LMFDB - Embedded newform 450.6.a.a.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 150)
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)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} -233.000 q^{7} -64.0000 q^{8} +O(q^{10})$

$q-4.00000 q^{2} +16.0000 q^{4} -233.000 q^{7} -64.0000 q^{8} +498.000 q^{11} -809.000 q^{13} +932.000 q^{14} +256.000 q^{16} -1002.00 q^{17} -1705.00 q^{19} -1992.00 q^{22} +1554.00 q^{23} +3236.00 q^{26} -3728.00 q^{28} -7830.00 q^{29} +977.000 q^{31} -1024.00 q^{32} +4008.00 q^{34} +4822.00 q^{37} +6820.00 q^{38} +8148.00 q^{41} -19469.0 q^{43} +7968.00 q^{44} -6216.00 q^{46} +8418.00 q^{47} +37482.0 q^{49} -12944.0 q^{52} +17664.0 q^{53} +14912.0 q^{56} +31320.0 q^{58} -35910.0 q^{59} +3527.00 q^{61} -3908.00 q^{62} +4096.00 q^{64} -57473.0 q^{67} -16032.0 q^{68} +7548.00 q^{71} +646.000 q^{73} -19288.0 q^{74} -27280.0 q^{76} -116034. q^{77} -22720.0 q^{79} -32592.0 q^{82} +11574.0 q^{83} +77876.0 q^{86} -31872.0 q^{88} +78960.0 q^{89} +188497. q^{91} +24864.0 q^{92} -33672.0 q^{94} -54593.0 q^{97} -149928. q^{98} +O(q^{100})$

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
$7$	−233.000	−1.79726	−0.898630	−	0.438708i	$-0.855436\pi$
$7$	−233.000	−1.79726	−0.898630	+	0.438708i	$0.855436\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	0	0
$11$	498.000	1.24093	0.620465	−	0.784234i	$-0.286944\pi$
$11$	498.000	1.24093	0.620465	+	0.784234i	$0.286944\pi$
$12$	0	0
$13$	−809.000	−1.32767	−0.663835	−	0.747879i	$-0.731072\pi$
$13$	−809.000	−1.32767	−0.663835	+	0.747879i	$0.731072\pi$
$14$	932.000	1.27085
$15$	0	0
$16$	256.000	0.250000
$17$	−1002.00	−0.840902	−0.420451	−	0.907315i	$-0.638128\pi$
$17$	−1002.00	−0.840902	−0.420451	+	0.907315i	$0.638128\pi$
$18$	0	0
$19$	−1705.00	−1.08353	−0.541764	−	0.840530i	$-0.682243\pi$
$19$	−1705.00	−1.08353	−0.541764	+	0.840530i	$0.682243\pi$
$20$	0	0
$21$	0	0
$22$	−1992.00	−0.877471
$23$	1554.00	0.612536	0.306268	−	0.951945i	$-0.400920\pi$
$23$	1554.00	0.612536	0.306268	+	0.951945i	$0.400920\pi$
$24$	0	0
$25$	0	0
$26$	3236.00	0.938804
$27$	0	0
$28$	−3728.00	−0.898630
$29$	−7830.00	−1.72889	−0.864444	−	0.502729i	$-0.832329\pi$
$29$	−7830.00	−1.72889	−0.864444	+	0.502729i	$0.832329\pi$
$30$	0	0
$31$	977.000	0.182596	0.0912978	−	0.995824i	$-0.470898\pi$
$31$	977.000	0.182596	0.0912978	+	0.995824i	$0.470898\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	4008.00	0.594608
$35$	0	0
$36$	0	0
$37$	4822.00	0.579059	0.289530	−	0.957169i	$-0.406501\pi$
$37$	4822.00	0.579059	0.289530	+	0.957169i	$0.406501\pi$
$38$	6820.00	0.766170
$39$	0	0
$40$	0	0
$41$	8148.00	0.756992	0.378496	−	0.925603i	$-0.376441\pi$
$41$	8148.00	0.756992	0.378496	+	0.925603i	$0.376441\pi$
$42$	0	0
$43$	−19469.0	−1.60573	−0.802865	−	0.596161i	$-0.796692\pi$
$43$	−19469.0	−1.60573	−0.802865	+	0.596161i	$0.796692\pi$
$44$	7968.00	0.620465
$45$	0	0
$46$	−6216.00	−0.433128
$47$	8418.00	0.555859	0.277929	−	0.960602i	$-0.410352\pi$
$47$	8418.00	0.555859	0.277929	+	0.960602i	$0.410352\pi$
$48$	0	0
$49$	37482.0	2.23014
$50$	0	0
$51$	0	0
$52$	−12944.0	−0.663835
$53$	17664.0	0.863773	0.431886	−	0.901928i	$-0.357848\pi$
$53$	17664.0	0.863773	0.431886	+	0.901928i	$0.357848\pi$
$54$	0	0
$55$	0	0
$56$	14912.0	0.635427
$57$	0	0
$58$	31320.0	1.22251
$59$	−35910.0	−1.34303	−0.671514	−	0.740991i	$-0.734356\pi$
$59$	−35910.0	−1.34303	−0.671514	+	0.740991i	$0.734356\pi$
$60$	0	0
$61$	3527.00	0.121361	0.0606807	−	0.998157i	$-0.480673\pi$
$61$	3527.00	0.121361	0.0606807	+	0.998157i	$0.480673\pi$
$62$	−3908.00	−0.129115
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	−57473.0	−1.56414	−0.782072	−	0.623188i	$-0.785837\pi$
$67$	−57473.0	−1.56414	−0.782072	+	0.623188i	$0.785837\pi$
$68$	−16032.0	−0.420451
$69$	0	0
$70$	0	0
$71$	7548.00	0.177699	0.0888497	−	0.996045i	$-0.471681\pi$
$71$	7548.00	0.177699	0.0888497	+	0.996045i	$0.471681\pi$
$72$	0	0
$73$	646.000	0.0141881	0.00709407	−	0.999975i	$-0.497742\pi$
$73$	646.000	0.0141881	0.00709407	+	0.999975i	$0.497742\pi$
$74$	−19288.0	−0.409457
$75$	0	0
$76$	−27280.0	−0.541764
$77$	−116034.	−2.23028
$78$	0	0
$79$	−22720.0	−0.409582	−0.204791	−	0.978806i	$-0.565651\pi$
$79$	−22720.0	−0.409582	−0.204791	+	0.978806i	$0.565651\pi$
$80$	0	0
$81$	0	0
$82$	−32592.0	−0.535274
$83$	11574.0	0.184412	0.0922058	−	0.995740i	$-0.470608\pi$
$83$	11574.0	0.184412	0.0922058	+	0.995740i	$0.470608\pi$
$84$	0	0
$85$	0	0
$86$	77876.0	1.13542
$87$	0	0
$88$	−31872.0	−0.438735
$89$	78960.0	1.05665	0.528326	−	0.849041i	$-0.322820\pi$
