LMFDB - Embedded newform 2112.4.a.bi.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2112,4,Mod(1,2112)] 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("2112.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,-10,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$124.612033932$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{97}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 24 $ x^2 - x - 24
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 66)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.42443$ of defining polynomial
Character	$\chi$	$=$	2112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -14.8489 q^{5} +30.5466 q^{7} +9.00000 q^{9} -11.0000 q^{11} -75.9420 q^{13} -44.5466 q^{15} -59.6977 q^{17} -128.489 q^{19} +91.6397 q^{21} +51.7557 q^{23} +95.4886 q^{25} +27.0000 q^{27} +227.093 q^{29} -221.582 q^{31} -33.0000 q^{33} -453.582 q^{35} +221.163 q^{37} -227.826 q^{39} +170.373 q^{41} +318.791 q^{43} -133.640 q^{45} +165.453 q^{47} +590.093 q^{49} -179.093 q^{51} +39.7100 q^{53} +163.337 q^{55} -385.466 q^{57} +363.279 q^{59} -456.058 q^{61} +274.919 q^{63} +1127.65 q^{65} +46.4183 q^{67} +155.267 q^{69} -323.267 q^{71} +1000.33 q^{73} +286.466 q^{75} -336.012 q^{77} +557.081 q^{79} +81.0000 q^{81} -688.141 q^{83} +886.443 q^{85} +681.279 q^{87} +860.418 q^{89} -2319.77 q^{91} -664.745 q^{93} +1907.91 q^{95} -521.420 q^{97} -99.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 10 q^{5} + 2 q^{7} + 18 q^{9} - 22 q^{11} - 14 q^{13} - 30 q^{15} - 80 q^{17} - 60 q^{19} + 6 q^{21} + 202 q^{23} - 6 q^{25} + 54 q^{27} + 336 q^{29} - 128 q^{31} - 66 q^{33} - 592 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 10 * q^5 + 2 * q^7 + 18 * q^9 - 22 * q^11 - 14 * q^13 - 30 * q^15 - 80 * q^17 - 60 * q^19 + 6 * q^21 + 202 * q^23 - 6 * q^25 + 54 * q^27 + 336 * q^29 - 128 * q^31 - 66 * q^33 - 592 * q^35 - 188 * q^37 - 42 * q^39 - 132 * q^41 + 480 * q^43 - 90 * q^45 + 390 * q^47 + 1062 * q^49 - 240 * q^51 - 610 * q^53 + 110 * q^55 - 180 * q^57 + 372 * q^59 - 1050 * q^61 + 18 * q^63 + 1428 * q^65 + 408 * q^67 + 606 * q^69 - 942 * q^71 + 740 * q^73 - 18 * q^75 - 22 * q^77 + 1646 * q^79 + 162 * q^81 - 352 * q^83 + 788 * q^85 + 1008 * q^87 + 2036 * q^89 - 4088 * q^91 - 384 * q^93 + 2240 * q^95 + 336 * q^97 - 198 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−14.8489	−1.32812	−0.664061	−	0.747678i	$-0.731169\pi$
$5$	−14.8489	−1.32812	−0.664061	+	0.747678i	$0.731169\pi$
$6$	0	0
$7$	30.5466	1.64936	0.824680	−	0.565600i	$-0.191355\pi$
$7$	30.5466	1.64936	0.824680	+	0.565600i	$0.191355\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−75.9420	−1.62019	−0.810097	−	0.586296i	$-0.800586\pi$
$13$	−75.9420	−1.62019	−0.810097	+	0.586296i	$0.800586\pi$
$14$	0	0
$15$	−44.5466	−0.766792
$16$	0	0
$17$	−59.6977	−0.851695	−0.425848	−	0.904795i	$-0.640024\pi$
$17$	−59.6977	−0.851695	−0.425848	+	0.904795i	$0.640024\pi$
$18$	0	0
$19$	−128.489	−1.55144	−0.775718	−	0.631079i	$-0.782612\pi$
$19$	−128.489	−1.55144	−0.775718	+	0.631079i	$0.782612\pi$
$20$	0	0
$21$	91.6397	0.952258
$22$	0	0
$23$	51.7557	0.469209	0.234605	−	0.972091i	$-0.424620\pi$
$23$	51.7557	0.469209	0.234605	+	0.972091i	$0.424620\pi$
$24$	0	0
$25$	95.4886	0.763909
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	227.093	1.45414	0.727071	−	0.686562i	$-0.240881\pi$
$29$	227.093	1.45414	0.727071	+	0.686562i	$0.240881\pi$
$30$	0	0
$31$	−221.582	−1.28378	−0.641891	−	0.766796i	$-0.721850\pi$
$31$	−221.582	−1.28378	−0.641891	+	0.766796i	$0.721850\pi$
$32$	0	0
$33$	−33.0000	−0.174078
$34$	0	0
$35$	−453.582	−2.19055
$36$	0	0
$37$	221.163	0.982677	0.491338	−	0.870969i	$-0.336508\pi$
$37$	221.163	0.982677	0.491338	+	0.870969i	$0.336508\pi$
$38$	0	0
$39$	−227.826	−0.935419
$40$	0	0
$41$	170.373	0.648969	0.324484	−	0.945891i	$-0.394809\pi$
$41$	170.373	0.648969	0.324484	+	0.945891i	$0.394809\pi$
$42$	0	0
$43$	318.791	1.13058	0.565292	−	0.824891i	$-0.308763\pi$
$43$	318.791	1.13058	0.565292	+	0.824891i	$0.308763\pi$
$44$	0	0
$45$	−133.640	−0.442707
$46$	0	0
$47$	165.453	0.513486	0.256743	−	0.966480i	$-0.417351\pi$
$47$	165.453	0.513486	0.256743	+	0.966480i	$0.417351\pi$
$48$	0	0
$49$	590.093	1.72039
$50$	0	0
$51$	−179.093	−0.491727
$52$	0	0
$53$	39.7100	0.102917	0.0514584	−	0.998675i	$-0.483613\pi$
$53$	39.7100	0.102917	0.0514584	+	0.998675i	$0.483613\pi$
$54$	0	0
$55$	163.337	0.400444
$56$	0	0
$57$	−385.466	−0.895723
$58$	0	0
$59$	363.279	0.801609	0.400805	−	0.916164i	$-0.368731\pi$
$59$	363.279	0.801609	0.400805	+	0.916164i	$0.368731\pi$
$60$	0	0
$61$	−456.058	−0.957250	−0.478625	−	0.878019i	$-0.658865\pi$
$61$	−456.058	−0.957250	−0.478625	+	0.878019i	$0.658865\pi$
$62$	0	0
$63$	274.919	0.549787
$64$	0	0
$65$	1127.65	2.15182
