LMFDB - Embedded newform 32.4.b.a.17.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [32,4,Mod(17,32)] 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("32.17"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	32.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.88806112018$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 2 $ x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$0.500000 - 1.32288i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.4.b.a.17.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.29150i q^{3} +10.5830i q^{5} +8.00000 q^{7} -1.00000 q^{9} -15.8745i q^{11} -52.9150i q^{13} -56.0000 q^{15} -14.0000 q^{17} -37.0405i q^{19} +42.3320i q^{21} +152.000 q^{23} +13.0000 q^{25} +137.579i q^{27} +158.745i q^{29} -224.000 q^{31} +84.0000 q^{33} +84.6640i q^{35} -243.409i q^{37} +280.000 q^{39} -70.0000 q^{41} -439.195i q^{43} -10.5830i q^{45} -336.000 q^{47} -279.000 q^{49} -74.0810i q^{51} -31.7490i q^{53} +168.000 q^{55} +196.000 q^{57} +534.442i q^{59} -95.2470i q^{61} -8.00000 q^{63} +560.000 q^{65} +174.620i q^{67} +804.308i q^{69} +72.0000 q^{71} -294.000 q^{73} +68.7895i q^{75} -126.996i q^{77} +464.000 q^{79} -755.000 q^{81} -545.025i q^{83} -148.162i q^{85} -840.000 q^{87} +266.000 q^{89} -423.320i q^{91} -1185.30i q^{93} +392.000 q^{95} +994.000 q^{97} +15.8745i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{7} - 2 q^{9} - 112 q^{15} - 28 q^{17} + 304 q^{23} + 26 q^{25} - 448 q^{31} + 168 q^{33} + 560 q^{39} - 140 q^{41} - 672 q^{47} - 558 q^{49} + 336 q^{55} + 392 q^{57} - 16 q^{63} + 1120 q^{65}+ \cdots + 1988 q^{97}+O(q^{100}) $ 2 * q + 16 * q^7 - 2 * q^9 - 112 * q^15 - 28 * q^17 + 304 * q^23 + 26 * q^25 - 448 * q^31 + 168 * q^33 + 560 * q^39 - 140 * q^41 - 672 * q^47 - 558 * q^49 + 336 * q^55 + 392 * q^57 - 16 * q^63 + 1120 * q^65 + 144 * q^71 - 588 * q^73 + 928 * q^79 - 1510 * q^81 - 1680 * q^87 + 532 * q^89 + 784 * q^95 + 1988 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	5.29150i	1.01835i	0.860663	+	0.509175i	$0.170049\pi$
$3$	5.29150i	1.01835i	−0.860663	+	0.509175i	$0.829951\pi$
$4$	0	0
$5$	10.5830i	0.946573i	0.880909	+	0.473286i	$0.156932\pi$
$5$	10.5830i	0.946573i	−0.880909	+	0.473286i	$0.843068\pi$
$6$	0	0
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	0	0
$9$	−1.00000	−0.0370370
$10$	0	0
$11$	− 15.8745i	− 0.435122i	−0.976047	−	0.217561i	$-0.930190\pi$
$11$	− 15.8745i	− 0.435122i	0.976047	−	0.217561i	$-0.0698101\pi$
$12$	0	0
$13$	− 52.9150i	− 1.12892i	−0.825460	−	0.564461i	$-0.809084\pi$
$13$	− 52.9150i	− 1.12892i	0.825460	−	0.564461i	$-0.190916\pi$
$14$	0	0
$15$	−56.0000	−0.963943
$16$	0	0
$17$	−14.0000	−0.199735	−0.0998676	−	0.995001i	$-0.531842\pi$
$17$	−14.0000	−0.199735	−0.0998676	+	0.995001i	$0.531842\pi$
$18$	0	0
$19$	− 37.0405i	− 0.447246i	−0.974676	−	0.223623i	$-0.928212\pi$
$19$	− 37.0405i	− 0.447246i	0.974676	−	0.223623i	$-0.0717885\pi$
$20$	0	0
$21$	42.3320i	0.439886i
$22$	0	0
$23$	152.000	1.37801	0.689004	−	0.724757i	$-0.258048\pi$
$23$	152.000	1.37801	0.689004	+	0.724757i	$0.258048\pi$
$24$	0	0
$25$	13.0000	0.104000
$26$	0	0
$27$	137.579i	0.980633i
$28$	0	0
$29$	158.745i	1.01649i	0.861212	+	0.508245i	$0.169706\pi$
$29$	158.745i	1.01649i	−0.861212	+	0.508245i	$0.830294\pi$
$30$	0	0
$31$	−224.000	−1.29779	−0.648897	−	0.760877i	$-0.724769\pi$
$31$	−224.000	−1.29779	−0.648897	+	0.760877i	$0.724769\pi$
$32$	0	0
$33$	84.0000	0.443107
$34$	0	0
$35$	84.6640i	0.408881i
$36$	0	0
$37$	− 243.409i	− 1.08152i	−0.841177	−	0.540760i	$-0.818137\pi$
$37$	− 243.409i	− 1.08152i	0.841177	−	0.540760i	$-0.181863\pi$
$38$	0	0
$39$	280.000	1.14964
$40$	0	0
$41$	−70.0000	−0.266638	−0.133319	−	0.991073i	$-0.542564\pi$
$41$	−70.0000	−0.266638	−0.133319	+	0.991073i	$0.542564\pi$
$42$	0	0
$43$	− 439.195i	− 1.55759i	−0.627276	−	0.778797i	$-0.715830\pi$
$43$	− 439.195i	− 1.55759i	0.627276	−	0.778797i	$-0.284170\pi$
$44$	0	0
$45$	− 10.5830i	− 0.0350583i
$46$	0	0
$47$	−336.000	−1.04278	−0.521390	−	0.853319i	$-0.674586\pi$
$47$	−336.000	−1.04278	−0.521390	+	0.853319i	$0.674586\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	− 74.0810i	− 0.203400i
$52$	0	0
$53$	− 31.7490i	− 0.0822842i	−0.999153	−	0.0411421i	$-0.986900\pi$
$53$	− 31.7490i	− 0.0822842i	0.999153	−	0.0411421i	$-0.0130996\pi$
$54$	0	0
$55$	168.000	0.411875
$56$	0	0
$57$	196.000	0.455453
$58$	0	0
$59$	534.442i	1.17929i	0.807661	+	0.589647i	$0.200733\pi$
$59$	534.442i	1.17929i	−0.807661	+	0.589647i	$0.799267\pi$
$60$	0	0
$61$	− 95.2470i	− 0.199920i	−0.994991	−	0.0999601i	$-0.968128\pi$
$61$	− 95.2470i	− 0.199920i	0.994991	−	0.0999601i	$-0.0318715\pi$
$62$	0	0
$63$	−8.00000	−0.0159985
$64$	0	0
$65$	560.000	1.06861
$66$	0	0
$67$	174.620i	0.318406i	0.987246	+	0.159203i	$0.0508924\pi$
$67$	174.620i	0.318406i	−0.987246	+	0.159203i	$0.949108\pi$
$68$	0	0
$69$	804.308i	1.40329i
$70$	0	0
$71$	72.0000	0.120350	0.0601748	−	0.998188i	$-0.480834\pi$
$71$	72.0000	0.120350	0.0601748	+	0.998188i	$0.480834\pi$
