Embedded newform 230.4.g.b.31.5

• Label: 230.4.g.b.31.5
• 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			31.5
Character	\(\chi\)	\(=\)	230.31
Dual form			230.4.g.b.141.5

$q$-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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

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

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
							\(3\)	2.61500	+	5.72604i	0.503256	+	1.10198i	0.975397	+	0.220455i	\(0.0707541\pi\)
−0.472141	+	0.881523i	\(0.656519\pi\)
\(5\)	−3.27430	−	3.77875i	−0.292863	−	0.337981i
							\(7\)	−5.58110	−	3.58676i	−0.301351	−	0.193667i	0.381223	−	0.924483i	\(-0.375503\pi\)
−0.682574	+	0.730817i	\(0.739139\pi\)
\(11\)	−5.29972	−	36.8603i	−0.145266	−	1.01035i	−0.923836	−	0.382789i	\(-0.874964\pi\)
							0.778570	−	0.627558i	\(-0.215945\pi\)
\(13\)	66.5430	−	42.7646i	1.41967	−	0.912366i	0.419681	−	0.907672i	\(-0.362142\pi\)
							0.999989	−	0.00469466i	\(-0.00149436\pi\)
\(17\)	−31.1062	+	9.13360i	−0.443786	+	0.130307i	−0.495988	−	0.868329i	\(-0.665194\pi\)
							0.0522024	+	0.998637i	\(0.483376\pi\)
\(19\)	93.5011	+	27.4544i	1.12898	+	0.331498i	0.792307	−	0.610123i	\(-0.208880\pi\)
							0.336673	+	0.941622i	\(0.390698\pi\)
\(23\)	18.7390	−	108.701i	0.169885	−	0.985464i
							\(29\)	129.524	−	38.0316i	0.829377	−	0.243527i	0.160628	−	0.987015i	\(-0.448648\pi\)
0.668749	+	0.743488i	\(0.266830\pi\)
\(31\)	10.5196	−	23.0346i	0.0609474	−	0.133456i	−0.876708	−	0.481024i	\(-0.840265\pi\)
							0.937655	+	0.347568i	\(0.112992\pi\)
\(37\)	40.1582	−	46.3450i	0.178431	−	0.205921i	−0.659488	−	0.751715i	\(-0.729227\pi\)
							0.837919	+	0.545795i	\(0.183772\pi\)
\(41\)	123.380	+	142.388i	0.469969	+	0.542373i	0.940403	−	0.340062i	\(-0.110448\pi\)
							−0.470434	+	0.882435i	\(0.655903\pi\)
\(43\)	−103.311	−	226.220i	−0.366391	−	0.802284i	−0.999599	−	0.0283025i	\(-0.990990\pi\)
							0.633209	−	0.773981i	\(-0.281737\pi\)
\(47\)	−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.31.5	✓	60
23.3	even	11	inner	230.4.g.b.141.5	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.31.5	✓	60	1.1	even	1	trivial
230.4.g.b.141.5	yes	60	23.3	even	11	inner