LMFDB - Embedded newform 16.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,6,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("16.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	16.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:	$2.56614111701$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+12.0000 q^{3} +54.0000 q^{5} +88.0000 q^{7} -99.0000 q^{9} +O(q^{10})$

$q+12.0000 q^{3} +54.0000 q^{5} +88.0000 q^{7} -99.0000 q^{9} -540.000 q^{11} -418.000 q^{13} +648.000 q^{15} +594.000 q^{17} -836.000 q^{19} +1056.00 q^{21} +4104.00 q^{23} -209.000 q^{25} -4104.00 q^{27} -594.000 q^{29} -4256.00 q^{31} -6480.00 q^{33} +4752.00 q^{35} -298.000 q^{37} -5016.00 q^{39} +17226.0 q^{41} +12100.0 q^{43} -5346.00 q^{45} +1296.00 q^{47} -9063.00 q^{49} +7128.00 q^{51} +19494.0 q^{53} -29160.0 q^{55} -10032.0 q^{57} +7668.00 q^{59} -34738.0 q^{61} -8712.00 q^{63} -22572.0 q^{65} -21812.0 q^{67} +49248.0 q^{69} +46872.0 q^{71} +67562.0 q^{73} -2508.00 q^{75} -47520.0 q^{77} +76912.0 q^{79} -25191.0 q^{81} -67716.0 q^{83} +32076.0 q^{85} -7128.00 q^{87} +29754.0 q^{89} -36784.0 q^{91} -51072.0 q^{93} -45144.0 q^{95} -122398. q^{97} +53460.0 q^{99} +O(q^{100})$

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
$3$	12.0000	0.769800	0.384900	−	0.922958i	$-0.374236\pi$
$3$	12.0000	0.769800	0.384900	+	0.922958i	$0.374236\pi$
$4$	0	0
$5$	54.0000	0.965981	0.482991	−	0.875625i	$-0.339550\pi$
$5$	54.0000	0.965981	0.482991	+	0.875625i	$0.339550\pi$
$6$	0	0
$7$	88.0000	0.678793	0.339397	−	0.940643i	$-0.389777\pi$
$7$	88.0000	0.678793	0.339397	+	0.940643i	$0.389777\pi$
$8$	0	0
$9$	−99.0000	−0.407407
$10$	0	0
$11$	−540.000	−1.34559	−0.672794	−	0.739830i	$-0.734906\pi$
$11$	−540.000	−1.34559	−0.672794	+	0.739830i	$0.734906\pi$
$12$	0	0
$13$	−418.000	−0.685990	−0.342995	−	0.939337i	$-0.611441\pi$
$13$	−418.000	−0.685990	−0.342995	+	0.939337i	$0.611441\pi$
$14$	0	0
$15$	648.000	0.743613
$16$	0	0
$17$	594.000	0.498499	0.249249	−	0.968439i	$-0.419816\pi$
$17$	594.000	0.498499	0.249249	+	0.968439i	$0.419816\pi$
$18$	0	0
$19$	−836.000	−0.531279	−0.265639	−	0.964072i	$-0.585583\pi$
$19$	−836.000	−0.531279	−0.265639	+	0.964072i	$0.585583\pi$
$20$	0	0
$21$	1056.00	0.522535
$22$	0	0
$23$	4104.00	1.61766	0.808831	−	0.588041i	$-0.200101\pi$
$23$	4104.00	1.61766	0.808831	+	0.588041i	$0.200101\pi$
$24$	0	0
$25$	−209.000	−0.0668800
$26$	0	0
$27$	−4104.00	−1.08342
$28$	0	0
$29$	−594.000	−0.131157	−0.0655785	−	0.997847i	$-0.520889\pi$
$29$	−594.000	−0.131157	−0.0655785	+	0.997847i	$0.520889\pi$
$30$	0	0
$31$	−4256.00	−0.795422	−0.397711	−	0.917511i	$-0.630195\pi$
$31$	−4256.00	−0.795422	−0.397711	+	0.917511i	$0.630195\pi$
$32$	0	0
$33$	−6480.00	−1.03583
$34$	0	0
$35$	4752.00	0.655702
$36$	0	0
$37$	−298.000	−0.0357859	−0.0178930	−	0.999840i	$-0.505696\pi$
$37$	−298.000	−0.0357859	−0.0178930	+	0.999840i	$0.505696\pi$
$38$	0	0
$39$	−5016.00	−0.528075
$40$	0	0
$41$	17226.0	1.60039	0.800193	−	0.599742i	$-0.204730\pi$
$41$	17226.0	1.60039	0.800193	+	0.599742i	$0.204730\pi$
$42$	0	0
$43$	12100.0	0.997963	0.498981	−	0.866613i	$-0.333708\pi$
$43$	12100.0	0.997963	0.498981	+	0.866613i	$0.333708\pi$
$44$	0	0
$45$	−5346.00	−0.393548
$46$	0	0
$47$	1296.00	0.0855777	0.0427888	−	0.999084i	$-0.486376\pi$
$47$	1296.00	0.0855777	0.0427888	+	0.999084i	$0.486376\pi$
$48$	0	0
$49$	−9063.00	−0.539240
$50$	0	0
$51$	7128.00	0.383745
$52$	0	0
$53$	19494.0	0.953260	0.476630	−	0.879104i	$-0.341858\pi$
$53$	19494.0	0.953260	0.476630	+	0.879104i	$0.341858\pi$
$54$	0	0
$55$	−29160.0	−1.29981
$56$	0	0
$57$	−10032.0	−0.408978
$58$	0	0
$59$	7668.00	0.286782	0.143391	−	0.989666i	$-0.454199\pi$
$59$	7668.00	0.286782	0.143391	+	0.989666i	$0.454199\pi$
$60$	0	0
$61$	−34738.0	−1.19531	−0.597655	−	0.801754i	$-0.703901\pi$
$61$	−34738.0	−1.19531	−0.597655	+	0.801754i	$0.703901\pi$
$62$	0	0
$63$	−8712.00	−0.276545
$64$	0	0
$65$	−22572.0	−0.662654
$66$	0	0
$67$	−21812.0	−0.593620	−0.296810	−	0.954937i	$-0.595923\pi$
$67$	−21812.0	−0.593620	−0.296810	+	0.954937i	$0.595923\pi$
$68$	0	0
$69$	49248.0	1.24528
$70$	0	0
$71$	46872.0	1.10349	0.551744	−	0.834014i	$-0.313963\pi$
$71$	46872.0	1.10349	0.551744	+	0.834014i	$0.313963\pi$
$72$	0	0
$73$	67562.0	1.48387	0.741934	−	0.670473i	$-0.233909\pi$
$73$	67562.0	1.48387	0.741934	+	0.670473i	$0.233909\pi$
$74$	0	0
$75$	−2508.00	−0.0514842
$76$	0	0
$77$	−47520.0	−0.913376
$78$	0	0
$79$	76912.0	1.38652	0.693260	−	0.720687i	$-0.256174\pi$
$79$	76912.0	1.38652	0.693260	+	0.720687i	$0.256174\pi$
$80$	0	0
$81$	−25191.0	−0.426612
$82$	0	0
$83$	−67716.0	−1.07894	−0.539468	−	0.842006i	$-0.681375\pi$
$83$	−67716.0	−1.07894	−0.539468	+	0.842006i	$0.681375\pi$
$84$	0	0
$85$	32076.0	0.481541
