LMFDB - Embedded newform 845.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,4,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("845.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	845.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:	$49.8566139549$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-5.00000 q^{2} +2.00000 q^{3} +17.0000 q^{4} +5.00000 q^{5} -10.0000 q^{6} +12.0000 q^{7} -45.0000 q^{8} -23.0000 q^{9} +O(q^{10})$

\(q-5.00000 q^{2} +2.00000 q^{3} +17.0000 q^{4} +5.00000 q^{5} -10.0000 q^{6} +12.0000 q^{7} -45.0000 q^{8} -23.0000 q^{9} -25.0000 q^{10} -14.0000 q^{11} +34.0000 q^{12} -60.0000 q^{14} +10.0000 q^{15} +89.0000 q^{16} +98.0000 q^{17} +115.000 q^{18} +26.0000 q^{19} +85.0000 q^{20} +24.0000 q^{21} +70.0000 q^{22} -114.000 q^{23} -90.0000 q^{24} +25.0000 q^{25} -100.000 q^{27} +204.000 q^{28} +58.0000 q^{29} -50.0000 q^{30} -306.000 q^{31} -85.0000 q^{32} -28.0000 q^{33} -490.000 q^{34} +60.0000 q^{35} -391.000 q^{36} -86.0000 q^{37} -130.000 q^{38} -225.000 q^{40} +374.000 q^{41} -120.000 q^{42} -314.000 q^{43} -238.000 q^{44} -115.000 q^{45} +570.000 q^{46} -620.000 q^{47} +178.000 q^{48} -199.000 q^{49} -125.000 q^{50} +196.000 q^{51} +362.000 q^{53} +500.000 q^{54} -70.0000 q^{55} -540.000 q^{56} +52.0000 q^{57} -290.000 q^{58} -266.000 q^{59} +170.000 q^{60} +634.000 q^{61} +1530.00 q^{62} -276.000 q^{63} -287.000 q^{64} +140.000 q^{66} -612.000 q^{67} +1666.00 q^{68} -228.000 q^{69} -300.000 q^{70} +686.000 q^{71} +1035.00 q^{72} -202.000 q^{73} +430.000 q^{74} +50.0000 q^{75} +442.000 q^{76} -168.000 q^{77} -516.000 q^{79} +445.000 q^{80} +421.000 q^{81} -1870.00 q^{82} -48.0000 q^{83} +408.000 q^{84} +490.000 q^{85} +1570.00 q^{86} +116.000 q^{87} +630.000 q^{88} +1230.00 q^{89} +575.000 q^{90} -1938.00 q^{92} -612.000 q^{93} +3100.00 q^{94} +130.000 q^{95} -170.000 q^{96} -350.000 q^{97} +995.000 q^{98} +322.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−5.00000	−1.76777	−0.883883	−	0.467707i	$-0.845080\pi$
$2$	−5.00000	−1.76777	−0.883883	+	0.467707i	$0.845080\pi$
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	17.0000	2.12500
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−10.0000	−0.680414
$7$	12.0000	0.647939	0.323970	−	0.946068i	$-0.394982\pi$
$7$	12.0000	0.647939	0.323970	+	0.946068i	$0.394982\pi$
$8$	−45.0000	−1.98874
$9$	−23.0000	−0.851852
$10$	−25.0000	−0.790569
$11$	−14.0000	−0.383742	−0.191871	−	0.981420i	$-0.561455\pi$
$11$	−14.0000	−0.383742	−0.191871	+	0.981420i	$0.561455\pi$
$12$	34.0000	0.817913
$13$	0	0
$13$	0	0
$14$	−60.0000	−1.14541
$15$	10.0000	0.172133
$16$	89.0000	1.39062
$17$	98.0000	1.39815	0.699073	−	0.715050i	$-0.253596\pi$
$17$	98.0000	1.39815	0.699073	+	0.715050i	$0.253596\pi$
$18$	115.000	1.50588
$19$	26.0000	0.313937	0.156969	−	0.987604i	$-0.449828\pi$
$19$	26.0000	0.313937	0.156969	+	0.987604i	$0.449828\pi$
$20$	85.0000	0.950329
$21$	24.0000	0.249392
$22$	70.0000	0.678366
$23$	−114.000	−1.03351	−0.516753	−	0.856134i	$-0.672859\pi$
$23$	−114.000	−1.03351	−0.516753	+	0.856134i	$0.672859\pi$
$24$	−90.0000	−0.765466
$25$	25.0000	0.200000
$26$	0	0
$27$	−100.000	−0.712778
$28$	204.000	1.37687
$29$	58.0000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	58.0000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−50.0000	−0.304290
$31$	−306.000	−1.77288	−0.886439	−	0.462845i	$-0.846829\pi$
$31$	−306.000	−1.77288	−0.886439	+	0.462845i	$0.846829\pi$
$32$	−85.0000	−0.469563
$33$	−28.0000	−0.147702
$34$	−490.000	−2.47160
$35$	60.0000	0.289767
$36$	−391.000	−1.81019
$37$	−86.0000	−0.382117	−0.191058	−	0.981579i	$-0.561192\pi$
$37$	−86.0000	−0.382117	−0.191058	+	0.981579i	$0.561192\pi$
$38$	−130.000	−0.554968
$39$	0	0
$40$	−225.000	−0.889391
$41$	374.000	1.42461	0.712305	−	0.701870i	$-0.247651\pi$
$41$	374.000	1.42461	0.712305	+	0.701870i	$0.247651\pi$
$42$	−120.000	−0.440867
$43$	−314.000	−1.11359	−0.556797	−	0.830649i	$-0.687970\pi$
$43$	−314.000	−1.11359	−0.556797	+	0.830649i	$0.687970\pi$
$44$	−238.000	−0.815451
$45$	−115.000	−0.380960
$46$	570.000	1.82700
$47$	−620.000	−1.92418	−0.962088	−	0.272738i	$-0.912071\pi$
$47$	−620.000	−1.92418	−0.962088	+	0.272738i	$0.912071\pi$
$48$	178.000	0.535252
$49$	−199.000	−0.580175
$50$	−125.000	−0.353553
$51$	196.000	0.538147
$52$	0	0
$53$	362.000	0.938199	0.469099	−	0.883145i	$-0.344579\pi$
$53$	362.000	0.938199	0.469099	+	0.883145i	$0.344579\pi$
$54$	500.000	1.26003
$55$	−70.0000	−0.171615
$56$	−540.000	−1.28858
$57$	52.0000	0.120835
$58$	−290.000	−0.656532
$59$	−266.000	−0.586953	−0.293477	−	0.955966i	$-0.594812\pi$
$59$	−266.000	−0.586953	−0.293477	+	0.955966i	$0.594812\pi$
$60$	170.000	0.365782
$61$	634.000	1.33074	0.665372	−	0.746512i	$-0.268273\pi$
$61$	634.000	1.33074	0.665372	+	0.746512i	$0.268273\pi$
$62$	1530.00	3.13404
$63$	−276.000	−0.551948
$64$	−287.000	−0.560547
$65$	0	0
$66$	140.000	0.261103
$67$	−612.000	−1.11594	−0.557968	−	0.829863i	$-0.688419\pi$
$67$	−612.000	−1.11594	−0.557968	+	0.829863i	$0.688419\pi$
$68$	1666.00	2.97106
$69$	−228.000	−0.397797
$70$	−300.000	−0.512241
$71$	686.000	1.14667	0.573333	−	0.819323i	$-0.305650\pi$
$71$	686.000	1.14667	0.573333	+	0.819323i	$0.305650\pi$
