LMFDB - Embedded newform 800.6.c.h.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(449,800)]

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, 1]))

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

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.c (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:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.6.c.h.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13.4164i q^{3} -138.636i q^{7} +63.0000 q^{9} +O(q^{10})$	$q+13.4164i q^{3} -138.636i q^{7} +63.0000 q^{9} -259.384 q^{11} -154.000i q^{13} +178.000i q^{17} +965.981 q^{19} +1860.00 q^{21} +2634.09i q^{23} +4105.42i q^{27} -4110.00 q^{29} -3157.33 q^{31} -3480.00i q^{33} +7442.00i q^{37} +2066.13 q^{39} +7270.00 q^{41} -17910.9i q^{43} +7410.33i q^{47} -2413.00 q^{49} -2388.12 q^{51} -32226.0i q^{53} +12960.0i q^{57} +34041.9 q^{59} +26770.0 q^{61} -8734.08i q^{63} -49806.2i q^{67} -35340.0 q^{69} -54103.9 q^{71} +18534.0i q^{73} +35960.0i q^{77} -86741.5 q^{79} -39771.0 q^{81} -78642.5i q^{83} -55141.4i q^{87} +107590. q^{89} -21350.0 q^{91} -42360.0i q^{93} -108838. i q^{97} -16341.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 252 q^{9}+O(q^{10}) $ 4 * q + 252 * q^9	$ 4 q + 252 q^{9} + 7440 q^{21} - 16440 q^{29} + 29080 q^{41} - 9652 q^{49} + 107080 q^{61} - 141360 q^{69} - 159084 q^{81} + 430360 q^{89}+O(q^{100}) $ 4 * q + 252 * q^9 + 7440 * q^21 - 16440 * q^29 + 29080 * q^41 - 9652 * q^49 + 107080 * q^61 - 141360 * q^69 - 159084 * q^81 + 430360 * q^89

Display 100 coefficients 10 coefficients

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
$3$	13.4164i	0.860663i	0.902671	+	0.430331i	$0.141603\pi$
$3$	13.4164i	0.860663i	−0.902671	+	0.430331i	$0.858397\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 138.636i	− 1.06938i	−0.845049	−	0.534689i	$-0.820429\pi$
$7$	− 138.636i	− 1.06938i	0.845049	−	0.534689i	$-0.179571\pi$
$8$	0	0
$9$	63.0000	0.259259
$10$	0	0
$11$	−259.384	−0.646340	−0.323170	−	0.946341i	$-0.604749\pi$
$11$	−259.384	−0.646340	−0.323170	+	0.946341i	$0.604749\pi$
$12$	0	0
$13$	− 154.000i	− 0.252733i	−0.991984	−	0.126367i	$-0.959668\pi$
$13$	− 154.000i	− 0.252733i	0.991984	−	0.126367i	$-0.0403316\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	178.000i	0.149382i	0.997207	+	0.0746909i	$0.0237970\pi$
$17$	178.000i	0.149382i	−0.997207	+	0.0746909i	$0.976203\pi$
$18$	0	0
$19$	965.981	0.613882	0.306941	−	0.951729i	$-0.400695\pi$
$19$	965.981	0.613882	0.306941	+	0.951729i	$0.400695\pi$
$20$	0	0
$21$	1860.00	0.920375
$22$	0	0
$23$	2634.09i	1.03827i	0.854692	+	0.519135i	$0.173746\pi$
$23$	2634.09i	1.03827i	−0.854692	+	0.519135i	$0.826254\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4105.42i	1.08380i
$28$	0	0
$29$	−4110.00	−0.907500	−0.453750	−	0.891129i	$-0.649914\pi$
$29$	−4110.00	−0.907500	−0.453750	+	0.891129i	$0.649914\pi$
$30$	0	0
$31$	−3157.33	−0.590086	−0.295043	−	0.955484i	$-0.595334\pi$
$31$	−3157.33	−0.590086	−0.295043	+	0.955484i	$0.595334\pi$
$32$	0	0
$33$	− 3480.00i	− 0.556281i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7442.00i	0.893687i	0.894612	+	0.446843i	$0.147452\pi$
$37$	7442.00i	0.893687i	−0.894612	+	0.446843i	$0.852548\pi$
$38$	0	0
$39$	2066.13	0.217518
$40$	0	0
$41$	7270.00	0.675421	0.337711	−	0.941250i	$-0.390347\pi$
$41$	7270.00	0.675421	0.337711	+	0.941250i	$0.390347\pi$
$42$	0	0
$43$	− 17910.9i	− 1.47722i	−0.674131	−	0.738612i	$-0.735482\pi$
$43$	− 17910.9i	− 1.47722i	0.674131	−	0.738612i	$-0.264518\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7410.33i	0.489320i	0.969609	+	0.244660i	$0.0786763\pi$
$47$	7410.33i	0.489320i	−0.969609	+	0.244660i	$0.921324\pi$
$48$	0	0
$49$	−2413.00	−0.143571
$50$	0	0
$51$	−2388.12	−0.128567
$52$	0	0
$53$	− 32226.0i	− 1.57586i	−0.615767	−	0.787928i	$-0.711154\pi$
$53$	− 32226.0i	− 1.57586i	0.615767	−	0.787928i	$-0.288846\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12960.0i	0.528345i
$58$	0	0
$59$	34041.9	1.27316	0.636581	−	0.771210i	$-0.280348\pi$
$59$	34041.9	1.27316	0.636581	+	0.771210i	$0.280348\pi$
$60$	0	0
$61$	26770.0	0.921136	0.460568	−	0.887624i	$-0.347646\pi$
$61$	26770.0	0.921136	0.460568	+	0.887624i	$0.347646\pi$
$62$	0	0
$63$	− 8734.08i	− 0.277246i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 49806.2i	− 1.35549i	−0.735297	−	0.677745i	$-0.762957\pi$
$67$	− 49806.2i	− 1.35549i	0.735297	−	0.677745i	$-0.237043\pi$
$68$	0	0
$69$	−35340.0	−0.893601
$70$	0	0
$71$	−54103.9	−1.27375	−0.636873	−	0.770969i	$-0.719772\pi$
$71$	−54103.9	−1.27375	−0.636873	+	0.770969i	$0.719772\pi$
$72$	0	0
$73$	18534.0i	0.407063i	0.979068	+	0.203532i	$0.0652420\pi$
$73$	18534.0i	0.407063i	−0.979068	+	0.203532i	$0.934758\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	35960.0i	0.691183i
$78$	0	0
