LMFDB - Embedded newform 3381.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, 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("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -3.00000 q^{15} -1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +3.00000 q^{20} +2.00000 q^{22} +1.00000 q^{23} +3.00000 q^{24} +4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.00000 q^{30} +10.0000 q^{31} -5.00000 q^{32} -2.00000 q^{33} -1.00000 q^{34} -1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{39} -9.00000 q^{40} +4.00000 q^{41} +4.00000 q^{43} +2.00000 q^{44} -3.00000 q^{45} -1.00000 q^{46} +7.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} +1.00000 q^{51} +1.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} -4.00000 q^{59} +3.00000 q^{60} -2.00000 q^{61} -10.0000 q^{62} +7.00000 q^{64} +3.00000 q^{65} +2.00000 q^{66} -3.00000 q^{67} -1.00000 q^{68} +1.00000 q^{69} -15.0000 q^{71} +3.00000 q^{72} -13.0000 q^{73} +2.00000 q^{74} +4.00000 q^{75} +1.00000 q^{78} +4.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} +6.00000 q^{83} -3.00000 q^{85} -4.00000 q^{86} -6.00000 q^{88} -6.00000 q^{89} +3.00000 q^{90} -1.00000 q^{92} +10.0000 q^{93} -7.00000 q^{94} -5.00000 q^{96} +16.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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	$-0.734058\pi$
$5$	−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
$11$	−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
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	−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
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	2.00000	0.426401
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	3.00000	0.612372
$25$	4.00000	0.800000
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	3.00000	0.547723
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	−5.00000	−0.883883
$33$	−2.00000	−0.348155
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	−9.00000	−1.42302
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	2.00000	0.301511
$45$	−3.00000	−0.447214
$46$	−1.00000	−0.147442
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	1.00000	0.140028
$52$	1.00000	0.138675
$53$	−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
$55$	6.00000	0.809040
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	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	$0.540868\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	7.00000	0.875000
$65$	3.00000	0.372104
$66$	2.00000	0.246183
$67$	−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
$69$	1.00000	0.120386
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	3.00000	0.353553
$73$	−13.0000	−1.52153	−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000	−1.52153	−0.760767	+	0.649025i	$0.775177\pi$
$74$	2.00000	0.232495
$75$	4.00000	0.461880
$76$	0	0
$77$	0	0
$78$	1.00000	0.113228
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−6.00000	−0.639602
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	−1.00000	−0.104257
$93$	10.0000	1.03695
$94$	−7.00000	−0.721995
$95$	0	0
$96$	−5.00000	−0.510310
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	−4.00000	−0.400000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	−1.00000	−0.0990148
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	9.00000	0.874157
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−1.00000	−0.0962250
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	−6.00000	−0.572078
$111$	−2.00000	−0.189832
