LMFDB - Embedded newform 400.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.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:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-26.0000 q^{3} -22.0000 q^{7} +433.000 q^{9} +O(q^{10})$

$q-26.0000 q^{3} -22.0000 q^{7} +433.000 q^{9} +768.000 q^{11} +46.0000 q^{13} -378.000 q^{17} -1100.00 q^{19} +572.000 q^{21} -1986.00 q^{23} -4940.00 q^{27} -5610.00 q^{29} +3988.00 q^{31} -19968.0 q^{33} +142.000 q^{37} -1196.00 q^{39} +1542.00 q^{41} -5026.00 q^{43} +24738.0 q^{47} -16323.0 q^{49} +9828.00 q^{51} +14166.0 q^{53} +28600.0 q^{57} -28380.0 q^{59} +5522.00 q^{61} -9526.00 q^{63} -24742.0 q^{67} +51636.0 q^{69} -42372.0 q^{71} +52126.0 q^{73} -16896.0 q^{77} +39640.0 q^{79} +23221.0 q^{81} -59826.0 q^{83} +145860. q^{87} +57690.0 q^{89} -1012.00 q^{91} -103688. q^{93} +144382. q^{97} +332544. q^{99} +O(q^{100})$

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
$3$	−26.0000	−1.66790	−0.833950	−	0.551839i	$-0.813926\pi$
$3$	−26.0000	−1.66790	−0.833950	+	0.551839i	$0.813926\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−22.0000	−0.169698	−0.0848492	−	0.996394i	$-0.527041\pi$
$7$	−22.0000	−0.169698	−0.0848492	+	0.996394i	$0.527041\pi$
$8$	0	0
$9$	433.000	1.78189
$10$	0	0
$11$	768.000	1.91372	0.956862	−	0.290541i	$-0.0938354\pi$
$11$	768.000	1.91372	0.956862	+	0.290541i	$0.0938354\pi$
$12$	0	0
$13$	46.0000	0.0754917	0.0377459	−	0.999287i	$-0.487982\pi$
$13$	46.0000	0.0754917	0.0377459	+	0.999287i	$0.487982\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−378.000	−0.317227	−0.158613	−	0.987341i	$-0.550702\pi$
$17$	−378.000	−0.317227	−0.158613	+	0.987341i	$0.550702\pi$
$18$	0	0
$19$	−1100.00	−0.699051	−0.349525	−	0.936927i	$-0.613657\pi$
$19$	−1100.00	−0.699051	−0.349525	+	0.936927i	$0.613657\pi$
$20$	0	0
$21$	572.000	0.283040
$22$	0	0
$23$	−1986.00	−0.782816	−0.391408	−	0.920217i	$-0.628012\pi$
$23$	−1986.00	−0.782816	−0.391408	+	0.920217i	$0.628012\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4940.00	−1.30412
$28$	0	0
$29$	−5610.00	−1.23870	−0.619352	−	0.785113i	$-0.712605\pi$
$29$	−5610.00	−1.23870	−0.619352	+	0.785113i	$0.712605\pi$
$30$	0	0
$31$	3988.00	0.745334	0.372667	−	0.927965i	$-0.378443\pi$
$31$	3988.00	0.745334	0.372667	+	0.927965i	$0.378443\pi$
$32$	0	0
$33$	−19968.0	−3.19190
$34$	0	0
$35$	0	0
$36$	0	0
$37$	142.000	0.0170523	0.00852617	−	0.999964i	$-0.497286\pi$
$37$	142.000	0.0170523	0.00852617	+	0.999964i	$0.497286\pi$
$38$	0	0
$39$	−1196.00	−0.125913
$40$	0	0
$41$	1542.00	0.143260	0.0716300	−	0.997431i	$-0.477180\pi$
$41$	1542.00	0.143260	0.0716300	+	0.997431i	$0.477180\pi$
$42$	0	0
$43$	−5026.00	−0.414526	−0.207263	−	0.978285i	$-0.566456\pi$
$43$	−5026.00	−0.414526	−0.207263	+	0.978285i	$0.566456\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	24738.0	1.63350	0.816752	−	0.576990i	$-0.195773\pi$
$47$	24738.0	1.63350	0.816752	+	0.576990i	$0.195773\pi$
$48$	0	0
$49$	−16323.0	−0.971202
$50$	0	0
$51$	9828.00	0.529102
$52$	0	0
$53$	14166.0	0.692720	0.346360	−	0.938102i	$-0.387418\pi$
$53$	14166.0	0.692720	0.346360	+	0.938102i	$0.387418\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	28600.0	1.16595
$58$	0	0
$59$	−28380.0	−1.06141	−0.530704	−	0.847557i	$-0.678072\pi$
$59$	−28380.0	−1.06141	−0.530704	+	0.847557i	$0.678072\pi$
$60$	0	0
$61$	5522.00	0.190008	0.0950040	−	0.995477i	$-0.469714\pi$
$61$	5522.00	0.190008	0.0950040	+	0.995477i	$0.469714\pi$
$62$	0	0
$63$	−9526.00	−0.302384
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−24742.0	−0.673361	−0.336680	−	0.941619i	$-0.609304\pi$
$67$	−24742.0	−0.673361	−0.336680	+	0.941619i	$0.609304\pi$
$68$	0	0
$69$	51636.0	1.30566
$70$	0	0
$71$	−42372.0	−0.997546	−0.498773	−	0.866733i	$-0.666216\pi$
$71$	−42372.0	−0.997546	−0.498773	+	0.866733i	$0.666216\pi$
$72$	0	0
$73$	52126.0	1.14485	0.572423	−	0.819958i	$-0.306003\pi$
$73$	52126.0	1.14485	0.572423	+	0.819958i	$0.306003\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−16896.0	−0.324756
$78$	0	0
$79$	39640.0	0.714605	0.357302	−	0.933989i	$-0.383697\pi$
$79$	39640.0	0.714605	0.357302	+	0.933989i	$0.383697\pi$
$80$	0	0
$81$	23221.0	0.393250
$82$	0	0
$83$	−59826.0	−0.953223	−0.476612	−	0.879114i	$-0.658135\pi$
$83$	−59826.0	−0.953223	−0.476612	+	0.879114i	$0.658135\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	145860.	2.06604
$88$	0	0
$89$	57690.0	0.772015	0.386007	−	0.922496i	$-0.373854\pi$
$89$	57690.0	0.772015	0.386007	+	0.922496i	$0.373854\pi$