$86$	0	0
$87$	−7128.00	−0.100965
$88$	0	0
$89$	29754.0	0.398172	0.199086	−	0.979982i	$-0.436203\pi$
$89$	29754.0	0.398172	0.199086	+	0.979982i	$0.436203\pi$
$90$	0	0
$91$	−36784.0	−0.465646
$92$	0	0
$93$	−51072.0	−0.612316
$94$	0	0
$95$	−45144.0	−0.513205
$96$	0	0
$97$	−122398.	−1.32082	−0.660412	−	0.750903i	$-0.729618\pi$
$97$	−122398.	−1.32082	−0.660412	+	0.750903i	$0.729618\pi$
$98$	0	0
$99$	53460.0	0.548202
$100$	0	0
$101$	11286.0	0.110087	0.0550436	−	0.998484i	$-0.482470\pi$
$101$	11286.0	0.110087	0.0550436	+	0.998484i	$0.482470\pi$
$102$	0	0
$103$	27256.0	0.253145	0.126572	−	0.991957i	$-0.459602\pi$
$103$	27256.0	0.253145	0.126572	+	0.991957i	$0.459602\pi$
$104$	0	0
$105$	57024.0	0.504759
$106$	0	0
$107$	−122364.	−1.03322	−0.516612	−	0.856220i	$-0.672807\pi$
$107$	−122364.	−1.03322	−0.516612	+	0.856220i	$0.672807\pi$
$108$	0	0
$109$	99902.0	0.805393	0.402697	−	0.915334i	$-0.368073\pi$
$109$	99902.0	0.805393	0.402697	+	0.915334i	$0.368073\pi$
$110$	0	0
$111$	−3576.00	−0.0275480
$112$	0	0
$113$	−29646.0	−0.218409	−0.109204	−	0.994019i	$-0.534830\pi$
$113$	−29646.0	−0.218409	−0.109204	+	0.994019i	$0.534830\pi$
$114$	0	0
$115$	221616.	1.56263
$116$	0	0
$117$	41382.0	0.279477
$118$	0	0
$119$	52272.0	0.338378
$120$	0	0
$121$	130549.	0.810607
$122$	0	0
$123$	206712.	1.23198
$124$	0	0
$125$	−180036.	−1.03059
$126$	0	0
$127$	−336512.	−1.85136	−0.925681	−	0.378305i	$-0.876507\pi$
$127$	−336512.	−1.85136	−0.925681	+	0.378305i	$0.876507\pi$
$128$	0	0
$129$	145200.	0.768232
$130$	0	0
$131$	−100980.	−0.514111	−0.257056	−	0.966397i	$-0.582752\pi$
$131$	−100980.	−0.514111	−0.257056	+	0.966397i	$0.582752\pi$
$132$	0	0
$133$	−73568.0	−0.360628
$134$	0	0
$135$	−221616.	−1.04657
$136$	0	0
$137$	−317142.	−1.44362	−0.721809	−	0.692092i	$-0.756689\pi$
$137$	−317142.	−1.44362	−0.721809	+	0.692092i	$0.756689\pi$
$138$	0	0
$139$	148324.	0.651140	0.325570	−	0.945518i	$-0.394444\pi$
$139$	148324.	0.651140	0.325570	+	0.945518i	$0.394444\pi$
$140$	0	0
$141$	15552.0	0.0658777
$142$	0	0
$143$	225720.	0.923060
$144$	0	0
$145$	−32076.0	−0.126695
$146$	0	0
$147$	−108756.	−0.415107
$148$	0	0
$149$	196614.	0.725519	0.362759	−	0.931883i	$-0.381835\pi$
$149$	196614.	0.725519	0.362759	+	0.931883i	$0.381835\pi$
$150$	0	0
$151$	−74360.0	−0.265398	−0.132699	−	0.991156i	$-0.542364\pi$
$151$	−74360.0	−0.265398	−0.132699	+	0.991156i	$0.542364\pi$
$152$	0	0
$153$	−58806.0	−0.203092
$154$	0	0
$155$	−229824.	−0.768362
$156$	0	0
$157$	120878.	0.391380	0.195690	−	0.980666i	$-0.437305\pi$
$157$	120878.	0.391380	0.195690	+	0.980666i	$0.437305\pi$
$158$	0	0
$159$	233928.	0.733820
$160$	0	0
$161$	361152.	1.09806
$162$	0	0
$163$	111340.	0.328233	0.164116	−	0.986441i	$-0.447523\pi$
$163$	111340.	0.328233	0.164116	+	0.986441i	$0.447523\pi$
$164$	0	0
$165$	−349920.	−1.00060
$166$	0	0
$167$	491832.	1.36466	0.682332	−	0.731043i	$-0.260966\pi$
$167$	491832.	1.36466	0.682332	+	0.731043i	$0.260966\pi$
$168$	0	0
$169$	−196569.	−0.529417
$170$	0	0
$171$	82764.0	0.216447
$172$	0	0
$173$	707454.	1.79714	0.898572	−	0.438826i	$-0.144605\pi$
$173$	707454.	1.79714	0.898572	+	0.438826i	$0.144605\pi$
$174$	0	0
$175$	−18392.0	−0.0453977
$176$	0	0
$177$	92016.0	0.220765
$178$	0	0
$179$	−493668.	−1.15160	−0.575801	−	0.817590i	$-0.695310\pi$
$179$	−493668.	−1.15160	−0.575801	+	0.817590i	$0.695310\pi$
$180$	0	0
$181$	−559450.	−1.26930	−0.634651	−	0.772799i	$-0.718856\pi$
$181$	−559450.	−1.26930	−0.634651	+	0.772799i	$0.718856\pi$
$182$	0	0
$183$	−416856.	−0.920149
$184$	0	0
$185$	−16092.0	−0.0345685
$186$	0	0
$187$	−320760.	−0.670774
$188$	0	0
$189$	−361152.	−0.735420
$190$	0	0
$191$	724032.	1.43607	0.718033	−	0.696009i	$-0.245043\pi$
$191$	724032.	1.43607	0.718033	+	0.696009i	$0.245043\pi$
$192$	0	0
$193$	7106.00	0.0137319	0.00686597	−	0.999976i	$-0.497814\pi$
$193$	7106.00	0.0137319	0.00686597	+	0.999976i	$0.497814\pi$
$194$	0	0
$195$	−270864.	−0.510111
$196$	0	0
$197$	−530442.	−0.973806	−0.486903	−	0.873456i	$-0.661873\pi$
$197$	−530442.	−0.973806	−0.486903	+	0.873456i	$0.661873\pi$
$198$	0	0
$199$	−56168.0	−0.100544	−0.0502720	−	0.998736i	$-0.516009\pi$
$199$	−56168.0	−0.100544	−0.0502720	+	0.998736i	$0.516009\pi$
$200$	0	0
$201$	−261744.	−0.456969
$202$	0	0
$203$	−52272.0	−0.0890285
$204$	0	0
$205$	930204.	1.54594
$206$	0	0
$207$	−406296.	−0.659047
$208$	0	0
$209$	451440.	0.714882
$210$	0	0
$211$	339196.	0.524499	0.262249	−	0.965000i	$-0.415536\pi$
$211$	339196.	0.524499	0.262249	+	0.965000i	$0.415536\pi$
$212$	0	0
$213$	562464.	0.849465
$214$	0	0
$215$	653400.	0.964013
$216$	0	0
$217$	−374528.	−0.539927
$218$	0	0
$219$	810744.	1.14228
$220$	0	0
$221$	−248292.	−0.341965
$222$	0	0
$223$	−779360.	−1.04948	−0.524742	−	0.851261i	$-0.675838\pi$
