LMFDB - Embedded newform 80.8.c.a.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [80,8,Mod(49,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.49"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	80.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,150] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$24.9908020387$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-29}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 29 $ x^2 + 29
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-5.38516i$ of defining polynomial
Character	$\chi$	$=$	80.49
Dual form			80.8.c.a.49.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-32.3110i q^{3} +(75.0000 - 269.258i) q^{5} +420.043i q^{7} +1143.00 q^{9} +6828.00 q^{11} -10145.7i q^{13} +(-8700.00 - 2423.32i) q^{15} +15681.6i q^{17} -6860.00 q^{19} +13572.0 q^{21} -29219.9i q^{23} +(-66875.0 - 40388.7i) q^{25} -107596. i q^{27} +25590.0 q^{29} -82112.0 q^{31} -220619. i q^{33} +(113100. + 31503.2i) q^{35} -223527. i q^{37} -327816. q^{39} -533118. q^{41} -708935. i q^{43} +(85725.0 - 307762. i) q^{45} -5826.75i q^{47} +647107. q^{49} +506688. q^{51} -589374. i q^{53} +(512100. - 1.83850e6i) q^{55} +221653. i q^{57} -1.43898e6 q^{59} +1.38102e6 q^{61} +480109. i q^{63} +(-2.73180e6 - 760924. i) q^{65} +2.71487e6i q^{67} -944124. q^{69} +481608. q^{71} +1.48618e6i q^{73} +(-1.30500e6 + 2.16080e6i) q^{75} +2.86805e6i q^{77} +1.05976e6 q^{79} -976779. q^{81} +2.60380e6i q^{83} +(4.22240e6 + 1.17612e6i) q^{85} -826838. i q^{87} +5.64417e6 q^{89} +4.26161e6 q^{91} +2.65312e6i q^{93} +(-514500. + 1.84711e6i) q^{95} +1.20091e7i q^{97} +7.80440e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 150 q^{5} + 2286 q^{9} + 13656 q^{11} - 17400 q^{15} - 13720 q^{19} + 27144 q^{21} - 133750 q^{25} + 51180 q^{29} - 164224 q^{31} + 226200 q^{35} - 655632 q^{39} - 1066236 q^{41} + 171450 q^{45} + 1294214 q^{49}+ \cdots + 15608808 q^{99}+O(q^{100}) $ 2 * q + 150 * q^5 + 2286 * q^9 + 13656 * q^11 - 17400 * q^15 - 13720 * q^19 + 27144 * q^21 - 133750 * q^25 + 51180 * q^29 - 164224 * q^31 + 226200 * q^35 - 655632 * q^39 - 1066236 * q^41 + 171450 * q^45 + 1294214 * q^49 + 1013376 * q^51 + 1024200 * q^55 - 2877960 * q^59 + 2762044 * q^61 - 5463600 * q^65 - 1888248 * q^69 + 963216 * q^71 - 2610000 * q^75 + 2119520 * q^79 - 1953558 * q^81 + 8444800 * q^85 + 11288340 * q^89 + 8523216 * q^91 - 1029000 * q^95 + 15608808 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	32.3110i		−	0.690917i	−0.938434	−	0.345458i	$-0.887723\pi$
$3$		−	32.3110i		−	0.690917i	0.938434	−	0.345458i	$-0.112277\pi$
$4$		0			0
$5$	75.0000	−	269.258i	0.268328	−	0.963328i
$5$	75.0000	−	269.258i	0.268328	−	0.963328i
$6$		0			0
$7$			420.043i			0.462861i	0.972851	+	0.231430i	$0.0743406\pi$
$7$			420.043i			0.462861i	−0.972851	+	0.231430i	$0.925659\pi$
$8$		0			0
$9$	1143.00			0.522634
$10$		0			0
$11$	6828.00			1.54675			0.773373	−	0.633951i	$-0.218568\pi$
$11$	6828.00			1.54675			0.773373	+	0.633951i	$0.218568\pi$
$12$		0			0
$13$		−	10145.7i		−	1.28079i	−0.768045	−	0.640395i	$-0.778771\pi$
$13$		−	10145.7i		−	1.28079i	0.768045	−	0.640395i	$-0.221229\pi$
$14$		0			0
$15$	−8700.00	−	2423.32i	−0.665579	−	0.185392i
$16$		0			0
$17$			15681.6i			0.774139i	0.922050	+	0.387070i	$0.126513\pi$
$17$			15681.6i			0.774139i	−0.922050	+	0.387070i	$0.873487\pi$
$18$		0			0
$19$	−6860.00			−0.229449			−0.114725	−	0.993397i	$-0.536599\pi$
$19$	−6860.00			−0.229449			−0.114725	+	0.993397i	$0.536599\pi$
$20$		0			0
$21$	13572.0			0.319798
$22$		0			0
$23$		−	29219.9i		−	0.500762i	−0.968147	−	0.250381i	$-0.919444\pi$
$23$		−	29219.9i		−	0.500762i	0.968147	−	0.250381i	$-0.0805559\pi$
$24$		0			0
$25$	−66875.0	−	40388.7i	−0.856000	−	0.516976i
$26$		0			0
$27$		−	107596.i		−	1.05201i
$28$		0			0
$29$	25590.0			0.194840			0.0974198	−	0.995243i	$-0.468941\pi$
$29$	25590.0			0.194840			0.0974198	+	0.995243i	$0.468941\pi$
$30$		0			0
$31$	−82112.0			−0.495040			−0.247520	−	0.968883i	$-0.579616\pi$
$31$	−82112.0			−0.495040			−0.247520	+	0.968883i	$0.579616\pi$
$32$		0			0
$33$		−	220619.i		−	1.06867i
$34$		0			0
$35$	113100.	+	31503.2i	0.445887	+	0.124199i
$36$		0			0
$37$		−	223527.i		−	0.725479i	−0.931891	−	0.362739i	$-0.881842\pi$
$37$		−	223527.i		−	0.725479i	0.931891	−	0.362739i	$-0.118158\pi$
$38$		0			0
$39$	−327816.			−0.884920
$40$		0			0
$41$	−533118.			−1.20804			−0.604018	−	0.796971i	$-0.706434\pi$
$41$	−533118.			−1.20804			−0.604018	+	0.796971i	$0.706434\pi$
$42$		0			0
$43$		−	708935.i		−	1.35978i	−0.733316	−	0.679888i	$-0.762029\pi$
$43$		−	708935.i		−	1.35978i	0.733316	−	0.679888i	$-0.237971\pi$
$44$		0			0
$45$	85725.0	−	307762.i	0.140237	−	0.503467i
$46$		0			0
$47$		−	5826.75i		−	0.00818623i	−0.999992	−	0.00409311i	$-0.998697\pi$
$47$		−	5826.75i		−	0.00818623i	0.999992	−	0.00409311i	$-0.00130288\pi$
$48$		0			0
$49$	647107.			0.785760
$50$		0			0
$51$	506688.			0.534866
$52$		0			0
$53$		−	589374.i		−	0.543783i	−0.962328	−	0.271891i	$-0.912351\pi$
$53$		−	589374.i		−	0.543783i	0.962328	−	0.271891i	$-0.0876491\pi$
$54$		0			0
$55$	512100.	−	1.83850e6i	0.415036	−	1.49002i
$56$		0			0
$57$			221653.i			0.158530i
$58$		0			0
$59$	−1.43898e6			−0.912164			−0.456082	−	0.889938i	$-0.650748\pi$
$59$	−1.43898e6			−0.912164			−0.456082	+	0.889938i	$0.650748\pi$
$60$		0			0
$61$	1.38102e6			0.779016			0.389508	−	0.921023i	$-0.372645\pi$
$61$	1.38102e6			0.779016			0.389508	+	0.921023i	$0.372645\pi$
$62$		0			0
$63$			480109.i			0.241907i
$64$		0			0
$65$	−2.73180e6	−	760924.i	−1.23382	−	0.343672i
$66$		0			0
$67$			2.71487e6i			1.10277i	0.834250	+	0.551387i	$0.185901\pi$
$67$			2.71487e6i			1.10277i	−0.834250	+	0.551387i	$0.814099\pi$
$68$		0			0
$69$	−944124.			−0.345985
$70$		0			0
$71$	481608.			0.159694			0.0798472	−	0.996807i	$-0.474557\pi$
