LMFDB - Embedded newform 1980.4.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1980,4,Mod(1,1980)] 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("1980.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1980, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1980.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,15,0,-5] 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:	$116.823781811$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.9192.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 18x + 30 $ x^3 - x^2 - 18*x + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.49274$ of defining polynomial
Character	$\chi$	$=$	1980.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} +2.58315 q^{7} -11.0000 q^{11} +34.5212 q^{13} -79.4737 q^{17} -10.0608 q^{19} +56.3760 q^{23} +25.0000 q^{25} -268.929 q^{29} +339.453 q^{31} +12.9158 q^{35} -383.783 q^{37} -315.319 q^{41} +500.759 q^{43} -262.012 q^{47} -336.327 q^{49} +532.464 q^{53} -55.0000 q^{55} -431.871 q^{59} +282.941 q^{61} +172.606 q^{65} -280.676 q^{67} -378.798 q^{71} -978.074 q^{73} -28.4147 q^{77} -59.4456 q^{79} -123.644 q^{83} -397.369 q^{85} +1000.67 q^{89} +89.1735 q^{91} -50.3039 q^{95} +871.521 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} - 5 q^{7} - 33 q^{11} + 2 q^{13} - 77 q^{17} + 171 q^{19} - 222 q^{23} + 75 q^{25} - 55 q^{29} + 181 q^{31} - 25 q^{35} + 317 q^{37} - 302 q^{41} - 188 q^{43} - 662 q^{47} - 268 q^{49} - 81 q^{53}+ \cdots - 1980 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 - 5 * q^7 - 33 * q^11 + 2 * q^13 - 77 * q^17 + 171 * q^19 - 222 * q^23 + 75 * q^25 - 55 * q^29 + 181 * q^31 - 25 * q^35 + 317 * q^37 - 302 * q^41 - 188 * q^43 - 662 * q^47 - 268 * q^49 - 81 * q^53 - 165 * q^55 + 42 * q^59 + 349 * q^61 + 10 * q^65 + 152 * q^67 - 927 * q^71 - 2378 * q^73 + 55 * q^77 - 1146 * q^79 + 458 * q^83 - 385 * q^85 + 875 * q^89 + 1982 * q^91 + 855 * q^95 - 1980 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	2.58315	0.139477	0.0697385	−	0.997565i	$-0.477783\pi$
$7$	2.58315	0.139477	0.0697385	+	0.997565i	$0.477783\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	34.5212	0.736496	0.368248	−	0.929728i	$-0.379958\pi$
$13$	34.5212	0.736496	0.368248	+	0.929728i	$0.379958\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−79.4737	−1.13384	−0.566918	−	0.823774i	$-0.691864\pi$
$17$	−79.4737	−1.13384	−0.566918	+	0.823774i	$0.691864\pi$
$18$	0	0
$19$	−10.0608	−0.121479	−0.0607395	−	0.998154i	$-0.519346\pi$
$19$	−10.0608	−0.121479	−0.0607395	+	0.998154i	$0.519346\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	56.3760	0.511096	0.255548	−	0.966796i	$-0.417744\pi$
$23$	56.3760	0.511096	0.255548	+	0.966796i	$0.417744\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−268.929	−1.72203	−0.861014	−	0.508580i	$-0.830170\pi$
$29$	−268.929	−1.72203	−0.861014	+	0.508580i	$0.830170\pi$
$30$	0	0
$31$	339.453	1.96670	0.983348	−	0.181731i	$-0.0581699\pi$
$31$	339.453	1.96670	0.983348	+	0.181731i	$0.0581699\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	12.9158	0.0623761
$36$	0	0
$37$	−383.783	−1.70523	−0.852616	−	0.522537i	$-0.824985\pi$
$37$	−383.783	−1.70523	−0.852616	+	0.522537i	$0.824985\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−315.319	−1.20109	−0.600543	−	0.799592i	$-0.705049\pi$
$41$	−315.319	−1.20109	−0.600543	+	0.799592i	$0.705049\pi$
$42$	0	0
$43$	500.759	1.77593	0.887966	−	0.459909i	$-0.152118\pi$
$43$	500.759	1.77593	0.887966	+	0.459909i	$0.152118\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−262.012	−0.813158	−0.406579	−	0.913616i	$-0.633278\pi$
$47$	−262.012	−0.813158	−0.406579	+	0.913616i	$0.633278\pi$
$48$	0	0
$49$	−336.327	−0.980546
$50$	0	0
$51$	0	0
$52$	0	0
$53$	532.464	1.37999	0.689996	−	0.723813i	$-0.257612\pi$
$53$	532.464	1.37999	0.689996	+	0.723813i	$0.257612\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−431.871	−0.952964	−0.476482	−	0.879184i	$-0.658088\pi$
$59$	−431.871	−0.952964	−0.476482	+	0.879184i	$0.658088\pi$
$60$	0	0
$61$	282.941	0.593883	0.296942	−	0.954896i	$-0.404033\pi$
$61$	282.941	0.593883	0.296942	+	0.954896i	$0.404033\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	172.606	0.329371
$66$	0	0
$67$	−280.676	−0.511791	−0.255895	−	0.966704i	$-0.582370\pi$
$67$	−280.676	−0.511791	−0.255895	+	0.966704i	$0.582370\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−378.798	−0.633170	−0.316585	−	0.948564i	$-0.602536\pi$
$71$	−378.798	−0.633170	−0.316585	+	0.948564i	$0.602536\pi$
$72$	0	0
$73$	−978.074	−1.56815	−0.784075	−	0.620666i	$-0.786862\pi$
