LMFDB - Embedded newform 1800.3.l.c.1601.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1800,3,Mod(1601,1800)] 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("1800.1601"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1800.l (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-4] 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:	no
Analytic conductor:	$49.0464475849$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 20x^{2} + 81 $ x^4 + 20*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1601.4
Root			$-2.37510i$ of defining polynomial
Character	$\chi$	$=$	1800.1601
Dual form			1800.3.l.c.1601.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+7.71780 q^{7} +1.41421i q^{11} +12.4356 q^{13} +10.9146i q^{17} -5.71780 q^{19} +17.5866i q^{23} -8.08619i q^{29} +45.1534 q^{31} +10.5644 q^{37} +62.4423i q^{41} -31.7178 q^{43} +4.02571i q^{47} +10.5644 q^{49} -62.4423i q^{53} +23.0265i q^{59} +22.1288 q^{61} +18.2822 q^{67} -32.5617i q^{71} -88.3068 q^{73} +10.9146i q^{77} +77.7424 q^{79} -80.0289i q^{83} +95.0040i q^{89} +95.9754 q^{91} +145.871 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 20 q^{13} + 12 q^{19} + 76 q^{31} + 112 q^{37} - 92 q^{43} + 112 q^{49} + 228 q^{61} + 108 q^{67} - 144 q^{73} + 32 q^{79} + 628 q^{91} + 444 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 20 * q^13 + 12 * q^19 + 76 * q^31 + 112 * q^37 - 92 * q^43 + 112 * q^49 + 228 * q^61 + 108 * q^67 - 144 * q^73 + 32 * q^79 + 628 * q^91 + 444 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.71780	1.10254	0.551271	−	0.834326i	$-0.314143\pi$
$7$	7.71780	1.10254	0.551271	+	0.834326i	$0.314143\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.41421i	0.128565i	0.997932	+	0.0642824i	$0.0204759\pi$
$11$	1.41421i	0.128565i	−0.997932	+	0.0642824i	$0.979524\pi$
$12$	0	0
$13$	12.4356	0.956584	0.478292	−	0.878201i	$-0.341256\pi$
$13$	12.4356	0.956584	0.478292	+	0.878201i	$0.341256\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	10.9146i	0.642036i	0.947073	+	0.321018i	$0.104025\pi$
$17$	10.9146i	0.642036i	−0.947073	+	0.321018i	$0.895975\pi$
$18$	0	0
$19$	−5.71780	−0.300937	−0.150468	−	0.988615i	$-0.548078\pi$
$19$	−5.71780	−0.300937	−0.150468	+	0.988615i	$0.548078\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	17.5866i	0.764634i	0.924031	+	0.382317i	$0.124874\pi$
$23$	17.5866i	0.764634i	−0.924031	+	0.382317i	$0.875126\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.08619i	− 0.278834i	−0.990234	−	0.139417i	$-0.955477\pi$
$29$	− 8.08619i	− 0.278834i	0.990234	−	0.139417i	$-0.0445229\pi$
$30$	0	0
$31$	45.1534	1.45656	0.728281	−	0.685279i	$-0.240320\pi$
$31$	45.1534	1.45656	0.728281	+	0.685279i	$0.240320\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.5644	0.285524	0.142762	−	0.989757i	$-0.454402\pi$
$37$	10.5644	0.285524	0.142762	+	0.989757i	$0.454402\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	62.4423i	1.52298i	0.648175	+	0.761492i	$0.275533\pi$
$41$	62.4423i	1.52298i	−0.648175	+	0.761492i	$0.724467\pi$
$42$	0	0
$43$	−31.7178	−0.737623	−0.368812	−	0.929504i	$-0.620235\pi$
$43$	−31.7178	−0.737623	−0.368812	+	0.929504i	$0.620235\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.02571i	0.0856534i	0.999083	+	0.0428267i	$0.0136363\pi$
$47$	4.02571i	0.0856534i	−0.999083	+	0.0428267i	$0.986364\pi$
$48$	0	0
$49$	10.5644	0.215600
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 62.4423i	− 1.17816i	−0.808076	−	0.589079i	$-0.799491\pi$
$53$	− 62.4423i	− 1.17816i	0.808076	−	0.589079i	$-0.200509\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	23.0265i	0.390280i	0.980775	+	0.195140i	$0.0625161\pi$
$59$	23.0265i	0.390280i	−0.980775	+	0.195140i	$0.937484\pi$
$60$	0	0
$61$	22.1288	0.362767	0.181384	−	0.983412i	$-0.441942\pi$
$61$	22.1288	0.362767	0.181384	+	0.983412i	$0.441942\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	18.2822	0.272869	0.136434	−	0.990649i	$-0.456436\pi$
$67$	18.2822	0.272869	0.136434	+	0.990649i	$0.456436\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 32.5617i	− 0.458615i	−0.973354	−	0.229308i	$-0.926354\pi$
$71$	− 32.5617i	− 0.458615i	0.973354	−	0.229308i	$-0.0736462\pi$
$72$	0	0
$73$	−88.3068	−1.20968	−0.604841	−	0.796346i	$-0.706763\pi$
$73$	−88.3068	−1.20968	−0.604841	+	0.796346i	$0.706763\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.9146i	0.141748i
