Embedded newform 105.3.w.a.38.3

• Label: 105.3.w.a.38.3
• Level: $105$
• Weight: $3$
• Character: 105.38
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	105.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(28\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			38.3
Character	\(\chi\)	\(=\)	105.38
Dual form			105.3.w.a.47.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.875634 + 3.26791i) q^{2} +(-1.74120 - 2.44299i) q^{3} +(-6.44841 - 3.72299i) q^{4} +(3.79351 - 3.25719i) q^{5} +(9.50814 - 3.55093i) q^{6} +(6.87171 - 1.33404i) q^{7} +(8.24376 - 8.24376i) q^{8} +(-2.93642 + 8.50749i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{3} - 16 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.875634	+	3.26791i	−0.437817	+	1.63396i	0.296416	+	0.955059i	\(0.404209\pi\)
							−0.734233	+	0.678897i	\(0.762458\pi\)
\(3\)	−1.74120	−	2.44299i	−0.580401	−	0.814331i
							\(5\)	3.79351	−	3.25719i	0.758701	−	0.651439i
\(7\)	6.87171	−	1.33404i	0.981672	−	0.190577i
							\(11\)	1.43950	+	0.831098i	0.130864	+	0.0755544i	0.564003	−	0.825773i	\(-0.309261\pi\)
−0.433139	+	0.901327i	\(0.642594\pi\)
\(13\)	15.7842	−	15.7842i	1.21417	−	1.21417i	0.244526	−	0.969643i	\(-0.421368\pi\)
							0.969643	−	0.244526i	\(-0.0786322\pi\)
\(17\)	−1.94906	−	7.27399i	−0.114651	−	0.427882i	0.884610	−	0.466332i	\(-0.154425\pi\)
							−0.999261	+	0.0384500i	\(0.987758\pi\)
\(19\)	6.28481	+	10.8856i	0.330779	+	0.572927i	0.982665	−	0.185391i	\(-0.0593551\pi\)
							−0.651885	+	0.758317i	\(0.726022\pi\)
\(23\)	19.2490	+	5.15777i	0.836915	+	0.224251i	0.651728	−	0.758453i	\(-0.274044\pi\)
							0.185187	+	0.982703i	\(0.440711\pi\)
\(29\)	−18.4846			−0.637401			−0.318700	−	0.947855i	\(-0.603246\pi\)
							−0.318700	+	0.947855i	\(0.603246\pi\)
\(31\)	14.8036	+	8.54685i	0.477535	+	0.275705i	0.719388	−	0.694608i	\(-0.244422\pi\)
							−0.241854	+	0.970313i	\(0.577755\pi\)
\(37\)	−16.4045	−	4.39557i	−0.443365	−	0.118799i	0.0302283	−	0.999543i	\(-0.490377\pi\)
							−0.473593	+	0.880744i	\(0.657043\pi\)
\(41\)	−57.3594			−1.39901			−0.699505	−	0.714628i	\(-0.746596\pi\)
							−0.699505	+	0.714628i	\(0.746596\pi\)
\(43\)	9.66086	+	9.66086i	0.224671	+	0.224671i	0.810462	−	0.585791i	\(-0.199216\pi\)
							−0.585791	+	0.810462i	\(0.699216\pi\)
\(47\)	−86.5523	−	23.1916i	−1.84154	−	0.493438i	−0.842558	−	0.538606i	\(-0.818951\pi\)
							−0.998979	+	0.0451672i	\(0.985618\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.w.a.38.3	yes	112
3.2	odd	2	inner	105.3.w.a.38.26	yes	112
5.2	odd	4	inner	105.3.w.a.17.3	✓	112
7.5	odd	6	inner	105.3.w.a.68.26	yes	112
15.2	even	4	inner	105.3.w.a.17.26	yes	112
21.5	even	6	inner	105.3.w.a.68.3	yes	112
35.12	even	12	inner	105.3.w.a.47.26	yes	112
105.47	odd	12	inner	105.3.w.a.47.3	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.w.a.17.3	✓	112	5.2	odd	4	inner
105.3.w.a.17.26	yes	112	15.2	even	4	inner
105.3.w.a.38.3	yes	112	1.1	even	1	trivial
105.3.w.a.38.26	yes	112	3.2	odd	2	inner
105.3.w.a.47.3	yes	112	105.47	odd	12	inner
105.3.w.a.47.26	yes	112	35.12	even	12	inner
105.3.w.a.68.3	yes	112	21.5	even	6	inner
105.3.w.a.68.26	yes	112	7.5	odd	6	inner