$72$	1035.00	1.69411
$73$	−202.000	−0.323867	−0.161934	−	0.986802i	$-0.551773\pi$
$73$	−202.000	−0.323867	−0.161934	+	0.986802i	$0.551773\pi$
$74$	430.000	0.675493
$75$	50.0000	0.0769800
$76$	442.000	0.667117
$77$	−168.000	−0.248641
$78$	0	0
$79$	−516.000	−0.734868	−0.367434	−	0.930050i	$-0.619764\pi$
$79$	−516.000	−0.734868	−0.367434	+	0.930050i	$0.619764\pi$
$80$	445.000	0.621906
$81$	421.000	0.577503
$82$	−1870.00	−2.51838
$83$	−48.0000	−0.0634781	−0.0317391	−	0.999496i	$-0.510105\pi$
$83$	−48.0000	−0.0634781	−0.0317391	+	0.999496i	$0.510105\pi$
$84$	408.000	0.529958
$85$	490.000	0.625270
$86$	1570.00	1.96858
$87$	116.000	0.142948
$88$	630.000	0.763162
$89$	1230.00	1.46494	0.732470	−	0.680799i	$-0.238367\pi$
$89$	1230.00	1.46494	0.732470	+	0.680799i	$0.238367\pi$
$90$	575.000	0.673448
$91$	0	0
$92$	−1938.00	−2.19620
$93$	−612.000	−0.682381
$94$	3100.00	3.40150
$95$	130.000	0.140397
$96$	−170.000	−0.180735
$97$	−350.000	−0.366362	−0.183181	−	0.983079i	$-0.558639\pi$
$97$	−350.000	−0.366362	−0.183181	+	0.983079i	$0.558639\pi$
$98$	995.000	1.02561
$99$	322.000	0.326891
$100$	425.000	0.425000
$101$	−1530.00	−1.50733	−0.753667	−	0.657257i	$-0.771717\pi$
$101$	−1530.00	−1.50733	−0.753667	+	0.657257i	$0.771717\pi$
$102$	−980.000	−0.951318
$103$	−58.0000	−0.0554846	−0.0277423	−	0.999615i	$-0.508832\pi$
$103$	−58.0000	−0.0554846	−0.0277423	+	0.999615i	$0.508832\pi$
$104$	0	0
$105$	120.000	0.111531
$106$	−1810.00	−1.65852
$107$	−186.000	−0.168050	−0.0840248	−	0.996464i	$-0.526777\pi$
$107$	−186.000	−0.168050	−0.0840248	+	0.996464i	$0.526777\pi$
$108$	−1700.00	−1.51465
$109$	1342.00	1.17927	0.589634	−	0.807670i	$-0.299272\pi$
$109$	1342.00	1.17927	0.589634	+	0.807670i	$0.299272\pi$
$110$	350.000	0.303374
$111$	−172.000	−0.147077
$112$	1068.00	0.901040
$113$	−1966.00	−1.63669	−0.818344	−	0.574729i	$-0.805108\pi$
$113$	−1966.00	−1.63669	−0.818344	+	0.574729i	$0.805108\pi$
$114$	−260.000	−0.213607
$115$	−570.000	−0.462198
$116$	986.000	0.789205
$117$	0	0
$118$	1330.00	1.03760
$119$	1176.00	0.905914
$120$	−450.000	−0.342327
$121$	−1135.00	−0.852742
$122$	−3170.00	−2.35245
$123$	748.000	0.548332
$124$	−5202.00	−3.76737
$125$	125.000	0.0894427
$126$	1380.00	0.975716
$127$	1738.00	1.21435	0.607175	−	0.794568i	$-0.292303\pi$
$127$	1738.00	1.21435	0.607175	+	0.794568i	$0.292303\pi$
$128$	2115.00	1.46048
$129$	−628.000	−0.428623
$130$	0	0
$131$	780.000	0.520221	0.260110	−	0.965579i	$-0.416241\pi$
$131$	780.000	0.520221	0.260110	+	0.965579i	$0.416241\pi$
$132$	−476.000	−0.313867
$133$	312.000	0.203412
$134$	3060.00	1.97271
$135$	−500.000	−0.318764
$136$	−4410.00	−2.78055
$137$	1074.00	0.669767	0.334883	−	0.942260i	$-0.391303\pi$
$137$	1074.00	0.669767	0.334883	+	0.942260i	$0.391303\pi$
$138$	1140.00	0.703212
$139$	416.000	0.253846	0.126923	−	0.991913i	$-0.459490\pi$
$139$	416.000	0.253846	0.126923	+	0.991913i	$0.459490\pi$
$140$	1020.00	0.615755
$141$	−1240.00	−0.740616
$142$	−3430.00	−2.02704
$143$	0	0
$144$	−2047.00	−1.18461
$145$	290.000	0.166091
$146$	1010.00	0.572522
$147$	−398.000	−0.223309
$148$	−1462.00	−0.811998
$149$	2054.00	1.12933	0.564665	−	0.825320i	$-0.309005\pi$
$149$	2054.00	1.12933	0.564665	+	0.825320i	$0.309005\pi$
$150$	−250.000	−0.136083
$151$	−2302.00	−1.24062	−0.620312	−	0.784355i	$-0.712994\pi$
$151$	−2302.00	−1.24062	−0.620312	+	0.784355i	$0.712994\pi$
$152$	−1170.00	−0.624339
$153$	−2254.00	−1.19101
$154$	840.000	0.439540
$155$	−1530.00	−0.792855
$156$	0	0
$157$	−2350.00	−1.19459	−0.597294	−	0.802022i	$-0.703758\pi$
$157$	−2350.00	−1.19459	−0.597294	+	0.802022i	$0.703758\pi$
$158$	2580.00	1.29907
$159$	724.000	0.361113
$160$	−425.000	−0.209995
$161$	−1368.00	−0.669649
$162$	−2105.00	−1.02089
$163$	−2924.00	−1.40506	−0.702532	−	0.711652i	$-0.747947\pi$
$163$	−2924.00	−1.40506	−0.702532	+	0.711652i	$0.747947\pi$
$164$	6358.00	3.02730
$165$	−140.000	−0.0660545
$166$	240.000	0.112215
$167$	−3036.00	−1.40678	−0.703391	−	0.710803i	$-0.748332\pi$
$167$	−3036.00	−1.40678	−0.703391	+	0.710803i	$0.748332\pi$
$168$	−1080.00	−0.495975
$169$	0	0
$170$	−2450.00	−1.10533
$171$	−598.000	−0.267428
$172$	−5338.00	−2.36639
$173$	2226.00	0.978264	0.489132	−	0.872210i	$-0.337314\pi$
$173$	2226.00	0.978264	0.489132	+	0.872210i	$0.337314\pi$
$174$	−580.000	−0.252699
$175$	300.000	0.129588
$176$	−1246.00	−0.533641
$177$	−532.000	−0.225918
$178$	−6150.00	−2.58967
$179$	−3244.00	−1.35457	−0.677285	−	0.735721i	$-0.736843\pi$
$179$	−3244.00	−1.35457	−0.677285	+	0.735721i	$0.736843\pi$
$180$	−1955.00	−0.809539
$181$	−1678.00	−0.689087	−0.344544	−	0.938770i	$-0.611966\pi$
$181$	−1678.00	−0.689087	−0.344544	+	0.938770i	$0.611966\pi$
$182$	0	0
$183$	1268.00	0.512204
$184$	5130.00	2.05537
$185$	−430.000	−0.170888
$186$	3060.00	1.20629
$187$	−1372.00	−0.536527
$188$	−10540.0	−4.08888
$189$	−1200.00	−0.461837
$190$	−650.000	−0.248189
$191$	−3968.00	−1.50322	−0.751608	−	0.659610i	$-0.770722\pi$
$191$	−3968.00	−1.50322	−0.751608	+	0.659610i	$0.770722\pi$
$192$	−574.000	−0.215755
$193$	−3998.00	−1.49110	−0.745550	−	0.666450i	$-0.767813\pi$
$193$	−3998.00	−1.49110	−0.745550	+	0.666450i	$0.767813\pi$
$194$	1750.00	0.647643
$195$	0	0
$196$	−3383.00	−1.23287
