LMFDB - Embedded newform 450.6.a.b.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)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} -164.000 q^{7} -64.0000 q^{8} +O(q^{10})$

$q-4.00000 q^{2} +16.0000 q^{4} -164.000 q^{7} -64.0000 q^{8} -720.000 q^{11} -698.000 q^{13} +656.000 q^{14} +256.000 q^{16} -2226.00 q^{17} +356.000 q^{19} +2880.00 q^{22} -1800.00 q^{23} +2792.00 q^{26} -2624.00 q^{28} -714.000 q^{29} +848.000 q^{31} -1024.00 q^{32} +8904.00 q^{34} +11302.0 q^{37} -1424.00 q^{38} -9354.00 q^{41} +5956.00 q^{43} -11520.0 q^{44} +7200.00 q^{46} -11160.0 q^{47} +10089.0 q^{49} -11168.0 q^{52} +14106.0 q^{53} +10496.0 q^{56} +2856.00 q^{58} -7920.00 q^{59} -13450.0 q^{61} -3392.00 q^{62} +4096.00 q^{64} +65476.0 q^{67} -35616.0 q^{68} -34560.0 q^{71} -86258.0 q^{73} -45208.0 q^{74} +5696.00 q^{76} +118080. q^{77} -108832. q^{79} +37416.0 q^{82} +10668.0 q^{83} -23824.0 q^{86} +46080.0 q^{88} -10818.0 q^{89} +114472. q^{91} -28800.0 q^{92} +44640.0 q^{94} -4418.00 q^{97} -40356.0 q^{98} +O(q^{100})$

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
$7$	−164.000	−1.26502	−0.632512	−	0.774551i	$-0.717976\pi$
$7$	−164.000	−1.26502	−0.632512	+	0.774551i	$0.717976\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	0	0
$11$	−720.000	−1.79412	−0.897059	−	0.441912i	$-0.854300\pi$
$11$	−720.000	−1.79412	−0.897059	+	0.441912i	$0.854300\pi$
$12$	0	0
$13$	−698.000	−1.14551	−0.572753	−	0.819728i	$-0.694124\pi$
$13$	−698.000	−1.14551	−0.572753	+	0.819728i	$0.694124\pi$
$14$	656.000	0.894507
$15$	0	0
$16$	256.000	0.250000
$17$	−2226.00	−1.86811	−0.934056	−	0.357127i	$-0.883756\pi$
$17$	−2226.00	−1.86811	−0.934056	+	0.357127i	$0.883756\pi$
$18$	0	0
$19$	356.000	0.226238	0.113119	−	0.993581i	$-0.463916\pi$
$19$	356.000	0.226238	0.113119	+	0.993581i	$0.463916\pi$
$20$	0	0
$21$	0	0
$22$	2880.00	1.26863
$23$	−1800.00	−0.709501	−0.354750	−	0.934961i	$-0.615434\pi$
$23$	−1800.00	−0.709501	−0.354750	+	0.934961i	$0.615434\pi$
$24$	0	0
$25$	0	0
$26$	2792.00	0.809994
$27$	0	0
$28$	−2624.00	−0.632512
$29$	−714.000	−0.157653	−0.0788267	−	0.996888i	$-0.525117\pi$
$29$	−714.000	−0.157653	−0.0788267	+	0.996888i	$0.525117\pi$
$30$	0	0
$31$	848.000	0.158486	0.0792431	−	0.996855i	$-0.474750\pi$
$31$	848.000	0.158486	0.0792431	+	0.996855i	$0.474750\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	8904.00	1.32095
$35$	0	0
$36$	0	0
$37$	11302.0	1.35722	0.678611	−	0.734498i	$-0.262582\pi$
$37$	11302.0	1.35722	0.678611	+	0.734498i	$0.262582\pi$
$38$	−1424.00	−0.159975
$39$	0	0
$40$	0	0
$41$	−9354.00	−0.869036	−0.434518	−	0.900663i	$-0.643081\pi$
$41$	−9354.00	−0.869036	−0.434518	+	0.900663i	$0.643081\pi$
$42$	0	0
$43$	5956.00	0.491228	0.245614	−	0.969368i	$-0.421010\pi$
$43$	5956.00	0.491228	0.245614	+	0.969368i	$0.421010\pi$
$44$	−11520.0	−0.897059
$45$	0	0
$46$	7200.00	0.501693
$47$	−11160.0	−0.736919	−0.368459	−	0.929644i	$-0.620115\pi$
$47$	−11160.0	−0.736919	−0.368459	+	0.929644i	$0.620115\pi$
$48$	0	0
$49$	10089.0	0.600286
$50$	0	0
$51$	0	0
$52$	−11168.0	−0.572753
$53$	14106.0	0.689786	0.344893	−	0.938642i	$-0.387915\pi$
$53$	14106.0	0.689786	0.344893	+	0.938642i	$0.387915\pi$
$54$	0	0
$55$	0	0
$56$	10496.0	0.447254
$57$	0	0
$58$	2856.00	0.111478
$59$	−7920.00	−0.296207	−0.148103	−	0.988972i	$-0.547317\pi$
$59$	−7920.00	−0.296207	−0.148103	+	0.988972i	$0.547317\pi$
$60$	0	0
$61$	−13450.0	−0.462805	−0.231402	−	0.972858i	$-0.574331\pi$
$61$	−13450.0	−0.462805	−0.231402	+	0.972858i	$0.574331\pi$
$62$	−3392.00	−0.112067
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	65476.0	1.78195	0.890974	−	0.454054i	$-0.150023\pi$
$67$	65476.0	1.78195	0.890974	+	0.454054i	$0.150023\pi$
$68$	−35616.0	−0.934056
$69$	0	0
$70$	0	0
$71$	−34560.0	−0.813632	−0.406816	−	0.913510i	$-0.633361\pi$
$71$	−34560.0	−0.813632	−0.406816	+	0.913510i	$0.633361\pi$
$72$	0	0
$73$	−86258.0	−1.89449	−0.947245	−	0.320511i	$-0.896145\pi$
$73$	−86258.0	−1.89449	−0.947245	+	0.320511i	$0.896145\pi$
$74$	−45208.0	−0.959701
$75$	0	0
$76$	5696.00	0.113119
$77$	118080.	2.26960
$78$	0	0
$79$	−108832.	−1.96195	−0.980977	−	0.194123i	$-0.937814\pi$
$79$	−108832.	−1.96195	−0.980977	+	0.194123i	$0.937814\pi$
$80$	0	0
$81$	0	0
$82$	37416.0	0.614501
$83$	10668.0	0.169976	0.0849880	−	0.996382i	$-0.472915\pi$
$83$	10668.0	0.169976	0.0849880	+	0.996382i	$0.472915\pi$
$84$	0	0
$85$	0	0
$86$	−23824.0	−0.347351
$87$	0	0
$88$	46080.0	0.634316
$89$	−10818.0	−0.144768	−0.0723839	−	0.997377i	$-0.523061\pi$
$89$	−10818.0	−0.144768	−0.0723839	+	0.997377i	$0.523061\pi$
$90$	0	0
