Embedded newform 845.2.bg.a.621.49

• Label: 845.2.bg.a.621.49
• Level: $845$
• Weight: $2$
• Character: 845.621
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $1440$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(1440\)
Relative dimension:	\(60\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			621.49
Character	\(\chi\)	\(=\)	845.621
Dual form			845.2.bg.a.381.49

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.47511 + 1.20439i) q^{2} +(-0.624972 - 0.0504530i) q^{3} +(0.325347 + 1.59365i) q^{4} +(-0.992709 + 0.120537i) q^{5} +(-0.861140 - 0.827136i) q^{6} +(-0.770730 - 1.21881i) q^{7} +(0.330523 - 0.629759i) q^{8} +(-2.57311 - 0.418170i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1440 q - 2 q^{3} - 58 q^{4} + 18 q^{6} + 6 q^{7} + 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{17}{78}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.47511	+	1.20439i	1.04306	+	0.851635i	0.989150	−	0.146911i	\(-0.0469331\pi\)
							0.0539134	+	0.998546i	\(0.482831\pi\)
\(3\)	−0.624972	−	0.0504530i	−0.360828	−	0.0291290i	−0.101276	−	0.994858i	\(-0.532293\pi\)
							−0.259551	+	0.965729i	\(0.583575\pi\)
\(5\)	−0.992709	+	0.120537i	−0.443953	+	0.0539056i
							\(7\)	−0.770730	−	1.21881i	−0.291309	−	0.460668i	0.667545	−	0.744570i	\(-0.267345\pi\)
−0.958853	+	0.283902i	\(0.908371\pi\)
\(11\)	−0.514666	−	3.16687i	−0.155178	−	0.954847i	−0.941496	−	0.337025i	\(-0.890579\pi\)
							0.786318	−	0.617822i	\(-0.211985\pi\)
\(13\)	3.18574	−	1.68850i	0.883567	−	0.468306i
							\(17\)	5.28583	−	3.34256i	1.28200	−	0.810689i	0.292649	−	0.956220i	\(-0.405463\pi\)
0.989354	+	0.145531i	\(0.0464889\pi\)
\(19\)	−3.88022	−	2.24024i	−0.890183	−	0.513947i	−0.0161806	−	0.999869i	\(-0.505151\pi\)
							−0.874002	+	0.485922i	\(0.838484\pi\)
\(23\)	1.39727	+	2.42015i	0.291351	+	0.504635i	0.974130	−	0.225990i	\(-0.0725618\pi\)
							−0.682778	+	0.730626i	\(0.739228\pi\)
\(29\)	−1.14147	+	1.39805i	−0.211966	+	0.259611i	−0.869634	−	0.493696i	\(-0.835645\pi\)
							0.657668	+	0.753308i	\(0.271543\pi\)
\(31\)	−1.18473	+	4.80666i	−0.212785	+	0.863301i	0.763324	+	0.646016i	\(0.223566\pi\)
							−0.976108	+	0.217285i	\(0.930280\pi\)
\(37\)	−8.32882	+	2.41247i	−1.36925	+	0.396608i	−0.879767	−	0.475405i	\(-0.842302\pi\)
							−0.489483	+	0.872013i	\(0.662814\pi\)
\(41\)	0.926263	−	11.4738i	0.144658	−	1.79191i	−0.364465	−	0.931217i	\(-0.618748\pi\)
							0.509123	−	0.860694i	\(-0.329970\pi\)
\(43\)	2.71475	−	9.37242i	0.413996	−	1.42928i	−0.434784	−	0.900535i	\(-0.643175\pi\)
							0.848780	−	0.528746i	\(-0.177338\pi\)
\(47\)	−4.93207	+	5.56716i	−0.719417	+	0.812054i	−0.988525	−	0.151056i	\(-0.951733\pi\)
							0.269108	+	0.963110i	\(0.413271\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bg.a.621.49	yes	1440
169.43	even	78	inner	845.2.bg.a.381.49	✓	1440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bg.a.381.49	✓	1440	169.43	even	78	inner
845.2.bg.a.621.49	yes	1440	1.1	even	1	trivial