Embedded newform 230.4.g.b.71.3

• Label: 230.4.g.b.71.3
• 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			71.3
Character	\(\chi\)	\(=\)	230.71
Dual form			230.4.g.b.81.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68251 - 1.08128i) q^{2} +(-0.259790 - 1.80688i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-4.79746 - 1.40866i) q^{5} +(-1.51665 + 3.32099i) q^{6} +(-3.50130 + 4.04072i) q^{7} +(1.13852 - 7.91857i) q^{8} +(22.7090 - 6.66796i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.259790	−	1.80688i	−0.0499966	−	0.347734i	−0.999430	−	0.0337649i	\(-0.989250\pi\)
0.949433	−	0.313969i	\(-0.101659\pi\)
\(5\)	−4.79746	−	1.40866i	−0.429098	−	0.125995i
							\(7\)	−3.50130	+	4.04072i	−0.189052	+	0.218178i	−0.842361	−	0.538914i	\(-0.818835\pi\)
0.653309	+	0.757092i	\(0.273380\pi\)
\(11\)	−20.4137	+	13.1191i	−0.559541	+	0.359595i	−0.789638	−	0.613572i	\(-0.789732\pi\)
							0.230097	+	0.973168i	\(0.426095\pi\)
\(13\)	−42.9889	−	49.6119i	−0.917153	−	1.05845i	−0.998093	−	0.0617317i	\(-0.980338\pi\)
							0.0809401	−	0.996719i	\(-0.474208\pi\)
\(17\)	−34.5899	+	75.7413i	−0.493487	+	1.08059i	0.485045	+	0.874489i	\(0.338803\pi\)
							−0.978532	+	0.206096i	\(0.933924\pi\)
\(19\)	30.0343	+	65.7660i	0.362650	+	0.794093i	0.999729	+	0.0232940i	\(0.00741537\pi\)
							−0.637079	+	0.770799i	\(0.719857\pi\)
\(23\)	47.3096	+	99.6434i	0.428901	+	0.903351i
							\(29\)	2.18701	−	4.78889i	0.0140041	−	0.0306646i	−0.902501	−	0.430688i	\(-0.858271\pi\)
0.916505	+	0.400024i	\(0.130998\pi\)
\(31\)	−11.1970	+	77.8766i	−0.0648721	+	0.451195i	0.931334	+	0.364167i	\(0.118646\pi\)
							−0.996206	+	0.0870284i	\(0.972263\pi\)
\(37\)	−280.098	+	82.2443i	−1.24454	+	0.365429i	−0.836718	−	0.547633i	\(-0.815529\pi\)
							−0.407819	+	0.913063i	\(0.633711\pi\)
\(41\)	394.291	+	115.774i	1.50190	+	0.440998i	0.926317	−	0.376744i	\(-0.122956\pi\)
							0.575584	+	0.817742i	\(0.304775\pi\)
\(43\)	62.0217	+	431.370i	0.219958	+	1.52984i	0.738183	+	0.674600i	\(0.235684\pi\)
							−0.518225	+	0.855245i	\(0.673407\pi\)
\(47\)	548.118			1.70109			0.850545	−	0.525902i	\(-0.176272\pi\)
							0.850545	+	0.525902i	\(0.176272\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.b.71.3	✓	60
23.12	even	11	inner	230.4.g.b.81.3	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.71.3	✓	60	1.1	even	1	trivial
230.4.g.b.81.3	yes	60	23.12	even	11	inner