LMFDB - Embedded newform 4998.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4998,2,Mod(1,4998)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4998, 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("4998.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4998.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:	$39.9092309302$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 714)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4998.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} -5.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -3.00000 q^{20} +6.00000 q^{22} +1.00000 q^{24} +4.00000 q^{25} +5.00000 q^{26} -1.00000 q^{27} +9.00000 q^{29} -3.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -4.00000 q^{38} +5.00000 q^{39} +3.00000 q^{40} +9.00000 q^{41} +8.00000 q^{43} -6.00000 q^{44} -3.00000 q^{45} -3.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} -1.00000 q^{51} -5.00000 q^{52} -12.0000 q^{53} +1.00000 q^{54} +18.0000 q^{55} -4.00000 q^{57} -9.00000 q^{58} -3.00000 q^{59} +3.00000 q^{60} -8.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} +15.0000 q^{65} -6.00000 q^{66} +8.00000 q^{67} +1.00000 q^{68} +12.0000 q^{71} -1.00000 q^{72} -2.00000 q^{73} -8.00000 q^{74} -4.00000 q^{75} +4.00000 q^{76} -5.00000 q^{78} +8.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} -9.00000 q^{83} -3.00000 q^{85} -8.00000 q^{86} -9.00000 q^{87} +6.00000 q^{88} -6.00000 q^{89} +3.00000 q^{90} -1.00000 q^{93} +3.00000 q^{94} -12.0000 q^{95} +1.00000 q^{96} -8.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.00000	0.948683
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−1.00000	−0.288675
$13$	−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
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	6.00000	1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	5.00000	0.980581
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	−3.00000	−0.547723
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	−1.00000	−0.176777
$33$	6.00000	1.04447
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−4.00000	−0.648886
$39$	5.00000	0.800641
$40$	3.00000	0.474342
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−6.00000	−0.904534
$45$	−3.00000	−0.447214
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−1.00000	−0.140028
$52$	−5.00000	−0.693375
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	1.00000	0.136083
$55$	18.0000	2.42712
$56$	0	0
$57$	−4.00000	−0.529813
$58$	−9.00000	−1.18176
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	3.00000	0.387298
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	15.0000	1.86052
$66$	−6.00000	−0.738549
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−8.00000	−0.929981
$75$	−4.00000	−0.461880
$76$	4.00000	0.458831
$77$	0	0
$78$	−5.00000	−0.566139
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−9.00000	−0.993884
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	−8.00000	−0.862662
$87$	−9.00000	−0.964901
$88$	6.00000	0.639602
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	0	0
$93$	−1.00000	−0.103695
$94$	3.00000	0.309426
$95$	−12.0000	−1.23117
$96$	1.00000	0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	4.00000	0.400000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	1.00000	0.0990148
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	12.0000	1.16554
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	−18.0000	−1.71623
$111$	−8.00000	−0.759326
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	9.00000	0.835629
$117$	−5.00000	−0.462250
$118$	3.00000	0.276172
$119$	0	0
$120$	−3.00000	−0.273861
$121$	25.0000	2.27273
$122$	8.00000	0.724286
$123$	−9.00000	−0.811503
$124$	1.00000	0.0898027
$125$	3.00000	0.268328
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	−15.0000	−1.31559
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	6.00000	0.522233
$133$	0	0
