LMFDB - Embedded newform 1638.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,4,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1638.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1638.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:	$96.6451285894$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} -3.00000 q^{5} +7.00000 q^{7} -8.00000 q^{8} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{4} -3.00000 q^{5} +7.00000 q^{7} -8.00000 q^{8} +6.00000 q^{10} +54.0000 q^{11} +13.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} +96.0000 q^{17} -151.000 q^{19} -12.0000 q^{20} -108.000 q^{22} -33.0000 q^{23} -116.000 q^{25} -26.0000 q^{26} +28.0000 q^{28} -183.000 q^{29} -331.000 q^{31} -32.0000 q^{32} -192.000 q^{34} -21.0000 q^{35} -88.0000 q^{37} +302.000 q^{38} +24.0000 q^{40} +42.0000 q^{41} +353.000 q^{43} +216.000 q^{44} +66.0000 q^{46} +465.000 q^{47} +49.0000 q^{49} +232.000 q^{50} +52.0000 q^{52} -195.000 q^{53} -162.000 q^{55} -56.0000 q^{56} +366.000 q^{58} -552.000 q^{59} +470.000 q^{61} +662.000 q^{62} +64.0000 q^{64} -39.0000 q^{65} +254.000 q^{67} +384.000 q^{68} +42.0000 q^{70} -132.000 q^{71} -943.000 q^{73} +176.000 q^{74} -604.000 q^{76} +378.000 q^{77} -727.000 q^{79} -48.0000 q^{80} -84.0000 q^{82} +1197.00 q^{83} -288.000 q^{85} -706.000 q^{86} -432.000 q^{88} -753.000 q^{89} +91.0000 q^{91} -132.000 q^{92} -930.000 q^{94} +453.000 q^{95} +1037.00 q^{97} -98.0000 q^{98} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−3.00000	−0.268328	−0.134164	−	0.990959i	$-0.542835\pi$
$5$	−3.00000	−0.268328	−0.134164	+	0.990959i	$0.542835\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	6.00000	0.189737
$11$	54.0000	1.48015	0.740073	−	0.672526i	$-0.234791\pi$
$11$	54.0000	1.48015	0.740073	+	0.672526i	$0.234791\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	96.0000	1.36961	0.684806	−	0.728725i	$-0.259887\pi$
$17$	96.0000	1.36961	0.684806	+	0.728725i	$0.259887\pi$
$18$	0	0
$19$	−151.000	−1.82325	−0.911626	−	0.411021i	$-0.865172\pi$
$19$	−151.000	−1.82325	−0.911626	+	0.411021i	$0.865172\pi$
$20$	−12.0000	−0.134164
$21$	0	0
$22$	−108.000	−1.04662
$23$	−33.0000	−0.299173	−0.149586	−	0.988749i	$-0.547794\pi$
$23$	−33.0000	−0.299173	−0.149586	+	0.988749i	$0.547794\pi$
$24$	0	0
$25$	−116.000	−0.928000
$26$	−26.0000	−0.196116
$27$	0	0
$28$	28.0000	0.188982
$29$	−183.000	−1.17180	−0.585901	−	0.810383i	$-0.699259\pi$
$29$	−183.000	−1.17180	−0.585901	+	0.810383i	$0.699259\pi$
$30$	0	0
$31$	−331.000	−1.91772	−0.958861	−	0.283877i	$-0.908379\pi$
$31$	−331.000	−1.91772	−0.958861	+	0.283877i	$0.908379\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−192.000	−0.968463
$35$	−21.0000	−0.101419
$36$	0	0
$37$	−88.0000	−0.391003	−0.195501	−	0.980703i	$-0.562633\pi$
$37$	−88.0000	−0.391003	−0.195501	+	0.980703i	$0.562633\pi$
$38$	302.000	1.28923
$39$	0	0
$40$	24.0000	0.0948683
$41$	42.0000	0.159983	0.0799914	−	0.996796i	$-0.474511\pi$
$41$	42.0000	0.159983	0.0799914	+	0.996796i	$0.474511\pi$
$42$	0	0
$43$	353.000	1.25191	0.625953	−	0.779860i	$-0.284710\pi$
$43$	353.000	1.25191	0.625953	+	0.779860i	$0.284710\pi$
$44$	216.000	0.740073
$45$	0	0
$46$	66.0000	0.211547
$47$	465.000	1.44313	0.721566	−	0.692345i	$-0.243423\pi$
$47$	465.000	1.44313	0.721566	+	0.692345i	$0.243423\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	232.000	0.656195
$51$	0	0
$52$	52.0000	0.138675
$53$	−195.000	−0.505383	−0.252692	−	0.967547i	$-0.581316\pi$
$53$	−195.000	−0.505383	−0.252692	+	0.967547i	$0.581316\pi$
$54$	0	0
$55$	−162.000	−0.397165
$56$	−56.0000	−0.133631
$57$	0	0
$58$	366.000	0.828589
$59$	−552.000	−1.21804	−0.609019	−	0.793155i	$-0.708437\pi$
$59$	−552.000	−1.21804	−0.609019	+	0.793155i	$0.708437\pi$
$60$	0	0
$61$	470.000	0.986514	0.493257	−	0.869884i	$-0.335806\pi$
$61$	470.000	0.986514	0.493257	+	0.869884i	$0.335806\pi$
$62$	662.000	1.35603
$63$	0	0
$64$	64.0000	0.125000
$65$	−39.0000	−0.0744208
$66$	0	0
$67$	254.000	0.463150	0.231575	−	0.972817i	$-0.425612\pi$
$67$	254.000	0.463150	0.231575	+	0.972817i	$0.425612\pi$
$68$	384.000	0.684806
$69$	0	0
$70$	42.0000	0.0717137
$71$	−132.000	−0.220641	−0.110321	−	0.993896i	$-0.535188\pi$
$71$	−132.000	−0.220641	−0.110321	+	0.993896i	$0.535188\pi$
$72$	0	0
$73$	−943.000	−1.51192	−0.755958	−	0.654621i	$-0.772828\pi$
$73$	−943.000	−1.51192	−0.755958	+	0.654621i	$0.772828\pi$
$74$	176.000	0.276481
$75$	0	0
$76$	−604.000	−0.911626
$77$	378.000	0.559443
$78$	0	0
$79$	−727.000	−1.03537	−0.517683	−	0.855573i	$-0.673205\pi$
$79$	−727.000	−1.03537	−0.517683	+	0.855573i	$0.673205\pi$
$80$	−48.0000	−0.0670820
$81$	0	0
$82$	−84.0000	−0.113125
