LMFDB - Embedded newform 8256.2.a.cu.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8256,2,Mod(1,8256)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8256.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8256 = 2^{6} \cdot 3 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8256.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,4,0,-4,0,3,0,1,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$65.9244919088$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 129)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	8256.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.12489 q^{5} -1.48486 q^{7} +1.00000 q^{9} -2.60975 q^{11} -3.00000 q^{13} +4.12489 q^{15} +0.484862 q^{17} -6.76491 q^{19} -1.48486 q^{21} -8.79518 q^{23} +12.0147 q^{25} +1.00000 q^{27} +2.12489 q^{29} -3.76491 q^{31} -2.60975 q^{33} -6.12489 q^{35} -5.28005 q^{37} -3.00000 q^{39} +0.734633 q^{41} -1.00000 q^{43} +4.12489 q^{45} +8.43521 q^{47} -4.79518 q^{49} +0.484862 q^{51} -3.76491 q^{53} -10.7649 q^{55} -6.76491 q^{57} +2.96972 q^{59} +11.2195 q^{61} -1.48486 q^{63} -12.3747 q^{65} +7.04496 q^{67} -8.79518 q^{69} -13.2195 q^{71} -5.03028 q^{73} +12.0147 q^{75} +3.87511 q^{77} -2.71995 q^{79} +1.00000 q^{81} -5.57947 q^{83} +2.00000 q^{85} +2.12489 q^{87} -18.2498 q^{89} +4.45459 q^{91} -3.76491 q^{93} -27.9045 q^{95} +6.21949 q^{97} -2.60975 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 4 q^{5} - 4 q^{7} + 3 q^{9} + q^{11} - 9 q^{13} + 4 q^{15} + q^{17} - 4 q^{19} - 4 q^{21} - 11 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} + 5 q^{31} + q^{33} - 10 q^{35} - 9 q^{39} - 15 q^{41}+ \cdots + q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 4 * q^5 - 4 * q^7 + 3 * q^9 + q^11 - 9 * q^13 + 4 * q^15 + q^17 - 4 * q^19 - 4 * q^21 - 11 * q^23 + 3 * q^25 + 3 * q^27 - 2 * q^29 + 5 * q^31 + q^33 - 10 * q^35 - 9 * q^39 - 15 * q^41 - 3 * q^43 + 4 * q^45 + 2 * q^47 + q^49 + q^51 + 5 * q^53 - 16 * q^55 - 4 * q^57 + 8 * q^59 + 16 * q^61 - 4 * q^63 - 12 * q^65 - 11 * q^67 - 11 * q^69 - 22 * q^71 - 16 * q^73 + 3 * q^75 + 20 * q^77 - 24 * q^79 + 3 * q^81 - 7 * q^83 + 6 * q^85 - 2 * q^87 - 38 * q^89 + 12 * q^91 + 5 * q^93 - 26 * q^95 + q^97 + q^99

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
$5$	4.12489	1.84470	0.922352	−	0.386350i	$-0.126264\pi$
$5$	4.12489	1.84470	0.922352	+	0.386350i	$0.126264\pi$
$6$	0	0
$7$	−1.48486	−0.561225	−0.280613	−	0.959821i	$-0.590538\pi$
$7$	−1.48486	−0.561225	−0.280613	+	0.959821i	$0.590538\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.60975	−0.786868	−0.393434	−	0.919353i	$-0.628713\pi$
$11$	−2.60975	−0.786868	−0.393434	+	0.919353i	$0.628713\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	4.12489	1.06504
$16$	0	0
$17$	0.484862	0.117596	0.0587981	−	0.998270i	$-0.481273\pi$
$17$	0.484862	0.117596	0.0587981	+	0.998270i	$0.481273\pi$
$18$	0	0
$19$	−6.76491	−1.55198	−0.775988	−	0.630747i	$-0.782748\pi$
$19$	−6.76491	−1.55198	−0.775988	+	0.630747i	$0.782748\pi$
$20$	0	0
$21$	−1.48486	−0.324023
$22$	0	0
$23$	−8.79518	−1.83392	−0.916961	−	0.398976i	$-0.869366\pi$
$23$	−8.79518	−1.83392	−0.916961	+	0.398976i	$0.869366\pi$
$24$	0	0
$25$	12.0147	2.40294
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.12489	0.394581	0.197291	−	0.980345i	$-0.436786\pi$
$29$	2.12489	0.394581	0.197291	+	0.980345i	$0.436786\pi$
$30$	0	0
$31$	−3.76491	−0.676198	−0.338099	−	0.941111i	$-0.609784\pi$
$31$	−3.76491	−0.676198	−0.338099	+	0.941111i	$0.609784\pi$
$32$	0	0
$33$	−2.60975	−0.454299
$34$	0	0
$35$	−6.12489	−1.03529
$36$	0	0
$37$	−5.28005	−0.868034	−0.434017	−	0.900905i	$-0.642904\pi$
$37$	−5.28005	−0.868034	−0.434017	+	0.900905i	$0.642904\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	0.734633	0.114730	0.0573652	−	0.998353i	$-0.481730\pi$
$41$	0.734633	0.114730	0.0573652	+	0.998353i	$0.481730\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	4.12489	0.614902
$46$	0	0
$47$	8.43521	1.23040	0.615201	−	0.788370i	$-0.289075\pi$
$47$	8.43521	1.23040	0.615201	+	0.788370i	$0.289075\pi$
$48$	0	0
$49$	−4.79518	−0.685026
$50$	0	0
$51$	0.484862	0.0678943
$52$	0	0
$53$	−3.76491	−0.517150	−0.258575	−	0.965991i	$-0.583253\pi$
$53$	−3.76491	−0.517150	−0.258575	+	0.965991i	$0.583253\pi$
$54$	0	0
$55$	−10.7649	−1.45154
$56$	0	0
$57$	−6.76491	−0.896034
$58$	0	0
$59$	2.96972	0.386625	0.193313	−	0.981137i	$-0.438077\pi$
$59$	2.96972	0.386625	0.193313	+	0.981137i	$0.438077\pi$
$60$	0	0
$61$	11.2195	1.43651	0.718255	−	0.695780i	$-0.244941\pi$
$61$	11.2195	1.43651	0.718255	+	0.695780i	$0.244941\pi$
$62$	0	0
$63$	−1.48486	−0.187075
$64$	0	0
$65$	−12.3747	−1.53489
$66$	0	0
$67$	7.04496	0.860678	0.430339	−	0.902667i	$-0.358394\pi$
$67$	7.04496	0.860678	0.430339	+	0.902667i	$0.358394\pi$
