LMFDB - Embedded newform 1050.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,2,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, 2, names="a")

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

chi := DirichletCharacter("1050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
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:	$8.38429221223$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} +7.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{21} -2.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} +7.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +9.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -7.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +8.00000 q^{38} -7.00000 q^{39} +11.0000 q^{41} -1.00000 q^{42} +3.00000 q^{43} -2.00000 q^{44} -5.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +7.00000 q^{51} +7.00000 q^{52} +3.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} -8.00000 q^{57} +9.00000 q^{58} +7.00000 q^{59} -5.00000 q^{61} +1.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -12.0000 q^{67} -7.00000 q^{68} +5.00000 q^{69} -4.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} +8.00000 q^{76} -2.00000 q^{77} -7.00000 q^{78} -6.00000 q^{79} +1.00000 q^{81} +11.0000 q^{82} -9.00000 q^{83} -1.00000 q^{84} +3.00000 q^{86} -9.00000 q^{87} -2.00000 q^{88} -10.0000 q^{89} +7.00000 q^{91} -5.00000 q^{92} -1.00000 q^{93} +4.00000 q^{94} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	7.00000	1.94145	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	7.00000	1.94145	0.970725	+	0.240192i	$0.0772105\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	1.00000	0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−2.00000	−0.426401
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	7.00000	1.37281
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	1.00000	0.176777
$33$	2.00000	0.348155
$34$	−7.00000	−1.20049
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	8.00000	1.29777
$39$	−7.00000	−1.12090
$40$	0	0
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	−1.00000	−0.154303
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−5.00000	−0.737210
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	0	0
$51$	7.00000	0.980196
$52$	7.00000	0.970725
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	−8.00000	−1.05963
$58$	9.00000	1.18176
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	1.00000	0.127000
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−7.00000	−0.848875
$69$	5.00000	0.601929
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	8.00000	0.917663
$77$	−2.00000	−0.227921
$78$	−7.00000	−0.792594
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	11.0000	1.21475
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	3.00000	0.323498
$87$	−9.00000	−0.964901
$88$	−2.00000	−0.213201
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	7.00000	0.733799
$92$	−5.00000	−0.521286
$93$	−1.00000	−0.103695
$94$	4.00000	0.412568
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	1.00000	0.101015
$99$	−2.00000	−0.201008
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	7.00000	0.693103
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	7.00000	0.686406
$105$	0	0
$106$	3.00000	0.291386
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	−1.00000	−0.0962250
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	1.00000	0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	9.00000	0.835629
$117$	7.00000	0.647150
$118$	7.00000	0.644402
$119$	−7.00000	−0.641689
