LMFDB - Embedded newform 1089.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,6,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, 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("1089.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1089.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:	$174.657979776$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-6.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +40.0000 q^{7} +168.000 q^{8} +O(q^{10})$

$q-6.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +40.0000 q^{7} +168.000 q^{8} +36.0000 q^{10} -638.000 q^{13} -240.000 q^{14} -1136.00 q^{16} +882.000 q^{17} +556.000 q^{19} -24.0000 q^{20} +840.000 q^{23} -3089.00 q^{25} +3828.00 q^{26} +160.000 q^{28} +4638.00 q^{29} +4400.00 q^{31} +1440.00 q^{32} -5292.00 q^{34} -240.000 q^{35} -2410.00 q^{37} -3336.00 q^{38} -1008.00 q^{40} -6870.00 q^{41} -9644.00 q^{43} -5040.00 q^{46} +18672.0 q^{47} -15207.0 q^{49} +18534.0 q^{50} -2552.00 q^{52} -33750.0 q^{53} +6720.00 q^{56} -27828.0 q^{58} +18084.0 q^{59} -39758.0 q^{61} -26400.0 q^{62} +27712.0 q^{64} +3828.00 q^{65} -23068.0 q^{67} +3528.00 q^{68} +1440.00 q^{70} +4248.00 q^{71} +41110.0 q^{73} +14460.0 q^{74} +2224.00 q^{76} -21920.0 q^{79} +6816.00 q^{80} +41220.0 q^{82} +82452.0 q^{83} -5292.00 q^{85} +57864.0 q^{86} +94086.0 q^{89} -25520.0 q^{91} +3360.00 q^{92} -112032. q^{94} -3336.00 q^{95} +49442.0 q^{97} +91242.0 q^{98} +O(q^{100})$

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 $

$2$	−6.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−6.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	0	0
$3$	0	0
$4$	4.00000	0.125000
$5$	−6.00000	−0.107331	−0.0536656	−	0.998559i	$-0.517091\pi$
$5$	−6.00000	−0.107331	−0.0536656	+	0.998559i	$0.517091\pi$
$6$	0	0
$7$	40.0000	0.308542	0.154271	−	0.988029i	$-0.450697\pi$
$7$	40.0000	0.308542	0.154271	+	0.988029i	$0.450697\pi$
$8$	168.000	0.928078
$9$	0	0
$10$	36.0000	0.113842
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−638.000	−1.04704	−0.523519	−	0.852014i	$-0.675381\pi$
$13$	−638.000	−1.04704	−0.523519	+	0.852014i	$0.675381\pi$
$14$	−240.000	−0.327259
$15$	0	0
$16$	−1136.00	−1.10938
$17$	882.000	0.740195	0.370098	−	0.928993i	$-0.379324\pi$
$17$	882.000	0.740195	0.370098	+	0.928993i	$0.379324\pi$
$18$	0	0
$19$	556.000	0.353338	0.176669	−	0.984270i	$-0.443468\pi$
$19$	556.000	0.353338	0.176669	+	0.984270i	$0.443468\pi$
$20$	−24.0000	−0.0134164
$21$	0	0
$22$	0	0
$23$	840.000	0.331100	0.165550	−	0.986201i	$-0.447060\pi$
$23$	840.000	0.331100	0.165550	+	0.986201i	$0.447060\pi$
$24$	0	0
$25$	−3089.00	−0.988480
$26$	3828.00	1.11055
$27$	0	0
$28$	160.000	0.0385678
$29$	4638.00	1.02408	0.512042	−	0.858960i	$-0.328889\pi$
$29$	4638.00	1.02408	0.512042	+	0.858960i	$0.328889\pi$
$30$	0	0
$31$	4400.00	0.822334	0.411167	−	0.911560i	$-0.365121\pi$
$31$	4400.00	0.822334	0.411167	+	0.911560i	$0.365121\pi$
$32$	1440.00	0.248592
$33$	0	0
$34$	−5292.00	−0.785096
$35$	−240.000	−0.0331162
$36$	0	0
$37$	−2410.00	−0.289409	−0.144705	−	0.989475i	$-0.546223\pi$
$37$	−2410.00	−0.289409	−0.144705	+	0.989475i	$0.546223\pi$
$38$	−3336.00	−0.374772
$39$	0	0
$40$	−1008.00	−0.0996117
$41$	−6870.00	−0.638259	−0.319130	−	0.947711i	$-0.603391\pi$
$41$	−6870.00	−0.638259	−0.319130	+	0.947711i	$0.603391\pi$
$42$	0	0
$43$	−9644.00	−0.795401	−0.397700	−	0.917515i	$-0.630192\pi$
$43$	−9644.00	−0.795401	−0.397700	+	0.917515i	$0.630192\pi$
$44$	0	0
$45$	0	0
$46$	−5040.00	−0.351185
$47$	18672.0	1.23295	0.616476	−	0.787374i	$-0.288560\pi$
$47$	18672.0	1.23295	0.616476	+	0.787374i	$0.288560\pi$
$48$	0	0
$49$	−15207.0	−0.904802
$50$	18534.0	1.04844
$51$	0	0
$52$	−2552.00	−0.130880
$53$	−33750.0	−1.65038	−0.825190	−	0.564855i	$-0.808932\pi$
$53$	−33750.0	−1.65038	−0.825190	+	0.564855i	$0.808932\pi$
$54$	0	0
$55$	0	0
$56$	6720.00	0.286351
$57$	0	0
$58$	−27828.0	−1.08621
$59$	18084.0	0.676339	0.338170	−	0.941085i	$-0.390192\pi$
$59$	18084.0	0.676339	0.338170	+	0.941085i	$0.390192\pi$
$60$	0	0
$61$	−39758.0	−1.36804	−0.684022	−	0.729462i	$-0.739771\pi$
$61$	−39758.0	−1.36804	−0.684022	+	0.729462i	$0.739771\pi$
$62$	−26400.0	−0.872217
$63$	0	0
$64$	27712.0	0.845703
$65$	3828.00	0.112380
$66$	0	0
$67$	−23068.0	−0.627802	−0.313901	−	0.949456i	$-0.601636\pi$
$67$	−23068.0	−0.627802	−0.313901	+	0.949456i	$0.601636\pi$
$68$	3528.00	0.0925244
$69$	0	0
$70$	1440.00	0.0351251
$71$	4248.00	0.100009	0.0500044	−	0.998749i	$-0.484076\pi$
$71$	4248.00	0.100009	0.0500044	+	0.998749i	$0.484076\pi$
$72$	0	0
$73$	41110.0	0.902901	0.451451	−	0.892296i	$-0.350907\pi$
$73$	41110.0	0.902901	0.451451	+	0.892296i	$0.350907\pi$
$74$	14460.0	0.306965
$75$	0	0
$76$	2224.00	0.0441673
$77$	0	0
$78$	0	0
$79$	−21920.0	−0.395160	−0.197580	−	0.980287i	$-0.563308\pi$
$79$	−21920.0	−0.395160	−0.197580	+	0.980287i	$0.563308\pi$
$80$	6816.00	0.119071
$81$	0	0
$82$	41220.0	0.676976
