LMFDB - Embedded newform 450.6.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.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:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} +4.00000 q^{7} +64.0000 q^{8} +O(q^{10})$

$q+4.00000 q^{2} +16.0000 q^{4} +4.00000 q^{7} +64.0000 q^{8} +500.000 q^{11} +288.000 q^{13} +16.0000 q^{14} +256.000 q^{16} +1516.00 q^{17} -1344.00 q^{19} +2000.00 q^{22} -4100.00 q^{23} +1152.00 q^{26} +64.0000 q^{28} +2646.00 q^{29} -5612.00 q^{31} +1024.00 q^{32} +6064.00 q^{34} +7288.00 q^{37} -5376.00 q^{38} +18986.0 q^{41} +2404.00 q^{43} +8000.00 q^{44} -16400.0 q^{46} +8900.00 q^{47} -16791.0 q^{49} +4608.00 q^{52} +39804.0 q^{53} +256.000 q^{56} +10584.0 q^{58} +28300.0 q^{59} +18290.0 q^{61} -22448.0 q^{62} +4096.00 q^{64} -65956.0 q^{67} +24256.0 q^{68} +28800.0 q^{71} +30808.0 q^{73} +29152.0 q^{74} -21504.0 q^{76} +2000.00 q^{77} +60228.0 q^{79} +75944.0 q^{82} -2468.00 q^{83} +9616.00 q^{86} +32000.0 q^{88} -22678.0 q^{89} +1152.00 q^{91} -65600.0 q^{92} +35600.0 q^{94} +36968.0 q^{97} -67164.0 q^{98} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000	0.0308542	0.0154271	−	0.999881i	$-0.495089\pi$
$7$	4.00000	0.0308542	0.0154271	+	0.999881i	$0.495089\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	0	0
$11$	500.000	1.24591	0.622957	−	0.782256i	$-0.285931\pi$
$11$	500.000	1.24591	0.622957	+	0.782256i	$0.285931\pi$
$12$	0	0
$13$	288.000	0.472644	0.236322	−	0.971675i	$-0.424058\pi$
$13$	288.000	0.472644	0.236322	+	0.971675i	$0.424058\pi$
$14$	16.0000	0.0218172
$15$	0	0
$16$	256.000	0.250000
$17$	1516.00	1.27226	0.636132	−	0.771581i	$-0.280534\pi$
$17$	1516.00	1.27226	0.636132	+	0.771581i	$0.280534\pi$
$18$	0	0
$19$	−1344.00	−0.854113	−0.427056	−	0.904225i	$-0.640449\pi$
$19$	−1344.00	−0.854113	−0.427056	+	0.904225i	$0.640449\pi$
$20$	0	0
$21$	0	0
$22$	2000.00	0.880995
$23$	−4100.00	−1.61609	−0.808043	−	0.589124i	$-0.799473\pi$
$23$	−4100.00	−1.61609	−0.808043	+	0.589124i	$0.799473\pi$
$24$	0	0
$25$	0	0
$26$	1152.00	0.334210
$27$	0	0
$28$	64.0000	0.0154271
$29$	2646.00	0.584245	0.292122	−	0.956381i	$-0.405639\pi$
$29$	2646.00	0.584245	0.292122	+	0.956381i	$0.405639\pi$
$30$	0	0
$31$	−5612.00	−1.04885	−0.524425	−	0.851457i	$-0.675720\pi$
$31$	−5612.00	−1.04885	−0.524425	+	0.851457i	$0.675720\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	6064.00	0.899626
$35$	0	0
$36$	0	0
$37$	7288.00	0.875193	0.437597	−	0.899171i	$-0.355830\pi$
$37$	7288.00	0.875193	0.437597	+	0.899171i	$0.355830\pi$
$38$	−5376.00	−0.603949
$39$	0	0
$40$	0	0
$41$	18986.0	1.76390	0.881950	−	0.471343i	$-0.156231\pi$
$41$	18986.0	1.76390	0.881950	+	0.471343i	$0.156231\pi$
$42$	0	0
$43$	2404.00	0.198273	0.0991364	−	0.995074i	$-0.468392\pi$
$43$	2404.00	0.198273	0.0991364	+	0.995074i	$0.468392\pi$
$44$	8000.00	0.622957
$45$	0	0
$46$	−16400.0	−1.14274
$47$	8900.00	0.587686	0.293843	−	0.955854i	$-0.405066\pi$
$47$	8900.00	0.587686	0.293843	+	0.955854i	$0.405066\pi$
$48$	0	0
$49$	−16791.0	−0.999048
$50$	0	0
$51$	0	0
$52$	4608.00	0.236322
$53$	39804.0	1.94642	0.973211	−	0.229913i	$-0.0738443\pi$
$53$	39804.0	1.94642	0.973211	+	0.229913i	$0.0738443\pi$
$54$	0	0
$55$	0	0
$56$	256.000	0.0109086
$57$	0	0
$58$	10584.0	0.413123
$59$	28300.0	1.05842	0.529208	−	0.848492i	$-0.322489\pi$
$59$	28300.0	1.05842	0.529208	+	0.848492i	$0.322489\pi$
$60$	0	0
$61$	18290.0	0.629345	0.314673	−	0.949200i	$-0.398105\pi$
$61$	18290.0	0.629345	0.314673	+	0.949200i	$0.398105\pi$
$62$	−22448.0	−0.741649
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	−65956.0	−1.79501	−0.897506	−	0.441002i	$-0.854623\pi$
$67$	−65956.0	−1.79501	−0.897506	+	0.441002i	$0.854623\pi$
$68$	24256.0	0.636132
$69$	0	0
$70$	0	0
$71$	28800.0	0.678026	0.339013	−	0.940782i	$-0.389907\pi$
$71$	28800.0	0.678026	0.339013	+	0.940782i	$0.389907\pi$
$72$	0	0
$73$	30808.0	0.676638	0.338319	−	0.941031i	$-0.390142\pi$
$73$	30808.0	0.676638	0.338319	+	0.941031i	$0.390142\pi$
$74$	29152.0	0.618855
$75$	0	0
$76$	−21504.0	−0.427056
$77$	2000.00	0.0384418
$78$	0	0
$79$	60228.0	1.08575	0.542876	−	0.839813i	$-0.317335\pi$
$79$	60228.0	1.08575	0.542876	+	0.839813i	$0.317335\pi$
$80$	0	0
$81$	0	0
$82$	75944.0	1.24727
$83$	−2468.00	−0.0393233	−0.0196616	−	0.999807i	$-0.506259\pi$
$83$	−2468.00	−0.0393233	−0.0196616	+	0.999807i	$0.506259\pi$
$84$	0	0
$85$	0	0
$86$	9616.00	0.140200
$87$	0	0
$88$	32000.0	0.440497
$89$	−22678.0	−0.303480	−0.151740	−	0.988420i	$-0.548488\pi$
$89$	−22678.0	−0.303480	−0.151740	+	0.988420i	$0.548488\pi$
$90$	0	0
