LMFDB - Embedded newform 3822.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, 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("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3822.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:	$30.5188236525$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3822.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -4.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +12.0000 q^{37} -4.00000 q^{38} +1.00000 q^{39} +2.00000 q^{40} -12.0000 q^{41} -8.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} -2.00000 q^{47} -1.00000 q^{48} +1.00000 q^{50} +6.00000 q^{51} -1.00000 q^{52} +8.00000 q^{53} +1.00000 q^{54} +8.00000 q^{55} -4.00000 q^{57} +4.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -14.0000 q^{67} -6.00000 q^{68} +8.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -12.0000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{78} +16.0000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} +12.0000 q^{85} +8.00000 q^{86} +4.00000 q^{87} +4.00000 q^{88} -4.00000 q^{89} +2.00000 q^{90} +4.00000 q^{93} +2.00000 q^{94} -8.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	−2.00000	−0.365148
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	12.0000	1.97279	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	12.0000	1.97279	0.986394	+	0.164399i	$0.0525685\pi$
$38$	−4.00000	−0.648886
$39$	1.00000	0.160128
$40$	2.00000	0.316228
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−4.00000	−0.603023
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	6.00000	0.840168
$52$	−1.00000	−0.138675
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	1.00000	0.136083
$55$	8.00000	1.07872
$56$	0	0
$57$	−4.00000	−0.529813
$58$	4.00000	0.525226
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	2.00000	0.258199
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−4.00000	−0.492366
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−12.0000	−1.39497
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	−1.00000	−0.113228
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	12.0000	1.32518
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	8.00000	0.862662
$87$	4.00000	0.428845
$88$	4.00000	0.426401
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	0	0
$93$	4.00000	0.414781
$94$	2.00000	0.206284
$95$	−8.00000	−0.820783
$96$	1.00000	0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	−6.00000	−0.594089
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−8.00000	−0.777029
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	−1.00000	−0.0962250
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	−8.00000	−0.762770
$111$	−12.0000	−1.13899
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−4.00000	−0.371391
$117$	−1.00000	−0.0924500
$118$	−4.00000	−0.368230
$119$	0	0
$120$	−2.00000	−0.182574
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	12.0000	1.08200
$124$	−4.00000	−0.359211
$125$	12.0000	1.07331
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	−2.00000	−0.175412
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	14.0000	1.20942
$135$	2.00000	0.172133
$136$	6.00000	0.514496
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	−8.00000	−0.671345
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	8.00000	0.664364
$146$	−2.00000	−0.165521
$147$	0	0
$148$	12.0000	0.986394
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	−1.00000	−0.0816497
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−4.00000	−0.324443
$153$	−6.00000	−0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	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	$0.369342\pi$
$158$	−16.0000	−1.27289
$159$	−8.00000	−0.634441
$160$	2.00000	0.158114
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	−12.0000	−0.937043
$165$	−8.00000	−0.622799
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−12.0000	−0.920358
$171$	4.00000	0.305888
$172$	−8.00000	−0.609994
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	−4.00000	−0.301511