$83$	82452.0	1.31373	0.656865	−	0.754008i	$-0.271882\pi$
$83$	82452.0	1.31373	0.656865	+	0.754008i	$0.271882\pi$
$84$	0	0
$85$	−5292.00	−0.0794461
$86$	57864.0	0.843650
$87$	0	0
$88$	0	0
$89$	94086.0	1.25907	0.629535	−	0.776972i	$-0.283245\pi$
$89$	94086.0	1.25907	0.629535	+	0.776972i	$0.283245\pi$
$90$	0	0
$91$	−25520.0	−0.323056
$92$	3360.00	0.0413875
$93$	0	0
$94$	−112032.	−1.30774
$95$	−3336.00	−0.0379243
$96$	0	0
$97$	49442.0	0.533540	0.266770	−	0.963760i	$-0.414044\pi$
$97$	49442.0	0.533540	0.266770	+	0.963760i	$0.414044\pi$
$98$	91242.0	0.959687
$99$	0	0
$100$	−12356.0	−0.123560
$101$	−143034.	−1.39520	−0.697599	−	0.716488i	$-0.745748\pi$
$101$	−143034.	−1.39520	−0.697599	+	0.716488i	$0.745748\pi$
$102$	0	0
$103$	53144.0	0.493584	0.246792	−	0.969068i	$-0.420624\pi$
$103$	53144.0	0.493584	0.246792	+	0.969068i	$0.420624\pi$
$104$	−107184.	−0.971732
$105$	0	0
$106$	202500.	1.75049
$107$	90828.0	0.766938	0.383469	−	0.923554i	$-0.374729\pi$
$107$	90828.0	0.766938	0.383469	+	0.923554i	$0.374729\pi$
$108$	0	0
$109$	61666.0	0.497141	0.248570	−	0.968614i	$-0.420039\pi$
$109$	61666.0	0.497141	0.248570	+	0.968614i	$0.420039\pi$
$110$	0	0
$111$	0	0
$112$	−45440.0	−0.342289
$113$	−10482.0	−0.0772232	−0.0386116	−	0.999254i	$-0.512294\pi$
$113$	−10482.0	−0.0772232	−0.0386116	+	0.999254i	$0.512294\pi$
$114$	0	0
$115$	−5040.00	−0.0355374
$116$	18552.0	0.128011
$117$	0	0
$118$	−108504.	−0.717366
$119$	35280.0	0.228382
$120$	0	0
$121$	0	0
$122$	238548.	1.45103
$123$	0	0
$124$	17600.0	0.102792
$125$	37284.0	0.213426
$126$	0	0
$127$	171088.	0.941261	0.470631	−	0.882330i	$-0.344026\pi$
$127$	171088.	0.941261	0.470631	+	0.882330i	$0.344026\pi$
$128$	−212352.	−1.14560
$129$	0	0
$130$	−22968.0	−0.119197
$131$	258468.	1.31592	0.657959	−	0.753054i	$-0.271420\pi$
$131$	258468.	1.31592	0.657959	+	0.753054i	$0.271420\pi$
$132$	0	0
$133$	22240.0	0.109020
$134$	138408.	0.665885
$135$	0	0
$136$	148176.	0.686959
$137$	−300234.	−1.36665	−0.683327	−	0.730113i	$-0.739468\pi$
$137$	−300234.	−1.36665	−0.683327	+	0.730113i	$0.739468\pi$
$138$	0	0
$139$	350164.	1.53721	0.768607	−	0.639721i	$-0.220950\pi$
$139$	350164.	1.53721	0.768607	+	0.639721i	$0.220950\pi$
$140$	−960.000	−0.00413953
$141$	0	0
$142$	−25488.0	−0.106075
$143$	0	0
$144$	0	0
$145$	−27828.0	−0.109916
$146$	−246660.	−0.957672
$147$	0	0
$148$	−9640.00	−0.0361762
$149$	−105258.	−0.388409	−0.194205	−	0.980961i	$-0.562213\pi$
$149$	−105258.	−0.388409	−0.194205	+	0.980961i	$0.562213\pi$
$150$	0	0
$151$	−396392.	−1.41476	−0.707380	−	0.706834i	$-0.750123\pi$
$151$	−396392.	−1.41476	−0.707380	+	0.706834i	$0.750123\pi$
$152$	93408.0	0.327925
$153$	0	0
$154$	0	0
$155$	−26400.0	−0.0882622
$156$	0	0
$157$	−137746.	−0.445995	−0.222997	−	0.974819i	$-0.571584\pi$
$157$	−137746.	−0.445995	−0.222997	+	0.974819i	$0.571584\pi$
$158$	131520.	0.419130
$159$	0	0
$160$	−8640.00	−0.0266817
$161$	33600.0	0.102159
$162$	0	0
$163$	352676.	1.03970	0.519849	−	0.854258i	$-0.325988\pi$
$163$	352676.	1.03970	0.519849	+	0.854258i	$0.325988\pi$
$164$	−27480.0	−0.0797824
$165$	0	0
$166$	−494712.	−1.39342
$167$	−217560.	−0.603654	−0.301827	−	0.953363i	$-0.597596\pi$
$167$	−217560.	−0.603654	−0.301827	+	0.953363i	$0.597596\pi$
$168$	0	0
$169$	35751.0	0.0962878
$170$	31752.0	0.0842653
$171$	0	0
$172$	−38576.0	−0.0994251
$173$	−163698.	−0.415842	−0.207921	−	0.978146i	$-0.566670\pi$
$173$	−163698.	−0.415842	−0.207921	+	0.978146i	$0.566670\pi$
$174$	0	0
$175$	−123560.	−0.304988
$176$	0	0
$177$	0	0
$178$	−564516.	−1.33545
$179$	−358740.	−0.836849	−0.418425	−	0.908252i	$-0.637418\pi$
$179$	−358740.	−0.836849	−0.418425	+	0.908252i	$0.637418\pi$
$180$	0	0
$181$	−507130.	−1.15060	−0.575298	−	0.817944i	$-0.695114\pi$
$181$	−507130.	−1.15060	−0.575298	+	0.817944i	$0.695114\pi$
$182$	153120.	0.342652
$183$	0	0
$184$	141120.	0.307287
$185$	14460.0	0.0310627
$186$	0	0
$187$	0	0
$188$	74688.0	0.154119
$189$	0	0
$190$	20016.0	0.0402247
$191$	648384.	1.28602	0.643012	−	0.765856i	$-0.277685\pi$
$191$	648384.	1.28602	0.643012	+	0.765856i	$0.277685\pi$
$192$	0	0
$193$	27838.0	0.0537954	0.0268977	−	0.999638i	$-0.491437\pi$
$193$	27838.0	0.0537954	0.0268977	+	0.999638i	$0.491437\pi$
$194$	−296652.	−0.565904
$195$	0	0
$196$	−60828.0	−0.113100
$197$	611046.	1.12178	0.560891	−	0.827890i	$-0.310459\pi$
$197$	611046.	1.12178	0.560891	+	0.827890i	$0.310459\pi$
$198$	0	0
$199$	879032.	1.57352	0.786760	−	0.617260i	$-0.211757\pi$
$199$	879032.	1.57352	0.786760	+	0.617260i	$0.211757\pi$
$200$	−518952.	−0.917386
$201$	0	0
$202$	858204.	1.47983
$203$	185520.	0.315973
$204$	0	0
$205$	41220.0	0.0685052
$206$	−318864.	−0.523525
$207$	0	0
$208$	724768.	1.16156
$209$	0	0
$210$	0	0
$211$	−48500.0	−0.0749956	−0.0374978	−	0.999297i	$-0.511939\pi$
$211$	−48500.0	−0.0749956	−0.0374978	+	0.999297i	$0.511939\pi$
$212$	−135000.	−0.206298
$213$	0	0
$214$	−544968.	−0.813461
$215$	57864.0	0.0853714
$216$	0	0
$217$	176000.	0.253725
$218$	−369996.	−0.527298
