LMFDB - Embedded newform 867.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,6,Mod(1,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, 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("867.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	867.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:	$139.052771778$
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	$\chi$	$=$	867.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} +54.0000 q^{6} +40.0000 q^{7} +168.000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} +54.0000 q^{6} +40.0000 q^{7} +168.000 q^{8} +81.0000 q^{9} +36.0000 q^{10} +564.000 q^{11} -36.0000 q^{12} +638.000 q^{13} -240.000 q^{14} +54.0000 q^{15} -1136.00 q^{16} -486.000 q^{18} -556.000 q^{19} -24.0000 q^{20} -360.000 q^{21} -3384.00 q^{22} +840.000 q^{23} -1512.00 q^{24} -3089.00 q^{25} -3828.00 q^{26} -729.000 q^{27} +160.000 q^{28} -4638.00 q^{29} -324.000 q^{30} -4400.00 q^{31} +1440.00 q^{32} -5076.00 q^{33} -240.000 q^{35} +324.000 q^{36} +2410.00 q^{37} +3336.00 q^{38} -5742.00 q^{39} -1008.00 q^{40} +6870.00 q^{41} +2160.00 q^{42} +9644.00 q^{43} +2256.00 q^{44} -486.000 q^{45} -5040.00 q^{46} -18672.0 q^{47} +10224.0 q^{48} -15207.0 q^{49} +18534.0 q^{50} +2552.00 q^{52} +33750.0 q^{53} +4374.00 q^{54} -3384.00 q^{55} +6720.00 q^{56} +5004.00 q^{57} +27828.0 q^{58} -18084.0 q^{59} +216.000 q^{60} -39758.0 q^{61} +26400.0 q^{62} +3240.00 q^{63} +27712.0 q^{64} -3828.00 q^{65} +30456.0 q^{66} -23068.0 q^{67} -7560.00 q^{69} +1440.00 q^{70} +4248.00 q^{71} +13608.0 q^{72} +41110.0 q^{73} -14460.0 q^{74} +27801.0 q^{75} -2224.00 q^{76} +22560.0 q^{77} +34452.0 q^{78} -21920.0 q^{79} +6816.00 q^{80} +6561.00 q^{81} -41220.0 q^{82} +82452.0 q^{83} -1440.00 q^{84} -57864.0 q^{86} +41742.0 q^{87} +94752.0 q^{88} -94086.0 q^{89} +2916.00 q^{90} +25520.0 q^{91} +3360.00 q^{92} +39600.0 q^{93} +112032. q^{94} +3336.00 q^{95} -12960.0 q^{96} -49442.0 q^{97} +91242.0 q^{98} +45684.0 q^{99} +O(q^{100})\)

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	$0.677932\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	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	$0.517091\pi$
$6$	54.0000	0.612372
$7$	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
$9$	81.0000	0.333333
$10$	36.0000	0.113842
$11$	564.000	1.40539	0.702696	−	0.711490i	$-0.251979\pi$
$11$	564.000	1.40539	0.702696	+	0.711490i	$0.251979\pi$
$12$	−36.0000	−0.0721688
$13$	638.000	1.04704	0.523519	−	0.852014i	$-0.324619\pi$
$13$	638.000	1.04704	0.523519	+	0.852014i	$0.324619\pi$
$14$	−240.000	−0.327259
$15$	54.0000	0.0619677
$16$	−1136.00	−1.10938
$17$	0	0
$17$	0	0
$18$	−486.000	−0.353553
$19$	−556.000	−0.353338	−0.176669	−	0.984270i	$-0.556532\pi$
$19$	−556.000	−0.353338	−0.176669	+	0.984270i	$0.556532\pi$
$20$	−24.0000	−0.0134164
$21$	−360.000	−0.178137
$22$	−3384.00	−1.49064
$23$	840.000	0.331100	0.165550	−	0.986201i	$-0.447060\pi$
$23$	840.000	0.331100	0.165550	+	0.986201i	$0.447060\pi$
$24$	−1512.00	−0.535826
$25$	−3089.00	−0.988480
$26$	−3828.00	−1.11055
$27$	−729.000	−0.192450
$28$	160.000	0.0385678
$29$	−4638.00	−1.02408	−0.512042	−	0.858960i	$-0.671111\pi$
$29$	−4638.00	−1.02408	−0.512042	+	0.858960i	$0.671111\pi$
$30$	−324.000	−0.0657267
$31$	−4400.00	−0.822334	−0.411167	−	0.911560i	$-0.634879\pi$
$31$	−4400.00	−0.822334	−0.411167	+	0.911560i	$0.634879\pi$
$32$	1440.00	0.248592
$33$	−5076.00	−0.811403
$34$	0	0
$35$	−240.000	−0.0331162
$36$	324.000	0.0416667
$37$	2410.00	0.289409	0.144705	−	0.989475i	$-0.453777\pi$
$37$	2410.00	0.289409	0.144705	+	0.989475i	$0.453777\pi$
$38$	3336.00	0.374772
$39$	−5742.00	−0.604507
$40$	−1008.00	−0.0996117
$41$	6870.00	0.638259	0.319130	−	0.947711i	$-0.396609\pi$
$41$	6870.00	0.638259	0.319130	+	0.947711i	$0.396609\pi$
$42$	2160.00	0.188943
$43$	9644.00	0.795401	0.397700	−	0.917515i	$-0.369808\pi$
$43$	9644.00	0.795401	0.397700	+	0.917515i	$0.369808\pi$
$44$	2256.00	0.175674
$45$	−486.000	−0.0357771
$46$	−5040.00	−0.351185
$47$	−18672.0	−1.23295	−0.616476	−	0.787374i	$-0.711440\pi$
$47$	−18672.0	−1.23295	−0.616476	+	0.787374i	$0.711440\pi$
$48$	10224.0	0.640498
$49$	−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	$-0.191068\pi$
$53$	33750.0	1.65038	0.825190	+	0.564855i	$0.191068\pi$
$54$	4374.00	0.204124
$55$	−3384.00	−0.150842
$56$	6720.00	0.286351
$57$	5004.00	0.204000
$58$	27828.0	1.08621
$59$	−18084.0	−0.676339	−0.338170	−	0.941085i	$-0.609808\pi$
$59$	−18084.0	−0.676339	−0.338170	+	0.941085i	$0.609808\pi$
$60$	216.000	0.00774597
$61$	−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
$63$	3240.00	0.102847
$64$	27712.0	0.845703
$65$	−3828.00	−0.112380
$66$	30456.0	0.860623
$67$	−23068.0	−0.627802	−0.313901	−	0.949456i	$-0.601636\pi$
$67$	−23068.0	−0.627802	−0.313901	+	0.949456i	$0.601636\pi$
$68$	0	0
$69$	−7560.00	−0.191161
$70$	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	$0.484076\pi$
$72$	13608.0	0.309359
$73$	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
