LMFDB - Embedded newform 18.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,8,Mod(1,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	18.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:	$5.62293045871$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	18.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +64.0000 q^{4} +210.000 q^{5} +1016.00 q^{7} +512.000 q^{8} +O(q^{10})$

\(q+8.00000 q^{2} +64.0000 q^{4} +210.000 q^{5} +1016.00 q^{7} +512.000 q^{8} +1680.00 q^{10} -1092.00 q^{11} +1382.00 q^{13} +8128.00 q^{14} +4096.00 q^{16} -14706.0 q^{17} -39940.0 q^{19} +13440.0 q^{20} -8736.00 q^{22} -68712.0 q^{23} -34025.0 q^{25} +11056.0 q^{26} +65024.0 q^{28} +102570. q^{29} +227552. q^{31} +32768.0 q^{32} -117648. q^{34} +213360. q^{35} +160526. q^{37} -319520. q^{38} +107520. q^{40} -10842.0 q^{41} -630748. q^{43} -69888.0 q^{44} -549696. q^{46} -472656. q^{47} +208713. q^{49} -272200. q^{50} +88448.0 q^{52} +1.49402e6 q^{53} -229320. q^{55} +520192. q^{56} +820560. q^{58} -2.64066e6 q^{59} +827702. q^{61} +1.82042e6 q^{62} +262144. q^{64} +290220. q^{65} -126004. q^{67} -941184. q^{68} +1.70688e6 q^{70} +1.41473e6 q^{71} +980282. q^{73} +1.28421e6 q^{74} -2.55616e6 q^{76} -1.10947e6 q^{77} -3.56680e6 q^{79} +860160. q^{80} -86736.0 q^{82} -5.67289e6 q^{83} -3.08826e6 q^{85} -5.04598e6 q^{86} -559104. q^{88} +1.19512e7 q^{89} +1.40411e6 q^{91} -4.39757e6 q^{92} -3.78125e6 q^{94} -8.38740e6 q^{95} +8.68215e6 q^{97} +1.66970e6 q^{98} +O(q^{100})\)

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	210.000	0.751319	0.375659	−	0.926758i	$-0.377416\pi$
$5$	210.000	0.751319	0.375659	+	0.926758i	$0.377416\pi$
$6$	0	0
$7$	1016.00	1.11957	0.559784	−	0.828638i	$-0.310884\pi$
$7$	1016.00	1.11957	0.559784	+	0.828638i	$0.310884\pi$
$8$	512.000	0.353553
$9$	0	0
$10$	1680.00	0.531263
$11$	−1092.00	−0.247371	−0.123685	−	0.992321i	$-0.539471\pi$
$11$	−1092.00	−0.247371	−0.123685	+	0.992321i	$0.539471\pi$
$12$	0	0
$13$	1382.00	0.174464	0.0872321	−	0.996188i	$-0.472198\pi$
$13$	1382.00	0.174464	0.0872321	+	0.996188i	$0.472198\pi$
$14$	8128.00	0.791654
$15$	0	0
$16$	4096.00	0.250000
$17$	−14706.0	−0.725978	−0.362989	−	0.931793i	$-0.618244\pi$
$17$	−14706.0	−0.725978	−0.362989	+	0.931793i	$0.618244\pi$
$18$	0	0
$19$	−39940.0	−1.33589	−0.667945	−	0.744211i	$-0.732826\pi$
$19$	−39940.0	−1.33589	−0.667945	+	0.744211i	$0.732826\pi$
$20$	13440.0	0.375659
$21$	0	0
$22$	−8736.00	−0.174917
$23$	−68712.0	−1.17757	−0.588783	−	0.808291i	$-0.700393\pi$
$23$	−68712.0	−1.17757	−0.588783	+	0.808291i	$0.700393\pi$
$24$	0	0
$25$	−34025.0	−0.435520
$26$	11056.0	0.123365
$27$	0	0
$28$	65024.0	0.559784
$29$	102570.	0.780957	0.390479	−	0.920612i	$-0.372310\pi$
$29$	102570.	0.780957	0.390479	+	0.920612i	$0.372310\pi$
$30$	0	0
$31$	227552.	1.37188	0.685938	−	0.727660i	$-0.259392\pi$
$31$	227552.	1.37188	0.685938	+	0.727660i	$0.259392\pi$
$32$	32768.0	0.176777
$33$	0	0
$34$	−117648.	−0.513344
$35$	213360.	0.841153
$36$	0	0
$37$	160526.	0.521002	0.260501	−	0.965474i	$-0.416112\pi$
$37$	160526.	0.521002	0.260501	+	0.965474i	$0.416112\pi$
$38$	−319520.	−0.944616
$39$	0	0
$40$	107520.	0.265631
$41$	−10842.0	−0.0245678	−0.0122839	−	0.999925i	$-0.503910\pi$
$41$	−10842.0	−0.0245678	−0.0122839	+	0.999925i	$0.503910\pi$
$42$	0	0
$43$	−630748.	−1.20981	−0.604904	−	0.796299i	$-0.706788\pi$
$43$	−630748.	−1.20981	−0.604904	+	0.796299i	$0.706788\pi$
$44$	−69888.0	−0.123685
$45$	0	0
$46$	−549696.	−0.832665
$47$	−472656.	−0.664053	−0.332026	−	0.943270i	$-0.607732\pi$
$47$	−472656.	−0.664053	−0.332026	+	0.943270i	$0.607732\pi$
$48$	0	0
$49$	208713.	0.253433
$50$	−272200.	−0.307959
$51$	0	0
$52$	88448.0	0.0872321
$53$	1.49402e6	1.37845	0.689224	−	0.724548i	$-0.257952\pi$
$53$	1.49402e6	1.37845	0.689224	+	0.724548i	$0.257952\pi$
$54$	0	0
$55$	−229320.	−0.185854
$56$	520192.	0.395827
$57$	0	0
$58$	820560.	0.552220
$59$	−2.64066e6	−1.67390	−0.836952	−	0.547277i	$-0.815665\pi$
$59$	−2.64066e6	−1.67390	−0.836952	+	0.547277i	$0.815665\pi$
$60$	0	0
$61$	827702.	0.466895	0.233448	−	0.972369i	$-0.424999\pi$
$61$	827702.	0.466895	0.233448	+	0.972369i	$0.424999\pi$
$62$	1.82042e6	0.970063
$63$	0	0
$64$	262144.	0.125000
$65$	290220.	0.131078
$66$	0	0
$67$	−126004.	−0.0511826	−0.0255913	−	0.999672i	$-0.508147\pi$
$67$	−126004.	−0.0511826	−0.0255913	+	0.999672i	$0.508147\pi$
$68$	−941184.	−0.362989
$69$	0	0
$70$	1.70688e6	0.594785
$71$	1.41473e6	0.469104	0.234552	−	0.972104i	$-0.424638\pi$
$71$	1.41473e6	0.469104	0.234552	+	0.972104i	$0.424638\pi$
$72$	0	0
$73$	980282.	0.294931	0.147466	−	0.989067i	$-0.452888\pi$
$73$	980282.	0.294931	0.147466	+	0.989067i	$0.452888\pi$
$74$	1.28421e6	0.368404
$75$	0	0
$76$	−2.55616e6	−0.667945
$77$	−1.10947e6	−0.276948
$78$	0	0
$79$	−3.56680e6	−0.813924	−0.406962	−	0.913445i	$-0.633412\pi$
$79$	−3.56680e6	−0.813924	−0.406962	+	0.913445i	$0.633412\pi$
$80$	860160.	0.187830
$81$	0	0
$82$	−86736.0	−0.0173720
$83$	−5.67289e6	−1.08901	−0.544504	−	0.838758i	$-0.683282\pi$
$83$	−5.67289e6	−1.08901	−0.544504	+	0.838758i	$0.683282\pi$
$84$	0	0
$85$	−3.08826e6	−0.545441
$86$	−5.04598e6	−0.855463