$134$	−8.00000	−0.691095
$135$	3.00000	0.258199
$136$	−1.00000	−0.0857493
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	−12.0000	−1.00702
$143$	30.0000	2.50873
$144$	1.00000	0.0833333
$145$	−27.0000	−2.24223
$146$	2.00000	0.165521
$147$	0	0
$148$	8.00000	0.657596
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	4.00000	0.326599
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−4.00000	−0.324443
$153$	1.00000	0.0808452
$154$	0	0
$155$	−3.00000	−0.240966
$156$	5.00000	0.400320
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	−8.00000	−0.636446
$159$	12.0000	0.951662
$160$	3.00000	0.237171
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	9.00000	0.702782
$165$	−18.0000	−1.40130
$166$	9.00000	0.698535
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	3.00000	0.230089
$171$	4.00000	0.305888
$172$	8.00000	0.609994
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	9.00000	0.682288
$175$	0	0
$176$	−6.00000	−0.452267
$177$	3.00000	0.225494
$178$	6.00000	0.449719
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	−3.00000	−0.223607
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	−24.0000	−1.76452
$186$	1.00000	0.0733236
$187$	−6.00000	−0.438763
$188$	−3.00000	−0.218797
$189$	0	0
$190$	12.0000	0.870572
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	−1.00000	−0.0721688
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	8.00000	0.574367
$195$	−15.0000	−1.07417
$196$	0	0
$197$	−21.0000	−1.49619	−0.748094	−	0.663593i	$-0.769031\pi$
$197$	−21.0000	−1.49619	−0.748094	+	0.663593i	$0.769031\pi$
$198$	6.00000	0.426401
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	−4.00000	−0.282843
$201$	−8.00000	−0.564276
$202$	18.0000	1.26648
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	−27.0000	−1.88576
$206$	−10.0000	−0.696733
$207$	0	0
$208$	−5.00000	−0.346688
$209$	−24.0000	−1.66011
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−12.0000	−0.824163
$213$	−12.0000	−0.822226
$214$	12.0000	0.820303
$215$	−24.0000	−1.63679
$216$	1.00000	0.0680414
$217$	0	0
$218$	4.00000	0.270914
$219$	2.00000	0.135147
$220$	18.0000	1.21356
$221$	−5.00000	−0.336336
$222$	8.00000	0.536925
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	−9.00000	−0.598671
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−4.00000	−0.264906
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	0	0
$231$	0	0
$232$	−9.00000	−0.590879
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	5.00000	0.326860
$235$	9.00000	0.587095
$236$	−3.00000	−0.195283
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	3.00000	0.193649
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−25.0000	−1.60706
$243$	−1.00000	−0.0641500
$244$	−8.00000	−0.512148
$245$	0	0
$246$	9.00000	0.573819
$247$	−20.0000	−1.27257
$248$	−1.00000	−0.0635001
$249$	9.00000	0.570352
$250$	−3.00000	−0.189737
$251$	9.00000	0.568075	0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000	0.568075	0.284037	+	0.958813i	$0.408326\pi$
$252$	0	0
$253$	0	0
$254$	4.00000	0.250982
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	8.00000	0.498058
$259$	0	0
$260$	15.0000	0.930261
$261$	9.00000	0.557086
$262$	18.0000	1.11204
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−6.00000	−0.369274
$265$	36.0000	2.21146
$266$	0	0
$267$	6.00000	0.367194
$268$	8.00000	0.488678
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	−3.00000	−0.182574
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	0	0
$275$	−24.0000	−1.44725
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−16.0000	−0.959616
$279$	1.00000	0.0598684
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	−3.00000	−0.178647
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	12.0000	0.712069
$285$	12.0000	0.710819
$286$	−30.0000	−1.77394
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	27.0000	1.58549
$291$	8.00000	0.468968
$292$	−2.00000	−0.117041