$89$	78960.0	1.05665	0.528326	+	0.849041i	$0.322820\pi$
$90$	0	0
$91$	188497.	2.38617
$92$	24864.0	0.306268
$93$	0	0
$94$	−33672.0	−0.393051
$95$	0	0
$96$	0	0
$97$	−54593.0	−0.589125	−0.294563	−	0.955632i	$-0.595174\pi$
$97$	−54593.0	−0.589125	−0.294563	+	0.955632i	$0.595174\pi$
$98$	−149928.	−1.57695
$99$	0	0
$100$	0	0
$101$	−105552.	−1.02959	−0.514793	−	0.857314i	$-0.672131\pi$
$101$	−105552.	−1.02959	−0.514793	+	0.857314i	$0.672131\pi$
$102$	0	0
$103$	177436.	1.64797	0.823984	−	0.566613i	$-0.191747\pi$
$103$	177436.	1.64797	0.823984	+	0.566613i	$0.191747\pi$
$104$	51776.0	0.469402
$105$	0	0
$106$	−70656.0	−0.610779
$107$	−183792.	−1.55191	−0.775956	−	0.630787i	$-0.782732\pi$
$107$	−183792.	−1.55191	−0.775956	+	0.630787i	$0.782732\pi$
$108$	0	0
$109$	169685.	1.36797	0.683986	−	0.729495i	$-0.260245\pi$
$109$	169685.	1.36797	0.683986	+	0.729495i	$0.260245\pi$
$110$	0	0
$111$	0	0
$112$	−59648.0	−0.449315
$113$	263484.	1.94115	0.970573	−	0.240808i	$-0.0774123\pi$
$113$	263484.	1.94115	0.970573	+	0.240808i	$0.0774123\pi$
$114$	0	0
$115$	0	0
$116$	−125280.	−0.864444
$117$	0	0
$118$	143640.	0.949665
$119$	233466.	1.51132
$120$	0	0
$121$	86953.0	0.539910
$122$	−14108.0	−0.0858155
$123$	0	0
$124$	15632.0	0.0912978
$125$	0	0
$126$	0	0
$127$	256912.	1.41343	0.706716	−	0.707497i	$-0.250176\pi$
$127$	256912.	1.41343	0.706716	+	0.707497i	$0.250176\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	0	0
$131$	219048.	1.11522	0.557611	−	0.830103i	$-0.311718\pi$
$131$	219048.	1.11522	0.557611	+	0.830103i	$0.311718\pi$
$132$	0	0
$133$	397265.	1.94738
$134$	229892.	1.10602
$135$	0	0
$136$	64128.0	0.297304
$137$	228678.	1.04093	0.520467	−	0.853882i	$-0.325758\pi$
$137$	228678.	1.04093	0.520467	+	0.853882i	$0.325758\pi$
$138$	0	0
$139$	339740.	1.49145	0.745727	−	0.666252i	$-0.232102\pi$
$139$	339740.	1.49145	0.745727	+	0.666252i	$0.232102\pi$
$140$	0	0
$141$	0	0
$142$	−30192.0	−0.125652
$143$	−402882.	−1.64755
$144$	0	0
$145$	0	0
$146$	−2584.00	−0.0100325
$147$	0	0
$148$	77152.0	0.289530
$149$	−306450.	−1.13082	−0.565411	−	0.824810i	$-0.691282\pi$
$149$	−306450.	−1.13082	−0.565411	+	0.824810i	$0.691282\pi$
$150$	0	0
$151$	198827.	0.709632	0.354816	−	0.934936i	$-0.384544\pi$
$151$	198827.	0.709632	0.354816	+	0.934936i	$0.384544\pi$
$152$	109120.	0.383085
$153$	0	0
$154$	464136.	1.57704
$155$	0	0
$156$	0	0
$157$	−361283.	−1.16976	−0.584882	−	0.811118i	$-0.698859\pi$
$157$	−361283.	−1.16976	−0.584882	+	0.811118i	$0.698859\pi$
$158$	90880.0	0.289618
$159$	0	0
$160$	0	0
$161$	−362082.	−1.10089
$162$	0	0
$163$	−338159.	−0.996901	−0.498450	−	0.866918i	$-0.666097\pi$
$163$	−338159.	−0.996901	−0.498450	+	0.866918i	$0.666097\pi$
$164$	130368.	0.378496
$165$	0	0
$166$	−46296.0	−0.130399
$167$	430248.	1.19379	0.596895	−	0.802320i	$-0.296401\pi$
$167$	430248.	1.19379	0.596895	+	0.802320i	$0.296401\pi$
$168$	0	0
$169$	283188.	0.762708
$170$	0	0
$171$	0	0
$172$	−311504.	−0.802865
$173$	603354.	1.53270	0.766350	−	0.642424i	$-0.222071\pi$
$173$	603354.	1.53270	0.766350	+	0.642424i	$0.222071\pi$
$174$	0	0
$175$	0	0
$176$	127488.	0.310233
$177$	0	0
$178$	−315840.	−0.747166
$179$	374370.	0.873310	0.436655	−	0.899629i	$-0.356163\pi$
$179$	374370.	0.873310	0.436655	+	0.899629i	$0.356163\pi$
$180$	0	0
$181$	−232423.	−0.527330	−0.263665	−	0.964614i	$-0.584931\pi$
$181$	−232423.	−0.527330	−0.263665	+	0.964614i	$0.584931\pi$
$182$	−753988.	−1.68728
$183$	0	0
$184$	−99456.0	−0.216564
$185$	0	0
$186$	0	0
$187$	−498996.	−1.04350
$188$	134688.	0.277929
$189$	0	0
$190$	0	0
$191$	846198.	1.67837	0.839187	−	0.543843i	$-0.183031\pi$
$191$	846198.	1.67837	0.839187	+	0.543843i	$0.183031\pi$
$192$	0	0
$193$	155581.	0.300651	0.150326	−	0.988637i	$-0.451968\pi$
$193$	155581.	0.300651	0.150326	+	0.988637i	$0.451968\pi$
$194$	218372.	0.416574
$195$	0	0
$196$	599712.	1.11507
$197$	−103482.	−0.189976	−0.0949881	−	0.995478i	$-0.530281\pi$
$197$	−103482.	−0.189976	−0.0949881	+	0.995478i	$0.530281\pi$
$198$	0	0
$199$	−140425.	−0.251369	−0.125685	−	0.992070i	$-0.540113\pi$
$199$	−140425.	−0.251369	−0.125685	+	0.992070i	$0.540113\pi$
$200$	0	0
$201$	0	0
$202$	422208.	0.728028
$203$	1.82439e6	3.10726
$204$	0	0
$205$	0	0
$206$	−709744.	−1.16529
$207$	0	0
$208$	−207104.	−0.331917
$209$	−849090.	−1.34458
$210$	0	0
$211$	−462673.	−0.715431	−0.357716	−	0.933831i	$-0.616444\pi$
$211$	−462673.	−0.715431	−0.357716	+	0.933831i	$0.616444\pi$
$212$	282624.	0.431886
$213$	0	0
$214$	735168.	1.09737
$215$	0	0
$216$	0	0
$217$	−227641.	−0.328172