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−5.00000	−0.452679
$123$	−11.0000	−0.991837
$124$	1.00000	0.0898027
$125$	0	0
$126$	1.00000	0.0890871
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	1.00000	0.0883883
$129$	−3.00000	−0.264135
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	2.00000	0.174078
$133$	8.00000	0.693688
$134$	−12.0000	−1.03664
$135$	0	0
$136$	−7.00000	−0.600245
$137$	16.0000	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$138$	5.00000	0.425628
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	−4.00000	−0.335673
$143$	−14.0000	−1.17074
$144$	1.00000	0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	−1.00000	−0.0824786
$148$	−2.00000	−0.164399
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	8.00000	0.648886
$153$	−7.00000	−0.565916
$154$	−2.00000	−0.161165
$155$	0	0
$156$	−7.00000	−0.560449
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−6.00000	−0.477334
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−5.00000	−0.394055
$162$	1.00000	0.0785674
$163$	−15.0000	−1.17489	−0.587445	−	0.809264i	$-0.699866\pi$
$163$	−15.0000	−1.17489	−0.587445	+	0.809264i	$0.699866\pi$
$164$	11.0000	0.858956
$165$	0	0
$166$	−9.00000	−0.698535
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	−1.00000	−0.0771517
$169$	36.0000	2.76923
$170$	0	0
$171$	8.00000	0.611775
$172$	3.00000	0.228748
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	−9.00000	−0.682288
$175$	0	0
$176$	−2.00000	−0.150756
$177$	−7.00000	−0.526152
$178$	−10.0000	−0.749532
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	7.00000	0.518875
$183$	5.00000	0.369611
$184$	−5.00000	−0.368605
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	14.0000	1.02378
$188$	4.00000	0.291730
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−1.00000	−0.0723575	−0.0361787	−	0.999345i	$-0.511519\pi$
$191$	−1.00000	−0.0723575	−0.0361787	+	0.999345i	$0.511519\pi$
$192$	−1.00000	−0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	−21.0000	−1.49619	−0.748094	−	0.663593i	$-0.769031\pi$
$197$	−21.0000	−1.49619	−0.748094	+	0.663593i	$0.769031\pi$
$198$	−2.00000	−0.142134
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	9.00000	0.631676
$204$	7.00000	0.490098
$205$	0	0
$206$	5.00000	0.348367
$207$	−5.00000	−0.347524
$208$	7.00000	0.485363
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	3.00000	0.206041
$213$	4.00000	0.274075
$214$	−2.00000	−0.136717
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	1.00000	0.0678844
$218$	4.00000	0.270914
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−49.0000	−3.29610
$222$	2.00000	0.134231
$223$	9.00000	0.602685	0.301342	−	0.953516i	$-0.402565\pi$
$223$	9.00000	0.602685	0.301342	+	0.953516i	$0.402565\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	−8.00000	−0.529813
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	9.00000	0.590879
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	7.00000	0.457604
$235$	0	0
$236$	7.00000	0.455661
$237$	6.00000	0.389742
$238$	−7.00000	−0.453743
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	−7.00000	−0.449977
$243$	−1.00000	−0.0641500
$244$	−5.00000	−0.320092
$245$	0	0
$246$	−11.0000	−0.701334
$247$	56.0000	3.56319
$248$	1.00000	0.0635001
$249$	9.00000	0.570352
$250$	0	0
$251$	−5.00000	−0.315597	−0.157799	−	0.987471i	$-0.550440\pi$
$251$	−5.00000	−0.315597	−0.157799	+	0.987471i	$0.550440\pi$
$252$	1.00000	0.0629941
$253$	10.0000	0.628695
$254$	−18.0000	−1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.00000	0.311891	0.155946	−	0.987766i	$-0.450158\pi$
