LMFDB - Embedded newform 2842.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2842,2,Mod(1,2842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2842, 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("2842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2842 = 2 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2842.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:	$22.6934842544$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 406)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} -3.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} +2.00000 q^{18} +4.00000 q^{19} +3.00000 q^{20} +3.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{26} +5.00000 q^{27} +1.00000 q^{29} +3.00000 q^{30} -5.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} -2.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} -3.00000 q^{40} -7.00000 q^{43} -3.00000 q^{44} -6.00000 q^{45} +6.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} +1.00000 q^{52} -9.00000 q^{53} -5.00000 q^{54} -9.00000 q^{55} -4.00000 q^{57} -1.00000 q^{58} -12.0000 q^{59} -3.00000 q^{60} +10.0000 q^{61} +5.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} -3.00000 q^{66} +2.00000 q^{67} +6.00000 q^{69} -12.0000 q^{71} +2.00000 q^{72} -8.00000 q^{73} -2.00000 q^{74} -4.00000 q^{75} +4.00000 q^{76} +1.00000 q^{78} +5.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} +7.00000 q^{86} -1.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} +6.00000 q^{90} -6.00000 q^{92} +5.00000 q^{93} -3.00000 q^{94} +12.0000 q^{95} +1.00000 q^{96} -8.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
$3$	−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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	−3.00000	−0.948683
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−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	$0.455716\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	2.00000	0.471405
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	3.00000	0.639602
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	−1.00000	−0.196116
$27$	5.00000	0.962250
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	3.00000	0.547723
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.00000	−0.648886
$39$	−1.00000	−0.160128
$40$	−3.00000	−0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	−3.00000	−0.452267
$45$	−6.00000	−0.894427
$46$	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
$49$	0	0
$50$	−4.00000	−0.565685
$51$	0	0
$52$	1.00000	0.138675
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	−5.00000	−0.680414
$55$	−9.00000	−1.21356
$56$	0	0
$57$	−4.00000	−0.529813
$58$	−1.00000	−0.131306
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−3.00000	−0.387298
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	5.00000	0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	−3.00000	−0.369274
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	2.00000	0.235702
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	−2.00000	−0.232495
$75$	−4.00000	−0.461880
$76$	4.00000	0.458831
$77$	0	0
$78$	1.00000	0.113228
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	0	0
$86$	7.00000	0.754829
$87$	−1.00000	−0.107211
$88$	3.00000	0.319801
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	6.00000	0.632456
$91$	0	0
$92$	−6.00000	−0.625543
$93$	5.00000	0.518476
$94$	−3.00000	−0.309426
$95$	12.0000	1.23117
$96$	1.00000	0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	4.00000	0.400000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	−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
$106$	9.00000	0.874157
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	5.00000	0.481125
$109$	−13.0000	−1.24517	−0.622587	−	0.782551i	$-0.713918\pi$
$109$	−13.0000	−1.24517	−0.622587	+	0.782551i	$0.713918\pi$
$110$	9.00000	0.858116
$111$	−2.00000	−0.189832
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	4.00000	0.374634
