LMFDB - Embedded newform 4009.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
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	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +3.00000 q^{5} -2.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +3.00000 q^{5} -2.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -2.00000 q^{12} +4.00000 q^{13} +6.00000 q^{15} -1.00000 q^{16} -1.00000 q^{18} -1.00000 q^{19} -3.00000 q^{20} -1.00000 q^{23} +6.00000 q^{24} +4.00000 q^{25} -4.00000 q^{26} -4.00000 q^{27} +3.00000 q^{29} -6.00000 q^{30} +4.00000 q^{31} -5.00000 q^{32} -1.00000 q^{36} +1.00000 q^{38} +8.00000 q^{39} +9.00000 q^{40} +10.0000 q^{41} +6.00000 q^{43} +3.00000 q^{45} +1.00000 q^{46} -2.00000 q^{48} -7.00000 q^{49} -4.00000 q^{50} -4.00000 q^{52} +6.00000 q^{53} +4.00000 q^{54} -2.00000 q^{57} -3.00000 q^{58} +11.0000 q^{59} -6.00000 q^{60} -4.00000 q^{62} +7.00000 q^{64} +12.0000 q^{65} +2.00000 q^{67} -2.00000 q^{69} -9.00000 q^{71} +3.00000 q^{72} -15.0000 q^{73} +8.00000 q^{75} +1.00000 q^{76} -8.00000 q^{78} +5.00000 q^{79} -3.00000 q^{80} -11.0000 q^{81} -10.0000 q^{82} -6.00000 q^{86} +6.00000 q^{87} +9.00000 q^{89} -3.00000 q^{90} +1.00000 q^{92} +8.00000 q^{93} -3.00000 q^{95} -10.0000 q^{96} +11.0000 q^{97} +7.00000 q^{98} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.00000	−0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−2.00000	−0.816497
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−2.00000	−0.577350
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	6.00000	1.54919
$16$	−1.00000	−0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−3.00000	−0.670820
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	6.00000	1.22474
$25$	4.00000	0.800000
$26$	−4.00000	−0.784465
$27$	−4.00000	−0.769800
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−6.00000	−1.09545
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−1.00000	−0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	1.00000	0.162221
$39$	8.00000	1.28103
$40$	9.00000	1.42302
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	1.00000	0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−2.00000	−0.288675
$49$	−7.00000	−1.00000
$50$	−4.00000	−0.565685
$51$	0	0
$52$	−4.00000	−0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	−3.00000	−0.393919
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	−6.00000	−0.774597
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	7.00000	0.875000
$65$	12.0000	1.48842
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	3.00000	0.353553
$73$	−15.0000	−1.75562	−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000	−1.75562	−0.877809	+	0.479012i	$0.840995\pi$
$74$	0	0
$75$	8.00000	0.923760
$76$	1.00000	0.114708
$77$	0	0
$78$	−8.00000	−0.905822
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	−3.00000	−0.335410
$81$	−11.0000	−1.22222
$82$	−10.0000	−1.10432
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−6.00000	−0.646997
$87$	6.00000	0.643268
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	1.00000	0.104257
$93$	8.00000	0.829561
$94$	0	0
$95$	−3.00000	−0.307794
$96$	−10.0000	−1.02062
$97$	11.0000	1.11688	0.558440	−	0.829545i	$-0.311400\pi$
$97$	11.0000	1.11688	0.558440	+	0.829545i	$0.311400\pi$
$98$	7.00000	0.707107
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−5.00000	−0.497519	−0.248759	−	0.968565i	$-0.580023\pi$
$101$	−5.00000	−0.497519	−0.248759	+	0.968565i	$0.580023\pi$
$102$	0	0
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	12.0000	1.17670
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−11.0000	−1.06341	−0.531705	−	0.846930i	$-0.678449\pi$
$107$	−11.0000	−1.06341	−0.531705	+	0.846930i	$0.678449\pi$
$108$	4.00000	0.384900
