Embedded newform 552.2.t.b.19.1

• Label: 552.2.t.b.19.1
• Level: $552$
• Weight: $2$
• Character: 552.19
• Analytic conductor: $4.408$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.40774219157\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			19.1
Character	\(\chi\)	\(=\)	552.19
Dual form			552.2.t.b.523.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40456 - 0.164932i) q^{2} +(-0.959493 + 0.281733i) q^{3} +(1.94559 + 0.463315i) q^{4} +(-1.02510 - 0.658795i) q^{5} +(1.39414 - 0.237460i) q^{6} +(-0.612113 - 4.25734i) q^{7} +(-2.65630 - 0.971645i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 4 q^{2} - 24 q^{3} - 4 q^{4} - 7 q^{6} + 4 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(185\)	\(277\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{15}{22}\right)\)	\(1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40456	−	0.164932i	−0.993176	−	0.116624i
							\(3\)	−0.959493	+	0.281733i	−0.553964	+	0.162658i
\(5\)	−1.02510	−	0.658795i	−0.458441	−	0.294622i								0.290964	−	0.956734i	\(-0.406024\pi\)
							−0.749405	+	0.662112i	\(0.769660\pi\)
\(7\)	−0.612113	−	4.25734i	−0.231357	−	1.60912i	−0.692243	−	0.721665i	\(-0.743377\pi\)
							0.460886	−	0.887459i	\(-0.347532\pi\)
\(11\)	1.75095	+	0.799634i	0.527933	+	0.241099i	0.661506	−	0.749940i	\(-0.269918\pi\)
							−0.133573	+	0.991039i	\(0.542645\pi\)
\(13\)	−2.37914	−	0.342068i	−0.659854	−	0.0948727i	−0.195751	−	0.980654i	\(-0.562714\pi\)
							−0.464103	+	0.885781i	\(0.653623\pi\)
\(17\)	−2.69157	+	2.33226i	−0.652801	+	0.565655i	−0.917037	−	0.398803i	\(-0.869426\pi\)
							0.264236	+	0.964458i	\(0.414880\pi\)
\(19\)	3.95458	+	3.42666i	0.907243	+	0.786130i	0.977398	−	0.211406i	\(-0.0678042\pi\)
							−0.0701558	+	0.997536i	\(0.522350\pi\)
\(23\)	−4.66973	+	1.09253i	−0.973706	+	0.227809i
							\(29\)	−3.39709	+	2.94360i	−0.630824	+	0.546612i	−0.910514	−	0.413478i	\(-0.864314\pi\)
0.279690	+	0.960090i	\(0.409768\pi\)
\(31\)	1.41352	−	4.81401i	0.253876	−	0.864621i	−0.729645	−	0.683826i	\(-0.760315\pi\)
							0.983521	−	0.180795i	\(-0.0578671\pi\)
\(37\)	−5.95436	+	3.82664i	−0.978891	+	0.629095i	−0.929164	−	0.369668i	\(-0.879471\pi\)
							−0.0497270	+	0.998763i	\(0.515835\pi\)
\(41\)	−6.71275	−	4.31402i	−1.04836	−	0.673737i	−0.101316	−	0.994854i	\(-0.532305\pi\)
							−0.947040	+	0.321117i	\(0.895942\pi\)
\(43\)	0.0911620	+	0.310469i	0.0139021	+	0.0473461i	0.966152	−	0.257974i	\(-0.0830548\pi\)
							−0.952250	+	0.305320i	\(0.901237\pi\)
\(47\)			6.43841i			0.939138i	0.882896	+	0.469569i	\(0.155591\pi\)
							−0.882896	+	0.469569i	\(0.844409\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.t.b.19.1	✓	240
8.3	odd	2	inner	552.2.t.b.19.17	yes	240
23.17	odd	22	inner	552.2.t.b.523.17	yes	240
184.155	even	22	inner	552.2.t.b.523.1	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.t.b.19.1	✓	240	1.1	even	1	trivial
552.2.t.b.19.17	yes	240	8.3	odd	2	inner
552.2.t.b.523.1	yes	240	184.155	even	22	inner
552.2.t.b.523.17	yes	240	23.17	odd	22	inner