LMFDB - Embedded newform 1005.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1005,2,Mod(1,1005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1005, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1005 = 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1005.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:	$8.02496540314$
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	$\chi$	$=$	1005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -6.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} +4.00000 q^{16} -3.00000 q^{17} -1.00000 q^{19} +2.00000 q^{20} +2.00000 q^{21} -9.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.00000 q^{28} +3.00000 q^{29} -4.00000 q^{31} -6.00000 q^{33} -2.00000 q^{35} -2.00000 q^{36} -7.00000 q^{37} +2.00000 q^{39} +6.00000 q^{41} -10.0000 q^{43} +12.0000 q^{44} -1.00000 q^{45} +9.00000 q^{47} +4.00000 q^{48} -3.00000 q^{49} -3.00000 q^{51} -4.00000 q^{52} -12.0000 q^{53} +6.00000 q^{55} -1.00000 q^{57} -15.0000 q^{59} +2.00000 q^{60} +8.00000 q^{61} +2.00000 q^{63} -8.00000 q^{64} -2.00000 q^{65} +1.00000 q^{67} +6.00000 q^{68} -9.00000 q^{69} -12.0000 q^{71} +11.0000 q^{73} +1.00000 q^{75} +2.00000 q^{76} -12.0000 q^{77} -4.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -4.00000 q^{84} +3.00000 q^{85} +3.00000 q^{87} +15.0000 q^{89} +4.00000 q^{91} +18.0000 q^{92} -4.00000 q^{93} +1.00000 q^{95} +2.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−2.00000	−1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−2.00000	−0.577350
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	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	$0.618522\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	2.00000	0.447214
$21$	2.00000	0.436436
$22$	0	0
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	−2.00000	−0.338062
$36$	−2.00000	−0.333333
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	12.0000	1.80907
$45$	−1.00000	−0.149071
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	4.00000	0.577350
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−3.00000	−0.420084
$52$	−4.00000	−0.554700
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−15.0000	−1.95283	−0.976417	−	0.215894i	$-0.930733\pi$
$59$	−15.0000	−1.95283	−0.976417	+	0.215894i	$0.930733\pi$
$60$	2.00000	0.258199
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	−8.00000	−1.00000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	6.00000	0.727607
$69$	−9.00000	−1.08347
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	2.00000	0.229416
$77$	−12.0000	−1.36753
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−4.00000	−0.436436
$85$	3.00000	0.325396
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	18.0000	1.87663
$93$	−4.00000	−0.414781
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	−2.00000	−0.200000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	−2.00000	−0.192450
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	8.00000	0.755929
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	9.00000	0.839254
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	6.00000	0.541002
$124$	8.00000	0.718421
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	12.0000	1.04447
$133$	−2.00000	−0.173422
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	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
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	4.00000	0.338062
$141$	9.00000	0.757937
$142$	0	0
$143$	−12.0000	−1.00349
$144$	4.00000	0.333333
$145$	−3.00000	−0.249136
$146$	0	0
$147$	−3.00000	−0.247436
$148$	14.0000	1.15079
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	4.00000	0.321288
$156$	−4.00000	−0.320256
$157$	−1.00000	−0.0798087	−0.0399043	−	0.999204i	$-0.512705\pi$
$157$	−1.00000	−0.0798087	−0.0399043	+	0.999204i	$0.512705\pi$
$158$	0	0
$159$	−12.0000	−0.951662
$160$	0	0
$161$	−18.0000	−1.41860
$162$	0	0
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	−12.0000	−0.937043
$165$	6.00000	0.467099
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	20.0000	1.52499
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	−24.0000	−1.80907
$177$	−15.0000	−1.12747
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	2.00000	0.149071
$181$	−25.0000	−1.85824	−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000	−1.85824	−0.929118	+	0.369784i	$0.879432\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	7.00000	0.514650