$79$	−86741.5	−1.56372	−0.781861	−	0.623453i	$-0.785729\pi$
$79$	−86741.5	−1.56372	−0.781861	+	0.623453i	$0.785729\pi$
$80$	0	0
$81$	−39771.0	−0.673525
$82$	0	0
$83$	− 78642.5i	− 1.25303i	−0.779409	−	0.626516i	$-0.784480\pi$
$83$	− 78642.5i	− 1.25303i	0.779409	−	0.626516i	$-0.215520\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 55141.4i	− 0.781052i
$88$	0	0
$89$	107590.	1.43978	0.719891	−	0.694087i	$-0.244192\pi$
$89$	107590.	1.43978	0.719891	+	0.694087i	$0.244192\pi$
$90$	0	0
$91$	−21350.0	−0.270268
$92$	0	0
$93$	− 42360.0i	− 0.507865i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 108838.i	− 1.17450i	−0.809407	−	0.587248i	$-0.800212\pi$
$97$	− 108838.i	− 1.17450i	0.809407	−	0.587248i	$-0.199788\pi$
$98$	0	0
$99$	−16341.2	−0.167570
$100$	0	0
$101$	59198.0	0.577436	0.288718	−	0.957414i	$-0.406771\pi$
$101$	59198.0	0.577436	0.288718	+	0.957414i	$0.406771\pi$
$102$	0	0
$103$	− 112908.i	− 1.04865i	−0.851517	−	0.524326i	$-0.824317\pi$
$103$	− 112908.i	− 1.04865i	0.851517	−	0.524326i	$-0.175683\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 40039.0i	− 0.338084i	−0.985609	−	0.169042i	$-0.945933\pi$
$107$	− 40039.0i	− 0.338084i	0.985609	−	0.169042i	$-0.0540673\pi$
$108$	0	0
$109$	139614.	1.12554	0.562772	−	0.826612i	$-0.309735\pi$
$109$	139614.	1.12554	0.562772	+	0.826612i	$0.309735\pi$
$110$	0	0
$111$	−99844.9	−0.769163
$112$	0	0
$113$	43046.0i	0.317130i	0.987349	+	0.158565i	$0.0506867\pi$
$113$	43046.0i	0.317130i	−0.987349	+	0.158565i	$0.949313\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 9702.00i	− 0.0655234i
$118$	0	0
$119$	24677.2	0.159746
$120$	0	0
$121$	−93771.0	−0.582244
$122$	0	0
$123$	97537.3i	0.581310i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	157370.i	0.865790i	0.901444	+	0.432895i	$0.142508\pi$
$127$	157370.i	0.865790i	−0.901444	+	0.432895i	$0.857492\pi$
$128$	0	0
$129$	240300.	1.27139
$130$	0	0
$131$	267729.	1.36307	0.681533	−	0.731787i	$-0.261314\pi$
$131$	267729.	1.36307	0.681533	+	0.731787i	$0.261314\pi$
$132$	0	0
$133$	− 133920.i	− 0.656472i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 112158.i	− 0.510539i	−0.966870	−	0.255269i	$-0.917836\pi$
$137$	− 112158.i	− 0.510539i	0.966870	−	0.255269i	$-0.0821642\pi$
$138$	0	0
$139$	147348.	0.646855	0.323428	−	0.946253i	$-0.395165\pi$
$139$	147348.	0.646855	0.323428	+	0.946253i	$0.395165\pi$
$140$	0	0
$141$	−99420.0	−0.421139
$142$	0	0
$143$	39945.1i	0.163352i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 32373.8i	− 0.123566i
$148$	0	0
$149$	174566.	0.644160	0.322080	−	0.946712i	$-0.395618\pi$
$149$	174566.	0.644160	0.322080	+	0.946712i	$0.395618\pi$
$150$	0	0
$151$	−345258.	−1.23226	−0.616128	−	0.787646i	$-0.711300\pi$
$151$	−345258.	−1.23226	−0.616128	+	0.787646i	$0.711300\pi$
$152$	0	0
$153$	11214.0i	0.0387286i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 26502.0i	− 0.0858083i	−0.999079	−	0.0429042i	$-0.986339\pi$
$157$	− 26502.0i	− 0.0858083i	0.999079	−	0.0429042i	$-0.0136610\pi$
$158$	0	0
$159$	432357.	1.35628
$160$	0	0
$161$	365180.	1.11030
$162$	0	0
$163$	141709.i	0.417760i	0.977941	+	0.208880i	$0.0669818\pi$
$163$	141709.i	0.417760i	−0.977941	+	0.208880i	$0.933018\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	411047.i	1.14051i	0.821466	+	0.570257i	$0.193156\pi$
$167$	411047.i	1.14051i	−0.821466	+	0.570257i	$0.806844\pi$
$168$	0	0
$169$	347577.	0.936126
$170$	0	0
$171$	60856.8	0.159155
$172$	0	0
$173$	− 595946.i	− 1.51388i	−0.653484	−	0.756940i	$-0.726693\pi$
$173$	− 595946.i	− 1.51388i	0.653484	−	0.756940i	$-0.273307\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	456720.i	1.09576i
$178$	0	0
$179$	−300939.	−0.702014	−0.351007	−	0.936373i	$-0.614161\pi$
$179$	−300939.	−0.702014	−0.351007	+	0.936373i	$0.614161\pi$
$180$	0	0
$181$	217022.	0.492388	0.246194	−	0.969221i	$-0.420820\pi$
$181$	217022.	0.492388	0.246194	+	0.969221i	$0.420820\pi$
$182$	0	0
$183$	359157.i	0.792788i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 46170.3i	− 0.0965515i
$188$	0	0
$189$	569160.	1.15899
$190$	0	0
$191$	916260.	1.81734	0.908668	−	0.417519i	$-0.137100\pi$
$191$	916260.	1.81734	0.908668	+	0.417519i	$0.137100\pi$
$192$	0	0
$193$	− 864114.i	− 1.66985i	−0.550363	−	0.834926i	$-0.685511\pi$
$193$	− 864114.i	− 1.66985i	0.550363	−	0.834926i	$-0.314489\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	432522.i	0.794040i	0.917810	+	0.397020i	$0.129956\pi$
$197$	432522.i	0.794040i	−0.917810	+	0.397020i	$0.870044\pi$
$198$	0	0
$199$	−795182.	−1.42342	−0.711711	−	0.702473i	$-0.752079\pi$
$199$	−795182.	−1.42342	−0.711711	+	0.702473i	$0.752079\pi$
$200$	0	0
$201$	668220.	1.16662
$202$	0	0
$203$	569795.i	0.970462i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	165948.i	0.269181i
$208$	0	0
$209$	−250560.	−0.396777
$210$	0	0
