LMFDB - Embedded newform 1050.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,4,Mod(1,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, 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("1050.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1050.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:	$61.9520055060$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +12.0000 q^{11} +12.0000 q^{12} +58.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -42.0000 q^{17} +18.0000 q^{18} -4.00000 q^{19} -21.0000 q^{21} +24.0000 q^{22} -24.0000 q^{23} +24.0000 q^{24} +116.000 q^{26} +27.0000 q^{27} -28.0000 q^{28} +294.000 q^{29} +128.000 q^{31} +32.0000 q^{32} +36.0000 q^{33} -84.0000 q^{34} +36.0000 q^{36} +58.0000 q^{37} -8.00000 q^{38} +174.000 q^{39} +282.000 q^{41} -42.0000 q^{42} -428.000 q^{43} +48.0000 q^{44} -48.0000 q^{46} -384.000 q^{47} +48.0000 q^{48} +49.0000 q^{49} -126.000 q^{51} +232.000 q^{52} +138.000 q^{53} +54.0000 q^{54} -56.0000 q^{56} -12.0000 q^{57} +588.000 q^{58} +468.000 q^{59} -250.000 q^{61} +256.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} +72.0000 q^{66} +556.000 q^{67} -168.000 q^{68} -72.0000 q^{69} +624.000 q^{71} +72.0000 q^{72} +958.000 q^{73} +116.000 q^{74} -16.0000 q^{76} -84.0000 q^{77} +348.000 q^{78} +632.000 q^{79} +81.0000 q^{81} +564.000 q^{82} -84.0000 q^{83} -84.0000 q^{84} -856.000 q^{86} +882.000 q^{87} +96.0000 q^{88} +810.000 q^{89} -406.000 q^{91} -96.0000 q^{92} +384.000 q^{93} -768.000 q^{94} +96.0000 q^{96} +790.000 q^{97} +98.0000 q^{98} +108.000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	12.0000	0.288675
$13$	58.0000	1.23741	0.618704	−	0.785624i	$-0.287658\pi$
$13$	58.0000	1.23741	0.618704	+	0.785624i	$0.287658\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−42.0000	−0.599206	−0.299603	−	0.954064i	$-0.596854\pi$
$17$	−42.0000	−0.599206	−0.299603	+	0.954064i	$0.596854\pi$
$18$	18.0000	0.235702
$19$	−4.00000	−0.0482980	−0.0241490	−	0.999708i	$-0.507688\pi$
$19$	−4.00000	−0.0482980	−0.0241490	+	0.999708i	$0.507688\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	24.0000	0.232583
$23$	−24.0000	−0.217580	−0.108790	−	0.994065i	$-0.534698\pi$
$23$	−24.0000	−0.217580	−0.108790	+	0.994065i	$0.534698\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	116.000	0.874980
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	294.000	1.88257	0.941283	−	0.337618i	$-0.109621\pi$
$29$	294.000	1.88257	0.941283	+	0.337618i	$0.109621\pi$
$30$	0	0
$31$	128.000	0.741596	0.370798	−	0.928714i	$-0.379084\pi$
$31$	128.000	0.741596	0.370798	+	0.928714i	$0.379084\pi$
$32$	32.0000	0.176777
$33$	36.0000	0.189903
$34$	−84.0000	−0.423702
$35$	0	0
$36$	36.0000	0.166667
$37$	58.0000	0.257707	0.128853	−	0.991664i	$-0.458870\pi$
$37$	58.0000	0.257707	0.128853	+	0.991664i	$0.458870\pi$
$38$	−8.00000	−0.0341519
$39$	174.000	0.714418
$40$	0	0
$41$	282.000	1.07417	0.537085	−	0.843528i	$-0.319525\pi$
$41$	282.000	1.07417	0.537085	+	0.843528i	$0.319525\pi$
$42$	−42.0000	−0.154303
$43$	−428.000	−1.51789	−0.758946	−	0.651153i	$-0.774286\pi$
$43$	−428.000	−1.51789	−0.758946	+	0.651153i	$0.774286\pi$
$44$	48.0000	0.164461
$45$	0	0
$46$	−48.0000	−0.153852
$47$	−384.000	−1.19175	−0.595874	−	0.803078i	$-0.703194\pi$
$47$	−384.000	−1.19175	−0.595874	+	0.803078i	$0.703194\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	0	0
$51$	−126.000	−0.345952
$52$	232.000	0.618704
$53$	138.000	0.357656	0.178828	−	0.983880i	$-0.442769\pi$
$53$	138.000	0.357656	0.178828	+	0.983880i	$0.442769\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	−56.0000	−0.133631
$57$	−12.0000	−0.0278849
$58$	588.000	1.33118
$59$	468.000	1.03268	0.516342	−	0.856382i	$-0.327293\pi$
$59$	468.000	1.03268	0.516342	+	0.856382i	$0.327293\pi$
$60$	0	0
$61$	−250.000	−0.524741	−0.262371	−	0.964967i	$-0.584504\pi$
$61$	−250.000	−0.524741	−0.262371	+	0.964967i	$0.584504\pi$
$62$	256.000	0.524388
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	0	0
$66$	72.0000	0.134282
$67$	556.000	1.01382	0.506912	−	0.861998i	$-0.330787\pi$
$67$	556.000	1.01382	0.506912	+	0.861998i	$0.330787\pi$
$68$	−168.000	−0.299603
$69$	−72.0000	−0.125620
$70$	0	0
$71$	624.000	1.04303	0.521515	−	0.853242i	$-0.325367\pi$
$71$	624.000	1.04303	0.521515	+	0.853242i	$0.325367\pi$
$72$	72.0000	0.117851
$73$	958.000	1.53596	0.767982	−	0.640471i	$-0.221261\pi$
$73$	958.000	1.53596	0.767982	+	0.640471i	$0.221261\pi$
$74$	116.000	0.182226
$75$	0	0
$76$	−16.0000	−0.0241490
$77$	−84.0000	−0.124321
$78$	348.000	0.505170
$79$	632.000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	632.000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	564.000	0.759553
$83$	−84.0000	−0.111087	−0.0555434	−	0.998456i	$-0.517689\pi$
$83$	−84.0000	−0.111087	−0.0555434	+	0.998456i	$0.517689\pi$
$84$	−84.0000	−0.109109
$85$	0	0
$86$	−856.000	−1.07331
$87$	882.000	1.08690
$88$	96.0000	0.116291
