LMFDB - Embedded newform 6027.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 861)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} -5.00000 q^{13} -3.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -3.00000 q^{20} +2.00000 q^{22} -3.00000 q^{23} -3.00000 q^{24} +4.00000 q^{25} +5.00000 q^{26} -1.00000 q^{27} +1.00000 q^{29} +3.00000 q^{30} +8.00000 q^{31} -5.00000 q^{32} +2.00000 q^{33} -2.00000 q^{34} -1.00000 q^{36} +1.00000 q^{37} -2.00000 q^{38} +5.00000 q^{39} +9.00000 q^{40} +1.00000 q^{41} -10.0000 q^{43} +2.00000 q^{44} +3.00000 q^{45} +3.00000 q^{46} +9.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -2.00000 q^{51} +5.00000 q^{52} -11.0000 q^{53} +1.00000 q^{54} -6.00000 q^{55} -2.00000 q^{57} -1.00000 q^{58} +4.00000 q^{59} +3.00000 q^{60} -2.00000 q^{61} -8.00000 q^{62} +7.00000 q^{64} -15.0000 q^{65} -2.00000 q^{66} -5.00000 q^{67} -2.00000 q^{68} +3.00000 q^{69} -6.00000 q^{71} +3.00000 q^{72} -4.00000 q^{73} -1.00000 q^{74} -4.00000 q^{75} -2.00000 q^{76} -5.00000 q^{78} +11.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -1.00000 q^{82} -14.0000 q^{83} +6.00000 q^{85} +10.0000 q^{86} -1.00000 q^{87} -6.00000 q^{88} -8.00000 q^{89} -3.00000 q^{90} +3.00000 q^{92} -8.00000 q^{93} -9.00000 q^{94} +6.00000 q^{95} +5.00000 q^{96} -1.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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
$7$	0	0
$7$	0	0
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	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	$0.743877\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.00000	−0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	2.00000	0.426401
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	−3.00000	−0.612372
$25$	4.00000	0.800000
$26$	5.00000	0.980581
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	3.00000	0.547723
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−5.00000	−0.883883
$33$	2.00000	0.348155
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	−2.00000	−0.324443
$39$	5.00000	0.800641
$40$	9.00000	1.42302
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	2.00000	0.301511
$45$	3.00000	0.447214
$46$	3.00000	0.442326
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−2.00000	−0.280056
$52$	5.00000	0.693375
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	1.00000	0.136083
$55$	−6.00000	−0.809040
$56$	0	0
$57$	−2.00000	−0.264906
$58$	−1.00000	−0.131306
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	3.00000	0.387298
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	−15.0000	−1.86052
$66$	−2.00000	−0.246183
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	−2.00000	−0.242536
$69$	3.00000	0.361158
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	3.00000	0.353553
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−1.00000	−0.116248
$75$	−4.00000	−0.461880
$76$	−2.00000	−0.229416
$77$	0	0
$78$	−5.00000	−0.566139
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−1.00000	−0.110432
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	10.0000	1.07833
$87$	−1.00000	−0.107211
$88$	−6.00000	−0.639602
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	3.00000	0.312772
$93$	−8.00000	−0.829561
$94$	−9.00000	−0.928279
$95$	6.00000	0.615587
$96$	5.00000	0.510310
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	−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	$0.254728\pi$
$102$	2.00000	0.198030
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	−15.0000	−1.47087
$105$	0	0
$106$	11.0000	1.06841
$107$	−11.0000	−1.06341	−0.531705	−	0.846930i	$-0.678449\pi$
$107$	−11.0000	−1.06341	−0.531705	+	0.846930i	$0.678449\pi$
$108$	1.00000	0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	6.00000	0.572078
