LMFDB - Embedded newform 1200.4.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$70.8022920069$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +13.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +13.0000 q^{7} +9.00000 q^{9} -6.00000 q^{11} +5.00000 q^{13} -78.0000 q^{17} -65.0000 q^{19} +39.0000 q^{21} -138.000 q^{23} +27.0000 q^{27} +66.0000 q^{29} -299.000 q^{31} -18.0000 q^{33} -214.000 q^{37} +15.0000 q^{39} +360.000 q^{41} -203.000 q^{43} -78.0000 q^{47} -174.000 q^{49} -234.000 q^{51} +636.000 q^{53} -195.000 q^{57} -786.000 q^{59} +467.000 q^{61} +117.000 q^{63} +217.000 q^{67} -414.000 q^{69} +360.000 q^{71} -286.000 q^{73} -78.0000 q^{77} -272.000 q^{79} +81.0000 q^{81} -498.000 q^{83} +198.000 q^{87} +65.0000 q^{91} -897.000 q^{93} -511.000 q^{97} -54.0000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	13.0000	0.701934	0.350967	−	0.936388i	$-0.385853\pi$
$7$	13.0000	0.701934	0.350967	+	0.936388i	$0.385853\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−6.00000	−0.164461	−0.0822304	−	0.996613i	$-0.526204\pi$
$11$	−6.00000	−0.164461	−0.0822304	+	0.996613i	$0.526204\pi$
$12$	0	0
$13$	5.00000	0.106673	0.0533366	−	0.998577i	$-0.483014\pi$
$13$	5.00000	0.106673	0.0533366	+	0.998577i	$0.483014\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−78.0000	−1.11281	−0.556405	−	0.830911i	$-0.687820\pi$
$17$	−78.0000	−1.11281	−0.556405	+	0.830911i	$0.687820\pi$
$18$	0	0
$19$	−65.0000	−0.784843	−0.392422	−	0.919785i	$-0.628363\pi$
$19$	−65.0000	−0.784843	−0.392422	+	0.919785i	$0.628363\pi$
$20$	0	0
$21$	39.0000	0.405262
$22$	0	0
$23$	−138.000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−138.000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	66.0000	0.422617	0.211308	−	0.977419i	$-0.432228\pi$
$29$	66.0000	0.422617	0.211308	+	0.977419i	$0.432228\pi$
$30$	0	0
$31$	−299.000	−1.73232	−0.866161	−	0.499765i	$-0.833420\pi$
$31$	−299.000	−1.73232	−0.866161	+	0.499765i	$0.833420\pi$
$32$	0	0
$33$	−18.0000	−0.0949514
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−214.000	−0.950848	−0.475424	−	0.879757i	$-0.657705\pi$
$37$	−214.000	−0.950848	−0.475424	+	0.879757i	$0.657705\pi$
$38$	0	0
$39$	15.0000	0.0615878
$40$	0	0
$41$	360.000	1.37128	0.685641	−	0.727940i	$-0.259522\pi$
$41$	360.000	1.37128	0.685641	+	0.727940i	$0.259522\pi$
$42$	0	0
$43$	−203.000	−0.719935	−0.359968	−	0.932965i	$-0.617212\pi$
$43$	−203.000	−0.719935	−0.359968	+	0.932965i	$0.617212\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−78.0000	−0.242074	−0.121037	−	0.992648i	$-0.538622\pi$
$47$	−78.0000	−0.242074	−0.121037	+	0.992648i	$0.538622\pi$
$48$	0	0
$49$	−174.000	−0.507289
$50$	0	0
$51$	−234.000	−0.642481
$52$	0	0
$53$	636.000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	636.000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−195.000	−0.453129
$58$	0	0
$59$	−786.000	−1.73438	−0.867191	−	0.497976i	$-0.834077\pi$
$59$	−786.000	−1.73438	−0.867191	+	0.497976i	$0.834077\pi$
$60$	0	0
$61$	467.000	0.980217	0.490108	−	0.871662i	$-0.336957\pi$
$61$	467.000	0.980217	0.490108	+	0.871662i	$0.336957\pi$
$62$	0	0
$63$	117.000	0.233978
$64$	0	0
$65$	0	0
$66$	0	0
$67$	217.000	0.395683	0.197842	−	0.980234i	$-0.436607\pi$
$67$	217.000	0.395683	0.197842	+	0.980234i	$0.436607\pi$
$68$	0	0
$69$	−414.000	−0.722315
$70$	0	0
$71$	360.000	0.601748	0.300874	−	0.953664i	$-0.402722\pi$
$71$	360.000	0.601748	0.300874	+	0.953664i	$0.402722\pi$
$72$	0	0
$73$	−286.000	−0.458545	−0.229272	−	0.973362i	$-0.573635\pi$
$73$	−286.000	−0.458545	−0.229272	+	0.973362i	$0.573635\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−78.0000	−0.115441
$78$	0	0
$79$	−272.000	−0.387372	−0.193686	−	0.981064i	$-0.562044\pi$
$79$	−272.000	−0.387372	−0.193686	+	0.981064i	$0.562044\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−498.000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−498.000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	198.000	0.243998
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	65.0000	0.0748775
$92$	0	0
$93$	−897.000	−1.00016
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−511.000	−0.534889	−0.267444	−	0.963573i	$-0.586179\pi$
$97$	−511.000	−0.534889	−0.267444	+	0.963573i	$0.586179\pi$
$98$	0	0
$99$	−54.0000	−0.0548202
$100$	0	0
