LMFDB - Embedded newform 4026.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4026,2,Mod(1,4026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4026, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4026.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4026 = 2 \cdot 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4026.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:	$32.1477718538$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4026.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +3.00000 q^{20} -4.00000 q^{21} +1.00000 q^{22} -9.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} -3.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -3.00000 q^{34} -12.0000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} -7.00000 q^{43} -1.00000 q^{44} +3.00000 q^{45} +9.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -4.00000 q^{50} +3.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -3.00000 q^{55} +4.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} +6.00000 q^{59} +3.00000 q^{60} +1.00000 q^{61} -2.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} +3.00000 q^{68} -9.00000 q^{69} +12.0000 q^{70} +3.00000 q^{71} -1.00000 q^{72} -7.00000 q^{73} +1.00000 q^{74} +4.00000 q^{75} -4.00000 q^{76} +4.00000 q^{77} -2.00000 q^{78} -4.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -3.00000 q^{83} -4.00000 q^{84} +9.00000 q^{85} +7.00000 q^{86} -6.00000 q^{87} +1.00000 q^{88} -15.0000 q^{89} -3.00000 q^{90} -8.00000 q^{91} -9.00000 q^{92} +2.00000 q^{93} +6.00000 q^{94} -12.0000 q^{95} -1.00000 q^{96} +11.0000 q^{97} -9.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−1.00000	−0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	4.00000	1.06904
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	3.00000	0.670820
$21$	−4.00000	−0.872872
$22$	1.00000	0.213201
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−3.00000	−0.547723
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−3.00000	−0.514496
$35$	−12.0000	−2.02837
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	4.00000	0.648886
$39$	2.00000	0.320256
$40$	−3.00000	−0.474342
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	4.00000	0.617213
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	−1.00000	−0.150756
$45$	3.00000	0.447214
$46$	9.00000	1.32698
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	−4.00000	−0.565685
$51$	3.00000	0.420084
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	−3.00000	−0.404520
$56$	4.00000	0.534522
$57$	−4.00000	−0.529813
$58$	6.00000	0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	3.00000	0.387298
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−2.00000	−0.254000
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	1.00000	0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	3.00000	0.363803
$69$	−9.00000	−1.08347
$70$	12.0000	1.43427
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	−1.00000	−0.117851
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	1.00000	0.116248
$75$	4.00000	0.461880
$76$	−4.00000	−0.458831
$77$	4.00000	0.455842
$78$	−2.00000	−0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	−4.00000	−0.436436
$85$	9.00000	0.976187
$86$	7.00000	0.754829
$87$	−6.00000	−0.643268
$88$	1.00000	0.106600
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	−3.00000	−0.316228
$91$	−8.00000	−0.838628
$92$	−9.00000	−0.938315
$93$	2.00000	0.207390
$94$	6.00000	0.618853
$95$	−12.0000	−1.23117
$96$	−1.00000	−0.102062
$97$	11.0000	1.11688	0.558440	−	0.829545i	$-0.311400\pi$
$97$	11.0000	1.11688	0.558440	+	0.829545i	$0.311400\pi$
$98$	−9.00000	−0.909137
$99$	−1.00000	−0.100504
$100$	4.00000	0.400000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	−3.00000	−0.297044
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	−2.00000	−0.196116