$197$	−4194.00	−1.51680	−0.758401	−	0.651788i	$-0.774019\pi$
$197$	−4194.00	−1.51680	−0.758401	+	0.651788i	$0.774019\pi$
$198$	−1610.00	−0.577867
$199$	4240.00	1.51038	0.755190	−	0.655506i	$-0.227545\pi$
$199$	4240.00	1.51038	0.755190	+	0.655506i	$0.227545\pi$
$200$	−1125.00	−0.397748
$201$	−1224.00	−0.429524
$202$	7650.00	2.66461
$203$	696.000	0.240639
$204$	3332.00	1.14356
$205$	1870.00	0.637105
$206$	290.000	0.0980838
$207$	2622.00	0.880394
$208$	0	0
$209$	−364.000	−0.120471
$210$	−600.000	−0.197162
$211$	596.000	0.194457	0.0972283	−	0.995262i	$-0.469002\pi$
$211$	596.000	0.194457	0.0972283	+	0.995262i	$0.469002\pi$
$212$	6154.00	1.99367
$213$	1372.00	0.441352
$214$	930.000	0.297072
$215$	−1570.00	−0.498014
$216$	4500.00	1.41753
$217$	−3672.00	−1.14872
$218$	−6710.00	−2.08467
$219$	−404.000	−0.124657
$220$	−1190.00	−0.364681
$221$	0	0
$222$	860.000	0.259997
$223$	−4268.00	−1.28164	−0.640822	−	0.767690i	$-0.721406\pi$
$223$	−4268.00	−1.28164	−0.640822	+	0.767690i	$0.721406\pi$
$224$	−1020.00	−0.304248
$225$	−575.000	−0.170370
$226$	9830.00	2.89328
$227$	5924.00	1.73211	0.866057	−	0.499946i	$-0.166647\pi$
$227$	5924.00	1.73211	0.866057	+	0.499946i	$0.166647\pi$
$228$	884.000	0.256773
$229$	750.000	0.216425	0.108213	−	0.994128i	$-0.465487\pi$
$229$	750.000	0.216425	0.108213	+	0.994128i	$0.465487\pi$
$230$	2850.00	0.817058
$231$	−336.000	−0.0957021
$232$	−2610.00	−0.738599
$233$	474.000	0.133274	0.0666369	−	0.997777i	$-0.478773\pi$
$233$	474.000	0.133274	0.0666369	+	0.997777i	$0.478773\pi$
$234$	0	0
$235$	−3100.00	−0.860518
$236$	−4522.00	−1.24728
$237$	−1032.00	−0.282851
$238$	−5880.00	−1.60144
$239$	−1598.00	−0.432494	−0.216247	−	0.976339i	$-0.569382\pi$
$239$	−1598.00	−0.432494	−0.216247	+	0.976339i	$0.569382\pi$
$240$	890.000	0.239372
$241$	−2410.00	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−2410.00	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	5675.00	1.50745
$243$	3542.00	0.935059
$244$	10778.0	2.82783
$245$	−995.000	−0.259462
$246$	−3740.00	−0.969324
$247$	0	0
$248$	13770.0	3.52579
$249$	−96.0000	−0.0244327
$250$	−625.000	−0.158114
$251$	−312.000	−0.0784592	−0.0392296	−	0.999230i	$-0.512490\pi$
$251$	−312.000	−0.0784592	−0.0392296	+	0.999230i	$0.512490\pi$
$252$	−4692.00	−1.17289
$253$	1596.00	0.396599
$254$	−8690.00	−2.14669
$255$	980.000	0.240667
$256$	−8279.00	−2.02124
$257$	−5886.00	−1.42863	−0.714316	−	0.699823i	$-0.753262\pi$
$257$	−5886.00	−1.42863	−0.714316	+	0.699823i	$0.753262\pi$
$258$	3140.00	0.757705
$259$	−1032.00	−0.247588
$260$	0	0
$261$	−1334.00	−0.316370
$262$	−3900.00	−0.919629
$263$	1258.00	0.294949	0.147475	−	0.989066i	$-0.452886\pi$
$263$	1258.00	0.294949	0.147475	+	0.989066i	$0.452886\pi$
$264$	1260.00	0.293741
$265$	1810.00	0.419575
$266$	−1560.00	−0.359585
$267$	2460.00	0.563856
$268$	−10404.0	−2.37136
$269$	2414.00	0.547153	0.273577	−	0.961850i	$-0.411793\pi$
$269$	2414.00	0.547153	0.273577	+	0.961850i	$0.411793\pi$
$270$	2500.00	0.563501
$271$	−598.000	−0.134044	−0.0670220	−	0.997751i	$-0.521350\pi$
$271$	−598.000	−0.134044	−0.0670220	+	0.997751i	$0.521350\pi$
$272$	8722.00	1.94430
$273$	0	0
$274$	−5370.00	−1.18399
$275$	−350.000	−0.0767483
$276$	−3876.00	−0.845318
$277$	−4710.00	−1.02165	−0.510824	−	0.859685i	$-0.670660\pi$
$277$	−4710.00	−1.02165	−0.510824	+	0.859685i	$0.670660\pi$
$278$	−2080.00	−0.448741
$279$	7038.00	1.51023
$280$	−2700.00	−0.576271
$281$	−4266.00	−0.905652	−0.452826	−	0.891599i	$-0.649584\pi$
$281$	−4266.00	−0.905652	−0.452826	+	0.891599i	$0.649584\pi$
$282$	6200.00	1.30924
$283$	−978.000	−0.205428	−0.102714	−	0.994711i	$-0.532753\pi$
$283$	−978.000	−0.205428	−0.102714	+	0.994711i	$0.532753\pi$
$284$	11662.0	2.43666
$285$	260.000	0.0540388
$286$	0	0
$287$	4488.00	0.923060
$288$	1955.00	0.399998
$289$	4691.00	0.954814
$290$	−1450.00	−0.293610
$291$	−700.000	−0.141013
$292$	−3434.00	−0.688218
$293$	882.000	0.175860	0.0879300	−	0.996127i	$-0.471975\pi$
$293$	882.000	0.175860	0.0879300	+	0.996127i	$0.471975\pi$
$294$	1990.00	0.394759
$295$	−1330.00	−0.262494
$296$	3870.00	0.759930
$297$	1400.00	0.273523
$298$	−10270.0	−1.99639
$299$	0	0
$300$	850.000	0.163583
$301$	−3768.00	−0.721541
$302$	11510.0	2.19313
$303$	−3060.00	−0.580173
$304$	2314.00	0.436569
$305$	3170.00	0.595127
$306$	11270.0	2.10543
$307$	−1816.00	−0.337605	−0.168802	−	0.985650i	$-0.553990\pi$
$307$	−1816.00	−0.337605	−0.168802	+	0.985650i	$0.553990\pi$
$308$	−2856.00	−0.528363
$309$	−116.000	−0.0213560
$310$	7650.00	1.40158
$311$	2292.00	0.417902	0.208951	−	0.977926i	$-0.432995\pi$
$311$	2292.00	0.417902	0.208951	+	0.977926i	$0.432995\pi$
$312$	0	0
$313$	8474.00	1.53028	0.765142	−	0.643862i	$-0.222669\pi$
$313$	8474.00	1.53028	0.765142	+	0.643862i	$0.222669\pi$
$314$	11750.0	2.11175
$315$	−1380.00	−0.246839
$316$	−8772.00	−1.56159
$317$	4586.00	0.812541	0.406270	−	0.913753i	$-0.366829\pi$
$317$	4586.00	0.812541	0.406270	+	0.913753i	$0.366829\pi$
$318$	−3620.00	−0.638363
$319$	−812.000	−0.142518
$320$	−1435.00	−0.250684
$321$	−372.000	−0.0646823
$322$	6840.00	1.18378
$323$	2548.00	0.438930
$324$	7157.00	1.22719
$325$	0	0
$326$	14620.0	2.48382
$327$	2684.00	0.453901
$328$	−16830.0	−2.83317
$329$	−7440.00	−1.24675
