LMFDB - Embedded newform 186.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [186,2,Mod(1,186)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(186, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("186.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 186 = 2 \cdot 3 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	186.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:	$1.48521747760$
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	$\chi$	$=$	186.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} +3.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +7.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} -3.00000 q^{22} +1.00000 q^{24} -4.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} -1.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -1.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -7.00000 q^{38} -3.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} +2.00000 q^{42} +6.00000 q^{43} +3.00000 q^{44} -1.00000 q^{45} -5.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} -1.00000 q^{51} +3.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} -2.00000 q^{56} -7.00000 q^{57} -4.00000 q^{58} +6.00000 q^{59} +1.00000 q^{60} +3.00000 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -3.00000 q^{65} +3.00000 q^{66} -3.00000 q^{67} +1.00000 q^{68} +2.00000 q^{70} +7.00000 q^{71} -1.00000 q^{72} -10.0000 q^{73} +10.0000 q^{74} +4.00000 q^{75} +7.00000 q^{76} +6.00000 q^{77} +3.00000 q^{78} -1.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +17.0000 q^{83} -2.00000 q^{84} -1.00000 q^{85} -6.00000 q^{86} -4.00000 q^{87} -3.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} +6.00000 q^{91} -1.00000 q^{93} +5.00000 q^{94} -7.00000 q^{95} +1.00000 q^{96} +5.00000 q^{97} +3.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	1.00000	0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.00000	−0.288675
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	−1.00000	−0.235702
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	−1.00000	−0.223607
$21$	−2.00000	−0.436436
$22$	−3.00000	−0.639602
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	−3.00000	−0.588348
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	−1.00000	−0.182574
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	−3.00000	−0.522233
$34$	−1.00000	−0.171499
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−7.00000	−1.13555
$39$	−3.00000	−0.480384
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	2.00000	0.308607
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	3.00000	0.452267
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−5.00000	−0.729325	−0.364662	−	0.931140i	$-0.618816\pi$
$47$	−5.00000	−0.729325	−0.364662	+	0.931140i	$0.618816\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	4.00000	0.565685
$51$	−1.00000	−0.140028
$52$	3.00000	0.416025
$53$	−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
$56$	−2.00000	−0.267261
$57$	−7.00000	−0.927173
$58$	−4.00000	−0.525226
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	1.00000	0.129099
$61$	3.00000	0.384111	0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000	0.384111	0.192055	+	0.981384i	$0.438485\pi$
$62$	−1.00000	−0.127000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−3.00000	−0.372104
$66$	3.00000	0.369274
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	2.00000	0.239046
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	−1.00000	−0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	10.0000	1.16248
$75$	4.00000	0.461880
$76$	7.00000	0.802955
$77$	6.00000	0.683763
$78$	3.00000	0.339683
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	17.0000	1.86599	0.932996	−	0.359886i	$-0.117184\pi$
$83$	17.0000	1.86599	0.932996	+	0.359886i	$0.117184\pi$
$84$	−2.00000	−0.218218
$85$	−1.00000	−0.108465
$86$	−6.00000	−0.646997
$87$	−4.00000	−0.428845
$88$	−3.00000	−0.319801
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	1.00000	0.105409
$91$	6.00000	0.628971
$92$	0	0
$93$	−1.00000	−0.103695
$94$	5.00000	0.515711
$95$	−7.00000	−0.718185