$218$	−678740.	−0.967302
$219$	0	0
$220$	0	0
$221$	810618.	1.11644
$222$	0	0
$223$	735271.	0.990114	0.495057	−	0.868860i	$-0.335147\pi$
$223$	735271.	0.990114	0.495057	+	0.868860i	$0.335147\pi$
$224$	238592.	0.317714
$225$	0	0
$226$	−1.05394e6	−1.37260
$227$	967188.	1.24579	0.622897	−	0.782304i	$-0.285956\pi$
$227$	967188.	1.24579	0.622897	+	0.782304i	$0.285956\pi$
$228$	0	0
$229$	−83695.0	−0.105466	−0.0527328	−	0.998609i	$-0.516793\pi$
$229$	−83695.0	−0.105466	−0.0527328	+	0.998609i	$0.516793\pi$
$230$	0	0
$231$	0	0
$232$	501120.	0.611254
$233$	−873876.	−1.05453	−0.527266	−	0.849700i	$-0.676783\pi$
$233$	−873876.	−1.05453	−0.527266	+	0.849700i	$0.676783\pi$
$234$	0	0
$235$	0	0
$236$	−574560.	−0.671514
$237$	0	0
$238$	−933864.	−1.06866
$239$	1.06056e6	1.20099	0.600497	−	0.799627i	$-0.294970\pi$
$239$	1.06056e6	1.20099	0.600497	+	0.799627i	$0.294970\pi$
$240$	0	0
$241$	−756823.	−0.839367	−0.419683	−	0.907671i	$-0.637859\pi$
$241$	−756823.	−0.839367	−0.419683	+	0.907671i	$0.637859\pi$
$242$	−347812.	−0.381774
$243$	0	0
$244$	56432.0	0.0606807
$245$	0	0
$246$	0	0
$247$	1.37934e6	1.43857
$248$	−62528.0	−0.0645573
$249$	0	0
$250$	0	0
$251$	635148.	0.636342	0.318171	−	0.948033i	$-0.396931\pi$
$251$	635148.	0.636342	0.318171	+	0.948033i	$0.396931\pi$
$252$	0	0
$253$	773892.	0.760115
$254$	−1.02765e6	−0.999448
$255$	0	0
$256$	65536.0	0.0625000
$257$	30708.0	0.0290014	0.0145007	−	0.999895i	$-0.495384\pi$
$257$	30708.0	0.0290014	0.0145007	+	0.999895i	$0.495384\pi$
$258$	0	0
$259$	−1.12353e6	−1.04072
$260$	0	0
$261$	0	0
$262$	−876192.	−0.788581
$263$	−189516.	−0.168949	−0.0844747	−	0.996426i	$-0.526921\pi$
$263$	−189516.	−0.168949	−0.0844747	+	0.996426i	$0.526921\pi$
$264$	0	0
$265$	0	0
$266$	−1.58906e6	−1.37701
$267$	0	0
$268$	−919568.	−0.782072
$269$	−1.10997e6	−0.935256	−0.467628	−	0.883925i	$-0.654891\pi$
$269$	−1.10997e6	−0.935256	−0.467628	+	0.883925i	$0.654891\pi$
$270$	0	0
$271$	211952.	0.175313	0.0876565	−	0.996151i	$-0.472062\pi$
$271$	211952.	0.175313	0.0876565	+	0.996151i	$0.472062\pi$
$272$	−256512.	−0.210226
$273$	0	0
$274$	−914712.	−0.736051
$275$	0	0
$276$	0	0
$277$	−1.04741e6	−0.820198	−0.410099	−	0.912041i	$-0.634506\pi$
$277$	−1.04741e6	−0.820198	−0.410099	+	0.912041i	$0.634506\pi$
$278$	−1.35896e6	−1.05462
$279$	0	0
$280$	0	0
$281$	−34002.0	−0.0256885	−0.0128442	−	0.999918i	$-0.504089\pi$
$281$	−34002.0	−0.0256885	−0.0128442	+	0.999918i	$0.504089\pi$
$282$	0	0
$283$	281101.	0.208639	0.104320	−	0.994544i	$-0.466733\pi$
$283$	281101.	0.208639	0.104320	+	0.994544i	$0.466733\pi$
$284$	120768.	0.0888497
$285$	0	0
$286$	1.61153e6	1.16499
$287$	−1.89848e6	−1.36051
$288$	0	0
$289$	−415853.	−0.292884
$290$	0	0
$291$	0	0
$292$	10336.0	0.00709407
$293$	−1.55851e6	−1.06057	−0.530285	−	0.847819i	$-0.677915\pi$
$293$	−1.55851e6	−1.06057	−0.530285	+	0.847819i	$0.677915\pi$
$294$	0	0
$295$	0	0
$296$	−308608.	−0.204728
$297$	0	0
$298$	1.22580e6	0.799611
$299$	−1.25719e6	−0.813245
$300$	0	0
$301$	4.53628e6	2.88591
$302$	−795308.	−0.501785
$303$	0	0
$304$	−436480.	−0.270882
$305$	0	0
$306$	0	0
$307$	839917.	0.508616	0.254308	−	0.967123i	$-0.418152\pi$
$307$	839917.	0.508616	0.254308	+	0.967123i	$0.418152\pi$
$308$	−1.85654e6	−1.11514
$309$	0	0
$310$	0	0
$311$	292698.	0.171601	0.0858003	−	0.996312i	$-0.472655\pi$
$311$	292698.	0.171601	0.0858003	+	0.996312i	$0.472655\pi$
$312$	0	0
$313$	−127859.	−0.0737684	−0.0368842	−	0.999320i	$-0.511743\pi$
$313$	−127859.	−0.0737684	−0.0368842	+	0.999320i	$0.511743\pi$
$314$	1.44513e6	0.827148
$315$	0	0
$316$	−363520.	−0.204791
$317$	648048.	0.362209	0.181104	−	0.983464i	$-0.442033\pi$
$317$	648048.	0.362209	0.181104	+	0.983464i	$0.442033\pi$
$318$	0	0
$319$	−3.89934e6	−2.14543
$320$	0	0
$321$	0	0
$322$	1.44833e6	0.778444
$323$	1.70841e6	0.911141
$324$	0	0
$325$	0	0
$326$	1.35264e6	0.704915
$327$	0	0
$328$	−521472.	−0.267637
$329$	−1.96139e6	−0.999022
$330$	0	0
$331$	−3.39315e6	−1.70229	−0.851144	−	0.524933i	$-0.824090\pi$
$331$	−3.39315e6	−1.70229	−0.851144	+	0.524933i	$0.824090\pi$
$332$	185184.	0.0922058
$333$	0	0
$334$	−1.72099e6	−0.844137
$335$	0	0
$336$	0	0
$337$	1.62085e6	0.777441	0.388720	−	0.921356i	$-0.372917\pi$
$337$	1.62085e6	0.777441	0.388720	+	0.921356i	$0.372917\pi$
$338$	−1.13275e6	−0.539316
$339$	0	0
$340$	0	0
$341$	486546.	0.226589
$342$	0	0
$343$	−4.81728e6	−2.21088
$344$	1.24602e6	0.567711
$345$	0	0
$346$	−2.41342e6	−1.08378
$347$	−1.28638e6	−0.573517	−0.286758	−	0.958003i	$-0.592578\pi$
$347$	−1.28638e6	−0.573517	−0.286758	+	0.958003i	$0.592578\pi$