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	2.00000	0.187317
$115$	−3.00000	−0.279751
$116$	−3.00000	−0.278543
$117$	4.00000	0.369800
$118$	−11.0000	−1.01263
$119$	0	0
$120$	18.0000	1.64317
$121$	−11.0000	−1.00000
$122$	0	0
$123$	20.0000	1.80334
$124$	−4.00000	−0.359211
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	3.00000	0.265165
$129$	12.0000	1.05654
$130$	−12.0000	−1.05247
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	0	0
$134$	−2.00000	−0.172774
$135$	−12.0000	−1.03280
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	2.00000	0.170251
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	0	0
$142$	9.00000	0.755263
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	9.00000	0.747409
$146$	15.0000	1.24141
$147$	−14.0000	−1.15470
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	−8.00000	−0.653197
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	0	0
$155$	12.0000	0.963863
$156$	−8.00000	−0.640513
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−5.00000	−0.397779
$159$	12.0000	0.951662
$160$	−15.0000	−1.18585
$161$	0	0
$162$	11.0000	0.864242
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	0	0
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−6.00000	−0.457496
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	0	0
$177$	22.0000	1.65362
$178$	−9.00000	−0.674579
$179$	21.0000	1.56961	0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000	1.56961	0.784807	+	0.619740i	$0.212762\pi$
$180$	−3.00000	−0.223607
$181$	3.00000	0.222988	0.111494	−	0.993765i	$-0.464436\pi$
$181$	3.00000	0.222988	0.111494	+	0.993765i	$0.464436\pi$
$182$	0	0
$183$	0	0
$184$	−3.00000	−0.221163
$185$	0	0
$186$	−8.00000	−0.586588
$187$	0	0
$188$	0	0
$189$	0	0
$190$	3.00000	0.217643
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	14.0000	1.01036
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−11.0000	−0.789754
$195$	24.0000	1.71868
$196$	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	$0.721573\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	12.0000	0.848528
$201$	4.00000	0.282138
$202$	5.00000	0.351799
$203$	0	0
$204$	0	0
$205$	30.0000	2.09529
$206$	−13.0000	−0.905753
$207$	−1.00000	−0.0695048
$208$	−4.00000	−0.277350
$209$	0	0
$210$	0	0
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−6.00000	−0.412082
$213$	−18.0000	−1.23334
$214$	11.0000	0.751945
$215$	18.0000	1.22759
$216$	−12.0000	−0.816497
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−30.0000	−2.02721
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	16.0000	1.06430
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	2.00000	0.132453
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	9.00000	0.590879
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	−11.0000	−0.716039
$237$	10.0000	0.649570
$238$	0	0
$239$	−13.0000	−0.840900	−0.420450	−	0.907316i	$-0.638128\pi$
$239$	−13.0000	−0.840900	−0.420450	+	0.907316i	$0.638128\pi$
$240$	−6.00000	−0.387298
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	11.0000	0.707107
$243$	−10.0000	−0.641500
$244$	0	0
$245$	−21.0000	−1.34164
$246$	−20.0000	−1.27515
$247$	−4.00000	−0.254514
$248$	12.0000	0.762001
$249$	0	0
$250$	3.00000	0.189737
$251$	−19.0000	−1.19927	−0.599635	−	0.800274i	$-0.704687\pi$
$251$	−19.0000	−1.19927	−0.599635	+	0.800274i	$0.704687\pi$
$252$	0	0
$253$	0	0
$254$	2.00000	0.125491
$255$	0	0
$256$	−17.0000	−1.06250
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	−12.0000	−0.747087
$259$	0	0
$260$	−12.0000	−0.744208
$261$	3.00000	0.185695
$262$	−15.0000	−0.926703
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	18.0000	1.10158
$268$	−2.00000	−0.122169
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	12.0000	0.730297
