LMFDB - Embedded newform 850.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [850,2,Mod(1,850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	850.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:	$6.78728417181$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 34)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +6.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +8.00000 q^{21} -6.00000 q^{22} -2.00000 q^{24} +2.00000 q^{26} -4.00000 q^{27} +4.00000 q^{28} -4.00000 q^{31} -1.00000 q^{32} +12.0000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} +4.00000 q^{38} -4.00000 q^{39} +6.00000 q^{41} -8.00000 q^{42} -8.00000 q^{43} +6.00000 q^{44} +2.00000 q^{48} +9.00000 q^{49} +2.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +4.00000 q^{54} -4.00000 q^{56} -8.00000 q^{57} -4.00000 q^{61} +4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -12.0000 q^{66} -8.00000 q^{67} +1.00000 q^{68} -1.00000 q^{72} -2.00000 q^{73} -4.00000 q^{74} -4.00000 q^{76} +24.0000 q^{77} +4.00000 q^{78} +8.00000 q^{79} -11.0000 q^{81} -6.00000 q^{82} +8.00000 q^{84} +8.00000 q^{86} -6.00000 q^{88} -6.00000 q^{89} -8.00000 q^{91} -8.00000 q^{93} -2.00000 q^{96} -14.0000 q^{97} -9.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	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	$0.589456\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	−6.00000	−1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	2.00000	0.392232
$27$	−4.00000	−0.769800
$28$	4.00000	0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	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
$33$	12.0000	2.08893
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	4.00000	0.648886
$39$	−4.00000	−0.640513
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−8.00000	−1.23443
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	2.00000	0.288675
$49$	9.00000	1.28571
$50$	0	0
$51$	2.00000	0.280056
$52$	−2.00000	−0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	−4.00000	−0.534522
$57$	−8.00000	−1.05963
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	4.00000	0.508001
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	−12.0000	−1.47710
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−4.00000	−0.458831
$77$	24.0000	2.73505
$78$	4.00000	0.452911
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−6.00000	−0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	8.00000	0.872872
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	−6.00000	−0.639602
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	0	0
$96$	−2.00000	−0.204124
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−9.00000	−0.909137
$99$	6.00000	0.603023
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	−2.00000	−0.198030
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	−4.00000	−0.384900
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	4.00000	0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	25.0000	2.27273
$122$	4.00000	0.362143
$123$	12.0000	1.08200
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−4.00000	−0.356348
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	−16.0000	−1.40872
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	12.0000	1.04447
$133$	−16.0000	−1.38738
$134$	8.00000	0.691095
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.0000	−1.00349
$144$	1.00000	0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	18.0000	1.48461
$148$	4.00000	0.328798
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000	0.324443
$153$	1.00000	0.0808452
$154$	−24.0000	−1.93398
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−8.00000	−0.636446
$159$	12.0000	0.951662
$160$	0	0
$161$	0	0
$162$	11.0000	0.864242
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−8.00000	−0.617213
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−8.00000	−0.609994
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	0	0
$178$	6.00000	0.449719
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	8.00000	0.592999
$183$	−8.00000	−0.591377
$184$	0	0
$185$	0	0
$186$	8.00000	0.586588
$187$	6.00000	0.438763
$188$	0	0
$189$	−16.0000	−1.16383
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	2.00000	0.144338
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	9.00000	0.642857
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	−6.00000	−0.426401
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	−18.0000	−1.26648
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	−16.0000	−1.11477