$87$	0	0
$88$	−559104.	−0.0874587
$89$	1.19512e7	1.79699	0.898496	−	0.438982i	$-0.144661\pi$
$89$	1.19512e7	1.79699	0.898496	+	0.438982i	$0.144661\pi$
$90$	0	0
$91$	1.40411e6	0.195325
$92$	−4.39757e6	−0.588783
$93$	0	0
$94$	−3.78125e6	−0.469556
$95$	−8.38740e6	−1.00368
$96$	0	0
$97$	8.68215e6	0.965886	0.482943	−	0.875652i	$-0.339568\pi$
$97$	8.68215e6	0.965886	0.482943	+	0.875652i	$0.339568\pi$
$98$	1.66970e6	0.179204
$99$	0	0
$100$	−2.17760e6	−0.217760
$101$	1.00795e7	0.973455	0.486727	−	0.873554i	$-0.338190\pi$
$101$	1.00795e7	0.973455	0.486727	+	0.873554i	$0.338190\pi$
$102$	0	0
$103$	3.74799e6	0.337962	0.168981	−	0.985619i	$-0.445952\pi$
$103$	3.74799e6	0.337962	0.168981	+	0.985619i	$0.445952\pi$
$104$	707584.	0.0616824
$105$	0	0
$106$	1.19521e7	0.974710
$107$	1.79856e7	1.41932	0.709661	−	0.704543i	$-0.248848\pi$
$107$	1.79856e7	1.41932	0.709661	+	0.704543i	$0.248848\pi$
$108$	0	0
$109$	1.22570e7	0.906552	0.453276	−	0.891370i	$-0.350255\pi$
$109$	1.22570e7	0.906552	0.453276	+	0.891370i	$0.350255\pi$
$110$	−1.83456e6	−0.131419
$111$	0	0
$112$	4.16154e6	0.279892
$113$	−1.65950e7	−1.08194	−0.540968	−	0.841043i	$-0.681942\pi$
$113$	−1.65950e7	−1.08194	−0.540968	+	0.841043i	$0.681942\pi$
$114$	0	0
$115$	−1.44295e7	−0.884727
$116$	6.56448e6	0.390479
$117$	0	0
$118$	−2.11253e7	−1.18363
$119$	−1.49413e7	−0.812782
$120$	0	0
$121$	−1.82947e7	−0.938808
$122$	6.62162e6	0.330145
$123$	0	0
$124$	1.45633e7	0.685938
$125$	−2.35515e7	−1.07853
$126$	0	0
$127$	1.16826e6	0.0506087	0.0253043	−	0.999680i	$-0.491945\pi$
$127$	1.16826e6	0.0506087	0.0253043	+	0.999680i	$0.491945\pi$
$128$	2.09715e6	0.0883883
$129$	0	0
$130$	2.32176e6	0.0926863
$131$	7.92383e6	0.307954	0.153977	−	0.988074i	$-0.450792\pi$
$131$	7.92383e6	0.307954	0.153977	+	0.988074i	$0.450792\pi$
$132$	0	0
$133$	−4.05790e7	−1.49562
$134$	−1.00803e6	−0.0361916
$135$	0	0
$136$	−7.52947e6	−0.256672
$137$	315654.	0.0104879	0.00524396	−	0.999986i	$-0.498331\pi$
$137$	315654.	0.0104879	0.00524396	+	0.999986i	$0.498331\pi$
$138$	0	0
$139$	3.92038e7	1.23816	0.619079	−	0.785329i	$-0.287506\pi$
$139$	3.92038e7	1.23816	0.619079	+	0.785329i	$0.287506\pi$
$140$	1.36550e7	0.420576
$141$	0	0
$142$	1.13178e7	0.331706
$143$	−1.50914e6	−0.0431573
$144$	0	0
$145$	2.15397e7	0.586748
$146$	7.84226e6	0.208548
$147$	0	0
$148$	1.02737e7	0.260501
$149$	2.18860e7	0.542020	0.271010	−	0.962577i	$-0.412642\pi$
$149$	2.18860e7	0.542020	0.271010	+	0.962577i	$0.412642\pi$
$150$	0	0
$151$	−2.94154e7	−0.695274	−0.347637	−	0.937629i	$-0.613016\pi$
$151$	−2.94154e7	−0.695274	−0.347637	+	0.937629i	$0.613016\pi$
$152$	−2.04493e7	−0.472308
$153$	0	0
$154$	−8.87578e6	−0.195832
$155$	4.77859e7	1.03072
$156$	0	0
$157$	6.05550e7	1.24882	0.624412	−	0.781095i	$-0.285339\pi$
$157$	6.05550e7	1.24882	0.624412	+	0.781095i	$0.285339\pi$
$158$	−2.85344e7	−0.575531
$159$	0	0
$160$	6.88128e6	0.132816
$161$	−6.98114e7	−1.31837
$162$	0	0
$163$	5.70853e7	1.03245	0.516223	−	0.856454i	$-0.327337\pi$
$163$	5.70853e7	1.03245	0.516223	+	0.856454i	$0.327337\pi$
$164$	−693888.	−0.0122839
$165$	0	0
$166$	−4.53831e7	−0.770045
$167$	8.77265e7	1.45755	0.728775	−	0.684754i	$-0.240090\pi$
$167$	8.77265e7	1.45755	0.728775	+	0.684754i	$0.240090\pi$
$168$	0	0
$169$	−6.08386e7	−0.969562
$170$	−2.47061e7	−0.385685
$171$	0	0
$172$	−4.03679e7	−0.604904
$173$	−8.56954e6	−0.125833	−0.0629167	−	0.998019i	$-0.520040\pi$
$173$	−8.56954e6	−0.125833	−0.0629167	+	0.998019i	$0.520040\pi$
$174$	0	0
$175$	−3.45694e7	−0.487594
$176$	−4.47283e6	−0.0618427
$177$	0	0
$178$	9.56095e7	1.27067
$179$	−1.88041e7	−0.245056	−0.122528	−	0.992465i	$-0.539100\pi$
$179$	−1.88041e7	−0.245056	−0.122528	+	0.992465i	$0.539100\pi$
$180$	0	0
$181$	−5.99625e7	−0.751631	−0.375816	−	0.926694i	$-0.622637\pi$
$181$	−5.99625e7	−0.751631	−0.375816	+	0.926694i	$0.622637\pi$
$182$	1.12329e7	0.138115
$183$	0	0
$184$	−3.51805e7	−0.416332
$185$	3.37105e7	0.391439
$186$	0	0
$187$	1.60590e7	0.179586
$188$	−3.02500e7	−0.332026
$189$	0	0
$190$	−6.70992e7	−0.709708
$191$	−9.39861e7	−0.975993	−0.487997	−	0.872845i	$-0.662272\pi$
$191$	−9.39861e7	−0.975993	−0.487997	+	0.872845i	$0.662272\pi$
$192$	0	0
$193$	−3.51946e7	−0.352391	−0.176196	−	0.984355i	$-0.556379\pi$
$193$	−3.51946e7	−0.352391	−0.176196	+	0.984355i	$0.556379\pi$
$194$	6.94572e7	0.682985
$195$	0	0
$196$	1.33576e7	0.126717
$197$	−1.02985e8	−0.959718	−0.479859	−	0.877346i	$-0.659312\pi$
$197$	−1.02985e8	−0.959718	−0.479859	+	0.877346i	$0.659312\pi$
$198$	0	0
$199$	8.36376e7	0.752342	0.376171	−	0.926550i	$-0.377240\pi$
$199$	8.36376e7	0.752342	0.376171	+	0.926550i	$0.377240\pi$
$200$	−1.74208e7	−0.153980
$201$	0	0
$202$	8.06363e7	0.688337
$203$	1.04211e8	0.874335
$204$	0	0
$205$	−2.27682e6	−0.0184582
$206$	2.99839e7	0.238975
$207$	0	0
$208$	5.66067e6	0.0436160
$209$	4.36145e7	0.330460
$210$	0	0
$211$	−9.74010e7	−0.713797	−0.356899	−	0.934143i	$-0.616166\pi$
$211$	−9.74010e7	−0.713797	−0.356899	+	0.934143i	$0.616166\pi$
$212$	9.56172e7	0.689224
$213$	0	0
$214$	1.43885e8	1.00361
$215$	−1.32457e8	−0.908951
$216$	0	0
$217$	2.31193e8	1.53591
$218$	9.80562e7	0.641029
$219$	0	0
