LMFDB - Embedded newform 850.2.a.b.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 170)
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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{12} +1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +2.00000 q^{18} -1.00000 q^{19} +2.00000 q^{21} +6.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} +5.00000 q^{27} -2.00000 q^{28} -3.00000 q^{29} +5.00000 q^{31} -1.00000 q^{32} -1.00000 q^{34} -2.00000 q^{36} -8.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} +6.00000 q^{41} -2.00000 q^{42} +10.0000 q^{43} -6.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{51} +1.00000 q^{52} +3.00000 q^{53} -5.00000 q^{54} +2.00000 q^{56} +1.00000 q^{57} +3.00000 q^{58} +3.00000 q^{59} +11.0000 q^{61} -5.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{67} +1.00000 q^{68} -6.00000 q^{69} +9.00000 q^{71} +2.00000 q^{72} -11.0000 q^{73} +8.00000 q^{74} -1.00000 q^{76} +1.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} -10.0000 q^{86} +3.00000 q^{87} +15.0000 q^{89} -2.00000 q^{91} +6.00000 q^{92} -5.00000 q^{93} -3.00000 q^{94} +1.00000 q^{96} +7.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.00000	0.471405
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	5.00000	0.962250
$28$	−2.00000	−0.377964
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	1.00000	0.162221
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−2.00000	−0.308607
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	0	0
$46$	−6.00000	−0.884652
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−1.00000	−0.140028
$52$	1.00000	0.138675
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	2.00000	0.267261
$57$	1.00000	0.132453
$58$	3.00000	0.393919
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	−5.00000	−0.635001
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	1.00000	0.121268
$69$	−6.00000	−0.722315
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	2.00000	0.235702
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	1.00000	0.113228
$79$	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
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−10.0000	−1.07833
$87$	3.00000	0.321634
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	6.00000	0.625543
$93$	−5.00000	−0.518476
$94$	−3.00000	−0.309426
$95$	0	0
$96$	1.00000	0.102062
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	1.00000	0.0990148
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−3.00000	−0.291386
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	5.00000	0.481125
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	−2.00000	−0.188982
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−3.00000	−0.278543
$117$	−2.00000	−0.184900
$118$	−3.00000	−0.276172
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−11.0000	−0.995893
$123$	−6.00000	−0.541002
$124$	5.00000	0.449013
$125$	0	0
$126$	−4.00000	−0.356348
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	−1.00000	−0.0883883
$129$	−10.0000	−0.880451
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	2.00000	0.172774
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	6.00000	0.510754
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	−9.00000	−0.755263
$143$	0	0
$144$	−2.00000	−0.166667
$145$	0	0
$146$	11.0000	0.910366
$147$	3.00000	0.247436
$148$	−8.00000	−0.657596
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	1.00000	0.0811107
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−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
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−12.0000	−0.945732
$162$	−1.00000	−0.0785674