$223$	−779360.	−1.04948	−0.524742	+	0.851261i	$0.675838\pi$
$224$	0	0
$225$	20691.0	0.0272474
$226$	0	0
$227$	744876.	0.959443	0.479722	−	0.877421i	$-0.340738\pi$
$227$	744876.	0.959443	0.479722	+	0.877421i	$0.340738\pi$
$228$	0	0
$229$	−272746.	−0.343692	−0.171846	−	0.985124i	$-0.554973\pi$
$229$	−272746.	−0.343692	−0.171846	+	0.985124i	$0.554973\pi$
$230$	0	0
$231$	−570240.	−0.703117
$232$	0	0
$233$	−153846.	−0.185651	−0.0928253	−	0.995682i	$-0.529590\pi$
$233$	−153846.	−0.185651	−0.0928253	+	0.995682i	$0.529590\pi$
$234$	0	0
$235$	69984.0	0.0826664
$236$	0	0
$237$	922944.	1.06734
$238$	0	0
$239$	−1.15474e6	−1.30764	−0.653820	−	0.756650i	$-0.726834\pi$
$239$	−1.15474e6	−1.30764	−0.653820	+	0.756650i	$0.726834\pi$
$240$	0	0
$241$	657074.	0.728738	0.364369	−	0.931255i	$-0.381285\pi$
$241$	657074.	0.728738	0.364369	+	0.931255i	$0.381285\pi$
$242$	0	0
$243$	694980.	0.755017
$244$	0	0
$245$	−489402.	−0.520895
$246$	0	0
$247$	349448.	0.364452
$248$	0	0
$249$	−812592.	−0.830566
$250$	0	0
$251$	−1.34190e6	−1.34442	−0.672211	−	0.740359i	$-0.734655\pi$
$251$	−1.34190e6	−1.34442	−0.672211	+	0.740359i	$0.734655\pi$
$252$	0	0
$253$	−2.21616e6	−2.17671
$254$	0	0
$255$	384912.	0.370690
$256$	0	0
$257$	132354.	0.124998	0.0624992	−	0.998045i	$-0.480093\pi$
$257$	132354.	0.124998	0.0624992	+	0.998045i	$0.480093\pi$
$258$	0	0
$259$	−26224.0	−0.0242912
$260$	0	0
$261$	58806.0	0.0534343
$262$	0	0
$263$	−943272.	−0.840906	−0.420453	−	0.907314i	$-0.638129\pi$
$263$	−943272.	−0.840906	−0.420453	+	0.907314i	$0.638129\pi$
$264$	0	0
$265$	1.05268e6	0.920831
$266$	0	0
$267$	357048.	0.306513
$268$	0	0
$269$	967518.	0.815227	0.407613	−	0.913155i	$-0.366361\pi$
$269$	967518.	0.815227	0.407613	+	0.913155i	$0.366361\pi$
$270$	0	0
$271$	518320.	0.428721	0.214360	−	0.976755i	$-0.431233\pi$
$271$	518320.	0.428721	0.214360	+	0.976755i	$0.431233\pi$
$272$	0	0
$273$	−441408.	−0.358454
$274$	0	0
$275$	112860.	0.0899929
$276$	0	0
$277$	2.22273e6	1.74055	0.870275	−	0.492566i	$-0.163941\pi$
$277$	2.22273e6	1.74055	0.870275	+	0.492566i	$0.163941\pi$
$278$	0	0
$279$	421344.	0.324061
$280$	0	0
$281$	−196614.	−0.148542	−0.0742709	−	0.997238i	$-0.523663\pi$
$281$	−196614.	−0.148542	−0.0742709	+	0.997238i	$0.523663\pi$
$282$	0	0
$283$	1.55228e6	1.15213	0.576067	−	0.817403i	$-0.304587\pi$
$283$	1.55228e6	1.15213	0.576067	+	0.817403i	$0.304587\pi$
$284$	0	0
$285$	−541728.	−0.395066
$286$	0	0
$287$	1.51589e6	1.08633
$288$	0	0
$289$	−1.06702e6	−0.751499
$290$	0	0
$291$	−1.46878e6	−1.01677
$292$	0	0
$293$	−1.07217e6	−0.729616	−0.364808	−	0.931083i	$-0.618865\pi$
$293$	−1.07217e6	−0.729616	−0.364808	+	0.931083i	$0.618865\pi$
$294$	0	0
$295$	414072.	0.277026
$296$	0	0
$297$	2.21616e6	1.45784
$298$	0	0
$299$	−1.71547e6	−1.10970
$300$	0	0
$301$	1.06480e6	0.677410
$302$	0	0
$303$	135432.	0.0847451
$304$	0	0
$305$	−1.87585e6	−1.15465
$306$	0	0
$307$	−1.58589e6	−0.960346	−0.480173	−	0.877174i	$-0.659426\pi$
$307$	−1.58589e6	−0.960346	−0.480173	+	0.877174i	$0.659426\pi$
$308$	0	0
$309$	327072.	0.194871
$310$	0	0
$311$	730728.	0.428405	0.214203	−	0.976789i	$-0.431285\pi$
$311$	730728.	0.428405	0.214203	+	0.976789i	$0.431285\pi$
$312$	0	0
$313$	584858.	0.337435	0.168717	−	0.985664i	$-0.446038\pi$
$313$	584858.	0.337435	0.168717	+	0.985664i	$0.446038\pi$
$314$	0	0
$315$	−470448.	−0.267138
$316$	0	0
$317$	−2.48287e6	−1.38773	−0.693865	−	0.720105i	$-0.744094\pi$
$317$	−2.48287e6	−1.38773	−0.693865	+	0.720105i	$0.744094\pi$
$318$	0	0
$319$	320760.	0.176483
$320$	0	0
$321$	−1.46837e6	−0.795376
$322$	0	0
$323$	−496584.	−0.264842
$324$	0	0
$325$	87362.0	0.0458790
$326$	0	0
$327$	1.19882e6	0.619992
$328$	0	0
$329$	114048.	0.0580895
$330$	0	0
$331$	−377948.	−0.189610	−0.0948052	−	0.995496i	$-0.530223\pi$
$331$	−377948.	−0.189610	−0.0948052	+	0.995496i	$0.530223\pi$
$332$	0	0
$333$	29502.0	0.0145794
$334$	0	0
$335$	−1.17785e6	−0.573426
$336$	0	0
$337$	639122.	0.306555	0.153278	−	0.988183i	$-0.451017\pi$
$337$	639122.	0.306555	0.153278	+	0.988183i	$0.451017\pi$
$338$	0	0
$339$	−355752.	−0.168131
$340$	0	0
$341$	2.29824e6	1.07031
$342$	0	0
$343$	−2.27656e6	−1.04483
$344$	0	0
$345$	2.65939e6	1.20291
$346$	0	0
$347$	2.90466e6	1.29501	0.647503	−	0.762063i	$-0.275813\pi$
$347$	2.90466e6	1.29501	0.647503	+	0.762063i	$0.275813\pi$
$348$	0	0
$349$	−3.99157e6	−1.75420	−0.877102	−	0.480304i	$-0.840526\pi$
$349$	−3.99157e6	−1.75420	−0.877102	+	0.480304i	$0.840526\pi$
$350$	0	0
$351$	1.71547e6	0.743217
$352$	0	0
$353$	1.42922e6	0.610466	0.305233	−	0.952278i	$-0.401266\pi$
$353$	1.42922e6	0.610466	0.305233	+	0.952278i	$0.401266\pi$
$354$	0	0
$355$	2.53109e6	1.06595
$356$	0	0
$357$	627264.	0.260483
$358$	0	0
$359$	−1.16186e6	−0.475794	−0.237897	−	0.971290i	$-0.576458\pi$