$257$	5.00000	0.311891	0.155946	+	0.987766i	$0.450158\pi$
$258$	−3.00000	−0.186772
$259$	−2.00000	−0.124274
$260$	0	0
$261$	9.00000	0.557086
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	8.00000	0.490511
$267$	10.0000	0.611990
$268$	−12.0000	−0.733017
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	−7.00000	−0.424437
$273$	−7.00000	−0.423659
$274$	16.0000	0.966595
$275$	0	0
$276$	5.00000	0.300965
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	8.00000	0.479808
$279$	1.00000	0.0598684
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	−4.00000	−0.238197
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	−14.0000	−0.827837
$287$	11.0000	0.649309
$288$	1.00000	0.0589256
$289$	32.0000	1.88235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	10.0000	0.585206
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	−2.00000	−0.116248
$297$	2.00000	0.116052
$298$	−3.00000	−0.173785
$299$	−35.0000	−2.02410
$300$	0	0
$301$	3.00000	0.172917
$302$	−22.0000	−1.26596
$303$	0	0
$304$	8.00000	0.458831
$305$	0	0
$306$	−7.00000	−0.400163
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	−2.00000	−0.113961
$309$	−5.00000	−0.284440
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	−7.00000	−0.396297
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−6.00000	−0.337526
$317$	23.0000	1.29181	0.645904	−	0.763418i	$-0.276480\pi$
$317$	23.0000	1.29181	0.645904	+	0.763418i	$0.276480\pi$
$318$	−3.00000	−0.168232
$319$	−18.0000	−1.00781
$320$	0	0
$321$	2.00000	0.111629
$322$	−5.00000	−0.278639
$323$	−56.0000	−3.11592
$324$	1.00000	0.0555556
$325$	0	0
$326$	−15.0000	−0.830773
$327$	−4.00000	−0.221201
$328$	11.0000	0.607373
$329$	4.00000	0.220527
$330$	0	0
$331$	25.0000	1.37412	0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000	1.37412	0.687062	+	0.726599i	$0.258900\pi$
$332$	−9.00000	−0.493939
$333$	−2.00000	−0.109599
$334$	2.00000	0.109435
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	36.0000	1.95814
$339$	14.0000	0.760376
$340$	0	0
$341$	−2.00000	−0.108306
$342$	8.00000	0.432590
$343$	1.00000	0.0539949
$344$	3.00000	0.161749
$345$	0	0
$346$	−16.0000	−0.860165
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	−9.00000	−0.482451
$349$	−7.00000	−0.374701	−0.187351	−	0.982293i	$-0.559990\pi$
$349$	−7.00000	−0.374701	−0.187351	+	0.982293i	$0.559990\pi$
$350$	0	0
$351$	−7.00000	−0.373632
$352$	−2.00000	−0.106600
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	−7.00000	−0.372046
$355$	0	0
$356$	−10.0000	−0.529999
$357$	7.00000	0.370479
$358$	−2.00000	−0.105703
$359$	19.0000	1.00278	0.501391	−	0.865221i	$-0.332822\pi$
$359$	19.0000	1.00278	0.501391	+	0.865221i	$0.332822\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	2.00000	0.105118
$363$	7.00000	0.367405
$364$	7.00000	0.366900
$365$	0	0
$366$	5.00000	0.261354
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	−5.00000	−0.260643
$369$	11.0000	0.572637
$370$	0	0
$371$	3.00000	0.155752
$372$	−1.00000	−0.0518476
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	14.0000	0.723923
$375$	0	0
$376$	4.00000	0.206284
$377$	63.0000	3.24467
$378$	−1.00000	−0.0514344
$379$	7.00000	0.359566	0.179783	−	0.983706i	$-0.442460\pi$
$379$	7.00000	0.359566	0.179783	+	0.983706i	$0.442460\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	−1.00000	−0.0511645
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	3.00000	0.152499
$388$	10.0000	0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	35.0000	1.77003
$392$	1.00000	0.0505076
$393$	0	0
$394$	−21.0000	−1.05796
$395$	0	0
$396$	−2.00000	−0.100504
$397$	37.0000	1.85698	0.928488	−	0.371361i	$-0.121109\pi$
