LMFDB - Embedded newform 800.6.a.s.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(1,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("800.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.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:	$128.307055850$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 226x^{3} + 455x^{2} + 9816x + 4656 $ x^5 - 2x^4 - 226x^3 + 455x^2 + 9816x + 4656
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$10.4009$ of defining polynomial
Character	$\chi$	$=$	800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+19.8018 q^{3} +160.939 q^{7} +149.111 q^{9} +O(q^{10})$	$q+19.8018 q^{3} +160.939 q^{7} +149.111 q^{9} -404.231 q^{11} -249.498 q^{13} -1251.45 q^{17} -905.566 q^{19} +3186.88 q^{21} +2150.12 q^{23} -1859.18 q^{27} -2792.21 q^{29} +3313.11 q^{31} -8004.49 q^{33} -13197.0 q^{37} -4940.51 q^{39} -5075.65 q^{41} -11840.0 q^{43} -1615.00 q^{47} +9094.33 q^{49} -24781.0 q^{51} -33580.6 q^{53} -17931.8 q^{57} +22150.2 q^{59} -43258.9 q^{61} +23997.7 q^{63} +44075.6 q^{67} +42576.2 q^{69} +38109.7 q^{71} +37225.1 q^{73} -65056.4 q^{77} -10876.2 q^{79} -73048.9 q^{81} -75737.5 q^{83} -55290.6 q^{87} +121533. q^{89} -40154.0 q^{91} +65605.4 q^{93} -157807. q^{97} -60275.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 110 q^{7} + 606 q^{9}+O(q^{10}) $ 5 * q - q^3 - 110 * q^7 + 606 * q^9	$ 5 q - q^{3} - 110 q^{7} + 606 q^{9} - 5 q^{11} - 280 q^{13} + 865 q^{17} - 4485 q^{19} - 418 q^{21} + 3946 q^{23} - 4987 q^{27} - 3252 q^{29} - 8250 q^{31} + 13465 q^{33} - 3210 q^{37} - 8800 q^{39} - 11415 q^{41} + 428 q^{43} + 9672 q^{47} + 26225 q^{49} - 2965 q^{51} - 30370 q^{53} + 29345 q^{57} + 39280 q^{59} + 34854 q^{61} - 87812 q^{63} + 68877 q^{67} - 40658 q^{69} - 51200 q^{71} + 41615 q^{73} - 57050 q^{77} + 550 q^{79} - 15219 q^{81} - 66223 q^{83} + 168860 q^{87} - 103829 q^{89} - 185280 q^{91} - 59870 q^{93} - 21150 q^{97} - 244390 q^{99}+O(q^{100}) $ 5 * q - q^3 - 110 * q^7 + 606 * q^9 - 5 * q^11 - 280 * q^13 + 865 * q^17 - 4485 * q^19 - 418 * q^21 + 3946 * q^23 - 4987 * q^27 - 3252 * q^29 - 8250 * q^31 + 13465 * q^33 - 3210 * q^37 - 8800 * q^39 - 11415 * q^41 + 428 * q^43 + 9672 * q^47 + 26225 * q^49 - 2965 * q^51 - 30370 * q^53 + 29345 * q^57 + 39280 * q^59 + 34854 * q^61 - 87812 * q^63 + 68877 * q^67 - 40658 * q^69 - 51200 * q^71 + 41615 * q^73 - 57050 * q^77 + 550 * q^79 - 15219 * q^81 - 66223 * q^83 + 168860 * q^87 - 103829 * q^89 - 185280 * q^91 - 59870 * q^93 - 21150 * q^97 - 244390 * q^99

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
$3$	19.8018	1.27028	0.635142	−	0.772395i	$-0.280941\pi$
$3$	19.8018	1.27028	0.635142	+	0.772395i	$0.280941\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	160.939	1.24141	0.620706	−	0.784043i	$-0.286846\pi$
$7$	160.939	1.24141	0.620706	+	0.784043i	$0.286846\pi$
$8$	0	0
$9$	149.111	0.613623
$10$	0	0
$11$	−404.231	−1.00727	−0.503637	−	0.863915i	$-0.668005\pi$
$11$	−404.231	−1.00727	−0.503637	+	0.863915i	$0.668005\pi$
$12$	0	0
$13$	−249.498	−0.409458	−0.204729	−	0.978819i	$-0.565631\pi$
$13$	−249.498	−0.409458	−0.204729	+	0.978819i	$0.565631\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1251.45	−1.05025	−0.525125	−	0.851025i	$-0.675981\pi$
$17$	−1251.45	−1.05025	−0.525125	+	0.851025i	$0.675981\pi$
$18$	0	0
$19$	−905.566	−0.575488	−0.287744	−	0.957707i	$-0.592905\pi$
$19$	−905.566	−0.575488	−0.287744	+	0.957707i	$0.592905\pi$
$20$	0	0
$21$	3186.88	1.57695
$22$	0	0
$23$	2150.12	0.847507	0.423754	−	0.905778i	$-0.360712\pi$
$23$	2150.12	0.847507	0.423754	+	0.905778i	$0.360712\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1859.18	−0.490808
$28$	0	0
$29$	−2792.21	−0.616527	−0.308264	−	0.951301i	$-0.599748\pi$
$29$	−2792.21	−0.616527	−0.308264	+	0.951301i	$0.599748\pi$
$30$	0	0
$31$	3313.11	0.619201	0.309600	−	0.950867i	$-0.399805\pi$
$31$	3313.11	0.619201	0.309600	+	0.950867i	$0.399805\pi$
$32$	0	0
$33$	−8004.49	−1.27952
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−13197.0	−1.58478	−0.792392	−	0.610013i	$-0.791164\pi$
$37$	−13197.0	−1.58478	−0.792392	+	0.610013i	$0.791164\pi$
$38$	0	0
$39$	−4940.51	−0.520128
$40$	0	0
$41$	−5075.65	−0.471555	−0.235777	−	0.971807i	$-0.575764\pi$
$41$	−5075.65	−0.471555	−0.235777	+	0.971807i	$0.575764\pi$
$42$	0	0
$43$	−11840.0	−0.976521	−0.488261	−	0.872698i	$-0.662368\pi$
$43$	−11840.0	−0.976521	−0.488261	+	0.872698i	$0.662368\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1615.00	−0.106642	−0.0533208	−	0.998577i	$-0.516981\pi$
$47$	−1615.00	−0.106642	−0.0533208	+	0.998577i	$0.516981\pi$
$48$	0	0
$49$	9094.33	0.541104
$50$	0	0
$51$	−24781.0	−1.33412
$52$	0	0
$53$	−33580.6	−1.64210	−0.821049	−	0.570857i	$-0.806611\pi$
$53$	−33580.6	−1.64210	−0.821049	+	0.570857i	$0.806611\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−17931.8	−0.731033
$58$	0	0
