Embedded newform 105.4.s.a.101.3

• Label: 105.4.s.a.101.3
• 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			101.3
Character	\(\chi\)	\(=\)	105.101
Dual form			105.4.s.a.26.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.31734 + 1.91526i) q^{2} +(4.42611 - 2.72205i) q^{3} +(3.33648 - 5.77895i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(-9.46943 + 17.5071i) q^{6} +(8.94735 + 16.2156i) q^{7} -5.08329i q^{8} +(12.1808 - 24.0962i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.31734	+	1.91526i	−1.17286	+	0.677148i	−0.954351	−	0.298688i	\(-0.903451\pi\)
							−0.218504	+	0.975836i	\(0.570118\pi\)
\(3\)	4.42611	−	2.72205i	0.851805	−	0.523860i
							\(5\)	−2.50000	−	4.33013i	−0.223607	−	0.387298i
\(7\)	8.94735	+	16.2156i	0.483112	+	0.875559i
							\(11\)	−12.9526	−	7.47816i	−0.355031	−	0.204977i	0.311868	−	0.950126i	\(-0.399045\pi\)
−0.666899	+	0.745148i	\(0.732379\pi\)
\(13\)		−	63.8636i		−	1.36251i	−0.732048	−	0.681253i	\(-0.761436\pi\)
							0.732048	−	0.681253i	\(-0.238564\pi\)
\(17\)	−11.0025	+	19.0569i	−0.156970	+	0.271881i	−0.933775	−	0.357861i	\(-0.883506\pi\)
							0.776804	+	0.629742i	\(0.216839\pi\)
\(19\)	137.229	−	79.2293i	1.65697	−	0.956655i	0.682875	−	0.730535i	\(-0.260729\pi\)
							0.974099	−	0.226120i	\(-0.0726041\pi\)
\(23\)	162.564	−	93.8566i	1.47378	−	0.850889i	0.474219	−	0.880407i	\(-0.342730\pi\)
							0.999564	+	0.0295172i	\(0.00939700\pi\)
\(29\)			179.589i			1.14996i	0.818167	+	0.574980i	\(0.194990\pi\)
							−0.818167	+	0.574980i	\(0.805010\pi\)
\(31\)	64.0984	+	37.0072i	0.371368	+	0.214410i	0.674056	−	0.738680i	\(-0.264551\pi\)
							−0.302688	+	0.953090i	\(0.597884\pi\)
\(37\)	−182.983	−	316.935i	−0.813031	−	1.40821i	−0.910733	−	0.412996i	\(-0.864482\pi\)
							0.0977017	−	0.995216i	\(-0.468851\pi\)
\(41\)	168.976			0.643650			0.321825	−	0.946799i	\(-0.395704\pi\)
							0.321825	+	0.946799i	\(0.395704\pi\)
\(43\)	−60.4265			−0.214301			−0.107151	−	0.994243i	\(-0.534173\pi\)
							−0.107151	+	0.994243i	\(0.534173\pi\)
\(47\)	−5.05040	−	8.74755i	−0.0156740	−	0.0271481i	0.858082	−	0.513513i	\(-0.171656\pi\)
							−0.873756	+	0.486365i	\(0.838323\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.a.101.3	yes	32
3.2	odd	2		105.4.s.b.101.14	yes	32
7.5	odd	6		105.4.s.b.26.14	yes	32
21.5	even	6	inner	105.4.s.a.26.3	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.3	✓	32	21.5	even	6	inner
105.4.s.a.101.3	yes	32	1.1	even	1	trivial
105.4.s.b.26.14	yes	32	7.5	odd	6
105.4.s.b.101.14	yes	32	3.2	odd	2