LMFDB - Embedded newform 222.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [222,2,Mod(1,222)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("222.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(222, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 222 = 2 \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	222.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1,1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.77267892487$
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	$\chi$	$=$	222.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +3.00000 q^{19} -3.00000 q^{21} +1.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -4.00000 q^{29} -6.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} +3.00000 q^{38} -1.00000 q^{39} -10.0000 q^{41} -3.00000 q^{42} +12.0000 q^{43} +1.00000 q^{44} -1.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -5.00000 q^{50} +3.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} -3.00000 q^{57} -4.00000 q^{58} +2.00000 q^{61} -6.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +2.00000 q^{67} -3.00000 q^{68} +1.00000 q^{69} +1.00000 q^{72} -3.00000 q^{73} -1.00000 q^{74} +5.00000 q^{75} +3.00000 q^{76} +3.00000 q^{77} -1.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} -10.0000 q^{82} +9.00000 q^{83} -3.00000 q^{84} +12.0000 q^{86} +4.00000 q^{87} +1.00000 q^{88} -3.00000 q^{89} +3.00000 q^{91} -1.00000 q^{92} +6.00000 q^{93} -6.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} +2.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.00000	−0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	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
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	1.00000	0.213201
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	−1.00000	−0.204124
$25$	−5.00000	−1.00000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	3.00000	0.566947
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	3.00000	0.486664
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	−3.00000	−0.462910
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	−5.00000	−0.707107
$51$	3.00000	0.420084
$52$	1.00000	0.138675
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	−3.00000	−0.397360
$58$	−4.00000	−0.525226
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−6.00000	−0.762001
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−3.00000	−0.363803
$69$	1.00000	0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	−1.00000	−0.116248
$75$	5.00000	0.577350
$76$	3.00000	0.344124
$77$	3.00000	0.341882
$78$	−1.00000	−0.113228
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	12.0000	1.29399
$87$	4.00000	0.428845
$88$	1.00000	0.106600
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	−1.00000	−0.104257
$93$	6.00000	0.622171
$94$	−6.00000	−0.618853
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	2.00000	0.202031
$99$	1.00000	0.100504
$100$	−5.00000	−0.500000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	3.00000	0.297044
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	−13.0000	−1.25676	−0.628379	−	0.777908i	$-0.716281\pi$
$107$	−13.0000	−1.25676	−0.628379	+	0.777908i	$0.716281\pi$
$108$	−1.00000	−0.0962250
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	3.00000	0.283473
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−3.00000	−0.280976
$115$	0	0
$116$	−4.00000	−0.371391
$117$	1.00000	0.0924500
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	−10.0000	−0.909091
$122$	2.00000	0.181071
$123$	10.0000	0.901670
$124$	−6.00000	−0.538816
$125$	0	0
$126$	3.00000	0.267261
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	1.00000	0.0883883
$129$	−12.0000	−1.05654
