LMFDB - Embedded newform 6027.2.a.b.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:	$0$
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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -6.00000 q^{11} +1.00000 q^{12} +7.00000 q^{13} +3.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} +3.00000 q^{20} +6.00000 q^{22} +5.00000 q^{23} -3.00000 q^{24} +4.00000 q^{25} -7.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} -3.00000 q^{30} -5.00000 q^{32} +6.00000 q^{33} -6.00000 q^{34} -1.00000 q^{36} +1.00000 q^{37} -6.00000 q^{38} -7.00000 q^{39} -9.00000 q^{40} -1.00000 q^{41} +6.00000 q^{43} +6.00000 q^{44} -3.00000 q^{45} -5.00000 q^{46} +5.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -6.00000 q^{51} -7.00000 q^{52} -9.00000 q^{53} +1.00000 q^{54} +18.0000 q^{55} -6.00000 q^{57} -3.00000 q^{58} +8.00000 q^{59} -3.00000 q^{60} +2.00000 q^{61} +7.00000 q^{64} -21.0000 q^{65} -6.00000 q^{66} +9.00000 q^{67} -6.00000 q^{68} -5.00000 q^{69} -10.0000 q^{71} +3.00000 q^{72} -4.00000 q^{73} -1.00000 q^{74} -4.00000 q^{75} -6.00000 q^{76} +7.00000 q^{78} +9.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +1.00000 q^{82} +2.00000 q^{83} -18.0000 q^{85} -6.00000 q^{86} -3.00000 q^{87} -18.0000 q^{88} -16.0000 q^{89} +3.00000 q^{90} -5.00000 q^{92} -5.00000 q^{94} -18.0000 q^{95} +5.00000 q^{96} -13.0000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	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
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	1.00000	0.288675
$13$	7.00000	1.94145	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	7.00000	1.94145	0.970725	+	0.240192i	$0.0772105\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−1.00000	−0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	6.00000	1.27920
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	−3.00000	−0.612372
$25$	4.00000	0.800000
$26$	−7.00000	−1.37281
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−3.00000	−0.547723
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−5.00000	−0.883883
$33$	6.00000	1.04447
$34$	−6.00000	−1.02899
$35$	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	$0.473806\pi$
$38$	−6.00000	−0.973329
$39$	−7.00000	−1.12090
$40$	−9.00000	−1.42302
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	6.00000	0.904534
$45$	−3.00000	−0.447214
$46$	−5.00000	−0.737210
$47$	5.00000	0.729325	0.364662	−	0.931140i	$-0.381184\pi$
$47$	5.00000	0.729325	0.364662	+	0.931140i	$0.381184\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−6.00000	−0.840168
$52$	−7.00000	−0.970725
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	1.00000	0.136083
$55$	18.0000	2.42712
$56$	0	0
$57$	−6.00000	−0.794719
$58$	−3.00000	−0.393919
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	−3.00000	−0.387298
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	7.00000	0.875000
$65$	−21.0000	−2.60473
$66$	−6.00000	−0.738549
$67$	9.00000	1.09952	0.549762	−	0.835321i	$-0.314718\pi$
$67$	9.00000	1.09952	0.549762	+	0.835321i	$0.314718\pi$
$68$	−6.00000	−0.727607
$69$	−5.00000	−0.601929
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	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
$76$	−6.00000	−0.688247
$77$	0	0
$78$	7.00000	0.792594
$79$	9.00000	1.01258	0.506290	−	0.862364i	$-0.331017\pi$
$79$	9.00000	1.01258	0.506290	+	0.862364i	$0.331017\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	1.00000	0.110432
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−18.0000	−1.95237
$86$	−6.00000	−0.646997
$87$	−3.00000	−0.321634
$88$	−18.0000	−1.91881
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	−5.00000	−0.521286
$93$	0	0
$94$	−5.00000	−0.515711
$95$	−18.0000	−1.84676
$96$	5.00000	0.510310