$96$	1.00000	0.102062
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	3.00000	0.303046
$99$	3.00000	0.301511
$100$	−4.00000	−0.400000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	1.00000	0.0990148
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	−3.00000	−0.294174
$105$	2.00000	0.195180
$106$	2.00000	0.194257
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	3.00000	0.286039
$111$	10.0000	0.949158
$112$	2.00000	0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	7.00000	0.655610
$115$	0	0
$116$	4.00000	0.371391
$117$	3.00000	0.277350
$118$	−6.00000	−0.552345
$119$	2.00000	0.183340
$120$	−1.00000	−0.0912871
$121$	−2.00000	−0.181818
$122$	−3.00000	−0.271607
$123$	6.00000	0.541002
$124$	1.00000	0.0898027
$125$	9.00000	0.804984
$126$	−2.00000	−0.178174
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	−1.00000	−0.0883883
$129$	−6.00000	−0.528271
$130$	3.00000	0.263117
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	−3.00000	−0.261116
$133$	14.0000	1.21395
$134$	3.00000	0.259161
$135$	1.00000	0.0860663
$136$	−1.00000	−0.0857493
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	−2.00000	−0.169031
$141$	5.00000	0.421076
$142$	−7.00000	−0.587427
$143$	9.00000	0.752618
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	10.0000	0.827606
$147$	3.00000	0.247436
$148$	−10.0000	−0.821995
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	−4.00000	−0.326599
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	−7.00000	−0.567775
$153$	1.00000	0.0808452
$154$	−6.00000	−0.483494
$155$	−1.00000	−0.0803219
$156$	−3.00000	−0.240192
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	1.00000	0.0795557
$159$	2.00000	0.158610
$160$	1.00000	0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	−6.00000	−0.468521
$165$	3.00000	0.233550
$166$	−17.0000	−1.31946
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	2.00000	0.154303
$169$	−4.00000	−0.307692
$170$	1.00000	0.0766965
$171$	7.00000	0.535303
$172$	6.00000	0.457496
$173$	13.0000	0.988372	0.494186	−	0.869356i	$-0.335466\pi$
$173$	13.0000	0.988372	0.494186	+	0.869356i	$0.335466\pi$
$174$	4.00000	0.303239
$175$	−8.00000	−0.604743
$176$	3.00000	0.226134
$177$	−6.00000	−0.450988
$178$	−6.00000	−0.449719
$179$	−11.0000	−0.822179	−0.411089	−	0.911595i	$-0.634852\pi$
$179$	−11.0000	−0.822179	−0.411089	+	0.911595i	$0.634852\pi$
$180$	−1.00000	−0.0745356
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−6.00000	−0.444750
$183$	−3.00000	−0.221766
$184$	0	0
$185$	10.0000	0.735215
$186$	1.00000	0.0733236
$187$	3.00000	0.219382
$188$	−5.00000	−0.364662
$189$	−2.00000	−0.145479
$190$	7.00000	0.507833
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	−1.00000	−0.0721688
$193$	19.0000	1.36765	0.683825	−	0.729646i	$-0.260315\pi$
$193$	19.0000	1.36765	0.683825	+	0.729646i	$0.260315\pi$
$194$	−5.00000	−0.358979
$195$	3.00000	0.214834
$196$	−3.00000	−0.214286
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	−3.00000	−0.213201
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	4.00000	0.282843
$201$	3.00000	0.211604
$202$	−14.0000	−0.985037
$203$	8.00000	0.561490
$204$	−1.00000	−0.0700140
$205$	6.00000	0.419058
$206$	14.0000	0.975426
$207$	0	0
$208$	3.00000	0.208013
$209$	21.0000	1.45260
$210$	−2.00000	−0.138013
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	−2.00000	−0.137361
$213$	−7.00000	−0.479632
$214$	16.0000	1.09374
$215$	−6.00000	−0.409197
$216$	1.00000	0.0680414
$217$	2.00000	0.135769
$218$	10.0000	0.677285
$219$	10.0000	0.675737
$220$	−3.00000	−0.202260
$221$	3.00000	0.201802
$222$	−10.0000	−0.671156
$223$	9.00000	0.602685	0.301342	−	0.953516i	$-0.402565\pi$
$223$	9.00000	0.602685	0.301342	+	0.953516i	$0.402565\pi$
$224$	−2.00000	−0.133631
$225$	−4.00000	−0.266667
$226$	6.00000	0.399114
$227$	−22.0000	−1.46019	−0.730096	−	0.683345i	$-0.760525\pi$
$227$	−22.0000	−1.46019	−0.730096	+	0.683345i	$0.760525\pi$
$228$	−7.00000	−0.463586
$229$	23.0000	1.51988	0.759941	−	0.649992i	$-0.225228\pi$
$229$	23.0000	1.51988	0.759941	+	0.649992i	$0.225228\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	−4.00000	−0.262613
