LMFDB - Embedded newform 693.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-5.00000 q^{2} +17.0000 q^{4} +6.00000 q^{5} +7.00000 q^{7} -45.0000 q^{8} +O(q^{10})$

\(q-5.00000 q^{2} +17.0000 q^{4} +6.00000 q^{5} +7.00000 q^{7} -45.0000 q^{8} -30.0000 q^{10} +11.0000 q^{11} +70.0000 q^{13} -35.0000 q^{14} +89.0000 q^{16} -126.000 q^{17} -80.0000 q^{19} +102.000 q^{20} -55.0000 q^{22} +200.000 q^{23} -89.0000 q^{25} -350.000 q^{26} +119.000 q^{28} -134.000 q^{29} -244.000 q^{31} -85.0000 q^{32} +630.000 q^{34} +42.0000 q^{35} -314.000 q^{37} +400.000 q^{38} -270.000 q^{40} -278.000 q^{41} -372.000 q^{43} +187.000 q^{44} -1000.00 q^{46} +84.0000 q^{47} +49.0000 q^{49} +445.000 q^{50} +1190.00 q^{52} -182.000 q^{53} +66.0000 q^{55} -315.000 q^{56} +670.000 q^{58} +756.000 q^{59} +694.000 q^{61} +1220.00 q^{62} -287.000 q^{64} +420.000 q^{65} +820.000 q^{67} -2142.00 q^{68} -210.000 q^{70} -160.000 q^{71} -2.00000 q^{73} +1570.00 q^{74} -1360.00 q^{76} +77.0000 q^{77} +40.0000 q^{79} +534.000 q^{80} +1390.00 q^{82} -760.000 q^{83} -756.000 q^{85} +1860.00 q^{86} -495.000 q^{88} +102.000 q^{89} +490.000 q^{91} +3400.00 q^{92} -420.000 q^{94} -480.000 q^{95} -862.000 q^{97} -245.000 q^{98} +O(q^{100})\)

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 $

$2$	−5.00000	−1.76777	−0.883883	−	0.467707i	$-0.845080\pi$
$2$	−5.00000	−1.76777	−0.883883	+	0.467707i	$0.845080\pi$
$3$	0	0
$3$	0	0
$4$	17.0000	2.12500
$5$	6.00000	0.536656	0.268328	−	0.963328i	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−45.0000	−1.98874
$9$	0	0
$10$	−30.0000	−0.948683
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	70.0000	1.49342	0.746712	−	0.665148i	$-0.231631\pi$
$13$	70.0000	1.49342	0.746712	+	0.665148i	$0.231631\pi$
$14$	−35.0000	−0.668153
$15$	0	0
$16$	89.0000	1.39062
$17$	−126.000	−1.79762	−0.898808	−	0.438342i	$-0.855566\pi$
$17$	−126.000	−1.79762	−0.898808	+	0.438342i	$0.855566\pi$
$18$	0	0
$19$	−80.0000	−0.965961	−0.482980	−	0.875631i	$-0.660446\pi$
$19$	−80.0000	−0.965961	−0.482980	+	0.875631i	$0.660446\pi$
$20$	102.000	1.14039
$21$	0	0
$22$	−55.0000	−0.533002
$23$	200.000	1.81317	0.906584	−	0.422025i	$-0.138680\pi$
$23$	200.000	1.81317	0.906584	+	0.422025i	$0.138680\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	−350.000	−2.64002
$27$	0	0
$28$	119.000	0.803175
$29$	−134.000	−0.858041	−0.429020	−	0.903295i	$-0.641141\pi$
$29$	−134.000	−0.858041	−0.429020	+	0.903295i	$0.641141\pi$
$30$	0	0
$31$	−244.000	−1.41367	−0.706834	−	0.707380i	$-0.749877\pi$
$31$	−244.000	−1.41367	−0.706834	+	0.707380i	$0.749877\pi$
$32$	−85.0000	−0.469563
$33$	0	0
$34$	630.000	3.17777
$35$	42.0000	0.202837
$36$	0	0
$37$	−314.000	−1.39517	−0.697585	−	0.716502i	$-0.745742\pi$
$37$	−314.000	−1.39517	−0.697585	+	0.716502i	$0.745742\pi$
$38$	400.000	1.70759
$39$	0	0
$40$	−270.000	−1.06727
$41$	−278.000	−1.05893	−0.529467	−	0.848330i	$-0.677608\pi$
$41$	−278.000	−1.05893	−0.529467	+	0.848330i	$0.677608\pi$
$42$	0	0
$43$	−372.000	−1.31929	−0.659645	−	0.751577i	$-0.729293\pi$
$43$	−372.000	−1.31929	−0.659645	+	0.751577i	$0.729293\pi$
$44$	187.000	0.640712
$45$	0	0
$46$	−1000.00	−3.20526
$47$	84.0000	0.260695	0.130347	−	0.991468i	$-0.458391\pi$
$47$	84.0000	0.260695	0.130347	+	0.991468i	$0.458391\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	445.000	1.25865
$51$	0	0
$52$	1190.00	3.17353
$53$	−182.000	−0.471691	−0.235845	−	0.971791i	$-0.575786\pi$
$53$	−182.000	−0.471691	−0.235845	+	0.971791i	$0.575786\pi$
$54$	0	0
$55$	66.0000	0.161808
$56$	−315.000	−0.751672
$57$	0	0
$58$	670.000	1.51682
$59$	756.000	1.66818	0.834092	−	0.551626i	$-0.185992\pi$
$59$	756.000	1.66818	0.834092	+	0.551626i	$0.185992\pi$
$60$	0	0
$61$	694.000	1.45668	0.728341	−	0.685215i	$-0.240292\pi$
$61$	694.000	1.45668	0.728341	+	0.685215i	$0.240292\pi$
$62$	1220.00	2.49903
$63$	0	0
$64$	−287.000	−0.560547
$65$	420.000	0.801455
$66$	0	0
$67$	820.000	1.49521	0.747604	−	0.664145i	$-0.231204\pi$
$67$	820.000	1.49521	0.747604	+	0.664145i	$0.231204\pi$
$68$	−2142.00	−3.81994
$69$	0	0
$70$	−210.000	−0.358569
$71$	−160.000	−0.267444	−0.133722	−	0.991019i	$-0.542693\pi$
$71$	−160.000	−0.267444	−0.133722	+	0.991019i	$0.542693\pi$
$72$	0	0
$73$	−2.00000	−0.00320661	−0.00160330	−	0.999999i	$-0.500510\pi$
$73$	−2.00000	−0.00320661	−0.00160330	+	0.999999i	$0.500510\pi$
$74$	1570.00	2.46634
$75$	0	0
$76$	−1360.00	−2.05267
$77$	77.0000	0.113961
$78$	0	0
$79$	40.0000	0.0569665	0.0284832	−	0.999594i	$-0.490932\pi$
$79$	40.0000	0.0569665	0.0284832	+	0.999594i	$0.490932\pi$
$80$	534.000	0.746288
$81$	0	0
$82$	1390.00	1.87195
