Embedded newform 230.3.i.a.109.8

• Label: 230.3.i.a.109.8
• 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			109.8
Character	\(\chi\)	\(=\)	230.109
Dual form			230.3.i.a.19.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 - 0.587486i) q^{2} +(0.540708 - 1.84148i) q^{3} +(1.30972 + 1.51150i) q^{4} +(4.67844 - 1.76413i) q^{5} +(-1.77742 + 2.05125i) q^{6} +(-0.0656966 + 0.456930i) q^{7} +(-0.796860 - 2.71386i) q^{8} +(4.47258 + 2.87436i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

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	\(-0.805451\pi\)
0.999200	−	0.0399834i	\(-0.0127305\pi\)
\(5\)	4.67844	−	1.76413i	0.935689	−	0.352826i
							\(7\)	−0.0656966	+	0.456930i	−0.00938523	+	0.0652757i	−0.993976	−	0.109600i	\(-0.965043\pi\)
0.984591	+	0.174875i	\(0.0559522\pi\)
\(11\)	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	0.716921	−	0.697155i	\(-0.245551\pi\)
							0.884292	+	0.466935i	\(0.154642\pi\)
\(17\)	−2.37645	+	2.74257i	−0.139791	+	0.161328i	−0.821328	−	0.570456i	\(-0.806766\pi\)
							0.681537	+	0.731784i	\(0.261312\pi\)
\(19\)	−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	\(-0.941599\pi\)
							−0.574400	−	0.818575i	\(-0.694765\pi\)
\(41\)	−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	−0.101339	−	0.994852i	\(-0.532313\pi\)
							−0.623109	+	0.782135i	\(0.714131\pi\)
\(47\)			60.7751i			1.29309i	0.762877	+	0.646543i	\(0.223786\pi\)
							−0.762877	+	0.646543i	\(0.776214\pi\)

(See \(a_n\) instead)

Twists

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

    

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