LMFDB - Embedded newform 198.6.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [198,6,Mod(1,198)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(198, 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("198.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 198 = 2 \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	198.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:	$31.7559963230$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	198.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} +51.0000 q^{5} -166.000 q^{7} +64.0000 q^{8} +O(q^{10})$

\(q+4.00000 q^{2} +16.0000 q^{4} +51.0000 q^{5} -166.000 q^{7} +64.0000 q^{8} +204.000 q^{10} +121.000 q^{11} +692.000 q^{13} -664.000 q^{14} +256.000 q^{16} +738.000 q^{17} +1424.00 q^{19} +816.000 q^{20} +484.000 q^{22} +1779.00 q^{23} -524.000 q^{25} +2768.00 q^{26} -2656.00 q^{28} +2064.00 q^{29} +6245.00 q^{31} +1024.00 q^{32} +2952.00 q^{34} -8466.00 q^{35} -14785.0 q^{37} +5696.00 q^{38} +3264.00 q^{40} -5304.00 q^{41} +17798.0 q^{43} +1936.00 q^{44} +7116.00 q^{46} +17184.0 q^{47} +10749.0 q^{49} -2096.00 q^{50} +11072.0 q^{52} +30726.0 q^{53} +6171.00 q^{55} -10624.0 q^{56} +8256.00 q^{58} +34989.0 q^{59} -45940.0 q^{61} +24980.0 q^{62} +4096.00 q^{64} +35292.0 q^{65} +25343.0 q^{67} +11808.0 q^{68} -33864.0 q^{70} -13311.0 q^{71} -53260.0 q^{73} -59140.0 q^{74} +22784.0 q^{76} -20086.0 q^{77} +77234.0 q^{79} +13056.0 q^{80} -21216.0 q^{82} -55014.0 q^{83} +37638.0 q^{85} +71192.0 q^{86} +7744.00 q^{88} -125415. q^{89} -114872. q^{91} +28464.0 q^{92} +68736.0 q^{94} +72624.0 q^{95} -88807.0 q^{97} +42996.0 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	51.0000	0.912316	0.456158	−	0.889899i	$-0.349225\pi$
$5$	51.0000	0.912316	0.456158	+	0.889899i	$0.349225\pi$
$6$	0	0
$7$	−166.000	−1.28045	−0.640226	−	0.768187i	$-0.721159\pi$
$7$	−166.000	−1.28045	−0.640226	+	0.768187i	$0.721159\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	204.000	0.645105
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	692.000	1.13566	0.567829	−	0.823146i	$-0.307783\pi$
$13$	692.000	1.13566	0.567829	+	0.823146i	$0.307783\pi$
$14$	−664.000	−0.905416
$15$	0	0
$16$	256.000	0.250000
$17$	738.000	0.619347	0.309674	−	0.950843i	$-0.399780\pi$
$17$	738.000	0.619347	0.309674	+	0.950843i	$0.399780\pi$
$18$	0	0
$19$	1424.00	0.904953	0.452476	−	0.891776i	$-0.350541\pi$
$19$	1424.00	0.904953	0.452476	+	0.891776i	$0.350541\pi$
$20$	816.000	0.456158
$21$	0	0
$22$	484.000	0.213201
$23$	1779.00	0.701223	0.350612	−	0.936521i	$-0.385974\pi$
$23$	1779.00	0.701223	0.350612	+	0.936521i	$0.385974\pi$
$24$	0	0
$25$	−524.000	−0.167680
$26$	2768.00	0.803032
$27$	0	0
$28$	−2656.00	−0.640226
$29$	2064.00	0.455737	0.227869	−	0.973692i	$-0.426824\pi$
$29$	2064.00	0.455737	0.227869	+	0.973692i	$0.426824\pi$
$30$	0	0
$31$	6245.00	1.16715	0.583577	−	0.812058i	$-0.301653\pi$
$31$	6245.00	1.16715	0.583577	+	0.812058i	$0.301653\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	2952.00	0.437944
$35$	−8466.00	−1.16818
$36$	0	0
$37$	−14785.0	−1.77549	−0.887743	−	0.460340i	$-0.847727\pi$
$37$	−14785.0	−1.77549	−0.887743	+	0.460340i	$0.847727\pi$
$38$	5696.00	0.639898
$39$	0	0
$40$	3264.00	0.322552
$41$	−5304.00	−0.492770	−0.246385	−	0.969172i	$-0.579243\pi$
$41$	−5304.00	−0.492770	−0.246385	+	0.969172i	$0.579243\pi$
$42$	0	0
$43$	17798.0	1.46791	0.733956	−	0.679197i	$-0.237672\pi$
$43$	17798.0	1.46791	0.733956	+	0.679197i	$0.237672\pi$
$44$	1936.00	0.150756
$45$	0	0
$46$	7116.00	0.495840
$47$	17184.0	1.13470	0.567348	−	0.823478i	$-0.307969\pi$
$47$	17184.0	1.13470	0.567348	+	0.823478i	$0.307969\pi$
$48$	0	0
$49$	10749.0	0.639555
$50$	−2096.00	−0.118568
$51$	0	0
$52$	11072.0	0.567829
$53$	30726.0	1.50251	0.751253	−	0.660014i	$-0.229450\pi$
$53$	30726.0	1.50251	0.751253	+	0.660014i	$0.229450\pi$
$54$	0	0
$55$	6171.00	0.275074
$56$	−10624.0	−0.452708
$57$	0	0
$58$	8256.00	0.322255
$59$	34989.0	1.30858	0.654292	−	0.756242i	$-0.272967\pi$
$59$	34989.0	1.30858	0.654292	+	0.756242i	$0.272967\pi$
$60$	0	0
$61$	−45940.0	−1.58076	−0.790381	−	0.612616i	$-0.790117\pi$
$61$	−45940.0	−1.58076	−0.790381	+	0.612616i	$0.790117\pi$
$62$	24980.0	0.825303
$63$	0	0
$64$	4096.00	0.125000
$65$	35292.0	1.03608
$66$	0	0
$67$	25343.0	0.689717	0.344859	−	0.938655i	$-0.387927\pi$
$67$	25343.0	0.689717	0.344859	+	0.938655i	$0.387927\pi$
$68$	11808.0	0.309674
$69$	0	0
$70$	−33864.0	−0.826025
$71$	−13311.0	−0.313375	−0.156688	−	0.987648i	$-0.550082\pi$
$71$	−13311.0	−0.313375	−0.156688	+	0.987648i	$0.550082\pi$
$72$	0	0
$73$	−53260.0	−1.16975	−0.584876	−	0.811123i	$-0.698857\pi$
$73$	−53260.0	−1.16975	−0.584876	+	0.811123i	$0.698857\pi$
$74$	−59140.0	−1.25546
$75$	0	0
$76$	22784.0	0.452476
$77$	−20086.0	−0.386071
$78$	0	0
$79$	77234.0	1.39233	0.696163	−	0.717884i	$-0.254889\pi$
$79$	77234.0	1.39233	0.696163	+	0.717884i	$0.254889\pi$
