LMFDB - Embedded newform 441.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-5.00000 q^{2} -7.00000 q^{4} +94.0000 q^{5} +195.000 q^{8} +O(q^{10})$

$q-5.00000 q^{2} -7.00000 q^{4} +94.0000 q^{5} +195.000 q^{8} -470.000 q^{10} -52.0000 q^{11} +770.000 q^{13} -751.000 q^{16} -2022.00 q^{17} -1732.00 q^{19} -658.000 q^{20} +260.000 q^{22} +576.000 q^{23} +5711.00 q^{25} -3850.00 q^{26} -5518.00 q^{29} -6336.00 q^{31} -2485.00 q^{32} +10110.0 q^{34} -7338.00 q^{37} +8660.00 q^{38} +18330.0 q^{40} -3262.00 q^{41} +5420.00 q^{43} +364.000 q^{44} -2880.00 q^{46} +864.000 q^{47} -28555.0 q^{50} -5390.00 q^{52} -4182.00 q^{53} -4888.00 q^{55} +27590.0 q^{58} -11220.0 q^{59} +45602.0 q^{61} +31680.0 q^{62} +36457.0 q^{64} +72380.0 q^{65} +1396.00 q^{67} +14154.0 q^{68} -18720.0 q^{71} -46362.0 q^{73} +36690.0 q^{74} +12124.0 q^{76} +97424.0 q^{79} -70594.0 q^{80} +16310.0 q^{82} -81228.0 q^{83} -190068. q^{85} -27100.0 q^{86} -10140.0 q^{88} -3182.00 q^{89} -4032.00 q^{92} -4320.00 q^{94} -162808. q^{95} -4914.00 q^{97} +O(q^{100})$

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 $

$2$	−5.00000	−0.883883	−0.441942	−	0.897044i	$-0.645710\pi$
$2$	−5.00000	−0.883883	−0.441942	+	0.897044i	$0.645710\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.218750
$5$	94.0000	1.68152	0.840762	−	0.541406i	$-0.182108\pi$
$5$	94.0000	1.68152	0.840762	+	0.541406i	$0.182108\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	195.000	1.07723
$9$	0	0
$10$	−470.000	−1.48627
$11$	−52.0000	−0.129575	−0.0647876	−	0.997899i	$-0.520637\pi$
$11$	−52.0000	−0.129575	−0.0647876	+	0.997899i	$0.520637\pi$
$12$	0	0
$13$	770.000	1.26367	0.631833	−	0.775104i	$-0.282303\pi$
$13$	770.000	1.26367	0.631833	+	0.775104i	$0.282303\pi$
$14$	0	0
$15$	0	0
$16$	−751.000	−0.733398
$17$	−2022.00	−1.69691	−0.848455	−	0.529267i	$-0.822467\pi$
$17$	−2022.00	−1.69691	−0.848455	+	0.529267i	$0.822467\pi$
$18$	0	0
$19$	−1732.00	−1.10069	−0.550344	−	0.834938i	$-0.685503\pi$
$19$	−1732.00	−1.10069	−0.550344	+	0.834938i	$0.685503\pi$
$20$	−658.000	−0.367833
$21$	0	0
$22$	260.000	0.114529
$23$	576.000	0.227040	0.113520	−	0.993536i	$-0.463787\pi$
$23$	576.000	0.227040	0.113520	+	0.993536i	$0.463787\pi$
$24$	0	0
$25$	5711.00	1.82752
$26$	−3850.00	−1.11693
$27$	0	0
$28$	0	0
$29$	−5518.00	−1.21839	−0.609196	−	0.793020i	$-0.708508\pi$
$29$	−5518.00	−1.21839	−0.609196	+	0.793020i	$0.708508\pi$
$30$	0	0
$31$	−6336.00	−1.18416	−0.592081	−	0.805879i	$-0.701693\pi$
$31$	−6336.00	−1.18416	−0.592081	+	0.805879i	$0.701693\pi$
$32$	−2485.00	−0.428994
$33$	0	0
$34$	10110.0	1.49987
$35$	0	0
$36$	0	0
$37$	−7338.00	−0.881198	−0.440599	−	0.897704i	$-0.645234\pi$
$37$	−7338.00	−0.881198	−0.440599	+	0.897704i	$0.645234\pi$
$38$	8660.00	0.972879
$39$	0	0
$40$	18330.0	1.81139
$41$	−3262.00	−0.303057	−0.151528	−	0.988453i	$-0.548420\pi$
$41$	−3262.00	−0.303057	−0.151528	+	0.988453i	$0.548420\pi$
$42$	0	0
$43$	5420.00	0.447021	0.223511	−	0.974701i	$-0.428248\pi$
$43$	5420.00	0.447021	0.223511	+	0.974701i	$0.428248\pi$
$44$	364.000	0.0283446
$45$	0	0
$46$	−2880.00	−0.200677
$47$	864.000	0.0570518	0.0285259	−	0.999593i	$-0.490919\pi$
$47$	864.000	0.0570518	0.0285259	+	0.999593i	$0.490919\pi$
$48$	0	0
$49$	0	0
$50$	−28555.0	−1.61531
$51$	0	0
$52$	−5390.00	−0.276427
$53$	−4182.00	−0.204500	−0.102250	−	0.994759i	$-0.532604\pi$
$53$	−4182.00	−0.204500	−0.102250	+	0.994759i	$0.532604\pi$
$54$	0	0
$55$	−4888.00	−0.217884
$56$	0	0
$57$	0	0
$58$	27590.0	1.07692
$59$	−11220.0	−0.419626	−0.209813	−	0.977741i	$-0.567286\pi$
$59$	−11220.0	−0.419626	−0.209813	+	0.977741i	$0.567286\pi$
$60$	0	0
$61$	45602.0	1.56913	0.784566	−	0.620046i	$-0.212886\pi$
$61$	45602.0	1.56913	0.784566	+	0.620046i	$0.212886\pi$
$62$	31680.0	1.04666
$63$	0	0
$64$	36457.0	1.11258
$65$	72380.0	2.12488
$66$	0	0
$67$	1396.00	0.0379925	0.0189963	−	0.999820i	$-0.493953\pi$
$67$	1396.00	0.0379925	0.0189963	+	0.999820i	$0.493953\pi$
$68$	14154.0	0.371199
$69$	0	0
$70$	0	0
$71$	−18720.0	−0.440717	−0.220359	−	0.975419i	$-0.570723\pi$
$71$	−18720.0	−0.440717	−0.220359	+	0.975419i	$0.570723\pi$
$72$	0	0
$73$	−46362.0	−1.01825	−0.509126	−	0.860692i	$-0.670031\pi$
$73$	−46362.0	−1.01825	−0.509126	+	0.860692i	$0.670031\pi$
$74$	36690.0	0.778876
$75$	0	0
$76$	12124.0	0.240775
$77$	0	0
$78$	0	0
$79$	97424.0	1.75630	0.878149	−	0.478387i	$-0.158778\pi$
$79$	97424.0	1.75630	0.878149	+	0.478387i	$0.158778\pi$
$80$	−70594.0	−1.23323
$81$	0	0
$82$	16310.0	0.267867
$83$	−81228.0	−1.29423	−0.647114	−	0.762394i	$-0.724024\pi$
$83$	−81228.0	−1.29423	−0.647114	+	0.762394i	$0.724024\pi$
$84$	0	0
$85$	−190068.	−2.85339
$86$	−27100.0	−0.395115
