LMFDB - Embedded newform 1274.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1274,4,Mod(1,1274)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1274.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1274, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1274 = 2 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1274.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$75.1684333473$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1274.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} -7.00000 q^{3} +4.00000 q^{4} +14.0000 q^{6} -8.00000 q^{8} +22.0000 q^{9} +39.0000 q^{11} -28.0000 q^{12} -13.0000 q^{13} +16.0000 q^{16} -24.0000 q^{17} -44.0000 q^{18} -38.0000 q^{19} -78.0000 q^{22} +39.0000 q^{23} +56.0000 q^{24} -125.000 q^{25} +26.0000 q^{26} +35.0000 q^{27} -96.0000 q^{29} -227.000 q^{31} -32.0000 q^{32} -273.000 q^{33} +48.0000 q^{34} +88.0000 q^{36} +425.000 q^{37} +76.0000 q^{38} +91.0000 q^{39} +105.000 q^{41} +344.000 q^{43} +156.000 q^{44} -78.0000 q^{46} -99.0000 q^{47} -112.000 q^{48} +250.000 q^{50} +168.000 q^{51} -52.0000 q^{52} -540.000 q^{53} -70.0000 q^{54} +266.000 q^{57} +192.000 q^{58} -114.000 q^{59} +565.000 q^{61} +454.000 q^{62} +64.0000 q^{64} +546.000 q^{66} -385.000 q^{67} -96.0000 q^{68} -273.000 q^{69} -156.000 q^{71} -176.000 q^{72} +673.000 q^{73} -850.000 q^{74} +875.000 q^{75} -152.000 q^{76} -182.000 q^{78} +749.000 q^{79} -839.000 q^{81} -210.000 q^{82} +1044.00 q^{83} -688.000 q^{86} +672.000 q^{87} -312.000 q^{88} +690.000 q^{89} +156.000 q^{92} +1589.00 q^{93} +198.000 q^{94} +224.000 q^{96} -317.000 q^{97} +858.000 q^{99} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−7.00000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	4.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	14.0000	0.952579
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	22.0000	0.814815
$10$	0	0
$11$	39.0000	1.06899	0.534497	−	0.845170i	$-0.320501\pi$
$11$	39.0000	1.06899	0.534497	+	0.845170i	$0.320501\pi$
$12$	−28.0000	−0.673575
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−24.0000	−0.342403	−0.171202	−	0.985236i	$-0.554765\pi$
$17$	−24.0000	−0.342403	−0.171202	+	0.985236i	$0.554765\pi$
$18$	−44.0000	−0.576161
$19$	−38.0000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−38.0000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	−78.0000	−0.755893
$23$	39.0000	0.353568	0.176784	−	0.984250i	$-0.443431\pi$
$23$	39.0000	0.353568	0.176784	+	0.984250i	$0.443431\pi$
$24$	56.0000	0.476290
$25$	−125.000	−1.00000
$26$	26.0000	0.196116
$27$	35.0000	0.249472
$28$	0	0
$29$	−96.0000	−0.614716	−0.307358	−	0.951594i	$-0.599445\pi$
$29$	−96.0000	−0.614716	−0.307358	+	0.951594i	$0.599445\pi$
$30$	0	0
$31$	−227.000	−1.31517	−0.657587	−	0.753378i	$-0.728423\pi$
$31$	−227.000	−1.31517	−0.657587	+	0.753378i	$0.728423\pi$
$32$	−32.0000	−0.176777
$33$	−273.000	−1.44010
$34$	48.0000	0.242116
$35$	0	0
$36$	88.0000	0.407407
$37$	425.000	1.88837	0.944183	−	0.329420i	$-0.106853\pi$
$37$	425.000	1.88837	0.944183	+	0.329420i	$0.106853\pi$
$38$	76.0000	0.324443
$39$	91.0000	0.373632
$40$	0	0
$41$	105.000	0.399957	0.199979	−	0.979800i	$-0.435913\pi$
$41$	105.000	0.399957	0.199979	+	0.979800i	$0.435913\pi$
$42$	0	0
$43$	344.000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	344.000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	156.000	0.534497
$45$	0	0
$46$	−78.0000	−0.250010
$47$	−99.0000	−0.307248	−0.153624	−	0.988129i	$-0.549094\pi$
$47$	−99.0000	−0.307248	−0.153624	+	0.988129i	$0.549094\pi$
$48$	−112.000	−0.336788
$49$	0	0
$50$	250.000	0.707107
$51$	168.000	0.461269
$52$	−52.0000	−0.138675
$53$	−540.000	−1.39952	−0.699761	−	0.714377i	$-0.746710\pi$
$53$	−540.000	−1.39952	−0.699761	+	0.714377i	$0.746710\pi$
$54$	−70.0000	−0.176404
$55$	0	0
$56$	0	0
$57$	266.000	0.618115
$58$	192.000	0.434670
$59$	−114.000	−0.251551	−0.125776	−	0.992059i	$-0.540142\pi$
$59$	−114.000	−0.251551	−0.125776	+	0.992059i	$0.540142\pi$
$60$	0	0
$61$	565.000	1.18592	0.592958	−	0.805234i	$-0.297960\pi$
$61$	565.000	1.18592	0.592958	+	0.805234i	$0.297960\pi$
$62$	454.000	0.929969
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	546.000	1.01830
$67$	−385.000	−0.702018	−0.351009	−	0.936372i	$-0.614161\pi$
$67$	−385.000	−0.702018	−0.351009	+	0.936372i	$0.614161\pi$
$68$	−96.0000	−0.171202
$69$	−273.000	−0.476309
$70$	0	0
$71$	−156.000	−0.260758	−0.130379	−	0.991464i	$-0.541619\pi$
$71$	−156.000	−0.260758	−0.130379	+	0.991464i	$0.541619\pi$
$72$	−176.000	−0.288081
$73$	673.000	1.07902	0.539512	−	0.841978i	$-0.318609\pi$
$73$	673.000	1.07902	0.539512	+	0.841978i	$0.318609\pi$
$74$	−850.000	−1.33528
$75$	875.000	1.34715
$76$	−152.000	−0.229416
$77$	0	0
$78$	−182.000	−0.264198
