Embedded newform 845.2.bn.a.392.3

• Label: 845.2.bn.a.392.3
• 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			392.3
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15694 + 2.43819i) q^{2} +(-2.20015 - 2.02969i) q^{3} +(-3.34140 - 4.09247i) q^{4} +(2.03926 + 0.917297i) q^{5} +(7.49422 - 3.01617i) q^{6} +(3.16219 + 0.915938i) q^{7} +(8.60334 - 2.12053i) q^{8} +(0.479606 + 5.94099i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15694	+	2.43819i	−0.818078	+	1.72406i	−0.138625	+	0.990345i	\(0.544268\pi\)
							−0.679454	+	0.733719i	\(0.737783\pi\)
\(3\)	−2.20015	−	2.02969i	−1.27026	−	1.17184i	−0.976394	−	0.215997i	\(-0.930700\pi\)
							−0.293863	−	0.955847i	\(-0.594941\pi\)
\(5\)	2.03926	+	0.917297i	0.911983	+	0.410228i
							\(7\)	3.16219	+	0.915938i	1.19519	+	0.346192i	0.815410	−	0.578883i	\(-0.196511\pi\)
0.379784	+	0.925075i	\(0.375998\pi\)
\(11\)	−0.123820	−	0.145567i	−0.0373330	−	0.0438900i	0.742730	−	0.669591i	\(-0.233531\pi\)
							−0.780063	+	0.625701i	\(0.784813\pi\)
\(13\)	1.79118	−	3.12916i	0.496785	−	0.867874i
							\(17\)	−2.71637	−	4.93164i	−0.658816	−	1.19610i	−0.969385	−	0.245547i	\(-0.921032\pi\)
0.310569	−	0.950551i	\(-0.399481\pi\)
\(19\)	2.53390	−	0.678957i	0.581317	−	0.155763i	0.0438352	−	0.999039i	\(-0.486042\pi\)
							0.537482	+	0.843275i	\(0.319376\pi\)
\(23\)	−1.40583	+	5.24664i	−0.293136	+	1.09400i	0.649550	+	0.760319i	\(0.274957\pi\)
							−0.942686	+	0.333681i	\(0.891709\pi\)
\(29\)	−2.67722	−	1.27036i	−0.497148	−	0.235900i	0.163573	−	0.986531i	\(-0.447698\pi\)
							−0.660721	+	0.750632i	\(0.729749\pi\)
\(31\)	2.16898	+	1.69929i	0.389560	+	0.305201i	0.792321	−	0.610104i	\(-0.208872\pi\)
							−0.402761	+	0.915305i	\(0.631949\pi\)
\(37\)	−5.51362	−	7.33730i	−0.906434	−	1.20624i	−0.977957	−	0.208806i	\(-0.933042\pi\)
							0.0715229	−	0.997439i	\(-0.477214\pi\)
\(41\)	3.84034	+	3.54281i	0.599760	+	0.553294i	0.918938	−	0.394401i	\(-0.129048\pi\)
							−0.319178	+	0.947695i	\(0.603407\pi\)
\(43\)	−0.547098	−	3.85524i	−0.0834316	−	0.587918i	−0.987151	−	0.159792i	\(-0.948918\pi\)
							0.903719	−	0.428126i	\(-0.140826\pi\)
\(47\)	6.68418	−	2.53498i	0.974988	−	0.369764i	0.184900	−	0.982757i	\(-0.440804\pi\)
							0.790088	+	0.612993i	\(0.210035\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.3	yes	4272
5.3	odd	4		845.2.bi.a.223.87	yes	4272
169.72	odd	156		845.2.bi.a.72.87	✓	4272
845.748	even	156	inner	845.2.bn.a.748.3	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.87	✓	4272	169.72	odd	156
845.2.bi.a.223.87	yes	4272	5.3	odd	4
845.2.bn.a.392.3	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.3	yes	4272	845.748	even	156	inner