$330$	700.000	0.116769
$331$	2046.00	0.339753	0.169877	−	0.985465i	$-0.445663\pi$
$331$	2046.00	0.339753	0.169877	+	0.985465i	$0.445663\pi$
$332$	−816.000	−0.134891
$333$	1978.00	0.325507
$334$	15180.0	2.48686
$335$	−3060.00	−0.499062
$336$	2136.00	0.346811
$337$	−1286.00	−0.207872	−0.103936	−	0.994584i	$-0.533144\pi$
$337$	−1286.00	−0.207872	−0.103936	+	0.994584i	$0.533144\pi$
$338$	0	0
$339$	−3932.00	−0.629961
$340$	8330.00	1.32870
$341$	4284.00	0.680327
$342$	2990.00	0.472751
$343$	−6504.00	−1.02386
$344$	14130.0	2.21465
$345$	−1140.00	−0.177900
$346$	−11130.0	−1.72934
$347$	−634.000	−0.0980833	−0.0490416	−	0.998797i	$-0.515617\pi$
$347$	−634.000	−0.0980833	−0.0490416	+	0.998797i	$0.515617\pi$
$348$	1972.00	0.303765
$349$	−2266.00	−0.347554	−0.173777	−	0.984785i	$-0.555597\pi$
$349$	−2266.00	−0.347554	−0.173777	+	0.984785i	$0.555597\pi$
$350$	−1500.00	−0.229081
$351$	0	0
$352$	1190.00	0.180191
$353$	−210.000	−0.0316634	−0.0158317	−	0.999875i	$-0.505040\pi$
$353$	−210.000	−0.0316634	−0.0158317	+	0.999875i	$0.505040\pi$
$354$	2660.00	0.399371
$355$	3430.00	0.512804
$356$	20910.0	3.11300
$357$	2352.00	0.348686
$358$	16220.0	2.39456
$359$	−162.000	−0.0238162	−0.0119081	−	0.999929i	$-0.503791\pi$
$359$	−162.000	−0.0238162	−0.0119081	+	0.999929i	$0.503791\pi$
$360$	5175.00	0.757629
$361$	−6183.00	−0.901443
$362$	8390.00	1.21815
$363$	−2270.00	−0.328221
$364$	0	0
$365$	−1010.00	−0.144838
$366$	−6340.00	−0.905457
$367$	678.000	0.0964341	0.0482170	−	0.998837i	$-0.484646\pi$
$367$	678.000	0.0964341	0.0482170	+	0.998837i	$0.484646\pi$
$368$	−10146.0	−1.43722
$369$	−8602.00	−1.21356
$370$	2150.00	0.302090
$371$	4344.00	0.607896
$372$	−10404.0	−1.45006
$373$	−6902.00	−0.958102	−0.479051	−	0.877787i	$-0.659019\pi$
$373$	−6902.00	−0.958102	−0.479051	+	0.877787i	$0.659019\pi$
$374$	6860.00	0.948455
$375$	250.000	0.0344265
$376$	27900.0	3.82668
$377$	0	0
$378$	6000.00	0.816420
$379$	−8998.00	−1.21952	−0.609758	−	0.792588i	$-0.708733\pi$
$379$	−8998.00	−1.21952	−0.609758	+	0.792588i	$0.708733\pi$
$380$	2210.00	0.298344
$381$	3476.00	0.467404
$382$	19840.0	2.65734
$383$	−132.000	−0.0176107	−0.00880533	−	0.999961i	$-0.502803\pi$
$383$	−132.000	−0.0176107	−0.00880533	+	0.999961i	$0.502803\pi$
$384$	4230.00	0.562139
$385$	−840.000	−0.111196
$386$	19990.0	2.63592
$387$	7222.00	0.948617
$388$	−5950.00	−0.778519
$389$	11694.0	1.52419	0.762094	−	0.647466i	$-0.224171\pi$
$389$	11694.0	1.52419	0.762094	+	0.647466i	$0.224171\pi$
$390$	0	0
$391$	−11172.0	−1.44499
$392$	8955.00	1.15382
$393$	1560.00	0.200233
$394$	20970.0	2.68135
$395$	−2580.00	−0.328643
$396$	5474.00	0.694644
$397$	−13678.0	−1.72917	−0.864583	−	0.502490i	$-0.832418\pi$
$397$	−13678.0	−1.72917	−0.864583	+	0.502490i	$0.832418\pi$
$398$	−21200.0	−2.67000
$399$	624.000	0.0782934
$400$	2225.00	0.278125
$401$	−8490.00	−1.05728	−0.528641	−	0.848845i	$-0.677298\pi$
$401$	−8490.00	−1.05728	−0.528641	+	0.848845i	$0.677298\pi$
$402$	6120.00	0.759298
$403$	0	0
$404$	−26010.0	−3.20308
$405$	2105.00	0.258267
$406$	−3480.00	−0.425393
$407$	1204.00	0.146634
$408$	−8820.00	−1.07023
$409$	9982.00	1.20679	0.603396	−	0.797442i	$-0.293814\pi$
$409$	9982.00	1.20679	0.603396	+	0.797442i	$0.293814\pi$
$410$	−9350.00	−1.12625
$411$	2148.00	0.257793
$412$	−986.000	−0.117905
$413$	−3192.00	−0.380310
$414$	−13110.0	−1.55633
$415$	−240.000	−0.0283883
$416$	0	0
$417$	832.000	0.0977056
$418$	1820.00	0.212964
$419$	−9848.00	−1.14823	−0.574113	−	0.818776i	$-0.694653\pi$
$419$	−9848.00	−1.14823	−0.574113	+	0.818776i	$0.694653\pi$
$420$	2040.00	0.237004
$421$	−7666.00	−0.887454	−0.443727	−	0.896162i	$-0.646344\pi$
$421$	−7666.00	−0.887454	−0.443727	+	0.896162i	$0.646344\pi$
$422$	−2980.00	−0.343754
$423$	14260.0	1.63911
$424$	−16290.0	−1.86583
$425$	2450.00	0.279629
$426$	−6860.00	−0.780207
$427$	7608.00	0.862241
$428$	−3162.00	−0.357105
$429$	0	0
$430$	7850.00	0.880374
$431$	8742.00	0.977001	0.488500	−	0.872564i	$-0.337544\pi$
$431$	8742.00	0.977001	0.488500	+	0.872564i	$0.337544\pi$
$432$	−8900.00	−0.991207
$433$	−5654.00	−0.627515	−0.313757	−	0.949503i	$-0.601588\pi$
$433$	−5654.00	−0.627515	−0.313757	+	0.949503i	$0.601588\pi$
$434$	18360.0	2.03066
$435$	580.000	0.0639284
$436$	22814.0	2.50595
$437$	−2964.00	−0.324456
$438$	2020.00	0.220364
$439$	6976.00	0.758420	0.379210	−	0.925311i	$-0.376196\pi$
$439$	6976.00	0.758420	0.379210	+	0.925311i	$0.376196\pi$
$440$	3150.00	0.341296
$441$	4577.00	0.494223
$442$	0	0
$443$	66.0000	0.00707845	0.00353923	−	0.999994i	$-0.498873\pi$
$443$	66.0000	0.00707845	0.00353923	+	0.999994i	$0.498873\pi$
$444$	−2924.00	−0.312538
$445$	6150.00	0.655141
$446$	21340.0	2.26565
$447$	4108.00	0.434679
$448$	−3444.00	−0.363200
$449$	−2426.00	−0.254989	−0.127494	−	0.991839i	$-0.540694\pi$
$449$	−2426.00	−0.254989	−0.127494	+	0.991839i	$0.540694\pi$
$450$	2875.00	0.301175
$451$	−5236.00	−0.546682
$452$	−33422.0	−3.47796
$453$	−4604.00	−0.477516
$454$	−29620.0	−3.06197
$455$	0	0
$456$	−2340.00	−0.240308
$457$	2186.00	0.223757	0.111878	−	0.993722i	$-0.464313\pi$
$457$	2186.00	0.223757	0.111878	+	0.993722i	$0.464313\pi$
$458$	−3750.00	−0.382590
$459$	−9800.00	−0.996568
$460$	−9690.00	−0.982171