$177$	−4.00000	−0.300658
$178$	4.00000	0.299813
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	−2.00000	−0.149071
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	−24.0000	−1.76452
$186$	−4.00000	−0.293294
$187$	24.0000	1.75505
$188$	−2.00000	−0.145865
$189$	0	0
$190$	8.00000	0.580381
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−2.00000	−0.143592
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	4.00000	0.284268
$199$	26.0000	1.84309	0.921546	−	0.388270i	$-0.126927\pi$
$199$	26.0000	1.84309	0.921546	+	0.388270i	$0.126927\pi$
$200$	1.00000	0.0707107
$201$	14.0000	0.987484
$202$	14.0000	0.985037
$203$	0	0
$204$	6.00000	0.420084
$205$	24.0000	1.67623
$206$	14.0000	0.975426
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	8.00000	0.549442
$213$	−8.00000	−0.548151
$214$	−10.0000	−0.683586
$215$	16.0000	1.09119
$216$	1.00000	0.0680414
$217$	0	0
$218$	8.00000	0.541828
$219$	−2.00000	−0.135147
$220$	8.00000	0.539360
$221$	6.00000	0.403604
$222$	12.0000	0.805387
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−10.0000	−0.665190
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	−4.00000	−0.264906
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	4.00000	0.262613
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	1.00000	0.0653720
$235$	4.00000	0.260931
$236$	4.00000	0.260378
$237$	−16.0000	−1.03931
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	2.00000	0.129099
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	0	0
$246$	−12.0000	−0.765092
$247$	−4.00000	−0.254514
$248$	4.00000	0.254000
$249$	0	0
$250$	−12.0000	−0.758947
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−8.00000	−0.498058
$259$	0	0
$260$	2.00000	0.124035
$261$	−4.00000	−0.247594
$262$	12.0000	0.741362
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	−4.00000	−0.246183
$265$	−16.0000	−0.982872
$266$	0	0
$267$	4.00000	0.244796
$268$	−14.0000	−0.855186
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	−2.00000	−0.121716
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	10.0000	0.604122
$275$	4.00000	0.241209
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−2.00000	−0.119098
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	8.00000	0.474713
$285$	8.00000	0.473879
$286$	−4.00000	−0.236525
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	−8.00000	−0.469776
$291$	−2.00000	−0.117242
$292$	2.00000	0.117041
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	−12.0000	−0.697486
$297$	4.00000	0.232104
$298$	−14.0000	−0.810998
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	−4.00000	−0.230174
$303$	14.0000	0.804279
$304$	4.00000	0.229416
$305$	20.0000	1.14520
$306$	6.00000	0.342997
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	−8.00000	−0.454369
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	−1.00000	−0.0566139
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	16.0000	0.900070
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	8.00000	0.448618
$319$	16.0000	0.895828
$320$	−2.00000	−0.111803
$321$	−10.0000	−0.558146
$322$	0	0
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	1.00000	0.0554700
$326$	18.0000	0.996928
$327$	8.00000	0.442401
$328$	12.0000	0.662589
$329$	0	0
$330$	8.00000	0.440386
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	0	0
$333$	12.0000	0.657596
$334$	−2.00000	−0.109435
$335$	28.0000	1.52980
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−1.00000	−0.0543928
$339$	−10.0000	−0.543125
$340$	12.0000	0.650791
$341$	16.0000	0.866449
$342$	−4.00000	−0.216295
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	−14.0000	−0.752645
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	4.00000	0.214423
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	4.00000	0.213201
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	4.00000	0.212598
$355$	−16.0000	−0.849192
$356$	−4.00000	−0.212000
$357$	0	0
$358$	6.00000	0.317110
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	2.00000	0.105409
