Embedded newform 230.4.g.b.41.3

• Label: 230.4.g.b.41.3
• Level: $230$
• Weight: $4$
• Character: 230.41
• Analytic conductor: $13.570$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.5704393013\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			41.3
Character	\(\chi\)	\(=\)	230.41
Dual form			230.4.g.b.101.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.91899 + 0.563465i) q^{2} +(0.460964 - 0.531981i) q^{3} +(3.36501 + 2.16256i) q^{4} +(-0.711574 + 4.94911i) q^{5} +(1.18434 - 0.761126i) q^{6} +(-1.15601 - 2.53131i) q^{7} +(5.23889 + 6.04600i) q^{8} +(3.77198 + 26.2347i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.91899	+	0.563465i	0.678464	+	0.199215i
							\(3\)	0.460964	−	0.531981i	0.0887126	−	0.102380i	−0.709658	−	0.704547i	\(-0.751150\pi\)
0.798370	+	0.602167i	\(0.205696\pi\)
\(5\)	−0.711574	+	4.94911i	−0.0636451	+	0.442662i
							\(7\)	−1.15601	−	2.53131i	−0.0624187	−	0.136678i	0.875852	−	0.482580i	\(-0.160300\pi\)
−0.938271	+	0.345902i	\(0.887573\pi\)
\(11\)	27.7442	−	8.14644i	0.760472	−	0.223295i	0.121570	−	0.992583i	\(-0.461207\pi\)
							0.638902	+	0.769288i	\(0.279389\pi\)
\(13\)	−9.09159	+	19.9078i	−0.193966	+	0.424725i	−0.981478	−	0.191573i	\(-0.938641\pi\)
							0.787513	+	0.616298i	\(0.211368\pi\)
\(17\)	−60.0930	+	38.6194i	−0.857335	+	0.550976i	−0.893855	−	0.448357i	\(-0.852009\pi\)
							0.0365193	+	0.999333i	\(0.488373\pi\)
\(19\)	82.7987	+	53.2115i	0.999754	+	0.642503i	0.934722	−	0.355380i	\(-0.115649\pi\)
							0.0650320	+	0.997883i	\(0.479285\pi\)
\(23\)	70.5303	+	84.8085i	0.639417	+	0.768860i
							\(29\)	9.45320	−	6.07520i	0.0605315	−	0.0389013i	−0.510025	−	0.860160i	\(-0.670364\pi\)
0.570557	+	0.821258i	\(0.306728\pi\)
\(31\)	−21.5608	−	24.8825i	−0.124917	−	0.144162i	0.689846	−	0.723956i	\(-0.257678\pi\)
							−0.814763	+	0.579794i	\(0.803133\pi\)
\(37\)	1.10983	+	7.71903i	0.00493121	+	0.0342973i	0.992138	−	0.125145i	\(-0.0399396\pi\)
							−0.987207	+	0.159442i	\(0.949030\pi\)
\(41\)	12.6386	−	87.9033i	0.0481418	−	0.334834i	−0.951489	−	0.307681i	\(-0.900447\pi\)
							0.999631	−	0.0271525i	\(-0.00864397\pi\)
\(43\)	−157.323	+	181.560i	−0.557943	+	0.643900i	−0.962715	−	0.270517i	\(-0.912805\pi\)
							0.404773	+	0.914417i	\(0.367351\pi\)
\(47\)	397.098			1.23240			0.616198	−	0.787591i	\(-0.288672\pi\)
							0.616198	+	0.787591i	\(0.288672\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.b.41.3	✓	60
23.9	even	11	inner	230.4.g.b.101.3	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.41.3	✓	60	1.1	even	1	trivial
230.4.g.b.101.3	yes	60	23.9	even	11	inner