$348$	0	0
$349$	−2.73055e6	−1.20001	−0.600007	−	0.799994i	$-0.704836\pi$
$349$	−2.73055e6	−1.20001	−0.600007	+	0.799994i	$0.704836\pi$
$350$	0	0
$351$	0	0
$352$	−509952.	−0.219368
$353$	−2.13649e6	−0.912564	−0.456282	−	0.889835i	$-0.650819\pi$
$353$	−2.13649e6	−0.912564	−0.456282	+	0.889835i	$0.650819\pi$
$354$	0	0
$355$	0	0
$356$	1.26336e6	0.528326
$357$	0	0
$358$	−1.49748e6	−0.617523
$359$	−3.26406e6	−1.33666	−0.668332	−	0.743863i	$-0.732991\pi$
$359$	−3.26406e6	−1.33666	−0.668332	+	0.743863i	$0.732991\pi$
$360$	0	0
$361$	430926.	0.174034
$362$	929692.	0.372879
$363$	0	0
$364$	3.01595e6	1.19308
$365$	0	0
$366$	0	0
$367$	4.15078e6	1.60866	0.804330	−	0.594183i	$-0.202524\pi$
$367$	4.15078e6	1.60866	0.804330	+	0.594183i	$0.202524\pi$
$368$	397824.	0.153134
$369$	0	0
$370$	0	0
$371$	−4.11571e6	−1.55242
$372$	0	0
$373$	2.14242e6	0.797320	0.398660	−	0.917099i	$-0.369475\pi$
$373$	2.14242e6	0.797320	0.398660	+	0.917099i	$0.369475\pi$
$374$	1.99598e6	0.737867
$375$	0	0
$376$	−538752.	−0.196526
$377$	6.33447e6	2.29539
$378$	0	0
$379$	−1.94699e6	−0.696253	−0.348126	−	0.937448i	$-0.613182\pi$
$379$	−1.94699e6	−0.696253	−0.348126	+	0.937448i	$0.613182\pi$
$380$	0	0
$381$	0	0
$382$	−3.38479e6	−1.18679
$383$	2.65052e6	0.923283	0.461641	−	0.887067i	$-0.347261\pi$
$383$	2.65052e6	0.923283	0.461641	+	0.887067i	$0.347261\pi$
$384$	0	0
$385$	0	0
$386$	−622324.	−0.212593
$387$	0	0
$388$	−873488.	−0.294563
$389$	−282540.	−0.0946686	−0.0473343	−	0.998879i	$-0.515073\pi$
$389$	−282540.	−0.0946686	−0.0473343	+	0.998879i	$0.515073\pi$
$390$	0	0
$391$	−1.55711e6	−0.515083
$392$	−2.39885e6	−0.788474
$393$	0	0
$394$	413928.	0.134333
$395$	0	0
$396$	0	0
$397$	2.62066e6	0.834515	0.417257	−	0.908788i	$-0.362991\pi$
$397$	2.62066e6	0.834515	0.417257	+	0.908788i	$0.362991\pi$
$398$	561700.	0.177745
$399$	0	0
$400$	0	0
$401$	286248.	0.0888959	0.0444479	−	0.999012i	$-0.485847\pi$
$401$	286248.	0.0888959	0.0444479	+	0.999012i	$0.485847\pi$
$402$	0	0
$403$	−790393.	−0.242427
$404$	−1.68883e6	−0.514793
$405$	0	0
$406$	−7.29756e6	−2.19716
$407$	2.40136e6	0.718572
$408$	0	0
$409$	4.12069e6	1.21804	0.609019	−	0.793155i	$-0.291563\pi$
$409$	4.12069e6	1.21804	0.609019	+	0.793155i	$0.291563\pi$
$410$	0	0
$411$	0	0
$412$	2.83898e6	0.823984
$413$	8.36703e6	2.41377
$414$	0	0
$415$	0	0
$416$	828416.	0.234701
$417$	0	0
$418$	3.39636e6	0.950765
$419$	2.37948e6	0.662136	0.331068	−	0.943607i	$-0.392591\pi$
$419$	2.37948e6	0.662136	0.331068	+	0.943607i	$0.392591\pi$
$420$	0	0
$421$	−741298.	−0.203839	−0.101920	−	0.994793i	$-0.532498\pi$
$421$	−741298.	−0.203839	−0.101920	+	0.994793i	$0.532498\pi$
$422$	1.85069e6	0.505886
$423$	0	0
$424$	−1.13050e6	−0.305390
$425$	0	0
$426$	0	0
$427$	−821791.	−0.218118
$428$	−2.94067e6	−0.775956
$429$	0	0
$430$	0	0
$431$	187398.	0.0485928	0.0242964	−	0.999705i	$-0.492265\pi$
$431$	187398.	0.0485928	0.0242964	+	0.999705i	$0.492265\pi$
$432$	0	0
$433$	−6.55110e6	−1.67917	−0.839585	−	0.543229i	$-0.817202\pi$
$433$	−6.55110e6	−1.67917	−0.839585	+	0.543229i	$0.817202\pi$
$434$	910564.	0.232052
$435$	0	0
$436$	2.71496e6	0.683986
$437$	−2.64957e6	−0.663700
$438$	0	0
$439$	270065.	0.0668817	0.0334408	−	0.999441i	$-0.489353\pi$
$439$	270065.	0.0668817	0.0334408	+	0.999441i	$0.489353\pi$
$440$	0	0
$441$	0	0
$442$	−3.24247e6	−0.789443
$443$	2.77934e6	0.672873	0.336436	−	0.941706i	$-0.390778\pi$
$443$	2.77934e6	0.672873	0.336436	+	0.941706i	$0.390778\pi$
$444$	0	0
$445$	0	0
$446$	−2.94108e6	−0.700116
$447$	0	0
$448$	−954368.	−0.224657
$449$	−7.86630e6	−1.84143	−0.920714	−	0.390238i	$-0.872393\pi$
$449$	−7.86630e6	−1.84143	−0.920714	+	0.390238i	$0.872393\pi$
$450$	0	0
$451$	4.05770e6	0.939375
$452$	4.21574e6	0.970573
$453$	0	0
$454$	−3.86875e6	−0.880909
$455$	0	0
$456$	0	0
$457$	−6.23356e6	−1.39619	−0.698097	−	0.716004i	$-0.745969\pi$
$457$	−6.23356e6	−1.39619	−0.698097	+	0.716004i	$0.745969\pi$
$458$	334780.	0.0745754
$459$	0	0
$460$	0	0
$461$	4.68305e6	1.02630	0.513152	−	0.858298i	$-0.328478\pi$
$461$	4.68305e6	1.02630	0.513152	+	0.858298i	$0.328478\pi$
$462$	0	0
$463$	382816.	0.0829923	0.0414961	−	0.999139i	$-0.486788\pi$
$463$	382816.	0.0829923	0.0414961	+	0.999139i	$0.486788\pi$
$464$	−2.00448e6	−0.432222
$465$	0	0
$466$	3.49550e6	0.745667
$467$	−1.93540e6	−0.410657	−0.205328	−	0.978693i	$-0.565826\pi$
$467$	−1.93540e6	−0.410657	−0.205328	+	0.978693i	$0.565826\pi$
$468$	0	0
$469$	1.33912e7	2.81117
$470$	0	0
$471$	0	0
$472$	2.29824e6	0.474832
$473$	−9.69556e6	−1.99260