$105$	−12.0000	−1.17108
$106$	−6.00000	−0.582772
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	1.00000	0.0962250
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	3.00000	0.286039
$111$	−1.00000	−0.0949158
$112$	−4.00000	−0.377964
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	4.00000	0.374634
$115$	−27.0000	−2.51776
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	−6.00000	−0.552345
$119$	−12.0000	−1.10004
$120$	−3.00000	−0.273861
$121$	1.00000	0.0909091
$122$	−1.00000	−0.0905357
$123$	−6.00000	−0.541002
$124$	2.00000	0.179605
$125$	−3.00000	−0.268328
$126$	4.00000	0.356348
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	−1.00000	−0.0883883
$129$	−7.00000	−0.616316
$130$	−6.00000	−0.526235
$131$	−21.0000	−1.83478	−0.917389	−	0.397991i	$-0.869707\pi$
$131$	−21.0000	−1.83478	−0.917389	+	0.397991i	$0.869707\pi$
$132$	−1.00000	−0.0870388
$133$	16.0000	1.38738
$134$	4.00000	0.345547
$135$	3.00000	0.258199
$136$	−3.00000	−0.257248
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	9.00000	0.766131
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	−12.0000	−1.01419
$141$	−6.00000	−0.505291
$142$	−3.00000	−0.251754
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	−18.0000	−1.49482
$146$	7.00000	0.579324
$147$	9.00000	0.742307
$148$	−1.00000	−0.0821995
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−4.00000	−0.326599
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	4.00000	0.324443
$153$	3.00000	0.242536
$154$	−4.00000	−0.322329
$155$	6.00000	0.481932
$156$	2.00000	0.160128
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	4.00000	0.318223
$159$	6.00000	0.475831
$160$	−3.00000	−0.237171
$161$	36.0000	2.83720
$162$	−1.00000	−0.0785674
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	−6.00000	−0.468521
$165$	−3.00000	−0.233550
$166$	3.00000	0.232845
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	4.00000	0.308607
$169$	−9.00000	−0.692308
$170$	−9.00000	−0.690268
$171$	−4.00000	−0.305888
$172$	−7.00000	−0.533745
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	6.00000	0.454859
$175$	−16.0000	−1.20949
$176$	−1.00000	−0.0753778
$177$	6.00000	0.450988
$178$	15.0000	1.12430
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	3.00000	0.223607
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	8.00000	0.592999
$183$	1.00000	0.0739221
$184$	9.00000	0.663489
$185$	−3.00000	−0.220564
$186$	−2.00000	−0.146647
$187$	−3.00000	−0.219382
$188$	−6.00000	−0.437595
$189$	−4.00000	−0.290957
$190$	12.0000	0.870572
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000	0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−11.0000	−0.789754
$195$	6.00000	0.429669
$196$	9.00000	0.642857
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	1.00000	0.0710669
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	−4.00000	−0.282843
$201$	−4.00000	−0.282138
$202$	12.0000	0.844317
$203$	24.0000	1.68447
$204$	3.00000	0.210042
$205$	−18.0000	−1.25717
$206$	−5.00000	−0.348367
$207$	−9.00000	−0.625543
$208$	2.00000	0.138675
$209$	4.00000	0.276686
$210$	12.0000	0.828079
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	6.00000	0.412082
$213$	3.00000	0.205557
$214$	−3.00000	−0.205076
$215$	−21.0000	−1.43219
$216$	−1.00000	−0.0680414
$217$	−8.00000	−0.543075
$218$	4.00000	0.270914
$219$	−7.00000	−0.473016
$220$	−3.00000	−0.202260
$221$	6.00000	0.403604
$222$	1.00000	0.0671156
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	4.00000	0.267261
$225$	4.00000	0.266667
$226$	−18.0000	−1.19734
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−4.00000	−0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	27.0000	1.78033
$231$	4.00000	0.263181
$232$	6.00000	0.393919
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−2.00000	−0.130744
$235$	−18.0000	−1.17419
