Embedded newform 825.2.ct.b.707.10

• Label: 825.2.ct.b.707.10
• Level: $825$
• Weight: $2$
• Character: 825.707
• Analytic conductor: $6.588$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.ct (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(20\) over \(\Q(\zeta_{20})\)
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			707.10
Character	\(\chi\)	\(=\)	825.707
Dual form			825.2.ct.b.818.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0347028 + 0.219105i) q^{2} +(1.31059 - 1.13241i) q^{3} +(1.85531 + 0.602827i) q^{4} +(0.202636 + 0.326455i) q^{6} +(2.12167 - 1.08105i) q^{7} +(-0.397890 + 0.780903i) q^{8} +(0.435295 - 2.96825i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{3} - 12 q^{6} + 20 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\)	\(376\)	\(551\)	\(727\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0347028	+	0.219105i	−0.0245386	+	0.154931i	−0.996915	−	0.0784872i	\(-0.974991\pi\)
							0.972377	+	0.233418i	\(0.0749910\pi\)
\(3\)	1.31059	−	1.13241i	0.756670	−	0.653797i
							\(5\)		0			0
\(7\)	2.12167	−	1.08105i	0.801918	−	0.408597i								−0.00446609	−	0.999990i	\(-0.501422\pi\)
							0.806384	+	0.591393i	\(0.201422\pi\)
\(11\)	−1.73437	+	2.82700i	−0.522934	+	0.852373i
							\(13\)	0.570086	−	3.59938i	0.158113	−	0.998289i	−0.773224	−	0.634133i	\(-0.781357\pi\)
0.931338	−	0.364156i	\(-0.118643\pi\)
\(17\)	−4.30376	+	0.681648i	−1.04381	+	0.165324i	−0.654718	−	0.755873i	\(-0.727213\pi\)
							−0.389096	+	0.921197i	\(0.627213\pi\)
\(19\)	6.07579	−	1.97414i	1.39388	−	0.452899i	0.486674	−	0.873584i	\(-0.338210\pi\)
							0.907207	+	0.420684i	\(0.138210\pi\)
\(23\)	1.82231	+	1.82231i	0.379978	+	0.379978i	0.871094	−	0.491116i	\(-0.163411\pi\)
							−0.491116	+	0.871094i	\(0.663411\pi\)
\(29\)	1.57307	−	4.84140i	0.292111	−	0.899026i	−0.692065	−	0.721835i	\(-0.743299\pi\)
							0.984177	−	0.177191i	\(-0.0567010\pi\)
\(31\)	−7.59054	+	5.51485i	−1.36330	+	0.990496i	−0.365073	+	0.930979i	\(0.618956\pi\)
							−0.998227	+	0.0595172i	\(0.981044\pi\)
\(37\)	−2.54716	+	1.29784i	−0.418750	+	0.213364i	−0.650653	−	0.759375i	\(-0.725505\pi\)
							0.231903	+	0.972739i	\(0.425505\pi\)
\(41\)	8.60356	−	2.79547i	1.34365	−	0.436578i	0.453099	−	0.891460i	\(-0.350319\pi\)
							0.890552	+	0.454882i	\(0.150319\pi\)
\(43\)	−5.81955	+	5.81955i	−0.887474	+	0.887474i	−0.994280	−	0.106806i	\(-0.965938\pi\)
							0.106806	+	0.994280i	\(0.465938\pi\)
\(47\)	2.20435	+	1.12317i	0.321538	+	0.163832i	0.607308	−	0.794466i	\(-0.292249\pi\)
							−0.285770	+	0.958298i	\(0.592249\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.ct.b.707.10		160
3.2	odd	2	inner	825.2.ct.b.707.11		160
5.2	odd	4		165.2.v.a.113.10	yes	160
5.3	odd	4	inner	825.2.ct.b.443.11		160
5.4	even	2		165.2.v.a.47.11	yes	160
11.4	even	5	inner	825.2.ct.b.257.10		160
15.2	even	4		165.2.v.a.113.11	yes	160
15.8	even	4	inner	825.2.ct.b.443.10		160
15.14	odd	2		165.2.v.a.47.10	✓	160
33.26	odd	10	inner	825.2.ct.b.257.11		160
55.4	even	10		165.2.v.a.92.11	yes	160
55.37	odd	20		165.2.v.a.158.10	yes	160
55.48	odd	20	inner	825.2.ct.b.818.11		160
165.59	odd	10		165.2.v.a.92.10	yes	160
165.92	even	20		165.2.v.a.158.11	yes	160
165.158	even	20	inner	825.2.ct.b.818.10		160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.v.a.47.10	✓	160	15.14	odd	2
165.2.v.a.47.11	yes	160	5.4	even	2
165.2.v.a.92.10	yes	160	165.59	odd	10
165.2.v.a.92.11	yes	160	55.4	even	10
165.2.v.a.113.10	yes	160	5.2	odd	4
165.2.v.a.113.11	yes	160	15.2	even	4
165.2.v.a.158.10	yes	160	55.37	odd	20
165.2.v.a.158.11	yes	160	165.92	even	20
825.2.ct.b.257.10		160	11.4	even	5	inner
825.2.ct.b.257.11		160	33.26	odd	10	inner
825.2.ct.b.443.10		160	15.8	even	4	inner
825.2.ct.b.443.11		160	5.3	odd	4	inner
825.2.ct.b.707.10		160	1.1	even	1	trivial
825.2.ct.b.707.11		160	3.2	odd	2	inner
825.2.ct.b.818.10		160	165.158	even	20	inner
825.2.ct.b.818.11		160	55.48	odd	20	inner