LMFDB - Embedded newform 43.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [43,2,Mod(1,43)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(43, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("43.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	43.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:	$0.343356728692$
Analytic rank:	$1$
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	$\chi$	$=$	43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} -4.00000 q^{5} +4.00000 q^{6} +1.00000 q^{9} +8.00000 q^{10} +3.00000 q^{11} -4.00000 q^{12} -5.00000 q^{13} +8.00000 q^{15} -4.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} -2.00000 q^{19} -8.00000 q^{20} -6.00000 q^{22} -1.00000 q^{23} +11.0000 q^{25} +10.0000 q^{26} +4.00000 q^{27} -6.00000 q^{29} -16.0000 q^{30} -1.00000 q^{31} +8.00000 q^{32} -6.00000 q^{33} +6.00000 q^{34} +2.00000 q^{36} +4.00000 q^{38} +10.0000 q^{39} +5.00000 q^{41} -1.00000 q^{43} +6.00000 q^{44} -4.00000 q^{45} +2.00000 q^{46} +4.00000 q^{47} +8.00000 q^{48} -7.00000 q^{49} -22.0000 q^{50} +6.00000 q^{51} -10.0000 q^{52} -5.00000 q^{53} -8.00000 q^{54} -12.0000 q^{55} +4.00000 q^{57} +12.0000 q^{58} -12.0000 q^{59} +16.0000 q^{60} +2.00000 q^{61} +2.00000 q^{62} -8.00000 q^{64} +20.0000 q^{65} +12.0000 q^{66} -3.00000 q^{67} -6.00000 q^{68} +2.00000 q^{69} +2.00000 q^{71} +2.00000 q^{73} -22.0000 q^{75} -4.00000 q^{76} -20.0000 q^{78} -8.00000 q^{79} +16.0000 q^{80} -11.0000 q^{81} -10.0000 q^{82} +15.0000 q^{83} +12.0000 q^{85} +2.00000 q^{86} +12.0000 q^{87} -4.00000 q^{89} +8.00000 q^{90} -2.00000 q^{92} +2.00000 q^{93} -8.00000 q^{94} +8.00000 q^{95} -16.0000 q^{96} +7.00000 q^{97} +14.0000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	2.00000	1.00000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	4.00000	1.63299
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	8.00000	2.52982
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−4.00000	−1.15470
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	8.00000	2.06559
$16$	−4.00000	−1.00000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−2.00000	−0.471405
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−8.00000	−1.78885
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	10.0000	1.96116
$27$	4.00000	0.769800
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−16.0000	−2.92119
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	8.00000	1.41421
$33$	−6.00000	−1.04447
$34$	6.00000	1.02899
$35$	0	0
$36$	2.00000	0.333333
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	4.00000	0.648886
$39$	10.0000	1.60128
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	6.00000	0.904534
$45$	−4.00000	−0.596285
$46$	2.00000	0.294884
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	8.00000	1.15470
$49$	−7.00000	−1.00000
$50$	−22.0000	−3.11127
$51$	6.00000	0.840168
$52$	−10.0000	−1.38675
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	−8.00000	−1.08866
$55$	−12.0000	−1.61808
$56$	0	0
$57$	4.00000	0.529813
$58$	12.0000	1.57568
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	16.0000	2.06559
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	−8.00000	−1.00000
$65$	20.0000	2.48069
$66$	12.0000	1.47710
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	−6.00000	−0.727607
$69$	2.00000	0.240772
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−22.0000	−2.54034
$76$	−4.00000	−0.458831
$77$	0	0
$78$	−20.0000	−2.26455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	16.0000	1.78885
$81$	−11.0000	−1.22222
$82$	−10.0000	−1.10432
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	2.00000	0.215666
$87$	12.0000	1.28654
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	8.00000	0.843274
