Embedded newform 61.6.g.a.3.9

• Label: 61.6.g.a.3.9
• Level: $61$
• Weight: $6$
• Character: 61.3
• Analytic conductor: $9.783$
• Analytic rank: $0$
• Dimension: $104$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	61.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.78341300859\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			3.9
Character	\(\chi\)	\(=\)	61.3
Dual form			61.6.g.a.41.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.94487 - 1.28177i) q^{2} +(-7.73018 + 23.7910i) q^{3} +(-11.9695 - 8.69633i) q^{4} +(-11.5628 - 8.40083i) q^{5} +(60.9891 - 83.9443i) q^{6} +(-211.654 + 68.7706i) q^{7} +(114.089 + 157.031i) q^{8} +(-309.667 - 224.986i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 5 q^{2} + 15 q^{3} + 465 q^{4} - 122 q^{5} + 445 q^{6} + 230 q^{7} - 5 q^{8} - 2841 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.94487	−	1.28177i	−0.697361	−	0.226586i	−0.0611806	−	0.998127i	\(-0.519487\pi\)
							−0.636180	+	0.771540i	\(0.719487\pi\)
\(3\)	−7.73018	+	23.7910i	−0.495891	+	1.52620i	0.319673	+	0.947528i	\(0.396427\pi\)
							−0.815564	+	0.578668i	\(0.803573\pi\)
\(5\)	−11.5628	−	8.40083i	−0.206841	−	0.150279i	0.479542	−	0.877519i	\(-0.340803\pi\)
							−0.686383	+	0.727240i	\(0.740803\pi\)
\(7\)	−211.654	+	68.7706i	−1.63261	+	0.530466i	−0.974869	−	0.222781i	\(-0.928487\pi\)
							−0.657738	+	0.753247i	\(0.728487\pi\)
\(11\)			44.8614i			0.111787i	0.998437	+	0.0558935i	\(0.0178007\pi\)
							−0.998437	+	0.0558935i	\(0.982199\pi\)
\(13\)	423.833			0.695563			0.347782	−	0.937576i	\(-0.386935\pi\)
							0.347782	+	0.937576i	\(0.386935\pi\)
\(17\)	1052.66	−	1448.87i	0.883421	−	1.21592i	−0.0920405	−	0.995755i	\(-0.529339\pi\)
							0.975462	−	0.220170i	\(-0.0706611\pi\)
\(19\)	−468.910	+	1443.16i	−0.297993	+	0.917127i	0.684207	+	0.729288i	\(0.260148\pi\)
							−0.982200	+	0.187839i	\(0.939852\pi\)
\(23\)	948.835	−	1305.96i	0.373999	−	0.514766i	−0.579983	−	0.814629i	\(-0.696941\pi\)
							0.953982	+	0.299863i	\(0.0969408\pi\)
\(29\)			2743.59i			0.605793i	0.953023	+	0.302897i	\(0.0979537\pi\)
							−0.953023	+	0.302897i	\(0.902046\pi\)
\(31\)	2618.60	−	850.834i	0.489401	−	0.159016i	−0.0539130	−	0.998546i	\(-0.517169\pi\)
							0.543314	+	0.839530i	\(0.317169\pi\)
\(37\)	−2227.86	+	723.875i	−0.267537	+	0.0869279i	−0.439713	−	0.898138i	\(-0.644920\pi\)
							0.172177	+	0.985066i	\(0.444920\pi\)
\(41\)	4804.11	+	14785.5i	0.446327	+	1.37365i	0.881022	+	0.473076i	\(0.156856\pi\)
							−0.434694	+	0.900578i	\(0.643144\pi\)
\(43\)	−6799.96	−	9359.35i	−0.560835	−	0.771924i	0.430597	−	0.902544i	\(-0.358303\pi\)
							−0.991432	+	0.130621i	\(0.958303\pi\)
\(47\)	−15204.8			−1.00401			−0.502004	−	0.864865i	\(-0.667404\pi\)
							−0.502004	+	0.864865i	\(0.667404\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.6.g.a.3.9	✓	104
61.41	even	10	inner	61.6.g.a.41.9	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.6.g.a.3.9	✓	104	1.1	even	1	trivial
61.6.g.a.41.9	yes	104	61.41	even	10	inner