$211$	453126.	0.700669	0.350334	−	0.936625i	$-0.386068\pi$
$211$	453126.	0.700669	0.350334	+	0.936625i	$0.386068\pi$
$212$	0	0
$213$	− 725880.i	− 1.09627i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	437720.i	0.631026i
$218$	0	0
$219$	−248660.	−0.350344
$220$	0	0
$221$	27412.0	0.0377537
$222$	0	0
$223$	− 430358.i	− 0.579519i	−0.957099	−	0.289760i	$-0.906425\pi$
$223$	− 430358.i	− 0.579519i	0.957099	−	0.289760i	$-0.0935754\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.12403e6i	− 1.44782i	−0.689896	−	0.723909i	$-0.742344\pi$
$227$	− 1.12403e6i	− 1.44782i	0.689896	−	0.723909i	$-0.257656\pi$
$228$	0	0
$229$	812410.	1.02373	0.511866	−	0.859065i	$-0.328954\pi$
$229$	812410.	1.02373	0.511866	+	0.859065i	$0.328954\pi$
$230$	0	0
$231$	−482454.	−0.594875
$232$	0	0
$233$	− 846194.i	− 1.02113i	−0.859840	−	0.510564i	$-0.829437\pi$
$233$	− 846194.i	− 1.02113i	0.859840	−	0.510564i	$-0.170563\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.16376e6i	− 1.34584i
$238$	0	0
$239$	−56688.8	−0.0641952	−0.0320976	−	0.999485i	$-0.510219\pi$
$239$	−56688.8	−0.0641952	−0.0320976	+	0.999485i	$0.510219\pi$
$240$	0	0
$241$	1.27571e6	1.41485	0.707423	−	0.706790i	$-0.249857\pi$
$241$	1.27571e6	1.41485	0.707423	+	0.706790i	$0.249857\pi$
$242$	0	0
$243$	464033.i	0.504119i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 148761.i	− 0.155148i
$248$	0	0
$249$	1.05510e6	1.07844
$250$	0	0
$251$	50937.6	0.0510334	0.0255167	−	0.999674i	$-0.491877\pi$
$251$	50937.6	0.0510334	0.0255167	+	0.999674i	$0.491877\pi$
$252$	0	0
$253$	− 683240.i	− 0.671076i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.64806e6i	1.55647i	0.627976	+	0.778233i	$0.283884\pi$
$257$	1.64806e6i	1.55647i	−0.627976	+	0.778233i	$0.716116\pi$
$258$	0	0
$259$	1.03173e6	0.955690
$260$	0	0
$261$	−258930.	−0.235278
$262$	0	0
$263$	522994.i	0.466238i	0.972448	+	0.233119i	$0.0748931\pi$
$263$	522994.i	0.466238i	−0.972448	+	0.233119i	$0.925107\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.44347e6i	1.23917i
$268$	0	0
$269$	1.93789e6	1.63285	0.816427	−	0.577448i	$-0.195951\pi$
$269$	1.93789e6	1.63285	0.816427	+	0.577448i	$0.195951\pi$
$270$	0	0
$271$	−1.00132e6	−0.828228	−0.414114	−	0.910225i	$-0.635908\pi$
$271$	−1.00132e6	−0.828228	−0.414114	+	0.910225i	$0.635908\pi$
$272$	0	0
$273$	− 286440.i	− 0.232609i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 445702.i	− 0.349016i	−0.984656	−	0.174508i	$-0.944167\pi$
$277$	− 445702.i	− 0.349016i	0.984656	−	0.174508i	$-0.0558335\pi$
$278$	0	0
$279$	−198912.	−0.152985
$280$	0	0
$281$	1.24647e6	0.941708	0.470854	−	0.882211i	$-0.343946\pi$
$281$	1.24647e6	0.941708	0.470854	+	0.882211i	$0.343946\pi$
$282$	0	0
$283$	− 2.27900e6i	− 1.69153i	−0.533557	−	0.845764i	$-0.679145\pi$
$283$	− 2.27900e6i	− 1.69153i	0.533557	−	0.845764i	$-0.320855\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.00789e6i	− 0.722281i
$288$	0	0
$289$	1.38817e6	0.977685
$290$	0	0
$291$	1.46021e6	1.01084
$292$	0	0
$293$	− 2.45427e6i	− 1.67014i	−0.550140	−	0.835072i	$-0.685426\pi$
$293$	− 2.45427e6i	− 1.67014i	0.550140	−	0.835072i	$-0.314574\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.06488e6i	− 0.700502i
$298$	0	0
$299$	405650.	0.262406
$300$	0	0
$301$	−2.48310e6	−1.57971
$302$	0	0
$303$	794225.i	0.496977i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20710.5i	0.0125413i	0.999980	+	0.00627067i	$0.00199603\pi$
$307$	20710.5i	0.0125413i	−0.999980	+	0.00627067i	$0.998004\pi$
$308$	0	0
$309$	1.51482e6	0.902537
$310$	0	0
$311$	47270.5	0.0277133	0.0138567	−	0.999904i	$-0.495589\pi$
$311$	47270.5	0.0277133	0.0138567	+	0.999904i	$0.495589\pi$
$312$	0	0
$313$	2.79169e6i	1.61067i	0.592822	+	0.805333i	$0.298014\pi$
$313$	2.79169e6i	1.61067i	−0.592822	+	0.805333i	$0.701986\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 471582.i	− 0.263578i	−0.991278	−	0.131789i	$-0.957928\pi$
$317$	− 471582.i	− 0.263578i	0.991278	−	0.131789i	$-0.0420721\pi$
$318$	0	0
$319$	1.06607e6	0.586554
$320$	0	0
$321$	537180.	0.290976
$322$	0	0
$323$	171945.i	0.0917028i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.87312e6i	0.968715i
$328$	0	0
$329$	1.02734e6	0.523268
$330$	0	0
$331$	−3.09092e6	−1.55066	−0.775331	−	0.631555i	$-0.782417\pi$
$331$	−3.09092e6	−1.55066	−0.775331	+	0.631555i	$0.782417\pi$
$332$	0	0
$333$	468846.i	0.231697i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 665838.i	− 0.319370i	−0.987168	−	0.159685i	$-0.948952\pi$
$337$	− 665838.i	− 0.319370i	0.987168	−	0.159685i	$-0.0510478\pi$
$338$	0	0
$339$	−577523.	−0.272942
$340$	0	0
$341$	818960.	0.381397
$342$	0	0
$343$	− 1.99553e6i	− 0.915847i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.80908e6i	0.806556i	0.915078	+	0.403278i	$0.132129\pi$
$347$	1.80908e6i	0.806556i	−0.915078	+	0.403278i	$0.867871\pi$
$348$	0	0
