LMFDB - Embedded newform 8032.2.a.i.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.18444 q^{3} -0.0964158 q^{5} -1.78364 q^{7} -1.59711 q^{9} +O(q^{10})$	$q+1.18444 q^{3} -0.0964158 q^{5} -1.78364 q^{7} -1.59711 q^{9} +1.62425 q^{11} -0.858788 q^{13} -0.114198 q^{15} -0.953026 q^{17} +3.91649 q^{19} -2.11260 q^{21} -1.65300 q^{23} -4.99070 q^{25} -5.44498 q^{27} +8.35557 q^{29} -6.03425 q^{31} +1.92382 q^{33} +0.171971 q^{35} +10.3889 q^{37} -1.01718 q^{39} -5.38101 q^{41} -1.55477 q^{43} +0.153987 q^{45} +0.729763 q^{47} -3.81864 q^{49} -1.12880 q^{51} +2.02803 q^{53} -0.156603 q^{55} +4.63883 q^{57} +13.8447 q^{59} +6.10292 q^{61} +2.84866 q^{63} +0.0828007 q^{65} -3.88352 q^{67} -1.95788 q^{69} +5.75853 q^{71} +12.8748 q^{73} -5.91117 q^{75} -2.89707 q^{77} +3.04097 q^{79} -1.65791 q^{81} -0.674764 q^{83} +0.0918867 q^{85} +9.89664 q^{87} -6.71352 q^{89} +1.53176 q^{91} -7.14719 q^{93} -0.377611 q^{95} +5.60829 q^{97} -2.59410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * 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
$3$	1.18444	0.683835	0.341917	−	0.939730i	$-0.388924\pi$
$3$	1.18444	0.683835	0.341917	+	0.939730i	$0.388924\pi$
$4$	0	0
$5$	−0.0964158	−0.0431184	−0.0215592	−	0.999768i	$-0.506863\pi$
$5$	−0.0964158	−0.0431184	−0.0215592	+	0.999768i	$0.506863\pi$
$6$	0	0
$7$	−1.78364	−0.674151	−0.337076	−	0.941478i	$-0.609438\pi$
$7$	−1.78364	−0.674151	−0.337076	+	0.941478i	$0.609438\pi$
$8$	0	0
$9$	−1.59711	−0.532370
$10$	0	0
$11$	1.62425	0.489729	0.244865	−	0.969557i	$-0.421256\pi$
$11$	1.62425	0.489729	0.244865	+	0.969557i	$0.421256\pi$
$12$	0	0
$13$	−0.858788	−0.238185	−0.119092	−	0.992883i	$-0.537998\pi$
$13$	−0.858788	−0.238185	−0.119092	+	0.992883i	$0.537998\pi$
$14$	0	0
$15$	−0.114198	−0.0294859
$16$	0	0
$17$	−0.953026	−0.231143	−0.115571	−	0.993299i	$-0.536870\pi$
$17$	−0.953026	−0.231143	−0.115571	+	0.993299i	$0.536870\pi$
$18$	0	0
$19$	3.91649	0.898504	0.449252	−	0.893405i	$-0.351691\pi$
$19$	3.91649	0.898504	0.449252	+	0.893405i	$0.351691\pi$
$20$	0	0
$21$	−2.11260	−0.461008
$22$	0	0
$23$	−1.65300	−0.344675	−0.172338	−	0.985038i	$-0.555132\pi$
$23$	−1.65300	−0.344675	−0.172338	+	0.985038i	$0.555132\pi$
$24$	0	0
$25$	−4.99070	−0.998141
$26$	0	0
$27$	−5.44498	−1.04789
$28$	0	0
$29$	8.35557	1.55159	0.775795	−	0.630985i	$-0.217349\pi$
$29$	8.35557	1.55159	0.775795	+	0.630985i	$0.217349\pi$
$30$	0	0
$31$	−6.03425	−1.08378	−0.541892	−	0.840448i	$-0.682292\pi$
$31$	−6.03425	−1.08378	−0.541892	+	0.840448i	$0.682292\pi$
$32$	0	0
$33$	1.92382	0.334894
$34$	0	0
$35$	0.171971	0.0290683
$36$	0	0
$37$	10.3889	1.70792	0.853962	−	0.520336i	$-0.174193\pi$
$37$	10.3889	1.70792	0.853962	+	0.520336i	$0.174193\pi$
$38$	0	0
$39$	−1.01718	−0.162879
$40$	0	0
$41$	−5.38101	−0.840373	−0.420186	−	0.907438i	$-0.638035\pi$
$41$	−5.38101	−0.840373	−0.420186	+	0.907438i	$0.638035\pi$
$42$	0	0
$43$	−1.55477	−0.237100	−0.118550	−	0.992948i	$-0.537825\pi$
$43$	−1.55477	−0.237100	−0.118550	+	0.992948i	$0.537825\pi$
$44$	0	0
$45$	0.153987	0.0229550
$46$	0	0
$47$	0.729763	0.106447	0.0532235	−	0.998583i	$-0.483050\pi$
$47$	0.729763	0.106447	0.0532235	+	0.998583i	$0.483050\pi$
$48$	0	0
$49$	−3.81864	−0.545520
$50$	0	0
$51$	−1.12880	−0.158063
$52$	0	0
$53$	2.02803	0.278571	0.139286	−	0.990252i	$-0.455519\pi$
$53$	2.02803	0.278571	0.139286	+	0.990252i	$0.455519\pi$
$54$	0	0
$55$	−0.156603	−0.0211164
$56$	0	0
$57$	4.63883	0.614428
$58$	0	0
$59$	13.8447	1.80242	0.901212	−	0.433379i	$-0.142679\pi$
$59$	13.8447	1.80242	0.901212	+	0.433379i	$0.142679\pi$
$60$	0	0
$61$	6.10292	0.781399	0.390699	−	0.920518i	$-0.372233\pi$
$61$	6.10292	0.781399	0.390699	+	0.920518i	$0.372233\pi$
$62$	0	0
$63$	2.84866	0.358898
$64$	0	0
$65$	0.0828007	0.0102702
$66$	0	0
$67$	−3.88352	−0.474448	−0.237224	−	0.971455i	$-0.576238\pi$
$67$	−3.88352	−0.474448	−0.237224	+	0.971455i	$0.576238\pi$
$68$	0	0
$69$	−1.95788	−0.235701
$70$	0	0
$71$	5.75853	0.683411	0.341706	−	0.939807i	$-0.388995\pi$
$71$	5.75853	0.683411	0.341706	+	0.939807i	$0.388995\pi$
$72$	0	0
$73$	12.8748	1.50688	0.753440	−	0.657517i	$-0.228393\pi$
$73$	12.8748	1.50688	0.753440	+	0.657517i	$0.228393\pi$
$74$	0	0
$75$	−5.91117	−0.682563
$76$	0	0
$77$	−2.89707	−0.330152
$78$	0	0
$79$	3.04097	0.342136	0.171068	−	0.985259i	$-0.445278\pi$
$79$	3.04097	0.342136	0.171068	+	0.985259i	$0.445278\pi$
$80$	0	0
$81$	−1.65791	−0.184212
$82$	0	0
$83$	−0.674764	−0.0740650	−0.0370325	−	0.999314i	$-0.511791\pi$
