LMFDB - Embedded newform 50.6.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,6,Mod(1,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

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

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

chi := DirichletCharacter("50.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	50.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:	$8.01919099065$
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	$\chi$	$=$	50.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +26.0000 q^{3} +16.0000 q^{4} +104.000 q^{6} +22.0000 q^{7} +64.0000 q^{8} +433.000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} +26.0000 q^{3} +16.0000 q^{4} +104.000 q^{6} +22.0000 q^{7} +64.0000 q^{8} +433.000 q^{9} -768.000 q^{11} +416.000 q^{12} +46.0000 q^{13} +88.0000 q^{14} +256.000 q^{16} -378.000 q^{17} +1732.00 q^{18} +1100.00 q^{19} +572.000 q^{21} -3072.00 q^{22} +1986.00 q^{23} +1664.00 q^{24} +184.000 q^{26} +4940.00 q^{27} +352.000 q^{28} -5610.00 q^{29} -3988.00 q^{31} +1024.00 q^{32} -19968.0 q^{33} -1512.00 q^{34} +6928.00 q^{36} +142.000 q^{37} +4400.00 q^{38} +1196.00 q^{39} +1542.00 q^{41} +2288.00 q^{42} +5026.00 q^{43} -12288.0 q^{44} +7944.00 q^{46} -24738.0 q^{47} +6656.00 q^{48} -16323.0 q^{49} -9828.00 q^{51} +736.000 q^{52} +14166.0 q^{53} +19760.0 q^{54} +1408.00 q^{56} +28600.0 q^{57} -22440.0 q^{58} +28380.0 q^{59} +5522.00 q^{61} -15952.0 q^{62} +9526.00 q^{63} +4096.00 q^{64} -79872.0 q^{66} +24742.0 q^{67} -6048.00 q^{68} +51636.0 q^{69} +42372.0 q^{71} +27712.0 q^{72} +52126.0 q^{73} +568.000 q^{74} +17600.0 q^{76} -16896.0 q^{77} +4784.00 q^{78} -39640.0 q^{79} +23221.0 q^{81} +6168.00 q^{82} +59826.0 q^{83} +9152.00 q^{84} +20104.0 q^{86} -145860. q^{87} -49152.0 q^{88} +57690.0 q^{89} +1012.00 q^{91} +31776.0 q^{92} -103688. q^{93} -98952.0 q^{94} +26624.0 q^{96} +144382. q^{97} -65292.0 q^{98} -332544. q^{99} +O(q^{100})\)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	26.0000	1.66790	0.833950	−	0.551839i	$-0.186074\pi$
$3$	26.0000	1.66790	0.833950	+	0.551839i	$0.186074\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	104.000	1.17938
$7$	22.0000	0.169698	0.0848492	−	0.996394i	$-0.472959\pi$
$7$	22.0000	0.169698	0.0848492	+	0.996394i	$0.472959\pi$
$8$	64.0000	0.353553
$9$	433.000	1.78189
$10$	0	0
$11$	−768.000	−1.91372	−0.956862	−	0.290541i	$-0.906165\pi$
$11$	−768.000	−1.91372	−0.956862	+	0.290541i	$0.906165\pi$
$12$	416.000	0.833950
$13$	46.0000	0.0754917	0.0377459	−	0.999287i	$-0.487982\pi$
$13$	46.0000	0.0754917	0.0377459	+	0.999287i	$0.487982\pi$
$14$	88.0000	0.119995
$15$	0	0
$16$	256.000	0.250000
$17$	−378.000	−0.317227	−0.158613	−	0.987341i	$-0.550702\pi$
$17$	−378.000	−0.317227	−0.158613	+	0.987341i	$0.550702\pi$
$18$	1732.00	1.25999
$19$	1100.00	0.699051	0.349525	−	0.936927i	$-0.386343\pi$
$19$	1100.00	0.699051	0.349525	+	0.936927i	$0.386343\pi$
$20$	0	0
$21$	572.000	0.283040
$22$	−3072.00	−1.35321
$23$	1986.00	0.782816	0.391408	−	0.920217i	$-0.371988\pi$
$23$	1986.00	0.782816	0.391408	+	0.920217i	$0.371988\pi$
$24$	1664.00	0.589692
$25$	0	0
$26$	184.000	0.0533807
$27$	4940.00	1.30412
$28$	352.000	0.0848492
$29$	−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	$-0.621557\pi$
$31$	−3988.00	−0.745334	−0.372667	+	0.927965i	$0.621557\pi$
$32$	1024.00	0.176777
$33$	−19968.0	−3.19190
$34$	−1512.00	−0.224313
$35$	0	0
$36$	6928.00	0.890947
$37$	142.000	0.0170523	0.00852617	−	0.999964i	$-0.497286\pi$
$37$	142.000	0.0170523	0.00852617	+	0.999964i	$0.497286\pi$
$38$	4400.00	0.494303
$39$	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	$0.477180\pi$
$42$	2288.00	0.200139
$43$	5026.00	0.414526	0.207263	−	0.978285i	$-0.433544\pi$
$43$	5026.00	0.414526	0.207263	+	0.978285i	$0.433544\pi$
$44$	−12288.0	−0.956862
$45$	0	0
$46$	7944.00	0.553534
$47$	−24738.0	−1.63350	−0.816752	−	0.576990i	$-0.804227\pi$
$47$	−24738.0	−1.63350	−0.816752	+	0.576990i	$0.804227\pi$
$48$	6656.00	0.416975
$49$	−16323.0	−0.971202
$50$	0	0
$51$	−9828.00	−0.529102
$52$	736.000	0.0377459
$53$	14166.0	0.692720	0.346360	−	0.938102i	$-0.387418\pi$
$53$	14166.0	0.692720	0.346360	+	0.938102i	$0.387418\pi$
$54$	19760.0	0.922152
$55$	0	0
$56$	1408.00	0.0599974
$57$	28600.0	1.16595
$58$	−22440.0	−0.875897
$59$	28380.0	1.06141	0.530704	−	0.847557i	$-0.321928\pi$
$59$	28380.0	1.06141	0.530704	+	0.847557i	$0.321928\pi$
$60$	0	0
$61$	5522.00	0.190008	0.0950040	−	0.995477i	$-0.469714\pi$
$61$	5522.00	0.190008	0.0950040	+	0.995477i	$0.469714\pi$
$62$	−15952.0	−0.527031
$63$	9526.00	0.302384
$64$	4096.00	0.125000
$65$	0	0
$66$	−79872.0	−2.25702
$67$	24742.0	0.673361	0.336680	−	0.941619i	$-0.390696\pi$
$67$	24742.0	0.673361	0.336680	+	0.941619i	$0.390696\pi$
$68$	−6048.00	−0.158613