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	−2.00000	−0.154303
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.00000	0.152944
$172$	10.0000	0.762493
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−3.00000	−0.227429
$175$	0	0
$176$	0	0
$177$	−3.00000	−0.225494
$178$	−15.0000	−1.12430
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	2.00000	0.148250
$183$	−11.0000	−0.813143
$184$	−6.00000	−0.442326
$185$	0	0
$186$	5.00000	0.366618
$187$	0	0
$188$	3.00000	0.218797
$189$	−10.0000	−0.727393
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−7.00000	−0.502571
$195$	0	0
$196$	−3.00000	−0.214286
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	−6.00000	−0.422159
$203$	6.00000	0.421117
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	8.00000	0.557386
$207$	−12.0000	−0.834058
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	3.00000	0.206041
$213$	−9.00000	−0.616670
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−5.00000	−0.340207
$217$	−10.0000	−0.678844
$218$	−11.0000	−0.745014
$219$	11.0000	0.743311
$220$	0	0
$221$	1.00000	0.0672673
$222$	−8.00000	−0.536925
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	9.00000	0.598671
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	1.00000	0.0662266
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	3.00000	0.196960
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	3.00000	0.195283
$237$	−8.00000	−0.519656
$238$	2.00000	0.129641
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	11.0000	0.707107
$243$	−16.0000	−1.02640
$244$	11.0000	0.704203
$245$	0	0
$246$	6.00000	0.382546
$247$	−1.00000	−0.0636285
$248$	−5.00000	−0.317500
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	11.0000	0.690201
$255$	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	$0.559920\pi$
$258$	10.0000	0.622573
$259$	16.0000	0.994192
$260$	0	0
$261$	6.00000	0.371391
$262$	18.0000	1.11204
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	−15.0000	−0.917985
$268$	−2.00000	−0.122169
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	1.00000	0.0606339
$273$	2.00000	0.121046
$274$	−18.0000	−1.08742
$275$	0	0
$276$	−6.00000	−0.361158
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−8.00000	−0.479808
$279$	−10.0000	−0.598684
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	3.00000	0.178647
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	9.00000	0.534052
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	2.00000	0.117851
$289$	1.00000	0.0588235
$290$	0	0
$291$	−7.00000	−0.410347
$292$	−11.0000	−0.643726
$293$	3.00000	0.175262	0.0876309	−	0.996153i	$-0.472070\pi$
$293$	3.00000	0.175262	0.0876309	+	0.996153i	$0.472070\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	8.00000	0.464991
$297$	0	0
$298$	12.0000	0.695141
$299$	6.00000	0.346989
$300$	0	0
$301$	−20.0000	−1.15278
$302$	22.0000	1.26596
$303$	−6.00000	−0.344691
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	2.00000	0.114332
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	1.00000	0.0566139
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	3.00000	0.168232
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	12.0000	0.668734
$323$	−1.00000	−0.0556415
$324$	1.00000	0.0555556
$325$	0	0
$326$	−16.0000	−0.886158
$327$	−11.0000	−0.608301
$328$	−6.00000	−0.331295
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−31.0000	−1.70391	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−31.0000	−1.70391	−0.851957	+	0.523612i	$0.824584\pi$
$332$	12.0000	0.658586
$333$	16.0000	0.876795
$334$	−18.0000	−0.984916
$335$	0	0
$336$	2.00000	0.109109
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	12.0000	0.652714
