Embedded newform 105.3.t.b.11.7

• Label: 105.3.t.b.11.7
• Level: $105$
• Weight: $3$
• Character: 105.11
• 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			11.7
Character	\(\chi\)	\(=\)	105.11
Dual form			105.3.t.b.86.7

$q$-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 \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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	\(-0.758651\pi\)
							0.232474	+	0.972603i	\(0.425318\pi\)
\(3\)	−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	−0.213909	−	0.976854i	\(-0.568620\pi\)
−0.952935	+	0.303176i	\(0.901953\pi\)
\(13\)	−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	\(-0.273019\pi\)
							−0.327933	+	0.944701i	\(0.606352\pi\)
\(19\)	3.55819	+	6.16296i	0.187273	+	0.324367i	0.944340	−	0.328971i	\(-0.106702\pi\)
							−0.757067	+	0.653337i	\(0.773368\pi\)
\(23\)	28.8602	−	16.6624i	1.25479	−	0.724453i	0.282733	−	0.959199i	\(-0.408759\pi\)
							0.972057	+	0.234746i	\(0.0754257\pi\)
\(29\)		−	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	−0.881749	−	0.471718i	\(-0.843634\pi\)
							0.849395	+	0.527758i	\(0.176967\pi\)
\(37\)	9.06147	+	15.6949i	0.244904	+	0.424187i	0.962105	−	0.272680i	\(-0.0879101\pi\)
							−0.717200	+	0.696867i	\(0.754577\pi\)
\(41\)			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	\(-0.642290\pi\)
							0.564792	+	0.825233i	\(0.308956\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.t.b.11.7	✓	36	1.1	even	1	trivial
105.3.t.b.11.12	yes	36	3.2	odd	2	inner
105.3.t.b.86.7	yes	36	21.2	odd	6	inner
105.3.t.b.86.12	yes	36	7.2	even	3	inner