Embedded newform 845.2.bn.a.392.6

• Label: 845.2.bn.a.392.6
• 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			392.6
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09888 + 2.31584i) q^{2} +(-0.410954 - 0.379115i) q^{3} +(-2.89068 - 3.54045i) q^{4} +(2.02492 + 0.948522i) q^{5} +(1.32956 - 0.535101i) q^{6} +(-2.48217 - 0.718968i) q^{7} +(6.39793 - 1.57695i) q^{8} +(-0.216245 - 2.67867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09888	+	2.31584i	−0.777025	+	1.63755i	−0.00663063	+	0.999978i	\(0.502111\pi\)
							−0.770394	+	0.637568i	\(0.779941\pi\)
\(3\)	−0.410954	−	0.379115i	−0.237264	−	0.218882i	0.550539	−	0.834810i	\(-0.314422\pi\)
							−0.787803	+	0.615928i	\(0.788781\pi\)
\(5\)	2.02492	+	0.948522i	0.905572	+	0.424192i
							\(7\)	−2.48217	−	0.718968i	−0.938171	−	0.271744i	−0.226302	−	0.974057i	\(-0.572664\pi\)
−0.711869	+	0.702313i	\(0.752151\pi\)
\(11\)	−1.92341	−	2.26122i	−0.579928	−	0.681784i	0.391210	−	0.920302i	\(-0.372057\pi\)
							−0.971138	+	0.238517i	\(0.923339\pi\)
\(13\)	1.57962	+	3.24111i	0.438107	+	0.898923i
							\(17\)	1.48827	+	2.70199i	0.360958	+	0.655330i	0.992802	−	0.119766i	\(-0.0382145\pi\)
−0.631844	+	0.775096i	\(0.717702\pi\)
\(19\)	−4.37599	+	1.17254i	−1.00392	+	0.269000i	−0.723087	−	0.690757i	\(-0.757278\pi\)
							−0.280833	+	0.959757i	\(0.590611\pi\)
\(23\)	0.123162	−	0.459648i	0.0256811	−	0.0958432i	−0.951896	−	0.306422i	\(-0.900868\pi\)
							0.977577	+	0.210579i	\(0.0675348\pi\)
\(29\)	4.54428	+	2.15629i	0.843851	+	0.400412i	0.800996	−	0.598669i	\(-0.204304\pi\)
							0.0428543	+	0.999081i	\(0.486355\pi\)
\(31\)	7.76579	+	6.08411i	1.39478	+	1.09274i	0.983436	+	0.181256i	\(0.0580162\pi\)
							0.411341	+	0.911482i	\(0.365061\pi\)
\(37\)	6.27720	+	8.35344i	1.03197	+	1.37330i	0.925400	+	0.378991i	\(0.123729\pi\)
							0.106565	+	0.994306i	\(0.466015\pi\)
\(41\)	4.82148	+	4.44794i	0.752989	+	0.694651i	0.958845	−	0.283931i	\(-0.0916386\pi\)
							−0.205855	+	0.978582i	\(0.565998\pi\)
\(43\)	0.0965385	+	0.680279i	0.0147220	+	0.103742i	0.996082	−	0.0884387i	\(-0.0281877\pi\)
							−0.981360	+	0.192180i	\(0.938444\pi\)
\(47\)	−1.51368	+	0.574065i	−0.220794	+	0.0837359i	−0.462521	−	0.886608i	\(-0.653055\pi\)
							0.241728	+	0.970344i	\(0.422286\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.6	yes	4272
5.3	odd	4		845.2.bi.a.223.84	yes	4272
169.72	odd	156		845.2.bi.a.72.84	✓	4272
845.748	even	156	inner	845.2.bn.a.748.6	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.84	✓	4272	169.72	odd	156
845.2.bi.a.223.84	yes	4272	5.3	odd	4
845.2.bn.a.392.6	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.6	yes	4272	845.748	even	156	inner