$233$	20.0000	1.31024	0.655122	−	0.755523i	$-0.272617\pi$
$233$	20.0000	1.31024	0.655122	+	0.755523i	$0.272617\pi$
$234$	−3.00000	−0.196116
$235$	5.00000	0.326164
$236$	6.00000	0.390567
$237$	1.00000	0.0649570
$238$	−2.00000	−0.129641
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	1.00000	0.0645497
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	3.00000	0.192055
$245$	3.00000	0.191663
$246$	−6.00000	−0.382546
$247$	21.0000	1.33620
$248$	−1.00000	−0.0635001
$249$	−17.0000	−1.07733
$250$	−9.00000	−0.569210
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	12.0000	0.752947
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	6.00000	0.373544
$259$	−20.0000	−1.24274
$260$	−3.00000	−0.186052
$261$	4.00000	0.247594
$262$	14.0000	0.864923
$263$	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
$265$	2.00000	0.122859
$266$	−14.0000	−0.858395
$267$	−6.00000	−0.367194
$268$	−3.00000	−0.183254
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	−1.00000	−0.0608581
$271$	−17.0000	−1.03268	−0.516338	−	0.856385i	$-0.672705\pi$
$271$	−17.0000	−1.03268	−0.516338	+	0.856385i	$0.672705\pi$
$272$	1.00000	0.0606339
$273$	−6.00000	−0.363137
$274$	−10.0000	−0.604122
$275$	−12.0000	−0.723627
$276$	0	0
$277$	27.0000	1.62227	0.811136	−	0.584857i	$-0.198849\pi$
$277$	27.0000	1.62227	0.811136	+	0.584857i	$0.198849\pi$
$278$	−16.0000	−0.959616
$279$	1.00000	0.0598684
$280$	2.00000	0.119523
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−5.00000	−0.297746
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	7.00000	0.415374
$285$	7.00000	0.414644
$286$	−9.00000	−0.532181
$287$	−12.0000	−0.708338
$288$	−1.00000	−0.0589256
$289$	−16.0000	−0.941176
$290$	4.00000	0.234888
$291$	−5.00000	−0.293105
$292$	−10.0000	−0.585206
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	−3.00000	−0.174964
$295$	−6.00000	−0.349334
$296$	10.0000	0.581238
$297$	−3.00000	−0.174078
$298$	21.0000	1.21650
$299$	0	0
$300$	4.00000	0.230940
$301$	12.0000	0.691669
$302$	−5.00000	−0.287718
$303$	−14.0000	−0.804279
$304$	7.00000	0.401478
$305$	−3.00000	−0.171780
$306$	−1.00000	−0.0571662
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	6.00000	0.341882
$309$	14.0000	0.796432
$310$	1.00000	0.0567962
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	3.00000	0.169842
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	20.0000	1.12867
$315$	−2.00000	−0.112687
$316$	−1.00000	−0.0562544
$317$	−23.0000	−1.29181	−0.645904	−	0.763418i	$-0.723520\pi$
$317$	−23.0000	−1.29181	−0.645904	+	0.763418i	$0.723520\pi$
$318$	−2.00000	−0.112154
$319$	12.0000	0.671871
$320$	−1.00000	−0.0559017
$321$	16.0000	0.893033
$322$	0	0
$323$	7.00000	0.389490
$324$	1.00000	0.0555556
$325$	−12.0000	−0.665640
$326$	1.00000	0.0553849
$327$	10.0000	0.553001
$328$	6.00000	0.331295
$329$	−10.0000	−0.551318
$330$	−3.00000	−0.165145
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	17.0000	0.932996
$333$	−10.0000	−0.547997
$334$	14.0000	0.766046
$335$	3.00000	0.163908
$336$	−2.00000	−0.109109
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	4.00000	0.217571
$339$	6.00000	0.325875
$340$	−1.00000	−0.0542326
$341$	3.00000	0.162459
$342$	−7.00000	−0.378517
$343$	−20.0000	−1.07990
$344$	−6.00000	−0.323498
$345$	0	0
$346$	−13.0000	−0.698884
$347$	−17.0000	−0.912608	−0.456304	−	0.889824i	$-0.650827\pi$
$347$	−17.0000	−0.912608	−0.456304	+	0.889824i	$0.650827\pi$
$348$	−4.00000	−0.214423
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	8.00000	0.427618
$351$	−3.00000	−0.160128
$352$	−3.00000	−0.159901
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	6.00000	0.318896
$355$	−7.00000	−0.371521
$356$	6.00000	0.317999
$357$	−2.00000	−0.105851
$358$	11.0000	0.581368
$359$	−5.00000	−0.263890	−0.131945	−	0.991257i	$-0.542122\pi$
$359$	−5.00000	−0.263890	−0.131945	+	0.991257i	$0.542122\pi$
$360$	1.00000	0.0527046
$361$	30.0000	1.57895
$362$	10.0000	0.525588
$363$	2.00000	0.104973
$364$	6.00000	0.314485
$365$	10.0000	0.523424
$366$	3.00000	0.156813
