Embedded newform 112.3.l.b.13.16

• Label: 112.3.l.b.13.16
• Level: $112$
• Weight: $3$
• Character: 112.13
• 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			13.16
Character	\(\chi\)	\(=\)	112.13
Dual form			112.3.l.b.69.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.159836 - 1.99360i) q^{2} +(1.79667 - 1.79667i) q^{3} +(-3.94890 - 0.637301i) q^{4} +(-3.85269 - 3.85269i) q^{5} +(-3.29468 - 3.86902i) q^{6} +(-6.95068 + 0.829499i) q^{7} +(-1.90170 + 7.77068i) q^{8} +2.54394i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 4 q^{2} - 8 q^{4} - 16 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.159836	−	1.99360i	0.0799182	−	0.996801i
							\(3\)	1.79667	−	1.79667i	0.598890	−	0.598890i	−0.341127	−	0.940017i	\(-0.610809\pi\)
0.940017	+	0.341127i	\(0.110809\pi\)
\(5\)	−3.85269	−	3.85269i	−0.770538	−	0.770538i	0.207663	−	0.978200i	\(-0.433414\pi\)
							−0.978200	+	0.207663i	\(0.933414\pi\)
\(7\)	−6.95068	+	0.829499i	−0.992954	+	0.118500i
							\(11\)	14.1386	−	14.1386i	1.28532	−	1.28532i	0.347729	−	0.937595i	\(-0.386953\pi\)
0.937595	−	0.347729i	\(-0.113047\pi\)
\(13\)	5.62959	−	5.62959i	0.433045	−	0.433045i	−0.456618	−	0.889663i	\(-0.650939\pi\)
							0.889663	+	0.456618i	\(0.150939\pi\)
\(17\)		−	23.7707i		−	1.39828i	−0.714987	−	0.699138i	\(-0.753567\pi\)
							0.714987	−	0.699138i	\(-0.246433\pi\)
\(19\)	−6.44465	+	6.44465i	−0.339192	+	0.339192i	−0.856063	−	0.516871i	\(-0.827097\pi\)
							0.516871	+	0.856063i	\(0.327097\pi\)
\(23\)			15.1330i			0.657958i	0.944337	+	0.328979i	\(0.106705\pi\)
							−0.944337	+	0.328979i	\(0.893295\pi\)
\(29\)	−10.0604	−	10.0604i	−0.346911	−	0.346911i	0.512047	−	0.858958i	\(-0.328887\pi\)
							−0.858958	+	0.512047i	\(0.828887\pi\)
\(31\)		−	22.6845i		−	0.731757i	−0.930663	−	0.365879i	\(-0.880768\pi\)
							0.930663	−	0.365879i	\(-0.119232\pi\)
\(37\)	31.9062	−	31.9062i	0.862331	−	0.862331i	−0.129278	−	0.991608i	\(-0.541266\pi\)
							0.991608	+	0.129278i	\(0.0412659\pi\)
\(41\)	49.4163			1.20528			0.602638	−	0.798015i	\(-0.294116\pi\)
							0.602638	+	0.798015i	\(0.294116\pi\)
\(43\)	−34.1717	+	34.1717i	−0.794691	+	0.794691i	−0.982253	−	0.187562i	\(-0.939942\pi\)
							0.187562	+	0.982253i	\(0.439942\pi\)
\(47\)			82.6364i			1.75822i	0.476618	+	0.879110i	\(0.341862\pi\)
							−0.476618	+	0.879110i	\(0.658138\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.3.l.b.13.16	yes	56
4.3	odd	2		448.3.l.b.209.9		56
7.6	odd	2	inner	112.3.l.b.13.15	✓	56
16.5	even	4	inner	112.3.l.b.69.15	yes	56
16.11	odd	4		448.3.l.b.433.20		56
28.27	even	2		448.3.l.b.209.20		56
112.27	even	4		448.3.l.b.433.9		56
112.69	odd	4	inner	112.3.l.b.69.16	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.15	✓	56	7.6	odd	2	inner
112.3.l.b.13.16	yes	56	1.1	even	1	trivial
112.3.l.b.69.15	yes	56	16.5	even	4	inner
112.3.l.b.69.16	yes	56	112.69	odd	4	inner
448.3.l.b.209.9		56	4.3	odd	2
448.3.l.b.209.20		56	28.27	even	2
448.3.l.b.433.9		56	112.27	even	4
448.3.l.b.433.20		56	16.11	odd	4