LMFDB - Embedded newform 8050.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8050,2,Mod(1,8050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8050, 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("8050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8050 = 2 \cdot 5^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8050.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:	$64.2795736271$
Analytic rank:	$1$
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	$\chi$	$=$	8050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -5.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} -2.00000 q^{18} +5.00000 q^{19} +1.00000 q^{21} -5.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} -5.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} -5.00000 q^{34} -2.00000 q^{36} +4.00000 q^{37} +5.00000 q^{38} +2.00000 q^{39} -5.00000 q^{41} +1.00000 q^{42} -8.00000 q^{43} -5.00000 q^{44} +1.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -5.00000 q^{51} +2.00000 q^{52} -4.00000 q^{53} -5.00000 q^{54} +1.00000 q^{56} +5.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +10.0000 q^{61} -4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +1.00000 q^{67} -5.00000 q^{68} +1.00000 q^{69} +2.00000 q^{71} -2.00000 q^{72} -15.0000 q^{73} +4.00000 q^{74} +5.00000 q^{76} -5.00000 q^{77} +2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -5.00000 q^{82} +3.00000 q^{83} +1.00000 q^{84} -8.00000 q^{86} -2.00000 q^{87} -5.00000 q^{88} -15.0000 q^{89} +2.00000 q^{91} +1.00000 q^{92} -4.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} +10.0000 q^{99} +O(q^{100})\)

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	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
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	−2.00000	−0.471405
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−5.00000	−1.06600
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	−5.00000	−0.962250
$28$	1.00000	0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	−5.00000	−0.870388
$34$	−5.00000	−0.857493
$35$	0	0
$36$	−2.00000	−0.333333
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	5.00000	0.811107
$39$	2.00000	0.320256
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	1.00000	0.154303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	1.00000	0.147442
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.00000	−0.700140
$52$	2.00000	0.277350
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	1.00000	0.133631
$57$	5.00000	0.662266
$58$	−2.00000	−0.262613
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−4.00000	−0.508001
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.00000	−0.615457
$67$	1.00000	0.122169	0.0610847	−	0.998133i	$-0.480544\pi$
$67$	1.00000	0.122169	0.0610847	+	0.998133i	$0.480544\pi$
$68$	−5.00000	−0.606339
$69$	1.00000	0.120386
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−2.00000	−0.235702
$73$	−15.0000	−1.75562	−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000	−1.75562	−0.877809	+	0.479012i	$0.840995\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	5.00000	0.573539
$77$	−5.00000	−0.569803
$78$	2.00000	0.226455
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.00000	−0.552158
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	−8.00000	−0.862662
$87$	−2.00000	−0.214423
$88$	−5.00000	−0.533002
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	1.00000	0.104257
$93$	−4.00000	−0.414781
$94$	−6.00000	−0.618853
$95$	0	0