$80$	13056.0	0.228079
$81$	0	0
$82$	−21216.0	−0.348441
$83$	−55014.0	−0.876553	−0.438276	−	0.898840i	$-0.644411\pi$
$83$	−55014.0	−0.876553	−0.438276	+	0.898840i	$0.644411\pi$
$84$	0	0
$85$	37638.0	0.565040
$86$	71192.0	1.03797
$87$	0	0
$88$	7744.00	0.106600
$89$	−125415.	−1.67832	−0.839159	−	0.543886i	$-0.816953\pi$
$89$	−125415.	−1.67832	−0.839159	+	0.543886i	$0.816953\pi$
$90$	0	0
$91$	−114872.	−1.45416
$92$	28464.0	0.350612
$93$	0	0
$94$	68736.0	0.802351
$95$	72624.0	0.825603
$96$	0	0
$97$	−88807.0	−0.958336	−0.479168	−	0.877723i	$-0.659062\pi$
$97$	−88807.0	−0.958336	−0.479168	+	0.877723i	$0.659062\pi$
$98$	42996.0	0.452234
$99$	0	0
$100$	−8384.00	−0.0838400
$101$	−1482.00	−0.0144559	−0.00722794	−	0.999974i	$-0.502301\pi$
$101$	−1482.00	−0.0144559	−0.00722794	+	0.999974i	$0.502301\pi$
$102$	0	0
$103$	−117496.	−1.09126	−0.545632	−	0.838025i	$-0.683710\pi$
$103$	−117496.	−1.09126	−0.545632	+	0.838025i	$0.683710\pi$
$104$	44288.0	0.401516
$105$	0	0
$106$	122904.	1.06243
$107$	79362.0	0.670121	0.335060	−	0.942197i	$-0.391243\pi$
$107$	79362.0	0.670121	0.335060	+	0.942197i	$0.391243\pi$
$108$	0	0
$109$	87842.0	0.708167	0.354084	−	0.935214i	$-0.384793\pi$
$109$	87842.0	0.708167	0.354084	+	0.935214i	$0.384793\pi$
$110$	24684.0	0.194506
$111$	0	0
$112$	−42496.0	−0.320113
$113$	47247.0	0.348079	0.174040	−	0.984739i	$-0.444318\pi$
$113$	47247.0	0.348079	0.174040	+	0.984739i	$0.444318\pi$
$114$	0	0
$115$	90729.0	0.639737
$116$	33024.0	0.227869
$117$	0	0
$118$	139956.	0.925308
$119$	−122508.	−0.793044
$120$	0	0
$121$	14641.0	0.0909091
$122$	−183760.	−1.11777
$123$	0	0
$124$	99920.0	0.583577
$125$	−186099.	−1.06529
$126$	0	0
$127$	−239416.	−1.31718	−0.658588	−	0.752504i	$-0.728846\pi$
$127$	−239416.	−1.31718	−0.658588	+	0.752504i	$0.728846\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	141168.	0.732618
$131$	98142.0	0.499662	0.249831	−	0.968289i	$-0.419625\pi$
$131$	98142.0	0.499662	0.249831	+	0.968289i	$0.419625\pi$
$132$	0	0
$133$	−236384.	−1.15875
$134$	101372.	0.487704
$135$	0	0
$136$	47232.0	0.218972
$137$	−400137.	−1.82141	−0.910704	−	0.413059i	$-0.864460\pi$
$137$	−400137.	−1.82141	−0.910704	+	0.413059i	$0.864460\pi$
$138$	0	0
$139$	205766.	0.903310	0.451655	−	0.892193i	$-0.350834\pi$
$139$	205766.	0.903310	0.451655	+	0.892193i	$0.350834\pi$
$140$	−135456.	−0.584088
$141$	0	0
$142$	−53244.0	−0.221590
$143$	83732.0	0.342414
$144$	0	0
$145$	105264.	0.415776
$146$	−213040.	−0.827140
$147$	0	0
$148$	−236560.	−0.887743
$149$	−87726.0	−0.323715	−0.161857	−	0.986814i	$-0.551748\pi$
$149$	−87726.0	−0.323715	−0.161857	+	0.986814i	$0.551748\pi$
$150$	0	0
$151$	−432778.	−1.54462	−0.772312	−	0.635243i	$-0.780900\pi$
$151$	−432778.	−1.54462	−0.772312	+	0.635243i	$0.780900\pi$
$152$	91136.0	0.319949
$153$	0	0
$154$	−80344.0	−0.272993
$155$	318495.	1.06481
$156$	0	0
$157$	−34075.0	−0.110328	−0.0551641	−	0.998477i	$-0.517568\pi$
$157$	−34075.0	−0.110328	−0.0551641	+	0.998477i	$0.517568\pi$
$158$	308936.	0.984523
$159$	0	0
$160$	52224.0	0.161276
$161$	−295314.	−0.897882
$162$	0	0
$163$	45020.0	0.132720	0.0663600	−	0.997796i	$-0.478861\pi$
$163$	45020.0	0.132720	0.0663600	+	0.997796i	$0.478861\pi$
$164$	−84864.0	−0.246385
$165$	0	0
$166$	−220056.	−0.619816
$167$	−482556.	−1.33893	−0.669463	−	0.742845i	$-0.733476\pi$
$167$	−482556.	−1.33893	−0.669463	+	0.742845i	$0.733476\pi$
$168$	0	0
$169$	107571.	0.289720
$170$	150552.	0.399544
$171$	0	0
$172$	284768.	0.733956
$173$	766254.	1.94651	0.973257	−	0.229719i	$-0.0737808\pi$
$173$	766254.	1.94651	0.973257	+	0.229719i	$0.0737808\pi$
$174$	0	0
$175$	86984.0	0.214706
$176$	30976.0	0.0753778
$177$	0	0
$178$	−501660.	−1.18675
$179$	−303399.	−0.707753	−0.353876	−	0.935292i	$-0.615137\pi$
$179$	−303399.	−0.707753	−0.353876	+	0.935292i	$0.615137\pi$
$180$	0	0
$181$	−285181.	−0.647030	−0.323515	−	0.946223i	$-0.604865\pi$
$181$	−285181.	−0.647030	−0.323515	+	0.946223i	$0.604865\pi$
$182$	−459488.	−1.02824
$183$	0	0
$184$	113856.	0.247920
$185$	−754035.	−1.61980
$186$	0	0
$187$	89298.0	0.186740
$188$	274944.	0.567348
$189$	0	0
$190$	290496.	0.583789
$191$	−767067.	−1.52142	−0.760711	−	0.649090i	$-0.775150\pi$
$191$	−767067.	−1.52142	−0.760711	+	0.649090i	$0.775150\pi$
$192$	0	0
$193$	411668.	0.795525	0.397763	−	0.917488i	$-0.369787\pi$
$193$	411668.	0.795525	0.397763	+	0.917488i	$0.369787\pi$
$194$	−355228.	−0.677646
$195$	0	0
$196$	171984.	0.319777
$197$	759258.	1.39387	0.696937	−	0.717132i	$-0.254545\pi$
$197$	759258.	1.39387	0.696937	+	0.717132i	$0.254545\pi$
$198$	0	0
$199$	−46600.0	−0.0834167	−0.0417084	−	0.999130i	$-0.513280\pi$
$199$	−46600.0	−0.0834167	−0.0417084	+	0.999130i	$0.513280\pi$
$200$	−33536.0	−0.0592838
$201$	0	0
$202$	−5928.00	−0.0102219
$203$	−342624.	−0.583549
$204$	0	0
$205$	−270504.	−0.449561
$206$	−469984.	−0.771641
$207$	0	0
$208$	177152.	0.283915
