LMFDB - Embedded newform 3381.2.a.d.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} +7.00000 q^{13} +1.00000 q^{15} -1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} +1.00000 q^{23} +3.00000 q^{24} -4.00000 q^{25} -7.00000 q^{26} +1.00000 q^{27} -8.00000 q^{29} -1.00000 q^{30} -6.00000 q^{31} -5.00000 q^{32} -2.00000 q^{33} +3.00000 q^{34} -1.00000 q^{36} +6.00000 q^{37} +8.00000 q^{38} +7.00000 q^{39} +3.00000 q^{40} -4.00000 q^{41} -4.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} -9.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} -3.00000 q^{51} -7.00000 q^{52} +3.00000 q^{53} -1.00000 q^{54} -2.00000 q^{55} -8.00000 q^{57} +8.00000 q^{58} +12.0000 q^{59} -1.00000 q^{60} -2.00000 q^{61} +6.00000 q^{62} +7.00000 q^{64} +7.00000 q^{65} +2.00000 q^{66} +1.00000 q^{67} +3.00000 q^{68} +1.00000 q^{69} +9.00000 q^{71} +3.00000 q^{72} +3.00000 q^{73} -6.00000 q^{74} -4.00000 q^{75} +8.00000 q^{76} -7.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -18.0000 q^{83} -3.00000 q^{85} +4.00000 q^{86} -8.00000 q^{87} -6.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} -1.00000 q^{92} -6.00000 q^{93} +9.00000 q^{94} -8.00000 q^{95} -5.00000 q^{96} -2.00000 q^{99} +O(q^{100})\)

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
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−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
$13$	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
$15$	1.00000	0.258199
$16$	−1.00000	−0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−1.00000	−0.235702
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−1.00000	−0.223607
$21$	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
$26$	−7.00000	−1.37281
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	−1.00000	−0.182574
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	−5.00000	−0.883883
$33$	−2.00000	−0.348155
$34$	3.00000	0.514496
$35$	0	0
$36$	−1.00000	−0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	8.00000	1.29777
$39$	7.00000	1.12090
$40$	3.00000	0.474342
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	2.00000	0.301511
$45$	1.00000	0.149071
$46$	−1.00000	−0.147442
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	4.00000	0.565685
$51$	−3.00000	−0.420084
$52$	−7.00000	−0.970725
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−1.00000	−0.136083
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−8.00000	−1.05963
$58$	8.00000	1.05045
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	−1.00000	−0.129099
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	7.00000	0.875000
$65$	7.00000	0.868243
$66$	2.00000	0.246183
$67$	1.00000	0.122169	0.0610847	−	0.998133i	$-0.480544\pi$
$67$	1.00000	0.122169	0.0610847	+	0.998133i	$0.480544\pi$
$68$	3.00000	0.363803
$69$	1.00000	0.120386
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	3.00000	0.353553
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	−6.00000	−0.697486
$75$	−4.00000	−0.461880
$76$	8.00000	0.917663
$77$	0	0
$78$	−7.00000	−0.792594
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	−18.0000	−1.97576	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	−18.0000	−1.97576	−0.987878	+	0.155230i	$0.950388\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	4.00000	0.431331
$87$	−8.00000	−0.857690
$88$	−6.00000	−0.639602
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−1.00000	−0.104257
$93$	−6.00000	−0.622171
$94$	9.00000	0.928279
$95$	−8.00000	−0.820783
$96$	−5.00000	−0.510310
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	4.00000	0.400000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	3.00000	0.297044
