Embedded newform 105.4.u.a.52.9

• Label: 105.4.u.a.52.9
• 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			52.9
Character	\(\chi\)	\(=\)	105.52
Dual form			105.4.u.a.103.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.514613 + 1.92056i) q^{2} +(-2.89778 + 0.776457i) q^{3} +(3.50448 + 2.02331i) q^{4} +(5.44490 - 9.76489i) q^{5} -5.96493i q^{6} +(14.3387 - 11.7218i) q^{7} +(-16.9369 + 16.9369i) q^{8} +(7.79423 - 4.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 48 q^{5} - 4 q^{7} + 168 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.514613	+	1.92056i	−0.181943	+	0.679021i	0.813321	+	0.581815i	\(0.197657\pi\)
							−0.995264	+	0.0972058i	\(0.969010\pi\)
\(3\)	−2.89778	+	0.776457i	−0.557678	+	0.149429i
							\(5\)	5.44490	−	9.76489i	0.487007	−	0.873398i
\(7\)	14.3387	−	11.7218i	0.774217	−	0.632920i
							\(11\)	25.8685	−	44.8056i	0.709059	−	1.22813i	−0.256148	−	0.966638i	\(-0.582453\pi\)
0.965207	−	0.261489i	\(-0.0842134\pi\)
\(13\)	23.1875	+	23.1875i	0.494697	+	0.494697i	0.909782	−	0.415086i	\(-0.136248\pi\)
							−0.415086	+	0.909782i	\(0.636248\pi\)
\(17\)	18.3267	+	68.3961i	0.261463	+	0.975793i	0.964380	+	0.264521i	\(0.0852139\pi\)
							−0.702917	+	0.711272i	\(0.748119\pi\)
\(19\)	54.2846	+	94.0236i	0.655460	+	1.13529i	0.981778	+	0.190030i	\(0.0608584\pi\)
							−0.326319	+	0.945260i	\(0.605808\pi\)
\(23\)	−7.58687	−	2.03289i	−0.0687814	−	0.0184299i	0.224264	−	0.974528i	\(-0.428002\pi\)
							−0.293046	+	0.956098i	\(0.594669\pi\)
\(29\)		−	265.694i		−	1.70132i	−0.525720	−	0.850658i	\(-0.676204\pi\)
							0.525720	−	0.850658i	\(-0.323796\pi\)
\(31\)	173.396	+	100.110i	1.00461	+	0.580010i	0.909608	−	0.415468i	\(-0.136382\pi\)
							0.0949984	+	0.995477i	\(0.469715\pi\)
\(37\)	−25.1866	+	93.9977i	−0.111910	+	0.417652i	−0.999037	−	0.0438720i	\(-0.986031\pi\)
							0.887128	+	0.461524i	\(0.152697\pi\)
\(41\)		−	271.303i		−	1.03342i	−0.856159	−	0.516712i	\(-0.827156\pi\)
							0.856159	−	0.516712i	\(-0.172844\pi\)
\(43\)	35.2525	−	35.2525i	0.125022	−	0.125022i	−0.641827	−	0.766849i	\(-0.721823\pi\)
							0.766849	+	0.641827i	\(0.221823\pi\)
\(47\)	−407.080	−	109.077i	−1.26338	−	0.338521i	−0.435887	−	0.900002i	\(-0.643565\pi\)
							−0.827490	+	0.561481i	\(0.810232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.u.a.52.9	✓	96
5.3	odd	4	inner	105.4.u.a.73.16	yes	96
7.5	odd	6	inner	105.4.u.a.82.16	yes	96
35.33	even	12	inner	105.4.u.a.103.9	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.u.a.52.9	✓	96	1.1	even	1	trivial
105.4.u.a.73.16	yes	96	5.3	odd	4	inner
105.4.u.a.82.16	yes	96	7.5	odd	6	inner
105.4.u.a.103.9	yes	96	35.33	even	12	inner