$83$	1197.00	1.58299	0.791493	−	0.611178i	$-0.209304\pi$
$83$	1197.00	1.58299	0.791493	+	0.611178i	$0.209304\pi$
$84$	0	0
$85$	−288.000	−0.367506
$86$	−706.000	−0.885232
$87$	0	0
$88$	−432.000	−0.523311
$89$	−753.000	−0.896830	−0.448415	−	0.893826i	$-0.648011\pi$
$89$	−753.000	−0.896830	−0.448415	+	0.893826i	$0.648011\pi$
$90$	0	0
$91$	91.0000	0.104828
$92$	−132.000	−0.149586
$93$	0	0
$94$	−930.000	−1.02045
$95$	453.000	0.489230
$96$	0	0
$97$	1037.00	1.08548	0.542739	−	0.839901i	$-0.317387\pi$
$97$	1037.00	1.08548	0.542739	+	0.839901i	$0.317387\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	−464.000	−0.464000
$101$	−1566.00	−1.54280	−0.771400	−	0.636350i	$-0.780443\pi$
$101$	−1566.00	−1.54280	−0.771400	+	0.636350i	$0.780443\pi$
$102$	0	0
$103$	416.000	0.397958	0.198979	−	0.980004i	$-0.436237\pi$
$103$	416.000	0.397958	0.198979	+	0.980004i	$0.436237\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	390.000	0.357360
$107$	660.000	0.596305	0.298152	−	0.954518i	$-0.403630\pi$
$107$	660.000	0.596305	0.298152	+	0.954518i	$0.403630\pi$
$108$	0	0
$109$	−250.000	−0.219685	−0.109842	−	0.993949i	$-0.535035\pi$
$109$	−250.000	−0.219685	−0.109842	+	0.993949i	$0.535035\pi$
$110$	324.000	0.280838
$111$	0	0
$112$	112.000	0.0944911
$113$	1515.00	1.26123	0.630616	−	0.776095i	$-0.282802\pi$
$113$	1515.00	1.26123	0.630616	+	0.776095i	$0.282802\pi$
$114$	0	0
$115$	99.0000	0.0802765
$116$	−732.000	−0.585901
$117$	0	0
$118$	1104.00	0.861283
$119$	672.000	0.517665
$120$	0	0
$121$	1585.00	1.19083
$122$	−940.000	−0.697571
$123$	0	0
$124$	−1324.00	−0.958861
$125$	723.000	0.517337
$126$	0	0
$127$	−1492.00	−1.04247	−0.521235	−	0.853413i	$-0.674528\pi$
$127$	−1492.00	−1.04247	−0.521235	+	0.853413i	$0.674528\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	78.0000	0.0526235
$131$	−1740.00	−1.16049	−0.580246	−	0.814441i	$-0.697044\pi$
$131$	−1740.00	−1.16049	−0.580246	+	0.814441i	$0.697044\pi$
$132$	0	0
$133$	−1057.00	−0.689124
$134$	−508.000	−0.327496
$135$	0	0
$136$	−768.000	−0.484231
$137$	−2820.00	−1.75860	−0.879302	−	0.476264i	$-0.841991\pi$
$137$	−2820.00	−1.75860	−0.879302	+	0.476264i	$0.841991\pi$
$138$	0	0
$139$	74.0000	0.0451554	0.0225777	−	0.999745i	$-0.492813\pi$
$139$	74.0000	0.0451554	0.0225777	+	0.999745i	$0.492813\pi$
$140$	−84.0000	−0.0507093
$141$	0	0
$142$	264.000	0.156017
$143$	702.000	0.410519
$144$	0	0
$145$	549.000	0.314427
$146$	1886.00	1.06909
$147$	0	0
$148$	−352.000	−0.195501
$149$	342.000	0.188038	0.0940192	−	0.995570i	$-0.470028\pi$
$149$	342.000	0.188038	0.0940192	+	0.995570i	$0.470028\pi$
$150$	0	0
$151$	−268.000	−0.144434	−0.0722170	−	0.997389i	$-0.523007\pi$
$151$	−268.000	−0.144434	−0.0722170	+	0.997389i	$0.523007\pi$
$152$	1208.00	0.644617
$153$	0	0
$154$	−756.000	−0.395586
$155$	993.000	0.514579
$156$	0	0
$157$	−2104.00	−1.06954	−0.534769	−	0.844998i	$-0.679601\pi$
$157$	−2104.00	−1.06954	−0.534769	+	0.844998i	$0.679601\pi$
$158$	1454.00	0.732114
$159$	0	0
$160$	96.0000	0.0474342
$161$	−231.000	−0.113077
$162$	0	0
$163$	−1312.00	−0.630453	−0.315226	−	0.949017i	$-0.602080\pi$
$163$	−1312.00	−0.630453	−0.315226	+	0.949017i	$0.602080\pi$
$164$	168.000	0.0799914
$165$	0	0
$166$	−2394.00	−1.11934
$167$	−2469.00	−1.14405	−0.572027	−	0.820235i	$-0.693843\pi$
$167$	−2469.00	−1.14405	−0.572027	+	0.820235i	$0.693843\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	576.000	0.259866
$171$	0	0
$172$	1412.00	0.625953
$173$	36.0000	0.0158210	0.00791049	−	0.999969i	$-0.497482\pi$
$173$	36.0000	0.0158210	0.00791049	+	0.999969i	$0.497482\pi$
$174$	0	0
$175$	−812.000	−0.350751
$176$	864.000	0.370037
$177$	0	0
$178$	1506.00	0.634154
$179$	−1809.00	−0.755369	−0.377684	−	0.925934i	$-0.623280\pi$
$179$	−1809.00	−0.755369	−0.377684	+	0.925934i	$0.623280\pi$
$180$	0	0
$181$	1190.00	0.488685	0.244343	−	0.969689i	$-0.421428\pi$
$181$	1190.00	0.488685	0.244343	+	0.969689i	$0.421428\pi$
$182$	−182.000	−0.0741249
$183$	0	0
$184$	264.000	0.105774
$185$	264.000	0.104917
$186$	0	0
$187$	5184.00	2.02723
$188$	1860.00	0.721566
$189$	0	0
$190$	−906.000	−0.345938
$191$	3384.00	1.28198	0.640989	−	0.767550i	$-0.278525\pi$
$191$	3384.00	1.28198	0.640989	+	0.767550i	$0.278525\pi$
$192$	0	0
$193$	110.000	0.0410258	0.0205129	−	0.999790i	$-0.493470\pi$
$193$	110.000	0.0410258	0.0205129	+	0.999790i	$0.493470\pi$
$194$	−2074.00	−0.767549
$195$	0	0
$196$	196.000	0.0714286
$197$	3966.00	1.43434	0.717172	−	0.696896i	$-0.245436\pi$
$197$	3966.00	1.43434	0.717172	+	0.696896i	$0.245436\pi$
$198$	0	0
$199$	−88.0000	−0.0313475	−0.0156738	−	0.999877i	$-0.504989\pi$
$199$	−88.0000	−0.0313475	−0.0156738	+	0.999877i	$0.504989\pi$
$200$	928.000	0.328098
$201$	0	0
$202$	3132.00	1.09092