$219$	0	0
$220$	0	0
$221$	−562716.	−0.775012
$222$	0	0
$223$	−999472.	−1.34589	−0.672943	−	0.739694i	$-0.734970\pi$
$223$	−999472.	−1.34589	−0.672943	+	0.739694i	$0.734970\pi$
$224$	57600.0	0.0767012
$225$	0	0
$226$	62892.0	0.0819076
$227$	606180.	0.780795	0.390397	−	0.920646i	$-0.372338\pi$
$227$	606180.	0.780795	0.390397	+	0.920646i	$0.372338\pi$
$228$	0	0
$229$	1.35993e6	1.71367	0.856834	−	0.515593i	$-0.172428\pi$
$229$	1.35993e6	1.71367	0.856834	+	0.515593i	$0.172428\pi$
$230$	30240.0	0.0376931
$231$	0	0
$232$	779184.	0.950430
$233$	−392886.	−0.474107	−0.237054	−	0.971497i	$-0.576182\pi$
$233$	−392886.	−0.474107	−0.237054	+	0.971497i	$0.576182\pi$
$234$	0	0
$235$	−112032.	−0.132334
$236$	72336.0	0.0845424
$237$	0	0
$238$	−211680.	−0.242235
$239$	−1.32514e6	−1.50060	−0.750301	−	0.661096i	$-0.770092\pi$
$239$	−1.32514e6	−1.50060	−0.750301	+	0.661096i	$0.770092\pi$
$240$	0	0
$241$	990094.	1.09808	0.549040	−	0.835796i	$-0.314994\pi$
$241$	990094.	1.09808	0.549040	+	0.835796i	$0.314994\pi$
$242$	0	0
$243$	0	0
$244$	−159032.	−0.171005
$245$	91242.0	0.0971135
$246$	0	0
$247$	−354728.	−0.369959
$248$	739200.	0.763190
$249$	0	0
$250$	−223704.	−0.226373
$251$	−147132.	−0.147409	−0.0737043	−	0.997280i	$-0.523482\pi$
$251$	−147132.	−0.147409	−0.0737043	+	0.997280i	$0.523482\pi$
$252$	0	0
$253$	0	0
$254$	−1.02653e6	−0.998358
$255$	0	0
$256$	387328.	0.369385
$257$	483582.	0.456707	0.228353	−	0.973578i	$-0.426666\pi$
$257$	483582.	0.456707	0.228353	+	0.973578i	$0.426666\pi$
$258$	0	0
$259$	−96400.0	−0.0892951
$260$	15312.0	0.0140475
$261$	0	0
$262$	−1.55081e6	−1.39574
$263$	813576.	0.725285	0.362643	−	0.931928i	$-0.381875\pi$
$263$	813576.	0.725285	0.362643	+	0.931928i	$0.381875\pi$
$264$	0	0
$265$	202500.	0.177137
$266$	−133440.	−0.115633
$267$	0	0
$268$	−92272.0	−0.0784753
$269$	461106.	0.388526	0.194263	−	0.980949i	$-0.437768\pi$
$269$	461106.	0.388526	0.194263	+	0.980949i	$0.437768\pi$
$270$	0	0
$271$	−1.67514e6	−1.38556	−0.692782	−	0.721147i	$-0.743615\pi$
$271$	−1.67514e6	−1.38556	−0.692782	+	0.721147i	$0.743615\pi$
$272$	−1.00195e6	−0.821154
$273$	0	0
$274$	1.80140e6	1.44956
$275$	0	0
$276$	0	0
$277$	−401126.	−0.314110	−0.157055	−	0.987590i	$-0.550200\pi$
$277$	−401126.	−0.314110	−0.157055	+	0.987590i	$0.550200\pi$
$278$	−2.10098e6	−1.63046
$279$	0	0
$280$	−40320.0	−0.0307344
$281$	−2.30977e6	−1.74503	−0.872514	−	0.488590i	$-0.837511\pi$
$281$	−2.30977e6	−1.74503	−0.872514	+	0.488590i	$0.837511\pi$
$282$	0	0
$283$	1.12877e6	0.837800	0.418900	−	0.908032i	$-0.362416\pi$
$283$	1.12877e6	0.837800	0.418900	+	0.908032i	$0.362416\pi$
$284$	16992.0	0.0125011
$285$	0	0
$286$	0	0
$287$	−274800.	−0.196930
$288$	0	0
$289$	−641933.	−0.452111
$290$	166968.	0.116584
$291$	0	0
$292$	164440.	0.112863
$293$	−938874.	−0.638908	−0.319454	−	0.947602i	$-0.603499\pi$
$293$	−938874.	−0.638908	−0.319454	+	0.947602i	$0.603499\pi$
$294$	0	0
$295$	−108504.	−0.0725923
$296$	−404880.	−0.268594
$297$	0	0
$298$	631548.	0.411970
$299$	−535920.	−0.346675
$300$	0	0
$301$	−385760.	−0.245415
$302$	2.37835e6	1.50058
$303$	0	0
$304$	−631616.	−0.391985
$305$	238548.	0.146834
$306$	0	0
$307$	−692948.	−0.419619	−0.209809	−	0.977742i	$-0.567284\pi$
$307$	−692948.	−0.419619	−0.209809	+	0.977742i	$0.567284\pi$
$308$	0	0
$309$	0	0
$310$	158400.	0.0936162
$311$	−2.94310e6	−1.72545	−0.862727	−	0.505670i	$-0.831245\pi$
$311$	−2.94310e6	−1.72545	−0.862727	+	0.505670i	$0.831245\pi$
$312$	0	0
$313$	885146.	0.510686	0.255343	−	0.966851i	$-0.417812\pi$
$313$	885146.	0.510686	0.255343	+	0.966851i	$0.417812\pi$
$314$	826476.	0.473049
$315$	0	0
$316$	−87680.0	−0.0493950
$317$	−2.50880e6	−1.40222	−0.701112	−	0.713051i	$-0.747313\pi$
$317$	−2.50880e6	−1.40222	−0.701112	+	0.713051i	$0.747313\pi$
$318$	0	0
$319$	0	0
$320$	−166272.	−0.0907704
$321$	0	0
$322$	−201600.	−0.108355
$323$	490392.	0.261539
$324$	0	0
$325$	1.97078e6	1.03498
$326$	−2.11606e6	−1.10277
$327$	0	0
$328$	−1.15416e6	−0.592354
$329$	746880.	0.380418
$330$	0	0
$331$	−216148.	−0.108438	−0.0542190	−	0.998529i	$-0.517267\pi$
$331$	−216148.	−0.108438	−0.0542190	+	0.998529i	$0.517267\pi$
$332$	329808.	0.164216
$333$	0	0
$334$	1.30536e6	0.640271
$335$	138408.	0.0673828
$336$	0	0
$337$	−3.25263e6	−1.56012	−0.780062	−	0.625702i	$-0.784813\pi$
$337$	−3.25263e6	−1.56012	−0.780062	+	0.625702i	$0.784813\pi$
$338$	−214506.	−0.102129
$339$	0	0
$340$	−21168.0	−0.00993076
$341$	0	0
$342$	0	0
$343$	−1.28056e6	−0.587712
$344$	−1.62019e6	−0.738194
$345$	0	0
$346$	982188.	0.441067
$347$	−2.93207e6	−1.30723	−0.653613	−	0.756829i	$-0.726747\pi$
$347$	−2.93207e6	−1.30723	−0.653613	+	0.756829i	$0.726747\pi$
$348$	0	0
$349$	−905198.	−0.397814	−0.198907	−	0.980018i	$-0.563739\pi$
$349$	−905198.	−0.397814	−0.198907	+	0.980018i	$0.563739\pi$
$350$	741360.	0.323489
$351$	0	0
$352$	0	0
$353$	−1.91786e6	−0.819181	−0.409590	−	0.912270i	$-0.634328\pi$
$353$	−1.91786e6	−0.819181	−0.409590	+	0.912270i	$0.634328\pi$
