Embedded newform 69.5.h.a.2.3

• Label: 69.5.h.a.2.3
• Level: $69$
• Weight: $5$
• Character: 69.2
• Analytic conductor: $7.133$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	69.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.13252745279\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(30\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			2.3
Character	\(\chi\)	\(=\)	69.2
Dual form			69.5.h.a.35.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.63251 + 5.65229i) q^{2} +(-6.90793 - 5.76893i) q^{3} +(-12.1066 - 26.5099i) q^{4} +(2.68502 - 9.14434i) q^{5} +(57.7008 - 18.0899i) q^{6} +(3.93980 - 4.54677i) q^{7} +(87.4108 + 12.5678i) q^{8} +(14.4389 + 79.7027i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 19 q^{3} + 234 q^{4} - 10 q^{6} - 18 q^{7} - 111 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.63251	+	5.65229i	−0.908127	+	1.41307i	0.00256998	+	0.999997i	\(0.499182\pi\)
							−0.910697	+	0.413076i	\(0.864454\pi\)
\(3\)	−6.90793	−	5.76893i	−0.767548	−	0.640992i
							\(5\)	2.68502	−	9.14434i	0.107401	−	0.365774i	−0.888202	−	0.459454i	\(-0.848045\pi\)
0.995602	+	0.0936806i	\(0.0298632\pi\)
\(7\)	3.93980	−	4.54677i	0.0804041	−	0.0927913i	−0.714122	−	0.700021i	\(-0.753174\pi\)
							0.794526	+	0.607230i	\(0.207719\pi\)
\(11\)	1.72982	+	2.69165i	0.0142960	+	0.0222451i	0.848328	−	0.529471i	\(-0.177610\pi\)
							−0.834032	+	0.551716i	\(0.813973\pi\)
\(13\)	63.8614	+	73.7000i	0.377878	+	0.436095i	0.912550	−	0.408965i	\(-0.134110\pi\)
							−0.534672	+	0.845060i	\(0.679565\pi\)
\(17\)	−173.318	−	79.1515i	−0.599715	−	0.273881i	0.0923377	−	0.995728i	\(-0.470566\pi\)
							−0.692052	+	0.721847i	\(0.743293\pi\)
\(19\)	126.133	+	276.193i	0.349400	+	0.765078i	0.999984	+	0.00564441i	\(0.00179668\pi\)
							−0.650585	+	0.759434i	\(0.725476\pi\)
\(23\)	−131.952	+	512.279i	−0.249437	+	0.968391i
							\(29\)	1179.79	+	538.794i	1.40285	+	0.640659i	0.965921	−	0.258839i	\(-0.0833398\pi\)
0.436926	+	0.899497i	\(0.356067\pi\)
\(31\)	−106.944	+	743.815i	−0.111285	+	0.774001i	0.855388	+	0.517987i	\(0.173318\pi\)
							−0.966673	+	0.256014i	\(0.917591\pi\)
\(37\)	2142.25	−	629.022i	1.56483	−	0.459476i	0.619340	−	0.785123i	\(-0.287400\pi\)
							0.945491	+	0.325647i	\(0.105582\pi\)
\(41\)	−14.1014	+	48.0250i	−0.00838870	+	0.0285693i	−0.963581	−	0.267417i	\(-0.913830\pi\)
							0.955192	+	0.295986i	\(0.0956482\pi\)
\(43\)	−121.574	−	845.567i	−0.0657513	−	0.457311i	−0.995925	−	0.0901898i	\(-0.971253\pi\)
							0.930173	−	0.367121i	\(-0.119656\pi\)
\(47\)		−	215.538i		−	0.0975728i	−0.998809	−	0.0487864i	\(-0.984465\pi\)
							0.998809	−	0.0487864i	\(-0.0155354\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.h.a.2.3	✓	300
3.2	odd	2	inner	69.5.h.a.2.28	yes	300
23.12	even	11	inner	69.5.h.a.35.28	yes	300
69.35	odd	22	inner	69.5.h.a.35.3	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.h.a.2.3	✓	300	1.1	even	1	trivial
69.5.h.a.2.28	yes	300	3.2	odd	2	inner
69.5.h.a.35.3	yes	300	69.35	odd	22	inner
69.5.h.a.35.28	yes	300	23.12	even	11	inner