$339$	9.00000	0.488813
$340$	0	0
$341$	0	0
$342$	−2.00000	−0.108148
$343$	20.0000	1.07990
$344$	−10.0000	−0.539164
$345$	0	0
$346$	18.0000	0.967686
$347$	15.0000	0.805242	0.402621	−	0.915367i	$-0.368099\pi$
$347$	15.0000	0.805242	0.402621	+	0.915367i	$0.368099\pi$
$348$	3.00000	0.160817
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	3.00000	0.159448
$355$	0	0
$356$	15.0000	0.794998
$357$	2.00000	0.105851
$358$	12.0000	0.634220
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	10.0000	0.525588
$363$	11.0000	0.577350
$364$	−2.00000	−0.104828
$365$	0	0
$366$	11.0000	0.574979
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	6.00000	0.312772
$369$	−12.0000	−0.624695
$370$	0	0
$371$	−6.00000	−0.311504
$372$	−5.00000	−0.259238
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	0	0
$376$	−3.00000	−0.154713
$377$	−3.00000	−0.154508
$378$	10.0000	0.514344
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	11.0000	0.563547
$382$	−12.0000	−0.613973
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	−20.0000	−1.01666
$388$	7.00000	0.355371
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	3.00000	0.151523
$393$	18.0000	0.907980
$394$	−24.0000	−1.20910
$395$	0	0
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	−11.0000	−0.551380
$399$	−2.00000	−0.100125
$400$	0	0
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	−2.00000	−0.0997509
$403$	5.00000	0.249068
$404$	6.00000	0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	0	0
$408$	1.00000	0.0495074
$409$	29.0000	1.43396	0.716979	−	0.697095i	$-0.245524\pi$
$409$	29.0000	1.43396	0.716979	+	0.697095i	$0.245524\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	−8.00000	−0.394132
$413$	−6.00000	−0.295241
$414$	12.0000	0.589768
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−8.00000	−0.391762
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	−2.00000	−0.0973585
$423$	−6.00000	−0.291730
$424$	−3.00000	−0.145693
$425$	0	0
$426$	9.00000	0.436051
$427$	−22.0000	−1.06465
$428$	12.0000	0.580042
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	5.00000	0.240563
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	10.0000	0.480015
$435$	0	0
$436$	11.0000	0.526804
$437$	−6.00000	−0.287019
$438$	−11.0000	−0.525600
$439$	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
$441$	6.00000	0.285714
$442$	−1.00000	−0.0475651
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	−1.00000	−0.0473514
$447$	12.0000	0.567581
$448$	−2.00000	−0.0944911
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	−9.00000	−0.423324
$453$	22.0000	1.03365
$454$	3.00000	0.140797
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	22.0000	1.02799
$459$	5.00000	0.233380
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	19.0000	0.883005	0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000	0.883005	0.441502	+	0.897260i	$0.354446\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	9.00000	0.416917
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	−2.00000	−0.0924500
$469$	4.00000	0.184703
$470$	0	0
$471$	14.0000	0.645086
$472$	−3.00000	−0.138086
$473$	0	0
$474$	8.00000	0.367452
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	−6.00000	−0.274721
$478$	−6.00000	−0.274434
$479$	27.0000	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$479$	27.0000	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	10.0000	0.455488
$483$	12.0000	0.546019
$484$	−11.0000	−0.500000
$485$	0	0
$486$	16.0000	0.725775
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	−11.0000	−0.497947
$489$	−16.0000	−0.723545
$490$	0	0
$491$	−39.0000	−1.76005	−0.880023	−	0.474932i	$-0.842473\pi$