$72$	0	0
$73$	−294.000	−0.471371	−0.235686	−	0.971829i	$-0.575734\pi$
$73$	−294.000	−0.471371	−0.235686	+	0.971829i	$0.575734\pi$
$74$	0	0
$75$	68.7895i	0.105908i
$76$	0	0
$77$	− 126.996i	− 0.187955i
$78$	0	0
$79$	464.000	0.660811	0.330406	−	0.943839i	$-0.392814\pi$
$79$	464.000	0.660811	0.330406	+	0.943839i	$0.392814\pi$
$80$	0	0
$81$	−755.000	−1.03567
$82$	0	0
$83$	− 545.025i	− 0.720774i	−0.932803	−	0.360387i	$-0.882645\pi$
$83$	− 545.025i	− 0.720774i	0.932803	−	0.360387i	$-0.117355\pi$
$84$	0	0
$85$	− 148.162i	− 0.189064i
$86$	0	0
$87$	−840.000	−1.03514
$88$	0	0
$89$	266.000	0.316808	0.158404	−	0.987374i	$-0.449365\pi$
$89$	266.000	0.316808	0.158404	+	0.987374i	$0.449365\pi$
$90$	0	0
$91$	− 423.320i	− 0.487649i
$92$	0	0
$93$	− 1185.30i	− 1.32161i
$94$	0	0
$95$	392.000	0.423351
$96$	0	0
$97$	994.000	1.04047	0.520234	−	0.854024i	$-0.325845\pi$
$97$	994.000	1.04047	0.520234	+	0.854024i	$0.325845\pi$
$98$	0	0
$99$	15.8745i	0.0161156i
$100$	0	0
$101$	− 751.393i	− 0.740262i	−0.928980	−	0.370131i	$-0.879313\pi$
$101$	− 751.393i	− 0.740262i	0.928980	−	0.370131i	$-0.120687\pi$
$102$	0	0
$103$	−1176.00	−1.12500	−0.562499	−	0.826798i	$-0.690160\pi$
$103$	−1176.00	−1.12500	−0.562499	+	0.826798i	$0.690160\pi$
$104$	0	0
$105$	−448.000	−0.416384
$106$	0	0
$107$	− 269.867i	− 0.243822i	−0.992541	−	0.121911i	$-0.961098\pi$
$107$	− 269.867i	− 0.243822i	0.992541	−	0.121911i	$-0.0389023\pi$
$108$	0	0
$109$	1894.36i	1.66465i	0.554290	+	0.832324i	$0.312990\pi$
$109$	1894.36i	1.66465i	−0.554290	+	0.832324i	$0.687010\pi$
$110$	0	0
$111$	1288.00	1.10137
$112$	0	0
$113$	−1710.00	−1.42357	−0.711784	−	0.702398i	$-0.752113\pi$
$113$	−1710.00	−1.42357	−0.711784	+	0.702398i	$0.752113\pi$
$114$	0	0
$115$	1608.62i	1.30439i
$116$	0	0
$117$	52.9150i	0.0418119i
$118$	0	0
$119$	−112.000	−0.0862775
$120$	0	0
$121$	1079.00	0.810669
$122$	0	0
$123$	− 370.405i	− 0.271531i
$124$	0	0
$125$	1460.45i	1.04502i
$126$	0	0
$127$	1664.00	1.16265	0.581323	−	0.813673i	$-0.302535\pi$
$127$	1664.00	1.16265	0.581323	+	0.813673i	$0.302535\pi$
$128$	0	0
$129$	2324.00	1.58618
$130$	0	0
$131$	− 672.021i	− 0.448204i	−0.974566	−	0.224102i	$-0.928055\pi$
$131$	− 672.021i	− 0.448204i	0.974566	−	0.224102i	$-0.0719449\pi$
$132$	0	0
$133$	− 296.324i	− 0.193192i
$134$	0	0
$135$	−1456.00	−0.928241
$136$	0	0
$137$	−1062.00	−0.662283	−0.331142	−	0.943581i	$-0.607434\pi$
$137$	−1062.00	−0.662283	−0.331142	+	0.943581i	$0.607434\pi$
$138$	0	0
$139$	2693.37i	1.64352i	0.569835	+	0.821759i	$0.307007\pi$
$139$	2693.37i	1.64352i	−0.569835	+	0.821759i	$0.692993\pi$
$140$	0	0
$141$	− 1777.94i	− 1.06191i
$142$	0	0
$143$	−840.000	−0.491219
$144$	0	0
$145$	−1680.00	−0.962182
$146$	0	0
$147$	− 1476.33i	− 0.828337i
$148$	0	0
$149$	− 793.725i	− 0.436406i	−0.975903	−	0.218203i	$-0.929980\pi$
$149$	− 793.725i	− 0.436406i	0.975903	−	0.218203i	$-0.0700195\pi$
$150$	0	0
$151$	−744.000	−0.400966	−0.200483	−	0.979697i	$-0.564251\pi$
$151$	−744.000	−0.400966	−0.200483	+	0.979697i	$0.564251\pi$
$152$	0	0
$153$	14.0000	0.00739760
$154$	0	0
$155$	− 2370.59i	− 1.22846i
$156$	0	0
$157$	− 179.911i	− 0.0914552i	−0.998954	−	0.0457276i	$-0.985439\pi$
$157$	− 179.911i	− 0.0914552i	0.998954	−	0.0457276i	$-0.0145606\pi$
$158$	0	0
$159$	168.000	0.0837941
$160$	0	0
$161$	1216.00	0.595244
$162$	0	0
$163$	− 1772.65i	− 0.851809i	−0.904768	−	0.425905i	$-0.859956\pi$
$163$	− 1772.65i	− 0.851809i	0.904768	−	0.425905i	$-0.140044\pi$
$164$	0	0
$165$	888.972i	0.419433i
$166$	0	0
$167$	1960.00	0.908200	0.454100	−	0.890951i	$-0.349961\pi$
$167$	1960.00	0.908200	0.454100	+	0.890951i	$0.349961\pi$
$168$	0	0
$169$	−603.000	−0.274465
$170$	0	0
$171$	37.0405i	0.0165647i
$172$	0	0
$173$	− 2000.19i	− 0.879026i	−0.898236	−	0.439513i	$-0.855151\pi$
$173$	− 2000.19i	− 0.879026i	0.898236	−	0.439513i	$-0.144849\pi$
$174$	0	0
$175$	104.000	0.0449238
$176$	0	0
$177$	−2828.00	−1.20094
$178$	0	0
$179$	3264.86i	1.36328i	0.731688	+	0.681639i	$0.238733\pi$
$179$	3264.86i	1.36328i	−0.731688	+	0.681639i	$0.761267\pi$
$180$	0	0
$181$	2338.84i	0.960469i	0.877140	+	0.480235i	$0.159448\pi$
$181$	2338.84i	0.960469i	−0.877140	+	0.480235i	$0.840552\pi$
$182$	0	0
$183$	504.000	0.203589
$184$	0	0
$185$	2576.00	1.02374
$186$	0	0
$187$	222.243i	0.0869092i
$188$	0	0
$189$	1100.63i	0.423594i
$190$	0	0
$191$	−3904.00	−1.47897	−0.739486	−	0.673172i	$-0.764931\pi$
$191$	−3904.00	−1.47897	−0.739486	+	0.673172i	$0.764931\pi$
$192$	0	0
$193$	3330.00	1.24196	0.620981	−	0.783826i	$-0.286734\pi$
$193$	3330.00	1.24196	0.620981	+	0.783826i	$0.286734\pi$
$194$	0	0
$195$	2963.24i	1.08822i
$196$	0	0
$197$	1195.88i	0.432502i	0.976338	+	0.216251i	$0.0693829\pi$
$197$	1195.88i	0.432502i	−0.976338	+	0.216251i	$0.930617\pi$
$198$	0	0
$199$	1736.00	0.618401	0.309200	−	0.950997i	$-0.399939\pi$
$199$	1736.00	0.618401	0.309200	+	0.950997i	$0.399939\pi$
$200$	0	0
$201$	−924.000	−0.324248
$202$	0	0
$203$	1269.96i	0.439083i
$204$	0	0
