Embedded newform 112.3.l.b.69.2

• Label: 112.3.l.b.69.2
• 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			69.2
Character	\(\chi\)	\(=\)	112.69
Dual form			112.3.l.b.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.97215 - 0.332609i) q^{2} +(3.31524 + 3.31524i) q^{3} +(3.77874 + 1.31191i) q^{4} +(4.66000 - 4.66000i) q^{5} +(-5.43547 - 7.64083i) q^{6} +(2.13987 - 6.66491i) q^{7} +(-7.01589 - 3.84412i) q^{8} +12.9816i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 4 q^{2} - 8 q^{4} - 16 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.97215	−	0.332609i	−0.986074	−	0.166304i
							\(3\)	3.31524	+	3.31524i	1.10508	+	1.10508i	0.993788	+	0.111293i	\(0.0354991\pi\)
0.111293	+	0.993788i	\(0.464501\pi\)
\(5\)	4.66000	−	4.66000i	0.932001	−	0.932001i	−0.0658299	−	0.997831i	\(-0.520969\pi\)
							0.997831	+	0.0658299i	\(0.0209695\pi\)
\(7\)	2.13987	−	6.66491i	0.305695	−	0.952129i
							\(11\)	8.29514	+	8.29514i	0.754103	+	0.754103i	0.975242	−	0.221139i	\(-0.0709774\pi\)
−0.221139	+	0.975242i	\(0.570977\pi\)
\(13\)	−13.1395	−	13.1395i	−1.01073	−	1.01073i	−0.999942	−	0.0107894i	\(-0.996566\pi\)
							−0.0107894	−	0.999942i	\(-0.503434\pi\)
\(17\)			11.5210i			0.677705i	0.940839	+	0.338853i	\(0.110039\pi\)
							−0.940839	+	0.338853i	\(0.889961\pi\)
\(19\)	−5.21978	−	5.21978i	−0.274725	−	0.274725i	0.556274	−	0.830999i	\(-0.312231\pi\)
							−0.830999	+	0.556274i	\(0.812231\pi\)
\(23\)			20.5622i			0.894007i	0.894532	+	0.447003i	\(0.147509\pi\)
							−0.894532	+	0.447003i	\(0.852491\pi\)
\(29\)	−39.7028	+	39.7028i	−1.36906	+	1.36906i	−0.507283	+	0.861780i	\(0.669350\pi\)
							−0.861780	+	0.507283i	\(0.830650\pi\)
\(31\)			12.7004i			0.409691i	0.978794	+	0.204846i	\(0.0656693\pi\)
							−0.978794	+	0.204846i	\(0.934331\pi\)
\(37\)	5.59936	+	5.59936i	0.151334	+	0.151334i	0.778714	−	0.627380i	\(-0.215873\pi\)
							−0.627380	+	0.778714i	\(0.715873\pi\)
\(41\)	−22.7338			−0.554482			−0.277241	−	0.960800i	\(-0.589420\pi\)
							−0.277241	+	0.960800i	\(0.589420\pi\)
\(43\)	−22.2151	−	22.2151i	−0.516631	−	0.516631i	0.399920	−	0.916550i	\(-0.369038\pi\)
							−0.916550	+	0.399920i	\(0.869038\pi\)
\(47\)		−	75.1143i		−	1.59818i	−0.601213	−	0.799089i	\(-0.705316\pi\)
							0.601213	−	0.799089i	\(-0.294684\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.3.l.b.69.2	yes	56
4.3	odd	2		448.3.l.b.433.4		56
7.6	odd	2	inner	112.3.l.b.69.1	yes	56
16.3	odd	4		448.3.l.b.209.25		56
16.13	even	4	inner	112.3.l.b.13.1	✓	56
28.27	even	2		448.3.l.b.433.25		56
112.13	odd	4	inner	112.3.l.b.13.2	yes	56
112.83	even	4		448.3.l.b.209.4		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.1	✓	56	16.13	even	4	inner
112.3.l.b.13.2	yes	56	112.13	odd	4	inner
112.3.l.b.69.1	yes	56	7.6	odd	2	inner
112.3.l.b.69.2	yes	56	1.1	even	1	trivial
448.3.l.b.209.4		56	112.83	even	4
448.3.l.b.209.25		56	16.3	odd	4
448.3.l.b.433.4		56	4.3	odd	2
448.3.l.b.433.25		56	28.27	even	2