Embedded newform 230.3.k.b.13.4

• Label: 230.3.k.b.13.4
• Level: $230$
• Weight: $3$
• Character: 230.13
• 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			13.4
Character	\(\chi\)	\(=\)	230.13
Dual form			230.3.k.b.177.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41061 + 0.100889i) q^{2} +(-0.526675 - 2.42109i) q^{3} +(1.97964 + 0.284630i) q^{4} +(1.87475 + 4.63522i) q^{5} +(-0.498673 - 3.46835i) q^{6} +(1.72156 - 0.940042i) q^{7} +(2.76379 + 0.601225i) q^{8} +(2.60241 - 1.18848i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41061	+	0.100889i	0.705305	+	0.0504444i
							\(3\)	−0.526675	−	2.42109i	−0.175558	−	0.807029i	−0.977719	−	0.209920i	\(-0.932680\pi\)
0.802160	−	0.597109i	\(-0.203684\pi\)
\(5\)	1.87475	+	4.63522i	0.374951	+	0.927045i
							\(7\)	1.72156	−	0.940042i	0.245937	−	0.134292i	−0.351553	−	0.936168i	\(-0.614346\pi\)
0.597490	+	0.801876i	\(0.296165\pi\)
\(11\)	2.86593	−	3.30746i	0.260539	−	0.300678i	−0.610376	−	0.792112i	\(-0.708982\pi\)
							0.870915	+	0.491434i	\(0.163527\pi\)
\(13\)	11.4674	+	6.26168i	0.882109	+	0.481668i	0.855396	−	0.517975i	\(-0.173314\pi\)
							0.0267131	+	0.999643i	\(0.491496\pi\)
\(17\)	−9.99859	−	7.48485i	−0.588152	−	0.440285i	0.263303	−	0.964713i	\(-0.415188\pi\)
							−0.851455	+	0.524428i	\(0.824279\pi\)
\(19\)	−0.916560	−	0.131781i	−0.0482400	−	0.00693587i	0.118153	−	0.992995i	\(-0.462303\pi\)
							−0.166393	+	0.986060i	\(0.553212\pi\)
\(23\)	20.2116	−	10.9769i	0.878765	−	0.477255i
							\(29\)	13.0868	−	1.88160i	0.451270	−	0.0648829i	0.0870673	−	0.996202i	\(-0.472250\pi\)
0.364203	+	0.931320i	\(0.381341\pi\)
\(31\)	3.80822	−	2.44739i	0.122846	−	0.0789482i	−0.477775	−	0.878482i	\(-0.658557\pi\)
							0.600621	+	0.799534i	\(0.294920\pi\)
\(37\)	−25.4199	−	9.48112i	−0.687024	−	0.256247i	−0.0183816	−	0.999831i	\(-0.505851\pi\)
							−0.668642	+	0.743584i	\(0.733124\pi\)
\(41\)	−25.0594	+	54.8725i	−0.611206	+	1.33835i	0.310540	+	0.950560i	\(0.399490\pi\)
							−0.921746	+	0.387794i	\(0.873237\pi\)
\(43\)	−7.03854	−	32.3557i	−0.163687	−	0.752457i	−0.983657	−	0.180051i	\(-0.942374\pi\)
							0.819970	−	0.572406i	\(-0.193990\pi\)
\(47\)	−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.13.4	✓	240
5.2	odd	4	inner	230.3.k.b.197.4	yes	240
23.16	even	11	inner	230.3.k.b.223.4	yes	240
115.62	odd	44	inner	230.3.k.b.177.4	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.13.4	✓	240	1.1	even	1	trivial
230.3.k.b.177.4	yes	240	115.62	odd	44	inner
230.3.k.b.197.4	yes	240	5.2	odd	4	inner
230.3.k.b.223.4	yes	240	23.16	even	11	inner