$71$	481608.			0.159694			0.0798472	+	0.996807i	$0.474557\pi$
$72$		0			0
$73$			1.48618e6i			0.447137i	0.974688	+	0.223568i	$0.0717706\pi$
$73$			1.48618e6i			0.447137i	−0.974688	+	0.223568i	$0.928229\pi$
$74$		0			0
$75$	−1.30500e6	+	2.16080e6i	−0.357187	+	0.591425i
$76$		0			0
$77$			2.86805e6i			0.715928i
$78$		0			0
$79$	1.05976e6			0.241831			0.120916	−	0.992663i	$-0.461417\pi$
$79$	1.05976e6			0.241831			0.120916	+	0.992663i	$0.461417\pi$
$80$		0			0
$81$	−976779.			−0.204220
$82$		0			0
$83$			2.60380e6i			0.499844i	0.968266	+	0.249922i	$0.0804050\pi$
$83$			2.60380e6i			0.499844i	−0.968266	+	0.249922i	$0.919595\pi$
$84$		0			0
$85$	4.22240e6	+	1.17612e6i	0.745750	+	0.207723i
$86$		0			0
$87$		−	826838.i		−	0.134618i
$88$		0			0
$89$	5.64417e6			0.848663			0.424331	−	0.905507i	$-0.360509\pi$
$89$	5.64417e6			0.848663			0.424331	+	0.905507i	$0.360509\pi$
$90$		0			0
$91$	4.26161e6			0.592828
$92$		0			0
$93$			2.65312e6i			0.342032i
$94$		0			0
$95$	−514500.	+	1.84711e6i	−0.0615677	+	0.221035i
$96$		0			0
$97$			1.20091e7i			1.33601i	0.744158	+	0.668004i	$0.232851\pi$
$97$			1.20091e7i			1.33601i	−0.744158	+	0.668004i	$0.767149\pi$
$98$		0			0
$99$	7.80440e6			0.808382
$100$		0			0
$101$	5.14270e6			0.496668			0.248334	−	0.968674i	$-0.420117\pi$
$101$	5.14270e6			0.496668			0.248334	+	0.968674i	$0.420117\pi$
$102$		0			0
$103$		−	3.48477e6i		−	0.314227i	−0.987581	−	0.157114i	$-0.949781\pi$
$103$		−	3.48477e6i		−	0.314227i	0.987581	−	0.157114i	$-0.0502189\pi$
$104$		0			0
$105$	1.01790e6	−	3.65437e6i	0.0858109	−	0.308071i
$106$		0			0
$107$			1.48640e7i			1.17299i	0.809954	+	0.586493i	$0.199492\pi$
$107$			1.48640e7i			1.17299i	−0.809954	+	0.586493i	$0.800508\pi$
$108$		0			0
$109$	−2.01124e7			−1.48755			−0.743773	−	0.668432i	$-0.766966\pi$
$109$	−2.01124e7			−1.48755			−0.743773	+	0.668432i	$0.766966\pi$
$110$		0			0
$111$	−7.22239e6			−0.501246
$112$		0			0
$113$		−	5.62633e6i		−	0.366818i	−0.983037	−	0.183409i	$-0.941287\pi$
$113$		−	5.62633e6i		−	0.366818i	0.983037	−	0.183409i	$-0.0587133\pi$
$114$		0			0
$115$	−7.86770e6	−	2.19149e6i	−0.482398	−	0.134369i
$116$		0			0
$117$		−	1.15965e7i		−	0.669384i
$118$		0			0
$119$	−6.58694e6			−0.358319
$120$		0			0
$121$	2.71344e7			1.39242
$122$		0			0
$123$			1.72256e7i			0.834653i
$124$		0			0
$125$	−1.58906e7	+	1.49775e7i	−0.727706	+	0.685889i
$126$		0			0
$127$		−	2.85360e7i		−	1.23618i	−0.786109	−	0.618088i	$-0.787908\pi$
$127$		−	2.85360e7i		−	1.23618i	0.786109	−	0.618088i	$-0.212092\pi$
$128$		0			0
$129$	−2.29064e7			−0.939492
$130$		0			0
$131$	3.33132e7			1.29469			0.647346	−	0.762196i	$-0.275879\pi$
$131$	3.33132e7			1.29469			0.647346	+	0.762196i	$0.275879\pi$
$132$		0			0
$133$		−	2.88149e6i		−	0.106203i
$134$		0			0
$135$	−2.89710e7	−	8.06967e6i	−1.01343	−	0.282285i
$136$		0			0
$137$		−	4.28099e7i		−	1.42240i	−0.702988	−	0.711202i	$-0.748151\pi$
$137$		−	4.28099e7i		−	1.42240i	0.702988	−	0.711202i	$-0.251849\pi$
$138$		0			0
$139$	−1.13808e7			−0.359436			−0.179718	−	0.983718i	$-0.557519\pi$
$139$	−1.13808e7			−0.359436			−0.179718	+	0.983718i	$0.557519\pi$
$140$		0			0
$141$	−188268.			−0.00565600
$142$		0			0
$143$		−	6.92745e7i		−	1.98106i
$144$		0			0
$145$	1.91925e6	−	6.89032e6i	0.0522810	−	0.187694i
$146$		0			0
$147$		−	2.09087e7i		−	0.542895i
$148$		0			0
$149$	4.00070e7			0.990794			0.495397	−	0.868667i	$-0.335023\pi$
$149$	4.00070e7			0.990794			0.495397	+	0.868667i	$0.335023\pi$
$150$		0			0
$151$	2.86594e7			0.677405			0.338703	−	0.940893i	$-0.390012\pi$
$151$	2.86594e7			0.677405			0.338703	+	0.940893i	$0.390012\pi$
$152$		0			0
$153$			1.79241e7i			0.404591i
$154$		0			0
$155$	−6.15840e6	+	2.21093e7i	−0.132833	+	0.476886i
$156$		0			0
$157$		−	3.01958e7i		−	0.622728i	−0.950291	−	0.311364i	$-0.899214\pi$
$157$		−	3.01958e7i		−	0.622728i	0.950291	−	0.311364i	$-0.100786\pi$
$158$		0			0
$159$	−1.90433e7			−0.375709
$160$		0			0
$161$	1.22736e7			0.231783
$162$		0			0
$163$			9.35416e7i			1.69180i	0.533345	+	0.845898i	$0.320935\pi$
$163$			9.35416e7i			1.69180i	−0.533345	+	0.845898i	$0.679065\pi$
$164$		0			0
$165$	−5.94036e7	−	1.65465e7i	−1.02948	−	0.286755i
$166$		0			0
$167$		−	5.73507e7i		−	0.952865i	−0.879211	−	0.476432i	$-0.841930\pi$
$167$		−	5.73507e7i		−	0.952865i	0.879211	−	0.476432i	$-0.158070\pi$
$168$		0			0
$169$	−4.01857e7			−0.640425
$170$		0			0
$171$	−7.84098e6			−0.119918
$172$		0			0
$173$		−	4.87192e7i		−	0.715383i	−0.933840	−	0.357691i	$-0.883564\pi$
$173$		−	4.87192e7i		−	0.715383i	0.933840	−	0.357691i	$-0.116436\pi$
$174$		0			0
$175$	1.69650e7	−	2.80904e7i	0.239288	−	0.396209i
$176$		0			0
$177$			4.64949e7i			0.630229i
$178$		0			0
$179$	−1.93505e7			−0.252178			−0.126089	−	0.992019i	$-0.540243\pi$
$179$	−1.93505e7			−0.252178			−0.126089	+	0.992019i	$0.540243\pi$
$180$		0			0
$181$	7.82617e7			0.981011			0.490506	−	0.871438i	$-0.336812\pi$
$181$	7.82617e7			0.981011			0.490506	+	0.871438i	$0.336812\pi$
$182$		0			0
$183$		−	4.46222e7i		−	0.538235i
$184$		0			0
$185$	−6.01866e7	−	1.67646e7i	−0.698874	−	0.194666i
$186$		0			0
$187$			1.07074e8i			1.19740i
$188$		0			0
$189$	4.51948e7			0.486936
$190$		0			0
$191$	1.19454e8			1.24046			0.620229	−	0.784420i	$-0.287040\pi$
$191$	1.19454e8			1.24046			0.620229	+	0.784420i	$0.287040\pi$
$192$		0			0
$193$			5.98469e7i			0.599227i	0.954061	+	0.299613i	$0.0968577\pi$
$193$			5.98469e7i			0.599227i	−0.954061	+	0.299613i	$0.903142\pi$
$194$		0			0
$195$	−2.45862e7	+	8.82672e7i	−0.237449	+	0.852468i
$196$		0			0
$197$			1.22964e8i			1.14590i	0.819589	+	0.572952i	$0.194202\pi$
$197$			1.22964e8i			1.14590i	−0.819589	+	0.572952i	$0.805798\pi$
$198$		0			0
$199$	−1.69053e8			−1.52067			−0.760337	−	0.649529i	$-0.774966\pi$
$199$	−1.69053e8			−1.52067			−0.760337	+	0.649529i	$0.774966\pi$
$200$		0			0
$201$	8.77200e7			0.761925
$202$		0			0
