LMFDB - Embedded newform 1110.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, 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("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.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:	$8.86339462436$
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	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} -5.00000 q^{13} -3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -1.00000 q^{20} -3.00000 q^{21} +3.00000 q^{22} -7.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +5.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} +6.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} -5.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} -1.00000 q^{38} +5.00000 q^{39} +1.00000 q^{40} +8.00000 q^{41} +3.00000 q^{42} +4.00000 q^{43} -3.00000 q^{44} -1.00000 q^{45} +7.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} -5.00000 q^{51} -5.00000 q^{52} -13.0000 q^{53} +1.00000 q^{54} +3.00000 q^{55} -3.00000 q^{56} -1.00000 q^{57} +6.00000 q^{58} +14.0000 q^{59} +1.00000 q^{60} -6.00000 q^{61} -6.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +5.00000 q^{65} -3.00000 q^{66} -2.00000 q^{67} +5.00000 q^{68} +7.00000 q^{69} +3.00000 q^{70} -14.0000 q^{71} -1.00000 q^{72} -13.0000 q^{73} +1.00000 q^{74} -1.00000 q^{75} +1.00000 q^{76} -9.00000 q^{77} -5.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} -17.0000 q^{83} -3.00000 q^{84} -5.00000 q^{85} -4.00000 q^{86} +6.00000 q^{87} +3.00000 q^{88} +7.00000 q^{89} +1.00000 q^{90} -15.0000 q^{91} -7.00000 q^{92} -6.00000 q^{93} +12.0000 q^{94} -1.00000 q^{95} +1.00000 q^{96} -8.00000 q^{97} -2.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−1.00000	−0.288675
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	−3.00000	−0.801784
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	−1.00000	−0.223607
$21$	−3.00000	−0.654654
$22$	3.00000	0.639602
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	5.00000	0.980581
$27$	−1.00000	−0.192450
$28$	3.00000	0.566947
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−1.00000	−0.182574
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	−5.00000	−0.857493
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−1.00000	−0.162221
$39$	5.00000	0.800641
$40$	1.00000	0.158114
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	3.00000	0.462910
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−3.00000	−0.452267
$45$	−1.00000	−0.149071
$46$	7.00000	1.03209
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	−5.00000	−0.700140
$52$	−5.00000	−0.693375
$53$	−13.0000	−1.78569	−0.892844	−	0.450367i	$-0.851293\pi$
$53$	−13.0000	−1.78569	−0.892844	+	0.450367i	$0.851293\pi$
$54$	1.00000	0.136083
$55$	3.00000	0.404520
$56$	−3.00000	−0.400892
$57$	−1.00000	−0.132453
$58$	6.00000	0.787839
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−6.00000	−0.762001
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	5.00000	0.620174
$66$	−3.00000	−0.369274
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	5.00000	0.606339
$69$	7.00000	0.842701
$70$	3.00000	0.358569
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	−1.00000	−0.117851
$73$	−13.0000	−1.52153	−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000	−1.52153	−0.760767	+	0.649025i	$0.775177\pi$
$74$	1.00000	0.116248
$75$	−1.00000	−0.115470
$76$	1.00000	0.114708
$77$	−9.00000	−1.02565
$78$	−5.00000	−0.566139
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−8.00000	−0.883452
$83$	−17.0000	−1.86599	−0.932996	−	0.359886i	$-0.882816\pi$
$83$	−17.0000	−1.86599	−0.932996	+	0.359886i	$0.882816\pi$
$84$	−3.00000	−0.327327
$85$	−5.00000	−0.542326
$86$	−4.00000	−0.431331
$87$	6.00000	0.643268
$88$	3.00000	0.319801
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	1.00000	0.105409
$91$	−15.0000	−1.57243
$92$	−7.00000	−0.729800
$93$	−6.00000	−0.622171
$94$	12.0000	1.23771
$95$	−1.00000	−0.102598
$96$	1.00000	0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	−2.00000	−0.202031
$99$	−3.00000	−0.301511
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	5.00000	0.495074
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	5.00000	0.490290