$59$	22150.2	0.828413	0.414206	−	0.910183i	$-0.364059\pi$
$59$	22150.2	0.828413	0.414206	+	0.910183i	$0.364059\pi$
$60$	0	0
$61$	−43258.9	−1.48851	−0.744254	−	0.667896i	$-0.767195\pi$
$61$	−43258.9	−1.48851	−0.744254	+	0.667896i	$0.767195\pi$
$62$	0	0
$63$	23997.7	0.761759
$64$	0	0
$65$	0	0
$66$	0	0
$67$	44075.6	1.19953	0.599766	−	0.800176i	$-0.295260\pi$
$67$	44075.6	1.19953	0.599766	+	0.800176i	$0.295260\pi$
$68$	0	0
$69$	42576.2	1.07658
$70$	0	0
$71$	38109.7	0.897200	0.448600	−	0.893733i	$-0.351923\pi$
$71$	38109.7	0.897200	0.448600	+	0.893733i	$0.351923\pi$
$72$	0	0
$73$	37225.1	0.817577	0.408788	−	0.912629i	$-0.365951\pi$
$73$	37225.1	0.817577	0.408788	+	0.912629i	$0.365951\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−65056.4	−1.25044
$78$	0	0
$79$	−10876.2	−0.196069	−0.0980345	−	0.995183i	$-0.531256\pi$
$79$	−10876.2	−0.196069	−0.0980345	+	0.995183i	$0.531256\pi$
$80$	0	0
$81$	−73048.9	−1.23709
$82$	0	0
$83$	−75737.5	−1.20675	−0.603373	−	0.797459i	$-0.706177\pi$
$83$	−75737.5	−1.20675	−0.603373	+	0.797459i	$0.706177\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−55290.6	−0.783165
$88$	0	0
$89$	121533.	1.62637	0.813183	−	0.582008i	$-0.197733\pi$
$89$	121533.	1.62637	0.813183	+	0.582008i	$0.197733\pi$
$90$	0	0
$91$	−40154.0	−0.508306
$92$	0	0
$93$	65605.4	0.786561
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−157807.	−1.70293	−0.851465	−	0.524411i	$-0.824286\pi$
$97$	−157807.	−1.70293	−0.851465	+	0.524411i	$0.824286\pi$
$98$	0	0
$99$	−60275.0	−0.618087
$100$	0	0
$101$	−32558.5	−0.317586	−0.158793	−	0.987312i	$-0.550760\pi$
$101$	−32558.5	−0.317586	−0.158793	+	0.987312i	$0.550760\pi$
$102$	0	0
$103$	1240.18	0.0115184	0.00575920	−	0.999983i	$-0.498167\pi$
$103$	1240.18	0.0115184	0.00575920	+	0.999983i	$0.498167\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	170351.	1.43842	0.719209	−	0.694794i	$-0.244504\pi$
$107$	170351.	1.43842	0.719209	+	0.694794i	$0.244504\pi$
$108$	0	0
$109$	127565.	1.02841	0.514204	−	0.857668i	$-0.328088\pi$
$109$	127565.	1.02841	0.514204	+	0.857668i	$0.328088\pi$
$110$	0	0
$111$	−261323.	−2.01313
$112$	0	0
$113$	−110843.	−0.816602	−0.408301	−	0.912847i	$-0.633879\pi$
$113$	−110843.	−0.816602	−0.408301	+	0.912847i	$0.633879\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−37202.8	−0.251253
$118$	0	0
$119$	−201408.	−1.30379
$120$	0	0
$121$	2351.45	0.0146007
$122$	0	0
$123$	−100507.	−0.599009
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−115536.	−0.635634	−0.317817	−	0.948152i	$-0.602950\pi$
$127$	−115536.	−0.635634	−0.317817	+	0.948152i	$0.602950\pi$
$128$	0	0
$129$	−234454.	−1.24046
$130$	0	0
$131$	29750.5	0.151466	0.0757332	−	0.997128i	$-0.475870\pi$
$131$	29750.5	0.151466	0.0757332	+	0.997128i	$0.475870\pi$
$132$	0	0
$133$	−145741.	−0.714417
$134$	0	0
$135$	0	0
$136$	0	0
$137$	238138.	1.08400	0.541998	−	0.840380i	$-0.317668\pi$
$137$	238138.	1.08400	0.541998	+	0.840380i	$0.317668\pi$
$138$	0	0
$139$	406539.	1.78470	0.892351	−	0.451342i	$-0.149055\pi$
$139$	406539.	1.78470	0.892351	+	0.451342i	$0.149055\pi$
$140$	0	0
$141$	−31979.8	−0.135465
$142$	0	0
$143$	100855.	0.412436
$144$	0	0
$145$	0	0
$146$	0	0
$147$	180084.	0.687356
$148$	0	0
$149$	−178599.	−0.659043	−0.329521	−	0.944148i	$-0.606887\pi$
$149$	−178599.	−0.659043	−0.329521	+	0.944148i	$0.606887\pi$
$150$	0	0
$151$	−432317.	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$151$	−432317.	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$152$	0	0
$153$	−186605.	−0.644458
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−198160.	−0.641603	−0.320802	−	0.947146i	$-0.603952\pi$
$157$	−198160.	−0.641603	−0.320802	+	0.947146i	$0.603952\pi$
$158$	0	0
$159$	−664957.	−2.08593
$160$	0	0
$161$	346038.	1.05211
$162$	0	0
$163$	−450882.	−1.32921	−0.664605	−	0.747195i	$-0.731400\pi$
$163$	−450882.	−1.32921	−0.664605	+	0.747195i	$0.731400\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	519661.	1.44188	0.720940	−	0.692998i	$-0.243710\pi$
$167$	519661.	1.44188	0.720940	+	0.692998i	$0.243710\pi$
$168$	0	0
$169$	−309044.	−0.832344
$170$	0	0
$171$	−135029.	−0.353133
$172$	0	0
$173$	−27351.8	−0.0694818	−0.0347409	−	0.999396i	$-0.511061\pi$
$173$	−27351.8	−0.0694818	−0.0347409	+	0.999396i	$0.511061\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	438613.	1.05232
$178$	0	0
$179$	100983.	0.235569	0.117784	−	0.993039i	$-0.462421\pi$
$179$	100983.	0.235569	0.117784	+	0.993039i	$0.462421\pi$
$180$	0	0
$181$	124851.	0.283267	0.141633	−	0.989919i	$-0.454765\pi$
$181$	124851.	0.283267	0.141633	+	0.989919i	$0.454765\pi$
$182$	0	0
$183$	−856604.	−1.89083
$184$	0	0
$185$	0	0
$186$	0	0
$187$	505876.	1.05789
$188$	0	0
$189$	−299214.	−0.609295
$190$	0	0
$191$	−829467.	−1.64519	−0.822594	−	0.568629i	$-0.807474\pi$
$191$	−829467.	−1.64519	−0.822594	+	0.568629i	$0.807474\pi$
$192$	0	0
$193$	712128.	1.37615	0.688073	−	0.725641i	$-0.258457\pi$