$66$	0	0
$67$	46.4183	0.0846402	0.0423201	−	0.999104i	$-0.486525\pi$
$67$	46.4183	0.0846402	0.0423201	+	0.999104i	$0.486525\pi$
$68$	0	0
$69$	155.267	0.270898
$70$	0	0
$71$	−323.267	−0.540349	−0.270174	−	0.962811i	$-0.587081\pi$
$71$	−323.267	−0.540349	−0.270174	+	0.962811i	$0.587081\pi$
$72$	0	0
$73$	1000.33	1.60383	0.801914	−	0.597440i	$-0.203815\pi$
$73$	1000.33	1.60383	0.801914	+	0.597440i	$0.203815\pi$
$74$	0	0
$75$	286.466	0.441043
$76$	0	0
$77$	−336.012	−0.497301
$78$	0	0
$79$	557.081	0.793373	0.396687	−	0.917954i	$-0.370160\pi$
$79$	557.081	0.793373	0.396687	+	0.917954i	$0.370160\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−688.141	−0.910039	−0.455020	−	0.890481i	$-0.650368\pi$
$83$	−688.141	−0.910039	−0.455020	+	0.890481i	$0.650368\pi$
$84$	0	0
$85$	886.443	1.13116
$86$	0	0
$87$	681.279	0.839550
$88$	0	0
$89$	860.418	1.02477	0.512383	−	0.858757i	$-0.328763\pi$
$89$	860.418	1.02477	0.512383	+	0.858757i	$0.328763\pi$
$90$	0	0
$91$	−2319.77	−2.67228
$92$	0	0
$93$	−664.745	−0.741192
$94$	0	0
$95$	1907.91	2.06050
$96$	0	0
$97$	−521.420	−0.545796	−0.272898	−	0.962043i	$-0.587982\pi$
$97$	−521.420	−0.545796	−0.272898	+	0.962043i	$0.587982\pi$
$98$	0	0
$99$	−99.0000	−0.100504
$100$	0	0
$101$	394.580	0.388734	0.194367	−	0.980929i	$-0.437735\pi$
$101$	394.580	0.388734	0.194367	+	0.980929i	$0.437735\pi$
$102$	0	0
$103$	−740.165	−0.708065	−0.354032	−	0.935233i	$-0.615190\pi$
$103$	−740.165	−0.708065	−0.354032	+	0.935233i	$0.615190\pi$
$104$	0	0
$105$	−1360.75	−1.26472
$106$	0	0
$107$	1350.47	1.22014	0.610068	−	0.792349i	$-0.291142\pi$
$107$	1350.47	1.22014	0.610068	+	0.792349i	$0.291142\pi$
$108$	0	0
$109$	−267.942	−0.235451	−0.117726	−	0.993046i	$-0.537560\pi$
$109$	−267.942	−0.235451	−0.117726	+	0.993046i	$0.537560\pi$
$110$	0	0
$111$	663.490	0.567349
$112$	0	0
$113$	−1464.47	−1.21916	−0.609582	−	0.792723i	$-0.708663\pi$
$113$	−1464.47	−1.21916	−0.609582	+	0.792723i	$0.708663\pi$
$114$	0	0
$115$	−768.513	−0.623167
$116$	0	0
$117$	−683.478	−0.540065
$118$	0	0
$119$	−1823.56	−1.40475
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	511.118	0.374682
$124$	0	0
$125$	438.211	0.313558
$126$	0	0
$127$	−1345.99	−0.940452	−0.470226	−	0.882546i	$-0.655828\pi$
$127$	−1345.99	−0.940452	−0.470226	+	0.882546i	$0.655828\pi$
$128$	0	0
$129$	956.373	0.652743
$130$	0	0
$131$	45.6731	0.0304617	0.0152308	−	0.999884i	$-0.495152\pi$
$131$	45.6731	0.0304617	0.0152308	+	0.999884i	$0.495152\pi$
$132$	0	0
$133$	−3924.89	−2.55888
$134$	0	0
$135$	−400.919	−0.255597
$136$	0	0
$137$	−300.815	−0.187594	−0.0937971	−	0.995591i	$-0.529900\pi$
$137$	−300.815	−0.187594	−0.0937971	+	0.995591i	$0.529900\pi$
$138$	0	0
$139$	−739.701	−0.451372	−0.225686	−	0.974200i	$-0.572462\pi$
$139$	−739.701	−0.451372	−0.225686	+	0.974200i	$0.572462\pi$
$140$	0	0
$141$	496.360	0.296462
$142$	0	0
$143$	835.362	0.488507
$144$	0	0
$145$	−3372.07	−1.93128
$146$	0	0
$147$	1770.28	0.993267
$148$	0	0
$149$	646.232	0.355311	0.177656	−	0.984093i	$-0.443149\pi$
$149$	646.232	0.355311	0.177656	+	0.984093i	$0.443149\pi$
$150$	0	0
$151$	−113.710	−0.0612821	−0.0306410	−	0.999530i	$-0.509755\pi$
$151$	−113.710	−0.0612821	−0.0306410	+	0.999530i	$0.509755\pi$
$152$	0	0
$153$	−537.279	−0.283898
$154$	0	0
$155$	3290.24	1.70502
$156$	0	0
$157$	681.230	0.346294	0.173147	−	0.984896i	$-0.444606\pi$
$157$	681.230	0.346294	0.173147	+	0.984896i	$0.444606\pi$
$158$	0	0
$159$	119.130	0.0594191
$160$	0	0
$161$	1580.96	0.773895
$162$	0	0
$163$	−875.072	−0.420497	−0.210248	−	0.977648i	$-0.567427\pi$
$163$	−875.072	−0.420497	−0.210248	+	0.977648i	$0.567427\pi$
$164$	0	0
$165$	490.012	0.231196
$166$	0	0
$167$	3797.09	1.75945	0.879724	−	0.475484i	$-0.157727\pi$
$167$	3797.09	1.75945	0.879724	+	0.475484i	$0.157727\pi$
$168$	0	0
$169$	3570.19	1.62503
$170$	0	0
$171$	−1156.40	−0.517146
$172$	0	0
$173$	2847.75	1.25150	0.625752	−	0.780022i	$-0.284792\pi$
$173$	2847.75	1.25150	0.625752	+	0.780022i	$0.284792\pi$
$174$	0	0
$175$	2916.85	1.25996
$176$	0	0
$177$	1089.84	0.462809
$178$	0	0
$179$	4067.05	1.69824	0.849121	−	0.528198i	$-0.177132\pi$
$179$	4067.05	1.69824	0.849121	+	0.528198i	$0.177132\pi$
$180$	0	0
$181$	93.3216	0.0383234	0.0191617	−	0.999816i	$-0.493900\pi$
$181$	93.3216	0.0383234	0.0191617	+	0.999816i	$0.493900\pi$
$182$	0	0
$183$	−1368.17	−0.552668
$184$	0	0
$185$	−3284.02	−1.30512
$186$	0	0
$187$	656.675	0.256796
$188$	0	0
$189$	824.757	0.317419
$190$	0	0
$191$	2716.86	1.02924	0.514620	−	0.857418i	$-0.327933\pi$
$191$	2716.86	1.02924	0.514620	+	0.857418i	$0.327933\pi$
$192$	0	0
$193$	1560.02	0.581828	0.290914	−	0.956749i	$-0.406041\pi$
$193$	1560.02	0.581828	0.290914	+	0.956749i	$0.406041\pi$
$194$	0	0
$195$	3382.96	1.24235
$196$	0	0
$197$	3274.21	1.18415	0.592076	−	0.805882i	$-0.298308\pi$
