LMFDB - Embedded newform 1089.6.a.c.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 11)
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-4.00000 q^{2} -16.0000 q^{4} +19.0000 q^{5} -10.0000 q^{7} +192.000 q^{8} -76.0000 q^{10} +1148.00 q^{13} +40.0000 q^{14} -256.000 q^{16} +686.000 q^{17} +384.000 q^{19} -304.000 q^{20} -3709.00 q^{23} -2764.00 q^{25} -4592.00 q^{26} +160.000 q^{28} -5424.00 q^{29} -6443.00 q^{31} -5120.00 q^{32} -2744.00 q^{34} -190.000 q^{35} +12063.0 q^{37} -1536.00 q^{38} +3648.00 q^{40} -1528.00 q^{41} +4026.00 q^{43} +14836.0 q^{46} -7168.00 q^{47} -16707.0 q^{49} +11056.0 q^{50} -18368.0 q^{52} +29862.0 q^{53} -1920.00 q^{56} +21696.0 q^{58} +6461.00 q^{59} +16980.0 q^{61} +25772.0 q^{62} +28672.0 q^{64} +21812.0 q^{65} +29999.0 q^{67} -10976.0 q^{68} +760.000 q^{70} -31023.0 q^{71} -1924.00 q^{73} -48252.0 q^{74} -6144.00 q^{76} -65138.0 q^{79} -4864.00 q^{80} +6112.00 q^{82} -102714. q^{83} +13034.0 q^{85} -16104.0 q^{86} -17415.0 q^{89} -11480.0 q^{91} +59344.0 q^{92} +28672.0 q^{94} +7296.00 q^{95} +66905.0 q^{97} +66828.0 q^{98} +O(q^{100})

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−4.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−4.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−16.0000	−0.500000
$5$	19.0000	0.339882	0.169941	−	0.985454i	$-0.445642\pi$
$5$	19.0000	0.339882	0.169941	+	0.985454i	$0.445642\pi$
$6$	0	0
$7$	−10.0000	−0.0771356	−0.0385678	−	0.999256i	$-0.512280\pi$
$7$	−10.0000	−0.0771356	−0.0385678	+	0.999256i	$0.512280\pi$
$8$	192.000	1.06066
$9$	0	0
$10$	−76.0000	−0.240333
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1148.00	1.88401	0.942006	−	0.335597i	$-0.108938\pi$
$13$	1148.00	1.88401	0.942006	+	0.335597i	$0.108938\pi$
$14$	40.0000	0.0545431
$15$	0	0
$16$	−256.000	−0.250000
$17$	686.000	0.575707	0.287854	−	0.957674i	$-0.407058\pi$
$17$	686.000	0.575707	0.287854	+	0.957674i	$0.407058\pi$
$18$	0	0
$19$	384.000	0.244032	0.122016	−	0.992528i	$-0.461064\pi$
$19$	384.000	0.244032	0.122016	+	0.992528i	$0.461064\pi$
$20$	−304.000	−0.169941
$21$	0	0
$22$	0	0
$23$	−3709.00	−1.46197	−0.730983	−	0.682396i	$-0.760938\pi$
$23$	−3709.00	−1.46197	−0.730983	+	0.682396i	$0.760938\pi$
$24$	0	0
$25$	−2764.00	−0.884480
$26$	−4592.00	−1.33220
$27$	0	0
$28$	160.000	0.0385678
$29$	−5424.00	−1.19764	−0.598818	−	0.800885i	$-0.704363\pi$
$29$	−5424.00	−1.19764	−0.598818	+	0.800885i	$0.704363\pi$
$30$	0	0
$31$	−6443.00	−1.20416	−0.602080	−	0.798436i	$-0.705661\pi$
$31$	−6443.00	−1.20416	−0.602080	+	0.798436i	$0.705661\pi$
$32$	−5120.00	−0.883883
$33$	0	0
$34$	−2744.00	−0.407087
$35$	−190.000	−0.0262170
$36$	0	0
$37$	12063.0	1.44861	0.724304	−	0.689481i	$-0.242161\pi$
$37$	12063.0	1.44861	0.724304	+	0.689481i	$0.242161\pi$
$38$	−1536.00	−0.172557
$39$	0	0
$40$	3648.00	0.360500
$41$	−1528.00	−0.141959	−0.0709796	−	0.997478i	$-0.522613\pi$
$41$	−1528.00	−0.141959	−0.0709796	+	0.997478i	$0.522613\pi$
$42$	0	0
$43$	4026.00	0.332049	0.166025	−	0.986122i	$-0.446907\pi$
$43$	4026.00	0.332049	0.166025	+	0.986122i	$0.446907\pi$
$44$	0	0
$45$	0	0
$46$	14836.0	1.03377
$47$	−7168.00	−0.473318	−0.236659	−	0.971593i	$-0.576052\pi$
$47$	−7168.00	−0.473318	−0.236659	+	0.971593i	$0.576052\pi$
$48$	0	0
$49$	−16707.0	−0.994050
$50$	11056.0	0.625422
$51$	0	0
$52$	−18368.0	−0.942006
$53$	29862.0	1.46026	0.730128	−	0.683310i	$-0.239460\pi$
$53$	29862.0	1.46026	0.730128	+	0.683310i	$0.239460\pi$
$54$	0	0
$55$	0	0
$56$	−1920.00	−0.0818147
$57$	0	0
$58$	21696.0	0.846856
$59$	6461.00	0.241640	0.120820	−	0.992674i	$-0.461448\pi$
$59$	6461.00	0.241640	0.120820	+	0.992674i	$0.461448\pi$
$60$	0	0
$61$	16980.0	0.584269	0.292135	−	0.956377i	$-0.405634\pi$
$61$	16980.0	0.584269	0.292135	+	0.956377i	$0.405634\pi$
$62$	25772.0	0.851469
$63$	0	0
$64$	28672.0	0.875000
$65$	21812.0	0.640342
$66$	0	0
$67$	29999.0	0.816432	0.408216	−	0.912885i	$-0.366151\pi$
$67$	29999.0	0.816432	0.408216	+	0.912885i	$0.366151\pi$
$68$	−10976.0	−0.287854
$69$	0	0
$70$	760.000	0.0185382
$71$	−31023.0	−0.730362	−0.365181	−	0.930937i	$-0.618993\pi$
$71$	−31023.0	−0.730362	−0.365181	+	0.930937i	$0.618993\pi$
$72$	0	0
$73$	−1924.00	−0.0422569	−0.0211285	−	0.999777i	$-0.506726\pi$
$73$	−1924.00	−0.0422569	−0.0211285	+	0.999777i	$0.506726\pi$
$74$	−48252.0	−1.02432
$75$	0	0
$76$	−6144.00	−0.122016
$77$	0	0
$78$	0	0
$79$	−65138.0	−1.17427	−0.587133	−	0.809490i	$-0.699744\pi$
$79$	−65138.0	−1.17427	−0.587133	+	0.809490i	$0.699744\pi$
$80$	−4864.00	−0.0849706
$81$	0	0
$82$	6112.00	0.100380
$83$	−102714.	−1.63657	−0.818285	−	0.574813i	$-0.805075\pi$
$83$	−102714.	−1.63657	−0.818285	+	0.574813i	$0.805075\pi$
$84$	0	0
$85$	13034.0	0.195673
$86$	−16104.0	−0.234794
$87$	0	0
$88$	0	0