$83$	−0.674764	−0.0740650	−0.0370325	+	0.999314i	$0.511791\pi$
$84$	0	0
$85$	0.0918867	0.00996651
$86$	0	0
$87$	9.89664	1.06103
$88$	0	0
$89$	−6.71352	−0.711632	−0.355816	−	0.934556i	$-0.615797\pi$
$89$	−6.71352	−0.711632	−0.355816	+	0.934556i	$0.615797\pi$
$90$	0	0
$91$	1.53176	0.160573
$92$	0	0
$93$	−7.14719	−0.741129
$94$	0	0
$95$	−0.377611	−0.0387421
$96$	0	0
$97$	5.60829	0.569436	0.284718	−	0.958611i	$-0.408100\pi$
$97$	5.60829	0.569436	0.284718	+	0.958611i	$0.408100\pi$
$98$	0	0
$99$	−2.59410	−0.260717
$100$	0	0
$101$	4.06667	0.404649	0.202325	−	0.979319i	$-0.435150\pi$
$101$	4.06667	0.404649	0.202325	+	0.979319i	$0.435150\pi$
$102$	0	0
$103$	15.6844	1.54543	0.772717	−	0.634751i	$-0.218897\pi$
$103$	15.6844	1.54543	0.772717	+	0.634751i	$0.218897\pi$
$104$	0	0
$105$	0.203688	0.0198779
$106$	0	0
$107$	−10.1974	−0.985822	−0.492911	−	0.870080i	$-0.664067\pi$
$107$	−10.1974	−0.985822	−0.492911	+	0.870080i	$0.664067\pi$
$108$	0	0
$109$	5.69714	0.545687	0.272843	−	0.962058i	$-0.412036\pi$
$109$	5.69714	0.545687	0.272843	+	0.962058i	$0.412036\pi$
$110$	0	0
$111$	12.3050	1.16794
$112$	0	0
$113$	−13.4804	−1.26813	−0.634063	−	0.773282i	$-0.718614\pi$
$113$	−13.4804	−1.26813	−0.634063	+	0.773282i	$0.718614\pi$
$114$	0	0
$115$	0.159376	0.0148619
$116$	0	0
$117$	1.37158	0.126802
$118$	0	0
$119$	1.69985	0.155825
$120$	0	0
$121$	−8.36182	−0.760165
$122$	0	0
$123$	−6.37346	−0.574676
$124$	0	0
$125$	0.963261	0.0861567
$126$	0	0
$127$	9.71551	0.862112	0.431056	−	0.902325i	$-0.358141\pi$
$127$	9.71551	0.862112	0.431056	+	0.902325i	$0.358141\pi$
$128$	0	0
$129$	−1.84152	−0.162137
$130$	0	0
$131$	18.1645	1.58704	0.793520	−	0.608544i	$-0.208246\pi$
$131$	18.1645	1.58704	0.793520	+	0.608544i	$0.208246\pi$
$132$	0	0
$133$	−6.98559	−0.605728
$134$	0	0
$135$	0.524982	0.0451833
$136$	0	0
$137$	−5.71646	−0.488390	−0.244195	−	0.969726i	$-0.578524\pi$
$137$	−5.71646	−0.488390	−0.244195	+	0.969726i	$0.578524\pi$
$138$	0	0
$139$	3.65541	0.310048	0.155024	−	0.987911i	$-0.450455\pi$
$139$	3.65541	0.310048	0.155024	+	0.987911i	$0.450455\pi$
$140$	0	0
$141$	0.864358	0.0727921
$142$	0	0
$143$	−1.39488	−0.116646
$144$	0	0
$145$	−0.805609	−0.0669022
$146$	0	0
$147$	−4.52294	−0.373046
$148$	0	0
$149$	−8.79306	−0.720355	−0.360178	−	0.932884i	$-0.617284\pi$
$149$	−8.79306	−0.720355	−0.360178	+	0.932884i	$0.617284\pi$
$150$	0	0
$151$	5.60952	0.456496	0.228248	−	0.973603i	$-0.426700\pi$
$151$	5.60952	0.456496	0.228248	+	0.973603i	$0.426700\pi$
$152$	0	0
$153$	1.52209	0.123053
$154$	0	0
$155$	0.581797	0.0467311
$156$	0	0
$157$	17.2266	1.37483	0.687416	−	0.726264i	$-0.258745\pi$
$157$	17.2266	1.37483	0.687416	+	0.726264i	$0.258745\pi$
$158$	0	0
$159$	2.40207	0.190497
$160$	0	0
$161$	2.94836	0.232363
$162$	0	0
$163$	22.9583	1.79823	0.899117	−	0.437708i	$-0.144210\pi$
$163$	22.9583	1.79823	0.899117	+	0.437708i	$0.144210\pi$
$164$	0	0
$165$	−0.185486	−0.0144401
$166$	0	0
$167$	3.32115	0.256999	0.128499	−	0.991710i	$-0.458984\pi$
$167$	3.32115	0.256999	0.128499	+	0.991710i	$0.458984\pi$
$168$	0	0
$169$	−12.2625	−0.943268
$170$	0	0
$171$	−6.25507	−0.478337
$172$	0	0
$173$	10.5899	0.805134	0.402567	−	0.915391i	$-0.368118\pi$
$173$	10.5899	0.805134	0.402567	+	0.915391i	$0.368118\pi$
$174$	0	0
$175$	8.90160	0.672898
$176$	0	0
$177$	16.3981	1.23256
$178$	0	0
$179$	−0.835805	−0.0624710	−0.0312355	−	0.999512i	$-0.509944\pi$
$179$	−0.835805	−0.0624710	−0.0312355	+	0.999512i	$0.509944\pi$
$180$	0	0
$181$	−1.08969	−0.0809959	−0.0404980	−	0.999180i	$-0.512894\pi$
$181$	−1.08969	−0.0809959	−0.0404980	+	0.999180i	$0.512894\pi$
$182$	0	0
$183$	7.22852	0.534348
$184$	0	0
$185$	−1.00165	−0.0736430
$186$	0	0
$187$	−1.54795	−0.113197
$188$	0	0
$189$	9.71187	0.706435
$190$	0	0
$191$	−20.0859	−1.45337	−0.726684	−	0.686972i	$-0.758940\pi$
$191$	−20.0859	−1.45337	−0.726684	+	0.686972i	$0.758940\pi$
$192$	0	0
$193$	−5.58846	−0.402266	−0.201133	−	0.979564i	$-0.564462\pi$
$193$	−5.58846	−0.402266	−0.201133	+	0.979564i	$0.564462\pi$
$194$	0	0
$195$	0.0980721	0.00702309
$196$	0	0
$197$	−2.19251	−0.156210	−0.0781050	−	0.996945i	$-0.524887\pi$
$197$	−2.19251	−0.156210	−0.0781050	+	0.996945i	$0.524887\pi$
$198$	0	0
$199$	4.87056	0.345265	0.172632	−	0.984986i	$-0.444773\pi$
$199$	4.87056	0.345265	0.172632	+	0.984986i	$0.444773\pi$
$200$	0	0
$201$	−4.59979	−0.324444
$202$	0	0
$203$	−14.9033	−1.04601
$204$	0	0
$205$	0.518814	0.0362356
$206$	0	0
$207$	2.64003	0.183495
$208$	0	0