$354$	0	0
$355$	−25488.0	−0.0107341
$356$	376344.	0.157384
$357$	0	0
$358$	2.15244e6	0.887613
$359$	−2.43698e6	−0.997968	−0.498984	−	0.866611i	$-0.666293\pi$
$359$	−2.43698e6	−0.997968	−0.498984	+	0.866611i	$0.666293\pi$
$360$	0	0
$361$	−2.16696e6	−0.875152
$362$	3.04278e6	1.22039
$363$	0	0
$364$	−102080.	−0.0403819
$365$	−246660.	−0.0969095
$366$	0	0
$367$	−984064.	−0.381380	−0.190690	−	0.981650i	$-0.561073\pi$
$367$	−984064.	−0.381380	−0.190690	+	0.981650i	$0.561073\pi$
$368$	−954240.	−0.367314
$369$	0	0
$370$	−86760.0	−0.0329470
$371$	−1.35000e6	−0.509212
$372$	0	0
$373$	−1.70365e6	−0.634029	−0.317015	−	0.948421i	$-0.602680\pi$
$373$	−1.70365e6	−0.634029	−0.317015	+	0.948421i	$0.602680\pi$
$374$	0	0
$375$	0	0
$376$	3.13690e6	1.14428
$377$	−2.95904e6	−1.07225
$378$	0	0
$379$	2.75654e6	0.985749	0.492874	−	0.870100i	$-0.335946\pi$
$379$	2.75654e6	0.985749	0.492874	+	0.870100i	$0.335946\pi$
$380$	−13344.0	−0.00474053
$381$	0	0
$382$	−3.89030e6	−1.36403
$383$	−456576.	−0.159044	−0.0795218	−	0.996833i	$-0.525339\pi$
$383$	−456576.	−0.159044	−0.0795218	+	0.996833i	$0.525339\pi$
$384$	0	0
$385$	0	0
$386$	−167028.	−0.0570586
$387$	0	0
$388$	197768.	0.0666925
$389$	2.00639e6	0.672268	0.336134	−	0.941814i	$-0.390881\pi$
$389$	2.00639e6	0.672268	0.336134	+	0.941814i	$0.390881\pi$
$390$	0	0
$391$	740880.	0.245079
$392$	−2.55478e6	−0.839726
$393$	0	0
$394$	−3.66628e6	−1.18983
$395$	131520.	0.0424130
$396$	0	0
$397$	−5.77040e6	−1.83751	−0.918755	−	0.394828i	$-0.870804\pi$
$397$	−5.77040e6	−1.83751	−0.918755	+	0.394828i	$0.870804\pi$
$398$	−5.27419e6	−1.66897
$399$	0	0
$400$	3.50910e6	1.09659
$401$	−3.00626e6	−0.933610	−0.466805	−	0.884360i	$-0.654595\pi$
$401$	−3.00626e6	−0.933610	−0.466805	+	0.884360i	$0.654595\pi$
$402$	0	0
$403$	−2.80720e6	−0.861015
$404$	−572136.	−0.174400
$405$	0	0
$406$	−1.11312e6	−0.335140
$407$	0	0
$408$	0	0
$409$	−1.53363e6	−0.453327	−0.226663	−	0.973973i	$-0.572782\pi$
$409$	−1.53363e6	−0.453327	−0.226663	+	0.973973i	$0.572782\pi$
$410$	−247320.	−0.0726607
$411$	0	0
$412$	212576.	0.0616980
$413$	723360.	0.208679
$414$	0	0
$415$	−494712.	−0.141004
$416$	−918720.	−0.260285
$417$	0	0
$418$	0	0
$419$	3.87376e6	1.07795	0.538973	−	0.842323i	$-0.318812\pi$
$419$	3.87376e6	1.07795	0.538973	+	0.842323i	$0.318812\pi$
$420$	0	0
$421$	−1.33307e6	−0.366561	−0.183281	−	0.983061i	$-0.558672\pi$
$421$	−1.33307e6	−0.366561	−0.183281	+	0.983061i	$0.558672\pi$
$422$	291000.	0.0795448
$423$	0	0
$424$	−5.67000e6	−1.53168
$425$	−2.72450e6	−0.731668
$426$	0	0
$427$	−1.59032e6	−0.422099
$428$	363312.	0.0958673
$429$	0	0
$430$	−347184.	−0.0905500
$431$	6.45192e6	1.67300	0.836500	−	0.547967i	$-0.184598\pi$
$431$	6.45192e6	1.67300	0.836500	+	0.547967i	$0.184598\pi$
$432$	0	0
$433$	−4.16577e6	−1.06777	−0.533883	−	0.845558i	$-0.679268\pi$
$433$	−4.16577e6	−1.06777	−0.533883	+	0.845558i	$0.679268\pi$
$434$	−1.05600e6	−0.269116
$435$	0	0
$436$	246664.	0.0621426
$437$	467040.	0.116990
$438$	0	0
$439$	−792680.	−0.196307	−0.0981537	−	0.995171i	$-0.531294\pi$
$439$	−792680.	−0.196307	−0.0981537	+	0.995171i	$0.531294\pi$
$440$	0	0
$441$	0	0
$442$	3.37630e6	0.822025
$443$	1.39981e6	0.338891	0.169446	−	0.985540i	$-0.445802\pi$
$443$	1.39981e6	0.338891	0.169446	+	0.985540i	$0.445802\pi$
$444$	0	0
$445$	−564516.	−0.135138
$446$	5.99683e6	1.42753
$447$	0	0
$448$	1.10848e6	0.260935
$449$	−2.99248e6	−0.700512	−0.350256	−	0.936654i	$-0.613905\pi$
$449$	−2.99248e6	−0.700512	−0.350256	+	0.936654i	$0.613905\pi$
$450$	0	0
$451$	0	0
$452$	−41928.0	−0.00965291
$453$	0	0
$454$	−3.63708e6	−0.828158
$455$	153120.	0.0346740
$456$	0	0
$457$	−6.29969e6	−1.41101	−0.705503	−	0.708707i	$-0.749279\pi$
$457$	−6.29969e6	−1.41101	−0.705503	+	0.708707i	$0.749279\pi$
$458$	−8.15956e6	−1.81762
$459$	0	0
$460$	−20160.0	−0.00444218
$461$	3.40318e6	0.745818	0.372909	−	0.927868i	$-0.378360\pi$
$461$	3.40318e6	0.745818	0.372909	+	0.927868i	$0.378360\pi$
$462$	0	0
$463$	−2.23034e6	−0.483524	−0.241762	−	0.970336i	$-0.577725\pi$
$463$	−2.23034e6	−0.483524	−0.241762	+	0.970336i	$0.577725\pi$
$464$	−5.26877e6	−1.13609
$465$	0	0
$466$	2.35732e6	0.502867
$467$	6.51409e6	1.38217	0.691085	−	0.722773i	$-0.257133\pi$
$467$	6.51409e6	1.38217	0.691085	+	0.722773i	$0.257133\pi$
$468$	0	0
$469$	−922720.	−0.193704
$470$	672192.	0.140362
$471$	0	0
$472$	3.03811e6	0.627695
$473$	0	0
$474$	0	0
$475$	−1.71748e6	−0.349268
$476$	141120.	0.0285477
$477$	0	0
$478$	7.95082e6	1.59163
$479$	2.39232e6	0.476410	0.238205	−	0.971215i	$-0.423441\pi$
$479$	2.39232e6	0.476410	0.238205	+	0.971215i	$0.423441\pi$
$480$	0	0
$481$	1.53758e6	0.303023
$482$	−5.94056e6	−1.16469
$483$	0	0
$484$	0	0
$485$	−296652.	−0.0572655
$486$	0	0
$487$	−6.13089e6	−1.17139	−0.585694	−	0.810532i	$-0.699178\pi$
$487$	−6.13089e6	−1.17139	−0.585694	+	0.810532i	$0.699178\pi$
$488$	−6.67934e6	−1.26965
$489$	0	0
$490$	−547452.	−0.103004