$79$	749.000	1.06670	0.533349	−	0.845896i	$-0.320933\pi$
$79$	749.000	1.06670	0.533349	+	0.845896i	$0.320933\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	−210.000	−0.282812
$83$	1044.00	1.38065	0.690325	−	0.723500i	$-0.257468\pi$
$83$	1044.00	1.38065	0.690325	+	0.723500i	$0.257468\pi$
$84$	0	0
$85$	0	0
$86$	−688.000	−0.862662
$87$	672.000	0.828115
$88$	−312.000	−0.377947
$89$	690.000	0.821796	0.410898	−	0.911681i	$-0.365215\pi$
$89$	690.000	0.821796	0.410898	+	0.911681i	$0.365215\pi$
$90$	0	0
$91$	0	0
$92$	156.000	0.176784
$93$	1589.00	1.77174
$94$	198.000	0.217257
$95$	0	0
$96$	224.000	0.238145
$97$	−317.000	−0.331819	−0.165910	−	0.986141i	$-0.553056\pi$
$97$	−317.000	−0.331819	−0.165910	+	0.986141i	$0.553056\pi$
$98$	0	0
$99$	858.000	0.871033
$100$	−500.000	−0.500000
$101$	663.000	0.653178	0.326589	−	0.945166i	$-0.394101\pi$
$101$	663.000	0.653178	0.326589	+	0.945166i	$0.394101\pi$
$102$	−336.000	−0.326166
$103$	646.000	0.617983	0.308992	−	0.951065i	$-0.400008\pi$
$103$	646.000	0.617983	0.308992	+	0.951065i	$0.400008\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	1080.00	0.989612
$107$	−744.000	−0.672198	−0.336099	−	0.941827i	$-0.609108\pi$
$107$	−744.000	−0.672198	−0.336099	+	0.941827i	$0.609108\pi$
$108$	140.000	0.124736
$109$	218.000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	218.000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2975.00	−2.54391
$112$	0	0
$113$	−1623.00	−1.35114	−0.675571	−	0.737295i	$-0.736103\pi$
$113$	−1623.00	−1.35114	−0.675571	+	0.737295i	$0.736103\pi$
$114$	−532.000	−0.437073
$115$	0	0
$116$	−384.000	−0.307358
$117$	−286.000	−0.225989
$118$	228.000	0.177874
$119$	0	0
$120$	0	0
$121$	190.000	0.142750
$122$	−1130.00	−0.838569
$123$	−735.000	−0.538803
$124$	−908.000	−0.657587
$125$	0	0
$126$	0	0
$127$	659.000	0.460447	0.230224	−	0.973138i	$-0.426054\pi$
$127$	659.000	0.460447	0.230224	+	0.973138i	$0.426054\pi$
$128$	−128.000	−0.0883883
$129$	−2408.00	−1.64351
$130$	0	0
$131$	216.000	0.144061	0.0720306	−	0.997402i	$-0.477052\pi$
$131$	216.000	0.144061	0.0720306	+	0.997402i	$0.477052\pi$
$132$	−1092.00	−0.720048
$133$	0	0
$134$	770.000	0.496402
$135$	0	0
$136$	192.000	0.121058
$137$	1842.00	1.14871	0.574353	−	0.818608i	$-0.305254\pi$
$137$	1842.00	1.14871	0.574353	+	0.818608i	$0.305254\pi$
$138$	546.000	0.336801
$139$	628.000	0.383211	0.191605	−	0.981472i	$-0.438631\pi$
$139$	628.000	0.383211	0.191605	+	0.981472i	$0.438631\pi$
$140$	0	0
$141$	693.000	0.413909
$142$	312.000	0.184384
$143$	−507.000	−0.296486
$144$	352.000	0.203704
$145$	0	0
$146$	−1346.00	−0.762985
$147$	0	0
$148$	1700.00	0.944183
$149$	321.000	0.176492	0.0882461	−	0.996099i	$-0.471874\pi$
$149$	321.000	0.176492	0.0882461	+	0.996099i	$0.471874\pi$
$150$	−1750.00	−0.952579
$151$	−1600.00	−0.862292	−0.431146	−	0.902282i	$-0.641891\pi$
$151$	−1600.00	−0.862292	−0.431146	+	0.902282i	$0.641891\pi$
$152$	304.000	0.162221
$153$	−528.000	−0.278995
$154$	0	0
$155$	0	0
$156$	364.000	0.186816
$157$	−1127.00	−0.572894	−0.286447	−	0.958096i	$-0.592474\pi$
$157$	−1127.00	−0.572894	−0.286447	+	0.958096i	$0.592474\pi$
$158$	−1498.00	−0.754269
$159$	3780.00	1.88537
$160$	0	0
$161$	0	0
$162$	1678.00	0.813803
$163$	−1204.00	−0.578556	−0.289278	−	0.957245i	$-0.593415\pi$
$163$	−1204.00	−0.578556	−0.289278	+	0.957245i	$0.593415\pi$
$164$	420.000	0.199979
$165$	0	0
$166$	−2088.00	−0.976266
$167$	600.000	0.278020	0.139010	−	0.990291i	$-0.455608\pi$
$167$	600.000	0.278020	0.139010	+	0.990291i	$0.455608\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−836.000	−0.373863
$172$	1376.00	0.609994
$173$	2154.00	0.946622	0.473311	−	0.880895i	$-0.343059\pi$
$173$	2154.00	0.946622	0.473311	+	0.880895i	$0.343059\pi$
$174$	−1344.00	−0.585565
$175$	0	0
$176$	624.000	0.267249
$177$	798.000	0.338878
$178$	−1380.00	−0.581098
$179$	−2850.00	−1.19005	−0.595025	−	0.803707i	$-0.702858\pi$
$179$	−2850.00	−1.19005	−0.595025	+	0.803707i	$0.702858\pi$
$180$	0	0
$181$	−4205.00	−1.72682	−0.863412	−	0.504499i	$-0.831677\pi$
$181$	−4205.00	−1.72682	−0.863412	+	0.504499i	$0.831677\pi$
$182$	0	0
$183$	−3955.00	−1.59761
$184$	−312.000	−0.125005
$185$	0	0
$186$	−3178.00	−1.25281
$187$	−936.000	−0.366027
$188$	−396.000	−0.153624
$189$	0	0
$190$	0	0
$191$	−4152.00	−1.57292	−0.786461	−	0.617640i	$-0.788089\pi$
$191$	−4152.00	−1.57292	−0.786461	+	0.617640i	$0.788089\pi$
$192$	−448.000	−0.168394
$193$	−3148.00	−1.17408	−0.587041	−	0.809557i	$-0.699707\pi$
$193$	−3148.00	−1.17408	−0.587041	+	0.809557i	$0.699707\pi$
$194$	634.000	0.234632
$195$	0	0
$196$	0	0
$197$	1173.00	0.424227	0.212114	−	0.977245i	$-0.431965\pi$
$197$	1173.00	0.424227	0.212114	+	0.977245i	$0.431965\pi$
$198$	−1716.00	−0.615913
$199$	−3512.00	−1.25105	−0.625525	−	0.780204i	$-0.715115\pi$
