Embedded newform 690.2.x.a.77.4

• Label: 690.2.x.a.77.4
• Level: $690$
• Weight: $2$
• Character: 690.77
• Analytic conductor: $5.510$
• Analytic rank: $0$
• Dimension: $960$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.x (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.50967773947\)
Analytic rank:	\(0\)
Dimension:	\(960\)
Relative dimension:	\(48\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			77.4
Character	\(\chi\)	\(=\)	690.77
Dual form			690.2.x.a.233.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.599278 - 0.800541i) q^{2} +(-1.54741 + 0.778153i) q^{3} +(-0.281733 + 0.959493i) q^{4} +(-1.72896 - 1.41799i) q^{5} +(1.55027 + 0.772436i) q^{6} +(0.112122 + 0.515415i) q^{7} +(0.936950 - 0.349464i) q^{8} +(1.78896 - 2.40824i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q + 8 q^{3} + 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(277\)	\(461\)	\(511\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.599278	−	0.800541i	−0.423753	−	0.566068i
							\(3\)	−1.54741	+	0.778153i	−0.893398	+	0.449267i
\(5\)	−1.72896	−	1.41799i	−0.773213	−	0.634147i
							\(7\)	0.112122	+	0.515415i	0.0423780	+	0.194808i	0.993469	−	0.114102i	\(-0.0363992\pi\)
−0.951091	+	0.308911i	\(0.900036\pi\)
\(11\)	−2.88284	+	0.414490i	−0.869209	+	0.124973i	−0.562467	−	0.826820i	\(-0.690148\pi\)
							−0.306741	+	0.951793i	\(0.599239\pi\)
\(13\)	−1.01637	+	4.67216i	−0.281889	+	1.29582i	0.590510	+	0.807030i	\(0.298927\pi\)
							−0.872399	+	0.488794i	\(0.837437\pi\)
\(17\)	0.572918	−	1.04922i	0.138953	−	0.254473i	−0.798931	−	0.601423i	\(-0.794601\pi\)
							0.937884	+	0.346950i	\(0.112782\pi\)
\(19\)	0.695429	−	2.36841i	0.159542	−	0.543351i	−0.840457	−	0.541879i	\(-0.817713\pi\)
							0.999999	−	0.00147230i	\(-0.000468647\pi\)
\(23\)	−4.20966	−	2.29756i	−0.877775	−	0.479073i
							\(29\)	8.53585	−	2.50635i	1.58507	−	0.465418i	0.633727	−	0.773557i	\(-0.281524\pi\)
0.951341	+	0.308139i	\(0.0997063\pi\)
\(31\)	0.803914	−	1.76033i	0.144387	−	0.316164i	−0.823597	−	0.567176i	\(-0.808036\pi\)
							0.967984	+	0.251012i	\(0.0807633\pi\)
\(37\)	10.7271	−	0.767219i	1.76353	−	0.126130i	0.848417	−	0.529329i	\(-0.177556\pi\)
							0.915112	+	0.403199i	\(0.132102\pi\)
\(41\)	−0.370153	+	0.320740i	−0.0578082	+	0.0500911i	−0.683286	−	0.730151i	\(-0.739450\pi\)
							0.625478	+	0.780242i	\(0.284904\pi\)
\(43\)	10.4363	+	3.89255i	1.59153	+	0.593608i	0.980242	−	0.197803i	\(-0.0633805\pi\)
							0.611284	+	0.791411i	\(0.290653\pi\)
\(47\)	2.35668	−	2.35668i	0.343758	−	0.343758i	−0.514020	−	0.857778i	\(-0.671844\pi\)
							0.857778	+	0.514020i	\(0.171844\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.2.x.a.77.4	✓	960
3.2	odd	2	inner	690.2.x.a.77.34	yes	960
5.3	odd	4	inner	690.2.x.a.353.8	yes	960
15.8	even	4	inner	690.2.x.a.353.45	yes	960
23.3	even	11	inner	690.2.x.a.647.45	yes	960
69.26	odd	22	inner	690.2.x.a.647.8	yes	960
115.3	odd	44	inner	690.2.x.a.233.34	yes	960
345.233	even	44	inner	690.2.x.a.233.4	yes	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.x.a.77.4	✓	960	1.1	even	1	trivial
690.2.x.a.77.34	yes	960	3.2	odd	2	inner
690.2.x.a.233.4	yes	960	345.233	even	44	inner
690.2.x.a.233.34	yes	960	115.3	odd	44	inner
690.2.x.a.353.8	yes	960	5.3	odd	4	inner
690.2.x.a.353.45	yes	960	15.8	even	4	inner
690.2.x.a.647.8	yes	960	69.26	odd	22	inner
690.2.x.a.647.45	yes	960	23.3	even	11	inner