$193$	712128.	1.37615	0.688073	+	0.725641i	$0.258457\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	970619.	1.78190	0.890950	−	0.454102i	$-0.150040\pi$
$197$	970619.	1.78190	0.890950	+	0.454102i	$0.150040\pi$
$198$	0	0
$199$	−516230.	−0.924082	−0.462041	−	0.886858i	$-0.652883\pi$
$199$	−516230.	−0.924082	−0.462041	+	0.886858i	$0.652883\pi$
$200$	0	0
$201$	872776.	1.52375
$202$	0	0
$203$	−449375.	−0.765365
$204$	0	0
$205$	0	0
$206$	0	0
$207$	320606.	0.520050
$208$	0	0
$209$	366057.	0.579674
$210$	0	0
$211$	−664118.	−1.02693	−0.513463	−	0.858112i	$-0.671638\pi$
$211$	−664118.	−1.02693	−0.513463	+	0.858112i	$0.671638\pi$
$212$	0	0
$213$	754640.	1.13970
$214$	0	0
$215$	0	0
$216$	0	0
$217$	533208.	0.768683
$218$	0	0
$219$	737123.	1.03856
$220$	0	0
$221$	312236.	0.430033
$222$	0	0
$223$	990032.	1.33317	0.666587	−	0.745427i	$-0.267754\pi$
$223$	990032.	1.33317	0.666587	+	0.745427i	$0.267754\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15566.7	−0.0200509	−0.0100254	−	0.999950i	$-0.503191\pi$
$227$	−15566.7	−0.0200509	−0.0100254	+	0.999950i	$0.503191\pi$
$228$	0	0
$229$	359522.	0.453040	0.226520	−	0.974006i	$-0.427265\pi$
$229$	359522.	0.453040	0.226520	+	0.974006i	$0.427265\pi$
$230$	0	0
$231$	−1.28823e6	−1.58842
$232$	0	0
$233$	1.21546e6	1.46673	0.733367	−	0.679833i	$-0.237948\pi$
$233$	1.21546e6	1.46673	0.733367	+	0.679833i	$0.237948\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−215368.	−0.249064
$238$	0	0
$239$	−1.00004e6	−1.13246	−0.566231	−	0.824247i	$-0.691599\pi$
$239$	−1.00004e6	−1.13246	−0.566231	+	0.824247i	$0.691599\pi$
$240$	0	0
$241$	−1.25841e6	−1.39566	−0.697831	−	0.716263i	$-0.745851\pi$
$241$	−1.25841e6	−1.39566	−0.697831	+	0.716263i	$0.745851\pi$
$242$	0	0
$243$	−994718.	−1.08065
$244$	0	0
$245$	0	0
$246$	0	0
$247$	225937.	0.235638
$248$	0	0
$249$	−1.49974e6	−1.53291
$250$	0	0
$251$	−990751.	−0.992614	−0.496307	−	0.868147i	$-0.665311\pi$
$251$	−990751.	−0.992614	−0.496307	+	0.868147i	$0.665311\pi$
$252$	0	0
$253$	−869145.	−0.853672
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.71639e6	1.62100	0.810498	−	0.585741i	$-0.199196\pi$
$257$	1.71639e6	1.62100	0.810498	+	0.585741i	$0.199196\pi$
$258$	0	0
$259$	−2.12391e6	−1.96737
$260$	0	0
$261$	−416347.	−0.378316
$262$	0	0
$263$	−1.59528e6	−1.42216	−0.711080	−	0.703112i	$-0.751793\pi$
$263$	−1.59528e6	−1.42216	−0.711080	+	0.703112i	$0.751793\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.40657e6	2.06595
$268$	0	0
$269$	895598.	0.754627	0.377313	−	0.926086i	$-0.376848\pi$
$269$	895598.	0.754627	0.377313	+	0.926086i	$0.376848\pi$
$270$	0	0
$271$	−648702.	−0.536565	−0.268282	−	0.963340i	$-0.586456\pi$
$271$	−648702.	−0.536565	−0.268282	+	0.963340i	$0.586456\pi$
$272$	0	0
$273$	−795120.	−0.645693
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.35736e6	−1.84597	−0.922987	−	0.384830i	$-0.874260\pi$
$277$	−2.35736e6	−1.84597	−0.922987	+	0.384830i	$0.874260\pi$
$278$	0	0
$279$	494019.	0.379956
$280$	0	0
$281$	586738.	0.443280	0.221640	−	0.975129i	$-0.428859\pi$
$281$	586738.	0.443280	0.221640	+	0.975129i	$0.428859\pi$
$282$	0	0
$283$	−1.42618e6	−1.05855	−0.529273	−	0.848452i	$-0.677535\pi$
$283$	−1.42618e6	−1.05855	−0.529273	+	0.848452i	$0.677535\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−816870.	−0.585394
$288$	0	0
$289$	146279.	0.103024
$290$	0	0
$291$	−3.12486e6	−2.16321
$292$	0	0
$293$	−931708.	−0.634032	−0.317016	−	0.948420i	$-0.602681\pi$
$293$	−931708.	−0.634032	−0.317016	+	0.948420i	$0.602681\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	751537.	0.494378
$298$	0	0
$299$	−536451.	−0.347018
$300$	0	0
$301$	−1.90552e6	−1.21227
$302$	0	0
$303$	−644716.	−0.403424
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.51715e6	0.918716	0.459358	−	0.888251i	$-0.348079\pi$
$307$	1.51715e6	0.918716	0.459358	+	0.888251i	$0.348079\pi$
$308$	0	0
$309$	24557.8	0.0146317
$310$	0	0
$311$	366257.	0.214726	0.107363	−	0.994220i	$-0.465759\pi$
$311$	366257.	0.214726	0.107363	+	0.994220i	$0.465759\pi$
$312$	0	0
$313$	−1.19487e6	−0.689381	−0.344691	−	0.938716i	$-0.612016\pi$
$313$	−1.19487e6	−0.689381	−0.344691	+	0.938716i	$0.612016\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.14809e6	−1.20062	−0.600309	−	0.799768i	$-0.704956\pi$
$317$	−2.14809e6	−1.20062	−0.600309	+	0.799768i	$0.704956\pi$
$318$	0	0
$319$	1.12870e6	0.621012
$320$	0	0
$321$	3.37325e6	1.82720
$322$	0	0
$323$	1.13327e6	0.604406
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.52601e6	1.30637
$328$	0	0
$329$	−259916.	−0.132386
$330$	0	0
$331$	442778.	0.222135	0.111067	−	0.993813i	$-0.464573\pi$
$331$	442778.	0.222135	0.111067	+	0.993813i	$0.464573\pi$
$332$	0	0
$333$	−1.96781e6	−0.972460
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.01258e6	−1.92464	−0.962319	−	0.271923i	$-0.912340\pi$