$78$	0	0
$79$	77.7424	0.984081	0.492040	−	0.870572i	$-0.336251\pi$
$79$	77.7424	0.984081	0.492040	+	0.870572i	$0.336251\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 80.0289i	− 0.964204i	−0.876115	−	0.482102i	$-0.839873\pi$
$83$	− 80.0289i	− 0.964204i	0.876115	−	0.482102i	$-0.160127\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	95.0040i	1.06746i	0.845655	+	0.533730i	$0.179210\pi$
$89$	95.0040i	1.06746i	−0.845655	+	0.533730i	$0.820790\pi$
$90$	0	0
$91$	95.9754	1.05467
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	145.871	1.50383	0.751913	−	0.659262i	$-0.229131\pi$
$97$	145.871	1.50383	0.751913	+	0.659262i	$0.229131\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	107.263i	1.06201i	0.847368	+	0.531006i	$0.178186\pi$
$101$	107.263i	1.06201i	−0.847368	+	0.531006i	$0.821814\pi$
$102$	0	0
$103$	−144.871	−1.40652	−0.703258	−	0.710935i	$-0.748272\pi$
$103$	−144.871	−1.40652	−0.703258	+	0.710935i	$0.748272\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	202.267i	1.89035i	0.326566	+	0.945174i	$0.394108\pi$
$107$	202.267i	1.89035i	−0.326566	+	0.945174i	$0.605892\pi$
$108$	0	0
$109$	148.742	1.36461	0.682305	−	0.731068i	$-0.260978\pi$
$109$	148.742	1.36461	0.682305	+	0.731068i	$0.260978\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 44.8210i	− 0.396646i	−0.980137	−	0.198323i	$-0.936451\pi$
$113$	− 44.8210i	− 0.396646i	0.980137	−	0.198323i	$-0.0635495\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	84.2368i	0.707872i
$120$	0	0
$121$	119.000	0.983471
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.5644	0.0831843	0.0415921	−	0.999135i	$-0.486757\pi$
$127$	10.5644	0.0831843	0.0415921	+	0.999135i	$0.486757\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	157.446i	1.20188i	0.799294	+	0.600940i	$0.205207\pi$
$131$	157.446i	1.20188i	−0.799294	+	0.600940i	$0.794793\pi$
$132$	0	0
$133$	−44.1288	−0.331796
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 183.301i	− 1.33797i	−0.743278	−	0.668983i	$-0.766730\pi$
$137$	− 183.301i	− 1.33797i	0.743278	−	0.668983i	$-0.233270\pi$
$138$	0	0
$139$	−67.1780	−0.483295	−0.241647	−	0.970364i	$-0.577688\pi$
$139$	−67.1780	−0.483295	−0.241647	+	0.970364i	$0.577688\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	17.5866i	0.122983i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 259.304i	− 1.74030i	−0.492788	−	0.870149i	$-0.664022\pi$
$149$	− 259.304i	− 1.74030i	0.492788	−	0.870149i	$-0.335978\pi$
$150$	0	0
$151$	262.025	1.73526	0.867631	−	0.497209i	$-0.165642\pi$
$151$	262.025	1.73526	0.867631	+	0.497209i	$0.165642\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	188.129	1.19827	0.599136	−	0.800647i	$-0.295511\pi$
$157$	188.129	1.19827	0.599136	+	0.800647i	$0.295511\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	135.730i	0.843042i
$162$	0	0
$163$	3.76697	0.0231102	0.0115551	−	0.999933i	$-0.496322\pi$
$163$	3.76697	0.0231102	0.0115551	+	0.999933i	$0.496322\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	262.029i	1.56903i	0.620108	+	0.784517i	$0.287089\pi$
$167$	262.029i	1.56903i	−0.620108	+	0.784517i	$0.712911\pi$
$168$	0	0
$169$	−14.3560	−0.0849465
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 109.910i	− 0.635315i	−0.948205	−	0.317658i	$-0.897104\pi$
$173$	− 109.910i	− 0.635315i	0.948205	−	0.317658i	$-0.102896\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	272.796i	1.52400i	0.647578	+	0.762000i	$0.275782\pi$
$179$	272.796i	1.52400i	−0.647578	+	0.762000i	$0.724218\pi$
$180$	0	0
$181$	320.534	1.77091	0.885453	−	0.464729i	$-0.153848\pi$
$181$	320.534	1.77091	0.885453	+	0.464729i	$0.153848\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−15.4356	−0.0825433
$188$	0	0
$189$	0	0
$190$	0	0
$191$	233.415i	1.22207i	0.791605	+	0.611033i	$0.209246\pi$
$191$	233.415i	1.22207i	−0.791605	+	0.611033i	$0.790754\pi$
$192$	0	0
$193$	−8.69321	−0.0450426	−0.0225213	−	0.999746i	$-0.507169\pi$
$193$	−8.69321	−0.0450426	−0.0225213	+	0.999746i	$0.507169\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	101.711i	0.516298i	0.966105	+	0.258149i	$0.0831126\pi$
$197$	101.711i	0.516298i	−0.966105	+	0.258149i	$0.916887\pi$
$198$	0	0
$199$	−128.638	−0.646423	−0.323211	−	0.946327i	$-0.604763\pi$
$199$	−128.638	−0.646423	−0.323211	+	0.946327i	$0.604763\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 62.4076i	− 0.307426i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 8.08619i	− 0.0386899i