$491$	−1.23589e6	−0.231354	−0.115677	−	0.993287i	$-0.536904\pi$
$491$	−1.23589e6	−0.231354	−0.115677	+	0.993287i	$0.536904\pi$
$492$	0	0
$493$	4.09072e6	0.758022
$494$	2.12837e6	0.392400
$495$	0	0
$496$	−4.99840e6	−0.912277
$497$	169920.	0.0308570
$498$	0	0
$499$	−9.85496e6	−1.77175	−0.885877	−	0.463921i	$-0.846442\pi$
$499$	−9.85496e6	−1.77175	−0.885877	+	0.463921i	$0.846442\pi$
$500$	149136.	0.0266783
$501$	0	0
$502$	882792.	0.156350
$503$	1.16777e6	0.205796	0.102898	−	0.994692i	$-0.467188\pi$
$503$	1.16777e6	0.205796	0.102898	+	0.994692i	$0.467188\pi$
$504$	0	0
$505$	858204.	0.149748
$506$	0	0
$507$	0	0
$508$	684352.	0.117658
$509$	−1.04941e6	−0.179535	−0.0897675	−	0.995963i	$-0.528612\pi$
$509$	−1.04941e6	−0.179535	−0.0897675	+	0.995963i	$0.528612\pi$
$510$	0	0
$511$	1.64440e6	0.278583
$512$	4.47130e6	0.753804
$513$	0	0
$514$	−2.90149e6	−0.484411
$515$	−318864.	−0.0529770
$516$	0	0
$517$	0	0
$518$	578400.	0.0947118
$519$	0	0
$520$	643104.	0.104297
$521$	9.61407e6	1.55172	0.775859	−	0.630906i	$-0.217317\pi$
$521$	9.61407e6	1.55172	0.775859	+	0.630906i	$0.217317\pi$
$522$	0	0
$523$	−6.96148e6	−1.11288	−0.556439	−	0.830888i	$-0.687833\pi$
$523$	−6.96148e6	−1.11288	−0.556439	+	0.830888i	$0.687833\pi$
$524$	1.03387e6	0.164490
$525$	0	0
$526$	−4.88146e6	−0.769281
$527$	3.88080e6	0.608688
$528$	0	0
$529$	−5.73074e6	−0.890373
$530$	−1.21500e6	−0.187883
$531$	0	0
$532$	88960.0	0.0136275
$533$	4.38306e6	0.668281
$534$	0	0
$535$	−544968.	−0.0823164
$536$	−3.87542e6	−0.582649
$537$	0	0
$538$	−2.76664e6	−0.412094
$539$	0	0
$540$	0	0
$541$	712690.	0.104691	0.0523453	−	0.998629i	$-0.483330\pi$
$541$	712690.	0.104691	0.0523453	+	0.998629i	$0.483330\pi$
$542$	1.00508e7	1.46961
$543$	0	0
$544$	1.27008e6	0.184007
$545$	−369996.	−0.0533588
$546$	0	0
$547$	3.62614e6	0.518175	0.259087	−	0.965854i	$-0.416578\pi$
$547$	3.62614e6	0.518175	0.259087	+	0.965854i	$0.416578\pi$
$548$	−1.20094e6	−0.170832
$549$	0	0
$550$	0	0
$551$	2.57873e6	0.361848
$552$	0	0
$553$	−876800.	−0.121924
$554$	2.40676e6	0.333164
$555$	0	0
$556$	1.40066e6	0.192152
$557$	4.84846e6	0.662165	0.331082	−	0.943602i	$-0.392586\pi$
$557$	4.84846e6	0.662165	0.331082	+	0.943602i	$0.392586\pi$
$558$	0	0
$559$	6.15287e6	0.832815
$560$	272640.	0.0367383
$561$	0	0
$562$	1.38586e7	1.85088
$563$	8.50405e6	1.13072	0.565360	−	0.824844i	$-0.308737\pi$
$563$	8.50405e6	1.13072	0.565360	+	0.824844i	$0.308737\pi$
$564$	0	0
$565$	62892.0	0.00828847
$566$	−6.77263e6	−0.888621
$567$	0	0
$568$	713664.	0.0928160
$569$	362874.	0.0469867	0.0234934	−	0.999724i	$-0.492521\pi$
$569$	362874.	0.0469867	0.0234934	+	0.999724i	$0.492521\pi$
$570$	0	0
$571$	−4.11024e6	−0.527566	−0.263783	−	0.964582i	$-0.584970\pi$
$571$	−4.11024e6	−0.527566	−0.263783	+	0.964582i	$0.584970\pi$
$572$	0	0
$573$	0	0
$574$	1.64880e6	0.208876
$575$	−2.59476e6	−0.327286
$576$	0	0
$577$	−7.87680e6	−0.984941	−0.492470	−	0.870329i	$-0.663906\pi$
$577$	−7.87680e6	−0.984941	−0.492470	+	0.870329i	$0.663906\pi$
$578$	3.85160e6	0.479536
$579$	0	0
$580$	−111312.	−0.0137395
$581$	3.29808e6	0.405341
$582$	0	0
$583$	0	0
$584$	6.90648e6	0.837963
$585$	0	0
$586$	5.63324e6	0.677664
$587$	−603948.	−0.0723443	−0.0361721	−	0.999346i	$-0.511516\pi$
$587$	−603948.	−0.0723443	−0.0361721	+	0.999346i	$0.511516\pi$
$588$	0	0
$589$	2.44640e6	0.290562
$590$	651024.	0.0769958
$591$	0	0
$592$	2.73776e6	0.321064
$593$	−5.39077e6	−0.629526	−0.314763	−	0.949170i	$-0.601925\pi$
$593$	−5.39077e6	−0.629526	−0.314763	+	0.949170i	$0.601925\pi$
$594$	0	0
$595$	−211680.	−0.0245125
$596$	−421032.	−0.0485511
$597$	0	0
$598$	3.21552e6	0.367704
$599$	−4.27999e6	−0.487389	−0.243695	−	0.969852i	$-0.578359\pi$
$599$	−4.27999e6	−0.487389	−0.243695	+	0.969852i	$0.578359\pi$
$600$	0	0
$601$	−1.02483e6	−0.115735	−0.0578674	−	0.998324i	$-0.518430\pi$
$601$	−1.02483e6	−0.115735	−0.0578674	+	0.998324i	$0.518430\pi$
$602$	2.31456e6	0.260302
$603$	0	0
$604$	−1.58557e6	−0.176845
$605$	0	0
$606$	0	0
$607$	−1.24342e7	−1.36976	−0.684882	−	0.728654i	$-0.740146\pi$
$607$	−1.24342e7	−1.36976	−0.684882	+	0.728654i	$0.740146\pi$
$608$	800640.	0.0878372
$609$	0	0
$610$	−1.43129e6	−0.155741
$611$	−1.19127e7	−1.29095
$612$	0	0
$613$	−4.21506e6	−0.453057	−0.226528	−	0.974005i	$-0.572738\pi$
$613$	−4.21506e6	−0.453057	−0.226528	+	0.974005i	$0.572738\pi$
$614$	4.15769e6	0.445073
$615$	0	0
$616$	0	0
$617$	4.40665e6	0.466010	0.233005	−	0.972476i	$-0.425144\pi$
$617$	4.40665e6	0.466010	0.233005	+	0.972476i	$0.425144\pi$
$618$	0	0
$619$	4.80168e6	0.503693	0.251847	−	0.967767i	$-0.418962\pi$
$619$	4.80168e6	0.503693	0.251847	+	0.967767i	$0.418962\pi$
$620$	−105600.	−0.0110328
$621$	0	0
$622$	1.76586e7	1.83012
$623$	3.76344e6	0.388477
$624$	0	0
$625$	9.42942e6	0.965573
$626$	−5.31088e6	−0.541664
$627$	0	0
$628$	−550984.	−0.0557494
$629$	−2.12562e6	−0.214220
$630$	0	0
$631$	8.30727e6	0.830587	0.415293	−	0.909688i	$-0.363679\pi$
$631$	8.30727e6	0.830587	0.415293	+	0.909688i	$0.363679\pi$