$83$	−760.000	−1.00507	−0.502535	−	0.864557i	$-0.667599\pi$
$83$	−760.000	−1.00507	−0.502535	+	0.864557i	$0.667599\pi$
$84$	0	0
$85$	−756.000	−0.964703
$86$	1860.00	2.33220
$87$	0	0
$88$	−495.000	−0.599627
$89$	102.000	0.121483	0.0607415	−	0.998154i	$-0.480653\pi$
$89$	102.000	0.121483	0.0607415	+	0.998154i	$0.480653\pi$
$90$	0	0
$91$	490.000	0.564461
$92$	3400.00	3.85298
$93$	0	0
$94$	−420.000	−0.460848
$95$	−480.000	−0.518389
$96$	0	0
$97$	−862.000	−0.902297	−0.451149	−	0.892449i	$-0.648986\pi$
$97$	−862.000	−0.902297	−0.451149	+	0.892449i	$0.648986\pi$
$98$	−245.000	−0.252538
$99$	0	0
$100$	−1513.00	−1.51300
$101$	202.000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	202.000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	140.000	0.133928	0.0669641	−	0.997755i	$-0.478669\pi$
$103$	140.000	0.133928	0.0669641	+	0.997755i	$0.478669\pi$
$104$	−3150.00	−2.97003
$105$	0	0
$106$	910.000	0.833840
$107$	−1556.00	−1.40583	−0.702917	−	0.711272i	$-0.748119\pi$
$107$	−1556.00	−1.40583	−0.702917	+	0.711272i	$0.748119\pi$
$108$	0	0
$109$	−394.000	−0.346223	−0.173112	−	0.984902i	$-0.555382\pi$
$109$	−394.000	−0.346223	−0.173112	+	0.984902i	$0.555382\pi$
$110$	−330.000	−0.286039
$111$	0	0
$112$	623.000	0.525607
$113$	−1554.00	−1.29370	−0.646850	−	0.762618i	$-0.723914\pi$
$113$	−1554.00	−1.29370	−0.646850	+	0.762618i	$0.723914\pi$
$114$	0	0
$115$	1200.00	0.973048
$116$	−2278.00	−1.82334
$117$	0	0
$118$	−3780.00	−2.94896
$119$	−882.000	−0.679435
$120$	0	0
$121$	121.000	0.0909091
$122$	−3470.00	−2.57507
$123$	0	0
$124$	−4148.00	−3.00404
$125$	−1284.00	−0.918756
$126$	0	0
$127$	−232.000	−0.162100	−0.0810499	−	0.996710i	$-0.525827\pi$
$127$	−232.000	−0.162100	−0.0810499	+	0.996710i	$0.525827\pi$
$128$	2115.00	1.46048
$129$	0	0
$130$	−2100.00	−1.41679
$131$	−112.000	−0.0746984	−0.0373492	−	0.999302i	$-0.511891\pi$
$131$	−112.000	−0.0746984	−0.0373492	+	0.999302i	$0.511891\pi$
$132$	0	0
$133$	−560.000	−0.365099
$134$	−4100.00	−2.64318
$135$	0	0
$136$	5670.00	3.57499
$137$	182.000	0.113499	0.0567493	−	0.998388i	$-0.481926\pi$
$137$	182.000	0.113499	0.0567493	+	0.998388i	$0.481926\pi$
$138$	0	0
$139$	−1504.00	−0.917753	−0.458876	−	0.888500i	$-0.651748\pi$
$139$	−1504.00	−0.917753	−0.458876	+	0.888500i	$0.651748\pi$
$140$	714.000	0.431029
$141$	0	0
$142$	800.000	0.472778
$143$	770.000	0.450284
$144$	0	0
$145$	−804.000	−0.460473
$146$	10.0000	0.00566853
$147$	0	0
$148$	−5338.00	−2.96474
$149$	3138.00	1.72534	0.862668	−	0.505771i	$-0.168792\pi$
$149$	3138.00	1.72534	0.862668	+	0.505771i	$0.168792\pi$
$150$	0	0
$151$	−2680.00	−1.44434	−0.722170	−	0.691716i	$-0.756855\pi$
$151$	−2680.00	−1.44434	−0.722170	+	0.691716i	$0.756855\pi$
$152$	3600.00	1.92104
$153$	0	0
$154$	−385.000	−0.201456
$155$	−1464.00	−0.758654
$156$	0	0
$157$	2002.00	1.01769	0.508844	−	0.860859i	$-0.330073\pi$
$157$	2002.00	1.01769	0.508844	+	0.860859i	$0.330073\pi$
$158$	−200.000	−0.100703
$159$	0	0
$160$	−510.000	−0.251994
$161$	1400.00	0.685313
$162$	0	0
$163$	−740.000	−0.355591	−0.177795	−	0.984067i	$-0.556896\pi$
$163$	−740.000	−0.355591	−0.177795	+	0.984067i	$0.556896\pi$
$164$	−4726.00	−2.25024
$165$	0	0
$166$	3800.00	1.77673
$167$	−3408.00	−1.57916	−0.789578	−	0.613651i	$-0.789700\pi$
$167$	−3408.00	−1.57916	−0.789578	+	0.613651i	$0.789700\pi$
$168$	0	0
$169$	2703.00	1.23031
$170$	3780.00	1.70537
$171$	0	0
$172$	−6324.00	−2.80349
$173$	−2374.00	−1.04331	−0.521653	−	0.853158i	$-0.674684\pi$
$173$	−2374.00	−1.04331	−0.521653	+	0.853158i	$0.674684\pi$
$174$	0	0
$175$	−623.000	−0.269111
$176$	979.000	0.419289
$177$	0	0
$178$	−510.000	−0.214753
$179$	1940.00	0.810069	0.405035	−	0.914301i	$-0.367259\pi$
$179$	1940.00	0.810069	0.405035	+	0.914301i	$0.367259\pi$
$180$	0	0
$181$	−1102.00	−0.452547	−0.226274	−	0.974064i	$-0.572654\pi$
$181$	−1102.00	−0.452547	−0.226274	+	0.974064i	$0.572654\pi$
$182$	−2450.00	−0.997836
$183$	0	0
$184$	−9000.00	−3.60592
$185$	−1884.00	−0.748727
$186$	0	0
$187$	−1386.00	−0.542002
$188$	1428.00	0.553977
$189$	0	0
$190$	2400.00	0.916391
$191$	112.000	0.0424295	0.0212148	−	0.999775i	$-0.493247\pi$
$191$	112.000	0.0424295	0.0212148	+	0.999775i	$0.493247\pi$
$192$	0	0
$193$	1058.00	0.394593	0.197297	−	0.980344i	$-0.436784\pi$
$193$	1058.00	0.394593	0.197297	+	0.980344i	$0.436784\pi$
$194$	4310.00	1.59505
$195$	0	0
$196$	833.000	0.303571
$197$	−2446.00	−0.884621	−0.442310	−	0.896862i	$-0.645841\pi$
$197$	−2446.00	−0.884621	−0.442310	+	0.896862i	$0.645841\pi$
$198$	0	0
$199$	4684.00	1.66854	0.834271	−	0.551354i	$-0.185889\pi$
$199$	4684.00	1.66854	0.834271	+	0.551354i	$0.185889\pi$
$200$	4005.00	1.41598
$201$	0	0