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	−10.0000	−0.519875
$371$	−4.00000	−0.207670
$372$	−1.00000	−0.0518476
$373$	−24.0000	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$373$	−24.0000	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$374$	−3.00000	−0.155126
$375$	−9.00000	−0.464758
$376$	5.00000	0.257855
$377$	12.0000	0.618031
$378$	2.00000	0.102869
$379$	−15.0000	−0.770498	−0.385249	−	0.922813i	$-0.625884\pi$
$379$	−15.0000	−0.770498	−0.385249	+	0.922813i	$0.625884\pi$
$380$	−7.00000	−0.359092
$381$	12.0000	0.614779
$382$	24.0000	1.22795
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	1.00000	0.0510310
$385$	−6.00000	−0.305788
$386$	−19.0000	−0.967075
$387$	6.00000	0.304997
$388$	5.00000	0.253837
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−3.00000	−0.151911
$391$	0	0
$392$	3.00000	0.151523
$393$	14.0000	0.706207
$394$	26.0000	1.30986
$395$	1.00000	0.0503155
$396$	3.00000	0.150756
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	−7.00000	−0.350878
$399$	−14.0000	−0.700877
$400$	−4.00000	−0.200000
$401$	−25.0000	−1.24844	−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000	−1.24844	−0.624220	+	0.781248i	$0.714583\pi$
$402$	−3.00000	−0.149626
$403$	3.00000	0.149441
$404$	14.0000	0.696526
$405$	−1.00000	−0.0496904
$406$	−8.00000	−0.397033
$407$	−30.0000	−1.48704
$408$	1.00000	0.0495074
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−6.00000	−0.296319
$411$	−10.0000	−0.493264
$412$	−14.0000	−0.689730
$413$	12.0000	0.590481
$414$	0	0
$415$	−17.0000	−0.834497
$416$	−3.00000	−0.147087
$417$	−16.0000	−0.783523
$418$	−21.0000	−1.02714
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	2.00000	0.0975900
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	−16.0000	−0.778868
$423$	−5.00000	−0.243108
$424$	2.00000	0.0971286
$425$	−4.00000	−0.194029
$426$	7.00000	0.339151
$427$	6.00000	0.290360
$428$	−16.0000	−0.773389
$429$	−9.00000	−0.434524
$430$	6.00000	0.289346
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−1.00000	−0.0481125
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	−2.00000	−0.0960031
$435$	4.00000	0.191785
$436$	−10.0000	−0.478913
$437$	0	0
$438$	−10.0000	−0.477818
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	3.00000	0.143019
$441$	−3.00000	−0.142857
$442$	−3.00000	−0.142695
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	10.0000	0.474579
$445$	−6.00000	−0.284427
$446$	−9.00000	−0.426162
$447$	21.0000	0.993266
$448$	2.00000	0.0944911
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	4.00000	0.188562
$451$	−18.0000	−0.847587
$452$	−6.00000	−0.282216
$453$	−5.00000	−0.234920
$454$	22.0000	1.03251
$455$	−6.00000	−0.281284
$456$	7.00000	0.327805
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−23.0000	−1.07472
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	6.00000	0.279145
$463$	19.0000	0.883005	0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000	0.883005	0.441502	+	0.897260i	$0.354446\pi$
$464$	4.00000	0.185695
$465$	1.00000	0.0463739
$466$	−20.0000	−0.926482
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	3.00000	0.138675
$469$	−6.00000	−0.277054
$470$	−5.00000	−0.230633
$471$	20.0000	0.921551
$472$	−6.00000	−0.276172
$473$	18.0000	0.827641
$474$	−1.00000	−0.0459315
$475$	−28.0000	−1.28473
$476$	2.00000	0.0916698
$477$	−2.00000	−0.0915737
$478$	8.00000	0.365911
$479$	−7.00000	−0.319838	−0.159919	−	0.987130i	$-0.551123\pi$
$479$	−7.00000	−0.319838	−0.159919	+	0.987130i	$0.551123\pi$
$480$	−1.00000	−0.0456435
$481$	−30.0000	−1.36788
$482$	18.0000	0.819878
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−5.00000	−0.227038
$486$	1.00000	0.0453609
$487$	7.00000	0.317200	0.158600	−	0.987343i	$-0.449302\pi$
$487$	7.00000	0.317200	0.158600	+	0.987343i	$0.449302\pi$
$488$	−3.00000	−0.135804
$489$	1.00000	0.0452216
$490$	−3.00000	−0.135526
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	6.00000	0.270501
$493$	4.00000	0.180151
$494$	−21.0000	−0.944835
$495$	−3.00000	−0.134840
$496$	1.00000	0.0449013
$497$	14.0000	0.627986
$498$	17.0000	0.761788
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	9.00000	0.402492