$101$	−1812.00	−1.78516	−0.892578	−	0.450893i	$-0.851106\pi$
$101$	−1812.00	−1.78516	−0.892578	+	0.450893i	$0.851106\pi$
$102$	0	0
$103$	1708.00	1.63392	0.816962	−	0.576691i	$-0.195656\pi$
$103$	1708.00	1.63392	0.816962	+	0.576691i	$0.195656\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1236.00	−1.11672	−0.558358	−	0.829600i	$-0.688568\pi$
$107$	−1236.00	−1.11672	−0.558358	+	0.829600i	$0.688568\pi$
$108$	0	0
$109$	−1543.00	−1.35590	−0.677948	−	0.735110i	$-0.737130\pi$
$109$	−1543.00	−1.35590	−0.677948	+	0.735110i	$0.737130\pi$
$110$	0	0
$111$	−642.000	−0.548972
$112$	0	0
$113$	1884.00	1.56842	0.784212	−	0.620494i	$-0.213068\pi$
$113$	1884.00	1.56842	0.784212	+	0.620494i	$0.213068\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	45.0000	0.0355577
$118$	0	0
$119$	−1014.00	−0.781120
$120$	0	0
$121$	−1295.00	−0.972953
$122$	0	0
$123$	1080.00	0.791710
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2072.00	−1.44772	−0.723859	−	0.689948i	$-0.757634\pi$
$127$	−2072.00	−1.44772	−0.723859	+	0.689948i	$0.757634\pi$
$128$	0	0
$129$	−609.000	−0.415655
$130$	0	0
$131$	2508.00	1.67271	0.836355	−	0.548188i	$-0.184682\pi$
$131$	2508.00	1.67271	0.836355	+	0.548188i	$0.184682\pi$
$132$	0	0
$133$	−845.000	−0.550908
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1566.00	0.976587	0.488293	−	0.872679i	$-0.337620\pi$
$137$	1566.00	0.976587	0.488293	+	0.872679i	$0.337620\pi$
$138$	0	0
$139$	196.000	0.119601	0.0598004	−	0.998210i	$-0.480954\pi$
$139$	196.000	0.119601	0.0598004	+	0.998210i	$0.480954\pi$
$140$	0	0
$141$	−234.000	−0.139761
$142$	0	0
$143$	−30.0000	−0.0175435
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−522.000	−0.292883
$148$	0	0
$149$	−1278.00	−0.702670	−0.351335	−	0.936250i	$-0.614272\pi$
$149$	−1278.00	−0.702670	−0.351335	+	0.936250i	$0.614272\pi$
$150$	0	0
$151$	−1385.00	−0.746422	−0.373211	−	0.927747i	$-0.621743\pi$
$151$	−1385.00	−0.746422	−0.373211	+	0.927747i	$0.621743\pi$
$152$	0	0
$153$	−702.000	−0.370937
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−481.000	−0.244509	−0.122255	−	0.992499i	$-0.539012\pi$
$157$	−481.000	−0.244509	−0.122255	+	0.992499i	$0.539012\pi$
$158$	0	0
$159$	1908.00	0.951662
$160$	0	0
$161$	−1794.00	−0.878180
$162$	0	0
$163$	2815.00	1.35269	0.676343	−	0.736587i	$-0.263564\pi$
$163$	2815.00	1.35269	0.676343	+	0.736587i	$0.263564\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1956.00	−0.906346	−0.453173	−	0.891423i	$-0.649708\pi$
$167$	−1956.00	−0.906346	−0.453173	+	0.891423i	$0.649708\pi$
$168$	0	0
$169$	−2172.00	−0.988621
$170$	0	0
$171$	−585.000	−0.261614
$172$	0	0
$173$	2382.00	1.04682	0.523411	−	0.852081i	$-0.324659\pi$
$173$	2382.00	1.04682	0.523411	+	0.852081i	$0.324659\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2358.00	−1.00135
$178$	0	0
$179$	−4386.00	−1.83142	−0.915712	−	0.401834i	$-0.868373\pi$
$179$	−4386.00	−1.83142	−0.915712	+	0.401834i	$0.868373\pi$
$180$	0	0
$181$	−2275.00	−0.934251	−0.467125	−	0.884191i	$-0.654710\pi$
$181$	−2275.00	−0.934251	−0.467125	+	0.884191i	$0.654710\pi$
$182$	0	0
$183$	1401.00	0.565928
$184$	0	0
$185$	0	0
$186$	0	0
$187$	468.000	0.183014
$188$	0	0
$189$	351.000	0.135087
$190$	0	0
$191$	−714.000	−0.270488	−0.135244	−	0.990812i	$-0.543182\pi$
$191$	−714.000	−0.270488	−0.135244	+	0.990812i	$0.543182\pi$
$192$	0	0
$193$	1547.00	0.576971	0.288486	−	0.957484i	$-0.406848\pi$
$193$	1547.00	0.576971	0.288486	+	0.957484i	$0.406848\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	438.000	0.158407	0.0792036	−	0.996858i	$-0.474762\pi$
$197$	438.000	0.158407	0.0792036	+	0.996858i	$0.474762\pi$
$198$	0	0
$199$	−437.000	−0.155669	−0.0778344	−	0.996966i	$-0.524801\pi$
$199$	−437.000	−0.155669	−0.0778344	+	0.996966i	$0.524801\pi$
$200$	0	0
$201$	651.000	0.228448
$202$	0	0
$203$	858.000	0.296649
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1242.00	−0.417029
$208$	0	0
$209$	390.000	0.129076
$210$	0	0
$211$	−1625.00	−0.530188	−0.265094	−	0.964223i	$-0.585403\pi$
$211$	−1625.00	−0.530188	−0.265094	+	0.964223i	$0.585403\pi$
$212$	0	0
$213$	1080.00	0.347420
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3887.00	−1.21598
$218$	0	0
$219$	−858.000	−0.264741
$220$	0	0
$221$	−390.000	−0.118707
$222$	0	0
$223$	−875.000	−0.262755	−0.131377	−	0.991332i	$-0.541940\pi$