$89$	810.000	0.964717	0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000	0.964717	0.482359	+	0.875974i	$0.339780\pi$
$90$	0	0
$91$	−406.000	−0.467696
$92$	−96.0000	−0.108790
$93$	384.000	0.428161
$94$	−768.000	−0.842693
$95$	0	0
$96$	96.0000	0.102062
$97$	790.000	0.826931	0.413466	−	0.910520i	$-0.364318\pi$
$97$	790.000	0.826931	0.413466	+	0.910520i	$0.364318\pi$
$98$	98.0000	0.101015
$99$	108.000	0.109640
$100$	0	0
$101$	−1890.00	−1.86200	−0.931000	−	0.365019i	$-0.881063\pi$
$101$	−1890.00	−1.86200	−0.931000	+	0.365019i	$0.881063\pi$
$102$	−252.000	−0.244625
$103$	−296.000	−0.283163	−0.141581	−	0.989927i	$-0.545219\pi$
$103$	−296.000	−0.283163	−0.141581	+	0.989927i	$0.545219\pi$
$104$	464.000	0.437490
$105$	0	0
$106$	276.000	0.252901
$107$	324.000	0.292731	0.146366	−	0.989231i	$-0.453242\pi$
$107$	324.000	0.292731	0.146366	+	0.989231i	$0.453242\pi$
$108$	108.000	0.0962250
$109$	854.000	0.750444	0.375222	−	0.926935i	$-0.377567\pi$
$109$	854.000	0.750444	0.375222	+	0.926935i	$0.377567\pi$
$110$	0	0
$111$	174.000	0.148787
$112$	−112.000	−0.0944911
$113$	−402.000	−0.334664	−0.167332	−	0.985901i	$-0.553515\pi$
$113$	−402.000	−0.334664	−0.167332	+	0.985901i	$0.553515\pi$
$114$	−24.0000	−0.0197176
$115$	0	0
$116$	1176.00	0.941283
$117$	522.000	0.412469
$118$	936.000	0.730219
$119$	294.000	0.226478
$120$	0	0
$121$	−1187.00	−0.891811
$122$	−500.000	−0.371048
$123$	846.000	0.620173
$124$	512.000	0.370798
$125$	0	0
$126$	−126.000	−0.0890871
$127$	−2192.00	−1.53156	−0.765782	−	0.643101i	$-0.777648\pi$
$127$	−2192.00	−1.53156	−0.765782	+	0.643101i	$0.777648\pi$
$128$	128.000	0.0883883
$129$	−1284.00	−0.876356
$130$	0	0
$131$	−1572.00	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−1572.00	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	144.000	0.0949514
$133$	28.0000	0.0182549
$134$	1112.00	0.716882
$135$	0	0
$136$	−336.000	−0.211851
$137$	198.000	0.123477	0.0617383	−	0.998092i	$-0.480336\pi$
$137$	198.000	0.123477	0.0617383	+	0.998092i	$0.480336\pi$
$138$	−144.000	−0.0888268
$139$	1364.00	0.832324	0.416162	−	0.909291i	$-0.363375\pi$
$139$	1364.00	0.832324	0.416162	+	0.909291i	$0.363375\pi$
$140$	0	0
$141$	−1152.00	−0.688056
$142$	1248.00	0.737534
$143$	696.000	0.407010
$144$	144.000	0.0833333
$145$	0	0
$146$	1916.00	1.08609
$147$	147.000	0.0824786
$148$	232.000	0.128853
$149$	1470.00	0.808236	0.404118	−	0.914707i	$-0.367579\pi$
$149$	1470.00	0.808236	0.404118	+	0.914707i	$0.367579\pi$
$150$	0	0
$151$	−1888.00	−1.01751	−0.508753	−	0.860913i	$-0.669893\pi$
$151$	−1888.00	−1.01751	−0.508753	+	0.860913i	$0.669893\pi$
$152$	−32.0000	−0.0170759
$153$	−378.000	−0.199735
$154$	−168.000	−0.0879080
$155$	0	0
$156$	696.000	0.357209
$157$	106.000	0.0538836	0.0269418	−	0.999637i	$-0.491423\pi$
$157$	106.000	0.0538836	0.0269418	+	0.999637i	$0.491423\pi$
$158$	1264.00	0.636446
$159$	414.000	0.206493
$160$	0	0
$161$	168.000	0.0822376
$162$	162.000	0.0785674
$163$	−1172.00	−0.563179	−0.281589	−	0.959535i	$-0.590862\pi$
$163$	−1172.00	−0.563179	−0.281589	+	0.959535i	$0.590862\pi$
$164$	1128.00	0.537085
$165$	0	0
$166$	−168.000	−0.0785502
$167$	−2664.00	−1.23441	−0.617205	−	0.786802i	$-0.711735\pi$
$167$	−2664.00	−1.23441	−0.617205	+	0.786802i	$0.711735\pi$
$168$	−168.000	−0.0771517
$169$	1167.00	0.531179
$170$	0	0
$171$	−36.0000	−0.0160993
$172$	−1712.00	−0.758946
$173$	1914.00	0.841149	0.420574	−	0.907258i	$-0.361829\pi$
$173$	1914.00	0.841149	0.420574	+	0.907258i	$0.361829\pi$
$174$	1764.00	0.768555
$175$	0	0
$176$	192.000	0.0822304
$177$	1404.00	0.596221
$178$	1620.00	0.682158
$179$	3300.00	1.37795	0.688976	−	0.724784i	$-0.258060\pi$
$179$	3300.00	1.37795	0.688976	+	0.724784i	$0.258060\pi$
$180$	0	0
$181$	302.000	0.124019	0.0620096	−	0.998076i	$-0.480249\pi$
$181$	302.000	0.124019	0.0620096	+	0.998076i	$0.480249\pi$
$182$	−812.000	−0.330711
$183$	−750.000	−0.302960
$184$	−192.000	−0.0769262
$185$	0	0
$186$	768.000	0.302755
$187$	−504.000	−0.197092
$188$	−1536.00	−0.595874
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−4248.00	−1.60929	−0.804645	−	0.593756i	$-0.797645\pi$
$191$	−4248.00	−1.60929	−0.804645	+	0.593756i	$0.797645\pi$
$192$	192.000	0.0721688
$193$	−2354.00	−0.877951	−0.438976	−	0.898499i	$-0.644659\pi$
$193$	−2354.00	−0.877951	−0.438976	+	0.898499i	$0.644659\pi$
$194$	1580.00	0.584729
$195$	0	0
$196$	196.000	0.0714286
$197$	3114.00	1.12621	0.563105	−	0.826385i	$-0.309607\pi$
$197$	3114.00	1.12621	0.563105	+	0.826385i	$0.309607\pi$
$198$	216.000	0.0775275
$199$	−3256.00	−1.15986	−0.579929	−	0.814667i	$-0.696920\pi$
$199$	−3256.00	−1.15986	−0.579929	+	0.814667i	$0.696920\pi$
$200$	0	0
$201$	1668.00	0.585332
$202$	−3780.00	−1.31663
$203$	−2058.00	−0.711543
$204$	−504.000	−0.172976
$205$	0	0
$206$	−592.000	−0.200226
$207$	−216.000	−0.0725268
$208$	928.000	0.309352
$209$	−48.0000	−0.0158863
$210$	0	0
$211$	5780.00	1.88584	0.942919	−	0.333024i	$-0.108069\pi$