$112$	0	0
$113$	7.00000	0.658505	0.329252	−	0.944242i	$-0.393203\pi$
$113$	7.00000	0.658505	0.329252	+	0.944242i	$0.393203\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	4.00000	0.368230
$119$	0	0
$120$	−9.00000	−0.821584
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	4.00000	0.360668
$124$	−10.0000	−0.898027
$125$	3.00000	0.268328
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	3.00000	0.265165
$129$	4.00000	0.352180
$130$	−3.00000	−0.263117
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	3.00000	0.259161
$135$	−3.00000	−0.258199
$136$	3.00000	0.257248
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	−1.00000	−0.0851257
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	7.00000	0.589506
$142$	15.0000	1.25877
$143$	2.00000	0.167248
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	13.0000	1.07589
$147$	0	0
$148$	2.00000	0.164399
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	−4.00000	−0.326599
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−30.0000	−2.40966
$156$	1.00000	0.0800641
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−4.00000	−0.318223
$159$	−9.00000	−0.713746
$160$	15.0000	1.18585
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	−4.00000	−0.312348
$165$	6.00000	0.467099
$166$	−6.00000	−0.465690
$167$	−19.0000	−1.47026	−0.735132	−	0.677924i	$-0.762880\pi$
$167$	−19.0000	−1.47026	−0.735132	+	0.677924i	$0.762880\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	3.00000	0.230089
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	−4.00000	−0.300658
$178$	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	$0.634852\pi$
$180$	3.00000	0.223607
$181$	−26.0000	−1.93256	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000	−1.93256	−0.966282	+	0.257485i	$0.917106\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	3.00000	0.221163
$185$	6.00000	0.441129
$186$	−10.0000	−0.733236
$187$	−2.00000	−0.146254
$188$	−7.00000	−0.510527
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	7.00000	0.505181
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	−16.0000	−1.14873
$195$	3.00000	0.214834
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	2.00000	0.142134
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	12.0000	0.848528
$201$	−3.00000	−0.211604
$202$	18.0000	1.26648
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	−12.0000	−0.838116
$206$	−11.0000	−0.766406
$207$	1.00000	0.0695048
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	9.00000	0.618123
$213$	−15.0000	−1.02778
$214$	−4.00000	−0.273434
$215$	−12.0000	−0.818393
$216$	3.00000	0.204124
$217$	0	0
$218$	−8.00000	−0.541828
$219$	−13.0000	−0.878459
$220$	−6.00000	−0.404520
$221$	−1.00000	−0.0672673
$222$	2.00000	0.134231
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	−7.00000	−0.465633
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	28.0000	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	1.00000	0.0653720
$235$	−21.0000	−1.36989
$236$	4.00000	0.260378
$237$	4.00000	0.259828
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	3.00000	0.193649
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	−4.00000	−0.255031
$247$	0	0
$248$	30.0000	1.90500
$249$	6.00000	0.380235
$250$	−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
$253$	−2.00000	−0.125739
$254$	−2.00000	−0.125491
$255$	−3.00000	−0.187867
$256$	−17.0000	−1.06250
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	−3.00000	−0.186052
$261$	0	0
$262$	3.00000	0.185341
