Embedded newform 105.4.s.a.101.9

• Label: 105.4.s.a.101.9
• 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			101.9
Character	\(\chi\)	\(=\)	105.101
Dual form			105.4.s.a.26.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.155491 - 0.0897728i) q^{2} +(5.17065 - 0.514148i) q^{3} +(-3.98388 + 6.90029i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(0.757834 - 0.544130i) q^{6} +(18.1774 - 3.54688i) q^{7} +2.86694i q^{8} +(26.4713 - 5.31696i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.155491	−	0.0897728i	0.0549744	−	0.0317395i	−0.472261	−	0.881459i	\(-0.656562\pi\)
							0.527235	+	0.849719i	\(0.323229\pi\)
\(3\)	5.17065	−	0.514148i	0.995093	−	0.0989478i
							\(5\)	−2.50000	−	4.33013i	−0.223607	−	0.387298i
\(7\)	18.1774	−	3.54688i	0.981490	−	0.191514i
							\(11\)	42.8268	+	24.7261i	1.17389	+	0.677745i	0.954593	−	0.297914i	\(-0.0962907\pi\)
0.219296	+	0.975658i	\(0.429624\pi\)
\(13\)			62.5979i			1.33550i	0.744385	+	0.667751i	\(0.232743\pi\)
							−0.744385	+	0.667751i	\(0.767257\pi\)
\(17\)	19.4432	−	33.6766i	0.277392	−	0.480457i	−0.693344	−	0.720607i	\(-0.743863\pi\)
							0.970736	+	0.240150i	\(0.0771966\pi\)
\(19\)	14.1784	−	8.18591i	0.171197	−	0.0988408i	−0.411953	−	0.911205i	\(-0.635153\pi\)
							0.583150	+	0.812364i	\(0.301820\pi\)
\(23\)	−125.793	+	72.6264i	−1.14042	+	0.658420i	−0.946534	−	0.322605i	\(-0.895442\pi\)
							−0.193883	+	0.981025i	\(0.562108\pi\)
\(29\)		−	246.319i		−	1.57725i	−0.614872	−	0.788627i	\(-0.710792\pi\)
							0.614872	−	0.788627i	\(-0.289208\pi\)
\(31\)	128.320	+	74.0854i	0.743449	+	0.429230i	0.823322	−	0.567575i	\(-0.192118\pi\)
							−0.0798733	+	0.996805i	\(0.525452\pi\)
\(37\)	−174.170	−	301.671i	−0.773875	−	1.34039i	−0.935424	−	0.353527i	\(-0.884982\pi\)
							0.161549	−	0.986865i	\(-0.448351\pi\)
\(41\)	−429.026			−1.63421			−0.817106	−	0.576488i	\(-0.804423\pi\)
							−0.817106	+	0.576488i	\(0.804423\pi\)
\(43\)	73.5626			0.260888			0.130444	−	0.991456i	\(-0.458360\pi\)
							0.130444	+	0.991456i	\(0.458360\pi\)
\(47\)	−124.129	−	214.998i	−0.385236	−	0.667249i	0.606566	−	0.795033i	\(-0.292547\pi\)
							−0.991802	+	0.127785i	\(0.959213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.a.101.9	yes	32
3.2	odd	2		105.4.s.b.101.8	yes	32
7.5	odd	6		105.4.s.b.26.8	yes	32
21.5	even	6	inner	105.4.s.a.26.9	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.9	✓	32	21.5	even	6	inner
105.4.s.a.101.9	yes	32	1.1	even	1	trivial
105.4.s.b.26.8	yes	32	7.5	odd	6
105.4.s.b.101.8	yes	32	3.2	odd	2