$207$	0	0
$208$	−2.00000	−0.138675
$209$	−24.0000	−1.66011
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−6.00000	−0.410152
$215$	0	0
$216$	4.00000	0.272166
$217$	−16.0000	−1.08615
$218$	16.0000	1.08366
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−2.00000	−0.134535
$222$	−8.00000	−0.536925
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−6.00000	−0.399114
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	−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
$231$	48.0000	3.15817
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	0	0
$237$	16.0000	1.03931
$238$	−4.00000	−0.259281
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−25.0000	−1.60706
$243$	−10.0000	−0.641500
$244$	−4.00000	−0.256074
$245$	0	0
$246$	−12.0000	−0.765092
$247$	8.00000	0.509028
$248$	4.00000	0.254000
$249$	0	0
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	16.0000	0.996116
$259$	16.0000	0.994192
$260$	0	0
$261$	0	0
$262$	6.00000	0.370681
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	16.0000	0.981023
$267$	−12.0000	−0.734388
$268$	−8.00000	−0.488678
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	1.00000	0.0606339
$273$	−16.0000	−0.968364
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−2.00000	−0.119952
$279$	−4.00000	−0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	0	0
$285$	0	0
$286$	12.0000	0.709575
$287$	24.0000	1.41668
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	−28.0000	−1.64139
$292$	−2.00000	−0.117041
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−18.0000	−1.04978
$295$	0	0
$296$	−4.00000	−0.232495
$297$	−24.0000	−1.39262
$298$	−6.00000	−0.347571
$299$	0	0
$300$	0	0
$301$	−32.0000	−1.84445
$302$	16.0000	0.920697
$303$	36.0000	2.06815
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	24.0000	1.36753
$309$	32.0000	1.82042
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	4.00000	0.226455
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	−12.0000	−0.672927
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	−4.00000	−0.222566
$324$	−11.0000	−0.611111
$325$	0	0
$326$	2.00000	0.110770
$327$	−32.0000	−1.76960
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	12.0000	0.656611
$335$	0	0
$336$	8.00000	0.436436
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	9.00000	0.489535
$339$	12.0000	0.651751
$340$	0	0
$341$	−24.0000	−1.29967
$342$	4.00000	0.216295
$343$	8.00000	0.431959
$344$	8.00000	0.431331
$345$	0	0
$346$	24.0000	1.29025
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	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
$356$	−6.00000	−0.317999
$357$	8.00000	0.423405
$358$	−12.0000	−0.634220
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	4.00000	0.210235
$363$	50.0000	2.62432
$364$	−8.00000	−0.419314
$365$	0	0
$366$	8.00000	0.418167
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	24.0000	1.24602
$372$	−8.00000	−0.414781
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	0	0
$377$	0	0
$378$	16.0000	0.822951
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	0	0
$381$	32.0000	1.63941
$382$	24.0000	1.22795
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	−10.0000	−0.508987
$387$	−8.00000	−0.406663
$388$	−14.0000	−0.710742
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	0	0
$392$	−9.00000	−0.454569
$393$	−12.0000	−0.605320
$394$	−12.0000	−0.604551
$395$	0	0
$396$	6.00000	0.301511
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	16.0000	0.802008
$399$	−32.0000	−1.60200
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	16.0000	0.798007
$403$	8.00000	0.398508
$404$	18.0000	0.895533
$405$	0	0
$406$	0	0
$407$	24.0000	1.18964
$408$	−2.00000	−0.0990148
$409$	−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
$411$	−12.0000	−0.591916
$412$	16.0000	0.788263
$413$	0	0
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	4.00000	0.195881
$418$	24.0000	1.17388
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	10.0000	0.486792
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	−16.0000	−0.774294
$428$	6.00000	0.290021
$429$	−24.0000	−1.15873
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	−4.00000	−0.192450
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	−16.0000	−0.766261
$437$	0	0
$438$	4.00000	0.191127
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	2.00000	0.0951303
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	8.00000	0.378811
$447$	12.0000	0.567581
$448$	4.00000	0.188982