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	−4.00000	−0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	6.00000	0.594089
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	21.0000	2.05922
$105$	0	0
$106$	9.00000	0.874157
$107$	−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	$0.469465\pi$
$110$	−18.0000	−1.71623
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	6.00000	0.561951
$115$	−15.0000	−1.39876
$116$	−3.00000	−0.278543
$117$	7.00000	0.647150
$118$	−8.00000	−0.736460
$119$	0	0
$120$	9.00000	0.821584
$121$	25.0000	2.27273
$122$	−2.00000	−0.181071
$123$	1.00000	0.0901670
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	3.00000	0.265165
$129$	−6.00000	−0.528271
$130$	21.0000	1.84182
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−9.00000	−0.777482
$135$	3.00000	0.258199
$136$	18.0000	1.54349
$137$	19.0000	1.62328	0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000	1.62328	0.811640	+	0.584158i	$0.198575\pi$
$138$	5.00000	0.425628
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	−5.00000	−0.421076
$142$	10.0000	0.839181
$143$	−42.0000	−3.51222
$144$	−1.00000	−0.0833333
$145$	−9.00000	−0.747409
$146$	4.00000	0.331042
$147$	0	0
$148$	−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	$0.420959\pi$
$150$	4.00000	0.326599
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	18.0000	1.45999
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	7.00000	0.560449
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−9.00000	−0.716002
$159$	9.00000	0.713746
$160$	15.0000	1.18585
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	1.00000	0.0780869
$165$	−18.0000	−1.40130
$166$	−2.00000	−0.155230
$167$	23.0000	1.77979	0.889897	−	0.456162i	$-0.150776\pi$
$167$	23.0000	1.77979	0.889897	+	0.456162i	$0.150776\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	18.0000	1.38054
$171$	6.00000	0.458831
$172$	−6.00000	−0.457496
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	6.00000	0.452267
$177$	−8.00000	−0.601317
$178$	16.0000	1.19925
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	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
$184$	15.0000	1.10581
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−36.0000	−2.63258
$188$	−5.00000	−0.364662
$189$	0	0
$190$	18.0000	1.30586
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	−7.00000	−0.505181
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	13.0000	0.933346
$195$	21.0000	1.50384
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	6.00000	0.426401
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	12.0000	0.848528
$201$	−9.00000	−0.634811
$202$	−6.00000	−0.422159
$203$	0	0
$204$	6.00000	0.420084
$205$	3.00000	0.209529
$206$	−7.00000	−0.487713
$207$	5.00000	0.347524
$208$	−7.00000	−0.485363
$209$	−36.0000	−2.49017
$210$	0	0
$211$	−21.0000	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$211$	−21.0000	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$212$	9.00000	0.618123
$213$	10.0000	0.685189
$214$	11.0000	0.751945
$215$	−18.0000	−1.22759
$216$	−3.00000	−0.204124
$217$	0	0
$218$	−2.00000	−0.135457
$219$	4.00000	0.270295
$220$	−18.0000	−1.21356
$221$	42.0000	2.82523
$222$	1.00000	0.0671156
$223$	−11.0000	−0.736614	−0.368307	−	0.929704i	$-0.620063\pi$
$223$	−11.0000	−0.736614	−0.368307	+	0.929704i	$0.620063\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	4.00000	0.266076
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	6.00000	0.397360
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	15.0000	0.989071
$231$	0	0
$232$	9.00000	0.590879
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−7.00000	−0.457604
$235$	−15.0000	−0.978492
$236$	−8.00000	−0.520756
$237$	−9.00000	−0.584613