$87$	0	0
$88$	−10140.0	−0.139583
$89$	−3182.00	−0.0425819	−0.0212910	−	0.999773i	$-0.506778\pi$
$89$	−3182.00	−0.0425819	−0.0212910	+	0.999773i	$0.506778\pi$
$90$	0	0
$91$	0	0
$92$	−4032.00	−0.0496651
$93$	0	0
$94$	−4320.00	−0.0504271
$95$	−162808.	−1.85083
$96$	0	0
$97$	−4914.00	−0.0530281	−0.0265140	−	0.999648i	$-0.508441\pi$
$97$	−4914.00	−0.0530281	−0.0265140	+	0.999648i	$0.508441\pi$
$98$	0	0
$99$	0	0
$100$	−39977.0	−0.399770
$101$	−166354.	−1.62267	−0.811334	−	0.584583i	$-0.801258\pi$
$101$	−166354.	−1.62267	−0.811334	+	0.584583i	$0.801258\pi$
$102$	0	0
$103$	−157160.	−1.45965	−0.729825	−	0.683634i	$-0.760399\pi$
$103$	−157160.	−1.45965	−0.729825	+	0.683634i	$0.760399\pi$
$104$	150150.	1.36126
$105$	0	0
$106$	20910.0	0.180755
$107$	6764.00	0.0571142	0.0285571	−	0.999592i	$-0.490909\pi$
$107$	6764.00	0.0571142	0.0285571	+	0.999592i	$0.490909\pi$
$108$	0	0
$109$	178398.	1.43821	0.719107	−	0.694899i	$-0.244551\pi$
$109$	178398.	1.43821	0.719107	+	0.694899i	$0.244551\pi$
$110$	24440.0	0.192584
$111$	0	0
$112$	0	0
$113$	45134.0	0.332512	0.166256	−	0.986083i	$-0.446832\pi$
$113$	45134.0	0.332512	0.166256	+	0.986083i	$0.446832\pi$
$114$	0	0
$115$	54144.0	0.381773
$116$	38626.0	0.266523
$117$	0	0
$118$	56100.0	0.370901
$119$	0	0
$120$	0	0
$121$	−158347.	−0.983210
$122$	−228010.	−1.38693
$123$	0	0
$124$	44352.0	0.259035
$125$	243084.	1.39149
$126$	0	0
$127$	−205056.	−1.12814	−0.564070	−	0.825727i	$-0.690765\pi$
$127$	−205056.	−1.12814	−0.564070	+	0.825727i	$0.690765\pi$
$128$	−102765.	−0.554396
$129$	0	0
$130$	−361900.	−1.87815
$131$	72964.0	0.371476	0.185738	−	0.982599i	$-0.440532\pi$
$131$	72964.0	0.371476	0.185738	+	0.982599i	$0.440532\pi$
$132$	0	0
$133$	0	0
$134$	−6980.00	−0.0335810
$135$	0	0
$136$	−394290.	−1.82797
$137$	94182.0	0.428713	0.214356	−	0.976756i	$-0.431235\pi$
$137$	94182.0	0.428713	0.214356	+	0.976756i	$0.431235\pi$
$138$	0	0
$139$	47796.0	0.209824	0.104912	−	0.994482i	$-0.466544\pi$
$139$	47796.0	0.209824	0.104912	+	0.994482i	$0.466544\pi$
$140$	0	0
$141$	0	0
$142$	93600.0	0.389543
$143$	−40040.0	−0.163740
$144$	0	0
$145$	−518692.	−2.04875
$146$	231810.	0.900016
$147$	0	0
$148$	51366.0	0.192762
$149$	124266.	0.458550	0.229275	−	0.973362i	$-0.426364\pi$
$149$	124266.	0.458550	0.229275	+	0.973362i	$0.426364\pi$
$150$	0	0
$151$	−446296.	−1.59287	−0.796436	−	0.604723i	$-0.793284\pi$
$151$	−446296.	−1.59287	−0.796436	+	0.604723i	$0.793284\pi$
$152$	−337740.	−1.18570
$153$	0	0
$154$	0	0
$155$	−595584.	−1.99119
$156$	0	0
$157$	159746.	0.517227	0.258613	−	0.965981i	$-0.416734\pi$
$157$	159746.	0.517227	0.258613	+	0.965981i	$0.416734\pi$
$158$	−487120.	−1.55236
$159$	0	0
$160$	−233590.	−0.721364
$161$	0	0
$162$	0	0
$163$	247252.	0.728905	0.364452	−	0.931222i	$-0.381256\pi$
$163$	247252.	0.728905	0.364452	+	0.931222i	$0.381256\pi$
$164$	22834.0	0.0662937
$165$	0	0
$166$	406140.	1.14395
$167$	−684488.	−1.89922	−0.949609	−	0.313438i	$-0.898519\pi$
$167$	−684488.	−1.89922	−0.949609	+	0.313438i	$0.898519\pi$
$168$	0	0
$169$	221607.	0.596852
$170$	950340.	2.52207
$171$	0	0
$172$	−37940.0	−0.0977859
$173$	−610474.	−1.55079	−0.775393	−	0.631479i	$-0.782448\pi$
$173$	−610474.	−1.55079	−0.775393	+	0.631479i	$0.782448\pi$
$174$	0	0
$175$	0	0
$176$	39052.0	0.0950302
$177$	0	0
$178$	15910.0	0.0376374
$179$	−662252.	−1.54487	−0.772433	−	0.635097i	$-0.780960\pi$
$179$	−662252.	−1.54487	−0.772433	+	0.635097i	$0.780960\pi$
$180$	0	0
$181$	−154630.	−0.350830	−0.175415	−	0.984495i	$-0.556127\pi$
$181$	−154630.	−0.350830	−0.175415	+	0.984495i	$0.556127\pi$
$182$	0	0
$183$	0	0
$184$	112320.	0.244575
$185$	−689772.	−1.48175
$186$	0	0
$187$	105144.	0.219877
$188$	−6048.00	−0.0124801
$189$	0	0
$190$	814040.	1.63592
$191$	−486904.	−0.965739	−0.482870	−	0.875692i	$-0.660406\pi$
$191$	−486904.	−0.965739	−0.482870	+	0.875692i	$0.660406\pi$
$192$	0	0
$193$	620546.	1.19917	0.599585	−	0.800311i	$-0.295332\pi$
$193$	620546.	1.19917	0.599585	+	0.800311i	$0.295332\pi$
$194$	24570.0	0.0468706
$195$	0	0
$196$	0	0
$197$	236570.	0.434304	0.217152	−	0.976138i	$-0.430323\pi$
$197$	236570.	0.434304	0.217152	+	0.976138i	$0.430323\pi$
$198$	0	0
$199$	−82104.0	−0.146971	−0.0734855	−	0.997296i	$-0.523412\pi$
$199$	−82104.0	−0.146971	−0.0734855	+	0.997296i	$0.523412\pi$
$200$	1.11364e6	1.96866
$201$	0	0
$202$	831770.	1.43425
$203$	0	0
$204$	0	0
$205$	−306628.	−0.509597
$206$	785800.	1.29016
$207$	0	0
$208$	−578270.	−0.926771
$209$	90064.0	0.142622
$210$	0	0
$211$	99892.0	0.154463	0.0772315	−	0.997013i	$-0.475392\pi$
$211$	99892.0	0.154463	0.0772315	+	0.997013i	$0.475392\pi$
$212$	29274.0	0.0447345
$213$	0	0
$214$	−33820.0	−0.0504823