$461$	−2370.00	−0.239440	−0.119720	−	0.992808i	$-0.538200\pi$
$461$	−2370.00	−0.239440	−0.119720	+	0.992808i	$0.538200\pi$
$462$	1680.00	0.169179
$463$	−11832.0	−1.18765	−0.593823	−	0.804596i	$-0.702382\pi$
$463$	−11832.0	−1.18765	−0.593823	+	0.804596i	$0.702382\pi$
$464$	5162.00	0.516465
$465$	−3060.00	−0.305170
$466$	−2370.00	−0.235597
$467$	−10070.0	−0.997824	−0.498912	−	0.866653i	$-0.666267\pi$
$467$	−10070.0	−0.997824	−0.498912	+	0.866653i	$0.666267\pi$
$468$	0	0
$469$	−7344.00	−0.723058
$470$	15500.0	1.52120
$471$	−4700.00	−0.459797
$472$	11970.0	1.16730
$473$	4396.00	0.427333
$474$	5160.00	0.500014
$475$	650.000	0.0627875
$476$	19992.0	1.92507
$477$	−8326.00	−0.799206
$478$	7990.00	0.764548
$479$	−15402.0	−1.46918	−0.734588	−	0.678513i	$-0.762625\pi$
$479$	−15402.0	−1.46918	−0.734588	+	0.678513i	$0.762625\pi$
$480$	−850.000	−0.0808271
$481$	0	0
$482$	12050.0	1.13872
$483$	−2736.00	−0.257748
$484$	−19295.0	−1.81208
$485$	−1750.00	−0.163842
$486$	−17710.0	−1.65297
$487$	760.000	0.0707164	0.0353582	−	0.999375i	$-0.488743\pi$
$487$	760.000	0.0707164	0.0353582	+	0.999375i	$0.488743\pi$
$488$	−28530.0	−2.64650
$489$	−5848.00	−0.540809
$490$	4975.00	0.458669
$491$	−13448.0	−1.23605	−0.618024	−	0.786159i	$-0.712067\pi$
$491$	−13448.0	−1.23605	−0.618024	+	0.786159i	$0.712067\pi$
$492$	12716.0	1.16521
$493$	5684.00	0.519259
$494$	0	0
$495$	1610.00	0.146190
$496$	−27234.0	−2.46541
$497$	8232.00	0.742969
$498$	480.000	0.0431914
$499$	170.000	0.0152510	0.00762550	−	0.999971i	$-0.497573\pi$
$499$	170.000	0.0152510	0.00762550	+	0.999971i	$0.497573\pi$
$500$	2125.00	0.190066
$501$	−6072.00	−0.541471
$502$	1560.00	0.138698
$503$	298.000	0.0264158	0.0132079	−	0.999913i	$-0.495796\pi$
$503$	298.000	0.0264158	0.0132079	+	0.999913i	$0.495796\pi$
$504$	12420.0	1.09768
$505$	−7650.00	−0.674100
$506$	−7980.00	−0.701095
$507$	0	0
$508$	29546.0	2.58050
$509$	14446.0	1.25797	0.628986	−	0.777417i	$-0.283470\pi$
$509$	14446.0	1.25797	0.628986	+	0.777417i	$0.283470\pi$
$510$	−4900.00	−0.425442
$511$	−2424.00	−0.209846
$512$	24475.0	2.11260
$513$	−2600.00	−0.223768
$514$	29430.0	2.52549
$515$	−290.000	−0.0248135
$516$	−10676.0	−0.910823
$517$	8680.00	0.738387
$518$	5160.00	0.437678
$519$	4452.00	0.376534
$520$	0	0
$521$	8278.00	0.696096	0.348048	−	0.937477i	$-0.386845\pi$
$521$	8278.00	0.696096	0.348048	+	0.937477i	$0.386845\pi$
$522$	6670.00	0.559268
$523$	−12950.0	−1.08272	−0.541361	−	0.840790i	$-0.682091\pi$
$523$	−12950.0	−1.08272	−0.541361	+	0.840790i	$0.682091\pi$
$524$	13260.0	1.10547
$525$	600.000	0.0498784
$526$	−6290.00	−0.521401
$527$	−29988.0	−2.47874
$528$	−2492.00	−0.205398
$529$	829.000	0.0681351
$530$	−9050.00	−0.741711
$531$	6118.00	0.499997
$532$	5304.00	0.432251
$533$	0	0
$534$	−12300.0	−0.996766
$535$	−930.000	−0.0751540
$536$	27540.0	2.21930
$537$	−6488.00	−0.521374
$538$	−12070.0	−0.967239
$539$	2786.00	0.222637
$540$	−8500.00	−0.677374
$541$	11838.0	0.940768	0.470384	−	0.882462i	$-0.344115\pi$
$541$	11838.0	0.940768	0.470384	+	0.882462i	$0.344115\pi$
$542$	2990.00	0.236958
$543$	−3356.00	−0.265230
$544$	−8330.00	−0.656518
$545$	6710.00	0.527385
$546$	0	0
$547$	11590.0	0.905946	0.452973	−	0.891524i	$-0.350363\pi$
$547$	11590.0	0.905946	0.452973	+	0.891524i	$0.350363\pi$
$548$	18258.0	1.42325
$549$	−14582.0	−1.13360
$550$	1750.00	0.135673
$551$	1508.00	0.116593
$552$	10260.0	0.791113
$553$	−6192.00	−0.476149
$554$	23550.0	1.80604
$555$	−860.000	−0.0657747
$556$	7072.00	0.539424
$557$	13170.0	1.00185	0.500925	−	0.865491i	$-0.332993\pi$
$557$	13170.0	1.00185	0.500925	+	0.865491i	$0.332993\pi$
$558$	−35190.0	−2.66973
$559$	0	0
$560$	5340.00	0.402957
$561$	−2744.00	−0.206509
$562$	21330.0	1.60098
$563$	−16074.0	−1.20327	−0.601633	−	0.798773i	$-0.705483\pi$
$563$	−16074.0	−1.20327	−0.601633	+	0.798773i	$0.705483\pi$
$564$	−21080.0	−1.57381
$565$	−9830.00	−0.731949
$566$	4890.00	0.363148
$567$	5052.00	0.374187
$568$	−30870.0	−2.28042
$569$	17622.0	1.29834	0.649168	−	0.760645i	$-0.275117\pi$
$569$	17622.0	1.29834	0.649168	+	0.760645i	$0.275117\pi$
$570$	−1300.00	−0.0955281
$571$	12872.0	0.943391	0.471696	−	0.881761i	$-0.343642\pi$
$571$	12872.0	0.943391	0.471696	+	0.881761i	$0.343642\pi$
$572$	0	0
$573$	−7936.00	−0.578588
$574$	−22440.0	−1.63176
$575$	−2850.00	−0.206701
$576$	6601.00	0.477503
$577$	1918.00	0.138384	0.0691918	−	0.997603i	$-0.477958\pi$
$577$	1918.00	0.138384	0.0691918	+	0.997603i	$0.477958\pi$
$578$	−23455.0	−1.68789
$579$	−7996.00	−0.573925
$580$	4930.00	0.352943
$581$	−576.000	−0.0411300
$582$	3500.00	0.249278
$583$	−5068.00	−0.360026
$584$	9090.00	0.644087
$585$	0	0
$586$	−4410.00	−0.310880
$587$	−20844.0	−1.46563	−0.732814	−	0.680429i	$-0.761794\pi$
$587$	−20844.0	−1.46563	−0.732814	+	0.680429i	$0.761794\pi$
$588$	−6766.00	−0.474533
$589$	−7956.00	−0.556573
$590$	6650.00	0.464027
$591$	−8388.00	−0.583818
$592$	−7654.00	−0.531381
$593$	16754.0	1.16021	0.580105	−	0.814542i	$-0.303012\pi$
$593$	16754.0	1.16021	0.580105	+	0.814542i	$0.303012\pi$
$594$	−7000.00	−0.483524
$595$	5880.00	0.405137
$596$	34918.0	2.39983
$597$	8480.00	0.581346
$598$	0	0
$599$	17140.0	1.16915	0.584575	−	0.811339i	$-0.301261\pi$
$599$	17140.0	1.16915	0.584575	+	0.811339i	$0.301261\pi$