$105$	3.00000	0.292770
$106$	13.0000	1.26267
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	−1.00000	−0.0962250
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	−3.00000	−0.286039
$111$	1.00000	0.0949158
$112$	3.00000	0.283473
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	1.00000	0.0936586
$115$	7.00000	0.652753
$116$	−6.00000	−0.557086
$117$	−5.00000	−0.462250
$118$	−14.0000	−1.28880
$119$	15.0000	1.37505
$120$	−1.00000	−0.0912871
$121$	−2.00000	−0.181818
$122$	6.00000	0.543214
$123$	−8.00000	−0.721336
$124$	6.00000	0.538816
$125$	−1.00000	−0.0894427
$126$	−3.00000	−0.267261
$127$	−3.00000	−0.266207	−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000	−0.266207	−0.133103	+	0.991102i	$0.542494\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	−5.00000	−0.438529
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	3.00000	0.261116
$133$	3.00000	0.260133
$134$	2.00000	0.172774
$135$	1.00000	0.0860663
$136$	−5.00000	−0.428746
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−7.00000	−0.595880
$139$	−18.0000	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$140$	−3.00000	−0.253546
$141$	12.0000	1.01058
$142$	14.0000	1.17485
$143$	15.0000	1.25436
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	13.0000	1.07589
$147$	−2.00000	−0.164957
$148$	−1.00000	−0.0821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	1.00000	0.0816497
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	−1.00000	−0.0811107
$153$	5.00000	0.404226
$154$	9.00000	0.725241
$155$	−6.00000	−0.481932
$156$	5.00000	0.400320
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	8.00000	0.636446
$159$	13.0000	1.03097
$160$	1.00000	0.0790569
$161$	−21.0000	−1.65503
$162$	−1.00000	−0.0785674
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	8.00000	0.624695
$165$	−3.00000	−0.233550
$166$	17.0000	1.31946
$167$	−1.00000	−0.0773823	−0.0386912	−	0.999251i	$-0.512319\pi$
$167$	−1.00000	−0.0773823	−0.0386912	+	0.999251i	$0.512319\pi$
$168$	3.00000	0.231455
$169$	12.0000	0.923077
$170$	5.00000	0.383482
$171$	1.00000	0.0764719
$172$	4.00000	0.304997
$173$	−19.0000	−1.44454	−0.722272	−	0.691609i	$-0.756902\pi$
$173$	−19.0000	−1.44454	−0.722272	+	0.691609i	$0.756902\pi$
$174$	−6.00000	−0.454859
$175$	3.00000	0.226779
$176$	−3.00000	−0.226134
$177$	−14.0000	−1.05230
$178$	−7.00000	−0.524672
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−1.00000	−0.0745356
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	15.0000	1.11187
$183$	6.00000	0.443533
$184$	7.00000	0.516047
$185$	1.00000	0.0735215
$186$	6.00000	0.439941
$187$	−15.0000	−1.09691
$188$	−12.0000	−0.875190
$189$	−3.00000	−0.218218
$190$	1.00000	0.0725476
$191$	−25.0000	−1.80894	−0.904468	−	0.426541i	$-0.859732\pi$
$191$	−25.0000	−1.80894	−0.904468	+	0.426541i	$0.859732\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	8.00000	0.574367
$195$	−5.00000	−0.358057
$196$	2.00000	0.142857
$197$	9.00000	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$198$	3.00000	0.213201
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	−1.00000	−0.0707107
$201$	2.00000	0.141069
$202$	10.0000	0.703598
$203$	−18.0000	−1.26335
$204$	−5.00000	−0.350070
$205$	−8.00000	−0.558744
$206$	−8.00000	−0.557386
$207$	−7.00000	−0.486534
$208$	−5.00000	−0.346688
$209$	−3.00000	−0.207514
$210$	−3.00000	−0.207020
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	−13.0000	−0.892844
$213$	14.0000	0.959264
$214$	3.00000	0.205076
$215$	−4.00000	−0.272798
$216$	1.00000	0.0680414
$217$	18.0000	1.22192
$218$	5.00000	0.338643
$219$	13.0000	0.878459
$220$	3.00000	0.202260
$221$	−25.0000	−1.68168
$222$	−1.00000	−0.0671156
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−3.00000	−0.200446
$225$	1.00000	0.0666667
$226$	−10.0000	−0.665190
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−1.00000	−0.0662266
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−7.00000	−0.461566
$231$	9.00000	0.592157
$232$	6.00000	0.393919
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	5.00000	0.326860
$235$	12.0000	0.782794
$236$	14.0000	0.911322
$237$	8.00000	0.519656
$238$	−15.0000	−0.972306