$220$	−1.46765e7	−0.0929271
$221$	−2.03237e7	−0.126657
$222$	0	0
$223$	−1.46457e7	−0.0884390	−0.0442195	−	0.999022i	$-0.514080\pi$
$223$	−1.46457e7	−0.0884390	−0.0442195	+	0.999022i	$0.514080\pi$
$224$	3.32923e7	0.197914
$225$	0	0
$226$	−1.32760e8	−0.765045
$227$	1.84541e8	1.04713	0.523567	−	0.851985i	$-0.324601\pi$
$227$	1.84541e8	1.04713	0.523567	+	0.851985i	$0.324601\pi$
$228$	0	0
$229$	−8.75461e6	−0.0481740	−0.0240870	−	0.999710i	$-0.507668\pi$
$229$	−8.75461e6	−0.0481740	−0.0240870	+	0.999710i	$0.507668\pi$
$230$	−1.15436e8	−0.625597
$231$	0	0
$232$	5.25158e7	0.276110
$233$	1.19556e8	0.619193	0.309597	−	0.950868i	$-0.399806\pi$
$233$	1.19556e8	0.619193	0.309597	+	0.950868i	$0.399806\pi$
$234$	0	0
$235$	−9.92578e7	−0.498915
$236$	−1.69002e8	−0.836952
$237$	0	0
$238$	−1.19530e8	−0.574723
$239$	−3.96209e8	−1.87729	−0.938646	−	0.344883i	$-0.887919\pi$
$239$	−3.96209e8	−1.87729	−0.938646	+	0.344883i	$0.887919\pi$
$240$	0	0
$241$	−2.56606e8	−1.18089	−0.590443	−	0.807080i	$-0.701047\pi$
$241$	−2.56606e8	−1.18089	−0.590443	+	0.807080i	$0.701047\pi$
$242$	−1.46358e8	−0.663837
$243$	0	0
$244$	5.29729e7	0.233448
$245$	4.38297e7	0.190409
$246$	0	0
$247$	−5.51971e7	−0.233065
$248$	1.16507e8	0.485031
$249$	0	0
$250$	−1.88412e8	−0.762638
$251$	7.34775e7	0.293290	0.146645	−	0.989189i	$-0.453153\pi$
$251$	7.34775e7	0.293290	0.146645	+	0.989189i	$0.453153\pi$
$252$	0	0
$253$	7.50335e7	0.291295
$254$	9.34605e6	0.0357857
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	2.02701e8	0.744886	0.372443	−	0.928055i	$-0.378520\pi$
$257$	2.02701e8	0.744886	0.372443	+	0.928055i	$0.378520\pi$
$258$	0	0
$259$	1.63094e8	0.583297
$260$	1.85741e7	0.0655391
$261$	0	0
$262$	6.33906e7	0.217756
$263$	−1.54254e8	−0.522867	−0.261434	−	0.965221i	$-0.584195\pi$
$263$	−1.54254e8	−0.522867	−0.261434	+	0.965221i	$0.584195\pi$
$264$	0	0
$265$	3.13744e8	1.03565
$266$	−3.24632e8	−1.05756
$267$	0	0
$268$	−8.06426e6	−0.0255913
$269$	6.24018e8	1.95463	0.977315	−	0.211793i	$-0.0679302\pi$
$269$	6.24018e8	1.95463	0.977315	+	0.211793i	$0.0679302\pi$
$270$	0	0
$271$	−3.87983e8	−1.18419	−0.592094	−	0.805869i	$-0.701698\pi$
$271$	−3.87983e8	−1.18419	−0.592094	+	0.805869i	$0.701698\pi$
$272$	−6.02358e7	−0.181494
$273$	0	0
$274$	2.52523e6	0.00741608
$275$	3.71553e7	0.107735
$276$	0	0
$277$	4.53952e8	1.28331	0.641654	−	0.766994i	$-0.278248\pi$
$277$	4.53952e8	1.28331	0.641654	+	0.766994i	$0.278248\pi$
$278$	3.13630e8	0.875510
$279$	0	0
$280$	1.09240e8	0.297392
$281$	−3.33770e8	−0.897377	−0.448689	−	0.893688i	$-0.648109\pi$
$281$	−3.33770e8	−0.897377	−0.448689	+	0.893688i	$0.648109\pi$
$282$	0	0
$283$	5.37695e8	1.41021	0.705104	−	0.709104i	$-0.250900\pi$
$283$	5.37695e8	1.41021	0.705104	+	0.709104i	$0.250900\pi$
$284$	9.05426e7	0.234552
$285$	0	0
$286$	−1.20732e7	−0.0305168
$287$	−1.10155e7	−0.0275053
$288$	0	0
$289$	−1.94072e8	−0.472956
$290$	1.72318e8	0.414894
$291$	0	0
$292$	6.27380e7	0.147466
$293$	−3.35600e8	−0.779445	−0.389722	−	0.920932i	$-0.627429\pi$
$293$	−3.35600e8	−0.779445	−0.389722	+	0.920932i	$0.627429\pi$
$294$	0	0
$295$	−5.54539e8	−1.25764
$296$	8.21893e7	0.184202
$297$	0	0
$298$	1.75088e8	0.383266
$299$	−9.49600e7	−0.205443
$300$	0	0
$301$	−6.40840e8	−1.35446
$302$	−2.35324e8	−0.491633
$303$	0	0
$304$	−1.63594e8	−0.333972
$305$	1.73817e8	0.350787
$306$	0	0
$307$	2.15029e8	0.424143	0.212072	−	0.977254i	$-0.431979\pi$
$307$	2.15029e8	0.424143	0.212072	+	0.977254i	$0.431979\pi$
$308$	−7.10062e7	−0.138474
$309$	0	0
$310$	3.82287e8	0.728826
$311$	−7.92062e8	−1.49313	−0.746565	−	0.665313i	$-0.768298\pi$
$311$	−7.92062e8	−1.49313	−0.746565	+	0.665313i	$0.768298\pi$
$312$	0	0
$313$	−1.18457e8	−0.218352	−0.109176	−	0.994022i	$-0.534821\pi$
$313$	−1.18457e8	−0.218352	−0.109176	+	0.994022i	$0.534821\pi$
$314$	4.84440e8	0.883051
$315$	0	0
$316$	−2.28275e8	−0.406962
$317$	5.07310e7	0.0894470	0.0447235	−	0.998999i	$-0.485759\pi$
$317$	5.07310e7	0.0894470	0.0447235	+	0.998999i	$0.485759\pi$
$318$	0	0
$319$	−1.12006e8	−0.193186
$320$	5.50502e7	0.0939149
$321$	0	0
$322$	−5.58491e8	−0.932225
$323$	5.87358e8	0.969826
$324$	0	0
$325$	−4.70226e7	−0.0759826
$326$	4.56682e8	0.730050
$327$	0	0
$328$	−5.55110e6	−0.00868602
$329$	−4.80218e8	−0.743453
$330$	0	0
$331$	2.73757e8	0.414923	0.207461	−	0.978243i	$-0.433480\pi$
$331$	2.73757e8	0.414923	0.207461	+	0.978243i	$0.433480\pi$
$332$	−3.63065e8	−0.544504
$333$	0	0
$334$	7.01812e8	1.03064
$335$	−2.64608e7	−0.0384545
$336$	0	0
$337$	−9.18512e7	−0.130732	−0.0653658	−	0.997861i	$-0.520821\pi$
$337$	−9.18512e7	−0.130732	−0.0653658	+	0.997861i	$0.520821\pi$
$338$	−4.86709e8	−0.685584
$339$	0	0
$340$	−1.97649e8	−0.272720
$341$	−2.48487e8	−0.339362
$342$	0	0
$343$	−6.24667e8	−0.835833
$344$	−3.22943e8	−0.427732
$345$	0	0
$346$	−6.85563e7	−0.0889777
$347$	1.36700e9	1.75637	0.878187	−	0.478318i	$-0.158753\pi$
$347$	1.36700e9	1.75637	0.878187	+	0.478318i	$0.158753\pi$
$348$	0	0
$349$	1.13143e9	1.42475	0.712377	−	0.701797i	$-0.247619\pi$
$349$	1.13143e9	1.42475	0.712377	+	0.701797i	$0.247619\pi$
$350$	−2.76555e8	−0.344781
$351$	0	0
$352$	−3.57827e7	−0.0437294
$353$	4.48395e7	0.0542562	0.0271281	−	0.999632i	$-0.491364\pi$