$210$	0	0
$211$	215.975	1.02358	0.511790	−	0.859111i	$-0.328983\pi$
$211$	215.975	1.02358	0.511790	+	0.859111i	$0.328983\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	348.485	1.60592
$218$	0	0
$219$	0	0
$220$	0	0
$221$	135.730i	0.614162i
$222$	0	0
$223$	−130.540	−0.585380	−0.292690	−	0.956207i	$-0.594550\pi$
$223$	−130.540	−0.585380	−0.292690	+	0.956207i	$0.594550\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	122.204i	0.538342i	0.963092	+	0.269171i	$0.0867497\pi$
$227$	122.204i	0.538342i	−0.963092	+	0.269171i	$0.913250\pi$
$228$	0	0
$229$	359.970	1.57192	0.785960	−	0.618278i	$-0.212169\pi$
$229$	359.970	1.57192	0.785960	+	0.618278i	$0.212169\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 12.2593i	− 0.0526150i	−0.999654	−	0.0263075i	$-0.991625\pi$
$233$	− 12.2593i	− 0.0526150i	0.999654	−	0.0263075i	$-0.00837490\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	62.4423i	0.261265i	0.991431	+	0.130632i	$0.0417008\pi$
$239$	62.4423i	0.261265i	−0.991431	+	0.130632i	$0.958299\pi$
$240$	0	0
$241$	−135.356	−0.561643	−0.280821	−	0.959760i	$-0.590607\pi$
$241$	−135.356	−0.561643	−0.280821	+	0.959760i	$0.590607\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−71.1042	−0.287871
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 359.714i	− 1.43312i	−0.697524	−	0.716561i	$-0.745715\pi$
$251$	− 359.714i	− 1.43312i	0.697524	−	0.716561i	$-0.254285\pi$
$252$	0	0
$253$	−24.8712	−0.0983051
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 264.710i	− 1.03000i	−0.857191	−	0.514999i	$-0.827792\pi$
$257$	− 264.710i	− 1.03000i	0.857191	−	0.514999i	$-0.172208\pi$
$258$	0	0
$259$	81.5339	0.314803
$260$	0	0
$261$	0	0
$262$	0	0
$263$	152.084i	0.578267i	0.957289	+	0.289134i	$0.0933672\pi$
$263$	152.084i	0.578267i	−0.957289	+	0.289134i	$0.906633\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 66.6154i	− 0.247641i	−0.992305	−	0.123821i	$-0.960485\pi$
$269$	− 66.6154i	− 0.247641i	0.992305	−	0.123821i	$-0.0395147\pi$
$270$	0	0
$271$	−81.5339	−0.300863	−0.150432	−	0.988620i	$-0.548066\pi$
$271$	−81.5339	−0.300863	−0.150432	+	0.988620i	$0.548066\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	125.822	0.454231	0.227116	−	0.973868i	$-0.427070\pi$
$277$	125.822	0.454231	0.227116	+	0.973868i	$0.427070\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	438.511i	1.56054i	0.625446	+	0.780268i	$0.284917\pi$
$281$	438.511i	1.56054i	−0.625446	+	0.780268i	$0.715083\pi$
$282$	0	0
$283$	−268.687	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$283$	−268.687	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	481.917i	1.67915i
$288$	0	0
$289$	169.871	0.587790
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 452.071i	− 1.54291i	−0.636286	−	0.771453i	$-0.719530\pi$
$293$	− 452.071i	− 1.54291i	0.636286	−	0.771453i	$-0.280470\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	218.700i	0.731437i
$300$	0	0
$301$	−244.792	−0.813261
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−194.896	−0.634840	−0.317420	−	0.948285i	$-0.602816\pi$
$307$	−194.896	−0.634840	−0.317420	+	0.948285i	$0.602816\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 298.686i	− 0.960404i	−0.877158	−	0.480202i	$-0.840563\pi$
$311$	− 298.686i	− 0.960404i	0.877158	−	0.480202i	$-0.159437\pi$
$312$	0	0
$313$	−203.405	−0.649857	−0.324928	−	0.945739i	$-0.605340\pi$
$313$	−203.405	−0.649857	−0.324928	+	0.945739i	$0.605340\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 449.356i	− 1.41753i	−0.705447	−	0.708763i	$-0.749254\pi$
$317$	− 449.356i	− 1.41753i	0.705447	−	0.708763i	$-0.250746\pi$
$318$	0	0
$319$	11.4356	0.0358483
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 62.4076i	− 0.193212i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	31.0696i	0.0944365i
$330$	0	0
$331$	533.632	1.61218	0.806091	−	0.591792i	$-0.201579\pi$
$331$	533.632	1.61218	0.806091	+	0.591792i	$0.201579\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−55.6627	−0.165171	−0.0825857	−	0.996584i	$-0.526318\pi$
$337$	−55.6627	−0.165171	−0.0825857	+	0.996584i	$0.526318\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	63.8565i	0.187263i
$342$	0	0
$343$	−296.638	−0.864834
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 104.392i	− 0.300841i	−0.988622	−	0.150420i	$-0.951937\pi$
$347$	− 104.392i	− 0.300841i	0.988622	−	0.150420i	$-0.0480627\pi$