$205$	− 740.810i	− 0.252392i
$206$	0	0
$207$	−152.000	−0.0510373
$208$	0	0
$209$	−588.000	−0.194607
$210$	0	0
$211$	− 2915.62i	− 0.951277i	−0.879641	−	0.475638i	$-0.842217\pi$
$211$	− 2915.62i	− 0.951277i	0.879641	−	0.475638i	$-0.157783\pi$
$212$	0	0
$213$	380.988i	0.122558i
$214$	0	0
$215$	4648.00	1.47438
$216$	0	0
$217$	−1792.00	−0.560594
$218$	0	0
$219$	− 1555.70i	− 0.480021i
$220$	0	0
$221$	740.810i	0.225486i
$222$	0	0
$223$	−1568.00	−0.470857	−0.235428	−	0.971892i	$-0.575649\pi$
$223$	−1568.00	−0.470857	−0.235428	+	0.971892i	$0.575649\pi$
$224$	0	0
$225$	−13.0000	−0.00385185
$226$	0	0
$227$	− 1264.67i	− 0.369775i	−0.982760	−	0.184888i	$-0.940808\pi$
$227$	− 1264.67i	− 0.369775i	0.982760	−	0.184888i	$-0.0591922\pi$
$228$	0	0
$229$	− 5153.92i	− 1.48725i	−0.668595	−	0.743626i	$-0.733104\pi$
$229$	− 5153.92i	− 1.48725i	0.668595	−	0.743626i	$-0.266896\pi$
$230$	0	0
$231$	672.000	0.191404
$232$	0	0
$233$	−838.000	−0.235619	−0.117809	−	0.993036i	$-0.537587\pi$
$233$	−838.000	−0.235619	−0.117809	+	0.993036i	$0.537587\pi$
$234$	0	0
$235$	− 3555.89i	− 0.987067i
$236$	0	0
$237$	2455.26i	0.672937i
$238$	0	0
$239$	−6288.00	−1.70183	−0.850914	−	0.525305i	$-0.823951\pi$
$239$	−6288.00	−1.70183	−0.850914	+	0.525305i	$0.823951\pi$
$240$	0	0
$241$	−2926.00	−0.782076	−0.391038	−	0.920375i	$-0.627884\pi$
$241$	−2926.00	−0.782076	−0.391038	+	0.920375i	$0.627884\pi$
$242$	0	0
$243$	− 280.450i	− 0.0740364i
$244$	0	0
$245$	− 2952.66i	− 0.769953i
$246$	0	0
$247$	−1960.00	−0.504906
$248$	0	0
$249$	2884.00	0.734000
$250$	0	0
$251$	5444.96i	1.36925i	0.728894	+	0.684627i	$0.240035\pi$
$251$	5444.96i	1.36925i	−0.728894	+	0.684627i	$0.759965\pi$
$252$	0	0
$253$	− 2412.93i	− 0.599602i
$254$	0	0
$255$	784.000	0.192533
$256$	0	0
$257$	2562.00	0.621841	0.310921	−	0.950436i	$-0.399363\pi$
$257$	2562.00	0.621841	0.310921	+	0.950436i	$0.399363\pi$
$258$	0	0
$259$	− 1947.27i	− 0.467172i
$260$	0	0
$261$	− 158.745i	− 0.0376478i
$262$	0	0
$263$	5896.00	1.38237	0.691184	−	0.722679i	$-0.257089\pi$
$263$	5896.00	1.38237	0.691184	+	0.722679i	$0.257089\pi$
$264$	0	0
$265$	336.000	0.0778880
$266$	0	0
$267$	1407.54i	0.322622i
$268$	0	0
$269$	5365.58i	1.21615i	0.793878	+	0.608077i	$0.208059\pi$
$269$	5365.58i	1.21615i	−0.793878	+	0.608077i	$0.791941\pi$
$270$	0	0
$271$	1680.00	0.376578	0.188289	−	0.982114i	$-0.439706\pi$
$271$	1680.00	0.376578	0.188289	+	0.982114i	$0.439706\pi$
$272$	0	0
$273$	2240.00	0.496597
$274$	0	0
$275$	− 206.369i	− 0.0452527i
$276$	0	0
$277$	1576.87i	0.342039i	0.985268	+	0.171019i	$0.0547061\pi$
$277$	1576.87i	0.342039i	−0.985268	+	0.171019i	$0.945294\pi$
$278$	0	0
$279$	224.000	0.0480664
$280$	0	0
$281$	−2742.00	−0.582114	−0.291057	−	0.956706i	$-0.594007\pi$
$281$	−2742.00	−0.582114	−0.291057	+	0.956706i	$0.594007\pi$
$282$	0	0
$283$	2989.70i	0.627983i	0.949426	+	0.313991i	$0.101666\pi$
$283$	2989.70i	0.627983i	−0.949426	+	0.313991i	$0.898334\pi$
$284$	0	0
$285$	2074.27i	0.431120i
$286$	0	0
$287$	−560.000	−0.115177
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	5259.75i	1.05956i
$292$	0	0
$293$	9238.96i	1.84214i	0.389401	+	0.921068i	$0.372682\pi$
$293$	9238.96i	1.84214i	−0.389401	+	0.921068i	$0.627318\pi$
$294$	0	0
$295$	−5656.00	−1.11629
$296$	0	0
$297$	2184.00	0.426695
$298$	0	0
$299$	− 8043.08i	− 1.55566i
$300$	0	0
$301$	− 3513.56i	− 0.672818i
$302$	0	0
$303$	3976.00	0.753846
$304$	0	0
$305$	1008.00	0.189239
$306$	0	0
$307$	2587.54i	0.481039i	0.970644	+	0.240520i	$0.0773178\pi$
$307$	2587.54i	0.481039i	−0.970644	+	0.240520i	$0.922682\pi$
$308$	0	0
$309$	− 6222.81i	− 1.14564i
$310$	0	0
$311$	2744.00	0.500315	0.250157	−	0.968205i	$-0.419518\pi$
$311$	2744.00	0.500315	0.250157	+	0.968205i	$0.419518\pi$
$312$	0	0
$313$	2282.00	0.412097	0.206048	−	0.978542i	$-0.433940\pi$
$313$	2282.00	0.412097	0.206048	+	0.978542i	$0.433940\pi$
$314$	0	0
$315$	− 84.6640i	− 0.0151437i
$316$	0	0
$317$	− 9577.62i	− 1.69695i	−0.529237	−	0.848474i	$-0.677522\pi$
$317$	− 9577.62i	− 1.69695i	0.529237	−	0.848474i	$-0.322478\pi$
$318$	0	0
$319$	2520.00	0.442298
$320$	0	0
$321$	1428.00	0.248297
$322$	0	0
$323$	518.567i	0.0893308i
$324$	0	0
$325$	− 687.895i	− 0.117408i
$326$	0	0
$327$	−10024.0	−1.69519
$328$	0	0
$329$	−2688.00	−0.450438
$330$	0	0
$331$	− 4249.08i	− 0.705590i	−0.935701	−	0.352795i	$-0.885231\pi$
$331$	− 4249.08i	− 0.705590i	0.935701	−	0.352795i	$-0.114769\pi$
$332$	0	0
$333$	243.409i	0.0400563i
$334$	0	0
$335$	−1848.00	−0.301394
$336$	0	0
$337$	6130.00	0.990868	0.495434	−	0.868646i	$-0.335009\pi$
$337$	6130.00	0.990868	0.495434	+	0.868646i	$0.335009\pi$
$338$	0	0
$339$	− 9048.47i	− 1.44969i
$340$	0	0
$341$	3555.89i	0.564699i
$342$	0	0
$343$	−4976.00	−0.783320
$344$	0	0
$345$	−8512.00	−1.32832
$346$	0	0
$347$	2481.71i	0.383935i	0.981401	+	0.191967i	$0.0614868\pi$
$347$	2481.71i	0.383935i	−0.981401	+	0.191967i	$0.938513\pi$
$348$	0	0
$349$	328.073i	0.0503191i	0.999683	+	0.0251595i	$0.00800937\pi$
$349$	328.073i	0.0503191i	−0.999683	+	0.0251595i	$0.991991\pi$
