LMFDB - Embedded newform 400.6.a.s.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,6,Mod(1,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-238] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 31 $ x^2 - 31
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.56776$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+11.1355 q^{3} -122.491 q^{7} -119.000 q^{9} +100.000 q^{11} +734.945 q^{13} -979.927 q^{17} +2244.00 q^{19} -1364.00 q^{21} +3418.61 q^{23} -4031.06 q^{27} -7854.00 q^{29} +2144.00 q^{31} +1113.55 q^{33} -10400.6 q^{37} +8184.00 q^{39} -7414.00 q^{41} -17761.2 q^{43} -9431.79 q^{47} -1803.00 q^{49} -10912.0 q^{51} -24253.2 q^{53} +24988.1 q^{57} -25972.0 q^{59} -3058.00 q^{61} +14576.4 q^{63} +58784.5 q^{67} +38068.0 q^{69} -37608.0 q^{71} +24008.2 q^{73} -12249.1 q^{77} -79728.0 q^{79} -15971.0 q^{81} -16291.3 q^{83} -87458.4 q^{87} -826.000 q^{89} -90024.0 q^{91} +23874.6 q^{93} +37593.5 q^{97} -11900.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 238 q^{9} + 200 q^{11} + 4488 q^{19} - 2728 q^{21} - 15708 q^{29} + 4288 q^{31} + 16368 q^{39} - 14828 q^{41} - 3606 q^{49} - 21824 q^{51} - 51944 q^{59} - 6116 q^{61} + 76136 q^{69} - 75216 q^{71}+ \cdots - 23800 q^{99}+O(q^{100}) $ 2 * q - 238 * q^9 + 200 * q^11 + 4488 * q^19 - 2728 * q^21 - 15708 * q^29 + 4288 * q^31 + 16368 * q^39 - 14828 * q^41 - 3606 * q^49 - 21824 * q^51 - 51944 * q^59 - 6116 * q^61 + 76136 * q^69 - 75216 * q^71 - 159456 * q^79 - 31942 * q^81 - 1652 * q^89 - 180048 * q^91 - 23800 * q^99

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
$3$	11.1355	0.714345	0.357172	−	0.934039i	$-0.383741\pi$
$3$	11.1355	0.714345	0.357172	+	0.934039i	$0.383741\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−122.491	−0.944840	−0.472420	−	0.881373i	$-0.656620\pi$
$7$	−122.491	−0.944840	−0.472420	+	0.881373i	$0.656620\pi$
$8$	0	0
$9$	−119.000	−0.489712
$10$	0	0
$11$	100.000	0.249183	0.124591	−	0.992208i	$-0.460238\pi$
$11$	100.000	0.249183	0.124591	+	0.992208i	$0.460238\pi$
$12$	0	0
$13$	734.945	1.20614	0.603068	−	0.797690i	$-0.293945\pi$
$13$	734.945	1.20614	0.603068	+	0.797690i	$0.293945\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−979.927	−0.822377	−0.411189	−	0.911550i	$-0.634886\pi$
$17$	−979.927	−0.822377	−0.411189	+	0.911550i	$0.634886\pi$
$18$	0	0
$19$	2244.00	1.42606	0.713032	−	0.701132i	$-0.247322\pi$
$19$	2244.00	1.42606	0.713032	+	0.701132i	$0.247322\pi$
$20$	0	0
$21$	−1364.00	−0.674942
$22$	0	0
$23$	3418.61	1.34750	0.673751	−	0.738958i	$-0.264682\pi$
$23$	3418.61	1.34750	0.673751	+	0.738958i	$0.264682\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4031.06	−1.06417
$28$	0	0
$29$	−7854.00	−1.73419	−0.867093	−	0.498146i	$-0.834015\pi$
$29$	−7854.00	−1.73419	−0.867093	+	0.498146i	$0.834015\pi$
$30$	0	0
$31$	2144.00	0.400701	0.200351	−	0.979724i	$-0.435792\pi$
$31$	2144.00	0.400701	0.200351	+	0.979724i	$0.435792\pi$
$32$	0	0
$33$	1113.55	0.178002
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10400.6	−1.24897	−0.624487	−	0.781035i	$-0.714692\pi$
$37$	−10400.6	−1.24897	−0.624487	+	0.781035i	$0.714692\pi$
$38$	0	0
$39$	8184.00	0.861597
$40$	0	0
$41$	−7414.00	−0.688800	−0.344400	−	0.938823i	$-0.611918\pi$
$41$	−7414.00	−0.688800	−0.344400	+	0.938823i	$0.611918\pi$
$42$	0	0
$43$	−17761.2	−1.46487	−0.732437	−	0.680835i	$-0.761617\pi$
$43$	−17761.2	−1.46487	−0.732437	+	0.680835i	$0.761617\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9431.79	−0.622801	−0.311401	−	0.950279i	$-0.600798\pi$
$47$	−9431.79	−0.622801	−0.311401	+	0.950279i	$0.600798\pi$
$48$	0	0
$49$	−1803.00	−0.107277
$50$	0	0
$51$	−10912.0	−0.587461
$52$	0	0
$53$	−24253.2	−1.18598	−0.592992	−	0.805208i	$-0.702054\pi$
$53$	−24253.2	−1.18598	−0.592992	+	0.805208i	$0.702054\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	24988.1	1.01870
$58$	0	0
$59$	−25972.0	−0.971349	−0.485675	−	0.874140i	$-0.661426\pi$
$59$	−25972.0	−0.971349	−0.485675	+	0.874140i	$0.661426\pi$
$60$	0	0
$61$	−3058.00	−0.105224	−0.0526118	−	0.998615i	$-0.516755\pi$
$61$	−3058.00	−0.105224	−0.0526118	+	0.998615i	$0.516755\pi$
$62$	0	0
$63$	14576.4	0.462700
$64$	0	0
$65$	0	0
$66$	0	0
$67$	58784.5	1.59984	0.799918	−	0.600109i	$-0.204876\pi$
$67$	58784.5	1.59984	0.799918	+	0.600109i	$0.204876\pi$
$68$	0	0
$69$	38068.0	0.962581
$70$	0	0
$71$	−37608.0	−0.885389	−0.442695	−	0.896672i	$-0.645977\pi$
$71$	−37608.0	−0.885389	−0.442695	+	0.896672i	$0.645977\pi$
$72$	0	0
$73$	24008.2	0.527294	0.263647	−	0.964619i	$-0.415075\pi$
$73$	24008.2	0.527294	0.263647	+	0.964619i	$0.415075\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12249.1	−0.235438
$78$	0	0
$79$	−79728.0	−1.43729	−0.718643	−	0.695379i	$-0.755236\pi$
$79$	−79728.0	−1.43729	−0.718643	+	0.695379i	$0.755236\pi$