$68$	0	0
$69$	−8.79518	−1.05882
$70$	0	0
$71$	−13.2195	−1.56887	−0.784433	−	0.620214i	$-0.787046\pi$
$71$	−13.2195	−1.56887	−0.784433	+	0.620214i	$0.787046\pi$
$72$	0	0
$73$	−5.03028	−0.588749	−0.294375	−	0.955690i	$-0.595111\pi$
$73$	−5.03028	−0.588749	−0.294375	+	0.955690i	$0.595111\pi$
$74$	0	0
$75$	12.0147	1.38734
$76$	0	0
$77$	3.87511	0.441610
$78$	0	0
$79$	−2.71995	−0.306019	−0.153009	−	0.988225i	$-0.548896\pi$
$79$	−2.71995	−0.306019	−0.153009	+	0.988225i	$0.548896\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.57947	−0.612427	−0.306213	−	0.951963i	$-0.599062\pi$
$83$	−5.57947	−0.612427	−0.306213	+	0.951963i	$0.599062\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	2.12489	0.227812
$88$	0	0
$89$	−18.2498	−1.93447	−0.967236	−	0.253879i	$-0.918293\pi$
$89$	−18.2498	−1.93447	−0.967236	+	0.253879i	$0.918293\pi$
$90$	0	0
$91$	4.45459	0.466967
$92$	0	0
$93$	−3.76491	−0.390403
$94$	0	0
$95$	−27.9045	−2.86294
$96$	0	0
$97$	6.21949	0.631494	0.315747	−	0.948843i	$-0.397745\pi$
$97$	6.21949	0.631494	0.315747	+	0.948843i	$0.397745\pi$
$98$	0	0
$99$	−2.60975	−0.262289
$100$	0	0
$101$	−7.76491	−0.772637	−0.386319	−	0.922365i	$-0.626253\pi$
$101$	−7.76491	−0.772637	−0.386319	+	0.922365i	$0.626253\pi$
$102$	0	0
$103$	−2.54541	−0.250807	−0.125404	−	0.992106i	$-0.540023\pi$
$103$	−2.54541	−0.250807	−0.125404	+	0.992106i	$0.540023\pi$
$104$	0	0
$105$	−6.12489	−0.597728
$106$	0	0
$107$	−3.15516	−0.305021	−0.152510	−	0.988302i	$-0.548736\pi$
$107$	−3.15516	−0.305021	−0.152510	+	0.988302i	$0.548736\pi$
$108$	0	0
$109$	9.24977	0.885967	0.442984	−	0.896530i	$-0.353920\pi$
$109$	9.24977	0.885967	0.442984	+	0.896530i	$0.353920\pi$
$110$	0	0
$111$	−5.28005	−0.501160
$112$	0	0
$113$	3.40493	0.320309	0.160155	−	0.987092i	$-0.448801\pi$
$113$	3.40493	0.320309	0.160155	+	0.987092i	$0.448801\pi$
$114$	0	0
$115$	−36.2791	−3.38305
$116$	0	0
$117$	−3.00000	−0.277350
$118$	0	0
$119$	−0.719953	−0.0659980
$120$	0	0
$121$	−4.18922	−0.380838
$122$	0	0
$123$	0.734633	0.0662396
$124$	0	0
$125$	28.9348	2.58800
$126$	0	0
$127$	9.04496	0.802610	0.401305	−	0.915944i	$-0.368557\pi$
$127$	9.04496	0.802610	0.401305	+	0.915944i	$0.368557\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	19.0596	1.66525	0.832624	−	0.553839i	$-0.186838\pi$
$131$	19.0596	1.66525	0.832624	+	0.553839i	$0.186838\pi$
$132$	0	0
$133$	10.0450	0.871008
$134$	0	0
$135$	4.12489	0.355014
$136$	0	0
$137$	−20.8742	−1.78340	−0.891702	−	0.452624i	$-0.850488\pi$
$137$	−20.8742	−1.78340	−0.891702	+	0.452624i	$0.850488\pi$
$138$	0	0
$139$	−13.2342	−1.12251	−0.561254	−	0.827644i	$-0.689681\pi$
$139$	−13.2342	−1.12251	−0.561254	+	0.827644i	$0.689681\pi$
$140$	0	0
$141$	8.43521	0.710373
$142$	0	0
$143$	7.82924	0.654714
$144$	0	0
$145$	8.76491	0.727886
$146$	0	0
$147$	−4.79518	−0.395500
$148$	0	0
$149$	11.2195	0.919137	0.459568	−	0.888142i	$-0.348004\pi$
$149$	11.2195	0.919137	0.459568	+	0.888142i	$0.348004\pi$
$150$	0	0
$151$	−7.52982	−0.612768	−0.306384	−	0.951908i	$-0.599119\pi$
$151$	−7.52982	−0.612768	−0.306384	+	0.951908i	$0.599119\pi$
$152$	0	0
$153$	0.484862	0.0391988
$154$	0	0
$155$	−15.5298	−1.24738
$156$	0	0
$157$	−24.8099	−1.98004	−0.990021	−	0.140917i	$-0.954995\pi$
$157$	−24.8099	−1.98004	−0.990021	+	0.140917i	$0.954995\pi$
$158$	0	0
$159$	−3.76491	−0.298577
$160$	0	0
$161$	13.0596	1.02924
$162$	0	0
$163$	−1.42431	−0.111561	−0.0557803	−	0.998443i	$-0.517765\pi$
$163$	−1.42431	−0.111561	−0.0557803	+	0.998443i	$0.517765\pi$
$164$	0	0
$165$	−10.7649	−0.838047
$166$	0	0
$167$	−8.35998	−0.646914	−0.323457	−	0.946243i	$-0.604845\pi$
$167$	−8.35998	−0.646914	−0.323457	+	0.946243i	$0.604845\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−6.76491	−0.517326
$172$	0	0
$173$	−12.4390	−0.945719	−0.472859	−	0.881138i	$-0.656778\pi$
$173$	−12.4390	−0.945719	−0.472859	+	0.881138i	$0.656778\pi$
$174$	0	0
$175$	−17.8401	−1.34859
$176$	0	0
$177$	2.96972	0.223218
$178$	0	0
$179$	2.24977	0.168156	0.0840779	−	0.996459i	$-0.473206\pi$
$179$	2.24977	0.168156	0.0840779	+	0.996459i	$0.473206\pi$
$180$	0	0
$181$	6.88601	0.511833	0.255917	−	0.966699i	$-0.417623\pi$
$181$	6.88601	0.511833	0.255917	+	0.966699i	$0.417623\pi$
$182$	0	0
$183$	11.2195	0.829369
$184$	0	0
$185$	−21.7796	−1.60127
$186$	0	0
$187$	−1.26537	−0.0925328
$188$	0	0
$189$	−1.48486	−0.108008
$190$	0	0
$191$	−6.47018	−0.468166	−0.234083	−	0.972217i	$-0.575209\pi$
$191$	−6.47018	−0.468166	−0.234083	+	0.972217i	$0.575209\pi$
$192$	0	0
$193$	2.23509	0.160885	0.0804427	−	0.996759i	$-0.474367\pi$
$193$	2.23509	0.160885	0.0804427	+	0.996759i	$0.474367\pi$
$194$	0	0
$195$	−12.3747	−0.886168
$196$	0	0
$197$	−7.77959	−0.554273	−0.277136	−	0.960831i	$-0.589385\pi$