$359$	−1.16186e6	−0.475794	−0.237897	+	0.971290i	$0.576458\pi$
$360$	0	0
$361$	−1.77720e6	−0.717743
$362$	0	0
$363$	1.56659e6	0.624005
$364$	0	0
$365$	3.64835e6	1.43339
$366$	0	0
$367$	1.08923e6	0.422139	0.211069	−	0.977471i	$-0.432305\pi$
$367$	1.08923e6	0.422139	0.211069	+	0.977471i	$0.432305\pi$
$368$	0	0
$369$	−1.70537e6	−0.652009
$370$	0	0
$371$	1.71547e6	0.647066
$372$	0	0
$373$	3.50577e6	1.30470	0.652350	−	0.757918i	$-0.273783\pi$
$373$	3.50577e6	1.30470	0.652350	+	0.757918i	$0.273783\pi$
$374$	0	0
$375$	−2.16043e6	−0.793346
$376$	0	0
$377$	248292.	0.0899724
$378$	0	0
$379$	−4.04385e6	−1.44610	−0.723048	−	0.690798i	$-0.757260\pi$
$379$	−4.04385e6	−1.44610	−0.723048	+	0.690798i	$0.757260\pi$
$380$	0	0
$381$	−4.03814e6	−1.42518
$382$	0	0
$383$	−5.18746e6	−1.80700	−0.903499	−	0.428591i	$-0.859010\pi$
$383$	−5.18746e6	−1.80700	−0.903499	+	0.428591i	$0.859010\pi$
$384$	0	0
$385$	−2.56608e6	−0.882304
$386$	0	0
$387$	−1.19790e6	−0.406577
$388$	0	0
$389$	−950346.	−0.318425	−0.159213	−	0.987244i	$-0.550896\pi$
$389$	−950346.	−0.318425	−0.159213	+	0.987244i	$0.550896\pi$
$390$	0	0
$391$	2.43778e6	0.806403
$392$	0	0
$393$	−1.21176e6	−0.395763
$394$	0	0
$395$	4.15325e6	1.33935
$396$	0	0
$397$	−520738.	−0.165822	−0.0829112	−	0.996557i	$-0.526422\pi$
$397$	−520738.	−0.165822	−0.0829112	+	0.996557i	$0.526422\pi$
$398$	0	0
$399$	−882816.	−0.277612
$400$	0	0
$401$	764370.	0.237379	0.118690	−	0.992931i	$-0.462131\pi$
$401$	764370.	0.237379	0.118690	+	0.992931i	$0.462131\pi$
$402$	0	0
$403$	1.77901e6	0.545651
$404$	0	0
$405$	−1.36031e6	−0.412099
$406$	0	0
$407$	160920.	0.0481531
$408$	0	0
$409$	2.64051e6	0.780511	0.390255	−	0.920707i	$-0.372387\pi$
$409$	2.64051e6	0.780511	0.390255	+	0.920707i	$0.372387\pi$
$410$	0	0
$411$	−3.80570e6	−1.11130
$412$	0	0
$413$	674784.	0.194666
$414$	0	0
$415$	−3.65666e6	−1.04223
$416$	0	0
$417$	1.77989e6	0.501248
$418$	0	0
$419$	4.98020e6	1.38584	0.692918	−	0.721016i	$-0.256325\pi$
$419$	4.98020e6	1.38584	0.692918	+	0.721016i	$0.256325\pi$
$420$	0	0
$421$	−237994.	−0.0654426	−0.0327213	−	0.999465i	$-0.510417\pi$
$421$	−237994.	−0.0654426	−0.0327213	+	0.999465i	$0.510417\pi$
$422$	0	0
$423$	−128304.	−0.0348650
$424$	0	0
$425$	−124146.	−0.0333396
$426$	0	0
$427$	−3.05694e6	−0.811368
$428$	0	0
$429$	2.70864e6	0.710572
$430$	0	0
$431$	3.88238e6	1.00671	0.503356	−	0.864079i	$-0.332098\pi$
$431$	3.88238e6	1.00671	0.503356	+	0.864079i	$0.332098\pi$
$432$	0	0
$433$	−66958.0	−0.0171626	−0.00858129	−	0.999963i	$-0.502732\pi$
$433$	−66958.0	−0.0171626	−0.00858129	+	0.999963i	$0.502732\pi$
$434$	0	0
$435$	−384912.	−0.0975300
$436$	0	0
$437$	−3.43094e6	−0.859429
$438$	0	0
$439$	6.50135e6	1.61006	0.805031	−	0.593233i	$-0.202149\pi$
$439$	6.50135e6	1.61006	0.805031	+	0.593233i	$0.202149\pi$
$440$	0	0
$441$	897237.	0.219690
$442$	0	0
$443$	4.60760e6	1.11549	0.557745	−	0.830012i	$-0.311667\pi$
$443$	4.60760e6	1.11549	0.557745	+	0.830012i	$0.311667\pi$
$444$	0	0
$445$	1.60672e6	0.384626
$446$	0	0
$447$	2.35937e6	0.558505
$448$	0	0
$449$	3.77671e6	0.884092	0.442046	−	0.896992i	$-0.354253\pi$
$449$	3.77671e6	0.884092	0.442046	+	0.896992i	$0.354253\pi$
$450$	0	0
$451$	−9.30204e6	−2.15346
$452$	0	0
$453$	−892320.	−0.204303
$454$	0	0
$455$	−1.98634e6	−0.449805
$456$	0	0
$457$	−3.18069e6	−0.712412	−0.356206	−	0.934407i	$-0.615930\pi$
$457$	−3.18069e6	−0.712412	−0.356206	+	0.934407i	$0.615930\pi$
$458$	0	0
$459$	−2.43778e6	−0.540085
$460$	0	0
$461$	6.68547e6	1.46514	0.732571	−	0.680691i	$-0.238320\pi$
$461$	6.68547e6	1.46514	0.732571	+	0.680691i	$0.238320\pi$
$462$	0	0
$463$	4.35122e6	0.943318	0.471659	−	0.881781i	$-0.343655\pi$
$463$	4.35122e6	0.943318	0.471659	+	0.881781i	$0.343655\pi$
$464$	0	0
$465$	−2.75789e6	−0.591486
$466$	0	0
$467$	−7.07994e6	−1.50223	−0.751117	−	0.660170i	$-0.770484\pi$
$467$	−7.07994e6	−1.50223	−0.751117	+	0.660170i	$0.770484\pi$
$468$	0	0
$469$	−1.91946e6	−0.402945
$470$	0	0
$471$	1.45054e6	0.301284
$472$	0	0
$473$	−6.53400e6	−1.34285
$474$	0	0
$475$	174724.	0.0355319
$476$	0	0
$477$	−1.92991e6	−0.388365
$478$	0	0
$479$	−3.22186e6	−0.641604	−0.320802	−	0.947146i	$-0.603952\pi$
$479$	−3.22186e6	−0.641604	−0.320802	+	0.947146i	$0.603952\pi$
$480$	0	0
$481$	124564.	0.0245488
$482$	0	0
$483$	4.33382e6	0.845286
$484$	0	0
$485$	−6.60949e6	−1.27589
$486$	0	0
$487$	−2.29710e6	−0.438891	−0.219446	−	0.975625i	$-0.570425\pi$
$487$	−2.29710e6	−0.438891	−0.219446	+	0.975625i	$0.570425\pi$
$488$	0	0
$489$	1.33608e6	0.252674
$490$	0	0
$491$	−2.82150e6	−0.528173	−0.264087	−	0.964499i	$-0.585070\pi$
$491$	−2.82150e6	−0.528173	−0.264087	+	0.964499i	$0.585070\pi$
$492$	0	0
$493$	−352836.	−0.0653816
$494$	0	0
$495$	2.88684e6	0.529553
