Embedded newform 224.3.w.a.43.19

• Label: 224.3.w.a.43.19
• Level: $224$
• Weight: $3$
• Character: 224.43
• Analytic conductor: $6.104$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.w (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			43.19
Character	\(\chi\)	\(=\)	224.43
Dual form			224.3.w.a.99.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.518278 + 1.93168i) q^{2} +(-0.266183 - 0.642623i) q^{3} +(-3.46277 - 2.00230i) q^{4} +(-0.747592 + 1.80485i) q^{5} +(1.37930 - 0.181123i) q^{6} +(1.87083 - 1.87083i) q^{7} +(5.66248 - 5.65123i) q^{8} +(6.02185 - 6.02185i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.518278	+	1.93168i	−0.259139	+	0.965840i
							\(3\)	−0.266183	−	0.642623i	−0.0887278	−	0.214208i	0.873286	−	0.487207i	\(-0.161984\pi\)
−0.962014	+	0.272999i	\(0.911984\pi\)
\(5\)	−0.747592	+	1.80485i	−0.149518	+	0.360969i	−0.980838	−	0.194826i	\(-0.937586\pi\)
							0.831320	+	0.555795i	\(0.187586\pi\)
\(7\)	1.87083	−	1.87083i	0.267261	−	0.267261i
							\(11\)	0.854581	−	2.06314i	0.0776892	−	0.187558i	−0.880263	−	0.474486i	\(-0.842634\pi\)
0.957952	+	0.286928i	\(0.0926339\pi\)
\(13\)	−2.40343	−	5.80239i	−0.184879	−	0.446338i	0.804081	−	0.594520i	\(-0.202658\pi\)
							−0.988960	+	0.148182i	\(0.952658\pi\)
\(17\)			18.1690i			1.06877i	0.845242	+	0.534384i	\(0.179456\pi\)
							−0.845242	+	0.534384i	\(0.820544\pi\)
\(19\)	34.1345	−	14.1390i	1.79655	−	0.744156i	0.808808	−	0.588073i	\(-0.200113\pi\)
							0.987744	−	0.156083i	\(-0.0498868\pi\)
\(23\)	6.06571	+	6.06571i	0.263727	+	0.263727i	0.826566	−	0.562839i	\(-0.190291\pi\)
							−0.562839	+	0.826566i	\(0.690291\pi\)
\(29\)	29.1695	−	12.0824i	1.00584	−	0.416635i	0.181908	−	0.983316i	\(-0.441773\pi\)
							0.823937	+	0.566681i	\(0.191773\pi\)
\(31\)		−	17.4712i		−	0.563586i	−0.959475	−	0.281793i	\(-0.909071\pi\)
							0.959475	−	0.281793i	\(-0.0909292\pi\)
\(37\)	5.11785	−	12.3556i	0.138320	−	0.333935i	−0.839507	−	0.543350i	\(-0.817156\pi\)
							0.977827	+	0.209415i	\(0.0671558\pi\)
\(41\)	−19.9681	+	19.9681i	−0.487028	+	0.487028i	−0.907367	−	0.420339i	\(-0.861911\pi\)
							0.420339	+	0.907367i	\(0.361911\pi\)
\(43\)	17.0934	−	41.2671i	0.397520	−	0.959699i	−0.590732	−	0.806868i	\(-0.701161\pi\)
							0.988252	−	0.152831i	\(-0.0488390\pi\)
\(47\)	35.7853			0.761389			0.380694	−	0.924701i	\(-0.375685\pi\)
							0.380694	+	0.924701i	\(0.375685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.43.19	✓	192
32.3	odd	8	inner	224.3.w.a.99.19	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.19	✓	192	1.1	even	1	trivial
224.3.w.a.99.19	yes	192	32.3	odd	8	inner