Embedded newform 845.2.bn.a.392.2

• Label: 845.2.bn.a.392.2
• Level: $845$
• Weight: $2$
• Character: 845.392
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $4272$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.bn (of order \(156\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(4272\)
Relative dimension:	\(89\) over \(\Q(\zeta_{156})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{156}]$

Embedding invariants

Embedding label			392.2
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17898 + 2.48464i) q^{2} +(1.41308 + 1.30360i) q^{3} +(-3.51856 - 4.30945i) q^{4} +(-2.13647 - 0.659909i) q^{5} +(-4.90496 + 1.97408i) q^{6} +(3.90636 + 1.13149i) q^{7} +(9.51521 - 2.34529i) q^{8} +(0.0560175 + 0.693902i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{61}{156}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17898	+	2.48464i	−0.833662	+	1.75691i	−0.205777	+	0.978599i	\(0.565972\pi\)
							−0.627885	+	0.778306i	\(0.716079\pi\)
\(3\)	1.41308	+	1.30360i	0.815841	+	0.752633i	0.971970	−	0.235103i	\(-0.0755428\pi\)
							−0.156130	+	0.987737i	\(0.549902\pi\)
\(5\)	−2.13647	−	0.659909i	−0.955460	−	0.295120i
							\(7\)	3.90636	+	1.13149i	1.47647	+	0.427663i	0.916224	−	0.400666i	\(-0.131221\pi\)
0.560241	+	0.828330i	\(0.310708\pi\)
\(11\)	−1.39275	−	1.63737i	−0.419930	−	0.493684i	0.510642	−	0.859793i	\(-0.329408\pi\)
							−0.930572	+	0.366109i	\(0.880690\pi\)
\(13\)	3.31066	−	1.42811i	0.918213	−	0.396088i
							\(17\)	0.355270	+	0.645002i	0.0861655	+	0.156436i	0.915502	−	0.402314i	\(-0.131794\pi\)
−0.829336	+	0.558750i	\(0.811281\pi\)
\(19\)	0.396179	−	0.106156i	0.0908898	−	0.0243538i	−0.213088	−	0.977033i	\(-0.568352\pi\)
							0.303977	+	0.952679i	\(0.401685\pi\)
\(23\)	−1.85655	+	6.92874i	−0.387117	+	1.44474i	0.447685	+	0.894191i	\(0.352249\pi\)
							−0.834802	+	0.550550i	\(0.814418\pi\)
\(29\)	4.88794	+	2.31936i	0.907667	+	0.430693i	0.824513	−	0.565843i	\(-0.191449\pi\)
							0.0831543	+	0.996537i	\(0.473501\pi\)
\(31\)	3.25707	+	2.55175i	0.584988	+	0.458309i	0.865059	−	0.501671i	\(-0.167281\pi\)
							−0.280071	+	0.959979i	\(0.590358\pi\)
\(37\)	6.19636	+	8.24586i	1.01868	+	1.35561i	0.933235	+	0.359267i	\(0.116973\pi\)
							0.0854406	+	0.996343i	\(0.472770\pi\)
\(41\)	−8.99591	−	8.29895i	−1.40493	−	1.29608i	−0.901992	−	0.431754i	\(-0.857895\pi\)
							−0.502934	−	0.864325i	\(-0.667746\pi\)
\(43\)	0.752529	+	5.30286i	0.114760	+	0.808678i	0.961133	+	0.276087i	\(0.0890379\pi\)
							−0.846373	+	0.532591i	\(0.821218\pi\)
\(47\)	2.89267	−	1.09704i	0.421939	−	0.160020i	−0.134491	−	0.990915i	\(-0.542940\pi\)
							0.556430	+	0.830894i	\(0.312171\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.2	yes	4272
5.3	odd	4		845.2.bi.a.223.88	yes	4272
169.72	odd	156		845.2.bi.a.72.88	✓	4272
845.748	even	156	inner	845.2.bn.a.748.2	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.88	✓	4272	169.72	odd	156
845.2.bi.a.223.88	yes	4272	5.3	odd	4
845.2.bn.a.392.2	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.2	yes	4272	845.748	even	156	inner