$501$	14.0000	0.625474
$502$	−28.0000	−1.24970
$503$	−5.00000	−0.222939	−0.111469	−	0.993768i	$-0.535556\pi$
$503$	−5.00000	−0.222939	−0.111469	+	0.993768i	$0.535556\pi$
$504$	−2.00000	−0.0890871
$505$	−14.0000	−0.622992
$506$	0	0
$507$	4.00000	0.177646
$508$	−12.0000	−0.532414
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	−1.00000	−0.0442807
$511$	−20.0000	−0.884748
$512$	−1.00000	−0.0441942
$513$	−7.00000	−0.309058
$514$	−12.0000	−0.529297
$515$	14.0000	0.616914
$516$	−6.00000	−0.264135
$517$	−15.0000	−0.659699
$518$	20.0000	0.878750
$519$	−13.0000	−0.570637
$520$	3.00000	0.131559
$521$	32.0000	1.40195	0.700973	−	0.713188i	$-0.252749\pi$
$521$	32.0000	1.40195	0.700973	+	0.713188i	$0.252749\pi$
$522$	−4.00000	−0.175075
$523$	42.0000	1.83653	0.918266	−	0.395964i	$-0.129590\pi$
$523$	42.0000	1.83653	0.918266	+	0.395964i	$0.129590\pi$
$524$	−14.0000	−0.611593
$525$	8.00000	0.349149
$526$	−24.0000	−1.04645
$527$	1.00000	0.0435607
$528$	−3.00000	−0.130558
$529$	−23.0000	−1.00000
$530$	−2.00000	−0.0868744
$531$	6.00000	0.260378
$532$	14.0000	0.606977
$533$	−18.0000	−0.779667
$534$	6.00000	0.259645
$535$	16.0000	0.691740
$536$	3.00000	0.129580
$537$	11.0000	0.474685
$538$	−2.00000	−0.0862261
$539$	−9.00000	−0.387657
$540$	1.00000	0.0430331
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	17.0000	0.730213
$543$	10.0000	0.429141
$544$	−1.00000	−0.0428746
$545$	10.0000	0.428353
$546$	6.00000	0.256776
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	10.0000	0.427179
$549$	3.00000	0.128037
$550$	12.0000	0.511682
$551$	28.0000	1.19284
$552$	0	0
$553$	−2.00000	−0.0850487
$554$	−27.0000	−1.14712
$555$	−10.0000	−0.424476
$556$	16.0000	0.678551
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	−1.00000	−0.0423334
$559$	18.0000	0.761319
$560$	−2.00000	−0.0845154
$561$	−3.00000	−0.126660
$562$	−6.00000	−0.253095
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	5.00000	0.210538
$565$	6.00000	0.252422
$566$	−21.0000	−0.882696
$567$	2.00000	0.0839921
$568$	−7.00000	−0.293713
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	−7.00000	−0.293198
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	9.00000	0.376309
$573$	24.0000	1.00261
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	−21.0000	−0.874241	−0.437121	−	0.899403i	$-0.644002\pi$
$577$	−21.0000	−0.874241	−0.437121	+	0.899403i	$0.644002\pi$
$578$	16.0000	0.665512
$579$	−19.0000	−0.789613
$580$	−4.00000	−0.166091
$581$	34.0000	1.41056
$582$	5.00000	0.207257
$583$	−6.00000	−0.248495
$584$	10.0000	0.413803
$585$	−3.00000	−0.124035
$586$	30.0000	1.23929
$587$	9.00000	0.371470	0.185735	−	0.982600i	$-0.440533\pi$
$587$	9.00000	0.371470	0.185735	+	0.982600i	$0.440533\pi$
$588$	3.00000	0.123718
$589$	7.00000	0.288430
$590$	6.00000	0.247016
$591$	26.0000	1.06950
$592$	−10.0000	−0.410997
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	3.00000	0.123091
$595$	−2.00000	−0.0819920
$596$	−21.0000	−0.860194
$597$	−7.00000	−0.286491
$598$	0	0
$599$	13.0000	0.531166	0.265583	−	0.964088i	$-0.414436\pi$
$599$	13.0000	0.531166	0.265583	+	0.964088i	$0.414436\pi$
$600$	−4.00000	−0.163299
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−12.0000	−0.489083
$603$	−3.00000	−0.122169
$604$	5.00000	0.203447
$605$	2.00000	0.0813116
$606$	14.0000	0.568711
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−7.00000	−0.283887
$609$	−8.00000	−0.324176
$610$	3.00000	0.121466
$611$	−15.0000	−0.606835
$612$	1.00000	0.0404226
$613$	−17.0000	−0.686624	−0.343312	−	0.939222i	$-0.611549\pi$
$613$	−17.0000	−0.686624	−0.343312	+	0.939222i	$0.611549\pi$
$614$	−16.0000	−0.645707
$615$	−6.00000	−0.241943
$616$	−6.00000	−0.241747
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	−14.0000	−0.563163
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−1.00000	−0.0401610
$621$	0	0
$622$	15.0000	0.601445
$623$	12.0000	0.480770
$624$	−3.00000	−0.120096
$625$	11.0000	0.440000
$626$	−10.0000	−0.399680
$627$	−21.0000	−0.838659
$628$	−20.0000	−0.798087
$629$	−10.0000	−0.398726
$630$	2.00000	0.0796819
