LMFDB - Embedded newform 320.6.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(1,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, 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("320.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.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:	$51.3228223402$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.4164 q^{3} +25.0000 q^{5} +138.636 q^{7} -63.0000 q^{9} +O(q^{10})$	$q-13.4164 q^{3} +25.0000 q^{5} +138.636 q^{7} -63.0000 q^{9} -259.384 q^{11} -154.000 q^{13} -335.410 q^{15} +178.000 q^{17} -965.981 q^{19} -1860.00 q^{21} +2634.09 q^{23} +625.000 q^{25} +4105.42 q^{27} -4110.00 q^{29} +3157.33 q^{31} +3480.00 q^{33} +3465.91 q^{35} -7442.00 q^{37} +2066.13 q^{39} +7270.00 q^{41} +17910.9 q^{43} -1575.00 q^{45} -7410.33 q^{47} +2413.00 q^{49} -2388.12 q^{51} -32226.0 q^{53} -6484.60 q^{55} +12960.0 q^{57} -34041.9 q^{59} -26770.0 q^{61} -8734.08 q^{63} -3850.00 q^{65} -49806.2 q^{67} -35340.0 q^{69} +54103.9 q^{71} -18534.0 q^{73} -8385.25 q^{75} -35960.0 q^{77} -86741.5 q^{79} -39771.0 q^{81} +78642.5 q^{83} +4450.00 q^{85} +55141.4 q^{87} -107590. q^{89} -21350.0 q^{91} -42360.0 q^{93} -24149.5 q^{95} -108838. q^{97} +16341.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 50 q^{5} - 126 q^{9}+O(q^{10}) $ 2 * q + 50 * q^5 - 126 * q^9	$ 2 q + 50 q^{5} - 126 q^{9} - 308 q^{13} + 356 q^{17} - 3720 q^{21} + 1250 q^{25} - 8220 q^{29} + 6960 q^{33} - 14884 q^{37} + 14540 q^{41} - 3150 q^{45} + 4826 q^{49} - 64452 q^{53} + 25920 q^{57} - 53540 q^{61} - 7700 q^{65} - 70680 q^{69} - 37068 q^{73} - 71920 q^{77} - 79542 q^{81} + 8900 q^{85} - 215180 q^{89} - 84720 q^{93} - 217676 q^{97}+O(q^{100}) $ 2 * q + 50 * q^5 - 126 * q^9 - 308 * q^13 + 356 * q^17 - 3720 * q^21 + 1250 * q^25 - 8220 * q^29 + 6960 * q^33 - 14884 * q^37 + 14540 * q^41 - 3150 * q^45 + 4826 * q^49 - 64452 * q^53 + 25920 * q^57 - 53540 * q^61 - 7700 * q^65 - 70680 * q^69 - 37068 * q^73 - 71920 * q^77 - 79542 * q^81 + 8900 * q^85 - 215180 * q^89 - 84720 * q^93 - 217676 * q^97

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
$3$	−13.4164	−0.860663	−0.430331	−	0.902671i	$-0.641603\pi$
$3$	−13.4164	−0.860663	−0.430331	+	0.902671i	$0.641603\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	138.636	1.06938	0.534689	−	0.845049i	$-0.320429\pi$
$7$	138.636	1.06938	0.534689	+	0.845049i	$0.320429\pi$
$8$	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
$13$	−154.000	−0.252733	−0.126367	−	0.991984i	$-0.540332\pi$
$13$	−154.000	−0.252733	−0.126367	+	0.991984i	$0.540332\pi$
$14$	0	0
$15$	−335.410	−0.384900
$16$	0	0
$17$	178.000	0.149382	0.0746909	−	0.997207i	$-0.476203\pi$
$17$	178.000	0.149382	0.0746909	+	0.997207i	$0.476203\pi$
$18$	0	0
$19$	−965.981	−0.613882	−0.306941	−	0.951729i	$-0.599305\pi$
$19$	−965.981	−0.613882	−0.306941	+	0.951729i	$0.599305\pi$
$20$	0	0
$21$	−1860.00	−0.920375
$22$	0	0
$23$	2634.09	1.03827	0.519135	−	0.854692i	$-0.326254\pi$
$23$	2634.09	1.03827	0.519135	+	0.854692i	$0.326254\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	4105.42	1.08380
$28$	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	$-0.404666\pi$
$31$	3157.33	0.590086	0.295043	+	0.955484i	$0.404666\pi$
$32$	0	0
$33$	3480.00	0.556281
$34$	0	0
$35$	3465.91	0.478241
$36$	0	0
$37$	−7442.00	−0.893687	−0.446843	−	0.894612i	$-0.647452\pi$
$37$	−7442.00	−0.893687	−0.446843	+	0.894612i	$0.647452\pi$
$38$	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
$43$	17910.9	1.47722	0.738612	−	0.674131i	$-0.235482\pi$
$43$	17910.9	1.47722	0.738612	+	0.674131i	$0.235482\pi$
$44$	0	0
$45$	−1575.00	−0.115944
$46$	0	0
$47$	−7410.33	−0.489320	−0.244660	−	0.969609i	$-0.578676\pi$
$47$	−7410.33	−0.489320	−0.244660	+	0.969609i	$0.578676\pi$
$48$	0	0
$49$	2413.00	0.143571
$50$	0	0
$51$	−2388.12	−0.128567
$52$	0	0
$53$	−32226.0	−1.57586	−0.787928	−	0.615767i	$-0.788846\pi$
$53$	−32226.0	−1.57586	−0.787928	+	0.615767i	$0.788846\pi$
$54$	0	0
$55$	−6484.60	−0.289052
$56$	0	0
$57$	12960.0	0.528345
$58$	0	0
$59$	−34041.9	−1.27316	−0.636581	−	0.771210i	$-0.719652\pi$
$59$	−34041.9	−1.27316	−0.636581	+	0.771210i	$0.719652\pi$
$60$	0	0
$61$	−26770.0	−0.921136	−0.460568	−	0.887624i	$-0.652354\pi$
