Embedded newform 224.3.w.a.43.5

• Label: 224.3.w.a.43.5
• 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			43.5
Character	\(\chi\)	\(=\)	224.43
Dual form			224.3.w.a.99.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86971 - 0.710051i) q^{2} +(0.216404 + 0.522445i) q^{3} +(2.99165 + 2.65518i) q^{4} +(2.53768 - 6.12651i) q^{5} +(-0.0336503 - 1.13048i) q^{6} +(1.87083 - 1.87083i) q^{7} +(-3.70822 - 7.08866i) q^{8} +(6.13784 - 6.13784i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86971	−	0.710051i	−0.934857	−	0.355026i
							\(3\)	0.216404	+	0.522445i	0.0721346	+	0.174148i	0.955834	−	0.293906i	\(-0.0949552\pi\)
−0.883700	+	0.468054i	\(0.844955\pi\)
\(5\)	2.53768	−	6.12651i	0.507536	−	1.22530i	−0.437761	−	0.899091i	\(-0.644228\pi\)
							0.945297	−	0.326210i	\(-0.105772\pi\)
\(7\)	1.87083	−	1.87083i	0.267261	−	0.267261i
							\(11\)	−7.96509	+	19.2294i	−0.724099	+	1.74813i	−0.0627773	+	0.998028i	\(0.519996\pi\)
−0.661322	+	0.750102i	\(0.730004\pi\)
\(13\)	−5.18638	−	12.5210i	−0.398952	−	0.963156i	−0.987915	−	0.154996i	\(-0.950464\pi\)
							0.588963	−	0.808160i	\(-0.299536\pi\)
\(17\)		−	4.24643i		−	0.249790i	−0.992170	−	0.124895i	\(-0.960141\pi\)
							0.992170	−	0.124895i	\(-0.0398594\pi\)
\(19\)	30.5012	−	12.6340i	1.60533	−	0.664948i	0.613170	−	0.789951i	\(-0.289894\pi\)
							0.992156	+	0.125003i	\(0.0398940\pi\)
\(23\)	−26.4678	−	26.4678i	−1.15078	−	1.15078i	−0.986397	−	0.164378i	\(-0.947438\pi\)
							−0.164378	−	0.986397i	\(-0.552562\pi\)
\(29\)	−18.1205	+	7.50575i	−0.624844	+	0.258819i	−0.672561	−	0.740042i	\(-0.734806\pi\)
							0.0477166	+	0.998861i	\(0.484806\pi\)
\(31\)		−	23.3942i		−	0.754653i	−0.926080	−	0.377326i	\(-0.876844\pi\)
							0.926080	−	0.377326i	\(-0.123156\pi\)
\(37\)	16.0518	−	38.7526i	0.433833	−	1.04737i	−0.544207	−	0.838951i	\(-0.683169\pi\)
							0.978040	−	0.208415i	\(-0.0668306\pi\)
\(41\)	11.2734	−	11.2734i	0.274961	−	0.274961i	−0.556132	−	0.831094i	\(-0.687715\pi\)
							0.831094	+	0.556132i	\(0.187715\pi\)
\(43\)	5.38077	−	12.9903i	0.125134	−	0.302101i	−0.848881	−	0.528584i	\(-0.822723\pi\)
							0.974015	+	0.226484i	\(0.0727230\pi\)
\(47\)	2.38320			0.0507063			0.0253532	−	0.999679i	\(-0.491929\pi\)
							0.0253532	+	0.999679i	\(0.491929\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.43.5	✓	192
32.3	odd	8	inner	224.3.w.a.99.5	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.5	✓	192	1.1	even	1	trivial
224.3.w.a.99.5	yes	192	32.3	odd	8	inner