$197$	3274.21	1.18415	0.592076	+	0.805882i	$0.298308\pi$
$198$	0	0
$199$	1422.58	0.506755	0.253378	−	0.967367i	$-0.418459\pi$
$199$	1422.58	0.506755	0.253378	+	0.967367i	$0.418459\pi$
$200$	0	0
$201$	139.255	0.0488671
$202$	0	0
$203$	6936.92	2.39840
$204$	0	0
$205$	−2529.84	−0.861910
$206$	0	0
$207$	465.801	0.156403
$208$	0	0
$209$	1413.37	0.467776
$210$	0	0
$211$	3241.59	1.05763	0.528816	−	0.848737i	$-0.322636\pi$
$211$	3241.59	1.05763	0.528816	+	0.848737i	$0.322636\pi$
$212$	0	0
$213$	−969.801	−0.311970
$214$	0	0
$215$	−4733.68	−1.50155
$216$	0	0
$217$	−6768.56	−2.11742
$218$	0	0
$219$	3000.98	0.925970
$220$	0	0
$221$	4533.56	1.37991
$222$	0	0
$223$	5127.63	1.53978	0.769892	−	0.638174i	$-0.220310\pi$
$223$	5127.63	1.53978	0.769892	+	0.638174i	$0.220310\pi$
$224$	0	0
$225$	859.397	0.254636
$226$	0	0
$227$	−4401.22	−1.28687	−0.643434	−	0.765502i	$-0.722491\pi$
$227$	−4401.22	−1.28687	−0.643434	+	0.765502i	$0.722491\pi$
$228$	0	0
$229$	−2566.02	−0.740470	−0.370235	−	0.928938i	$-0.620723\pi$
$229$	−2566.02	−0.740470	−0.370235	+	0.928938i	$0.620723\pi$
$230$	0	0
$231$	−1008.04	−0.287117
$232$	0	0
$233$	−1318.14	−0.370619	−0.185310	−	0.982680i	$-0.559329\pi$
$233$	−1318.14	−0.370619	−0.185310	+	0.982680i	$0.559329\pi$
$234$	0	0
$235$	−2456.79	−0.681973
$236$	0	0
$237$	1671.24	0.458054
$238$	0	0
$239$	−94.8998	−0.0256843	−0.0128422	−	0.999918i	$-0.504088\pi$
$239$	−94.8998	−0.0256843	−0.0128422	+	0.999918i	$0.504088\pi$
$240$	0	0
$241$	1608.86	0.430025	0.215013	−	0.976611i	$-0.431021\pi$
$241$	1608.86	0.430025	0.215013	+	0.976611i	$0.431021\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−8762.21	−2.28489
$246$	0	0
$247$	9757.68	2.51363
$248$	0	0
$249$	−2064.42	−0.525411
$250$	0	0
$251$	121.557	0.0305682	0.0152841	−	0.999883i	$-0.495135\pi$
$251$	121.557	0.0305682	0.0152841	+	0.999883i	$0.495135\pi$
$252$	0	0
$253$	−569.313	−0.141472
$254$	0	0
$255$	2659.33	0.653073
$256$	0	0
$257$	7834.01	1.90145	0.950724	−	0.310037i	$-0.100341\pi$
$257$	7834.01	1.90145	0.950724	+	0.310037i	$0.100341\pi$
$258$	0	0
$259$	6755.79	1.62079
$260$	0	0
$261$	2043.84	0.484714
$262$	0	0
$263$	−3098.24	−0.726408	−0.363204	−	0.931710i	$-0.618317\pi$
$263$	−3098.24	−0.726408	−0.363204	+	0.931710i	$0.618317\pi$
$264$	0	0
$265$	−589.648	−0.136686
$266$	0	0
$267$	2581.25	0.591649
$268$	0	0
$269$	4276.28	0.969254	0.484627	−	0.874721i	$-0.338955\pi$
$269$	4276.28	0.969254	0.484627	+	0.874721i	$0.338955\pi$
$270$	0	0
$271$	−2036.53	−0.456496	−0.228248	−	0.973603i	$-0.573300\pi$
$271$	−2036.53	−0.456496	−0.228248	+	0.973603i	$0.573300\pi$
$272$	0	0
$273$	−6959.30	−1.54284
$274$	0	0
$275$	−1050.37	−0.230327
$276$	0	0
$277$	−7614.43	−1.65165	−0.825825	−	0.563927i	$-0.809290\pi$
$277$	−7614.43	−1.65165	−0.825825	+	0.563927i	$0.809290\pi$
$278$	0	0
$279$	−1994.24	−0.427927
$280$	0	0
$281$	1135.83	0.241132	0.120566	−	0.992705i	$-0.461529\pi$
$281$	1135.83	0.241132	0.120566	+	0.992705i	$0.461529\pi$
$282$	0	0
$283$	7932.11	1.66613	0.833065	−	0.553175i	$-0.186584\pi$
$283$	7932.11	1.66613	0.833065	+	0.553175i	$0.186584\pi$
$284$	0	0
$285$	5723.73	1.18963
$286$	0	0
$287$	5204.30	1.07038
$288$	0	0
$289$	−1349.18	−0.274615
$290$	0	0
$291$	−1564.26	−0.315115
$292$	0	0
$293$	9328.00	1.85989	0.929945	−	0.367699i	$-0.119854\pi$
$293$	9328.00	1.85989	0.929945	+	0.367699i	$0.119854\pi$
$294$	0	0
$295$	−5394.28	−1.06464
$296$	0	0
$297$	−297.000	−0.0580259
$298$	0	0
$299$	−3930.43	−0.760210
$300$	0	0
$301$	9737.97	1.86474
$302$	0	0
$303$	1183.74	0.224436
$304$	0	0
$305$	6771.94	1.27134
$306$	0	0
$307$	−5303.69	−0.985986	−0.492993	−	0.870033i	$-0.664097\pi$
$307$	−5303.69	−0.985986	−0.492993	+	0.870033i	$0.664097\pi$
$308$	0	0
$309$	−2220.50	−0.408801
$310$	0	0
$311$	−1475.45	−0.269020	−0.134510	−	0.990912i	$-0.542946\pi$
$311$	−1475.45	−0.269020	−0.134510	+	0.990912i	$0.542946\pi$
$312$	0	0
$313$	−5888.33	−1.06335	−0.531675	−	0.846949i	$-0.678437\pi$
$313$	−5888.33	−1.06335	−0.531675	+	0.846949i	$0.678437\pi$
$314$	0	0
$315$	−4082.24	−0.730184
$316$	0	0
$317$	6164.47	1.09221	0.546106	−	0.837716i	$-0.316110\pi$
$317$	6164.47	1.09221	0.546106	+	0.837716i	$0.316110\pi$
$318$	0	0
$319$	−2498.02	−0.438441
$320$	0	0
$321$	4051.40	0.704446
$322$	0	0
$323$	7670.47	1.32135
$324$	0	0
$325$	−7251.59	−1.23768
$326$	0	0
$327$	−803.826	−0.135938
$328$	0	0
$329$	5054.04	0.846924
$330$	0	0
$331$	−6305.27	−1.04704	−0.523518	−	0.852015i	$-0.675381\pi$
$331$	−6305.27	−1.04704	−0.523518	+	0.852015i	$0.675381\pi$
$332$	0	0
$333$	1990.47	0.327559
$334$	0	0
$335$	−689.258	−0.112413
$336$	0	0
$337$	−9751.27	−1.57622	−0.788109	−	0.615535i	$-0.788940\pi$
$337$	−9751.27	−1.57622	−0.788109	+	0.615535i	$0.788940\pi$
$338$	0	0
$339$	−4393.40	−0.703885
$340$	0	0