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	21.0000	2.05922
$105$	0	0
$106$	−3.00000	−0.291386
$107$	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
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	2.00000	0.190693
$111$	6.00000	0.569495
$112$	0	0
$113$	−5.00000	−0.470360	−0.235180	−	0.971952i	$-0.575568\pi$
$113$	−5.00000	−0.470360	−0.235180	+	0.971952i	$0.575568\pi$
$114$	8.00000	0.749269
$115$	1.00000	0.0932505
$116$	8.00000	0.742781
$117$	7.00000	0.647150
$118$	−12.0000	−1.10469
$119$	0	0
$120$	3.00000	0.273861
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	−4.00000	−0.360668
$124$	6.00000	0.538816
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	3.00000	0.265165
$129$	−4.00000	−0.352180
$130$	−7.00000	−0.613941
$131$	−11.0000	−0.961074	−0.480537	−	0.876974i	$-0.659558\pi$
$131$	−11.0000	−0.961074	−0.480537	+	0.876974i	$0.659558\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	−1.00000	−0.0863868
$135$	1.00000	0.0860663
$136$	−9.00000	−0.771744
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	−1.00000	−0.0851257
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	−9.00000	−0.755263
$143$	−14.0000	−1.17074
$144$	−1.00000	−0.0833333
$145$	−8.00000	−0.664364
$146$	−3.00000	−0.248282
$147$	0	0
$148$	−6.00000	−0.493197
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	4.00000	0.326599
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	−24.0000	−1.94666
$153$	−3.00000	−0.242536
$154$	0	0
$155$	−6.00000	−0.481932
$156$	−7.00000	−0.560449
$157$	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
$159$	3.00000	0.237915
$160$	−5.00000	−0.395285
$161$	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
$165$	−2.00000	−0.155700
$166$	18.0000	1.39707
$167$	−11.0000	−0.851206	−0.425603	−	0.904910i	$-0.639938\pi$
$167$	−11.0000	−0.851206	−0.425603	+	0.904910i	$0.639938\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	3.00000	0.230089
$171$	−8.00000	−0.611775
$172$	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	$0.708120\pi$
$174$	8.00000	0.606478
$175$	0	0
$176$	2.00000	0.150756
$177$	12.0000	0.901975
$178$	−10.0000	−0.749532
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	−1.00000	−0.0745356
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	3.00000	0.221163
$185$	6.00000	0.441129
$186$	6.00000	0.439941
$187$	6.00000	0.438763
$188$	9.00000	0.656392
$189$	0	0
$190$	8.00000	0.580381
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	7.00000	0.505181
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	0	0
$195$	7.00000	0.501280
$196$	0	0
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	2.00000	0.142134
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	−12.0000	−0.848528
$201$	1.00000	0.0705346
$202$	10.0000	0.703598
$203$	0	0
$204$	3.00000	0.210042
$205$	−4.00000	−0.279372
$206$	1.00000	0.0696733
$207$	1.00000	0.0695048
$208$	−7.00000	−0.485363
$209$	16.0000	1.10674
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	−3.00000	−0.206041
$213$	9.00000	0.616670
$214$	−4.00000	−0.273434
$215$	−4.00000	−0.272798
$216$	3.00000	0.204124
$217$	0	0
$218$	0	0
$219$	3.00000	0.202721
$220$	2.00000	0.134840
$221$	−21.0000	−1.41261
$222$	−6.00000	−0.402694
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	5.00000	0.332595
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	8.00000	0.529813
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−24.0000	−1.57568
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	−7.00000	−0.457604
$235$	−9.00000	−0.587095
