Embedded newform 230.3.k.a.13.8

• Label: 230.3.k.a.13.8
• 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			13.8
Character	\(\chi\)	\(=\)	230.13
Dual form			230.3.k.a.177.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41061 - 0.100889i) q^{2} +(0.313117 + 1.43938i) q^{3} +(1.97964 + 0.284630i) q^{4} +(0.477701 + 4.97713i) q^{5} +(-0.296470 - 2.06199i) q^{6} +(7.52566 - 4.10932i) q^{7} +(-2.76379 - 0.601225i) q^{8} +(6.21292 - 2.83735i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	0.313117	+	1.43938i	0.104372	+	0.479793i	0.999427	+	0.0338366i	\(0.0107726\pi\)
−0.895055	+	0.445956i	\(0.852864\pi\)
\(5\)	0.477701	+	4.97713i	0.0955403	+	0.995426i
							\(7\)	7.52566	−	4.10932i	1.07509	−	0.587046i	0.158702	−	0.987327i	\(-0.449269\pi\)
0.916393	+	0.400280i	\(0.131087\pi\)
\(11\)	−2.29584	+	2.64954i	−0.208713	+	0.240867i	−0.850448	−	0.526059i	\(-0.823669\pi\)
							0.641736	+	0.766926i	\(0.278215\pi\)
\(13\)	−3.54537	−	1.93592i	−0.272721	−	0.148917i	0.337069	−	0.941480i	\(-0.390564\pi\)
							−0.609790	+	0.792563i	\(0.708746\pi\)
\(17\)	21.8246	+	16.3377i	1.28380	+	0.961042i	0.999998	+	0.00215956i	\(0.000687410\pi\)
							0.283804	+	0.958882i	\(0.408403\pi\)
\(19\)	−18.6310	−	2.67874i	−0.980581	−	0.140986i	−0.366661	−	0.930355i	\(-0.619499\pi\)
							−0.613920	+	0.789368i	\(0.710408\pi\)
\(23\)	12.1044	+	19.5572i	0.526279	+	0.850312i
							\(29\)	18.0860	−	2.60038i	0.623656	−	0.0896682i	0.176760	−	0.984254i	\(-0.443438\pi\)
0.446896	+	0.894586i	\(0.352529\pi\)
\(31\)	2.62788	−	1.68884i	0.0847704	−	0.0544786i	−0.497569	−	0.867424i	\(-0.665774\pi\)
							0.582339	+	0.812946i	\(0.302137\pi\)
\(37\)	7.34042	+	2.73783i	0.198390	+	0.0739955i	0.446695	−	0.894686i	\(-0.352601\pi\)
							−0.248305	+	0.968682i	\(0.579874\pi\)
\(41\)	−7.26517	+	15.9085i	−0.177199	+	0.388012i	−0.977302	−	0.211851i	\(-0.932051\pi\)
							0.800103	+	0.599863i	\(0.204778\pi\)
\(43\)	8.94198	+	41.1056i	0.207953	+	0.955944i	0.956243	+	0.292573i	\(0.0945114\pi\)
							−0.748290	+	0.663371i	\(0.769125\pi\)
\(47\)	−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.13.8	✓	240
5.2	odd	4	inner	230.3.k.a.197.8	yes	240
23.16	even	11	inner	230.3.k.a.223.8	yes	240
115.62	odd	44	inner	230.3.k.a.177.8	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.a.13.8	✓	240	1.1	even	1	trivial
230.3.k.a.177.8	yes	240	115.62	odd	44	inner
230.3.k.a.197.8	yes	240	5.2	odd	4	inner
230.3.k.a.223.8	yes	240	23.16	even	11	inner