$271$	29.0000	1.76162	0.880812	−	0.473466i	$-0.156997\pi$
$271$	29.0000	1.76162	0.880812	+	0.473466i	$0.156997\pi$
$272$	0	0
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	2.00000	0.120386
$277$	−3.00000	−0.180253	−0.0901263	−	0.995930i	$-0.528727\pi$
$277$	−3.00000	−0.180253	−0.0901263	+	0.995930i	$0.528727\pi$
$278$	−14.0000	−0.839664
$279$	4.00000	0.239474
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	0	0
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	9.00000	0.534052
$285$	−6.00000	−0.355409
$286$	0	0
$287$	0	0
$288$	−5.00000	−0.294628
$289$	−17.0000	−1.00000
$290$	−9.00000	−0.528498
$291$	22.0000	1.28966
$292$	15.0000	0.877809
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	14.0000	0.816497
$295$	33.0000	1.92133
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.00000	−0.231326
$300$	−8.00000	−0.461880
$301$	0	0
$302$	−9.00000	−0.517892
$303$	−10.0000	−0.574485
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	26.0000	1.47909
$310$	−12.0000	−0.681554
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	24.0000	1.35873
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−5.00000	−0.281272
$317$	−9.00000	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$318$	−12.0000	−0.672927
$319$	0	0
$320$	21.0000	1.17394
$321$	−22.0000	−1.22792
$322$	0	0
$323$	0	0
$324$	11.0000	0.611111
$325$	16.0000	0.887520
$326$	−6.00000	−0.332309
$327$	20.0000	1.10600
$328$	30.0000	1.65647
$329$	0	0
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	0	0
$334$	24.0000	1.31322
$335$	6.00000	0.327815
$336$	0	0
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	−3.00000	−0.163178
$339$	−32.0000	−1.73800
$340$	0	0
$341$	0	0
$342$	1.00000	0.0540738
$343$	0	0
$344$	18.0000	0.970495
$345$	−6.00000	−0.323029
$346$	22.0000	1.18273
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	−6.00000	−0.321634
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	−22.0000	−1.16929
$355$	−27.0000	−1.43301
$356$	−9.00000	−0.476999
$357$	0	0
$358$	−21.0000	−1.10988
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	9.00000	0.474342
$361$	1.00000	0.0526316
$362$	−3.00000	−0.157676
$363$	−22.0000	−1.15470
$364$	0	0
$365$	−45.0000	−2.35541
$366$	0	0
$367$	−3.00000	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$368$	1.00000	0.0521286
$369$	10.0000	0.520579
$370$	0	0
$371$	0	0
$372$	−8.00000	−0.414781
$373$	−23.0000	−1.19089	−0.595447	−	0.803394i	$-0.703025\pi$
$373$	−23.0000	−1.19089	−0.595447	+	0.803394i	$0.703025\pi$
$374$	0	0
$375$	−6.00000	−0.309839
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	3.00000	0.153897
$381$	−4.00000	−0.204926
$382$	4.00000	0.204658
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	6.00000	0.306186
$385$	0	0
$386$	16.0000	0.814379
$387$	6.00000	0.304997
$388$	−11.0000	−0.558440
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	−24.0000	−1.21529
$391$	0	0
$392$	−21.0000	−1.06066
$393$	30.0000	1.51330
$394$	18.0000	0.906827
$395$	15.0000	0.754732
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	0	0
$400$	−4.00000	−0.200000
$401$	17.0000	0.848939	0.424470	−	0.905442i	$-0.360461\pi$
$401$	17.0000	0.848939	0.424470	+	0.905442i	$0.360461\pi$
$402$	−4.00000	−0.199502
$403$	16.0000	0.797017
$404$	5.00000	0.248759
$405$	−33.0000	−1.63978
$406$	0	0
$407$	0	0
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	−30.0000	−1.48159
$411$	20.0000	0.986527
$412$	−13.0000	−0.640464
$413$	0	0
$414$	1.00000	0.0491473
$415$	0	0
$416$	−20.0000	−0.980581
$417$	28.0000	1.37117
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	−1.00000	−0.0486792
$423$	0	0
$424$	18.0000	0.874157
$425$	0	0
$426$	18.0000	0.872103
$427$	0	0
$428$	11.0000	0.531705
$429$	0	0
$430$	−18.0000	−0.868037