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	−1.00000	−0.0870388
$133$	9.00000	0.780399
$134$	2.00000	0.172774
$135$	0	0
$136$	−3.00000	−0.257248
$137$	20.0000	1.70872	0.854358	−	0.519685i	$-0.173951\pi$
$137$	20.0000	1.70872	0.854358	+	0.519685i	$0.173951\pi$
$138$	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
$141$	6.00000	0.505291
$142$	0	0
$143$	1.00000	0.0836242
$144$	1.00000	0.0833333
$145$	0	0
$146$	−3.00000	−0.248282
$147$	−2.00000	−0.164957
$148$	−1.00000	−0.0821995
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	5.00000	0.408248
$151$	1.00000	0.0813788	0.0406894	−	0.999172i	$-0.487045\pi$
$151$	1.00000	0.0813788	0.0406894	+	0.999172i	$0.487045\pi$
$152$	3.00000	0.243332
$153$	−3.00000	−0.242536
$154$	3.00000	0.241747
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	14.0000	1.11378
$159$	1.00000	0.0793052
$160$	0	0
$161$	−3.00000	−0.236433
$162$	1.00000	0.0785674
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	9.00000	0.698535
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	−3.00000	−0.231455
$169$	−12.0000	−0.923077
$170$	0	0
$171$	3.00000	0.229416
$172$	12.0000	0.914991
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	4.00000	0.303239
$175$	−15.0000	−1.13389
$176$	1.00000	0.0753778
$177$	0	0
$178$	−3.00000	−0.224860
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	3.00000	0.222375
$183$	−2.00000	−0.147844
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	6.00000	0.439941
$187$	−3.00000	−0.219382
$188$	−6.00000	−0.437595
$189$	−3.00000	−0.218218
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	−1.00000	−0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	2.00000	0.142857
$197$	13.0000	0.926212	0.463106	−	0.886303i	$-0.346735\pi$
$197$	13.0000	0.926212	0.463106	+	0.886303i	$0.346735\pi$
$198$	1.00000	0.0710669
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−5.00000	−0.353553
$201$	−2.00000	−0.141069
$202$	−6.00000	−0.422159
$203$	−12.0000	−0.842235
$204$	3.00000	0.210042
$205$	0	0
$206$	−2.00000	−0.139347
$207$	−1.00000	−0.0695048
$208$	1.00000	0.0693375
$209$	3.00000	0.207514
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−1.00000	−0.0686803
$213$	0	0
$214$	−13.0000	−0.888662
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−18.0000	−1.22192
$218$	11.0000	0.745014
$219$	3.00000	0.202721
$220$	0	0
$221$	−3.00000	−0.201802
$222$	1.00000	0.0671156
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	3.00000	0.200446
$225$	−5.00000	−0.333333
$226$	18.0000	1.19734
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−3.00000	−0.198680
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	−4.00000	−0.262613
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	0	0
$237$	−14.0000	−0.909398
$238$	−9.00000	−0.583383
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	−10.0000	−0.642824
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	10.0000	0.637577
$247$	3.00000	0.190885
$248$	−6.00000	−0.381000
$249$	−9.00000	−0.570352
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	3.00000	0.188982
$253$	−1.00000	−0.0628695
$254$	−7.00000	−0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	29.0000	1.80897	0.904485	−	0.426505i	$-0.140255\pi$
$257$	29.0000	1.80897	0.904485	+	0.426505i	$0.140255\pi$
$258$	−12.0000	−0.747087
$259$	−3.00000	−0.186411
$260$	0	0
$261$	−4.00000	−0.247594
$262$	10.0000	0.617802
$263$	26.0000	1.60323	0.801614	−	0.597841i	$-0.203975\pi$
$263$	26.0000	1.60323	0.801614	+	0.597841i	$0.203975\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	9.00000	0.551825
$267$	3.00000	0.183597
$268$	2.00000	0.122169
$269$	−7.00000	−0.426798	−0.213399	−	0.976965i	$-0.568453\pi$
$269$	−7.00000	−0.426798	−0.213399	+	0.976965i	$0.568453\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	−3.00000	−0.181902