$600$	−2250.00	−0.153093
$601$	−12550.0	−0.851789	−0.425894	−	0.904773i	$-0.640041\pi$
$601$	−12550.0	−0.851789	−0.425894	+	0.904773i	$0.640041\pi$
$602$	18840.0	1.27552
$603$	14076.0	0.950612
$604$	−39134.0	−2.63632
$605$	−5675.00	−0.381358
$606$	15300.0	1.02561
$607$	−3434.00	−0.229624	−0.114812	−	0.993387i	$-0.536627\pi$
$607$	−3434.00	−0.229624	−0.114812	+	0.993387i	$0.536627\pi$
$608$	−2210.00	−0.147413
$609$	1392.00	0.0926218
$610$	−15850.0	−1.05205
$611$	0	0
$612$	−38318.0	−2.53090
$613$	6110.00	0.402578	0.201289	−	0.979532i	$-0.435487\pi$
$613$	6110.00	0.402578	0.201289	+	0.979532i	$0.435487\pi$
$614$	9080.00	0.596806
$615$	3740.00	0.245222
$616$	7560.00	0.494482
$617$	−9702.00	−0.633043	−0.316522	−	0.948585i	$-0.602515\pi$
$617$	−9702.00	−0.633043	−0.316522	+	0.948585i	$0.602515\pi$
$618$	580.000	0.0377525
$619$	18574.0	1.20606	0.603031	−	0.797718i	$-0.293960\pi$
$619$	18574.0	1.20606	0.603031	+	0.797718i	$0.293960\pi$
$620$	−26010.0	−1.68482
$621$	11400.0	0.736661
$622$	−11460.0	−0.738753
$623$	14760.0	0.949192
$624$	0	0
$625$	625.000	0.0400000
$626$	−42370.0	−2.70518
$627$	−728.000	−0.0463692
$628$	−39950.0	−2.53850
$629$	−8428.00	−0.534255
$630$	6900.00	0.436353
$631$	3530.00	0.222705	0.111353	−	0.993781i	$-0.464482\pi$
$631$	3530.00	0.222705	0.111353	+	0.993781i	$0.464482\pi$
$632$	23220.0	1.46146
$633$	1192.00	0.0748464
$634$	−22930.0	−1.43638
$635$	8690.00	0.543074
$636$	12308.0	0.767365
$637$	0	0
$638$	4060.00	0.251939
$639$	−15778.0	−0.976789
$640$	10575.0	0.653146
$641$	21330.0	1.31433	0.657164	−	0.753748i	$-0.271756\pi$
$641$	21330.0	1.31433	0.657164	+	0.753748i	$0.271756\pi$
$642$	1860.00	0.114343
$643$	18768.0	1.15107	0.575535	−	0.817777i	$-0.304794\pi$
$643$	18768.0	1.15107	0.575535	+	0.817777i	$0.304794\pi$
$644$	−23256.0	−1.42300
$645$	−3140.00	−0.191686
$646$	−12740.0	−0.775927
$647$	5618.00	0.341370	0.170685	−	0.985326i	$-0.445402\pi$
$647$	5618.00	0.341370	0.170685	+	0.985326i	$0.445402\pi$
$648$	−18945.0	−1.14850
$649$	3724.00	0.225239
$650$	0	0
$651$	−7344.00	−0.442141
$652$	−49708.0	−2.98576
$653$	15730.0	0.942668	0.471334	−	0.881955i	$-0.343773\pi$
$653$	15730.0	0.942668	0.471334	+	0.881955i	$0.343773\pi$
$654$	−13420.0	−0.802391
$655$	3900.00	0.232650
$656$	33286.0	1.98110
$657$	4646.00	0.275887
$658$	37200.0	2.20396
$659$	−2080.00	−0.122952	−0.0614759	−	0.998109i	$-0.519581\pi$
$659$	−2080.00	−0.122952	−0.0614759	+	0.998109i	$0.519581\pi$
$660$	−2380.00	−0.140366
$661$	−25066.0	−1.47497	−0.737484	−	0.675364i	$-0.763986\pi$
$661$	−25066.0	−1.47497	−0.737484	+	0.675364i	$0.763986\pi$
$662$	−10230.0	−0.600605
$663$	0	0
$664$	2160.00	0.126241
$665$	1560.00	0.0909687
$666$	−9890.00	−0.575420
$667$	−6612.00	−0.383835
$668$	−51612.0	−2.98941
$669$	−8536.00	−0.493305
$670$	15300.0	0.882225
$671$	−8876.00	−0.510662
$672$	−2040.00	−0.117105
$673$	12642.0	0.724091	0.362046	−	0.932160i	$-0.382078\pi$
$673$	12642.0	0.724091	0.362046	+	0.932160i	$0.382078\pi$
$674$	6430.00	0.367469
$675$	−2500.00	−0.142556
$676$	0	0
$677$	11490.0	0.652284	0.326142	−	0.945321i	$-0.394251\pi$
$677$	11490.0	0.652284	0.326142	+	0.945321i	$0.394251\pi$
$678$	19660.0	1.11363
$679$	−4200.00	−0.237380
$680$	−22050.0	−1.24350
$681$	11848.0	0.666691
$682$	−21420.0	−1.20266
$683$	−13716.0	−0.768416	−0.384208	−	0.923247i	$-0.625525\pi$
$683$	−13716.0	−0.768416	−0.384208	+	0.923247i	$0.625525\pi$
$684$	−10166.0	−0.568285
$685$	5370.00	0.299529
$686$	32520.0	1.80994
$687$	1500.00	0.0833021
$688$	−27946.0	−1.54859
$689$	0	0
$690$	5700.00	0.314486
$691$	15350.0	0.845067	0.422534	−	0.906347i	$-0.361141\pi$
$691$	15350.0	0.845067	0.422534	+	0.906347i	$0.361141\pi$
$692$	37842.0	2.07881
$693$	3864.00	0.211806
$694$	3170.00	0.173388
$695$	2080.00	0.113524
$696$	−5220.00	−0.284287
$697$	36652.0	1.99181
$698$	11330.0	0.614394
$699$	948.000	0.0512971
$700$	5100.00	0.275374
$701$	−30098.0	−1.62166	−0.810832	−	0.585280i	$-0.800985\pi$
$701$	−30098.0	−1.62166	−0.810832	+	0.585280i	$0.800985\pi$
$702$	0	0
$703$	−2236.00	−0.119961
$704$	4018.00	0.215105
$705$	−6200.00	−0.331213
$706$	1050.00	0.0559735
$707$	−18360.0	−0.976660
$708$	−9044.00	−0.480077
$709$	32270.0	1.70934	0.854672	−	0.519168i	$-0.173758\pi$
$709$	32270.0	1.70934	0.854672	+	0.519168i	$0.173758\pi$
$710$	−17150.0	−0.906518
$711$	11868.0	0.625998
$712$	−55350.0	−2.91338
$713$	34884.0	1.83228
$714$	−11760.0	−0.616396
$715$	0	0
$716$	−55148.0	−2.87846
$717$	−3196.00	−0.166467
$718$	810.000	0.0421016
$719$	−1632.00	−0.0846500	−0.0423250	−	0.999104i	$-0.513476\pi$
$719$	−1632.00	−0.0846500	−0.0423250	+	0.999104i	$0.513476\pi$
$720$	−10235.0	−0.529772
$721$	−696.000	−0.0359506
$722$	30915.0	1.59354
$723$	−4820.00	−0.247936
$724$	−28526.0	−1.46431
$725$	1450.00	0.0742781
$726$	11350.0	0.580218
$727$	29354.0	1.49750	0.748748	−	0.662855i	$-0.230655\pi$
$727$	29354.0	1.49750	0.748748	+	0.662855i	$0.230655\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	5050.00	0.256040
$731$	−30772.0	−1.55697
$732$	21556.0	1.08843
$733$	−18650.0	−0.939773	−0.469886	−	0.882727i	$-0.655705\pi$
$733$	−18650.0	−0.939773	−0.469886	+	0.882727i	$0.655705\pi$
$734$	−3390.00	−0.170473
$735$	−1990.00	−0.0998670
$736$	9690.00	0.485296