$73$	−978.074	−1.56815	−0.784075	+	0.620666i	$0.786862\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−28.4147	−0.0420539
$78$	0	0
$79$	−59.4456	−0.0846602	−0.0423301	−	0.999104i	$-0.513478\pi$
$79$	−59.4456	−0.0846602	−0.0423301	+	0.999104i	$0.513478\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−123.644	−0.163515	−0.0817573	−	0.996652i	$-0.526053\pi$
$83$	−123.644	−0.163515	−0.0817573	+	0.996652i	$0.526053\pi$
$84$	0	0
$85$	−397.369	−0.507067
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1000.67	1.19180	0.595902	−	0.803057i	$-0.296795\pi$
$89$	1000.67	1.19180	0.595902	+	0.803057i	$0.296795\pi$
$90$	0	0
$91$	89.1735	0.102724
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−50.3039	−0.0543270
$96$	0	0
$97$	871.521	0.912264	0.456132	−	0.889912i	$-0.349234\pi$
$97$	871.521	0.912264	0.456132	+	0.889912i	$0.349234\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	853.553	0.840908	0.420454	−	0.907314i	$-0.361871\pi$
$101$	853.553	0.840908	0.420454	+	0.907314i	$0.361871\pi$
$102$	0	0
$103$	−1629.96	−1.55927	−0.779635	−	0.626234i	$-0.784595\pi$
$103$	−1629.96	−1.55927	−0.779635	+	0.626234i	$0.784595\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1020.59	−0.922096	−0.461048	−	0.887375i	$-0.652526\pi$
$107$	−1020.59	−0.922096	−0.461048	+	0.887375i	$0.652526\pi$
$108$	0	0
$109$	784.501	0.689372	0.344686	−	0.938718i	$-0.387985\pi$
$109$	784.501	0.689372	0.344686	+	0.938718i	$0.387985\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	579.296	0.482261	0.241131	−	0.970493i	$-0.422482\pi$
$113$	579.296	0.482261	0.241131	+	0.970493i	$0.422482\pi$
$114$	0	0
$115$	281.880	0.228569
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−205.293	−0.158144
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	587.679	0.410615	0.205307	−	0.978698i	$-0.434181\pi$
$127$	587.679	0.410615	0.205307	+	0.978698i	$0.434181\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2801.91	−1.86874	−0.934368	−	0.356311i	$-0.884035\pi$
$131$	−2801.91	−1.86874	−0.934368	+	0.356311i	$0.884035\pi$
$132$	0	0
$133$	−25.9885	−0.0169435
$134$	0	0
$135$	0	0
$136$	0	0
$137$	397.947	0.248167	0.124084	−	0.992272i	$-0.460401\pi$
$137$	397.947	0.248167	0.124084	+	0.992272i	$0.460401\pi$
$138$	0	0
$139$	−1966.17	−1.19977	−0.599887	−	0.800085i	$-0.704788\pi$
$139$	−1966.17	−1.19977	−0.599887	+	0.800085i	$0.704788\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−379.733	−0.222062
$144$	0	0
$145$	−1344.64	−0.770115
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2816.65	−1.54865	−0.774324	−	0.632789i	$-0.781910\pi$
$149$	−2816.65	−1.54865	−0.774324	+	0.632789i	$0.781910\pi$
$150$	0	0
$151$	−441.553	−0.237967	−0.118984	−	0.992896i	$-0.537964\pi$
$151$	−441.553	−0.237967	−0.118984	+	0.992896i	$0.537964\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1697.27	0.879534
$156$	0	0
$157$	2126.91	1.08119	0.540593	−	0.841284i	$-0.318200\pi$
$157$	2126.91	1.08119	0.540593	+	0.841284i	$0.318200\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	145.628	0.0712861
$162$	0	0
$163$	404.767	0.194502	0.0972510	−	0.995260i	$-0.468995\pi$
$163$	404.767	0.194502	0.0972510	+	0.995260i	$0.468995\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−135.283	−0.0626855	−0.0313428	−	0.999509i	$-0.509978\pi$
$167$	−135.283	−0.0626855	−0.0313428	+	0.999509i	$0.509978\pi$
$168$	0	0
$169$	−1005.29	−0.457573
$170$	0	0
$171$	0	0
$172$	0	0
$173$	748.565	0.328973	0.164487	−	0.986379i	$-0.447403\pi$
$173$	748.565	0.328973	0.164487	+	0.986379i	$0.447403\pi$
$174$	0	0
$175$	64.5788	0.0278954
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3648.69	−1.52355	−0.761776	−	0.647840i	$-0.775672\pi$
$179$	−3648.69	−1.52355	−0.761776	+	0.647840i	$0.775672\pi$
$180$	0	0
$181$	2571.34	1.05595	0.527974	−	0.849261i	$-0.322952\pi$
$181$	2571.34	1.05595	0.527974	+	0.849261i	$0.322952\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1918.92	−0.762603
$186$	0	0
$187$	874.211	0.341864
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−81.4965	−0.0308737	−0.0154369	−	0.999881i	$-0.504914\pi$
$191$	−81.4965	−0.0308737	−0.0154369	+	0.999881i	$0.504914\pi$
$192$	0	0
$193$	−2866.69	−1.06917	−0.534583	−	0.845116i	$-0.679531\pi$
$193$	−2866.69	−1.06917	−0.534583	+	0.845116i	$0.679531\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5380.96	−1.94608	−0.973039	−	0.230642i	$-0.925917\pi$
$197$	−5380.96	−1.94608	−0.973039	+	0.230642i	$0.925917\pi$
$198$	0	0
$199$	2712.01	0.966077	0.483039	−	0.875599i	$-0.339533\pi$
