Embedded newform 845.2.bf.a.74.6

• Label: 845.2.bf.a.74.6
• Level: $845$
• Weight: $2$
• Character: 845.74
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $2160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.bf (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(2160\)
Relative dimension:	\(90\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			74.6
Character	\(\chi\)	\(=\)	845.74
Dual form			845.2.bf.a.354.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.208627 - 2.58432i) q^{2} +(0.0546105 + 0.0863596i) q^{3} +(-4.66106 + 0.757496i) q^{4} +(-1.18206 - 1.89809i) q^{5} +(0.211787 - 0.159148i) q^{6} +(2.25722 - 2.16809i) q^{7} +(1.68908 + 6.85285i) q^{8} +(1.28160 - 2.70092i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2160 q - 140 q^{4} - 20 q^{5} - 42 q^{6} - 142 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{38}{39}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.208627	−	2.58432i	−0.147522	−	1.82739i	−0.472119	−	0.881535i	\(-0.656511\pi\)
							0.324598	−	0.945852i	\(-0.394771\pi\)
\(3\)	0.0546105	+	0.0863596i	0.0315294	+	0.0498597i	0.860587	−	0.509303i	\(-0.170097\pi\)
							−0.829058	+	0.559163i	\(0.811123\pi\)
\(5\)	−1.18206	−	1.89809i	−0.528632	−	0.848851i
							\(7\)	2.25722	−	2.16809i	0.853151	−	0.819462i	−0.131983	−	0.991252i	\(-0.542134\pi\)
0.985134	+	0.171790i	\(0.0549550\pi\)
\(11\)	−2.59986	−	5.47909i	−0.783888	−	1.65201i	−0.757922	−	0.652345i	\(-0.773785\pi\)
							−0.0259659	−	0.999663i	\(-0.508266\pi\)
\(13\)	3.41988	+	1.14211i	0.948504	+	0.316765i
							\(17\)	2.61077	−	2.50768i	0.633204	−	0.608201i	−0.305805	−	0.952094i	\(-0.598925\pi\)
0.939009	+	0.343894i	\(0.111746\pi\)
\(19\)	−0.715385	−	1.23908i	−0.164120	−	0.284265i	0.772222	−	0.635353i	\(-0.219145\pi\)
							−0.936343	+	0.351088i	\(0.885812\pi\)
\(23\)	3.90743	+	2.25595i	0.814755	+	0.470399i	0.848604	−	0.529028i	\(-0.177443\pi\)
							−0.0338494	+	0.999427i	\(0.510777\pi\)
\(29\)	−6.75248	+	0.545116i	−1.25390	+	0.101226i	−0.689507	−	0.724279i	\(-0.742173\pi\)
							−0.564396	+	0.825504i	\(0.690891\pi\)
\(31\)	0.473052	+	3.89593i	0.0849626	+	0.699730i	0.971494	+	0.237062i	\(0.0761846\pi\)
							−0.886532	+	0.462668i	\(0.846892\pi\)
\(37\)	2.76012	−	6.47824i	0.453761	−	1.06502i	−0.523318	−	0.852138i	\(-0.675306\pi\)
							0.977079	−	0.212878i	\(-0.0682838\pi\)
\(41\)	−2.87689	+	1.81924i	−0.449295	+	0.284117i	−0.739889	−	0.672729i	\(-0.765122\pi\)
							0.290593	+	0.956847i	\(0.406147\pi\)
\(43\)	3.94326	+	9.25517i	0.601342	+	1.41140i	0.890882	+	0.454235i	\(0.150087\pi\)
							−0.289540	+	0.957166i	\(0.593502\pi\)
\(47\)	−5.17952	+	1.96433i	−0.755511	+	0.286528i	−0.702132	−	0.712047i	\(-0.747768\pi\)
							−0.0533790	+	0.998574i	\(0.516999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bf.a.74.6	✓	2160
5.4	even	2	inner	845.2.bf.a.74.85	yes	2160
169.16	even	39	inner	845.2.bf.a.354.85	yes	2160
845.354	even	78	inner	845.2.bf.a.354.6	yes	2160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bf.a.74.6	✓	2160	1.1	even	1	trivial
845.2.bf.a.74.85	yes	2160	5.4	even	2	inner
845.2.bf.a.354.6	yes	2160	845.354	even	78	inner
845.2.bf.a.354.85	yes	2160	169.16	even	39	inner