$115$	−18.0000	−1.67851
$116$	1.00000	0.0928477
$117$	−2.00000	−0.184900
$118$	12.0000	1.10469
$119$	0	0
$120$	3.00000	0.273861
$121$	−2.00000	−0.181818
$122$	−10.0000	−0.905357
$123$	0	0
$124$	−5.00000	−0.449013
$125$	−3.00000	−0.268328
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	7.00000	0.616316
$130$	−3.00000	−0.263117
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	3.00000	0.261116
$133$	0	0
$134$	−2.00000	−0.172774
$135$	15.0000	1.29099
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−6.00000	−0.510754
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	12.0000	1.00702
$143$	−3.00000	−0.250873
$144$	−2.00000	−0.166667
$145$	3.00000	0.249136
$146$	8.00000	0.662085
$147$	0	0
$148$	2.00000	0.164399
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	4.00000	0.326599
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	0	0
$155$	−15.0000	−1.20483
$156$	−1.00000	−0.0800641
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−5.00000	−0.397779
$159$	9.00000	0.713746
$160$	−3.00000	−0.237171
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−7.00000	−0.548282	−0.274141	−	0.961689i	$-0.588394\pi$
$163$	−7.00000	−0.548282	−0.274141	+	0.961689i	$0.588394\pi$
$164$	0	0
$165$	9.00000	0.700649
$166$	12.0000	0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−8.00000	−0.611775
$172$	−7.00000	−0.533745
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	−3.00000	−0.226134
$177$	12.0000	0.901975
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−6.00000	−0.447214
$181$	25.0000	1.85824	0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000	1.85824	0.929118	+	0.369784i	$0.120568\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	6.00000	0.442326
$185$	6.00000	0.441129
$186$	−5.00000	−0.366618
$187$	0	0
$188$	3.00000	0.218797
$189$	0	0
$190$	−12.0000	−0.870572
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	−1.00000	−0.0721688
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	8.00000	0.574367
$195$	−3.00000	−0.214834
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−6.00000	−0.426401
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	−4.00000	−0.282843
$201$	−2.00000	−0.141069
$202$	−12.0000	−0.844317
$203$	0	0
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	12.0000	0.834058
$208$	1.00000	0.0693375
$209$	−12.0000	−0.830057
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−9.00000	−0.618123
$213$	12.0000	0.822226
$214$	−18.0000	−1.23045
$215$	−21.0000	−1.43219
$216$	−5.00000	−0.340207
$217$	0	0
$218$	13.0000	0.880471
$219$	8.00000	0.540590
$220$	−9.00000	−0.606780
$221$	0	0
$222$	2.00000	0.134231
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	−12.0000	−0.798228
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	−4.00000	−0.264906
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	18.0000	1.18688
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	2.00000	0.130744
$235$	9.00000	0.587095
$236$	−12.0000	−0.781133
$237$	−5.00000	−0.324785
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	−3.00000	−0.193649
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	2.00000	0.128565
$243$	−16.0000	−1.02640
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	5.00000	0.317500
$249$	12.0000	0.760469
$250$	3.00000	0.189737
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	−7.00000	−0.435801
$259$	0	0
$260$	3.00000	0.186052
$261$	−2.00000	−0.123797
$262$	0	0
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	−3.00000	−0.184637
$265$	−27.0000	−1.65860
$266$	0	0
$267$	6.00000	0.367194
$268$	2.00000	0.122169
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	−15.0000	−0.912871
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	0	0
$273$	0	0
$274$	6.00000	0.362473
$275$	−12.0000	−0.723627
$276$	6.00000	0.361158
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−10.0000	−0.599760
$279$	10.0000	0.598684