$737$	8568.00	0.428231
$738$	43010.0	2.14528
$739$	28550.0	1.42115	0.710574	−	0.703622i	$-0.248435\pi$
$739$	28550.0	1.42115	0.710574	+	0.703622i	$0.248435\pi$
$740$	−7310.00	−0.363136
$741$	0	0
$742$	−21720.0	−1.07462
$743$	27556.0	1.36061	0.680304	−	0.732930i	$-0.261848\pi$
$743$	27556.0	1.36061	0.680304	+	0.732930i	$0.261848\pi$
$744$	27540.0	1.35708
$745$	10270.0	0.505052
$746$	34510.0	1.69370
$747$	1104.00	0.0540740
$748$	−23324.0	−1.14012
$749$	−2232.00	−0.108886
$750$	−1250.00	−0.0608581
$751$	−23732.0	−1.15312	−0.576560	−	0.817055i	$-0.695605\pi$
$751$	−23732.0	−1.15312	−0.576560	+	0.817055i	$0.695605\pi$
$752$	−55180.0	−2.67581
$753$	−624.000	−0.0301990
$754$	0	0
$755$	−11510.0	−0.554824
$756$	−20400.0	−0.981403
$757$	−30806.0	−1.47908	−0.739540	−	0.673113i	$-0.764957\pi$
$757$	−30806.0	−1.47908	−0.739540	+	0.673113i	$0.764957\pi$
$758$	44990.0	2.15582
$759$	3192.00	0.152651
$760$	−5850.00	−0.279213
$761$	−24642.0	−1.17381	−0.586907	−	0.809655i	$-0.699654\pi$
$761$	−24642.0	−1.17381	−0.586907	+	0.809655i	$0.699654\pi$
$762$	−17380.0	−0.826261
$763$	16104.0	0.764094
$764$	−67456.0	−3.19434
$765$	−11270.0	−0.532638
$766$	660.000	0.0311316
$767$	0	0
$768$	−16558.0	−0.777976
$769$	14566.0	0.683047	0.341524	−	0.939873i	$-0.389057\pi$
$769$	14566.0	0.683047	0.341524	+	0.939873i	$0.389057\pi$
$770$	4200.00	0.196568
$771$	−11772.0	−0.549881
$772$	−67966.0	−3.16859
$773$	1386.00	0.0644902	0.0322451	−	0.999480i	$-0.489734\pi$
$773$	1386.00	0.0644902	0.0322451	+	0.999480i	$0.489734\pi$
$774$	−36110.0	−1.67693
$775$	−7650.00	−0.354576
$776$	15750.0	0.728598
$777$	−2064.00	−0.0952968
$778$	−58470.0	−2.69441
$779$	9724.00	0.447238
$780$	0	0
$781$	−9604.00	−0.440023
$782$	55860.0	2.55441
$783$	−5800.00	−0.264719
$784$	−17711.0	−0.806806
$785$	−11750.0	−0.534236
$786$	−7800.00	−0.353965
$787$	26984.0	1.22221	0.611103	−	0.791551i	$-0.290726\pi$
$787$	26984.0	1.22221	0.611103	+	0.791551i	$0.290726\pi$
$788$	−71298.0	−3.22321
$789$	2516.00	0.113526
$790$	12900.0	0.580964
$791$	−23592.0	−1.06047
$792$	−14490.0	−0.650101
$793$	0	0
$794$	68390.0	3.05676
$795$	3620.00	0.161495
$796$	72080.0	3.20956
$797$	−29438.0	−1.30834	−0.654170	−	0.756347i	$-0.726982\pi$
$797$	−29438.0	−1.30834	−0.654170	+	0.756347i	$0.726982\pi$
$798$	−3120.00	−0.138405
$799$	−60760.0	−2.69028
$800$	−2125.00	−0.0939126
$801$	−28290.0	−1.24791
$802$	42450.0	1.86903
$803$	2828.00	0.124281
$804$	−20808.0	−0.912738
$805$	−6840.00	−0.299476
$806$	0	0
$807$	4828.00	0.210599
$808$	68850.0	2.99769
$809$	40462.0	1.75843	0.879214	−	0.476427i	$-0.158068\pi$
$809$	40462.0	1.75843	0.879214	+	0.476427i	$0.158068\pi$
$810$	−10525.0	−0.456557
$811$	1830.00	0.0792355	0.0396178	−	0.999215i	$-0.487386\pi$
$811$	1830.00	0.0792355	0.0396178	+	0.999215i	$0.487386\pi$
$812$	11832.0	0.511357
$813$	−1196.00	−0.0515935
$814$	−6020.00	−0.259215
$815$	−14620.0	−0.628364
$816$	17444.0	0.748360
$817$	−8164.00	−0.349599
$818$	−49910.0	−2.13333
$819$	0	0
$820$	31790.0	1.35385
$821$	42726.0	1.81626	0.908129	−	0.418691i	$-0.137511\pi$
$821$	42726.0	1.81626	0.908129	+	0.418691i	$0.137511\pi$
$822$	−10740.0	−0.455718
$823$	11798.0	0.499699	0.249850	−	0.968285i	$-0.419619\pi$
$823$	11798.0	0.499699	0.249850	+	0.968285i	$0.419619\pi$
$824$	2610.00	0.110344
$825$	−700.000	−0.0295405
$826$	15960.0	0.672300
$827$	22528.0	0.947249	0.473625	−	0.880727i	$-0.342945\pi$
$827$	22528.0	0.947249	0.473625	+	0.880727i	$0.342945\pi$
$828$	44574.0	1.87084
$829$	−8934.00	−0.374295	−0.187148	−	0.982332i	$-0.559924\pi$
$829$	−8934.00	−0.374295	−0.187148	+	0.982332i	$0.559924\pi$
$830$	1200.00	0.0501839
$831$	−9420.00	−0.393232
$832$	0	0
$833$	−19502.0	−0.811170
$834$	−4160.00	−0.172721
$835$	−15180.0	−0.629132
$836$	−6188.00	−0.256001
$837$	30600.0	1.26367
$838$	49240.0	2.02979
$839$	13454.0	0.553616	0.276808	−	0.960925i	$-0.410723\pi$
$839$	13454.0	0.553616	0.276808	+	0.960925i	$0.410723\pi$
$840$	−5400.00	−0.221807
$841$	−21025.0	−0.862069
$842$	38330.0	1.56881
$843$	−8532.00	−0.348586
$844$	10132.0	0.413220
$845$	0	0
$846$	−71300.0	−2.89757
$847$	−13620.0	−0.552525
$848$	32218.0	1.30468
$849$	−1956.00	−0.0790692
$850$	−12250.0	−0.494319
$851$	9804.00	0.394920
$852$	23324.0	0.937872
$853$	21282.0	0.854258	0.427129	−	0.904191i	$-0.359525\pi$
$853$	21282.0	0.854258	0.427129	+	0.904191i	$0.359525\pi$
$854$	−38040.0	−1.52424
$855$	−2990.00	−0.119597
$856$	8370.00	0.334206
$857$	12674.0	0.505176	0.252588	−	0.967574i	$-0.418718\pi$
$857$	12674.0	0.505176	0.252588	+	0.967574i	$0.418718\pi$
$858$	0	0
$859$	19000.0	0.754682	0.377341	−	0.926074i	$-0.376839\pi$
$859$	19000.0	0.754682	0.377341	+	0.926074i	$0.376839\pi$
$860$	−26690.0	−1.05828
$861$	8976.00	0.355286
$862$	−43710.0	−1.72711
$863$	6908.00	0.272481	0.136240	−	0.990676i	$-0.456498\pi$
$863$	6908.00	0.272481	0.136240	+	0.990676i	$0.456498\pi$
$864$	8500.00	0.334694
$865$	11130.0	0.437493
$866$	28270.0	1.10930
$867$	9382.00	0.367508
$868$	−62424.0	−2.44102
$869$	7224.00	0.281999
$870$	−2900.00	−0.113011
$871$	0	0
$872$	−60390.0	−2.34526
$873$	8050.00	0.312086
$874$	14820.0	0.573563
$875$	1500.00	0.0579534
$876$	−6868.00	−0.264895
$877$	13014.0	0.501085	0.250543	−	0.968106i	$-0.419391\pi$
