Embedded newform 105.4.s.a.101.13

• Label: 105.4.s.a.101.13
• Level: $105$
• Weight: $4$
• Character: 105.101
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			101.13
Character	\(\chi\)	\(=\)	105.101
Dual form			105.4.s.a.26.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.53626 - 2.04166i) q^{2} +(-2.57442 - 4.51357i) q^{3} +(4.33676 - 7.51148i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(-18.3190 - 10.7051i) q^{6} +(0.627588 - 18.5096i) q^{7} -2.75017i q^{8} +(-13.7447 + 23.2397i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.53626	−	2.04166i	1.25026	−	0.721836i	0.279096	−	0.960263i	\(-0.409965\pi\)
							0.971160	+	0.238427i	\(0.0766318\pi\)
\(3\)	−2.57442	−	4.51357i	−0.495448	−	0.868638i
							\(5\)	−2.50000	−	4.33013i	−0.223607	−	0.387298i
\(7\)	0.627588	−	18.5096i	0.0338866	−	0.999426i
							\(11\)	−7.33279	−	4.23359i	−0.200993	−	0.116043i	0.396126	−	0.918196i	\(-0.370354\pi\)
−0.597118	+	0.802153i	\(0.703688\pi\)
\(13\)		−	5.50408i		−	0.117427i	−0.998275	−	0.0587137i	\(-0.981300\pi\)
							0.998275	−	0.0587137i	\(-0.0186999\pi\)
\(17\)	60.2944	−	104.433i	0.860208	−	1.48992i	−0.0115198	−	0.999934i	\(-0.503667\pi\)
							0.871728	−	0.489990i	\(-0.163000\pi\)
\(19\)	23.3904	−	13.5044i	0.282427	−	0.163059i	−0.352095	−	0.935964i	\(-0.614530\pi\)
							0.634522	+	0.772905i	\(0.281197\pi\)
\(23\)	−2.98204	+	1.72168i	−0.0270347	+	0.0156085i	−0.513456	−	0.858116i	\(-0.671635\pi\)
							0.486422	+	0.873724i	\(0.338302\pi\)
\(29\)		−	72.7598i		−	0.465902i	−0.972489	−	0.232951i	\(-0.925162\pi\)
							0.972489	−	0.232951i	\(-0.0748381\pi\)
\(31\)	242.287	+	139.885i	1.40374	+	0.810452i	0.994775	−	0.102095i	\(-0.0325546\pi\)
							0.408970	+	0.912548i	\(0.365888\pi\)
\(37\)	44.8944	+	77.7594i	0.199476	+	0.345502i	0.948359	−	0.317200i	\(-0.102743\pi\)
							−0.748883	+	0.662702i	\(0.769409\pi\)
\(41\)	−221.251			−0.842770			−0.421385	−	0.906882i	\(-0.638456\pi\)
							−0.421385	+	0.906882i	\(0.638456\pi\)
\(43\)	495.285			1.75652			0.878259	−	0.478185i	\(-0.158705\pi\)
							0.878259	+	0.478185i	\(0.158705\pi\)
\(47\)	−100.415	−	173.923i	−0.311638	−	0.539773i	0.667079	−	0.744987i	\(-0.267544\pi\)
							−0.978717	+	0.205214i	\(0.934211\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.a.101.13	yes	32
3.2	odd	2		105.4.s.b.101.4	yes	32
7.5	odd	6		105.4.s.b.26.4	yes	32
21.5	even	6	inner	105.4.s.a.26.13	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.13	✓	32	21.5	even	6	inner
105.4.s.a.101.13	yes	32	1.1	even	1	trivial
105.4.s.b.26.4	yes	32	7.5	odd	6
105.4.s.b.101.4	yes	32	3.2	odd	2