Embedded newform 845.2.bn.a.392.12

• Label: 845.2.bn.a.392.12
• Level: $845$
• Weight: $2$
• Character: 845.392
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $4272$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.bn (of order \(156\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(4272\)
Relative dimension:	\(89\) over \(\Q(\zeta_{156})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{156}]$

Embedding invariants

Embedding label			392.12
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.930503 + 1.96099i) q^{2} +(0.189079 + 0.174430i) q^{3} +(-1.71477 - 2.10021i) q^{4} +(-2.11338 - 0.730492i) q^{5} +(-0.517995 + 0.208475i) q^{6} +(1.06359 + 0.308073i) q^{7} +(1.49913 - 0.369502i) q^{8} +(-0.236075 - 2.92431i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{61}{156}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.930503	+	1.96099i	−0.657965	+	1.38663i	0.250272	+	0.968176i	\(0.419480\pi\)
							−0.908237	+	0.418456i	\(0.862571\pi\)
\(3\)	0.189079	+	0.174430i	0.109165	+	0.100707i	0.730771	−	0.682623i	\(-0.239161\pi\)
							−0.621606	+	0.783330i	\(0.713520\pi\)
\(5\)	−2.11338	−	0.730492i	−0.945133	−	0.326686i
							\(7\)	1.06359	+	0.308073i	0.402000	+	0.116441i	0.473058	−	0.881031i	\(-0.343150\pi\)
−0.0710575	+	0.997472i	\(0.522637\pi\)
\(11\)	2.20671	+	2.59429i	0.665349	+	0.782208i	0.986313	−	0.164883i	\(-0.0527247\pi\)
							−0.320964	+	0.947091i	\(0.604007\pi\)
\(13\)	−3.58490	−	0.385318i	−0.994273	−	0.106868i
							\(17\)	0.0735955	+	0.133615i	0.0178495	+	0.0324063i	0.884693	−	0.466174i	\(-0.154368\pi\)
−0.866844	+	0.498580i	\(0.833855\pi\)
\(19\)	5.69696	−	1.52649i	1.30697	−	0.350202i	0.462889	−	0.886416i	\(-0.346813\pi\)
							0.844082	+	0.536214i	\(0.180146\pi\)
\(23\)	0.410544	−	1.53217i	0.0856043	−	0.319480i	−0.909824	−	0.414995i	\(-0.863783\pi\)
							0.995428	+	0.0955155i	\(0.0304500\pi\)
\(29\)	3.99143	+	1.89396i	0.741190	+	0.351699i	0.761561	−	0.648094i	\(-0.224433\pi\)
							−0.0203709	+	0.999792i	\(0.506485\pi\)
\(31\)	3.70097	+	2.89953i	0.664715	+	0.520771i	0.891412	−	0.453195i	\(-0.149716\pi\)
							−0.226697	+	0.973965i	\(0.572793\pi\)
\(37\)	−4.52357	−	6.01978i	−0.743671	−	0.989646i	−0.999736	−	0.0229864i	\(-0.992683\pi\)
							0.256065	−	0.966660i	\(-0.417574\pi\)
\(41\)	8.64461	+	7.97486i	1.35006	+	1.24546i	0.943305	+	0.331928i	\(0.107699\pi\)
							0.406756	+	0.913537i	\(0.366660\pi\)
\(43\)	0.768301	+	5.41400i	0.117165	+	0.825627i	0.958496	+	0.285107i	\(0.0920290\pi\)
							−0.841331	+	0.540520i	\(0.818227\pi\)
\(47\)	3.90102	−	1.47946i	0.569022	−	0.215802i	−0.0532788	−	0.998580i	\(-0.516967\pi\)
							0.622301	+	0.782778i	\(0.286198\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.12	yes	4272
5.3	odd	4		845.2.bi.a.223.78	yes	4272
169.72	odd	156		845.2.bi.a.72.78	✓	4272
845.748	even	156	inner	845.2.bn.a.748.12	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.78	✓	4272	169.72	odd	156
845.2.bi.a.223.78	yes	4272	5.3	odd	4
845.2.bn.a.392.12	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.12	yes	4272	845.748	even	156	inner