Embedded newform 230.3.i.a.109.7

• Label: 230.3.i.a.109.7
• Level: $230$
• Weight: $3$
• Character: 230.109
• Analytic conductor: $6.267$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	230.i (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26704608029\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			109.7
Character	\(\chi\)	\(=\)	230.109
Dual form			230.3.i.a.19.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 - 0.587486i) q^{2} +(0.271363 - 0.924178i) q^{3} +(1.30972 + 1.51150i) q^{4} +(-2.58664 - 4.27894i) q^{5} +(-0.892026 + 1.02945i) q^{6} +(-1.64158 + 11.4174i) q^{7} +(-0.796860 - 2.71386i) q^{8} +(6.79082 + 4.36419i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28641	−	0.587486i	−0.643207	−	0.293743i
							\(3\)	0.271363	−	0.924178i	0.0904544	−	0.308059i	−0.901822	−	0.432108i	\(-0.857770\pi\)
0.992276	+	0.124049i	\(0.0395880\pi\)
\(5\)	−2.58664	−	4.27894i	−0.517328	−	0.855788i
							\(7\)	−1.64158	+	11.4174i	−0.234511	+	1.63106i	0.443687	+	0.896182i	\(0.353670\pi\)
−0.678198	+	0.734879i	\(0.737239\pi\)
\(11\)	12.0406	−	5.49877i	1.09460	−	0.499888i	0.215492	−	0.976506i	\(-0.430864\pi\)
							0.879111	+	0.476617i	\(0.158137\pi\)
\(13\)	2.01624	−	0.289891i	0.155095	−	0.0222993i	−0.0643297	−	0.997929i	\(-0.520491\pi\)
							0.219425	+	0.975629i	\(0.429582\pi\)
\(17\)	−1.75412	+	2.02436i	−0.103184	+	0.119080i	−0.804992	−	0.593285i	\(-0.797831\pi\)
							0.701809	+	0.712365i	\(0.252376\pi\)
\(19\)	15.6925	−	13.5976i	0.825920	−	0.715663i	−0.135492	−	0.990778i	\(-0.543261\pi\)
							0.961411	+	0.275115i	\(0.0887160\pi\)
\(23\)	14.0502	−	18.2097i	0.610879	−	0.791724i
							\(29\)	−1.93917	+	2.23792i	−0.0668679	+	0.0771697i	−0.788198	−	0.615422i	\(-0.788986\pi\)
0.721330	+	0.692592i	\(0.243531\pi\)
\(31\)	11.8816	−	3.48876i	0.383279	−	0.112541i	−0.0844166	−	0.996431i	\(-0.526903\pi\)
							0.467695	+	0.883890i	\(0.345084\pi\)
\(37\)	51.5349	+	33.1195i	1.39284	+	0.895121i	0.999703	−	0.0243610i	\(-0.00775511\pi\)
							0.393132	+	0.919482i	\(0.371391\pi\)
\(41\)	49.8663	−	32.0471i	1.21625	−	0.781637i	0.234557	−	0.972102i	\(-0.424636\pi\)
							0.981694	+	0.190466i	\(0.0609997\pi\)
\(43\)	−13.5483	−	3.97814i	−0.315077	−	0.0925150i	0.120371	−	0.992729i	\(-0.461592\pi\)
							−0.435448	+	0.900214i	\(0.643410\pi\)
\(47\)		−	0.939964i		−	0.0199992i	−0.999950	−	0.00999961i	\(-0.996817\pi\)
							0.999950	−	0.00999961i	\(-0.00318303\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.i.a.109.7	yes	240
5.4	even	2	inner	230.3.i.a.109.18	yes	240
23.19	odd	22	inner	230.3.i.a.19.18	yes	240
115.19	odd	22	inner	230.3.i.a.19.7	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.7	✓	240	115.19	odd	22	inner
230.3.i.a.19.18	yes	240	23.19	odd	22	inner
230.3.i.a.109.7	yes	240	1.1	even	1	trivial
230.3.i.a.109.18	yes	240	5.4	even	2	inner