$186$	0	0
$187$	18.0000	1.31629
$188$	−18.0000	−1.31278
$189$	2.00000	0.145479
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−8.00000	−0.577350
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	6.00000	0.428571
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	6.00000	0.421117
$204$	6.00000	0.420084
$205$	−6.00000	−0.419058
$206$	0	0
$207$	−9.00000	−0.625543
$208$	8.00000	0.554700
$209$	6.00000	0.415029
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	24.0000	1.64833
$213$	−12.0000	−0.822226
$214$	0	0
$215$	10.0000	0.681994
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	11.0000	0.743311
$220$	−12.0000	−0.809040
$221$	−6.00000	−0.403604
$222$	0	0
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	2.00000	0.132453
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	30.0000	1.95283
$237$	−4.00000	−0.259828
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	−4.00000	−0.258199
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−16.0000	−1.02430
$245$	3.00000	0.191663
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−4.00000	−0.251976
$253$	54.0000	3.39495
$254$	0	0
$255$	3.00000	0.187867
$256$	16.0000	1.00000
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	0	0
$259$	−14.0000	−0.869918
$260$	4.00000	0.248069
$261$	3.00000	0.185695
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	15.0000	0.917985
$268$	−2.00000	−0.122169
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−12.0000	−0.727607
$273$	4.00000	0.242091
$274$	0	0
$275$	−6.00000	−0.361814
$276$	18.0000	1.08347
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	24.0000	1.42414
$285$	1.00000	0.0592349
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	2.00000	0.117242
$292$	−22.0000	−1.28745
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	15.0000	0.873334
$296$	0	0
$297$	−6.00000	−0.348155
$298$	0	0
$299$	−18.0000	−1.04097
$300$	−2.00000	−0.115470
$301$	−20.0000	−1.15278
$302$	0	0
$303$	0	0
$304$	−4.00000	−0.229416
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	24.0000	1.36753
$309$	−4.00000	−0.227552
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	8.00000	0.450035
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	8.00000	0.447214
$321$	15.0000	0.837218
$322$	0	0
$323$	3.00000	0.166924
$324$	−2.00000	−0.111111
$325$	2.00000	0.110940
$326$	0	0
$327$	−10.0000	−0.553001
$328$	0	0
$329$	18.0000	0.992372
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	24.0000	1.31717
$333$	−7.00000	−0.383598
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	8.00000	0.436436
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	−6.00000	−0.325396
$341$	24.0000	1.29967
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	9.00000	0.484544
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	−6.00000	−0.321634
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	−30.0000	−1.59000
$357$	−6.00000	−0.317554
$358$	0	0
$359$	−33.0000	−1.74167	−0.870837	−	0.491572i	$-0.836422\pi$
$359$	−33.0000	−1.74167	−0.870837	+	0.491572i	$0.836422\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	25.0000	1.31216
$364$	−8.00000	−0.419314
$365$	−11.0000	−0.575766
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	−36.0000	−1.87663
$369$	6.00000	0.312348
$370$	0	0
$371$	−24.0000	−1.24602
$372$	8.00000	0.414781
$373$	−28.0000	−1.44979	−0.724893	−	0.688862i	$-0.758111\pi$
$373$	−28.0000	−1.44979	−0.724893	+	0.688862i	$0.758111\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	−2.00000	−0.102598
$381$	17.0000	0.870936
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	−10.0000	−0.508329
$388$	−4.00000	−0.203069
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	27.0000	1.36545
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	4.00000	0.201262
$396$	12.0000	0.603023
$397$	−13.0000	−0.652451	−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000	−0.652451	−0.326226	+	0.945292i	$0.605777\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	4.00000	0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	42.0000	2.08186
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	8.00000	0.394132
$413$	−30.0000	−1.47620
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	4.00000	0.195180
$421$	11.0000	0.536107	0.268054	−	0.963404i	$-0.413620\pi$
$421$	11.0000	0.536107	0.268054	+	0.963404i	$0.413620\pi$
$422$	0	0