$397$	37.0000	1.85698	0.928488	+	0.371361i	$0.121109\pi$
$398$	−16.0000	−0.802008
$399$	−8.00000	−0.400501
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	12.0000	0.598506
$403$	7.00000	0.348695
$404$	0	0
$405$	0	0
$406$	9.00000	0.446663
$407$	4.00000	0.198273
$408$	7.00000	0.346552
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−16.0000	−0.789222
$412$	5.00000	0.246332
$413$	7.00000	0.344447
$414$	−5.00000	−0.245737
$415$	0	0
$416$	7.00000	0.343203
$417$	−8.00000	−0.391762
$418$	−16.0000	−0.782586
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−13.0000	−0.632830
$423$	4.00000	0.194487
$424$	3.00000	0.145693
$425$	0	0
$426$	4.00000	0.193801
$427$	−5.00000	−0.241967
$428$	−2.00000	−0.0966736
$429$	14.0000	0.675926
$430$	0	0
$431$	39.0000	1.87856	0.939282	−	0.343146i	$-0.111493\pi$
$431$	39.0000	1.87856	0.939282	+	0.343146i	$0.111493\pi$
$432$	−1.00000	−0.0481125
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	1.00000	0.0480015
$435$	0	0
$436$	4.00000	0.191565
$437$	−40.0000	−1.91346
$438$	−10.0000	−0.477818
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−49.0000	−2.33069
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	9.00000	0.426162
$447$	3.00000	0.141895
$448$	1.00000	0.0472456
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	−22.0000	−1.03594
$452$	−14.0000	−0.658505
$453$	22.0000	1.03365
$454$	−3.00000	−0.140797
$455$	0	0
$456$	−8.00000	−0.374634
$457$	7.00000	0.327446	0.163723	−	0.986506i	$-0.447650\pi$
$457$	7.00000	0.327446	0.163723	+	0.986506i	$0.447650\pi$
$458$	−14.0000	−0.654177
$459$	7.00000	0.326732
$460$	0	0
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	2.00000	0.0930484
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	16.0000	0.741186
$467$	−29.0000	−1.34196	−0.670980	−	0.741475i	$-0.734126\pi$
$467$	−29.0000	−1.34196	−0.670980	+	0.741475i	$0.734126\pi$
$468$	7.00000	0.323575
$469$	−12.0000	−0.554109
$470$	0	0
$471$	10.0000	0.460776
$472$	7.00000	0.322201
$473$	−6.00000	−0.275880
$474$	6.00000	0.275589
$475$	0	0
$476$	−7.00000	−0.320844
$477$	3.00000	0.137361
$478$	8.00000	0.365911
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	−30.0000	−1.36646
$483$	5.00000	0.227508
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	6.00000	0.271886	0.135943	−	0.990717i	$-0.456594\pi$
$487$	6.00000	0.271886	0.135943	+	0.990717i	$0.456594\pi$
$488$	−5.00000	−0.226339
$489$	15.0000	0.678323
$490$	0	0
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	−11.0000	−0.495918
$493$	−63.0000	−2.83738
$494$	56.0000	2.51956
$495$	0	0
$496$	1.00000	0.0449013
$497$	−4.00000	−0.179425
$498$	9.00000	0.403300
$499$	−23.0000	−1.02962	−0.514811	−	0.857304i	$-0.672138\pi$
$499$	−23.0000	−1.02962	−0.514811	+	0.857304i	$0.672138\pi$
$500$	0	0
$501$	−2.00000	−0.0893534
$502$	−5.00000	−0.223161
$503$	−26.0000	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	10.0000	0.444554
$507$	−36.0000	−1.59882
$508$	−18.0000	−0.798621
$509$	−28.0000	−1.24108	−0.620539	−	0.784176i	$-0.713086\pi$
$509$	−28.0000	−1.24108	−0.620539	+	0.784176i	$0.713086\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	1.00000	0.0441942
$513$	−8.00000	−0.353209
$514$	5.00000	0.220541
$515$	0	0
$516$	−3.00000	−0.132068
$517$	−8.00000	−0.351840
$518$	−2.00000	−0.0878750
$519$	16.0000	0.702322
$520$	0	0
$521$	9.00000	0.394297	0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000	0.394297	0.197149	+	0.980374i	$0.436832\pi$
$522$	9.00000	0.393919
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	0	0
$525$	0	0
$526$	−9.00000	−0.392419