$209$	6.36135	0.440024
$210$	0	0
$211$	16.0812	1.10708	0.553538	−	0.832824i	$-0.313278\pi$
$211$	16.0812	1.10708	0.553538	+	0.832824i	$0.313278\pi$
$212$	0	0
$213$	6.82061	0.467340
$214$	0	0
$215$	0.149904	0.0102234
$216$	0	0
$217$	10.7629	0.730634
$218$	0	0
$219$	15.2494	1.03046
$220$	0	0
$221$	0.818446	0.0550547
$222$	0	0
$223$	−19.2728	−1.29060	−0.645300	−	0.763929i	$-0.723268\pi$
$223$	−19.2728	−1.29060	−0.645300	+	0.763929i	$0.723268\pi$
$224$	0	0
$225$	7.97071	0.531380
$226$	0	0
$227$	7.06360	0.468827	0.234414	−	0.972137i	$-0.424683\pi$
$227$	7.06360	0.468827	0.234414	+	0.972137i	$0.424683\pi$
$228$	0	0
$229$	5.73764	0.379154	0.189577	−	0.981866i	$-0.439288\pi$
$229$	5.73764	0.379154	0.189577	+	0.981866i	$0.439288\pi$
$230$	0	0
$231$	−3.43139	−0.225769
$232$	0	0
$233$	11.8217	0.774468	0.387234	−	0.921981i	$-0.373430\pi$
$233$	11.8217	0.774468	0.387234	+	0.921981i	$0.373430\pi$
$234$	0	0
$235$	−0.0703607	−0.00458983
$236$	0	0
$237$	3.60184	0.233964
$238$	0	0
$239$	15.6301	1.01102	0.505512	−	0.862819i	$-0.331303\pi$
$239$	15.6301	1.01102	0.505512	+	0.862819i	$0.331303\pi$
$240$	0	0
$241$	−4.63865	−0.298802	−0.149401	−	0.988777i	$-0.547735\pi$
$241$	−4.63865	−0.298802	−0.149401	+	0.988777i	$0.547735\pi$
$242$	0	0
$243$	14.3713	0.921917
$244$	0	0
$245$	0.368177	0.0235220
$246$	0	0
$247$	−3.36343	−0.214010
$248$	0	0
$249$	−0.799215	−0.0506482
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−2.68489	−0.168797
$254$	0	0
$255$	0.108834	0.00681544
$256$	0	0
$257$	29.6870	1.85183	0.925913	−	0.377736i	$-0.123297\pi$
$257$	29.6870	1.85183	0.925913	+	0.377736i	$0.123297\pi$
$258$	0	0
$259$	−18.5300	−1.15140
$260$	0	0
$261$	−13.3448	−0.826021
$262$	0	0
$263$	0.874103	0.0538995	0.0269497	−	0.999637i	$-0.491421\pi$
$263$	0.874103	0.0538995	0.0269497	+	0.999637i	$0.491421\pi$
$264$	0	0
$265$	−0.195534	−0.0120116
$266$	0	0
$267$	−7.95174	−0.486638
$268$	0	0
$269$	26.4502	1.61269	0.806347	−	0.591442i	$-0.201441\pi$
$269$	26.4502	1.61269	0.806347	+	0.591442i	$0.201441\pi$
$270$	0	0
$271$	−0.882910	−0.0536329	−0.0268165	−	0.999640i	$-0.508537\pi$
$271$	−0.882910	−0.0536329	−0.0268165	+	0.999640i	$0.508537\pi$
$272$	0	0
$273$	1.81428	0.109805
$274$	0	0
$275$	−8.10614	−0.488819
$276$	0	0
$277$	−13.1092	−0.787656	−0.393828	−	0.919184i	$-0.628849\pi$
$277$	−13.1092	−0.787656	−0.393828	+	0.919184i	$0.628849\pi$
$278$	0	0
$279$	9.63737	0.576974
$280$	0	0
$281$	12.0045	0.716127	0.358064	−	0.933697i	$-0.383437\pi$
$281$	12.0045	0.716127	0.358064	+	0.933697i	$0.383437\pi$
$282$	0	0
$283$	−7.87011	−0.467829	−0.233915	−	0.972257i	$-0.575154\pi$
$283$	−7.87011	−0.467829	−0.233915	+	0.972257i	$0.575154\pi$
$284$	0	0
$285$	−0.447256	−0.0264932
$286$	0	0
$287$	9.59777	0.566538
$288$	0	0
$289$	−16.0917	−0.946573
$290$	0	0
$291$	6.64267	0.389400
$292$	0	0
$293$	−12.2654	−0.716550	−0.358275	−	0.933616i	$-0.616635\pi$
$293$	−12.2654	−0.716550	−0.358275	+	0.933616i	$0.616635\pi$
$294$	0	0
$295$	−1.33485	−0.0777177
$296$	0	0
$297$	−8.84401	−0.513181
$298$	0	0
$299$	1.41958	0.0820964
$300$	0	0
$301$	2.77314	0.159841
$302$	0	0
$303$	4.81672	0.276713
$304$	0	0
$305$	−0.588418	−0.0336927
$306$	0	0
$307$	11.1991	0.639168	0.319584	−	0.947558i	$-0.396457\pi$
$307$	11.1991	0.639168	0.319584	+	0.947558i	$0.396457\pi$
$308$	0	0
$309$	18.5772	1.05682
$310$	0	0
$311$	13.1411	0.745163	0.372581	−	0.928000i	$-0.378473\pi$
$311$	13.1411	0.745163	0.372581	+	0.928000i	$0.378473\pi$
$312$	0	0
$313$	6.65713	0.376283	0.188142	−	0.982142i	$-0.439754\pi$
$313$	6.65713	0.376283	0.188142	+	0.982142i	$0.439754\pi$
$314$	0	0
$315$	−0.274656	−0.0154751
$316$	0	0
$317$	−5.47753	−0.307649	−0.153824	−	0.988098i	$-0.549159\pi$
$317$	−5.47753	−0.307649	−0.153824	+	0.988098i	$0.549159\pi$
$318$	0	0
$319$	13.5715	0.759859
$320$	0	0
$321$	−12.0782	−0.674140
$322$	0	0
$323$	−3.73251	−0.207683
$324$	0	0
$325$	4.28595	0.237742
$326$	0	0
$327$	6.74790	0.373160
$328$	0	0
$329$	−1.30163	−0.0717613
$330$	0	0
$331$	−5.88079	−0.323237	−0.161619	−	0.986853i	$-0.551671\pi$
$331$	−5.88079	−0.323237	−0.161619	+	0.986853i	$0.551671\pi$
$332$	0	0
$333$	−16.5922	−0.909248
$334$	0	0
$335$	0.374433	0.0204575
$336$	0	0
$337$	1.95874	0.106699	0.0533497	−	0.998576i	$-0.483010\pi$
$337$	1.95874	0.106699	0.0533497	+	0.998576i	$0.483010\pi$
$338$	0	0
$339$	−15.9666	−0.867188
$340$	0	0
$341$	−9.80112	−0.530761
$342$	0	0
$343$	19.2965	1.04191
$344$	0	0
$345$	0.188770	0.0101630
$346$	0	0
