LMFDB - Embedded newform 7800.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.56155 q^{7} +1.00000 q^{9} -1.43845 q^{11} -1.00000 q^{13} -5.68466 q^{17} -5.12311 q^{19} +2.56155 q^{21} +1.43845 q^{23} +1.00000 q^{27} -2.00000 q^{29} +1.12311 q^{31} -1.43845 q^{33} +10.8078 q^{37} -1.00000 q^{39} -9.68466 q^{41} -6.24621 q^{43} +1.12311 q^{47} -0.438447 q^{49} -5.68466 q^{51} +0.561553 q^{53} -5.12311 q^{57} -8.00000 q^{59} +1.68466 q^{61} +2.56155 q^{63} -2.24621 q^{67} +1.43845 q^{69} -7.68466 q^{71} +0.246211 q^{73} -3.68466 q^{77} -8.80776 q^{79} +1.00000 q^{81} +8.00000 q^{83} -2.00000 q^{87} -2.31534 q^{89} -2.56155 q^{91} +1.12311 q^{93} -2.80776 q^{97} -1.43845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + q^7 + 2 * q^9	$ 2 q + 2 q^{3} + q^{7} + 2 q^{9} - 7 q^{11} - 2 q^{13} + q^{17} - 2 q^{19} + q^{21} + 7 q^{23} + 2 q^{27} - 4 q^{29} - 6 q^{31} - 7 q^{33} + q^{37} - 2 q^{39} - 7 q^{41} + 4 q^{43} - 6 q^{47} - 5 q^{49} + q^{51} - 3 q^{53} - 2 q^{57} - 16 q^{59} - 9 q^{61} + q^{63} + 12 q^{67} + 7 q^{69} - 3 q^{71} - 16 q^{73} + 5 q^{77} + 3 q^{79} + 2 q^{81} + 16 q^{83} - 4 q^{87} - 17 q^{89} - q^{91} - 6 q^{93} + 15 q^{97} - 7 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^7 + 2 * q^9 - 7 * q^11 - 2 * q^13 + q^17 - 2 * q^19 + q^21 + 7 * q^23 + 2 * q^27 - 4 * q^29 - 6 * q^31 - 7 * q^33 + q^37 - 2 * q^39 - 7 * q^41 + 4 * q^43 - 6 * q^47 - 5 * q^49 + q^51 - 3 * q^53 - 2 * q^57 - 16 * q^59 - 9 * q^61 + q^63 + 12 * q^67 + 7 * q^69 - 3 * q^71 - 16 * q^73 + 5 * q^77 + 3 * q^79 + 2 * q^81 + 16 * q^83 - 4 * q^87 - 17 * q^89 - q^91 - 6 * q^93 + 15 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.43845	−0.433708	−0.216854	−	0.976204i	$-0.569580\pi$
$11$	−1.43845	−0.433708	−0.216854	+	0.976204i	$0.569580\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.68466	−1.37873	−0.689366	−	0.724413i	$-0.742111\pi$
$17$	−5.68466	−1.37873	−0.689366	+	0.724413i	$0.742111\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	0	0
$21$	2.56155	0.558977
$22$	0	0
$23$	1.43845	0.299937	0.149968	−	0.988691i	$-0.452083\pi$
$23$	1.43845	0.299937	0.149968	+	0.988691i	$0.452083\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	1.12311	0.201716	0.100858	−	0.994901i	$-0.467841\pi$
$31$	1.12311	0.201716	0.100858	+	0.994901i	$0.467841\pi$
$32$	0	0
$33$	−1.43845	−0.250402
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.8078	1.77679	0.888393	−	0.459084i	$-0.151822\pi$
$37$	10.8078	1.77679	0.888393	+	0.459084i	$0.151822\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−9.68466	−1.51249	−0.756245	−	0.654289i	$-0.772968\pi$
$41$	−9.68466	−1.51249	−0.756245	+	0.654289i	$0.772968\pi$
$42$	0	0
$43$	−6.24621	−0.952538	−0.476269	−	0.879300i	$-0.658011\pi$
$43$	−6.24621	−0.952538	−0.476269	+	0.879300i	$0.658011\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.12311	0.163822	0.0819109	−	0.996640i	$-0.473898\pi$
$47$	1.12311	0.163822	0.0819109	+	0.996640i	$0.473898\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	−5.68466	−0.796011
$52$	0	0
$53$	0.561553	0.0771352	0.0385676	−	0.999256i	$-0.487721\pi$
$53$	0.561553	0.0771352	0.0385676	+	0.999256i	$0.487721\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.12311	−0.678572
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	1.68466	0.215698	0.107849	−	0.994167i	$-0.465604\pi$
$61$	1.68466	0.215698	0.107849	+	0.994167i	$0.465604\pi$
$62$	0	0
$63$	2.56155	0.322725
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.24621	−0.274418	−0.137209	−	0.990542i	$-0.543813\pi$
$67$	−2.24621	−0.274418	−0.137209	+	0.990542i	$0.543813\pi$
$68$	0	0
$69$	1.43845	0.173169
$70$	0	0
$71$	−7.68466	−0.912001	−0.456001	−	0.889979i	$-0.650719\pi$
$71$	−7.68466	−0.912001	−0.456001	+	0.889979i	$0.650719\pi$
$72$	0	0
$73$	0.246211	0.0288168	0.0144084	−	0.999896i	$-0.495413\pi$
$73$	0.246211	0.0288168	0.0144084	+	0.999896i	$0.495413\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.68466	−0.419906
$78$	0	0
$79$	−8.80776	−0.990951	−0.495475	−	0.868622i	$-0.665006\pi$
$79$	−8.80776	−0.990951	−0.495475	+	0.868622i	$0.665006\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−2.31534	−0.245426	−0.122713	−	0.992442i	$-0.539159\pi$
$89$	−2.31534	−0.245426	−0.122713	+	0.992442i	$0.539159\pi$
$90$	0	0