$199$	2712.01	0.966077	0.483039	+	0.875599i	$0.339533\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−694.684	−0.240184
$204$	0	0
$205$	−1576.59	−0.537142
$206$	0	0
$207$	0	0
$208$	0	0
$209$	110.668	0.0366273
$210$	0	0
$211$	2930.90	0.956263	0.478132	−	0.878288i	$-0.341314\pi$
$211$	2930.90	0.956263	0.478132	+	0.878288i	$0.341314\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2503.80	0.794221
$216$	0	0
$217$	876.859	0.274309
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2743.53	−0.835066
$222$	0	0
$223$	1351.61	0.405877	0.202938	−	0.979192i	$-0.434951\pi$
$223$	1351.61	0.405877	0.202938	+	0.979192i	$0.434951\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	695.418	0.203333	0.101666	−	0.994819i	$-0.467583\pi$
$227$	695.418	0.203333	0.101666	+	0.994819i	$0.467583\pi$
$228$	0	0
$229$	−1658.73	−0.478655	−0.239328	−	0.970939i	$-0.576927\pi$
$229$	−1658.73	−0.478655	−0.239328	+	0.970939i	$0.576927\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	375.154	0.105481	0.0527407	−	0.998608i	$-0.483204\pi$
$233$	375.154	0.105481	0.0527407	+	0.998608i	$0.483204\pi$
$234$	0	0
$235$	−1310.06	−0.363655
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2578.11	0.697757	0.348878	−	0.937168i	$-0.386563\pi$
$239$	2578.11	0.697757	0.348878	+	0.937168i	$0.386563\pi$
$240$	0	0
$241$	568.506	0.151953	0.0759765	−	0.997110i	$-0.475793\pi$
$241$	568.506	0.151953	0.0759765	+	0.997110i	$0.475793\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1681.64	−0.438514
$246$	0	0
$247$	−347.310	−0.0894688
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7451.23	−1.87377	−0.936887	−	0.349631i	$-0.886307\pi$
$251$	−7451.23	−1.87377	−0.936887	+	0.349631i	$0.886307\pi$
$252$	0	0
$253$	−620.136	−0.154101
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1526.81	0.370583	0.185291	−	0.982684i	$-0.440677\pi$
$257$	1526.81	0.370583	0.185291	+	0.982684i	$0.440677\pi$
$258$	0	0
$259$	−991.371	−0.237841
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4702.46	−1.10253	−0.551266	−	0.834330i	$-0.685855\pi$
$263$	−4702.46	−1.10253	−0.551266	+	0.834330i	$0.685855\pi$
$264$	0	0
$265$	2662.32	0.617151
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4325.43	0.980395	0.490197	−	0.871611i	$-0.336925\pi$
$269$	4325.43	0.980395	0.490197	+	0.871611i	$0.336925\pi$
$270$	0	0
$271$	−5124.00	−1.14856	−0.574282	−	0.818658i	$-0.694719\pi$
$271$	−5124.00	−1.14856	−0.574282	+	0.818658i	$0.694719\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−275.000	−0.0603023
$276$	0	0
$277$	−6606.24	−1.43296	−0.716481	−	0.697607i	$-0.754248\pi$
$277$	−6606.24	−1.43296	−0.716481	+	0.697607i	$0.754248\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−572.782	−0.121599	−0.0607995	−	0.998150i	$-0.519365\pi$
$281$	−572.782	−0.121599	−0.0607995	+	0.998150i	$0.519365\pi$
$282$	0	0
$283$	−5538.08	−1.16327	−0.581634	−	0.813450i	$-0.697586\pi$
$283$	−5538.08	−1.16327	−0.581634	+	0.813450i	$0.697586\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−814.517	−0.167524
$288$	0	0
$289$	1403.08	0.285584
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5982.32	1.19280	0.596401	−	0.802687i	$-0.296597\pi$
$293$	5982.32	1.19280	0.596401	+	0.802687i	$0.296597\pi$
$294$	0	0
$295$	−2159.36	−0.426178
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1946.17	0.376420
$300$	0	0
$301$	1293.54	0.247702
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1414.70	0.265593
$306$	0	0
$307$	−9253.87	−1.72035	−0.860173	−	0.510002i	$-0.829645\pi$
$307$	−9253.87	−1.72035	−0.860173	+	0.510002i	$0.829645\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4508.58	−0.822051	−0.411026	−	0.911624i	$-0.634829\pi$
$311$	−4508.58	−0.822051	−0.411026	+	0.911624i	$0.634829\pi$
$312$	0	0
$313$	−4658.04	−0.841175	−0.420587	−	0.907252i	$-0.638176\pi$
$313$	−4658.04	−0.841175	−0.420587	+	0.907252i	$0.638176\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3784.24	−0.670485	−0.335243	−	0.942132i	$-0.608818\pi$
$317$	−3784.24	−0.670485	−0.335243	+	0.942132i	$0.608818\pi$
$318$	0	0
$319$	2958.22	0.519211
$320$	0	0
$321$	0	0
$322$	0	0
$323$	799.567	0.137737
$324$	0	0
$325$	863.030	0.147299
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−676.818	−0.113417
$330$	0	0
$331$	−8278.92	−1.37478	−0.687388	−	0.726291i	$-0.741243\pi$
$331$	−8278.92	−1.37478	−0.687388	+	0.726291i	$0.741243\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1403.38	−0.228880
$336$	0	0
$337$	−2647.22	−0.427903	−0.213951	−	0.976844i	$-0.568633\pi$
$337$	−2647.22	−0.427903	−0.213951	+	0.976844i	$0.568633\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3733.99	−0.592981
