Embedded newform 230.4.g.b.141.5

• Label: 230.4.g.b.141.5
• Level: $230$
• Weight: $4$
• Character: 230.141
• 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			141.5
Character	\(\chi\)	\(=\)	230.141
Dual form			230.4.g.b.31.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.284630 + 1.97964i) q^{2} +(2.61500 - 5.72604i) q^{3} +(-3.83797 + 1.12693i) q^{4} +(-3.27430 + 3.77875i) q^{5} +(12.0798 + 3.54696i) q^{6} +(-5.58110 + 3.58676i) q^{7} +(-3.32332 - 7.27706i) q^{8} +(-8.26814 - 9.54194i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{8}{11}\right)\)

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
							\(3\)	2.61500	−	5.72604i	0.503256	−	1.10198i	−0.472141	−	0.881523i	\(-0.656519\pi\)
0.975397	−	0.220455i	\(-0.0707541\pi\)
\(5\)	−3.27430	+	3.77875i	−0.292863	+	0.337981i
							\(7\)	−5.58110	+	3.58676i	−0.301351	+	0.193667i	−0.682574	−	0.730817i	\(-0.739139\pi\)
0.381223	+	0.924483i	\(0.375503\pi\)
\(11\)	−5.29972	+	36.8603i	−0.145266	+	1.01035i	0.778570	+	0.627558i	\(0.215945\pi\)
							−0.923836	+	0.382789i	\(0.874964\pi\)
\(13\)	66.5430	+	42.7646i	1.41967	+	0.912366i	0.999989	+	0.00469466i	\(0.00149436\pi\)
							0.419681	+	0.907672i	\(0.362142\pi\)
\(17\)	−31.1062	−	9.13360i	−0.443786	−	0.130307i	0.0522024	−	0.998637i	\(-0.483376\pi\)
							−0.495988	+	0.868329i	\(0.665194\pi\)
\(19\)	93.5011	−	27.4544i	1.12898	−	0.331498i	0.336673	−	0.941622i	\(-0.390698\pi\)
							0.792307	+	0.610123i	\(0.208880\pi\)
\(23\)	18.7390	+	108.701i	0.169885	+	0.985464i
							\(29\)	129.524	+	38.0316i	0.829377	+	0.243527i	0.668749	−	0.743488i	\(-0.266830\pi\)
0.160628	+	0.987015i	\(0.448648\pi\)
\(31\)	10.5196	+	23.0346i	0.0609474	+	0.133456i	0.937655	−	0.347568i	\(-0.112992\pi\)
							−0.876708	+	0.481024i	\(0.840265\pi\)
\(37\)	40.1582	+	46.3450i	0.178431	+	0.205921i	0.837919	−	0.545795i	\(-0.183772\pi\)
							−0.659488	+	0.751715i	\(0.729227\pi\)
\(41\)	123.380	−	142.388i	0.469969	−	0.542373i	−0.470434	−	0.882435i	\(-0.655903\pi\)
							0.940403	+	0.340062i	\(0.110448\pi\)
\(43\)	−103.311	+	226.220i	−0.366391	+	0.802284i	0.633209	+	0.773981i	\(0.281737\pi\)
							−0.999599	+	0.0283025i	\(0.990990\pi\)
\(47\)	−169.871			−0.527197			−0.263598	−	0.964633i	\(-0.584909\pi\)
							−0.263598	+	0.964633i	\(0.584909\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.b.141.5	yes	60
23.8	even	11	inner	230.4.g.b.31.5	✓	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.31.5	✓	60	23.8	even	11	inner
230.4.g.b.141.5	yes	60	1.1	even	1	trivial