Embedded newform 845.2.bg.a.621.16

• Label: 845.2.bg.a.621.16
• 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			621.16
Character	\(\chi\)	\(=\)	845.621
Dual form			845.2.bg.a.381.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20281 - 0.982063i) q^{2} +(2.59387 + 0.209399i) q^{3} +(0.0822494 + 0.402884i) q^{4} +(-0.992709 + 0.120537i) q^{5} +(-2.91429 - 2.79921i) q^{6} +(-2.51239 - 3.97303i) q^{7} +(-1.14652 + 2.18451i) q^{8} +(3.72318 + 0.605075i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1440 q - 2 q^{3} - 58 q^{4} + 18 q^{6} + 6 q^{7} + 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20281	−	0.982063i	−0.850514	−	0.694423i	0.103356	−	0.994644i	\(-0.467042\pi\)
							−0.953870	+	0.300221i	\(0.902939\pi\)
\(3\)	2.59387	+	0.209399i	1.49757	+	0.120897i	0.801898	−	0.597460i	\(-0.203824\pi\)
							0.695675	+	0.718357i	\(0.255106\pi\)
\(5\)	−0.992709	+	0.120537i	−0.443953	+	0.0539056i
							\(7\)	−2.51239	−	3.97303i	−0.949595	−	1.50166i	−0.862866	−	0.505432i	\(-0.831333\pi\)
−0.0867286	−	0.996232i	\(-0.527641\pi\)
\(11\)	0.517016	+	3.18133i	0.155886	+	0.959206i	0.940642	+	0.339401i	\(0.110224\pi\)
							−0.784756	+	0.619805i	\(0.787212\pi\)
\(13\)	−2.44943	−	2.64580i	−0.679350	−	0.733814i
							\(17\)	4.74833	−	3.00266i	1.15164	−	0.728253i	0.184584	−	0.982817i	\(-0.440906\pi\)
0.967056	+	0.254564i	\(0.0819320\pi\)
\(19\)	−5.09077	−	2.93916i	−1.16790	−	0.674289i	−0.214718	−	0.976676i	\(-0.568883\pi\)
							−0.953185	+	0.302387i	\(0.902217\pi\)
\(23\)	−2.26429	−	3.92186i	−0.472137	−	0.817765i	0.527355	−	0.849645i	\(-0.323184\pi\)
							−0.999492	+	0.0318804i	\(0.989850\pi\)
\(29\)	−5.51400	+	6.75343i	−1.02392	+	1.25408i	−0.0571730	+	0.998364i	\(0.518209\pi\)
							−0.966752	+	0.255716i	\(0.917689\pi\)
\(31\)	−0.479834	+	1.94676i	−0.0861807	+	0.349649i	−0.998163	−	0.0605798i	\(-0.980705\pi\)
							0.911983	+	0.410229i	\(0.134551\pi\)
\(37\)	−6.71204	+	1.94416i	−1.10345	+	0.319619i	−0.779467	−	0.626444i	\(-0.784510\pi\)
							−0.323985	+	0.946062i	\(0.605023\pi\)
\(41\)	0.805318	−	9.97566i	0.125770	−	1.55794i	−0.558995	−	0.829171i	\(-0.688813\pi\)
							0.684764	−	0.728765i	\(-0.259905\pi\)
\(43\)	1.15387	−	3.98361i	0.175963	−	0.607495i	−0.823244	−	0.567687i	\(-0.807838\pi\)
							0.999207	−	0.0398075i	\(-0.0126745\pi\)
\(47\)	−2.25339	+	2.54355i	−0.328690	+	0.371015i	−0.889504	−	0.456927i	\(-0.848950\pi\)
							0.560814	+	0.827942i	\(0.310488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bg.a.621.16	yes	1440
169.43	even	78	inner	845.2.bg.a.381.16	✓	1440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bg.a.381.16	✓	1440	169.43	even	78	inner
845.2.bg.a.621.16	yes	1440	1.1	even	1	trivial