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	−20.0000	−1.88144	−0.940721	−	0.339182i	$-0.889850\pi$
$113$	−20.0000	−1.88144	−0.940721	+	0.339182i	$0.889850\pi$
$114$	2.00000	0.187317
$115$	−9.00000	−0.839254
$116$	−1.00000	−0.0928477
$117$	−5.00000	−0.462250
$118$	−4.00000	−0.368230
$119$	0	0
$120$	−9.00000	−0.821584
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	−1.00000	−0.0901670
$124$	−8.00000	−0.718421
$125$	−3.00000	−0.268328
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	3.00000	0.265165
$129$	10.0000	0.880451
$130$	15.0000	1.31559
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	5.00000	0.431934
$135$	−3.00000	−0.258199
$136$	6.00000	0.514496
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−3.00000	−0.255377
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	6.00000	0.503509
$143$	10.0000	0.836242
$144$	−1.00000	−0.0833333
$145$	3.00000	0.249136
$146$	4.00000	0.331042
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	4.00000	0.326599
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	6.00000	0.486664
$153$	2.00000	0.161690
$154$	0	0
$155$	24.0000	1.92773
$156$	−5.00000	−0.400320
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−11.0000	−0.875113
$159$	11.0000	0.872357
$160$	−15.0000	−1.18585
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−1.00000	−0.0780869
$165$	6.00000	0.467099
$166$	14.0000	1.08661
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	−6.00000	−0.460179
$171$	2.00000	0.152944
$172$	10.0000	0.762493
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	2.00000	0.150756
$177$	−4.00000	−0.300658
$178$	8.00000	0.599625
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	−3.00000	−0.223607
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	−9.00000	−0.663489
$185$	3.00000	0.220564
$186$	8.00000	0.586588
$187$	−4.00000	−0.292509
$188$	−9.00000	−0.656392
$189$	0	0
$190$	−6.00000	−0.435286
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	−7.00000	−0.505181
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	1.00000	0.0717958
$195$	15.0000	1.07417
$196$	0	0
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	2.00000	0.142134
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	12.0000	0.848528
$201$	5.00000	0.352673
$202$	−14.0000	−0.985037
$203$	0	0
$204$	2.00000	0.140028
$205$	3.00000	0.209529
$206$	−9.00000	−0.627060
$207$	−3.00000	−0.208514
$208$	5.00000	0.346688
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−7.00000	−0.481900	−0.240950	−	0.970538i	$-0.577459\pi$
$211$	−7.00000	−0.481900	−0.240950	+	0.970538i	$0.577459\pi$
$212$	11.0000	0.755483
$213$	6.00000	0.411113
$214$	11.0000	0.751945
$215$	−30.0000	−2.04598
$216$	−3.00000	−0.204124
$217$	0	0
$218$	−2.00000	−0.135457
$219$	4.00000	0.270295
$220$	6.00000	0.404520
$221$	−10.0000	−0.672673
$222$	1.00000	0.0671156
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	20.0000	1.33038
$227$	−21.0000	−1.39382	−0.696909	−	0.717159i	$-0.745442\pi$
$227$	−21.0000	−1.39382	−0.696909	+	0.717159i	$0.745442\pi$
$228$	2.00000	0.132453
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	9.00000	0.593442
$231$	0	0
$232$	3.00000	0.196960
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	5.00000	0.326860
$235$	27.0000	1.76129
$236$	−4.00000	−0.260378
$237$	−11.0000	−0.714527
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	3.00000	0.193649
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	1.00000	0.0637577
$247$	−10.0000	−0.636285
$248$	24.0000	1.52400
$249$	14.0000	0.887214
$250$	3.00000	0.189737
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	−8.00000	−0.501965
$255$	−6.00000	−0.375735
$256$	−17.0000	−1.06250
$257$	4.00000	0.249513	0.124757	−	0.992187i	$-0.460185\pi$