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	1.00000	0.0645497
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	−2.00000	−0.127775
$246$	8.00000	0.510061
$247$	−5.00000	−0.318142
$248$	−6.00000	−0.381000
$249$	17.0000	1.07733
$250$	1.00000	0.0632456
$251$	26.0000	1.64111	0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000	1.64111	0.820553	+	0.571571i	$0.193666\pi$
$252$	3.00000	0.188982
$253$	21.0000	1.32026
$254$	3.00000	0.188237
$255$	5.00000	0.313112
$256$	1.00000	0.0625000
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	4.00000	0.249029
$259$	−3.00000	−0.186411
$260$	5.00000	0.310087
$261$	−6.00000	−0.371391
$262$	−12.0000	−0.741362
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	−3.00000	−0.184637
$265$	13.0000	0.798584
$266$	−3.00000	−0.183942
$267$	−7.00000	−0.428393
$268$	−2.00000	−0.122169
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	−1.00000	−0.0608581
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	5.00000	0.303170
$273$	15.0000	0.907841
$274$	6.00000	0.362473
$275$	−3.00000	−0.180907
$276$	7.00000	0.421350
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	18.0000	1.07957
$279$	6.00000	0.359211
$280$	3.00000	0.179284
$281$	−21.0000	−1.25275	−0.626377	−	0.779520i	$-0.715463\pi$
$281$	−21.0000	−1.25275	−0.626377	+	0.779520i	$0.715463\pi$
$282$	−12.0000	−0.714590
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	−14.0000	−0.830747
$285$	1.00000	0.0592349
$286$	−15.0000	−0.886969
$287$	24.0000	1.41668
$288$	−1.00000	−0.0589256
$289$	8.00000	0.470588
$290$	−6.00000	−0.352332
$291$	8.00000	0.468968
$292$	−13.0000	−0.760767
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	2.00000	0.116642
$295$	−14.0000	−0.815112
$296$	1.00000	0.0581238
$297$	3.00000	0.174078
$298$	−6.00000	−0.347571
$299$	35.0000	2.02410
$300$	−1.00000	−0.0577350
$301$	12.0000	0.691669
$302$	−19.0000	−1.09333
$303$	10.0000	0.574485
$304$	1.00000	0.0573539
$305$	6.00000	0.343559
$306$	−5.00000	−0.285831
$307$	6.00000	0.342438	0.171219	−	0.985233i	$-0.445229\pi$
$307$	6.00000	0.342438	0.171219	+	0.985233i	$0.445229\pi$
$308$	−9.00000	−0.512823
$309$	−8.00000	−0.455104
$310$	6.00000	0.340777
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	−5.00000	−0.283069
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	8.00000	0.451466
$315$	−3.00000	−0.169031
$316$	−8.00000	−0.450035
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	−13.0000	−0.729004
$319$	18.0000	1.00781
$320$	−1.00000	−0.0559017
$321$	3.00000	0.167444
$322$	21.0000	1.17028
$323$	5.00000	0.278207
$324$	1.00000	0.0555556
$325$	−5.00000	−0.277350
$326$	−9.00000	−0.498464
$327$	5.00000	0.276501
$328$	−8.00000	−0.441726
$329$	−36.0000	−1.98474
$330$	3.00000	0.165145
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−17.0000	−0.932996
$333$	−1.00000	−0.0547997
$334$	1.00000	0.0547176
$335$	2.00000	0.109272
$336$	−3.00000	−0.163663
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	−12.0000	−0.652714
$339$	−10.0000	−0.543125
$340$	−5.00000	−0.271163
$341$	−18.0000	−0.974755
$342$	−1.00000	−0.0540738
$343$	−15.0000	−0.809924
$344$	−4.00000	−0.215666
$345$	−7.00000	−0.376867
$346$	19.0000	1.02145
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	6.00000	0.321634
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	−3.00000	−0.160357
$351$	5.00000	0.266880
$352$	3.00000	0.159901
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	14.0000	0.744092
$355$	14.0000	0.743043
$356$	7.00000	0.370999
$357$	−15.0000	−0.793884
$358$	4.00000	0.211407
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	1.00000	0.0527046
$361$	−18.0000	−0.947368
$362$	14.0000	0.735824
$363$	2.00000	0.104973
$364$	−15.0000	−0.786214
$365$	13.0000	0.680451
$366$	−6.00000	−0.313625
$367$	−21.0000	−1.09619	−0.548096	−	0.836416i	$-0.684647\pi$
$367$	−21.0000	−1.09619	−0.548096	+	0.836416i	$0.684647\pi$
$368$	−7.00000	−0.364900
$369$	8.00000	0.416463
$370$	−1.00000	−0.0519875
$371$	−39.0000	−2.02478
$372$	−6.00000	−0.311086
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	15.0000	0.775632
$375$	1.00000	0.0516398
$376$	12.0000	0.618853
$377$	30.0000	1.54508
$378$	3.00000	0.154303
