LMFDB - Embedded newform 222.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [222,2,Mod(1,222)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 222 = 2 \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	222.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:	$1.77267892487$
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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} +2.00000 q^{20} +4.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} +2.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} -8.00000 q^{38} -6.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} -8.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} +6.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -8.00000 q^{55} -8.00000 q^{57} +6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} -4.00000 q^{66} -12.0000 q^{67} +6.00000 q^{68} -1.00000 q^{72} +10.0000 q^{73} -1.00000 q^{74} +1.00000 q^{75} +8.00000 q^{76} +6.00000 q^{78} -12.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} +12.0000 q^{85} +8.00000 q^{86} +6.00000 q^{87} +4.00000 q^{88} -10.0000 q^{89} -2.00000 q^{90} -4.00000 q^{93} -8.00000 q^{94} +16.0000 q^{95} +1.00000 q^{96} -6.00000 q^{97} +7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	1.00000	0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−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
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−1.00000	−0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	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
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	2.00000	0.365148
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−8.00000	−1.29777
$39$	−6.00000	−0.960769
$40$	−2.00000	−0.316228
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	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
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	−6.00000	−0.840168
$52$	6.00000	0.832050
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	−8.00000	−1.07872
$56$	0	0
$57$	−8.00000	−1.05963
$58$	6.00000	0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	−2.00000	−0.258199
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	12.0000	1.48842
$66$	−4.00000	−0.492366
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−1.00000	−0.116248
$75$	1.00000	0.115470
$76$	8.00000	0.917663
$77$	0	0
$78$	6.00000	0.679366
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	8.00000	0.862662
$87$	6.00000	0.643268
$88$	4.00000	0.426401
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	0	0
$93$	−4.00000	−0.414781
$94$	−8.00000	−0.825137
$95$	16.0000	1.64157
$96$	1.00000	0.102062
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	7.00000	0.707107
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	6.00000	0.594089
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	8.00000	0.762770
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	−6.00000	−0.557086
$117$	6.00000	0.554700
$118$	4.00000	0.368230
$119$	0	0
$120$	2.00000	0.182574
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	6.00000	0.541002
$124$	4.00000	0.359211
$125$	−12.0000	−1.07331
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	−12.0000	−1.05247
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	12.0000	1.03664
$135$	−2.00000	−0.172133
$136$	−6.00000	−0.514496
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−24.0000	−2.00698
$144$	1.00000	0.0833333
$145$	−12.0000	−0.996546
$146$	−10.0000	−0.827606
$147$	7.00000	0.577350
$148$	1.00000	0.0821995
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−1.00000	−0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−8.00000	−0.648886
$153$	6.00000	0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	−6.00000	−0.480384
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	12.0000	0.954669
$159$	−6.00000	−0.475831
$160$	−2.00000	−0.158114
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−6.00000	−0.468521
$165$	8.00000	0.622799
$166$	4.00000	0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−12.0000	−0.920358
