Embedded newform 230.4.g.b.71.1

• Label: 230.4.g.b.71.1
• Level: $230$
• Weight: $4$
• Character: 230.71
• 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			71.1
Character	\(\chi\)	\(=\)	230.71
Dual form			230.4.g.b.81.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68251 - 1.08128i) q^{2} +(-1.10005 - 7.65101i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-4.79746 - 1.40866i) q^{5} +(-6.42205 + 14.0623i) q^{6} +(-11.9276 + 13.7651i) q^{7} +(1.13852 - 7.91857i) q^{8} +(-31.4215 + 9.22619i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68251	−	1.08128i	−0.594856	−	0.382291i
							\(3\)	−1.10005	−	7.65101i	−0.211705	−	1.47244i	−0.767463	−	0.641093i	\(-0.778481\pi\)
0.555758	−	0.831344i	\(-0.312428\pi\)
\(5\)	−4.79746	−	1.40866i	−0.429098	−	0.125995i
							\(7\)	−11.9276	+	13.7651i	−0.644027	+	0.743247i	−0.980082	−	0.198595i	\(-0.936362\pi\)
0.336054	+	0.941843i	\(0.390907\pi\)
\(11\)	10.1501	−	6.52307i	0.278215	−	0.178798i	−0.394090	−	0.919072i	\(-0.628940\pi\)
							0.672306	+	0.740273i	\(0.265304\pi\)
\(13\)	59.7776	+	68.9870i	1.27533	+	1.47181i	0.809712	+	0.586828i	\(0.199623\pi\)
							0.465621	+	0.884984i	\(0.345831\pi\)
\(17\)	−23.5175	+	51.4961i	−0.335519	+	0.734684i	−0.999919	−	0.0126926i	\(-0.995960\pi\)
							0.664400	+	0.747377i	\(0.268687\pi\)
\(19\)	−18.4094	−	40.3109i	−0.222284	−	0.486735i	0.765329	−	0.643639i	\(-0.222576\pi\)
							−0.987614	+	0.156904i	\(0.949849\pi\)
\(23\)	−80.7033	+	75.1930i	−0.731643	+	0.681688i
							\(29\)	109.743	−	240.303i	0.702715	−	1.53873i	−0.133934	−	0.990990i	\(-0.542761\pi\)
0.836649	−	0.547740i	\(-0.184512\pi\)
\(31\)	−20.9445	+	145.672i	−0.121346	+	0.843983i	0.834687	+	0.550725i	\(0.185649\pi\)
							−0.956033	+	0.293258i	\(0.905260\pi\)
\(37\)	184.817	−	54.2670i	0.821180	−	0.241120i	0.155956	−	0.987764i	\(-0.450154\pi\)
							0.665224	+	0.746644i	\(0.268336\pi\)
\(41\)	217.535	+	63.8741i	0.828617	+	0.243304i	0.668423	−	0.743782i	\(-0.266970\pi\)
							0.160194	+	0.987086i	\(0.448788\pi\)
\(43\)	55.5704	+	386.501i	0.197079	+	1.37072i	0.812704	+	0.582677i	\(0.197995\pi\)
							−0.615624	+	0.788040i	\(0.711096\pi\)
\(47\)	−352.008			−1.09246			−0.546230	−	0.837635i	\(-0.683938\pi\)
							−0.546230	+	0.837635i	\(0.683938\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.b.71.1	✓	60
23.12	even	11	inner	230.4.g.b.81.1	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.71.1	✓	60	1.1	even	1	trivial
230.4.g.b.81.1	yes	60	23.12	even	11	inner