$223$	−875.000	−0.262755	−0.131377	+	0.991332i	$0.541940\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4680.00	1.36838	0.684191	−	0.729303i	$-0.260156\pi$
$227$	4680.00	1.36838	0.684191	+	0.729303i	$0.260156\pi$
$228$	0	0
$229$	1469.00	0.423905	0.211953	−	0.977280i	$-0.432018\pi$
$229$	1469.00	0.423905	0.211953	+	0.977280i	$0.432018\pi$
$230$	0	0
$231$	−234.000	−0.0666497
$232$	0	0
$233$	3960.00	1.11343	0.556713	−	0.830705i	$-0.312062\pi$
$233$	3960.00	1.11343	0.556713	+	0.830705i	$0.312062\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−816.000	−0.223649
$238$	0	0
$239$	−2652.00	−0.717756	−0.358878	−	0.933385i	$-0.616841\pi$
$239$	−2652.00	−0.717756	−0.358878	+	0.933385i	$0.616841\pi$
$240$	0	0
$241$	5753.00	1.53769	0.768845	−	0.639435i	$-0.220832\pi$
$241$	5753.00	1.53769	0.768845	+	0.639435i	$0.220832\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−325.000	−0.0837217
$248$	0	0
$249$	−1494.00	−0.380235
$250$	0	0
$251$	−3588.00	−0.902281	−0.451141	−	0.892453i	$-0.648983\pi$
$251$	−3588.00	−0.902281	−0.451141	+	0.892453i	$0.648983\pi$
$252$	0	0
$253$	828.000	0.205755
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−564.000	−0.136892	−0.0684462	−	0.997655i	$-0.521804\pi$
$257$	−564.000	−0.136892	−0.0684462	+	0.997655i	$0.521804\pi$
$258$	0	0
$259$	−2782.00	−0.667433
$260$	0	0
$261$	594.000	0.140872
$262$	0	0
$263$	−1248.00	−0.292604	−0.146302	−	0.989240i	$-0.546737\pi$
$263$	−1248.00	−0.292604	−0.146302	+	0.989240i	$0.546737\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7266.00	−1.64690	−0.823450	−	0.567390i	$-0.807953\pi$
$269$	−7266.00	−1.64690	−0.823450	+	0.567390i	$0.807953\pi$
$270$	0	0
$271$	−3224.00	−0.722672	−0.361336	−	0.932436i	$-0.617679\pi$
$271$	−3224.00	−0.722672	−0.361336	+	0.932436i	$0.617679\pi$
$272$	0	0
$273$	195.000	0.0432305
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−151.000	−0.0327535	−0.0163767	−	0.999866i	$-0.505213\pi$
$277$	−151.000	−0.0327535	−0.0163767	+	0.999866i	$0.505213\pi$
$278$	0	0
$279$	−2691.00	−0.577441
$280$	0	0
$281$	7566.00	1.60623	0.803113	−	0.595826i	$-0.203175\pi$
$281$	7566.00	1.60623	0.803113	+	0.595826i	$0.203175\pi$
$282$	0	0
$283$	−1469.00	−0.308562	−0.154281	−	0.988027i	$-0.549306\pi$
$283$	−1469.00	−0.308562	−0.154281	+	0.988027i	$0.549306\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4680.00	0.962549
$288$	0	0
$289$	1171.00	0.238347
$290$	0	0
$291$	−1533.00	−0.308818
$292$	0	0
$293$	−306.000	−0.0610127	−0.0305063	−	0.999535i	$-0.509712\pi$
$293$	−306.000	−0.0610127	−0.0305063	+	0.999535i	$0.509712\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−162.000	−0.0316505
$298$	0	0
$299$	−690.000	−0.133457
$300$	0	0
$301$	−2639.00	−0.505347
$302$	0	0
$303$	−5436.00	−1.03066
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4661.00	−0.866506	−0.433253	−	0.901272i	$-0.642634\pi$
$307$	−4661.00	−0.866506	−0.433253	+	0.901272i	$0.642634\pi$
$308$	0	0
$309$	5124.00	0.943347
$310$	0	0
$311$	−2514.00	−0.458379	−0.229189	−	0.973382i	$-0.573608\pi$
$311$	−2514.00	−0.458379	−0.229189	+	0.973382i	$0.573608\pi$
$312$	0	0
$313$	6707.00	1.21119	0.605594	−	0.795774i	$-0.292935\pi$
$313$	6707.00	1.21119	0.605594	+	0.795774i	$0.292935\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4632.00	0.820691	0.410345	−	0.911930i	$-0.365408\pi$
$317$	4632.00	0.820691	0.410345	+	0.911930i	$0.365408\pi$
$318$	0	0
$319$	−396.000	−0.0695039
$320$	0	0
$321$	−3708.00	−0.644736
$322$	0	0
$323$	5070.00	0.873382
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4629.00	−0.782827
$328$	0	0
$329$	−1014.00	−0.169920
$330$	0	0
$331$	988.000	0.164065	0.0820323	−	0.996630i	$-0.473859\pi$
$331$	988.000	0.164065	0.0820323	+	0.996630i	$0.473859\pi$
$332$	0	0
$333$	−1926.00	−0.316949
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9079.00	−1.46755	−0.733775	−	0.679392i	$-0.762244\pi$
$337$	−9079.00	−1.46755	−0.733775	+	0.679392i	$0.762244\pi$
$338$	0	0
$339$	5652.00	0.905530
$340$	0	0
$341$	1794.00	0.284899
$342$	0	0
$343$	−6721.00	−1.05802
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5994.00	0.927305	0.463652	−	0.886017i	$-0.346539\pi$
$347$	5994.00	0.927305	0.463652	+	0.886017i	$0.346539\pi$
$348$	0	0
$349$	−286.000	−0.0438660	−0.0219330	−	0.999759i	$-0.506982\pi$
$349$	−286.000	−0.0438660	−0.0219330	+	0.999759i	$0.506982\pi$
$350$	0	0