$203$	−1281.00	−0.442899
$204$	0	0
$205$	−126.000	−0.0429279
$206$	−832.000	−0.281399
$207$	0	0
$208$	208.000	0.0693375
$209$	−8154.00	−2.69868
$210$	0	0
$211$	−331.000	−0.107995	−0.0539976	−	0.998541i	$-0.517196\pi$
$211$	−331.000	−0.107995	−0.0539976	+	0.998541i	$0.517196\pi$
$212$	−780.000	−0.252692
$213$	0	0
$214$	−1320.00	−0.421651
$215$	−1059.00	−0.335922
$216$	0	0
$217$	−2317.00	−0.724830
$218$	500.000	0.155341
$219$	0	0
$220$	−648.000	−0.198583
$221$	1248.00	0.379862
$222$	0	0
$223$	−4777.00	−1.43449	−0.717246	−	0.696820i	$-0.754597\pi$
$223$	−4777.00	−1.43449	−0.717246	+	0.696820i	$0.754597\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−3030.00	−0.891826
$227$	−3396.00	−0.992953	−0.496477	−	0.868050i	$-0.665373\pi$
$227$	−3396.00	−0.992953	−0.496477	+	0.868050i	$0.665373\pi$
$228$	0	0
$229$	−5182.00	−1.49535	−0.747677	−	0.664062i	$-0.768831\pi$
$229$	−5182.00	−1.49535	−0.747677	+	0.664062i	$0.768831\pi$
$230$	−198.000	−0.0567641
$231$	0	0
$232$	1464.00	0.414294
$233$	5001.00	1.40612	0.703061	−	0.711130i	$-0.251816\pi$
$233$	5001.00	1.40612	0.703061	+	0.711130i	$0.251816\pi$
$234$	0	0
$235$	−1395.00	−0.387233
$236$	−2208.00	−0.609019
$237$	0	0
$238$	−1344.00	−0.366044
$239$	−2340.00	−0.633314	−0.316657	−	0.948540i	$-0.602560\pi$
$239$	−2340.00	−0.633314	−0.316657	+	0.948540i	$0.602560\pi$
$240$	0	0
$241$	5645.00	1.50882	0.754412	−	0.656402i	$-0.227922\pi$
$241$	5645.00	1.50882	0.754412	+	0.656402i	$0.227922\pi$
$242$	−3170.00	−0.842047
$243$	0	0
$244$	1880.00	0.493257
$245$	−147.000	−0.0383326
$246$	0	0
$247$	−1963.00	−0.505679
$248$	2648.00	0.678017
$249$	0	0
$250$	−1446.00	−0.365812
$251$	642.000	0.161445	0.0807225	−	0.996737i	$-0.474277\pi$
$251$	642.000	0.161445	0.0807225	+	0.996737i	$0.474277\pi$
$252$	0	0
$253$	−1782.00	−0.442820
$254$	2984.00	0.737137
$255$	0	0
$256$	256.000	0.0625000
$257$	−2886.00	−0.700481	−0.350241	−	0.936660i	$-0.613900\pi$
$257$	−2886.00	−0.700481	−0.350241	+	0.936660i	$0.613900\pi$
$258$	0	0
$259$	−616.000	−0.147785
$260$	−156.000	−0.0372104
$261$	0	0
$262$	3480.00	0.820592
$263$	−7587.00	−1.77884	−0.889419	−	0.457093i	$-0.848891\pi$
$263$	−7587.00	−1.77884	−0.889419	+	0.457093i	$0.848891\pi$
$264$	0	0
$265$	585.000	0.135609
$266$	2114.00	0.487284
$267$	0	0
$268$	1016.00	0.231575
$269$	−120.000	−0.0271990	−0.0135995	−	0.999908i	$-0.504329\pi$
$269$	−120.000	−0.0271990	−0.0135995	+	0.999908i	$0.504329\pi$
$270$	0	0
$271$	56.0000	0.0125526	0.00627631	−	0.999980i	$-0.498002\pi$
$271$	56.0000	0.0125526	0.00627631	+	0.999980i	$0.498002\pi$
$272$	1536.00	0.342403
$273$	0	0
$274$	5640.00	1.24352
$275$	−6264.00	−1.37358
$276$	0	0
$277$	−475.000	−0.103032	−0.0515162	−	0.998672i	$-0.516405\pi$
$277$	−475.000	−0.103032	−0.0515162	+	0.998672i	$0.516405\pi$
$278$	−148.000	−0.0319297
$279$	0	0
$280$	168.000	0.0358569
$281$	−918.000	−0.194887	−0.0974436	−	0.995241i	$-0.531067\pi$
$281$	−918.000	−0.194887	−0.0974436	+	0.995241i	$0.531067\pi$
$282$	0	0
$283$	−7252.00	−1.52327	−0.761637	−	0.648004i	$-0.775604\pi$
$283$	−7252.00	−1.52327	−0.761637	+	0.648004i	$0.775604\pi$
$284$	−528.000	−0.110321
$285$	0	0
$286$	−1404.00	−0.290281
$287$	294.000	0.0604678
$288$	0	0
$289$	4303.00	0.875840
$290$	−1098.00	−0.222334
$291$	0	0
$292$	−3772.00	−0.755958
$293$	525.000	0.104679	0.0523393	−	0.998629i	$-0.483332\pi$
$293$	525.000	0.104679	0.0523393	+	0.998629i	$0.483332\pi$
$294$	0	0
$295$	1656.00	0.326834
$296$	704.000	0.138240
$297$	0	0
$298$	−684.000	−0.132963
$299$	−429.000	−0.0829756
$300$	0	0
$301$	2471.00	0.473176
$302$	536.000	0.102130
$303$	0	0
$304$	−2416.00	−0.455813
$305$	−1410.00	−0.264709
$306$	0	0
$307$	−2077.00	−0.386126	−0.193063	−	0.981186i	$-0.561842\pi$
$307$	−2077.00	−0.386126	−0.193063	+	0.981186i	$0.561842\pi$
$308$	1512.00	0.279721
$309$	0	0
$310$	−1986.00	−0.363862
$311$	−1842.00	−0.335853	−0.167926	−	0.985800i	$-0.553707\pi$
$311$	−1842.00	−0.335853	−0.167926	+	0.985800i	$0.553707\pi$
$312$	0	0
$313$	−1150.00	−0.207674	−0.103837	−	0.994594i	$-0.533112\pi$
$313$	−1150.00	−0.207674	−0.103837	+	0.994594i	$0.533112\pi$
$314$	4208.00	0.756278
$315$	0	0
$316$	−2908.00	−0.517683
$317$	1896.00	0.335931	0.167965	−	0.985793i	$-0.446280\pi$
$317$	1896.00	0.335931	0.167965	+	0.985793i	$0.446280\pi$
$318$	0	0
$319$	−9882.00	−1.73444
$320$	−192.000	−0.0335410
$321$	0	0
$322$	462.000	0.0799573
$323$	−14496.0	−2.49715
$324$	0	0
$325$	−1508.00	−0.257381
$326$	2624.00	0.445797
$327$	0	0
$328$	−336.000	−0.0565625
$329$	3255.00	0.545453
$330$	0	0
$331$	4934.00	0.819327	0.409663	−	0.912237i	$-0.365646\pi$
$331$	4934.00	0.819327	0.409663	+	0.912237i	$0.365646\pi$
$332$	4788.00	0.791493
$333$	0	0