$353$	4.48395e7	0.0542562	0.0271281	+	0.999632i	$0.491364\pi$
$354$	0	0
$355$	2.97093e8	0.352446
$356$	7.64876e8	0.898496
$357$	0	0
$358$	−1.50432e8	−0.173281
$359$	−3.98281e8	−0.454317	−0.227158	−	0.973858i	$-0.572943\pi$
$359$	−3.98281e8	−0.454317	−0.227158	+	0.973858i	$0.572943\pi$
$360$	0	0
$361$	7.01332e8	0.784600
$362$	−4.79700e8	−0.531483
$363$	0	0
$364$	8.98632e7	0.0976623
$365$	2.05859e8	0.221588
$366$	0	0
$367$	1.63472e9	1.72628	0.863140	−	0.504964i	$-0.168494\pi$
$367$	1.63472e9	1.72628	0.863140	+	0.504964i	$0.168494\pi$
$368$	−2.81444e8	−0.294391
$369$	0	0
$370$	2.69684e8	0.276789
$371$	1.51792e9	1.54327
$372$	0	0
$373$	−1.54633e9	−1.54284	−0.771421	−	0.636325i	$-0.780454\pi$
$373$	−1.54633e9	−1.54284	−0.771421	+	0.636325i	$0.780454\pi$
$374$	1.28472e8	0.126986
$375$	0	0
$376$	−2.42000e8	−0.234778
$377$	1.41752e8	0.136249
$378$	0	0
$379$	−1.05688e9	−0.997216	−0.498608	−	0.866828i	$-0.666155\pi$
$379$	−1.05688e9	−0.997216	−0.498608	+	0.866828i	$0.666155\pi$
$380$	−5.36794e8	−0.501839
$381$	0	0
$382$	−7.51889e8	−0.690132
$383$	−2.24910e8	−0.204556	−0.102278	−	0.994756i	$-0.532613\pi$
$383$	−2.24910e8	−0.204556	−0.102278	+	0.994756i	$0.532613\pi$
$384$	0	0
$385$	−2.32989e8	−0.208077
$386$	−2.81556e8	−0.249178
$387$	0	0
$388$	5.55657e8	0.482943
$389$	−1.01788e9	−0.876746	−0.438373	−	0.898793i	$-0.644445\pi$
$389$	−1.01788e9	−0.876746	−0.438373	+	0.898793i	$0.644445\pi$
$390$	0	0
$391$	1.01048e9	0.854887
$392$	1.06861e8	0.0896021
$393$	0	0
$394$	−8.23883e8	−0.678623
$395$	−7.49028e8	−0.611517
$396$	0	0
$397$	−1.47565e9	−1.18363	−0.591817	−	0.806072i	$-0.701589\pi$
$397$	−1.47565e9	−1.18363	−0.591817	+	0.806072i	$0.701589\pi$
$398$	6.69100e8	0.531986
$399$	0	0
$400$	−1.39366e8	−0.108880
$401$	−2.74912e8	−0.212906	−0.106453	−	0.994318i	$-0.533949\pi$
$401$	−2.74912e8	−0.212906	−0.106453	+	0.994318i	$0.533949\pi$
$402$	0	0
$403$	3.14477e8	0.239343
$404$	6.45090e8	0.486727
$405$	0	0
$406$	8.33689e8	0.618248
$407$	−1.75294e8	−0.128881
$408$	0	0
$409$	−1.63427e9	−1.18112	−0.590558	−	0.806995i	$-0.701092\pi$
$409$	−1.63427e9	−1.18112	−0.590558	+	0.806995i	$0.701092\pi$
$410$	−1.82146e7	−0.0130519
$411$	0	0
$412$	2.39871e8	0.168981
$413$	−2.68291e9	−1.87405
$414$	0	0
$415$	−1.19131e9	−0.818192
$416$	4.52854e7	0.0308412
$417$	0	0
$418$	3.48916e8	0.233670
$419$	1.11280e9	0.739039	0.369519	−	0.929223i	$-0.379522\pi$
$419$	1.11280e9	0.739039	0.369519	+	0.929223i	$0.379522\pi$
$420$	0	0
$421$	9.22528e8	0.602549	0.301274	−	0.953537i	$-0.402588\pi$
$421$	9.22528e8	0.602549	0.301274	+	0.953537i	$0.402588\pi$
$422$	−7.79208e8	−0.504731
$423$	0	0
$424$	7.64937e8	0.487355
$425$	5.00372e8	0.316178
$426$	0	0
$427$	8.40945e8	0.522721
$428$	1.15108e9	0.709661
$429$	0	0
$430$	−1.05966e9	−0.642726
$431$	9.81508e8	0.590505	0.295252	−	0.955419i	$-0.404596\pi$
$431$	9.81508e8	0.590505	0.295252	+	0.955419i	$0.404596\pi$
$432$	0	0
$433$	2.84998e9	1.68707	0.843537	−	0.537071i	$-0.180469\pi$
$433$	2.84998e9	1.68707	0.843537	+	0.537071i	$0.180469\pi$
$434$	1.84954e9	1.08605
$435$	0	0
$436$	7.84450e8	0.453276
$437$	2.74436e9	1.57310
$438$	0	0
$439$	−1.05622e9	−0.595838	−0.297919	−	0.954591i	$-0.596292\pi$
$439$	−1.05622e9	−0.595838	−0.297919	+	0.954591i	$0.596292\pi$
$440$	−1.17412e8	−0.0657094
$441$	0	0
$442$	−1.62590e8	−0.0895601
$443$	−1.82325e9	−0.996401	−0.498201	−	0.867062i	$-0.666006\pi$
$443$	−1.82325e9	−0.996401	−0.498201	+	0.867062i	$0.666006\pi$
$444$	0	0
$445$	2.50975e9	1.35011
$446$	−1.17166e8	−0.0625358
$447$	0	0
$448$	2.66338e8	0.139946
$449$	−1.84846e9	−0.963713	−0.481856	−	0.876250i	$-0.660037\pi$
$449$	−1.84846e9	−0.963713	−0.481856	+	0.876250i	$0.660037\pi$
$450$	0	0
$451$	1.18395e7	0.00607735
$452$	−1.06208e9	−0.540968
$453$	0	0
$454$	1.47633e9	0.740435
$455$	2.94864e8	0.146751
$456$	0	0
$457$	−2.98066e9	−1.46085	−0.730425	−	0.682993i	$-0.760678\pi$
$457$	−2.98066e9	−1.46085	−0.730425	+	0.682993i	$0.760678\pi$
$458$	−7.00369e7	−0.0340642
$459$	0	0
$460$	−9.23489e8	−0.442364
$461$	2.52781e9	1.20169	0.600843	−	0.799367i	$-0.294832\pi$
$461$	2.52781e9	1.20169	0.600843	+	0.799367i	$0.294832\pi$
$462$	0	0
$463$	−8.90291e8	−0.416868	−0.208434	−	0.978036i	$-0.566837\pi$
$463$	−8.90291e8	−0.416868	−0.208434	+	0.978036i	$0.566837\pi$
$464$	4.20127e8	0.195239
$465$	0	0
$466$	9.56450e8	0.437836
$467$	−2.65667e9	−1.20706	−0.603529	−	0.797341i	$-0.706239\pi$
$467$	−2.65667e9	−1.20706	−0.603529	+	0.797341i	$0.706239\pi$
$468$	0	0
$469$	−1.28020e8	−0.0573024
$470$	−7.94062e8	−0.352786
$471$	0	0
$472$	−1.35202e9	−0.591814
$473$	6.88777e8	0.299271
$474$	0	0
$475$	1.35896e9	0.581806
$476$	−9.56243e8	−0.406391
$477$	0	0
$478$	−3.16967e9	−1.32745
$479$	−1.30093e9	−0.540855	−0.270428	−	0.962740i	$-0.587165\pi$
$479$	−1.30093e9	−0.540855	−0.270428	+	0.962740i	$0.587165\pi$
$480$	0	0
$481$	2.21847e8	0.0908962
$482$	−2.05285e9	−0.835012
$483$	0	0
$484$	−1.17086e9	−0.469404
$485$	1.82325e9	0.725689
$486$	0	0
$487$	−1.07447e9	−0.421542	−0.210771	−	0.977535i	$-0.567598\pi$
$487$	−1.07447e9	−0.421542	−0.210771	+	0.977535i	$0.567598\pi$
$488$	4.23783e8	0.165072
$489$	0	0