$215$	509480.	0.751677
$216$	0	0
$217$	0	0
$218$	−891990.	−1.27121
$219$	0	0
$220$	34216.0	0.0476620
$221$	−1.55694e6	−2.14433
$222$	0	0
$223$	186704.	0.251415	0.125708	−	0.992067i	$-0.459880\pi$
$223$	186704.	0.251415	0.125708	+	0.992067i	$0.459880\pi$
$224$	0	0
$225$	0	0
$226$	−225670.	−0.293902
$227$	336372.	0.433267	0.216633	−	0.976253i	$-0.430492\pi$
$227$	336372.	0.433267	0.216633	+	0.976253i	$0.430492\pi$
$228$	0	0
$229$	926314.	1.16727	0.583633	−	0.812018i	$-0.301631\pi$
$229$	926314.	1.16727	0.583633	+	0.812018i	$0.301631\pi$
$230$	−270720.	−0.337443
$231$	0	0
$232$	−1.07601e6	−1.31249
$233$	−1.25711e6	−1.51700	−0.758499	−	0.651675i	$-0.774067\pi$
$233$	−1.25711e6	−1.51700	−0.758499	+	0.651675i	$0.774067\pi$
$234$	0	0
$235$	81216.0	0.0959339
$236$	78540.0	0.0917933
$237$	0	0
$238$	0	0
$239$	347016.	0.392966	0.196483	−	0.980507i	$-0.437048\pi$
$239$	347016.	0.392966	0.196483	+	0.980507i	$0.437048\pi$
$240$	0	0
$241$	−99170.0	−0.109986	−0.0549930	−	0.998487i	$-0.517514\pi$
$241$	−99170.0	−0.109986	−0.0549930	+	0.998487i	$0.517514\pi$
$242$	791735.	0.869043
$243$	0	0
$244$	−319214.	−0.343247
$245$	0	0
$246$	0	0
$247$	−1.33364e6	−1.39090
$248$	−1.23552e6	−1.27562
$249$	0	0
$250$	−1.21542e6	−1.22992
$251$	344428.	0.345076	0.172538	−	0.985003i	$-0.444803\pi$
$251$	344428.	0.345076	0.172538	+	0.985003i	$0.444803\pi$
$252$	0	0
$253$	−29952.0	−0.0294188
$254$	1.02528e6	0.997145
$255$	0	0
$256$	−652799.	−0.622558
$257$	295130.	0.278728	0.139364	−	0.990241i	$-0.455494\pi$
$257$	295130.	0.278728	0.139364	+	0.990241i	$0.455494\pi$
$258$	0	0
$259$	0	0
$260$	−506660.	−0.464818
$261$	0	0
$262$	−364820.	−0.328341
$263$	1.27246e6	1.13437	0.567187	−	0.823589i	$-0.308032\pi$
$263$	1.27246e6	1.13437	0.567187	+	0.823589i	$0.308032\pi$
$264$	0	0
$265$	−393108.	−0.343872
$266$	0	0
$267$	0	0
$268$	−9772.00	−0.00831087
$269$	276774.	0.233209	0.116604	−	0.993178i	$-0.462799\pi$
$269$	276774.	0.233209	0.116604	+	0.993178i	$0.462799\pi$
$270$	0	0
$271$	1.28994e6	1.06695	0.533476	−	0.845815i	$-0.320885\pi$
$271$	1.28994e6	1.06695	0.533476	+	0.845815i	$0.320885\pi$
$272$	1.51852e6	1.24451
$273$	0	0
$274$	−470910.	−0.378932
$275$	−296972.	−0.236801
$276$	0	0
$277$	1.71655e6	1.34418	0.672089	−	0.740470i	$-0.265397\pi$
$277$	1.71655e6	1.34418	0.672089	+	0.740470i	$0.265397\pi$
$278$	−238980.	−0.185460
$279$	0	0
$280$	0	0
$281$	1.47218e6	1.11223	0.556116	−	0.831104i	$-0.312291\pi$
$281$	1.47218e6	1.11223	0.556116	+	0.831104i	$0.312291\pi$
$282$	0	0
$283$	−1.02881e6	−0.763607	−0.381804	−	0.924244i	$-0.624697\pi$
$283$	−1.02881e6	−0.763607	−0.381804	+	0.924244i	$0.624697\pi$
$284$	131040.	0.0964069
$285$	0	0
$286$	200200.	0.144727
$287$	0	0
$288$	0	0
$289$	2.66863e6	1.87950
$290$	2.59346e6	1.81086
$291$	0	0
$292$	324534.	0.222742
$293$	−1.18607e6	−0.807123	−0.403562	−	0.914952i	$-0.632228\pi$
$293$	−1.18607e6	−0.807123	−0.403562	+	0.914952i	$0.632228\pi$
$294$	0	0
$295$	−1.05468e6	−0.705612
$296$	−1.43091e6	−0.949255
$297$	0	0
$298$	−621330.	−0.405305
$299$	443520.	0.286903
$300$	0	0
$301$	0	0
$302$	2.23148e6	1.40791
$303$	0	0
$304$	1.30073e6	0.807242
$305$	4.28659e6	2.63853
$306$	0	0
$307$	−1.51892e6	−0.919788	−0.459894	−	0.887974i	$-0.652113\pi$
$307$	−1.51892e6	−0.919788	−0.459894	+	0.887974i	$0.652113\pi$
$308$	0	0
$309$	0	0
$310$	2.97792e6	1.75998
$311$	212808.	0.124763	0.0623817	−	0.998052i	$-0.480130\pi$
$311$	212808.	0.124763	0.0623817	+	0.998052i	$0.480130\pi$
$312$	0	0
$313$	1894.00	0.00109275	0.000546373	−	1.00000i	$-0.499826\pi$
$313$	1894.00	0.00109275	0.000546373		1.00000i	$0.499826\pi$
$314$	−798730.	−0.457168
$315$	0	0
$316$	−681968.	−0.384190
$317$	1.57898e6	0.882527	0.441263	−	0.897378i	$-0.354530\pi$
$317$	1.57898e6	0.882527	0.441263	+	0.897378i	$0.354530\pi$
$318$	0	0
$319$	286936.	0.157873
$320$	3.42696e6	1.87083
$321$	0	0
$322$	0	0
$323$	3.50210e6	1.86777
$324$	0	0
$325$	4.39747e6	2.30938
$326$	−1.23626e6	−0.644267
$327$	0	0
$328$	−636090.	−0.326463
$329$	0	0
$330$	0	0
$331$	−3.39471e6	−1.70307	−0.851535	−	0.524298i	$-0.824328\pi$
$331$	−3.39471e6	−1.70307	−0.851535	+	0.524298i	$0.824328\pi$
$332$	568596.	0.283112
$333$	0	0
$334$	3.42244e6	1.67869
$335$	131224.	0.0638853
$336$	0	0
$337$	2.02731e6	0.972403	0.486201	−	0.873847i	$-0.338382\pi$
$337$	2.02731e6	0.972403	0.486201	+	0.873847i	$0.338382\pi$
$338$	−1.10804e6	−0.527548
$339$	0	0
$340$	1.33048e6	0.624180
$341$	329472.	0.153438
$342$	0	0
$343$	0	0
$344$	1.05690e6	0.481546
$345$	0	0
$346$	3.05237e6	1.37071
$347$	−3.48885e6	−1.55546	−0.777730	−	0.628598i	$-0.783629\pi$
$347$	−3.48885e6	−1.55546	−0.777730	+	0.628598i	$0.783629\pi$
$348$	0	0