$203$			1.07489e7i			0.0901836i
$204$		0			0
$205$	−3.99838e7	+	1.43546e8i	−0.324150	+	1.16373i
$206$		0			0
$207$		−	3.33984e7i		−	0.261715i
$208$		0			0
$209$	−4.68401e7			−0.354900
$210$		0			0
$211$	2.67605e8			1.96113			0.980565	−	0.196195i	$-0.0628586\pi$
$211$	2.67605e8			1.96113			0.980565	+	0.196195i	$0.0628586\pi$
$212$		0			0
$213$		−	1.55612e7i		−	0.110336i
$214$		0			0
$215$	−1.90887e8	−	5.31702e7i	−1.30991	−	0.364866i
$216$		0			0
$217$		−	3.44906e7i		−	0.229135i
$218$		0			0
$219$	4.80198e7			0.308934
$220$		0			0
$221$	1.59100e8			0.991510
$222$		0			0
$223$			1.49333e8i			0.901753i	0.892586	+	0.450877i	$0.148889\pi$
$223$			1.49333e8i			0.901753i	−0.892586	+	0.450877i	$0.851111\pi$
$224$		0			0
$225$	−7.64381e7	−	4.61643e7i	−0.447374	−	0.270189i
$226$		0			0
$227$			2.34185e8i			1.32883i	0.747365	+	0.664414i	$0.231319\pi$
$227$			2.34185e8i			1.32883i	−0.747365	+	0.664414i	$0.768681\pi$
$228$		0			0
$229$	−1.31882e8			−0.725706			−0.362853	−	0.931846i	$-0.618197\pi$
$229$	−1.31882e8			−0.725706			−0.362853	+	0.931846i	$0.618197\pi$
$230$		0			0
$231$	9.26696e7			0.494647
$232$		0			0
$233$			1.83419e8i			0.949948i	0.880000	+	0.474974i	$0.157542\pi$
$233$			1.83419e8i			0.949948i	−0.880000	+	0.474974i	$0.842458\pi$
$234$		0			0
$235$	−1.56890e6	−	437006.i	−0.00788602	−	0.00219660i
$236$		0			0
$237$		−	3.42419e7i		−	0.167085i
$238$		0			0
$239$	1.05117e8			0.498058			0.249029	−	0.968496i	$-0.419889\pi$
$239$	1.05117e8			0.498058			0.249029	+	0.968496i	$0.419889\pi$
$240$		0			0
$241$	1.94216e8			0.893770			0.446885	−	0.894591i	$-0.352533\pi$
$241$	1.94216e8			0.893770			0.446885	+	0.894591i	$0.352533\pi$
$242$		0			0
$243$		−	2.03751e8i		−	0.910914i
$244$		0			0
$245$	4.85330e7	−	1.74239e8i	0.210841	−	0.756944i
$246$		0			0
$247$			6.95992e7i			0.293876i
$248$		0			0
$249$	8.41314e7			0.345351
$250$		0			0
$251$	−2.02689e8			−0.809045			−0.404523	−	0.914528i	$-0.632562\pi$
$251$	−2.02689e8			−0.809045			−0.404523	+	0.914528i	$0.632562\pi$
$252$		0			0
$253$		−	1.99514e8i		−	0.774552i
$254$		0			0
$255$	3.80016e7	−	1.36430e8i	0.143520	−	0.515251i
$256$		0			0
$257$		−	1.34593e7i		−	0.0494603i	−0.999694	−	0.0247301i	$-0.992127\pi$
$257$		−	1.34593e7i		−	0.0494603i	0.999694	−	0.0247301i	$-0.00787265\pi$
$258$		0			0
$259$	9.38911e7			0.335796
$260$		0			0
$261$	2.92494e7			0.101830
$262$		0			0
$263$		−	1.42205e8i		−	0.482026i	−0.970522	−	0.241013i	$-0.922520\pi$
$263$		−	1.42205e8i		−	0.482026i	0.970522	−	0.241013i	$-0.0774797\pi$
$264$		0			0
$265$	−1.58694e8	−	4.42030e7i	−0.523841	−	0.145912i
$266$		0			0
$267$		−	1.82369e8i		−	0.586355i
$268$		0			0
$269$	−5.07548e8			−1.58981			−0.794903	−	0.606737i	$-0.792478\pi$
$269$	−5.07548e8			−1.58981			−0.794903	+	0.606737i	$0.792478\pi$
$270$		0			0
$271$	−1.12836e8			−0.344393			−0.172197	−	0.985063i	$-0.555086\pi$
$271$	−1.12836e8			−0.344393			−0.172197	+	0.985063i	$0.555086\pi$
$272$		0			0
$273$		−	1.37697e8i		−	0.409595i
$274$		0			0
$275$	−4.56622e8	−	2.75774e8i	−1.32401	−	0.799630i
$276$		0			0
$277$			5.10728e8i			1.44381i	0.691991	+	0.721906i	$0.256734\pi$
$277$			5.10728e8i			1.44381i	−0.691991	+	0.721906i	$0.743266\pi$
$278$		0			0
$279$	−9.38540e7			−0.258725
$280$		0			0
$281$	−1.70459e8			−0.458297			−0.229148	−	0.973391i	$-0.573594\pi$
$281$	−1.70459e8			−0.458297			−0.229148	+	0.973391i	$0.573594\pi$
$282$		0			0
$283$			1.62144e8i			0.425253i	0.977134	+	0.212626i	$0.0682017\pi$
$283$			1.62144e8i			0.425253i	−0.977134	+	0.212626i	$0.931798\pi$
$284$		0			0
$285$	5.96820e7	+	1.66240e7i	0.152717	+	0.0425382i
$286$		0			0
$287$		−	2.23932e8i		−	0.559153i
$288$		0			0
$289$	1.64426e8			0.400708
$290$		0			0
$291$	3.88026e8			0.923071
$292$		0			0
$293$		−	3.85845e8i		−	0.896141i	−0.893998	−	0.448070i	$-0.852111\pi$
$293$		−	3.85845e8i		−	0.896141i	0.893998	−	0.448070i	$-0.147889\pi$
$294$		0			0
$295$	−1.07924e8	+	3.87457e8i	−0.244759	+	0.878712i
$296$		0			0
$297$		−	7.34663e8i		−	1.62720i
$298$		0			0
$299$	−2.96455e8			−0.641371
$300$		0			0
$301$	2.97783e8			0.629387
$302$		0			0
$303$		−	1.66166e8i		−	0.343157i
$304$		0			0
$305$	1.03577e8	−	3.71852e8i	0.209032	−	0.750447i
$306$		0			0
$307$		−	6.37817e8i		−	1.25809i	−0.777369	−	0.629045i	$-0.783446\pi$
$307$		−	6.37817e8i		−	1.25809i	0.777369	−	0.629045i	$-0.216554\pi$
$308$		0			0
$309$	−1.12596e8			−0.217105
$310$		0			0
$311$	7.27817e8			1.37202			0.686010	−	0.727592i	$-0.259360\pi$
$311$	7.27817e8			1.37202			0.686010	+	0.727592i	$0.259360\pi$
$312$		0			0
$313$			4.46869e8i			0.823711i	0.911249	+	0.411855i	$0.135119\pi$
$313$			4.46869e8i			0.823711i	−0.911249	+	0.411855i	$0.864881\pi$
$314$		0			0
$315$	1.29273e8	+	3.60082e7i	0.233035	+	0.0649104i
$316$		0			0
$317$		−	5.91083e8i		−	1.04218i	−0.853503	−	0.521088i	$-0.825526\pi$
$317$		−	5.91083e8i		−	1.04218i	0.853503	−	0.521088i	$-0.174474\pi$
$318$		0			0
$319$	1.74729e8			0.301367
$320$		0			0
$321$	4.80271e8			0.810436
$322$		0			0
$323$		−	1.07576e8i		−	0.177626i
$324$		0			0
$325$	−4.09770e8	+	6.78490e8i	−0.662138	+	1.09636i
$326$		0			0
$327$			6.49851e8i			1.02777i
$328$		0			0
$329$	2.44748e6			0.00378908
$330$		0			0
$331$	−5.84868e8			−0.886462			−0.443231	−	0.896407i	$-0.646168\pi$
$331$	−5.84868e8			−0.886462			−0.443231	+	0.896407i	$0.646168\pi$
$332$		0			0
$333$		−	2.55492e8i		−	0.379160i
$334$		0			0
$335$	7.31000e8	+	2.03615e8i	1.06233	+	0.295905i
$336$		0			0
$337$			7.39373e8i			1.05235i	0.850377	+	0.526174i	$0.176374\pi$
$337$			7.39373e8i			1.05235i	−0.850377	+	0.526174i	$0.823626\pi$
$338$		0			0
$339$	−1.81792e8			−0.253441
$340$		0			0
$341$	−5.60661e8			−0.765702
$342$		0			0
$343$			6.17736e8i			0.826558i
$344$		0			0
$345$	−7.08093e7	+	2.54213e8i	−0.0928375	+	0.333297i
$346$		0			0
$347$		−	3.70870e8i		−	0.476506i	−0.971203	−	0.238253i	$-0.923425\pi$
