Embedded newform 845.2.w.a.51.19

• Label: 845.2.w.a.51.19
• Level: $845$
• Weight: $2$
• Character: 845.51
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $744$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.w (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(744\)
Relative dimension:	\(62\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			51.19
Character	\(\chi\)	\(=\)	845.51
Dual form			845.2.w.a.116.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.802563 + 0.905906i) q^{2} +(1.01575 - 2.67831i) q^{3} +(0.0645146 + 0.531325i) q^{4} +(-0.239316 + 0.970942i) q^{5} +(1.61110 + 3.06969i) q^{6} +(-0.865640 - 0.597508i) q^{7} +(-2.52519 - 1.74301i) q^{8} +(-3.89607 - 3.45162i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 744 q + 4 q^{3} + 66 q^{4} - 62 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{19}{26}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.802563	+	0.905906i	−0.567497	+	0.640572i	−0.960273	−	0.279062i	\(-0.909976\pi\)
							0.392776	+	0.919634i	\(0.371515\pi\)
\(3\)	1.01575	−	2.67831i	0.586443	−	1.54632i	−0.232588	−	0.972575i	\(-0.574719\pi\)
							0.819032	−	0.573748i	\(-0.194511\pi\)
\(5\)	−0.239316	+	0.970942i	−0.107025	+	0.434218i
							\(7\)	−0.865640	−	0.597508i	−0.327181	−	0.225837i	0.393138	−	0.919479i	\(-0.371389\pi\)
−0.720319	+	0.693643i	\(0.756005\pi\)
\(11\)	−0.408150	−	0.460707i	−0.123062	−	0.138908i	0.683750	−	0.729716i	\(-0.260348\pi\)
							−0.806812	+	0.590808i	\(0.798809\pi\)
\(13\)	−3.19149	+	1.67761i	−0.885161	+	0.465285i
							\(17\)	−2.33809	+	3.38731i	−0.567070	+	0.821543i	−0.996676	−	0.0814735i	\(-0.974037\pi\)
0.429605	+	0.903017i	\(0.358653\pi\)
\(19\)			4.00100i			0.917893i	0.888464	+	0.458946i	\(0.151773\pi\)
							−0.888464	+	0.458946i	\(0.848227\pi\)
\(23\)	−7.48570			−1.56088			−0.780438	−	0.625233i	\(-0.785004\pi\)
							−0.780438	+	0.625233i	\(0.785004\pi\)
\(29\)	−2.05391	−	1.81961i	−0.381402	−	0.337893i	0.450539	−	0.892757i	\(-0.351232\pi\)
							−0.831941	+	0.554864i	\(0.812770\pi\)
\(31\)	1.63672	+	3.11851i	0.293964	+	0.560102i	0.986984	−	0.160816i	\(-0.0514127\pi\)
							−0.693020	+	0.720918i	\(0.743720\pi\)
\(37\)	2.61508	+	4.98262i	0.429917	+	0.819138i	0.999997	−	0.00245123i	\(-0.000780252\pi\)
							−0.570080	+	0.821589i	\(0.693088\pi\)
\(41\)	9.68808	+	3.67421i	1.51302	+	0.573814i	0.964908	−	0.262588i	\(-0.0845759\pi\)
							0.548116	+	0.836402i	\(0.315345\pi\)
\(43\)	−9.81884	−	5.15333i	−1.49736	−	0.785875i	−0.500551	−	0.865707i	\(-0.666869\pi\)
							−0.996808	+	0.0798319i	\(0.974562\pi\)
\(47\)	−7.56685	−	0.918782i	−1.10374	−	0.134018i	−0.451675	−	0.892182i	\(-0.649173\pi\)
							−0.652064	+	0.758164i	\(0.726096\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.w.a.51.19	✓	744
169.116	even	26	inner	845.2.w.a.116.19	yes	744

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.w.a.51.19	✓	744	1.1	even	1	trivial
845.2.w.a.116.19	yes	744	169.116	even	26	inner