Embedded newform 105.4.x.a.2.20

• Label: 105.4.x.a.2.20
• Level: $105$
• Weight: $4$
• Character: 105.2
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.20
Character	\(\chi\)	\(=\)	105.2
Dual form			105.4.x.a.53.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.189195 - 0.706085i) q^{2} +(-1.86875 + 4.84848i) q^{3} +(6.46544 - 3.73282i) q^{4} +(-2.65237 - 10.8612i) q^{5} +(3.77700 + 0.402189i) q^{6} +(15.8305 + 9.61223i) q^{7} +(-7.99404 - 7.99404i) q^{8} +(-20.0155 - 18.1212i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.189195	−	0.706085i	−0.0668905	−	0.249639i	0.924382	−	0.381468i	\(-0.124582\pi\)
							−0.991272	+	0.131830i	\(0.957915\pi\)
\(3\)	−1.86875	+	4.84848i	−0.359641	+	0.933091i
							\(5\)	−2.65237	−	10.8612i	−0.237235	−	0.971452i
\(7\)	15.8305	+	9.61223i	0.854767	+	0.519011i
							\(11\)	−3.09441	+	1.78656i	−0.0848181	+	0.0489697i	−0.541809	−	0.840502i	\(-0.682260\pi\)
0.456991	+	0.889471i	\(0.348927\pi\)
\(13\)	53.9865	−	53.9865i	1.15178	−	1.15178i	0.165587	−	0.986195i	\(-0.447048\pi\)
							0.986195	−	0.165587i	\(-0.0529520\pi\)
\(17\)	70.9536	+	19.0120i	1.01228	+	0.271240i	0.726582	−	0.687080i	\(-0.241108\pi\)
							0.285698	+	0.958320i	\(0.407774\pi\)
\(19\)	55.1554	+	31.8440i	0.665974	+	0.384500i	0.794550	−	0.607199i	\(-0.207707\pi\)
							−0.128575	+	0.991700i	\(0.541040\pi\)
\(23\)	122.300	−	32.7701i	1.10875	−	0.297088i	0.342426	−	0.939545i	\(-0.388751\pi\)
							0.766322	+	0.642457i	\(0.222085\pi\)
\(29\)	−222.285			−1.42336			−0.711678	−	0.702506i	\(-0.752064\pi\)
							−0.711678	+	0.702506i	\(0.752064\pi\)
\(31\)	−47.0629	−	81.5153i	−0.272669	−	0.472277i	0.696875	−	0.717192i	\(-0.254573\pi\)
							−0.969544	+	0.244916i	\(0.921240\pi\)
\(37\)	−250.495	+	67.1200i	−1.11301	+	0.298229i	−0.768050	−	0.640390i	\(-0.778773\pi\)
							−0.344955	+	0.938619i	\(0.612106\pi\)
\(41\)			28.6039i			0.108956i	0.998515	+	0.0544779i	\(0.0173494\pi\)
							−0.998515	+	0.0544779i	\(0.982651\pi\)
\(43\)	−233.065	+	233.065i	−0.826559	+	0.826559i	−0.987039	−	0.160480i	\(-0.948696\pi\)
							0.160480	+	0.987039i	\(0.448696\pi\)
\(47\)	110.541	+	412.544i	0.343064	+	1.28033i	0.894857	+	0.446353i	\(0.147277\pi\)
							−0.551793	+	0.833981i	\(0.686056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.x.a.2.20	✓	176
3.2	odd	2	inner	105.4.x.a.2.25	yes	176
5.3	odd	4	inner	105.4.x.a.23.20	yes	176
7.4	even	3	inner	105.4.x.a.32.25	yes	176
15.8	even	4	inner	105.4.x.a.23.25	yes	176
21.11	odd	6	inner	105.4.x.a.32.20	yes	176
35.18	odd	12	inner	105.4.x.a.53.25	yes	176
105.53	even	12	inner	105.4.x.a.53.20	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.x.a.2.20	✓	176	1.1	even	1	trivial
105.4.x.a.2.25	yes	176	3.2	odd	2	inner
105.4.x.a.23.20	yes	176	5.3	odd	4	inner
105.4.x.a.23.25	yes	176	15.8	even	4	inner
105.4.x.a.32.20	yes	176	21.11	odd	6	inner
105.4.x.a.32.25	yes	176	7.4	even	3	inner
105.4.x.a.53.20	yes	176	105.53	even	12	inner
105.4.x.a.53.25	yes	176	35.18	odd	12	inner