LMFDB - Embedded newform 9338.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9338,2,Mod(1,9338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9338, 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("9338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9338 = 2 \cdot 7 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9338.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:	$74.5643054075$
Analytic rank:	$2$
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	$\chi$	$=$	9338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -6.00000 q^{11} -2.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} -2.00000 q^{20} -2.00000 q^{21} -6.00000 q^{22} -1.00000 q^{23} -2.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} +1.00000 q^{29} +4.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} +12.0000 q^{33} -6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -8.00000 q^{37} -8.00000 q^{38} +4.00000 q^{39} -2.00000 q^{40} -4.00000 q^{41} -2.00000 q^{42} -2.00000 q^{43} -6.00000 q^{44} -2.00000 q^{45} -1.00000 q^{46} -2.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +12.0000 q^{51} -2.00000 q^{52} +2.00000 q^{53} +4.00000 q^{54} +12.0000 q^{55} +1.00000 q^{56} +16.0000 q^{57} +1.00000 q^{58} +4.00000 q^{60} -2.00000 q^{61} -6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +12.0000 q^{66} -4.00000 q^{67} -6.00000 q^{68} +2.00000 q^{69} -2.00000 q^{70} +1.00000 q^{72} +4.00000 q^{73} -8.00000 q^{74} +2.00000 q^{75} -8.00000 q^{76} -6.00000 q^{77} +4.00000 q^{78} +10.0000 q^{79} -2.00000 q^{80} -11.0000 q^{81} -4.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} +12.0000 q^{85} -2.00000 q^{86} -2.00000 q^{87} -6.00000 q^{88} +2.00000 q^{89} -2.00000 q^{90} -2.00000 q^{91} -1.00000 q^{92} +12.0000 q^{93} -2.00000 q^{94} +16.0000 q^{95} -2.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−2.00000	−0.816497
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−2.00000	−0.577350
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	1.00000	0.267261
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	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	$0.869927\pi$
$20$	−2.00000	−0.447214
$21$	−2.00000	−0.436436
$22$	−6.00000	−1.27920
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−2.00000	−0.408248
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	4.00000	0.769800
$28$	1.00000	0.188982
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	4.00000	0.730297
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.00000	0.176777
$33$	12.0000	2.08893
$34$	−6.00000	−1.02899
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−8.00000	−1.29777
$39$	4.00000	0.640513
$40$	−2.00000	−0.316228
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	−2.00000	−0.308607
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−6.00000	−0.904534
$45$	−2.00000	−0.298142
$46$	−1.00000	−0.147442
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	12.0000	1.68034
$52$	−2.00000	−0.277350
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	4.00000	0.544331
$55$	12.0000	1.61808
$56$	1.00000	0.133631
$57$	16.0000	2.11925
$58$	1.00000	0.131306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	4.00000	0.516398
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−6.00000	−0.762001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	12.0000	1.47710
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−6.00000	−0.727607
$69$	2.00000	0.240772
$70$	−2.00000	−0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−8.00000	−0.929981
$75$	2.00000	0.230940
$76$	−8.00000	−0.917663
$77$	−6.00000	−0.683763
$78$	4.00000	0.452911
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−2.00000	−0.223607
$81$	−11.0000	−1.22222