$199$	−3512.00	−1.25105	−0.625525	+	0.780204i	$0.715115\pi$
$200$	1000.00	0.353553
$201$	2695.00	0.945725
$202$	−1326.00	−0.461867
$203$	0	0
$204$	672.000	0.230634
$205$	0	0
$206$	−1292.00	−0.436980
$207$	858.000	0.288092
$208$	−208.000	−0.0693375
$209$	−1482.00	−0.490488
$210$	0	0
$211$	−3418.00	−1.11519	−0.557594	−	0.830114i	$-0.688276\pi$
$211$	−3418.00	−1.11519	−0.557594	+	0.830114i	$0.688276\pi$
$212$	−2160.00	−0.699761
$213$	1092.00	0.351280
$214$	1488.00	0.475316
$215$	0	0
$216$	−280.000	−0.0882018
$217$	0	0
$218$	−436.000	−0.135457
$219$	−4711.00	−1.45361
$220$	0	0
$221$	312.000	0.0949656
$222$	5950.00	1.79882
$223$	−4241.00	−1.27354	−0.636768	−	0.771056i	$-0.719729\pi$
$223$	−4241.00	−1.27354	−0.636768	+	0.771056i	$0.719729\pi$
$224$	0	0
$225$	−2750.00	−0.814815
$226$	3246.00	0.955401
$227$	−888.000	−0.259642	−0.129821	−	0.991537i	$-0.541440\pi$
$227$	−888.000	−0.259642	−0.129821	+	0.991537i	$0.541440\pi$
$228$	1064.00	0.309058
$229$	−4700.00	−1.35627	−0.678133	−	0.734940i	$-0.737211\pi$
$229$	−4700.00	−1.35627	−0.678133	+	0.734940i	$0.737211\pi$
$230$	0	0
$231$	0	0
$232$	768.000	0.217335
$233$	6363.00	1.78907	0.894536	−	0.446995i	$-0.147506\pi$
$233$	6363.00	1.78907	0.894536	+	0.446995i	$0.147506\pi$
$234$	572.000	0.159798
$235$	0	0
$236$	−456.000	−0.125776
$237$	−5243.00	−1.43700
$238$	0	0
$239$	−3078.00	−0.833051	−0.416526	−	0.909124i	$-0.636752\pi$
$239$	−3078.00	−0.833051	−0.416526	+	0.909124i	$0.636752\pi$
$240$	0	0
$241$	−3674.00	−0.982005	−0.491002	−	0.871158i	$-0.663369\pi$
$241$	−3674.00	−0.982005	−0.491002	+	0.871158i	$0.663369\pi$
$242$	−380.000	−0.100939
$243$	4928.00	1.30095
$244$	2260.00	0.592958
$245$	0	0
$246$	1470.00	0.380991
$247$	494.000	0.127257
$248$	1816.00	0.464984
$249$	−7308.00	−1.85994
$250$	0	0
$251$	345.000	0.0867578	0.0433789	−	0.999059i	$-0.486188\pi$
$251$	345.000	0.0867578	0.0433789	+	0.999059i	$0.486188\pi$
$252$	0	0
$253$	1521.00	0.377962
$254$	−1318.00	−0.325585
$255$	0	0
$256$	256.000	0.0625000
$257$	−6888.00	−1.67184	−0.835918	−	0.548855i	$-0.815064\pi$
$257$	−6888.00	−1.67184	−0.835918	+	0.548855i	$0.815064\pi$
$258$	4816.00	1.16214
$259$	0	0
$260$	0	0
$261$	−2112.00	−0.500879
$262$	−432.000	−0.101867
$263$	−1248.00	−0.292604	−0.146302	−	0.989240i	$-0.546737\pi$
$263$	−1248.00	−0.292604	−0.146302	+	0.989240i	$0.546737\pi$
$264$	2184.00	0.509151
$265$	0	0
$266$	0	0
$267$	−4830.00	−1.10708
$268$	−1540.00	−0.351009
$269$	2253.00	0.510661	0.255331	−	0.966854i	$-0.417816\pi$
$269$	2253.00	0.510661	0.255331	+	0.966854i	$0.417816\pi$
$270$	0	0
$271$	−1397.00	−0.313143	−0.156571	−	0.987667i	$-0.550044\pi$
$271$	−1397.00	−0.313143	−0.156571	+	0.987667i	$0.550044\pi$
$272$	−384.000	−0.0856008
$273$	0	0
$274$	−3684.00	−0.812258
$275$	−4875.00	−1.06899
$276$	−1092.00	−0.238155
$277$	−2302.00	−0.499328	−0.249664	−	0.968333i	$-0.580320\pi$
$277$	−2302.00	−0.499328	−0.249664	+	0.968333i	$0.580320\pi$
$278$	−1256.00	−0.270971
$279$	−4994.00	−1.07162
$280$	0	0
$281$	8532.00	1.81130	0.905652	−	0.424022i	$-0.139382\pi$
$281$	8532.00	1.81130	0.905652	+	0.424022i	$0.139382\pi$
$282$	−1386.00	−0.292678
$283$	3769.00	0.791674	0.395837	−	0.918321i	$-0.370454\pi$
$283$	3769.00	0.791674	0.395837	+	0.918321i	$0.370454\pi$
$284$	−624.000	−0.130379
$285$	0	0
$286$	1014.00	0.209647
$287$	0	0
$288$	−704.000	−0.144040
$289$	−4337.00	−0.882760
$290$	0	0
$291$	2219.00	0.447011
$292$	2692.00	0.539512
$293$	6150.00	1.22623	0.613117	−	0.789992i	$-0.289915\pi$
$293$	6150.00	1.22623	0.613117	+	0.789992i	$0.289915\pi$
$294$	0	0
$295$	0	0
$296$	−3400.00	−0.667638
$297$	1365.00	0.266685
$298$	−642.000	−0.124799
$299$	−507.000	−0.0980621
$300$	3500.00	0.673575
$301$	0	0
$302$	3200.00	0.609733
$303$	−4641.00	−0.879929
$304$	−608.000	−0.114708
$305$	0	0
$306$	1056.00	0.197279
$307$	−4592.00	−0.853678	−0.426839	−	0.904328i	$-0.640373\pi$
$307$	−4592.00	−0.853678	−0.426839	+	0.904328i	$0.640373\pi$
$308$	0	0
$309$	−4522.00	−0.832516
$310$	0	0
$311$	−6498.00	−1.18478	−0.592392	−	0.805650i	$-0.701816\pi$
$311$	−6498.00	−1.18478	−0.592392	+	0.805650i	$0.701816\pi$
$312$	−728.000	−0.132099
$313$	−5762.00	−1.04054	−0.520268	−	0.854003i	$-0.674168\pi$
$313$	−5762.00	−1.04054	−0.520268	+	0.854003i	$0.674168\pi$
$314$	2254.00	0.405097
$315$	0	0
$316$	2996.00	0.533349
$317$	2565.00	0.454463	0.227231	−	0.973841i	$-0.427033\pi$
$317$	2565.00	0.454463	0.227231	+	0.973841i	$0.427033\pi$
$318$	−7560.00	−1.33316
$319$	−3744.00	−0.657128
$320$	0	0
$321$	5208.00	0.905552
$322$	0	0
$323$	912.000	0.157105
$324$	−3356.00	−0.575446
$325$	1625.00	0.277350
$326$	2408.00	0.409101
$327$	−1526.00	−0.258067
$328$	−840.000	−0.141406
$329$	0	0
$330$	0	0
$331$	−745.000	−0.123713	−0.0618563	−	0.998085i	$-0.519702\pi$
$331$	−745.000	−0.123713	−0.0618563	+	0.998085i	$0.519702\pi$
