Embedded newform 69.6.e.a.13.8

• Label: 69.6.e.a.13.8
• 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			13.8
Character	\(\chi\)	\(=\)	69.13
Dual form			69.6.e.a.16.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.85485 + 3.29467i) q^{2} +(-7.57128 - 4.86577i) q^{3} +(1.84938 - 12.8627i) q^{4} +(-20.1797 + 44.1873i) q^{5} +(-5.58376 - 38.8359i) q^{6} +(0.521273 + 0.153060i) q^{7} +(165.016 - 106.049i) q^{8} +(33.6486 + 73.6802i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 8 q^{2} + 90 q^{3} - 148 q^{4} + 6 q^{5} + 171 q^{6} + 118 q^{7} + 1616 q^{8} - 810 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.85485	+	3.29467i	0.504671	+	0.582421i	0.949726	−	0.313082i	\(-0.101361\pi\)
							−0.445055	+	0.895503i	\(0.646816\pi\)
\(3\)	−7.57128	−	4.86577i	−0.485698	−	0.312139i
							\(5\)	−20.1797	+	44.1873i	−0.360985	+	0.790446i	0.638793	+	0.769378i	\(0.279434\pi\)
−0.999778	+	0.0210677i	\(0.993293\pi\)
\(7\)	0.521273	+	0.153060i	0.00402087	+	0.00118063i	0.283742	−	0.958901i	\(-0.408424\pi\)
							−0.279722	+	0.960081i	\(0.590242\pi\)
\(11\)	349.810	−	403.703i	0.871667	−	1.00596i	−0.128232	−	0.991744i	\(-0.540930\pi\)
							0.999899	−	0.0142135i	\(-0.00452446\pi\)
\(13\)	864.214	−	253.756i	1.41828	−	0.416445i	0.519361	−	0.854555i	\(-0.326170\pi\)
							0.898922	+	0.438109i	\(0.144352\pi\)
\(17\)	−295.024	−	2051.94i	−0.247591	−	1.72203i	−0.612055	−	0.790815i	\(-0.709657\pi\)
							0.364464	−	0.931218i	\(-0.381252\pi\)
\(19\)	−135.236	+	940.590i	−0.0859428	+	0.597746i	0.900650	+	0.434545i	\(0.143091\pi\)
							−0.986593	+	0.163200i	\(0.947818\pi\)
\(23\)	1996.78	−	1565.00i	0.787064	−	0.616871i
							\(29\)	1023.89	+	7121.34i	0.226079	+	1.57241i	0.714393	+	0.699745i	\(0.246703\pi\)
−0.488314	+	0.872668i	\(0.662388\pi\)
\(31\)	−5628.48	+	3617.21i	−1.05193	+	0.676035i	−0.947909	−	0.318540i	\(-0.896807\pi\)
							−0.104021	+	0.994575i	\(0.533171\pi\)
\(37\)	−3865.49	−	8464.24i	−0.464195	−	1.01644i	−0.986511	−	0.163693i	\(-0.947659\pi\)
							0.522317	−	0.852752i	\(-0.325068\pi\)
\(41\)	7080.32	−	15503.7i	0.657799	−	1.44038i	−0.226760	−	0.973951i	\(-0.572813\pi\)
							0.884559	−	0.466428i	\(-0.154459\pi\)
\(43\)	16948.7	+	10892.3i	1.39786	+	0.898353i	0.999819	−	0.0190135i	\(-0.00605254\pi\)
							0.398045	+	0.917366i	\(0.369689\pi\)
\(47\)	−18346.9			−1.21149			−0.605743	−	0.795660i	\(-0.707124\pi\)
							−0.605743	+	0.795660i	\(0.707124\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.e.a.13.8	✓	100
23.16	even	11	inner	69.6.e.a.16.8	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.e.a.13.8	✓	100	1.1	even	1	trivial
69.6.e.a.16.8	yes	100	23.16	even	11	inner