Embedded newform 861.2.bz.a.4.9

• Label: 861.2.bz.a.4.9
• Level: $861$
• Weight: $2$
• Character: 861.4
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $448$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.bz (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(448\)
Relative dimension:	\(56\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.9
Character	\(\chi\)	\(=\)	861.4
Dual form			861.2.bz.a.646.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.00458 - 0.892498i) q^{2} +(0.866025 + 0.500000i) q^{3} +(1.88354 + 2.09188i) q^{4} +(-3.19213 + 0.678507i) q^{5} +(-1.28977 - 1.77522i) q^{6} +(-1.47599 + 2.19578i) q^{7} +(-0.552564 - 1.70062i) q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 448 q + 56 q^{4} - 24 q^{8} + 224 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/861\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(493\)	\(575\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00458	−	0.892498i	−1.41745	−	0.631091i	−0.452084	−	0.891975i	\(-0.649319\pi\)
							−0.965370	+	0.260884i	\(0.915986\pi\)
\(3\)	0.866025	+	0.500000i	0.500000	+	0.288675i
							\(5\)	−3.19213	+	0.678507i	−1.42756	+	0.303438i	−0.855938	−	0.517078i	\(-0.827020\pi\)
−0.571624	+	0.820516i	\(0.693686\pi\)
\(7\)	−1.47599	+	2.19578i	−0.557873	+	0.829926i
							\(11\)	−1.28525	+	6.04665i	−0.387519	+	1.82313i	0.160951	+	0.986962i	\(0.448544\pi\)
−0.548470	+	0.836170i	\(0.684790\pi\)
\(13\)	1.29690	+	1.78502i	0.359694	+	0.495077i	0.950064	−	0.312057i	\(-0.101018\pi\)
							−0.590369	+	0.807133i	\(0.701018\pi\)
\(17\)	0.894829	−	4.20984i	0.217028	−	1.02104i	−0.725839	−	0.687865i	\(-0.758548\pi\)
							0.942866	−	0.333171i	\(-0.108119\pi\)
\(19\)	1.56702	+	0.164701i	0.359500	+	0.0377849i	0.282557	−	0.959251i	\(-0.408817\pi\)
							0.0769428	+	0.997036i	\(0.475484\pi\)
\(23\)	−1.78997	−	0.796946i	−0.373235	−	0.166175i	0.211542	−	0.977369i	\(-0.432151\pi\)
							−0.584777	+	0.811194i	\(0.698818\pi\)
\(29\)	−5.87029	−	1.90737i	−1.09009	−	0.354190i	−0.291806	−	0.956477i	\(-0.594256\pi\)
							−0.798280	+	0.602287i	\(0.794256\pi\)
\(31\)	2.33479	+	0.496275i	0.419341	+	0.0891336i	0.412749	−	0.910845i	\(-0.364569\pi\)
							0.00659138	+	0.999978i	\(0.497902\pi\)
\(37\)	−7.09043	+	1.50712i	−1.16566	+	0.247769i	−0.749803	−	0.661661i	\(-0.769852\pi\)
							−0.415857	+	0.909430i	\(0.636518\pi\)
\(41\)	−5.89962	−	2.48888i	−0.921365	−	0.388698i
							\(43\)	4.56281	−	3.31507i	0.695822	−	0.505544i	−0.182747	−	0.983160i	\(-0.558499\pi\)
0.878569	+	0.477616i	\(0.158499\pi\)
\(47\)	−1.31798	+	2.96023i	−0.192247	+	0.431794i	−0.983791	−	0.179318i	\(-0.942611\pi\)
							0.791544	+	0.611112i	\(0.209278\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	861.2.bz.a.4.9	✓	448
7.2	even	3	inner	861.2.bz.a.373.48	yes	448
41.31	even	10	inner	861.2.bz.a.277.48	yes	448
287.72	even	30	inner	861.2.bz.a.646.9	yes	448

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.bz.a.4.9	✓	448	1.1	even	1	trivial
861.2.bz.a.277.48	yes	448	41.31	even	10	inner
861.2.bz.a.373.48	yes	448	7.2	even	3	inner
861.2.bz.a.646.9	yes	448	287.72	even	30	inner