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	3.00000	0.178647
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	−12.0000	−0.712069
$285$	−12.0000	−0.710819
$286$	3.00000	0.177394
$287$	0	0
$288$	2.00000	0.117851
$289$	−17.0000	−1.00000
$290$	−3.00000	−0.176166
$291$	8.00000	0.468968
$292$	−8.00000	−0.468165
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	−36.0000	−2.09600
$296$	−2.00000	−0.116248
$297$	−15.0000	−0.870388
$298$	−9.00000	−0.521356
$299$	−6.00000	−0.346989
$300$	−4.00000	−0.230940
$301$	0	0
$302$	10.0000	0.575435
$303$	−12.0000	−0.689382
$304$	4.00000	0.229416
$305$	30.0000	1.71780
$306$	0	0
$307$	−23.0000	−1.31268	−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000	−1.31268	−0.656340	+	0.754466i	$0.727896\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	15.0000	0.851943
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	1.00000	0.0566139
$313$	25.0000	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$313$	25.0000	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	5.00000	0.281272
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−9.00000	−0.504695
$319$	−3.00000	−0.167968
$320$	3.00000	0.167705
$321$	−18.0000	−1.00466
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	7.00000	0.387694
$327$	13.0000	0.718902
$328$	0	0
$329$	0	0
$330$	−9.00000	−0.495434
$331$	−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
$333$	−4.00000	−0.219199
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	12.0000	0.652714
$339$	−12.0000	−0.651751
$340$	0	0
$341$	15.0000	0.812296
$342$	8.00000	0.432590
$343$	0	0
$344$	7.00000	0.377415
$345$	18.0000	0.969087
$346$	−18.0000	−0.967686
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	−1.00000	−0.0536056
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	3.00000	0.159901
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−12.0000	−0.637793
$355$	−36.0000	−1.91068
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	21.0000	1.10834	0.554169	−	0.832404i	$-0.313036\pi$
$359$	21.0000	1.10834	0.554169	+	0.832404i	$0.313036\pi$
$360$	6.00000	0.316228
$361$	−3.00000	−0.157895
$362$	−25.0000	−1.31397
$363$	2.00000	0.104973
$364$	0	0
$365$	−24.0000	−1.25622
$366$	10.0000	0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	−6.00000	−0.311925
$371$	0	0
$372$	5.00000	0.259238
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	−3.00000	−0.154713
$377$	1.00000	0.0515026
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	12.0000	0.615587
$381$	−8.00000	−0.409852
$382$	24.0000	1.22795
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	4.00000	0.203595
$387$	14.0000	0.711660
$388$	−8.00000	−0.406138
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	3.00000	0.151911
$391$	0	0
$392$	0	0
$393$	0	0
$394$	18.0000	0.906827
$395$	15.0000	0.754732
$396$	6.00000	0.301511
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	4.00000	0.200000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	2.00000	0.0997509
$403$	−5.00000	−0.249068
$404$	12.0000	0.597022
$405$	3.00000	0.149071
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	−8.00000	−0.394132
$413$	0	0
$414$	−12.0000	−0.589768
$415$	−36.0000	−1.76717
$416$	−1.00000	−0.0490290
$417$	−10.0000	−0.489702
$418$	12.0000	0.586939
$419$	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
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	−5.00000	−0.243396
$423$	−6.00000	−0.291730
$424$	9.00000	0.437079
$425$	0	0
$426$	−12.0000	−0.581402
$427$	0	0
$428$	18.0000	0.870063
$429$	3.00000	0.144841
$430$	21.0000	1.01271
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	5.00000	0.240563
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	−13.0000	−0.622587
$437$	−24.0000	−1.14808
$438$	−8.00000	−0.382255
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	9.00000	0.429058
$441$	0	0
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−2.00000	−0.0949158
$445$	−18.0000	−0.853282
$446$	8.00000	0.378811
$447$	−9.00000	−0.425685