$171$	8.00000	0.611775
$172$	−8.00000	−0.609994
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−4.00000	−0.301511
$177$	4.00000	0.300658
$178$	10.0000	0.749532
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	2.00000	0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	2.00000	0.147043
$186$	4.00000	0.293294
$187$	−24.0000	−1.75505
$188$	8.00000	0.583460
$189$	0	0
$190$	−16.0000	−1.16076
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	−1.00000	−0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	6.00000	0.430775
$195$	−12.0000	−0.859338
$196$	−7.00000	−0.500000
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	4.00000	0.284268
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	1.00000	0.0707107
$201$	12.0000	0.846415
$202$	2.00000	0.140720
$203$	0	0
$204$	−6.00000	−0.420084
$205$	−12.0000	−0.838116
$206$	−4.00000	−0.278693
$207$	0	0
$208$	6.00000	0.416025
$209$	−32.0000	−2.21349
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	4.00000	0.273434
$215$	−16.0000	−1.09119
$216$	1.00000	0.0680414
$217$	0	0
$218$	10.0000	0.677285
$219$	−10.0000	−0.675737
$220$	−8.00000	−0.539360
$221$	36.0000	2.42162
$222$	1.00000	0.0671156
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−14.0000	−0.931266
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−8.00000	−0.529813
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	−6.00000	−0.392232
$235$	16.0000	1.04372
$236$	−4.00000	−0.260378
$237$	12.0000	0.779484
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	−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
$244$	−2.00000	−0.128037
$245$	−14.0000	−0.894427
$246$	−6.00000	−0.382546
$247$	48.0000	3.05417
$248$	−4.00000	−0.254000
$249$	4.00000	0.253490
$250$	12.0000	0.758947
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	0	0
$254$	−16.0000	−1.00393
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	−8.00000	−0.498058
$259$	0	0
$260$	12.0000	0.744208
$261$	−6.00000	−0.371391
$262$	−20.0000	−1.23560
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−4.00000	−0.246183
$265$	12.0000	0.737154
$266$	0	0
$267$	10.0000	0.611990
$268$	−12.0000	−0.733017
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	2.00000	0.121716
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−18.0000	−1.08742
$275$	4.00000	0.241209
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	4.00000	0.239904
$279$	4.00000	0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	8.00000	0.476393
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	0	0
$285$	−16.0000	−0.947758
$286$	24.0000	1.41915
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	12.0000	0.704664
$291$	6.00000	0.351726
$292$	10.0000	0.585206
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	−7.00000	−0.408248
$295$	−8.00000	−0.465778
$296$	−1.00000	−0.0581238
$297$	4.00000	0.232104
$298$	18.0000	1.04271
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	0	0
$303$	2.00000	0.114897
$304$	8.00000	0.458831
$305$	−4.00000	−0.229039
$306$	−6.00000	−0.342997
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	−8.00000	−0.454369
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	6.00000	0.339683
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	6.00000	0.336463
$319$	24.0000	1.34374
$320$	2.00000	0.111803
$321$	4.00000	0.223258
$322$	0	0
$323$	48.0000	2.67079
$324$	1.00000	0.0555556
$325$	−6.00000	−0.332820
$326$	8.00000	0.443079
$327$	10.0000	0.553001
$328$	6.00000	0.331295
$329$	0	0
$330$	−8.00000	−0.440386
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	−4.00000	−0.219529
$333$	1.00000	0.0547997
$334$	8.00000	0.437741
$335$	−24.0000	−1.31126
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	−23.0000	−1.25104
$339$	−14.0000	−0.760376
$340$	12.0000	0.650791
$341$	−16.0000	−0.866449
$342$	−8.00000	−0.432590
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	2.00000	0.107521
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	6.00000	0.321634
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	4.00000	0.213201
