Embedded newform 99.2.m.b.4.8

• Label: 99.2.m.b.4.8
• Level: $99$
• Weight: $2$
• Character: 99.4
• Analytic conductor: $0.791$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	99.m (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.790518980011\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(9\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			4.8
Character	\(\chi\)	\(=\)	99.4
Dual form			99.2.m.b.25.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.209960 - 1.99763i) q^{2} +(1.72741 + 0.126717i) q^{3} +(-1.99016 - 0.423021i) q^{4} +(-0.000569867 - 0.00542192i) q^{5} +(0.615820 - 3.42412i) q^{6} +(-2.56130 + 2.84461i) q^{7} +(-0.0214877 + 0.0661323i) q^{8} +(2.96789 + 0.437784i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - q^{2} - 12 q^{3} + 11 q^{4} - 8 q^{5} - 7 q^{6} - 2 q^{7} + 6 q^{8} - 22 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.209960	−	1.99763i	0.148464	−	1.41254i	−0.625952	−	0.779862i	\(-0.715289\pi\)
							0.774415	−	0.632677i	\(-0.218044\pi\)
\(3\)	1.72741	+	0.126717i	0.997320	+	0.0731601i
							\(5\)	−0.000569867	−	0.00542192i	−0.000254852	−	0.00242476i	0.994394	−	0.105741i	\(-0.0337214\pi\)
−0.994649	+	0.103316i	\(0.967055\pi\)
\(7\)	−2.56130	+	2.84461i	−0.968081	+	1.07516i	0.0290584	+	0.999578i	\(0.490749\pi\)
							−0.997139	+	0.0755851i	\(0.975918\pi\)
\(11\)	−3.21382	+	0.819382i	−0.969002	+	0.247053i
							\(13\)	−3.22798	+	1.43719i	−0.895281	+	0.398605i	−0.802201	−	0.597054i	\(-0.796338\pi\)
−0.0930798	+	0.995659i	\(0.529671\pi\)
\(17\)	3.70012	−	2.68830i	0.897412	−	0.652008i	−0.0403882	−	0.999184i	\(-0.512859\pi\)
							0.937800	+	0.347176i	\(0.112859\pi\)
\(19\)	−0.211664	+	0.651435i	−0.0485591	+	0.149449i	−0.972396	−	0.233337i	\(-0.925035\pi\)
							0.923837	+	0.382786i	\(0.125035\pi\)
\(23\)	0.760697	−	1.31757i	0.158616	−	0.274731i	−0.775754	−	0.631036i	\(-0.782630\pi\)
							0.934370	+	0.356304i	\(0.115963\pi\)
\(29\)	0.242622	−	0.269459i	0.0450538	−	0.0500373i	−0.720197	−	0.693770i	\(-0.755948\pi\)
							0.765250	+	0.643733i	\(0.222615\pi\)
\(31\)	3.91776	−	1.74430i	0.703650	−	0.313285i	−0.0235388	−	0.999723i	\(-0.507493\pi\)
							0.727188	+	0.686438i	\(0.240827\pi\)
\(37\)	1.90653	+	5.86771i	0.313432	+	0.964645i	0.976395	+	0.215993i	\(0.0692988\pi\)
							−0.662963	+	0.748652i	\(0.730701\pi\)
\(41\)	−4.46840	−	4.96266i	−0.697846	−	0.775037i	0.285185	−	0.958472i	\(-0.407945\pi\)
							−0.983032	+	0.183435i	\(0.941278\pi\)
\(43\)	−3.65705	−	6.33419i	−0.557694	−	0.965955i	−0.997688	−	0.0679544i	\(-0.978353\pi\)
							0.439994	−	0.898001i	\(-0.354981\pi\)
\(47\)	−6.47537	+	1.37638i	−0.944530	+	0.200766i	−0.654338	−	0.756202i	\(-0.727053\pi\)
							−0.290192	+	0.956968i	\(0.593719\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.2.m.b.4.8	✓	72
3.2	odd	2		297.2.n.b.37.2		72
9.2	odd	6		297.2.n.b.235.8		72
9.4	even	3		891.2.f.f.730.8		36
9.5	odd	6		891.2.f.e.730.2		36
9.7	even	3	inner	99.2.m.b.70.2	yes	72
11.3	even	5	inner	99.2.m.b.58.2	yes	72
11.5	even	5		1089.2.e.p.364.16		36
11.6	odd	10		1089.2.e.o.364.3		36
33.14	odd	10		297.2.n.b.91.8		72
99.5	odd	30		9801.2.a.cp.1.16		18
99.14	odd	30		891.2.f.e.487.2		36
99.16	even	15		1089.2.e.p.727.16		36
99.25	even	15	inner	99.2.m.b.25.8	yes	72
99.47	odd	30		297.2.n.b.289.2		72
99.49	even	15		9801.2.a.cm.1.3		18
99.50	even	30		9801.2.a.cn.1.3		18
99.58	even	15		891.2.f.f.487.8		36
99.61	odd	30		1089.2.e.o.727.3		36
99.94	odd	30		9801.2.a.co.1.16		18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.m.b.4.8	✓	72	1.1	even	1	trivial
99.2.m.b.25.8	yes	72	99.25	even	15	inner
99.2.m.b.58.2	yes	72	11.3	even	5	inner
99.2.m.b.70.2	yes	72	9.7	even	3	inner
297.2.n.b.37.2		72	3.2	odd	2
297.2.n.b.91.8		72	33.14	odd	10
297.2.n.b.235.8		72	9.2	odd	6
297.2.n.b.289.2		72	99.47	odd	30
891.2.f.e.487.2		36	99.14	odd	30
891.2.f.e.730.2		36	9.5	odd	6
891.2.f.f.487.8		36	99.58	even	15
891.2.f.f.730.8		36	9.4	even	3
1089.2.e.o.364.3		36	11.6	odd	10
1089.2.e.o.727.3		36	99.61	odd	30
1089.2.e.p.364.16		36	11.5	even	5
1089.2.e.p.727.16		36	99.16	even	15
9801.2.a.cm.1.3		18	99.49	even	15
9801.2.a.cn.1.3		18	99.50	even	30
9801.2.a.co.1.16		18	99.94	odd	30
9801.2.a.cp.1.16		18	99.5	odd	30