$347$		−	3.70870e8i		−	0.476506i	0.971203	−	0.238253i	$-0.0765747\pi$
$348$		0			0
$349$	1.13274e9			1.42640			0.713199	−	0.700962i	$-0.247246\pi$
$349$	1.13274e9			1.42640			0.713199	+	0.700962i	$0.247246\pi$
$350$		0			0
$351$	−1.09163e9			−1.34741
$352$		0			0
$353$			8.32858e8i			1.00777i	0.863772	+	0.503883i	$0.168096\pi$
$353$			8.32858e8i			1.00777i	−0.863772	+	0.503883i	$0.831904\pi$
$354$		0			0
$355$	3.61206e7	−	1.29677e8i	0.0428505	−	0.153838i
$356$		0			0
$357$			2.12831e8i			0.247569i
$358$		0			0
$359$	−6.75318e8			−0.770332			−0.385166	−	0.922847i	$-0.625856\pi$
$359$	−6.75318e8			−0.770332			−0.385166	+	0.922847i	$0.625856\pi$
$360$		0			0
$361$	−8.46812e8			−0.947353
$362$		0			0
$363$		−	8.76740e8i		−	0.962050i
$364$		0			0
$365$	4.00165e8	+	1.11463e8i	0.430739	+	0.119979i
$366$		0			0
$367$			1.80237e9i			1.90333i	0.307141	+	0.951664i	$0.400628\pi$
$367$			1.80237e9i			1.90333i	−0.307141	+	0.951664i	$0.599372\pi$
$368$		0			0
$369$	−6.09354e8			−0.631360
$370$		0			0
$371$	2.47562e8			0.251696
$372$		0			0
$373$			9.29928e8i			0.927830i	0.885880	+	0.463915i	$0.153556\pi$
$373$			9.29928e8i			0.927830i	−0.885880	+	0.463915i	$0.846444\pi$
$374$		0			0
$375$	4.83938e8	+	5.13442e8i	0.473893	+	0.502784i
$376$		0			0
$377$		−	2.59627e8i		−	0.249549i
$378$		0			0
$379$	1.43545e9			1.35441			0.677206	−	0.735794i	$-0.263191\pi$
$379$	1.43545e9			1.35441			0.677206	+	0.735794i	$0.263191\pi$
$380$		0			0
$381$	−9.22026e8			−0.854094
$382$		0			0
$383$			1.57707e9i			1.43435i	0.696894	+	0.717174i	$0.254565\pi$
$383$			1.57707e9i			1.43435i	−0.696894	+	0.717174i	$0.745435\pi$
$384$		0			0
$385$	7.72247e8	+	2.15104e8i	0.689674	+	0.192104i
$386$		0			0
$387$		−	8.10313e8i		−	0.710664i
$388$		0			0
$389$	2.24425e9			1.93307			0.966537	−	0.256528i	$-0.0825785\pi$
$389$	2.24425e9			1.93307			0.966537	+	0.256528i	$0.0825785\pi$
$390$		0			0
$391$	4.58215e8			0.387660
$392$		0			0
$393$		−	1.07638e9i		−	0.894525i
$394$		0			0
$395$	7.94820e7	−	2.85349e8i	0.0648902	−	0.232963i
$396$		0			0
$397$		−	2.26641e8i		−	0.181791i	−0.995860	−	0.0908956i	$-0.971027\pi$
$397$		−	2.26641e8i		−	0.181791i	0.995860	−	0.0908956i	$-0.0289729\pi$
$398$		0			0
$399$	−9.31039e7			−0.0733775
$400$		0			0
$401$	−9.11721e8			−0.706085			−0.353042	−	0.935607i	$-0.614853\pi$
$401$	−9.11721e8			−0.706085			−0.353042	+	0.935607i	$0.614853\pi$
$402$		0			0
$403$			8.33080e8i			0.634043i
$404$		0			0
$405$	−7.32584e7	+	2.63006e8i	−0.0547980	+	0.196731i
$406$		0			0
$407$		−	1.52625e9i		−	1.12213i
$408$		0			0
$409$	−2.55215e8			−0.184449			−0.0922243	−	0.995738i	$-0.529398\pi$
$409$	−2.55215e8			−0.184449			−0.0922243	+	0.995738i	$0.529398\pi$
$410$		0			0
$411$	−1.38323e9			−0.982763
$412$		0			0
$413$		−	6.04433e8i		−	0.422205i
$414$		0			0
$415$	7.01095e8	+	1.95285e8i	0.481514	+	0.134122i
$416$		0			0
$417$			3.67726e8i			0.248341i
$418$		0			0
$419$	−2.96316e8			−0.196791			−0.0983957	−	0.995147i	$-0.531371\pi$
$419$	−2.96316e8			−0.196791			−0.0983957	+	0.995147i	$0.531371\pi$
$420$		0			0
$421$	1.06676e9			0.696754			0.348377	−	0.937354i	$-0.386733\pi$
$421$	1.06676e9			0.696754			0.348377	+	0.937354i	$0.386733\pi$
$422$		0			0
$423$		−	6.65997e6i		−	0.00427840i
$424$		0			0
$425$	6.33360e8	−	1.04871e9i	0.400211	−	0.662663i
$426$		0			0
$427$			5.80088e8i			0.360576i
$428$		0			0
$429$	−2.23833e9			−1.36875
$430$		0			0
$431$	−9.53169e7			−0.0573455			−0.0286728	−	0.999589i	$-0.509128\pi$
$431$	−9.53169e7			−0.0573455			−0.0286728	+	0.999589i	$0.509128\pi$
$432$		0			0
$433$			1.89973e9i			1.12456i	0.826946	+	0.562281i	$0.190076\pi$
$433$			1.89973e9i			1.12456i	−0.826946	+	0.562281i	$0.809924\pi$
$434$		0			0
$435$	−2.22633e8	−	6.20129e7i	−0.129681	−	0.0361218i
$436$		0			0
$437$			2.00449e8i			0.114899i
$438$		0			0
$439$	1.11226e9			0.627450			0.313725	−	0.949514i	$-0.398423\pi$
$439$	1.11226e9			0.627450			0.313725	+	0.949514i	$0.398423\pi$
$440$		0			0
$441$	7.39643e8			0.410665
$442$		0			0
$443$		−	3.22249e9i		−	1.76108i	−0.473972	−	0.880540i	$-0.657180\pi$
$443$		−	3.22249e9i		−	1.76108i	0.473972	−	0.880540i	$-0.342820\pi$
$444$		0			0
$445$	4.23313e8	−	1.51974e9i	0.227720	−	0.817540i
$446$		0			0
$447$		−	1.29266e9i		−	0.684557i
$448$		0			0
$449$	−7.29482e7			−0.0380323			−0.0190161	−	0.999819i	$-0.506053\pi$
$449$	−7.29482e7			−0.0380323			−0.0190161	+	0.999819i	$0.506053\pi$
$450$		0			0
$451$	−3.64013e9			−1.86853
$452$		0			0
$453$		−	9.26015e8i		−	0.468031i
$454$		0			0
$455$	3.19621e8	−	1.14747e9i	0.159072	−	0.571087i
$456$		0			0
$457$			2.45286e8i			0.120217i	0.998192	+	0.0601085i	$0.0191447\pi$
$457$			2.45286e8i			0.120217i	−0.998192	+	0.0601085i	$0.980855\pi$
$458$		0			0
$459$	1.68727e9			0.814405
$460$		0			0
$461$	−3.25654e9			−1.54812			−0.774058	−	0.633115i	$-0.781776\pi$
$461$	−3.25654e9			−1.54812			−0.774058	+	0.633115i	$0.781776\pi$
$462$		0			0
$463$		−	5.48463e8i		−	0.256811i	−0.991722	−	0.128406i	$-0.959014\pi$
$463$		−	5.48463e8i		−	0.256811i	0.991722	−	0.128406i	$-0.0409859\pi$
$464$		0			0
$465$	7.14374e8	+	1.98984e8i	0.329489	+	0.0917768i
$466$		0			0
$467$			1.31891e9i			0.599245i	0.954058	+	0.299623i	$0.0968607\pi$
$467$			1.31891e9i			0.599245i	−0.954058	+	0.299623i	$0.903139\pi$
$468$		0			0
$469$	−1.14036e9			−0.510431
$470$		0			0
$471$	−9.75658e8			−0.430253
$472$		0			0
$473$		−	4.84061e9i		−	2.10323i
$474$		0			0
$475$	4.58762e8	+	2.77067e8i	0.196408	+	0.118620i
$476$		0			0
$477$		−	6.73654e8i		−	0.284199i
$478$		0			0
$479$	−1.59989e9			−0.665144			−0.332572	−	0.943078i	$-0.607916\pi$
$479$	−1.59989e9			−0.665144			−0.332572	+	0.943078i	$0.607916\pi$
$480$		0			0
$481$	−2.26783e9			−0.929187
$482$		0			0
$483$		−	3.96573e8i		−	0.160143i
$484$		0			0
$485$	3.23355e9	+	9.00682e8i	1.28701	+	0.358489i
$486$		0			0