$209$	172304.	0.272854
$210$	0	0
$211$	−932428.	−1.44181	−0.720907	−	0.693032i	$-0.756274\pi$
$211$	−932428.	−1.44181	−0.720907	+	0.693032i	$0.756274\pi$
$212$	491616.	0.751253
$213$	0	0
$214$	317448.	0.473847
$215$	907698.	1.33920
$216$	0	0
$217$	−1.03667e6	−1.49448
$218$	351368.	0.500750
$219$	0	0
$220$	98736.0	0.137537
$221$	510696.	0.703367
$222$	0	0
$223$	169745.	0.228578	0.114289	−	0.993448i	$-0.463541\pi$
$223$	169745.	0.228578	0.114289	+	0.993448i	$0.463541\pi$
$224$	−169984.	−0.226354
$225$	0	0
$226$	188988.	0.246129
$227$	−198078.	−0.255136	−0.127568	−	0.991830i	$-0.540717\pi$
$227$	−198078.	−0.255136	−0.127568	+	0.991830i	$0.540717\pi$
$228$	0	0
$229$	−849997.	−1.07110	−0.535548	−	0.844505i	$-0.679895\pi$
$229$	−849997.	−1.07110	−0.535548	+	0.844505i	$0.679895\pi$
$230$	362916.	0.452362
$231$	0	0
$232$	132096.	0.161128
$233$	401832.	0.484903	0.242451	−	0.970164i	$-0.422048\pi$
$233$	401832.	0.484903	0.242451	+	0.970164i	$0.422048\pi$
$234$	0	0
$235$	876384.	1.03520
$236$	559824.	0.654292
$237$	0	0
$238$	−490032.	−0.560766
$239$	−855174.	−0.968411	−0.484206	−	0.874954i	$-0.660891\pi$
$239$	−855174.	−0.968411	−0.484206	+	0.874954i	$0.660891\pi$
$240$	0	0
$241$	1.12546e6	1.24821	0.624107	−	0.781339i	$-0.285463\pi$
$241$	1.12546e6	1.24821	0.624107	+	0.781339i	$0.285463\pi$
$242$	58564.0	0.0642824
$243$	0	0
$244$	−735040.	−0.790381
$245$	548199.	0.583476
$246$	0	0
$247$	985408.	1.02772
$248$	399680.	0.412651
$249$	0	0
$250$	−744396.	−0.753276
$251$	1.19751e6	1.19976	0.599882	−	0.800088i	$-0.295214\pi$
$251$	1.19751e6	1.19976	0.599882	+	0.800088i	$0.295214\pi$
$252$	0	0
$253$	215259.	0.211427
$254$	−957664.	−0.931384
$255$	0	0
$256$	65536.0	0.0625000
$257$	−37758.0	−0.0356596	−0.0178298	−	0.999841i	$-0.505676\pi$
$257$	−37758.0	−0.0356596	−0.0178298	+	0.999841i	$0.505676\pi$
$258$	0	0
$259$	2.45431e6	2.27342
$260$	564672.	0.518040
$261$	0	0
$262$	392568.	0.353315
$263$	631254.	0.562749	0.281375	−	0.959598i	$-0.409210\pi$
$263$	631254.	0.562749	0.281375	+	0.959598i	$0.409210\pi$
$264$	0	0
$265$	1.56703e6	1.37076
$266$	−945536.	−0.819359
$267$	0	0
$268$	405488.	0.344859
$269$	1.08034e6	0.910292	0.455146	−	0.890417i	$-0.349587\pi$
$269$	1.08034e6	0.910292	0.455146	+	0.890417i	$0.349587\pi$
$270$	0	0
$271$	−816100.	−0.675025	−0.337513	−	0.941321i	$-0.609586\pi$
$271$	−816100.	−0.675025	−0.337513	+	0.941321i	$0.609586\pi$
$272$	188928.	0.154837
$273$	0	0
$274$	−1.60055e6	−1.28793
$275$	−63404.0	−0.0505574
$276$	0	0
$277$	1.68820e6	1.32198	0.660989	−	0.750396i	$-0.270137\pi$
$277$	1.68820e6	1.32198	0.660989	+	0.750396i	$0.270137\pi$
$278$	823064.	0.638736
$279$	0	0
$280$	−541824.	−0.413012
$281$	879042.	0.664116	0.332058	−	0.943259i	$-0.392257\pi$
$281$	879042.	0.664116	0.332058	+	0.943259i	$0.392257\pi$
$282$	0	0
$283$	1.54027e6	1.14322	0.571611	−	0.820525i	$-0.306319\pi$
$283$	1.54027e6	1.14322	0.571611	+	0.820525i	$0.306319\pi$
$284$	−212976.	−0.156688
$285$	0	0
$286$	334928.	0.242123
$287$	880464.	0.630967
$288$	0	0
$289$	−875213.	−0.616409
$290$	421056.	0.293998
$291$	0	0
$292$	−852160.	−0.584876
$293$	−720840.	−0.490535	−0.245267	−	0.969455i	$-0.578876\pi$
$293$	−720840.	−0.490535	−0.245267	+	0.969455i	$0.578876\pi$
$294$	0	0
$295$	1.78444e6	1.19384
$296$	−946240.	−0.627729
$297$	0	0
$298$	−350904.	−0.228901
$299$	1.23107e6	0.796350
$300$	0	0
$301$	−2.95447e6	−1.87959
$302$	−1.73111e6	−1.09221
$303$	0	0
$304$	364544.	0.226238
$305$	−2.34294e6	−1.44215
$306$	0	0
$307$	−1.03905e6	−0.629201	−0.314601	−	0.949224i	$-0.601871\pi$
$307$	−1.03905e6	−0.629201	−0.314601	+	0.949224i	$0.601871\pi$
$308$	−321376.	−0.193035
$309$	0	0
$310$	1.27398e6	0.752937
$311$	1.25135e6	0.733630	0.366815	−	0.930294i	$-0.380448\pi$
$311$	1.25135e6	0.733630	0.366815	+	0.930294i	$0.380448\pi$
$312$	0	0
$313$	−1.44336e6	−0.832749	−0.416375	−	0.909193i	$-0.636699\pi$
$313$	−1.44336e6	−0.832749	−0.416375	+	0.909193i	$0.636699\pi$
$314$	−136300.	−0.0780139
$315$	0	0
$316$	1.23574e6	0.696163
$317$	2.01208e6	1.12460	0.562298	−	0.826934i	$-0.309917\pi$
$317$	2.01208e6	1.12460	0.562298	+	0.826934i	$0.309917\pi$
$318$	0	0
$319$	249744.	0.137410
$320$	208896.	0.114039
$321$	0	0
$322$	−1.18126e6	−0.634899
$323$	1.05091e6	0.560480
$324$	0	0
$325$	−362608.	−0.190427
$326$	180080.	0.0938472
$327$	0	0
$328$	−339456.	−0.174220
$329$	−2.85254e6	−1.45292
$330$	0	0
$331$	2.01734e6	1.01207	0.506033	−	0.862514i	$-0.331112\pi$
$331$	2.01734e6	1.01207	0.506033	+	0.862514i	$0.331112\pi$
$332$	−880224.	−0.438276
$333$	0	0
$334$	−1.93022e6	−0.946764
$335$	1.29249e6	0.629240
$336$	0	0
$337$	264122.	0.126686	0.0633432	−	0.997992i	$-0.479824\pi$
$337$	264122.	0.126686	0.0633432	+	0.997992i	$0.479824\pi$
$338$	430284.	0.204863
$339$	0	0
$340$	602208.	0.282520
$341$	755645.	0.351910
$342$	0	0
$343$	1.00563e6	0.461532
$344$	1.13907e6	0.518985
$345$	0	0