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	6.00000	0.282216
$453$	−32.0000	−1.50349
$454$	−6.00000	−0.281594
$455$	0	0
$456$	8.00000	0.374634
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	−14.0000	−0.654177
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	−48.0000	−2.23316
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	0	0
$466$	18.0000	0.833834
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−2.00000	−0.0924500
$469$	−32.0000	−1.47762
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	−48.0000	−2.20704
$474$	−16.0000	−0.734904
$475$	0	0
$476$	4.00000	0.183340
$477$	6.00000	0.274721
$478$	−24.0000	−1.09773
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	10.0000	0.455488
$483$	0	0
$484$	25.0000	1.13636
$485$	0	0
$486$	10.0000	0.453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	4.00000	0.181071
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	12.0000	0.541002
$493$	0	0
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	24.0000	1.07117
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	0	0
$507$	−18.0000	−0.799408
$508$	16.0000	0.709885
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	−1.00000	−0.0441942
$513$	16.0000	0.706417
$514$	6.00000	0.264649
$515$	0	0
$516$	−16.0000	−0.704361
$517$	0	0
$518$	−16.0000	−0.703000
$519$	−48.0000	−2.10697
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	24.0000	1.04645
$527$	−4.00000	−0.174243
$528$	12.0000	0.522233
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	−16.0000	−0.693688
$533$	−12.0000	−0.519778
$534$	12.0000	0.519291
$535$	0	0
$536$	8.00000	0.345547
$537$	24.0000	1.03568
$538$	24.0000	1.03471
$539$	54.0000	2.32594
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−8.00000	−0.343629
$543$	−8.00000	−0.343313
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	16.0000	0.684737
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−6.00000	−0.256307
$549$	−4.00000	−0.170716
$550$	0	0
$551$	0	0
$552$	0	0
$553$	32.0000	1.36078
$554$	8.00000	0.339887
$555$	0	0
$556$	2.00000	0.0848189
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	4.00000	0.169334
$559$	16.0000	0.676728
$560$	0	0
$561$	12.0000	0.506640
$562$	−6.00000	−0.253095
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	14.0000	0.588464
$567$	−44.0000	−1.84783
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	−12.0000	−0.501745
$573$	−48.0000	−2.00523
$574$	−24.0000	−1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−1.00000	−0.0415945
$579$	20.0000	0.831172
$580$	0	0
$581$	0	0
$582$	28.0000	1.16064
$583$	36.0000	1.49097
$584$	2.00000	0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	18.0000	0.742307
$589$	16.0000	0.659269
$590$	0	0
$591$	24.0000	0.987228
$592$	4.00000	0.164399
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	24.0000	0.984732
$595$	0	0
$596$	6.00000	0.245770
$597$	−32.0000	−1.30967
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	32.0000	1.30422
$603$	−8.00000	−0.325785
$604$	−16.0000	−0.651031
$605$	0	0
$606$	−36.0000	−1.46240
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	0	0
$611$	0	0
$612$	1.00000	0.0404226
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	−24.0000	−0.966988
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	−32.0000	−1.28723
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	0	0
$621$	0	0
$622$	−12.0000	−0.481156
$623$	−24.0000	−0.961540
$624$	−4.00000	−0.160128
$625$	0	0
$626$	−34.0000	−1.35891
$627$	−48.0000	−1.91694
$628$	−14.0000	−0.558661
$629$	4.00000	0.159490
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−8.00000	−0.318223
$633$	−20.0000	−0.794929
$634$	−12.0000	−0.476581
$635$	0	0
$636$	12.0000	0.475831
$637$	−18.0000	−0.713186
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	−12.0000	−0.473602
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	11.0000	0.432121
$649$	0	0
$650$	0	0
$651$	−32.0000	−1.25418
$652$	−2.00000	−0.0783260
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	32.0000	1.25130
$655$	0	0
$656$	6.00000	0.234261
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	16.0000	0.621858
$663$	−4.00000	−0.155347
$664$	0	0
$665$	0	0
$666$	−4.00000	−0.154997
$667$	0	0
$668$	−12.0000	−0.464294
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−24.0000	−0.926510
$672$	−8.00000	−0.308607
$673$	−38.0000	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	−9.00000	−0.346154
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	−12.0000	−0.460857
$679$	−56.0000	−2.14908
$680$	0	0