$82$	−4.00000	−0.441726
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−2.00000	−0.218218
$85$	12.0000	1.30158
$86$	−2.00000	−0.215666
$87$	−2.00000	−0.214423
$88$	−6.00000	−0.639602
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	−2.00000	−0.210819
$91$	−2.00000	−0.209657
$92$	−1.00000	−0.104257
$93$	12.0000	1.24434
$94$	−2.00000	−0.206284
$95$	16.0000	1.64157
$96$	−2.00000	−0.204124
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	1.00000	0.101015
$99$	−6.00000	−0.603023
$100$	−1.00000	−0.100000
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	12.0000	1.18818
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−2.00000	−0.196116
$105$	4.00000	0.390360
$106$	2.00000	0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	4.00000	0.384900
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	12.0000	1.14416
$111$	16.0000	1.51865
$112$	1.00000	0.0944911
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	16.0000	1.49854
$115$	2.00000	0.186501
$116$	1.00000	0.0928477
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−6.00000	−0.550019
$120$	4.00000	0.365148
$121$	25.0000	2.27273
$122$	−2.00000	−0.181071
$123$	8.00000	0.721336
$124$	−6.00000	−0.538816
$125$	12.0000	1.07331
$126$	1.00000	0.0890871
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	4.00000	0.350823
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	12.0000	1.04447
$133$	−8.00000	−0.693688
$134$	−4.00000	−0.345547
$135$	−8.00000	−0.688530
$136$	−6.00000	−0.514496
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	2.00000	0.170251
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−2.00000	−0.169031
$141$	4.00000	0.336861
$142$	0	0
$143$	12.0000	1.00349
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	4.00000	0.331042
$147$	−2.00000	−0.164957
$148$	−8.00000	−0.657596
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	2.00000	0.163299
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−8.00000	−0.648886
$153$	−6.00000	−0.485071
$154$	−6.00000	−0.483494
$155$	12.0000	0.963863
$156$	4.00000	0.320256
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	10.0000	0.795557
$159$	−4.00000	−0.317221
$160$	−2.00000	−0.158114
$161$	−1.00000	−0.0788110
$162$	−11.0000	−0.864242
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−4.00000	−0.312348
$165$	−24.0000	−1.86840
$166$	−4.00000	−0.310460
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−2.00000	−0.154303
$169$	−9.00000	−0.692308
$170$	12.0000	0.920358
$171$	−8.00000	−0.611775
$172$	−2.00000	−0.152499
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	−2.00000	−0.151620
$175$	−1.00000	−0.0755929
$176$	−6.00000	−0.452267
$177$	0	0
$178$	2.00000	0.149906
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−2.00000	−0.149071
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	−2.00000	−0.148250
$183$	4.00000	0.295689
$184$	−1.00000	−0.0737210
$185$	16.0000	1.17634
$186$	12.0000	0.879883
$187$	36.0000	2.63258
$188$	−2.00000	−0.145865
$189$	4.00000	0.290957
$190$	16.0000	1.16076
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	−2.00000	−0.144338
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−2.00000	−0.143592
$195$	−8.00000	−0.572892
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−6.00000	−0.426401
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−1.00000	−0.0707107
$201$	8.00000	0.564276
$202$	−16.0000	−1.12576
$203$	1.00000	0.0701862
$204$	12.0000	0.840168
$205$	8.00000	0.558744
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	−2.00000	−0.138675
$209$	48.0000	3.32023
$210$	4.00000	0.276026
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−12.0000	−0.820303
$215$	4.00000	0.272798
$216$	4.00000	0.272166
$217$	−6.00000	−0.407307
$218$	10.0000	0.677285
$219$	−8.00000	−0.540590