$334$	4938.00	0.808968
$335$	−762.000	−0.124276
$336$	0	0
$337$	−5839.00	−0.943830	−0.471915	−	0.881644i	$-0.656437\pi$
$337$	−5839.00	−0.943830	−0.471915	+	0.881644i	$0.656437\pi$
$338$	−338.000	−0.0543928
$339$	0	0
$340$	−1152.00	−0.183753
$341$	−17874.0	−2.83851
$342$	0	0
$343$	343.000	0.0539949
$344$	−2824.00	−0.442616
$345$	0	0
$346$	−72.0000	−0.0111871
$347$	456.000	0.0705457	0.0352729	−	0.999378i	$-0.488770\pi$
$347$	456.000	0.0705457	0.0352729	+	0.999378i	$0.488770\pi$
$348$	0	0
$349$	−6451.00	−0.989439	−0.494719	−	0.869053i	$-0.664729\pi$
$349$	−6451.00	−0.989439	−0.494719	+	0.869053i	$0.664729\pi$
$350$	1624.00	0.248018
$351$	0	0
$352$	−1728.00	−0.261655
$353$	9342.00	1.40857	0.704284	−	0.709918i	$-0.251268\pi$
$353$	9342.00	1.40857	0.704284	+	0.709918i	$0.251268\pi$
$354$	0	0
$355$	396.000	0.0592042
$356$	−3012.00	−0.448415
$357$	0	0
$358$	3618.00	0.534126
$359$	−8328.00	−1.22433	−0.612166	−	0.790729i	$-0.709701\pi$
$359$	−8328.00	−1.22433	−0.612166	+	0.790729i	$0.709701\pi$
$360$	0	0
$361$	15942.0	2.32425
$362$	−2380.00	−0.345553
$363$	0	0
$364$	364.000	0.0524142
$365$	2829.00	0.405689
$366$	0	0
$367$	−2230.00	−0.317180	−0.158590	−	0.987345i	$-0.550695\pi$
$367$	−2230.00	−0.317180	−0.158590	+	0.987345i	$0.550695\pi$
$368$	−528.000	−0.0747932
$369$	0	0
$370$	−528.000	−0.0741876
$371$	−1365.00	−0.191017
$372$	0	0
$373$	−178.000	−0.0247091	−0.0123545	−	0.999924i	$-0.503933\pi$
$373$	−178.000	−0.0247091	−0.0123545	+	0.999924i	$0.503933\pi$
$374$	−10368.0	−1.43347
$375$	0	0
$376$	−3720.00	−0.510224
$377$	−2379.00	−0.324999
$378$	0	0
$379$	−6874.00	−0.931646	−0.465823	−	0.884878i	$-0.654242\pi$
$379$	−6874.00	−0.931646	−0.465823	+	0.884878i	$0.654242\pi$
$380$	1812.00	0.244615
$381$	0	0
$382$	−6768.00	−0.906495
$383$	10548.0	1.40725	0.703626	−	0.710570i	$-0.251563\pi$
$383$	10548.0	1.40725	0.703626	+	0.710570i	$0.251563\pi$
$384$	0	0
$385$	−1134.00	−0.150114
$386$	−220.000	−0.0290096
$387$	0	0
$388$	4148.00	0.542739
$389$	−6054.00	−0.789075	−0.394537	−	0.918880i	$-0.629095\pi$
$389$	−6054.00	−0.789075	−0.394537	+	0.918880i	$0.629095\pi$
$390$	0	0
$391$	−3168.00	−0.409751
$392$	−392.000	−0.0505076
$393$	0	0
$394$	−7932.00	−1.01423
$395$	2181.00	0.277818
$396$	0	0
$397$	−10195.0	−1.28885	−0.644424	−	0.764669i	$-0.722903\pi$
$397$	−10195.0	−1.28885	−0.644424	+	0.764669i	$0.722903\pi$
$398$	176.000	0.0221660
$399$	0	0
$400$	−1856.00	−0.232000
$401$	4512.00	0.561892	0.280946	−	0.959724i	$-0.409352\pi$
$401$	4512.00	0.561892	0.280946	+	0.959724i	$0.409352\pi$
$402$	0	0
$403$	−4303.00	−0.531880
$404$	−6264.00	−0.771400
$405$	0	0
$406$	2562.00	0.313177
$407$	−4752.00	−0.578742
$408$	0	0
$409$	−9511.00	−1.14985	−0.574925	−	0.818206i	$-0.694969\pi$
$409$	−9511.00	−1.14985	−0.574925	+	0.818206i	$0.694969\pi$
$410$	252.000	0.0303546
$411$	0	0
$412$	1664.00	0.198979
$413$	−3864.00	−0.460375
$414$	0	0
$415$	−3591.00	−0.424760
$416$	−416.000	−0.0490290
$417$	0	0
$418$	16308.0	1.90825
$419$	5190.00	0.605127	0.302563	−	0.953129i	$-0.402158\pi$
$419$	5190.00	0.605127	0.302563	+	0.953129i	$0.402158\pi$
$420$	0	0
$421$	−4192.00	−0.485286	−0.242643	−	0.970116i	$-0.578014\pi$
$421$	−4192.00	−0.485286	−0.242643	+	0.970116i	$0.578014\pi$
$422$	662.000	0.0763641
$423$	0	0
$424$	1560.00	0.178680
$425$	−11136.0	−1.27100
$426$	0	0
$427$	3290.00	0.372867
$428$	2640.00	0.298152
$429$	0	0
$430$	2118.00	0.237533
$431$	9918.00	1.10843	0.554215	−	0.832374i	$-0.313019\pi$
$431$	9918.00	1.10843	0.554215	+	0.832374i	$0.313019\pi$
$432$	0	0
$433$	−5092.00	−0.565141	−0.282570	−	0.959247i	$-0.591187\pi$
$433$	−5092.00	−0.565141	−0.282570	+	0.959247i	$0.591187\pi$
$434$	4634.00	0.512533
$435$	0	0
$436$	−1000.00	−0.109842
$437$	4983.00	0.545467
$438$	0	0
$439$	290.000	0.0315283	0.0157642	−	0.999876i	$-0.494982\pi$
$439$	290.000	0.0315283	0.0157642	+	0.999876i	$0.494982\pi$
$440$	1296.00	0.140419
$441$	0	0
$442$	−2496.00	−0.268603
$443$	3867.00	0.414733	0.207366	−	0.978263i	$-0.433511\pi$
$443$	3867.00	0.414733	0.207366	+	0.978263i	$0.433511\pi$
$444$	0	0
$445$	2259.00	0.240645
$446$	9554.00	1.01434
$447$	0	0
$448$	448.000	0.0472456
$449$	4008.00	0.421268	0.210634	−	0.977565i	$-0.432447\pi$
$449$	4008.00	0.421268	0.210634	+	0.977565i	$0.432447\pi$
$450$	0	0
$451$	2268.00	0.236798
$452$	6060.00	0.630616
$453$	0	0
$454$	6792.00	0.702124
$455$	−273.000	−0.0281284
$456$	0	0
$457$	−7936.00	−0.812320	−0.406160	−	0.913802i	$-0.633133\pi$
$457$	−7936.00	−0.812320	−0.406160	+	0.913802i	$0.633133\pi$
$458$	10364.0	1.05738
$459$	0	0
$460$	396.000	0.0401383
$461$	−9942.00	−1.00444	−0.502218	−	0.864741i	$-0.667483\pi$
$461$	−9942.00	−1.00444	−0.502218	+	0.864741i	$0.667483\pi$