$347$	34.8167	1.86906	0.934530	−	0.355884i	$-0.115820\pi$
$347$	34.8167	1.86906	0.934530	+	0.355884i	$0.115820\pi$
$348$	0	0
$349$	13.2040	0.706794	0.353397	−	0.935473i	$-0.385027\pi$
$349$	13.2040	0.706794	0.353397	+	0.935473i	$0.385027\pi$
$350$	0	0
$351$	4.67609	0.249591
$352$	0	0
$353$	−21.1865	−1.12764	−0.563822	−	0.825896i	$-0.690670\pi$
$353$	−21.1865	−1.12764	−0.563822	+	0.825896i	$0.690670\pi$
$354$	0	0
$355$	−0.555213	−0.0294676
$356$	0	0
$357$	2.01337	0.106559
$358$	0	0
$359$	27.7531	1.46475	0.732375	−	0.680901i	$-0.238412\pi$
$359$	27.7531	1.46475	0.732375	+	0.680901i	$0.238412\pi$
$360$	0	0
$361$	−3.66111	−0.192690
$362$	0	0
$363$	−9.90404	−0.519827
$364$	0	0
$365$	−1.24133	−0.0649743
$366$	0	0
$367$	28.1780	1.47088	0.735439	−	0.677590i	$-0.236976\pi$
$367$	28.1780	1.47088	0.735439	+	0.677590i	$0.236976\pi$
$368$	0	0
$369$	8.59407	0.447389
$370$	0	0
$371$	−3.61727	−0.187799
$372$	0	0
$373$	25.5941	1.32521	0.662606	−	0.748968i	$-0.269450\pi$
$373$	25.5941	1.32521	0.662606	+	0.748968i	$0.269450\pi$
$374$	0	0
$375$	1.14092	0.0589169
$376$	0	0
$377$	−7.17566	−0.369565
$378$	0	0
$379$	22.5626	1.15896	0.579482	−	0.814985i	$-0.303255\pi$
$379$	22.5626	1.15896	0.579482	+	0.814985i	$0.303255\pi$
$380$	0	0
$381$	11.5074	0.589542
$382$	0	0
$383$	24.4383	1.24874	0.624369	−	0.781129i	$-0.285356\pi$
$383$	24.4383	1.24874	0.624369	+	0.781129i	$0.285356\pi$
$384$	0	0
$385$	0.279323	0.0142356
$386$	0	0
$387$	2.48314	0.126225
$388$	0	0
$389$	−5.77115	−0.292609	−0.146304	−	0.989240i	$-0.546738\pi$
$389$	−5.77115	−0.292609	−0.146304	+	0.989240i	$0.546738\pi$
$390$	0	0
$391$	1.57535	0.0796691
$392$	0	0
$393$	21.5147	1.08527
$394$	0	0
$395$	−0.293198	−0.0147524
$396$	0	0
$397$	−5.10890	−0.256408	−0.128204	−	0.991748i	$-0.540921\pi$
$397$	−5.10890	−0.256408	−0.128204	+	0.991748i	$0.540921\pi$
$398$	0	0
$399$	−8.27399	−0.414218
$400$	0	0
$401$	−9.25316	−0.462081	−0.231040	−	0.972944i	$-0.574213\pi$
$401$	−9.25316	−0.462081	−0.231040	+	0.972944i	$0.574213\pi$
$402$	0	0
$403$	5.18214	0.258141
$404$	0	0
$405$	0.159848	0.00794292
$406$	0	0
$407$	16.8741	0.836420
$408$	0	0
$409$	−38.9488	−1.92589	−0.962947	−	0.269689i	$-0.913079\pi$
$409$	−38.9488	−1.92589	−0.962947	+	0.269689i	$0.913079\pi$
$410$	0	0
$411$	−6.77078	−0.333978
$412$	0	0
$413$	−24.6939	−1.21511
$414$	0	0
$415$	0.0650579	0.00319357
$416$	0	0
$417$	4.32960	0.212021
$418$	0	0
$419$	8.87515	0.433579	0.216790	−	0.976218i	$-0.430441\pi$
$419$	8.87515	0.433579	0.216790	+	0.976218i	$0.430441\pi$
$420$	0	0
$421$	−30.7574	−1.49903	−0.749513	−	0.661990i	$-0.769712\pi$
$421$	−30.7574	−1.49903	−0.749513	+	0.661990i	$0.769712\pi$
$422$	0	0
$423$	−1.16551	−0.0566692
$424$	0	0
$425$	4.75627	0.230713
$426$	0	0
$427$	−10.8854	−0.526781
$428$	0	0
$429$	−1.65215	−0.0797666
$430$	0	0
$431$	−15.6613	−0.754379	−0.377190	−	0.926136i	$-0.623109\pi$
$431$	−15.6613	−0.754379	−0.377190	+	0.926136i	$0.623109\pi$
$432$	0	0
$433$	−15.3036	−0.735443	−0.367722	−	0.929936i	$-0.619862\pi$
$433$	−15.3036	−0.735443	−0.367722	+	0.929936i	$0.619862\pi$
$434$	0	0
$435$	−0.954192	−0.0457500
$436$	0	0
$437$	−6.47397	−0.309692
$438$	0	0
$439$	−13.3173	−0.635600	−0.317800	−	0.948158i	$-0.602944\pi$
$439$	−13.3173	−0.635600	−0.317800	+	0.948158i	$0.602944\pi$
$440$	0	0
$441$	6.09879	0.290419
$442$	0	0
$443$	−2.15838	−0.102548	−0.0512738	−	0.998685i	$-0.516328\pi$
$443$	−2.15838	−0.102548	−0.0512738	+	0.998685i	$0.516328\pi$
$444$	0	0
$445$	0.647289	0.0306845
$446$	0	0
$447$	−10.4148	−0.492604
$448$	0	0
$449$	3.77092	0.177961	0.0889804	−	0.996033i	$-0.471639\pi$
$449$	3.77092	0.177961	0.0889804	+	0.996033i	$0.471639\pi$
$450$	0	0
$451$	−8.74010	−0.411555
$452$	0	0
$453$	6.64412	0.312168
$454$	0	0
$455$	−0.147686	−0.00692364
$456$	0	0
$457$	18.6042	0.870269	0.435135	−	0.900365i	$-0.356701\pi$
$457$	18.6042	0.870269	0.435135	+	0.900365i	$0.356701\pi$
$458$	0	0
$459$	5.18921	0.242212
$460$	0	0
$461$	−14.6008	−0.680025	−0.340013	−	0.940421i	$-0.610431\pi$
$461$	−14.6008	−0.680025	−0.340013	+	0.940421i	$0.610431\pi$
$462$	0	0
$463$	11.1881	0.519955	0.259978	−	0.965615i	$-0.416285\pi$
$463$	11.1881	0.519955	0.259978	+	0.965615i	$0.416285\pi$
$464$	0	0
$465$	0.689102	0.0319563
$466$	0	0
$467$	−36.5475	−1.69122	−0.845608	−	0.533804i	$-0.820762\pi$
$467$	−36.5475	−1.69122	−0.845608	+	0.533804i	$0.820762\pi$
$468$	0	0
$469$	6.92679	0.319850
$470$	0	0
$471$	20.4038	0.940157
$472$	0	0
$473$	−2.52533	−0.116115
$474$	0	0
$475$	−19.5460	−0.896834
$476$	0	0