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	8.00000	0.377124
$451$	0	0
$452$	12.0000	0.564433
$453$	10.0000	0.469841
$454$	18.0000	0.844782
$455$	0	0
$456$	4.00000	0.187317
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	2.00000	0.0934539
$459$	0	0
$460$	−18.0000	−0.839254
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	1.00000	0.0464238
$465$	15.0000	0.695608
$466$	21.0000	0.972806
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	−9.00000	−0.415139
$471$	−4.00000	−0.184310
$472$	12.0000	0.552345
$473$	21.0000	0.965581
$474$	5.00000	0.229658
$475$	16.0000	0.734130
$476$	0	0
$477$	18.0000	0.824163
$478$	24.0000	1.09773
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	3.00000	0.136931
$481$	2.00000	0.0911922
$482$	−1.00000	−0.0455488
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−24.0000	−1.08978
$486$	16.0000	0.725775
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−10.0000	−0.452679
$489$	7.00000	0.316551
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	0	0
$494$	−4.00000	−0.179969
$495$	18.0000	0.809040
$496$	−5.00000	−0.224507
$497$	0	0
$498$	−12.0000	−0.537733
$499$	38.0000	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$499$	38.0000	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	−15.0000	−0.669483
$503$	−39.0000	−1.73892	−0.869462	−	0.494000i	$-0.835534\pi$
$503$	−39.0000	−1.73892	−0.869462	+	0.494000i	$0.835534\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	−18.0000	−0.800198
$507$	12.0000	0.532939
$508$	8.00000	0.354943
$509$	39.0000	1.72864	0.864322	−	0.502938i	$-0.167748\pi$
$509$	39.0000	1.72864	0.864322	+	0.502938i	$0.167748\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	20.0000	0.883022
$514$	27.0000	1.19092
$515$	−24.0000	−1.05757
$516$	7.00000	0.308158
$517$	−9.00000	−0.395820
$518$	0	0
$519$	−18.0000	−0.790112
$520$	−3.00000	−0.131559
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	2.00000	0.0875376
$523$	10.0000	0.437269	0.218635	−	0.975807i	$-0.429840\pi$
$523$	10.0000	0.437269	0.218635	+	0.975807i	$0.429840\pi$
$524$	0	0
$525$	0	0
$526$	−3.00000	−0.130806
$527$	0	0
$528$	3.00000	0.130558
$529$	13.0000	0.565217
$530$	27.0000	1.17281
$531$	24.0000	1.04151
$532$	0	0
$533$	0	0
$534$	−6.00000	−0.259645
$535$	54.0000	2.33462
$536$	−2.00000	−0.0863868
$537$	12.0000	0.517838
$538$	0	0
$539$	0	0
$540$	15.0000	0.645497
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	11.0000	0.472490
$543$	−25.0000	−1.07285
$544$	0	0
$545$	−39.0000	−1.67058
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	−6.00000	−0.256307
$549$	−20.0000	−0.853579
$550$	12.0000	0.511682
$551$	4.00000	0.170406
$552$	−6.00000	−0.255377
$553$	0	0
$554$	22.0000	0.934690
$555$	−6.00000	−0.254686
$556$	10.0000	0.424094
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	−10.0000	−0.423334
$559$	−7.00000	−0.296068
$560$	0	0
$561$	0	0
$562$	−3.00000	−0.126547
$563$	−15.0000	−0.632175	−0.316087	−	0.948730i	$-0.602369\pi$
$563$	−15.0000	−0.632175	−0.316087	+	0.948730i	$0.602369\pi$
$564$	−3.00000	−0.126323
$565$	36.0000	1.51453
$566$	−22.0000	−0.924729
$567$	0	0
$568$	12.0000	0.503509
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	12.0000	0.502625
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	−3.00000	−0.125436
$573$	24.0000	1.00261
$574$	0	0
$575$	−24.0000	−1.00087
$576$	−2.00000	−0.0833333
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	17.0000	0.707107
$579$	4.00000	0.166234
$580$	3.00000	0.124568
$581$	0	0
$582$	−8.00000	−0.331611
$583$	27.0000	1.11823
$584$	8.00000	0.331042
$585$	−6.00000	−0.248069
$586$	18.0000	0.743573
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	36.0000	1.48210
$591$	18.0000	0.740421
$592$	2.00000	0.0821995
$593$	−33.0000	−1.35515	−0.677574	−	0.735455i	$-0.736969\pi$
$593$	−33.0000	−1.35515	−0.677574	+	0.735455i	$0.736969\pi$
$594$	15.0000	0.615457
$595$	0	0
$596$	9.00000	0.368654
$597$	20.0000	0.818546