$236$	−12.0000	−0.781133
$237$	−4.00000	−0.259828
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−1.00000	−0.0645497
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	4.00000	0.255031
$247$	−56.0000	−3.56319
$248$	−18.0000	−1.14300
$249$	−18.0000	−1.14070
$250$	9.00000	0.569210
$251$	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
$254$	6.00000	0.376473
$255$	−3.00000	−0.187867
$256$	−17.0000	−1.06250
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	−7.00000	−0.434122
$261$	−8.00000	−0.495188
$262$	11.0000	0.679582
$263$	−32.0000	−1.97320	−0.986602	−	0.163144i	$-0.947836\pi$
$263$	−32.0000	−1.97320	−0.986602	+	0.163144i	$0.947836\pi$
$264$	−6.00000	−0.369274
$265$	3.00000	0.184289
$266$	0	0
$267$	10.0000	0.611990
$268$	−1.00000	−0.0610847
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	−1.00000	−0.0608581
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−3.00000	−0.181237
$275$	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	$0.372286\pi$
$278$	−6.00000	−0.359856
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−25.0000	−1.49137	−0.745687	−	0.666296i	$-0.767879\pi$
$281$	−25.0000	−1.49137	−0.745687	+	0.666296i	$0.767879\pi$
$282$	9.00000	0.535942
$283$	19.0000	1.12943	0.564716	−	0.825285i	$-0.308986\pi$
$283$	19.0000	1.12943	0.564716	+	0.825285i	$0.308986\pi$
$284$	−9.00000	−0.534052
$285$	−8.00000	−0.473879
$286$	14.0000	0.827837
$287$	0	0
$288$	−5.00000	−0.294628
$289$	−8.00000	−0.470588
$290$	8.00000	0.469776
$291$	0	0
$292$	−3.00000	−0.175562
$293$	23.0000	1.34367	0.671837	−	0.740699i	$-0.265505\pi$
$293$	23.0000	1.34367	0.671837	+	0.740699i	$0.265505\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	18.0000	1.04623
$297$	−2.00000	−0.116052
$298$	15.0000	0.868927
$299$	7.00000	0.404820
$300$	4.00000	0.230940
$301$	0	0
$302$	4.00000	0.230174
$303$	−10.0000	−0.574485
$304$	8.00000	0.458831
$305$	−2.00000	−0.114520
$306$	3.00000	0.171499
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	6.00000	0.340777
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	21.0000	1.18889
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	4.00000	0.225018
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−3.00000	−0.168232
$319$	16.0000	0.895828
$320$	7.00000	0.391312
$321$	4.00000	0.223258
$322$	0	0
$323$	24.0000	1.33540
$324$	−1.00000	−0.0555556
$325$	−28.0000	−1.55316
$326$	20.0000	1.10770
$327$	0	0
$328$	−12.0000	−0.662589
$329$	0	0
$330$	2.00000	0.110096
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	18.0000	0.987878
$333$	6.00000	0.328798
$334$	11.0000	0.601893
$335$	1.00000	0.0546358
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	−36.0000	−1.95814
$339$	−5.00000	−0.271563
$340$	3.00000	0.162698
$341$	12.0000	0.649836
$342$	8.00000	0.432590
$343$	0	0
$344$	−12.0000	−0.646997
$345$	1.00000	0.0538382
$346$	16.0000	0.860165
$347$	−13.0000	−0.697877	−0.348938	−	0.937146i	$-0.613458\pi$
$347$	−13.0000	−0.697877	−0.348938	+	0.937146i	$0.613458\pi$
$348$	8.00000	0.428845
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	7.00000	0.373632
$352$	10.0000	0.533002
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	−12.0000	−0.637793
$355$	9.00000	0.477670
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−5.00000	−0.264258
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	3.00000	0.158114
$361$	45.0000	2.36842
$362$	−22.0000	−1.15629
$363$	−7.00000	−0.367405
$364$	0	0
$365$	3.00000	0.157027
$366$	2.00000	0.104542
$367$	−15.0000	−0.782994	−0.391497	−	0.920179i	$-0.628043\pi$
$367$	−15.0000	−0.782994	−0.391497	+	0.920179i	$0.628043\pi$