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	9.00000	0.524000
$296$	−8.00000	−0.464991
$297$	6.00000	0.348155
$298$	−18.0000	−1.04271
$299$	0	0
$300$	−4.00000	−0.230940
$301$	0	0
$302$	−8.00000	−0.460348
$303$	18.0000	1.03407
$304$	4.00000	0.229416
$305$	24.0000	1.37424
$306$	−1.00000	−0.0571662
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	0	0
$309$	−10.0000	−0.568880
$310$	3.00000	0.170389
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	−5.00000	−0.283069
$313$	4.00000	0.226093	0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000	0.226093	0.113047	+	0.993590i	$0.463939\pi$
$314$	5.00000	0.282166
$315$	0	0
$316$	8.00000	0.450035
$317$	−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
$319$	−54.0000	−3.02342
$320$	−3.00000	−0.167705
$321$	12.0000	0.669775
$322$	0	0
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	−20.0000	−1.10940
$326$	16.0000	0.886158
$327$	4.00000	0.221201
$328$	−9.00000	−0.496942
$329$	0	0
$330$	18.0000	0.990867
$331$	−34.0000	−1.86881	−0.934405	−	0.356214i	$-0.884068\pi$
$331$	−34.0000	−1.86881	−0.934405	+	0.356214i	$0.884068\pi$
$332$	−9.00000	−0.493939
$333$	8.00000	0.438397
$334$	6.00000	0.328305
$335$	−24.0000	−1.31126
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−12.0000	−0.652714
$339$	−9.00000	−0.488813
$340$	−3.00000	−0.162698
$341$	−6.00000	−0.324918
$342$	−4.00000	−0.216295
$343$	0	0
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−9.00000	−0.483843
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−9.00000	−0.482451
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	6.00000	0.319801
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	−3.00000	−0.159448
$355$	−36.0000	−1.91068
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−3.00000	−0.158555
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	3.00000	0.158114
$361$	−3.00000	−0.157895
$362$	8.00000	0.420471
$363$	−25.0000	−1.31216
$364$	0	0
$365$	6.00000	0.314054
$366$	−8.00000	−0.418167
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	9.00000	0.468521
$370$	24.0000	1.24770
$371$	0	0
$372$	−1.00000	−0.0518476
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	6.00000	0.310253
$375$	−3.00000	−0.154919
$376$	3.00000	0.154713
$377$	−45.0000	−2.31762
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	−12.0000	−0.615587
$381$	4.00000	0.204926
$382$	9.00000	0.460480
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−26.0000	−1.32337
$387$	8.00000	0.406663
$388$	−8.00000	−0.406138
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	15.0000	0.759555
$391$	0	0
$392$	0	0
$393$	18.0000	0.907980
$394$	21.0000	1.05796
$395$	−24.0000	−1.20757
$396$	−6.00000	−0.301511
$397$	−32.0000	−1.60603	−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000	−1.60603	−0.803017	+	0.595956i	$0.796773\pi$
$398$	−7.00000	−0.350878
$399$	0	0
$400$	4.00000	0.200000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	8.00000	0.399004
$403$	−5.00000	−0.249068
$404$	−18.0000	−0.895533
$405$	−3.00000	−0.149071
$406$	0	0
$407$	−48.0000	−2.37927
$408$	1.00000	0.0495074
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	27.0000	1.33343
$411$	0	0
$412$	10.0000	0.492665
$413$	0	0
$414$	0	0
$415$	27.0000	1.32538
$416$	5.00000	0.245145
$417$	−16.0000	−0.783523
$418$	24.0000	1.17388
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	5.00000	0.243685	0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000	0.243685	0.121843	+	0.992549i	$0.461120\pi$
$422$	−5.00000	−0.243396
$423$	−3.00000	−0.145865
$424$	12.0000	0.582772
$425$	4.00000	0.194029
$426$	12.0000	0.581402
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−30.0000	−1.44841
$430$	24.0000	1.15738
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−1.00000	−0.0481125
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	27.0000	1.29455
$436$	−4.00000	−0.191565
$437$	0	0
$438$	−2.00000	−0.0955637