$91$	0	0
$92$	−2.00000	−0.208514
$93$	2.00000	0.207390
$94$	−8.00000	−0.825137
$95$	8.00000	0.820783
$96$	−16.0000	−1.63299
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	14.0000	1.41421
$99$	3.00000	0.301511
$100$	22.0000	2.20000
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	−12.0000	−1.18818
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	0	0
$106$	10.0000	0.971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	8.00000	0.769800
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	24.0000	2.28831
$111$	0	0
$112$	0	0
$113$	−20.0000	−1.88144	−0.940721	−	0.339182i	$-0.889850\pi$
$113$	−20.0000	−1.88144	−0.940721	+	0.339182i	$0.889850\pi$
$114$	−8.00000	−0.749269
$115$	4.00000	0.373002
$116$	−12.0000	−1.11417
$117$	−5.00000	−0.462250
$118$	24.0000	2.20938
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−4.00000	−0.362143
$123$	−10.0000	−0.901670
$124$	−2.00000	−0.179605
$125$	−24.0000	−2.14663
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	−40.0000	−3.50823
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	−12.0000	−1.04447
$133$	0	0
$134$	6.00000	0.518321
$135$	−16.0000	−1.37706
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	−4.00000	−0.340503
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	−4.00000	−0.335673
$143$	−15.0000	−1.25436
$144$	−4.00000	−0.333333
$145$	24.0000	1.99309
$146$	−4.00000	−0.331042
$147$	14.0000	1.15470
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	44.0000	3.59258
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	4.00000	0.321288
$156$	20.0000	1.60128
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	16.0000	1.27289
$159$	10.0000	0.793052
$160$	−32.0000	−2.52982
$161$	0	0
$162$	22.0000	1.72848
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	10.0000	0.780869
$165$	24.0000	1.86840
$166$	−30.0000	−2.32845
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	−24.0000	−1.84072
$171$	−2.00000	−0.152944
$172$	−2.00000	−0.152499
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−24.0000	−1.81944
$175$	0	0
$176$	−12.0000	−0.904534
$177$	24.0000	1.80395
$178$	8.00000	0.599625
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−8.00000	−0.596285
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	0	0
$186$	−4.00000	−0.293294
$187$	−9.00000	−0.658145
$188$	8.00000	0.583460
$189$	0	0
$190$	−16.0000	−1.16076
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	16.0000	1.15470
$193$	3.00000	0.215945	0.107972	−	0.994154i	$-0.465564\pi$
$193$	3.00000	0.215945	0.107972	+	0.994154i	$0.465564\pi$
$194$	−14.0000	−1.00514
$195$	−40.0000	−2.86446
$196$	−14.0000	−1.00000
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−6.00000	−0.426401
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	18.0000	1.26648
$203$	0	0
$204$	12.0000	0.840168
$205$	−20.0000	−1.39686
$206$	−2.00000	−0.139347
$207$	−1.00000	−0.0695048
$208$	20.0000	1.38675
$209$	−6.00000	−0.415029
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	−10.0000	−0.686803
$213$	−4.00000	−0.274075
$214$	24.0000	1.64061
$215$	4.00000	0.272798
$216$	0	0
$217$	0	0
$218$	−14.0000	−0.948200
$219$	−4.00000	−0.270295
$220$	−24.0000	−1.61808
$221$	15.0000	1.00901
$222$	0	0
$223$	−28.0000	−1.87502	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−28.0000	−1.87502	−0.937509	+	0.347960i	$0.886874\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	40.0000	2.66076
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	8.00000	0.529813
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	10.0000	0.653720
$235$	−16.0000	−1.04372
$236$	−24.0000	−1.56227
$237$	16.0000	1.03931
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	−32.0000	−2.06559
