LMFDB - Embedded newform 4026.2.a.c.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:	$0$
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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +4.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +3.00000 q^{20} -4.00000 q^{21} -1.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +2.00000 q^{29} -3.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -3.00000 q^{34} -12.0000 q^{35} +1.00000 q^{36} -5.00000 q^{37} +4.00000 q^{38} -6.00000 q^{39} -3.00000 q^{40} +10.0000 q^{41} +4.00000 q^{42} +9.00000 q^{43} +1.00000 q^{44} +3.00000 q^{45} -3.00000 q^{46} +10.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -4.00000 q^{50} +3.00000 q^{51} -6.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +3.00000 q^{55} +4.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +6.00000 q^{59} +3.00000 q^{60} +1.00000 q^{61} -2.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -18.0000 q^{65} -1.00000 q^{66} -12.0000 q^{67} +3.00000 q^{68} +3.00000 q^{69} +12.0000 q^{70} +15.0000 q^{71} -1.00000 q^{72} -11.0000 q^{73} +5.00000 q^{74} +4.00000 q^{75} -4.00000 q^{76} -4.00000 q^{77} +6.00000 q^{78} +12.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +9.00000 q^{83} -4.00000 q^{84} +9.00000 q^{85} -9.00000 q^{86} +2.00000 q^{87} -1.00000 q^{88} +13.0000 q^{89} -3.00000 q^{90} +24.0000 q^{91} +3.00000 q^{92} +2.00000 q^{93} -10.0000 q^{94} -12.0000 q^{95} -1.00000 q^{96} -5.00000 q^{97} -9.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	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
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−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
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	4.00000	0.648886
$39$	−6.00000	−0.960769
$40$	−3.00000	−0.474342
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	4.00000	0.617213
$43$	9.00000	1.37249	0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000	1.37249	0.686244	+	0.727372i	$0.259258\pi$
$44$	1.00000	0.150756
$45$	3.00000	0.447214
$46$	−3.00000	−0.442326
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	−4.00000	−0.565685
$51$	3.00000	0.420084
$52$	−6.00000	−0.832050
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−1.00000	−0.136083
$55$	3.00000	0.404520
$56$	4.00000	0.534522
$57$	−4.00000	−0.529813
$58$	−2.00000	−0.262613
$59$	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
$65$	−18.0000	−2.23263
$66$	−1.00000	−0.123091
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	3.00000	0.363803
$69$	3.00000	0.361158
$70$	12.0000	1.43427
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	−1.00000	−0.117851
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	5.00000	0.581238
$75$	4.00000	0.461880
$76$	−4.00000	−0.458831
$77$	−4.00000	−0.455842
$78$	6.00000	0.679366
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	−4.00000	−0.436436
$85$	9.00000	0.976187
$86$	−9.00000	−0.970495
$87$	2.00000	0.214423
$88$	−1.00000	−0.106600
$89$	13.0000	1.37800	0.688999	−	0.724763i	$-0.258051\pi$
$89$	13.0000	1.37800	0.688999	+	0.724763i	$0.258051\pi$
$90$	−3.00000	−0.316228
$91$	24.0000	2.51588
$92$	3.00000	0.312772
$93$	2.00000	0.207390
$94$	−10.0000	−1.03142
$95$	−12.0000	−1.23117
$96$	−1.00000	−0.102062
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	−9.00000	−0.909137
$99$	1.00000	0.100504
$100$	4.00000	0.400000
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	−3.00000	−0.297044
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	6.00000	0.588348
$105$	−12.0000	−1.17108
$106$	2.00000	0.194257
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	1.00000	0.0962250
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	−3.00000	−0.286039
$111$	−5.00000	−0.474579
$112$	−4.00000	−0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	4.00000	0.374634
$115$	9.00000	0.839254
$116$	2.00000	0.185695
$117$	−6.00000	−0.554700
$118$	−6.00000	−0.552345
$119$	−12.0000	−1.10004
$120$	−3.00000	−0.273861