$202$	−1010.00	−0.351799
$203$	−938.000	−0.324309
$204$	0	0
$205$	−1668.00	−0.568284
$206$	−700.000	−0.236754
$207$	0	0
$208$	6230.00	2.07679
$209$	−880.000	−0.291248
$210$	0	0
$211$	68.0000	0.0221863	0.0110932	−	0.999938i	$-0.496469\pi$
$211$	68.0000	0.0221863	0.0110932	+	0.999938i	$0.496469\pi$
$212$	−3094.00	−1.00234
$213$	0	0
$214$	7780.00	2.48519
$215$	−2232.00	−0.708005
$216$	0	0
$217$	−1708.00	−0.534316
$218$	1970.00	0.612042
$219$	0	0
$220$	1122.00	0.343842
$221$	−8820.00	−2.68460
$222$	0	0
$223$	−2196.00	−0.659440	−0.329720	−	0.944079i	$-0.606954\pi$
$223$	−2196.00	−0.659440	−0.329720	+	0.944079i	$0.606954\pi$
$224$	−595.000	−0.177478
$225$	0	0
$226$	7770.00	2.28696
$227$	−4672.00	−1.36604	−0.683021	−	0.730399i	$-0.739334\pi$
$227$	−4672.00	−1.36604	−0.683021	+	0.730399i	$0.739334\pi$
$228$	0	0
$229$	−574.000	−0.165638	−0.0828188	−	0.996565i	$-0.526392\pi$
$229$	−574.000	−0.165638	−0.0828188	+	0.996565i	$0.526392\pi$
$230$	−6000.00	−1.72012
$231$	0	0
$232$	6030.00	1.70642
$233$	−218.000	−0.0612947	−0.0306473	−	0.999530i	$-0.509757\pi$
$233$	−218.000	−0.0612947	−0.0306473	+	0.999530i	$0.509757\pi$
$234$	0	0
$235$	504.000	0.139904
$236$	12852.0	3.54489
$237$	0	0
$238$	4410.00	1.20108
$239$	−4296.00	−1.16270	−0.581350	−	0.813654i	$-0.697475\pi$
$239$	−4296.00	−1.16270	−0.581350	+	0.813654i	$0.697475\pi$
$240$	0	0
$241$	3398.00	0.908234	0.454117	−	0.890942i	$-0.349955\pi$
$241$	3398.00	0.908234	0.454117	+	0.890942i	$0.349955\pi$
$242$	−605.000	−0.160706
$243$	0	0
$244$	11798.0	3.09545
$245$	294.000	0.0766652
$246$	0	0
$247$	−5600.00	−1.44259
$248$	10980.0	2.81141
$249$	0	0
$250$	6420.00	1.62415
$251$	1188.00	0.298749	0.149374	−	0.988781i	$-0.452274\pi$
$251$	1188.00	0.298749	0.149374	+	0.988781i	$0.452274\pi$
$252$	0	0
$253$	2200.00	0.546691
$254$	1160.00	0.286555
$255$	0	0
$256$	−8279.00	−2.02124
$257$	−1490.00	−0.361648	−0.180824	−	0.983515i	$-0.557877\pi$
$257$	−1490.00	−0.361648	−0.180824	+	0.983515i	$0.557877\pi$
$258$	0	0
$259$	−2198.00	−0.527325
$260$	7140.00	1.70309
$261$	0	0
$262$	560.000	0.132049
$263$	−7032.00	−1.64871	−0.824357	−	0.566070i	$-0.808463\pi$
$263$	−7032.00	−1.64871	−0.824357	+	0.566070i	$0.808463\pi$
$264$	0	0
$265$	−1092.00	−0.253136
$266$	2800.00	0.645410
$267$	0	0
$268$	13940.0	3.17732
$269$	4270.00	0.967831	0.483915	−	0.875115i	$-0.339214\pi$
$269$	4270.00	0.967831	0.483915	+	0.875115i	$0.339214\pi$
$270$	0	0
$271$	−1592.00	−0.356853	−0.178426	−	0.983953i	$-0.557101\pi$
$271$	−1592.00	−0.356853	−0.178426	+	0.983953i	$0.557101\pi$
$272$	−11214.0	−2.49981
$273$	0	0
$274$	−910.000	−0.200639
$275$	−979.000	−0.214676
$276$	0	0
$277$	782.000	0.169624	0.0848120	−	0.996397i	$-0.472971\pi$
$277$	782.000	0.169624	0.0848120	+	0.996397i	$0.472971\pi$
$278$	7520.00	1.62237
$279$	0	0
$280$	−1890.00	−0.403390
$281$	1590.00	0.337550	0.168775	−	0.985655i	$-0.446019\pi$
$281$	1590.00	0.337550	0.168775	+	0.985655i	$0.446019\pi$
$282$	0	0
$283$	−4384.00	−0.920854	−0.460427	−	0.887698i	$-0.652304\pi$
$283$	−4384.00	−0.920854	−0.460427	+	0.887698i	$0.652304\pi$
$284$	−2720.00	−0.568318
$285$	0	0
$286$	−3850.00	−0.795997
$287$	−1946.00	−0.400240
$288$	0	0
$289$	10963.0	2.23143
$290$	4020.00	0.814009
$291$	0	0
$292$	−34.0000	−0.00681404
$293$	−702.000	−0.139970	−0.0699851	−	0.997548i	$-0.522295\pi$
$293$	−702.000	−0.139970	−0.0699851	+	0.997548i	$0.522295\pi$
$294$	0	0
$295$	4536.00	0.895241
$296$	14130.0	2.77463
$297$	0	0
$298$	−15690.0	−3.04999
$299$	14000.0	2.70783
$300$	0	0
$301$	−2604.00	−0.498645
$302$	13400.0	2.55326
$303$	0	0
$304$	−7120.00	−1.34329
$305$	4164.00	0.781738
$306$	0	0
$307$	10248.0	1.90516	0.952580	−	0.304288i	$-0.0984184\pi$
$307$	10248.0	1.90516	0.952580	+	0.304288i	$0.0984184\pi$
$308$	1309.00	0.242166
$309$	0	0
$310$	7320.00	1.34112
$311$	1220.00	0.222443	0.111222	−	0.993796i	$-0.464524\pi$
$311$	1220.00	0.222443	0.111222	+	0.993796i	$0.464524\pi$
$312$	0	0
$313$	−4638.00	−0.837557	−0.418778	−	0.908088i	$-0.637542\pi$
$313$	−4638.00	−0.837557	−0.418778	+	0.908088i	$0.637542\pi$
$314$	−10010.0	−1.79903
$315$	0	0
$316$	680.000	0.121054
$317$	7554.00	1.33841	0.669203	−	0.743079i	$-0.266636\pi$
$317$	7554.00	1.33841	0.669203	+	0.743079i	$0.266636\pi$
$318$	0	0
$319$	−1474.00	−0.258709
$320$	−1722.00	−0.300821
$321$	0	0
$322$	−7000.00	−1.21147
$323$	10080.0	1.73643
$324$	0	0
$325$	−6230.00	−1.06332
$326$	3700.00	0.628601
$327$	0	0
$328$	12510.0	2.10594
$329$	588.000	0.0985334
$330$	0	0
$331$	−8004.00	−1.32912	−0.664561	−	0.747234i	$-0.731382\pi$
$331$	−8004.00	−1.32912	−0.664561	+	0.747234i	$0.731382\pi$
$332$	−12920.0	−2.13577
$333$	0	0