$80$	0	0
$81$	−15971.0	−0.270470
$82$	0	0
$83$	−16291.3	−0.259573	−0.129787	−	0.991542i	$-0.541429\pi$
$83$	−16291.3	−0.259573	−0.129787	+	0.991542i	$0.541429\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−87458.4	−1.23881
$88$	0	0
$89$	−826.000	−0.0110536	−0.00552682	−	0.999985i	$-0.501759\pi$
$89$	−826.000	−0.0110536	−0.00552682	+	0.999985i	$0.501759\pi$
$90$	0	0
$91$	−90024.0	−1.13961
$92$	0	0
$93$	23874.6	0.286239
$94$	0	0
$95$	0	0
$96$	0	0
$97$	37593.5	0.405680	0.202840	−	0.979212i	$-0.434983\pi$
$97$	37593.5	0.405680	0.202840	+	0.979212i	$0.434983\pi$
$98$	0	0
$99$	−11900.0	−0.122028
$100$	0	0
$101$	−143594.	−1.40066	−0.700330	−	0.713819i	$-0.746964\pi$
$101$	−143594.	−1.40066	−0.700330	+	0.713819i	$0.746964\pi$
$102$	0	0
$103$	111834.	1.03868	0.519339	−	0.854568i	$-0.326178\pi$
$103$	111834.	1.03868	0.519339	+	0.854568i	$0.326178\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	92235.6	0.778824	0.389412	−	0.921064i	$-0.372678\pi$
$107$	92235.6	0.778824	0.389412	+	0.921064i	$0.372678\pi$
$108$	0	0
$109$	−106238.	−0.856473	−0.428236	−	0.903667i	$-0.640865\pi$
$109$	−106238.	−0.856473	−0.428236	+	0.903667i	$0.640865\pi$
$110$	0	0
$111$	−115816.	−0.892198
$112$	0	0
$113$	113048.	0.832849	0.416425	−	0.909170i	$-0.363283\pi$
$113$	113048.	0.832849	0.416425	+	0.909170i	$0.363283\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−87458.4	−0.590659
$118$	0	0
$119$	120032.	0.777015
$120$	0	0
$121$	−151051.	−0.937908
$122$	0	0
$123$	−82558.8	−0.492040
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−51568.6	−0.283711	−0.141856	−	0.989887i	$-0.545307\pi$
$127$	−51568.6	−0.283711	−0.141856	+	0.989887i	$0.545307\pi$
$128$	0	0
$129$	−197780.	−1.04642
$130$	0	0
$131$	89100.0	0.453628	0.226814	−	0.973938i	$-0.427169\pi$
$131$	89100.0	0.453628	0.226814	+	0.973938i	$0.427169\pi$
$132$	0	0
$133$	−274869.	−1.34740
$134$	0	0
$135$	0	0
$136$	0	0
$137$	38350.8	0.174571	0.0872856	−	0.996183i	$-0.472181\pi$
$137$	38350.8	0.174571	0.0872856	+	0.996183i	$0.472181\pi$
$138$	0	0
$139$	134684.	0.591261	0.295630	−	0.955302i	$-0.404470\pi$
$139$	134684.	0.591261	0.295630	+	0.955302i	$0.404470\pi$
$140$	0	0
$141$	−105028.	−0.444895
$142$	0	0
$143$	73494.5	0.300549
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−20077.4	−0.0766325
$148$	0	0
$149$	−248006.	−0.915159	−0.457579	−	0.889169i	$-0.651283\pi$
$149$	−248006.	−0.915159	−0.457579	+	0.889169i	$0.651283\pi$
$150$	0	0
$151$	−313720.	−1.11970	−0.559848	−	0.828596i	$-0.689140\pi$
$151$	−313720.	−1.11970	−0.559848	+	0.828596i	$0.689140\pi$
$152$	0	0
$153$	116611.	0.402728
$154$	0	0
$155$	0	0
$156$	0	0
$157$	245583.	0.795150	0.397575	−	0.917570i	$-0.369852\pi$
$157$	245583.	0.795150	0.397575	+	0.917570i	$0.369852\pi$
$158$	0	0
$159$	−270072.	−0.847202
$160$	0	0
$161$	−418748.	−1.27317
$162$	0	0
$163$	−397483.	−1.17179	−0.585894	−	0.810388i	$-0.699257\pi$
$163$	−397483.	−1.17179	−0.585894	+	0.810388i	$0.699257\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	189983.	0.527138	0.263569	−	0.964641i	$-0.415100\pi$
$167$	189983.	0.527138	0.263569	+	0.964641i	$0.415100\pi$
$168$	0	0
$169$	168851.	0.454765
$170$	0	0
$171$	−267036.	−0.698360
$172$	0	0
$173$	81088.9	0.205990	0.102995	−	0.994682i	$-0.467157\pi$
$173$	81088.9	0.205990	0.102995	+	0.994682i	$0.467157\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−289212.	−0.693878
$178$	0	0
$179$	−142108.	−0.331502	−0.165751	−	0.986168i	$-0.553005\pi$
$179$	−142108.	−0.331502	−0.165751	+	0.986168i	$0.553005\pi$
$180$	0	0
$181$	250790.	0.569002	0.284501	−	0.958676i	$-0.408172\pi$
$181$	250790.	0.569002	0.284501	+	0.958676i	$0.408172\pi$
$182$	0	0
$183$	−34052.4	−0.0751659
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−97992.7	−0.204922
$188$	0	0
$189$	493768.	1.00547
$190$	0	0
$191$	209472.	0.415473	0.207736	−	0.978185i	$-0.433390\pi$
$191$	209472.	0.415473	0.207736	+	0.978185i	$0.433390\pi$
$192$	0	0
$193$	−356693.	−0.689289	−0.344645	−	0.938733i	$-0.612001\pi$
$193$	−356693.	−0.689289	−0.344645	+	0.938733i	$0.612001\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−86478.5	−0.158761	−0.0793803	−	0.996844i	$-0.525294\pi$
$197$	−86478.5	−0.158761	−0.0793803	+	0.996844i	$0.525294\pi$
$198$	0	0
$199$	−749208.	−1.34113	−0.670563	−	0.741852i	$-0.733947\pi$
$199$	−749208.	−1.34113	−0.670563	+	0.741852i	$0.733947\pi$
$200$	0	0
$201$	654596.	1.14283
$202$	0	0
$203$	962043.	1.63853
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−406814.	−0.659888
$208$	0	0
$209$	224400.	0.355351
$210$	0	0
$211$	−287364.	−0.444351	−0.222176	−	0.975007i	$-0.571316\pi$
$211$	−287364.	−0.444351	−0.222176	+	0.975007i	$0.571316\pi$
$212$	0	0
$213$	−418785.	−0.632473
