Embedded newform 105.3.t.b.86.7

• Label: 105.3.t.b.86.7
• 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			86.7
Character	\(\chi\)	\(=\)	105.86
Dual form			105.3.t.b.11.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.987174 - 0.569945i) q^{2} +(-0.202464 + 2.99316i) q^{3} +(-1.35032 - 2.33883i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(1.90581 - 2.83938i) q^{6} +(2.61061 + 6.49498i) q^{7} +7.63801i q^{8} +(-8.91802 - 1.21201i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 4 q^{3} + 36 q^{4} - 24 q^{6} - 58 q^{7} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.987174	−	0.569945i	−0.493587	−	0.284973i	0.232474	−	0.972603i	\(-0.425318\pi\)
							−0.726061	+	0.687630i	\(0.758651\pi\)
\(3\)	−0.202464	+	2.99316i	−0.0674880	+	0.997720i
							\(5\)	−1.93649	−	1.11803i	−0.387298	−	0.223607i
\(7\)	2.61061	+	6.49498i	0.372944	+	0.927854i
							\(11\)	−12.8353	+	7.41045i	−1.16684	+	0.673678i	−0.952935	−	0.303176i	\(-0.901953\pi\)
−0.213909	+	0.976854i	\(0.568620\pi\)
\(13\)	−23.9894			−1.84534			−0.922671	−	0.385589i	\(-0.873998\pi\)
							−0.922671	+	0.385589i	\(0.873998\pi\)
\(17\)	5.54599	−	3.20198i	0.326235	−	0.188352i	−0.327933	−	0.944701i	\(-0.606352\pi\)
							0.654168	+	0.756349i	\(0.273019\pi\)
\(19\)	3.55819	−	6.16296i	0.187273	−	0.324367i	−0.757067	−	0.653337i	\(-0.773368\pi\)
							0.944340	+	0.328971i	\(0.106702\pi\)
\(23\)	28.8602	+	16.6624i	1.25479	+	0.724453i	0.972057	−	0.234746i	\(-0.0754257\pi\)
							0.282733	+	0.959199i	\(0.408759\pi\)
\(29\)			32.6179i			1.12476i	0.826880	+	0.562378i	\(0.190113\pi\)
							−0.826880	+	0.562378i	\(0.809887\pi\)
\(31\)	−1.00299	−	1.73724i	−0.0323546	−	0.0560398i	0.849395	−	0.527758i	\(-0.176967\pi\)
							−0.881749	+	0.471718i	\(0.843634\pi\)
\(37\)	9.06147	−	15.6949i	0.244904	−	0.424187i	−0.717200	−	0.696867i	\(-0.754577\pi\)
							0.962105	+	0.272680i	\(0.0879101\pi\)
\(41\)		−	40.8628i		−	0.996654i	−0.866989	−	0.498327i	\(-0.833948\pi\)
							0.866989	−	0.498327i	\(-0.166052\pi\)
\(43\)	−29.7063			−0.690844			−0.345422	−	0.938448i	\(-0.612264\pi\)
							−0.345422	+	0.938448i	\(0.612264\pi\)
\(47\)	6.22822	+	3.59586i	0.132515	+	0.0765077i	0.564792	−	0.825233i	\(-0.308956\pi\)
							−0.432277	+	0.901741i	\(0.642290\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.t.b.86.7	yes	36
3.2	odd	2	inner	105.3.t.b.86.12	yes	36
7.4	even	3	inner	105.3.t.b.11.12	yes	36
21.11	odd	6	inner	105.3.t.b.11.7	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.t.b.11.7	✓	36	21.11	odd	6	inner
105.3.t.b.11.12	yes	36	7.4	even	3	inner
105.3.t.b.86.7	yes	36	1.1	even	1	trivial
105.3.t.b.86.12	yes	36	3.2	odd	2	inner