$69$	51636.0	1.30566
$70$	0	0
$71$	42372.0	0.997546	0.498773	−	0.866733i	$-0.333784\pi$
$71$	42372.0	0.997546	0.498773	+	0.866733i	$0.333784\pi$
$72$	27712.0	0.629994
$73$	52126.0	1.14485	0.572423	−	0.819958i	$-0.306003\pi$
$73$	52126.0	1.14485	0.572423	+	0.819958i	$0.306003\pi$
$74$	568.000	0.0120578
$75$	0	0
$76$	17600.0	0.349525
$77$	−16896.0	−0.324756
$78$	4784.00	0.0890337
$79$	−39640.0	−0.714605	−0.357302	−	0.933989i	$-0.616303\pi$
$79$	−39640.0	−0.714605	−0.357302	+	0.933989i	$0.616303\pi$
$80$	0	0
$81$	23221.0	0.393250
$82$	6168.00	0.101300
$83$	59826.0	0.953223	0.476612	−	0.879114i	$-0.341865\pi$
$83$	59826.0	0.953223	0.476612	+	0.879114i	$0.341865\pi$
$84$	9152.00	0.141520
$85$	0	0
$86$	20104.0	0.293114
$87$	−145860.	−2.06604
$88$	−49152.0	−0.676604
$89$	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
$92$	31776.0	0.391408
$93$	−103688.	−1.24314
$94$	−98952.0	−1.15506
$95$	0	0
$96$	26624.0	0.294846
$97$	144382.	1.55806	0.779029	−	0.626988i	$-0.215712\pi$
$97$	144382.	1.55806	0.779029	+	0.626988i	$0.215712\pi$
$98$	−65292.0	−0.686744
$99$	−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	$0.741922\pi$
$102$	−39312.0	−0.374132
$103$	−139814.	−1.29855	−0.649273	−	0.760555i	$-0.724927\pi$
$103$	−139814.	−1.29855	−0.649273	+	0.760555i	$0.724927\pi$
$104$	2944.00	0.0266904
$105$	0	0
$106$	56664.0	0.489827
$107$	−86418.0	−0.729701	−0.364850	−	0.931066i	$-0.618880\pi$
$107$	−86418.0	−0.729701	−0.364850	+	0.931066i	$0.618880\pi$
$108$	79040.0	0.652060
$109$	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
$112$	5632.00	0.0424246
$113$	28806.0	0.212220	0.106110	−	0.994354i	$-0.466160\pi$
$113$	28806.0	0.212220	0.106110	+	0.994354i	$0.466160\pi$
$114$	114400.	0.824449
$115$	0	0
$116$	−89760.0	−0.619352
$117$	19918.0	0.134518
$118$	113520.	0.750529
$119$	−8316.00	−0.0538328
$120$	0	0
$121$	428773.	2.66234
$122$	22088.0	0.134356
$123$	40092.0	0.238943
$124$	−63808.0	−0.372667
$125$	0	0
$126$	38104.0	0.213818
$127$	216502.	1.19111	0.595556	−	0.803314i	$-0.296932\pi$
$127$	216502.	1.19111	0.595556	+	0.803314i	$0.296932\pi$
$128$	16384.0	0.0883883
$129$	130676.	0.691388
$130$	0	0
$131$	−244608.	−1.24535	−0.622676	−	0.782479i	$-0.713955\pi$
$131$	−244608.	−1.24535	−0.622676	+	0.782479i	$0.713955\pi$
$132$	−319488.	−1.59595
$133$	24200.0	0.118628
$134$	98968.0	0.476138
$135$	0	0
$136$	−24192.0	−0.112157
$137$	239502.	1.09020	0.545102	−	0.838370i	$-0.316491\pi$
$137$	239502.	1.09020	0.545102	+	0.838370i	$0.316491\pi$
$138$	206544.	0.923241
$139$	30860.0	0.135475	0.0677375	−	0.997703i	$-0.478422\pi$
$139$	30860.0	0.135475	0.0677375	+	0.997703i	$0.478422\pi$
$140$	0	0
$141$	−643188.	−2.72452
$142$	169488.	0.705372
$143$	−35328.0	−0.144470
$144$	110848.	0.445473
$145$	0	0
$146$	208504.	0.809529
$147$	−424398.	−1.61987
$148$	2272.00	0.00852617
$149$	−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	$-0.492926\pi$
$151$	12452.0	0.0444423	0.0222212	+	0.999753i	$0.492926\pi$
$152$	70400.0	0.247152
$153$	−163674.	−0.565264
$154$	−67584.0	−0.229637
$155$	0	0
$156$	19136.0	0.0629564
$157$	6022.00	0.0194981	0.00974903	−	0.999952i	$-0.496897\pi$
$157$	6022.00	0.0194981	0.00974903	+	0.999952i	$0.496897\pi$
$158$	−158560.	−0.505302
$159$	368316.	1.15539
$160$	0	0
$161$	43692.0	0.132843
$162$	92884.0	0.278070
$163$	500866.	1.47656	0.738282	−	0.674492i	$-0.235637\pi$
$163$	500866.	1.47656	0.738282	+	0.674492i	$0.235637\pi$
$164$	24672.0	0.0716300
$165$	0	0
$166$	239304.	0.674031
$167$	−555258.	−1.54065	−0.770324	−	0.637652i	$-0.779906\pi$
$167$	−555258.	−1.54065	−0.770324	+	0.637652i	$0.779906\pi$
$168$	36608.0	0.100070
$169$	−369177.	−0.994301
$170$	0	0
$171$	476300.	1.24563
$172$	80416.0	0.207263
$173$	−417354.	−1.06020	−0.530102	−	0.847934i	$-0.677846\pi$
$173$	−417354.	−1.06020	−0.530102	+	0.847934i	$0.677846\pi$
$174$	−583440.	−1.46091
$175$	0	0
$176$	−196608.	−0.478431
$177$	737880.	1.77032
$178$	230760.	0.545897
$179$	−52380.0	−0.122189	−0.0610946	−	0.998132i	$-0.519459\pi$
$179$	−52380.0	−0.122189	−0.0610946	+	0.998132i	$0.519459\pi$
$180$	0	0
$181$	546662.	1.24029	0.620144	−	0.784488i	$-0.287074\pi$
$181$	546662.	1.24029	0.620144	+	0.784488i	$0.287074\pi$
$182$	4048.00	0.00905862
$183$	143572.	0.316914
$184$	127104.	0.276767
$185$	0	0
$186$	−414752.	−0.879035
$187$	290304.	0.607084
$188$	−395808.	−0.816752
$189$	108680.	0.221307
$190$	0	0
$191$	−452028.	−0.896565	−0.448283	−	0.893892i	$-0.647964\pi$
$191$	−452028.	−0.896565	−0.448283	+	0.893892i	$0.647964\pi$
$192$	106496.	0.208488
$193$	−485594.	−0.938383	−0.469191	−	0.883097i	$-0.655455\pi$
$193$	−485594.	−0.938383	−0.469191	+	0.883097i	$0.655455\pi$
$194$	577528.	1.10171
$195$	0	0
$196$	−261168.	−0.485601
$197$	−1.01018e6	−1.85452	−0.927262	−	0.374414i	$-0.877844\pi$