$236$	6.00000	0.390567
$237$	−4.00000	−0.259828
$238$	12.0000	0.777844
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	3.00000	0.193649
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	1.00000	0.0640184
$245$	27.0000	1.72497
$246$	6.00000	0.382546
$247$	−8.00000	−0.509028
$248$	−2.00000	−0.127000
$249$	−3.00000	−0.190117
$250$	3.00000	0.189737
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−4.00000	−0.251976
$253$	9.00000	0.565825
$254$	−17.0000	−1.06667
$255$	9.00000	0.563602
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	7.00000	0.435801
$259$	4.00000	0.248548
$260$	6.00000	0.372104
$261$	−6.00000	−0.371391
$262$	21.0000	1.29738
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	1.00000	0.0615457
$265$	18.0000	1.10573
$266$	−16.0000	−0.981023
$267$	−15.0000	−0.917985
$268$	−4.00000	−0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−3.00000	−0.182574
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	3.00000	0.181902
$273$	−8.00000	−0.484182
$274$	6.00000	0.362473
$275$	−4.00000	−0.241209
$276$	−9.00000	−0.541736
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	13.0000	0.779688
$279$	2.00000	0.119737
$280$	12.0000	0.717137
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	6.00000	0.357295
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	3.00000	0.178017
$285$	−12.0000	−0.710819
$286$	2.00000	0.118262
$287$	24.0000	1.41668
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	18.0000	1.05700
$291$	11.0000	0.644831
$292$	−7.00000	−0.409644
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	−9.00000	−0.524891
$295$	18.0000	1.04800
$296$	1.00000	0.0581238
$297$	−1.00000	−0.0580259
$298$	−18.0000	−1.04271
$299$	−18.0000	−1.04097
$300$	4.00000	0.230940
$301$	28.0000	1.61389
$302$	10.0000	0.575435
$303$	−12.0000	−0.689382
$304$	−4.00000	−0.229416
$305$	3.00000	0.171780
$306$	−3.00000	−0.171499
$307$	11.0000	0.627803	0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000	0.627803	0.313902	+	0.949456i	$0.398364\pi$
$308$	4.00000	0.227921
$309$	5.00000	0.284440
$310$	−6.00000	−0.340777
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−2.00000	−0.113228
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	22.0000	1.24153
$315$	−12.0000	−0.676123
$316$	−4.00000	−0.225018
$317$	−9.00000	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$318$	−6.00000	−0.336463
$319$	6.00000	0.335936
$320$	3.00000	0.167705
$321$	3.00000	0.167444
$322$	−36.0000	−2.00620
$323$	−12.0000	−0.667698
$324$	1.00000	0.0555556
$325$	8.00000	0.443760
$326$	22.0000	1.21847
$327$	−4.00000	−0.221201
$328$	6.00000	0.331295
$329$	24.0000	1.32316
$330$	3.00000	0.165145
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	−3.00000	−0.164646
$333$	−1.00000	−0.0547997
$334$	−6.00000	−0.328305
$335$	−12.0000	−0.655630
$336$	−4.00000	−0.218218
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	9.00000	0.489535
$339$	18.0000	0.977626
$340$	9.00000	0.488094
$341$	−2.00000	−0.108306
$342$	4.00000	0.216295
$343$	−8.00000	−0.431959
$344$	7.00000	0.377415
$345$	−27.0000	−1.45363
$346$	0	0
$347$	−9.00000	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$347$	−9.00000	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$348$	−6.00000	−0.321634
$349$	35.0000	1.87351	0.936754	−	0.349990i	$-0.113815\pi$
$349$	35.0000	1.87351	0.936754	+	0.349990i	$0.113815\pi$
$350$	16.0000	0.855236
$351$	2.00000	0.106752
$352$	1.00000	0.0533002
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−6.00000	−0.318896
$355$	9.00000	0.477670
$356$	−15.0000	−0.794998
$357$	−12.0000	−0.635107
$358$	−15.0000	−0.792775
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	−3.00000	−0.158114
$361$	−3.00000	−0.157895
$362$	7.00000	0.367912
$363$	1.00000	0.0524864
$364$	−8.00000	−0.419314
$365$	−21.0000	−1.09919
$366$	−1.00000	−0.0522708