$61$	−26770.0	−0.921136	−0.460568	+	0.887624i	$0.652354\pi$
$62$	0	0
$63$	−8734.08	−0.277246
$64$	0	0
$65$	−3850.00	−0.113026
$66$	0	0
$67$	−49806.2	−1.35549	−0.677745	−	0.735297i	$-0.737043\pi$
$67$	−49806.2	−1.35549	−0.677745	+	0.735297i	$0.737043\pi$
$68$	0	0
$69$	−35340.0	−0.893601
$70$	0	0
$71$	54103.9	1.27375	0.636873	−	0.770969i	$-0.280228\pi$
$71$	54103.9	1.27375	0.636873	+	0.770969i	$0.280228\pi$
$72$	0	0
$73$	−18534.0	−0.407063	−0.203532	−	0.979068i	$-0.565242\pi$
$73$	−18534.0	−0.407063	−0.203532	+	0.979068i	$0.565242\pi$
$74$	0	0
$75$	−8385.25	−0.172133
$76$	0	0
$77$	−35960.0	−0.691183
$78$	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
$83$	78642.5	1.25303	0.626516	−	0.779409i	$-0.284480\pi$
$83$	78642.5	1.25303	0.626516	+	0.779409i	$0.284480\pi$
$84$	0	0
$85$	4450.00	0.0668056
$86$	0	0
$87$	55141.4	0.781052
$88$	0	0
$89$	−107590.	−1.43978	−0.719891	−	0.694087i	$-0.755808\pi$
$89$	−107590.	−1.43978	−0.719891	+	0.694087i	$0.755808\pi$
$90$	0	0
$91$	−21350.0	−0.270268
$92$	0	0
$93$	−42360.0	−0.507865
$94$	0	0
$95$	−24149.5	−0.274536
$96$	0	0
$97$	−108838.	−1.17450	−0.587248	−	0.809407i	$-0.699788\pi$
$97$	−108838.	−1.17450	−0.587248	+	0.809407i	$0.699788\pi$
$98$	0	0
$99$	16341.2	0.167570
$100$	0	0
$101$	−59198.0	−0.577436	−0.288718	−	0.957414i	$-0.593229\pi$
$101$	−59198.0	−0.577436	−0.288718	+	0.957414i	$0.593229\pi$
$102$	0	0
$103$	−112908.	−1.04865	−0.524326	−	0.851517i	$-0.675683\pi$
$103$	−112908.	−1.04865	−0.524326	+	0.851517i	$0.675683\pi$
$104$	0	0
$105$	−46500.0	−0.411604
$106$	0	0
$107$	−40039.0	−0.338084	−0.169042	−	0.985609i	$-0.554067\pi$
$107$	−40039.0	−0.338084	−0.169042	+	0.985609i	$0.554067\pi$
$108$	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
$113$	−43046.0	−0.317130	−0.158565	−	0.987349i	$-0.550687\pi$
$113$	−43046.0	−0.317130	−0.158565	+	0.987349i	$0.550687\pi$
$114$	0	0
$115$	65852.2	0.464329
$116$	0	0
$117$	9702.00	0.0655234
$118$	0	0
$119$	24677.2	0.159746
$120$	0	0
$121$	−93771.0	−0.582244
$122$	0	0
$123$	−97537.3	−0.581310
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−157370.	−0.865790	−0.432895	−	0.901444i	$-0.642508\pi$
$127$	−157370.	−0.865790	−0.432895	+	0.901444i	$0.642508\pi$
$128$	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.	−0.656472
$134$	0	0
$135$	102636.	0.484689
$136$	0	0
$137$	−112158.	−0.510539	−0.255269	−	0.966870i	$-0.582164\pi$
$137$	−112158.	−0.510539	−0.255269	+	0.966870i	$0.582164\pi$
$138$	0	0
$139$	−147348.	−0.646855	−0.323428	−	0.946253i	$-0.604835\pi$
$139$	−147348.	−0.646855	−0.323428	+	0.946253i	$0.604835\pi$
$140$	0	0
$141$	99420.0	0.421139
$142$	0	0
$143$	39945.1	0.163352
$144$	0	0
$145$	−102750.	−0.405847
$146$	0	0
$147$	−32373.8	−0.123566
$148$	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	$-0.288700\pi$
$151$	345258.	1.23226	0.616128	+	0.787646i	$0.288700\pi$
$152$	0	0
$153$	−11214.0	−0.0387286
$154$	0	0
$155$	78933.2	0.263895
$156$	0	0
$157$	26502.0	0.0858083	0.0429042	−	0.999079i	$-0.486339\pi$
$157$	26502.0	0.0858083	0.0429042	+	0.999079i	$0.486339\pi$
$158$	0	0
$159$	432357.	1.35628
$160$	0	0
$161$	365180.	1.11030
$162$	0	0
$163$	−141709.	−0.417760	−0.208880	−	0.977941i	$-0.566982\pi$
$163$	−141709.	−0.417760	−0.208880	+	0.977941i	$0.566982\pi$
$164$	0	0
$165$	87000.0	0.248777
$166$	0	0
$167$	−411047.	−1.14051	−0.570257	−	0.821466i	$-0.693156\pi$
$167$	−411047.	−1.14051	−0.570257	+	0.821466i	$0.693156\pi$
$168$	0	0
$169$	−347577.	−0.936126
$170$	0	0
$171$	60856.8	0.159155
$172$	0	0
$173$	−595946.	−1.51388	−0.756940	−	0.653484i	$-0.773307\pi$
$173$	−595946.	−1.51388	−0.756940	+	0.653484i	$0.773307\pi$
$174$	0	0
$175$	86647.6	0.213876
$176$	0	0
$177$	456720.	1.09576
$178$	0	0
$179$	300939.	0.702014	0.351007	−	0.936373i	$-0.385839\pi$
$179$	300939.	0.702014	0.351007	+	0.936373i	$0.385839\pi$
$180$	0	0
$181$	−217022.	−0.492388	−0.246194	−	0.969221i	$-0.579180\pi$
$181$	−217022.	−0.492388	−0.246194	+	0.969221i	$0.579180\pi$
$182$	0	0
$183$	359157.	0.792788
$184$	0	0
$185$	−186050.	−0.399669
$186$	0	0
$187$	−46170.3	−0.0965515
$188$	0	0
$189$	569160.	1.15899
$190$	0	0
$191$	−916260.	−1.81734	−0.908668	−	0.417519i	$-0.862900\pi$
$191$	−916260.	−1.81734	−0.908668	+	0.417519i	$0.862900\pi$
$192$	0	0
$193$	864114.	1.66985	0.834926	−	0.550363i	$-0.185511\pi$