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	−6.00000	−0.369274
$265$	27.0000	1.65860
$266$	0	0
$267$	−6.00000	−0.367194
$268$	3.00000	0.183254
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	3.00000	0.182574
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	17.0000	1.02701
$275$	−8.00000	−0.482418
$276$	−1.00000	−0.0601929
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	10.0000	0.599760
$279$	10.0000	0.598684
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	−7.00000	−0.416844
$283$	31.0000	1.84276	0.921379	−	0.388664i	$-0.127063\pi$
$283$	31.0000	1.84276	0.921379	+	0.388664i	$0.127063\pi$
$284$	15.0000	0.890086
$285$	0	0
$286$	−2.00000	−0.118262
$287$	0	0
$288$	−5.00000	−0.294628
$289$	−16.0000	−0.941176
$290$	0	0
$291$	16.0000	0.937937
$292$	13.0000	0.760767
$293$	−13.0000	−0.759468	−0.379734	−	0.925096i	$-0.623985\pi$
$293$	−13.0000	−0.759468	−0.379734	+	0.925096i	$0.623985\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	−6.00000	−0.348743
$297$	−2.00000	−0.116052
$298$	−5.00000	−0.289642
$299$	−1.00000	−0.0578315
$300$	−4.00000	−0.230940
$301$	0	0
$302$	−4.00000	−0.230174
$303$	−18.0000	−1.03407
$304$	0	0
$305$	6.00000	0.343559
$306$	−1.00000	−0.0571662
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	11.0000	0.625768
$310$	30.0000	1.70389
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	−3.00000	−0.169842
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	9.00000	0.504695
$319$	0	0
$320$	−21.0000	−1.17394
$321$	4.00000	0.223258
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	−4.00000	−0.221880
$326$	20.0000	1.10770
$327$	8.00000	0.442401
$328$	12.0000	0.662589
$329$	0	0
$330$	−6.00000	−0.330289
$331$	−34.0000	−1.86881	−0.934405	−	0.356214i	$-0.884068\pi$
$331$	−34.0000	−1.86881	−0.934405	+	0.356214i	$0.884068\pi$
$332$	−6.00000	−0.329293
$333$	−2.00000	−0.109599
$334$	19.0000	1.03963
$335$	9.00000	0.491723
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	12.0000	0.652714
$339$	7.00000	0.380188
$340$	3.00000	0.162698
$341$	−20.0000	−1.08306
$342$	0	0
$343$	0	0
$344$	12.0000	0.646997
$345$	−3.00000	−0.161515
$346$	16.0000	0.860165
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	0	0
$349$	−5.00000	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	10.0000	0.533002
$353$	−4.00000	−0.212899	−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000	−0.212899	−0.106449	+	0.994318i	$0.533948\pi$
$354$	4.00000	0.212598
$355$	45.0000	2.38835
$356$	6.00000	0.317999
$357$	0	0
$358$	11.0000	0.581368
$359$	22.0000	1.16112	0.580558	−	0.814219i	$-0.302835\pi$
$359$	22.0000	1.16112	0.580558	+	0.814219i	$0.302835\pi$
$360$	−9.00000	−0.474342
$361$	−19.0000	−1.00000
$362$	26.0000	1.36653
$363$	−7.00000	−0.367405
$364$	0	0
$365$	39.0000	2.04135
$366$	2.00000	0.104542
$367$	−19.0000	−0.991792	−0.495896	−	0.868382i	$-0.665160\pi$
$367$	−19.0000	−0.991792	−0.495896	+	0.868382i	$0.665160\pi$
$368$	−1.00000	−0.0521286
$369$	4.00000	0.208232
$370$	−6.00000	−0.311925
$371$	0	0
$372$	−10.0000	−0.518476
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	2.00000	0.103418
$375$	3.00000	0.154919
$376$	21.0000	1.08299
$377$	0	0
$378$	0	0
$379$	−19.0000	−0.975964	−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000	−0.975964	−0.487982	+	0.872854i	$0.662267\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	11.0000	0.559885
$387$	4.00000	0.203331
$388$	−16.0000	−0.812277
$389$	−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
$391$	1.00000	0.0505722
$392$	0	0
$393$	−3.00000	−0.151330