$90$	0	0
$91$	−1012.00	−0.0128108
$92$	0	0
$93$	−103688.	−1.24314
$94$	0	0
$95$	0	0
$96$	0	0
$97$	144382.	1.55806	0.779029	−	0.626988i	$-0.215712\pi$
$97$	144382.	1.55806	0.779029	+	0.626988i	$0.215712\pi$
$98$	0	0
$99$	332544.	3.41005
$100$	0	0
$101$	−141258.	−1.37787	−0.688937	−	0.724821i	$-0.741922\pi$
$101$	−141258.	−1.37787	−0.688937	+	0.724821i	$0.741922\pi$
$102$	0	0
$103$	139814.	1.29855	0.649273	−	0.760555i	$-0.275073\pi$
$103$	139814.	1.29855	0.649273	+	0.760555i	$0.275073\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	86418.0	0.729701	0.364850	−	0.931066i	$-0.381120\pi$
$107$	86418.0	0.729701	0.364850	+	0.931066i	$0.381120\pi$
$108$	0	0
$109$	218450.	1.76111	0.880554	−	0.473947i	$-0.157171\pi$
$109$	218450.	1.76111	0.880554	+	0.473947i	$0.157171\pi$
$110$	0	0
$111$	−3692.00	−0.0284416
$112$	0	0
$113$	28806.0	0.212220	0.106110	−	0.994354i	$-0.466160\pi$
$113$	28806.0	0.212220	0.106110	+	0.994354i	$0.466160\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	19918.0	0.134518
$118$	0	0
$119$	8316.00	0.0538328
$120$	0	0
$121$	428773.	2.66234
$122$	0	0
$123$	−40092.0	−0.238943
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−216502.	−1.19111	−0.595556	−	0.803314i	$-0.703068\pi$
$127$	−216502.	−1.19111	−0.595556	+	0.803314i	$0.703068\pi$
$128$	0	0
$129$	130676.	0.691388
$130$	0	0
$131$	244608.	1.24535	0.622676	−	0.782479i	$-0.286045\pi$
$131$	244608.	1.24535	0.622676	+	0.782479i	$0.286045\pi$
$132$	0	0
$133$	24200.0	0.118628
$134$	0	0
$135$	0	0
$136$	0	0
$137$	239502.	1.09020	0.545102	−	0.838370i	$-0.316491\pi$
$137$	239502.	1.09020	0.545102	+	0.838370i	$0.316491\pi$
$138$	0	0
$139$	−30860.0	−0.135475	−0.0677375	−	0.997703i	$-0.521578\pi$
$139$	−30860.0	−0.135475	−0.0677375	+	0.997703i	$0.521578\pi$
$140$	0	0
$141$	−643188.	−2.72452
$142$	0	0
$143$	35328.0	0.144470
$144$	0	0
$145$	0	0
$146$	0	0
$147$	424398.	1.61987
$148$	0	0
$149$	−100950.	−0.372512	−0.186256	−	0.982501i	$-0.559635\pi$
$149$	−100950.	−0.372512	−0.186256	+	0.982501i	$0.559635\pi$
$150$	0	0
$151$	−12452.0	−0.0444423	−0.0222212	−	0.999753i	$-0.507074\pi$
$151$	−12452.0	−0.0444423	−0.0222212	+	0.999753i	$0.507074\pi$
$152$	0	0
$153$	−163674.	−0.565264
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6022.00	0.0194981	0.00974903	−	0.999952i	$-0.496897\pi$
$157$	6022.00	0.0194981	0.00974903	+	0.999952i	$0.496897\pi$
$158$	0	0
$159$	−368316.	−1.15539
$160$	0	0
$161$	43692.0	0.132843
$162$	0	0
$163$	−500866.	−1.47656	−0.738282	−	0.674492i	$-0.764363\pi$
$163$	−500866.	−1.47656	−0.738282	+	0.674492i	$0.764363\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	555258.	1.54065	0.770324	−	0.637652i	$-0.220094\pi$
$167$	555258.	1.54065	0.770324	+	0.637652i	$0.220094\pi$
$168$	0	0
$169$	−369177.	−0.994301
$170$	0	0
$171$	−476300.	−1.24563
$172$	0	0
$173$	−417354.	−1.06020	−0.530102	−	0.847934i	$-0.677846\pi$
$173$	−417354.	−1.06020	−0.530102	+	0.847934i	$0.677846\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	737880.	1.77032
$178$	0	0
$179$	52380.0	0.122189	0.0610946	−	0.998132i	$-0.480541\pi$
$179$	52380.0	0.122189	0.0610946	+	0.998132i	$0.480541\pi$
$180$	0	0
$181$	546662.	1.24029	0.620144	−	0.784488i	$-0.287074\pi$
$181$	546662.	1.24029	0.620144	+	0.784488i	$0.287074\pi$
$182$	0	0
$183$	−143572.	−0.316914
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−290304.	−0.607084
$188$	0	0
$189$	108680.	0.221307
$190$	0	0
$191$	452028.	0.896565	0.448283	−	0.893892i	$-0.352036\pi$
$191$	452028.	0.896565	0.448283	+	0.893892i	$0.352036\pi$
$192$	0	0
$193$	−485594.	−0.938383	−0.469191	−	0.883097i	$-0.655455\pi$
$193$	−485594.	−0.938383	−0.469191	+	0.883097i	$0.655455\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.01018e6	−1.85452	−0.927262	−	0.374414i	$-0.877844\pi$
$197$	−1.01018e6	−1.85452	−0.927262	+	0.374414i	$0.877844\pi$
$198$	0	0
$199$	807640.	1.44572	0.722862	−	0.690993i	$-0.242826\pi$
$199$	807640.	1.44572	0.722862	+	0.690993i	$0.242826\pi$
$200$	0	0
$201$	643292.	1.12310
$202$	0	0
$203$	123420.	0.210206
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−859938.	−1.39489
$208$	0	0
$209$	−844800.	−1.33779
$210$	0	0
$211$	−149552.	−0.231252	−0.115626	−	0.993293i	$-0.536887\pi$
$211$	−149552.	−0.231252	−0.115626	+	0.993293i	$0.536887\pi$
$212$	0	0
$213$	1.10167e6	1.66381
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−87736.0	−0.126482
$218$	0	0
$219$	−1.35528e6	−1.90949
$220$	0	0
$221$	−17388.0	−0.0239480