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	4.00000	0.257130
$243$	10.0000	0.641500
$244$	4.00000	0.256074
$245$	28.0000	1.78885
$246$	20.0000	1.27515
$247$	10.0000	0.636285
$248$	0	0
$249$	−30.0000	−1.90117
$250$	48.0000	3.03579
$251$	−23.0000	−1.45175	−0.725874	−	0.687828i	$-0.758564\pi$
$251$	−23.0000	−1.45175	−0.725874	+	0.687828i	$0.758564\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	−2.00000	−0.125491
$255$	−24.0000	−1.50294
$256$	16.0000	1.00000
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	40.0000	2.48069
$261$	−6.00000	−0.371391
$262$	−16.0000	−0.988483
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	20.0000	1.22859
$266$	0	0
$267$	8.00000	0.489592
$268$	−6.00000	−0.366508
$269$	−25.0000	−1.52428	−0.762138	−	0.647414i	$-0.775850\pi$
$269$	−25.0000	−1.52428	−0.762138	+	0.647414i	$0.775850\pi$
$270$	32.0000	1.94746
$271$	23.0000	1.39715	0.698575	−	0.715537i	$-0.253818\pi$
$271$	23.0000	1.39715	0.698575	+	0.715537i	$0.253818\pi$
$272$	12.0000	0.727607
$273$	0	0
$274$	−12.0000	−0.724947
$275$	33.0000	1.98997
$276$	4.00000	0.240772
$277$	−32.0000	−1.92269	−0.961347	−	0.275340i	$-0.911209\pi$
$277$	−32.0000	−1.92269	−0.961347	+	0.275340i	$0.911209\pi$
$278$	−38.0000	−2.27909
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	16.0000	0.952786
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	4.00000	0.237356
$285$	−16.0000	−0.947758
$286$	30.0000	1.77394
$287$	0	0
$288$	8.00000	0.471405
$289$	−8.00000	−0.470588
$290$	−48.0000	−2.81866
$291$	−14.0000	−0.820695
$292$	4.00000	0.234082
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	−28.0000	−1.63299
$295$	48.0000	2.79467
$296$	0	0
$297$	12.0000	0.696311
$298$	−24.0000	−1.39028
$299$	5.00000	0.289157
$300$	−44.0000	−2.54034
$301$	0	0
$302$	40.0000	2.30174
$303$	18.0000	1.03407
$304$	8.00000	0.458831
$305$	−8.00000	−0.458079
$306$	6.00000	0.342997
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	0	0
$309$	−2.00000	−0.113776
$310$	−8.00000	−0.454369
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	20.0000	1.12867
$315$	0	0
$316$	−16.0000	−0.900070
$317$	9.00000	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$318$	−20.0000	−1.12154
$319$	−18.0000	−1.00781
$320$	32.0000	1.78885
$321$	24.0000	1.33955
$322$	0	0
$323$	6.00000	0.333849
$324$	−22.0000	−1.22222
$325$	−55.0000	−3.05085
$326$	−28.0000	−1.55078
$327$	−14.0000	−0.774202
$328$	0	0
$329$	0	0
$330$	−48.0000	−2.64231
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	30.0000	1.64646
$333$	0	0
$334$	18.0000	0.984916
$335$	12.0000	0.655630
$336$	0	0
$337$	−3.00000	−0.163420	−0.0817102	−	0.996656i	$-0.526038\pi$
$337$	−3.00000	−0.163420	−0.0817102	+	0.996656i	$0.526038\pi$
$338$	−24.0000	−1.30543
$339$	40.0000	2.17250
$340$	24.0000	1.30158
$341$	−3.00000	−0.162459
$342$	4.00000	0.216295
$343$	0	0
$344$	0	0
$345$	−8.00000	−0.430706
$346$	−12.0000	−0.645124
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	24.0000	1.28654
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	−20.0000	−1.06752
$352$	24.0000	1.27920
$353$	−31.0000	−1.64996	−0.824982	−	0.565159i	$-0.808815\pi$
$353$	−31.0000	−1.64996	−0.824982	+	0.565159i	$0.808815\pi$
$354$	−48.0000	−2.55117
$355$	−8.00000	−0.424596
$356$	−8.00000	−0.423999
$357$	0	0
$358$	−40.0000	−2.11407
$359$	19.0000	1.00278	0.501391	−	0.865221i	$-0.332822\pi$
$359$	19.0000	1.00278	0.501391	+	0.865221i	$0.332822\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−20.0000	−1.05118
$363$	4.00000	0.209946
$364$	0	0
$365$	−8.00000	−0.418739
$366$	8.00000	0.418167
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	4.00000	0.208514
$369$	5.00000	0.260290
$370$	0	0
$371$	0	0
$372$	4.00000	0.207390