$257$	4.00000	0.249513	0.124757	+	0.992187i	$0.460185\pi$
$258$	−10.0000	−0.622573
$259$	0	0
$260$	15.0000	0.930261
$261$	1.00000	0.0618984
$262$	2.00000	0.123560
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	6.00000	0.369274
$265$	−33.0000	−2.02717
$266$	0	0
$267$	8.00000	0.489592
$268$	5.00000	0.305424
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	3.00000	0.182574
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−9.00000	−0.543710
$275$	−8.00000	−0.482418
$276$	−3.00000	−0.180579
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	−11.0000	−0.659736
$279$	8.00000	0.478947
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	9.00000	0.535942
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	6.00000	0.356034
$285$	−6.00000	−0.355409
$286$	−10.0000	−0.591312
$287$	0	0
$288$	−5.00000	−0.294628
$289$	−13.0000	−0.764706
$290$	−3.00000	−0.176166
$291$	1.00000	0.0586210
$292$	4.00000	0.234082
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	3.00000	0.174371
$297$	2.00000	0.116052
$298$	6.00000	0.347571
$299$	15.0000	0.867472
$300$	4.00000	0.230940
$301$	0	0
$302$	−16.0000	−0.920697
$303$	−14.0000	−0.804279
$304$	−2.00000	−0.114708
$305$	−6.00000	−0.343559
$306$	−2.00000	−0.114332
$307$	1.00000	0.0570730	0.0285365	−	0.999593i	$-0.490915\pi$
$307$	1.00000	0.0570730	0.0285365	+	0.999593i	$0.490915\pi$
$308$	0	0
$309$	−9.00000	−0.511992
$310$	−24.0000	−1.36311
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	15.0000	0.849208
$313$	21.0000	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$313$	21.0000	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−11.0000	−0.618798
$317$	−7.00000	−0.393159	−0.196580	−	0.980488i	$-0.562983\pi$
$317$	−7.00000	−0.393159	−0.196580	+	0.980488i	$0.562983\pi$
$318$	−11.0000	−0.616849
$319$	−2.00000	−0.111979
$320$	21.0000	1.17394
$321$	11.0000	0.613960
$322$	0	0
$323$	4.00000	0.222566
$324$	−1.00000	−0.0555556
$325$	−20.0000	−1.10940
$326$	12.0000	0.664619
$327$	−2.00000	−0.110600
$328$	3.00000	0.165647
$329$	0	0
$330$	−6.00000	−0.330289
$331$	−33.0000	−1.81384	−0.906922	−	0.421299i	$-0.861574\pi$
$331$	−33.0000	−1.81384	−0.906922	+	0.421299i	$0.861574\pi$
$332$	14.0000	0.768350
$333$	1.00000	0.0547997
$334$	−3.00000	−0.164153
$335$	−15.0000	−0.819538
$336$	0	0
$337$	−29.0000	−1.57973	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000	−1.57973	−0.789865	+	0.613280i	$0.789850\pi$
$338$	−12.0000	−0.652714
$339$	20.0000	1.08625
$340$	−6.00000	−0.325396
$341$	−16.0000	−0.866449
$342$	−2.00000	−0.108148
$343$	0	0
$344$	−30.0000	−1.61749
$345$	9.00000	0.484544
$346$	9.00000	0.483843
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	1.00000	0.0536056
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	10.0000	0.533002
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	4.00000	0.212598
$355$	−18.0000	−0.955341
$356$	8.00000	0.423999
$357$	0	0
$358$	2.00000	0.105703
$359$	31.0000	1.63612	0.818059	−	0.575135i	$-0.195050\pi$
$359$	31.0000	1.63612	0.818059	+	0.575135i	$0.195050\pi$
$360$	9.00000	0.474342
$361$	−15.0000	−0.789474
$362$	−6.00000	−0.315353
$363$	7.00000	0.367405
$364$	0	0
$365$	−12.0000	−0.628109
$366$	−2.00000	−0.104542
$367$	−23.0000	−1.20059	−0.600295	−	0.799779i	$-0.704950\pi$
$367$	−23.0000	−1.20059	−0.600295	+	0.799779i	$0.704950\pi$
$368$	3.00000	0.156386
$369$	1.00000	0.0520579
$370$	−3.00000	−0.155963
$371$	0	0
$372$	8.00000	0.414781
$373$	1.00000	0.0517780	0.0258890	−	0.999665i	$-0.491758\pi$
$373$	1.00000	0.0517780	0.0258890	+	0.999665i	$0.491758\pi$
$374$	4.00000	0.206835
$375$	3.00000	0.154919
$376$	27.0000	1.39242
$377$	−5.00000	−0.257513
$378$	0	0
$379$	−30.0000	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$380$	−6.00000	−0.307794
$381$	−8.00000	−0.409852