$341$	2437.40	0.387075
$342$	0	0
$343$	7547.85	1.18818
$344$	0	0
$345$	−2305.54	−0.359786
$346$	0	0
$347$	528.091	0.0816986	0.0408493	−	0.999165i	$-0.486994\pi$
$347$	528.091	0.0816986	0.0408493	+	0.999165i	$0.486994\pi$
$348$	0	0
$349$	−1249.01	−0.191570	−0.0957851	−	0.995402i	$-0.530536\pi$
$349$	−1249.01	−0.191570	−0.0957851	+	0.995402i	$0.530536\pi$
$350$	0	0
$351$	−2050.43	−0.311806
$352$	0	0
$353$	−7643.17	−1.15242	−0.576211	−	0.817301i	$-0.695469\pi$
$353$	−7643.17	−1.15242	−0.576211	+	0.817301i	$0.695469\pi$
$354$	0	0
$355$	4800.15	0.717649
$356$	0	0
$357$	−5470.68	−0.811034
$358$	0	0
$359$	−4503.57	−0.662088	−0.331044	−	0.943615i	$-0.607401\pi$
$359$	−4503.57	−0.662088	−0.331044	+	0.943615i	$0.607401\pi$
$360$	0	0
$361$	9650.31	1.40696
$362$	0	0
$363$	363.000	0.0524864
$364$	0	0
$365$	−14853.7	−2.13008
$366$	0	0
$367$	−2435.78	−0.346448	−0.173224	−	0.984882i	$-0.555418\pi$
$367$	−2435.78	−0.346448	−0.173224	+	0.984882i	$0.555418\pi$
$368$	0	0
$369$	1533.35	0.216323
$370$	0	0
$371$	1213.01	0.169747
$372$	0	0
$373$	4559.47	0.632923	0.316462	−	0.948605i	$-0.397505\pi$
$373$	4559.47	0.632923	0.316462	+	0.948605i	$0.397505\pi$
$374$	0	0
$375$	1314.63	0.181033
$376$	0	0
$377$	−17245.9	−2.35599
$378$	0	0
$379$	11267.6	1.52712	0.763562	−	0.645734i	$-0.223449\pi$
$379$	11267.6	1.52712	0.763562	+	0.645734i	$0.223449\pi$
$380$	0	0
$381$	−4037.97	−0.542970
$382$	0	0
$383$	10053.0	1.34121	0.670607	−	0.741813i	$-0.266034\pi$
$383$	10053.0	1.34121	0.670607	+	0.741813i	$0.266034\pi$
$384$	0	0
$385$	4989.40	0.660476
$386$	0	0
$387$	2869.12	0.376862
$388$	0	0
$389$	−12206.5	−1.59099	−0.795493	−	0.605963i	$-0.792788\pi$
$389$	−12206.5	−1.59099	−0.795493	+	0.605963i	$0.792788\pi$
$390$	0	0
$391$	−3089.70	−0.399623
$392$	0	0
$393$	137.019	0.0175870
$394$	0	0
$395$	−8272.01	−1.05370
$396$	0	0
$397$	−10461.8	−1.32258	−0.661288	−	0.750132i	$-0.729990\pi$
$397$	−10461.8	−1.32258	−0.661288	+	0.750132i	$0.729990\pi$
$398$	0	0
$399$	−11774.7	−1.47737
$400$	0	0
$401$	2482.85	0.309197	0.154598	−	0.987977i	$-0.450592\pi$
$401$	2482.85	0.309197	0.154598	+	0.987977i	$0.450592\pi$
$402$	0	0
$403$	16827.4	2.07998
$404$	0	0
$405$	−1202.76	−0.147569
$406$	0	0
$407$	−2432.80	−0.296288
$408$	0	0
$409$	11488.8	1.38896	0.694478	−	0.719514i	$-0.255635\pi$
$409$	11488.8	1.38896	0.694478	+	0.719514i	$0.255635\pi$
$410$	0	0
$411$	−902.446	−0.108308
$412$	0	0
$413$	11096.9	1.32214
$414$	0	0
$415$	10218.1	1.20864
$416$	0	0
$417$	−2219.10	−0.260599
$418$	0	0
$419$	−10355.4	−1.20738	−0.603691	−	0.797218i	$-0.706304\pi$
$419$	−10355.4	−1.20738	−0.603691	+	0.797218i	$0.706304\pi$
$420$	0	0
$421$	−1613.90	−0.186833	−0.0934166	−	0.995627i	$-0.529779\pi$
$421$	−1613.90	−0.186833	−0.0934166	+	0.995627i	$0.529779\pi$
$422$	0	0
$423$	1489.08	0.171162
$424$	0	0
$425$	−5700.45	−0.650618
$426$	0	0
$427$	−13931.0	−1.57885
$428$	0	0
$429$	2506.09	0.282040
$430$	0	0
$431$	−11675.3	−1.30483	−0.652414	−	0.757863i	$-0.726244\pi$
$431$	−11675.3	−1.30483	−0.652414	+	0.757863i	$0.726244\pi$
$432$	0	0
$433$	−237.852	−0.0263983	−0.0131991	−	0.999913i	$-0.504202\pi$
$433$	−237.852	−0.0263983	−0.0131991	+	0.999913i	$0.504202\pi$
$434$	0	0
$435$	−10116.2	−1.11502
$436$	0	0
$437$	−6650.02	−0.727948
$438$	0	0
$439$	10351.3	1.12538	0.562688	−	0.826670i	$-0.309767\pi$
$439$	10351.3	1.12538	0.562688	+	0.826670i	$0.309767\pi$
$440$	0	0
$441$	5310.84	0.573463
$442$	0	0
$443$	13124.1	1.40755	0.703775	−	0.710423i	$-0.251496\pi$
$443$	13124.1	1.40755	0.703775	+	0.710423i	$0.251496\pi$
$444$	0	0
$445$	−12776.2	−1.36101
$446$	0	0
$447$	1938.70	0.205139
$448$	0	0
$449$	−5724.96	−0.601732	−0.300866	−	0.953666i	$-0.597276\pi$
$449$	−5724.96	−0.601732	−0.300866	+	0.953666i	$0.597276\pi$
$450$	0	0
$451$	−1874.10	−0.195672
$452$	0	0
$453$	−341.130	−0.0353812
$454$	0	0
$455$	34445.9	3.54912
$456$	0	0
$457$	12331.4	1.26223	0.631117	−	0.775688i	$-0.282597\pi$
$457$	12331.4	1.26223	0.631117	+	0.775688i	$0.282597\pi$
$458$	0	0
$459$	−1611.84	−0.163909
$460$	0	0
$461$	−15334.6	−1.54925	−0.774623	−	0.632423i	$-0.782061\pi$
$461$	−15334.6	−1.54925	−0.774623	+	0.632423i	$0.782061\pi$
$462$	0	0
$463$	574.120	0.0576276	0.0288138	−	0.999585i	$-0.490827\pi$
$463$	574.120	0.0576276	0.0288138	+	0.999585i	$0.490827\pi$
$464$	0	0
$465$	9870.71	0.984394
$466$	0	0
$467$	−4939.95	−0.489494	−0.244747	−	0.969587i	$-0.578705\pi$
$467$	−4939.95	−0.489494	−0.244747	+	0.969587i	$0.578705\pi$
$468$	0	0
$469$	1417.92	0.139602
$470$	0	0
$471$	2043.69	0.199933
$472$	0	0
$473$	−3506.70	−0.340884
$474$	0	0
$475$	−12269.2	−1.18516
$476$	0	0
$477$	357.390	0.0343056
$478$	0	0
$479$	−11656.5	−1.11190	−0.555949	−	0.831217i	$-0.687645\pi$
$479$	−11656.5	−1.11190	−0.555949	+	0.831217i	$0.687645\pi$
$480$	0	0
$481$	−16795.6	−1.59213