$632$	−3.68256e6	−0.366739
$633$	0	0
$634$	1.50528e7	1.48728
$635$	−1.02653e6	−0.101027
$636$	0	0
$637$	9.70207e6	0.947361
$638$	0	0
$639$	0	0
$640$	1.27411e6	0.122958
$641$	−1.76956e7	−1.70107	−0.850534	−	0.525921i	$-0.823721\pi$
$641$	−1.76956e7	−1.70107	−0.850534	+	0.525921i	$0.823721\pi$
$642$	0	0
$643$	−1.28394e7	−1.22466	−0.612330	−	0.790602i	$-0.709768\pi$
$643$	−1.28394e7	−1.22466	−0.612330	+	0.790602i	$0.709768\pi$
$644$	134400.	0.0127698
$645$	0	0
$646$	−2.94235e6	−0.277404
$647$	2.08468e7	1.95785	0.978924	−	0.204226i	$-0.0654678\pi$
$647$	2.08468e7	1.95785	0.978924	+	0.204226i	$0.0654678\pi$
$648$	0	0
$649$	0	0
$650$	−1.18247e7	−1.09776
$651$	0	0
$652$	1.41070e6	0.129962
$653$	−1.29632e7	−1.18968	−0.594841	−	0.803843i	$-0.702785\pi$
$653$	−1.29632e7	−1.18968	−0.594841	+	0.803843i	$0.702785\pi$
$654$	0	0
$655$	−1.55081e6	−0.141239
$656$	7.80432e6	0.708069
$657$	0	0
$658$	−4.48128e6	−0.403494
$659$	−5.66862e6	−0.508468	−0.254234	−	0.967143i	$-0.581823\pi$
$659$	−5.66862e6	−0.508468	−0.254234	+	0.967143i	$0.581823\pi$
$660$	0	0
$661$	−3.11430e6	−0.277240	−0.138620	−	0.990346i	$-0.544267\pi$
$661$	−3.11430e6	−0.277240	−0.138620	+	0.990346i	$0.544267\pi$
$662$	1.29689e6	0.115016
$663$	0	0
$664$	1.38519e7	1.21924
$665$	−133440.	−0.0117012
$666$	0	0
$667$	3.89592e6	0.339075
$668$	−870240.	−0.0754567
$669$	0	0
$670$	−830448.	−0.0714703
$671$	0	0
$672$	0	0
$673$	−105890.	−0.00901192	−0.00450596	−	0.999990i	$-0.501434\pi$
$673$	−105890.	−0.00901192	−0.00450596	+	0.999990i	$0.501434\pi$
$674$	1.95158e7	1.65476
$675$	0	0
$676$	143004.	0.0120360
$677$	−1.60910e7	−1.34931	−0.674656	−	0.738132i	$-0.735708\pi$
$677$	−1.60910e7	−1.34931	−0.674656	+	0.738132i	$0.735708\pi$
$678$	0	0
$679$	1.97768e6	0.164620
$680$	−889056.	−0.0737321
$681$	0	0
$682$	0	0
$683$	−1.60780e7	−1.31880	−0.659402	−	0.751791i	$-0.729190\pi$
$683$	−1.60780e7	−1.31880	−0.659402	+	0.751791i	$0.729190\pi$
$684$	0	0
$685$	1.80140e6	0.146685
$686$	7.68336e6	0.623363
$687$	0	0
$688$	1.09556e7	0.882398
$689$	2.15325e7	1.72801
$690$	0	0
$691$	−165964.	−0.0132227	−0.00661133	−	0.999978i	$-0.502104\pi$
$691$	−165964.	−0.0132227	−0.00661133	+	0.999978i	$0.502104\pi$
$692$	−654792.	−0.0519802
$693$	0	0
$694$	1.75924e7	1.38652
$695$	−2.10098e6	−0.164991
$696$	0	0
$697$	−6.05934e6	−0.472436
$698$	5.43119e6	0.421945
$699$	0	0
$700$	−494240.	−0.0381235
$701$	1.77248e7	1.36234	0.681171	−	0.732124i	$-0.261471\pi$
$701$	1.77248e7	1.36234	0.681171	+	0.732124i	$0.261471\pi$
$702$	0	0
$703$	−1.33996e6	−0.102259
$704$	0	0
$705$	0	0
$706$	1.15071e7	0.868872
$707$	−5.72136e6	−0.430478
$708$	0	0
$709$	−1.06023e7	−0.792112	−0.396056	−	0.918226i	$-0.629621\pi$
$709$	−1.06023e7	−0.792112	−0.396056	+	0.918226i	$0.629621\pi$
$710$	152928.	0.0113852
$711$	0	0
$712$	1.58064e7	1.16852
$713$	3.69600e6	0.272275
$714$	0	0
$715$	0	0
$716$	−1.43496e6	−0.104606
$717$	0	0
$718$	1.46219e7	1.05850
$719$	−9.03211e6	−0.651579	−0.325790	−	0.945442i	$-0.605630\pi$
$719$	−9.03211e6	−0.651579	−0.325790	+	0.945442i	$0.605630\pi$
$720$	0	0
$721$	2.12576e6	0.152292
$722$	1.30018e7	0.928239
$723$	0	0
$724$	−2.02852e6	−0.143825
$725$	−1.43268e7	−1.01229
$726$	0	0
$727$	1.87575e7	1.31625	0.658127	−	0.752907i	$-0.271349\pi$
$727$	1.87575e7	1.31625	0.658127	+	0.752907i	$0.271349\pi$
$728$	−4.28736e6	−0.299821
$729$	0	0
$730$	1.47996e6	0.102788
$731$	−8.50601e6	−0.588752
$732$	0	0
$733$	1.17773e7	0.809626	0.404813	−	0.914399i	$-0.367337\pi$
$733$	1.17773e7	0.809626	0.404813	+	0.914399i	$0.367337\pi$
$734$	5.90438e6	0.404515
$735$	0	0
$736$	1.20960e6	0.0823090
$737$	0	0
$738$	0	0
$739$	−5.88948e6	−0.396703	−0.198352	−	0.980131i	$-0.563559\pi$
$739$	−5.88948e6	−0.396703	−0.198352	+	0.980131i	$0.563559\pi$
$740$	57840.0	0.00388284
$741$	0	0
$742$	8.10000e6	0.540101
$743$	−1.00476e7	−0.667712	−0.333856	−	0.942624i	$-0.608350\pi$
$743$	−1.00476e7	−0.667712	−0.333856	+	0.942624i	$0.608350\pi$
$744$	0	0
$745$	631548.	0.0416884
$746$	1.02219e7	0.672490
$747$	0	0
$748$	0	0
$749$	3.63312e6	0.236633
$750$	0	0
$751$	4.81530e6	0.311547	0.155773	−	0.987793i	$-0.450213\pi$
$751$	4.81530e6	0.311547	0.155773	+	0.987793i	$0.450213\pi$
$752$	−2.12114e7	−1.36781
$753$	0	0
$754$	1.77543e7	1.13730
$755$	2.37835e6	0.151848
$756$	0	0
$757$	3.12973e6	0.198503	0.0992516	−	0.995062i	$-0.468355\pi$
$757$	3.12973e6	0.198503	0.0992516	+	0.995062i	$0.468355\pi$
$758$	−1.65392e7	−1.04554
$759$	0	0
$760$	−560448.	−0.0351967
$761$	−1.17773e7	−0.737197	−0.368599	−	0.929589i	$-0.620162\pi$
$761$	−1.17773e7	−0.737197	−0.368599	+	0.929589i	$0.620162\pi$
$762$	0	0
$763$	2.46664e6	0.153389
$764$	2.59354e6	0.160753
$765$	0	0
$766$	2.73946e6	0.168691
$767$	−1.15376e7	−0.708152
$768$	0	0
$769$	1.49376e6	0.0910887	0.0455443	−	0.998962i	$-0.485498\pi$
$769$	1.49376e6	0.0910887	0.0455443	+	0.998962i	$0.485498\pi$
$770$	0	0
$771$	0	0
$772$	111352.	0.00672442
$773$	2.25125e7	1.35511	0.677555	−	0.735472i	$-0.263040\pi$