$193$	864114.	1.66985	0.834926	+	0.550363i	$0.185511\pi$
$194$	0	0
$195$	51653.2	0.0972771
$196$	0	0
$197$	−432522.	−0.794040	−0.397020	−	0.917810i	$-0.629956\pi$
$197$	−432522.	−0.794040	−0.397020	+	0.917810i	$0.629956\pi$
$198$	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.	−0.970462
$204$	0	0
$205$	181750.	0.302058
$206$	0	0
$207$	−165948.	−0.269181
$208$	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.	−1.09627
$214$	0	0
$215$	447773.	0.660635
$216$	0	0
$217$	437720.	0.631026
$218$	0	0
$219$	248660.	0.350344
$220$	0	0
$221$	−27412.0	−0.0377537
$222$	0	0
$223$	−430358.	−0.579519	−0.289760	−	0.957099i	$-0.593575\pi$
$223$	−430358.	−0.579519	−0.289760	+	0.957099i	$0.593575\pi$
$224$	0	0
$225$	−39375.0	−0.0518519
$226$	0	0
$227$	−1.12403e6	−1.44782	−0.723909	−	0.689896i	$-0.757656\pi$
$227$	−1.12403e6	−1.44782	−0.723909	+	0.689896i	$0.757656\pi$
$228$	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.	1.02113	0.510564	−	0.859840i	$-0.329437\pi$
$233$	846194.	1.02113	0.510564	+	0.859840i	$0.329437\pi$
$234$	0	0
$235$	−185258.	−0.218831
$236$	0	0
$237$	1.16376e6	1.34584
$238$	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.	−0.504119
$244$	0	0
$245$	60325.0	0.0642070
$246$	0	0
$247$	148761.	0.155148
$248$	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.	−0.671076
$254$	0	0
$255$	−59703.0	−0.0574971
$256$	0	0
$257$	1.64806e6	1.55647	0.778233	−	0.627976i	$-0.216116\pi$
$257$	1.64806e6	1.55647	0.778233	+	0.627976i	$0.216116\pi$
$258$	0	0
$259$	−1.03173e6	−0.955690
$260$	0	0
$261$	258930.	0.235278
$262$	0	0
$263$	522994.	0.466238	0.233119	−	0.972448i	$-0.425107\pi$
$263$	522994.	0.466238	0.233119	+	0.972448i	$0.425107\pi$
$264$	0	0
$265$	−805650.	−0.704745
$266$	0	0
$267$	1.44347e6	1.23917
$268$	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	$-0.364092\pi$
$271$	1.00132e6	0.828228	0.414114	+	0.910225i	$0.364092\pi$
$272$	0	0
$273$	286440.	0.232609
$274$	0	0
$275$	−162115.	−0.129268
$276$	0	0
$277$	445702.	0.349016	0.174508	−	0.984656i	$-0.444167\pi$
$277$	445702.	0.349016	0.174508	+	0.984656i	$0.444167\pi$
$278$	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
$283$	2.27900e6	1.69153	0.845764	−	0.533557i	$-0.179145\pi$
$283$	2.27900e6	1.69153	0.845764	+	0.533557i	$0.179145\pi$
$284$	0	0
$285$	324000.	0.236283
$286$	0	0
$287$	1.00789e6	0.722281
$288$	0	0
$289$	−1.38817e6	−0.977685
$290$	0	0
$291$	1.46021e6	1.01084
$292$	0	0
$293$	−2.45427e6	−1.67014	−0.835072	−	0.550140i	$-0.814574\pi$
$293$	−2.45427e6	−1.67014	−0.835072	+	0.550140i	$0.814574\pi$
$294$	0	0
$295$	−851047.	−0.569375
$296$	0	0
$297$	−1.06488e6	−0.700502
$298$	0	0
$299$	−405650.	−0.262406
$300$	0	0
$301$	2.48310e6	1.57971
$302$	0	0
$303$	794225.	0.496977
$304$	0	0
$305$	−669250.	−0.411945
$306$	0	0
$307$	20710.5	0.0125413	0.00627067	−	0.999980i	$-0.498004\pi$
$307$	20710.5	0.0125413	0.00627067	+	0.999980i	$0.498004\pi$
$308$	0	0
$309$	1.51482e6	0.902537
$310$	0	0
$311$	−47270.5	−0.0277133	−0.0138567	−	0.999904i	$-0.504411\pi$
$311$	−47270.5	−0.0277133	−0.0138567	+	0.999904i	$0.504411\pi$
$312$	0	0
$313$	−2.79169e6	−1.61067	−0.805333	−	0.592822i	$-0.798014\pi$
$313$	−2.79169e6	−1.61067	−0.805333	+	0.592822i	$0.798014\pi$
$314$	0	0
$315$	−218352.	−0.123988
$316$	0	0
$317$	471582.	0.263578	0.131789	−	0.991278i	$-0.457928\pi$
$317$	471582.	0.263578	0.131789	+	0.991278i	$0.457928\pi$
$318$	0	0
$319$	1.06607e6	0.586554
$320$	0	0
$321$	537180.	0.290976
$322$	0	0
$323$	−171945.	−0.0917028
$324$	0	0
$325$	−96250.0	−0.0505466
$326$	0	0
$327$	−1.87312e6	−0.968715
$328$	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.	0.231697
$334$	0	0
$335$	−1.24515e6	−0.606193
$336$	0	0
$337$	−665838.	−0.319370	−0.159685	−	0.987168i	$-0.551048\pi$
$337$	−665838.	−0.319370	−0.159685	+	0.987168i	$0.551048\pi$
$338$	0	0
$339$	577523.	0.272942
$340$	0	0
$341$	−818960.	−0.381397
$342$	0	0
$343$	−1.99553e6	−0.915847
$344$	0	0
$345$	−883500.	−0.399631
$346$	0	0
$347$	1.80908e6	0.806556	0.403278	−	0.915078i	$-0.367871\pi$
$347$	1.80908e6	0.806556	0.403278	+	0.915078i	$0.367871\pi$
$348$	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
$353$	1.14535e6	0.489215	0.244608	−	0.969622i	$-0.421341\pi$