$91$	−2.56155	−0.268524
$92$	0	0
$93$	1.12311	0.116461
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.80776	−0.285085	−0.142543	−	0.989789i	$-0.545528\pi$
$97$	−2.80776	−0.285085	−0.142543	+	0.989789i	$0.545528\pi$
$98$	0	0
$99$	−1.43845	−0.144569
$100$	0	0
$101$	−7.12311	−0.708776	−0.354388	−	0.935099i	$-0.615311\pi$
$101$	−7.12311	−0.708776	−0.354388	+	0.935099i	$0.615311\pi$
$102$	0	0
$103$	2.24621	0.221326	0.110663	−	0.993858i	$-0.464703\pi$
$103$	2.24621	0.221326	0.110663	+	0.993858i	$0.464703\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.80776	−0.464784	−0.232392	−	0.972622i	$-0.574655\pi$
$107$	−4.80776	−0.464784	−0.232392	+	0.972622i	$0.574655\pi$
$108$	0	0
$109$	−4.87689	−0.467122	−0.233561	−	0.972342i	$-0.575038\pi$
$109$	−4.87689	−0.467122	−0.233561	+	0.972342i	$0.575038\pi$
$110$	0	0
$111$	10.8078	1.02583
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−14.5616	−1.33486
$120$	0	0
$121$	−8.93087	−0.811897
$122$	0	0
$123$	−9.68466	−0.873236
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.36932	−0.653921	−0.326961	−	0.945038i	$-0.606024\pi$
$127$	−7.36932	−0.653921	−0.326961	+	0.945038i	$0.606024\pi$
$128$	0	0
$129$	−6.24621	−0.549948
$130$	0	0
$131$	−6.87689	−0.600837	−0.300419	−	0.953807i	$-0.597126\pi$
$131$	−6.87689	−0.600837	−0.300419	+	0.953807i	$0.597126\pi$
$132$	0	0
$133$	−13.1231	−1.13792
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.246211	0.0210352	0.0105176	−	0.999945i	$-0.496652\pi$
$137$	0.246211	0.0210352	0.0105176	+	0.999945i	$0.496652\pi$
$138$	0	0
$139$	−5.43845	−0.461283	−0.230642	−	0.973039i	$-0.574082\pi$
$139$	−5.43845	−0.461283	−0.230642	+	0.973039i	$0.574082\pi$
$140$	0	0
$141$	1.12311	0.0945826
$142$	0	0
$143$	1.43845	0.120289
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.438447	−0.0361625
$148$	0	0
$149$	−16.5616	−1.35678	−0.678388	−	0.734704i	$-0.737321\pi$
$149$	−16.5616	−1.35678	−0.678388	+	0.734704i	$0.737321\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	−5.68466	−0.459577
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.4924	1.79509	0.897545	−	0.440922i	$-0.145349\pi$
$157$	22.4924	1.79509	0.897545	+	0.440922i	$0.145349\pi$
$158$	0	0
$159$	0.561553	0.0445340
$160$	0	0
$161$	3.68466	0.290392
$162$	0	0
$163$	−9.43845	−0.739276	−0.369638	−	0.929176i	$-0.620518\pi$
$163$	−9.43845	−0.739276	−0.369638	+	0.929176i	$0.620518\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.2462	1.72146	0.860732	−	0.509059i	$-0.170006\pi$
$167$	22.2462	1.72146	0.860732	+	0.509059i	$0.170006\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.12311	−0.391774
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	−14.8769	−1.11195	−0.555976	−	0.831199i	$-0.687655\pi$
$179$	−14.8769	−1.11195	−0.555976	+	0.831199i	$0.687655\pi$
$180$	0	0
$181$	19.9309	1.48145	0.740725	−	0.671808i	$-0.234482\pi$
$181$	19.9309	1.48145	0.740725	+	0.671808i	$0.234482\pi$
$182$	0	0
$183$	1.68466	0.124534
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.17708	0.597967
$188$	0	0
$189$	2.56155	0.186326
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−21.6847	−1.56090	−0.780448	−	0.625221i	$-0.785009\pi$
$193$	−21.6847	−1.56090	−0.780448	+	0.625221i	$0.785009\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.3693	1.23751	0.618756	−	0.785583i	$-0.287637\pi$
$197$	17.3693	1.23751	0.618756	+	0.785583i	$0.287637\pi$
$198$	0	0
$199$	18.2462	1.29344	0.646720	−	0.762728i	$-0.276140\pi$
$199$	18.2462	1.29344	0.646720	+	0.762728i	$0.276140\pi$
$200$	0	0
$201$	−2.24621	−0.158436
$202$	0	0
$203$	−5.12311	−0.359572
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.43845	0.0999790
$208$	0	0
$209$	7.36932	0.509746
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	−7.68466	−0.526544
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.87689	0.195296
$218$	0	0
$219$	0.246211	0.0166374
$220$	0	0
$221$	5.68466	0.382392
$222$	0	0
$223$	−14.2462	−0.953997	−0.476998	−	0.878904i	$-0.658275\pi$
$223$	−14.2462	−0.953997	−0.476998	+	0.878904i	$0.658275\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.630683	−0.0418599	−0.0209300	−	0.999781i	$-0.506663\pi$
$227$	−0.630683	−0.0418599	−0.0209300	+	0.999781i	$0.506663\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−3.68466	−0.242433
$232$	0	0