$96$	1.00000	0.102062
$97$	−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
$99$	10.0000	1.00504
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−5.00000	−0.495074
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−4.00000	−0.388514
$107$	11.0000	1.06341	0.531705	−	0.846930i	$-0.321551\pi$
$107$	11.0000	1.06341	0.531705	+	0.846930i	$0.321551\pi$
$108$	−5.00000	−0.481125
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	1.00000	0.0944911
$113$	−13.0000	−1.22294	−0.611469	−	0.791269i	$-0.709421\pi$
$113$	−13.0000	−1.22294	−0.611469	+	0.791269i	$0.709421\pi$
$114$	5.00000	0.468293
$115$	0	0
$116$	−2.00000	−0.185695
$117$	−4.00000	−0.369800
$118$	4.00000	0.368230
$119$	−5.00000	−0.458349
$120$	0	0
$121$	14.0000	1.27273
$122$	10.0000	0.905357
$123$	−5.00000	−0.450835
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−2.00000	−0.178174
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	1.00000	0.0883883
$129$	−8.00000	−0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−5.00000	−0.435194
$133$	5.00000	0.433555
$134$	1.00000	0.0863868
$135$	0	0
$136$	−5.00000	−0.428746
$137$	1.00000	0.0854358	0.0427179	−	0.999087i	$-0.486398\pi$
$137$	1.00000	0.0854358	0.0427179	+	0.999087i	$0.486398\pi$
$138$	1.00000	0.0851257
$139$	17.0000	1.44192	0.720961	−	0.692976i	$-0.243701\pi$
$139$	17.0000	1.44192	0.720961	+	0.692976i	$0.243701\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	2.00000	0.167836
$143$	−10.0000	−0.836242
$144$	−2.00000	−0.166667
$145$	0	0
$146$	−15.0000	−1.24141
$147$	1.00000	0.0824786
$148$	4.00000	0.328798
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	5.00000	0.405554
$153$	10.0000	0.808452
$154$	−5.00000	−0.402911
$155$	0	0
$156$	2.00000	0.160128
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	−10.0000	−0.795557
$159$	−4.00000	−0.317221
$160$	0	0
$161$	1.00000	0.0788110
$162$	1.00000	0.0785674
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	−5.00000	−0.390434
$165$	0	0
$166$	3.00000	0.232845
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	1.00000	0.0771517
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−10.0000	−0.764719
$172$	−8.00000	−0.609994
$173$	−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
$175$	0	0
$176$	−5.00000	−0.376889
$177$	4.00000	0.300658
$178$	−15.0000	−1.12430
$179$	−17.0000	−1.27064	−0.635320	−	0.772249i	$-0.719132\pi$
$179$	−17.0000	−1.27064	−0.635320	+	0.772249i	$0.719132\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	2.00000	0.148250
$183$	10.0000	0.739221
$184$	1.00000	0.0737210
$185$	0	0
$186$	−4.00000	−0.293294
$187$	25.0000	1.82818
$188$	−6.00000	−0.437595
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	1.00000	0.0721688
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	10.0000	0.710669
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−2.00000	−0.140372
$204$	−5.00000	−0.350070
$205$	0	0
$206$	−16.0000	−1.11477
$207$	−2.00000	−0.139010
$208$	2.00000	0.138675
$209$	−25.0000	−1.72929
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	−4.00000	−0.274721
$213$	2.00000	0.137038
$214$	11.0000	0.751945
$215$	0	0
$216$	−5.00000	−0.340207
$217$	−4.00000	−0.271538
$218$	−2.00000	−0.135457
$219$	−15.0000	−1.01361
$220$	0	0
$221$	−10.0000	−0.672673
$222$	4.00000	0.268462
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−13.0000	−0.864747
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	5.00000	0.331133
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	−2.00000	−0.131306
