	\(0.106433\pi\)
\(19\)	−6.35510	−	0.363398i	−1.45796	−	0.0833691i	−0.689973	−	0.723835i	\(-0.742378\pi\)
							−0.767986	+	0.640466i	\(0.778741\pi\)
\(23\)	0.114333	−	0.795204i	0.0238401	−	0.165812i	−0.974423	−	0.224721i	\(-0.927853\pi\)
							0.998263	+	0.0589090i	\(0.0187622\pi\)
\(29\)	−1.21671	+	3.12510i	−0.225938	+	0.580317i	−0.998508	−	0.0546129i	\(-0.982608\pi\)
							0.772570	+	0.634930i	\(0.218971\pi\)
\(31\)	−3.37902	−	1.42799i	−0.606890	−	0.256474i	0.0640959	−	0.997944i	\(-0.479584\pi\)
							−0.670986	+	0.741470i	\(0.734129\pi\)
\(37\)	0.849231	−	9.88742i	0.139613	−	1.62548i	−0.506500	−	0.862240i	\(-0.669061\pi\)
							0.646113	−	0.763242i	\(-0.276393\pi\)
\(41\)	−5.98010	+	3.37712i	−0.933934	+	0.527417i	−0.882008	−	0.471234i	\(-0.843809\pi\)
							−0.0519258	+	0.998651i	\(0.516536\pi\)
\(43\)	7.96743	−	2.33945i	1.21502	−	0.356762i	0.389443	−	0.921050i	\(-0.372667\pi\)
							0.825578	+	0.564288i	\(0.190849\pi\)
\(47\)	1.10534	−	5.45509i	0.161231	−	0.795706i	−0.815082	−	0.579345i	\(-0.803308\pi\)
							0.976313	−	0.216361i	\(-0.0694189\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.b.16.9	✓	880
121.53	even	55	inner	605.2.s.b.416.9	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.b.16.9	✓	880	1.1	even	1	trivial
605.2.s.b.416.9	yes	880	121.53	even	55	inner