$89$	−17415.0	−0.233050	−0.116525	−	0.993188i	$-0.537175\pi$
$89$	−17415.0	−0.233050	−0.116525	+	0.993188i	$0.537175\pi$
$90$	0	0
$91$	−11480.0	−0.145324
$92$	59344.0	0.730983
$93$	0	0
$94$	28672.0	0.334687
$95$	7296.00	0.0829422
$96$	0	0
$97$	66905.0	0.721987	0.360993	−	0.932568i	$-0.382438\pi$
$97$	66905.0	0.721987	0.360993	+	0.932568i	$0.382438\pi$
$98$	66828.0	0.702900
$99$	0	0
$100$	44224.0	0.442240
$101$	96730.0	0.943534	0.471767	−	0.881723i	$-0.343616\pi$
$101$	96730.0	0.943534	0.471767	+	0.881723i	$0.343616\pi$
$102$	0	0
$103$	−95704.0	−0.888868	−0.444434	−	0.895812i	$-0.646595\pi$
$103$	−95704.0	−0.888868	−0.444434	+	0.895812i	$0.646595\pi$
$104$	220416.	1.99830
$105$	0	0
$106$	−119448.	−1.03256
$107$	−32658.0	−0.275759	−0.137880	−	0.990449i	$-0.544029\pi$
$107$	−32658.0	−0.275759	−0.137880	+	0.990449i	$0.544029\pi$
$108$	0	0
$109$	185438.	1.49497	0.747485	−	0.664279i	$-0.231261\pi$
$109$	185438.	1.49497	0.747485	+	0.664279i	$0.231261\pi$
$110$	0	0
$111$	0	0
$112$	2560.00	0.0192839
$113$	−72849.0	−0.536695	−0.268347	−	0.963322i	$-0.586478\pi$
$113$	−72849.0	−0.536695	−0.268347	+	0.963322i	$0.586478\pi$
$114$	0	0
$115$	−70471.0	−0.496896
$116$	86784.0	0.598818
$117$	0	0
$118$	−25844.0	−0.170866
$119$	−6860.00	−0.0444075
$120$	0	0
$121$	0	0
$122$	−67920.0	−0.413141
$123$	0	0
$124$	103088.	0.602080
$125$	−111891.	−0.640501
$126$	0	0
$127$	78184.0	0.430139	0.215069	−	0.976599i	$-0.431002\pi$
$127$	78184.0	0.430139	0.215069	+	0.976599i	$0.431002\pi$
$128$	49152.0	0.265165
$129$	0	0
$130$	−87248.0	−0.452790
$131$	−462.000	−0.00235214	−0.00117607	−	0.999999i	$-0.500374\pi$
$131$	−462.000	−0.00235214	−0.00117607	+	0.999999i	$0.500374\pi$
$132$	0	0
$133$	−3840.00	−0.0188236
$134$	−119996.	−0.577304
$135$	0	0
$136$	131712.	0.610630
$137$	−296233.	−1.34844	−0.674221	−	0.738530i	$-0.735520\pi$
$137$	−296233.	−1.34844	−0.674221	+	0.738530i	$0.735520\pi$
$138$	0	0
$139$	399818.	1.75519	0.877597	−	0.479398i	$-0.159145\pi$
$139$	399818.	1.75519	0.877597	+	0.479398i	$0.159145\pi$
$140$	3040.00	0.0131085
$141$	0	0
$142$	124092.	0.516444
$143$	0	0
$144$	0	0
$145$	−103056.	−0.407055
$146$	7696.00	0.0298802
$147$	0	0
$148$	−193008.	−0.724304
$149$	72670.0	0.268157	0.134079	−	0.990971i	$-0.457193\pi$
$149$	72670.0	0.268157	0.134079	+	0.990971i	$0.457193\pi$
$150$	0	0
$151$	303082.	1.08173	0.540864	−	0.841110i	$-0.318098\pi$
$151$	303082.	1.08173	0.540864	+	0.841110i	$0.318098\pi$
$152$	73728.0	0.258835
$153$	0	0
$154$	0	0
$155$	−122417.	−0.409272
$156$	0	0
$157$	−532987.	−1.72571	−0.862854	−	0.505453i	$-0.831326\pi$
$157$	−532987.	−1.72571	−0.862854	+	0.505453i	$0.831326\pi$
$158$	260552.	0.830332
$159$	0	0
$160$	−97280.0	−0.300416
$161$	37090.0	0.112770
$162$	0	0
$163$	282076.	0.831567	0.415783	−	0.909464i	$-0.363507\pi$
$163$	282076.	0.831567	0.415783	+	0.909464i	$0.363507\pi$
$164$	24448.0	0.0709796
$165$	0	0
$166$	410856.	1.15723
$167$	−573588.	−1.59151	−0.795754	−	0.605620i	$-0.792925\pi$
$167$	−573588.	−1.59151	−0.795754	+	0.605620i	$0.792925\pi$
$168$	0	0
$169$	946611.	2.54950
$170$	−52136.0	−0.138362
$171$	0	0
$172$	−64416.0	−0.166025
$173$	−386286.	−0.981282	−0.490641	−	0.871362i	$-0.663237\pi$
$173$	−386286.	−0.981282	−0.490641	+	0.871362i	$0.663237\pi$
$174$	0	0
$175$	27640.0	0.0682249
$176$	0	0
$177$	0	0
$178$	69660.0	0.164791
$179$	−545079.	−1.27153	−0.635765	−	0.771882i	$-0.719315\pi$
$179$	−545079.	−1.27153	−0.635765	+	0.771882i	$0.719315\pi$
$180$	0	0
$181$	−279485.	−0.634106	−0.317053	−	0.948408i	$-0.602693\pi$
$181$	−279485.	−0.634106	−0.317053	+	0.948408i	$0.602693\pi$
$182$	45920.0	0.102760
$183$	0	0
$184$	−712128.	−1.55065
$185$	229197.	0.492356
$186$	0	0
$187$	0	0
$188$	114688.	0.236659
$189$	0	0
$190$	−29184.0	−0.0586490
$191$	444437.	0.881509	0.440755	−	0.897628i	$-0.354711\pi$
$191$	444437.	0.881509	0.440755	+	0.897628i	$0.354711\pi$
$192$	0	0
$193$	18476.0	0.0357038	0.0178519	−	0.999841i	$-0.494317\pi$
$193$	18476.0	0.0357038	0.0178519	+	0.999841i	$0.494317\pi$
$194$	−267620.	−0.510522
$195$	0	0
$196$	267312.	0.497025
$197$	270182.	0.496010	0.248005	−	0.968759i	$-0.420225\pi$
$197$	270182.	0.496010	0.248005	+	0.968759i	$0.420225\pi$
$198$	0	0
$199$	43320.0	0.0775453	0.0387727	−	0.999248i	$-0.487655\pi$
$199$	43320.0	0.0775453	0.0387727	+	0.999248i	$0.487655\pi$
$200$	−530688.	−0.938133
$201$	0	0
$202$	−386920.	−0.667180
$203$	54240.0	0.0923803
$204$	0	0
$205$	−29032.0	−0.0482494
$206$	382816.	0.628524
$207$	0	0
$208$	−293888.	−0.471003
$209$	0	0
$210$	0	0
$211$	−1.02968e6	−1.59220	−0.796100	−	0.605165i	$-0.793107\pi$
$211$	−1.02968e6	−1.59220	−0.796100	+	0.605165i	$0.793107\pi$
$212$	−477792.	−0.730128
$213$	0	0
$214$	130632.	0.194991
$215$	76494.0	0.112858
$216$	0	0