$75$	27801.0	0.570699
$76$	−2224.00	−0.0441673
$77$	22560.0	0.433623
$78$	34452.0	0.641177
$79$	−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
$81$	6561.00	0.111111
$82$	−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	$0.271882\pi$
$84$	−1440.00	−0.0222671
$85$	0	0
$86$	−57864.0	−0.843650
$87$	41742.0	0.591255
$88$	94752.0	1.30431
$89$	−94086.0	−1.25907	−0.629535	−	0.776972i	$-0.716755\pi$
$89$	−94086.0	−1.25907	−0.629535	+	0.776972i	$0.716755\pi$
$90$	2916.00	0.0379473
$91$	25520.0	0.323056
$92$	3360.00	0.0413875
$93$	39600.0	0.474775
$94$	112032.	1.30774
$95$	3336.00	0.0379243
$96$	−12960.0	−0.143525
$97$	−49442.0	−0.533540	−0.266770	−	0.963760i	$-0.585956\pi$
$97$	−49442.0	−0.533540	−0.266770	+	0.963760i	$0.585956\pi$
$98$	91242.0	0.959687
$99$	45684.0	0.468464
$100$	−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
$105$	2160.00	0.0191197
$106$	−202500.	−1.75049
$107$	−90828.0	−0.766938	−0.383469	−	0.923554i	$-0.625271\pi$
$107$	−90828.0	−0.766938	−0.383469	+	0.923554i	$0.625271\pi$
$108$	−2916.00	−0.0240563
$109$	61666.0	0.497141	0.248570	−	0.968614i	$-0.420039\pi$
$109$	61666.0	0.497141	0.248570	+	0.968614i	$0.420039\pi$
$110$	20304.0	0.159993
$111$	−21690.0	−0.167091
$112$	−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	$0.512294\pi$
$114$	−30024.0	−0.216375
$115$	−5040.00	−0.0355374
$116$	−18552.0	−0.128011
$117$	51678.0	0.349013
$118$	108504.	0.717366
$119$	0	0
$120$	9072.00	0.0575109
$121$	157045.	0.975126
$122$	238548.	1.45103
$123$	−61830.0	−0.368499
$124$	−17600.0	−0.102792
$125$	37284.0	0.213426
$126$	−19440.0	−0.109086
$127$	−171088.	−0.941261	−0.470631	−	0.882330i	$-0.655974\pi$
$127$	−171088.	−0.941261	−0.470631	+	0.882330i	$0.655974\pi$
$128$	−212352.	−1.14560
$129$	−86796.0	−0.459225
$130$	22968.0	0.119197
$131$	−258468.	−1.31592	−0.657959	−	0.753054i	$-0.728580\pi$
$131$	−258468.	−1.31592	−0.657959	+	0.753054i	$0.728580\pi$
$132$	−20304.0	−0.101425
$133$	−22240.0	−0.109020
$134$	138408.	0.665885
$135$	4374.00	0.0206559
$136$	0	0
$137$	300234.	1.36665	0.683327	−	0.730113i	$-0.260532\pi$
$137$	300234.	1.36665	0.683327	+	0.730113i	$0.260532\pi$
$138$	45360.0	0.202757
$139$	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
$141$	168048.	0.711845
$142$	−25488.0	−0.106075
$143$	359832.	1.47150
$144$	−92016.0	−0.369792
$145$	27828.0	0.109916
$146$	−246660.	−0.957672
$147$	136863.	0.522387
$148$	9640.00	0.0361762
$149$	−105258.	−0.388409	−0.194205	−	0.980961i	$-0.562213\pi$
$149$	−105258.	−0.388409	−0.194205	+	0.980961i	$0.562213\pi$
$150$	−166806.	−0.605318
$151$	396392.	1.41476	0.707380	−	0.706834i	$-0.249877\pi$
$151$	396392.	1.41476	0.707380	+	0.706834i	$0.249877\pi$
$152$	−93408.0	−0.327925
$153$	0	0
$154$	−135360.	−0.459927
$155$	26400.0	0.0882622
$156$	−22968.0	−0.0755634
$157$	−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
$159$	−303750.	−0.952848
$160$	−8640.00	−0.0266817
$161$	33600.0	0.102159
$162$	−39366.0	−0.117851
$163$	−352676.	−1.03970	−0.519849	−	0.854258i	$-0.674012\pi$
$163$	−352676.	−1.03970	−0.519849	+	0.854258i	$0.674012\pi$
$164$	27480.0	0.0797824
$165$	30456.0	0.0870889
$166$	−494712.	−1.39342
$167$	217560.	0.603654	0.301827	−	0.953363i	$-0.402404\pi$
$167$	217560.	0.603654	0.301827	+	0.953363i	$0.402404\pi$
$168$	−60480.0	−0.165325
$169$	35751.0	0.0962878
$170$	0	0
$171$	−45036.0	−0.117779
$172$	38576.0	0.0994251
$173$	163698.	0.415842	0.207921	−	0.978146i	$-0.433330\pi$
$173$	163698.	0.415842	0.207921	+	0.978146i	$0.433330\pi$
$174$	−250452.	−0.627121
$175$	−123560.	−0.304988
$176$	−640704.	−1.55911
$177$	162756.	0.390485
$178$	564516.	1.33545
$179$	358740.	0.836849	0.418425	−	0.908252i	$-0.362582\pi$
$179$	358740.	0.836849	0.418425	+	0.908252i	$0.362582\pi$
$180$	−1944.00	−0.00447214
$181$	507130.	1.15060	0.575298	−	0.817944i	$-0.304886\pi$
$181$	507130.	1.15060	0.575298	+	0.817944i	$0.304886\pi$
$182$	−153120.	−0.342652
$183$	357822.	0.789840
$184$	141120.	0.307287
$185$	−14460.0	−0.0310627
$186$	−237600.	−0.503575
$187$	0	0
$188$	−74688.0	−0.154119
$189$	−29160.0	−0.0593790
$190$	−20016.0	−0.0402247
$191$	−648384.	−1.28602	−0.643012	−	0.765856i	$-0.722315\pi$
$191$	−648384.	−1.28602	−0.643012	+	0.765856i	$0.722315\pi$
$192$	−249408.	−0.488267
$193$	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
$195$	34452.0	0.0648826
$196$	−60828.0	−0.113100
$197$	−611046.	−1.12178	−0.560891	−	0.827890i	$-0.689541\pi$
$197$	−611046.	−1.12178	−0.560891	+	0.827890i	$0.689541\pi$
$198$	−274104.	−0.496881
$199$	−879032.	−1.57352	−0.786760	−	0.617260i	$-0.788243\pi$
$199$	−879032.	−1.57352	−0.786760	+	0.617260i	$0.788243\pi$
$200$	−518952.	−0.917386
$201$	207612.	0.362462
$202$	858204.	1.47983
$203$	−185520.	−0.315973
$204$	0	0
$205$	−41220.0	−0.0685052
$206$	−318864.	−0.523525
$207$	68040.0	0.110367
$208$	−724768.	−1.16156
$209$	−313584.	−0.496579
$210$	−12960.0	−0.0202795
$211$	−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
$213$	−38232.0	−0.0577402
$214$	544968.	0.813461
$215$	−57864.0	−0.0853714