$233$	−6.31534	−0.413732	−0.206866	−	0.978369i	$-0.566326\pi$
$233$	−6.31534	−0.413732	−0.206866	+	0.978369i	$0.566326\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.80776	−0.572126
$238$	0	0
$239$	−15.6847	−1.01456	−0.507278	−	0.861782i	$-0.669348\pi$
$239$	−15.6847	−1.01456	−0.507278	+	0.861782i	$0.669348\pi$
$240$	0	0
$241$	28.2462	1.81950	0.909749	−	0.415158i	$-0.136274\pi$
$241$	28.2462	1.81950	0.909749	+	0.415158i	$0.136274\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.12311	0.325975
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	−24.4924	−1.54595	−0.772974	−	0.634438i	$-0.781232\pi$
$251$	−24.4924	−1.54595	−0.772974	+	0.634438i	$0.781232\pi$
$252$	0	0
$253$	−2.06913	−0.130085
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.4924	1.15353	0.576763	−	0.816912i	$-0.304316\pi$
$257$	18.4924	1.15353	0.576763	+	0.816912i	$0.304316\pi$
$258$	0	0
$259$	27.6847	1.72024
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	4.49242	0.277015	0.138507	−	0.990361i	$-0.455770\pi$
$263$	4.49242	0.277015	0.138507	+	0.990361i	$0.455770\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.31534	−0.141697
$268$	0	0
$269$	−4.87689	−0.297349	−0.148675	−	0.988886i	$-0.547501\pi$
$269$	−4.87689	−0.297349	−0.148675	+	0.988886i	$0.547501\pi$
$270$	0	0
$271$	19.3693	1.17660	0.588301	−	0.808642i	$-0.299797\pi$
$271$	19.3693	1.17660	0.588301	+	0.808642i	$0.299797\pi$
$272$	0	0
$273$	−2.56155	−0.155032
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.4924	−0.630429	−0.315214	−	0.949021i	$-0.602076\pi$
$277$	−10.4924	−0.630429	−0.315214	+	0.949021i	$0.602076\pi$
$278$	0	0
$279$	1.12311	0.0672386
$280$	0	0
$281$	20.2462	1.20779	0.603894	−	0.797065i	$-0.293615\pi$
$281$	20.2462	1.20779	0.603894	+	0.797065i	$0.293615\pi$
$282$	0	0
$283$	18.7386	1.11390	0.556948	−	0.830547i	$-0.311972\pi$
$283$	18.7386	1.11390	0.556948	+	0.830547i	$0.311972\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.8078	−1.46436
$288$	0	0
$289$	15.3153	0.900902
$290$	0	0
$291$	−2.80776	−0.164594
$292$	0	0
$293$	−23.6155	−1.37963	−0.689817	−	0.723984i	$-0.742309\pi$
$293$	−23.6155	−1.37963	−0.689817	+	0.723984i	$0.742309\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.43845	−0.0834672
$298$	0	0
$299$	−1.43845	−0.0831875
$300$	0	0
$301$	−16.0000	−0.922225
$302$	0	0
$303$	−7.12311	−0.409212
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9.43845	−0.538681	−0.269340	−	0.963045i	$-0.586806\pi$
$307$	−9.43845	−0.538681	−0.269340	+	0.963045i	$0.586806\pi$
$308$	0	0
$309$	2.24621	0.127782
$310$	0	0
$311$	22.7386	1.28939	0.644695	−	0.764440i	$-0.276984\pi$
$311$	22.7386	1.28939	0.644695	+	0.764440i	$0.276984\pi$
$312$	0	0
$313$	−24.7386	−1.39831	−0.699155	−	0.714970i	$-0.746440\pi$
$313$	−24.7386	−1.39831	−0.699155	+	0.714970i	$0.746440\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.1231	−0.624736	−0.312368	−	0.949961i	$-0.601122\pi$
$317$	−11.1231	−0.624736	−0.312368	+	0.949961i	$0.601122\pi$
$318$	0	0
$319$	2.87689	0.161075
$320$	0	0
$321$	−4.80776	−0.268343
$322$	0	0
$323$	29.1231	1.62045
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4.87689	−0.269693
$328$	0	0
$329$	2.87689	0.158608
$330$	0	0
$331$	26.2462	1.44262	0.721311	−	0.692611i	$-0.243540\pi$
$331$	26.2462	1.44262	0.721311	+	0.692611i	$0.243540\pi$
$332$	0	0
$333$	10.8078	0.592262
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.1231	0.605914	0.302957	−	0.953004i	$-0.402026\pi$
$337$	11.1231	0.605914	0.302957	+	0.953004i	$0.402026\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−1.61553	−0.0874858
$342$	0	0
$343$	−19.0540	−1.02882
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−31.0540	−1.66706	−0.833532	−	0.552471i	$-0.813685\pi$
$347$	−31.0540	−1.66706	−0.833532	+	0.552471i	$0.813685\pi$
$348$	0	0
$349$	−25.3693	−1.35799	−0.678994	−	0.734144i	$-0.737584\pi$
$349$	−25.3693	−1.35799	−0.678994	+	0.734144i	$0.737584\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−12.2462	−0.651800	−0.325900	−	0.945404i	$-0.605667\pi$
$353$	−12.2462	−0.651800	−0.325900	+	0.945404i	$0.605667\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−14.5616	−0.770679
$358$	0	0
$359$	14.2462	0.751886	0.375943	−	0.926643i	$-0.377319\pi$