$211$	5780.00	1.88584	0.942919	+	0.333024i	$0.108069\pi$
$212$	552.000	0.178828
$213$	1872.00	0.602194
$214$	648.000	0.206992
$215$	0	0
$216$	216.000	0.0680414
$217$	−896.000	−0.280297
$218$	1708.00	0.530644
$219$	2874.00	0.886790
$220$	0	0
$221$	−2436.00	−0.741462
$222$	348.000	0.105208
$223$	1456.00	0.437224	0.218612	−	0.975812i	$-0.429847\pi$
$223$	1456.00	0.437224	0.218612	+	0.975812i	$0.429847\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−804.000	−0.236643
$227$	2604.00	0.761381	0.380691	−	0.924702i	$-0.375686\pi$
$227$	2604.00	0.761381	0.380691	+	0.924702i	$0.375686\pi$
$228$	−48.0000	−0.0139424
$229$	−4354.00	−1.25642	−0.628211	−	0.778043i	$-0.716212\pi$
$229$	−4354.00	−1.25642	−0.628211	+	0.778043i	$0.716212\pi$
$230$	0	0
$231$	−252.000	−0.0717765
$232$	2352.00	0.665588
$233$	2070.00	0.582018	0.291009	−	0.956720i	$-0.406009\pi$
$233$	2070.00	0.582018	0.291009	+	0.956720i	$0.406009\pi$
$234$	1044.00	0.291660
$235$	0	0
$236$	1872.00	0.516342
$237$	1896.00	0.519656
$238$	588.000	0.160144
$239$	744.000	0.201361	0.100681	−	0.994919i	$-0.467898\pi$
$239$	744.000	0.201361	0.100681	+	0.994919i	$0.467898\pi$
$240$	0	0
$241$	−1726.00	−0.461334	−0.230667	−	0.973033i	$-0.574091\pi$
$241$	−1726.00	−0.461334	−0.230667	+	0.973033i	$0.574091\pi$
$242$	−2374.00	−0.630605
$243$	243.000	0.0641500
$244$	−1000.00	−0.262371
$245$	0	0
$246$	1692.00	0.438528
$247$	−232.000	−0.0597644
$248$	1024.00	0.262194
$249$	−252.000	−0.0641359
$250$	0	0
$251$	−6060.00	−1.52392	−0.761960	−	0.647624i	$-0.775763\pi$
$251$	−6060.00	−1.52392	−0.761960	+	0.647624i	$0.775763\pi$
$252$	−252.000	−0.0629941
$253$	−288.000	−0.0715668
$254$	−4384.00	−1.08298
$255$	0	0
$256$	256.000	0.0625000
$257$	150.000	0.0364076	0.0182038	−	0.999834i	$-0.494205\pi$
$257$	150.000	0.0364076	0.0182038	+	0.999834i	$0.494205\pi$
$258$	−2568.00	−0.619677
$259$	−406.000	−0.0974039
$260$	0	0
$261$	2646.00	0.627522
$262$	−3144.00	−0.741362
$263$	3288.00	0.770900	0.385450	−	0.922729i	$-0.374046\pi$
$263$	3288.00	0.770900	0.385450	+	0.922729i	$0.374046\pi$
$264$	288.000	0.0671408
$265$	0	0
$266$	56.0000	0.0129082
$267$	2430.00	0.556980
$268$	2224.00	0.506912
$269$	−6378.00	−1.44563	−0.722813	−	0.691043i	$-0.757151\pi$
$269$	−6378.00	−1.44563	−0.722813	+	0.691043i	$0.757151\pi$
$270$	0	0
$271$	−1600.00	−0.358646	−0.179323	−	0.983790i	$-0.557391\pi$
$271$	−1600.00	−0.358646	−0.179323	+	0.983790i	$0.557391\pi$
$272$	−672.000	−0.149801
$273$	−1218.00	−0.270025
$274$	396.000	0.0873111
$275$	0	0
$276$	−288.000	−0.0628100
$277$	−4934.00	−1.07024	−0.535118	−	0.844777i	$-0.679733\pi$
$277$	−4934.00	−1.07024	−0.535118	+	0.844777i	$0.679733\pi$
$278$	2728.00	0.588542
$279$	1152.00	0.247199
$280$	0	0
$281$	−630.000	−0.133746	−0.0668730	−	0.997761i	$-0.521302\pi$
$281$	−630.000	−0.133746	−0.0668730	+	0.997761i	$0.521302\pi$
$282$	−2304.00	−0.486529
$283$	5380.00	1.13006	0.565031	−	0.825069i	$-0.308864\pi$
$283$	5380.00	1.13006	0.565031	+	0.825069i	$0.308864\pi$
$284$	2496.00	0.521515
$285$	0	0
$286$	1392.00	0.287800
$287$	−1974.00	−0.405998
$288$	288.000	0.0589256
$289$	−3149.00	−0.640953
$290$	0	0
$291$	2370.00	0.477429
$292$	3832.00	0.767982
$293$	−1470.00	−0.293100	−0.146550	−	0.989203i	$-0.546817\pi$
$293$	−1470.00	−0.293100	−0.146550	+	0.989203i	$0.546817\pi$
$294$	294.000	0.0583212
$295$	0	0
$296$	464.000	0.0911130
$297$	324.000	0.0633010
$298$	2940.00	0.571509
$299$	−1392.00	−0.269236
$300$	0	0
$301$	2996.00	0.573710
$302$	−3776.00	−0.719485
$303$	−5670.00	−1.07503
$304$	−64.0000	−0.0120745
$305$	0	0
$306$	−756.000	−0.141234
$307$	1036.00	0.192598	0.0962991	−	0.995352i	$-0.469299\pi$
$307$	1036.00	0.192598	0.0962991	+	0.995352i	$0.469299\pi$
$308$	−336.000	−0.0621603
$309$	−888.000	−0.163484
$310$	0	0
$311$	−216.000	−0.0393834	−0.0196917	−	0.999806i	$-0.506268\pi$
$311$	−216.000	−0.0393834	−0.0196917	+	0.999806i	$0.506268\pi$
$312$	1392.00	0.252585
$313$	1534.00	0.277019	0.138509	−	0.990361i	$-0.455769\pi$
$313$	1534.00	0.277019	0.138509	+	0.990361i	$0.455769\pi$
$314$	212.000	0.0381014
$315$	0	0
$316$	2528.00	0.450035
$317$	6546.00	1.15981	0.579905	−	0.814684i	$-0.303090\pi$
$317$	6546.00	1.15981	0.579905	+	0.814684i	$0.303090\pi$
$318$	828.000	0.146012
$319$	3528.00	0.619217
$320$	0	0
$321$	972.000	0.169009
$322$	336.000	0.0581508
$323$	168.000	0.0289405
$324$	324.000	0.0555556
$325$	0	0
$326$	−2344.00	−0.398227
$327$	2562.00	0.433269
$328$	2256.00	0.379777
$329$	2688.00	0.450438
$330$	0	0
$331$	92.0000	0.0152773	0.00763864	−	0.999971i	$-0.497569\pi$
$331$	92.0000	0.0152773	0.00763864	+	0.999971i	$0.497569\pi$
$332$	−336.000	−0.0555434
$333$	522.000	0.0859022
$334$	−5328.00	−0.872860
$335$	0	0
$336$	−336.000	−0.0545545
$337$	−8546.00	−1.38140	−0.690698	−	0.723144i	$-0.742696\pi$
$337$	−8546.00	−1.38140	−0.690698	+	0.723144i	$0.742696\pi$
$338$	2334.00	0.375600
$339$	−1206.00	−0.193218
$340$	0	0
$341$	1536.00	0.243927