$220$	12.0000	0.809040
$221$	12.0000	0.807207
$222$	16.0000	1.07385
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	1.00000	0.0668153
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	16.0000	1.05963
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	2.00000	0.131876
$231$	12.0000	0.789542
$232$	1.00000	0.0656532
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−2.00000	−0.130744
$235$	4.00000	0.260931
$236$	0	0
$237$	−20.0000	−1.29914
$238$	−6.00000	−0.388922
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	4.00000	0.258199
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	25.0000	1.60706
$243$	10.0000	0.641500
$244$	−2.00000	−0.128037
$245$	−2.00000	−0.127775
$246$	8.00000	0.510061
$247$	16.0000	1.01806
$248$	−6.00000	−0.381000
$249$	8.00000	0.506979
$250$	12.0000	0.758947
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	1.00000	0.0629941
$253$	6.00000	0.377217
$254$	20.0000	1.25491
$255$	−24.0000	−1.50294
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	4.00000	0.249029
$259$	−8.00000	−0.497096
$260$	4.00000	0.248069
$261$	1.00000	0.0618984
$262$	−10.0000	−0.617802
$263$	−2.00000	−0.123325	−0.0616626	−	0.998097i	$-0.519640\pi$
$263$	−2.00000	−0.123325	−0.0616626	+	0.998097i	$0.519640\pi$
$264$	12.0000	0.738549
$265$	−4.00000	−0.245718
$266$	−8.00000	−0.490511
$267$	−4.00000	−0.244796
$268$	−4.00000	−0.244339
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	−8.00000	−0.486864
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	−6.00000	−0.363803
$273$	4.00000	0.242091
$274$	−8.00000	−0.483298
$275$	6.00000	0.361814
$276$	2.00000	0.120386
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	−2.00000	−0.119523
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	4.00000	0.238197
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	−32.0000	−1.89552
$286$	12.0000	0.709575
$287$	−4.00000	−0.236113
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	−2.00000	−0.117444
$291$	4.00000	0.234484
$292$	4.00000	0.234082
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	−8.00000	−0.464991
$297$	−24.0000	−1.39262
$298$	14.0000	0.810998
$299$	2.00000	0.115663
$300$	2.00000	0.115470
$301$	−2.00000	−0.115278
$302$	−16.0000	−0.920697
$303$	32.0000	1.83835
$304$	−8.00000	−0.458831
$305$	4.00000	0.229039
$306$	−6.00000	−0.342997
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	−6.00000	−0.341882
$309$	0	0
$310$	12.0000	0.681554
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	4.00000	0.226455
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−18.0000	−1.01580
$315$	−2.00000	−0.112687
$316$	10.0000	0.562544
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−4.00000	−0.224309
$319$	−6.00000	−0.335936
$320$	−2.00000	−0.111803
$321$	24.0000	1.33955
$322$	−1.00000	−0.0557278
$323$	48.0000	2.67079
$324$	−11.0000	−0.611111
$325$	2.00000	0.110940
$326$	−16.0000	−0.886158
$327$	−20.0000	−1.10600
$328$	−4.00000	−0.220863
$329$	−2.00000	−0.110264
$330$	−24.0000	−1.32116
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	−4.00000	−0.219529
$333$	−8.00000	−0.438397
$334$	−12.0000	−0.656611
$335$	8.00000	0.437087
$336$	−2.00000	−0.109109
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	12.0000	0.650791
$341$	36.0000	1.94951
$342$	−8.00000	−0.432590
$343$	1.00000	0.0539949
$344$	−2.00000	−0.107833
$345$	−4.00000	−0.215353
$346$	−2.00000	−0.107521
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−2.00000	−0.107211
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	−1.00000	−0.0534522
$351$	−8.00000	−0.427008
$352$	−6.00000	−0.319801
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	2.00000	0.106000
$357$	12.0000	0.635107
$358$	−12.0000	−0.634220