$349$	2.36181e6	1.03796	0.518981	−	0.854786i	$-0.326312\pi$
$349$	2.36181e6	1.03796	0.518981	+	0.854786i	$0.326312\pi$
$350$	0	0
$351$	632235.	0.273912
$352$	0	0
$353$	− 1.14535e6i	− 0.489215i	−0.969622	−	0.244608i	$-0.921341\pi$
$353$	− 1.14535e6i	− 0.489215i	0.969622	−	0.244608i	$-0.0786591\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	331080.i	0.137487i
$358$	0	0
$359$	−767275.	−0.314206	−0.157103	−	0.987582i	$-0.550216\pi$
$359$	−767275.	−0.314206	−0.157103	+	0.987582i	$0.550216\pi$
$360$	0	0
$361$	−1.54298e6	−0.623149
$362$	0	0
$363$	− 1.25807e6i	− 0.501116i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.99537e6i	− 1.54843i	−0.632923	−	0.774215i	$-0.718145\pi$
$367$	− 3.99537e6i	− 1.54843i	0.632923	−	0.774215i	$-0.281855\pi$
$368$	0	0
$369$	458010.	0.175109
$370$	0	0
$371$	−4.46769e6	−1.68519
$372$	0	0
$373$	− 4.68131e6i	− 1.74219i	−0.491117	−	0.871094i	$-0.663411\pi$
$373$	− 4.68131e6i	− 1.74219i	0.491117	−	0.871094i	$-0.336589\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	632940.i	0.229356i
$378$	0	0
$379$	337271.	0.120609	0.0603046	−	0.998180i	$-0.480793\pi$
$379$	337271.	0.120609	0.0603046	+	0.998180i	$0.480793\pi$
$380$	0	0
$381$	−2.11134e6	−0.745153
$382$	0	0
$383$	− 598689.i	− 0.208547i	−0.994549	−	0.104274i	$-0.966748\pi$
$383$	− 598689.i	− 0.208547i	0.994549	−	0.104274i	$-0.0332518\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.12839e6i	− 0.382984i
$388$	0	0
$389$	−3.97243e6	−1.33101	−0.665506	−	0.746393i	$-0.731784\pi$
$389$	−3.97243e6	−1.33101	−0.665506	+	0.746393i	$0.731784\pi$
$390$	0	0
$391$	−468868.	−0.155099
$392$	0	0
$393$	3.59196e6i	1.17314i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 179398.i	− 0.0571270i	−0.999592	−	0.0285635i	$-0.990907\pi$
$397$	− 179398.i	− 0.0571270i	0.999592	−	0.0285635i	$-0.00909328\pi$
$398$	0	0
$399$	1.79673e6	0.565001
$400$	0	0
$401$	6.14504e6	1.90838	0.954188	−	0.299208i	$-0.0967224\pi$
$401$	6.14504e6	1.90838	0.954188	+	0.299208i	$0.0967224\pi$
$402$	0	0
$403$	486229.i	0.149134i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.93033e6i	− 0.577626i
$408$	0	0
$409$	2.64503e6	0.781847	0.390923	−	0.920423i	$-0.372156\pi$
$409$	2.64503e6	0.781847	0.390923	+	0.920423i	$0.372156\pi$
$410$	0	0
$411$	1.50476e6	0.439402
$412$	0	0
$413$	− 4.71944e6i	− 1.36149i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.97688e6i	0.556724i
$418$	0	0
$419$	−2.15984e6	−0.601018	−0.300509	−	0.953779i	$-0.597157\pi$
$419$	−2.15984e6	−0.601018	−0.300509	+	0.953779i	$0.597157\pi$
$420$	0	0
$421$	1.47209e6	0.404789	0.202395	−	0.979304i	$-0.435128\pi$
$421$	1.47209e6	0.404789	0.202395	+	0.979304i	$0.435128\pi$
$422$	0	0
$423$	466851.i	0.126861i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3.71129e6i	− 0.985043i
$428$	0	0
$429$	−535920.	−0.140591
$430$	0	0
$431$	6.63748e6	1.72112	0.860558	−	0.509352i	$-0.170115\pi$
$431$	6.63748e6	1.72112	0.860558	+	0.509352i	$0.170115\pi$
$432$	0	0
$433$	6.27853e6i	1.60931i	0.593746	+	0.804653i	$0.297649\pi$
$433$	6.27853e6i	1.60931i	−0.593746	+	0.804653i	$0.702351\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.54448e6i	0.637376i
$438$	0	0
$439$	−2.44021e6	−0.604319	−0.302160	−	0.953257i	$-0.597708\pi$
$439$	−2.44021e6	−0.604319	−0.302160	+	0.953257i	$0.597708\pi$
$440$	0	0
$441$	−152019.	−0.0372221
$442$	0	0
$443$	− 2.30646e6i	− 0.558390i	−0.960234	−	0.279195i	$-0.909932\pi$
$443$	− 2.30646e6i	− 0.558390i	0.960234	−	0.279195i	$-0.0900675\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.34205e6i	0.554405i
$448$	0	0
$449$	−7.60241e6	−1.77965	−0.889826	−	0.456300i	$-0.849175\pi$
$449$	−7.60241e6	−1.77965	−0.889826	+	0.456300i	$0.849175\pi$
$450$	0	0
$451$	−1.88572e6	−0.436552
$452$	0	0
$453$	− 4.63212e6i	− 1.06056i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1.56938e6i	− 0.351510i	−0.984434	−	0.175755i	$-0.943763\pi$
$457$	− 1.56938e6i	− 0.351510i	0.984434	−	0.175755i	$-0.0562367\pi$
$458$	0	0
$459$	−730765.	−0.161900
$460$	0	0
$461$	−562602.	−0.123296	−0.0616480	−	0.998098i	$-0.519636\pi$
$461$	−562602.	−0.123296	−0.0616480	+	0.998098i	$0.519636\pi$
$462$	0	0
$463$	− 4.87382e6i	− 1.05662i	−0.849053	−	0.528308i	$-0.822827\pi$
$463$	− 4.87382e6i	− 1.05662i	0.849053	−	0.528308i	$-0.177173\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.11862e6i	1.08608i	0.839708	+	0.543039i	$0.182726\pi$
$467$	5.11862e6i	1.08608i	−0.839708	+	0.543039i	$0.817274\pi$
$468$	0	0
$469$	−6.90494e6	−1.44953
$470$	0	0
$471$	355562.	0.0738521
$472$	0	0
$473$	4.64580e6i	0.954790i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.03024e6i	− 0.408555i
$478$	0	0
$479$	4.00179e6	0.796922	0.398461	−	0.917185i	$-0.369544\pi$
$479$	4.00179e6	0.796922	0.398461	+	0.917185i	$0.369544\pi$
$480$	0	0
$481$	1.14607e6	0.225864
$482$	0	0