$348$	0	0
$349$	−304.197	−0.871624	−0.435812	−	0.900038i	$-0.643539\pi$
$349$	−304.197	−0.871624	−0.435812	+	0.900038i	$0.643539\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 20.2676i	− 0.0574154i	−0.999588	−	0.0287077i	$-0.990861\pi$
$353$	− 20.2676i	− 0.0574154i	0.999588	−	0.0287077i	$-0.00913919\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 187.327i	− 0.521802i	−0.965366	−	0.260901i	$-0.915980\pi$
$359$	− 187.327i	− 0.521802i	0.965366	−	0.260901i	$-0.0840196\pi$
$360$	0	0
$361$	−328.307	−0.909437
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−454.945	−1.23963	−0.619816	−	0.784747i	$-0.712793\pi$
$367$	−454.945	−1.23963	−0.619816	+	0.784747i	$0.712793\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 481.917i	− 1.29897i
$372$	0	0
$373$	464.534	1.24540	0.622700	−	0.782461i	$-0.286036\pi$
$373$	464.534	1.24540	0.622700	+	0.782461i	$0.286036\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 100.557i	− 0.266728i
$378$	0	0
$379$	−210.233	−0.554705	−0.277352	−	0.960768i	$-0.589457\pi$
$379$	−210.233	−0.554705	−0.277352	+	0.960768i	$0.589457\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	259.348i	0.677148i	0.940940	+	0.338574i	$0.109944\pi$
$383$	259.348i	0.677148i	−0.940940	+	0.338574i	$0.890056\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 17.6214i	− 0.0452991i	−0.999743	−	0.0226496i	$-0.992790\pi$
$389$	− 17.6214i	− 0.0452991i	0.999743	−	0.0226496i	$-0.00721020\pi$
$390$	0	0
$391$	−191.951	−0.490923
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−62.4356	−0.157269	−0.0786343	−	0.996904i	$-0.525056\pi$
$397$	−62.4356	−0.157269	−0.0786343	+	0.996904i	$0.525056\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	32.5617i	0.0812012i	0.999175	+	0.0406006i	$0.0129271\pi$
$401$	32.5617i	0.0812012i	−0.999175	+	0.0406006i	$0.987073\pi$
$402$	0	0
$403$	561.509	1.39332
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.9403i	0.0367084i
$408$	0	0
$409$	−81.5644	−0.199424	−0.0997120	−	0.995016i	$-0.531792\pi$
$409$	−81.5644	−0.199424	−0.0997120	+	0.995016i	$0.531792\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	177.714i	0.430300i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	62.4423i	0.149027i	0.997220	+	0.0745135i	$0.0237404\pi$
$419$	62.4423i	0.149027i	−0.997220	+	0.0745135i	$0.976260\pi$
$420$	0	0
$421$	−582.663	−1.38400	−0.691999	−	0.721899i	$-0.743270\pi$
$421$	−582.663	−1.38400	−0.691999	+	0.721899i	$0.743270\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	170.786	0.399966
$428$	0	0
$429$	0	0
$430$	0	0
$431$	208.896i	0.484678i	0.970192	+	0.242339i	$0.0779146\pi$
$431$	208.896i	0.484678i	−0.970192	+	0.242339i	$0.922085\pi$
$432$	0	0
$433$	574.019	1.32568	0.662839	−	0.748762i	$-0.269351\pi$
$433$	574.019	1.32568	0.662839	+	0.748762i	$0.269351\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 100.557i	− 0.230107i
$438$	0	0
$439$	29.7670	0.0678063	0.0339032	−	0.999425i	$-0.489206\pi$
$439$	29.7670	0.0678063	0.0339032	+	0.999425i	$0.489206\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	504.901i	1.13973i	0.821738	+	0.569865i	$0.193005\pi$
$443$	504.901i	1.13973i	−0.821738	+	0.569865i	$0.806995\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 829.372i	− 1.84715i	−0.383414	−	0.923576i	$-0.625252\pi$
$449$	− 829.372i	− 1.84715i	0.383414	−	0.923576i	$-0.374748\pi$
$450$	0	0
$451$	−88.3068	−0.195802
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	174.712	0.382302	0.191151	−	0.981561i	$-0.438778\pi$
$457$	174.712	0.382302	0.191151	+	0.981561i	$0.438778\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	172.387i	0.373941i	0.982366	+	0.186970i	$0.0598668\pi$
$461$	172.387i	0.373941i	−0.982366	+	0.186970i	$0.940133\pi$
$462$	0	0
$463$	39.2763	0.0848301	0.0424150	−	0.999100i	$-0.486495\pi$
$463$	39.2763	0.0848301	0.0424150	+	0.999100i	$0.486495\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	168.361i	0.360516i	0.983619	+	0.180258i	$0.0576933\pi$
$467$	168.361i	0.360516i	−0.983619	+	0.180258i	$0.942307\pi$
$468$	0	0
$469$	141.098	0.300849
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 44.8557i	− 0.0948324i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	446.631i	0.932425i	0.884673	+	0.466212i	$0.154382\pi$
$479$	446.631i	0.932425i	−0.884673	+	0.466212i	$0.845618\pi$
$480$	0	0
$481$	131.375	0.273128
