Embedded newform 69.8.e.b.58.11

• Label: 69.8.e.b.58.11
• Level: $69$
• Weight: $8$
• Character: 69.58
• Analytic conductor: $21.555$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.5545667584\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(14\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			58.11
Character	\(\chi\)	\(=\)	69.58
Dual form			69.8.e.b.25.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.6588 - 7.49264i) q^{2} +(3.84250 - 26.7252i) q^{3} +(26.6142 - 58.2769i) q^{4} +(70.9340 - 20.8281i) q^{5} +(-155.443 - 340.373i) q^{6} +(-924.953 - 1067.45i) q^{7} +(126.097 + 877.024i) q^{8} +(-699.470 - 205.383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 12 q^{2} + 378 q^{3} - 740 q^{4} + 2 q^{5} - 1755 q^{6} - 394 q^{7} - 1612 q^{8} - 10206 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{10}{11}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.6588	−	7.49264i	1.03050	−	0.662262i	0.0878799	−	0.996131i	\(-0.471991\pi\)
							0.942619	+	0.333869i	\(0.108354\pi\)
\(3\)	3.84250	−	26.7252i	0.0821655	−	0.571474i
							\(5\)	70.9340	−	20.8281i	0.253781	−	0.0745168i	−0.152368	−	0.988324i	\(-0.548690\pi\)
0.406149	+	0.913807i	\(0.366872\pi\)
\(7\)	−924.953	−	1067.45i	−1.01924	−	1.17627i	−0.984233	−	0.176879i	\(-0.943400\pi\)
							−0.0350074	−	0.999387i	\(-0.511145\pi\)
\(11\)	−3220.53	−	2069.71i	−0.729546	−	0.468851i	0.122400	−	0.992481i	\(-0.460941\pi\)
							−0.851946	+	0.523630i	\(0.824577\pi\)
\(13\)	−3762.86	+	4342.57i	−0.475025	+	0.548208i	−0.941802	−	0.336167i	\(-0.890869\pi\)
							0.466778	+	0.884375i	\(0.345415\pi\)
\(17\)	−8138.60	−	17821.0i	−0.401771	−	0.879756i	−0.997088	−	0.0762644i	\(-0.975701\pi\)
							0.595317	−	0.803491i	\(-0.297027\pi\)
\(19\)	13476.2	−	29508.7i	0.450744	−	0.986991i	−0.538757	−	0.842461i	\(-0.681106\pi\)
							0.989500	−	0.144530i	\(-0.0461669\pi\)
\(23\)	−52370.1	−	25733.2i	−0.897503	−	0.441007i
							\(29\)	18796.4	+	41158.3i	0.143114	+	0.313375i	0.967592	−	0.252518i	\(-0.0812587\pi\)
−0.824478	+	0.565893i	\(0.808531\pi\)
\(31\)	3163.39	+	22001.9i	0.0190716	+	0.132646i	0.997133	−	0.0756720i	\(-0.0241102\pi\)
							−0.978061	+	0.208318i	\(0.933201\pi\)
\(37\)	285649.	+	83874.1i	0.927100	+	0.272221i	0.710222	−	0.703978i	\(-0.248595\pi\)
							0.216878	+	0.976199i	\(0.430413\pi\)
\(41\)	−9828.74	+	2885.98i	−0.0222717	+	0.00653957i	−0.292849	−	0.956159i	\(-0.594603\pi\)
							0.270578	+	0.962698i	\(0.412785\pi\)
\(43\)	49930.8	−	347277.i	0.0957700	−	0.666095i	−0.884223	−	0.467064i	\(-0.845312\pi\)
							0.979993	−	0.199030i	\(-0.0637793\pi\)
\(47\)	825025.			1.15911			0.579555	−	0.814933i	\(-0.303227\pi\)
							0.579555	+	0.814933i	\(0.303227\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.e.b.58.11	yes	140
23.2	even	11	inner	69.8.e.b.25.11	✓	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.e.b.25.11	✓	140	23.2	even	11	inner
69.8.e.b.58.11	yes	140	1.1	even	1	trivial