Embedded newform 230.4.j.a.9.12

• Label: 230.4.j.a.9.12
• Level: $230$
• Weight: $4$
• Character: 230.9
• Analytic conductor: $13.570$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	230.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.5704393013\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(36\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			9.12
Character	\(\chi\)	\(=\)	230.9
Dual form			230.4.j.a.179.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.563465 - 1.91899i) q^{2} +(2.02304 - 1.75298i) q^{3} +(-3.36501 + 2.16256i) q^{4} +(-11.1730 - 0.405517i) q^{5} +(-4.50385 - 2.89445i) q^{6} +(8.71258 + 3.97890i) q^{7} +(6.04600 + 5.23889i) q^{8} +(-2.82272 + 19.6325i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + 144 q^{4} + 32 q^{5} + 8 q^{6} + 224 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.563465	−	1.91899i	−0.199215	−	0.678464i
							\(3\)	2.02304	−	1.75298i	0.389335	−	0.337361i	−0.438096	−	0.898928i	\(-0.644347\pi\)
0.827431	+	0.561568i	\(0.189802\pi\)
\(5\)	−11.1730	−	0.405517i	−0.999342	−	0.0362705i
							\(7\)	8.71258	+	3.97890i	0.470435	+	0.214840i	0.636504	−	0.771274i	\(-0.280380\pi\)
−0.166069	+	0.986114i	\(0.553107\pi\)
\(11\)	47.7207	+	14.0121i	1.30803	+	0.384073i	0.860159	−	0.510025i	\(-0.170364\pi\)
							0.447872	+	0.894098i	\(0.352182\pi\)
\(13\)	17.0736	−	7.79723i	0.364258	−	0.166351i	−0.224874	−	0.974388i	\(-0.572197\pi\)
							0.589132	+	0.808037i	\(0.299470\pi\)
\(17\)	59.5577	−	92.6736i	0.849698	−	1.32216i	−0.0954240	−	0.995437i	\(-0.530421\pi\)
							0.945122	−	0.326719i	\(-0.105943\pi\)
\(19\)	−38.1274	+	24.5030i	−0.460369	+	0.295861i	−0.750193	−	0.661219i	\(-0.770040\pi\)
							0.289824	+	0.957080i	\(0.406403\pi\)
\(23\)	9.71673	−	109.875i	0.0880904	−	0.996112i
							\(29\)	2.40399	+	1.54495i	0.0153934	+	0.00989275i	0.548315	−	0.836272i	\(-0.315270\pi\)
−0.532921	+	0.846165i	\(0.678906\pi\)
\(31\)	−5.53826	+	6.39149i	−0.0320871	+	0.0370305i	−0.771566	−	0.636149i	\(-0.780526\pi\)
							0.739479	+	0.673179i	\(0.235072\pi\)
\(37\)	331.302	+	47.6340i	1.47205	+	0.211648i	0.831179	−	0.556005i	\(-0.187666\pi\)
							0.640866	+	0.767653i	\(0.278575\pi\)
\(41\)	41.7412	+	290.316i	0.158997	+	1.10585i	0.900486	+	0.434884i	\(0.143211\pi\)
							−0.741489	+	0.670965i	\(0.765880\pi\)
\(43\)	−50.5861	+	43.8331i	−0.179402	+	0.155453i	−0.739933	−	0.672681i	\(-0.765143\pi\)
							0.560530	+	0.828134i	\(0.310597\pi\)
\(47\)		−	396.732i		−	1.23126i	−0.788034	−	0.615632i	\(-0.788901\pi\)
							0.788034	−	0.615632i	\(-0.211099\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.j.a.9.12	✓	360
5.4	even	2	inner	230.4.j.a.9.25	yes	360
23.18	even	11	inner	230.4.j.a.179.25	yes	360
115.64	even	22	inner	230.4.j.a.179.12	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.j.a.9.12	✓	360	1.1	even	1	trivial
230.4.j.a.9.25	yes	360	5.4	even	2	inner
230.4.j.a.179.12	yes	360	115.64	even	22	inner
230.4.j.a.179.25	yes	360	23.18	even	11	inner