$216$	−122472.	−0.178609
$217$	−176000.	−0.253725
$218$	−369996.	−0.527298
$219$	−369990.	−0.521290
$220$	−13536.0	−0.0188553
$221$	0	0
$222$	130140.	0.177226
$223$	−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
$225$	−250209.	−0.329493
$226$	62892.0	0.0819076
$227$	−606180.	−0.780795	−0.390397	−	0.920646i	$-0.627662\pi$
$227$	−606180.	−0.780795	−0.390397	+	0.920646i	$0.627662\pi$
$228$	20016.0	0.0255000
$229$	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
$231$	−203040.	−0.250352
$232$	−779184.	−0.950430
$233$	392886.	0.474107	0.237054	−	0.971497i	$-0.423818\pi$
$233$	392886.	0.474107	0.237054	+	0.971497i	$0.423818\pi$
$234$	−310068.	−0.370184
$235$	112032.	0.132334
$236$	−72336.0	−0.0845424
$237$	197280.	0.228146
$238$	0	0
$239$	−1.32514e6	−1.50060	−0.750301	−	0.661096i	$-0.770092\pi$
$239$	−1.32514e6	−1.50060	−0.750301	+	0.661096i	$0.770092\pi$
$240$	−61344.0	−0.0687455
$241$	990094.	1.09808	0.549040	−	0.835796i	$-0.314994\pi$
$241$	990094.	1.09808	0.549040	+	0.835796i	$0.314994\pi$
$242$	−942270.	−1.03428
$243$	−59049.0	−0.0641500
$244$	−159032.	−0.171005
$245$	91242.0	0.0971135
$246$	370980.	0.390852
$247$	−354728.	−0.369959
$248$	−739200.	−0.763190
$249$	−742068.	−0.758482
$250$	−223704.	−0.226373
$251$	147132.	0.147409	0.0737043	−	0.997280i	$-0.476518\pi$
$251$	147132.	0.147409	0.0737043	+	0.997280i	$0.476518\pi$
$252$	12960.0	0.0128559
$253$	473760.	0.465326
$254$	1.02653e6	0.998358
$255$	0	0
$256$	387328.	0.369385
$257$	−483582.	−0.456707	−0.228353	−	0.973578i	$-0.573334\pi$
$257$	−483582.	−0.456707	−0.228353	+	0.973578i	$0.573334\pi$
$258$	520776.	0.487082
$259$	96400.0	0.0892951
$260$	−15312.0	−0.0140475
$261$	−375678.	−0.341361
$262$	1.55081e6	1.39574
$263$	813576.	0.725285	0.362643	−	0.931928i	$-0.381875\pi$
$263$	813576.	0.725285	0.362643	+	0.931928i	$0.381875\pi$
$264$	−852768.	−0.753045
$265$	−202500.	−0.177137
$266$	133440.	0.115633
$267$	846774.	0.726925
$268$	−92272.0	−0.0784753
$269$	461106.	0.388526	0.194263	−	0.980949i	$-0.437768\pi$
$269$	461106.	0.388526	0.194263	+	0.980949i	$0.437768\pi$
$270$	−26244.0	−0.0219089
$271$	1.67514e6	1.38556	0.692782	−	0.721147i	$-0.256385\pi$
$271$	1.67514e6	1.38556	0.692782	+	0.721147i	$0.256385\pi$
$272$	0	0
$273$	−229680.	−0.186516
$274$	−1.80140e6	−1.44956
$275$	−1.74220e6	−1.38920
$276$	−30240.0	−0.0238951
$277$	−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
$279$	−356400.	−0.274111
$280$	−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	$0.837511\pi$
$282$	−1.00829e6	−0.755026
$283$	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
$285$	−30024.0	−0.0218956
$286$	−2.15899e6	−1.56076
$287$	274800.	0.196930
$288$	116640.	0.0828641
$289$	0	0
$290$	−166968.	−0.116584
$291$	444978.	0.308039
$292$	164440.	0.112863
$293$	−938874.	−0.638908	−0.319454	−	0.947602i	$-0.603499\pi$
$293$	−938874.	−0.638908	−0.319454	+	0.947602i	$0.603499\pi$
$294$	−821178.	−0.554076
$295$	108504.	0.0725923
$296$	404880.	0.268594
$297$	−411156.	−0.270468
$298$	631548.	0.411970
$299$	535920.	0.346675
$300$	111204.	0.0713374
$301$	385760.	0.245415
$302$	−2.37835e6	−1.50058
$303$	1.28731e6	0.805518
$304$	631616.	0.391985
$305$	238548.	0.146834
$306$	0	0
$307$	692948.	0.419619	0.209809	−	0.977742i	$-0.432716\pi$
$307$	692948.	0.419619	0.209809	+	0.977742i	$0.432716\pi$
$308$	90240.0	0.0542029
$309$	−478296.	−0.284971
$310$	−158400.	−0.0936162
$311$	−2.94310e6	−1.72545	−0.862727	−	0.505670i	$-0.831245\pi$
$311$	−2.94310e6	−1.72545	−0.862727	+	0.505670i	$0.831245\pi$
$312$	−964656.	−0.561030
$313$	−885146.	−0.510686	−0.255343	−	0.966851i	$-0.582188\pi$
$313$	−885146.	−0.510686	−0.255343	+	0.966851i	$0.582188\pi$
$314$	826476.	0.473049
$315$	−19440.0	−0.0110387
$316$	−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	$0.747313\pi$
$318$	1.82250e6	1.01065
$319$	−2.61583e6	−1.43924
$320$	−166272.	−0.0907704
$321$	817452.	0.442792
$322$	−201600.	−0.108355
$323$	0	0
$324$	26244.0	0.0138889
$325$	−1.97078e6	−1.03498
$326$	2.11606e6	1.10277
$327$	−554994.	−0.287024
$328$	1.15416e6	0.592354
$329$	−746880.	−0.380418
$330$	−182736.	−0.0923718
$331$	−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
$333$	195210.	0.0964698
$334$	−1.30536e6	−0.640271
$335$	138408.	0.0673828
$336$	408960.	0.197621
$337$	−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
$339$	94338.0	0.0445849
$340$	0	0
$341$	−2.48160e6	−1.15570
$342$	270216.	0.124924
$343$	−1.28056e6	−0.587712
$344$	1.62019e6	0.738194
$345$	45360.0	0.0205175
$346$	−982188.	−0.441067
$347$	2.93207e6	1.30723	0.653613	−	0.756829i	$-0.273253\pi$
$347$	2.93207e6	1.30723	0.653613	+	0.756829i	$0.273253\pi$
$348$	166968.	0.0739069
$349$	905198.	0.397814	0.198907	−	0.980018i	$-0.436261\pi$
$349$	905198.	0.397814	0.198907	+	0.980018i	$0.436261\pi$
$350$	741360.	0.323489
$351$	−465102.	−0.201502
$352$	812160.	0.349369
$353$	1.91786e6	0.819181	0.409590	−	0.912270i	$-0.365672\pi$
$353$	1.91786e6	0.819181	0.409590	+	0.912270i	$0.365672\pi$
$354$	−976536.	−0.414171