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	4.00000	0.192450
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	18.0000	0.863034
$436$	−10.0000	−0.478913
$437$	1.00000	0.0478365
$438$	30.0000	1.43346
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	27.0000	1.27992
$446$	−14.0000	−0.662919
$447$	0	0
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	16.0000	0.752577
$453$	18.0000	0.845714
$454$	−7.00000	−0.328526
$455$	0	0
$456$	−6.00000	−0.280976
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	26.0000	1.21490
$459$	0	0
$460$	3.00000	0.139876
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	−13.0000	−0.604161	−0.302081	−	0.953282i	$-0.597681\pi$
$463$	−13.0000	−0.604161	−0.302081	+	0.953282i	$0.597681\pi$
$464$	−3.00000	−0.139272
$465$	24.0000	1.11297
$466$	14.0000	0.648537
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	−4.00000	−0.184900
$469$	0	0
$470$	0	0
$471$	28.0000	1.29017
$472$	33.0000	1.51895
$473$	0	0
$474$	−10.0000	−0.459315
$475$	−4.00000	−0.183533
$476$	0	0
$477$	6.00000	0.274721
$478$	13.0000	0.594606
$479$	35.0000	1.59919	0.799595	−	0.600539i	$-0.205047\pi$
$479$	35.0000	1.59919	0.799595	+	0.600539i	$0.205047\pi$
$480$	−30.0000	−1.36931
$481$	0	0
$482$	12.0000	0.546585
$483$	0	0
$484$	11.0000	0.500000
$485$	33.0000	1.49845
$486$	10.0000	0.453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	21.0000	0.948683
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	−20.0000	−0.901670
$493$	0	0
$494$	4.00000	0.179969
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	−9.00000	−0.402895	−0.201448	−	0.979499i	$-0.564565\pi$
$499$	−9.00000	−0.402895	−0.201448	+	0.979499i	$0.564565\pi$
$500$	3.00000	0.134164
$501$	−48.0000	−2.14448
$502$	19.0000	0.848012
$503$	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
$505$	−15.0000	−0.667491
$506$	0	0
$507$	6.00000	0.266469
$508$	2.00000	0.0887357
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	4.00000	0.176604
$514$	−20.0000	−0.882162
$515$	39.0000	1.71855
$516$	−12.0000	−0.528271
$517$	0	0
$518$	0	0
$519$	−44.0000	−1.93139
$520$	36.0000	1.57870
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	−3.00000	−0.131306
$523$	7.00000	0.306089	0.153044	−	0.988219i	$-0.451092\pi$
$523$	7.00000	0.306089	0.153044	+	0.988219i	$0.451092\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	−18.0000	−0.784837
$527$	0	0
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−18.0000	−0.781870
$531$	11.0000	0.477359
$532$	0	0
$533$	40.0000	1.73259
$534$	−18.0000	−0.778936
$535$	−33.0000	−1.42671
$536$	6.00000	0.259161
$537$	42.0000	1.81243
$538$	−24.0000	−1.03471
$539$	0	0
$540$	12.0000	0.516398
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	−29.0000	−1.24566
$543$	6.00000	0.257485
$544$	0	0
$545$	30.0000	1.28506
$546$	0	0
$547$	−35.0000	−1.49649	−0.748246	−	0.663421i	$-0.769104\pi$
$547$	−35.0000	−1.49649	−0.748246	+	0.663421i	$0.769104\pi$
$548$	−10.0000	−0.427179
$549$	0	0
$550$	0	0
$551$	−3.00000	−0.127804
$552$	−6.00000	−0.255377
$553$	0	0
$554$	3.00000	0.127458
$555$	0	0
$556$	−14.0000	−0.593732
$557$	40.0000	1.69485	0.847427	−	0.530912i	$-0.178150\pi$
$557$	40.0000	1.69485	0.847427	+	0.530912i	$0.178150\pi$
$558$	−4.00000	−0.169334
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	4.00000	0.168730
$563$	42.0000	1.77009	0.885044	−	0.465506i	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	0	0
$565$	−48.0000	−2.01938
$566$	−11.0000	−0.462364
$567$	0	0
$568$	−27.0000	−1.13289
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	6.00000	0.251312
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	−4.00000	−0.166812
$576$	7.00000	0.291667
$577$	−40.0000	−1.66522	−0.832611	−	0.553858i	$-0.813155\pi$
$577$	−40.0000	−1.66522	−0.832611	+	0.553858i	$0.813155\pi$
