Embedded newform 105.4.u.a.52.13

• Label: 105.4.u.a.52.13
• Level: $105$
• Weight: $4$
• Character: 105.52
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			52.13
Character	\(\chi\)	\(=\)	105.52
Dual form			105.4.u.a.103.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.121038 - 0.451720i) q^{2} +(-2.89778 + 0.776457i) q^{3} +(6.73880 + 3.89065i) q^{4} +(-10.1913 - 4.59752i) q^{5} +1.40296i q^{6} +(16.4236 + 8.55956i) q^{7} +(5.21859 - 5.21859i) q^{8} +(7.79423 - 4.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 48 q^{5} - 4 q^{7} + 168 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.121038	−	0.451720i	0.0427934	−	0.159707i	−0.941223	−	0.337787i	\(-0.890322\pi\)
							0.984016	+	0.178080i	\(0.0569884\pi\)
\(3\)	−2.89778	+	0.776457i	−0.557678	+	0.149429i
							\(5\)	−10.1913	−	4.59752i	−0.911539	−	0.411214i
\(7\)	16.4236	+	8.55956i	0.886790	+	0.462173i
							\(11\)	−29.3651	+	50.8618i	−0.804900	+	1.39413i	0.111458	+	0.993769i	\(0.464448\pi\)
−0.916358	+	0.400359i	\(0.868885\pi\)
\(13\)	45.9249	+	45.9249i	0.979791	+	0.979791i	0.999800	−	0.0200092i	\(-0.00636954\pi\)
							−0.0200092	+	0.999800i	\(0.506370\pi\)
\(17\)	11.3974	+	42.5357i	0.162605	+	0.606848i	0.998334	+	0.0577064i	\(0.0183787\pi\)
							−0.835729	+	0.549142i	\(0.814955\pi\)
\(19\)	−29.2541	−	50.6695i	−0.353229	−	0.611810i	0.633584	−	0.773674i	\(-0.281583\pi\)
							−0.986813	+	0.161863i	\(0.948250\pi\)
\(23\)	123.394	+	33.0632i	1.11867	+	0.299746i	0.770344	−	0.637628i	\(-0.220084\pi\)
							0.348324	+	0.937374i	\(0.386751\pi\)
\(29\)			85.9804i			0.550557i	0.961364	+	0.275279i	\(0.0887701\pi\)
							−0.961364	+	0.275279i	\(0.911230\pi\)
\(31\)	−233.242	−	134.663i	−1.35134	−	0.780197i	−0.362903	−	0.931827i	\(-0.618215\pi\)
							−0.988437	+	0.151630i	\(0.951548\pi\)
\(37\)	30.1113	−	112.377i	0.133791	−	0.499315i	−0.866209	−	0.499682i	\(-0.833450\pi\)
							1.00000		0.000367163i	\(0.000116872\pi\)
\(41\)		−	311.430i		−	1.18627i	−0.805102	−	0.593136i	\(-0.797890\pi\)
							0.805102	−	0.593136i	\(-0.202110\pi\)
\(43\)	62.6395	−	62.6395i	0.222150	−	0.222150i	−0.587253	−	0.809403i	\(-0.699791\pi\)
							0.809403	+	0.587253i	\(0.199791\pi\)
\(47\)	−299.192	−	80.1684i	−0.928547	−	0.248803i	−0.237312	−	0.971434i	\(-0.576266\pi\)
							−0.691235	+	0.722630i	\(0.742933\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.u.a.52.13	✓	96
5.3	odd	4	inner	105.4.u.a.73.12	yes	96
7.5	odd	6	inner	105.4.u.a.82.12	yes	96
35.33	even	12	inner	105.4.u.a.103.13	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.u.a.52.13	✓	96	1.1	even	1	trivial
105.4.u.a.73.12	yes	96	5.3	odd	4	inner
105.4.u.a.82.12	yes	96	7.5	odd	6	inner
105.4.u.a.103.13	yes	96	35.33	even	12	inner