$491$	−39.0000	−1.76005	−0.880023	+	0.474932i	$0.842473\pi$
$492$	−6.00000	−0.270501
$493$	−3.00000	−0.135113
$494$	1.00000	0.0449921
$495$	0	0
$496$	5.00000	0.224507
$497$	−18.0000	−0.807410
$498$	12.0000	0.537733
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	12.0000	0.535586
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	0	0
$507$	12.0000	0.532939
$508$	−11.0000	−0.488046
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	22.0000	0.973223
$512$	−1.00000	−0.0441942
$513$	−5.00000	−0.220755
$514$	6.00000	0.264649
$515$	0	0
$516$	−10.0000	−0.440225
$517$	0	0
$518$	−16.0000	−0.703000
$519$	18.0000	0.790112
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	−6.00000	−0.262613
$523$	−2.00000	−0.0874539	−0.0437269	−	0.999044i	$-0.513923\pi$
$523$	−2.00000	−0.0874539	−0.0437269	+	0.999044i	$0.513923\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	9.00000	0.392419
$527$	5.00000	0.217803
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−6.00000	−0.260378
$532$	2.00000	0.0867110
$533$	6.00000	0.259889
$534$	15.0000	0.649113
$535$	0	0
$536$	2.00000	0.0863868
$537$	12.0000	0.517838
$538$	−15.0000	−0.646696
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−32.0000	−1.37452
$543$	10.0000	0.429141
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	31.0000	1.32546	0.662732	−	0.748857i	$-0.269397\pi$
$547$	31.0000	1.32546	0.662732	+	0.748857i	$0.269397\pi$
$548$	18.0000	0.768922
$549$	−22.0000	−0.938937
$550$	0	0
$551$	3.00000	0.127804
$552$	6.00000	0.255377
$553$	−16.0000	−0.680389
$554$	−10.0000	−0.424859
$555$	0	0
$556$	8.00000	0.339276
$557$	−39.0000	−1.65248	−0.826242	−	0.563316i	$-0.809525\pi$
$557$	−39.0000	−1.65248	−0.826242	+	0.563316i	$0.809525\pi$
$558$	10.0000	0.423334
$559$	10.0000	0.422955
$560$	0	0
$561$	0	0
$562$	−27.0000	−1.13893
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	−3.00000	−0.126323
$565$	0	0
$566$	−13.0000	−0.546431
$567$	−2.00000	−0.0839921
$568$	−9.00000	−0.377632
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	0	0
$571$	38.0000	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$571$	38.0000	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	12.0000	0.500870
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	−1.00000	−0.0415945
$579$	2.00000	0.0831172
$580$	0	0
$581$	−24.0000	−0.995688
$582$	7.00000	0.290159
$583$	0	0
$584$	11.0000	0.455183
$585$	0	0
$586$	−3.00000	−0.123929
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	3.00000	0.123718
$589$	−5.00000	−0.206021
$590$	0	0
$591$	−24.0000	−0.987228
$592$	−8.00000	−0.328798
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	−11.0000	−0.450200
$598$	−6.00000	−0.245358
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	20.0000	0.815139
$603$	4.00000	0.162893
$604$	−22.0000	−0.895167
$605$	0	0
$606$	6.00000	0.243733
$607$	−26.0000	−1.05531	−0.527654	−	0.849460i	$-0.676928\pi$
$607$	−26.0000	−1.05531	−0.527654	+	0.849460i	$0.676928\pi$
$608$	1.00000	0.0405554
$609$	−6.00000	−0.243132
$610$	0	0
$611$	3.00000	0.121367
$612$	−2.00000	−0.0808452
$613$	1.00000	0.0403896	0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000	0.0403896	0.0201948	+	0.999796i	$0.493571\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	39.0000	1.57008	0.785040	−	0.619445i	$-0.212642\pi$
$617$	39.0000	1.57008	0.785040	+	0.619445i	$0.212642\pi$
$618$	−8.00000	−0.321807
$619$	−46.0000	−1.84890	−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000	−1.84890	−0.924448	+	0.381308i	$0.875474\pi$
$620$	0	0
$621$	30.0000	1.20386
$622$	0	0
$623$	−30.0000	−1.20192
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	14.0000	0.559553
$627$	0	0