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	−3.00000	−0.193649
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	−25.0000	−1.60706
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−1.00000	−0.0637577
$247$	42.0000	2.67240
$248$	0	0
$249$	−2.00000	−0.126745
$250$	−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
$253$	−30.0000	−1.88608
$254$	20.0000	1.25491
$255$	18.0000	1.12720
$256$	−17.0000	−1.06250
$257$	−16.0000	−0.998053	−0.499026	−	0.866587i	$-0.666309\pi$
$257$	−16.0000	−0.998053	−0.499026	+	0.866587i	$0.666309\pi$
$258$	6.00000	0.373544
$259$	0	0
$260$	21.0000	1.30236
$261$	3.00000	0.185695
$262$	−6.00000	−0.370681
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	18.0000	1.10782
$265$	27.0000	1.65860
$266$	0	0
$267$	16.0000	0.979184
$268$	−9.00000	−0.549762
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	−3.00000	−0.182574
$271$	−31.0000	−1.88312	−0.941558	−	0.336851i	$-0.890638\pi$
$271$	−31.0000	−1.88312	−0.941558	+	0.336851i	$0.890638\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−19.0000	−1.14783
$275$	−24.0000	−1.44725
$276$	5.00000	0.300965
$277$	−3.00000	−0.180253	−0.0901263	−	0.995930i	$-0.528727\pi$
$277$	−3.00000	−0.180253	−0.0901263	+	0.995930i	$0.528727\pi$
$278$	−13.0000	−0.779688
$279$	0	0
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	5.00000	0.297746
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	10.0000	0.593391
$285$	18.0000	1.06623
$286$	42.0000	2.48351
$287$	0	0
$288$	−5.00000	−0.294628
$289$	19.0000	1.11765
$290$	9.00000	0.528498
$291$	13.0000	0.762073
$292$	4.00000	0.234082
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	3.00000	0.174371
$297$	6.00000	0.348155
$298$	−6.00000	−0.347571
$299$	35.0000	2.02410
$300$	4.00000	0.230940
$301$	0	0
$302$	−8.00000	−0.460348
$303$	−6.00000	−0.344691
$304$	−6.00000	−0.344124
$305$	−6.00000	−0.343559
$306$	−6.00000	−0.342997
$307$	−9.00000	−0.513657	−0.256829	−	0.966457i	$-0.582678\pi$
$307$	−9.00000	−0.513657	−0.256829	+	0.966457i	$0.582678\pi$
$308$	0	0
$309$	−7.00000	−0.398216
$310$	0	0
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	−21.0000	−1.18889
$313$	9.00000	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−9.00000	−0.506290
$317$	27.0000	1.51647	0.758236	−	0.651981i	$-0.226062\pi$
$317$	27.0000	1.51647	0.758236	+	0.651981i	$0.226062\pi$
$318$	−9.00000	−0.504695
$319$	−18.0000	−1.00781
$320$	−21.0000	−1.17394
$321$	11.0000	0.613960
$322$	0	0
$323$	36.0000	2.00309
$324$	−1.00000	−0.0555556
$325$	28.0000	1.55316
$326$	16.0000	0.886158
$327$	−2.00000	−0.110600
$328$	−3.00000	−0.165647
$329$	0	0
$330$	18.0000	0.990867
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	−2.00000	−0.109764
$333$	1.00000	0.0547997
$334$	−23.0000	−1.25850
$335$	−27.0000	−1.47517
$336$	0	0
$337$	11.0000	0.599208	0.299604	−	0.954064i	$-0.403145\pi$
$337$	11.0000	0.599208	0.299604	+	0.954064i	$0.403145\pi$
$338$	−36.0000	−1.95814
$339$	4.00000	0.217250
$340$	18.0000	0.976187
$341$	0	0
$342$	−6.00000	−0.324443
$343$	0	0
$344$	18.0000	0.970495
$345$	15.0000	0.807573
$346$	15.0000	0.806405
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	3.00000	0.160817
$349$	12.0000	0.642345	0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000	0.642345	0.321173	+	0.947021i	$0.395923\pi$
$350$	0	0
$351$	−7.00000	−0.373632
$352$	30.0000	1.59901
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	8.00000	0.425195
$355$	30.0000	1.59223
$356$	16.0000	0.847998
$357$	0	0
$358$	−6.00000	−0.317110
$359$	−33.0000	−1.74167	−0.870837	−	0.491572i	$-0.836422\pi$
$359$	−33.0000	−1.74167	−0.870837	+	0.491572i	$0.836422\pi$
$360$	−9.00000	−0.474342
$361$	17.0000	0.894737
$362$	−6.00000	−0.315353
$363$	−25.0000	−1.31216
$364$	0	0
$365$	12.0000	0.628109
$366$	2.00000	0.104542