$773$	2.25125e7	1.35511	0.677555	+	0.735472i	$0.263040\pi$
$774$	0	0
$775$	−1.35916e7	−0.812861
$776$	8.30626e6	0.495166
$777$	0	0
$778$	−1.20384e7	−0.713048
$779$	−3.81972e6	−0.225521
$780$	0	0
$781$	0	0
$782$	−4.44528e6	−0.259945
$783$	0	0
$784$	1.72752e7	1.00376
$785$	826476.	0.0478692
$786$	0	0
$787$	−1.19547e7	−0.688022	−0.344011	−	0.938966i	$-0.611786\pi$
$787$	−1.19547e7	−0.688022	−0.344011	+	0.938966i	$0.611786\pi$
$788$	2.44418e6	0.140223
$789$	0	0
$790$	−789120.	−0.0449858
$791$	−419280.	−0.0238266
$792$	0	0
$793$	2.53656e7	1.43239
$794$	3.46224e7	1.94897
$795$	0	0
$796$	3.51613e6	0.196690
$797$	−540798.	−0.0301571	−0.0150785	−	0.999886i	$-0.504800\pi$
$797$	−540798.	−0.0301571	−0.0150785	+	0.999886i	$0.504800\pi$
$798$	0	0
$799$	1.64687e7	0.912625
$800$	−4.44816e6	−0.245728
$801$	0	0
$802$	1.80375e7	0.990243
$803$	0	0
$804$	0	0
$805$	−201600.	−0.0109648
$806$	1.68432e7	0.913244
$807$	0	0
$808$	−2.40297e7	−1.29485
$809$	−6.14223e6	−0.329955	−0.164978	−	0.986297i	$-0.552755\pi$
$809$	−6.14223e6	−0.329955	−0.164978	+	0.986297i	$0.552755\pi$
$810$	0	0
$811$	3.16734e7	1.69100	0.845499	−	0.533977i	$-0.179303\pi$
$811$	3.16734e7	1.69100	0.845499	+	0.533977i	$0.179303\pi$
$812$	742080.	0.0394967
$813$	0	0
$814$	0	0
$815$	−2.11606e6	−0.111592
$816$	0	0
$817$	−5.36206e6	−0.281046
$818$	9.20176e6	0.480825
$819$	0	0
$820$	164880.	0.00856315
$821$	2.66175e7	1.37819	0.689095	−	0.724671i	$-0.258008\pi$
$821$	2.66175e7	1.37819	0.689095	+	0.724671i	$0.258008\pi$
$822$	0	0
$823$	3.62817e7	1.86719	0.933593	−	0.358335i	$-0.116655\pi$
$823$	3.62817e7	1.86719	0.933593	+	0.358335i	$0.116655\pi$
$824$	8.92819e6	0.458084
$825$	0	0
$826$	−4.34016e6	−0.221338
$827$	1.09033e6	0.0554364	0.0277182	−	0.999616i	$-0.491176\pi$
$827$	1.09033e6	0.0554364	0.0277182	+	0.999616i	$0.491176\pi$
$828$	0	0
$829$	−1.03016e7	−0.520620	−0.260310	−	0.965525i	$-0.583825\pi$
$829$	−1.03016e7	−0.520620	−0.260310	+	0.965525i	$0.583825\pi$
$830$	2.96827e6	0.149558
$831$	0	0
$832$	−1.76803e7	−0.885483
$833$	−1.34126e7	−0.669730
$834$	0	0
$835$	1.30536e6	0.0647909
$836$	0	0
$837$	0	0
$838$	−2.32425e7	−1.14333
$839$	1.96134e7	0.961940	0.480970	−	0.876737i	$-0.340285\pi$
$839$	1.96134e7	0.961940	0.480970	+	0.876737i	$0.340285\pi$
$840$	0	0
$841$	999895.	0.0487489
$842$	7.99840e6	0.388797
$843$	0	0
$844$	−194000.	−0.00937445
$845$	−214506.	−0.0103347
$846$	0	0
$847$	0	0
$848$	3.83400e7	1.83089
$849$	0	0
$850$	1.63470e7	0.776051
$851$	−2.02440e6	−0.0958236
$852$	0	0
$853$	−3.27565e7	−1.54143	−0.770717	−	0.637178i	$-0.780102\pi$
$853$	−3.27565e7	−1.54143	−0.770717	+	0.637178i	$0.780102\pi$
$854$	9.54192e6	0.447704
$855$	0	0
$856$	1.52591e7	0.711778
$857$	−2.57953e7	−1.19974	−0.599872	−	0.800096i	$-0.704782\pi$
$857$	−2.57953e7	−1.19974	−0.599872	+	0.800096i	$0.704782\pi$
$858$	0	0
$859$	−1.98548e7	−0.918085	−0.459043	−	0.888414i	$-0.651807\pi$
$859$	−1.98548e7	−0.918085	−0.459043	+	0.888414i	$0.651807\pi$
$860$	231456.	0.0106714
$861$	0	0
$862$	−3.87115e7	−1.77448
$863$	673056.	0.0307627	0.0153813	−	0.999882i	$-0.495104\pi$
$863$	673056.	0.0307627	0.0153813	+	0.999882i	$0.495104\pi$
$864$	0	0
$865$	982188.	0.0446328
$866$	2.49946e7	1.13254
$867$	0	0
$868$	704000.	0.0317156
$869$	0	0
$870$	0	0
$871$	1.47174e7	0.657333
$872$	1.03599e7	0.461385
$873$	0	0
$874$	−2.80224e6	−0.124087
$875$	1.49136e6	0.0658510
$876$	0	0
$877$	−5.32115e6	−0.233618	−0.116809	−	0.993154i	$-0.537267\pi$
$877$	−5.32115e6	−0.233618	−0.116809	+	0.993154i	$0.537267\pi$
$878$	4.75608e6	0.208215
$879$	0	0
$880$	0	0
$881$	−2.78891e7	−1.21058	−0.605291	−	0.796004i	$-0.706943\pi$
$881$	−2.78891e7	−1.21058	−0.605291	+	0.796004i	$0.706943\pi$
$882$	0	0
$883$	−2.83786e7	−1.22487	−0.612435	−	0.790521i	$-0.709810\pi$
$883$	−2.83786e7	−1.22487	−0.612435	+	0.790521i	$0.709810\pi$
$884$	−2.25086e6	−0.0968765
$885$	0	0
$886$	−8.39887e6	−0.359448
$887$	4.22678e7	1.80385	0.901925	−	0.431893i	$-0.142154\pi$
$887$	4.22678e7	1.80385	0.901925	+	0.431893i	$0.142154\pi$
$888$	0	0
$889$	6.84352e6	0.290419
$890$	3.38710e6	0.143335
$891$	0	0
$892$	−3.99789e6	−0.168236
$893$	1.03816e7	0.435649
$894$	0	0
$895$	2.15244e6	0.0898201
$896$	−8.49408e6	−0.353465
$897$	0	0
$898$	1.79549e7	0.743005
$899$	2.04072e7	0.842140
$900$	0	0
$901$	−2.97675e7	−1.22160
$902$	0	0
$903$	0	0
$904$	−1.76098e6	−0.0716692
$905$	3.04278e6	0.123495
$906$	0	0
$907$	3.19526e7	1.28970	0.644849	−	0.764310i	$-0.276920\pi$
$907$	3.19526e7	1.28970	0.644849	+	0.764310i	$0.276920\pi$
$908$	2.42472e6	0.0975994
$909$	0	0
$910$	−918720.	−0.0367773
$911$	1.16429e7	0.464800	0.232400	−	0.972620i	$-0.425342\pi$
$911$	1.16429e7	0.464800	0.232400	+	0.972620i	$0.425342\pi$
$912$	0	0
$913$	0	0
$914$	3.77981e7	1.49660
$915$	0	0
$916$	5.43970e6	0.214208
$917$	1.03387e7	0.406016
$918$	0	0
$919$	−1.39844e6	−0.0546204	−0.0273102	−	0.999627i	$-0.508694\pi$
$919$	−1.39844e6	−0.0546204	−0.0273102	+	0.999627i	$0.508694\pi$
