Embedded newform 845.2.w.a.116.11

• Label: 845.2.w.a.116.11
• 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			116.11
Character	\(\chi\)	\(=\)	845.116
Dual form			845.2.w.a.51.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31163 - 1.48052i) q^{2} +(-0.177927 - 0.469154i) q^{3} +(-0.230503 + 1.89836i) q^{4} +(0.239316 + 0.970942i) q^{5} +(-0.461219 + 0.878780i) q^{6} +(0.291911 - 0.201492i) q^{7} +(-0.142749 + 0.0985325i) q^{8} +(2.05708 - 1.82242i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 744 q + 4 q^{3} + 66 q^{4} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31163	−	1.48052i	−0.927460	−	1.04689i	−0.998777	−	0.0494326i	\(-0.984259\pi\)
							0.0713172	−	0.997454i	\(-0.477280\pi\)
\(3\)	−0.177927	−	0.469154i	−0.102726	−	0.270866i	0.873793	−	0.486298i	\(-0.161653\pi\)
							−0.976519	+	0.215432i	\(0.930884\pi\)
\(5\)	0.239316	+	0.970942i	0.107025	+	0.434218i
							\(7\)	0.291911	−	0.201492i	0.110332	−	0.0761567i	−0.511621	−	0.859211i	\(-0.670955\pi\)
0.621953	+	0.783054i	\(0.286339\pi\)
\(11\)	−1.41394	+	1.59601i	−0.426318	+	0.481214i	−0.921944	−	0.387322i	\(-0.873400\pi\)
							0.495626	+	0.868536i	\(0.334939\pi\)
\(13\)	0.417424	−	3.58131i	0.115773	−	0.993276i
							\(17\)	0.185364	+	0.268546i	0.0449573	+	0.0651319i	0.844818	−	0.535054i	\(-0.179709\pi\)
−0.799861	+	0.600186i	\(0.795093\pi\)
\(19\)			7.21210i			1.65457i	0.561783	+	0.827285i	\(0.310116\pi\)
							−0.561783	+	0.827285i	\(0.689884\pi\)
\(23\)	3.14483			0.655743			0.327872	−	0.944722i	\(-0.393669\pi\)
							0.327872	+	0.944722i	\(0.393669\pi\)
\(29\)	6.33528	−	5.61257i	1.17643	−	1.04223i	0.178129	−	0.984007i	\(-0.442996\pi\)
							0.998304	−	0.0582209i	\(-0.0185428\pi\)
\(31\)	4.90368	−	9.34318i	0.880726	−	1.67808i	0.157448	−	0.987527i	\(-0.449673\pi\)
							0.723278	−	0.690557i	\(-0.242634\pi\)
\(37\)	2.20446	−	4.20025i	0.362411	−	0.690517i	−0.634165	−	0.773198i	\(-0.718656\pi\)
							0.996576	+	0.0826809i	\(0.0263482\pi\)
\(41\)	−1.19966	+	0.454972i	−0.187356	+	0.0710547i	−0.446499	−	0.894784i	\(-0.647329\pi\)
							0.259143	+	0.965839i	\(0.416560\pi\)
\(43\)	6.40876	−	3.36358i	0.977326	−	0.512940i	0.101119	−	0.994874i	\(-0.467758\pi\)
							0.876208	+	0.481934i	\(0.160065\pi\)
\(47\)	−5.83750	+	0.708801i	−0.851487	+	0.103389i	−0.534632	−	0.845085i	\(-0.679550\pi\)
							−0.316855	+	0.948474i	\(0.602627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.w.a.116.11	yes	744
169.51	even	26	inner	845.2.w.a.51.11	✓	744

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.w.a.51.11	✓	744	169.51	even	26	inner
845.2.w.a.116.11	yes	744	1.1	even	1	trivial