$877$	13014.0	0.501085	0.250543	+	0.968106i	$0.419391\pi$
$878$	−34880.0	−1.34071
$879$	1764.00	0.0676886
$880$	−6230.00	−0.238651
$881$	17318.0	0.662268	0.331134	−	0.943584i	$-0.392569\pi$
$881$	17318.0	0.662268	0.331134	+	0.943584i	$0.392569\pi$
$882$	−22885.0	−0.873671
$883$	42038.0	1.60214	0.801071	−	0.598569i	$-0.204264\pi$
$883$	42038.0	1.60214	0.801071	+	0.598569i	$0.204264\pi$
$884$	0	0
$885$	−2660.00	−0.101034
$886$	−330.000	−0.0125131
$887$	−24882.0	−0.941889	−0.470945	−	0.882163i	$-0.656087\pi$
$887$	−24882.0	−0.941889	−0.470945	+	0.882163i	$0.656087\pi$
$888$	7740.00	0.292497
$889$	20856.0	0.786825
$890$	−30750.0	−1.15814
$891$	−5894.00	−0.221612
$892$	−72556.0	−2.72349
$893$	−16120.0	−0.604071
$894$	−20540.0	−0.768412
$895$	−16220.0	−0.605782
$896$	25380.0	0.946302
$897$	0	0
$898$	12130.0	0.450761
$899$	−17748.0	−0.658430
$900$	−9775.00	−0.362037
$901$	35476.0	1.31174
$902$	26180.0	0.966406
$903$	−7536.00	−0.277721
$904$	88470.0	3.25494
$905$	−8390.00	−0.308169
$906$	23020.0	0.844137
$907$	24018.0	0.879277	0.439639	−	0.898175i	$-0.355106\pi$
$907$	24018.0	0.879277	0.439639	+	0.898175i	$0.355106\pi$
$908$	100708.	3.68074
$909$	35190.0	1.28402
$910$	0	0
$911$	9584.00	0.348553	0.174277	−	0.984697i	$-0.444241\pi$
$911$	9584.00	0.348553	0.174277	+	0.984697i	$0.444241\pi$
$912$	4628.00	0.168036
$913$	672.000	0.0243592
$914$	−10930.0	−0.395550
$915$	6340.00	0.229064
$916$	12750.0	0.459904
$917$	9360.00	0.337071
$918$	49000.0	1.76170
$919$	49252.0	1.76787	0.883936	−	0.467609i	$-0.154884\pi$
$919$	49252.0	1.76787	0.883936	+	0.467609i	$0.154884\pi$
$920$	25650.0	0.919191
$921$	−3632.00	−0.129944
$922$	11850.0	0.423274
$923$	0	0
$924$	−5712.00	−0.203367
$925$	−2150.00	−0.0764233
$926$	59160.0	2.09948
$927$	1334.00	0.0472646
$928$	−4930.00	−0.174391
$929$	48614.0	1.71687	0.858436	−	0.512921i	$-0.171437\pi$
$929$	48614.0	1.71687	0.858436	+	0.512921i	$0.171437\pi$
$930$	15300.0	0.539470
$931$	−5174.00	−0.182139
$932$	8058.00	0.283207
$933$	4584.00	0.160850
$934$	50350.0	1.76392
$935$	−6860.00	−0.239942
$936$	0	0
$937$	−37326.0	−1.30137	−0.650687	−	0.759346i	$-0.725519\pi$
$937$	−37326.0	−1.30137	−0.650687	+	0.759346i	$0.725519\pi$
$938$	36720.0	1.27820
$939$	16948.0	0.589006
$940$	−52700.0	−1.82860
$941$	−10474.0	−0.362851	−0.181425	−	0.983405i	$-0.558071\pi$
$941$	−10474.0	−0.362851	−0.181425	+	0.983405i	$0.558071\pi$
$942$	23500.0	0.812815
$943$	−42636.0	−1.47234
$944$	−23674.0	−0.816232
$945$	−6000.00	−0.206540
$946$	−21980.0	−0.755424
$947$	15096.0	0.518009	0.259004	−	0.965876i	$-0.416606\pi$
$947$	15096.0	0.518009	0.259004	+	0.965876i	$0.416606\pi$
$948$	−17544.0	−0.601058
$949$	0	0
$950$	−3250.00	−0.110994
$951$	9172.00	0.312747
$952$	−52920.0	−1.80163
$953$	−3262.00	−0.110878	−0.0554389	−	0.998462i	$-0.517656\pi$
$953$	−3262.00	−0.110878	−0.0554389	+	0.998462i	$0.517656\pi$
$954$	41630.0	1.41281
$955$	−19840.0	−0.672259
$956$	−27166.0	−0.919049
$957$	−1624.00	−0.0548552
$958$	77010.0	2.59716
$959$	12888.0	0.433968
$960$	−2870.00	−0.0964884
$961$	63845.0	2.14310
$962$	0	0
$963$	4278.00	0.143153
$964$	−40970.0	−1.36883
$965$	−19990.0	−0.666840
$966$	13680.0	0.455638
$967$	−48872.0	−1.62525	−0.812625	−	0.582786i	$-0.801962\pi$
$967$	−48872.0	−1.62525	−0.812625	+	0.582786i	$0.801962\pi$
$968$	51075.0	1.69588
$969$	5096.00	0.168944
$970$	8750.00	0.289635
$971$	43932.0	1.45195	0.725976	−	0.687720i	$-0.241388\pi$
$971$	43932.0	1.45195	0.725976	+	0.687720i	$0.241388\pi$
$972$	60214.0	1.98700
$973$	4992.00	0.164477
$974$	−3800.00	−0.125010
$975$	0	0
$976$	56426.0	1.85057
$977$	3406.00	0.111533	0.0557664	−	0.998444i	$-0.482240\pi$
$977$	3406.00	0.111533	0.0557664	+	0.998444i	$0.482240\pi$
$978$	29240.0	0.956025
$979$	−17220.0	−0.562159
$980$	−16915.0	−0.551357
$981$	−30866.0	−1.00456
$982$	67240.0	2.18505
$983$	−35496.0	−1.15173	−0.575863	−	0.817546i	$-0.695334\pi$
$983$	−35496.0	−1.15173	−0.575863	+	0.817546i	$0.695334\pi$
$984$	−33660.0	−1.09049
$985$	−20970.0	−0.678335
$986$	−28420.0	−0.917928
$987$	−14880.0	−0.479874
$988$	0	0
$989$	35796.0	1.15091
$990$	−8050.00	−0.258430
$991$	−20464.0	−0.655964	−0.327982	−	0.944684i	$-0.606369\pi$
$991$	−20464.0	−0.655964	−0.327982	+	0.944684i	$0.606369\pi$
$992$	26010.0	0.832478
$993$	4092.00	0.130771
$994$	−41160.0	−1.31340
$995$	21200.0	0.675462
$996$	−1632.00	−0.0519196
$997$	−30318.0	−0.963070	−0.481535	−	0.876427i	$-0.659921\pi$
$997$	−30318.0	−0.963070	−0.481535	+	0.876427i	$0.659921\pi$
$998$	−850.000	−0.0269602
$999$	8600.00	0.272364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.4.a.a.1.1		1
13.12	even	2		65.4.a.a.1.1	✓	1
39.38	odd	2		585.4.a.a.1.1		1
52.51	odd	2		1040.4.a.a.1.1		1
65.12	odd	4		325.4.b.a.274.2		2
65.38	odd	4		325.4.b.a.274.1		2
65.64	even	2		325.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.4.a.a.1.1	✓	1	13.12	even	2
325.4.a.a.1.1		1	65.64	even	2
325.4.b.a.274.1		2	65.38	odd	4
325.4.b.a.274.2		2	65.12	odd	4
585.4.a.a.1.1		1	39.38	odd	2
845.4.a.a.1.1		1	1.1	even	1	trivial
1040.4.a.a.1.1		1	52.51	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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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