LMFDB - Embedded newform 800.6.f.d.49.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(49,800)]

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, 1, 1]))

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

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$128.307055850$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.10
Character	$\chi$	$=$	800.49
Dual form			800.6.f.d.49.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-18.2848 q^{3} +2.25573i q^{7} +91.3327 q^{9} +419.261i q^{11} +106.889 q^{13} +849.098i q^{17} +335.414i q^{19} -41.2454i q^{21} +3541.78i q^{23} +2773.20 q^{27} -5208.57i q^{29} -5637.83 q^{31} -7666.09i q^{33} -61.9860 q^{37} -1954.44 q^{39} +16286.1 q^{41} -2417.19 q^{43} +22781.9i q^{47} +16801.9 q^{49} -15525.6i q^{51} -13667.5 q^{53} -6132.98i q^{57} +23407.1i q^{59} -33444.7i q^{61} +206.022i q^{63} +66162.9 q^{67} -64760.7i q^{69} +51421.4 q^{71} -21271.6i q^{73} -945.738 q^{77} +38418.7 q^{79} -72901.2 q^{81} -93166.7 q^{83} +95237.5i q^{87} +60678.0 q^{89} +241.112i q^{91} +103086. q^{93} +157428. i q^{97} +38292.2i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 3240 q^{9} - 14320 q^{31} + 44904 q^{39} - 11608 q^{41} - 125304 q^{49} + 15448 q^{71} - 15560 q^{79} + 193968 q^{81} + 6320 q^{89}+O(q^{100}) $ 40 * q + 3240 * q^9 - 14320 * q^31 + 44904 * q^39 - 11608 * q^41 - 125304 * q^49 + 15448 * q^71 - 15560 * q^79 + 193968 * q^81 + 6320 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−18.2848	−1.17297	−0.586484	−	0.809961i	$-0.699488\pi$
$3$	−18.2848	−1.17297	−0.586484	+	0.809961i	$0.699488\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.25573i	0.0173997i	0.999962	+	0.00869984i	$0.00276928\pi$
$7$	2.25573i	0.0173997i	−0.999962	+	0.00869984i	$0.997231\pi$
$8$	0	0
$9$	91.3327	0.375855
$10$	0	0
$11$	419.261i	1.04473i	0.852723	+	0.522363i	$0.174950\pi$
$11$	419.261i	1.04473i	−0.852723	+	0.522363i	$0.825050\pi$
$12$	0	0
$13$	106.889	0.175418	0.0877090	−	0.996146i	$-0.472045\pi$
$13$	106.889	0.175418	0.0877090	+	0.996146i	$0.472045\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	849.098i	0.712584i	0.934375	+	0.356292i	$0.115959\pi$
$17$	849.098i	0.712584i	−0.934375	+	0.356292i	$0.884041\pi$
$18$	0	0
$19$	335.414i	0.213156i	0.994304	+	0.106578i	$0.0339894\pi$
$19$	335.414i	0.213156i	−0.994304	+	0.106578i	$0.966011\pi$
$20$	0	0
$21$	− 41.2454i	− 0.0204093i
$22$	0	0
$23$	3541.78i	1.39605i	0.716071	+	0.698027i	$0.245938\pi$
$23$	3541.78i	1.39605i	−0.716071	+	0.698027i	$0.754062\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2773.20	0.732103
$28$	0	0
$29$	− 5208.57i	− 1.15007i	−0.818129	−	0.575034i	$-0.804989\pi$
$29$	− 5208.57i	− 1.15007i	0.818129	−	0.575034i	$-0.195011\pi$
$30$	0	0
$31$	−5637.83	−1.05368	−0.526839	−	0.849965i	$-0.676623\pi$
$31$	−5637.83	−1.05368	−0.526839	+	0.849965i	$0.676623\pi$
$32$	0	0
$33$	− 7666.09i	− 1.22543i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−61.9860	−0.00744370	−0.00372185	−	0.999993i	$-0.501185\pi$
$37$	−61.9860	−0.00744370	−0.00372185	+	0.999993i	$0.501185\pi$
$38$	0	0
$39$	−1954.44	−0.205760
$40$	0	0
$41$	16286.1	1.51306	0.756531	−	0.653958i	$-0.226893\pi$
$41$	16286.1	1.51306	0.756531	+	0.653958i	$0.226893\pi$
$42$	0	0
$43$	−2417.19	−0.199361	−0.0996803	−	0.995020i	$-0.531782\pi$
$43$	−2417.19	−0.199361	−0.0996803	+	0.995020i	$0.531782\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	22781.9i	1.50434i	0.658970	+	0.752170i	$0.270993\pi$
$47$	22781.9i	1.50434i	−0.658970	+	0.752170i	$0.729007\pi$
$48$	0	0
$49$	16801.9	0.999697
$50$	0	0
$51$	− 15525.6i	− 0.835838i
$52$	0	0
$53$	−13667.5	−0.668344	−0.334172	−	0.942512i	$-0.608457\pi$
$53$	−13667.5	−0.668344	−0.334172	+	0.942512i	$0.608457\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6132.98i	− 0.250025i
$58$	0	0
$59$	23407.1i	0.875423i	0.899116	+	0.437711i	$0.144211\pi$
$59$	23407.1i	0.875423i	−0.899116	+	0.437711i	$0.855789\pi$
$60$	0	0
$61$	− 33444.7i	− 1.15081i	−0.817869	−	0.575405i	$-0.804845\pi$
$61$	− 33444.7i	− 1.15081i	0.817869	−	0.575405i	$-0.195155\pi$
$62$	0	0
$63$	206.022i	0.00653975i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	66162.9	1.80064	0.900321	−	0.435226i	$-0.143332\pi$
$67$	66162.9	1.80064	0.900321	+	0.435226i	$0.143332\pi$
$68$	0	0
$69$	− 64760.7i	− 1.63753i
$70$	0	0
$71$	51421.4	1.21059	0.605296	−	0.796001i	$-0.293055\pi$
$71$	51421.4	1.21059	0.605296	+	0.796001i	$0.293055\pi$
$72$	0	0
$73$	− 21271.6i	− 0.467189i	−0.972334	−	0.233595i	$-0.924951\pi$
$73$	− 21271.6i	− 0.467189i	0.972334	−	0.233595i	$-0.0750489\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−945.738	−0.0181779
$78$	0	0
$79$	38418.7	0.692589	0.346294	−	0.938126i	$-0.387440\pi$
$79$	38418.7	0.692589	0.346294	+	0.938126i	$0.387440\pi$
$80$	0	0
$81$	−72901.2	−1.23459
$82$	0	0