$273$	−3.00000	−0.181568
$274$	20.0000	1.20824
$275$	−5.00000	−0.301511
$276$	1.00000	0.0601929
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	−10.0000	−0.599760
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−31.0000	−1.84930	−0.924652	−	0.380812i	$-0.875644\pi$
$281$	−31.0000	−1.84930	−0.924652	+	0.380812i	$0.875644\pi$
$282$	6.00000	0.357295
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	0	0
$285$	0	0
$286$	1.00000	0.0591312
$287$	−30.0000	−1.77084
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	10.0000	0.586210
$292$	−3.00000	−0.175562
$293$	31.0000	1.81104	0.905520	−	0.424304i	$-0.139481\pi$
$293$	31.0000	1.81104	0.905520	+	0.424304i	$0.139481\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	−1.00000	−0.0580259
$298$	18.0000	1.04271
$299$	−1.00000	−0.0578315
$300$	5.00000	0.288675
$301$	36.0000	2.07501
$302$	1.00000	0.0575435
$303$	6.00000	0.344691
$304$	3.00000	0.172062
$305$	0	0
$306$	−3.00000	−0.171499
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	3.00000	0.170941
$309$	2.00000	0.113776
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−1.00000	−0.0566139
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	14.0000	0.787562
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	1.00000	0.0560772
$319$	−4.00000	−0.223957
$320$	0	0
$321$	13.0000	0.725589
$322$	−3.00000	−0.167183
$323$	−9.00000	−0.500773
$324$	1.00000	0.0555556
$325$	−5.00000	−0.277350
$326$	17.0000	0.941543
$327$	−11.0000	−0.608301
$328$	−10.0000	−0.552158
$329$	−18.0000	−0.992372
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	9.00000	0.493939
$333$	−1.00000	−0.0547997
$334$	21.0000	1.14907
$335$	0	0
$336$	−3.00000	−0.163663
$337$	21.0000	1.14394	0.571971	−	0.820274i	$-0.306179\pi$
$337$	21.0000	1.14394	0.571971	+	0.820274i	$0.306179\pi$
$338$	−12.0000	−0.652714
$339$	−18.0000	−0.977626
$340$	0	0
$341$	−6.00000	−0.324918
$342$	3.00000	0.162221
$343$	−15.0000	−0.809924
$344$	12.0000	0.646997
$345$	0	0
$346$	1.00000	0.0537603
$347$	22.0000	1.18102	0.590511	−	0.807030i	$-0.298926\pi$
$347$	22.0000	1.18102	0.590511	+	0.807030i	$0.298926\pi$
$348$	4.00000	0.214423
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	−15.0000	−0.801784
$351$	−1.00000	−0.0533761
$352$	1.00000	0.0533002
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	0	0
$356$	−3.00000	−0.159000
$357$	9.00000	0.476331
$358$	−24.0000	−1.26844
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−20.0000	−1.05118
$363$	10.0000	0.524864
$364$	3.00000	0.157243
$365$	0	0
$366$	−2.00000	−0.104542
$367$	−21.0000	−1.09619	−0.548096	−	0.836416i	$-0.684647\pi$
$367$	−21.0000	−1.09619	−0.548096	+	0.836416i	$0.684647\pi$
$368$	−1.00000	−0.0521286
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−3.00000	−0.155752
$372$	6.00000	0.311086
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	−3.00000	−0.155126
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−4.00000	−0.206010
$378$	−3.00000	−0.154303
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	3.00000	0.153493
$383$	1.00000	0.0510976	0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000	0.0510976	0.0255488	+	0.999674i	$0.491867\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−18.0000	−0.916176
$387$	12.0000	0.609994
$388$	−10.0000	−0.507673
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	2.00000	0.101015
$393$	−10.0000	−0.504433
$394$	13.0000	0.654931
$395$	0	0
$396$	1.00000	0.0502519
$397$	−12.0000	−0.602263	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	−12.0000	−0.602263	−0.301131	+	0.953583i	$0.597364\pi$
$398$	−16.0000	−0.802008
$399$	−9.00000	−0.450564
$400$	−5.00000	−0.250000
$401$	−11.0000	−0.549314	−0.274657	−	0.961542i	$-0.588564\pi$