$353$	1.14535e6	0.489215	0.244608	+	0.969622i	$0.421341\pi$
$354$	0	0
$355$	1.35260e6	0.569636
$356$	0	0
$357$	−331080.	−0.137487
$358$	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
$363$	1.25807e6	0.501116
$364$	0	0
$365$	−463350.	−0.182044
$366$	0	0
$367$	3.99537e6	1.54843	0.774215	−	0.632923i	$-0.218145\pi$
$367$	3.99537e6	1.54843	0.774215	+	0.632923i	$0.218145\pi$
$368$	0	0
$369$	−458010.	−0.175109
$370$	0	0
$371$	−4.46769e6	−1.68519
$372$	0	0
$373$	−4.68131e6	−1.74219	−0.871094	−	0.491117i	$-0.836589\pi$
$373$	−4.68131e6	−1.74219	−0.871094	+	0.491117i	$0.836589\pi$
$374$	0	0
$375$	−209631.	−0.0769800
$376$	0	0
$377$	632940.	0.229356
$378$	0	0
$379$	−337271.	−0.120609	−0.0603046	−	0.998180i	$-0.519207\pi$
$379$	−337271.	−0.120609	−0.0603046	+	0.998180i	$0.519207\pi$
$380$	0	0
$381$	2.11134e6	0.745153
$382$	0	0
$383$	−598689.	−0.208547	−0.104274	−	0.994549i	$-0.533252\pi$
$383$	−598689.	−0.208547	−0.104274	+	0.994549i	$0.533252\pi$
$384$	0	0
$385$	−899000.	−0.309106
$386$	0	0
$387$	−1.12839e6	−0.382984
$388$	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
$393$	−3.59196e6	−1.17314
$394$	0	0
$395$	−2.16854e6	−0.699318
$396$	0	0
$397$	179398.	0.0571270	0.0285635	−	0.999592i	$-0.490907\pi$
$397$	179398.	0.0571270	0.0285635	+	0.999592i	$0.490907\pi$
$398$	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.	−0.149134
$404$	0	0
$405$	−994275.	−0.301210
$406$	0	0
$407$	1.93033e6	0.577626
$408$	0	0
$409$	−2.64503e6	−0.781847	−0.390923	−	0.920423i	$-0.627844\pi$
$409$	−2.64503e6	−0.781847	−0.390923	+	0.920423i	$0.627844\pi$
$410$	0	0
$411$	1.50476e6	0.439402
$412$	0	0
$413$	−4.71944e6	−1.36149
$414$	0	0
$415$	1.96606e6	0.560373
$416$	0	0
$417$	1.97688e6	0.556724
$418$	0	0
$419$	2.15984e6	0.601018	0.300509	−	0.953779i	$-0.402843\pi$
$419$	2.15984e6	0.601018	0.300509	+	0.953779i	$0.402843\pi$
$420$	0	0
$421$	−1.47209e6	−0.404789	−0.202395	−	0.979304i	$-0.564872\pi$
$421$	−1.47209e6	−0.404789	−0.202395	+	0.979304i	$0.564872\pi$
$422$	0	0
$423$	466851.	0.126861
$424$	0	0
$425$	111250.	0.0298764
$426$	0	0
$427$	−3.71129e6	−0.985043
$428$	0	0
$429$	−535920.	−0.140591
$430$	0	0
$431$	−6.63748e6	−1.72112	−0.860558	−	0.509352i	$-0.829885\pi$
$431$	−6.63748e6	−1.72112	−0.860558	+	0.509352i	$0.829885\pi$
$432$	0	0
$433$	−6.27853e6	−1.60931	−0.804653	−	0.593746i	$-0.797649\pi$
$433$	−6.27853e6	−1.60931	−0.804653	+	0.593746i	$0.797649\pi$
$434$	0	0
$435$	1.37854e6	0.349297
$436$	0	0
$437$	−2.54448e6	−0.637376
$438$	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
$443$	2.30646e6	0.558390	0.279195	−	0.960234i	$-0.409932\pi$
$443$	2.30646e6	0.558390	0.279195	+	0.960234i	$0.409932\pi$
$444$	0	0
$445$	−2.68975e6	−0.643890
$446$	0	0
$447$	−2.34205e6	−0.554405
$448$	0	0
$449$	7.60241e6	1.77965	0.889826	−	0.456300i	$-0.150825\pi$
$449$	7.60241e6	1.77965	0.889826	+	0.456300i	$0.150825\pi$
$450$	0	0
$451$	−1.88572e6	−0.436552
$452$	0	0
$453$	−4.63212e6	−1.06056
$454$	0	0
$455$	−533749.	−0.120867
$456$	0	0
$457$	−1.56938e6	−0.351510	−0.175755	−	0.984434i	$-0.556237\pi$
$457$	−1.56938e6	−0.351510	−0.175755	+	0.984434i	$0.556237\pi$
$458$	0	0
$459$	730765.	0.161900
$460$	0	0
$461$	562602.	0.123296	0.0616480	−	0.998098i	$-0.480364\pi$
$461$	562602.	0.123296	0.0616480	+	0.998098i	$0.480364\pi$
$462$	0	0
$463$	−4.87382e6	−1.05662	−0.528308	−	0.849053i	$-0.677173\pi$
$463$	−4.87382e6	−1.05662	−0.528308	+	0.849053i	$0.677173\pi$
$464$	0	0
$465$	−1.05900e6	−0.227124
$466$	0	0
$467$	5.11862e6	1.08608	0.543039	−	0.839708i	$-0.317274\pi$
$467$	5.11862e6	1.08608	0.543039	+	0.839708i	$0.317274\pi$
$468$	0	0
$469$	−6.90494e6	−1.44953
$470$	0	0
$471$	−355562.	−0.0738521
$472$	0	0
$473$	−4.64580e6	−0.954790
$474$	0	0
$475$	−603738.	−0.122776
$476$	0	0
$477$	2.03024e6	0.408555
$478$	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
$483$	−4.89940e6	−0.955598
$484$	0	0
$485$	−2.72095e6	−0.525250
$486$	0	0
$487$	1.68175e6	0.321321	0.160661	−	0.987010i	$-0.448638\pi$
$487$	1.68175e6	0.321321	0.160661	+	0.987010i	$0.448638\pi$
$488$	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.	−0.135564
$494$	0	0
$495$	408530.	0.0749395
$496$	0	0
$497$	7.50076e6	1.36212
$498$	0	0
$499$	7.98867e6	1.43623	0.718113	−	0.695926i	$-0.245006\pi$
$499$	7.98867e6	1.43623	0.718113	+	0.695926i	$0.245006\pi$