$382$	−6.00000	−0.306987
$383$	11.0000	0.562074	0.281037	−	0.959697i	$-0.409322\pi$
$383$	11.0000	0.562074	0.281037	+	0.959697i	$0.409322\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	18.0000	0.916176
$387$	−10.0000	−0.508329
$388$	1.00000	0.0507673
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	−15.0000	−0.759555
$391$	−6.00000	−0.303433
$392$	0	0
$393$	2.00000	0.100887
$394$	−14.0000	−0.705310
$395$	33.0000	1.66041
$396$	2.00000	0.100504
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	14.0000	0.701757
$399$	0	0
$400$	−4.00000	−0.200000
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	−5.00000	−0.249377
$403$	−40.0000	−1.99254
$404$	−14.0000	−0.696526
$405$	3.00000	0.149071
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	−6.00000	−0.297044
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	−3.00000	−0.148159
$411$	−9.00000	−0.443937
$412$	−9.00000	−0.443398
$413$	0	0
$414$	3.00000	0.147442
$415$	−42.0000	−2.06170
$416$	25.0000	1.22573
$417$	−11.0000	−0.538672
$418$	4.00000	0.195646
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−38.0000	−1.85201	−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000	−1.85201	−0.926003	+	0.377515i	$0.876779\pi$
$422$	7.00000	0.340755
$423$	9.00000	0.437595
$424$	−33.0000	−1.60262
$425$	8.00000	0.388057
$426$	−6.00000	−0.290701
$427$	0	0
$428$	11.0000	0.531705
$429$	−10.0000	−0.482805
$430$	30.0000	1.44673
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	1.00000	0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	−2.00000	−0.0957826
$437$	−6.00000	−0.287019
$438$	−4.00000	−0.191127
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	−18.0000	−0.858116
$441$	0	0
$442$	10.0000	0.475651
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	1.00000	0.0474579
$445$	−24.0000	−1.13771
$446$	−19.0000	−0.899676
$447$	6.00000	0.283790
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	−4.00000	−0.188562
$451$	−2.00000	−0.0941763
$452$	20.0000	0.940721
$453$	−16.0000	−0.751746
$454$	21.0000	0.985579
$455$	0	0
$456$	−6.00000	−0.280976
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	15.0000	0.700904
$459$	−2.00000	−0.0933520
$460$	9.00000	0.419627
$461$	−33.0000	−1.53696	−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000	−1.53696	−0.768482	+	0.639872i	$0.778987\pi$
$462$	0	0
$463$	−13.0000	−0.604161	−0.302081	−	0.953282i	$-0.597681\pi$
$463$	−13.0000	−0.604161	−0.302081	+	0.953282i	$0.597681\pi$
$464$	−1.00000	−0.0464238
$465$	−24.0000	−1.11297
$466$	−14.0000	−0.648537
$467$	−30.0000	−1.38823	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000	−1.38823	−0.694117	+	0.719862i	$0.744205\pi$
$468$	5.00000	0.231125
$469$	0	0
$470$	−27.0000	−1.24542
$471$	−2.00000	−0.0921551
$472$	12.0000	0.552345
$473$	20.0000	0.919601
$474$	11.0000	0.505247
$475$	8.00000	0.367065
$476$	0	0
$477$	−11.0000	−0.503655
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	15.0000	0.684653
$481$	−5.00000	−0.227980
$482$	26.0000	1.18427
$483$	0	0
$484$	7.00000	0.318182
$485$	−3.00000	−0.136223
$486$	1.00000	0.0453609
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	−6.00000	−0.271607
$489$	12.0000	0.542659
$490$	0	0
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	1.00000	0.0450835
$493$	2.00000	0.0900755
$494$	10.0000	0.449921
$495$	−6.00000	−0.269680
$496$	−8.00000	−0.359211
$497$	0	0
$498$	−14.0000	−0.627355
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	3.00000	0.134164
$501$	−3.00000	−0.134030
$502$	−10.0000	−0.446322
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	0	0
$505$	42.0000	1.86898
$506$	−6.00000	−0.266733
$507$	−12.0000	−0.532939
$508$	−8.00000	−0.354943
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	6.00000	0.265684
$511$	0	0
$512$	11.0000	0.486136