$83$	−93166.7	−1.48445	−0.742225	−	0.670151i	$-0.766229\pi$
$83$	−93166.7	−1.48445	−0.742225	+	0.670151i	$0.766229\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	95237.5i	1.34899i
$88$	0	0
$89$	60678.0	0.812000	0.406000	−	0.913873i	$-0.366923\pi$
$89$	60678.0	0.812000	0.406000	+	0.913873i	$0.366923\pi$
$90$	0	0
$91$	241.112i	0.00305222i
$92$	0	0
$93$	103086.	1.23593
$94$	0	0
$95$	0	0
$96$	0	0
$97$	157428.i	1.69884i	0.527721	+	0.849418i	$0.323047\pi$
$97$	157428.i	1.69884i	−0.527721	+	0.849418i	$0.676953\pi$
$98$	0	0
$99$	38292.2i	0.392665i
$100$	0	0
$101$	− 124508.i	− 1.21449i	−0.794516	−	0.607243i	$-0.792275\pi$
$101$	− 124508.i	− 1.21449i	0.794516	−	0.607243i	$-0.207725\pi$
$102$	0	0
$103$	− 10346.4i	− 0.0960938i	−0.998845	−	0.0480469i	$-0.984700\pi$
$103$	− 10346.4i	− 0.0960938i	0.998845	−	0.0480469i	$-0.0152997\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	70765.9	0.597537	0.298769	−	0.954326i	$-0.403424\pi$
$107$	70765.9	0.597537	0.298769	+	0.954326i	$0.403424\pi$
$108$	0	0
$109$	− 128453.i	− 1.03557i	−0.855511	−	0.517784i	$-0.826757\pi$
$109$	− 128453.i	− 1.03557i	0.855511	−	0.517784i	$-0.173243\pi$
$110$	0	0
$111$	1133.40	0.00873123
$112$	0	0
$113$	115196.i	0.848672i	0.905505	+	0.424336i	$0.139492\pi$
$113$	115196.i	0.848672i	−0.905505	+	0.424336i	$0.860508\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9762.45	0.0659317
$118$	0	0
$119$	−1915.33	−0.0123987
$120$	0	0
$121$	−14728.7	−0.0914537
$122$	0	0
$123$	−297787.	−1.77477
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 146026.i	− 0.803378i	−0.915776	−	0.401689i	$-0.868423\pi$
$127$	− 146026.i	− 0.803378i	0.915776	−	0.401689i	$-0.131577\pi$
$128$	0	0
$129$	44197.7	0.233844
$130$	0	0
$131$	12998.0i	0.0661757i	0.999452	+	0.0330878i	$0.0105341\pi$
$131$	12998.0i	0.0661757i	−0.999452	+	0.0330878i	$0.989466\pi$
$132$	0	0
$133$	−756.603	−0.00370885
$134$	0	0
$135$	0	0
$136$	0	0
$137$	189204.i	0.861249i	0.902531	+	0.430624i	$0.141707\pi$
$137$	189204.i	0.861249i	−0.902531	+	0.430624i	$0.858293\pi$
$138$	0	0
$139$	44334.4i	0.194627i	0.995254	+	0.0973136i	$0.0310250\pi$
$139$	44334.4i	0.194627i	−0.995254	+	0.0973136i	$0.968975\pi$
$140$	0	0
$141$	− 416562.i	− 1.76454i
$142$	0	0
$143$	44814.3i	0.183264i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−307219.	−1.17261
$148$	0	0
$149$	352240.i	1.29979i	0.760024	+	0.649895i	$0.225187\pi$
$149$	352240.i	1.29979i	−0.760024	+	0.649895i	$0.774813\pi$
$150$	0	0
$151$	−446273.	−1.59279	−0.796395	−	0.604776i	$-0.793262\pi$
$151$	−446273.	−1.59279	−0.796395	+	0.604776i	$0.793262\pi$
$152$	0	0
$153$	77550.5i	0.267828i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−383316.	−1.24110	−0.620552	−	0.784165i	$-0.713091\pi$
$157$	−383316.	−1.24110	−0.620552	+	0.784165i	$0.713091\pi$
$158$	0	0
$159$	249907.	0.783946
$160$	0	0
$161$	−7989.29	−0.0242909
$162$	0	0
$163$	−405200.	−1.19454	−0.597270	−	0.802040i	$-0.703748\pi$
$163$	−405200.	−1.19454	−0.597270	+	0.802040i	$0.703748\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	193042.i	0.535625i	0.963471	+	0.267812i	$0.0863008\pi$
$167$	193042.i	0.535625i	−0.963471	+	0.267812i	$0.913699\pi$
$168$	0	0
$169$	−359868.	−0.969229
$170$	0	0
$171$	30634.3i	0.0801157i
$172$	0	0
$173$	599397.	1.52265	0.761323	−	0.648372i	$-0.224550\pi$
$173$	599397.	1.52265	0.761323	+	0.648372i	$0.224550\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 427994.i	− 1.02684i
$178$	0	0
$179$	421278.i	0.982736i	0.870952	+	0.491368i	$0.163503\pi$
$179$	421278.i	0.982736i	−0.870952	+	0.491368i	$0.836497\pi$
$180$	0	0
$181$	− 183362.i	− 0.416018i	−0.978127	−	0.208009i	$-0.933302\pi$
$181$	− 183362.i	− 0.416018i	0.978127	−	0.208009i	$-0.0666983\pi$
$182$	0	0
$183$	611529.i	1.34986i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−355994.	−0.744455
$188$	0	0
$189$	6255.58i	0.0127384i
$190$	0	0
$191$	−982533.	−1.94878	−0.974392	−	0.224856i	$-0.927809\pi$
$191$	−982533.	−1.94878	−0.974392	+	0.224856i	$0.927809\pi$
$192$	0	0
$193$	164978.i	0.318810i	0.987213	+	0.159405i	$0.0509576\pi$
$193$	164978.i	0.318810i	−0.987213	+	0.159405i	$0.949042\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−506605.	−0.930044	−0.465022	−	0.885299i	$-0.653954\pi$
$197$	−506605.	−0.930044	−0.465022	+	0.885299i	$0.653954\pi$
$198$	0	0
$199$	475645.	0.851433	0.425716	−	0.904857i	$-0.360022\pi$
$199$	475645.	0.851433	0.425716	+	0.904857i	$0.360022\pi$
$200$	0	0
$201$	−1.20977e6	−2.11210
$202$	0	0
$203$	11749.1	0.0200108
$204$	0	0
$205$	0	0
$206$	0	0
$207$	323481.i	0.524714i
$208$	0	0
$209$	−140626.	−0.222690
$210$	0	0
$211$	− 260466.i	− 0.402759i	−0.979513	−	0.201379i	$-0.935458\pi$
$211$	− 260466.i	− 0.402759i	0.979513	−	0.201379i	$-0.0645424\pi$
$212$	0	0
$213$	−940228.	−1.41999
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 12717.4i	− 0.0183337i