$500$	0	0
$501$	5.51478e6	0.981598
$502$	0	0
$503$	8.07650e6	1.42332	0.711661	−	0.702523i	$-0.247943\pi$
$503$	8.07650e6	1.42332	0.711661	+	0.702523i	$0.247943\pi$
$504$	0	0
$505$	−1.47995e6	−0.258237
$506$	0	0
$507$	4.66323e6	0.805689
$508$	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
$513$	−3.96576e6	−0.665324
$514$	0	0
$515$	−2.82270e6	−0.468972
$516$	0	0
$517$	1.92212e6	0.316267
$518$	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
$523$	−7.72888e6	−1.23556	−0.617778	−	0.786352i	$-0.711967\pi$
$523$	−7.72888e6	−1.23556	−0.617778	+	0.786352i	$0.711967\pi$
$524$	0	0
$525$	−1.16250e6	−0.184075
$526$	0	0
$527$	562004.	0.0881481
$528$	0	0
$529$	502077.	0.0780066
$530$	0	0
$531$	2.14464e6	0.330079
$532$	0	0
$533$	−1.11958e6	−0.170701
$534$	0	0
$535$	−1.00098e6	−0.151196
$536$	0	0
$537$	−4.03752e6	−0.604198
$538$	0	0
$539$	−625893.	−0.0927958
$540$	0	0
$541$	9.83660e6	1.44495	0.722474	−	0.691399i	$-0.243005\pi$
$541$	9.83660e6	1.44495	0.722474	+	0.691399i	$0.243005\pi$
$542$	0	0
$543$	2.91166e6	0.423780
$544$	0	0
$545$	3.49035e6	0.503359
$546$	0	0
$547$	3.45608e6	0.493873	0.246937	−	0.969032i	$-0.420576\pi$
$547$	3.45608e6	0.493873	0.246937	+	0.969032i	$0.420576\pi$
$548$	0	0
$549$	1.68651e6	0.238813
$550$	0	0
$551$	3.97018e6	0.557098
$552$	0	0
$553$	−1.20255e7	−1.67221
$554$	0	0
$555$	2.49612e6	0.343980
$556$	0	0
$557$	7.05760e6	0.963871	0.481936	−	0.876207i	$-0.339934\pi$
$557$	7.05760e6	0.963871	0.481936	+	0.876207i	$0.339934\pi$
$558$	0	0
$559$	−2.75828e6	−0.373344
$560$	0	0
$561$	619440.	0.0830983
$562$	0	0
$563$	−2.32495e6	−0.309131	−0.154566	−	0.987983i	$-0.549398\pi$
$563$	−2.32495e6	−0.309131	−0.154566	+	0.987983i	$0.549398\pi$
$564$	0	0
$565$	−1.07615e6	−0.141825
$566$	0	0
$567$	−5.51370e6	−0.720254
$568$	0	0
$569$	1.01947e7	1.32006	0.660029	−	0.751240i	$-0.270544\pi$
$569$	1.01947e7	1.32006	0.660029	+	0.751240i	$0.270544\pi$
$570$	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
$573$	1.22929e7	1.56411
$574$	0	0
$575$	1.64631e6	0.207654
$576$	0	0
$577$	1.46358e6	0.183011	0.0915053	−	0.995805i	$-0.470832\pi$
$577$	1.46358e6	0.183011	0.0915053	+	0.995805i	$0.470832\pi$
$578$	0	0
$579$	−1.15933e7	−1.43718
$580$	0	0
$581$	1.09027e7	1.33997
$582$	0	0
$583$	8.35891e6	1.01854
$584$	0	0
$585$	242550.	0.0293030
$586$	0	0
$587$	−5.18425e6	−0.620998	−0.310499	−	0.950574i	$-0.600496\pi$
$587$	−5.18425e6	−0.620998	−0.310499	+	0.950574i	$0.600496\pi$
$588$	0	0
$589$	−3.04992e6	−0.362243
$590$	0	0
$591$	5.80289e6	0.683401
$592$	0	0
$593$	−1.02722e7	−1.19957	−0.599785	−	0.800161i	$-0.704747\pi$
$593$	−1.02722e7	−1.19957	−0.599785	+	0.800161i	$0.704747\pi$
$594$	0	0
$595$	616931.	0.0714405
$596$	0	0
$597$	1.06685e7	1.22509
$598$	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
$603$	3.13779e6	0.351423
$604$	0	0
$605$	−2.34427e6	−0.260387
$606$	0	0
$607$	−1.04800e7	−1.15449	−0.577245	−	0.816571i	$-0.695872\pi$
$607$	−1.04800e7	−1.15449	−0.577245	+	0.816571i	$0.695872\pi$
$608$	0	0
$609$	7.64460e6	0.835240
$610$	0	0
$611$	1.14119e6	0.123667
$612$	0	0
$613$	3.07977e6	0.331029	0.165515	−	0.986207i	$-0.447071\pi$
$613$	3.07977e6	0.331029	0.165515	+	0.986207i	$0.447071\pi$
$614$	0	0
$615$	−2.43843e6	−0.259970
$616$	0	0
$617$	1.15522e6	0.122166	0.0610831	−	0.998133i	$-0.480545\pi$
$617$	1.15522e6	0.122166	0.0610831	+	0.998133i	$0.480545\pi$
$618$	0	0
$619$	−1.76853e7	−1.85517	−0.927587	−	0.373606i	$-0.878121\pi$
$619$	−1.76853e7	−1.85517	−0.927587	+	0.373606i	$0.878121\pi$
$620$	0	0
$621$	1.08140e7	1.12528
$622$	0	0
$623$	−1.49159e7	−1.53967
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−3.36162e6	−0.341491
$628$	0	0
$629$	−1.32468e6	−0.133501
$630$	0	0
$631$	−5.42541e6	−0.542449	−0.271225	−	0.962516i	$-0.587429\pi$
$631$	−5.42541e6	−0.542449	−0.271225	+	0.962516i	$0.587429\pi$
$632$	0	0
$633$	−6.07932e6	−0.603039
$634$	0	0
$635$	−3.93425e6	−0.387193
$636$	0	0
$637$	−371602.	−0.0362852
$638$	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
$643$	9.57873e6	0.913651	0.456826	−	0.889556i	$-0.348986\pi$
$643$	9.57873e6	0.913651	0.456826	+	0.889556i	$0.348986\pi$
$644$	0	0
$645$	−6.00750e6	−0.568584
$646$	0	0
$647$	1.28130e7	1.20334	0.601671	−	0.798744i	$-0.294502\pi$