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−262620.	−0.378599
$218$	0	0
$219$	267344.	0.376669
$220$	0	0
$221$	−720192.	−0.991899
$222$	0	0
$223$	−1.18866e6	−1.60065	−0.800325	−	0.599567i	$-0.795340\pi$
$223$	−1.18866e6	−1.60065	−0.800325	+	0.599567i	$0.795340\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−978334.	−1.26015	−0.630075	−	0.776534i	$-0.716976\pi$
$227$	−978334.	−1.26015	−0.630075	+	0.776534i	$0.716976\pi$
$228$	0	0
$229$	506474.	0.638217	0.319109	−	0.947718i	$-0.396617\pi$
$229$	506474.	0.638217	0.319109	+	0.947718i	$0.396617\pi$
$230$	0	0
$231$	−136400.	−0.168184
$232$	0	0
$233$	1.55465e6	1.87605	0.938024	−	0.346571i	$-0.112654\pi$
$233$	1.55465e6	1.87605	0.938024	+	0.346571i	$0.112654\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−887813.	−1.02672
$238$	0	0
$239$	374704.	0.424320	0.212160	−	0.977235i	$-0.431950\pi$
$239$	374704.	0.424320	0.212160	+	0.977235i	$0.431950\pi$
$240$	0	0
$241$	843634.	0.935646	0.467823	−	0.883822i	$-0.345039\pi$
$241$	843634.	0.935646	0.467823	+	0.883822i	$0.345039\pi$
$242$	0	0
$243$	801702.	0.870959
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.64922e6	1.72003
$248$	0	0
$249$	−181412.	−0.185425
$250$	0	0
$251$	1.72050e6	1.72373	0.861867	−	0.507134i	$-0.169295\pi$
$251$	1.72050e6	1.72373	0.861867	+	0.507134i	$0.169295\pi$
$252$	0	0
$253$	341861.	0.335775
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.55220e6	1.46594	0.732969	−	0.680262i	$-0.238134\pi$
$257$	1.55220e6	1.46594	0.732969	+	0.680262i	$0.238134\pi$
$258$	0	0
$259$	1.27398e6	1.18008
$260$	0	0
$261$	934626.	0.849252
$262$	0	0
$263$	407772.	0.363520	0.181760	−	0.983343i	$-0.441821\pi$
$263$	407772.	0.363520	0.181760	+	0.983343i	$0.441821\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9197.95	−0.00789610
$268$	0	0
$269$	−1.82710e6	−1.53951	−0.769754	−	0.638340i	$-0.779621\pi$
$269$	−1.82710e6	−1.53951	−0.769754	+	0.638340i	$0.779621\pi$
$270$	0	0
$271$	616880.	0.510243	0.255122	−	0.966909i	$-0.417884\pi$
$271$	616880.	0.510243	0.255122	+	0.966909i	$0.417884\pi$
$272$	0	0
$273$	−1.00246e6	−0.814071
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.83712e6	1.43859	0.719296	−	0.694704i	$-0.244465\pi$
$277$	1.83712e6	1.43859	0.719296	+	0.694704i	$0.244465\pi$
$278$	0	0
$279$	−255136.	−0.196228
$280$	0	0
$281$	−1.22093e6	−0.922415	−0.461208	−	0.887292i	$-0.652584\pi$
$281$	−1.22093e6	−0.922415	−0.461208	+	0.887292i	$0.652584\pi$
$282$	0	0
$283$	688766.	0.511217	0.255609	−	0.966780i	$-0.417724\pi$
$283$	688766.	0.511217	0.255609	+	0.966780i	$0.417724\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	908147.	0.650806
$288$	0	0
$289$	−459601.	−0.323695
$290$	0	0
$291$	418624.	0.289796
$292$	0	0
$293$	−856211.	−0.582655	−0.291328	−	0.956623i	$-0.594097\pi$
$293$	−856211.	−0.582655	−0.291328	+	0.956623i	$0.594097\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−403106.	−0.265172
$298$	0	0
$299$	2.51249e6	1.62527
$300$	0	0
$301$	2.17558e6	1.38407
$302$	0	0
$303$	−1.59900e6	−1.00055
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.09617e6	0.663792	0.331896	−	0.943316i	$-0.392312\pi$
$307$	1.09617e6	0.663792	0.331896	+	0.943316i	$0.392312\pi$
$308$	0	0
$309$	1.24533e6	0.741974
$310$	0	0
$311$	2.12465e6	1.24562	0.622811	−	0.782373i	$-0.285991\pi$
$311$	2.12465e6	1.24562	0.622811	+	0.782373i	$0.285991\pi$
$312$	0	0
$313$	−294824.	−0.170099	−0.0850496	−	0.996377i	$-0.527105\pi$
$313$	−294824.	−0.170099	−0.0850496	+	0.996377i	$0.527105\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.53153e6	−1.41493	−0.707465	−	0.706749i	$-0.750161\pi$
$317$	−2.53153e6	−1.41493	−0.707465	+	0.706749i	$0.750161\pi$
$318$	0	0
$319$	−785400.	−0.432130
$320$	0	0
$321$	1.02709e6	0.556348
$322$	0	0
$323$	−2.19896e6	−1.17276
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.18302e6	−0.611817
$328$	0	0
$329$	1.15531e6	0.588448
$330$	0	0
$331$	1.17021e6	0.587076	0.293538	−	0.955947i	$-0.405167\pi$
$331$	1.17021e6	0.587076	0.293538	+	0.955947i	$0.405167\pi$
$332$	0	0
$333$	1.23767e6	0.611637
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.86872e6	0.896333	0.448167	−	0.893950i	$-0.352077\pi$
$337$	1.86872e6	0.896333	0.448167	+	0.893950i	$0.352077\pi$
$338$	0	0
$339$	1.25885e6	0.594941
$340$	0	0
$341$	214400.	0.0998479
$342$	0	0
$343$	2.27955e6	1.04620
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.63342e6	−0.728237	−0.364119	−	0.931353i	$-0.618630\pi$
$347$	−1.63342e6	−0.728237	−0.364119	+	0.931353i	$0.618630\pi$
$348$	0	0
$349$	2.00629e6	0.881719	0.440859	−	0.897576i	$-0.354674\pi$
$349$	2.00629e6	0.881719	0.440859	+	0.897576i	$0.354674\pi$
$350$	0	0
$351$	−2.96261e6	−1.28353
$352$	0	0
$353$	−1.80859e6	−0.772508	−0.386254	−	0.922392i	$-0.626231\pi$