$527$	−7.00000	−0.304925
$528$	2.00000	0.0870388
$529$	2.00000	0.0869565
$530$	0	0
$531$	7.00000	0.303774
$532$	8.00000	0.346844
$533$	77.0000	3.33524
$534$	10.0000	0.432742
$535$	0	0
$536$	−12.0000	−0.518321
$537$	2.00000	0.0863064
$538$	−6.00000	−0.258678
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	12.0000	0.515444
$543$	−2.00000	−0.0858282
$544$	−7.00000	−0.300123
$545$	0	0
$546$	−7.00000	−0.299572
$547$	−35.0000	−1.49649	−0.748246	−	0.663421i	$-0.769104\pi$
$547$	−35.0000	−1.49649	−0.748246	+	0.663421i	$0.769104\pi$
$548$	16.0000	0.683486
$549$	−5.00000	−0.213395
$550$	0	0
$551$	72.0000	3.06730
$552$	5.00000	0.212814
$553$	−6.00000	−0.255146
$554$	2.00000	0.0849719
$555$	0	0
$556$	8.00000	0.339276
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	1.00000	0.0423334
$559$	21.0000	0.888205
$560$	0	0
$561$	−14.0000	−0.591080
$562$	−8.00000	−0.337460
$563$	27.0000	1.13791	0.568957	−	0.822367i	$-0.307347\pi$
$563$	27.0000	1.13791	0.568957	+	0.822367i	$0.307347\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	−16.0000	−0.672530
$567$	1.00000	0.0419961
$568$	−4.00000	−0.167836
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	31.0000	1.29731	0.648655	−	0.761083i	$-0.275332\pi$
$571$	31.0000	1.29731	0.648655	+	0.761083i	$0.275332\pi$
$572$	−14.0000	−0.585369
$573$	1.00000	0.0417756
$574$	11.0000	0.459131
$575$	0	0
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	32.0000	1.33102
$579$	−10.0000	−0.415586
$580$	0	0
$581$	−9.00000	−0.373383
$582$	−10.0000	−0.414513
$583$	−6.00000	−0.248495
$584$	10.0000	0.413803
$585$	0	0
$586$	24.0000	0.991431
$587$	39.0000	1.60970	0.804851	−	0.593477i	$-0.202245\pi$
$587$	39.0000	1.60970	0.804851	+	0.593477i	$0.202245\pi$
$588$	−1.00000	−0.0412393
$589$	8.00000	0.329634
$590$	0	0
$591$	21.0000	0.863825
$592$	−2.00000	−0.0821995
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	−3.00000	−0.122885
$597$	16.0000	0.654836
$598$	−35.0000	−1.43126
$599$	−1.00000	−0.0408589	−0.0204294	−	0.999791i	$-0.506503\pi$
$599$	−1.00000	−0.0408589	−0.0204294	+	0.999791i	$0.506503\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	3.00000	0.122271
$603$	−12.0000	−0.488678
$604$	−22.0000	−0.895167
$605$	0	0
$606$	0	0
$607$	−36.0000	−1.46119	−0.730597	−	0.682808i	$-0.760758\pi$
$607$	−36.0000	−1.46119	−0.730597	+	0.682808i	$0.760758\pi$
$608$	8.00000	0.324443
$609$	−9.00000	−0.364698
$610$	0	0
$611$	28.0000	1.13276
$612$	−7.00000	−0.282958
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	−5.00000	−0.201129
$619$	−38.0000	−1.52735	−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000	−1.52735	−0.763674	+	0.645601i	$0.776607\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	2.00000	0.0801927
$623$	−10.0000	−0.400642
$624$	−7.00000	−0.280224
$625$	0	0
$626$	10.0000	0.399680
$627$	16.0000	0.638978
$628$	−10.0000	−0.399043
$629$	14.0000	0.558217
$630$	0	0
$631$	42.0000	1.67199	0.835997	−	0.548734i	$-0.184890\pi$
$631$	42.0000	1.67199	0.835997	+	0.548734i	$0.184890\pi$
$632$	−6.00000	−0.238667
$633$	13.0000	0.516704
$634$	23.0000	0.913447
$635$	0	0
$636$	−3.00000	−0.118958
$637$	7.00000	0.277350
$638$	−18.0000	−0.712627
$639$	−4.00000	−0.158238
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	2.00000	0.0789337
$643$	10.0000	0.394362	0.197181	−	0.980367i	$-0.436821\pi$
$643$	10.0000	0.394362	0.197181	+	0.980367i	$0.436821\pi$
$644$	−5.00000	−0.197028
$645$	0	0
$646$	−56.0000	−2.20329
$647$	46.0000	1.80845	0.904223	−	0.427060i	$-0.140451\pi$
$647$	46.0000	1.80845	0.904223	+	0.427060i	$0.140451\pi$
$648$	1.00000	0.0392837
$649$	−14.0000	−0.549548