$462$	0	0
$463$	15914.0	1.59738	0.798689	−	0.601744i	$-0.205527\pi$
$463$	15914.0	1.59738	0.798689	+	0.601744i	$0.205527\pi$
$464$	−2928.00	−0.292950
$465$	0	0
$466$	−10002.0	−0.994278
$467$	−10434.0	−1.03389	−0.516946	−	0.856018i	$-0.672931\pi$
$467$	−10434.0	−1.03389	−0.516946	+	0.856018i	$0.672931\pi$
$468$	0	0
$469$	1778.00	0.175054
$470$	2790.00	0.273815
$471$	0	0
$472$	4416.00	0.430642
$473$	19062.0	1.85301
$474$	0	0
$475$	17516.0	1.69198
$476$	2688.00	0.258833
$477$	0	0
$478$	4680.00	0.447821
$479$	7155.00	0.682506	0.341253	−	0.939972i	$-0.389149\pi$
$479$	7155.00	0.682506	0.341253	+	0.939972i	$0.389149\pi$
$480$	0	0
$481$	−1144.00	−0.108445
$482$	−11290.0	−1.06690
$483$	0	0
$484$	6340.00	0.595417
$485$	−3111.00	−0.291264
$486$	0	0
$487$	−16882.0	−1.57083	−0.785417	−	0.618967i	$-0.787552\pi$
$487$	−16882.0	−1.57083	−0.785417	+	0.618967i	$0.787552\pi$
$488$	−3760.00	−0.348785
$489$	0	0
$490$	294.000	0.0271052
$491$	10140.0	0.932000	0.466000	−	0.884785i	$-0.345695\pi$
$491$	10140.0	0.932000	0.466000	+	0.884785i	$0.345695\pi$
$492$	0	0
$493$	−17568.0	−1.60491
$494$	3926.00	0.357569
$495$	0	0
$496$	−5296.00	−0.479430
$497$	−924.000	−0.0833945
$498$	0	0
$499$	−9304.00	−0.834678	−0.417339	−	0.908751i	$-0.637037\pi$
$499$	−9304.00	−0.834678	−0.417339	+	0.908751i	$0.637037\pi$
$500$	2892.00	0.258668
$501$	0	0
$502$	−1284.00	−0.114159
$503$	5898.00	0.522821	0.261410	−	0.965228i	$-0.415812\pi$
$503$	5898.00	0.522821	0.261410	+	0.965228i	$0.415812\pi$
$504$	0	0
$505$	4698.00	0.413977
$506$	3564.00	0.313121
$507$	0	0
$508$	−5968.00	−0.521235
$509$	3693.00	0.321590	0.160795	−	0.986988i	$-0.448594\pi$
$509$	3693.00	0.321590	0.160795	+	0.986988i	$0.448594\pi$
$510$	0	0
$511$	−6601.00	−0.571450
$512$	−512.000	−0.0441942
$513$	0	0
$514$	5772.00	0.495315
$515$	−1248.00	−0.106783
$516$	0	0
$517$	25110.0	2.13605
$518$	1232.00	0.104500
$519$	0	0
$520$	312.000	0.0263117
$521$	−9924.00	−0.834507	−0.417254	−	0.908790i	$-0.637007\pi$
$521$	−9924.00	−0.834507	−0.417254	+	0.908790i	$0.637007\pi$
$522$	0	0
$523$	19442.0	1.62551	0.812753	−	0.582609i	$-0.197968\pi$
$523$	19442.0	1.62551	0.812753	+	0.582609i	$0.197968\pi$
$524$	−6960.00	−0.580246
$525$	0	0
$526$	15174.0	1.25783
$527$	−31776.0	−2.62654
$528$	0	0
$529$	−11078.0	−0.910496
$530$	−1170.00	−0.0958897
$531$	0	0
$532$	−4228.00	−0.344562
$533$	546.000	0.0443713
$534$	0	0
$535$	−1980.00	−0.160005
$536$	−2032.00	−0.163748
$537$	0	0
$538$	240.000	0.0192326
$539$	2646.00	0.211450
$540$	0	0
$541$	−23416.0	−1.86087	−0.930437	−	0.366453i	$-0.880572\pi$
$541$	−23416.0	−1.86087	−0.930437	+	0.366453i	$0.880572\pi$
$542$	−112.000	−0.00887604
$543$	0	0
$544$	−3072.00	−0.242116
$545$	750.000	0.0589477
$546$	0	0
$547$	−5065.00	−0.395912	−0.197956	−	0.980211i	$-0.563430\pi$
$547$	−5065.00	−0.395912	−0.197956	+	0.980211i	$0.563430\pi$
$548$	−11280.0	−0.879302
$549$	0	0
$550$	12528.0	0.971265
$551$	27633.0	2.13649
$552$	0	0
$553$	−5089.00	−0.391331
$554$	950.000	0.0728549
$555$	0	0
$556$	296.000	0.0225777
$557$	−8424.00	−0.640819	−0.320410	−	0.947279i	$-0.603821\pi$
$557$	−8424.00	−0.640819	−0.320410	+	0.947279i	$0.603821\pi$
$558$	0	0
$559$	4589.00	0.347216
$560$	−336.000	−0.0253546
$561$	0	0
$562$	1836.00	0.137806
$563$	−11316.0	−0.847092	−0.423546	−	0.905875i	$-0.639215\pi$
$563$	−11316.0	−0.847092	−0.423546	+	0.905875i	$0.639215\pi$
$564$	0	0
$565$	−4545.00	−0.338424
$566$	14504.0	1.07712
$567$	0	0
$568$	1056.00	0.0780084
$569$	18849.0	1.38874	0.694368	−	0.719620i	$-0.255684\pi$
$569$	18849.0	1.38874	0.694368	+	0.719620i	$0.255684\pi$
$570$	0	0
$571$	9281.00	0.680206	0.340103	−	0.940388i	$-0.389538\pi$
$571$	9281.00	0.680206	0.340103	+	0.940388i	$0.389538\pi$
$572$	2808.00	0.205259
$573$	0	0
$574$	−588.000	−0.0427572
$575$	3828.00	0.277632
$576$	0	0
$577$	−16306.0	−1.17648	−0.588239	−	0.808687i	$-0.700179\pi$
$577$	−16306.0	−1.17648	−0.588239	+	0.808687i	$0.700179\pi$
$578$	−8606.00	−0.619312
$579$	0	0
$580$	2196.00	0.157214
$581$	8379.00	0.598312
$582$	0	0
$583$	−10530.0	−0.748041
$584$	7544.00	0.534543
$585$	0	0
$586$	−1050.00	−0.0740189
$587$	−22971.0	−1.61519	−0.807593	−	0.589740i	$-0.799230\pi$
$587$	−22971.0	−1.61519	−0.807593	+	0.589740i	$0.799230\pi$
$588$	0	0
$589$	49981.0	3.49649
$590$	−3312.00	−0.231107
$591$	0	0
$592$	−1408.00	−0.0977507
$593$	−10983.0	−0.760570	−0.380285	−	0.924869i	$-0.624174\pi$
$593$	−10983.0	−0.760570	−0.380285	+	0.924869i	$0.624174\pi$
$594$	0	0
$595$	−2016.00	−0.138904
$596$	1368.00	0.0940192
$597$	0	0
$598$	858.000	0.0586726
$599$	−4209.00	−0.287104	−0.143552	−	0.989643i	$-0.545852\pi$
$599$	−4209.00	−0.287104	−0.143552	+	0.989643i	$0.545852\pi$
$600$	0	0
