Embedded newform 69.7.f.a.34.20

• Label: 69.7.f.a.34.20
• Level: $69$
• Weight: $7$
• Character: 69.34
• Analytic conductor: $15.874$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	69.f (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.8737317698\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			34.20
Character	\(\chi\)	\(=\)	69.34
Dual form			69.7.f.a.67.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.66106 - 5.56613i) q^{2} +(-2.21847 + 15.4298i) q^{3} +(17.4457 - 38.2007i) q^{4} +(-44.7289 - 152.333i) q^{5} +(66.6699 + 145.987i) q^{6} +(-478.159 + 414.327i) q^{7} +(32.2403 + 224.237i) q^{8} +(-233.157 - 68.4610i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{9}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.66106	−	5.56613i	1.08263	−	0.695766i	0.127469	−	0.991843i	\(-0.459315\pi\)
							0.955164	+	0.296077i	\(0.0956784\pi\)
\(3\)	−2.21847	+	15.4298i	−0.0821655	+	0.571474i
							\(5\)	−44.7289	−	152.333i	−0.357831	−	1.21866i	−0.920093	−	0.391701i	\(-0.871887\pi\)
0.562262	−	0.826959i	\(-0.309931\pi\)
\(7\)	−478.159	+	414.327i	−1.39405	+	1.20795i	−0.443930	+	0.896061i	\(0.646416\pi\)
							−0.950119	+	0.311889i	\(0.899038\pi\)
\(11\)	−466.757	+	726.288i	−0.350681	+	0.545671i	−0.971122	−	0.238582i	\(-0.923317\pi\)
							0.620441	+	0.784253i	\(0.286954\pi\)
\(13\)	569.051	−	656.720i	0.259013	−	0.298917i	−0.611317	−	0.791386i	\(-0.709360\pi\)
							0.870330	+	0.492469i	\(0.163905\pi\)
\(17\)	−3176.88	+	1450.83i	−0.646628	+	0.295305i	−0.711603	−	0.702582i	\(-0.752030\pi\)
							0.0649752	+	0.997887i	\(0.479303\pi\)
\(19\)	−6762.53	−	3088.35i	−0.985936	−	0.450262i	−0.143847	−	0.989600i	\(-0.545947\pi\)
							−0.842089	+	0.539338i	\(0.818674\pi\)
\(23\)	−11324.0	+	4450.03i	−0.930715	+	0.365746i
							\(29\)	8338.62	+	18259.0i	0.341901	+	0.748658i	0.999991	−	0.00428111i	\(-0.00136272\pi\)
−0.658090	+	0.752939i	\(0.728635\pi\)
\(31\)	234.241	+	1629.18i	0.00786282	+	0.0546871i	0.993375	−	0.114917i	\(-0.0366602\pi\)
							−0.985512	+	0.169604i	\(0.945751\pi\)
\(37\)	21339.0	−	72673.9i	0.421277	−	1.43474i	−0.426548	−	0.904465i	\(-0.640271\pi\)
							0.847826	−	0.530275i	\(-0.177911\pi\)
\(41\)	−4421.67	+	1298.32i	−0.0641556	+	0.0188378i	−0.313653	−	0.949538i	\(-0.601553\pi\)
							0.249497	+	0.968376i	\(0.419735\pi\)
\(43\)	−1327.36	−	190.845i	−0.0166949	−	0.00240036i	0.133962	−	0.990986i	\(-0.457230\pi\)
							−0.150657	+	0.988586i	\(0.548139\pi\)
\(47\)	−12444.2			−0.119859			−0.0599297	−	0.998203i	\(-0.519088\pi\)
							−0.0599297	+	0.998203i	\(0.519088\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.f.a.34.20	✓	240
23.21	odd	22	inner	69.7.f.a.67.20	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.f.a.34.20	✓	240	1.1	even	1	trivial
69.7.f.a.67.20	yes	240	23.21	odd	22	inner