LMFDB - Embedded newform 8034.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1518129839$
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	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +3.00000 q^{21} -2.00000 q^{22} -1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.00000 q^{28} +7.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +3.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -3.00000 q^{42} +2.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +13.0000 q^{47} +1.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} -1.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -3.00000 q^{56} +4.00000 q^{57} -7.00000 q^{58} -5.00000 q^{59} +1.00000 q^{60} +4.00000 q^{61} +4.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} -2.00000 q^{66} -1.00000 q^{67} -3.00000 q^{70} -1.00000 q^{72} +3.00000 q^{73} -2.00000 q^{74} -4.00000 q^{75} +4.00000 q^{76} +6.00000 q^{77} +1.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -1.00000 q^{83} +3.00000 q^{84} -2.00000 q^{86} +7.00000 q^{87} -2.00000 q^{88} -8.00000 q^{89} -1.00000 q^{90} -3.00000 q^{91} -4.00000 q^{93} -13.0000 q^{94} +4.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} -2.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.00000	−0.801784
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	3.00000	0.654654
$22$	−2.00000	−0.426401
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	3.00000	0.566947
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	−1.00000	−0.182574
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	0	0
$35$	3.00000	0.507093
$36$	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	$0.447432\pi$
$38$	−4.00000	−0.648886
$39$	−1.00000	−0.160128
$40$	−1.00000	−0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−3.00000	−0.462910
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	2.00000	0.301511
$45$	1.00000	0.149071
$46$	0	0
$47$	13.0000	1.89624	0.948122	−	0.317905i	$-0.102979\pi$
$47$	13.0000	1.89624	0.948122	+	0.317905i	$0.102979\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	4.00000	0.565685
$51$	0	0
$52$	−1.00000	−0.138675
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	−1.00000	−0.136083
$55$	2.00000	0.269680
$56$	−3.00000	−0.400892
$57$	4.00000	0.529813
$58$	−7.00000	−0.919145
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	1.00000	0.129099
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	4.00000	0.508001
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−2.00000	−0.246183
$67$	−1.00000	−0.122169	−0.0610847	−	0.998133i	$-0.519456\pi$
$67$	−1.00000	−0.122169	−0.0610847	+	0.998133i	$0.519456\pi$
$68$	0	0
$69$	0	0
$70$	−3.00000	−0.358569
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	−2.00000	−0.232495
$75$	−4.00000	−0.461880
$76$	4.00000	0.458831
$77$	6.00000	0.683763
$78$	1.00000	0.113228
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	−2.00000	−0.215666
$87$	7.00000	0.750479
$88$	−2.00000	−0.213201
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	−1.00000	−0.105409
$91$	−3.00000	−0.314485
$92$	0	0
$93$	−4.00000	−0.414781
$94$	−13.0000	−1.34085
$95$	4.00000	0.410391
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−2.00000	−0.202031
$99$	2.00000	0.201008
$100$	−4.00000	−0.400000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	1.00000	0.0980581
$105$	3.00000	0.292770
$106$	−2.00000	−0.194257
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	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	$0.469465\pi$
$110$	−2.00000	−0.190693