$474$	0	0
$475$	0	0
$476$	3.73546e6	0.755660
$477$	0	0
$478$	−4.24224e6	−0.849230
$479$	5.56917e6	1.10905	0.554526	−	0.832167i	$-0.312900\pi$
$479$	5.56917e6	1.10905	0.554526	+	0.832167i	$0.312900\pi$
$480$	0	0
$481$	−3.90100e6	−0.768799
$482$	3.02729e6	0.593522
$483$	0	0
$484$	1.39125e6	0.269955
$485$	0	0
$486$	0	0
$487$	−4.22450e6	−0.807148	−0.403574	−	0.914947i	$-0.632232\pi$
$487$	−4.22450e6	−0.807148	−0.403574	+	0.914947i	$0.632232\pi$
$488$	−225728.	−0.0429078
$489$	0	0
$490$	0	0
$491$	−6.96295e6	−1.30344	−0.651718	−	0.758461i	$-0.725951\pi$
$491$	−6.96295e6	−1.30344	−0.651718	+	0.758461i	$0.725951\pi$
$492$	0	0
$493$	7.84566e6	1.45383
$494$	−5.51738e6	−1.01722
$495$	0	0
$496$	250112.	0.0456489
$497$	−1.75868e6	−0.319372
$498$	0	0
$499$	−9.29582e6	−1.67123	−0.835616	−	0.549315i	$-0.814889\pi$
$499$	−9.29582e6	−1.67123	−0.835616	+	0.549315i	$0.814889\pi$
$500$	0	0
$501$	0	0
$502$	−2.54059e6	−0.449962
$503$	6.34136e6	1.11754	0.558770	−	0.829323i	$-0.311274\pi$
$503$	6.34136e6	1.11754	0.558770	+	0.829323i	$0.311274\pi$
$504$	0	0
$505$	0	0
$506$	−3.09557e6	−0.537482
$507$	0	0
$508$	4.11059e6	0.706716
$509$	7.38309e6	1.26312	0.631559	−	0.775328i	$-0.282415\pi$
$509$	7.38309e6	1.26312	0.631559	+	0.775328i	$0.282415\pi$
$510$	0	0
$511$	−150518.	−0.0254998
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−122832.	−0.0205071
$515$	0	0
$516$	0	0
$517$	4.19216e6	0.689782
$518$	4.49410e6	0.735900
$519$	0	0
$520$	0	0
$521$	−4.19620e6	−0.677270	−0.338635	−	0.940918i	$-0.609965\pi$
$521$	−4.19620e6	−0.677270	−0.338635	+	0.940918i	$0.609965\pi$
$522$	0	0
$523$	3.57942e6	0.572214	0.286107	−	0.958198i	$-0.407639\pi$
$523$	3.57942e6	0.572214	0.286107	+	0.958198i	$0.407639\pi$
$524$	3.50477e6	0.557611
$525$	0	0
$526$	758064.	0.119465
$527$	−978954.	−0.153545
$528$	0	0
$529$	−4.02143e6	−0.624800
$530$	0	0
$531$	0	0
$532$	6.35624e6	0.973691
$533$	−6.59173e6	−1.00504
$534$	0	0
$535$	0	0
$536$	3.67827e6	0.553009
$537$	0	0
$538$	4.43988e6	0.661326
$539$	1.86660e7	2.76745
$540$	0	0
$541$	4.95548e6	0.727934	0.363967	−	0.931412i	$-0.381422\pi$
$541$	4.95548e6	0.727934	0.363967	+	0.931412i	$0.381422\pi$
$542$	−847808.	−0.123965
$543$	0	0
$544$	1.02605e6	0.148652
$545$	0	0
$546$	0	0
$547$	5.31803e6	0.759946	0.379973	−	0.924998i	$-0.375933\pi$
$547$	5.31803e6	0.759946	0.379973	+	0.924998i	$0.375933\pi$
$548$	3.65885e6	0.520467
$549$	0	0
$550$	0	0
$551$	1.33502e7	1.87330
$552$	0	0
$553$	5.29376e6	0.736125
$554$	4.18965e6	0.579967
$555$	0	0
$556$	5.43584e6	0.745727
$557$	−4.25794e6	−0.581516	−0.290758	−	0.956797i	$-0.593907\pi$
$557$	−4.25794e6	−0.581516	−0.290758	+	0.956797i	$0.593907\pi$
$558$	0	0
$559$	1.57504e7	2.13188
$560$	0	0
$561$	0	0
$562$	136008.	0.0181645
$563$	−4.37617e6	−0.581866	−0.290933	−	0.956743i	$-0.593966\pi$
$563$	−4.37617e6	−0.581866	−0.290933	+	0.956743i	$0.593966\pi$
$564$	0	0
$565$	0	0
$566$	−1.12440e6	−0.147530
$567$	0	0
$568$	−483072.	−0.0628262
$569$	−3.57717e6	−0.463190	−0.231595	−	0.972812i	$-0.574394\pi$
$569$	−3.57717e6	−0.463190	−0.231595	+	0.972812i	$0.574394\pi$
$570$	0	0
$571$	1.94693e6	0.249896	0.124948	−	0.992163i	$-0.460124\pi$
$571$	1.94693e6	0.249896	0.124948	+	0.992163i	$0.460124\pi$
$572$	−6.44611e6	−0.823773
$573$	0	0
$574$	7.59394e6	0.962027
$575$	0	0
$576$	0	0
$577$	−6.95576e6	−0.869772	−0.434886	−	0.900486i	$-0.643211\pi$
$577$	−6.95576e6	−0.869772	−0.434886	+	0.900486i	$0.643211\pi$
$578$	1.66341e6	0.207100
$579$	0	0
$580$	0	0
$581$	−2.69674e6	−0.331436
$582$	0	0
$583$	8.79667e6	1.07188
$584$	−41344.0	−0.00501626
$585$	0	0
$586$	6.23402e6	0.749936
$587$	2.41853e6	0.289705	0.144852	−	0.989453i	$-0.453729\pi$
$587$	2.41853e6	0.289705	0.144852	+	0.989453i	$0.453729\pi$
$588$	0	0
$589$	−1.66578e6	−0.197848
$590$	0	0
$591$	0	0
$592$	1.23443e6	0.144765
$593$	−9.58396e6	−1.11920	−0.559600	−	0.828763i	$-0.689045\pi$
$593$	−9.58396e6	−1.11920	−0.559600	+	0.828763i	$0.689045\pi$
$594$	0	0
$595$	0	0
$596$	−4.90320e6	−0.565411
$597$	0	0
$598$	5.02874e6	0.575051
$599$	1.52070e6	0.173172	0.0865858	−	0.996244i	$-0.472404\pi$
$599$	1.52070e6	0.173172	0.0865858	+	0.996244i	$0.472404\pi$
$600$	0	0
$601$	3.88283e6	0.438492	0.219246	−	0.975670i	$-0.429640\pi$
$601$	3.88283e6	0.438492	0.219246	+	0.975670i	$0.429640\pi$
$602$	−1.81451e7	−2.04065
$603$	0	0
$604$	3.18123e6	0.354816
$605$	0	0
$606$	0	0
$607$	107992.	0.0118965	0.00594826	−	0.999982i	$-0.498107\pi$
$607$	107992.	0.0118965	0.00594826	+	0.999982i	$0.498107\pi$
$608$	1.74592e6	0.191543
$609$	0	0
$610$	0	0