$91$	1152.00	0.0145831
$92$	−65600.0	−0.808043
$93$	0	0
$94$	35600.0	0.415557
$95$	0	0
$96$	0	0
$97$	36968.0	0.398930	0.199465	−	0.979905i	$-0.436080\pi$
$97$	36968.0	0.398930	0.199465	+	0.979905i	$0.436080\pi$
$98$	−67164.0	−0.706434
$99$	0	0
$100$	0	0
$101$	−167918.	−1.63792	−0.818962	−	0.573848i	$-0.805450\pi$
$101$	−167918.	−1.63792	−0.818962	+	0.573848i	$0.805450\pi$
$102$	0	0
$103$	154364.	1.43368	0.716841	−	0.697236i	$-0.245587\pi$
$103$	154364.	1.43368	0.716841	+	0.697236i	$0.245587\pi$
$104$	18432.0	0.167105
$105$	0	0
$106$	159216.	1.37633
$107$	136788.	1.15502	0.577509	−	0.816385i	$-0.304025\pi$
$107$	136788.	1.15502	0.577509	+	0.816385i	$0.304025\pi$
$108$	0	0
$109$	−53810.0	−0.433807	−0.216904	−	0.976193i	$-0.569596\pi$
$109$	−53810.0	−0.433807	−0.216904	+	0.976193i	$0.569596\pi$
$110$	0	0
$111$	0	0
$112$	1024.00	0.00771356
$113$	82692.0	0.609211	0.304605	−	0.952479i	$-0.401475\pi$
$113$	82692.0	0.609211	0.304605	+	0.952479i	$0.401475\pi$
$114$	0	0
$115$	0	0
$116$	42336.0	0.292122
$117$	0	0
$118$	113200.	0.748413
$119$	6064.00	0.0392547
$120$	0	0
$121$	88949.0	0.552303
$122$	73160.0	0.445014
$123$	0	0
$124$	−89792.0	−0.524425
$125$	0	0
$126$	0	0
$127$	211780.	1.16513	0.582567	−	0.812783i	$-0.302048\pi$
$127$	211780.	1.16513	0.582567	+	0.812783i	$0.302048\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	0	0
$131$	−169500.	−0.862962	−0.431481	−	0.902122i	$-0.642009\pi$
$131$	−169500.	−0.862962	−0.431481	+	0.902122i	$0.642009\pi$
$132$	0	0
$133$	−5376.00	−0.0263530
$134$	−263824.	−1.26927
$135$	0	0
$136$	97024.0	0.449813
$137$	252036.	1.14726	0.573629	−	0.819115i	$-0.305535\pi$
$137$	252036.	1.14726	0.573629	+	0.819115i	$0.305535\pi$
$138$	0	0
$139$	192016.	0.842947	0.421474	−	0.906841i	$-0.361513\pi$
$139$	192016.	0.842947	0.421474	+	0.906841i	$0.361513\pi$
$140$	0	0
$141$	0	0
$142$	115200.	0.479437
$143$	144000.	0.588874
$144$	0	0
$145$	0	0
$146$	123232.	0.478455
$147$	0	0
$148$	116608.	0.437597
$149$	−235694.	−0.869727	−0.434863	−	0.900496i	$-0.643203\pi$
$149$	−235694.	−0.869727	−0.434863	+	0.900496i	$0.643203\pi$
$150$	0	0
$151$	−371492.	−1.32589	−0.662944	−	0.748669i	$-0.730693\pi$
$151$	−371492.	−1.32589	−0.662944	+	0.748669i	$0.730693\pi$
$152$	−86016.0	−0.301975
$153$	0	0
$154$	8000.00	0.0271824
$155$	0	0
$156$	0	0
$157$	−264952.	−0.857863	−0.428932	−	0.903337i	$-0.641110\pi$
$157$	−264952.	−0.857863	−0.428932	+	0.903337i	$0.641110\pi$
$158$	240912.	0.767743
$159$	0	0
$160$	0	0
$161$	−16400.0	−0.0498631
$162$	0	0
$163$	403124.	1.18842	0.594210	−	0.804310i	$-0.297465\pi$
$163$	403124.	1.18842	0.594210	+	0.804310i	$0.297465\pi$
$164$	303776.	0.881950
$165$	0	0
$166$	−9872.00	−0.0278058
$167$	−261900.	−0.726682	−0.363341	−	0.931656i	$-0.618364\pi$
$167$	−261900.	−0.726682	−0.363341	+	0.931656i	$0.618364\pi$
$168$	0	0
$169$	−288349.	−0.776608
$170$	0	0
$171$	0	0
$172$	38464.0	0.0991364
$173$	326228.	0.828716	0.414358	−	0.910114i	$-0.364006\pi$
$173$	326228.	0.828716	0.414358	+	0.910114i	$0.364006\pi$
$174$	0	0
$175$	0	0
$176$	128000.	0.311479
$177$	0	0
$178$	−90712.0	−0.214593
$179$	109516.	0.255473	0.127736	−	0.991808i	$-0.459229\pi$
$179$	109516.	0.255473	0.127736	+	0.991808i	$0.459229\pi$
$180$	0	0
$181$	−53146.0	−0.120580	−0.0602898	−	0.998181i	$-0.519202\pi$
$181$	−53146.0	−0.120580	−0.0602898	+	0.998181i	$0.519202\pi$
$182$	4608.00	0.0103118
$183$	0	0
$184$	−262400.	−0.571372
$185$	0	0
$186$	0	0
$187$	758000.	1.58513
$188$	142400.	0.293843
$189$	0	0
$190$	0	0
$191$	−232056.	−0.460267	−0.230133	−	0.973159i	$-0.573916\pi$
$191$	−232056.	−0.460267	−0.230133	+	0.973159i	$0.573916\pi$
$192$	0	0
$193$	1.03067e6	1.99172	0.995858	−	0.0909274i	$-0.0289831\pi$
$193$	1.03067e6	1.99172	0.995858	+	0.0909274i	$0.0289831\pi$
$194$	147872.	0.282086
$195$	0	0
$196$	−268656.	−0.499524
$197$	−522796.	−0.959769	−0.479884	−	0.877332i	$-0.659321\pi$
$197$	−522796.	−0.959769	−0.479884	+	0.877332i	$0.659321\pi$
$198$	0	0
$199$	−215292.	−0.385385	−0.192693	−	0.981259i	$-0.561722\pi$
$199$	−215292.	−0.385385	−0.192693	+	0.981259i	$0.561722\pi$
$200$	0	0
$201$	0	0
$202$	−671672.	−1.15819
$203$	10584.0	0.0180264
$204$	0	0
$205$	0	0
$206$	617456.	1.01377
$207$	0	0
$208$	73728.0	0.118161
$209$	−672000.	−1.06415
$210$	0	0
$211$	−1.03008e6	−1.59281	−0.796407	−	0.604762i	$-0.793268\pi$
$211$	−1.03008e6	−1.59281	−0.796407	+	0.604762i	$0.793268\pi$
$212$	636864.	0.973211
$213$	0	0
$214$	547152.	0.816721
$215$	0	0
$216$	0	0
$217$	−22448.0	−0.0323615
$218$	−215240.	−0.306748
$219$	0	0
$220$	0	0
$221$	436608.	0.601327
$222$	0	0
