LMFDB - Embedded newform 7350.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, 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("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7350.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:	$58.6900454856$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1050)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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
$7$	0	0
$7$	0	0
$8$	−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	$0.0772105\pi$
$14$	0	0
$15$	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	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−7.00000	−1.37281
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	−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	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	8.00000	1.29777
$39$	−7.00000	−1.12090
$40$	0	0
$41$	−11.0000	−1.71791	−0.858956	−	0.512050i	$-0.828886\pi$
$41$	−11.0000	−1.71791	−0.858956	+	0.512050i	$0.828886\pi$
$42$	0	0
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	−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
$49$	0	0
$50$	0	0
$51$	7.00000	0.980196
$52$	7.00000	0.970725
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	8.00000	1.05963
$58$	−9.00000	−1.18176
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−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
$77$	0	0
$78$	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	$0.664443\pi$
$84$	0	0
$85$	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	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	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	$0.330506\pi$
$98$	0	0
$99$	−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	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	−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
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	9.00000	0.835629
$117$	7.00000	0.647150
$118$	7.00000	0.644402
$119$	0	0
$120$	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
$126$	0	0
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	−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
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	7.00000	0.600245
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	5.00000	0.425628
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	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
$147$	0	0
$148$	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
$154$	0	0
$155$	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
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	15.0000	1.17489	0.587445	−	0.809264i	$-0.300134\pi$
$163$	15.0000	1.17489	0.587445	+	0.809264i	$0.300134\pi$
$164$	−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	$0.475344\pi$
$168$	0	0
$169$	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	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−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
$189$	0	0
$190$	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	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	0	0
$197$	21.0000	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$198$	2.00000	0.142134
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	0	0
$204$	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
$217$	0	0
$218$	−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	$0.402565\pi$
$224$	0	0
$225$	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	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	−9.00000	−0.590879
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	−7.00000	−0.457604
$235$	0	0
$236$	−7.00000	−0.455661
$237$	6.00000	0.389742
$238$	0	0
$239$	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	$-0.0829523\pi$
$241$	30.0000	1.93247	0.966235	+	0.257663i	$0.0829523\pi$
$242$	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	$-0.449560\pi$
$251$	5.00000	0.315597	0.157799	+	0.987471i	$0.449560\pi$
$252$	0	0
$253$	−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
$259$	0	0
$260$	0	0
$261$	9.00000	0.557086
$262$	0	0
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	0	0
$267$	−10.0000	−0.611990
$268$	12.0000	0.733017
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	−7.00000	−0.424437
$273$	0	0
$274$	16.0000	0.966595
$275$	0	0
$276$	−5.00000	−0.300965
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	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
$287$	0	0
$288$	−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	$0.252716\pi$
$294$	0	0
$295$	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
$301$	0	0
$302$	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	$0.716046\pi$
$308$	0	0
$309$	−5.00000	−0.284440
$310$	0	0
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	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	$-0.723520\pi$
$317$	−23.0000	−1.29181	−0.645904	+	0.763418i	$0.723520\pi$
$318$	−3.00000	−0.168232
$319$	−18.0000	−1.00781
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	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
$329$	0	0
$330$	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
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	−36.0000	−1.95814
$339$	−14.0000	−0.760376
$340$	0	0
$341$	2.00000	0.108306
$342$	8.00000	0.432590
$343$	0	0