$217$	64430.0	0.0928835
$218$	−741752.	−1.05710
$219$	0	0
$220$	0	0
$221$	787528.	1.08464
$222$	0	0
$223$	461281.	0.621160	0.310580	−	0.950547i	$-0.399477\pi$
$223$	461281.	0.621160	0.310580	+	0.950547i	$0.399477\pi$
$224$	51200.0	0.0681789
$225$	0	0
$226$	291396.	0.379501
$227$	−855570.	−1.10202	−0.551012	−	0.834497i	$-0.685758\pi$
$227$	−855570.	−1.10202	−0.551012	+	0.834497i	$0.685758\pi$
$228$	0	0
$229$	−665805.	−0.838993	−0.419497	−	0.907757i	$-0.637793\pi$
$229$	−665805.	−0.838993	−0.419497	+	0.907757i	$0.637793\pi$
$230$	281884.	0.351359
$231$	0	0
$232$	−1.04141e6	−1.27028
$233$	1.20934e6	1.45934	0.729671	−	0.683798i	$-0.239673\pi$
$233$	1.20934e6	1.45934	0.729671	+	0.683798i	$0.239673\pi$
$234$	0	0
$235$	−136192.	−0.160873
$236$	−103376.	−0.120820
$237$	0	0
$238$	27440.0	0.0314009
$239$	−571482.	−0.647154	−0.323577	−	0.946202i	$-0.604886\pi$
$239$	−571482.	−0.647154	−0.323577	+	0.946202i	$0.604886\pi$
$240$	0	0
$241$	267080.	0.296209	0.148105	−	0.988972i	$-0.452683\pi$
$241$	267080.	0.296209	0.148105	+	0.988972i	$0.452683\pi$
$242$	0	0
$243$	0	0
$244$	−271680.	−0.292135
$245$	−317433.	−0.337860
$246$	0	0
$247$	440832.	0.459760
$248$	−1.23706e6	−1.27720
$249$	0	0
$250$	447564.	0.452903
$251$	−1.38737e6	−1.38998	−0.694988	−	0.719022i	$-0.744590\pi$
$251$	−1.38737e6	−1.38998	−0.694988	+	0.719022i	$0.744590\pi$
$252$	0	0
$253$	0	0
$254$	−312736.	−0.304154
$255$	0	0
$256$	−1.11411e6	−1.06250
$257$	885922.	0.836686	0.418343	−	0.908289i	$-0.362611\pi$
$257$	885922.	0.836686	0.418343	+	0.908289i	$0.362611\pi$
$258$	0	0
$259$	−120630.	−0.111739
$260$	−348992.	−0.320171
$261$	0	0
$262$	1848.00	0.00166322
$263$	1.44687e6	1.28986	0.644928	−	0.764243i	$-0.276887\pi$
$263$	1.44687e6	1.28986	0.644928	+	0.764243i	$0.276887\pi$
$264$	0	0
$265$	567378.	0.496315
$266$	15360.0	0.0133103
$267$	0	0
$268$	−479984.	−0.408216
$269$	353878.	0.298176	0.149088	−	0.988824i	$-0.452366\pi$
$269$	353878.	0.298176	0.149088	+	0.988824i	$0.452366\pi$
$270$	0	0
$271$	−525260.	−0.434461	−0.217231	−	0.976120i	$-0.569702\pi$
$271$	−525260.	−0.434461	−0.217231	+	0.976120i	$0.569702\pi$
$272$	−175616.	−0.143927
$273$	0	0
$274$	1.18493e6	0.953492
$275$	0	0
$276$	0	0
$277$	595610.	0.466404	0.233202	−	0.972428i	$-0.425080\pi$
$277$	595610.	0.466404	0.233202	+	0.972428i	$0.425080\pi$
$278$	−1.59927e6	−1.24111
$279$	0	0
$280$	−36480.0	−0.0278074
$281$	732318.	0.553266	0.276633	−	0.960976i	$-0.410781\pi$
$281$	732318.	0.553266	0.276633	+	0.960976i	$0.410781\pi$
$282$	0	0
$283$	−2.23380e6	−1.65798	−0.828989	−	0.559264i	$-0.811084\pi$
$283$	−2.23380e6	−1.65798	−0.828989	+	0.559264i	$0.811084\pi$
$284$	496368.	0.365181
$285$	0	0
$286$	0	0
$287$	15280.0	0.0109501
$288$	0	0
$289$	−949261.	−0.668561
$290$	412224.	0.287831
$291$	0	0
$292$	30784.0	0.0211285
$293$	−1.53108e6	−1.04191	−0.520953	−	0.853585i	$-0.674423\pi$
$293$	−1.53108e6	−1.04191	−0.520953	+	0.853585i	$0.674423\pi$
$294$	0	0
$295$	122759.	0.0821293
$296$	2.31610e6	1.53648
$297$	0	0
$298$	−290680.	−0.189616
$299$	−4.25793e6	−2.75436
$300$	0	0
$301$	−40260.0	−0.0256128
$302$	−1.21233e6	−0.764897
$303$	0	0
$304$	−98304.0	−0.0610081
$305$	322620.	0.198583
$306$	0	0
$307$	1.14268e6	0.691956	0.345978	−	0.938243i	$-0.387547\pi$
$307$	1.14268e6	0.691956	0.345978	+	0.938243i	$0.387547\pi$
$308$	0	0
$309$	0	0
$310$	489668.	0.289399
$311$	−586956.	−0.344116	−0.172058	−	0.985087i	$-0.555042\pi$
$311$	−586956.	−0.344116	−0.172058	+	0.985087i	$0.555042\pi$
$312$	0	0
$313$	−233857.	−0.134924	−0.0674621	−	0.997722i	$-0.521490\pi$
$313$	−233857.	−0.134924	−0.0674621	+	0.997722i	$0.521490\pi$
$314$	2.13195e6	1.22026
$315$	0	0
$316$	1.04221e6	0.587133
$317$	935503.	0.522874	0.261437	−	0.965221i	$-0.415804\pi$
$317$	935503.	0.522874	0.261437	+	0.965221i	$0.415804\pi$
$318$	0	0
$319$	0	0
$320$	544768.	0.297397
$321$	0	0
$322$	−148360.	−0.0797402
$323$	263424.	0.140491
$324$	0	0
$325$	−3.17307e6	−1.66637
$326$	−1.12830e6	−0.588007
$327$	0	0
$328$	−293376.	−0.150571
$329$	71680.0	0.0365097
$330$	0	0
$331$	−1.05823e6	−0.530897	−0.265449	−	0.964125i	$-0.585520\pi$
$331$	−1.05823e6	−0.530897	−0.265449	+	0.964125i	$0.585520\pi$
$332$	1.64342e6	0.818285
$333$	0	0
$334$	2.29435e6	1.12537
$335$	569981.	0.277491
$336$	0	0
$337$	−506186.	−0.242793	−0.121396	−	0.992604i	$-0.538737\pi$
$337$	−506186.	−0.242793	−0.121396	+	0.992604i	$0.538737\pi$
$338$	−3.78644e6	−1.80277
$339$	0	0
$340$	−208544.	−0.0978364
$341$	0	0
$342$	0	0
$343$	335140.	0.153812
$344$	772992.	0.352192
$345$	0	0
$346$	1.54514e6	0.693871
$347$	467636.	0.208490	0.104245	−	0.994552i	$-0.466757\pi$
$347$	467636.	0.208490	0.104245	+	0.994552i	$0.466757\pi$
$348$	0	0
$349$	−304470.	−0.133808	−0.0669038	−	0.997759i	$-0.521312\pi$
