Embedded newform 105.3.k.c.62.5

• Label: 105.3.k.c.62.5
• Level: $105$
• Weight: $3$
• Character: 105.62
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	\( x^{16} + 433x^{8} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			62.5
Root			\(-0.817327 - 1.97320i\) of defining polynomial
Character	\(\chi\)	\(=\)	105.62
Dual form			105.3.k.c.83.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{2} +(-2.99611 - 0.152778i) q^{3} +3.00000i q^{4} +(4.24762 - 2.63775i) q^{5} +(-2.22660 + 2.01054i) q^{6} +(4.33402 + 5.49694i) q^{7} +(4.94975 + 4.94975i) q^{8} +(8.95332 + 0.915476i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.707107	−	0.707107i	0.353553	−	0.353553i	−0.507877	−	0.861430i	\(-0.669569\pi\)
							0.861430	+	0.507877i	\(0.169569\pi\)
\(3\)	−2.99611	−	0.152778i	−0.998702	−	0.0509259i
							\(5\)	4.24762	−	2.63775i	0.849524	−	0.527550i
\(7\)	4.33402	+	5.49694i	0.619145	+	0.785277i
							\(11\)		−	13.9031i		−	1.26392i	−0.775003	−	0.631958i	\(-0.782252\pi\)
0.775003	−	0.631958i	\(-0.217748\pi\)
\(13\)	14.6307	+	14.6307i	1.12543	+	1.12543i	0.990910	+	0.134524i	\(0.0429507\pi\)
							0.134524	+	0.990910i	\(0.457049\pi\)
\(17\)	4.86435	+	4.86435i	0.286138	+	0.286138i	0.835551	−	0.549413i	\(-0.185149\pi\)
							−0.549413	+	0.835551i	\(0.685149\pi\)
\(19\)	−21.7515			−1.14481			−0.572407	−	0.819970i	\(-0.693990\pi\)
							−0.572407	+	0.819970i	\(0.693990\pi\)
\(23\)	1.77282	+	1.77282i	0.0770791	+	0.0770791i	0.744595	−	0.667516i	\(-0.232643\pi\)
							−0.667516	+	0.744595i	\(0.732643\pi\)
\(29\)	−28.0452			−0.967076			−0.483538	−	0.875323i	\(-0.660648\pi\)
							−0.483538	+	0.875323i	\(0.660648\pi\)
\(31\)		−	17.2472i		−	0.556362i	−0.960529	−	0.278181i	\(-0.910268\pi\)
							0.960529	−	0.278181i	\(-0.0897315\pi\)
\(37\)	−6.50714	−	6.50714i	−0.175869	−	0.175869i	0.613683	−	0.789552i	\(-0.289687\pi\)
							−0.789552	+	0.613683i	\(0.789687\pi\)
\(41\)	26.7192			0.651687			0.325844	−	0.945424i	\(-0.394352\pi\)
							0.325844	+	0.945424i	\(0.394352\pi\)
\(43\)	33.1548	−	33.1548i	0.771041	−	0.771041i	−0.207248	−	0.978289i	\(-0.566451\pi\)
							0.978289	+	0.207248i	\(0.0664506\pi\)
\(47\)	−18.5656	−	18.5656i	−0.395012	−	0.395012i	0.481457	−	0.876470i	\(-0.340108\pi\)
							−0.876470	+	0.481457i	\(0.840108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.k.c.62.5	yes	16
3.2	odd	2	inner	105.3.k.c.62.2	✓	16
5.3	odd	4	inner	105.3.k.c.83.3	yes	16
7.6	odd	2	inner	105.3.k.c.62.8	yes	16
15.8	even	4	inner	105.3.k.c.83.8	yes	16
21.20	even	2	inner	105.3.k.c.62.3	yes	16
35.13	even	4	inner	105.3.k.c.83.2	yes	16
105.83	odd	4	inner	105.3.k.c.83.5	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.k.c.62.2	✓	16	3.2	odd	2	inner
105.3.k.c.62.3	yes	16	21.20	even	2	inner
105.3.k.c.62.5	yes	16	1.1	even	1	trivial
105.3.k.c.62.8	yes	16	7.6	odd	2	inner
105.3.k.c.83.2	yes	16	35.13	even	4	inner
105.3.k.c.83.3	yes	16	5.3	odd	4	inner
105.3.k.c.83.5	yes	16	105.83	odd	4	inner
105.3.k.c.83.8	yes	16	15.8	even	4	inner