$121$	1.00000	0.0909091
$122$	−1.00000	−0.0905357
$123$	10.0000	0.901670
$124$	2.00000	0.179605
$125$	−3.00000	−0.268328
$126$	4.00000	0.356348
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	−1.00000	−0.0883883
$129$	9.00000	0.792406
$130$	18.0000	1.57870
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	1.00000	0.0870388
$133$	16.0000	1.38738
$134$	12.0000	1.03664
$135$	3.00000	0.258199
$136$	−3.00000	−0.257248
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−3.00000	−0.255377
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	−12.0000	−1.01419
$141$	10.0000	0.842152
$142$	−15.0000	−1.25877
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	11.0000	0.910366
$147$	9.00000	0.742307
$148$	−5.00000	−0.410997
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	−4.00000	−0.326599
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	4.00000	0.324443
$153$	3.00000	0.242536
$154$	4.00000	0.322329
$155$	6.00000	0.481932
$156$	−6.00000	−0.480384
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−12.0000	−0.954669
$159$	−2.00000	−0.158610
$160$	−3.00000	−0.237171
$161$	−12.0000	−0.945732
$162$	−1.00000	−0.0785674
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	10.0000	0.780869
$165$	3.00000	0.233550
$166$	−9.00000	−0.698535
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	4.00000	0.308607
$169$	23.0000	1.76923
$170$	−9.00000	−0.690268
$171$	−4.00000	−0.305888
$172$	9.00000	0.686244
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	−2.00000	−0.151620
$175$	−16.0000	−1.20949
$176$	1.00000	0.0753778
$177$	6.00000	0.450988
$178$	−13.0000	−0.974391
$179$	−17.0000	−1.27064	−0.635320	−	0.772249i	$-0.719132\pi$
$179$	−17.0000	−1.27064	−0.635320	+	0.772249i	$0.719132\pi$
$180$	3.00000	0.223607
$181$	−3.00000	−0.222988	−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000	−0.222988	−0.111494	+	0.993765i	$0.535564\pi$
$182$	−24.0000	−1.77900
$183$	1.00000	0.0739221
$184$	−3.00000	−0.221163
$185$	−15.0000	−1.10282
$186$	−2.00000	−0.146647
$187$	3.00000	0.219382
$188$	10.0000	0.729325
$189$	−4.00000	−0.290957
$190$	12.0000	0.870572
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.00000	0.0721688
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	5.00000	0.358979
$195$	−18.0000	−1.28901
$196$	9.00000	0.642857
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−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
$201$	−12.0000	−0.846415
$202$	−4.00000	−0.281439
$203$	−8.00000	−0.561490
$204$	3.00000	0.210042
$205$	30.0000	2.09529
$206$	−13.0000	−0.905753
$207$	3.00000	0.208514
$208$	−6.00000	−0.416025
$209$	−4.00000	−0.276686
$210$	12.0000	0.828079
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−2.00000	−0.137361
$213$	15.0000	1.02778
$214$	−15.0000	−1.02538
$215$	27.0000	1.84138
$216$	−1.00000	−0.0680414
$217$	−8.00000	−0.543075
$218$	−12.0000	−0.812743
$219$	−11.0000	−0.743311
$220$	3.00000	0.202260
$221$	−18.0000	−1.21081
$222$	5.00000	0.335578
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	4.00000	0.267261
$225$	4.00000	0.266667
$226$	6.00000	0.399114
$227$	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
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−9.00000	−0.593442
$231$	−4.00000	−0.263181
$232$	−2.00000	−0.131306
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	6.00000	0.392232
$235$	30.0000	1.95698
$236$	6.00000	0.390567
$237$	12.0000	0.779484
$238$	12.0000	0.777844
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	3.00000	0.193649
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	1.00000	0.0640184
$245$	27.0000	1.72497
$246$	−10.0000	−0.637577
$247$	24.0000	1.52708
$248$	−2.00000	−0.127000
$249$	9.00000	0.570352
$250$	3.00000	0.189737
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	−4.00000	−0.251976
$253$	3.00000	0.188608
$254$	−5.00000	−0.313728
$255$	9.00000	0.563602
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−9.00000	−0.560316
$259$	20.0000	1.24274
$260$	−18.0000	−1.11631
$261$	2.00000	0.123797
