Embedded newform 690.2.n.a.89.9

• Label: 690.2.n.a.89.9
• Level: $690$
• Weight: $2$
• Character: 690.89
• Analytic conductor: $5.510$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.n (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.50967773947\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			89.9
Character	\(\chi\)	\(=\)	690.89
Dual form			690.2.n.a.659.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.142315 - 0.989821i) q^{2} +(-0.763868 - 1.55451i) q^{3} +(-0.959493 + 0.281733i) q^{4} +(-1.25961 - 1.84754i) q^{5} +(-1.42998 + 0.977323i) q^{6} +(3.27194 - 2.10275i) q^{7} +(0.415415 + 0.909632i) q^{8} +(-1.83301 + 2.37488i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 24 q^{2} + 2 q^{3} - 24 q^{4} + 2 q^{6} - 24 q^{8} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\)	\(277\)	\(461\)	\(511\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.142315	−	0.989821i	−0.100632	−	0.699909i
							\(3\)	−0.763868	−	1.55451i	−0.441019	−	0.897498i
\(5\)	−1.25961	−	1.84754i	−0.563314	−	0.826243i
							\(7\)	3.27194	−	2.10275i	1.23668	−	0.794765i	0.251761	−	0.967790i	\(-0.418990\pi\)
0.984917	+	0.173025i	\(0.0553541\pi\)
\(11\)	0.440681	−	3.06500i	0.132870	−	0.924133i	−0.808917	−	0.587923i	\(-0.799946\pi\)
							0.941787	−	0.336210i	\(-0.109145\pi\)
\(13\)	3.74156	−	5.82197i	1.03772	−	1.61472i	0.282485	−	0.959272i	\(-0.408841\pi\)
							0.755236	−	0.655453i	\(-0.227522\pi\)
\(17\)	−1.14788	+	3.90933i	−0.278403	+	0.948153i	0.694992	+	0.719018i	\(0.255408\pi\)
							−0.973395	+	0.229135i	\(0.926410\pi\)
\(19\)	−1.77219	−	6.03553i	−0.406569	−	1.38465i	−0.867600	−	0.497262i	\(-0.834339\pi\)
							0.461032	−	0.887384i	\(-0.347479\pi\)
\(23\)	−4.26358	−	2.19588i	−0.889018	−	0.457873i
							\(29\)	−1.15504	+	3.93371i	−0.214486	+	0.730471i	0.780017	+	0.625758i	\(0.215210\pi\)
−0.994503	+	0.104713i	\(0.966608\pi\)
\(31\)	4.59962	+	10.0718i	0.826116	+	1.80894i	0.509824	+	0.860278i	\(0.329710\pi\)
							0.316291	+	0.948662i	\(0.397562\pi\)
\(37\)	2.13622	+	2.46533i	0.351193	+	0.405298i	0.903670	−	0.428230i	\(-0.140863\pi\)
							−0.552477	+	0.833528i	\(0.686317\pi\)
\(41\)	4.29664	+	3.72306i	0.671023	+	0.581445i	0.922302	−	0.386470i	\(-0.126306\pi\)
							−0.251279	+	0.967915i	\(0.580851\pi\)
\(43\)	−0.812374	+	1.77885i	−0.123886	+	0.271272i	−0.961406	−	0.275135i	\(-0.911278\pi\)
							0.837520	+	0.546407i	\(0.184005\pi\)
\(47\)	0.0592923			0.00864867			0.00432434	−	0.999991i	\(-0.498624\pi\)
							0.00432434	+	0.999991i	\(0.498624\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.2.n.a.89.9	✓	240
3.2	odd	2		690.2.n.b.89.3	yes	240
5.4	even	2		690.2.n.b.89.16	yes	240
15.14	odd	2	inner	690.2.n.a.89.22	yes	240
23.15	odd	22	inner	690.2.n.a.659.22	yes	240
69.38	even	22		690.2.n.b.659.16	yes	240
115.84	odd	22		690.2.n.b.659.3	yes	240
345.314	even	22	inner	690.2.n.a.659.9	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.n.a.89.9	✓	240	1.1	even	1	trivial
690.2.n.a.89.22	yes	240	15.14	odd	2	inner
690.2.n.a.659.9	yes	240	345.314	even	22	inner
690.2.n.a.659.22	yes	240	23.15	odd	22	inner
690.2.n.b.89.3	yes	240	3.2	odd	2
690.2.n.b.89.16	yes	240	5.4	even	2
690.2.n.b.659.3	yes	240	115.84	odd	22
690.2.n.b.659.16	yes	240	69.38	even	22