$349$	−304470.	−0.133808	−0.0669038	+	0.997759i	$0.521312\pi$
$350$	−110560.	−0.0482423
$351$	0	0
$352$	0	0
$353$	−2.51868e6	−1.07581	−0.537906	−	0.843005i	$-0.680785\pi$
$353$	−2.51868e6	−1.07581	−0.537906	+	0.843005i	$0.680785\pi$
$354$	0	0
$355$	−589437.	−0.248237
$356$	278640.	0.116525
$357$	0	0
$358$	2.18032e6	0.899108
$359$	−3.01841e6	−1.23607	−0.618034	−	0.786151i	$-0.712071\pi$
$359$	−3.01841e6	−1.23607	−0.618034	+	0.786151i	$0.712071\pi$
$360$	0	0
$361$	−2.32864e6	−0.940448
$362$	1.11794e6	0.448381
$363$	0	0
$364$	183680.	0.0726622
$365$	−36556.0	−0.0143624
$366$	0	0
$367$	994429.	0.385397	0.192699	−	0.981258i	$-0.438276\pi$
$367$	994429.	0.385397	0.192699	+	0.981258i	$0.438276\pi$
$368$	949504.	0.365491
$369$	0	0
$370$	−916788.	−0.348149
$371$	−298620.	−0.112638
$372$	0	0
$373$	−1.72896e6	−0.643446	−0.321723	−	0.946834i	$-0.604262\pi$
$373$	−1.72896e6	−0.643446	−0.321723	+	0.946834i	$0.604262\pi$
$374$	0	0
$375$	0	0
$376$	−1.37626e6	−0.502030
$377$	−6.22675e6	−2.25636
$378$	0	0
$379$	454765.	0.162626	0.0813128	−	0.996689i	$-0.474089\pi$
$379$	454765.	0.162626	0.0813128	+	0.996689i	$0.474089\pi$
$380$	−116736.	−0.0414711
$381$	0	0
$382$	−1.77775e6	−0.623321
$383$	−2.27557e6	−0.792673	−0.396336	−	0.918105i	$-0.629719\pi$
$383$	−2.27557e6	−0.792673	−0.396336	+	0.918105i	$0.629719\pi$
$384$	0	0
$385$	0	0
$386$	−73904.0	−0.0252464
$387$	0	0
$388$	−1.07048e6	−0.360993
$389$	−389781.	−0.130601	−0.0653005	−	0.997866i	$-0.520801\pi$
$389$	−389781.	−0.130601	−0.0653005	+	0.997866i	$0.520801\pi$
$390$	0	0
$391$	−2.54437e6	−0.841665
$392$	−3.20774e6	−1.05435
$393$	0	0
$394$	−1.08073e6	−0.350732
$395$	−1.23762e6	−0.399112
$396$	0	0
$397$	−1.61933e6	−0.515655	−0.257827	−	0.966191i	$-0.583007\pi$
$397$	−1.61933e6	−0.515655	−0.257827	+	0.966191i	$0.583007\pi$
$398$	−173280.	−0.0548328
$399$	0	0
$400$	707584.	0.221120
$401$	5.54368e6	1.72162	0.860810	−	0.508927i	$-0.169958\pi$
$401$	5.54368e6	1.72162	0.860810	+	0.508927i	$0.169958\pi$
$402$	0	0
$403$	−7.39656e6	−2.26865
$404$	−1.54768e6	−0.471767
$405$	0	0
$406$	−216960.	−0.0653228
$407$	0	0
$408$	0	0
$409$	2.70493e6	0.799553	0.399776	−	0.916613i	$-0.369088\pi$
$409$	2.70493e6	0.799553	0.399776	+	0.916613i	$0.369088\pi$
$410$	116128.	0.0341175
$411$	0	0
$412$	1.53126e6	0.444434
$413$	−64610.0	−0.0186391
$414$	0	0
$415$	−1.95157e6	−0.556241
$416$	−5.87776e6	−1.66525
$417$	0	0
$418$	0	0
$419$	−3.37337e6	−0.938705	−0.469353	−	0.883011i	$-0.655513\pi$
$419$	−3.37337e6	−0.938705	−0.469353	+	0.883011i	$0.655513\pi$
$420$	0	0
$421$	−4.52551e6	−1.24441	−0.622204	−	0.782855i	$-0.713762\pi$
$421$	−4.52551e6	−1.24441	−0.622204	+	0.782855i	$0.713762\pi$
$422$	4.11874e6	1.12586
$423$	0	0
$424$	5.73350e6	1.54884
$425$	−1.89610e6	−0.509202
$426$	0	0
$427$	−169800.	−0.0450680
$428$	522528.	0.137880
$429$	0	0
$430$	−305976.	−0.0798024
$431$	−684534.	−0.177501	−0.0887507	−	0.996054i	$-0.528287\pi$
$431$	−684534.	−0.177501	−0.0887507	+	0.996054i	$0.528287\pi$
$432$	0	0
$433$	−4.22591e6	−1.08318	−0.541589	−	0.840643i	$-0.682177\pi$
$433$	−4.22591e6	−1.08318	−0.541589	+	0.840643i	$0.682177\pi$
$434$	−257720.	−0.0656786
$435$	0	0
$436$	−2.96701e6	−0.747485
$437$	−1.42426e6	−0.356767
$438$	0	0
$439$	2.09185e6	0.518047	0.259023	−	0.965871i	$-0.416599\pi$
$439$	2.09185e6	0.518047	0.259023	+	0.965871i	$0.416599\pi$
$440$	0	0
$441$	0	0
$442$	−3.15011e6	−0.766956
$443$	−1.56284e6	−0.378361	−0.189180	−	0.981942i	$-0.560583\pi$
$443$	−1.56284e6	−0.378361	−0.189180	+	0.981942i	$0.560583\pi$
$444$	0	0
$445$	−330885.	−0.0792095
$446$	−1.84512e6	−0.439226
$447$	0	0
$448$	−286720.	−0.0674937
$449$	3.00449e6	0.703324	0.351662	−	0.936127i	$-0.385617\pi$
$449$	3.00449e6	0.703324	0.351662	+	0.936127i	$0.385617\pi$
$450$	0	0
$451$	0	0
$452$	1.16558e6	0.268347
$453$	0	0
$454$	3.42228e6	0.779248
$455$	−218120.	−0.0493932
$456$	0	0
$457$	2.44552e6	0.547747	0.273874	−	0.961766i	$-0.411695\pi$
$457$	2.44552e6	0.547747	0.273874	+	0.961766i	$0.411695\pi$
$458$	2.66322e6	0.593258
$459$	0	0
$460$	1.12754e6	0.248448
$461$	7.79104e6	1.70743	0.853715	−	0.520741i	$-0.174344\pi$
$461$	7.79104e6	1.70743	0.853715	+	0.520741i	$0.174344\pi$
$462$	0	0
$463$	−1.05196e6	−0.228059	−0.114029	−	0.993477i	$-0.536376\pi$
$463$	−1.05196e6	−0.228059	−0.114029	+	0.993477i	$0.536376\pi$
$464$	1.38854e6	0.299409
$465$	0	0
$466$	−4.83734e6	−1.03191
$467$	−3.97003e6	−0.842369	−0.421184	−	0.906975i	$-0.638385\pi$
$467$	−3.97003e6	−0.842369	−0.421184	+	0.906975i	$0.638385\pi$
$468$	0	0
$469$	−299990.	−0.0629759
$470$	544768.	0.113754
$471$	0	0
$472$	1.24051e6	0.256298
$473$	0	0
$474$	0	0
$475$	−1.06138e6	−0.215842
$476$	109760.	0.0222038
$477$	0	0
$478$	2.28593e6	0.457607
$479$	−8.53908e6	−1.70048	−0.850241	−	0.526393i	$-0.823544\pi$