$91$	114472.	1.44909
$92$	−28800.0	−0.354750
$93$	0	0
$94$	44640.0	0.521080
$95$	0	0
$96$	0	0
$97$	−4418.00	−0.0476756	−0.0238378	−	0.999716i	$-0.507589\pi$
$97$	−4418.00	−0.0476756	−0.0238378	+	0.999716i	$0.507589\pi$
$98$	−40356.0	−0.424466
$99$	0	0
$100$	0	0
$101$	102942.	1.00413	0.502064	−	0.864830i	$-0.332574\pi$
$101$	102942.	1.00413	0.502064	+	0.864830i	$0.332574\pi$
$102$	0	0
$103$	69436.0	0.644899	0.322449	−	0.946587i	$-0.395494\pi$
$103$	69436.0	0.644899	0.322449	+	0.946587i	$0.395494\pi$
$104$	44672.0	0.404997
$105$	0	0
$106$	−56424.0	−0.487752
$107$	17412.0	0.147024	0.0735122	−	0.997294i	$-0.476579\pi$
$107$	17412.0	0.147024	0.0735122	+	0.997294i	$0.476579\pi$
$108$	0	0
$109$	−203770.	−1.64276	−0.821380	−	0.570382i	$-0.806795\pi$
$109$	−203770.	−1.64276	−0.821380	+	0.570382i	$0.806795\pi$
$110$	0	0
$111$	0	0
$112$	−41984.0	−0.316256
$113$	−212202.	−1.56334	−0.781670	−	0.623692i	$-0.785632\pi$
$113$	−212202.	−1.56334	−0.781670	+	0.623692i	$0.785632\pi$
$114$	0	0
$115$	0	0
$116$	−11424.0	−0.0788267
$117$	0	0
$118$	31680.0	0.209450
$119$	365064.	2.36321
$120$	0	0
$121$	357349.	2.21886
$122$	53800.0	0.327252
$123$	0	0
$124$	13568.0	0.0792431
$125$	0	0
$126$	0	0
$127$	−6140.00	−0.0337800	−0.0168900	−	0.999857i	$-0.505377\pi$
$127$	−6140.00	−0.0337800	−0.0168900	+	0.999857i	$0.505377\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	0	0
$131$	205920.	1.04838	0.524192	−	0.851600i	$-0.324367\pi$
$131$	205920.	1.04838	0.524192	+	0.851600i	$0.324367\pi$
$132$	0	0
$133$	−58384.0	−0.286197
$134$	−261904.	−1.26003
$135$	0	0
$136$	142464.	0.660477
$137$	230334.	1.04847	0.524236	−	0.851573i	$-0.324351\pi$
$137$	230334.	1.04847	0.524236	+	0.851573i	$0.324351\pi$
$138$	0	0
$139$	260756.	1.14471	0.572357	−	0.820004i	$-0.306029\pi$
$139$	260756.	1.14471	0.572357	+	0.820004i	$0.306029\pi$
$140$	0	0
$141$	0	0
$142$	138240.	0.575324
$143$	502560.	2.05517
$144$	0	0
$145$	0	0
$146$	345032.	1.33961
$147$	0	0
$148$	180832.	0.678611
$149$	29526.0	0.108953	0.0544765	−	0.998515i	$-0.482651\pi$
$149$	29526.0	0.108953	0.0544765	+	0.998515i	$0.482651\pi$
$150$	0	0
$151$	125168.	0.446736	0.223368	−	0.974734i	$-0.428295\pi$
$151$	125168.	0.446736	0.223368	+	0.974734i	$0.428295\pi$
$152$	−22784.0	−0.0799873
$153$	0	0
$154$	−472320.	−1.60485
$155$	0	0
$156$	0	0
$157$	43222.0	0.139944	0.0699722	−	0.997549i	$-0.477709\pi$
$157$	43222.0	0.139944	0.0699722	+	0.997549i	$0.477709\pi$
$158$	435328.	1.38731
$159$	0	0
$160$	0	0
$161$	295200.	0.897536
$162$	0	0
$163$	293476.	0.865174	0.432587	−	0.901592i	$-0.357601\pi$
$163$	293476.	0.865174	0.432587	+	0.901592i	$0.357601\pi$
$164$	−149664.	−0.434518
$165$	0	0
$166$	−42672.0	−0.120191
$167$	322200.	0.893993	0.446997	−	0.894536i	$-0.352494\pi$
$167$	322200.	0.893993	0.446997	+	0.894536i	$0.352494\pi$
$168$	0	0
$169$	115911.	0.312182
$170$	0	0
$171$	0	0
$172$	95296.0	0.245614
$173$	−261918.	−0.665350	−0.332675	−	0.943042i	$-0.607951\pi$
$173$	−261918.	−0.665350	−0.332675	+	0.943042i	$0.607951\pi$
$174$	0	0
$175$	0	0
$176$	−184320.	−0.448529
$177$	0	0
$178$	43272.0	0.102366
$179$	−623544.	−1.45457	−0.727285	−	0.686336i	$-0.759218\pi$
$179$	−623544.	−1.45457	−0.727285	+	0.686336i	$0.759218\pi$
$180$	0	0
$181$	−61186.0	−0.138821	−0.0694106	−	0.997588i	$-0.522112\pi$
$181$	−61186.0	−0.138821	−0.0694106	+	0.997588i	$0.522112\pi$
$182$	−457888.	−1.02466
$183$	0	0
$184$	115200.	0.250846
$185$	0	0
$186$	0	0
$187$	1.60272e6	3.35161
$188$	−178560.	−0.368459
$189$	0	0
$190$	0	0
$191$	−737256.	−1.46229	−0.731147	−	0.682220i	$-0.761015\pi$
$191$	−737256.	−1.46229	−0.731147	+	0.682220i	$0.761015\pi$
$192$	0	0
$193$	−539162.	−1.04190	−0.520950	−	0.853587i	$-0.674422\pi$
$193$	−539162.	−1.04190	−0.520950	+	0.853587i	$0.674422\pi$
$194$	17672.0	0.0337118
$195$	0	0
$196$	161424.	0.300143
$197$	−651174.	−1.19545	−0.597725	−	0.801701i	$-0.703929\pi$
$197$	−651174.	−1.19545	−0.597725	+	0.801701i	$0.703929\pi$
$198$	0	0
$199$	157328.	0.281626	0.140813	−	0.990036i	$-0.455028\pi$
$199$	157328.	0.281626	0.140813	+	0.990036i	$0.455028\pi$
$200$	0	0
$201$	0	0
$202$	−411768.	−0.710026
$203$	117096.	0.199435
$204$	0	0
$205$	0	0
$206$	−277744.	−0.456012
$207$	0	0
$208$	−178688.	−0.286376
$209$	−256320.	−0.405898
$210$	0	0
$211$	707180.	1.09351	0.546756	−	0.837292i	$-0.315862\pi$
$211$	707180.	1.09351	0.546756	+	0.837292i	$0.315862\pi$
$212$	225696.	0.344893
$213$	0	0
$214$	−69648.0	−0.103962
$215$	0	0
$216$	0	0
$217$	−139072.	−0.200489
$218$	815080.	1.16161
$219$	0	0
$220$	0	0
$221$	1.55375e6	2.13993
$222$	0	0