$650$	0	0
$651$	−1.00000	−0.0391931
$652$	−15.0000	−0.587445
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	11.0000	0.429478
$657$	10.0000	0.390137
$658$	4.00000	0.155936
$659$	34.0000	1.32445	0.662226	−	0.749304i	$-0.269612\pi$
$659$	34.0000	1.32445	0.662226	+	0.749304i	$0.269612\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	25.0000	0.971653
$663$	49.0000	1.90300
$664$	−9.00000	−0.349268
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−45.0000	−1.74241
$668$	2.00000	0.0773823
$669$	−9.00000	−0.347960
$670$	0	0
$671$	10.0000	0.386046
$672$	−1.00000	−0.0385758
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	36.0000	1.38462
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	14.0000	0.537667
$679$	10.0000	0.383765
$680$	0	0
$681$	3.00000	0.114960
$682$	−2.00000	−0.0765840
$683$	38.0000	1.45403	0.727015	−	0.686622i	$-0.240907\pi$
$683$	38.0000	1.45403	0.727015	+	0.686622i	$0.240907\pi$
$684$	8.00000	0.305888
$685$	0	0
$686$	1.00000	0.0381802
$687$	14.0000	0.534133
$688$	3.00000	0.114374
$689$	21.0000	0.800036
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	−16.0000	−0.608229
$693$	−2.00000	−0.0759737
$694$	−6.00000	−0.227757
$695$	0	0
$696$	−9.00000	−0.341144
$697$	−77.0000	−2.91658
$698$	−7.00000	−0.264954
$699$	−16.0000	−0.605176
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	−7.00000	−0.264198
$703$	−16.0000	−0.603451
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	30.0000	1.12906
$707$	0	0
$708$	−7.00000	−0.263076
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	−10.0000	−0.374766
$713$	−5.00000	−0.187251
$714$	7.00000	0.261968
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	−8.00000	−0.298765
$718$	19.0000	0.709074
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	5.00000	0.186210
$722$	45.0000	1.67473
$723$	30.0000	1.11571
$724$	2.00000	0.0743294
$725$	0	0
$726$	7.00000	0.259794
$727$	21.0000	0.778847	0.389423	−	0.921059i	$-0.372674\pi$
$727$	21.0000	0.778847	0.389423	+	0.921059i	$0.372674\pi$
$728$	7.00000	0.259437
$729$	1.00000	0.0370370
$730$	0	0
$731$	−21.0000	−0.776713
$732$	5.00000	0.184805
$733$	45.0000	1.66211	0.831056	−	0.556188i	$-0.187737\pi$
$733$	45.0000	1.66211	0.831056	+	0.556188i	$0.187737\pi$
$734$	−17.0000	−0.627481
$735$	0	0
$736$	−5.00000	−0.184302
$737$	24.0000	0.884051
$738$	11.0000	0.404916
$739$	15.0000	0.551784	0.275892	−	0.961189i	$-0.411027\pi$
$739$	15.0000	0.551784	0.275892	+	0.961189i	$0.411027\pi$
$740$	0	0
$741$	−56.0000	−2.05721
$742$	3.00000	0.110133
$743$	−39.0000	−1.43077	−0.715386	−	0.698730i	$-0.753749\pi$
$743$	−39.0000	−1.43077	−0.715386	+	0.698730i	$0.753749\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−4.00000	−0.146450
$747$	−9.00000	−0.329293
$748$	14.0000	0.511891
$749$	−2.00000	−0.0730784
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	4.00000	0.145865
$753$	5.00000	0.182210
$754$	63.0000	2.29432
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	7.00000	0.254251
$759$	−10.0000	−0.362977
$760$	0	0
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	18.0000	0.652071
$763$	4.00000	0.144810
$764$	−1.00000	−0.0361787
$765$	0	0
$766$	0	0
$767$	49.0000	1.76929
$768$	−1.00000	−0.0360844
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	−5.00000	−0.180071
$772$	10.0000	0.359908
$773$	28.0000	1.00709	0.503545	−	0.863969i	$-0.332029\pi$
$773$	28.0000	1.00709	0.503545	+	0.863969i	$0.332029\pi$
$774$	3.00000	0.107833
$775$	0	0
$776$	10.0000	0.358979
$777$	2.00000	0.0717496
$778$	6.00000	0.215110
$779$	88.0000	3.15293
$780$	0	0
$781$	8.00000	0.286263