$513$	−2.00000	−0.0883022
$514$	−4.00000	−0.176432
$515$	27.0000	1.18976
$516$	−10.0000	−0.440225
$517$	−18.0000	−0.791639
$518$	0	0
$519$	9.00000	0.395056
$520$	−45.0000	−1.97338
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	−1.00000	−0.0437688
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	2.00000	0.0873704
$525$	0	0
$526$	−30.0000	−1.30806
$527$	16.0000	0.696971
$528$	−2.00000	−0.0870388
$529$	−14.0000	−0.608696
$530$	33.0000	1.43343
$531$	4.00000	0.173585
$532$	0	0
$533$	−5.00000	−0.216574
$534$	−8.00000	−0.346194
$535$	−33.0000	−1.42671
$536$	−15.0000	−0.647901
$537$	2.00000	0.0863064
$538$	−15.0000	−0.646696
$539$	0	0
$540$	3.00000	0.129099
$541$	39.0000	1.67674	0.838370	−	0.545101i	$-0.183509\pi$
$541$	39.0000	1.67674	0.838370	+	0.545101i	$0.183509\pi$
$542$	−7.00000	−0.300676
$543$	−6.00000	−0.257485
$544$	−10.0000	−0.428746
$545$	6.00000	0.257012
$546$	0	0
$547$	19.0000	0.812381	0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000	0.812381	0.406191	+	0.913788i	$0.366857\pi$
$548$	−9.00000	−0.384461
$549$	−2.00000	−0.0853579
$550$	8.00000	0.341121
$551$	2.00000	0.0852029
$552$	9.00000	0.383065
$553$	0	0
$554$	−13.0000	−0.552317
$555$	−3.00000	−0.127343
$556$	−11.0000	−0.466504
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	−8.00000	−0.338667
$559$	50.0000	2.11477
$560$	0	0
$561$	4.00000	0.168880
$562$	15.0000	0.632737
$563$	−31.0000	−1.30649	−0.653247	−	0.757145i	$-0.726594\pi$
$563$	−31.0000	−1.30649	−0.653247	+	0.757145i	$0.726594\pi$
$564$	9.00000	0.378968
$565$	−60.0000	−2.52422
$566$	20.0000	0.840663
$567$	0	0
$568$	−18.0000	−0.755263
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	6.00000	0.251312
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−10.0000	−0.418121
$573$	−6.00000	−0.250654
$574$	0	0
$575$	−12.0000	−0.500435
$576$	7.00000	0.291667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	13.0000	0.540729
$579$	18.0000	0.748054
$580$	−3.00000	−0.124568
$581$	0	0
$582$	−1.00000	−0.0414513
$583$	22.0000	0.911147
$584$	−12.0000	−0.496564
$585$	−15.0000	−0.620174
$586$	14.0000	0.578335
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	−12.0000	−0.494032
$591$	−14.0000	−0.575883
$592$	−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	$0.579245\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	6.00000	0.245770
$597$	14.0000	0.572982
$598$	−15.0000	−0.613396
$599$	3.00000	0.122577	0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000	0.122577	0.0612883	+	0.998120i	$0.480479\pi$
$600$	−12.0000	−0.489898
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	−5.00000	−0.203616
$604$	−16.0000	−0.651031
$605$	−21.0000	−0.853771
$606$	14.0000	0.568711
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	−10.0000	−0.405554
$609$	0	0
$610$	6.00000	0.242933
$611$	−45.0000	−1.82051
$612$	−2.00000	−0.0808452
$613$	−45.0000	−1.81753	−0.908766	−	0.417305i	$-0.862975\pi$
$613$	−45.0000	−1.81753	−0.908766	+	0.417305i	$0.862975\pi$
$614$	−1.00000	−0.0403567
$615$	−3.00000	−0.120972
$616$	0	0
$617$	8.00000	0.322068	0.161034	−	0.986949i	$-0.448517\pi$
$617$	8.00000	0.322068	0.161034	+	0.986949i	$0.448517\pi$
$618$	9.00000	0.362033
$619$	45.0000	1.80870	0.904351	−	0.426789i	$-0.140355\pi$
$619$	45.0000	1.80870	0.904351	+	0.426789i	$0.140355\pi$
$620$	−24.0000	−0.963863
$621$	3.00000	0.120386
$622$	−15.0000	−0.601445
$623$	0	0
$624$	−5.00000	−0.200160
$625$	−29.0000	−1.16000
$626$	−21.0000	−0.839329
$627$	4.00000	0.159745
$628$	−2.00000	−0.0798087
$629$	2.00000	0.0797452
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	33.0000	1.31267
$633$	7.00000	0.278225
$634$	7.00000	0.278006
$635$	24.0000	0.952411
$636$	−11.0000	−0.436178
$637$	0	0
$638$	2.00000	0.0791808
$639$	−6.00000	−0.237356
