Embedded newform 224.3.w.a.43.7

• Label: 224.3.w.a.43.7
• 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			43.7
Character	\(\chi\)	\(=\)	224.43
Dual form			224.3.w.a.99.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80532 + 0.860701i) q^{2} +(1.98167 + 4.78418i) q^{3} +(2.51839 - 3.10769i) q^{4} +(1.28939 - 3.11286i) q^{5} +(-7.69530 - 6.93136i) q^{6} +(1.87083 - 1.87083i) q^{7} +(-1.87171 + 7.77796i) q^{8} +(-12.5974 + 12.5974i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80532	+	0.860701i	−0.902662	+	0.430351i
							\(3\)	1.98167	+	4.78418i	0.660557	+	1.59473i	0.796933	+	0.604068i	\(0.206454\pi\)
−0.136376	+	0.990657i	\(0.543546\pi\)
\(5\)	1.28939	−	3.11286i	0.257878	−	0.622572i	−0.740920	−	0.671594i	\(-0.765610\pi\)
							0.998798	+	0.0490211i	\(0.0156101\pi\)
\(7\)	1.87083	−	1.87083i	0.267261	−	0.267261i
							\(11\)	−2.68304	+	6.47744i	−0.243913	+	0.588858i	−0.997665	−	0.0683012i	\(-0.978242\pi\)
0.753752	+	0.657159i	\(0.228242\pi\)
\(13\)	4.80288	+	11.5952i	0.369452	+	0.891937i	0.993840	+	0.110823i	\(0.0353485\pi\)
							−0.624388	+	0.781115i	\(0.714651\pi\)
\(17\)			32.2298i			1.89587i	0.318463	+	0.947935i	\(0.396833\pi\)
							−0.318463	+	0.947935i	\(0.603167\pi\)
\(19\)	−2.14744	+	0.889498i	−0.113023	+	0.0468157i	−0.438478	−	0.898742i	\(-0.644482\pi\)
							0.325455	+	0.945557i	\(0.394482\pi\)
\(23\)	−20.4827	−	20.4827i	−0.890552	−	0.890552i	0.104023	−	0.994575i	\(-0.466828\pi\)
							−0.994575	+	0.104023i	\(0.966828\pi\)
\(29\)	6.33351	−	2.62343i	0.218397	−	0.0904630i	−0.270803	−	0.962635i	\(-0.587289\pi\)
							0.489199	+	0.872172i	\(0.337289\pi\)
\(31\)			2.54417i			0.0820701i	0.999158	+	0.0410351i	\(0.0130655\pi\)
							−0.999158	+	0.0410351i	\(0.986934\pi\)
\(37\)	2.41415	−	5.82827i	0.0652473	−	0.157521i	−0.887893	−	0.460051i	\(-0.847831\pi\)
							0.953140	+	0.302530i	\(0.0978312\pi\)
\(41\)	−23.5727	+	23.5727i	−0.574945	+	0.574945i	−0.933506	−	0.358561i	\(-0.883267\pi\)
							0.358561	+	0.933506i	\(0.383267\pi\)
\(43\)	22.2031	−	53.6031i	0.516352	−	1.24658i	−0.423778	−	0.905766i	\(-0.639296\pi\)
							0.940130	−	0.340817i	\(-0.110704\pi\)
\(47\)	79.1930			1.68496			0.842479	−	0.538730i	\(-0.181096\pi\)
							0.842479	+	0.538730i	\(0.181096\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.43.7	✓	192
32.3	odd	8	inner	224.3.w.a.99.7	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.7	✓	192	1.1	even	1	trivial
224.3.w.a.99.7	yes	192	32.3	odd	8	inner