$349$	−965566.	−0.424344	−0.212172	−	0.977232i	$-0.568054\pi$
$349$	−965566.	−0.424344	−0.212172	+	0.977232i	$0.568054\pi$
$350$	0	0
$351$	0	0
$352$	129220.	0.0555870
$353$	1.15393e6	0.492882	0.246441	−	0.969158i	$-0.420739\pi$
$353$	1.15393e6	0.492882	0.246441	+	0.969158i	$0.420739\pi$
$354$	0	0
$355$	−1.75968e6	−0.741076
$356$	22274.0	0.00931479
$357$	0	0
$358$	3.31126e6	1.36548
$359$	−1.61110e6	−0.659762	−0.329881	−	0.944022i	$-0.607009\pi$
$359$	−1.61110e6	−0.659762	−0.329881	+	0.944022i	$0.607009\pi$
$360$	0	0
$361$	523725.	0.211512
$362$	773150.	0.310093
$363$	0	0
$364$	0	0
$365$	−4.35803e6	−1.71221
$366$	0	0
$367$	−3.67747e6	−1.42523	−0.712614	−	0.701557i	$-0.752489\pi$
$367$	−3.67747e6	−1.42523	−0.712614	+	0.701557i	$0.752489\pi$
$368$	−432576.	−0.166511
$369$	0	0
$370$	3.44886e6	1.30970
$371$	0	0
$372$	0	0
$373$	649766.	0.241816	0.120908	−	0.992664i	$-0.461419\pi$
$373$	649766.	0.241816	0.120908	+	0.992664i	$0.461419\pi$
$374$	−525720.	−0.194346
$375$	0	0
$376$	168480.	0.0614580
$377$	−4.24886e6	−1.53964
$378$	0	0
$379$	320700.	0.114683	0.0573417	−	0.998355i	$-0.481738\pi$
$379$	320700.	0.114683	0.0573417	+	0.998355i	$0.481738\pi$
$380$	1.13966e6	0.404869
$381$	0	0
$382$	2.43452e6	0.853601
$383$	−2.36189e6	−0.822740	−0.411370	−	0.911469i	$-0.634950\pi$
$383$	−2.36189e6	−0.822740	−0.411370	+	0.911469i	$0.634950\pi$
$384$	0	0
$385$	0	0
$386$	−3.10273e6	−1.05993
$387$	0	0
$388$	34398.0	0.0115999
$389$	3.53390e6	1.18408	0.592039	−	0.805910i	$-0.298323\pi$
$389$	3.53390e6	1.18408	0.592039	+	0.805910i	$0.298323\pi$
$390$	0	0
$391$	−1.16467e6	−0.385267
$392$	0	0
$393$	0	0
$394$	−1.18285e6	−0.383874
$395$	9.15786e6	2.95326
$396$	0	0
$397$	−4.04811e6	−1.28907	−0.644534	−	0.764575i	$-0.722949\pi$
$397$	−4.04811e6	−1.28907	−0.644534	+	0.764575i	$0.722949\pi$
$398$	410520.	0.129905
$399$	0	0
$400$	−4.28896e6	−1.34030
$401$	−2.07645e6	−0.644853	−0.322426	−	0.946595i	$-0.604498\pi$
$401$	−2.07645e6	−0.644853	−0.322426	+	0.946595i	$0.604498\pi$
$402$	0	0
$403$	−4.87872e6	−1.49638
$404$	1.16448e6	0.354959
$405$	0	0
$406$	0	0
$407$	381576.	0.114181
$408$	0	0
$409$	−2.57431e6	−0.760945	−0.380472	−	0.924792i	$-0.624239\pi$
$409$	−2.57431e6	−0.760945	−0.380472	+	0.924792i	$0.624239\pi$
$410$	1.53314e6	0.450425
$411$	0	0
$412$	1.10012e6	0.319299
$413$	0	0
$414$	0	0
$415$	−7.63543e6	−2.17627
$416$	−1.91345e6	−0.542105
$417$	0	0
$418$	−450320.	−0.126061
$419$	848148.	0.236013	0.118007	−	0.993013i	$-0.462350\pi$
$419$	848148.	0.236013	0.118007	+	0.993013i	$0.462350\pi$
$420$	0	0
$421$	1.43682e6	0.395092	0.197546	−	0.980294i	$-0.436703\pi$
$421$	1.43682e6	0.395092	0.197546	+	0.980294i	$0.436703\pi$
$422$	−499460.	−0.136527
$423$	0	0
$424$	−815490.	−0.220295
$425$	−1.15476e7	−3.10114
$426$	0	0
$427$	0	0
$428$	−47348.0	−0.0124937
$429$	0	0
$430$	−2.54740e6	−0.664394
$431$	−2.35438e6	−0.610496	−0.305248	−	0.952273i	$-0.598739\pi$
$431$	−2.35438e6	−0.610496	−0.305248	+	0.952273i	$0.598739\pi$
$432$	0	0
$433$	3.78808e6	0.970955	0.485478	−	0.874249i	$-0.338646\pi$
$433$	3.78808e6	0.970955	0.485478	+	0.874249i	$0.338646\pi$
$434$	0	0
$435$	0	0
$436$	−1.24879e6	−0.314609
$437$	−997632.	−0.249900
$438$	0	0
$439$	3.64322e6	0.902245	0.451123	−	0.892462i	$-0.351024\pi$
$439$	3.64322e6	0.902245	0.451123	+	0.892462i	$0.351024\pi$
$440$	−953160.	−0.234711
$441$	0	0
$442$	7.78470e6	1.89534
$443$	−2.48389e6	−0.601345	−0.300672	−	0.953728i	$-0.597211\pi$
$443$	−2.48389e6	−0.601345	−0.300672	+	0.953728i	$0.597211\pi$
$444$	0	0
$445$	−299108.	−0.0716025
$446$	−933520.	−0.222222
$447$	0	0
$448$	0	0
$449$	2.63177e6	0.616074	0.308037	−	0.951374i	$-0.400328\pi$
$449$	2.63177e6	0.616074	0.308037	+	0.951374i	$0.400328\pi$
$450$	0	0
$451$	169624.	0.0392686
$452$	−315938.	−0.0727371
$453$	0	0
$454$	−1.68186e6	−0.382957
$455$	0	0
$456$	0	0
$457$	−1.16130e6	−0.260109	−0.130054	−	0.991507i	$-0.541515\pi$
$457$	−1.16130e6	−0.260109	−0.130054	+	0.991507i	$0.541515\pi$
$458$	−4.63157e6	−1.03173
$459$	0	0
$460$	−379008.	−0.0835129
$461$	2.81385e6	0.616663	0.308332	−	0.951279i	$-0.400229\pi$
$461$	2.81385e6	0.616663	0.308332	+	0.951279i	$0.400229\pi$
$462$	0	0
$463$	6.84299e6	1.48352	0.741760	−	0.670665i	$-0.233991\pi$
$463$	6.84299e6	1.48352	0.741760	+	0.670665i	$0.233991\pi$
$464$	4.14402e6	0.893566
$465$	0	0
$466$	6.28557e6	1.34085
$467$	3.34314e6	0.709353	0.354676	−	0.934989i	$-0.384591\pi$
$467$	3.34314e6	0.709353	0.354676	+	0.934989i	$0.384591\pi$
$468$	0	0
$469$	0	0
$470$	−406080.	−0.0847944
$471$	0	0
$472$	−2.18790e6	−0.452035
$473$	−281840.	−0.0579228
$474$	0	0
$475$	−9.89145e6	−2.01153
$476$	0	0
$477$	0	0
$478$	−1.73508e6	−0.347336