$353$	−1.80859e6	−0.772508	−0.386254	+	0.922392i	$0.626231\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.33662e6	0.555057
$358$	0	0
$359$	−4.50674e6	−1.84555	−0.922777	−	0.385334i	$-0.874086\pi$
$359$	−4.50674e6	−1.84555	−0.922777	+	0.385334i	$0.874086\pi$
$360$	0	0
$361$	2.55944e6	1.03366
$362$	0	0
$363$	−1.68203e6	−0.669989
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3.02796e6	−1.17351	−0.586753	−	0.809766i	$-0.699594\pi$
$367$	−3.02796e6	−1.17351	−0.586753	+	0.809766i	$0.699594\pi$
$368$	0	0
$369$	882266.	0.337313
$370$	0	0
$371$	2.97079e6	1.12057
$372$	0	0
$373$	1.16342e6	0.432976	0.216488	−	0.976285i	$-0.430540\pi$
$373$	1.16342e6	0.432976	0.216488	+	0.976285i	$0.430540\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.77226e6	−2.09167
$378$	0	0
$379$	−832052.	−0.297545	−0.148772	−	0.988871i	$-0.547532\pi$
$379$	−832052.	−0.297545	−0.148772	+	0.988871i	$0.547532\pi$
$380$	0	0
$381$	−574244.	−0.202667
$382$	0	0
$383$	2.86948e6	0.999554	0.499777	−	0.866154i	$-0.333415\pi$
$383$	2.86948e6	0.999554	0.499777	+	0.866154i	$0.333415\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.11358e6	0.717366
$388$	0	0
$389$	−311926.	−0.104515	−0.0522574	−	0.998634i	$-0.516642\pi$
$389$	−311926.	−0.104515	−0.0522574	+	0.998634i	$0.516642\pi$
$390$	0	0
$391$	−3.34998e6	−1.10816
$392$	0	0
$393$	992176.	0.324046
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.95619e6	−0.941362	−0.470681	−	0.882304i	$-0.655992\pi$
$397$	−2.95619e6	−0.941362	−0.470681	+	0.882304i	$0.655992\pi$
$398$	0	0
$399$	−3.06082e6	−0.962509
$400$	0	0
$401$	2770.00	0.000860238 0	0.000430119	−	1.00000i	$-0.499863\pi$
$401$	2770.00	0.000860238 0	0.000430119		1.00000i	$0.499863\pi$
$402$	0	0
$403$	1.57572e6	0.483300
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.04006e6	−0.311223
$408$	0	0
$409$	1.97985e6	0.585225	0.292613	−	0.956231i	$-0.405475\pi$
$409$	1.97985e6	0.585225	0.292613	+	0.956231i	$0.405475\pi$
$410$	0	0
$411$	427056.	0.124704
$412$	0	0
$413$	3.18133e6	0.917770
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.49978e6	0.422364
$418$	0	0
$419$	5.10120e6	1.41951	0.709754	−	0.704450i	$-0.248806\pi$
$419$	5.10120e6	1.41951	0.709754	+	0.704450i	$0.248806\pi$
$420$	0	0
$421$	−2.43223e6	−0.668806	−0.334403	−	0.942430i	$-0.608535\pi$
$421$	−2.43223e6	−0.668806	−0.334403	+	0.942430i	$0.608535\pi$
$422$	0	0
$423$	1.12238e6	0.304993
$424$	0	0
$425$	0	0
$426$	0	0
$427$	374577.	0.0994194
$428$	0	0
$429$	818400.	0.214695
$430$	0	0
$431$	−918896.	−0.238272	−0.119136	−	0.992878i	$-0.538012\pi$
$431$	−918896.	−0.238272	−0.119136	+	0.992878i	$0.538012\pi$
$432$	0	0
$433$	−2.05455e6	−0.526619	−0.263310	−	0.964711i	$-0.584814\pi$
$433$	−2.05455e6	−0.526619	−0.263310	+	0.964711i	$0.584814\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.67135e6	1.92162
$438$	0	0
$439$	676632.	0.167568	0.0837840	−	0.996484i	$-0.473299\pi$
$439$	676632.	0.167568	0.0837840	+	0.996484i	$0.473299\pi$
$440$	0	0
$441$	214557.	0.0525347
$442$	0	0
$443$	−2.53092e6	−0.612729	−0.306365	−	0.951914i	$-0.599113\pi$
$443$	−2.53092e6	−0.612729	−0.306365	+	0.951914i	$0.599113\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.76168e6	−0.653739
$448$	0	0
$449$	−5.17619e6	−1.21170	−0.605849	−	0.795579i	$-0.707167\pi$
$449$	−5.17619e6	−1.21170	−0.605849	+	0.795579i	$0.707167\pi$
$450$	0	0
$451$	−741400.	−0.171637
$452$	0	0
$453$	−3.49344e6	−0.799848
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.11274e6	−0.697191	−0.348596	−	0.937273i	$-0.613341\pi$
$457$	−3.11274e6	−0.697191	−0.348596	+	0.937273i	$0.613341\pi$
$458$	0	0
$459$	3.95014e6	0.875147
$460$	0	0
$461$	−2.64957e6	−0.580662	−0.290331	−	0.956926i	$-0.593765\pi$
$461$	−2.64957e6	−0.580662	−0.290331	+	0.956926i	$0.593765\pi$
$462$	0	0
$463$	−2.59165e6	−0.561854	−0.280927	−	0.959729i	$-0.590642\pi$
$463$	−2.59165e6	−0.561854	−0.280927	+	0.959729i	$0.590642\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.62135e6	1.40493	0.702465	−	0.711719i	$-0.252083\pi$
$467$	6.62135e6	1.40493	0.702465	+	0.711719i	$0.252083\pi$
$468$	0	0
$469$	−7.20056e6	−1.51159
$470$	0	0
$471$	2.73470e6	0.568011
$472$	0	0
$473$	−1.77612e6	−0.365022
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.88613e6	0.580791
$478$	0	0
$479$	−6.89322e6	−1.37272	−0.686362	−	0.727260i	$-0.740793\pi$
$479$	−6.89322e6	−1.37272	−0.686362	+	0.727260i	$0.740793\pi$
$480$	0	0
$481$	−7.64386e6	−1.50643
$482$	0	0
$483$	−4.66298e6	−0.909485
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.65370e6	1.08021	0.540107	−	0.841596i	$-0.318384\pi$
$487$	5.65370e6	1.08021	0.540107	+	0.841596i	$0.318384\pi$
$488$	0	0
$489$	−4.42618e6	−0.837061
$490$	0	0
$491$	−5.88390e6	−1.10144	−0.550721	−	0.834689i	$-0.685647\pi$
$491$	−5.88390e6	−1.10144	−0.550721	+	0.834689i	$0.685647\pi$
