Embedded newform 224.3.w.a.99.7

• Label: 224.3.w.a.99.7
• 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			99.7
Character	\(\chi\)	\(=\)	224.99
Dual form			224.3.w.a.43.7

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80532	−	0.860701i	−0.902662	−	0.430351i
							\(3\)	1.98167	−	4.78418i	0.660557	−	1.59473i	−0.136376	−	0.990657i	\(-0.543546\pi\)
0.796933	−	0.604068i	\(-0.206454\pi\)
\(5\)	1.28939	+	3.11286i	0.257878	+	0.622572i	0.998798	−	0.0490211i	\(-0.0156101\pi\)
							−0.740920	+	0.671594i	\(0.765610\pi\)
\(7\)	1.87083	+	1.87083i	0.267261	+	0.267261i
							\(11\)	−2.68304	−	6.47744i	−0.243913	−	0.588858i	0.753752	−	0.657159i	\(-0.228242\pi\)
−0.997665	+	0.0683012i	\(0.978242\pi\)
\(13\)	4.80288	−	11.5952i	0.369452	−	0.891937i	−0.624388	−	0.781115i	\(-0.714651\pi\)
							0.993840	−	0.110823i	\(-0.0353485\pi\)
\(17\)		−	32.2298i		−	1.89587i	−0.318463	−	0.947935i	\(-0.603167\pi\)
							0.318463	−	0.947935i	\(-0.396833\pi\)
\(19\)	−2.14744	−	0.889498i	−0.113023	−	0.0468157i	0.325455	−	0.945557i	\(-0.394482\pi\)
							−0.438478	+	0.898742i	\(0.644482\pi\)
\(23\)	−20.4827	+	20.4827i	−0.890552	+	0.890552i	−0.994575	−	0.104023i	\(-0.966828\pi\)
							0.104023	+	0.994575i	\(0.466828\pi\)
\(29\)	6.33351	+	2.62343i	0.218397	+	0.0904630i	0.489199	−	0.872172i	\(-0.337289\pi\)
							−0.270803	+	0.962635i	\(0.587289\pi\)
\(31\)		−	2.54417i		−	0.0820701i	−0.999158	−	0.0410351i	\(-0.986934\pi\)
							0.999158	−	0.0410351i	\(-0.0130655\pi\)
\(37\)	2.41415	+	5.82827i	0.0652473	+	0.157521i	0.953140	−	0.302530i	\(-0.0978312\pi\)
							−0.887893	+	0.460051i	\(0.847831\pi\)
\(41\)	−23.5727	−	23.5727i	−0.574945	−	0.574945i	0.358561	−	0.933506i	\(-0.383267\pi\)
							−0.933506	+	0.358561i	\(0.883267\pi\)
\(43\)	22.2031	+	53.6031i	0.516352	+	1.24658i	0.940130	+	0.340817i	\(0.110704\pi\)
							−0.423778	+	0.905766i	\(0.639296\pi\)
\(47\)	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.99.7	yes	192
32.11	odd	8	inner	224.3.w.a.43.7	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.7	✓	192	32.11	odd	8	inner
224.3.w.a.99.7	yes	192	1.1	even	1	trivial