$479$	−4.28248e6	−0.852818	−0.426409	−	0.904530i	$-0.640222\pi$
$479$	−4.28248e6	−0.852818	−0.426409	+	0.904530i	$0.640222\pi$
$480$	0	0
$481$	−5.65026e6	−1.11354
$482$	495850.	0.0972149
$483$	0	0
$484$	1.10843e6	0.215077
$485$	−461916.	−0.0891679
$486$	0	0
$487$	−8.93175e6	−1.70653	−0.853266	−	0.521477i	$-0.825381\pi$
$487$	−8.93175e6	−1.70653	−0.853266	+	0.521477i	$0.825381\pi$
$488$	8.89239e6	1.69032
$489$	0	0
$490$	0	0
$491$	−2.75306e6	−0.515361	−0.257681	−	0.966230i	$-0.582958\pi$
$491$	−2.75306e6	−0.515361	−0.257681	+	0.966230i	$0.582958\pi$
$492$	0	0
$493$	1.11574e7	2.06750
$494$	6.66820e6	1.22939
$495$	0	0
$496$	4.75834e6	0.868462
$497$	0	0
$498$	0	0
$499$	4.80408e6	0.863693	0.431846	−	0.901947i	$-0.357862\pi$
$499$	4.80408e6	0.863693	0.431846	+	0.901947i	$0.357862\pi$
$500$	−1.70159e6	−0.304389
$501$	0	0
$502$	−1.72214e6	−0.305007
$503$	−6.02465e6	−1.06172	−0.530862	−	0.847458i	$-0.678132\pi$
$503$	−6.02465e6	−1.06172	−0.530862	+	0.847458i	$0.678132\pi$
$504$	0	0
$505$	−1.56373e7	−2.72855
$506$	149760.	0.0260028
$507$	0	0
$508$	1.43539e6	0.246781
$509$	−8.42987e6	−1.44220	−0.721101	−	0.692830i	$-0.756364\pi$
$509$	−8.42987e6	−1.44220	−0.721101	+	0.692830i	$0.756364\pi$
$510$	0	0
$511$	0	0
$512$	6.55248e6	1.10466
$513$	0	0
$514$	−1.47565e6	−0.246363
$515$	−1.47730e7	−2.45444
$516$	0	0
$517$	−44928.0	−0.00739249
$518$	0	0
$519$	0	0
$520$	1.41141e7	2.28899
$521$	9.25058e6	1.49305	0.746525	−	0.665357i	$-0.231721\pi$
$521$	9.25058e6	1.49305	0.746525	+	0.665357i	$0.231721\pi$
$522$	0	0
$523$	−5.84494e6	−0.934385	−0.467192	−	0.884156i	$-0.654734\pi$
$523$	−5.84494e6	−0.934385	−0.467192	+	0.884156i	$0.654734\pi$
$524$	−510748.	−0.0812603
$525$	0	0
$526$	−6.36232e6	−1.00265
$527$	1.28114e7	2.00942
$528$	0	0
$529$	−6.10457e6	−0.948453
$530$	1.96554e6	0.303943
$531$	0	0
$532$	0	0
$533$	−2.51174e6	−0.382963
$534$	0	0
$535$	635816.	0.0960389
$536$	272220.	0.0409268
$537$	0	0
$538$	−1.38387e6	−0.206129
$539$	0	0
$540$	0	0
$541$	9.22533e6	1.35515	0.677577	−	0.735452i	$-0.263030\pi$
$541$	9.22533e6	1.35515	0.677577	+	0.735452i	$0.263030\pi$
$542$	−6.44968e6	−0.943061
$543$	0	0
$544$	5.02467e6	0.727965
$545$	1.67694e7	2.41839
$546$	0	0
$547$	−6.44337e6	−0.920757	−0.460378	−	0.887723i	$-0.652286\pi$
$547$	−6.44337e6	−0.920757	−0.460378	+	0.887723i	$0.652286\pi$
$548$	−659274.	−0.0937809
$549$	0	0
$550$	1.48486e6	0.209305
$551$	9.55718e6	1.34107
$552$	0	0
$553$	0	0
$554$	−8.58275e6	−1.18810
$555$	0	0
$556$	−334572.	−0.0458989
$557$	−3.74213e6	−0.511070	−0.255535	−	0.966800i	$-0.582252\pi$
$557$	−3.74213e6	−0.511070	−0.255535	+	0.966800i	$0.582252\pi$
$558$	0	0
$559$	4.17340e6	0.564886
$560$	0	0
$561$	0	0
$562$	−7.36091e6	−0.983084
$563$	−1.46384e7	−1.94635	−0.973176	−	0.230060i	$-0.926108\pi$
$563$	−1.46384e7	−1.94635	−0.973176	+	0.230060i	$0.926108\pi$
$564$	0	0
$565$	4.24260e6	0.559127
$566$	5.14406e6	0.674940
$567$	0	0
$568$	−3.65040e6	−0.474755
$569$	−1.41805e7	−1.83616	−0.918078	−	0.396400i	$-0.870259\pi$
$569$	−1.41805e7	−1.83616	−0.918078	+	0.396400i	$0.870259\pi$
$570$	0	0
$571$	−1.25160e6	−0.160648	−0.0803242	−	0.996769i	$-0.525596\pi$
$571$	−1.25160e6	−0.160648	−0.0803242	+	0.996769i	$0.525596\pi$
$572$	280280.	0.0358181
$573$	0	0
$574$	0	0
$575$	3.28954e6	0.414921
$576$	0	0
$577$	−5.94378e6	−0.743230	−0.371615	−	0.928387i	$-0.621196\pi$
$577$	−5.94378e6	−0.743230	−0.371615	+	0.928387i	$0.621196\pi$
$578$	−1.33431e7	−1.66126
$579$	0	0
$580$	3.63084e6	0.448165
$581$	0	0
$582$	0	0
$583$	217464.	0.0264982
$584$	−9.04059e6	−1.09689
$585$	0	0
$586$	5.93033e6	0.713403
$587$	−6.46192e6	−0.774046	−0.387023	−	0.922070i	$-0.626497\pi$
$587$	−6.46192e6	−0.774046	−0.387023	+	0.922070i	$0.626497\pi$
$588$	0	0
$589$	1.09740e7	1.30339
$590$	5.27340e6	0.623678
$591$	0	0
$592$	5.51084e6	0.646269
$593$	−2.34605e6	−0.273969	−0.136984	−	0.990573i	$-0.543741\pi$
$593$	−2.34605e6	−0.273969	−0.136984	+	0.990573i	$0.543741\pi$
$594$	0	0
$595$	0	0
$596$	−869862.	−0.100308
$597$	0	0
$598$	−2.21760e6	−0.253589
$599$	1.34959e7	1.53686	0.768432	−	0.639931i	$-0.221037\pi$
$599$	1.34959e7	1.53686	0.768432	+	0.639931i	$0.221037\pi$
$600$	0	0
$601$	−3.87849e6	−0.438002	−0.219001	−	0.975725i	$-0.570280\pi$
$601$	−3.87849e6	−0.438002	−0.219001	+	0.975725i	$0.570280\pi$
$602$	0	0
$603$	0	0
$604$	3.12407e6	0.348441
$605$	−1.48846e7	−1.65329
$606$	0	0
$607$	533488.	0.0587696	0.0293848	−	0.999568i	$-0.490645\pi$
$607$	533488.	0.0587696	0.0293848	+	0.999568i	$0.490645\pi$
$608$	4.30402e6	0.472188
$609$	0	0
$610$	−2.14329e7	−2.33215
$611$	665280.	0.0720944
$612$	0	0
$613$	5.14610e6	0.553130	0.276565	−	0.960995i	$-0.410804\pi$