$344$	3.00000	0.161749
$345$	0	0
$346$	16.0000	0.860165
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	−9.00000	−0.482451
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	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
$357$	0	0
$358$	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
$364$	0	0
$365$	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
$371$	0	0
$372$	1.00000	0.0518476
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	−14.0000	−0.723923
$375$	0	0
$376$	−4.00000	−0.206284
$377$	63.0000	3.24467
$378$	0	0
$379$	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
$392$	0	0
$393$	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
$399$	0	0
$400$	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
$406$	0	0
$407$	−4.00000	−0.198273
$408$	−7.00000	−0.346552
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	16.0000	0.789222
$412$	5.00000	0.246332
$413$	0	0
$414$	−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	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	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
$427$	0	0
$428$	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	$0.453949\pi$
$434$	0	0
$435$	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	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	0	0
$441$	0	0
$442$	49.0000	2.33069
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−9.00000	−0.426162
$447$	3.00000	0.141895
$448$	0	0
$449$	−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	$-0.552350\pi$
$457$	−7.00000	−0.327446	−0.163723	+	0.986506i	$0.552350\pi$
$458$	−14.0000	−0.654177
$459$	7.00000	0.326732
$460$	0	0
$461$	−8.00000	−0.372597	−0.186299	−	0.982493i	$-0.559649\pi$
$461$	−8.00000	−0.372597	−0.186299	+	0.982493i	$0.559649\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	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
$469$	0	0
$470$	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
$476$	0	0
$477$	−3.00000	−0.137361
$478$	−8.00000	−0.365911
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	14.0000	0.638345
$482$	−30.0000	−1.36646
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	−6.00000	−0.271886	−0.135943	−	0.990717i	$-0.543406\pi$
$487$	−6.00000	−0.271886	−0.135943	+	0.990717i	$0.543406\pi$
$488$	−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
$497$	0	0
$498$	−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	$0.696807\pi$
$504$	0	0
$505$	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	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	0	0
$511$	0	0
$512$	−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
$518$	0	0
$519$	16.0000	0.702322
$520$	0	0
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	−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
$532$	0	0
$533$	−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
$539$	0	0
$540$	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
$546$	0	0
$547$	35.0000	1.49649	0.748246	−	0.663421i	$-0.230896\pi$
$547$	35.0000	1.49649	0.748246	+	0.663421i	$0.230896\pi$
$548$	−16.0000	−0.683486
$549$	5.00000	0.213395
$550$	0	0
$551$	−72.0000	−3.06730
$552$	5.00000	0.212814
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	−8.00000	−0.339276
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	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
$567$	0	0
$568$	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
$574$	0	0
$575$	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
$581$	0	0
$582$	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	$0.202245\pi$
$588$	0	0
$589$	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	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	0	0
$603$	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
$609$	0	0
$610$	0	0
$611$	28.0000	1.13276
$612$	−7.00000	−0.282958
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	22.0000	0.887848
$615$	0	0
$616$	0	0
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	5.00000	0.201129
$619$	38.0000	1.52735	0.763674	−	0.645601i	$-0.223393\pi$
$619$	38.0000	1.52735	0.763674	+	0.645601i	$0.223393\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	2.00000	0.0801927
$623$	0	0
$624$	−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
$637$	0	0
$638$	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	$0.436821\pi$
$644$	0	0
$645$	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
$651$	0	0
$652$	15.0000	0.587445
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	−11.0000	−0.429478
$657$	10.0000	0.390137
$658$	0	0
$659$	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	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−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
$672$	0	0
$673$	13.0000	0.501113	0.250557	−	0.968102i	$-0.419386\pi$
$673$	13.0000	0.501113	0.250557	+	0.968102i	$0.419386\pi$
$674$	−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
$679$	0	0
$680$	0	0
$681$	3.00000	0.114960
$682$	−2.00000	−0.0765840
$683$	−38.0000	−1.45403	−0.727015	−	0.686622i	$-0.759093\pi$
$683$	−38.0000	−1.45403	−0.727015	+	0.686622i	$0.759093\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	0	0
$687$	−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	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	−16.0000	−0.608229
$693$	0	0
