Embedded newform 112.3.l.b.69.9

• Label: 112.3.l.b.69.9
• Level: $112$
• Weight: $3$
• Character: 112.69
• Analytic conductor: $3.052$
• Analytic rank: $0$
• Dimension: $56$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	112.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.05177896084\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			69.9
Character	\(\chi\)	\(=\)	112.69
Dual form			112.3.l.b.13.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08539 - 1.67986i) q^{2} +(-2.28266 - 2.28266i) q^{3} +(-1.64387 + 3.64660i) q^{4} +(-2.90674 + 2.90674i) q^{5} +(-1.35698 + 6.31211i) q^{6} +(2.26451 + 6.62359i) q^{7} +(7.91002 - 1.19649i) q^{8} +1.42104i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 4 q^{2} - 8 q^{4} - 16 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08539	−	1.67986i	−0.542693	−	0.839931i
							\(3\)	−2.28266	−	2.28266i	−0.760885	−	0.760885i	0.215597	−	0.976482i	\(-0.430830\pi\)
−0.976482	+	0.215597i	\(0.930830\pi\)
\(5\)	−2.90674	+	2.90674i	−0.581348	+	0.581348i	−0.935274	−	0.353925i	\(-0.884847\pi\)
							0.353925	+	0.935274i	\(0.384847\pi\)
\(7\)	2.26451	+	6.62359i	0.323501	+	0.946228i
							\(11\)	9.96314	+	9.96314i	0.905740	+	0.905740i	0.995925	−	0.0901854i	\(-0.0287460\pi\)
−0.0901854	+	0.995925i	\(0.528746\pi\)
\(13\)	−15.5346	−	15.5346i	−1.19497	−	1.19497i	−0.975655	−	0.219310i	\(-0.929619\pi\)
							−0.219310	−	0.975655i	\(-0.570381\pi\)
\(17\)			32.2316i			1.89597i	0.318309	+	0.947987i	\(0.396885\pi\)
							−0.318309	+	0.947987i	\(0.603115\pi\)
\(19\)	14.4296	+	14.4296i	0.759454	+	0.759454i	0.976223	−	0.216769i	\(-0.0695518\pi\)
							−0.216769	+	0.976223i	\(0.569552\pi\)
\(23\)			16.0348i			0.697165i	0.937278	+	0.348583i	\(0.113337\pi\)
							−0.937278	+	0.348583i	\(0.886663\pi\)
\(29\)	−10.0884	+	10.0884i	−0.347877	+	0.347877i	−0.859318	−	0.511441i	\(-0.829112\pi\)
							0.511441	+	0.859318i	\(0.329112\pi\)
\(31\)			3.11658i			0.100535i	0.998736	+	0.0502674i	\(0.0160073\pi\)
							−0.998736	+	0.0502674i	\(0.983993\pi\)
\(37\)	−36.8766	−	36.8766i	−0.996664	−	0.996664i	0.00333020	−	0.999994i	\(-0.498940\pi\)
							−0.999994	+	0.00333020i	\(0.998940\pi\)
\(41\)	39.5924			0.965668			0.482834	−	0.875712i	\(-0.339607\pi\)
							0.482834	+	0.875712i	\(0.339607\pi\)
\(43\)	−19.5881	−	19.5881i	−0.455536	−	0.455536i	0.441651	−	0.897187i	\(-0.354393\pi\)
							−0.897187	+	0.441651i	\(0.854393\pi\)
\(47\)		−	20.9684i		−	0.446136i	−0.974803	−	0.223068i	\(-0.928393\pi\)
							0.974803	−	0.223068i	\(-0.0716072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.3.l.b.69.9	yes	56
4.3	odd	2		448.3.l.b.433.22		56
7.6	odd	2	inner	112.3.l.b.69.10	yes	56
16.3	odd	4		448.3.l.b.209.7		56
16.13	even	4	inner	112.3.l.b.13.10	yes	56
28.27	even	2		448.3.l.b.433.7		56
112.13	odd	4	inner	112.3.l.b.13.9	✓	56
112.83	even	4		448.3.l.b.209.22		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.9	✓	56	112.13	odd	4	inner
112.3.l.b.13.10	yes	56	16.13	even	4	inner
112.3.l.b.69.9	yes	56	1.1	even	1	trivial
112.3.l.b.69.10	yes	56	7.6	odd	2	inner
448.3.l.b.209.7		56	16.3	odd	4
448.3.l.b.209.22		56	112.83	even	4
448.3.l.b.433.7		56	28.27	even	2
448.3.l.b.433.22		56	4.3	odd	2