Embedded newform 69.6.e.a.58.5

• Label: 69.6.e.a.58.5
• 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			58.5
Character	\(\chi\)	\(=\)	69.58
Dual form			69.6.e.a.25.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.415809 + 0.267224i) q^{2} +(1.28083 - 8.90839i) q^{3} +(-13.1918 + 28.8860i) q^{4} +(52.4771 - 15.4087i) q^{5} +(1.84796 + 4.04646i) q^{6} +(-115.686 - 133.509i) q^{7} +(-4.48473 - 31.1920i) q^{8} +(-77.7189 - 22.8203i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 8 q^{2} + 90 q^{3} - 148 q^{4} + 6 q^{5} + 171 q^{6} + 118 q^{7} + 1616 q^{8} - 810 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.415809	+	0.267224i	−0.0735054	+	0.0472391i	−0.576877	−	0.816831i	\(-0.695729\pi\)
							0.503372	+	0.864070i	\(0.332093\pi\)
\(3\)	1.28083	−	8.90839i	0.0821655	−	0.571474i
							\(5\)	52.4771	−	15.4087i	0.938738	−	0.275638i	0.223647	−	0.974670i	\(-0.428204\pi\)
0.715091	+	0.699032i	\(0.246385\pi\)
\(7\)	−115.686	−	133.509i	−0.892350	−	1.02983i	−0.999367	−	0.0355695i	\(-0.988676\pi\)
							0.107017	−	0.994257i	\(-0.465870\pi\)
\(11\)	409.436	+	263.128i	1.02024	+	0.655671i	0.940025	−	0.341106i	\(-0.110802\pi\)
							0.0802190	+	0.996777i	\(0.474438\pi\)
\(13\)	245.699	−	283.552i	0.403224	−	0.465345i	−0.517430	−	0.855726i	\(-0.673111\pi\)
							0.920653	+	0.390381i	\(0.127657\pi\)
\(17\)	−761.738	−	1667.97i	−0.639268	−	1.39980i	−0.900641	−	0.434564i	\(-0.856902\pi\)
							0.261373	−	0.965238i	\(-0.415825\pi\)
\(19\)	1092.35	−	2391.91i	0.694189	−	1.52006i	−0.152689	−	0.988274i	\(-0.548793\pi\)
							0.846878	−	0.531787i	\(-0.178479\pi\)
\(23\)	313.580	−	2517.54i	0.123603	−	0.992332i
							\(29\)	−1385.81	−	3034.49i	−0.305990	−	0.670025i	0.692698	−	0.721228i	\(-0.256422\pi\)
−0.998688	+	0.0512028i	\(0.983695\pi\)
\(31\)	675.542	+	4698.49i	0.126255	+	0.878121i	0.950242	+	0.311513i	\(0.100836\pi\)
							−0.823987	+	0.566608i	\(0.808255\pi\)
\(37\)	5821.44	+	1709.33i	0.699079	+	0.205268i	0.611912	−	0.790926i	\(-0.290400\pi\)
							0.0871669	+	0.996194i	\(0.472219\pi\)
\(41\)	7946.41	−	2333.28i	0.738264	−	0.216774i	0.109083	−	0.994033i	\(-0.465209\pi\)
							0.629181	+	0.777259i	\(0.283390\pi\)
\(43\)	−2485.54	+	17287.3i	−0.204998	+	1.42579i	0.584178	+	0.811626i	\(0.301417\pi\)
							−0.789176	+	0.614167i	\(0.789492\pi\)
\(47\)	−5180.25			−0.342063			−0.171032	−	0.985266i	\(-0.554710\pi\)
							−0.171032	+	0.985266i	\(0.554710\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.e.a.58.5	yes	100
23.2	even	11	inner	69.6.e.a.25.5	✓	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.e.a.25.5	✓	100	23.2	even	11	inner
69.6.e.a.58.5	yes	100	1.1	even	1	trivial