Embedded newform 600.2.bq.a.67.32

• Label: 600.2.bq.a.67.32
• Level: $600$
• Weight: $2$
• Character: 600.67
• Analytic conductor: $4.791$
• Analytic rank: $0$
• Dimension: $480$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.bq (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.79102412128\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			67.32
Character	\(\chi\)	\(=\)	600.67
Dual form			600.2.bq.a.403.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.137625 + 1.40750i) q^{2} +(-0.156434 + 0.987688i) q^{3} +(-1.96212 + 0.387414i) q^{4} +(-1.86666 - 1.23110i) q^{5} +(-1.41170 - 0.0842514i) q^{6} +(2.13082 - 2.13082i) q^{7} +(-0.815321 - 2.70837i) q^{8} +(-0.951057 - 0.309017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 12 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\)	\(151\)	\(301\)	\(401\)	\(577\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.137625	+	1.40750i	0.0973153	+	0.995254i
							\(3\)	−0.156434	+	0.987688i	−0.0903175	+	0.570242i
\(5\)	−1.86666	−	1.23110i	−0.834794	−	0.550563i
							\(7\)	2.13082	−	2.13082i	0.805375	−	0.805375i	−0.178555	−	0.983930i	\(-0.557142\pi\)
0.983930	+	0.178555i	\(0.0571423\pi\)
\(11\)	−0.0873587	−	0.268863i	−0.0263397	−	0.0810651i	0.937023	−	0.349269i	\(-0.113570\pi\)
							−0.963362	+	0.268204i	\(0.913570\pi\)
\(13\)	2.99402	−	1.52553i	0.830391	−	0.423105i	0.0135080	−	0.999909i	\(-0.495700\pi\)
							0.816883	+	0.576803i	\(0.195700\pi\)
\(17\)	−0.216929	−	1.36963i	−0.0526129	−	0.332185i	−0.999929	−	0.0118856i	\(-0.996217\pi\)
							0.947316	−	0.320299i	\(-0.103783\pi\)
\(19\)	−0.0549118	+	0.0755796i	−0.0125976	+	0.0173392i	−0.815269	−	0.579082i	\(-0.803411\pi\)
							0.802672	+	0.596421i	\(0.203411\pi\)
\(23\)	1.84517	−	3.62135i	0.384744	−	0.755103i	−0.614689	−	0.788770i	\(-0.710718\pi\)
							0.999433	+	0.0336667i	\(0.0107185\pi\)
\(29\)	−2.28992	+	1.66373i	−0.425228	+	0.308946i	−0.779738	−	0.626106i	\(-0.784648\pi\)
							0.354510	+	0.935052i	\(0.384648\pi\)
\(31\)	5.56242	−	7.65602i	0.999041	−	1.37506i	0.0731285	−	0.997323i	\(-0.476702\pi\)
							0.925912	−	0.377739i	\(-0.123298\pi\)
\(37\)	0.630361	+	1.23715i	0.103631	+	0.203387i	0.937000	−	0.349331i	\(-0.113591\pi\)
							−0.833369	+	0.552717i	\(0.813591\pi\)
\(41\)	2.92379	−	8.99849i	0.456619	−	1.40533i	−0.412605	−	0.910910i	\(-0.635381\pi\)
							0.869224	−	0.494418i	\(-0.164619\pi\)
\(43\)	−5.49497	+	5.49497i	−0.837975	+	0.837975i	−0.988592	−	0.150617i	\(-0.951874\pi\)
							0.150617	+	0.988592i	\(0.451874\pi\)
\(47\)	−0.289550	+	1.82814i	−0.0422352	+	0.266662i	−0.999765	−	0.0216714i	\(-0.993101\pi\)
							0.957530	+	0.288334i	\(0.0931012\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.2.bq.a.67.32	yes	480
8.3	odd	2	inner	600.2.bq.a.67.11	✓	480
25.3	odd	20	inner	600.2.bq.a.403.11	yes	480
200.3	even	20	inner	600.2.bq.a.403.32	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.bq.a.67.11	✓	480	8.3	odd	2	inner
600.2.bq.a.67.32	yes	480	1.1	even	1	trivial
600.2.bq.a.403.11	yes	480	25.3	odd	20	inner
600.2.bq.a.403.32	yes	480	200.3	even	20	inner