$368$	−1.00000	−0.0521286
$369$	−4.00000	−0.208232
$370$	−6.00000	−0.311925
$371$	0	0
$372$	6.00000	0.311086
$373$	38.0000	1.96757	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	38.0000	1.96757	0.983783	+	0.179364i	$0.0574041\pi$
$374$	−6.00000	−0.310253
$375$	−9.00000	−0.464758
$376$	−27.0000	−1.39242
$377$	−56.0000	−2.88415
$378$	0	0
$379$	9.00000	0.462299	0.231149	−	0.972918i	$-0.425751\pi$
$379$	9.00000	0.462299	0.231149	+	0.972918i	$0.425751\pi$
$380$	8.00000	0.410391
$381$	−6.00000	−0.307389
$382$	8.00000	0.409316
$383$	−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
$386$	19.0000	0.967075
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−7.00000	−0.354459
$391$	−3.00000	−0.151717
$392$	0	0
$393$	−11.0000	−0.554877
$394$	26.0000	1.30986
$395$	−4.00000	−0.201262
$396$	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	$0.323324\pi$
$398$	−24.0000	−1.20301
$399$	0	0
$400$	4.00000	0.200000
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	−1.00000	−0.0498755
$403$	−42.0000	−2.09217
$404$	10.0000	0.497519
$405$	1.00000	0.0496904
$406$	0	0
$407$	−12.0000	−0.594818
$408$	−9.00000	−0.445566
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	4.00000	0.197546
$411$	3.00000	0.147979
$412$	1.00000	0.0492665
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	−18.0000	−0.883585
$416$	−35.0000	−1.71602
$417$	6.00000	0.293821
$418$	−16.0000	−0.782586
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	14.0000	0.681509
$423$	−9.00000	−0.437595
$424$	9.00000	0.437079
$425$	12.0000	0.582086
$426$	−9.00000	−0.436051
$427$	0	0
$428$	−4.00000	−0.193347
$429$	−14.0000	−0.675926
$430$	4.00000	0.192897
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.00000	−0.0481125
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	0	0
$437$	−8.00000	−0.382692
$438$	−3.00000	−0.143346
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	−6.00000	−0.286039
$441$	0	0
$442$	21.0000	0.998868
$443$	−3.00000	−0.142534	−0.0712672	−	0.997457i	$-0.522704\pi$
$443$	−3.00000	−0.142534	−0.0712672	+	0.997457i	$0.522704\pi$
$444$	−6.00000	−0.284747
$445$	10.0000	0.474045
$446$	20.0000	0.947027
$447$	−15.0000	−0.709476
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	4.00000	0.188562
$451$	8.00000	0.376705
$452$	5.00000	0.235180
$453$	−4.00000	−0.187936
$454$	−18.0000	−0.844782
$455$	0	0
$456$	−24.0000	−1.12390
$457$	16.0000	0.748448	0.374224	−	0.927338i	$-0.377909\pi$
$457$	16.0000	0.748448	0.374224	+	0.927338i	$0.377909\pi$
$458$	−20.0000	−0.934539
$459$	−3.00000	−0.140028
$460$	−1.00000	−0.0466252
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	8.00000	0.371391
$465$	−6.00000	−0.278243
$466$	−24.0000	−1.11178
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−7.00000	−0.323575
$469$	0	0
$470$	9.00000	0.415139
$471$	10.0000	0.460776
$472$	36.0000	1.65703
$473$	8.00000	0.367840
$474$	4.00000	0.183726
$475$	32.0000	1.46826
$476$	0	0
$477$	3.00000	0.137361
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	−5.00000	−0.228218
$481$	42.0000	1.91504
$482$	−14.0000	−0.637683
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−6.00000	−0.271607
$489$	−20.0000	−0.904431
$490$	0	0
$491$	−7.00000	−0.315906	−0.157953	−	0.987447i	$-0.550489\pi$
$491$	−7.00000	−0.315906	−0.157953	+	0.987447i	$0.550489\pi$
$492$	4.00000	0.180334
$493$	24.0000	1.08091
$494$	56.0000	2.51956
$495$	−2.00000	−0.0898933
$496$	6.00000	0.269408
$497$	0	0
$498$	18.0000	0.806599
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	9.00000	0.402492
$501$	−11.0000	−0.491444
$502$	−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