$496$	0	0
$497$	4.12474e6	0.749040
$498$	0	0
$499$	4.13628e6	0.743634	0.371817	−	0.928306i	$-0.378735\pi$
$499$	4.13628e6	0.743634	0.371817	+	0.928306i	$0.378735\pi$
$500$	0	0
$501$	5.90198e6	1.05052
$502$	0	0
$503$	−8.33263e6	−1.46846	−0.734230	−	0.678901i	$-0.762457\pi$
$503$	−8.33263e6	−1.46846	−0.734230	+	0.678901i	$0.762457\pi$
$504$	0	0
$505$	609444.	0.106342
$506$	0	0
$507$	−2.35883e6	−0.407546
$508$	0	0
$509$	4.34101e6	0.742670	0.371335	−	0.928499i	$-0.378900\pi$
$509$	4.34101e6	0.742670	0.371335	+	0.928499i	$0.378900\pi$
$510$	0	0
$511$	5.94546e6	1.00724
$512$	0	0
$513$	3.43094e6	0.575599
$514$	0	0
$515$	1.47182e6	0.244533
$516$	0	0
$517$	−699840.	−0.115152
$518$	0	0
$519$	8.48945e6	1.38344
$520$	0	0
$521$	−6.74185e6	−1.08814	−0.544070	−	0.839040i	$-0.683117\pi$
$521$	−6.74185e6	−1.08814	−0.544070	+	0.839040i	$0.683117\pi$
$522$	0	0
$523$	7.72196e6	1.23445	0.617224	−	0.786787i	$-0.288257\pi$
$523$	7.72196e6	1.23445	0.617224	+	0.786787i	$0.288257\pi$
$524$	0	0
$525$	−220704.	−0.0349472
$526$	0	0
$527$	−2.52806e6	−0.396517
$528$	0	0
$529$	1.04065e7	1.61683
$530$	0	0
$531$	−759132.	−0.116837
$532$	0	0
$533$	−7.20047e6	−1.09785
$534$	0	0
$535$	−6.60766e6	−0.998075
$536$	0	0
$537$	−5.92402e6	−0.886504
$538$	0	0
$539$	4.89402e6	0.725594
$540$	0	0
$541$	−682066.	−0.100192	−0.0500960	−	0.998744i	$-0.515953\pi$
$541$	−682066.	−0.100192	−0.0500960	+	0.998744i	$0.515953\pi$
$542$	0	0
$543$	−6.71340e6	−0.977109
$544$	0	0
$545$	5.39471e6	0.777995
$546$	0	0
$547$	−2.15772e6	−0.308337	−0.154169	−	0.988045i	$-0.549270\pi$
$547$	−2.15772e6	−0.308337	−0.154169	+	0.988045i	$0.549270\pi$
$548$	0	0
$549$	3.43906e6	0.486978
$550$	0	0
$551$	496584.	0.0696809
$552$	0	0
$553$	6.76826e6	0.941161
$554$	0	0
$555$	−193104.	−0.0266109
$556$	0	0
$557$	−2.67597e6	−0.365463	−0.182731	−	0.983163i	$-0.558494\pi$
$557$	−2.67597e6	−0.365463	−0.182731	+	0.983163i	$0.558494\pi$
$558$	0	0
$559$	−5.05780e6	−0.684592
$560$	0	0
$561$	−3.84912e6	−0.516362
$562$	0	0
$563$	3.55331e6	0.472457	0.236228	−	0.971698i	$-0.424089\pi$
$563$	3.55331e6	0.472457	0.236228	+	0.971698i	$0.424089\pi$
$564$	0	0
$565$	−1.60088e6	−0.210979
$566$	0	0
$567$	−2.21681e6	−0.289581
$568$	0	0
$569$	−1.29225e7	−1.67327	−0.836633	−	0.547764i	$-0.815479\pi$
$569$	−1.29225e7	−1.67327	−0.836633	+	0.547764i	$0.815479\pi$
$570$	0	0
$571$	6.08357e6	0.780851	0.390426	−	0.920634i	$-0.372328\pi$
$571$	6.08357e6	0.780851	0.390426	+	0.920634i	$0.372328\pi$
$572$	0	0
$573$	8.68838e6	1.10548
$574$	0	0
$575$	−857736.	−0.108189
$576$	0	0
$577$	−1.58241e7	−1.97869	−0.989347	−	0.145579i	$-0.953495\pi$
$577$	−1.58241e7	−1.97869	−0.989347	+	0.145579i	$0.953495\pi$
$578$	0	0
$579$	85272.0	0.0105709
$580$	0	0
$581$	−5.95901e6	−0.732375
$582$	0	0
$583$	−1.05268e7	−1.28269
$584$	0	0
$585$	2.23463e6	0.269970
$586$	0	0
$587$	−4.60220e6	−0.551278	−0.275639	−	0.961261i	$-0.588889\pi$
$587$	−4.60220e6	−0.551278	−0.275639	+	0.961261i	$0.588889\pi$
$588$	0	0
$589$	3.55802e6	0.422590
$590$	0	0
$591$	−6.36530e6	−0.749636
$592$	0	0
$593$	8.61122e6	1.00561	0.502803	−	0.864401i	$-0.332302\pi$
$593$	8.61122e6	1.00561	0.502803	+	0.864401i	$0.332302\pi$
$594$	0	0
$595$	2.82269e6	0.326867
$596$	0	0
$597$	−674016.	−0.0773988
$598$	0	0
$599$	7.98228e6	0.908992	0.454496	−	0.890749i	$-0.349819\pi$
$599$	7.98228e6	0.908992	0.454496	+	0.890749i	$0.349819\pi$
$600$	0	0
$601$	1.01740e7	1.14896	0.574481	−	0.818518i	$-0.305204\pi$
$601$	1.01740e7	1.14896	0.574481	+	0.818518i	$0.305204\pi$
$602$	0	0
$603$	2.15939e6	0.241845
$604$	0	0
$605$	7.04965e6	0.783031
$606$	0	0
$607$	9.95843e6	1.09703	0.548516	−	0.836140i	$-0.315193\pi$
$607$	9.95843e6	1.09703	0.548516	+	0.836140i	$0.315193\pi$
$608$	0	0
$609$	−627264.	−0.0685342
$610$	0	0
$611$	−541728.	−0.0587054
$612$	0	0
$613$	4.19586e6	0.450993	0.225497	−	0.974244i	$-0.427600\pi$
$613$	4.19586e6	0.450993	0.225497	+	0.974244i	$0.427600\pi$
$614$	0	0
$615$	1.11624e7	1.19007
$616$	0	0
$617$	9.12551e6	0.965038	0.482519	−	0.875885i	$-0.339722\pi$
$617$	9.12551e6	0.965038	0.482519	+	0.875885i	$0.339722\pi$
$618$	0	0
$619$	−6.45734e6	−0.677372	−0.338686	−	0.940900i	$-0.609982\pi$
$619$	−6.45734e6	−0.677372	−0.338686	+	0.940900i	$0.609982\pi$
$620$	0	0
$621$	−1.68428e7	−1.75261
$622$	0	0
$623$	2.61835e6	0.270276
$624$	0	0
$625$	−9.06882e6	−0.928647
$626$	0	0
$627$	5.41728e6	0.550316
$628$	0	0
$629$	−177012.	−0.0178392
$630$	0	0
$631$	1.40514e7	1.40490	0.702450	−	0.711733i	$-0.252090\pi$
$631$	1.40514e7	1.40490	0.702450	+	0.711733i	$0.252090\pi$
$632$	0	0
$633$	4.07035e6	0.403759
$634$	0	0
$635$	−1.81716e7	−1.78838
$636$	0	0
$637$	3.78833e6	0.369913
$638$	0	0
$639$	−4.64033e6	−0.449569
$640$	0	0
$641$	8.47168e6	0.814375	0.407188	−	0.913345i	$-0.366510\pi$