$332$	4176.00	0.690325
$333$	9350.00	1.53867
$334$	−1200.00	−0.196590
$335$	0	0
$336$	0	0
$337$	8345.00	1.34891	0.674453	−	0.738318i	$-0.264380\pi$
$337$	8345.00	1.34891	0.674453	+	0.738318i	$0.264380\pi$
$338$	−338.000	−0.0543928
$339$	11361.0	1.82019
$340$	0	0
$341$	−8853.00	−1.40591
$342$	1672.00	0.264361
$343$	0	0
$344$	−2752.00	−0.431331
$345$	0	0
$346$	−4308.00	−0.669363
$347$	−1776.00	−0.274757	−0.137378	−	0.990519i	$-0.543868\pi$
$347$	−1776.00	−0.274757	−0.137378	+	0.990519i	$0.543868\pi$
$348$	2688.00	0.414057
$349$	8602.00	1.31935	0.659677	−	0.751549i	$-0.270693\pi$
$349$	8602.00	1.31935	0.659677	+	0.751549i	$0.270693\pi$
$350$	0	0
$351$	−455.000	−0.0691912
$352$	−1248.00	−0.188973
$353$	6477.00	0.976589	0.488295	−	0.872679i	$-0.337619\pi$
$353$	6477.00	0.976589	0.488295	+	0.872679i	$0.337619\pi$
$354$	−1596.00	−0.239623
$355$	0	0
$356$	2760.00	0.410898
$357$	0	0
$358$	5700.00	0.841493
$359$	7920.00	1.16435	0.582175	−	0.813064i	$-0.302202\pi$
$359$	7920.00	1.16435	0.582175	+	0.813064i	$0.302202\pi$
$360$	0	0
$361$	−5415.00	−0.789474
$362$	8410.00	1.22105
$363$	−1330.00	−0.192305
$364$	0	0
$365$	0	0
$366$	7910.00	1.12968
$367$	−3404.00	−0.484162	−0.242081	−	0.970256i	$-0.577830\pi$
$367$	−3404.00	−0.484162	−0.242081	+	0.970256i	$0.577830\pi$
$368$	624.000	0.0883920
$369$	2310.00	0.325891
$370$	0	0
$371$	0	0
$372$	6356.00	0.885869
$373$	10604.0	1.47200	0.735998	−	0.676984i	$-0.236713\pi$
$373$	10604.0	1.47200	0.735998	+	0.676984i	$0.236713\pi$
$374$	1872.00	0.258820
$375$	0	0
$376$	792.000	0.108628
$377$	1248.00	0.170491
$378$	0	0
$379$	−11680.0	−1.58301	−0.791506	−	0.611162i	$-0.790702\pi$
$379$	−11680.0	−1.58301	−0.791506	+	0.611162i	$0.790702\pi$
$380$	0	0
$381$	−4613.00	−0.620292
$382$	8304.00	1.11222
$383$	−8133.00	−1.08506	−0.542529	−	0.840037i	$-0.682533\pi$
$383$	−8133.00	−1.08506	−0.542529	+	0.840037i	$0.682533\pi$
$384$	896.000	0.119072
$385$	0	0
$386$	6296.00	0.830202
$387$	7568.00	0.994065
$388$	−1268.00	−0.165910
$389$	2556.00	0.333147	0.166574	−	0.986029i	$-0.446730\pi$
$389$	2556.00	0.333147	0.166574	+	0.986029i	$0.446730\pi$
$390$	0	0
$391$	−936.000	−0.121063
$392$	0	0
$393$	−1512.00	−0.194072
$394$	−2346.00	−0.299974
$395$	0	0
$396$	3432.00	0.435516
$397$	−12620.0	−1.59541	−0.797707	−	0.603045i	$-0.793954\pi$
$397$	−12620.0	−1.59541	−0.797707	+	0.603045i	$0.793954\pi$
$398$	7024.00	0.884626
$399$	0	0
$400$	−2000.00	−0.250000
$401$	2064.00	0.257036	0.128518	−	0.991707i	$-0.458978\pi$
$401$	2064.00	0.257036	0.128518	+	0.991707i	$0.458978\pi$
$402$	−5390.00	−0.668728
$403$	2951.00	0.364764
$404$	2652.00	0.326589
$405$	0	0
$406$	0	0
$407$	16575.0	2.01865
$408$	−1344.00	−0.163083
$409$	5974.00	0.722238	0.361119	−	0.932520i	$-0.382395\pi$
$409$	5974.00	0.722238	0.361119	+	0.932520i	$0.382395\pi$
$410$	0	0
$411$	−12894.0	−1.54748
$412$	2584.00	0.308992
$413$	0	0
$414$	−1716.00	−0.203712
$415$	0	0
$416$	416.000	0.0490290
$417$	−4396.00	−0.516242
$418$	2964.00	0.346828
$419$	−12459.0	−1.45265	−0.726327	−	0.687349i	$-0.758774\pi$
$419$	−12459.0	−1.45265	−0.726327	+	0.687349i	$0.758774\pi$
$420$	0	0
$421$	−2221.00	−0.257114	−0.128557	−	0.991702i	$-0.541034\pi$
$421$	−2221.00	−0.257114	−0.128557	+	0.991702i	$0.541034\pi$
$422$	6836.00	0.788558
$423$	−2178.00	−0.250350
$424$	4320.00	0.494806
$425$	3000.00	0.342403
$426$	−2184.00	−0.248392
$427$	0	0
$428$	−2976.00	−0.336099
$429$	3549.00	0.399411
$430$	0	0
$431$	−15162.0	−1.69450	−0.847248	−	0.531197i	$-0.821742\pi$
$431$	−15162.0	−1.69450	−0.847248	+	0.531197i	$0.821742\pi$
$432$	560.000	0.0623681
$433$	10978.0	1.21840	0.609202	−	0.793015i	$-0.291490\pi$
$433$	10978.0	1.21840	0.609202	+	0.793015i	$0.291490\pi$
$434$	0	0
$435$	0	0
$436$	872.000	0.0957826
$437$	−1482.00	−0.162228
$438$	9422.00	1.02786
$439$	3274.00	0.355944	0.177972	−	0.984036i	$-0.443046\pi$
$439$	3274.00	0.355944	0.177972	+	0.984036i	$0.443046\pi$
$440$	0	0
$441$	0	0
$442$	−624.000	−0.0671508
$443$	3888.00	0.416985	0.208493	−	0.978024i	$-0.433144\pi$
$443$	3888.00	0.416985	0.208493	+	0.978024i	$0.433144\pi$
$444$	−11900.0	−1.27196
$445$	0	0
$446$	8482.00	0.900525
$447$	−2247.00	−0.237762
$448$	0	0
$449$	−11262.0	−1.18371	−0.591856	−	0.806044i	$-0.701605\pi$
$449$	−11262.0	−1.18371	−0.591856	+	0.806044i	$0.701605\pi$
$450$	5500.00	0.576161
$451$	4095.00	0.427552
$452$	−6492.00	−0.675571
$453$	11200.0	1.16164
$454$	1776.00	0.183594
$455$	0	0
$456$	−2128.00	−0.218537
$457$	−9718.00	−0.994724	−0.497362	−	0.867543i	$-0.665698\pi$
$457$	−9718.00	−0.994724	−0.497362	+	0.867543i	$0.665698\pi$
$458$	9400.00	0.959024
$459$	−840.000	−0.0854201
$460$	0	0
$461$	13656.0	1.37966	0.689830	−	0.723971i	$-0.257685\pi$
$461$	13656.0	1.37966	0.689830	+	0.723971i	$0.257685\pi$
$462$	0	0