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	18.0000	0.930758
$375$	48.0000	2.47871
$376$	0	0
$377$	30.0000	1.54508
$378$	0	0
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	16.0000	0.820783
$381$	−2.00000	−0.102463
$382$	32.0000	1.63726
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	0	0
$386$	−6.00000	−0.305392
$387$	−1.00000	−0.0508329
$388$	14.0000	0.710742
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	80.0000	4.05096
$391$	3.00000	0.151717
$392$	0	0
$393$	−16.0000	−0.807093
$394$	−4.00000	−0.201517
$395$	32.0000	1.61009
$396$	6.00000	0.301511
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	−28.0000	−1.40351
$399$	0	0
$400$	−44.0000	−2.20000
$401$	5.00000	0.249688	0.124844	−	0.992176i	$-0.460157\pi$
$401$	5.00000	0.249688	0.124844	+	0.992176i	$0.460157\pi$
$402$	−12.0000	−0.598506
$403$	5.00000	0.249068
$404$	−18.0000	−0.895533
$405$	44.0000	2.18638
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	40.0000	1.97546
$411$	−12.0000	−0.591916
$412$	2.00000	0.0985329
$413$	0	0
$414$	2.00000	0.0982946
$415$	−60.0000	−2.94528
$416$	−40.0000	−1.96116
$417$	−38.0000	−1.86087
$418$	12.0000	0.586939
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−4.00000	−0.194717
$423$	4.00000	0.194487
$424$	0	0
$425$	−33.0000	−1.60074
$426$	8.00000	0.387601
$427$	0	0
$428$	−24.0000	−1.16008
$429$	30.0000	1.44841
$430$	−8.00000	−0.385794
$431$	−21.0000	−1.01153	−0.505767	−	0.862670i	$-0.668791\pi$
$431$	−21.0000	−1.01153	−0.505767	+	0.862670i	$0.668791\pi$
$432$	−16.0000	−0.769800
$433$	−12.0000	−0.576683	−0.288342	−	0.957528i	$-0.593104\pi$
$433$	−12.0000	−0.576683	−0.288342	+	0.957528i	$0.593104\pi$
$434$	0	0
$435$	−48.0000	−2.30142
$436$	14.0000	0.670478
$437$	2.00000	0.0956730
$438$	8.00000	0.382255
$439$	17.0000	0.811366	0.405683	−	0.914014i	$-0.367034\pi$
$439$	17.0000	0.811366	0.405683	+	0.914014i	$0.367034\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	−30.0000	−1.42695
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	16.0000	0.758473
$446$	56.0000	2.65168
$447$	−24.0000	−1.13516
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	−22.0000	−1.03709
$451$	15.0000	0.706322
$452$	−40.0000	−1.88144
$453$	40.0000	1.87936
$454$	8.00000	0.375459
$455$	0	0
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	30.0000	1.40181
$459$	−12.0000	−0.560112
$460$	8.00000	0.373002
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	24.0000	1.11417
$465$	−8.00000	−0.370991
$466$	−12.0000	−0.555889
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	−10.0000	−0.462250
$469$	0	0
$470$	32.0000	1.47605
$471$	20.0000	0.921551
$472$	0	0
$473$	−3.00000	−0.137940
$474$	−32.0000	−1.46981
$475$	−22.0000	−1.00943
$476$	0	0
$477$	−5.00000	−0.228934
$478$	−32.0000	−1.46365
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	64.0000	2.92119
$481$	0	0
$482$	24.0000	1.09317
$483$	0	0
$484$	−4.00000	−0.181818
$485$	−28.0000	−1.27141
$486$	−20.0000	−0.907218
$487$	36.0000	1.63132	0.815658	−	0.578535i	$-0.196375\pi$
$487$	36.0000	1.63132	0.815658	+	0.578535i	$0.196375\pi$
$488$	0	0
$489$	−28.0000	−1.26620
$490$	−56.0000	−2.52982
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	−20.0000	−0.901670
$493$	18.0000	0.810679
$494$	−20.0000	−0.899843
$495$	−12.0000	−0.539360
$496$	4.00000	0.179605
$497$	0	0
$498$	60.0000	2.68866
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	−48.0000	−2.14663
$501$	18.0000	0.804181
$502$	46.0000	2.05308
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	6.00000	0.266733
$507$	−24.0000	−1.06588
$508$	2.00000	0.0887357
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	48.0000	2.12548
$511$	0	0
$512$	−32.0000	−1.41421
$513$	−8.00000	−0.353209
$514$	48.0000	2.11719