$197$	−7.77959	−0.554273	−0.277136	+	0.960831i	$0.589385\pi$
$198$	0	0
$199$	−6.90917	−0.489778	−0.244889	−	0.969551i	$-0.578752\pi$
$199$	−6.90917	−0.489778	−0.244889	+	0.969551i	$0.578752\pi$
$200$	0	0
$201$	7.04496	0.496913
$202$	0	0
$203$	−3.15516	−0.221449
$204$	0	0
$205$	3.03028	0.211644
$206$	0	0
$207$	−8.79518	−0.611308
$208$	0	0
$209$	17.6547	1.22120
$210$	0	0
$211$	3.42431	0.235739	0.117870	−	0.993029i	$-0.462394\pi$
$211$	3.42431	0.235739	0.117870	+	0.993029i	$0.462394\pi$
$212$	0	0
$213$	−13.2195	−0.905785
$214$	0	0
$215$	−4.12489	−0.281315
$216$	0	0
$217$	5.59037	0.379499
$218$	0	0
$219$	−5.03028	−0.339915
$220$	0	0
$221$	−1.45459	−0.0978460
$222$	0	0
$223$	9.73463	0.651879	0.325940	−	0.945391i	$-0.394319\pi$
$223$	9.73463	0.651879	0.325940	+	0.945391i	$0.394319\pi$
$224$	0	0
$225$	12.0147	0.800979
$226$	0	0
$227$	3.09083	0.205145	0.102573	−	0.994726i	$-0.467293\pi$
$227$	3.09083	0.205145	0.102573	+	0.994726i	$0.467293\pi$
$228$	0	0
$229$	−3.20482	−0.211780	−0.105890	−	0.994378i	$-0.533769\pi$
$229$	−3.20482	−0.211780	−0.105890	+	0.994378i	$0.533769\pi$
$230$	0	0
$231$	3.87511	0.254964
$232$	0	0
$233$	11.0946	0.726832	0.363416	−	0.931627i	$-0.381610\pi$
$233$	11.0946	0.726832	0.363416	+	0.931627i	$0.381610\pi$
$234$	0	0
$235$	34.7943	2.26973
$236$	0	0
$237$	−2.71995	−0.176680
$238$	0	0
$239$	−17.0946	−1.10576	−0.552879	−	0.833261i	$-0.686471\pi$
$239$	−17.0946	−1.10576	−0.552879	+	0.833261i	$0.686471\pi$
$240$	0	0
$241$	22.5601	1.45322	0.726612	−	0.687048i	$-0.241094\pi$
$241$	22.5601	1.45322	0.726612	+	0.687048i	$0.241094\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−19.7796	−1.26367
$246$	0	0
$247$	20.2947	1.29132
$248$	0	0
$249$	−5.57947	−0.353585
$250$	0	0
$251$	15.6400	0.987190	0.493595	−	0.869692i	$-0.335683\pi$
$251$	15.6400	0.987190	0.493595	+	0.869692i	$0.335683\pi$
$252$	0	0
$253$	22.9532	1.44306
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	−17.5336	−1.09372	−0.546858	−	0.837225i	$-0.684176\pi$
$257$	−17.5336	−1.09372	−0.546858	+	0.837225i	$0.684176\pi$
$258$	0	0
$259$	7.84014	0.487163
$260$	0	0
$261$	2.12489	0.131527
$262$	0	0
$263$	6.37088	0.392845	0.196422	−	0.980519i	$-0.437068\pi$
$263$	6.37088	0.392845	0.196422	+	0.980519i	$0.437068\pi$
$264$	0	0
$265$	−15.5298	−0.953989
$266$	0	0
$267$	−18.2498	−1.11687
$268$	0	0
$269$	−29.5445	−1.80136	−0.900680	−	0.434483i	$-0.856931\pi$
$269$	−29.5445	−1.80136	−0.900680	+	0.434483i	$0.856931\pi$
$270$	0	0
$271$	11.8255	0.718346	0.359173	−	0.933271i	$-0.383059\pi$
$271$	11.8255	0.718346	0.359173	+	0.933271i	$0.383059\pi$
$272$	0	0
$273$	4.45459	0.269604
$274$	0	0
$275$	−31.3553	−1.89079
$276$	0	0
$277$	15.7796	0.948104	0.474052	−	0.880497i	$-0.342791\pi$
$277$	15.7796	0.948104	0.474052	+	0.880497i	$0.342791\pi$
$278$	0	0
$279$	−3.76491	−0.225399
$280$	0	0
$281$	−18.0147	−1.07467	−0.537333	−	0.843370i	$-0.680568\pi$
$281$	−18.0147	−1.07467	−0.537333	+	0.843370i	$0.680568\pi$
$282$	0	0
$283$	5.64380	0.335489	0.167745	−	0.985830i	$-0.446352\pi$
$283$	5.64380	0.335489	0.167745	+	0.985830i	$0.446352\pi$
$284$	0	0
$285$	−27.9045	−1.65292
$286$	0	0
$287$	−1.09083	−0.0643896
$288$	0	0
$289$	−16.7649	−0.986171
$290$	0	0
$291$	6.21949	0.364593
$292$	0	0
$293$	−1.34060	−0.0783186	−0.0391593	−	0.999233i	$-0.512468\pi$
$293$	−1.34060	−0.0783186	−0.0391593	+	0.999233i	$0.512468\pi$
$294$	0	0
$295$	12.2498	0.713209
$296$	0	0
$297$	−2.60975	−0.151433
$298$	0	0
$299$	26.3856	1.52592
$300$	0	0
$301$	1.48486	0.0855860
$302$	0	0
$303$	−7.76491	−0.446082
$304$	0	0
$305$	46.2791	2.64994
$306$	0	0
$307$	−0.424310	−0.0242166	−0.0121083	−	0.999927i	$-0.503854\pi$
$307$	−0.424310	−0.0242166	−0.0121083	+	0.999927i	$0.503854\pi$
$308$	0	0
$309$	−2.54541	−0.144804
$310$	0	0
$311$	−0.110206	−0.00624919	−0.00312460	−	0.999995i	$-0.500995\pi$
$311$	−0.110206	−0.00624919	−0.00312460	+	0.999995i	$0.500995\pi$
$312$	0	0
$313$	−2.96972	−0.167859	−0.0839294	−	0.996472i	$-0.526747\pi$
$313$	−2.96972	−0.167859	−0.0839294	+	0.996472i	$0.526747\pi$
$314$	0	0
$315$	−6.12489	−0.345098
$316$	0	0
$317$	12.4243	0.697819	0.348909	−	0.937156i	$-0.386552\pi$
$317$	12.4243	0.697819	0.348909	+	0.937156i	$0.386552\pi$
$318$	0	0
$319$	−5.54541	−0.310484
$320$	0	0
$321$	−3.15516	−0.176104
$322$	0	0
$323$	−3.28005	−0.182507
$324$	0	0
$325$	−36.0440	−1.99936
$326$	0	0
$327$	9.24977	0.511513
$328$	0	0
$329$	−12.5251	−0.690532
$330$	0	0
$331$	31.2938	1.72006	0.860032	−	0.510241i	$-0.170444\pi$
$331$	31.2938	1.72006	0.860032	+	0.510241i	$0.170444\pi$
$332$	0	0
$333$	−5.28005	−0.289345
$334$	0	0
$335$	29.0596	1.58770
$336$	0	0
$337$	−14.8401	−0.808394	−0.404197	−	0.914672i	$-0.632449\pi$
$337$	−14.8401	−0.808394	−0.404197	+	0.914672i	$0.632449\pi$
$338$	0	0
$339$	3.40493	0.184931
$340$	0	0