$222$	0	0
$223$	−443506.	−0.597224	−0.298612	−	0.954375i	$-0.596524\pi$
$223$	−443506.	−0.597224	−0.298612	+	0.954375i	$0.596524\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	420018.	0.541007	0.270504	−	0.962719i	$-0.412810\pi$
$227$	420018.	0.541007	0.270504	+	0.962719i	$0.412810\pi$
$228$	0	0
$229$	1.05875e6	1.33415	0.667075	−	0.744990i	$-0.267546\pi$
$229$	1.05875e6	1.33415	0.667075	+	0.744990i	$0.267546\pi$
$230$	0	0
$231$	439296.	0.541661
$232$	0	0
$233$	1.27345e6	1.53671	0.768353	−	0.640026i	$-0.221077\pi$
$233$	1.27345e6	1.53671	0.768353	+	0.640026i	$0.221077\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1.03064e6	−1.19189
$238$	0	0
$239$	370680.	0.419763	0.209882	−	0.977727i	$-0.432692\pi$
$239$	370680.	0.419763	0.209882	+	0.977727i	$0.432692\pi$
$240$	0	0
$241$	−561298.	−0.622517	−0.311258	−	0.950325i	$-0.600750\pi$
$241$	−561298.	−0.622517	−0.311258	+	0.950325i	$0.600750\pi$
$242$	0	0
$243$	596674.	0.648219
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−50600.0	−0.0527726
$248$	0	0
$249$	1.55548e6	1.58988
$250$	0	0
$251$	−577152.	−0.578237	−0.289119	−	0.957293i	$-0.593362\pi$
$251$	−577152.	−0.578237	−0.289119	+	0.957293i	$0.593362\pi$
$252$	0	0
$253$	−1.52525e6	−1.49809
$254$	0	0
$255$	0	0
$256$	0	0
$257$	651462.	0.615257	0.307628	−	0.951507i	$-0.400465\pi$
$257$	651462.	0.615257	0.307628	+	0.951507i	$0.400465\pi$
$258$	0	0
$259$	−3124.00	−0.00289375
$260$	0	0
$261$	−2.42913e6	−2.20724
$262$	0	0
$263$	917574.	0.817997	0.408999	−	0.912535i	$-0.365878\pi$
$263$	917574.	0.817997	0.408999	+	0.912535i	$0.365878\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.49994e6	−1.28764
$268$	0	0
$269$	−735390.	−0.619637	−0.309818	−	0.950796i	$-0.600268\pi$
$269$	−735390.	−0.619637	−0.309818	+	0.950796i	$0.600268\pi$
$270$	0	0
$271$	1.12131e6	0.927474	0.463737	−	0.885973i	$-0.346508\pi$
$271$	1.12131e6	0.927474	0.463737	+	0.885973i	$0.346508\pi$
$272$	0	0
$273$	26312.0	0.0213672
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.66034e6	1.30016	0.650082	−	0.759864i	$-0.274735\pi$
$277$	1.66034e6	1.30016	0.650082	+	0.759864i	$0.274735\pi$
$278$	0	0
$279$	1.72680e6	1.32811
$280$	0	0
$281$	1.45210e6	1.09706	0.548531	−	0.836130i	$-0.315187\pi$
$281$	1.45210e6	1.09706	0.548531	+	0.836130i	$0.315187\pi$
$282$	0	0
$283$	309014.	0.229357	0.114679	−	0.993403i	$-0.463416\pi$
$283$	309014.	0.229357	0.114679	+	0.993403i	$0.463416\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−33924.0	−0.0243110
$288$	0	0
$289$	−1.27697e6	−0.899367
$290$	0	0
$291$	−3.75393e6	−2.59869
$292$	0	0
$293$	1.59301e6	1.08405	0.542024	−	0.840363i	$-0.317658\pi$
$293$	1.59301e6	1.08405	0.542024	+	0.840363i	$0.317658\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.79392e6	−2.49573
$298$	0	0
$299$	−91356.0	−0.0590961
$300$	0	0
$301$	110572.	0.0703443
$302$	0	0
$303$	3.67271e6	2.29816
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.24726e6	0.755284	0.377642	−	0.925952i	$-0.376735\pi$
$307$	1.24726e6	0.755284	0.377642	+	0.925952i	$0.376735\pi$
$308$	0	0
$309$	−3.63516e6	−2.16585
$310$	0	0
$311$	665988.	0.390450	0.195225	−	0.980758i	$-0.437456\pi$
$311$	665988.	0.390450	0.195225	+	0.980758i	$0.437456\pi$
$312$	0	0
$313$	591286.	0.341143	0.170572	−	0.985345i	$-0.445439\pi$
$313$	591286.	0.341143	0.170572	+	0.985345i	$0.445439\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	516342.	0.288595	0.144298	−	0.989534i	$-0.453908\pi$
$317$	516342.	0.288595	0.144298	+	0.989534i	$0.453908\pi$
$318$	0	0
$319$	−4.30848e6	−2.37054
$320$	0	0
$321$	−2.24687e6	−1.21707
$322$	0	0
$323$	415800.	0.221757
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.67970e6	−2.93735
$328$	0	0
$329$	−544236.	−0.277203
$330$	0	0
$331$	3.29577e6	1.65343	0.826717	−	0.562619i	$-0.190206\pi$
$331$	3.29577e6	1.65343	0.826717	+	0.562619i	$0.190206\pi$
$332$	0	0
$333$	61486.0	0.0303854
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.91098e6	−0.916602	−0.458301	−	0.888797i	$-0.651542\pi$
$337$	−1.91098e6	−0.916602	−0.458301	+	0.888797i	$0.651542\pi$
$338$	0	0
$339$	−748956.	−0.353962
$340$	0	0
$341$	3.06278e6	1.42636
$342$	0	0
$343$	728860.	0.334510
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.42006e6	1.07895	0.539476	−	0.842001i	$-0.318622\pi$
$347$	2.42006e6	1.07895	0.539476	+	0.842001i	$0.318622\pi$
$348$	0	0
$349$	2.50727e6	1.10189	0.550944	−	0.834542i	$-0.314268\pi$
$349$	2.50727e6	1.10189	0.550944	+	0.834542i	$0.314268\pi$
$350$	0	0
$351$	−227240.	−0.0984503