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	4.00000	0.260378
$237$	−10.0000	−0.649570
$238$	−5.00000	−0.324102
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	14.0000	0.899954
$243$	16.0000	1.02640
$244$	10.0000	0.640184
$245$	0	0
$246$	−5.00000	−0.318788
$247$	10.0000	0.636285
$248$	−4.00000	−0.254000
$249$	3.00000	0.190117
$250$	0	0
$251$	−5.00000	−0.315597	−0.157799	−	0.987471i	$-0.550440\pi$
$251$	−5.00000	−0.315597	−0.157799	+	0.987471i	$0.550440\pi$
$252$	−2.00000	−0.125988
$253$	−5.00000	−0.314347
$254$	6.00000	0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−8.00000	−0.498058
$259$	4.00000	0.248548
$260$	0	0
$261$	4.00000	0.247594
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	−5.00000	−0.307729
$265$	0	0
$266$	5.00000	0.306570
$267$	−15.0000	−0.917985
$268$	1.00000	0.0610847
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−5.00000	−0.303170
$273$	2.00000	0.121046
$274$	1.00000	0.0604122
$275$	0	0
$276$	1.00000	0.0601929
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	17.0000	1.01959
$279$	8.00000	0.478947
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	−6.00000	−0.357295
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	−10.0000	−0.591312
$287$	−5.00000	−0.295141
$288$	−2.00000	−0.117851
$289$	8.00000	0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	−15.0000	−0.877809
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	4.00000	0.232495
$297$	25.0000	1.45065
$298$	8.00000	0.463428
$299$	2.00000	0.115663
$300$	0	0
$301$	−8.00000	−0.461112
$302$	−24.0000	−1.38104
$303$	0	0
$304$	5.00000	0.286770
$305$	0	0
$306$	10.0000	0.571662
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	−5.00000	−0.284901
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	2.00000	0.113228
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	8.00000	0.451466
$315$	0	0
$316$	−10.0000	−0.562544
$317$	20.0000	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$317$	20.0000	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$318$	−4.00000	−0.224309
$319$	10.0000	0.559893
$320$	0	0
$321$	11.0000	0.613960
$322$	1.00000	0.0557278
$323$	−25.0000	−1.39104
$324$	1.00000	0.0555556
$325$	0	0
$326$	17.0000	0.941543
$327$	−2.00000	−0.110600
$328$	−5.00000	−0.276079
$329$	−6.00000	−0.330791
$330$	0	0
$331$	19.0000	1.04433	0.522167	−	0.852843i	$-0.325124\pi$
$331$	19.0000	1.04433	0.522167	+	0.852843i	$0.325124\pi$
$332$	3.00000	0.164646
$333$	−8.00000	−0.438397
$334$	−24.0000	−1.31322
$335$	0	0
$336$	1.00000	0.0545545
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	−9.00000	−0.489535
$339$	−13.0000	−0.706063
$340$	0	0
$341$	20.0000	1.08306
$342$	−10.0000	−0.540738
$343$	1.00000	0.0539949
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−2.00000	−0.107521
$347$	23.0000	1.23470	0.617352	−	0.786687i	$-0.288205\pi$
$347$	23.0000	1.23470	0.617352	+	0.786687i	$0.288205\pi$
$348$	−2.00000	−0.107211
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	−10.0000	−0.533761
$352$	−5.00000	−0.266501
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	−15.0000	−0.794998
$357$	−5.00000	−0.264628
$358$	−17.0000	−0.898478
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−14.0000	−0.735824
$363$	14.0000	0.734809
$364$	2.00000	0.104828
$365$	0	0
$366$	10.0000	0.522708
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	1.00000	0.0521286
$369$	10.0000	0.520579
$370$	0	0
$371$	−4.00000	−0.207670
$372$	−4.00000	−0.207390
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	25.0000	1.29272