$197$	−1.01018e6	−1.85452	−0.927262	+	0.374414i	$0.877844\pi$
$198$	−1.33018e6	−2.41127
$199$	−807640.	−1.44572	−0.722862	−	0.690993i	$-0.757174\pi$
$199$	−807640.	−1.44572	−0.722862	+	0.690993i	$0.757174\pi$
$200$	0	0
$201$	643292.	1.12310
$202$	−565032.	−0.974304
$203$	−123420.	−0.210206
$204$	−157248.	−0.264551
$205$	0	0
$206$	−559256.	−0.918211
$207$	859938.	1.39489
$208$	11776.0	0.0188729
$209$	−844800.	−1.33779
$210$	0	0
$211$	149552.	0.231252	0.115626	−	0.993293i	$-0.463113\pi$
$211$	149552.	0.231252	0.115626	+	0.993293i	$0.463113\pi$
$212$	226656.	0.346360
$213$	1.10167e6	1.66381
$214$	−345672.	−0.515976
$215$	0	0
$216$	316160.	0.461076
$217$	−87736.0	−0.126482
$218$	873800.	1.24529
$219$	1.35528e6	1.90949
$220$	0	0
$221$	−17388.0	−0.0239480
$222$	14768.0	0.0201113
$223$	443506.	0.597224	0.298612	−	0.954375i	$-0.403476\pi$
$223$	443506.	0.597224	0.298612	+	0.954375i	$0.403476\pi$
$224$	22528.0	0.0299987
$225$	0	0
$226$	115224.	0.150062
$227$	−420018.	−0.541007	−0.270504	−	0.962719i	$-0.587190\pi$
$227$	−420018.	−0.541007	−0.270504	+	0.962719i	$0.587190\pi$
$228$	457600.	0.582974
$229$	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
$232$	−359040.	−0.437948
$233$	1.27345e6	1.53671	0.768353	−	0.640026i	$-0.221077\pi$
$233$	1.27345e6	1.53671	0.768353	+	0.640026i	$0.221077\pi$
$234$	79672.0	0.0951187
$235$	0	0
$236$	454080.	0.530704
$237$	−1.03064e6	−1.19189
$238$	−33264.0	−0.0380655
$239$	−370680.	−0.419763	−0.209882	−	0.977727i	$-0.567308\pi$
$239$	−370680.	−0.419763	−0.209882	+	0.977727i	$0.567308\pi$
$240$	0	0
$241$	−561298.	−0.622517	−0.311258	−	0.950325i	$-0.600750\pi$
$241$	−561298.	−0.622517	−0.311258	+	0.950325i	$0.600750\pi$
$242$	1.71509e6	1.88256
$243$	−596674.	−0.648219
$244$	88352.0	0.0950040
$245$	0	0
$246$	160368.	0.168958
$247$	50600.0	0.0527726
$248$	−255232.	−0.263515
$249$	1.55548e6	1.58988
$250$	0	0
$251$	577152.	0.578237	0.289119	−	0.957293i	$-0.406638\pi$
$251$	577152.	0.578237	0.289119	+	0.957293i	$0.406638\pi$
$252$	152416.	0.151192
$253$	−1.52525e6	−1.49809
$254$	866008.	0.842243
$255$	0	0
$256$	65536.0	0.0625000
$257$	651462.	0.615257	0.307628	−	0.951507i	$-0.400465\pi$
$257$	651462.	0.615257	0.307628	+	0.951507i	$0.400465\pi$
$258$	522704.	0.488885
$259$	3124.00	0.00289375
$260$	0	0
$261$	−2.42913e6	−2.20724
$262$	−978432.	−0.880597
$263$	−917574.	−0.817997	−0.408999	−	0.912535i	$-0.634122\pi$
$263$	−917574.	−0.817997	−0.408999	+	0.912535i	$0.634122\pi$
$264$	−1.27795e6	−1.12851
$265$	0	0
$266$	96800.0	0.0838825
$267$	1.49994e6	1.28764
$268$	395872.	0.336680
$269$	−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	$-0.653492\pi$
$271$	−1.12131e6	−0.927474	−0.463737	+	0.885973i	$0.653492\pi$
$272$	−96768.0	−0.0793066
$273$	26312.0	0.0213672
$274$	958008.	0.770891
$275$	0	0
$276$	826176.	0.652830
$277$	1.66034e6	1.30016	0.650082	−	0.759864i	$-0.274735\pi$
$277$	1.66034e6	1.30016	0.650082	+	0.759864i	$0.274735\pi$
$278$	123440.	0.0957952
$279$	−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	$0.315187\pi$
$282$	−2.57275e6	−1.92653
$283$	−309014.	−0.229357	−0.114679	−	0.993403i	$-0.536584\pi$
$283$	−309014.	−0.229357	−0.114679	+	0.993403i	$0.536584\pi$
$284$	677952.	0.498773
$285$	0	0
$286$	−141312.	−0.102156
$287$	33924.0	0.0243110
$288$	443392.	0.314997
$289$	−1.27697e6	−0.899367
$290$	0	0
$291$	3.75393e6	2.59869
$292$	834016.	0.572423
$293$	1.59301e6	1.08405	0.542024	−	0.840363i	$-0.317658\pi$
$293$	1.59301e6	1.08405	0.542024	+	0.840363i	$0.317658\pi$
$294$	−1.69759e6	−1.14542
$295$	0	0
$296$	9088.00	0.00602891
$297$	−3.79392e6	−2.49573
$298$	−403800.	−0.263406
$299$	91356.0	0.0590961
$300$	0	0
$301$	110572.	0.0703443
$302$	49808.0	0.0314255
$303$	−3.67271e6	−2.29816
$304$	281600.	0.174763
$305$	0	0
$306$	−654696.	−0.399702
$307$	−1.24726e6	−0.755284	−0.377642	−	0.925952i	$-0.623265\pi$
$307$	−1.24726e6	−0.755284	−0.377642	+	0.925952i	$0.623265\pi$
$308$	−270336.	−0.162378
$309$	−3.63516e6	−2.16585
$310$	0	0
$311$	−665988.	−0.390450	−0.195225	−	0.980758i	$-0.562544\pi$
$311$	−665988.	−0.390450	−0.195225	+	0.980758i	$0.562544\pi$
$312$	76544.0	0.0445169
$313$	591286.	0.341143	0.170572	−	0.985345i	$-0.445439\pi$
$313$	591286.	0.341143	0.170572	+	0.985345i	$0.445439\pi$
$314$	24088.0	0.0137872
$315$	0	0
$316$	−634240.	−0.357302
$317$	516342.	0.288595	0.144298	−	0.989534i	$-0.453908\pi$
$317$	516342.	0.288595	0.144298	+	0.989534i	$0.453908\pi$
$318$	1.47326e6	0.816983
$319$	4.30848e6	2.37054
$320$	0	0
$321$	−2.24687e6	−1.21707
$322$	174768.	0.0939339
$323$	−415800.	−0.221757
$324$	371536.	0.196625
$325$	0	0
$326$	2.00346e6	1.04409
$327$	5.67970e6	2.93735
$328$	98688.0	0.0506500
$329$	−544236.	−0.277203
$330$	0	0
$331$	−3.29577e6	−1.65343	−0.826717	−	0.562619i	$-0.809794\pi$
