LMFDB - Embedded newform 2850.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(1,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, 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("2850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.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:	$22.7573645761$
Analytic rank:	$0$
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	$\chi$	$=$	2850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} +2.00000 q^{21} -3.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +3.00000 q^{29} -7.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -1.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} -2.00000 q^{42} -10.0000 q^{43} +3.00000 q^{44} -3.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +6.00000 q^{51} +2.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} -2.00000 q^{56} +1.00000 q^{57} -3.00000 q^{58} +6.00000 q^{59} -1.00000 q^{61} +7.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} +5.00000 q^{67} +6.00000 q^{68} +3.00000 q^{69} -1.00000 q^{72} -1.00000 q^{73} -2.00000 q^{74} +1.00000 q^{76} +6.00000 q^{77} -2.00000 q^{78} -13.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +15.0000 q^{83} +2.00000 q^{84} +10.0000 q^{86} +3.00000 q^{87} -3.00000 q^{88} -15.0000 q^{89} +4.00000 q^{91} +3.00000 q^{92} -7.00000 q^{93} -12.0000 q^{94} -1.00000 q^{96} +8.00000 q^{97} +3.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	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	$0.410544\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	−3.00000	−0.639602
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−1.00000	−0.162221
$39$	2.00000	0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−2.00000	−0.308607
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−3.00000	−0.442326
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	6.00000	0.840168
$52$	2.00000	0.277350
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	1.00000	0.132453
$58$	−3.00000	−0.393919
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	7.00000	0.889001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	6.00000	0.727607
$69$	3.00000	0.361158
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	1.00000	0.114708
$77$	6.00000	0.683763
$78$	−2.00000	−0.226455
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	10.0000	1.07833
$87$	3.00000	0.321634
$88$	−3.00000	−0.319801
$89$	−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
$91$	4.00000	0.419314
$92$	3.00000	0.312772
$93$	−7.00000	−0.725866
$94$	−12.0000	−1.23771
$95$	0	0
$96$	−1.00000	−0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	3.00000	0.303046
$99$	3.00000	0.301511
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−6.00000	−0.594089
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	9.00000	0.874157
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	1.00000	0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	2.00000	0.188982
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	3.00000	0.278543
$117$	2.00000	0.184900
$118$	−6.00000	−0.552345
$119$	12.0000	1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	1.00000	0.0905357
$123$	−6.00000	−0.541002
$124$	−7.00000	−0.628619
$125$	0	0
$126$	−2.00000	−0.178174
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	−1.00000	−0.0883883
$129$	−10.0000	−0.880451
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	3.00000	0.261116
$133$	2.00000	0.173422
$134$	−5.00000	−0.431934
$135$	0	0
$136$	−6.00000	−0.514496
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	−3.00000	−0.255377
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	6.00000	0.501745