$351$	135.000	0.0205293
$352$	0	0
$353$	−11466.0	−1.72882	−0.864410	−	0.502787i	$-0.832308\pi$
$353$	−11466.0	−1.72882	−0.864410	+	0.502787i	$0.832308\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3042.00	−0.450980
$358$	0	0
$359$	624.000	0.0917367	0.0458683	−	0.998947i	$-0.485395\pi$
$359$	624.000	0.0917367	0.0458683	+	0.998947i	$0.485395\pi$
$360$	0	0
$361$	−2634.00	−0.384021
$362$	0	0
$363$	−3885.00	−0.561734
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8477.00	−1.20571	−0.602855	−	0.797851i	$-0.705970\pi$
$367$	−8477.00	−1.20571	−0.602855	+	0.797851i	$0.705970\pi$
$368$	0	0
$369$	3240.00	0.457094
$370$	0	0
$371$	8268.00	1.15702
$372$	0	0
$373$	−11257.0	−1.56264	−0.781321	−	0.624130i	$-0.785454\pi$
$373$	−11257.0	−1.56264	−0.781321	+	0.624130i	$0.785454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	330.000	0.0450819
$378$	0	0
$379$	1213.00	0.164400	0.0822000	−	0.996616i	$-0.473805\pi$
$379$	1213.00	0.164400	0.0822000	+	0.996616i	$0.473805\pi$
$380$	0	0
$381$	−6216.00	−0.835841
$382$	0	0
$383$	11076.0	1.47769	0.738847	−	0.673873i	$-0.235370\pi$
$383$	11076.0	1.47769	0.738847	+	0.673873i	$0.235370\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1827.00	−0.239978
$388$	0	0
$389$	−8016.00	−1.04480	−0.522400	−	0.852700i	$-0.674963\pi$
$389$	−8016.00	−1.04480	−0.522400	+	0.852700i	$0.674963\pi$
$390$	0	0
$391$	10764.0	1.39222
$392$	0	0
$393$	7524.00	0.965739
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6851.00	0.866100	0.433050	−	0.901370i	$-0.357437\pi$
$397$	6851.00	0.866100	0.433050	+	0.901370i	$0.357437\pi$
$398$	0	0
$399$	−2535.00	−0.318067
$400$	0	0
$401$	12636.0	1.57360	0.786798	−	0.617211i	$-0.211738\pi$
$401$	12636.0	1.57360	0.786798	+	0.617211i	$0.211738\pi$
$402$	0	0
$403$	−1495.00	−0.184792
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1284.00	0.156377
$408$	0	0
$409$	10829.0	1.30919	0.654596	−	0.755979i	$-0.272839\pi$
$409$	10829.0	1.30919	0.654596	+	0.755979i	$0.272839\pi$
$410$	0	0
$411$	4698.00	0.563833
$412$	0	0
$413$	−10218.0	−1.21742
$414$	0	0
$415$	0	0
$416$	0	0
$417$	588.000	0.0690515
$418$	0	0
$419$	6252.00	0.728950	0.364475	−	0.931213i	$-0.381248\pi$
$419$	6252.00	0.728950	0.364475	+	0.931213i	$0.381248\pi$
$420$	0	0
$421$	−15730.0	−1.82098	−0.910491	−	0.413529i	$-0.864296\pi$
$421$	−15730.0	−1.82098	−0.910491	+	0.413529i	$0.864296\pi$
$422$	0	0
$423$	−702.000	−0.0806913
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6071.00	0.688047
$428$	0	0
$429$	−90.0000	−0.0101288
$430$	0	0
$431$	10062.0	1.12452	0.562262	−	0.826959i	$-0.309931\pi$
$431$	10062.0	1.12452	0.562262	+	0.826959i	$0.309931\pi$
$432$	0	0
$433$	707.000	0.0784671	0.0392335	−	0.999230i	$-0.487508\pi$
$433$	707.000	0.0784671	0.0392335	+	0.999230i	$0.487508\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8970.00	0.981907
$438$	0	0
$439$	12493.0	1.35822	0.679110	−	0.734037i	$-0.262366\pi$
$439$	12493.0	1.35822	0.679110	+	0.734037i	$0.262366\pi$
$440$	0	0
$441$	−1566.00	−0.169096
$442$	0	0
$443$	5928.00	0.635774	0.317887	−	0.948129i	$-0.397027\pi$
$443$	5928.00	0.635774	0.317887	+	0.948129i	$0.397027\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3834.00	−0.405687
$448$	0	0
$449$	−2028.00	−0.213156	−0.106578	−	0.994304i	$-0.533989\pi$
$449$	−2028.00	−0.213156	−0.106578	+	0.994304i	$0.533989\pi$
$450$	0	0
$451$	−2160.00	−0.225522
$452$	0	0
$453$	−4155.00	−0.430947
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17030.0	1.74317	0.871586	−	0.490242i	$-0.163092\pi$
$457$	17030.0	1.74317	0.871586	+	0.490242i	$0.163092\pi$
$458$	0	0
$459$	−2106.00	−0.214160
$460$	0	0
$461$	−1044.00	−0.105475	−0.0527374	−	0.998608i	$-0.516795\pi$
$461$	−1044.00	−0.105475	−0.0527374	+	0.998608i	$0.516795\pi$
$462$	0	0
$463$	−9704.00	−0.974046	−0.487023	−	0.873389i	$-0.661917\pi$
$463$	−9704.00	−0.974046	−0.487023	+	0.873389i	$0.661917\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4194.00	0.415579	0.207789	−	0.978174i	$-0.433373\pi$
$467$	4194.00	0.415579	0.207789	+	0.978174i	$0.433373\pi$
$468$	0	0
$469$	2821.00	0.277743
$470$	0	0
$471$	−1443.00	−0.141168
$472$	0	0
$473$	1218.00	0.118401
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5724.00	0.549442
$478$	0	0
$479$	12174.0	1.16126	0.580631	−	0.814167i	$-0.302806\pi$
$479$	12174.0	1.16126	0.580631	+	0.814167i	$0.302806\pi$