$331$	−3.29577e6	−1.65343	−0.826717	+	0.562619i	$0.809794\pi$
$332$	957216.	0.476612
$333$	61486.0	0.0303854
$334$	−2.22103e6	−1.08940
$335$	0	0
$336$	146432.	0.0707600
$337$	−1.91098e6	−0.916602	−0.458301	−	0.888797i	$-0.651542\pi$
$337$	−1.91098e6	−0.916602	−0.458301	+	0.888797i	$0.651542\pi$
$338$	−1.47671e6	−0.703077
$339$	748956.	0.353962
$340$	0	0
$341$	3.06278e6	1.42636
$342$	1.90520e6	0.880796
$343$	−728860.	−0.334510
$344$	321664.	0.146557
$345$	0	0
$346$	−1.66942e6	−0.749677
$347$	−2.42006e6	−1.07895	−0.539476	−	0.842001i	$-0.681378\pi$
$347$	−2.42006e6	−1.07895	−0.539476	+	0.842001i	$0.681378\pi$
$348$	−2.33376e6	−1.03302
$349$	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
$352$	−786432.	−0.338302
$353$	413166.	0.176477	0.0882384	−	0.996099i	$-0.471876\pi$
$353$	413166.	0.176477	0.0882384	+	0.996099i	$0.471876\pi$
$354$	2.95152e6	1.25181
$355$	0	0
$356$	923040.	0.386007
$357$	−216216.	−0.0897878
$358$	−209520.	−0.0864008
$359$	1.73772e6	0.711613	0.355806	−	0.934560i	$-0.384206\pi$
$359$	1.73772e6	0.711613	0.355806	+	0.934560i	$0.384206\pi$
$360$	0	0
$361$	−1.26610e6	−0.511328
$362$	2.18665e6	0.877016
$363$	1.11481e7	4.44052
$364$	16192.0	0.00640541
$365$	0	0
$366$	574288.	0.224092
$367$	−1.16098e6	−0.449944	−0.224972	−	0.974365i	$-0.572229\pi$
$367$	−1.16098e6	−0.449944	−0.224972	+	0.974365i	$0.572229\pi$
$368$	508416.	0.195704
$369$	667686.	0.255274
$370$	0	0
$371$	311652.	0.117553
$372$	−1.65901e6	−0.621572
$373$	−343754.	−0.127931	−0.0639655	−	0.997952i	$-0.520375\pi$
$373$	−343754.	−0.127931	−0.0639655	+	0.997952i	$0.520375\pi$
$374$	1.16122e6	0.429273
$375$	0	0
$376$	−1.58323e6	−0.577531
$377$	−258060.	−0.0935120
$378$	434720.	0.156488
$379$	573140.	0.204957	0.102478	−	0.994735i	$-0.467323\pi$
$379$	573140.	0.204957	0.102478	+	0.994735i	$0.467323\pi$
$380$	0	0
$381$	5.62905e6	1.98666
$382$	−1.80811e6	−0.633967
$383$	2.88055e6	1.00341	0.501704	−	0.865039i	$-0.332707\pi$
$383$	2.88055e6	1.00341	0.501704	+	0.865039i	$0.332707\pi$
$384$	425984.	0.147423
$385$	0	0
$386$	−1.94238e6	−0.663537
$387$	2.17626e6	0.738640
$388$	2.31011e6	0.779029
$389$	−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
$392$	−1.04467e6	−0.343372
$393$	−6.35981e6	−2.07712
$394$	−4.04071e6	−1.31135
$395$	0	0
$396$	−5.32070e6	−1.70503
$397$	−885458.	−0.281963	−0.140981	−	0.990012i	$-0.545026\pi$
$397$	−885458.	−0.281963	−0.140981	+	0.990012i	$0.545026\pi$
$398$	−3.23056e6	−1.02228
$399$	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	$0.698053\pi$
$402$	2.57317e6	0.794151
$403$	−183448.	−0.0562666
$404$	−2.26013e6	−0.688937
$405$	0	0
$406$	−493680.	−0.148638
$407$	−109056.	−0.0326335
$408$	−628992.	−0.187066
$409$	−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
$412$	−2.23702e6	−0.649273
$413$	624360.	0.180119
$414$	3.43975e6	0.986339
$415$	0	0
$416$	47104.0	0.0133452
$417$	802360.	0.225959
$418$	−3.37920e6	−0.945961
$419$	−2.99166e6	−0.832486	−0.416243	−	0.909253i	$-0.636654\pi$
$419$	−2.99166e6	−0.832486	−0.416243	+	0.909253i	$0.636654\pi$
$420$	0	0
$421$	3.96660e6	1.09072	0.545360	−	0.838202i	$-0.316393\pi$
$421$	3.96660e6	1.09072	0.545360	+	0.838202i	$0.316393\pi$
$422$	598208.	0.163520
$423$	−1.07116e7	−2.91073
$424$	906624.	0.244913
$425$	0	0
$426$	4.40669e6	1.17649
$427$	121484.	0.0322440
$428$	−1.38269e6	−0.364850
$429$	−918528.	−0.240962
$430$	0	0
$431$	−5.17115e6	−1.34089	−0.670446	−	0.741958i	$-0.733897\pi$
$431$	−5.17115e6	−1.34089	−0.670446	+	0.741958i	$0.733897\pi$
$432$	1.26464e6	0.326030
$433$	4.53485e6	1.16237	0.581183	−	0.813773i	$-0.302590\pi$
$433$	4.53485e6	1.16237	0.581183	+	0.813773i	$0.302590\pi$
$434$	−350944.	−0.0894362
$435$	0	0
$436$	3.49520e6	0.880554
$437$	2.18460e6	0.547228
$438$	5.42110e6	1.35021
$439$	−1.08220e6	−0.268007	−0.134004	−	0.990981i	$-0.542783\pi$
$439$	−1.08220e6	−0.268007	−0.134004	+	0.990981i	$0.542783\pi$
$440$	0	0
$441$	−7.06786e6	−1.73058
$442$	−69552.0	−0.0169338
$443$	1.08079e6	0.261656	0.130828	−	0.991405i	$-0.458236\pi$
$443$	1.08079e6	0.261656	0.130828	+	0.991405i	$0.458236\pi$
$444$	59072.0	0.0142208
$445$	0	0
$446$	1.77402e6	0.422301
$447$	−2.62470e6	−0.621314
$448$	90112.0	0.0212123
$449$	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
$452$	460896.	0.106110
$453$	323752.	0.0741254
$454$	−1.68007e6	−0.382550
$455$	0	0
$456$	1.83040e6	0.412225
$457$	−1.59046e6	−0.356231	−0.178115	−	0.984010i	$-0.557000\pi$
$457$	−1.59046e6	−0.356231	−0.178115	+	0.984010i	$0.557000\pi$
$458$	4.23500e6	0.943387
$459$	−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	$0.345617\pi$
$462$	−1.75718e6	−0.383012
$463$	−3.26605e6	−0.708061	−0.354031	−	0.935234i	$-0.615189\pi$
$463$	−3.26605e6	−0.708061	−0.354031	+	0.935234i	$0.615189\pi$
$464$	−1.43616e6	−0.309676