$483$	4.89940e6i	0.955598i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.68175e6i	− 0.321321i	−0.987010	−	0.160661i	$-0.948638\pi$
$487$	− 1.68175e6i	− 0.321321i	0.987010	−	0.160661i	$-0.0513625\pi$
$488$	0	0
$489$	−1.90122e6	−0.359551
$490$	0	0
$491$	−115927.	−0.0217010	−0.0108505	−	0.999941i	$-0.503454\pi$
$491$	−115927.	−0.0217010	−0.0108505	+	0.999941i	$0.503454\pi$
$492$	0	0
$493$	− 731580.i	− 0.135564i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.50076e6i	1.36212i
$498$	0	0
$499$	−7.98867e6	−1.43623	−0.718113	−	0.695926i	$-0.754994\pi$
$499$	−7.98867e6	−1.43623	−0.718113	+	0.695926i	$0.754994\pi$
$500$	0	0
$501$	−5.51478e6	−0.981598
$502$	0	0
$503$	8.07650e6i	1.42332i	0.702523	+	0.711661i	$0.252057\pi$
$503$	8.07650e6i	1.42332i	−0.702523	+	0.711661i	$0.747943\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.66323e6i	0.805689i
$508$	0	0
$509$	−2.83427e6	−0.484894	−0.242447	−	0.970165i	$-0.577950\pi$
$509$	−2.83427e6	−0.484894	−0.242447	+	0.970165i	$0.577950\pi$
$510$	0	0
$511$	2.56948e6	0.435305
$512$	0	0
$513$	3.96576e6i	0.665324i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.92212e6i	− 0.316267i
$518$	0	0
$519$	7.99545e6	1.30294
$520$	0	0
$521$	2.97526e6	0.480209	0.240105	−	0.970747i	$-0.422818\pi$
$521$	2.97526e6	0.480209	0.240105	+	0.970747i	$0.422818\pi$
$522$	0	0
$523$	7.72888e6i	1.23556i	0.786352	+	0.617778i	$0.211967\pi$
$523$	7.72888e6i	1.23556i	−0.786352	+	0.617778i	$0.788033\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 562004.i	− 0.0881481i
$528$	0	0
$529$	−502077.	−0.0780066
$530$	0	0
$531$	2.14464e6	0.330079
$532$	0	0
$533$	− 1.11958e6i	− 0.170701i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.03752e6i	− 0.604198i
$538$	0	0
$539$	625893.	0.0927958
$540$	0	0
$541$	−9.83660e6	−1.44495	−0.722474	−	0.691399i	$-0.756995\pi$
$541$	−9.83660e6	−1.44495	−0.722474	+	0.691399i	$0.756995\pi$
$542$	0	0
$543$	2.91166e6i	0.423780i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.45608e6i	0.493873i	0.969032	+	0.246937i	$0.0794239\pi$
$547$	3.45608e6i	0.493873i	−0.969032	+	0.246937i	$0.920576\pi$
$548$	0	0
$549$	1.68651e6	0.238813
$550$	0	0
$551$	−3.97018e6	−0.557098
$552$	0	0
$553$	1.20255e7i	1.67221i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.05760e6i	− 0.963871i	−0.876207	−	0.481936i	$-0.839934\pi$
$557$	− 7.05760e6i	− 0.963871i	0.876207	−	0.481936i	$-0.160066\pi$
$558$	0	0
$559$	−2.75828e6	−0.373344
$560$	0	0
$561$	619440.	0.0830983
$562$	0	0
$563$	2.32495e6i	0.309131i	0.987983	+	0.154566i	$0.0493978\pi$
$563$	2.32495e6i	0.309131i	−0.987983	+	0.154566i	$0.950602\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.51370e6i	0.720254i
$568$	0	0
$569$	−1.01947e7	−1.32006	−0.660029	−	0.751240i	$-0.729456\pi$
$569$	−1.01947e7	−1.32006	−0.660029	+	0.751240i	$0.729456\pi$
$570$	0	0
$571$	1.21297e7	1.55689	0.778446	−	0.627712i	$-0.216008\pi$
$571$	1.21297e7	1.55689	0.778446	+	0.627712i	$0.216008\pi$
$572$	0	0
$573$	1.22929e7i	1.56411i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.46358e6i	0.183011i	0.995805	+	0.0915053i	$0.0291678\pi$
$577$	1.46358e6i	0.183011i	−0.995805	+	0.0915053i	$0.970832\pi$
$578$	0	0
$579$	1.15933e7	1.43718
$580$	0	0
$581$	−1.09027e7	−1.33997
$582$	0	0
$583$	8.35891e6i	1.01854i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.18425e6i	− 0.620998i	−0.950574	−	0.310499i	$-0.899504\pi$
$587$	− 5.18425e6i	− 0.620998i	0.950574	−	0.310499i	$-0.100496\pi$
$588$	0	0
$589$	−3.04992e6	−0.362243
$590$	0	0
$591$	−5.80289e6	−0.683401
$592$	0	0
$593$	1.02722e7i	1.19957i	0.800161	+	0.599785i	$0.204747\pi$
$593$	1.02722e7i	1.19957i	−0.800161	+	0.599785i	$0.795253\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 1.06685e7i	− 1.22509i
$598$	0	0
$599$	−1.39289e7	−1.58617	−0.793085	−	0.609111i	$-0.791526\pi$
$599$	−1.39289e7	−1.58617	−0.793085	+	0.609111i	$0.791526\pi$
$600$	0	0
$601$	−4.81441e6	−0.543697	−0.271848	−	0.962340i	$-0.587635\pi$
$601$	−4.81441e6	−0.543697	−0.271848	+	0.962340i	$0.587635\pi$
$602$	0	0
$603$	− 3.13779e6i	− 0.351423i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.04800e7i	1.15449i	0.816571	+	0.577245i	$0.195872\pi$
$607$	1.04800e7i	1.15449i	−0.816571	+	0.577245i	$0.804128\pi$
$608$	0	0
$609$	−7.64460e6	−0.835240
$610$	0	0
$611$	1.14119e6	0.123667
$612$	0	0
$613$	3.07977e6i	0.331029i	0.986207	+	0.165515i	$0.0529285\pi$
$613$	3.07977e6i	0.331029i	−0.986207	+	0.165515i	$0.947071\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.15522e6i	0.122166i	0.998133	+	0.0610831i	$0.0194555\pi$
$617$	1.15522e6i	0.122166i	−0.998133	+	0.0610831i	$0.980545\pi$
$618$	0	0
$619$	1.76853e7	1.85517	0.927587	−	0.373606i	$-0.121879\pi$
$619$	1.76853e7	1.85517	0.927587	+	0.373606i	$0.121879\pi$
$620$	0	0
$621$	−1.08140e7	−1.12528
$622$	0	0