$337$	−4.01258e6	−1.92464	−0.962319	+	0.271923i	$0.912340\pi$
$338$	0	0
$339$	−2.19488e6	−1.03732
$340$	0	0
$341$	−1.33926e6	−0.623705
$342$	0	0
$343$	−1.24127e6	−0.569679
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.81191e6	−0.807816	−0.403908	−	0.914800i	$-0.632348\pi$
$347$	−1.81191e6	−0.807816	−0.403908	+	0.914800i	$0.632348\pi$
$348$	0	0
$349$	3.69069e6	1.62197	0.810987	−	0.585065i	$-0.198931\pi$
$349$	3.69069e6	1.62197	0.810987	+	0.585065i	$0.198931\pi$
$350$	0	0
$351$	463862.	0.200965
$352$	0	0
$353$	119137.	0.0508876	0.0254438	−	0.999676i	$-0.491900\pi$
$353$	119137.	0.0508876	0.0254438	+	0.999676i	$0.491900\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.98823e6	−1.65619
$358$	0	0
$359$	3.65995e6	1.49879	0.749393	−	0.662126i	$-0.230346\pi$
$359$	3.65995e6	1.49879	0.749393	+	0.662126i	$0.230346\pi$
$360$	0	0
$361$	−1.65605e6	−0.668814
$362$	0	0
$363$	46563.0	0.0185470
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.04946e6	0.406726	0.203363	−	0.979103i	$-0.434813\pi$
$367$	1.04946e6	0.406726	0.203363	+	0.979103i	$0.434813\pi$
$368$	0	0
$369$	−756833.	−0.289357
$370$	0	0
$371$	−5.40443e6	−2.03852
$372$	0	0
$373$	−150748.	−0.0561020	−0.0280510	−	0.999606i	$-0.508930\pi$
$373$	−150748.	−0.0561020	−0.0280510	+	0.999606i	$0.508930\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	696651.	0.252442
$378$	0	0
$379$	−416359.	−0.148892	−0.0744458	−	0.997225i	$-0.523719\pi$
$379$	−416359.	−0.148892	−0.0744458	+	0.997225i	$0.523719\pi$
$380$	0	0
$381$	−2.28781e6	−0.807436
$382$	0	0
$383$	−2.08322e6	−0.725669	−0.362834	−	0.931854i	$-0.618191\pi$
$383$	−2.08322e6	−0.725669	−0.362834	+	0.931854i	$0.618191\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.76547e6	−0.599216
$388$	0	0
$389$	1.37343e6	0.460186	0.230093	−	0.973169i	$-0.426097\pi$
$389$	1.37343e6	0.460186	0.230093	+	0.973169i	$0.426097\pi$
$390$	0	0
$391$	−2.69078e6	−0.890094
$392$	0	0
$393$	589113.	0.192405
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.99424e6	−0.635040	−0.317520	−	0.948252i	$-0.602850\pi$
$397$	−1.99424e6	−0.635040	−0.317520	+	0.948252i	$0.602850\pi$
$398$	0	0
$399$	−2.88593e6	−0.907513
$400$	0	0
$401$	−23001.8	−0.00714333	−0.00357166	−	0.999994i	$-0.501137\pi$
$401$	−23001.8	−0.00714333	−0.00357166	+	0.999994i	$0.501137\pi$
$402$	0	0
$403$	−826615.	−0.253537
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.33462e6	1.59631
$408$	0	0
$409$	2.27406e6	0.672193	0.336097	−	0.941828i	$-0.390893\pi$
$409$	2.27406e6	0.672193	0.336097	+	0.941828i	$0.390893\pi$
$410$	0	0
$411$	4.71556e6	1.37698
$412$	0	0
$413$	3.56482e6	1.02840
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.05020e6	2.26708
$418$	0	0
$419$	−3.94280e6	−1.09716	−0.548579	−	0.836099i	$-0.684831\pi$
$419$	−3.94280e6	−1.09716	−0.548579	+	0.836099i	$0.684831\pi$
$420$	0	0
$421$	−4.20370e6	−1.15592	−0.577958	−	0.816067i	$-0.696150\pi$
$421$	−4.20370e6	−1.15592	−0.577958	+	0.816067i	$0.696150\pi$
$422$	0	0
$423$	−240813.	−0.0654378
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.96205e6	−1.84785
$428$	0	0
$429$	1.99711e6	0.523911
$430$	0	0
$431$	5.90469e6	1.53110	0.765550	−	0.643376i	$-0.222467\pi$
$431$	5.90469e6	1.53110	0.765550	+	0.643376i	$0.222467\pi$
$432$	0	0
$433$	3.98952e6	1.02259	0.511294	−	0.859406i	$-0.329166\pi$
$433$	3.98952e6	1.02259	0.511294	+	0.859406i	$0.329166\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.94708e6	−0.487730
$438$	0	0
$439$	−10775.1	−0.00266846	−0.00133423	−	0.999999i	$-0.500425\pi$
$439$	−10775.1	−0.00266846	−0.00133423	+	0.999999i	$0.500425\pi$
$440$	0	0
$441$	1.35606e6	0.332034
$442$	0	0
$443$	6.72453e6	1.62799	0.813996	−	0.580870i	$-0.197288\pi$
$443$	6.72453e6	1.62799	0.813996	+	0.580870i	$0.197288\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.53658e6	−0.837172
$448$	0	0
$449$	−6.39615e6	−1.49728	−0.748640	−	0.662977i	$-0.769293\pi$
$449$	−6.39615e6	−1.49728	−0.748640	+	0.662977i	$0.769293\pi$
$450$	0	0
$451$	2.05173e6	0.474985
$452$	0	0
$453$	−8.56065e6	−1.96002
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.40846e6	1.65935	0.829674	−	0.558248i	$-0.188526\pi$
$457$	7.40846e6	1.65935	0.829674	+	0.558248i	$0.188526\pi$
$458$	0	0
$459$	2.32668e6	0.515471
$460$	0	0
$461$	−4.34835e6	−0.952954	−0.476477	−	0.879187i	$-0.658086\pi$
$461$	−4.34835e6	−0.952954	−0.476477	+	0.879187i	$0.658086\pi$
$462$	0	0
$463$	2.15517e6	0.467229	0.233614	−	0.972329i	$-0.424945\pi$
$463$	2.15517e6	0.467229	0.233614	+	0.972329i	$0.424945\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.33954e6	0.284226	0.142113	−	0.989850i	$-0.454610\pi$
$467$	1.33954e6	0.284226	0.142113	+	0.989850i	$0.454610\pi$
$468$	0	0
$469$	7.09349e6	1.48911
$470$	0	0
$471$	−3.92392e6	−0.815019
$472$	0	0
$473$	4.78610e6	0.983624
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.00723e6	−1.00763
$478$	0	0
$479$	9.34745e6	1.86146	0.930731	−	0.365704i	$-0.119172\pi$