$144$	1.00000	0.0833333
$145$	0	0
$146$	1.00000	0.0827606
$147$	−3.00000	−0.247436
$148$	2.00000	0.164399
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	−1.00000	−0.0811107
$153$	6.00000	0.485071
$154$	−6.00000	−0.483494
$155$	0	0
$156$	2.00000	0.160128
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	13.0000	1.03422
$159$	−9.00000	−0.713746
$160$	0	0
$161$	6.00000	0.472866
$162$	−1.00000	−0.0785674
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−15.0000	−1.16423
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	−2.00000	−0.154303
$169$	−9.00000	−0.692308
$170$	0	0
$171$	1.00000	0.0764719
$172$	−10.0000	−0.762493
$173$	−3.00000	−0.228086	−0.114043	−	0.993476i	$-0.536380\pi$
$173$	−3.00000	−0.228086	−0.114043	+	0.993476i	$0.536380\pi$
$174$	−3.00000	−0.227429
$175$	0	0
$176$	3.00000	0.226134
$177$	6.00000	0.450988
$178$	15.0000	1.12430
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−4.00000	−0.296500
$183$	−1.00000	−0.0739221
$184$	−3.00000	−0.221163
$185$	0	0
$186$	7.00000	0.513265
$187$	18.0000	1.31629
$188$	12.0000	0.875190
$189$	2.00000	0.145479
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	1.00000	0.0721688
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−3.00000	−0.213201
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	−6.00000	−0.422159
$203$	6.00000	0.421117
$204$	6.00000	0.420084
$205$	0	0
$206$	7.00000	0.487713
$207$	3.00000	0.208514
$208$	2.00000	0.138675
$209$	3.00000	0.207514
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	−9.00000	−0.618123
$213$	0	0
$214$	0	0
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−14.0000	−0.950382
$218$	−2.00000	−0.135457
$219$	−1.00000	−0.0675737
$220$	0	0
$221$	12.0000	0.807207
$222$	−2.00000	−0.134231
$223$	5.00000	0.334825	0.167412	−	0.985887i	$-0.446459\pi$
$223$	5.00000	0.334825	0.167412	+	0.985887i	$0.446459\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−9.00000	−0.598671
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	1.00000	0.0662266
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	−3.00000	−0.196960
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	6.00000	0.390567
$237$	−13.0000	−0.844441
$238$	−12.0000	−0.777844
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	2.00000	0.128565
$243$	1.00000	0.0641500
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	6.00000	0.382546
$247$	2.00000	0.127257
$248$	7.00000	0.444500
$249$	15.0000	0.950586
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	2.00000	0.125988
$253$	9.00000	0.565825
$254$	−11.0000	−0.690201
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.00000	−0.561405	−0.280702	−	0.959795i	$-0.590567\pi$
$257$	−9.00000	−0.561405	−0.280702	+	0.959795i	$0.590567\pi$
$258$	10.0000	0.622573
$259$	4.00000	0.248548
$260$	0	0
$261$	3.00000	0.185695
$262$	15.0000	0.926703
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	−2.00000	−0.122628
$267$	−15.0000	−0.917985
$268$	5.00000	0.305424
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	6.00000	0.363803
$273$	4.00000	0.242091
$274$	−12.0000	−0.724947
$275$	0	0
$276$	3.00000	0.180579
$277$	11.0000	0.660926	0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$278$	4.00000	0.239904
$279$	−7.00000	−0.419079
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	−12.0000	−0.714590
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	0	0
$285$	0	0
$286$	−6.00000	−0.354787
$287$	−12.0000	−0.708338
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	8.00000	0.468968
$292$	−1.00000	−0.0585206
$293$	−15.0000	−0.876309	−0.438155	−	0.898900i	$-0.644368\pi$