$611$	−6.81016e6	−0.737997
$612$	0	0
$613$	−7.49923e6	−0.806057	−0.403028	−	0.915187i	$-0.632042\pi$
$613$	−7.49923e6	−0.806057	−0.403028	+	0.915187i	$0.632042\pi$
$614$	−3.35967e6	−0.359646
$615$	0	0
$616$	7.42618e6	0.788521
$617$	−1.22695e6	−0.129752	−0.0648761	−	0.997893i	$-0.520665\pi$
$617$	−1.22695e6	−0.129752	−0.0648761	+	0.997893i	$0.520665\pi$
$618$	0	0
$619$	9.51340e6	0.997950	0.498975	−	0.866616i	$-0.333710\pi$
$619$	9.51340e6	0.997950	0.498975	+	0.866616i	$0.333710\pi$
$620$	0	0
$621$	0	0
$622$	−1.17079e6	−0.121340
$623$	−1.83977e7	−1.89908
$624$	0	0
$625$	0	0
$626$	511436.	0.0521621
$627$	0	0
$628$	−5.78053e6	−0.584882
$629$	−4.83164e6	−0.486932
$630$	0	0
$631$	−1.56449e7	−1.56423	−0.782114	−	0.623135i	$-0.785859\pi$
$631$	−1.56449e7	−1.56423	−0.782114	+	0.623135i	$0.785859\pi$
$632$	1.45408e6	0.144809
$633$	0	0
$634$	−2.59219e6	−0.256120
$635$	0	0
$636$	0	0
$637$	−3.03229e7	−2.96089
$638$	1.55974e7	1.51705
$639$	0	0
$640$	0	0
$641$	9.60395e6	0.923219	0.461609	−	0.887083i	$-0.347272\pi$
$641$	9.60395e6	0.923219	0.461609	+	0.887083i	$0.347272\pi$
$642$	0	0
$643$	8.24396e6	0.786336	0.393168	−	0.919467i	$-0.371379\pi$
$643$	8.24396e6	0.786336	0.393168	+	0.919467i	$0.371379\pi$
$644$	−5.79331e6	−0.550443
$645$	0	0
$646$	−6.83364e6	−0.644274
$647$	−4.07353e6	−0.382570	−0.191285	−	0.981535i	$-0.561265\pi$
$647$	−4.07353e6	−0.382570	−0.191285	+	0.981535i	$0.561265\pi$
$648$	0	0
$649$	−1.78832e7	−1.66661
$650$	0	0
$651$	0	0
$652$	−5.41054e6	−0.498450
$653$	1.68193e7	1.54357	0.771783	−	0.635886i	$-0.219365\pi$
$653$	1.68193e7	1.54357	0.771783	+	0.635886i	$0.219365\pi$
$654$	0	0
$655$	0	0
$656$	2.08589e6	0.189248
$657$	0	0
$658$	7.84558e6	0.706415
$659$	−2.87826e6	−0.258176	−0.129088	−	0.991633i	$-0.541205\pi$
$659$	−2.87826e6	−0.258176	−0.129088	+	0.991633i	$0.541205\pi$
$660$	0	0
$661$	1.33386e7	1.18743	0.593713	−	0.804677i	$-0.297661\pi$
$661$	1.33386e7	1.18743	0.593713	+	0.804677i	$0.297661\pi$
$662$	1.35726e7	1.20370
$663$	0	0
$664$	−740736.	−0.0651993
$665$	0	0
$666$	0	0
$667$	−1.21678e7	−1.05901
$668$	6.88397e6	0.596895
$669$	0	0
$670$	0	0
$671$	1.75645e6	0.150601
$672$	0	0
$673$	−6.37345e6	−0.542422	−0.271211	−	0.962520i	$-0.587424\pi$
$673$	−6.37345e6	−0.542422	−0.271211	+	0.962520i	$0.587424\pi$
$674$	−6.48339e6	−0.549734
$675$	0	0
$676$	4.53101e6	0.381354
$677$	−2.27210e7	−1.90526	−0.952632	−	0.304126i	$-0.901636\pi$
$677$	−2.27210e7	−1.90526	−0.952632	+	0.304126i	$0.901636\pi$
$678$	0	0
$679$	1.27202e7	1.05881
$680$	0	0
$681$	0	0
$682$	−1.94618e6	−0.160222
$683$	−1.53612e7	−1.26001	−0.630003	−	0.776593i	$-0.716946\pi$
$683$	−1.53612e7	−1.26001	−0.630003	+	0.776593i	$0.716946\pi$
$684$	0	0
$685$	0	0
$686$	1.92691e7	1.56333
$687$	0	0
$688$	−4.98406e6	−0.401432
$689$	−1.42902e7	−1.14680
$690$	0	0
$691$	8.05035e6	0.641386	0.320693	−	0.947183i	$-0.396084\pi$
$691$	8.05035e6	0.641386	0.320693	+	0.947183i	$0.396084\pi$
$692$	9.65366e6	0.766350
$693$	0	0
$694$	5.14553e6	0.405538
$695$	0	0
$696$	0	0
$697$	−8.16430e6	−0.636556
$698$	1.09222e7	0.848539
$699$	0	0
$700$	0	0
$701$	−7.27840e6	−0.559424	−0.279712	−	0.960084i	$-0.590239\pi$
$701$	−7.27840e6	−0.559424	−0.279712	+	0.960084i	$0.590239\pi$
$702$	0	0
$703$	−8.22151e6	−0.627427
$704$	2.03981e6	0.155116
$705$	0	0
$706$	8.54594e6	0.645280
$707$	2.45936e7	1.85044
$708$	0	0
$709$	1.37498e7	1.02726	0.513630	−	0.858012i	$-0.328300\pi$
$709$	1.37498e7	1.02726	0.513630	+	0.858012i	$0.328300\pi$
$710$	0	0
$711$	0	0
$712$	−5.05344e6	−0.373583
$713$	1.51826e6	0.111846
$714$	0	0
$715$	0	0
$716$	5.98992e6	0.436655
$717$	0	0
$718$	1.30562e7	0.945164
$719$	9.05583e6	0.653290	0.326645	−	0.945147i	$-0.394082\pi$
$719$	9.05583e6	0.653290	0.326645	+	0.945147i	$0.394082\pi$
$720$	0	0
$721$	−4.13426e7	−2.96183
$722$	−1.72370e6	−0.123061
$723$	0	0
$724$	−3.71877e6	−0.263665
$725$	0	0
$726$	0	0
$727$	2.20979e7	1.55065	0.775327	−	0.631560i	$-0.217585\pi$
$727$	2.20979e7	1.55065	0.775327	+	0.631560i	$0.217585\pi$
$728$	−1.20638e7	−0.843638
$729$	0	0
$730$	0	0
$731$	1.95079e7	1.35026
$732$	0	0
$733$	−2.07337e6	−0.142534	−0.0712669	−	0.997457i	$-0.522704\pi$
$733$	−2.07337e6	−0.142534	−0.0712669	+	0.997457i	$0.522704\pi$
$734$	−1.66031e7	−1.13749
$735$	0	0
$736$	−1.59130e6	−0.108282
$737$	−2.86216e7	−1.94100
$738$	0	0
$739$	1.48849e7	1.00262	0.501310	−	0.865268i	$-0.332852\pi$
$739$	1.48849e7	1.00262	0.501310	+	0.865268i	$0.332852\pi$
$740$	0	0
$741$	0	0
$742$	1.64628e7	1.09773
$743$	2.58856e7	1.72023	0.860116	−	0.510099i	$-0.170391\pi$