$334$	17040.0	2.79158
$335$	4920.00	0.802413
$336$	0	0
$337$	2746.00	0.443870	0.221935	−	0.975061i	$-0.428763\pi$
$337$	2746.00	0.443870	0.221935	+	0.975061i	$0.428763\pi$
$338$	−13515.0	−2.17491
$339$	0	0
$340$	−12852.0	−2.04999
$341$	−2684.00	−0.426237
$342$	0	0
$343$	343.000	0.0539949
$344$	16740.0	2.62372
$345$	0	0
$346$	11870.0	1.84432
$347$	−1444.00	−0.223395	−0.111697	−	0.993742i	$-0.535629\pi$
$347$	−1444.00	−0.223395	−0.111697	+	0.993742i	$0.535629\pi$
$348$	0	0
$349$	−9522.00	−1.46046	−0.730231	−	0.683201i	$-0.760587\pi$
$349$	−9522.00	−1.46046	−0.730231	+	0.683201i	$0.760587\pi$
$350$	3115.00	0.475725
$351$	0	0
$352$	−935.000	−0.141579
$353$	−642.000	−0.0967995	−0.0483997	−	0.998828i	$-0.515412\pi$
$353$	−642.000	−0.0967995	−0.0483997	+	0.998828i	$0.515412\pi$
$354$	0	0
$355$	−960.000	−0.143525
$356$	1734.00	0.258151
$357$	0	0
$358$	−9700.00	−1.43201
$359$	−5160.00	−0.758592	−0.379296	−	0.925275i	$-0.623834\pi$
$359$	−5160.00	−0.758592	−0.379296	+	0.925275i	$0.623834\pi$
$360$	0	0
$361$	−459.000	−0.0669194
$362$	5510.00	0.799998
$363$	0	0
$364$	8330.00	1.19948
$365$	−12.0000	−0.00172085
$366$	0	0
$367$	−8900.00	−1.26588	−0.632938	−	0.774203i	$-0.718151\pi$
$367$	−8900.00	−1.26588	−0.632938	+	0.774203i	$0.718151\pi$
$368$	17800.0	2.52144
$369$	0	0
$370$	9420.00	1.32357
$371$	−1274.00	−0.178282
$372$	0	0
$373$	−2898.00	−0.402286	−0.201143	−	0.979562i	$-0.564466\pi$
$373$	−2898.00	−0.402286	−0.201143	+	0.979562i	$0.564466\pi$
$374$	6930.00	0.958133
$375$	0	0
$376$	−3780.00	−0.518454
$377$	−9380.00	−1.28142
$378$	0	0
$379$	13156.0	1.78306	0.891528	−	0.452965i	$-0.149634\pi$
$379$	13156.0	1.78306	0.891528	+	0.452965i	$0.149634\pi$
$380$	−8160.00	−1.10158
$381$	0	0
$382$	−560.000	−0.0750055
$383$	6220.00	0.829836	0.414918	−	0.909859i	$-0.363810\pi$
$383$	6220.00	0.829836	0.414918	+	0.909859i	$0.363810\pi$
$384$	0	0
$385$	462.000	0.0611577
$386$	−5290.00	−0.697549
$387$	0	0
$388$	−14654.0	−1.91738
$389$	−6062.00	−0.790117	−0.395059	−	0.918656i	$-0.629276\pi$
$389$	−6062.00	−0.790117	−0.395059	+	0.918656i	$0.629276\pi$
$390$	0	0
$391$	−25200.0	−3.25938
$392$	−2205.00	−0.284105
$393$	0	0
$394$	12230.0	1.56380
$395$	240.000	0.0305714
$396$	0	0
$397$	−990.000	−0.125155	−0.0625777	−	0.998040i	$-0.519932\pi$
$397$	−990.000	−0.125155	−0.0625777	+	0.998040i	$0.519932\pi$
$398$	−23420.0	−2.94959
$399$	0	0
$400$	−7921.00	−0.990125
$401$	4158.00	0.517807	0.258904	−	0.965903i	$-0.416639\pi$
$401$	4158.00	0.517807	0.258904	+	0.965903i	$0.416639\pi$
$402$	0	0
$403$	−17080.0	−2.11120
$404$	3434.00	0.422891
$405$	0	0
$406$	4690.00	0.573302
$407$	−3454.00	−0.420660
$408$	0	0
$409$	−8554.00	−1.03415	−0.517076	−	0.855940i	$-0.672980\pi$
$409$	−8554.00	−1.03415	−0.517076	+	0.855940i	$0.672980\pi$
$410$	8340.00	1.00459
$411$	0	0
$412$	2380.00	0.284598
$413$	5292.00	0.630514
$414$	0	0
$415$	−4560.00	−0.539377
$416$	−5950.00	−0.701257
$417$	0	0
$418$	4400.00	0.514859
$419$	9364.00	1.09179	0.545897	−	0.837853i	$-0.316189\pi$
$419$	9364.00	1.09179	0.545897	+	0.837853i	$0.316189\pi$
$420$	0	0
$421$	−7266.00	−0.841148	−0.420574	−	0.907258i	$-0.638171\pi$
$421$	−7266.00	−0.841148	−0.420574	+	0.907258i	$0.638171\pi$
$422$	−340.000	−0.0392202
$423$	0	0
$424$	8190.00	0.938070
$425$	11214.0	1.27990
$426$	0	0
$427$	4858.00	0.550574
$428$	−26452.0	−2.98740
$429$	0	0
$430$	11160.0	1.25159
$431$	3760.00	0.420215	0.210108	−	0.977678i	$-0.432619\pi$
$431$	3760.00	0.420215	0.210108	+	0.977678i	$0.432619\pi$
$432$	0	0
$433$	7378.00	0.818855	0.409427	−	0.912343i	$-0.365729\pi$
$433$	7378.00	0.818855	0.409427	+	0.912343i	$0.365729\pi$
$434$	8540.00	0.944546
$435$	0	0
$436$	−6698.00	−0.735725
$437$	−16000.0	−1.75145
$438$	0	0
$439$	6400.00	0.695798	0.347899	−	0.937532i	$-0.386895\pi$
$439$	6400.00	0.695798	0.347899	+	0.937532i	$0.386895\pi$
$440$	−2970.00	−0.321794
$441$	0	0
$442$	44100.0	4.74575
$443$	−260.000	−0.0278848	−0.0139424	−	0.999903i	$-0.504438\pi$
$443$	−260.000	−0.0278848	−0.0139424	+	0.999903i	$0.504438\pi$
$444$	0	0
$445$	612.000	0.0651946
$446$	10980.0	1.16574
$447$	0	0
$448$	−2009.00	−0.211867
$449$	−8034.00	−0.844427	−0.422214	−	0.906496i	$-0.638747\pi$
$449$	−8034.00	−0.844427	−0.422214	+	0.906496i	$0.638747\pi$
$450$	0	0
$451$	−3058.00	−0.319281
$452$	−26418.0	−2.74911
$453$	0	0
$454$	23360.0	2.41484
$455$	2940.00	0.302922
$456$	0	0
$457$	5994.00	0.613539	0.306770	−	0.951784i	$-0.400752\pi$
$457$	5994.00	0.613539	0.306770	+	0.951784i	$0.400752\pi$
$458$	2870.00	0.292808
$459$	0	0
$460$	20400.0	2.06773
$461$	258.000	0.0260656	0.0130328	−	0.999915i	$-0.495851\pi$
$461$	258.000	0.0260656	0.0130328	+	0.999915i	$0.495851\pi$
