Embedded newform 230.3.i.a.109.17

• Label: 230.3.i.a.109.17
• 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			109.17
Character	\(\chi\)	\(=\)	230.109
Dual form			230.3.i.a.19.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28641 + 0.587486i) q^{2} +(-0.540708 + 1.84148i) q^{3} +(1.30972 + 1.51150i) q^{4} +(-3.54821 + 3.52282i) q^{5} +(-1.77742 + 2.05125i) q^{6} +(0.0656966 - 0.456930i) q^{7} +(0.796860 + 2.71386i) q^{8} +(4.47258 + 2.87436i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.540708	+	1.84148i	−0.180236	+	0.613828i	0.818964	+	0.573845i	\(0.194549\pi\)
−0.999200	+	0.0399834i	\(0.987270\pi\)
\(5\)	−3.54821	+	3.52282i	−0.709641	+	0.704563i
							\(7\)	0.0656966	−	0.456930i	0.00938523	−	0.0652757i	−0.984591	−	0.174875i	\(-0.944048\pi\)
0.993976	+	0.109600i	\(0.0349569\pi\)
\(11\)	0.696679	−	0.318163i	0.0633344	−	0.0289239i	−0.383496	−	0.923543i	\(-0.625280\pi\)
							0.446830	+	0.894619i	\(0.352553\pi\)
\(13\)	−20.8158	+	2.99285i	−1.60121	+	0.230220i	−0.884292	−	0.466935i	\(-0.845358\pi\)
							−0.716921	+	0.697155i	\(0.754449\pi\)
\(17\)	2.37645	−	2.74257i	0.139791	−	0.161328i	−0.681537	−	0.731784i	\(-0.738688\pi\)
							0.821328	+	0.570456i	\(0.193234\pi\)
\(19\)	−22.1306	+	19.1762i	−1.16477	+	1.00928i	−0.165030	+	0.986288i	\(0.552772\pi\)
							−0.999736	+	0.0229870i	\(0.992682\pi\)
\(23\)	9.91399	+	20.7536i	0.431043	+	0.902331i
							\(29\)	26.9700	−	31.1250i	0.930000	−	1.07328i	−0.0671431	−	0.997743i	\(-0.521388\pi\)
0.997143	−	0.0755340i	\(-0.0240661\pi\)
\(31\)	35.8114	−	10.5152i	1.15521	−	0.339199i	0.352639	−	0.935760i	\(-0.385284\pi\)
							0.802568	+	0.596560i	\(0.203466\pi\)
\(37\)	57.6318	+	37.0377i	1.55762	+	1.00102i	0.983216	+	0.182444i	\(0.0584009\pi\)
							0.574400	+	0.818575i	\(0.305235\pi\)
\(41\)	−30.9951	+	19.9193i	−0.755978	+	0.485837i	−0.860983	−	0.508634i	\(-0.830151\pi\)
							0.105005	+	0.994472i	\(0.466514\pi\)
\(43\)	31.1513	+	9.14684i	0.724449	+	0.212717i	0.623109	−	0.782135i	\(-0.285869\pi\)
							0.101339	+	0.994852i	\(0.467687\pi\)
\(47\)		−	60.7751i		−	1.29309i	−0.762877	−	0.646543i	\(-0.776214\pi\)
							0.762877	−	0.646543i	\(-0.223786\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.i.a.109.17	yes	240
5.4	even	2	inner	230.3.i.a.109.8	yes	240
23.19	odd	22	inner	230.3.i.a.19.8	✓	240
115.19	odd	22	inner	230.3.i.a.19.17	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.8	✓	240	23.19	odd	22	inner
230.3.i.a.19.17	yes	240	115.19	odd	22	inner
230.3.i.a.109.8	yes	240	5.4	even	2	inner
230.3.i.a.109.17	yes	240	1.1	even	1	trivial