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	−1.00000	−0.0512989
$381$	3.00000	0.153695
$382$	25.0000	1.27911
$383$	3.00000	0.153293	0.0766464	−	0.997058i	$-0.475579\pi$
$383$	3.00000	0.153293	0.0766464	+	0.997058i	$0.475579\pi$
$384$	1.00000	0.0510310
$385$	9.00000	0.458682
$386$	14.0000	0.712581
$387$	4.00000	0.203331
$388$	−8.00000	−0.406138
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	5.00000	0.253185
$391$	−35.0000	−1.77003
$392$	−2.00000	−0.101015
$393$	−12.0000	−0.605320
$394$	−9.00000	−0.453413
$395$	8.00000	0.402524
$396$	−3.00000	−0.150756
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−16.0000	−0.802008
$399$	−3.00000	−0.150188
$400$	1.00000	0.0500000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	−2.00000	−0.0997509
$403$	−30.0000	−1.49441
$404$	−10.0000	−0.497519
$405$	−1.00000	−0.0496904
$406$	18.0000	0.893325
$407$	3.00000	0.148704
$408$	5.00000	0.247537
$409$	−40.0000	−1.97787	−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−40.0000	−1.97787	−0.988936	+	0.148340i	$0.952607\pi$
$410$	8.00000	0.395092
$411$	6.00000	0.295958
$412$	8.00000	0.394132
$413$	42.0000	2.06668
$414$	7.00000	0.344031
$415$	17.0000	0.834497
$416$	5.00000	0.245145
$417$	18.0000	0.881464
$418$	3.00000	0.146735
$419$	−5.00000	−0.244266	−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000	−0.244266	−0.122133	+	0.992514i	$0.538973\pi$
$420$	3.00000	0.146385
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−22.0000	−1.07094
$423$	−12.0000	−0.583460
$424$	13.0000	0.631336
$425$	5.00000	0.242536
$426$	−14.0000	−0.678302
$427$	−18.0000	−0.871081
$428$	−3.00000	−0.145010
$429$	−15.0000	−0.724207
$430$	4.00000	0.192897
$431$	17.0000	0.818861	0.409431	−	0.912341i	$-0.365727\pi$
$431$	17.0000	0.818861	0.409431	+	0.912341i	$0.365727\pi$
$432$	−1.00000	−0.0481125
$433$	33.0000	1.58588	0.792939	−	0.609301i	$-0.208550\pi$
$433$	33.0000	1.58588	0.792939	+	0.609301i	$0.208550\pi$
$434$	−18.0000	−0.864028
$435$	−6.00000	−0.287678
$436$	−5.00000	−0.239457
$437$	−7.00000	−0.334855
$438$	−13.0000	−0.621164
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	−3.00000	−0.143019
$441$	2.00000	0.0952381
$442$	25.0000	1.18913
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	1.00000	0.0474579
$445$	−7.00000	−0.331832
$446$	−16.0000	−0.757622
$447$	−6.00000	−0.283790
$448$	3.00000	0.141737
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	−1.00000	−0.0471405
$451$	−24.0000	−1.13012
$452$	10.0000	0.470360
$453$	−19.0000	−0.892698
$454$	−12.0000	−0.563188
$455$	15.0000	0.703211
$456$	1.00000	0.0468293
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	−14.0000	−0.654177
$459$	−5.00000	−0.233380
$460$	7.00000	0.326377
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	−9.00000	−0.418718
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−6.00000	−0.278543
$465$	6.00000	0.278243
$466$	−22.0000	−1.01913
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−5.00000	−0.231125
$469$	−6.00000	−0.277054
$470$	−12.0000	−0.553519
$471$	8.00000	0.368621
$472$	−14.0000	−0.644402
$473$	−12.0000	−0.551761
$474$	−8.00000	−0.367452
$475$	1.00000	0.0458831
$476$	15.0000	0.687524
$477$	−13.0000	−0.595229
$478$	−20.0000	−0.914779
$479$	−11.0000	−0.502603	−0.251301	−	0.967909i	$-0.580859\pi$
$479$	−11.0000	−0.502603	−0.251301	+	0.967909i	$0.580859\pi$
$480$	−1.00000	−0.0456435
$481$	5.00000	0.227980
$482$	14.0000	0.637683
$483$	21.0000	0.955533
$484$	−2.00000	−0.0909091
$485$	8.00000	0.363261
$486$	1.00000	0.0453609
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	6.00000	0.271607
$489$	−9.00000	−0.406994
$490$	2.00000	0.0903508
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	−8.00000	−0.360668
$493$	−30.0000	−1.35113
$494$	5.00000	0.224961
$495$	3.00000	0.134840
$496$	6.00000	0.269408
$497$	−42.0000	−1.88396
$498$	−17.0000	−0.761788
$499$	−17.0000	−0.761025	−0.380512	−	0.924776i	$-0.624252\pi$
$499$	−17.0000	−0.761025	−0.380512	+	0.924776i	$0.624252\pi$
$500$	−1.00000	−0.0447214
$501$	1.00000	0.0446767
$502$	−26.0000	−1.16044
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	−3.00000	−0.133631
$505$	10.0000	0.444994