$394$	18.0000	0.906827
$395$	−12.0000	−0.603786
$396$	2.00000	0.100504
$397$	21.0000	1.05396	0.526980	−	0.849878i	$-0.323324\pi$
$397$	21.0000	1.05396	0.526980	+	0.849878i	$0.323324\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−11.0000	−0.549314	−0.274657	−	0.961542i	$-0.588564\pi$
$401$	−11.0000	−0.549314	−0.274657	+	0.961542i	$0.588564\pi$
$402$	3.00000	0.149626
$403$	−10.0000	−0.498135
$404$	18.0000	0.895533
$405$	−3.00000	−0.149071
$406$	0	0
$407$	4.00000	0.198273
$408$	3.00000	0.148522
$409$	31.0000	1.53285	0.766426	−	0.642333i	$-0.222033\pi$
$409$	31.0000	1.53285	0.766426	+	0.642333i	$0.222033\pi$
$410$	12.0000	0.592638
$411$	−17.0000	−0.838548
$412$	−11.0000	−0.541931
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	−18.0000	−0.883585
$416$	5.00000	0.245145
$417$	−10.0000	−0.489702
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	22.0000	1.07094
$423$	7.00000	0.340352
$424$	−27.0000	−1.31124
$425$	4.00000	0.194029
$426$	15.0000	0.726752
$427$	0	0
$428$	−4.00000	−0.193347
$429$	2.00000	0.0965609
$430$	12.0000	0.578691
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	−1.00000	−0.0481125
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	0	0
$435$	0	0
$436$	−8.00000	−0.383131
$437$	0	0
$438$	13.0000	0.621164
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	18.0000	0.858116
$441$	0	0
$442$	1.00000	0.0475651
$443$	−11.0000	−0.522626	−0.261313	−	0.965254i	$-0.584155\pi$
$443$	−11.0000	−0.522626	−0.261313	+	0.965254i	$0.584155\pi$
$444$	2.00000	0.0949158
$445$	18.0000	0.853282
$446$	−4.00000	−0.189405
$447$	5.00000	0.236492
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	−4.00000	−0.188562
$451$	−8.00000	−0.376705
$452$	−7.00000	−0.329252
$453$	4.00000	0.187936
$454$	14.0000	0.657053
$455$	0	0
$456$	0	0
$457$	16.0000	0.748448	0.374224	−	0.927338i	$-0.377909\pi$
$457$	16.0000	0.748448	0.374224	+	0.927338i	$0.377909\pi$
$458$	−28.0000	−1.30835
$459$	1.00000	0.0466760
$460$	3.00000	0.139876
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	0	0
$465$	−30.0000	−1.39122
$466$	24.0000	1.11178
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	21.0000	0.968658
$471$	10.0000	0.460776
$472$	−12.0000	−0.552345
$473$	−8.00000	−0.367840
$474$	−4.00000	−0.183726
$475$	0	0
$476$	0	0
$477$	−9.00000	−0.412082
$478$	16.0000	0.731823
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	15.0000	0.684653
$481$	2.00000	0.0911922
$482$	−6.00000	−0.273293
$483$	0	0
$484$	7.00000	0.318182
$485$	−48.0000	−2.17957
$486$	−1.00000	−0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	−6.00000	−0.271607
$489$	−20.0000	−0.904431
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	−4.00000	−0.180334
$493$	0	0
$494$	0	0
$495$	6.00000	0.269680
$496$	−10.0000	−0.449013
$497$	0	0
$498$	−6.00000	−0.268866
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	−3.00000	−0.134164
$501$	−19.0000	−0.848857
$502$	−10.0000	−0.446322
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	54.0000	2.40297
$506$	2.00000	0.0889108
$507$	−12.0000	−0.532939
$508$	−2.00000	−0.0887357
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	3.00000	0.132842
$511$	0	0
$512$	11.0000	0.486136
$513$	0	0
$514$	18.0000	0.793946
$515$	−33.0000	−1.45415
$516$	−4.00000	−0.176090
$517$	−14.0000	−0.615719
$518$	0	0
$519$	−16.0000	−0.702322
$520$	9.00000	0.394676
$521$	−29.0000	−1.27051	−0.635257	−	0.772301i	$-0.719106\pi$
$521$	−29.0000	−1.27051	−0.635257	+	0.772301i	$0.719106\pi$
$522$	0	0
$523$	17.0000	0.743358	0.371679	−	0.928361i	$-0.378782\pi$