$341$	9.82546	0.532079
$342$	0	0
$343$	17.5142	0.945679
$344$	0	0
$345$	−36.2791	−1.95320
$346$	0	0
$347$	17.0984	0.917890	0.458945	−	0.888465i	$-0.348228\pi$
$347$	17.0984	0.917890	0.458945	+	0.888465i	$0.348228\pi$
$348$	0	0
$349$	12.3784	0.662603	0.331301	−	0.943525i	$-0.392512\pi$
$349$	12.3784	0.662603	0.331301	+	0.943525i	$0.392512\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	0	0
$353$	15.4546	0.822565	0.411282	−	0.911508i	$-0.365081\pi$
$353$	15.4546	0.822565	0.411282	+	0.911508i	$0.365081\pi$
$354$	0	0
$355$	−54.5289	−2.89409
$356$	0	0
$357$	−0.719953	−0.0381040
$358$	0	0
$359$	−15.3553	−0.810421	−0.405210	−	0.914223i	$-0.632802\pi$
$359$	−15.3553	−0.810421	−0.405210	+	0.914223i	$0.632802\pi$
$360$	0	0
$361$	26.7640	1.40863
$362$	0	0
$363$	−4.18922	−0.219877
$364$	0	0
$365$	−20.7493	−1.08607
$366$	0	0
$367$	0.530734	0.0277041	0.0138521	−	0.999904i	$-0.495591\pi$
$367$	0.530734	0.0277041	0.0138521	+	0.999904i	$0.495591\pi$
$368$	0	0
$369$	0.734633	0.0382435
$370$	0	0
$371$	5.59037	0.290238
$372$	0	0
$373$	6.02936	0.312188	0.156094	−	0.987742i	$-0.450110\pi$
$373$	6.02936	0.312188	0.156094	+	0.987742i	$0.450110\pi$
$374$	0	0
$375$	28.9348	1.49418
$376$	0	0
$377$	−6.37466	−0.328312
$378$	0	0
$379$	11.8936	0.610932	0.305466	−	0.952203i	$-0.401188\pi$
$379$	11.8936	0.610932	0.305466	+	0.952203i	$0.401188\pi$
$380$	0	0
$381$	9.04496	0.463387
$382$	0	0
$383$	−17.7796	−0.908495	−0.454247	−	0.890876i	$-0.650092\pi$
$383$	−17.7796	−0.908495	−0.454247	+	0.890876i	$0.650092\pi$
$384$	0	0
$385$	15.9844	0.814641
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	24.4646	1.24040	0.620201	−	0.784443i	$-0.287051\pi$
$389$	24.4646	1.24040	0.620201	+	0.784443i	$0.287051\pi$
$390$	0	0
$391$	−4.26445	−0.215663
$392$	0	0
$393$	19.0596	0.961431
$394$	0	0
$395$	−11.2195	−0.564514
$396$	0	0
$397$	38.5739	1.93597	0.967983	−	0.251015i	$-0.0807644\pi$
$397$	38.5739	1.93597	0.967983	+	0.251015i	$0.0807644\pi$
$398$	0	0
$399$	10.0450	0.502877
$400$	0	0
$401$	−4.51422	−0.225429	−0.112715	−	0.993627i	$-0.535955\pi$
$401$	−4.51422	−0.225429	−0.112715	+	0.993627i	$0.535955\pi$
$402$	0	0
$403$	11.2947	0.562630
$404$	0	0
$405$	4.12489	0.204967
$406$	0	0
$407$	13.7796	0.683029
$408$	0	0
$409$	17.7190	0.876150	0.438075	−	0.898938i	$-0.355660\pi$
$409$	17.7190	0.876150	0.438075	+	0.898938i	$0.355660\pi$
$410$	0	0
$411$	−20.8742	−1.02965
$412$	0	0
$413$	−4.40963	−0.216984
$414$	0	0
$415$	−23.0147	−1.12975
$416$	0	0
$417$	−13.2342	−0.648080
$418$	0	0
$419$	16.4390	0.803097	0.401549	−	0.915838i	$-0.368472\pi$
$419$	16.4390	0.803097	0.401549	+	0.915838i	$0.368472\pi$
$420$	0	0
$421$	−23.9083	−1.16522	−0.582609	−	0.812753i	$-0.697968\pi$
$421$	−23.9083	−1.16522	−0.582609	+	0.812753i	$0.697968\pi$
$422$	0	0
$423$	8.43521	0.410134
$424$	0	0
$425$	5.82546	0.282576
$426$	0	0
$427$	−16.6594	−0.806205
$428$	0	0
$429$	7.82924	0.377999
$430$	0	0
$431$	15.6088	0.751851	0.375925	−	0.926650i	$-0.377325\pi$
$431$	15.6088	0.751851	0.375925	+	0.926650i	$0.377325\pi$
$432$	0	0
$433$	−10.9092	−0.524261	−0.262131	−	0.965032i	$-0.584425\pi$
$433$	−10.9092	−0.524261	−0.262131	+	0.965032i	$0.584425\pi$
$434$	0	0
$435$	8.76491	0.420245
$436$	0	0
$437$	59.4986	2.84621
$438$	0	0
$439$	−19.8255	−0.946218	−0.473109	−	0.881004i	$-0.656868\pi$
$439$	−19.8255	−0.946218	−0.473109	+	0.881004i	$0.656868\pi$
$440$	0	0
$441$	−4.79518	−0.228342
$442$	0	0
$443$	−0.314104	−0.0149235	−0.00746177	−	0.999972i	$-0.502375\pi$
$443$	−0.314104	−0.0149235	−0.00746177	+	0.999972i	$0.502375\pi$
$444$	0	0
$445$	−75.2782	−3.56853
$446$	0	0
$447$	11.2195	0.530664
$448$	0	0
$449$	10.2536	0.483895	0.241948	−	0.970289i	$-0.422214\pi$
$449$	10.2536	0.483895	0.241948	+	0.970289i	$0.422214\pi$
$450$	0	0
$451$	−1.91721	−0.0902777
$452$	0	0
$453$	−7.52982	−0.353782
$454$	0	0
$455$	18.3747	0.861417
$456$	0	0
$457$	3.21949	0.150602	0.0753008	−	0.997161i	$-0.476008\pi$
$457$	3.21949	0.150602	0.0753008	+	0.997161i	$0.476008\pi$
$458$	0	0
$459$	0.484862	0.0226314
$460$	0	0
$461$	16.1892	0.754007	0.377004	−	0.926212i	$-0.376954\pi$
$461$	16.1892	0.754007	0.377004	+	0.926212i	$0.376954\pi$
$462$	0	0
$463$	−14.8860	−0.691812	−0.345906	−	0.938269i	$-0.612428\pi$
$463$	−14.8860	−0.691812	−0.345906	+	0.938269i	$0.612428\pi$
$464$	0	0
$465$	−15.5298	−0.720178
$466$	0	0
$467$	−8.22041	−0.380395	−0.190198	−	0.981746i	$-0.560913\pi$
$467$	−8.22041	−0.380395	−0.190198	+	0.981746i	$0.560913\pi$
$468$	0	0
$469$	−10.4608	−0.483034
$470$	0	0
$471$	−24.8099	−1.14318
$472$	0	0
$473$	2.60975	0.119996
$474$	0	0
$475$	−81.2782	−3.72930
$476$	0	0
$477$	−3.76491	−0.172383
$478$	0	0
$479$	−9.41961	−0.430393	−0.215197	−	0.976571i	$-0.569039\pi$
$479$	−9.41961	−0.430393	−0.215197	+	0.976571i	$0.569039\pi$