$506$	−21.0000	−0.933564
$507$	−12.0000	−0.532939
$508$	−3.00000	−0.133103
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	−5.00000	−0.221404
$511$	−39.0000	−1.72526
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	7.00000	0.308757
$515$	−8.00000	−0.352522
$516$	−4.00000	−0.176090
$517$	36.0000	1.58328
$518$	3.00000	0.131812
$519$	19.0000	0.834007
$520$	−5.00000	−0.219265
$521$	−32.0000	−1.40195	−0.700973	−	0.713188i	$-0.747251\pi$
$521$	−32.0000	−1.40195	−0.700973	+	0.713188i	$0.747251\pi$
$522$	6.00000	0.262613
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	12.0000	0.524222
$525$	−3.00000	−0.130931
$526$	−24.0000	−1.04645
$527$	30.0000	1.30682
$528$	3.00000	0.130558
$529$	26.0000	1.13043
$530$	−13.0000	−0.564684
$531$	14.0000	0.607548
$532$	3.00000	0.130066
$533$	−40.0000	−1.73259
$534$	7.00000	0.302920
$535$	3.00000	0.129701
$536$	2.00000	0.0863868
$537$	4.00000	0.172613
$538$	5.00000	0.215565
$539$	−6.00000	−0.258438
$540$	1.00000	0.0430331
$541$	−11.0000	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$542$	20.0000	0.859074
$543$	14.0000	0.600798
$544$	−5.00000	−0.214373
$545$	5.00000	0.214176
$546$	−15.0000	−0.641941
$547$	−3.00000	−0.128271	−0.0641354	−	0.997941i	$-0.520429\pi$
$547$	−3.00000	−0.128271	−0.0641354	+	0.997941i	$0.520429\pi$
$548$	−6.00000	−0.256307
$549$	−6.00000	−0.256074
$550$	3.00000	0.127920
$551$	−6.00000	−0.255609
$552$	−7.00000	−0.297940
$553$	−24.0000	−1.02058
$554$	−13.0000	−0.552317
$555$	−1.00000	−0.0424476
$556$	−18.0000	−0.763370
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	−6.00000	−0.254000
$559$	−20.0000	−0.845910
$560$	−3.00000	−0.126773
$561$	15.0000	0.633300
$562$	21.0000	0.885832
$563$	30.0000	1.26435	0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000	1.26435	0.632175	+	0.774826i	$0.282163\pi$
$564$	12.0000	0.505291
$565$	−10.0000	−0.420703
$566$	−5.00000	−0.210166
$567$	3.00000	0.125988
$568$	14.0000	0.587427
$569$	33.0000	1.38343	0.691716	−	0.722170i	$-0.256855\pi$
$569$	33.0000	1.38343	0.691716	+	0.722170i	$0.256855\pi$
$570$	−1.00000	−0.0418854
$571$	−18.0000	−0.753277	−0.376638	−	0.926360i	$-0.622920\pi$
$571$	−18.0000	−0.753277	−0.376638	+	0.926360i	$0.622920\pi$
$572$	15.0000	0.627182
$573$	25.0000	1.04439
$574$	−24.0000	−1.00174
$575$	−7.00000	−0.291920
$576$	1.00000	0.0416667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	−8.00000	−0.332756
$579$	14.0000	0.581820
$580$	6.00000	0.249136
$581$	−51.0000	−2.11584
$582$	−8.00000	−0.331611
$583$	39.0000	1.61521
$584$	13.0000	0.537944
$585$	5.00000	0.206725
$586$	21.0000	0.867502
$587$	−40.0000	−1.65098	−0.825488	−	0.564419i	$-0.809100\pi$
$587$	−40.0000	−1.65098	−0.825488	+	0.564419i	$0.809100\pi$
$588$	−2.00000	−0.0824786
$589$	6.00000	0.247226
$590$	14.0000	0.576371
$591$	−9.00000	−0.370211
$592$	−1.00000	−0.0410997
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	−3.00000	−0.123091
$595$	−15.0000	−0.614940
$596$	6.00000	0.245770
$597$	−16.0000	−0.654836
$598$	−35.0000	−1.43126
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	1.00000	0.0408248
$601$	11.0000	0.448699	0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000	0.448699	0.224350	+	0.974509i	$0.427974\pi$
$602$	−12.0000	−0.489083
$603$	−2.00000	−0.0814463
$604$	19.0000	0.773099
$605$	2.00000	0.0813116
$606$	−10.0000	−0.406222
$607$	26.0000	1.05531	0.527654	−	0.849460i	$-0.323072\pi$
$607$	26.0000	1.05531	0.527654	+	0.849460i	$0.323072\pi$
$608$	−1.00000	−0.0405554
$609$	18.0000	0.729397
$610$	−6.00000	−0.242933
$611$	60.0000	2.42734
$612$	5.00000	0.202113
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	−6.00000	−0.242140
$615$	8.00000	0.322591
$616$	9.00000	0.362620
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	8.00000	0.321807
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	−6.00000	−0.240966
$621$	7.00000	0.280900
$622$	−32.0000	−1.28308
$623$	21.0000	0.841347
$624$	5.00000	0.200160
$625$	1.00000	0.0400000
$626$	20.0000	0.799361
$627$	3.00000	0.119808
$628$	−8.00000	−0.319235
$629$	−5.00000	−0.199363
$630$	3.00000	0.119523
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	8.00000	0.318223
$633$	−22.0000	−0.874421