$367$	−25.0000	−1.30499	−0.652495	−	0.757793i	$-0.726278\pi$
$367$	−25.0000	−1.30499	−0.652495	+	0.757793i	$0.726278\pi$
$368$	−5.00000	−0.260643
$369$	−1.00000	−0.0520579
$370$	3.00000	0.155963
$371$	0	0
$372$	0	0
$373$	25.0000	1.29445	0.647225	−	0.762299i	$-0.275929\pi$
$373$	25.0000	1.29445	0.647225	+	0.762299i	$0.275929\pi$
$374$	36.0000	1.86152
$375$	−3.00000	−0.154919
$376$	15.0000	0.773566
$377$	21.0000	1.08156
$378$	0	0
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	18.0000	0.923381
$381$	20.0000	1.02463
$382$	18.0000	0.920960
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−2.00000	−0.101797
$387$	6.00000	0.304997
$388$	13.0000	0.659975
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	−21.0000	−1.06338
$391$	30.0000	1.51717
$392$	0	0
$393$	−6.00000	−0.302660
$394$	−2.00000	−0.100759
$395$	−27.0000	−1.35852
$396$	6.00000	0.301511
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−6.00000	−0.300753
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	9.00000	0.448879
$403$	0	0
$404$	−6.00000	−0.298511
$405$	−3.00000	−0.149071
$406$	0	0
$407$	−6.00000	−0.297409
$408$	−18.0000	−0.891133
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	−3.00000	−0.148159
$411$	−19.0000	−0.937201
$412$	−7.00000	−0.344865
$413$	0	0
$414$	−5.00000	−0.245737
$415$	−6.00000	−0.294528
$416$	−35.0000	−1.71602
$417$	−13.0000	−0.636613
$418$	36.0000	1.76082
$419$	22.0000	1.07477	0.537385	−	0.843337i	$-0.319412\pi$
$419$	22.0000	1.07477	0.537385	+	0.843337i	$0.319412\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	21.0000	1.02226
$423$	5.00000	0.243108
$424$	−27.0000	−1.31124
$425$	24.0000	1.16417
$426$	−10.0000	−0.484502
$427$	0	0
$428$	11.0000	0.531705
$429$	42.0000	2.02778
$430$	18.0000	0.868037
$431$	−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
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	−2.00000	−0.0957826
$437$	30.0000	1.43509
$438$	−4.00000	−0.191127
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	54.0000	2.57435
$441$	0	0
$442$	−42.0000	−1.99774
$443$	−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
$445$	48.0000	2.27542
$446$	11.0000	0.520865
$447$	−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
$451$	6.00000	0.282529
$452$	4.00000	0.188144
$453$	−8.00000	−0.375873
$454$	−7.00000	−0.328526
$455$	0	0
$456$	−18.0000	−0.842927
$457$	20.0000	0.935561	0.467780	−	0.883845i	$-0.345054\pi$
$457$	20.0000	0.935561	0.467780	+	0.883845i	$0.345054\pi$
$458$	−5.00000	−0.233635
$459$	−6.00000	−0.280056
$460$	15.0000	0.699379
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	1.00000	0.0464739	0.0232370	−	0.999730i	$-0.492603\pi$
$463$	1.00000	0.0464739	0.0232370	+	0.999730i	$0.492603\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	6.00000	0.277945
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	−7.00000	−0.323575
$469$	0	0
$470$	15.0000	0.691898
$471$	−2.00000	−0.0921551
$472$	24.0000	1.10469
$473$	−36.0000	−1.65528
$474$	9.00000	0.413384
$475$	24.0000	1.10120
$476$	0	0
$477$	−9.00000	−0.412082
$478$	−24.0000	−1.09773
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−15.0000	−0.684653
$481$	7.00000	0.319173
$482$	−18.0000	−0.819878
$483$	0	0
$484$	−25.0000	−1.13636
$485$	39.0000	1.77090
$486$	1.00000	0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	6.00000	0.271607
$489$	16.0000	0.723545
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	−1.00000	−0.0450835
$493$	18.0000	0.810679
$494$	−42.0000	−1.88967
$495$	18.0000	0.809040
$496$	0	0
$497$	0	0
$498$	2.00000	0.0896221
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	−3.00000	−0.134164
$501$	−23.0000	−1.02756
$502$	−10.0000	−0.446322