$111$	2.00000	0.189832
$112$	3.00000	0.283473
$113$	19.0000	1.78737	0.893685	−	0.448695i	$-0.148111\pi$
$113$	19.0000	1.78737	0.893685	+	0.448695i	$0.148111\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	7.00000	0.649934
$117$	−1.00000	−0.0924500
$118$	5.00000	0.460287
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−7.00000	−0.636364
$122$	−4.00000	−0.362143
$123$	−6.00000	−0.541002
$124$	−4.00000	−0.359211
$125$	−9.00000	−0.804984
$126$	−3.00000	−0.267261
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	−1.00000	−0.0883883
$129$	2.00000	0.176090
$130$	1.00000	0.0877058
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	2.00000	0.174078
$133$	12.0000	1.04053
$134$	1.00000	0.0863868
$135$	1.00000	0.0860663
$136$	0	0
$137$	13.0000	1.11066	0.555332	−	0.831628i	$-0.312591\pi$
$137$	13.0000	1.11066	0.555332	+	0.831628i	$0.312591\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	3.00000	0.253546
$141$	13.0000	1.09480
$142$	0	0
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	7.00000	0.581318
$146$	−3.00000	−0.248282
$147$	2.00000	0.164957
$148$	2.00000	0.164399
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	4.00000	0.326599
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	−6.00000	−0.483494
$155$	−4.00000	−0.321288
$156$	−1.00000	−0.0800641
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	4.00000	0.318223
$159$	2.00000	0.158610
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	−6.00000	−0.468521
$165$	2.00000	0.155700
$166$	1.00000	0.0776151
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	−3.00000	−0.231455
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.00000	0.305888
$172$	2.00000	0.152499
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	−7.00000	−0.530669
$175$	−12.0000	−0.907115
$176$	2.00000	0.150756
$177$	−5.00000	−0.375823
$178$	8.00000	0.599625
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	1.00000	0.0745356
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	3.00000	0.222375
$183$	4.00000	0.295689
$184$	0	0
$185$	2.00000	0.147043
$186$	4.00000	0.293294
$187$	0	0
$188$	13.0000	0.948122
$189$	3.00000	0.218218
$190$	−4.00000	−0.290191
$191$	22.0000	1.59186	0.795932	−	0.605386i	$-0.206981\pi$
$191$	22.0000	1.59186	0.795932	+	0.605386i	$0.206981\pi$
$192$	1.00000	0.0721688
$193$	17.0000	1.22369	0.611843	−	0.790979i	$-0.290428\pi$
$193$	17.0000	1.22369	0.611843	+	0.790979i	$0.290428\pi$
$194$	−14.0000	−1.00514
$195$	−1.00000	−0.0716115
$196$	2.00000	0.142857
$197$	−1.00000	−0.0712470	−0.0356235	−	0.999365i	$-0.511342\pi$
$197$	−1.00000	−0.0712470	−0.0356235	+	0.999365i	$0.511342\pi$
$198$	−2.00000	−0.142134
$199$	9.00000	0.637993	0.318997	−	0.947756i	$-0.396654\pi$
$199$	9.00000	0.637993	0.318997	+	0.947756i	$0.396654\pi$
$200$	4.00000	0.282843
$201$	−1.00000	−0.0705346
$202$	−14.0000	−0.985037
$203$	21.0000	1.47391
$204$	0	0
$205$	−6.00000	−0.419058
$206$	1.00000	0.0696733
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	8.00000	0.553372
$210$	−3.00000	−0.207020
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−18.0000	−1.23045
$215$	2.00000	0.136399
$216$	−1.00000	−0.0680414
$217$	−12.0000	−0.814613
$218$	−2.00000	−0.135457
$219$	3.00000	0.202721
$220$	2.00000	0.134840
$221$	0	0
$222$	−2.00000	−0.134231
$223$	−27.0000	−1.80805	−0.904027	−	0.427476i	$-0.859403\pi$
$223$	−27.0000	−1.80805	−0.904027	+	0.427476i	$0.859403\pi$
$224$	−3.00000	−0.200446
$225$	−4.00000	−0.266667
$226$	−19.0000	−1.26386
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	4.00000	0.264906