$401$	−11.0000	−0.549314	−0.274657	+	0.961542i	$0.588564\pi$
$402$	−2.00000	−0.0997509
$403$	−6.00000	−0.298881
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−12.0000	−0.595550
$407$	−1.00000	−0.0495682
$408$	3.00000	0.148522
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	−20.0000	−0.986527
$412$	−2.00000	−0.0985329
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	1.00000	0.0490290
$417$	10.0000	0.489702
$418$	3.00000	0.146735
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	−8.00000	−0.389434
$423$	−6.00000	−0.291730
$424$	−1.00000	−0.0485643
$425$	15.0000	0.727607
$426$	0	0
$427$	6.00000	0.290360
$428$	−13.0000	−0.628379
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	5.00000	0.240842	0.120421	−	0.992723i	$-0.461576\pi$
$431$	5.00000	0.240842	0.120421	+	0.992723i	$0.461576\pi$
$432$	−1.00000	−0.0481125
$433$	7.00000	0.336399	0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000	0.336399	0.168199	+	0.985753i	$0.446205\pi$
$434$	−18.0000	−0.864028
$435$	0	0
$436$	11.0000	0.526804
$437$	−3.00000	−0.143509
$438$	3.00000	0.143346
$439$	34.0000	1.62273	0.811366	−	0.584539i	$-0.198725\pi$
$439$	34.0000	1.62273	0.811366	+	0.584539i	$0.198725\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−3.00000	−0.142695
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	1.00000	0.0474579
$445$	0	0
$446$	16.0000	0.757622
$447$	−18.0000	−0.851371
$448$	3.00000	0.141737
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−5.00000	−0.235702
$451$	−10.0000	−0.470882
$452$	18.0000	0.846649
$453$	−1.00000	−0.0469841
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−3.00000	−0.140488
$457$	−20.0000	−0.935561	−0.467780	−	0.883845i	$-0.654946\pi$
$457$	−20.0000	−0.935561	−0.467780	+	0.883845i	$0.654946\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	−3.00000	−0.139573
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	1.00000	0.0462250
$469$	6.00000	0.277054
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	12.0000	0.551761
$474$	−14.0000	−0.643041
$475$	−15.0000	−0.688247
$476$	−9.00000	−0.412514
$477$	−1.00000	−0.0457869
$478$	−16.0000	−0.731823
$479$	−19.0000	−0.868132	−0.434066	−	0.900881i	$-0.642922\pi$
$479$	−19.0000	−0.868132	−0.434066	+	0.900881i	$0.642922\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	8.00000	0.364390
$483$	3.00000	0.136505
$484$	−10.0000	−0.454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−6.00000	−0.271886	−0.135943	−	0.990717i	$-0.543406\pi$
$487$	−6.00000	−0.271886	−0.135943	+	0.990717i	$0.543406\pi$
$488$	2.00000	0.0905357
$489$	−17.0000	−0.768767
$490$	0	0
$491$	−11.0000	−0.496423	−0.248212	−	0.968706i	$-0.579843\pi$
$491$	−11.0000	−0.496423	−0.248212	+	0.968706i	$0.579843\pi$
$492$	10.0000	0.450835
$493$	12.0000	0.540453
$494$	3.00000	0.134976
$495$	0	0
$496$	−6.00000	−0.269408
$497$	0	0
$498$	−9.00000	−0.403300
$499$	−15.0000	−0.671492	−0.335746	−	0.941953i	$-0.608988\pi$
$499$	−15.0000	−0.671492	−0.335746	+	0.941953i	$0.608988\pi$
$500$	0	0
$501$	−21.0000	−0.938211
$502$	−18.0000	−0.803379
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	−1.00000	−0.0444554
$507$	12.0000	0.532939
$508$	−7.00000	−0.310575
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	−9.00000	−0.398137
$512$	1.00000	0.0441942
$513$	−3.00000	−0.132453
$514$	29.0000	1.27914
$515$	0	0
$516$	−12.0000	−0.528271
$517$	−6.00000	−0.263880
$518$	−3.00000	−0.131812
$519$	−1.00000	−0.0438951
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−4.00000	−0.175075
$523$	32.0000	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$524$	10.0000	0.436852
$525$	15.0000	0.654654
$526$	26.0000	1.13365
$527$	18.0000	0.784092
$528$	−1.00000	−0.0435194
$529$	−22.0000	−0.956522
$530$	0	0
$531$	0	0