$359$	14.2462	0.751886	0.375943	+	0.926643i	$0.377319\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	−8.93087	−0.468749
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.24621	0.117251	0.0586256	−	0.998280i	$-0.481328\pi$
$367$	2.24621	0.117251	0.0586256	+	0.998280i	$0.481328\pi$
$368$	0	0
$369$	−9.68466	−0.504163
$370$	0	0
$371$	1.43845	0.0746805
$372$	0	0
$373$	23.1231	1.19727	0.598635	−	0.801022i	$-0.295710\pi$
$373$	23.1231	1.19727	0.598635	+	0.801022i	$0.295710\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	2.87689	0.147776	0.0738881	−	0.997267i	$-0.476459\pi$
$379$	2.87689	0.147776	0.0738881	+	0.997267i	$0.476459\pi$
$380$	0	0
$381$	−7.36932	−0.377542
$382$	0	0
$383$	−27.3693	−1.39851	−0.699253	−	0.714874i	$-0.746484\pi$
$383$	−27.3693	−1.39851	−0.699253	+	0.714874i	$0.746484\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.24621	−0.317513
$388$	0	0
$389$	−21.8617	−1.10843	−0.554217	−	0.832372i	$-0.686982\pi$
$389$	−21.8617	−1.10843	−0.554217	+	0.832372i	$0.686982\pi$
$390$	0	0
$391$	−8.17708	−0.413533
$392$	0	0
$393$	−6.87689	−0.346893
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13.1922	−0.662099	−0.331050	−	0.943613i	$-0.607403\pi$
$397$	−13.1922	−0.662099	−0.331050	+	0.943613i	$0.607403\pi$
$398$	0	0
$399$	−13.1231	−0.656977
$400$	0	0
$401$	12.2462	0.611547	0.305773	−	0.952104i	$-0.401085\pi$
$401$	12.2462	0.611547	0.305773	+	0.952104i	$0.401085\pi$
$402$	0	0
$403$	−1.12311	−0.0559459
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.5464	−0.770606
$408$	0	0
$409$	14.4924	0.716604	0.358302	−	0.933606i	$-0.383356\pi$
$409$	14.4924	0.716604	0.358302	+	0.933606i	$0.383356\pi$
$410$	0	0
$411$	0.246211	0.0121447
$412$	0	0
$413$	−20.4924	−1.00837
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5.43845	−0.266322
$418$	0	0
$419$	9.12311	0.445693	0.222846	−	0.974854i	$-0.428465\pi$
$419$	9.12311	0.445693	0.222846	+	0.974854i	$0.428465\pi$
$420$	0	0
$421$	−37.8617	−1.84527	−0.922634	−	0.385676i	$-0.873968\pi$
$421$	−37.8617	−1.84527	−0.922634	+	0.385676i	$0.873968\pi$
$422$	0	0
$423$	1.12311	0.0546073
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.31534	0.208834
$428$	0	0
$429$	1.43845	0.0694489
$430$	0	0
$431$	30.2462	1.45691	0.728454	−	0.685094i	$-0.240239\pi$
$431$	30.2462	1.45691	0.728454	+	0.685094i	$0.240239\pi$
$432$	0	0
$433$	−10.6307	−0.510878	−0.255439	−	0.966825i	$-0.582220\pi$
$433$	−10.6307	−0.510878	−0.255439	+	0.966825i	$0.582220\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.36932	−0.352522
$438$	0	0
$439$	−21.9309	−1.04670	−0.523352	−	0.852117i	$-0.675319\pi$
$439$	−21.9309	−1.04670	−0.523352	+	0.852117i	$0.675319\pi$
$440$	0	0
$441$	−0.438447	−0.0208784
$442$	0	0
$443$	15.6847	0.745201	0.372600	−	0.927992i	$-0.378466\pi$
$443$	15.6847	0.745201	0.372600	+	0.927992i	$0.378466\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−16.5616	−0.783335
$448$	0	0
$449$	−1.68466	−0.0795039	−0.0397520	−	0.999210i	$-0.512657\pi$
$449$	−1.68466	−0.0795039	−0.0397520	+	0.999210i	$0.512657\pi$
$450$	0	0
$451$	13.9309	0.655979
$452$	0	0
$453$	4.00000	0.187936
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.4384	1.47063	0.735314	−	0.677726i	$-0.237035\pi$
$457$	31.4384	1.47063	0.735314	+	0.677726i	$0.237035\pi$
$458$	0	0
$459$	−5.68466	−0.265337
$460$	0	0
$461$	−11.4384	−0.532742	−0.266371	−	0.963871i	$-0.585825\pi$
$461$	−11.4384	−0.532742	−0.266371	+	0.963871i	$0.585825\pi$
$462$	0	0
$463$	10.5616	0.490837	0.245418	−	0.969417i	$-0.421075\pi$
$463$	10.5616	0.490837	0.245418	+	0.969417i	$0.421075\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.1771	0.933684	0.466842	−	0.884341i	$-0.345392\pi$
$467$	20.1771	0.933684	0.466842	+	0.884341i	$0.345392\pi$
$468$	0	0
$469$	−5.75379	−0.265685
$470$	0	0
$471$	22.4924	1.03640
$472$	0	0
$473$	8.98485	0.413124
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.561553	0.0257117
$478$	0	0
$479$	−35.5464	−1.62416	−0.812078	−	0.583549i	$-0.801664\pi$
$479$	−35.5464	−1.62416	−0.812078	+	0.583549i	$0.801664\pi$
$480$	0	0
$481$	−10.8078	−0.492792
$482$	0	0
$483$	3.68466	0.167658
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.68466	0.348225	0.174113	−	0.984726i	$-0.444294\pi$
$487$	7.68466	0.348225	0.174113	+	0.984726i	$0.444294\pi$
$488$	0	0
$489$	−9.43845	−0.426821
$490$	0	0