$480$	0	0
$481$	15.8401	0.722248
$482$	0	0
$483$	13.0596	0.594234
$484$	0	0
$485$	25.6547	1.16492
$486$	0	0
$487$	6.71995	0.304510	0.152255	−	0.988341i	$-0.451346\pi$
$487$	6.71995	0.304510	0.152255	+	0.988341i	$0.451346\pi$
$488$	0	0
$489$	−1.42431	−0.0644095
$490$	0	0
$491$	−36.2186	−1.63452	−0.817261	−	0.576268i	$-0.804508\pi$
$491$	−36.2186	−1.63452	−0.817261	+	0.576268i	$0.804508\pi$
$492$	0	0
$493$	1.03028	0.0464013
$494$	0	0
$495$	−10.7649	−0.483847
$496$	0	0
$497$	19.6291	0.880487
$498$	0	0
$499$	−29.8245	−1.33513	−0.667565	−	0.744552i	$-0.732663\pi$
$499$	−29.8245	−1.33513	−0.667565	+	0.744552i	$0.732663\pi$
$500$	0	0
$501$	−8.35998	−0.373496
$502$	0	0
$503$	22.2498	0.992068	0.496034	−	0.868303i	$-0.334789\pi$
$503$	22.2498	0.992068	0.496034	+	0.868303i	$0.334789\pi$
$504$	0	0
$505$	−32.0294	−1.42529
$506$	0	0
$507$	−4.00000	−0.177646
$508$	0	0
$509$	−1.26537	−0.0560864	−0.0280432	−	0.999607i	$-0.508928\pi$
$509$	−1.26537	−0.0560864	−0.0280432	+	0.999607i	$0.508928\pi$
$510$	0	0
$511$	7.46927	0.330421
$512$	0	0
$513$	−6.76491	−0.298678
$514$	0	0
$515$	−10.4995	−0.462665
$516$	0	0
$517$	−22.0138	−0.968164
$518$	0	0
$519$	−12.4390	−0.546011
$520$	0	0
$521$	21.4343	0.939053	0.469527	−	0.882918i	$-0.344425\pi$
$521$	21.4343	0.939053	0.469527	+	0.882918i	$0.344425\pi$
$522$	0	0
$523$	5.11399	0.223619	0.111810	−	0.993730i	$-0.464335\pi$
$523$	5.11399	0.223619	0.111810	+	0.993730i	$0.464335\pi$
$524$	0	0
$525$	−17.8401	−0.778608
$526$	0	0
$527$	−1.82546	−0.0795183
$528$	0	0
$529$	54.3553	2.36327
$530$	0	0
$531$	2.96972	0.128875
$532$	0	0
$533$	−2.20390	−0.0954614
$534$	0	0
$535$	−13.0147	−0.562674
$536$	0	0
$537$	2.24977	0.0970848
$538$	0	0
$539$	12.5142	0.539026
$540$	0	0
$541$	44.2333	1.90174	0.950868	−	0.309596i	$-0.100194\pi$
$541$	44.2333	1.90174	0.950868	+	0.309596i	$0.100194\pi$
$542$	0	0
$543$	6.88601	0.295507
$544$	0	0
$545$	38.1542	1.63435
$546$	0	0
$547$	31.0743	1.32864	0.664321	−	0.747447i	$-0.268721\pi$
$547$	31.0743	1.32864	0.664321	+	0.747447i	$0.268721\pi$
$548$	0	0
$549$	11.2195	0.478836
$550$	0	0
$551$	−14.3747	−0.612381
$552$	0	0
$553$	4.03875	0.171745
$554$	0	0
$555$	−21.7796	−0.924492
$556$	0	0
$557$	20.7952	0.881120	0.440560	−	0.897723i	$-0.354780\pi$
$557$	20.7952	0.881120	0.440560	+	0.897723i	$0.354780\pi$
$558$	0	0
$559$	3.00000	0.126886
$560$	0	0
$561$	−1.26537	−0.0534238
$562$	0	0
$563$	4.82454	0.203330	0.101665	−	0.994819i	$-0.467583\pi$
$563$	4.82454	0.203330	0.101665	+	0.994819i	$0.467583\pi$
$564$	0	0
$565$	14.0450	0.590876
$566$	0	0
$567$	−1.48486	−0.0623583
$568$	0	0
$569$	25.0450	1.04994	0.524970	−	0.851121i	$-0.324077\pi$
$569$	25.0450	1.04994	0.524970	+	0.851121i	$0.324077\pi$
$570$	0	0
$571$	15.4849	0.648021	0.324011	−	0.946053i	$-0.394969\pi$
$571$	15.4849	0.648021	0.324011	+	0.946053i	$0.394969\pi$
$572$	0	0
$573$	−6.47018	−0.270296
$574$	0	0
$575$	−105.671	−4.40680
$576$	0	0
$577$	−18.8704	−0.785586	−0.392793	−	0.919627i	$-0.628491\pi$
$577$	−18.8704	−0.785586	−0.392793	+	0.919627i	$0.628491\pi$
$578$	0	0
$579$	2.23509	0.0928872
$580$	0	0
$581$	8.28474	0.343709
$582$	0	0
$583$	9.82546	0.406929
$584$	0	0
$585$	−12.3747	−0.511629
$586$	0	0
$587$	−37.1883	−1.53493	−0.767463	−	0.641094i	$-0.778481\pi$
$587$	−37.1883	−1.53493	−0.767463	+	0.641094i	$0.778481\pi$
$588$	0	0
$589$	25.4693	1.04944
$590$	0	0
$591$	−7.77959	−0.320010
$592$	0	0
$593$	9.87133	0.405367	0.202684	−	0.979244i	$-0.435034\pi$
$593$	9.87133	0.405367	0.202684	+	0.979244i	$0.435034\pi$
$594$	0	0
$595$	−2.96972	−0.121747
$596$	0	0
$597$	−6.90917	−0.282774
$598$	0	0
$599$	41.2598	1.68583	0.842914	−	0.538048i	$-0.180838\pi$
$599$	41.2598	1.68583	0.842914	+	0.538048i	$0.180838\pi$
$600$	0	0
$601$	−44.3397	−1.80865	−0.904327	−	0.426841i	$-0.859626\pi$
$601$	−44.3397	−1.80865	−0.904327	+	0.426841i	$0.859626\pi$
$602$	0	0
$603$	7.04496	0.286893
$604$	0	0
$605$	−17.2800	−0.702534
$606$	0	0
$607$	5.76399	0.233953	0.116977	−	0.993135i	$-0.462680\pi$
$607$	5.76399	0.233953	0.116977	+	0.993135i	$0.462680\pi$
$608$	0	0
$609$	−3.15516	−0.127854
$610$	0	0
$611$	−25.3056	−1.02376
$612$	0	0
$613$	−26.8780	−1.08559	−0.542796	−	0.839865i	$-0.682634\pi$
$613$	−26.8780	−1.08559	−0.542796	+	0.839865i	$0.682634\pi$
$614$	0	0
$615$	3.03028	0.122193
$616$	0	0
$617$	18.3250	0.737737	0.368868	−	0.929482i	$-0.379745\pi$
$617$	18.3250	0.737737	0.368868	+	0.929482i	$0.379745\pi$
$618$	0	0
$619$	−26.9697	−1.08400	−0.542002	−	0.840377i	$-0.682334\pi$
$619$	−26.9697	−1.08400	−0.542002	+	0.840377i	$0.682334\pi$
$620$	0	0
$621$	−8.79518	−0.352939
$622$	0	0
$623$	27.0984	1.08567
$624$	0	0
$625$	59.2791	2.37117
$626$	0	0
$627$	17.6547	0.705061
$628$	0	0
$629$	−2.56009	−0.102078
$630$	0	0