$515$	−4.00000	−0.176261
$516$	4.00000	0.176090
$517$	12.0000	0.527759
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	12.0000	0.525226
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	36.0000	1.56967
$527$	3.00000	0.130682
$528$	24.0000	1.04447
$529$	−22.0000	−0.956522
$530$	−40.0000	−1.73749
$531$	−12.0000	−0.520756
$532$	0	0
$533$	−25.0000	−1.08287
$534$	−16.0000	−0.692388
$535$	48.0000	2.07522
$536$	0	0
$537$	−40.0000	−1.72613
$538$	50.0000	2.15565
$539$	−21.0000	−0.904534
$540$	−32.0000	−1.37706
$541$	1.00000	0.0429934	0.0214967	−	0.999769i	$-0.493157\pi$
$541$	1.00000	0.0429934	0.0214967	+	0.999769i	$0.493157\pi$
$542$	−46.0000	−1.97587
$543$	−20.0000	−0.858282
$544$	−24.0000	−1.02899
$545$	−28.0000	−1.19939
$546$	0	0
$547$	−29.0000	−1.23995	−0.619975	−	0.784621i	$-0.712857\pi$
$547$	−29.0000	−1.23995	−0.619975	+	0.784621i	$0.712857\pi$
$548$	12.0000	0.512615
$549$	2.00000	0.0853579
$550$	−66.0000	−2.81425
$551$	12.0000	0.511217
$552$	0	0
$553$	0	0
$554$	64.0000	2.71910
$555$	0	0
$556$	38.0000	1.61156
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	2.00000	0.0846668
$559$	5.00000	0.211477
$560$	0	0
$561$	18.0000	0.759961
$562$	−38.0000	−1.60293
$563$	37.0000	1.55936	0.779682	−	0.626176i	$-0.215381\pi$
$563$	37.0000	1.55936	0.779682	+	0.626176i	$0.215381\pi$
$564$	−16.0000	−0.673722
$565$	80.0000	3.36563
$566$	−42.0000	−1.76539
$567$	0	0
$568$	0	0
$569$	7.00000	0.293455	0.146728	−	0.989177i	$-0.453126\pi$
$569$	7.00000	0.293455	0.146728	+	0.989177i	$0.453126\pi$
$570$	32.0000	1.34033
$571$	−14.0000	−0.585882	−0.292941	−	0.956131i	$-0.594634\pi$
$571$	−14.0000	−0.585882	−0.292941	+	0.956131i	$0.594634\pi$
$572$	−30.0000	−1.25436
$573$	32.0000	1.33682
$574$	0	0
$575$	−11.0000	−0.458732
$576$	−8.00000	−0.333333
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	16.0000	0.665512
$579$	−6.00000	−0.249351
$580$	48.0000	1.99309
$581$	0	0
$582$	28.0000	1.16064
$583$	−15.0000	−0.621237
$584$	0	0
$585$	20.0000	0.826898
$586$	52.0000	2.14810
$587$	2.00000	0.0825488	0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000	0.0825488	0.0412744	+	0.999148i	$0.486858\pi$
$588$	28.0000	1.15470
$589$	2.00000	0.0824086
$590$	−96.0000	−3.95226
$591$	−4.00000	−0.164538
$592$	0	0
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	−24.0000	−0.984732
$595$	0	0
$596$	24.0000	0.983078
$597$	−28.0000	−1.14596
$598$	−10.0000	−0.408930
$599$	−1.00000	−0.0408589	−0.0204294	−	0.999791i	$-0.506503\pi$
$599$	−1.00000	−0.0408589	−0.0204294	+	0.999791i	$0.506503\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	−3.00000	−0.122169
$604$	−40.0000	−1.62758
$605$	8.00000	0.325246
$606$	−36.0000	−1.46240
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	−16.0000	−0.648886
$609$	0	0
$610$	16.0000	0.647821
$611$	−20.0000	−0.809113
$612$	−6.00000	−0.242536
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	14.0000	0.564994
$615$	40.0000	1.61296
$616$	0	0
$617$	−21.0000	−0.845428	−0.422714	−	0.906263i	$-0.638923\pi$
$617$	−21.0000	−0.845428	−0.422714	+	0.906263i	$0.638923\pi$
$618$	4.00000	0.160904
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	8.00000	0.321288
$621$	−4.00000	−0.160514
$622$	−30.0000	−1.20289
$623$	0	0
$624$	−40.0000	−1.60128
$625$	41.0000	1.64000
$626$	−44.0000	−1.75859
$627$	12.0000	0.479234
$628$	−20.0000	−0.798087
$629$	0	0
$630$	0	0
$631$	−6.00000	−0.238856	−0.119428	−	0.992843i	$-0.538106\pi$
$631$	−6.00000	−0.238856	−0.119428	+	0.992843i	$0.538106\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	−18.0000	−0.714871
$635$	−4.00000	−0.158735
$636$	20.0000	0.793052
$637$	35.0000	1.38675
$638$	36.0000	1.42525
$639$	2.00000	0.0791188
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	−48.0000	−1.89441