$218$	0	0
$219$	388946.i	0.547998i
$220$	0	0
$221$	90759.2i	0.125000i
$222$	0	0
$223$	624180.i	0.840519i	0.907404	+	0.420260i	$0.138061\pi$
$223$	624180.i	0.840519i	−0.907404	+	0.420260i	$0.861939\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−656014.	−0.844984	−0.422492	−	0.906367i	$-0.638845\pi$
$227$	−656014.	−0.844984	−0.422492	+	0.906367i	$0.638845\pi$
$228$	0	0
$229$	807777.i	1.01790i	0.860798	+	0.508948i	$0.169965\pi$
$229$	807777.i	1.01790i	−0.860798	+	0.508948i	$0.830035\pi$
$230$	0	0
$231$	17292.6	0.0213221
$232$	0	0
$233$	1.30452e6i	1.57421i	0.616822	+	0.787103i	$0.288420\pi$
$233$	1.30452e6i	1.57421i	−0.616822	+	0.787103i	$0.711580\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−702478.	−0.812385
$238$	0	0
$239$	−1.21861e6	−1.37997	−0.689985	−	0.723824i	$-0.742383\pi$
$239$	−1.21861e6	−1.37997	−0.689985	+	0.723824i	$0.742383\pi$
$240$	0	0
$241$	−983578.	−1.09085	−0.545426	−	0.838159i	$-0.683632\pi$
$241$	−983578.	−1.09085	−0.545426	+	0.838159i	$0.683632\pi$
$242$	0	0
$243$	659093.	0.716030
$244$	0	0
$245$	0	0
$246$	0	0
$247$	35852.1i	0.0373914i
$248$	0	0
$249$	1.70353e6	1.74121
$250$	0	0
$251$	− 1.60324e6i	− 1.60625i	−0.595809	−	0.803126i	$-0.703168\pi$
$251$	− 1.60324e6i	− 1.60625i	0.595809	−	0.803126i	$-0.296832\pi$
$252$	0	0
$253$	−1.48493e6	−1.45849
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.07709e6i	1.01723i	0.860993	+	0.508617i	$0.169843\pi$
$257$	1.07709e6i	1.01723i	−0.860993	+	0.508617i	$0.830157\pi$
$258$	0	0
$259$	− 139.823i	0 0.000129518i
$260$	0	0
$261$	− 475713.i	− 0.432259i
$262$	0	0
$263$	− 812889.i	− 0.724673i	−0.932047	−	0.362336i	$-0.881979\pi$
$263$	− 812889.i	− 0.724673i	0.932047	−	0.362336i	$-0.118021\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.10948e6	−0.952451
$268$	0	0
$269$	− 113957.i	− 0.0960201i	−0.998847	−	0.0480101i	$-0.984712\pi$
$269$	− 113957.i	− 0.0960201i	0.998847	−	0.0480101i	$-0.0152880\pi$
$270$	0	0
$271$	−372728.	−0.308297	−0.154148	−	0.988048i	$-0.549263\pi$
$271$	−372728.	−0.308297	−0.154148	+	0.988048i	$0.549263\pi$
$272$	0	0
$273$	− 4408.68i	− 0.00358015i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−914263.	−0.715932	−0.357966	−	0.933735i	$-0.616530\pi$
$277$	−914263.	−0.715932	−0.357966	+	0.933735i	$0.616530\pi$
$278$	0	0
$279$	−514918.	−0.396030
$280$	0	0
$281$	−511184.	−0.386199	−0.193100	−	0.981179i	$-0.561854\pi$
$281$	−511184.	−0.386199	−0.193100	+	0.981179i	$0.561854\pi$
$282$	0	0
$283$	533078.	0.395662	0.197831	−	0.980236i	$-0.436610\pi$
$283$	533078.	0.395662	0.197831	+	0.980236i	$0.436610\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	36736.9i	0.0263268i
$288$	0	0
$289$	698889.	0.492225
$290$	0	0
$291$	− 2.87853e6i	− 1.99268i
$292$	0	0
$293$	−2.54648e6	−1.73289	−0.866445	−	0.499273i	$-0.833600\pi$
$293$	−2.54648e6	−1.73289	−0.866445	+	0.499273i	$0.833600\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.16269e6i	0.764847i
$298$	0	0
$299$	378577.i	0.244893i
$300$	0	0
$301$	− 5452.51i	− 0.00346881i
$302$	0	0
$303$	2.27659e6i	1.42455i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	740673.	0.448518	0.224259	−	0.974530i	$-0.428004\pi$
$307$	740673.	0.448518	0.224259	+	0.974530i	$0.428004\pi$
$308$	0	0
$309$	189181.i	0.112715i
$310$	0	0
$311$	−1.08876e6	−0.638310	−0.319155	−	0.947702i	$-0.603399\pi$
$311$	−1.08876e6	−0.638310	−0.319155	+	0.947702i	$0.603399\pi$
$312$	0	0
$313$	− 1.08277e6i	− 0.624705i	−0.949966	−	0.312353i	$-0.898883\pi$
$313$	− 1.08277e6i	− 0.624705i	0.949966	−	0.312353i	$-0.101117\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.93243e6	1.63900	0.819501	−	0.573078i	$-0.194251\pi$
$317$	2.93243e6	1.63900	0.819501	+	0.573078i	$0.194251\pi$
$318$	0	0
$319$	2.18375e6	1.20151
$320$	0	0
$321$	−1.29394e6	−0.700892
$322$	0	0
$323$	−284800.	−0.151892
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.34874e6i	1.21469i
$328$	0	0
$329$	−51389.8	−0.0261750
$330$	0	0
$331$	− 357214.i	− 0.179209i	−0.995977	−	0.0896043i	$-0.971440\pi$
$331$	− 357214.i	− 0.179209i	0.995977	−	0.0896043i	$-0.0285602\pi$
$332$	0	0
$333$	−5661.34	−0.00279775
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.45195e6i	0.696429i	0.937415	+	0.348215i	$0.113212\pi$
$337$	1.45195e6i	0.696429i	−0.937415	+	0.348215i	$0.886788\pi$
$338$	0	0
$339$	− 2.10633e6i	− 0.995465i
$340$	0	0
$341$	− 2.36372e6i	− 1.10081i
$342$	0	0
$343$	75812.5i	0.0347941i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	393735.	0.175541	0.0877707	−	0.996141i	$-0.472026\pi$
$347$	393735.	0.175541	0.0877707	+	0.996141i	$0.472026\pi$
$348$	0	0
$349$	1.73875e6i	0.764140i	0.924133	+	0.382070i	$0.124789\pi$
$349$	1.73875e6i	0.764140i	−0.924133	+	0.382070i	$0.875211\pi$
$350$	0	0
$351$	296424.	0.128424
$352$	0	0
$353$	2.05124e6i	0.876154i	0.898937	+	0.438077i	$0.144340\pi$