$482$	0	0
$483$	4742.88	0.446808
$484$	0	0
$485$	7742.49	0.724883
$486$	0	0
$487$	18250.7	1.69818	0.849092	−	0.528244i	$-0.177149\pi$
$487$	18250.7	1.69818	0.849092	+	0.528244i	$0.177149\pi$
$488$	0	0
$489$	−2625.22	−0.242774
$490$	0	0
$491$	11076.7	1.01809	0.509046	−	0.860740i	$-0.329998\pi$
$491$	11076.7	1.01809	0.509046	+	0.860740i	$0.329998\pi$
$492$	0	0
$493$	−13556.9	−1.23849
$494$	0	0
$495$	1470.04	0.133481
$496$	0	0
$497$	−9874.70	−0.891229
$498$	0	0
$499$	10128.9	0.908682	0.454341	−	0.890828i	$-0.349875\pi$
$499$	10128.9	0.908682	0.454341	+	0.890828i	$0.349875\pi$
$500$	0	0
$501$	11391.3	1.01582
$502$	0	0
$503$	−2321.33	−0.205772	−0.102886	−	0.994693i	$-0.532808\pi$
$503$	−2321.33	−0.205772	−0.102886	+	0.994693i	$0.532808\pi$
$504$	0	0
$505$	−5859.06	−0.516287
$506$	0	0
$507$	10710.6	0.938211
$508$	0	0
$509$	21841.5	1.90198	0.950989	−	0.309223i	$-0.100069\pi$
$509$	21841.5	1.90198	0.950989	+	0.309223i	$0.100069\pi$
$510$	0	0
$511$	30556.6	2.64529
$512$	0	0
$513$	−3469.19	−0.298574
$514$	0	0
$515$	10990.6	0.940396
$516$	0	0
$517$	−1819.99	−0.154822
$518$	0	0
$519$	8543.24	0.722556
$520$	0	0
$521$	18027.6	1.51593	0.757967	−	0.652293i	$-0.226193\pi$
$521$	18027.6	1.51593	0.757967	+	0.652293i	$0.226193\pi$
$522$	0	0
$523$	−22833.2	−1.90904	−0.954518	−	0.298154i	$-0.903629\pi$
$523$	−22833.2	−1.90904	−0.954518	+	0.298154i	$0.903629\pi$
$524$	0	0
$525$	8750.55	0.727438
$526$	0	0
$527$	13227.9	1.09339
$528$	0	0
$529$	−9488.35	−0.779843
$530$	0	0
$531$	3269.51	0.267203
$532$	0	0
$533$	−12938.4	−1.05146
$534$	0	0
$535$	−20052.9	−1.62049
$536$	0	0
$537$	12201.1	0.980481
$538$	0	0
$539$	−6491.02	−0.518717
$540$	0	0
$541$	9807.37	0.779393	0.389697	−	0.920943i	$-0.372580\pi$
$541$	9807.37	0.779393	0.389697	+	0.920943i	$0.372580\pi$
$542$	0	0
$543$	279.965	0.0221260
$544$	0	0
$545$	3978.63	0.312708
$546$	0	0
$547$	6911.43	0.540240	0.270120	−	0.962827i	$-0.412937\pi$
$547$	6911.43	0.540240	0.270120	+	0.962827i	$0.412937\pi$
$548$	0	0
$549$	−4104.52	−0.319083
$550$	0	0
$551$	−29178.9	−2.25601
$552$	0	0
$553$	17016.9	1.30856
$554$	0	0
$555$	−9852.07	−0.753509
$556$	0	0
$557$	−7061.58	−0.537179	−0.268589	−	0.963255i	$-0.586557\pi$
$557$	−7061.58	−0.537179	−0.268589	+	0.963255i	$0.586557\pi$
$558$	0	0
$559$	−24209.6	−1.83177
$560$	0	0
$561$	1970.02	0.148261
$562$	0	0
$563$	−5314.28	−0.397815	−0.198908	−	0.980018i	$-0.563739\pi$
$563$	−5314.28	−0.397815	−0.198908	+	0.980018i	$0.563739\pi$
$564$	0	0
$565$	21745.7	1.61920
$566$	0	0
$567$	2474.27	0.183262
$568$	0	0
$569$	2050.43	0.151070	0.0755348	−	0.997143i	$-0.475934\pi$
$569$	2050.43	0.151070	0.0755348	+	0.997143i	$0.475934\pi$
$570$	0	0
$571$	12369.2	0.906545	0.453272	−	0.891372i	$-0.350257\pi$
$571$	12369.2	0.906545	0.453272	+	0.891372i	$0.350257\pi$
$572$	0	0
$573$	8150.57	0.594232
$574$	0	0
$575$	4942.08	0.358433
$576$	0	0
$577$	1400.34	0.101034	0.0505172	−	0.998723i	$-0.483913\pi$
$577$	1400.34	0.101034	0.0505172	+	0.998723i	$0.483913\pi$
$578$	0	0
$579$	4680.06	0.335918
$580$	0	0
$581$	−21020.3	−1.50098
$582$	0	0
$583$	−436.810	−0.0310306
$584$	0	0
$585$	10148.9	0.717272
$586$	0	0
$587$	111.701	0.00785418	0.00392709	−	0.999992i	$-0.498750\pi$
$587$	111.701	0.00785418	0.00392709	+	0.999992i	$0.498750\pi$
$588$	0	0
$589$	28470.7	1.99171
$590$	0	0
$591$	9822.64	0.683671
$592$	0	0
$593$	11987.1	0.830103	0.415051	−	0.909798i	$-0.363764\pi$
$593$	11987.1	0.830103	0.415051	+	0.909798i	$0.363764\pi$
$594$	0	0
$595$	27077.8	1.86568
$596$	0	0
$597$	4267.75	0.292575
$598$	0	0
$599$	2042.69	0.139336	0.0696680	−	0.997570i	$-0.477806\pi$
$599$	2042.69	0.139336	0.0696680	+	0.997570i	$0.477806\pi$
$600$	0	0
$601$	−21039.4	−1.42798	−0.713989	−	0.700156i	$-0.753114\pi$
$601$	−21039.4	−1.42798	−0.713989	+	0.700156i	$0.753114\pi$
$602$	0	0
$603$	417.764	0.0282134
$604$	0	0
$605$	−1796.71	−0.120738
$606$	0	0
$607$	−8431.97	−0.563827	−0.281913	−	0.959440i	$-0.590969\pi$
$607$	−8431.97	−0.563827	−0.281913	+	0.959440i	$0.590969\pi$
$608$	0	0
$609$	20810.8	1.38472
$610$	0	0
$611$	−12564.9	−0.831948
$612$	0	0
$613$	18283.0	1.20464	0.602320	−	0.798254i	$-0.294243\pi$
$613$	18283.0	1.20464	0.602320	+	0.798254i	$0.294243\pi$
$614$	0	0
$615$	−7589.51	−0.497624
$616$	0	0
$617$	19029.7	1.24167	0.620833	−	0.783942i	$-0.286794\pi$
$617$	19029.7	1.24167	0.620833	+	0.783942i	$0.286794\pi$
$618$	0	0
$619$	501.940	0.0325924	0.0162962	−	0.999867i	$-0.494813\pi$
$619$	501.940	0.0325924	0.0162962	+	0.999867i	$0.494813\pi$
$620$	0	0
$621$	1397.40	0.0902994
$622$	0	0
$623$	26282.8	1.69021
$624$	0	0
$625$	−18443.0	−1.18035
$626$	0	0
$627$	4240.12	0.270071
$628$	0	0
$629$	−13203.0	−0.836941
$630$	0	0
$631$	9253.88	0.583821	0.291910	−	0.956446i	$-0.405709\pi$
$631$	9253.88	0.583821	0.291910	+	0.956446i	$0.405709\pi$