$505$	−10.0000	−0.444994
$506$	2.00000	0.0889108
$507$	36.0000	1.59882
$508$	6.00000	0.266207
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	3.00000	0.132842
$511$	0	0
$512$	11.0000	0.486136
$513$	−8.00000	−0.353209
$514$	10.0000	0.441081
$515$	−1.00000	−0.0440653
$516$	4.00000	0.176090
$517$	18.0000	0.791639
$518$	0	0
$519$	−16.0000	−0.702322
$520$	21.0000	0.920911
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	8.00000	0.350150
$523$	37.0000	1.61790	0.808949	−	0.587879i	$-0.200037\pi$
$523$	37.0000	1.61790	0.808949	+	0.587879i	$0.200037\pi$
$524$	11.0000	0.480537
$525$	0	0
$526$	32.0000	1.39527
$527$	18.0000	0.784092
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	−3.00000	−0.130312
$531$	12.0000	0.520756
$532$	0	0
$533$	−28.0000	−1.21281
$534$	−10.0000	−0.432742
$535$	4.00000	0.172935
$536$	3.00000	0.129580
$537$	5.00000	0.215766
$538$	12.0000	0.517357
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	−19.0000	−0.816874	−0.408437	−	0.912787i	$-0.633926\pi$
$541$	−19.0000	−0.816874	−0.408437	+	0.912787i	$0.633926\pi$
$542$	2.00000	0.0859074
$543$	22.0000	0.944110
$544$	15.0000	0.643120
$545$	0	0
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	−3.00000	−0.128154
$549$	−2.00000	−0.0853579
$550$	−8.00000	−0.341121
$551$	64.0000	2.72649
$552$	3.00000	0.127688
$553$	0	0
$554$	−13.0000	−0.552317
$555$	6.00000	0.254686
$556$	−6.00000	−0.254457
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	6.00000	0.254000
$559$	−28.0000	−1.18427
$560$	0	0
$561$	6.00000	0.253320
$562$	25.0000	1.05456
$563$	16.0000	0.674320	0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$564$	9.00000	0.378968
$565$	−5.00000	−0.210352
$566$	−19.0000	−0.798630
$567$	0	0
$568$	27.0000	1.13289
$569$	13.0000	0.544988	0.272494	−	0.962157i	$-0.412151\pi$
$569$	13.0000	0.544988	0.272494	+	0.962157i	$0.412151\pi$
$570$	8.00000	0.335083
$571$	11.0000	0.460336	0.230168	−	0.973151i	$-0.426072\pi$
$571$	11.0000	0.460336	0.230168	+	0.973151i	$0.426072\pi$
$572$	14.0000	0.585369
$573$	−8.00000	−0.334205
$574$	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	$0.513255\pi$
$578$	8.00000	0.332756
$579$	−19.0000	−0.789613
$580$	8.00000	0.332182
$581$	0	0
$582$	0	0
$583$	−6.00000	−0.248495
$584$	9.00000	0.372423
$585$	7.00000	0.289414
$586$	−23.0000	−0.950121
$587$	19.0000	0.784214	0.392107	−	0.919920i	$-0.371746\pi$
$587$	19.0000	0.784214	0.392107	+	0.919920i	$0.371746\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	−12.0000	−0.494032
$591$	−26.0000	−1.06950
$592$	−6.00000	−0.246598
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	15.0000	0.614424
$597$	24.0000	0.982255
$598$	−7.00000	−0.286251
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	−12.0000	−0.489898
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	4.00000	0.162758
$605$	−7.00000	−0.284590
$606$	10.0000	0.406222
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	40.0000	1.62221
$609$	0	0
$610$	2.00000	0.0809776
$611$	−63.0000	−2.54871
$612$	3.00000	0.121268
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	−4.00000	−0.161427
$615$	−4.00000	−0.161296
$616$	0	0
$617$	39.0000	1.57008	0.785040	−	0.619445i	$-0.212642\pi$
$617$	39.0000	1.57008	0.785040	+	0.619445i	$0.212642\pi$
$618$	1.00000	0.0402259
$619$	25.0000	1.00483	0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000	1.00483	0.502417	+	0.864625i	$0.332444\pi$
$620$	6.00000	0.240966
$621$	1.00000	0.0401286
$622$	27.0000	1.08260
$623$	0	0
$624$	−7.00000	−0.280224
$625$	11.0000	0.440000
$626$	−8.00000	−0.319744
$627$	16.0000	0.638978
$628$	−10.0000	−0.399043