$601$	−12958.0	−0.879481	−0.439740	−	0.898125i	$-0.644930\pi$
$601$	−12958.0	−0.879481	−0.439740	+	0.898125i	$0.644930\pi$
$602$	−4942.00	−0.334586
$603$	0	0
$604$	−1072.00	−0.0722170
$605$	−4755.00	−0.319534
$606$	0	0
$607$	−11302.0	−0.755740	−0.377870	−	0.925859i	$-0.623343\pi$
$607$	−11302.0	−0.755740	−0.377870	+	0.925859i	$0.623343\pi$
$608$	4832.00	0.322308
$609$	0	0
$610$	2820.00	0.187178
$611$	6045.00	0.400253
$612$	0	0
$613$	−8188.00	−0.539495	−0.269747	−	0.962931i	$-0.586940\pi$
$613$	−8188.00	−0.539495	−0.269747	+	0.962931i	$0.586940\pi$
$614$	4154.00	0.273032
$615$	0	0
$616$	−3024.00	−0.197793
$617$	7482.00	0.488191	0.244096	−	0.969751i	$-0.421509\pi$
$617$	7482.00	0.488191	0.244096	+	0.969751i	$0.421509\pi$
$618$	0	0
$619$	18380.0	1.19346	0.596732	−	0.802440i	$-0.296465\pi$
$619$	18380.0	1.19346	0.596732	+	0.802440i	$0.296465\pi$
$620$	3972.00	0.257289
$621$	0	0
$622$	3684.00	0.237484
$623$	−5271.00	−0.338970
$624$	0	0
$625$	12331.0	0.789184
$626$	2300.00	0.146847
$627$	0	0
$628$	−8416.00	−0.534769
$629$	−8448.00	−0.535523
$630$	0	0
$631$	27470.0	1.73306	0.866532	−	0.499121i	$-0.166344\pi$
$631$	27470.0	1.73306	0.866532	+	0.499121i	$0.166344\pi$
$632$	5816.00	0.366057
$633$	0	0
$634$	−3792.00	−0.237539
$635$	4476.00	0.279724
$636$	0	0
$637$	637.000	0.0396214
$638$	19764.0	1.22643
$639$	0	0
$640$	384.000	0.0237171
$641$	−9417.00	−0.580264	−0.290132	−	0.956987i	$-0.593699\pi$
$641$	−9417.00	−0.580264	−0.290132	+	0.956987i	$0.593699\pi$
$642$	0	0
$643$	2792.00	0.171238	0.0856188	−	0.996328i	$-0.472713\pi$
$643$	2792.00	0.171238	0.0856188	+	0.996328i	$0.472713\pi$
$644$	−924.000	−0.0565384
$645$	0	0
$646$	28992.0	1.76575
$647$	21402.0	1.30046	0.650231	−	0.759736i	$-0.274672\pi$
$647$	21402.0	1.30046	0.650231	+	0.759736i	$0.274672\pi$
$648$	0	0
$649$	−29808.0	−1.80288
$650$	3016.00	0.181996
$651$	0	0
$652$	−5248.00	−0.315226
$653$	7518.00	0.450539	0.225270	−	0.974296i	$-0.427674\pi$
$653$	7518.00	0.450539	0.225270	+	0.974296i	$0.427674\pi$
$654$	0	0
$655$	5220.00	0.311393
$656$	672.000	0.0399957
$657$	0	0
$658$	−6510.00	−0.385693
$659$	23097.0	1.36530	0.682649	−	0.730746i	$-0.260828\pi$
$659$	23097.0	1.36530	0.682649	+	0.730746i	$0.260828\pi$
$660$	0	0
$661$	5033.00	0.296159	0.148079	−	0.988975i	$-0.452691\pi$
$661$	5033.00	0.296159	0.148079	+	0.988975i	$0.452691\pi$
$662$	−9868.00	−0.579352
$663$	0	0
$664$	−9576.00	−0.559670
$665$	3171.00	0.184911
$666$	0	0
$667$	6039.00	0.350571
$668$	−9876.00	−0.572027
$669$	0	0
$670$	1524.00	0.0878765
$671$	25380.0	1.46018
$672$	0	0
$673$	−14983.0	−0.858176	−0.429088	−	0.903263i	$-0.641165\pi$
$673$	−14983.0	−0.858176	−0.429088	+	0.903263i	$0.641165\pi$
$674$	11678.0	0.667388
$675$	0	0
$676$	676.000	0.0384615
$677$	−5034.00	−0.285779	−0.142889	−	0.989739i	$-0.545639\pi$
$677$	−5034.00	−0.285779	−0.142889	+	0.989739i	$0.545639\pi$
$678$	0	0
$679$	7259.00	0.410272
$680$	2304.00	0.129933
$681$	0	0
$682$	35748.0	2.00713
$683$	12048.0	0.674969	0.337485	−	0.941331i	$-0.390424\pi$
$683$	12048.0	0.674969	0.337485	+	0.941331i	$0.390424\pi$
$684$	0	0
$685$	8460.00	0.471883
$686$	−686.000	−0.0381802
$687$	0	0
$688$	5648.00	0.312977
$689$	−2535.00	−0.140168
$690$	0	0
$691$	27191.0	1.49695	0.748476	−	0.663161i	$-0.230786\pi$
$691$	27191.0	1.49695	0.748476	+	0.663161i	$0.230786\pi$
$692$	144.000	0.00791049
$693$	0	0
$694$	−912.000	−0.0498834
$695$	−222.000	−0.0121165
$696$	0	0
$697$	4032.00	0.219115
$698$	12902.0	0.699639
$699$	0	0
$700$	−3248.00	−0.175376
$701$	21015.0	1.13228	0.566138	−	0.824310i	$-0.308437\pi$
$701$	21015.0	1.13228	0.566138	+	0.824310i	$0.308437\pi$
$702$	0	0
$703$	13288.0	0.712897
$704$	3456.00	0.185018
$705$	0	0
$706$	−18684.0	−0.996008
$707$	−10962.0	−0.583124
$708$	0	0
$709$	−30382.0	−1.60934	−0.804669	−	0.593724i	$-0.797657\pi$
$709$	−30382.0	−1.60934	−0.804669	+	0.593724i	$0.797657\pi$
$710$	−792.000	−0.0418637
$711$	0	0
$712$	6024.00	0.317077
$713$	10923.0	0.573730
$714$	0	0
$715$	−2106.00	−0.110154
$716$	−7236.00	−0.377684
$717$	0	0
$718$	16656.0	0.865733
$719$	−20322.0	−1.05408	−0.527039	−	0.849841i	$-0.676698\pi$
$719$	−20322.0	−1.05408	−0.527039	+	0.849841i	$0.676698\pi$
$720$	0	0
$721$	2912.00	0.150414
$722$	−31884.0	−1.64349
$723$	0	0
$724$	4760.00	0.244343
$725$	21228.0	1.08743
$726$	0	0
$727$	27938.0	1.42526	0.712629	−	0.701541i	$-0.247504\pi$
$727$	27938.0	1.42526	0.712629	+	0.701541i	$0.247504\pi$
$728$	−728.000	−0.0370625
$729$	0	0
$730$	−5658.00	−0.286866
$731$	33888.0	1.71463
$732$	0	0
$733$	11117.0	0.560185	0.280093	−	0.959973i	$-0.409635\pi$
$733$	11117.0	0.560185	0.280093	+	0.959973i	$0.409635\pi$
$734$	4460.00	0.224280
$735$	0	0
$736$	1056.00	0.0528868