$465$	0	0
$466$	5.09378e6	1.08662
$467$	601542.	0.127636	0.0638181	−	0.997962i	$-0.479672\pi$
$467$	601542.	0.127636	0.0638181	+	0.997962i	$0.479672\pi$
$468$	318688.	0.0672591
$469$	544324.	0.114268
$470$	0	0
$471$	156572.	0.0325208
$472$	1.81632e6	0.375264
$473$	−3.85997e6	−0.793288
$474$	−4.12256e6	−0.842793
$475$	0	0
$476$	−133056.	−0.0269164
$477$	6.13388e6	1.23435
$478$	−1.48272e6	−0.296817
$479$	−4.57932e6	−0.911931	−0.455966	−	0.889997i	$-0.650706\pi$
$479$	−4.57932e6	−0.911931	−0.455966	+	0.889997i	$0.650706\pi$
$480$	0	0
$481$	6532.00	0.00128731
$482$	−2.24519e6	−0.440186
$483$	1.13599e6	0.221568
$484$	6.86037e6	1.33117
$485$	0	0
$486$	−2.38670e6	−0.458360
$487$	−7.05226e6	−1.34743	−0.673714	−	0.738992i	$-0.735302\pi$
$487$	−7.05226e6	−1.34743	−0.673714	+	0.738992i	$0.735302\pi$
$488$	353408.	0.0671780
$489$	1.30225e7	2.46276
$490$	0	0
$491$	−2.62349e6	−0.491106	−0.245553	−	0.969383i	$-0.578970\pi$
$491$	−2.62349e6	−0.491106	−0.245553	+	0.969383i	$0.578970\pi$
$492$	641472.	0.119472
$493$	2.12058e6	0.392950
$494$	202400.	0.0373158
$495$	0	0
$496$	−1.02093e6	−0.186333
$497$	932184.	0.169282
$498$	6.22190e6	1.12422
$499$	−3.61234e6	−0.649437	−0.324719	−	0.945811i	$-0.605270\pi$
$499$	−3.61234e6	−0.649437	−0.324719	+	0.945811i	$0.605270\pi$
$500$	0	0
$501$	−1.44367e7	−2.56965
$502$	2.30861e6	0.408875
$503$	−9.15629e6	−1.61361	−0.806807	−	0.590815i	$-0.798806\pi$
$503$	−9.15629e6	−1.61361	−0.806807	+	0.590815i	$0.798806\pi$
$504$	609664.	0.106909
$505$	0	0
$506$	−6.10099e6	−1.05931
$507$	−9.59860e6	−1.65840
$508$	3.46403e6	0.595556
$509$	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
$512$	262144.	0.0441942
$513$	5.43400e6	0.911646
$514$	2.60585e6	0.435052
$515$	0	0
$516$	2.09082e6	0.345694
$517$	1.89988e7	3.12608
$518$	12496.0	0.00204619
$519$	−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	$0.344656\pi$
$522$	−9.71652e6	−1.56075
$523$	8.17067e6	1.30618	0.653090	−	0.757280i	$-0.273472\pi$
$523$	8.17067e6	1.30618	0.653090	+	0.757280i	$0.273472\pi$
$524$	−3.91373e6	−0.622676
$525$	0	0
$526$	−3.67030e6	−0.578411
$527$	1.50746e6	0.236440
$528$	−5.11181e6	−0.797976
$529$	−2.49215e6	−0.387199
$530$	0	0
$531$	1.22885e7	1.89132
$532$	387200.	0.0593139
$533$	70932.0	0.0108149
$534$	5.99976e6	0.910502
$535$	0	0
$536$	1.58349e6	0.238069
$537$	−1.36188e6	−0.203800
$538$	−2.94156e6	−0.438149
$539$	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	$0.519121\pi$
$542$	−4.48523e6	−0.655823
$543$	1.42132e7	2.06868
$544$	−387072.	−0.0560783
$545$	0	0
$546$	105248.	0.0151089
$547$	3.50750e6	0.501221	0.250611	−	0.968088i	$-0.419369\pi$
$547$	3.50750e6	0.501221	0.250611	+	0.968088i	$0.419369\pi$
$548$	3.83203e6	0.545102
$549$	2.39103e6	0.338574
$550$	0	0
$551$	−6.17100e6	−0.865918
$552$	3.30470e6	0.461620
$553$	−872080.	−0.121267
$554$	6.64137e6	0.919355
$555$	0	0
$556$	493760.	0.0677375
$557$	−9.61490e6	−1.31313	−0.656563	−	0.754271i	$-0.727991\pi$
$557$	−9.61490e6	−1.31313	−0.656563	+	0.754271i	$0.727991\pi$
$558$	−6.90722e6	−0.939112
$559$	231196.	0.0312933
$560$	0	0
$561$	7.54790e6	1.01256
$562$	5.80841e6	0.775740
$563$	−2.01941e6	−0.268506	−0.134253	−	0.990947i	$-0.542864\pi$
$563$	−2.01941e6	−0.268506	−0.134253	+	0.990947i	$0.542864\pi$
$564$	−1.02910e7	−1.36226
$565$	0	0
$566$	−1.23606e6	−0.162180
$567$	510862.	0.0667338
$568$	2.71181e6	0.352686
$569$	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	$-0.315290\pi$
$571$	8.54295e6	1.09652	0.548261	+	0.836307i	$0.315290\pi$
$572$	−565248.	−0.0722352
$573$	−1.17527e7	−1.49538
$574$	135696.	0.0171905
$575$	0	0
$576$	1.77357e6	0.222737
$577$	2.31458e6	0.289423	0.144711	−	0.989474i	$-0.453775\pi$
$577$	2.31458e6	0.289423	0.144711	+	0.989474i	$0.453775\pi$
$578$	−5.10789e6	−0.635949
$579$	−1.26254e7	−1.56513
$580$	0	0
$581$	1.31617e6	0.161760
$582$	1.50157e7	1.83755
$583$	−1.08795e7	−1.32568
$584$	3.33606e6	0.404764
$585$	0	0
$586$	6.37202e6	0.766537
$587$	−928338.	−0.111202	−0.0556008	−	0.998453i	$-0.517707\pi$
$587$	−928338.	−0.111202	−0.0556008	+	0.998453i	$0.517707\pi$
$588$	−6.79037e6	−0.809935
$589$	−4.38680e6	−0.521026
$590$	0	0
$591$	−2.62646e7	−3.09316
$592$	36352.0	0.00426309
$593$	909486.	0.106209	0.0531043	−	0.998589i	$-0.483088\pi$
$593$	909486.	0.106209	0.0531043	+	0.998589i	$0.483088\pi$
$594$	−1.51757e7	−1.76475
$595$	0	0
$596$	−1.61520e6	−0.186256
$597$	−2.09986e7	−2.41132
$598$	365424.	0.0417873
$599$	−8.51136e6	−0.969241	−0.484621	−	0.874724i	$-0.661042\pi$
$599$	−8.51136e6	−0.969241	−0.484621	+	0.874724i	$0.661042\pi$
$600$	0	0
$601$	6.12498e6	0.691701	0.345851	−	0.938290i	$-0.387590\pi$
$601$	6.12498e6	0.691701	0.345851	+	0.938290i	$0.387590\pi$
$602$	442288.	0.0497409
$603$	1.07133e7	1.19986
$604$	199232.	0.0222212
$605$	0	0
$606$	−1.46908e7	−1.62504