$262$	−15.0000	−0.926703
$263$	−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
$265$	−6.00000	−0.368577
$266$	−16.0000	−0.981023
$267$	13.0000	0.795587
$268$	−12.0000	−0.733017
$269$	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
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	3.00000	0.181902
$273$	24.0000	1.45255
$274$	−2.00000	−0.120824
$275$	4.00000	0.241209
$276$	3.00000	0.180579
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	5.00000	0.299880
$279$	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	$0.442725\pi$
$282$	−10.0000	−0.595491
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	15.0000	0.890086
$285$	−12.0000	−0.710819
$286$	6.00000	0.354787
$287$	−40.0000	−2.36113
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	−6.00000	−0.352332
$291$	−5.00000	−0.293105
$292$	−11.0000	−0.643726
$293$	−3.00000	−0.175262	−0.0876309	−	0.996153i	$-0.527930\pi$
$293$	−3.00000	−0.175262	−0.0876309	+	0.996153i	$0.527930\pi$
$294$	−9.00000	−0.524891
$295$	18.0000	1.04800
$296$	5.00000	0.290619
$297$	1.00000	0.0580259
$298$	14.0000	0.810998
$299$	−18.0000	−1.04097
$300$	4.00000	0.230940
$301$	−36.0000	−2.07501
$302$	2.00000	0.115087
$303$	4.00000	0.229794
$304$	−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
$309$	13.0000	0.739544
$310$	−6.00000	−0.340777
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	6.00000	0.339683
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	−10.0000	−0.564333
$315$	−12.0000	−0.676123
$316$	12.0000	0.675053
$317$	−1.00000	−0.0561656	−0.0280828	−	0.999606i	$-0.508940\pi$
$317$	−1.00000	−0.0561656	−0.0280828	+	0.999606i	$0.508940\pi$
$318$	2.00000	0.112154
$319$	2.00000	0.111979
$320$	3.00000	0.167705
$321$	15.0000	0.837218
$322$	12.0000	0.668734
$323$	−12.0000	−0.667698
$324$	1.00000	0.0555556
$325$	−24.0000	−1.33128
$326$	−10.0000	−0.553849
$327$	12.0000	0.663602
$328$	−10.0000	−0.552158
$329$	−40.0000	−2.20527
$330$	−3.00000	−0.165145
$331$	15.0000	0.824475	0.412237	−	0.911077i	$-0.364747\pi$
$331$	15.0000	0.824475	0.412237	+	0.911077i	$0.364747\pi$
$332$	9.00000	0.493939
$333$	−5.00000	−0.273998
$334$	18.0000	0.984916
$335$	−36.0000	−1.96689
$336$	−4.00000	−0.218218
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	−23.0000	−1.25104
$339$	−6.00000	−0.325875
$340$	9.00000	0.488094
$341$	2.00000	0.108306
$342$	4.00000	0.216295
$343$	−8.00000	−0.431959
$344$	−9.00000	−0.485247
$345$	9.00000	0.484544
$346$	16.0000	0.860165
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	2.00000	0.107211
$349$	−21.0000	−1.12410	−0.562052	−	0.827102i	$-0.689988\pi$
$349$	−21.0000	−1.12410	−0.562052	+	0.827102i	$0.689988\pi$
$350$	16.0000	0.855236
$351$	−6.00000	−0.320256
$352$	−1.00000	−0.0533002
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	−6.00000	−0.318896
$355$	45.0000	2.38835
$356$	13.0000	0.688999
$357$	−12.0000	−0.635107
$358$	17.0000	0.898478
$359$	27.0000	1.42501	0.712503	−	0.701669i	$-0.247562\pi$
$359$	27.0000	1.42501	0.712503	+	0.701669i	$0.247562\pi$
$360$	−3.00000	−0.158114
$361$	−3.00000	−0.157895
$362$	3.00000	0.157676
$363$	1.00000	0.0524864
$364$	24.0000	1.25794
$365$	−33.0000	−1.72730
$366$	−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	$0.760850\pi$
$368$	3.00000	0.156386
$369$	10.0000	0.520579
$370$	15.0000	0.779813
$371$	8.00000	0.415339
$372$	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
$376$	−10.0000	−0.515711
$377$	−12.0000	−0.618031
$378$	4.00000	0.205738
$379$	−18.0000	−0.924598	−0.462299	−	0.886724i	$-0.652975\pi$
$379$	−18.0000	−0.924598	−0.462299	+	0.886724i	$0.652975\pi$
$380$	−12.0000	−0.615587
$381$	5.00000	0.256158
$382$	12.0000	0.613973
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	−1.00000	−0.0510310
$385$	−12.0000	−0.611577
$386$	−6.00000	−0.305392
$387$	9.00000	0.457496
$388$	−5.00000	−0.253837
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	18.0000	0.911465
$391$	9.00000	0.455150
$392$	−9.00000	−0.454569
$393$	15.0000	0.756650
$394$	−2.00000	−0.100759