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−275.619	−0.565954	−0.282977	−	0.959127i	$-0.591322\pi$
$487$	−275.619	−0.565954	−0.282977	+	0.959127i	$0.591322\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 883.771i	− 1.79994i	−0.435951	−	0.899970i	$-0.643588\pi$
$491$	− 883.771i	− 1.79994i	0.435951	−	0.899970i	$-0.356412\pi$
$492$	0	0
$493$	88.2576	0.179022
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 251.304i	− 0.505643i
$498$	0	0
$499$	−958.884	−1.92161	−0.960806	−	0.277223i	$-0.910586\pi$
$499$	−958.884	−1.92161	−0.960806	+	0.277223i	$0.910586\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	791.448i	1.57346i	0.617300	+	0.786728i	$0.288226\pi$
$503$	791.448i	1.57346i	−0.617300	+	0.786728i	$0.711774\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 374.697i	− 0.736144i	−0.929797	−	0.368072i	$-0.880018\pi$
$509$	− 374.697i	− 0.736144i	0.929797	−	0.368072i	$-0.119982\pi$
$510$	0	0
$511$	−681.534	−1.33373
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.69321	−0.0110120
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 590.447i	− 1.13330i	−0.823960	−	0.566648i	$-0.808240\pi$
$521$	− 590.447i	− 1.13330i	0.823960	−	0.566648i	$-0.191760\pi$
$522$	0	0
$523$	532.797	1.01873	0.509367	−	0.860550i	$-0.329880\pi$
$523$	532.797	1.01873	0.509367	+	0.860550i	$0.329880\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	492.832i	0.935165i
$528$	0	0
$529$	219.712	0.415334
$530$	0	0
$531$	0	0
$532$	0	0
$533$	776.508i	1.45686i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	14.9403i	0.0277186i
$540$	0	0
$541$	−531.822	−0.983035	−0.491518	−	0.870868i	$-0.663558\pi$
$541$	−531.822	−0.983035	−0.491518	+	0.870868i	$0.663558\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−633.583	−1.15829	−0.579144	−	0.815226i	$-0.696613\pi$
$547$	−633.583	−1.15829	−0.579144	+	0.815226i	$0.696613\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	46.2352i	0.0839114i
$552$	0	0
$553$	600.000	1.08499
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 916.142i	− 1.64478i	−0.568925	−	0.822390i	$-0.692640\pi$
$557$	− 916.142i	− 1.64478i	0.568925	−	0.822390i	$-0.307360\pi$
$558$	0	0
$559$	−394.430	−0.705599
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 1076.49i	− 1.91207i	−0.293255	−	0.956034i	$-0.594738\pi$
$563$	− 1076.49i	− 1.91207i	0.293255	−	0.956034i	$-0.405262\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	282.374i	0.496264i	0.968726	+	0.248132i	$0.0798166\pi$
$569$	282.374i	0.496264i	−0.968726	+	0.248132i	$0.920183\pi$
$570$	0	0
$571$	−739.043	−1.29430	−0.647148	−	0.762364i	$-0.724039\pi$
$571$	−739.043	−1.29430	−0.647148	+	0.762364i	$0.724039\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−570.068	−0.987986	−0.493993	−	0.869466i	$-0.664463\pi$
$577$	−570.068	−0.987986	−0.493993	+	0.869466i	$0.664463\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 617.647i	− 1.06308i
$582$	0	0
$583$	88.3068	0.151470
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 771.145i	− 1.31371i	−0.754019	−	0.656853i	$-0.771887\pi$
$587$	− 771.145i	− 1.31371i	0.754019	−	0.656853i	$-0.228113\pi$
$588$	0	0
$589$	−258.178	−0.438333
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 552.403i	− 0.931539i	−0.884906	−	0.465770i	$-0.845778\pi$
$593$	− 552.403i	− 0.931539i	0.884906	−	0.465770i	$-0.154222\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 943.532i	− 1.57518i	−0.616200	−	0.787589i	$-0.711329\pi$
$599$	− 943.532i	− 1.57518i	0.616200	−	0.787589i	$-0.288671\pi$
$600$	0	0
$601$	−256.337	−0.426518	−0.213259	−	0.976996i	$-0.568408\pi$
$601$	−256.337	−0.426518	−0.213259	+	0.976996i	$0.568408\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−960.810	−1.58288	−0.791442	−	0.611245i	$-0.790669\pi$
$607$	−960.810	−1.58288	−0.791442	+	0.611245i	$0.790669\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	50.0621i	0.0819347i
$612$	0	0
$613$	−657.424	−1.07247	−0.536235	−	0.844069i	$-0.680154\pi$
$613$	−657.424	−1.07247	−0.536235	+	0.844069i	$0.680154\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	86.9609i	0.140942i	0.997514	+	0.0704708i	$0.0224501\pi$
$617$	86.9609i	0.140942i	−0.997514	+	0.0704708i	$0.977550\pi$
$618$	0	0
$619$	126.528	0.204407	0.102204	−	0.994764i	$-0.467411\pi$
$619$	126.528	0.204407	0.102204	+	0.994764i	$0.467411\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	733.222i	1.17692i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	115.306i	0.183317i