$223$	−456020.	−0.614075	−0.307038	−	0.951697i	$-0.599338\pi$
$223$	−456020.	−0.614075	−0.307038	+	0.951697i	$0.599338\pi$
$224$	4096.00	0.00545431
$225$	0	0
$226$	330768.	0.430777
$227$	−434252.	−0.559342	−0.279671	−	0.960096i	$-0.590225\pi$
$227$	−434252.	−0.559342	−0.279671	+	0.960096i	$0.590225\pi$
$228$	0	0
$229$	−722710.	−0.910700	−0.455350	−	0.890313i	$-0.650486\pi$
$229$	−722710.	−0.910700	−0.455350	+	0.890313i	$0.650486\pi$
$230$	0	0
$231$	0	0
$232$	169344.	0.206562
$233$	−565348.	−0.682223	−0.341111	−	0.940023i	$-0.610803\pi$
$233$	−565348.	−0.682223	−0.341111	+	0.940023i	$0.610803\pi$
$234$	0	0
$235$	0	0
$236$	452800.	0.529208
$237$	0	0
$238$	24256.0	0.0277573
$239$	−324904.	−0.367926	−0.183963	−	0.982933i	$-0.558893\pi$
$239$	−324904.	−0.367926	−0.183963	+	0.982933i	$0.558893\pi$
$240$	0	0
$241$	915262.	1.01509	0.507543	−	0.861626i	$-0.330554\pi$
$241$	915262.	1.01509	0.507543	+	0.861626i	$0.330554\pi$
$242$	355796.	0.390537
$243$	0	0
$244$	292640.	0.314673
$245$	0	0
$246$	0	0
$247$	−387072.	−0.403691
$248$	−359168.	−0.370825
$249$	0	0
$250$	0	0
$251$	−1.36708e6	−1.36965	−0.684823	−	0.728709i	$-0.740121\pi$
$251$	−1.36708e6	−1.36965	−0.684823	+	0.728709i	$0.740121\pi$
$252$	0	0
$253$	−2.05000e6	−2.01350
$254$	847120.	0.823874
$255$	0	0
$256$	65536.0	0.0625000
$257$	892932.	0.843307	0.421653	−	0.906757i	$-0.361450\pi$
$257$	892932.	0.843307	0.421653	+	0.906757i	$0.361450\pi$
$258$	0	0
$259$	29152.0	0.0270034
$260$	0	0
$261$	0	0
$262$	−678000.	−0.610206
$263$	−1.86650e6	−1.66394	−0.831972	−	0.554818i	$-0.812788\pi$
$263$	−1.86650e6	−1.66394	−0.831972	+	0.554818i	$0.812788\pi$
$264$	0	0
$265$	0	0
$266$	−21504.0	−0.0186344
$267$	0	0
$268$	−1.05530e6	−0.897506
$269$	1.37227e6	1.15627	0.578133	−	0.815943i	$-0.303782\pi$
$269$	1.37227e6	1.15627	0.578133	+	0.815943i	$0.303782\pi$
$270$	0	0
$271$	458644.	0.379361	0.189680	−	0.981846i	$-0.439255\pi$
$271$	458644.	0.379361	0.189680	+	0.981846i	$0.439255\pi$
$272$	388096.	0.318066
$273$	0	0
$274$	1.00814e6	0.811234
$275$	0	0
$276$	0	0
$277$	985408.	0.771643	0.385822	−	0.922573i	$-0.373918\pi$
$277$	985408.	0.771643	0.385822	+	0.922573i	$0.373918\pi$
$278$	768064.	0.596054
$279$	0	0
$280$	0	0
$281$	−165798.	−0.125260	−0.0626302	−	0.998037i	$-0.519949\pi$
$281$	−165798.	−0.125260	−0.0626302	+	0.998037i	$0.519949\pi$
$282$	0	0
$283$	1.66471e6	1.23558	0.617792	−	0.786342i	$-0.288028\pi$
$283$	1.66471e6	1.23558	0.617792	+	0.786342i	$0.288028\pi$
$284$	460800.	0.339013
$285$	0	0
$286$	576000.	0.416397
$287$	75944.0	0.0544238
$288$	0	0
$289$	878399.	0.618653
$290$	0	0
$291$	0	0
$292$	492928.	0.338319
$293$	2.55104e6	1.73600	0.867998	−	0.496567i	$-0.165406\pi$
$293$	2.55104e6	1.73600	0.867998	+	0.496567i	$0.165406\pi$
$294$	0	0
$295$	0	0
$296$	466432.	0.309428
$297$	0	0
$298$	−942776.	−0.614990
$299$	−1.18080e6	−0.763833
$300$	0	0
$301$	9616.00	0.00611756
$302$	−1.48597e6	−0.937545
$303$	0	0
$304$	−344064.	−0.213528
$305$	0	0
$306$	0	0
$307$	−736020.	−0.445701	−0.222851	−	0.974853i	$-0.571536\pi$
$307$	−736020.	−0.445701	−0.222851	+	0.974853i	$0.571536\pi$
$308$	32000.0	0.0192209
$309$	0	0
$310$	0	0
$311$	−1.71660e6	−1.00639	−0.503197	−	0.864172i	$-0.667843\pi$
$311$	−1.71660e6	−1.00639	−0.503197	+	0.864172i	$0.667843\pi$
$312$	0	0
$313$	−2.83851e6	−1.63768	−0.818842	−	0.574020i	$-0.805383\pi$
$313$	−2.83851e6	−1.63768	−0.818842	+	0.574020i	$0.805383\pi$
$314$	−1.05981e6	−0.606601
$315$	0	0
$316$	963648.	0.542876
$317$	−1.27605e6	−0.713215	−0.356607	−	0.934254i	$-0.616067\pi$
$317$	−1.27605e6	−0.713215	−0.356607	+	0.934254i	$0.616067\pi$
$318$	0	0
$319$	1.32300e6	0.727919
$320$	0	0
$321$	0	0
$322$	−65600.0	−0.0352585
$323$	−2.03750e6	−1.08666
$324$	0	0
$325$	0	0
$326$	1.61250e6	0.840339
$327$	0	0
$328$	1.21510e6	0.623633
$329$	35600.0	0.0181326
$330$	0	0
$331$	443992.	0.222744	0.111372	−	0.993779i	$-0.464476\pi$
$331$	443992.	0.222744	0.111372	+	0.993779i	$0.464476\pi$
$332$	−39488.0	−0.0196616
$333$	0	0
$334$	−1.04760e6	−0.513842
$335$	0	0
$336$	0	0
$337$	−2.71326e6	−1.30142	−0.650708	−	0.759328i	$-0.725528\pi$
$337$	−2.71326e6	−1.30142	−0.650708	+	0.759328i	$0.725528\pi$
$338$	−1.15340e6	−0.549145
$339$	0	0
$340$	0	0
$341$	−2.80600e6	−1.30678
$342$	0	0
$343$	−134392.	−0.0616791
$344$	153856.	0.0701001
$345$	0	0
$346$	1.30491e6	0.585991
$347$	−1.31051e6	−0.584273	−0.292137	−	0.956377i	$-0.594366\pi$
$347$	−1.31051e6	−0.584273	−0.292137	+	0.956377i	$0.594366\pi$
$348$	0	0
$349$	−298910.	−0.131364	−0.0656821	−	0.997841i	$-0.520922\pi$
