Embedded newform 69.7.h.a.2.4

• Label: 69.7.h.a.2.4
• Level: $69$
• Weight: $7$
• Character: 69.2
• Analytic conductor: $15.874$
• Analytic rank: $0$
• Dimension: $460$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	69.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.8737317698\)
Analytic rank:	\(0\)
Dimension:	\(460\)
Relative dimension:	\(46\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			2.4
Character	\(\chi\)	\(=\)	69.2
Dual form			69.7.h.a.35.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.89511 + 12.2850i) q^{2} +(23.0414 - 14.0746i) q^{3} +(-62.0027 - 135.767i) q^{4} +(-55.1270 + 187.745i) q^{5} +(-9.00771 + 394.185i) q^{6} +(221.214 - 255.295i) q^{7} +(1232.32 + 177.181i) q^{8} +(332.812 - 648.596i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 460 q + 23 q^{3} + 1258 q^{4} - 106 q^{6} - 18 q^{7} - 921 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.89511	+	12.2850i	−0.986889	+	1.53563i	−0.149688	+	0.988733i	\(0.547827\pi\)
							−0.837201	+	0.546896i	\(0.815809\pi\)
\(3\)	23.0414	−	14.0746i	0.853385	−	0.521281i
							\(5\)	−55.1270	+	187.745i	−0.441016	+	1.50196i	0.376697	+	0.926337i	\(0.377060\pi\)
−0.817713	+	0.575626i	\(0.804758\pi\)
\(7\)	221.214	−	255.295i	0.644939	−	0.744299i	−0.335301	−	0.942111i	\(-0.608838\pi\)
							0.980240	+	0.197812i	\(0.0633835\pi\)
\(11\)	900.655	+	1401.45i	0.676676	+	1.05293i	0.994496	+	0.104777i	\(0.0334128\pi\)
							−0.317820	+	0.948151i	\(0.602951\pi\)
\(13\)	1264.91	+	1459.78i	0.575744	+	0.664444i	0.966684	−	0.255972i	\(-0.0823956\pi\)
							−0.390940	+	0.920416i	\(0.627850\pi\)
\(17\)	−7113.02	−	3248.41i	−1.44780	−	0.661186i	−0.472347	−	0.881413i	\(-0.656593\pi\)
							−0.975449	+	0.220227i	\(0.929320\pi\)
\(19\)	4776.51	+	10459.1i	0.696386	+	1.52487i	0.844299	+	0.535872i	\(0.180017\pi\)
							−0.147913	+	0.989000i	\(0.547256\pi\)
\(23\)	−4281.21	+	11388.9i	−0.351871	+	0.936049i
							\(29\)	−14008.8	−	6397.60i	−0.574389	−	0.262315i	0.106972	−	0.994262i	\(-0.465884\pi\)
−0.681361	+	0.731947i	\(0.738612\pi\)
\(31\)	−582.999	+	4054.85i	−0.0195696	+	0.136110i	−0.997264	−	0.0739231i	\(-0.976448\pi\)
							0.977694	+	0.210033i	\(0.0673571\pi\)
\(37\)	−29858.8	+	8767.34i	−0.589478	+	0.173086i	−0.562850	−	0.826559i	\(-0.690295\pi\)
							−0.0266279	+	0.999645i	\(0.508477\pi\)
\(41\)	−617.923	+	2104.45i	−0.00896567	+	0.0305343i	−0.963854	−	0.266430i	\(-0.914156\pi\)
							0.954889	+	0.296964i	\(0.0959741\pi\)
\(43\)	−2242.04	−	15593.7i	−0.0281993	−	0.196130i	0.970852	−	0.239680i	\(-0.0770424\pi\)
							−0.999051	+	0.0435496i	\(0.986133\pi\)
\(47\)			109036.i			1.05021i	0.851038	+	0.525104i	\(0.175974\pi\)
							−0.851038	+	0.525104i	\(0.824026\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.h.a.2.4	✓	460
3.2	odd	2	inner	69.7.h.a.2.43	yes	460
23.12	even	11	inner	69.7.h.a.35.43	yes	460
69.35	odd	22	inner	69.7.h.a.35.4	yes	460

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.h.a.2.4	✓	460	1.1	even	1	trivial
69.7.h.a.2.43	yes	460	3.2	odd	2	inner
69.7.h.a.35.4	yes	460	69.35	odd	22	inner
69.7.h.a.35.43	yes	460	23.12	even	11	inner