Embedded newform 112.2.j.d.83.1

• Label: 112.2.j.d.83.1
• 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			83.1
Root			\(-1.40927 + 0.118126i\) of defining polynomial
Character	\(\chi\)	\(=\)	112.83
Dual form			112.2.j.d.27.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41298 - 0.0591148i) q^{2} +(-2.23450 - 2.23450i) q^{3} +(1.99301 + 0.167056i) q^{4} +(-0.584038 - 0.584038i) q^{5} +(3.02521 + 3.28940i) q^{6} +(-1.82596 + 1.91465i) q^{7} +(-2.80620 - 0.353863i) q^{8} +6.98602i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} - 8 q^{4} + 8 q^{7} - 16 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41298	−	0.0591148i	−0.999126	−	0.0418005i
							\(3\)	−2.23450	−	2.23450i	−1.29009	−	1.29009i	−0.934728	−	0.355364i	\(-0.884357\pi\)
−0.355364	−	0.934728i	\(-0.615643\pi\)
\(5\)	−0.584038	−	0.584038i	−0.261190	−	0.261190i	0.564347	−	0.825537i	\(-0.309128\pi\)
							−0.825537	+	0.564347i	\(0.809128\pi\)
\(7\)	−1.82596	+	1.91465i	−0.690146	+	0.723670i
							\(11\)	−2.00000	−	2.00000i	−0.603023	−	0.603023i	0.338091	−	0.941113i	\(-0.390219\pi\)
−0.941113	+	0.338091i	\(0.890219\pi\)
\(13\)	−2.91203	+	2.91203i	−0.807651	+	0.807651i	−0.984278	−	0.176627i	\(-0.943481\pi\)
							0.176627	+	0.984278i	\(0.443481\pi\)
\(17\)		−	2.66123i		−	0.645442i	−0.946494	−	0.322721i	\(-0.895402\pi\)
							0.946494	−	0.322721i	\(-0.104598\pi\)
\(19\)	−0.319854	−	0.319854i	−0.0733795	−	0.0733795i	0.669465	−	0.742844i	\(-0.266524\pi\)
							−0.742844	+	0.669465i	\(0.766524\pi\)
\(23\)	−3.27830			−0.683572			−0.341786	−	0.939778i	\(-0.611032\pi\)
							−0.341786	+	0.939778i	\(0.611032\pi\)
\(29\)	−2.04184	−	2.04184i	−0.379160	−	0.379160i	0.491639	−	0.870799i	\(-0.336398\pi\)
							−0.870799	+	0.491639i	\(0.836398\pi\)
\(31\)	−2.52312			−0.453166			−0.226583	−	0.973992i	\(-0.572756\pi\)
							−0.226583	+	0.973992i	\(0.572756\pi\)
\(37\)	1.70773	−	1.70773i	0.280748	−	0.280748i	−0.552659	−	0.833407i	\(-0.686387\pi\)
							0.833407	+	0.552659i	\(0.186387\pi\)
\(41\)	−11.9895			−1.87245			−0.936224	−	0.351405i	\(-0.885704\pi\)
							−0.936224	+	0.351405i	\(0.885704\pi\)
\(43\)	−3.27830	−	3.27830i	−0.499936	−	0.499936i	0.411482	−	0.911418i	\(-0.365011\pi\)
							−0.911418	+	0.411482i	\(0.865011\pi\)
\(47\)	9.96799			1.45398			0.726991	−	0.686647i	\(-0.240918\pi\)
							0.726991	+	0.686647i	\(0.240918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.2.j.d.83.1	yes	16
4.3	odd	2		448.2.j.d.111.8		16
7.2	even	3		784.2.w.e.227.6		32
7.3	odd	6		784.2.w.e.19.6		32
7.4	even	3		784.2.w.e.19.5		32
7.5	odd	6		784.2.w.e.227.5		32
7.6	odd	2	inner	112.2.j.d.83.2	yes	16
8.3	odd	2		896.2.j.g.223.1		16
8.5	even	2		896.2.j.h.223.8		16
16.3	odd	4		896.2.j.h.671.1		16
16.5	even	4		448.2.j.d.335.1		16
16.11	odd	4	inner	112.2.j.d.27.2	yes	16
16.13	even	4		896.2.j.g.671.8		16
28.27	even	2		448.2.j.d.111.1		16
56.13	odd	2		896.2.j.h.223.1		16
56.27	even	2		896.2.j.g.223.8		16
112.11	odd	12		784.2.w.e.411.5		32
112.13	odd	4		896.2.j.g.671.1		16
112.27	even	4	inner	112.2.j.d.27.1	✓	16
112.59	even	12		784.2.w.e.411.6		32
112.69	odd	4		448.2.j.d.335.8		16
112.75	even	12		784.2.w.e.619.5		32
112.83	even	4		896.2.j.h.671.8		16
112.107	odd	12		784.2.w.e.619.6		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.j.d.27.1	✓	16	112.27	even	4	inner
112.2.j.d.27.2	yes	16	16.11	odd	4	inner
112.2.j.d.83.1	yes	16	1.1	even	1	trivial
112.2.j.d.83.2	yes	16	7.6	odd	2	inner
448.2.j.d.111.1		16	28.27	even	2
448.2.j.d.111.8		16	4.3	odd	2
448.2.j.d.335.1		16	16.5	even	4
448.2.j.d.335.8		16	112.69	odd	4
784.2.w.e.19.5		32	7.4	even	3
784.2.w.e.19.6		32	7.3	odd	6
784.2.w.e.227.5		32	7.5	odd	6
784.2.w.e.227.6		32	7.2	even	3
784.2.w.e.411.5		32	112.11	odd	12
784.2.w.e.411.6		32	112.59	even	12
784.2.w.e.619.5		32	112.75	even	12
784.2.w.e.619.6		32	112.107	odd	12
896.2.j.g.223.1		16	8.3	odd	2
896.2.j.g.223.8		16	56.27	even	2
896.2.j.g.671.1		16	112.13	odd	4
896.2.j.g.671.8		16	16.13	even	4
896.2.j.h.223.1		16	56.13	odd	2
896.2.j.h.223.8		16	8.5	even	2
896.2.j.h.671.1		16	16.3	odd	4
896.2.j.h.671.8		16	112.83	even	4