$293$	−15.0000	−0.876309	−0.438155	+	0.898900i	$0.644368\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	3.00000	0.174078
$298$	−12.0000	−0.695141
$299$	6.00000	0.346989
$300$	0	0
$301$	−20.0000	−1.15278
$302$	−20.0000	−1.15087
$303$	6.00000	0.344691
$304$	1.00000	0.0573539
$305$	0	0
$306$	−6.00000	−0.342997
$307$	−19.0000	−1.08439	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−19.0000	−1.08439	−0.542194	+	0.840254i	$0.682406\pi$
$308$	6.00000	0.341882
$309$	−7.00000	−0.398216
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−2.00000	−0.113228
$313$	−13.0000	−0.734803	−0.367402	−	0.930062i	$-0.619753\pi$
$313$	−13.0000	−0.734803	−0.367402	+	0.930062i	$0.619753\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−13.0000	−0.731307
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	9.00000	0.504695
$319$	9.00000	0.503903
$320$	0	0
$321$	0	0
$322$	−6.00000	−0.334367
$323$	6.00000	0.333849
$324$	1.00000	0.0555556
$325$	0	0
$326$	22.0000	1.21847
$327$	2.00000	0.110600
$328$	6.00000	0.331295
$329$	24.0000	1.32316
$330$	0	0
$331$	5.00000	0.274825	0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000	0.274825	0.137412	+	0.990514i	$0.456121\pi$
$332$	15.0000	0.823232
$333$	2.00000	0.109599
$334$	18.0000	0.984916
$335$	0	0
$336$	2.00000	0.109109
$337$	32.0000	1.74315	0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000	1.74315	0.871576	+	0.490261i	$0.163099\pi$
$338$	9.00000	0.489535
$339$	9.00000	0.488813
$340$	0	0
$341$	−21.0000	−1.13721
$342$	−1.00000	−0.0540738
$343$	−20.0000	−1.07990
$344$	10.0000	0.539164
$345$	0	0
$346$	3.00000	0.161281
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	3.00000	0.160817
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−3.00000	−0.159901
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	−15.0000	−0.794998
$357$	12.0000	0.635107
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	10.0000	0.525588
$363$	−2.00000	−0.104973
$364$	4.00000	0.209657
$365$	0	0
$366$	1.00000	0.0522708
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	3.00000	0.156386
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−18.0000	−0.934513
$372$	−7.00000	−0.362933
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	−12.0000	−0.618853
$377$	6.00000	0.309016
$378$	−2.00000	−0.102869
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	11.0000	0.563547
$382$	−3.00000	−0.153493
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−20.0000	−1.01797
$387$	−10.0000	−0.508329
$388$	8.00000	0.406138
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	3.00000	0.151523
$393$	−15.0000	−0.756650
$394$	−6.00000	−0.302276
$395$	0	0
$396$	3.00000	0.150756
$397$	17.0000	0.853206	0.426603	−	0.904439i	$-0.359710\pi$
$397$	17.0000	0.853206	0.426603	+	0.904439i	$0.359710\pi$
$398$	−2.00000	−0.100251
$399$	2.00000	0.100125
$400$	0	0
$401$	9.00000	0.449439	0.224719	−	0.974424i	$-0.427853\pi$
$401$	9.00000	0.449439	0.224719	+	0.974424i	$0.427853\pi$
$402$	−5.00000	−0.249377
$403$	−14.0000	−0.697390
$404$	6.00000	0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	6.00000	0.297409
$408$	−6.00000	−0.297044
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	−7.00000	−0.344865
$413$	12.0000	0.590481
$414$	−3.00000	−0.147442
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	−4.00000	−0.195881
$418$	−3.00000	−0.146735
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	13.0000	0.632830
$423$	12.0000	0.583460
$424$	9.00000	0.437079
$425$	0	0
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	1.00000	0.0481125
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	14.0000	0.672022
$435$	0	0
$436$	2.00000	0.0957826
$437$	3.00000	0.143509
$438$	1.00000	0.0477818
