Embedded newform 845.2.bf.a.29.29

• Label: 845.2.bf.a.29.29
• 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			29.29
Character	\(\chi\)	\(=\)	845.29
Dual form			845.2.bf.a.204.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.786350 + 0.755300i) q^{2} +(1.73763 + 2.31236i) q^{3} +(-0.0326627 + 0.810518i) q^{4} +(-1.81815 + 1.30167i) q^{5} +(-3.11291 - 0.505897i) q^{6} +(4.58979 - 0.937011i) q^{7} +(-2.03255 - 2.29427i) q^{8} +(-1.49302 + 5.15450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2160 q - 140 q^{4} - 20 q^{5} - 42 q^{6} - 142 q^{9}+O(q^{10}) \)

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	−0.917202	−	0.398423i	\(-0.869558\pi\)
							0.361168	+	0.932501i	\(0.382378\pi\)
\(3\)	1.73763	+	2.31236i	1.00322	+	1.33504i	0.941523	+	0.336949i	\(0.109395\pi\)
							0.0616972	+	0.998095i	\(0.480349\pi\)
\(5\)	−1.81815	+	1.30167i	−0.813101	+	0.582123i
							\(7\)	4.58979	−	0.937011i	1.73478	−	0.354157i	0.774364	−	0.632741i	\(-0.218070\pi\)
0.960412	+	0.278584i	\(0.0898649\pi\)
\(11\)	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	−0.669356	−	0.742941i	\(-0.733430\pi\)
−0.324586	+	0.945856i	\(0.605225\pi\)
\(19\)	−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	−0.522153	−	0.852852i	\(-0.674871\pi\)
							0.477515	+	0.878624i	\(0.341538\pi\)
\(29\)	−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	0.976649	−	0.214841i	\(-0.0689233\pi\)
							−0.0151443	+	0.999885i	\(0.504821\pi\)
\(41\)	−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	−0.807018	−	0.590527i	\(-0.798920\pi\)
							0.417168	+	0.908829i	\(0.363023\pi\)
\(47\)	4.79252	−	9.13137i	0.699060	−	1.33195i	−0.234003	−	0.972236i	\(-0.575183\pi\)
							0.933063	−	0.359712i	\(-0.117125\pi\)

(See \(a_n\) instead)

Twists

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

    

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