$353$	2.05124e6i	0.876154i	−0.898937	+	0.438077i	$0.855660\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	35021.4	0.0145433
$358$	0	0
$359$	3.56728e6	1.46084	0.730418	−	0.683000i	$-0.239325\pi$
$359$	3.56728e6	1.46084	0.730418	+	0.683000i	$0.239325\pi$
$360$	0	0
$361$	2.36360e6	0.954564
$362$	0	0
$363$	269311.	0.107272
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.72721e6i	1.44450i	0.691631	+	0.722251i	$0.256893\pi$
$367$	3.72721e6i	1.44450i	−0.691631	+	0.722251i	$0.743107\pi$
$368$	0	0
$369$	1.48745e6	0.568691
$370$	0	0
$371$	− 30830.2i	− 0.0116290i
$372$	0	0
$373$	−1.71134e6	−0.636891	−0.318446	−	0.947941i	$-0.603161\pi$
$373$	−1.71134e6	−0.636891	−0.318446	+	0.947941i	$0.603161\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 556738.i	− 0.201743i
$378$	0	0
$379$	− 3.60174e6i	− 1.28799i	−0.765028	−	0.643997i	$-0.777275\pi$
$379$	− 3.60174e6i	− 1.28799i	0.765028	−	0.643997i	$-0.222725\pi$
$380$	0	0
$381$	2.67005e6i	0.942337i
$382$	0	0
$383$	− 3.80875e6i	− 1.32674i	−0.748293	−	0.663369i	$-0.769126\pi$
$383$	− 3.80875e6i	− 1.32674i	0.748293	−	0.663369i	$-0.230874\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−220768.	−0.0749306
$388$	0	0
$389$	431800.i	0.144680i	0.997380	+	0.0723401i	$0.0230467\pi$
$389$	431800.i	0.144680i	−0.997380	+	0.0723401i	$0.976953\pi$
$390$	0	0
$391$	−3.00732e6	−0.994805
$392$	0	0
$393$	− 237665.i	− 0.0776220i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.28465e6	0.727518	0.363759	−	0.931493i	$-0.381493\pi$
$397$	2.28465e6	0.727518	0.363759	+	0.931493i	$0.381493\pi$
$398$	0	0
$399$	13834.3	0.00435036
$400$	0	0
$401$	1.63473e6	0.507674	0.253837	−	0.967247i	$-0.418307\pi$
$401$	1.63473e6	0.507674	0.253837	+	0.967247i	$0.418307\pi$
$402$	0	0
$403$	−602621.	−0.184834
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 25988.3i	− 0.00777663i
$408$	0	0
$409$	−5.21834e6	−1.54250	−0.771248	−	0.636534i	$-0.780367\pi$
$409$	−5.21834e6	−1.54250	−0.771248	+	0.636534i	$0.780367\pi$
$410$	0	0
$411$	− 3.45955e6i	− 1.01022i
$412$	0	0
$413$	−52800.0	−0.0152321
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 810644.i	− 0.228292i
$418$	0	0
$419$	666477.i	0.185460i	0.995691	+	0.0927299i	$0.0295593\pi$
$419$	666477.i	0.185460i	−0.995691	+	0.0927299i	$0.970441\pi$
$420$	0	0
$421$	− 6.82007e6i	− 1.87536i	−0.347504	−	0.937678i	$-0.612971\pi$
$421$	− 6.82007e6i	− 1.87536i	0.347504	−	0.937678i	$-0.387029\pi$
$422$	0	0
$423$	2.08073e6i	0.565413i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	75442.2	0.0200237
$428$	0	0
$429$	− 819419.i	− 0.214963i
$430$	0	0
$431$	1.38707e6	0.359671	0.179836	−	0.983697i	$-0.442443\pi$
$431$	1.38707e6	0.359671	0.179836	+	0.983697i	$0.442443\pi$
$432$	0	0
$433$	− 1.33169e6i	− 0.341336i	−0.985329	−	0.170668i	$-0.945407\pi$
$433$	− 1.33169e6i	− 0.341336i	0.985329	−	0.170668i	$-0.0545926\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.18797e6	−0.297577
$438$	0	0
$439$	−2.78241e6	−0.689064	−0.344532	−	0.938775i	$-0.611962\pi$
$439$	−2.78241e6	−0.689064	−0.344532	+	0.938775i	$0.611962\pi$
$440$	0	0
$441$	1.53456e6	0.375741
$442$	0	0
$443$	−4.63595e6	−1.12235	−0.561177	−	0.827696i	$-0.689651\pi$
$443$	−4.63595e6	−1.12235	−0.561177	+	0.827696i	$0.689651\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 6.44063e6i	− 1.52461i
$448$	0	0
$449$	−1.18197e6	−0.276688	−0.138344	−	0.990384i	$-0.544178\pi$
$449$	−1.18197e6	−0.276688	−0.138344	+	0.990384i	$0.544178\pi$
$450$	0	0
$451$	6.82811e6i	1.58074i
$452$	0	0
$453$	8.16001e6	1.86829
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.55240e6i	0.347708i	0.984771	+	0.173854i	$0.0556220\pi$
$457$	1.55240e6i	0.347708i	−0.984771	+	0.173854i	$0.944378\pi$
$458$	0	0
$459$	2.35472e6i	0.521684i
$460$	0	0
$461$	7.43437e6i	1.62926i	0.579978	+	0.814632i	$0.303061\pi$
$461$	7.43437e6i	1.62926i	−0.579978	+	0.814632i	$0.696939\pi$
$462$	0	0
$463$	20179.5i	0.00437480i	0.999998	+	0.00218740i	$0.000696271\pi$
$463$	20179.5i	0.00437480i	−0.999998	+	0.00218740i	$0.999304\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.58698e6	−0.761092	−0.380546	−	0.924762i	$-0.624264\pi$
$467$	−3.58698e6	−0.761092	−0.380546	+	0.924762i	$0.624264\pi$
$468$	0	0
$469$	149245.i	0.0313306i
$470$	0	0
$471$	7.00885e6	1.45578
$472$	0	0
$473$	− 1.01343e6i	− 0.208277i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.24829e6	−0.251200
$478$	0	0
$479$	183482.	0.0365388	0.0182694	−	0.999833i	$-0.494184\pi$
$479$	183482.	0.0365388	0.0182694	+	0.999833i	$0.494184\pi$
$480$	0	0
$481$	−6625.61	−0.00130576
$482$	0	0
$483$	146082.	0.0284925
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 6.68968e6i	− 1.27815i	−0.769143	−	0.639077i	$-0.779317\pi$
$487$	− 6.68968e6i	− 1.27815i	0.769143	−	0.639077i	$-0.220683\pi$
$488$	0	0
$489$	7.40899e6	1.40116
$490$	0	0
$491$	9.90038e6i	1.85331i	0.375913	+	0.926655i	$0.377329\pi$