$342$	0	0
$343$	−1754.81	−0.276241
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1963.50	0.303764	0.151882	−	0.988399i	$-0.451467\pi$
$347$	1963.50	0.303764	0.151882	+	0.988399i	$0.451467\pi$
$348$	0	0
$349$	1747.45	0.268020	0.134010	−	0.990980i	$-0.457215\pi$
$349$	1747.45	0.268020	0.134010	+	0.990980i	$0.457215\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4841.13	−0.729936	−0.364968	−	0.931020i	$-0.618920\pi$
$353$	−4841.13	−0.729936	−0.364968	+	0.931020i	$0.618920\pi$
$354$	0	0
$355$	−1893.99	−0.283162
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6201.44	0.911697	0.455849	−	0.890057i	$-0.349336\pi$
$359$	6201.44	0.911697	0.455849	+	0.890057i	$0.349336\pi$
$360$	0	0
$361$	−6757.78	−0.985243
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4890.37	−0.701298
$366$	0	0
$367$	−10313.7	−1.46695	−0.733474	−	0.679718i	$-0.762102\pi$
$367$	−10313.7	−1.46695	−0.733474	+	0.679718i	$0.762102\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1375.44	0.192477
$372$	0	0
$373$	12895.3	1.79006	0.895030	−	0.446006i	$-0.147154\pi$
$373$	12895.3	1.79006	0.895030	+	0.446006i	$0.147154\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9283.74	−1.26827
$378$	0	0
$379$	−2379.46	−0.322492	−0.161246	−	0.986914i	$-0.551551\pi$
$379$	−2379.46	−0.322492	−0.161246	+	0.986914i	$0.551551\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3610.56	−0.481699	−0.240850	−	0.970562i	$-0.577426\pi$
$383$	−3610.56	−0.481699	−0.240850	+	0.970562i	$0.577426\pi$
$384$	0	0
$385$	−142.073	−0.0188071
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5477.55	−0.713940	−0.356970	−	0.934116i	$-0.616190\pi$
$389$	−5477.55	−0.713940	−0.356970	+	0.934116i	$0.616190\pi$
$390$	0	0
$391$	−4480.41	−0.579499
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−297.228	−0.0378612
$396$	0	0
$397$	8582.17	1.08495	0.542477	−	0.840071i	$-0.317486\pi$
$397$	8582.17	1.08495	0.542477	+	0.840071i	$0.317486\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5550.01	−0.691157	−0.345579	−	0.938390i	$-0.612317\pi$
$401$	−5550.01	−0.691157	−0.345579	+	0.938390i	$0.612317\pi$
$402$	0	0
$403$	11718.3	1.44847
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4221.62	0.514147
$408$	0	0
$409$	−7395.83	−0.894133	−0.447066	−	0.894501i	$-0.647531\pi$
$409$	−7395.83	−0.894133	−0.447066	+	0.894501i	$0.647531\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1115.59	−0.132917
$414$	0	0
$415$	−618.221	−0.0731260
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1735.97	−0.202405	−0.101203	−	0.994866i	$-0.532269\pi$
$419$	−1735.97	−0.202405	−0.101203	+	0.994866i	$0.532269\pi$
$420$	0	0
$421$	9567.68	1.10760	0.553801	−	0.832649i	$-0.313177\pi$
$421$	9567.68	1.10760	0.553801	+	0.832649i	$0.313177\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1986.84	−0.226767
$426$	0	0
$427$	730.880	0.0828331
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10941.3	1.22280	0.611398	−	0.791323i	$-0.290608\pi$
$431$	10941.3	1.22280	0.611398	+	0.791323i	$0.290608\pi$
$432$	0	0
$433$	−3968.84	−0.440486	−0.220243	−	0.975445i	$-0.570685\pi$
$433$	−3968.84	−0.440486	−0.220243	+	0.975445i	$0.570685\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−567.186	−0.0620873
$438$	0	0
$439$	8216.72	0.893309	0.446654	−	0.894707i	$-0.352615\pi$
$439$	8216.72	0.893309	0.446654	+	0.894707i	$0.352615\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14007.9	−1.50234	−0.751171	−	0.660107i	$-0.770511\pi$
$443$	−14007.9	−1.50234	−0.751171	+	0.660107i	$0.770511\pi$
$444$	0	0
$445$	5003.34	0.532991
$446$	0	0
$447$	0	0
$448$	0	0
$449$	498.251	0.0523695	0.0261848	−	0.999657i	$-0.491664\pi$
$449$	498.251	0.0523695	0.0261848	+	0.999657i	$0.491664\pi$
$450$	0	0
$451$	3468.51	0.362141
$452$	0	0
$453$	0	0
$454$	0	0
$455$	445.867	0.0459397
$456$	0	0
$457$	7588.15	0.776715	0.388357	−	0.921509i	$-0.373043\pi$
$457$	7588.15	0.776715	0.388357	+	0.921509i	$0.373043\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14500.5	1.46498	0.732490	−	0.680778i	$-0.238358\pi$
$461$	14500.5	1.46498	0.732490	+	0.680778i	$0.238358\pi$
$462$	0	0
$463$	6161.30	0.618445	0.309222	−	0.950990i	$-0.399931\pi$
$463$	6161.30	0.618445	0.309222	+	0.950990i	$0.399931\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8877.74	0.879684	0.439842	−	0.898075i	$-0.355034\pi$
$467$	8877.74	0.879684	0.439842	+	0.898075i	$0.355034\pi$
$468$	0	0
$469$	−725.028	−0.0713831
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5508.35	−0.535464
$474$	0	0
$475$	−251.519	−0.0242958