$477$	−3.23899	−0.148303
$478$	0	0
$479$	30.8728	1.41061	0.705307	−	0.708902i	$-0.250809\pi$
$479$	30.8728	1.41061	0.705307	+	0.708902i	$0.250809\pi$
$480$	0	0
$481$	−8.92185	−0.406802
$482$	0	0
$483$	3.49214	0.158898
$484$	0	0
$485$	−0.540728	−0.0245532
$486$	0	0
$487$	14.4283	0.653808	0.326904	−	0.945058i	$-0.393995\pi$
$487$	14.4283	0.653808	0.326904	+	0.945058i	$0.393995\pi$
$488$	0	0
$489$	27.1927	1.22970
$490$	0	0
$491$	23.7144	1.07022	0.535108	−	0.844784i	$-0.320271\pi$
$491$	23.7144	1.07022	0.535108	+	0.844784i	$0.320271\pi$
$492$	0	0
$493$	−7.96307	−0.358639
$494$	0	0
$495$	0.250113	0.0112417
$496$	0	0
$497$	−10.2711	−0.460723
$498$	0	0
$499$	−10.6859	−0.478367	−0.239184	−	0.970974i	$-0.576880\pi$
$499$	−10.6859	−0.478367	−0.239184	+	0.970974i	$0.576880\pi$
$500$	0	0
$501$	3.93370	0.175745
$502$	0	0
$503$	−37.1975	−1.65855	−0.829276	−	0.558839i	$-0.811247\pi$
$503$	−37.1975	−1.65855	−0.829276	+	0.558839i	$0.811247\pi$
$504$	0	0
$505$	−0.392091	−0.0174478
$506$	0	0
$507$	−14.5241	−0.645039
$508$	0	0
$509$	−26.2569	−1.16382	−0.581908	−	0.813255i	$-0.697694\pi$
$509$	−26.2569	−1.16382	−0.581908	+	0.813255i	$0.697694\pi$
$510$	0	0
$511$	−22.9639	−1.01586
$512$	0	0
$513$	−21.3252	−0.941532
$514$	0	0
$515$	−1.51223	−0.0666367
$516$	0	0
$517$	1.18532	0.0521302
$518$	0	0
$519$	12.5430	0.550579
$520$	0	0
$521$	−35.2655	−1.54501	−0.772505	−	0.635009i	$-0.780996\pi$
$521$	−35.2655	−1.54501	−0.772505	+	0.635009i	$0.780996\pi$
$522$	0	0
$523$	−18.6121	−0.813851	−0.406925	−	0.913461i	$-0.633399\pi$
$523$	−18.6121	−0.813851	−0.406925	+	0.913461i	$0.633399\pi$
$524$	0	0
$525$	10.5434	0.460151
$526$	0	0
$527$	5.75080	0.250509
$528$	0	0
$529$	−20.2676	−0.881199
$530$	0	0
$531$	−22.1115	−0.959557
$532$	0	0
$533$	4.62114	0.200164
$534$	0	0
$535$	0.983193	0.0425071
$536$	0	0
$537$	−0.989957	−0.0427198
$538$	0	0
$539$	−6.20242	−0.267157
$540$	0	0
$541$	16.1991	0.696452	0.348226	−	0.937411i	$-0.386784\pi$
$541$	16.1991	0.696452	0.348226	+	0.937411i	$0.386784\pi$
$542$	0	0
$543$	−1.29067	−0.0553878
$544$	0	0
$545$	−0.549294	−0.0235292
$546$	0	0
$547$	−18.1454	−0.775840	−0.387920	−	0.921693i	$-0.626806\pi$
$547$	−18.1454	−0.775840	−0.387920	+	0.921693i	$0.626806\pi$
$548$	0	0
$549$	−9.74704	−0.415993
$550$	0	0
$551$	32.7245	1.39411
$552$	0	0
$553$	−5.42399	−0.230651
$554$	0	0
$555$	−1.18639	−0.0503596
$556$	0	0
$557$	10.2921	0.436090	0.218045	−	0.975939i	$-0.430032\pi$
$557$	10.2921	0.436090	0.218045	+	0.975939i	$0.430032\pi$
$558$	0	0
$559$	1.33522	0.0564736
$560$	0	0
$561$	−1.83345	−0.0774082
$562$	0	0
$563$	40.3004	1.69846	0.849229	−	0.528024i	$-0.177067\pi$
$563$	40.3004	1.69846	0.849229	+	0.528024i	$0.177067\pi$
$564$	0	0
$565$	1.29972	0.0546796
$566$	0	0
$567$	2.95710	0.124187
$568$	0	0
$569$	−17.0428	−0.714471	−0.357235	−	0.934014i	$-0.616281\pi$
$569$	−17.0428	−0.714471	−0.357235	+	0.934014i	$0.616281\pi$
$570$	0	0
$571$	23.5579	0.985867	0.492933	−	0.870067i	$-0.335925\pi$
$571$	23.5579	0.985867	0.492933	+	0.870067i	$0.335925\pi$
$572$	0	0
$573$	−23.7905	−0.993863
$574$	0	0
$575$	8.24965	0.344034
$576$	0	0
$577$	29.2162	1.21629	0.608143	−	0.793828i	$-0.291915\pi$
$577$	29.2162	1.21629	0.608143	+	0.793828i	$0.291915\pi$
$578$	0	0
$579$	−6.61918	−0.275084
$580$	0	0
$581$	1.20353	0.0499310
$582$	0	0
$583$	3.29402	0.136425
$584$	0	0
$585$	−0.132242	−0.00546753
$586$	0	0
$587$	−37.6621	−1.55448	−0.777241	−	0.629204i	$-0.783381\pi$
$587$	−37.6621	−1.55448	−0.777241	+	0.629204i	$0.783381\pi$
$588$	0	0
$589$	−23.6331	−0.973784
$590$	0	0
$591$	−2.59689	−0.106822
$592$	0	0
$593$	−13.6068	−0.558763	−0.279381	−	0.960180i	$-0.590129\pi$
$593$	−13.6068	−0.558763	−0.279381	+	0.960180i	$0.590129\pi$
$594$	0	0
$595$	−0.163892	−0.00671893
$596$	0	0
$597$	5.76887	0.236104
$598$	0	0
$599$	2.60782	0.106552	0.0532762	−	0.998580i	$-0.483034\pi$
$599$	2.60782	0.106552	0.0532762	+	0.998580i	$0.483034\pi$
$600$	0	0
$601$	15.9829	0.651955	0.325978	−	0.945377i	$-0.394307\pi$
$601$	15.9829	0.651955	0.325978	+	0.945377i	$0.394307\pi$
$602$	0	0
$603$	6.20242	0.252582
$604$	0	0
$605$	0.806211	0.0327771
$606$	0	0
$607$	11.5886	0.470366	0.235183	−	0.971951i	$-0.424431\pi$
$607$	11.5886	0.470366	0.235183	+	0.971951i	$0.424431\pi$
$608$	0	0
$609$	−17.6520	−0.715296
$610$	0	0
$611$	−0.626712	−0.0253540
$612$	0	0
$613$	−17.5988	−0.710807	−0.355403	−	0.934713i	$-0.615657\pi$
$613$	−17.5988	−0.710807	−0.355403	+	0.934713i	$0.615657\pi$
$614$	0	0