$479$	−8.53908e6	−1.70048	−0.850241	+	0.526393i	$0.823544\pi$
$480$	0	0
$481$	1.38483e7	2.72919
$482$	−1.06832e6	−0.209452
$483$	0	0
$484$	0	0
$485$	1.27120e6	0.245391
$486$	0	0
$487$	−1.86487e6	−0.356308	−0.178154	−	0.984003i	$-0.557013\pi$
$487$	−1.86487e6	−0.356308	−0.178154	+	0.984003i	$0.557013\pi$
$488$	3.26016e6	0.619711
$489$	0	0
$490$	1.26973e6	0.238903
$491$	5.15727e6	0.965420	0.482710	−	0.875780i	$-0.339653\pi$
$491$	5.15727e6	0.965420	0.482710	+	0.875780i	$0.339653\pi$
$492$	0	0
$493$	−3.72086e6	−0.689488
$494$	−1.76333e6	−0.325099
$495$	0	0
$496$	1.64941e6	0.301040
$497$	310230.	0.0563369
$498$	0	0
$499$	4.53340e6	0.815029	0.407514	−	0.913199i	$-0.366396\pi$
$499$	4.53340e6	0.815029	0.407514	+	0.913199i	$0.366396\pi$
$500$	1.79026e6	0.320251
$501$	0	0
$502$	5.54947e6	0.982861
$503$	1.71163e6	0.301641	0.150821	−	0.988561i	$-0.451808\pi$
$503$	1.71163e6	0.301641	0.150821	+	0.988561i	$0.451808\pi$
$504$	0	0
$505$	1.83787e6	0.320691
$506$	0	0
$507$	0	0
$508$	−1.25094e6	−0.215069
$509$	−9.73822e6	−1.66604	−0.833019	−	0.553244i	$-0.813390\pi$
$509$	−9.73822e6	−1.66604	−0.833019	+	0.553244i	$0.813390\pi$
$510$	0	0
$511$	19240.0	0.00325951
$512$	2.88358e6	0.486136
$513$	0	0
$514$	−3.54369e6	−0.591627
$515$	−1.81838e6	−0.302110
$516$	0	0
$517$	0	0
$518$	482520.	0.0790116
$519$	0	0
$520$	4.18790e6	0.679185
$521$	−4.30279e6	−0.694474	−0.347237	−	0.937777i	$-0.612880\pi$
$521$	−4.30279e6	−0.694474	−0.347237	+	0.937777i	$0.612880\pi$
$522$	0	0
$523$	−2.62280e6	−0.419287	−0.209643	−	0.977778i	$-0.567230\pi$
$523$	−2.62280e6	−0.419287	−0.209643	+	0.977778i	$0.567230\pi$
$524$	7392.00	0.00117607
$525$	0	0
$526$	−5.78750e6	−0.912066
$527$	−4.41990e6	−0.693243
$528$	0	0
$529$	7.32034e6	1.13734
$530$	−2.26951e6	−0.350948
$531$	0	0
$532$	61440.0	0.00941179
$533$	−1.75414e6	−0.267453
$534$	0	0
$535$	−620502.	−0.0937257
$536$	5.75981e6	0.865956
$537$	0	0
$538$	−1.41551e6	−0.210842
$539$	0	0
$540$	0	0
$541$	2.49634e6	0.366700	0.183350	−	0.983048i	$-0.441306\pi$
$541$	2.49634e6	0.366700	0.183350	+	0.983048i	$0.441306\pi$
$542$	2.10104e6	0.307211
$543$	0	0
$544$	−3.51232e6	−0.508858
$545$	3.52332e6	0.508114
$546$	0	0
$547$	−1.14323e7	−1.63368	−0.816838	−	0.576868i	$-0.804275\pi$
$547$	−1.14323e7	−1.63368	−0.816838	+	0.576868i	$0.804275\pi$
$548$	4.73973e6	0.674221
$549$	0	0
$550$	0	0
$551$	−2.08282e6	−0.292262
$552$	0	0
$553$	651380.	0.0905778
$554$	−2.38244e6	−0.329798
$555$	0	0
$556$	−6.39709e6	−0.877597
$557$	−9.81529e6	−1.34049	−0.670247	−	0.742138i	$-0.733812\pi$
$557$	−9.81529e6	−1.34049	−0.670247	+	0.742138i	$0.733812\pi$
$558$	0	0
$559$	4.62185e6	0.625585
$560$	48640.0	0.00655426
$561$	0	0
$562$	−2.92927e6	−0.391218
$563$	−8.19192e6	−1.08922	−0.544609	−	0.838690i	$-0.683322\pi$
$563$	−8.19192e6	−1.08922	−0.544609	+	0.838690i	$0.683322\pi$
$564$	0	0
$565$	−1.38413e6	−0.182413
$566$	8.93522e6	1.17237
$567$	0	0
$568$	−5.95642e6	−0.774665
$569$	−7.54286e6	−0.976687	−0.488344	−	0.872651i	$-0.662399\pi$
$569$	−7.54286e6	−0.976687	−0.488344	+	0.872651i	$0.662399\pi$
$570$	0	0
$571$	8.69400e6	1.11591	0.557956	−	0.829871i	$-0.311586\pi$
$571$	8.69400e6	1.11591	0.557956	+	0.829871i	$0.311586\pi$
$572$	0	0
$573$	0	0
$574$	−61120.0	−0.00774290
$575$	1.02517e7	1.29308
$576$	0	0
$577$	2.03379e6	0.254312	0.127156	−	0.991883i	$-0.459415\pi$
$577$	2.03379e6	0.254312	0.127156	+	0.991883i	$0.459415\pi$
$578$	3.79704e6	0.472744
$579$	0	0
$580$	1.64890e6	0.203528
$581$	1.02714e6	0.126238
$582$	0	0
$583$	0	0
$584$	−369408.	−0.0448202
$585$	0	0
$586$	6.12432e6	0.736739
$587$	−3.51780e6	−0.421381	−0.210691	−	0.977553i	$-0.567571\pi$
$587$	−3.51780e6	−0.421381	−0.210691	+	0.977553i	$0.567571\pi$
$588$	0	0
$589$	−2.47411e6	−0.293854
$590$	−491036.	−0.0580742
$591$	0	0
$592$	−3.08813e6	−0.362152
$593$	−8.34535e6	−0.974558	−0.487279	−	0.873246i	$-0.662011\pi$
$593$	−8.34535e6	−0.974558	−0.487279	+	0.873246i	$0.662011\pi$
$594$	0	0
$595$	−130340.	−0.0150933
$596$	−1.16272e6	−0.134079
$597$	0	0
$598$	1.70317e7	1.94763
$599$	−6.15022e6	−0.700364	−0.350182	−	0.936682i	$-0.613880\pi$
$599$	−6.15022e6	−0.700364	−0.350182	+	0.936682i	$0.613880\pi$
$600$	0	0
$601$	6.86232e6	0.774970	0.387485	−	0.921876i	$-0.373344\pi$
$601$	6.86232e6	0.774970	0.387485	+	0.921876i	$0.373344\pi$
$602$	161040.	0.0181110
$603$	0	0
$604$	−4.84931e6	−0.540864
$605$	0	0
$606$	0	0
$607$	9.45536e6	1.04161	0.520807	−	0.853675i	$-0.325631\pi$
$607$	9.45536e6	1.04161	0.520807	+	0.853675i	$0.325631\pi$
$608$	−1.96608e6	−0.215696
$609$	0	0
$610$	−1.29048e6	−0.140419
$611$	−8.22886e6	−0.891737
$612$	0	0
$613$	4.63658e6	0.498363	0.249182	−	0.968457i	$-0.419838\pi$
$613$	4.63658e6	0.498363	0.249182	+	0.968457i	$0.419838\pi$
$614$	−4.57072e6	−0.489287