$395$	36.0000	1.81136
$396$	1.00000	0.0502519
$397$	25.0000	1.25471	0.627357	−	0.778732i	$-0.284137\pi$
$397$	25.0000	1.25471	0.627357	+	0.778732i	$0.284137\pi$
$398$	25.0000	1.25314
$399$	16.0000	0.801002
$400$	4.00000	0.200000
$401$	25.0000	1.24844	0.624220	−	0.781248i	$-0.285417\pi$
$401$	25.0000	1.24844	0.624220	+	0.781248i	$0.285417\pi$
$402$	12.0000	0.598506
$403$	−12.0000	−0.597763
$404$	4.00000	0.199007
$405$	3.00000	0.149071
$406$	8.00000	0.397033
$407$	−5.00000	−0.247841
$408$	−3.00000	−0.148522
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	−30.0000	−1.48159
$411$	2.00000	0.0986527
$412$	13.0000	0.640464
$413$	−24.0000	−1.18096
$414$	−3.00000	−0.147442
$415$	27.0000	1.32538
$416$	6.00000	0.294174
$417$	−5.00000	−0.244851
$418$	4.00000	0.195646
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	−12.0000	−0.585540
$421$	−17.0000	−0.828529	−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000	−0.828529	−0.414265	+	0.910156i	$0.635961\pi$
$422$	8.00000	0.389434
$423$	10.0000	0.486217
$424$	2.00000	0.0971286
$425$	12.0000	0.582086
$426$	−15.0000	−0.726752
$427$	−4.00000	−0.193574
$428$	15.0000	0.725052
$429$	−6.00000	−0.289683
$430$	−27.0000	−1.30206
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	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
$435$	6.00000	0.287678
$436$	12.0000	0.574696
$437$	−12.0000	−0.574038
$438$	11.0000	0.525600
$439$	23.0000	1.09773	0.548865	−	0.835911i	$-0.315060\pi$
$439$	23.0000	1.09773	0.548865	+	0.835911i	$0.315060\pi$
$440$	−3.00000	−0.143019
$441$	9.00000	0.428571
$442$	18.0000	0.856173
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	−5.00000	−0.237289
$445$	39.0000	1.84878
$446$	−12.0000	−0.568216
$447$	−14.0000	−0.662177
$448$	−4.00000	−0.188982
$449$	42.0000	1.98210	0.991051	−	0.133482i	$-0.0426157\pi$
$449$	42.0000	1.98210	0.991051	+	0.133482i	$0.0426157\pi$
$450$	−4.00000	−0.188562
$451$	10.0000	0.470882
$452$	−6.00000	−0.282216
$453$	−2.00000	−0.0939682
$454$	−18.0000	−0.844782
$455$	72.0000	3.37541
$456$	4.00000	0.187317
$457$	−24.0000	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$458$	10.0000	0.467269
$459$	3.00000	0.140028
$460$	9.00000	0.419627
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	4.00000	0.186097
$463$	−11.0000	−0.511213	−0.255607	−	0.966781i	$-0.582275\pi$
$463$	−11.0000	−0.511213	−0.255607	+	0.966781i	$0.582275\pi$
$464$	2.00000	0.0928477
$465$	6.00000	0.278243
$466$	18.0000	0.833834
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	−6.00000	−0.277350
$469$	48.0000	2.21643
$470$	−30.0000	−1.38380
$471$	10.0000	0.460776
$472$	−6.00000	−0.276172
$473$	9.00000	0.413820
$474$	−12.0000	−0.551178
$475$	−16.0000	−0.734130
$476$	−12.0000	−0.550019
$477$	−2.00000	−0.0915737
$478$	8.00000	0.365911
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	−3.00000	−0.136931
$481$	30.0000	1.36788
$482$	17.0000	0.774329
$483$	−12.0000	−0.546019
$484$	1.00000	0.0454545
$485$	−15.0000	−0.681115
$486$	−1.00000	−0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−1.00000	−0.0452679
$489$	10.0000	0.452216
$490$	−27.0000	−1.21974
$491$	32.0000	1.44414	0.722070	−	0.691820i	$-0.243191\pi$
$491$	32.0000	1.44414	0.722070	+	0.691820i	$0.243191\pi$
$492$	10.0000	0.450835
$493$	6.00000	0.270226
$494$	−24.0000	−1.07981
$495$	3.00000	0.134840
$496$	2.00000	0.0898027
$497$	−60.0000	−2.69137
$498$	−9.00000	−0.403300
$499$	−9.00000	−0.402895	−0.201448	−	0.979499i	$-0.564565\pi$
$499$	−9.00000	−0.402895	−0.201448	+	0.979499i	$0.564565\pi$
$500$	−3.00000	−0.134164
$501$	−18.0000	−0.804181
$502$	−16.0000	−0.714115
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	4.00000	0.178174
$505$	12.0000	0.533993
$506$	−3.00000	−0.133366
$507$	23.0000	1.02147
$508$	5.00000	0.221839
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	−9.00000	−0.398527
$511$	44.0000	1.94645
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	18.0000	0.793946
$515$	39.0000	1.71855
$516$	9.00000	0.396203
$517$	10.0000	0.439799