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	−2.00000	−0.105409
$361$	45.0000	2.36842
$362$	10.0000	0.525588
$363$	−5.00000	−0.262432
$364$	0	0
$365$	20.0000	1.04685
$366$	−2.00000	−0.104542
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	−2.00000	−0.103975
$371$	0	0
$372$	−4.00000	−0.207390
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	24.0000	1.24101
$375$	12.0000	0.619677
$376$	−8.00000	−0.412568
$377$	−36.0000	−1.85409
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	16.0000	0.820783
$381$	−16.0000	−0.819705
$382$	−16.0000	−0.818631
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	−8.00000	−0.406663
$388$	−6.00000	−0.304604
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	12.0000	0.607644
$391$	0	0
$392$	7.00000	0.353553
$393$	−20.0000	−1.00887
$394$	18.0000	0.906827
$395$	−24.0000	−1.20757
$396$	−4.00000	−0.201008
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−12.0000	−0.598506
$403$	24.0000	1.19553
$404$	−2.00000	−0.0995037
$405$	2.00000	0.0993808
$406$	0	0
$407$	−4.00000	−0.198273
$408$	6.00000	0.297044
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	12.0000	0.592638
$411$	−18.0000	−0.887875
$412$	4.00000	0.197066
$413$	0	0
$414$	0	0
$415$	−8.00000	−0.392705
$416$	−6.00000	−0.294174
$417$	4.00000	0.195881
$418$	32.0000	1.56517
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	12.0000	0.584151
$423$	8.00000	0.388973
$424$	−6.00000	−0.291386
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	−4.00000	−0.193347
$429$	24.0000	1.15873
$430$	16.0000	0.771589
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	−1.00000	−0.0481125
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	−10.0000	−0.478913
$437$	0	0
$438$	10.0000	0.477818
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	8.00000	0.381385
$441$	−7.00000	−0.333333
$442$	−36.0000	−1.71235
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	−1.00000	−0.0474579
$445$	−20.0000	−0.948091
$446$	16.0000	0.757622
$447$	18.0000	0.851371
$448$	0	0
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	1.00000	0.0471405
$451$	24.0000	1.13012
$452$	14.0000	0.658505
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	8.00000	0.374634
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−22.0000	−1.02799
$459$	−6.00000	−0.280056
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	−6.00000	−0.278543
$465$	−8.00000	−0.370991
$466$	22.0000	1.01913
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	6.00000	0.277350
$469$	0	0
$470$	−16.0000	−0.738025
$471$	2.00000	0.0921551
$472$	4.00000	0.184115
$473$	32.0000	1.47136
$474$	−12.0000	−0.551178
$475$	−8.00000	−0.367065
$476$	0	0
$477$	6.00000	0.274721
$478$	8.00000	0.365911
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	2.00000	0.0912871
$481$	6.00000	0.273576
$482$	−26.0000	−1.18427
$483$	0	0
$484$	5.00000	0.227273
$485$	−12.0000	−0.544892
$486$	1.00000	0.0453609
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	2.00000	0.0905357
$489$	8.00000	0.361773
$490$	14.0000	0.632456
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	6.00000	0.270501
$493$	−36.0000	−1.62136
$494$	−48.0000	−2.15962
$495$	−8.00000	−0.359573
$496$	4.00000	0.179605
$497$	0	0
$498$	−4.00000	−0.179244
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	−12.0000	−0.536656
$501$	8.00000	0.357414
$502$	−20.0000	−0.892644
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	−23.0000	−1.02147
$508$	16.0000	0.709885
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	12.0000	0.531369
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−8.00000	−0.353209
$514$	2.00000	0.0882162
$515$	8.00000	0.352522
$516$	8.00000	0.352180
$517$	−32.0000	−1.40736
$518$	0	0
$519$	2.00000	0.0877903
$520$	−12.0000	−0.526235
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	6.00000	0.262613
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	0	0
$527$	24.0000	1.04546
$528$	4.00000	0.174078
$529$	−23.0000	−1.00000
$530$	−12.0000	−0.521247