$694$	−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
$714$	0	0
$715$	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	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	0	0
$722$	−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	$0.372674\pi$
$728$	0	0
$729$	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
$742$	0	0
$743$	39.0000	1.43077	0.715386	−	0.698730i	$-0.246251\pi$
$743$	39.0000	1.43077	0.715386	+	0.698730i	$0.246251\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−4.00000	−0.146450
$747$	−9.00000	−0.329293
$748$	14.0000	0.511891
$749$	0	0
$750$	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
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	−7.00000	−0.254251
$759$	10.0000	0.362977
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	18.0000	0.652071
$763$	0	0
$764$	−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	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	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
$777$	0	0
$778$	−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
$784$	0	0
$785$	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
$791$	0	0
$792$	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	$0.678307\pi$
$798$	0	0
$799$	−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	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	0	0
$813$	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
$819$	0	0
$820$	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	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	−5.00000	−0.174183
$825$	0	0
$826$	0	0
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	5.00000	0.173762
$829$	19.0000	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$829$	19.0000	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	7.00000	0.242681
$833$	0	0
$834$	−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	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	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
$847$	0	0
$848$	−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	$0.704510\pi$
$854$	0	0
$855$	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	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	0	0
$861$	0	0
$862$	−39.0000	−1.32835
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−6.00000	−0.203888
$867$	−32.0000	−1.08678
$868$	0	0
$869$	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	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	−19.0000	−0.641219
$879$	−24.0000	−0.809500
$880$	0	0
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	0	0
$883$	23.0000	0.774012	0.387006	−	0.922077i	$-0.373509\pi$
$883$	23.0000	0.774012	0.387006	+	0.922077i	$0.373509\pi$
$884$	−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
$889$	0	0
$890$	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
$896$	0	0
$897$	−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
$903$	0	0
$904$	−14.0000	−0.465633
$905$	0	0
$906$	−22.0000	−0.730901
$907$	39.0000	1.29497	0.647487	−	0.762077i	$-0.275820\pi$
$907$	39.0000	1.29497	0.647487	+	0.762077i	$0.275820\pi$
$908$	−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
$924$	0	0
$925$	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	$-0.536633\pi$
$929$	−7.00000	−0.229663	−0.114831	+	0.993385i	$0.536633\pi$
$930$	0	0
$931$	0	0
$932$	−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	$0.531246\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	−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	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	6.00000	0.194871
$949$	70.0000	2.27230
$950$	0	0
$951$	23.0000	0.745826
$952$	0	0
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	3.00000	0.0971286
$955$	0	0
$956$	8.00000	0.258738
$957$	18.0000	0.581857
$958$	−12.0000	−0.387702
$959$	0	0
$960$	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
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	7.00000	0.224989
$969$	−56.0000	−1.79898
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	6.00000	0.192252
$975$	0	0
$976$	5.00000	0.160046
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	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
$987$	0	0
$988$	−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
$994$	0	0
$995$	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	7350.2.a.i.1.1		1
5.4	even	2		7350.2.a.cm.1.1		1
7.6	odd	2		1050.2.a.f.1.1	✓	1
21.20	even	2		3150.2.a.bd.1.1		1
28.27	even	2		8400.2.a.bb.1.1		1
35.13	even	4		1050.2.g.b.799.2		2
35.27	even	4		1050.2.g.b.799.1		2
35.34	odd	2		1050.2.a.n.1.1	yes	1
105.62	odd	4		3150.2.g.p.2899.2		2
105.83	odd	4		3150.2.g.p.2899.1		2
105.104	even	2		3150.2.a.r.1.1		1
140.139	even	2		8400.2.a.cb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.2.a.f.1.1	✓	1	7.6	odd	2
1050.2.a.n.1.1	yes	1	35.34	odd	2
1050.2.g.b.799.1		2	35.27	even	4
1050.2.g.b.799.2		2	35.13	even	4
3150.2.a.r.1.1		1	105.104	even	2
3150.2.a.bd.1.1		1	21.20	even	2
3150.2.g.p.2899.1		2	105.83	odd	4
3150.2.g.p.2899.2		2	105.62	odd	4
7350.2.a.i.1.1		1	1.1	even	1	trivial
7350.2.a.cm.1.1		1	5.4	even	2
8400.2.a.bb.1.1		1	28.27	even	2
8400.2.a.cb.1.1		1	140.139	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:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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