$342$	−72.0000	−0.0113840
$343$	−343.000	−0.0539949
$344$	−3424.00	−0.536656
$345$	0	0
$346$	3828.00	0.594782
$347$	−6588.00	−1.01920	−0.509600	−	0.860411i	$-0.670207\pi$
$347$	−6588.00	−1.01920	−0.509600	+	0.860411i	$0.670207\pi$
$348$	3528.00	0.543450
$349$	−6874.00	−1.05432	−0.527159	−	0.849767i	$-0.676743\pi$
$349$	−6874.00	−1.05432	−0.527159	+	0.849767i	$0.676743\pi$
$350$	0	0
$351$	1566.00	0.238139
$352$	384.000	0.0581456
$353$	−7530.00	−1.13536	−0.567679	−	0.823250i	$-0.692159\pi$
$353$	−7530.00	−1.13536	−0.567679	+	0.823250i	$0.692159\pi$
$354$	2808.00	0.421592
$355$	0	0
$356$	3240.00	0.482359
$357$	882.000	0.130757
$358$	6600.00	0.974360
$359$	9264.00	1.36194	0.680968	−	0.732313i	$-0.261559\pi$
$359$	9264.00	1.36194	0.680968	+	0.732313i	$0.261559\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	604.000	0.0876948
$363$	−3561.00	−0.514887
$364$	−1624.00	−0.233848
$365$	0	0
$366$	−1500.00	−0.214225
$367$	−12944.0	−1.84107	−0.920533	−	0.390665i	$-0.872245\pi$
$367$	−12944.0	−1.84107	−0.920533	+	0.390665i	$0.872245\pi$
$368$	−384.000	−0.0543951
$369$	2538.00	0.358057
$370$	0	0
$371$	−966.000	−0.135181
$372$	1536.00	0.214080
$373$	−7862.00	−1.09136	−0.545682	−	0.837992i	$-0.683729\pi$
$373$	−7862.00	−1.09136	−0.545682	+	0.837992i	$0.683729\pi$
$374$	−1008.00	−0.139365
$375$	0	0
$376$	−3072.00	−0.421347
$377$	17052.0	2.32950
$378$	−378.000	−0.0514344
$379$	−2980.00	−0.403885	−0.201942	−	0.979397i	$-0.564725\pi$
$379$	−2980.00	−0.403885	−0.201942	+	0.979397i	$0.564725\pi$
$380$	0	0
$381$	−6576.00	−0.884249
$382$	−8496.00	−1.13794
$383$	−9840.00	−1.31280	−0.656398	−	0.754415i	$-0.727921\pi$
$383$	−9840.00	−1.31280	−0.656398	+	0.754415i	$0.727921\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−4708.00	−0.620805
$387$	−3852.00	−0.505964
$388$	3160.00	0.413466
$389$	6318.00	0.823484	0.411742	−	0.911300i	$-0.364920\pi$
$389$	6318.00	0.823484	0.411742	+	0.911300i	$0.364920\pi$
$390$	0	0
$391$	1008.00	0.130375
$392$	392.000	0.0505076
$393$	−4716.00	−0.605320
$394$	6228.00	0.796351
$395$	0	0
$396$	432.000	0.0548202
$397$	10906.0	1.37873	0.689366	−	0.724413i	$-0.257889\pi$
$397$	10906.0	1.37873	0.689366	+	0.724413i	$0.257889\pi$
$398$	−6512.00	−0.820143
$399$	84.0000	0.0105395
$400$	0	0
$401$	−9294.00	−1.15741	−0.578704	−	0.815538i	$-0.696441\pi$
$401$	−9294.00	−1.15741	−0.578704	+	0.815538i	$0.696441\pi$
$402$	3336.00	0.413892
$403$	7424.00	0.917657
$404$	−7560.00	−0.931000
$405$	0	0
$406$	−4116.00	−0.503137
$407$	696.000	0.0847652
$408$	−1008.00	−0.122312
$409$	−6166.00	−0.745450	−0.372725	−	0.927942i	$-0.621577\pi$
$409$	−6166.00	−0.745450	−0.372725	+	0.927942i	$0.621577\pi$
$410$	0	0
$411$	594.000	0.0712892
$412$	−1184.00	−0.141581
$413$	−3276.00	−0.390318
$414$	−432.000	−0.0512842
$415$	0	0
$416$	1856.00	0.218745
$417$	4092.00	0.480542
$418$	−96.0000	−0.0112333
$419$	−10740.0	−1.25223	−0.626114	−	0.779732i	$-0.715355\pi$
$419$	−10740.0	−1.25223	−0.626114	+	0.779732i	$0.715355\pi$
$420$	0	0
$421$	13454.0	1.55750	0.778750	−	0.627334i	$-0.215854\pi$
$421$	13454.0	1.55750	0.778750	+	0.627334i	$0.215854\pi$
$422$	11560.0	1.33349
$423$	−3456.00	−0.397249
$424$	1104.00	0.126450
$425$	0	0
$426$	3744.00	0.425815
$427$	1750.00	0.198334
$428$	1296.00	0.146366
$429$	2088.00	0.234987
$430$	0	0
$431$	10152.0	1.13458	0.567291	−	0.823518i	$-0.307992\pi$
$431$	10152.0	1.13458	0.567291	+	0.823518i	$0.307992\pi$
$432$	432.000	0.0481125
$433$	−15818.0	−1.75558	−0.877788	−	0.479049i	$-0.840982\pi$
$433$	−15818.0	−1.75558	−0.877788	+	0.479049i	$0.840982\pi$
$434$	−1792.00	−0.198200
$435$	0	0
$436$	3416.00	0.375222
$437$	96.0000	0.0105087
$438$	5748.00	0.627055
$439$	−7432.00	−0.807995	−0.403998	−	0.914760i	$-0.632380\pi$
$439$	−7432.00	−0.807995	−0.403998	+	0.914760i	$0.632380\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	−4872.00	−0.524293
$443$	−5916.00	−0.634487	−0.317243	−	0.948344i	$-0.602757\pi$
$443$	−5916.00	−0.634487	−0.317243	+	0.948344i	$0.602757\pi$
$444$	696.000	0.0743935
$445$	0	0
$446$	2912.00	0.309164
$447$	4410.00	0.466635
$448$	−448.000	−0.0472456
$449$	2466.00	0.259193	0.129597	−	0.991567i	$-0.458632\pi$
$449$	2466.00	0.259193	0.129597	+	0.991567i	$0.458632\pi$
$450$	0	0
$451$	3384.00	0.353318
$452$	−1608.00	−0.167332
$453$	−5664.00	−0.587457
$454$	5208.00	0.538378
$455$	0	0
$456$	−96.0000	−0.00985880
$457$	−13994.0	−1.43241	−0.716205	−	0.697890i	$-0.754123\pi$
$457$	−13994.0	−1.43241	−0.716205	+	0.697890i	$0.754123\pi$
$458$	−8708.00	−0.888424
$459$	−1134.00	−0.115317
$460$	0	0
$461$	−3978.00	−0.401896	−0.200948	−	0.979602i	$-0.564402\pi$
$461$	−3978.00	−0.401896	−0.200948	+	0.979602i	$0.564402\pi$
$462$	−504.000	−0.0507537
$463$	−10352.0	−1.03909	−0.519545	−	0.854443i	$-0.673898\pi$
$463$	−10352.0	−1.03909	−0.519545	+	0.854443i	$0.673898\pi$
$464$	4704.00	0.470642
$465$	0	0
$466$	4140.00	0.411549
$467$	−1428.00	−0.141499	−0.0707494	−	0.997494i	$-0.522539\pi$
