Embedded newform 230.4.g.b.81.5

• Label: 230.4.g.b.81.5
• Level: $230$
• Weight: $4$
• Character: 230.81
• 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			81.5
Character	\(\chi\)	\(=\)	230.81
Dual form			230.4.g.b.71.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68251 + 1.08128i) q^{2} +(0.839469 - 5.83864i) q^{3} +(1.66166 - 3.63853i) q^{4} +(-4.79746 + 1.40866i) q^{5} +(4.90080 + 10.7312i) q^{6} +(5.10412 + 5.89047i) q^{7} +(1.13852 + 7.91857i) q^{8} +(-7.47864 - 2.19593i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{10}{11}\right)\)

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
							\(3\)	0.839469	−	5.83864i	0.161556	−	1.12365i	−0.734146	−	0.678991i	\(-0.762417\pi\)
0.895702	−	0.444655i	\(-0.146674\pi\)
\(5\)	−4.79746	+	1.40866i	−0.429098	+	0.125995i
							\(7\)	5.10412	+	5.89047i	0.275597	+	0.318055i	0.876627	−	0.481171i	\(-0.159788\pi\)
−0.601030	+	0.799226i	\(0.705243\pi\)
\(11\)	32.9368	+	21.1672i	0.902801	+	0.580195i	0.907620	−	0.419793i	\(-0.137897\pi\)
							−0.00481864	+	0.999988i	\(0.501534\pi\)
\(13\)	4.12196	−	4.75700i	0.0879405	−	0.101489i	−0.710074	−	0.704127i	\(-0.751338\pi\)
							0.798014	+	0.602639i	\(0.205884\pi\)
\(17\)	53.3867	+	116.901i	0.761658	+	1.66780i	0.744206	+	0.667951i	\(0.232828\pi\)
							0.0174523	+	0.999848i	\(0.494444\pi\)
\(19\)	34.0453	−	74.5488i	0.411081	−	0.900141i	−0.584945	−	0.811073i	\(-0.698884\pi\)
							0.996026	−	0.0890678i	\(-0.0283888\pi\)
\(23\)	−28.9900	−	106.426i	−0.262818	−	0.964845i
							\(29\)	−41.6960	−	91.3016i	−0.266992	−	0.584630i	0.727888	−	0.685696i	\(-0.240502\pi\)
−0.994880	+	0.101066i	\(0.967775\pi\)
\(31\)	−16.5320	−	114.983i	−0.0957817	−	0.666176i	−0.979985	−	0.199073i	\(-0.936207\pi\)
							0.884203	−	0.467103i	\(-0.154702\pi\)
\(37\)	149.431	+	43.8768i	0.663952	+	0.194954i	0.596308	−	0.802756i	\(-0.296634\pi\)
							0.0676440	+	0.997710i	\(0.478452\pi\)
\(41\)	320.732	−	94.1754i	1.22170	−	0.358725i	0.393591	−	0.919285i	\(-0.371232\pi\)
							0.828113	+	0.560561i	\(0.189414\pi\)
\(43\)	−20.3111	+	141.267i	−0.0720328	+	0.500999i	0.921583	+	0.388182i	\(0.126897\pi\)
							−0.993616	+	0.112818i	\(0.964012\pi\)
\(47\)	51.3169			0.159263			0.0796313	−	0.996824i	\(-0.474626\pi\)
							0.0796313	+	0.996824i	\(0.474626\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.b.81.5	yes	60
23.2	even	11	inner	230.4.g.b.71.5	✓	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.71.5	✓	60	23.2	even	11	inner
230.4.g.b.81.5	yes	60	1.1	even	1	trivial