Embedded newform 67.5.h.a.2.1

• Label: 67.5.h.a.2.1
• Level: $67$
• Weight: $5$
• Character: 67.2
• Analytic conductor: $6.926$
• Analytic rank: $0$
• Dimension: $440$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 67 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	67.h (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.92578752662\)
Analytic rank:	\(0\)
Dimension:	\(440\)
Relative dimension:	\(22\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			2.1
Character	\(\chi\)	\(=\)	67.2
Dual form			67.5.h.a.34.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.67522 - 0.365616i) q^{2} +(-4.26164 + 14.5138i) q^{3} +(42.8478 + 4.09147i) q^{4} +(-28.7699 - 24.9292i) q^{5} +(38.0155 - 109.839i) q^{6} +(4.03136 + 7.81975i) q^{7} +(-205.679 - 29.5722i) q^{8} +(-124.347 - 79.9132i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 440 q - 19 q^{2} - 22 q^{3} - 191 q^{4} - 22 q^{5} - 53 q^{6} - 16 q^{7} - 517 q^{8} + 840 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/67\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.67522	−	0.365616i	−1.91881	−	0.0914040i	−0.946160	−	0.323699i	\(-0.895073\pi\)
							−0.972645	+	0.232295i	\(0.925376\pi\)
\(3\)	−4.26164	+	14.5138i	−0.473515	+	1.61264i	0.283310	+	0.959028i	\(0.408567\pi\)
							−0.756826	+	0.653617i	\(0.773251\pi\)
\(5\)	−28.7699	−	24.9292i	−1.15079	−	0.997169i	−0.999963	−	0.00860949i	\(-0.997259\pi\)
							−0.150831	−	0.988559i	\(-0.548195\pi\)
\(7\)	4.03136	+	7.81975i	0.0822727	+	0.159587i	0.926383	−	0.376583i	\(-0.122901\pi\)
							−0.844110	+	0.536170i	\(0.819871\pi\)
\(11\)	84.4006	−	29.2113i	0.697525	−	0.241416i	0.0447839	−	0.998997i	\(-0.485740\pi\)
							0.652741	+	0.757581i	\(0.273619\pi\)
\(13\)	−115.908	−	147.389i	−0.685848	−	0.872127i	0.311085	−	0.950382i	\(-0.399308\pi\)
							−0.996933	+	0.0782552i	\(0.975065\pi\)
\(17\)	−162.203	+	15.4885i	−0.561257	+	0.0535935i	−0.371829	−	0.928301i	\(-0.621269\pi\)
							−0.189428	+	0.981895i	\(0.560663\pi\)
\(19\)	545.842	+	281.401i	1.51203	+	0.779505i	0.996980	−	0.0776564i	\(-0.0247437\pi\)
							0.515048	+	0.857161i	\(0.327774\pi\)
\(23\)	121.186	+	499.535i	0.229085	+	0.944300i	0.963949	+	0.266086i	\(0.0857305\pi\)
							−0.734865	+	0.678214i	\(0.762754\pi\)
\(29\)	150.426	−	260.546i	0.178866	−	0.309805i	−0.762627	−	0.646839i	\(-0.776091\pi\)
							0.941492	+	0.337034i	\(0.109424\pi\)
\(31\)	−121.399	+	154.371i	−0.126326	+	0.160636i	−0.845102	−	0.534606i	\(-0.820460\pi\)
							0.718776	+	0.695242i	\(0.244703\pi\)
\(37\)	−188.984	−	327.329i	−0.138045	−	0.239101i	0.788712	−	0.614763i	\(-0.210749\pi\)
							−0.926757	+	0.375663i	\(0.877415\pi\)
\(41\)	634.832	−	452.062i	0.377652	−	0.268925i	−0.375431	−	0.926850i	\(-0.622505\pi\)
							0.753083	+	0.657926i	\(0.228566\pi\)
\(43\)	71.3194	+	32.5705i	0.0385719	+	0.0176152i	0.434608	−	0.900620i	\(-0.356887\pi\)
							−0.396036	+	0.918235i	\(0.629614\pi\)
\(47\)	1206.34	−	1150.24i	0.546101	−	0.520706i	−0.366031	−	0.930603i	\(-0.619284\pi\)
							0.912132	+	0.409896i	\(0.134435\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	67.5.h.a.2.1	✓	440
67.34	odd	66	inner	67.5.h.a.34.1	yes	440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.5.h.a.2.1	✓	440	1.1	even	1	trivial
67.5.h.a.34.1	yes	440	67.34	odd	66	inner