$423$	9.00000	0.437595
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	16.0000	0.774294
$428$	−30.0000	−1.45010
$429$	−12.0000	−0.579365
$430$	0	0
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	4.00000	0.192450
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	20.0000	0.957826
$437$	9.00000	0.430528
$438$	0	0
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	14.0000	0.664411
$445$	−15.0000	−0.711068
$446$	0	0
$447$	−3.00000	−0.141895
$448$	−16.0000	−0.755929
$449$	3.00000	0.141579	0.0707894	−	0.997491i	$-0.477448\pi$
$449$	3.00000	0.141579	0.0707894	+	0.997491i	$0.477448\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	24.0000	1.12887
$453$	−1.00000	−0.0469841
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	11.0000	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$457$	11.0000	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$458$	0	0
$459$	−3.00000	−0.140028
$460$	−18.0000	−0.839254
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	12.0000	0.557086
$465$	4.00000	0.185496
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−4.00000	−0.184900
$469$	2.00000	0.0923514
$470$	0	0
$471$	−1.00000	−0.0460776
$472$	0	0
$473$	60.0000	2.75880
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	12.0000	0.550019
$477$	−12.0000	−0.549442
$478$	0	0
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	0	0
$483$	−18.0000	−0.819028
$484$	−50.0000	−2.27273
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	5.00000	0.226108
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	−12.0000	−0.541002
$493$	−9.00000	−0.405340
$494$	0	0
$495$	6.00000	0.269680
$496$	−16.0000	−0.718421
$497$	−24.0000	−1.07655
$498$	0	0
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	2.00000	0.0894427
$501$	0	0
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.00000	−0.399704
$508$	−34.0000	−1.50851
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	22.0000	0.973223
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	4.00000	0.176261
$516$	20.0000	0.880451
$517$	−54.0000	−2.37492
$518$	0	0
$519$	21.0000	0.921798
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	−25.0000	−1.09317	−0.546587	−	0.837402i	$-0.684073\pi$
$523$	−25.0000	−1.09317	−0.546587	+	0.837402i	$0.684073\pi$
$524$	−24.0000	−1.04844
$525$	2.00000	0.0872872
$526$	0	0
$527$	12.0000	0.522728
$528$	−24.0000	−1.04447
$529$	58.0000	2.52174
$530$	0	0
$531$	−15.0000	−0.650945
$532$	4.00000	0.173422
$533$	12.0000	0.519778
$534$	0	0
$535$	−15.0000	−0.648507
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18.0000	0.775315
$540$	2.00000	0.0860663
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	0	0
$543$	−25.0000	−1.07285
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−36.0000	−1.53784
$549$	8.00000	0.341432
$550$	0	0
$551$	−3.00000	−0.127804
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	7.00000	0.297133
$556$	−40.0000	−1.69638
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	−8.00000	−0.338062
$561$	18.0000	0.759961
$562$	0	0
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	−18.0000	−0.757937
$565$	12.0000	0.504844
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−27.0000	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$569$	−27.0000	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	24.0000	1.00349
$573$	−12.0000	−0.501307
$574$	0	0
$575$	−9.00000	−0.375326
$576$	−8.00000	−0.333333
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	−13.0000	−0.540262
$580$	6.00000	0.249136
$581$	−24.0000	−0.995688
$582$	0	0
$583$	72.0000	2.98194
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	6.00000	0.247436
$589$	4.00000	0.164817
$590$	0	0
$591$	−12.0000	−0.493614
$592$	−28.0000	−1.15079
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	6.00000	0.245770
$597$	11.0000	0.450200
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	2.00000	0.0813788
$605$	−25.0000	−1.01639
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	18.0000	0.728202
$612$	6.00000	0.242536
$613$	−7.00000	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$613$	−7.00000	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	33.0000	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$617$	33.0000	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−8.00000	−0.321288
$621$	−9.00000	−0.361158
$622$	0	0
$623$	30.0000	1.20192
$624$	8.00000	0.320256