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−36.0000	−1.55933
$534$	−10.0000	−0.432742
$535$	−8.00000	−0.345870
$536$	12.0000	0.518321
$537$	−12.0000	−0.517838
$538$	10.0000	0.431131
$539$	28.0000	1.20605
$540$	−2.00000	−0.0860663
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	−24.0000	−1.03089
$543$	10.0000	0.429141
$544$	−6.00000	−0.257248
$545$	−20.0000	−0.856706
$546$	0	0
$547$	24.0000	1.02617	0.513083	−	0.858339i	$-0.328503\pi$
$547$	24.0000	1.02617	0.513083	+	0.858339i	$0.328503\pi$
$548$	18.0000	0.768922
$549$	−2.00000	−0.0853579
$550$	−4.00000	−0.170561
$551$	−48.0000	−2.04487
$552$	0	0
$553$	0	0
$554$	18.0000	0.764747
$555$	−2.00000	−0.0848953
$556$	−4.00000	−0.169638
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	−4.00000	−0.169334
$559$	−48.0000	−2.03018
$560$	0	0
$561$	24.0000	1.01328
$562$	−6.00000	−0.253095
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	−8.00000	−0.336861
$565$	28.0000	1.17797
$566$	8.00000	0.336265
$567$	0	0
$568$	0	0
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	16.0000	0.670166
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	−24.0000	−1.00349
$573$	−16.0000	−0.668410
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−19.0000	−0.790296
$579$	−2.00000	−0.0831172
$580$	−12.0000	−0.498273
$581$	0	0
$582$	−6.00000	−0.248708
$583$	−24.0000	−0.993978
$584$	−10.0000	−0.413803
$585$	12.0000	0.496139
$586$	26.0000	1.07405
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	7.00000	0.288675
$589$	32.0000	1.31854
$590$	8.00000	0.329355
$591$	18.0000	0.740421
$592$	1.00000	0.0410997
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−18.0000	−0.737309
$597$	20.0000	0.818546
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	−1.00000	−0.0408248
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	10.0000	0.406558
$606$	−2.00000	−0.0812444
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	4.00000	0.161955
$611$	48.0000	1.94187
$612$	6.00000	0.242536
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	4.00000	0.161427
$615$	12.0000	0.483887
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	4.00000	0.160904
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	−24.0000	−0.962312
$623$	0	0
$624$	−6.00000	−0.240192
$625$	−19.0000	−0.760000
$626$	−10.0000	−0.399680
$627$	32.0000	1.27796
$628$	−2.00000	−0.0798087
$629$	6.00000	0.239236
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	12.0000	0.477334
$633$	12.0000	0.476957
$634$	2.00000	0.0794301
$635$	32.0000	1.26988
$636$	−6.00000	−0.237915
$637$	−42.0000	−1.66410
$638$	−24.0000	−0.950169
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	−4.00000	−0.157867
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	16.0000	0.629999
$646$	−48.0000	−1.88853
$647$	40.0000	1.57256	0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000	1.57256	0.786281	+	0.617869i	$0.212004\pi$
$648$	−1.00000	−0.0392837
$649$	16.0000	0.628055
$650$	6.00000	0.235339
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	−10.0000	−0.391031
$655$	40.0000	1.56293
$656$	−6.00000	−0.234261
$657$	10.0000	0.390137
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	8.00000	0.311400
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	32.0000	1.24372
$663$	−36.0000	−1.39812
$664$	4.00000	0.155230
$665$	0	0
$666$	−1.00000	−0.0387492
$667$	0	0
$668$	−8.00000	−0.309529
$669$	16.0000	0.618596
$670$	24.0000	0.927201
$671$	8.00000	0.308837
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	30.0000	1.15556
$675$	1.00000	0.0384900
$676$	23.0000	0.884615
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	14.0000	0.537667
$679$	0	0
$680$	−12.0000	−0.460179
$681$	−12.0000	−0.459841
$682$	16.0000	0.612672
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	8.00000	0.305888
$685$	36.0000	1.37549
$686$	0	0
$687$	−22.0000	−0.839352
$688$	−8.00000	−0.304997
$689$	36.0000	1.37149
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	12.0000	0.455514
$695$	−8.00000	−0.303457
$696$	−6.00000	−0.227429
$697$	−36.0000	−1.36360
$698$	2.00000	0.0757011
