Embedded newform 112.2.j.d.83.8

• Label: 112.2.j.d.83.8
• Level: $112$
• Weight: $2$
• Character: 112.83
• Analytic conductor: $0.894$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	112.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.894324502638\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			83.8
Root			\(-0.517174 - 1.31626i\) of defining polynomial
Character	\(\chi\)	\(=\)	112.83
Dual form			112.2.j.d.27.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.796431 + 1.16863i) q^{2} +(1.50619 + 1.50619i) q^{3} +(-0.731395 + 1.86147i) q^{4} +(-2.54054 - 2.54054i) q^{5} +(-0.560603 + 2.95975i) q^{6} +(2.59286 - 0.526369i) q^{7} +(-2.75787 + 0.627801i) q^{8} +1.53721i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} - 8 q^{4} + 8 q^{7} - 16 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(17\)	\(85\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.796431	+	1.16863i	0.563162	+	0.826347i
							\(3\)	1.50619	+	1.50619i	0.869599	+	0.869599i	0.992428	−	0.122829i	\(-0.0391967\pi\)
−0.122829	+	0.992428i	\(0.539197\pi\)
\(5\)	−2.54054	−	2.54054i	−1.13616	−	1.13616i	−0.989132	−	0.147031i	\(-0.953028\pi\)
							−0.147031	−	0.989132i	\(-0.546972\pi\)
\(7\)	2.59286	−	0.526369i	0.980010	−	0.198949i
							\(11\)	−2.00000	−	2.00000i	−0.603023	−	0.603023i	0.338091	−	0.941113i	\(-0.390219\pi\)
−0.941113	+	0.338091i	\(0.890219\pi\)
\(13\)	1.17573	−	1.17573i	0.326090	−	0.326090i	−0.525008	−	0.851098i	\(-0.675938\pi\)
							0.851098	+	0.525008i	\(0.175938\pi\)
\(17\)			6.13381i			1.48767i	0.668364	+	0.743834i	\(0.266995\pi\)
							−0.668364	+	0.743834i	\(0.733005\pi\)
\(19\)	0.979820	+	0.979820i	0.224786	+	0.224786i	0.810510	−	0.585724i	\(-0.199190\pi\)
							−0.585724	+	0.810510i	\(0.699190\pi\)
\(23\)	0.207188			0.0432017			0.0216009	−	0.999767i	\(-0.493124\pi\)
							0.0216009	+	0.999767i	\(0.493124\pi\)
\(29\)	−3.46733	−	3.46733i	−0.643868	−	0.643868i	0.307636	−	0.951504i	\(-0.400462\pi\)
							−0.951504	+	0.307636i	\(0.900462\pi\)
\(31\)	−5.74198			−1.03129			−0.515645	−	0.856802i	\(-0.672448\pi\)
							−0.515645	+	0.856802i	\(0.672448\pi\)
\(37\)	−0.255601	+	0.255601i	−0.0420206	+	0.0420206i	−0.727805	−	0.685784i	\(-0.759459\pi\)
							0.685784	+	0.727805i	\(0.259459\pi\)
\(41\)	−6.75794			−1.05541			−0.527707	−	0.849427i	\(-0.676948\pi\)
							−0.527707	+	0.849427i	\(0.676948\pi\)
\(43\)	0.207188	+	0.207188i	0.0315959	+	0.0315959i	0.722728	−	0.691132i	\(-0.242888\pi\)
							−0.691132	+	0.722728i	\(0.742888\pi\)
\(47\)	10.9321			1.59461			0.797307	−	0.603575i	\(-0.206257\pi\)
							0.797307	+	0.603575i	\(0.206257\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.2.j.d.83.8	yes	16
4.3	odd	2		448.2.j.d.111.2		16
7.2	even	3		784.2.w.e.227.3		32
7.3	odd	6		784.2.w.e.19.1		32
7.4	even	3		784.2.w.e.19.2		32
7.5	odd	6		784.2.w.e.227.4		32
7.6	odd	2	inner	112.2.j.d.83.7	yes	16
8.3	odd	2		896.2.j.g.223.7		16
8.5	even	2		896.2.j.h.223.2		16
16.3	odd	4		896.2.j.h.671.7		16
16.5	even	4		448.2.j.d.335.7		16
16.11	odd	4	inner	112.2.j.d.27.7	✓	16
16.13	even	4		896.2.j.g.671.2		16
28.27	even	2		448.2.j.d.111.7		16
56.13	odd	2		896.2.j.h.223.7		16
56.27	even	2		896.2.j.g.223.2		16
112.11	odd	12		784.2.w.e.411.4		32
112.13	odd	4		896.2.j.g.671.7		16
112.27	even	4	inner	112.2.j.d.27.8	yes	16
112.59	even	12		784.2.w.e.411.3		32
112.69	odd	4		448.2.j.d.335.2		16
112.75	even	12		784.2.w.e.619.2		32
112.83	even	4		896.2.j.h.671.2		16
112.107	odd	12		784.2.w.e.619.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.j.d.27.7	✓	16	16.11	odd	4	inner
112.2.j.d.27.8	yes	16	112.27	even	4	inner
112.2.j.d.83.7	yes	16	7.6	odd	2	inner
112.2.j.d.83.8	yes	16	1.1	even	1	trivial
448.2.j.d.111.2		16	4.3	odd	2
448.2.j.d.111.7		16	28.27	even	2
448.2.j.d.335.2		16	112.69	odd	4
448.2.j.d.335.7		16	16.5	even	4
784.2.w.e.19.1		32	7.3	odd	6
784.2.w.e.19.2		32	7.4	even	3
784.2.w.e.227.3		32	7.2	even	3
784.2.w.e.227.4		32	7.5	odd	6
784.2.w.e.411.3		32	112.59	even	12
784.2.w.e.411.4		32	112.11	odd	12
784.2.w.e.619.1		32	112.107	odd	12
784.2.w.e.619.2		32	112.75	even	12
896.2.j.g.223.2		16	56.27	even	2
896.2.j.g.223.7		16	8.3	odd	2
896.2.j.g.671.2		16	16.13	even	4
896.2.j.g.671.7		16	112.13	odd	4
896.2.j.h.223.2		16	8.5	even	2
896.2.j.h.223.7		16	56.13	odd	2
896.2.j.h.671.2		16	112.83	even	4
896.2.j.h.671.7		16	16.3	odd	4