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	−9.00000	−0.469157
$369$	−6.00000	−0.312348
$370$	3.00000	0.155963
$371$	−24.0000	−1.24602
$372$	2.00000	0.103695
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	3.00000	0.155126
$375$	−3.00000	−0.154919
$376$	6.00000	0.309426
$377$	−12.0000	−0.618031
$378$	4.00000	0.205738
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	−12.0000	−0.615587
$381$	17.0000	0.870936
$382$	−12.0000	−0.613973
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−1.00000	−0.0510310
$385$	12.0000	0.611577
$386$	−14.0000	−0.712581
$387$	−7.00000	−0.355830
$388$	11.0000	0.558440
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−6.00000	−0.303822
$391$	−27.0000	−1.36545
$392$	−9.00000	−0.454569
$393$	−21.0000	−1.05931
$394$	−18.0000	−0.906827
$395$	−12.0000	−0.603786
$396$	−1.00000	−0.0502519
$397$	−19.0000	−0.953583	−0.476791	−	0.879017i	$-0.658200\pi$
$397$	−19.0000	−0.953583	−0.476791	+	0.879017i	$0.658200\pi$
$398$	25.0000	1.25314
$399$	16.0000	0.801002
$400$	4.00000	0.200000
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	4.00000	0.199502
$403$	4.00000	0.199254
$404$	−12.0000	−0.597022
$405$	3.00000	0.149071
$406$	−24.0000	−1.19110
$407$	1.00000	0.0495682
$408$	−3.00000	−0.148522
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	18.0000	0.888957
$411$	−6.00000	−0.295958
$412$	5.00000	0.246332
$413$	−24.0000	−1.18096
$414$	9.00000	0.442326
$415$	−9.00000	−0.441793
$416$	−2.00000	−0.0980581
$417$	−13.0000	−0.636613
$418$	−4.00000	−0.195646
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	−12.0000	−0.585540
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	−8.00000	−0.389434
$423$	−6.00000	−0.291730
$424$	−6.00000	−0.291386
$425$	12.0000	0.582086
$426$	−3.00000	−0.145350
$427$	−4.00000	−0.193574
$428$	3.00000	0.145010
$429$	−2.00000	−0.0965609
$430$	21.0000	1.01271
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	1.00000	0.0481125
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	8.00000	0.384012
$435$	−18.0000	−0.863034
$436$	−4.00000	−0.191565
$437$	36.0000	1.72211
$438$	7.00000	0.334473
$439$	−37.0000	−1.76591	−0.882957	−	0.469454i	$-0.844451\pi$
$439$	−37.0000	−1.76591	−0.882957	+	0.469454i	$0.844451\pi$
$440$	3.00000	0.143019
$441$	9.00000	0.428571
$442$	−6.00000	−0.285391
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	−1.00000	−0.0474579
$445$	−45.0000	−2.13320
$446$	−20.0000	−0.947027
$447$	18.0000	0.851371
$448$	−4.00000	−0.188982
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	−4.00000	−0.188562
$451$	6.00000	0.282529
$452$	18.0000	0.846649
$453$	−10.0000	−0.469841
$454$	−18.0000	−0.844782
$455$	−24.0000	−1.12514
$456$	4.00000	0.187317
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	−14.0000	−0.654177
$459$	3.00000	0.140028
$460$	−27.0000	−1.25888
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	−4.00000	−0.186097
$463$	5.00000	0.232370	0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000	0.232370	0.116185	+	0.993228i	$0.462933\pi$
$464$	−6.00000	−0.278543
$465$	6.00000	0.278243
$466$	18.0000	0.833834
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	2.00000	0.0924500
$469$	16.0000	0.738811
$470$	18.0000	0.830278
$471$	−22.0000	−1.01371
$472$	−6.00000	−0.276172
$473$	7.00000	0.321860
$474$	4.00000	0.183726
$475$	−16.0000	−0.734130
$476$	−12.0000	−0.550019
$477$	6.00000	0.274721
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	−3.00000	−0.136931
$481$	−2.00000	−0.0911922
$482$	13.0000	0.592134
$483$	36.0000	1.63806
$484$	1.00000	0.0454545
$485$	33.0000	1.49845
$486$	−1.00000	−0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−1.00000	−0.0452679
$489$	−22.0000	−0.994874
$490$	−27.0000	−1.21974
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	−6.00000	−0.270501
$493$	−18.0000	−0.810679