$346$	3.06502e6	1.37639
$347$	1.71049e6	0.762601	0.381300	−	0.924451i	$-0.375476\pi$
$347$	1.71049e6	0.762601	0.381300	+	0.924451i	$0.375476\pi$
$348$	0	0
$349$	218822.	0.0961673	0.0480836	−	0.998843i	$-0.484689\pi$
$349$	218822.	0.0961673	0.0480836	+	0.998843i	$0.484689\pi$
$350$	347936.	0.151820
$351$	0	0
$352$	123904.	0.0533002
$353$	−3.68192e6	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$353$	−3.68192e6	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$354$	0	0
$355$	−678861.	−0.285897
$356$	−2.00664e6	−0.839159
$357$	0	0
$358$	−1.21360e6	−0.500457
$359$	−1.88528e6	−0.772042	−0.386021	−	0.922490i	$-0.626151\pi$
$359$	−1.88528e6	−0.772042	−0.386021	+	0.922490i	$0.626151\pi$
$360$	0	0
$361$	−448323.	−0.181060
$362$	−1.14072e6	−0.457519
$363$	0	0
$364$	−1.83795e6	−0.727078
$365$	−2.71626e6	−1.06718
$366$	0	0
$367$	−3.11666e6	−1.20788	−0.603940	−	0.797029i	$-0.706404\pi$
$367$	−3.11666e6	−1.20788	−0.603940	+	0.797029i	$0.706404\pi$
$368$	455424.	0.175306
$369$	0	0
$370$	−3.01614e6	−1.14537
$371$	−5.10052e6	−1.92389
$372$	0	0
$373$	1.39441e6	0.518943	0.259471	−	0.965751i	$-0.416452\pi$
$373$	1.39441e6	0.518943	0.259471	+	0.965751i	$0.416452\pi$
$374$	357192.	0.132045
$375$	0	0
$376$	1.09978e6	0.401176
$377$	1.42829e6	0.517562
$378$	0	0
$379$	−4.26036e6	−1.52352	−0.761759	−	0.647860i	$-0.775664\pi$
$379$	−4.26036e6	−1.52352	−0.761759	+	0.647860i	$0.775664\pi$
$380$	1.16198e6	0.412801
$381$	0	0
$382$	−3.06827e6	−1.07581
$383$	−201765.	−0.0702828	−0.0351414	−	0.999382i	$-0.511188\pi$
$383$	−201765.	−0.0702828	−0.0351414	+	0.999382i	$0.511188\pi$
$384$	0	0
$385$	−1.02439e6	−0.352218
$386$	1.64667e6	0.562521
$387$	0	0
$388$	−1.42091e6	−0.479168
$389$	−1.94882e6	−0.652977	−0.326489	−	0.945201i	$-0.605865\pi$
$389$	−1.94882e6	−0.652977	−0.326489	+	0.945201i	$0.605865\pi$
$390$	0	0
$391$	1.31290e6	0.434301
$392$	687936.	0.226117
$393$	0	0
$394$	3.03703e6	0.985618
$395$	3.93893e6	1.27024
$396$	0	0
$397$	−1.46826e6	−0.467548	−0.233774	−	0.972291i	$-0.575108\pi$
$397$	−1.46826e6	−0.467548	−0.233774	+	0.972291i	$0.575108\pi$
$398$	−186400.	−0.0589845
$399$	0	0
$400$	−134144.	−0.0419200
$401$	−2.24618e6	−0.697563	−0.348781	−	0.937204i	$-0.613404\pi$
$401$	−2.24618e6	−0.697563	−0.348781	+	0.937204i	$0.613404\pi$
$402$	0	0
$403$	4.32154e6	1.32549
$404$	−23712.0	−0.00722794
$405$	0	0
$406$	−1.37050e6	−0.412632
$407$	−1.78898e6	−0.535329
$408$	0	0
$409$	−3.61488e6	−1.06853	−0.534263	−	0.845318i	$-0.679411\pi$
$409$	−3.61488e6	−1.06853	−0.534263	+	0.845318i	$0.679411\pi$
$410$	−1.08202e6	−0.317888
$411$	0	0
$412$	−1.87994e6	−0.545632
$413$	−5.80817e6	−1.67558
$414$	0	0
$415$	−2.80571e6	−0.799693
$416$	708608.	0.200758
$417$	0	0
$418$	689216.	0.192937
$419$	3.81239e6	1.06087	0.530435	−	0.847726i	$-0.322029\pi$
$419$	3.81239e6	1.06087	0.530435	+	0.847726i	$0.322029\pi$
$420$	0	0
$421$	1.97346e6	0.542655	0.271327	−	0.962487i	$-0.412537\pi$
$421$	1.97346e6	0.542655	0.271327	+	0.962487i	$0.412537\pi$
$422$	−3.72971e6	−1.01952
$423$	0	0
$424$	1.96646e6	0.531216
$425$	−386712.	−0.103852
$426$	0	0
$427$	7.62604e6	2.02409
$428$	1.26979e6	0.335060
$429$	0	0
$430$	3.63079e6	0.946957
$431$	2.08359e6	0.540280	0.270140	−	0.962821i	$-0.412930\pi$
$431$	2.08359e6	0.540280	0.270140	+	0.962821i	$0.412930\pi$
$432$	0	0
$433$	−72691.0	−0.0186321	−0.00931603	−	0.999957i	$-0.502965\pi$
$433$	−72691.0	−0.0186321	−0.00931603	+	0.999957i	$0.502965\pi$
$434$	−4.14668e6	−1.05676
$435$	0	0
$436$	1.40547e6	0.354084
$437$	2.53330e6	0.634574
$438$	0	0
$439$	594392.	0.147201	0.0736007	−	0.997288i	$-0.476551\pi$
$439$	594392.	0.147201	0.0736007	+	0.997288i	$0.476551\pi$
$440$	394944.	0.0972532
$441$	0	0
$442$	2.04278e6	0.497355
$443$	−4.56651e6	−1.10554	−0.552770	−	0.833334i	$-0.686429\pi$
$443$	−4.56651e6	−1.10554	−0.552770	+	0.833334i	$0.686429\pi$
$444$	0	0
$445$	−6.39616e6	−1.53116
$446$	678980.	0.161629
$447$	0	0
$448$	−679936.	−0.160056
$449$	5.44382e6	1.27435	0.637174	−	0.770720i	$-0.280103\pi$
$449$	5.44382e6	1.27435	0.637174	+	0.770720i	$0.280103\pi$
$450$	0	0
$451$	−641784.	−0.148576
$452$	755952.	0.174040
$453$	0	0
$454$	−792312.	−0.180408
$455$	−5.85847e6	−1.32665
$456$	0	0
$457$	6.70312e6	1.50137	0.750683	−	0.660662i	$-0.229724\pi$
$457$	6.70312e6	1.50137	0.750683	+	0.660662i	$0.229724\pi$
$458$	−3.39999e6	−0.757380
$459$	0	0
$460$	1.45166e6	0.319869
$461$	1.25994e6	0.276120	0.138060	−	0.990424i	$-0.455913\pi$
$461$	1.25994e6	0.276120	0.138060	+	0.990424i	$0.455913\pi$
$462$	0	0
$463$	−5.02308e6	−1.08897	−0.544487	−	0.838769i	$-0.683276\pi$
$463$	−5.02308e6	−1.08897	−0.544487	+	0.838769i	$0.683276\pi$
$464$	528384.	0.113934
$465$	0	0
$466$	1.60733e6	0.342878
$467$	2.35660e6	0.500028	0.250014	−	0.968242i	$-0.419565\pi$
$467$	2.35660e6	0.500028	0.250014	+	0.968242i	$0.419565\pi$
$468$	0	0
$469$	−4.20694e6	−0.883149
$470$	3.50554e6	0.731998
$471$	0	0