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	1.00000	0.0397779
$633$	−16.0000	−0.635943
$634$	23.0000	0.913447
$635$	12.0000	0.476205
$636$	2.00000	0.0793052
$637$	−9.00000	−0.356593
$638$	−12.0000	−0.475085
$639$	7.00000	0.276916
$640$	1.00000	0.0395285
$641$	1.00000	0.0394976	0.0197488	−	0.999805i	$-0.493713\pi$
$641$	1.00000	0.0394976	0.0197488	+	0.999805i	$0.493713\pi$
$642$	−16.0000	−0.631470
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	−7.00000	−0.275411
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	−1.00000	−0.0392837
$649$	18.0000	0.706562
$650$	12.0000	0.470679
$651$	−2.00000	−0.0783862
$652$	−1.00000	−0.0391630
$653$	−1.00000	−0.0391330	−0.0195665	−	0.999809i	$-0.506229\pi$
$653$	−1.00000	−0.0391330	−0.0195665	+	0.999809i	$0.506229\pi$
$654$	−10.0000	−0.391031
$655$	14.0000	0.547025
$656$	−6.00000	−0.234261
$657$	−10.0000	−0.390137
$658$	10.0000	0.389841
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	3.00000	0.116775
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	28.0000	1.08825
$663$	−3.00000	−0.116510
$664$	−17.0000	−0.659728
$665$	−14.0000	−0.542897
$666$	10.0000	0.387492
$667$	0	0
$668$	−14.0000	−0.541676
$669$	−9.00000	−0.347960
$670$	−3.00000	−0.115900
$671$	9.00000	0.347441
$672$	2.00000	0.0771517
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	−8.00000	−0.308148
$675$	4.00000	0.153960
$676$	−4.00000	−0.153846
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	−6.00000	−0.230429
$679$	10.0000	0.383765
$680$	1.00000	0.0383482
$681$	22.0000	0.843042
$682$	−3.00000	−0.114876
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	7.00000	0.267652
$685$	−10.0000	−0.382080
$686$	20.0000	0.763604
$687$	−23.0000	−0.877505
$688$	6.00000	0.228748
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−11.0000	−0.418460	−0.209230	−	0.977866i	$-0.567096\pi$
$691$	−11.0000	−0.418460	−0.209230	+	0.977866i	$0.567096\pi$
$692$	13.0000	0.494186
$693$	6.00000	0.227921
$694$	17.0000	0.645311
$695$	−16.0000	−0.606915
$696$	4.00000	0.151620
$697$	−6.00000	−0.227266
$698$	−16.0000	−0.605609
$699$	−20.0000	−0.756469
$700$	−8.00000	−0.302372
$701$	−41.0000	−1.54855	−0.774274	−	0.632850i	$-0.781885\pi$
$701$	−41.0000	−1.54855	−0.774274	+	0.632850i	$0.781885\pi$
$702$	3.00000	0.113228
$703$	−70.0000	−2.64010
$704$	3.00000	0.113067
$705$	−5.00000	−0.188311
$706$	−3.00000	−0.112906
$707$	28.0000	1.05305
$708$	−6.00000	−0.225494
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	7.00000	0.262705
$711$	−1.00000	−0.0375029
$712$	−6.00000	−0.224860
$713$	0	0
$714$	2.00000	0.0748481
$715$	−9.00000	−0.336581
$716$	−11.0000	−0.411089
$717$	8.00000	0.298765
$718$	5.00000	0.186598
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	−1.00000	−0.0372678
$721$	−28.0000	−1.04277
$722$	−30.0000	−1.11648
$723$	18.0000	0.669427
$724$	−10.0000	−0.371647
$725$	−16.0000	−0.594225
$726$	−2.00000	−0.0742270
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	−10.0000	−0.370117
$731$	6.00000	0.221918
$732$	−3.00000	−0.110883
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−7.00000	−0.258375
$735$	−3.00000	−0.110657
$736$	0	0
$737$	−9.00000	−0.331519
$738$	6.00000	0.220863
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	10.0000	0.367607
$741$	−21.0000	−0.771454
$742$	4.00000	0.146845
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	1.00000	0.0366618
$745$	21.0000	0.769380
$746$	24.0000	0.878702
$747$	17.0000	0.621997
$748$	3.00000	0.109691
$749$	−32.0000	−1.16925
$750$	9.00000	0.328634
$751$	30.0000	1.09472	0.547358	−	0.836899i	$-0.315634\pi$
$751$	30.0000	1.09472	0.547358	+	0.836899i	$0.315634\pi$
$752$	−5.00000	−0.182331
$753$	−28.0000	−1.02038
$754$	−12.0000	−0.437014
$755$	−5.00000	−0.181969
$756$	−2.00000	−0.0727393
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	15.0000	0.544825
$759$	0	0
$760$	7.00000	0.253917
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	−12.0000	−0.434714
$763$	−20.0000	−0.724049
$764$	−24.0000	−0.868290
$765$	−1.00000	−0.0361551
$766$	18.0000	0.650366
$767$	18.0000	0.649942
$768$	−1.00000	−0.0360844