$494$	8.00000	0.359937
$495$	−3.00000	−0.134840
$496$	2.00000	0.0898027
$497$	−12.0000	−0.538274
$498$	3.00000	0.134433
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	−3.00000	−0.134164
$501$	6.00000	0.268060
$502$	0	0
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	4.00000	0.178174
$505$	−36.0000	−1.60198
$506$	−9.00000	−0.400099
$507$	−9.00000	−0.399704
$508$	17.0000	0.754253
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	−9.00000	−0.398527
$511$	28.0000	1.23865
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	−6.00000	−0.264649
$515$	15.0000	0.660979
$516$	−7.00000	−0.308158
$517$	6.00000	0.263880
$518$	−4.00000	−0.175750
$519$	0	0
$520$	−6.00000	−0.263117
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	6.00000	0.262613
$523$	−7.00000	−0.306089	−0.153044	−	0.988219i	$-0.548908\pi$
$523$	−7.00000	−0.306089	−0.153044	+	0.988219i	$0.548908\pi$
$524$	−21.0000	−0.917389
$525$	−16.0000	−0.698297
$526$	6.00000	0.261612
$527$	6.00000	0.261364
$528$	−1.00000	−0.0435194
$529$	58.0000	2.52174
$530$	−18.0000	−0.781870
$531$	6.00000	0.260378
$532$	16.0000	0.693688
$533$	−12.0000	−0.519778
$534$	15.0000	0.649113
$535$	9.00000	0.389104
$536$	4.00000	0.172774
$537$	15.0000	0.647298
$538$	−18.0000	−0.776035
$539$	−9.00000	−0.387657
$540$	3.00000	0.129099
$541$	5.00000	0.214967	0.107483	−	0.994207i	$-0.465721\pi$
$541$	5.00000	0.214967	0.107483	+	0.994207i	$0.465721\pi$
$542$	−11.0000	−0.472490
$543$	−7.00000	−0.300399
$544$	−3.00000	−0.128624
$545$	−12.0000	−0.514024
$546$	8.00000	0.342368
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−6.00000	−0.256307
$549$	1.00000	0.0426790
$550$	4.00000	0.170561
$551$	24.0000	1.02243
$552$	9.00000	0.383065
$553$	16.0000	0.680389
$554$	22.0000	0.934690
$555$	−3.00000	−0.127343
$556$	−13.0000	−0.551323
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	−2.00000	−0.0846668
$559$	−14.0000	−0.592137
$560$	−12.0000	−0.507093
$561$	−3.00000	−0.126660
$562$	−6.00000	−0.253095
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	−6.00000	−0.252646
$565$	54.0000	2.27180
$566$	16.0000	0.672530
$567$	−4.00000	−0.167984
$568$	−3.00000	−0.125877
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	12.0000	0.502625
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	−2.00000	−0.0836242
$573$	12.0000	0.501307
$574$	−24.0000	−1.00174
$575$	−36.0000	−1.50130
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	8.00000	0.332756
$579$	14.0000	0.581820
$580$	−18.0000	−0.747409
$581$	12.0000	0.497844
$582$	−11.0000	−0.455965
$583$	−6.00000	−0.248495
$584$	7.00000	0.289662
$585$	6.00000	0.248069
$586$	−9.00000	−0.371787
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	9.00000	0.371154
$589$	−8.00000	−0.329634
$590$	−18.0000	−0.741048
$591$	18.0000	0.740421
$592$	−1.00000	−0.0410997
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	1.00000	0.0410305
$595$	−36.0000	−1.47586
$596$	18.0000	0.737309
$597$	−25.0000	−1.02318
$598$	18.0000	0.736075
$599$	45.0000	1.83865	0.919325	−	0.393499i	$-0.128735\pi$
$599$	45.0000	1.83865	0.919325	+	0.393499i	$0.128735\pi$
$600$	−4.00000	−0.163299
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	−28.0000	−1.14119
$603$	−4.00000	−0.162893
$604$	−10.0000	−0.406894
$605$	3.00000	0.121967
$606$	12.0000	0.487467
$607$	35.0000	1.42061	0.710303	−	0.703896i	$-0.248558\pi$
$607$	35.0000	1.42061	0.710303	+	0.703896i	$0.248558\pi$
$608$	4.00000	0.162221
$609$	24.0000	0.972529
$610$	−3.00000	−0.121466
$611$	−12.0000	−0.485468
$612$	3.00000	0.121268
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	−11.0000	−0.443924
$615$	−18.0000	−0.725830
$616$	−4.00000	−0.161165
$617$	−3.00000	−0.120775	−0.0603877	−	0.998175i	$-0.519234\pi$
$617$	−3.00000	−0.120775	−0.0603877	+	0.998175i	$0.519234\pi$
$618$	−5.00000	−0.201129