$647$	1.28130e7	1.20334	0.601671	+	0.798744i	$0.294502\pi$
$648$	0	0
$649$	8.82992e6	0.822896
$650$	0	0
$651$	−5.87263e6	−0.543100
$652$	0	0
$653$	1.58665e6	0.145613	0.0728064	−	0.997346i	$-0.476804\pi$
$653$	1.58665e6	0.145613	0.0728064	+	0.997346i	$0.476804\pi$
$654$	0	0
$655$	6.69322e6	0.609582
$656$	0	0
$657$	1.16764e6	0.105535
$658$	0	0
$659$	1.95559e7	1.75414	0.877071	−	0.480361i	$-0.159494\pi$
$659$	1.95559e7	1.75414	0.877071	+	0.480361i	$0.159494\pi$
$660$	0	0
$661$	7.46471e6	0.664522	0.332261	−	0.943188i	$-0.392189\pi$
$661$	7.46471e6	0.664522	0.332261	+	0.943188i	$0.392189\pi$
$662$	0	0
$663$	367771.	0.0324933
$664$	0	0
$665$	−3.34800e6	−0.293583
$666$	0	0
$667$	−1.08261e7	−0.942231
$668$	0	0
$669$	5.77386e6	0.498771
$670$	0	0
$671$	6.94371e6	0.595367
$672$	0	0
$673$	−1.09694e7	−0.933568	−0.466784	−	0.884371i	$-0.654587\pi$
$673$	−1.09694e7	−0.933568	−0.466784	+	0.884371i	$0.654587\pi$
$674$	0	0
$675$	2.56589e6	0.216760
$676$	0	0
$677$	6.52708e6	0.547327	0.273664	−	0.961825i	$-0.411764\pi$
$677$	6.52708e6	0.547327	0.273664	+	0.961825i	$0.411764\pi$
$678$	0	0
$679$	−1.50889e7	−1.25598
$680$	0	0
$681$	1.50805e7	1.24608
$682$	0	0
$683$	1.54389e7	1.26638	0.633191	−	0.773995i	$-0.281745\pi$
$683$	1.54389e7	1.26638	0.633191	+	0.773995i	$0.281745\pi$
$684$	0	0
$685$	−2.80395e6	−0.228320
$686$	0	0
$687$	−1.08996e7	−0.881089
$688$	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
$693$	2.26548e6	0.179196
$694$	0	0
$695$	−3.68370e6	−0.289282
$696$	0	0
$697$	1.29406e6	0.100896
$698$	0	0
$699$	−1.13529e7	−0.878847
$700$	0	0
$701$	3.31891e6	0.255094	0.127547	−	0.991833i	$-0.459290\pi$
$701$	3.31891e6	0.255094	0.127547	+	0.991833i	$0.459290\pi$
$702$	0	0
$703$	7.18883e6	0.548618
$704$	0	0
$705$	2.48550e6	0.188339
$706$	0	0
$707$	−8.20699e6	−0.617497
$708$	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
$713$	8.31668e6	0.612669
$714$	0	0
$715$	998628.	0.0730531
$716$	0	0
$717$	760560.	0.0552504
$718$	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
$723$	−1.71154e7	−1.21771
$724$	0	0
$725$	−2.56875e6	−0.181500
$726$	0	0
$727$	1.14545e7	0.803783	0.401891	−	0.915687i	$-0.368353\pi$
$727$	1.14545e7	0.803783	0.401891	+	0.915687i	$0.368353\pi$
$728$	0	0
$729$	1.58900e7	1.10740
$730$	0	0
$731$	3.18814e6	0.220670
$732$	0	0
$733$	−1.13267e7	−0.778650	−0.389325	−	0.921101i	$-0.627292\pi$
$733$	−1.13267e7	−0.778650	−0.389325	+	0.921101i	$0.627292\pi$
$734$	0	0
$735$	−809345.	−0.0552606
$736$	0	0
$737$	1.29189e7	0.876108
$738$	0	0
$739$	1.31055e7	0.882756	0.441378	−	0.897321i	$-0.354490\pi$
$739$	1.31055e7	0.882756	0.441378	+	0.897321i	$0.354490\pi$
$740$	0	0
$741$	−1.99584e6	−0.133530
$742$	0	0
$743$	−4.16787e6	−0.276976	−0.138488	−	0.990364i	$-0.544224\pi$
$743$	−4.16787e6	−0.276976	−0.138488	+	0.990364i	$0.544224\pi$
$744$	0	0
$745$	4.36415e6	0.288077
$746$	0	0
$747$	−4.95448e6	−0.324860
$748$	0	0
$749$	−5.55086e6	−0.361539
$750$	0	0
$751$	2.89308e7	1.87180	0.935902	−	0.352260i	$-0.114587\pi$
$751$	2.89308e7	1.87180	0.935902	+	0.352260i	$0.114587\pi$
$752$	0	0
$753$	−683400.	−0.0439225
$754$	0	0
$755$	8.63145e6	0.551082
$756$	0	0
$757$	−8.32870e6	−0.528247	−0.264124	−	0.964489i	$-0.585083\pi$
$757$	−8.32870e6	−0.528247	−0.264124	+	0.964489i	$0.585083\pi$
$758$	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
$763$	1.93556e7	1.20363
$764$	0	0
$765$	−280350.	−0.0173200
$766$	0	0
$767$	5.24245e6	0.321770
$768$	0	0
$769$	1.32141e6	0.0805790	0.0402895	−	0.999188i	$-0.487172\pi$
$769$	1.32141e6	0.0805790	0.0402895	+	0.999188i	$0.487172\pi$
$770$	0	0
$771$	−2.21110e7	−1.33959
$772$	0	0
$773$	7.24897e6	0.436342	0.218171	−	0.975911i	$-0.429991\pi$
$773$	7.24897e6	0.436342	0.218171	+	0.975911i	$0.429991\pi$
$774$	0	0
$775$	1.97333e6	0.118017
$776$	0	0
$777$	1.38421e7	0.822527
$778$	0	0
$779$	−7.02268e6	−0.414629
$780$	0	0
$781$	−1.40337e7	−0.823273
$782$	0	0
$783$	−1.68733e7	−0.983547
$784$	0	0
$785$	662550.	0.0383747
$786$	0	0
$787$	−4.98037e6	−0.286632	−0.143316	−	0.989677i	$-0.545777\pi$
$787$	−4.98037e6	−0.286632	−0.143316	+	0.989677i	$0.545777\pi$
$788$	0	0
$789$	−7.01670e6	−0.401273
$790$	0	0
$791$	−5.96773e6	−0.339132
$792$	0	0
$793$	4.12258e6	0.232802
$794$	0	0
$795$	1.08089e7	0.606548
$796$	0	0