$480$	0	0
$481$	−1070.00	−0.101430
$482$	0	0
$483$	−5382.00	−0.507018
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1651.00	0.153622	0.0768110	−	0.997046i	$-0.475526\pi$
$487$	1651.00	0.153622	0.0768110	+	0.997046i	$0.475526\pi$
$488$	0	0
$489$	8445.00	0.780974
$490$	0	0
$491$	6456.00	0.593391	0.296696	−	0.954972i	$-0.404115\pi$
$491$	6456.00	0.593391	0.296696	+	0.954972i	$0.404115\pi$
$492$	0	0
$493$	−5148.00	−0.470293
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4680.00	0.422388
$498$	0	0
$499$	559.000	0.0501489	0.0250744	−	0.999686i	$-0.492018\pi$
$499$	559.000	0.0501489	0.0250744	+	0.999686i	$0.492018\pi$
$500$	0	0
$501$	−5868.00	−0.523279
$502$	0	0
$503$	14208.0	1.25945	0.629725	−	0.776818i	$-0.283168\pi$
$503$	14208.0	1.25945	0.629725	+	0.776818i	$0.283168\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6516.00	−0.570781
$508$	0	0
$509$	−8082.00	−0.703789	−0.351894	−	0.936040i	$-0.614462\pi$
$509$	−8082.00	−0.703789	−0.351894	+	0.936040i	$0.614462\pi$
$510$	0	0
$511$	−3718.00	−0.321868
$512$	0	0
$513$	−1755.00	−0.151043
$514$	0	0
$515$	0	0
$516$	0	0
$517$	468.000	0.0398116
$518$	0	0
$519$	7146.00	0.604383
$520$	0	0
$521$	20502.0	1.72401	0.862005	−	0.506900i	$-0.169209\pi$
$521$	20502.0	1.72401	0.862005	+	0.506900i	$0.169209\pi$
$522$	0	0
$523$	−2069.00	−0.172985	−0.0864924	−	0.996253i	$-0.527566\pi$
$523$	−2069.00	−0.172985	−0.0864924	+	0.996253i	$0.527566\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23322.0	1.92775
$528$	0	0
$529$	6877.00	0.565217
$530$	0	0
$531$	−7074.00	−0.578127
$532$	0	0
$533$	1800.00	0.146279
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−13158.0	−1.05737
$538$	0	0
$539$	1044.00	0.0834291
$540$	0	0
$541$	1553.00	0.123417	0.0617086	−	0.998094i	$-0.480345\pi$
$541$	1553.00	0.123417	0.0617086	+	0.998094i	$0.480345\pi$
$542$	0	0
$543$	−6825.00	−0.539390
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12584.0	−0.983643	−0.491822	−	0.870696i	$-0.663669\pi$
$547$	−12584.0	−0.983643	−0.491822	+	0.870696i	$0.663669\pi$
$548$	0	0
$549$	4203.00	0.326739
$550$	0	0
$551$	−4290.00	−0.331688
$552$	0	0
$553$	−3536.00	−0.271910
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20754.0	1.57877	0.789385	−	0.613898i	$-0.210399\pi$
$557$	20754.0	1.57877	0.789385	+	0.613898i	$0.210399\pi$
$558$	0	0
$559$	−1015.00	−0.0767977
$560$	0	0
$561$	1404.00	0.105663
$562$	0	0
$563$	−14370.0	−1.07571	−0.537854	−	0.843038i	$-0.680765\pi$
$563$	−14370.0	−1.07571	−0.537854	+	0.843038i	$0.680765\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1053.00	0.0779927
$568$	0	0
$569$	15774.0	1.16218	0.581090	−	0.813839i	$-0.302626\pi$
$569$	15774.0	1.16218	0.581090	+	0.813839i	$0.302626\pi$
$570$	0	0
$571$	3055.00	0.223902	0.111951	−	0.993714i	$-0.464290\pi$
$571$	3055.00	0.223902	0.111951	+	0.993714i	$0.464290\pi$
$572$	0	0
$573$	−2142.00	−0.156166
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−25021.0	−1.80526	−0.902632	−	0.430412i	$-0.858368\pi$
$577$	−25021.0	−1.80526	−0.902632	+	0.430412i	$0.858368\pi$
$578$	0	0
$579$	4641.00	0.333115
$580$	0	0
$581$	−6474.00	−0.462284
$582$	0	0
$583$	−3816.00	−0.271085
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18828.0	1.32388	0.661938	−	0.749559i	$-0.269734\pi$
$587$	18828.0	1.32388	0.661938	+	0.749559i	$0.269734\pi$
$588$	0	0
$589$	19435.0	1.35960
$590$	0	0
$591$	1314.00	0.0914564
$592$	0	0
$593$	19968.0	1.38278	0.691389	−	0.722483i	$-0.256999\pi$
$593$	19968.0	1.38278	0.691389	+	0.722483i	$0.256999\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1311.00	−0.0898755
$598$	0	0
$599$	−9684.00	−0.660563	−0.330282	−	0.943882i	$-0.607144\pi$
$599$	−9684.00	−0.660563	−0.330282	+	0.943882i	$0.607144\pi$
$600$	0	0
$601$	23243.0	1.57754	0.788770	−	0.614688i	$-0.210718\pi$
$601$	23243.0	1.57754	0.788770	+	0.614688i	$0.210718\pi$
$602$	0	0
$603$	1953.00	0.131894
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22984.0	1.53689	0.768445	−	0.639916i	$-0.221031\pi$
$607$	22984.0	1.53689	0.768445	+	0.639916i	$0.221031\pi$
$608$	0	0
$609$	2574.00	0.171271
$610$	0	0
$611$	−390.000	−0.0258228
$612$	0	0
$613$	13754.0	0.906230	0.453115	−	0.891452i	$-0.350313\pi$
$613$	13754.0	0.906230	0.453115	+	0.891452i	$0.350313\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9468.00	0.617775	0.308888	−	0.951099i	$-0.400043\pi$
