Embedded newform 230.3.k.b.3.6

• Label: 230.3.k.b.3.6
• Level: $230$
• Weight: $3$
• Character: 230.3
• Analytic conductor: $6.267$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	230.k (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26704608029\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			3.6
Character	\(\chi\)	\(=\)	230.3
Dual form			230.3.k.b.77.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13214 + 0.847507i) q^{2} +(-1.33310 + 0.497219i) q^{3} +(0.563465 + 1.91899i) q^{4} +(-1.71782 - 4.69565i) q^{5} +(-1.93064 - 0.566888i) q^{6} +(-0.622339 + 2.86085i) q^{7} +(-0.988434 + 2.65009i) q^{8} +(-5.27183 + 4.56806i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{8}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.13214	+	0.847507i	0.566068	+	0.423753i
							\(3\)	−1.33310	+	0.497219i	−0.444366	+	0.165740i	−0.561679	−	0.827355i	\(-0.689844\pi\)
0.117314	+	0.993095i	\(0.462572\pi\)
\(5\)	−1.71782	−	4.69565i	−0.343564	−	0.939129i
							\(7\)	−0.622339	+	2.86085i	−0.0889056	+	0.408692i	−0.999994	−	0.00351900i	\(-0.998880\pi\)
0.911088	+	0.412211i	\(0.135244\pi\)
\(11\)	−2.20005	+	15.3017i	−0.200004	+	1.39106i	0.604259	+	0.796788i	\(0.293469\pi\)
							−0.804263	+	0.594273i	\(0.797440\pi\)
\(13\)	4.86573	+	22.3674i	0.374287	+	1.72057i	0.650530	+	0.759480i	\(0.274547\pi\)
							−0.276244	+	0.961088i	\(0.589090\pi\)
\(17\)	−8.17637	+	4.46464i	−0.480963	+	0.262626i	−0.701390	−	0.712778i	\(-0.747437\pi\)
							0.220427	+	0.975403i	\(0.429255\pi\)
\(19\)	−3.74654	−	12.7595i	−0.197186	−	0.671554i	−0.997415	−	0.0718592i	\(-0.977107\pi\)
							0.800229	−	0.599695i	\(-0.204711\pi\)
\(23\)	−2.85592	−	22.8220i	−0.124170	−	0.992261i
							\(29\)	5.92995	−	20.1956i	0.204481	−	0.696398i	−0.791842	−	0.610726i	\(-0.790878\pi\)
0.996323	−	0.0856728i	\(-0.0273040\pi\)
\(31\)	14.6279	+	32.0306i	0.471867	+	1.03324i	0.984620	+	0.174709i	\(0.0558984\pi\)
							−0.512753	+	0.858536i	\(0.671374\pi\)
\(37\)	−11.5400	−	0.825358i	−0.311892	−	0.0223070i	−0.0854843	−	0.996340i	\(-0.527244\pi\)
							−0.226408	+	0.974033i	\(0.572698\pi\)
\(41\)	−32.5358	+	37.5483i	−0.793556	+	0.915813i	−0.998010	−	0.0630634i	\(-0.979913\pi\)
							0.204453	+	0.978876i	\(0.434458\pi\)
\(43\)	38.5976	−	14.3962i	0.897618	−	0.334794i	0.142046	−	0.989860i	\(-0.454632\pi\)
							0.755572	+	0.655066i	\(0.227359\pi\)
\(47\)	28.9224	−	28.9224i	0.615370	−	0.615370i	−0.328971	−	0.944340i	\(-0.606702\pi\)
							0.944340	+	0.328971i	\(0.106702\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.3.6	✓	240
5.2	odd	4	inner	230.3.k.b.187.7	yes	240
23.8	even	11	inner	230.3.k.b.123.7	yes	240
115.77	odd	44	inner	230.3.k.b.77.6	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.3.6	✓	240	1.1	even	1	trivial
230.3.k.b.77.6	yes	240	115.77	odd	44	inner
230.3.k.b.123.7	yes	240	23.8	even	11	inner
230.3.k.b.187.7	yes	240	5.2	odd	4	inner