Embedded newform 69.3.f.a.19.1

• Label: 69.3.f.a.19.1
• Level: $69$
• Weight: $3$
• Character: 69.19
• Analytic conductor: $1.880$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.88011382409\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(8\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			19.1
Character	\(\chi\)	\(=\)	69.19
Dual form			69.3.f.a.40.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23384 - 2.70174i) q^{2} +(-1.66189 + 0.487975i) q^{3} +(-3.15759 + 3.64405i) q^{4} +(-4.82215 + 7.50341i) q^{5} +(3.36890 + 3.88791i) q^{6} +(1.37721 - 0.198013i) q^{7} +(2.34192 + 0.687649i) q^{8} +(2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 4 q^{2} - 12 q^{6} - 4 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{15}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23384	−	2.70174i	−0.616922	−	1.35087i	−0.917737	−	0.397188i	\(-0.869986\pi\)
							0.300815	−	0.953682i	\(-0.402741\pi\)
\(3\)	−1.66189	+	0.487975i	−0.553964	+	0.162658i
							\(5\)	−4.82215	+	7.50341i	−0.964430	+	1.50068i	−0.101827	+	0.994802i	\(0.532469\pi\)
−0.862603	+	0.505881i	\(0.831168\pi\)
\(7\)	1.37721	−	0.198013i	0.196744	−	0.0282875i	−0.0432382	−	0.999065i	\(-0.513767\pi\)
							0.239982	+	0.970777i	\(0.422858\pi\)
\(11\)	−10.2281	−	4.67100i	−0.929825	−	0.424637i	−0.107854	−	0.994167i	\(-0.534398\pi\)
							−0.821971	+	0.569530i	\(0.807125\pi\)
\(13\)	−1.94283	+	13.5127i	−0.149449	+	1.03944i	0.767676	+	0.640838i	\(0.221413\pi\)
							−0.917125	+	0.398600i	\(0.869496\pi\)
\(17\)	−6.74692	+	5.84624i	−0.396878	+	0.343897i	−0.830322	−	0.557284i	\(-0.811844\pi\)
							0.433444	+	0.901180i	\(0.357298\pi\)
\(19\)	−17.6434	−	15.2881i	−0.928601	−	0.804638i	0.0524024	−	0.998626i	\(-0.483312\pi\)
							−0.981004	+	0.193988i	\(0.937858\pi\)
\(23\)	12.7617	+	19.1347i	0.554859	+	0.831945i
							\(29\)	−25.0191	−	28.8736i	−0.862728	−	0.995641i	−0.999987	−	0.00510621i	\(-0.998375\pi\)
0.137259	−	0.990535i	\(-0.456171\pi\)
\(31\)	−0.186054	−	0.0546305i	−0.00600175	−	0.00176227i	0.278730	−	0.960369i	\(-0.410086\pi\)
							−0.284732	+	0.958607i	\(0.591905\pi\)
\(37\)	21.7883	+	33.9032i	0.588872	+	0.916303i	0.999989	+	0.00472006i	\(0.00150245\pi\)
							−0.411117	+	0.911583i	\(0.634861\pi\)
\(41\)	33.0747	+	21.2558i	0.806700	+	0.518435i	0.877796	−	0.479035i	\(-0.159013\pi\)
							−0.0710957	+	0.997469i	\(0.522650\pi\)
\(43\)	−0.468547	−	1.59573i	−0.0108964	−	0.0371099i	0.953865	−	0.300235i	\(-0.0970650\pi\)
							−0.964762	+	0.263125i	\(0.915247\pi\)
\(47\)	89.9970			1.91483			0.957415	−	0.288716i	\(-0.0932282\pi\)
							0.957415	+	0.288716i	\(0.0932282\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.3.f.a.19.1	✓	80
3.2	odd	2		207.3.j.b.19.8		80
23.17	odd	22	inner	69.3.f.a.40.1	yes	80
69.17	even	22		207.3.j.b.109.8		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.3.f.a.19.1	✓	80	1.1	even	1	trivial
69.3.f.a.40.1	yes	80	23.17	odd	22	inner
207.3.j.b.19.8		80	3.2	odd	2
207.3.j.b.109.8		80	69.17	even	22