$359$	−14.0000	−0.738892	−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000	−0.738892	−0.369446	+	0.929252i	$0.620452\pi$
$360$	−2.00000	−0.105409
$361$	45.0000	2.36842
$362$	−6.00000	−0.315353
$363$	−50.0000	−2.62432
$364$	−2.00000	−0.104828
$365$	−8.00000	−0.418739
$366$	4.00000	0.209083
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−1.00000	−0.0521286
$369$	−4.00000	−0.208232
$370$	16.0000	0.831800
$371$	2.00000	0.103835
$372$	12.0000	0.622171
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	36.0000	1.86152
$375$	−24.0000	−1.23935
$376$	−2.00000	−0.103142
$377$	−2.00000	−0.103005
$378$	4.00000	0.205738
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	16.0000	0.820783
$381$	−40.0000	−2.04926
$382$	6.00000	0.306987
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−2.00000	−0.102062
$385$	12.0000	0.611577
$386$	14.0000	0.712581
$387$	−2.00000	−0.101666
$388$	−2.00000	−0.101535
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	−8.00000	−0.405096
$391$	6.00000	0.303433
$392$	1.00000	0.0505076
$393$	20.0000	1.00887
$394$	−6.00000	−0.302276
$395$	−20.0000	−1.00631
$396$	−6.00000	−0.301511
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−8.00000	−0.401004
$399$	16.0000	0.801002
$400$	−1.00000	−0.0500000
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	8.00000	0.399004
$403$	12.0000	0.597763
$404$	−16.0000	−0.796030
$405$	22.0000	1.09319
$406$	1.00000	0.0496292
$407$	48.0000	2.37927
$408$	12.0000	0.594089
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	8.00000	0.395092
$411$	16.0000	0.789222
$412$	0	0
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	8.00000	0.392705
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	48.0000	2.34776
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	4.00000	0.195180
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	8.00000	0.389434
$423$	−2.00000	−0.0972433
$424$	2.00000	0.0971286
$425$	6.00000	0.291043
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	−12.0000	−0.580042
$429$	−24.0000	−1.15873
$430$	4.00000	0.192897
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	4.00000	0.192450
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	−6.00000	−0.288009
$435$	4.00000	0.191785
$436$	10.0000	0.478913
$437$	8.00000	0.382692
$438$	−8.00000	−0.382255
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	12.0000	0.572078
$441$	1.00000	0.0476190
$442$	12.0000	0.570782
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	16.0000	0.759326
$445$	−4.00000	−0.189618
$446$	4.00000	0.189405
$447$	−28.0000	−1.32435
$448$	1.00000	0.0472456
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	−1.00000	−0.0471405
$451$	24.0000	1.13012
$452$	0	0
$453$	32.0000	1.50349
$454$	−12.0000	−0.563188
$455$	4.00000	0.187523
$456$	16.0000	0.749269
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−22.0000	−1.02799
$459$	−24.0000	−1.12022
$460$	2.00000	0.0932505
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	12.0000	0.558291
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	1.00000	0.0464238
$465$	−24.0000	−1.11297
$466$	−10.0000	−0.463241
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	−2.00000	−0.0924500
$469$	−4.00000	−0.184703
$470$	4.00000	0.184506
$471$	36.0000	1.65879
$472$	0	0
$473$	12.0000	0.551761
$474$	−20.0000	−0.918630
$475$	8.00000	0.367065
$476$	−6.00000	−0.275010
$477$	2.00000	0.0915737
$478$	24.0000	1.09773
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	4.00000	0.182574
$481$	16.0000	0.729537
$482$	18.0000	0.819878
$483$	2.00000	0.0910032
$484$	25.0000	1.13636
$485$	4.00000	0.181631
$486$	10.0000	0.453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−2.00000	−0.0905357
$489$	32.0000	1.44709
$490$	−2.00000	−0.0903508
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	8.00000	0.360668