$615$	0.614502	0.0247791
$616$	0	0
$617$	16.3705	0.659050	0.329525	−	0.944147i	$-0.393111\pi$
$617$	16.3705	0.659050	0.329525	+	0.944147i	$0.393111\pi$
$618$	0	0
$619$	−32.8326	−1.31965	−0.659826	−	0.751418i	$-0.729370\pi$
$619$	−32.8326	−1.31965	−0.659826	+	0.751418i	$0.729370\pi$
$620$	0	0
$621$	9.00058	0.361181
$622$	0	0
$623$	11.9745	0.479747
$624$	0	0
$625$	24.8606	0.994426
$626$	0	0
$627$	7.53461	0.300903
$628$	0	0
$629$	−9.90088	−0.394774
$630$	0	0
$631$	−12.2055	−0.485893	−0.242947	−	0.970040i	$-0.578114\pi$
$631$	−12.2055	−0.485893	−0.242947	+	0.970040i	$0.578114\pi$
$632$	0	0
$633$	19.0472	0.757056
$634$	0	0
$635$	−0.936729	−0.0371729
$636$	0	0
$637$	3.27940	0.129935
$638$	0	0
$639$	−9.19700	−0.363828
$640$	0	0
$641$	26.6110	1.05107	0.525536	−	0.850771i	$-0.323865\pi$
$641$	26.6110	1.05107	0.525536	+	0.850771i	$0.323865\pi$
$642$	0	0
$643$	−6.53573	−0.257744	−0.128872	−	0.991661i	$-0.541136\pi$
$643$	−6.53573	−0.257744	−0.128872	+	0.991661i	$0.541136\pi$
$644$	0	0
$645$	0.177552	0.00699110
$646$	0	0
$647$	38.2364	1.50323	0.751614	−	0.659604i	$-0.229276\pi$
$647$	38.2364	1.50323	0.751614	+	0.659604i	$0.229276\pi$
$648$	0	0
$649$	22.4872	0.882700
$650$	0	0
$651$	12.7480	0.499633
$652$	0	0
$653$	27.6139	1.08062	0.540308	−	0.841467i	$-0.318308\pi$
$653$	27.6139	1.08062	0.540308	+	0.841467i	$0.318308\pi$
$654$	0	0
$655$	−1.75134	−0.0684307
$656$	0	0
$657$	−20.5625	−0.802218
$658$	0	0
$659$	34.1044	1.32852	0.664260	−	0.747501i	$-0.268747\pi$
$659$	34.1044	1.32852	0.664260	+	0.747501i	$0.268747\pi$
$660$	0	0
$661$	16.2049	0.630296	0.315148	−	0.949043i	$-0.397946\pi$
$661$	16.2049	0.630296	0.315148	+	0.949043i	$0.397946\pi$
$662$	0	0
$663$	0.969398	0.0376483
$664$	0	0
$665$	0.673521	0.0261180
$666$	0	0
$667$	−13.8118	−0.534795
$668$	0	0
$669$	−22.8274	−0.882557
$670$	0	0
$671$	9.91266	0.382674
$672$	0	0
$673$	−26.4944	−1.02128	−0.510642	−	0.859794i	$-0.670592\pi$
$673$	−26.4944	−1.02128	−0.510642	+	0.859794i	$0.670592\pi$
$674$	0	0
$675$	27.1743	1.04594
$676$	0	0
$677$	−11.5794	−0.445031	−0.222515	−	0.974929i	$-0.571427\pi$
$677$	−11.5794	−0.445031	−0.222515	+	0.974929i	$0.571427\pi$
$678$	0	0
$679$	−10.0032	−0.383886
$680$	0	0
$681$	8.36638	0.320600
$682$	0	0
$683$	42.5758	1.62912	0.814559	−	0.580080i	$-0.196979\pi$
$683$	42.5758	1.62912	0.814559	+	0.580080i	$0.196979\pi$
$684$	0	0
$685$	0.551157	0.0210586
$686$	0	0
$687$	6.79587	0.259279
$688$	0	0
$689$	−1.74165	−0.0663515
$690$	0	0
$691$	−46.2441	−1.75921	−0.879604	−	0.475706i	$-0.842193\pi$
$691$	−46.2441	−1.75921	−0.879604	+	0.475706i	$0.842193\pi$
$692$	0	0
$693$	4.62694	0.175763
$694$	0	0
$695$	−0.352439	−0.0133688
$696$	0	0
$697$	5.12824	0.194246
$698$	0	0
$699$	14.0021	0.529608
$700$	0	0
$701$	−14.2632	−0.538714	−0.269357	−	0.963040i	$-0.586811\pi$
$701$	−14.2632	−0.538714	−0.269357	+	0.963040i	$0.586811\pi$
$702$	0	0
$703$	40.6880	1.53458
$704$	0	0
$705$	−0.0833378	−0.00313868
$706$	0	0
$707$	−7.25347	−0.272795
$708$	0	0
$709$	−3.21541	−0.120757	−0.0603786	−	0.998176i	$-0.519231\pi$
$709$	−3.21541	−0.120757	−0.0603786	+	0.998176i	$0.519231\pi$
$710$	0	0
$711$	−4.85677	−0.182143
$712$	0	0
$713$	9.97464	0.373553
$714$	0	0
$715$	0.134489	0.00502960
$716$	0	0
$717$	18.5128	0.691374
$718$	0	0
$719$	−12.8475	−0.479130	−0.239565	−	0.970880i	$-0.577005\pi$
$719$	−12.8475	−0.479130	−0.239565	+	0.970880i	$0.577005\pi$
$720$	0	0
$721$	−27.9753	−1.04186
$722$	0	0
$723$	−5.49419	−0.204331
$724$	0	0
$725$	−41.7002	−1.54871
$726$	0	0
$727$	−0.359341	−0.0133272	−0.00666362	−	0.999978i	$-0.502121\pi$
$727$	−0.359341	−0.0133272	−0.00666362	+	0.999978i	$0.502121\pi$
$728$	0	0
$729$	21.9956	0.814651
$730$	0	0
$731$	1.48173	0.0548039
$732$	0	0
$733$	22.2255	0.820917	0.410458	−	0.911879i	$-0.365369\pi$
$733$	22.2255	0.820917	0.410458	+	0.911879i	$0.365369\pi$
$734$	0	0
$735$	0.436082	0.0160851
$736$	0	0
$737$	−6.30780	−0.232351
$738$	0	0
$739$	13.0202	0.478956	0.239478	−	0.970902i	$-0.423024\pi$
$739$	13.0202	0.478956	0.239478	+	0.970902i	$0.423024\pi$
$740$	0	0
$741$	−3.98377	−0.146347
$742$	0	0
$743$	−14.7695	−0.541841	−0.270920	−	0.962602i	$-0.587328\pi$
$743$	−14.7695	−0.541841	−0.270920	+	0.962602i	$0.587328\pi$
$744$	0	0
$745$	0.847789	0.0310606
$746$	0	0
$747$	1.07767	0.0394300
$748$	0	0
$749$	18.1885	0.664593
$750$	0	0
$751$	21.7701	0.794404	0.397202	−	0.917731i	$-0.369981\pi$
$751$	21.7701	0.794404	0.397202	+	0.917731i	$0.369981\pi$
$752$	0	0
$753$	1.18444	0.0431633
$754$	0	0