$355$	−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	$0.666293\pi$
$360$	−81648.0	−0.0332039
$361$	−2.16696e6	−0.875152
$362$	−3.04278e6	−1.22039
$363$	−1.41340e6	−0.562989
$364$	102080.	0.0403819
$365$	−246660.	−0.0969095
$366$	−2.14693e6	−0.837752
$367$	984064.	0.381380	0.190690	−	0.981650i	$-0.438927\pi$
$367$	984064.	0.381380	0.190690	+	0.981650i	$0.438927\pi$
$368$	−954240.	−0.367314
$369$	556470.	0.212753
$370$	86760.0	0.0329470
$371$	1.35000e6	0.509212
$372$	158400.	0.0593469
$373$	1.70365e6	0.634029	0.317015	−	0.948421i	$-0.397320\pi$
$373$	1.70365e6	0.634029	0.317015	+	0.948421i	$0.397320\pi$
$374$	0	0
$375$	−335556.	−0.123222
$376$	−3.13690e6	−1.14428
$377$	−2.95904e6	−1.07225
$378$	174960.	0.0629810
$379$	−2.75654e6	−0.985749	−0.492874	−	0.870100i	$-0.664054\pi$
$379$	−2.75654e6	−0.985749	−0.492874	+	0.870100i	$0.664054\pi$
$380$	13344.0	0.00474053
$381$	1.53979e6	0.543438
$382$	3.89030e6	1.36403
$383$	456576.	0.159044	0.0795218	−	0.996833i	$-0.474661\pi$
$383$	456576.	0.159044	0.0795218	+	0.996833i	$0.474661\pi$
$384$	1.91117e6	0.661410
$385$	−135360.	−0.0465413
$386$	−167028.	−0.0570586
$387$	781164.	0.265134
$388$	−197768.	−0.0666925
$389$	−2.00639e6	−0.672268	−0.336134	−	0.941814i	$-0.609119\pi$
$389$	−2.00639e6	−0.672268	−0.336134	+	0.941814i	$0.609119\pi$
$390$	−206712.	−0.0688183
$391$	0	0
$392$	−2.55478e6	−0.839726
$393$	2.32621e6	0.759745
$394$	3.66628e6	1.18983
$395$	131520.	0.0424130
$396$	182736.	0.0585580
$397$	5.77040e6	1.83751	0.918755	−	0.394828i	$-0.129196\pi$
$397$	5.77040e6	1.83751	0.918755	+	0.394828i	$0.129196\pi$
$398$	5.27419e6	1.66897
$399$	200160.	0.0629427
$400$	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	$0.654595\pi$
$402$	−1.24567e6	−0.384449
$403$	−2.80720e6	−0.861015
$404$	−572136.	−0.174400
$405$	−39366.0	−0.0119257
$406$	1.11312e6	0.335140
$407$	1.35924e6	0.406734
$408$	0	0
$409$	1.53363e6	0.453327	0.226663	−	0.973973i	$-0.427218\pi$
$409$	1.53363e6	0.453327	0.226663	+	0.973973i	$0.427218\pi$
$410$	247320.	0.0726607
$411$	−2.70211e6	−0.789038
$412$	212576.	0.0616980
$413$	−723360.	−0.208679
$414$	−408240.	−0.117062
$415$	−494712.	−0.141004
$416$	918720.	0.260285
$417$	−3.15148e6	−0.887511
$418$	1.88150e6	0.526701
$419$	3.87376e6	1.07795	0.538973	−	0.842323i	$-0.318812\pi$
$419$	3.87376e6	1.07795	0.538973	+	0.842323i	$0.318812\pi$
$420$	8640.00	0.00238996
$421$	−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
$423$	−1.51243e6	−0.410984
$424$	5.67000e6	1.53168
$425$	0	0
$426$	229392.	0.0612427
$427$	−1.59032e6	−0.422099
$428$	−363312.	−0.0958673
$429$	−3.23849e6	−0.849570
$430$	347184.	0.0905500
$431$	−6.45192e6	−1.67300	−0.836500	−	0.547967i	$-0.815402\pi$
$431$	−6.45192e6	−1.67300	−0.836500	+	0.547967i	$0.815402\pi$
$432$	828144.	0.213499
$433$	−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
$435$	−250452.	−0.0634602
$436$	246664.	0.0621426
$437$	−467040.	−0.116990
$438$	2.21994e6	0.552912
$439$	−792680.	−0.196307	−0.0981537	−	0.995171i	$-0.531294\pi$
$439$	−792680.	−0.196307	−0.0981537	+	0.995171i	$0.531294\pi$
$440$	−568512.	−0.139994
$441$	−1.23177e6	−0.301601
$442$	0	0
$443$	−1.39981e6	−0.338891	−0.169446	−	0.985540i	$-0.554198\pi$
$443$	−1.39981e6	−0.338891	−0.169446	+	0.985540i	$0.554198\pi$
$444$	−86760.0	−0.0208863
$445$	564516.	0.135138
$446$	5.99683e6	1.42753
$447$	947322.	0.224248
$448$	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	$0.613905\pi$
$450$	1.50125e6	0.349480
$451$	3.87468e6	0.897004
$452$	−41928.0	−0.00965291
$453$	−3.56753e6	−0.816812
$454$	3.63708e6	0.828158
$455$	−153120.	−0.0346740
$456$	840672.	0.189328
$457$	6.29969e6	1.41101	0.705503	−	0.708707i	$-0.250721\pi$
$457$	6.29969e6	1.41101	0.705503	+	0.708707i	$0.250721\pi$
$458$	−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	$0.378360\pi$
$462$	1.21824e6	0.265539
$463$	−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
$465$	−237600.	−0.0509582
$466$	−2.35732e6	−0.502867
$467$	−6.51409e6	−1.38217	−0.691085	−	0.722773i	$-0.742867\pi$
$467$	−6.51409e6	−1.38217	−0.691085	+	0.722773i	$0.742867\pi$
$468$	206712.	0.0436266
$469$	−922720.	−0.193704
$470$	−672192.	−0.140362
$471$	1.23971e6	0.257495
$472$	−3.03811e6	−0.627695
$473$	5.43922e6	1.11785
$474$	−1.18368e6	−0.241985
$475$	1.71748e6	0.349268
$476$	0	0
$477$	2.73375e6	0.550127
$478$	7.95082e6	1.59163
$479$	−2.39232e6	−0.476410	−0.238205	−	0.971215i	$-0.576559\pi$
$479$	−2.39232e6	−0.476410	−0.238205	+	0.971215i	$0.576559\pi$
$480$	77760.0	0.0154047
$481$	1.53758e6	0.303023
$482$	−5.94056e6	−1.16469
$483$	−302400.	−0.0589812
$484$	628180.	0.121891
$485$	296652.	0.0572655
$486$	354294.	0.0680414
$487$	6.13089e6	1.17139	0.585694	−	0.810532i	$-0.300822\pi$
$487$	6.13089e6	1.17139	0.585694	+	0.810532i	$0.300822\pi$
$488$	−6.67934e6	−1.26965
$489$	3.17408e6	0.600269
$490$	−547452.	−0.103004
$491$	−1.23589e6	−0.231354	−0.115677	−	0.993287i	$-0.536904\pi$
$491$	−1.23589e6	−0.231354	−0.115677	+	0.993287i	$0.536904\pi$
$492$	−247320.	−0.0460624
$493$	0	0
$494$	2.12837e6	0.392400