$523$	17.0000	0.743358	0.371679	+	0.928361i	$0.378782\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	16.0000	0.697633
$527$	10.0000	0.435607
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	−27.0000	−1.17281
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−4.00000	−0.173259
$534$	6.00000	0.259645
$535$	−12.0000	−0.518805
$536$	−9.00000	−0.388741
$537$	−11.0000	−0.474685
$538$	−4.00000	−0.172452
$539$	0	0
$540$	3.00000	0.129099
$541$	5.00000	0.214967	0.107483	−	0.994207i	$-0.465721\pi$
$541$	5.00000	0.214967	0.107483	+	0.994207i	$0.465721\pi$
$542$	−14.0000	−0.601351
$543$	−26.0000	−1.11577
$544$	−5.00000	−0.214373
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	17.0000	0.726204
$549$	−2.00000	−0.0853579
$550$	8.00000	0.341121
$551$	0	0
$552$	3.00000	0.127688
$553$	0	0
$554$	−13.0000	−0.552317
$555$	6.00000	0.254686
$556$	10.0000	0.424094
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	−10.0000	−0.423334
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	−27.0000	−1.13893
$563$	40.0000	1.68580	0.842900	−	0.538071i	$-0.180847\pi$
$563$	40.0000	1.68580	0.842900	+	0.538071i	$0.180847\pi$
$564$	−7.00000	−0.294753
$565$	−21.0000	−0.883477
$566$	−31.0000	−1.30303
$567$	0	0
$568$	−45.0000	−1.88816
$569$	33.0000	1.38343	0.691716	−	0.722170i	$-0.256855\pi$
$569$	33.0000	1.38343	0.691716	+	0.722170i	$0.256855\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	0	0
$575$	4.00000	0.166812
$576$	7.00000	0.291667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	16.0000	0.665512
$579$	−11.0000	−0.457144
$580$	0	0
$581$	0	0
$582$	−16.0000	−0.663221
$583$	18.0000	0.745484
$584$	−39.0000	−1.61383
$585$	3.00000	0.124035
$586$	13.0000	0.537025
$587$	−37.0000	−1.52715	−0.763577	−	0.645717i	$-0.776559\pi$
$587$	−37.0000	−1.52715	−0.763577	+	0.645717i	$0.776559\pi$
$588$	0	0
$589$	0	0
$590$	−12.0000	−0.494032
$591$	−18.0000	−0.740421
$592$	2.00000	0.0821995
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	−5.00000	−0.204808
$597$	16.0000	0.654836
$598$	1.00000	0.0408930
$599$	39.0000	1.59350	0.796748	−	0.604311i	$-0.206552\pi$
$599$	39.0000	1.59350	0.796748	+	0.604311i	$0.206552\pi$
$600$	12.0000	0.489898
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	0	0
$603$	−3.00000	−0.122169
$604$	−4.00000	−0.162758
$605$	21.0000	0.853771
$606$	18.0000	0.731200
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	0	0
$609$	0	0
$610$	−6.00000	−0.242933
$611$	−7.00000	−0.283190
$612$	−1.00000	−0.0404226
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	20.0000	0.807134
$615$	−12.0000	−0.483887
$616$	0	0
$617$	−29.0000	−1.16750	−0.583748	−	0.811935i	$-0.698414\pi$
$617$	−29.0000	−1.16750	−0.583748	+	0.811935i	$0.698414\pi$
$618$	−11.0000	−0.442485
$619$	−27.0000	−1.08522	−0.542611	−	0.839984i	$-0.682564\pi$
$619$	−27.0000	−1.08522	−0.542611	+	0.839984i	$0.682564\pi$
$620$	30.0000	1.20483
$621$	1.00000	0.0401286
$622$	−21.0000	−0.842023
$623$	0	0
$624$	1.00000	0.0400320
$625$	−29.0000	−1.16000
$626$	8.00000	0.319744
$627$	0	0
$628$	−10.0000	−0.399043
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−5.00000	−0.199047	−0.0995234	−	0.995035i	$-0.531732\pi$
$631$	−5.00000	−0.199047	−0.0995234	+	0.995035i	$0.531732\pi$
$632$	12.0000	0.477334
$633$	−22.0000	−0.874421
$634$	−10.0000	−0.397151
$635$	−6.00000	−0.238103
$636$	9.00000	0.356873
$637$	0	0
$638$	0	0
$639$	−15.0000	−0.593391
$640$	−9.00000	−0.355756
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	−4.00000	−0.157867
