Embedded newform 105.3.w.a.17.4

• Label: 105.3.w.a.17.4
• Level: $105$
• Weight: $3$
• Character: 105.17
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	105.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(28\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			17.4
Character	\(\chi\)	\(=\)	105.17
Dual form			105.3.w.a.68.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.02468 - 0.810462i) q^{2} +(-0.664693 - 2.92544i) q^{3} +(5.02776 + 2.90278i) q^{4} +(-2.83293 + 4.12001i) q^{5} +(-0.360468 + 9.38723i) q^{6} +(-5.96475 - 3.66357i) q^{7} +(-3.99791 - 3.99791i) q^{8} +(-8.11637 + 3.88904i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 6 q^{3} - 16 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.02468	−	0.810462i	−1.51234	−	0.405231i	−0.595130	−	0.803630i	\(-0.702899\pi\)
							−0.917212	+	0.398399i	\(0.869566\pi\)
\(3\)	−0.664693	−	2.92544i	−0.221564	−	0.975146i
							\(5\)	−2.83293	+	4.12001i	−0.566586	+	0.824002i
\(7\)	−5.96475	−	3.66357i	−0.852108	−	0.523367i
							\(11\)	10.3392	+	5.96936i	0.939930	+	0.542669i	0.889938	−	0.456081i	\(-0.150747\pi\)
0.0499915	+	0.998750i	\(0.484081\pi\)
\(13\)	7.11430	+	7.11430i	0.547254	+	0.547254i	0.925645	−	0.378392i	\(-0.123523\pi\)
							−0.378392	+	0.925645i	\(0.623523\pi\)
\(17\)	3.02956	−	0.811768i	0.178209	−	0.0477511i	−0.168611	−	0.985683i	\(-0.553928\pi\)
							0.346820	+	0.937932i	\(0.387261\pi\)
\(19\)	17.2142	+	29.8159i	0.906013	+	1.56926i	0.819552	+	0.573005i	\(0.194222\pi\)
							0.0864609	+	0.996255i	\(0.472444\pi\)
\(23\)	−2.51441	+	9.38391i	−0.109322	+	0.407996i	−0.998800	−	0.0489831i	\(-0.984402\pi\)
							0.889477	+	0.456979i	\(0.151069\pi\)
\(29\)	−26.5784			−0.916496			−0.458248	−	0.888824i	\(-0.651523\pi\)
							−0.458248	+	0.888824i	\(0.651523\pi\)
\(31\)	0.677558	+	0.391188i	0.0218567	+	0.0126190i	0.510889	−	0.859647i	\(-0.329316\pi\)
							−0.489032	+	0.872266i	\(0.662650\pi\)
\(37\)	−7.62362	+	28.4517i	−0.206044	+	0.768965i	0.783085	+	0.621914i	\(0.213645\pi\)
							−0.989129	+	0.147051i	\(0.953022\pi\)
\(41\)	−43.0909			−1.05100			−0.525499	−	0.850794i	\(-0.676121\pi\)
							−0.525499	+	0.850794i	\(0.676121\pi\)
\(43\)	−40.7104	+	40.7104i	−0.946754	+	0.946754i	−0.998652	−	0.0518979i	\(-0.983473\pi\)
							0.0518979	+	0.998652i	\(0.483473\pi\)
\(47\)	−7.76382	+	28.9750i	−0.165188	+	0.616489i	0.832828	+	0.553531i	\(0.186720\pi\)
							−0.998016	+	0.0629579i	\(0.979947\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.w.a.17.4	✓	112
3.2	odd	2	inner	105.3.w.a.17.25	yes	112
5.3	odd	4	inner	105.3.w.a.38.4	yes	112
7.5	odd	6	inner	105.3.w.a.47.25	yes	112
15.8	even	4	inner	105.3.w.a.38.25	yes	112
21.5	even	6	inner	105.3.w.a.47.4	yes	112
35.33	even	12	inner	105.3.w.a.68.25	yes	112
105.68	odd	12	inner	105.3.w.a.68.4	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.w.a.17.4	✓	112	1.1	even	1	trivial
105.3.w.a.17.25	yes	112	3.2	odd	2	inner
105.3.w.a.38.4	yes	112	5.3	odd	4	inner
105.3.w.a.38.25	yes	112	15.8	even	4	inner
105.3.w.a.47.4	yes	112	21.5	even	6	inner
105.3.w.a.47.25	yes	112	7.5	odd	6	inner
105.3.w.a.68.4	yes	112	105.68	odd	12	inner
105.3.w.a.68.25	yes	112	35.33	even	12	inner