$640$	9.00000	0.355756
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	−11.0000	−0.434135
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	30.0000	1.18125
$646$	−4.00000	−0.157378
$647$	−26.0000	−1.02217	−0.511083	−	0.859532i	$-0.670755\pi$
$647$	−26.0000	−1.02217	−0.511083	+	0.859532i	$0.670755\pi$
$648$	3.00000	0.117851
$649$	−8.00000	−0.314027
$650$	20.0000	0.784465
$651$	0	0
$652$	12.0000	0.469956
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	2.00000	0.0782062
$655$	−6.00000	−0.234439
$656$	−1.00000	−0.0390434
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−14.0000	−0.545363	−0.272681	−	0.962104i	$-0.587910\pi$
$659$	−14.0000	−0.545363	−0.272681	+	0.962104i	$0.587910\pi$
$660$	−6.00000	−0.233550
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	33.0000	1.28258
$663$	10.0000	0.388368
$664$	−42.0000	−1.62992
$665$	0	0
$666$	−1.00000	−0.0387492
$667$	−3.00000	−0.116160
$668$	−3.00000	−0.116073
$669$	−19.0000	−0.734582
$670$	15.0000	0.579501
$671$	4.00000	0.154418
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	29.0000	1.11704
$675$	−4.00000	−0.153960
$676$	−12.0000	−0.461538
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−20.0000	−0.768095
$679$	0	0
$680$	18.0000	0.690268
$681$	21.0000	0.804722
$682$	16.0000	0.612672
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	−2.00000	−0.0764719
$685$	27.0000	1.03162
$686$	0	0
$687$	15.0000	0.572286
$688$	10.0000	0.381246
$689$	55.0000	2.09533
$690$	−9.00000	−0.342624
$691$	48.0000	1.82601	0.913003	−	0.407953i	$-0.133757\pi$
$691$	48.0000	1.82601	0.913003	+	0.407953i	$0.133757\pi$
$692$	9.00000	0.342129
$693$	0	0
$694$	18.0000	0.683271
$695$	33.0000	1.25176
$696$	−3.00000	−0.113715
$697$	2.00000	0.0757554
$698$	20.0000	0.757011
$699$	−14.0000	−0.529529
$700$	0	0
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	−5.00000	−0.188713
$703$	2.00000	0.0754314
$704$	−14.0000	−0.527645
$705$	−27.0000	−1.01688
$706$	14.0000	0.526897
$707$	0	0
$708$	4.00000	0.150329
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	18.0000	0.675528
$711$	11.0000	0.412532
$712$	−24.0000	−0.899438
$713$	−24.0000	−0.898807
$714$	0	0
$715$	30.0000	1.12194
$716$	2.00000	0.0747435
$717$	0	0
$718$	−31.0000	−1.15691
$719$	44.0000	1.64092	0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000	1.64092	0.820462	+	0.571702i	$0.193717\pi$
$720$	−3.00000	−0.111803
$721$	0	0
$722$	15.0000	0.558242
$723$	26.0000	0.966950
$724$	−6.00000	−0.222988
$725$	4.00000	0.148556
$726$	−7.00000	−0.259794
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	12.0000	0.444140
$731$	−20.0000	−0.739727
$732$	−2.00000	−0.0739221
$733$	−44.0000	−1.62518	−0.812589	−	0.582838i	$-0.801942\pi$
$733$	−44.0000	−1.62518	−0.812589	+	0.582838i	$0.801942\pi$
$734$	23.0000	0.848945
$735$	0	0
$736$	15.0000	0.552907
$737$	10.0000	0.368355
$738$	−1.00000	−0.0368105
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	−3.00000	−0.110282
$741$	10.0000	0.367359
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	−24.0000	−0.879883
$745$	−18.0000	−0.659469
$746$	−1.00000	−0.0366126
$747$	−14.0000	−0.512233
$748$	4.00000	0.146254
$749$	0	0
$750$	−3.00000	−0.109545
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	−9.00000	−0.328196
$753$	−10.0000	−0.364420
$754$	5.00000	0.182089
$755$	48.0000	1.74690
$756$	0	0
$757$	46.0000	1.67190	0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000	1.67190	0.835949	+	0.548807i	$0.184918\pi$
$758$	30.0000	1.08965
$759$	−6.00000	−0.217786
$760$	18.0000	0.652929
$761$	9.00000	0.326250	0.163125	−	0.986605i	$-0.447843\pi$
$761$	9.00000	0.326250	0.163125	+	0.986605i	$0.447843\pi$
$762$	8.00000	0.289809
$763$	0	0
$764$	−6.00000	−0.217072
$765$	6.00000	0.216930
$766$	−11.0000	−0.397446