$578$	17.0000	0.707107
$579$	−32.0000	−1.32987
$580$	−9.00000	−0.373705
$581$	0	0
$582$	−22.0000	−0.911929
$583$	0	0
$584$	−45.0000	−1.86211
$585$	12.0000	0.496139
$586$	−24.0000	−0.991431
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	14.0000	0.577350
$589$	−4.00000	−0.164817
$590$	−33.0000	−1.35859
$591$	−36.0000	−1.48084
$592$	0	0
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	4.00000	0.163572
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	24.0000	0.979796
$601$	−40.0000	−1.63163	−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000	−1.63163	−0.815817	+	0.578310i	$0.803712\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	−9.00000	−0.366205
$605$	−33.0000	−1.34164
$606$	10.0000	0.406222
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	0	0
$615$	60.0000	2.41943
$616$	0	0
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	−26.0000	−1.04587
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	−12.0000	−0.481932
$621$	4.00000	0.160514
$622$	18.0000	0.721734
$623$	0	0
$624$	−8.00000	−0.320256
$625$	−29.0000	−1.16000
$626$	−20.0000	−0.799361
$627$	0	0
$628$	−14.0000	−0.558661
$629$	0	0
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	15.0000	0.596668
$633$	2.00000	0.0794929
$634$	9.00000	0.357436
$635$	−6.00000	−0.238103
$636$	−12.0000	−0.475831
$637$	−28.0000	−1.10940
$638$	0	0
$639$	−9.00000	−0.356034
$640$	9.00000	0.355756
$641$	−23.0000	−0.908445	−0.454223	−	0.890888i	$-0.650083\pi$
$641$	−23.0000	−0.908445	−0.454223	+	0.890888i	$0.650083\pi$
$642$	22.0000	0.868271
$643$	−21.0000	−0.828159	−0.414080	−	0.910241i	$-0.635896\pi$
$643$	−21.0000	−0.828159	−0.414080	+	0.910241i	$0.635896\pi$
$644$	0	0
$645$	36.0000	1.41750
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	−33.0000	−1.29636
$649$	0	0
$650$	−16.0000	−0.627572
$651$	0	0
$652$	−6.00000	−0.234978
$653$	−33.0000	−1.29139	−0.645695	−	0.763596i	$-0.723432\pi$
$653$	−33.0000	−1.29139	−0.645695	+	0.763596i	$0.723432\pi$
$654$	−20.0000	−0.782062
$655$	45.0000	1.75830
$656$	−10.0000	−0.390434
$657$	−15.0000	−0.585206
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	−47.0000	−1.82809	−0.914044	−	0.405615i	$-0.867057\pi$
$661$	−47.0000	−1.82809	−0.914044	+	0.405615i	$0.867057\pi$
$662$	−12.0000	−0.466393
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.00000	−0.116160
$668$	24.0000	0.928588
$669$	28.0000	1.08254
$670$	−6.00000	−0.231800
$671$	0	0
$672$	0	0
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	28.0000	1.07852
$675$	−16.0000	−0.615840
$676$	−3.00000	−0.115385
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	32.0000	1.22895
$679$	0	0
$680$	0	0
$681$	14.0000	0.536481
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	1.00000	0.0382360
$685$	30.0000	1.14624
$686$	0	0
$687$	−52.0000	−1.98392
$688$	−6.00000	−0.228748
$689$	24.0000	0.914327
$690$	6.00000	0.228416
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	22.0000	0.836315
$693$	0	0
$694$	−28.0000	−1.06287
$695$	42.0000	1.59315
$696$	18.0000	0.682288
$697$	0	0
$698$	−20.0000	−0.757011
$699$	−28.0000	−1.05906
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	16.0000	0.603881
$703$	0	0
$704$	0	0
$705$	0	0
$706$	2.00000	0.0752710
$707$	0	0
$708$	−22.0000	−0.826811
$709$	−43.0000	−1.61490	−0.807449	−	0.589937i	$-0.799153\pi$
$709$	−43.0000	−1.61490	−0.807449	+	0.589937i	$0.799153\pi$
$710$	27.0000	1.01329
$711$	5.00000	0.187515
$712$	27.0000	1.01187
$713$	−4.00000	−0.149801
$714$	0	0
$715$	0	0
$716$	−21.0000	−0.784807
$717$	−26.0000	−0.970988
$718$	36.0000	1.34351
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	−3.00000	−0.111803
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	−24.0000	−0.892570
$724$	−3.00000	−0.111494
$725$	12.0000	0.445669