$492$	0	0
$493$	7.69634e6	1.42616
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.60663e6	0.836552
$498$	0	0
$499$	6.72080e6	1.20829	0.604143	−	0.796876i	$-0.293515\pi$
$499$	6.72080e6	1.20829	0.604143	+	0.796876i	$0.293515\pi$
$500$	0	0
$501$	2.11556e6	0.376558
$502$	0	0
$503$	−469262.	−0.0826981	−0.0413491	−	0.999145i	$-0.513166\pi$
$503$	−469262.	−0.0826981	−0.0413491	+	0.999145i	$0.513166\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.88025e6	0.324859
$508$	0	0
$509$	−294414.	−0.0503691	−0.0251845	−	0.999683i	$-0.508017\pi$
$509$	−294414.	−0.0503691	−0.0251845	+	0.999683i	$0.508017\pi$
$510$	0	0
$511$	−2.94078e6	−0.498208
$512$	0	0
$513$	−9.04570e6	−1.51757
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−943179.	−0.155191
$518$	0	0
$519$	902968.	0.147148
$520$	0	0
$521$	−7.10025e6	−1.14599	−0.572993	−	0.819560i	$-0.694218\pi$
$521$	−7.10025e6	−1.14599	−0.572993	+	0.819560i	$0.694218\pi$
$522$	0	0
$523$	5.96567e6	0.953685	0.476843	−	0.878989i	$-0.341781\pi$
$523$	5.96567e6	0.953685	0.476843	+	0.878989i	$0.341781\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.10096e6	−0.329528
$528$	0	0
$529$	5.25053e6	0.815763
$530$	0	0
$531$	3.09067e6	0.475681
$532$	0	0
$533$	−5.44888e6	−0.830786
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.58245e6	−0.236807
$538$	0	0
$539$	−180300.	−0.0267315
$540$	0	0
$541$	−2.72367e6	−0.400093	−0.200046	−	0.979786i	$-0.564109\pi$
$541$	−2.72367e6	−0.400093	−0.200046	+	0.979786i	$0.564109\pi$
$542$	0	0
$543$	2.79268e6	0.406463
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−9.22148e6	−1.31775	−0.658874	−	0.752254i	$-0.728967\pi$
$547$	−9.22148e6	−1.31775	−0.658874	+	0.752254i	$0.728967\pi$
$548$	0	0
$549$	363902.	0.0515292
$550$	0	0
$551$	−1.76244e7	−2.47306
$552$	0	0
$553$	9.76595e6	1.35801
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.42852e6	−0.468240	−0.234120	−	0.972208i	$-0.575221\pi$
$557$	−3.42852e6	−0.468240	−0.234120	+	0.972208i	$0.575221\pi$
$558$	0	0
$559$	−1.30535e7	−1.76684
$560$	0	0
$561$	−1.09120e6	−0.146385
$562$	0	0
$563$	5.77899e6	0.768389	0.384195	−	0.923252i	$-0.374479\pi$
$563$	5.77899e6	0.768389	0.384195	+	0.923252i	$0.374479\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.95630e6	0.255551
$568$	0	0
$569$	−3.89257e6	−0.504029	−0.252015	−	0.967723i	$-0.581093\pi$
$569$	−3.89257e6	−0.504029	−0.252015	+	0.967723i	$0.581093\pi$
$570$	0	0
$571$	5.06277e6	0.649828	0.324914	−	0.945744i	$-0.394665\pi$
$571$	5.06277e6	0.649828	0.324914	+	0.945744i	$0.394665\pi$
$572$	0	0
$573$	2.33258e6	0.296791
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.30075e6	−0.412737	−0.206368	−	0.978474i	$-0.566164\pi$
$577$	−3.30075e6	−0.412737	−0.206368	+	0.978474i	$0.566164\pi$
$578$	0	0
$579$	−3.97197e6	−0.492390
$580$	0	0
$581$	1.99553e6	0.245255
$582$	0	0
$583$	−2.42532e6	−0.295527
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.16997e6	−0.619288	−0.309644	−	0.950853i	$-0.600210\pi$
$587$	−5.16997e6	−0.619288	−0.309644	+	0.950853i	$0.600210\pi$
$588$	0	0
$589$	4.81114e6	0.571425
$590$	0	0
$591$	−962984.	−0.113410
$592$	0	0
$593$	1.58484e7	1.85075	0.925374	−	0.379055i	$-0.123751\pi$
$593$	1.58484e7	1.85075	0.925374	+	0.379055i	$0.123751\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.34283e6	−0.958026
$598$	0	0
$599$	−1.66146e7	−1.89201	−0.946004	−	0.324156i	$-0.894920\pi$
$599$	−1.66146e7	−1.89201	−0.946004	+	0.324156i	$0.894920\pi$
$600$	0	0
$601$	7.88249e6	0.890179	0.445089	−	0.895486i	$-0.353172\pi$
$601$	7.88249e6	0.890179	0.445089	+	0.895486i	$0.353172\pi$
$602$	0	0
$603$	−6.99535e6	−0.783459
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−782594.	−0.0862114	−0.0431057	−	0.999071i	$-0.513725\pi$
$607$	−782594.	−0.0862114	−0.0431057	+	0.999071i	$0.513725\pi$
$608$	0	0
$609$	1.07129e7	1.17047
$610$	0	0
$611$	−6.93185e6	−0.751183
$612$	0	0
$613$	−2.41233e6	−0.259290	−0.129645	−	0.991560i	$-0.541384\pi$
$613$	−2.41233e6	−0.259290	−0.129645	+	0.991560i	$0.541384\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.66355e6	1.02194	0.510968	−	0.859600i	$-0.329287\pi$
$617$	9.66355e6	1.02194	0.510968	+	0.859600i	$0.329287\pi$
$618$	0	0
$619$	1.80036e7	1.88857	0.944283	−	0.329134i	$-0.106757\pi$
$619$	1.80036e7	1.88857	0.944283	+	0.329134i	$0.106757\pi$
$620$	0	0
$621$	−1.37806e7	−1.43397
$622$	0	0
$623$	101177.	0.0104439
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.49881e6	0.253843
$628$	0	0
$629$	1.01918e7	1.02713
$630$	0	0
$631$	4.80081e6	0.480000	0.240000	−	0.970773i	$-0.422853\pi$
$631$	4.80081e6	0.480000	0.240000	+	0.970773i	$0.422853\pi$
$632$	0	0
$633$	−3.19995e6	−0.317420
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.32511e6	−0.129390
$638$	0	0
$639$	4.47535e6	0.433586
