Embedded newform 224.3.w.a.43.12

• Label: 224.3.w.a.43.12
• 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			43.12
Character	\(\chi\)	\(=\)	224.43
Dual form			224.3.w.a.99.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.49125 - 1.33273i) q^{2} +(-1.51777 - 3.66421i) q^{3} +(0.447645 + 3.97487i) q^{4} +(2.18754 - 5.28118i) q^{5} +(-2.62005 + 7.48703i) q^{6} +(1.87083 - 1.87083i) q^{7} +(4.62989 - 6.52412i) q^{8} +(-4.75887 + 4.75887i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.49125	−	1.33273i	−0.745624	−	0.666367i
							\(3\)	−1.51777	−	3.66421i	−0.505922	−	1.22140i	−0.946212	−	0.323548i	\(-0.895124\pi\)
0.440290	−	0.897856i	\(-0.354876\pi\)
\(5\)	2.18754	−	5.28118i	0.437507	−	1.05624i	−0.539300	−	0.842114i	\(-0.681311\pi\)
							0.976807	−	0.214122i	\(-0.0686889\pi\)
\(7\)	1.87083	−	1.87083i	0.267261	−	0.267261i
							\(11\)	7.75926	−	18.7325i	0.705388	−	1.70296i	−0.00582822	−	0.999983i	\(-0.501855\pi\)
0.711216	−	0.702974i	\(-0.248145\pi\)
\(13\)	−5.88041	−	14.1966i	−0.452339	−	1.09204i	−0.971430	−	0.237325i	\(-0.923730\pi\)
							0.519091	−	0.854719i	\(-0.326270\pi\)
\(17\)			24.0748i			1.41616i	0.706130	+	0.708082i	\(0.250439\pi\)
							−0.706130	+	0.708082i	\(0.749561\pi\)
\(19\)	1.53905	−	0.637494i	0.0810024	−	0.0335523i	−0.341814	−	0.939768i	\(-0.611041\pi\)
							0.422817	+	0.906215i	\(0.361041\pi\)
\(23\)	28.3097	+	28.3097i	1.23085	+	1.23085i	0.963636	+	0.267218i	\(0.0861045\pi\)
							0.267218	+	0.963636i	\(0.413895\pi\)
\(29\)	−13.0610	+	5.41004i	−0.450379	+	0.186553i	−0.596331	−	0.802738i	\(-0.703376\pi\)
							0.145952	+	0.989292i	\(0.453376\pi\)
\(31\)		−	33.1552i		−	1.06952i	−0.845004	−	0.534761i	\(-0.820402\pi\)
							0.845004	−	0.534761i	\(-0.179598\pi\)
\(37\)	−27.7286	+	66.9428i	−0.749422	+	1.80926i	−0.187111	+	0.982339i	\(0.559912\pi\)
							−0.562311	+	0.826926i	\(0.690088\pi\)
\(41\)	−8.73947	+	8.73947i	−0.213158	+	0.213158i	−0.805607	−	0.592450i	\(-0.798161\pi\)
							0.592450	+	0.805607i	\(0.298161\pi\)
\(43\)	6.34896	−	15.3277i	0.147650	−	0.356459i	−0.832700	−	0.553725i	\(-0.813206\pi\)
							0.980350	+	0.197265i	\(0.0632060\pi\)
\(47\)	17.6271			0.375046			0.187523	−	0.982260i	\(-0.439954\pi\)
							0.187523	+	0.982260i	\(0.439954\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.43.12	✓	192
32.3	odd	8	inner	224.3.w.a.99.12	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.12	✓	192	1.1	even	1	trivial
224.3.w.a.99.12	yes	192	32.3	odd	8	inner