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−4.00000	−0.206010
$378$	−5.00000	−0.257172
$379$	−15.0000	−0.770498	−0.385249	−	0.922813i	$-0.625884\pi$
$379$	−15.0000	−0.770498	−0.385249	+	0.922813i	$0.625884\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	−18.0000	−0.920960
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−11.0000	−0.559885
$387$	16.0000	0.813326
$388$	−2.00000	−0.101535
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	−5.00000	−0.252861
$392$	1.00000	0.0505076
$393$	0	0
$394$	22.0000	1.10834
$395$	0	0
$396$	10.0000	0.502519
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	−2.00000	−0.100251
$399$	5.00000	0.250313
$400$	0	0
$401$	33.0000	1.64794	0.823971	−	0.566632i	$-0.191754\pi$
$401$	33.0000	1.64794	0.823971	+	0.566632i	$0.191754\pi$
$402$	1.00000	0.0498755
$403$	−8.00000	−0.398508
$404$	0	0
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	−20.0000	−0.991363
$408$	−5.00000	−0.247537
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	0	0
$411$	1.00000	0.0493264
$412$	−16.0000	−0.788263
$413$	4.00000	0.196827
$414$	−2.00000	−0.0982946
$415$	0	0
$416$	2.00000	0.0980581
$417$	17.0000	0.832494
$418$	−25.0000	−1.22279
$419$	−23.0000	−1.12362	−0.561812	−	0.827265i	$-0.689895\pi$
$419$	−23.0000	−1.12362	−0.561812	+	0.827265i	$0.689895\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	13.0000	0.632830
$423$	12.0000	0.583460
$424$	−4.00000	−0.194257
$425$	0	0
$426$	2.00000	0.0969003
$427$	10.0000	0.483934
$428$	11.0000	0.531705
$429$	−10.0000	−0.482805
$430$	0	0
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	−5.00000	−0.240563
$433$	−29.0000	−1.39365	−0.696826	−	0.717241i	$-0.745405\pi$
$433$	−29.0000	−1.39365	−0.696826	+	0.717241i	$0.745405\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	5.00000	0.239182
$438$	−15.0000	−0.716728
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	−10.0000	−0.475651
$443$	1.00000	0.0475114	0.0237557	−	0.999718i	$-0.492438\pi$
$443$	1.00000	0.0475114	0.0237557	+	0.999718i	$0.492438\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	10.0000	0.473514
$447$	8.00000	0.378387
$448$	1.00000	0.0472456
$449$	41.0000	1.93491	0.967455	−	0.253044i	$-0.0814317\pi$
$449$	41.0000	1.93491	0.967455	+	0.253044i	$0.0814317\pi$
$450$	0	0
$451$	25.0000	1.17720
$452$	−13.0000	−0.611469
$453$	−24.0000	−1.12762
$454$	−12.0000	−0.563188
$455$	0	0
$456$	5.00000	0.234146
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	2.00000	0.0934539
$459$	25.0000	1.16690
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	−5.00000	−0.232621
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−18.0000	−0.833834
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	−4.00000	−0.184900
$469$	1.00000	0.0461757
$470$	0	0
$471$	8.00000	0.368621
$472$	4.00000	0.184115
$473$	40.0000	1.83920
$474$	−10.0000	−0.459315
$475$	0	0
$476$	−5.00000	−0.229175
$477$	8.00000	0.366295
$478$	−8.00000	−0.365911
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	7.00000	0.318841
$483$	1.00000	0.0455016
$484$	14.0000	0.636364
$485$	0	0
$486$	16.0000	0.725775
$487$	30.0000	1.35943	0.679715	−	0.733476i	$-0.262104\pi$
$487$	30.0000	1.35943	0.679715	+	0.733476i	$0.262104\pi$
$488$	10.0000	0.452679
$489$	17.0000	0.768767
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	−5.00000	−0.225417
$493$	10.0000	0.450377
$494$	10.0000	0.449921
$495$	0	0
$496$	−4.00000	−0.179605
$497$	2.00000	0.0897123
$498$	3.00000	0.134433