$491$	−40.4924	−1.82740	−0.913699	−	0.406392i	$-0.866787\pi$
$491$	−40.4924	−1.82740	−0.913699	+	0.406392i	$0.866787\pi$
$492$	0	0
$493$	11.3693	0.512048
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−19.6847	−0.882978
$498$	0	0
$499$	19.8617	0.889134	0.444567	−	0.895746i	$-0.353358\pi$
$499$	19.8617	0.889134	0.444567	+	0.895746i	$0.353358\pi$
$500$	0	0
$501$	22.2462	0.993887
$502$	0	0
$503$	30.7386	1.37057	0.685284	−	0.728276i	$-0.259678\pi$
$503$	30.7386	1.37057	0.685284	+	0.728276i	$0.259678\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−30.3153	−1.34370	−0.671852	−	0.740685i	$-0.734501\pi$
$509$	−30.3153	−1.34370	−0.671852	+	0.740685i	$0.734501\pi$
$510$	0	0
$511$	0.630683	0.0278998
$512$	0	0
$513$	−5.12311	−0.226191
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.61553	−0.0710508
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−30.2462	−1.32257	−0.661287	−	0.750133i	$-0.729990\pi$
$523$	−30.2462	−1.32257	−0.661287	+	0.750133i	$0.729990\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.38447	−0.278112
$528$	0	0
$529$	−20.9309	−0.910038
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	9.68466	0.419489
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−14.8769	−0.641985
$538$	0	0
$539$	0.630683	0.0271654
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	19.9309	0.855315
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−42.7386	−1.82737	−0.913686	−	0.406421i	$-0.866777\pi$
$547$	−42.7386	−1.82737	−0.913686	+	0.406421i	$0.866777\pi$
$548$	0	0
$549$	1.68466	0.0718995
$550$	0	0
$551$	10.2462	0.436503
$552$	0	0
$553$	−22.5616	−0.959415
$554$	0	0
$555$	0	0
$556$	0	0
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	0	0
$559$	6.24621	0.264187
$560$	0	0
$561$	8.17708	0.345237
$562$	0	0
$563$	21.4384	0.903523	0.451761	−	0.892139i	$-0.350796\pi$
$563$	21.4384	0.903523	0.451761	+	0.892139i	$0.350796\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.56155	0.107575
$568$	0	0
$569$	7.12311	0.298616	0.149308	−	0.988791i	$-0.452295\pi$
$569$	7.12311	0.298616	0.149308	+	0.988791i	$0.452295\pi$
$570$	0	0
$571$	21.4384	0.897171	0.448586	−	0.893740i	$-0.351928\pi$
$571$	21.4384	0.897171	0.448586	+	0.893740i	$0.351928\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−33.5464	−1.39655	−0.698277	−	0.715827i	$-0.746050\pi$
$577$	−33.5464	−1.39655	−0.698277	+	0.715827i	$0.746050\pi$
$578$	0	0
$579$	−21.6847	−0.901184
$580$	0	0
$581$	20.4924	0.850169
$582$	0	0
$583$	−0.807764	−0.0334542
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.36932	−0.304164	−0.152082	−	0.988368i	$-0.548598\pi$
$587$	−7.36932	−0.304164	−0.152082	+	0.988368i	$0.548598\pi$
$588$	0	0
$589$	−5.75379	−0.237081
$590$	0	0
$591$	17.3693	0.714478
$592$	0	0
$593$	−35.6155	−1.46255	−0.731277	−	0.682080i	$-0.761075\pi$
$593$	−35.6155	−1.46255	−0.731277	+	0.682080i	$0.761075\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18.2462	0.746768
$598$	0	0
$599$	30.7386	1.25595	0.627973	−	0.778235i	$-0.283885\pi$
$599$	30.7386	1.25595	0.627973	+	0.778235i	$0.283885\pi$
$600$	0	0
$601$	0.561553	0.0229062	0.0114531	−	0.999934i	$-0.496354\pi$
$601$	0.561553	0.0229062	0.0114531	+	0.999934i	$0.496354\pi$
$602$	0	0
$603$	−2.24621	−0.0914728
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22.7386	0.922933	0.461466	−	0.887158i	$-0.347324\pi$
$607$	22.7386	0.922933	0.461466	+	0.887158i	$0.347324\pi$
$608$	0	0
$609$	−5.12311	−0.207599
$610$	0	0
$611$	−1.12311	−0.0454360
$612$	0	0
$613$	23.9309	0.966559	0.483279	−	0.875466i	$-0.339446\pi$
$613$	23.9309	0.966559	0.483279	+	0.875466i	$0.339446\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.2462	0.654048	0.327024	−	0.945016i	$-0.393954\pi$
$617$	16.2462	0.654048	0.327024	+	0.945016i	$0.393954\pi$
$618$	0	0
$619$	31.3693	1.26084	0.630420	−	0.776255i	$-0.282883\pi$
$619$	31.3693	1.26084	0.630420	+	0.776255i	$0.282883\pi$
$620$	0	0
$621$	1.43845	0.0577229
$622$	0	0
$623$	−5.93087	−0.237615
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.36932	0.294302
$628$	0	0
$629$	−61.4384	−2.44971
$630$	0	0
$631$	9.75379	0.388292	0.194146	−	0.980973i	$-0.437806\pi$
$631$	9.75379	0.388292	0.194146	+	0.980973i	$0.437806\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.438447	0.0173719
$638$	0	0
$639$	−7.68466	−0.304000
$640$	0	0
