Embedded newform 105.4.u.a.52.1

• Label: 105.4.u.a.52.1
• 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			52.1
Character	\(\chi\)	\(=\)	105.52
Dual form			105.4.u.a.103.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41620 + 5.28532i) q^{2} +(-2.89778 + 0.776457i) q^{3} +(-19.0007 - 10.9701i) q^{4} +(11.0616 - 1.62545i) q^{5} -16.4153i q^{6} +(0.587638 - 18.5109i) q^{7} +(53.9362 - 53.9362i) q^{8} +(7.79423 - 4.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 48 q^{5} - 4 q^{7} + 168 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41620	+	5.28532i	−0.500701	+	1.86864i	−0.00528365	+	0.999986i	\(0.501682\pi\)
							−0.495417	+	0.868655i	\(0.664985\pi\)
\(3\)	−2.89778	+	0.776457i	−0.557678	+	0.149429i
							\(5\)	11.0616	−	1.62545i	0.989375	−	0.145385i
\(7\)	0.587638	−	18.5109i	0.0317295	−	0.999496i
							\(11\)	−5.58669	+	9.67644i	−0.153132	+	0.265232i	−0.932377	−	0.361487i	\(-0.882269\pi\)
0.779245	+	0.626719i	\(0.215603\pi\)
\(13\)	−19.9526	−	19.9526i	−0.425681	−	0.425681i	0.461473	−	0.887154i	\(-0.347321\pi\)
							−0.887154	+	0.461473i	\(0.847321\pi\)
\(17\)	−29.9098	−	111.625i	−0.426718	−	1.59253i	−0.760143	−	0.649755i	\(-0.774871\pi\)
							0.333426	−	0.942776i	\(-0.391795\pi\)
\(19\)	−12.4107	−	21.4960i	−0.149853	−	0.259554i	0.781320	−	0.624131i	\(-0.214547\pi\)
							−0.931173	+	0.364577i	\(0.881214\pi\)
\(23\)	78.4671	+	21.0252i	0.711371	+	0.190611i	0.596318	−	0.802748i	\(-0.296630\pi\)
							0.115053	+	0.993359i	\(0.463296\pi\)
\(29\)			82.1833i			0.526243i	0.964763	+	0.263121i	\(0.0847520\pi\)
							−0.964763	+	0.263121i	\(0.915248\pi\)
\(31\)	98.9454	+	57.1262i	0.573262	+	0.330973i	0.758451	−	0.651730i	\(-0.225956\pi\)
							−0.185189	+	0.982703i	\(0.559290\pi\)
\(37\)	84.1735	−	314.140i	0.374001	−	1.39579i	−0.480798	−	0.876831i	\(-0.659653\pi\)
							0.854799	−	0.518959i	\(-0.173680\pi\)
\(41\)		−	326.134i		−	1.24228i	−0.783699	−	0.621141i	\(-0.786670\pi\)
							0.783699	−	0.621141i	\(-0.213330\pi\)
\(43\)	−15.6299	+	15.6299i	−0.0554311	+	0.0554311i	−0.734279	−	0.678848i	\(-0.762480\pi\)
							0.678848	+	0.734279i	\(0.262480\pi\)
\(47\)	270.331	+	72.4350i	0.838975	+	0.224803i	0.652625	−	0.757681i	\(-0.273668\pi\)
							0.186350	+	0.982483i	\(0.440334\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.u.a.52.1	✓	96
5.3	odd	4	inner	105.4.u.a.73.24	yes	96
7.5	odd	6	inner	105.4.u.a.82.24	yes	96
35.33	even	12	inner	105.4.u.a.103.1	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.u.a.52.1	✓	96	1.1	even	1	trivial
105.4.u.a.73.24	yes	96	5.3	odd	4	inner
105.4.u.a.82.24	yes	96	7.5	odd	6	inner
105.4.u.a.103.1	yes	96	35.33	even	12	inner