$490$	3.50638e8	0.134640
$491$	7.83344e8	0.298653	0.149327	−	0.988788i	$-0.452289\pi$
$491$	7.83344e8	0.298653	0.149327	+	0.988788i	$0.452289\pi$
$492$	0	0
$493$	−1.50839e9	−0.566958
$494$	−4.41577e8	−0.164802
$495$	0	0
$496$	9.32053e8	0.342969
$497$	1.43736e9	0.525193
$498$	0	0
$499$	−6.23188e8	−0.224526	−0.112263	−	0.993679i	$-0.535810\pi$
$499$	−6.23188e8	−0.224526	−0.112263	+	0.993679i	$0.535810\pi$
$500$	−1.50730e9	−0.539267
$501$	0	0
$502$	5.87820e8	0.207387
$503$	2.70927e9	0.949215	0.474607	−	0.880198i	$-0.342590\pi$
$503$	2.70927e9	0.949215	0.474607	+	0.880198i	$0.342590\pi$
$504$	0	0
$505$	2.11670e9	0.731375
$506$	6.00268e8	0.205977
$507$	0	0
$508$	7.47684e7	0.0253043
$509$	−3.49943e9	−1.17621	−0.588106	−	0.808784i	$-0.700126\pi$
$509$	−3.49943e9	−1.17621	−0.588106	+	0.808784i	$0.700126\pi$
$510$	0	0
$511$	9.95967e8	0.330196
$512$	1.34218e8	0.0441942
$513$	0	0
$514$	1.62161e9	0.526714
$515$	7.87078e8	0.253918
$516$	0	0
$517$	5.16140e8	0.164267
$518$	1.30476e9	0.412453
$519$	0	0
$520$	1.48593e8	0.0463432
$521$	1.37683e9	0.426530	0.213265	−	0.976994i	$-0.431590\pi$
$521$	1.37683e9	0.426530	0.213265	+	0.976994i	$0.431590\pi$
$522$	0	0
$523$	−2.86154e9	−0.874669	−0.437334	−	0.899299i	$-0.644077\pi$
$523$	−2.86154e9	−0.874669	−0.437334	+	0.899299i	$0.644077\pi$
$524$	5.07125e8	0.153977
$525$	0	0
$526$	−1.23403e9	−0.369723
$527$	−3.34638e9	−0.995951
$528$	0	0
$529$	1.31651e9	0.386661
$530$	2.50995e9	0.732318
$531$	0	0
$532$	−2.59706e9	−0.747810
$533$	−1.49836e7	−0.00428620
$534$	0	0
$535$	3.77697e9	1.06636
$536$	−6.45140e7	−0.0180958
$537$	0	0
$538$	4.99215e9	1.38213
$539$	−2.27915e8	−0.0626919
$540$	0	0
$541$	5.34467e9	1.45121	0.725605	−	0.688111i	$-0.241560\pi$
$541$	5.34467e9	1.45121	0.725605	+	0.688111i	$0.241560\pi$
$542$	−3.10387e9	−0.837347
$543$	0	0
$544$	−4.81886e8	−0.128336
$545$	2.57398e9	0.681109
$546$	0	0
$547$	−3.37135e9	−0.880740	−0.440370	−	0.897816i	$-0.645153\pi$
$547$	−3.37135e9	−0.880740	−0.440370	+	0.897816i	$0.645153\pi$
$548$	2.02019e7	0.00524396
$549$	0	0
$550$	2.97242e8	0.0761801
$551$	−4.09665e9	−1.04327
$552$	0	0
$553$	−3.62387e9	−0.911244
$554$	3.63162e9	0.907436
$555$	0	0
$556$	2.50904e9	0.619079
$557$	5.61106e9	1.37579	0.687894	−	0.725811i	$-0.258535\pi$
$557$	5.61106e9	1.37579	0.687894	+	0.725811i	$0.258535\pi$
$558$	0	0
$559$	−8.71694e8	−0.211068
$560$	8.73923e8	0.210288
$561$	0	0
$562$	−2.67016e9	−0.634542
$563$	−6.69690e9	−1.58159	−0.790795	−	0.612081i	$-0.790333\pi$
$563$	−6.69690e9	−1.58159	−0.790795	+	0.612081i	$0.790333\pi$
$564$	0	0
$565$	−3.48494e9	−0.812879
$566$	4.30156e9	0.997168
$567$	0	0
$568$	7.24341e8	0.165853
$569$	−1.96850e9	−0.447964	−0.223982	−	0.974593i	$-0.571906\pi$
$569$	−1.96850e9	−0.447964	−0.223982	+	0.974593i	$0.571906\pi$
$570$	0	0
$571$	1.02926e9	0.231365	0.115682	−	0.993286i	$-0.463094\pi$
$571$	1.02926e9	0.231365	0.115682	+	0.993286i	$0.463094\pi$
$572$	−9.65852e7	−0.0215787
$573$	0	0
$574$	−8.81238e7	−0.0194492
$575$	2.33793e9	0.512853
$576$	0	0
$577$	3.31179e9	0.717708	0.358854	−	0.933394i	$-0.383168\pi$
$577$	3.31179e9	0.717708	0.358854	+	0.933394i	$0.383168\pi$
$578$	−1.55258e9	−0.334431
$579$	0	0
$580$	1.37854e9	0.293374
$581$	−5.76366e9	−1.21922
$582$	0	0
$583$	−1.63147e9	−0.340988
$584$	5.01904e8	0.104274
$585$	0	0
$586$	−2.68480e9	−0.551151
$587$	5.59411e8	0.114156	0.0570778	−	0.998370i	$-0.481822\pi$
$587$	5.59411e8	0.114156	0.0570778	+	0.998370i	$0.481822\pi$
$588$	0	0
$589$	−9.08843e9	−1.83267
$590$	−4.43631e9	−0.889282
$591$	0	0
$592$	6.57514e8	0.130250
$593$	3.02459e9	0.595628	0.297814	−	0.954624i	$-0.403742\pi$
$593$	3.02459e9	0.595628	0.297814	+	0.954624i	$0.403742\pi$
$594$	0	0
$595$	−3.13767e9	−0.610658
$596$	1.40071e9	0.271010
$597$	0	0
$598$	−7.59680e8	−0.145270
$599$	5.63246e9	1.07079	0.535395	−	0.844602i	$-0.320163\pi$
$599$	5.63246e9	1.07079	0.535395	+	0.844602i	$0.320163\pi$
$600$	0	0
$601$	3.40792e8	0.0640366	0.0320183	−	0.999487i	$-0.489807\pi$
$601$	3.40792e8	0.0640366	0.0320183	+	0.999487i	$0.489807\pi$
$602$	−5.12672e9	−0.957749
$603$	0	0
$604$	−1.88259e9	−0.347637
$605$	−3.84189e9	−0.705344
$606$	0	0
$607$	3.85420e9	0.699477	0.349739	−	0.936847i	$-0.386270\pi$
$607$	3.85420e9	0.699477	0.349739	+	0.936847i	$0.386270\pi$
$608$	−1.30875e9	−0.236154
$609$	0	0
$610$	1.39054e9	0.248044
$611$	−6.53211e8	−0.115853
$612$	0	0
$613$	9.22245e9	1.61709	0.808545	−	0.588434i	$-0.200255\pi$
$613$	9.22245e9	1.61709	0.808545	+	0.588434i	$0.200255\pi$
$614$	1.72023e9	0.299915
$615$	0	0
$616$	−5.68050e8	−0.0979160
$617$	−6.53611e9	−1.12027	−0.560133	−	0.828402i	$-0.689250\pi$
$617$	−6.53611e9	−1.12027	−0.560133	+	0.828402i	$0.689250\pi$
$618$	0	0
$619$	1.36559e9	0.231420	0.115710	−	0.993283i	$-0.463086\pi$
$619$	1.36559e9	0.231420	0.115710	+	0.993283i	$0.463086\pi$
$620$	3.05830e9	0.515358
$621$	0	0
$622$	−6.33649e9	−1.05580
$623$	1.21424e10	2.01186
$624$	0	0
$625$	−2.28761e9	−0.374802
$626$	−9.47659e8	−0.154398
$627$	0	0
$628$	3.87552e9	0.624412
$629$	−2.36070e9	−0.378236
$630$	0	0
$631$	1.54079e9	0.244141	0.122070	−	0.992521i	$-0.461047\pi$
$631$	1.54079e9	0.244141	0.122070	+	0.992521i	$0.461047\pi$