$518$	−20.0000	−0.878750
$519$	−16.0000	−0.702322
$520$	18.0000	0.789352
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−2.00000	−0.0875376
$523$	25.0000	1.09317	0.546587	−	0.837402i	$-0.315927\pi$
$523$	25.0000	1.09317	0.546587	+	0.837402i	$0.315927\pi$
$524$	15.0000	0.655278
$525$	−16.0000	−0.698297
$526$	6.00000	0.261612
$527$	6.00000	0.261364
$528$	1.00000	0.0435194
$529$	−14.0000	−0.608696
$530$	6.00000	0.260623
$531$	6.00000	0.260378
$532$	16.0000	0.693688
$533$	−60.0000	−2.59889
$534$	−13.0000	−0.562565
$535$	45.0000	1.94552
$536$	12.0000	0.518321
$537$	−17.0000	−0.733604
$538$	−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	$0.465721\pi$
$542$	25.0000	1.07384
$543$	−3.00000	−0.128742
$544$	−3.00000	−0.128624
$545$	36.0000	1.54207
$546$	−24.0000	−1.02711
$547$	40.0000	1.71028	0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000	1.71028	0.855138	+	0.518400i	$0.173472\pi$
$548$	2.00000	0.0854358
$549$	1.00000	0.0426790
$550$	−4.00000	−0.170561
$551$	−8.00000	−0.340811
$552$	−3.00000	−0.127688
$553$	−48.0000	−2.04117
$554$	−10.0000	−0.424859
$555$	−15.0000	−0.636715
$556$	−5.00000	−0.212047
$557$	8.00000	0.338971	0.169485	−	0.985533i	$-0.445789\pi$
$557$	8.00000	0.338971	0.169485	+	0.985533i	$0.445789\pi$
$558$	−2.00000	−0.0846668
$559$	−54.0000	−2.28396
$560$	−12.0000	−0.507093
$561$	3.00000	0.126660
$562$	−6.00000	−0.253095
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	10.0000	0.421076
$565$	−18.0000	−0.757266
$566$	−16.0000	−0.672530
$567$	−4.00000	−0.167984
$568$	−15.0000	−0.629386
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	12.0000	0.502625
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	−6.00000	−0.250873
$573$	−12.0000	−0.501307
$574$	40.0000	1.66957
$575$	12.0000	0.500435
$576$	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
$579$	6.00000	0.249351
$580$	6.00000	0.249136
$581$	−36.0000	−1.49353
$582$	5.00000	0.207257
$583$	−2.00000	−0.0828315
$584$	11.0000	0.455183
$585$	−18.0000	−0.744208
$586$	3.00000	0.123929
$587$	−22.0000	−0.908037	−0.454019	−	0.890992i	$-0.650010\pi$
$587$	−22.0000	−0.908037	−0.454019	+	0.890992i	$0.650010\pi$
$588$	9.00000	0.371154
$589$	−8.00000	−0.329634
$590$	−18.0000	−0.741048
$591$	2.00000	0.0822690
$592$	−5.00000	−0.205499
$593$	17.0000	0.698106	0.349053	−	0.937103i	$-0.386503\pi$
$593$	17.0000	0.698106	0.349053	+	0.937103i	$0.386503\pi$
$594$	−1.00000	−0.0410305
$595$	−36.0000	−1.47586
$596$	−14.0000	−0.573462
$597$	−25.0000	−1.02318
$598$	18.0000	0.736075
$599$	−39.0000	−1.59350	−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000	−1.59350	−0.796748	+	0.604311i	$0.793448\pi$
$600$	−4.00000	−0.163299
$601$	31.0000	1.26452	0.632258	−	0.774758i	$-0.282128\pi$
$601$	31.0000	1.26452	0.632258	+	0.774758i	$0.282128\pi$
$602$	36.0000	1.46725
$603$	−12.0000	−0.488678
$604$	−2.00000	−0.0813788
$605$	3.00000	0.121967
$606$	−4.00000	−0.162489
$607$	7.00000	0.284121	0.142061	−	0.989858i	$-0.454627\pi$
$607$	7.00000	0.284121	0.142061	+	0.989858i	$0.454627\pi$
$608$	4.00000	0.162221
$609$	−8.00000	−0.324176
$610$	−3.00000	−0.121466
$611$	−60.0000	−2.42734
$612$	3.00000	0.121268
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	−11.0000	−0.443924
$615$	30.0000	1.20972
$616$	4.00000	0.161165
$617$	17.0000	0.684394	0.342197	−	0.939628i	$-0.388829\pi$
$617$	17.0000	0.684394	0.342197	+	0.939628i	$0.388829\pi$
$618$	−13.0000	−0.522937
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	6.00000	0.240966
$621$	3.00000	0.120386
$622$	−24.0000	−0.962312
$623$	−52.0000	−2.08334
$624$	−6.00000	−0.240192
$625$	−29.0000	−1.16000
$626$	28.0000	1.11911
$627$	−4.00000	−0.159745
$628$	10.0000	0.399043
$629$	−15.0000	−0.598089
$630$	12.0000	0.478091
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	−12.0000	−0.477334
$633$	−8.00000	−0.317971
$634$	1.00000	0.0397151
$635$	15.0000	0.595257
$636$	−2.00000	−0.0793052
$637$	−54.0000	−2.13956
$638$	−2.00000	−0.0791808
$639$	15.0000	0.593391
$640$	−3.00000	−0.118585