$630$	0	0
$631$	−218.957	−0.347000	−0.173500	−	0.984834i	$-0.555508\pi$
$631$	−218.957	−0.347000	−0.173500	+	0.984834i	$0.555508\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	131.375	0.206240
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 819.793i	− 1.27893i	−0.768821	−	0.639464i	$-0.779156\pi$
$641$	− 819.793i	− 1.27893i	0.768821	−	0.639464i	$-0.220844\pi$
$642$	0	0
$643$	17.3373	0.0269631	0.0134815	−	0.999909i	$-0.495709\pi$
$643$	17.3373	0.0269631	0.0134815	+	0.999909i	$0.495709\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 694.909i	− 1.07405i	−0.843567	−	0.537024i	$-0.819549\pi$
$647$	− 694.909i	− 1.07405i	0.843567	−	0.537024i	$-0.180451\pi$
$648$	0	0
$649$	−32.5644	−0.0501763
$650$	0	0
$651$	0	0
$652$	0	0
$653$	362.429i	0.555022i	0.960722	+	0.277511i	$0.0895095\pi$
$653$	362.429i	0.555022i	−0.960722	+	0.277511i	$0.910490\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1130.86i	1.71602i	0.513631	+	0.858011i	$0.328300\pi$
$659$	1130.86i	1.71602i	−0.513631	+	0.858011i	$0.671700\pi$
$660$	0	0
$661$	−120.000	−0.181543	−0.0907716	−	0.995872i	$-0.528933\pi$
$661$	−120.000	−0.181543	−0.0907716	+	0.995872i	$0.528933\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	142.208	0.213206
$668$	0	0
$669$	0	0
$670$	0	0
$671$	31.2949i	0.0466391i
$672$	0	0
$673$	−159.276	−0.236666	−0.118333	−	0.992974i	$-0.537755\pi$
$673$	−159.276	−0.236666	−0.118333	+	0.992974i	$0.537755\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 415.449i	− 0.613662i	−0.951764	−	0.306831i	$-0.900731\pi$
$677$	− 415.449i	− 0.613662i	0.951764	−	0.306831i	$-0.0992687\pi$
$678$	0	0
$679$	1125.80	1.65803
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1138.90i	− 1.66750i	−0.552142	−	0.833750i	$-0.686190\pi$
$683$	− 1138.90i	− 1.66750i	0.552142	−	0.833750i	$-0.313810\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 776.508i	− 1.12701i
$690$	0	0
$691$	−712.037	−1.03044	−0.515222	−	0.857056i	$-0.672291\pi$
$691$	−712.037	−1.03044	−0.515222	+	0.857056i	$0.672291\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−681.534	−0.977811
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36.5095i	0.0520821i	0.999661	+	0.0260410i	$0.00829006\pi$
$701$	36.5095i	0.0520821i	−0.999661	+	0.0260410i	$0.991710\pi$
$702$	0	0
$703$	−60.4051	−0.0859248
$704$	0	0
$705$	0	0
$706$	0	0
$707$	827.836i	1.17091i
$708$	0	0
$709$	−748.681	−1.05597	−0.527984	−	0.849254i	$-0.677052\pi$
$709$	−748.681	−1.05597	−0.527984	+	0.849254i	$0.677052\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	794.094i	1.11374i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 889.176i	− 1.23668i	−0.785909	−	0.618342i	$-0.787805\pi$
$719$	− 889.176i	− 1.23668i	0.785909	−	0.618342i	$-0.212195\pi$
$720$	0	0
$721$	−1118.09	−1.55074
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	99.8161	0.137299	0.0686493	−	0.997641i	$-0.478131\pi$
$727$	99.8161	0.137299	0.0686493	+	0.997641i	$0.478131\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 346.188i	− 0.473581i
$732$	0	0
$733$	352.417	0.480787	0.240394	−	0.970676i	$-0.422724\pi$
$733$	352.417	0.480787	0.240394	+	0.970676i	$0.422724\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.8549i	0.0350813i
$738$	0	0
$739$	−501.129	−0.678117	−0.339059	−	0.940765i	$-0.610109\pi$
$739$	−501.129	−0.678117	−0.339059	+	0.940765i	$0.610109\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	959.618i	1.29155i	0.763530	+	0.645773i	$0.223465\pi$
$743$	959.618i	1.29155i	−0.763530	+	0.645773i	$0.776535\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1561.06i	2.08419i
$750$	0	0
$751$	−772.822	−1.02906	−0.514529	−	0.857473i	$-0.672033\pi$
$751$	−772.822	−1.02906	−0.514529	+	0.857473i	$0.672033\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−159.417	−0.210590	−0.105295	−	0.994441i	$-0.533579\pi$
$757$	−159.417	−0.210590	−0.105295	+	0.994441i	$0.533579\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 399.173i	− 0.524537i	−0.964995	−	0.262268i	$-0.915529\pi$
$761$	− 399.173i	− 0.524537i	0.964995	−	0.262268i	$-0.0844706\pi$
$762$	0	0
$763$	1147.96	1.50454
$764$	0	0
$765$	0	0
$766$	0	0
$767$	286.348i	0.373336i
$768$	0	0
$769$	102.632	0.133462	0.0667310	−	0.997771i	$-0.478743\pi$
$769$	102.632	0.133462	0.0667310	+	0.997771i	$0.478743\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 142.506i	− 0.184354i	−0.995743	−	0.0921772i	$-0.970617\pi$