$767$	−20.0000	−0.722158
$768$	17.0000	0.613435
$769$	−28.0000	−1.00971	−0.504853	−	0.863205i	$-0.668453\pi$
$769$	−28.0000	−1.00971	−0.504853	+	0.863205i	$0.668453\pi$
$770$	0	0
$771$	−4.00000	−0.144056
$772$	18.0000	0.647834
$773$	16.0000	0.575480	0.287740	−	0.957709i	$-0.407096\pi$
$773$	16.0000	0.575480	0.287740	+	0.957709i	$0.407096\pi$
$774$	10.0000	0.359443
$775$	32.0000	1.14947
$776$	−3.00000	−0.107694
$777$	0	0
$778$	2.00000	0.0717035
$779$	2.00000	0.0716574
$780$	−15.0000	−0.537086
$781$	12.0000	0.429394
$782$	6.00000	0.214560
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	6.00000	0.214149
$786$	−2.00000	−0.0713376
$787$	1.00000	0.0356462	0.0178231	−	0.999841i	$-0.494326\pi$
$787$	1.00000	0.0356462	0.0178231	+	0.999841i	$0.494326\pi$
$788$	−14.0000	−0.498729
$789$	−30.0000	−1.06803
$790$	−33.0000	−1.17409
$791$	0	0
$792$	−6.00000	−0.213201
$793$	10.0000	0.355110
$794$	−14.0000	−0.496841
$795$	33.0000	1.17039
$796$	14.0000	0.496217
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	−20.0000	−0.707107
$801$	−8.00000	−0.282666
$802$	−32.0000	−1.12996
$803$	8.00000	0.282314
$804$	−5.00000	−0.176336
$805$	0	0
$806$	40.0000	1.40894
$807$	−15.0000	−0.528025
$808$	42.0000	1.47755
$809$	25.0000	0.878953	0.439477	−	0.898254i	$-0.355164\pi$
$809$	25.0000	0.878953	0.439477	+	0.898254i	$0.355164\pi$
$810$	−3.00000	−0.105409
$811$	31.0000	1.08856	0.544279	−	0.838905i	$-0.316803\pi$
$811$	31.0000	1.08856	0.544279	+	0.838905i	$0.316803\pi$
$812$	0	0
$813$	−7.00000	−0.245501
$814$	2.00000	0.0701000
$815$	−36.0000	−1.26102
$816$	2.00000	0.0700140
$817$	−20.0000	−0.699711
$818$	32.0000	1.11885
$819$	0	0
$820$	−3.00000	−0.104765
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	9.00000	0.313911
$823$	−56.0000	−1.95204	−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000	−1.95204	−0.976019	+	0.217687i	$0.930149\pi$
$824$	27.0000	0.940590
$825$	8.00000	0.278524
$826$	0	0
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	3.00000	0.104257
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	42.0000	1.45784
$831$	−13.0000	−0.450965
$832$	−35.0000	−1.21341
$833$	0	0
$834$	11.0000	0.380899
$835$	9.00000	0.311458
$836$	4.00000	0.138343
$837$	−8.00000	−0.276520
$838$	30.0000	1.03633
$839$	−55.0000	−1.89881	−0.949405	−	0.314053i	$-0.898313\pi$
$839$	−55.0000	−1.89881	−0.949405	+	0.314053i	$0.898313\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	38.0000	1.30957
$843$	15.0000	0.516627
$844$	7.00000	0.240950
$845$	36.0000	1.23844
$846$	−9.00000	−0.309426
$847$	0	0
$848$	11.0000	0.377742
$849$	20.0000	0.686398
$850$	−8.00000	−0.274398
$851$	−3.00000	−0.102839
$852$	−6.00000	−0.205557
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	−33.0000	−1.12792
$857$	21.0000	0.717346	0.358673	−	0.933463i	$-0.383229\pi$
$857$	21.0000	0.717346	0.358673	+	0.933463i	$0.383229\pi$
$858$	10.0000	0.341394
$859$	−3.00000	−0.102359	−0.0511793	−	0.998689i	$-0.516298\pi$
$859$	−3.00000	−0.102359	−0.0511793	+	0.998689i	$0.516298\pi$
$860$	30.0000	1.02299
$861$	0	0
$862$	−16.0000	−0.544962
$863$	51.0000	1.73606	0.868030	−	0.496512i	$-0.165386\pi$
$863$	51.0000	1.73606	0.868030	+	0.496512i	$0.165386\pi$
$864$	5.00000	0.170103
$865$	−27.0000	−0.918028
$866$	26.0000	0.883516
$867$	13.0000	0.441503
$868$	0	0
$869$	−22.0000	−0.746299
$870$	3.00000	0.101710
$871$	25.0000	0.847093
$872$	6.00000	0.203186
$873$	−1.00000	−0.0338449
$874$	6.00000	0.202953
$875$	0	0
$876$	−4.00000	−0.135147
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−20.0000	−0.674967
$879$	14.0000	0.472208
$880$	6.00000	0.202260
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	0	0
$883$	1.00000	0.0336527	0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000	0.0336527	0.0168263	+	0.999858i	$0.494644\pi$
