Embedded newform 69.6.e.a.58.6

• Label: 69.6.e.a.58.6
• Level: $69$
• Weight: $6$
• Character: 69.58
• 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			58.6
Character	\(\chi\)	\(=\)	69.58
Dual form			69.6.e.a.25.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.74214 - 1.11960i) q^{2} +(1.28083 - 8.90839i) q^{3} +(-11.5117 + 25.2072i) q^{4} +(1.84930 - 0.543004i) q^{5} +(-7.74249 - 16.9537i) q^{6} +(-46.8839 - 54.1069i) q^{7} +(17.5980 + 122.397i) q^{8} +(-77.7189 - 22.8203i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 8 q^{2} + 90 q^{3} - 148 q^{4} + 6 q^{5} + 171 q^{6} + 118 q^{7} + 1616 q^{8} - 810 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{10}{11}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.74214	−	1.11960i	0.307970	−	0.197920i	−0.377522	−	0.926001i	\(-0.623224\pi\)
							0.685492	+	0.728081i	\(0.259587\pi\)
\(3\)	1.28083	−	8.90839i	0.0821655	−	0.571474i
							\(5\)	1.84930	−	0.543004i	0.0330813	−	0.00971355i	−0.265150	−	0.964207i	\(-0.585421\pi\)
0.298231	+	0.954494i	\(0.403603\pi\)
\(7\)	−46.8839	−	54.1069i	−0.361642	−	0.417357i	0.545547	−	0.838080i	\(-0.316322\pi\)
							−0.907189	+	0.420723i	\(0.861776\pi\)
\(11\)	−573.077	−	368.294i	−1.42801	−	0.917726i	−0.999902	−	0.0139976i	\(-0.995544\pi\)
							−0.428107	−	0.903728i	\(-0.640819\pi\)
\(13\)	−538.968	+	622.002i	−0.884514	+	1.02078i	0.115110	+	0.993353i	\(0.463278\pi\)
							−0.999624	+	0.0274307i	\(0.991267\pi\)
\(17\)	99.4271	+	217.715i	0.0834415	+	0.182712i	0.946754	−	0.321959i	\(-0.104341\pi\)
							−0.863312	+	0.504670i	\(0.831614\pi\)
\(19\)	−683.089	+	1495.76i	−0.434104	+	0.950554i	0.558539	+	0.829478i	\(0.311362\pi\)
							−0.992643	+	0.121077i	\(0.961365\pi\)
\(23\)	1963.13	+	1607.00i	0.773801	+	0.633428i
							\(29\)	−1269.48	−	2779.76i	−0.280304	−	0.613780i	0.716147	−	0.697949i	\(-0.245904\pi\)
−0.996452	+	0.0841687i	\(0.973177\pi\)
\(31\)	−1498.57	−	10422.8i	−0.280073	−	1.94795i	−0.316438	−	0.948613i	\(-0.602487\pi\)
							0.0363646	−	0.999339i	\(-0.488422\pi\)
\(37\)	6800.83	+	1996.91i	0.816691	+	0.239802i	0.663291	−	0.748362i	\(-0.269159\pi\)
							0.153401	+	0.988164i	\(0.450978\pi\)
\(41\)	−14215.7	+	4174.11i	−1.32071	+	0.387797i	−0.864750	−	0.502202i	\(-0.832523\pi\)
							−0.455963	+	0.889999i	\(0.650705\pi\)
\(43\)	793.806	−	5521.04i	0.0654701	−	0.455355i	−0.930546	−	0.366176i	\(-0.880667\pi\)
							0.996016	−	0.0891785i	\(-0.0284242\pi\)
\(47\)	21178.1			1.39843			0.699217	−	0.714910i	\(-0.253532\pi\)
							0.699217	+	0.714910i	\(0.253532\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.e.a.58.6	yes	100
23.2	even	11	inner	69.6.e.a.25.6	✓	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.e.a.25.6	✓	100	23.2	even	11	inner
69.6.e.a.58.6	yes	100	1.1	even	1	trivial