Embedded newform 78.3.j.a.17.6

• Label: 78.3.j.a.17.6
• Level: $78$
• Weight: $3$
• Character: 78.17
• Analytic conductor: $2.125$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	78.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.12534606201\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	x^20 + 2*x^18 - 12*x^17 - 51*x^16 - 180*x^15 + 1136*x^14 + 144*x^13 + 6481*x^12 - 16152*x^11 - 55134*x^10 - 145368*x^9 + 524961*x^8 + 104976*x^7 + 7453296*x^6 - 10628820*x^5 - 27103491*x^4 - 57395628*x^3 + 86093442*x^2 + 3486784401
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.6
Root			\(2.96586 - 0.451313i\) of defining polynomial
Character	\(\chi\)	\(=\)	78.17
Dual form			78.3.j.a.23.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.707107 - 1.22474i) q^{2} +(-2.96586 + 0.451313i) q^{3} +(-1.00000 - 1.73205i) q^{4} -8.26315 q^{5} +(-1.54444 + 3.95155i) q^{6} +(-3.50129 + 2.02147i) q^{7} -2.82843 q^{8} +(8.59263 - 2.67706i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{4} + 18 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(53\)	\(67\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.707107	−	1.22474i	0.353553	−	0.612372i
							\(3\)	−2.96586	+	0.451313i	−0.988619	+	0.150438i
\(5\)	−8.26315			−1.65263										−0.826315	−	0.563208i	\(-0.809567\pi\)
							−0.826315	+	0.563208i	\(0.809567\pi\)
\(7\)	−3.50129	+	2.02147i	−0.500184	+	0.288781i	−0.728789	−	0.684738i	\(-0.759917\pi\)
							0.228606	+	0.973519i	\(0.426583\pi\)
\(11\)	4.05794	−	7.02856i	0.368904	−	0.638960i	−0.620491	−	0.784214i	\(-0.713067\pi\)
							0.989394	+	0.145254i	\(0.0463999\pi\)
\(13\)	−8.22547	−	10.0669i	−0.632728	−	0.774374i
							\(17\)	−19.4862	+	11.2504i	−1.14625	+	0.661786i	−0.947970	−	0.318360i	\(-0.896868\pi\)
−0.198277	+	0.980146i	\(0.563535\pi\)
\(19\)	8.90745	−	5.14272i	0.468813	−	0.270669i	−0.246930	−	0.969033i	\(-0.579422\pi\)
							0.715743	+	0.698364i	\(0.246088\pi\)
\(23\)	7.58981	+	4.38198i	0.329992	+	0.190521i	0.655837	−	0.754902i	\(-0.272316\pi\)
							−0.325846	+	0.945423i	\(0.605649\pi\)
\(29\)	−19.0444	−	10.9953i	−0.656703	−	0.379148i	0.134317	−	0.990938i	\(-0.457116\pi\)
							−0.791020	+	0.611791i	\(0.790449\pi\)
\(31\)		−	40.2922i		−	1.29975i	−0.760042	−	0.649874i	\(-0.774822\pi\)
							0.760042	−	0.649874i	\(-0.225178\pi\)
\(37\)	−42.8127	−	24.7179i	−1.15710	−	0.668052i	−0.206493	−	0.978448i	\(-0.566205\pi\)
							−0.950607	+	0.310396i	\(0.899538\pi\)
\(41\)	−31.3280	+	54.2618i	−0.764099	+	1.32346i	0.176623	+	0.984279i	\(0.443483\pi\)
							−0.940722	+	0.339179i	\(0.889851\pi\)
\(43\)	33.3480	+	57.7605i	0.775535	+	1.34327i	0.934493	+	0.355981i	\(0.115853\pi\)
							−0.158958	+	0.987285i	\(0.550813\pi\)
\(47\)	27.6271			0.587810			0.293905	−	0.955835i	\(-0.405045\pi\)
							0.293905	+	0.955835i	\(0.405045\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.3.j.a.17.6	yes	20
3.2	odd	2	inner	78.3.j.a.17.4	✓	20
13.10	even	6	inner	78.3.j.a.23.4	yes	20
39.23	odd	6	inner	78.3.j.a.23.6	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.3.j.a.17.4	✓	20	3.2	odd	2	inner
78.3.j.a.17.6	yes	20	1.1	even	1	trivial
78.3.j.a.23.4	yes	20	13.10	even	6	inner
78.3.j.a.23.6	yes	20	39.23	odd	6	inner