Embedded newform 69.8.e.a.13.2

• Label: 69.8.e.a.13.2
• Level: $69$
• Weight: $8$
• Character: 69.13
• 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			13.2
Character	\(\chi\)	\(=\)	69.13
Dual form			69.8.e.a.16.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.6508 - 13.4458i) q^{2} +(22.7138 + 14.5973i) q^{3} +(-26.8307 + 186.612i) q^{4} +(71.8962 - 157.431i) q^{5} +(-68.3631 - 475.476i) q^{6} +(-54.4916 - 16.0002i) q^{7} +(905.962 - 582.226i) q^{8} +(302.838 + 663.122i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 16 q^{2} - 378 q^{3} - 740 q^{4} - 6 q^{5} + 1647 q^{6} + 978 q^{7} - 824 q^{8} - 10206 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{7}{11}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.6508	−	13.4458i	−1.02980	−	1.18845i	−0.981862	−	0.189599i	\(-0.939281\pi\)
							−0.0479359	−	0.998850i	\(-0.515264\pi\)
\(3\)	22.7138	+	14.5973i	0.485698	+	0.312139i
							\(5\)	71.8962	−	157.431i	0.257224	−	0.563241i	−0.736328	−	0.676625i	\(-0.763442\pi\)
0.993551	+	0.113384i	\(0.0361691\pi\)
\(7\)	−54.4916	−	16.0002i	−0.0600464	−	0.0176312i	0.251571	−	0.967839i	\(-0.419053\pi\)
							−0.311618	+	0.950208i	\(0.600871\pi\)
\(11\)	−4263.67	+	4920.54i	−0.965850	+	1.11465i	0.0275121	+	0.999621i	\(0.491242\pi\)
							−0.993362	+	0.115029i	\(0.963304\pi\)
\(13\)	15003.8	−	4405.50i	1.89408	−	0.556152i	0.901798	−	0.432158i	\(-0.142248\pi\)
							0.992283	−	0.123994i	\(-0.0395704\pi\)
\(17\)	−2367.93	−	16469.3i	−0.116896	−	0.813027i	−0.960941	−	0.276755i	\(-0.910741\pi\)
							0.844045	−	0.536273i	\(-0.180168\pi\)
\(19\)	7847.69	−	54581.9i	0.262485	−	1.82562i	−0.251541	−	0.967847i	\(-0.580937\pi\)
							0.514025	−	0.857775i	\(-0.328154\pi\)
\(23\)	16966.8	+	55829.7i	0.290773	+	0.956792i
							\(29\)	−11813.6	−	82165.2i	−0.0899473	−	0.625597i	−0.984071	−	0.177777i	\(-0.943110\pi\)
0.894124	−	0.447820i	\(-0.147800\pi\)
\(31\)	−54017.0	+	34714.6i	−0.325660	+	0.209289i	−0.693247	−	0.720700i	\(-0.743820\pi\)
							0.367587	+	0.929989i	\(0.380184\pi\)
\(37\)	−171392.	−	375297.i	−0.556270	−	1.21806i	−0.953792	−	0.300469i	\(-0.902857\pi\)
							0.397522	−	0.917593i	\(-0.369870\pi\)
\(41\)	318553.	−	697533.i	0.721834	−	1.58060i	−0.0894813	−	0.995989i	\(-0.528521\pi\)
							0.811316	−	0.584608i	\(-0.198752\pi\)
\(43\)	−56940.2	−	36593.2i	−0.109214	−	0.0701877i	0.484893	−	0.874573i	\(-0.338858\pi\)
							−0.594108	+	0.804385i	\(0.702495\pi\)
\(47\)	−908968.			−1.27704			−0.638522	−	0.769603i	\(-0.720454\pi\)
							−0.638522	+	0.769603i	\(0.720454\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.e.a.13.2	✓	140
23.16	even	11	inner	69.8.e.a.16.2	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.e.a.13.2	✓	140	1.1	even	1	trivial
69.8.e.a.16.2	yes	140	23.16	even	11	inner