$623$	− 1.49159e7i	− 1.53967i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.36162e6i	− 0.341491i
$628$	0	0
$629$	−1.32468e6	−0.133501
$630$	0	0
$631$	5.42541e6	0.542449	0.271225	−	0.962516i	$-0.412571\pi$
$631$	5.42541e6	0.542449	0.271225	+	0.962516i	$0.412571\pi$
$632$	0	0
$633$	6.07932e6i	0.603039i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	371602.i	0.0362852i
$638$	0	0
$639$	−3.40855e6	−0.330230
$640$	0	0
$641$	1.16005e7	1.11515	0.557573	−	0.830128i	$-0.311733\pi$
$641$	1.16005e7	1.11515	0.557573	+	0.830128i	$0.311733\pi$
$642$	0	0
$643$	− 9.57873e6i	− 0.913651i	−0.889556	−	0.456826i	$-0.848986\pi$
$643$	− 9.57873e6i	− 0.913651i	0.889556	−	0.456826i	$-0.151014\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.28130e7i	− 1.20334i	−0.798744	−	0.601671i	$-0.794502\pi$
$647$	− 1.28130e7i	− 1.20334i	0.798744	−	0.601671i	$-0.205498\pi$
$648$	0	0
$649$	−8.82992e6	−0.822896
$650$	0	0
$651$	−5.87263e6	−0.543100
$652$	0	0
$653$	1.58665e6i	0.145613i	0.997346	+	0.0728064i	$0.0231955\pi$
$653$	1.58665e6i	0.145613i	−0.997346	+	0.0728064i	$0.976804\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.16764e6i	0.105535i
$658$	0	0
$659$	−1.95559e7	−1.75414	−0.877071	−	0.480361i	$-0.840506\pi$
$659$	−1.95559e7	−1.75414	−0.877071	+	0.480361i	$0.840506\pi$
$660$	0	0
$661$	−7.46471e6	−0.664522	−0.332261	−	0.943188i	$-0.607811\pi$
$661$	−7.46471e6	−0.664522	−0.332261	+	0.943188i	$0.607811\pi$
$662$	0	0
$663$	367771.i	0.0324933i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.08261e7i	− 0.942231i
$668$	0	0
$669$	5.77386e6	0.498771
$670$	0	0
$671$	−6.94371e6	−0.595367
$672$	0	0
$673$	1.09694e7i	0.933568i	0.884371	+	0.466784i	$0.154587\pi$
$673$	1.09694e7i	0.933568i	−0.884371	+	0.466784i	$0.845413\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.52708e6i	− 0.547327i	−0.961825	−	0.273664i	$-0.911764\pi$
$677$	− 6.52708e6i	− 0.547327i	0.961825	−	0.273664i	$-0.0882355\pi$
$678$	0	0
$679$	−1.50889e7	−1.25598
$680$	0	0
$681$	1.50805e7	1.24608
$682$	0	0
$683$	− 1.54389e7i	− 1.26638i	−0.773995	−	0.633191i	$-0.781745\pi$
$683$	− 1.54389e7i	− 1.26638i	0.773995	−	0.633191i	$-0.218255\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.08996e7i	0.881089i
$688$	0	0
$689$	−4.96280e6	−0.398271
$690$	0	0
$691$	4.78757e6	0.381435	0.190717	−	0.981645i	$-0.438919\pi$
$691$	4.78757e6	0.381435	0.190717	+	0.981645i	$0.438919\pi$
$692$	0	0
$693$	2.26548e6i	0.179196i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.29406e6i	0.100896i
$698$	0	0
$699$	1.13529e7	0.878847
$700$	0	0
$701$	−3.31891e6	−0.255094	−0.127547	−	0.991833i	$-0.540710\pi$
$701$	−3.31891e6	−0.255094	−0.127547	+	0.991833i	$0.540710\pi$
$702$	0	0
$703$	7.18883e6i	0.548618i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8.20699e6i	− 0.617497i
$708$	0	0
$709$	−1.60044e7	−1.19570	−0.597852	−	0.801607i	$-0.703979\pi$
$709$	−1.60044e7	−1.19570	−0.597852	+	0.801607i	$0.703979\pi$
$710$	0	0
$711$	−5.46472e6	−0.405409
$712$	0	0
$713$	− 8.31668e6i	− 0.612669i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 760560.i	− 0.0552504i
$718$	0	0
$719$	1.56006e7	1.12543	0.562715	−	0.826651i	$-0.309757\pi$
$719$	1.56006e7	1.12543	0.562715	+	0.826651i	$0.309757\pi$
$720$	0	0
$721$	−1.56531e7	−1.12141
$722$	0	0
$723$	1.71154e7i	1.21771i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.14545e7i	− 0.803783i	−0.915687	−	0.401891i	$-0.868353\pi$
$727$	− 1.14545e7i	− 0.803783i	0.915687	−	0.401891i	$-0.131647\pi$
$728$	0	0
$729$	−1.58900e7	−1.10740
$730$	0	0
$731$	3.18814e6	0.220670
$732$	0	0
$733$	− 1.13267e7i	− 0.778650i	−0.921101	−	0.389325i	$-0.872708\pi$
$733$	− 1.13267e7i	− 0.778650i	0.921101	−	0.389325i	$-0.127292\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.29189e7i	0.876108i
$738$	0	0
$739$	−1.31055e7	−0.882756	−0.441378	−	0.897321i	$-0.645510\pi$
$739$	−1.31055e7	−0.882756	−0.441378	+	0.897321i	$0.645510\pi$
$740$	0	0
$741$	1.99584e6	0.133530
$742$	0	0
$743$	− 4.16787e6i	− 0.276976i	−0.990364	−	0.138488i	$-0.955776\pi$
$743$	− 4.16787e6i	− 0.276976i	0.990364	−	0.138488i	$-0.0442242\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.95448e6i	− 0.324860i
$748$	0	0
$749$	−5.55086e6	−0.361539
$750$	0	0
$751$	−2.89308e7	−1.87180	−0.935902	−	0.352260i	$-0.885413\pi$
$751$	−2.89308e7	−1.87180	−0.935902	+	0.352260i	$0.885413\pi$
$752$	0	0
$753$	683400.i	0.0439225i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.32870e6i	0.528247i	0.964489	+	0.264124i	$0.0850827\pi$
$757$	8.32870e6i	0.528247i	−0.964489	+	0.264124i	$0.914917\pi$
$758$	0	0
$759$	9.16663e6	0.577571
$760$	0	0
$761$	4.48292e6	0.280608	0.140304	−	0.990108i	$-0.455192\pi$
$761$	4.48292e6	0.280608	0.140304	+	0.990108i	$0.455192\pi$
$762$	0	0
$763$	− 1.93556e7i	− 1.20363i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.24245e6i	− 0.321770i
$768$	0	0