$503$	21.0000	0.936344	0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000	0.936344	0.468172	+	0.883637i	$0.344913\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	30.0000	1.33366
$507$	−36.0000	−1.59882
$508$	20.0000	0.887357
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	−18.0000	−0.797053
$511$	0	0
$512$	11.0000	0.486136
$513$	−6.00000	−0.264906
$514$	16.0000	0.705730
$515$	−21.0000	−0.925371
$516$	6.00000	0.264135
$517$	−30.0000	−1.31940
$518$	0	0
$519$	15.0000	0.658427
$520$	−63.0000	−2.76273
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	−3.00000	−0.131306
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	14.0000	0.610429
$527$	0	0
$528$	−6.00000	−0.261116
$529$	2.00000	0.0869565
$530$	−27.0000	−1.17281
$531$	8.00000	0.347170
$532$	0	0
$533$	−7.00000	−0.303204
$534$	−16.0000	−0.692388
$535$	33.0000	1.42671
$536$	27.0000	1.16622
$537$	−6.00000	−0.258919
$538$	−9.00000	−0.388018
$539$	0	0
$540$	−3.00000	−0.129099
$541$	−9.00000	−0.386940	−0.193470	−	0.981106i	$-0.561974\pi$
$541$	−9.00000	−0.386940	−0.193470	+	0.981106i	$0.561974\pi$
$542$	31.0000	1.33156
$543$	−6.00000	−0.257485
$544$	−30.0000	−1.28624
$545$	−6.00000	−0.257012
$546$	0	0
$547$	9.00000	0.384812	0.192406	−	0.981315i	$-0.438371\pi$
$547$	9.00000	0.384812	0.192406	+	0.981315i	$0.438371\pi$
$548$	−19.0000	−0.811640
$549$	2.00000	0.0853579
$550$	24.0000	1.02336
$551$	18.0000	0.766826
$552$	−15.0000	−0.638442
$553$	0	0
$554$	3.00000	0.127458
$555$	3.00000	0.127343
$556$	−13.0000	−0.551323
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	42.0000	1.77641
$560$	0	0
$561$	36.0000	1.51992
$562$	−3.00000	−0.126547
$563$	−19.0000	−0.800755	−0.400377	−	0.916350i	$-0.631121\pi$
$563$	−19.0000	−0.800755	−0.400377	+	0.916350i	$0.631121\pi$
$564$	5.00000	0.210538
$565$	12.0000	0.504844
$566$	20.0000	0.840663
$567$	0	0
$568$	−30.0000	−1.25877
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	−18.0000	−0.753937
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	42.0000	1.75611
$573$	18.0000	0.751961
$574$	0	0
$575$	20.0000	0.834058
$576$	7.00000	0.291667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−19.0000	−0.790296
$579$	−2.00000	−0.0831172
$580$	9.00000	0.373705
$581$	0	0
$582$	−13.0000	−0.538867
$583$	54.0000	2.23645
$584$	−12.0000	−0.496564
$585$	−21.0000	−0.868243
$586$	−6.00000	−0.247858
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	0	0
$590$	24.0000	0.988064
$591$	−2.00000	−0.0822690
$592$	−1.00000	−0.0410997
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	−6.00000	−0.246183
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−6.00000	−0.245564
$598$	−35.0000	−1.43126
$599$	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
$601$	−39.0000	−1.59084	−0.795422	−	0.606057i	$-0.792751\pi$
$601$	−39.0000	−1.59084	−0.795422	+	0.606057i	$0.792751\pi$
$602$	0	0
$603$	9.00000	0.366508
$604$	−8.00000	−0.325515
$605$	−75.0000	−3.04918
$606$	6.00000	0.243733
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	−30.0000	−1.21666
$609$	0	0
$610$	6.00000	0.242933
$611$	35.0000	1.41595
$612$	−6.00000	−0.242536
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	9.00000	0.363210
$615$	−3.00000	−0.120972
$616$	0	0
$617$	4.00000	0.161034	0.0805170	−	0.996753i	$-0.474343\pi$
$617$	4.00000	0.161034	0.0805170	+	0.996753i	$0.474343\pi$
$618$	7.00000	0.281581
$619$	35.0000	1.40677	0.703384	−	0.710810i	$-0.251671\pi$
$619$	35.0000	1.40677	0.703384	+	0.710810i	$0.251671\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	21.0000	0.842023
$623$	0	0
$624$	7.00000	0.280224
$625$	−29.0000	−1.16000
$626$	−9.00000	−0.359712
$627$	36.0000	1.43770
$628$	−2.00000	−0.0798087
$629$	6.00000	0.239236
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	27.0000	1.07400
$633$	21.0000	0.834675
$634$	−27.0000	−1.07231
$635$	60.0000	2.38103
