Embedded newform 224.3.w.a.99.12

• Label: 224.3.w.a.99.12
• 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			99.12
Character	\(\chi\)	\(=\)	224.99
Dual form			224.3.w.a.43.12

$q$-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 \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.49125	+	1.33273i	−0.745624	+	0.666367i
							\(3\)	−1.51777	+	3.66421i	−0.505922	+	1.22140i	0.440290	+	0.897856i	\(0.354876\pi\)
−0.946212	+	0.323548i	\(0.895124\pi\)
\(5\)	2.18754	+	5.28118i	0.437507	+	1.05624i	0.976807	+	0.214122i	\(0.0686889\pi\)
							−0.539300	+	0.842114i	\(0.681311\pi\)
\(7\)	1.87083	+	1.87083i	0.267261	+	0.267261i
							\(11\)	7.75926	+	18.7325i	0.705388	+	1.70296i	0.711216	+	0.702974i	\(0.248145\pi\)
−0.00582822	+	0.999983i	\(0.501855\pi\)
\(13\)	−5.88041	+	14.1966i	−0.452339	+	1.09204i	0.519091	+	0.854719i	\(0.326270\pi\)
							−0.971430	+	0.237325i	\(0.923730\pi\)
\(17\)		−	24.0748i		−	1.41616i	−0.706130	−	0.708082i	\(-0.749561\pi\)
							0.706130	−	0.708082i	\(-0.250439\pi\)
\(19\)	1.53905	+	0.637494i	0.0810024	+	0.0335523i	0.422817	−	0.906215i	\(-0.361041\pi\)
							−0.341814	+	0.939768i	\(0.611041\pi\)
\(23\)	28.3097	−	28.3097i	1.23085	−	1.23085i	0.267218	−	0.963636i	\(-0.413895\pi\)
							0.963636	−	0.267218i	\(-0.0861045\pi\)
\(29\)	−13.0610	−	5.41004i	−0.450379	−	0.186553i	0.145952	−	0.989292i	\(-0.453376\pi\)
							−0.596331	+	0.802738i	\(0.703376\pi\)
\(31\)			33.1552i			1.06952i	0.845004	+	0.534761i	\(0.179598\pi\)
							−0.845004	+	0.534761i	\(0.820402\pi\)
\(37\)	−27.7286	−	66.9428i	−0.749422	−	1.80926i	−0.562311	−	0.826926i	\(-0.690088\pi\)
							−0.187111	−	0.982339i	\(-0.559912\pi\)
\(41\)	−8.73947	−	8.73947i	−0.213158	−	0.213158i	0.592450	−	0.805607i	\(-0.298161\pi\)
							−0.805607	+	0.592450i	\(0.798161\pi\)
\(43\)	6.34896	+	15.3277i	0.147650	+	0.356459i	0.980350	−	0.197265i	\(-0.0632060\pi\)
							−0.832700	+	0.553725i	\(0.813206\pi\)
\(47\)	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.99.12	yes	192
32.11	odd	8	inner	224.3.w.a.43.12	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.12	✓	192	32.11	odd	8	inner
224.3.w.a.99.12	yes	192	1.1	even	1	trivial