Embedded newform 69.4.e.b.58.2

• Label: 69.4.e.b.58.2
• Level: $69$
• Weight: $4$
• Character: 69.58
• Analytic conductor: $4.071$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.07113179040\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			58.2
Character	\(\chi\)	\(=\)	69.58
Dual form			69.4.e.b.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.39601 + 1.53982i) q^{2} +(0.426945 - 2.96946i) q^{3} +(0.0464818 - 0.101781i) q^{4} +(-1.22325 + 0.359177i) q^{5} +(3.54948 + 7.77227i) q^{6} +(-1.84765 - 2.13230i) q^{7} +(-3.19731 - 22.2377i) q^{8} +(-8.63544 - 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 4 q^{2} + 18 q^{3} - 28 q^{4} - 6 q^{5} + 21 q^{6} - 4 q^{7} - 52 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.39601	+	1.53982i	−0.847116	+	0.544409i	−0.890674	−	0.454642i	\(-0.849767\pi\)
							0.0435579	+	0.999051i	\(0.486131\pi\)
\(3\)	0.426945	−	2.96946i	0.0821655	−	0.571474i
							\(5\)	−1.22325	+	0.359177i	−0.109410	+	0.0321258i	−0.335979	−	0.941869i	\(-0.609067\pi\)
0.226569	+	0.973995i	\(0.427249\pi\)
\(7\)	−1.84765	−	2.13230i	−0.0997638	−	0.115134i	0.703672	−	0.710525i	\(-0.251542\pi\)
							−0.803436	+	0.595391i	\(0.796997\pi\)
\(11\)	−46.9103	−	30.1474i	−1.28582	−	0.826345i	−0.294224	−	0.955736i	\(-0.595061\pi\)
							−0.991594	+	0.129391i	\(0.958698\pi\)
\(13\)	41.6560	−	48.0736i	0.888716	−	1.02563i	−0.110779	−	0.993845i	\(-0.535335\pi\)
							0.999495	−	0.0317875i	\(-0.0101200\pi\)
\(17\)	−43.6303	−	95.5370i	−0.622464	−	1.36301i	−0.913713	−	0.406360i	\(-0.866798\pi\)
							0.291249	−	0.956647i	\(-0.405929\pi\)
\(19\)	21.0526	−	46.0988i	0.254200	−	0.556621i	−0.738910	−	0.673804i	\(-0.764659\pi\)
							0.993110	+	0.117183i	\(0.0373865\pi\)
\(23\)	−41.4645	+	102.214i	−0.375911	+	0.926656i
							\(29\)	−25.6108	−	56.0799i	−0.163993	−	0.359095i	0.809739	−	0.586790i	\(-0.199609\pi\)
−0.973733	+	0.227694i	\(0.926881\pi\)
\(31\)	25.6432	+	178.352i	0.148569	+	1.03332i	0.918564	+	0.395272i	\(0.129350\pi\)
							−0.769995	+	0.638050i	\(0.779741\pi\)
\(37\)	−6.66337	−	1.95654i	−0.0296068	−	0.00869334i	0.266896	−	0.963725i	\(-0.414002\pi\)
							−0.296502	+	0.955032i	\(0.595820\pi\)
\(41\)	−182.253	+	53.5143i	−0.694223	+	0.203842i	−0.609762	−	0.792585i	\(-0.708735\pi\)
							−0.0844608	+	0.996427i	\(0.526917\pi\)
\(43\)	23.4578	−	163.152i	0.0831924	−	0.578616i	−0.905002	−	0.425408i	\(-0.860130\pi\)
							0.988194	−	0.153208i	\(-0.0489604\pi\)
\(47\)	347.894			1.07969			0.539846	−	0.841764i	\(-0.318482\pi\)
							0.539846	+	0.841764i	\(0.318482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.e.b.58.2	yes	60
3.2	odd	2		207.4.i.b.127.5		60
23.2	even	11	inner	69.4.e.b.25.2	✓	60
23.5	odd	22		1587.4.a.v.1.8		30
23.18	even	11		1587.4.a.w.1.8		30
69.2	odd	22		207.4.i.b.163.5		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.e.b.25.2	✓	60	23.2	even	11	inner
69.4.e.b.58.2	yes	60	1.1	even	1	trivial
207.4.i.b.127.5		60	3.2	odd	2
207.4.i.b.163.5		60	69.2	odd	22
1587.4.a.v.1.8		30	23.5	odd	22
1587.4.a.w.1.8		30	23.18	even	11