$491$	9.90038e6i	1.85331i	−0.375913	+	0.926655i	$0.622671\pi$
$492$	0	0
$493$	4.42259e6	0.819520
$494$	0	0
$495$	0	0
$496$	0	0
$497$	115993.i	0.0210639i
$498$	0	0
$499$	5.95907e6i	1.07134i	0.844428	+	0.535670i	$0.179941\pi$
$499$	5.95907e6i	1.07134i	−0.844428	+	0.535670i	$0.820059\pi$
$500$	0	0
$501$	− 3.52973e6i	− 0.628271i
$502$	0	0
$503$	− 3.69353e6i	− 0.650910i	−0.945557	−	0.325455i	$-0.894482\pi$
$503$	− 3.69353e6i	− 0.650910i	0.945557	−	0.325455i	$-0.105518\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.58010e6	1.13687
$508$	0	0
$509$	− 7.94222e6i	− 1.35877i	−0.733780	−	0.679387i	$-0.762246\pi$
$509$	− 7.94222e6i	− 1.35877i	0.733780	−	0.679387i	$-0.237754\pi$
$510$	0	0
$511$	47982.9	0.00812894
$512$	0	0
$513$	930172.i	0.156052i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9.55157e6	−1.57162
$518$	0	0
$519$	−1.09598e7	−1.78602
$520$	0	0
$521$	−1.06312e7	−1.71588	−0.857940	−	0.513749i	$-0.828256\pi$
$521$	−1.06312e7	−1.71588	−0.857940	+	0.513749i	$0.828256\pi$
$522$	0	0
$523$	3.13913e6	0.501828	0.250914	−	0.968009i	$-0.419269\pi$
$523$	3.13913e6	0.501828	0.250914	+	0.968009i	$0.419269\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.78708e6i	− 0.750834i
$528$	0	0
$529$	−6.10788e6	−0.948967
$530$	0	0
$531$	2.13783e6i	0.329032i
$532$	0	0
$533$	1.74080e6	0.265418
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 7.70298e6i	− 1.15272i
$538$	0	0
$539$	7.04438e6i	1.04441i
$540$	0	0
$541$	1.20044e7i	1.76339i	0.471821	+	0.881694i	$0.343597\pi$
$541$	1.20044e7i	1.76339i	−0.471821	+	0.881694i	$0.656403\pi$
$542$	0	0
$543$	3.35273e6i	0.487976i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.93035e6	1.41905	0.709523	−	0.704682i	$-0.248910\pi$
$547$	9.93035e6	1.41905	0.709523	+	0.704682i	$0.248910\pi$
$548$	0	0
$549$	− 3.05460e6i	− 0.432537i
$550$	0	0
$551$	1.74703e6	0.245144
$552$	0	0
$553$	86662.2i	0.0120508i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.72764e6	1.19195	0.595976	−	0.803002i	$-0.296765\pi$
$557$	8.72764e6	1.19195	0.595976	+	0.803002i	$0.296765\pi$
$558$	0	0
$559$	−258370.	−0.0349714
$560$	0	0
$561$	6.50926e6	0.873222
$562$	0	0
$563$	2.13884e6	0.284386	0.142193	−	0.989839i	$-0.454585\pi$
$563$	2.13884e6	0.284386	0.142193	+	0.989839i	$0.454585\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 164445.i	− 0.0214814i
$568$	0	0
$569$	−2.98366e6	−0.386339	−0.193169	−	0.981165i	$-0.561877\pi$
$569$	−2.98366e6	−0.386339	−0.193169	+	0.981165i	$0.561877\pi$
$570$	0	0
$571$	8.50264e6i	1.09135i	0.837997	+	0.545674i	$0.183726\pi$
$571$	8.50264e6i	1.09135i	−0.837997	+	0.545674i	$0.816274\pi$
$572$	0	0
$573$	1.79654e7	2.28586
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.55926e7i	− 1.94975i	−0.222756	−	0.974874i	$-0.571505\pi$
$577$	− 1.55926e7i	− 1.94975i	0.222756	−	0.974874i	$-0.428495\pi$
$578$	0	0
$579$	− 3.01658e6i	− 0.373954i
$580$	0	0
$581$	− 210159.i	− 0.0258289i
$582$	0	0
$583$	− 5.73025e6i	− 0.698236i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.01432e7	−1.21501	−0.607507	−	0.794314i	$-0.707830\pi$
$587$	−1.01432e7	−1.21501	−0.607507	+	0.794314i	$0.707830\pi$
$588$	0	0
$589$	− 1.89101e6i	− 0.224598i
$590$	0	0
$591$	9.26315e6	1.09091
$592$	0	0
$593$	− 1.44098e7i	− 1.68276i	−0.540442	−	0.841381i	$-0.681743\pi$
$593$	− 1.44098e7i	− 1.68276i	0.540442	−	0.841381i	$-0.318257\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.69706e6	−0.998704
$598$	0	0
$599$	−3.13415e6	−0.356905	−0.178452	−	0.983949i	$-0.557109\pi$
$599$	−3.13415e6	−0.356905	−0.178452	+	0.983949i	$0.557109\pi$
$600$	0	0
$601$	1.57244e7	1.77578	0.887889	−	0.460058i	$-0.152171\pi$
$601$	1.57244e7	1.77578	0.887889	+	0.460058i	$0.152171\pi$
$602$	0	0
$603$	6.04283e6	0.676780
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.03887e7i	1.14443i	0.820102	+	0.572217i	$0.193917\pi$
$607$	1.03887e7i	1.14443i	−0.820102	+	0.572217i	$0.806083\pi$
$608$	0	0
$609$	−214830.	−0.0234721
$610$	0	0
$611$	2.43513e6i	0.263888i
$612$	0	0
$613$	−1.46355e7	−1.57310	−0.786551	−	0.617525i	$-0.788135\pi$
$613$	−1.46355e7	−1.57310	−0.786551	+	0.617525i	$0.788135\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.06227e6i	− 0.641095i	−0.947233	−	0.320547i	$-0.896133\pi$
$617$	− 6.06227e6i	− 0.641095i	0.947233	−	0.320547i	$-0.103867\pi$
$618$	0	0
$619$	888836.i	0.0932384i	0.998913	+	0.0466192i	$0.0148447\pi$
$619$	888836.i	0.0932384i	−0.998913	+	0.0466192i	$0.985155\pi$
$620$	0	0
$621$	9.82207e6i	1.02205i
$622$	0	0
$623$	136873.i	0.0141285i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.57132e6	0.261208
$628$	0	0
$629$	− 52632.2i	− 0.00530426i
$630$	0	0
$631$	−5.86047e6	−0.585948	−0.292974	−	0.956120i	$-0.594645\pi$
$631$	−5.86047e6	−0.585948	−0.292974	+	0.956120i	$0.594645\pi$
$632$	0	0
$633$	4.76256e6i	0.472423i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.79594e6	0.175365