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−12.0000	−0.570782
$443$	27.0000	1.28281	0.641404	−	0.767203i	$-0.278352\pi$
$443$	27.0000	1.28281	0.641404	+	0.767203i	$0.278352\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	−5.00000	−0.236757
$447$	12.0000	0.567581
$448$	2.00000	0.0944911
$449$	−33.0000	−1.55737	−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000	−1.55737	−0.778683	+	0.627417i	$0.784112\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	9.00000	0.423324
$453$	20.0000	0.939682
$454$	18.0000	0.844782
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	13.0000	0.607450
$459$	6.00000	0.280056
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	−6.00000	−0.279145
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−39.0000	−1.80470	−0.902352	−	0.430999i	$-0.858161\pi$
$467$	−39.0000	−1.80470	−0.902352	+	0.430999i	$0.858161\pi$
$468$	2.00000	0.0924500
$469$	10.0000	0.461757
$470$	0	0
$471$	14.0000	0.645086
$472$	−6.00000	−0.276172
$473$	−30.0000	−1.37940
$474$	13.0000	0.597110
$475$	0	0
$476$	12.0000	0.550019
$477$	−9.00000	−0.412082
$478$	0	0
$479$	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
$481$	4.00000	0.182384
$482$	10.0000	0.455488
$483$	6.00000	0.273009
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	1.00000	0.0452679
$489$	−22.0000	−0.994874
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−6.00000	−0.270501
$493$	18.0000	0.810679
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	−7.00000	−0.314309
$497$	0	0
$498$	−15.0000	−0.672166
$499$	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
$501$	−18.0000	−0.804181
$502$	−24.0000	−1.07117
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	−9.00000	−0.400099
$507$	−9.00000	−0.399704
$508$	11.0000	0.488046
$509$	33.0000	1.46270	0.731350	−	0.682003i	$-0.238891\pi$
$509$	33.0000	1.46270	0.731350	+	0.682003i	$0.238891\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	9.00000	0.396973
$515$	0	0
$516$	−10.0000	−0.440225
$517$	36.0000	1.58328
$518$	−4.00000	−0.175750
$519$	−3.00000	−0.131685
$520$	0	0
$521$	−15.0000	−0.657162	−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000	−0.657162	−0.328581	+	0.944476i	$0.606570\pi$
$522$	−3.00000	−0.131306
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	21.0000	0.915644
$527$	−42.0000	−1.82955
$528$	3.00000	0.130558
$529$	−14.0000	−0.608696
$530$	0	0
$531$	6.00000	0.260378
$532$	2.00000	0.0867110
$533$	−12.0000	−0.519778
$534$	15.0000	0.649113
$535$	0	0
$536$	−5.00000	−0.215967
$537$	0	0
$538$	−30.0000	−1.29339
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−43.0000	−1.84871	−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000	−1.84871	−0.924357	+	0.381528i	$0.875398\pi$
$542$	−20.0000	−0.859074
$543$	−10.0000	−0.429141
$544$	−6.00000	−0.257248
$545$	0	0
$546$	−4.00000	−0.171184
$547$	11.0000	0.470326	0.235163	−	0.971956i	$-0.424438\pi$
$547$	11.0000	0.470326	0.235163	+	0.971956i	$0.424438\pi$
$548$	12.0000	0.512615
$549$	−1.00000	−0.0426790
$550$	0	0
$551$	3.00000	0.127804
$552$	−3.00000	−0.127688
$553$	−26.0000	−1.10563
$554$	−11.0000	−0.467345
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	7.00000	0.296334
$559$	−20.0000	−0.845910
$560$	0	0
$561$	18.0000	0.759961
$562$	−3.00000	−0.126547
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	−8.00000	−0.336265
$567$	2.00000	0.0839921
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	6.00000	0.250873
$573$	3.00000	0.125327
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	−19.0000	−0.790296
$579$	20.0000	0.831172
$580$	0	0
$581$	30.0000	1.24461
$582$	−8.00000	−0.331611
$583$	−27.0000	−1.11823
$584$	1.00000	0.0413803
$585$	0	0