$493$	−6.00000	−0.270226
$494$	16.0000	0.719874
$495$	12.0000	0.539360
$496$	−6.00000	−0.269408
$497$	0	0
$498$	8.00000	0.358489
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	12.0000	0.536656
$501$	24.0000	1.07224
$502$	8.00000	0.357057
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	1.00000	0.0445435
$505$	32.0000	1.42398
$506$	6.00000	0.266733
$507$	18.0000	0.799408
$508$	20.0000	0.887357
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	−24.0000	−1.06274
$511$	4.00000	0.176950
$512$	1.00000	0.0441942
$513$	−32.0000	−1.41283
$514$	30.0000	1.32324
$515$	0	0
$516$	4.00000	0.176090
$517$	12.0000	0.527759
$518$	−8.00000	−0.351500
$519$	4.00000	0.175581
$520$	4.00000	0.175412
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	1.00000	0.0437688
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	−10.0000	−0.436852
$525$	2.00000	0.0872872
$526$	−2.00000	−0.0872041
$527$	36.0000	1.56818
$528$	12.0000	0.522233
$529$	1.00000	0.0434783
$530$	−4.00000	−0.173749
$531$	0	0
$532$	−8.00000	−0.346844
$533$	8.00000	0.346518
$534$	−4.00000	−0.173097
$535$	24.0000	1.03761
$536$	−4.00000	−0.172774
$537$	24.0000	1.03568
$538$	0	0
$539$	−6.00000	−0.258438
$540$	−8.00000	−0.344265
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	−22.0000	−0.944981
$543$	12.0000	0.514969
$544$	−6.00000	−0.257248
$545$	−20.0000	−0.856706
$546$	4.00000	0.171184
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−8.00000	−0.341743
$549$	−2.00000	−0.0853579
$550$	6.00000	0.255841
$551$	−8.00000	−0.340811
$552$	2.00000	0.0851257
$553$	10.0000	0.425243
$554$	10.0000	0.424859
$555$	−32.0000	−1.35832
$556$	0	0
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	−6.00000	−0.254000
$559$	4.00000	0.169182
$560$	−2.00000	−0.0845154
$561$	−72.0000	−3.03984
$562$	−30.0000	−1.26547
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	4.00000	0.168430
$565$	0	0
$566$	−20.0000	−0.840663
$567$	−11.0000	−0.461957
$568$	0	0
$569$	4.00000	0.167689	0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000	0.167689	0.0838444	+	0.996479i	$0.473280\pi$
$570$	−32.0000	−1.34033
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	12.0000	0.501745
$573$	−12.0000	−0.501307
$574$	−4.00000	−0.166957
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	12.0000	0.499567	0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000	0.499567	0.249783	+	0.968302i	$0.419641\pi$
$578$	19.0000	0.790296
$579$	−28.0000	−1.16364
$580$	−2.00000	−0.0830455
$581$	−4.00000	−0.165948
$582$	4.00000	0.165805
$583$	−12.0000	−0.496989
$584$	4.00000	0.165521
$585$	4.00000	0.165380
$586$	−22.0000	−0.908812
$587$	−40.0000	−1.65098	−0.825488	−	0.564419i	$-0.809100\pi$
$587$	−40.0000	−1.65098	−0.825488	+	0.564419i	$0.809100\pi$
$588$	−2.00000	−0.0824786
$589$	48.0000	1.97781
$590$	0	0
$591$	12.0000	0.493614
$592$	−8.00000	−0.328798
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	−24.0000	−0.984732
$595$	12.0000	0.491952
$596$	14.0000	0.573462
$597$	16.0000	0.654836
$598$	2.00000	0.0817861
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	2.00000	0.0816497
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	−2.00000	−0.0815139
$603$	−4.00000	−0.162893
$604$	−16.0000	−0.651031
$605$	−50.0000	−2.03279
$606$	32.0000	1.29991
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	−8.00000	−0.324443
$609$	−2.00000	−0.0810441
$610$	4.00000	0.161955
$611$	4.00000	0.161823
$612$	−6.00000	−0.242536
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−2.00000	−0.0807134
$615$	−16.0000	−0.645182
$616$	−6.00000	−0.241747
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	12.0000	0.481932
$621$	−4.00000	−0.160514
$622$	−18.0000	−0.721734
$623$	2.00000	0.0801283
$624$	4.00000	0.160128
$625$	−19.0000	−0.760000
