Embedded newform 224.3.w.a.99.1

• Label: 224.3.w.a.99.1
• Level: $224$
• Weight: $3$
• Character: 224.99
• 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			99.1
Character	\(\chi\)	\(=\)	224.99
Dual form			224.3.w.a.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99992 + 0.0176912i) q^{2} +(-0.954366 + 2.30404i) q^{3} +(3.99937 - 0.0707620i) q^{4} +(-0.872291 - 2.10590i) q^{5} +(1.86790 - 4.62479i) q^{6} +(-1.87083 - 1.87083i) q^{7} +(-7.99718 + 0.212272i) q^{8} +(1.96616 + 1.96616i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99992	+	0.0176912i	−0.999961	+	0.00884560i
							\(3\)	−0.954366	+	2.30404i	−0.318122	+	0.768015i	0.681232	+	0.732068i	\(0.261445\pi\)
−0.999354	+	0.0359467i	\(0.988555\pi\)
\(5\)	−0.872291	−	2.10590i	−0.174458	−	0.421179i	0.812329	−	0.583199i	\(-0.198199\pi\)
							−0.986788	+	0.162020i	\(0.948199\pi\)
\(7\)	−1.87083	−	1.87083i	−0.267261	−	0.267261i
							\(11\)	2.92618	+	7.06442i	0.266016	+	0.642220i	0.999289	−	0.0377153i	\(-0.0120080\pi\)
−0.733272	+	0.679935i	\(0.762008\pi\)
\(13\)	−5.89149	+	14.2233i	−0.453192	+	1.09410i	0.517910	+	0.855435i	\(0.326710\pi\)
							−0.971102	+	0.238666i	\(0.923290\pi\)
\(17\)		−	17.6716i		−	1.03950i	−0.854317	−	0.519752i	\(-0.826024\pi\)
							0.854317	−	0.519752i	\(-0.173976\pi\)
\(19\)	−19.0646	−	7.89681i	−1.00340	−	0.415621i	−0.180356	−	0.983601i	\(-0.557725\pi\)
							−0.823042	+	0.567980i	\(0.807725\pi\)
\(23\)	−29.6806	+	29.6806i	−1.29046	+	1.29046i	−0.355962	+	0.934501i	\(0.615847\pi\)
							−0.934501	+	0.355962i	\(0.884153\pi\)
\(29\)	−45.9685	−	19.0408i	−1.58512	−	0.656578i	−0.595906	−	0.803054i	\(-0.703207\pi\)
							−0.989214	+	0.146476i	\(0.953207\pi\)
\(31\)			35.6063i			1.14859i	0.818649	+	0.574295i	\(0.194724\pi\)
							−0.818649	+	0.574295i	\(0.805276\pi\)
\(37\)	17.8954	+	43.2034i	0.483660	+	1.16766i	0.957858	+	0.287241i	\(0.0927380\pi\)
							−0.474198	+	0.880418i	\(0.657262\pi\)
\(41\)	−54.6502	−	54.6502i	−1.33293	−	1.33293i	−0.902735	−	0.430197i	\(-0.858444\pi\)
							−0.430197	−	0.902735i	\(-0.641556\pi\)
\(43\)	−7.16417	−	17.2958i	−0.166609	−	0.402229i	0.818420	−	0.574621i	\(-0.194850\pi\)
							−0.985028	+	0.172392i	\(0.944850\pi\)
\(47\)	−26.6258			−0.566506			−0.283253	−	0.959045i	\(-0.591414\pi\)
							−0.283253	+	0.959045i	\(0.591414\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.99.1	yes	192
32.11	odd	8	inner	224.3.w.a.43.1	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.1	✓	192	32.11	odd	8	inner
224.3.w.a.99.1	yes	192	1.1	even	1	trivial