$638$	0	0
$639$	4.69645e6	0.455007
$640$	0	0
$641$	6.88679e6	0.662021	0.331011	−	0.943627i	$-0.392610\pi$
$641$	6.88679e6	0.662021	0.331011	+	0.943627i	$0.392610\pi$
$642$	0	0
$643$	−7.66403e6	−0.731021	−0.365510	−	0.930807i	$-0.619106\pi$
$643$	−7.66403e6	−0.731021	−0.365510	+	0.930807i	$0.619106\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.93185e6i	− 0.369263i	−0.982808	−	0.184631i	$-0.940891\pi$
$647$	− 3.93185e6i	− 0.369263i	0.982808	−	0.184631i	$-0.0591092\pi$
$648$	0	0
$649$	−9.81369e6	−0.914577
$650$	0	0
$651$	232535.i	0.0215048i
$652$	0	0
$653$	5.55582e6	0.509876	0.254938	−	0.966957i	$-0.417945\pi$
$653$	5.55582e6	0.509876	0.254938	+	0.966957i	$0.417945\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.94279e6i	− 0.175595i
$658$	0	0
$659$	2.17299e6i	0.194914i	0.995240	+	0.0974572i	$0.0310709\pi$
$659$	2.17299e6i	0.194914i	−0.995240	+	0.0974572i	$0.968929\pi$
$660$	0	0
$661$	− 6.93458e6i	− 0.617329i	−0.951171	−	0.308665i	$-0.900118\pi$
$661$	− 6.93458e6i	− 0.617329i	0.951171	−	0.308665i	$-0.0998821\pi$
$662$	0	0
$663$	− 1.65951e6i	− 0.146621i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.84476e7	1.60556
$668$	0	0
$669$	− 1.14130e7i	− 0.985902i
$670$	0	0
$671$	1.40221e7	1.20228
$672$	0	0
$673$	− 7.72402e6i	− 0.657364i	−0.944441	−	0.328682i	$-0.893396\pi$
$673$	− 7.72402e6i	− 0.657364i	0.944441	−	0.328682i	$-0.106604\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.38586e7	1.16211	0.581054	−	0.813865i	$-0.302640\pi$
$677$	1.38586e7	1.16211	0.581054	+	0.813865i	$0.302640\pi$
$678$	0	0
$679$	−355113.	−0.0295592
$680$	0	0
$681$	1.19951e7	0.991139
$682$	0	0
$683$	778871.	0.0638872	0.0319436	−	0.999490i	$-0.489830\pi$
$683$	778871.	0.0638872	0.0319436	+	0.999490i	$0.489830\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.47700e7i	− 1.19396i
$688$	0	0
$689$	−1.46090e6	−0.117239
$690$	0	0
$691$	6.98899e6i	0.556826i	0.960461	+	0.278413i	$0.0898084\pi$
$691$	6.98899e6i	0.556826i	−0.960461	+	0.278413i	$0.910192\pi$
$692$	0	0
$693$	−86376.8	−0.00683225
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.38285e7i	1.07818i
$698$	0	0
$699$	− 2.38529e7i	− 1.84649i
$700$	0	0
$701$	4.14587e6i	0.318655i	0.987226	+	0.159328i	$0.0509326\pi$
$701$	4.14587e6i	0.318655i	−0.987226	+	0.159328i	$0.949067\pi$
$702$	0	0
$703$	− 20791.0i	− 0.00158667i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	280855.	0.0211317
$708$	0	0
$709$	1.52511e7i	1.13943i	0.821843	+	0.569714i	$0.192946\pi$
$709$	1.52511e7i	1.13943i	−0.821843	+	0.569714i	$0.807054\pi$
$710$	0	0
$711$	3.50889e6	0.260313
$712$	0	0
$713$	− 1.99680e7i	− 1.47099i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.22820e7	1.61866
$718$	0	0
$719$	−1.60151e6	−0.115534	−0.0577668	−	0.998330i	$-0.518398\pi$
$719$	−1.60151e6	−0.115534	−0.0577668	+	0.998330i	$0.518398\pi$
$720$	0	0
$721$	23338.6	0.00167200
$722$	0	0
$723$	1.79845e7	1.27954
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.15453e6i	0.431876i	0.976407	+	0.215938i	$0.0692809\pi$
$727$	6.15453e6i	0.431876i	−0.976407	+	0.215938i	$0.930719\pi$
$728$	0	0
$729$	5.66362e6	0.394708
$730$	0	0
$731$	− 2.05243e6i	− 0.142061i
$732$	0	0
$733$	7.15476e6	0.491853	0.245926	−	0.969289i	$-0.420908\pi$
$733$	7.15476e6	0.491853	0.245926	+	0.969289i	$0.420908\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.77395e7i	1.88118i
$738$	0	0
$739$	− 2.74026e7i	− 1.84578i	−0.385063	−	0.922890i	$-0.625820\pi$
$739$	− 2.74026e7i	− 1.84578i	0.385063	−	0.922890i	$-0.374180\pi$
$740$	0	0
$741$	− 655547.i	− 0.0438589i
$742$	0	0
$743$	− 1.20016e7i	− 0.797569i	−0.917045	−	0.398785i	$-0.869432\pi$
$743$	− 1.20016e7i	− 0.797569i	0.917045	−	0.398785i	$-0.130568\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.50916e6	−0.557937
$748$	0	0
$749$	159629.i	0.0103970i
$750$	0	0
$751$	−3.38961e6	−0.219305	−0.109653	−	0.993970i	$-0.534974\pi$
$751$	−3.38961e6	−0.219305	−0.109653	+	0.993970i	$0.534974\pi$
$752$	0	0
$753$	2.93148e7i	1.88408i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.47336e6	0.220298	0.110149	−	0.993915i	$-0.464867\pi$
$757$	3.47336e6	0.220298	0.110149	+	0.993915i	$0.464867\pi$
$758$	0	0
$759$	2.71516e7	1.71077
$760$	0	0
$761$	−1.16257e6	−0.0727710	−0.0363855	−	0.999338i	$-0.511584\pi$
$761$	−1.16257e6	−0.0727710	−0.0363855	+	0.999338i	$0.511584\pi$
$762$	0	0
$763$	289755.	0.0180186
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.50196e6i	0.153565i
$768$	0	0
$769$	1.71387e7	1.04511	0.522556	−	0.852605i	$-0.324979\pi$
$769$	1.71387e7	1.04511	0.522556	+	0.852605i	$0.324979\pi$
$770$	0	0
$771$	− 1.96944e7i	− 1.19318i
$772$	0	0
$773$	−3.17392e7	−1.91050	−0.955250	−	0.295800i	$-0.904414\pi$
$773$	−3.17392e7	−1.91050	−0.955250	+	0.295800i	$0.904414\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2556.64i	0 0.000151921i