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	−5.00000	−0.262432
$364$	0	0
$365$	−4.00000	−0.209370
$366$	−10.0000	−0.522708
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	24.0000	1.24770
$371$	0	0
$372$	4.00000	0.207390
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−24.0000	−1.24101
$375$	−12.0000	−0.619677
$376$	2.00000	0.103142
$377$	4.00000	0.206010
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	−8.00000	−0.410391
$381$	−8.00000	−0.409852
$382$	12.0000	0.613973
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−8.00000	−0.406663
$388$	2.00000	0.101535
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	2.00000	0.101274
$391$	0	0
$392$	0	0
$393$	12.0000	0.605320
$394$	6.00000	0.302276
$395$	−32.0000	−1.61009
$396$	−4.00000	−0.201008
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	−26.0000	−1.30326
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	−14.0000	−0.698257
$403$	4.00000	0.199254
$404$	−14.0000	−0.696526
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−48.0000	−2.37927
$408$	−6.00000	−0.297044
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	−24.0000	−1.18528
$411$	10.0000	0.493264
$412$	−14.0000	−0.689730
$413$	0	0
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	16.0000	0.782586
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	16.0000	0.778868
$423$	−2.00000	−0.0972433
$424$	−8.00000	−0.388514
$425$	6.00000	0.291043
$426$	8.00000	0.387601
$427$	0	0
$428$	10.0000	0.483368
$429$	−4.00000	−0.193122
$430$	−16.0000	−0.771589
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	−1.00000	−0.0481125
$433$	−32.0000	−1.53782	−0.768911	−	0.639356i	$-0.779201\pi$
$433$	−32.0000	−1.53782	−0.768911	+	0.639356i	$0.779201\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	−8.00000	−0.383131
$437$	0	0
$438$	2.00000	0.0955637
$439$	30.0000	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$440$	−8.00000	−0.381385
$441$	0	0
$442$	−6.00000	−0.285391
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	−12.0000	−0.569495
$445$	8.00000	0.379236
$446$	−20.0000	−0.947027
$447$	−14.0000	−0.662177
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	1.00000	0.0471405
$451$	48.0000	2.26023
$452$	10.0000	0.470360
$453$	−4.00000	−0.187936
$454$	−8.00000	−0.375459
$455$	0	0
$456$	4.00000	0.187317
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	2.00000	0.0934539
$459$	6.00000	0.280056
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−4.00000	−0.185695
$465$	−8.00000	−0.370991
$466$	−14.0000	−0.648537
$467$	−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
$470$	−4.00000	−0.184506
$471$	−10.0000	−0.460776
$472$	−4.00000	−0.184115
$473$	32.0000	1.47136
$474$	16.0000	0.734904
$475$	−4.00000	−0.183533
$476$	0	0
$477$	8.00000	0.366295
$478$	−24.0000	−1.09773
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	−2.00000	−0.0912871
$481$	−12.0000	−0.547153
$482$	−26.0000	−1.18427
$483$	0	0
$484$	5.00000	0.227273
$485$	−4.00000	−0.181631
$486$	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	$0.381918\pi$
$488$	10.0000	0.452679
$489$	18.0000	0.813988
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	12.0000	0.541002
$493$	24.0000	1.08091
$494$	4.00000	0.179969
$495$	8.00000	0.359573
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	−42.0000	−1.88018	−0.940089	−	0.340929i	$-0.889258\pi$
$499$	−42.0000	−1.88018	−0.940089	+	0.340929i	$0.889258\pi$
$500$	12.0000	0.536656
$501$	−2.00000	−0.0893534
$502$	−4.00000	−0.178529
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	8.00000	0.354943
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	12.0000	0.531369
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	−2.00000	−0.0882162
$515$	28.0000	1.23383
$516$	8.00000	0.352180
$517$	8.00000	0.351840
$518$	0	0
$519$	−14.0000	−0.614532
$520$	−2.00000	−0.0877058
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	4.00000	0.175075
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	4.00000	0.174408