$223$	530740.	0.714693	0.357347	−	0.933972i	$-0.383681\pi$
$223$	530740.	0.714693	0.357347	+	0.933972i	$0.383681\pi$
$224$	167936.	0.223627
$225$	0	0
$226$	848808.	1.10545
$227$	120372.	0.155046	0.0775230	−	0.996991i	$-0.475299\pi$
$227$	120372.	0.155046	0.0775230	+	0.996991i	$0.475299\pi$
$228$	0	0
$229$	772310.	0.973202	0.486601	−	0.873624i	$-0.338237\pi$
$229$	772310.	0.973202	0.486601	+	0.873624i	$0.338237\pi$
$230$	0	0
$231$	0	0
$232$	45696.0	0.0557389
$233$	8838.00	0.0106651	0.00533254	−	0.999986i	$-0.498303\pi$
$233$	8838.00	0.0106651	0.00533254	+	0.999986i	$0.498303\pi$
$234$	0	0
$235$	0	0
$236$	−126720.	−0.148103
$237$	0	0
$238$	−1.46026e6	−1.67104
$239$	775416.	0.878092	0.439046	−	0.898465i	$-0.355317\pi$
$239$	775416.	0.878092	0.439046	+	0.898465i	$0.355317\pi$
$240$	0	0
$241$	−373438.	−0.414167	−0.207084	−	0.978323i	$-0.566397\pi$
$241$	−373438.	−0.414167	−0.207084	+	0.978323i	$0.566397\pi$
$242$	−1.42940e6	−1.56897
$243$	0	0
$244$	−215200.	−0.231402
$245$	0	0
$246$	0	0
$247$	−248488.	−0.259157
$248$	−54272.0	−0.0560334
$249$	0	0
$250$	0	0
$251$	−71976.0	−0.0721113	−0.0360557	−	0.999350i	$-0.511479\pi$
$251$	−71976.0	−0.0721113	−0.0360557	+	0.999350i	$0.511479\pi$
$252$	0	0
$253$	1.29600e6	1.27293
$254$	24560.0	0.0238860
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.59356e6	1.50500	0.752498	−	0.658595i	$-0.228849\pi$
$257$	1.59356e6	1.50500	0.752498	+	0.658595i	$0.228849\pi$
$258$	0	0
$259$	−1.85353e6	−1.71692
$260$	0	0
$261$	0	0
$262$	−823680.	−0.741319
$263$	−2.05452e6	−1.83156	−0.915780	−	0.401681i	$-0.868426\pi$
$263$	−2.05452e6	−1.83156	−0.915780	+	0.401681i	$0.868426\pi$
$264$	0	0
$265$	0	0
$266$	233536.	0.202372
$267$	0	0
$268$	1.04762e6	0.890974
$269$	258486.	0.217799	0.108900	−	0.994053i	$-0.465267\pi$
$269$	258486.	0.217799	0.108900	+	0.994053i	$0.465267\pi$
$270$	0	0
$271$	−1.98398e6	−1.64102	−0.820509	−	0.571634i	$-0.806310\pi$
$271$	−1.98398e6	−1.64102	−0.820509	+	0.571634i	$0.806310\pi$
$272$	−569856.	−0.467028
$273$	0	0
$274$	−921336.	−0.741381
$275$	0	0
$276$	0	0
$277$	−1.61326e6	−1.26329	−0.631647	−	0.775256i	$-0.717621\pi$
$277$	−1.61326e6	−1.26329	−0.631647	+	0.775256i	$0.717621\pi$
$278$	−1.04302e6	−0.809436
$279$	0	0
$280$	0	0
$281$	−1.37882e6	−1.04170	−0.520848	−	0.853649i	$-0.674384\pi$
$281$	−1.37882e6	−1.04170	−0.520848	+	0.853649i	$0.674384\pi$
$282$	0	0
$283$	−1.45831e6	−1.08239	−0.541194	−	0.840898i	$-0.682028\pi$
$283$	−1.45831e6	−1.08239	−0.541194	+	0.840898i	$0.682028\pi$
$284$	−552960.	−0.406816
$285$	0	0
$286$	−2.01024e6	−1.45322
$287$	1.53406e6	1.09935
$288$	0	0
$289$	3.53522e6	2.48984
$290$	0	0
$291$	0	0
$292$	−1.38013e6	−0.947245
$293$	−988134.	−0.672430	−0.336215	−	0.941785i	$-0.609147\pi$
$293$	−988134.	−0.672430	−0.336215	+	0.941785i	$0.609147\pi$
$294$	0	0
$295$	0	0
$296$	−723328.	−0.479851
$297$	0	0
$298$	−118104.	−0.0770414
$299$	1.25640e6	0.812737
$300$	0	0
$301$	−976784.	−0.621416
$302$	−500672.	−0.315890
$303$	0	0
$304$	91136.0	0.0565596
$305$	0	0
$306$	0	0
$307$	393820.	0.238480	0.119240	−	0.992865i	$-0.461954\pi$
$307$	393820.	0.238480	0.119240	+	0.992865i	$0.461954\pi$
$308$	1.88928e6	1.13480
$309$	0	0
$310$	0	0
$311$	−1.55448e6	−0.911348	−0.455674	−	0.890147i	$-0.650602\pi$
$311$	−1.55448e6	−0.911348	−0.455674	+	0.890147i	$0.650602\pi$
$312$	0	0
$313$	−1.76050e6	−1.01572	−0.507861	−	0.861439i	$-0.669564\pi$
$313$	−1.76050e6	−1.01572	−0.507861	+	0.861439i	$0.669564\pi$
$314$	−172888.	−0.0989557
$315$	0	0
$316$	−1.74131e6	−0.980977
$317$	2.37112e6	1.32527	0.662637	−	0.748941i	$-0.269437\pi$
$317$	2.37112e6	1.32527	0.662637	+	0.748941i	$0.269437\pi$
$318$	0	0
$319$	514080.	0.282849
$320$	0	0
$321$	0	0
$322$	−1.18080e6	−0.634653
$323$	−792456.	−0.422638
$324$	0	0
$325$	0	0
$326$	−1.17390e6	−0.611771
$327$	0	0
$328$	598656.	0.307251
$329$	1.83024e6	0.932220
$330$	0	0
$331$	−980068.	−0.491684	−0.245842	−	0.969310i	$-0.579064\pi$
$331$	−980068.	−0.491684	−0.245842	+	0.969310i	$0.579064\pi$
$332$	170688.	0.0849880
$333$	0	0
$334$	−1.28880e6	−0.632149
$335$	0	0
$336$	0	0
$337$	−905834.	−0.434484	−0.217242	−	0.976118i	$-0.569706\pi$
$337$	−905834.	−0.434484	−0.217242	+	0.976118i	$0.569706\pi$
$338$	−463644.	−0.220746
$339$	0	0
$340$	0	0
$341$	−610560.	−0.284343
$342$	0	0
$343$	1.10175e6	0.505648
$344$	−381184.	−0.173675
$345$	0	0
$346$	1.04767e6	0.470473
$347$	−2.95069e6	−1.31553	−0.657764	−	0.753224i	$-0.728498\pi$
$347$	−2.95069e6	−1.31553	−0.657764	+	0.753224i	$0.728498\pi$
$348$	0	0
$349$	−2.15761e6	−0.948221	−0.474110	−	0.880465i	$-0.657230\pi$