$607$	4.51646e6	0.497538	0.248769	−	0.968563i	$-0.419974\pi$
$607$	4.51646e6	0.497538	0.248769	+	0.968563i	$0.419974\pi$
$608$	1.12640e6	0.123576
$609$	−3.20892e6	−0.350603
$610$	0	0
$611$	−1.13795e6	−0.123316
$612$	−2.61878e6	−0.282632
$613$	−9.63979e6	−1.03614	−0.518068	−	0.855340i	$-0.673349\pi$
$613$	−9.63979e6	−1.03614	−0.518068	+	0.855340i	$0.673349\pi$
$614$	−4.98903e6	−0.534067
$615$	0	0
$616$	−1.08134e6	−0.114819
$617$	9.92650e6	1.04974	0.524872	−	0.851181i	$-0.324113\pi$
$617$	9.92650e6	1.04974	0.524872	+	0.851181i	$0.324113\pi$
$618$	−1.45407e7	−1.53149
$619$	7.63322e6	0.800721	0.400360	−	0.916358i	$-0.368885\pi$
$619$	7.63322e6	0.800721	0.400360	+	0.916358i	$0.368885\pi$
$620$	0	0
$621$	9.81084e6	1.02089
$622$	−2.66395e6	−0.276090
$623$	1.26918e6	0.131010
$624$	306176.	0.0314782
$625$	0	0
$626$	2.36514e6	0.241225
$627$	−2.19648e7	−2.23130
$628$	96352.0	0.00974903
$629$	−53676.0	−0.00540946
$630$	0	0
$631$	1.80314e7	1.80284	0.901418	−	0.432949i	$-0.142527\pi$
$631$	1.80314e7	1.80284	0.901418	+	0.432949i	$0.142527\pi$
$632$	−2.53696e6	−0.252651
$633$	3.88835e6	0.385706
$634$	2.06537e6	0.204068
$635$	0	0
$636$	5.89306e6	0.577694
$637$	−750858.	−0.0733178
$638$	1.72339e7	1.67623
$639$	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	$0.352460\pi$
$642$	−8.98747e6	−0.860597
$643$	1.38332e7	1.31946	0.659730	−	0.751503i	$-0.270671\pi$
$643$	1.38332e7	1.31946	0.659730	+	0.751503i	$0.270671\pi$
$644$	699072.	0.0664213
$645$	0	0
$646$	−1.66320e6	−0.156806
$647$	1.48997e7	1.39932	0.699658	−	0.714478i	$-0.253336\pi$
$647$	1.48997e7	1.39932	0.699658	+	0.714478i	$0.253336\pi$
$648$	1.48614e6	0.139035
$649$	−2.17958e7	−2.03124
$650$	0	0
$651$	−2.28114e6	−0.210959
$652$	8.01386e6	0.738282
$653$	1.93306e7	1.77403	0.887016	−	0.461738i	$-0.152774\pi$
$653$	1.93306e7	1.77403	0.887016	+	0.461738i	$0.152774\pi$
$654$	2.27188e7	2.07702
$655$	0	0
$656$	394752.	0.0358150
$657$	2.25706e7	2.03999
$658$	−2.17694e6	−0.196012
$659$	−4.06110e6	−0.364276	−0.182138	−	0.983273i	$-0.558302\pi$
$659$	−4.06110e6	−0.364276	−0.182138	+	0.983273i	$0.558302\pi$
$660$	0	0
$661$	−1.35152e7	−1.20315	−0.601575	−	0.798816i	$-0.705460\pi$
$661$	−1.35152e7	−1.20315	−0.601575	+	0.798816i	$0.705460\pi$
$662$	−1.31831e7	−1.16915
$663$	−452088.	−0.0399429
$664$	3.82886e6	0.337015
$665$	0	0
$666$	245944.	0.0214858
$667$	−1.11415e7	−0.969678
$668$	−8.88413e6	−0.770324
$669$	1.15312e7	0.996111
$670$	0	0
$671$	−4.24090e6	−0.363623
$672$	585728.	0.0500349
$673$	−1.43520e7	−1.22144	−0.610722	−	0.791845i	$-0.709121\pi$
$673$	−1.43520e7	−1.22144	−0.610722	+	0.791845i	$0.709121\pi$
$674$	−7.64391e6	−0.648136
$675$	0	0
$676$	−5.90683e6	−0.497150
$677$	−1.89530e6	−0.158930	−0.0794650	−	0.996838i	$-0.525321\pi$
$677$	−1.89530e6	−0.158930	−0.0794650	+	0.996838i	$0.525321\pi$
$678$	2.99582e6	0.250289
$679$	3.17640e6	0.264400
$680$	0	0
$681$	−1.09205e7	−0.902347
$682$	1.22511e7	1.00859
$683$	−2.91641e6	−0.239220	−0.119610	−	0.992821i	$-0.538164\pi$
$683$	−2.91641e6	−0.239220	−0.119610	+	0.992821i	$0.538164\pi$
$684$	7.62080e6	0.622817
$685$	0	0
$686$	−2.91544e6	−0.236534
$687$	2.75275e7	2.22523
$688$	1.28666e6	0.103631
$689$	651636.	0.0522946
$690$	0	0
$691$	1.44278e7	1.14949	0.574743	−	0.818334i	$-0.305102\pi$
$691$	1.44278e7	1.14949	0.574743	+	0.818334i	$0.305102\pi$
$692$	−6.67766e6	−0.530102
$693$	−7.31597e6	−0.578680
$694$	−9.68023e6	−0.762934
$695$	0	0
$696$	−9.33504e6	−0.730454
$697$	−582876.	−0.0454458
$698$	1.00291e7	0.779153
$699$	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	$0.708754\pi$
$702$	908960.	0.0696149
$703$	156200.	0.0119205
$704$	−3.14573e6	−0.239216
$705$	0	0
$706$	1.65266e6	0.124788
$707$	−3.10768e6	−0.233823
$708$	1.18061e7	0.885162
$709$	−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
$712$	3.69216e6	0.272948
$713$	−7.92017e6	−0.583459
$714$	−864864.	−0.0634896
$715$	0	0
$716$	−838080.	−0.0610946
$717$	−9.63768e6	−0.700123
$718$	6.95088e6	0.503186
$719$	2.12677e7	1.53426	0.767130	−	0.641492i	$-0.221684\pi$
$719$	2.12677e7	1.53426	0.767130	+	0.641492i	$0.221684\pi$
$720$	0	0
$721$	−3.07591e6	−0.220361
$722$	−5.06440e6	−0.361564
$723$	−1.45937e7	−1.03830
$724$	8.74659e6	0.620144
$725$	0	0
$726$	4.45924e7	3.13992
$727$	−1.55009e7	−1.08773	−0.543863	−	0.839174i	$-0.683039\pi$
$727$	−1.55009e7	−1.08773	−0.543863	+	0.839174i	$0.683039\pi$
$728$	64768.0	0.00452931
$729$	−2.11562e7	−1.47441
$730$	0	0
$731$	−1.89983e6	−0.131499
$732$	2.29715e6	0.158457
$733$	1.21850e7	0.837653	0.418827	−	0.908066i	$-0.362441\pi$
$733$	1.21850e7	0.837653	0.418827	+	0.908066i	$0.362441\pi$
$734$	−4.64391e6	−0.318159
$735$	0	0
$736$	2.03366e6	0.138384
$737$	−1.90019e7	−1.28863
$738$	2.67074e6	0.180506
$739$	−2.90282e7	−1.95528	−0.977641	−	0.210282i	$-0.932562\pi$