$641$	8.47168e6	0.814375	0.407188	+	0.913345i	$0.366510\pi$
$642$	0	0
$643$	−488564.	−0.0466009	−0.0233004	−	0.999729i	$-0.507417\pi$
$643$	−488564.	−0.0466009	−0.0233004	+	0.999729i	$0.507417\pi$
$644$	0	0
$645$	7.84080e6	0.742098
$646$	0	0
$647$	−2.48119e6	−0.233023	−0.116512	−	0.993189i	$-0.537171\pi$
$647$	−2.48119e6	−0.233023	−0.116512	+	0.993189i	$0.537171\pi$
$648$	0	0
$649$	−4.14072e6	−0.385891
$650$	0	0
$651$	−4.49434e6	−0.415636
$652$	0	0
$653$	−5.29130e6	−0.485601	−0.242800	−	0.970076i	$-0.578066\pi$
$653$	−5.29130e6	−0.485601	−0.242800	+	0.970076i	$0.578066\pi$
$654$	0	0
$655$	−5.45292e6	−0.496622
$656$	0	0
$657$	−6.68864e6	−0.604539
$658$	0	0
$659$	−4.72468e6	−0.423798	−0.211899	−	0.977292i	$-0.567965\pi$
$659$	−4.72468e6	−0.423798	−0.211899	+	0.977292i	$0.567965\pi$
$660$	0	0
$661$	−6.17420e6	−0.549639	−0.274819	−	0.961496i	$-0.588618\pi$
$661$	−6.17420e6	−0.549639	−0.274819	+	0.961496i	$0.588618\pi$
$662$	0	0
$663$	−2.97950e6	−0.263245
$664$	0	0
$665$	−3.97267e6	−0.348360
$666$	0	0
$667$	−2.43778e6	−0.212168
$668$	0	0
$669$	−9.35232e6	−0.807893
$670$	0	0
$671$	1.87585e7	1.60839
$672$	0	0
$673$	−9.40925e6	−0.800787	−0.400394	−	0.916343i	$-0.631127\pi$
$673$	−9.40925e6	−0.800787	−0.400394	+	0.916343i	$0.631127\pi$
$674$	0	0
$675$	857736.	0.0724593
$676$	0	0
$677$	1.50086e7	1.25854	0.629272	−	0.777185i	$-0.283353\pi$
$677$	1.50086e7	1.25854	0.629272	+	0.777185i	$0.283353\pi$
$678$	0	0
$679$	−1.07710e7	−0.896567
$680$	0	0
$681$	8.93851e6	0.738580
$682$	0	0
$683$	1.29707e7	1.06393	0.531963	−	0.846768i	$-0.321455\pi$
$683$	1.29707e7	1.06393	0.531963	+	0.846768i	$0.321455\pi$
$684$	0	0
$685$	−1.71257e7	−1.39451
$686$	0	0
$687$	−3.27295e6	−0.264574
$688$	0	0
$689$	−8.14849e6	−0.653927
$690$	0	0
$691$	−2.26556e7	−1.80501	−0.902506	−	0.430677i	$-0.858275\pi$
$691$	−2.26556e7	−1.80501	−0.902506	+	0.430677i	$0.858275\pi$
$692$	0	0
$693$	4.70448e6	0.372116
$694$	0	0
$695$	8.00950e6	0.628989
$696$	0	0
$697$	1.02322e7	0.797791
$698$	0	0
$699$	−1.84615e6	−0.142914
$700$	0	0
$701$	1.90169e7	1.46166	0.730828	−	0.682562i	$-0.239134\pi$
$701$	1.90169e7	1.46166	0.730828	+	0.682562i	$0.239134\pi$
$702$	0	0
$703$	249128.	0.0190123
$704$	0	0
$705$	839808.	0.0636366
$706$	0	0
$707$	993168.	0.0747264
$708$	0	0
$709$	1.51311e7	1.13046	0.565231	−	0.824933i	$-0.308787\pi$
$709$	1.51311e7	1.13046	0.565231	+	0.824933i	$0.308787\pi$
$710$	0	0
$711$	−7.61429e6	−0.564879
$712$	0	0
$713$	−1.74666e7	−1.28672
$714$	0	0
$715$	1.21889e7	0.891659
$716$	0	0
$717$	−1.38568e7	−1.00662
$718$	0	0
$719$	1.50323e7	1.08443	0.542217	−	0.840238i	$-0.317585\pi$
$719$	1.50323e7	1.08443	0.542217	+	0.840238i	$0.317585\pi$
$720$	0	0
$721$	2.39853e6	0.171833
$722$	0	0
$723$	7.88489e6	0.560983
$724$	0	0
$725$	124146.	0.00877178
$726$	0	0
$727$	7.41230e6	0.520136	0.260068	−	0.965590i	$-0.416255\pi$
$727$	7.41230e6	0.520136	0.260068	+	0.965590i	$0.416255\pi$
$728$	0	0
$729$	1.44612e7	1.00782
$730$	0	0
$731$	7.18740e6	0.497483
$732$	0	0
$733$	−2.77928e6	−0.191061	−0.0955306	−	0.995426i	$-0.530455\pi$
$733$	−2.77928e6	−0.191061	−0.0955306	+	0.995426i	$0.530455\pi$
$734$	0	0
$735$	−5.87282e6	−0.400985
$736$	0	0
$737$	1.17785e7	0.798768
$738$	0	0
$739$	1.21046e7	0.815342	0.407671	−	0.913129i	$-0.366341\pi$
$739$	1.21046e7	0.815342	0.407671	+	0.913129i	$0.366341\pi$
$740$	0	0
$741$	4.19338e6	0.280555
$742$	0	0
$743$	−4.46926e6	−0.297005	−0.148502	−	0.988912i	$-0.547445\pi$
$743$	−4.46926e6	−0.297005	−0.148502	+	0.988912i	$0.547445\pi$
$744$	0	0
$745$	1.06172e7	0.700838
$746$	0	0
$747$	6.70388e6	0.439567
$748$	0	0
$749$	−1.07680e7	−0.701345
$750$	0	0
$751$	−2.88463e7	−1.86634	−0.933168	−	0.359442i	$-0.882967\pi$
$751$	−2.88463e7	−1.86634	−0.933168	+	0.359442i	$0.882967\pi$
$752$	0	0
$753$	−1.61028e7	−1.03494
$754$	0	0
$755$	−4.01544e6	−0.256369
$756$	0	0
$757$	9.60868e6	0.609430	0.304715	−	0.952444i	$-0.401439\pi$
$757$	9.60868e6	0.609430	0.304715	+	0.952444i	$0.401439\pi$
$758$	0	0
$759$	−2.65939e7	−1.67563
$760$	0	0
$761$	4.54588e6	0.284549	0.142274	−	0.989827i	$-0.454558\pi$
$761$	4.54588e6	0.284549	0.142274	+	0.989827i	$0.454558\pi$
$762$	0	0
$763$	8.79138e6	0.546696
$764$	0	0
$765$	−3.17552e6	−0.196183
$766$	0	0
$767$	−3.20522e6	−0.196730
$768$	0	0
$769$	−2.15923e7	−1.31669	−0.658345	−	0.752716i	$-0.728743\pi$
$769$	−2.15923e7	−1.31669	−0.658345	+	0.752716i	$0.728743\pi$
$770$	0	0
$771$	1.58825e6	0.0962238
$772$	0	0
$773$	−1.48400e7	−0.893276	−0.446638	−	0.894715i	$-0.647379\pi$
$773$	−1.48400e7	−0.893276	−0.446638	+	0.894715i	$0.647379\pi$
$774$	0	0
$775$	889504.	0.0531978
$776$	0	0
$777$	−314688.	−0.0186994
$778$	0	0
$779$	−1.44009e7	−0.850251
$780$	0	0
$781$	−2.53109e7	−1.48484
$782$	0	0