$487$			1.95948e9i			0.768759i	0.923175	+	0.384380i	$0.125585\pi$
$487$			1.95948e9i			0.768759i	−0.923175	+	0.384380i	$0.874415\pi$
$488$		0			0
$489$	3.02242e9			1.16889
$490$		0			0
$491$	−2.38785e8			−0.0910376			−0.0455188	−	0.998963i	$-0.514494\pi$
$491$	−2.38785e8			−0.0910376			−0.0455188	+	0.998963i	$0.514494\pi$
$492$		0			0
$493$			4.01292e8i			0.150833i
$494$		0			0
$495$	5.85330e8	−	2.10140e9i	0.216912	−	0.778737i
$496$		0			0
$497$			2.02296e8i			0.0739163i
$498$		0			0
$499$	3.06642e9			1.10479			0.552394	−	0.833583i	$-0.313714\pi$
$499$	3.06642e9			1.10479			0.552394	+	0.833583i	$0.313714\pi$
$500$		0			0
$501$	−1.85306e9			−0.658350
$502$		0			0
$503$		−	1.60348e9i		−	0.561793i	−0.959738	−	0.280897i	$-0.909368\pi$
$503$		−	1.60348e9i		−	0.561793i	0.959738	−	0.280897i	$-0.0906318\pi$
$504$		0			0
$505$	3.85703e8	−	1.38471e9i	0.133270	−	0.478454i
$506$		0			0
$507$			1.29844e9i			0.442480i
$508$		0			0
$509$	−1.21742e9			−0.409192			−0.204596	−	0.978847i	$-0.565588\pi$
$509$	−1.21742e9			−0.409192			−0.204596	+	0.978847i	$0.565588\pi$
$510$		0			0
$511$	−6.24258e8			−0.206962
$512$		0			0
$513$			7.38106e8i			0.241384i
$514$		0			0
$515$	−9.38304e8	−	2.61358e8i	−0.302704	−	0.0843161i
$516$		0			0
$517$		−	3.97850e7i		−	0.0126620i
$518$		0			0
$519$	−1.57416e9			−0.494270
$520$		0			0
$521$	−2.08635e9			−0.646331			−0.323166	−	0.946342i	$-0.604747\pi$
$521$	−2.08635e9			−0.646331			−0.323166	+	0.946342i	$0.604747\pi$
$522$		0			0
$523$			4.28922e9i			1.31106i	0.755169	+	0.655531i	$0.227555\pi$
$523$			4.28922e9i			1.31106i	−0.755169	+	0.655531i	$0.772445\pi$
$524$		0			0
$525$	−9.07628e8	−	5.48156e8i	−0.273747	−	0.165328i
$526$		0			0
$527$		−	1.28765e9i		−	0.383230i
$528$		0			0
$529$	2.55102e9			0.749237
$530$		0			0
$531$	−1.64475e9			−0.476727
$532$		0			0
$533$			5.40883e9i			1.54724i
$534$		0			0
$535$	4.00226e9	+	1.11480e9i	1.12997	+	0.314745i
$536$		0			0
$537$			6.25235e8i			0.174234i
$538$		0			0
$539$	4.41845e9			1.21537
$540$		0			0
$541$	4.91116e9			1.33350			0.666751	−	0.745280i	$-0.267684\pi$
$541$	4.91116e9			1.33350			0.666751	+	0.745280i	$0.267684\pi$
$542$		0			0
$543$		−	2.52871e9i		−	0.677797i
$544$		0			0
$545$	−1.50843e9	+	5.41542e9i	−0.399151	+	1.43299i
$546$		0			0
$547$		−	1.76451e9i		−	0.460965i	−0.973077	−	0.230482i	$-0.925970\pi$
$547$		−	1.76451e9i		−	0.460965i	0.973077	−	0.230482i	$-0.0740304\pi$
$548$		0			0
$549$	1.57851e9			0.407140
$550$		0			0
$551$	−1.75547e8			−0.0447058
$552$		0			0
$553$			4.45145e8i			0.111934i
$554$		0			0
$555$	−5.41679e8	+	1.94469e9i	−0.134498	+	0.482864i
$556$		0			0
$557$		−	4.13406e8i		−	0.101364i	−0.998715	−	0.0506820i	$-0.983860\pi$
$557$		−	4.13406e8i		−	0.101364i	0.998715	−	0.0506820i	$-0.0161395\pi$
$558$		0			0
$559$	−7.19261e9			−1.74159
$560$		0			0
$561$	3.45967e9			0.827302
$562$		0			0
$563$			7.57073e9i			1.78796i	0.448104	+	0.893982i	$0.352100\pi$
$563$			7.57073e9i			1.78796i	−0.448104	+	0.893982i	$0.647900\pi$
$564$		0			0
$565$	−1.51494e9	−	4.21975e8i	−0.353366	−	0.0984277i
$566$		0			0
$567$		−	4.10289e8i		−	0.0945255i
$568$		0			0
$569$	−8.90287e8			−0.202599			−0.101299	−	0.994856i	$-0.532300\pi$
$569$	−8.90287e8			−0.202599			−0.101299	+	0.994856i	$0.532300\pi$
$570$		0			0
$571$	4.96089e9			1.11515			0.557575	−	0.830126i	$-0.311732\pi$
$571$	4.96089e9			1.11515			0.557575	+	0.830126i	$0.311732\pi$
$572$		0			0
$573$		−	3.85966e9i		−	0.857054i
$574$		0			0
$575$	−1.18016e9	+	1.95408e9i	−0.258882	+	0.428652i
$576$		0			0
$577$			1.53066e9i			0.331713i	0.986150	+	0.165856i	$0.0530388\pi$
$577$			1.53066e9i			0.331713i	−0.986150	+	0.165856i	$0.946961\pi$
$578$		0			0
$579$	1.93371e9			0.414016
$580$		0			0
$581$	−1.09371e9			−0.231358
$582$		0			0
$583$		−	4.02425e9i		−	0.841094i
$584$		0			0
$585$	−3.12245e9	−	8.69736e8i	−0.644836	−	0.179615i
$586$		0			0
$587$			4.39564e9i			0.896992i	0.893785	+	0.448496i	$0.148040\pi$
$587$			4.39564e9i			0.896992i	−0.893785	+	0.448496i	$0.851960\pi$
$588$		0			0
$589$	5.63288e8			0.113587
$590$		0			0
$591$	3.97310e9			0.791724
$592$		0			0
$593$			3.32990e9i			0.655753i	0.944721	+	0.327877i	$0.106333\pi$
$593$			3.32990e9i			0.655753i	−0.944721	+	0.327877i	$0.893667\pi$
$594$		0			0
$595$	−4.94021e8	+	1.77359e9i	−0.0961470	+	0.345178i
$596$		0			0
$597$			5.46226e9i			1.05066i
$598$		0			0
$599$	−4.53030e9			−0.861258			−0.430629	−	0.902529i	$-0.641708\pi$
$599$	−4.53030e9			−0.861258			−0.430629	+	0.902529i	$0.641708\pi$
$600$		0			0
$601$	−4.70479e9			−0.884056			−0.442028	−	0.897001i	$-0.645741\pi$
$601$	−4.70479e9			−0.884056			−0.442028	+	0.897001i	$0.645741\pi$
$602$		0			0
$603$			3.10309e9i			0.576347i
$604$		0			0
$605$	2.03508e9	−	7.30616e9i	0.373627	−	1.34136i
$606$		0			0
$607$		−	2.24429e9i		−	0.407303i	−0.979043	−	0.203652i	$-0.934719\pi$
$607$		−	2.24429e9i		−	0.407303i	0.979043	−	0.203652i	$-0.0652810\pi$
$608$		0			0
$609$	3.47307e8			0.0623094
$610$		0			0
$611$	−5.91162e7			−0.0104848
$612$		0			0
$613$			8.74415e9i			1.53323i	0.642109	+	0.766613i	$0.278059\pi$
$613$			8.74415e9i			1.53323i	−0.642109	+	0.766613i	$0.721941\pi$
$614$		0			0
$615$	4.63813e9	+	1.29192e9i	0.804044	+	0.223961i
$616$		0			0
$617$		−	4.49031e9i		−	0.769623i	−0.922995	−	0.384812i	$-0.874266\pi$
$617$		−	4.49031e9i		−	0.769623i	0.922995	−	0.384812i	$-0.125734\pi$
$618$		0			0
$619$	−3.74101e9			−0.633974			−0.316987	−	0.948430i	$-0.602671\pi$
$619$	−3.74101e9			−0.633974			−0.316987	+	0.948430i	$0.602671\pi$
$620$		0			0
$621$	−3.14393e9			−0.526808
$622$		0			0
$623$			2.37079e9i			0.392813i
$624$		0			0
$625$	2.84102e9	+	5.40199e9i	0.465472	+	0.885063i
$626$		0			0
$627$			1.51345e9i			0.245206i
$628$		0			0
$629$	3.50527e9			0.561622
$630$		0			0
$631$	−1.93545e9			−0.306675			−0.153337	−	0.988174i	$-0.549002\pi$
$631$	−1.93545e9			−0.306675			−0.153337	+	0.988174i	$0.549002\pi$