$467$	−1428.00	−0.141499	−0.0707494	+	0.997494i	$0.522539\pi$
$468$	2088.00	0.206235
$469$	−3892.00	−0.383189
$470$	0	0
$471$	318.000	0.0311097
$472$	3744.00	0.365109
$473$	−5136.00	−0.499268
$474$	3792.00	0.367452
$475$	0	0
$476$	1176.00	0.113239
$477$	1242.00	0.119219
$478$	1488.00	0.142384
$479$	8832.00	0.842473	0.421236	−	0.906951i	$-0.361596\pi$
$479$	8832.00	0.842473	0.421236	+	0.906951i	$0.361596\pi$
$480$	0	0
$481$	3364.00	0.318888
$482$	−3452.00	−0.326212
$483$	504.000	0.0474799
$484$	−4748.00	−0.445905
$485$	0	0
$486$	486.000	0.0453609
$487$	9064.00	0.843386	0.421693	−	0.906739i	$-0.361436\pi$
$487$	9064.00	0.843386	0.421693	+	0.906739i	$0.361436\pi$
$488$	−2000.00	−0.185524
$489$	−3516.00	−0.325151
$490$	0	0
$491$	12780.0	1.17465	0.587325	−	0.809351i	$-0.300181\pi$
$491$	12780.0	1.17465	0.587325	+	0.809351i	$0.300181\pi$
$492$	3384.00	0.310086
$493$	−12348.0	−1.12804
$494$	−464.000	−0.0422598
$495$	0	0
$496$	2048.00	0.185399
$497$	−4368.00	−0.394229
$498$	−504.000	−0.0453510
$499$	−15628.0	−1.40201	−0.701007	−	0.713154i	$-0.747266\pi$
$499$	−15628.0	−1.40201	−0.701007	+	0.713154i	$0.747266\pi$
$500$	0	0
$501$	−7992.00	−0.712687
$502$	−12120.0	−1.07757
$503$	−3096.00	−0.274441	−0.137220	−	0.990541i	$-0.543817\pi$
$503$	−3096.00	−0.274441	−0.137220	+	0.990541i	$0.543817\pi$
$504$	−504.000	−0.0445435
$505$	0	0
$506$	−576.000	−0.0506054
$507$	3501.00	0.306676
$508$	−8768.00	−0.765782
$509$	17190.0	1.49692	0.748461	−	0.663179i	$-0.230793\pi$
$509$	17190.0	1.49692	0.748461	+	0.663179i	$0.230793\pi$
$510$	0	0
$511$	−6706.00	−0.580540
$512$	512.000	0.0441942
$513$	−108.000	−0.00929496
$514$	300.000	0.0257440
$515$	0	0
$516$	−5136.00	−0.438178
$517$	−4608.00	−0.391992
$518$	−812.000	−0.0688750
$519$	5742.00	0.485637
$520$	0	0
$521$	17034.0	1.43239	0.716193	−	0.697902i	$-0.245883\pi$
$521$	17034.0	1.43239	0.716193	+	0.697902i	$0.245883\pi$
$522$	5292.00	0.443725
$523$	10516.0	0.879221	0.439610	−	0.898189i	$-0.355116\pi$
$523$	10516.0	0.879221	0.439610	+	0.898189i	$0.355116\pi$
$524$	−6288.00	−0.524222
$525$	0	0
$526$	6576.00	0.545109
$527$	−5376.00	−0.444369
$528$	576.000	0.0474757
$529$	−11591.0	−0.952659
$530$	0	0
$531$	4212.00	0.344228
$532$	112.000	0.00912747
$533$	16356.0	1.32919
$534$	4860.00	0.393844
$535$	0	0
$536$	4448.00	0.358441
$537$	9900.00	0.795562
$538$	−12756.0	−1.02221
$539$	588.000	0.0469888
$540$	0	0
$541$	−538.000	−0.0427549	−0.0213775	−	0.999771i	$-0.506805\pi$
$541$	−538.000	−0.0427549	−0.0213775	+	0.999771i	$0.506805\pi$
$542$	−3200.00	−0.253601
$543$	906.000	0.0716025
$544$	−1344.00	−0.105926
$545$	0	0
$546$	−2436.00	−0.190936
$547$	1420.00	0.110996	0.0554980	−	0.998459i	$-0.482325\pi$
$547$	1420.00	0.110996	0.0554980	+	0.998459i	$0.482325\pi$
$548$	792.000	0.0617383
$549$	−2250.00	−0.174914
$550$	0	0
$551$	−1176.00	−0.0909243
$552$	−576.000	−0.0444134
$553$	−4424.00	−0.340195
$554$	−9868.00	−0.756771
$555$	0	0
$556$	5456.00	0.416162
$557$	17106.0	1.30126	0.650632	−	0.759393i	$-0.274504\pi$
$557$	17106.0	1.30126	0.650632	+	0.759393i	$0.274504\pi$
$558$	2304.00	0.174796
$559$	−24824.0	−1.87825
$560$	0	0
$561$	−1512.00	−0.113791
$562$	−1260.00	−0.0945728
$563$	−3252.00	−0.243438	−0.121719	−	0.992565i	$-0.538841\pi$
$563$	−3252.00	−0.243438	−0.121719	+	0.992565i	$0.538841\pi$
$564$	−4608.00	−0.344028
$565$	0	0
$566$	10760.0	0.799075
$567$	−567.000	−0.0419961
$568$	4992.00	0.368767
$569$	−7254.00	−0.534453	−0.267226	−	0.963634i	$-0.586107\pi$
$569$	−7254.00	−0.534453	−0.267226	+	0.963634i	$0.586107\pi$
$570$	0	0
$571$	−24676.0	−1.80851	−0.904254	−	0.426994i	$-0.859572\pi$
$571$	−24676.0	−1.80851	−0.904254	+	0.426994i	$0.859572\pi$
$572$	2784.00	0.203505
$573$	−12744.0	−0.929124
$574$	−3948.00	−0.287084
$575$	0	0
$576$	576.000	0.0416667
$577$	−5162.00	−0.372438	−0.186219	−	0.982508i	$-0.559623\pi$
$577$	−5162.00	−0.372438	−0.186219	+	0.982508i	$0.559623\pi$
$578$	−6298.00	−0.453222
$579$	−7062.00	−0.506885
$580$	0	0
$581$	588.000	0.0419868
$582$	4740.00	0.337593
$583$	1656.00	0.117641
$584$	7664.00	0.543046
$585$	0	0
$586$	−2940.00	−0.207253
$587$	17556.0	1.23444	0.617218	−	0.786792i	$-0.288260\pi$
$587$	17556.0	1.23444	0.617218	+	0.786792i	$0.288260\pi$
$588$	588.000	0.0412393
$589$	−512.000	−0.0358176
$590$	0	0
$591$	9342.00	0.650217
$592$	928.000	0.0644266
$593$	−23274.0	−1.61172	−0.805859	−	0.592108i	$-0.798296\pi$
$593$	−23274.0	−1.61172	−0.805859	+	0.592108i	$0.798296\pi$
$594$	648.000	0.0447605
$595$	0	0
$596$	5880.00	0.404118
$597$	−9768.00	−0.669644
$598$	−2784.00	−0.190378
$599$	12000.0	0.818542	0.409271	−	0.912413i	$-0.365783\pi$
$599$	12000.0	0.818542	0.409271	+	0.912413i	$0.365783\pi$
$600$	0	0
$601$	−24742.0	−1.67928	−0.839640	−	0.543143i	$-0.817234\pi$
$601$	−24742.0	−1.67928	−0.839640	+	0.543143i	$0.817234\pi$
$602$	5992.00	0.405674
$603$	5004.00	0.337941
$604$	−7552.00	−0.508753
$605$	0	0
$606$	−11340.0	−0.760158