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11430.1	−1.09030	−0.545151	−	0.838338i	$-0.683528\pi$
$479$	−11430.1	−1.09030	−0.545151	+	0.838338i	$0.683528\pi$
$480$	0	0
$481$	−13248.7	−1.25590
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4357.61	0.407977
$486$	0	0
$487$	2226.06	0.207130	0.103565	−	0.994623i	$-0.466975\pi$
$487$	2226.06	0.207130	0.103565	+	0.994623i	$0.466975\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17598.2	1.61750	0.808751	−	0.588151i	$-0.200144\pi$
$491$	17598.2	1.61750	0.808751	+	0.588151i	$0.200144\pi$
$492$	0	0
$493$	21372.8	1.95250
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−978.493	−0.0883127
$498$	0	0
$499$	−11902.9	−1.06783	−0.533917	−	0.845537i	$-0.679280\pi$
$499$	−11902.9	−1.06783	−0.533917	+	0.845537i	$0.679280\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16272.4	−1.44245	−0.721225	−	0.692701i	$-0.756421\pi$
$503$	−16272.4	−1.44245	−0.721225	+	0.692701i	$0.756421\pi$
$504$	0	0
$505$	4267.76	0.376065
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6901.76	−0.601012	−0.300506	−	0.953780i	$-0.597156\pi$
$509$	−6901.76	−0.601012	−0.300506	+	0.953780i	$0.597156\pi$
$510$	0	0
$511$	−2526.51	−0.218721
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8149.81	−0.697327
$516$	0	0
$517$	2882.14	0.245176
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3365.48	0.283003	0.141501	−	0.989938i	$-0.454807\pi$
$521$	3365.48	0.283003	0.141501	+	0.989938i	$0.454807\pi$
$522$	0	0
$523$	−11385.1	−0.951885	−0.475943	−	0.879476i	$-0.657893\pi$
$523$	−11385.1	−0.951885	−0.475943	+	0.879476i	$0.657893\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26977.6	−2.22991
$528$	0	0
$529$	−8988.75	−0.738781
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10885.2	−0.884596
$534$	0	0
$535$	−5102.96	−0.412374
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3699.60	0.295646
$540$	0	0
$541$	14502.0	1.15247	0.576237	−	0.817283i	$-0.304521\pi$
$541$	14502.0	1.15247	0.576237	+	0.817283i	$0.304521\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3922.50	0.308297
$546$	0	0
$547$	−14644.8	−1.14473	−0.572365	−	0.819999i	$-0.693974\pi$
$547$	−14644.8	−1.14473	−0.572365	+	0.819999i	$0.693974\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2705.63	0.209190
$552$	0	0
$553$	−153.557	−0.0118082
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12772.0	−0.971576	−0.485788	−	0.874077i	$-0.661467\pi$
$557$	−12772.0	−0.971576	−0.485788	+	0.874077i	$0.661467\pi$
$558$	0	0
$559$	17286.8	1.30797
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14317.3	−1.07176	−0.535881	−	0.844294i	$-0.680020\pi$
$563$	−14317.3	−1.07176	−0.535881	+	0.844294i	$0.680020\pi$
$564$	0	0
$565$	2896.48	0.215674
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18360.3	1.35273	0.676365	−	0.736567i	$-0.263554\pi$
$569$	18360.3	1.35273	0.676365	+	0.736567i	$0.263554\pi$
$570$	0	0
$571$	6538.27	0.479191	0.239595	−	0.970873i	$-0.422985\pi$
$571$	6538.27	0.479191	0.239595	+	0.970873i	$0.422985\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1409.40	0.102219
$576$	0	0
$577$	−18025.1	−1.30051	−0.650254	−	0.759717i	$-0.725338\pi$
$577$	−18025.1	−1.30051	−0.650254	+	0.759717i	$0.725338\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−319.392	−0.0228066
$582$	0	0
$583$	−5857.11	−0.416083
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19171.0	1.34799	0.673996	−	0.738735i	$-0.264577\pi$
$587$	19171.0	1.34799	0.673996	+	0.738735i	$0.264577\pi$
$588$	0	0
$589$	−3415.16	−0.238912
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3606.30	0.249736	0.124868	−	0.992173i	$-0.460149\pi$
$593$	3606.30	0.249736	0.124868	+	0.992173i	$0.460149\pi$
$594$	0	0
$595$	−1026.46	−0.0707242
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9376.08	−0.639560	−0.319780	−	0.947492i	$-0.603609\pi$
$599$	−9376.08	−0.639560	−0.319780	+	0.947492i	$0.603609\pi$
$600$	0	0
$601$	−20485.2	−1.39036	−0.695182	−	0.718833i	$-0.744676\pi$
$601$	−20485.2	−1.39036	−0.695182	+	0.718833i	$0.744676\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	605.000	0.0406558
$606$	0	0
$607$	−13502.5	−0.902883	−0.451441	−	0.892301i	$-0.649090\pi$
$607$	−13502.5	−0.902883	−0.451441	+	0.892301i	$0.649090\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9044.98	−0.598888
$612$	0	0
$613$	18714.1	1.23304	0.616520	−	0.787339i	$-0.288542\pi$
$613$	18714.1	1.23304	0.616520	+	0.787339i	$0.288542\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1645.47	−0.107365	−0.0536824	−	0.998558i	$-0.517096\pi$
$617$	−1645.47	−0.107365	−0.0536824	+	0.998558i	$0.517096\pi$