$641$	25.3693	1.00203	0.501014	−	0.865439i	$-0.332961\pi$
$641$	25.3693	1.00203	0.501014	+	0.865439i	$0.332961\pi$
$642$	0	0
$643$	8.80776	0.347344	0.173672	−	0.984804i	$-0.444437\pi$
$643$	8.80776	0.347344	0.173672	+	0.984804i	$0.444437\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.6847	−0.459371	−0.229686	−	0.973265i	$-0.573770\pi$
$647$	−11.6847	−0.459371	−0.229686	+	0.973265i	$0.573770\pi$
$648$	0	0
$649$	11.5076	0.451712
$650$	0	0
$651$	2.87689	0.112754
$652$	0	0
$653$	−40.2462	−1.57496	−0.787478	−	0.616343i	$-0.788614\pi$
$653$	−40.2462	−1.57496	−0.787478	+	0.616343i	$0.788614\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.246211	0.00960562
$658$	0	0
$659$	−24.4924	−0.954089	−0.477045	−	0.878879i	$-0.658292\pi$
$659$	−24.4924	−0.954089	−0.477045	+	0.878879i	$0.658292\pi$
$660$	0	0
$661$	−39.1231	−1.52171	−0.760856	−	0.648920i	$-0.775221\pi$
$661$	−39.1231	−1.52171	−0.760856	+	0.648920i	$0.775221\pi$
$662$	0	0
$663$	5.68466	0.220774
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.87689	−0.111394
$668$	0	0
$669$	−14.2462	−0.550790
$670$	0	0
$671$	−2.42329	−0.0935502
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−35.9309	−1.38094	−0.690468	−	0.723363i	$-0.742595\pi$
$677$	−35.9309	−1.38094	−0.690468	+	0.723363i	$0.742595\pi$
$678$	0	0
$679$	−7.19224	−0.276013
$680$	0	0
$681$	−0.630683	−0.0241678
$682$	0	0
$683$	28.4924	1.09023	0.545116	−	0.838361i	$-0.316486\pi$
$683$	28.4924	1.09023	0.545116	+	0.838361i	$0.316486\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	−0.561553	−0.0213935
$690$	0	0
$691$	37.1231	1.41223	0.706115	−	0.708097i	$-0.250446\pi$
$691$	37.1231	1.41223	0.706115	+	0.708097i	$0.250446\pi$
$692$	0	0
$693$	−3.68466	−0.139969
$694$	0	0
$695$	0	0
$696$	0	0
$697$	55.0540	2.08532
$698$	0	0
$699$	−6.31534	−0.238868
$700$	0	0
$701$	5.36932	0.202796	0.101398	−	0.994846i	$-0.467668\pi$
$701$	5.36932	0.202796	0.101398	+	0.994846i	$0.467668\pi$
$702$	0	0
$703$	−55.3693	−2.08829
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18.2462	−0.686219
$708$	0	0
$709$	8.87689	0.333379	0.166689	−	0.986009i	$-0.446692\pi$
$709$	8.87689	0.333379	0.166689	+	0.986009i	$0.446692\pi$
$710$	0	0
$711$	−8.80776	−0.330317
$712$	0	0
$713$	1.61553	0.0605020
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.6847	−0.585754
$718$	0	0
$719$	26.2462	0.978819	0.489409	−	0.872054i	$-0.337212\pi$
$719$	26.2462	0.978819	0.489409	+	0.872054i	$0.337212\pi$
$720$	0	0
$721$	5.75379	0.214282
$722$	0	0
$723$	28.2462	1.05049
$724$	0	0
$725$	0	0
$726$	0	0
$727$	47.3693	1.75683	0.878415	−	0.477898i	$-0.158601\pi$
$727$	47.3693	1.75683	0.878415	+	0.477898i	$0.158601\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	35.5076	1.31330
$732$	0	0
$733$	−32.4233	−1.19758	−0.598791	−	0.800905i	$-0.704352\pi$
$733$	−32.4233	−1.19758	−0.598791	+	0.800905i	$0.704352\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.23106	0.119017
$738$	0	0
$739$	−27.8617	−1.02491	−0.512455	−	0.858714i	$-0.671264\pi$
$739$	−27.8617	−1.02491	−0.512455	+	0.858714i	$0.671264\pi$
$740$	0	0
$741$	5.12311	0.188202
$742$	0	0
$743$	17.1231	0.628186	0.314093	−	0.949392i	$-0.398300\pi$
$743$	17.1231	0.628186	0.314093	+	0.949392i	$0.398300\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	−12.3153	−0.449993
$750$	0	0
$751$	−19.6847	−0.718303	−0.359152	−	0.933279i	$-0.616934\pi$
$751$	−19.6847	−0.718303	−0.359152	+	0.933279i	$0.616934\pi$
$752$	0	0
$753$	−24.4924	−0.892553
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23.1231	0.840424	0.420212	−	0.907426i	$-0.361956\pi$
$757$	23.1231	0.840424	0.420212	+	0.907426i	$0.361956\pi$
$758$	0	0
$759$	−2.06913	−0.0751047
$760$	0	0
$761$	−10.4924	−0.380350	−0.190175	−	0.981750i	$-0.560906\pi$
$761$	−10.4924	−0.380350	−0.190175	+	0.981750i	$0.560906\pi$
$762$	0	0
$763$	−12.4924	−0.452256
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−15.6155	−0.563110	−0.281555	−	0.959545i	$-0.590850\pi$
$769$	−15.6155	−0.563110	−0.281555	+	0.959545i	$0.590850\pi$
$770$	0	0
$771$	18.4924	0.665988
$772$	0	0
$773$	12.8769	0.463150	0.231575	−	0.972817i	$-0.425612\pi$
$773$	12.8769	0.463150	0.231575	+	0.972817i	$0.425612\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	27.6847	0.993182
$778$	0	0
$779$	49.6155	1.77766
$780$	0	0
