Embedded newform 845.2.w.a.116.62

• Label: 845.2.w.a.116.62
• 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			116.62
Character	\(\chi\)	\(=\)	845.116
Dual form			845.2.w.a.51.62

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.86658 + 2.10693i) q^{2} +(0.565480 + 1.49105i) q^{3} +(-0.713975 + 5.88011i) q^{4} +(0.239316 + 0.970942i) q^{5} +(-2.08603 + 3.97459i) q^{6} +(2.22342 - 1.53472i) q^{7} +(-9.08858 + 6.27339i) q^{8} +(0.342074 - 0.303051i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 744 q + 4 q^{3} + 66 q^{4} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.86658	+	2.10693i	1.31987	+	1.48983i	0.733227	+	0.679984i	\(0.238013\pi\)
							0.586646	+	0.809844i	\(0.300448\pi\)
\(3\)	0.565480	+	1.49105i	0.326480	+	0.860857i	0.993292	+	0.115631i	\(0.0368891\pi\)
							−0.666812	+	0.745226i	\(0.732342\pi\)
\(5\)	0.239316	+	0.970942i	0.107025	+	0.434218i
							\(7\)	2.22342	−	1.53472i	0.840374	−	0.580068i	−0.0682571	−	0.997668i	\(-0.521744\pi\)
0.908631	+	0.417599i	\(0.137128\pi\)
\(11\)	−0.730746	+	0.824842i	−0.220328	+	0.248699i	−0.848274	−	0.529558i	\(-0.822358\pi\)
							0.627945	+	0.778258i	\(0.283896\pi\)
\(13\)	1.85017	−	3.09465i	0.513143	−	0.858303i
							\(17\)	−2.12003	−	3.07139i	−0.514182	−	0.744921i	0.476786	−	0.879019i	\(-0.341802\pi\)
−0.990968	+	0.134098i	\(0.957186\pi\)
\(19\)			0.508602i			0.116681i	0.998297	+	0.0583406i	\(0.0185809\pi\)
							−0.998297	+	0.0583406i	\(0.981419\pi\)
\(23\)	4.79087			0.998965			0.499482	−	0.866324i	\(-0.333524\pi\)
							0.499482	+	0.866324i	\(0.333524\pi\)
\(29\)	3.25424	−	2.88301i	0.604297	−	0.535361i	−0.304546	−	0.952497i	\(-0.598505\pi\)
							0.908844	+	0.417137i	\(0.136966\pi\)
\(31\)	−2.33020	+	4.43983i	−0.418516	+	0.797416i	−0.999868	−	0.0162360i	\(-0.994832\pi\)
							0.581352	+	0.813652i	\(0.302524\pi\)
\(37\)	2.72265	−	5.18758i	0.447602	−	0.852834i	−0.552215	−	0.833702i	\(-0.686217\pi\)
							0.999817	−	0.0191321i	\(-0.00609032\pi\)
\(41\)	9.54298	−	3.61918i	1.49036	−	0.565220i	0.530995	−	0.847375i	\(-0.321818\pi\)
							0.959369	+	0.282155i	\(0.0910492\pi\)
\(43\)	−7.22984	+	3.79451i	−1.10254	+	0.578658i	−0.914941	−	0.403588i	\(-0.867763\pi\)
							−0.187599	+	0.982246i	\(0.560071\pi\)
\(47\)	−12.1417	+	1.47427i	−1.77105	+	0.215044i	−0.940003	−	0.341165i	\(-0.889178\pi\)
							−0.831042	+	0.556209i	\(0.812255\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.w.a.116.62	yes	744
169.51	even	26	inner	845.2.w.a.51.62	✓	744

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.w.a.51.62	✓	744	169.51	even	26	inner
845.2.w.a.116.62	yes	744	1.1	even	1	trivial