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	3.00000	0.117851
$649$	8.00000	0.314027
$650$	4.00000	0.156893
$651$	0	0
$652$	20.0000	0.783260
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−8.00000	−0.312825
$655$	9.00000	0.351659
$656$	−4.00000	−0.156174
$657$	−13.0000	−0.507178
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	−6.00000	−0.233550
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	34.0000	1.32145
$663$	−1.00000	−0.0388368
$664$	18.0000	0.698535
$665$	0	0
$666$	2.00000	0.0774984
$667$	0	0
$668$	19.0000	0.735132
$669$	4.00000	0.154649
$670$	−9.00000	−0.347700
$671$	4.00000	0.154418
$672$	0	0
$673$	50.0000	1.92736	0.963679	−	0.267063i	$-0.0860531\pi$
$673$	50.0000	1.92736	0.963679	+	0.267063i	$0.0860531\pi$
$674$	12.0000	0.462223
$675$	4.00000	0.153960
$676$	12.0000	0.461538
$677$	−11.0000	−0.422764	−0.211382	−	0.977403i	$-0.567796\pi$
$677$	−11.0000	−0.422764	−0.211382	+	0.977403i	$0.567796\pi$
$678$	−7.00000	−0.268833
$679$	0	0
$680$	−9.00000	−0.345134
$681$	−14.0000	−0.536481
$682$	20.0000	0.765840
$683$	41.0000	1.56882	0.784411	−	0.620242i	$-0.212966\pi$
$683$	41.0000	1.56882	0.784411	+	0.620242i	$0.212966\pi$
$684$	0	0
$685$	51.0000	1.94861
$686$	0	0
$687$	28.0000	1.06827
$688$	−4.00000	−0.152499
$689$	9.00000	0.342873
$690$	3.00000	0.114208
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	16.0000	0.608229
$693$	0	0
$694$	−3.00000	−0.113878
$695$	30.0000	1.13796
$696$	0	0
$697$	4.00000	0.151511
$698$	5.00000	0.189253
$699$	−24.0000	−0.907763
$700$	0	0
$701$	−39.0000	−1.47301	−0.736505	−	0.676432i	$-0.763525\pi$
$701$	−39.0000	−1.47301	−0.736505	+	0.676432i	$0.763525\pi$
$702$	1.00000	0.0377426
$703$	0	0
$704$	−14.0000	−0.527645
$705$	−21.0000	−0.790906
$706$	4.00000	0.150542
$707$	0	0
$708$	4.00000	0.150329
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	−45.0000	−1.68882
$711$	4.00000	0.150012
$712$	−18.0000	−0.674579
$713$	10.0000	0.374503
$714$	0	0
$715$	−6.00000	−0.224387
$716$	11.0000	0.411089
$717$	−16.0000	−0.597531
$718$	−22.0000	−0.821033
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	19.0000	0.707107
$723$	6.00000	0.223142
$724$	26.0000	0.966282
$725$	0	0
$726$	7.00000	0.259794
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−39.0000	−1.44345
$731$	4.00000	0.147945
$732$	2.00000	0.0739221
$733$	−24.0000	−0.886460	−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000	−0.886460	−0.443230	+	0.896408i	$0.646168\pi$
$734$	19.0000	0.701303
$735$	0	0
$736$	−5.00000	−0.184302
$737$	6.00000	0.221013
$738$	−4.00000	−0.147242
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	0	0
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	30.0000	1.09985
$745$	−15.0000	−0.549557
$746$	26.0000	0.951928
$747$	6.00000	0.219529
$748$	2.00000	0.0731272
$749$	0	0
$750$	−3.00000	−0.109545
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−7.00000	−0.255264
$753$	10.0000	0.364420
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	12.0000	0.436147	0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000	0.436147	0.218074	+	0.975932i	$0.430023\pi$
$758$	19.0000	0.690111
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	−2.00000	−0.0724524
$763$	0	0
$764$	0	0
$765$	−3.00000	−0.108465
$766$	6.00000	0.216789
$767$	4.00000	0.144432
$768$	−17.0000	−0.613435
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	11.0000	0.395899
$773$	9.00000	0.323708	0.161854	−	0.986815i	$-0.448253\pi$
$773$	9.00000	0.323708	0.161854	+	0.986815i	$0.448253\pi$
$774$	−4.00000	−0.143777