$629$	−18.0000	−0.717707
$630$	0	0
$631$	−1.00000	−0.0398094	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000	−0.0398094	−0.0199047	+	0.999802i	$0.506336\pi$
$632$	−12.0000	−0.477334
$633$	−14.0000	−0.556450
$634$	6.00000	0.238290
$635$	−6.00000	−0.238103
$636$	−3.00000	−0.118958
$637$	0	0
$638$	−16.0000	−0.633446
$639$	9.00000	0.356034
$640$	3.00000	0.118585
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	−4.00000	−0.157867
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	−24.0000	−0.944267
$647$	−40.0000	−1.57256	−0.786281	−	0.617869i	$-0.787996\pi$
$647$	−40.0000	−1.57256	−0.786281	+	0.617869i	$0.787996\pi$
$648$	3.00000	0.117851
$649$	−24.0000	−0.942082
$650$	28.0000	1.09825
$651$	0	0
$652$	20.0000	0.783260
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	−11.0000	−0.429806
$656$	4.00000	0.156174
$657$	3.00000	0.117041
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	2.00000	0.0778499
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	10.0000	0.388661
$663$	−21.0000	−0.815572
$664$	−54.0000	−2.09561
$665$	0	0
$666$	−6.00000	−0.232495
$667$	−8.00000	−0.309761
$668$	11.0000	0.425603
$669$	−20.0000	−0.773245
$670$	−1.00000	−0.0386334
$671$	4.00000	0.154418
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−4.00000	−0.154074
$675$	−4.00000	−0.153960
$676$	−36.0000	−1.38462
$677$	−39.0000	−1.49889	−0.749446	−	0.662066i	$-0.769680\pi$
$677$	−39.0000	−1.49889	−0.749446	+	0.662066i	$0.769680\pi$
$678$	5.00000	0.192024
$679$	0	0
$680$	−9.00000	−0.345134
$681$	18.0000	0.689761
$682$	−12.0000	−0.459504
$683$	−15.0000	−0.573959	−0.286980	−	0.957937i	$-0.592651\pi$
$683$	−15.0000	−0.573959	−0.286980	+	0.957937i	$0.592651\pi$
$684$	8.00000	0.305888
$685$	3.00000	0.114624
$686$	0	0
$687$	20.0000	0.763048
$688$	4.00000	0.152499
$689$	21.0000	0.800036
$690$	−1.00000	−0.0380693
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	16.0000	0.608229
$693$	0	0
$694$	13.0000	0.493473
$695$	6.00000	0.227593
$696$	−24.0000	−0.909718
$697$	12.0000	0.454532
$698$	−19.0000	−0.719161
$699$	24.0000	0.907763
$700$	0	0
$701$	45.0000	1.69963	0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000	1.69963	0.849813	+	0.527084i	$0.176715\pi$
$702$	−7.00000	−0.264198
$703$	−48.0000	−1.81035
$704$	−14.0000	−0.527645
$705$	−9.00000	−0.338960
$706$	−4.00000	−0.150542
$707$	0	0
$708$	−12.0000	−0.450988
$709$	−24.0000	−0.901339	−0.450669	−	0.892691i	$-0.648815\pi$
$709$	−24.0000	−0.901339	−0.450669	+	0.892691i	$0.648815\pi$
$710$	−9.00000	−0.337764
$711$	−4.00000	−0.150012
$712$	30.0000	1.12430
$713$	−6.00000	−0.224702
$714$	0	0
$715$	−14.0000	−0.523570
$716$	−5.00000	−0.186859
$717$	0	0
$718$	10.0000	0.373197
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−45.0000	−1.67473
$723$	14.0000	0.520666
$724$	−22.0000	−0.817624
$725$	32.0000	1.18845
$726$	7.00000	0.259794
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−3.00000	−0.111035
$731$	12.0000	0.443836
$732$	2.00000	0.0739221
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	15.0000	0.553660
$735$	0	0
$736$	−5.00000	−0.184302
$737$	−2.00000	−0.0736709
$738$	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
$741$	−56.0000	−2.05721
$742$	0	0
$743$	54.0000	1.98107	0.990534	−	0.137268i	$-0.0438322\pi$
$743$	54.0000	1.98107	0.990534	+	0.137268i	$0.0438322\pi$
$744$	−18.0000	−0.659912
$745$	−15.0000	−0.549557
$746$	−38.0000	−1.39128
$747$	−18.0000	−0.658586
$748$	−6.00000	−0.219382
$749$	0	0
$750$	9.00000	0.328634
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	9.00000	0.328196
$753$	10.0000	0.364420
$754$	56.0000	2.03940