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.00000	0.239617
$628$	2.00000	0.0798087
$629$	21.0000	0.837325
$630$	0	0
$631$	−10.0000	−0.398094	−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000	−0.398094	−0.199047	+	0.979990i	$0.563785\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	−17.0000	−0.674624
$636$	24.0000	0.951662
$637$	−6.00000	−0.237729
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	36.0000	1.41860
$645$	10.0000	0.393750
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	90.0000	3.53281
$650$	0	0
$651$	−8.00000	−0.313545
$652$	−10.0000	−0.391630
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	24.0000	0.937043
$657$	11.0000	0.429151
$658$	0	0
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	−12.0000	−0.467099
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	−6.00000	−0.233021
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	−27.0000	−1.04544
$668$	0	0
$669$	−7.00000	−0.270636
$670$	0	0
$671$	−48.0000	−1.85302
$672$	0	0
$673$	20.0000	0.770943	0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000	0.770943	0.385472	+	0.922720i	$0.374039\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	18.0000	0.692308
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	−3.00000	−0.114960
$682$	0	0
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	2.00000	0.0764719
$685$	−18.0000	−0.687745
$686$	0	0
$687$	26.0000	0.991962
$688$	−40.0000	−1.52499
$689$	−24.0000	−0.914327
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	−42.0000	−1.59660
$693$	−12.0000	−0.455842
$694$	0	0
$695$	−20.0000	−0.758643
$696$	0	0
$697$	−18.0000	−0.681799
$698$	0	0
$699$	−18.0000	−0.680823
$700$	−4.00000	−0.151186
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	7.00000	0.264010
$704$	48.0000	1.80907
$705$	−9.00000	−0.338960
$706$	0	0
$707$	0	0
$708$	30.0000	1.12747
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	36.0000	1.34821
$714$	0	0
$715$	12.0000	0.448775
$716$	0	0
$717$	12.0000	0.448148
$718$	0	0
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	−4.00000	−0.149071
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−7.00000	−0.260333
$724$	50.0000	1.85824
$725$	3.00000	0.111417
$726$	0	0
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.0000	1.10959
$732$	−16.0000	−0.591377
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	−6.00000	−0.221013
$738$	0	0
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	−14.0000	−0.514650
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	3.00000	0.109911
$746$	0	0
$747$	−12.0000	−0.439057
$748$	−36.0000	−1.31629
$749$	30.0000	1.09618
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	36.0000	1.31278
$753$	−12.0000	−0.437304
$754$	0	0
$755$	1.00000	0.0363937
$756$	−4.00000	−0.145479
$757$	−52.0000	−1.88997	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000	−1.88997	−0.944986	+	0.327111i	$0.893925\pi$
$758$	0	0
$759$	54.0000	1.96008
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	24.0000	0.868290
$765$	3.00000	0.108465
$766$	0	0
$767$	−30.0000	−1.08324
$768$	16.0000	0.577350
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	0	0
$771$	−15.0000	−0.540212
$772$	26.0000	0.935760
$773$	51.0000	1.83434	0.917171	−	0.398493i	$-0.130467\pi$
$773$	51.0000	1.83434	0.917171	+	0.398493i	$0.130467\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−14.0000	−0.502247
$778$	0	0
$779$	−6.00000	−0.214972
$780$	4.00000	0.143223
$781$	72.0000	2.57636
$782$	0	0
$783$	3.00000	0.107211
$784$	−12.0000	−0.428571
$785$	1.00000	0.0356915
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	24.0000	0.854965
$789$	12.0000	0.427211
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	16.0000	0.568177
$794$	0	0
$795$	12.0000	0.425596
$796$	−22.0000	−0.779769
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	0	0
$801$	15.0000	0.529999
$802$	0	0
$803$	−66.0000	−2.32909
$804$	−2.00000	−0.0705346
$805$	18.0000	0.634417
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	−12.0000	−0.421117
$813$	20.0000	0.701431
$814$	0	0
$815$	−5.00000	−0.175142
$816$	−12.0000	−0.420084
$817$	10.0000	0.349856
$818$	0	0
$819$	4.00000	0.139771
$820$	12.0000	0.419058
$821$	3.00000	0.104701	0.0523504	−	0.998629i	$-0.483329\pi$
$821$	3.00000	0.104701	0.0523504	+	0.998629i	$0.483329\pi$
$822$	0	0
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	−33.0000	−1.14752	−0.573761	−	0.819023i	$-0.694516\pi$