$783$	2.43778e6	0.142098
$784$	0	0
$785$	6.52741e6	0.378065
$786$	0	0
$787$	2.48785e7	1.43182	0.715909	−	0.698194i	$-0.246013\pi$
$787$	2.48785e7	1.43182	0.715909	+	0.698194i	$0.246013\pi$
$788$	0	0
$789$	−1.13193e7	−0.647330
$790$	0	0
$791$	−2.60885e6	−0.148254
$792$	0	0
$793$	1.45205e7	0.819970
$794$	0	0
$795$	1.26321e7	0.708856
$796$	0	0
$797$	3.16080e7	1.76259	0.881294	−	0.472568i	$-0.156673\pi$
$797$	3.16080e7	1.76259	0.881294	+	0.472568i	$0.156673\pi$
$798$	0	0
$799$	769824.	0.0426604
$800$	0	0
$801$	−2.94565e6	−0.162218
$802$	0	0
$803$	−3.64835e7	−1.99668
$804$	0	0
$805$	1.95022e7	1.06070
$806$	0	0
$807$	1.16102e7	0.627562
$808$	0	0
$809$	−3.10009e6	−0.166534	−0.0832669	−	0.996527i	$-0.526535\pi$
$809$	−3.10009e6	−0.166534	−0.0832669	+	0.996527i	$0.526535\pi$
$810$	0	0
$811$	−1.87180e6	−0.0999328	−0.0499664	−	0.998751i	$-0.515911\pi$
$811$	−1.87180e6	−0.0999328	−0.0499664	+	0.998751i	$0.515911\pi$
$812$	0	0
$813$	6.21984e6	0.330030
$814$	0	0
$815$	6.01236e6	0.317067
$816$	0	0
$817$	−1.01156e7	−0.530196
$818$	0	0
$819$	3.64162e6	0.189707
$820$	0	0
$821$	−2.00184e7	−1.03650	−0.518252	−	0.855228i	$-0.673417\pi$
$821$	−2.00184e7	−1.03650	−0.518252	+	0.855228i	$0.673417\pi$
$822$	0	0
$823$	−1.53118e7	−0.787999	−0.394000	−	0.919111i	$-0.628909\pi$
$823$	−1.53118e7	−0.787999	−0.394000	+	0.919111i	$0.628909\pi$
$824$	0	0
$825$	1.35432e6	0.0692766
$826$	0	0
$827$	−9.59310e6	−0.487748	−0.243874	−	0.969807i	$-0.578418\pi$
$827$	−9.59310e6	−0.487748	−0.243874	+	0.969807i	$0.578418\pi$
$828$	0	0
$829$	2.52209e7	1.27460	0.637302	−	0.770615i	$-0.280051\pi$
$829$	2.52209e7	1.27460	0.637302	+	0.770615i	$0.280051\pi$
$830$	0	0
$831$	2.66727e7	1.33988
$832$	0	0
$833$	−5.38342e6	−0.268810
$834$	0	0
$835$	2.65589e7	1.31824
$836$	0	0
$837$	1.74666e7	0.861778
$838$	0	0
$839$	1.77623e7	0.871154	0.435577	−	0.900151i	$-0.356544\pi$
$839$	1.77623e7	0.871154	0.435577	+	0.900151i	$0.356544\pi$
$840$	0	0
$841$	−2.01583e7	−0.982798
$842$	0	0
$843$	−2.35937e6	−0.114348
$844$	0	0
$845$	−1.06147e7	−0.511407
$846$	0	0
$847$	1.14883e7	0.550234
$848$	0	0
$849$	1.86273e7	0.886913
$850$	0	0
$851$	−1.22299e6	−0.0578895
$852$	0	0
$853$	−486970.	−0.0229155	−0.0114578	−	0.999934i	$-0.503647\pi$
$853$	−486970.	−0.0229155	−0.0114578	+	0.999934i	$0.503647\pi$
$854$	0	0
$855$	4.46926e6	0.209084
$856$	0	0
$857$	−1.92634e6	−0.0895945	−0.0447972	−	0.998996i	$-0.514264\pi$
$857$	−1.92634e6	−0.0895945	−0.0447972	+	0.998996i	$0.514264\pi$
$858$	0	0
$859$	−2.23538e7	−1.03364	−0.516820	−	0.856094i	$-0.672884\pi$
$859$	−2.23538e7	−1.03364	−0.516820	+	0.856094i	$0.672884\pi$
$860$	0	0
$861$	1.81907e7	0.836258
$862$	0	0
$863$	−1.85838e7	−0.849390	−0.424695	−	0.905337i	$-0.639619\pi$
$863$	−1.85838e7	−0.849390	−0.424695	+	0.905337i	$0.639619\pi$
$864$	0	0
$865$	3.82025e7	1.73601
$866$	0	0
$867$	−1.28043e7	−0.578504
$868$	0	0
$869$	−4.15325e7	−1.86569
$870$	0	0
$871$	9.11742e6	0.407217
$872$	0	0
$873$	1.21174e7	0.538114
$874$	0	0
$875$	−1.58432e7	−0.699555
$876$	0	0
$877$	−2.91048e7	−1.27781	−0.638905	−	0.769286i	$-0.720612\pi$
$877$	−2.91048e7	−1.27781	−0.638905	+	0.769286i	$0.720612\pi$
$878$	0	0
$879$	−1.28660e7	−0.561659
$880$	0	0
$881$	−3.14696e6	−0.136600	−0.0683001	−	0.997665i	$-0.521758\pi$
$881$	−3.14696e6	−0.136600	−0.0683001	+	0.997665i	$0.521758\pi$
$882$	0	0
$883$	−1.59995e7	−0.690566	−0.345283	−	0.938499i	$-0.612217\pi$
$883$	−1.59995e7	−0.690566	−0.345283	+	0.938499i	$0.612217\pi$
$884$	0	0
$885$	4.96886e6	0.213255
$886$	0	0
$887$	3.45874e7	1.47608	0.738039	−	0.674758i	$-0.235752\pi$
$887$	3.45874e7	1.47608	0.738039	+	0.674758i	$0.235752\pi$
$888$	0	0
$889$	−2.96131e7	−1.25669
$890$	0	0
$891$	1.36031e7	0.574044
$892$	0	0
$893$	−1.08346e6	−0.0454656
$894$	0	0
$895$	−2.66581e7	−1.11243
$896$	0	0
$897$	−2.05857e7	−0.854248
$898$	0	0
$899$	2.52806e6	0.104325
$900$	0	0
$901$	1.15794e7	0.475199
$902$	0	0
$903$	1.27776e7	0.521471
$904$	0	0
$905$	−3.02103e7	−1.22612
$906$	0	0
$907$	−1.74396e7	−0.703914	−0.351957	−	0.936016i	$-0.614484\pi$
$907$	−1.74396e7	−0.703914	−0.351957	+	0.936016i	$0.614484\pi$
$908$	0	0
$909$	−1.11731e6	−0.0448503
$910$	0	0
$911$	2.59589e6	0.103631	0.0518155	−	0.998657i	$-0.483499\pi$
$911$	2.59589e6	0.103631	0.0518155	+	0.998657i	$0.483499\pi$
$912$	0	0
$913$	3.65666e7	1.45180
$914$	0	0
$915$	−2.25102e7	−0.888847
$916$	0	0
$917$	−8.88624e6	−0.348975
$918$	0	0
$919$	1.76411e7	0.689028	0.344514	−	0.938781i	$-0.388044\pi$
$919$	1.76411e7	0.689028	0.344514	+	0.938781i	$0.388044\pi$
$920$	0	0
$921$	−1.90307e7	−0.739275
$922$	0	0
$923$	−1.95925e7	−0.756982
$924$	0	0
$925$	62282.0	0.00239336
$926$	0	0
$927$	−2.69834e6	−0.103133
$928$	0	0
$929$	3.96785e7	1.50840	0.754199	−	0.656646i	$-0.228025\pi$
