Embedded newform 230.4.g.b.31.4

• Label: 230.4.g.b.31.4
• Level: $230$
• Weight: $4$
• Character: 230.31
• 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			31.4
Character	\(\chi\)	\(=\)	230.31
Dual form			230.4.g.b.141.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.284630 - 1.97964i) q^{2} +(-0.654798 - 1.43381i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(-3.27430 - 3.77875i) q^{5} +(-3.02480 + 0.888162i) q^{6} +(7.64693 + 4.91439i) q^{7} +(-3.32332 + 7.27706i) q^{8} +(16.0542 - 18.5275i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.284630	−	1.97964i	0.100632	−	0.699909i
							\(3\)	−0.654798	−	1.43381i	−0.126016	−	0.275936i	0.836100	−	0.548577i	\(-0.184830\pi\)
−0.962116	+	0.272641i	\(0.912103\pi\)
\(5\)	−3.27430	−	3.77875i	−0.292863	−	0.337981i
							\(7\)	7.64693	+	4.91439i	0.412896	+	0.265352i	0.730556	−	0.682853i	\(-0.239261\pi\)
−0.317661	+	0.948204i	\(0.602897\pi\)
\(11\)	−7.09604	−	49.3540i	−0.194503	−	1.35280i	−0.819906	−	0.572498i	\(-0.805974\pi\)
							0.625403	−	0.780302i	\(-0.284935\pi\)
\(13\)	−16.5482	+	10.6349i	−0.353050	+	0.226891i	−0.705129	−	0.709079i	\(-0.749111\pi\)
							0.352079	+	0.935970i	\(0.385475\pi\)
\(17\)	−60.9101	+	17.8848i	−0.868992	+	0.255159i	−0.685686	−	0.727897i	\(-0.740498\pi\)
							−0.183305	+	0.983056i	\(0.558680\pi\)
\(19\)	−136.808	−	40.1705i	−1.65189	−	0.485039i	−0.682566	−	0.730824i	\(-0.739136\pi\)
							−0.969324	+	0.245785i	\(0.920954\pi\)
\(23\)	56.0224	+	95.0184i	0.507891	+	0.861422i
							\(29\)	−67.9581	+	19.9543i	−0.435155	+	0.127773i	−0.491971	−	0.870612i	\(-0.663723\pi\)
0.0568158	+	0.998385i	\(0.481905\pi\)
\(31\)	−95.7530	+	209.670i	−0.554766	+	1.21477i	0.399754	+	0.916622i	\(0.369095\pi\)
							−0.954520	+	0.298146i	\(0.903632\pi\)
\(37\)	78.7276	−	90.8565i	0.349804	−	0.403695i	−0.553394	−	0.832920i	\(-0.686668\pi\)
							0.903198	+	0.429224i	\(0.141213\pi\)
\(41\)	8.01352	+	9.24810i	0.0305244	+	0.0352271i	0.770807	−	0.637069i	\(-0.219854\pi\)
							−0.740282	+	0.672296i	\(0.765308\pi\)
\(43\)	−207.436	−	454.222i	−0.735669	−	1.61089i	−0.790550	−	0.612397i	\(-0.790205\pi\)
							0.0548812	−	0.998493i	\(-0.482522\pi\)
\(47\)	476.063			1.47747			0.738734	−	0.673998i	\(-0.235424\pi\)
							0.738734	+	0.673998i	\(0.235424\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.g.b.31.4	✓	60
23.3	even	11	inner	230.4.g.b.141.4	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.g.b.31.4	✓	60	1.1	even	1	trivial
230.4.g.b.141.4	yes	60	23.3	even	11	inner