$349$	−2.15761e6	−0.948221	−0.474110	+	0.880465i	$0.657230\pi$
$350$	0	0
$351$	0	0
$352$	737280.	0.317158
$353$	1.28873e6	0.550461	0.275230	−	0.961378i	$-0.411246\pi$
$353$	1.28873e6	0.550461	0.275230	+	0.961378i	$0.411246\pi$
$354$	0	0
$355$	0	0
$356$	−173088.	−0.0723839
$357$	0	0
$358$	2.49418e6	1.02854
$359$	2.26946e6	0.929367	0.464683	−	0.885477i	$-0.346168\pi$
$359$	2.26946e6	0.929367	0.464683	+	0.885477i	$0.346168\pi$
$360$	0	0
$361$	−2.34936e6	−0.948816
$362$	244744.	0.0981614
$363$	0	0
$364$	1.83155e6	0.724546
$365$	0	0
$366$	0	0
$367$	−1.04659e6	−0.405612	−0.202806	−	0.979219i	$-0.565006\pi$
$367$	−1.04659e6	−0.405612	−0.202806	+	0.979219i	$0.565006\pi$
$368$	−460800.	−0.177375
$369$	0	0
$370$	0	0
$371$	−2.31338e6	−0.872595
$372$	0	0
$373$	−1.79827e6	−0.669243	−0.334621	−	0.942353i	$-0.608608\pi$
$373$	−1.79827e6	−0.669243	−0.334621	+	0.942353i	$0.608608\pi$
$374$	−6.41088e6	−2.36995
$375$	0	0
$376$	714240.	0.260540
$377$	498372.	0.180593
$378$	0	0
$379$	−2.18412e6	−0.781051	−0.390525	−	0.920592i	$-0.627707\pi$
$379$	−2.18412e6	−0.781051	−0.390525	+	0.920592i	$0.627707\pi$
$380$	0	0
$381$	0	0
$382$	2.94902e6	1.03400
$383$	1.78452e6	0.621619	0.310810	−	0.950472i	$-0.399400\pi$
$383$	1.78452e6	0.621619	0.310810	+	0.950472i	$0.399400\pi$
$384$	0	0
$385$	0	0
$386$	2.15665e6	0.736734
$387$	0	0
$388$	−70688.0	−0.0238378
$389$	1.10953e6	0.371761	0.185880	−	0.982572i	$-0.440486\pi$
$389$	1.10953e6	0.371761	0.185880	+	0.982572i	$0.440486\pi$
$390$	0	0
$391$	4.00680e6	1.32543
$392$	−645696.	−0.212233
$393$	0	0
$394$	2.60470e6	0.845311
$395$	0	0
$396$	0	0
$397$	−3.89568e6	−1.24053	−0.620265	−	0.784392i	$-0.712975\pi$
$397$	−3.89568e6	−1.24053	−0.620265	+	0.784392i	$0.712975\pi$
$398$	−629312.	−0.199140
$399$	0	0
$400$	0	0
$401$	1.20673e6	0.374755	0.187378	−	0.982288i	$-0.440001\pi$
$401$	1.20673e6	0.374755	0.187378	+	0.982288i	$0.440001\pi$
$402$	0	0
$403$	−591904.	−0.181547
$404$	1.64707e6	0.502064
$405$	0	0
$406$	−468384.	−0.141022
$407$	−8.13744e6	−2.43502
$408$	0	0
$409$	5.61363e6	1.65934	0.829670	−	0.558255i	$-0.188529\pi$
$409$	5.61363e6	1.65934	0.829670	+	0.558255i	$0.188529\pi$
$410$	0	0
$411$	0	0
$412$	1.11098e6	0.322449
$413$	1.29888e6	0.374709
$414$	0	0
$415$	0	0
$416$	714752.	0.202499
$417$	0	0
$418$	1.02528e6	0.287013
$419$	−1.15056e6	−0.320165	−0.160083	−	0.987104i	$-0.551176\pi$
$419$	−1.15056e6	−0.320165	−0.160083	+	0.987104i	$0.551176\pi$
$420$	0	0
$421$	−3.83089e6	−1.05340	−0.526701	−	0.850050i	$-0.676571\pi$
$421$	−3.83089e6	−1.05340	−0.526701	+	0.850050i	$0.676571\pi$
$422$	−2.82872e6	−0.773230
$423$	0	0
$424$	−902784.	−0.243876
$425$	0	0
$426$	0	0
$427$	2.20580e6	0.585459
$428$	278592.	0.0735122
$429$	0	0
$430$	0	0
$431$	−155520.	−0.0403267	−0.0201634	−	0.999797i	$-0.506419\pi$
$431$	−155520.	−0.0403267	−0.0201634	+	0.999797i	$0.506419\pi$
$432$	0	0
$433$	−4.14391e6	−1.06216	−0.531081	−	0.847321i	$-0.678214\pi$
$433$	−4.14391e6	−1.06216	−0.531081	+	0.847321i	$0.678214\pi$
$434$	556288.	0.141767
$435$	0	0
$436$	−3.26032e6	−0.821380
$437$	−640800.	−0.160516
$438$	0	0
$439$	6.23653e6	1.54448	0.772239	−	0.635332i	$-0.219137\pi$
$439$	6.23653e6	1.54448	0.772239	+	0.635332i	$0.219137\pi$
$440$	0	0
$441$	0	0
$442$	−6.21499e6	−1.51316
$443$	4.52507e6	1.09551	0.547754	−	0.836639i	$-0.315483\pi$
$443$	4.52507e6	1.09551	0.547754	+	0.836639i	$0.315483\pi$
$444$	0	0
$445$	0	0
$446$	−2.12296e6	−0.505364
$447$	0	0
$448$	−671744.	−0.158128
$449$	−2.56463e6	−0.600357	−0.300178	−	0.953883i	$-0.597046\pi$
$449$	−2.56463e6	−0.600357	−0.300178	+	0.953883i	$0.597046\pi$
$450$	0	0
$451$	6.73488e6	1.55915
$452$	−3.39523e6	−0.781670
$453$	0	0
$454$	−481488.	−0.109634
$455$	0	0
$456$	0	0
$457$	5.53409e6	1.23953	0.619763	−	0.784789i	$-0.287229\pi$
$457$	5.53409e6	1.23953	0.619763	+	0.784789i	$0.287229\pi$
$458$	−3.08924e6	−0.688158
$459$	0	0
$460$	0	0
$461$	7.19211e6	1.57617	0.788087	−	0.615564i	$-0.211072\pi$
$461$	7.19211e6	1.57617	0.788087	+	0.615564i	$0.211072\pi$
$462$	0	0
$463$	1.13936e6	0.247006	0.123503	−	0.992344i	$-0.460587\pi$
$463$	1.13936e6	0.247006	0.123503	+	0.992344i	$0.460587\pi$
$464$	−182784.	−0.0394133
$465$	0	0
$466$	−35352.0	−0.00754135
$467$	7.36168e6	1.56201	0.781006	−	0.624523i	$-0.214707\pi$
$467$	7.36168e6	1.56201	0.781006	+	0.624523i	$0.214707\pi$
$468$	0	0
$469$	−1.07381e7	−2.25421
$470$	0	0
$471$	0	0
$472$	506880.	0.104725
$473$	−4.28832e6	−0.881321
$474$	0	0
$475$	0	0
$476$	5.84102e6	1.18160
$477$	0	0
$478$	−3.10166e6	−0.620905
$479$	1.36226e6	0.271283	0.135641	−	0.990758i	$-0.456690\pi$
$479$	1.36226e6	0.271283	0.135641	+	0.990758i	$0.456690\pi$