$479$	9.34745e6	1.86146	0.930731	+	0.365704i	$0.119172\pi$
$480$	0	0
$481$	3.29262e6	0.648902
$482$	0	0
$483$	6.85217e6	1.33647
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.51352e6	0.671305	0.335652	−	0.941986i	$-0.391043\pi$
$487$	3.51352e6	0.671305	0.335652	+	0.941986i	$0.391043\pi$
$488$	0	0
$489$	−8.92826e6	−1.68847
$490$	0	0
$491$	5.22485e6	0.978070	0.489035	−	0.872264i	$-0.337349\pi$
$491$	5.22485e6	0.978070	0.489035	+	0.872264i	$0.337349\pi$
$492$	0	0
$493$	3.49432e6	0.647508
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.13333e6	1.11380
$498$	0	0
$499$	1.86243e6	0.334834	0.167417	−	0.985886i	$-0.446457\pi$
$499$	1.86243e6	0.334834	0.167417	+	0.985886i	$0.446457\pi$
$500$	0	0
$501$	1.02902e7	1.83160
$502$	0	0
$503$	−6.06780e6	−1.06933	−0.534664	−	0.845065i	$-0.679562\pi$
$503$	−6.06780e6	−1.06933	−0.534664	+	0.845065i	$0.679562\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.11961e6	−1.05731
$508$	0	0
$509$	−6.24842e6	−1.06899	−0.534497	−	0.845170i	$-0.679499\pi$
$509$	−6.24842e6	−1.06899	−0.534497	+	0.845170i	$0.679499\pi$
$510$	0	0
$511$	5.99096e6	1.01495
$512$	0	0
$513$	1.68361e6	0.282454
$514$	0	0
$515$	0	0
$516$	0	0
$517$	652831.	0.107417
$518$	0	0
$519$	−541615.	−0.0882617
$520$	0	0
$521$	9.23593e6	1.49069	0.745343	−	0.666681i	$-0.232286\pi$
$521$	9.23593e6	1.49069	0.745343	+	0.666681i	$0.232286\pi$
$522$	0	0
$523$	1.05161e6	0.168113	0.0840566	−	0.996461i	$-0.473212\pi$
$523$	1.05161e6	0.168113	0.0840566	+	0.996461i	$0.473212\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.14620e6	−0.650315
$528$	0	0
$529$	−1.81332e6	−0.281732
$530$	0	0
$531$	3.30282e6	0.508334
$532$	0	0
$533$	1.26637e6	0.193082
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.99965e6	0.299239
$538$	0	0
$539$	−3.67621e6	−0.545040
$540$	0	0
$541$	4.34911e6	0.638862	0.319431	−	0.947610i	$-0.396508\pi$
$541$	4.34911e6	0.638862	0.319431	+	0.947610i	$0.396508\pi$
$542$	0	0
$543$	2.47227e6	0.359830
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.98339e6	0.426326	0.213163	−	0.977017i	$-0.431624\pi$
$547$	2.98339e6	0.426326	0.213163	+	0.977017i	$0.431624\pi$
$548$	0	0
$549$	−6.45036e6	−0.913384
$550$	0	0
$551$	2.52853e6	0.354804
$552$	0	0
$553$	−1.75040e6	−0.243402
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.74611e6	0.648186	0.324093	−	0.946025i	$-0.394941\pi$
$557$	4.74611e6	0.648186	0.324093	+	0.946025i	$0.394941\pi$
$558$	0	0
$559$	2.95407e6	0.399844
$560$	0	0
$561$	1.00172e7	1.34382
$562$	0	0
$563$	−4.22516e6	−0.561787	−0.280894	−	0.959739i	$-0.590631\pi$
$563$	−4.22516e6	−0.561787	−0.280894	+	0.959739i	$0.590631\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.17564e7	−1.53574
$568$	0	0
$569$	1.95043e6	0.252551	0.126275	−	0.991995i	$-0.459698\pi$
$569$	1.95043e6	0.252551	0.126275	+	0.991995i	$0.459698\pi$
$570$	0	0
$571$	1.24377e7	1.59643	0.798215	−	0.602372i	$-0.205778\pi$
$571$	1.24377e7	1.59643	0.798215	+	0.602372i	$0.205778\pi$
$572$	0	0
$573$	−1.64249e7	−2.08986
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.21199e7	1.51552	0.757758	−	0.652536i	$-0.226295\pi$
$577$	1.21199e7	1.51552	0.757758	+	0.652536i	$0.226295\pi$
$578$	0	0
$579$	1.41014e7	1.74810
$580$	0	0
$581$	−1.21891e7	−1.49807
$582$	0	0
$583$	1.35743e7	1.65404
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.40591e6	−0.168407	−0.0842036	−	0.996449i	$-0.526835\pi$
$587$	−1.40591e6	−0.168407	−0.0842036	+	0.996449i	$0.526835\pi$
$588$	0	0
$589$	−3.00024e6	−0.356342
$590$	0	0
$591$	1.92200e7	2.26352
$592$	0	0
$593$	8.82785e6	1.03090	0.515452	−	0.856919i	$-0.327624\pi$
$593$	8.82785e6	1.03090	0.515452	+	0.856919i	$0.327624\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.02223e7	−1.17385
$598$	0	0
$599$	−4.21509e6	−0.479999	−0.239999	−	0.970773i	$-0.577147\pi$
$599$	−4.21509e6	−0.479999	−0.239999	+	0.970773i	$0.577147\pi$
$600$	0	0
$601$	9.00063e6	1.01645	0.508226	−	0.861224i	$-0.330302\pi$
$601$	9.00063e6	1.01645	0.508226	+	0.861224i	$0.330302\pi$
$602$	0	0
$603$	6.57214e6	0.736061
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.74038e7	−1.91722	−0.958611	−	0.284721i	$-0.908099\pi$
$607$	−1.74038e7	−1.91722	−0.958611	+	0.284721i	$0.908099\pi$
$608$	0	0
$609$	−8.89842e6	−0.972231
$610$	0	0
$611$	402939.	0.0436653
$612$	0	0
$613$	7.31051e6	0.785772	0.392886	−	0.919587i	$-0.371477\pi$
$613$	7.31051e6	0.785772	0.392886	+	0.919587i	$0.371477\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.06016e6	0.323617	0.161809	−	0.986822i	$-0.448267\pi$
$617$	3.06016e6	0.323617	0.161809	+	0.986822i	$0.448267\pi$
$618$	0	0
$619$	1.94950e6	0.204502	0.102251	−	0.994759i	$-0.467396\pi$
$619$	1.94950e6	0.204502	0.102251	+	0.994759i	$0.467396\pi$
$620$	0	0
$621$	−3.99746e6	−0.415963
$622$	0	0
$623$	1.95593e7	2.01899
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.24859e6	0.736351
$628$	0	0
$629$	1.65154e7	1.66442