$462$	0	0
$463$	−16976.0	−1.70398	−0.851989	−	0.523560i	$-0.824604\pi$
$463$	−16976.0	−1.70398	−0.851989	+	0.523560i	$0.824604\pi$
$464$	−11926.0	−1.19321
$465$	0	0
$466$	1090.00	0.108355
$467$	−15676.0	−1.55332	−0.776658	−	0.629922i	$-0.783087\pi$
$467$	−15676.0	−1.55332	−0.776658	+	0.629922i	$0.783087\pi$
$468$	0	0
$469$	5740.00	0.565136
$470$	−2520.00	−0.247317
$471$	0	0
$472$	−34020.0	−3.31758
$473$	−4092.00	−0.397781
$474$	0	0
$475$	7120.00	0.687764
$476$	−14994.0	−1.44380
$477$	0	0
$478$	21480.0	2.05538
$479$	−2576.00	−0.245721	−0.122861	−	0.992424i	$-0.539207\pi$
$479$	−2576.00	−0.245721	−0.122861	+	0.992424i	$0.539207\pi$
$480$	0	0
$481$	−21980.0	−2.08358
$482$	−16990.0	−1.60555
$483$	0	0
$484$	2057.00	0.193182
$485$	−5172.00	−0.484224
$486$	0	0
$487$	−1520.00	−0.141433	−0.0707164	−	0.997496i	$-0.522529\pi$
$487$	−1520.00	−0.141433	−0.0707164	+	0.997496i	$0.522529\pi$
$488$	−31230.0	−2.89696
$489$	0	0
$490$	−1470.00	−0.135526
$491$	13572.0	1.24745	0.623723	−	0.781646i	$-0.285619\pi$
$491$	13572.0	1.24745	0.623723	+	0.781646i	$0.285619\pi$
$492$	0	0
$493$	16884.0	1.54243
$494$	28000.0	2.55016
$495$	0	0
$496$	−21716.0	−1.96588
$497$	−1120.00	−0.101084
$498$	0	0
$499$	4372.00	0.392220	0.196110	−	0.980582i	$-0.437169\pi$
$499$	4372.00	0.392220	0.196110	+	0.980582i	$0.437169\pi$
$500$	−21828.0	−1.95236
$501$	0	0
$502$	−5940.00	−0.528118
$503$	−2192.00	−0.194307	−0.0971535	−	0.995269i	$-0.530974\pi$
$503$	−2192.00	−0.194307	−0.0971535	+	0.995269i	$0.530974\pi$
$504$	0	0
$505$	1212.00	0.106799
$506$	−11000.0	−0.966422
$507$	0	0
$508$	−3944.00	−0.344462
$509$	6030.00	0.525098	0.262549	−	0.964919i	$-0.415437\pi$
$509$	6030.00	0.525098	0.262549	+	0.964919i	$0.415437\pi$
$510$	0	0
$511$	−14.0000	−0.00121198
$512$	24475.0	2.11260
$513$	0	0
$514$	7450.00	0.639310
$515$	840.000	0.0718734
$516$	0	0
$517$	924.000	0.0786025
$518$	10990.0	0.932187
$519$	0	0
$520$	−18900.0	−1.59388
$521$	−1850.00	−0.155566	−0.0777831	−	0.996970i	$-0.524784\pi$
$521$	−1850.00	−0.155566	−0.0777831	+	0.996970i	$0.524784\pi$
$522$	0	0
$523$	−5104.00	−0.426735	−0.213367	−	0.976972i	$-0.568443\pi$
$523$	−5104.00	−0.426735	−0.213367	+	0.976972i	$0.568443\pi$
$524$	−1904.00	−0.158734
$525$	0	0
$526$	35160.0	2.91454
$527$	30744.0	2.54123
$528$	0	0
$529$	27833.0	2.28758
$530$	5460.00	0.447485
$531$	0	0
$532$	−9520.00	−0.775835
$533$	−19460.0	−1.58144
$534$	0	0
$535$	−9336.00	−0.754449
$536$	−36900.0	−2.97358
$537$	0	0
$538$	−21350.0	−1.71090
$539$	539.000	0.0430730
$540$	0	0
$541$	−11706.0	−0.930278	−0.465139	−	0.885238i	$-0.653996\pi$
$541$	−11706.0	−0.930278	−0.465139	+	0.885238i	$0.653996\pi$
$542$	7960.00	0.630833
$543$	0	0
$544$	10710.0	0.844095
$545$	−2364.00	−0.185803
$546$	0	0
$547$	1676.00	0.131007	0.0655033	−	0.997852i	$-0.479135\pi$
$547$	1676.00	0.131007	0.0655033	+	0.997852i	$0.479135\pi$
$548$	3094.00	0.241185
$549$	0	0
$550$	4895.00	0.379497
$551$	10720.0	0.828834
$552$	0	0
$553$	280.000	0.0215313
$554$	−3910.00	−0.299856
$555$	0	0
$556$	−25568.0	−1.95022
$557$	22362.0	1.70109	0.850546	−	0.525901i	$-0.176272\pi$
$557$	22362.0	1.70109	0.850546	+	0.525901i	$0.176272\pi$
$558$	0	0
$559$	−26040.0	−1.97026
$560$	3738.00	0.282070
$561$	0	0
$562$	−7950.00	−0.596709
$563$	6816.00	0.510231	0.255116	−	0.966911i	$-0.417886\pi$
$563$	6816.00	0.510231	0.255116	+	0.966911i	$0.417886\pi$
$564$	0	0
$565$	−9324.00	−0.694272
$566$	21920.0	1.62786
$567$	0	0
$568$	7200.00	0.531876
$569$	17206.0	1.26769	0.633843	−	0.773462i	$-0.281477\pi$
$569$	17206.0	1.26769	0.633843	+	0.773462i	$0.281477\pi$
$570$	0	0
$571$	18164.0	1.33124	0.665621	−	0.746290i	$-0.268167\pi$
$571$	18164.0	1.33124	0.665621	+	0.746290i	$0.268167\pi$
$572$	13090.0	0.956854
$573$	0	0
$574$	9730.00	0.707530
$575$	−17800.0	−1.29098
$576$	0	0
$577$	3610.00	0.260461	0.130231	−	0.991484i	$-0.458428\pi$
$577$	3610.00	0.260461	0.130231	+	0.991484i	$0.458428\pi$
$578$	−54815.0	−3.94464
$579$	0	0
$580$	−13668.0	−0.978505
$581$	−5320.00	−0.379881
$582$	0	0
$583$	−2002.00	−0.142220
$584$	90.0000	0.00637710
$585$	0	0
$586$	3510.00	0.247435
$587$	7236.00	0.508793	0.254397	−	0.967100i	$-0.418123\pi$
$587$	7236.00	0.508793	0.254397	+	0.967100i	$0.418123\pi$
$588$	0	0
$589$	19520.0	1.36555
$590$	−22680.0	−1.58258
$591$	0	0
$592$	−27946.0	−1.94016
$593$	1570.00	0.108722	0.0543610	−	0.998521i	$-0.482688\pi$
$593$	1570.00	0.108722	0.0543610	+	0.998521i	$0.482688\pi$
$594$	0	0
$595$	−5292.00	−0.364623
$596$	53346.0	3.66634
$597$	0	0
$598$	−70000.0	−4.78681
$599$	−11360.0	−0.774887	−0.387443	−	0.921894i	$-0.626642\pi$
$599$	−11360.0	−0.774887	−0.387443	+	0.921894i	$0.626642\pi$