$699$	22.0000	0.832116
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	6.00000	0.226455
$703$	8.00000	0.301726
$704$	−4.00000	−0.150756
$705$	−16.0000	−0.602595
$706$	−30.0000	−1.12906
$707$	0	0
$708$	4.00000	0.150329
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	10.0000	0.374766
$713$	0	0
$714$	0	0
$715$	−48.0000	−1.79510
$716$	12.0000	0.448461
$717$	8.00000	0.298765
$718$	8.00000	0.298557
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	−45.0000	−1.67473
$723$	−26.0000	−0.966950
$724$	−10.0000	−0.371647
$725$	6.00000	0.222834
$726$	5.00000	0.185567
$727$	−52.0000	−1.92857	−0.964287	−	0.264861i	$-0.914674\pi$
$727$	−52.0000	−1.92857	−0.964287	+	0.264861i	$0.914674\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−20.0000	−0.740233
$731$	−48.0000	−1.77534
$732$	2.00000	0.0739221
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−32.0000	−1.18114
$735$	14.0000	0.516398
$736$	0	0
$737$	48.0000	1.76810
$738$	6.00000	0.220863
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	2.00000	0.0735215
$741$	−48.0000	−1.76332
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	4.00000	0.146647
$745$	−36.0000	−1.31894
$746$	10.0000	0.366126
$747$	−4.00000	−0.146352
$748$	−24.0000	−0.877527
$749$	0	0
$750$	−12.0000	−0.438178
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	8.00000	0.291730
$753$	−20.0000	−0.728841
$754$	36.0000	1.31104
$755$	0	0
$756$	0	0
$757$	46.0000	1.67190	0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000	1.67190	0.835949	+	0.548807i	$0.184918\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	−16.0000	−0.580381
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	16.0000	0.579619
$763$	0	0
$764$	16.0000	0.578860
$765$	12.0000	0.433861
$766$	−8.00000	−0.289052
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	2.00000	0.0719816
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	8.00000	0.287554
$775$	−4.00000	−0.143684
$776$	6.00000	0.215387
$777$	0	0
$778$	22.0000	0.788738
$779$	−48.0000	−1.71978
$780$	−12.0000	−0.429669
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	−7.00000	−0.250000
$785$	−4.00000	−0.142766
$786$	20.0000	0.713376
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	24.0000	0.853882
$791$	0	0
$792$	4.00000	0.142134
$793$	−12.0000	−0.426132
$794$	−14.0000	−0.496841
$795$	−12.0000	−0.425596
$796$	−20.0000	−0.708881
$797$	26.0000	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	1.00000	0.0353553
$801$	−10.0000	−0.353333
$802$	−6.00000	−0.211867
$803$	−40.0000	−1.41157
$804$	12.0000	0.423207
$805$	0	0
$806$	−24.0000	−0.845364
$807$	10.0000	0.352017
$808$	2.00000	0.0703598
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	−2.00000	−0.0702728
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	4.00000	0.140200
$815$	−16.0000	−0.560456
$816$	−6.00000	−0.210042
$817$	−64.0000	−2.23908
$818$	30.0000	1.04893
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	18.0000	0.627822
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−4.00000	−0.139347
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−52.0000	−1.80822	−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−52.0000	−1.80822	−0.904109	+	0.427303i	$0.859464\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	8.00000	0.277684
$831$	18.0000	0.624413
$832$	6.00000	0.208013
$833$	−42.0000	−1.45521
$834$	−4.00000	−0.138509
$835$	−16.0000	−0.553703
$836$	−32.0000	−1.10674
$837$	−4.00000	−0.138260
$838$	4.00000	0.138178
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	34.0000	1.17172
$843$	−6.00000	−0.206651
$844$	−12.0000	−0.413057
$845$	46.0000	1.58245
$846$	−8.00000	−0.275046
$847$	0	0
$848$	6.00000	0.206041
$849$	8.00000	0.274559
$850$	6.00000	0.205798
$851$	0	0
$852$	0	0
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	4.00000	0.136717
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	−24.0000	−0.819346
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−16.0000	−0.545595
$861$	0	0
$862$	32.0000	1.08992
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	1.00000	0.0340207