$615$	0	0
$616$	0	0
$617$	−6.05704e6	−0.640542	−0.320271	−	0.947326i	$-0.603774\pi$
$617$	−6.05704e6	−0.640542	−0.320271	+	0.947326i	$0.603774\pi$
$618$	0	0
$619$	−5.63994e6	−0.591626	−0.295813	−	0.955246i	$-0.595591\pi$
$619$	−5.63994e6	−0.591626	−0.295813	+	0.955246i	$0.595591\pi$
$620$	1.95867e6	0.204636
$621$	0	0
$622$	2.34782e6	0.243327
$623$	174150.	0.0179764
$624$	0	0
$625$	6.51157e6	0.666785
$626$	935428.	0.0954057
$627$	0	0
$628$	8.52779e6	0.862854
$629$	8.27522e6	0.833975
$630$	0	0
$631$	1.12616e6	0.112597	0.0562987	−	0.998414i	$-0.482070\pi$
$631$	1.12616e6	0.112597	0.0562987	+	0.998414i	$0.482070\pi$
$632$	−1.25065e7	−1.24550
$633$	0	0
$634$	−3.74201e6	−0.369728
$635$	1.48550e6	0.146197
$636$	0	0
$637$	−1.91796e7	−1.87280
$638$	0	0
$639$	0	0
$640$	933888.	0.0901249
$641$	1.42020e7	1.36522	0.682611	−	0.730782i	$-0.260844\pi$
$641$	1.42020e7	1.36522	0.682611	+	0.730782i	$0.260844\pi$
$642$	0	0
$643$	1.60794e6	0.153371	0.0766853	−	0.997055i	$-0.475566\pi$
$643$	1.60794e6	0.153371	0.0766853	+	0.997055i	$0.475566\pi$
$644$	−593440.	−0.0563848
$645$	0	0
$646$	−1.05370e6	−0.0993423
$647$	−3.10236e6	−0.291361	−0.145680	−	0.989332i	$-0.546537\pi$
$647$	−3.10236e6	−0.291361	−0.145680	+	0.989332i	$0.546537\pi$
$648$	0	0
$649$	0	0
$650$	1.26923e7	1.17830
$651$	0	0
$652$	−4.51322e6	−0.415783
$653$	−6.88852e6	−0.632183	−0.316091	−	0.948729i	$-0.602371\pi$
$653$	−6.88852e6	−0.632183	−0.316091	+	0.948729i	$0.602371\pi$
$654$	0	0
$655$	−8778.00	−0.000799452 0
$656$	391168.	0.0354898
$657$	0	0
$658$	−286720.	−0.0258163
$659$	−1.24134e7	−1.11347	−0.556735	−	0.830690i	$-0.687946\pi$
$659$	−1.24134e7	−1.11347	−0.556735	+	0.830690i	$0.687946\pi$
$660$	0	0
$661$	−8.10994e6	−0.721961	−0.360980	−	0.932573i	$-0.617558\pi$
$661$	−8.10994e6	−0.721961	−0.360980	+	0.932573i	$0.617558\pi$
$662$	4.23292e6	0.375401
$663$	0	0
$664$	−1.97211e7	−1.73584
$665$	−72960.0	−0.00639780
$666$	0	0
$667$	2.01176e7	1.75090
$668$	9.17741e6	0.795754
$669$	0	0
$670$	−2.27992e6	−0.196216
$671$	0	0
$672$	0	0
$673$	−1.78063e7	−1.51543	−0.757717	−	0.652584i	$-0.773685\pi$
$673$	−1.78063e7	−1.51543	−0.757717	+	0.652584i	$0.773685\pi$
$674$	2.02474e6	0.171680
$675$	0	0
$676$	−1.51458e7	−1.27475
$677$	1.55179e7	1.30125	0.650626	−	0.759398i	$-0.274507\pi$
$677$	1.55179e7	1.30125	0.650626	+	0.759398i	$0.274507\pi$
$678$	0	0
$679$	−669050.	−0.0556909
$680$	2.50253e6	0.207542
$681$	0	0
$682$	0	0
$683$	2.18106e6	0.178902	0.0894510	−	0.995991i	$-0.471489\pi$
$683$	2.18106e6	0.178902	0.0894510	+	0.995991i	$0.471489\pi$
$684$	0	0
$685$	−5.62843e6	−0.458311
$686$	−1.34056e6	−0.108762
$687$	0	0
$688$	−1.03066e6	−0.0830123
$689$	3.42816e7	2.75114
$690$	0	0
$691$	2.29892e7	1.83159	0.915795	−	0.401647i	$-0.131562\pi$
$691$	2.29892e7	1.83159	0.915795	+	0.401647i	$0.131562\pi$
$692$	6.18058e6	0.490641
$693$	0	0
$694$	−1.87054e6	−0.147424
$695$	7.59654e6	0.596560
$696$	0	0
$697$	−1.04821e6	−0.0817270
$698$	1.21788e6	0.0946163
$699$	0	0
$700$	−442240.	−0.0341125
$701$	2.34092e6	0.179925	0.0899626	−	0.995945i	$-0.471325\pi$
$701$	2.34092e6	0.179925	0.0899626	+	0.995945i	$0.471325\pi$
$702$	0	0
$703$	4.63219e6	0.353507
$704$	0	0
$705$	0	0
$706$	1.00747e7	0.760715
$707$	−967300.	−0.0727801
$708$	0	0
$709$	−1.92694e7	−1.43964	−0.719820	−	0.694161i	$-0.755775\pi$
$709$	−1.92694e7	−1.43964	−0.719820	+	0.694161i	$0.755775\pi$
$710$	2.35775e6	0.175530
$711$	0	0
$712$	−3.34368e6	−0.247186
$713$	2.38971e7	1.76044
$714$	0	0
$715$	0	0
$716$	8.72126e6	0.635765
$717$	0	0
$718$	1.20736e7	0.874032
$719$	2.14665e7	1.54860	0.774300	−	0.632819i	$-0.218102\pi$
$719$	2.14665e7	1.54860	0.774300	+	0.632819i	$0.218102\pi$
$720$	0	0
$721$	957040.	0.0685633
$722$	9.31457e6	0.664997
$723$	0	0
$724$	4.47176e6	0.317053
$725$	1.49919e7	1.05928
$726$	0	0
$727$	−1.67705e7	−1.17682	−0.588411	−	0.808562i	$-0.700246\pi$
$727$	−1.67705e7	−1.17682	−0.588411	+	0.808562i	$0.700246\pi$
$728$	−2.20416e6	−0.154140
$729$	0	0
$730$	146224.	0.0101557
$731$	2.76184e6	0.191163
$732$	0	0
$733$	1.75373e7	1.20560	0.602798	−	0.797894i	$-0.294052\pi$
$733$	1.75373e7	1.20560	0.602798	+	0.797894i	$0.294052\pi$
$734$	−3.97772e6	−0.272517
$735$	0	0
$736$	1.89901e7	1.29221
$737$	0	0
$738$	0	0
$739$	−1.47387e7	−0.992766	−0.496383	−	0.868104i	$-0.665339\pi$
$739$	−1.47387e7	−0.992766	−0.496383	+	0.868104i	$0.665339\pi$
$740$	−3.66715e6	−0.246178
$741$	0	0
$742$	1.19448e6	0.0796469
$743$	−4.80946e6	−0.319613	−0.159806	−	0.987148i	$-0.551087\pi$
$743$	−4.80946e6	−0.319613	−0.159806	+	0.987148i	$0.551087\pi$
$744$	0	0
$745$	1.38073e6	0.0911419
$746$	6.91583e6	0.454985
$747$	0	0
$748$	0	0
$749$	326580.	0.0212709
$750$	0	0
$751$	8.29317e6	0.536563	0.268282	−	0.963341i	$-0.413544\pi$