$349$	−298910.	−0.131364	−0.0656821	+	0.997841i	$0.520922\pi$
$350$	0	0
$351$	0	0
$352$	512000.	0.220249
$353$	737996.	0.315223	0.157611	−	0.987501i	$-0.449621\pi$
$353$	737996.	0.315223	0.157611	+	0.987501i	$0.449621\pi$
$354$	0	0
$355$	0	0
$356$	−362848.	−0.151740
$357$	0	0
$358$	438064.	0.180647
$359$	2.34074e6	0.958557	0.479278	−	0.877663i	$-0.340898\pi$
$359$	2.34074e6	0.958557	0.479278	+	0.877663i	$0.340898\pi$
$360$	0	0
$361$	−669763.	−0.270491
$362$	−212584.	−0.0852627
$363$	0	0
$364$	18432.0	0.00729154
$365$	0	0
$366$	0	0
$367$	−127292.	−0.0493328	−0.0246664	−	0.999696i	$-0.507852\pi$
$367$	−127292.	−0.0493328	−0.0246664	+	0.999696i	$0.507852\pi$
$368$	−1.04960e6	−0.404021
$369$	0	0
$370$	0	0
$371$	159216.	0.0600554
$372$	0	0
$373$	−4.03870e6	−1.50303	−0.751517	−	0.659713i	$-0.770678\pi$
$373$	−4.03870e6	−1.50303	−0.751517	+	0.659713i	$0.770678\pi$
$374$	3.03200e6	1.12086
$375$	0	0
$376$	569600.	0.207778
$377$	762048.	0.276140
$378$	0	0
$379$	1.01214e6	0.361944	0.180972	−	0.983488i	$-0.442076\pi$
$379$	1.01214e6	0.361944	0.180972	+	0.983488i	$0.442076\pi$
$380$	0	0
$381$	0	0
$382$	−928224.	−0.325458
$383$	−2.37610e6	−0.827690	−0.413845	−	0.910347i	$-0.635814\pi$
$383$	−2.37610e6	−0.827690	−0.413845	+	0.910347i	$0.635814\pi$
$384$	0	0
$385$	0	0
$386$	4.12269e6	1.40836
$387$	0	0
$388$	591488.	0.199465
$389$	−1.42497e6	−0.477456	−0.238728	−	0.971087i	$-0.576730\pi$
$389$	−1.42497e6	−0.477456	−0.238728	+	0.971087i	$0.576730\pi$
$390$	0	0
$391$	−6.21560e6	−2.05609
$392$	−1.07462e6	−0.353217
$393$	0	0
$394$	−2.09118e6	−0.678659
$395$	0	0
$396$	0	0
$397$	−1.69345e6	−0.539257	−0.269628	−	0.962964i	$-0.586901\pi$
$397$	−1.69345e6	−0.539257	−0.269628	+	0.962964i	$0.586901\pi$
$398$	−861168.	−0.272509
$399$	0	0
$400$	0	0
$401$	2.84501e6	0.883532	0.441766	−	0.897130i	$-0.354352\pi$
$401$	2.84501e6	0.883532	0.441766	+	0.897130i	$0.354352\pi$
$402$	0	0
$403$	−1.61626e6	−0.495733
$404$	−2.68669e6	−0.818962
$405$	0	0
$406$	42336.0	0.0127466
$407$	3.64400e6	1.09042
$408$	0	0
$409$	−1.89069e6	−0.558873	−0.279436	−	0.960164i	$-0.590148\pi$
$409$	−1.89069e6	−0.558873	−0.279436	+	0.960164i	$0.590148\pi$
$410$	0	0
$411$	0	0
$412$	2.46982e6	0.716841
$413$	113200.	0.0326566
$414$	0	0
$415$	0	0
$416$	294912.	0.0835524
$417$	0	0
$418$	−2.68800e6	−0.752469
$419$	−4.60930e6	−1.28263	−0.641313	−	0.767280i	$-0.721610\pi$
$419$	−4.60930e6	−1.28263	−0.641313	+	0.767280i	$0.721610\pi$
$420$	0	0
$421$	−6.04151e6	−1.66127	−0.830635	−	0.556817i	$-0.812022\pi$
$421$	−6.04151e6	−1.66127	−0.830635	+	0.556817i	$0.812022\pi$
$422$	−4.12032e6	−1.12629
$423$	0	0
$424$	2.54746e6	0.688164
$425$	0	0
$426$	0	0
$427$	73160.0	0.0194180
$428$	2.18861e6	0.577509
$429$	0	0
$430$	0	0
$431$	3800.00	0.000985350 0	0.000492675	−	1.00000i	$-0.499843\pi$
$431$	3800.00	0.000985350 0	0.000492675		1.00000i	$0.499843\pi$
$432$	0	0
$433$	−250736.	−0.0642683	−0.0321342	−	0.999484i	$-0.510230\pi$
$433$	−250736.	−0.0642683	−0.0321342	+	0.999484i	$0.510230\pi$
$434$	−89792.0	−0.0228830
$435$	0	0
$436$	−860960.	−0.216904
$437$	5.51040e6	1.38032
$438$	0	0
$439$	−3.58873e6	−0.888750	−0.444375	−	0.895841i	$-0.646574\pi$
$439$	−3.58873e6	−0.888750	−0.444375	+	0.895841i	$0.646574\pi$
$440$	0	0
$441$	0	0
$442$	1.74643e6	0.425203
$443$	−1.41479e6	−0.342517	−0.171258	−	0.985226i	$-0.554783\pi$
$443$	−1.41479e6	−0.342517	−0.171258	+	0.985226i	$0.554783\pi$
$444$	0	0
$445$	0	0
$446$	−1.82408e6	−0.434217
$447$	0	0
$448$	16384.0	0.00385678
$449$	829806.	0.194250	0.0971249	−	0.995272i	$-0.469035\pi$
$449$	829806.	0.194250	0.0971249	+	0.995272i	$0.469035\pi$
$450$	0	0
$451$	9.49300e6	2.19767
$452$	1.32307e6	0.304605
$453$	0	0
$454$	−1.73701e6	−0.395514
$455$	0	0
$456$	0	0
$457$	−4.68198e6	−1.04867	−0.524335	−	0.851512i	$-0.675686\pi$
$457$	−4.68198e6	−1.04867	−0.524335	+	0.851512i	$0.675686\pi$
$458$	−2.89084e6	−0.643962
$459$	0	0
$460$	0	0
$461$	141930.	0.0311044	0.0155522	−	0.999879i	$-0.495049\pi$
$461$	141930.	0.0311044	0.0155522	+	0.999879i	$0.495049\pi$
$462$	0	0
$463$	−727476.	−0.157713	−0.0788563	−	0.996886i	$-0.525127\pi$
$463$	−727476.	−0.157713	−0.0788563	+	0.996886i	$0.525127\pi$
$464$	677376.	0.146061
$465$	0	0
$466$	−2.26139e6	−0.482404
$467$	−4.47640e6	−0.949809	−0.474905	−	0.880037i	$-0.657517\pi$
$467$	−4.47640e6	−0.949809	−0.474905	+	0.880037i	$0.657517\pi$
$468$	0	0
$469$	−263824.	−0.0553837
$470$	0	0
$471$	0	0
$472$	1.81120e6	0.374207
$473$	1.20200e6	0.247031
$474$	0	0
$475$	0	0
$476$	97024.0	0.0196274
$477$	0	0
$478$	−1.29962e6	−0.260163
$479$	1.32718e6	0.264297	0.132149	−	0.991230i	$-0.457812\pi$