$726$	22.0000	0.816497
$727$	−5.00000	−0.185440	−0.0927199	−	0.995692i	$-0.529556\pi$
$727$	−5.00000	−0.185440	−0.0927199	+	0.995692i	$0.529556\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	45.0000	1.66552
$731$	0	0
$732$	0	0
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	3.00000	0.110732
$735$	−42.0000	−1.54919
$736$	5.00000	0.184302
$737$	0	0
$738$	−10.0000	−0.368105
$739$	−5.00000	−0.183928	−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000	−0.183928	−0.0919640	+	0.995762i	$0.529314\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	24.0000	0.879883
$745$	0	0
$746$	23.0000	0.842090
$747$	0	0
$748$	0	0
$749$	0	0
$750$	6.00000	0.219089
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	−38.0000	−1.38480
$754$	−12.0000	−0.437014
$755$	27.0000	0.982631
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−22.0000	−0.799076
$759$	0	0
$760$	−9.00000	−0.326464
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	4.00000	0.144905
$763$	0	0
$764$	4.00000	0.144715
$765$	0	0
$766$	1.00000	0.0361315
$767$	44.0000	1.58875
$768$	−34.0000	−1.22687
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	40.0000	1.44056
$772$	16.0000	0.575853
$773$	17.0000	0.611448	0.305724	−	0.952120i	$-0.401102\pi$
$773$	17.0000	0.611448	0.305724	+	0.952120i	$0.401102\pi$
$774$	−6.00000	−0.215666
$775$	16.0000	0.574737
$776$	33.0000	1.18463
$777$	0	0
$778$	−22.0000	−0.788738
$779$	−10.0000	−0.358287
$780$	−24.0000	−0.859338
$781$	0	0
$782$	0	0
$783$	−12.0000	−0.428845
$784$	7.00000	0.250000
$785$	42.0000	1.49904
$786$	−30.0000	−1.07006
$787$	−9.00000	−0.320815	−0.160408	−	0.987051i	$-0.551281\pi$
$787$	−9.00000	−0.320815	−0.160408	+	0.987051i	$0.551281\pi$
$788$	18.0000	0.641223
$789$	36.0000	1.28163
$790$	−15.0000	−0.533676
$791$	0	0
$792$	0	0
$793$	0	0
$794$	22.0000	0.780751
$795$	36.0000	1.27679
$796$	0	0
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	0	0
$800$	−20.0000	−0.707107
$801$	9.00000	0.317999
$802$	−17.0000	−0.600291
$803$	0	0
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−16.0000	−0.563576
$807$	48.0000	1.68968
$808$	−15.0000	−0.527698
$809$	−7.00000	−0.246107	−0.123053	−	0.992400i	$-0.539269\pi$
$809$	−7.00000	−0.246107	−0.123053	+	0.992400i	$0.539269\pi$
$810$	33.0000	1.15950
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	0	0
$813$	58.0000	2.03415
$814$	0	0
$815$	18.0000	0.630512
$816$	0	0
$817$	−6.00000	−0.209913
$818$	−25.0000	−0.874105
$819$	0	0
$820$	−30.0000	−1.04765
$821$	1.00000	0.0349002	0.0174501	−	0.999848i	$-0.494445\pi$
$821$	1.00000	0.0349002	0.0174501	+	0.999848i	$0.494445\pi$
$822$	−20.0000	−0.697580
$823$	−3.00000	−0.104573	−0.0522867	−	0.998632i	$-0.516651\pi$
$823$	−3.00000	−0.104573	−0.0522867	+	0.998632i	$0.516651\pi$
$824$	39.0000	1.35863
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	1.00000	0.0347524
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	28.0000	0.970725
$833$	0	0
$834$	−28.0000	−0.969561
$835$	−72.0000	−2.49166
$836$	0	0
$837$	−16.0000	−0.553041
$838$	20.0000	0.690889
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	14.0000	0.482472
$843$	−8.00000	−0.275535
$844$	−1.00000	−0.0344214
$845$	9.00000	0.309609
$846$	0	0
$847$	0	0
$848$	−6.00000	−0.206041
$849$	22.0000	0.755038
$850$	0	0
$851$	0	0
$852$	18.0000	0.616670
$853$	−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
$855$	−3.00000	−0.102598
$856$	−33.0000	−1.12792
$857$	−34.0000	−1.16142	−0.580709	−	0.814111i	$-0.697225\pi$
$857$	−34.0000	−1.16142	−0.580709	+	0.814111i	$0.697225\pi$
$858$	0	0
$859$	−35.0000	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$859$	−35.0000	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$860$	−18.0000	−0.613795
$861$	0	0
$862$	24.0000	0.817443