$751$	8.29317e6	0.536563	0.268282	+	0.963341i	$0.413544\pi$
$752$	1.83501e6	0.118330
$753$	0	0
$754$	2.49070e7	1.59549
$755$	5.75856e6	0.367660
$756$	0	0
$757$	−352294.	−0.0223442	−0.0111721	−	0.999938i	$-0.503556\pi$
$757$	−352294.	−0.0223442	−0.0111721	+	0.999938i	$0.503556\pi$
$758$	−1.81906e6	−0.114994
$759$	0	0
$760$	1.40083e6	0.0879735
$761$	1.68985e7	1.05776	0.528878	−	0.848698i	$-0.322613\pi$
$761$	1.68985e7	1.05776	0.528878	+	0.848698i	$0.322613\pi$
$762$	0	0
$763$	−1.85438e6	−0.115315
$764$	−7.11099e6	−0.440755
$765$	0	0
$766$	9.10229e6	0.560504
$767$	7.41723e6	0.455253
$768$	0	0
$769$	36652.0	0.00223502	0.00111751	−	0.999999i	$-0.499644\pi$
$769$	36652.0	0.00223502	0.00111751	+	0.999999i	$0.499644\pi$
$770$	0	0
$771$	0	0
$772$	−295616.	−0.0178519
$773$	3.17462e7	1.91093	0.955463	−	0.295113i	$-0.0953571\pi$
$773$	3.17462e7	1.91093	0.955463	+	0.295113i	$0.0953571\pi$
$774$	0	0
$775$	1.78085e7	1.06505
$776$	1.28458e7	0.765783
$777$	0	0
$778$	1.55912e6	0.0923489
$779$	−586752.	−0.0346426
$780$	0	0
$781$	0	0
$782$	1.01775e7	0.595147
$783$	0	0
$784$	4.27699e6	0.248513
$785$	−1.01268e7	−0.586538
$786$	0	0
$787$	−2.01985e7	−1.16247	−0.581236	−	0.813735i	$-0.697431\pi$
$787$	−2.01985e7	−1.16247	−0.581236	+	0.813735i	$0.697431\pi$
$788$	−4.32291e6	−0.248005
$789$	0	0
$790$	4.95049e6	0.282215
$791$	728490.	0.0413983
$792$	0	0
$793$	1.94930e7	1.10077
$794$	6.47732e6	0.364623
$795$	0	0
$796$	−693120.	−0.0387727
$797$	−1.55660e7	−0.868023	−0.434011	−	0.900907i	$-0.642902\pi$
$797$	−1.55660e7	−0.868023	−0.434011	+	0.900907i	$0.642902\pi$
$798$	0	0
$799$	−4.91725e6	−0.272493
$800$	1.41517e7	0.781777
$801$	0	0
$802$	−2.21747e7	−1.21737
$803$	0	0
$804$	0	0
$805$	704710.	0.0383284
$806$	2.95863e7	1.60418
$807$	0	0
$808$	1.85722e7	1.00077
$809$	−2.91667e7	−1.56681	−0.783404	−	0.621513i	$-0.786518\pi$
$809$	−2.91667e7	−1.56681	−0.783404	+	0.621513i	$0.786518\pi$
$810$	0	0
$811$	1.65215e7	0.882057	0.441029	−	0.897493i	$-0.354614\pi$
$811$	1.65215e7	0.882057	0.441029	+	0.897493i	$0.354614\pi$
$812$	−867840.	−0.0461902
$813$	0	0
$814$	0	0
$815$	5.35944e6	0.282635
$816$	0	0
$817$	1.54598e6	0.0810307
$818$	−1.08197e7	−0.565369
$819$	0	0
$820$	464512.	0.0241247
$821$	5.56614e6	0.288202	0.144101	−	0.989563i	$-0.453971\pi$
$821$	5.56614e6	0.288202	0.144101	+	0.989563i	$0.453971\pi$
$822$	0	0
$823$	1.18801e7	0.611391	0.305696	−	0.952129i	$-0.401111\pi$
$823$	1.18801e7	0.611391	0.305696	+	0.952129i	$0.401111\pi$
$824$	−1.83752e7	−0.942786
$825$	0	0
$826$	258440.	0.0131798
$827$	−1.32856e7	−0.675489	−0.337745	−	0.941238i	$-0.609664\pi$
$827$	−1.32856e7	−0.675489	−0.337745	+	0.941238i	$0.609664\pi$
$828$	0	0
$829$	−653987.	−0.0330509	−0.0165254	−	0.999863i	$-0.505260\pi$
$829$	−653987.	−0.0330509	−0.0165254	+	0.999863i	$0.505260\pi$
$830$	7.80626e6	0.393322
$831$	0	0
$832$	3.29155e7	1.64851
$833$	−1.14610e7	−0.572282
$834$	0	0
$835$	−1.08982e7	−0.540926
$836$	0	0
$837$	0	0
$838$	1.34935e7	0.663765
$839$	−2.47747e7	−1.21508	−0.607538	−	0.794290i	$-0.707843\pi$
$839$	−2.47747e7	−1.21508	−0.607538	+	0.794290i	$0.707843\pi$
$840$	0	0
$841$	8.90863e6	0.434331
$842$	1.81021e7	0.879929
$843$	0	0
$844$	1.64749e7	0.796100
$845$	1.79856e7	0.866530
$846$	0	0
$847$	0	0
$848$	−7.64467e6	−0.365064
$849$	0	0
$850$	7.58442e6	0.360060
$851$	−4.47417e7	−2.11782
$852$	0	0
$853$	−2.71291e7	−1.27662	−0.638311	−	0.769779i	$-0.720367\pi$
$853$	−2.71291e7	−1.27662	−0.638311	+	0.769779i	$0.720367\pi$
$854$	679200.	0.0318679
$855$	0	0
$856$	−6.27034e6	−0.292487
$857$	−2.84232e7	−1.32197	−0.660984	−	0.750400i	$-0.729861\pi$
$857$	−2.84232e7	−1.32197	−0.660984	+	0.750400i	$0.729861\pi$
$858$	0	0
$859$	2.65922e7	1.22962	0.614810	−	0.788675i	$-0.289233\pi$
$859$	2.65922e7	1.22962	0.614810	+	0.788675i	$0.289233\pi$
$860$	−1.22390e6	−0.0564289
$861$	0	0
$862$	2.73814e6	0.125512
$863$	2.22500e7	1.01696	0.508479	−	0.861074i	$-0.330208\pi$
$863$	2.22500e7	1.01696	0.508479	+	0.861074i	$0.330208\pi$
$864$	0	0
$865$	−7.33943e6	−0.333520
$866$	1.69036e7	0.765923
$867$	0	0
$868$	−1.03088e6	−0.0464418
$869$	0	0
$870$	0	0
$871$	3.44389e7	1.53817
$872$	3.56041e7	1.58566
$873$	0	0
$874$	5.69702e6	0.252272
$875$	1.11891e6	0.0494055
$876$	0	0
$877$	−2.83428e7	−1.24435	−0.622176	−	0.782877i	$-0.713751\pi$
$877$	−2.83428e7	−1.24435	−0.622176	+	0.782877i	$0.713751\pi$
$878$	−8.36739e6	−0.366314
$879$	0	0
$880$	0	0
$881$	−3.66445e7	−1.59063	−0.795315	−	0.606196i	$-0.792695\pi$
$881$	−3.66445e7	−1.59063	−0.795315	+	0.606196i	$0.792695\pi$
$882$	0	0
$883$	1.68772e7	0.728447	0.364223	−	0.931312i	$-0.381334\pi$
$883$	1.68772e7	0.728447	0.364223	+	0.931312i	$0.381334\pi$
$884$	−1.26004e7	−0.542320
$885$	0	0
$886$	6.25137e6	0.267541
$887$	2.73941e6	0.116909	0.0584544	−	0.998290i	$-0.481383\pi$