$479$	1.32718e6	0.264297	0.132149	+	0.991230i	$0.457812\pi$
$480$	0	0
$481$	2.09894e6	0.413655
$482$	3.66105e6	0.717774
$483$	0	0
$484$	1.42318e6	0.276152
$485$	0	0
$486$	0	0
$487$	4.11647e6	0.786507	0.393253	−	0.919430i	$-0.371350\pi$
$487$	4.11647e6	0.786507	0.393253	+	0.919430i	$0.371350\pi$
$488$	1.17056e6	0.222507
$489$	0	0
$490$	0	0
$491$	6.12316e6	1.14623	0.573115	−	0.819475i	$-0.305735\pi$
$491$	6.12316e6	1.14623	0.573115	+	0.819475i	$0.305735\pi$
$492$	0	0
$493$	4.01134e6	0.743313
$494$	−1.54829e6	−0.285453
$495$	0	0
$496$	−1.43667e6	−0.262213
$497$	115200.	0.0209200
$498$	0	0
$499$	−7.90490e6	−1.42117	−0.710584	−	0.703613i	$-0.751569\pi$
$499$	−7.90490e6	−1.42117	−0.710584	+	0.703613i	$0.751569\pi$
$500$	0	0
$501$	0	0
$502$	−5.46830e6	−0.968486
$503$	3.97628e6	0.700741	0.350370	−	0.936611i	$-0.386056\pi$
$503$	3.97628e6	0.700741	0.350370	+	0.936611i	$0.386056\pi$
$504$	0	0
$505$	0	0
$506$	−8.20000e6	−1.42376
$507$	0	0
$508$	3.38848e6	0.582567
$509$	−781914.	−0.133772	−0.0668859	−	0.997761i	$-0.521306\pi$
$509$	−781914.	−0.133772	−0.0668859	+	0.997761i	$0.521306\pi$
$510$	0	0
$511$	123232.	0.0208772
$512$	262144.	0.0441942
$513$	0	0
$514$	3.57173e6	0.596308
$515$	0	0
$516$	0	0
$517$	4.45000e6	0.732207
$518$	116608.	0.0190943
$519$	0	0
$520$	0	0
$521$	−5.82694e6	−0.940472	−0.470236	−	0.882541i	$-0.655831\pi$
$521$	−5.82694e6	−0.940472	−0.470236	+	0.882541i	$0.655831\pi$
$522$	0	0
$523$	9.78938e6	1.56495	0.782476	−	0.622681i	$-0.213957\pi$
$523$	9.78938e6	1.56495	0.782476	+	0.622681i	$0.213957\pi$
$524$	−2.71200e6	−0.431481
$525$	0	0
$526$	−7.46600e6	−1.17659
$527$	−8.50779e6	−1.33441
$528$	0	0
$529$	1.03737e7	1.61173
$530$	0	0
$531$	0	0
$532$	−86016.0	−0.0131765
$533$	5.46797e6	0.833696
$534$	0	0
$535$	0	0
$536$	−4.22118e6	−0.634633
$537$	0	0
$538$	5.48906e6	0.817603
$539$	−8.39550e6	−1.24473
$540$	0	0
$541$	4.76059e6	0.699307	0.349653	−	0.936879i	$-0.386299\pi$
$541$	4.76059e6	0.699307	0.349653	+	0.936879i	$0.386299\pi$
$542$	1.83458e6	0.268249
$543$	0	0
$544$	1.55238e6	0.224906
$545$	0	0
$546$	0	0
$547$	1.16595e6	0.166614	0.0833069	−	0.996524i	$-0.473452\pi$
$547$	1.16595e6	0.166614	0.0833069	+	0.996524i	$0.473452\pi$
$548$	4.03258e6	0.573629
$549$	0	0
$550$	0	0
$551$	−3.55622e6	−0.499011
$552$	0	0
$553$	240912.	0.0335001
$554$	3.94163e6	0.545634
$555$	0	0
$556$	3.07226e6	0.421474
$557$	−1.61293e6	−0.220282	−0.110141	−	0.993916i	$-0.535130\pi$
$557$	−1.61293e6	−0.220282	−0.110141	+	0.993916i	$0.535130\pi$
$558$	0	0
$559$	692352.	0.0937125
$560$	0	0
$561$	0	0
$562$	−663192.	−0.0885724
$563$	3.40603e6	0.452874	0.226437	−	0.974026i	$-0.427292\pi$
$563$	3.40603e6	0.452874	0.226437	+	0.974026i	$0.427292\pi$
$564$	0	0
$565$	0	0
$566$	6.65883e6	0.873689
$567$	0	0
$568$	1.84320e6	0.239719
$569$	1.44009e7	1.86470	0.932350	−	0.361557i	$-0.117755\pi$
$569$	1.44009e7	1.86470	0.932350	+	0.361557i	$0.117755\pi$
$570$	0	0
$571$	4.74772e6	0.609389	0.304695	−	0.952450i	$-0.401446\pi$
$571$	4.74772e6	0.609389	0.304695	+	0.952450i	$0.401446\pi$
$572$	2.30400e6	0.294437
$573$	0	0
$574$	303776.	0.0384834
$575$	0	0
$576$	0	0
$577$	1.09094e7	1.36415	0.682074	−	0.731283i	$-0.261078\pi$
$577$	1.09094e7	1.36415	0.682074	+	0.731283i	$0.261078\pi$
$578$	3.51360e6	0.437454
$579$	0	0
$580$	0	0
$581$	−9872.00	−0.00121329
$582$	0	0
$583$	1.99020e7	2.42508
$584$	1.97171e6	0.239228
$585$	0	0
$586$	1.02042e7	1.22754
$587$	8.53223e6	1.02204	0.511019	−	0.859569i	$-0.329268\pi$
$587$	8.53223e6	1.02204	0.511019	+	0.859569i	$0.329268\pi$
$588$	0	0
$589$	7.54253e6	0.895836
$590$	0	0
$591$	0	0
$592$	1.86573e6	0.218798
$593$	−4.63182e6	−0.540897	−0.270449	−	0.962734i	$-0.587172\pi$
$593$	−4.63182e6	−0.540897	−0.270449	+	0.962734i	$0.587172\pi$
$594$	0	0
$595$	0	0
$596$	−3.77110e6	−0.434863
$597$	0	0
$598$	−4.72320e6	−0.540111
$599$	−6.27598e6	−0.714684	−0.357342	−	0.933974i	$-0.616317\pi$
$599$	−6.27598e6	−0.714684	−0.357342	+	0.933974i	$0.616317\pi$
$600$	0	0
$601$	7.71988e6	0.871815	0.435907	−	0.899992i	$-0.356428\pi$
$601$	7.71988e6	0.871815	0.435907	+	0.899992i	$0.356428\pi$
$602$	38464.0	0.00432577
$603$	0	0
$604$	−5.94387e6	−0.662944
$605$	0	0
$606$	0	0
$607$	6.06160e6	0.667753	0.333876	−	0.942617i	$-0.391643\pi$
$607$	6.06160e6	0.667753	0.333876	+	0.942617i	$0.391643\pi$
$608$	−1.37626e6	−0.150987
$609$	0	0
$610$	0	0
$611$	2.56320e6	0.277766
$612$	0	0
$613$	3.66489e6	0.393921	0.196961	−	0.980411i	$-0.436893\pi$
$613$	3.66489e6	0.393921	0.196961	+	0.980411i	$0.436893\pi$
$614$	−2.94408e6	−0.315158