$352$	0	0
$353$	413166.	0.176477	0.0882384	−	0.996099i	$-0.471876\pi$
$353$	413166.	0.176477	0.0882384	+	0.996099i	$0.471876\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−216216.	−0.0897878
$358$	0	0
$359$	−1.73772e6	−0.711613	−0.355806	−	0.934560i	$-0.615794\pi$
$359$	−1.73772e6	−0.711613	−0.355806	+	0.934560i	$0.615794\pi$
$360$	0	0
$361$	−1.26610e6	−0.511328
$362$	0	0
$363$	−1.11481e7	−4.44052
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.16098e6	0.449944	0.224972	−	0.974365i	$-0.427771\pi$
$367$	1.16098e6	0.449944	0.224972	+	0.974365i	$0.427771\pi$
$368$	0	0
$369$	667686.	0.255274
$370$	0	0
$371$	−311652.	−0.117553
$372$	0	0
$373$	−343754.	−0.127931	−0.0639655	−	0.997952i	$-0.520375\pi$
$373$	−343754.	−0.127931	−0.0639655	+	0.997952i	$0.520375\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−258060.	−0.0935120
$378$	0	0
$379$	−573140.	−0.204957	−0.102478	−	0.994735i	$-0.532677\pi$
$379$	−573140.	−0.204957	−0.102478	+	0.994735i	$0.532677\pi$
$380$	0	0
$381$	5.62905e6	1.98666
$382$	0	0
$383$	−2.88055e6	−1.00341	−0.501704	−	0.865039i	$-0.667293\pi$
$383$	−2.88055e6	−1.00341	−0.501704	+	0.865039i	$0.667293\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.17626e6	−0.738640
$388$	0	0
$389$	−3.08559e6	−1.03387	−0.516933	−	0.856026i	$-0.672926\pi$
$389$	−3.08559e6	−1.03387	−0.516933	+	0.856026i	$0.672926\pi$
$390$	0	0
$391$	750708.	0.248330
$392$	0	0
$393$	−6.35981e6	−2.07712
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−885458.	−0.281963	−0.140981	−	0.990012i	$-0.545026\pi$
$397$	−885458.	−0.281963	−0.140981	+	0.990012i	$0.545026\pi$
$398$	0	0
$399$	−629200.	−0.197859
$400$	0	0
$401$	−3.75344e6	−1.16565	−0.582825	−	0.812598i	$-0.698053\pi$
$401$	−3.75344e6	−1.16565	−0.582825	+	0.812598i	$0.698053\pi$
$402$	0	0
$403$	183448.	0.0562666
$404$	0	0
$405$	0	0
$406$	0	0
$407$	109056.	0.0326335
$408$	0	0
$409$	−1.94653e6	−0.575377	−0.287689	−	0.957724i	$-0.592887\pi$
$409$	−1.94653e6	−0.575377	−0.287689	+	0.957724i	$0.592887\pi$
$410$	0	0
$411$	−6.22705e6	−1.81835
$412$	0	0
$413$	624360.	0.180119
$414$	0	0
$415$	0	0
$416$	0	0
$417$	802360.	0.225959
$418$	0	0
$419$	2.99166e6	0.832486	0.416243	−	0.909253i	$-0.363346\pi$
$419$	2.99166e6	0.832486	0.416243	+	0.909253i	$0.363346\pi$
$420$	0	0
$421$	3.96660e6	1.09072	0.545360	−	0.838202i	$-0.316393\pi$
$421$	3.96660e6	1.09072	0.545360	+	0.838202i	$0.316393\pi$
$422$	0	0
$423$	1.07116e7	2.91073
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−121484.	−0.0322440
$428$	0	0
$429$	−918528.	−0.240962
$430$	0	0
$431$	5.17115e6	1.34089	0.670446	−	0.741958i	$-0.266103\pi$
$431$	5.17115e6	1.34089	0.670446	+	0.741958i	$0.266103\pi$
$432$	0	0
$433$	4.53485e6	1.16237	0.581183	−	0.813773i	$-0.302590\pi$
$433$	4.53485e6	1.16237	0.581183	+	0.813773i	$0.302590\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.18460e6	0.547228
$438$	0	0
$439$	1.08220e6	0.268007	0.134004	−	0.990981i	$-0.457217\pi$
$439$	1.08220e6	0.268007	0.134004	+	0.990981i	$0.457217\pi$
$440$	0	0
$441$	−7.06786e6	−1.73058
$442$	0	0
$443$	−1.08079e6	−0.261656	−0.130828	−	0.991405i	$-0.541764\pi$
$443$	−1.08079e6	−0.261656	−0.130828	+	0.991405i	$0.541764\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.62470e6	0.621314
$448$	0	0
$449$	2.61783e6	0.612810	0.306405	−	0.951901i	$-0.400874\pi$
$449$	2.61783e6	0.612810	0.306405	+	0.951901i	$0.400874\pi$
$450$	0	0
$451$	1.18426e6	0.274160
$452$	0	0
$453$	323752.	0.0741254
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.59046e6	−0.356231	−0.178115	−	0.984010i	$-0.557000\pi$
$457$	−1.59046e6	−0.356231	−0.178115	+	0.984010i	$0.557000\pi$
$458$	0	0
$459$	1.86732e6	0.413701
$460$	0	0
$461$	4.25470e6	0.932431	0.466216	−	0.884671i	$-0.345617\pi$
$461$	4.25470e6	0.932431	0.466216	+	0.884671i	$0.345617\pi$
$462$	0	0
$463$	3.26605e6	0.708061	0.354031	−	0.935234i	$-0.384811\pi$
$463$	3.26605e6	0.708061	0.354031	+	0.935234i	$0.384811\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−601542.	−0.127636	−0.0638181	−	0.997962i	$-0.520328\pi$
$467$	−601542.	−0.127636	−0.0638181	+	0.997962i	$0.520328\pi$
$468$	0	0
$469$	544324.	0.114268
$470$	0	0
$471$	−156572.	−0.0325208
$472$	0	0
$473$	−3.85997e6	−0.793288
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.13388e6	1.23435
$478$	0	0
$479$	4.57932e6	0.911931	0.455966	−	0.889997i	$-0.349294\pi$
$479$	4.57932e6	0.911931	0.455966	+	0.889997i	$0.349294\pi$
$480$	0	0
$481$	6532.00	0.00128731
$482$	0	0