$600$	0	0
$601$	−2682.00	−0.182032	−0.0910159	−	0.995849i	$-0.529011\pi$
$601$	−2682.00	−0.182032	−0.0910159	+	0.995849i	$0.529011\pi$
$602$	13020.0	0.881488
$603$	0	0
$604$	−45560.0	−3.06922
$605$	726.000	0.0487869
$606$	0	0
$607$	5488.00	0.366970	0.183485	−	0.983022i	$-0.441262\pi$
$607$	5488.00	0.366970	0.183485	+	0.983022i	$0.441262\pi$
$608$	6800.00	0.453580
$609$	0	0
$610$	−20820.0	−1.38193
$611$	5880.00	0.389328
$612$	0	0
$613$	23870.0	1.57276	0.786379	−	0.617745i	$-0.211954\pi$
$613$	23870.0	1.57276	0.786379	+	0.617745i	$0.211954\pi$
$614$	−51240.0	−3.36788
$615$	0	0
$616$	−3465.00	−0.226638
$617$	−13242.0	−0.864024	−0.432012	−	0.901868i	$-0.642196\pi$
$617$	−13242.0	−0.864024	−0.432012	+	0.901868i	$0.642196\pi$
$618$	0	0
$619$	21924.0	1.42359	0.711793	−	0.702389i	$-0.247883\pi$
$619$	21924.0	1.42359	0.711793	+	0.702389i	$0.247883\pi$
$620$	−24888.0	−1.61214
$621$	0	0
$622$	−6100.00	−0.393228
$623$	714.000	0.0459162
$624$	0	0
$625$	3421.00	0.218944
$626$	23190.0	1.48061
$627$	0	0
$628$	34034.0	2.16259
$629$	39564.0	2.50798
$630$	0	0
$631$	−19168.0	−1.20930	−0.604648	−	0.796493i	$-0.706686\pi$
$631$	−19168.0	−1.20930	−0.604648	+	0.796493i	$0.706686\pi$
$632$	−1800.00	−0.113291
$633$	0	0
$634$	−37770.0	−2.36599
$635$	−1392.00	−0.0869919
$636$	0	0
$637$	3430.00	0.213346
$638$	7370.00	0.457337
$639$	0	0
$640$	12690.0	0.783775
$641$	−25122.0	−1.54799	−0.773993	−	0.633194i	$-0.781744\pi$
$641$	−25122.0	−1.54799	−0.773993	+	0.633194i	$0.781744\pi$
$642$	0	0
$643$	18708.0	1.14739	0.573695	−	0.819069i	$-0.305510\pi$
$643$	18708.0	1.14739	0.573695	+	0.819069i	$0.305510\pi$
$644$	23800.0	1.45629
$645$	0	0
$646$	−50400.0	−3.06960
$647$	−24396.0	−1.48239	−0.741195	−	0.671290i	$-0.765740\pi$
$647$	−24396.0	−1.48239	−0.741195	+	0.671290i	$0.765740\pi$
$648$	0	0
$649$	8316.00	0.502976
$650$	31150.0	1.87970
$651$	0	0
$652$	−12580.0	−0.755630
$653$	31386.0	1.88090	0.940451	−	0.339929i	$-0.110403\pi$
$653$	31386.0	1.88090	0.940451	+	0.339929i	$0.110403\pi$
$654$	0	0
$655$	−672.000	−0.0400873
$656$	−24742.0	−1.47258
$657$	0	0
$658$	−2940.00	−0.174184
$659$	−16284.0	−0.962571	−0.481286	−	0.876564i	$-0.659830\pi$
$659$	−16284.0	−0.962571	−0.481286	+	0.876564i	$0.659830\pi$
$660$	0	0
$661$	−21646.0	−1.27372	−0.636862	−	0.770978i	$-0.719768\pi$
$661$	−21646.0	−1.27372	−0.636862	+	0.770978i	$0.719768\pi$
$662$	40020.0	2.34958
$663$	0	0
$664$	34200.0	1.99882
$665$	−3360.00	−0.195933
$666$	0	0
$667$	−26800.0	−1.55577
$668$	−57936.0	−3.35571
$669$	0	0
$670$	−24600.0	−1.41848
$671$	7634.00	0.439206
$672$	0	0
$673$	19978.0	1.14427	0.572136	−	0.820159i	$-0.306115\pi$
$673$	19978.0	1.14427	0.572136	+	0.820159i	$0.306115\pi$
$674$	−13730.0	−0.784659
$675$	0	0
$676$	45951.0	2.61442
$677$	−6654.00	−0.377746	−0.188873	−	0.982002i	$-0.560483\pi$
$677$	−6654.00	−0.377746	−0.188873	+	0.982002i	$0.560483\pi$
$678$	0	0
$679$	−6034.00	−0.341036
$680$	34020.0	1.91854
$681$	0	0
$682$	13420.0	0.753487
$683$	−2164.00	−0.121234	−0.0606172	−	0.998161i	$-0.519307\pi$
$683$	−2164.00	−0.121234	−0.0606172	+	0.998161i	$0.519307\pi$
$684$	0	0
$685$	1092.00	0.0609097
$686$	−1715.00	−0.0954504
$687$	0	0
$688$	−33108.0	−1.83464
$689$	−12740.0	−0.704434
$690$	0	0
$691$	−3788.00	−0.208542	−0.104271	−	0.994549i	$-0.533251\pi$
$691$	−3788.00	−0.208542	−0.104271	+	0.994549i	$0.533251\pi$
$692$	−40358.0	−2.21702
$693$	0	0
$694$	7220.00	0.394910
$695$	−9024.00	−0.492518
$696$	0	0
$697$	35028.0	1.90356
$698$	47610.0	2.58176
$699$	0	0
$700$	−10591.0	−0.571860
$701$	−28262.0	−1.52274	−0.761370	−	0.648317i	$-0.775473\pi$
$701$	−28262.0	−1.52274	−0.761370	+	0.648317i	$0.775473\pi$
$702$	0	0
$703$	25120.0	1.34768
$704$	−3157.00	−0.169011
$705$	0	0
$706$	3210.00	0.171119
$707$	1414.00	0.0752177
$708$	0	0
$709$	6150.00	0.325766	0.162883	−	0.986645i	$-0.447921\pi$
$709$	6150.00	0.325766	0.162883	+	0.986645i	$0.447921\pi$
$710$	4800.00	0.253719
$711$	0	0
$712$	−4590.00	−0.241598
$713$	−48800.0	−2.56322
$714$	0	0
$715$	4620.00	0.241648
$716$	32980.0	1.72140
$717$	0	0
$718$	25800.0	1.34101
$719$	−14532.0	−0.753758	−0.376879	−	0.926262i	$-0.623003\pi$
$719$	−14532.0	−0.753758	−0.376879	+	0.926262i	$0.623003\pi$
$720$	0	0
$721$	980.000	0.0506201
$722$	2295.00	0.118298
$723$	0	0
$724$	−18734.0	−0.961662
$725$	11926.0	0.610925
$726$	0	0
$727$	−22700.0	−1.15804	−0.579021	−	0.815313i	$-0.696565\pi$
$727$	−22700.0	−1.15804	−0.579021	+	0.815313i	$0.696565\pi$
$728$	−22050.0	−1.12257
$729$	0	0
$730$	60.0000	0.00304205
$731$	46872.0	2.37158
$732$	0	0
$733$	33782.0	1.70227	0.851137	−	0.524944i	$-0.175914\pi$