$641$	27.0000	1.06644	0.533218	−	0.845978i	$-0.320983\pi$
$641$	27.0000	1.06644	0.533218	+	0.845978i	$0.320983\pi$
$642$	−15.0000	−0.592003
$643$	−21.0000	−0.828159	−0.414080	−	0.910241i	$-0.635896\pi$
$643$	−21.0000	−0.828159	−0.414080	+	0.910241i	$0.635896\pi$
$644$	−12.0000	−0.472866
$645$	27.0000	1.06312
$646$	12.0000	0.472134
$647$	−15.0000	−0.589711	−0.294855	−	0.955542i	$-0.595271\pi$
$647$	−15.0000	−0.589711	−0.294855	+	0.955542i	$0.595271\pi$
$648$	−1.00000	−0.0392837
$649$	6.00000	0.235521
$650$	24.0000	0.941357
$651$	−8.00000	−0.313545
$652$	10.0000	0.391630
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	−12.0000	−0.469237
$655$	45.0000	1.75830
$656$	10.0000	0.390434
$657$	−11.0000	−0.429151
$658$	40.0000	1.55936
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	3.00000	0.116775
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	−15.0000	−0.582992
$663$	−18.0000	−0.699062
$664$	−9.00000	−0.349268
$665$	48.0000	1.86136
$666$	5.00000	0.193746
$667$	6.00000	0.232321
$668$	−18.0000	−0.696441
$669$	12.0000	0.463947
$670$	36.0000	1.39080
$671$	1.00000	0.0386046
$672$	4.00000	0.154303
$673$	4.00000	0.154189	0.0770943	−	0.997024i	$-0.475436\pi$
$673$	4.00000	0.154189	0.0770943	+	0.997024i	$0.475436\pi$
$674$	−8.00000	−0.308148
$675$	4.00000	0.153960
$676$	23.0000	0.884615
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	6.00000	0.230429
$679$	20.0000	0.767530
$680$	−9.00000	−0.345134
$681$	18.0000	0.689761
$682$	−2.00000	−0.0765840
$683$	41.0000	1.56882	0.784411	−	0.620242i	$-0.212966\pi$
$683$	41.0000	1.56882	0.784411	+	0.620242i	$0.212966\pi$
$684$	−4.00000	−0.152944
$685$	6.00000	0.229248
$686$	8.00000	0.305441
$687$	−10.0000	−0.381524
$688$	9.00000	0.343122
$689$	12.0000	0.457164
$690$	−9.00000	−0.342624
$691$	−48.0000	−1.82601	−0.913003	−	0.407953i	$-0.866243\pi$
$691$	−48.0000	−1.82601	−0.913003	+	0.407953i	$0.866243\pi$
$692$	−16.0000	−0.608229
$693$	−4.00000	−0.151947
$694$	−3.00000	−0.113878
$695$	−15.0000	−0.568982
$696$	−2.00000	−0.0758098
$697$	30.0000	1.13633
$698$	21.0000	0.794862
$699$	−18.0000	−0.680823
$700$	−16.0000	−0.604743
$701$	40.0000	1.51078	0.755390	−	0.655276i	$-0.227448\pi$
$701$	40.0000	1.51078	0.755390	+	0.655276i	$0.227448\pi$
$702$	6.00000	0.226455
$703$	20.0000	0.754314
$704$	1.00000	0.0376889
$705$	30.0000	1.12987
$706$	18.0000	0.677439
$707$	−16.0000	−0.601742
$708$	6.00000	0.225494
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	−45.0000	−1.68882
$711$	12.0000	0.450035
$712$	−13.0000	−0.487196
$713$	6.00000	0.224702
$714$	12.0000	0.449089
$715$	−18.0000	−0.673162
$716$	−17.0000	−0.635320
$717$	−8.00000	−0.298765
$718$	−27.0000	−1.00763
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	3.00000	0.111803
$721$	−52.0000	−1.93658
$722$	3.00000	0.111648
$723$	−17.0000	−0.632237
$724$	−3.00000	−0.111494
$725$	8.00000	0.297113
$726$	−1.00000	−0.0371135
$727$	20.0000	0.741759	0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000	0.741759	0.370879	+	0.928681i	$0.379056\pi$
$728$	−24.0000	−0.889499
$729$	1.00000	0.0370370
$730$	33.0000	1.22138
$731$	27.0000	0.998631
$732$	1.00000	0.0369611
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	28.0000	1.03350
$735$	27.0000	0.995910
$736$	−3.00000	−0.110581
$737$	−12.0000	−0.442026
$738$	−10.0000	−0.368105
$739$	−27.0000	−0.993211	−0.496606	−	0.867976i	$-0.665420\pi$
$739$	−27.0000	−0.993211	−0.496606	+	0.867976i	$0.665420\pi$
$740$	−15.0000	−0.551411
$741$	24.0000	0.881662
$742$	−8.00000	−0.293689
$743$	−21.0000	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$744$	−2.00000	−0.0733236
$745$	−42.0000	−1.53876
$746$	19.0000	0.695639
$747$	9.00000	0.329293
$748$	3.00000	0.109691
$749$	−60.0000	−2.19235
$750$	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	$0.435679\pi$
$752$	10.0000	0.364662
$753$	16.0000	0.583072
$754$	12.0000	0.437014
$755$	−6.00000	−0.218362
$756$	−4.00000	−0.145479
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	18.0000	0.653789
$759$	3.00000	0.108893
$760$	12.0000	0.435286