$636$	−9.00000	−0.356873
$637$	0	0
$638$	18.0000	0.712627
$639$	−10.0000	−0.395594
$640$	−9.00000	−0.355756
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−11.0000	−0.434135
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	0	0
$645$	18.0000	0.708749
$646$	−36.0000	−1.41640
$647$	−10.0000	−0.393141	−0.196570	−	0.980490i	$-0.562980\pi$
$647$	−10.0000	−0.393141	−0.196570	+	0.980490i	$0.562980\pi$
$648$	3.00000	0.117851
$649$	−48.0000	−1.88416
$650$	−28.0000	−1.09825
$651$	0	0
$652$	16.0000	0.626608
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	2.00000	0.0782062
$655$	−18.0000	−0.703318
$656$	1.00000	0.0390434
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−38.0000	−1.48027	−0.740135	−	0.672458i	$-0.765238\pi$
$659$	−38.0000	−1.48027	−0.740135	+	0.672458i	$0.765238\pi$
$660$	18.0000	0.700649
$661$	28.0000	1.08907	0.544537	−	0.838737i	$-0.316705\pi$
$661$	28.0000	1.08907	0.544537	+	0.838737i	$0.316705\pi$
$662$	19.0000	0.738456
$663$	−42.0000	−1.63114
$664$	6.00000	0.232845
$665$	0	0
$666$	−1.00000	−0.0387492
$667$	15.0000	0.580802
$668$	−23.0000	−0.889897
$669$	11.0000	0.425285
$670$	27.0000	1.04310
$671$	−12.0000	−0.463255
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	−11.0000	−0.423704
$675$	−4.00000	−0.153960
$676$	−36.0000	−1.38462
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	−4.00000	−0.153619
$679$	0	0
$680$	−54.0000	−2.07081
$681$	−7.00000	−0.268241
$682$	0	0
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	−6.00000	−0.229416
$685$	−57.0000	−2.17786
$686$	0	0
$687$	−5.00000	−0.190762
$688$	−6.00000	−0.228748
$689$	−63.0000	−2.40011
$690$	−15.0000	−0.571040
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	15.0000	0.570214
$693$	0	0
$694$	18.0000	0.683271
$695$	−39.0000	−1.47935
$696$	−9.00000	−0.341144
$697$	−6.00000	−0.227266
$698$	−12.0000	−0.454207
$699$	6.00000	0.226941
$700$	0	0
$701$	40.0000	1.51078	0.755390	−	0.655276i	$-0.227448\pi$
$701$	40.0000	1.51078	0.755390	+	0.655276i	$0.227448\pi$
$702$	7.00000	0.264198
$703$	6.00000	0.226294
$704$	−42.0000	−1.58293
$705$	15.0000	0.564933
$706$	−30.0000	−1.12906
$707$	0	0
$708$	8.00000	0.300658
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−30.0000	−1.12588
$711$	9.00000	0.337526
$712$	−48.0000	−1.79888
$713$	0	0
$714$	0	0
$715$	126.000	4.71213
$716$	−6.00000	−0.224231
$717$	−24.0000	−0.896296
$718$	33.0000	1.23155
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	−17.0000	−0.632674
$723$	−18.0000	−0.669427
$724$	−6.00000	−0.222988
$725$	12.0000	0.445669
$726$	25.0000	0.927837
$727$	20.0000	0.741759	0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000	0.741759	0.370879	+	0.928681i	$0.379056\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−12.0000	−0.444140
$731$	36.0000	1.33151
$732$	2.00000	0.0739221
$733$	28.0000	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$734$	25.0000	0.922767
$735$	0	0
$736$	−25.0000	−0.921512
$737$	−54.0000	−1.98912
$738$	1.00000	0.0368105
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	3.00000	0.110282
$741$	−42.0000	−1.54291
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	−25.0000	−0.915315
$747$	2.00000	0.0731762
$748$	36.0000	1.31629
$749$	0	0
$750$	3.00000	0.109545
$751$	−37.0000	−1.35015	−0.675075	−	0.737749i	$-0.735889\pi$
$751$	−37.0000	−1.35015	−0.675075	+	0.737749i	$0.735889\pi$
$752$	−5.00000	−0.182331
$753$	−10.0000	−0.364420
$754$	−21.0000	−0.764775
$755$	−24.0000	−0.873449
$756$	0	0
$757$	−30.0000	−1.09037	−0.545184	−	0.838316i	$-0.683540\pi$
$757$	−30.0000	−1.09037	−0.545184	+	0.838316i	$0.683540\pi$
$758$	−22.0000	−0.799076
$759$	30.0000	1.08893
$760$	−54.0000	−1.95879
$761$	47.0000	1.70375	0.851874	−	0.523746i	$-0.175466\pi$
$761$	47.0000	1.70375	0.851874	+	0.523746i	$0.175466\pi$