$781$	11.0540	0.395542
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−30.7386	−1.09571	−0.547857	−	0.836572i	$-0.684556\pi$
$787$	−30.7386	−1.09571	−0.547857	+	0.836572i	$0.684556\pi$
$788$	0	0
$789$	4.49242	0.159934
$790$	0	0
$791$	−5.12311	−0.182157
$792$	0	0
$793$	−1.68466	−0.0598240
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.4233	0.723430	0.361715	−	0.932289i	$-0.382191\pi$
$797$	20.4233	0.723430	0.361715	+	0.932289i	$0.382191\pi$
$798$	0	0
$799$	−6.38447	−0.225866
$800$	0	0
$801$	−2.31534	−0.0818086
$802$	0	0
$803$	−0.354162	−0.0124981
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.87689	−0.171675
$808$	0	0
$809$	14.4924	0.509526	0.254763	−	0.967003i	$-0.418002\pi$
$809$	14.4924	0.509526	0.254763	+	0.967003i	$0.418002\pi$
$810$	0	0
$811$	−24.9848	−0.877337	−0.438668	−	0.898649i	$-0.644550\pi$
$811$	−24.9848	−0.877337	−0.438668	+	0.898649i	$0.644550\pi$
$812$	0	0
$813$	19.3693	0.679312
$814$	0	0
$815$	0	0
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	−2.56155	−0.0895079
$820$	0	0
$821$	−45.6847	−1.59441	−0.797203	−	0.603712i	$-0.793688\pi$
$821$	−45.6847	−1.59441	−0.797203	+	0.603712i	$0.793688\pi$
$822$	0	0
$823$	44.4924	1.55091	0.775454	−	0.631404i	$-0.217521\pi$
$823$	44.4924	1.55091	0.775454	+	0.631404i	$0.217521\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−47.3693	−1.64719	−0.823596	−	0.567177i	$-0.808036\pi$
$827$	−47.3693	−1.64719	−0.823596	+	0.567177i	$0.808036\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	−10.4924	−0.363978
$832$	0	0
$833$	2.49242	0.0863573
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.12311	0.0388202
$838$	0	0
$839$	−11.1922	−0.386399	−0.193199	−	0.981160i	$-0.561886\pi$
$839$	−11.1922	−0.386399	−0.193199	+	0.981160i	$0.561886\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	20.2462	0.697317
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−22.8769	−0.786059
$848$	0	0
$849$	18.7386	0.643108
$850$	0	0
$851$	15.5464	0.532924
$852$	0	0
$853$	35.4384	1.21339	0.606695	−	0.794935i	$-0.292495\pi$
$853$	35.4384	1.21339	0.606695	+	0.794935i	$0.292495\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−51.7926	−1.76920	−0.884601	−	0.466349i	$-0.845569\pi$
$857$	−51.7926	−1.76920	−0.884601	+	0.466349i	$0.845569\pi$
$858$	0	0
$859$	52.8078	1.80178	0.900889	−	0.434050i	$-0.142916\pi$
$859$	52.8078	1.80178	0.900889	+	0.434050i	$0.142916\pi$
$860$	0	0
$861$	−24.8078	−0.845446
$862$	0	0
$863$	55.8617	1.90156	0.950778	−	0.309873i	$-0.100287\pi$
$863$	55.8617	1.90156	0.950778	+	0.309873i	$0.100287\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15.3153	0.520136
$868$	0	0
$869$	12.6695	0.429783
$870$	0	0
$871$	2.24621	0.0761100
$872$	0	0
$873$	−2.80776	−0.0950284
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	0	0
$879$	−23.6155	−0.796532
$880$	0	0
$881$	−8.24621	−0.277822	−0.138911	−	0.990305i	$-0.544360\pi$
$881$	−8.24621	−0.277822	−0.138911	+	0.990305i	$0.544360\pi$
$882$	0	0
$883$	0.492423	0.0165713	0.00828567	−	0.999966i	$-0.497363\pi$
$883$	0.492423	0.0165713	0.00828567	+	0.999966i	$0.497363\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.0540	0.371156	0.185578	−	0.982630i	$-0.440584\pi$
$887$	11.0540	0.371156	0.185578	+	0.982630i	$0.440584\pi$
$888$	0	0
$889$	−18.8769	−0.633111
$890$	0	0
$891$	−1.43845	−0.0481898
$892$	0	0
$893$	−5.75379	−0.192543
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.43845	−0.0480284
$898$	0	0
$899$	−2.24621	−0.0749153
$900$	0	0
$901$	−3.19224	−0.106349
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38.8769	1.29089	0.645443	−	0.763808i	$-0.276673\pi$
$907$	38.8769	1.29089	0.645443	+	0.763808i	$0.276673\pi$
$908$	0	0
$909$	−7.12311	−0.236259
$910$	0	0
$911$	34.2462	1.13463	0.567314	−	0.823502i	$-0.307983\pi$
$911$	34.2462	1.13463	0.567314	+	0.823502i	$0.307983\pi$
$912$	0	0
$913$	−11.5076	−0.380845
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.6155	−0.581716
$918$	0	0
$919$	−32.8078	−1.08223	−0.541114	−	0.840949i	$-0.681997\pi$
$919$	−32.8078	−1.08223	−0.541114	+	0.840949i	$0.681997\pi$
$920$	0	0
$921$	−9.43845	−0.311007
$922$	0	0
$923$	7.68466	0.252944
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.24621	0.0737753
$928$	0	0
$929$	−35.3002	−1.15816	−0.579081	−	0.815270i	$-0.696588\pi$