$739$	−2.90282e7	−1.95528	−0.977641	+	0.210282i	$0.932562\pi$
$740$	0	0
$741$	1.31560e6	0.0880194
$742$	1.24661e6	0.0831228
$743$	−1.61145e7	−1.07089	−0.535445	−	0.844570i	$-0.679856\pi$
$743$	−1.61145e7	−1.07089	−0.535445	+	0.844570i	$0.679856\pi$
$744$	−6.63603e6	−0.439517
$745$	0	0
$746$	−1.37502e6	−0.0904609
$747$	2.59047e7	1.69854
$748$	4.64486e6	0.303542
$749$	−1.90120e6	−0.123829
$750$	0	0
$751$	−2.92431e6	−0.189201	−0.0946005	−	0.995515i	$-0.530157\pi$
$751$	−2.92431e6	−0.189201	−0.0946005	+	0.995515i	$0.530157\pi$
$752$	−6.33293e6	−0.408376
$753$	1.50060e7	0.964442
$754$	−1.03224e6	−0.0661230
$755$	0	0
$756$	1.73888e6	0.110653
$757$	−2.60325e7	−1.65111	−0.825557	−	0.564319i	$-0.809139\pi$
$757$	−2.60325e7	−1.65111	−0.825557	+	0.564319i	$0.809139\pi$
$758$	2.29256e6	0.144926
$759$	−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	$0.329286\pi$
$762$	2.25162e7	1.40478
$763$	4.80590e6	0.298857
$764$	−7.23245e6	−0.448283
$765$	0	0
$766$	1.15222e7	0.709517
$767$	1.30548e6	0.0801275
$768$	1.70394e6	0.104244
$769$	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
$772$	−7.76950e6	−0.469191
$773$	1.90592e7	1.14725	0.573624	−	0.819119i	$-0.305537\pi$
$773$	1.90592e7	1.14725	0.573624	+	0.819119i	$0.305537\pi$
$774$	8.70503e6	0.522298
$775$	0	0
$776$	9.24045e6	0.550857
$777$	81224.0	0.00482649
$778$	−1.23424e7	−0.731054
$779$	1.69620e6	0.100146
$780$	0	0
$781$	−3.25417e7	−1.90903
$782$	−3.00283e6	−0.175596
$783$	−2.77134e7	−1.61542
$784$	−4.17869e6	−0.242801
$785$	0	0
$786$	−2.54392e7	−1.46875
$787$	1.73411e7	0.998021	0.499011	−	0.866596i	$-0.333697\pi$
$787$	1.73411e7	0.998021	0.499011	+	0.866596i	$0.333697\pi$
$788$	−1.61628e7	−0.927262
$789$	−2.38569e7	−1.36434
$790$	0	0
$791$	633732.	0.0360134
$792$	−2.12828e7	−1.20564
$793$	254012.	0.0143440
$794$	−3.54183e6	−0.199378
$795$	0	0
$796$	−1.29222e7	−0.722862
$797$	2.58169e7	1.43965	0.719827	−	0.694153i	$-0.244221\pi$
$797$	2.58169e7	1.43965	0.719827	+	0.694153i	$0.244221\pi$
$798$	2.51680e6	0.139908
$799$	9.35096e6	0.518190
$800$	0	0
$801$	2.49798e7	1.37565
$802$	−1.50138e7	−0.824239
$803$	−4.00328e7	−2.19092
$804$	1.02927e7	0.561549
$805$	0	0
$806$	−733792.	−0.0397865
$807$	−1.91201e7	−1.03349
$808$	−9.04051e6	−0.487152
$809$	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	$-0.728485\pi$
$811$	−2.46396e7	−1.31547	−0.657735	+	0.753249i	$0.728485\pi$
$812$	−1.97472e6	−0.105103
$813$	−2.91540e7	−1.54693
$814$	−436224.	−0.0230754
$815$	0	0
$816$	−2.51597e6	−0.132276
$817$	5.52860e6	0.289774
$818$	−7.78612e6	−0.406853
$819$	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	$0.404837\pi$
$822$	2.49082e7	1.28577
$823$	1.30783e7	0.673057	0.336529	−	0.941673i	$-0.390747\pi$
$823$	1.30783e7	0.673057	0.336529	+	0.941673i	$0.390747\pi$
$824$	−8.94810e6	−0.459106
$825$	0	0
$826$	2.49744e6	0.127363
$827$	3.57188e7	1.81607	0.908037	−	0.418891i	$-0.137581\pi$
$827$	3.57188e7	1.81607	0.908037	+	0.418891i	$0.137581\pi$
$828$	1.37590e7	0.697447
$829$	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
$832$	188416.	0.00943647
$833$	6.17009e6	0.308091
$834$	3.20944e6	0.159777
$835$	0	0
$836$	−1.35168e7	−0.668895
$837$	−1.97007e7	−0.972005
$838$	−1.19666e7	−0.588657
$839$	−2.55497e7	−1.25309	−0.626543	−	0.779387i	$-0.715531\pi$
$839$	−2.55497e7	−1.25309	−0.626543	+	0.779387i	$0.715531\pi$
$840$	0	0
$841$	1.09610e7	0.534390
$842$	1.58664e7	0.771256
$843$	3.77547e7	1.82979
$844$	2.39283e6	0.115626
$845$	0	0
$846$	−4.28462e7	−2.05820
$847$	9.43301e6	0.451795
$848$	3.62650e6	0.173180
$849$	−8.03436e6	−0.382545
$850$	0	0
$851$	282012.	0.0133488
$852$	1.76268e7	0.831904
$853$	2.22953e7	1.04916	0.524579	−	0.851362i	$-0.324223\pi$
$853$	2.22953e7	1.04916	0.524579	+	0.851362i	$0.324223\pi$
$854$	485936.	0.0228000
$855$	0	0
$856$	−5.53075e6	−0.257988
$857$	−1.96872e7	−0.915656	−0.457828	−	0.889041i	$-0.651372\pi$
$857$	−1.96872e7	−0.915656	−0.457828	+	0.889041i	$0.651372\pi$
$858$	−3.67411e6	−0.170386
$859$	6.77582e6	0.313313	0.156657	−	0.987653i	$-0.449928\pi$
$859$	6.77582e6	0.313313	0.156657	+	0.987653i	$0.449928\pi$
$860$	0	0
$861$	882024.	0.0405483
$862$	−2.06846e7	−0.948154
$863$	2.63804e7	1.20574	0.602871	−	0.797839i	$-0.294023\pi$
$863$	2.63804e7	1.20574	0.602871	+	0.797839i	$0.294023\pi$
$864$	5.05856e6	0.230538
$865$	0	0
$866$	1.81394e7	0.821917
$867$	−3.32013e7	−1.50006
$868$	−1.40378e6	−0.0632410
$869$	3.04435e7	1.36756
$870$	0	0
$871$	1.13813e6	0.0508332
$872$	1.39808e7	0.622645
$873$	6.25174e7	2.77629
$874$	8.73840e6	0.386949
$875$	0	0
$876$	2.16844e7	0.954745
$877$	−2.95161e7	−1.29587	−0.647934	−	0.761697i	$-0.724367\pi$
$877$	−2.95161e7	−1.29587	−0.647934	+	0.761697i	$0.724367\pi$
$878$	−4.32880e6	−0.189510
$879$	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	$0.604502\pi$
