Embedded newform 69.6.e.a.13.1

• Label: 69.6.e.a.13.1
• Level: $69$
• Weight: $6$
• Character: 69.13
• Analytic conductor: $11.066$
• Analytic rank: $0$
• Dimension: $100$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0664835671\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Relative dimension:	\(10\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			13.1
Character	\(\chi\)	\(=\)	69.13
Dual form			69.6.e.a.16.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.00907 - 8.08889i) q^{2} +(-7.57128 - 4.86577i) q^{3} +(-11.7491 + 81.7169i) q^{4} +(31.1021 - 68.1041i) q^{5} +(13.7089 + 95.3478i) q^{6} +(113.192 + 33.2362i) q^{7} +(455.220 - 292.552i) q^{8} +(33.6486 + 73.6802i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 8 q^{2} + 90 q^{3} - 148 q^{4} + 6 q^{5} + 171 q^{6} + 118 q^{7} + 1616 q^{8} - 810 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{7}{11}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.00907	−	8.08889i	−1.23904	−	1.42993i	−0.864436	−	0.502743i	\(-0.832324\pi\)
							−0.374604	−	0.927185i	\(-0.622221\pi\)
\(3\)	−7.57128	−	4.86577i	−0.485698	−	0.312139i
							\(5\)	31.1021	−	68.1041i	0.556371	−	1.21828i	−0.397371	−	0.917658i	\(-0.630077\pi\)
0.953742	−	0.300625i	\(-0.0971954\pi\)
\(7\)	113.192	+	33.2362i	0.873113	+	0.256369i	0.687439	−	0.726242i	\(-0.258735\pi\)
							0.185674	+	0.982611i	\(0.440553\pi\)
\(11\)	370.856	−	427.991i	0.924110	−	1.06648i	−0.0734936	−	0.997296i	\(-0.523415\pi\)
							0.997604	−	0.0691844i	\(-0.0220397\pi\)
\(13\)	1027.18	−	301.608i	1.68574	−	0.494977i	0.708248	−	0.705964i	\(-0.249486\pi\)
							0.977489	+	0.210987i	\(0.0676677\pi\)
\(17\)	81.8440	+	569.238i	0.0686854	+	0.477718i	0.994912	+	0.100750i	\(0.0321242\pi\)
							−0.926226	+	0.376968i	\(0.876967\pi\)
\(19\)	−179.976	+	1251.76i	−0.114375	+	0.795493i	0.849203	+	0.528066i	\(0.177083\pi\)
							−0.963578	+	0.267427i	\(0.913826\pi\)
\(23\)	−426.728	+	2500.85i	−0.168202	+	0.985753i
							\(29\)	−383.465	−	2667.06i	−0.0846703	−	0.588895i	−0.987347	−	0.158575i	\(-0.949310\pi\)
0.902677	−	0.430320i	\(-0.141599\pi\)
\(31\)	7160.59	−	4601.83i	1.33827	−	0.860055i	0.341465	−	0.939895i	\(-0.389077\pi\)
							0.996808	+	0.0798391i	\(0.0254407\pi\)
\(37\)	−1670.29	−	3657.42i	−0.200580	−	0.439209i	0.782436	−	0.622732i	\(-0.213977\pi\)
							−0.983015	+	0.183523i	\(0.941250\pi\)
\(41\)	2054.35	−	4498.41i	0.190860	−	0.417926i	−0.789875	−	0.613268i	\(-0.789855\pi\)
							0.980735	+	0.195342i	\(0.0625818\pi\)
\(43\)	−6708.04	−	4311.00i	−0.553254	−	0.355555i	0.233949	−	0.972249i	\(-0.424835\pi\)
							−0.787203	+	0.616694i	\(0.788472\pi\)
\(47\)	−20531.2			−1.35572			−0.677860	−	0.735191i	\(-0.737092\pi\)
							−0.677860	+	0.735191i	\(0.737092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.e.a.13.1	✓	100
23.16	even	11	inner	69.6.e.a.16.1	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.e.a.13.1	✓	100	1.1	even	1	trivial
69.6.e.a.16.1	yes	100	23.16	even	11	inner