$827$	−33.0000	−1.14752	−0.573761	+	0.819023i	$0.694516\pi$
$828$	18.0000	0.625543
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	−16.0000	−0.554700
$833$	9.00000	0.311832
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	−4.00000	−0.138260
$838$	0	0
$839$	−39.0000	−1.34643	−0.673215	−	0.739447i	$-0.735087\pi$
$839$	−39.0000	−1.34643	−0.673215	+	0.739447i	$0.735087\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	8.00000	0.275371
$845$	9.00000	0.309609
$846$	0	0
$847$	50.0000	1.71802
$848$	−48.0000	−1.64833
$849$	11.0000	0.377519
$850$	0	0
$851$	63.0000	2.15961
$852$	24.0000	0.822226
$853$	−37.0000	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	−20.0000	−0.681994
$861$	12.0000	0.408959
$862$	0	0
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	0	0
$865$	−21.0000	−0.714021
$866$	0	0
$867$	−8.00000	−0.271694
$868$	16.0000	0.543075
$869$	24.0000	0.814144
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	−22.0000	−0.743311
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	0	0
$879$	30.0000	1.01187
$880$	24.0000	0.809040
$881$	−57.0000	−1.92038	−0.960189	−	0.279350i	$-0.909881\pi$
$881$	−57.0000	−1.92038	−0.960189	+	0.279350i	$0.909881\pi$
$882$	0	0
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	12.0000	0.403604
$885$	15.0000	0.504219
$886$	0	0
$887$	15.0000	0.503651	0.251825	−	0.967773i	$-0.418969\pi$
$887$	15.0000	0.503651	0.251825	+	0.967773i	$0.418969\pi$
$888$	0	0
$889$	34.0000	1.14032
$890$	0	0
$891$	−6.00000	−0.201008
$892$	14.0000	0.468755
$893$	−9.00000	−0.301174
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−18.0000	−0.601003
$898$	0	0
$899$	−12.0000	−0.400222
$900$	−2.00000	−0.0666667
$901$	36.0000	1.19933
$902$	0	0
$903$	−20.0000	−0.665558
$904$	0	0
$905$	25.0000	0.831028
$906$	0	0
$907$	−1.00000	−0.0332045	−0.0166022	−	0.999862i	$-0.505285\pi$
$907$	−1.00000	−0.0332045	−0.0166022	+	0.999862i	$0.505285\pi$
$908$	6.00000	0.199117
$909$	0	0
$910$	0	0
$911$	21.0000	0.695761	0.347881	−	0.937539i	$-0.386901\pi$
$911$	21.0000	0.695761	0.347881	+	0.937539i	$0.386901\pi$
$912$	−4.00000	−0.132453
$913$	72.0000	2.38285
$914$	0	0
$915$	−8.00000	−0.264472
$916$	−52.0000	−1.71813
$917$	24.0000	0.792550
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	−7.00000	−0.230658
$922$	0	0
$923$	−24.0000	−0.789970
$924$	24.0000	0.789542
$925$	−7.00000	−0.230159
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	36.0000	1.17922
$933$	18.0000	0.589294
$934$	0	0
$935$	−18.0000	−0.588663
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	18.0000	0.587095
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	0	0
$943$	−54.0000	−1.75848
$944$	−60.0000	−1.95283
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	8.00000	0.259828
$949$	22.0000	0.714150
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	3.00000	0.0971795	0.0485898	−	0.998819i	$-0.484527\pi$
$953$	3.00000	0.0971795	0.0485898	+	0.998819i	$0.484527\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	−24.0000	−0.776215
$957$	−18.0000	−0.581857
$958$	0	0
$959$	36.0000	1.16250
$960$	8.00000	0.258199
$961$	−15.0000	−0.483871
$962$	0	0
$963$	15.0000	0.483368
$964$	14.0000	0.450910
$965$	13.0000	0.418485
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	3.00000	0.0963739
$970$	0	0
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	−2.00000	−0.0641500
$973$	40.0000	1.28234
$974$	0	0
$975$	2.00000	0.0640513
$976$	32.0000	1.02430
$977$	−15.0000	−0.479893	−0.239946	−	0.970786i	$-0.577130\pi$
$977$	−15.0000	−0.479893	−0.239946	+	0.970786i	$0.577130\pi$
$978$	0	0
$979$	−90.0000	−2.87641
$980$	−6.00000	−0.191663
$981$	−10.0000	−0.319275
$982$	0	0
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	18.0000	0.572946
$988$	4.00000	0.127257
$989$	90.0000	2.86183
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	0	0
$993$	−22.0000	−0.698149
$994$	0	0
$995$	−11.0000	−0.348723
$996$	24.0000	0.760469
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1005.2.a.a.1.1	✓	1
3.2	odd	2		3015.2.a.c.1.1		1
5.4	even	2		5025.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1005.2.a.a.1.1	✓	1	1.1	even	1	trivial
3015.2.a.c.1.1		1	3.2	odd	2
5025.2.a.d.1.1		1	5.4	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:	\( N \)	\(=\)	\( 1005 = 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1005.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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