$480$	0	0
$481$	−7.88880e6	−1.55471
$482$	1.49375e6	0.292861
$483$	0	0
$484$	5.71758e6	1.10943
$485$	0	0
$486$	0	0
$487$	−606428.	−0.115866	−0.0579331	−	0.998320i	$-0.518451\pi$
$487$	−606428.	−0.115866	−0.0579331	+	0.998320i	$0.518451\pi$
$488$	860800.	0.163626
$489$	0	0
$490$	0	0
$491$	5.84278e6	1.09374	0.546872	−	0.837216i	$-0.315819\pi$
$491$	5.84278e6	1.09374	0.546872	+	0.837216i	$0.315819\pi$
$492$	0	0
$493$	1.58936e6	0.294514
$494$	993952.	0.183252
$495$	0	0
$496$	217088.	0.0396216
$497$	5.66784e6	1.02926
$498$	0	0
$499$	−1.15044e6	−0.206830	−0.103415	−	0.994638i	$-0.532977\pi$
$499$	−1.15044e6	−0.206830	−0.103415	+	0.994638i	$0.532977\pi$
$500$	0	0
$501$	0	0
$502$	287904.	0.0509904
$503$	−869664.	−0.153261	−0.0766305	−	0.997060i	$-0.524416\pi$
$503$	−869664.	−0.153261	−0.0766305	+	0.997060i	$0.524416\pi$
$504$	0	0
$505$	0	0
$506$	−5.18400e6	−0.900096
$507$	0	0
$508$	−98240.0	−0.0168900
$509$	−1.43495e6	−0.245495	−0.122748	−	0.992438i	$-0.539171\pi$
$509$	−1.43495e6	−0.245495	−0.122748	+	0.992438i	$0.539171\pi$
$510$	0	0
$511$	1.41463e7	2.39657
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−6.37423e6	−1.06419
$515$	0	0
$516$	0	0
$517$	8.03520e6	1.32212
$518$	7.41411e6	1.21404
$519$	0	0
$520$	0	0
$521$	−1.04371e7	−1.68456	−0.842281	−	0.539038i	$-0.818788\pi$
$521$	−1.04371e7	−1.68456	−0.842281	+	0.539038i	$0.818788\pi$
$522$	0	0
$523$	7.75942e6	1.24044	0.620219	−	0.784429i	$-0.287044\pi$
$523$	7.75942e6	1.24044	0.620219	+	0.784429i	$0.287044\pi$
$524$	3.29472e6	0.524192
$525$	0	0
$526$	8.21808e6	1.29511
$527$	−1.88765e6	−0.296070
$528$	0	0
$529$	−3.19634e6	−0.496609
$530$	0	0
$531$	0	0
$532$	−934144.	−0.143098
$533$	6.52909e6	0.995485
$534$	0	0
$535$	0	0
$536$	−4.19046e6	−0.630014
$537$	0	0
$538$	−1.03394e6	−0.154007
$539$	−7.26408e6	−1.07698
$540$	0	0
$541$	−1.10233e6	−0.161927	−0.0809633	−	0.996717i	$-0.525800\pi$
$541$	−1.10233e6	−0.161927	−0.0809633	+	0.996717i	$0.525800\pi$
$542$	7.93590e6	1.16037
$543$	0	0
$544$	2.27942e6	0.330239
$545$	0	0
$546$	0	0
$547$	−2.48263e6	−0.354767	−0.177384	−	0.984142i	$-0.556763\pi$
$547$	−2.48263e6	−0.354767	−0.177384	+	0.984142i	$0.556763\pi$
$548$	3.68534e6	0.524236
$549$	0	0
$550$	0	0
$551$	−254184.	−0.0356672
$552$	0	0
$553$	1.78484e7	2.48192
$554$	6.45303e6	0.893284
$555$	0	0
$556$	4.17210e6	0.572357
$557$	5.73568e6	0.783334	0.391667	−	0.920107i	$-0.371898\pi$
$557$	5.73568e6	0.783334	0.391667	+	0.920107i	$0.371898\pi$
$558$	0	0
$559$	−4.15729e6	−0.562705
$560$	0	0
$561$	0	0
$562$	5.51527e6	0.736591
$563$	517092.	0.0687538	0.0343769	−	0.999409i	$-0.489055\pi$
$563$	517092.	0.0687538	0.0343769	+	0.999409i	$0.489055\pi$
$564$	0	0
$565$	0	0
$566$	5.83323e6	0.765364
$567$	0	0
$568$	2.21184e6	0.287662
$569$	6.72766e6	0.871131	0.435566	−	0.900157i	$-0.356548\pi$
$569$	6.72766e6	0.871131	0.435566	+	0.900157i	$0.356548\pi$
$570$	0	0
$571$	−1.03290e7	−1.32577	−0.662883	−	0.748723i	$-0.730668\pi$
$571$	−1.03290e7	−1.32577	−0.662883	+	0.748723i	$0.730668\pi$
$572$	8.04096e6	1.02759
$573$	0	0
$574$	−6.13622e6	−0.777359
$575$	0	0
$576$	0	0
$577$	−9.25834e6	−1.15769	−0.578847	−	0.815436i	$-0.696497\pi$
$577$	−9.25834e6	−1.15769	−0.578847	+	0.815436i	$0.696497\pi$
$578$	−1.41409e7	−1.76058
$579$	0	0
$580$	0	0
$581$	−1.74955e6	−0.215024
$582$	0	0
$583$	−1.01563e7	−1.23756
$584$	5.52051e6	0.669803
$585$	0	0
$586$	3.95254e6	0.475479
$587$	−9.57155e6	−1.14653	−0.573267	−	0.819369i	$-0.694324\pi$
$587$	−9.57155e6	−1.14653	−0.573267	+	0.819369i	$0.694324\pi$
$588$	0	0
$589$	301888.	0.0358557
$590$	0	0
$591$	0	0
$592$	2.89331e6	0.339306
$593$	−1.12388e7	−1.31245	−0.656226	−	0.754564i	$-0.727848\pi$
$593$	−1.12388e7	−1.31245	−0.656226	+	0.754564i	$0.727848\pi$
$594$	0	0
$595$	0	0
$596$	472416.	0.0544765
$597$	0	0
$598$	−5.02560e6	−0.574692
$599$	−3.72670e6	−0.424382	−0.212191	−	0.977228i	$-0.568060\pi$
$599$	−3.72670e6	−0.424382	−0.212191	+	0.977228i	$0.568060\pi$
$600$	0	0
$601$	6.74734e6	0.761985	0.380992	−	0.924578i	$-0.375582\pi$
$601$	6.74734e6	0.761985	0.380992	+	0.924578i	$0.375582\pi$
$602$	3.90714e6	0.439407
$603$	0	0
$604$	2.00269e6	0.223368
$605$	0	0
$606$	0	0
$607$	6.83384e6	0.752823	0.376411	−	0.926453i	$-0.377158\pi$
$607$	6.83384e6	0.752823	0.376411	+	0.926453i	$0.377158\pi$
$608$	−364544.	−0.0399936
$609$	0	0
$610$	0	0
$611$	7.78968e6	0.844144
$612$	0	0
$613$	433222.	0.0465650	0.0232825	−	0.999729i	$-0.492588\pi$
$613$	433222.	0.0465650	0.0232825	+	0.999729i	$0.492588\pi$
$614$	−1.57528e6	−0.168631
$615$	0	0
$616$	−7.55712e6	−0.802425
