Embedded newform 78.2.e.b.61.1

• Label: 78.2.e.b.61.1
• Level: $78$
• Weight: $2$
• Character: 78.61
• Analytic conductor: $0.623$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	78.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.622833135766\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			61.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	78.61
Dual form			78.2.e.b.55.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} -1.00000 q^{5} +(-0.500000 - 0.866025i) q^{6} +(1.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{3} - q^{4} - 2 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\)	\(53\)	\(67\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500000	−	0.866025i	0.353553	−	0.612372i
							\(3\)	0.500000	−	0.866025i	0.288675	−	0.500000i
\(5\)	−1.00000			−0.447214										−0.223607	−	0.974679i	\(-0.571783\pi\)
							−0.223607	+	0.974679i	\(0.571783\pi\)
\(7\)	1.00000	+	1.73205i	0.377964	+	0.654654i	0.990766	−	0.135583i	\(-0.0432908\pi\)
							−0.612801	+	0.790237i	\(0.709957\pi\)
\(11\)	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	\(-0.930824\pi\)
							0.674967	+	0.737848i	\(0.264158\pi\)
\(13\)	2.50000	+	2.59808i	0.693375	+	0.720577i
							\(17\)	−2.50000	−	4.33013i	−0.606339	−	1.05021i	−0.991838	−	0.127502i	\(-0.959304\pi\)
0.385499	−	0.922708i	\(-0.374029\pi\)
\(19\)	1.00000	+	1.73205i	0.229416	+	0.397360i	0.957635	−	0.287984i	\(-0.0929851\pi\)
							−0.728219	+	0.685344i	\(0.759652\pi\)
\(23\)	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	\(0.381789\pi\)
							−0.988436	+	0.151642i	\(0.951544\pi\)
\(29\)	4.50000	−	7.79423i	0.835629	−	1.44735i	−0.0578882	−	0.998323i	\(-0.518437\pi\)
							0.893517	−	0.449029i	\(-0.148230\pi\)
\(31\)	−4.00000			−0.718421			−0.359211	−	0.933257i	\(-0.616954\pi\)
							−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	5.50000	−	9.52628i	0.904194	−	1.56611i	0.0821995	−	0.996616i	\(-0.473806\pi\)
							0.821995	−	0.569495i	\(-0.192861\pi\)
\(41\)	−2.50000	+	4.33013i	−0.390434	+	0.676252i	−0.992507	−	0.122189i	\(-0.961009\pi\)
							0.602072	+	0.798441i	\(0.294342\pi\)
\(43\)	−5.00000	−	8.66025i	−0.762493	−	1.32068i	−0.941562	−	0.336840i	\(-0.890642\pi\)
							0.179069	−	0.983836i	\(-0.442691\pi\)
\(47\)	2.00000			0.291730			0.145865	−	0.989305i	\(-0.453403\pi\)
							0.145865	+	0.989305i	\(0.453403\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.2.e.b.61.1	yes	2
3.2	odd	2		234.2.h.b.217.1		2
4.3	odd	2		624.2.q.b.529.1		2
5.2	odd	4		1950.2.z.b.1699.2		4
5.3	odd	4		1950.2.z.b.1699.1		4
5.4	even	2		1950.2.i.b.451.1		2
12.11	even	2		1872.2.t.i.1153.1		2
13.2	odd	12		1014.2.i.e.361.2		4
13.3	even	3	inner	78.2.e.b.55.1	✓	2
13.4	even	6		1014.2.a.e.1.1		1
13.5	odd	4		1014.2.i.e.823.1		4
13.6	odd	12		1014.2.b.a.337.2		2
13.7	odd	12		1014.2.b.a.337.1		2
13.8	odd	4		1014.2.i.e.823.2		4
13.9	even	3		1014.2.a.a.1.1		1
13.10	even	6		1014.2.e.d.991.1		2
13.11	odd	12		1014.2.i.e.361.1		4
13.12	even	2		1014.2.e.d.529.1		2
39.17	odd	6		3042.2.a.d.1.1		1
39.20	even	12		3042.2.b.d.1351.2		2
39.29	odd	6		234.2.h.b.55.1		2
39.32	even	12		3042.2.b.d.1351.1		2
39.35	odd	6		3042.2.a.m.1.1		1
52.3	odd	6		624.2.q.b.289.1		2
52.35	odd	6		8112.2.a.x.1.1		1
52.43	odd	6		8112.2.a.bb.1.1		1
65.3	odd	12		1950.2.z.b.1849.2		4
65.29	even	6		1950.2.i.b.601.1		2
65.42	odd	12		1950.2.z.b.1849.1		4
156.107	even	6		1872.2.t.i.289.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.e.b.55.1	✓	2	13.3	even	3	inner
78.2.e.b.61.1	yes	2	1.1	even	1	trivial
234.2.h.b.55.1		2	39.29	odd	6
234.2.h.b.217.1		2	3.2	odd	2
624.2.q.b.289.1		2	52.3	odd	6
624.2.q.b.529.1		2	4.3	odd	2
1014.2.a.a.1.1		1	13.9	even	3
1014.2.a.e.1.1		1	13.4	even	6
1014.2.b.a.337.1		2	13.7	odd	12
1014.2.b.a.337.2		2	13.6	odd	12
1014.2.e.d.529.1		2	13.12	even	2
1014.2.e.d.991.1		2	13.10	even	6
1014.2.i.e.361.1		4	13.11	odd	12
1014.2.i.e.361.2		4	13.2	odd	12
1014.2.i.e.823.1		4	13.5	odd	4
1014.2.i.e.823.2		4	13.8	odd	4
1872.2.t.i.289.1		2	156.107	even	6
1872.2.t.i.1153.1		2	12.11	even	2
1950.2.i.b.451.1		2	5.4	even	2
1950.2.i.b.601.1		2	65.29	even	6
1950.2.z.b.1699.1		4	5.3	odd	4
1950.2.z.b.1699.2		4	5.2	odd	4
1950.2.z.b.1849.1		4	65.42	odd	12
1950.2.z.b.1849.2		4	65.3	odd	12
3042.2.a.d.1.1		1	39.17	odd	6
3042.2.a.m.1.1		1	39.35	odd	6
3042.2.b.d.1351.1		2	39.32	even	12
3042.2.b.d.1351.2		2	39.20	even	12
8112.2.a.x.1.1		1	52.35	odd	6
8112.2.a.bb.1.1		1	52.43	odd	6