Embedded newform 845.2.bf.a.29.62

• Label: 845.2.bf.a.29.62
• Level: $845$
• Weight: $2$
• Character: 845.29
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $2160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.bf (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(2160\)
Relative dimension:	\(90\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			29.62
Character	\(\chi\)	\(=\)	845.29
Dual form			845.2.bf.a.204.62

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.786350 - 0.755300i) q^{2} +(-1.73763 - 2.31236i) q^{3} +(-0.0326627 + 0.810518i) q^{4} +(1.86180 - 1.23842i) q^{5} +(-3.11291 - 0.505897i) q^{6} +(-4.58979 + 0.937011i) q^{7} +(2.03255 + 2.29427i) q^{8} +(-1.49302 + 5.15450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2160 q - 140 q^{4} - 20 q^{5} - 42 q^{6} - 142 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{10}{39}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.786350	−	0.755300i	0.556034	−	0.534078i	−0.361168	−	0.932501i	\(-0.617622\pi\)
							0.917202	+	0.398423i	\(0.130442\pi\)
\(3\)	−1.73763	−	2.31236i	−1.00322	−	1.33504i	−0.941523	−	0.336949i	\(-0.890605\pi\)
							−0.0616972	−	0.998095i	\(-0.519651\pi\)
\(5\)	1.86180	−	1.23842i	0.832624	−	0.553838i
							\(7\)	−4.58979	+	0.937011i	−1.73478	+	0.354157i	−0.960412	−	0.278584i	\(-0.910135\pi\)
−0.774364	+	0.632741i	\(0.781930\pi\)
\(11\)	1.14845	+	3.96490i	0.346270	+	1.19546i	0.925007	+	0.379949i	\(0.124058\pi\)
							−0.578737	+	0.815514i	\(0.696454\pi\)
\(13\)	1.10445	+	3.43223i	0.306319	+	0.951929i
							\(17\)	4.09813	−	0.836639i	0.993942	−	0.202915i	0.324586	−	0.945856i	\(-0.394775\pi\)
0.669356	+	0.742941i	\(0.266570\pi\)
\(19\)	−2.21627	+	3.83869i	−0.508448	+	0.880657i	0.491505	+	0.870875i	\(0.336447\pi\)
							−0.999952	+	0.00978204i	\(0.996886\pi\)
\(23\)	0.214074	−	0.123596i	0.0446375	−	0.0257715i	−0.477515	−	0.878624i	\(-0.658462\pi\)
							0.522153	+	0.852852i	\(0.325129\pi\)
\(29\)	−2.08665	−	2.17244i	−0.387482	−	0.403411i	0.498681	−	0.866786i	\(-0.333818\pi\)
							−0.886163	+	0.463375i	\(0.846638\pi\)
\(31\)	1.35308	+	3.56777i	0.243020	+	0.640791i	0.999919	−	0.0127409i	\(-0.00405568\pi\)
							−0.756899	+	0.653532i	\(0.773286\pi\)
\(37\)	−5.84861	−	4.77524i	−0.961505	−	0.785044i	0.0151443	−	0.999885i	\(-0.495179\pi\)
							−0.976649	+	0.214841i	\(0.931077\pi\)
\(41\)	−6.76045	+	5.08015i	−1.05581	+	0.793386i	−0.979041	−	0.203663i	\(-0.934715\pi\)
							−0.0767643	+	0.997049i	\(0.524459\pi\)
\(43\)	2.55642	−	2.08725i	0.389850	−	0.318303i	−0.417168	−	0.908829i	\(-0.636977\pi\)
							0.807018	+	0.590527i	\(0.201080\pi\)
\(47\)	−4.79252	+	9.13137i	−0.699060	+	1.33195i	0.234003	+	0.972236i	\(0.424817\pi\)
							−0.933063	+	0.359712i	\(0.882875\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bf.a.29.62	yes	2160
5.4	even	2	inner	845.2.bf.a.29.29	✓	2160
169.35	even	39	inner	845.2.bf.a.204.29	yes	2160
845.204	even	78	inner	845.2.bf.a.204.62	yes	2160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bf.a.29.29	✓	2160	5.4	even	2	inner
845.2.bf.a.29.62	yes	2160	1.1	even	1	trivial
845.2.bf.a.204.29	yes	2160	169.35	even	39	inner
845.2.bf.a.204.62	yes	2160	845.204	even	78	inner