$618$	0	0
$619$	18300.6	1.18831	0.594153	−	0.804352i	$-0.297487\pi$
$619$	18300.6	1.18831	0.594153	+	0.804352i	$0.297487\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2584.87	0.166229
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	30500.7	1.93345
$630$	0	0
$631$	18410.5	1.16151	0.580753	−	0.814080i	$-0.302758\pi$
$631$	18410.5	1.16151	0.580753	+	0.814080i	$0.302758\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2938.40	0.183633
$636$	0	0
$637$	−11610.4	−0.722169
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30686.4	1.89086	0.945428	−	0.325830i	$-0.105644\pi$
$641$	30686.4	1.89086	0.945428	+	0.325830i	$0.105644\pi$
$642$	0	0
$643$	25030.0	1.53513	0.767565	−	0.640972i	$-0.221468\pi$
$643$	25030.0	1.53513	0.767565	+	0.640972i	$0.221468\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13695.3	0.832174	0.416087	−	0.909325i	$-0.363401\pi$
$647$	13695.3	0.832174	0.416087	+	0.909325i	$0.363401\pi$
$648$	0	0
$649$	4750.58	0.287329
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28392.5	1.70150	0.850752	−	0.525567i	$-0.176147\pi$
$653$	28392.5	1.70150	0.850752	+	0.525567i	$0.176147\pi$
$654$	0	0
$655$	−14009.6	−0.835724
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15499.0	0.916170	0.458085	−	0.888908i	$-0.348535\pi$
$659$	15499.0	0.916170	0.458085	+	0.888908i	$0.348535\pi$
$660$	0	0
$661$	2935.93	0.172760	0.0863800	−	0.996262i	$-0.472470\pi$
$661$	2935.93	0.172760	0.0863800	+	0.996262i	$0.472470\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−129.943	−0.00757737
$666$	0	0
$667$	−15161.1	−0.880122
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3112.35	−0.179063
$672$	0	0
$673$	−491.693	−0.0281625	−0.0140813	−	0.999901i	$-0.504482\pi$
$673$	−491.693	−0.0281625	−0.0140813	+	0.999901i	$0.504482\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9447.10	−0.536310	−0.268155	−	0.963376i	$-0.586414\pi$
$677$	−9447.10	−0.536310	−0.268155	+	0.963376i	$0.586414\pi$
$678$	0	0
$679$	2251.27	0.127240
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28871.5	1.61748	0.808740	−	0.588166i	$-0.200150\pi$
$683$	28871.5	1.61748	0.808740	+	0.588166i	$0.200150\pi$
$684$	0	0
$685$	1989.73	0.110984
$686$	0	0
$687$	0	0
$688$	0	0
$689$	18381.3	1.01636
$690$	0	0
$691$	−12916.1	−0.711072	−0.355536	−	0.934663i	$-0.615702\pi$
$691$	−12916.1	−0.711072	−0.355536	+	0.934663i	$0.615702\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9830.86	−0.536555
$696$	0	0
$697$	25059.6	1.36183
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8780.97	−0.473114	−0.236557	−	0.971618i	$-0.576019\pi$
$701$	−8780.97	−0.473114	−0.236557	+	0.971618i	$0.576019\pi$
$702$	0	0
$703$	3861.16	0.207150
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2204.86	0.117287
$708$	0	0
$709$	−7254.51	−0.384272	−0.192136	−	0.981368i	$-0.561541\pi$
$709$	−7254.51	−0.384272	−0.192136	+	0.981368i	$0.561541\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19137.0	1.00517
$714$	0	0
$715$	−1898.67	−0.0993092
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20157.6	−1.04555	−0.522776	−	0.852470i	$-0.675104\pi$
$719$	−20157.6	−1.04555	−0.522776	+	0.852470i	$0.675104\pi$
$720$	0	0
$721$	−4210.44	−0.217482
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6723.22	−0.344406
$726$	0	0
$727$	−28724.5	−1.46538	−0.732690	−	0.680562i	$-0.761736\pi$
$727$	−28724.5	−1.46538	−0.732690	+	0.680562i	$0.761736\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−39797.2	−2.01362
$732$	0	0
$733$	7895.01	0.397829	0.198915	−	0.980017i	$-0.436258\pi$
$733$	7895.01	0.397829	0.198915	+	0.980017i	$0.436258\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3087.43	0.154311
$738$	0	0
$739$	14605.4	0.727018	0.363509	−	0.931591i	$-0.381579\pi$
$739$	14605.4	0.727018	0.363509	+	0.931591i	$0.381579\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14464.8	−0.714217	−0.357108	−	0.934063i	$-0.616237\pi$
$743$	−14464.8	−0.714217	−0.357108	+	0.934063i	$0.616237\pi$
$744$	0	0
$745$	−14083.2	−0.692576
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2636.34	−0.128611
$750$	0	0
$751$	6387.83	0.310380	0.155190	−	0.987885i	$-0.450401\pi$
$751$	6387.83	0.310380	0.155190	+	0.987885i	$0.450401\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2207.76	−0.106422
$756$	0	0
$757$	−8597.36	−0.412783	−0.206391	−	0.978470i	$-0.566172\pi$
$757$	−8597.36	−0.412783	−0.206391	+	0.978470i	$0.566172\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	16824.0	0.801403	0.400702	−	0.916209i	$-0.368766\pi$
$761$	16824.0	0.801403	0.400702	+	0.916209i	$0.368766\pi$