$532$	9.00000	0.390199
$533$	−10.0000	−0.433148
$534$	3.00000	0.129823
$535$	0	0
$536$	2.00000	0.0863868
$537$	24.0000	1.03568
$538$	−7.00000	−0.301791
$539$	2.00000	0.0861461
$540$	0	0
$541$	−27.0000	−1.16082	−0.580410	−	0.814324i	$-0.697108\pi$
$541$	−27.0000	−1.16082	−0.580410	+	0.814324i	$0.697108\pi$
$542$	32.0000	1.37452
$543$	20.0000	0.858282
$544$	−3.00000	−0.128624
$545$	0	0
$546$	−3.00000	−0.128388
$547$	−23.0000	−0.983409	−0.491704	−	0.870762i	$-0.663626\pi$
$547$	−23.0000	−0.983409	−0.491704	+	0.870762i	$0.663626\pi$
$548$	20.0000	0.854358
$549$	2.00000	0.0853579
$550$	−5.00000	−0.213201
$551$	−12.0000	−0.511217
$552$	1.00000	0.0425628
$553$	42.0000	1.78602
$554$	−1.00000	−0.0424859
$555$	0	0
$556$	−10.0000	−0.424094
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−6.00000	−0.254000
$559$	12.0000	0.507546
$560$	0	0
$561$	3.00000	0.126660
$562$	−31.0000	−1.30766
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	13.0000	0.546431
$567$	3.00000	0.125988
$568$	0	0
$569$	19.0000	0.796521	0.398261	−	0.917272i	$-0.369614\pi$
$569$	19.0000	0.796521	0.398261	+	0.917272i	$0.369614\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	1.00000	0.0418121
$573$	−3.00000	−0.125327
$574$	−30.0000	−1.25218
$575$	5.00000	0.208514
$576$	1.00000	0.0416667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	−8.00000	−0.332756
$579$	18.0000	0.748054
$580$	0	0
$581$	27.0000	1.12015
$582$	10.0000	0.414513
$583$	−1.00000	−0.0414158
$584$	−3.00000	−0.124141
$585$	0	0
$586$	31.0000	1.28060
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	−2.00000	−0.0824786
$589$	−18.0000	−0.741677
$590$	0	0
$591$	−13.0000	−0.534749
$592$	−1.00000	−0.0410997
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	18.0000	0.737309
$597$	16.0000	0.654836
$598$	−1.00000	−0.0408930
$599$	34.0000	1.38920	0.694601	−	0.719395i	$-0.255581\pi$
$599$	34.0000	1.38920	0.694601	+	0.719395i	$0.255581\pi$
$600$	5.00000	0.204124
$601$	−1.00000	−0.0407909	−0.0203954	−	0.999792i	$-0.506493\pi$
$601$	−1.00000	−0.0407909	−0.0203954	+	0.999792i	$0.506493\pi$
$602$	36.0000	1.46725
$603$	2.00000	0.0814463
$604$	1.00000	0.0406894
$605$	0	0
$606$	6.00000	0.243733
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	3.00000	0.121666
$609$	12.0000	0.486265
$610$	0	0
$611$	−6.00000	−0.242734
$612$	−3.00000	−0.121268
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	3.00000	0.120873
$617$	−32.0000	−1.28827	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	−32.0000	−1.28827	−0.644136	+	0.764911i	$0.722783\pi$
$618$	2.00000	0.0804518
$619$	−18.0000	−0.723481	−0.361741	−	0.932279i	$-0.617817\pi$
$619$	−18.0000	−0.723481	−0.361741	+	0.932279i	$0.617817\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	8.00000	0.320771
$623$	−9.00000	−0.360577
$624$	−1.00000	−0.0400320
$625$	25.0000	1.00000
$626$	30.0000	1.19904
$627$	−3.00000	−0.119808
$628$	−4.00000	−0.159617
$629$	3.00000	0.119618
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	14.0000	0.556890
$633$	8.00000	0.317971
$634$	18.0000	0.714871
$635$	0	0
$636$	1.00000	0.0396526
$637$	2.00000	0.0792429
$638$	−4.00000	−0.158362
$639$	0	0
$640$	0	0
$641$	−32.0000	−1.26392	−0.631962	−	0.774999i	$-0.717750\pi$
$641$	−32.0000	−1.26392	−0.631962	+	0.774999i	$0.717750\pi$
$642$	13.0000	0.513069
$643$	31.0000	1.22252	0.611260	−	0.791430i	$-0.290663\pi$
$643$	31.0000	1.22252	0.611260	+	0.791430i	$0.290663\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	−9.00000	−0.354100
$647$	31.0000	1.21874	0.609368	−	0.792888i	$-0.291423\pi$
$647$	31.0000	1.21874	0.609368	+	0.792888i	$0.291423\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−5.00000	−0.196116
$651$	18.0000	0.705476
$652$	17.0000	0.665771
$653$	−8.00000	−0.313064	−0.156532	−	0.987673i	$-0.550031\pi$