$586$	15.0000	0.619644
$587$	−45.0000	−1.85735	−0.928674	−	0.370896i	$-0.879051\pi$
$587$	−45.0000	−1.85735	−0.928674	+	0.370896i	$0.879051\pi$
$588$	−3.00000	−0.123718
$589$	−7.00000	−0.288430
$590$	0	0
$591$	6.00000	0.246807
$592$	2.00000	0.0821995
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	12.0000	0.491539
$597$	2.00000	0.0818546
$598$	−6.00000	−0.245358
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	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
$603$	5.00000	0.203616
$604$	20.0000	0.813788
$605$	0	0
$606$	−6.00000	−0.243733
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	−1.00000	−0.0405554
$609$	6.00000	0.243132
$610$	0	0
$611$	24.0000	0.970936
$612$	6.00000	0.242536
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	19.0000	0.766778
$615$	0	0
$616$	−6.00000	−0.241747
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	7.00000	0.281581
$619$	−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
$621$	3.00000	0.120386
$622$	−12.0000	−0.481156
$623$	−30.0000	−1.20192
$624$	2.00000	0.0800641
$625$	0	0
$626$	13.0000	0.519584
$627$	3.00000	0.119808
$628$	14.0000	0.558661
$629$	12.0000	0.478471
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	13.0000	0.517112
$633$	−13.0000	−0.516704
$634$	3.00000	0.119145
$635$	0	0
$636$	−9.00000	−0.356873
$637$	−6.00000	−0.237729
$638$	−9.00000	−0.356313
$639$	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	$0.615682\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	−6.00000	−0.236067
$647$	−15.0000	−0.589711	−0.294855	−	0.955542i	$-0.595271\pi$
$647$	−15.0000	−0.589711	−0.294855	+	0.955542i	$0.595271\pi$
$648$	−1.00000	−0.0392837
$649$	18.0000	0.706562
$650$	0	0
$651$	−14.0000	−0.548703
$652$	−22.0000	−0.861586
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	−1.00000	−0.0390137
$658$	−24.0000	−0.935617
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	−5.00000	−0.194331
$663$	12.0000	0.466041
$664$	−15.0000	−0.582113
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	9.00000	0.348481
$668$	−18.0000	−0.696441
$669$	5.00000	0.193311
$670$	0	0
$671$	−3.00000	−0.115814
$672$	−2.00000	−0.0771517
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	−32.0000	−1.23259
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−33.0000	−1.26829	−0.634147	−	0.773213i	$-0.718648\pi$
$677$	−33.0000	−1.26829	−0.634147	+	0.773213i	$0.718648\pi$
$678$	−9.00000	−0.345643
$679$	16.0000	0.614024
$680$	0	0
$681$	−18.0000	−0.689761
$682$	21.0000	0.804132
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	20.0000	0.763604
$687$	−13.0000	−0.495981
$688$	−10.0000	−0.381246
$689$	−18.0000	−0.685745
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−3.00000	−0.114043
$693$	6.00000	0.227921
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−3.00000	−0.113715
$697$	−36.0000	−1.36360
$698$	19.0000	0.719161
$699$	6.00000	0.226941
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	−2.00000	−0.0754851
$703$	2.00000	0.0754314
$704$	3.00000	0.113067
$705$	0	0
$706$	6.00000	0.225813
$707$	12.0000	0.451306
$708$	6.00000	0.225494
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	−13.0000	−0.487538
$712$	15.0000	0.562149
$713$	−21.0000	−0.786456
$714$	−12.0000	−0.449089
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	−1.00000	−0.0372161
$723$	−10.0000	−0.371904
$724$	−10.0000	−0.371647
$725$	0	0
$726$	2.00000	0.0742270
$727$	−34.0000	−1.26099	−0.630495	−	0.776193i	$-0.717148\pi$
$727$	−34.0000	−1.26099	−0.630495	+	0.776193i	$0.717148\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	0	0
$731$	−60.0000	−2.21918
$732$	−1.00000	−0.0369611