$681$	12.0000	0.459841
$682$	24.0000	0.919007
$683$	−30.0000	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−8.00000	−0.305441
$687$	28.0000	1.06827
$688$	−8.00000	−0.304997
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−24.0000	−0.912343
$693$	24.0000	0.911685
$694$	18.0000	0.683271
$695$	0	0
$696$	0	0
$697$	6.00000	0.227266
$698$	−26.0000	−0.984115
$699$	−36.0000	−1.36165
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	−8.00000	−0.301941
$703$	−16.0000	−0.603451
$704$	6.00000	0.226134
$705$	0	0
$706$	6.00000	0.225813
$707$	72.0000	2.70784
$708$	0	0
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	6.00000	0.224860
$713$	0	0
$714$	−8.00000	−0.299392
$715$	0	0
$716$	12.0000	0.448461
$717$	48.0000	1.79259
$718$	−24.0000	−0.895672
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	64.0000	2.38348
$722$	3.00000	0.111648
$723$	−20.0000	−0.743808
$724$	−4.00000	−0.148659
$725$	0	0
$726$	−50.0000	−1.85567
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	8.00000	0.296500
$729$	13.0000	0.481481
$730$	0	0
$731$	−8.00000	−0.295891
$732$	−8.00000	−0.295689
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	0	0
$737$	−48.0000	−1.76810
$738$	−6.00000	−0.220863
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	−24.0000	−0.881068
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−22.0000	−0.805477
$747$	0	0
$748$	6.00000	0.219382
$749$	24.0000	0.876941
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	−48.0000	−1.74922
$754$	0	0
$755$	0	0
$756$	−16.0000	−0.581914
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	−14.0000	−0.508503
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−32.0000	−1.15924
$763$	−64.0000	−2.31696
$764$	−24.0000	−0.868290
$765$	0	0
$766$	−24.0000	−0.867155
$767$	0	0
$768$	2.00000	0.0721688
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	10.0000	0.359908
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	14.0000	0.502571
$777$	32.0000	1.14799
$778$	−30.0000	−1.07555
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	12.0000	0.428026
$787$	46.0000	1.63972	0.819861	−	0.572562i	$-0.194050\pi$
$787$	46.0000	1.63972	0.819861	+	0.572562i	$0.194050\pi$
$788$	12.0000	0.427482
$789$	−48.0000	−1.70885
$790$	0	0
$791$	24.0000	0.853342
$792$	−6.00000	−0.213201
$793$	8.00000	0.284088
$794$	20.0000	0.709773
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	32.0000	1.13279
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	−30.0000	−1.05934
$803$	−12.0000	−0.423471
$804$	−16.0000	−0.564276
$805$	0	0
$806$	−8.00000	−0.281788
$807$	−48.0000	−1.68968
$808$	−18.0000	−0.633238
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	−24.0000	−0.841200
$815$	0	0
$816$	2.00000	0.0700140
$817$	32.0000	1.11954
$818$	10.0000	0.349642
$819$	−8.00000	−0.279543
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	12.0000	0.418548
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	0	0
$827$	−6.00000	−0.208640	−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000	−0.208640	−0.104320	+	0.994544i	$0.533267\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	−2.00000	−0.0693375
$833$	9.00000	0.311832
$834$	−4.00000	−0.138509
$835$	0	0
$836$	−24.0000	−0.830057
$837$	16.0000	0.553041
$838$	30.0000	1.03633
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−2.00000	−0.0689246
$843$	12.0000	0.413302
$844$	−10.0000	−0.344214
$845$	0	0
$846$	0	0
$847$	100.000	3.43604
$848$	6.00000	0.206041
$849$	−28.0000	−0.960958
$850$	0	0
$851$	0	0
$852$	0	0
$853$	40.0000	1.36957	0.684787	−	0.728743i	$-0.259895\pi$
$853$	40.0000	1.36957	0.684787	+	0.728743i	$0.259895\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	−6.00000	−0.205076
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	24.0000	0.819346
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	0	0
$861$	48.0000	1.63584
$862$	−24.0000	−0.817443
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	2.00000	0.0679628
$867$	2.00000	0.0679236
$868$	−16.0000	−0.543075
$869$	48.0000	1.62829
$870$	0	0
$871$	16.0000	0.542139
$872$	16.0000	0.541828
$873$	−14.0000	−0.473828
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	−8.00000	−0.269987
$879$	−12.0000	−0.404750
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−9.00000	−0.303046
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−24.0000	−0.806296
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−8.00000	−0.268462
$889$	64.0000	2.14649
$890$	0	0
$891$	−66.0000	−2.21108
$892$	−8.00000	−0.267860
$893$	0	0
$894$	−12.0000	−0.401340