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	−5.00000	−0.223161
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	−5.00000	−0.222277
$507$	−9.00000	−0.399704
$508$	6.00000	0.266207
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−15.0000	−0.663561
$512$	1.00000	0.0441942
$513$	−25.0000	−1.10378
$514$	18.0000	0.793946
$515$	0	0
$516$	−8.00000	−0.352180
$517$	30.0000	1.31940
$518$	4.00000	0.175750
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−25.0000	−1.09527	−0.547635	−	0.836717i	$-0.684472\pi$
$521$	−25.0000	−1.09527	−0.547635	+	0.836717i	$0.684472\pi$
$522$	4.00000	0.175075
$523$	21.0000	0.918266	0.459133	−	0.888368i	$-0.348160\pi$
$523$	21.0000	0.918266	0.459133	+	0.888368i	$0.348160\pi$
$524$	0	0
$525$	0	0
$526$	18.0000	0.784837
$527$	20.0000	0.871214
$528$	−5.00000	−0.217597
$529$	1.00000	0.0434783
$530$	0	0
$531$	−8.00000	−0.347170
$532$	5.00000	0.216777
$533$	−10.0000	−0.433148
$534$	−15.0000	−0.649113
$535$	0	0
$536$	1.00000	0.0431934
$537$	−17.0000	−0.733604
$538$	2.00000	0.0862261
$539$	−5.00000	−0.215365
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	2.00000	0.0859074
$543$	−14.0000	−0.600798
$544$	−5.00000	−0.214373
$545$	0	0
$546$	2.00000	0.0855921
$547$	23.0000	0.983409	0.491704	−	0.870762i	$-0.336374\pi$
$547$	23.0000	0.983409	0.491704	+	0.870762i	$0.336374\pi$
$548$	1.00000	0.0427179
$549$	−20.0000	−0.853579
$550$	0	0
$551$	−10.0000	−0.426014
$552$	1.00000	0.0425628
$553$	−10.0000	−0.425243
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	17.0000	0.720961
$557$	−16.0000	−0.677942	−0.338971	−	0.940797i	$-0.610079\pi$
$557$	−16.0000	−0.677942	−0.338971	+	0.940797i	$0.610079\pi$
$558$	8.00000	0.338667
$559$	−16.0000	−0.676728
$560$	0	0
$561$	25.0000	1.05550
$562$	22.0000	0.928014
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−5.00000	−0.210166
$567$	1.00000	0.0419961
$568$	2.00000	0.0839181
$569$	−47.0000	−1.97034	−0.985171	−	0.171574i	$-0.945115\pi$
$569$	−47.0000	−1.97034	−0.985171	+	0.171574i	$0.945115\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	−10.0000	−0.418121
$573$	−18.0000	−0.751961
$574$	−5.00000	−0.208696
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	8.00000	0.332756
$579$	−11.0000	−0.457144
$580$	0	0
$581$	3.00000	0.124461
$582$	−2.00000	−0.0829027
$583$	20.0000	0.828315
$584$	−15.0000	−0.620704
$585$	0	0
$586$	−26.0000	−1.07405
$587$	7.00000	0.288921	0.144460	−	0.989511i	$-0.453855\pi$
$587$	7.00000	0.288921	0.144460	+	0.989511i	$0.453855\pi$
$588$	1.00000	0.0412393
$589$	−20.0000	−0.824086
$590$	0	0
$591$	22.0000	0.904959
$592$	4.00000	0.164399
$593$	33.0000	1.35515	0.677574	−	0.735455i	$-0.263031\pi$
$593$	33.0000	1.35515	0.677574	+	0.735455i	$0.263031\pi$
$594$	25.0000	1.02576
$595$	0	0
$596$	8.00000	0.327693
$597$	−2.00000	−0.0818546
$598$	2.00000	0.0817861
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	−8.00000	−0.326056
$603$	−2.00000	−0.0814463
$604$	−24.0000	−0.976546
$605$	0	0
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	5.00000	0.202777
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	−12.0000	−0.485468
$612$	10.0000	0.404226
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	−5.00000	−0.201456
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−16.0000	−0.643614
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	−10.0000	−0.400963
$623$	−15.0000	−0.600962
$624$	2.00000	0.0800641
$625$	0	0
$626$	−10.0000	−0.399680
$627$	−25.0000	−0.998404
$628$	8.00000	0.319235
$629$	−20.0000	−0.797452