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	−12.0000	−0.472134
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	110.000	4.31455
$651$	0	0
$652$	28.0000	1.09656
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	28.0000	1.09489
$655$	−32.0000	−1.25034
$656$	−20.0000	−0.780869
$657$	2.00000	0.0780274
$658$	0	0
$659$	−19.0000	−0.740135	−0.370067	−	0.929005i	$-0.620665\pi$
$659$	−19.0000	−0.740135	−0.370067	+	0.929005i	$0.620665\pi$
$660$	48.0000	1.86840
$661$	31.0000	1.20576	0.602880	−	0.797832i	$-0.294020\pi$
$661$	31.0000	1.20576	0.602880	+	0.797832i	$0.294020\pi$
$662$	52.0000	2.02104
$663$	−30.0000	−1.16510
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.00000	0.232321
$668$	−18.0000	−0.696441
$669$	56.0000	2.16509
$670$	−24.0000	−0.927201
$671$	6.00000	0.231627
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	6.00000	0.231111
$675$	44.0000	1.69356
$676$	24.0000	0.923077
$677$	34.0000	1.30673	0.653363	−	0.757045i	$-0.273358\pi$
$677$	34.0000	1.30673	0.653363	+	0.757045i	$0.273358\pi$
$678$	−80.0000	−3.07238
$679$	0	0
$680$	0	0
$681$	8.00000	0.306561
$682$	6.00000	0.229752
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	−4.00000	−0.152944
$685$	−24.0000	−0.916993
$686$	0	0
$687$	30.0000	1.14457
$688$	4.00000	0.152499
$689$	25.0000	0.952424
$690$	16.0000	0.609110
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	−56.0000	−2.12573
$695$	−76.0000	−2.88284
$696$	0	0
$697$	−15.0000	−0.568166
$698$	−28.0000	−1.05982
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	40.0000	1.50970
$703$	0	0
$704$	−24.0000	−0.904534
$705$	32.0000	1.20519
$706$	62.0000	2.33340
$707$	0	0
$708$	48.0000	1.80395
$709$	−1.00000	−0.0375558	−0.0187779	−	0.999824i	$-0.505978\pi$
$709$	−1.00000	−0.0375558	−0.0187779	+	0.999824i	$0.505978\pi$
$710$	16.0000	0.600469
$711$	−8.00000	−0.300023
$712$	0	0
$713$	1.00000	0.0374503
$714$	0	0
$715$	60.0000	2.24387
$716$	40.0000	1.49487
$717$	−32.0000	−1.19506
$718$	−38.0000	−1.41815
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	16.0000	0.596285
$721$	0	0
$722$	30.0000	1.11648
$723$	24.0000	0.892570
$724$	20.0000	0.743294
$725$	−66.0000	−2.45118
$726$	−8.00000	−0.296908
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	16.0000	0.592187
$731$	3.00000	0.110959
$732$	−8.00000	−0.295689
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	64.0000	2.36228
$735$	−56.0000	−2.06559
$736$	−8.00000	−0.294884
$737$	−9.00000	−0.331519
$738$	−10.0000	−0.368105
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	0	0
$741$	−20.0000	−0.734718
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−48.0000	−1.75858
$746$	−64.0000	−2.34321
$747$	15.0000	0.548821
$748$	−18.0000	−0.658145
$749$	0	0
$750$	−96.0000	−3.50542
$751$	−6.00000	−0.218943	−0.109472	−	0.993990i	$-0.534916\pi$
$751$	−6.00000	−0.218943	−0.109472	+	0.993990i	$0.534916\pi$
$752$	−16.0000	−0.583460
$753$	46.0000	1.67633
$754$	−60.0000	−2.18507
$755$	80.0000	2.91150
$756$	0	0
$757$	28.0000	1.01768	0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000	1.01768	0.508839	+	0.860862i	$0.330075\pi$
$758$	−22.0000	−0.799076
$759$	6.00000	0.217786
$760$	0	0
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	4.00000	0.144905
$763$	0	0
$764$	−32.0000	−1.15772
$765$	12.0000	0.433861
$766$	−64.0000	−2.31241
$767$	60.0000	2.16647
$768$	−32.0000	−1.15470
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	48.0000	1.72868
$772$	6.00000	0.215945
$773$	−4.00000	−0.143870	−0.0719350	−	0.997409i	$-0.522917\pi$
$773$	−4.00000	−0.143870	−0.0719350	+	0.997409i	$0.522917\pi$
$774$	2.00000	0.0718885
$775$	−11.0000	−0.395132
$776$	0	0
$777$	0	0
$778$	−12.0000	−0.430221
$779$	−10.0000	−0.358287