$797$	−1.48673e7	−0.829061	−0.414530	−	0.910036i	$-0.636054\pi$
$797$	−1.48673e7	−0.829061	−0.414530	+	0.910036i	$0.636054\pi$
$798$	0	0
$799$	−1.31904e6	−0.0730955
$800$	0	0
$801$	6.77817e6	0.373277
$802$	0	0
$803$	4.80742e6	0.263101
$804$	0	0
$805$	9.12950e6	0.496543
$806$	0	0
$807$	−2.59995e7	−1.40534
$808$	0	0
$809$	−684390.	−0.0367648	−0.0183824	−	0.999831i	$-0.505852\pi$
$809$	−684390.	−0.0367648	−0.0183824	+	0.999831i	$0.505852\pi$
$810$	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
$813$	−1.34341e7	−0.712825
$814$	0	0
$815$	−3.54271e6	−0.186828
$816$	0	0
$817$	−1.73016e7	−0.906841
$818$	0	0
$819$	1.34505e6	0.0700694
$820$	0	0
$821$	−7.17265e6	−0.371383	−0.185691	−	0.982608i	$-0.559452\pi$
$821$	−7.17265e6	−0.371383	−0.185691	+	0.982608i	$0.559452\pi$
$822$	0	0
$823$	1.32939e6	0.0684153	0.0342077	−	0.999415i	$-0.489109\pi$
$823$	1.32939e6	0.0684153	0.0342077	+	0.999415i	$0.489109\pi$
$824$	0	0
$825$	2.17500e6	0.111256
$826$	0	0
$827$	1.92964e7	0.981099	0.490550	−	0.871413i	$-0.336796\pi$
$827$	1.92964e7	0.981099	0.490550	+	0.871413i	$0.336796\pi$
$828$	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.	0.0214469
$834$	0	0
$835$	−1.02762e7	−0.510054
$836$	0	0
$837$	1.29622e7	0.639534
$838$	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
$843$	−1.67231e7	−0.810493
$844$	0	0
$845$	−8.68942e6	−0.418648
$846$	0	0
$847$	−1.30001e7	−0.622640
$848$	0	0
$849$	−3.05761e7	−1.45584
$850$	0	0
$851$	−1.96029e7	−0.927889
$852$	0	0
$853$	−1.86921e7	−0.879599	−0.439800	−	0.898096i	$-0.644951\pi$
$853$	−1.86921e7	−0.879599	−0.439800	+	0.898096i	$0.644951\pi$
$854$	0	0
$855$	1.52142e6	0.0711761
$856$	0	0
$857$	−2.24473e7	−1.04403	−0.522013	−	0.852937i	$-0.674819\pi$
$857$	−2.24473e7	−1.04403	−0.522013	+	0.852937i	$0.674819\pi$
$858$	0	0
$859$	−3.24436e6	−0.150019	−0.0750094	−	0.997183i	$-0.523899\pi$
$859$	−3.24436e6	−0.150019	−0.0750094	+	0.997183i	$0.523899\pi$
$860$	0	0
$861$	−1.35222e7	−0.621641
$862$	0	0
$863$	−1.01542e6	−0.0464108	−0.0232054	−	0.999731i	$-0.507387\pi$
$863$	−1.01542e6	−0.0464108	−0.0232054	+	0.999731i	$0.507387\pi$
$864$	0	0
$865$	−1.48987e7	−0.677028
$866$	0	0
$867$	1.86243e7	0.841457
$868$	0	0
$869$	2.24994e7	1.01070
$870$	0	0
$871$	7.67015e6	0.342577
$872$	0	0
$873$	6.85679e6	0.304499
$874$	0	0
$875$	2.16619e6	0.0956482
$876$	0	0
$877$	−3.96764e7	−1.74194	−0.870970	−	0.491337i	$-0.836508\pi$
$877$	−3.96764e7	−1.74194	−0.870970	+	0.491337i	$0.836508\pi$
$878$	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
$883$	−3.11179e7	−1.34310	−0.671549	−	0.740960i	$-0.734371\pi$
$883$	−3.11179e7	−1.34310	−0.671549	+	0.740960i	$0.734371\pi$
$884$	0	0
$885$	1.14180e7	0.490040
$886$	0	0
$887$	1.87514e7	0.800249	0.400124	−	0.916461i	$-0.368967\pi$
$887$	1.87514e7	0.800249	0.400124	+	0.916461i	$0.368967\pi$
$888$	0	0
$889$	−2.18172e7	−0.925858
$890$	0	0
$891$	1.03160e7	0.435327
$892$	0	0
$893$	7.15824e6	0.300385
$894$	0	0
$895$	7.52347e6	0.313950
$896$	0	0
$897$	5.44236e6	0.225843
$898$	0	0
$899$	−1.29766e7	−0.535503
$900$	0	0
$901$	−5.73623e6	−0.235404
$902$	0	0
$903$	−3.33143e7	−1.35960
$904$	0	0
$905$	−5.42555e6	−0.220203
$906$	0	0
$907$	−5.07504e6	−0.204843	−0.102421	−	0.994741i	$-0.532659\pi$
$907$	−5.07504e6	−0.204843	−0.102421	+	0.994741i	$0.532659\pi$
$908$	0	0
$909$	3.72947e6	0.149706
$910$	0	0
$911$	−2.83599e7	−1.13216	−0.566082	−	0.824349i	$-0.691541\pi$
$911$	−2.83599e7	−1.13216	−0.566082	+	0.824349i	$0.691541\pi$
$912$	0	0
$913$	−2.03986e7	−0.809885
$914$	0	0
$915$	8.97893e6	0.354545
$916$	0	0
$917$	3.71169e7	1.45763
$918$	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
$923$	−8.33200e6	−0.321918
$924$	0	0
$925$	−4.65125e6	−0.178737
$926$	0	0
$927$	7.11321e6	0.271873
$928$	0	0
$929$	−3.82212e7	−1.45300	−0.726500	−	0.687167i	$-0.758854\pi$
$929$	−3.82212e7	−1.45300	−0.726500	+	0.687167i	$0.758854\pi$
$930$	0	0
$931$	−2.33091e6	−0.0881357
$932$	0	0
$933$	634200.	0.0238519
$934$	0	0
$935$	−1.15426e6	−0.0431791
$936$	0	0
$937$	−9.37422e6	−0.348808	−0.174404	−	0.984674i	$-0.555800\pi$
$937$	−9.37422e6	−0.348808	−0.174404	+	0.984674i	$0.555800\pi$
$938$	0	0
$939$	3.74544e7	1.38624
$940$	0	0
$941$	−3.04898e6	−0.112249	−0.0561243	−	0.998424i	$-0.517874\pi$
$941$	−3.04898e6	−0.112249	−0.0561243	+	0.998424i	$0.517874\pi$