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	−5.00000	−0.181131
$763$	−48.0000	−1.73772
$764$	−12.0000	−0.434145
$765$	9.00000	0.325396
$766$	16.0000	0.578103
$767$	−36.0000	−1.29988
$768$	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
$771$	−18.0000	−0.648254
$772$	6.00000	0.215945
$773$	31.0000	1.11499	0.557496	−	0.830179i	$-0.311762\pi$
$773$	31.0000	1.11499	0.557496	+	0.830179i	$0.311762\pi$
$774$	−9.00000	−0.323498
$775$	8.00000	0.287368
$776$	5.00000	0.179490
$777$	20.0000	0.717496
$778$	−30.0000	−1.07555
$779$	−40.0000	−1.43315
$780$	−18.0000	−0.644503
$781$	15.0000	0.536742
$782$	−9.00000	−0.321839
$783$	2.00000	0.0714742
$784$	9.00000	0.321429
$785$	30.0000	1.07075
$786$	−15.0000	−0.535032
$787$	48.0000	1.71102	0.855508	−	0.517790i	$-0.173245\pi$
$787$	48.0000	1.71102	0.855508	+	0.517790i	$0.173245\pi$
$788$	2.00000	0.0712470
$789$	−6.00000	−0.213606
$790$	−36.0000	−1.28082
$791$	24.0000	0.853342
$792$	−1.00000	−0.0355335
$793$	−6.00000	−0.213066
$794$	−25.0000	−0.887217
$795$	−6.00000	−0.212798
$796$	−25.0000	−0.886102
$797$	−54.0000	−1.91278	−0.956389	−	0.292096i	$-0.905647\pi$
$797$	−54.0000	−1.91278	−0.956389	+	0.292096i	$0.905647\pi$
$798$	−16.0000	−0.566394
$799$	30.0000	1.06132
$800$	−4.00000	−0.141421
$801$	13.0000	0.459332
$802$	−25.0000	−0.882781
$803$	−11.0000	−0.388182
$804$	−12.0000	−0.423207
$805$	−36.0000	−1.26883
$806$	12.0000	0.422682
$807$	18.0000	0.633630
$808$	−4.00000	−0.140720
$809$	−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
$811$	−23.0000	−0.807639	−0.403820	−	0.914839i	$-0.632318\pi$
$811$	−23.0000	−0.807639	−0.403820	+	0.914839i	$0.632318\pi$
$812$	−8.00000	−0.280745
$813$	−25.0000	−0.876788
$814$	5.00000	0.175250
$815$	30.0000	1.05085
$816$	3.00000	0.105021
$817$	−36.0000	−1.25948
$818$	−24.0000	−0.839140
$819$	24.0000	0.838628
$820$	30.0000	1.04765
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	−2.00000	−0.0697580
$823$	−38.0000	−1.32460	−0.662298	−	0.749240i	$-0.730419\pi$
$823$	−38.0000	−1.32460	−0.662298	+	0.749240i	$0.730419\pi$
$824$	−13.0000	−0.452876
$825$	4.00000	0.139262
$826$	24.0000	0.835067
$827$	−43.0000	−1.49526	−0.747628	−	0.664117i	$-0.768807\pi$
$827$	−43.0000	−1.49526	−0.747628	+	0.664117i	$0.768807\pi$
$828$	3.00000	0.104257
$829$	50.0000	1.73657	0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	−27.0000	−0.937184
$831$	10.0000	0.346896
$832$	−6.00000	−0.208013
$833$	27.0000	0.935495
$834$	5.00000	0.173136
$835$	−54.0000	−1.86875
$836$	−4.00000	−0.138343
$837$	2.00000	0.0691301
$838$	−30.0000	−1.03633
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	12.0000	0.414039
$841$	−25.0000	−0.862069
$842$	17.0000	0.585859
$843$	6.00000	0.206651
$844$	−8.00000	−0.275371
$845$	69.0000	2.37367
$846$	−10.0000	−0.343807
$847$	−4.00000	−0.137442
$848$	−2.00000	−0.0686803
$849$	16.0000	0.549119
$850$	−12.0000	−0.411597
$851$	−15.0000	−0.514193
$852$	15.0000	0.513892
$853$	−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
$856$	−15.0000	−0.512689
$857$	52.0000	1.77629	0.888143	−	0.459567i	$-0.151995\pi$
$857$	52.0000	1.77629	0.888143	+	0.459567i	$0.151995\pi$
$858$	6.00000	0.204837
$859$	−24.0000	−0.818869	−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000	−0.818869	−0.409435	+	0.912339i	$0.634274\pi$
$860$	27.0000	0.920692
$861$	−40.0000	−1.36320
$862$	2.00000	0.0681203
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	−1.00000	−0.0340207
$865$	−48.0000	−1.63205
$866$	−8.00000	−0.271851
$867$	−8.00000	−0.271694
$868$	−8.00000	−0.271538
$869$	12.0000	0.407072
$870$	−6.00000	−0.203419
$871$	72.0000	2.43963
$872$	−12.0000	−0.406371
$873$	−5.00000	−0.169224
$874$	12.0000	0.405906
$875$	12.0000	0.405674
$876$	−11.0000	−0.371656
$877$	−31.0000	−1.04680	−0.523398	−	0.852088i	$-0.675336\pi$
$877$	−31.0000	−1.04680	−0.523398	+	0.852088i	$0.675336\pi$
$878$	−23.0000	−0.776212
$879$	−3.00000	−0.101187
$880$	3.00000	0.101130
$881$	−10.0000	−0.336909	−0.168454	−	0.985709i	$-0.553878\pi$
