Embedded newform 230.3.k.b.3.3

• Label: 230.3.k.b.3.3
• Level: $230$
• Weight: $3$
• Character: 230.3
• Analytic conductor: $6.267$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	230.k (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26704608029\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			3.3
Character	\(\chi\)	\(=\)	230.3
Dual form			230.3.k.b.77.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13214 + 0.847507i) q^{2} +(-3.78586 + 1.41205i) q^{3} +(0.563465 + 1.91899i) q^{4} +(-0.916589 + 4.91527i) q^{5} +(-5.48284 - 1.60991i) q^{6} +(0.413787 - 1.90215i) q^{7} +(-0.988434 + 2.65009i) q^{8} +(5.53710 - 4.79793i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{8}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.13214	+	0.847507i	0.566068	+	0.423753i
							\(3\)	−3.78586	+	1.41205i	−1.26195	+	0.470684i	−0.889316	−	0.457293i	\(-0.848819\pi\)
−0.372637	+	0.927977i	\(0.621546\pi\)
\(5\)	−0.916589	+	4.91527i	−0.183318	+	0.983054i
							\(7\)	0.413787	−	1.90215i	0.0591124	−	0.271735i	−0.938099	−	0.346368i	\(-0.887415\pi\)
0.997211	+	0.0746327i	\(0.0237784\pi\)
\(11\)	−1.34509	+	9.35529i	−0.122281	+	0.850481i	0.832681	+	0.553753i	\(0.186805\pi\)
							−0.954962	+	0.296728i	\(0.904104\pi\)
\(13\)	−4.27229	−	19.6394i	−0.328638	−	1.51072i	−0.784193	−	0.620517i	\(-0.786923\pi\)
							0.455555	−	0.890208i	\(-0.349441\pi\)
\(17\)	−28.0398	+	15.3109i	−1.64940	+	0.900639i	−0.660712	+	0.750640i	\(0.729745\pi\)
							−0.988686	+	0.150000i	\(0.952073\pi\)
\(19\)	−8.01522	−	27.2973i	−0.421854	−	1.43670i	−0.847007	−	0.531581i	\(-0.821598\pi\)
							0.425153	−	0.905121i	\(-0.360220\pi\)
\(23\)	20.8034	−	9.80910i	0.904496	−	0.426483i
							\(29\)	−9.52470	+	32.4381i	−0.328438	+	1.11856i	0.615418	+	0.788201i	\(0.288987\pi\)
−0.943856	+	0.330356i	\(0.892831\pi\)
\(31\)	14.7749	+	32.3524i	0.476608	+	1.04363i	0.983382	+	0.181548i	\(0.0581107\pi\)
							−0.506774	+	0.862079i	\(0.669162\pi\)
\(37\)	3.31324	+	0.236968i	0.0895471	+	0.00640453i	0.116041	−	0.993244i	\(-0.462980\pi\)
							−0.0264937	+	0.999649i	\(0.508434\pi\)
\(41\)	−42.3569	+	48.8825i	−1.03309	+	1.19225i	−0.0520160	+	0.998646i	\(0.516565\pi\)
							−0.981079	+	0.193609i	\(0.937981\pi\)
\(43\)	−32.3351	+	12.0604i	−0.751980	+	0.280474i	−0.696082	−	0.717962i	\(-0.745075\pi\)
							−0.0558982	+	0.998436i	\(0.517802\pi\)
\(47\)	−20.1736	+	20.1736i	−0.429225	+	0.429225i	−0.888364	−	0.459139i	\(-0.848158\pi\)
							0.459139	+	0.888364i	\(0.348158\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.3.3	✓	240
5.2	odd	4	inner	230.3.k.b.187.10	yes	240
23.8	even	11	inner	230.3.k.b.123.10	yes	240
115.77	odd	44	inner	230.3.k.b.77.3	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.3.3	✓	240	1.1	even	1	trivial
230.3.k.b.77.3	yes	240	115.77	odd	44	inner
230.3.k.b.123.10	yes	240	23.8	even	11	inner
230.3.k.b.187.10	yes	240	5.2	odd	4	inner