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	6.00000	0.216225
$771$	−12.0000	−0.432169
$772$	19.0000	0.683825
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	−6.00000	−0.215666
$775$	−4.00000	−0.143684
$776$	−5.00000	−0.179490
$777$	20.0000	0.717496
$778$	6.00000	0.215110
$779$	−42.0000	−1.50481
$780$	3.00000	0.107417
$781$	21.0000	0.751439
$782$	0	0
$783$	−4.00000	−0.142948
$784$	−3.00000	−0.107143
$785$	20.0000	0.713831
$786$	−14.0000	−0.499363
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	−26.0000	−0.926212
$789$	−24.0000	−0.854423
$790$	−1.00000	−0.0355784
$791$	−12.0000	−0.426671
$792$	−3.00000	−0.106600
$793$	9.00000	0.319599
$794$	−10.0000	−0.354887
$795$	−2.00000	−0.0709327
$796$	7.00000	0.248108
$797$	−48.0000	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	14.0000	0.495595
$799$	−5.00000	−0.176887
$800$	4.00000	0.141421
$801$	6.00000	0.212000
$802$	25.0000	0.882781
$803$	−30.0000	−1.05868
$804$	3.00000	0.105802
$805$	0	0
$806$	−3.00000	−0.105670
$807$	−2.00000	−0.0704033
$808$	−14.0000	−0.492518
$809$	−43.0000	−1.51180	−0.755900	−	0.654687i	$-0.772800\pi$
$809$	−43.0000	−1.51180	−0.755900	+	0.654687i	$0.772800\pi$
$810$	1.00000	0.0351364
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	8.00000	0.280745
$813$	17.0000	0.596216
$814$	30.0000	1.05150
$815$	1.00000	0.0350285
$816$	−1.00000	−0.0350070
$817$	42.0000	1.46939
$818$	22.0000	0.769212
$819$	6.00000	0.209657
$820$	6.00000	0.209529
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	10.0000	0.348790
$823$	17.0000	0.592583	0.296291	−	0.955098i	$-0.404250\pi$
$823$	17.0000	0.592583	0.296291	+	0.955098i	$0.404250\pi$
$824$	14.0000	0.487713
$825$	12.0000	0.417786
$826$	−12.0000	−0.417533
$827$	45.0000	1.56480	0.782402	−	0.622774i	$-0.213994\pi$
$827$	45.0000	1.56480	0.782402	+	0.622774i	$0.213994\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	17.0000	0.590079
$831$	−27.0000	−0.936620
$832$	3.00000	0.104006
$833$	−3.00000	−0.103944
$834$	16.0000	0.554035
$835$	14.0000	0.484490
$836$	21.0000	0.726300
$837$	−1.00000	−0.0345651
$838$	−12.0000	−0.414533
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	−2.00000	−0.0690066
$841$	−13.0000	−0.448276
$842$	2.00000	0.0689246
$843$	−6.00000	−0.206651
$844$	16.0000	0.550743
$845$	4.00000	0.137604
$846$	5.00000	0.171904
$847$	−4.00000	−0.137442
$848$	−2.00000	−0.0686803
$849$	−21.0000	−0.720718
$850$	4.00000	0.137199
$851$	0	0
$852$	−7.00000	−0.239816
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	−6.00000	−0.205316
$855$	−7.00000	−0.239395
$856$	16.0000	0.546869
$857$	−28.0000	−0.956462	−0.478231	−	0.878234i	$-0.658722\pi$
$857$	−28.0000	−0.956462	−0.478231	+	0.878234i	$0.658722\pi$
$858$	9.00000	0.307255
$859$	42.0000	1.43302	0.716511	−	0.697576i	$-0.245738\pi$
$859$	42.0000	1.43302	0.716511	+	0.697576i	$0.245738\pi$
$860$	−6.00000	−0.204598
$861$	12.0000	0.408959
$862$	24.0000	0.817443
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	1.00000	0.0340207
$865$	−13.0000	−0.442013
$866$	−24.0000	−0.815553
$867$	16.0000	0.543388
$868$	2.00000	0.0678844
$869$	−3.00000	−0.101768
$870$	−4.00000	−0.135613
$871$	−9.00000	−0.304953
$872$	10.0000	0.338643
$873$	5.00000	0.169224
$874$	0	0
$875$	18.0000	0.608511
$876$	10.0000	0.337869
$877$	−4.00000	−0.135070	−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000	−0.135070	−0.0675352	+	0.997717i	$0.521513\pi$
$878$	32.0000	1.07995
$879$	30.0000	1.01187
$880$	−3.00000	−0.101130
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	3.00000	0.101015
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	3.00000	0.100901
$885$	6.00000	0.201688
$886$	−30.0000	−1.00787
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−10.0000	−0.335578
$889$	−24.0000	−0.804934
$890$	6.00000	0.201120
$891$	3.00000	0.100504
$892$	9.00000	0.301342
$893$	−35.0000	−1.17123
$894$	−21.0000	−0.702345
$895$	11.0000	0.367689
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−27.0000	−0.901002
$899$	4.00000	0.133407
$900$	−4.00000	−0.133333
$901$	−2.00000	−0.0666297
$902$	18.0000	0.599334
