Embedded newform 112.2.v.a.3.9

• Label: 112.2.v.a.3.9
• Level: $112$
• Weight: $2$
• Character: 112.3
• Analytic conductor: $0.894$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	112.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.894324502638\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			3.9
Character	\(\chi\)	\(=\)	112.3
Dual form			112.2.v.a.75.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.187641 + 1.40171i) q^{2} +(0.198087 - 0.0530773i) q^{3} +(-1.92958 + 0.526037i) q^{4} +(-0.487744 + 1.82029i) q^{5} +(0.111568 + 0.267701i) q^{6} +(1.84933 + 1.89208i) q^{7} +(-1.09942 - 2.60601i) q^{8} +(-2.56165 + 1.47897i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{2} - 6 q^{3} - 4 q^{4} - 6 q^{5} - 8 q^{7} + 4 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(17\)	\(85\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.187641	+	1.40171i	0.132682	+	0.991159i
							\(3\)	0.198087	−	0.0530773i	0.114366	−	0.0306442i	−0.201182	−	0.979554i	\(-0.564478\pi\)
0.315548	+	0.948910i	\(0.397812\pi\)
\(5\)	−0.487744	+	1.82029i	−0.218126	+	0.814057i	0.766917	+	0.641746i	\(0.221790\pi\)
							−0.985043	+	0.172310i	\(0.944877\pi\)
\(7\)	1.84933	+	1.89208i	0.698982	+	0.715140i
							\(11\)	4.07225	−	1.09116i	1.22783	−	0.328996i	0.414096	−	0.910233i	\(-0.364098\pi\)
0.813734	+	0.581237i	\(0.197431\pi\)
\(13\)	2.63146	−	2.63146i	0.729835	−	0.729835i	−0.240752	−	0.970587i	\(-0.577394\pi\)
							0.970587	+	0.240752i	\(0.0773940\pi\)
\(17\)	−5.54801	−	3.20315i	−1.34559	−	0.776877i	−0.357969	−	0.933733i	\(-0.616531\pi\)
							−0.987621	+	0.156856i	\(0.949864\pi\)
\(19\)	1.29610	−	4.83712i	0.297346	−	1.10971i	−0.641990	−	0.766713i	\(-0.721891\pi\)
							0.939336	−	0.342998i	\(-0.111442\pi\)
\(23\)	2.51654	+	4.35877i	0.524735	+	0.908867i	0.999585	+	0.0288006i	\(0.00916880\pi\)
							−0.474850	+	0.880066i	\(0.657498\pi\)
\(29\)	0.0380909	+	0.0380909i	0.00707330	+	0.00707330i	0.710635	−	0.703561i	\(-0.248408\pi\)
							−0.703561	+	0.710635i	\(0.748408\pi\)
\(31\)	2.32199	−	4.02180i	0.417041	−	0.722337i	−0.578599	−	0.815612i	\(-0.696400\pi\)
							0.995640	+	0.0932755i	\(0.0297337\pi\)
\(37\)	1.30517	+	0.349719i	0.214569	+	0.0574935i	0.364502	−	0.931203i	\(-0.381239\pi\)
							−0.149933	+	0.988696i	\(0.547906\pi\)
\(41\)	−11.2477			−1.75659			−0.878295	−	0.478120i	\(-0.841319\pi\)
							−0.878295	+	0.478120i	\(0.841319\pi\)
\(43\)	−3.36902	−	3.36902i	−0.513770	−	0.513770i	0.401909	−	0.915679i	\(-0.368347\pi\)
							−0.915679	+	0.401909i	\(0.868347\pi\)
\(47\)	1.66673	+	2.88687i	0.243118	+	0.421093i	0.961601	−	0.274452i	\(-0.0884964\pi\)
							−0.718483	+	0.695545i	\(0.755163\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.2.v.a.3.9	✓	56
4.3	odd	2		448.2.z.a.367.6		56
7.2	even	3		784.2.w.f.19.11		56
7.3	odd	6		784.2.j.a.195.4		56
7.4	even	3		784.2.j.a.195.3		56
7.5	odd	6	inner	112.2.v.a.19.11	yes	56
7.6	odd	2		784.2.w.f.227.9		56
8.3	odd	2		896.2.z.a.479.9		56
8.5	even	2		896.2.z.b.479.6		56
16.3	odd	4		896.2.z.b.31.6		56
16.5	even	4		448.2.z.a.143.6		56
16.11	odd	4	inner	112.2.v.a.59.11	yes	56
16.13	even	4		896.2.z.a.31.9		56
28.19	even	6		448.2.z.a.47.6		56
56.5	odd	6		896.2.z.b.607.6		56
56.19	even	6		896.2.z.a.607.9		56
112.5	odd	12		448.2.z.a.271.6		56
112.11	odd	12		784.2.j.a.587.4		56
112.19	even	12		896.2.z.b.159.6		56
112.27	even	4		784.2.w.f.619.11		56
112.59	even	12		784.2.j.a.587.3		56
112.61	odd	12		896.2.z.a.159.9		56
112.75	even	12	inner	112.2.v.a.75.9	yes	56
112.107	odd	12		784.2.w.f.411.9		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.v.a.3.9	✓	56	1.1	even	1	trivial
112.2.v.a.19.11	yes	56	7.5	odd	6	inner
112.2.v.a.59.11	yes	56	16.11	odd	4	inner
112.2.v.a.75.9	yes	56	112.75	even	12	inner
448.2.z.a.47.6		56	28.19	even	6
448.2.z.a.143.6		56	16.5	even	4
448.2.z.a.271.6		56	112.5	odd	12
448.2.z.a.367.6		56	4.3	odd	2
784.2.j.a.195.3		56	7.4	even	3
784.2.j.a.195.4		56	7.3	odd	6
784.2.j.a.587.3		56	112.59	even	12
784.2.j.a.587.4		56	112.11	odd	12
784.2.w.f.19.11		56	7.2	even	3
784.2.w.f.227.9		56	7.6	odd	2
784.2.w.f.411.9		56	112.107	odd	12
784.2.w.f.619.11		56	112.27	even	4
896.2.z.a.31.9		56	16.13	even	4
896.2.z.a.159.9		56	112.61	odd	12
896.2.z.a.479.9		56	8.3	odd	2
896.2.z.a.607.9		56	56.19	even	6
896.2.z.b.31.6		56	16.3	odd	4
896.2.z.b.159.6		56	112.19	even	12
896.2.z.b.479.6		56	8.5	even	2
896.2.z.b.607.6		56	56.5	odd	6