Embedded newform 45.6.e.b.31.6

• Label: 45.6.e.b.31.6
• Level: $45$
• Weight: $6$
• Character: 45.31
• Analytic conductor: $7.217$
• Analytic rank: $0$
• Dimension: $22$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	45.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.21727189158\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Relative dimension:	\(11\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			31.6
Character	\(\chi\)	\(=\)	45.31
Dual form			45.6.e.b.16.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.567163 - 0.982355i) q^{2} +(-10.4949 + 11.5264i) q^{3} +(15.3567 - 26.5985i) q^{4} +(-12.5000 + 21.6506i) q^{5} +(17.2753 + 3.77232i) q^{6} +(6.11924 + 10.5988i) q^{7} -71.1373 q^{8} +(-22.7162 - 241.936i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 4 q^{2} - 11 q^{3} - 208 q^{4} - 275 q^{5} + 167 q^{6} - 225 q^{7} + 624 q^{8} + 473 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.567163	−	0.982355i	−0.100261	−	0.173657i	0.811531	−	0.584309i	\(-0.198635\pi\)
							−0.911792	+	0.410652i	\(0.865301\pi\)
\(3\)	−10.4949	+	11.5264i	−0.673245	+	0.739419i
							\(5\)	−12.5000	+	21.6506i	−0.223607	+	0.387298i
\(7\)	6.11924	+	10.5988i	0.0472011	+	0.0817548i								0.888661	−	0.458565i	\(-0.151636\pi\)
							−0.841460	+	0.540320i	\(0.818303\pi\)
\(11\)	−313.531	−	543.052i	−0.781266	−	1.35319i	−0.931205	−	0.364497i	\(-0.881241\pi\)
							0.149938	−	0.988695i	\(-0.452092\pi\)
\(13\)	523.811	−	907.267i	0.859639	−	1.48894i	−0.0126343	−	0.999920i	\(-0.504022\pi\)
							0.872273	−	0.489019i	\(-0.162645\pi\)
\(17\)	37.7027			0.0316410			0.0158205	−	0.999875i	\(-0.494964\pi\)
							0.0158205	+	0.999875i	\(0.494964\pi\)
\(19\)	−1205.25			−0.765940			−0.382970	−	0.923761i	\(-0.625099\pi\)
							−0.382970	+	0.923761i	\(0.625099\pi\)
\(23\)	−664.407	+	1150.79i	−0.261887	+	0.453602i	−0.966743	−	0.255748i	\(-0.917678\pi\)
							0.704856	+	0.709350i	\(0.251012\pi\)
\(29\)	1284.81	+	2225.35i	0.283690	+	0.491365i	0.972291	−	0.233776i	\(-0.0751082\pi\)
							−0.688601	+	0.725140i	\(0.741775\pi\)
\(31\)	3034.56	−	5256.01i	0.567141	−	0.982317i	−0.429706	−	0.902969i	\(-0.641383\pi\)
							0.996847	−	0.0793482i	\(-0.0252839\pi\)
\(37\)	−2321.14			−0.278739			−0.139369	−	0.990240i	\(-0.544508\pi\)
							−0.139369	+	0.990240i	\(0.544508\pi\)
\(41\)	−9411.95	+	16302.0i	−0.874420	+	1.51454i	−0.0170408	+	0.999855i	\(0.505425\pi\)
							−0.857379	+	0.514685i	\(0.827909\pi\)
\(43\)	1413.63	+	2448.48i	0.116591	+	0.201941i	0.918415	−	0.395619i	\(-0.129470\pi\)
							−0.801824	+	0.597561i	\(0.796137\pi\)
\(47\)	−10151.0	−	17582.1i	−0.670295	−	1.16099i	−0.977820	−	0.209446i	\(-0.932834\pi\)
							0.307525	−	0.951540i	\(-0.400499\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.6.e.b.31.6	yes	22
3.2	odd	2		135.6.e.b.91.6		22
9.2	odd	6		135.6.e.b.46.6		22
9.4	even	3		405.6.a.j.1.6		11
9.5	odd	6		405.6.a.i.1.6		11
9.7	even	3	inner	45.6.e.b.16.6	✓	22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.6.e.b.16.6	✓	22	9.7	even	3	inner
45.6.e.b.31.6	yes	22	1.1	even	1	trivial
135.6.e.b.46.6		22	9.2	odd	6
135.6.e.b.91.6		22	3.2	odd	2
405.6.a.i.1.6		11	9.5	odd	6
405.6.a.j.1.6		11	9.4	even	3