$920$	−846720.	−0.0329815
$921$	0	0
$922$	−2.04191e7	−0.791059
$923$	−2.71022e6	−0.104713
$924$	0	0
$925$	7.44449e6	0.286075
$926$	1.33820e7	0.512854
$927$	0	0
$928$	6.67872e6	0.254579
$929$	1.66792e7	0.634067	0.317033	−	0.948414i	$-0.397313\pi$
$929$	1.66792e7	0.634067	0.317033	+	0.948414i	$0.397313\pi$
$930$	0	0
$931$	−8.45509e6	−0.319701
$932$	−1.57154e6	−0.0592634
$933$	0	0
$934$	−3.90846e7	−1.46601
$935$	0	0
$936$	0	0
$937$	2.47956e7	0.922625	0.461312	−	0.887238i	$-0.347379\pi$
$937$	2.47956e7	0.922625	0.461312	+	0.887238i	$0.347379\pi$
$938$	5.53632e6	0.205454
$939$	0	0
$940$	−448128.	−0.0165418
$941$	2.79574e7	1.02925	0.514627	−	0.857414i	$-0.327930\pi$
$941$	2.79574e7	1.02925	0.514627	+	0.857414i	$0.327930\pi$
$942$	0	0
$943$	−5.77080e6	−0.211328
$944$	−2.05434e7	−0.750314
$945$	0	0
$946$	0	0
$947$	−7.64936e6	−0.277173	−0.138586	−	0.990350i	$-0.544256\pi$
$947$	−7.64936e6	−0.277173	−0.138586	+	0.990350i	$0.544256\pi$
$948$	0	0
$949$	−2.62282e7	−0.945372
$950$	1.03049e7	0.370455
$951$	0	0
$952$	5.92704e6	0.211956
$953$	−4.62179e7	−1.64846	−0.824228	−	0.566257i	$-0.808391\pi$
$953$	−4.62179e7	−1.64846	−0.824228	+	0.566257i	$0.808391\pi$
$954$	0	0
$955$	−3.89030e6	−0.138031
$956$	−5.30054e6	−0.187575
$957$	0	0
$958$	−1.43539e7	−0.505309
$959$	−1.20094e7	−0.421671
$960$	0	0
$961$	−9.26915e6	−0.323766
$962$	−9.22548e6	−0.321404
$963$	0	0
$964$	3.96038e6	0.137260
$965$	−167028.	−0.00577392
$966$	0	0
$967$	−2.08557e7	−0.717229	−0.358615	−	0.933486i	$-0.616751\pi$
$967$	−2.08557e7	−0.717229	−0.358615	+	0.933486i	$0.616751\pi$
$968$	0	0
$969$	0	0
$970$	1.77991e6	0.0607392
$971$	4.58152e7	1.55941	0.779707	−	0.626144i	$-0.215368\pi$
$971$	4.58152e7	1.55941	0.779707	+	0.626144i	$0.215368\pi$
$972$	0	0
$973$	1.40066e7	0.474296
$974$	3.67853e7	1.24245
$975$	0	0
$976$	4.51651e7	1.51767
$977$	1.09544e6	0.0367157	0.0183578	−	0.999831i	$-0.494156\pi$
$977$	1.09544e6	0.0367157	0.0183578	+	0.999831i	$0.494156\pi$
$978$	0	0
$979$	0	0
$980$	364968.	0.0121392
$981$	0	0
$982$	7.41535e6	0.245388
$983$	−5.25817e7	−1.73561	−0.867803	−	0.496909i	$-0.834468\pi$
$983$	−5.25817e7	−1.73561	−0.867803	+	0.496909i	$0.834468\pi$
$984$	0	0
$985$	−3.66628e6	−0.120402
$986$	−2.45443e7	−0.804004
$987$	0	0
$988$	−1.41891e6	−0.0462448
$989$	−8.10096e6	−0.263358
$990$	0	0
$991$	−4.90389e7	−1.58620	−0.793098	−	0.609094i	$-0.791533\pi$
$991$	−4.90389e7	−1.58620	−0.793098	+	0.609094i	$0.791533\pi$
$992$	6.33600e6	0.204426
$993$	0	0
$994$	−1.01952e6	−0.0327288
$995$	−5.27419e6	−0.168888
$996$	0	0
$997$	−3.05461e6	−0.0973237	−0.0486618	−	0.998815i	$-0.515496\pi$
$997$	−3.05461e6	−0.0973237	−0.0486618	+	0.998815i	$0.515496\pi$
$998$	5.91297e7	1.87923
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.b.1.1		1
3.2	odd	2		363.6.a.d.1.1		1
11.10	odd	2		9.6.a.a.1.1		1
33.32	even	2		3.6.a.a.1.1	✓	1
44.43	even	2		144.6.a.f.1.1		1
55.32	even	4		225.6.b.b.199.2		2
55.43	even	4		225.6.b.b.199.1		2
55.54	odd	2		225.6.a.a.1.1		1
77.76	even	2		441.6.a.i.1.1		1
88.21	odd	2		576.6.a.s.1.1		1
88.43	even	2		576.6.a.t.1.1		1
99.32	even	6		81.6.c.c.55.1		2
99.43	odd	6		81.6.c.a.28.1		2
99.65	even	6		81.6.c.c.28.1		2
99.76	odd	6		81.6.c.a.55.1		2
132.131	odd	2		48.6.a.a.1.1		1
165.32	odd	4		75.6.b.b.49.1		2
165.98	odd	4		75.6.b.b.49.2		2
165.164	even	2		75.6.a.e.1.1		1
231.32	even	6		147.6.e.h.79.1		2
231.65	even	6		147.6.e.h.67.1		2
231.131	odd	6		147.6.e.k.67.1		2
231.164	odd	6		147.6.e.k.79.1		2
231.230	odd	2		147.6.a.a.1.1		1
264.131	odd	2		192.6.a.l.1.1		1
264.197	even	2		192.6.a.d.1.1		1
429.428	even	2		507.6.a.b.1.1		1
528.131	odd	4		768.6.d.h.385.1		2
528.197	even	4		768.6.d.k.385.1		2
528.395	odd	4		768.6.d.h.385.2		2
528.461	even	4		768.6.d.k.385.2		2
561.560	even	2		867.6.a.a.1.1		1
627.626	odd	2		1083.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.6.a.a.1.1	✓	1	33.32	even	2
9.6.a.a.1.1		1	11.10	odd	2
48.6.a.a.1.1		1	132.131	odd	2
75.6.a.e.1.1		1	165.164	even	2
75.6.b.b.49.1		2	165.32	odd	4
75.6.b.b.49.2		2	165.98	odd	4
81.6.c.a.28.1		2	99.43	odd	6
81.6.c.a.55.1		2	99.76	odd	6
81.6.c.c.28.1		2	99.65	even	6
81.6.c.c.55.1		2	99.32	even	6
144.6.a.f.1.1		1	44.43	even	2
147.6.a.a.1.1		1	231.230	odd	2
147.6.e.h.67.1		2	231.65	even	6
147.6.e.h.79.1		2	231.32	even	6
147.6.e.k.67.1		2	231.131	odd	6
147.6.e.k.79.1		2	231.164	odd	6
192.6.a.d.1.1		1	264.197	even	2
192.6.a.l.1.1		1	264.131	odd	2
225.6.a.a.1.1		1	55.54	odd	2
225.6.b.b.199.1		2	55.43	even	4
225.6.b.b.199.2		2	55.32	even	4
363.6.a.d.1.1		1	3.2	odd	2
441.6.a.i.1.1		1	77.76	even	2
507.6.a.b.1.1		1	429.428	even	2
576.6.a.s.1.1		1	88.21	odd	2
576.6.a.t.1.1		1	88.43	even	2
768.6.d.h.385.1		2	528.131	odd	4
768.6.d.h.385.2		2	528.395	odd	4
768.6.d.k.385.1		2	528.197	even	4
768.6.d.k.385.2		2	528.461	even	4
867.6.a.a.1.1		1	561.560	even	2
1083.6.a.c.1.1		1	627.626	odd	2
1089.6.a.b.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more