$762$	0	0
$763$	2026.48	0.0961516
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14908.7	−0.701854
$768$	0	0
$769$	693.217	0.0325072	0.0162536	−	0.999868i	$-0.494826\pi$
$769$	693.217	0.0325072	0.0162536	+	0.999868i	$0.494826\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3443.03	−0.160203	−0.0801016	−	0.996787i	$-0.525524\pi$
$773$	−3443.03	−0.160203	−0.0801016	+	0.996787i	$0.525524\pi$
$774$	0	0
$775$	8486.33	0.393339
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3172.35	0.145907
$780$	0	0
$781$	4166.78	0.190908
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10634.6	0.483521
$786$	0	0
$787$	32124.1	1.45502	0.727510	−	0.686097i	$-0.240678\pi$
$787$	32124.1	1.45502	0.727510	+	0.686097i	$0.240678\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1496.41	0.0672644
$792$	0	0
$793$	9767.46	0.437393
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1639.99	−0.0728875	−0.0364437	−	0.999336i	$-0.511603\pi$
$797$	−1639.99	−0.0728875	−0.0364437	+	0.999336i	$0.511603\pi$
$798$	0	0
$799$	20823.1	0.921988
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10758.8	0.472815
$804$	0	0
$805$	728.138	0.0318801
$806$	0	0
$807$	0	0
$808$	0	0
$809$	11315.3	0.491748	0.245874	−	0.969302i	$-0.420925\pi$
$809$	11315.3	0.491748	0.245874	+	0.969302i	$0.420925\pi$
$810$	0	0
$811$	18827.5	0.815194	0.407597	−	0.913162i	$-0.366367\pi$
$811$	18827.5	0.815194	0.407597	+	0.913162i	$0.366367\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2023.84	0.0869840
$816$	0	0
$817$	−5038.02	−0.215738
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19897.4	−0.845827	−0.422914	−	0.906170i	$-0.638993\pi$
$821$	−19897.4	−0.845827	−0.422914	+	0.906170i	$0.638993\pi$
$822$	0	0
$823$	19067.4	0.807593	0.403796	−	0.914849i	$-0.367690\pi$
$823$	19067.4	0.807593	0.403796	+	0.914849i	$0.367690\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−756.691	−0.0318171	−0.0159085	−	0.999873i	$-0.505064\pi$
$827$	−756.691	−0.0318171	−0.0159085	+	0.999873i	$0.505064\pi$
$828$	0	0
$829$	12953.8	0.542707	0.271354	−	0.962480i	$-0.412529\pi$
$829$	12953.8	0.542707	0.271354	+	0.962480i	$0.412529\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	26729.2	1.11178
$834$	0	0
$835$	−676.413	−0.0280338
$836$	0	0
$837$	0	0
$838$	0	0
$839$	42201.8	1.73655	0.868276	−	0.496081i	$-0.165228\pi$
$839$	42201.8	1.73655	0.868276	+	0.496081i	$0.165228\pi$
$840$	0	0
$841$	47933.7	1.96538
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−5026.44	−0.204633
$846$	0	0
$847$	312.561	0.0126797
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−21636.2	−0.871537
$852$	0	0
$853$	−25803.7	−1.03576	−0.517879	−	0.855454i	$-0.673278\pi$
$853$	−25803.7	−1.03576	−0.517879	+	0.855454i	$0.673278\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46039.3	1.83509	0.917546	−	0.397630i	$-0.130167\pi$
$857$	46039.3	1.83509	0.917546	+	0.397630i	$0.130167\pi$
$858$	0	0
$859$	−11554.7	−0.458953	−0.229476	−	0.973314i	$-0.573701\pi$
$859$	−11554.7	−0.458953	−0.229476	+	0.973314i	$0.573701\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	43136.8	1.70150	0.850751	−	0.525570i	$-0.176148\pi$
$863$	43136.8	1.70150	0.850751	+	0.525570i	$0.176148\pi$
$864$	0	0
$865$	3742.83	0.147121
$866$	0	0
$867$	0	0
$868$	0	0
$869$	653.902	0.0255260
$870$	0	0
$871$	−9689.26	−0.376932
$872$	0	0
$873$	0	0
$874$	0	0
$875$	322.894	0.0124752
$876$	0	0
$877$	−516.000	−0.0198678	−0.00993392	−	0.999951i	$-0.503162\pi$
$877$	−516.000	−0.0198678	−0.00993392	+	0.999951i	$0.503162\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4638.97	−0.177402	−0.0887009	−	0.996058i	$-0.528272\pi$
$881$	−4638.97	−0.177402	−0.0887009	+	0.996058i	$0.528272\pi$
$882$	0	0
$883$	−44049.7	−1.67881	−0.839406	−	0.543505i	$-0.817097\pi$
$883$	−44049.7	−1.67881	−0.839406	+	0.543505i	$0.817097\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13050.4	0.494012	0.247006	−	0.969014i	$-0.420553\pi$
$887$	13050.4	0.494012	0.247006	+	0.969014i	$0.420553\pi$
$888$	0	0
$889$	1518.06	0.0572714
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2636.05	0.0987816
$894$	0	0
$895$	−18243.4	−0.681353
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−91288.8	−3.38671
$900$	0	0
$901$	−42316.9	−1.56468
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12856.7	0.472234
$906$	0	0
$907$	3907.95	0.143067	0.0715333	−	0.997438i	$-0.477211\pi$
$907$	3907.95	0.143067	0.0715333	+	0.997438i	$0.477211\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8041.96	−0.292472	−0.146236	−	0.989250i	$-0.546716\pi$
$911$	−8041.96	−0.292472	−0.146236	+	0.989250i	$0.546716\pi$