$626$	22.0000	0.879297
$627$	−96.0000	−3.83387
$628$	−18.0000	−0.718278
$629$	48.0000	1.91389
$630$	−2.00000	−0.0796819
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	10.0000	0.397779
$633$	−16.0000	−0.635943
$634$	−6.00000	−0.238290
$635$	−40.0000	−1.58735
$636$	−4.00000	−0.158610
$637$	−2.00000	−0.0792429
$638$	−6.00000	−0.237542
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	24.0000	0.947204
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	−1.00000	−0.0394055
$645$	−8.00000	−0.315000
$646$	48.0000	1.88853
$647$	20.0000	0.786281	0.393141	−	0.919478i	$-0.371389\pi$
$647$	20.0000	0.786281	0.393141	+	0.919478i	$0.371389\pi$
$648$	−11.0000	−0.432121
$649$	0	0
$650$	2.00000	0.0784465
$651$	12.0000	0.470317
$652$	−16.0000	−0.626608
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	−20.0000	−0.782062
$655$	20.0000	0.781465
$656$	−4.00000	−0.156174
$657$	4.00000	0.156055
$658$	−2.00000	−0.0779681
$659$	−42.0000	−1.63609	−0.818044	−	0.575156i	$-0.804941\pi$
$659$	−42.0000	−1.63609	−0.818044	+	0.575156i	$0.804941\pi$
$660$	−24.0000	−0.934199
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	−16.0000	−0.621858
$663$	−24.0000	−0.932083
$664$	−4.00000	−0.155230
$665$	16.0000	0.620453
$666$	−8.00000	−0.309994
$667$	−1.00000	−0.0387202
$668$	−12.0000	−0.464294
$669$	−8.00000	−0.309298
$670$	8.00000	0.309067
$671$	12.0000	0.463255
$672$	−2.00000	−0.0771517
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−12.0000	−0.462223
$675$	−4.00000	−0.153960
$676$	−9.00000	−0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	12.0000	0.460179
$681$	24.0000	0.919682
$682$	36.0000	1.37851
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−8.00000	−0.305888
$685$	16.0000	0.611329
$686$	1.00000	0.0381802
$687$	44.0000	1.67870
$688$	−2.00000	−0.0762493
$689$	−4.00000	−0.152388
$690$	−4.00000	−0.152277
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	−2.00000	−0.0760286
$693$	−6.00000	−0.227921
$694$	12.0000	0.455514
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	24.0000	0.909065
$698$	26.0000	0.984115
$699$	20.0000	0.756469
$700$	−1.00000	−0.0377964
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	−8.00000	−0.301941
$703$	64.0000	2.41381
$704$	−6.00000	−0.226134
$705$	−8.00000	−0.301297
$706$	−6.00000	−0.225813
$707$	−16.0000	−0.601742
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	2.00000	0.0749532
$713$	6.00000	0.224702
$714$	12.0000	0.449089
$715$	−24.0000	−0.897549
$716$	−12.0000	−0.448461
$717$	−48.0000	−1.79259
$718$	−14.0000	−0.522475
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	45.0000	1.67473
$723$	−36.0000	−1.33885
$724$	−6.00000	−0.222988
$725$	−1.00000	−0.0371391
$726$	−50.0000	−1.85567
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	−2.00000	−0.0741249
$729$	13.0000	0.481481
$730$	−8.00000	−0.296093
$731$	12.0000	0.443836
$732$	4.00000	0.147844
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	4.00000	0.147542
$736$	−1.00000	−0.0368605
$737$	24.0000	0.884051
$738$	−4.00000	−0.147242
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	16.0000	0.588172
$741$	−32.0000	−1.17555
$742$	2.00000	0.0734223
$743$	−38.0000	−1.39408	−0.697042	−	0.717030i	$-0.745501\pi$
$743$	−38.0000	−1.39408	−0.697042	+	0.717030i	$0.745501\pi$
$744$	12.0000	0.439941
$745$	−28.0000	−1.02584
$746$	14.0000	0.512576
$747$	−4.00000	−0.146352
$748$	36.0000	1.31629
$749$	−12.0000	−0.438470
$750$	−24.0000	−0.876356
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	−2.00000	−0.0729325
$753$	−16.0000	−0.583072
$754$	−2.00000	−0.0728357
$755$	32.0000	1.16460
$756$	4.00000	0.145479
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	34.0000	1.23494
$759$	−12.0000	−0.435572
$760$	16.0000	0.580381