$755$	−4.00000	−0.145575
$756$	0	0
$757$	−4.00000	−0.145382	−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000	−0.145382	−0.0726912	+	0.997354i	$0.523159\pi$
$758$	−9.00000	−0.326895
$759$	−2.00000	−0.0725954
$760$	−24.0000	−0.870572
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	6.00000	0.217357
$763$	0	0
$764$	8.00000	0.289430
$765$	−3.00000	−0.108465
$766$	6.00000	0.216789
$767$	84.0000	3.03306
$768$	−17.0000	−0.613435
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−10.0000	−0.360141
$772$	19.0000	0.683825
$773$	29.0000	1.04306	0.521529	−	0.853234i	$-0.325362\pi$
$773$	29.0000	1.04306	0.521529	+	0.853234i	$0.325362\pi$
$774$	4.00000	0.143777
$775$	24.0000	0.862105
$776$	0	0
$777$	0	0
$778$	6.00000	0.215110
$779$	32.0000	1.14652
$780$	−7.00000	−0.250640
$781$	−18.0000	−0.644091
$782$	3.00000	0.107280
$783$	−8.00000	−0.285897
$784$	0	0
$785$	10.0000	0.356915
$786$	11.0000	0.392357
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	26.0000	0.926212
$789$	−32.0000	−1.13923
$790$	4.00000	0.142314
$791$	0	0
$792$	−6.00000	−0.213201
$793$	−14.0000	−0.497155
$794$	−21.0000	−0.745262
$795$	3.00000	0.106399
$796$	−24.0000	−0.850657
$797$	7.00000	0.247953	0.123976	−	0.992285i	$-0.460435\pi$
$797$	7.00000	0.247953	0.123976	+	0.992285i	$0.460435\pi$
$798$	0	0
$799$	27.0000	0.955191
$800$	20.0000	0.707107
$801$	10.0000	0.353333
$802$	15.0000	0.529668
$803$	−6.00000	−0.211735
$804$	−1.00000	−0.0352673
$805$	0	0
$806$	42.0000	1.47939
$807$	−12.0000	−0.422420
$808$	−30.0000	−1.05540
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	−1.00000	−0.0351364
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	0	0
$813$	−2.00000	−0.0701431
$814$	12.0000	0.420600
$815$	−20.0000	−0.700569
$816$	3.00000	0.105021
$817$	32.0000	1.11954
$818$	−23.0000	−0.804176
$819$	0	0
$820$	4.00000	0.139686
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	−3.00000	−0.104637
$823$	−18.0000	−0.627441	−0.313720	−	0.949515i	$-0.601575\pi$
$823$	−18.0000	−0.627441	−0.313720	+	0.949515i	$0.601575\pi$
$824$	−3.00000	−0.104510
$825$	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
$829$	−43.0000	−1.49345	−0.746726	−	0.665132i	$-0.768375\pi$
$829$	−43.0000	−1.49345	−0.746726	+	0.665132i	$0.768375\pi$
$830$	18.0000	0.624789
$831$	13.0000	0.450965
$832$	49.0000	1.69877
$833$	0	0
$834$	−6.00000	−0.207763
$835$	−11.0000	−0.380671
$836$	−16.0000	−0.553372
$837$	−6.00000	−0.207390
$838$	12.0000	0.414533
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−8.00000	−0.275698
$843$	−25.0000	−0.861046
$844$	14.0000	0.481900
$845$	36.0000	1.23844
$846$	9.00000	0.309426
$847$	0	0
$848$	−3.00000	−0.103020
$849$	19.0000	0.652078
$850$	−12.0000	−0.411597
$851$	6.00000	0.205677
$852$	−9.00000	−0.308335
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	12.0000	0.410152
$857$	34.0000	1.16142	0.580709	−	0.814111i	$-0.302775\pi$
$857$	34.0000	1.16142	0.580709	+	0.814111i	$0.302775\pi$
$858$	14.0000	0.477952
$859$	56.0000	1.91070	0.955348	−	0.295484i	$-0.0954809\pi$
$859$	56.0000	1.91070	0.955348	+	0.295484i	$0.0954809\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	0	0
$863$	−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
$865$	−16.0000	−0.544016
$866$	38.0000	1.29129
$867$	−8.00000	−0.271694
$868$	0	0
$869$	8.00000	0.271381
$870$	8.00000	0.271225
$871$	7.00000	0.237186
$872$	0	0
$873$	0	0
$874$	8.00000	0.270604
$875$	0	0
$876$	−3.00000	−0.101361
$877$	−29.0000	−0.979260	−0.489630	−	0.871930i	$-0.662868\pi$
$877$	−29.0000	−0.979260	−0.489630	+	0.871930i	$0.662868\pi$
$878$	−6.00000	−0.202490
$879$	23.0000	0.775771
