Embedded newform 845.2.w.a.116.36

• Label: 845.2.w.a.116.36
• 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			116.36
Character	\(\chi\)	\(=\)	845.116
Dual form			845.2.w.a.51.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.257906 + 0.291115i) q^{2} +(0.0778804 + 0.205354i) q^{3} +(0.222841 - 1.83526i) q^{4} +(-0.239316 - 0.970942i) q^{5} +(-0.0396958 + 0.0756341i) q^{6} +(-3.51088 + 2.42338i) q^{7} +(1.23190 - 0.850322i) q^{8} +(2.20943 - 1.95738i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 744 q + 4 q^{3} + 66 q^{4} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.257906	+	0.291115i	0.182367	+	0.205850i	0.832584	−	0.553899i	\(-0.186861\pi\)
							−0.650217	+	0.759748i	\(0.725322\pi\)
\(3\)	0.0778804	+	0.205354i	0.0449643	+	0.118561i	0.955617	−	0.294611i	\(-0.0951901\pi\)
							−0.910653	+	0.413172i	\(0.864421\pi\)
\(5\)	−0.239316	−	0.970942i	−0.107025	−	0.434218i
							\(7\)	−3.51088	+	2.42338i	−1.32699	+	0.915952i	−0.999581	−	0.0289317i	\(-0.990789\pi\)
−0.327405	+	0.944884i	\(0.606174\pi\)
\(11\)	−3.80483	+	4.29476i	−1.14720	+	1.29492i	−0.198015	+	0.980199i	\(0.563449\pi\)
							−0.949184	+	0.314721i	\(0.898089\pi\)
\(13\)	−3.05981	−	1.90724i	−0.848638	−	0.528974i
							\(17\)	−2.36484	−	3.42607i	−0.573559	−	0.830944i	0.423633	−	0.905834i	\(-0.360755\pi\)
−0.997192	+	0.0748904i	\(0.976139\pi\)
\(19\)		−	0.348715i		−	0.0800007i	−0.999200	−	0.0400004i	\(-0.987264\pi\)
							0.999200	−	0.0400004i	\(-0.0127359\pi\)
\(23\)	0.700424			0.146048			0.0730242	−	0.997330i	\(-0.476735\pi\)
							0.0730242	+	0.997330i	\(0.476735\pi\)
\(29\)	1.20578	−	1.06823i	0.223907	−	0.198364i	−0.543710	−	0.839273i	\(-0.682981\pi\)
							0.767617	+	0.640909i	\(0.221442\pi\)
\(31\)	0.781755	−	1.48951i	0.140407	−	0.267524i	−0.805091	−	0.593152i	\(-0.797883\pi\)
							0.945498	+	0.325628i	\(0.105576\pi\)
\(37\)	−4.02460	+	7.66824i	−0.661641	+	1.26065i	0.291050	+	0.956708i	\(0.405995\pi\)
							−0.952690	+	0.303943i	\(0.901697\pi\)
\(41\)	−8.15659	+	3.09339i	−1.27384	+	0.483106i	−0.896531	−	0.442980i	\(-0.853921\pi\)
							−0.377313	+	0.926086i	\(0.623152\pi\)
\(43\)	−9.86392	+	5.17698i	−1.50423	+	0.789483i	−0.997365	−	0.0725412i	\(-0.976889\pi\)
							−0.506868	+	0.862024i	\(0.669197\pi\)
\(47\)	−6.90697	+	0.838659i	−1.00749	+	0.122331i	−0.607608	−	0.794237i	\(-0.707871\pi\)
							−0.399878	+	0.916568i	\(0.630948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.w.a.116.36	yes	744
169.51	even	26	inner	845.2.w.a.51.36	✓	744

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.w.a.51.36	✓	744	169.51	even	26	inner
845.2.w.a.116.36	yes	744	1.1	even	1	trivial