$472$	2.23930e6	0.462654
$473$	2.15356e6	0.442592
$474$	0	0
$475$	−746176.	−0.151743
$476$	−1.96013e6	−0.396522
$477$	0	0
$478$	−3.42070e6	−0.684770
$479$	6.72258e6	1.33874	0.669371	−	0.742928i	$-0.266563\pi$
$479$	6.72258e6	1.33874	0.669371	+	0.742928i	$0.266563\pi$
$480$	0	0
$481$	−1.02312e7	−2.01634
$482$	4.50186e6	0.882620
$483$	0	0
$484$	234256.	0.0454545
$485$	−4.52916e6	−0.874305
$486$	0	0
$487$	1.96001e6	0.374487	0.187243	−	0.982314i	$-0.440045\pi$
$487$	1.96001e6	0.374487	0.187243	+	0.982314i	$0.440045\pi$
$488$	−2.94016e6	−0.558884
$489$	0	0
$490$	2.19280e6	0.412580
$491$	579624.	0.108503	0.0542516	−	0.998527i	$-0.482723\pi$
$491$	579624.	0.108503	0.0542516	+	0.998527i	$0.482723\pi$
$492$	0	0
$493$	1.52323e6	0.282260
$494$	3.94163e6	0.726706
$495$	0	0
$496$	1.59872e6	0.291789
$497$	2.20963e6	0.401262
$498$	0	0
$499$	1.36905e6	0.246132	0.123066	−	0.992398i	$-0.460727\pi$
$499$	1.36905e6	0.246132	0.123066	+	0.992398i	$0.460727\pi$
$500$	−2.97758e6	−0.532646
$501$	0	0
$502$	4.79005e6	0.848361
$503$	−1.83343e6	−0.323105	−0.161552	−	0.986864i	$-0.551650\pi$
$503$	−1.83343e6	−0.323105	−0.161552	+	0.986864i	$0.551650\pi$
$504$	0	0
$505$	−75582.0	−0.0131883
$506$	861036.	0.149501
$507$	0	0
$508$	−3.83066e6	−0.658588
$509$	1.71266e6	0.293006	0.146503	−	0.989210i	$-0.453198\pi$
$509$	1.71266e6	0.293006	0.146503	+	0.989210i	$0.453198\pi$
$510$	0	0
$511$	8.84116e6	1.49781
$512$	262144.	0.0441942
$513$	0	0
$514$	−151032.	−0.0252151
$515$	−5.99230e6	−0.995578
$516$	0	0
$517$	2.07926e6	0.342124
$518$	9.81724e6	1.60755
$519$	0	0
$520$	2.25869e6	0.366309
$521$	789435.	0.127415	0.0637077	−	0.997969i	$-0.479707\pi$
$521$	789435.	0.127415	0.0637077	+	0.997969i	$0.479707\pi$
$522$	0	0
$523$	627392.	0.100296	0.0501481	−	0.998742i	$-0.484031\pi$
$523$	627392.	0.100296	0.0501481	+	0.998742i	$0.484031\pi$
$524$	1.57027e6	0.249831
$525$	0	0
$526$	2.52502e6	0.397924
$527$	4.60881e6	0.722873
$528$	0	0
$529$	−3.27150e6	−0.508286
$530$	6.26810e6	0.969274
$531$	0	0
$532$	−3.78214e6	−0.579374
$533$	−3.67037e6	−0.559618
$534$	0	0
$535$	4.04746e6	0.611362
$536$	1.62195e6	0.243852
$537$	0	0
$538$	4.32137e6	0.643673
$539$	1.30063e6	0.192833
$540$	0	0
$541$	3.20895e6	0.471379	0.235689	−	0.971828i	$-0.424265\pi$
$541$	3.20895e6	0.471379	0.235689	+	0.971828i	$0.424265\pi$
$542$	−3.26440e6	−0.477315
$543$	0	0
$544$	755712.	0.109486
$545$	4.47994e6	0.646072
$546$	0	0
$547$	3.42658e6	0.489658	0.244829	−	0.969566i	$-0.421268\pi$
$547$	3.42658e6	0.489658	0.244829	+	0.969566i	$0.421268\pi$
$548$	−6.40219e6	−0.910704
$549$	0	0
$550$	−253616.	−0.0357495
$551$	2.93914e6	0.412421
$552$	0	0
$553$	−1.28208e7	−1.78280
$554$	6.75279e6	0.934779
$555$	0	0
$556$	3.29226e6	0.451655
$557$	−1.05198e7	−1.43672	−0.718358	−	0.695674i	$-0.755106\pi$
$557$	−1.05198e7	−1.43672	−0.718358	+	0.695674i	$0.755106\pi$
$558$	0	0
$559$	1.23162e7	1.66705
$560$	−2.16730e6	−0.292044
$561$	0	0
$562$	3.51617e6	0.469601
$563$	−5.47288e6	−0.727687	−0.363844	−	0.931460i	$-0.618536\pi$
$563$	−5.47288e6	−0.727687	−0.363844	+	0.931460i	$0.618536\pi$
$564$	0	0
$565$	2.40960e6	0.317558
$566$	6.16107e6	0.808379
$567$	0	0
$568$	−851904.	−0.110795
$569$	1.17787e7	1.52516	0.762580	−	0.646893i	$-0.223932\pi$
$569$	1.17787e7	1.52516	0.762580	+	0.646893i	$0.223932\pi$
$570$	0	0
$571$	−8.35628e6	−1.07256	−0.536281	−	0.844039i	$-0.680171\pi$
$571$	−8.35628e6	−1.07256	−0.536281	+	0.844039i	$0.680171\pi$
$572$	1.33971e6	0.171207
$573$	0	0
$574$	3.52186e6	0.446161
$575$	−932196.	−0.117581
$576$	0	0
$577$	−1.37758e7	−1.72258	−0.861288	−	0.508117i	$-0.830342\pi$
$577$	−1.37758e7	−1.72258	−0.861288	+	0.508117i	$0.830342\pi$
$578$	−3.50085e6	−0.435867
$579$	0	0
$580$	1.68422e6	0.207888
$581$	9.13232e6	1.12238
$582$	0	0
$583$	3.71785e6	0.453023
$584$	−3.40864e6	−0.413570
$585$	0	0
$586$	−2.88336e6	−0.346860
$587$	1.27093e7	1.52239	0.761196	−	0.648522i	$-0.224612\pi$
$587$	1.27093e7	1.52239	0.761196	+	0.648522i	$0.224612\pi$
$588$	0	0
$589$	8.89288e6	1.05622
$590$	7.13776e6	0.844173
$591$	0	0
$592$	−3.78496e6	−0.443871
$593$	−1.00825e6	−0.117742	−0.0588711	−	0.998266i	$-0.518750\pi$
$593$	−1.00825e6	−0.117742	−0.0588711	+	0.998266i	$0.518750\pi$
$594$	0	0
$595$	−6.24791e6	−0.723506
$596$	−1.40362e6	−0.161857
$597$	0	0
$598$	4.92427e6	0.563105
$599$	−1.05100e7	−1.19684	−0.598421	−	0.801182i	$-0.704205\pi$
$599$	−1.05100e7	−1.19684	−0.598421	+	0.801182i	$0.704205\pi$
$600$	0	0
$601$	−199390.	−0.0225173	−0.0112587	−	0.999937i	$-0.503584\pi$
$601$	−199390.	−0.0225173	−0.0112587	+	0.999937i	$0.503584\pi$
$602$	−1.18179e7	−1.32907
$603$	0	0
$604$	−6.92445e6	−0.772312
$605$	746691.	0.0829378
$606$	0	0
$607$	16190.0	0.00178351	0.000891754	−	1.00000i	$-0.499716\pi$
$607$	16190.0	0.00178351	0.000891754		1.00000i	$0.499716\pi$
$608$	1.45818e6	0.159975
$609$	0	0
$610$	−9.37176e6	−1.01976
$611$	1.18913e7	1.28863
