Embedded newform 230.4.g.b.31.1

• Label: 230.4.g.b.31.1
• Level: $230$
• Weight: $4$
• Character: 230.31
• 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			31.1
Character	\(\chi\)	\(=\)	230.31
Dual form			230.4.g.b.141.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.284630 - 1.97964i) q^{2} +(-3.91607 - 8.57500i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(-3.27430 - 3.77875i) q^{5} +(-18.0901 + 5.31172i) q^{6} +(13.7433 + 8.83226i) q^{7} +(-3.32332 + 7.27706i) q^{8} +(-40.5137 + 46.7553i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.284630	−	1.97964i	0.100632	−	0.699909i
							\(3\)	−3.91607	−	8.57500i	−0.753648	−	1.65026i	−0.759697	−	0.650277i	\(-0.774653\pi\)
0.00604904	−	0.999982i	\(-0.498075\pi\)
\(5\)	−3.27430	−	3.77875i	−0.292863	−	0.337981i
							\(7\)	13.7433	+	8.83226i	0.742066	+	0.476897i	0.856249	−	0.516563i	\(-0.172789\pi\)
−0.114183	+	0.993460i	\(0.536425\pi\)
\(11\)	−1.47476	−	10.2572i	−0.0404233	−	0.281150i	0.959577	−	0.281447i	\(-0.0908145\pi\)
							−1.00000		0.000297315i	\(0.999905\pi\)
\(13\)	−48.3807	+	31.0924i	−1.03218	+	0.663344i	−0.943041	−	0.332675i	\(-0.892049\pi\)
							−0.0891417	+	0.996019i	\(0.528412\pi\)
\(17\)	−52.8175	+	15.5086i	−0.753537	+	0.221259i	−0.635872	−	0.771795i	\(-0.719359\pi\)
							−0.117666	+	0.993053i	\(0.537541\pi\)
\(19\)	59.8281	+	17.5671i	0.722395	+	0.212114i	0.622206	−	0.782854i	\(-0.286237\pi\)
							0.100190	+	0.994968i	\(0.468055\pi\)
\(23\)	−110.080	+	7.02854i	−0.997968	+	0.0637197i
							\(29\)	238.597	−	70.0584i	1.52781	−	0.448604i	0.593427	−	0.804887i	\(-0.297774\pi\)
0.934378	+	0.356283i	\(0.115956\pi\)
\(31\)	65.6512	−	143.756i	0.380365	−	0.832883i	−0.618525	−	0.785765i	\(-0.712269\pi\)
							0.998889	−	0.0471172i	\(-0.0150034\pi\)
\(37\)	−233.984	+	270.032i	−1.03964	+	1.19981i	−0.0601774	+	0.998188i	\(0.519167\pi\)
							−0.979463	+	0.201622i	\(0.935379\pi\)
\(41\)	−320.284	−	369.627i	−1.22000	−	1.40795i	−0.884907	−	0.465768i	\(-0.845778\pi\)
							−0.335092	−	0.942185i	\(-0.608767\pi\)
\(43\)	53.6591	+	117.497i	0.190301	+	0.416701i	0.980600	−	0.196021i	\(-0.0628020\pi\)
							−0.790299	+	0.612722i	\(0.790075\pi\)
\(47\)	−126.988			−0.394109			−0.197054	−	0.980393i	\(-0.563138\pi\)
							−0.197054	+	0.980393i	\(0.563138\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.b.31.1	✓	60
23.3	even	11	inner	230.4.g.b.141.1	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.31.1	✓	60	1.1	even	1	trivial
230.4.g.b.141.1	yes	60	23.3	even	11	inner