Embedded newform 105.3.t.b.86.12

• Label: 105.3.t.b.86.12
• 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			86.12
Character	\(\chi\)	\(=\)	105.86
Dual form			105.3.t.b.11.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.987174 + 0.569945i) q^{2} +(2.69338 + 1.32124i) q^{3} +(-1.35032 - 2.33883i) q^{4} +(1.93649 + 1.11803i) q^{5} +(1.90581 + 2.83938i) q^{6} +(2.61061 + 6.49498i) q^{7} -7.63801i q^{8} +(5.50864 + 7.11722i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 4 q^{3} + 36 q^{4} - 24 q^{6} - 58 q^{7} - 2 q^{9}+O(q^{10}) \)

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	0.726061	−	0.687630i	\(-0.241349\pi\)
							−0.232474	+	0.972603i	\(0.574682\pi\)
\(3\)	2.69338	+	1.32124i	0.897795	+	0.440414i
							\(5\)	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	0.213909	−	0.976854i	\(-0.431380\pi\)
0.952935	+	0.303176i	\(0.0980470\pi\)
\(13\)	−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	−0.654168	−	0.756349i	\(-0.726981\pi\)
							0.327933	+	0.944701i	\(0.393648\pi\)
\(19\)	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	−0.282733	−	0.959199i	\(-0.591241\pi\)
							−0.972057	+	0.234746i	\(0.924574\pi\)
\(29\)		−	32.6179i		−	1.12476i	−0.826880	−	0.562378i	\(-0.809887\pi\)
							0.826880	−	0.562378i	\(-0.190113\pi\)
\(31\)	−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	\(0.166052\pi\)
							−0.866989	+	0.498327i	\(0.833948\pi\)
\(43\)	−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	0.432277	−	0.901741i	\(-0.357710\pi\)
							−0.564792	+	0.825233i	\(0.691044\pi\)

(See \(a_n\) instead)

Twists

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

    

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