$640$	0	0
$641$	1.44950e7	1.39340	0.696698	−	0.717365i	$-0.254652\pi$
$641$	1.44950e7	1.39340	0.696698	+	0.717365i	$0.254652\pi$
$642$	0	0
$643$	1.82430e7	1.74008	0.870041	−	0.492979i	$-0.164092\pi$
$643$	1.82430e7	1.74008	0.870041	+	0.492979i	$0.164092\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.64592e6	0.905905	0.452953	−	0.891535i	$-0.350371\pi$
$647$	9.64592e6	0.905905	0.452953	+	0.891535i	$0.350371\pi$
$648$	0	0
$649$	−2.59720e6	−0.242044
$650$	0	0
$651$	−2.92442e6	−0.270450
$652$	0	0
$653$	−1.92807e7	−1.76945	−0.884726	−	0.466111i	$-0.845655\pi$
$653$	−1.92807e7	−1.76945	−0.884726	+	0.466111i	$0.845655\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.85698e6	−0.258222
$658$	0	0
$659$	−9.70592e6	−0.870609	−0.435304	−	0.900283i	$-0.643359\pi$
$659$	−9.70592e6	−0.870609	−0.435304	+	0.900283i	$0.643359\pi$
$660$	0	0
$661$	−4.28396e6	−0.381366	−0.190683	−	0.981652i	$-0.561070\pi$
$661$	−4.28396e6	−0.381366	−0.190683	+	0.981652i	$0.561070\pi$
$662$	0	0
$663$	−8.01972e6	−0.708558
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.68497e7	−2.33682
$668$	0	0
$669$	−1.32364e7	−1.14342
$670$	0	0
$671$	−305800.	−0.0262199
$672$	0	0
$673$	−1.30585e7	−1.11136	−0.555681	−	0.831395i	$-0.687542\pi$
$673$	−1.30585e7	−1.11136	−0.555681	+	0.831395i	$0.687542\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.42565e6	−0.706532	−0.353266	−	0.935523i	$-0.614929\pi$
$677$	−8.42565e6	−0.706532	−0.353266	+	0.935523i	$0.614929\pi$
$678$	0	0
$679$	−4.60486e6	−0.383303
$680$	0	0
$681$	−1.08943e7	−0.900182
$682$	0	0
$683$	−1.64100e7	−1.34603	−0.673017	−	0.739627i	$-0.735002\pi$
$683$	−1.64100e7	−1.34603	−0.673017	+	0.739627i	$0.735002\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.63986e6	0.455907
$688$	0	0
$689$	−1.78248e7	−1.43046
$690$	0	0
$691$	1.12139e7	0.893428	0.446714	−	0.894677i	$-0.352594\pi$
$691$	1.12139e7	0.893428	0.446714	+	0.894677i	$0.352594\pi$
$692$	0	0
$693$	1.45764e6	0.115297
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.26518e6	0.566453
$698$	0	0
$699$	1.73119e7	1.34014
$700$	0	0
$701$	2.04707e7	1.57339	0.786696	−	0.617340i	$-0.211790\pi$
$701$	2.04707e7	1.57339	0.786696	+	0.617340i	$0.211790\pi$
$702$	0	0
$703$	−2.33389e7	−1.78112
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.75889e7	1.32340
$708$	0	0
$709$	−81654.0	−0.00610045	−0.00305023	−	0.999995i	$-0.500971\pi$
$709$	−81654.0	−0.00610045	−0.00305023	+	0.999995i	$0.500971\pi$
$710$	0	0
$711$	9.48763e6	0.703856
$712$	0	0
$713$	7.32949e6	0.539946
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.17253e6	0.303111
$718$	0	0
$719$	8.61006e6	0.621132	0.310566	−	0.950552i	$-0.399481\pi$
$719$	8.61006e6	0.621132	0.310566	+	0.950552i	$0.399481\pi$
$720$	0	0
$721$	−1.36987e7	−0.981386
$722$	0	0
$723$	9.39431e6	0.668373
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.17682e7	0.825796	0.412898	−	0.910777i	$-0.364517\pi$
$727$	1.17682e7	0.825796	0.412898	+	0.910777i	$0.364517\pi$
$728$	0	0
$729$	1.28083e7	0.892635
$730$	0	0
$731$	1.74046e7	1.20468
$732$	0	0
$733$	−3.93759e6	−0.270689	−0.135344	−	0.990799i	$-0.543214\pi$
$733$	−3.93759e6	−0.270689	−0.135344	+	0.990799i	$0.543214\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.87845e6	0.398652
$738$	0	0
$739$	2.30602e7	1.55329	0.776643	−	0.629941i	$-0.216921\pi$
$739$	2.30602e7	1.55329	0.776643	+	0.629941i	$0.216921\pi$
$740$	0	0
$741$	1.83649e7	1.22869
$742$	0	0
$743$	1.72675e6	0.114751	0.0573757	−	0.998353i	$-0.481727\pi$
$743$	1.72675e6	0.114751	0.0573757	+	0.998353i	$0.481727\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.93866e6	0.127116
$748$	0	0
$749$	−1.12980e7	−0.735864
$750$	0	0
$751$	2.58030e6	0.166944	0.0834720	−	0.996510i	$-0.473399\pi$
$751$	2.58030e6	0.166944	0.0834720	+	0.996510i	$0.473399\pi$
$752$	0	0
$753$	1.91587e7	1.23134
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.75878e6	0.365251	0.182625	−	0.983183i	$-0.441540\pi$
$757$	5.75878e6	0.365251	0.182625	+	0.983183i	$0.441540\pi$
$758$	0	0
$759$	3.80680e6	0.239859
$760$	0	0
$761$	1.40499e7	0.879450	0.439725	−	0.898133i	$-0.355076\pi$
$761$	1.40499e7	0.879450	0.439725	+	0.898133i	$0.355076\pi$
$762$	0	0
$763$	1.30132e7	0.809230
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.90880e7	−1.17158
$768$	0	0
$769$	−5.59898e6	−0.341423	−0.170712	−	0.985321i	$-0.554607\pi$
$769$	−5.59898e6	−0.341423	−0.170712	+	0.985321i	$0.554607\pi$
$770$	0	0
$771$	1.72846e7	1.04719
$772$	0	0
$773$	6.34625e6	0.382004	0.191002	−	0.981590i	$-0.438826\pi$
$773$	6.34625e6	0.382004	0.191002	+	0.981590i	$0.438826\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.41864e7	0.842984
$778$	0	0
$779$	−1.66370e7	−0.982272
$780$	0	0
$781$	−3.76080e6	−0.220624
$782$	0	0
$783$	3.16600e7	1.84547
