Embedded newform 230.3.k.a.223.8

• Label: 230.3.k.a.223.8
• 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			223.8
Character	\(\chi\)	\(=\)	230.223
Dual form			230.3.k.a.197.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.100889 + 1.41061i) q^{2} +(1.43938 + 0.313117i) q^{3} +(-1.97964 + 0.284630i) q^{4} +(-4.07429 - 2.89830i) q^{5} +(-0.296470 + 2.06199i) q^{6} +(4.10932 - 7.52566i) q^{7} +(-0.601225 - 2.76379i) q^{8} +(-6.21292 - 2.83735i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8}+O(q^{10}) \)

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
							\(3\)	1.43938	+	0.313117i	0.479793	+	0.104372i	0.445956	−	0.895055i	\(-0.352864\pi\)
0.0338366	+	0.999427i	\(0.489227\pi\)
\(5\)	−4.07429	−	2.89830i	−0.814858	−	0.579661i
							\(7\)	4.10932	−	7.52566i	0.587046	−	1.07509i	−0.400280	−	0.916393i	\(-0.631087\pi\)
0.987327	−	0.158702i	\(-0.0507309\pi\)
\(11\)	−2.29584	−	2.64954i	−0.208713	−	0.240867i	0.641736	−	0.766926i	\(-0.278215\pi\)
							−0.850448	+	0.526059i	\(0.823669\pi\)
\(13\)	−1.93592	−	3.54537i	−0.148917	−	0.272721i	0.792563	−	0.609790i	\(-0.208746\pi\)
							−0.941480	+	0.337069i	\(0.890564\pi\)
\(17\)	−16.3377	−	21.8246i	−0.961042	−	1.28380i	−0.958882	−	0.283804i	\(-0.908403\pi\)
							−0.00215956	−	0.999998i	\(-0.500687\pi\)
\(19\)	18.6310	−	2.67874i	0.980581	−	0.140986i	0.366661	−	0.930355i	\(-0.380501\pi\)
							0.613920	+	0.789368i	\(0.289592\pi\)
\(23\)	19.5572	+	12.1044i	0.850312	+	0.526279i
							\(29\)	−18.0860	−	2.60038i	−0.623656	−	0.0896682i	−0.176760	−	0.984254i	\(-0.556562\pi\)
−0.446896	+	0.894586i	\(0.647471\pi\)
\(31\)	2.62788	+	1.68884i	0.0847704	+	0.0544786i	0.582339	−	0.812946i	\(-0.302137\pi\)
							−0.497569	+	0.867424i	\(0.665774\pi\)
\(37\)	−2.73783	−	7.34042i	−0.0739955	−	0.198390i	0.894686	−	0.446695i	\(-0.147399\pi\)
							−0.968682	+	0.248305i	\(0.920126\pi\)
\(41\)	−7.26517	−	15.9085i	−0.177199	−	0.388012i	0.800103	−	0.599863i	\(-0.204778\pi\)
							−0.977302	+	0.211851i	\(0.932051\pi\)
\(43\)	41.1056	+	8.94198i	0.955944	+	0.207953i	0.663371	−	0.748290i	\(-0.269125\pi\)
							0.292573	+	0.956243i	\(0.405489\pi\)
\(47\)	−3.60338	+	3.60338i	−0.0766678	+	0.0766678i	−0.744401	−	0.667733i	\(-0.767265\pi\)
							0.667733	+	0.744401i	\(0.267265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.a.223.8	yes	240
5.2	odd	4	inner	230.3.k.a.177.8	yes	240
23.13	even	11	inner	230.3.k.a.13.8	✓	240
115.82	odd	44	inner	230.3.k.a.197.8	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.a.13.8	✓	240	23.13	even	11	inner
230.3.k.a.177.8	yes	240	5.2	odd	4	inner
230.3.k.a.197.8	yes	240	115.82	odd	44	inner
230.3.k.a.223.8	yes	240	1.1	even	1	trivial