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−10.0000	−0.397779
$633$	13.0000	0.516704
$634$	20.0000	0.794301
$635$	0	0
$636$	−4.00000	−0.158610
$637$	2.00000	0.0792429
$638$	10.0000	0.395904
$639$	−4.00000	−0.158238
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	11.0000	0.434135
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	−25.0000	−0.983612
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	1.00000	0.0392837
$649$	−20.0000	−0.785069
$650$	0	0
$651$	−4.00000	−0.156772
$652$	17.0000	0.665771
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−5.00000	−0.195217
$657$	30.0000	1.17041
$658$	−6.00000	−0.233904
$659$	49.0000	1.90877	0.954384	−	0.298580i	$-0.0965131\pi$
$659$	49.0000	1.90877	0.954384	+	0.298580i	$0.0965131\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	19.0000	0.738456
$663$	−10.0000	−0.388368
$664$	3.00000	0.116423
$665$	0	0
$666$	−8.00000	−0.309994
$667$	−2.00000	−0.0774403
$668$	−24.0000	−0.928588
$669$	10.0000	0.386622
$670$	0	0
$671$	−50.0000	−1.93023
$672$	1.00000	0.0385758
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	23.0000	0.885927
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	−13.0000	−0.499262
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−12.0000	−0.459841
$682$	20.0000	0.765840
$683$	−41.0000	−1.56882	−0.784411	−	0.620242i	$-0.787034\pi$
$683$	−41.0000	−1.56882	−0.784411	+	0.620242i	$0.787034\pi$
$684$	−10.0000	−0.382360
$685$	0	0
$686$	1.00000	0.0381802
$687$	2.00000	0.0763048
$688$	−8.00000	−0.304997
$689$	−8.00000	−0.304776
$690$	0	0
$691$	21.0000	0.798878	0.399439	−	0.916760i	$-0.369205\pi$
$691$	21.0000	0.798878	0.399439	+	0.916760i	$0.369205\pi$
$692$	−2.00000	−0.0760286
$693$	10.0000	0.379869
$694$	23.0000	0.873068
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	25.0000	0.946943
$698$	28.0000	1.05982
$699$	−18.0000	−0.680823
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	−10.0000	−0.377426
$703$	20.0000	0.754314
$704$	−5.00000	−0.188445
$705$	0	0
$706$	14.0000	0.526897
$707$	0	0
$708$	4.00000	0.150329
$709$	−36.0000	−1.35201	−0.676004	−	0.736898i	$-0.736290\pi$
$709$	−36.0000	−1.35201	−0.676004	+	0.736898i	$0.736290\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	−15.0000	−0.562149
$713$	−4.00000	−0.149801
$714$	−5.00000	−0.187120
$715$	0	0
$716$	−17.0000	−0.635320
$717$	−8.00000	−0.298765
$718$	−18.0000	−0.671754
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	6.00000	0.223297
$723$	7.00000	0.260333
$724$	−14.0000	−0.520306
$725$	0	0
$726$	14.0000	0.519589
$727$	−6.00000	−0.222528	−0.111264	−	0.993791i	$-0.535490\pi$
$727$	−6.00000	−0.222528	−0.111264	+	0.993791i	$0.535490\pi$
$728$	2.00000	0.0741249
$729$	13.0000	0.481481
$730$	0	0
$731$	40.0000	1.47945
$732$	10.0000	0.369611
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	1.00000	0.0368605
$737$	−5.00000	−0.184177
$738$	10.0000	0.368105
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	−4.00000	−0.146845
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	4.00000	0.146450
$747$	−6.00000	−0.219529
$748$	25.0000	0.914091
$749$	11.0000	0.401931
$750$	0	0
$751$	−42.0000	−1.53260	−0.766301	−	0.642482i	$-0.777905\pi$
$751$	−42.0000	−1.53260	−0.766301	+	0.642482i	$0.777905\pi$
$752$	−6.00000	−0.218797
$753$	−5.00000	−0.182210
$754$	−4.00000	−0.145671
$755$	0	0
$756$	−5.00000	−0.181848
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−15.0000	−0.544825
$759$	−5.00000	−0.181489
$760$	0	0
$761$	19.0000	0.688749	0.344375	−	0.938832i	$-0.388091\pi$
