Embedded newform 69.6.g.a.5.4

• Label: 69.6.g.a.5.4
• 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			5.4
Character	\(\chi\)	\(=\)	69.5
Dual form			69.6.g.a.14.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60390 + 8.86808i) q^{2} +(14.0479 + 6.75686i) q^{3} +(-44.9425 - 28.8828i) q^{4} +(14.8404 - 103.218i) q^{5} +(-96.4999 + 106.984i) q^{6} +(66.5201 - 30.3787i) q^{7} +(149.641 - 129.665i) q^{8} +(151.690 + 189.840i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 380 q - 11 q^{3} + 578 q^{4} + 64 q^{6} - 22 q^{7} + 433 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60390	+	8.86808i	−0.460310	+	1.56767i	0.323225	+	0.946322i	\(0.395233\pi\)
							−0.783535	+	0.621348i	\(0.786585\pi\)
\(3\)	14.0479	+	6.75686i	0.901176	+	0.433453i
							\(5\)	14.8404	−	103.218i	0.265474	−	1.84641i	−0.224246	−	0.974532i	\(-0.571992\pi\)
0.489720	−	0.871880i	\(-0.337099\pi\)
\(7\)	66.5201	−	30.3787i	0.513107	−	0.234328i	−0.141997	−	0.989867i	\(-0.545352\pi\)
							0.655104	+	0.755539i	\(0.272625\pi\)
\(11\)	516.449	−	151.643i	1.28690	−	0.377869i	0.434461	−	0.900691i	\(-0.356939\pi\)
							0.852442	+	0.522822i	\(0.175121\pi\)
\(13\)	−188.245	+	412.199i	−0.308934	+	0.676470i	−0.998876	−	0.0473937i	\(-0.984908\pi\)
							0.689943	+	0.723864i	\(0.257636\pi\)
\(17\)	396.926	−	255.089i	0.333110	−	0.214077i	−0.363384	−	0.931639i	\(-0.618379\pi\)
							0.696494	+	0.717563i	\(0.254742\pi\)
\(19\)	595.162	−	926.089i	0.378226	−	0.588530i	−0.598996	−	0.800752i	\(-0.704433\pi\)
							0.977222	+	0.212222i	\(0.0680698\pi\)
\(23\)	2536.99	−	1.39766i	1.00000	−	0.000550913i
							\(29\)	−1764.28	−	2745.27i	−0.389558	−	0.606164i	0.589981	−	0.807417i	\(-0.299135\pi\)
−0.979539	+	0.201253i	\(0.935499\pi\)
\(31\)	5241.45	+	6048.96i	0.979597	+	1.13052i	0.991437	+	0.130585i	\(0.0416857\pi\)
							−0.0118400	+	0.999930i	\(0.503769\pi\)
\(37\)	−3987.30	+	573.287i	−0.478823	+	0.0688443i	−0.377501	−	0.926009i	\(-0.623216\pi\)
							−0.101322	+	0.994854i	\(0.532307\pi\)
\(41\)	−267.213	−	38.4194i	−0.0248255	−	0.00356936i	0.129891	−	0.991528i	\(-0.458537\pi\)
							−0.154716	+	0.987959i	\(0.549446\pi\)
\(43\)	−7040.44	−	6100.58i	−0.580669	−	0.503153i	0.314274	−	0.949332i	\(-0.398239\pi\)
							−0.894943	+	0.446180i	\(0.852784\pi\)
\(47\)		−	24740.0i		−	1.63363i	−0.576898	−	0.816816i	\(-0.695737\pi\)
							0.576898	−	0.816816i	\(-0.304263\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.g.a.5.4	✓	380
3.2	odd	2	inner	69.6.g.a.5.35	yes	380
23.14	odd	22	inner	69.6.g.a.14.35	yes	380
69.14	even	22	inner	69.6.g.a.14.4	yes	380

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.g.a.5.4	✓	380	1.1	even	1	trivial
69.6.g.a.5.35	yes	380	3.2	odd	2	inner
69.6.g.a.14.4	yes	380	69.14	even	22	inner
69.6.g.a.14.35	yes	380	23.14	odd	22	inner