$613$	5.14610e6	0.553130	0.276565	+	0.960995i	$0.410804\pi$
$614$	7.59458e6	0.812986
$615$	0	0
$616$	0	0
$617$	2.37860e6	0.251541	0.125770	−	0.992059i	$-0.459860\pi$
$617$	2.37860e6	0.251541	0.125770	+	0.992059i	$0.459860\pi$
$618$	0	0
$619$	−1.60023e7	−1.67863	−0.839317	−	0.543642i	$-0.817045\pi$
$619$	−1.60023e7	−1.67863	−0.839317	+	0.543642i	$0.817045\pi$
$620$	4.16909e6	0.435574
$621$	0	0
$622$	−1.06404e6	−0.110276
$623$	0	0
$624$	0	0
$625$	5.00302e6	0.512309
$626$	−9470.00	−0.000965860 0
$627$	0	0
$628$	−1.11822e6	−0.113143
$629$	1.48374e7	1.49531
$630$	0	0
$631$	1.23459e7	1.23439	0.617193	−	0.786812i	$-0.288270\pi$
$631$	1.23459e7	1.23439	0.617193	+	0.786812i	$0.288270\pi$
$632$	1.89977e7	1.89194
$633$	0	0
$634$	−7.89489e6	−0.780051
$635$	−1.92753e7	−1.89699
$636$	0	0
$637$	0	0
$638$	−1.43468e6	−0.139541
$639$	0	0
$640$	−9.65991e6	−0.932230
$641$	3.43755e6	0.330449	0.165224	−	0.986256i	$-0.447165\pi$
$641$	3.43755e6	0.330449	0.165224	+	0.986256i	$0.447165\pi$
$642$	0	0
$643$	1.62191e7	1.54703	0.773515	−	0.633778i	$-0.218497\pi$
$643$	1.62191e7	1.54703	0.773515	+	0.633778i	$0.218497\pi$
$644$	0	0
$645$	0	0
$646$	−1.75105e7	−1.65089
$647$	1.19929e7	1.12632	0.563160	−	0.826348i	$-0.309585\pi$
$647$	1.19929e7	1.12632	0.563160	+	0.826348i	$0.309585\pi$
$648$	0	0
$649$	583440.	0.0543731
$650$	−2.19873e7	−2.04122
$651$	0	0
$652$	−1.73076e6	−0.159448
$653$	−1.58009e6	−0.145011	−0.0725053	−	0.997368i	$-0.523099\pi$
$653$	−1.58009e6	−0.145011	−0.0725053	+	0.997368i	$0.523099\pi$
$654$	0	0
$655$	6.85862e6	0.624645
$656$	2.44976e6	0.222262
$657$	0	0
$658$	0	0
$659$	−6.98358e6	−0.626419	−0.313209	−	0.949684i	$-0.601404\pi$
$659$	−6.98358e6	−0.626419	−0.313209	+	0.949684i	$0.601404\pi$
$660$	0	0
$661$	−3.69602e6	−0.329027	−0.164513	−	0.986375i	$-0.552605\pi$
$661$	−3.69602e6	−0.329027	−0.164513	+	0.986375i	$0.552605\pi$
$662$	1.69735e7	1.50532
$663$	0	0
$664$	−1.58395e7	−1.39418
$665$	0	0
$666$	0	0
$667$	−3.17837e6	−0.276624
$668$	4.79142e6	0.415454
$669$	0	0
$670$	−656120.	−0.0564672
$671$	−2.37130e6	−0.203320
$672$	0	0
$673$	1.84688e6	0.157182	0.0785908	−	0.996907i	$-0.474958\pi$
$673$	1.84688e6	0.157182	0.0785908	+	0.996907i	$0.474958\pi$
$674$	−1.01366e7	−0.859491
$675$	0	0
$676$	−1.55125e6	−0.130561
$677$	−7.68501e6	−0.644426	−0.322213	−	0.946667i	$-0.604427\pi$
$677$	−7.68501e6	−0.644426	−0.322213	+	0.946667i	$0.604427\pi$
$678$	0	0
$679$	0	0
$680$	−3.70633e7	−3.07377
$681$	0	0
$682$	−1.64736e6	−0.135621
$683$	−7.12180e6	−0.584168	−0.292084	−	0.956393i	$-0.594349\pi$
$683$	−7.12180e6	−0.584168	−0.292084	+	0.956393i	$0.594349\pi$
$684$	0	0
$685$	8.85311e6	0.720891
$686$	0	0
$687$	0	0
$688$	−4.07042e6	−0.327845
$689$	−3.22014e6	−0.258420
$690$	0	0
$691$	3.23787e6	0.257967	0.128983	−	0.991647i	$-0.458829\pi$
$691$	3.23787e6	0.257967	0.128983	+	0.991647i	$0.458829\pi$
$692$	4.27332e6	0.339234
$693$	0	0
$694$	1.74443e7	1.37485
$695$	4.49282e6	0.352823
$696$	0	0
$697$	6.59576e6	0.514260
$698$	4.82783e6	0.375071
$699$	0	0
$700$	0	0
$701$	7.39163e6	0.568127	0.284063	−	0.958805i	$-0.408317\pi$
$701$	7.39163e6	0.568127	0.284063	+	0.958805i	$0.408317\pi$
$702$	0	0
$703$	1.27094e7	0.969923
$704$	−1.89576e6	−0.144163
$705$	0	0
$706$	−5.76965e6	−0.435650
$707$	0	0
$708$	0	0
$709$	−5.33361e6	−0.398479	−0.199240	−	0.979951i	$-0.563847\pi$
$709$	−5.33361e6	−0.398479	−0.199240	+	0.979951i	$0.563847\pi$
$710$	8.79840e6	0.655025
$711$	0	0
$712$	−620490.	−0.0458706
$713$	−3.64954e6	−0.268852
$714$	0	0
$715$	−3.76376e6	−0.275332
$716$	4.63576e6	0.337939
$717$	0	0
$718$	8.05552e6	0.583153
$719$	−1.14564e7	−0.826468	−0.413234	−	0.910625i	$-0.635601\pi$
$719$	−1.14564e7	−0.826468	−0.413234	+	0.910625i	$0.635601\pi$
$720$	0	0
$721$	0	0
$722$	−2.61862e6	−0.186952
$723$	0	0
$724$	1.08241e6	0.0767442
$725$	−3.15133e7	−2.22663
$726$	0	0
$727$	2.49540e7	1.75107	0.875536	−	0.483153i	$-0.160508\pi$
$727$	2.49540e7	1.75107	0.875536	+	0.483153i	$0.160508\pi$
$728$	0	0
$729$	0	0
$730$	2.17901e7	1.51340
$731$	−1.09592e7	−0.758555
$732$	0	0
$733$	1.43398e7	0.985789	0.492894	−	0.870089i	$-0.335939\pi$
$733$	1.43398e7	0.985789	0.492894	+	0.870089i	$0.335939\pi$
$734$	1.83874e7	1.25974
$735$	0	0
$736$	−1.43136e6	−0.0973990
$737$	−72592.0	−0.00492289
$738$	0	0
$739$	922932.	0.0621668	0.0310834	−	0.999517i	$-0.490104\pi$
$739$	922932.	0.0621668	0.0310834	+	0.999517i	$0.490104\pi$
$740$	4.82840e6	0.324134
$741$	0	0
$742$	0	0
$743$	9.38995e6	0.624010	0.312005	−	0.950081i	$-0.398999\pi$
$743$	9.38995e6	0.624010	0.312005	+	0.950081i	$0.398999\pi$
$744$	0	0
$745$	1.16810e7	0.771062
$746$	−3.24883e6	−0.213737
$747$	0	0
$748$	−736008.	−0.0480982
$749$	0	0
$750$	0	0