$617$	9468.00	0.617775	0.308888	+	0.951099i	$0.400043\pi$
$618$	0	0
$619$	28099.0	1.82455	0.912273	−	0.409582i	$-0.134326\pi$
$619$	28099.0	1.82455	0.912273	+	0.409582i	$0.134326\pi$
$620$	0	0
$621$	−3726.00	−0.240772
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1170.00	0.0745220
$628$	0	0
$629$	16692.0	1.05811
$630$	0	0
$631$	−14807.0	−0.934164	−0.467082	−	0.884214i	$-0.654695\pi$
$631$	−14807.0	−0.934164	−0.467082	+	0.884214i	$0.654695\pi$
$632$	0	0
$633$	−4875.00	−0.306104
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−870.000	−0.0541141
$638$	0	0
$639$	3240.00	0.200583
$640$	0	0
$641$	20844.0	1.28438	0.642191	−	0.766545i	$-0.278026\pi$
$641$	20844.0	1.28438	0.642191	+	0.766545i	$0.278026\pi$
$642$	0	0
$643$	−3692.00	−0.226436	−0.113218	−	0.993570i	$-0.536116\pi$
$643$	−3692.00	−0.226436	−0.113218	+	0.993570i	$0.536116\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20004.0	1.21552	0.607758	−	0.794123i	$-0.292069\pi$
$647$	20004.0	1.21552	0.607758	+	0.794123i	$0.292069\pi$
$648$	0	0
$649$	4716.00	0.285238
$650$	0	0
$651$	−11661.0	−0.702044
$652$	0	0
$653$	−17862.0	−1.07044	−0.535218	−	0.844714i	$-0.679770\pi$
$653$	−17862.0	−1.07044	−0.535218	+	0.844714i	$0.679770\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2574.00	−0.152848
$658$	0	0
$659$	−18552.0	−1.09664	−0.548318	−	0.836270i	$-0.684732\pi$
$659$	−18552.0	−1.09664	−0.548318	+	0.836270i	$0.684732\pi$
$660$	0	0
$661$	−12382.0	−0.728599	−0.364300	−	0.931282i	$-0.618692\pi$
$661$	−12382.0	−0.728599	−0.364300	+	0.931282i	$0.618692\pi$
$662$	0	0
$663$	−1170.00	−0.0685355
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9108.00	−0.528730
$668$	0	0
$669$	−2625.00	−0.151702
$670$	0	0
$671$	−2802.00	−0.161207
$672$	0	0
$673$	−1690.00	−0.0967975	−0.0483987	−	0.998828i	$-0.515412\pi$
$673$	−1690.00	−0.0967975	−0.0483987	+	0.998828i	$0.515412\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5682.00	0.322566	0.161283	−	0.986908i	$-0.448437\pi$
$677$	5682.00	0.322566	0.161283	+	0.986908i	$0.448437\pi$
$678$	0	0
$679$	−6643.00	−0.375456
$680$	0	0
$681$	14040.0	0.790035
$682$	0	0
$683$	−31512.0	−1.76541	−0.882704	−	0.469930i	$-0.844279\pi$
$683$	−31512.0	−1.76541	−0.882704	+	0.469930i	$0.844279\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4407.00	0.244742
$688$	0	0
$689$	3180.00	0.175832
$690$	0	0
$691$	−20648.0	−1.13674	−0.568370	−	0.822773i	$-0.692426\pi$
$691$	−20648.0	−1.13674	−0.568370	+	0.822773i	$0.692426\pi$
$692$	0	0
$693$	−702.000	−0.0384802
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−28080.0	−1.52598
$698$	0	0
$699$	11880.0	0.642837
$700$	0	0
$701$	−11874.0	−0.639764	−0.319882	−	0.947457i	$-0.603643\pi$
$701$	−11874.0	−0.639764	−0.319882	+	0.947457i	$0.603643\pi$
$702$	0	0
$703$	13910.0	0.746267
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−23556.0	−1.25306
$708$	0	0
$709$	−10699.0	−0.566727	−0.283363	−	0.959013i	$-0.591450\pi$
$709$	−10699.0	−0.566727	−0.283363	+	0.959013i	$0.591450\pi$
$710$	0	0
$711$	−2448.00	−0.129124
$712$	0	0
$713$	41262.0	2.16728
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7956.00	−0.414396
$718$	0	0
$719$	5598.00	0.290362	0.145181	−	0.989405i	$-0.453624\pi$
$719$	5598.00	0.290362	0.145181	+	0.989405i	$0.453624\pi$
$720$	0	0
$721$	22204.0	1.14691
$722$	0	0
$723$	17259.0	0.887786
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−22859.0	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$727$	−22859.0	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	15834.0	0.801151
$732$	0	0
$733$	−31642.0	−1.59444	−0.797220	−	0.603689i	$-0.793697\pi$
$733$	−31642.0	−1.59444	−0.797220	+	0.603689i	$0.793697\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1302.00	−0.0650743
$738$	0	0
$739$	12220.0	0.608281	0.304141	−	0.952627i	$-0.401631\pi$
$739$	12220.0	0.608281	0.304141	+	0.952627i	$0.401631\pi$
$740$	0	0
$741$	−975.000	−0.0483367
$742$	0	0
$743$	−17892.0	−0.883437	−0.441719	−	0.897154i	$-0.645631\pi$
$743$	−17892.0	−0.883437	−0.441719	+	0.897154i	$0.645631\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−4482.00	−0.219529
$748$	0	0
$749$	−16068.0	−0.783861
$750$	0	0
$751$	16648.0	0.808914	0.404457	−	0.914557i	$-0.367461\pi$
$751$	16648.0	0.808914	0.404457	+	0.914557i	$0.367461\pi$