$773$	− 142.506i	− 0.184354i	0.995743	−	0.0921772i	$-0.0293826\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 357.033i	− 0.458322i
$780$	0	0
$781$	46.0492	0.0589618
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	163.203	0.207373	0.103687	−	0.994610i	$-0.466936\pi$
$787$	163.203	0.207373	0.103687	+	0.994610i	$0.466936\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 345.919i	− 0.437319i
$792$	0	0
$793$	275.185	0.347018
$794$	0	0
$795$	0	0
$796$	0	0
$797$	689.547i	0.865178i	0.901591	+	0.432589i	$0.142400\pi$
$797$	689.547i	0.865178i	−0.901591	+	0.432589i	$0.857600\pi$
$798$	0	0
$799$	−43.9391	−0.0549926
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 124.885i	− 0.155523i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 802.172i	− 0.991560i	−0.868448	−	0.495780i	$-0.834882\pi$
$809$	− 802.172i	− 0.991560i	0.868448	−	0.495780i	$-0.165118\pi$
$810$	0	0
$811$	1329.29	1.63907	0.819537	−	0.573026i	$-0.194231\pi$
$811$	1329.29	1.63907	0.819537	+	0.573026i	$0.194231\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	181.356	0.221978
$818$	0	0
$819$	0	0
$820$	0	0
$821$	206.363i	0.251355i	0.992071	+	0.125678i	$0.0401105\pi$
$821$	206.363i	0.251355i	−0.992071	+	0.125678i	$0.959890\pi$
$822$	0	0
$823$	−1048.33	−1.27379	−0.636896	−	0.770949i	$-0.719782\pi$
$823$	−1048.33	−1.27379	−0.636896	+	0.770949i	$0.719782\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1301.71i	− 1.57402i	−0.616943	−	0.787008i	$-0.711629\pi$
$827$	− 1301.71i	− 1.57402i	0.616943	−	0.787008i	$-0.288371\pi$
$828$	0	0
$829$	−91.2271	−0.110045	−0.0550224	−	0.998485i	$-0.517523\pi$
$829$	−91.2271	−0.110045	−0.0550224	+	0.998485i	$0.517523\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	115.306i	0.138423i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 59.7182i	− 0.0711778i	−0.999367	−	0.0355889i	$-0.988669\pi$
$839$	− 59.7182i	− 0.0711778i	0.999367	−	0.0355889i	$-0.0113307\pi$
$840$	0	0
$841$	775.614	0.922252
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	918.418	1.08432
$848$	0	0
$849$	0	0
$850$	0	0
$851$	185.792i	0.218322i
$852$	0	0
$853$	436.681	0.511936	0.255968	−	0.966685i	$-0.417606\pi$
$853$	436.681	0.511936	0.255968	+	0.966685i	$0.417606\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	594.543i	0.693749i	0.937912	+	0.346874i	$0.112757\pi$
$857$	594.543i	0.693749i	−0.937912	+	0.346874i	$0.887243\pi$
$858$	0	0
$859$	−376.037	−0.437762	−0.218881	−	0.975752i	$-0.570241\pi$
$859$	−376.037	−0.437762	−0.218881	+	0.975752i	$0.570241\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	104.582i	0.121185i	0.998163	+	0.0605923i	$0.0192989\pi$
$863$	104.582i	0.121185i	−0.998163	+	0.0605923i	$0.980701\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	109.944i	0.126518i
$870$	0	0
$871$	227.350	0.261022
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	117.159	0.133591	0.0667955	−	0.997767i	$-0.478722\pi$
$877$	117.159	0.133591	0.0667955	+	0.997767i	$0.478722\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	840.096i	0.953571i	0.879020	+	0.476785i	$0.158198\pi$
$881$	840.096i	0.953571i	−0.879020	+	0.476785i	$0.841802\pi$
$882$	0	0
$883$	1557.04	1.76336	0.881678	−	0.471852i	$-0.156414\pi$
$883$	1557.04	1.76336	0.881678	+	0.471852i	$0.156414\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 661.002i	− 0.745211i	−0.927990	−	0.372606i	$-0.878464\pi$
$887$	− 661.002i	− 0.745211i	0.927990	−	0.372606i	$-0.121536\pi$
$888$	0	0
$889$	81.5339	0.0917142
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 23.0182i	− 0.0257763i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 365.119i	− 0.406139i
$900$	0	0
$901$	681.534	0.756419
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	889.030	0.980188	0.490094	−	0.871670i	$-0.336962\pi$
$907$	889.030	0.980188	0.490094	+	0.871670i	$0.336962\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1391.62i	1.52758i	0.645468	+	0.763788i	$0.276662\pi$
$911$	1391.62i	1.52758i	−0.645468	+	0.763788i	$0.723338\pi$
$912$	0	0
$913$	113.178	0.123963
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1215.14i	1.32512i
$918$	0	0
$919$	450.233	0.489916	0.244958	−	0.969534i	$-0.421226\pi$
$919$	450.233	0.489916	0.244958	+	0.969534i	$0.421226\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 404.924i	− 0.438704i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1254.55i	1.35044i	0.737618	+	0.675218i	$0.235950\pi$
