Embedded newform 69.4.g.a.5.11

• Label: 69.4.g.a.5.11
• Level: $69$
• Weight: $4$
• Character: 69.5
• Analytic conductor: $4.071$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.07113179040\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.11
Character	\(\chi\)	\(=\)	69.5
Dual form			69.4.g.a.14.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0243207 + 0.0828288i) q^{2} +(0.514506 - 5.17062i) q^{3} +(6.72376 + 4.32110i) q^{4} +(2.06531 - 14.3646i) q^{5} +(0.415763 + 0.168369i) q^{6} +(-11.4770 + 5.24135i) q^{7} +(-1.04336 + 0.904078i) q^{8} +(-26.4706 - 5.32063i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 5 q^{3} + 50 q^{4} - 32 q^{6} - 22 q^{7} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0243207	+	0.0828288i	−0.00859867	+	0.0292844i	−0.963681	−	0.267058i	\(-0.913949\pi\)
							0.955082	+	0.296342i	\(0.0957668\pi\)
\(3\)	0.514506	−	5.17062i	0.0990167	−	0.995086i
							\(5\)	2.06531	−	14.3646i	0.184727	−	1.28481i	−0.660673	−	0.750673i	\(-0.729729\pi\)
0.845401	−	0.534133i	\(-0.179362\pi\)
\(7\)	−11.4770	+	5.24135i	−0.619698	+	0.283006i	−0.700421	−	0.713730i	\(-0.747004\pi\)
							0.0807231	+	0.996737i	\(0.474277\pi\)
\(11\)	35.0737	−	10.2986i	0.961373	−	0.282285i	0.236859	−	0.971544i	\(-0.423882\pi\)
							0.724515	+	0.689259i	\(0.242064\pi\)
\(13\)	34.0483	−	74.5555i	0.726409	−	1.59061i	−0.0782906	−	0.996931i	\(-0.524946\pi\)
							0.804699	−	0.593683i	\(-0.202327\pi\)
\(17\)	−59.1088	+	37.9869i	−0.843293	+	0.541952i	−0.889476	−	0.456981i	\(-0.848931\pi\)
							0.0461829	+	0.998933i	\(0.485294\pi\)
\(19\)	5.28284	−	8.22026i	0.0637877	−	0.0992557i	−0.807895	−	0.589327i	\(-0.799393\pi\)
							0.871682	+	0.490071i	\(0.163029\pi\)
\(23\)	38.5867	+	103.335i	0.349821	+	0.936817i
							\(29\)	120.291	+	187.177i	0.770261	+	1.19855i	0.975535	+	0.219845i	\(0.0705551\pi\)
−0.205274	+	0.978705i	\(0.565809\pi\)
\(31\)	66.4182	+	76.6507i	0.384808	+	0.444092i	0.914798	−	0.403912i	\(-0.132350\pi\)
							−0.529990	+	0.848004i	\(0.677804\pi\)
\(37\)	−293.223	+	42.1592i	−1.30286	+	0.187322i	−0.758578	−	0.651583i	\(-0.774105\pi\)
							−0.544278	+	0.838905i	\(0.683196\pi\)
\(41\)	147.849	+	21.2575i	0.563173	+	0.0809721i	0.418021	−	0.908437i	\(-0.362724\pi\)
							0.145152	+	0.989409i	\(0.453633\pi\)
\(43\)	266.142	+	230.614i	0.943868	+	0.817867i	0.983418	−	0.181351i	\(-0.0580470\pi\)
							−0.0395501	+	0.999218i	\(0.512592\pi\)
\(47\)		−	569.598i		−	1.76775i	−0.467721	−	0.883876i	\(-0.654925\pi\)
							0.467721	−	0.883876i	\(-0.345075\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.g.a.5.11	✓	220
3.2	odd	2	inner	69.4.g.a.5.12	yes	220
23.14	odd	22	inner	69.4.g.a.14.12	yes	220
69.14	even	22	inner	69.4.g.a.14.11	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.g.a.5.11	✓	220	1.1	even	1	trivial
69.4.g.a.5.12	yes	220	3.2	odd	2	inner
69.4.g.a.14.11	yes	220	69.14	even	22	inner
69.4.g.a.14.12	yes	220	23.14	odd	22	inner