$350$	0	0
$351$	7280.00	1.10706
$352$	0	0
$353$	−10206.0	−1.53884	−0.769420	−	0.638743i	$-0.779455\pi$
$353$	−10206.0	−1.53884	−0.769420	+	0.638743i	$0.779455\pi$
$354$	0	0
$355$	761.976i	0.113920i
$356$	0	0
$357$	− 592.648i	− 0.0878607i
$358$	0	0
$359$	3176.00	0.466916	0.233458	−	0.972367i	$-0.424996\pi$
$359$	3176.00	0.466916	0.233458	+	0.972367i	$0.424996\pi$
$360$	0	0
$361$	5487.00	0.799971
$362$	0	0
$363$	5709.53i	0.825545i
$364$	0	0
$365$	− 3111.40i	− 0.446187i
$366$	0	0
$367$	11760.0	1.67266	0.836331	−	0.548225i	$-0.184696\pi$
$367$	11760.0	1.67266	0.836331	+	0.548225i	$0.184696\pi$
$368$	0	0
$369$	70.0000	0.00987549
$370$	0	0
$371$	− 253.992i	− 0.0355434i
$372$	0	0
$373$	10974.6i	1.52344i	0.647908	+	0.761719i	$0.275644\pi$
$373$	10974.6i	1.52344i	−0.647908	+	0.761719i	$0.724356\pi$
$374$	0	0
$375$	−7728.00	−1.06419
$376$	0	0
$377$	8400.00	1.14754
$378$	0	0
$379$	3074.36i	0.416674i	0.978057	+	0.208337i	$0.0668051\pi$
$379$	3074.36i	0.416674i	−0.978057	+	0.208337i	$0.933195\pi$
$380$	0	0
$381$	8805.06i	1.18398i
$382$	0	0
$383$	−2688.00	−0.358617	−0.179309	−	0.983793i	$-0.557386\pi$
$383$	−2688.00	−0.358617	−0.179309	+	0.983793i	$0.557386\pi$
$384$	0	0
$385$	1344.00	0.177913
$386$	0	0
$387$	439.195i	0.0576887i
$388$	0	0
$389$	− 10487.8i	− 1.36697i	−0.729966	−	0.683484i	$-0.760464\pi$
$389$	− 10487.8i	− 1.36697i	0.729966	−	0.683484i	$-0.239536\pi$
$390$	0	0
$391$	−2128.00	−0.275237
$392$	0	0
$393$	3556.00	0.456429
$394$	0	0
$395$	4910.51i	0.625506i
$396$	0	0
$397$	5704.24i	0.721127i	0.932735	+	0.360564i	$0.117416\pi$
$397$	5704.24i	0.721127i	−0.932735	+	0.360564i	$0.882584\pi$
$398$	0	0
$399$	1568.00	0.196737
$400$	0	0
$401$	12402.0	1.54445	0.772227	−	0.635346i	$-0.219143\pi$
$401$	12402.0	1.54445	0.772227	+	0.635346i	$0.219143\pi$
$402$	0	0
$403$	11853.0i	1.46511i
$404$	0	0
$405$	− 7990.17i	− 0.980333i
$406$	0	0
$407$	−3864.00	−0.470593
$408$	0	0
$409$	−12278.0	−1.48437	−0.742186	−	0.670194i	$-0.766211\pi$
$409$	−12278.0	−1.48437	−0.742186	+	0.670194i	$0.766211\pi$
$410$	0	0
$411$	− 5619.58i	− 0.674436i
$412$	0	0
$413$	4275.53i	0.509407i
$414$	0	0
$415$	5768.00	0.682265
$416$	0	0
$417$	−14252.0	−1.67368
$418$	0	0
$419$	− 8207.12i	− 0.956907i	−0.878113	−	0.478454i	$-0.841198\pi$
$419$	− 8207.12i	− 0.956907i	0.878113	−	0.478454i	$-0.158802\pi$
$420$	0	0
$421$	1449.87i	0.167844i	0.996472	+	0.0839221i	$0.0267447\pi$
$421$	1449.87i	0.167844i	−0.996472	+	0.0839221i	$0.973255\pi$
$422$	0	0
$423$	336.000	0.0386215
$424$	0	0
$425$	−182.000	−0.0207725
$426$	0	0
$427$	− 761.976i	− 0.0863574i
$428$	0	0
$429$	− 4444.86i	− 0.500233i
$430$	0	0
$431$	−7632.00	−0.852948	−0.426474	−	0.904500i	$-0.640244\pi$
$431$	−7632.00	−0.852948	−0.426474	+	0.904500i	$0.640244\pi$
$432$	0	0
$433$	3794.00	0.421081	0.210540	−	0.977585i	$-0.432478\pi$
$433$	3794.00	0.421081	0.210540	+	0.977585i	$0.432478\pi$
$434$	0	0
$435$	− 8889.72i	− 0.979838i
$436$	0	0
$437$	− 5630.16i	− 0.616309i
$438$	0	0
$439$	1848.00	0.200912	0.100456	−	0.994942i	$-0.467970\pi$
$439$	1848.00	0.200912	0.100456	+	0.994942i	$0.467970\pi$
$440$	0	0
$441$	279.000	0.0301263
$442$	0	0
$443$	− 12334.5i	− 1.32287i	−0.750004	−	0.661433i	$-0.769949\pi$
$443$	− 12334.5i	− 1.32287i	0.750004	−	0.661433i	$-0.230051\pi$
$444$	0	0
$445$	2815.08i	0.299882i
$446$	0	0
$447$	4200.00	0.444414
$448$	0	0
$449$	−3582.00	−0.376492	−0.188246	−	0.982122i	$-0.560280\pi$
$449$	−3582.00	−0.376492	−0.188246	+	0.982122i	$0.560280\pi$
$450$	0	0
$451$	1111.22i	0.116020i
$452$	0	0
$453$	− 3936.88i	− 0.408324i
$454$	0	0
$455$	4480.00	0.461595
$456$	0	0
$457$	2714.00	0.277802	0.138901	−	0.990306i	$-0.455643\pi$
$457$	2714.00	0.277802	0.138901	+	0.990306i	$0.455643\pi$
$458$	0	0
$459$	− 1926.11i	− 0.195867i
$460$	0	0
$461$	− 8349.99i	− 0.843596i	−0.906690	−	0.421798i	$-0.861399\pi$
$461$	− 8349.99i	− 0.843596i	0.906690	−	0.421798i	$-0.138601\pi$
$462$	0	0
$463$	−2224.00	−0.223236	−0.111618	−	0.993751i	$-0.535603\pi$
$463$	−2224.00	−0.223236	−0.111618	+	0.993751i	$0.535603\pi$
$464$	0	0
$465$	12544.0	1.25100
$466$	0	0
$467$	10292.0i	1.01982i	0.860228	+	0.509910i	$0.170321\pi$
$467$	10292.0i	1.01982i	−0.860228	+	0.509910i	$0.829679\pi$
$468$	0	0
$469$	1396.96i	0.137538i
$470$	0	0
$471$	952.000	0.0931334
$472$	0	0
$473$	−6972.00	−0.677744
$474$	0	0
$475$	− 481.527i	− 0.0465136i
$476$	0	0
$477$	31.7490i	0.00304756i
$478$	0	0
$479$	−17696.0	−1.68800	−0.843999	−	0.536345i	$-0.819805\pi$
$479$	−17696.0	−1.68800	−0.843999	+	0.536345i	$0.819805\pi$
$480$	0	0
$481$	−12880.0	−1.22095
$482$	0	0
$483$	6434.47i	0.606166i
$484$	0	0
$485$	10519.5i	0.984879i
$486$	0	0
$487$	−1304.00	−0.121334	−0.0606672	−	0.998158i	$-0.519323\pi$
$487$	−1304.00	−0.121334	−0.0606672	+	0.998158i	$0.519323\pi$
$488$	0	0
$489$	9380.00	0.867440
$490$	0	0
$491$	16662.9i	1.53154i	0.643112	+	0.765772i	$0.277643\pi$
$491$	16662.9i	1.53154i	−0.643112	+	0.765772i	$0.722357\pi$
$492$	0	0
$493$	− 2222.43i	− 0.203029i