$778$	0	0
$779$	5.46258e6i	0.322518i
$780$	0	0
$781$	2.15590e7i	1.26474i
$782$	0	0
$783$	− 1.44444e7i	− 0.841968i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.67032e6	−0.153683	−0.0768416	−	0.997043i	$-0.524484\pi$
$787$	−2.67032e6	−0.153683	−0.0768416	+	0.997043i	$0.524484\pi$
$788$	0	0
$789$	1.48635e7i	0.850018i
$790$	0	0
$791$	−259850.	−0.0147666
$792$	0	0
$793$	− 3.57487e6i	− 0.201873i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.37393e7	1.88144	0.940719	−	0.339188i	$-0.110152\pi$
$797$	3.37393e7	1.88144	0.940719	+	0.339188i	$0.110152\pi$
$798$	0	0
$799$	−1.93441e7	−1.07197
$800$	0	0
$801$	5.54189e6	0.305194
$802$	0	0
$803$	8.91835e6	0.488085
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.08369e6i	0.112629i
$808$	0	0
$809$	−1.25443e6	−0.0673870	−0.0336935	−	0.999432i	$-0.510727\pi$
$809$	−1.25443e6	−0.0673870	−0.0336935	+	0.999432i	$0.510727\pi$
$810$	0	0
$811$	− 1.12827e7i	− 0.602365i	−0.953567	−	0.301182i	$-0.902619\pi$
$811$	− 1.12827e7i	− 0.602365i	0.953567	−	0.301182i	$-0.0973813\pi$
$812$	0	0
$813$	6.81525e6	0.361622
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 810760.i	− 0.0424949i
$818$	0	0
$819$	22021.4i	0.00114719i
$820$	0	0
$821$	− 1.21845e7i	− 0.630886i	−0.948945	−	0.315443i	$-0.897847\pi$
$821$	− 1.21845e7i	− 0.630886i	0.948945	−	0.315443i	$-0.102153\pi$
$822$	0	0
$823$	− 2.12650e7i	− 1.09437i	−0.837010	−	0.547187i	$-0.815698\pi$
$823$	− 2.12650e7i	− 1.09437i	0.837010	−	0.547187i	$-0.184302\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.06316e6	0.308273	0.154136	−	0.988050i	$-0.450740\pi$
$827$	6.06316e6	0.308273	0.154136	+	0.988050i	$0.450740\pi$
$828$	0	0
$829$	− 727353.i	− 0.0367586i	−0.999831	−	0.0183793i	$-0.994149\pi$
$829$	− 727353.i	− 0.0367586i	0.999831	−	0.0183793i	$-0.00585064\pi$
$830$	0	0
$831$	1.67171e7	0.839765
$832$	0	0
$833$	1.42665e7i	0.712368i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.56348e7	−0.771401
$838$	0	0
$839$	−1.36723e7	−0.670560	−0.335280	−	0.942118i	$-0.608831\pi$
$839$	−1.36723e7	−0.670560	−0.335280	+	0.942118i	$0.608831\pi$
$840$	0	0
$841$	−6.61809e6	−0.322658
$842$	0	0
$843$	9.34688e6	0.452999
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 33223.9i	− 0.00159126i
$848$	0	0
$849$	−9.74720e6	−0.464099
$850$	0	0
$851$	− 219541.i	− 0.0103918i
$852$	0	0
$853$	−3.66971e7	−1.72687	−0.863434	−	0.504462i	$-0.831691\pi$
$853$	−3.66971e7	−1.72687	−0.863434	+	0.504462i	$0.831691\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.73498e7i	− 0.806941i	−0.914993	−	0.403470i	$-0.867804\pi$
$857$	− 1.73498e7i	− 0.806941i	0.914993	−	0.403470i	$-0.132196\pi$
$858$	0	0
$859$	− 1.90085e7i	− 0.878954i	−0.898254	−	0.439477i	$-0.855164\pi$
$859$	− 1.90085e7i	− 0.878954i	0.898254	−	0.439477i	$-0.144836\pi$
$860$	0	0
$861$	− 671726.i	− 0.0308805i
$862$	0	0
$863$	3.46920e7i	1.58563i	0.609460	+	0.792816i	$0.291386\pi$
$863$	3.46920e7i	1.58563i	−0.609460	+	0.792816i	$0.708614\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.27790e7	−0.577364
$868$	0	0
$869$	1.61075e7i	0.723566i
$870$	0	0
$871$	7.07207e6	0.315865
$872$	0	0
$873$	1.43783e7i	0.638515i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.99003e7	−1.75177	−0.875885	−	0.482520i	$-0.839721\pi$
$877$	−3.99003e7	−1.75177	−0.875885	+	0.482520i	$0.839721\pi$
$878$	0	0
$879$	4.65618e7	2.03262
$880$	0	0
$881$	−3.50875e7	−1.52304	−0.761522	−	0.648139i	$-0.775548\pi$
$881$	−3.50875e7	−1.52304	−0.761522	+	0.648139i	$0.775548\pi$
$882$	0	0
$883$	−1.47881e7	−0.638279	−0.319139	−	0.947708i	$-0.603394\pi$
$883$	−1.47881e7	−0.638279	−0.319139	+	0.947708i	$0.603394\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.17291e7i	0.500560i	0.968173	+	0.250280i	$0.0805228\pi$
$887$	1.17291e7i	0.500560i	−0.968173	+	0.250280i	$0.919477\pi$
$888$	0	0
$889$	329394.	0.0139785
$890$	0	0
$891$	− 3.05646e7i	− 1.28981i
$892$	0	0
$893$	−7.64139e6	−0.320659
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 6.92219e6i	− 0.287252i
$898$	0	0
$899$	2.93651e7i	1.21180i
$900$	0	0
$901$	− 1.16051e7i	− 0.476251i
$902$	0	0
$903$	99697.9i	0.00406880i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.36136e7	−0.549483	−0.274742	−	0.961518i	$-0.588592\pi$
$907$	−1.36136e7	−0.549483	−0.274742	+	0.961518i	$0.588592\pi$
$908$	0	0
$909$	− 1.13716e7i	− 0.456470i
$910$	0	0
$911$	−1.26888e7	−0.506551	−0.253276	−	0.967394i	$-0.581508\pi$
$911$	−1.26888e7	−0.506551	−0.253276	+	0.967394i	$0.581508\pi$
$912$	0	0
$913$	− 3.90611e7i	− 1.55084i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−29319.9	−0.00115144
$918$	0	0
$919$	−4.25253e7	−1.66096	−0.830479	−	0.557050i	$-0.811933\pi$
$919$	−4.25253e7	−1.66096	−0.830479	+	0.557050i	$0.811933\pi$
$920$	0	0
$921$	−1.35430e7	−0.526098
$922$	0	0
$923$	5.49637e6	0.212360
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 944963.i	− 0.0361173i