$751$	−408032.	−0.0263994	−0.0131997	−	0.999913i	$-0.504202\pi$
$751$	−408032.	−0.0263994	−0.0131997	+	0.999913i	$0.504202\pi$
$752$	−648864.	−0.0418417
$753$	0	0
$754$	2.12443e7	1.36086
$755$	−4.19518e7	−2.67845
$756$	0	0
$757$	2.59605e7	1.64654	0.823271	−	0.567649i	$-0.192147\pi$
$757$	2.59605e7	1.64654	0.823271	+	0.567649i	$0.192147\pi$
$758$	−1.60350e6	−0.101367
$759$	0	0
$760$	−3.17476e7	−1.99378
$761$	1.83554e7	1.14895	0.574477	−	0.818521i	$-0.305206\pi$
$761$	1.83554e7	1.14895	0.574477	+	0.818521i	$0.305206\pi$
$762$	0	0
$763$	0	0
$764$	3.40833e6	0.211255
$765$	0	0
$766$	1.18094e7	0.727206
$767$	−8.63940e6	−0.530268
$768$	0	0
$769$	747166.	0.0455618	0.0227809	−	0.999740i	$-0.492748\pi$
$769$	747166.	0.0455618	0.0227809	+	0.999740i	$0.492748\pi$
$770$	0	0
$771$	0	0
$772$	−4.34382e6	−0.262318
$773$	2.02692e7	1.22008	0.610038	−	0.792372i	$-0.291154\pi$
$773$	2.02692e7	1.22008	0.610038	+	0.792372i	$0.291154\pi$
$774$	0	0
$775$	−3.61849e7	−2.16408
$776$	−958230.	−0.0571236
$777$	0	0
$778$	−1.76695e7	−1.04659
$779$	5.64978e6	0.333571
$780$	0	0
$781$	973440.	0.0571060
$782$	5.82336e6	0.340531
$783$	0	0
$784$	0	0
$785$	1.50161e7	0.869729
$786$	0	0
$787$	4.69982e6	0.270486	0.135243	−	0.990812i	$-0.456819\pi$
$787$	4.69982e6	0.270486	0.135243	+	0.990812i	$0.456819\pi$
$788$	−1.65599e6	−0.0950041
$789$	0	0
$790$	−4.57893e7	−2.61033
$791$	0	0
$792$	0	0
$793$	3.51135e7	1.98286
$794$	2.02406e7	1.13939
$795$	0	0
$796$	574728.	0.0321499
$797$	584710.	0.0326058	0.0163029	−	0.999867i	$-0.494810\pi$
$797$	584710.	0.0326058	0.0163029	+	0.999867i	$0.494810\pi$
$798$	0	0
$799$	−1.74701e6	−0.0968117
$800$	−1.41918e7	−0.783996
$801$	0	0
$802$	1.03822e7	0.569975
$803$	2.41082e6	0.131940
$804$	0	0
$805$	0	0
$806$	2.43936e7	1.32263
$807$	0	0
$808$	−3.24390e7	−1.74799
$809$	1.64013e7	0.881061	0.440531	−	0.897738i	$-0.354790\pi$
$809$	1.64013e7	0.881061	0.440531	+	0.897738i	$0.354790\pi$
$810$	0	0
$811$	304948.	0.0162807	0.00814036	−	0.999967i	$-0.497409\pi$
$811$	304948.	0.0162807	0.00814036	+	0.999967i	$0.497409\pi$
$812$	0	0
$813$	0	0
$814$	−1.90788e6	−0.100923
$815$	2.32417e7	1.22567
$816$	0	0
$817$	−9.38744e6	−0.492031
$818$	1.28716e7	0.672587
$819$	0	0
$820$	2.14640e6	0.111474
$821$	−3.43428e7	−1.77819	−0.889095	−	0.457722i	$-0.848665\pi$
$821$	−3.43428e7	−1.77819	−0.889095	+	0.457722i	$0.848665\pi$
$822$	0	0
$823$	1.56684e7	0.806351	0.403176	−	0.915123i	$-0.367906\pi$
$823$	1.56684e7	0.806351	0.403176	+	0.915123i	$0.367906\pi$
$824$	−3.06462e7	−1.57238
$825$	0	0
$826$	0	0
$827$	2.96886e7	1.50948	0.754738	−	0.656026i	$-0.227764\pi$
$827$	2.96886e7	1.50948	0.754738	+	0.656026i	$0.227764\pi$
$828$	0	0
$829$	2.30708e7	1.16594	0.582970	−	0.812494i	$-0.301890\pi$
$829$	2.30708e7	1.16594	0.582970	+	0.812494i	$0.301890\pi$
$830$	3.81772e7	1.92357
$831$	0	0
$832$	2.80719e7	1.40593
$833$	0	0
$834$	0	0
$835$	−6.43419e7	−3.19358
$836$	−630448.	−0.0311985
$837$	0	0
$838$	−4.24074e6	−0.208608
$839$	2.32642e7	1.14100	0.570498	−	0.821299i	$-0.306750\pi$
$839$	2.32642e7	1.14100	0.570498	+	0.821299i	$0.306750\pi$
$840$	0	0
$841$	9.93718e6	0.484477
$842$	−7.18411e6	−0.349215
$843$	0	0
$844$	−699244.	−0.0337888
$845$	2.08311e7	1.00362
$846$	0	0
$847$	0	0
$848$	3.14068e6	0.149980
$849$	0	0
$850$	5.77382e7	2.74104
$851$	−4.22669e6	−0.200067
$852$	0	0
$853$	−1.91515e7	−0.901219	−0.450610	−	0.892721i	$-0.648793\pi$
$853$	−1.91515e7	−0.901219	−0.450610	+	0.892721i	$0.648793\pi$
$854$	0	0
$855$	0	0
$856$	1.31898e6	0.0615253
$857$	−5.34683e6	−0.248682	−0.124341	−	0.992240i	$-0.539682\pi$
$857$	−5.34683e6	−0.248682	−0.124341	+	0.992240i	$0.539682\pi$
$858$	0	0
$859$	−3.95858e7	−1.83045	−0.915223	−	0.402948i	$-0.867986\pi$
$859$	−3.95858e7	−1.83045	−0.915223	+	0.402948i	$0.867986\pi$
$860$	−3.56636e6	−0.164429
$861$	0	0
$862$	1.17719e7	0.539607
$863$	2.50284e7	1.14395	0.571973	−	0.820272i	$-0.306178\pi$
$863$	2.50284e7	1.14395	0.571973	+	0.820272i	$0.306178\pi$
$864$	0	0
$865$	−5.73846e7	−2.60768
$866$	−1.89404e7	−0.858211
$867$	0	0
$868$	0	0
$869$	−5.06605e6	−0.227573
$870$	0	0
$871$	1.07492e6	0.0480099
$872$	3.47876e7	1.54929
$873$	0	0
$874$	4.98816e6	0.220883
$875$	0	0
$876$	0	0
$877$	−5.02589e6	−0.220655	−0.110328	−	0.993895i	$-0.535190\pi$
$877$	−5.02589e6	−0.220655	−0.110328	+	0.993895i	$0.535190\pi$
$878$	−1.82161e7	−0.797480
$879$	0	0
$880$	3.67089e6	0.159795
$881$	−2.60490e7	−1.13071	−0.565356	−	0.824847i	$-0.691261\pi$
$881$	−2.60490e7	−1.13071	−0.565356	+	0.824847i	$0.691261\pi$
$882$	0	0
$883$	−6.82462e6	−0.294562	−0.147281	−	0.989095i	$-0.547052\pi$
$883$	−6.82462e6	−0.294562	−0.147281	+	0.989095i	$0.547052\pi$
$884$	1.08986e7	0.469072
