Embedded newform 230.4.g.b.41.5

• Label: 230.4.g.b.41.5
• 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			41.5
Character	\(\chi\)	\(=\)	230.41
Dual form			230.4.g.b.101.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.91899 + 0.563465i) q^{2} +(3.92901 - 4.53431i) q^{3} +(3.36501 + 2.16256i) q^{4} +(-0.711574 + 4.94911i) q^{5} +(10.0946 - 6.48743i) q^{6} +(-9.52105 - 20.8482i) q^{7} +(5.23889 + 6.04600i) q^{8} +(-1.28042 - 8.90551i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9}+O(q^{10}) \)

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
							\(3\)	3.92901	−	4.53431i	0.756137	−	0.872629i	−0.239011	−	0.971017i	\(-0.576823\pi\)
0.995148	+	0.0983878i	\(0.0313686\pi\)
\(5\)	−0.711574	+	4.94911i	−0.0636451	+	0.442662i
							\(7\)	−9.52105	−	20.8482i	−0.514089	−	1.12570i	−0.971629	−	0.236508i	\(-0.923997\pi\)
0.457541	−	0.889189i	\(-0.348730\pi\)
\(11\)	13.9340	−	4.09140i	0.381934	−	0.112146i	−0.0851292	−	0.996370i	\(-0.527130\pi\)
							0.467063	+	0.884224i	\(0.345312\pi\)
\(13\)	37.0096	−	81.0397i	0.789586	−	1.72895i	0.111752	−	0.993736i	\(-0.464354\pi\)
							0.677834	−	0.735215i	\(-0.262919\pi\)
\(17\)	77.8174	−	50.0102i	1.11021	−	0.713486i	0.148868	−	0.988857i	\(-0.452437\pi\)
							0.961338	+	0.275371i	\(0.0888007\pi\)
\(19\)	53.9221	+	34.6536i	0.651083	+	0.418426i	0.824062	−	0.566500i	\(-0.191703\pi\)
							−0.172979	+	0.984926i	\(0.555339\pi\)
\(23\)	−110.267	+	2.84616i	−0.999667	+	0.0258029i
							\(29\)	−107.264	+	68.9345i	−0.686844	+	0.441408i	−0.836962	−	0.547260i	\(-0.815671\pi\)
0.150119	+	0.988668i	\(0.452034\pi\)
\(31\)	−66.3076	−	76.5231i	−0.384168	−	0.443353i	0.530423	−	0.847733i	\(-0.322033\pi\)
							−0.914591	+	0.404380i	\(0.867487\pi\)
\(37\)	29.8403	+	207.544i	0.132587	+	0.922161i	0.942165	+	0.335149i	\(0.108787\pi\)
							−0.809578	+	0.587012i	\(0.800304\pi\)
\(41\)	−72.2486	+	502.500i	−0.275203	+	1.91408i	0.115026	+	0.993362i	\(0.463305\pi\)
							−0.390229	+	0.920718i	\(0.627604\pi\)
\(43\)	308.570	−	356.109i	1.09434	−	1.26293i	0.131947	−	0.991257i	\(-0.457877\pi\)
							0.962389	−	0.271675i	\(-0.0875775\pi\)
\(47\)	−581.613			−1.80504			−0.902521	−	0.430647i	\(-0.858285\pi\)
							−0.902521	+	0.430647i	\(0.858285\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.b.41.5	✓	60
23.9	even	11	inner	230.4.g.b.101.5	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.41.5	✓	60	1.1	even	1	trivial
230.4.g.b.101.5	yes	60	23.9	even	11	inner