$439$	19.0000	0.906821	0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	−18.0000	−0.858116
$441$	0	0
$442$	5.00000	0.237826
$443$	−15.0000	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$444$	−8.00000	−0.379663
$445$	18.0000	0.853282
$446$	8.00000	0.378811
$447$	−18.0000	−0.851371
$448$	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
$451$	−54.0000	−2.54276
$452$	9.00000	0.423324
$453$	−8.00000	−0.375873
$454$	−18.0000	−0.844782
$455$	0	0
$456$	4.00000	0.187317
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	−13.0000	−0.607450
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	9.00000	0.417815
$465$	3.00000	0.139122
$466$	−3.00000	−0.138972
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	−5.00000	−0.231125
$469$	0	0
$470$	−9.00000	−0.415139
$471$	5.00000	0.230388
$472$	3.00000	0.138086
$473$	−48.0000	−2.20704
$474$	8.00000	0.367452
$475$	16.0000	0.734130
$476$	0	0
$477$	−12.0000	−0.549442
$478$	15.0000	0.686084
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	−3.00000	−0.136931
$481$	−40.0000	−1.82384
$482$	26.0000	1.18427
$483$	0	0
$484$	25.0000	1.13636
$485$	24.0000	1.08978
$486$	1.00000	0.0453609
$487$	23.0000	1.04223	0.521115	−	0.853487i	$-0.325516\pi$
$487$	23.0000	1.04223	0.521115	+	0.853487i	$0.325516\pi$
$488$	8.00000	0.362143
$489$	16.0000	0.723545
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−9.00000	−0.405751
$493$	9.00000	0.405340
$494$	20.0000	0.899843
$495$	18.0000	0.809040
$496$	1.00000	0.0449013
$497$	0	0
$498$	−9.00000	−0.403300
$499$	11.0000	0.492428	0.246214	−	0.969216i	$-0.420813\pi$
$499$	11.0000	0.492428	0.246214	+	0.969216i	$0.420813\pi$
$500$	3.00000	0.134164
$501$	6.00000	0.268060
$502$	−9.00000	−0.401690
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	54.0000	2.40297
$506$	0	0
$507$	−12.0000	−0.532939
$508$	−4.00000	−0.177471
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	−3.00000	−0.132842
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	−30.0000	−1.32324
$515$	−30.0000	−1.32196
$516$	−8.00000	−0.352180
$517$	18.0000	0.791639
$518$	0	0
$519$	−9.00000	−0.395056
$520$	−15.0000	−0.657794
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−9.00000	−0.393919
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	24.0000	1.04645
$527$	1.00000	0.0435607
$528$	6.00000	0.261116
$529$	−23.0000	−1.00000
$530$	−36.0000	−1.56374
$531$	−3.00000	−0.130189
$532$	0	0
$533$	−45.0000	−1.94917
$534$	−6.00000	−0.259645
$535$	36.0000	1.55642
$536$	−8.00000	−0.345547
$537$	−3.00000	−0.129460
$538$	−15.0000	−0.646696
$539$	0	0
$540$	3.00000	0.129099
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	2.00000	0.0859074
$543$	8.00000	0.343313
$544$	−1.00000	−0.0428746
$545$	12.0000	0.514024
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	24.0000	1.02336
$551$	36.0000	1.53365
$552$	0	0
$553$	0	0
$554$	28.0000	1.18961
$555$	24.0000	1.01874
$556$	16.0000	0.678551
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	−1.00000	−0.0423334
$559$	−40.0000	−1.69182
$560$	0	0
$561$	6.00000	0.253320
$562$	12.0000	0.506189
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	3.00000	0.126323
$565$	−27.0000	−1.13590
$566$	5.00000	0.210166
$567$	0	0
$568$	−12.0000	−0.503509
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	−12.0000	−0.502625
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	30.0000	1.25436
$573$	9.00000	0.375980
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−41.0000	−1.70685	−0.853426	−	0.521214i	$-0.825479\pi$
$577$	−41.0000	−1.70685	−0.853426	+	0.521214i	$0.825479\pi$
$578$	−1.00000	−0.0415945
$579$	−26.0000	−1.08052
$580$	−27.0000	−1.12111
$581$	0	0
$582$	−8.00000	−0.331611
$583$	72.0000	2.98194
$584$	2.00000	0.0827606
$585$	15.0000	0.620174
$586$	−12.0000	−0.495715
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	−9.00000	−0.370524
$591$	21.0000	0.863825
$592$	8.00000	0.328798
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	−6.00000	−0.246183