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	−7.00000	−0.459573
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	1.00000	0.0653720
$235$	13.0000	0.848026
$236$	−5.00000	−0.325472
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	1.00000	0.0645497
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	4.00000	0.256074
$245$	2.00000	0.127775
$246$	6.00000	0.382546
$247$	−4.00000	−0.254514
$248$	4.00000	0.254000
$249$	−1.00000	−0.0633724
$250$	9.00000	0.569210
$251$	1.00000	0.0631194	0.0315597	−	0.999502i	$-0.489953\pi$
$251$	1.00000	0.0631194	0.0315597	+	0.999502i	$0.489953\pi$
$252$	3.00000	0.188982
$253$	0	0
$254$	20.0000	1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	−2.00000	−0.124515
$259$	6.00000	0.372822
$260$	−1.00000	−0.0620174
$261$	7.00000	0.433289
$262$	−2.00000	−0.123560
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	−2.00000	−0.123091
$265$	2.00000	0.122859
$266$	−12.0000	−0.735767
$267$	−8.00000	−0.489592
$268$	−1.00000	−0.0610847
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	−1.00000	−0.0608581
$271$	30.0000	1.82237	0.911185	−	0.411997i	$-0.135169\pi$
$271$	30.0000	1.82237	0.911185	+	0.411997i	$0.135169\pi$
$272$	0	0
$273$	−3.00000	−0.181568
$274$	−13.0000	−0.785359
$275$	−8.00000	−0.482418
$276$	0	0
$277$	−31.0000	−1.86261	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−31.0000	−1.86261	−0.931305	+	0.364241i	$0.881328\pi$
$278$	4.00000	0.239904
$279$	−4.00000	−0.239474
$280$	−3.00000	−0.179284
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	−13.0000	−0.774139
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	2.00000	0.118262
$287$	−18.0000	−1.06251
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	−7.00000	−0.411054
$291$	14.0000	0.820695
$292$	3.00000	0.175562
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	−2.00000	−0.116642
$295$	−5.00000	−0.291111
$296$	−2.00000	−0.116248
$297$	2.00000	0.116052
$298$	18.0000	1.04271
$299$	0	0
$300$	−4.00000	−0.230940
$301$	6.00000	0.345834
$302$	−2.00000	−0.115087
$303$	14.0000	0.804279
$304$	4.00000	0.229416
$305$	4.00000	0.229039
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	6.00000	0.341882
$309$	−1.00000	−0.0568880
$310$	4.00000	0.227185
$311$	−35.0000	−1.98467	−0.992334	−	0.123585i	$-0.960561\pi$
$311$	−35.0000	−1.98467	−0.992334	+	0.123585i	$0.960561\pi$
$312$	1.00000	0.0566139
$313$	−17.0000	−0.960897	−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000	−0.960897	−0.480448	+	0.877023i	$0.659526\pi$
$314$	13.0000	0.733632
$315$	3.00000	0.169031
$316$	−4.00000	−0.225018
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−2.00000	−0.112154
$319$	14.0000	0.783850
$320$	1.00000	0.0559017
$321$	18.0000	1.00466
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	2.00000	0.110770
$327$	2.00000	0.110600
$328$	6.00000	0.331295
$329$	39.0000	2.15014
$330$	−2.00000	−0.110096
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	−1.00000	−0.0548821
$333$	2.00000	0.109599
$334$	18.0000	0.984916
$335$	−1.00000	−0.0546358
$336$	3.00000	0.163663
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−1.00000	−0.0543928
$339$	19.0000	1.03194
$340$	0	0
$341$	−8.00000	−0.433224
$342$	−4.00000	−0.216295
$343$	−15.0000	−0.809924
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−22.0000	−1.18273
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	7.00000	0.375239
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	12.0000	0.641427
$351$	−1.00000	−0.0533761
$352$	−2.00000	−0.106600