$653$	−8.00000	−0.313064	−0.156532	+	0.987673i	$0.550031\pi$
$654$	−11.0000	−0.430134
$655$	0	0
$656$	−10.0000	−0.390434
$657$	−3.00000	−0.117041
$658$	−18.0000	−0.701713
$659$	−32.0000	−1.24654	−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000	−1.24654	−0.623272	+	0.782006i	$0.714197\pi$
$660$	0	0
$661$	1.00000	0.0388955	0.0194477	−	0.999811i	$-0.493809\pi$
$661$	1.00000	0.0388955	0.0194477	+	0.999811i	$0.493809\pi$
$662$	28.0000	1.08825
$663$	3.00000	0.116510
$664$	9.00000	0.349268
$665$	0	0
$666$	−1.00000	−0.0387492
$667$	4.00000	0.154881
$668$	21.0000	0.812514
$669$	−16.0000	−0.618596
$670$	0	0
$671$	2.00000	0.0772091
$672$	−3.00000	−0.115728
$673$	−51.0000	−1.96591	−0.982953	−	0.183858i	$-0.941141\pi$
$673$	−51.0000	−1.96591	−0.982953	+	0.183858i	$0.941141\pi$
$674$	21.0000	0.808890
$675$	5.00000	0.192450
$676$	−12.0000	−0.461538
$677$	13.0000	0.499631	0.249815	−	0.968294i	$-0.419630\pi$
$677$	13.0000	0.499631	0.249815	+	0.968294i	$0.419630\pi$
$678$	−18.0000	−0.691286
$679$	−30.0000	−1.15129
$680$	0	0
$681$	12.0000	0.459841
$682$	−6.00000	−0.229752
$683$	−2.00000	−0.0765279	−0.0382639	−	0.999268i	$-0.512183\pi$
$683$	−2.00000	−0.0765279	−0.0382639	+	0.999268i	$0.512183\pi$
$684$	3.00000	0.114708
$685$	0	0
$686$	−15.0000	−0.572703
$687$	0	0
$688$	12.0000	0.457496
$689$	−1.00000	−0.0380970
$690$	0	0
$691$	24.0000	0.913003	0.456502	−	0.889723i	$-0.349102\pi$
$691$	24.0000	0.913003	0.456502	+	0.889723i	$0.349102\pi$
$692$	1.00000	0.0380143
$693$	3.00000	0.113961
$694$	22.0000	0.835109
$695$	0	0
$696$	4.00000	0.151620
$697$	30.0000	1.13633
$698$	10.0000	0.378506
$699$	6.00000	0.226941
$700$	−15.0000	−0.566947
$701$	−16.0000	−0.604312	−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000	−0.604312	−0.302156	+	0.953259i	$0.597706\pi$
$702$	−1.00000	−0.0377426
$703$	−3.00000	−0.113147
$704$	1.00000	0.0376889
$705$	0	0
$706$	2.00000	0.0752710
$707$	−18.0000	−0.676960
$708$	0	0
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	−3.00000	−0.112430
$713$	6.00000	0.224702
$714$	9.00000	0.336817
$715$	0	0
$716$	−24.0000	−0.896922
$717$	16.0000	0.597531
$718$	30.0000	1.11959
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	−10.0000	−0.372161
$723$	−8.00000	−0.297523
$724$	−20.0000	−0.743294
$725$	20.0000	0.742781
$726$	10.0000	0.371135
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	3.00000	0.111187
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.0000	−1.33151
$732$	−2.00000	−0.0739221
$733$	18.0000	0.664845	0.332423	−	0.943131i	$-0.392134\pi$
$733$	18.0000	0.664845	0.332423	+	0.943131i	$0.392134\pi$
$734$	−21.0000	−0.775124
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	2.00000	0.0736709
$738$	−10.0000	−0.368105
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	0	0
$741$	−3.00000	−0.110208
$742$	−3.00000	−0.110133
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	20.0000	0.732252
$747$	9.00000	0.329293
$748$	−3.00000	−0.109691
$749$	−39.0000	−1.42503
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	−6.00000	−0.218797
$753$	18.0000	0.655956
$754$	−4.00000	−0.145671
$755$	0	0
$756$	−3.00000	−0.109109
$757$	3.00000	0.109037	0.0545184	−	0.998513i	$-0.482638\pi$
$757$	3.00000	0.109037	0.0545184	+	0.998513i	$0.482638\pi$
$758$	−4.00000	−0.145287
$759$	1.00000	0.0362977
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	7.00000	0.253583
$763$	33.0000	1.19468
$764$	3.00000	0.108536
$765$	0	0
$766$	1.00000	0.0361315
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	0	0
$771$	−29.0000	−1.04441
$772$	−18.0000	−0.647834
$773$	−35.0000	−1.25886	−0.629431	−	0.777056i	$-0.716712\pi$
$773$	−35.0000	−1.25886	−0.629431	+	0.777056i	$0.716712\pi$
$774$	12.0000	0.431331
$775$	30.0000	1.07763