$631$	−15.1514	−0.603167	−0.301583	−	0.953440i	$-0.597515\pi$
$631$	−15.1514	−0.603167	−0.301583	+	0.953440i	$0.597515\pi$
$632$	0	0
$633$	3.42431	0.136104
$634$	0	0
$635$	37.3094	1.48058
$636$	0	0
$637$	14.3856	0.569976
$638$	0	0
$639$	−13.2195	−0.522955
$640$	0	0
$641$	−4.06811	−0.160681	−0.0803404	−	0.996767i	$-0.525601\pi$
$641$	−4.06811	−0.160681	−0.0803404	+	0.996767i	$0.525601\pi$
$642$	0	0
$643$	12.6594	0.499238	0.249619	−	0.968344i	$-0.419695\pi$
$643$	12.6594	0.499238	0.249619	+	0.968344i	$0.419695\pi$
$644$	0	0
$645$	−4.12489	−0.162417
$646$	0	0
$647$	−0.128666	−0.00505840	−0.00252920	−	0.999997i	$-0.500805\pi$
$647$	−0.128666	−0.00505840	−0.00252920	+	0.999997i	$0.500805\pi$
$648$	0	0
$649$	−7.75023	−0.304223
$650$	0	0
$651$	5.59037	0.219104
$652$	0	0
$653$	37.7834	1.47858	0.739289	−	0.673389i	$-0.235162\pi$
$653$	37.7834	1.47858	0.739289	+	0.673389i	$0.235162\pi$
$654$	0	0
$655$	78.6188	3.07189
$656$	0	0
$657$	−5.03028	−0.196250
$658$	0	0
$659$	−29.9154	−1.16534	−0.582669	−	0.812710i	$-0.697991\pi$
$659$	−29.9154	−1.16534	−0.582669	+	0.812710i	$0.697991\pi$
$660$	0	0
$661$	−20.7796	−0.808232	−0.404116	−	0.914708i	$-0.632421\pi$
$661$	−20.7796	−0.808232	−0.404116	+	0.914708i	$0.632421\pi$
$662$	0	0
$663$	−1.45459	−0.0564914
$664$	0	0
$665$	41.4343	1.60675
$666$	0	0
$667$	−18.6888	−0.723632
$668$	0	0
$669$	9.73463	0.376363
$670$	0	0
$671$	−29.2800	−1.13034
$672$	0	0
$673$	−48.7181	−1.87795	−0.938973	−	0.343992i	$-0.888221\pi$
$673$	−48.7181	−1.87795	−0.938973	+	0.343992i	$0.888221\pi$
$674$	0	0
$675$	12.0147	0.462445
$676$	0	0
$677$	46.2498	1.77752	0.888762	−	0.458370i	$-0.151566\pi$
$677$	46.2498	1.77752	0.888762	+	0.458370i	$0.151566\pi$
$678$	0	0
$679$	−9.23509	−0.354410
$680$	0	0
$681$	3.09083	0.118441
$682$	0	0
$683$	4.17454	0.159734	0.0798671	−	0.996806i	$-0.474550\pi$
$683$	4.17454	0.159734	0.0798671	+	0.996806i	$0.474550\pi$
$684$	0	0
$685$	−86.1037	−3.28985
$686$	0	0
$687$	−3.20482	−0.122271
$688$	0	0
$689$	11.2947	0.430295
$690$	0	0
$691$	−25.5904	−0.973504	−0.486752	−	0.873540i	$-0.661818\pi$
$691$	−25.5904	−0.973504	−0.486752	+	0.873540i	$0.661818\pi$
$692$	0	0
$693$	3.87511	0.147203
$694$	0	0
$695$	−54.5895	−2.07070
$696$	0	0
$697$	0.356195	0.0134919
$698$	0	0
$699$	11.0946	0.419637
$700$	0	0
$701$	15.0303	0.567686	0.283843	−	0.958871i	$-0.408391\pi$
$701$	15.0303	0.567686	0.283843	+	0.958871i	$0.408391\pi$
$702$	0	0
$703$	35.7190	1.34717
$704$	0	0
$705$	34.7943	1.31043
$706$	0	0
$707$	11.5298	0.433623
$708$	0	0
$709$	−3.12110	−0.117216	−0.0586078	−	0.998281i	$-0.518666\pi$
$709$	−3.12110	−0.117216	−0.0586078	+	0.998281i	$0.518666\pi$
$710$	0	0
$711$	−2.71995	−0.102006
$712$	0	0
$713$	33.1131	1.24009
$714$	0	0
$715$	32.2947	1.20775
$716$	0	0
$717$	−17.0946	−0.638410
$718$	0	0
$719$	12.5895	0.469507	0.234754	−	0.972055i	$-0.424572\pi$
$719$	12.5895	0.469507	0.234754	+	0.972055i	$0.424572\pi$
$720$	0	0
$721$	3.77959	0.140759
$722$	0	0
$723$	22.5601	0.839019
$724$	0	0
$725$	25.5298	0.948154
$726$	0	0
$727$	−19.4537	−0.721497	−0.360748	−	0.932663i	$-0.617479\pi$
$727$	−19.4537	−0.721497	−0.360748	+	0.932663i	$0.617479\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.484862	−0.0179333
$732$	0	0
$733$	−25.9394	−0.958095	−0.479048	−	0.877789i	$-0.659018\pi$
$733$	−25.9394	−0.958095	−0.479048	+	0.877789i	$0.659018\pi$
$734$	0	0
$735$	−19.7796	−0.729581
$736$	0	0
$737$	−18.3856	−0.677241
$738$	0	0
$739$	−49.4225	−1.81804	−0.909018	−	0.416758i	$-0.863166\pi$
$739$	−49.4225	−1.81804	−0.909018	+	0.416758i	$0.863166\pi$
$740$	0	0
$741$	20.2947	0.745545
$742$	0	0
$743$	30.5677	1.12142	0.560709	−	0.828013i	$-0.310529\pi$
$743$	30.5677	1.12142	0.560709	+	0.828013i	$0.310529\pi$
$744$	0	0
$745$	46.2791	1.69554
$746$	0	0
$747$	−5.57947	−0.204142
$748$	0	0
$749$	4.68498	0.171185
$750$	0	0
$751$	−2.73555	−0.0998216	−0.0499108	−	0.998754i	$-0.515894\pi$
$751$	−2.73555	−0.0998216	−0.0499108	+	0.998754i	$0.515894\pi$
$752$	0	0
$753$	15.6400	0.569954
$754$	0	0
$755$	−31.0596	−1.13038
$756$	0	0
$757$	2.68876	0.0977247	0.0488623	−	0.998806i	$-0.484440\pi$
$757$	2.68876	0.0977247	0.0488623	+	0.998806i	$0.484440\pi$
$758$	0	0
$759$	22.9532	0.833149
$760$	0	0
$761$	−42.4002	−1.53701	−0.768504	−	0.639845i	$-0.778998\pi$
$761$	−42.4002	−1.53701	−0.768504	+	0.639845i	$0.778998\pi$
$762$	0	0
$763$	−13.7346	−0.497227
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	−8.90917	−0.321692
$768$	0	0
$769$	−35.1046	−1.26590	−0.632952	−	0.774191i	$-0.718157\pi$
$769$	−35.1046	−1.26590	−0.632952	+	0.774191i	$0.718157\pi$
$770$	0	0
$771$	−17.5336	−0.631457
$772$	0	0
$773$	15.0946	0.542915	0.271458	−	0.962450i	$-0.412494\pi$
$773$	15.0946	0.542915	0.271458	+	0.962450i	$0.412494\pi$
$774$	0	0
$775$	−45.2342	−1.62486
$776$	0	0