$929$	3.96785e7	1.50840	0.754199	+	0.656646i	$0.228025\pi$
$930$	0	0
$931$	7.57667e6	0.286486
$932$	0	0
$933$	8.76874e6	0.329787
$934$	0	0
$935$	−1.73210e7	−0.647955
$936$	0	0
$937$	3.93413e7	1.46386	0.731930	−	0.681380i	$-0.238620\pi$
$937$	3.93413e7	1.46386	0.731930	+	0.681380i	$0.238620\pi$
$938$	0	0
$939$	7.01830e6	0.259757
$940$	0	0
$941$	4.62506e7	1.70272	0.851361	−	0.524581i	$-0.175778\pi$
$941$	4.62506e7	1.70272	0.851361	+	0.524581i	$0.175778\pi$
$942$	0	0
$943$	7.06955e7	2.58888
$944$	0	0
$945$	−1.95022e7	−0.710402
$946$	0	0
$947$	3.79025e7	1.37339	0.686693	−	0.726947i	$-0.259062\pi$
$947$	3.79025e7	1.37339	0.686693	+	0.726947i	$0.259062\pi$
$948$	0	0
$949$	−2.82409e7	−1.01792
$950$	0	0
$951$	−2.97944e7	−1.06828
$952$	0	0
$953$	−2.66462e7	−0.950394	−0.475197	−	0.879879i	$-0.657623\pi$
$953$	−2.66462e7	−0.950394	−0.475197	+	0.879879i	$0.657623\pi$
$954$	0	0
$955$	3.90977e7	1.38721
$956$	0	0
$957$	3.84912e6	0.135857
$958$	0	0
$959$	−2.79085e7	−0.979918
$960$	0	0
$961$	−1.05156e7	−0.367304
$962$	0	0
$963$	1.21140e7	0.420943
$964$	0	0
$965$	383724.	0.0132648
$966$	0	0
$967$	−4.09790e7	−1.40927	−0.704637	−	0.709568i	$-0.748890\pi$
$967$	−4.09790e7	−1.40927	−0.704637	+	0.709568i	$0.748890\pi$
$968$	0	0
$969$	−5.95901e6	−0.203875
$970$	0	0
$971$	2.72034e7	0.925922	0.462961	−	0.886379i	$-0.346787\pi$
$971$	2.72034e7	0.925922	0.462961	+	0.886379i	$0.346787\pi$
$972$	0	0
$973$	1.30525e7	0.441990
$974$	0	0
$975$	1.04834e6	0.0353177
$976$	0	0
$977$	2.53555e7	0.849839	0.424919	−	0.905231i	$-0.360302\pi$
$977$	2.53555e7	0.849839	0.424919	+	0.905231i	$0.360302\pi$
$978$	0	0
$979$	−1.60672e7	−0.535775
$980$	0	0
$981$	−9.89030e6	−0.328123
$982$	0	0
$983$	−1.19139e7	−0.393252	−0.196626	−	0.980479i	$-0.562998\pi$
$983$	−1.19139e7	−0.393252	−0.196626	+	0.980479i	$0.562998\pi$
$984$	0	0
$985$	−2.86439e7	−0.940678
$986$	0	0
$987$	1.36858e6	0.0447173
$988$	0	0
$989$	4.96584e7	1.61437
$990$	0	0
$991$	−2.91931e7	−0.944268	−0.472134	−	0.881527i	$-0.656516\pi$
$991$	−2.91931e7	−0.944268	−0.472134	+	0.881527i	$0.656516\pi$
$992$	0	0
$993$	−4.53538e6	−0.145962
$994$	0	0
$995$	−3.03307e6	−0.0971237
$996$	0	0
$997$	−1.73001e7	−0.551201	−0.275601	−	0.961272i	$-0.588877\pi$
$997$	−1.73001e7	−0.551201	−0.275601	+	0.961272i	$0.588877\pi$
$998$	0	0
$999$	1.22299e6	0.0387713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.6.a.b.1.1		1
3.2	odd	2		144.6.a.c.1.1		1
4.3	odd	2		4.6.a.a.1.1	✓	1
5.2	odd	4		400.6.c.f.49.1		2
5.3	odd	4		400.6.c.f.49.2		2
5.4	even	2		400.6.a.d.1.1		1
7.6	odd	2		784.6.a.d.1.1		1
8.3	odd	2		64.6.a.f.1.1		1
8.5	even	2		64.6.a.b.1.1		1
12.11	even	2		36.6.a.a.1.1		1
16.3	odd	4		256.6.b.g.129.1		2
16.5	even	4		256.6.b.c.129.1		2
16.11	odd	4		256.6.b.g.129.2		2
16.13	even	4		256.6.b.c.129.2		2
20.3	even	4		100.6.c.b.49.1		2
20.7	even	4		100.6.c.b.49.2		2
20.19	odd	2		100.6.a.b.1.1		1
24.5	odd	2		576.6.a.bd.1.1		1
24.11	even	2		576.6.a.bc.1.1		1
28.3	even	6		196.6.e.d.177.1		2
28.11	odd	6		196.6.e.g.177.1		2
28.19	even	6		196.6.e.d.165.1		2
28.23	odd	6		196.6.e.g.165.1		2
28.27	even	2		196.6.a.e.1.1		1
36.7	odd	6		324.6.e.a.109.1		2
36.11	even	6		324.6.e.d.109.1		2
36.23	even	6		324.6.e.d.217.1		2
36.31	odd	6		324.6.e.a.217.1		2
44.43	even	2		484.6.a.a.1.1		1
52.31	even	4		676.6.d.a.337.1		2
52.47	even	4		676.6.d.a.337.2		2
52.51	odd	2		676.6.a.a.1.1		1
60.23	odd	4		900.6.d.a.649.2		2
60.47	odd	4		900.6.d.a.649.1		2
60.59	even	2		900.6.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.6.a.a.1.1	✓	1	4.3	odd	2
16.6.a.b.1.1		1	1.1	even	1	trivial
36.6.a.a.1.1		1	12.11	even	2
64.6.a.b.1.1		1	8.5	even	2
64.6.a.f.1.1		1	8.3	odd	2
100.6.a.b.1.1		1	20.19	odd	2
100.6.c.b.49.1		2	20.3	even	4
100.6.c.b.49.2		2	20.7	even	4
144.6.a.c.1.1		1	3.2	odd	2
196.6.a.e.1.1		1	28.27	even	2
196.6.e.d.165.1		2	28.19	even	6
196.6.e.d.177.1		2	28.3	even	6
196.6.e.g.165.1		2	28.23	odd	6
196.6.e.g.177.1		2	28.11	odd	6
256.6.b.c.129.1		2	16.5	even	4
256.6.b.c.129.2		2	16.13	even	4
256.6.b.g.129.1		2	16.3	odd	4
256.6.b.g.129.2		2	16.11	odd	4
324.6.e.a.109.1		2	36.7	odd	6
324.6.e.a.217.1		2	36.31	odd	6
324.6.e.d.109.1		2	36.11	even	6
324.6.e.d.217.1		2	36.23	even	6
400.6.a.d.1.1		1	5.4	even	2
400.6.c.f.49.1		2	5.2	odd	4
400.6.c.f.49.2		2	5.3	odd	4
484.6.a.a.1.1		1	44.43	even	2
576.6.a.bc.1.1		1	24.11	even	2
576.6.a.bd.1.1		1	24.5	odd	2
676.6.a.a.1.1		1	52.51	odd	2
676.6.d.a.337.1		2	52.31	even	4
676.6.d.a.337.2		2	52.47	even	4
784.6.a.d.1.1		1	7.6	odd	2
900.6.a.h.1.1		1	60.59	even	2
900.6.d.a.649.1		2	60.47	odd	4
900.6.d.a.649.2		2	60.23	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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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