$865$	−4.00000	−0.136004
$866$	−18.0000	−0.611665
$867$	−19.0000	−0.645274
$868$	0	0
$869$	48.0000	1.62829
$870$	−12.0000	−0.406838
$871$	−72.0000	−2.43963
$872$	10.0000	0.338643
$873$	−6.00000	−0.203069
$874$	0	0
$875$	0	0
$876$	−10.0000	−0.337869
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	−28.0000	−0.944954
$879$	26.0000	0.876958
$880$	−8.00000	−0.269680
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	7.00000	0.235702
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	36.0000	1.21081
$885$	8.00000	0.268917
$886$	−36.0000	−1.20944
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	1.00000	0.0335578
$889$	0	0
$890$	20.0000	0.670402
$891$	−4.00000	−0.134005
$892$	−16.0000	−0.535720
$893$	64.0000	2.14168
$894$	−18.0000	−0.602010
$895$	24.0000	0.802232
$896$	0	0
$897$	0	0
$898$	34.0000	1.13459
$899$	−24.0000	−0.800445
$900$	−1.00000	−0.0333333
$901$	36.0000	1.19933
$902$	−24.0000	−0.799113
$903$	0	0
$904$	−14.0000	−0.465633
$905$	−20.0000	−0.664822
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	12.0000	0.398234
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	−8.00000	−0.264906
$913$	16.0000	0.529523
$914$	−18.0000	−0.595387
$915$	4.00000	0.132236
$916$	22.0000	0.726900
$917$	0	0
$918$	6.00000	0.198030
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	−2.00000	−0.0658665
$923$	0	0
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	20.0000	0.657241
$927$	4.00000	0.131377
$928$	6.00000	0.196960
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	8.00000	0.262330
$931$	−56.0000	−1.83533
$932$	−22.0000	−0.720634
$933$	−24.0000	−0.785725
$934$	12.0000	0.392652
$935$	−48.0000	−1.56977
$936$	−6.00000	−0.196116
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	16.0000	0.521862
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−2.00000	−0.0651635
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−32.0000	−1.04041
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	12.0000	0.389742
$949$	60.0000	1.94768
$950$	8.00000	0.259554
$951$	2.00000	0.0648544
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	−6.00000	−0.194257
$955$	32.0000	1.03550
$956$	−8.00000	−0.258738
$957$	−24.0000	−0.775810
$958$	16.0000	0.516937
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	−15.0000	−0.483871
$962$	−6.00000	−0.193448
$963$	−4.00000	−0.128898
$964$	26.0000	0.837404
$965$	4.00000	0.128765
$966$	0	0
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	−5.00000	−0.160706
$969$	−48.0000	−1.54198
$970$	12.0000	0.385297
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	20.0000	0.640841
$975$	6.00000	0.192154
$976$	−2.00000	−0.0640184
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−8.00000	−0.255812
$979$	40.0000	1.27841
$980$	−14.0000	−0.447214
$981$	−10.0000	−0.319275
$982$	12.0000	0.382935
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	−6.00000	−0.191273
$985$	−36.0000	−1.14706
$986$	36.0000	1.14647
$987$	0	0
$988$	48.0000	1.52708
$989$	0	0
$990$	8.00000	0.254257
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	−4.00000	−0.127000
$993$	32.0000	1.01549
$994$	0	0
$995$	−40.0000	−1.26809
$996$	4.00000	0.126745
$997$	54.0000	1.71020	0.855099	−	0.518465i	$-0.173497\pi$
$997$	54.0000	1.71020	0.855099	+	0.518465i	$0.173497\pi$
$998$	−24.0000	−0.759707
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	222.2.a.b.1.1	✓	1
3.2	odd	2		666.2.a.f.1.1		1
4.3	odd	2		1776.2.a.k.1.1		1
5.4	even	2		5550.2.a.bl.1.1		1
8.3	odd	2		7104.2.a.c.1.1		1
8.5	even	2		7104.2.a.q.1.1		1
12.11	even	2		5328.2.a.e.1.1		1
37.36	even	2		8214.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.2.a.b.1.1	✓	1	1.1	even	1	trivial
666.2.a.f.1.1		1	3.2	odd	2
1776.2.a.k.1.1		1	4.3	odd	2
5328.2.a.e.1.1		1	12.11	even	2
5550.2.a.bl.1.1		1	5.4	even	2
7104.2.a.c.1.1		1	8.3	odd	2
7104.2.a.q.1.1		1	8.5	even	2
8214.2.a.f.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 $\F_{q}$

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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