$743$	2.58856e7	1.72023	0.860116	+	0.510099i	$0.170391\pi$
$744$	0	0
$745$	0	0
$746$	−8.56968e6	−0.563790
$747$	0	0
$748$	−7.98394e6	−0.521751
$749$	4.28235e7	2.78919
$750$	0	0
$751$	−7.98645e6	−0.516718	−0.258359	−	0.966049i	$-0.583182\pi$
$751$	−7.98645e6	−0.516718	−0.258359	+	0.966049i	$0.583182\pi$
$752$	2.15501e6	0.138965
$753$	0	0
$754$	−2.53379e7	−1.62309
$755$	0	0
$756$	0	0
$757$	1.15570e7	0.733000	0.366500	−	0.930418i	$-0.380556\pi$
$757$	1.15570e7	0.733000	0.366500	+	0.930418i	$0.380556\pi$
$758$	7.78798e6	0.492325
$759$	0	0
$760$	0	0
$761$	−3.16705e6	−0.198241	−0.0991205	−	0.995075i	$-0.531603\pi$
$761$	−3.16705e6	−0.198241	−0.0991205	+	0.995075i	$0.531603\pi$
$762$	0	0
$763$	−3.95366e7	−2.45860
$764$	1.35392e7	0.839187
$765$	0	0
$766$	−1.06021e7	−0.652860
$767$	2.90512e7	1.78310
$768$	0	0
$769$	8.21560e6	0.500983	0.250492	−	0.968119i	$-0.419408\pi$
$769$	8.21560e6	0.500983	0.250492	+	0.968119i	$0.419408\pi$
$770$	0	0
$771$	0	0
$772$	2.48930e6	0.150326
$773$	1.19708e7	0.720567	0.360284	−	0.932843i	$-0.382680\pi$
$773$	1.19708e7	0.720567	0.360284	+	0.932843i	$0.382680\pi$
$774$	0	0
$775$	0	0
$776$	3.49395e6	0.208287
$777$	0	0
$778$	1.13016e6	0.0669408
$779$	−1.38923e7	−0.820223
$780$	0	0
$781$	3.75890e6	0.220513
$782$	6.22843e6	0.364218
$783$	0	0
$784$	9.59539e6	0.557536
$785$	0	0
$786$	0	0
$787$	1.71154e7	0.985032	0.492516	−	0.870304i	$-0.336077\pi$
$787$	1.71154e7	0.985032	0.492516	+	0.870304i	$0.336077\pi$
$788$	−1.65571e6	−0.0949881
$789$	0	0
$790$	0	0
$791$	−6.13918e7	−3.48874
$792$	0	0
$793$	−2.85334e6	−0.161128
$794$	−1.04826e7	−0.590091
$795$	0	0
$796$	−2.24680e6	−0.125685
$797$	−2.80753e7	−1.56559	−0.782797	−	0.622277i	$-0.786208\pi$
$797$	−2.80753e7	−1.56559	−0.782797	+	0.622277i	$0.786208\pi$
$798$	0	0
$799$	−8.43484e6	−0.467423
$800$	0	0
$801$	0	0
$802$	−1.14499e6	−0.0628589
$803$	321708.	0.0176065
$804$	0	0
$805$	0	0
$806$	3.16157e6	0.171422
$807$	0	0
$808$	6.75533e6	0.364014
$809$	−9.90816e6	−0.532257	−0.266129	−	0.963938i	$-0.585745\pi$
$809$	−9.90816e6	−0.532257	−0.266129	+	0.963938i	$0.585745\pi$
$810$	0	0
$811$	5.72573e6	0.305688	0.152844	−	0.988250i	$-0.451157\pi$
$811$	5.72573e6	0.305688	0.152844	+	0.988250i	$0.451157\pi$
$812$	2.91902e7	1.55363
$813$	0	0
$814$	−9.60542e6	−0.508107
$815$	0	0
$816$	0	0
$817$	3.31946e7	1.73985
$818$	−1.64827e7	−0.861284
$819$	0	0
$820$	0	0
$821$	5.96570e6	0.308890	0.154445	−	0.988001i	$-0.450641\pi$
$821$	5.96570e6	0.308890	0.154445	+	0.988001i	$0.450641\pi$
$822$	0	0
$823$	1.97778e7	1.01784	0.508918	−	0.860815i	$-0.330046\pi$
$823$	1.97778e7	1.01784	0.508918	+	0.860815i	$0.330046\pi$
$824$	−1.13559e7	−0.582645
$825$	0	0
$826$	−3.34681e7	−1.70679
$827$	2.04045e7	1.03744	0.518719	−	0.854945i	$-0.326409\pi$
$827$	2.04045e7	1.03744	0.518719	+	0.854945i	$0.326409\pi$
$828$	0	0
$829$	−2.77183e7	−1.40081	−0.700406	−	0.713745i	$-0.746998\pi$
$829$	−2.77183e7	−1.40081	−0.700406	+	0.713745i	$0.746998\pi$
$830$	0	0
$831$	0	0
$832$	−3.31366e6	−0.165959
$833$	−3.75570e7	−1.87533
$834$	0	0
$835$	0	0
$836$	−1.35854e7	−0.672292
$837$	0	0
$838$	−9.51792e6	−0.468201
$839$	7.79841e6	0.382473	0.191237	−	0.981544i	$-0.438750\pi$
$839$	7.79841e6	0.382473	0.191237	+	0.981544i	$0.438750\pi$
$840$	0	0
$841$	4.07978e7	1.98905
$842$	2.96519e6	0.144136
$843$	0	0
$844$	−7.40277e6	−0.357716
$845$	0	0
$846$	0	0
$847$	−2.02600e7	−0.970358
$848$	4.52198e6	0.215943
$849$	0	0
$850$	0	0
$851$	7.49339e6	0.354694
$852$	0	0
$853$	−6.22554e6	−0.292957	−0.146479	−	0.989214i	$-0.546794\pi$
$853$	−6.22554e6	−0.292957	−0.146479	+	0.989214i	$0.546794\pi$
$854$	3.28716e6	0.154233
$855$	0	0
$856$	1.17627e7	0.548684
$857$	3.39757e7	1.58022	0.790108	−	0.612968i	$-0.210024\pi$
$857$	3.39757e7	1.58022	0.790108	+	0.612968i	$0.210024\pi$
$858$	0	0
$859$	−1.47282e7	−0.681033	−0.340516	−	0.940239i	$-0.610602\pi$
$859$	−1.47282e7	−0.681033	−0.340516	+	0.940239i	$0.610602\pi$
$860$	0	0
$861$	0	0
$862$	−749592.	−0.0343603
$863$	−1.88594e7	−0.861988	−0.430994	−	0.902355i	$-0.641837\pi$
$863$	−1.88594e7	−0.861988	−0.430994	+	0.902355i	$0.641837\pi$
$864$	0	0
$865$	0	0
$866$	2.62044e7	1.18735
$867$	0	0
$868$	−3.64226e6	−0.164086
$869$	−1.13146e7	−0.508263
$870$	0	0
$871$	4.64957e7	2.07667
$872$	−1.08598e7	−0.483651
$873$	0	0
$874$	1.05983e7	0.469307
$875$	0	0
$876$	0	0
$877$	1.82112e7	0.799540	0.399770	−	0.916615i	$-0.369090\pi$
$877$	1.82112e7	0.799540	0.399770	+	0.916615i	$0.369090\pi$
$878$	−1.08026e6	−0.0472925
$879$	0	0
$880$	0	0
$881$	7.65425e6	0.332248	0.166124	−	0.986105i	$-0.446875\pi$
