Embedded newform 230.4.g.b.101.1

• Label: 230.4.g.b.101.1
• Level: $230$
• Weight: $4$
• Character: 230.101
• 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			101.1
Character	\(\chi\)	\(=\)	230.101
Dual form			230.4.g.b.41.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.91899 - 0.563465i) q^{2} +(-4.61458 - 5.32551i) q^{3} +(3.36501 - 2.16256i) q^{4} +(-0.711574 - 4.94911i) q^{5} +(-11.8561 - 7.61942i) q^{6} +(10.2472 - 22.4383i) q^{7} +(5.23889 - 6.04600i) q^{8} +(-3.22420 + 22.4248i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{11}\right)\)

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
							\(3\)	−4.61458	−	5.32551i	−0.888077	−	1.02489i	−0.999516	−	0.0311237i	\(-0.990091\pi\)
0.111439	−	0.993771i	\(-0.464454\pi\)
\(5\)	−0.711574	−	4.94911i	−0.0636451	−	0.442662i
							\(7\)	10.2472	−	22.4383i	0.553298	−	1.21155i	−0.401927	−	0.915672i	\(-0.631659\pi\)
0.955225	−	0.295881i	\(-0.0956133\pi\)
\(11\)	−2.47957	−	0.728066i	−0.0679652	−	0.0199564i	0.247573	−	0.968869i	\(-0.420367\pi\)
							−0.315538	+	0.948913i	\(0.602185\pi\)
\(13\)	−2.22139	−	4.86418i	−0.0473926	−	0.103775i	0.884455	−	0.466626i	\(-0.154531\pi\)
							−0.931847	+	0.362851i	\(0.881803\pi\)
\(17\)	−16.7875	−	10.7886i	−0.239503	−	0.153919i	0.415384	−	0.909646i	\(-0.363647\pi\)
							−0.654887	+	0.755727i	\(0.727284\pi\)
\(19\)	−22.8522	+	14.6862i	−0.275930	+	0.177329i	−0.671286	−	0.741199i	\(-0.734258\pi\)
							0.395356	+	0.918528i	\(0.370621\pi\)
\(23\)	63.7291	+	90.0311i	0.577758	+	0.816208i
							\(29\)	−187.429	−	120.453i	−1.20016	−	0.771296i	−0.221176	−	0.975234i	\(-0.570990\pi\)
−0.978984	+	0.203938i	\(0.934626\pi\)
\(31\)	−94.2181	+	108.734i	−0.545873	+	0.629971i	−0.959917	−	0.280286i	\(-0.909571\pi\)
							0.414043	+	0.910257i	\(0.364116\pi\)
\(37\)	54.9345	−	382.078i	0.244086	−	1.69765i	−0.387112	−	0.922033i	\(-0.626527\pi\)
							0.631198	−	0.775622i	\(-0.282564\pi\)
\(41\)	18.6734	+	129.877i	0.0711292	+	0.494715i	0.993980	+	0.109558i	\(0.0349435\pi\)
							−0.922851	+	0.385157i	\(0.874147\pi\)
\(43\)	214.474	+	247.516i	0.760626	+	0.877809i	0.995553	−	0.0942039i	\(-0.0300306\pi\)
							−0.234927	+	0.972013i	\(0.575485\pi\)
\(47\)	45.1359			0.140080			0.0700399	−	0.997544i	\(-0.477687\pi\)
							0.0700399	+	0.997544i	\(0.477687\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.b.101.1	yes	60
23.18	even	11	inner	230.4.g.b.41.1	✓	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.41.1	✓	60	23.18	even	11	inner
230.4.g.b.101.1	yes	60	1.1	even	1	trivial