$912$	0	0
$913$	1360.09	0.0493015
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7237.77	−0.260646
$918$	0	0
$919$	33169.8	1.19061	0.595306	−	0.803499i	$-0.297031\pi$
$919$	33169.8	1.19061	0.595306	+	0.803499i	$0.297031\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−13076.6	−0.466327
$924$	0	0
$925$	−9594.59	−0.341047
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34436.9	1.21619	0.608093	−	0.793866i	$-0.291935\pi$
$929$	34436.9	1.21619	0.608093	+	0.793866i	$0.291935\pi$
$930$	0	0
$931$	3383.71	0.119116
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4371.06	0.152886
$936$	0	0
$937$	754.654	0.0263111	0.0131555	−	0.999913i	$-0.495812\pi$
$937$	754.654	0.0263111	0.0131555	+	0.999913i	$0.495812\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2229.10	0.0772227	0.0386114	−	0.999254i	$-0.487707\pi$
$941$	2229.10	0.0772227	0.0386114	+	0.999254i	$0.487707\pi$
$942$	0	0
$943$	−17776.4	−0.613870
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10205.1	0.350181	0.175090	−	0.984552i	$-0.443978\pi$
$947$	10205.1	0.350181	0.175090	+	0.984552i	$0.443978\pi$
$948$	0	0
$949$	−33764.3	−1.15494
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−56745.2	−1.92881	−0.964405	−	0.264429i	$-0.914817\pi$
$953$	−56745.2	−1.92881	−0.964405	+	0.264429i	$0.914817\pi$
$954$	0	0
$955$	−407.482	−0.0138071
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1027.96	0.0346136
$960$	0	0
$961$	85437.5	2.86790
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−14333.5	−0.478146
$966$	0	0
$967$	−20517.1	−0.682301	−0.341150	−	0.940009i	$-0.610817\pi$
$967$	−20517.1	−0.682301	−0.341150	+	0.940009i	$0.610817\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17453.6	0.576840	0.288420	−	0.957504i	$-0.406870\pi$
$971$	17453.6	0.576840	0.288420	+	0.957504i	$0.406870\pi$
$972$	0	0
$973$	−5078.92	−0.167341
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14560.1	−0.476786	−0.238393	−	0.971169i	$-0.576621\pi$
$977$	−14560.1	−0.476786	−0.238393	+	0.971169i	$0.576621\pi$
$978$	0	0
$979$	−11007.3	−0.359342
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22528.9	0.730987	0.365494	−	0.930814i	$-0.380900\pi$
$983$	22528.9	0.730987	0.365494	+	0.930814i	$0.380900\pi$
$984$	0	0
$985$	−26904.8	−0.870312
$986$	0	0
$987$	0	0
$988$	0	0
$989$	28230.8	0.907671
$990$	0	0
$991$	32132.9	1.03001	0.515003	−	0.857189i	$-0.327791\pi$
$991$	32132.9	1.03001	0.515003	+	0.857189i	$0.327791\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13560.1	0.432043
$996$	0	0
$997$	47649.6	1.51362	0.756809	−	0.653636i	$-0.226757\pi$
$997$	47649.6	1.51362	0.756809	+	0.653636i	$0.226757\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1980.4.a.l.1.2		3
3.2	odd	2		220.4.a.f.1.2	✓	3
12.11	even	2		880.4.a.w.1.2		3
15.2	even	4		1100.4.b.h.749.4		6
15.8	even	4		1100.4.b.h.749.3		6
15.14	odd	2		1100.4.a.i.1.2		3
33.32	even	2		2420.4.a.i.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.4.a.f.1.2	✓	3	3.2	odd	2
880.4.a.w.1.2		3	12.11	even	2
1100.4.a.i.1.2		3	15.14	odd	2
1100.4.b.h.749.3		6	15.8	even	4
1100.4.b.h.749.4		6	15.2	even	4
1980.4.a.l.1.2		3	1.1	even	1	trivial
2420.4.a.i.1.2		3	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1980.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +2.58315 q^{7} -11.0000 q^{11} +34.5212 q^{13} -79.4737 q^{17} -10.0608 q^{19} +56.3760 q^{23} +25.0000 q^{25} -268.929 q^{29} +339.453 q^{31} +12.9158 q^{35} -383.783 q^{37} -315.319 q^{41} +500.759 q^{43} -262.012 q^{47} -336.327 q^{49} +532.464 q^{53} -55.0000 q^{55} -431.871 q^{59} +282.941 q^{61} +172.606 q^{65} -280.676 q^{67} -378.798 q^{71} -978.074 q^{73} -28.4147 q^{77} -59.4456 q^{79} -123.644 q^{83} -397.369 q^{85} +1000.67 q^{89} +89.1735 q^{91} -50.3039 q^{95} +871.521 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} - 5 q^{7} - 33 q^{11} + 2 q^{13} - 77 q^{17} + 171 q^{19} - 222 q^{23} + 75 q^{25} - 55 q^{29} + 181 q^{31} - 25 q^{35} + 317 q^{37} - 302 q^{41} - 188 q^{43} - 662 q^{47} - 268 q^{49} - 81 q^{53}+ \cdots - 1980 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 - 5 * q^7 - 33 * q^11 + 2 * q^13 - 77 * q^17 + 171 * q^19 - 222 * q^23 + 75 * q^25 - 55 * q^29 + 181 * q^31 - 25 * q^35 + 317 * q^37 - 302 * q^41 - 188 * q^43 - 662 * q^47 - 268 * q^49 - 81 * q^53 - 165 * q^55 + 42 * q^59 + 349 * q^61 + 10 * q^65 + 152 * q^67 - 927 * q^71 - 2378 * q^73 + 55 * q^77 - 1146 * q^79 + 458 * q^83 - 385 * q^85 + 875 * q^89 + 1982 * q^91 + 855 * q^95 - 1980 * q^97

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