$733$	33782.0	1.70227	0.851137	+	0.524944i	$0.175914\pi$
$734$	44500.0	2.23777
$735$	0	0
$736$	−17000.0	−0.851397
$737$	9020.00	0.450822
$738$	0	0
$739$	−36372.0	−1.81051	−0.905254	−	0.424870i	$-0.860320\pi$
$739$	−36372.0	−1.81051	−0.905254	+	0.424870i	$0.860320\pi$
$740$	−32028.0	−1.59104
$741$	0	0
$742$	6370.00	0.315162
$743$	9168.00	0.452680	0.226340	−	0.974048i	$-0.427324\pi$
$743$	9168.00	0.452680	0.226340	+	0.974048i	$0.427324\pi$
$744$	0	0
$745$	18828.0	0.925912
$746$	14490.0	0.711148
$747$	0	0
$748$	−23562.0	−1.15175
$749$	−10892.0	−0.531355
$750$	0	0
$751$	−25096.0	−1.21940	−0.609698	−	0.792634i	$-0.708709\pi$
$751$	−25096.0	−1.21940	−0.609698	+	0.792634i	$0.708709\pi$
$752$	7476.00	0.362529
$753$	0	0
$754$	46900.0	2.26525
$755$	−16080.0	−0.775114
$756$	0	0
$757$	−27690.0	−1.32947	−0.664736	−	0.747078i	$-0.731456\pi$
$757$	−27690.0	−1.32947	−0.664736	+	0.747078i	$0.731456\pi$
$758$	−65780.0	−3.15203
$759$	0	0
$760$	21600.0	1.03094
$761$	−846.000	−0.0402989	−0.0201495	−	0.999797i	$-0.506414\pi$
$761$	−846.000	−0.0402989	−0.0201495	+	0.999797i	$0.506414\pi$
$762$	0	0
$763$	−2758.00	−0.130860
$764$	1904.00	0.0901627
$765$	0	0
$766$	−31100.0	−1.46696
$767$	52920.0	2.49130
$768$	0	0
$769$	7854.00	0.368300	0.184150	−	0.982898i	$-0.441047\pi$
$769$	7854.00	0.368300	0.184150	+	0.982898i	$0.441047\pi$
$770$	−2310.00	−0.108112
$771$	0	0
$772$	17986.0	0.838511
$773$	4830.00	0.224739	0.112369	−	0.993667i	$-0.464156\pi$
$773$	4830.00	0.224739	0.112369	+	0.993667i	$0.464156\pi$
$774$	0	0
$775$	21716.0	1.00653
$776$	38790.0	1.79443
$777$	0	0
$778$	30310.0	1.39674
$779$	22240.0	1.02289
$780$	0	0
$781$	−1760.00	−0.0806373
$782$	126000.	5.76183
$783$	0	0
$784$	4361.00	0.198661
$785$	12012.0	0.546149
$786$	0	0
$787$	8032.00	0.363799	0.181900	−	0.983317i	$-0.441775\pi$
$787$	8032.00	0.363799	0.181900	+	0.983317i	$0.441775\pi$
$788$	−41582.0	−1.87982
$789$	0	0
$790$	−1200.00	−0.0540431
$791$	−10878.0	−0.488972
$792$	0	0
$793$	48580.0	2.17544
$794$	4950.00	0.221245
$795$	0	0
$796$	79628.0	3.54565
$797$	26486.0	1.17714	0.588571	−	0.808445i	$-0.299691\pi$
$797$	26486.0	1.17714	0.588571	+	0.808445i	$0.299691\pi$
$798$	0	0
$799$	−10584.0	−0.468630
$800$	7565.00	0.334329
$801$	0	0
$802$	−20790.0	−0.915362
$803$	−22.0000	−0.000966828 0
$804$	0	0
$805$	8400.00	0.367778
$806$	85400.0	3.73212
$807$	0	0
$808$	−9090.00	−0.395774
$809$	9582.00	0.416422	0.208211	−	0.978084i	$-0.433236\pi$
$809$	9582.00	0.416422	0.208211	+	0.978084i	$0.433236\pi$
$810$	0	0
$811$	37760.0	1.63494	0.817468	−	0.575974i	$-0.195377\pi$
$811$	37760.0	1.63494	0.817468	+	0.575974i	$0.195377\pi$
$812$	−15946.0	−0.689156
$813$	0	0
$814$	17270.0	0.743628
$815$	−4440.00	−0.190830
$816$	0	0
$817$	29760.0	1.27438
$818$	42770.0	1.82814
$819$	0	0
$820$	−28356.0	−1.20760
$821$	30754.0	1.30733	0.653667	−	0.756782i	$-0.273230\pi$
$821$	30754.0	1.30733	0.653667	+	0.756782i	$0.273230\pi$
$822$	0	0
$823$	−23216.0	−0.983304	−0.491652	−	0.870792i	$-0.663607\pi$
$823$	−23216.0	−0.983304	−0.491652	+	0.870792i	$0.663607\pi$
$824$	−6300.00	−0.266348
$825$	0	0
$826$	−26460.0	−1.11460
$827$	4948.00	0.208052	0.104026	−	0.994575i	$-0.466828\pi$
$827$	4948.00	0.208052	0.104026	+	0.994575i	$0.466828\pi$
$828$	0	0
$829$	11162.0	0.467638	0.233819	−	0.972280i	$-0.424878\pi$
$829$	11162.0	0.467638	0.233819	+	0.972280i	$0.424878\pi$
$830$	22800.0	0.953493
$831$	0	0
$832$	−20090.0	−0.837134
$833$	−6174.00	−0.256802
$834$	0	0
$835$	−20448.0	−0.847464
$836$	−14960.0	−0.618902
$837$	0	0
$838$	−46820.0	−1.93004
$839$	−18492.0	−0.760923	−0.380462	−	0.924797i	$-0.624235\pi$
$839$	−18492.0	−0.760923	−0.380462	+	0.924797i	$0.624235\pi$
$840$	0	0
$841$	−6433.00	−0.263766
$842$	36330.0	1.48695
$843$	0	0
$844$	1156.00	0.0471459
$845$	16218.0	0.660256
$846$	0	0
$847$	847.000	0.0343604
$848$	−16198.0	−0.655945
$849$	0	0
$850$	−56070.0	−2.26257
$851$	−62800.0	−2.52968
$852$	0	0
$853$	−30994.0	−1.24410	−0.622048	−	0.782979i	$-0.713699\pi$
$853$	−30994.0	−1.24410	−0.622048	+	0.782979i	$0.713699\pi$
$854$	−24290.0	−0.973287
$855$	0	0
$856$	70020.0	2.79583
$857$	−20366.0	−0.811773	−0.405886	−	0.913924i	$-0.633037\pi$
$857$	−20366.0	−0.811773	−0.405886	+	0.913924i	$0.633037\pi$
$858$	0	0
$859$	2556.00	0.101525	0.0507623	−	0.998711i	$-0.483835\pi$
$859$	2556.00	0.101525	0.0507623	+	0.998711i	$0.483835\pi$
$860$	−37944.0	−1.50451
$861$	0	0
$862$	−18800.0	−0.742843
$863$	−49568.0	−1.95517	−0.977587	−	0.210534i	$-0.932480\pi$
$863$	−49568.0	−1.95517	−0.977587	+	0.210534i	$0.932480\pi$
$864$	0	0
$865$	−14244.0	−0.559897
$866$	−36890.0	−1.44754
$867$	0	0
$868$	−29036.0	−1.13542
$869$	440.000	0.0171760