$598$	6.00000	0.245358
$599$	−39.0000	−1.59350	−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000	−1.59350	−0.796748	+	0.604311i	$0.793448\pi$
$600$	4.00000	0.163299
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	−10.0000	−0.406894
$605$	−6.00000	−0.243935
$606$	12.0000	0.487467
$607$	−35.0000	−1.42061	−0.710303	−	0.703896i	$-0.751442\pi$
$607$	−35.0000	−1.42061	−0.710303	+	0.703896i	$0.751442\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−30.0000	−1.21466
$611$	3.00000	0.121367
$612$	0	0
$613$	−25.0000	−1.00974	−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000	−1.00974	−0.504870	+	0.863195i	$0.668460\pi$
$614$	23.0000	0.928204
$615$	0	0
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	−8.00000	−0.321807
$619$	37.0000	1.48716	0.743578	−	0.668649i	$-0.233127\pi$
$619$	37.0000	1.48716	0.743578	+	0.668649i	$0.233127\pi$
$620$	−15.0000	−0.602414
$621$	−30.0000	−1.20386
$622$	−12.0000	−0.481156
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	−29.0000	−1.16000
$626$	−25.0000	−0.999201
$627$	12.0000	0.479234
$628$	4.00000	0.159617
$629$	0	0
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	−5.00000	−0.198889
$633$	−5.00000	−0.198732
$634$	18.0000	0.714871
$635$	24.0000	0.952411
$636$	9.00000	0.356873
$637$	0	0
$638$	3.00000	0.118771
$639$	24.0000	0.949425
$640$	−3.00000	−0.118585
$641$	48.0000	1.89589	0.947943	−	0.318440i	$-0.103159\pi$
$641$	48.0000	1.89589	0.947943	+	0.318440i	$0.103159\pi$
$642$	18.0000	0.710403
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	21.0000	0.826874
$646$	0	0
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	−1.00000	−0.0392837
$649$	36.0000	1.41312
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−7.00000	−0.274141
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	−13.0000	−0.508340
$655$	0	0
$656$	0	0
$657$	16.0000	0.624219
$658$	0	0
$659$	39.0000	1.51922	0.759612	−	0.650376i	$-0.225389\pi$
$659$	39.0000	1.51922	0.759612	+	0.650376i	$0.225389\pi$
$660$	9.00000	0.350325
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	31.0000	1.20485
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	4.00000	0.154997
$667$	−6.00000	−0.232321
$668$	0	0
$669$	8.00000	0.309298
$670$	−6.00000	−0.231800
$671$	−30.0000	−1.15814
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	−26.0000	−1.00148
$675$	20.0000	0.769800
$676$	−12.0000	−0.461538
$677$	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
$679$	0	0
$680$	0	0
$681$	18.0000	0.689761
$682$	−15.0000	−0.574380
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	−8.00000	−0.305888
$685$	−18.0000	−0.687745
$686$	0	0
$687$	2.00000	0.0763048
$688$	−7.00000	−0.266872
$689$	−9.00000	−0.342873
$690$	−18.0000	−0.685248
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−6.00000	−0.227757
$695$	30.0000	1.13796
$696$	1.00000	0.0379049
$697$	0	0
$698$	−19.0000	−0.719161
$699$	21.0000	0.794293
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	−5.00000	−0.188713
$703$	8.00000	0.301726
$704$	−3.00000	−0.113067
$705$	−9.00000	−0.338960
$706$	−18.0000	−0.677439
$707$	0	0
$708$	12.0000	0.450988
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	36.0000	1.35106
$711$	−10.0000	−0.375029
$712$	6.00000	0.224860
$713$	30.0000	1.12351
$714$	0	0
$715$	−9.00000	−0.336581
$716$	−12.0000	−0.448461
$717$	24.0000	0.896296
$718$	−21.0000	−0.783713
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−6.00000	−0.223607
$721$	0	0
$722$	3.00000	0.111648
$723$	−1.00000	−0.0371904
$724$	25.0000	0.929118
$725$	4.00000	0.148556
$726$	−2.00000	−0.0742270
$727$	52.0000	1.92857	0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000	1.92857	0.964287	+	0.264861i	$0.0853260\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	24.0000	0.888280
$731$	0	0
$732$	−10.0000	−0.369611
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	6.00000	0.221163
$737$	−6.00000	−0.221013
$738$	0	0
$739$	−13.0000	−0.478213	−0.239106	−	0.970993i	$-0.576854\pi$