$762$	−20.0000	−0.724524
$763$	0	0
$764$	18.0000	0.651217
$765$	−18.0000	−0.650791
$766$	−15.0000	−0.541972
$767$	56.0000	2.02204
$768$	17.0000	0.613435
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	16.0000	0.576226
$772$	−2.00000	−0.0719816
$773$	16.0000	0.575480	0.287740	−	0.957709i	$-0.407096\pi$
$773$	16.0000	0.575480	0.287740	+	0.957709i	$0.407096\pi$
$774$	−6.00000	−0.215666
$775$	0	0
$776$	−39.0000	−1.40002
$777$	0	0
$778$	14.0000	0.501924
$779$	−6.00000	−0.214972
$780$	−21.0000	−0.751921
$781$	60.0000	2.14697
$782$	−30.0000	−1.07280
$783$	−3.00000	−0.107211
$784$	0	0
$785$	−6.00000	−0.214149
$786$	6.00000	0.214013
$787$	39.0000	1.39020	0.695100	−	0.718913i	$-0.255360\pi$
$787$	39.0000	1.39020	0.695100	+	0.718913i	$0.255360\pi$
$788$	−2.00000	−0.0712470
$789$	14.0000	0.498413
$790$	27.0000	0.960617
$791$	0	0
$792$	−18.0000	−0.639602
$793$	14.0000	0.497155
$794$	2.00000	0.0709773
$795$	−27.0000	−0.957591
$796$	−6.00000	−0.212664
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	0	0
$799$	30.0000	1.06132
$800$	−20.0000	−0.707107
$801$	−16.0000	−0.565332
$802$	20.0000	0.706225
$803$	24.0000	0.846942
$804$	9.00000	0.317406
$805$	0	0
$806$	0	0
$807$	−9.00000	−0.316815
$808$	18.0000	0.633238
$809$	−13.0000	−0.457056	−0.228528	−	0.973537i	$-0.573391\pi$
$809$	−13.0000	−0.457056	−0.228528	+	0.973537i	$0.573391\pi$
$810$	3.00000	0.105409
$811$	41.0000	1.43970	0.719852	−	0.694127i	$-0.244209\pi$
$811$	41.0000	1.43970	0.719852	+	0.694127i	$0.244209\pi$
$812$	0	0
$813$	31.0000	1.08722
$814$	6.00000	0.210300
$815$	48.0000	1.68137
$816$	6.00000	0.210042
$817$	36.0000	1.25948
$818$	−32.0000	−1.11885
$819$	0	0
$820$	−3.00000	−0.104765
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	19.0000	0.662701
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	21.0000	0.731570
$825$	24.0000	0.835573
$826$	0	0
$827$	−32.0000	−1.11275	−0.556375	−	0.830932i	$-0.687808\pi$
$827$	−32.0000	−1.11275	−0.556375	+	0.830932i	$0.687808\pi$
$828$	−5.00000	−0.173762
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	6.00000	0.208263
$831$	3.00000	0.104069
$832$	49.0000	1.69877
$833$	0	0
$834$	13.0000	0.450153
$835$	−69.0000	−2.38784
$836$	36.0000	1.24509
$837$	0	0
$838$	−22.0000	−0.759977
$839$	−27.0000	−0.932144	−0.466072	−	0.884747i	$-0.654331\pi$
$839$	−27.0000	−0.932144	−0.466072	+	0.884747i	$0.654331\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	2.00000	0.0689246
$843$	−3.00000	−0.103325
$844$	21.0000	0.722850
$845$	−108.000	−3.71531
$846$	−5.00000	−0.171904
$847$	0	0
$848$	9.00000	0.309061
$849$	20.0000	0.686398
$850$	−24.0000	−0.823193
$851$	5.00000	0.171398
$852$	−10.0000	−0.342594
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	0	0
$855$	−18.0000	−0.615587
$856$	−33.0000	−1.12792
$857$	−21.0000	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$857$	−21.0000	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$858$	−42.0000	−1.43386
$859$	11.0000	0.375315	0.187658	−	0.982235i	$-0.439910\pi$
$859$	11.0000	0.375315	0.187658	+	0.982235i	$0.439910\pi$
$860$	18.0000	0.613795
$861$	0	0
$862$	24.0000	0.817443
$863$	11.0000	0.374444	0.187222	−	0.982318i	$-0.440052\pi$
$863$	11.0000	0.374444	0.187222	+	0.982318i	$0.440052\pi$
$864$	5.00000	0.170103
$865$	45.0000	1.53005
$866$	−14.0000	−0.475739
$867$	−19.0000	−0.645274
$868$	0	0
$869$	−54.0000	−1.83182
$870$	−9.00000	−0.305129
$871$	63.0000	2.13467
$872$	6.00000	0.203186
$873$	−13.0000	−0.439983
$874$	−30.0000	−1.01477
$875$	0	0
$876$	−4.00000	−0.135147
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	−4.00000	−0.134993
$879$	−6.00000	−0.202375
$880$	−18.0000	−0.606780
$881$	41.0000	1.38133	0.690663	−	0.723177i	$-0.257319\pi$
$881$	41.0000	1.38133	0.690663	+	0.723177i	$0.257319\pi$