$615$	0	0
$616$	128000.	0.0135912
$617$	−9.32522e6	−0.986157	−0.493079	−	0.869985i	$-0.664129\pi$
$617$	−9.32522e6	−0.986157	−0.493079	+	0.869985i	$0.664129\pi$
$618$	0	0
$619$	−7.40162e6	−0.776426	−0.388213	−	0.921570i	$-0.626907\pi$
$619$	−7.40162e6	−0.776426	−0.388213	+	0.921570i	$0.626907\pi$
$620$	0	0
$621$	0	0
$622$	−6.86640e6	−0.711628
$623$	−90712.0	−0.00936364
$624$	0	0
$625$	0	0
$626$	−1.13540e7	−1.15802
$627$	0	0
$628$	−4.23923e6	−0.428932
$629$	1.10486e7	1.11348
$630$	0	0
$631$	160052.	0.0160025	0.00800125	−	0.999968i	$-0.497453\pi$
$631$	160052.	0.0160025	0.00800125	+	0.999968i	$0.497453\pi$
$632$	3.85459e6	0.383871
$633$	0	0
$634$	−5.10421e6	−0.504319
$635$	0	0
$636$	0	0
$637$	−4.83581e6	−0.472194
$638$	5.29200e6	0.514717
$639$	0	0
$640$	0	0
$641$	1.69565e7	1.63002	0.815008	−	0.579450i	$-0.196732\pi$
$641$	1.69565e7	1.63002	0.815008	+	0.579450i	$0.196732\pi$
$642$	0	0
$643$	−1.10128e7	−1.05044	−0.525219	−	0.850967i	$-0.676016\pi$
$643$	−1.10128e7	−1.05044	−0.525219	+	0.850967i	$0.676016\pi$
$644$	−262400.	−0.0249315
$645$	0	0
$646$	−8.15002e6	−0.768382
$647$	3.33848e6	0.313537	0.156768	−	0.987635i	$-0.449892\pi$
$647$	3.33848e6	0.313537	0.156768	+	0.987635i	$0.449892\pi$
$648$	0	0
$649$	1.41500e7	1.31870
$650$	0	0
$651$	0	0
$652$	6.44998e6	0.594210
$653$	−4.76181e6	−0.437008	−0.218504	−	0.975836i	$-0.570118\pi$
$653$	−4.76181e6	−0.437008	−0.218504	+	0.975836i	$0.570118\pi$
$654$	0	0
$655$	0	0
$656$	4.86042e6	0.440975
$657$	0	0
$658$	142400.	0.0128217
$659$	−798188.	−0.0715965	−0.0357982	−	0.999359i	$-0.511397\pi$
$659$	−798188.	−0.0715965	−0.0357982	+	0.999359i	$0.511397\pi$
$660$	0	0
$661$	−1.54048e7	−1.37136	−0.685682	−	0.727901i	$-0.740496\pi$
$661$	−1.54048e7	−1.37136	−0.685682	+	0.727901i	$0.740496\pi$
$662$	1.77597e6	0.157503
$663$	0	0
$664$	−157952.	−0.0139029
$665$	0	0
$666$	0	0
$667$	−1.08486e7	−0.944189
$668$	−4.19040e6	−0.363341
$669$	0	0
$670$	0	0
$671$	9.14500e6	0.784111
$672$	0	0
$673$	976704.	0.0831238	0.0415619	−	0.999136i	$-0.486767\pi$
$673$	976704.	0.0831238	0.0415619	+	0.999136i	$0.486767\pi$
$674$	−1.08530e7	−0.920240
$675$	0	0
$676$	−4.61358e6	−0.388304
$677$	1.93885e7	1.62582	0.812911	−	0.582388i	$-0.197881\pi$
$677$	1.93885e7	1.62582	0.812911	+	0.582388i	$0.197881\pi$
$678$	0	0
$679$	147872.	0.0123087
$680$	0	0
$681$	0	0
$682$	−1.12240e7	−0.924031
$683$	−5.25573e6	−0.431103	−0.215552	−	0.976492i	$-0.569155\pi$
$683$	−5.25573e6	−0.431103	−0.215552	+	0.976492i	$0.569155\pi$
$684$	0	0
$685$	0	0
$686$	−537568.	−0.0436137
$687$	0	0
$688$	615424.	0.0495682
$689$	1.14636e7	0.919965
$690$	0	0
$691$	−5.45034e6	−0.434238	−0.217119	−	0.976145i	$-0.569666\pi$
$691$	−5.45034e6	−0.434238	−0.217119	+	0.976145i	$0.569666\pi$
$692$	5.21965e6	0.414358
$693$	0	0
$694$	−5.24203e6	−0.413144
$695$	0	0
$696$	0	0
$697$	2.87828e7	2.24414
$698$	−1.19564e6	−0.0928885
$699$	0	0
$700$	0	0
$701$	4.43961e6	0.341232	0.170616	−	0.985338i	$-0.445424\pi$
$701$	4.43961e6	0.341232	0.170616	+	0.985338i	$0.445424\pi$
$702$	0	0
$703$	−9.79507e6	−0.747514
$704$	2.04800e6	0.155739
$705$	0	0
$706$	2.95198e6	0.222896
$707$	−671672.	−0.0505369
$708$	0	0
$709$	4.55918e6	0.340621	0.170310	−	0.985390i	$-0.445523\pi$
$709$	4.55918e6	0.340621	0.170310	+	0.985390i	$0.445523\pi$
$710$	0	0
$711$	0	0
$712$	−1.45139e6	−0.107296
$713$	2.30092e7	1.69503
$714$	0	0
$715$	0	0
$716$	1.75226e6	0.127736
$717$	0	0
$718$	9.36298e6	0.677802
$719$	2.06630e7	1.49063	0.745317	−	0.666710i	$-0.232298\pi$
$719$	2.06630e7	1.49063	0.745317	+	0.666710i	$0.232298\pi$
$720$	0	0
$721$	617456.	0.0442352
$722$	−2.67905e6	−0.191266
$723$	0	0
$724$	−850336.	−0.0602898
$725$	0	0
$726$	0	0
$727$	−5.48161e6	−0.384656	−0.192328	−	0.981331i	$-0.561604\pi$
$727$	−5.48161e6	−0.384656	−0.192328	+	0.981331i	$0.561604\pi$
$728$	73728.0	0.00515589
$729$	0	0
$730$	0	0
$731$	3.64446e6	0.252255
$732$	0	0
$733$	8.55579e6	0.588166	0.294083	−	0.955780i	$-0.404986\pi$
$733$	8.55579e6	0.588166	0.294083	+	0.955780i	$0.404986\pi$
$734$	−509168.	−0.0348836
$735$	0	0
$736$	−4.19840e6	−0.285686
$737$	−3.29780e7	−2.23643
$738$	0	0
$739$	−5.29119e6	−0.356404	−0.178202	−	0.983994i	$-0.557028\pi$
$739$	−5.29119e6	−0.356404	−0.178202	+	0.983994i	$0.557028\pi$
$740$	0	0
$741$	0	0
$742$	636864.	0.0424656
$743$	−2.36432e6	−0.157121	−0.0785606	−	0.996909i	$-0.525032\pi$
$743$	−2.36432e6	−0.157121	−0.0785606	+	0.996909i	$0.525032\pi$
$744$	0	0
$745$	0	0
$746$	−1.61548e7	−1.06281
$747$	0	0
$748$	1.21280e7	0.792566
$749$	547152.	0.0356372
$750$	0	0
$751$	−8.79694e6	−0.569157	−0.284578	−	0.958653i	$-0.591854\pi$