$737$	13716.0	0.685530
$738$	0	0
$739$	12314.0	0.612961	0.306480	−	0.951877i	$-0.400849\pi$
$739$	12314.0	0.612961	0.306480	+	0.951877i	$0.400849\pi$
$740$	1056.00	0.0524586
$741$	0	0
$742$	2730.00	0.135069
$743$	−25968.0	−1.28220	−0.641099	−	0.767458i	$-0.721521\pi$
$743$	−25968.0	−1.28220	−0.641099	+	0.767458i	$0.721521\pi$
$744$	0	0
$745$	−1026.00	−0.0504560
$746$	356.000	0.0174720
$747$	0	0
$748$	20736.0	1.01361
$749$	4620.00	0.225382
$750$	0	0
$751$	35093.0	1.70514	0.852571	−	0.522611i	$-0.175042\pi$
$751$	35093.0	1.70514	0.852571	+	0.522611i	$0.175042\pi$
$752$	7440.00	0.360783
$753$	0	0
$754$	4758.00	0.229809
$755$	804.000	0.0387557
$756$	0	0
$757$	12701.0	0.609809	0.304905	−	0.952383i	$-0.401375\pi$
$757$	12701.0	0.609809	0.304905	+	0.952383i	$0.401375\pi$
$758$	13748.0	0.658773
$759$	0	0
$760$	−3624.00	−0.172969
$761$	−35019.0	−1.66812	−0.834059	−	0.551675i	$-0.813989\pi$
$761$	−35019.0	−1.66812	−0.834059	+	0.551675i	$0.813989\pi$
$762$	0	0
$763$	−1750.00	−0.0830331
$764$	13536.0	0.640989
$765$	0	0
$766$	−21096.0	−0.995078
$767$	−7176.00	−0.337823
$768$	0	0
$769$	25247.0	1.18391	0.591957	−	0.805969i	$-0.298355\pi$
$769$	25247.0	1.18391	0.591957	+	0.805969i	$0.298355\pi$
$770$	2268.00	0.106147
$771$	0	0
$772$	440.000	0.0205129
$773$	−3162.00	−0.147127	−0.0735635	−	0.997291i	$-0.523437\pi$
$773$	−3162.00	−0.147127	−0.0735635	+	0.997291i	$0.523437\pi$
$774$	0	0
$775$	38396.0	1.77965
$776$	−8296.00	−0.383775
$777$	0	0
$778$	12108.0	0.557960
$779$	−6342.00	−0.291689
$780$	0	0
$781$	−7128.00	−0.326581
$782$	6336.00	0.289738
$783$	0	0
$784$	784.000	0.0357143
$785$	6312.00	0.286987
$786$	0	0
$787$	15833.0	0.717135	0.358568	−	0.933504i	$-0.383265\pi$
$787$	15833.0	0.717135	0.358568	+	0.933504i	$0.383265\pi$
$788$	15864.0	0.717172
$789$	0	0
$790$	−4362.00	−0.196447
$791$	10605.0	0.476701
$792$	0	0
$793$	6110.00	0.273610
$794$	20390.0	0.911353
$795$	0	0
$796$	−352.000	−0.0156738
$797$	20946.0	0.930923	0.465461	−	0.885068i	$-0.345888\pi$
$797$	20946.0	0.930923	0.465461	+	0.885068i	$0.345888\pi$
$798$	0	0
$799$	44640.0	1.97653
$800$	3712.00	0.164049
$801$	0	0
$802$	−9024.00	−0.397317
$803$	−50922.0	−2.23786
$804$	0	0
$805$	693.000	0.0303417
$806$	8606.00	0.376096
$807$	0	0
$808$	12528.0	0.545462
$809$	−1953.00	−0.0848749	−0.0424375	−	0.999099i	$-0.513512\pi$
$809$	−1953.00	−0.0848749	−0.0424375	+	0.999099i	$0.513512\pi$
$810$	0	0
$811$	−45412.0	−1.96625	−0.983126	−	0.182928i	$-0.941443\pi$
$811$	−45412.0	−1.96625	−0.983126	+	0.182928i	$0.941443\pi$
$812$	−5124.00	−0.221450
$813$	0	0
$814$	9504.00	0.409232
$815$	3936.00	0.169168
$816$	0	0
$817$	−53303.0	−2.28254
$818$	19022.0	0.813067
$819$	0	0
$820$	−504.000	−0.0214640
$821$	−5766.00	−0.245109	−0.122555	−	0.992462i	$-0.539109\pi$
$821$	−5766.00	−0.245109	−0.122555	+	0.992462i	$0.539109\pi$
$822$	0	0
$823$	−14560.0	−0.616682	−0.308341	−	0.951276i	$-0.599774\pi$
$823$	−14560.0	−0.616682	−0.308341	+	0.951276i	$0.599774\pi$
$824$	−3328.00	−0.140699
$825$	0	0
$826$	7728.00	0.325535
$827$	−21240.0	−0.893092	−0.446546	−	0.894761i	$-0.647346\pi$
$827$	−21240.0	−0.893092	−0.446546	+	0.894761i	$0.647346\pi$
$828$	0	0
$829$	−21922.0	−0.918435	−0.459217	−	0.888324i	$-0.651870\pi$
$829$	−21922.0	−0.918435	−0.459217	+	0.888324i	$0.651870\pi$
$830$	7182.00	0.300350
$831$	0	0
$832$	832.000	0.0346688
$833$	4704.00	0.195659
$834$	0	0
$835$	7407.00	0.306982
$836$	−32616.0	−1.34934
$837$	0	0
$838$	−10380.0	−0.427889
$839$	−11040.0	−0.454283	−0.227141	−	0.973862i	$-0.572938\pi$
$839$	−11040.0	−0.454283	−0.227141	+	0.973862i	$0.572938\pi$
$840$	0	0
$841$	9100.00	0.373119
$842$	8384.00	0.343149
$843$	0	0
$844$	−1324.00	−0.0539976
$845$	−507.000	−0.0206406
$846$	0	0
$847$	11095.0	0.450093
$848$	−3120.00	−0.126346
$849$	0	0
$850$	22272.0	0.898733
$851$	2904.00	0.116977
$852$	0	0
$853$	−27493.0	−1.10357	−0.551783	−	0.833987i	$-0.686052\pi$
$853$	−27493.0	−1.10357	−0.551783	+	0.833987i	$0.686052\pi$
$854$	−6580.00	−0.263657
$855$	0	0
$856$	−5280.00	−0.210826
$857$	13314.0	0.530686	0.265343	−	0.964154i	$-0.414515\pi$
$857$	13314.0	0.530686	0.265343	+	0.964154i	$0.414515\pi$
$858$	0	0
$859$	19550.0	0.776528	0.388264	−	0.921548i	$-0.373075\pi$
$859$	19550.0	0.776528	0.388264	+	0.921548i	$0.373075\pi$
$860$	−4236.00	−0.167961
$861$	0	0
$862$	−19836.0	−0.783778
$863$	−10740.0	−0.423631	−0.211816	−	0.977310i	$-0.567938\pi$
$863$	−10740.0	−0.423631	−0.211816	+	0.977310i	$0.567938\pi$
$864$	0	0
$865$	−108.000	−0.00424521
$866$	10184.0	0.399615
$867$	0	0
$868$	−9268.00	−0.362415
$869$	−39258.0	−1.53249
$870$	0	0
$871$	3302.00	0.128455
$872$	2000.00	0.0776704
$873$	0	0
$874$	−9966.00	−0.385704
$875$	5061.00	0.195535
