Embedded newform 230.3.i.a.109.9

• Label: 230.3.i.a.109.9
• 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			109.9
Character	\(\chi\)	\(=\)	230.109
Dual form			230.3.i.a.19.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 - 0.587486i) q^{2} +(0.576935 - 1.96486i) q^{3} +(1.30972 + 1.51150i) q^{4} +(1.83597 + 4.65072i) q^{5} +(-1.89650 + 2.18868i) q^{6} +(-0.457318 + 3.18071i) q^{7} +(-0.796860 - 2.71386i) q^{8} +(4.04347 + 2.59858i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

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
							\(3\)	0.576935	−	1.96486i	0.192312	−	0.654953i	−0.805723	−	0.592292i	\(-0.798223\pi\)
0.998035	−	0.0626608i	\(-0.0199586\pi\)
\(5\)	1.83597	+	4.65072i	0.367194	+	0.930144i
							\(7\)	−0.457318	+	3.18071i	−0.0653311	+	0.454388i	0.930729	+	0.365709i	\(0.119173\pi\)
−0.996060	+	0.0886786i	\(0.971736\pi\)
\(11\)	−18.0924	+	8.26254i	−1.64477	+	0.751140i	−0.999913	−	0.0131666i	\(-0.995809\pi\)
							−0.644853	+	0.764306i	\(0.723082\pi\)
\(13\)	−14.1697	+	2.03730i	−1.08998	+	0.156716i	−0.663791	−	0.747918i	\(-0.731053\pi\)
							−0.426190	+	0.904634i	\(0.640144\pi\)
\(17\)	−8.56409	+	9.88349i	−0.503770	+	0.581382i	−0.949493	−	0.313789i	\(-0.898401\pi\)
							0.445722	+	0.895171i	\(0.352947\pi\)
\(19\)	−4.81195	+	4.16958i	−0.253261	+	0.219452i	−0.772233	−	0.635339i	\(-0.780860\pi\)
							0.518972	+	0.854791i	\(0.326315\pi\)
\(23\)	22.0451	+	6.55838i	0.958484	+	0.285147i
							\(29\)	−15.7498	+	18.1762i	−0.543096	+	0.626766i	−0.959260	−	0.282523i	\(-0.908829\pi\)
0.416165	+	0.909289i	\(0.363374\pi\)
\(31\)	50.2447	−	14.7532i	1.62080	−	0.475909i	0.659565	−	0.751647i	\(-0.270740\pi\)
							0.961233	+	0.275738i	\(0.0889222\pi\)
\(37\)	23.8013	+	15.2962i	0.643279	+	0.413410i	0.821205	−	0.570634i	\(-0.193302\pi\)
							−0.177926	+	0.984044i	\(0.556939\pi\)
\(41\)	30.5899	−	19.6590i	0.746096	−	0.479487i	−0.111530	−	0.993761i	\(-0.535575\pi\)
							0.857626	+	0.514274i	\(0.171939\pi\)
\(43\)	−77.6196	−	22.7912i	−1.80511	−	0.530027i	−0.806946	−	0.590626i	\(-0.798881\pi\)
							−0.998162	+	0.0605984i	\(0.980699\pi\)
\(47\)			61.7989i			1.31487i	0.753511	+	0.657435i	\(0.228359\pi\)
							−0.753511	+	0.657435i	\(0.771641\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.i.a.109.9	yes	240
5.4	even	2	inner	230.3.i.a.109.16	yes	240
23.19	odd	22	inner	230.3.i.a.19.16	yes	240
115.19	odd	22	inner	230.3.i.a.19.9	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.9	✓	240	115.19	odd	22	inner
230.3.i.a.19.16	yes	240	23.19	odd	22	inner
230.3.i.a.109.9	yes	240	1.1	even	1	trivial
230.3.i.a.109.16	yes	240	5.4	even	2	inner