Embedded newform 224.3.w.a.99.15

• Label: 224.3.w.a.99.15
• 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			99.15
Character	\(\chi\)	\(=\)	224.99
Dual form			224.3.w.a.43.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03112 - 1.71371i) q^{2} +(0.406888 - 0.982314i) q^{3} +(-1.87359 + 3.53407i) q^{4} +(0.691237 + 1.66879i) q^{5} +(-2.10295 + 0.315595i) q^{6} +(-1.87083 - 1.87083i) q^{7} +(7.98826 - 0.433260i) q^{8} +(5.56458 + 5.56458i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03112	−	1.71371i	−0.515559	−	0.856854i
							\(3\)	0.406888	−	0.982314i	0.135629	−	0.327438i	−0.841443	−	0.540346i	\(-0.818293\pi\)
0.977072	+	0.212908i	\(0.0682934\pi\)
\(5\)	0.691237	+	1.66879i	0.138247	+	0.333759i	0.977807	−	0.209509i	\(-0.0671867\pi\)
							−0.839559	+	0.543268i	\(0.817187\pi\)
\(7\)	−1.87083	−	1.87083i	−0.267261	−	0.267261i
							\(11\)	−6.83275	−	16.4957i	−0.621159	−	1.49961i	−0.850344	−	0.526228i	\(-0.823606\pi\)
0.229184	−	0.973383i	\(-0.426394\pi\)
\(13\)	7.71888	−	18.6350i	0.593760	−	1.43346i	−0.286085	−	0.958204i	\(-0.592354\pi\)
							0.879846	−	0.475260i	\(-0.157646\pi\)
\(17\)		−	1.25394i		−	0.0737609i	−0.999320	−	0.0368805i	\(-0.988258\pi\)
							0.999320	−	0.0368805i	\(-0.0117421\pi\)
\(19\)	−8.00708	−	3.31664i	−0.421425	−	0.174560i	0.161885	−	0.986810i	\(-0.448243\pi\)
							−0.583310	+	0.812250i	\(0.698243\pi\)
\(23\)	4.46320	−	4.46320i	0.194052	−	0.194052i	−0.603392	−	0.797444i	\(-0.706185\pi\)
							0.797444	+	0.603392i	\(0.206185\pi\)
\(29\)	−15.0132	−	6.21865i	−0.517695	−	0.214436i	0.108509	−	0.994095i	\(-0.465392\pi\)
							−0.626204	+	0.779659i	\(0.715392\pi\)
\(31\)		−	1.05830i		−	0.0341388i	−0.999854	−	0.0170694i	\(-0.994566\pi\)
							0.999854	−	0.0170694i	\(-0.00543362\pi\)
\(37\)	17.3032	+	41.7737i	0.467655	+	1.12902i	0.965184	+	0.261572i	\(0.0842407\pi\)
							−0.497529	+	0.867447i	\(0.665759\pi\)
\(41\)	−41.0423	−	41.0423i	−1.00103	−	1.00103i	−0.999999	−	0.00103201i	\(-0.999672\pi\)
							−0.00103201	−	0.999999i	\(-0.500328\pi\)
\(43\)	−18.8034	−	45.3955i	−0.437289	−	1.05571i	−0.976881	−	0.213782i	\(-0.931422\pi\)
							0.539592	−	0.841926i	\(-0.318578\pi\)
\(47\)	−16.4636			−0.350290			−0.175145	−	0.984543i	\(-0.556039\pi\)
							−0.175145	+	0.984543i	\(0.556039\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.99.15	yes	192
32.11	odd	8	inner	224.3.w.a.43.15	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.15	✓	192	32.11	odd	8	inner
224.3.w.a.99.15	yes	192	1.1	even	1	trivial