$782$	35.0000	1.25160
$783$	−9.00000	−0.321634
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	26.0000	0.926800	0.463400	−	0.886149i	$-0.346629\pi$
$787$	26.0000	0.926800	0.463400	+	0.886149i	$0.346629\pi$
$788$	−21.0000	−0.748094
$789$	9.00000	0.320408
$790$	0	0
$791$	−14.0000	−0.497783
$792$	−2.00000	−0.0710669
$793$	−35.0000	−1.24289
$794$	37.0000	1.31308
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	−8.00000	−0.283197
$799$	−28.0000	−0.990569
$800$	0	0
$801$	−10.0000	−0.353333
$802$	−12.0000	−0.423735
$803$	−20.0000	−0.705785
$804$	12.0000	0.423207
$805$	0	0
$806$	7.00000	0.246564
$807$	6.00000	0.211210
$808$	0	0
$809$	−44.0000	−1.54696	−0.773479	−	0.633822i	$-0.781485\pi$
$809$	−44.0000	−1.54696	−0.773479	+	0.633822i	$0.781485\pi$
$810$	0	0
$811$	−6.00000	−0.210688	−0.105344	−	0.994436i	$-0.533594\pi$
$811$	−6.00000	−0.210688	−0.105344	+	0.994436i	$0.533594\pi$
$812$	9.00000	0.315838
$813$	−12.0000	−0.420858
$814$	4.00000	0.140200
$815$	0	0
$816$	7.00000	0.245049
$817$	24.0000	0.839654
$818$	10.0000	0.349642
$819$	7.00000	0.244600
$820$	0	0
$821$	34.0000	1.18661	0.593304	−	0.804978i	$-0.297823\pi$
$821$	34.0000	1.18661	0.593304	+	0.804978i	$0.297823\pi$
$822$	−16.0000	−0.558064
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	5.00000	0.174183
$825$	0	0
$826$	7.00000	0.243561
$827$	−6.00000	−0.208640	−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000	−0.208640	−0.104320	+	0.994544i	$0.533267\pi$
$828$	−5.00000	−0.173762
$829$	−19.0000	−0.659897	−0.329949	−	0.943999i	$-0.607031\pi$
$829$	−19.0000	−0.659897	−0.329949	+	0.943999i	$0.607031\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	7.00000	0.242681
$833$	−7.00000	−0.242536
$834$	−8.00000	−0.277017
$835$	0	0
$836$	−16.0000	−0.553372
$837$	−1.00000	−0.0345651
$838$	−3.00000	−0.103633
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	10.0000	0.344623
$843$	8.00000	0.275535
$844$	−13.0000	−0.447478
$845$	0	0
$846$	4.00000	0.137523
$847$	−7.00000	−0.240523
$848$	3.00000	0.103020
$849$	16.0000	0.549119
$850$	0	0
$851$	10.0000	0.342796
$852$	4.00000	0.137038
$853$	−35.0000	−1.19838	−0.599189	−	0.800608i	$-0.704510\pi$
$853$	−35.0000	−1.19838	−0.599189	+	0.800608i	$0.704510\pi$
$854$	−5.00000	−0.171096
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	14.0000	0.477952
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	0	0
$861$	−11.0000	−0.374879
$862$	39.0000	1.32835
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	6.00000	0.203888
$867$	−32.0000	−1.08678
$868$	1.00000	0.0339422
$869$	12.0000	0.407072
$870$	0	0
$871$	−84.0000	−2.84623
$872$	4.00000	0.135457
$873$	10.0000	0.338449
$874$	−40.0000	−1.35302
$875$	0	0
$876$	−10.0000	−0.337869
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	−19.0000	−0.641219
$879$	−24.0000	−0.809500
$880$	0	0
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	1.00000	0.0336718
$883$	−23.0000	−0.774012	−0.387006	−	0.922077i	$-0.626491\pi$
$883$	−23.0000	−0.774012	−0.387006	+	0.922077i	$0.626491\pi$
$884$	−49.0000	−1.64805
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−14.0000	−0.470074	−0.235037	−	0.971986i	$-0.575521\pi$
$887$	−14.0000	−0.470074	−0.235037	+	0.971986i	$0.575521\pi$
$888$	2.00000	0.0671156
$889$	−18.0000	−0.603701
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	9.00000	0.301342
$893$	32.0000	1.07084
$894$	3.00000	0.100335
$895$	0	0
$896$	1.00000	0.0334077
$897$	35.0000	1.16862
$898$	−20.0000	−0.667409
$899$	9.00000	0.300167
$900$	0	0
$901$	−21.0000	−0.699611
$902$	−22.0000	−0.732520
$903$	−3.00000	−0.0998337
$904$	−14.0000	−0.465633
