Embedded newform 845.2.bn.a.392.35

• Label: 845.2.bn.a.392.35
• 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			392.35
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.290988 + 0.613244i) q^{2} +(-0.720135 - 0.664343i) q^{3} +(0.973496 + 1.19232i) q^{4} +(2.08155 + 0.816805i) q^{5} +(0.616955 - 0.248303i) q^{6} +(-2.00427 - 0.580543i) q^{7} +(-2.33257 + 0.574927i) q^{8} +(-0.164156 - 2.03344i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.290988	+	0.613244i	−0.205760	+	0.433629i	−0.979928	−	0.199350i	\(-0.936117\pi\)
							0.774169	+	0.632979i	\(0.218168\pi\)
\(3\)	−0.720135	−	0.664343i	−0.415770	−	0.383558i	0.442490	−	0.896774i	\(-0.354095\pi\)
							−0.858260	+	0.513215i	\(0.828454\pi\)
\(5\)	2.08155	+	0.816805i	0.930895	+	0.365286i
							\(7\)	−2.00427	−	0.580543i	−0.757541	−	0.219424i	−0.123099	−	0.992394i	\(-0.539283\pi\)
−0.634443	+	0.772970i	\(0.718770\pi\)
\(11\)	2.18827	+	2.57261i	0.659788	+	0.775670i	0.985477	−	0.169807i	\(-0.0543143\pi\)
							−0.325689	+	0.945477i	\(0.605596\pi\)
\(13\)	0.949129	−	3.47838i	0.263241	−	0.964730i
							\(17\)	3.58223	+	6.50364i	0.868819	+	1.57736i	0.815478	+	0.578788i	\(0.196474\pi\)
0.0533409	+	0.998576i	\(0.483013\pi\)
\(19\)	4.53174	−	1.21428i	1.03965	−	0.278574i	0.301683	−	0.953408i	\(-0.402452\pi\)
							0.737969	+	0.674835i	\(0.235785\pi\)
\(23\)	0.0674363	−	0.251676i	0.0140614	−	0.0524780i	−0.958539	−	0.284963i	\(-0.908019\pi\)
							0.972600	+	0.232485i	\(0.0746855\pi\)
\(29\)	−0.133065	−	0.0631400i	−0.0247095	−	0.0117248i	0.416298	−	0.909228i	\(-0.363327\pi\)
							−0.441007	+	0.897504i	\(0.645379\pi\)
\(31\)	2.36682	+	1.85428i	0.425093	+	0.333039i	0.806359	−	0.591426i	\(-0.201435\pi\)
							−0.381266	+	0.924465i	\(0.624512\pi\)
\(37\)	−0.737739	−	0.981753i	−0.121284	−	0.161399i	0.734717	−	0.678373i	\(-0.237315\pi\)
							−0.856001	+	0.516974i	\(0.827058\pi\)
\(41\)	4.70697	+	4.34230i	0.735106	+	0.678153i	0.954744	−	0.297428i	\(-0.0961288\pi\)
							−0.219638	+	0.975581i	\(0.570488\pi\)
\(43\)	0.657943	+	4.63633i	0.100335	+	0.707034i	0.974984	+	0.222276i	\(0.0713487\pi\)
							−0.874648	+	0.484758i	\(0.838908\pi\)
\(47\)	−8.45319	+	3.20587i	−1.23302	+	0.467624i	−0.883114	−	0.469159i	\(-0.844557\pi\)
							−0.349910	+	0.936783i	\(0.613788\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.35	yes	4272
5.3	odd	4		845.2.bi.a.223.55	yes	4272
169.72	odd	156		845.2.bi.a.72.55	✓	4272
845.748	even	156	inner	845.2.bn.a.748.35	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.55	✓	4272	169.72	odd	156
845.2.bi.a.223.55	yes	4272	5.3	odd	4
845.2.bn.a.392.35	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.35	yes	4272	845.748	even	156	inner