$494$	0	0
$495$	−168.000	−0.0152546
$496$	0	0
$497$	576.000	0.0519862
$498$	0	0
$499$	3095.53i	0.277705i	0.990313	+	0.138853i	$0.0443414\pi$
$499$	3095.53i	0.277705i	−0.990313	+	0.138853i	$0.955659\pi$
$500$	0	0
$501$	10371.3i	0.924865i
$502$	0	0
$503$	19320.0	1.71260	0.856298	−	0.516481i	$-0.172758\pi$
$503$	19320.0	1.71260	0.856298	+	0.516481i	$0.172758\pi$
$504$	0	0
$505$	7952.00	0.700712
$506$	0	0
$507$	− 3190.78i	− 0.279502i
$508$	0	0
$509$	4476.61i	0.389828i	0.980820	+	0.194914i	$0.0624427\pi$
$509$	4476.61i	0.389828i	−0.980820	+	0.194914i	$0.937557\pi$
$510$	0	0
$511$	−2352.00	−0.203613
$512$	0	0
$513$	5096.00	0.438585
$514$	0	0
$515$	− 12445.6i	− 1.06489i
$516$	0	0
$517$	5333.83i	0.453737i
$518$	0	0
$519$	10584.0	0.895156
$520$	0	0
$521$	−2982.00	−0.250756	−0.125378	−	0.992109i	$-0.540014\pi$
$521$	−2982.00	−0.250756	−0.125378	+	0.992109i	$0.540014\pi$
$522$	0	0
$523$	2016.06i	0.168559i	0.996442	+	0.0842794i	$0.0268588\pi$
$523$	2016.06i	0.168559i	−0.996442	+	0.0842794i	$0.973141\pi$
$524$	0	0
$525$	550.316i	0.0457481i
$526$	0	0
$527$	3136.00	0.259215
$528$	0	0
$529$	10937.0	0.898907
$530$	0	0
$531$	− 534.442i	− 0.0436776i
$532$	0	0
$533$	3704.05i	0.301014i
$534$	0	0
$535$	2856.00	0.230796
$536$	0	0
$537$	−17276.0	−1.38830
$538$	0	0
$539$	4428.99i	0.353933i
$540$	0	0
$541$	− 15419.4i	− 1.22539i	−0.790321	−	0.612693i	$-0.790086\pi$
$541$	− 15419.4i	− 1.22539i	0.790321	−	0.612693i	$-0.209914\pi$
$542$	0	0
$543$	−12376.0	−0.978094
$544$	0	0
$545$	−20048.0	−1.57571
$546$	0	0
$547$	− 12609.7i	− 0.985649i	−0.870129	−	0.492824i	$-0.835965\pi$
$547$	− 12609.7i	− 0.985649i	0.870129	−	0.492824i	$-0.164035\pi$
$548$	0	0
$549$	95.2470i	0.00740445i
$550$	0	0
$551$	5880.00	0.454621
$552$	0	0
$553$	3712.00	0.285444
$554$	0	0
$555$	13630.9i	1.04252i
$556$	0	0
$557$	7143.53i	0.543413i	0.962380	+	0.271706i	$0.0875880\pi$
$557$	7143.53i	0.543413i	−0.962380	+	0.271706i	$0.912412\pi$
$558$	0	0
$559$	−23240.0	−1.75840
$560$	0	0
$561$	−1176.00	−0.0885040
$562$	0	0
$563$	− 7572.14i	− 0.566834i	−0.958997	−	0.283417i	$-0.908532\pi$
$563$	− 7572.14i	− 0.566834i	0.958997	−	0.283417i	$-0.0914681\pi$
$564$	0	0
$565$	− 18096.9i	− 1.34751i
$566$	0	0
$567$	−6040.00	−0.447365
$568$	0	0
$569$	15594.0	1.14892	0.574459	−	0.818533i	$-0.305212\pi$
$569$	15594.0	1.14892	0.574459	+	0.818533i	$0.305212\pi$
$570$	0	0
$571$	− 16737.0i	− 1.22666i	−0.789827	−	0.613330i	$-0.789830\pi$
$571$	− 16737.0i	− 1.22666i	0.789827	−	0.613330i	$-0.210170\pi$
$572$	0	0
$573$	− 20658.0i	− 1.50611i
$574$	0	0
$575$	1976.00	0.143313
$576$	0	0
$577$	6594.00	0.475757	0.237879	−	0.971295i	$-0.423548\pi$
$577$	6594.00	0.475757	0.237879	+	0.971295i	$0.423548\pi$
$578$	0	0
$579$	17620.7i	1.26475i
$580$	0	0
$581$	− 4360.20i	− 0.311345i
$582$	0	0
$583$	−504.000	−0.0358037
$584$	0	0
$585$	−560.000	−0.0395780
$586$	0	0
$587$	− 23213.8i	− 1.63226i	−0.577868	−	0.816130i	$-0.696115\pi$
$587$	− 23213.8i	− 1.63226i	0.577868	−	0.816130i	$-0.303885\pi$
$588$	0	0
$589$	8297.08i	0.580433i
$590$	0	0
$591$	−6328.00	−0.440438
$592$	0	0
$593$	14322.0	0.991794	0.495897	−	0.868381i	$-0.334839\pi$
$593$	14322.0	0.991794	0.495897	+	0.868381i	$0.334839\pi$
$594$	0	0
$595$	− 1185.30i	− 0.0816679i
$596$	0	0
$597$	9186.05i	0.629749i
$598$	0	0
$599$	16088.0	1.09739	0.548696	−	0.836022i	$-0.315124\pi$
$599$	16088.0	1.09739	0.548696	+	0.836022i	$0.315124\pi$
$600$	0	0
$601$	−21238.0	−1.44146	−0.720729	−	0.693217i	$-0.756193\pi$
$601$	−21238.0	−1.44146	−0.720729	+	0.693217i	$0.756193\pi$
$602$	0	0
$603$	− 174.620i	− 0.0117928i
$604$	0	0
$605$	11419.1i	0.767357i
$606$	0	0
$607$	13664.0	0.913681	0.456841	−	0.889549i	$-0.348981\pi$
$607$	13664.0	0.913681	0.456841	+	0.889549i	$0.348981\pi$
$608$	0	0
$609$	−6720.00	−0.447140
$610$	0	0
$611$	17779.4i	1.17722i
$612$	0	0
$613$	− 20393.5i	− 1.34369i	−0.740690	−	0.671846i	$-0.765501\pi$
$613$	− 20393.5i	− 1.34369i	0.740690	−	0.671846i	$-0.234499\pi$
$614$	0	0
$615$	3920.00	0.257024
$616$	0	0
$617$	−3782.00	−0.246771	−0.123385	−	0.992359i	$-0.539375\pi$
$617$	−3782.00	−0.246771	−0.123385	+	0.992359i	$0.539375\pi$
$618$	0	0
$619$	5825.94i	0.378295i	0.981949	+	0.189147i	$0.0605724\pi$
$619$	5825.94i	0.378295i	−0.981949	+	0.189147i	$0.939428\pi$
$620$	0	0
$621$	20912.0i	1.35132i
$622$	0	0
$623$	2128.00	0.136848
$624$	0	0
$625$	−13831.0	−0.885184
$626$	0	0
$627$	− 3111.40i	− 0.198178i
$628$	0	0
$629$	3407.73i	0.216017i
$630$	0	0
$631$	−2056.00	−0.129712	−0.0648558	−	0.997895i	$-0.520659\pi$
$631$	−2056.00	−0.129712	−0.0648558	+	0.997895i	$0.520659\pi$
$632$	0	0
$633$	15428.0	0.968733
$634$	0	0
$635$	17610.1i	1.10053i
$636$	0	0
$637$	14763.3i	0.918278i
$638$	0	0
$639$	−72.0000	−0.00445740
$640$	0	0
$641$	11842.0	0.729689	0.364845	−	0.931068i	$-0.381122\pi$
$641$	11842.0	0.729689	0.364845	+	0.931068i	$0.381122\pi$
$642$	0	0
$643$	− 16250.2i	− 0.996649i	−0.866991	−	0.498325i	$-0.833949\pi$
$643$	− 16250.2i	− 0.996649i	0.866991	−	0.498325i	$-0.166051\pi$