$630$	0	0
$631$	−5.34043e6	−0.533953	−0.266976	−	0.963703i	$-0.586025\pi$
$631$	−5.34043e6	−0.533953	−0.266976	+	0.963703i	$0.586025\pi$
$632$	0	0
$633$	−1.31507e7	−1.30449
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.26902e6	−0.221559
$638$	0	0
$639$	5.68255e6	0.550543
$640$	0	0
$641$	−1.70992e7	−1.64373	−0.821864	−	0.569684i	$-0.807066\pi$
$641$	−1.70992e7	−1.64373	−0.821864	+	0.569684i	$0.807066\pi$
$642$	0	0
$643$	−1.21597e7	−1.15983	−0.579917	−	0.814676i	$-0.696915\pi$
$643$	−1.21597e7	−1.15983	−0.579917	+	0.814676i	$0.696915\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.68331e6	−0.158089	−0.0790447	−	0.996871i	$-0.525187\pi$
$647$	−1.68331e6	−0.158089	−0.0790447	+	0.996871i	$0.525187\pi$
$648$	0	0
$649$	−8.95377e6	−0.834439
$650$	0	0
$651$	1.05585e7	0.976446
$652$	0	0
$653$	1.29171e7	1.18545	0.592726	−	0.805404i	$-0.298052\pi$
$653$	1.29171e7	1.18545	0.592726	+	0.805404i	$0.298052\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.55065e6	0.501684
$658$	0	0
$659$	−3.20405e6	−0.287399	−0.143700	−	0.989621i	$-0.545900\pi$
$659$	−3.20405e6	−0.287399	−0.143700	+	0.989621i	$0.545900\pi$
$660$	0	0
$661$	−1.82464e7	−1.62433	−0.812165	−	0.583428i	$-0.801711\pi$
$661$	−1.82464e7	−1.62433	−0.812165	+	0.583428i	$0.801711\pi$
$662$	0	0
$663$	6.18282e6	0.546264
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.00358e6	−0.522511
$668$	0	0
$669$	1.96044e7	1.69351
$670$	0	0
$671$	1.74866e7	1.49934
$672$	0	0
$673$	1.17328e7	0.998538	0.499269	−	0.866447i	$-0.333602\pi$
$673$	1.17328e7	0.998538	0.499269	+	0.866447i	$0.333602\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.27923e7	−1.07270	−0.536348	−	0.843997i	$-0.680197\pi$
$677$	−1.27923e7	−1.07270	−0.536348	+	0.843997i	$0.680197\pi$
$678$	0	0
$679$	−2.53973e7	−2.11404
$680$	0	0
$681$	−308249.	−0.0254703
$682$	0	0
$683$	1.34398e7	1.10240	0.551202	−	0.834372i	$-0.314169\pi$
$683$	1.34398e7	1.10240	0.551202	+	0.834372i	$0.314169\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.11918e6	0.575490
$688$	0	0
$689$	8.37831e6	0.672370
$690$	0	0
$691$	−1.79328e7	−1.42874	−0.714371	−	0.699767i	$-0.753287\pi$
$691$	−1.79328e7	−1.42874	−0.714371	+	0.699767i	$0.753287\pi$
$692$	0	0
$693$	−9.70060e6	−0.767300
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.35194e6	0.495250
$698$	0	0
$699$	2.40683e7	1.86317
$700$	0	0
$701$	2.36981e6	0.182146	0.0910728	−	0.995844i	$-0.470970\pi$
$701$	2.36981e6	0.182146	0.0910728	+	0.995844i	$0.470970\pi$
$702$	0	0
$703$	1.19507e7	0.912023
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.23993e6	−0.394255
$708$	0	0
$709$	−1.86295e6	−0.139183	−0.0695913	−	0.997576i	$-0.522170\pi$
$709$	−1.86295e6	−0.139183	−0.0695913	+	0.997576i	$0.522170\pi$
$710$	0	0
$711$	−1.62175e6	−0.120313
$712$	0	0
$713$	7.12358e6	0.524777
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.98026e7	−1.43855
$718$	0	0
$719$	1.14819e7	0.828308	0.414154	−	0.910207i	$-0.364077\pi$
$719$	1.14819e7	0.828308	0.414154	+	0.910207i	$0.364077\pi$
$720$	0	0
$721$	199594.	0.0142991
$722$	0	0
$723$	−2.49188e7	−1.77289
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.08210e7	−1.46105	−0.730524	−	0.682887i	$-0.760724\pi$
$727$	−2.08210e7	−1.46105	−0.730524	+	0.682887i	$0.760724\pi$
$728$	0	0
$729$	−1.94630e6	−0.135641
$730$	0	0
$731$	1.48172e7	1.02559
$732$	0	0
$733$	8.47296e6	0.582472	0.291236	−	0.956651i	$-0.405934\pi$
$733$	8.47296e6	0.582472	0.291236	+	0.956651i	$0.405934\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.78167e7	−1.20826
$738$	0	0
$739$	1.57784e7	1.06280	0.531402	−	0.847120i	$-0.321665\pi$
$739$	1.57784e7	1.06280	0.531402	+	0.847120i	$0.321665\pi$
$740$	0	0
$741$	4.47396e6	0.299327
$742$	0	0
$743$	−1.54066e7	−1.02384	−0.511922	−	0.859032i	$-0.671066\pi$
$743$	−1.54066e7	−1.02384	−0.511922	+	0.859032i	$0.671066\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.12933e7	−0.740487
$748$	0	0
$749$	2.74161e7	1.78567
$750$	0	0
$751$	1.60309e7	1.03719	0.518594	−	0.855021i	$-0.326456\pi$
$751$	1.60309e7	1.03719	0.518594	+	0.855021i	$0.326456\pi$
$752$	0	0
$753$	−1.96186e7	−1.26090
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.65387e6	0.104896	0.0524482	−	0.998624i	$-0.483298\pi$
$757$	1.65387e6	0.104896	0.0524482	+	0.998624i	$0.483298\pi$
$758$	0	0
$759$	−1.72106e7	−1.08441
$760$	0	0
$761$	−1.00105e7	−0.626605	−0.313302	−	0.949653i	$-0.601435\pi$
$761$	−1.00105e7	−0.626605	−0.313302	+	0.949653i	$0.601435\pi$
$762$	0	0
$763$	2.05302e7	1.27668
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.52643e6	−0.339200
$768$	0	0
$769$	3.06629e7	1.86981	0.934906	−	0.354895i	$-0.115483\pi$
$769$	3.06629e7	1.86981	0.934906	+	0.354895i	$0.115483\pi$
$770$	0	0
$771$	3.39875e7	2.05913
$772$	0	0
$773$	−2.29138e7	−1.37927	−0.689633	−	0.724159i	$-0.742228\pi$
$773$	−2.29138e7	−1.37927	−0.689633	+	0.724159i	$0.742228\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.20571e7	−2.49912