$632$		0			0
$633$		−	8.64660e9i		−	1.35498i
$634$		0			0
$635$	−7.68355e9	−	2.14020e9i	−1.19084	−	0.331701i
$636$		0			0
$637$		−	6.56532e9i		−	1.00639i
$638$		0			0
$639$	5.50478e8			0.0834616
$640$		0			0
$641$	5.89076e9			0.883422			0.441711	−	0.897157i	$-0.354372\pi$
$641$	5.89076e9			0.883422			0.441711	+	0.897157i	$0.354372\pi$
$642$		0			0
$643$		−	3.16008e9i		−	0.468770i	−0.972144	−	0.234385i	$-0.924692\pi$
$643$		−	3.16008e9i		−	0.468770i	0.972144	−	0.234385i	$-0.0753076\pi$
$644$		0			0
$645$	−1.71798e9	+	6.16774e9i	−0.252092	+	0.905038i
$646$		0			0
$647$		−	1.27557e10i		−	1.85157i	−0.378054	−	0.925783i	$-0.623407\pi$
$647$		−	1.27557e10i		−	1.85157i	0.378054	−	0.925783i	$-0.376593\pi$
$648$		0			0
$649$	−9.82536e9			−1.41089
$650$		0			0
$651$	−1.11442e9			−0.158313
$652$		0			0
$653$		−	4.43892e9i		−	0.623852i	−0.950106	−	0.311926i	$-0.899026\pi$
$653$		−	4.43892e9i		−	0.623852i	0.950106	−	0.311926i	$-0.100974\pi$
$654$		0			0
$655$	2.49849e9	−	8.96985e9i	0.347402	−	1.24721i
$656$		0			0
$657$			1.69870e9i			0.233689i
$658$		0			0
$659$	1.08526e10			1.47719			0.738595	−	0.674149i	$-0.235489\pi$
$659$	1.08526e10			1.47719			0.738595	+	0.674149i	$0.235489\pi$
$660$		0			0
$661$	8.49307e9			1.14382			0.571912	−	0.820315i	$-0.306202\pi$
$661$	8.49307e9			1.14382			0.571912	+	0.820315i	$0.306202\pi$
$662$		0			0
$663$		−	5.14068e9i		−	0.685051i
$664$		0			0
$665$	−7.75866e8	−	2.16112e8i	−0.102308	−	0.0284973i
$666$		0			0
$667$		−	7.47737e8i		−	0.0975683i
$668$		0			0
$669$	4.82509e9			0.623037
$670$		0			0
$671$	9.42962e9			1.20494
$672$		0			0
$673$		−	4.20188e9i		−	0.531362i	−0.964061	−	0.265681i	$-0.914403\pi$
$673$		−	4.20188e9i		−	0.531362i	0.964061	−	0.265681i	$-0.0855969\pi$
$674$		0			0
$675$	−4.34565e9	+	7.19546e9i	−0.543866	+	0.900524i
$676$		0			0
$677$		−	6.56755e9i		−	0.813472i	−0.913546	−	0.406736i	$-0.866667\pi$
$677$		−	6.56755e9i		−	0.813472i	0.913546	−	0.406736i	$-0.133333\pi$
$678$		0			0
$679$	−5.04433e9			−0.618386
$680$		0			0
$681$	7.56676e9			0.918110
$682$		0			0
$683$		−	6.31484e9i		−	0.758386i	−0.925318	−	0.379193i	$-0.876202\pi$
$683$		−	6.31484e9i		−	0.758386i	0.925318	−	0.379193i	$-0.123798\pi$
$684$		0			0
$685$	−1.15269e10	−	3.21075e9i	−1.37024	−	0.381671i
$686$		0			0
$687$			4.26123e9i			0.501403i
$688$		0			0
$689$	−5.97958e9			−0.696472
$690$		0			0
$691$	−3.76447e9			−0.434041			−0.217020	−	0.976167i	$-0.569634\pi$
$691$	−3.76447e9			−0.434041			−0.217020	+	0.976167i	$0.569634\pi$
$692$		0			0
$693$			3.27818e9i			0.374168i
$694$		0			0
$695$	−8.53562e8	+	3.06438e9i	−0.0964468	+	0.346255i
$696$		0			0
$697$		−	8.36014e9i		−	0.935188i
$698$		0			0
$699$	5.92646e9			0.656335
$700$		0			0
$701$	1.97083e9			0.216090			0.108045	−	0.994146i	$-0.465541\pi$
$701$	1.97083e9			0.216090			0.108045	+	0.994146i	$0.465541\pi$
$702$		0			0
$703$			1.53340e9i			0.166461i
$704$		0			0
$705$	−1.41201e7	+	5.06927e7i	−0.00151766	+	0.00544858i
$706$		0			0
$707$			2.16016e9i			0.229888i
$708$		0			0
$709$	−9.62853e9			−1.01461			−0.507304	−	0.861767i	$-0.669358\pi$
$709$	−9.62853e9			−1.01461			−0.507304	+	0.861767i	$0.669358\pi$
$710$		0			0
$711$	1.21131e9			0.126389
$712$		0			0
$713$			2.39930e9i			0.247897i
$714$		0			0
$715$	−1.86527e10	−	5.19559e9i	−1.90841	−	0.531574i
$716$		0			0
$717$		−	3.39643e9i		−	0.344117i
$718$		0			0
$719$	−1.89490e10			−1.90123			−0.950614	−	0.310376i	$-0.899545\pi$
$719$	−1.89490e10			−1.90123			−0.950614	+	0.310376i	$0.899545\pi$
$720$		0			0
$721$	1.46375e9			0.145444
$722$		0			0
$723$		−	6.27532e9i		−	0.617521i
$724$		0			0
$725$	−1.71133e9	−	1.03355e9i	−0.166783	−	0.100727i
$726$		0			0
$727$			1.44446e9i			0.139423i	0.997567	+	0.0697116i	$0.0222079\pi$
$727$			1.44446e9i			0.139423i	−0.997567	+	0.0697116i	$0.977792\pi$
$728$		0			0
$729$	−8.71961e9			−0.833586
$730$		0			0
$731$	1.11172e10			1.05266
$732$		0			0
$733$		−	1.38939e10i		−	1.30305i	−0.758627	−	0.651525i	$-0.774129\pi$
$733$		−	1.38939e10i		−	1.30305i	0.758627	−	0.651525i	$-0.225871\pi$
$734$		0			0
$735$	−5.62983e9	−	1.56815e9i	−0.522986	−	0.145674i
$736$		0			0
$737$			1.85371e10i			1.70571i
$738$		0			0
$739$	1.19008e10			1.08473			0.542366	−	0.840142i	$-0.317529\pi$
$739$	1.19008e10			1.08473			0.542366	+	0.840142i	$0.317529\pi$
$740$		0			0
$741$	2.24882e9			0.203044
$742$		0			0
$743$			1.57512e10i			1.40882i	0.709796	+	0.704408i	$0.248787\pi$
$743$			1.57512e10i			1.40882i	−0.709796	+	0.704408i	$0.751213\pi$
$744$		0			0
$745$	3.00052e9	−	1.07722e10i	0.265858	−	0.954459i
$746$		0			0
$747$			2.97615e9i			0.261235i
$748$		0			0
$749$	−6.24352e9			−0.542929
$750$		0			0
$751$	1.60645e10			1.38397			0.691984	−	0.721912i	$-0.256737\pi$
$751$	1.60645e10			1.38397			0.691984	+	0.721912i	$0.256737\pi$
$752$		0			0
$753$			6.54909e9i			0.558983i
$754$		0			0
$755$	2.14946e9	−	7.71679e9i	0.181767	−	0.652563i
$756$		0			0
$757$			1.88969e10i			1.58327i	0.610993	+	0.791636i	$0.290771\pi$
$757$			1.88969e10i			1.58327i	−0.610993	+	0.791636i	$0.709229\pi$
$758$		0			0
$759$	−6.44648e9			−0.535151
$760$		0			0
$761$	1.01100e10			0.831584			0.415792	−	0.909460i	$-0.363504\pi$
$761$	1.01100e10			0.831584			0.415792	+	0.909460i	$0.363504\pi$
$762$		0			0
$763$		−	8.44806e9i		−	0.688527i
$764$		0			0
$765$	4.82620e9	+	1.34431e9i	0.389754	+	0.108563i
$766$		0			0
$767$			1.45994e10i			1.16829i
$768$		0			0
$769$	2.00677e10			1.59132			0.795658	−	0.605746i	$-0.207125\pi$
$769$	2.00677e10			1.59132			0.795658	+	0.605746i	$0.207125\pi$
$770$		0			0
$771$	−4.34883e8			−0.0341729
$772$		0			0
$773$			2.15770e10i			1.68020i	0.542428	+	0.840102i	$0.317505\pi$
$773$			2.15770e10i			1.68020i	−0.542428	+	0.840102i	$0.682495\pi$
$774$		0			0
$775$	5.49124e9	+	3.31640e9i	0.423755	+	0.255924i
$776$		0			0
$777$		−	3.03371e9i		−	0.232007i