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	20.0000	0.680414
$865$	−66.0000	−2.24407
$866$	26.0000	0.883516
$867$	−34.0000	−1.15470
$868$	0	0
$869$	0	0
$870$	−18.0000	−0.610257
$871$	8.00000	0.271070
$872$	30.0000	1.01593
$873$	11.0000	0.372294
$874$	−1.00000	−0.0338255
$875$	0	0
$876$	30.0000	1.01361
$877$	−3.00000	−0.101303	−0.0506514	−	0.998716i	$-0.516130\pi$
$877$	−3.00000	−0.101303	−0.0506514	+	0.998716i	$0.516130\pi$
$878$	−14.0000	−0.472477
$879$	48.0000	1.61900
$880$	0	0
$881$	23.0000	0.774890	0.387445	−	0.921893i	$-0.373358\pi$
$881$	23.0000	0.774890	0.387445	+	0.921893i	$0.373358\pi$
$882$	7.00000	0.235702
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	0	0
$885$	66.0000	2.21857
$886$	−18.0000	−0.604722
$887$	9.00000	0.302190	0.151095	−	0.988519i	$-0.451720\pi$
$887$	9.00000	0.302190	0.151095	+	0.988519i	$0.451720\pi$
$888$	0	0
$889$	0	0
$890$	−27.0000	−0.905042
$891$	0	0
$892$	−14.0000	−0.468755
$893$	0	0
$894$	0	0
$895$	63.0000	2.10586
$896$	0	0
$897$	−8.00000	−0.267112
$898$	−30.0000	−1.00111
$899$	12.0000	0.400222
$900$	−4.00000	−0.133333
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−48.0000	−1.59646
$905$	9.00000	0.299170
$906$	−18.0000	−0.598010
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	−7.00000	−0.232303
$909$	−5.00000	−0.165840
$910$	0	0
$911$	−46.0000	−1.52405	−0.762024	−	0.647549i	$-0.775794\pi$
$911$	−46.0000	−1.52405	−0.762024	+	0.647549i	$0.775794\pi$
$912$	2.00000	0.0662266
$913$	0	0
$914$	−22.0000	−0.727695
$915$	0	0
$916$	26.0000	0.859064
$917$	0	0
$918$	0	0
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	−9.00000	−0.296721
$921$	0	0
$922$	42.0000	1.38320
$923$	−36.0000	−1.18495
$924$	0	0
$925$	0	0
$926$	13.0000	0.427207
$927$	13.0000	0.426976
$928$	−15.0000	−0.492399
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	−24.0000	−0.786991
$931$	7.00000	0.229416
$932$	14.0000	0.458585
$933$	−36.0000	−1.17859
$934$	−6.00000	−0.196326
$935$	0	0
$936$	12.0000	0.392232
$937$	−5.00000	−0.163343	−0.0816714	−	0.996659i	$-0.526026\pi$
$937$	−5.00000	−0.163343	−0.0816714	+	0.996659i	$0.526026\pi$
$938$	0	0
$939$	40.0000	1.30535
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	−28.0000	−0.912289
$943$	−10.0000	−0.325645
$944$	−11.0000	−0.358020
$945$	0	0
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−10.0000	−0.324785
$949$	−60.0000	−1.94768
$950$	4.00000	0.129777
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−16.0000	−0.518291	−0.259145	−	0.965838i	$-0.583441\pi$
$953$	−16.0000	−0.518291	−0.259145	+	0.965838i	$0.583441\pi$
$954$	−6.00000	−0.194257
$955$	−12.0000	−0.388311
$956$	13.0000	0.420450
$957$	0	0
$958$	−35.0000	−1.13080
$959$	0	0
$960$	42.0000	1.35554
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−11.0000	−0.354470
$964$	12.0000	0.386494
$965$	−48.0000	−1.54517
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	−33.0000	−1.06066
$969$	0	0
$970$	−33.0000	−1.05957
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	8.00000	0.256337
$975$	32.0000	1.02482
$976$	0	0
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	−12.0000	−0.383718
$979$	0	0
$980$	21.0000	0.670820
$981$	10.0000	0.319275
$982$	28.0000	0.893516
$983$	−33.0000	−1.05254	−0.526268	−	0.850319i	$-0.676409\pi$
$983$	−33.0000	−1.05254	−0.526268	+	0.850319i	$0.676409\pi$
$984$	60.0000	1.91273
$985$	−54.0000	−1.72058
$986$	0	0
$987$	0	0
$988$	4.00000	0.127257
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	−20.0000	−0.635001
$993$	24.0000	0.761617
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	9.00000	0.284890
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.a.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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