$780$	−80.0000	−2.86446
$781$	6.00000	0.214697
$782$	−6.00000	−0.214560
$783$	−24.0000	−0.857690
$784$	28.0000	1.00000
$785$	40.0000	1.42766
$786$	32.0000	1.14140
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	4.00000	0.142494
$789$	36.0000	1.28163
$790$	−64.0000	−2.27702
$791$	0	0
$792$	0	0
$793$	−10.0000	−0.355110
$794$	12.0000	0.425864
$795$	−40.0000	−1.41865
$796$	28.0000	0.992434
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	88.0000	3.11127
$801$	−4.00000	−0.141333
$802$	−10.0000	−0.353112
$803$	6.00000	0.211735
$804$	12.0000	0.423207
$805$	0	0
$806$	−10.0000	−0.352235
$807$	50.0000	1.76008
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	−88.0000	−3.09200
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	0	0
$813$	−46.0000	−1.61329
$814$	0	0
$815$	−56.0000	−1.96159
$816$	−24.0000	−0.840168
$817$	2.00000	0.0699711
$818$	48.0000	1.67828
$819$	0	0
$820$	−40.0000	−1.39686
$821$	49.0000	1.71011	0.855056	−	0.518536i	$-0.173523\pi$
$821$	49.0000	1.71011	0.855056	+	0.518536i	$0.173523\pi$
$822$	24.0000	0.837096
$823$	−1.00000	−0.0348578	−0.0174289	−	0.999848i	$-0.505548\pi$
$823$	−1.00000	−0.0348578	−0.0174289	+	0.999848i	$0.505548\pi$
$824$	0	0
$825$	−66.0000	−2.29783
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−2.00000	−0.0695048
$829$	44.0000	1.52818	0.764092	−	0.645108i	$-0.223188\pi$
$829$	44.0000	1.52818	0.764092	+	0.645108i	$0.223188\pi$
$830$	120.000	4.16526
$831$	64.0000	2.22014
$832$	40.0000	1.38675
$833$	21.0000	0.727607
$834$	76.0000	2.63166
$835$	36.0000	1.24583
$836$	−12.0000	−0.415029
$837$	−4.00000	−0.138260
$838$	56.0000	1.93449
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	20.0000	0.689246
$843$	−38.0000	−1.30879
$844$	4.00000	0.137686
$845$	−48.0000	−1.65125
$846$	−8.00000	−0.275046
$847$	0	0
$848$	20.0000	0.686803
$849$	−42.0000	−1.44144
$850$	66.0000	2.26378
$851$	0	0
$852$	−8.00000	−0.274075
$853$	−29.0000	−0.992941	−0.496471	−	0.868054i	$-0.665371\pi$
$853$	−29.0000	−0.992941	−0.496471	+	0.868054i	$0.665371\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	−60.0000	−2.04837
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	42.0000	1.43053
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	32.0000	1.08866
$865$	−24.0000	−0.816024
$866$	24.0000	0.815553
$867$	16.0000	0.543388
$868$	0	0
$869$	−24.0000	−0.814144
$870$	96.0000	3.25470
$871$	15.0000	0.508256
$872$	0	0
$873$	7.00000	0.236914
$874$	−4.00000	−0.135302
$875$	0	0
$876$	−8.00000	−0.270295
$877$	41.0000	1.38447	0.692236	−	0.721671i	$-0.256626\pi$
$877$	41.0000	1.38447	0.692236	+	0.721671i	$0.256626\pi$
$878$	−34.0000	−1.14744
$879$	52.0000	1.75392
$880$	48.0000	1.61808
$881$	37.0000	1.24656	0.623281	−	0.781998i	$-0.285799\pi$
$881$	37.0000	1.24656	0.623281	+	0.781998i	$0.285799\pi$
$882$	14.0000	0.471405
$883$	31.0000	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$883$	31.0000	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$884$	30.0000	1.00901
$885$	−96.0000	−3.22700
$886$	8.00000	0.268765
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	0	0
$889$	0	0
$890$	−32.0000	−1.07264
$891$	−33.0000	−1.10554
$892$	−56.0000	−1.87502
$893$	−8.00000	−0.267710
$894$	48.0000	1.60536
$895$	−80.0000	−2.67411
$896$	0	0
$897$	−10.0000	−0.333890
$898$	−60.0000	−2.00223
$899$	6.00000	0.200111
$900$	22.0000	0.733333
$901$	15.0000	0.499722
$902$	−30.0000	−0.998891
$903$	0	0
$904$	0	0
$905$	−40.0000	−1.32964
$906$	−80.0000	−2.65782
$907$	47.0000	1.56061	0.780305	−	0.625400i	$-0.215064\pi$
$907$	47.0000	1.56061	0.780305	+	0.625400i	$0.215064\pi$
$908$	−8.00000	−0.265489
$909$	−9.00000	−0.298511
$910$	0	0
$911$	−22.0000	−0.728893	−0.364446	−	0.931224i	$-0.618742\pi$
$911$	−22.0000	−0.728893	−0.364446	+	0.931224i	$0.618742\pi$