$495$	−274104.	−0.0502808
$496$	4.99840e6	0.912277
$497$	169920.	0.0308570
$498$	4.45241e6	0.804492
$499$	9.85496e6	1.77175	0.885877	−	0.463921i	$-0.153558\pi$
$499$	9.85496e6	1.77175	0.885877	+	0.463921i	$0.153558\pi$
$500$	149136.	0.0266783
$501$	−1.95804e6	−0.348520
$502$	−882792.	−0.156350
$503$	−1.16777e6	−0.205796	−0.102898	−	0.994692i	$-0.532812\pi$
$503$	−1.16777e6	−0.205796	−0.102898	+	0.994692i	$0.532812\pi$
$504$	544320.	0.0954504
$505$	858204.	0.149748
$506$	−2.84256e6	−0.493552
$507$	−321759.	−0.0555918
$508$	−684352.	−0.117658
$509$	1.04941e6	0.179535	0.0897675	−	0.995963i	$-0.471388\pi$
$509$	1.04941e6	0.179535	0.0897675	+	0.995963i	$0.471388\pi$
$510$	0	0
$511$	1.64440e6	0.278583
$512$	4.47130e6	0.753804
$513$	405324.	0.0680000
$514$	2.90149e6	0.484411
$515$	−318864.	−0.0529770
$516$	−347184.	−0.0574031
$517$	−1.05310e7	−1.73278
$518$	−578400.	−0.0947118
$519$	−1.47328e6	−0.240086
$520$	−643104.	−0.104297
$521$	9.61407e6	1.55172	0.775859	−	0.630906i	$-0.217317\pi$
$521$	9.61407e6	1.55172	0.775859	+	0.630906i	$0.217317\pi$
$522$	2.25407e6	0.362069
$523$	6.96148e6	1.11288	0.556439	−	0.830888i	$-0.312167\pi$
$523$	6.96148e6	1.11288	0.556439	+	0.830888i	$0.312167\pi$
$524$	−1.03387e6	−0.164490
$525$	1.11204e6	0.176085
$526$	−4.88146e6	−0.769281
$527$	0	0
$528$	5.76634e6	0.900151
$529$	−5.73074e6	−0.890373
$530$	1.21500e6	0.187883
$531$	−1.46480e6	−0.225446
$532$	−88960.0	−0.0136275
$533$	4.38306e6	0.668281
$534$	−5.08064e6	−0.771020
$535$	544968.	0.0823164
$536$	−3.87542e6	−0.582649
$537$	−3.22866e6	−0.483155
$538$	−2.76664e6	−0.412094
$539$	−8.57675e6	−1.27160
$540$	17496.0	0.00258199
$541$	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
$543$	−4.56417e6	−0.664297
$544$	0	0
$545$	−369996.	−0.0533588
$546$	1.37808e6	0.197830
$547$	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
$549$	−3.22040e6	−0.456015
$550$	1.04532e7	1.47347
$551$	2.57873e6	0.361848
$552$	−1.27008e6	−0.177412
$553$	−876800.	−0.121924
$554$	2.40676e6	0.333164
$555$	130140.	0.0179340
$556$	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	$0.392586\pi$
$558$	2.13840e6	0.290739
$559$	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	$0.308737\pi$
$564$	672192.	0.0889807
$565$	62892.0	0.00828847
$566$	−6.77263e6	−0.888621
$567$	262440.	0.0342825
$568$	713664.	0.0928160
$569$	362874.	0.0469867	0.0234934	−	0.999724i	$-0.492521\pi$
$569$	362874.	0.0469867	0.0234934	+	0.999724i	$0.492521\pi$
$570$	180144.	0.0232238
$571$	−4.11024e6	−0.527566	−0.263783	−	0.964582i	$-0.584970\pi$
$571$	−4.11024e6	−0.527566	−0.263783	+	0.964582i	$0.584970\pi$
$572$	1.43933e6	0.183937
$573$	5.83546e6	0.742486
$574$	−1.64880e6	−0.208876
$575$	−2.59476e6	−0.327286
$576$	2.24467e6	0.281901
$577$	−7.87680e6	−0.984941	−0.492470	−	0.870329i	$-0.663906\pi$
$577$	−7.87680e6	−0.984941	−0.492470	+	0.870329i	$0.663906\pi$
$578$	0	0
$579$	−250542.	−0.0310588
$580$	111312.	0.0137395
$581$	3.29808e6	0.405341
$582$	−2.66987e6	−0.326725
$583$	1.90350e7	2.31943
$584$	6.90648e6	0.837963
$585$	−310068.	−0.0374600
$586$	5.63324e6	0.677664
$587$	603948.	0.0723443	0.0361721	−	0.999346i	$-0.488484\pi$
$587$	603948.	0.0723443	0.0361721	+	0.999346i	$0.488484\pi$
$588$	547452.	0.0652984
$589$	2.44640e6	0.290562
$590$	−651024.	−0.0769958
$591$	5.49941e6	0.647661
$592$	−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	$0.601925\pi$
$594$	2.46694e6	0.286874
$595$	0	0
$596$	−421032.	−0.0485511
$597$	7.91129e6	0.908472
$598$	−3.21552e6	−0.367704
$599$	4.27999e6	0.487389	0.243695	−	0.969852i	$-0.421641\pi$
$599$	4.27999e6	0.487389	0.243695	+	0.969852i	$0.421641\pi$
$600$	4.67057e6	0.529653
$601$	−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
$603$	−1.86851e6	−0.209267
$604$	1.58557e6	0.176845
$605$	−942270.	−0.104661
$606$	−7.72384e6	−0.854381
$607$	−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
$609$	1.66968e6	0.182427
$610$	−1.43129e6	−0.155741
$611$	−1.19127e7	−1.29095
$612$	0	0
$613$	4.21506e6	0.453057	0.226528	−	0.974005i	$-0.427262\pi$
$613$	4.21506e6	0.453057	0.226528	+	0.974005i	$0.427262\pi$
$614$	−4.15769e6	−0.445073
$615$	370980.	0.0395515
$616$	3.79008e6	0.402436
$617$	4.40665e6	0.466010	0.233005	−	0.972476i	$-0.425144\pi$
$617$	4.40665e6	0.466010	0.233005	+	0.972476i	$0.425144\pi$
$618$	2.86978e6	0.302257
$619$	−4.80168e6	−0.503693	−0.251847	−	0.967767i	$-0.581038\pi$
$619$	−4.80168e6	−0.503693	−0.251847	+	0.967767i	$0.581038\pi$
$620$	105600.	0.0110328
$621$	−612360.	−0.0637203
$622$	1.76586e7	1.83012
$623$	−3.76344e6	−0.388477
$624$	6.52291e6	0.670625
$625$	9.42942e6	0.965573
$626$	5.31088e6	0.541664
$627$	2.82226e6	0.286700
$628$	−550984.	−0.0557494
$629$	0	0
$630$	116640.	0.0117084
$631$	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
$633$	436500.	0.0432987
$634$	1.50528e7	1.48728
$635$	1.02653e6	0.101027
$636$	−1.21500e6	−0.119106
$637$	−9.70207e6	−0.947361
$638$	1.56950e7	1.52654
$639$	344088.	0.0333363
$640$	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	$0.823721\pi$