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	6.00000	0.240966
$621$	−9.00000	−0.361158
$622$	0	0
$623$	60.0000	2.40385
$624$	2.00000	0.0800641
$625$	−29.0000	−1.16000
$626$	4.00000	0.159872
$627$	4.00000	0.159745
$628$	−22.0000	−0.877896
$629$	−3.00000	−0.119618
$630$	12.0000	0.478091
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	4.00000	0.159111
$633$	8.00000	0.317971
$634$	9.00000	0.357436
$635$	51.0000	2.02387
$636$	6.00000	0.237915
$637$	18.0000	0.713186
$638$	−6.00000	−0.237542
$639$	3.00000	0.118678
$640$	−3.00000	−0.118585
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	−3.00000	−0.118401
$643$	23.0000	0.907031	0.453516	−	0.891248i	$-0.350170\pi$
$643$	23.0000	0.907031	0.453516	+	0.891248i	$0.350170\pi$
$644$	36.0000	1.41860
$645$	−21.0000	−0.826874
$646$	12.0000	0.472134
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	−1.00000	−0.0392837
$649$	−6.00000	−0.235521
$650$	−8.00000	−0.313786
$651$	−8.00000	−0.313545
$652$	−22.0000	−0.861586
$653$	48.0000	1.87839	0.939193	−	0.343391i	$-0.111576\pi$
$653$	48.0000	1.87839	0.939193	+	0.343391i	$0.111576\pi$
$654$	4.00000	0.156412
$655$	−63.0000	−2.46161
$656$	−6.00000	−0.234261
$657$	−7.00000	−0.273096
$658$	−24.0000	−0.935617
$659$	45.0000	1.75295	0.876476	−	0.481446i	$-0.159888\pi$
$659$	45.0000	1.75295	0.876476	+	0.481446i	$0.159888\pi$
$660$	−3.00000	−0.116775
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	13.0000	0.505259
$663$	6.00000	0.233021
$664$	3.00000	0.116423
$665$	48.0000	1.86136
$666$	1.00000	0.0387492
$667$	54.0000	2.09089
$668$	6.00000	0.232147
$669$	20.0000	0.773245
$670$	12.0000	0.463600
$671$	−1.00000	−0.0386046
$672$	4.00000	0.154303
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	16.0000	0.616297
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−18.0000	−0.691286
$679$	−44.0000	−1.68857
$680$	−9.00000	−0.345134
$681$	18.0000	0.689761
$682$	2.00000	0.0765840
$683$	−15.0000	−0.573959	−0.286980	−	0.957937i	$-0.592651\pi$
$683$	−15.0000	−0.573959	−0.286980	+	0.957937i	$0.592651\pi$
$684$	−4.00000	−0.152944
$685$	−18.0000	−0.687745
$686$	8.00000	0.305441
$687$	14.0000	0.534133
$688$	−7.00000	−0.266872
$689$	12.0000	0.457164
$690$	27.0000	1.02787
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	0	0
$693$	4.00000	0.151947
$694$	9.00000	0.341635
$695$	−39.0000	−1.47935
$696$	6.00000	0.227429
$697$	−18.0000	−0.681799
$698$	−35.0000	−1.32477
$699$	−18.0000	−0.680823
$700$	−16.0000	−0.604743
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	−2.00000	−0.0754851
$703$	4.00000	0.150863
$704$	−1.00000	−0.0376889
$705$	−18.0000	−0.677919
$706$	−6.00000	−0.225813
$707$	48.0000	1.80523
$708$	6.00000	0.225494
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	−9.00000	−0.337764
$711$	−4.00000	−0.150012
$712$	15.0000	0.562149
$713$	−18.0000	−0.674105
$714$	12.0000	0.449089
$715$	−6.00000	−0.224387
$716$	15.0000	0.560576
$717$	0	0
$718$	−3.00000	−0.111959
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	3.00000	0.111803
$721$	−20.0000	−0.744839
$722$	3.00000	0.111648
$723$	−13.0000	−0.483475
$724$	−7.00000	−0.260153
$725$	−24.0000	−0.891338
$726$	−1.00000	−0.0371135
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	8.00000	0.296500
$729$	1.00000	0.0370370
$730$	21.0000	0.777245
$731$	−21.0000	−0.776713
$732$	1.00000	0.0369611
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	28.0000	1.03350
$735$	27.0000	0.995910
$736$	9.00000	0.331744
$737$	4.00000	0.147342
$738$	6.00000	0.220863
$739$	5.00000	0.183928	0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000	0.183928	0.0919640	+	0.995762i	$0.470686\pi$
$740$	−3.00000	−0.110282
$741$	−8.00000	−0.293887
$742$	24.0000	0.881068
$743$	51.0000	1.87101	0.935504	−	0.353315i	$-0.114946\pi$
$743$	51.0000	1.87101	0.935504	+	0.353315i	$0.114946\pi$
$744$	−2.00000	−0.0733236