$752$	0	0
$753$	−10764.0	−0.520932
$754$	0	0
$755$	0	0
$756$	0	0
$757$	31733.0	1.52359	0.761794	−	0.647820i	$-0.224319\pi$
$757$	31733.0	1.52359	0.761794	+	0.647820i	$0.224319\pi$
$758$	0	0
$759$	2484.00	0.118792
$760$	0	0
$761$	−10068.0	−0.479586	−0.239793	−	0.970824i	$-0.577080\pi$
$761$	−10068.0	−0.479586	−0.239793	+	0.970824i	$0.577080\pi$
$762$	0	0
$763$	−20059.0	−0.951749
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3930.00	−0.185012
$768$	0	0
$769$	24323.0	1.14058	0.570292	−	0.821442i	$-0.306830\pi$
$769$	24323.0	1.14058	0.570292	+	0.821442i	$0.306830\pi$
$770$	0	0
$771$	−1692.00	−0.0790349
$772$	0	0
$773$	−2184.00	−0.101621	−0.0508105	−	0.998708i	$-0.516180\pi$
$773$	−2184.00	−0.101621	−0.0508105	+	0.998708i	$0.516180\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8346.00	−0.385342
$778$	0	0
$779$	−23400.0	−1.07624
$780$	0	0
$781$	−2160.00	−0.0989640
$782$	0	0
$783$	1782.00	0.0813327
$784$	0	0
$785$	0	0
$786$	0	0
$787$	37531.0	1.69992	0.849959	−	0.526849i	$-0.176627\pi$
$787$	37531.0	1.69992	0.849959	+	0.526849i	$0.176627\pi$
$788$	0	0
$789$	−3744.00	−0.168935
$790$	0	0
$791$	24492.0	1.10093
$792$	0	0
$793$	2335.00	0.104563
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28644.0	1.27305	0.636526	−	0.771255i	$-0.280371\pi$
$797$	28644.0	1.27305	0.636526	+	0.771255i	$0.280371\pi$
$798$	0	0
$799$	6084.00	0.269382
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1716.00	0.0754126
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−21798.0	−0.950838
$808$	0	0
$809$	−17316.0	−0.752532	−0.376266	−	0.926512i	$-0.622792\pi$
$809$	−17316.0	−0.752532	−0.376266	+	0.926512i	$0.622792\pi$
$810$	0	0
$811$	−425.000	−0.0184017	−0.00920084	−	0.999958i	$-0.502929\pi$
$811$	−425.000	−0.0184017	−0.00920084	+	0.999958i	$0.502929\pi$
$812$	0	0
$813$	−9672.00	−0.417235
$814$	0	0
$815$	0	0
$816$	0	0
$817$	13195.0	0.565036
$818$	0	0
$819$	585.000	0.0249592
$820$	0	0
$821$	−25194.0	−1.07098	−0.535491	−	0.844541i	$-0.679874\pi$
$821$	−25194.0	−1.07098	−0.535491	+	0.844541i	$0.679874\pi$
$822$	0	0
$823$	33961.0	1.43840	0.719202	−	0.694801i	$-0.244508\pi$
$823$	33961.0	1.43840	0.719202	+	0.694801i	$0.244508\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1956.00	0.0822452	0.0411226	−	0.999154i	$-0.486907\pi$
$827$	1956.00	0.0822452	0.0411226	+	0.999154i	$0.486907\pi$
$828$	0	0
$829$	−47302.0	−1.98174	−0.990872	−	0.134803i	$-0.956960\pi$
$829$	−47302.0	−1.98174	−0.990872	+	0.134803i	$0.956960\pi$
$830$	0	0
$831$	−453.000	−0.0189102
$832$	0	0
$833$	13572.0	0.564516
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8073.00	−0.333386
$838$	0	0
$839$	−30726.0	−1.26434	−0.632169	−	0.774831i	$-0.717835\pi$
$839$	−30726.0	−1.26434	−0.632169	+	0.774831i	$0.717835\pi$
$840$	0	0
$841$	−20033.0	−0.821395
$842$	0	0
$843$	22698.0	0.927355
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−16835.0	−0.682949
$848$	0	0
$849$	−4407.00	−0.178148
$850$	0	0
$851$	29532.0	1.18959
$852$	0	0
$853$	−34489.0	−1.38439	−0.692193	−	0.721713i	$-0.743355\pi$
$853$	−34489.0	−1.38439	−0.692193	+	0.721713i	$0.743355\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8484.00	0.338166	0.169083	−	0.985602i	$-0.445919\pi$
$857$	8484.00	0.338166	0.169083	+	0.985602i	$0.445919\pi$
$858$	0	0
$859$	520.000	0.0206544	0.0103272	−	0.999947i	$-0.496713\pi$
$859$	520.000	0.0206544	0.0103272	+	0.999947i	$0.496713\pi$
$860$	0	0
$861$	14040.0	0.555728
$862$	0	0
$863$	6864.00	0.270745	0.135373	−	0.990795i	$-0.456777\pi$
$863$	6864.00	0.270745	0.135373	+	0.990795i	$0.456777\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3513.00	0.137610
$868$	0	0
$869$	1632.00	0.0637075
$870$	0	0
$871$	1085.00	0.0422088
$872$	0	0
$873$	−4599.00	−0.178296
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38857.0	−1.49613	−0.748066	−	0.663624i	$-0.769017\pi$
$877$	−38857.0	−1.49613	−0.748066	+	0.663624i	$0.769017\pi$
$878$	0	0
$879$	−918.000	−0.0352257
$880$	0	0
$881$	−10080.0	−0.385475	−0.192738	−	0.981250i	$-0.561737\pi$
$881$	−10080.0	−0.385475	−0.192738	+	0.981250i	$0.561737\pi$
$882$	0	0
$883$	34549.0	1.31672	0.658362	−	0.752702i	$-0.271250\pi$
$883$	34549.0	1.31672	0.658362	+	0.752702i	$0.271250\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18798.0	0.711584	0.355792	−	0.934565i	$-0.384211\pi$