$607$	4096.00	0.273890	0.136945	−	0.990579i	$-0.456272\pi$
$607$	4096.00	0.273890	0.136945	+	0.990579i	$0.456272\pi$
$608$	−128.000	−0.00853797
$609$	−6174.00	−0.410810
$610$	0	0
$611$	−22272.0	−1.47468
$612$	−1512.00	−0.0998676
$613$	28474.0	1.87611	0.938054	−	0.346489i	$-0.112626\pi$
$613$	28474.0	1.87611	0.938054	+	0.346489i	$0.112626\pi$
$614$	2072.00	0.136187
$615$	0	0
$616$	−672.000	−0.0439540
$617$	5238.00	0.341773	0.170886	−	0.985291i	$-0.445337\pi$
$617$	5238.00	0.341773	0.170886	+	0.985291i	$0.445337\pi$
$618$	−1776.00	−0.115601
$619$	−21388.0	−1.38878	−0.694391	−	0.719598i	$-0.744326\pi$
$619$	−21388.0	−1.38878	−0.694391	+	0.719598i	$0.744326\pi$
$620$	0	0
$621$	−648.000	−0.0418733
$622$	−432.000	−0.0278483
$623$	−5670.00	−0.364629
$624$	2784.00	0.178604
$625$	0	0
$626$	3068.00	0.195882
$627$	−144.000	−0.00917194
$628$	424.000	0.0269418
$629$	−2436.00	−0.154419
$630$	0	0
$631$	−18304.0	−1.15479	−0.577394	−	0.816466i	$-0.695930\pi$
$631$	−18304.0	−1.15479	−0.577394	+	0.816466i	$0.695930\pi$
$632$	5056.00	0.318223
$633$	17340.0	1.08879
$634$	13092.0	0.820110
$635$	0	0
$636$	1656.00	0.103246
$637$	2842.00	0.176773
$638$	7056.00	0.437852
$639$	5616.00	0.347677
$640$	0	0
$641$	5634.00	0.347160	0.173580	−	0.984820i	$-0.444466\pi$
$641$	5634.00	0.347160	0.173580	+	0.984820i	$0.444466\pi$
$642$	1944.00	0.119507
$643$	8476.00	0.519846	0.259923	−	0.965629i	$-0.416303\pi$
$643$	8476.00	0.519846	0.259923	+	0.965629i	$0.416303\pi$
$644$	672.000	0.0411188
$645$	0	0
$646$	336.000	0.0204640
$647$	−22728.0	−1.38104	−0.690518	−	0.723316i	$-0.742617\pi$
$647$	−22728.0	−1.38104	−0.690518	+	0.723316i	$0.742617\pi$
$648$	648.000	0.0392837
$649$	5616.00	0.339672
$650$	0	0
$651$	−2688.00	−0.161830
$652$	−4688.00	−0.281589
$653$	11682.0	0.700080	0.350040	−	0.936735i	$-0.386168\pi$
$653$	11682.0	0.700080	0.350040	+	0.936735i	$0.386168\pi$
$654$	5124.00	0.306367
$655$	0	0
$656$	4512.00	0.268543
$657$	8622.00	0.511988
$658$	5376.00	0.318508
$659$	−19644.0	−1.16119	−0.580593	−	0.814194i	$-0.697179\pi$
$659$	−19644.0	−1.16119	−0.580593	+	0.814194i	$0.697179\pi$
$660$	0	0
$661$	2318.00	0.136399	0.0681995	−	0.997672i	$-0.478275\pi$
$661$	2318.00	0.136399	0.0681995	+	0.997672i	$0.478275\pi$
$662$	184.000	0.0108027
$663$	−7308.00	−0.428083
$664$	−672.000	−0.0392751
$665$	0	0
$666$	1044.00	0.0607420
$667$	−7056.00	−0.409609
$668$	−10656.0	−0.617205
$669$	4368.00	0.252431
$670$	0	0
$671$	−3000.00	−0.172599
$672$	−672.000	−0.0385758
$673$	29086.0	1.66595	0.832974	−	0.553312i	$-0.186636\pi$
$673$	29086.0	1.66595	0.832974	+	0.553312i	$0.186636\pi$
$674$	−17092.0	−0.976794
$675$	0	0
$676$	4668.00	0.265589
$677$	−15630.0	−0.887311	−0.443656	−	0.896197i	$-0.646319\pi$
$677$	−15630.0	−0.887311	−0.443656	+	0.896197i	$0.646319\pi$
$678$	−2412.00	−0.136626
$679$	−5530.00	−0.312551
$680$	0	0
$681$	7812.00	0.439584
$682$	3072.00	0.172482
$683$	20292.0	1.13683	0.568413	−	0.822744i	$-0.307558\pi$
$683$	20292.0	1.13683	0.568413	+	0.822744i	$0.307558\pi$
$684$	−144.000	−0.00804967
$685$	0	0
$686$	−686.000	−0.0381802
$687$	−13062.0	−0.725395
$688$	−6848.00	−0.379473
$689$	8004.00	0.442566
$690$	0	0
$691$	−532.000	−0.0292883	−0.0146442	−	0.999893i	$-0.504662\pi$
$691$	−532.000	−0.0292883	−0.0146442	+	0.999893i	$0.504662\pi$
$692$	7656.00	0.420574
$693$	−756.000	−0.0414402
$694$	−13176.0	−0.720683
$695$	0	0
$696$	7056.00	0.384277
$697$	−11844.0	−0.643649
$698$	−13748.0	−0.745515
$699$	6210.00	0.336028
$700$	0	0
$701$	24486.0	1.31929	0.659646	−	0.751577i	$-0.270706\pi$
$701$	24486.0	1.31929	0.659646	+	0.751577i	$0.270706\pi$
$702$	3132.00	0.168390
$703$	−232.000	−0.0124467
$704$	768.000	0.0411152
$705$	0	0
$706$	−15060.0	−0.802820
$707$	13230.0	0.703770
$708$	5616.00	0.298110
$709$	30926.0	1.63815	0.819076	−	0.573684i	$-0.194486\pi$
$709$	30926.0	1.63815	0.819076	+	0.573684i	$0.194486\pi$
$710$	0	0
$711$	5688.00	0.300023
$712$	6480.00	0.341079
$713$	−3072.00	−0.161357
$714$	1764.00	0.0924594
$715$	0	0
$716$	13200.0	0.688976
$717$	2232.00	0.116256
$718$	18528.0	0.963035
$719$	27504.0	1.42660	0.713300	−	0.700858i	$-0.247199\pi$
$719$	27504.0	1.42660	0.713300	+	0.700858i	$0.247199\pi$
$720$	0	0
$721$	2072.00	0.107025
$722$	−13686.0	−0.705457
$723$	−5178.00	−0.266351
$724$	1208.00	0.0620096
$725$	0	0
$726$	−7122.00	−0.364080
$727$	−29192.0	−1.48923	−0.744616	−	0.667493i	$-0.767367\pi$
$727$	−29192.0	−1.48923	−0.744616	+	0.667493i	$0.767367\pi$
$728$	−3248.00	−0.165356
$729$	729.000	0.0370370
$730$	0	0
$731$	17976.0	0.909530
$732$	−3000.00	−0.151480
$733$	−22118.0	−1.11453	−0.557263	−	0.830336i	$-0.688148\pi$
$733$	−22118.0	−1.11453	−0.557263	+	0.830336i	$0.688148\pi$
$734$	−25888.0	−1.30183
$735$	0	0
$736$	−768.000	−0.0384631
$737$	6672.00	0.333468
$738$	5076.00	0.253184
$739$	−23836.0	−1.18650	−0.593249	−	0.805019i	$-0.702155\pi$
$739$	−23836.0	−1.18650	−0.593249	+	0.805019i	$0.702155\pi$
$740$	0	0
$741$	−696.000	−0.0345050