$745$	54.0000	1.97841
$746$	19.0000	0.695639
$747$	−3.00000	−0.109764
$748$	−3.00000	−0.109691
$749$	−12.0000	−0.438470
$750$	3.00000	0.109545
$751$	11.0000	0.401396	0.200698	−	0.979653i	$-0.435679\pi$
$751$	11.0000	0.401396	0.200698	+	0.979653i	$0.435679\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	12.0000	0.437014
$755$	−30.0000	−1.09181
$756$	−4.00000	−0.145479
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	34.0000	1.23494
$759$	9.00000	0.326679
$760$	12.0000	0.435286
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	−17.0000	−0.615845
$763$	16.0000	0.579239
$764$	12.0000	0.434145
$765$	9.00000	0.325396
$766$	−24.0000	−0.867155
$767$	12.0000	0.433295
$768$	1.00000	0.0360844
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	−12.0000	−0.432450
$771$	6.00000	0.216085
$772$	14.0000	0.503871
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	7.00000	0.251610
$775$	8.00000	0.287368
$776$	−11.0000	−0.394877
$777$	4.00000	0.143499
$778$	−6.00000	−0.215110
$779$	24.0000	0.859889
$780$	6.00000	0.214834
$781$	−3.00000	−0.107348
$782$	27.0000	0.965518
$783$	−6.00000	−0.214423
$784$	9.00000	0.321429
$785$	−66.0000	−2.35564
$786$	21.0000	0.749045
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	18.0000	0.641223
$789$	−6.00000	−0.213606
$790$	12.0000	0.426941
$791$	−72.0000	−2.56003
$792$	1.00000	0.0355335
$793$	2.00000	0.0710221
$794$	19.0000	0.674285
$795$	18.0000	0.638394
$796$	−25.0000	−0.886102
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	−16.0000	−0.566394
$799$	−18.0000	−0.636794
$800$	−4.00000	−0.141421
$801$	−15.0000	−0.529999
$802$	3.00000	0.105934
$803$	7.00000	0.247025
$804$	−4.00000	−0.141069
$805$	108.000	3.80650
$806$	−4.00000	−0.140894
$807$	18.0000	0.633630
$808$	12.0000	0.422159
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	−3.00000	−0.105409
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	24.0000	0.842235
$813$	11.0000	0.385787
$814$	−1.00000	−0.0350500
$815$	−66.0000	−2.31188
$816$	3.00000	0.105021
$817$	28.0000	0.979596
$818$	−32.0000	−1.11885
$819$	−8.00000	−0.279543
$820$	−18.0000	−0.628587
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	6.00000	0.209274
$823$	−22.0000	−0.766872	−0.383436	−	0.923567i	$-0.625259\pi$
$823$	−22.0000	−0.766872	−0.383436	+	0.923567i	$0.625259\pi$
$824$	−5.00000	−0.174183
$825$	−4.00000	−0.139262
$826$	24.0000	0.835067
$827$	−15.0000	−0.521601	−0.260801	−	0.965393i	$-0.583986\pi$
$827$	−15.0000	−0.521601	−0.260801	+	0.965393i	$0.583986\pi$
$828$	−9.00000	−0.312772
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	9.00000	0.312395
$831$	−22.0000	−0.763172
$832$	2.00000	0.0693375
$833$	27.0000	0.935495
$834$	13.0000	0.450153
$835$	18.0000	0.622916
$836$	4.00000	0.138343
$837$	2.00000	0.0691301
$838$	18.0000	0.621800
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	12.0000	0.414039
$841$	7.00000	0.241379
$842$	13.0000	0.448010
$843$	6.00000	0.206651
$844$	8.00000	0.275371
$845$	−27.0000	−0.928828
$846$	6.00000	0.206284
$847$	−4.00000	−0.137442
$848$	6.00000	0.206041
$849$	−16.0000	−0.549119
$850$	−12.0000	−0.411597
$851$	9.00000	0.308516
$852$	3.00000	0.102778
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	4.00000	0.136877
$855$	−12.0000	−0.410391
$856$	−3.00000	−0.102538
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	2.00000	0.0682789
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	−21.0000	−0.716094
$861$	24.0000	0.817918
$862$	18.0000	0.613082
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−8.00000	−0.271851
$867$	−8.00000	−0.271694
$868$	−8.00000	−0.271538
$869$	4.00000	0.135691
$870$	18.0000	0.610257
$871$	−8.00000	−0.271070
$872$	4.00000	0.135457
$873$	11.0000	0.372294
$874$	−36.0000	−1.21772
$875$	12.0000	0.405674
$876$	−7.00000	−0.236508
$877$	41.0000	1.38447	0.692236	−	0.721671i	$-0.256626\pi$