$755$	−0.540846	−0.0196834
$756$	0	0
$757$	−4.14768	−0.150750	−0.0753751	−	0.997155i	$-0.524015\pi$
$757$	−4.14768	−0.150750	−0.0753751	+	0.997155i	$0.524015\pi$
$758$	0	0
$759$	−3.18008	−0.115430
$760$	0	0
$761$	−46.4511	−1.68385	−0.841926	−	0.539593i	$-0.818578\pi$
$761$	−46.4511	−1.68385	−0.841926	+	0.539593i	$0.818578\pi$
$762$	0	0
$763$	−10.1616	−0.367875
$764$	0	0
$765$	−0.146753	−0.00530587
$766$	0	0
$767$	−11.8896	−0.429310
$768$	0	0
$769$	23.4277	0.844824	0.422412	−	0.906404i	$-0.361184\pi$
$769$	23.4277	0.844824	0.422412	+	0.906404i	$0.361184\pi$
$770$	0	0
$771$	35.1624	1.26634
$772$	0	0
$773$	−44.8646	−1.61367	−0.806834	−	0.590779i	$-0.798821\pi$
$773$	−44.8646	−1.61367	−0.806834	+	0.590779i	$0.798821\pi$
$774$	0	0
$775$	30.1152	1.08177
$776$	0	0
$777$	−21.9476	−0.787366
$778$	0	0
$779$	−21.0747	−0.755078
$780$	0	0
$781$	9.35328	0.334687
$782$	0	0
$783$	−45.4960	−1.62589
$784$	0	0
$785$	−1.66091	−0.0592806
$786$	0	0
$787$	22.1150	0.788316	0.394158	−	0.919043i	$-0.371036\pi$
$787$	22.1150	0.788316	0.394158	+	0.919043i	$0.371036\pi$
$788$	0	0
$789$	1.03532	0.0368583
$790$	0	0
$791$	24.0441	0.854908
$792$	0	0
$793$	−5.24111	−0.186117
$794$	0	0
$795$	−0.231598	−0.00821392
$796$	0	0
$797$	−38.0995	−1.34955	−0.674777	−	0.738022i	$-0.735760\pi$
$797$	−38.0995	−1.34955	−0.674777	+	0.738022i	$0.735760\pi$
$798$	0	0
$799$	−0.695483	−0.0246044
$800$	0	0
$801$	10.7222	0.378852
$802$	0	0
$803$	20.9118	0.737963
$804$	0	0
$805$	−0.284268	−0.0100191
$806$	0	0
$807$	31.3285	1.10282
$808$	0	0
$809$	−40.8987	−1.43792	−0.718961	−	0.695051i	$-0.755382\pi$
$809$	−40.8987	−1.43792	−0.718961	+	0.695051i	$0.755382\pi$
$810$	0	0
$811$	−34.5210	−1.21220	−0.606099	−	0.795389i	$-0.707266\pi$
$811$	−34.5210	−1.21220	−0.606099	+	0.795389i	$0.707266\pi$
$812$	0	0
$813$	−1.04575	−0.0366761
$814$	0	0
$815$	−2.21354	−0.0775371
$816$	0	0
$817$	−6.08923	−0.213035
$818$	0	0
$819$	−2.44640	−0.0854841
$820$	0	0
$821$	2.44878	0.0854630	0.0427315	−	0.999087i	$-0.486394\pi$
$821$	2.44878	0.0854630	0.0427315	+	0.999087i	$0.486394\pi$
$822$	0	0
$823$	37.5966	1.31053	0.655267	−	0.755397i	$-0.272556\pi$
$823$	37.5966	1.31053	0.655267	+	0.755397i	$0.272556\pi$
$824$	0	0
$825$	−9.60121	−0.334271
$826$	0	0
$827$	23.7983	0.827550	0.413775	−	0.910379i	$-0.364210\pi$
$827$	23.7983	0.827550	0.413775	+	0.910379i	$0.364210\pi$
$828$	0	0
$829$	−33.5768	−1.16617	−0.583086	−	0.812411i	$-0.698155\pi$
$829$	−33.5768	−1.16617	−0.583086	+	0.812411i	$0.698155\pi$
$830$	0	0
$831$	−15.5270	−0.538626
$832$	0	0
$833$	3.63926	0.126093
$834$	0	0
$835$	−0.320212	−0.0110814
$836$	0	0
$837$	32.8564	1.13568
$838$	0	0
$839$	8.68484	0.299834	0.149917	−	0.988699i	$-0.452099\pi$
$839$	8.68484	0.299834	0.149917	+	0.988699i	$0.452099\pi$
$840$	0	0
$841$	40.8156	1.40743
$842$	0	0
$843$	14.2185	0.489713
$844$	0	0
$845$	1.18230	0.0406722
$846$	0	0
$847$	14.9144	0.512466
$848$	0	0
$849$	−9.32164	−0.319918
$850$	0	0
$851$	−17.1729	−0.588679
$852$	0	0
$853$	−7.50910	−0.257107	−0.128553	−	0.991703i	$-0.541033\pi$
$853$	−7.50910	−0.257107	−0.128553	+	0.991703i	$0.541033\pi$
$854$	0	0
$855$	0.603087	0.0206251
$856$	0	0
$857$	−32.0177	−1.09370	−0.546852	−	0.837230i	$-0.684174\pi$
$857$	−32.0177	−1.09370	−0.546852	+	0.837230i	$0.684174\pi$
$858$	0	0
$859$	−2.63522	−0.0899124	−0.0449562	−	0.998989i	$-0.514315\pi$
$859$	−2.63522	−0.0899124	−0.0449562	+	0.998989i	$0.514315\pi$
$860$	0	0
$861$	11.3679	0.387418
$862$	0	0
$863$	−46.7028	−1.58978	−0.794891	−	0.606752i	$-0.792472\pi$
$863$	−46.7028	−1.58978	−0.794891	+	0.606752i	$0.792472\pi$
$864$	0	0
$865$	−1.02103	−0.0347161
$866$	0	0
$867$	−19.0596	−0.647299
$868$	0	0
$869$	4.93929	0.167554
$870$	0	0
$871$	3.33512	0.113006
$872$	0	0
$873$	−8.95706	−0.303151
$874$	0	0
$875$	−1.71811	−0.0580826
$876$	0	0
$877$	40.0021	1.35078	0.675388	−	0.737463i	$-0.263976\pi$
$877$	40.0021	1.35078	0.675388	+	0.737463i	$0.263976\pi$
$878$	0	0
$879$	−14.5275	−0.490001
$880$	0	0
$881$	49.9427	1.68261	0.841307	−	0.540558i	$-0.181787\pi$
$881$	49.9427	1.68261	0.841307	+	0.540558i	$0.181787\pi$
$882$	0	0
$883$	−1.50369	−0.0506033	−0.0253017	−	0.999680i	$-0.508055\pi$
$883$	−1.50369	−0.0506033	−0.0253017	+	0.999680i	$0.508055\pi$
$884$	0	0
$885$	−1.58104	−0.0531460
$886$	0	0
$887$	28.1256	0.944364	0.472182	−	0.881501i	$-0.343467\pi$
$887$	28.1256	0.944364	0.472182	+	0.881501i	$0.343467\pi$
$888$	0	0
$889$	−17.3289	−0.581194
$890$	0	0
$891$	−2.69285	−0.0902139
$892$	0	0