$739$	−13.0000	−0.478213	−0.239106	+	0.970993i	$0.576854\pi$
$740$	6.00000	0.220564
$741$	−4.00000	−0.146944
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	−5.00000	−0.183309
$745$	27.0000	0.989203
$746$	19.0000	0.695639
$747$	24.0000	0.878114
$748$	0	0
$749$	0	0
$750$	−3.00000	−0.109545
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	3.00000	0.109399
$753$	−15.0000	−0.546630
$754$	−1.00000	−0.0364179
$755$	−30.0000	−1.09181
$756$	0	0
$757$	−16.0000	−0.581530	−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000	−0.581530	−0.290765	+	0.956795i	$0.593910\pi$
$758$	−20.0000	−0.726433
$759$	−18.0000	−0.653359
$760$	−12.0000	−0.435286
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	8.00000	0.289809
$763$	0	0
$764$	−24.0000	−0.868290
$765$	0	0
$766$	6.00000	0.216789
$767$	−12.0000	−0.433295
$768$	−1.00000	−0.0360844
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	27.0000	0.972381
$772$	−4.00000	−0.143963
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	−14.0000	−0.503220
$775$	−20.0000	−0.718421
$776$	8.00000	0.287183
$777$	0	0
$778$	−36.0000	−1.29066
$779$	0	0
$780$	−3.00000	−0.107417
$781$	36.0000	1.28818
$782$	0	0
$783$	5.00000	0.178685
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−18.0000	−0.641223
$789$	−3.00000	−0.106803
$790$	−15.0000	−0.533676
$791$	0	0
$792$	−6.00000	−0.213201
$793$	10.0000	0.355110
$794$	−13.0000	−0.461353
$795$	27.0000	0.957591
$796$	−20.0000	−0.708881
$797$	−48.0000	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	0	0
$799$	0	0
$800$	−4.00000	−0.141421
$801$	12.0000	0.423999
$802$	−15.0000	−0.529668
$803$	24.0000	0.846942
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	5.00000	0.176117
$807$	0	0
$808$	−12.0000	−0.422159
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	−3.00000	−0.105409
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	11.0000	0.385787
$814$	6.00000	0.210300
$815$	−21.0000	−0.735598
$816$	0	0
$817$	−28.0000	−0.979596
$818$	−16.0000	−0.559427
$819$	0	0
$820$	0	0
$821$	−39.0000	−1.36111	−0.680555	−	0.732697i	$-0.738261\pi$
$821$	−39.0000	−1.36111	−0.680555	+	0.732697i	$0.738261\pi$
$822$	−6.00000	−0.209274
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	8.00000	0.278693
$825$	12.0000	0.417786
$826$	0	0
$827$	21.0000	0.730242	0.365121	−	0.930960i	$-0.381028\pi$
$827$	21.0000	0.730242	0.365121	+	0.930960i	$0.381028\pi$
$828$	12.0000	0.417029
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	36.0000	1.24958
$831$	22.0000	0.763172
$832$	1.00000	0.0346688
$833$	0	0
$834$	10.0000	0.346272
$835$	0	0
$836$	−12.0000	−0.415029
$837$	−25.0000	−0.864126
$838$	−12.0000	−0.414533
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−26.0000	−0.896019
$843$	−3.00000	−0.103325
$844$	5.00000	0.172107
$845$	−36.0000	−1.23844
$846$	6.00000	0.206284
$847$	0	0
$848$	−9.00000	−0.309061
$849$	−22.0000	−0.755038
$850$	0	0
$851$	−12.0000	−0.411355
$852$	12.0000	0.411113
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	0	0
$855$	−24.0000	−0.820783
$856$	−18.0000	−0.615227
$857$	−51.0000	−1.74213	−0.871063	−	0.491171i	$-0.836569\pi$
$857$	−51.0000	−1.74213	−0.871063	+	0.491171i	$0.836569\pi$
$858$	−3.00000	−0.102418
$859$	31.0000	1.05771	0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000	1.05771	0.528853	+	0.848713i	$0.322622\pi$
$860$	−21.0000	−0.716094
$861$	0	0
$862$	12.0000	0.408722
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	−5.00000	−0.170103
$865$	54.0000	1.83606
$866$	−10.0000	−0.339814
$867$	17.0000	0.577350
$868$	0	0
$869$	−15.0000	−0.508840
$870$	3.00000	0.101710
$871$	2.00000	0.0677674
$872$	13.0000	0.440236
$873$	16.0000	0.541518
$874$	24.0000	0.811812
$875$	0	0
$876$	8.00000	0.270295
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	−10.0000	−0.337484
$879$	18.0000	0.607125
$880$	−9.00000	−0.303390
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	0	0
$885$	36.0000	1.21013