$483$	−1.13599e6	−0.221568
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.05226e6	1.34743	0.673714	−	0.738992i	$-0.264698\pi$
$487$	7.05226e6	1.34743	0.673714	+	0.738992i	$0.264698\pi$
$488$	0	0
$489$	1.30225e7	2.46276
$490$	0	0
$491$	2.62349e6	0.491106	0.245553	−	0.969383i	$-0.421030\pi$
$491$	2.62349e6	0.491106	0.245553	+	0.969383i	$0.421030\pi$
$492$	0	0
$493$	2.12058e6	0.392950
$494$	0	0
$495$	0	0
$496$	0	0
$497$	932184.	0.169282
$498$	0	0
$499$	3.61234e6	0.649437	0.324719	−	0.945811i	$-0.394730\pi$
$499$	3.61234e6	0.649437	0.324719	+	0.945811i	$0.394730\pi$
$500$	0	0
$501$	−1.44367e7	−2.56965
$502$	0	0
$503$	9.15629e6	1.61361	0.806807	−	0.590815i	$-0.201194\pi$
$503$	9.15629e6	1.61361	0.806807	+	0.590815i	$0.201194\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.59860e6	1.65840
$508$	0	0
$509$	7.26159e6	1.24233	0.621165	−	0.783679i	$-0.286660\pi$
$509$	7.26159e6	1.24233	0.621165	+	0.783679i	$0.286660\pi$
$510$	0	0
$511$	−1.14677e6	−0.194279
$512$	0	0
$513$	5.43400e6	0.911646
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.89988e7	3.12608
$518$	0	0
$519$	1.08512e7	1.76831
$520$	0	0
$521$	5.81020e6	0.937771	0.468886	−	0.883259i	$-0.344656\pi$
$521$	5.81020e6	0.937771	0.468886	+	0.883259i	$0.344656\pi$
$522$	0	0
$523$	−8.17067e6	−1.30618	−0.653090	−	0.757280i	$-0.726528\pi$
$523$	−8.17067e6	−1.30618	−0.653090	+	0.757280i	$0.726528\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.50746e6	−0.236440
$528$	0	0
$529$	−2.49215e6	−0.387199
$530$	0	0
$531$	−1.22885e7	−1.89132
$532$	0	0
$533$	70932.0	0.0108149
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.36188e6	−0.203800
$538$	0	0
$539$	−1.25361e7	−1.85861
$540$	0	0
$541$	−817378.	−0.120069	−0.0600343	−	0.998196i	$-0.519121\pi$
$541$	−817378.	−0.120069	−0.0600343	+	0.998196i	$0.519121\pi$
$542$	0	0
$543$	−1.42132e7	−2.06868
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3.50750e6	−0.501221	−0.250611	−	0.968088i	$-0.580631\pi$
$547$	−3.50750e6	−0.501221	−0.250611	+	0.968088i	$0.580631\pi$
$548$	0	0
$549$	2.39103e6	0.338574
$550$	0	0
$551$	6.17100e6	0.865918
$552$	0	0
$553$	−872080.	−0.121267
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.61490e6	−1.31313	−0.656563	−	0.754271i	$-0.727991\pi$
$557$	−9.61490e6	−1.31313	−0.656563	+	0.754271i	$0.727991\pi$
$558$	0	0
$559$	−231196.	−0.0312933
$560$	0	0
$561$	7.54790e6	1.01256
$562$	0	0
$563$	2.01941e6	0.268506	0.134253	−	0.990947i	$-0.457136\pi$
$563$	2.01941e6	0.268506	0.134253	+	0.990947i	$0.457136\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−510862.	−0.0667338
$568$	0	0
$569$	1.37859e6	0.178507	0.0892533	−	0.996009i	$-0.471552\pi$
$569$	1.37859e6	0.178507	0.0892533	+	0.996009i	$0.471552\pi$
$570$	0	0
$571$	−8.54295e6	−1.09652	−0.548261	−	0.836307i	$-0.684710\pi$
$571$	−8.54295e6	−1.09652	−0.548261	+	0.836307i	$0.684710\pi$
$572$	0	0
$573$	−1.17527e7	−1.49538
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.31458e6	0.289423	0.144711	−	0.989474i	$-0.453775\pi$
$577$	2.31458e6	0.289423	0.144711	+	0.989474i	$0.453775\pi$
$578$	0	0
$579$	1.26254e7	1.56513
$580$	0	0
$581$	1.31617e6	0.161760
$582$	0	0
$583$	1.08795e7	1.32568
$584$	0	0
$585$	0	0
$586$	0	0
$587$	928338.	0.111202	0.0556008	−	0.998453i	$-0.482293\pi$
$587$	928338.	0.111202	0.0556008	+	0.998453i	$0.482293\pi$
$588$	0	0
$589$	−4.38680e6	−0.521026
$590$	0	0
$591$	2.62646e7	3.09316
$592$	0	0
$593$	909486.	0.106209	0.0531043	−	0.998589i	$-0.483088\pi$
$593$	909486.	0.106209	0.0531043	+	0.998589i	$0.483088\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.09986e7	−2.41132
$598$	0	0
$599$	8.51136e6	0.969241	0.484621	−	0.874724i	$-0.338958\pi$
$599$	8.51136e6	0.969241	0.484621	+	0.874724i	$0.338958\pi$
$600$	0	0
$601$	6.12498e6	0.691701	0.345851	−	0.938290i	$-0.387590\pi$
$601$	6.12498e6	0.691701	0.345851	+	0.938290i	$0.387590\pi$
$602$	0	0
$603$	−1.07133e7	−1.19986
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.51646e6	−0.497538	−0.248769	−	0.968563i	$-0.580026\pi$
$607$	−4.51646e6	−0.497538	−0.248769	+	0.968563i	$0.580026\pi$
$608$	0	0
$609$	−3.20892e6	−0.350603
$610$	0	0
$611$	1.13795e6	0.123316
$612$	0	0
$613$	−9.63979e6	−1.03614	−0.518068	−	0.855340i	$-0.673349\pi$
$613$	−9.63979e6	−1.03614	−0.518068	+	0.855340i	$0.673349\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.92650e6	1.04974	0.524872	−	0.851181i	$-0.324113\pi$
$617$	9.92650e6	1.04974	0.524872	+	0.851181i	$0.324113\pi$
$618$	0	0
$619$	−7.63322e6	−0.800721	−0.400360	−	0.916358i	$-0.631115\pi$