$905$	0	0
$906$	22.0000	0.730901
$907$	−39.0000	−1.29497	−0.647487	−	0.762077i	$-0.724180\pi$
$907$	−39.0000	−1.29497	−0.647487	+	0.762077i	$0.724180\pi$
$908$	−3.00000	−0.0995585
$909$	0	0
$910$	0	0
$911$	−39.0000	−1.29213	−0.646064	−	0.763283i	$-0.723586\pi$
$911$	−39.0000	−1.29213	−0.646064	+	0.763283i	$0.723586\pi$
$912$	−8.00000	−0.264906
$913$	18.0000	0.595713
$914$	7.00000	0.231539
$915$	0	0
$916$	−14.0000	−0.462573
$917$	0	0
$918$	7.00000	0.231034
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	22.0000	0.724925
$922$	8.00000	0.263466
$923$	−28.0000	−0.921631
$924$	2.00000	0.0657952
$925$	0	0
$926$	8.00000	0.262896
$927$	5.00000	0.164222
$928$	9.00000	0.295439
$929$	7.00000	0.229663	0.114831	−	0.993385i	$-0.463367\pi$
$929$	7.00000	0.229663	0.114831	+	0.993385i	$0.463367\pi$
$930$	0	0
$931$	8.00000	0.262189
$932$	16.0000	0.524097
$933$	−2.00000	−0.0654771
$934$	−29.0000	−0.948909
$935$	0	0
$936$	7.00000	0.228802
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	−12.0000	−0.391814
$939$	−10.0000	−0.326338
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	10.0000	0.325818
$943$	−55.0000	−1.79105
$944$	7.00000	0.227831
$945$	0	0
$946$	−6.00000	−0.195077
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	6.00000	0.194871
$949$	70.0000	2.27230
$950$	0	0
$951$	−23.0000	−0.745826
$952$	−7.00000	−0.226871
$953$	−48.0000	−1.55487	−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000	−1.55487	−0.777436	+	0.628962i	$0.783480\pi$
$954$	3.00000	0.0971286
$955$	0	0
$956$	8.00000	0.258738
$957$	18.0000	0.581857
$958$	−12.0000	−0.387702
$959$	16.0000	0.516667
$960$	0	0
$961$	−30.0000	−0.967742
$962$	−14.0000	−0.451378
$963$	−2.00000	−0.0644491
$964$	−30.0000	−0.966235
$965$	0	0
$966$	5.00000	0.160872
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	−7.00000	−0.224989
$969$	56.0000	1.79898
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−1.00000	−0.0320750
$973$	8.00000	0.256468
$974$	6.00000	0.192252
$975$	0	0
$976$	−5.00000	−0.160046
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	15.0000	0.479647
$979$	20.0000	0.639203
$980$	0	0
$981$	4.00000	0.127710
$982$	−42.0000	−1.34027
$983$	32.0000	1.02064	0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000	1.02064	0.510321	+	0.859984i	$0.329527\pi$
$984$	−11.0000	−0.350667
$985$	0	0
$986$	−63.0000	−2.00633
$987$	−4.00000	−0.127321
$988$	56.0000	1.78160
$989$	−15.0000	−0.476972
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	1.00000	0.0317500
$993$	−25.0000	−0.793351
$994$	−4.00000	−0.126872
$995$	0	0
$996$	9.00000	0.285176
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	−23.0000	−0.728052
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.a.n.1.1	yes	1
3.2	odd	2		3150.2.a.r.1.1		1
4.3	odd	2		8400.2.a.cb.1.1		1
5.2	odd	4		1050.2.g.b.799.2		2
5.3	odd	4		1050.2.g.b.799.1		2
5.4	even	2		1050.2.a.f.1.1	✓	1
7.6	odd	2		7350.2.a.cm.1.1		1
15.2	even	4		3150.2.g.p.2899.1		2
15.8	even	4		3150.2.g.p.2899.2		2
15.14	odd	2		3150.2.a.bd.1.1		1
20.19	odd	2		8400.2.a.bb.1.1		1
35.34	odd	2		7350.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.2.a.f.1.1	✓	1	5.4	even	2
1050.2.a.n.1.1	yes	1	1.1	even	1	trivial
1050.2.g.b.799.1		2	5.3	odd	4
1050.2.g.b.799.2		2	5.2	odd	4
3150.2.a.r.1.1		1	3.2	odd	2
3150.2.a.bd.1.1		1	15.14	odd	2
3150.2.g.p.2899.1		2	15.2	even	4
3150.2.g.p.2899.2		2	15.8	even	4
7350.2.a.i.1.1		1	35.34	odd	2
7350.2.a.cm.1.1		1	7.6	odd	2
8400.2.a.bb.1.1		1	20.19	odd	2
8400.2.a.cb.1.1		1	4.3	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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