Embedded newform 113.2.i.a.9.2

• Label: 113.2.i.a.9.2
• Level: $113$
• Weight: $2$
• Character: 113.9
• Analytic conductor: $0.902$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	113.i (of order \(56\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.902309542840\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(9\) over \(\Q(\zeta_{56})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{56}]$

Embedding invariants

Embedding label			9.2
Character	\(\chi\)	\(=\)	113.9
Dual form			113.2.i.a.88.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.72874 - 1.37862i) q^{2} +(0.800598 + 0.0449606i) q^{3} +(0.642891 + 2.81669i) q^{4} +(0.220921 + 3.93387i) q^{5} +(-1.32204 - 1.18145i) q^{6} +(3.01720 + 1.45301i) q^{7} +(0.853007 - 1.77129i) q^{8} +(-2.34220 - 0.263903i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 28 q^{2} - 24 q^{3} + 16 q^{4} - 24 q^{5} - 20 q^{6} - 12 q^{7} - 28 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/113\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{56}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.72874	−	1.37862i	−1.22240	−	0.974832i	−0.222401	−	0.974955i	\(-0.571390\pi\)
							−1.00000		0.000122733i	\(0.999961\pi\)
\(3\)	0.800598	+	0.0449606i	0.462225	+	0.0259580i	0.285660	−	0.958331i	\(-0.407787\pi\)
							0.176565	+	0.984289i	\(0.443501\pi\)
\(5\)	0.220921	+	3.93387i	0.0987990	+	1.75928i	0.521724	+	0.853114i	\(0.325289\pi\)
							−0.422925	+	0.906164i	\(0.638997\pi\)
\(7\)	3.01720	+	1.45301i	1.14039	+	0.549184i	0.906134	−	0.422990i	\(-0.139019\pi\)
							0.234259	+	0.972174i	\(0.424734\pi\)
\(11\)	0.430096	+	0.684494i	0.129679	+	0.206383i	0.905252	−	0.424875i	\(-0.139682\pi\)
							−0.775573	+	0.631258i	\(0.782539\pi\)
\(13\)	−1.33850	−	3.82521i	−0.371233	−	1.06092i	−0.966338	−	0.257275i	\(-0.917175\pi\)
							0.595105	−	0.803648i	\(-0.297110\pi\)
\(17\)	3.41242	+	0.983100i	0.827633	+	0.238437i	0.663666	−	0.748029i	\(-0.269000\pi\)
							0.163967	+	0.986466i	\(0.447571\pi\)
\(19\)	1.75893	−	1.57188i	0.403527	−	0.360614i	−0.439725	−	0.898133i	\(-0.644924\pi\)
							0.843252	+	0.537519i	\(0.180638\pi\)
\(23\)	−3.16577	+	3.54249i	−0.660108	+	0.738661i	−0.977802	−	0.209532i	\(-0.932806\pi\)
							0.317694	+	0.948193i	\(0.397092\pi\)
\(29\)	2.21733	−	7.69654i	0.411749	−	1.42921i	−0.436544	−	0.899683i	\(-0.643798\pi\)
							0.848292	−	0.529528i	\(-0.177631\pi\)
\(31\)	3.17515	−	1.11103i	0.570274	−	0.199547i	−0.0297113	−	0.999559i	\(-0.509459\pi\)
							0.599985	+	0.800011i	\(0.295173\pi\)
\(37\)	8.42703	+	5.97929i	1.38539	+	0.982990i	0.998137	+	0.0610182i	\(0.0194348\pi\)
							0.387258	+	0.921972i	\(0.373422\pi\)
\(41\)	2.38288	−	3.79233i	0.372144	−	0.592263i	−0.607180	−	0.794565i	\(-0.707699\pi\)
							0.979323	+	0.202301i	\(0.0648420\pi\)
\(43\)	−3.99100	+	2.20575i	−0.608622	+	0.336373i	−0.757849	−	0.652429i	\(-0.773750\pi\)
							0.149228	+	0.988803i	\(0.452321\pi\)
\(47\)	1.71666	−	10.1035i	0.250401	−	1.47375i	−0.529745	−	0.848157i	\(-0.677712\pi\)
							0.780146	−	0.625597i	\(-0.215145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	113.2.i.a.9.2	✓	216
113.88	even	56	inner	113.2.i.a.88.2	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
113.2.i.a.9.2	✓	216	1.1	even	1	trivial
113.2.i.a.88.2	yes	216	113.88	even	56	inner