$619$	−7.63322e6	−0.800721	−0.400360	+	0.916358i	$0.631115\pi$
$620$	0	0
$621$	9.81084e6	1.02089
$622$	0	0
$623$	−1.26918e6	−0.131010
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.19648e7	2.23130
$628$	0	0
$629$	−53676.0	−0.00540946
$630$	0	0
$631$	−1.80314e7	−1.80284	−0.901418	−	0.432949i	$-0.857473\pi$
$631$	−1.80314e7	−1.80284	−0.901418	+	0.432949i	$0.857473\pi$
$632$	0	0
$633$	3.88835e6	0.385706
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−750858.	−0.0733178
$638$	0	0
$639$	−1.83471e7	−1.77752
$640$	0	0
$641$	9.30190e6	0.894184	0.447092	−	0.894488i	$-0.352460\pi$
$641$	9.30190e6	0.894184	0.447092	+	0.894488i	$0.352460\pi$
$642$	0	0
$643$	−1.38332e7	−1.31946	−0.659730	−	0.751503i	$-0.729329\pi$
$643$	−1.38332e7	−1.31946	−0.659730	+	0.751503i	$0.729329\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.48997e7	−1.39932	−0.699658	−	0.714478i	$-0.746664\pi$
$647$	−1.48997e7	−1.39932	−0.699658	+	0.714478i	$0.746664\pi$
$648$	0	0
$649$	−2.17958e7	−2.03124
$650$	0	0
$651$	2.28114e6	0.210959
$652$	0	0
$653$	1.93306e7	1.77403	0.887016	−	0.461738i	$-0.152774\pi$
$653$	1.93306e7	1.77403	0.887016	+	0.461738i	$0.152774\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.25706e7	2.03999
$658$	0	0
$659$	4.06110e6	0.364276	0.182138	−	0.983273i	$-0.441698\pi$
$659$	4.06110e6	0.364276	0.182138	+	0.983273i	$0.441698\pi$
$660$	0	0
$661$	−1.35152e7	−1.20315	−0.601575	−	0.798816i	$-0.705460\pi$
$661$	−1.35152e7	−1.20315	−0.601575	+	0.798816i	$0.705460\pi$
$662$	0	0
$663$	452088.	0.0399429
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.11415e7	0.969678
$668$	0	0
$669$	1.15312e7	0.996111
$670$	0	0
$671$	4.24090e6	0.363623
$672$	0	0
$673$	−1.43520e7	−1.22144	−0.610722	−	0.791845i	$-0.709121\pi$
$673$	−1.43520e7	−1.22144	−0.610722	+	0.791845i	$0.709121\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.89530e6	−0.158930	−0.0794650	−	0.996838i	$-0.525321\pi$
$677$	−1.89530e6	−0.158930	−0.0794650	+	0.996838i	$0.525321\pi$
$678$	0	0
$679$	−3.17640e6	−0.264400
$680$	0	0
$681$	−1.09205e7	−0.902347
$682$	0	0
$683$	2.91641e6	0.239220	0.119610	−	0.992821i	$-0.461836\pi$
$683$	2.91641e6	0.239220	0.119610	+	0.992821i	$0.461836\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.75275e7	−2.22523
$688$	0	0
$689$	651636.	0.0522946
$690$	0	0
$691$	−1.44278e7	−1.14949	−0.574743	−	0.818334i	$-0.694898\pi$
$691$	−1.44278e7	−1.14949	−0.574743	+	0.818334i	$0.694898\pi$
$692$	0	0
$693$	−7.31597e6	−0.578680
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−582876.	−0.0454458
$698$	0	0
$699$	−3.31096e7	−2.56307
$700$	0	0
$701$	−1.58679e7	−1.21962	−0.609811	−	0.792547i	$-0.708754\pi$
$701$	−1.58679e7	−1.21962	−0.609811	+	0.792547i	$0.708754\pi$
$702$	0	0
$703$	−156200.	−0.0119205
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.10768e6	0.233823
$708$	0	0
$709$	−301810.	−0.0225485	−0.0112743	−	0.999936i	$-0.503589\pi$
$709$	−301810.	−0.0225485	−0.0112743	+	0.999936i	$0.503589\pi$
$710$	0	0
$711$	1.71641e7	1.27335
$712$	0	0
$713$	−7.92017e6	−0.583459
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.63768e6	−0.700123
$718$	0	0
$719$	−2.12677e7	−1.53426	−0.767130	−	0.641492i	$-0.778316\pi$
$719$	−2.12677e7	−1.53426	−0.767130	+	0.641492i	$0.778316\pi$
$720$	0	0
$721$	−3.07591e6	−0.220361
$722$	0	0
$723$	1.45937e7	1.03830
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.55009e7	1.08773	0.543863	−	0.839174i	$-0.316961\pi$
$727$	1.55009e7	1.08773	0.543863	+	0.839174i	$0.316961\pi$
$728$	0	0
$729$	−2.11562e7	−1.47441
$730$	0	0
$731$	1.89983e6	0.131499
$732$	0	0
$733$	1.21850e7	0.837653	0.418827	−	0.908066i	$-0.362441\pi$
$733$	1.21850e7	0.837653	0.418827	+	0.908066i	$0.362441\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.90019e7	−1.28863
$738$	0	0
$739$	2.90282e7	1.95528	0.977641	−	0.210282i	$-0.0674382\pi$
$739$	2.90282e7	1.95528	0.977641	+	0.210282i	$0.0674382\pi$
$740$	0	0
$741$	1.31560e6	0.0880194
$742$	0	0
$743$	1.61145e7	1.07089	0.535445	−	0.844570i	$-0.320144\pi$
$743$	1.61145e7	1.07089	0.535445	+	0.844570i	$0.320144\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.59047e7	−1.69854
$748$	0	0
$749$	−1.90120e6	−0.123829
$750$	0	0
$751$	2.92431e6	0.189201	0.0946005	−	0.995515i	$-0.469843\pi$
$751$	2.92431e6	0.189201	0.0946005	+	0.995515i	$0.469843\pi$
$752$	0	0
$753$	1.50060e7	0.964442
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.60325e7	−1.65111	−0.825557	−	0.564319i	$-0.809139\pi$