$942$	0	0
$943$	1.91498e7	0.701270
$944$	0	0
$945$	1.42290e7	0.518316
$946$	0	0
$947$	−2.95948e7	−1.07236	−0.536180	−	0.844104i	$-0.680133\pi$
$947$	−2.95948e7	−1.07236	−0.536180	+	0.844104i	$0.680133\pi$
$948$	0	0
$949$	2.85424e6	0.102878
$950$	0	0
$951$	−6.32694e6	−0.226852
$952$	0	0
$953$	1.58338e7	0.564744	0.282372	−	0.959305i	$-0.408879\pi$
$953$	1.58338e7	0.564744	0.282372	+	0.959305i	$0.408879\pi$
$954$	0	0
$955$	−2.29065e7	−0.812738
$956$	0	0
$957$	−1.43028e7	−0.504825
$958$	0	0
$959$	−1.55492e7	−0.545960
$960$	0	0
$961$	−1.86604e7	−0.651798
$962$	0	0
$963$	2.52246e6	0.0876513
$964$	0	0
$965$	2.16028e7	0.746780
$966$	0	0
$967$	8.69009e6	0.298853	0.149427	−	0.988773i	$-0.452257\pi$
$967$	8.69009e6	0.298853	0.149427	+	0.988773i	$0.452257\pi$
$968$	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
$973$	−2.04278e7	−0.691733
$974$	0	0
$975$	1.29133e6	0.0435036
$976$	0	0
$977$	−4.34090e7	−1.45493	−0.727467	−	0.686143i	$-0.759303\pi$
$977$	−4.34090e7	−1.45493	−0.727467	+	0.686143i	$0.759303\pi$
$978$	0	0
$979$	2.79071e7	0.930590
$980$	0	0
$981$	−8.79568e6	−0.291808
$982$	0	0
$983$	4.64172e7	1.53213	0.766065	−	0.642763i	$-0.222212\pi$
$983$	4.64172e7	1.53213	0.766065	+	0.642763i	$0.222212\pi$
$984$	0	0
$985$	−1.08131e7	−0.355106
$986$	0	0
$987$	1.37832e7	0.450358
$988$	0	0
$989$	4.71789e7	1.53376
$990$	0	0
$991$	−4.64577e7	−1.50271	−0.751353	−	0.659901i	$-0.770598\pi$
$991$	−4.64577e7	−1.50271	−0.751353	+	0.659901i	$0.770598\pi$
$992$	0	0
$993$	4.14690e7	1.33460
$994$	0	0
$995$	−1.98795e7	−0.636574
$996$	0	0
$997$	−1.35968e7	−0.433211	−0.216605	−	0.976259i	$-0.569499\pi$
$997$	−1.35968e7	−0.433211	−0.216605	+	0.976259i	$0.569499\pi$
$998$	0	0
$999$	−3.05525e7	−0.968576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.a.t.1.1		2
4.3	odd	2	inner	320.6.a.t.1.2		2
8.3	odd	2		160.6.a.b.1.1	✓	2
8.5	even	2		160.6.a.b.1.2	yes	2
40.3	even	4		800.6.c.h.449.1		4
40.13	odd	4		800.6.c.h.449.4		4
40.19	odd	2		800.6.a.i.1.2		2
40.27	even	4		800.6.c.h.449.3		4
40.29	even	2		800.6.a.i.1.1		2
40.37	odd	4		800.6.c.h.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.b.1.1	✓	2	8.3	odd	2
160.6.a.b.1.2	yes	2	8.5	even	2
320.6.a.t.1.1		2	1.1	even	1	trivial
320.6.a.t.1.2		2	4.3	odd	2	inner
800.6.a.i.1.1		2	40.29	even	2
800.6.a.i.1.2		2	40.19	odd	2
800.6.c.h.449.1		4	40.3	even	4
800.6.c.h.449.2		4	40.37	odd	4
800.6.c.h.449.3		4	40.27	even	4
800.6.c.h.449.4		4	40.13	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

\(f(q)\)	\(=\)	\(q-13.4164 q^{3} +25.0000 q^{5} +138.636 q^{7} -63.0000 q^{9} +O(q^{10})\)	\(q-13.4164 q^{3} +25.0000 q^{5} +138.636 q^{7} -63.0000 q^{9} -259.384 q^{11} -154.000 q^{13} -335.410 q^{15} +178.000 q^{17} -965.981 q^{19} -1860.00 q^{21} +2634.09 q^{23} +625.000 q^{25} +4105.42 q^{27} -4110.00 q^{29} +3157.33 q^{31} +3480.00 q^{33} +3465.91 q^{35} -7442.00 q^{37} +2066.13 q^{39} +7270.00 q^{41} +17910.9 q^{43} -1575.00 q^{45} -7410.33 q^{47} +2413.00 q^{49} -2388.12 q^{51} -32226.0 q^{53} -6484.60 q^{55} +12960.0 q^{57} -34041.9 q^{59} -26770.0 q^{61} -8734.08 q^{63} -3850.00 q^{65} -49806.2 q^{67} -35340.0 q^{69} +54103.9 q^{71} -18534.0 q^{73} -8385.25 q^{75} -35960.0 q^{77} -86741.5 q^{79} -39771.0 q^{81} +78642.5 q^{83} +4450.00 q^{85} +55141.4 q^{87} -107590. q^{89} -21350.0 q^{91} -42360.0 q^{93} -24149.5 q^{95} -108838. q^{97} +16341.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 50 q^{5} - 126 q^{9}+O(q^{10}) \) 2 * q + 50 * q^5 - 126 * q^9	\( 2 q + 50 q^{5} - 126 q^{9} - 308 q^{13} + 356 q^{17} - 3720 q^{21} + 1250 q^{25} - 8220 q^{29} + 6960 q^{33} - 14884 q^{37} + 14540 q^{41} - 3150 q^{45} + 4826 q^{49} - 64452 q^{53} + 25920 q^{57} - 53540 q^{61} - 7700 q^{65} - 70680 q^{69} - 37068 q^{73} - 71920 q^{77} - 79542 q^{81} + 8900 q^{85} - 215180 q^{89} - 84720 q^{93} - 217676 q^{97}+O(q^{100}) \) 2 * q + 50 * q^5 - 126 * q^9 - 308 * q^13 + 356 * q^17 - 3720 * q^21 + 1250 * q^25 - 8220 * q^29 + 6960 * q^33 - 14884 * q^37 + 14540 * q^41 - 3150 * q^45 + 4826 * q^49 - 64452 * q^53 + 25920 * q^57 - 53540 * q^61 - 7700 * q^65 - 70680 * q^69 - 37068 * q^73 - 71920 * q^77 - 79542 * q^81 + 8900 * q^85 - 215180 * q^89 - 84720 * q^93 - 217676 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more