$751$	−8.79694e6	−0.569157	−0.284578	+	0.958653i	$0.591854\pi$
$752$	2.27840e6	0.146922
$753$	0	0
$754$	3.04819e6	0.195260
$755$	0	0
$756$	0	0
$757$	−2.95808e7	−1.87616	−0.938079	−	0.346421i	$-0.887397\pi$
$757$	−2.95808e7	−1.87616	−0.938079	+	0.346421i	$0.887397\pi$
$758$	4.04854e6	0.255933
$759$	0	0
$760$	0	0
$761$	1.26296e7	0.790549	0.395274	−	0.918563i	$-0.370649\pi$
$761$	1.26296e7	0.790549	0.395274	+	0.918563i	$0.370649\pi$
$762$	0	0
$763$	−215240.	−0.0133848
$764$	−3.71290e6	−0.230133
$765$	0	0
$766$	−9.50440e6	−0.585265
$767$	8.15040e6	0.500254
$768$	0	0
$769$	−2.32186e7	−1.41586	−0.707929	−	0.706283i	$-0.750370\pi$
$769$	−2.32186e7	−1.41586	−0.707929	+	0.706283i	$0.750370\pi$
$770$	0	0
$771$	0	0
$772$	1.64908e7	0.995858
$773$	−1.73201e7	−1.04256	−0.521280	−	0.853386i	$-0.674545\pi$
$773$	−1.73201e7	−1.04256	−0.521280	+	0.853386i	$0.674545\pi$
$774$	0	0
$775$	0	0
$776$	2.36595e6	0.141043
$777$	0	0
$778$	−5.69990e6	−0.337612
$779$	−2.55172e7	−1.50657
$780$	0	0
$781$	1.44000e7	0.844763
$782$	−2.48624e7	−1.45387
$783$	0	0
$784$	−4.29850e6	−0.249762
$785$	0	0
$786$	0	0
$787$	−556676.	−0.0320380	−0.0160190	−	0.999872i	$-0.505099\pi$
$787$	−556676.	−0.0320380	−0.0160190	+	0.999872i	$0.505099\pi$
$788$	−8.36474e6	−0.479884
$789$	0	0
$790$	0	0
$791$	330768.	0.0187967
$792$	0	0
$793$	5.26752e6	0.297456
$794$	−6.77379e6	−0.381312
$795$	0	0
$796$	−3.44467e6	−0.192693
$797$	3.00562e6	0.167606	0.0838028	−	0.996482i	$-0.473293\pi$
$797$	3.00562e6	0.167606	0.0838028	+	0.996482i	$0.473293\pi$
$798$	0	0
$799$	1.34924e7	0.747691
$800$	0	0
$801$	0	0
$802$	1.13800e7	0.624751
$803$	1.54040e7	0.843033
$804$	0	0
$805$	0	0
$806$	−6.46502e6	−0.350536
$807$	0	0
$808$	−1.07468e7	−0.579094
$809$	−2.23153e6	−0.119876	−0.0599378	−	0.998202i	$-0.519090\pi$
$809$	−2.23153e6	−0.119876	−0.0599378	+	0.998202i	$0.519090\pi$
$810$	0	0
$811$	2.24862e7	1.20051	0.600253	−	0.799810i	$-0.295067\pi$
$811$	2.24862e7	1.20051	0.600253	+	0.799810i	$0.295067\pi$
$812$	169344.	0.00901322
$813$	0	0
$814$	1.45760e7	0.771041
$815$	0	0
$816$	0	0
$817$	−3.23098e6	−0.169347
$818$	−7.56278e6	−0.395183
$819$	0	0
$820$	0	0
$821$	1.65921e7	0.859098	0.429549	−	0.903044i	$-0.358673\pi$
$821$	1.65921e7	0.859098	0.429549	+	0.903044i	$0.358673\pi$
$822$	0	0
$823$	−1.47544e7	−0.759316	−0.379658	−	0.925127i	$-0.623958\pi$
$823$	−1.47544e7	−0.759316	−0.379658	+	0.925127i	$0.623958\pi$
$824$	9.87930e6	0.506883
$825$	0	0
$826$	452800.	0.0230917
$827$	−3.39475e6	−0.172601	−0.0863006	−	0.996269i	$-0.527505\pi$
$827$	−3.39475e6	−0.172601	−0.0863006	+	0.996269i	$0.527505\pi$
$828$	0	0
$829$	−509442.	−0.0257459	−0.0128730	−	0.999917i	$-0.504098\pi$
$829$	−509442.	−0.0257459	−0.0128730	+	0.999917i	$0.504098\pi$
$830$	0	0
$831$	0	0
$832$	1.17965e6	0.0590805
$833$	−2.54552e7	−1.27105
$834$	0	0
$835$	0	0
$836$	−1.07520e7	−0.532076
$837$	0	0
$838$	−1.84372e7	−0.906953
$839$	−4.00609e7	−1.96479	−0.982394	−	0.186819i	$-0.940182\pi$
$839$	−4.00609e7	−1.96479	−0.982394	+	0.186819i	$0.940182\pi$
$840$	0	0
$841$	−1.35098e7	−0.658658
$842$	−2.41660e7	−1.17470
$843$	0	0
$844$	−1.64813e7	−0.796407
$845$	0	0
$846$	0	0
$847$	355796.	0.0170409
$848$	1.01898e7	0.486606
$849$	0	0
$850$	0	0
$851$	−2.98808e7	−1.41439
$852$	0	0
$853$	−9.67506e6	−0.455283	−0.227641	−	0.973745i	$-0.573101\pi$
$853$	−9.67506e6	−0.455283	−0.227641	+	0.973745i	$0.573101\pi$
$854$	292640.	0.0137306
$855$	0	0
$856$	8.75443e6	0.408360
$857$	3.27535e7	1.52337	0.761686	−	0.647946i	$-0.224372\pi$
$857$	3.27535e7	1.52337	0.761686	+	0.647946i	$0.224372\pi$
$858$	0	0
$859$	−2.17420e7	−1.00535	−0.502675	−	0.864476i	$-0.667651\pi$
$859$	−2.17420e7	−1.00535	−0.502675	+	0.864476i	$0.667651\pi$
$860$	0	0
$861$	0	0
$862$	15200.0	0.000696748 0
$863$	2.08744e7	0.954087	0.477043	−	0.878880i	$-0.341708\pi$
$863$	2.08744e7	0.954087	0.477043	+	0.878880i	$0.341708\pi$
$864$	0	0
$865$	0	0
$866$	−1.00294e6	−0.0454446
$867$	0	0
$868$	−359168.	−0.0161807
$869$	3.01140e7	1.35275
$870$	0	0
$871$	−1.89953e7	−0.848401
$872$	−3.44384e6	−0.153374
$873$	0	0
$874$	2.20416e7	0.976033
$875$	0	0
$876$	0	0
$877$	−3.96804e7	−1.74212	−0.871058	−	0.491181i	$-0.836566\pi$
$877$	−3.96804e7	−1.74212	−0.871058	+	0.491181i	$0.836566\pi$
$878$	−1.43549e7	−0.628441
$879$	0	0
$880$	0	0
$881$	−2.60742e7	−1.13180	−0.565902	−	0.824472i	$-0.691472\pi$
$881$	−2.60742e7	−1.13180	−0.565902	+	0.824472i	$0.691472\pi$
$882$	0	0
$883$	−4.10486e7	−1.77172	−0.885862	−	0.463949i	$-0.846432\pi$
$883$	−4.10486e7	−1.77172	−0.885862	+	0.463949i	$0.846432\pi$