$742$	−1932.00	−0.0955875
$743$	33576.0	1.65785	0.828926	−	0.559358i	$-0.188952\pi$
$743$	33576.0	1.65785	0.828926	+	0.559358i	$0.188952\pi$
$744$	3072.00	0.151378
$745$	0	0
$746$	−15724.0	−0.771711
$747$	−756.000	−0.0370289
$748$	−2016.00	−0.0985458
$749$	−2268.00	−0.110642
$750$	0	0
$751$	22040.0	1.07091	0.535453	−	0.844565i	$-0.320141\pi$
$751$	22040.0	1.07091	0.535453	+	0.844565i	$0.320141\pi$
$752$	−6144.00	−0.297937
$753$	−18180.0	−0.879835
$754$	34104.0	1.64721
$755$	0	0
$756$	−756.000	−0.0363696
$757$	−2774.00	−0.133187	−0.0665936	−	0.997780i	$-0.521213\pi$
$757$	−2774.00	−0.133187	−0.0665936	+	0.997780i	$0.521213\pi$
$758$	−5960.00	−0.285590
$759$	−864.000	−0.0413191
$760$	0	0
$761$	9594.00	0.457007	0.228503	−	0.973543i	$-0.426617\pi$
$761$	9594.00	0.457007	0.228503	+	0.973543i	$0.426617\pi$
$762$	−13152.0	−0.625258
$763$	−5978.00	−0.283641
$764$	−16992.0	−0.804645
$765$	0	0
$766$	−19680.0	−0.928286
$767$	27144.0	1.27785
$768$	768.000	0.0360844
$769$	29330.0	1.37538	0.687690	−	0.726005i	$-0.258625\pi$
$769$	29330.0	1.37538	0.687690	+	0.726005i	$0.258625\pi$
$770$	0	0
$771$	450.000	0.0210199
$772$	−9416.00	−0.438976
$773$	33090.0	1.53967	0.769835	−	0.638243i	$-0.220339\pi$
$773$	33090.0	1.53967	0.769835	+	0.638243i	$0.220339\pi$
$774$	−7704.00	−0.357771
$775$	0	0
$776$	6320.00	0.292364
$777$	−1218.00	−0.0562362
$778$	12636.0	0.582291
$779$	−1128.00	−0.0518804
$780$	0	0
$781$	7488.00	0.343075
$782$	2016.00	0.0921893
$783$	7938.00	0.362300
$784$	784.000	0.0357143
$785$	0	0
$786$	−9432.00	−0.428026
$787$	−32564.0	−1.47494	−0.737472	−	0.675377i	$-0.763981\pi$
$787$	−32564.0	−1.47494	−0.737472	+	0.675377i	$0.763981\pi$
$788$	12456.0	0.563105
$789$	9864.00	0.445079
$790$	0	0
$791$	2814.00	0.126491
$792$	864.000	0.0387638
$793$	−14500.0	−0.649319
$794$	21812.0	0.974910
$795$	0	0
$796$	−13024.0	−0.579929
$797$	−14646.0	−0.650926	−0.325463	−	0.945555i	$-0.605520\pi$
$797$	−14646.0	−0.650926	−0.325463	+	0.945555i	$0.605520\pi$
$798$	168.000	0.00745255
$799$	16128.0	0.714102
$800$	0	0
$801$	7290.00	0.321572
$802$	−18588.0	−0.818410
$803$	11496.0	0.505212
$804$	6672.00	0.292666
$805$	0	0
$806$	14848.0	0.648882
$807$	−19134.0	−0.834633
$808$	−15120.0	−0.658317
$809$	41850.0	1.81875	0.909374	−	0.415979i	$-0.136561\pi$
$809$	41850.0	1.81875	0.909374	+	0.415979i	$0.136561\pi$
$810$	0	0
$811$	−39004.0	−1.68880	−0.844399	−	0.535714i	$-0.820042\pi$
$811$	−39004.0	−1.68880	−0.844399	+	0.535714i	$0.820042\pi$
$812$	−8232.00	−0.355772
$813$	−4800.00	−0.207064
$814$	1392.00	0.0599381
$815$	0	0
$816$	−2016.00	−0.0864879
$817$	1712.00	0.0733113
$818$	−12332.0	−0.527113
$819$	−3654.00	−0.155899
$820$	0	0
$821$	−13458.0	−0.572092	−0.286046	−	0.958216i	$-0.592341\pi$
$821$	−13458.0	−0.572092	−0.286046	+	0.958216i	$0.592341\pi$
$822$	1188.00	0.0504091
$823$	40696.0	1.72366	0.861831	−	0.507196i	$-0.169318\pi$
$823$	40696.0	1.72366	0.861831	+	0.507196i	$0.169318\pi$
$824$	−2368.00	−0.100113
$825$	0	0
$826$	−6552.00	−0.275997
$827$	−22140.0	−0.930935	−0.465467	−	0.885065i	$-0.654114\pi$
$827$	−22140.0	−0.930935	−0.465467	+	0.885065i	$0.654114\pi$
$828$	−864.000	−0.0362634
$829$	−20698.0	−0.867155	−0.433577	−	0.901116i	$-0.642749\pi$
$829$	−20698.0	−0.867155	−0.433577	+	0.901116i	$0.642749\pi$
$830$	0	0
$831$	−14802.0	−0.617901
$832$	3712.00	0.154676
$833$	−2058.00	−0.0856008
$834$	8184.00	0.339795
$835$	0	0
$836$	−192.000	−0.00794313
$837$	3456.00	0.142720
$838$	−21480.0	−0.885459
$839$	−15480.0	−0.636983	−0.318492	−	0.947926i	$-0.603176\pi$
$839$	−15480.0	−0.636983	−0.318492	+	0.947926i	$0.603176\pi$
$840$	0	0
$841$	62047.0	2.54406
$842$	26908.0	1.10132
$843$	−1890.00	−0.0772183
$844$	23120.0	0.942919
$845$	0	0
$846$	−6912.00	−0.280898
$847$	8309.00	0.337073
$848$	2208.00	0.0894140
$849$	16140.0	0.652442
$850$	0	0
$851$	−1392.00	−0.0560719
$852$	7488.00	0.301097
$853$	−47486.0	−1.90608	−0.953042	−	0.302838i	$-0.902066\pi$
$853$	−47486.0	−1.90608	−0.953042	+	0.302838i	$0.902066\pi$
$854$	3500.00	0.140243
$855$	0	0
$856$	2592.00	0.103496
$857$	15438.0	0.615347	0.307673	−	0.951492i	$-0.400450\pi$
$857$	15438.0	0.615347	0.307673	+	0.951492i	$0.400450\pi$
$858$	4176.00	0.166161
$859$	−15100.0	−0.599773	−0.299887	−	0.953975i	$-0.596949\pi$
$859$	−15100.0	−0.599773	−0.299887	+	0.953975i	$0.596949\pi$
$860$	0	0
$861$	−5922.00	−0.234403
$862$	20304.0	0.802270
$863$	7056.00	0.278319	0.139159	−	0.990270i	$-0.455560\pi$
$863$	7056.00	0.278319	0.139159	+	0.990270i	$0.455560\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	−31636.0	−1.24138
$867$	−9447.00	−0.370054
$868$	−3584.00	−0.140148
$869$	7584.00	0.296052
$870$	0	0
$871$	32248.0	1.25451
$872$	6832.00	0.265322
$873$	7110.00	0.275644
$874$	192.000	0.00743077
$875$	0	0
$876$	11496.0	0.443395
$877$	−23726.0	−0.913535	−0.456767	−	0.889586i	$-0.650993\pi$
$877$	−23726.0	−0.913535	−0.456767	+	0.889586i	$0.650993\pi$
$878$	−14864.0	−0.571339