$870$	0	0
$871$	57400.0	2.23298
$872$	17730.0	0.688548
$873$	0	0
$874$	80000.0	3.09616
$875$	−8988.00	−0.347257
$876$	0	0
$877$	−44346.0	−1.70748	−0.853739	−	0.520701i	$-0.825671\pi$
$877$	−44346.0	−1.70748	−0.853739	+	0.520701i	$0.825671\pi$
$878$	−32000.0	−1.23001
$879$	0	0
$880$	5874.00	0.225014
$881$	39254.0	1.50114	0.750568	−	0.660793i	$-0.229780\pi$
$881$	39254.0	1.50114	0.750568	+	0.660793i	$0.229780\pi$
$882$	0	0
$883$	−9972.00	−0.380050	−0.190025	−	0.981779i	$-0.560857\pi$
$883$	−9972.00	−0.380050	−0.190025	+	0.981779i	$0.560857\pi$
$884$	−149940.	−5.70478
$885$	0	0
$886$	1300.00	0.0492939
$887$	904.000	0.0342202	0.0171101	−	0.999854i	$-0.494553\pi$
$887$	904.000	0.0342202	0.0171101	+	0.999854i	$0.494553\pi$
$888$	0	0
$889$	−1624.00	−0.0612680
$890$	−3060.00	−0.115249
$891$	0	0
$892$	−37332.0	−1.40131
$893$	−6720.00	−0.251821
$894$	0	0
$895$	11640.0	0.434729
$896$	14805.0	0.552009
$897$	0	0
$898$	40170.0	1.49275
$899$	32696.0	1.21298
$900$	0	0
$901$	22932.0	0.847920
$902$	15290.0	0.564414
$903$	0	0
$904$	69930.0	2.57283
$905$	−6612.00	−0.242862
$906$	0	0
$907$	−24324.0	−0.890480	−0.445240	−	0.895411i	$-0.646882\pi$
$907$	−24324.0	−0.890480	−0.445240	+	0.895411i	$0.646882\pi$
$908$	−79424.0	−2.90284
$909$	0	0
$910$	−14700.0	−0.535495
$911$	49968.0	1.81725	0.908624	−	0.417614i	$-0.137134\pi$
$911$	49968.0	1.81725	0.908624	+	0.417614i	$0.137134\pi$
$912$	0	0
$913$	−8360.00	−0.303040
$914$	−29970.0	−1.08459
$915$	0	0
$916$	−9758.00	−0.351980
$917$	−784.000	−0.0282333
$918$	0	0
$919$	26072.0	0.935839	0.467919	−	0.883771i	$-0.345004\pi$
$919$	26072.0	0.935839	0.467919	+	0.883771i	$0.345004\pi$
$920$	−54000.0	−1.93514
$921$	0	0
$922$	−1290.00	−0.0460780
$923$	−11200.0	−0.399407
$924$	0	0
$925$	27946.0	0.993361
$926$	84880.0	3.01224
$927$	0	0
$928$	11390.0	0.402904
$929$	−8802.00	−0.310855	−0.155427	−	0.987847i	$-0.549675\pi$
$929$	−8802.00	−0.310855	−0.155427	+	0.987847i	$0.549675\pi$
$930$	0	0
$931$	−3920.00	−0.137994
$932$	−3706.00	−0.130251
$933$	0	0
$934$	78380.0	2.74590
$935$	−8316.00	−0.290869
$936$	0	0
$937$	−9506.00	−0.331427	−0.165714	−	0.986174i	$-0.552993\pi$
$937$	−9506.00	−0.331427	−0.165714	+	0.986174i	$0.552993\pi$
$938$	−28700.0	−0.999028
$939$	0	0
$940$	8568.00	0.297295
$941$	23594.0	0.817367	0.408684	−	0.912676i	$-0.365988\pi$
$941$	23594.0	0.817367	0.408684	+	0.912676i	$0.365988\pi$
$942$	0	0
$943$	−55600.0	−1.92003
$944$	67284.0	2.31982
$945$	0	0
$946$	20460.0	0.703184
$947$	−3684.00	−0.126414	−0.0632069	−	0.998000i	$-0.520133\pi$
$947$	−3684.00	−0.126414	−0.0632069	+	0.998000i	$0.520133\pi$
$948$	0	0
$949$	−140.000	−0.00478882
$950$	−35600.0	−1.21581
$951$	0	0
$952$	39690.0	1.35122
$953$	15950.0	0.542152	0.271076	−	0.962558i	$-0.412620\pi$
$953$	15950.0	0.542152	0.271076	+	0.962558i	$0.412620\pi$
$954$	0	0
$955$	672.000	0.0227701
$956$	−73032.0	−2.47074
$957$	0	0
$958$	12880.0	0.434378
$959$	1274.00	0.0428984
$960$	0	0
$961$	29745.0	0.998456
$962$	109900.	3.68328
$963$	0	0
$964$	57766.0	1.93000
$965$	6348.00	0.211761
$966$	0	0
$967$	−16312.0	−0.542460	−0.271230	−	0.962515i	$-0.587430\pi$
$967$	−16312.0	−0.542460	−0.271230	+	0.962515i	$0.587430\pi$
$968$	−5445.00	−0.180794
$969$	0	0
$970$	25860.0	0.855994
$971$	−22148.0	−0.731991	−0.365995	−	0.930617i	$-0.619271\pi$
$971$	−22148.0	−0.731991	−0.365995	+	0.930617i	$0.619271\pi$
$972$	0	0
$973$	−10528.0	−0.346878
$974$	7600.00	0.250020
$975$	0	0
$976$	61766.0	2.02570
$977$	−32306.0	−1.05789	−0.528946	−	0.848655i	$-0.677413\pi$
$977$	−32306.0	−1.05789	−0.528946	+	0.848655i	$0.677413\pi$
$978$	0	0
$979$	1122.00	0.0366285
$980$	4998.00	0.162914
$981$	0	0
$982$	−67860.0	−2.20519
$983$	30268.0	0.982095	0.491047	−	0.871133i	$-0.336614\pi$
$983$	30268.0	0.982095	0.491047	+	0.871133i	$0.336614\pi$
$984$	0	0
$985$	−14676.0	−0.474737
$986$	−84420.0	−2.72665
$987$	0	0
$988$	−95200.0	−3.06550
$989$	−74400.0	−2.39210
$990$	0	0
$991$	7328.00	0.234896	0.117448	−	0.993079i	$-0.462529\pi$
$991$	7328.00	0.234896	0.117448	+	0.993079i	$0.462529\pi$
$992$	20740.0	0.663806
$993$	0	0
$994$	5600.00	0.178693
$995$	28104.0	0.895434
$996$	0	0
$997$	18406.0	0.584678	0.292339	−	0.956315i	$-0.405566\pi$
$997$	18406.0	0.584678	0.292339	+	0.956315i	$0.405566\pi$
$998$	−21860.0	−0.693353
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.a.1.1		1
3.2	odd	2		231.4.a.e.1.1	✓	1
21.20	even	2		1617.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.e.1.1	✓	1	3.2	odd	2
693.4.a.a.1.1		1	1.1	even	1	trivial
1617.4.a.g.1.1		1	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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