$884$	6.98573e6	0.300664
$885$	0	0
$886$	−5.65915e6	−0.242196
$887$	1.37553e7	0.587031	0.293515	−	0.955954i	$-0.405175\pi$
$887$	1.37553e7	0.587031	0.293515	+	0.955954i	$0.405175\pi$
$888$	0	0
$889$	847120.	0.0359493
$890$	0	0
$891$	0	0
$892$	−7.29632e6	−0.307038
$893$	−1.19616e7	−0.501950
$894$	0	0
$895$	0	0
$896$	65536.0	0.00272716
$897$	0	0
$898$	3.31922e6	0.137355
$899$	−1.48494e7	−0.612785
$900$	0	0
$901$	6.03429e7	2.47636
$902$	3.79720e7	1.55399
$903$	0	0
$904$	5.29229e6	0.215388
$905$	0	0
$906$	0	0
$907$	5.86936e6	0.236904	0.118452	−	0.992960i	$-0.462207\pi$
$907$	5.86936e6	0.236904	0.118452	+	0.992960i	$0.462207\pi$
$908$	−6.94803e6	−0.279671
$909$	0	0
$910$	0	0
$911$	−4.63982e7	−1.85227	−0.926137	−	0.377188i	$-0.876891\pi$
$911$	−4.63982e7	−1.85227	−0.926137	+	0.377188i	$0.876891\pi$
$912$	0	0
$913$	−1.23400e6	−0.0489935
$914$	−1.87279e7	−0.741521
$915$	0	0
$916$	−1.15634e7	−0.455350
$917$	−678000.	−0.0266260
$918$	0	0
$919$	2.27859e7	0.889975	0.444988	−	0.895537i	$-0.353208\pi$
$919$	2.27859e7	0.889975	0.444988	+	0.895537i	$0.353208\pi$
$920$	0	0
$921$	0	0
$922$	567720.	0.0219941
$923$	8.29440e6	0.320465
$924$	0	0
$925$	0	0
$926$	−2.90990e6	−0.111520
$927$	0	0
$928$	2.70950e6	0.103281
$929$	−2.70352e7	−1.02775	−0.513877	−	0.857864i	$-0.671791\pi$
$929$	−2.70352e7	−1.02775	−0.513877	+	0.857864i	$0.671791\pi$
$930$	0	0
$931$	2.25671e7	0.853300
$932$	−9.04557e6	−0.341111
$933$	0	0
$934$	−1.79056e7	−0.671616
$935$	0	0
$936$	0	0
$937$	2.86149e7	1.06474	0.532370	−	0.846512i	$-0.321301\pi$
$937$	2.86149e7	1.06474	0.532370	+	0.846512i	$0.321301\pi$
$938$	−1.05530e6	−0.0391622
$939$	0	0
$940$	0	0
$941$	3.67892e7	1.35440	0.677200	−	0.735799i	$-0.263193\pi$
$941$	3.67892e7	1.35440	0.677200	+	0.735799i	$0.263193\pi$
$942$	0	0
$943$	−7.78426e7	−2.85061
$944$	7.24480e6	0.264604
$945$	0	0
$946$	4.80800e6	0.174677
$947$	7.96828e6	0.288728	0.144364	−	0.989525i	$-0.453886\pi$
$947$	7.96828e6	0.288728	0.144364	+	0.989525i	$0.453886\pi$
$948$	0	0
$949$	8.87270e6	0.319809
$950$	0	0
$951$	0	0
$952$	388096.	0.0138786
$953$	−4.82202e7	−1.71987	−0.859937	−	0.510400i	$-0.829497\pi$
$953$	−4.82202e7	−1.71987	−0.859937	+	0.510400i	$0.829497\pi$
$954$	0	0
$955$	0	0
$956$	−5.19846e6	−0.183963
$957$	0	0
$958$	5.30874e6	0.186886
$959$	1.00814e6	0.0353978
$960$	0	0
$961$	2.86539e6	0.100087
$962$	8.39578e6	0.292498
$963$	0	0
$964$	1.46442e7	0.507543
$965$	0	0
$966$	0	0
$967$	−4.83510e7	−1.66280	−0.831398	−	0.555678i	$-0.812459\pi$
$967$	−4.83510e7	−1.66280	−0.831398	+	0.555678i	$0.812459\pi$
$968$	5.69274e6	0.195269
$969$	0	0
$970$	0	0
$971$	4.05515e7	1.38025	0.690127	−	0.723688i	$-0.257555\pi$
$971$	4.05515e7	1.38025	0.690127	+	0.723688i	$0.257555\pi$
$972$	0	0
$973$	768064.	0.0260085
$974$	1.64659e7	0.556144
$975$	0	0
$976$	4.68224e6	0.157336
$977$	−4.34929e7	−1.45775	−0.728874	−	0.684648i	$-0.759956\pi$
$977$	−4.34929e7	−1.45775	−0.728874	+	0.684648i	$0.759956\pi$
$978$	0	0
$979$	−1.13390e7	−0.378110
$980$	0	0
$981$	0	0
$982$	2.44926e7	0.810507
$983$	−3.34896e6	−0.110542	−0.0552709	−	0.998471i	$-0.517602\pi$
$983$	−3.34896e6	−0.110542	−0.0552709	+	0.998471i	$0.517602\pi$
$984$	0	0
$985$	0	0
$986$	1.60453e7	0.525602
$987$	0	0
$988$	−6.19315e6	−0.201846
$989$	−9.85640e6	−0.320426
$990$	0	0
$991$	−5.55726e7	−1.79753	−0.898766	−	0.438429i	$-0.855535\pi$
$991$	−5.55726e7	−1.79753	−0.898766	+	0.438429i	$0.855535\pi$
$992$	−5.74669e6	−0.185412
$993$	0	0
$994$	460800.	0.0147927
$995$	0	0
$996$	0	0
$997$	1.27342e7	0.405726	0.202863	−	0.979207i	$-0.434975\pi$
$997$	1.27342e7	0.405726	0.202863	+	0.979207i	$0.434975\pi$
$998$	−3.16196e7	−1.00492
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.s.1.1		1
3.2	odd	2		150.6.a.f.1.1		1
5.2	odd	4		90.6.c.b.19.2		2
5.3	odd	4		90.6.c.b.19.1		2
5.4	even	2		450.6.a.f.1.1		1
15.2	even	4		30.6.c.a.19.1	✓	2
15.8	even	4		30.6.c.a.19.2	yes	2
15.14	odd	2		150.6.a.j.1.1		1
20.3	even	4		720.6.f.g.289.1		2
20.7	even	4		720.6.f.g.289.2		2
60.23	odd	4		240.6.f.a.49.1		2
60.47	odd	4		240.6.f.a.49.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.6.c.a.19.1	✓	2	15.2	even	4
30.6.c.a.19.2	yes	2	15.8	even	4
90.6.c.b.19.1		2	5.3	odd	4
90.6.c.b.19.2		2	5.2	odd	4
150.6.a.f.1.1		1	3.2	odd	2
150.6.a.j.1.1		1	15.14	odd	2
240.6.f.a.49.1		2	60.23	odd	4
240.6.f.a.49.2		2	60.47	odd	4
450.6.a.f.1.1		1	5.4	even	2
450.6.a.s.1.1		1	1.1	even	1	trivial
720.6.f.g.289.1		2	20.3	even	4
720.6.f.g.289.2		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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