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.73688e7	0.999617	0.499809	−	0.866136i	$-0.333404\pi$
$787$	1.73688e7	0.999617	0.499809	+	0.866136i	$0.333404\pi$
$788$	0	0
$789$	4.54076e6	0.259678
$790$	0	0
$791$	−1.38473e7	−0.786909
$792$	0	0
$793$	−2.24746e6	−0.126914
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.06932e6	0.449978	0.224989	−	0.974361i	$-0.427765\pi$
$797$	8.06932e6	0.449978	0.224989	+	0.974361i	$0.427765\pi$
$798$	0	0
$799$	9.24246e6	0.512178
$800$	0	0
$801$	98294.0	0.00541310
$802$	0	0
$803$	2.40082e6	0.131393
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.03457e7	−1.09974
$808$	0	0
$809$	1.94554e7	1.04513	0.522564	−	0.852600i	$-0.324976\pi$
$809$	1.94554e7	1.04513	0.522564	+	0.852600i	$0.324976\pi$
$810$	0	0
$811$	2.85204e6	0.152266	0.0761330	−	0.997098i	$-0.475743\pi$
$811$	2.85204e6	0.152266	0.0761330	+	0.997098i	$0.475743\pi$
$812$	0	0
$813$	6.86928e6	0.364490
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.98561e7	−2.08900
$818$	0	0
$819$	1.07129e7	0.558079
$820$	0	0
$821$	−5.48431e6	−0.283965	−0.141982	−	0.989869i	$-0.545348\pi$
$821$	−5.48431e6	−0.283965	−0.141982	+	0.989869i	$0.545348\pi$
$822$	0	0
$823$	−6.57652e6	−0.338452	−0.169226	−	0.985577i	$-0.554127\pi$
$823$	−6.57652e6	−0.338452	−0.169226	+	0.985577i	$0.554127\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.19715e6	0.213398	0.106699	−	0.994291i	$-0.465972\pi$
$827$	4.19715e6	0.213398	0.106699	+	0.994291i	$0.465972\pi$
$828$	0	0
$829$	2.78889e7	1.40943	0.704717	−	0.709488i	$-0.251074\pi$
$829$	2.78889e7	1.40943	0.704717	+	0.709488i	$0.251074\pi$
$830$	0	0
$831$	2.04573e7	1.02765
$832$	0	0
$833$	1.76681e6	0.0882220
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.64260e6	−0.426413
$838$	0	0
$839$	−1.27752e7	−0.626560	−0.313280	−	0.949661i	$-0.601428\pi$
$839$	−1.27752e7	−0.626560	−0.313280	+	0.949661i	$0.601428\pi$
$840$	0	0
$841$	4.11742e7	2.00740
$842$	0	0
$843$	−1.35957e7	−0.658922
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.85024e7	0.886173
$848$	0	0
$849$	7.66977e6	0.365185
$850$	0	0
$851$	−3.55555e7	−1.68300
$852$	0	0
$853$	5.37122e6	0.252755	0.126378	−	0.991982i	$-0.459665\pi$
$853$	5.37122e6	0.252755	0.126378	+	0.991982i	$0.459665\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.61585e7	−0.751535	−0.375767	−	0.926714i	$-0.622621\pi$
$857$	−1.61585e7	−0.751535	−0.375767	+	0.926714i	$0.622621\pi$
$858$	0	0
$859$	−3.34643e7	−1.54739	−0.773693	−	0.633561i	$-0.781593\pi$
$859$	−3.34643e7	−1.54739	−0.773693	+	0.633561i	$0.781593\pi$
$860$	0	0
$861$	1.01127e7	0.464899
$862$	0	0
$863$	2.51788e7	1.15082	0.575412	−	0.817864i	$-0.304842\pi$
$863$	2.51788e7	1.15082	0.575412	+	0.817864i	$0.304842\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−5.11790e6	−0.231230
$868$	0	0
$869$	−7.97280e6	−0.358147
$870$	0	0
$871$	4.32033e7	1.92962
$872$	0	0
$873$	−4.47363e6	−0.198666
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.46327e7	1.08146	0.540732	−	0.841195i	$-0.318147\pi$
$877$	2.46327e7	1.08146	0.540732	+	0.841195i	$0.318147\pi$
$878$	0	0
$879$	−9.53436e6	−0.416217
$880$	0	0
$881$	−1.10564e7	−0.479926	−0.239963	−	0.970782i	$-0.577135\pi$
$881$	−1.10564e7	−0.479926	−0.239963	+	0.970782i	$0.577135\pi$
$882$	0	0
$883$	−3.74671e7	−1.61714	−0.808572	−	0.588398i	$-0.799759\pi$
$883$	−3.74671e7	−1.61714	−0.808572	+	0.588398i	$0.799759\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.96341e7	−1.26469	−0.632343	−	0.774689i	$-0.717906\pi$
$887$	−2.96341e7	−1.26469	−0.632343	+	0.774689i	$0.717906\pi$
$888$	0	0
$889$	6.31668e6	0.268062
$890$	0	0
$891$	−1.59710e6	−0.0673966
$892$	0	0
$893$	−2.11649e7	−0.888154
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.79779e7	1.16100
$898$	0	0
$899$	−1.68390e7	−0.694891
$900$	0	0
$901$	2.37663e7	0.975327
$902$	0	0
$903$	2.42262e7	0.988705
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.94430e6	0.0784773	0.0392387	−	0.999230i	$-0.487507\pi$
$907$	1.94430e6	0.0784773	0.0392387	+	0.999230i	$0.487507\pi$
$908$	0	0
$909$	1.70877e7	0.685920
$910$	0	0
$911$	−1.28064e7	−0.511245	−0.255623	−	0.966777i	$-0.582280\pi$
$911$	−1.28064e7	−0.511245	−0.255623	+	0.966777i	$0.582280\pi$
$912$	0	0
$913$	−1.62913e6	−0.0646812
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.09139e7	−0.428606
$918$	0	0
$919$	3.58404e7	1.39986	0.699929	−	0.714213i	$-0.253215\pi$
$919$	3.58404e7	1.39986	0.699929	+	0.714213i	$0.253215\pi$
$920$	0	0
$921$	1.22064e7	0.474176
$922$	0	0
$923$	−2.76398e7	−1.06790
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.33083e7	−0.508653
$928$	0	0
$929$	−8.47792e6	−0.322292	−0.161146	−	0.986931i	$-0.551519\pi$
$929$	−8.47792e6	−0.322292	−0.161146	+	0.986931i	$0.551519\pi$
$930$	0	0
$931$	−4.04593e6	−0.152983