$634$	−26.0000	−1.03259
$635$	3.00000	0.119051
$636$	13.0000	0.515484
$637$	−10.0000	−0.396214
$638$	−18.0000	−0.712627
$639$	−14.0000	−0.553831
$640$	1.00000	0.0395285
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	−3.00000	−0.118401
$643$	7.00000	0.276053	0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000	0.276053	0.138027	+	0.990429i	$0.455924\pi$
$644$	−21.0000	−0.827516
$645$	4.00000	0.157500
$646$	−5.00000	−0.196722
$647$	−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
$649$	−42.0000	−1.64864
$650$	5.00000	0.196116
$651$	−18.0000	−0.705476
$652$	9.00000	0.352467
$653$	20.0000	0.782660	0.391330	−	0.920250i	$-0.372015\pi$
$653$	20.0000	0.782660	0.391330	+	0.920250i	$0.372015\pi$
$654$	−5.00000	−0.195515
$655$	−12.0000	−0.468879
$656$	8.00000	0.312348
$657$	−13.0000	−0.507178
$658$	36.0000	1.40343
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	−3.00000	−0.116775
$661$	33.0000	1.28355	0.641776	−	0.766892i	$-0.278198\pi$
$661$	33.0000	1.28355	0.641776	+	0.766892i	$0.278198\pi$
$662$	−20.0000	−0.777322
$663$	25.0000	0.970920
$664$	17.0000	0.659728
$665$	−3.00000	−0.116335
$666$	1.00000	0.0387492
$667$	42.0000	1.62625
$668$	−1.00000	−0.0386912
$669$	−16.0000	−0.618596
$670$	−2.00000	−0.0772667
$671$	18.0000	0.694882
$672$	3.00000	0.115728
$673$	15.0000	0.578208	0.289104	−	0.957298i	$-0.406643\pi$
$673$	15.0000	0.578208	0.289104	+	0.957298i	$0.406643\pi$
$674$	5.00000	0.192593
$675$	−1.00000	−0.0384900
$676$	12.0000	0.461538
$677$	41.0000	1.57576	0.787879	−	0.615830i	$-0.211179\pi$
$677$	41.0000	1.57576	0.787879	+	0.615830i	$0.211179\pi$
$678$	10.0000	0.384048
$679$	−24.0000	−0.921035
$680$	5.00000	0.191741
$681$	−12.0000	−0.459841
$682$	18.0000	0.689256
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	1.00000	0.0382360
$685$	6.00000	0.229248
$686$	15.0000	0.572703
$687$	−14.0000	−0.534133
$688$	4.00000	0.152499
$689$	65.0000	2.47630
$690$	7.00000	0.266485
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	−19.0000	−0.722272
$693$	−9.00000	−0.341882
$694$	12.0000	0.455514
$695$	18.0000	0.682779
$696$	−6.00000	−0.227429
$697$	40.0000	1.51511
$698$	16.0000	0.605609
$699$	−22.0000	−0.832116
$700$	3.00000	0.113389
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	−5.00000	−0.188713
$703$	−1.00000	−0.0377157
$704$	−3.00000	−0.113067
$705$	−12.0000	−0.451946
$706$	−2.00000	−0.0752710
$707$	−30.0000	−1.12827
$708$	−14.0000	−0.526152
$709$	5.00000	0.187779	0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000	0.187779	0.0938895	+	0.995583i	$0.470070\pi$
$710$	−14.0000	−0.525411
$711$	−8.00000	−0.300023
$712$	−7.00000	−0.262336
$713$	−42.0000	−1.57291
$714$	15.0000	0.561361
$715$	−15.0000	−0.560968
$716$	−4.00000	−0.149487
$717$	−20.0000	−0.746914
$718$	−24.0000	−0.895672
$719$	−10.0000	−0.372937	−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000	−0.372937	−0.186469	+	0.982461i	$0.559704\pi$
$720$	−1.00000	−0.0372678
$721$	24.0000	0.893807
$722$	18.0000	0.669891
$723$	14.0000	0.520666
$724$	−14.0000	−0.520306
$725$	−6.00000	−0.222834
$726$	−2.00000	−0.0742270
$727$	−30.0000	−1.11264	−0.556319	−	0.830969i	$-0.687787\pi$
$727$	−30.0000	−1.11264	−0.556319	+	0.830969i	$0.687787\pi$
$728$	15.0000	0.555937
$729$	1.00000	0.0370370
$730$	−13.0000	−0.481152
$731$	20.0000	0.739727
$732$	6.00000	0.221766
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	21.0000	0.775124
$735$	2.00000	0.0737711
$736$	7.00000	0.258023
$737$	6.00000	0.221013
$738$	−8.00000	−0.294484
$739$	−38.0000	−1.39785	−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000	−1.39785	−0.698926	+	0.715194i	$0.746338\pi$
$740$	1.00000	0.0367607
$741$	5.00000	0.183680
$742$	39.0000	1.43174
$743$	−34.0000	−1.24734	−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000	−1.24734	−0.623670	+	0.781688i	$0.714359\pi$
$744$	6.00000	0.219971
$745$	−6.00000	−0.219823
$746$	−6.00000	−0.219676
$747$	−17.0000	−0.621997
$748$	−15.0000	−0.548454
$749$	−9.00000	−0.328853
$750$	−1.00000	−0.0365148
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	−12.0000	−0.437595
$753$	−26.0000	−0.947493
$754$	−30.0000	−1.09254
$755$	−19.0000	−0.691481
$756$	−3.00000	−0.109109