$929$	−35.3002	−1.15816	−0.579081	+	0.815270i	$0.696588\pi$
$930$	0	0
$931$	2.24621	0.0736166
$932$	0	0
$933$	22.7386	0.744429
$934$	0	0
$935$	0	0
$936$	0	0
$937$	48.2462	1.57614	0.788068	−	0.615589i	$-0.211082\pi$
$937$	48.2462	1.57614	0.788068	+	0.615589i	$0.211082\pi$
$938$	0	0
$939$	−24.7386	−0.807315
$940$	0	0
$941$	−49.5464	−1.61517	−0.807583	−	0.589754i	$-0.799225\pi$
$941$	−49.5464	−1.61517	−0.807583	+	0.589754i	$0.799225\pi$
$942$	0	0
$943$	−13.9309	−0.453652
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.26137	0.0409889	0.0204944	−	0.999790i	$-0.493476\pi$
$947$	1.26137	0.0409889	0.0204944	+	0.999790i	$0.493476\pi$
$948$	0	0
$949$	−0.246211	−0.00799236
$950$	0	0
$951$	−11.1231	−0.360691
$952$	0	0
$953$	−9.19224	−0.297766	−0.148883	−	0.988855i	$-0.547568\pi$
$953$	−9.19224	−0.297766	−0.148883	+	0.988855i	$0.547568\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.87689	0.0929968
$958$	0	0
$959$	0.630683	0.0203658
$960$	0	0
$961$	−29.7386	−0.959311
$962$	0	0
$963$	−4.80776	−0.154928
$964$	0	0
$965$	0	0
$966$	0	0
$967$	25.7538	0.828186	0.414093	−	0.910235i	$-0.364099\pi$
$967$	25.7538	0.828186	0.414093	+	0.910235i	$0.364099\pi$
$968$	0	0
$969$	29.1231	0.935569
$970$	0	0
$971$	24.4924	0.785999	0.393000	−	0.919539i	$-0.371437\pi$
$971$	24.4924	0.785999	0.393000	+	0.919539i	$0.371437\pi$
$972$	0	0
$973$	−13.9309	−0.446603
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.8617	−0.443476	−0.221738	−	0.975106i	$-0.571173\pi$
$977$	−13.8617	−0.443476	−0.221738	+	0.975106i	$0.571173\pi$
$978$	0	0
$979$	3.33050	0.106443
$980$	0	0
$981$	−4.87689	−0.155707
$982$	0	0
$983$	−12.9848	−0.414152	−0.207076	−	0.978325i	$-0.566395\pi$
$983$	−12.9848	−0.414152	−0.207076	+	0.978325i	$0.566395\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.87689	0.0915726
$988$	0	0
$989$	−8.98485	−0.285701
$990$	0	0
$991$	3.05398	0.0970127	0.0485064	−	0.998823i	$-0.484554\pi$
$991$	3.05398	0.0970127	0.0485064	+	0.998823i	$0.484554\pi$
$992$	0	0
$993$	26.2462	0.832898
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−19.1231	−0.605635	−0.302817	−	0.953049i	$-0.597927\pi$
$997$	−19.1231	−0.605635	−0.302817	+	0.953049i	$0.597927\pi$
$998$	0	0
$999$	10.8078	0.341943

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bd.1.2		2
5.4	even	2		1560.2.a.o.1.1	✓	2
15.14	odd	2		4680.2.a.y.1.1		2
20.19	odd	2		3120.2.a.bg.1.2		2
60.59	even	2		9360.2.a.ci.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.o.1.1	✓	2	5.4	even	2
3120.2.a.bg.1.2		2	20.19	odd	2
4680.2.a.y.1.1		2	15.14	odd	2
7800.2.a.bd.1.2		2	1.1	even	1	trivial
9360.2.a.ci.1.2		2	60.59	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.56155 q^{7} +1.00000 q^{9} -1.43845 q^{11} -1.00000 q^{13} -5.68466 q^{17} -5.12311 q^{19} +2.56155 q^{21} +1.43845 q^{23} +1.00000 q^{27} -2.00000 q^{29} +1.12311 q^{31} -1.43845 q^{33} +10.8078 q^{37} -1.00000 q^{39} -9.68466 q^{41} -6.24621 q^{43} +1.12311 q^{47} -0.438447 q^{49} -5.68466 q^{51} +0.561553 q^{53} -5.12311 q^{57} -8.00000 q^{59} +1.68466 q^{61} +2.56155 q^{63} -2.24621 q^{67} +1.43845 q^{69} -7.68466 q^{71} +0.246211 q^{73} -3.68466 q^{77} -8.80776 q^{79} +1.00000 q^{81} +8.00000 q^{83} -2.00000 q^{87} -2.31534 q^{89} -2.56155 q^{91} +1.12311 q^{93} -2.80776 q^{97} -1.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + q^7 + 2 * q^9	\( 2 q + 2 q^{3} + q^{7} + 2 q^{9} - 7 q^{11} - 2 q^{13} + q^{17} - 2 q^{19} + q^{21} + 7 q^{23} + 2 q^{27} - 4 q^{29} - 6 q^{31} - 7 q^{33} + q^{37} - 2 q^{39} - 7 q^{41} + 4 q^{43} - 6 q^{47} - 5 q^{49} + q^{51} - 3 q^{53} - 2 q^{57} - 16 q^{59} - 9 q^{61} + q^{63} + 12 q^{67} + 7 q^{69} - 3 q^{71} - 16 q^{73} + 5 q^{77} + 3 q^{79} + 2 q^{81} + 16 q^{83} - 4 q^{87} - 17 q^{89} - q^{91} - 6 q^{93} + 15 q^{97} - 7 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^7 + 2 * q^9 - 7 * q^11 - 2 * q^13 + q^17 - 2 * q^19 + q^21 + 7 * q^23 + 2 * q^27 - 4 * q^29 - 6 * q^31 - 7 * q^33 + q^37 - 2 * q^39 - 7 * q^41 + 4 * q^43 - 6 * q^47 - 5 * q^49 + q^51 - 3 * q^53 - 2 * q^57 - 16 * q^59 - 9 * q^61 + q^63 + 12 * q^67 + 7 * q^69 - 3 * q^71 - 16 * q^73 + 5 * q^77 + 3 * q^79 + 2 * q^81 + 16 * q^83 - 4 * q^87 - 17 * q^89 - q^91 - 6 * q^93 + 15 * q^97 - 7 * q^99

Introduction

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