Embedded newform 105.4.s.a.26.16

• Label: 105.4.s.a.26.16
• Level: $105$
• Weight: $4$
• Character: 105.26
• 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			26.16
Character	\(\chi\)	\(=\)	105.26
Dual form			105.4.s.a.101.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.69077 + 2.70822i) q^{2} +(-2.85230 - 4.34332i) q^{3} +(10.6689 + 18.4790i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(-1.61684 - 28.0981i) q^{6} +(14.5222 + 11.4937i) q^{7} +72.2429i q^{8} +(-10.7288 + 24.7769i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.69077	+	2.70822i	1.65844	+	0.957499i	0.973437	+	0.228955i	\(0.0735310\pi\)
							0.685000	+	0.728543i	\(0.259802\pi\)
\(3\)	−2.85230	−	4.34332i	−0.548925	−	0.835871i
							\(5\)	−2.50000	+	4.33013i	−0.223607	+	0.387298i
\(7\)	14.5222	+	11.4937i	0.784125	+	0.620602i
							\(11\)	38.4696	−	22.2104i	1.05446	−	0.608790i	0.130562	−	0.991440i	\(-0.458322\pi\)
0.923893	+	0.382650i	\(0.124989\pi\)
\(13\)		−	22.9471i		−	0.489568i	−0.969578	−	0.244784i	\(-0.921283\pi\)
							0.969578	−	0.244784i	\(-0.0787171\pi\)
\(17\)	−32.3140	−	55.9695i	−0.461018	−	0.798506i	0.537994	−	0.842949i	\(-0.319182\pi\)
							−0.999012	+	0.0444424i	\(0.985849\pi\)
\(19\)	−15.6960	−	9.06207i	−0.189521	−	0.109420i	0.402237	−	0.915535i	\(-0.368233\pi\)
							−0.591758	+	0.806115i	\(0.701566\pi\)
\(23\)	−76.2989	−	44.0512i	−0.691714	−	0.399361i	0.112540	−	0.993647i	\(-0.464101\pi\)
							−0.804254	+	0.594286i	\(0.797435\pi\)
\(29\)		−	308.783i		−	1.97722i	−0.150485	−	0.988612i	\(-0.548084\pi\)
							0.150485	−	0.988612i	\(-0.451916\pi\)
\(31\)	−82.5212	+	47.6436i	−0.478105	+	0.276034i	−0.719626	−	0.694362i	\(-0.755687\pi\)
							0.241522	+	0.970395i	\(0.422354\pi\)
\(37\)	−83.8935	+	145.308i	−0.372757	+	0.645634i	−0.989989	−	0.141147i	\(-0.954921\pi\)
							0.617232	+	0.786782i	\(0.288254\pi\)
\(41\)	271.265			1.03328			0.516639	−	0.856203i	\(-0.327183\pi\)
							0.516639	+	0.856203i	\(0.327183\pi\)
\(43\)	225.930			0.801257			0.400629	−	0.916241i	\(-0.368792\pi\)
							0.400629	+	0.916241i	\(0.368792\pi\)
\(47\)	−31.1914	+	54.0251i	−0.0968029	+	0.167667i	−0.910360	−	0.413818i	\(-0.864195\pi\)
							0.813557	+	0.581486i	\(0.197528\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.a.26.16	✓	32
3.2	odd	2		105.4.s.b.26.1	yes	32
7.3	odd	6		105.4.s.b.101.1	yes	32
21.17	even	6	inner	105.4.s.a.101.16	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.16	✓	32	1.1	even	1	trivial
105.4.s.a.101.16	yes	32	21.17	even	6	inner
105.4.s.b.26.1	yes	32	3.2	odd	2
105.4.s.b.101.1	yes	32	7.3	odd	6