$757$	−2.60325e7	−1.65111	−0.825557	+	0.564319i	$0.809139\pi$
$758$	0	0
$759$	3.96564e7	2.49867
$760$	0	0
$761$	1.63263e7	1.02194	0.510970	−	0.859598i	$-0.329286\pi$
$761$	1.63263e7	1.02194	0.510970	+	0.859598i	$0.329286\pi$
$762$	0	0
$763$	−4.80590e6	−0.298857
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.30548e6	−0.0801275
$768$	0	0
$769$	2.58132e7	1.57408	0.787040	−	0.616902i	$-0.211612\pi$
$769$	2.58132e7	1.57408	0.787040	+	0.616902i	$0.211612\pi$
$770$	0	0
$771$	−1.69380e7	−1.02619
$772$	0	0
$773$	1.90592e7	1.14725	0.573624	−	0.819119i	$-0.305537\pi$
$773$	1.90592e7	1.14725	0.573624	+	0.819119i	$0.305537\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	81224.0	0.00482649
$778$	0	0
$779$	−1.69620e6	−0.100146
$780$	0	0
$781$	−3.25417e7	−1.90903
$782$	0	0
$783$	2.77134e7	1.61542
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.73411e7	−0.998021	−0.499011	−	0.866596i	$-0.666303\pi$
$787$	−1.73411e7	−0.998021	−0.499011	+	0.866596i	$0.666303\pi$
$788$	0	0
$789$	−2.38569e7	−1.36434
$790$	0	0
$791$	−633732.	−0.0360134
$792$	0	0
$793$	254012.	0.0143440
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.58169e7	1.43965	0.719827	−	0.694153i	$-0.244221\pi$
$797$	2.58169e7	1.43965	0.719827	+	0.694153i	$0.244221\pi$
$798$	0	0
$799$	−9.35096e6	−0.518190
$800$	0	0
$801$	2.49798e7	1.37565
$802$	0	0
$803$	4.00328e7	2.19092
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.91201e7	1.03349
$808$	0	0
$809$	8.88489e6	0.477288	0.238644	−	0.971107i	$-0.423297\pi$
$809$	8.88489e6	0.477288	0.238644	+	0.971107i	$0.423297\pi$
$810$	0	0
$811$	2.46396e7	1.31547	0.657735	−	0.753249i	$-0.271515\pi$
$811$	2.46396e7	1.31547	0.657735	+	0.753249i	$0.271515\pi$
$812$	0	0
$813$	−2.91540e7	−1.54693
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.52860e6	0.289774
$818$	0	0
$819$	−438196.	−0.0228275
$820$	0	0
$821$	1.13768e7	0.589062	0.294531	−	0.955642i	$-0.404837\pi$
$821$	1.13768e7	0.589062	0.294531	+	0.955642i	$0.404837\pi$
$822$	0	0
$823$	−1.30783e7	−0.673057	−0.336529	−	0.941673i	$-0.609253\pi$
$823$	−1.30783e7	−0.673057	−0.336529	+	0.941673i	$0.609253\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.57188e7	−1.81607	−0.908037	−	0.418891i	$-0.862419\pi$
$827$	−3.57188e7	−1.81607	−0.908037	+	0.418891i	$0.862419\pi$
$828$	0	0
$829$	1.61880e7	0.818103	0.409052	−	0.912511i	$-0.365860\pi$
$829$	1.61880e7	0.818103	0.409052	+	0.912511i	$0.365860\pi$
$830$	0	0
$831$	−4.31689e7	−2.16854
$832$	0	0
$833$	6.17009e6	0.308091
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.97007e7	−0.972005
$838$	0	0
$839$	2.55497e7	1.25309	0.626543	−	0.779387i	$-0.284469\pi$
$839$	2.55497e7	1.25309	0.626543	+	0.779387i	$0.284469\pi$
$840$	0	0
$841$	1.09610e7	0.534390
$842$	0	0
$843$	−3.77547e7	−1.82979
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9.43301e6	−0.451795
$848$	0	0
$849$	−8.03436e6	−0.382545
$850$	0	0
$851$	−282012.	−0.0133488
$852$	0	0
$853$	2.22953e7	1.04916	0.524579	−	0.851362i	$-0.324223\pi$
$853$	2.22953e7	1.04916	0.524579	+	0.851362i	$0.324223\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.96872e7	−0.915656	−0.457828	−	0.889041i	$-0.651372\pi$
$857$	−1.96872e7	−0.915656	−0.457828	+	0.889041i	$0.651372\pi$
$858$	0	0
$859$	−6.77582e6	−0.313313	−0.156657	−	0.987653i	$-0.550072\pi$
$859$	−6.77582e6	−0.313313	−0.156657	+	0.987653i	$0.550072\pi$
$860$	0	0
$861$	882024.	0.0405483
$862$	0	0
$863$	−2.63804e7	−1.20574	−0.602871	−	0.797839i	$-0.705977\pi$
$863$	−2.63804e7	−1.20574	−0.602871	+	0.797839i	$0.705977\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.32013e7	1.50006
$868$	0	0
$869$	3.04435e7	1.36756
$870$	0	0
$871$	−1.13813e6	−0.0508332
$872$	0	0
$873$	6.25174e7	2.77629
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.95161e7	−1.29587	−0.647934	−	0.761697i	$-0.724367\pi$
$877$	−2.95161e7	−1.29587	−0.647934	+	0.761697i	$0.724367\pi$
$878$	0	0
$879$	−4.14182e7	−1.80808
$880$	0	0
$881$	−1.48565e7	−0.644877	−0.322438	−	0.946590i	$-0.604502\pi$
$881$	−1.48565e7	−0.644877	−0.322438	+	0.946590i	$0.604502\pi$
$882$	0	0
$883$	−1.45340e7	−0.627313	−0.313656	−	0.949537i	$-0.601554\pi$
$883$	−1.45340e7	−0.627313	−0.313656	+	0.949537i	$0.601554\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.72028e7	−0.734160	−0.367080	−	0.930189i	$-0.619642\pi$
$887$	−1.72028e7	−0.734160	−0.367080	+	0.930189i	$0.619642\pi$
$888$	0	0
$889$	4.76304e6	0.202130
$890$	0	0
$891$	1.78337e7	0.752572
$892$	0	0
$893$	−2.72118e7	−1.14190