$769$	−1.32141e6	−0.0805790	−0.0402895	−	0.999188i	$-0.512828\pi$
$769$	−1.32141e6	−0.0805790	−0.0402895	+	0.999188i	$0.512828\pi$
$770$	0	0
$771$	−2.21110e7	−1.33959
$772$	0	0
$773$	7.24897e6i	0.436342i	0.975911	+	0.218171i	$0.0700091\pi$
$773$	7.24897e6i	0.436342i	−0.975911	+	0.218171i	$0.929991\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.38421e7i	0.822527i
$778$	0	0
$779$	7.02268e6	0.414629
$780$	0	0
$781$	1.40337e7	0.823273
$782$	0	0
$783$	− 1.68733e7i	− 0.983547i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 4.98037e6i	− 0.286632i	−0.989677	−	0.143316i	$-0.954223\pi$
$787$	− 4.98037e6i	− 0.286632i	0.989677	−	0.143316i	$-0.0457766\pi$
$788$	0	0
$789$	−7.01670e6	−0.401273
$790$	0	0
$791$	5.96773e6	0.339132
$792$	0	0
$793$	− 4.12258e6i	− 0.232802i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.48673e7i	0.829061i	0.910036	+	0.414530i	$0.136054\pi$
$797$	1.48673e7i	0.829061i	−0.910036	+	0.414530i	$0.863946\pi$
$798$	0	0
$799$	−1.31904e6	−0.0730955
$800$	0	0
$801$	6.77817e6	0.373277
$802$	0	0
$803$	− 4.80742e6i	− 0.263101i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.59995e7i	1.40534i
$808$	0	0
$809$	684390.	0.0367648	0.0183824	−	0.999831i	$-0.494148\pi$
$809$	684390.	0.0367648	0.0183824	+	0.999831i	$0.494148\pi$
$810$	0	0
$811$	3.27890e7	1.75056	0.875279	−	0.483618i	$-0.160677\pi$
$811$	3.27890e7	1.75056	0.875279	+	0.483618i	$0.160677\pi$
$812$	0	0
$813$	− 1.34341e7i	− 0.712825i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.73016e7i	− 0.906841i
$818$	0	0
$819$	−1.34505e6	−0.0700694
$820$	0	0
$821$	7.17265e6	0.371383	0.185691	−	0.982608i	$-0.440548\pi$
$821$	7.17265e6	0.371383	0.185691	+	0.982608i	$0.440548\pi$
$822$	0	0
$823$	1.32939e6i	0.0684153i	0.999415	+	0.0342077i	$0.0108908\pi$
$823$	1.32939e6i	0.0684153i	−0.999415	+	0.0342077i	$0.989109\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.92964e7i	0.981099i	0.871413	+	0.490550i	$0.163204\pi$
$827$	1.92964e7i	0.981099i	−0.871413	+	0.490550i	$0.836796\pi$
$828$	0	0
$829$	−7.10811e6	−0.359226	−0.179613	−	0.983737i	$-0.557485\pi$
$829$	−7.10811e6	−0.359226	−0.179613	+	0.983737i	$0.557485\pi$
$830$	0	0
$831$	5.97972e6	0.300385
$832$	0	0
$833$	− 429514.i	− 0.0214469i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.29622e7i	− 0.639534i
$838$	0	0
$839$	−8.24538e6	−0.404395	−0.202198	−	0.979345i	$-0.564808\pi$
$839$	−8.24538e6	−0.404395	−0.202198	+	0.979345i	$0.564808\pi$
$840$	0	0
$841$	−3.61905e6	−0.176443
$842$	0	0
$843$	1.67231e7i	0.810493i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.30001e7i	0.622640i
$848$	0	0
$849$	3.05761e7	1.45584
$850$	0	0
$851$	−1.96029e7	−0.927889
$852$	0	0
$853$	− 1.86921e7i	− 0.879599i	−0.898096	−	0.439800i	$-0.855049\pi$
$853$	− 1.86921e7i	− 0.879599i	0.898096	−	0.439800i	$-0.144951\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.24473e7i	− 1.04403i	−0.852937	−	0.522013i	$-0.825181\pi$
$857$	− 2.24473e7i	− 1.04403i	0.852937	−	0.522013i	$-0.174819\pi$
$858$	0	0
$859$	3.24436e6	0.150019	0.0750094	−	0.997183i	$-0.476101\pi$
$859$	3.24436e6	0.150019	0.0750094	+	0.997183i	$0.476101\pi$
$860$	0	0
$861$	1.35222e7	0.621641
$862$	0	0
$863$	− 1.01542e6i	− 0.0464108i	−0.999731	−	0.0232054i	$-0.992613\pi$
$863$	− 1.01542e6i	− 0.0464108i	0.999731	−	0.0232054i	$-0.00738717\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.86243e7i	0.841457i
$868$	0	0
$869$	2.24994e7	1.01070
$870$	0	0
$871$	−7.67015e6	−0.342577
$872$	0	0
$873$	− 6.85679e6i	− 0.304499i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.96764e7i	1.74194i	0.491337	+	0.870970i	$0.336508\pi$
$877$	3.96764e7i	1.74194i	−0.491337	+	0.870970i	$0.663492\pi$
$878$	0	0
$879$	3.29275e7	1.43743
$880$	0	0
$881$	−2.44584e7	−1.06167	−0.530833	−	0.847477i	$-0.678121\pi$
$881$	−2.44584e7	−1.06167	−0.530833	+	0.847477i	$0.678121\pi$
$882$	0	0
$883$	3.11179e7i	1.34310i	0.740960	+	0.671549i	$0.234371\pi$
$883$	3.11179e7i	1.34310i	−0.740960	+	0.671549i	$0.765629\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.87514e7i	− 0.800249i	−0.916461	−	0.400124i	$-0.868967\pi$
$887$	− 1.87514e7i	− 0.800249i	0.916461	−	0.400124i	$-0.131033\pi$
$888$	0	0
$889$	2.18172e7	0.925858
$890$	0	0
$891$	1.03160e7	0.435327
$892$	0	0
$893$	7.15824e6i	0.300385i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.44236e6i	0.225843i
$898$	0	0
$899$	1.29766e7	0.535503
$900$	0	0
$901$	5.73623e6	0.235404
$902$	0	0
$903$	− 3.33143e7i	− 1.35960i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 5.07504e6i	− 0.204843i	−0.994741	−	0.102421i	$-0.967341\pi$
$907$	− 5.07504e6i	− 0.204843i	0.994741	−	0.102421i	$-0.0326590\pi$
$908$	0	0
$909$	3.72947e6	0.149706
$910$	0	0
$911$	2.83599e7	1.13216	0.566082	−	0.824349i	$-0.308459\pi$
$911$	2.83599e7	1.13216	0.566082	+	0.824349i	$0.308459\pi$
$912$	0	0