$880$	2.00000	0.0674200
$881$	3.00000	0.101073	0.0505363	−	0.998722i	$-0.483907\pi$
$881$	3.00000	0.101073	0.0505363	+	0.998722i	$0.483907\pi$
$882$	0	0
$883$	−40.0000	−1.34611	−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000	−1.34611	−0.673054	+	0.739594i	$0.735018\pi$
$884$	21.0000	0.706306
$885$	12.0000	0.403376
$886$	3.00000	0.100787
$887$	−40.0000	−1.34307	−0.671534	−	0.740973i	$-0.734364\pi$
$887$	−40.0000	−1.34307	−0.671534	+	0.740973i	$0.734364\pi$
$888$	18.0000	0.604040
$889$	0	0
$890$	−10.0000	−0.335201
$891$	−2.00000	−0.0670025
$892$	20.0000	0.669650
$893$	72.0000	2.40939
$894$	15.0000	0.501675
$895$	5.00000	0.167132
$896$	0	0
$897$	7.00000	0.233723
$898$	−6.00000	−0.200223
$899$	48.0000	1.60089
$900$	4.00000	0.133333
$901$	−9.00000	−0.299833
$902$	−8.00000	−0.266371
$903$	0	0
$904$	−15.0000	−0.498893
$905$	22.0000	0.731305
$906$	4.00000	0.132891
$907$	47.0000	1.56061	0.780305	−	0.625400i	$-0.215064\pi$
$907$	47.0000	1.56061	0.780305	+	0.625400i	$0.215064\pi$
$908$	−18.0000	−0.597351
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	8.00000	0.264906
$913$	36.0000	1.19143
$914$	−16.0000	−0.529233
$915$	−2.00000	−0.0661180
$916$	−20.0000	−0.660819
$917$	0	0
$918$	3.00000	0.0990148
$919$	35.0000	1.15454	0.577272	−	0.816552i	$-0.304117\pi$
$919$	35.0000	1.15454	0.577272	+	0.816552i	$0.304117\pi$
$920$	3.00000	0.0989071
$921$	4.00000	0.131804
$922$	10.0000	0.329332
$923$	63.0000	2.07367
$924$	0	0
$925$	−24.0000	−0.789115
$926$	8.00000	0.262896
$927$	−1.00000	−0.0328443
$928$	40.0000	1.31306
$929$	56.0000	1.83730	0.918650	−	0.395072i	$-0.129280\pi$
$929$	56.0000	1.83730	0.918650	+	0.395072i	$0.129280\pi$
$930$	6.00000	0.196748
$931$	0	0
$932$	−24.0000	−0.786146
$933$	−27.0000	−0.883940
$934$	12.0000	0.392652
$935$	6.00000	0.196221
$936$	21.0000	0.686406
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	8.00000	0.261070
$940$	9.00000	0.293548
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	−10.0000	−0.325818
$943$	−4.00000	−0.130258
$944$	−12.0000	−0.390567
$945$	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
$949$	21.0000	0.681689
$950$	−32.0000	−1.03822
$951$	−6.00000	−0.194563
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	−3.00000	−0.0971286
$955$	−8.00000	−0.258874
$956$	0	0
$957$	16.0000	0.517207
$958$	−8.00000	−0.258468
$959$	0	0
$960$	7.00000	0.225924
$961$	5.00000	0.161290
$962$	−42.0000	−1.35413
$963$	4.00000	0.128898
$964$	−14.0000	−0.450910
$965$	−19.0000	−0.611632
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	−21.0000	−0.674966
$969$	24.0000	0.770991
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	8.00000	0.256337
$975$	−28.0000	−0.896718
$976$	2.00000	0.0640184
$977$	−45.0000	−1.43968	−0.719839	−	0.694141i	$-0.755784\pi$
$977$	−45.0000	−1.43968	−0.719839	+	0.694141i	$0.755784\pi$
$978$	20.0000	0.639529
$979$	−20.0000	−0.639203
$980$	0	0
$981$	0	0
$982$	7.00000	0.223379
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	−12.0000	−0.382546
$985$	−26.0000	−0.828429
$986$	−24.0000	−0.764316
$987$	0	0
$988$	56.0000	1.78160
$989$	−4.00000	−0.127193
$990$	2.00000	0.0635642
$991$	14.0000	0.444725	0.222362	−	0.974964i	$-0.428623\pi$
$991$	14.0000	0.444725	0.222362	+	0.974964i	$0.428623\pi$
$992$	30.0000	0.952501
$993$	−10.0000	−0.317340
$994$	0	0
$995$	24.0000	0.760851
$996$	18.0000	0.570352
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	20.0000	0.633089
$999$	6.00000	0.189832

Twists

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

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

Downloads

Learn more