$932$	0	0
$933$	2.36591e7	0.889803
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.60566e7	0.597454	0.298727	−	0.954339i	$-0.403438\pi$
$937$	1.60566e7	0.597454	0.298727	+	0.954339i	$0.403438\pi$
$938$	0	0
$939$	−3.28302e6	−0.121509
$940$	0	0
$941$	−4.24675e7	−1.56344	−0.781722	−	0.623627i	$-0.785658\pi$
$941$	−4.24675e7	−1.56344	−0.781722	+	0.623627i	$0.785658\pi$
$942$	0	0
$943$	−2.53456e7	−0.928159
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.78877e7	1.01050	0.505252	−	0.862972i	$-0.331400\pi$
$947$	2.78877e7	1.01050	0.505252	+	0.862972i	$0.331400\pi$
$948$	0	0
$949$	1.76447e7	0.635988
$950$	0	0
$951$	−2.81899e7	−1.01075
$952$	0	0
$953$	−4.77895e7	−1.70451	−0.852257	−	0.523123i	$-0.824767\pi$
$953$	−4.77895e7	−1.70451	−0.852257	+	0.523123i	$0.824767\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8.74584e6	−0.308690
$958$	0	0
$959$	−4.69762e6	−0.164942
$960$	0	0
$961$	−2.40324e7	−0.839439
$962$	0	0
$963$	−1.09760e7	−0.381399
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.51588e7	1.55302	0.776509	−	0.630107i	$-0.216989\pi$
$967$	4.51588e7	1.55302	0.776509	+	0.630107i	$0.216989\pi$
$968$	0	0
$969$	−2.44865e7	−0.837756
$970$	0	0
$971$	2.70901e7	0.922067	0.461033	−	0.887383i	$-0.347479\pi$
$971$	2.70901e7	0.922067	0.461033	+	0.887383i	$0.347479\pi$
$972$	0	0
$973$	−1.64976e7	−0.558647
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.34630e7	−1.45675	−0.728373	−	0.685181i	$-0.759723\pi$
$977$	−4.34630e7	−1.45675	−0.728373	+	0.685181i	$0.759723\pi$
$978$	0	0
$979$	−82600.0	−0.00275438
$980$	0	0
$981$	1.26423e7	0.419425
$982$	0	0
$983$	6.56037e6	0.216543	0.108272	−	0.994121i	$-0.465468\pi$
$983$	6.56037e6	0.216543	0.108272	+	0.994121i	$0.465468\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.28650e7	0.420355
$988$	0	0
$989$	−6.07185e7	−1.97392
$990$	0	0
$991$	−1.72942e7	−0.559391	−0.279696	−	0.960089i	$-0.590234\pi$
$991$	−1.72942e7	−0.559391	−0.279696	+	0.960089i	$0.590234\pi$
$992$	0	0
$993$	1.30309e7	0.419375
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.24107e7	0.714031	0.357015	−	0.934098i	$-0.383794\pi$
$997$	2.24107e7	0.714031	0.357015	+	0.934098i	$0.383794\pi$
$998$	0	0
$999$	4.19254e7	1.32912

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.s.1.2		2
4.3	odd	2		100.6.a.d.1.1		2
5.2	odd	4		80.6.c.b.49.1		2
5.3	odd	4		80.6.c.b.49.2		2
5.4	even	2	inner	400.6.a.s.1.1		2
12.11	even	2		900.6.a.q.1.2		2
15.2	even	4		720.6.f.d.289.1		2
15.8	even	4		720.6.f.d.289.2		2
20.3	even	4		20.6.c.a.9.1	✓	2
20.7	even	4		20.6.c.a.9.2	yes	2
20.19	odd	2		100.6.a.d.1.2		2
40.3	even	4		320.6.c.e.129.2		2
40.13	odd	4		320.6.c.d.129.1		2
40.27	even	4		320.6.c.e.129.1		2
40.37	odd	4		320.6.c.d.129.2		2
60.23	odd	4		180.6.d.b.109.2		2
60.47	odd	4		180.6.d.b.109.1		2
60.59	even	2		900.6.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.6.c.a.9.1	✓	2	20.3	even	4
20.6.c.a.9.2	yes	2	20.7	even	4
80.6.c.b.49.1		2	5.2	odd	4
80.6.c.b.49.2		2	5.3	odd	4
100.6.a.d.1.1		2	4.3	odd	2
100.6.a.d.1.2		2	20.19	odd	2
180.6.d.b.109.1		2	60.47	odd	4
180.6.d.b.109.2		2	60.23	odd	4
320.6.c.d.129.1		2	40.13	odd	4
320.6.c.d.129.2		2	40.37	odd	4
320.6.c.e.129.1		2	40.27	even	4
320.6.c.e.129.2		2	40.3	even	4
400.6.a.s.1.1		2	5.4	even	2	inner
400.6.a.s.1.2		2	1.1	even	1	trivial
720.6.f.d.289.1		2	15.2	even	4
720.6.f.d.289.2		2	15.8	even	4
900.6.a.q.1.1		2	60.59	even	2
900.6.a.q.1.2		2	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q+11.1355 q^{3} -122.491 q^{7} -119.000 q^{9} +100.000 q^{11} +734.945 q^{13} -979.927 q^{17} +2244.00 q^{19} -1364.00 q^{21} +3418.61 q^{23} -4031.06 q^{27} -7854.00 q^{29} +2144.00 q^{31} +1113.55 q^{33} -10400.6 q^{37} +8184.00 q^{39} -7414.00 q^{41} -17761.2 q^{43} -9431.79 q^{47} -1803.00 q^{49} -10912.0 q^{51} -24253.2 q^{53} +24988.1 q^{57} -25972.0 q^{59} -3058.00 q^{61} +14576.4 q^{63} +58784.5 q^{67} +38068.0 q^{69} -37608.0 q^{71} +24008.2 q^{73} -12249.1 q^{77} -79728.0 q^{79} -15971.0 q^{81} -16291.3 q^{83} -87458.4 q^{87} -826.000 q^{89} -90024.0 q^{91} +23874.6 q^{93} +37593.5 q^{97} -11900.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 238 q^{9} + 200 q^{11} + 4488 q^{19} - 2728 q^{21} - 15708 q^{29} + 4288 q^{31} + 16368 q^{39} - 14828 q^{41} - 3606 q^{49} - 21824 q^{51} - 51944 q^{59} - 6116 q^{61} + 76136 q^{69} - 75216 q^{71}+ \cdots - 23800 q^{99}+O(q^{100}) \) 2 * q - 238 * q^9 + 200 * q^11 + 4488 * q^19 - 2728 * q^21 - 15708 * q^29 + 4288 * q^31 + 16368 * q^39 - 14828 * q^41 - 3606 * q^49 - 21824 * q^51 - 51944 * q^59 - 6116 * q^61 + 76136 * q^69 - 75216 * q^71 - 159456 * q^79 - 31942 * q^81 - 1652 * q^89 - 180048 * q^91 - 23800 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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