Embedded newform 845.2.w.a.116.57

• Label: 845.2.w.a.116.57
• Level: $845$
• Weight: $2$
• Character: 845.116
• 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			116.57
Character	\(\chi\)	\(=\)	845.116
Dual form			845.2.w.a.51.57

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.66406 + 1.87834i) q^{2} +(-0.181218 - 0.477833i) q^{3} +(-0.517979 + 4.26594i) q^{4} +(0.239316 + 0.970942i) q^{5} +(0.595975 - 1.13553i) q^{6} +(-2.76717 + 1.91004i) q^{7} +(-4.74440 + 3.27482i) q^{8} +(2.05005 - 1.81618i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 744 q + 4 q^{3} + 66 q^{4} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.66406	+	1.87834i	1.17667	+	1.32819i	0.933658	+	0.358165i	\(0.116597\pi\)
							0.243013	+	0.970023i	\(0.421864\pi\)
\(3\)	−0.181218	−	0.477833i	−0.104626	−	0.275877i	0.872472	−	0.488663i	\(-0.162515\pi\)
							−0.977099	+	0.212786i	\(0.931746\pi\)
\(5\)	0.239316	+	0.970942i	0.107025	+	0.434218i
							\(7\)	−2.76717	+	1.91004i	−1.04589	+	0.721927i	−0.961583	−	0.274513i	\(-0.911483\pi\)
−0.0843080	+	0.996440i	\(0.526868\pi\)
\(11\)	−2.39844	+	2.70727i	−0.723156	+	0.816274i	−0.989027	−	0.147735i	\(-0.952802\pi\)
							0.265871	+	0.964008i	\(0.414340\pi\)
\(13\)	−1.97890	+	3.01396i	−0.548848	+	0.835922i
							\(17\)	−1.61313	−	2.33702i	−0.391241	−	0.566810i	0.577027	−	0.816725i	\(-0.304213\pi\)
−0.968268	+	0.249915i	\(0.919597\pi\)
\(19\)		−	3.07070i		−	0.704468i	−0.935912	−	0.352234i	\(-0.885422\pi\)
							0.935912	−	0.352234i	\(-0.114578\pi\)
\(23\)	9.46517			1.97362			0.986812	−	0.161868i	\(-0.0517520\pi\)
							0.986812	+	0.161868i	\(0.0517520\pi\)
\(29\)	−3.54753	+	3.14284i	−0.658760	+	0.583610i	−0.924847	−	0.380338i	\(-0.875807\pi\)
							0.266087	+	0.963949i	\(0.414269\pi\)
\(31\)	−2.83537	+	5.40235i	−0.509248	+	0.970291i	0.486096	+	0.873905i	\(0.338420\pi\)
							−0.995344	+	0.0963854i	\(0.969272\pi\)
\(37\)	0.0356409	−	0.0679080i	0.00585932	−	0.0111640i	−0.882509	−	0.470296i	\(-0.844147\pi\)
							0.888368	+	0.459132i	\(0.151839\pi\)
\(41\)	−8.69030	+	3.29579i	−1.35720	+	0.514717i	−0.922548	−	0.385883i	\(-0.873897\pi\)
							−0.434649	+	0.900600i	\(0.643127\pi\)
\(43\)	4.38294	−	2.30035i	0.668393	−	0.350800i	−0.0961553	−	0.995366i	\(-0.530655\pi\)
							0.764548	+	0.644567i	\(0.222962\pi\)
\(47\)	9.82139	−	1.19253i	1.43260	−	0.173949i	0.632844	−	0.774279i	\(-0.281887\pi\)
							0.799752	+	0.600330i	\(0.204964\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.w.a.116.57	yes	744
169.51	even	26	inner	845.2.w.a.51.57	✓	744

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.w.a.51.57	✓	744	169.51	even	26	inner
845.2.w.a.116.57	yes	744	1.1	even	1	trivial