Embedded newform 78.3.j.a.17.5

• Label: 78.3.j.a.17.5
• 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			17.5
Root			\(-2.56516 + 1.55562i\) of defining polynomial
Character	\(\chi\)	\(=\)	78.17
Dual form			78.3.j.a.23.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{2} +(2.56516 - 1.55562i) q^{3} +(-1.00000 - 1.73205i) q^{4} +1.88727 q^{5} +(0.0914001 + 4.24166i) q^{6} +(4.14217 - 2.39148i) q^{7} +2.82843 q^{8} +(4.16008 - 7.98084i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{4} + 18 q^{7} - 4 q^{9}+O(q^{10}) \)

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
							\(3\)	2.56516	−	1.55562i	0.855053	−	0.518541i
\(5\)	1.88727			0.377453										0.188727	−	0.982030i	\(-0.439564\pi\)
							0.188727	+	0.982030i	\(0.439564\pi\)
\(7\)	4.14217	−	2.39148i	0.591738	−	0.341640i	−0.174046	−	0.984737i	\(-0.555684\pi\)
							0.765784	+	0.643097i	\(0.222351\pi\)
\(11\)	−2.50053	+	4.33104i	−0.227321	+	0.393731i	−0.957013	−	0.290044i	\(-0.906330\pi\)
							0.729692	+	0.683776i	\(0.239663\pi\)
\(13\)	7.59115	+	10.5534i	0.583934	+	0.811801i
							\(17\)	−15.3219	+	8.84609i	−0.901287	+	0.520358i	−0.877618	−	0.479362i	\(-0.840868\pi\)
−0.0236695	+	0.999720i	\(0.507535\pi\)
\(19\)	8.89286	−	5.13430i	0.468045	−	0.270226i	−0.247376	−	0.968920i	\(-0.579568\pi\)
							0.715421	+	0.698694i	\(0.246235\pi\)
\(23\)	−30.3221	−	17.5065i	−1.31835	−	0.761152i	−0.334890	−	0.942257i	\(-0.608699\pi\)
							−0.983464	+	0.181106i	\(0.942032\pi\)
\(29\)	−5.98051	−	3.45285i	−0.206224	−	0.119064i	0.393331	−	0.919397i	\(-0.371323\pi\)
							−0.599555	+	0.800333i	\(0.704656\pi\)
\(31\)			30.9538i			0.998509i	0.866455	+	0.499255i	\(0.166393\pi\)
							−0.866455	+	0.499255i	\(0.833607\pi\)
\(37\)	−35.4649	−	20.4757i	−0.958510	−	0.553396i	−0.0627962	−	0.998026i	\(-0.520002\pi\)
							−0.895714	+	0.444630i	\(0.853335\pi\)
\(41\)	−6.44861	+	11.1693i	−0.157283	+	0.272423i	−0.933888	−	0.357565i	\(-0.883607\pi\)
							0.776605	+	0.629988i	\(0.216940\pi\)
\(43\)	18.5527	+	32.1343i	0.431459	+	0.747308i	0.996999	−	0.0774121i	\(-0.0246657\pi\)
							−0.565540	+	0.824721i	\(0.691332\pi\)
\(47\)	79.5796			1.69318			0.846591	−	0.532244i	\(-0.178651\pi\)
							0.846591	+	0.532244i	\(0.178651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.3.j.a.17.5	✓	20
3.2	odd	2	inner	78.3.j.a.17.8	yes	20
13.10	even	6	inner	78.3.j.a.23.8	yes	20
39.23	odd	6	inner	78.3.j.a.23.5	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.3.j.a.17.5	✓	20	1.1	even	1	trivial
78.3.j.a.17.8	yes	20	3.2	odd	2	inner
78.3.j.a.23.5	yes	20	39.23	odd	6	inner
78.3.j.a.23.8	yes	20	13.10	even	6	inner