$632$	0	0
$633$	9724.77	0.610624
$634$	0	0
$635$	19986.4	1.24904
$636$	0	0
$637$	−44812.9	−2.78736
$638$	0	0
$639$	−2909.40	−0.180116
$640$	0	0
$641$	−12102.1	−0.745715	−0.372857	−	0.927889i	$-0.621622\pi$
$641$	−12102.1	−0.745715	−0.372857	+	0.927889i	$0.621622\pi$
$642$	0	0
$643$	−12793.4	−0.784636	−0.392318	−	0.919830i	$-0.628327\pi$
$643$	−12793.4	−0.784636	−0.392318	+	0.919830i	$0.628327\pi$
$644$	0	0
$645$	−14201.0	−0.866923
$646$	0	0
$647$	−7463.21	−0.453492	−0.226746	−	0.973954i	$-0.572809\pi$
$647$	−7463.21	−0.453492	−0.226746	+	0.973954i	$0.572809\pi$
$648$	0	0
$649$	−3996.07	−0.241694
$650$	0	0
$651$	−20305.7	−1.22249
$652$	0	0
$653$	23225.5	1.39186	0.695931	−	0.718109i	$-0.254992\pi$
$653$	23225.5	1.39186	0.695931	+	0.718109i	$0.254992\pi$
$654$	0	0
$655$	−678.193	−0.0404568
$656$	0	0
$657$	9002.94	0.534609
$658$	0	0
$659$	10460.5	0.618336	0.309168	−	0.951007i	$-0.399950\pi$
$659$	10460.5	0.618336	0.309168	+	0.951007i	$0.399950\pi$
$660$	0	0
$661$	24559.0	1.44513	0.722567	−	0.691301i	$-0.242962\pi$
$661$	24559.0	1.44513	0.722567	+	0.691301i	$0.242962\pi$
$662$	0	0
$663$	13600.7	0.796693
$664$	0	0
$665$	58280.1	3.39850
$666$	0	0
$667$	11753.4	0.682297
$668$	0	0
$669$	15382.9	0.888995
$670$	0	0
$671$	5016.64	0.288622
$672$	0	0
$673$	−15621.9	−0.894771	−0.447386	−	0.894341i	$-0.647645\pi$
$673$	−15621.9	−0.894771	−0.447386	+	0.894341i	$0.647645\pi$
$674$	0	0
$675$	2578.19	0.147014
$676$	0	0
$677$	−1854.68	−0.105290	−0.0526448	−	0.998613i	$-0.516765\pi$
$677$	−1854.68	−0.105290	−0.0526448	+	0.998613i	$0.516765\pi$
$678$	0	0
$679$	−15927.6	−0.900214
$680$	0	0
$681$	−13203.6	−0.742973
$682$	0	0
$683$	−26118.0	−1.46322	−0.731608	−	0.681726i	$-0.761230\pi$
$683$	−26118.0	−1.46322	−0.731608	+	0.681726i	$0.761230\pi$
$684$	0	0
$685$	4466.77	0.249148
$686$	0	0
$687$	−7698.07	−0.427511
$688$	0	0
$689$	−3015.66	−0.166745
$690$	0	0
$691$	−1474.59	−0.0811812	−0.0405906	−	0.999176i	$-0.512924\pi$
$691$	−1474.59	−0.0811812	−0.0405906	+	0.999176i	$0.512924\pi$
$692$	0	0
$693$	−3024.11	−0.165767
$694$	0	0
$695$	10983.7	0.599477
$696$	0	0
$697$	−10170.9	−0.552724
$698$	0	0
$699$	−3954.42	−0.213977
$700$	0	0
$701$	8135.59	0.438341	0.219170	−	0.975687i	$-0.429665\pi$
$701$	8135.59	0.438341	0.219170	+	0.975687i	$0.429665\pi$
$702$	0	0
$703$	−28417.0	−1.52456
$704$	0	0
$705$	−7370.38	−0.393737
$706$	0	0
$707$	12053.1	0.641163
$708$	0	0
$709$	−3033.27	−0.160673	−0.0803363	−	0.996768i	$-0.525599\pi$
$709$	−3033.27	−0.160673	−0.0803363	+	0.996768i	$0.525599\pi$
$710$	0	0
$711$	5013.73	0.264458
$712$	0	0
$713$	−11468.1	−0.602362
$714$	0	0
$715$	−12404.2	−0.648797
$716$	0	0
$717$	−284.699	−0.0148289
$718$	0	0
$719$	22899.2	1.18776	0.593879	−	0.804555i	$-0.297596\pi$
$719$	22899.2	1.18776	0.593879	+	0.804555i	$0.297596\pi$
$720$	0	0
$721$	−22609.5	−1.16785
$722$	0	0
$723$	4826.59	0.248275
$724$	0	0
$725$	21684.8	1.11083
$726$	0	0
$727$	−23641.5	−1.20607	−0.603036	−	0.797714i	$-0.706042\pi$
$727$	−23641.5	−1.20607	−0.603036	+	0.797714i	$0.706042\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−19031.1	−0.962914
$732$	0	0
$733$	−23557.1	−1.18704	−0.593521	−	0.804819i	$-0.702263\pi$
$733$	−23557.1	−1.18704	−0.593521	+	0.804819i	$0.702263\pi$
$734$	0	0
$735$	−26286.6	−1.31918
$736$	0	0
$737$	−510.601	−0.0255200
$738$	0	0
$739$	15019.6	0.747638	0.373819	−	0.927502i	$-0.378048\pi$
$739$	15019.6	0.747638	0.373819	+	0.927502i	$0.378048\pi$
$740$	0	0
$741$	29273.0	1.45124
$742$	0	0
$743$	−15212.8	−0.751148	−0.375574	−	0.926792i	$-0.622554\pi$
$743$	−15212.8	−0.751148	−0.375574	+	0.926792i	$0.622554\pi$
$744$	0	0
$745$	−9595.81	−0.471897
$746$	0	0
$747$	−6193.27	−0.303346
$748$	0	0
$749$	41252.2	2.01244
$750$	0	0
$751$	−26157.1	−1.27095	−0.635477	−	0.772120i	$-0.719197\pi$
$751$	−26157.1	−1.27095	−0.635477	+	0.772120i	$0.719197\pi$
$752$	0	0
$753$	364.671	0.0176486
$754$	0	0
$755$	1688.46	0.0813901
$756$	0	0
$757$	−17192.0	−0.825434	−0.412717	−	0.910859i	$-0.635420\pi$
$757$	−17192.0	−0.825434	−0.412717	+	0.910859i	$0.635420\pi$
$758$	0	0
$759$	−1707.94	−0.0816788
$760$	0	0
$761$	−30304.9	−1.44356	−0.721781	−	0.692121i	$-0.756676\pi$
$761$	−30304.9	−1.44356	−0.721781	+	0.692121i	$0.756676\pi$
$762$	0	0
$763$	−8184.71	−0.388344
$764$	0	0
$765$	7977.99	0.377052
$766$	0	0
$767$	−27588.2	−1.29876
$768$	0	0
$769$	−20955.4	−0.982665	−0.491332	−	0.870972i	$-0.663490\pi$
$769$	−20955.4	−0.982665	−0.491332	+	0.870972i	$0.663490\pi$
$770$	0	0
$771$	23502.0	1.09780
$772$	0	0
$773$	9629.74	0.448070	0.224035	−	0.974581i	$-0.428077\pi$
$773$	9629.74	0.448070	0.224035	+	0.974581i	$0.428077\pi$
$774$	0	0
$775$	−21158.5	−0.980692
$776$	0	0
$777$	20267.4	0.935762
$778$	0	0
$779$	−21890.9	−1.00683
$780$	0	0
$781$	3555.94	0.162921
