Embedded newform 230.4.g.b.41.6

• Label: 230.4.g.b.41.6
• Level: $230$
• Weight: $4$
• Character: 230.41
• Analytic conductor: $13.570$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.5704393013\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			41.6
Character	\(\chi\)	\(=\)	230.41
Dual form			230.4.g.b.101.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.91899 + 0.563465i) q^{2} +(6.53866 - 7.54601i) q^{3} +(3.36501 + 2.16256i) q^{4} +(-0.711574 + 4.94911i) q^{5} +(16.7995 - 10.7964i) q^{6} +(14.9302 + 32.6926i) q^{7} +(5.23889 + 6.04600i) q^{8} +(-10.3458 - 71.9564i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.91899	+	0.563465i	0.678464	+	0.199215i
							\(3\)	6.53866	−	7.54601i	1.25837	−	1.45223i	0.419627	−	0.907697i	\(-0.362161\pi\)
0.838738	−	0.544535i	\(-0.183294\pi\)
\(5\)	−0.711574	+	4.94911i	−0.0636451	+	0.442662i
							\(7\)	14.9302	+	32.6926i	0.806156	+	1.76524i	0.623116	+	0.782130i	\(0.285867\pi\)
0.183040	+	0.983105i	\(0.441406\pi\)
\(11\)	46.9001	−	13.7711i	1.28554	−	0.377468i	0.433597	−	0.901107i	\(-0.357244\pi\)
							0.851940	+	0.523639i	\(0.175426\pi\)
\(13\)	6.38830	−	13.9884i	0.136292	−	0.298438i	−0.829163	−	0.559006i	\(-0.811183\pi\)
							0.965455	+	0.260569i	\(0.0839100\pi\)
\(17\)	−68.0814	+	43.7533i	−0.971304	+	0.624219i	−0.927104	−	0.374804i	\(-0.877710\pi\)
							−0.0441996	+	0.999023i	\(0.514074\pi\)
\(19\)	−34.4385	−	22.1323i	−0.415828	−	0.267236i	0.315955	−	0.948774i	\(-0.397675\pi\)
							−0.731783	+	0.681538i	\(0.761311\pi\)
\(23\)	−91.8947	+	61.0111i	−0.833103	+	0.553118i
							\(29\)	129.611	−	83.2958i	0.829935	−	0.533367i	−0.0553222	−	0.998469i	\(-0.517619\pi\)
0.885257	+	0.465102i	\(0.153982\pi\)
\(31\)	−119.954	−	138.434i	−0.694978	−	0.802048i	0.293086	−	0.956086i	\(-0.405318\pi\)
							−0.988065	+	0.154038i	\(0.950772\pi\)
\(37\)	−4.51784	−	31.4223i	−0.0200737	−	0.139616i	0.977320	−	0.211768i	\(-0.0679222\pi\)
							−0.997394	+	0.0721525i	\(0.977013\pi\)
\(41\)	54.5599	−	379.472i	0.207825	−	1.44545i	−0.572410	−	0.819967i	\(-0.693992\pi\)
							0.780235	−	0.625486i	\(-0.215099\pi\)
\(43\)	−25.7030	+	29.6629i	−0.0911552	+	0.105199i	−0.799495	−	0.600673i	\(-0.794899\pi\)
							0.708339	+	0.705872i	\(0.249445\pi\)
\(47\)	−145.863			−0.452687			−0.226343	−	0.974048i	\(-0.572677\pi\)
							−0.226343	+	0.974048i	\(0.572677\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.b.41.6	✓	60
23.9	even	11	inner	230.4.g.b.101.6	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.41.6	✓	60	1.1	even	1	trivial
230.4.g.b.101.6	yes	60	23.9	even	11	inner