Embedded newform 105.3.w.a.68.3

• Label: 105.3.w.a.68.3
• Level: $105$
• Weight: $3$
• Character: 105.68
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $112$
• 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			68.3
Character	\(\chi\)	\(=\)	105.68
Dual form			105.3.w.a.17.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.26791 + 0.875634i) q^{2} +(-2.44299 - 1.74120i) q^{3} +(6.44841 - 3.72299i) q^{4} +(-4.71757 - 1.65668i) q^{5} +(9.50814 + 3.55093i) q^{6} +(1.33404 - 6.87171i) q^{7} +(-8.24376 + 8.24376i) q^{8} +(2.93642 + 8.50749i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{3} - 16 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.26791	+	0.875634i	−1.63396	+	0.437817i	−0.955059	−	0.296416i	\(-0.904209\pi\)
							−0.678897	+	0.734233i	\(0.737542\pi\)
\(3\)	−2.44299	−	1.74120i	−0.814331	−	0.580401i
							\(5\)	−4.71757	−	1.65668i	−0.943513	−	0.331335i
\(7\)	1.33404	−	6.87171i	0.190577	−	0.981672i
							\(11\)	1.43950	−	0.831098i	0.130864	−	0.0755544i	−0.433139	−	0.901327i	\(-0.642594\pi\)
0.564003	+	0.825773i	\(0.309261\pi\)
\(13\)	−15.7842	+	15.7842i	−1.21417	+	1.21417i	−0.244526	+	0.969643i	\(0.578632\pi\)
							−0.969643	+	0.244526i	\(0.921368\pi\)
\(17\)	7.27399	+	1.94906i	0.427882	+	0.114651i	0.466332	−	0.884610i	\(-0.345575\pi\)
							−0.0384500	+	0.999261i	\(0.512242\pi\)
\(19\)	−6.28481	+	10.8856i	−0.330779	+	0.572927i	−0.982665	−	0.185391i	\(-0.940645\pi\)
							0.651885	+	0.758317i	\(0.273978\pi\)
\(23\)	5.15777	+	19.2490i	0.224251	+	0.836915i	0.982703	+	0.185187i	\(0.0592890\pi\)
							−0.758453	+	0.651728i	\(0.774044\pi\)
\(29\)	18.4846			0.637401			0.318700	−	0.947855i	\(-0.396754\pi\)
							0.318700	+	0.947855i	\(0.396754\pi\)
\(31\)	14.8036	−	8.54685i	0.477535	−	0.275705i	−0.241854	−	0.970313i	\(-0.577755\pi\)
							0.719388	+	0.694608i	\(0.244422\pi\)
\(37\)	4.39557	+	16.4045i	0.118799	+	0.443365i	0.999543	−	0.0302283i	\(-0.00962343\pi\)
							−0.880744	+	0.473593i	\(0.842957\pi\)
\(41\)	−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	\(0.699216\pi\)
\(47\)	23.1916	+	86.5523i	0.493438	+	1.84154i	0.538606	+	0.842558i	\(0.318951\pi\)
							−0.0451672	+	0.998979i	\(0.514382\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.w.a.17.3	✓	112	105.17	odd	12	inner
105.3.w.a.17.26	yes	112	35.17	even	12	inner
105.3.w.a.38.3	yes	112	21.17	even	6	inner
105.3.w.a.38.26	yes	112	7.3	odd	6	inner
105.3.w.a.47.3	yes	112	5.2	odd	4	inner
105.3.w.a.47.26	yes	112	15.2	even	4	inner
105.3.w.a.68.3	yes	112	1.1	even	1	trivial
105.3.w.a.68.26	yes	112	3.2	odd	2	inner