$782$	0	0
$783$	6131.51	0.279850
$784$	0	0
$785$	−10115.5	−0.459920
$786$	0	0
$787$	21006.2	0.951451	0.475725	−	0.879594i	$-0.342186\pi$
$787$	21006.2	0.951451	0.475725	+	0.879594i	$0.342186\pi$
$788$	0	0
$789$	−9294.71	−0.419392
$790$	0	0
$791$	−44734.5	−2.01084
$792$	0	0
$793$	34634.0	1.55093
$794$	0	0
$795$	−1768.95	−0.0789158
$796$	0	0
$797$	15614.7	0.693980	0.346990	−	0.937869i	$-0.387204\pi$
$797$	15614.7	0.693980	0.346990	+	0.937869i	$0.387204\pi$
$798$	0	0
$799$	−9877.19	−0.437334
$800$	0	0
$801$	7743.76	0.341589
$802$	0	0
$803$	−11003.6	−0.483572
$804$	0	0
$805$	−23475.4	−1.02783
$806$	0	0
$807$	12828.8	0.559599
$808$	0	0
$809$	22261.0	0.967437	0.483718	−	0.875224i	$-0.339286\pi$
$809$	22261.0	0.967437	0.483718	+	0.875224i	$0.339286\pi$
$810$	0	0
$811$	38071.9	1.64844	0.824220	−	0.566270i	$-0.191614\pi$
$811$	38071.9	1.64844	0.824220	+	0.566270i	$0.191614\pi$
$812$	0	0
$813$	−6109.60	−0.263558
$814$	0	0
$815$	12993.8	0.558471
$816$	0	0
$817$	−40961.0	−1.75403
$818$	0	0
$819$	−20877.9	−0.890761
$820$	0	0
$821$	13867.3	0.589489	0.294745	−	0.955576i	$-0.404765\pi$
$821$	13867.3	0.589489	0.294745	+	0.955576i	$0.404765\pi$
$822$	0	0
$823$	27046.5	1.14554	0.572770	−	0.819716i	$-0.305869\pi$
$823$	27046.5	1.14554	0.572770	+	0.819716i	$0.305869\pi$
$824$	0	0
$825$	−3151.12	−0.132979
$826$	0	0
$827$	5121.88	0.215363	0.107681	−	0.994185i	$-0.465657\pi$
$827$	5121.88	0.215363	0.107681	+	0.994185i	$0.465657\pi$
$828$	0	0
$829$	−36023.1	−1.50921	−0.754605	−	0.656179i	$-0.772171\pi$
$829$	−36023.1	−1.50921	−0.754605	+	0.656179i	$0.772171\pi$
$830$	0	0
$831$	−22843.3	−0.953580
$832$	0	0
$833$	−35227.2	−1.46525
$834$	0	0
$835$	−56382.5	−2.33676
$836$	0	0
$837$	−5982.71	−0.247064
$838$	0	0
$839$	6408.98	0.263722	0.131861	−	0.991268i	$-0.457905\pi$
$839$	6408.98	0.263722	0.131861	+	0.991268i	$0.457905\pi$
$840$	0	0
$841$	27182.3	1.11453
$842$	0	0
$843$	3407.49	0.139217
$844$	0	0
$845$	−53013.2	−2.15824
$846$	0	0
$847$	3696.14	0.149942
$848$	0	0
$849$	23796.3	0.961940
$850$	0	0
$851$	11446.5	0.461081
$852$	0	0
$853$	36368.6	1.45983	0.729915	−	0.683538i	$-0.239559\pi$
$853$	36368.6	1.45983	0.729915	+	0.683538i	$0.239559\pi$
$854$	0	0
$855$	17171.2	0.686833
$856$	0	0
$857$	−3239.65	−0.129130	−0.0645649	−	0.997914i	$-0.520566\pi$
$857$	−3239.65	−0.129130	−0.0645649	+	0.997914i	$0.520566\pi$
$858$	0	0
$859$	833.399	0.0331027	0.0165513	−	0.999863i	$-0.494731\pi$
$859$	833.399	0.0331027	0.0165513	+	0.999863i	$0.494731\pi$
$860$	0	0
$861$	15612.9	0.617986
$862$	0	0
$863$	32232.9	1.27140	0.635701	−	0.771936i	$-0.280711\pi$
$863$	32232.9	1.27140	0.635701	+	0.771936i	$0.280711\pi$
$864$	0	0
$865$	−42285.8	−1.66215
$866$	0	0
$867$	−4047.55	−0.158549
$868$	0	0
$869$	−6127.89	−0.239211
$870$	0	0
$871$	−3525.10	−0.137134
$872$	0	0
$873$	−4692.78	−0.181932
$874$	0	0
$875$	13385.8	0.517170
$876$	0	0
$877$	−4725.68	−0.181955	−0.0909777	−	0.995853i	$-0.528999\pi$
$877$	−4725.68	−0.181955	−0.0909777	+	0.995853i	$0.528999\pi$
$878$	0	0
$879$	27984.0	1.07381
$880$	0	0
$881$	−10434.2	−0.399022	−0.199511	−	0.979896i	$-0.563935\pi$
$881$	−10434.2	−0.399022	−0.199511	+	0.979896i	$0.563935\pi$
$882$	0	0
$883$	−12139.1	−0.462641	−0.231321	−	0.972878i	$-0.574305\pi$
$883$	−12139.1	−0.462641	−0.231321	+	0.972878i	$0.574305\pi$
$884$	0	0
$885$	−16182.9	−0.614667
$886$	0	0
$887$	−5811.15	−0.219977	−0.109988	−	0.993933i	$-0.535081\pi$
$887$	−5811.15	−0.219977	−0.109988	+	0.993933i	$0.535081\pi$
$888$	0	0
$889$	−41115.4	−1.55114
$890$	0	0
$891$	−891.000	−0.0335013
$892$	0	0
$893$	−21258.9	−0.796642
$894$	0	0
$895$	−60391.0	−2.25547
$896$	0	0
$897$	−11791.3	−0.438907
$898$	0	0
$899$	−50319.7	−1.86680
$900$	0	0
$901$	−2370.60	−0.0876538
$902$	0	0
$903$	29213.9	1.07661
$904$	0	0
$905$	−1385.72	−0.0508982
$906$	0	0
$907$	−16197.5	−0.592976	−0.296488	−	0.955037i	$-0.595816\pi$
$907$	−16197.5	−0.592976	−0.296488	+	0.955037i	$0.595816\pi$
$908$	0	0
$909$	3551.22	0.129578
$910$	0	0
$911$	−40670.5	−1.47911	−0.739557	−	0.673093i	$-0.764965\pi$
$911$	−40670.5	−1.47911	−0.739557	+	0.673093i	$0.764965\pi$
$912$	0	0
$913$	7569.55	0.274387
$914$	0	0
$915$	20315.8	0.734011
$916$	0	0
$917$	1395.16	0.0502422
$918$	0	0
$919$	−14069.6	−0.505021	−0.252510	−	0.967594i	$-0.581256\pi$
$919$	−14069.6	−0.505021	−0.252510	+	0.967594i	$0.581256\pi$
$920$	0	0
$921$	−15911.1	−0.569259
$922$	0	0
$923$	24549.6	0.875470
$924$	0	0
$925$	21118.6	0.750675
$926$	0	0
$927$	−6661.49	−0.236022
$928$	0	0
$929$	13382.9	0.472635	0.236318	−	0.971676i	$-0.424059\pi$
$929$	13382.9	0.472635	0.236318	+	0.971676i	$0.424059\pi$
$930$	0	0
$931$	−75820.2	−2.66907
$932$	0	0
$933$	−4426.35	−0.155319
$934$	0	0
$935$	−9750.87	−0.341056
$936$	0	0
$937$	36799.2	1.28301	0.641503	−	0.767120i	$-0.278311\pi$