$881$	−10.0000	−0.336909	−0.168454	+	0.985709i	$0.553878\pi$
$882$	−9.00000	−0.303046
$883$	57.0000	1.91820	0.959101	−	0.283063i	$-0.0913505\pi$
$883$	57.0000	1.91820	0.959101	+	0.283063i	$0.0913505\pi$
$884$	−18.0000	−0.605406
$885$	18.0000	0.605063
$886$	−39.0000	−1.31023
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	5.00000	0.167789
$889$	−20.0000	−0.670778
$890$	−39.0000	−1.30728
$891$	1.00000	0.0335013
$892$	12.0000	0.401790
$893$	−40.0000	−1.33855
$894$	14.0000	0.468230
$895$	−51.0000	−1.70474
$896$	4.00000	0.133631
$897$	−18.0000	−0.601003
$898$	−42.0000	−1.40156
$899$	4.00000	0.133407
$900$	4.00000	0.133333
$901$	−6.00000	−0.199889
$902$	−10.0000	−0.332964
$903$	−36.0000	−1.19800
$904$	6.00000	0.199557
$905$	−9.00000	−0.299170
$906$	2.00000	0.0664455
$907$	−11.0000	−0.365249	−0.182625	−	0.983183i	$-0.558459\pi$
$907$	−11.0000	−0.365249	−0.182625	+	0.983183i	$0.558459\pi$
$908$	18.0000	0.597351
$909$	4.00000	0.132672
$910$	−72.0000	−2.38678
$911$	−46.0000	−1.52405	−0.762024	−	0.647549i	$-0.775794\pi$
$911$	−46.0000	−1.52405	−0.762024	+	0.647549i	$0.775794\pi$
$912$	−4.00000	−0.132453
$913$	9.00000	0.297857
$914$	24.0000	0.793849
$915$	3.00000	0.0991769
$916$	−10.0000	−0.330409
$917$	−60.0000	−1.98137
$918$	−3.00000	−0.0990148
$919$	−39.0000	−1.28649	−0.643246	−	0.765660i	$-0.722413\pi$
$919$	−39.0000	−1.28649	−0.643246	+	0.765660i	$0.722413\pi$
$920$	−9.00000	−0.296721
$921$	11.0000	0.362462
$922$	9.00000	0.296399
$923$	−90.0000	−2.96239
$924$	−4.00000	−0.131590
$925$	−20.0000	−0.657596
$926$	11.0000	0.361482
$927$	13.0000	0.426976
$928$	−2.00000	−0.0656532
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	−6.00000	−0.196748
$931$	−36.0000	−1.17985
$932$	−18.0000	−0.589610
$933$	24.0000	0.785725
$934$	−6.00000	−0.196326
$935$	9.00000	0.294331
$936$	6.00000	0.196116
$937$	29.0000	0.947389	0.473694	−	0.880689i	$-0.342920\pi$
$937$	29.0000	0.947389	0.473694	+	0.880689i	$0.342920\pi$
$938$	−48.0000	−1.56726
$939$	−28.0000	−0.913745
$940$	30.0000	0.978492
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−10.0000	−0.325818
$943$	30.0000	0.976934
$944$	6.00000	0.195283
$945$	−12.0000	−0.390360
$946$	−9.00000	−0.292615
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	12.0000	0.389742
$949$	66.0000	2.14245
$950$	16.0000	0.519109
$951$	−1.00000	−0.0324272
$952$	12.0000	0.388922
$953$	−1.00000	−0.0323932	−0.0161966	−	0.999869i	$-0.505156\pi$
$953$	−1.00000	−0.0323932	−0.0161966	+	0.999869i	$0.505156\pi$
$954$	2.00000	0.0647524
$955$	−36.0000	−1.16493
$956$	−8.00000	−0.258738
$957$	2.00000	0.0646508
$958$	28.0000	0.904639
$959$	−8.00000	−0.258333
$960$	3.00000	0.0968246
$961$	−27.0000	−0.870968
$962$	−30.0000	−0.967239
$963$	15.0000	0.483368
$964$	−17.0000	−0.547533
$965$	18.0000	0.579441
$966$	12.0000	0.386094
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	−1.00000	−0.0321412
$969$	−12.0000	−0.385496
$970$	15.0000	0.481621
$971$	45.0000	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$972$	1.00000	0.0320750
$973$	20.0000	0.641171
$974$	32.0000	1.02535
$975$	−24.0000	−0.768615
$976$	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	$0.592969\pi$
$978$	−10.0000	−0.319765
$979$	13.0000	0.415482
$980$	27.0000	0.862483
$981$	12.0000	0.383131
$982$	−32.0000	−1.02116
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	−10.0000	−0.318788
$985$	6.00000	0.191176
$986$	−6.00000	−0.191079
$987$	−40.0000	−1.27321
$988$	24.0000	0.763542
$989$	27.0000	0.858550
$990$	−3.00000	−0.0953463
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	−2.00000	−0.0635001
$993$	15.0000	0.476011
$994$	60.0000	1.90308
$995$	−75.0000	−2.37766
$996$	9.00000	0.285176
$997$	−39.0000	−1.23514	−0.617571	−	0.786515i	$-0.711883\pi$
$997$	−39.0000	−1.23514	−0.617571	+	0.786515i	$0.711883\pi$
$998$	9.00000	0.284890
$999$	−5.00000	−0.158193

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.c.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

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