$887$	2.73941e6	0.116909	0.0584544	+	0.998290i	$0.481383\pi$
$888$	0	0
$889$	−781840.	−0.0331790
$890$	1.32354e6	0.0560095
$891$	0	0
$892$	−7.38050e6	−0.310580
$893$	−2.75251e6	−0.115505
$894$	0	0
$895$	−1.03565e7	−0.432171
$896$	−491520.	−0.0204537
$897$	0	0
$898$	−1.20180e7	−0.497325
$899$	3.49468e7	1.44214
$900$	0	0
$901$	2.04853e7	0.840681
$902$	0	0
$903$	0	0
$904$	−1.39870e7	−0.569251
$905$	−5.31022e6	−0.215522
$906$	0	0
$907$	−3.13286e7	−1.26451	−0.632255	−	0.774760i	$-0.717871\pi$
$907$	−3.13286e7	−1.26451	−0.632255	+	0.774760i	$0.717871\pi$
$908$	1.36891e7	0.551012
$909$	0	0
$910$	872480.	0.0349263
$911$	2.49762e7	0.997081	0.498541	−	0.866866i	$-0.333869\pi$
$911$	2.49762e7	0.997081	0.498541	+	0.866866i	$0.333869\pi$
$912$	0	0
$913$	0	0
$914$	−9.78206e6	−0.387316
$915$	0	0
$916$	1.06529e7	0.419497
$917$	4620.00	0.000181434 0
$918$	0	0
$919$	1.10613e7	0.432032	0.216016	−	0.976390i	$-0.430694\pi$
$919$	1.10613e7	0.432032	0.216016	+	0.976390i	$0.430694\pi$
$920$	−1.35304e7	−0.527038
$921$	0	0
$922$	−3.11641e7	−1.20734
$923$	−3.56144e7	−1.37601
$924$	0	0
$925$	−3.33421e7	−1.28127
$926$	4.20784e6	0.161262
$927$	0	0
$928$	2.77709e7	1.05857
$929$	2.01739e7	0.766919	0.383460	−	0.923558i	$-0.374733\pi$
$929$	2.01739e7	0.766919	0.383460	+	0.923558i	$0.374733\pi$
$930$	0	0
$931$	−6.41549e6	−0.242580
$932$	−1.93494e7	−0.729671
$933$	0	0
$934$	1.58801e7	0.595645
$935$	0	0
$936$	0	0
$937$	−9.10734e6	−0.338877	−0.169439	−	0.985541i	$-0.554195\pi$
$937$	−9.10734e6	−0.338877	−0.169439	+	0.985541i	$0.554195\pi$
$938$	1.19996e6	0.0445307
$939$	0	0
$940$	2.17907e6	0.0804363
$941$	−3.67709e7	−1.35372	−0.676861	−	0.736110i	$-0.736660\pi$
$941$	−3.67709e7	−1.35372	−0.676861	+	0.736110i	$0.736660\pi$
$942$	0	0
$943$	5.66735e6	0.207540
$944$	−1.65402e6	−0.0604101
$945$	0	0
$946$	0	0
$947$	4.95743e7	1.79631	0.898156	−	0.439677i	$-0.144907\pi$
$947$	4.95743e7	1.79631	0.898156	+	0.439677i	$0.144907\pi$
$948$	0	0
$949$	−2.20875e6	−0.0796125
$950$	4.24550e6	0.152623
$951$	0	0
$952$	−1.31712e6	−0.0471013
$953$	3.53787e7	1.26186	0.630928	−	0.775841i	$-0.282674\pi$
$953$	3.53787e7	1.26186	0.630928	+	0.775841i	$0.282674\pi$
$954$	0	0
$955$	8.44430e6	0.299609
$956$	9.14371e6	0.323577
$957$	0	0
$958$	3.41563e7	1.20242
$959$	2.96233e6	0.104013
$960$	0	0
$961$	1.28831e7	0.449999
$962$	−5.53933e7	−1.92983
$963$	0	0
$964$	−4.27328e6	−0.148105
$965$	351044.	0.0121351
$966$	0	0
$967$	−2.78059e7	−0.956248	−0.478124	−	0.878292i	$-0.658683\pi$
$967$	−2.78059e7	−0.956248	−0.478124	+	0.878292i	$0.658683\pi$
$968$	0	0
$969$	0	0
$970$	−5.08478e6	−0.173517
$971$	−1.56835e7	−0.533821	−0.266910	−	0.963721i	$-0.586003\pi$
$971$	−1.56835e7	−0.533821	−0.266910	+	0.963721i	$0.586003\pi$
$972$	0	0
$973$	−3.99818e6	−0.135388
$974$	7.45947e6	0.251948
$975$	0	0
$976$	−4.34688e6	−0.146067
$977$	−2.01140e7	−0.674157	−0.337079	−	0.941476i	$-0.609439\pi$
$977$	−2.01140e7	−0.674157	−0.337079	+	0.941476i	$0.609439\pi$
$978$	0	0
$979$	0	0
$980$	5.07893e6	0.168930
$981$	0	0
$982$	−2.06291e7	−0.682655
$983$	2.09269e7	0.690750	0.345375	−	0.938465i	$-0.387752\pi$
$983$	2.09269e7	0.690750	0.345375	+	0.938465i	$0.387752\pi$
$984$	0	0
$985$	5.13346e6	0.168585
$986$	1.48835e7	0.487541
$987$	0	0
$988$	−7.05331e6	−0.229880
$989$	−1.49324e7	−0.485445
$990$	0	0
$991$	−3.18663e7	−1.03074	−0.515368	−	0.856969i	$-0.672345\pi$
$991$	−3.18663e7	−1.03074	−0.515368	+	0.856969i	$0.672345\pi$
$992$	3.29882e7	1.06434
$993$	0	0
$994$	−1.24092e6	−0.0398362
$995$	823080.	0.0263563
$996$	0	0
$997$	−1.38913e6	−0.0442595	−0.0221297	−	0.999755i	$-0.507045\pi$
$997$	−1.38913e6	−0.0442595	−0.0221297	+	0.999755i	$0.507045\pi$
$998$	−1.81336e7	−0.576313
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.c.1.1		1
3.2	odd	2		121.6.a.b.1.1		1
11.10	odd	2		99.6.a.c.1.1		1
33.32	even	2		11.6.a.a.1.1	✓	1
132.131	odd	2		176.6.a.c.1.1		1
165.32	odd	4		275.6.b.a.199.1		2
165.98	odd	4		275.6.b.a.199.2		2
165.164	even	2		275.6.a.a.1.1		1
231.230	odd	2		539.6.a.c.1.1		1
264.131	odd	2		704.6.a.c.1.1		1
264.197	even	2		704.6.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.a.a.1.1	✓	1	33.32	even	2
99.6.a.c.1.1		1	11.10	odd	2
121.6.a.b.1.1		1	3.2	odd	2
176.6.a.c.1.1		1	132.131	odd	2
275.6.a.a.1.1		1	165.164	even	2
275.6.b.a.199.1		2	165.32	odd	4
275.6.b.a.199.2		2	165.98	odd	4
539.6.a.c.1.1		1	231.230	odd	2
704.6.a.c.1.1		1	264.131	odd	2
704.6.a.h.1.1		1	264.197	even	2
1089.6.a.c.1.1		1	1.1	even	1	trivial

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Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1089.6.a.c.1.1

Level	$1089$
Weight	$6$
Character	1089.1
Self dual	yes
Analytic conductor	$174.658$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$