Embedded newform 552.2.bb.a.85.25

• Label: 552.2.bb.a.85.25
• Level: $552$
• Weight: $2$
• Character: 552.85
• Analytic conductor: $4.408$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.40774219157\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(48\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			85.25
Character	\(\chi\)	\(=\)	552.85
Dual form			552.2.bb.a.13.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0519801 + 1.41326i) q^{2} +(0.540641 + 0.841254i) q^{3} +(-1.99460 - 0.146923i) q^{4} +(2.13541 - 0.975208i) q^{5} +(-1.21701 + 0.720336i) q^{6} +(0.595694 - 0.174912i) q^{7} +(0.311319 - 2.81124i) q^{8} +(-0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{2} + 4 q^{6} + 8 q^{7} + 4 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{4}{11}\right)\)	\(1\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0519801	+	1.41326i	−0.0367555	+	0.999324i
							\(3\)	0.540641	+	0.841254i	0.312139	+	0.485698i
\(5\)	2.13541	−	0.975208i	0.954983	−	0.436126i								0.123912	−	0.992293i	\(-0.460456\pi\)
							0.831070	+	0.556167i	\(0.187729\pi\)
\(7\)	0.595694	−	0.174912i	0.225151	−	0.0661104i	−0.167211	−	0.985921i	\(-0.553476\pi\)
							0.392362	+	0.919811i	\(0.371658\pi\)
\(11\)	−1.11157	+	0.963180i	−0.335151	+	0.290410i	−0.806135	−	0.591732i	\(-0.798444\pi\)
							0.470984	+	0.882142i	\(0.343899\pi\)
\(13\)	−0.995277	+	3.38960i	−0.276040	+	0.940106i	0.698447	+	0.715662i	\(0.253875\pi\)
							−0.974487	+	0.224444i	\(0.927943\pi\)
\(17\)	−1.11608	+	7.76251i	−0.270689	+	1.88268i	0.170632	+	0.985335i	\(0.445419\pi\)
							−0.441321	+	0.897349i	\(0.645490\pi\)
\(19\)	6.09868	−	0.876857i	1.39913	−	0.201165i	0.598885	−	0.800835i	\(-0.295611\pi\)
							0.800247	+	0.599670i	\(0.204701\pi\)
\(23\)	4.79557	+	0.0498387i	0.999946	+	0.0103921i
							\(29\)	1.45547	+	0.209266i	0.270275	+	0.0388596i	0.276120	−	0.961123i	\(-0.410951\pi\)
−0.00584489	+	0.999983i	\(0.501860\pi\)
\(31\)	5.62873	+	3.61737i	1.01095	+	0.649698i	0.937639	−	0.347611i	\(-0.113007\pi\)
							0.0733112	+	0.997309i	\(0.476643\pi\)
\(37\)	−9.90037	−	4.52135i	−1.62761	−	0.743305i	−0.628215	−	0.778040i	\(-0.716214\pi\)
							−0.999397	+	0.0347345i	\(0.988941\pi\)
\(41\)	−1.33535	−	2.92401i	−0.208546	−	0.456653i	0.776237	−	0.630442i	\(-0.217126\pi\)
							−0.984783	+	0.173789i	\(0.944399\pi\)
\(43\)	−5.83694	−	9.08245i	−0.890124	−	1.38506i	−0.922670	−	0.385590i	\(-0.873998\pi\)
							0.0325457	−	0.999470i	\(-0.489639\pi\)
\(47\)	10.8617			1.58435			0.792173	−	0.610297i	\(-0.208950\pi\)
							0.792173	+	0.610297i	\(0.208950\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.bb.a.85.25	yes	480
8.5	even	2	inner	552.2.bb.a.85.2	yes	480
23.13	even	11	inner	552.2.bb.a.13.2	✓	480
184.13	even	22	inner	552.2.bb.a.13.25	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.bb.a.13.2	✓	480	23.13	even	11	inner
552.2.bb.a.13.25	yes	480	184.13	even	22	inner
552.2.bb.a.85.2	yes	480	8.5	even	2	inner
552.2.bb.a.85.25	yes	480	1.1	even	1	trivial