$463$	12008.0	1.20531	0.602656	−	0.798001i	$-0.294109\pi$
$463$	12008.0	1.20531	0.602656	+	0.798001i	$0.294109\pi$
$464$	−1536.00	−0.153679
$465$	0	0
$466$	−12726.0	−1.26507
$467$	−2268.00	−0.224733	−0.112367	−	0.993667i	$-0.535843\pi$
$467$	−2268.00	−0.224733	−0.112367	+	0.993667i	$0.535843\pi$
$468$	−1144.00	−0.112994
$469$	0	0
$470$	0	0
$471$	7889.00	0.771775
$472$	912.000	0.0889369
$473$	13416.0	1.30416
$474$	10486.0	1.01611
$475$	4750.00	0.458831
$476$	0	0
$477$	−11880.0	−1.14035
$478$	6156.00	0.589056
$479$	10536.0	1.00501	0.502507	−	0.864573i	$-0.332411\pi$
$479$	10536.0	1.00501	0.502507	+	0.864573i	$0.332411\pi$
$480$	0	0
$481$	−5525.00	−0.523739
$482$	7348.00	0.694382
$483$	0	0
$484$	760.000	0.0713749
$485$	0	0
$486$	−9856.00	−0.919912
$487$	12566.0	1.16924	0.584620	−	0.811307i	$-0.301244\pi$
$487$	12566.0	1.16924	0.584620	+	0.811307i	$0.301244\pi$
$488$	−4520.00	−0.419284
$489$	8428.00	0.779402
$490$	0	0
$491$	−12444.0	−1.14377	−0.571884	−	0.820335i	$-0.693787\pi$
$491$	−12444.0	−1.14377	−0.571884	+	0.820335i	$0.693787\pi$
$492$	−2940.00	−0.269401
$493$	2304.00	0.210481
$494$	−988.000	−0.0899843
$495$	0	0
$496$	−3632.00	−0.328794
$497$	0	0
$498$	14616.0	1.31518
$499$	−6091.00	−0.546434	−0.273217	−	0.961952i	$-0.588088\pi$
$499$	−6091.00	−0.546434	−0.273217	+	0.961952i	$0.588088\pi$
$500$	0	0
$501$	−4200.00	−0.374535
$502$	−690.000	−0.0613470
$503$	−9204.00	−0.815877	−0.407938	−	0.913009i	$-0.633752\pi$
$503$	−9204.00	−0.815877	−0.407938	+	0.913009i	$0.633752\pi$
$504$	0	0
$505$	0	0
$506$	−3042.00	−0.267260
$507$	−1183.00	−0.103627
$508$	2636.00	0.230224
$509$	−21522.0	−1.87416	−0.937078	−	0.349119i	$-0.886481\pi$
$509$	−21522.0	−1.87416	−0.937078	+	0.349119i	$0.886481\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−1330.00	−0.114466
$514$	13776.0	1.18217
$515$	0	0
$516$	−9632.00	−0.821754
$517$	−3861.00	−0.328446
$518$	0	0
$519$	−15078.0	−1.27524
$520$	0	0
$521$	12474.0	1.04894	0.524468	−	0.851430i	$-0.324264\pi$
$521$	12474.0	1.04894	0.524468	+	0.851430i	$0.324264\pi$
$522$	4224.00	0.354175
$523$	−16607.0	−1.38848	−0.694238	−	0.719745i	$-0.744259\pi$
$523$	−16607.0	−1.38848	−0.694238	+	0.719745i	$0.744259\pi$
$524$	864.000	0.0720306
$525$	0	0
$526$	2496.00	0.206903
$527$	5448.00	0.450320
$528$	−4368.00	−0.360024
$529$	−10646.0	−0.874990
$530$	0	0
$531$	−2508.00	−0.204968
$532$	0	0
$533$	−1365.00	−0.110928
$534$	9660.00	0.782826
$535$	0	0
$536$	3080.00	0.248201
$537$	19950.0	1.60318
$538$	−4506.00	−0.361092
$539$	0	0
$540$	0	0
$541$	−19690.0	−1.56477	−0.782384	−	0.622797i	$-0.785996\pi$
$541$	−19690.0	−1.56477	−0.782384	+	0.622797i	$0.785996\pi$
$542$	2794.00	0.221425
$543$	29435.0	2.32629
$544$	768.000	0.0605289
$545$	0	0
$546$	0	0
$547$	−19960.0	−1.56020	−0.780099	−	0.625656i	$-0.784831\pi$
$547$	−19960.0	−1.56020	−0.780099	+	0.625656i	$0.784831\pi$
$548$	7368.00	0.574353
$549$	12430.0	0.966301
$550$	9750.00	0.755893
$551$	3648.00	0.282051
$552$	2184.00	0.168401
$553$	0	0
$554$	4604.00	0.353078
$555$	0	0
$556$	2512.00	0.191605
$557$	19137.0	1.45576	0.727882	−	0.685702i	$-0.240505\pi$
$557$	19137.0	1.45576	0.727882	+	0.685702i	$0.240505\pi$
$558$	9988.00	0.757752
$559$	−4472.00	−0.338364
$560$	0	0
$561$	6552.00	0.493094
$562$	−17064.0	−1.28079
$563$	−6711.00	−0.502371	−0.251186	−	0.967939i	$-0.580820\pi$
$563$	−6711.00	−0.502371	−0.251186	+	0.967939i	$0.580820\pi$
$564$	2772.00	0.206954
$565$	0	0
$566$	−7538.00	−0.559798
$567$	0	0
$568$	1248.00	0.0921918
$569$	−5595.00	−0.412222	−0.206111	−	0.978529i	$-0.566081\pi$
$569$	−5595.00	−0.412222	−0.206111	+	0.978529i	$0.566081\pi$
$570$	0	0
$571$	−21274.0	−1.55918	−0.779588	−	0.626293i	$-0.784571\pi$
$571$	−21274.0	−1.55918	−0.779588	+	0.626293i	$0.784571\pi$
$572$	−2028.00	−0.148243
$573$	29064.0	2.11896
$574$	0	0
$575$	−4875.00	−0.353568
$576$	1408.00	0.101852
$577$	15694.0	1.13232	0.566161	−	0.824295i	$-0.308428\pi$
$577$	15694.0	1.13232	0.566161	+	0.824295i	$0.308428\pi$
$578$	8674.00	0.624206
$579$	22036.0	1.58167
$580$	0	0
$581$	0	0
$582$	−4438.00	−0.316084
$583$	−21060.0	−1.49608
$584$	−5384.00	−0.381492
$585$	0	0
$586$	−12300.0	−0.867079
$587$	−21054.0	−1.48039	−0.740197	−	0.672390i	$-0.765268\pi$
$587$	−21054.0	−1.48039	−0.740197	+	0.672390i	$0.765268\pi$
$588$	0	0
$589$	8626.00	0.603443
$590$	0	0
$591$	−8211.00	−0.571498
$592$	6800.00	0.472092
$593$	17910.0	1.24026	0.620131	−	0.784498i	$-0.287079\pi$
$593$	17910.0	1.24026	0.620131	+	0.784498i	$0.287079\pi$
$594$	−2730.00	−0.188575
$595$	0	0
$596$	1284.00	0.0882461
$597$	24584.0	1.68535
$598$	1014.00	0.0693404
$599$	3213.00	0.219165	0.109582	−	0.993978i	$-0.465049\pi$
$599$	3213.00	0.219165	0.109582	+	0.993978i	$0.465049\pi$
$600$	−7000.00	−0.476290
$601$	−15158.0	−1.02880	−0.514399	−	0.857551i	$-0.671985\pi$