$617$	−5.17569e6	−0.547338	−0.273669	−	0.961824i	$-0.588237\pi$
$617$	−5.17569e6	−0.547338	−0.273669	+	0.961824i	$0.588237\pi$
$618$	0	0
$619$	−151996.	−0.0159443	−0.00797215	−	0.999968i	$-0.502538\pi$
$619$	−151996.	−0.0159443	−0.00797215	+	0.999968i	$0.502538\pi$
$620$	0	0
$621$	0	0
$622$	6.21792e6	0.644420
$623$	1.77415e6	0.183135
$624$	0	0
$625$	0	0
$626$	7.04199e6	0.718224
$627$	0	0
$628$	691552.	0.0699722
$629$	−2.51583e7	−2.53544
$630$	0	0
$631$	1.05635e7	1.05617	0.528086	−	0.849191i	$-0.322910\pi$
$631$	1.05635e7	1.05617	0.528086	+	0.849191i	$0.322910\pi$
$632$	6.96525e6	0.693656
$633$	0	0
$634$	−9.48449e6	−0.937110
$635$	0	0
$636$	0	0
$637$	−7.04212e6	−0.687630
$638$	−2.05632e6	−0.200004
$639$	0	0
$640$	0	0
$641$	5.53755e6	0.532320	0.266160	−	0.963929i	$-0.414245\pi$
$641$	5.53755e6	0.532320	0.266160	+	0.963929i	$0.414245\pi$
$642$	0	0
$643$	−8.89132e6	−0.848084	−0.424042	−	0.905642i	$-0.639389\pi$
$643$	−8.89132e6	−0.848084	−0.424042	+	0.905642i	$0.639389\pi$
$644$	4.72320e6	0.448768
$645$	0	0
$646$	3.16982e6	0.298850
$647$	2.29474e6	0.215512	0.107756	−	0.994177i	$-0.465633\pi$
$647$	2.29474e6	0.215512	0.107756	+	0.994177i	$0.465633\pi$
$648$	0	0
$649$	5.70240e6	0.531430
$650$	0	0
$651$	0	0
$652$	4.69562e6	0.432587
$653$	−1.36338e7	−1.25122	−0.625608	−	0.780137i	$-0.715149\pi$
$653$	−1.36338e7	−1.25122	−0.625608	+	0.780137i	$0.715149\pi$
$654$	0	0
$655$	0	0
$656$	−2.39462e6	−0.217259
$657$	0	0
$658$	−7.32096e6	−0.659179
$659$	1.31234e7	1.17715	0.588576	−	0.808442i	$-0.299689\pi$
$659$	1.31234e7	1.17715	0.588576	+	0.808442i	$0.299689\pi$
$660$	0	0
$661$	1.78522e7	1.58923	0.794616	−	0.607112i	$-0.207672\pi$
$661$	1.78522e7	1.58923	0.794616	+	0.607112i	$0.207672\pi$
$662$	3.92027e6	0.347673
$663$	0	0
$664$	−682752.	−0.0600956
$665$	0	0
$666$	0	0
$667$	1.28520e6	0.111855
$668$	5.15520e6	0.446997
$669$	0	0
$670$	0	0
$671$	9.68400e6	0.830326
$672$	0	0
$673$	−1.32471e7	−1.12741	−0.563707	−	0.825975i	$-0.690625\pi$
$673$	−1.32471e7	−1.12741	−0.563707	+	0.825975i	$0.690625\pi$
$674$	3.62334e6	0.307227
$675$	0	0
$676$	1.85458e6	0.156091
$677$	1.04491e7	0.876205	0.438103	−	0.898925i	$-0.355651\pi$
$677$	1.04491e7	0.876205	0.438103	+	0.898925i	$0.355651\pi$
$678$	0	0
$679$	724552.	0.0603108
$680$	0	0
$681$	0	0
$682$	2.44224e6	0.201061
$683$	−613308.	−0.0503068	−0.0251534	−	0.999684i	$-0.508007\pi$
$683$	−613308.	−0.0503068	−0.0251534	+	0.999684i	$0.508007\pi$
$684$	0	0
$685$	0	0
$686$	−4.40701e6	−0.357547
$687$	0	0
$688$	1.52474e6	0.122807
$689$	−9.84599e6	−0.790153
$690$	0	0
$691$	−2.13992e7	−1.70491	−0.852457	−	0.522798i	$-0.824888\pi$
$691$	−2.13992e7	−1.70491	−0.852457	+	0.522798i	$0.824888\pi$
$692$	−4.19069e6	−0.332675
$693$	0	0
$694$	1.18028e7	0.930219
$695$	0	0
$696$	0	0
$697$	2.08220e7	1.62346
$698$	8.63044e6	0.670493
$699$	0	0
$700$	0	0
$701$	−1.09778e7	−0.843765	−0.421883	−	0.906650i	$-0.638631\pi$
$701$	−1.09778e7	−0.843765	−0.421883	+	0.906650i	$0.638631\pi$
$702$	0	0
$703$	4.02351e6	0.307056
$704$	−2.94912e6	−0.224265
$705$	0	0
$706$	−5.15494e6	−0.389235
$707$	−1.68825e7	−1.27025
$708$	0	0
$709$	1.69732e6	0.126808	0.0634041	−	0.997988i	$-0.479804\pi$
$709$	1.69732e6	0.126808	0.0634041	+	0.997988i	$0.479804\pi$
$710$	0	0
$711$	0	0
$712$	692352.	0.0511831
$713$	−1.52640e6	−0.112446
$714$	0	0
$715$	0	0
$716$	−9.97670e6	−0.727285
$717$	0	0
$718$	−9.07786e6	−0.657162
$719$	−3.71304e6	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$719$	−3.71304e6	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$720$	0	0
$721$	−1.13875e7	−0.815813
$722$	9.39745e6	0.670914
$723$	0	0
$724$	−978976.	−0.0694106
$725$	0	0
$726$	0	0
$727$	−1.38067e6	−0.0968843	−0.0484421	−	0.998826i	$-0.515426\pi$
$727$	−1.38067e6	−0.0968843	−0.0484421	+	0.998826i	$0.515426\pi$
$728$	−7.32621e6	−0.512331
$729$	0	0
$730$	0	0
$731$	−1.32581e7	−0.917670
$732$	0	0
$733$	−1.38156e7	−0.949751	−0.474876	−	0.880053i	$-0.657507\pi$
$733$	−1.38156e7	−0.949751	−0.474876	+	0.880053i	$0.657507\pi$
$734$	4.18635e6	0.286811
$735$	0	0
$736$	1.84320e6	0.125423
$737$	−4.71427e7	−3.19702
$738$	0	0
$739$	4.59463e6	0.309485	0.154742	−	0.987955i	$-0.450545\pi$
$739$	4.59463e6	0.309485	0.154742	+	0.987955i	$0.450545\pi$
$740$	0	0
$741$	0	0
$742$	9.25354e6	0.617018
$743$	8.51174e6	0.565648	0.282824	−	0.959172i	$-0.408729\pi$
$743$	8.51174e6	0.565648	0.282824	+	0.959172i	$0.408729\pi$
$744$	0	0
$745$	0	0
$746$	7.19310e6	0.473226
$747$	0	0
$748$	2.56435e7	1.67581
$749$	−2.85557e6	−0.185989
$750$	0	0
$751$	1.71224e7	1.10781	0.553904	−	0.832580i	$-0.313137\pi$