$881$	7.65425e6	0.332248	0.166124	+	0.986105i	$0.446875\pi$
$882$	0	0
$883$	597451.	0.0257870	0.0128935	−	0.999917i	$-0.495896\pi$
$883$	597451.	0.0257870	0.0128935	+	0.999917i	$0.495896\pi$
$884$	1.29699e7	0.558220
$885$	0	0
$886$	−1.11174e7	−0.475793
$887$	2.06888e6	0.0882929	0.0441465	−	0.999025i	$-0.485943\pi$
$887$	2.06888e6	0.0882929	0.0441465	+	0.999025i	$0.485943\pi$
$888$	0	0
$889$	−5.98605e7	−2.54031
$890$	0	0
$891$	0	0
$892$	1.17643e7	0.495057
$893$	−1.43527e7	−0.602289
$894$	0	0
$895$	0	0
$896$	3.81747e6	0.158857
$897$	0	0
$898$	3.14652e7	1.30209
$899$	−7.64991e6	−0.315687
$900$	0	0
$901$	−1.76993e7	−0.726348
$902$	−1.62308e7	−0.664238
$903$	0	0
$904$	−1.68630e7	−0.686299
$905$	0	0
$906$	0	0
$907$	−7.83331e6	−0.316175	−0.158087	−	0.987425i	$-0.550533\pi$
$907$	−7.83331e6	−0.316175	−0.158087	+	0.987425i	$0.550533\pi$
$908$	1.54750e7	0.622897
$909$	0	0
$910$	0	0
$911$	−4.08133e7	−1.62932	−0.814659	−	0.579940i	$-0.803076\pi$
$911$	−4.08133e7	−1.62932	−0.814659	+	0.579940i	$0.803076\pi$
$912$	0	0
$913$	5.76385e6	0.228842
$914$	2.49342e7	0.987258
$915$	0	0
$916$	−1.33912e6	−0.0527328
$917$	−5.10382e7	−2.00434
$918$	0	0
$919$	6.21579e6	0.242777	0.121389	−	0.992605i	$-0.461265\pi$
$919$	6.21579e6	0.242777	0.121389	+	0.992605i	$0.461265\pi$
$920$	0	0
$921$	0	0
$922$	−1.87322e7	−0.725707
$923$	−6.10633e6	−0.235926
$924$	0	0
$925$	0	0
$926$	−1.53126e6	−0.0586844
$927$	0	0
$928$	8.01792e6	0.305627
$929$	4.64847e6	0.176714	0.0883570	−	0.996089i	$-0.471838\pi$
$929$	4.64847e6	0.176714	0.0883570	+	0.996089i	$0.471838\pi$
$930$	0	0
$931$	−6.39068e7	−2.41642
$932$	−1.39820e7	−0.527266
$933$	0	0
$934$	7.74161e6	0.290378
$935$	0	0
$936$	0	0
$937$	1.33603e7	0.497127	0.248563	−	0.968616i	$-0.420042\pi$
$937$	1.33603e7	0.497127	0.248563	+	0.968616i	$0.420042\pi$
$938$	−5.35648e7	−1.98780
$939$	0	0
$940$	0	0
$941$	−3.32569e7	−1.22436	−0.612178	−	0.790720i	$-0.709706\pi$
$941$	−3.32569e7	−1.22436	−0.612178	+	0.790720i	$0.709706\pi$
$942$	0	0
$943$	1.26620e7	0.463685
$944$	−9.19296e6	−0.335757
$945$	0	0
$946$	3.87822e7	1.40898
$947$	4.21530e7	1.52740	0.763702	−	0.645569i	$-0.223380\pi$
$947$	4.21530e7	1.52740	0.763702	+	0.645569i	$0.223380\pi$
$948$	0	0
$949$	−522614.	−0.0188372
$950$	0	0
$951$	0	0
$952$	−1.49418e7	−0.534332
$953$	2.24691e7	0.801406	0.400703	−	0.916208i	$-0.368766\pi$
$953$	2.24691e7	0.801406	0.400703	+	0.916208i	$0.368766\pi$
$954$	0	0
$955$	0	0
$956$	1.69690e7	0.600497
$957$	0	0
$958$	−2.22767e7	−0.784218
$959$	−5.32820e7	−1.87083
$960$	0	0
$961$	−2.76746e7	−0.966659
$962$	1.56040e7	0.543623
$963$	0	0
$964$	−1.21092e7	−0.419683
$965$	0	0
$966$	0	0
$967$	1.75000e7	0.601826	0.300913	−	0.953652i	$-0.402709\pi$
$967$	1.75000e7	0.601826	0.300913	+	0.953652i	$0.402709\pi$
$968$	−5.56499e6	−0.190887
$969$	0	0
$970$	0	0
$971$	−5.42920e7	−1.84794	−0.923970	−	0.382465i	$-0.875075\pi$
$971$	−5.42920e7	−1.84794	−0.923970	+	0.382465i	$0.875075\pi$
$972$	0	0
$973$	−7.91594e7	−2.68053
$974$	1.68980e7	0.570740
$975$	0	0
$976$	902912.	0.0303404
$977$	2.55925e7	0.857782	0.428891	−	0.903356i	$-0.358904\pi$
$977$	2.55925e7	0.857782	0.428891	+	0.903356i	$0.358904\pi$
$978$	0	0
$979$	3.93221e7	1.31123
$980$	0	0
$981$	0	0
$982$	2.78518e7	0.921668
$983$	−4.82488e6	−0.159258	−0.0796292	−	0.996825i	$-0.525374\pi$
$983$	−4.82488e6	−0.159258	−0.0796292	+	0.996825i	$0.525374\pi$
$984$	0	0
$985$	0	0
$986$	−3.13826e7	−1.02801
$987$	0	0
$988$	2.20695e7	0.719284
$989$	−3.02548e7	−0.983567
$990$	0	0
$991$	1.18448e6	0.0383127	0.0191563	−	0.999817i	$-0.493902\pi$
$991$	1.18448e6	0.0383127	0.0191563	+	0.999817i	$0.493902\pi$
$992$	−1.00045e6	−0.0322786
$993$	0	0
$994$	7.03474e6	0.225830
$995$	0	0
$996$	0	0
$997$	2.58178e7	0.822585	0.411293	−	0.911503i	$-0.365077\pi$
$997$	2.58178e7	0.822585	0.411293	+	0.911503i	$0.365077\pi$
$998$	3.71833e7	1.18174
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.a.1.1		1
3.2	odd	2		150.6.a.l.1.1	yes	1
5.2	odd	4		450.6.c.n.199.1		2
5.3	odd	4		450.6.c.n.199.2		2
5.4	even	2		450.6.a.x.1.1		1
15.2	even	4		150.6.c.e.49.2		2
15.8	even	4		150.6.c.e.49.1		2
15.14	odd	2		150.6.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.a.c.1.1	✓	1	15.14	odd	2
150.6.a.l.1.1	yes	1	3.2	odd	2
150.6.c.e.49.1		2	15.8	even	4
150.6.c.e.49.2		2	15.2	even	4
450.6.a.a.1.1		1	1.1	even	1	trivial
450.6.a.x.1.1		1	5.4	even	2
450.6.c.n.199.1		2	5.2	odd	4
450.6.c.n.199.2		2	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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