Embedded newform 69.6.g.a.5.18

• Label: 69.6.g.a.5.18
• Level: $69$
• Weight: $6$
• Character: 69.5
• Analytic conductor: $11.066$
• Analytic rank: $0$
• Dimension: $380$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0664835671\)
Analytic rank:	\(0\)
Dimension:	\(380\)
Relative dimension:	\(38\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.18
Character	\(\chi\)	\(=\)	69.5
Dual form			69.6.g.a.14.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.235789 + 0.803025i) q^{2} +(-13.0330 + 8.55220i) q^{3} +(26.3309 + 16.9218i) q^{4} +(11.5441 - 80.2911i) q^{5} +(-3.79458 - 12.4824i) q^{6} +(-106.024 + 48.4193i) q^{7} +(-40.0374 + 34.6926i) q^{8} +(96.7196 - 222.922i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 380 q - 11 q^{3} + 578 q^{4} + 64 q^{6} - 22 q^{7} + 433 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.235789	+	0.803025i	−0.0416820	+	0.141956i	−0.977705	−	0.209983i	\(-0.932659\pi\)
							0.936023	+	0.351939i	\(0.114477\pi\)
\(3\)	−13.0330	+	8.55220i	−0.836069	+	0.548624i
							\(5\)	11.5441	−	80.2911i	0.206507	−	1.43629i	−0.577932	−	0.816085i	\(-0.696140\pi\)
0.784439	−	0.620206i	\(-0.212951\pi\)
\(7\)	−106.024	+	48.4193i	−0.817819	+	0.373485i	−0.779985	−	0.625799i	\(-0.784773\pi\)
							−0.0378341	+	0.999284i	\(0.512046\pi\)
\(11\)	−172.379	+	50.6151i	−0.429540	+	0.126124i	−0.489354	−	0.872085i	\(-0.662767\pi\)
							0.0598141	+	0.998210i	\(0.480949\pi\)
\(13\)	388.772	−	851.291i	0.638023	−	1.39708i	−0.263634	−	0.964623i	\(-0.584921\pi\)
							0.901657	−	0.432452i	\(-0.142352\pi\)
\(17\)	1678.49	−	1078.70i	1.40863	−	0.905272i	0.408659	−	0.912687i	\(-0.365997\pi\)
							0.999973	+	0.00741512i	\(0.00236033\pi\)
\(19\)	333.132	−	518.363i	0.211706	−	0.329420i	−0.719119	−	0.694887i	\(-0.755454\pi\)
							0.930824	+	0.365467i	\(0.119091\pi\)
\(23\)	2470.53	−	576.908i	0.973802	−	0.227398i
							\(29\)	−4002.90	−	6228.63i	−0.883851	−	1.37530i	−0.926534	−	0.376211i	\(-0.877227\pi\)
0.0426827	−	0.999089i	\(-0.486410\pi\)
\(31\)	910.557	+	1050.84i	0.170178	+	0.196396i	0.834432	−	0.551111i	\(-0.185796\pi\)
							−0.664254	+	0.747507i	\(0.731251\pi\)
\(37\)	1491.37	−	214.427i	0.179094	−	0.0257499i	−0.0521840	−	0.998637i	\(-0.516618\pi\)
							0.231278	+	0.972888i	\(0.425709\pi\)
\(41\)	4451.89	+	640.085i	0.413604	+	0.0594673i	0.345976	−	0.938243i	\(-0.387548\pi\)
							0.0676279	+	0.997711i	\(0.478457\pi\)
\(43\)	−6817.43	−	5907.34i	−0.562276	−	0.487215i	0.326725	−	0.945119i	\(-0.394055\pi\)
							−0.889002	+	0.457904i	\(0.848600\pi\)
\(47\)		−	8832.46i		−	0.583226i	−0.956536	−	0.291613i	\(-0.905808\pi\)
							0.956536	−	0.291613i	\(-0.0941920\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.g.a.5.18	✓	380
3.2	odd	2	inner	69.6.g.a.5.21	yes	380
23.14	odd	22	inner	69.6.g.a.14.21	yes	380
69.14	even	22	inner	69.6.g.a.14.18	yes	380

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.g.a.5.18	✓	380	1.1	even	1	trivial
69.6.g.a.5.21	yes	380	3.2	odd	2	inner
69.6.g.a.14.18	yes	380	69.14	even	22	inner
69.6.g.a.14.21	yes	380	23.14	odd	22	inner