$882$	−2.82714e7	−1.22370
$883$	1.45340e7	0.627313	0.313656	−	0.949537i	$-0.398446\pi$
$883$	1.45340e7	0.627313	0.313656	+	0.949537i	$0.398446\pi$
$884$	−278208.	−0.0119740
$885$	0	0
$886$	4.32314e6	0.185019
$887$	1.72028e7	0.734160	0.367080	−	0.930189i	$-0.380358\pi$
$887$	1.72028e7	0.734160	0.367080	+	0.930189i	$0.380358\pi$
$888$	236288.	0.0100556
$889$	4.76304e6	0.202130
$890$	0	0
$891$	−1.78337e7	−0.752572
$892$	7.09610e6	0.298612
$893$	−2.72118e7	−1.14190
$894$	−1.04988e7	−0.439335
$895$	0	0
$896$	360448.	0.0149994
$897$	2.37526e6	0.0985665
$898$	1.04713e7	0.433322
$899$	2.23727e7	0.923249
$900$	0	0
$901$	−5.35475e6	−0.219749
$902$	−4.73702e6	−0.193860
$903$	2.87487e6	0.117327
$904$	1.84358e6	0.0750312
$905$	0	0
$906$	1.29501e6	0.0524146
$907$	3.44434e7	1.39023	0.695116	−	0.718897i	$-0.255353\pi$
$907$	3.44434e7	1.39023	0.695116	+	0.718897i	$0.255353\pi$
$908$	−6.72029e6	−0.270504
$909$	−6.11647e7	−2.45522
$910$	0	0
$911$	−983748.	−0.0392724	−0.0196362	−	0.999807i	$-0.506251\pi$
$911$	−983748.	−0.0392724	−0.0196362	+	0.999807i	$0.506251\pi$
$912$	7.32160e6	0.291487
$913$	−4.59464e7	−1.82421
$914$	−6.36183e6	−0.251893
$915$	0	0
$916$	1.69400e7	0.667075
$917$	−5.38138e6	−0.211334
$918$	−7.46928e6	−0.292531
$919$	3.08857e7	1.20634	0.603168	−	0.797614i	$-0.293905\pi$
$919$	3.08857e7	1.20634	0.603168	+	0.797614i	$0.293905\pi$
$920$	0	0
$921$	−3.24287e7	−1.25974
$922$	1.70188e7	0.659328
$923$	1.94911e6	0.0753065
$924$	−7.02874e6	−0.270830
$925$	0	0
$926$	−1.30642e7	−0.500675
$927$	−6.05395e7	−2.31387
$928$	−5.74464e6	−0.218974
$929$	−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
$932$	2.03751e7	0.768353
$933$	−1.73157e7	−0.651232
$934$	2.40617e6	0.0902524
$935$	0	0
$936$	1.27475e6	0.0475594
$937$	−1.52520e7	−0.567515	−0.283757	−	0.958896i	$-0.591581\pi$
$937$	−1.52520e7	−0.567515	−0.283757	+	0.958896i	$0.591581\pi$
$938$	2.17730e6	0.0807998
$939$	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	$0.479586\pi$
$942$	626288.	0.0229957
$943$	3.06241e6	0.112146
$944$	7.26528e6	0.265352
$945$	0	0
$946$	−1.54399e7	−0.560939
$947$	2.54010e7	0.920398	0.460199	−	0.887816i	$-0.347778\pi$
$947$	2.54010e7	0.920398	0.460199	+	0.887816i	$0.347778\pi$
$948$	−1.64902e7	−0.595945
$949$	2.39780e6	0.0864265
$950$	0	0
$951$	1.34249e7	0.481348
$952$	−532224.	−0.0190328
$953$	4.97352e7	1.77391	0.886955	−	0.461856i	$-0.152816\pi$
$953$	4.97352e7	1.77391	0.886955	+	0.461856i	$0.152816\pi$
$954$	2.45355e7	0.872819
$955$	0	0
$956$	−5.93088e6	−0.209882
$957$	1.12020e8	3.95383
$958$	−1.83173e7	−0.644833
$959$	5.26904e6	0.185006
$960$	0	0
$961$	−1.27250e7	−0.444477
$962$	26128.0	0.000910266 0
$963$	−3.74190e7	−1.30025
$964$	−8.98077e6	−0.311258
$965$	0	0
$966$	4.54397e6	0.156672
$967$	−3.05173e7	−1.04949	−0.524747	−	0.851258i	$-0.675840\pi$
$967$	−3.05173e7	−1.04949	−0.524747	+	0.851258i	$0.675840\pi$
$968$	2.74415e7	0.941280
$969$	−1.08108e7	−0.369869
$970$	0	0
$971$	3.19854e7	1.08869	0.544344	−	0.838862i	$-0.316779\pi$
$971$	3.19854e7	1.08869	0.544344	+	0.838862i	$0.316779\pi$
$972$	−9.54678e6	−0.324109
$973$	678920.	0.0229899
$974$	−2.82090e7	−0.952776
$975$	0	0
$976$	1.41363e6	0.0475020
$977$	−2.90786e6	−0.0974623	−0.0487312	−	0.998812i	$-0.515518\pi$
$977$	−2.90786e6	−0.0974623	−0.0487312	+	0.998812i	$0.515518\pi$
$978$	5.20901e7	1.74144
$979$	−4.43059e7	−1.47742
$980$	0	0
$981$	9.45888e7	3.13810
$982$	−1.04940e7	−0.347264
$983$	−3.49621e7	−1.15402	−0.577010	−	0.816737i	$-0.695781\pi$
$983$	−3.49621e7	−1.15402	−0.577010	+	0.816737i	$0.695781\pi$
$984$	2.56589e6	0.0844792
$985$	0	0
$986$	8.48232e6	0.277858
$987$	−1.41501e7	−0.462347
$988$	809600.	0.0263863
$989$	9.98164e6	0.324497
$990$	0	0
$991$	3.00465e6	0.0971874	0.0485937	−	0.998819i	$-0.484526\pi$
$991$	3.00465e6	0.0971874	0.0485937	+	0.998819i	$0.484526\pi$
$992$	−4.08371e6	−0.131758
$993$	−8.56900e7	−2.75776
$994$	3.72874e6	0.119700
$995$	0	0
$996$	2.48876e7	0.794941
$997$	−3.20789e7	−1.02207	−0.511035	−	0.859560i	$-0.670738\pi$
$997$	−3.20789e7	−1.02207	−0.511035	+	0.859560i	$0.670738\pi$
$998$	−1.44494e7	−0.459222
$999$	701480.	0.0222383

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.a.1.1	✓	1	5.4	even	2
50.6.a.g.1.1		1	1.1	even	1	trivial
50.6.b.d.49.1		2	5.3	odd	4
50.6.b.d.49.2		2	5.2	odd	4
80.6.a.h.1.1		1	20.19	odd	2
90.6.a.f.1.1		1	15.14	odd	2
320.6.a.a.1.1		1	40.19	odd	2
320.6.a.p.1.1		1	40.29	even	2
400.6.a.a.1.1		1	4.3	odd	2
400.6.c.a.49.1		2	20.3	even	4
400.6.c.a.49.2		2	20.7	even	4
450.6.a.h.1.1		1	3.2	odd	2
450.6.c.o.199.1		2	15.2	even	4
450.6.c.o.199.2		2	15.8	even	4
490.6.a.j.1.1		1	35.34	odd	2
720.6.a.r.1.1		1	60.59	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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