$527$	24.0000	1.04546
$528$	4.00000	0.174078
$529$	−23.0000	−1.00000
$530$	16.0000	0.694996
$531$	4.00000	0.173585
$532$	0	0
$533$	12.0000	0.519778
$534$	−4.00000	−0.173097
$535$	−20.0000	−0.864675
$536$	14.0000	0.604708
$537$	6.00000	0.258919
$538$	14.0000	0.603583
$539$	0	0
$540$	2.00000	0.0860663
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	20.0000	0.859074
$543$	−2.00000	−0.0858282
$544$	6.00000	0.257248
$545$	16.0000	0.685365
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	−10.0000	−0.427179
$549$	−10.0000	−0.426790
$550$	−4.00000	−0.170561
$551$	−16.0000	−0.681623
$552$	0	0
$553$	0	0
$554$	2.00000	0.0849719
$555$	24.0000	1.01874
$556$	0	0
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	4.00000	0.169334
$559$	8.00000	0.338364
$560$	0	0
$561$	−24.0000	−1.01328
$562$	−18.0000	−0.759284
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	2.00000	0.0842152
$565$	−20.0000	−0.841406
$566$	4.00000	0.168133
$567$	0	0
$568$	−8.00000	−0.335673
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	−8.00000	−0.335083
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	4.00000	0.167248
$573$	12.0000	0.501307
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−19.0000	−0.790296
$579$	−14.0000	−0.581820
$580$	8.00000	0.332182
$581$	0	0
$582$	2.00000	0.0829027
$583$	−32.0000	−1.32530
$584$	−2.00000	−0.0827606
$585$	2.00000	0.0826898
$586$	−30.0000	−1.23929
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	8.00000	0.329355
$591$	6.00000	0.246807
$592$	12.0000	0.493197
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	14.0000	0.573462
$597$	−26.0000	−1.06411
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	−1.00000	−0.0408248
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	0	0
$603$	−14.0000	−0.570124
$604$	4.00000	0.162758
$605$	−10.0000	−0.406558
$606$	−14.0000	−0.568711
$607$	−42.0000	−1.70473	−0.852364	−	0.522949i	$-0.824832\pi$
$607$	−42.0000	−1.70473	−0.852364	+	0.522949i	$0.824832\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−20.0000	−0.809776
$611$	2.00000	0.0809113
$612$	−6.00000	−0.242536
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	−20.0000	−0.807134
$615$	−24.0000	−0.967773
$616$	0	0
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	−14.0000	−0.563163
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	28.0000	1.12270
$623$	0	0
$624$	1.00000	0.0400320
$625$	−19.0000	−0.760000
$626$	0	0
$627$	16.0000	0.638978
$628$	10.0000	0.399043
$629$	−72.0000	−2.87083
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	−16.0000	−0.636446
$633$	16.0000	0.635943
$634$	−18.0000	−0.714871
$635$	−16.0000	−0.634941
$636$	−8.00000	−0.317221
$637$	0	0
$638$	−16.0000	−0.633446
$639$	8.00000	0.316475
$640$	2.00000	0.0790569
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	10.0000	0.394669
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	24.0000	0.944267
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	−1.00000	−0.0392837
$649$	−16.0000	−0.628055
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	−18.0000	−0.704934
$653$	−16.0000	−0.626128	−0.313064	−	0.949732i	$-0.601356\pi$
$653$	−16.0000	−0.626128	−0.313064	+	0.949732i	$0.601356\pi$
$654$	−8.00000	−0.312825
$655$	24.0000	0.937758
$656$	−12.0000	−0.468521
$657$	2.00000	0.0780274
$658$	0	0
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	−8.00000	−0.311400
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−14.0000	−0.544125
$663$	−6.00000	−0.233021
$664$	0	0
$665$	0	0
$666$	−12.0000	−0.464991
$667$	0	0
$668$	2.00000	0.0773823
$669$	−20.0000	−0.773245
$670$	−28.0000	−1.08173
$671$	40.0000	1.54418
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	−14.0000	−0.539260
$675$	1.00000	0.0384900
$676$	1.00000	0.0384615
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	10.0000	0.384048
$679$	0	0
$680$	−12.0000	−0.460179
$681$	−8.00000	−0.306561
$682$	−16.0000	−0.612672
$683$	40.0000	1.53056	0.765279	−	0.643699i	$-0.222601\pi$
$683$	40.0000	1.53056	0.765279	+	0.643699i	$0.222601\pi$