$775$	40.0000	1.43684
$776$	48.0000	1.72310
$777$	0	0
$778$	6.00000	0.215110
$779$	0	0
$780$	−3.00000	−0.107417
$781$	30.0000	1.07348
$782$	−1.00000	−0.0357599
$783$	0	0
$784$	0	0
$785$	−30.0000	−1.07075
$786$	3.00000	0.107006
$787$	51.0000	1.81795	0.908977	−	0.416847i	$-0.136865\pi$
$787$	51.0000	1.81795	0.908977	+	0.416847i	$0.136865\pi$
$788$	18.0000	0.641223
$789$	−16.0000	−0.569615
$790$	12.0000	0.426941
$791$	0	0
$792$	−6.00000	−0.213201
$793$	2.00000	0.0710221
$794$	−21.0000	−0.745262
$795$	27.0000	0.957591
$796$	−16.0000	−0.567105
$797$	19.0000	0.673015	0.336507	−	0.941681i	$-0.390754\pi$
$797$	19.0000	0.673015	0.336507	+	0.941681i	$0.390754\pi$
$798$	0	0
$799$	7.00000	0.247642
$800$	−20.0000	−0.707107
$801$	−6.00000	−0.212000
$802$	11.0000	0.388424
$803$	26.0000	0.917520
$804$	3.00000	0.105802
$805$	0	0
$806$	10.0000	0.352235
$807$	4.00000	0.140807
$808$	−54.0000	−1.89971
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	3.00000	0.105409
$811$	−30.0000	−1.05344	−0.526721	−	0.850038i	$-0.676579\pi$
$811$	−30.0000	−1.05344	−0.526721	+	0.850038i	$0.676579\pi$
$812$	0	0
$813$	14.0000	0.491001
$814$	−4.00000	−0.140200
$815$	60.0000	2.10171
$816$	−1.00000	−0.0350070
$817$	0	0
$818$	−31.0000	−1.08389
$819$	0	0
$820$	12.0000	0.419058
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	17.0000	0.592943
$823$	−26.0000	−0.906303	−0.453152	−	0.891434i	$-0.649700\pi$
$823$	−26.0000	−0.906303	−0.453152	+	0.891434i	$0.649700\pi$
$824$	33.0000	1.14961
$825$	−8.00000	−0.278524
$826$	0	0
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	−1.00000	−0.0347524
$829$	−3.00000	−0.104194	−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000	−0.104194	−0.0520972	+	0.998642i	$0.516591\pi$
$830$	18.0000	0.624789
$831$	13.0000	0.450965
$832$	−7.00000	−0.242681
$833$	0	0
$834$	10.0000	0.346272
$835$	57.0000	1.97257
$836$	0	0
$837$	10.0000	0.345651
$838$	−12.0000	−0.414533
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−8.00000	−0.275698
$843$	27.0000	0.929929
$844$	22.0000	0.757271
$845$	36.0000	1.23844
$846$	−7.00000	−0.240665
$847$	0	0
$848$	9.00000	0.309061
$849$	31.0000	1.06392
$850$	−4.00000	−0.137199
$851$	−2.00000	−0.0685591
$852$	15.0000	0.513892
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	−46.0000	−1.57133	−0.785665	−	0.618652i	$-0.787679\pi$
$857$	−46.0000	−1.57133	−0.785665	+	0.618652i	$0.787679\pi$
$858$	−2.00000	−0.0682789
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	12.0000	0.409197
$861$	0	0
$862$	32.0000	1.08992
$863$	−21.0000	−0.714848	−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714848	−0.357424	+	0.933942i	$0.616345\pi$
$864$	−5.00000	−0.170103
$865$	48.0000	1.63205
$866$	22.0000	0.747590
$867$	−16.0000	−0.543388
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	3.00000	0.101651
$872$	24.0000	0.812743
$873$	16.0000	0.541518
$874$	0	0
$875$	0	0
$876$	13.0000	0.439229
$877$	19.0000	0.641584	0.320792	−	0.947150i	$-0.396051\pi$
$877$	19.0000	0.641584	0.320792	+	0.947150i	$0.396051\pi$
$878$	−22.0000	−0.742464
$879$	−13.0000	−0.438479
$880$	−6.00000	−0.202260
$881$	−9.00000	−0.303218	−0.151609	−	0.988441i	$-0.548445\pi$
$881$	−9.00000	−0.303218	−0.151609	+	0.988441i	$0.548445\pi$
$882$	0	0
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	1.00000	0.0336336
$885$	12.0000	0.403376
$886$	11.0000	0.369552
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	−6.00000	−0.201347
$889$	0	0
$890$	−18.0000	−0.603361
$891$	−2.00000	−0.0670025
$892$	−4.00000	−0.133930
$893$	0	0