$612$	0	0
$613$	−1.15253e7	−1.23880	−0.619402	−	0.785074i	$-0.712625\pi$
$613$	−1.15253e7	−1.23880	−0.619402	+	0.785074i	$0.712625\pi$
$614$	−4.15619e6	−0.444913
$615$	0	0
$616$	−1.28550e6	−0.136497
$617$	−1.69974e7	−1.79750	−0.898751	−	0.438459i	$-0.855524\pi$
$617$	−1.69974e7	−1.79750	−0.898751	+	0.438459i	$0.855524\pi$
$618$	0	0
$619$	−1.84875e7	−1.93933	−0.969663	−	0.244445i	$-0.921394\pi$
$619$	−1.84875e7	−1.93933	−0.969663	+	0.244445i	$0.921394\pi$
$620$	5.09592e6	0.532407
$621$	0	0
$622$	5.00539e6	0.518755
$623$	2.08189e7	2.14901
$624$	0	0
$625$	−7.85355e6	−0.804203
$626$	−5.77344e6	−0.588842
$627$	0	0
$628$	−545200.	−0.0551641
$629$	−1.09113e7	−1.09964
$630$	0	0
$631$	−4.54281e6	−0.454204	−0.227102	−	0.973871i	$-0.572925\pi$
$631$	−4.54281e6	−0.454204	−0.227102	+	0.973871i	$0.572925\pi$
$632$	4.94298e6	0.492261
$633$	0	0
$634$	8.04832e6	0.795210
$635$	−1.22102e7	−1.20168
$636$	0	0
$637$	7.43831e6	0.726316
$638$	998976.	0.0971635
$639$	0	0
$640$	835584.	0.0806381
$641$	−1.84286e7	−1.77153	−0.885764	−	0.464136i	$-0.846365\pi$
$641$	−1.84286e7	−1.77153	−0.885764	+	0.464136i	$0.846365\pi$
$642$	0	0
$643$	9.66604e6	0.921979	0.460989	−	0.887406i	$-0.347495\pi$
$643$	9.66604e6	0.921979	0.460989	+	0.887406i	$0.347495\pi$
$644$	−4.72502e6	−0.448941
$645$	0	0
$646$	4.20365e6	0.396319
$647$	4.51430e6	0.423965	0.211982	−	0.977273i	$-0.432008\pi$
$647$	4.51430e6	0.423965	0.211982	+	0.977273i	$0.432008\pi$
$648$	0	0
$649$	4.23367e6	0.394553
$650$	−1.45043e6	−0.134652
$651$	0	0
$652$	720320.	0.0663600
$653$	5.37235e6	0.493039	0.246519	−	0.969138i	$-0.420713\pi$
$653$	5.37235e6	0.493039	0.246519	+	0.969138i	$0.420713\pi$
$654$	0	0
$655$	5.00524e6	0.455850
$656$	−1.35782e6	−0.123192
$657$	0	0
$658$	−1.14102e7	−1.02737
$659$	−9.87956e6	−0.886184	−0.443092	−	0.896476i	$-0.646119\pi$
$659$	−9.87956e6	−0.886184	−0.443092	+	0.896476i	$0.646119\pi$
$660$	0	0
$661$	1.08052e7	0.961898	0.480949	−	0.876748i	$-0.340292\pi$
$661$	1.08052e7	0.961898	0.480949	+	0.876748i	$0.340292\pi$
$662$	8.06935e6	0.715638
$663$	0	0
$664$	−3.52090e6	−0.309908
$665$	−1.20556e7	−1.05714
$666$	0	0
$667$	3.67186e6	0.319574
$668$	−7.72090e6	−0.669463
$669$	0	0
$670$	5.16997e6	0.444940
$671$	−5.55874e6	−0.476618
$672$	0	0
$673$	1.13275e7	0.964042	0.482021	−	0.876160i	$-0.339903\pi$
$673$	1.13275e7	0.964042	0.482021	+	0.876160i	$0.339903\pi$
$674$	1.05649e6	0.0895808
$675$	0	0
$676$	1.72114e6	0.144860
$677$	1.20595e7	1.01125	0.505624	−	0.862754i	$-0.331262\pi$
$677$	1.20595e7	1.01125	0.505624	+	0.862754i	$0.331262\pi$
$678$	0	0
$679$	1.47420e7	1.22710
$680$	2.40883e6	0.199772
$681$	0	0
$682$	3.02258e6	0.248838
$683$	5.14166e6	0.421747	0.210873	−	0.977513i	$-0.432369\pi$
$683$	5.14166e6	0.421747	0.210873	+	0.977513i	$0.432369\pi$
$684$	0	0
$685$	−2.04070e7	−1.66170
$686$	4.02251e6	0.326353
$687$	0	0
$688$	4.55629e6	0.366978
$689$	2.12624e7	1.70633
$690$	0	0
$691$	1.31243e7	1.04563	0.522817	−	0.852445i	$-0.324881\pi$
$691$	1.31243e7	1.04563	0.522817	+	0.852445i	$0.324881\pi$
$692$	1.22601e7	0.973257
$693$	0	0
$694$	6.84197e6	0.539240
$695$	1.04941e7	0.824104
$696$	0	0
$697$	−3.91435e6	−0.305195
$698$	875288.	0.0680005
$699$	0	0
$700$	1.39174e6	0.107353
$701$	−3.65956e6	−0.281277	−0.140638	−	0.990061i	$-0.544916\pi$
$701$	−3.65956e6	−0.281277	−0.140638	+	0.990061i	$0.544916\pi$
$702$	0	0
$703$	−2.10538e7	−1.60673
$704$	495616.	0.0376889
$705$	0	0
$706$	−1.47277e7	−1.11204
$707$	246012.	0.0185101
$708$	0	0
$709$	1.02252e7	0.763935	0.381968	−	0.924176i	$-0.375247\pi$
$709$	1.02252e7	0.763935	0.381968	+	0.924176i	$0.375247\pi$
$710$	−2.71544e6	−0.202160
$711$	0	0
$712$	−8.02656e6	−0.593375
$713$	1.11099e7	0.818436
$714$	0	0
$715$	4.27033e6	0.312390
$716$	−4.85438e6	−0.353876
$717$	0	0
$718$	−7.54114e6	−0.545916
$719$	−2.41683e7	−1.74351	−0.871753	−	0.489945i	$-0.837017\pi$
$719$	−2.41683e7	−1.74351	−0.871753	+	0.489945i	$0.837017\pi$
$720$	0	0
$721$	1.95043e7	1.39731
$722$	−1.79329e6	−0.128029
$723$	0	0
$724$	−4.56290e6	−0.323515
$725$	−1.08154e6	−0.0764181
$726$	0	0
$727$	1.68246e7	1.18062	0.590310	−	0.807177i	$-0.299006\pi$
$727$	1.68246e7	1.18062	0.590310	+	0.807177i	$0.299006\pi$
$728$	−7.35181e6	−0.514121
$729$	0	0
$730$	−1.08650e7	−0.754613
$731$	1.31349e7	0.909147
$732$	0	0
$733$	−5.04168e6	−0.346590	−0.173295	−	0.984870i	$-0.555441\pi$
$733$	−5.04168e6	−0.346590	−0.173295	+	0.984870i	$0.555441\pi$
$734$	−1.24666e7	−0.854101
$735$	0	0
$736$	1.82170e6	0.123960
$737$	3.06650e6	0.207958
$738$	0	0
$739$	−6.26375e6	−0.421913	−0.210957	−	0.977495i	$-0.567658\pi$
$739$	−6.26375e6	−0.421913	−0.210957	+	0.977495i	$0.567658\pi$
$740$	−1.20646e7	−0.809901
$741$	0	0
$742$	−2.04021e7	−1.36039
$743$	−3.63976e6	−0.241880	−0.120940	−	0.992660i	$-0.538591\pi$
$743$	−3.63976e6	−0.241880	−0.120940	+	0.992660i	$0.538591\pi$
$744$	0	0
$745$	−4.47403e6	−0.295330