$778$		0			0
$779$	3.65719e9			0.277183
$780$		0			0
$781$	3.28842e9			0.247007
$782$		0			0
$783$		−	2.75337e9i		−	0.204974i
$784$		0			0
$785$	−8.13048e9	−	2.26469e9i	−0.599891	−	0.167095i
$786$		0			0
$787$		−	2.32055e10i		−	1.69699i	−0.529205	−	0.848494i	$-0.677510\pi$
$787$		−	2.32055e10i		−	1.69699i	0.529205	−	0.848494i	$-0.322490\pi$
$788$		0			0
$789$	−4.59479e9			−0.333040
$790$		0			0
$791$	2.36330e9			0.169786
$792$		0			0
$793$		−	1.40114e10i		−	0.997756i
$794$		0			0
$795$	−1.42824e9	+	5.12755e9i	−0.100813	+	0.361931i
$796$		0			0
$797$		−	6.77123e8i		−	0.0473766i	−0.999719	−	0.0236883i	$-0.992459\pi$
$797$		−	6.77123e8i		−	0.0473766i	0.999719	−	0.0236883i	$-0.00754092\pi$
$798$		0			0
$799$	9.13727e7			0.00633728
$800$		0			0
$801$	6.45129e9			0.443540
$802$		0			0
$803$			1.01476e10i			0.691607i
$804$		0			0
$805$	9.20521e8	−	3.30477e9i	0.0621939	−	0.223283i
$806$		0			0
$807$			1.63994e10i			1.09842i
$808$		0			0
$809$	5.84504e9			0.388122			0.194061	−	0.980990i	$-0.437834\pi$
$809$	5.84504e9			0.388122			0.194061	+	0.980990i	$0.437834\pi$
$810$		0			0
$811$	−1.91491e10			−1.26060			−0.630299	−	0.776353i	$-0.717068\pi$
$811$	−1.91491e10			−1.26060			−0.630299	+	0.776353i	$0.717068\pi$
$812$		0			0
$813$			3.64584e9i			0.237947i
$814$		0			0
$815$	2.51868e10	+	7.01562e9i	1.62975	+	0.453957i
$816$		0			0
$817$			4.86330e9i			0.311999i
$818$		0			0
$819$	4.87102e9			0.309832
$820$		0			0
$821$	6.17006e9			0.389124			0.194562	−	0.980890i	$-0.437671\pi$
$821$	6.17006e9			0.389124			0.194562	+	0.980890i	$0.437671\pi$
$822$		0			0
$823$			2.25285e10i			1.40875i	0.709830	+	0.704373i	$0.248772\pi$
$823$			2.25285e10i			1.40875i	−0.709830	+	0.704373i	$0.751228\pi$
$824$		0			0
$825$	−8.91054e9	+	1.47539e10i	−0.552478	+	0.914784i
$826$		0			0
$827$			6.08545e9i			0.374131i	0.982347	+	0.187065i	$0.0598976\pi$
$827$			6.08545e9i			0.374131i	−0.982347	+	0.187065i	$0.940102\pi$
$828$		0			0
$829$	−4.81588e9			−0.293586			−0.146793	−	0.989167i	$-0.546895\pi$
$829$	−4.81588e9			−0.293586			−0.146793	+	0.989167i	$0.546895\pi$
$830$		0			0
$831$	1.65021e10			0.997554
$832$		0			0
$833$			1.01477e10i			0.608288i
$834$		0			0
$835$	−1.54422e10	−	4.30130e9i	−0.917921	−	0.255680i
$836$		0			0
$837$			8.83489e9i			0.520789i
$838$		0			0
$839$	−2.92635e10			−1.71064			−0.855320	−	0.518100i	$-0.826640\pi$
$839$	−2.92635e10			−1.71064			−0.855320	+	0.518100i	$0.826640\pi$
$840$		0			0
$841$	−1.65950e10			−0.962038
$842$		0			0
$843$			5.50769e9i			0.316645i
$844$		0			0
$845$	−3.01393e9	+	1.08203e10i	−0.171844	+	0.616939i
$846$		0			0
$847$			1.13976e10i			0.644499i
$848$		0			0
$849$	5.23902e9			0.293814
$850$		0			0
$851$	−6.53145e9			−0.363292
$852$		0			0
$853$		−	2.17155e10i		−	1.19797i	−0.800759	−	0.598987i	$-0.795570\pi$
$853$		−	2.17155e10i		−	1.19797i	0.800759	−	0.598987i	$-0.204430\pi$
$854$		0			0
$855$	−5.88074e8	+	2.11125e9i	−0.0321773	+	0.115520i
$856$		0			0
$857$		−	4.01757e9i		−	0.218037i	−0.994040	−	0.109019i	$-0.965229\pi$
$857$		−	4.01757e9i		−	0.218037i	0.994040	−	0.109019i	$-0.0347708\pi$
$858$		0			0
$859$	1.62487e10			0.874666			0.437333	−	0.899300i	$-0.355923\pi$
$859$	1.62487e10			0.874666			0.437333	+	0.899300i	$0.355923\pi$
$860$		0			0
$861$	−7.23548e9			−0.386328
$862$		0			0
$863$		−	1.95958e10i		−	1.03783i	−0.854826	−	0.518914i	$-0.826336\pi$
$863$		−	1.95958e10i		−	1.03783i	0.854826	−	0.518914i	$-0.173664\pi$
$864$		0			0
$865$	−1.31180e10	−	3.65394e9i	−0.689148	−	0.191957i
$866$		0			0
$867$		−	5.31277e9i		−	0.276856i
$868$		0			0
$869$	7.23604e9			0.374052
$870$		0			0
$871$	2.75441e10			1.41242
$872$		0			0
$873$			1.37264e10i			0.698243i
$874$		0			0
$875$	−6.29119e9	−	6.67474e9i	−0.317471	−	0.336827i
$876$		0			0
$877$		−	3.67840e10i		−	1.84145i	−0.390207	−	0.920727i	$-0.627597\pi$
$877$		−	3.67840e10i		−	1.84145i	0.390207	−	0.920727i	$-0.372403\pi$
$878$		0			0
$879$	−1.24670e10			−0.619159
$880$		0			0
$881$	1.48378e9			0.0731062			0.0365531	−	0.999332i	$-0.488362\pi$
$881$	1.48378e9			0.0731062			0.0365531	+	0.999332i	$0.488362\pi$
$882$		0			0
$883$		−	2.36597e10i		−	1.15650i	−0.815859	−	0.578250i	$-0.803736\pi$
$883$		−	2.36597e10i		−	1.15650i	0.815859	−	0.578250i	$-0.196264\pi$
$884$		0			0
$885$	1.25191e10	+	3.48712e9i	0.607117	+	0.169108i
$886$		0			0
$887$		−	4.21269e9i		−	0.202687i	−0.994851	−	0.101344i	$-0.967686\pi$
$887$		−	4.21269e9i		−	0.202687i	0.994851	−	0.101344i	$-0.0323142\pi$
$888$		0			0
$889$	1.19863e10			0.572177
$890$		0			0
$891$	−6.66945e9			−0.315877
$892$		0			0
$893$			3.99715e7i			0.00187832i
$894$		0			0
$895$	−1.45129e9	+	5.21029e9i	−0.0676665	+	0.242930i
$896$		0			0
$897$			9.57875e9i			0.443134i
$898$		0			0
$899$	−2.10125e9			−0.0964535
$900$		0			0
$901$	9.24233e9			0.420964
$902$		0			0
$903$		−	9.62167e9i		−	0.434854i
$904$		0			0
$905$	5.86962e9	−	2.10726e10i	0.263233	−	0.945035i
$906$		0			0
$907$		−	1.32662e10i		−	0.590367i	−0.955441	−	0.295184i	$-0.904619\pi$
$907$		−	1.32662e10i		−	0.590367i	0.955441	−	0.295184i	$-0.0953808\pi$
$908$		0			0
$909$	5.87811e9			0.259576
$910$		0			0
$911$	−7.15727e9			−0.313641			−0.156821	−	0.987627i	$-0.550124\pi$
$911$	−7.15727e9			−0.313641			−0.156821	+	0.987627i	$0.550124\pi$
$912$		0			0
$913$			1.77788e10i			0.773132i
$914$		0			0
$915$	−1.20149e10	−	3.34666e9i	−0.518497	−	0.144424i
$916$		0			0
$917$			1.39930e10i			0.599263i
$918$		0			0
$919$	−9.78152e9			−0.415721			−0.207860	−	0.978158i	$-0.566650\pi$
$919$	−9.78152e9			−0.415721			−0.207860	+	0.978158i	$0.566650\pi$
$920$		0			0
$921$	−2.06085e10			−0.869236
$922$		0			0
$923$		−	4.88623e9i		−	0.204535i
$924$		0			0
$925$	−9.02799e9	+	1.49484e10i	−0.375055	+	0.621010i
$926$		0			0
$927$		−	3.98309e9i		−	0.164226i
$928$		0			0
$929$	−2.92073e10			−1.19519			−0.597594	−	0.801799i	$-0.703876\pi$