$877$	41.0000	1.38447	0.692236	+	0.721671i	$0.256626\pi$
$878$	37.0000	1.24869
$879$	9.00000	0.303562
$880$	−3.00000	−0.101130
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	−9.00000	−0.303046
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	6.00000	0.201802
$885$	18.0000	0.605063
$886$	−39.0000	−1.31023
$887$	27.0000	0.906571	0.453286	−	0.891365i	$-0.350252\pi$
$887$	27.0000	0.906571	0.453286	+	0.891365i	$0.350252\pi$
$888$	1.00000	0.0335578
$889$	−68.0000	−2.28065
$890$	45.0000	1.50840
$891$	−1.00000	−0.0335013
$892$	20.0000	0.669650
$893$	24.0000	0.803129
$894$	−18.0000	−0.602010
$895$	45.0000	1.50418
$896$	4.00000	0.133631
$897$	−18.0000	−0.601003
$898$	6.00000	0.200223
$899$	−12.0000	−0.400222
$900$	4.00000	0.133333
$901$	18.0000	0.599667
$902$	−6.00000	−0.199778
$903$	28.0000	0.931782
$904$	−18.0000	−0.598671
$905$	−21.0000	−0.698064
$906$	10.0000	0.332228
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	18.0000	0.597351
$909$	−12.0000	−0.398015
$910$	24.0000	0.795592
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	−4.00000	−0.132453
$913$	3.00000	0.0992855
$914$	16.0000	0.529233
$915$	3.00000	0.0991769
$916$	14.0000	0.462573
$917$	84.0000	2.77392
$918$	−3.00000	−0.0990148
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	27.0000	0.890164
$921$	11.0000	0.362462
$922$	21.0000	0.691598
$923$	6.00000	0.197492
$924$	4.00000	0.131590
$925$	−4.00000	−0.131519
$926$	−5.00000	−0.164310
$927$	5.00000	0.164222
$928$	6.00000	0.196960
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	−6.00000	−0.196748
$931$	−36.0000	−1.17985
$932$	−18.0000	−0.589610
$933$	0	0
$934$	18.0000	0.588978
$935$	−9.00000	−0.294331
$936$	−2.00000	−0.0653720
$937$	41.0000	1.33941	0.669706	−	0.742627i	$-0.266420\pi$
$937$	41.0000	1.33941	0.669706	+	0.742627i	$0.266420\pi$
$938$	−16.0000	−0.522419
$939$	−4.00000	−0.130535
$940$	−18.0000	−0.587095
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	22.0000	0.716799
$943$	54.0000	1.75848
$944$	6.00000	0.195283
$945$	−12.0000	−0.390360
$946$	−7.00000	−0.227590
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−4.00000	−0.129914
$949$	−14.0000	−0.454459
$950$	16.0000	0.519109
$951$	−9.00000	−0.291845
$952$	12.0000	0.388922
$953$	−9.00000	−0.291539	−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000	−0.291539	−0.145769	+	0.989319i	$0.546566\pi$
$954$	−6.00000	−0.194257
$955$	36.0000	1.16493
$956$	0	0
$957$	6.00000	0.193952
$958$	12.0000	0.387702
$959$	24.0000	0.775000
$960$	3.00000	0.0968246
$961$	−27.0000	−0.870968
$962$	2.00000	0.0644826
$963$	3.00000	0.0966736
$964$	−13.0000	−0.418702
$965$	42.0000	1.35203
$966$	−36.0000	−1.15828
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	−1.00000	−0.0321412
$969$	−12.0000	−0.385496
$970$	−33.0000	−1.05957
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	1.00000	0.0320750
$973$	52.0000	1.66704
$974$	−32.0000	−1.02535
$975$	8.00000	0.256205
$976$	1.00000	0.0320092
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	22.0000	0.703482
$979$	15.0000	0.479402
$980$	27.0000	0.862483
$981$	−4.00000	−0.127710
$982$	24.0000	0.765871
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	6.00000	0.191273
$985$	54.0000	1.72058
$986$	18.0000	0.573237
$987$	24.0000	0.763928
$988$	−8.00000	−0.254514
$989$	63.0000	2.00328
$990$	3.00000	0.0953463
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−2.00000	−0.0635001
$993$	−13.0000	−0.412543
$994$	12.0000	0.380617
$995$	−75.0000	−2.37766
$996$	−3.00000	−0.0950586
$997$	17.0000	0.538395	0.269198	−	0.963085i	$-0.413241\pi$
$997$	17.0000	0.538395	0.269198	+	0.963085i	$0.413241\pi$
$998$	13.0000	0.411508
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.b.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4026.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more