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−40.0000	−1.44905
$763$	10.0000	0.362024
$764$	6.00000	0.217072
$765$	12.0000	0.433861
$766$	−24.0000	−0.867155
$767$	0	0
$768$	−2.00000	−0.0721688
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	12.0000	0.432450
$771$	−60.0000	−2.16085
$772$	14.0000	0.503871
$773$	−50.0000	−1.79838	−0.899188	−	0.437564i	$-0.855842\pi$
$773$	−50.0000	−1.79838	−0.899188	+	0.437564i	$0.855842\pi$
$774$	−2.00000	−0.0718885
$775$	6.00000	0.215526
$776$	−2.00000	−0.0717958
$777$	16.0000	0.573997
$778$	24.0000	0.860442
$779$	32.0000	1.14652
$780$	−8.00000	−0.286446
$781$	0	0
$782$	6.00000	0.214560
$783$	4.00000	0.142948
$784$	1.00000	0.0357143
$785$	36.0000	1.28490
$786$	20.0000	0.713376
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−6.00000	−0.213741
$789$	4.00000	0.142404
$790$	−20.0000	−0.711568
$791$	0	0
$792$	−6.00000	−0.213201
$793$	4.00000	0.142044
$794$	−18.0000	−0.638796
$795$	8.00000	0.283731
$796$	−8.00000	−0.283552
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	16.0000	0.566394
$799$	12.0000	0.424529
$800$	−1.00000	−0.0353553
$801$	2.00000	0.0706665
$802$	−22.0000	−0.776847
$803$	−24.0000	−0.846942
$804$	8.00000	0.282138
$805$	2.00000	0.0704907
$806$	12.0000	0.422682
$807$	0	0
$808$	−16.0000	−0.562878
$809$	14.0000	0.492214	0.246107	−	0.969243i	$-0.420849\pi$
$809$	14.0000	0.492214	0.246107	+	0.969243i	$0.420849\pi$
$810$	22.0000	0.773001
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	1.00000	0.0350931
$813$	44.0000	1.54315
$814$	48.0000	1.68240
$815$	32.0000	1.12091
$816$	12.0000	0.420084
$817$	16.0000	0.559769
$818$	12.0000	0.419570
$819$	−2.00000	−0.0698857
$820$	8.00000	0.279372
$821$	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
$823$	−12.0000	−0.418294	−0.209147	−	0.977884i	$-0.567069\pi$
$823$	−12.0000	−0.418294	−0.209147	+	0.977884i	$0.567069\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−6.00000	−0.208640	−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000	−0.208640	−0.104320	+	0.994544i	$0.533267\pi$
$828$	−1.00000	−0.0347524
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	8.00000	0.277684
$831$	−20.0000	−0.693792
$832$	−2.00000	−0.0693375
$833$	−6.00000	−0.207888
$834$	0	0
$835$	24.0000	0.830554
$836$	48.0000	1.66011
$837$	−24.0000	−0.829561
$838$	−20.0000	−0.690889
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	4.00000	0.138013
$841$	1.00000	0.0344828
$842$	28.0000	0.964944
$843$	60.0000	2.06651
$844$	8.00000	0.275371
$845$	18.0000	0.619219
$846$	−2.00000	−0.0687614
$847$	25.0000	0.859010
$848$	2.00000	0.0686803
$849$	40.0000	1.37280
$850$	6.00000	0.205798
$851$	8.00000	0.274236
$852$	0	0
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	−2.00000	−0.0684386
$855$	16.0000	0.547188
$856$	−12.0000	−0.410152
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	−24.0000	−0.819346
$859$	−34.0000	−1.16007	−0.580033	−	0.814593i	$-0.696960\pi$
$859$	−34.0000	−1.16007	−0.580033	+	0.814593i	$0.696960\pi$
$860$	4.00000	0.136399
$861$	8.00000	0.272639
$862$	−12.0000	−0.408722
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	4.00000	0.136083
$865$	4.00000	0.136004
$866$	−34.0000	−1.15537
$867$	−38.0000	−1.29055
$868$	−6.00000	−0.203653
$869$	−60.0000	−2.03536
$870$	4.00000	0.135613
$871$	8.00000	0.271070
$872$	10.0000	0.338643
$873$	−2.00000	−0.0676897
$874$	8.00000	0.270604
$875$	12.0000	0.405674
$876$	−8.00000	−0.270295
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	24.0000	0.809961
$879$	44.0000	1.48408
$880$	12.0000	0.404520
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	1.00000	0.0336718
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	20.0000	0.671913
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	16.0000	0.536925
