Embedded newform 98.4.g.a.11.5

• Label: 98.4.g.a.11.5
• Level: $98$
• Weight: $4$
• Character: 98.11
• Analytic conductor: $5.782$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	98.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.78218718056\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(7\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			11.5
Character	\(\chi\)	\(=\)	98.11
Dual form			98.4.g.a.9.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.46610 - 1.36035i) q^{2} +(2.92450 - 0.440798i) q^{3} +(0.298920 - 3.98882i) q^{4} +(6.56225 - 16.7203i) q^{5} +(3.68799 - 4.62459i) q^{6} +(9.69922 + 15.7774i) q^{7} +(-4.98792 - 6.25465i) q^{8} +(-17.4420 + 5.38016i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 14 q^{2} - 8 q^{3} + 28 q^{4} + 7 q^{5} + 10 q^{6} + 20 q^{7} + 112 q^{8} + 189 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.46610	−	1.36035i	0.518346	−	0.480955i
							\(3\)	2.92450	−	0.440798i	0.562821	−	0.0848316i	0.138531	−	0.990358i	\(-0.455762\pi\)
0.424290	+	0.905526i	\(0.360524\pi\)
\(5\)	6.56225	−	16.7203i	0.586945	−	1.49551i	−0.260952	−	0.965352i	\(-0.584037\pi\)
							0.847897	−	0.530160i	\(-0.177868\pi\)
\(7\)	9.69922	+	15.7774i	0.523709	+	0.851897i
							\(11\)	−8.26651	−	2.54988i	−0.226586	−	0.0698925i	0.179383	−	0.983779i	\(-0.442590\pi\)
−0.405969	+	0.913887i	\(0.633066\pi\)
\(13\)	−5.60095	−	24.5394i	−0.119494	−	0.523538i	−0.998875	−	0.0474193i	\(-0.984900\pi\)
							0.879381	−	0.476119i	\(-0.157957\pi\)
\(17\)	72.0309	−	49.1099i	1.02765	−	0.700641i	0.0726846	−	0.997355i	\(-0.476843\pi\)
							0.954966	+	0.296714i	\(0.0958910\pi\)
\(19\)	29.9680	+	51.9060i	0.361849	+	0.626740i	0.988265	−	0.152748i	\(-0.0488124\pi\)
							−0.626417	+	0.779488i	\(0.715479\pi\)
\(23\)	126.942	+	86.5474i	1.15083	+	0.784626i	0.979670	−	0.200617i	\(-0.0642946\pi\)
							0.171165	+	0.985242i	\(0.445247\pi\)
\(29\)	−64.8089	−	31.2103i	−0.414990	−	0.199849i	0.214726	−	0.976674i	\(-0.431114\pi\)
							−0.629715	+	0.776826i	\(0.716829\pi\)
\(31\)	−119.848	+	207.583i	−0.694365	+	1.20268i	0.276029	+	0.961149i	\(0.410981\pi\)
							−0.970394	+	0.241527i	\(0.922352\pi\)
\(37\)	20.7613	+	277.040i	0.0922469	+	1.23095i	0.830729	+	0.556676i	\(0.187924\pi\)
							−0.738483	+	0.674272i	\(0.764457\pi\)
\(41\)	150.059	+	188.168i	0.571593	+	0.716755i	0.980653	−	0.195752i	\(-0.0627149\pi\)
							−0.409060	+	0.912507i	\(0.634143\pi\)
\(43\)	75.2498	−	94.3603i	0.266872	−	0.334647i	−0.630281	−	0.776367i	\(-0.717060\pi\)
							0.897153	+	0.441720i	\(0.145632\pi\)
\(47\)	−106.588	+	98.8988i	−0.330796	+	0.306934i	−0.828031	−	0.560682i	\(-0.810539\pi\)
							0.497235	+	0.867616i	\(0.334349\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.g.a.11.5	yes	84
49.9	even	21	inner	98.4.g.a.9.5	✓	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.g.a.9.5	✓	84	49.9	even	21	inner
98.4.g.a.11.5	yes	84	1.1	even	1	trivial