$928$	0	0
$929$	−4.25450e6	−0.161737	−0.0808686	−	0.996725i	$-0.525769\pi$
$929$	−4.25450e6	−0.161737	−0.0808686	+	0.996725i	$0.525769\pi$
$930$	0	0
$931$	5.63560e6i	0.213092i
$932$	0	0
$933$	1.99078e7	0.748718
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.07844e7i	0.401279i	0.979665	+	0.200640i	$0.0643020\pi$
$937$	1.07844e7i	0.401279i	−0.979665	+	0.200640i	$0.935698\pi$
$938$	0	0
$939$	1.97982e7i	0.732760i
$940$	0	0
$941$	112960.i	0.00415862i	0.999998	+	0.00207931i	$0.000661865\pi$
$941$	112960.i	0.00415862i	−0.999998	+	0.00207931i	$0.999338\pi$
$942$	0	0
$943$	5.76817e7i	2.11232i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.62952e7	−0.590454	−0.295227	−	0.955427i	$-0.595395\pi$
$947$	−1.62952e7	−0.590454	−0.295227	+	0.955427i	$0.595395\pi$
$948$	0	0
$949$	− 2.27370e6i	− 0.0819534i
$950$	0	0
$951$	−5.36188e7	−1.92250
$952$	0	0
$953$	1.07203e7i	0.382363i	0.981555	+	0.191181i	$0.0612318\pi$
$953$	1.07203e7i	0.382363i	−0.981555	+	0.191181i	$0.938768\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.99294e7	−1.40933
$958$	0	0
$959$	−426792.	−0.0149855
$960$	0	0
$961$	3.15601e6	0.110238
$962$	0	0
$963$	6.46324e6	0.224587
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.92793e7i	1.35082i	0.737442	+	0.675411i	$0.236034\pi$
$967$	3.92793e7i	1.35082i	−0.737442	+	0.675411i	$0.763966\pi$
$968$	0	0
$969$	5.20750e6	0.178164
$970$	0	0
$971$	2.76288e7i	0.940405i	0.882559	+	0.470202i	$0.155819\pi$
$971$	2.76288e7i	0.940405i	−0.882559	+	0.470202i	$0.844181\pi$
$972$	0	0
$973$	−100006.	−0.00338645
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.56548e7i	− 0.859868i	−0.902860	−	0.429934i	$-0.858537\pi$
$977$	− 2.56548e7i	− 0.859868i	0.902860	−	0.429934i	$-0.141463\pi$
$978$	0	0
$979$	2.54399e7i	0.848318i
$980$	0	0
$981$	− 1.17320e7i	− 0.389223i
$982$	0	0
$983$	− 5.21860e6i	− 0.172254i	−0.996284	−	0.0861271i	$-0.972551\pi$
$983$	− 5.21860e6i	− 0.172254i	0.996284	−	0.0861271i	$-0.0274491\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	939650.	0.0307025
$988$	0	0
$989$	− 8.56115e6i	− 0.278318i
$990$	0	0
$991$	−4.76772e7	−1.54215	−0.771075	−	0.636744i	$-0.780281\pi$
$991$	−4.76772e7	−1.54215	−0.771075	+	0.636744i	$0.780281\pi$
$992$	0	0
$993$	6.53158e6i	0.210206i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.96368e7	−0.944263	−0.472132	−	0.881528i	$-0.656515\pi$
$997$	−2.96368e7	−0.944263	−0.472132	+	0.881528i	$0.656515\pi$
$998$	0	0
$999$	−171900.	−0.00544955

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.f.d.49.10		40
4.3	odd	2		200.6.f.d.149.11		40
5.2	odd	4		800.6.d.b.401.16		20
5.3	odd	4		800.6.d.d.401.5		20
5.4	even	2	inner	800.6.f.d.49.31		40
8.3	odd	2		200.6.f.d.149.29		40
8.5	even	2	inner	800.6.f.d.49.32		40
20.3	even	4		200.6.d.c.101.6	yes	20
20.7	even	4		200.6.d.d.101.15	yes	20
20.19	odd	2		200.6.f.d.149.30		40
40.3	even	4		200.6.d.c.101.5	✓	20
40.13	odd	4		800.6.d.d.401.16		20
40.19	odd	2		200.6.f.d.149.12		40
40.27	even	4		200.6.d.d.101.16	yes	20
40.29	even	2	inner	800.6.f.d.49.9		40
40.37	odd	4		800.6.d.b.401.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.6.d.c.101.5	✓	20	40.3	even	4
200.6.d.c.101.6	yes	20	20.3	even	4
200.6.d.d.101.15	yes	20	20.7	even	4
200.6.d.d.101.16	yes	20	40.27	even	4
200.6.f.d.149.11		40	4.3	odd	2
200.6.f.d.149.12		40	40.19	odd	2
200.6.f.d.149.29		40	8.3	odd	2
200.6.f.d.149.30		40	20.19	odd	2
800.6.d.b.401.5		20	40.37	odd	4
800.6.d.b.401.16		20	5.2	odd	4
800.6.d.d.401.5		20	5.3	odd	4
800.6.d.d.401.16		20	40.13	odd	4
800.6.f.d.49.9		40	40.29	even	2	inner
800.6.f.d.49.10		40	1.1	even	1	trivial
800.6.f.d.49.31		40	5.4	even	2	inner
800.6.f.d.49.32		40	8.5	even	2	inner

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-18.2848 q^{3} +2.25573i q^{7} +91.3327 q^{9} +419.261i q^{11} +106.889 q^{13} +849.098i q^{17} +335.414i q^{19} -41.2454i q^{21} +3541.78i q^{23} +2773.20 q^{27} -5208.57i q^{29} -5637.83 q^{31} -7666.09i q^{33} -61.9860 q^{37} -1954.44 q^{39} +16286.1 q^{41} -2417.19 q^{43} +22781.9i q^{47} +16801.9 q^{49} -15525.6i q^{51} -13667.5 q^{53} -6132.98i q^{57} +23407.1i q^{59} -33444.7i q^{61} +206.022i q^{63} +66162.9 q^{67} -64760.7i q^{69} +51421.4 q^{71} -21271.6i q^{73} -945.738 q^{77} +38418.7 q^{79} -72901.2 q^{81} -93166.7 q^{83} +95237.5i q^{87} +60678.0 q^{89} +241.112i q^{91} +103086. q^{93} +157428. i q^{97} +38292.2i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 3240 q^{9} - 14320 q^{31} + 44904 q^{39} - 11608 q^{41} - 125304 q^{49} + 15448 q^{71} - 15560 q^{79} + 193968 q^{81} + 6320 q^{89}+O(q^{100}) \) 40 * q + 3240 * q^9 - 14320 * q^31 + 44904 * q^39 - 11608 * q^41 - 125304 * q^49 + 15448 * q^71 - 15560 * q^79 + 193968 * q^81 + 6320 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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