$876$	0	0
$877$	14456.0	0.556607	0.278304	−	0.960493i	$-0.410228\pi$
$877$	14456.0	0.556607	0.278304	+	0.960493i	$0.410228\pi$
$878$	−580.000	−0.0222939
$879$	0	0
$880$	−2592.00	−0.0992913
$881$	9798.00	0.374691	0.187346	−	0.982294i	$-0.440012\pi$
$881$	9798.00	0.374691	0.187346	+	0.982294i	$0.440012\pi$
$882$	0	0
$883$	45308.0	1.72677	0.863384	−	0.504548i	$-0.168341\pi$
$883$	45308.0	1.72677	0.863384	+	0.504548i	$0.168341\pi$
$884$	4992.00	0.189931
$885$	0	0
$886$	−7734.00	−0.293261
$887$	−24684.0	−0.934394	−0.467197	−	0.884153i	$-0.654736\pi$
$887$	−24684.0	−0.934394	−0.467197	+	0.884153i	$0.654736\pi$
$888$	0	0
$889$	−10444.0	−0.394016
$890$	−4518.00	−0.170161
$891$	0	0
$892$	−19108.0	−0.717246
$893$	−70215.0	−2.63119
$894$	0	0
$895$	5427.00	0.202687
$896$	−896.000	−0.0334077
$897$	0	0
$898$	−8016.00	−0.297881
$899$	60573.0	2.24719
$900$	0	0
$901$	−18720.0	−0.692179
$902$	−4536.00	−0.167442
$903$	0	0
$904$	−12120.0	−0.445913
$905$	−3570.00	−0.131128
$906$	0	0
$907$	−13615.0	−0.498433	−0.249216	−	0.968448i	$-0.580173\pi$
$907$	−13615.0	−0.498433	−0.249216	+	0.968448i	$0.580173\pi$
$908$	−13584.0	−0.496477
$909$	0	0
$910$	546.000	0.0198898
$911$	15165.0	0.551525	0.275762	−	0.961226i	$-0.411070\pi$
$911$	15165.0	0.551525	0.275762	+	0.961226i	$0.411070\pi$
$912$	0	0
$913$	64638.0	2.34305
$914$	15872.0	0.574397
$915$	0	0
$916$	−20728.0	−0.747677
$917$	−12180.0	−0.438625
$918$	0	0
$919$	−25576.0	−0.918035	−0.459018	−	0.888427i	$-0.651799\pi$
$919$	−25576.0	−0.918035	−0.459018	+	0.888427i	$0.651799\pi$
$920$	−792.000	−0.0283820
$921$	0	0
$922$	19884.0	0.710244
$923$	−1716.00	−0.0611948
$924$	0	0
$925$	10208.0	0.362851
$926$	−31828.0	−1.12952
$927$	0	0
$928$	5856.00	0.207147
$929$	−24759.0	−0.874399	−0.437199	−	0.899365i	$-0.644030\pi$
$929$	−24759.0	−0.874399	−0.437199	+	0.899365i	$0.644030\pi$
$930$	0	0
$931$	−7399.00	−0.260464
$932$	20004.0	0.703061
$933$	0	0
$934$	20868.0	0.731073
$935$	−15552.0	−0.543962
$936$	0	0
$937$	−12976.0	−0.452409	−0.226205	−	0.974080i	$-0.572632\pi$
$937$	−12976.0	−0.452409	−0.226205	+	0.974080i	$0.572632\pi$
$938$	−3556.00	−0.123782
$939$	0	0
$940$	−5580.00	−0.193617
$941$	4575.00	0.158492	0.0792459	−	0.996855i	$-0.474749\pi$
$941$	4575.00	0.158492	0.0792459	+	0.996855i	$0.474749\pi$
$942$	0	0
$943$	−1386.00	−0.0478625
$944$	−8832.00	−0.304510
$945$	0	0
$946$	−38124.0	−1.31027
$947$	53346.0	1.83053	0.915265	−	0.402852i	$-0.131981\pi$
$947$	53346.0	1.83053	0.915265	+	0.402852i	$0.131981\pi$
$948$	0	0
$949$	−12259.0	−0.419330
$950$	−35032.0	−1.19641
$951$	0	0
$952$	−5376.00	−0.183022
$953$	−52809.0	−1.79502	−0.897509	−	0.440997i	$-0.854625\pi$
$953$	−52809.0	−1.79502	−0.897509	+	0.440997i	$0.854625\pi$
$954$	0	0
$955$	−10152.0	−0.343991
$956$	−9360.00	−0.316657
$957$	0	0
$958$	−14310.0	−0.482605
$959$	−19740.0	−0.664690
$960$	0	0
$961$	79770.0	2.67765
$962$	2288.00	0.0766820
$963$	0	0
$964$	22580.0	0.754412
$965$	−330.000	−0.0110084
$966$	0	0
$967$	7346.00	0.244293	0.122147	−	0.992512i	$-0.461022\pi$
$967$	7346.00	0.244293	0.122147	+	0.992512i	$0.461022\pi$
$968$	−12680.0	−0.421023
$969$	0	0
$970$	6222.00	0.205955
$971$	−37170.0	−1.22847	−0.614234	−	0.789124i	$-0.710535\pi$
$971$	−37170.0	−1.22847	−0.614234	+	0.789124i	$0.710535\pi$
$972$	0	0
$973$	518.000	0.0170671
$974$	33764.0	1.11075
$975$	0	0
$976$	7520.00	0.246628
$977$	29358.0	0.961357	0.480678	−	0.876897i	$-0.340390\pi$
$977$	29358.0	0.961357	0.480678	+	0.876897i	$0.340390\pi$
$978$	0	0
$979$	−40662.0	−1.32744
$980$	−588.000	−0.0191663
$981$	0	0
$982$	−20280.0	−0.659023
$983$	−21777.0	−0.706590	−0.353295	−	0.935512i	$-0.614939\pi$
$983$	−21777.0	−0.706590	−0.353295	+	0.935512i	$0.614939\pi$
$984$	0	0
$985$	−11898.0	−0.384875
$986$	35136.0	1.13485
$987$	0	0
$988$	−7852.00	−0.252839
$989$	−11649.0	−0.374537
$990$	0	0
$991$	10388.0	0.332983	0.166491	−	0.986043i	$-0.446756\pi$
$991$	10388.0	0.332983	0.166491	+	0.986043i	$0.446756\pi$
$992$	10592.0	0.339008
$993$	0	0
$994$	1848.00	0.0589688
$995$	264.000	0.00841142
$996$	0	0
$997$	27776.0	0.882322	0.441161	−	0.897428i	$-0.354567\pi$
$997$	27776.0	0.882322	0.441161	+	0.897428i	$0.354567\pi$
$998$	18608.0	0.590206
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.4.a.d.1.1		1
3.2	odd	2		182.4.a.b.1.1	✓	1
12.11	even	2		1456.4.a.g.1.1		1
21.20	even	2		1274.4.a.i.1.1		1
39.38	odd	2		2366.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.4.a.b.1.1	✓	1	3.2	odd	2
1274.4.a.i.1.1		1	21.20	even	2
1456.4.a.g.1.1		1	12.11	even	2
1638.4.a.d.1.1		1	1.1	even	1	trivial
2366.4.a.a.1.1		1	39.38	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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