$884$	10.0000	0.336336
$885$	−12.0000	−0.403376
$886$	24.0000	0.806296
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	−3.00000	−0.100673
$889$	0	0
$890$	24.0000	0.804482
$891$	−2.00000	−0.0670025
$892$	−19.0000	−0.636167
$893$	18.0000	0.602347
$894$	−6.00000	−0.200670
$895$	−6.00000	−0.200558
$896$	0	0
$897$	−15.0000	−0.500835
$898$	0	0
$899$	8.00000	0.266815
$900$	−4.00000	−0.133333
$901$	−22.0000	−0.732926
$902$	2.00000	0.0665927
$903$	0	0
$904$	−60.0000	−1.99557
$905$	18.0000	0.598340
$906$	16.0000	0.531564
$907$	38.0000	1.26177	0.630885	−	0.775877i	$-0.282692\pi$
$907$	38.0000	1.26177	0.630885	+	0.775877i	$0.282692\pi$
$908$	21.0000	0.696909
$909$	14.0000	0.464351
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	2.00000	0.0662266
$913$	28.0000	0.926665
$914$	16.0000	0.529233
$915$	6.00000	0.198354
$916$	15.0000	0.495614
$917$	0	0
$918$	2.00000	0.0660098
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	−27.0000	−0.890164
$921$	−1.00000	−0.0329511
$922$	33.0000	1.08680
$923$	30.0000	0.987462
$924$	0	0
$925$	4.00000	0.131519
$926$	13.0000	0.427207
$927$	9.00000	0.295599
$928$	−5.00000	−0.164133
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	24.0000	0.786991
$931$	0	0
$932$	−14.0000	−0.458585
$933$	−15.0000	−0.491078
$934$	30.0000	0.981630
$935$	−12.0000	−0.392442
$936$	−15.0000	−0.490290
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	−21.0000	−0.685309
$940$	−27.0000	−0.880643
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	2.00000	0.0651635
$943$	−3.00000	−0.0976934
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−20.0000	−0.650256
$947$	−21.0000	−0.682408	−0.341204	−	0.939989i	$-0.610835\pi$
$947$	−21.0000	−0.682408	−0.341204	+	0.939989i	$0.610835\pi$
$948$	11.0000	0.357263
$949$	20.0000	0.649227
$950$	−8.00000	−0.259554
$951$	7.00000	0.226991
$952$	0	0
$953$	58.0000	1.87880	0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000	1.87880	0.939402	+	0.342817i	$0.111381\pi$
$954$	11.0000	0.356138
$955$	18.0000	0.582466
$956$	0	0
$957$	2.00000	0.0646508
$958$	0	0
$959$	0	0
$960$	−21.0000	−0.677772
$961$	33.0000	1.06452
$962$	5.00000	0.161206
$963$	−11.0000	−0.354470
$964$	26.0000	0.837404
$965$	−54.0000	−1.73832
$966$	0	0
$967$	49.0000	1.57573	0.787867	−	0.615846i	$-0.211185\pi$
$967$	49.0000	1.57573	0.787867	+	0.615846i	$0.211185\pi$
$968$	−21.0000	−0.674966
$969$	−4.00000	−0.128499
$970$	3.00000	0.0963242
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−4.00000	−0.128168
$975$	20.0000	0.640513
$976$	2.00000	0.0640184
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	−12.0000	−0.383718
$979$	16.0000	0.511362
$980$	0	0
$981$	2.00000	0.0638551
$982$	32.0000	1.02116
$983$	−52.0000	−1.65854	−0.829271	−	0.558846i	$-0.811244\pi$
$983$	−52.0000	−1.65854	−0.829271	+	0.558846i	$0.811244\pi$
$984$	−3.00000	−0.0956365
$985$	42.0000	1.33823
$986$	−2.00000	−0.0636930
$987$	0	0
$988$	10.0000	0.318142
$989$	30.0000	0.953945
$990$	6.00000	0.190693
$991$	−47.0000	−1.49300	−0.746502	−	0.665383i	$-0.768268\pi$
$991$	−47.0000	−1.49300	−0.746502	+	0.665383i	$0.768268\pi$
$992$	−40.0000	−1.27000
$993$	33.0000	1.04722
$994$	0	0
$995$	−42.0000	−1.33149
$996$	−14.0000	−0.443607
$997$	−29.0000	−0.918439	−0.459220	−	0.888323i	$-0.651871\pi$
$997$	−29.0000	−0.918439	−0.459220	+	0.888323i	$0.651871\pi$
$998$	16.0000	0.506471
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.c.1.1		1
7.6	odd	2		861.2.a.b.1.1	✓	1
21.20	even	2		2583.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.a.b.1.1	✓	1	7.6	odd	2
2583.2.a.f.1.1		1	21.20	even	2
6027.2.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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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