Embedded newform 230.3.k.b.223.4

• Label: 230.3.k.b.223.4
• Level: $230$
• Weight: $3$
• Character: 230.223
• 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			223.4
Character	\(\chi\)	\(=\)	230.223
Dual form			230.3.k.b.197.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.100889 - 1.41061i) q^{2} +(-2.42109 - 0.526675i) q^{3} +(-1.97964 + 0.284630i) q^{4} +(-4.73077 - 1.61858i) q^{5} +(-0.498673 + 3.46835i) q^{6} +(0.940042 - 1.72156i) q^{7} +(0.601225 + 2.76379i) q^{8} +(-2.60241 - 1.18848i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.100889	−	1.41061i	−0.0504444	−	0.705305i
							\(3\)	−2.42109	−	0.526675i	−0.807029	−	0.175558i	−0.209920	−	0.977719i	\(-0.567320\pi\)
−0.597109	+	0.802160i	\(0.703684\pi\)
\(5\)	−4.73077	−	1.61858i	−0.946154	−	0.323716i
							\(7\)	0.940042	−	1.72156i	0.134292	−	0.245937i	−0.801876	−	0.597490i	\(-0.796165\pi\)
0.936168	+	0.351553i	\(0.114346\pi\)
\(11\)	2.86593	+	3.30746i	0.260539	+	0.300678i	0.870915	−	0.491434i	\(-0.163527\pi\)
							−0.610376	+	0.792112i	\(0.708982\pi\)
\(13\)	6.26168	+	11.4674i	0.481668	+	0.882109i	0.999643	+	0.0267131i	\(0.00850405\pi\)
							−0.517975	+	0.855396i	\(0.673314\pi\)
\(17\)	7.48485	+	9.99859i	0.440285	+	0.588152i	0.964713	−	0.263303i	\(-0.0848118\pi\)
							−0.524428	+	0.851455i	\(0.675721\pi\)
\(19\)	0.916560	−	0.131781i	0.0482400	−	0.00693587i	−0.118153	−	0.992995i	\(-0.537697\pi\)
							0.166393	+	0.986060i	\(0.446788\pi\)
\(23\)	−10.9769	+	20.2116i	−0.477255	+	0.878765i
							\(29\)	−13.0868	−	1.88160i	−0.451270	−	0.0648829i	−0.0870673	−	0.996202i	\(-0.527750\pi\)
−0.364203	+	0.931320i	\(0.618659\pi\)
\(31\)	3.80822	+	2.44739i	0.122846	+	0.0789482i	0.600621	−	0.799534i	\(-0.294920\pi\)
							−0.477775	+	0.878482i	\(0.658557\pi\)
\(37\)	9.48112	+	25.4199i	0.256247	+	0.687024i	0.999831	+	0.0183816i	\(0.00585138\pi\)
							−0.743584	+	0.668642i	\(0.766876\pi\)
\(41\)	−25.0594	−	54.8725i	−0.611206	−	1.33835i	−0.921746	−	0.387794i	\(-0.873237\pi\)
							0.310540	−	0.950560i	\(-0.399490\pi\)
\(43\)	−32.3557	−	7.03854i	−0.752457	−	0.163687i	−0.180051	−	0.983657i	\(-0.557626\pi\)
							−0.572406	+	0.819970i	\(0.693990\pi\)
\(47\)	−36.2557	+	36.2557i	−0.771398	+	0.771398i	−0.978351	−	0.206953i	\(-0.933645\pi\)
							0.206953	+	0.978351i	\(0.433645\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.223.4	yes	240
5.2	odd	4	inner	230.3.k.b.177.4	yes	240
23.13	even	11	inner	230.3.k.b.13.4	✓	240
115.82	odd	44	inner	230.3.k.b.197.4	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.13.4	✓	240	23.13	even	11	inner
230.3.k.b.177.4	yes	240	5.2	odd	4	inner
230.3.k.b.197.4	yes	240	115.82	odd	44	inner
230.3.k.b.223.4	yes	240	1.1	even	1	trivial