$632$	−1.82620e9	−0.287766
$633$	0	0
$634$	4.05848e8	0.0632486
$635$	2.45334e8	0.0380233
$636$	0	0
$637$	2.88441e8	0.0442150
$638$	−8.96052e8	−0.136603
$639$	0	0
$640$	4.40402e8	0.0664078
$641$	4.54018e9	0.680879	0.340440	−	0.940266i	$-0.389424\pi$
$641$	4.54018e9	0.680879	0.340440	+	0.940266i	$0.389424\pi$
$642$	0	0
$643$	1.14054e10	1.69189	0.845944	−	0.533272i	$-0.179038\pi$
$643$	1.14054e10	1.69189	0.845944	+	0.533272i	$0.179038\pi$
$644$	−4.46793e9	−0.659183
$645$	0	0
$646$	4.69886e9	0.685770
$647$	1.26393e10	1.83468	0.917338	−	0.398109i	$-0.130334\pi$
$647$	1.26393e10	1.83468	0.917338	+	0.398109i	$0.130334\pi$
$648$	0	0
$649$	2.88360e9	0.414075
$650$	−3.76180e8	−0.0537278
$651$	0	0
$652$	3.65346e9	0.516223
$653$	1.05004e10	1.47575	0.737873	−	0.674940i	$-0.235830\pi$
$653$	1.05004e10	1.47575	0.737873	+	0.674940i	$0.235830\pi$
$654$	0	0
$655$	1.66400e9	0.231371
$656$	−4.44088e7	−0.00614194
$657$	0	0
$658$	−3.84175e9	−0.525700
$659$	−9.64818e9	−1.31325	−0.656624	−	0.754219i	$-0.728016\pi$
$659$	−9.64818e9	−1.31325	−0.656624	+	0.754219i	$0.728016\pi$
$660$	0	0
$661$	−6.58299e9	−0.886580	−0.443290	−	0.896378i	$-0.646189\pi$
$661$	−6.58299e9	−0.886580	−0.443290	+	0.896378i	$0.646189\pi$
$662$	2.19006e9	0.293395
$663$	0	0
$664$	−2.90452e9	−0.385023
$665$	−8.52160e9	−1.12369
$666$	0	0
$667$	−7.04779e9	−0.919629
$668$	5.61450e9	0.728775
$669$	0	0
$670$	−2.11687e8	−0.0271914
$671$	−9.03851e8	−0.115496
$672$	0	0
$673$	−8.54649e9	−1.08077	−0.540387	−	0.841416i	$-0.681722\pi$
$673$	−8.54649e9	−1.08077	−0.540387	+	0.841416i	$0.681722\pi$
$674$	−7.34810e8	−0.0924411
$675$	0	0
$676$	−3.89367e9	−0.484781
$677$	−8.71305e9	−1.07922	−0.539610	−	0.841915i	$-0.681428\pi$
$677$	−8.71305e9	−1.07922	−0.539610	+	0.841915i	$0.681428\pi$
$678$	0	0
$679$	8.82106e9	1.08138
$680$	−1.58119e9	−0.192842
$681$	0	0
$682$	−1.98789e9	−0.239965
$683$	−1.46109e10	−1.75470	−0.877351	−	0.479849i	$-0.840692\pi$
$683$	−1.46109e10	−1.75470	−0.877351	+	0.479849i	$0.840692\pi$
$684$	0	0
$685$	6.62873e7	0.00787977
$686$	−4.99734e9	−0.591023
$687$	0	0
$688$	−2.58354e9	−0.302452
$689$	2.06473e9	0.240490
$690$	0	0
$691$	−1.47348e10	−1.69891	−0.849454	−	0.527662i	$-0.823069\pi$
$691$	−1.47348e10	−1.69891	−0.849454	+	0.527662i	$0.823069\pi$
$692$	−5.48451e8	−0.0629167
$693$	0	0
$694$	1.09360e10	1.24194
$695$	8.23279e9	0.930252
$696$	0	0
$697$	1.59442e8	0.0178357
$698$	9.05146e9	1.00745
$699$	0	0
$700$	−2.21244e9	−0.243797
$701$	−1.31502e9	−0.144185	−0.0720923	−	0.997398i	$-0.522968\pi$
$701$	−1.31502e9	−0.144185	−0.0720923	+	0.997398i	$0.522968\pi$
$702$	0	0
$703$	−6.41141e9	−0.696001
$704$	−2.86261e8	−0.0309213
$705$	0	0
$706$	3.58716e8	0.0383649
$707$	1.02408e10	1.08985
$708$	0	0
$709$	6.64028e8	0.0699721	0.0349860	−	0.999388i	$-0.488861\pi$
$709$	6.64028e8	0.0699721	0.0349860	+	0.999388i	$0.488861\pi$
$710$	2.37674e9	0.249217
$711$	0	0
$712$	6.11901e9	0.635333
$713$	−1.56356e10	−1.61547
$714$	0	0
$715$	−3.16920e8	−0.0324249
$716$	−1.20346e9	−0.122528
$717$	0	0
$718$	−3.18624e9	−0.321250
$719$	−4.95034e9	−0.496689	−0.248344	−	0.968672i	$-0.579886\pi$
$719$	−4.95034e9	−0.496689	−0.248344	+	0.968672i	$0.579886\pi$
$720$	0	0
$721$	3.80796e9	0.378372
$722$	5.61065e9	0.554796
$723$	0	0
$724$	−3.83760e9	−0.375816
$725$	−3.48994e9	−0.340123
$726$	0	0
$727$	8.81101e9	0.850463	0.425231	−	0.905085i	$-0.360193\pi$
$727$	8.81101e9	0.850463	0.425231	+	0.905085i	$0.360193\pi$
$728$	7.18905e8	0.0690577
$729$	0	0
$730$	1.64687e9	0.156686
$731$	9.27578e9	0.878293
$732$	0	0
$733$	−1.49414e8	−0.0140129	−0.00700643	−	0.999975i	$-0.502230\pi$
$733$	−1.49414e8	−0.0140129	−0.00700643	+	0.999975i	$0.502230\pi$
$734$	1.30777e10	1.22066
$735$	0	0
$736$	−2.25155e9	−0.208166
$737$	1.37596e8	0.0126611
$738$	0	0
$739$	−4.70806e9	−0.429127	−0.214564	−	0.976710i	$-0.568833\pi$
$739$	−4.70806e9	−0.429127	−0.214564	+	0.976710i	$0.568833\pi$
$740$	2.15747e9	0.195719
$741$	0	0
$742$	1.21434e10	1.09125
$743$	−1.69676e9	−0.151761	−0.0758805	−	0.997117i	$-0.524177\pi$
$743$	−1.69676e9	−0.151761	−0.0758805	+	0.997117i	$0.524177\pi$
$744$	0	0
$745$	4.59607e9	0.407230
$746$	−1.23707e10	−1.09095
$747$	0	0
$748$	1.02777e9	0.0897928
$749$	1.82733e10	1.58903
$750$	0	0
$751$	1.06650e10	0.918800	0.459400	−	0.888229i	$-0.348064\pi$
$751$	1.06650e10	0.918800	0.459400	+	0.888229i	$0.348064\pi$
$752$	−1.93600e9	−0.166013
$753$	0	0
$754$	1.13401e9	0.0963427
$755$	−6.17724e9	−0.522373
$756$	0	0
$757$	6.22876e9	0.521874	0.260937	−	0.965356i	$-0.415968\pi$
$757$	6.22876e9	0.521874	0.260937	+	0.965356i	$0.415968\pi$
$758$	−8.45506e9	−0.705138
$759$	0	0
$760$	−4.29435e9	−0.354854
$761$	8.38334e9	0.689558	0.344779	−	0.938684i	$-0.387954\pi$
$761$	8.38334e9	0.689558	0.344779	+	0.938684i	$0.387954\pi$
$762$	0	0
$763$	1.24531e10	1.01495
$764$	−6.01511e9	−0.487997
$765$	0	0
$766$	−1.79928e9	−0.144643
$767$	−3.64939e9	−0.292036
$768$	0	0
$769$	−1.18649e10	−0.940852	−0.470426	−	0.882439i	$-0.655900\pi$
$769$	−1.18649e10	−0.940852	−0.470426	+	0.882439i	$0.655900\pi$
$770$	−1.86391e9	−0.147132
$771$	0	0
$772$	−2.25245e9	−0.176196
$773$	−5.56680e9	−0.433488	−0.216744	−	0.976228i	$-0.569544\pi$