$595$	0	0
$596$	18.0000	0.737309
$597$	−7.00000	−0.286491
$598$	0	0
$599$	−9.00000	−0.367730	−0.183865	−	0.982952i	$-0.558861\pi$
$599$	−9.00000	−0.367730	−0.183865	+	0.982952i	$0.558861\pi$
$600$	4.00000	0.163299
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	8.00000	0.325515
$605$	−75.0000	−3.04918
$606$	−18.0000	−0.731200
$607$	−11.0000	−0.446476	−0.223238	−	0.974764i	$-0.571663\pi$
$607$	−11.0000	−0.446476	−0.223238	+	0.974764i	$0.571663\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−24.0000	−0.971732
$611$	15.0000	0.606835
$612$	1.00000	0.0404226
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	8.00000	0.322854
$615$	27.0000	1.08875
$616$	0	0
$617$	−33.0000	−1.32853	−0.664265	−	0.747497i	$-0.731255\pi$
$617$	−33.0000	−1.32853	−0.664265	+	0.747497i	$0.731255\pi$
$618$	10.0000	0.402259
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	−3.00000	−0.120483
$621$	0	0
$622$	6.00000	0.240578
$623$	0	0
$624$	5.00000	0.200160
$625$	−29.0000	−1.16000
$626$	−4.00000	−0.159872
$627$	24.0000	0.958468
$628$	−5.00000	−0.199522
$629$	8.00000	0.318981
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	−8.00000	−0.318223
$633$	−5.00000	−0.198732
$634$	9.00000	0.357436
$635$	12.0000	0.476205
$636$	12.0000	0.475831
$637$	0	0
$638$	54.0000	2.13788
$639$	12.0000	0.474713
$640$	3.00000	0.118585
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−12.0000	−0.473602
$643$	−11.0000	−0.433798	−0.216899	−	0.976194i	$-0.569594\pi$
$643$	−11.0000	−0.433798	−0.216899	+	0.976194i	$0.569594\pi$
$644$	0	0
$645$	24.0000	0.944999
$646$	−4.00000	−0.157378
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	−1.00000	−0.0392837
$649$	18.0000	0.706562
$650$	20.0000	0.784465
$651$	0	0
$652$	−16.0000	−0.626608
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−4.00000	−0.156412
$655$	54.0000	2.10995
$656$	9.00000	0.351391
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	−33.0000	−1.28550	−0.642749	−	0.766077i	$-0.722206\pi$
$659$	−33.0000	−1.28550	−0.642749	+	0.766077i	$0.722206\pi$
$660$	−18.0000	−0.700649
$661$	13.0000	0.505641	0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000	0.505641	0.252821	+	0.967513i	$0.418642\pi$
$662$	34.0000	1.32145
$663$	5.00000	0.194184
$664$	9.00000	0.349268
$665$	0	0
$666$	−8.00000	−0.309994
$667$	0	0
$668$	−6.00000	−0.232147
$669$	8.00000	0.309298
$670$	24.0000	0.927201
$671$	48.0000	1.85302
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	−2.00000	−0.0770371
$675$	−4.00000	−0.153960
$676$	12.0000	0.461538
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	9.00000	0.345643
$679$	0	0
$680$	3.00000	0.115045
$681$	−18.0000	−0.689761
$682$	6.00000	0.229752
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	−13.0000	−0.495981
$688$	8.00000	0.304997
$689$	60.0000	2.28582
$690$	0	0
$691$	1.00000	0.0380418	0.0190209	−	0.999819i	$-0.493945\pi$
$691$	1.00000	0.0380418	0.0190209	+	0.999819i	$0.493945\pi$
$692$	9.00000	0.342129
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−48.0000	−1.82074
$696$	9.00000	0.341144
$697$	9.00000	0.340899
$698$	−1.00000	−0.0378506
$699$	−3.00000	−0.113470
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	−5.00000	−0.188713
$703$	32.0000	1.20690
$704$	−6.00000	−0.226134
$705$	−9.00000	−0.338960
$706$	24.0000	0.903252
$707$	0	0
$708$	3.00000	0.112747
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	36.0000	1.35106
$711$	8.00000	0.300023
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	−90.0000	−3.36581
$716$	3.00000	0.112115
$717$	15.0000	0.560185
$718$	−3.00000	−0.111959
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	−3.00000	−0.111803
$721$	0	0
$722$	3.00000	0.111648
$723$	26.0000	0.966950
$724$	−8.00000	−0.297318
$725$	36.0000	1.33701
$726$	25.0000	0.927837
$727$	−50.0000	−1.85440	−0.927199	−	0.374570i	$-0.877790\pi$