$644$	0	0
$645$	24594.9i	1.50143i
$646$	0	0
$647$	−19320.0	−1.17395	−0.586976	−	0.809604i	$-0.699682\pi$
$647$	−19320.0	−1.17395	−0.586976	+	0.809604i	$0.699682\pi$
$648$	0	0
$649$	8484.00	0.513137
$650$	0	0
$651$	− 9482.37i	− 0.570881i
$652$	0	0
$653$	2317.68i	0.138894i	0.997586	+	0.0694470i	$0.0221235\pi$
$653$	2317.68i	0.138894i	−0.997586	+	0.0694470i	$0.977877\pi$
$654$	0	0
$655$	7112.00	0.424258
$656$	0	0
$657$	294.000	0.0174582
$658$	0	0
$659$	27732.8i	1.63932i	0.572847	+	0.819662i	$0.305839\pi$
$659$	27732.8i	1.63932i	−0.572847	+	0.819662i	$0.694161\pi$
$660$	0	0
$661$	− 22467.7i	− 1.32208i	−0.750352	−	0.661039i	$-0.770116\pi$
$661$	− 22467.7i	− 1.32208i	0.750352	−	0.661039i	$-0.229884\pi$
$662$	0	0
$663$	−3920.00	−0.229623
$664$	0	0
$665$	3136.00	0.182870
$666$	0	0
$667$	24129.3i	1.40073i
$668$	0	0
$669$	− 8297.08i	− 0.479497i
$670$	0	0
$671$	−1512.00	−0.0869897
$672$	0	0
$673$	−10078.0	−0.577234	−0.288617	−	0.957445i	$-0.593195\pi$
$673$	−10078.0	−0.577234	−0.288617	+	0.957445i	$0.593195\pi$
$674$	0	0
$675$	1788.53i	0.101986i
$676$	0	0
$677$	− 16160.2i	− 0.917413i	−0.888588	−	0.458707i	$-0.848313\pi$
$677$	− 16160.2i	− 0.917413i	0.888588	−	0.458707i	$-0.151687\pi$
$678$	0	0
$679$	7952.00	0.449440
$680$	0	0
$681$	6692.00	0.376561
$682$	0	0
$683$	− 16356.0i	− 0.916320i	−0.888870	−	0.458160i	$-0.848509\pi$
$683$	− 16356.0i	− 0.916320i	0.888870	−	0.458160i	$-0.151491\pi$
$684$	0	0
$685$	− 11239.2i	− 0.626899i
$686$	0	0
$687$	27272.0	1.51454
$688$	0	0
$689$	−1680.00	−0.0928925
$690$	0	0
$691$	− 29246.1i	− 1.61009i	−0.593211	−	0.805047i	$-0.702140\pi$
$691$	− 29246.1i	− 1.61009i	0.593211	−	0.805047i	$-0.297860\pi$
$692$	0	0
$693$	126.996i	0.00696130i
$694$	0	0
$695$	−28504.0	−1.55571
$696$	0	0
$697$	980.000	0.0532570
$698$	0	0
$699$	− 4434.28i	− 0.239943i
$700$	0	0
$701$	− 2465.84i	− 0.132858i	−0.997791	−	0.0664290i	$-0.978839\pi$
$701$	− 2465.84i	− 0.132858i	0.997791	−	0.0664290i	$-0.0211606\pi$
$702$	0	0
$703$	−9016.00	−0.483705
$704$	0	0
$705$	18816.0	1.00518
$706$	0	0
$707$	− 6011.15i	− 0.319763i
$708$	0	0
$709$	31674.9i	1.67782i	0.544267	+	0.838912i	$0.316808\pi$
$709$	31674.9i	1.67782i	−0.544267	+	0.838912i	$0.683192\pi$
$710$	0	0
$711$	−464.000	−0.0244745
$712$	0	0
$713$	−34048.0	−1.78837
$714$	0	0
$715$	− 8889.72i	− 0.464975i
$716$	0	0
$717$	− 33273.0i	− 1.73306i
$718$	0	0
$719$	9296.00	0.482173	0.241086	−	0.970504i	$-0.422496\pi$
$719$	9296.00	0.482173	0.241086	+	0.970504i	$0.422496\pi$
$720$	0	0
$721$	−9408.00	−0.485953
$722$	0	0
$723$	− 15482.9i	− 0.796427i
$724$	0	0
$725$	2063.69i	0.105715i
$726$	0	0
$727$	−21672.0	−1.10560	−0.552799	−	0.833315i	$-0.686440\pi$
$727$	−21672.0	−1.10560	−0.552799	+	0.833315i	$0.686440\pi$
$728$	0	0
$729$	−18901.0	−0.960270
$730$	0	0
$731$	6148.73i	0.311106i
$732$	0	0
$733$	9471.79i	0.477283i	0.971108	+	0.238642i	$0.0767021\pi$
$733$	9471.79i	0.477283i	−0.971108	+	0.238642i	$0.923298\pi$
$734$	0	0
$735$	15624.0	0.784082
$736$	0	0
$737$	2772.00	0.138545
$738$	0	0
$739$	6863.08i	0.341627i	0.985303	+	0.170814i	$0.0546396\pi$
$739$	6863.08i	0.341627i	−0.985303	+	0.170814i	$0.945360\pi$
$740$	0	0
$741$	− 10371.3i	− 0.514171i
$742$	0	0
$743$	−17432.0	−0.860724	−0.430362	−	0.902656i	$-0.641614\pi$
$743$	−17432.0	−0.860724	−0.430362	+	0.902656i	$0.641614\pi$
$744$	0	0
$745$	8400.00	0.413090
$746$	0	0
$747$	545.025i	0.0266953i
$748$	0	0
$749$	− 2158.93i	− 0.105321i
$750$	0	0
$751$	11632.0	0.565190	0.282595	−	0.959239i	$-0.408805\pi$
$751$	11632.0	0.565190	0.282595	+	0.959239i	$0.408805\pi$
$752$	0	0
$753$	−28812.0	−1.39438
$754$	0	0
$755$	− 7873.76i	− 0.379543i
$756$	0	0
$757$	16731.7i	0.803336i	0.915785	+	0.401668i	$0.131569\pi$
$757$	16731.7i	0.803336i	−0.915785	+	0.401668i	$0.868431\pi$
$758$	0	0
$759$	12768.0	0.610605
$760$	0	0
$761$	39466.0	1.87995	0.939975	−	0.341244i	$-0.110848\pi$
$761$	39466.0	1.87995	0.939975	+	0.341244i	$0.110848\pi$
$762$	0	0
$763$	15154.9i	0.719060i
$764$	0	0
$765$	148.162i	0.00700237i
$766$	0	0
$767$	28280.0	1.33133
$768$	0	0
$769$	35266.0	1.65374	0.826869	−	0.562395i	$-0.190120\pi$
$769$	35266.0	1.65374	0.826869	+	0.562395i	$0.190120\pi$
$770$	0	0
$771$	13556.8i	0.633252i
$772$	0	0
$773$	− 16244.9i	− 0.755872i	−0.925832	−	0.377936i	$-0.876634\pi$
$773$	− 16244.9i	− 0.755872i	0.925832	−	0.377936i	$-0.123366\pi$
$774$	0	0
$775$	−2912.00	−0.134970
$776$	0	0
$777$	10304.0	0.475745
$778$	0	0
$779$	2592.84i	0.119253i
$780$	0	0
$781$	− 1142.96i	− 0.0523668i
$782$	0	0
$783$	−21840.0	−0.996805
$784$	0	0
$785$	1904.00	0.0865690
$786$	0	0
$787$	34844.5i	1.57824i	0.614240	+	0.789119i	$0.289463\pi$
$787$	34844.5i	1.57824i	−0.614240	+	0.789119i	$0.710537\pi$
$788$	0	0
$789$	31198.7i	1.40774i
$790$	0	0
$791$	−13680.0	−0.614924
$792$	0	0
$793$	−5040.00	−0.225694
$794$	0	0
$795$	1777.94i	0.0793172i
$796$	0	0
$797$	− 2550.50i	− 0.113354i	−0.998393	−	0.0566772i	$-0.981949\pi$
$797$	− 2550.50i	− 0.113354i	0.998393	−	0.0566772i	$-0.0180506\pi$