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.37526e6	0.0985665
$898$	0	0
$899$	−2.23727e7	−0.923249
$900$	0	0
$901$	−5.35475e6	−0.219749
$902$	0	0
$903$	−2.87487e6	−0.117327
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−3.44434e7	−1.39023	−0.695116	−	0.718897i	$-0.744647\pi$
$907$	−3.44434e7	−1.39023	−0.695116	+	0.718897i	$0.744647\pi$
$908$	0	0
$909$	−6.11647e7	−2.45522
$910$	0	0
$911$	983748.	0.0392724	0.0196362	−	0.999807i	$-0.493749\pi$
$911$	983748.	0.0392724	0.0196362	+	0.999807i	$0.493749\pi$
$912$	0	0
$913$	−4.59464e7	−1.82421
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.38138e6	−0.211334
$918$	0	0
$919$	−3.08857e7	−1.20634	−0.603168	−	0.797614i	$-0.706095\pi$
$919$	−3.08857e7	−1.20634	−0.603168	+	0.797614i	$0.706095\pi$
$920$	0	0
$921$	−3.24287e7	−1.25974
$922$	0	0
$923$	−1.94911e6	−0.0753065
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.05395e7	2.31387
$928$	0	0
$929$	−3.20874e7	−1.21982	−0.609909	−	0.792472i	$-0.708794\pi$
$929$	−3.20874e7	−1.21982	−0.609909	+	0.792472i	$0.708794\pi$
$930$	0	0
$931$	1.79553e7	0.678920
$932$	0	0
$933$	−1.73157e7	−0.651232
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.52520e7	−0.567515	−0.283757	−	0.958896i	$-0.591581\pi$
$937$	−1.52520e7	−0.567515	−0.283757	+	0.958896i	$0.591581\pi$
$938$	0	0
$939$	−1.53734e7	−0.568993
$940$	0	0
$941$	3.48166e6	0.128178	0.0640889	−	0.997944i	$-0.479586\pi$
$941$	3.48166e6	0.128178	0.0640889	+	0.997944i	$0.479586\pi$
$942$	0	0
$943$	−3.06241e6	−0.112146
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.54010e7	−0.920398	−0.460199	−	0.887816i	$-0.652222\pi$
$947$	−2.54010e7	−0.920398	−0.460199	+	0.887816i	$0.652222\pi$
$948$	0	0
$949$	2.39780e6	0.0864265
$950$	0	0
$951$	−1.34249e7	−0.481348
$952$	0	0
$953$	4.97352e7	1.77391	0.886955	−	0.461856i	$-0.152816\pi$
$953$	4.97352e7	1.77391	0.886955	+	0.461856i	$0.152816\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.12020e8	3.95383
$958$	0	0
$959$	−5.26904e6	−0.185006
$960$	0	0
$961$	−1.27250e7	−0.444477
$962$	0	0
$963$	3.74190e7	1.30025
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.05173e7	1.04949	0.524747	−	0.851258i	$-0.324160\pi$
$967$	3.05173e7	1.04949	0.524747	+	0.851258i	$0.324160\pi$
$968$	0	0
$969$	−1.08108e7	−0.369869
$970$	0	0
$971$	−3.19854e7	−1.08869	−0.544344	−	0.838862i	$-0.683221\pi$
$971$	−3.19854e7	−1.08869	−0.544344	+	0.838862i	$0.683221\pi$
$972$	0	0
$973$	678920.	0.0229899
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.90786e6	−0.0974623	−0.0487312	−	0.998812i	$-0.515518\pi$
$977$	−2.90786e6	−0.0974623	−0.0487312	+	0.998812i	$0.515518\pi$
$978$	0	0
$979$	4.43059e7	1.47742
$980$	0	0
$981$	9.45888e7	3.13810
$982$	0	0
$983$	3.49621e7	1.15402	0.577010	−	0.816737i	$-0.304219\pi$
$983$	3.49621e7	1.15402	0.577010	+	0.816737i	$0.304219\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.41501e7	0.462347
$988$	0	0
$989$	9.98164e6	0.324497
$990$	0	0
$991$	−3.00465e6	−0.0971874	−0.0485937	−	0.998819i	$-0.515474\pi$
$991$	−3.00465e6	−0.0971874	−0.0485937	+	0.998819i	$0.515474\pi$
$992$	0	0
$993$	−8.56900e7	−2.75776
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.20789e7	−1.02207	−0.511035	−	0.859560i	$-0.670738\pi$
$997$	−3.20789e7	−1.02207	−0.511035	+	0.859560i	$0.670738\pi$
$998$	0	0
$999$	−701480.	−0.0222383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.a.1.1		1
4.3	odd	2		50.6.a.g.1.1		1
5.2	odd	4		400.6.c.a.49.2		2
5.3	odd	4		400.6.c.a.49.1		2
5.4	even	2		80.6.a.h.1.1		1
12.11	even	2		450.6.a.h.1.1		1
15.14	odd	2		720.6.a.r.1.1		1
20.3	even	4		50.6.b.d.49.1		2
20.7	even	4		50.6.b.d.49.2		2
20.19	odd	2		10.6.a.a.1.1	✓	1
40.19	odd	2		320.6.a.p.1.1		1
40.29	even	2		320.6.a.a.1.1		1
60.23	odd	4		450.6.c.o.199.2		2
60.47	odd	4		450.6.c.o.199.1		2
60.59	even	2		90.6.a.f.1.1		1
140.139	even	2		490.6.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.a.1.1	✓	1	20.19	odd	2
50.6.a.g.1.1		1	4.3	odd	2
50.6.b.d.49.1		2	20.3	even	4
50.6.b.d.49.2		2	20.7	even	4
80.6.a.h.1.1		1	5.4	even	2
90.6.a.f.1.1		1	60.59	even	2
320.6.a.a.1.1		1	40.29	even	2
320.6.a.p.1.1		1	40.19	odd	2
400.6.a.a.1.1		1	1.1	even	1	trivial
400.6.c.a.49.1		2	5.3	odd	4
400.6.c.a.49.2		2	5.2	odd	4
450.6.a.h.1.1		1	12.11	even	2
450.6.c.o.199.1		2	60.47	odd	4
450.6.c.o.199.2		2	60.23	odd	4
490.6.a.j.1.1		1	140.139	even	2
720.6.a.r.1.1		1	15.14	odd	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 $\F_{q}$

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more