$777$	7.84014	0.281263
$778$	0	0
$779$	−4.96972	−0.178059
$780$	0	0
$781$	34.4995	1.23449
$782$	0	0
$783$	2.12489	0.0759372
$784$	0	0
$785$	−102.338	−3.65259
$786$	0	0
$787$	41.1202	1.46578	0.732888	−	0.680349i	$-0.238172\pi$
$787$	41.1202	1.46578	0.732888	+	0.680349i	$0.238172\pi$
$788$	0	0
$789$	6.37088	0.226809
$790$	0	0
$791$	−5.05585	−0.179765
$792$	0	0
$793$	−33.6585	−1.19525
$794$	0	0
$795$	−15.5298	−0.550786
$796$	0	0
$797$	−2.78807	−0.0987584	−0.0493792	−	0.998780i	$-0.515724\pi$
$797$	−2.78807	−0.0987584	−0.0493792	+	0.998780i	$0.515724\pi$
$798$	0	0
$799$	4.08991	0.144691
$800$	0	0
$801$	−18.2498	−0.644824
$802$	0	0
$803$	13.1277	0.463268
$804$	0	0
$805$	53.8695	1.89865
$806$	0	0
$807$	−29.5445	−1.04002
$808$	0	0
$809$	−8.27913	−0.291079	−0.145539	−	0.989352i	$-0.546492\pi$
$809$	−8.27913	−0.291079	−0.145539	+	0.989352i	$0.546492\pi$
$810$	0	0
$811$	47.1807	1.65674	0.828370	−	0.560181i	$-0.189269\pi$
$811$	47.1807	1.65674	0.828370	+	0.560181i	$0.189269\pi$
$812$	0	0
$813$	11.8255	0.414737
$814$	0	0
$815$	−5.87511	−0.205796
$816$	0	0
$817$	6.76491	0.236674
$818$	0	0
$819$	4.45459	0.155656
$820$	0	0
$821$	−38.7640	−1.35287	−0.676436	−	0.736501i	$-0.736477\pi$
$821$	−38.7640	−1.35287	−0.676436	+	0.736501i	$0.736477\pi$
$822$	0	0
$823$	−36.1433	−1.25988	−0.629939	−	0.776645i	$-0.716920\pi$
$823$	−36.1433	−1.25988	−0.629939	+	0.776645i	$0.716920\pi$
$824$	0	0
$825$	−31.3553	−1.09165
$826$	0	0
$827$	30.4958	1.06044	0.530221	−	0.847860i	$-0.322109\pi$
$827$	30.4958	1.06044	0.530221	+	0.847860i	$0.322109\pi$
$828$	0	0
$829$	−38.5601	−1.33925	−0.669624	−	0.742701i	$-0.733545\pi$
$829$	−38.5601	−1.33925	−0.669624	+	0.742701i	$0.733545\pi$
$830$	0	0
$831$	15.7796	0.547388
$832$	0	0
$833$	−2.32500	−0.0805566
$834$	0	0
$835$	−34.4839	−1.19337
$836$	0	0
$837$	−3.76491	−0.130134
$838$	0	0
$839$	44.2791	1.52869	0.764343	−	0.644810i	$-0.223064\pi$
$839$	44.2791	1.52869	0.764343	+	0.644810i	$0.223064\pi$
$840$	0	0
$841$	−24.4849	−0.844306
$842$	0	0
$843$	−18.0147	−0.620459
$844$	0	0
$845$	−16.4995	−0.567601
$846$	0	0
$847$	6.22041	0.213736
$848$	0	0
$849$	5.64380	0.193695
$850$	0	0
$851$	46.4390	1.59191
$852$	0	0
$853$	−29.2716	−1.00224	−0.501120	−	0.865378i	$-0.667078\pi$
$853$	−29.2716	−1.00224	−0.501120	+	0.865378i	$0.667078\pi$
$854$	0	0
$855$	−27.9045	−0.954313
$856$	0	0
$857$	6.37088	0.217625	0.108812	−	0.994062i	$-0.465295\pi$
$857$	6.37088	0.217625	0.108812	+	0.994062i	$0.465295\pi$
$858$	0	0
$859$	32.4839	1.10834	0.554169	−	0.832404i	$-0.313036\pi$
$859$	32.4839	1.10834	0.554169	+	0.832404i	$0.313036\pi$
$860$	0	0
$861$	−1.09083	−0.0371753
$862$	0	0
$863$	−32.2791	−1.09879	−0.549397	−	0.835561i	$-0.685143\pi$
$863$	−32.2791	−1.09879	−0.549397	+	0.835561i	$0.685143\pi$
$864$	0	0
$865$	−51.3094	−1.74457
$866$	0	0
$867$	−16.7649	−0.569366
$868$	0	0
$869$	7.09839	0.240796
$870$	0	0
$871$	−21.1349	−0.716128
$872$	0	0
$873$	6.21949	0.210498
$874$	0	0
$875$	−42.9641	−1.45245
$876$	0	0
$877$	0.674081	0.0227621	0.0113810	−	0.999935i	$-0.496377\pi$
$877$	0.674081	0.0227621	0.0113810	+	0.999935i	$0.496377\pi$
$878$	0	0
$879$	−1.34060	−0.0452173
$880$	0	0
$881$	1.01560	0.0342163	0.0171082	−	0.999854i	$-0.494554\pi$
$881$	1.01560	0.0342163	0.0171082	+	0.999854i	$0.494554\pi$
$882$	0	0
$883$	−31.6732	−1.06589	−0.532943	−	0.846151i	$-0.678914\pi$
$883$	−31.6732	−1.06589	−0.532943	+	0.846151i	$0.678914\pi$
$884$	0	0
$885$	12.2498	0.411772
$886$	0	0
$887$	−40.8686	−1.37223	−0.686116	−	0.727492i	$-0.740686\pi$
$887$	−40.8686	−1.37223	−0.686116	+	0.727492i	$0.740686\pi$
$888$	0	0
$889$	−13.4305	−0.450445
$890$	0	0
$891$	−2.60975	−0.0874298
$892$	0	0
$893$	−57.0634	−1.90955
$894$	0	0
$895$	9.28005	0.310198
$896$	0	0
$897$	26.3856	0.880988
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−1.82546	−0.0608149
$902$	0	0
$903$	1.48486	0.0494131
$904$	0	0
$905$	28.4040	0.944181
$906$	0	0
$907$	−16.0828	−0.534020	−0.267010	−	0.963694i	$-0.586036\pi$
$907$	−16.0828	−0.534020	−0.267010	+	0.963694i	$0.586036\pi$
$908$	0	0
$909$	−7.76491	−0.257546
$910$	0	0
$911$	−37.1277	−1.23010	−0.615049	−	0.788489i	$-0.710864\pi$
$911$	−37.1277	−1.23010	−0.615049	+	0.788489i	$0.710864\pi$
$912$	0	0
$913$	14.5610	0.481899
$914$	0	0
$915$	46.2791	1.52994
$916$	0	0
$917$	−28.3009	−0.934579
$918$	0	0
$919$	−27.3940	−0.903646	−0.451823	−	0.892108i	$-0.649226\pi$
$919$	−27.3940	−0.903646	−0.451823	+	0.892108i	$0.649226\pi$
$920$	0	0
$921$	−0.424310	−0.0139815
$922$	0	0
$923$	39.6585	1.30537
$924$	0	0
$925$	−63.4381	−2.08583
$926$	0	0
$927$	−2.54541	−0.0836024
$928$	0	0
$929$	30.5639	1.00277	0.501384	−	0.865225i	$-0.332824\pi$
$929$	30.5639	1.00277	0.501384	+	0.865225i	$0.332824\pi$
$930$	0	0
$931$	32.4390	1.06314
$932$	0	0