$798$	0	0
$799$	4704.00	0.208280
$800$	0	0
$801$	−266.000	−0.0117336
$802$	0	0
$803$	4667.11i	0.205104i
$804$	0	0
$805$	12868.9i	0.563441i
$806$	0	0
$807$	−28392.0	−1.23847
$808$	0	0
$809$	−24390.0	−1.05996	−0.529979	−	0.848010i	$-0.677800\pi$
$809$	−24390.0	−1.05996	−0.529979	+	0.848010i	$0.677800\pi$
$810$	0	0
$811$	− 9582.91i	− 0.414922i	−0.978243	−	0.207461i	$-0.933480\pi$
$811$	− 9582.91i	− 0.414922i	0.978243	−	0.207461i	$-0.0665200\pi$
$812$	0	0
$813$	8889.72i	0.383489i
$814$	0	0
$815$	18760.0	0.806300
$816$	0	0
$817$	−16268.0	−0.696628
$818$	0	0
$819$	423.320i	0.0180611i
$820$	0	0
$821$	8773.31i	0.372948i	0.982460	+	0.186474i	$0.0597061\pi$
$821$	8773.31i	0.372948i	−0.982460	+	0.186474i	$0.940294\pi$
$822$	0	0
$823$	21688.0	0.918586	0.459293	−	0.888285i	$-0.348103\pi$
$823$	21688.0	0.918586	0.459293	+	0.888285i	$0.348103\pi$
$824$	0	0
$825$	1092.00	0.0460831
$826$	0	0
$827$	− 19446.3i	− 0.817670i	−0.912608	−	0.408835i	$-0.865935\pi$
$827$	− 19446.3i	− 0.817670i	0.912608	−	0.408835i	$-0.134065\pi$
$828$	0	0
$829$	19546.8i	0.818925i	0.912327	+	0.409462i	$0.134284\pi$
$829$	19546.8i	0.818925i	−0.912327	+	0.409462i	$0.865716\pi$
$830$	0	0
$831$	−8344.00	−0.348315
$832$	0	0
$833$	3906.00	0.162467
$834$	0	0
$835$	20742.7i	0.859677i
$836$	0	0
$837$	− 30817.7i	− 1.27266i
$838$	0	0
$839$	18760.0	0.771951	0.385976	−	0.922509i	$-0.373865\pi$
$839$	18760.0	0.771951	0.385976	+	0.922509i	$0.373865\pi$
$840$	0	0
$841$	−811.000	−0.0332527
$842$	0	0
$843$	− 14509.3i	− 0.592796i
$844$	0	0
$845$	− 6381.55i	− 0.259801i
$846$	0	0
$847$	8632.00	0.350176
$848$	0	0
$849$	−15820.0	−0.639506
$850$	0	0
$851$	− 36998.2i	− 1.49034i
$852$	0	0
$853$	− 28732.9i	− 1.15333i	−0.816979	−	0.576667i	$-0.804353\pi$
$853$	− 28732.9i	− 1.15333i	0.816979	−	0.576667i	$-0.195647\pi$
$854$	0	0
$855$	−392.000	−0.0156797
$856$	0	0
$857$	8778.00	0.349884	0.174942	−	0.984579i	$-0.444026\pi$
$857$	8778.00	0.349884	0.174942	+	0.984579i	$0.444026\pi$
$858$	0	0
$859$	− 5646.03i	− 0.224261i	−0.993693	−	0.112130i	$-0.964233\pi$
$859$	− 5646.03i	− 0.224261i	0.993693	−	0.112130i	$-0.0357675\pi$
$860$	0	0
$861$	− 2963.24i	− 0.117290i
$862$	0	0
$863$	9312.00	0.367305	0.183652	−	0.982991i	$-0.441208\pi$
$863$	9312.00	0.367305	0.183652	+	0.982991i	$0.441208\pi$
$864$	0	0
$865$	21168.0	0.832062
$866$	0	0
$867$	− 24960.0i	− 0.977724i
$868$	0	0
$869$	− 7365.77i	− 0.287534i
$870$	0	0
$871$	9240.00	0.359455
$872$	0	0
$873$	−994.000	−0.0385359
$874$	0	0
$875$	11683.6i	0.451405i
$876$	0	0
$877$	− 137.579i	− 0.00529728i	−0.999996	−	0.00264864i	$-0.999157\pi$
$877$	− 137.579i	− 0.00529728i	0.999996	−	0.00264864i	$-0.000843089\pi$
$878$	0	0
$879$	−48888.0	−1.87594
$880$	0	0
$881$	−31150.0	−1.19123	−0.595613	−	0.803272i	$-0.703091\pi$
$881$	−31150.0	−1.19123	−0.595613	+	0.803272i	$0.703091\pi$
$882$	0	0
$883$	12577.9i	0.479366i	0.970851	+	0.239683i	$0.0770435\pi$
$883$	12577.9i	0.479366i	−0.970851	+	0.239683i	$0.922957\pi$
$884$	0	0
$885$	− 29928.7i	− 1.13677i
$886$	0	0
$887$	−37128.0	−1.40545	−0.702726	−	0.711460i	$-0.748034\pi$
$887$	−37128.0	−1.40545	−0.702726	+	0.711460i	$0.748034\pi$
$888$	0	0
$889$	13312.0	0.502216
$890$	0	0
$891$	11985.3i	0.450641i
$892$	0	0
$893$	12445.6i	0.466379i
$894$	0	0
$895$	−34552.0	−1.29044
$896$	0	0
$897$	42560.0	1.58421
$898$	0	0
$899$	− 35558.9i	− 1.31919i
$900$	0	0
$901$	444.486i	0.0164351i
$902$	0	0
$903$	18592.0	0.685164
$904$	0	0
$905$	−24752.0	−0.909154
$906$	0	0
$907$	35204.4i	1.28880i	0.764688	+	0.644400i	$0.222893\pi$
$907$	35204.4i	1.28880i	−0.764688	+	0.644400i	$0.777107\pi$
$908$	0	0
$909$	751.393i	0.0274171i
$910$	0	0
$911$	10512.0	0.382303	0.191152	−	0.981561i	$-0.438778\pi$
$911$	10512.0	0.382303	0.191152	+	0.981561i	$0.438778\pi$
$912$	0	0
$913$	−8652.00	−0.313625
$914$	0	0
$915$	5333.83i	0.192712i
$916$	0	0
$917$	− 5376.17i	− 0.193606i
$918$	0	0
$919$	46104.0	1.65488	0.827438	−	0.561557i	$-0.189798\pi$
$919$	46104.0	1.65488	0.827438	+	0.561557i	$0.189798\pi$
$920$	0	0
$921$	−13692.0	−0.489866
$922$	0	0
$923$	− 3809.88i	− 0.135865i
$924$	0	0
$925$	− 3164.32i	− 0.112478i
$926$	0	0
$927$	1176.00	0.0416666
$928$	0	0
$929$	−5726.00	−0.202222	−0.101111	−	0.994875i	$-0.532240\pi$
$929$	−5726.00	−0.202222	−0.101111	+	0.994875i	$0.532240\pi$
$930$	0	0
$931$	10334.3i	0.363795i
$932$	0	0
$933$	14519.9i	0.509496i
$934$	0	0
$935$	−2352.00	−0.0822659
$936$	0	0
$937$	1274.00	0.0444181	0.0222091	−	0.999753i	$-0.492930\pi$
$937$	1274.00	0.0444181	0.0222091	+	0.999753i	$0.492930\pi$
$938$	0	0
$939$	12075.2i	0.419659i
$940$	0	0
$941$	26446.9i	0.916201i	0.888900	+	0.458101i	$0.151470\pi$
$941$	26446.9i	0.916201i	−0.888900	+	0.458101i	$0.848530\pi$
$942$	0	0
$943$	−10640.0	−0.367430
$944$	0	0
$945$	−11648.0	−0.400962
$946$	0	0
$947$	23922.9i	0.820897i	0.911884	+	0.410448i	$0.134628\pi$
$947$	23922.9i	0.820897i	−0.911884	+	0.410448i	$0.865372\pi$
$948$	0	0
$949$	15557.0i	0.532141i
$950$	0	0
$951$	50680.0	1.72809
$952$	0	0