$733$	5.00000	0.184679	0.0923396	−	0.995728i	$-0.470565\pi$
$733$	5.00000	0.184679	0.0923396	+	0.995728i	$0.470565\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−3.00000	−0.110581
$737$	15.0000	0.552532
$738$	6.00000	0.220863
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	18.0000	0.660801
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	7.00000	0.256632
$745$	0	0
$746$	22.0000	0.805477
$747$	15.0000	0.548821
$748$	18.0000	0.658145
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	12.0000	0.437595
$753$	24.0000	0.874609
$754$	−6.00000	−0.218507
$755$	0	0
$756$	2.00000	0.0727393
$757$	29.0000	1.05402	0.527011	−	0.849858i	$-0.323312\pi$
$757$	29.0000	1.05402	0.527011	+	0.849858i	$0.323312\pi$
$758$	−20.0000	−0.726433
$759$	9.00000	0.326679
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−11.0000	−0.398488
$763$	4.00000	0.144810
$764$	3.00000	0.108536
$765$	0	0
$766$	−12.0000	−0.433578
$767$	12.0000	0.433295
$768$	1.00000	0.0360844
$769$	11.0000	0.396670	0.198335	−	0.980134i	$-0.436447\pi$
$769$	11.0000	0.396670	0.198335	+	0.980134i	$0.436447\pi$
$770$	0	0
$771$	−9.00000	−0.324127
$772$	20.0000	0.719816
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	10.0000	0.359443
$775$	0	0
$776$	−8.00000	−0.287183
$777$	4.00000	0.143499
$778$	6.00000	0.215110
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	−18.0000	−0.643679
$783$	3.00000	0.107211
$784$	−3.00000	−0.107143
$785$	0	0
$786$	15.0000	0.535032
$787$	−49.0000	−1.74666	−0.873331	−	0.487128i	$-0.838045\pi$
$787$	−49.0000	−1.74666	−0.873331	+	0.487128i	$0.838045\pi$
$788$	6.00000	0.213741
$789$	−21.0000	−0.747620
$790$	0	0
$791$	18.0000	0.640006
$792$	−3.00000	−0.106600
$793$	−2.00000	−0.0710221
$794$	−17.0000	−0.603307
$795$	0	0
$796$	2.00000	0.0708881
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	−2.00000	−0.0707992
$799$	72.0000	2.54718
$800$	0	0
$801$	−15.0000	−0.529999
$802$	−9.00000	−0.317801
$803$	−3.00000	−0.105868
$804$	5.00000	0.176336
$805$	0	0
$806$	14.0000	0.493129
$807$	30.0000	1.05605
$808$	−6.00000	−0.211079
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	41.0000	1.43970	0.719852	−	0.694127i	$-0.244209\pi$
$811$	41.0000	1.43970	0.719852	+	0.694127i	$0.244209\pi$
$812$	6.00000	0.210559
$813$	20.0000	0.701431
$814$	−6.00000	−0.210300
$815$	0	0
$816$	6.00000	0.210042
$817$	−10.0000	−0.349856
$818$	34.0000	1.18878
$819$	4.00000	0.139771
$820$	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
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	7.00000	0.243857
$825$	0	0
$826$	−12.0000	−0.417533
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	3.00000	0.104257
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	11.0000	0.381586
$832$	2.00000	0.0693375
$833$	−18.0000	−0.623663
$834$	4.00000	0.138509
$835$	0	0
$836$	3.00000	0.103757
$837$	−7.00000	−0.241955
$838$	−12.0000	−0.414533
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	10.0000	0.344623
$843$	3.00000	0.103325
$844$	−13.0000	−0.447478
$845$	0	0
$846$	−12.0000	−0.412568
$847$	−4.00000	−0.137442
$848$	−9.00000	−0.309061
$849$	8.00000	0.274559
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	−6.00000	−0.204837
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	−18.0000	−0.613082
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	16.0000	0.543702
$867$	19.0000	0.645274
$868$	−14.0000	−0.475191
$869$	−39.0000	−1.32298
$870$	0	0
$871$	10.0000	0.338837
$872$	−2.00000	−0.0677285
$873$	8.00000	0.270759
$874$	−3.00000	−0.101477
$875$	0	0
$876$	−1.00000	−0.0337869
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−35.0000	−1.18119
$879$	−15.0000	−0.505937
$880$	0	0
