Embedded newform 105.4.q.b.4.12

• Label: 105.4.q.b.4.12
• Level: $105$
• Weight: $4$
• Character: 105.4
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4.12
Character	\(\chi\)	\(=\)	105.4
Dual form			105.4.q.b.79.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.755385 - 0.436122i) q^{2} +(2.59808 + 1.50000i) q^{3} +(-3.61960 + 6.26932i) q^{4} +(10.8209 - 2.81204i) q^{5} +2.61673 q^{6} +(-10.7711 + 15.0659i) q^{7} +13.2923i q^{8} +(4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 62 q^{4} - 4 q^{5} + 108 q^{6} + 198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.755385	−	0.436122i	0.267069	−	0.154192i	−0.360486	−	0.932765i	\(-0.617389\pi\)
							0.627555	+	0.778572i	\(0.284056\pi\)
\(3\)	2.59808	+	1.50000i	0.500000	+	0.288675i
							\(5\)	10.8209	−	2.81204i	0.967853	−	0.251516i
\(7\)	−10.7711	+	15.0659i	−0.581587	+	0.813484i
							\(11\)	10.5369	−	18.2505i	0.288818	−	0.500248i	−0.684710	−	0.728816i	\(-0.740071\pi\)
0.973528	+	0.228568i	\(0.0734043\pi\)
\(13\)			62.2537i			1.32816i	0.747661	+	0.664080i	\(0.231177\pi\)
							−0.747661	+	0.664080i	\(0.768823\pi\)
\(17\)	35.9206	+	20.7387i	0.512472	+	0.295876i	0.733849	−	0.679313i	\(-0.237722\pi\)
							−0.221377	+	0.975188i	\(0.571055\pi\)
\(19\)	11.2121	+	19.4200i	0.135381	+	0.234487i	0.925743	−	0.378154i	\(-0.123441\pi\)
							−0.790362	+	0.612640i	\(0.790108\pi\)
\(23\)	33.8044	−	19.5170i	0.306466	−	0.176938i	−0.338878	−	0.940830i	\(-0.610047\pi\)
							0.645344	+	0.763892i	\(0.276714\pi\)
\(29\)	59.0992			0.378429			0.189214	−	0.981936i	\(-0.439406\pi\)
							0.189214	+	0.981936i	\(0.439406\pi\)
\(31\)	152.722	−	264.522i	0.884829	−	1.53257i	0.0389194	−	0.999242i	\(-0.487608\pi\)
							0.845910	−	0.533326i	\(-0.179058\pi\)
\(37\)	−138.335	+	79.8678i	−0.614653	+	0.354870i	−0.774784	−	0.632226i	\(-0.782141\pi\)
							0.160132	+	0.987096i	\(0.448808\pi\)
\(41\)	−497.810			−1.89622			−0.948108	−	0.317949i	\(-0.897006\pi\)
							−0.948108	+	0.317949i	\(0.897006\pi\)
\(43\)		−	257.461i		−	0.913080i	−0.889703	−	0.456540i	\(-0.849088\pi\)
							0.889703	−	0.456540i	\(-0.150912\pi\)
\(47\)	446.810	−	257.966i	1.38668	−	0.800600i	0.393741	−	0.919221i	\(-0.371181\pi\)
							0.992940	+	0.118621i	\(0.0378473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.q.b.4.12	yes	44
5.4	even	2	inner	105.4.q.b.4.11	✓	44
7.2	even	3	inner	105.4.q.b.79.11	yes	44
35.9	even	6	inner	105.4.q.b.79.12	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.q.b.4.11	✓	44	5.4	even	2	inner
105.4.q.b.4.12	yes	44	1.1	even	1	trivial
105.4.q.b.79.11	yes	44	7.2	even	3	inner
105.4.q.b.79.12	yes	44	35.9	even	6	inner