Embedded newform 230.4.g.b.71.2

• Label: 230.4.g.b.71.2
• 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			71.2
Character	\(\chi\)	\(=\)	230.71
Dual form			230.4.g.b.81.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68251 - 1.08128i) q^{2} +(-0.719442 - 5.00383i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-4.79746 - 1.40866i) q^{5} +(-4.20008 + 9.19689i) q^{6} +(9.79868 - 11.3083i) q^{7} +(1.13852 - 7.91857i) q^{8} +(1.38563 - 0.406859i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.719442	−	5.00383i	−0.138457	−	0.962987i	−0.934047	−	0.357151i	\(-0.883748\pi\)
0.795590	−	0.605835i	\(-0.207161\pi\)
\(5\)	−4.79746	−	1.40866i	−0.429098	−	0.125995i
							\(7\)	9.79868	−	11.3083i	0.529079	−	0.610590i	−0.426802	−	0.904345i	\(-0.640360\pi\)
0.955881	+	0.293756i	\(0.0949052\pi\)
\(11\)	33.3817	−	21.4531i	0.914996	−	0.588032i	0.00379453	−	0.999993i	\(-0.498792\pi\)
							0.911202	+	0.411960i	\(0.135156\pi\)
\(13\)	−17.7157	−	20.4450i	−0.377957	−	0.436185i	0.534619	−	0.845093i	\(-0.320455\pi\)
							−0.912576	+	0.408908i	\(0.865910\pi\)
\(17\)	1.56041	−	3.41682i	0.0222621	−	0.0487472i	−0.898174	−	0.439639i	\(-0.855106\pi\)
							0.920436	+	0.390892i	\(0.127834\pi\)
\(19\)	−38.8393	−	85.0461i	−0.468965	−	1.02689i	−0.985352	−	0.170535i	\(-0.945450\pi\)
							0.516386	−	0.856356i	\(-0.327277\pi\)
\(23\)	71.3386	−	84.1297i	0.646744	−	0.762707i
							\(29\)	−122.698	+	268.672i	−0.785672	+	1.72038i	−0.0970304	+	0.995281i	\(0.530934\pi\)
−0.688642	+	0.725101i	\(0.741793\pi\)
\(31\)	−34.3519	+	238.923i	−0.199025	+	1.38425i	0.608095	+	0.793864i	\(0.291934\pi\)
							−0.807120	+	0.590387i	\(0.798975\pi\)
\(37\)	153.661	−	45.1191i	0.682751	−	0.200474i	0.0780777	−	0.996947i	\(-0.475122\pi\)
							0.604673	+	0.796473i	\(0.293304\pi\)
\(41\)	−473.706	−	139.093i	−1.80440	−	0.529820i	−0.806306	−	0.591499i	\(-0.798536\pi\)
							−0.998096	+	0.0616791i	\(0.980354\pi\)
\(43\)	−40.6616	−	282.808i	−0.144205	−	1.00297i	−0.925483	−	0.378788i	\(-0.876341\pi\)
							0.781278	−	0.624183i	\(-0.214568\pi\)
\(47\)	−206.521			−0.640940			−0.320470	−	0.947259i	\(-0.603841\pi\)
							−0.320470	+	0.947259i	\(0.603841\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.b.71.2	✓	60
23.12	even	11	inner	230.4.g.b.81.2	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.71.2	✓	60	1.1	even	1	trivial
230.4.g.b.81.2	yes	60	23.12	even	11	inner