$601$	−15158.0	−1.02880	−0.514399	+	0.857551i	$0.671985\pi$
$602$	0	0
$603$	−8470.00	−0.572015
$604$	−6400.00	−0.431146
$605$	0	0
$606$	9282.00	0.622204
$607$	26206.0	1.75234	0.876169	−	0.482005i	$-0.160091\pi$
$607$	26206.0	1.75234	0.876169	+	0.482005i	$0.160091\pi$
$608$	1216.00	0.0811107
$609$	0	0
$610$	0	0
$611$	1287.00	0.0852151
$612$	−2112.00	−0.139498
$613$	−6145.00	−0.404885	−0.202442	−	0.979294i	$-0.564888\pi$
$613$	−6145.00	−0.404885	−0.202442	+	0.979294i	$0.564888\pi$
$614$	9184.00	0.603642
$615$	0	0
$616$	0	0
$617$	9474.00	0.618167	0.309083	−	0.951035i	$-0.399978\pi$
$617$	9474.00	0.618167	0.309083	+	0.951035i	$0.399978\pi$
$618$	9044.00	0.588678
$619$	14326.0	0.930227	0.465114	−	0.885251i	$-0.346013\pi$
$619$	14326.0	0.930227	0.465114	+	0.885251i	$0.346013\pi$
$620$	0	0
$621$	1365.00	0.0882054
$622$	12996.0	0.837769
$623$	0	0
$624$	1456.00	0.0934081
$625$	15625.0	1.00000
$626$	11524.0	0.735769
$627$	10374.0	0.660762
$628$	−4508.00	−0.286447
$629$	−10200.0	−0.646583
$630$	0	0
$631$	−15478.0	−0.976497	−0.488248	−	0.872705i	$-0.662364\pi$
$631$	−15478.0	−0.976497	−0.488248	+	0.872705i	$0.662364\pi$
$632$	−5992.00	−0.377134
$633$	23926.0	1.50233
$634$	−5130.00	−0.321354
$635$	0	0
$636$	15120.0	0.942684
$637$	0	0
$638$	7488.00	0.464659
$639$	−3432.00	−0.212469
$640$	0	0
$641$	17439.0	1.07457	0.537285	−	0.843401i	$-0.319450\pi$
$641$	17439.0	1.07457	0.537285	+	0.843401i	$0.319450\pi$
$642$	−10416.0	−0.640322
$643$	−30296.0	−1.85810	−0.929049	−	0.369956i	$-0.879373\pi$
$643$	−30296.0	−1.85810	−0.929049	+	0.369956i	$0.879373\pi$
$644$	0	0
$645$	0	0
$646$	−1824.00	−0.111090
$647$	17124.0	1.04052	0.520258	−	0.854009i	$-0.325836\pi$
$647$	17124.0	1.04052	0.520258	+	0.854009i	$0.325836\pi$
$648$	6712.00	0.406902
$649$	−4446.00	−0.268907
$650$	−3250.00	−0.196116
$651$	0	0
$652$	−4816.00	−0.289278
$653$	27120.0	1.62525	0.812625	−	0.582788i	$-0.198038\pi$
$653$	27120.0	1.62525	0.812625	+	0.582788i	$0.198038\pi$
$654$	3052.00	0.182481
$655$	0	0
$656$	1680.00	0.0999893
$657$	14806.0	0.879204
$658$	0	0
$659$	−138.000	−0.00815739	−0.00407869	−	0.999992i	$-0.501298\pi$
$659$	−138.000	−0.00815739	−0.00407869	+	0.999992i	$0.501298\pi$
$660$	0	0
$661$	−29720.0	−1.74883	−0.874413	−	0.485182i	$-0.838753\pi$
$661$	−29720.0	−1.74883	−0.874413	+	0.485182i	$0.838753\pi$
$662$	1490.00	0.0874781
$663$	−2184.00	−0.127933
$664$	−8352.00	−0.488133
$665$	0	0
$666$	−18700.0	−1.08800
$667$	−3744.00	−0.217344
$668$	2400.00	0.139010
$669$	29687.0	1.71564
$670$	0	0
$671$	22035.0	1.26774
$672$	0	0
$673$	−32929.0	−1.88606	−0.943031	−	0.332705i	$-0.892039\pi$
$673$	−32929.0	−1.88606	−0.943031	+	0.332705i	$0.892039\pi$
$674$	−16690.0	−0.953820
$675$	−4375.00	−0.249472
$676$	676.000	0.0384615
$677$	−33021.0	−1.87459	−0.937297	−	0.348532i	$-0.886680\pi$
$677$	−33021.0	−1.87459	−0.937297	+	0.348532i	$0.886680\pi$
$678$	−22722.0	−1.28707
$679$	0	0
$680$	0	0
$681$	6216.00	0.349776
$682$	17706.0	0.994132
$683$	18519.0	1.03750	0.518748	−	0.854927i	$-0.326398\pi$
$683$	18519.0	1.03750	0.518748	+	0.854927i	$0.326398\pi$
$684$	−3344.00	−0.186931
$685$	0	0
$686$	0	0
$687$	32900.0	1.82709
$688$	5504.00	0.304997
$689$	7020.00	0.388158
$690$	0	0
$691$	−15248.0	−0.839452	−0.419726	−	0.907651i	$-0.637874\pi$
$691$	−15248.0	−0.839452	−0.419726	+	0.907651i	$0.637874\pi$
$692$	8616.00	0.473311
$693$	0	0
$694$	3552.00	0.194283
$695$	0	0
$696$	−5376.00	−0.292783
$697$	−2520.00	−0.136947
$698$	−17204.0	−0.932924
$699$	−44541.0	−2.41015
$700$	0	0
$701$	−7740.00	−0.417027	−0.208513	−	0.978020i	$-0.566862\pi$
$701$	−7740.00	−0.417027	−0.208513	+	0.978020i	$0.566862\pi$
$702$	910.000	0.0489256
$703$	−16150.0	−0.866442
$704$	2496.00	0.133624
$705$	0	0
$706$	−12954.0	−0.690553
$707$	0	0
$708$	3192.00	0.169439
$709$	29747.0	1.57570	0.787851	−	0.615867i	$-0.211194\pi$
$709$	29747.0	1.57570	0.787851	+	0.615867i	$0.211194\pi$
$710$	0	0
$711$	16478.0	0.869161
$712$	−5520.00	−0.290549
$713$	−8853.00	−0.465003
$714$	0	0
$715$	0	0
$716$	−11400.0	−0.595025
$717$	21546.0	1.12225
$718$	−15840.0	−0.823320
$719$	−10266.0	−0.532486	−0.266243	−	0.963906i	$-0.585782\pi$
$719$	−10266.0	−0.532486	−0.266243	+	0.963906i	$0.585782\pi$
$720$	0	0
$721$	0	0
$722$	10830.0	0.558242
$723$	25718.0	1.32291
$724$	−16820.0	−0.863412
$725$	12000.0	0.614716
$726$	2660.00	0.135981
$727$	8026.00	0.409447	0.204723	−	0.978820i	$-0.434371\pi$
$727$	8026.00	0.409447	0.204723	+	0.978820i	$0.434371\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	0	0
$731$	−8256.00	−0.417728
$732$	−15820.0	−0.798803
$733$	−13268.0	−0.668574	−0.334287	−	0.942471i	$-0.608495\pi$
$733$	−13268.0	−0.668574	−0.334287	+	0.942471i	$0.608495\pi$
$734$	6808.00	0.342354
$735$	0	0
$736$	−1248.00	−0.0625026
$737$	−15015.0	−0.750454