$773$	−5.56680e9	−0.433488	−0.216744	+	0.976228i	$0.569544\pi$
$774$	0	0
$775$	−7.74246e9	−0.597479
$776$	4.44526e9	0.341492
$777$	0	0
$778$	−8.14306e9	−0.619953
$779$	4.33029e8	0.0328198
$780$	0	0
$781$	−1.54488e9	−0.116042
$782$	8.08383e9	0.604496
$783$	0	0
$784$	8.54888e8	0.0633583
$785$	1.27165e10	0.938264
$786$	0	0
$787$	1.34611e8	0.00984395	0.00492198	−	0.999988i	$-0.498433\pi$
$787$	1.34611e8	0.00984395	0.00492198	+	0.999988i	$0.498433\pi$
$788$	−6.59106e9	−0.479859
$789$	0	0
$790$	−5.99222e9	−0.432408
$791$	−1.68605e10	−1.21130
$792$	0	0
$793$	1.14388e9	0.0814565
$794$	−1.18052e10	−0.836955
$795$	0	0
$796$	5.35280e9	0.376171
$797$	7.41548e9	0.518842	0.259421	−	0.965764i	$-0.416468\pi$
$797$	7.41548e9	0.518842	0.259421	+	0.965764i	$0.416468\pi$
$798$	0	0
$799$	6.95088e9	0.482088
$800$	−1.11493e9	−0.0769898
$801$	0	0
$802$	−2.19930e9	−0.150548
$803$	−1.07047e9	−0.0729574
$804$	0	0
$805$	−1.46604e10	−0.990513
$806$	2.51581e9	0.169241
$807$	0	0
$808$	5.16072e9	0.344168
$809$	1.41542e10	0.939863	0.469932	−	0.882703i	$-0.344279\pi$
$809$	1.41542e10	0.939863	0.469932	+	0.882703i	$0.344279\pi$
$810$	0	0
$811$	−2.63708e10	−1.73600	−0.868001	−	0.496563i	$-0.834595\pi$
$811$	−2.63708e10	−1.73600	−0.868001	+	0.496563i	$0.834595\pi$
$812$	6.66951e9	0.437168
$813$	0	0
$814$	−1.40236e9	−0.0911324
$815$	1.19879e10	0.775697
$816$	0	0
$817$	2.51921e10	1.61617
$818$	−1.30742e10	−0.835176
$819$	0	0
$820$	−1.45716e8	−0.00922912
$821$	−8.06264e9	−0.508483	−0.254241	−	0.967141i	$-0.581826\pi$
$821$	−8.06264e9	−0.508483	−0.254241	+	0.967141i	$0.581826\pi$
$822$	0	0
$823$	−2.34202e10	−1.46451	−0.732253	−	0.681033i	$-0.761531\pi$
$823$	−2.34202e10	−1.46451	−0.732253	+	0.681033i	$0.761531\pi$
$824$	1.91897e9	0.119488
$825$	0	0
$826$	−2.14633e10	−1.32515
$827$	−5.55722e9	−0.341655	−0.170828	−	0.985301i	$-0.554644\pi$
$827$	−5.55722e9	−0.341655	−0.170828	+	0.985301i	$0.554644\pi$
$828$	0	0
$829$	2.84256e10	1.73288	0.866440	−	0.499281i	$-0.166403\pi$
$829$	2.84256e10	1.73288	0.866440	+	0.499281i	$0.166403\pi$
$830$	−9.53046e9	−0.578549
$831$	0	0
$832$	3.62283e8	0.0218080
$833$	−3.06933e9	−0.183987
$834$	0	0
$835$	1.84226e10	1.09508
$836$	2.79133e9	0.165230
$837$	0	0
$838$	8.90238e9	0.522579
$839$	−1.04036e10	−0.608156	−0.304078	−	0.952647i	$-0.598348\pi$
$839$	−1.04036e10	−0.608156	−0.304078	+	0.952647i	$0.598348\pi$
$840$	0	0
$841$	−6.72927e9	−0.390105
$842$	7.38023e9	0.426066
$843$	0	0
$844$	−6.23367e9	−0.356899
$845$	−1.27761e10	−0.728450
$846$	0	0
$847$	−1.85874e10	−1.05106
$848$	6.11950e9	0.344612
$849$	0	0
$850$	4.00297e9	0.223572
$851$	−1.10301e10	−0.613514
$852$	0	0
$853$	−1.80580e10	−0.996205	−0.498102	−	0.867118i	$-0.665970\pi$
$853$	−1.80580e10	−0.996205	−0.498102	+	0.867118i	$0.665970\pi$
$854$	6.72756e9	0.369620
$855$	0	0
$856$	9.20861e9	0.501806
$857$	6.34034e9	0.344096	0.172048	−	0.985089i	$-0.444962\pi$
$857$	6.34034e9	0.344096	0.172048	+	0.985089i	$0.444962\pi$
$858$	0	0
$859$	1.21489e10	0.653973	0.326987	−	0.945029i	$-0.393967\pi$
$859$	1.21489e10	0.653973	0.326987	+	0.945029i	$0.393967\pi$
$860$	−8.47725e9	−0.454476
$861$	0	0
$862$	7.85206e9	0.417550
$863$	2.87111e10	1.52059	0.760295	−	0.649578i	$-0.225054\pi$
$863$	2.87111e10	1.52059	0.760295	+	0.649578i	$0.225054\pi$
$864$	0	0
$865$	−1.79960e9	−0.0945411
$866$	2.27998e10	1.19294
$867$	0	0
$868$	1.47963e10	0.767954
$869$	3.89495e9	0.201341
$870$	0	0
$871$	−1.74138e8	−0.00892953
$872$	6.27560e9	0.320514
$873$	0	0
$874$	2.19549e10	1.11235
$875$	−2.39283e10	−1.20749
$876$	0	0
$877$	2.46021e10	1.23161	0.615806	−	0.787898i	$-0.288831\pi$
$877$	2.46021e10	1.23161	0.615806	+	0.787898i	$0.288831\pi$
$878$	−8.44975e9	−0.421321
$879$	0	0
$880$	−9.39295e8	−0.0464636
$881$	1.25378e10	0.617738	0.308869	−	0.951105i	$-0.400049\pi$
$881$	1.25378e10	0.617738	0.308869	+	0.951105i	$0.400049\pi$
$882$	0	0
$883$	1.93097e10	0.943873	0.471937	−	0.881633i	$-0.343555\pi$
$883$	1.93097e10	0.943873	0.471937	+	0.881633i	$0.343555\pi$
$884$	−1.30072e9	−0.0633286
$885$	0	0
$886$	−1.45860e10	−0.704562
$887$	−3.20268e10	−1.54092	−0.770462	−	0.637486i	$-0.779974\pi$
$887$	−3.20268e10	−1.54092	−0.770462	+	0.637486i	$0.779974\pi$
$888$	0	0
$889$	1.18695e9	0.0566599
$890$	2.00780e10	0.954675
$891$	0	0
$892$	−9.37327e8	−0.0442195
$893$	1.88779e10	0.887101
$894$	0	0
$895$	−3.94885e9	−0.184115
$896$	2.13071e9	0.0989568
$897$	0	0
$898$	−1.47877e10	−0.681448
$899$	2.33400e10	1.07138
$900$	0	0
$901$	−2.19710e10	−1.00072
$902$	9.47157e7	0.00429733
$903$	0	0
$904$	−8.49662e9	−0.382522
$905$	−1.25921e10	−0.564715
$906$	0	0
$907$	2.33703e9	0.104002	0.0520008	−	0.998647i	$-0.483440\pi$
$907$	2.33703e9	0.104002	0.0520008	+	0.998647i	$0.483440\pi$
$908$	1.18106e10	0.523567
$909$	0	0
$910$	2.35891e9	0.103769
$911$	−2.20343e10	−0.965573	−0.482786	−	0.875738i	$-0.660375\pi$
$911$	−2.20343e10	−0.965573	−0.482786	+	0.875738i	$0.660375\pi$
$912$	0	0
$913$	6.19480e9	0.269389
$914$	−2.38453e10	−1.03298
$915$	0	0
$916$	−5.60295e8	−0.0240870
$917$	8.05061e9	0.344775
$918$	0	0
$919$	−1.43277e10	−0.608938	−0.304469	−	0.952522i	$-0.598479\pi$
