Embedded newform 845.2.bg.a.621.10

• Label: 845.2.bg.a.621.10
• Level: $845$
• Weight: $2$
• Character: 845.621
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $1440$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(1440\)
Relative dimension:	\(60\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			621.10
Character	\(\chi\)	\(=\)	845.621
Dual form			845.2.bg.a.381.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.70743 - 1.39407i) q^{2} +(0.230040 + 0.0185708i) q^{3} +(0.571824 + 2.80098i) q^{4} +(-0.992709 + 0.120537i) q^{5} +(-0.366888 - 0.352401i) q^{6} +(0.886165 + 1.40136i) q^{7} +(0.879678 - 1.67609i) q^{8} +(-2.90858 - 0.472690i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1440 q - 2 q^{3} - 58 q^{4} + 18 q^{6} + 6 q^{7} + 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{17}{78}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.70743	−	1.39407i	−1.20733	−	0.985758i	−0.999972	−	0.00749392i	\(-0.997615\pi\)
							−0.207363	−	0.978264i	\(-0.566488\pi\)
\(3\)	0.230040	+	0.0185708i	0.132814	+	0.0107218i	0.146695	−	0.989182i	\(-0.453137\pi\)
							−0.0138809	+	0.999904i	\(0.504419\pi\)
\(5\)	−0.992709	+	0.120537i	−0.443953	+	0.0539056i
							\(7\)	0.886165	+	1.40136i	0.334939	+	0.529663i	0.970097	−	0.242719i	\(-0.0780393\pi\)
−0.635158	+	0.772382i	\(0.719065\pi\)
\(11\)	−0.460735	−	2.83501i	−0.138917	−	0.854789i	−0.959179	−	0.282801i	\(-0.908736\pi\)
							0.820262	−	0.571988i	\(-0.193828\pi\)
\(13\)	−0.0234712	−	3.60547i	−0.00650973	−	0.999979i
							\(17\)	−0.0813147	+	0.0514203i	−0.0197217	+	0.0124713i	−0.544290	−	0.838897i	\(-0.683201\pi\)
0.524569	+	0.851368i	\(0.324227\pi\)
\(19\)	5.29562	+	3.05743i	1.21490	+	0.701423i	0.963823	−	0.266545i	\(-0.0858820\pi\)
							0.251077	+	0.967967i	\(0.419215\pi\)
\(23\)	−1.36618	−	2.36629i	−0.284868	−	0.493406i	0.687709	−	0.725986i	\(-0.258616\pi\)
							−0.972577	+	0.232581i	\(0.925283\pi\)
\(29\)	−4.28796	+	5.25179i	−0.796253	+	0.975233i	−0.999993	−	0.00379181i	\(-0.998793\pi\)
							0.203739	+	0.979025i	\(0.434690\pi\)
\(31\)	−0.548222	+	2.22422i	−0.0984635	+	0.399482i	−0.999406	−	0.0344558i	\(-0.989030\pi\)
							0.900943	+	0.433938i	\(0.142876\pi\)
\(37\)	−7.28983	+	2.11152i	−1.19844	+	0.347132i	−0.816659	−	0.577120i	\(-0.804176\pi\)
							−0.381781	+	0.924253i	\(0.624689\pi\)
\(41\)	−0.144323	+	1.78777i	−0.0225395	+	0.279202i	0.975663	+	0.219275i	\(0.0703691\pi\)
							−0.998203	+	0.0599277i	\(0.980913\pi\)
\(43\)	−2.01964	+	6.97261i	−0.307992	+	1.06331i	0.645964	+	0.763368i	\(0.276456\pi\)
							−0.953956	+	0.299945i	\(0.903032\pi\)
\(47\)	−7.77731	+	8.77877i	−1.13444	+	1.28052i	−0.179484	+	0.983761i	\(0.557443\pi\)
							−0.954954	+	0.296754i	\(0.904096\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bg.a.621.10	yes	1440
169.43	even	78	inner	845.2.bg.a.381.10	✓	1440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bg.a.381.10	✓	1440	169.43	even	78	inner
845.2.bg.a.621.10	yes	1440	1.1	even	1	trivial