Embedded newform 67.5.h.a.2.17

• Label: 67.5.h.a.2.17
• 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			2.17
Character	\(\chi\)	\(=\)	67.2
Dual form			67.5.h.a.34.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.80072 + 0.181051i) q^{2} +(1.10242 - 3.75449i) q^{3} +(-1.51483 - 0.144649i) q^{4} +(-14.6109 - 12.6604i) q^{5} +(4.86973 - 14.0702i) q^{6} +(-42.6315 - 82.6937i) q^{7} +(-65.9921 - 9.48824i) q^{8} +(55.2607 + 35.5139i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 440 q - 19 q^{2} - 22 q^{3} - 191 q^{4} - 22 q^{5} - 53 q^{6} - 16 q^{7} - 517 q^{8} + 840 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.80072	+	0.181051i	0.950181	+	0.0452627i	0.516946	−	0.856018i	\(-0.327069\pi\)
							0.433235	+	0.901281i	\(0.357372\pi\)
\(3\)	1.10242	−	3.75449i	0.122491	−	0.417165i	−0.875302	−	0.483577i	\(-0.839337\pi\)
							0.997792	+	0.0664125i	\(0.0211553\pi\)
\(5\)	−14.6109	−	12.6604i	−0.584436	−	0.506417i	0.311709	−	0.950178i	\(-0.399098\pi\)
							−0.896145	+	0.443761i	\(0.853644\pi\)
\(7\)	−42.6315	−	82.6937i	−0.870032	−	1.68763i	−0.714315	−	0.699824i	\(-0.753262\pi\)
							−0.155716	−	0.987802i	\(-0.549769\pi\)
\(11\)	130.509	−	45.1695i	1.07859	−	0.373302i	0.270713	−	0.962660i	\(-0.412740\pi\)
							0.807872	+	0.589358i	\(0.200619\pi\)
\(13\)	22.3762	+	28.4536i	0.132404	+	0.168365i	0.847734	−	0.530421i	\(-0.177966\pi\)
							−0.715331	+	0.698786i	\(0.753724\pi\)
\(17\)	−228.530	+	21.8220i	−0.790763	+	0.0755087i	−0.482618	−	0.875831i	\(-0.660314\pi\)
							−0.308144	+	0.951340i	\(0.599708\pi\)
\(19\)	407.280	+	209.968i	1.12820	+	0.581628i	0.918217	−	0.396077i	\(-0.129629\pi\)
							0.209983	+	0.977705i	\(0.432659\pi\)
\(23\)	−88.4022	−	364.399i	−0.167112	−	0.688844i	−0.992286	−	0.123967i	\(-0.960438\pi\)
							0.825175	−	0.564878i	\(-0.191077\pi\)
\(29\)	−122.725	+	212.566i	−0.145927	+	0.252753i	−0.929719	−	0.368271i	\(-0.879950\pi\)
							0.783791	+	0.621024i	\(0.213283\pi\)
\(31\)	−50.6821	+	64.4476i	−0.0527389	+	0.0670630i	−0.811707	−	0.584065i	\(-0.801461\pi\)
							0.758968	+	0.651128i	\(0.225704\pi\)
\(37\)	−418.866	−	725.497i	−0.305965	−	0.529947i	0.671511	−	0.740995i	\(-0.265646\pi\)
							−0.977476	+	0.211048i	\(0.932312\pi\)
\(41\)	321.630	−	229.031i	0.191332	−	0.136247i	−0.480377	−	0.877062i	\(-0.659500\pi\)
							0.671709	+	0.740815i	\(0.265561\pi\)
\(43\)	2083.19	+	951.361i	1.12666	+	0.514527i	0.889496	−	0.456943i	\(-0.151056\pi\)
							0.237161	+	0.971470i	\(0.423783\pi\)
\(47\)	1883.02	−	1795.45i	0.852429	−	0.812789i	−0.131506	−	0.991315i	\(-0.541981\pi\)
							0.983935	+	0.178526i	\(0.0571329\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	67.5.h.a.2.17	✓	440
67.34	odd	66	inner	67.5.h.a.34.17	yes	440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.5.h.a.2.17	✓	440	1.1	even	1	trivial
67.5.h.a.34.17	yes	440	67.34	odd	66	inner