$919$	−1.43277e10	−0.608938	−0.304469	+	0.952522i	$0.598479\pi$
$920$	−7.38791e9	−0.312798
$921$	0	0
$922$	2.02225e10	0.849720
$923$	1.95515e9	0.0818418
$924$	0	0
$925$	−5.46190e9	−0.226907
$926$	−7.12233e9	−0.294770
$927$	0	0
$928$	3.36101e9	0.138055
$929$	−1.31280e10	−0.537208	−0.268604	−	0.963251i	$-0.586562\pi$
$929$	−1.31280e10	−0.537208	−0.268604	+	0.963251i	$0.586562\pi$
$930$	0	0
$931$	−8.33600e9	−0.338558
$932$	7.65160e9	0.309597
$933$	0	0
$934$	−2.12533e10	−0.853519
$935$	3.37238e9	0.134926
$936$	0	0
$937$	−3.87626e10	−1.53930	−0.769652	−	0.638463i	$-0.779571\pi$
$937$	−3.87626e10	−1.53930	−0.769652	+	0.638463i	$0.779571\pi$
$938$	−1.02416e9	−0.0405189
$939$	0	0
$940$	−6.35250e9	−0.249458
$941$	−2.06279e10	−0.807035	−0.403517	−	0.914972i	$-0.632212\pi$
$941$	−2.06279e10	−0.807035	−0.403517	+	0.914972i	$0.632212\pi$
$942$	0	0
$943$	7.44976e8	0.0289302
$944$	−1.08161e10	−0.418476
$945$	0	0
$946$	5.51021e9	0.211617
$947$	2.11705e10	0.810040	0.405020	−	0.914308i	$-0.367264\pi$
$947$	2.11705e10	0.810040	0.405020	+	0.914308i	$0.367264\pi$
$948$	0	0
$949$	1.35475e9	0.0514550
$950$	1.08717e10	0.411399
$951$	0	0
$952$	−7.64994e9	−0.287362
$953$	−2.14876e10	−0.804196	−0.402098	−	0.915597i	$-0.631719\pi$
$953$	−2.14876e10	−0.804196	−0.402098	+	0.915597i	$0.631719\pi$
$954$	0	0
$955$	−1.97371e10	−0.733282
$956$	−2.53574e10	−0.938646
$957$	0	0
$958$	−1.04075e10	−0.382442
$959$	3.20704e8	0.0117419
$960$	0	0
$961$	2.42673e10	0.882043
$962$	1.77478e9	0.0642733
$963$	0	0
$964$	−1.64228e10	−0.590443
$965$	−7.39086e9	−0.264758
$966$	0	0
$967$	3.92625e10	1.39632	0.698161	−	0.715941i	$-0.254002\pi$
$967$	3.92625e10	1.39632	0.698161	+	0.715941i	$0.254002\pi$
$968$	−9.36689e9	−0.331919
$969$	0	0
$970$	1.45860e10	0.513139
$971$	5.62647e10	1.97228	0.986140	−	0.165917i	$-0.0530585\pi$
$971$	5.62647e10	1.97228	0.986140	+	0.165917i	$0.0530585\pi$
$972$	0	0
$973$	3.98310e10	1.38620
$974$	−8.59573e9	−0.298076
$975$	0	0
$976$	3.39027e9	0.116724
$977$	8.43437e9	0.289349	0.144674	−	0.989479i	$-0.453787\pi$
$977$	8.43437e9	0.289349	0.144674	+	0.989479i	$0.453787\pi$
$978$	0	0
$979$	−1.30507e10	−0.444523
$980$	2.80510e9	0.0952045
$981$	0	0
$982$	6.26675e9	0.211180
$983$	2.24230e10	0.752932	0.376466	−	0.926430i	$-0.377139\pi$
$983$	2.24230e10	0.752932	0.376466	+	0.926430i	$0.377139\pi$
$984$	0	0
$985$	−2.16269e10	−0.721054
$986$	−1.20672e10	−0.400900
$987$	0	0
$988$	−3.53261e9	−0.116532
$989$	4.33400e10	1.42463
$990$	0	0
$991$	3.46728e10	1.13170	0.565849	−	0.824509i	$-0.308548\pi$
$991$	3.46728e10	1.13170	0.565849	+	0.824509i	$0.308548\pi$
$992$	7.45642e9	0.242516
$993$	0	0
$994$	1.14989e10	0.371368
$995$	1.75639e10	0.565249
$996$	0	0
$997$	−2.96474e10	−0.947444	−0.473722	−	0.880674i	$-0.657090\pi$
$997$	−2.96474e10	−0.947444	−0.473722	+	0.880674i	$0.657090\pi$
$998$	−4.98550e9	−0.158764
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.8.a.b.1.1		1
3.2	odd	2		2.8.a.a.1.1	✓	1
4.3	odd	2		144.8.a.i.1.1		1
5.2	odd	4		450.8.c.g.199.2		2
5.3	odd	4		450.8.c.g.199.1		2
5.4	even	2		450.8.a.c.1.1		1
8.3	odd	2		576.8.a.f.1.1		1
8.5	even	2		576.8.a.g.1.1		1
9.2	odd	6		162.8.c.l.109.1		2
9.4	even	3		162.8.c.a.55.1		2
9.5	odd	6		162.8.c.l.55.1		2
9.7	even	3		162.8.c.a.109.1		2
12.11	even	2		16.8.a.b.1.1		1
15.2	even	4		50.8.b.c.49.1		2
15.8	even	4		50.8.b.c.49.2		2
15.14	odd	2		50.8.a.g.1.1		1
21.2	odd	6		98.8.c.d.67.1		2
21.5	even	6		98.8.c.e.67.1		2
21.11	odd	6		98.8.c.d.79.1		2
21.17	even	6		98.8.c.e.79.1		2
21.20	even	2		98.8.a.a.1.1		1
24.5	odd	2		64.8.a.c.1.1		1
24.11	even	2		64.8.a.e.1.1		1
33.32	even	2		242.8.a.e.1.1		1
39.5	even	4		338.8.b.d.337.2		2
39.8	even	4		338.8.b.d.337.1		2
39.38	odd	2		338.8.a.d.1.1		1
48.5	odd	4		256.8.b.b.129.1		2
48.11	even	4		256.8.b.f.129.2		2
48.29	odd	4		256.8.b.b.129.2		2
48.35	even	4		256.8.b.f.129.1		2
51.50	odd	2		578.8.a.b.1.1		1
60.23	odd	4		400.8.c.j.49.1		2
60.47	odd	4		400.8.c.j.49.2		2
60.59	even	2		400.8.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.8.a.a.1.1	✓	1	3.2	odd	2
16.8.a.b.1.1		1	12.11	even	2
18.8.a.b.1.1		1	1.1	even	1	trivial
50.8.a.g.1.1		1	15.14	odd	2
50.8.b.c.49.1		2	15.2	even	4
50.8.b.c.49.2		2	15.8	even	4
64.8.a.c.1.1		1	24.5	odd	2
64.8.a.e.1.1		1	24.11	even	2
98.8.a.a.1.1		1	21.20	even	2
98.8.c.d.67.1		2	21.2	odd	6
98.8.c.d.79.1		2	21.11	odd	6
98.8.c.e.67.1		2	21.5	even	6
98.8.c.e.79.1		2	21.17	even	6
144.8.a.i.1.1		1	4.3	odd	2
162.8.c.a.55.1		2	9.4	even	3
162.8.c.a.109.1		2	9.7	even	3
162.8.c.l.55.1		2	9.5	odd	6
162.8.c.l.109.1		2	9.2	odd	6
242.8.a.e.1.1		1	33.32	even	2
256.8.b.b.129.1		2	48.5	odd	4
256.8.b.b.129.2		2	48.29	odd	4
256.8.b.f.129.1		2	48.35	even	4
256.8.b.f.129.2		2	48.11	even	4
338.8.a.d.1.1		1	39.38	odd	2
338.8.b.d.337.1		2	39.8	even	4
338.8.b.d.337.2		2	39.5	even	4
400.8.a.l.1.1		1	60.59	even	2
400.8.c.j.49.1		2	60.23	odd	4
400.8.c.j.49.2		2	60.47	odd	4
450.8.a.c.1.1		1	5.4	even	2
450.8.c.g.199.1		2	5.3	odd	4
450.8.c.g.199.2		2	5.2	odd	4
576.8.a.f.1.1		1	8.3	odd	2
576.8.a.g.1.1		1	8.5	even	2
578.8.a.b.1.1		1	51.50	odd	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:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more