Embedded newform 552.2.bf.a.5.6

• Label: 552.2.bf.a.5.6
• Level: $552$
• Weight: $2$
• Character: 552.5
• Analytic conductor: $4.408$
• Analytic rank: $0$
• Dimension: $920$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.bf (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.40774219157\)
Analytic rank:	\(0\)
Dimension:	\(920\)
Relative dimension:	\(92\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	552.5
Dual form			552.2.bf.a.221.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39862 - 0.209415i) q^{2} +(1.28405 + 1.16242i) q^{3} +(1.91229 + 0.585785i) q^{4} +(1.33695 + 0.192224i) q^{5} +(-1.55247 - 1.89469i) q^{6} +(3.52408 - 1.60939i) q^{7} +(-2.55190 - 1.21976i) q^{8} +(0.297559 + 2.98521i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 920 q - 22 q^{4} - 2 q^{6} - 44 q^{7} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/552\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)	\(-1\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39862	−	0.209415i	−0.988976	−	0.148079i
							\(3\)	1.28405	+	1.16242i	0.741346	+	0.671124i
\(5\)	1.33695	+	0.192224i	0.597901	+	0.0859651i								0.434619	−	0.900614i	\(-0.356883\pi\)
							0.163282	+	0.986580i	\(0.447792\pi\)
\(7\)	3.52408	−	1.60939i	1.33198	−	0.608294i	0.383032	−	0.923735i	\(-0.374880\pi\)
							0.948945	+	0.315441i	\(0.102153\pi\)
\(11\)	1.20569	+	4.10619i	0.363528	+	1.23806i	0.914858	+	0.403775i	\(0.132302\pi\)
							−0.551331	+	0.834287i	\(0.685880\pi\)
\(13\)	−0.0966927	−	0.0441581i	−0.0268177	−	0.0122472i	0.401961	−	0.915657i	\(-0.368329\pi\)
							−0.428779	+	0.903410i	\(0.641056\pi\)
\(17\)	4.59195	−	2.95107i	1.11371	−	0.715739i	0.151613	−	0.988440i	\(-0.451553\pi\)
							0.962099	+	0.272701i	\(0.0879169\pi\)
\(19\)	−6.18573	−	3.97533i	−1.41910	−	0.912003i	−0.999992	−	0.00406864i	\(-0.998705\pi\)
							−0.419113	−	0.907934i	\(-0.637659\pi\)
\(23\)	−0.114499	−	4.79446i	−0.0238746	−	0.999715i
							\(29\)	−2.18782	+	1.40603i	−0.406268	+	0.261093i	−0.727777	−	0.685814i	\(-0.759446\pi\)
0.321509	+	0.946907i	\(0.395810\pi\)
\(31\)	3.06302	+	3.53491i	0.550134	+	0.634889i	0.960914	−	0.276846i	\(-0.0892891\pi\)
							−0.410780	+	0.911734i	\(0.634744\pi\)
\(37\)	0.0427923	+	0.297627i	0.00703502	+	0.0489296i	0.993037	−	0.117807i	\(-0.0375863\pi\)
							−0.986002	+	0.166736i	\(0.946677\pi\)
\(41\)	−4.00438	−	0.575743i	−0.625379	−	0.0899159i	−0.177663	−	0.984091i	\(-0.556854\pi\)
							−0.447717	+	0.894176i	\(0.647763\pi\)
\(43\)	−1.90676	+	2.20052i	−0.290779	+	0.335576i	−0.882278	−	0.470729i	\(-0.843991\pi\)
							0.591499	+	0.806306i	\(0.298536\pi\)
\(47\)			5.79702i			0.845583i	0.906227	+	0.422791i	\(0.138950\pi\)
							−0.906227	+	0.422791i	\(0.861050\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bf.a.5.6	✓	920
3.2	odd	2	inner	552.2.bf.a.5.87	yes	920
8.5	even	2	inner	552.2.bf.a.5.82	yes	920
23.14	odd	22	inner	552.2.bf.a.221.11	yes	920
24.5	odd	2	inner	552.2.bf.a.5.11	yes	920
69.14	even	22	inner	552.2.bf.a.221.82	yes	920
184.37	odd	22	inner	552.2.bf.a.221.87	yes	920
552.221	even	22	inner	552.2.bf.a.221.6	yes	920

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bf.a.5.6	✓	920	1.1	even	1	trivial
552.2.bf.a.5.11	yes	920	24.5	odd	2	inner
552.2.bf.a.5.82	yes	920	8.5	even	2	inner
552.2.bf.a.5.87	yes	920	3.2	odd	2	inner
552.2.bf.a.221.6	yes	920	552.221	even	22	inner
552.2.bf.a.221.11	yes	920	23.14	odd	22	inner
552.2.bf.a.221.82	yes	920	69.14	even	22	inner
552.2.bf.a.221.87	yes	920	184.37	odd	22	inner