$778$	0	0
$779$	4.59634e6	0.271374
$780$	0	0
$781$	−1.54051e7	−0.903727
$782$	0	0
$783$	5.19121e6	0.302597
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.27348e7	1.88396	0.941982	−	0.335663i	$-0.108960\pi$
$787$	3.27348e7	1.88396	0.941982	+	0.335663i	$0.108960\pi$
$788$	0	0
$789$	−3.15894e7	−1.80655
$790$	0	0
$791$	−1.78389e7	−1.01374
$792$	0	0
$793$	1.07930e7	0.609481
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.16718e6	−0.232379	−0.116189	−	0.993227i	$-0.537068\pi$
$797$	−4.16718e6	−0.232379	−0.116189	+	0.993227i	$0.537068\pi$
$798$	0	0
$799$	2.02109e6	0.112000
$800$	0	0
$801$	1.81218e7	0.997976
$802$	0	0
$803$	−1.50475e7	−0.823524
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.77344e7	0.958591
$808$	0	0
$809$	1.28521e6	0.0690405	0.0345202	−	0.999404i	$-0.489010\pi$
$809$	1.28521e6	0.0690405	0.0345202	+	0.999404i	$0.489010\pi$
$810$	0	0
$811$	−6.16431e6	−0.329103	−0.164552	−	0.986368i	$-0.552618\pi$
$811$	−6.16431e6	−0.329103	−0.164552	+	0.986368i	$0.552618\pi$
$812$	0	0
$813$	−1.28455e7	−0.681590
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.07219e7	0.561976
$818$	0	0
$819$	−5.98738e6	−0.311908
$820$	0	0
$821$	1.50757e7	0.780581	0.390291	−	0.920692i	$-0.372375\pi$
$821$	1.50757e7	0.780581	0.390291	+	0.920692i	$0.372375\pi$
$822$	0	0
$823$	7.55088e6	0.388596	0.194298	−	0.980943i	$-0.437757\pi$
$823$	7.55088e6	0.388596	0.194298	+	0.980943i	$0.437757\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.02532e6	−0.204662	−0.102331	−	0.994750i	$-0.532630\pi$
$827$	−4.02532e6	−0.204662	−0.102331	+	0.994750i	$0.532630\pi$
$828$	0	0
$829$	−1.48566e7	−0.750813	−0.375407	−	0.926860i	$-0.622497\pi$
$829$	−1.48566e7	−0.750813	−0.375407	+	0.926860i	$0.622497\pi$
$830$	0	0
$831$	−4.66798e7	−2.34491
$832$	0	0
$833$	−1.13811e7	−0.568294
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6.15966e6	−0.303909
$838$	0	0
$839$	9.59062e6	0.470372	0.235186	−	0.971950i	$-0.424430\pi$
$839$	9.59062e6	0.470372	0.235186	+	0.971950i	$0.424430\pi$
$840$	0	0
$841$	−1.27147e7	−0.619894
$842$	0	0
$843$	1.16185e7	0.563092
$844$	0	0
$845$	0	0
$846$	0	0
$847$	378440.	0.0181255
$848$	0	0
$849$	−2.82410e7	−1.34465
$850$	0	0
$851$	−2.83751e7	−1.34311
$852$	0	0
$853$	−6.24756e6	−0.293994	−0.146997	−	0.989137i	$-0.546961\pi$
$853$	−6.24756e6	−0.293994	−0.146997	+	0.989137i	$0.546961\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.19127e7	1.01916	0.509581	−	0.860423i	$-0.329800\pi$
$857$	2.19127e7	1.01916	0.509581	+	0.860423i	$0.329800\pi$
$858$	0	0
$859$	−8.90797e6	−0.411904	−0.205952	−	0.978562i	$-0.566029\pi$
$859$	−8.90797e6	−0.411904	−0.205952	+	0.978562i	$0.566029\pi$
$860$	0	0
$861$	−1.61755e7	−0.743617
$862$	0	0
$863$	−1.28474e7	−0.587204	−0.293602	−	0.955928i	$-0.594854\pi$
$863$	−1.28474e7	−0.587204	−0.293602	+	0.955928i	$0.594854\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.89659e6	0.130870
$868$	0	0
$869$	4.39649e6	0.197495
$870$	0	0
$871$	−1.09968e7	−0.491158
$872$	0	0
$873$	−2.35307e7	−1.04496
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.36870e7	1.03995	0.519973	−	0.854183i	$-0.325942\pi$
$877$	2.36870e7	1.03995	0.519973	+	0.854183i	$0.325942\pi$
$878$	0	0
$879$	−1.84495e7	−0.805401
$880$	0	0
$881$	−1.77488e7	−0.770423	−0.385212	−	0.922828i	$-0.625872\pi$
$881$	−1.77488e7	−0.770423	−0.385212	+	0.922828i	$0.625872\pi$
$882$	0	0
$883$	293447.	0.0126656	0.00633282	−	0.999980i	$-0.497984\pi$
$883$	293447.	0.0126656	0.00633282	+	0.999980i	$0.497984\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.50694e7	0.643112	0.321556	−	0.946891i	$-0.395794\pi$
$887$	1.50694e7	0.643112	0.321556	+	0.946891i	$0.395794\pi$
$888$	0	0
$889$	−1.85942e7	−0.789083
$890$	0	0
$891$	2.95286e7	1.24609
$892$	0	0
$893$	1.46249e6	0.0613710
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.06227e7	−0.440812
$898$	0	0
$899$	−9.25088e6	−0.381754
$900$	0	0
$901$	4.20246e7	1.72461
$902$	0	0
$903$	−3.77327e7	−1.53992
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.62189e6	−0.186553	−0.0932764	−	0.995640i	$-0.529734\pi$
$907$	−4.62189e6	−0.186553	−0.0932764	+	0.995640i	$0.529734\pi$
$908$	0	0
$909$	−4.85481e6	−0.194878
$910$	0	0
$911$	9.02270e6	0.360197	0.180099	−	0.983649i	$-0.442358\pi$
$911$	9.02270e6	0.360197	0.180099	+	0.983649i	$0.442358\pi$
$912$	0	0
$913$	3.06154e7	1.21552
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.78801e6	0.188032
$918$	0	0
$919$	−1.81033e7	−0.707081	−0.353540	−	0.935419i	$-0.615022\pi$
$919$	−1.81033e7	−0.707081	−0.353540	+	0.935419i	$0.615022\pi$
$920$	0	0
$921$	3.00422e7	1.16703
$922$	0	0
$923$	−9.50830e6	−0.367366
$924$	0	0
$925$	0	0
$926$	0	0
$927$	184924.	0.00706797
$928$	0	0
$929$	3.59953e7	1.36838	0.684190	−	0.729304i	$-0.260156\pi$
$929$	3.59953e7	1.36838	0.684190	+	0.729304i	$0.260156\pi$
$930$	0	0
