Embedded newform 143.4.q.a.3.7

• Label: 143.4.q.a.3.7
• Level: $143$
• Weight: $4$
• Character: 143.3
• Analytic conductor: $8.437$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 143 = 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	143.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43727313082\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(40\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	143.3
Dual form			143.4.q.a.48.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.96685 - 1.76615i) q^{2} +(-6.00386 + 6.66796i) q^{3} +(7.26353 + 8.06697i) q^{4} +(-4.50154 - 3.27056i) q^{5} +(35.5930 - 15.8470i) q^{6} +(-18.3573 - 20.3879i) q^{7} +(-3.83117 - 11.7911i) q^{8} +(-5.59310 - 53.2148i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 3 q^{2} - 3 q^{3} + 149 q^{4} - 12 q^{5} - 35 q^{6} - 3 q^{7} - 44 q^{8} + 369 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/143\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.96685	−	1.76615i	−1.40249	−	0.624430i	−0.440561	−	0.897722i	\(-0.645221\pi\)
							−0.961931	+	0.273293i	\(0.911887\pi\)
\(3\)	−6.00386	+	6.66796i	−1.15544	+	1.28325i	−0.202778	+	0.979225i	\(0.564997\pi\)
							−0.952664	+	0.304024i	\(0.901670\pi\)
\(5\)	−4.50154	−	3.27056i	−0.402630	−	0.292528i	0.367982	−	0.929833i	\(-0.380049\pi\)
							−0.770611	+	0.637305i	\(0.780049\pi\)
\(7\)	−18.3573	−	20.3879i	−0.991203	−	1.10084i	−0.994902	−	0.100849i	\(-0.967844\pi\)
							0.00369861	−	0.999993i	\(-0.498823\pi\)
\(11\)	−4.96984	−	36.1428i	−0.136224	−	0.990678i
							\(13\)	−39.1370	+	25.7933i	−0.834973	+	0.550291i
\(17\)	−32.8234	+	14.6139i	−0.468285	+	0.208494i								−0.627293	−	0.778783i	\(-0.715837\pi\)
							0.159008	+	0.987277i	\(0.449170\pi\)
\(19\)	−11.8328	−	2.51513i	−0.142875	−	0.0303690i	0.135919	−	0.990720i	\(-0.456601\pi\)
							−0.278794	+	0.960351i	\(0.589935\pi\)
\(23\)	79.9557	+	138.487i	0.724866	+	1.25551i	0.959029	+	0.283308i	\(0.0914318\pi\)
							−0.234163	+	0.972197i	\(0.575235\pi\)
\(29\)	−128.200	+	27.2497i	−0.820900	+	0.174488i	−0.599171	−	0.800621i	\(-0.704503\pi\)
							−0.221728	+	0.975108i	\(0.571170\pi\)
\(31\)	67.4582	−	49.0112i	0.390834	−	0.283957i	−0.374963	−	0.927040i	\(-0.622345\pi\)
							0.765797	+	0.643082i	\(0.222345\pi\)
\(37\)	231.276	−	49.1591i	1.02761	−	0.218425i	0.336886	−	0.941546i	\(-0.390626\pi\)
							0.690721	+	0.723121i	\(0.257293\pi\)
\(41\)	−171.262	+	190.206i	−0.652357	+	0.724516i	−0.975049	−	0.221988i	\(-0.928745\pi\)
							0.322692	+	0.946504i	\(0.395412\pi\)
\(43\)	−94.7164	+	164.054i	−0.335910	+	0.581813i	−0.983659	−	0.180041i	\(-0.942377\pi\)
							0.647749	+	0.761853i	\(0.275710\pi\)
\(47\)	59.6887	+	183.703i	0.185244	+	0.570124i	0.999952	−	0.00974769i	\(-0.00310284\pi\)
							−0.814708	+	0.579871i	\(0.803103\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	143.4.q.a.3.7	✓	320
11.4	even	5	inner	143.4.q.a.81.34	yes	320
13.9	even	3	inner	143.4.q.a.113.34	yes	320
143.48	even	15	inner	143.4.q.a.48.7	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.q.a.3.7	✓	320	1.1	even	1	trivial
143.4.q.a.48.7	yes	320	143.48	even	15	inner
143.4.q.a.81.34	yes	320	11.4	even	5	inner
143.4.q.a.113.34	yes	320	13.9	even	3	inner