$684$	4.00000	0.152944
$685$	20.0000	0.764161
$686$	0	0
$687$	2.00000	0.0763048
$688$	−8.00000	−0.304997
$689$	−8.00000	−0.304776
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	−30.0000	−1.13878
$695$	0	0
$696$	−4.00000	−0.151620
$697$	72.0000	2.72719
$698$	−6.00000	−0.227103
$699$	−14.0000	−0.529529
$700$	0	0
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	−1.00000	−0.0377426
$703$	48.0000	1.81035
$704$	−4.00000	−0.150756
$705$	−4.00000	−0.150649
$706$	−24.0000	−0.903252
$707$	0	0
$708$	−4.00000	−0.150329
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	16.0000	0.600469
$711$	16.0000	0.600047
$712$	4.00000	0.149906
$713$	0	0
$714$	0	0
$715$	−8.00000	−0.299183
$716$	−6.00000	−0.224231
$717$	−24.0000	−0.896296
$718$	−16.0000	−0.597115
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	3.00000	0.111648
$723$	−26.0000	−0.966950
$724$	2.00000	0.0743294
$725$	4.00000	0.148556
$726$	5.00000	0.185567
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	48.0000	1.77534
$732$	10.0000	0.369611
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	0	0
$737$	56.0000	2.06279
$738$	12.0000	0.441726
$739$	−14.0000	−0.514998	−0.257499	−	0.966279i	$-0.582898\pi$
$739$	−14.0000	−0.514998	−0.257499	+	0.966279i	$0.582898\pi$
$740$	−24.0000	−0.882258
$741$	4.00000	0.146944
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−4.00000	−0.146647
$745$	−28.0000	−1.02584
$746$	−10.0000	−0.366126
$747$	0	0
$748$	24.0000	0.877527
$749$	0	0
$750$	12.0000	0.438178
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	−2.00000	−0.0729325
$753$	−4.00000	−0.145768
$754$	−4.00000	−0.145671
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	6.00000	0.217930
$759$	0	0
$760$	8.00000	0.290191
$761$	28.0000	1.01500	0.507500	−	0.861652i	$-0.330570\pi$
$761$	28.0000	1.01500	0.507500	+	0.861652i	$0.330570\pi$
$762$	8.00000	0.289809
$763$	0	0
$764$	−12.0000	−0.434145
$765$	12.0000	0.433861
$766$	18.0000	0.650366
$767$	−4.00000	−0.144432
$768$	−1.00000	−0.0360844
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	14.0000	0.503871
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	8.00000	0.287554
$775$	4.00000	0.143684
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	36.0000	1.29066
$779$	−48.0000	−1.71978
$780$	−2.00000	−0.0716115
$781$	−32.0000	−1.14505
$782$	0	0
$783$	4.00000	0.142948
$784$	0	0
$785$	−20.0000	−0.713831
$786$	−12.0000	−0.428026
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−6.00000	−0.213741
$789$	4.00000	0.142404
$790$	32.0000	1.13851
$791$	0	0
$792$	4.00000	0.142134
$793$	10.0000	0.355110
$794$	−22.0000	−0.780751
$795$	16.0000	0.567462
$796$	26.0000	0.921546
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	1.00000	0.0353553
$801$	−4.00000	−0.141333
$802$	30.0000	1.05934
$803$	−8.00000	−0.282314
$804$	14.0000	0.493742
$805$	0	0
$806$	−4.00000	−0.140894
$807$	14.0000	0.492823
$808$	14.0000	0.492518
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	2.00000	0.0702728
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	48.0000	1.68240
$815$	36.0000	1.26102
$816$	6.00000	0.210042
$817$	−32.0000	−1.11954
$818$	−18.0000	−0.629355
$819$	0	0
$820$	24.0000	0.838116
$821$	−34.0000	−1.18661	−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000	−1.18661	−0.593304	+	0.804978i	$0.702177\pi$
$822$	−10.0000	−0.348790
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	14.0000	0.487713
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−24.0000	−0.834562	−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000	−0.834562	−0.417281	+	0.908778i	$0.637017\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	0	0
$835$	−4.00000	−0.138426
$836$	−16.0000	−0.553372
$837$	4.00000	0.138260
$838$	20.0000	0.690889
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−18.0000	−0.619953
$844$	−16.0000	−0.550743
$845$	−2.00000	−0.0688021
$846$	2.00000	0.0687614
$847$	0	0
$848$	8.00000	0.274721
$849$	4.00000	0.137280
$850$	−6.00000	−0.205798
$851$	0	0
$852$	−8.00000	−0.274075