$937$	36799.2	1.28301	0.641503	+	0.767120i	$0.278311\pi$
$938$	0	0
$939$	−17665.0	−0.613925
$940$	0	0
$941$	4426.99	0.153364	0.0766821	−	0.997056i	$-0.475567\pi$
$941$	4426.99	0.153364	0.0766821	+	0.997056i	$0.475567\pi$
$942$	0	0
$943$	8817.75	0.304502
$944$	0	0
$945$	−12246.7	−0.421572
$946$	0	0
$947$	−55046.7	−1.88889	−0.944445	−	0.328671i	$-0.893399\pi$
$947$	−55046.7	−1.88889	−0.944445	+	0.328671i	$0.893399\pi$
$948$	0	0
$949$	−75966.8	−2.59851
$950$	0	0
$951$	18493.4	0.630589
$952$	0	0
$953$	33050.3	1.12340	0.561702	−	0.827340i	$-0.310147\pi$
$953$	33050.3	1.12340	0.561702	+	0.827340i	$0.310147\pi$
$954$	0	0
$955$	−40342.2	−1.36696
$956$	0	0
$957$	−7494.07	−0.253134
$958$	0	0
$959$	−9188.88	−0.309410
$960$	0	0
$961$	19307.5	0.648097
$962$	0	0
$963$	12154.2	0.406712
$964$	0	0
$965$	−23164.5	−0.772738
$966$	0	0
$967$	9330.42	0.310285	0.155143	−	0.987892i	$-0.450416\pi$
$967$	9330.42	0.310285	0.155143	+	0.987892i	$0.450416\pi$
$968$	0	0
$969$	23011.4	0.762883
$970$	0	0
$971$	22309.0	0.737313	0.368656	−	0.929566i	$-0.379818\pi$
$971$	22309.0	0.737313	0.368656	+	0.929566i	$0.379818\pi$
$972$	0	0
$973$	−22595.3	−0.744474
$974$	0	0
$975$	−21754.8	−0.714575
$976$	0	0
$977$	4832.65	0.158250	0.0791250	−	0.996865i	$-0.474787\pi$
$977$	4832.65	0.158250	0.0791250	+	0.996865i	$0.474787\pi$
$978$	0	0
$979$	−9464.60	−0.308979
$980$	0	0
$981$	−2411.48	−0.0784838
$982$	0	0
$983$	−37471.4	−1.21582	−0.607911	−	0.794005i	$-0.707992\pi$
$983$	−37471.4	−1.21582	−0.607911	+	0.794005i	$0.707992\pi$
$984$	0	0
$985$	−48618.3	−1.57270
$986$	0	0
$987$	15162.1	0.488972
$988$	0	0
$989$	16499.2	0.530481
$990$	0	0
$991$	9717.06	0.311476	0.155738	−	0.987798i	$-0.450224\pi$
$991$	9717.06	0.311476	0.155738	+	0.987798i	$0.450224\pi$
$992$	0	0
$993$	−18915.8	−0.604506
$994$	0	0
$995$	−21123.7	−0.673033
$996$	0	0
$997$	−47624.5	−1.51282	−0.756411	−	0.654096i	$-0.773049\pi$
$997$	−47624.5	−1.51282	−0.756411	+	0.654096i	$0.773049\pi$
$998$	0	0
$999$	5971.41	0.189116

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.4.a.bi.1.1		2
4.3	odd	2		2112.4.a.bb.1.1		2
8.3	odd	2		66.4.a.c.1.2	✓	2
8.5	even	2		528.4.a.n.1.2		2
24.5	odd	2		1584.4.a.ba.1.1		2
24.11	even	2		198.4.a.h.1.1		2
40.3	even	4		1650.4.c.u.199.2		4
40.19	odd	2		1650.4.a.s.1.2		2
40.27	even	4		1650.4.c.u.199.3		4
88.43	even	2		726.4.a.o.1.2		2
264.131	odd	2		2178.4.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.4.a.c.1.2	✓	2	8.3	odd	2
198.4.a.h.1.1		2	24.11	even	2
528.4.a.n.1.2		2	8.5	even	2
726.4.a.o.1.2		2	88.43	even	2
1584.4.a.ba.1.1		2	24.5	odd	2
1650.4.a.s.1.2		2	40.19	odd	2
1650.4.c.u.199.2		4	40.3	even	4
1650.4.c.u.199.3		4	40.27	even	4
2112.4.a.bb.1.1		2	4.3	odd	2
2112.4.a.bi.1.1		2	1.1	even	1	trivial
2178.4.a.bf.1.1		2	264.131	odd	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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2112.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -14.8489 q^{5} +30.5466 q^{7} +9.00000 q^{9} -11.0000 q^{11} -75.9420 q^{13} -44.5466 q^{15} -59.6977 q^{17} -128.489 q^{19} +91.6397 q^{21} +51.7557 q^{23} +95.4886 q^{25} +27.0000 q^{27} +227.093 q^{29} -221.582 q^{31} -33.0000 q^{33} -453.582 q^{35} +221.163 q^{37} -227.826 q^{39} +170.373 q^{41} +318.791 q^{43} -133.640 q^{45} +165.453 q^{47} +590.093 q^{49} -179.093 q^{51} +39.7100 q^{53} +163.337 q^{55} -385.466 q^{57} +363.279 q^{59} -456.058 q^{61} +274.919 q^{63} +1127.65 q^{65} +46.4183 q^{67} +155.267 q^{69} -323.267 q^{71} +1000.33 q^{73} +286.466 q^{75} -336.012 q^{77} +557.081 q^{79} +81.0000 q^{81} -688.141 q^{83} +886.443 q^{85} +681.279 q^{87} +860.418 q^{89} -2319.77 q^{91} -664.745 q^{93} +1907.91 q^{95} -521.420 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 10 q^{5} + 2 q^{7} + 18 q^{9} - 22 q^{11} - 14 q^{13} - 30 q^{15} - 80 q^{17} - 60 q^{19} + 6 q^{21} + 202 q^{23} - 6 q^{25} + 54 q^{27} + 336 q^{29} - 128 q^{31} - 66 q^{33} - 592 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 10 * q^5 + 2 * q^7 + 18 * q^9 - 22 * q^11 - 14 * q^13 - 30 * q^15 - 80 * q^17 - 60 * q^19 + 6 * q^21 + 202 * q^23 - 6 * q^25 + 54 * q^27 + 336 * q^29 - 128 * q^31 - 66 * q^33 - 592 * q^35 - 188 * q^37 - 42 * q^39 - 132 * q^41 + 480 * q^43 - 90 * q^45 + 390 * q^47 + 1062 * q^49 - 240 * q^51 - 610 * q^53 + 110 * q^55 - 180 * q^57 + 372 * q^59 - 1050 * q^61 + 18 * q^63 + 1428 * q^65 + 408 * q^67 + 606 * q^69 - 942 * q^71 + 740 * q^73 - 18 * q^75 - 22 * q^77 + 1646 * q^79 + 162 * q^81 - 352 * q^83 + 788 * q^85 + 1008 * q^87 + 2036 * q^89 - 4088 * q^91 - 384 * q^93 + 2240 * q^95 + 336 * q^97 - 198 * q^99

Introduction

L-functions

Modular forms

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Database

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