$642$	−4.90471e6	−0.469652
$643$	1.28394e7	1.22466	0.612330	−	0.790602i	$-0.290232\pi$
$643$	1.28394e7	1.22466	0.612330	+	0.790602i	$0.290232\pi$
$644$	134400.	0.0127698
$645$	520776.	0.0492892
$646$	0	0
$647$	−2.08468e7	−1.95785	−0.978924	−	0.204226i	$-0.934532\pi$
$647$	−2.08468e7	−1.95785	−0.978924	+	0.204226i	$0.934532\pi$
$648$	1.10225e6	0.103120
$649$	−1.01994e7	−0.950521
$650$	1.18247e7	1.09776
$651$	1.58400e6	0.146488
$652$	−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	$0.702785\pi$
$654$	3.32996e6	0.304435
$655$	1.55081e6	0.141239
$656$	−7.80432e6	−0.708069
$657$	3.32991e6	0.300967
$658$	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	$0.581823\pi$
$660$	121824.	0.0108861
$661$	−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
$666$	−1.17126e6	−0.102322
$667$	−3.89592e6	−0.339075
$668$	870240.	0.0754567
$669$	8.99525e6	0.777048
$670$	−830448.	−0.0714703
$671$	−2.24235e7	−1.92264
$672$	−518400.	−0.0442835
$673$	−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
$675$	2.25188e6	0.190233
$676$	143004.	0.0120360
$677$	1.60910e7	1.34931	0.674656	−	0.738132i	$-0.264292\pi$
$677$	1.60910e7	1.34931	0.674656	+	0.738132i	$0.264292\pi$
$678$	−566028.	−0.0472894
$679$	−1.97768e6	−0.164620
$680$	0	0
$681$	5.45562e6	0.450792
$682$	1.48896e7	1.22581
$683$	−1.60780e7	−1.31880	−0.659402	−	0.751791i	$-0.729190\pi$
$683$	−1.60780e7	−1.31880	−0.659402	+	0.751791i	$0.729190\pi$
$684$	−180144.	−0.0147224
$685$	−1.80140e6	−0.146685
$686$	7.68336e6	0.623363
$687$	−1.22393e7	−0.989386
$688$	−1.09556e7	−0.882398
$689$	2.15325e7	1.72801
$690$	−272160.	−0.0217621
$691$	165964.	0.0132227	0.00661133	−	0.999978i	$-0.497896\pi$
$691$	165964.	0.0132227	0.00661133	+	0.999978i	$0.497896\pi$
$692$	654792.	0.0519802
$693$	1.82736e6	0.144541
$694$	−1.75924e7	−1.38652
$695$	−2.10098e6	−0.164991
$696$	7.01266e6	0.548731
$697$	0	0
$698$	−5.43119e6	−0.421945
$699$	−3.53597e6	−0.273726
$700$	−494240.	−0.0381235
$701$	1.77248e7	1.36234	0.681171	−	0.732124i	$-0.261471\pi$
$701$	1.77248e7	1.36234	0.681171	+	0.732124i	$0.261471\pi$
$702$	2.79061e6	0.213726
$703$	−1.33996e6	−0.102259
$704$	1.56296e7	1.18854
$705$	−1.00829e6	−0.0764032
$706$	−1.15071e7	−0.868872
$707$	−5.72136e6	−0.430478
$708$	651024.	0.0488106
$709$	1.06023e7	0.792112	0.396056	−	0.918226i	$-0.370379\pi$
$709$	1.06023e7	0.792112	0.396056	+	0.918226i	$0.370379\pi$
$710$	152928.	0.0113852
$711$	−1.77552e6	−0.131720
$712$	−1.58064e7	−1.16852
$713$	−3.69600e6	−0.272275
$714$	0	0
$715$	−2.15899e6	−0.157938
$716$	1.43496e6	0.104606
$717$	1.19262e7	0.866373
$718$	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	$0.605630\pi$
$720$	552096.	0.0396902
$721$	2.12576e6	0.152292
$722$	1.30018e7	0.928239
$723$	−8.91085e6	−0.633977
$724$	2.02852e6	0.143825
$725$	1.43268e7	1.01229
$726$	8.48043e6	0.597140
$727$	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
$729$	531441.	0.0370370
$730$	1.47996e6	0.102788
$731$	0	0
$732$	1.43129e6	0.0987300
$733$	−1.17773e7	−0.809626	−0.404813	−	0.914399i	$-0.632663\pi$
$733$	−1.17773e7	−0.809626	−0.404813	+	0.914399i	$0.632663\pi$
$734$	−5.90438e6	−0.404515
$735$	−821178.	−0.0560685
$736$	1.20960e6	0.0823090
$737$	−1.30104e7	−0.882308
$738$	−3.33882e6	−0.225659
$739$	5.88948e6	0.396703	0.198352	−	0.980131i	$-0.436441\pi$
$739$	5.88948e6	0.396703	0.198352	+	0.980131i	$0.436441\pi$
$740$	−57840.0	−0.00388284
$741$	3.19255e6	0.213596
$742$	−8.10000e6	−0.540101
$743$	1.00476e7	0.667712	0.333856	−	0.942624i	$-0.391650\pi$
$743$	1.00476e7	0.667712	0.333856	+	0.942624i	$0.391650\pi$
$744$	6.65280e6	0.440628
$745$	631548.	0.0416884
$746$	−1.02219e7	−0.672490
$747$	6.67861e6	0.437910
$748$	0	0
$749$	−3.63312e6	−0.236633
$750$	2.01334e6	0.130696
$751$	−4.81530e6	−0.311547	−0.155773	−	0.987793i	$-0.549787\pi$
$751$	−4.81530e6	−0.311547	−0.155773	+	0.987793i	$0.549787\pi$
$752$	2.12114e7	1.36781
$753$	−1.32419e6	−0.0851064
$754$	1.77543e7	1.13730
$755$	−2.37835e6	−0.151848
$756$	−116640.	−0.00742238
$757$	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
$759$	−4.26384e6	−0.268656
$760$	560448.	0.0351967
$761$	−1.17773e7	−0.737197	−0.368599	−	0.929589i	$-0.620162\pi$
$761$	−1.17773e7	−0.737197	−0.368599	+	0.929589i	$0.620162\pi$
$762$	−9.23875e6	−0.576403
$763$	2.46664e6	0.153389
$764$	−2.59354e6	−0.160753
$765$	0	0
$766$	−2.73946e6	−0.168691
$767$	−1.15376e7	−0.708152
$768$	−3.48595e6	−0.213264
$769$	−1.49376e6	−0.0910887	−0.0455443	−	0.998962i	$-0.514502\pi$
$769$	−1.49376e6	−0.0910887	−0.0455443	+	0.998962i	$0.514502\pi$
$770$	812160.	0.0493645
$771$	4.35224e6	0.263680
$772$	111352.	0.00672442
$773$	−2.25125e7	−1.35511	−0.677555	−	0.735472i	$-0.736960\pi$
$773$	−2.25125e7	−1.35511	−0.677555	+	0.735472i	$0.736960\pi$
$774$	−4.68698e6	−0.281217
$775$	1.35916e7	0.812861
$776$	−8.30626e6	−0.495166
$777$	−867600.	−0.0515545
$778$	1.20384e7	0.713048
$779$	−3.81972e6	−0.225521
$780$	137808.	0.00811032
$781$	2.39587e6	0.140552
$782$	0	0
$783$	3.38110e6	0.197085
$784$	1.72752e7	1.00376