$931$	−8.23551e6	−0.311398
$932$	0	0
$933$	7.25253e6	0.272763
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.79708e7	1.04077	0.520387	−	0.853931i	$-0.325788\pi$
$937$	2.79708e7	1.04077	0.520387	+	0.853931i	$0.325788\pi$
$938$	0	0
$939$	−2.36605e7	−0.875710
$940$	0	0
$941$	−1.35773e7	−0.499851	−0.249926	−	0.968265i	$-0.580406\pi$
$941$	−1.35773e7	−0.499851	−0.249926	+	0.968265i	$0.580406\pi$
$942$	0	0
$943$	−1.09133e7	−0.399646
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.32801e6	−0.193059	−0.0965296	−	0.995330i	$-0.530774\pi$
$947$	−5.32801e6	−0.193059	−0.0965296	+	0.995330i	$0.530774\pi$
$948$	0	0
$949$	−9.28759e6	−0.334763
$950$	0	0
$951$	−4.25360e7	−1.52513
$952$	0	0
$953$	1.02144e7	0.364316	0.182158	−	0.983269i	$-0.441692\pi$
$953$	1.02144e7	0.364316	0.182158	+	0.983269i	$0.441692\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.23502e7	0.788862
$958$	0	0
$959$	3.83257e7	1.34569
$960$	0	0
$961$	−1.76525e7	−0.616591
$962$	0	0
$963$	2.54011e7	0.882647
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.17005e7	0.746285	0.373142	−	0.927774i	$-0.378280\pi$
$967$	2.17005e7	0.746285	0.373142	+	0.927774i	$0.378280\pi$
$968$	0	0
$969$	2.24408e7	0.767767
$970$	0	0
$971$	−4.93228e7	−1.67880	−0.839401	−	0.543512i	$-0.817094\pi$
$971$	−4.93228e7	−1.67880	−0.839401	+	0.543512i	$0.817094\pi$
$972$	0	0
$973$	6.54280e7	2.21555
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.04782e7	0.686366	0.343183	−	0.939269i	$-0.388495\pi$
$977$	2.04782e7	0.686366	0.343183	+	0.939269i	$0.388495\pi$
$978$	0	0
$979$	−4.91273e7	−1.63820
$980$	0	0
$981$	1.90213e7	0.631055
$982$	0	0
$983$	−1.58306e7	−0.522533	−0.261267	−	0.965267i	$-0.584140\pi$
$983$	−1.58306e7	−0.522533	−0.261267	+	0.965267i	$0.584140\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.14679e6	−0.168168
$988$	0	0
$989$	−2.54575e7	−0.827609
$990$	0	0
$991$	1.75681e7	0.568250	0.284125	−	0.958787i	$-0.408297\pi$
$991$	1.75681e7	0.568250	0.284125	+	0.958787i	$0.408297\pi$
$992$	0	0
$993$	8.76779e6	0.282174
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.21502e7	1.34296	0.671478	−	0.741025i	$-0.265660\pi$
$997$	4.21502e7	1.34296	0.671478	+	0.741025i	$0.265660\pi$
$998$	0	0
$999$	2.45355e7	0.777825

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.a.s.1.4	✓	5
4.3	odd	2		800.6.a.v.1.2	yes	5
5.2	odd	4		800.6.c.n.449.3		10
5.3	odd	4		800.6.c.n.449.8		10
5.4	even	2		800.6.a.u.1.2	yes	5
20.3	even	4		800.6.c.o.449.3		10
20.7	even	4		800.6.c.o.449.8		10
20.19	odd	2		800.6.a.t.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.6.a.s.1.4	✓	5	1.1	even	1	trivial
800.6.a.t.1.4	yes	5	20.19	odd	2
800.6.a.u.1.2	yes	5	5.4	even	2
800.6.a.v.1.2	yes	5	4.3	odd	2
800.6.c.n.449.3		10	5.2	odd	4
800.6.c.n.449.8		10	5.3	odd	4
800.6.c.o.449.3		10	20.3	even	4
800.6.c.o.449.8		10	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q+19.8018 q^{3} +160.939 q^{7} +149.111 q^{9} +O(q^{10})\)	\(q+19.8018 q^{3} +160.939 q^{7} +149.111 q^{9} -404.231 q^{11} -249.498 q^{13} -1251.45 q^{17} -905.566 q^{19} +3186.88 q^{21} +2150.12 q^{23} -1859.18 q^{27} -2792.21 q^{29} +3313.11 q^{31} -8004.49 q^{33} -13197.0 q^{37} -4940.51 q^{39} -5075.65 q^{41} -11840.0 q^{43} -1615.00 q^{47} +9094.33 q^{49} -24781.0 q^{51} -33580.6 q^{53} -17931.8 q^{57} +22150.2 q^{59} -43258.9 q^{61} +23997.7 q^{63} +44075.6 q^{67} +42576.2 q^{69} +38109.7 q^{71} +37225.1 q^{73} -65056.4 q^{77} -10876.2 q^{79} -73048.9 q^{81} -75737.5 q^{83} -55290.6 q^{87} +121533. q^{89} -40154.0 q^{91} +65605.4 q^{93} -157807. q^{97} -60275.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 110 q^{7} + 606 q^{9}+O(q^{10}) \) 5 * q - q^3 - 110 * q^7 + 606 * q^9	\( 5 q - q^{3} - 110 q^{7} + 606 q^{9} - 5 q^{11} - 280 q^{13} + 865 q^{17} - 4485 q^{19} - 418 q^{21} + 3946 q^{23} - 4987 q^{27} - 3252 q^{29} - 8250 q^{31} + 13465 q^{33} - 3210 q^{37} - 8800 q^{39} - 11415 q^{41} + 428 q^{43} + 9672 q^{47} + 26225 q^{49} - 2965 q^{51} - 30370 q^{53} + 29345 q^{57} + 39280 q^{59} + 34854 q^{61} - 87812 q^{63} + 68877 q^{67} - 40658 q^{69} - 51200 q^{71} + 41615 q^{73} - 57050 q^{77} + 550 q^{79} - 15219 q^{81} - 66223 q^{83} + 168860 q^{87} - 103829 q^{89} - 185280 q^{91} - 59870 q^{93} - 21150 q^{97} - 244390 q^{99}+O(q^{100}) \) 5 * q - q^3 - 110 * q^7 + 606 * q^9 - 5 * q^11 - 280 * q^13 + 865 * q^17 - 4485 * q^19 - 418 * q^21 + 3946 * q^23 - 4987 * q^27 - 3252 * q^29 - 8250 * q^31 + 13465 * q^33 - 3210 * q^37 - 8800 * q^39 - 11415 * q^41 + 428 * q^43 + 9672 * q^47 + 26225 * q^49 - 2965 * q^51 - 30370 * q^53 + 29345 * q^57 + 39280 * q^59 + 34854 * q^61 - 87812 * q^63 + 68877 * q^67 - 40658 * q^69 - 51200 * q^71 + 41615 * q^73 - 57050 * q^77 + 550 * q^79 - 15219 * q^81 - 66223 * q^83 + 168860 * q^87 - 103829 * q^89 - 185280 * q^91 - 59870 * q^93 - 21150 * q^97 - 244390 * q^99

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