$757$	17.0000	0.617876	0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000	0.617876	0.308938	+	0.951082i	$0.400027\pi$
$758$	−6.00000	−0.217930
$759$	−21.0000	−0.762252
$760$	1.00000	0.0362738
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	−3.00000	−0.108679
$763$	−15.0000	−0.543036
$764$	−25.0000	−0.904468
$765$	−5.00000	−0.180775
$766$	−3.00000	−0.108394
$767$	−70.0000	−2.52755
$768$	−1.00000	−0.0360844
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	−9.00000	−0.324337
$771$	7.00000	0.252099
$772$	−14.0000	−0.503871
$773$	37.0000	1.33080	0.665399	−	0.746488i	$-0.268262\pi$
$773$	37.0000	1.33080	0.665399	+	0.746488i	$0.268262\pi$
$774$	−4.00000	−0.143777
$775$	6.00000	0.215526
$776$	8.00000	0.287183
$777$	3.00000	0.107624
$778$	−6.00000	−0.215110
$779$	8.00000	0.286630
$780$	−5.00000	−0.179029
$781$	42.0000	1.50288
$782$	35.0000	1.25160
$783$	6.00000	0.214423
$784$	2.00000	0.0714286
$785$	8.00000	0.285532
$786$	12.0000	0.428026
$787$	46.0000	1.63972	0.819861	−	0.572562i	$-0.194050\pi$
$787$	46.0000	1.63972	0.819861	+	0.572562i	$0.194050\pi$
$788$	9.00000	0.320612
$789$	−24.0000	−0.854423
$790$	−8.00000	−0.284627
$791$	30.0000	1.06668
$792$	3.00000	0.106600
$793$	30.0000	1.06533
$794$	0	0
$795$	−13.0000	−0.461062
$796$	16.0000	0.567105
$797$	−54.0000	−1.91278	−0.956389	−	0.292096i	$-0.905647\pi$
$797$	−54.0000	−1.91278	−0.956389	+	0.292096i	$0.905647\pi$
$798$	3.00000	0.106199
$799$	−60.0000	−2.12265
$800$	−1.00000	−0.0353553
$801$	7.00000	0.247333
$802$	−15.0000	−0.529668
$803$	39.0000	1.37628
$804$	2.00000	0.0705346
$805$	21.0000	0.740153
$806$	30.0000	1.05670
$807$	5.00000	0.176008
$808$	10.0000	0.351799
$809$	19.0000	0.668004	0.334002	−	0.942572i	$-0.391601\pi$
$809$	19.0000	0.668004	0.334002	+	0.942572i	$0.391601\pi$
$810$	1.00000	0.0351364
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	−18.0000	−0.631676
$813$	20.0000	0.701431
$814$	−3.00000	−0.105150
$815$	−9.00000	−0.315256
$816$	−5.00000	−0.175035
$817$	4.00000	0.139942
$818$	40.0000	1.39857
$819$	−15.0000	−0.524142
$820$	−8.00000	−0.279372
$821$	13.0000	0.453703	0.226852	−	0.973929i	$-0.427157\pi$
$821$	13.0000	0.453703	0.226852	+	0.973929i	$0.427157\pi$
$822$	−6.00000	−0.209274
$823$	−27.0000	−0.941161	−0.470580	−	0.882357i	$-0.655955\pi$
$823$	−27.0000	−0.941161	−0.470580	+	0.882357i	$0.655955\pi$
$824$	−8.00000	−0.278693
$825$	3.00000	0.104447
$826$	−42.0000	−1.46137
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	−7.00000	−0.243267
$829$	31.0000	1.07667	0.538337	−	0.842729i	$-0.319053\pi$
$829$	31.0000	1.07667	0.538337	+	0.842729i	$0.319053\pi$
$830$	−17.0000	−0.590079
$831$	−13.0000	−0.450965
$832$	−5.00000	−0.173344
$833$	10.0000	0.346479
$834$	−18.0000	−0.623289
$835$	1.00000	0.0346064
$836$	−3.00000	−0.103757
$837$	−6.00000	−0.207390
$838$	5.00000	0.172722
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	−3.00000	−0.103510
$841$	7.00000	0.241379
$842$	22.0000	0.758170
$843$	21.0000	0.723278
$844$	22.0000	0.757271
$845$	−12.0000	−0.412813
$846$	12.0000	0.412568
$847$	−6.00000	−0.206162
$848$	−13.0000	−0.446422
$849$	−5.00000	−0.171600
$850$	−5.00000	−0.171499
$851$	7.00000	0.239957
$852$	14.0000	0.479632
$853$	−25.0000	−0.855984	−0.427992	−	0.903783i	$-0.640779\pi$
$853$	−25.0000	−0.855984	−0.427992	+	0.903783i	$0.640779\pi$
$854$	18.0000	0.615947
$855$	−1.00000	−0.0341993
$856$	3.00000	0.102538
$857$	39.0000	1.33221	0.666107	−	0.745856i	$-0.267959\pi$
$857$	39.0000	1.33221	0.666107	+	0.745856i	$0.267959\pi$
$858$	15.0000	0.512092
$859$	3.00000	0.102359	0.0511793	−	0.998689i	$-0.483702\pi$
$859$	3.00000	0.102359	0.0511793	+	0.998689i	$0.483702\pi$
$860$	−4.00000	−0.136399
$861$	−24.0000	−0.817918
$862$	−17.0000	−0.579022
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	1.00000	0.0340207
$865$	19.0000	0.646019
$866$	−33.0000	−1.12139
$867$	−8.00000	−0.271694
$868$	18.0000	0.610960
$869$	24.0000	0.814144
$870$	6.00000	0.203419
$871$	10.0000	0.338837
$872$	5.00000	0.169321
$873$	−8.00000	−0.270759
$874$	7.00000	0.236779
$875$	−3.00000	−0.101419
$876$	13.0000	0.439229
$877$	−12.0000	−0.405211	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	−12.0000	−0.405211	−0.202606	+	0.979260i	$0.564941\pi$