$628$	−14.0000	−0.558661
$629$	−8.00000	−0.318981
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	−8.00000	−0.318223
$633$	−2.00000	−0.0794929
$634$	0	0
$635$	0	0
$636$	−3.00000	−0.118958
$637$	−3.00000	−0.118864
$638$	0	0
$639$	−18.0000	−0.712069
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	12.0000	0.473602
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	1.00000	0.0393445
$647$	33.0000	1.29736	0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000	1.29736	0.648682	+	0.761060i	$0.275321\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	0	0
$651$	10.0000	0.391931
$652$	16.0000	0.626608
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	11.0000	0.430134
$655$	0	0
$656$	6.00000	0.234261
$657$	22.0000	0.858302
$658$	6.00000	0.233904
$659$	9.00000	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	31.0000	1.20485
$663$	−1.00000	−0.0388368
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−16.0000	−0.619987
$667$	−18.0000	−0.696963
$668$	18.0000	0.696441
$669$	−1.00000	−0.0386622
$670$	0	0
$671$	0	0
$672$	−2.00000	−0.0771517
$673$	−23.0000	−0.886585	−0.443292	−	0.896377i	$-0.646190\pi$
$673$	−23.0000	−0.886585	−0.443292	+	0.896377i	$0.646190\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	−12.0000	−0.461538
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−9.00000	−0.345643
$679$	−14.0000	−0.537271
$680$	0	0
$681$	3.00000	0.114960
$682$	0	0
$683$	3.00000	0.114792	0.0573959	−	0.998351i	$-0.481720\pi$
$683$	3.00000	0.114792	0.0573959	+	0.998351i	$0.481720\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	−20.0000	−0.763604
$687$	22.0000	0.839352
$688$	10.0000	0.381246
$689$	3.00000	0.114291
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−15.0000	−0.569392
$695$	0	0
$696$	−3.00000	−0.113715
$697$	6.00000	0.227266
$698$	28.0000	1.05982
$699$	9.00000	0.340411
$700$	0	0
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	−5.00000	−0.188713
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	−12.0000	−0.451306
$708$	−3.00000	−0.112747
$709$	41.0000	1.53979	0.769894	−	0.638172i	$-0.220309\pi$
$709$	41.0000	1.53979	0.769894	+	0.638172i	$0.220309\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	−15.0000	−0.562149
$713$	30.0000	1.12351
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−6.00000	−0.224074
$718$	−30.0000	−1.11959
$719$	−39.0000	−1.45445	−0.727227	−	0.686397i	$-0.759191\pi$
$719$	−39.0000	−1.45445	−0.727227	+	0.686397i	$0.759191\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	18.0000	0.669891
$723$	10.0000	0.371904
$724$	−10.0000	−0.371647
$725$	0	0
$726$	−11.0000	−0.408248
$727$	19.0000	0.704671	0.352335	−	0.935874i	$-0.385388\pi$
$727$	19.0000	0.704671	0.352335	+	0.935874i	$0.385388\pi$
$728$	2.00000	0.0741249
$729$	13.0000	0.481481
$730$	0	0
$731$	10.0000	0.369863
$732$	−11.0000	−0.406572
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−22.0000	−0.812035
$735$	0	0
$736$	−6.00000	−0.221163
$737$	0	0
$738$	12.0000	0.441726
$739$	29.0000	1.06678	0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000	1.06678	0.533391	+	0.845869i	$0.320917\pi$
$740$	0	0
$741$	1.00000	0.0367359
$742$	6.00000	0.220267
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	5.00000	0.183309
$745$	0	0
$746$	−22.0000	−0.805477
$747$	−24.0000	−0.878114
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	3.00000	0.109399
$753$	12.0000	0.437304
$754$	3.00000	0.109254
$755$	0	0
$756$	−10.0000	−0.363696
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	−8.00000	−0.290573
$759$	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	$0.465315\pi$
$762$	−11.0000	−0.398488
$763$	−22.0000	−0.796453