$776$	−10.0000	−0.358979
$777$	3.00000	0.107624
$778$	30.0000	1.07555
$779$	−30.0000	−1.07486
$780$	0	0
$781$	0	0
$782$	3.00000	0.107280
$783$	4.00000	0.142948
$784$	2.00000	0.0714286
$785$	0	0
$786$	−10.0000	−0.356688
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	13.0000	0.463106
$789$	−26.0000	−0.925625
$790$	0	0
$791$	54.0000	1.92002
$792$	1.00000	0.0355335
$793$	2.00000	0.0710221
$794$	−12.0000	−0.425864
$795$	0	0
$796$	−16.0000	−0.567105
$797$	16.0000	0.566749	0.283375	−	0.959009i	$-0.408546\pi$
$797$	16.0000	0.566749	0.283375	+	0.959009i	$0.408546\pi$
$798$	−9.00000	−0.318597
$799$	18.0000	0.636794
$800$	−5.00000	−0.176777
$801$	−3.00000	−0.106000
$802$	−11.0000	−0.388424
$803$	−3.00000	−0.105868
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	−6.00000	−0.211341
$807$	7.00000	0.246412
$808$	−6.00000	−0.211079
$809$	21.0000	0.738321	0.369160	−	0.929366i	$-0.379645\pi$
$809$	21.0000	0.738321	0.369160	+	0.929366i	$0.379645\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−12.0000	−0.421117
$813$	−32.0000	−1.12229
$814$	−1.00000	−0.0350500
$815$	0	0
$816$	3.00000	0.105021
$817$	36.0000	1.25948
$818$	−6.00000	−0.209785
$819$	3.00000	0.104828
$820$	0	0
$821$	−45.0000	−1.57051	−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000	−1.57051	−0.785255	+	0.619172i	$0.787468\pi$
$822$	−20.0000	−0.697580
$823$	29.0000	1.01088	0.505438	−	0.862863i	$-0.331331\pi$
$823$	29.0000	1.01088	0.505438	+	0.862863i	$0.331331\pi$
$824$	−2.00000	−0.0696733
$825$	5.00000	0.174078
$826$	0	0
$827$	2.00000	0.0695468	0.0347734	−	0.999395i	$-0.488929\pi$
$827$	2.00000	0.0695468	0.0347734	+	0.999395i	$0.488929\pi$
$828$	−1.00000	−0.0347524
$829$	−17.0000	−0.590434	−0.295217	−	0.955430i	$-0.595392\pi$
$829$	−17.0000	−0.590434	−0.295217	+	0.955430i	$0.595392\pi$
$830$	0	0
$831$	1.00000	0.0346896
$832$	1.00000	0.0346688
$833$	−6.00000	−0.207888
$834$	10.0000	0.346272
$835$	0	0
$836$	3.00000	0.103757
$837$	6.00000	0.207390
$838$	−21.0000	−0.725433
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	26.0000	0.896019
$843$	31.0000	1.06770
$844$	−8.00000	−0.275371
$845$	0	0
$846$	−6.00000	−0.206284
$847$	−30.0000	−1.03081
$848$	−1.00000	−0.0343401
$849$	−13.0000	−0.446159
$850$	15.0000	0.514496
$851$	1.00000	0.0342796
$852$	0	0
$853$	−35.0000	−1.19838	−0.599189	−	0.800608i	$-0.704510\pi$
$853$	−35.0000	−1.19838	−0.599189	+	0.800608i	$0.704510\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	−13.0000	−0.444331
$857$	−45.0000	−1.53717	−0.768585	−	0.639747i	$-0.779039\pi$
$857$	−45.0000	−1.53717	−0.768585	+	0.639747i	$0.779039\pi$
$858$	−1.00000	−0.0341394
$859$	49.0000	1.67186	0.835929	−	0.548837i	$-0.184929\pi$
$859$	49.0000	1.67186	0.835929	+	0.548837i	$0.184929\pi$
$860$	0	0
$861$	30.0000	1.02240
$862$	5.00000	0.170301
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	7.00000	0.237870
$867$	8.00000	0.271694
$868$	−18.0000	−0.610960
$869$	14.0000	0.474917
$870$	0	0
$871$	2.00000	0.0677674
$872$	11.0000	0.372507
$873$	−10.0000	−0.338449
$874$	−3.00000	−0.101477
$875$	0	0
$876$	3.00000	0.101361
$877$	−30.0000	−1.01303	−0.506514	−	0.862232i	$-0.669066\pi$
$877$	−30.0000	−1.01303	−0.506514	+	0.862232i	$0.669066\pi$
$878$	34.0000	1.14744
$879$	−31.0000	−1.04560
$880$	0	0
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	2.00000	0.0673435
$883$	21.0000	0.706706	0.353353	−	0.935490i	$-0.385041\pi$
$883$	21.0000	0.706706	0.353353	+	0.935490i	$0.385041\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	1.00000	0.0335578
$889$	−21.0000	−0.704317
$890$	0	0
$891$	1.00000	0.0335013
$892$	16.0000	0.535720
$893$	−18.0000	−0.602347
$894$	−18.0000	−0.602010
$895$	0	0
$896$	3.00000	0.100223
$897$	1.00000	0.0333890
$898$	6.00000	0.200223