$879$	−4410.00	−0.169221
$880$	0	0
$881$	−38190.0	−1.46045	−0.730223	−	0.683208i	$-0.760584\pi$
$881$	−38190.0	−1.46045	−0.730223	+	0.683208i	$0.760584\pi$
$882$	882.000	0.0336718
$883$	46732.0	1.78104	0.890519	−	0.454946i	$-0.150341\pi$
$883$	46732.0	1.78104	0.890519	+	0.454946i	$0.150341\pi$
$884$	−9744.00	−0.370731
$885$	0	0
$886$	−11832.0	−0.448650
$887$	−13800.0	−0.522389	−0.261194	−	0.965286i	$-0.584116\pi$
$887$	−13800.0	−0.522389	−0.261194	+	0.965286i	$0.584116\pi$
$888$	1392.00	0.0526041
$889$	15344.0	0.578877
$890$	0	0
$891$	972.000	0.0365468
$892$	5824.00	0.218612
$893$	1536.00	0.0575591
$894$	8820.00	0.329961
$895$	0	0
$896$	−896.000	−0.0334077
$897$	−4176.00	−0.155443
$898$	4932.00	0.183277
$899$	37632.0	1.39610
$900$	0	0
$901$	−5796.00	−0.214309
$902$	6768.00	0.249833
$903$	8988.00	0.331231
$904$	−3216.00	−0.118321
$905$	0	0
$906$	−11328.0	−0.415395
$907$	5668.00	0.207500	0.103750	−	0.994603i	$-0.466916\pi$
$907$	5668.00	0.207500	0.103750	+	0.994603i	$0.466916\pi$
$908$	10416.0	0.380691
$909$	−17010.0	−0.620667
$910$	0	0
$911$	−1464.00	−0.0532431	−0.0266216	−	0.999646i	$-0.508475\pi$
$911$	−1464.00	−0.0532431	−0.0266216	+	0.999646i	$0.508475\pi$
$912$	−192.000	−0.00697122
$913$	−1008.00	−0.0365388
$914$	−27988.0	−1.01287
$915$	0	0
$916$	−17416.0	−0.628211
$917$	11004.0	0.396275
$918$	−2268.00	−0.0815416
$919$	−28000.0	−1.00504	−0.502522	−	0.864565i	$-0.667594\pi$
$919$	−28000.0	−1.00504	−0.502522	+	0.864565i	$0.667594\pi$
$920$	0	0
$921$	3108.00	0.111197
$922$	−7956.00	−0.284183
$923$	36192.0	1.29065
$924$	−1008.00	−0.0358883
$925$	0	0
$926$	−20704.0	−0.734747
$927$	−2664.00	−0.0943875
$928$	9408.00	0.332794
$929$	−45870.0	−1.61996	−0.809982	−	0.586455i	$-0.800523\pi$
$929$	−45870.0	−1.61996	−0.809982	+	0.586455i	$0.800523\pi$
$930$	0	0
$931$	−196.000	−0.00689972
$932$	8280.00	0.291009
$933$	−648.000	−0.0227380
$934$	−2856.00	−0.100055
$935$	0	0
$936$	4176.00	0.145830
$937$	−7058.00	−0.246078	−0.123039	−	0.992402i	$-0.539264\pi$
$937$	−7058.00	−0.246078	−0.123039	+	0.992402i	$0.539264\pi$
$938$	−7784.00	−0.270956
$939$	4602.00	0.159937
$940$	0	0
$941$	35670.0	1.23572	0.617858	−	0.786290i	$-0.288001\pi$
$941$	35670.0	1.23572	0.617858	+	0.786290i	$0.288001\pi$
$942$	636.000	0.0219979
$943$	−6768.00	−0.233718
$944$	7488.00	0.258171
$945$	0	0
$946$	−10272.0	−0.353035
$947$	15996.0	0.548891	0.274446	−	0.961603i	$-0.411506\pi$
$947$	15996.0	0.548891	0.274446	+	0.961603i	$0.411506\pi$
$948$	7584.00	0.259828
$949$	55564.0	1.90062
$950$	0	0
$951$	19638.0	0.669617
$952$	2352.00	0.0800722
$953$	31110.0	1.05745	0.528726	−	0.848793i	$-0.322670\pi$
$953$	31110.0	1.05745	0.528726	+	0.848793i	$0.322670\pi$
$954$	2484.00	0.0843003
$955$	0	0
$956$	2976.00	0.100681
$957$	10584.0	0.357505
$958$	17664.0	0.595718
$959$	−1386.00	−0.0466697
$960$	0	0
$961$	−13407.0	−0.450035
$962$	6728.00	0.225488
$963$	2916.00	0.0975771
$964$	−6904.00	−0.230667
$965$	0	0
$966$	1008.00	0.0335734
$967$	−5768.00	−0.191816	−0.0959082	−	0.995390i	$-0.530576\pi$
$967$	−5768.00	−0.191816	−0.0959082	+	0.995390i	$0.530576\pi$
$968$	−9496.00	−0.315303
$969$	504.000	0.0167088
$970$	0	0
$971$	−39324.0	−1.29966	−0.649829	−	0.760081i	$-0.725159\pi$
$971$	−39324.0	−1.29966	−0.649829	+	0.760081i	$0.725159\pi$
$972$	972.000	0.0320750
$973$	−9548.00	−0.314589
$974$	18128.0	0.596364
$975$	0	0
$976$	−4000.00	−0.131185
$977$	8334.00	0.272905	0.136453	−	0.990647i	$-0.456430\pi$
$977$	8334.00	0.272905	0.136453	+	0.990647i	$0.456430\pi$
$978$	−7032.00	−0.229917
$979$	9720.00	0.317316
$980$	0	0
$981$	7686.00	0.250148
$982$	25560.0	0.830603
$983$	17352.0	0.563014	0.281507	−	0.959559i	$-0.409166\pi$
$983$	17352.0	0.563014	0.281507	+	0.959559i	$0.409166\pi$
$984$	6768.00	0.219264
$985$	0	0
$986$	−24696.0	−0.797648
$987$	8064.00	0.260061
$988$	−928.000	−0.0298822
$989$	10272.0	0.330263
$990$	0	0
$991$	35384.0	1.13422	0.567109	−	0.823643i	$-0.308062\pi$
$991$	35384.0	1.13422	0.567109	+	0.823643i	$0.308062\pi$
$992$	4096.00	0.131097
$993$	276.000	0.00882034
$994$	−8736.00	−0.278762
$995$	0	0
$996$	−1008.00	−0.0320680
$997$	48706.0	1.54718	0.773588	−	0.633689i	$-0.218460\pi$
$997$	48706.0	1.54718	0.773588	+	0.633689i	$0.218460\pi$
$998$	−31256.0	−0.991374
$999$	1566.00	0.0495956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.4.a.u.1.1		1
5.2	odd	4		1050.4.g.p.799.2		2
5.3	odd	4		1050.4.g.p.799.1		2
5.4	even	2		210.4.a.c.1.1	✓	1
15.14	odd	2		630.4.a.q.1.1		1
20.19	odd	2		1680.4.a.t.1.1		1
35.34	odd	2		1470.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.4.a.c.1.1	✓	1	5.4	even	2
630.4.a.q.1.1		1	15.14	odd	2
1050.4.a.u.1.1		1	1.1	even	1	trivial
1050.4.g.p.799.1		2	5.3	odd	4
1050.4.g.p.799.2		2	5.2	odd	4
1470.4.a.k.1.1		1	35.34	odd	2
1680.4.a.t.1.1		1	20.19	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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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