$761$	19.0000	0.688749	0.344375	+	0.938832i	$0.388091\pi$
$762$	6.00000	0.217357
$763$	−2.00000	−0.0724049
$764$	−18.0000	−0.651217
$765$	0	0
$766$	−6.00000	−0.216789
$767$	8.00000	0.288863
$768$	1.00000	0.0360844
$769$	−41.0000	−1.47850	−0.739249	−	0.673432i	$-0.764819\pi$
$769$	−41.0000	−1.47850	−0.739249	+	0.673432i	$0.764819\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	−11.0000	−0.395899
$773$	20.0000	0.719350	0.359675	−	0.933078i	$-0.382888\pi$
$773$	20.0000	0.719350	0.359675	+	0.933078i	$0.382888\pi$
$774$	16.0000	0.575108
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	4.00000	0.143499
$778$	−24.0000	−0.860442
$779$	−25.0000	−0.895718
$780$	0	0
$781$	−10.0000	−0.357828
$782$	−5.00000	−0.178800
$783$	10.0000	0.357371
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	22.0000	0.783718
$789$	18.0000	0.640817
$790$	0	0
$791$	−13.0000	−0.462227
$792$	10.0000	0.355335
$793$	20.0000	0.710221
$794$	−10.0000	−0.354887
$795$	0	0
$796$	−2.00000	−0.0708881
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	5.00000	0.176998
$799$	30.0000	1.06132
$800$	0	0
$801$	30.0000	1.06000
$802$	33.0000	1.16527
$803$	75.0000	2.64669
$804$	1.00000	0.0352673
$805$	0	0
$806$	−8.00000	−0.281788
$807$	2.00000	0.0704033
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−2.00000	−0.0701862
$813$	2.00000	0.0701431
$814$	−20.0000	−0.701000
$815$	0	0
$816$	−5.00000	−0.175035
$817$	−40.0000	−1.39942
$818$	19.0000	0.664319
$819$	−4.00000	−0.139771
$820$	0	0
$821$	16.0000	0.558404	0.279202	−	0.960232i	$-0.409930\pi$
$821$	16.0000	0.558404	0.279202	+	0.960232i	$0.409930\pi$
$822$	1.00000	0.0348790
$823$	18.0000	0.627441	0.313720	−	0.949515i	$-0.398425\pi$
$823$	18.0000	0.627441	0.313720	+	0.949515i	$0.398425\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	4.00000	0.139178
$827$	−39.0000	−1.35616	−0.678081	−	0.734987i	$-0.737188\pi$
$827$	−39.0000	−1.35616	−0.678081	+	0.734987i	$0.737188\pi$
$828$	−2.00000	−0.0695048
$829$	36.0000	1.25033	0.625166	−	0.780492i	$-0.285031\pi$
$829$	36.0000	1.25033	0.625166	+	0.780492i	$0.285031\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	2.00000	0.0693375
$833$	−5.00000	−0.173240
$834$	17.0000	0.588662
$835$	0	0
$836$	−25.0000	−0.864643
$837$	20.0000	0.691301
$838$	−23.0000	−0.794522
$839$	−14.0000	−0.483334	−0.241667	−	0.970359i	$-0.577694\pi$
$839$	−14.0000	−0.483334	−0.241667	+	0.970359i	$0.577694\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	8.00000	0.275698
$843$	22.0000	0.757720
$844$	13.0000	0.447478
$845$	0	0
$846$	12.0000	0.412568
$847$	14.0000	0.481046
$848$	−4.00000	−0.137361
$849$	−5.00000	−0.171600
$850$	0	0
$851$	4.00000	0.137118
$852$	2.00000	0.0685189
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	11.0000	0.375972
$857$	19.0000	0.649028	0.324514	−	0.945881i	$-0.394799\pi$
$857$	19.0000	0.649028	0.324514	+	0.945881i	$0.394799\pi$
$858$	−10.0000	−0.341394
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	0	0
$861$	−5.00000	−0.170400
$862$	−28.0000	−0.953684
$863$	−20.0000	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−29.0000	−0.985460
$867$	8.00000	0.271694
$868$	−4.00000	−0.135769
$869$	50.0000	1.69613
$870$	0	0
$871$	2.00000	0.0677674
$872$	−2.00000	−0.0677285
$873$	4.00000	0.135379
$874$	5.00000	0.169128
$875$	0	0
$876$	−15.0000	−0.506803
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	24.0000	0.809961
$879$	−26.0000	−0.876958
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	−2.00000	−0.0673435