$894$	−5.00000	−0.167225
$895$	33.0000	1.10307
$896$	0	0
$897$	−1.00000	−0.0333890
$898$	−14.0000	−0.467186
$899$	0	0
$900$	−4.00000	−0.133333
$901$	−9.00000	−0.299833
$902$	8.00000	0.266371
$903$	0	0
$904$	21.0000	0.698450
$905$	78.0000	2.59281
$906$	−4.00000	−0.132891
$907$	−21.0000	−0.697294	−0.348647	−	0.937254i	$-0.613359\pi$
$907$	−21.0000	−0.697294	−0.348647	+	0.937254i	$0.613359\pi$
$908$	14.0000	0.464606
$909$	−18.0000	−0.597022
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	−16.0000	−0.529233
$915$	6.00000	0.198354
$916$	−28.0000	−0.925146
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	−49.0000	−1.61636	−0.808180	−	0.588935i	$-0.799547\pi$
$919$	−49.0000	−1.61636	−0.808180	+	0.588935i	$0.799547\pi$
$920$	−9.00000	−0.296721
$921$	−20.0000	−0.659022
$922$	42.0000	1.38320
$923$	15.0000	0.493731
$924$	0	0
$925$	−8.00000	−0.263038
$926$	−40.0000	−1.31448
$927$	11.0000	0.361287
$928$	0	0
$929$	24.0000	0.787414	0.393707	−	0.919236i	$-0.371192\pi$
$929$	24.0000	0.787414	0.393707	+	0.919236i	$0.371192\pi$
$930$	30.0000	0.983739
$931$	0	0
$932$	24.0000	0.786146
$933$	21.0000	0.687509
$934$	28.0000	0.916188
$935$	6.00000	0.196221
$936$	−3.00000	−0.0980581
$937$	−8.00000	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$937$	−8.00000	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$938$	0	0
$939$	−8.00000	−0.261070
$940$	21.0000	0.684944
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	−10.0000	−0.325818
$943$	4.00000	0.130258
$944$	4.00000	0.130189
$945$	0	0
$946$	8.00000	0.260102
$947$	47.0000	1.52729	0.763647	−	0.645634i	$-0.223407\pi$
$947$	47.0000	1.52729	0.763647	+	0.645634i	$0.223407\pi$
$948$	−4.00000	−0.129914
$949$	13.0000	0.421998
$950$	0	0
$951$	10.0000	0.324272
$952$	0	0
$953$	−14.0000	−0.453504	−0.226752	−	0.973952i	$-0.572811\pi$
$953$	−14.0000	−0.453504	−0.226752	+	0.973952i	$0.572811\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	−16.0000	−0.516937
$959$	0	0
$960$	−21.0000	−0.677772
$961$	69.0000	2.22581
$962$	−2.00000	−0.0644826
$963$	4.00000	0.128898
$964$	−6.00000	−0.193247
$965$	33.0000	1.06231
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	48.0000	1.54119
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	−4.00000	−0.128103
$976$	2.00000	0.0640184
$977$	39.0000	1.24772	0.623860	−	0.781536i	$-0.285563\pi$
$977$	39.0000	1.24772	0.623860	+	0.781536i	$0.285563\pi$
$978$	20.0000	0.639529
$979$	12.0000	0.383522
$980$	0	0
$981$	8.00000	0.255420
$982$	−9.00000	−0.287202
$983$	−14.0000	−0.446531	−0.223265	−	0.974758i	$-0.571672\pi$
$983$	−14.0000	−0.446531	−0.223265	+	0.974758i	$0.571672\pi$
$984$	12.0000	0.382546
$985$	54.0000	1.72058
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.00000	0.127193
$990$	−6.00000	−0.190693
$991$	−50.0000	−1.58830	−0.794151	−	0.607720i	$-0.792084\pi$
$991$	−50.0000	−1.58830	−0.794151	+	0.607720i	$0.792084\pi$
$992$	−50.0000	−1.58750
$993$	−34.0000	−1.07896
$994$	0	0
$995$	−48.0000	−1.52170
$996$	−6.00000	−0.190117
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	−36.0000	−1.13956
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.c.1.1		1
7.2	even	3		483.2.i.d.277.1	✓	2
7.4	even	3		483.2.i.d.415.1	yes	2
7.6	odd	2		3381.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.i.d.277.1	✓	2	7.2	even	3
483.2.i.d.415.1	yes	2	7.4	even	3
3381.2.a.b.1.1		1	7.6	odd	2
3381.2.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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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