$738$	−4620.00	−0.230440
$739$	−8080.00	−0.402202	−0.201101	−	0.979570i	$-0.564452\pi$
$739$	−8080.00	−0.402202	−0.201101	+	0.979570i	$0.564452\pi$
$740$	0	0
$741$	−3458.00	−0.171434
$742$	0	0
$743$	−27096.0	−1.33789	−0.668947	−	0.743310i	$-0.733255\pi$
$743$	−27096.0	−1.33789	−0.668947	+	0.743310i	$0.733255\pi$
$744$	−12712.0	−0.626404
$745$	0	0
$746$	−21208.0	−1.04086
$747$	22968.0	1.12497
$748$	−3744.00	−0.183014
$749$	0	0
$750$	0	0
$751$	25067.0	1.21799	0.608993	−	0.793175i	$-0.291574\pi$
$751$	25067.0	1.21799	0.608993	+	0.793175i	$0.291574\pi$
$752$	−1584.00	−0.0768119
$753$	−2415.00	−0.116876
$754$	−2496.00	−0.120556
$755$	0	0
$756$	0	0
$757$	−6442.00	−0.309298	−0.154649	−	0.987969i	$-0.549425\pi$
$757$	−6442.00	−0.309298	−0.154649	+	0.987969i	$0.549425\pi$
$758$	23360.0	1.11936
$759$	−10647.0	−0.509172
$760$	0	0
$761$	−26511.0	−1.26284	−0.631421	−	0.775440i	$-0.717528\pi$
$761$	−26511.0	−1.26284	−0.631421	+	0.775440i	$0.717528\pi$
$762$	9226.00	0.438612
$763$	0	0
$764$	−16608.0	−0.786461
$765$	0	0
$766$	16266.0	0.767251
$767$	1482.00	0.0697678
$768$	−1792.00	−0.0841969
$769$	17665.0	0.828370	0.414185	−	0.910193i	$-0.364067\pi$
$769$	17665.0	0.828370	0.414185	+	0.910193i	$0.364067\pi$
$770$	0	0
$771$	48216.0	2.25221
$772$	−12592.0	−0.587041
$773$	−36258.0	−1.68708	−0.843538	−	0.537070i	$-0.819531\pi$
$773$	−36258.0	−1.68708	−0.843538	+	0.537070i	$0.819531\pi$
$774$	−15136.0	−0.702910
$775$	28375.0	1.31517
$776$	2536.00	0.117316
$777$	0	0
$778$	−5112.00	−0.235571
$779$	−3990.00	−0.183513
$780$	0	0
$781$	−6084.00	−0.278749
$782$	1872.00	0.0856043
$783$	−3360.00	−0.153355
$784$	0	0
$785$	0	0
$786$	3024.00	0.137230
$787$	−21296.0	−0.964575	−0.482287	−	0.876013i	$-0.660194\pi$
$787$	−21296.0	−0.964575	−0.482287	+	0.876013i	$0.660194\pi$
$788$	4692.00	0.212114
$789$	8736.00	0.394182
$790$	0	0
$791$	0	0
$792$	−6864.00	−0.307957
$793$	−7345.00	−0.328914
$794$	25240.0	1.12813
$795$	0	0
$796$	−14048.0	−0.625525
$797$	28947.0	1.28652	0.643259	−	0.765648i	$-0.277582\pi$
$797$	28947.0	1.28652	0.643259	+	0.765648i	$0.277582\pi$
$798$	0	0
$799$	2376.00	0.105203
$800$	4000.00	0.176777
$801$	15180.0	0.669612
$802$	−4128.00	−0.181752
$803$	26247.0	1.15347
$804$	10780.0	0.472862
$805$	0	0
$806$	−5902.00	−0.257927
$807$	−15771.0	−0.687937
$808$	−5304.00	−0.230933
$809$	−20418.0	−0.887341	−0.443670	−	0.896190i	$-0.646324\pi$
$809$	−20418.0	−0.887341	−0.443670	+	0.896190i	$0.646324\pi$
$810$	0	0
$811$	14524.0	0.628861	0.314431	−	0.949280i	$-0.398186\pi$
$811$	14524.0	0.628861	0.314431	+	0.949280i	$0.398186\pi$
$812$	0	0
$813$	9779.00	0.421851
$814$	−33150.0	−1.42740
$815$	0	0
$816$	2688.00	0.115317
$817$	−13072.0	−0.559769
$818$	−11948.0	−0.510699
$819$	0	0
$820$	0	0
$821$	−7710.00	−0.327748	−0.163874	−	0.986481i	$-0.552399\pi$
$821$	−7710.00	−0.327748	−0.163874	+	0.986481i	$0.552399\pi$
$822$	25788.0	1.09423
$823$	2531.00	0.107199	0.0535997	−	0.998563i	$-0.482931\pi$
$823$	2531.00	0.107199	0.0535997	+	0.998563i	$0.482931\pi$
$824$	−5168.00	−0.218490
$825$	34125.0	1.44010
$826$	0	0
$827$	24516.0	1.03084	0.515420	−	0.856938i	$-0.327636\pi$
$827$	24516.0	1.03084	0.515420	+	0.856938i	$0.327636\pi$
$828$	3432.00	0.144046
$829$	15586.0	0.652985	0.326492	−	0.945200i	$-0.394133\pi$
$829$	15586.0	0.652985	0.326492	+	0.945200i	$0.394133\pi$
$830$	0	0
$831$	16114.0	0.672670
$832$	−832.000	−0.0346688
$833$	0	0
$834$	8792.00	0.365038
$835$	0	0
$836$	−5928.00	−0.245244
$837$	−7945.00	−0.328100
$838$	24918.0	1.02718
$839$	24915.0	1.02522	0.512611	−	0.858621i	$-0.328678\pi$
$839$	24915.0	1.02522	0.512611	+	0.858621i	$0.328678\pi$
$840$	0	0
$841$	−15173.0	−0.622125
$842$	4442.00	0.181807
$843$	−59724.0	−2.44010
$844$	−13672.0	−0.557594
$845$	0	0
$846$	4356.00	0.177024
$847$	0	0
$848$	−8640.00	−0.349881
$849$	−26383.0	−1.06650
$850$	−6000.00	−0.242116
$851$	16575.0	0.667666
$852$	4368.00	0.175640
$853$	20500.0	0.822868	0.411434	−	0.911439i	$-0.365028\pi$
$853$	20500.0	0.822868	0.411434	+	0.911439i	$0.365028\pi$
$854$	0	0
$855$	0	0
$856$	5952.00	0.237658
$857$	−26694.0	−1.06400	−0.532001	−	0.846744i	$-0.678560\pi$
$857$	−26694.0	−1.06400	−0.532001	+	0.846744i	$0.678560\pi$
$858$	−7098.00	−0.282426
$859$	−20801.0	−0.826218	−0.413109	−	0.910682i	$-0.635557\pi$
$859$	−20801.0	−0.826218	−0.413109	+	0.910682i	$0.635557\pi$
$860$	0	0
$861$	0	0
$862$	30324.0	1.19819
$863$	−37404.0	−1.47537	−0.737687	−	0.675143i	$-0.764082\pi$
$863$	−37404.0	−1.47537	−0.737687	+	0.675143i	$0.764082\pi$
$864$	−1120.00	−0.0441009
$865$	0	0
$866$	−21956.0	−0.861542
$867$	30359.0	1.18921
$868$	0	0
$869$	29211.0	1.14029
$870$	0	0
$871$	5005.00	0.194705
$872$	−1744.00	−0.0677285
$873$	−6974.00	−0.270371
$874$	2964.00	0.114713
$875$	0	0
$876$	−18844.0	−0.726803