$727$	−50.0000	−1.85440	−0.927199	+	0.374570i	$0.877790\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	8.00000	0.295891
$732$	8.00000	0.295689
$733$	10.0000	0.369358	0.184679	−	0.982799i	$-0.440875\pi$
$733$	10.0000	0.369358	0.184679	+	0.982799i	$0.440875\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	0	0
$737$	−48.0000	−1.76810
$738$	−9.00000	−0.331295
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	−24.0000	−0.882258
$741$	20.0000	0.734718
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	1.00000	0.0366618
$745$	−54.0000	−1.97841
$746$	1.00000	0.0366126
$747$	−9.00000	−0.329293
$748$	−6.00000	−0.219382
$749$	0	0
$750$	3.00000	0.109545
$751$	47.0000	1.71505	0.857527	−	0.514439i	$-0.172000\pi$
$751$	47.0000	1.71505	0.857527	+	0.514439i	$0.172000\pi$
$752$	−3.00000	−0.109399
$753$	−9.00000	−0.327978
$754$	45.0000	1.63880
$755$	−24.0000	−0.873449
$756$	0	0
$757$	−25.0000	−0.908640	−0.454320	−	0.890838i	$-0.650118\pi$
$757$	−25.0000	−0.908640	−0.454320	+	0.890838i	$0.650118\pi$
$758$	−8.00000	−0.290573
$759$	0	0
$760$	12.0000	0.435286
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	−4.00000	−0.144905
$763$	0	0
$764$	−9.00000	−0.325609
$765$	−3.00000	−0.108465
$766$	0	0
$767$	15.0000	0.541619
$768$	−1.00000	−0.0360844
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	26.0000	0.935760
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	−8.00000	−0.287554
$775$	4.00000	0.143684
$776$	8.00000	0.287183
$777$	0	0
$778$	6.00000	0.215110
$779$	36.0000	1.28983
$780$	−15.0000	−0.537086
$781$	−72.0000	−2.57636
$782$	0	0
$783$	−9.00000	−0.321634
$784$	0	0
$785$	15.0000	0.535373
$786$	−18.0000	−0.642039
$787$	−5.00000	−0.178231	−0.0891154	−	0.996021i	$-0.528404\pi$
$787$	−5.00000	−0.178231	−0.0891154	+	0.996021i	$0.528404\pi$
$788$	−21.0000	−0.748094
$789$	24.0000	0.854423
$790$	24.0000	0.853882
$791$	0	0
$792$	6.00000	0.213201
$793$	40.0000	1.42044
$794$	32.0000	1.13564
$795$	−36.0000	−1.27679
$796$	7.00000	0.248108
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	−3.00000	−0.106132
$800$	−4.00000	−0.141421
$801$	−6.00000	−0.212000
$802$	30.0000	1.05934
$803$	12.0000	0.423471
$804$	−8.00000	−0.282138
$805$	0	0
$806$	5.00000	0.176117
$807$	−15.0000	−0.528025
$808$	18.0000	0.633238
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	3.00000	0.105409
$811$	43.0000	1.50993	0.754967	−	0.655763i	$-0.227653\pi$
$811$	43.0000	1.50993	0.754967	+	0.655763i	$0.227653\pi$
$812$	0	0
$813$	2.00000	0.0701431
$814$	48.0000	1.68240
$815$	48.0000	1.68137
$816$	−1.00000	−0.0350070
$817$	32.0000	1.11954
$818$	2.00000	0.0699284
$819$	0	0
$820$	−27.0000	−0.942881
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	−10.0000	−0.348367
$825$	24.0000	0.835573
$826$	0	0
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	0	0
$829$	22.0000	0.764092	0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000	0.764092	0.382046	+	0.924143i	$0.375220\pi$
$830$	−27.0000	−0.937184
$831$	28.0000	0.971309
$832$	−5.00000	−0.173344
$833$	0	0
$834$	16.0000	0.554035
$835$	18.0000	0.622916
$836$	−24.0000	−0.830057
$837$	−1.00000	−0.0345651
$838$	−36.0000	−1.24360
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−5.00000	−0.172311
$843$	12.0000	0.413302
$844$	5.00000	0.172107
$845$	−36.0000	−1.23844
$846$	3.00000	0.103142
$847$	0	0
$848$	−12.0000	−0.412082
$849$	5.00000	0.171600
$850$	−4.00000	−0.137199
$851$	0	0
$852$	−12.0000	−0.411113
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	12.0000	0.410152
$857$	51.0000	1.74213	0.871063	−	0.491171i	$-0.163431\pi$
$857$	51.0000	1.74213	0.871063	+	0.491171i	$0.163431\pi$
$858$	30.0000	1.02418
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	−24.0000	−0.818393
$861$	0	0
$862$	−12.0000	−0.408722
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	1.00000	0.0340207
$865$	−27.0000	−0.918028
$866$	11.0000	0.373795
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−48.0000	−1.62829
$870$	−27.0000	−0.915386
