Embedded newform 112.2.j.d.83.4

• Label: 112.2.j.d.83.4
• Level: $112$
• Weight: $2$
• Character: 112.83
• Analytic conductor: $0.894$
• Analytic rank: $0$
• Dimension: $16$
• 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			83.4
Root			\(0.944649 - 1.05244i\) of defining polynomial
Character	\(\chi\)	\(=\)	112.83
Dual form			112.2.j.d.27.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.576222 + 1.29150i) q^{2} +(1.28999 + 1.28999i) q^{3} +(-1.33594 - 1.48838i) q^{4} +(0.599312 + 0.599312i) q^{5} +(-2.40933 + 0.922696i) q^{6} +(-0.152445 + 2.64136i) q^{7} +(2.69204 - 0.867721i) q^{8} +0.328129i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} - 8 q^{4} + 8 q^{7} - 16 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.576222	+	1.29150i	−0.407451	+	0.913227i
							\(3\)	1.28999	+	1.28999i	0.744774	+	0.744774i	0.973493	−	0.228719i	\(-0.0734536\pi\)
−0.228719	+	0.973493i	\(0.573454\pi\)
\(5\)	0.599312	+	0.599312i	0.268021	+	0.268021i	0.828302	−	0.560282i	\(-0.189307\pi\)
							−0.560282	+	0.828302i	\(0.689307\pi\)
\(7\)	−0.152445	+	2.64136i	−0.0576187	+	0.998339i
							\(11\)	−2.00000	−	2.00000i	−0.603023	−	0.603023i	0.338091	−	0.941113i	\(-0.390219\pi\)
−0.941113	+	0.338091i	\(0.890219\pi\)
\(13\)	−1.00197	+	1.00197i	−0.277897	+	0.277897i	−0.832269	−	0.554372i	\(-0.812959\pi\)
							0.554372	+	0.832269i	\(0.312959\pi\)
\(17\)		−	6.48134i		−	1.57195i	−0.618255	−	0.785977i	\(-0.712160\pi\)
							0.618255	−	0.785977i	\(-0.287840\pi\)
\(19\)	3.93134	+	3.93134i	0.901912	+	0.901912i	0.995601	−	0.0936897i	\(-0.0298662\pi\)
							−0.0936897	+	0.995601i	\(0.529866\pi\)
\(23\)	4.40731			0.918988			0.459494	−	0.888181i	\(-0.348031\pi\)
							0.459494	+	0.888181i	\(0.348031\pi\)
\(29\)	0.241319	+	0.241319i	0.0448119	+	0.0448119i	0.729158	−	0.684346i	\(-0.239912\pi\)
							−0.684346	+	0.729158i	\(0.739912\pi\)
\(31\)	−3.38529			−0.608017			−0.304008	−	0.952669i	\(-0.598325\pi\)
							−0.304008	+	0.952669i	\(0.598325\pi\)
\(37\)	2.73544	−	2.73544i	0.449704	−	0.449704i	−0.445552	−	0.895256i	\(-0.646993\pi\)
							0.895256	+	0.445552i	\(0.146993\pi\)
\(41\)	−4.88941			−0.763597			−0.381799	−	0.924246i	\(-0.624695\pi\)
							−0.381799	+	0.924246i	\(0.624695\pi\)
\(43\)	4.40731	+	4.40731i	0.672109	+	0.672109i	0.958202	−	0.286093i	\(-0.0923566\pi\)
							−0.286093	+	0.958202i	\(0.592357\pi\)
\(47\)	−9.45461			−1.37910			−0.689548	−	0.724240i	\(-0.742191\pi\)
							−0.689548	+	0.724240i	\(0.742191\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.2.j.d.83.4	yes	16
4.3	odd	2		448.2.j.d.111.3		16
7.2	even	3		784.2.w.e.227.7		32
7.3	odd	6		784.2.w.e.19.3		32
7.4	even	3		784.2.w.e.19.4		32
7.5	odd	6		784.2.w.e.227.8		32
7.6	odd	2	inner	112.2.j.d.83.3	yes	16
8.3	odd	2		896.2.j.g.223.6		16
8.5	even	2		896.2.j.h.223.3		16
16.3	odd	4		896.2.j.h.671.6		16
16.5	even	4		448.2.j.d.335.6		16
16.11	odd	4	inner	112.2.j.d.27.3	✓	16
16.13	even	4		896.2.j.g.671.3		16
28.27	even	2		448.2.j.d.111.6		16
56.13	odd	2		896.2.j.h.223.6		16
56.27	even	2		896.2.j.g.223.3		16
112.11	odd	12		784.2.w.e.411.8		32
112.13	odd	4		896.2.j.g.671.6		16
112.27	even	4	inner	112.2.j.d.27.4	yes	16
112.59	even	12		784.2.w.e.411.7		32
112.69	odd	4		448.2.j.d.335.3		16
112.75	even	12		784.2.w.e.619.4		32
112.83	even	4		896.2.j.h.671.3		16
112.107	odd	12		784.2.w.e.619.3		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.j.d.27.3	✓	16	16.11	odd	4	inner
112.2.j.d.27.4	yes	16	112.27	even	4	inner
112.2.j.d.83.3	yes	16	7.6	odd	2	inner
112.2.j.d.83.4	yes	16	1.1	even	1	trivial
448.2.j.d.111.3		16	4.3	odd	2
448.2.j.d.111.6		16	28.27	even	2
448.2.j.d.335.3		16	112.69	odd	4
448.2.j.d.335.6		16	16.5	even	4
784.2.w.e.19.3		32	7.3	odd	6
784.2.w.e.19.4		32	7.4	even	3
784.2.w.e.227.7		32	7.2	even	3
784.2.w.e.227.8		32	7.5	odd	6
784.2.w.e.411.7		32	112.59	even	12
784.2.w.e.411.8		32	112.11	odd	12
784.2.w.e.619.3		32	112.107	odd	12
784.2.w.e.619.4		32	112.75	even	12
896.2.j.g.223.3		16	56.27	even	2
896.2.j.g.223.6		16	8.3	odd	2
896.2.j.g.671.3		16	16.13	even	4
896.2.j.g.671.6		16	112.13	odd	4
896.2.j.h.223.3		16	8.5	even	2
896.2.j.h.223.6		16	56.13	odd	2
896.2.j.h.671.3		16	112.83	even	4
896.2.j.h.671.6		16	16.3	odd	4