$889$	20.0000	0.670778
$890$	−4.00000	−0.134080
$891$	66.0000	2.21108
$892$	4.00000	0.133930
$893$	16.0000	0.535420
$894$	−28.0000	−0.936460
$895$	24.0000	0.802232
$896$	1.00000	0.0334077
$897$	−4.00000	−0.133556
$898$	−14.0000	−0.467186
$899$	−6.00000	−0.200111
$900$	−1.00000	−0.0333333
$901$	−12.0000	−0.399778
$902$	24.0000	0.799113
$903$	4.00000	0.133112
$904$	0	0
$905$	12.0000	0.398893
$906$	32.0000	1.06313
$907$	−18.0000	−0.597680	−0.298840	−	0.954303i	$-0.596600\pi$
$907$	−18.0000	−0.597680	−0.298840	+	0.954303i	$0.596600\pi$
$908$	−12.0000	−0.398234
$909$	−16.0000	−0.530687
$910$	4.00000	0.132599
$911$	22.0000	0.728893	0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000	0.728893	0.364446	+	0.931224i	$0.381258\pi$
$912$	16.0000	0.529813
$913$	24.0000	0.794284
$914$	−22.0000	−0.727695
$915$	−8.00000	−0.264472
$916$	−22.0000	−0.726900
$917$	−10.0000	−0.330229
$918$	−24.0000	−0.792118
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	2.00000	0.0659380
$921$	4.00000	0.131804
$922$	−24.0000	−0.790398
$923$	0	0
$924$	12.0000	0.394771
$925$	8.00000	0.263038
$926$	16.0000	0.525793
$927$	0	0
$928$	1.00000	0.0328266
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	−24.0000	−0.786991
$931$	−8.00000	−0.262189
$932$	−10.0000	−0.327561
$933$	36.0000	1.17859
$934$	8.00000	0.261768
$935$	−72.0000	−2.35465
$936$	−2.00000	−0.0653720
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	−4.00000	−0.130605
$939$	−44.0000	−1.43589
$940$	4.00000	0.130466
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	36.0000	1.17294
$943$	4.00000	0.130258
$944$	0	0
$945$	−8.00000	−0.260240
$946$	12.0000	0.390154
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	−20.0000	−0.649570
$949$	−8.00000	−0.259691
$950$	8.00000	0.259554
$951$	12.0000	0.389127
$952$	−6.00000	−0.194461
$953$	−46.0000	−1.49009	−0.745043	−	0.667016i	$-0.767571\pi$
$953$	−46.0000	−1.49009	−0.745043	+	0.667016i	$0.767571\pi$
$954$	2.00000	0.0647524
$955$	−12.0000	−0.388311
$956$	24.0000	0.776215
$957$	12.0000	0.387905
$958$	0	0
$959$	−8.00000	−0.258333
$960$	4.00000	0.129099
$961$	5.00000	0.161290
$962$	16.0000	0.515861
$963$	−12.0000	−0.386695
$964$	18.0000	0.579741
$965$	−28.0000	−0.901352
$966$	2.00000	0.0643489
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	25.0000	0.803530
$969$	−96.0000	−3.08396
$970$	4.00000	0.128432
$971$	−56.0000	−1.79713	−0.898563	−	0.438845i	$-0.855388\pi$
$971$	−56.0000	−1.79713	−0.898563	+	0.438845i	$0.855388\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	−32.0000	−1.02535
$975$	−4.00000	−0.128103
$976$	−2.00000	−0.0640184
$977$	−26.0000	−0.831814	−0.415907	−	0.909407i	$-0.636536\pi$
$977$	−26.0000	−0.831814	−0.415907	+	0.909407i	$0.636536\pi$
$978$	32.0000	1.02325
$979$	−12.0000	−0.383522
$980$	−2.00000	−0.0638877
$981$	10.0000	0.319275
$982$	−4.00000	−0.127645
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	8.00000	0.255031
$985$	12.0000	0.382352
$986$	−6.00000	−0.191079
$987$	4.00000	0.127321
$988$	16.0000	0.509028
$989$	2.00000	0.0635963
$990$	12.0000	0.381385
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	−6.00000	−0.190500
$993$	32.0000	1.01549
$994$	0	0
$995$	16.0000	0.507234
$996$	8.00000	0.253490
$997$	−12.0000	−0.380044	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	−12.0000	−0.380044	−0.190022	+	0.981780i	$0.560856\pi$
$998$	−12.0000	−0.379853
$999$	−32.0000	−1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9338.2.a.f.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9338.2.a.f.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9338 = 2 \cdot 7 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9338.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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