Embedded newform 69.5.h.a.2.11

• Label: 69.5.h.a.2.11
• Level: $69$
• Weight: $5$
• Character: 69.2
• Analytic conductor: $7.133$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	69.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.13252745279\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(30\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			2.11
Character	\(\chi\)	\(=\)	69.2
Dual form			69.5.h.a.35.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18989 + 1.85151i) q^{2} +(-8.40626 - 3.21477i) q^{3} +(4.63439 + 10.1479i) q^{4} +(-4.40450 + 15.0003i) q^{5} +(15.9547 - 11.7391i) q^{6} +(8.45761 - 9.76060i) q^{7} +(-59.1593 - 8.50582i) q^{8} +(60.3306 + 54.0483i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 19 q^{3} + 234 q^{4} - 10 q^{6} - 18 q^{7} - 111 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18989	+	1.85151i	−0.297473	+	0.462878i	−0.957528	−	0.288340i	\(-0.906897\pi\)
							0.660055	+	0.751218i	\(0.270533\pi\)
\(3\)	−8.40626	−	3.21477i	−0.934029	−	0.357196i
							\(5\)	−4.40450	+	15.0003i	−0.176180	+	0.600014i	0.823295	+	0.567614i	\(0.192133\pi\)
−0.999475	+	0.0324001i	\(0.989685\pi\)
\(7\)	8.45761	−	9.76060i	0.172604	−	0.199196i	−0.662856	−	0.748747i	\(-0.730656\pi\)
							0.835460	+	0.549551i	\(0.185201\pi\)
\(11\)	−66.9597	−	104.191i	−0.553386	−	0.861086i	0.446038	−	0.895014i	\(-0.352835\pi\)
							−0.999424	+	0.0339282i	\(0.989198\pi\)
\(13\)	−125.667	−	145.027i	−0.743591	−	0.858149i	0.250340	−	0.968158i	\(-0.419458\pi\)
							−0.993931	+	0.110009i	\(0.964912\pi\)
\(17\)	−190.659	−	87.0711i	−0.659720	−	0.301284i	0.0572789	−	0.998358i	\(-0.481758\pi\)
							−0.716999	+	0.697074i	\(0.754485\pi\)
\(19\)	−197.393	−	432.230i	−0.546794	−	1.19731i	−0.958263	−	0.285888i	\(-0.907711\pi\)
							0.411469	−	0.911424i	\(-0.365016\pi\)
\(23\)	510.010	−	140.466i	0.964102	−	0.265532i
							\(29\)	−537.015	−	245.247i	−0.638544	−	0.291613i	0.0697140	−	0.997567i	\(-0.477791\pi\)
−0.708258	+	0.705954i	\(0.750519\pi\)
\(31\)	−207.347	+	1442.13i	−0.215762	+	1.50066i	0.537682	+	0.843148i	\(0.319300\pi\)
							−0.753444	+	0.657512i	\(0.771609\pi\)
\(37\)	−2192.66	+	643.823i	−1.60165	+	0.470287i	−0.956005	−	0.293349i	\(-0.905230\pi\)
							−0.645646	+	0.763637i	\(0.723412\pi\)
\(41\)	−72.6048	+	247.269i	−0.0431914	+	0.147097i	−0.978266	−	0.207355i	\(-0.933514\pi\)
							0.935074	+	0.354452i	\(0.115333\pi\)
\(43\)	71.5682	+	497.768i	0.0387064	+	0.269209i	0.999980	−	0.00638822i	\(-0.00203345\pi\)
							−0.961273	+	0.275597i	\(0.911124\pi\)
\(47\)		−	2678.07i		−	1.21234i	−0.795334	−	0.606171i	\(-0.792705\pi\)
							0.795334	−	0.606171i	\(-0.207295\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.h.a.2.11	✓	300
3.2	odd	2	inner	69.5.h.a.2.20	yes	300
23.12	even	11	inner	69.5.h.a.35.20	yes	300
69.35	odd	22	inner	69.5.h.a.35.11	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.h.a.2.11	✓	300	1.1	even	1	trivial
69.5.h.a.2.20	yes	300	3.2	odd	2	inner
69.5.h.a.35.11	yes	300	69.35	odd	22	inner
69.5.h.a.35.20	yes	300	23.12	even	11	inner