$877$	−20581.0	−0.792441	−0.396221	−	0.918155i	$-0.629678\pi$
$877$	−20581.0	−0.792441	−0.396221	+	0.918155i	$0.629678\pi$
$878$	−6548.00	−0.251691
$879$	−43050.0	−1.65192
$880$	0	0
$881$	34314.0	1.31222	0.656111	−	0.754664i	$-0.272200\pi$
$881$	34314.0	1.31222	0.656111	+	0.754664i	$0.272200\pi$
$882$	0	0
$883$	−12058.0	−0.459552	−0.229776	−	0.973244i	$-0.573799\pi$
$883$	−12058.0	−0.459552	−0.229776	+	0.973244i	$0.573799\pi$
$884$	1248.00	0.0474828
$885$	0	0
$886$	−7776.00	−0.294853
$887$	−20406.0	−0.772454	−0.386227	−	0.922404i	$-0.626222\pi$
$887$	−20406.0	−0.772454	−0.386227	+	0.922404i	$0.626222\pi$
$888$	23800.0	0.899410
$889$	0	0
$890$	0	0
$891$	−32721.0	−1.23030
$892$	−16964.0	−0.636768
$893$	3762.00	0.140975
$894$	4494.00	0.168123
$895$	0	0
$896$	0	0
$897$	3549.00	0.132104
$898$	22524.0	0.837011
$899$	21792.0	0.808458
$900$	−11000.0	−0.407407
$901$	12960.0	0.479201
$902$	−8190.00	−0.302325
$903$	0	0
$904$	12984.0	0.477701
$905$	0	0
$906$	−22400.0	−0.821402
$907$	−2914.00	−0.106679	−0.0533395	−	0.998576i	$-0.516987\pi$
$907$	−2914.00	−0.106679	−0.0533395	+	0.998576i	$0.516987\pi$
$908$	−3552.00	−0.129821
$909$	14586.0	0.532219
$910$	0	0
$911$	−1044.00	−0.0379685	−0.0189842	−	0.999820i	$-0.506043\pi$
$911$	−1044.00	−0.0379685	−0.0189842	+	0.999820i	$0.506043\pi$
$912$	4256.00	0.154529
$913$	40716.0	1.47591
$914$	19436.0	0.703376
$915$	0	0
$916$	−18800.0	−0.678133
$917$	0	0
$918$	1680.00	0.0604012
$919$	20693.0	0.742763	0.371381	−	0.928480i	$-0.378884\pi$
$919$	20693.0	0.742763	0.371381	+	0.928480i	$0.378884\pi$
$920$	0	0
$921$	32144.0	1.15003
$922$	−27312.0	−0.975567
$923$	2028.00	0.0723212
$924$	0	0
$925$	−53125.0	−1.88837
$926$	−24016.0	−0.852284
$927$	14212.0	0.503542
$928$	3072.00	0.108667
$929$	52653.0	1.85951	0.929757	−	0.368173i	$-0.120017\pi$
$929$	52653.0	1.85951	0.929757	+	0.368173i	$0.120017\pi$
$930$	0	0
$931$	0	0
$932$	25452.0	0.894536
$933$	45486.0	1.59608
$934$	4536.00	0.158911
$935$	0	0
$936$	2288.00	0.0798992
$937$	−34868.0	−1.21568	−0.607838	−	0.794061i	$-0.707963\pi$
$937$	−34868.0	−1.21568	−0.607838	+	0.794061i	$0.707963\pi$
$938$	0	0
$939$	40334.0	1.40176
$940$	0	0
$941$	−25542.0	−0.884852	−0.442426	−	0.896805i	$-0.645882\pi$
$941$	−25542.0	−0.884852	−0.442426	+	0.896805i	$0.645882\pi$
$942$	−15778.0	−0.545727
$943$	4095.00	0.141412
$944$	−1824.00	−0.0628879
$945$	0	0
$946$	−26832.0	−0.922181
$947$	−672.000	−0.0230592	−0.0115296	−	0.999934i	$-0.503670\pi$
$947$	−672.000	−0.0230592	−0.0115296	+	0.999934i	$0.503670\pi$
$948$	−20972.0	−0.718501
$949$	−8749.00	−0.299267
$950$	−9500.00	−0.324443
$951$	−17955.0	−0.612230
$952$	0	0
$953$	52278.0	1.77697	0.888484	−	0.458908i	$-0.151759\pi$
$953$	52278.0	1.77697	0.888484	+	0.458908i	$0.151759\pi$
$954$	23760.0	0.806351
$955$	0	0
$956$	−12312.0	−0.416526
$957$	26208.0	0.885250
$958$	−21072.0	−0.710653
$959$	0	0
$960$	0	0
$961$	21738.0	0.729683
$962$	11050.0	0.370339
$963$	−16368.0	−0.547717
$964$	−14696.0	−0.491002
$965$	0	0
$966$	0	0
$967$	758.000	0.0252075	0.0126037	−	0.999921i	$-0.495988\pi$
$967$	758.000	0.0252075	0.0126037	+	0.999921i	$0.495988\pi$
$968$	−1520.00	−0.0504697
$969$	−6384.00	−0.211645
$970$	0	0
$971$	−27285.0	−0.901769	−0.450884	−	0.892582i	$-0.648891\pi$
$971$	−27285.0	−0.901769	−0.450884	+	0.892582i	$0.648891\pi$
$972$	19712.0	0.650476
$973$	0	0
$974$	−25132.0	−0.826777
$975$	−11375.0	−0.373632
$976$	9040.00	0.296479
$977$	18786.0	0.615166	0.307583	−	0.951521i	$-0.400480\pi$
$977$	18786.0	0.615166	0.307583	+	0.951521i	$0.400480\pi$
$978$	−16856.0	−0.551120
$979$	26910.0	0.878496
$980$	0	0
$981$	4796.00	0.156090
$982$	24888.0	0.808766
$983$	−37152.0	−1.20546	−0.602729	−	0.797946i	$-0.705920\pi$
$983$	−37152.0	−1.20546	−0.602729	+	0.797946i	$0.705920\pi$
$984$	5880.00	0.190495
$985$	0	0
$986$	−4608.00	−0.148832
$987$	0	0
$988$	1976.00	0.0636285
$989$	13416.0	0.431349
$990$	0	0
$991$	−17143.0	−0.549511	−0.274755	−	0.961514i	$-0.588597\pi$
$991$	−17143.0	−0.549511	−0.274755	+	0.961514i	$0.588597\pi$
$992$	7264.00	0.232492
$993$	5215.00	0.166660
$994$	0	0
$995$	0	0
$996$	−29232.0	−0.929971
$997$	14137.0	0.449070	0.224535	−	0.974466i	$-0.427914\pi$
$997$	14137.0	0.449070	0.224535	+	0.974466i	$0.427914\pi$
$998$	12182.0	0.386387
$999$	14875.0	0.471095

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1274.4.a.a.1.1		1
7.6	odd	2		182.4.a.a.1.1	✓	1
21.20	even	2		1638.4.a.j.1.1		1
28.27	even	2		1456.4.a.b.1.1		1
91.90	odd	2		2366.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.4.a.a.1.1	✓	1	7.6	odd	2
1274.4.a.a.1.1		1	1.1	even	1	trivial
1456.4.a.b.1.1		1	28.27	even	2
1638.4.a.j.1.1		1	21.20	even	2
2366.4.a.g.1.1		1	91.90	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1274.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more