$886$	36.0000	1.20944
$887$	15.0000	0.503651	0.251825	−	0.967773i	$-0.418969\pi$
$887$	15.0000	0.503651	0.251825	+	0.967773i	$0.418969\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	18.0000	0.603361
$891$	−3.00000	−0.100504
$892$	−8.00000	−0.267860
$893$	12.0000	0.401565
$894$	9.00000	0.301005
$895$	−36.0000	−1.20335
$896$	0	0
$897$	6.00000	0.200334
$898$	−12.0000	−0.400445
$899$	−5.00000	−0.166759
$900$	−8.00000	−0.266667
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−12.0000	−0.399114
$905$	75.0000	2.49308
$906$	−10.0000	−0.332228
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	−18.0000	−0.597351
$909$	−24.0000	−0.796030
$910$	0	0
$911$	−39.0000	−1.29213	−0.646064	−	0.763283i	$-0.723586\pi$
$911$	−39.0000	−1.29213	−0.646064	+	0.763283i	$0.723586\pi$
$912$	−4.00000	−0.132453
$913$	36.0000	1.19143
$914$	34.0000	1.12462
$915$	−30.0000	−0.991769
$916$	−2.00000	−0.0660819
$917$	0	0
$918$	0	0
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	18.0000	0.593442
$921$	23.0000	0.757876
$922$	24.0000	0.790398
$923$	−12.0000	−0.394985
$924$	0	0
$925$	8.00000	0.263038
$926$	−8.00000	−0.262896
$927$	16.0000	0.525509
$928$	−1.00000	−0.0328266
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	−15.0000	−0.491869
$931$	0	0
$932$	−21.0000	−0.687878
$933$	−12.0000	−0.392862
$934$	3.00000	0.0981630
$935$	0	0
$936$	2.00000	0.0653720
$937$	46.0000	1.50275	0.751377	−	0.659873i	$-0.229390\pi$
$937$	46.0000	1.50275	0.751377	+	0.659873i	$0.229390\pi$
$938$	0	0
$939$	−25.0000	−0.815844
$940$	9.00000	0.293548
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	4.00000	0.130327
$943$	0	0
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−21.0000	−0.682769
$947$	9.00000	0.292461	0.146230	−	0.989251i	$-0.453286\pi$
$947$	9.00000	0.292461	0.146230	+	0.989251i	$0.453286\pi$
$948$	−5.00000	−0.162392
$949$	−8.00000	−0.259691
$950$	−16.0000	−0.519109
$951$	18.0000	0.583690
$952$	0	0
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	−18.0000	−0.582772
$955$	−72.0000	−2.32987
$956$	−24.0000	−0.776215
$957$	3.00000	0.0969762
$958$	15.0000	0.484628
$959$	0	0
$960$	−3.00000	−0.0968246
$961$	−6.00000	−0.193548
$962$	−2.00000	−0.0644826
$963$	−36.0000	−1.16008
$964$	1.00000	0.0322078
$965$	−12.0000	−0.386294
$966$	0	0
$967$	23.0000	0.739630	0.369815	−	0.929105i	$-0.379421\pi$
$967$	23.0000	0.739630	0.369815	+	0.929105i	$0.379421\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	24.0000	0.770594
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−16.0000	−0.513200
$973$	0	0
$974$	−20.0000	−0.640841
$975$	−4.00000	−0.128103
$976$	10.0000	0.320092
$977$	21.0000	0.671850	0.335925	−	0.941889i	$-0.390951\pi$
$977$	21.0000	0.671850	0.335925	+	0.941889i	$0.390951\pi$
$978$	−7.00000	−0.223835
$979$	18.0000	0.575282
$980$	0	0
$981$	26.0000	0.830116
$982$	−3.00000	−0.0957338
$983$	3.00000	0.0956851	0.0478426	−	0.998855i	$-0.484765\pi$
$983$	3.00000	0.0956851	0.0478426	+	0.998855i	$0.484765\pi$
$984$	0	0
$985$	−54.0000	−1.72058
$986$	0	0
$987$	0	0
$988$	4.00000	0.127257
$989$	42.0000	1.33552
$990$	−18.0000	−0.572078
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	5.00000	0.158750
$993$	31.0000	0.983755
$994$	0	0
$995$	−60.0000	−1.90213
$996$	12.0000	0.380235
$997$	58.0000	1.83688	0.918439	−	0.395562i	$-0.129450\pi$
$997$	58.0000	1.83688	0.918439	+	0.395562i	$0.129450\pi$
$998$	−38.0000	−1.20287
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.b.1.1		1
7.6	odd	2		406.2.a.b.1.1	✓	1
21.20	even	2		3654.2.a.w.1.1		1
28.27	even	2		3248.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
406.2.a.b.1.1	✓	1	7.6	odd	2
2842.2.a.b.1.1		1	1.1	even	1	trivial
3248.2.a.f.1.1		1	28.27	even	2
3654.2.a.w.1.1		1	21.20	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:	\( N \)	\(=\)	\( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2842.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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