$885$	0	0
$886$	1.24195e7	0.531519
$887$	2.33835e7	0.997931	0.498965	−	0.866622i	$-0.333713\pi$
$887$	2.33835e7	0.997931	0.498965	+	0.866622i	$0.333713\pi$
$888$	0	0
$889$	0	0
$890$	1.49554e6	0.0632882
$891$	0	0
$892$	−1.30693e6	−0.0549971
$893$	−1.49645e6	−0.0627961
$894$	0	0
$895$	−6.22517e7	−2.59773
$896$	0	0
$897$	0	0
$898$	−1.31589e7	−0.544537
$899$	3.49620e7	1.44277
$900$	0	0
$901$	8.45600e6	0.347019
$902$	−848120.	−0.0347089
$903$	0	0
$904$	8.80113e6	0.358193
$905$	−1.45352e7	−0.589930
$906$	0	0
$907$	3.95959e7	1.59820	0.799102	−	0.601196i	$-0.205309\pi$
$907$	3.95959e7	1.59820	0.799102	+	0.601196i	$0.205309\pi$
$908$	−2.35460e6	−0.0947771
$909$	0	0
$910$	0	0
$911$	4.67570e6	0.186660	0.0933300	−	0.995635i	$-0.470249\pi$
$911$	4.67570e6	0.186660	0.0933300	+	0.995635i	$0.470249\pi$
$912$	0	0
$913$	4.22386e6	0.167700
$914$	5.80651e6	0.229906
$915$	0	0
$916$	−6.48420e6	−0.255339
$917$	0	0
$918$	0	0
$919$	−4.92594e6	−0.192398	−0.0961990	−	0.995362i	$-0.530669\pi$
$919$	−4.92594e6	−0.192398	−0.0961990	+	0.995362i	$0.530669\pi$
$920$	1.05581e7	0.411259
$921$	0	0
$922$	−1.40692e7	−0.545058
$923$	−1.44144e7	−0.556919
$924$	0	0
$925$	−4.19073e7	−1.61041
$926$	−3.42150e7	−1.31126
$927$	0	0
$928$	1.37122e7	0.522683
$929$	−3.23688e7	−1.23052	−0.615258	−	0.788326i	$-0.710948\pi$
$929$	−3.23688e7	−1.23052	−0.615258	+	0.788326i	$0.710948\pi$
$930$	0	0
$931$	0	0
$932$	8.79980e6	0.331843
$933$	0	0
$934$	−1.67157e7	−0.626985
$935$	9.88354e6	0.369729
$936$	0	0
$937$	3.32337e7	1.23660	0.618301	−	0.785941i	$-0.287821\pi$
$937$	3.32337e7	1.23660	0.618301	+	0.785941i	$0.287821\pi$
$938$	0	0
$939$	0	0
$940$	−568512.	−0.0209855
$941$	−2.66426e7	−0.980852	−0.490426	−	0.871483i	$-0.663159\pi$
$941$	−2.66426e7	−0.980852	−0.490426	+	0.871483i	$0.663159\pi$
$942$	0	0
$943$	−1.87891e6	−0.0688061
$944$	8.42622e6	0.307753
$945$	0	0
$946$	1.40920e6	0.0511970
$947$	−3.14663e7	−1.14017	−0.570086	−	0.821585i	$-0.693090\pi$
$947$	−3.14663e7	−1.14017	−0.570086	+	0.821585i	$0.693090\pi$
$948$	0	0
$949$	−3.56987e7	−1.28673
$950$	4.94573e7	1.77796
$951$	0	0
$952$	0	0
$953$	1.34516e7	0.479779	0.239890	−	0.970800i	$-0.422889\pi$
$953$	1.34516e7	0.479779	0.239890	+	0.970800i	$0.422889\pi$
$954$	0	0
$955$	−4.57690e7	−1.62391
$956$	−2.42911e6	−0.0859613
$957$	0	0
$958$	2.14124e7	0.753792
$959$	0	0
$960$	0	0
$961$	1.15157e7	0.402238
$962$	2.82513e7	0.984239
$963$	0	0
$964$	694190.	0.0240595
$965$	5.83313e7	2.01643
$966$	0	0
$967$	−2.84963e7	−0.979992	−0.489996	−	0.871725i	$-0.663002\pi$
$967$	−2.84963e7	−0.979992	−0.489996	+	0.871725i	$0.663002\pi$
$968$	−3.08777e7	−1.05915
$969$	0	0
$970$	2.30958e6	0.0788141
$971$	1.81858e7	0.618990	0.309495	−	0.950901i	$-0.399840\pi$
$971$	1.81858e7	0.618990	0.309495	+	0.950901i	$0.399840\pi$
$972$	0	0
$973$	0	0
$974$	4.46588e7	1.50837
$975$	0	0
$976$	−3.42471e7	−1.15080
$977$	−3.20941e7	−1.07569	−0.537847	−	0.843042i	$-0.680762\pi$
$977$	−3.20941e7	−1.07569	−0.537847	+	0.843042i	$0.680762\pi$
$978$	0	0
$979$	165464.	0.00551756
$980$	0	0
$981$	0	0
$982$	1.37653e7	0.455519
$983$	1.56154e7	0.515429	0.257715	−	0.966221i	$-0.417031\pi$
$983$	1.56154e7	0.515429	0.257715	+	0.966221i	$0.417031\pi$
$984$	0	0
$985$	2.22376e7	0.730293
$986$	−5.57870e7	−1.82743
$987$	0	0
$988$	9.33548e6	0.304260
$989$	3.12192e6	0.101492
$990$	0	0
$991$	4.84499e7	1.56714	0.783572	−	0.621301i	$-0.213395\pi$
$991$	4.84499e7	1.56714	0.783572	+	0.621301i	$0.213395\pi$
$992$	1.57450e7	0.507998
$993$	0	0
$994$	0	0
$995$	−7.71778e6	−0.247135
$996$	0	0
$997$	4.54336e7	1.44757	0.723784	−	0.690027i	$-0.242401\pi$
$997$	4.54336e7	1.44757	0.723784	+	0.690027i	$0.242401\pi$
$998$	−2.40204e7	−0.763404
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.c.1.1		1
3.2	odd	2		147.6.a.f.1.1		1
7.6	odd	2		63.6.a.b.1.1		1
21.2	odd	6		147.6.e.d.67.1		2
21.5	even	6		147.6.e.c.67.1		2
21.11	odd	6		147.6.e.d.79.1		2
21.17	even	6		147.6.e.c.79.1		2
21.20	even	2		21.6.a.c.1.1	✓	1
28.27	even	2		1008.6.a.a.1.1		1
84.83	odd	2		336.6.a.i.1.1		1
105.62	odd	4		525.6.d.c.274.2		2
105.83	odd	4		525.6.d.c.274.1		2
105.104	even	2		525.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.c.1.1	✓	1	21.20	even	2
63.6.a.b.1.1		1	7.6	odd	2
147.6.a.f.1.1		1	3.2	odd	2
147.6.e.c.67.1		2	21.5	even	6
147.6.e.c.79.1		2	21.17	even	6
147.6.e.d.67.1		2	21.2	odd	6
147.6.e.d.79.1		2	21.11	odd	6
336.6.a.i.1.1		1	84.83	odd	2
441.6.a.c.1.1		1	1.1	even	1	trivial
525.6.a.b.1.1		1	105.104	even	2
525.6.d.c.274.1		2	105.83	odd	4
525.6.d.c.274.2		2	105.62	odd	4
1008.6.a.a.1.1		1	28.27	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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