$929$	−2.92073e10			−1.19519			−0.597594	+	0.801799i	$0.703876\pi$
$930$		0			0
$931$	−4.43915e9			−0.180292
$932$		0			0
$933$		−	2.35165e10i		−	0.947952i
$934$		0			0
$935$	2.88305e10	+	8.03055e9i	1.15349	+	0.321295i
$936$		0			0
$937$		−	3.72053e10i		−	1.47746i	−0.674000	−	0.738731i	$-0.735425\pi$
$937$		−	3.72053e10i		−	1.47746i	0.674000	−	0.738731i	$-0.264575\pi$
$938$		0			0
$939$	1.44388e10			0.569116
$940$		0			0
$941$	6.20016e9			0.242571			0.121286	−	0.992618i	$-0.461298\pi$
$941$	6.20016e9			0.242571			0.121286	+	0.992618i	$0.461298\pi$
$942$		0			0
$943$			1.55777e10i			0.604938i
$944$		0			0
$945$	3.38961e9	−	1.21691e10i	0.130659	−	0.469079i
$946$		0			0
$947$		−	1.27543e10i		−	0.488015i	−0.969773	−	0.244008i	$-0.921538\pi$
$947$		−	1.27543e10i		−	0.488015i	0.969773	−	0.244008i	$-0.0784622\pi$
$948$		0			0
$949$	1.50782e10			0.572689
$950$		0			0
$951$	−1.90985e10			−0.720057
$952$		0			0
$953$		−	3.08638e10i		−	1.15511i	−0.816351	−	0.577556i	$-0.804006\pi$
$953$		−	3.08638e10i		−	1.15511i	0.816351	−	0.577556i	$-0.195994\pi$
$954$		0			0
$955$	8.95902e9	−	3.21639e10i	0.332850	−	1.19497i
$956$		0			0
$957$		−	5.64565e9i		−	0.208220i
$958$		0			0
$959$	1.79820e10			0.658375
$960$		0			0
$961$	−2.07702e10			−0.754935
$962$		0			0
$963$			1.69896e10i			0.613042i
$964$		0			0
$965$	1.61143e10	+	4.48852e9i	0.577252	+	0.160789i
$966$		0			0
$967$		−	3.32539e10i		−	1.18263i	−0.806439	−	0.591317i	$-0.798608\pi$
$967$		−	3.32539e10i		−	1.18263i	0.806439	−	0.591317i	$-0.201392\pi$
$968$		0			0
$969$	−3.47588e9			−0.122725
$970$		0			0
$971$	−4.45095e10			−1.56022			−0.780108	−	0.625644i	$-0.784836\pi$
$971$	−4.45095e10			−1.56022			−0.780108	+	0.625644i	$0.784836\pi$
$972$		0			0
$973$		−	4.78043e9i		−	0.166369i
$974$		0			0
$975$	2.19227e10	+	1.32401e10i	0.757492	+	0.457482i
$976$		0			0
$977$		−	6.80736e9i		−	0.233533i	−0.993159	−	0.116766i	$-0.962747\pi$
$977$		−	6.80736e9i		−	0.233533i	0.993159	−	0.116766i	$-0.0372529\pi$
$978$		0			0
$979$	3.85384e10			1.31267
$980$		0			0
$981$	−2.29884e10			−0.777442
$982$		0			0
$983$		−	6.37498e8i		−	0.0214063i	−0.999943	−	0.0107031i	$-0.996593\pi$
$983$		−	6.37498e8i		−	0.0214063i	0.999943	−	0.0107031i	$-0.00340698\pi$
$984$		0			0
$985$	3.31092e10	+	9.22234e9i	1.10388	+	0.307478i
$986$		0			0
$987$		−	7.90806e7i		−	0.00261794i
$988$		0			0
$989$	−2.07150e10			−0.680924
$990$		0			0
$991$	−5.60147e10			−1.82828			−0.914142	−	0.405393i	$-0.867135\pi$
$991$	−5.60147e10			−1.82828			−0.914142	+	0.405393i	$0.867135\pi$
$992$		0			0
$993$			1.88977e10i			0.612472i
$994$		0			0
$995$	−1.26789e10	+	4.55188e10i	−0.408040	+	1.46491i
$996$		0			0
$997$			8.97443e9i			0.286796i	0.989665	+	0.143398i	$0.0458030\pi$
$997$			8.97443e9i			0.286796i	−0.989665	+	0.143398i	$0.954197\pi$
$998$		0			0
$999$	−2.40506e10			−0.763214

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.8.c.a.49.1		2
4.3	odd	2		5.8.b.a.4.2	yes	2
5.2	odd	4		400.8.a.y.1.1		2
5.3	odd	4		400.8.a.y.1.2		2
5.4	even	2	inner	80.8.c.a.49.2		2
8.3	odd	2		320.8.c.d.129.1		2
8.5	even	2		320.8.c.c.129.2		2
12.11	even	2		45.8.b.a.19.1		2
20.3	even	4		25.8.a.d.1.2		2
20.7	even	4		25.8.a.d.1.1		2
20.19	odd	2		5.8.b.a.4.1	✓	2
40.19	odd	2		320.8.c.d.129.2		2
40.29	even	2		320.8.c.c.129.1		2
60.23	odd	4		225.8.a.n.1.1		2
60.47	odd	4		225.8.a.n.1.2		2
60.59	even	2		45.8.b.a.19.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.8.b.a.4.1	✓	2	20.19	odd	2
5.8.b.a.4.2	yes	2	4.3	odd	2
25.8.a.d.1.1		2	20.7	even	4
25.8.a.d.1.2		2	20.3	even	4
45.8.b.a.19.1		2	12.11	even	2
45.8.b.a.19.2		2	60.59	even	2
80.8.c.a.49.1		2	1.1	even	1	trivial
80.8.c.a.49.2		2	5.4	even	2	inner
225.8.a.n.1.1		2	60.23	odd	4
225.8.a.n.1.2		2	60.47	odd	4
320.8.c.c.129.1		2	40.29	even	2
320.8.c.c.129.2		2	8.5	even	2
320.8.c.d.129.1		2	8.3	odd	2
320.8.c.d.129.2		2	40.19	odd	2
400.8.a.y.1.1		2	5.2	odd	4
400.8.a.y.1.2		2	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	80.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-32.3110i q^{3} +(75.0000 - 269.258i) q^{5} +420.043i q^{7} +1143.00 q^{9} +6828.00 q^{11} -10145.7i q^{13} +(-8700.00 - 2423.32i) q^{15} +15681.6i q^{17} -6860.00 q^{19} +13572.0 q^{21} -29219.9i q^{23} +(-66875.0 - 40388.7i) q^{25} -107596. i q^{27} +25590.0 q^{29} -82112.0 q^{31} -220619. i q^{33} +(113100. + 31503.2i) q^{35} -223527. i q^{37} -327816. q^{39} -533118. q^{41} -708935. i q^{43} +(85725.0 - 307762. i) q^{45} -5826.75i q^{47} +647107. q^{49} +506688. q^{51} -589374. i q^{53} +(512100. - 1.83850e6i) q^{55} +221653. i q^{57} -1.43898e6 q^{59} +1.38102e6 q^{61} +480109. i q^{63} +(-2.73180e6 - 760924. i) q^{65} +2.71487e6i q^{67} -944124. q^{69} +481608. q^{71} +1.48618e6i q^{73} +(-1.30500e6 + 2.16080e6i) q^{75} +2.86805e6i q^{77} +1.05976e6 q^{79} -976779. q^{81} +2.60380e6i q^{83} +(4.22240e6 + 1.17612e6i) q^{85} -826838. i q^{87} +5.64417e6 q^{89} +4.26161e6 q^{91} +2.65312e6i q^{93} +(-514500. + 1.84711e6i) q^{95} +1.20091e7i q^{97} +7.80440e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 150 q^{5} + 2286 q^{9} + 13656 q^{11} - 17400 q^{15} - 13720 q^{19} + 27144 q^{21} - 133750 q^{25} + 51180 q^{29} - 164224 q^{31} + 226200 q^{35} - 655632 q^{39} - 1066236 q^{41} + 171450 q^{45} + 1294214 q^{49}+ \cdots + 15608808 q^{99}+O(q^{100}) \) 2 * q + 150 * q^5 + 2286 * q^9 + 13656 * q^11 - 17400 * q^15 - 13720 * q^19 + 27144 * q^21 - 133750 * q^25 + 51180 * q^29 - 164224 * q^31 + 226200 * q^35 - 655632 * q^39 - 1066236 * q^41 + 171450 * q^45 + 1294214 * q^49 + 1013376 * q^51 + 1024200 * q^55 - 2877960 * q^59 + 2762044 * q^61 - 5463600 * q^65 - 1888248 * q^69 + 963216 * q^71 - 2610000 * q^75 + 2119520 * q^79 - 1953558 * q^81 + 8444800 * q^85 + 11288340 * q^89 + 8523216 * q^91 - 1029000 * q^95 + 15608808 * q^99

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