Embedded newform 845.2.bn.a.392.26

• Label: 845.2.bn.a.392.26
• 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			392.26
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.558037 + 1.17604i) q^{2} +(1.97698 + 1.82382i) q^{3} +(0.193229 + 0.236662i) q^{4} +(-2.15942 + 0.580452i) q^{5} +(-3.24811 + 1.30725i) q^{6} +(2.78610 + 0.807002i) q^{7} +(-2.91394 + 0.718222i) q^{8} +(0.340757 + 4.22103i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.558037	+	1.17604i	−0.394592	+	0.831585i	0.604788	+	0.796386i	\(0.293258\pi\)
							−0.999380	+	0.0351988i	\(0.988794\pi\)
\(3\)	1.97698	+	1.82382i	1.14141	+	1.05298i	0.997993	+	0.0633185i	\(0.0201684\pi\)
							0.143418	+	0.989662i	\(0.454191\pi\)
\(5\)	−2.15942	+	0.580452i	−0.965720	+	0.259586i
							\(7\)	2.78610	+	0.807002i	1.05305	+	0.305018i	0.759215	−	0.650840i	\(-0.225583\pi\)
0.293830	+	0.955858i	\(0.405070\pi\)
\(11\)	2.74928	+	3.23215i	0.828938	+	0.974529i	0.999942	−	0.0107417i	\(-0.00341925\pi\)
							−0.171005	+	0.985270i	\(0.554701\pi\)
\(13\)	3.38531	−	1.24084i	0.938915	−	0.344148i
							\(17\)	1.99033	+	3.61351i	0.482727	+	0.876404i	0.999791	+	0.0204570i	\(0.00651212\pi\)
−0.517064	+	0.855947i	\(0.672975\pi\)
\(19\)	0.428563	−	0.114833i	0.0983192	−	0.0263446i	−0.209324	−	0.977846i	\(-0.567126\pi\)
							0.307643	+	0.951502i	\(0.400460\pi\)
\(23\)	1.22146	−	4.55854i	0.254692	−	0.950522i	−0.713570	−	0.700584i	\(-0.752923\pi\)
							0.968262	−	0.249938i	\(-0.0804103\pi\)
\(29\)	−3.49803	−	1.65984i	−0.649568	−	0.308224i	0.0752497	−	0.997165i	\(-0.476025\pi\)
							−0.724817	+	0.688941i	\(0.758076\pi\)
\(31\)	−5.83750	−	4.57339i	−1.04845	−	0.821405i	−0.0641517	−	0.997940i	\(-0.520434\pi\)
							−0.984294	+	0.176535i	\(0.943511\pi\)
\(37\)	−5.36678	−	7.14189i	−0.882294	−	1.17412i	−0.983797	−	0.179284i	\(-0.942622\pi\)
							0.101504	−	0.994835i	\(-0.467635\pi\)
\(41\)	1.11944	+	1.03271i	0.174828	+	0.161283i	0.760662	−	0.649148i	\(-0.224874\pi\)
							−0.585835	+	0.810431i	\(0.699233\pi\)
\(43\)	−0.431221	−	3.03869i	−0.0657606	−	0.463396i	−0.995468	−	0.0950968i	\(-0.969684\pi\)
							0.929707	−	0.368299i	\(-0.120060\pi\)
\(47\)	−9.98196	+	3.78566i	−1.45602	+	0.552195i	−0.950528	−	0.310638i	\(-0.899457\pi\)
							−0.505490	+	0.862833i	\(0.668688\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.26	yes	4272
5.3	odd	4		845.2.bi.a.223.64	yes	4272
169.72	odd	156		845.2.bi.a.72.64	✓	4272
845.748	even	156	inner	845.2.bn.a.748.26	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.64	✓	4272	169.72	odd	156
845.2.bi.a.223.64	yes	4272	5.3	odd	4
845.2.bn.a.392.26	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.26	yes	4272	845.748	even	156	inner