$912$	−16.0000	−0.529813
$913$	45.0000	1.48928
$914$	36.0000	1.19077
$915$	16.0000	0.528944
$916$	−30.0000	−0.991228
$917$	0	0
$918$	24.0000	0.792118
$919$	−49.0000	−1.61636	−0.808180	−	0.588935i	$-0.799547\pi$
$919$	−49.0000	−1.61636	−0.808180	+	0.588935i	$0.799547\pi$
$920$	0	0
$921$	14.0000	0.461316
$922$	−60.0000	−1.97599
$923$	−10.0000	−0.329154
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	1.00000	0.0328443
$928$	−48.0000	−1.57568
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	16.0000	0.524661
$931$	14.0000	0.458831
$932$	12.0000	0.393073
$933$	−30.0000	−0.982156
$934$	−12.0000	−0.392652
$935$	36.0000	1.17733
$936$	0	0
$937$	32.0000	1.04539	0.522697	−	0.852518i	$-0.324926\pi$
$937$	32.0000	1.04539	0.522697	+	0.852518i	$0.324926\pi$
$938$	0	0
$939$	−44.0000	−1.43589
$940$	−32.0000	−1.04372
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	−40.0000	−1.30327
$943$	−5.00000	−0.162822
$944$	48.0000	1.56227
$945$	0	0
$946$	6.00000	0.195077
$947$	−33.0000	−1.07236	−0.536178	−	0.844105i	$-0.680132\pi$
$947$	−33.0000	−1.07236	−0.536178	+	0.844105i	$0.680132\pi$
$948$	32.0000	1.03931
$949$	−10.0000	−0.324614
$950$	44.0000	1.42755
$951$	−18.0000	−0.583690
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	10.0000	0.323762
$955$	64.0000	2.07099
$956$	32.0000	1.03495
$957$	36.0000	1.16371
$958$	−42.0000	−1.35696
$959$	0	0
$960$	−64.0000	−2.06559
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−12.0000	−0.386695
$964$	−24.0000	−0.772988
$965$	−12.0000	−0.386294
$966$	0	0
$967$	37.0000	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$967$	37.0000	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	56.0000	1.79805
$971$	−13.0000	−0.417190	−0.208595	−	0.978002i	$-0.566889\pi$
$971$	−13.0000	−0.417190	−0.208595	+	0.978002i	$0.566889\pi$
$972$	20.0000	0.641500
$973$	0	0
$974$	−72.0000	−2.30703
$975$	110.000	3.52282
$976$	−8.00000	−0.256074
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	56.0000	1.79068
$979$	−12.0000	−0.383522
$980$	56.0000	1.78885
$981$	7.00000	0.223493
$982$	12.0000	0.382935
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	−8.00000	−0.254901
$986$	−36.0000	−1.14647
$987$	0	0
$988$	20.0000	0.636285
$989$	1.00000	0.0317982
$990$	24.0000	0.762770
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	−8.00000	−0.254000
$993$	52.0000	1.65017
$994$	0	0
$995$	−56.0000	−1.77532
$996$	−60.0000	−1.90117
$997$	4.00000	0.126681	0.0633406	−	0.997992i	$-0.479825\pi$
$997$	4.00000	0.126681	0.0633406	+	0.997992i	$0.479825\pi$
$998$	16.0000	0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.2.a.a.1.1	✓	1
3.2	odd	2		387.2.a.e.1.1		1
4.3	odd	2		688.2.a.b.1.1		1
5.2	odd	4		1075.2.b.b.474.1		2
5.3	odd	4		1075.2.b.b.474.2		2
5.4	even	2		1075.2.a.h.1.1		1
7.6	odd	2		2107.2.a.a.1.1		1
8.3	odd	2		2752.2.a.b.1.1		1
8.5	even	2		2752.2.a.f.1.1		1
11.10	odd	2		5203.2.a.a.1.1		1
12.11	even	2		6192.2.a.ba.1.1		1
13.12	even	2		7267.2.a.a.1.1		1
15.14	odd	2		9675.2.a.b.1.1		1
43.42	odd	2		1849.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.a.a.1.1	✓	1	1.1	even	1	trivial
387.2.a.e.1.1		1	3.2	odd	2
688.2.a.b.1.1		1	4.3	odd	2
1075.2.a.h.1.1		1	5.4	even	2
1075.2.b.b.474.1		2	5.2	odd	4
1075.2.b.b.474.2		2	5.3	odd	4
1849.2.a.d.1.1		1	43.42	odd	2
2107.2.a.a.1.1		1	7.6	odd	2
2752.2.a.b.1.1		1	8.3	odd	2
2752.2.a.f.1.1		1	8.5	even	2
5203.2.a.a.1.1		1	11.10	odd	2
6192.2.a.ba.1.1		1	12.11	even	2
7267.2.a.a.1.1		1	13.12	even	2
9675.2.a.b.1.1		1	15.14	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:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	43.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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