$929$	1254.55i	1.35044i	−0.737618	+	0.675218i	$0.764050\pi$
$930$	0	0
$931$	−60.4051	−0.0648820
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1553.09	−1.65751	−0.828755	−	0.559612i	$-0.810950\pi$
$937$	−1553.09	−1.65751	−0.828755	+	0.559612i	$0.810950\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	21.7166i	0.0230782i	0.999933	+	0.0115391i	$0.00367309\pi$
$941$	21.7166i	0.0230782i	−0.999933	+	0.0115391i	$0.996327\pi$
$942$	0	0
$943$	−1098.15	−1.16453
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1760.83i	− 1.85938i	−0.368340	−	0.929691i	$-0.620074\pi$
$947$	− 1760.83i	− 1.85938i	0.368340	−	0.929691i	$-0.379926\pi$
$948$	0	0
$949$	−1098.15	−1.15716
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 495.322i	− 0.519751i	−0.965642	−	0.259875i	$-0.916319\pi$
$953$	− 495.322i	− 0.519751i	0.965642	−	0.259875i	$-0.0836815\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1414.68i	− 1.47516i
$960$	0	0
$961$	1077.83	1.12157
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	486.773	0.503385	0.251692	−	0.967807i	$-0.419013\pi$
$967$	486.773	0.503385	0.251692	+	0.967807i	$0.419013\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 568.757i	− 0.585744i	−0.956152	−	0.292872i	$-0.905389\pi$
$971$	− 568.757i	− 0.585744i	0.956152	−	0.292872i	$-0.0946110\pi$
$972$	0	0
$973$	−518.466	−0.532853
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1486.55i	1.52154i	0.649020	+	0.760771i	$0.275179\pi$
$977$	1486.55i	1.52154i	−0.649020	+	0.760771i	$0.724821\pi$
$978$	0	0
$979$	−134.356	−0.137238
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1612.78i	1.64067i	0.571885	+	0.820334i	$0.306212\pi$
$983$	1612.78i	1.64067i	−0.571885	+	0.820334i	$0.693788\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 557.808i	− 0.564012i
$990$	0	0
$991$	−381.043	−0.384504	−0.192252	−	0.981346i	$-0.561579\pi$
$991$	−381.043	−0.384504	−0.192252	+	0.981346i	$0.561579\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1469.52	−1.47394	−0.736972	−	0.675923i	$-0.763745\pi$
$997$	−1469.52	−1.47394	−0.736972	+	0.675923i	$0.763745\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.3.l.c.1601.4	yes	4
3.2	odd	2	inner	1800.3.l.c.1601.3	✓	4
4.3	odd	2		3600.3.l.t.1601.1		4
5.2	odd	4		1800.3.c.c.449.6		8
5.3	odd	4		1800.3.c.c.449.4		8
5.4	even	2		1800.3.l.e.1601.2	yes	4
12.11	even	2		3600.3.l.t.1601.2		4
15.2	even	4		1800.3.c.c.449.5		8
15.8	even	4		1800.3.c.c.449.3		8
15.14	odd	2		1800.3.l.e.1601.1	yes	4
20.3	even	4		3600.3.c.j.449.5		8
20.7	even	4		3600.3.c.j.449.3		8
20.19	odd	2		3600.3.l.p.1601.3		4
60.23	odd	4		3600.3.c.j.449.6		8
60.47	odd	4		3600.3.c.j.449.4		8
60.59	even	2		3600.3.l.p.1601.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1800.3.c.c.449.3		8	15.8	even	4
1800.3.c.c.449.4		8	5.3	odd	4
1800.3.c.c.449.5		8	15.2	even	4
1800.3.c.c.449.6		8	5.2	odd	4
1800.3.l.c.1601.3	✓	4	3.2	odd	2	inner
1800.3.l.c.1601.4	yes	4	1.1	even	1	trivial
1800.3.l.e.1601.1	yes	4	15.14	odd	2
1800.3.l.e.1601.2	yes	4	5.4	even	2
3600.3.c.j.449.3		8	20.7	even	4
3600.3.c.j.449.4		8	60.47	odd	4
3600.3.c.j.449.5		8	20.3	even	4
3600.3.c.j.449.6		8	60.23	odd	4
3600.3.l.p.1601.3		4	20.19	odd	2
3600.3.l.p.1601.4		4	60.59	even	2
3600.3.l.t.1601.1		4	4.3	odd	2
3600.3.l.t.1601.2		4	12.11	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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1800.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.71780 q^{7} +1.41421i q^{11} +12.4356 q^{13} +10.9146i q^{17} -5.71780 q^{19} +17.5866i q^{23} -8.08619i q^{29} +45.1534 q^{31} +10.5644 q^{37} +62.4423i q^{41} -31.7178 q^{43} +4.02571i q^{47} +10.5644 q^{49} -62.4423i q^{53} +23.0265i q^{59} +22.1288 q^{61} +18.2822 q^{67} -32.5617i q^{71} -88.3068 q^{73} +10.9146i q^{77} +77.7424 q^{79} -80.0289i q^{83} +95.0040i q^{89} +95.9754 q^{91} +145.871 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 20 q^{13} + 12 q^{19} + 76 q^{31} + 112 q^{37} - 92 q^{43} + 112 q^{49} + 228 q^{61} + 108 q^{67} - 144 q^{73} + 32 q^{79} + 628 q^{91} + 444 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 20 * q^13 + 12 * q^19 + 76 * q^31 + 112 * q^37 - 92 * q^43 + 112 * q^49 + 228 * q^61 + 108 * q^67 - 144 * q^73 + 32 * q^79 + 628 * q^91 + 444 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more