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	3.00000	0.101015
$883$	56.0000	1.88455	0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000	1.88455	0.942275	+	0.334840i	$0.108682\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−27.0000	−0.907083
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	−2.00000	−0.0671156
$889$	22.0000	0.737856
$890$	0	0
$891$	3.00000	0.100504
$892$	5.00000	0.167412
$893$	12.0000	0.401565
$894$	−12.0000	−0.401340
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	6.00000	0.200334
$898$	33.0000	1.10122
$899$	−21.0000	−0.700389
$900$	0	0
$901$	−54.0000	−1.79900
$902$	18.0000	0.599334
$903$	−20.0000	−0.665558
$904$	−9.00000	−0.299336
$905$	0	0
$906$	−20.0000	−0.664455
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	−18.0000	−0.597351
$909$	6.00000	0.199007
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	1.00000	0.0331133
$913$	45.0000	1.48928
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	−13.0000	−0.429532
$917$	−30.0000	−0.990687
$918$	−6.00000	−0.198030
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−19.0000	−0.626071
$922$	12.0000	0.395199
$923$	0	0
$924$	6.00000	0.197386
$925$	0	0
$926$	16.0000	0.525793
$927$	−7.00000	−0.229910
$928$	−3.00000	−0.0984798
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	6.00000	0.196537
$933$	12.0000	0.392862
$934$	39.0000	1.27612
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−10.0000	−0.326512
$939$	−13.0000	−0.424239
$940$	0	0
$941$	−51.0000	−1.66255	−0.831276	−	0.555860i	$-0.812389\pi$
$941$	−51.0000	−1.66255	−0.831276	+	0.555860i	$0.812389\pi$
$942$	−14.0000	−0.456145
$943$	−18.0000	−0.586161
$944$	6.00000	0.195283
$945$	0	0
$946$	30.0000	0.975384
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−13.0000	−0.422220
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	−3.00000	−0.0972817
$952$	−12.0000	−0.388922
$953$	−33.0000	−1.06897	−0.534487	−	0.845176i	$-0.679495\pi$
$953$	−33.0000	−1.06897	−0.534487	+	0.845176i	$0.679495\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	0	0
$957$	9.00000	0.290929
$958$	−27.0000	−0.872330
$959$	24.0000	0.775000
$960$	0	0
$961$	18.0000	0.580645
$962$	−4.00000	−0.128965
$963$	0	0
$964$	−10.0000	−0.322078
$965$	0	0
$966$	−6.00000	−0.193047
$967$	38.0000	1.22200	0.610999	−	0.791632i	$-0.290768\pi$
$967$	38.0000	1.22200	0.610999	+	0.791632i	$0.290768\pi$
$968$	2.00000	0.0642824
$969$	6.00000	0.192748
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	1.00000	0.0320750
$973$	−8.00000	−0.256468
$974$	40.0000	1.28168
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	22.0000	0.703482
$979$	−45.0000	−1.43821
$980$	0	0
$981$	2.00000	0.0638551
$982$	12.0000	0.382935
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	−18.0000	−0.573237
$987$	24.0000	0.763928
$988$	2.00000	0.0636285
$989$	−30.0000	−0.953945
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	7.00000	0.222250
$993$	5.00000	0.158670
$994$	0	0
$995$	0	0
$996$	15.0000	0.475293
$997$	−19.0000	−0.601736	−0.300868	−	0.953666i	$-0.597276\pi$
$997$	−19.0000	−0.601736	−0.300868	+	0.953666i	$0.597276\pi$
$998$	−14.0000	−0.443162
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.a.o.1.1	✓	1
3.2	odd	2		8550.2.a.be.1.1		1
5.2	odd	4		2850.2.d.h.799.1		2
5.3	odd	4		2850.2.d.h.799.2		2
5.4	even	2		2850.2.a.r.1.1	yes	1
15.14	odd	2		8550.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.o.1.1	✓	1	1.1	even	1	trivial
2850.2.a.r.1.1	yes	1	5.4	even	2
2850.2.d.h.799.1		2	5.2	odd	4
2850.2.d.h.799.2		2	5.3	odd	4
8550.2.a.d.1.1		1	15.14	odd	2
8550.2.a.be.1.1		1	3.2	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:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more