$751$	1.71224e7	1.10781	0.553904	+	0.832580i	$0.313137\pi$
$752$	−2.85696e6	−0.184230
$753$	0	0
$754$	−1.99349e6	−0.127698
$755$	0	0
$756$	0	0
$757$	1.22018e6	0.0773900	0.0386950	−	0.999251i	$-0.487680\pi$
$757$	1.22018e6	0.0773900	0.0386950	+	0.999251i	$0.487680\pi$
$758$	8.73650e6	0.552286
$759$	0	0
$760$	0	0
$761$	−1.20327e7	−0.753182	−0.376591	−	0.926380i	$-0.622904\pi$
$761$	−1.20327e7	−0.753182	−0.376591	+	0.926380i	$0.622904\pi$
$762$	0	0
$763$	3.34183e7	2.07813
$764$	−1.17961e7	−0.731147
$765$	0	0
$766$	−7.13808e6	−0.439551
$767$	5.52816e6	0.339307
$768$	0	0
$769$	−1.88952e7	−1.15222	−0.576110	−	0.817372i	$-0.695430\pi$
$769$	−1.88952e7	−1.15222	−0.576110	+	0.817372i	$0.695430\pi$
$770$	0	0
$771$	0	0
$772$	−8.62659e6	−0.520950
$773$	1.72115e7	1.03602	0.518012	−	0.855373i	$-0.326672\pi$
$773$	1.72115e7	1.03602	0.518012	+	0.855373i	$0.326672\pi$
$774$	0	0
$775$	0	0
$776$	282752.	0.0168559
$777$	0	0
$778$	−4.43810e6	−0.262875
$779$	−3.33002e6	−0.196609
$780$	0	0
$781$	2.48832e7	1.45975
$782$	−1.60272e7	−0.937218
$783$	0	0
$784$	2.58278e6	0.150071
$785$	0	0
$786$	0	0
$787$	−1.32970e7	−0.765274	−0.382637	−	0.923899i	$-0.624984\pi$
$787$	−1.32970e7	−0.765274	−0.382637	+	0.923899i	$0.624984\pi$
$788$	−1.04188e7	−0.597725
$789$	0	0
$790$	0	0
$791$	3.48011e7	1.97766
$792$	0	0
$793$	9.38810e6	0.530145
$794$	1.55827e7	0.877187
$795$	0	0
$796$	2.51725e6	0.140813
$797$	2.15632e7	1.20245	0.601227	−	0.799078i	$-0.294679\pi$
$797$	2.15632e7	1.20245	0.601227	+	0.799078i	$0.294679\pi$
$798$	0	0
$799$	2.48422e7	1.37665
$800$	0	0
$801$	0	0
$802$	−4.82690e6	−0.264992
$803$	6.21058e7	3.39894
$804$	0	0
$805$	0	0
$806$	2.36762e6	0.128373
$807$	0	0
$808$	−6.58829e6	−0.355013
$809$	−2.65355e7	−1.42547	−0.712733	−	0.701436i	$-0.752543\pi$
$809$	−2.65355e7	−1.42547	−0.712733	+	0.701436i	$0.752543\pi$
$810$	0	0
$811$	−1.40015e7	−0.747518	−0.373759	−	0.927526i	$-0.621931\pi$
$811$	−1.40015e7	−0.747518	−0.373759	+	0.927526i	$0.621931\pi$
$812$	1.87354e6	0.0997176
$813$	0	0
$814$	3.25498e7	1.72182
$815$	0	0
$816$	0	0
$817$	2.12034e6	0.111135
$818$	−2.24545e7	−1.17333
$819$	0	0
$820$	0	0
$821$	−1.32286e7	−0.684944	−0.342472	−	0.939528i	$-0.611264\pi$
$821$	−1.32286e7	−0.684944	−0.342472	+	0.939528i	$0.611264\pi$
$822$	0	0
$823$	7.25818e6	0.373532	0.186766	−	0.982404i	$-0.440199\pi$
$823$	7.25818e6	0.373532	0.186766	+	0.982404i	$0.440199\pi$
$824$	−4.44390e6	−0.228006
$825$	0	0
$826$	−5.19552e6	−0.264959
$827$	1.84527e7	0.938204	0.469102	−	0.883144i	$-0.344578\pi$
$827$	1.84527e7	0.938204	0.469102	+	0.883144i	$0.344578\pi$
$828$	0	0
$829$	−2.43640e7	−1.23130	−0.615649	−	0.788021i	$-0.711106\pi$
$829$	−2.43640e7	−1.23130	−0.615649	+	0.788021i	$0.711106\pi$
$830$	0	0
$831$	0	0
$832$	−2.85901e6	−0.143188
$833$	−2.24581e7	−1.12140
$834$	0	0
$835$	0	0
$836$	−4.10112e6	−0.202949
$837$	0	0
$838$	4.60224e6	0.226391
$839$	1.55793e7	0.764089	0.382045	−	0.924144i	$-0.375220\pi$
$839$	1.55793e7	0.764089	0.382045	+	0.924144i	$0.375220\pi$
$840$	0	0
$841$	−2.00014e7	−0.975145
$842$	1.53236e7	0.744868
$843$	0	0
$844$	1.13149e7	0.546756
$845$	0	0
$846$	0	0
$847$	−5.86052e7	−2.80691
$848$	3.61114e6	0.172446
$849$	0	0
$850$	0	0
$851$	−2.03436e7	−0.962950
$852$	0	0
$853$	−3.08062e7	−1.44966	−0.724829	−	0.688929i	$-0.758081\pi$
$853$	−3.08062e7	−1.44966	−0.724829	+	0.688929i	$0.758081\pi$
$854$	−8.82320e6	−0.413982
$855$	0	0
$856$	−1.11437e6	−0.0519810
$857$	−2.02084e6	−0.0939897	−0.0469949	−	0.998895i	$-0.514964\pi$
$857$	−2.02084e6	−0.0939897	−0.0469949	+	0.998895i	$0.514964\pi$
$858$	0	0
$859$	−2.24790e7	−1.03943	−0.519714	−	0.854340i	$-0.673961\pi$
$859$	−2.24790e7	−1.03943	−0.519714	+	0.854340i	$0.673961\pi$
$860$	0	0
$861$	0	0
$862$	622080.	0.0285153
$863$	−9.20942e6	−0.420926	−0.210463	−	0.977602i	$-0.567497\pi$
$863$	−9.20942e6	−0.420926	−0.210463	+	0.977602i	$0.567497\pi$
$864$	0	0
$865$	0	0
$866$	1.65757e7	0.751062
$867$	0	0
$868$	−2.22515e6	−0.100244
$869$	7.83590e7	3.51998
$870$	0	0
$871$	−4.57022e7	−2.04123
$872$	1.30413e7	0.580803
$873$	0	0
$874$	2.56320e6	0.113502
$875$	0	0
$876$	0	0
$877$	5.36258e6	0.235437	0.117719	−	0.993047i	$-0.462442\pi$
$877$	5.36258e6	0.235437	0.117719	+	0.993047i	$0.462442\pi$
$878$	−2.49461e7	−1.09211
$879$	0	0
$880$	0	0
$881$	−1.38347e7	−0.600525	−0.300263	−	0.953857i	$-0.597074\pi$
$881$	−1.38347e7	−0.600525	−0.300263	+	0.953857i	$0.597074\pi$
$882$	0	0
$883$	1.66004e6	0.0716499	0.0358250	−	0.999358i	$-0.488594\pi$
$883$	1.66004e6	0.0716499	0.0358250	+	0.999358i	$0.488594\pi$
$884$	2.48600e7	1.06997
$885$	0	0
$886$	−1.81003e7	−0.774642
