Embedded newform 105.3.t.b.86.6

• Label: 105.3.t.b.86.6
• Level: $105$
• Weight: $3$
• Character: 105.86
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	105.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			86.6
Character	\(\chi\)	\(=\)	105.86
Dual form			105.3.t.b.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50527 - 0.869067i) q^{2} +(-1.65926 - 2.49937i) q^{3} +(-0.489445 - 0.847743i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(0.325501 + 5.20423i) q^{6} +(2.93407 - 6.35541i) q^{7} +8.65398i q^{8} +(-3.49374 + 8.29420i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 4 q^{3} + 36 q^{4} - 24 q^{6} - 58 q^{7} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50527	−	0.869067i	−0.752634	−	0.434534i	0.0740107	−	0.997257i	\(-0.476420\pi\)
							−0.826645	+	0.562724i	\(0.809753\pi\)
\(3\)	−1.65926	−	2.49937i	−0.553085	−	0.833125i
							\(5\)	−1.93649	−	1.11803i	−0.387298	−	0.223607i
\(7\)	2.93407	−	6.35541i	0.419153	−	0.907916i
							\(11\)	−11.0674	+	6.38976i	−1.00613	+	0.580887i	−0.910055	−	0.414487i	\(-0.863961\pi\)
−0.0960710	+	0.995374i	\(0.530628\pi\)
\(13\)	0.690184			0.0530911			0.0265455	−	0.999648i	\(-0.491549\pi\)
							0.0265455	+	0.999648i	\(0.491549\pi\)
\(17\)	12.5056	−	7.22012i	0.735625	−	0.424713i	−0.0848518	−	0.996394i	\(-0.527042\pi\)
							0.820476	+	0.571681i	\(0.193708\pi\)
\(19\)	−13.8399	+	23.9714i	−0.728415	+	1.26165i	0.229138	+	0.973394i	\(0.426409\pi\)
							−0.957553	+	0.288258i	\(0.906924\pi\)
\(23\)	−37.3582	−	21.5687i	−1.62427	−	0.937771i	−0.985760	−	0.168156i	\(-0.946219\pi\)
							−0.638507	−	0.769616i	\(-0.720448\pi\)
\(29\)		−	26.6616i		−	0.919366i	−0.888083	−	0.459683i	\(-0.847963\pi\)
							0.888083	−	0.459683i	\(-0.152037\pi\)
\(31\)	−8.94214	−	15.4882i	−0.288456	−	0.499621i	0.684985	−	0.728557i	\(-0.259809\pi\)
							−0.973441	+	0.228936i	\(0.926475\pi\)
\(37\)	17.5794	−	30.4484i	0.475119	−	0.822930i	−0.524475	−	0.851426i	\(-0.675738\pi\)
							0.999594	+	0.0284956i	\(0.00907164\pi\)
\(41\)			15.0963i			0.368202i	0.982907	+	0.184101i	\(0.0589373\pi\)
							−0.982907	+	0.184101i	\(0.941063\pi\)
\(43\)	−23.1680			−0.538791			−0.269395	−	0.963030i	\(-0.586824\pi\)
							−0.269395	+	0.963030i	\(0.586824\pi\)
\(47\)	−38.1508	−	22.0264i	−0.811719	−	0.468646i	0.0358338	−	0.999358i	\(-0.488591\pi\)
							−0.847552	+	0.530712i	\(0.821925\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.t.b.86.6	yes	36
3.2	odd	2	inner	105.3.t.b.86.13	yes	36
7.4	even	3	inner	105.3.t.b.11.13	yes	36
21.11	odd	6	inner	105.3.t.b.11.6	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.t.b.11.6	✓	36	21.11	odd	6	inner
105.3.t.b.11.13	yes	36	7.4	even	3	inner
105.3.t.b.86.6	yes	36	1.1	even	1	trivial
105.3.t.b.86.13	yes	36	3.2	odd	2	inner