$764$	12.0000	0.434145
$765$	0	0
$766$	−9.00000	−0.325183
$767$	3.00000	0.108324
$768$	−1.00000	−0.0360844
$769$	−31.0000	−1.11789	−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000	−1.11789	−0.558944	+	0.829205i	$0.688793\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−2.00000	−0.0719816
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	20.0000	0.718885
$775$	0	0
$776$	−7.00000	−0.251285
$777$	−16.0000	−0.573997
$778$	−12.0000	−0.430221
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	−6.00000	−0.214560
$783$	−15.0000	−0.536056
$784$	−3.00000	−0.107143
$785$	0	0
$786$	−18.0000	−0.642039
$787$	7.00000	0.249523	0.124762	−	0.992187i	$-0.460183\pi$
$787$	7.00000	0.249523	0.124762	+	0.992187i	$0.460183\pi$
$788$	24.0000	0.854965
$789$	9.00000	0.320408
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	11.0000	0.390621
$794$	20.0000	0.709773
$795$	0	0
$796$	11.0000	0.389885
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	2.00000	0.0707992
$799$	3.00000	0.106132
$800$	0	0
$801$	−30.0000	−1.06000
$802$	−36.0000	−1.27120
$803$	0	0
$804$	2.00000	0.0705346
$805$	0	0
$806$	−5.00000	−0.176117
$807$	−15.0000	−0.528025
$808$	−6.00000	−0.211079
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	6.00000	0.210559
$813$	−32.0000	−1.12229
$814$	0	0
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	−10.0000	−0.349856
$818$	−29.0000	−1.01396
$819$	4.00000	0.139771
$820$	0	0
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	18.0000	0.627822
$823$	10.0000	0.348578	0.174289	−	0.984695i	$-0.444237\pi$
$823$	10.0000	0.348578	0.174289	+	0.984695i	$0.444237\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	6.00000	0.208767
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−12.0000	−0.417029
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	1.00000	0.0346688
$833$	−3.00000	−0.103944
$834$	8.00000	0.277017
$835$	0	0
$836$	0	0
$837$	25.0000	0.864126
$838$	−6.00000	−0.207267
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	4.00000	0.137849
$843$	−27.0000	−0.929929
$844$	2.00000	0.0688428
$845$	0	0
$846$	6.00000	0.206284
$847$	22.0000	0.755929
$848$	3.00000	0.103020
$849$	−13.0000	−0.446159
$850$	0	0
$851$	−48.0000	−1.64542
$852$	−9.00000	−0.308335
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	22.0000	0.752825
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−33.0000	−1.12726	−0.563629	−	0.826028i	$-0.690595\pi$
$857$	−33.0000	−1.12726	−0.563629	+	0.826028i	$0.690595\pi$
$858$	0	0
$859$	−1.00000	−0.0341196	−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000	−0.0341196	−0.0170598	+	0.999854i	$0.505431\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	24.0000	0.817443
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−34.0000	−1.15537
$867$	−1.00000	−0.0339618
$868$	−10.0000	−0.339422
$869$	0	0
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	−11.0000	−0.372507
$873$	−14.0000	−0.473828
$874$	6.00000	0.202953
$875$	0	0
$876$	11.0000	0.371656
$877$	40.0000	1.35070	0.675352	−	0.737496i	$-0.263992\pi$
$877$	40.0000	1.35070	0.675352	+	0.737496i	$0.263992\pi$
$878$	−8.00000	−0.269987
$879$	−3.00000	−0.101187
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−6.00000	−0.202031
$883$	−38.0000	−1.27880	−0.639401	−	0.768874i	$-0.720818\pi$
$883$	−38.0000	−1.27880	−0.639401	+	0.768874i	$0.720818\pi$
$884$	1.00000	0.0336336
$885$	0	0
$886$	12.0000	0.403148
$887$	30.0000	1.00730	0.503651	−	0.863907i	$-0.331990\pi$
$887$	30.0000	1.00730	0.503651	+	0.863907i	$0.331990\pi$
$888$	−8.00000	−0.268462
$889$	22.0000	0.737856
$890$	0	0
$891$	0	0
$892$	1.00000	0.0334825
$893$	−3.00000	−0.100391
$894$	−12.0000	−0.401340
$895$	0	0