$746$	5.57766e6	0.366948
$747$	0	0
$748$	1.42877e6	0.0933701
$749$	−1.31741e7	−0.858057
$750$	0	0
$751$	−1.87370e7	−1.21227	−0.606135	−	0.795362i	$-0.707281\pi$
$751$	−1.87370e7	−1.21227	−0.606135	+	0.795362i	$0.707281\pi$
$752$	4.39910e6	0.283674
$753$	0	0
$754$	5.71315e6	0.365972
$755$	−2.20717e7	−1.40918
$756$	0	0
$757$	489242.	0.0310302	0.0155151	−	0.999880i	$-0.495061\pi$
$757$	489242.	0.0310302	0.0155151	+	0.999880i	$0.495061\pi$
$758$	−1.70414e7	−1.07729
$759$	0	0
$760$	4.64794e6	0.291895
$761$	−1.46969e7	−0.919952	−0.459976	−	0.887931i	$-0.652142\pi$
$761$	−1.46969e7	−0.919952	−0.459976	+	0.887931i	$0.652142\pi$
$762$	0	0
$763$	−1.45818e7	−0.906774
$764$	−1.22731e7	−0.760711
$765$	0	0
$766$	−807060.	−0.0496974
$767$	2.42124e7	1.48610
$768$	0	0
$769$	2.42072e7	1.47615	0.738073	−	0.674721i	$-0.235736\pi$
$769$	2.42072e7	1.47615	0.738073	+	0.674721i	$0.235736\pi$
$770$	−4.09754e6	−0.249056
$771$	0	0
$772$	6.58669e6	0.397763
$773$	−1.35260e7	−0.814181	−0.407091	−	0.913388i	$-0.633457\pi$
$773$	−1.35260e7	−0.814181	−0.407091	+	0.913388i	$0.633457\pi$
$774$	0	0
$775$	−3.27238e6	−0.195708
$776$	−5.68365e6	−0.338823
$777$	0	0
$778$	−7.79528e6	−0.461725
$779$	−7.55290e6	−0.445933
$780$	0	0
$781$	−1.61063e6	−0.0944862
$782$	5.25161e6	0.307097
$783$	0	0
$784$	2.75174e6	0.159889
$785$	−1.73782e6	−0.100654
$786$	0	0
$787$	1.42094e7	0.817786	0.408893	−	0.912582i	$-0.365915\pi$
$787$	1.42094e7	0.817786	0.408893	+	0.912582i	$0.365915\pi$
$788$	1.21481e7	0.696937
$789$	0	0
$790$	1.57557e7	0.898196
$791$	−7.84300e6	−0.445698
$792$	0	0
$793$	−3.17905e7	−1.79521
$794$	−5.87303e6	−0.330606
$795$	0	0
$796$	−745600.	−0.0417084
$797$	7.93333e6	0.442395	0.221197	−	0.975229i	$-0.429003\pi$
$797$	7.93333e6	0.442395	0.221197	+	0.975229i	$0.429003\pi$
$798$	0	0
$799$	1.26818e7	0.702771
$800$	−536576.	−0.0296419
$801$	0	0
$802$	−8.98471e6	−0.493251
$803$	−6.44446e6	−0.352694
$804$	0	0
$805$	−1.50610e7	−0.819152
$806$	1.72862e7	0.937262
$807$	0	0
$808$	−94848.0	−0.00511093
$809$	1.04685e7	0.562359	0.281180	−	0.959655i	$-0.409274\pi$
$809$	1.04685e7	0.562359	0.281180	+	0.959655i	$0.409274\pi$
$810$	0	0
$811$	1.19147e7	0.636110	0.318055	−	0.948072i	$-0.396970\pi$
$811$	1.19147e7	0.636110	0.318055	+	0.948072i	$0.396970\pi$
$812$	−5.48198e6	−0.291775
$813$	0	0
$814$	−7.15594e6	−0.378535
$815$	2.29602e6	0.121083
$816$	0	0
$817$	2.53444e7	1.32839
$818$	−1.44595e7	−0.755562
$819$	0	0
$820$	−4.32806e6	−0.224781
$821$	1.86112e6	0.0963645	0.0481822	−	0.998839i	$-0.484657\pi$
$821$	1.86112e6	0.0963645	0.0481822	+	0.998839i	$0.484657\pi$
$822$	0	0
$823$	2.30153e7	1.18445	0.592225	−	0.805773i	$-0.298250\pi$
$823$	2.30153e7	1.18445	0.592225	+	0.805773i	$0.298250\pi$
$824$	−7.51974e6	−0.385820
$825$	0	0
$826$	−2.32327e7	−1.18481
$827$	1.68351e7	0.855959	0.427980	−	0.903788i	$-0.359225\pi$
$827$	1.68351e7	0.855959	0.427980	+	0.903788i	$0.359225\pi$
$828$	0	0
$829$	−2.35299e7	−1.18914	−0.594570	−	0.804044i	$-0.702678\pi$
$829$	−2.35299e7	−1.18914	−0.594570	+	0.804044i	$0.702678\pi$
$830$	−1.12229e7	−0.565468
$831$	0	0
$832$	2.83443e6	0.141957
$833$	7.93276e6	0.396106
$834$	0	0
$835$	−2.46104e7	−1.22152
$836$	2.75686e6	0.136427
$837$	0	0
$838$	1.52496e7	0.750148
$839$	−2.91549e7	−1.42990	−0.714952	−	0.699173i	$-0.753552\pi$
$839$	−2.91549e7	−1.42990	−0.714952	+	0.699173i	$0.753552\pi$
$840$	0	0
$841$	−1.62511e7	−0.792303
$842$	7.89385e6	0.383715
$843$	0	0
$844$	−1.49188e7	−0.720907
$845$	5.48612e6	0.264316
$846$	0	0
$847$	−2.43041e6	−0.116405
$848$	7.86586e6	0.375627
$849$	0	0
$850$	−1.54685e6	−0.0734345
$851$	−2.63025e7	−1.24501
$852$	0	0
$853$	−9.49052e6	−0.446599	−0.223299	−	0.974750i	$-0.571683\pi$
$853$	−9.49052e6	−0.446599	−0.223299	+	0.974750i	$0.571683\pi$
$854$	3.05042e7	1.43125
$855$	0	0
$856$	5.07917e6	0.236924
$857$	1.81553e6	0.0844405	0.0422203	−	0.999108i	$-0.486557\pi$
$857$	1.81553e6	0.0844405	0.0422203	+	0.999108i	$0.486557\pi$
$858$	0	0
$859$	−1.07812e7	−0.498522	−0.249261	−	0.968436i	$-0.580188\pi$
$859$	−1.07812e7	−0.498522	−0.249261	+	0.968436i	$0.580188\pi$
$860$	1.45232e7	0.669600
$861$	0	0
$862$	8.33436e6	0.382036
$863$	2.83355e7	1.29510	0.647550	−	0.762023i	$-0.275794\pi$
$863$	2.83355e7	1.29510	0.647550	+	0.762023i	$0.275794\pi$
$864$	0	0
$865$	3.90790e7	1.77584
$866$	−290764.	−0.0131749
$867$	0	0
$868$	−1.65867e7	−0.747242
$869$	9.34531e6	0.419802
$870$	0	0
$871$	1.75374e7	0.783283
$872$	5.62189e6	0.250375
$873$	0	0
$874$	1.01332e7	0.448712
$875$	3.08924e7	1.36406
$876$	0	0
$877$	−2.68919e7	−1.18065	−0.590326	−	0.807165i	$-0.701001\pi$
$877$	−2.68919e7	−1.18065	−0.590326	+	0.807165i	$0.701001\pi$
$878$	2.37757e6	0.104087
$879$	0	0
$880$	1.57978e6	0.0687684
$881$	1.92132e7	0.833989	0.416995	−	0.908909i	$-0.363083\pi$
$881$	1.92132e7	0.833989	0.416995	+	0.908909i	$0.363083\pi$
$882$	0	0
$883$	1.15931e7	0.500378	0.250189	−	0.968197i	$-0.419507\pi$