$913$	2.03986e7i	0.809885i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3.71169e7i	− 1.45763i
$918$	0	0
$919$	9.58570e6	0.374399	0.187200	−	0.982322i	$-0.440059\pi$
$919$	9.58570e6	0.374399	0.187200	+	0.982322i	$0.440059\pi$
$920$	0	0
$921$	−277860.	−0.0107939
$922$	0	0
$923$	8.33200e6i	0.321918i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 7.11321e6i	− 0.271873i
$928$	0	0
$929$	3.82212e7	1.45300	0.726500	−	0.687167i	$-0.241146\pi$
$929$	3.82212e7	1.45300	0.726500	+	0.687167i	$0.241146\pi$
$930$	0	0
$931$	−2.33091e6	−0.0881357
$932$	0	0
$933$	634200.i	0.0238519i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 9.37422e6i	− 0.348808i	−0.984674	−	0.174404i	$-0.944200\pi$
$937$	− 9.37422e6i	− 0.348808i	0.984674	−	0.174404i	$-0.0557998\pi$
$938$	0	0
$939$	−3.74544e7	−1.38624
$940$	0	0
$941$	3.04898e6	0.112249	0.0561243	−	0.998424i	$-0.482126\pi$
$941$	3.04898e6	0.112249	0.0561243	+	0.998424i	$0.482126\pi$
$942$	0	0
$943$	1.91498e7i	0.701270i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2.95948e7i	− 1.07236i	−0.844104	−	0.536180i	$-0.819867\pi$
$947$	− 2.95948e7i	− 1.07236i	0.844104	−	0.536180i	$-0.180133\pi$
$948$	0	0
$949$	2.85424e6	0.102878
$950$	0	0
$951$	6.32694e6	0.226852
$952$	0	0
$953$	− 1.58338e7i	− 0.564744i	−0.959305	−	0.282372i	$-0.908879\pi$
$953$	− 1.58338e7i	− 0.564744i	0.959305	−	0.282372i	$-0.0911212\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.43028e7i	0.504825i
$958$	0	0
$959$	−1.55492e7	−0.545960
$960$	0	0
$961$	−1.86604e7	−0.651798
$962$	0	0
$963$	− 2.52246e6i	− 0.0876513i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 8.69009e6i	− 0.298853i	−0.988773	−	0.149427i	$-0.952257\pi$
$967$	− 8.69009e6i	− 0.298853i	0.988773	−	0.149427i	$-0.0477428\pi$
$968$	0	0
$969$	−2.30688e6	−0.0789252
$970$	0	0
$971$	−2.81474e7	−0.958054	−0.479027	−	0.877800i	$-0.659010\pi$
$971$	−2.81474e7	−0.958054	−0.479027	+	0.877800i	$0.659010\pi$
$972$	0	0
$973$	− 2.04278e7i	− 0.691733i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.34090e7i	− 1.45493i	−0.686143	−	0.727467i	$-0.740697\pi$
$977$	− 4.34090e7i	− 1.45493i	0.686143	−	0.727467i	$-0.259303\pi$
$978$	0	0
$979$	−2.79071e7	−0.930590
$980$	0	0
$981$	8.79568e6	0.291808
$982$	0	0
$983$	4.64172e7i	1.53213i	0.642763	+	0.766065i	$0.277788\pi$
$983$	4.64172e7i	1.53213i	−0.642763	+	0.766065i	$0.722212\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.37832e7i	0.450358i
$988$	0	0
$989$	4.71789e7	1.53376
$990$	0	0
$991$	4.64577e7	1.50271	0.751353	−	0.659901i	$-0.229402\pi$
$991$	4.64577e7	1.50271	0.751353	+	0.659901i	$0.229402\pi$
$992$	0	0
$993$	− 4.14690e7i	− 1.33460i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.35968e7i	0.433211i	0.976259	+	0.216605i	$0.0694985\pi$
$997$	1.35968e7i	0.433211i	−0.976259	+	0.216605i	$0.930501\pi$
$998$	0	0
$999$	−3.05525e7	−0.968576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.c.h.449.3		4
4.3	odd	2	inner	800.6.c.h.449.2		4
5.2	odd	4		800.6.a.i.1.2		2
5.3	odd	4		160.6.a.b.1.1	✓	2
5.4	even	2	inner	800.6.c.h.449.1		4
20.3	even	4		160.6.a.b.1.2	yes	2
20.7	even	4		800.6.a.i.1.1		2
20.19	odd	2	inner	800.6.c.h.449.4		4
40.3	even	4		320.6.a.t.1.1		2
40.13	odd	4		320.6.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.b.1.1	✓	2	5.3	odd	4
160.6.a.b.1.2	yes	2	20.3	even	4
320.6.a.t.1.1		2	40.3	even	4
320.6.a.t.1.2		2	40.13	odd	4
800.6.a.i.1.1		2	20.7	even	4
800.6.a.i.1.2		2	5.2	odd	4
800.6.c.h.449.1		4	5.4	even	2	inner
800.6.c.h.449.2		4	4.3	odd	2	inner
800.6.c.h.449.3		4	1.1	even	1	trivial
800.6.c.h.449.4		4	20.19	odd	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+13.4164i q^{3} -138.636i q^{7} +63.0000 q^{9} +O(q^{10})\)	\(q+13.4164i q^{3} -138.636i q^{7} +63.0000 q^{9} -259.384 q^{11} -154.000i q^{13} +178.000i q^{17} +965.981 q^{19} +1860.00 q^{21} +2634.09i q^{23} +4105.42i q^{27} -4110.00 q^{29} -3157.33 q^{31} -3480.00i q^{33} +7442.00i q^{37} +2066.13 q^{39} +7270.00 q^{41} -17910.9i q^{43} +7410.33i q^{47} -2413.00 q^{49} -2388.12 q^{51} -32226.0i q^{53} +12960.0i q^{57} +34041.9 q^{59} +26770.0 q^{61} -8734.08i q^{63} -49806.2i q^{67} -35340.0 q^{69} -54103.9 q^{71} +18534.0i q^{73} +35960.0i q^{77} -86741.5 q^{79} -39771.0 q^{81} -78642.5i q^{83} -55141.4i q^{87} +107590. q^{89} -21350.0 q^{91} -42360.0i q^{93} -108838. i q^{97} -16341.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 252 q^{9}+O(q^{10}) \) 4 * q + 252 * q^9	\( 4 q + 252 q^{9} + 7440 q^{21} - 16440 q^{29} + 29080 q^{41} - 9652 q^{49} + 107080 q^{61} - 141360 q^{69} - 159084 q^{81} + 430360 q^{89}+O(q^{100}) \) 4 * q + 252 * q^9 + 7440 * q^21 - 16440 * q^29 + 29080 * q^41 - 9652 * q^49 + 107080 * q^61 - 141360 * q^69 - 159084 * q^81 + 430360 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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