Embedded newform 845.2.w.a.116.19

• Label: 845.2.w.a.116.19
• Level: $845$
• Weight: $2$
• Character: 845.116
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $744$
• 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			116.19
Character	\(\chi\)	\(=\)	845.116
Dual form			845.2.w.a.51.19

$q$-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 \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{7}{26}\right)\)	\(1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.w.a.51.19	✓	744	169.51	even	26	inner
845.2.w.a.116.19	yes	744	1.1	even	1	trivial