Embedded newform 230.4.g.a.31.3

• Label: 230.4.g.a.31.3
• Level: $230$
• Weight: $4$
• Character: 230.31
• Analytic conductor: $13.570$
• Analytic rank: $0$
• Dimension: $50$
• 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:	\(50\)
Relative dimension:	\(5\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			31.3
Character	\(\chi\)	\(=\)	230.31
Dual form			230.4.g.a.141.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.284630 + 1.97964i) q^{2} +(0.678933 + 1.48666i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(3.27430 + 3.77875i) q^{5} +(-3.13629 + 0.920899i) q^{6} +(-9.83778 - 6.32236i) q^{7} +(3.32332 - 7.27706i) q^{8} +(15.9320 - 18.3866i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 10 q^{2} - 5 q^{3} - 20 q^{4} + 25 q^{5} - 10 q^{6} - 19 q^{7} - 40 q^{8} - 126 q^{9}+O(q^{10}) \)

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
							\(3\)	0.678933	+	1.48666i	0.130661	+	0.286107i	0.963643	−	0.267192i	\(-0.0860958\pi\)
−0.832983	+	0.553299i	\(0.813369\pi\)
\(5\)	3.27430	+	3.77875i	0.292863	+	0.337981i
							\(7\)	−9.83778	−	6.32236i	−0.531190	−	0.341375i	0.247394	−	0.968915i	\(-0.420426\pi\)
−0.778585	+	0.627540i	\(0.784062\pi\)
\(11\)	−8.04090	−	55.9257i	−0.220402	−	1.53293i	−0.736522	−	0.676414i	\(-0.763533\pi\)
							0.516120	−	0.856516i	\(-0.327376\pi\)
\(13\)	8.98951	−	5.77721i	0.191788	−	0.123255i	−0.441223	−	0.897398i	\(-0.645455\pi\)
							0.633011	+	0.774143i	\(0.281819\pi\)
\(17\)	−14.1725	+	4.16141i	−0.202196	+	0.0593701i	−0.381262	−	0.924467i	\(-0.624510\pi\)
							0.179066	+	0.983837i	\(0.442692\pi\)
\(19\)	61.9714	+	18.1964i	0.748274	+	0.219713i	0.633569	−	0.773686i	\(-0.281589\pi\)
							0.114706	+	0.993400i	\(0.463408\pi\)
\(23\)	−73.9699	−	81.8257i	−0.670600	−	0.741819i
							\(29\)	66.8079	−	19.6166i	0.427790	−	0.125611i	−0.0607478	−	0.998153i	\(-0.519349\pi\)
0.488538	+	0.872543i	\(0.337530\pi\)
\(31\)	108.612	−	237.828i	0.629270	−	1.37791i	−0.279311	−	0.960201i	\(-0.590106\pi\)
							0.908581	−	0.417708i	\(-0.137167\pi\)
\(37\)	−20.0498	+	23.1387i	−0.0890856	+	0.102810i	−0.798542	−	0.601939i	\(-0.794395\pi\)
							0.709457	+	0.704749i	\(0.248940\pi\)
\(41\)	303.056	+	349.745i	1.15437	+	1.33222i	0.934197	+	0.356759i	\(0.116118\pi\)
							0.220177	+	0.975460i	\(0.429336\pi\)
\(43\)	−62.9738	−	137.893i	−0.223335	−	0.489036i	0.764484	−	0.644643i	\(-0.222994\pi\)
							−0.987819	+	0.155607i	\(0.950267\pi\)
\(47\)	310.848			0.964721			0.482360	−	0.875973i	\(-0.339780\pi\)
							0.482360	+	0.875973i	\(0.339780\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.a.31.3	✓	50
23.3	even	11	inner	230.4.g.a.141.3	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.a.31.3	✓	50	1.1	even	1	trivial
230.4.g.a.141.3	yes	50	23.3	even	11	inner