$785$	826476.	0.0478692
$786$	−1.39573e7	−0.805831
$787$	−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
$789$	−7.32218e6	−0.418744
$790$	−789120.	−0.0449858
$791$	−419280.	−0.0238266
$792$	7.67491e6	0.434771
$793$	−2.53656e7	−1.43239
$794$	−3.46224e7	−1.94897
$795$	1.82250e6	0.102270
$796$	−3.51613e6	−0.196690
$797$	540798.	0.0301571	0.0150785	−	0.999886i	$-0.495200\pi$
$797$	540798.	0.0301571	0.0150785	+	0.999886i	$0.495200\pi$
$798$	−1.20096e6	−0.0667608
$799$	0	0
$800$	−4.44816e6	−0.245728
$801$	−7.62097e6	−0.419690
$802$	1.80375e7	0.990243
$803$	2.31860e7	1.26893
$804$	830448.	0.0453077
$805$	−201600.	−0.0109648
$806$	1.68432e7	0.913244
$807$	−4.14995e6	−0.224316
$808$	−2.40297e7	−1.29485
$809$	6.14223e6	0.329955	0.164978	−	0.986297i	$-0.447245\pi$
$809$	6.14223e6	0.329955	0.164978	+	0.986297i	$0.447245\pi$
$810$	236196.	0.0126491
$811$	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
$813$	−1.50762e7	−0.799956
$814$	−8.15544e6	−0.431406
$815$	2.11606e6	0.111592
$816$	0	0
$817$	−5.36206e6	−0.281046
$818$	−9.20176e6	−0.480825
$819$	2.06712e6	0.107685
$820$	−164880.	−0.00856315
$821$	−2.66175e7	−1.37819	−0.689095	−	0.724671i	$-0.741992\pi$
$821$	−2.66175e7	−1.37819	−0.689095	+	0.724671i	$0.741992\pi$
$822$	1.62126e7	0.836901
$823$	−3.62817e7	−1.86719	−0.933593	−	0.358335i	$-0.883345\pi$
$823$	−3.62817e7	−1.86719	−0.933593	+	0.358335i	$0.883345\pi$
$824$	8.92819e6	0.458084
$825$	1.56798e7	0.802056
$826$	4.34016e6	0.221338
$827$	−1.09033e6	−0.0554364	−0.0277182	−	0.999616i	$-0.508824\pi$
$827$	−1.09033e6	−0.0554364	−0.0277182	+	0.999616i	$0.508824\pi$
$828$	272160.	0.0137958
$829$	−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
$831$	3.61013e6	0.181351
$832$	1.76803e7	0.885483
$833$	0	0
$834$	1.89089e7	0.941348
$835$	−1.30536e6	−0.0647909
$836$	−1.25434e6	−0.0620724
$837$	3.20760e6	0.158258
$838$	−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	$0.340285\pi$
$840$	362880.	0.0177445
$841$	999895.	0.0487489
$842$	7.99840e6	0.388797
$843$	2.07879e7	1.00749
$844$	−194000.	−0.00937445
$845$	−214506.	−0.0103347
$846$	9.07459e6	0.435914
$847$	6.28180e6	0.300868
$848$	−3.83400e7	−1.83089
$849$	−1.01589e7	−0.483704
$850$	0	0
$851$	2.02440e6	0.0958236
$852$	−152928.	−0.00721752
$853$	−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
$855$	270216.	0.0126414
$856$	−1.52591e7	−0.711778
$857$	2.57953e7	1.19974	0.599872	−	0.800096i	$-0.295218\pi$
$857$	2.57953e7	1.19974	0.599872	+	0.800096i	$0.295218\pi$
$858$	1.94309e7	0.901105
$859$	−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
$861$	−2.47320e6	−0.113698
$862$	3.87115e7	1.77448
$863$	−673056.	−0.0307627	−0.0153813	−	0.999882i	$-0.504896\pi$
$863$	−673056.	−0.0307627	−0.0153813	+	0.999882i	$0.504896\pi$
$864$	−1.04976e6	−0.0478416
$865$	−982188.	−0.0446328
$866$	2.49946e7	1.13254
$867$	0	0
$868$	−704000.	−0.0317156
$869$	−1.23629e7	−0.555354
$870$	1.50271e6	0.0673097
$871$	−1.47174e7	−0.657333
$872$	1.03599e7	0.461385
$873$	−4.00480e6	−0.177847
$874$	2.80224e6	0.124087
$875$	1.49136e6	0.0658510
$876$	−1.47996e6	−0.0651613
$877$	−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
$879$	8.44987e6	0.368874
$880$	3.84422e6	0.167341
$881$	−2.78891e7	−1.21058	−0.605291	−	0.796004i	$-0.706943\pi$
$881$	−2.78891e7	−1.21058	−0.605291	+	0.796004i	$0.706943\pi$
$882$	7.39060e6	0.319896
$883$	−2.83786e7	−1.22487	−0.612435	−	0.790521i	$-0.709810\pi$
$883$	−2.83786e7	−1.22487	−0.612435	+	0.790521i	$0.709810\pi$
$884$	0	0
$885$	−976536.	−0.0419112
$886$	8.39887e6	0.359448
$887$	−4.22678e7	−1.80385	−0.901925	−	0.431893i	$-0.857846\pi$
$887$	−4.22678e7	−1.80385	−0.901925	+	0.431893i	$0.857846\pi$
$888$	−3.64392e6	−0.155073
$889$	−6.84352e6	−0.290419
$890$	−3.38710e6	−0.143335
$891$	3.70040e6	0.156155
$892$	−3.99789e6	−0.168236
$893$	1.03816e7	0.435649
$894$	−5.68393e6	−0.237851
$895$	−2.15244e6	−0.0898201
$896$	−8.49408e6	−0.353465
$897$	−4.82328e6	−0.200153
$898$	1.79549e7	0.743005
$899$	2.04072e7	0.842140
$900$	−1.00084e6	−0.0411867
$901$	0	0
$902$	−2.32481e7	−0.951417
$903$	−3.47184e6	−0.141690
$904$	−1.76098e6	−0.0716692
$905$	−3.04278e6	−0.123495
$906$	2.14052e7	0.866360
$907$	−3.19526e7	−1.28970	−0.644849	−	0.764310i	$-0.723080\pi$
$907$	−3.19526e7	−1.28970	−0.644849	+	0.764310i	$0.723080\pi$
$908$	−2.42472e6	−0.0975994
$909$	−1.15858e7	−0.465066
$910$	918720.	0.0367773
$911$	1.16429e7	0.464800	0.232400	−	0.972620i	$-0.425342\pi$
$911$	1.16429e7	0.464800	0.232400	+	0.972620i	$0.425342\pi$
$912$	−5.68454e6	−0.226312
$913$	4.65029e7	1.84630
$914$	−3.77981e7	−1.49660
$915$	−2.14693e6	−0.0847746
$916$	5.43970e6	0.214208
$917$	−1.03387e7	−0.406016
$918$	0	0
$919$	1.39844e6	0.0546204	0.0273102	−	0.999627i	$-0.491306\pi$
$919$	1.39844e6	0.0546204	0.0273102	+	0.999627i	$0.491306\pi$
$920$	−846720.	−0.0329815
$921$	−6.23653e6	−0.242267
$922$	−2.04191e7	−0.791059
$923$	2.71022e6	0.104713
$924$	−812160.	−0.0312940
$925$	−7.44449e6	−0.286075
$926$	1.33820e7	0.512854
$927$	4.30466e6	0.164528