$878$	12.0000	0.404980
$879$	21.0000	0.708312
$880$	3.00000	0.101130
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	−2.00000	−0.0673435
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	−25.0000	−0.840841
$885$	14.0000	0.470605
$886$	−20.0000	−0.671913
$887$	−10.0000	−0.335767	−0.167884	−	0.985807i	$-0.553693\pi$
$887$	−10.0000	−0.335767	−0.167884	+	0.985807i	$0.553693\pi$
$888$	−1.00000	−0.0335578
$889$	−9.00000	−0.301850
$890$	7.00000	0.234641
$891$	−3.00000	−0.100504
$892$	16.0000	0.535720
$893$	−12.0000	−0.401565
$894$	6.00000	0.200670
$895$	4.00000	0.133705
$896$	−3.00000	−0.100223
$897$	−35.0000	−1.16862
$898$	22.0000	0.734150
$899$	−36.0000	−1.20067
$900$	1.00000	0.0333333
$901$	−65.0000	−2.16546
$902$	24.0000	0.799113
$903$	−12.0000	−0.399335
$904$	−10.0000	−0.332595
$905$	14.0000	0.465376
$906$	19.0000	0.631233
$907$	13.0000	0.431658	0.215829	−	0.976431i	$-0.430755\pi$
$907$	13.0000	0.431658	0.215829	+	0.976431i	$0.430755\pi$
$908$	12.0000	0.398234
$909$	−10.0000	−0.331679
$910$	−15.0000	−0.497245
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−1.00000	−0.0331133
$913$	51.0000	1.68785
$914$	−28.0000	−0.926158
$915$	−6.00000	−0.198354
$916$	14.0000	0.462573
$917$	36.0000	1.18882
$918$	5.00000	0.165025
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	−7.00000	−0.230783
$921$	−6.00000	−0.197707
$922$	10.0000	0.329332
$923$	70.0000	2.30408
$924$	9.00000	0.296078
$925$	−1.00000	−0.0328798
$926$	8.00000	0.262896
$927$	8.00000	0.262754
$928$	6.00000	0.196960
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	−6.00000	−0.196748
$931$	2.00000	0.0655474
$932$	22.0000	0.720634
$933$	−32.0000	−1.04763
$934$	12.0000	0.392652
$935$	15.0000	0.490552
$936$	5.00000	0.163430
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	6.00000	0.195907
$939$	20.0000	0.652675
$940$	12.0000	0.391397
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−8.00000	−0.260654
$943$	−56.0000	−1.82361
$944$	14.0000	0.455661
$945$	3.00000	0.0975900
$946$	12.0000	0.390154
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	8.00000	0.259828
$949$	65.0000	2.10999
$950$	−1.00000	−0.0324443
$951$	−26.0000	−0.843108
$952$	−15.0000	−0.486153
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	13.0000	0.420891
$955$	25.0000	0.808981
$956$	20.0000	0.646846
$957$	−18.0000	−0.581857
$958$	11.0000	0.355394
$959$	−18.0000	−0.581250
$960$	1.00000	0.0322749
$961$	5.00000	0.161290
$962$	−5.00000	−0.161206
$963$	−3.00000	−0.0966736
$964$	−14.0000	−0.450910
$965$	14.0000	0.450676
$966$	−21.0000	−0.675664
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	2.00000	0.0642824
$969$	−5.00000	−0.160623
$970$	−8.00000	−0.256865
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	−1.00000	−0.0320750
$973$	−54.0000	−1.73116
$974$	−22.0000	−0.704925
$975$	5.00000	0.160128
$976$	−6.00000	−0.192055
$977$	−15.0000	−0.479893	−0.239946	−	0.970786i	$-0.577130\pi$
$977$	−15.0000	−0.479893	−0.239946	+	0.970786i	$0.577130\pi$
$978$	9.00000	0.287788
$979$	−21.0000	−0.671163
$980$	−2.00000	−0.0638877
$981$	−5.00000	−0.159638
$982$	−9.00000	−0.287202
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	8.00000	0.255031
$985$	−9.00000	−0.286764
$986$	30.0000	0.955395
$987$	36.0000	1.14589
$988$	−5.00000	−0.159071
$989$	−28.0000	−0.890348
$990$	−3.00000	−0.0953463
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	−6.00000	−0.190500
$993$	−20.0000	−0.634681
$994$	42.0000	1.33216
$995$	−16.0000	−0.507234
$996$	17.0000	0.538666
$997$	21.0000	0.665077	0.332538	−	0.943090i	$-0.392095\pi$
$997$	21.0000	0.665077	0.332538	+	0.943090i	$0.392095\pi$
$998$	17.0000	0.538126
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.a.c.1.1	✓	1
3.2	odd	2		3330.2.a.z.1.1		1
4.3	odd	2		8880.2.a.p.1.1		1
5.4	even	2		5550.2.a.bg.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.c.1.1	✓	1	1.1	even	1	trivial
3330.2.a.z.1.1		1	3.2	odd	2
5550.2.a.bg.1.1		1	5.4	even	2
8880.2.a.p.1.1		1	4.3	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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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