$933$	−0.110206	−0.00360797
$934$	0	0
$935$	−5.21949	−0.170696
$936$	0	0
$937$	−20.4702	−0.668732	−0.334366	−	0.942443i	$-0.608522\pi$
$937$	−20.4702	−0.668732	−0.334366	+	0.942443i	$0.608522\pi$
$938$	0	0
$939$	−2.96972	−0.0969133
$940$	0	0
$941$	−1.89358	−0.0617288	−0.0308644	−	0.999524i	$-0.509826\pi$
$941$	−1.89358	−0.0617288	−0.0308644	+	0.999524i	$0.509826\pi$
$942$	0	0
$943$	−6.46123	−0.210407
$944$	0	0
$945$	−6.12489	−0.199243
$946$	0	0
$947$	−33.1311	−1.07662	−0.538308	−	0.842748i	$-0.680936\pi$
$947$	−33.1311	−1.07662	−0.538308	+	0.842748i	$0.680936\pi$
$948$	0	0
$949$	15.0908	0.489869
$950$	0	0
$951$	12.4243	0.402886
$952$	0	0
$953$	−29.5942	−0.958649	−0.479324	−	0.877638i	$-0.659118\pi$
$953$	−29.5942	−0.958649	−0.479324	+	0.877638i	$0.659118\pi$
$954$	0	0
$955$	−26.6888	−0.863628
$956$	0	0
$957$	−5.54541	−0.179258
$958$	0	0
$959$	30.9953	1.00089
$960$	0	0
$961$	−16.8255	−0.542757
$962$	0	0
$963$	−3.15516	−0.101674
$964$	0	0
$965$	9.21949	0.296786
$966$	0	0
$967$	−15.4234	−0.495983	−0.247991	−	0.968762i	$-0.579770\pi$
$967$	−15.4234	−0.495983	−0.247991	+	0.968762i	$0.579770\pi$
$968$	0	0
$969$	−3.28005	−0.105370
$970$	0	0
$971$	−25.6438	−0.822949	−0.411474	−	0.911421i	$-0.634986\pi$
$971$	−25.6438	−0.822949	−0.411474	+	0.911421i	$0.634986\pi$
$972$	0	0
$973$	19.6509	0.629980
$974$	0	0
$975$	−36.0440	−1.15433
$976$	0	0
$977$	11.1202	0.355766	0.177883	−	0.984052i	$-0.443075\pi$
$977$	11.1202	0.355766	0.177883	+	0.984052i	$0.443075\pi$
$978$	0	0
$979$	47.6273	1.52217
$980$	0	0
$981$	9.24977	0.295322
$982$	0	0
$983$	36.9697	1.17915	0.589576	−	0.807713i	$-0.299295\pi$
$983$	36.9697	1.17915	0.589576	+	0.807713i	$0.299295\pi$
$984$	0	0
$985$	−32.0899	−1.02247
$986$	0	0
$987$	−12.5251	−0.398679
$988$	0	0
$989$	8.79518	0.279671
$990$	0	0
$991$	−9.70527	−0.308298	−0.154149	−	0.988048i	$-0.549264\pi$
$991$	−9.70527	−0.308298	−0.154149	+	0.988048i	$0.549264\pi$
$992$	0	0
$993$	31.2938	0.993079
$994$	0	0
$995$	−28.4995	−0.903496
$996$	0	0
$997$	23.2413	0.736059	0.368030	−	0.929814i	$-0.380033\pi$
$997$	23.2413	0.736059	0.368030	+	0.929814i	$0.380033\pi$
$998$	0	0
$999$	−5.28005	−0.167053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8256.2.a.cu.1.3		3
4.3	odd	2		8256.2.a.cr.1.3		3
8.3	odd	2		129.2.a.d.1.3	✓	3
8.5	even	2		2064.2.a.x.1.1		3
24.5	odd	2		6192.2.a.bw.1.3		3
24.11	even	2		387.2.a.i.1.1		3
40.19	odd	2		3225.2.a.t.1.1		3
56.27	even	2		6321.2.a.p.1.3		3
120.59	even	2		9675.2.a.bq.1.3		3
344.171	even	2		5547.2.a.p.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.a.d.1.3	✓	3	8.3	odd	2
387.2.a.i.1.1		3	24.11	even	2
2064.2.a.x.1.1		3	8.5	even	2
3225.2.a.t.1.1		3	40.19	odd	2
5547.2.a.p.1.1		3	344.171	even	2
6192.2.a.bw.1.3		3	24.5	odd	2
6321.2.a.p.1.3		3	56.27	even	2
8256.2.a.cr.1.3		3	4.3	odd	2
8256.2.a.cu.1.3		3	1.1	even	1	trivial
9675.2.a.bq.1.3		3	120.59	even	2

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:	\( N \)	\(=\)	\( 8256 = 2^{6} \cdot 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8256.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.12489 q^{5} -1.48486 q^{7} +1.00000 q^{9} -2.60975 q^{11} -3.00000 q^{13} +4.12489 q^{15} +0.484862 q^{17} -6.76491 q^{19} -1.48486 q^{21} -8.79518 q^{23} +12.0147 q^{25} +1.00000 q^{27} +2.12489 q^{29} -3.76491 q^{31} -2.60975 q^{33} -6.12489 q^{35} -5.28005 q^{37} -3.00000 q^{39} +0.734633 q^{41} -1.00000 q^{43} +4.12489 q^{45} +8.43521 q^{47} -4.79518 q^{49} +0.484862 q^{51} -3.76491 q^{53} -10.7649 q^{55} -6.76491 q^{57} +2.96972 q^{59} +11.2195 q^{61} -1.48486 q^{63} -12.3747 q^{65} +7.04496 q^{67} -8.79518 q^{69} -13.2195 q^{71} -5.03028 q^{73} +12.0147 q^{75} +3.87511 q^{77} -2.71995 q^{79} +1.00000 q^{81} -5.57947 q^{83} +2.00000 q^{85} +2.12489 q^{87} -18.2498 q^{89} +4.45459 q^{91} -3.76491 q^{93} -27.9045 q^{95} +6.21949 q^{97} -2.60975 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 4 q^{5} - 4 q^{7} + 3 q^{9} + q^{11} - 9 q^{13} + 4 q^{15} + q^{17} - 4 q^{19} - 4 q^{21} - 11 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} + 5 q^{31} + q^{33} - 10 q^{35} - 9 q^{39} - 15 q^{41}+ \cdots + q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 4 * q^5 - 4 * q^7 + 3 * q^9 + q^11 - 9 * q^13 + 4 * q^15 + q^17 - 4 * q^19 - 4 * q^21 - 11 * q^23 + 3 * q^25 + 3 * q^27 - 2 * q^29 + 5 * q^31 + q^33 - 10 * q^35 - 9 * q^39 - 15 * q^41 - 3 * q^43 + 4 * q^45 + 2 * q^47 + q^49 + q^51 + 5 * q^53 - 16 * q^55 - 4 * q^57 + 8 * q^59 + 16 * q^61 - 4 * q^63 - 12 * q^65 - 11 * q^67 - 11 * q^69 - 22 * q^71 - 16 * q^73 + 3 * q^75 + 20 * q^77 - 24 * q^79 + 3 * q^81 - 7 * q^83 + 6 * q^85 - 2 * q^87 - 38 * q^89 + 12 * q^91 + 5 * q^93 - 26 * q^95 + q^97 + q^99

Introduction

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