$903$	−12.0000	−0.399335
$904$	6.00000	0.199557
$905$	10.0000	0.332411
$906$	5.00000	0.166114
$907$	23.0000	0.763702	0.381851	−	0.924224i	$-0.375287\pi$
$907$	23.0000	0.763702	0.381851	+	0.924224i	$0.375287\pi$
$908$	−22.0000	−0.730096
$909$	14.0000	0.464351
$910$	6.00000	0.198898
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	−7.00000	−0.231793
$913$	51.0000	1.68785
$914$	−10.0000	−0.330771
$915$	3.00000	0.0991769
$916$	23.0000	0.759941
$917$	−28.0000	−0.924641
$918$	1.00000	0.0330049
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	−10.0000	−0.329332
$923$	21.0000	0.691223
$924$	−6.00000	−0.197386
$925$	40.0000	1.31519
$926$	−19.0000	−0.624379
$927$	−14.0000	−0.459820
$928$	−4.00000	−0.131306
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	−1.00000	−0.0327913
$931$	−21.0000	−0.688247
$932$	20.0000	0.655122
$933$	15.0000	0.491078
$934$	−4.00000	−0.130884
$935$	−3.00000	−0.0981105
$936$	−3.00000	−0.0980581
$937$	33.0000	1.07806	0.539032	−	0.842286i	$-0.318790\pi$
$937$	33.0000	1.07806	0.539032	+	0.842286i	$0.318790\pi$
$938$	6.00000	0.195907
$939$	−10.0000	−0.326338
$940$	5.00000	0.163082
$941$	40.0000	1.30396	0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000	1.30396	0.651981	+	0.758235i	$0.273938\pi$
$942$	−20.0000	−0.651635
$943$	0	0
$944$	6.00000	0.195283
$945$	2.00000	0.0650600
$946$	−18.0000	−0.585230
$947$	−35.0000	−1.13735	−0.568674	−	0.822563i	$-0.692543\pi$
$947$	−35.0000	−1.13735	−0.568674	+	0.822563i	$0.692543\pi$
$948$	1.00000	0.0324785
$949$	−30.0000	−0.973841
$950$	28.0000	0.908440
$951$	23.0000	0.745826
$952$	−2.00000	−0.0648204
$953$	13.0000	0.421111	0.210556	−	0.977582i	$-0.432473\pi$
$953$	13.0000	0.421111	0.210556	+	0.977582i	$0.432473\pi$
$954$	2.00000	0.0647524
$955$	24.0000	0.776622
$956$	−8.00000	−0.258738
$957$	−12.0000	−0.387905
$958$	7.00000	0.226160
$959$	20.0000	0.645834
$960$	1.00000	0.0322749
$961$	1.00000	0.0322581
$962$	30.0000	0.967239
$963$	−16.0000	−0.515593
$964$	−18.0000	−0.579741
$965$	−19.0000	−0.611632
$966$	0	0
$967$	−5.00000	−0.160789	−0.0803946	−	0.996763i	$-0.525618\pi$
$967$	−5.00000	−0.160789	−0.0803946	+	0.996763i	$0.525618\pi$
$968$	2.00000	0.0642824
$969$	−7.00000	−0.224872
$970$	5.00000	0.160540
$971$	−34.0000	−1.09111	−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000	−1.09111	−0.545556	+	0.838074i	$0.683681\pi$
$972$	−1.00000	−0.0320750
$973$	32.0000	1.02587
$974$	−7.00000	−0.224294
$975$	12.0000	0.384308
$976$	3.00000	0.0960277
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	−1.00000	−0.0319765
$979$	18.0000	0.575282
$980$	3.00000	0.0958315
$981$	−10.0000	−0.319275
$982$	−28.0000	−0.893516
$983$	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
$985$	26.0000	0.828429
$986$	−4.00000	−0.127386
$987$	10.0000	0.318304
$988$	21.0000	0.668099
$989$	0	0
$990$	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	$0.602897\pi$
$992$	−1.00000	−0.0317500
$993$	28.0000	0.888553
$994$	−14.0000	−0.444053
$995$	−7.00000	−0.221915
$996$	−17.0000	−0.538666
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	−24.0000	−0.759707
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	186.2.a.a.1.1	✓	1
3.2	odd	2		558.2.a.h.1.1		1
4.3	odd	2		1488.2.a.j.1.1		1
5.2	odd	4		4650.2.d.z.3349.1		2
5.3	odd	4		4650.2.d.z.3349.2		2
5.4	even	2		4650.2.a.bo.1.1		1
7.6	odd	2		9114.2.a.n.1.1		1
8.3	odd	2		5952.2.a.g.1.1		1
8.5	even	2		5952.2.a.z.1.1		1
12.11	even	2		4464.2.a.m.1.1		1
31.30	odd	2		5766.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
186.2.a.a.1.1	✓	1	1.1	even	1	trivial
558.2.a.h.1.1		1	3.2	odd	2
1488.2.a.j.1.1		1	4.3	odd	2
4464.2.a.m.1.1		1	12.11	even	2
4650.2.a.bo.1.1		1	5.4	even	2
4650.2.d.z.3349.1		2	5.2	odd	4
4650.2.d.z.3349.2		2	5.3	odd	4
5766.2.a.d.1.1		1	31.30	odd	2
5952.2.a.g.1.1		1	8.3	odd	2
5952.2.a.z.1.1		1	8.5	even	2
9114.2.a.n.1.1		1	7.6	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:	\( N \)	\(=\)	\( 186 = 2 \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	186.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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