$882$	0	0
$883$	−13.0000	−0.437485	−0.218742	−	0.975783i	$-0.570195\pi$
$883$	−13.0000	−0.437485	−0.218742	+	0.975783i	$0.570195\pi$
$884$	−42.0000	−1.41261
$885$	24.0000	0.806751
$886$	24.0000	0.806296
$887$	−33.0000	−1.10803	−0.554016	−	0.832506i	$-0.686905\pi$
$887$	−33.0000	−1.10803	−0.554016	+	0.832506i	$0.686905\pi$
$888$	−3.00000	−0.100673
$889$	0	0
$890$	−48.0000	−1.60896
$891$	−6.00000	−0.201008
$892$	11.0000	0.368307
$893$	30.0000	1.00391
$894$	6.00000	0.200670
$895$	−18.0000	−0.601674
$896$	0	0
$897$	−35.0000	−1.16862
$898$	0	0
$899$	0	0
$900$	−4.00000	−0.133333
$901$	−54.0000	−1.79900
$902$	−6.00000	−0.199778
$903$	0	0
$904$	−12.0000	−0.399114
$905$	−18.0000	−0.598340
$906$	8.00000	0.265782
$907$	26.0000	0.863316	0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000	0.863316	0.431658	+	0.902037i	$0.357929\pi$
$908$	−7.00000	−0.232303
$909$	6.00000	0.199007
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	6.00000	0.198680
$913$	−12.0000	−0.397142
$914$	−20.0000	−0.661541
$915$	6.00000	0.198354
$916$	−5.00000	−0.165205
$917$	0	0
$918$	6.00000	0.198030
$919$	−47.0000	−1.55039	−0.775193	−	0.631724i	$-0.782348\pi$
$919$	−47.0000	−1.55039	−0.775193	+	0.631724i	$0.782348\pi$
$920$	−45.0000	−1.48361
$921$	9.00000	0.296560
$922$	15.0000	0.493999
$923$	−70.0000	−2.30408
$924$	0	0
$925$	4.00000	0.131519
$926$	−1.00000	−0.0328620
$927$	7.00000	0.229910
$928$	−15.0000	−0.492399
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	0	0
$932$	6.00000	0.196537
$933$	21.0000	0.687509
$934$	6.00000	0.196326
$935$	108.000	3.53198
$936$	21.0000	0.686406
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−9.00000	−0.293704
$940$	15.0000	0.489246
$941$	47.0000	1.53216	0.766078	−	0.642747i	$-0.222206\pi$
$941$	47.0000	1.53216	0.766078	+	0.642747i	$0.222206\pi$
$942$	2.00000	0.0651635
$943$	−5.00000	−0.162822
$944$	−8.00000	−0.260378
$945$	0	0
$946$	36.0000	1.17046
$947$	59.0000	1.91724	0.958621	−	0.284685i	$-0.0918889\pi$
$947$	59.0000	1.91724	0.958621	+	0.284685i	$0.0918889\pi$
$948$	9.00000	0.292306
$949$	−28.0000	−0.908918
$950$	−24.0000	−0.778663
$951$	−27.0000	−0.875535
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	9.00000	0.291386
$955$	54.0000	1.74740
$956$	−24.0000	−0.776215
$957$	18.0000	0.581857
$958$	24.0000	0.775405
$959$	0	0
$960$	21.0000	0.677772
$961$	−31.0000	−1.00000
$962$	−7.00000	−0.225689
$963$	−11.0000	−0.354470
$964$	−18.0000	−0.579741
$965$	−6.00000	−0.193147
$966$	0	0
$967$	−5.00000	−0.160789	−0.0803946	−	0.996763i	$-0.525618\pi$
$967$	−5.00000	−0.160789	−0.0803946	+	0.996763i	$0.525618\pi$
$968$	75.0000	2.41059
$969$	−36.0000	−1.15649
$970$	−39.0000	−1.25221
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−32.0000	−1.02535
$975$	−28.0000	−0.896718
$976$	−2.00000	−0.0640184
$977$	−10.0000	−0.319928	−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000	−0.319928	−0.159964	+	0.987123i	$0.551138\pi$
$978$	−16.0000	−0.511624
$979$	96.0000	3.06817
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	3.00000	0.0956365
$985$	−6.00000	−0.191176
$986$	−18.0000	−0.573237
$987$	0	0
$988$	−42.0000	−1.33620
$989$	30.0000	0.953945
$990$	−18.0000	−0.572078
$991$	−61.0000	−1.93773	−0.968864	−	0.247592i	$-0.920361\pi$
$991$	−61.0000	−1.93773	−0.968864	+	0.247592i	$0.920361\pi$
$992$	0	0
$993$	19.0000	0.602947
$994$	0	0
$995$	−18.0000	−0.570638
$996$	2.00000	0.0633724
$997$	−9.00000	−0.285033	−0.142516	−	0.989792i	$-0.545519\pi$
$997$	−9.00000	−0.285033	−0.142516	+	0.989792i	$0.545519\pi$
$998$	−24.0000	−0.759707
$999$	−1.00000	−0.0316386

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
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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