$883$	1.15931e7	0.500378	0.250189	+	0.968197i	$0.419507\pi$
$884$	8.17114e6	0.351683
$885$	0	0
$886$	−1.82660e7	−0.781735
$887$	−1.31857e7	−0.562721	−0.281361	−	0.959602i	$-0.590786\pi$
$887$	−1.31857e7	−0.562721	−0.281361	+	0.959602i	$0.590786\pi$
$888$	0	0
$889$	3.97431e7	1.68658
$890$	−2.55847e7	−1.08269
$891$	0	0
$892$	2.71592e6	0.114289
$893$	2.44700e7	1.02685
$894$	0	0
$895$	−1.54733e7	−0.645694
$896$	−2.71974e6	−0.113177
$897$	0	0
$898$	2.17753e7	0.901100
$899$	1.28897e7	0.531916
$900$	0	0
$901$	2.26758e7	0.930573
$902$	−2.56714e6	−0.105059
$903$	0	0
$904$	3.02381e6	0.123065
$905$	−1.45442e7	−0.590295
$906$	0	0
$907$	2.98195e6	0.120360	0.0601800	−	0.998188i	$-0.480833\pi$
$907$	2.98195e6	0.120360	0.0601800	+	0.998188i	$0.480833\pi$
$908$	−3.16925e6	−0.127568
$909$	0	0
$910$	−2.34339e7	−0.938082
$911$	−2.96579e7	−1.18398	−0.591989	−	0.805946i	$-0.701657\pi$
$911$	−2.96579e7	−1.18398	−0.591989	+	0.805946i	$0.701657\pi$
$912$	0	0
$913$	−6.65669e6	−0.264291
$914$	2.68125e7	1.06163
$915$	0	0
$916$	−1.36000e7	−0.535548
$917$	−1.62916e7	−0.639793
$918$	0	0
$919$	3.18057e7	1.24227	0.621135	−	0.783704i	$-0.286672\pi$
$919$	3.18057e7	1.24227	0.621135	+	0.783704i	$0.286672\pi$
$920$	5.80666e6	0.226181
$921$	0	0
$922$	5.03976e6	0.195246
$923$	−9.21121e6	−0.355887
$924$	0	0
$925$	7.74734e6	0.297713
$926$	−2.00923e7	−0.770021
$927$	0	0
$928$	2.11354e6	0.0805638
$929$	2.33444e7	0.887451	0.443725	−	0.896163i	$-0.353657\pi$
$929$	2.33444e7	0.887451	0.443725	+	0.896163i	$0.353657\pi$
$930$	0	0
$931$	1.53066e7	0.578767
$932$	6.42931e6	0.242451
$933$	0	0
$934$	9.42642e6	0.353573
$935$	4.55420e6	0.170366
$936$	0	0
$937$	2.07372e7	0.771616	0.385808	−	0.922579i	$-0.373923\pi$
$937$	2.07372e7	0.771616	0.385808	+	0.922579i	$0.373923\pi$
$938$	−1.68278e7	−0.624481
$939$	0	0
$940$	1.40221e7	0.517601
$941$	−2.69193e7	−0.991036	−0.495518	−	0.868598i	$-0.665022\pi$
$941$	−2.69193e7	−0.991036	−0.495518	+	0.868598i	$0.665022\pi$
$942$	0	0
$943$	−9.43582e6	−0.345542
$944$	8.95718e6	0.327146
$945$	0	0
$946$	8.61423e6	0.312960
$947$	−1.01896e7	−0.369216	−0.184608	−	0.982812i	$-0.559102\pi$
$947$	−1.01896e7	−0.369216	−0.184608	+	0.982812i	$0.559102\pi$
$948$	0	0
$949$	−3.68559e7	−1.32844
$950$	−2.98470e6	−0.107298
$951$	0	0
$952$	−7.84051e6	−0.280383
$953$	−1.03924e7	−0.370665	−0.185333	−	0.982676i	$-0.559336\pi$
$953$	−1.03924e7	−0.370665	−0.185333	+	0.982676i	$0.559336\pi$
$954$	0	0
$955$	−3.91204e7	−1.38802
$956$	−1.36828e7	−0.484206
$957$	0	0
$958$	2.68903e7	0.946634
$959$	6.64227e7	2.33222
$960$	0	0
$961$	1.03709e7	0.362249
$962$	−4.09249e7	−1.42577
$963$	0	0
$964$	1.80074e7	0.624107
$965$	2.09951e7	0.725770
$966$	0	0
$967$	−8.18877e6	−0.281613	−0.140806	−	0.990037i	$-0.544970\pi$
$967$	−8.18877e6	−0.281613	−0.140806	+	0.990037i	$0.544970\pi$
$968$	937024.	0.0321412
$969$	0	0
$970$	−1.81166e7	−0.618227
$971$	1.73274e7	0.589775	0.294887	−	0.955532i	$-0.404718\pi$
$971$	1.73274e7	0.589775	0.294887	+	0.955532i	$0.404718\pi$
$972$	0	0
$973$	−3.41572e7	−1.15664
$974$	7.84005e6	0.264802
$975$	0	0
$976$	−1.17606e7	−0.395190
$977$	438963.	0.0147127	0.00735634	−	0.999973i	$-0.497658\pi$
$977$	438963.	0.0147127	0.00735634	+	0.999973i	$0.497658\pi$
$978$	0	0
$979$	−1.51752e7	−0.506032
$980$	8.77118e6	0.291738
$981$	0	0
$982$	2.31850e6	0.0767234
$983$	2.79124e7	0.921326	0.460663	−	0.887575i	$-0.347612\pi$
$983$	2.79124e7	0.921326	0.460663	+	0.887575i	$0.347612\pi$
$984$	0	0
$985$	3.87222e7	1.27165
$986$	6.09293e6	0.199588
$987$	0	0
$988$	1.57665e7	0.513859
$989$	3.16626e7	1.02933
$990$	0	0
$991$	−4.26846e7	−1.38066	−0.690331	−	0.723494i	$-0.742535\pi$
$991$	−4.26846e7	−1.38066	−0.690331	+	0.723494i	$0.742535\pi$
$992$	6.39488e6	0.206326
$993$	0	0
$994$	8.83850e6	0.283735
$995$	−2.37660e6	−0.0761024
$996$	0	0
$997$	−2.21044e7	−0.704273	−0.352137	−	0.935949i	$-0.614545\pi$
$997$	−2.21044e7	−0.704273	−0.352137	+	0.935949i	$0.614545\pi$
$998$	5.47621e6	0.174042
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	198.6.a.i.1.1		1
3.2	odd	2		22.6.a.b.1.1	✓	1
12.11	even	2		176.6.a.b.1.1		1
15.2	even	4		550.6.b.f.199.1		2
15.8	even	4		550.6.b.f.199.2		2
15.14	odd	2		550.6.a.f.1.1		1
21.20	even	2		1078.6.a.a.1.1		1
24.5	odd	2		704.6.a.e.1.1		1
24.11	even	2		704.6.a.f.1.1		1
33.32	even	2		242.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.6.a.b.1.1	✓	1	3.2	odd	2
176.6.a.b.1.1		1	12.11	even	2
198.6.a.i.1.1		1	1.1	even	1	trivial
242.6.a.d.1.1		1	33.32	even	2
550.6.a.f.1.1		1	15.14	odd	2
550.6.b.f.199.1		2	15.2	even	4
550.6.b.f.199.2		2	15.8	even	4
704.6.a.e.1.1		1	24.5	odd	2
704.6.a.f.1.1		1	24.11	even	2
1078.6.a.a.1.1		1	21.20	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:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	198.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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