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	18.0000	0.600668
$899$	0	0
$900$	0	0
$901$	6.00000	0.199889
$902$	−36.0000	−1.19867
$903$	−64.0000	−2.12979
$904$	−6.00000	−0.199557
$905$	0	0
$906$	32.0000	1.06313
$907$	58.0000	1.92586	0.962929	−	0.269754i	$-0.0869425\pi$
$907$	58.0000	1.92586	0.962929	+	0.269754i	$0.0869425\pi$
$908$	6.00000	0.199117
$909$	18.0000	0.597022
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−8.00000	−0.264906
$913$	0	0
$914$	26.0000	0.860004
$915$	0	0
$916$	14.0000	0.462573
$917$	−24.0000	−0.792550
$918$	4.00000	0.132020
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	−40.0000	−1.31804
$922$	6.00000	0.197599
$923$	0	0
$924$	48.0000	1.57908
$925$	0	0
$926$	−16.0000	−0.525793
$927$	16.0000	0.525509
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	−18.0000	−0.589610
$933$	24.0000	0.785725
$934$	12.0000	0.392652
$935$	0	0
$936$	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	$0.103712\pi$
$938$	32.0000	1.04484
$939$	68.0000	2.21910
$940$	0	0
$941$	−36.0000	−1.17357	−0.586783	−	0.809744i	$-0.699606\pi$
$941$	−36.0000	−1.17357	−0.586783	+	0.809744i	$0.699606\pi$
$942$	28.0000	0.912289
$943$	0	0
$944$	0	0
$945$	0	0
$946$	48.0000	1.56061
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	16.0000	0.519656
$949$	4.00000	0.129845
$950$	0	0
$951$	24.0000	0.778253
$952$	−4.00000	−0.129641
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	8.00000	0.257930
$963$	6.00000	0.193347
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	−25.0000	−0.803530
$969$	−8.00000	−0.256997
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	−10.0000	−0.320750
$973$	8.00000	0.256468
$974$	8.00000	0.256337
$975$	0	0
$976$	−4.00000	−0.128037
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	4.00000	0.127906
$979$	−36.0000	−1.15056
$980$	0	0
$981$	−16.0000	−0.510841
$982$	12.0000	0.382935
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	−12.0000	−0.382546
$985$	0	0
$986$	0	0
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	4.00000	0.127000
$993$	−32.0000	−1.01549
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−14.0000	−0.443162
$999$	−16.0000	−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.a.e.1.1		1
3.2	odd	2		7650.2.a.ci.1.1		1
4.3	odd	2		6800.2.a.b.1.1		1
5.2	odd	4		850.2.c.b.749.1		2
5.3	odd	4		850.2.c.b.749.2		2
5.4	even	2		34.2.a.a.1.1	✓	1
15.14	odd	2		306.2.a.a.1.1		1
20.19	odd	2		272.2.a.d.1.1		1
35.34	odd	2		1666.2.a.m.1.1		1
40.19	odd	2		1088.2.a.d.1.1		1
40.29	even	2		1088.2.a.l.1.1		1
55.54	odd	2		4114.2.a.a.1.1		1
60.59	even	2		2448.2.a.k.1.1		1
65.64	even	2		5746.2.a.b.1.1		1
85.4	even	4		578.2.b.a.577.2		2
85.9	even	8		578.2.c.e.251.1		4
85.14	odd	16		578.2.d.e.179.2		8
85.19	even	8		578.2.c.e.327.1		4
85.24	odd	16		578.2.d.e.423.2		8
85.29	odd	16		578.2.d.e.399.1		8
85.39	odd	16		578.2.d.e.399.2		8
85.44	odd	16		578.2.d.e.423.1		8
85.49	even	8		578.2.c.e.327.2		4
85.54	odd	16		578.2.d.e.179.1		8
85.59	even	8		578.2.c.e.251.2		4
85.64	even	4		578.2.b.a.577.1		2
85.74	odd	16		578.2.d.e.155.1		8
85.79	odd	16		578.2.d.e.155.2		8
85.84	even	2		578.2.a.a.1.1		1
120.29	odd	2		9792.2.a.y.1.1		1
120.59	even	2		9792.2.a.bj.1.1		1
255.254	odd	2		5202.2.a.d.1.1		1
340.339	odd	2		4624.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
34.2.a.a.1.1	✓	1	5.4	even	2
272.2.a.d.1.1		1	20.19	odd	2
306.2.a.a.1.1		1	15.14	odd	2
578.2.a.a.1.1		1	85.84	even	2
578.2.b.a.577.1		2	85.64	even	4
578.2.b.a.577.2		2	85.4	even	4
578.2.c.e.251.1		4	85.9	even	8
578.2.c.e.251.2		4	85.59	even	8
578.2.c.e.327.1		4	85.19	even	8
578.2.c.e.327.2		4	85.49	even	8
578.2.d.e.155.1		8	85.74	odd	16
578.2.d.e.155.2		8	85.79	odd	16
578.2.d.e.179.1		8	85.54	odd	16
578.2.d.e.179.2		8	85.14	odd	16
578.2.d.e.399.1		8	85.29	odd	16
578.2.d.e.399.2		8	85.39	odd	16
578.2.d.e.423.1		8	85.44	odd	16
578.2.d.e.423.2		8	85.24	odd	16
850.2.a.e.1.1		1	1.1	even	1	trivial
850.2.c.b.749.1		2	5.2	odd	4
850.2.c.b.749.2		2	5.3	odd	4
1088.2.a.d.1.1		1	40.19	odd	2
1088.2.a.l.1.1		1	40.29	even	2
1666.2.a.m.1.1		1	35.34	odd	2
2448.2.a.k.1.1		1	60.59	even	2
4114.2.a.a.1.1		1	55.54	odd	2
4624.2.a.a.1.1		1	340.339	odd	2
5202.2.a.d.1.1		1	255.254	odd	2
5746.2.a.b.1.1		1	65.64	even	2
6800.2.a.b.1.1		1	4.3	odd	2
7650.2.a.ci.1.1		1	3.2	odd	2
9792.2.a.y.1.1		1	120.29	odd	2
9792.2.a.bj.1.1		1	120.59	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:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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