$887$	18798.0	0.711584	0.355792	+	0.934565i	$0.384211\pi$
$888$	0	0
$889$	−26936.0	−1.01620
$890$	0	0
$891$	−486.000	−0.0182734
$892$	0	0
$893$	5070.00	0.189990
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2070.00	−0.0770516
$898$	0	0
$899$	−19734.0	−0.732109
$900$	0	0
$901$	−49608.0	−1.83428
$902$	0	0
$903$	−7917.00	−0.291762
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−50276.0	−1.84056	−0.920280	−	0.391261i	$-0.872039\pi$
$907$	−50276.0	−1.84056	−0.920280	+	0.391261i	$0.872039\pi$
$908$	0	0
$909$	−16308.0	−0.595052
$910$	0	0
$911$	20502.0	0.745622	0.372811	−	0.927907i	$-0.378394\pi$
$911$	20502.0	0.745622	0.372811	+	0.927907i	$0.378394\pi$
$912$	0	0
$913$	2988.00	0.108311
$914$	0	0
$915$	0	0
$916$	0	0
$917$	32604.0	1.17413
$918$	0	0
$919$	−37817.0	−1.35742	−0.678709	−	0.734407i	$-0.737460\pi$
$919$	−37817.0	−1.35742	−0.678709	+	0.734407i	$0.737460\pi$
$920$	0	0
$921$	−13983.0	−0.500277
$922$	0	0
$923$	1800.00	0.0641904
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15372.0	0.544642
$928$	0	0
$929$	−11778.0	−0.415957	−0.207978	−	0.978133i	$-0.566688\pi$
$929$	−11778.0	−0.415957	−0.207978	+	0.978133i	$0.566688\pi$
$930$	0	0
$931$	11310.0	0.398142
$932$	0	0
$933$	−7542.00	−0.264645
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18409.0	−0.641831	−0.320916	−	0.947108i	$-0.603991\pi$
$937$	−18409.0	−0.641831	−0.320916	+	0.947108i	$0.603991\pi$
$938$	0	0
$939$	20121.0	0.699280
$940$	0	0
$941$	−9330.00	−0.323219	−0.161610	−	0.986855i	$-0.551669\pi$
$941$	−9330.00	−0.323219	−0.161610	+	0.986855i	$0.551669\pi$
$942$	0	0
$943$	−49680.0	−1.71559
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−49146.0	−1.68641	−0.843205	−	0.537592i	$-0.819334\pi$
$947$	−49146.0	−1.68641	−0.843205	+	0.537592i	$0.819334\pi$
$948$	0	0
$949$	−1430.00	−0.0489144
$950$	0	0
$951$	13896.0	0.473826
$952$	0	0
$953$	−5928.00	−0.201497	−0.100749	−	0.994912i	$-0.532124\pi$
$953$	−5928.00	−0.201497	−0.100749	+	0.994912i	$0.532124\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1188.00	−0.0401281
$958$	0	0
$959$	20358.0	0.685500
$960$	0	0
$961$	59610.0	2.00094
$962$	0	0
$963$	−11124.0	−0.372239
$964$	0	0
$965$	0	0
$966$	0	0
$967$	9520.00	0.316590	0.158295	−	0.987392i	$-0.449400\pi$
$967$	9520.00	0.316590	0.158295	+	0.987392i	$0.449400\pi$
$968$	0	0
$969$	15210.0	0.504247
$970$	0	0
$971$	−12558.0	−0.415042	−0.207521	−	0.978231i	$-0.566539\pi$
$971$	−12558.0	−0.415042	−0.207521	+	0.978231i	$0.566539\pi$
$972$	0	0
$973$	2548.00	0.0839518
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−46206.0	−1.51306	−0.756531	−	0.653958i	$-0.773107\pi$
$977$	−46206.0	−1.51306	−0.756531	+	0.653958i	$0.773107\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−13887.0	−0.451965
$982$	0	0
$983$	32676.0	1.06023	0.530113	−	0.847927i	$-0.322149\pi$
$983$	32676.0	1.06023	0.530113	+	0.847927i	$0.322149\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3042.00	−0.0981033
$988$	0	0
$989$	28014.0	0.900701
$990$	0	0
$991$	8137.00	0.260828	0.130414	−	0.991460i	$-0.458369\pi$
$991$	8137.00	0.260828	0.130414	+	0.991460i	$0.458369\pi$
$992$	0	0
$993$	2964.00	0.0947228
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18578.0	0.590142	0.295071	−	0.955475i	$-0.404657\pi$
$997$	18578.0	0.590142	0.295071	+	0.955475i	$0.404657\pi$
$998$	0	0
$999$	−5778.00	−0.182991

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.bi.1.1		1
4.3	odd	2		300.4.a.a.1.1	✓	1
5.2	odd	4		1200.4.f.k.49.1		2
5.3	odd	4		1200.4.f.k.49.2		2
5.4	even	2		1200.4.a.d.1.1		1
12.11	even	2		900.4.a.f.1.1		1
20.3	even	4		300.4.d.c.49.1		2
20.7	even	4		300.4.d.c.49.2		2
20.19	odd	2		300.4.a.h.1.1	yes	1
60.23	odd	4		900.4.d.e.649.2		2
60.47	odd	4		900.4.d.e.649.1		2
60.59	even	2		900.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.4.a.a.1.1	✓	1	4.3	odd	2
300.4.a.h.1.1	yes	1	20.19	odd	2
300.4.d.c.49.1		2	20.3	even	4
300.4.d.c.49.2		2	20.7	even	4
900.4.a.f.1.1		1	12.11	even	2
900.4.a.l.1.1		1	60.59	even	2
900.4.d.e.649.1		2	60.47	odd	4
900.4.d.e.649.2		2	60.23	odd	4
1200.4.a.d.1.1		1	5.4	even	2
1200.4.a.bi.1.1		1	1.1	even	1	trivial
1200.4.f.k.49.1		2	5.2	odd	4
1200.4.f.k.49.2		2	5.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)

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L-functions

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