$899$	24.0000	0.800445
$900$	−5.00000	−0.166667
$901$	3.00000	0.0999445
$902$	−10.0000	−0.332964
$903$	−36.0000	−1.19800
$904$	18.0000	0.598671
$905$	0	0
$906$	−1.00000	−0.0332228
$907$	5.00000	0.166022	0.0830111	−	0.996549i	$-0.473546\pi$
$907$	5.00000	0.166022	0.0830111	+	0.996549i	$0.473546\pi$
$908$	−12.0000	−0.398234
$909$	−6.00000	−0.199007
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	−3.00000	−0.0993399
$913$	9.00000	0.297857
$914$	−20.0000	−0.661541
$915$	0	0
$916$	0	0
$917$	30.0000	0.990687
$918$	3.00000	0.0990148
$919$	22.0000	0.725713	0.362857	−	0.931845i	$-0.381802\pi$
$919$	22.0000	0.725713	0.362857	+	0.931845i	$0.381802\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	30.0000	0.987997
$923$	0	0
$924$	−3.00000	−0.0986928
$925$	5.00000	0.164399
$926$	−26.0000	−0.854413
$927$	−2.00000	−0.0656886
$928$	−4.00000	−0.131306
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	−6.00000	−0.196537
$933$	−8.00000	−0.261908
$934$	−6.00000	−0.196326
$935$	0	0
$936$	1.00000	0.0326860
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	6.00000	0.195907
$939$	−30.0000	−0.979013
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	4.00000	0.130327
$943$	10.0000	0.325645
$944$	0	0
$945$	0	0
$946$	12.0000	0.390154
$947$	10.0000	0.324956	0.162478	−	0.986712i	$-0.448051\pi$
$947$	10.0000	0.324956	0.162478	+	0.986712i	$0.448051\pi$
$948$	−14.0000	−0.454699
$949$	−3.00000	−0.0973841
$950$	−15.0000	−0.486664
$951$	−18.0000	−0.583690
$952$	−9.00000	−0.291692
$953$	−28.0000	−0.907009	−0.453504	−	0.891254i	$-0.649826\pi$
$953$	−28.0000	−0.907009	−0.453504	+	0.891254i	$0.649826\pi$
$954$	−1.00000	−0.0323762
$955$	0	0
$956$	−16.0000	−0.517477
$957$	4.00000	0.129302
$958$	−19.0000	−0.613862
$959$	60.0000	1.93750
$960$	0	0
$961$	5.00000	0.161290
$962$	−1.00000	−0.0322413
$963$	−13.0000	−0.418919
$964$	8.00000	0.257663
$965$	0	0
$966$	3.00000	0.0965234
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	−10.0000	−0.321412
$969$	9.00000	0.289122
$970$	0	0
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	−1.00000	−0.0320750
$973$	−30.0000	−0.961756
$974$	−6.00000	−0.192252
$975$	5.00000	0.160128
$976$	2.00000	0.0640184
$977$	−43.0000	−1.37569	−0.687846	−	0.725857i	$-0.741444\pi$
$977$	−43.0000	−1.37569	−0.687846	+	0.725857i	$0.741444\pi$
$978$	−17.0000	−0.543600
$979$	−3.00000	−0.0958804
$980$	0	0
$981$	11.0000	0.351203
$982$	−11.0000	−0.351024
$983$	46.0000	1.46717	0.733586	−	0.679597i	$-0.237845\pi$
$983$	46.0000	1.46717	0.733586	+	0.679597i	$0.237845\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	12.0000	0.382158
$987$	18.0000	0.572946
$988$	3.00000	0.0954427
$989$	−12.0000	−0.381578
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	−6.00000	−0.190500
$993$	−28.0000	−0.888553
$994$	0	0
$995$	0	0
$996$	−9.00000	−0.285176
$997$	27.0000	0.855099	0.427549	−	0.903992i	$-0.359377\pi$
$997$	27.0000	0.855099	0.427549	+	0.903992i	$0.359377\pi$
$998$	−15.0000	−0.474817
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	222.2.a.d.1.1	✓	1
3.2	odd	2		666.2.a.b.1.1		1
4.3	odd	2		1776.2.a.h.1.1		1
5.4	even	2		5550.2.a.n.1.1		1
8.3	odd	2		7104.2.a.f.1.1		1
8.5	even	2		7104.2.a.v.1.1		1
12.11	even	2		5328.2.a.h.1.1		1
37.36	even	2		8214.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.2.a.d.1.1	✓	1	1.1	even	1	trivial
666.2.a.b.1.1		1	3.2	odd	2
1776.2.a.h.1.1		1	4.3	odd	2
5328.2.a.h.1.1		1	12.11	even	2
5550.2.a.n.1.1		1	5.4	even	2
7104.2.a.f.1.1		1	8.3	odd	2
7104.2.a.v.1.1		1	8.5	even	2
8214.2.a.b.1.1		1	37.36	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 222 = 2 \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	222.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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