$893$	2.85811	0.0956430
$894$	0	0
$895$	0.0805847	0.00269365
$896$	0	0
$897$	1.68140	0.0561403
$898$	0	0
$899$	−50.4196	−1.68159
$900$	0	0
$901$	−1.93276	−0.0643897
$902$	0	0
$903$	3.28461	0.109305
$904$	0	0
$905$	0.105063	0.00349242
$906$	0	0
$907$	−45.9048	−1.52424	−0.762122	−	0.647434i	$-0.775842\pi$
$907$	−45.9048	−1.52424	−0.762122	+	0.647434i	$0.775842\pi$
$908$	0	0
$909$	−6.49493	−0.215423
$910$	0	0
$911$	−16.2283	−0.537669	−0.268834	−	0.963186i	$-0.586638\pi$
$911$	−16.2283	−0.537669	−0.268834	+	0.963186i	$0.586638\pi$
$912$	0	0
$913$	−1.09598	−0.0362718
$914$	0	0
$915$	−0.696943	−0.0230402
$916$	0	0
$917$	−32.3989	−1.06990
$918$	0	0
$919$	−30.5482	−1.00769	−0.503847	−	0.863793i	$-0.668082\pi$
$919$	−30.5482	−1.00769	−0.503847	+	0.863793i	$0.668082\pi$
$920$	0	0
$921$	13.2647	0.437086
$922$	0	0
$923$	−4.94535	−0.162778
$924$	0	0
$925$	−51.8479	−1.70475
$926$	0	0
$927$	−25.0498	−0.822743
$928$	0	0
$929$	−21.1985	−0.695500	−0.347750	−	0.937587i	$-0.613054\pi$
$929$	−21.1985	−0.695500	−0.347750	+	0.937587i	$0.613054\pi$
$930$	0	0
$931$	−14.9557	−0.490152
$932$	0	0
$933$	15.5648	0.509568
$934$	0	0
$935$	0.149247	0.00488089
$936$	0	0
$937$	−9.78162	−0.319552	−0.159776	−	0.987153i	$-0.551077\pi$
$937$	−9.78162	−0.319552	−0.159776	+	0.987153i	$0.551077\pi$
$938$	0	0
$939$	7.88495	0.257316
$940$	0	0
$941$	44.0032	1.43446	0.717232	−	0.696835i	$-0.245409\pi$
$941$	44.0032	1.43446	0.717232	+	0.696835i	$0.245409\pi$
$942$	0	0
$943$	8.89483	0.289656
$944$	0	0
$945$	−0.936378	−0.0304604
$946$	0	0
$947$	20.1163	0.653690	0.326845	−	0.945078i	$-0.394014\pi$
$947$	20.1163	0.653690	0.326845	+	0.945078i	$0.394014\pi$
$948$	0	0
$949$	−11.0567	−0.358916
$950$	0	0
$951$	−6.48779	−0.210381
$952$	0	0
$953$	13.5500	0.438927	0.219464	−	0.975621i	$-0.429569\pi$
$953$	13.5500	0.438927	0.219464	+	0.975621i	$0.429569\pi$
$954$	0	0
$955$	1.93660	0.0626670
$956$	0	0
$957$	16.0746	0.519618
$958$	0	0
$959$	10.1961	0.329249
$960$	0	0
$961$	5.41221	0.174587
$962$	0	0
$963$	16.2864	0.524823
$964$	0	0
$965$	0.538816	0.0173451
$966$	0	0
$967$	−34.3648	−1.10510	−0.552549	−	0.833481i	$-0.686345\pi$
$967$	−34.3648	−1.10510	−0.552549	+	0.833481i	$0.686345\pi$
$968$	0	0
$969$	−4.42093	−0.142021
$970$	0	0
$971$	33.6312	1.07928	0.539638	−	0.841897i	$-0.318561\pi$
$971$	33.6312	1.07928	0.539638	+	0.841897i	$0.318561\pi$
$972$	0	0
$973$	−6.51992	−0.209019
$974$	0	0
$975$	5.07644	0.162576
$976$	0	0
$977$	6.55141	0.209598	0.104799	−	0.994493i	$-0.466580\pi$
$977$	6.55141	0.209598	0.104799	+	0.994493i	$0.466580\pi$
$978$	0	0
$979$	−10.9044	−0.348507
$980$	0	0
$981$	−9.09896	−0.290507
$982$	0	0
$983$	41.7573	1.33185	0.665925	−	0.746019i	$-0.268037\pi$
$983$	41.7573	1.33185	0.665925	+	0.746019i	$0.268037\pi$
$984$	0	0
$985$	0.211393	0.00673553
$986$	0	0
$987$	−1.54170	−0.0490729
$988$	0	0
$989$	2.57004	0.0817224
$990$	0	0
$991$	−17.5299	−0.556855	−0.278428	−	0.960457i	$-0.589813\pi$
$991$	−17.5299	−0.556855	−0.278428	+	0.960457i	$0.589813\pi$
$992$	0	0
$993$	−6.96542	−0.221041
$994$	0	0
$995$	−0.469599	−0.0148873
$996$	0	0
$997$	33.1176	1.04884	0.524422	−	0.851458i	$-0.324281\pi$
$997$	33.1176	1.04884	0.524422	+	0.851458i	$0.324281\pi$
$998$	0	0
$999$	−56.5674	−1.78971

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.i.1.20	yes	30
4.3	odd	2		8032.2.a.h.1.11	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.11	✓	30	4.3	odd	2
8032.2.a.i.1.20	yes	30	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q+1.18444 q^{3} -0.0964158 q^{5} -1.78364 q^{7} -1.59711 q^{9} +O(q^{10})\)	\(q+1.18444 q^{3} -0.0964158 q^{5} -1.78364 q^{7} -1.59711 q^{9} +1.62425 q^{11} -0.858788 q^{13} -0.114198 q^{15} -0.953026 q^{17} +3.91649 q^{19} -2.11260 q^{21} -1.65300 q^{23} -4.99070 q^{25} -5.44498 q^{27} +8.35557 q^{29} -6.03425 q^{31} +1.92382 q^{33} +0.171971 q^{35} +10.3889 q^{37} -1.01718 q^{39} -5.38101 q^{41} -1.55477 q^{43} +0.153987 q^{45} +0.729763 q^{47} -3.81864 q^{49} -1.12880 q^{51} +2.02803 q^{53} -0.156603 q^{55} +4.63883 q^{57} +13.8447 q^{59} +6.10292 q^{61} +2.84866 q^{63} +0.0828007 q^{65} -3.88352 q^{67} -1.95788 q^{69} +5.75853 q^{71} +12.8748 q^{73} -5.91117 q^{75} -2.89707 q^{77} +3.04097 q^{79} -1.65791 q^{81} -0.674764 q^{83} +0.0918867 q^{85} +9.89664 q^{87} -6.71352 q^{89} +1.53176 q^{91} -7.14719 q^{93} -0.377611 q^{95} +5.60829 q^{97} -2.59410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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