Embedded newform 105.3.t.b.86.2

• Label: 105.3.t.b.86.2
• 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			86.2
Character	\(\chi\)	\(=\)	105.86
Dual form			105.3.t.b.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.61597 - 1.51033i) q^{2} +(-0.469377 + 2.96305i) q^{3} +(2.56220 + 4.43785i) q^{4} +(1.93649 + 1.11803i) q^{5} +(5.70307 - 7.04234i) q^{6} +(-4.99419 - 4.90490i) q^{7} -3.39641i q^{8} +(-8.55937 - 2.78158i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 4 q^{3} + 36 q^{4} - 24 q^{6} - 58 q^{7} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.61597	−	1.51033i	−1.30798	−	0.755165i	−0.326225	−	0.945292i	\(-0.605777\pi\)
							−0.981760	+	0.190127i	\(0.939110\pi\)
\(3\)	−0.469377	+	2.96305i	−0.156459	+	0.987684i
							\(5\)	1.93649	+	1.11803i	0.387298	+	0.223607i
\(7\)	−4.99419	−	4.90490i	−0.713456	−	0.700700i
							\(11\)	1.06414	−	0.614380i	0.0967398	−	0.0558528i	−0.450850	−	0.892600i	\(-0.648879\pi\)
0.547589	+	0.836747i	\(0.315546\pi\)
\(13\)	−18.8197			−1.44767			−0.723834	−	0.689974i	\(-0.757622\pi\)
							−0.723834	+	0.689974i	\(0.757622\pi\)
\(17\)	−16.5849	+	9.57529i	−0.975581	+	0.563252i	−0.900933	−	0.433958i	\(-0.857117\pi\)
							−0.0746481	+	0.997210i	\(0.523783\pi\)
\(19\)	−10.0788	+	17.4570i	−0.530464	+	0.918791i	0.468904	+	0.883249i	\(0.344649\pi\)
							−0.999368	+	0.0355421i	\(0.988684\pi\)
\(23\)	−16.4001	−	9.46858i	−0.713046	−	0.411677i	0.0991417	−	0.995073i	\(-0.468390\pi\)
							−0.812188	+	0.583396i	\(0.801724\pi\)
\(29\)		−	31.9618i		−	1.10213i	−0.834462	−	0.551066i	\(-0.814221\pi\)
							0.834462	−	0.551066i	\(-0.185779\pi\)
\(31\)	−14.6778	−	25.4227i	−0.473477	−	0.820087i	0.526062	−	0.850446i	\(-0.323668\pi\)
							−0.999539	+	0.0303595i	\(0.990335\pi\)
\(37\)	−9.97462	+	17.2765i	−0.269584	+	0.466934i	−0.968754	−	0.248022i	\(-0.920220\pi\)
							0.699170	+	0.714955i	\(0.253553\pi\)
\(41\)			58.4356i			1.42526i	0.701541	+	0.712629i	\(0.252496\pi\)
							−0.701541	+	0.712629i	\(0.747504\pi\)
\(43\)	57.9352			1.34733			0.673666	−	0.739036i	\(-0.264719\pi\)
							0.673666	+	0.739036i	\(0.264719\pi\)
\(47\)	41.1569	+	23.7620i	0.875680	+	0.505574i	0.869232	−	0.494405i	\(-0.164614\pi\)
							0.00644823	+	0.999979i	\(0.497947\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.t.b.86.2	yes	36
3.2	odd	2	inner	105.3.t.b.86.17	yes	36
7.4	even	3	inner	105.3.t.b.11.17	yes	36
21.11	odd	6	inner	105.3.t.b.11.2	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.t.b.11.2	✓	36	21.11	odd	6	inner
105.3.t.b.11.17	yes	36	7.4	even	3	inner
105.3.t.b.86.2	yes	36	1.1	even	1	trivial
105.3.t.b.86.17	yes	36	3.2	odd	2	inner