$953$	38250.0	1.30015	0.650073	−	0.759872i	$-0.274738\pi$
$953$	38250.0	1.30015	0.650073	+	0.759872i	$0.274738\pi$
$954$	0	0
$955$	− 41316.1i	− 1.39995i
$956$	0	0
$957$	13334.6i	0.450414i
$958$	0	0
$959$	−8496.00	−0.286079
$960$	0	0
$961$	20385.0	0.684267
$962$	0	0
$963$	269.867i	0.00903046i
$964$	0	0
$965$	35241.4i	1.17561i
$966$	0	0
$967$	−4664.00	−0.155103	−0.0775513	−	0.996988i	$-0.524710\pi$
$967$	−4664.00	−0.155103	−0.0775513	+	0.996988i	$0.524710\pi$
$968$	0	0
$969$	−2744.00	−0.0909701
$970$	0	0
$971$	30971.2i	1.02360i	0.859106	+	0.511798i	$0.171020\pi$
$971$	30971.2i	1.02360i	−0.859106	+	0.511798i	$0.828980\pi$
$972$	0	0
$973$	21547.0i	0.709933i
$974$	0	0
$975$	3640.00	0.119562
$976$	0	0
$977$	−4814.00	−0.157639	−0.0788196	−	0.996889i	$-0.525115\pi$
$977$	−4814.00	−0.157639	−0.0788196	+	0.996889i	$0.525115\pi$
$978$	0	0
$979$	− 4222.62i	− 0.137850i
$980$	0	0
$981$	− 1894.36i	− 0.0616536i
$982$	0	0
$983$	12376.0	0.401560	0.200780	−	0.979636i	$-0.435652\pi$
$983$	12376.0	0.401560	0.200780	+	0.979636i	$0.435652\pi$
$984$	0	0
$985$	−12656.0	−0.409395
$986$	0	0
$987$	− 14223.6i	− 0.458704i
$988$	0	0
$989$	− 66757.6i	− 2.14638i
$990$	0	0
$991$	−45344.0	−1.45348	−0.726740	−	0.686912i	$-0.758966\pi$
$991$	−45344.0	−1.45348	−0.726740	+	0.686912i	$0.758966\pi$
$992$	0	0
$993$	22484.0	0.718538
$994$	0	0
$995$	18372.1i	0.585361i
$996$	0	0
$997$	26002.4i	0.825984i	0.910735	+	0.412992i	$0.135516\pi$
$997$	26002.4i	0.825984i	−0.910735	+	0.412992i	$0.864484\pi$
$998$	0	0
$999$	33488.0	1.06057

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.b.a.17.2		2
3.2	odd	2		288.4.d.a.145.1		2
4.3	odd	2		8.4.b.a.5.2	yes	2
5.2	odd	4		800.4.f.a.49.4		4
5.3	odd	4		800.4.f.a.49.1		4
5.4	even	2		800.4.d.a.401.1		2
8.3	odd	2		8.4.b.a.5.1	✓	2
8.5	even	2	inner	32.4.b.a.17.1		2
12.11	even	2		72.4.d.b.37.1		2
16.3	odd	4		256.4.a.l.1.2		2
16.5	even	4		256.4.a.j.1.2		2
16.11	odd	4		256.4.a.l.1.1		2
16.13	even	4		256.4.a.j.1.1		2
20.3	even	4		200.4.f.a.149.4		4
20.7	even	4		200.4.f.a.149.1		4
20.19	odd	2		200.4.d.a.101.1		2
24.5	odd	2		288.4.d.a.145.2		2
24.11	even	2		72.4.d.b.37.2		2
40.3	even	4		200.4.f.a.149.2		4
40.13	odd	4		800.4.f.a.49.3		4
40.19	odd	2		200.4.d.a.101.2		2
40.27	even	4		200.4.f.a.149.3		4
40.29	even	2		800.4.d.a.401.2		2
40.37	odd	4		800.4.f.a.49.2		4
48.5	odd	4		2304.4.a.v.1.2		2
48.11	even	4		2304.4.a.bn.1.2		2
48.29	odd	4		2304.4.a.v.1.1		2
48.35	even	4		2304.4.a.bn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.4.b.a.5.1	✓	2	8.3	odd	2
8.4.b.a.5.2	yes	2	4.3	odd	2
32.4.b.a.17.1		2	8.5	even	2	inner
32.4.b.a.17.2		2	1.1	even	1	trivial
72.4.d.b.37.1		2	12.11	even	2
72.4.d.b.37.2		2	24.11	even	2
200.4.d.a.101.1		2	20.19	odd	2
200.4.d.a.101.2		2	40.19	odd	2
200.4.f.a.149.1		4	20.7	even	4
200.4.f.a.149.2		4	40.3	even	4
200.4.f.a.149.3		4	40.27	even	4
200.4.f.a.149.4		4	20.3	even	4
256.4.a.j.1.1		2	16.13	even	4
256.4.a.j.1.2		2	16.5	even	4
256.4.a.l.1.1		2	16.11	odd	4
256.4.a.l.1.2		2	16.3	odd	4
288.4.d.a.145.1		2	3.2	odd	2
288.4.d.a.145.2		2	24.5	odd	2
800.4.d.a.401.1		2	5.4	even	2
800.4.d.a.401.2		2	40.29	even	2
800.4.f.a.49.1		4	5.3	odd	4
800.4.f.a.49.2		4	40.37	odd	4
800.4.f.a.49.3		4	40.13	odd	4
800.4.f.a.49.4		4	5.2	odd	4
2304.4.a.v.1.1		2	48.29	odd	4
2304.4.a.v.1.2		2	48.5	odd	4
2304.4.a.bn.1.1		2	48.35	even	4
2304.4.a.bn.1.2		2	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	32.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.29150i q^{3} +10.5830i q^{5} +8.00000 q^{7} -1.00000 q^{9} -15.8745i q^{11} -52.9150i q^{13} -56.0000 q^{15} -14.0000 q^{17} -37.0405i q^{19} +42.3320i q^{21} +152.000 q^{23} +13.0000 q^{25} +137.579i q^{27} +158.745i q^{29} -224.000 q^{31} +84.0000 q^{33} +84.6640i q^{35} -243.409i q^{37} +280.000 q^{39} -70.0000 q^{41} -439.195i q^{43} -10.5830i q^{45} -336.000 q^{47} -279.000 q^{49} -74.0810i q^{51} -31.7490i q^{53} +168.000 q^{55} +196.000 q^{57} +534.442i q^{59} -95.2470i q^{61} -8.00000 q^{63} +560.000 q^{65} +174.620i q^{67} +804.308i q^{69} +72.0000 q^{71} -294.000 q^{73} +68.7895i q^{75} -126.996i q^{77} +464.000 q^{79} -755.000 q^{81} -545.025i q^{83} -148.162i q^{85} -840.000 q^{87} +266.000 q^{89} -423.320i q^{91} -1185.30i q^{93} +392.000 q^{95} +994.000 q^{97} +15.8745i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{7} - 2 q^{9} - 112 q^{15} - 28 q^{17} + 304 q^{23} + 26 q^{25} - 448 q^{31} + 168 q^{33} + 560 q^{39} - 140 q^{41} - 672 q^{47} - 558 q^{49} + 336 q^{55} + 392 q^{57} - 16 q^{63} + 1120 q^{65}+ \cdots + 1988 q^{97}+O(q^{100}) \) 2 * q + 16 * q^7 - 2 * q^9 - 112 * q^15 - 28 * q^17 + 304 * q^23 + 26 * q^25 - 448 * q^31 + 168 * q^33 + 560 * q^39 - 140 * q^41 - 672 * q^47 - 558 * q^49 + 336 * q^55 + 392 * q^57 - 16 * q^63 + 1120 * q^65 + 144 * q^71 - 588 * q^73 + 928 * q^79 - 1510 * q^81 - 1680 * q^87 + 532 * q^89 + 784 * q^95 + 1988 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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