$883$	−37.0000	−1.24515	−0.622575	−	0.782560i	$-0.713913\pi$
$883$	−37.0000	−1.24515	−0.622575	+	0.782560i	$0.713913\pi$
$884$	−10.0000	−0.336336
$885$	0	0
$886$	1.00000	0.0335957
$887$	26.0000	0.872995	0.436497	−	0.899706i	$-0.356219\pi$
$887$	26.0000	0.872995	0.436497	+	0.899706i	$0.356219\pi$
$888$	4.00000	0.134231
$889$	6.00000	0.201234
$890$	0	0
$891$	−5.00000	−0.167506
$892$	10.0000	0.334825
$893$	−30.0000	−1.00391
$894$	8.00000	0.267560
$895$	0	0
$896$	1.00000	0.0334077
$897$	2.00000	0.0667781
$898$	41.0000	1.36819
$899$	8.00000	0.266815
$900$	0	0
$901$	20.0000	0.666297
$902$	25.0000	0.832409
$903$	−8.00000	−0.266223
$904$	−13.0000	−0.432374
$905$	0	0
$906$	−24.0000	−0.797347
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	5.00000	0.165567
$913$	−15.0000	−0.496428
$914$	5.00000	0.165385
$915$	0	0
$916$	2.00000	0.0660819
$917$	0	0
$918$	25.0000	0.825123
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	−7.00000	−0.230658
$922$	−18.0000	−0.592798
$923$	4.00000	0.131662
$924$	−5.00000	−0.164488
$925$	0	0
$926$	−20.0000	−0.657241
$927$	32.0000	1.05102
$928$	−2.00000	−0.0656532
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	−18.0000	−0.589610
$933$	−10.0000	−0.327385
$934$	36.0000	1.17796
$935$	0	0
$936$	−4.00000	−0.130744
$937$	−43.0000	−1.40475	−0.702374	−	0.711808i	$-0.747877\pi$
$937$	−43.0000	−1.40475	−0.702374	+	0.711808i	$0.747877\pi$
$938$	1.00000	0.0326512
$939$	−10.0000	−0.326338
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	8.00000	0.260654
$943$	−5.00000	−0.162822
$944$	4.00000	0.130189
$945$	0	0
$946$	40.0000	1.30051
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−10.0000	−0.324785
$949$	−30.0000	−0.973841
$950$	0	0
$951$	20.0000	0.648544
$952$	−5.00000	−0.162051
$953$	45.0000	1.45769	0.728846	−	0.684677i	$-0.240057\pi$
$953$	45.0000	1.45769	0.728846	+	0.684677i	$0.240057\pi$
$954$	8.00000	0.259010
$955$	0	0
$956$	−8.00000	−0.258738
$957$	10.0000	0.323254
$958$	−18.0000	−0.581554
$959$	1.00000	0.0322917
$960$	0	0
$961$	−15.0000	−0.483871
$962$	8.00000	0.257930
$963$	−22.0000	−0.708940
$964$	7.00000	0.225455
$965$	0	0
$966$	1.00000	0.0321745
$967$	−38.0000	−1.22200	−0.610999	−	0.791632i	$-0.709232\pi$
$967$	−38.0000	−1.22200	−0.610999	+	0.791632i	$0.709232\pi$
$968$	14.0000	0.449977
$969$	−25.0000	−0.803116
$970$	0	0
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	16.0000	0.513200
$973$	17.0000	0.544995
$974$	30.0000	0.961262
$975$	0	0
$976$	10.0000	0.320092
$977$	−47.0000	−1.50366	−0.751832	−	0.659355i	$-0.770829\pi$
$977$	−47.0000	−1.50366	−0.751832	+	0.659355i	$0.770829\pi$
$978$	17.0000	0.543600
$979$	75.0000	2.39701
$980$	0	0
$981$	4.00000	0.127710
$982$	−36.0000	−1.14881
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	−5.00000	−0.159394
$985$	0	0
$986$	10.0000	0.318465
$987$	−6.00000	−0.190982
$988$	10.0000	0.318142
$989$	−8.00000	−0.254385
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	−4.00000	−0.127000
$993$	19.0000	0.602947
$994$	2.00000	0.0634361
$995$	0	0
$996$	3.00000	0.0950586
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	20.0000	0.633089
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8050.2.a.s.1.1	yes	1
5.4	even	2		8050.2.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8050.2.a.c.1.1	✓	1	5.4	even	2
8050.2.a.s.1.1	yes	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:	\( N \)	\(=\)	\( 8050 = 2 \cdot 5^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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