$896$	2.00000	0.0668153
$897$	−6.00000	−0.200334
$898$	0	0
$899$	−15.0000	−0.500278
$900$	0	0
$901$	3.00000	0.0999445
$902$	0	0
$903$	20.0000	0.665558
$904$	9.00000	0.299336
$905$	0	0
$906$	−22.0000	−0.730901
$907$	−47.0000	−1.56061	−0.780305	−	0.625400i	$-0.784936\pi$
$907$	−47.0000	−1.56061	−0.780305	+	0.625400i	$0.784936\pi$
$908$	−3.00000	−0.0995585
$909$	−12.0000	−0.398015
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	1.00000	0.0331133
$913$	0	0
$914$	26.0000	0.860004
$915$	0	0
$916$	−22.0000	−0.726900
$917$	36.0000	1.18882
$918$	−5.00000	−0.165025
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	−30.0000	−0.987997
$923$	9.00000	0.296239
$924$	0	0
$925$	0	0
$926$	−19.0000	−0.624379
$927$	16.0000	0.525509
$928$	3.00000	0.0984798
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	−9.00000	−0.294805
$933$	0	0
$934$	24.0000	0.785304
$935$	0	0
$936$	2.00000	0.0653720
$937$	−44.0000	−1.43742	−0.718709	−	0.695311i	$-0.755266\pi$
$937$	−44.0000	−1.43742	−0.718709	+	0.695311i	$0.755266\pi$
$938$	−4.00000	−0.130605
$939$	14.0000	0.456873
$940$	0	0
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	−14.0000	−0.456145
$943$	36.0000	1.17232
$944$	3.00000	0.0976417
$945$	0	0
$946$	0	0
$947$	−51.0000	−1.65728	−0.828639	−	0.559784i	$-0.810884\pi$
$947$	−51.0000	−1.65728	−0.828639	+	0.559784i	$0.810884\pi$
$948$	−8.00000	−0.259828
$949$	−11.0000	−0.357075
$950$	0	0
$951$	0	0
$952$	2.00000	0.0648204
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	6.00000	0.194054
$957$	0	0
$958$	−27.0000	−0.872330
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−6.00000	−0.193548
$962$	8.00000	0.257930
$963$	−24.0000	−0.773389
$964$	−10.0000	−0.322078
$965$	0	0
$966$	−12.0000	−0.386094
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	11.0000	0.353553
$969$	1.00000	0.0321246
$970$	0	0
$971$	−21.0000	−0.673922	−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000	−0.673922	−0.336961	+	0.941519i	$0.609399\pi$
$972$	−16.0000	−0.513200
$973$	−16.0000	−0.512936
$974$	−22.0000	−0.704925
$975$	0	0
$976$	11.0000	0.352101
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	16.0000	0.511624
$979$	0	0
$980$	0	0
$981$	−22.0000	−0.702406
$982$	39.0000	1.24454
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	3.00000	0.0955395
$987$	6.00000	0.190982
$988$	−1.00000	−0.0318142
$989$	60.0000	1.90789
$990$	0	0
$991$	29.0000	0.921215	0.460608	−	0.887604i	$-0.347632\pi$
$991$	29.0000	0.921215	0.460608	+	0.887604i	$0.347632\pi$
$992$	−5.00000	−0.158750
$993$	31.0000	0.983755
$994$	18.0000	0.570925
$995$	0	0
$996$	−12.0000	−0.380235
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	22.0000	0.696398
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.a.b.1.1		1
3.2	odd	2		7650.2.a.bo.1.1		1
4.3	odd	2		6800.2.a.t.1.1		1
5.2	odd	4		850.2.c.e.749.1		2
5.3	odd	4		850.2.c.e.749.2		2
5.4	even	2		170.2.a.e.1.1	✓	1
15.14	odd	2		1530.2.a.g.1.1		1
20.19	odd	2		1360.2.a.d.1.1		1
35.34	odd	2		8330.2.a.q.1.1		1
40.19	odd	2		5440.2.a.r.1.1		1
40.29	even	2		5440.2.a.k.1.1		1
85.4	even	4		2890.2.b.b.2311.1		2
85.64	even	4		2890.2.b.b.2311.2		2
85.84	even	2		2890.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.e.1.1	✓	1	5.4	even	2
850.2.a.b.1.1		1	1.1	even	1	trivial
850.2.c.e.749.1		2	5.2	odd	4
850.2.c.e.749.2		2	5.3	odd	4
1360.2.a.d.1.1		1	20.19	odd	2
1530.2.a.g.1.1		1	15.14	odd	2
2890.2.a.n.1.1		1	85.84	even	2
2890.2.b.b.2311.1		2	85.4	even	4
2890.2.b.b.2311.2		2	85.64	even	4
5440.2.a.k.1.1		1	40.29	even	2
5440.2.a.r.1.1		1	40.19	odd	2
6800.2.a.t.1.1		1	4.3	odd	2
7650.2.a.bo.1.1		1	3.2	odd	2
8330.2.a.q.1.1		1	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more