$928$	−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	$0.397313\pi$
$930$	1.42560e6	0.0540493
$931$	8.45509e6	0.319701
$932$	1.57154e6	0.0592634
$933$	2.64879e7	0.996191
$934$	3.90846e7	1.46601
$935$	0	0
$936$	8.68190e6	0.323911
$937$	−2.47956e7	−0.922625	−0.461312	−	0.887238i	$-0.652621\pi$
$937$	−2.47956e7	−0.922625	−0.461312	+	0.887238i	$0.652621\pi$
$938$	5.53632e6	0.205454
$939$	7.96631e6	0.294845
$940$	448128.	0.0165418
$941$	−2.79574e7	−1.02925	−0.514627	−	0.857414i	$-0.672070\pi$
$941$	−2.79574e7	−1.02925	−0.514627	+	0.857414i	$0.672070\pi$
$942$	−7.43828e6	−0.273115
$943$	5.77080e6	0.211328
$944$	2.05434e7	0.750314
$945$	174960.	0.00637322
$946$	−3.26353e7	−1.18566
$947$	−7.64936e6	−0.277173	−0.138586	−	0.990350i	$-0.544256\pi$
$947$	−7.64936e6	−0.277173	−0.138586	+	0.990350i	$0.544256\pi$
$948$	789120.	0.0285182
$949$	2.62282e7	0.945372
$950$	−1.03049e7	−0.370455
$951$	2.25792e7	0.809575
$952$	0	0
$953$	−4.62179e7	−1.64846	−0.824228	−	0.566257i	$-0.808391\pi$
$953$	−4.62179e7	−1.64846	−0.824228	+	0.566257i	$0.808391\pi$
$954$	−1.64025e7	−0.583498
$955$	3.89030e6	0.138031
$956$	−5.30054e6	−0.187575
$957$	2.35425e7	0.830945
$958$	1.43539e7	0.505309
$959$	1.20094e7	0.421671
$960$	1.49645e6	0.0524063
$961$	−9.26915e6	−0.323766
$962$	−9.22548e6	−0.321404
$963$	−7.35707e6	−0.255646
$964$	3.96038e6	0.137260
$965$	−167028.	−0.00577392
$966$	1.81440e6	0.0625591
$967$	2.08557e7	0.717229	0.358615	−	0.933486i	$-0.383249\pi$
$967$	2.08557e7	0.717229	0.358615	+	0.933486i	$0.383249\pi$
$968$	2.63836e7	0.904993
$969$	0	0
$970$	−1.77991e6	−0.0607392
$971$	−4.58152e7	−1.55941	−0.779707	−	0.626144i	$-0.784632\pi$
$971$	−4.58152e7	−1.55941	−0.779707	+	0.626144i	$0.784632\pi$
$972$	−236196.	−0.00801875
$973$	1.40066e7	0.474296
$974$	−3.67853e7	−1.24245
$975$	1.77370e7	0.597544
$976$	4.51651e7	1.51767
$977$	−1.09544e6	−0.0367157	−0.0183578	−	0.999831i	$-0.505844\pi$
$977$	−1.09544e6	−0.0367157	−0.0183578	+	0.999831i	$0.505844\pi$
$978$	−1.90445e7	−0.636682
$979$	−5.30645e7	−1.76949
$980$	364968.	0.0121392
$981$	4.99495e6	0.165714
$982$	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	$0.834468\pi$
$984$	−1.03874e7	−0.341996
$985$	3.66628e6	0.120402
$986$	0	0
$987$	6.72192e6	0.219634
$988$	−1.41891e6	−0.0462448
$989$	8.10096e6	0.263358
$990$	1.64462e6	0.0533309
$991$	4.90389e7	1.58620	0.793098	−	0.609094i	$-0.208467\pi$
$991$	4.90389e7	1.58620	0.793098	+	0.609094i	$0.208467\pi$
$992$	−6.33600e6	−0.204426
$993$	1.94533e6	0.0626067
$994$	−1.01952e6	−0.0327288
$995$	5.27419e6	0.168888
$996$	−2.96827e6	−0.0948103
$997$	−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
$999$	−1.75689e6	−0.0556969

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.6.a.a.1.1		1
17.16	even	2		3.6.a.a.1.1	✓	1
51.50	odd	2		9.6.a.a.1.1		1
68.67	odd	2		48.6.a.a.1.1		1
85.33	odd	4		75.6.b.b.49.2		2
85.67	odd	4		75.6.b.b.49.1		2
85.84	even	2		75.6.a.e.1.1		1
119.16	even	6		147.6.e.h.67.1		2
119.33	odd	6		147.6.e.k.67.1		2
119.67	even	6		147.6.e.h.79.1		2
119.101	odd	6		147.6.e.k.79.1		2
119.118	odd	2		147.6.a.a.1.1		1
136.67	odd	2		192.6.a.l.1.1		1
136.101	even	2		192.6.a.d.1.1		1
153.16	even	6		81.6.c.c.28.1		2
153.50	odd	6		81.6.c.a.55.1		2
153.67	even	6		81.6.c.c.55.1		2
153.101	odd	6		81.6.c.a.28.1		2
187.186	odd	2		363.6.a.d.1.1		1
204.203	even	2		144.6.a.f.1.1		1
221.220	even	2		507.6.a.b.1.1		1
255.152	even	4		225.6.b.b.199.2		2
255.203	even	4		225.6.b.b.199.1		2
255.254	odd	2		225.6.a.a.1.1		1
272.67	odd	4		768.6.d.h.385.1		2
272.101	even	4		768.6.d.k.385.1		2
272.203	odd	4		768.6.d.h.385.2		2
272.237	even	4		768.6.d.k.385.2		2
323.322	odd	2		1083.6.a.c.1.1		1
357.356	even	2		441.6.a.i.1.1		1
408.101	odd	2		576.6.a.s.1.1		1
408.203	even	2		576.6.a.t.1.1		1
561.560	even	2		1089.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.6.a.a.1.1	✓	1	17.16	even	2
9.6.a.a.1.1		1	51.50	odd	2
48.6.a.a.1.1		1	68.67	odd	2
75.6.a.e.1.1		1	85.84	even	2
75.6.b.b.49.1		2	85.67	odd	4
75.6.b.b.49.2		2	85.33	odd	4
81.6.c.a.28.1		2	153.101	odd	6
81.6.c.a.55.1		2	153.50	odd	6
81.6.c.c.28.1		2	153.16	even	6
81.6.c.c.55.1		2	153.67	even	6
144.6.a.f.1.1		1	204.203	even	2
147.6.a.a.1.1		1	119.118	odd	2
147.6.e.h.67.1		2	119.16	even	6
147.6.e.h.79.1		2	119.67	even	6
147.6.e.k.67.1		2	119.33	odd	6
147.6.e.k.79.1		2	119.101	odd	6
192.6.a.d.1.1		1	136.101	even	2
192.6.a.l.1.1		1	136.67	odd	2
225.6.a.a.1.1		1	255.254	odd	2
225.6.b.b.199.1		2	255.203	even	4
225.6.b.b.199.2		2	255.152	even	4
363.6.a.d.1.1		1	187.186	odd	2
441.6.a.i.1.1		1	357.356	even	2
507.6.a.b.1.1		1	221.220	even	2
576.6.a.s.1.1		1	408.101	odd	2
576.6.a.t.1.1		1	408.203	even	2
768.6.d.h.385.1		2	272.67	odd	4
768.6.d.h.385.2		2	272.203	odd	4
768.6.d.k.385.1		2	272.101	even	4
768.6.d.k.385.2		2	272.237	even	4
867.6.a.a.1.1		1	1.1	even	1	trivial
1083.6.a.c.1.1		1	323.322	odd	2
1089.6.a.b.1.1		1	561.560	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:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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