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	−10.0000	−0.341793
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	4.00000	0.136558
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	16.0000	0.545595
$861$	0	0
$862$	−16.0000	−0.544962
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	1.00000	0.0340207
$865$	−28.0000	−0.952029
$866$	32.0000	1.08740
$867$	−19.0000	−0.645274
$868$	0	0
$869$	−64.0000	−2.17105
$870$	8.00000	0.271225
$871$	14.0000	0.474372
$872$	8.00000	0.270914
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	−30.0000	−1.01245
$879$	−30.0000	−1.01187
$880$	8.00000	0.269680
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	0	0
$883$	−56.0000	−1.88455	−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000	−1.88455	−0.942275	+	0.334840i	$0.891318\pi$
$884$	6.00000	0.201802
$885$	8.00000	0.268917
$886$	−6.00000	−0.201574
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	12.0000	0.402694
$889$	0	0
$890$	−8.00000	−0.268161
$891$	−4.00000	−0.134005
$892$	20.0000	0.669650
$893$	−8.00000	−0.267710
$894$	14.0000	0.468230
$895$	12.0000	0.401116
$896$	0	0
$897$	0	0
$898$	−30.0000	−1.00111
$899$	16.0000	0.533630
$900$	−1.00000	−0.0333333
$901$	−48.0000	−1.59911
$902$	−48.0000	−1.59823
$903$	0	0
$904$	−10.0000	−0.332595
$905$	−4.00000	−0.132964
$906$	4.00000	0.132891
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	8.00000	0.265489
$909$	−14.0000	−0.464351
$910$	0	0
$911$	−56.0000	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	−38.0000	−1.25693
$915$	−20.0000	−0.661180
$916$	−2.00000	−0.0660819
$917$	0	0
$918$	−6.00000	−0.198030
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	6.00000	0.197599
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−12.0000	−0.394558
$926$	4.00000	0.131448
$927$	−14.0000	−0.459820
$928$	4.00000	0.131306
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	8.00000	0.262330
$931$	0	0
$932$	14.0000	0.458585
$933$	28.0000	0.916679
$934$	28.0000	0.916188
$935$	−48.0000	−1.56977
$936$	1.00000	0.0326860
$937$	−48.0000	−1.56809	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	−48.0000	−1.56809	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	0	0
$940$	4.00000	0.130466
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	10.0000	0.325818
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	−32.0000	−1.04041
$947$	−40.0000	−1.29983	−0.649913	−	0.760009i	$-0.725195\pi$
$947$	−40.0000	−1.29983	−0.649913	+	0.760009i	$0.725195\pi$
$948$	−16.0000	−0.519656
$949$	−2.00000	−0.0649227
$950$	4.00000	0.129777
$951$	−18.0000	−0.583690
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	−8.00000	−0.259010
$955$	24.0000	0.776622
$956$	24.0000	0.776215
$957$	−16.0000	−0.517207
$958$	6.00000	0.193851
$959$	0	0
$960$	2.00000	0.0645497
$961$	−15.0000	−0.483871
$962$	12.0000	0.386896
$963$	10.0000	0.322245
$964$	26.0000	0.837404
$965$	−28.0000	−0.901352
$966$	0	0
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	−5.00000	−0.160706
$969$	24.0000	0.770991
$970$	4.00000	0.128432
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	−1.00000	−0.0320256
$976$	−10.0000	−0.320092
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	−18.0000	−0.575577
$979$	16.0000	0.511362
$980$	0	0
$981$	−8.00000	−0.255420
$982$	6.00000	0.191468
$983$	−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
$985$	12.0000	0.382352
$986$	−24.0000	−0.764316
$987$	0	0
$988$	−4.00000	−0.127257
$989$	0	0
$990$	−8.00000	−0.254257
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	4.00000	0.127000
$993$	−14.0000	−0.444277
$994$	0	0
$995$	−52.0000	−1.64851
$996$	0	0
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	42.0000	1.32949
$999$	−12.0000	−0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.b.1.1	✓	1
7.6	odd	2		3822.2.a.p.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3822.2.a.b.1.1	✓	1	1.1	even	1	trivial
3822.2.a.p.1.1	yes	1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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