$887$	−8.10612e6	−0.345943	−0.172971	−	0.984927i	$-0.555337\pi$
$887$	−8.10612e6	−0.345943	−0.172971	+	0.984927i	$0.555337\pi$
$888$	0	0
$889$	1.00696e6	0.0427325
$890$	0	0
$891$	0	0
$892$	8.49184e6	0.357347
$893$	−3.97296e6	−0.166719
$894$	0	0
$895$	0	0
$896$	2.68698e6	0.111813
$897$	0	0
$898$	1.02585e7	0.424516
$899$	−605472.	−0.0249859
$900$	0	0
$901$	−3.14000e7	−1.28860
$902$	−2.69395e7	−1.10249
$903$	0	0
$904$	1.35809e7	0.552724
$905$	0	0
$906$	0	0
$907$	−4.05360e6	−0.163615	−0.0818073	−	0.996648i	$-0.526069\pi$
$907$	−4.05360e6	−0.163615	−0.0818073	+	0.996648i	$0.526069\pi$
$908$	1.92595e6	0.0775230
$909$	0	0
$910$	0	0
$911$	1.38233e7	0.551844	0.275922	−	0.961180i	$-0.411017\pi$
$911$	1.38233e7	0.551844	0.275922	+	0.961180i	$0.411017\pi$
$912$	0	0
$913$	−7.68096e6	−0.304957
$914$	−2.21363e7	−0.876477
$915$	0	0
$916$	1.23570e7	0.486601
$917$	−3.37709e7	−1.32623
$918$	0	0
$919$	−3.78443e7	−1.47813	−0.739063	−	0.673636i	$-0.764732\pi$
$919$	−3.78443e7	−1.47813	−0.739063	+	0.673636i	$0.764732\pi$
$920$	0	0
$921$	0	0
$922$	−2.87684e7	−1.11452
$923$	2.41229e7	0.932019
$924$	0	0
$925$	0	0
$926$	−4.55742e6	−0.174659
$927$	0	0
$928$	731136.	0.0278694
$929$	2.72822e7	1.03715	0.518574	−	0.855033i	$-0.326463\pi$
$929$	2.72822e7	1.03715	0.518574	+	0.855033i	$0.326463\pi$
$930$	0	0
$931$	3.59168e6	0.135808
$932$	141408.	0.00533254
$933$	0	0
$934$	−2.94467e7	−1.10451
$935$	0	0
$936$	0	0
$937$	−4.32666e7	−1.60992	−0.804958	−	0.593332i	$-0.797812\pi$
$937$	−4.32666e7	−1.60992	−0.804958	+	0.593332i	$0.797812\pi$
$938$	4.29523e7	1.59397
$939$	0	0
$940$	0	0
$941$	−8.50106e6	−0.312967	−0.156484	−	0.987681i	$-0.550016\pi$
$941$	−8.50106e6	−0.312967	−0.156484	+	0.987681i	$0.550016\pi$
$942$	0	0
$943$	1.68372e7	0.616582
$944$	−2.02752e6	−0.0740517
$945$	0	0
$946$	1.71533e7	0.623188
$947$	7.10456e6	0.257432	0.128716	−	0.991681i	$-0.458914\pi$
$947$	7.10456e6	0.257432	0.128716	+	0.991681i	$0.458914\pi$
$948$	0	0
$949$	6.02081e7	2.17015
$950$	0	0
$951$	0	0
$952$	−2.33641e7	−0.835520
$953$	−5.39741e7	−1.92510	−0.962551	−	0.271102i	$-0.912612\pi$
$953$	−5.39741e7	−1.92510	−0.962551	+	0.271102i	$0.912612\pi$
$954$	0	0
$955$	0	0
$956$	1.24067e7	0.439046
$957$	0	0
$958$	−5.44906e6	−0.191826
$959$	−3.77748e7	−1.32634
$960$	0	0
$961$	−2.79100e7	−0.974882
$962$	3.15552e7	1.09934
$963$	0	0
$964$	−5.97501e6	−0.207084
$965$	0	0
$966$	0	0
$967$	−3.64583e7	−1.25381	−0.626903	−	0.779097i	$-0.715678\pi$
$967$	−3.64583e7	−1.25381	−0.626903	+	0.779097i	$0.715678\pi$
$968$	−2.28703e7	−0.784484
$969$	0	0
$970$	0	0
$971$	5.20286e7	1.77090	0.885450	−	0.464734i	$-0.153850\pi$
$971$	5.20286e7	1.77090	0.885450	+	0.464734i	$0.153850\pi$
$972$	0	0
$973$	−4.27640e7	−1.44809
$974$	2.42571e6	0.0819298
$975$	0	0
$976$	−3.44320e6	−0.115701
$977$	127902.	0.00428688	0.00214344	−	0.999998i	$-0.499318\pi$
$977$	127902.	0.00428688	0.00214344	+	0.999998i	$0.499318\pi$
$978$	0	0
$979$	7.78896e6	0.259730
$980$	0	0
$981$	0	0
$982$	−2.33711e7	−0.773393
$983$	−1.57667e7	−0.520422	−0.260211	−	0.965552i	$-0.583792\pi$
$983$	−1.57667e7	−0.520422	−0.260211	+	0.965552i	$0.583792\pi$
$984$	0	0
$985$	0	0
$986$	−6.35746e6	−0.208253
$987$	0	0
$988$	−3.97581e6	−0.129579
$989$	−1.07208e7	−0.348527
$990$	0	0
$991$	2.99415e7	0.968479	0.484239	−	0.874936i	$-0.339096\pi$
$991$	2.99415e7	0.968479	0.484239	+	0.874936i	$0.339096\pi$
$992$	−868352.	−0.0280167
$993$	0	0
$994$	−2.26714e7	−0.727799
$995$	0	0
$996$	0	0
$997$	5.07440e7	1.61676	0.808382	−	0.588659i	$-0.200344\pi$
$997$	5.07440e7	1.61676	0.808382	+	0.588659i	$0.200344\pi$
$998$	4.60178e6	0.146251
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.b.1.1		1
3.2	odd	2		150.6.a.h.1.1		1
5.2	odd	4		450.6.c.b.199.1		2
5.3	odd	4		450.6.c.b.199.2		2
5.4	even	2		90.6.a.g.1.1		1
15.2	even	4		150.6.c.d.49.2		2
15.8	even	4		150.6.c.d.49.1		2
15.14	odd	2		30.6.a.a.1.1	✓	1
20.19	odd	2		720.6.a.m.1.1		1
60.59	even	2		240.6.a.a.1.1		1
120.29	odd	2		960.6.a.n.1.1		1
120.59	even	2		960.6.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.6.a.a.1.1	✓	1	15.14	odd	2
90.6.a.g.1.1		1	5.4	even	2
150.6.a.h.1.1		1	3.2	odd	2
150.6.c.d.49.1		2	15.8	even	4
150.6.c.d.49.2		2	15.2	even	4
240.6.a.a.1.1		1	60.59	even	2
450.6.a.b.1.1		1	1.1	even	1	trivial
450.6.c.b.199.1		2	5.2	odd	4
450.6.c.b.199.2		2	5.3	odd	4
720.6.a.m.1.1		1	20.19	odd	2
960.6.a.n.1.1		1	120.29	odd	2
960.6.a.u.1.1		1	120.59	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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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