$871$	−40.0000	−1.35535
$872$	4.00000	0.135457
$873$	−8.00000	−0.270759
$874$	0	0
$875$	0	0
$876$	2.00000	0.0675737
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−19.0000	−0.641219
$879$	−12.0000	−0.404750
$880$	18.0000	0.606780
$881$	−3.00000	−0.101073	−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	−5.00000	−0.168168
$885$	−9.00000	−0.302532
$886$	15.0000	0.503935
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	8.00000	0.268462
$889$	0	0
$890$	−18.0000	−0.603361
$891$	−6.00000	−0.201008
$892$	−8.00000	−0.267860
$893$	−12.0000	−0.401565
$894$	18.0000	0.602010
$895$	−9.00000	−0.300837
$896$	0	0
$897$	0	0
$898$	−30.0000	−1.00111
$899$	9.00000	0.300167
$900$	4.00000	0.133333
$901$	−12.0000	−0.399778
$902$	54.0000	1.79800
$903$	0	0
$904$	−9.00000	−0.299336
$905$	24.0000	0.797787
$906$	8.00000	0.265782
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	18.0000	0.597351
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	−4.00000	−0.132453
$913$	54.0000	1.78714
$914$	−17.0000	−0.562310
$915$	−24.0000	−0.793416
$916$	13.0000	0.429532
$917$	0	0
$918$	1.00000	0.0330049
$919$	−46.0000	−1.51740	−0.758700	−	0.651440i	$-0.774165\pi$
$919$	−46.0000	−1.51740	−0.758700	+	0.651440i	$0.774165\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	12.0000	0.395199
$923$	−60.0000	−1.97492
$924$	0	0
$925$	32.0000	1.05215
$926$	10.0000	0.328620
$927$	10.0000	0.328443
$928$	−9.00000	−0.295439
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	−3.00000	−0.0983739
$931$	0	0
$932$	3.00000	0.0982683
$933$	6.00000	0.196431
$934$	−27.0000	−0.883467
$935$	18.0000	0.588663
$936$	5.00000	0.163430
$937$	46.0000	1.50275	0.751377	−	0.659873i	$-0.229390\pi$
$937$	46.0000	1.50275	0.751377	+	0.659873i	$0.229390\pi$
$938$	0	0
$939$	−4.00000	−0.130535
$940$	9.00000	0.293548
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−5.00000	−0.162909
$943$	0	0
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	48.0000	1.56061
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	−8.00000	−0.259828
$949$	10.0000	0.324614
$950$	−16.0000	−0.519109
$951$	9.00000	0.291845
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	12.0000	0.388514
$955$	27.0000	0.873699
$956$	−15.0000	−0.485135
$957$	54.0000	1.74557
$958$	−18.0000	−0.581554
$959$	0	0
$960$	3.00000	0.0968246
$961$	−30.0000	−0.967742
$962$	40.0000	1.28965
$963$	−12.0000	−0.386695
$964$	−26.0000	−0.837404
$965$	−78.0000	−2.51091
$966$	0	0
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	−25.0000	−0.803530
$969$	−4.00000	−0.128499
$970$	−24.0000	−0.770594
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−23.0000	−0.736968
$975$	20.0000	0.640513
$976$	−8.00000	−0.256074
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	−16.0000	−0.511624
$979$	36.0000	1.15056
$980$	0	0
$981$	−4.00000	−0.127710
$982$	−12.0000	−0.382935
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	9.00000	0.286910
$985$	63.0000	2.00735
$986$	−9.00000	−0.286618
$987$	0	0
$988$	−20.0000	−0.636285
$989$	0	0
$990$	−18.0000	−0.572078
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	−1.00000	−0.0317500
$993$	34.0000	1.07896
$994$	0	0
$995$	−21.0000	−0.665745
$996$	9.00000	0.285176
$997$	−32.0000	−1.01345	−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000	−1.01345	−0.506725	+	0.862108i	$0.669144\pi$
$998$	−11.0000	−0.348199
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4998.2.a.a.1.1		1
7.3	odd	6		714.2.i.g.205.1	✓	2
7.5	odd	6		714.2.i.g.613.1	yes	2
7.6	odd	2		4998.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.i.g.205.1	✓	2	7.3	odd	6
714.2.i.g.613.1	yes	2	7.5	odd	6
4998.2.a.a.1.1		1	1.1	even	1	trivial
4998.2.a.w.1.1		1	7.6	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:	\( N \)	\(=\)	\( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4998.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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