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	5.00000	0.265747
$355$	0	0
$356$	−8.00000	−0.423999
$357$	0	0
$358$	4.00000	0.211407
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	6.00000	0.315353
$363$	−7.00000	−0.367405
$364$	−3.00000	−0.157243
$365$	3.00000	0.157027
$366$	−4.00000	−0.209083
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	−2.00000	−0.103975
$371$	6.00000	0.311504
$372$	−4.00000	−0.207390
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	−13.0000	−0.670424
$377$	−7.00000	−0.360518
$378$	−3.00000	−0.154303
$379$	1.00000	0.0513665	0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000	0.0513665	0.0256833	+	0.999670i	$0.491824\pi$
$380$	4.00000	0.205196
$381$	−20.0000	−1.02463
$382$	−22.0000	−1.12562
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	−1.00000	−0.0510310
$385$	6.00000	0.305788
$386$	−17.0000	−0.865277
$387$	2.00000	0.101666
$388$	14.0000	0.710742
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	1.00000	0.0506370
$391$	0	0
$392$	−2.00000	−0.101015
$393$	2.00000	0.100887
$394$	1.00000	0.0503793
$395$	−4.00000	−0.201262
$396$	2.00000	0.100504
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	−9.00000	−0.451129
$399$	12.0000	0.600751
$400$	−4.00000	−0.200000
$401$	33.0000	1.64794	0.823971	−	0.566632i	$-0.191754\pi$
$401$	33.0000	1.64794	0.823971	+	0.566632i	$0.191754\pi$
$402$	1.00000	0.0498755
$403$	4.00000	0.199254
$404$	14.0000	0.696526
$405$	1.00000	0.0496904
$406$	−21.0000	−1.04221
$407$	4.00000	0.198273
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	6.00000	0.296319
$411$	13.0000	0.641243
$412$	−1.00000	−0.0492665
$413$	−15.0000	−0.738102
$414$	0	0
$415$	−1.00000	−0.0490881
$416$	1.00000	0.0490290
$417$	−4.00000	−0.195881
$418$	−8.00000	−0.391293
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	3.00000	0.146385
$421$	37.0000	1.80327	0.901635	−	0.432498i	$-0.142368\pi$
$421$	37.0000	1.80327	0.901635	+	0.432498i	$0.142368\pi$
$422$	−10.0000	−0.486792
$423$	13.0000	0.632082
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	0	0
$427$	12.0000	0.580721
$428$	18.0000	0.870063
$429$	−2.00000	−0.0965609
$430$	−2.00000	−0.0964486
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	1.00000	0.0481125
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	12.0000	0.576018
$435$	7.00000	0.335624
$436$	2.00000	0.0957826
$437$	0	0
$438$	−3.00000	−0.143346
$439$	13.0000	0.620456	0.310228	−	0.950662i	$-0.399595\pi$
$439$	13.0000	0.620456	0.310228	+	0.950662i	$0.399595\pi$
$440$	−2.00000	−0.0953463
$441$	2.00000	0.0952381
$442$	0	0
$443$	33.0000	1.56788	0.783939	−	0.620838i	$-0.213208\pi$
$443$	33.0000	1.56788	0.783939	+	0.620838i	$0.213208\pi$
$444$	2.00000	0.0949158
$445$	−8.00000	−0.379236
$446$	27.0000	1.27849
$447$	−18.0000	−0.851371
$448$	3.00000	0.141737
$449$	42.0000	1.98210	0.991051	−	0.133482i	$-0.0426157\pi$
$449$	42.0000	1.98210	0.991051	+	0.133482i	$0.0426157\pi$
$450$	4.00000	0.188562
$451$	−12.0000	−0.565058
$452$	19.0000	0.893685
$453$	2.00000	0.0939682
$454$	−24.0000	−1.12638
$455$	−3.00000	−0.140642
$456$	−4.00000	−0.187317
$457$	9.00000	0.421002	0.210501	−	0.977594i	$-0.432490\pi$
$457$	9.00000	0.421002	0.210501	+	0.977594i	$0.432490\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	−6.00000	−0.279145
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	7.00000	0.324967
$465$	−4.00000	−0.185496
$466$	−6.00000	−0.277945
$467$	−14.0000	−0.647843	−0.323921	−	0.946084i	$-0.605001\pi$
$467$	−14.0000	−0.647843	−0.323921	+	0.946084i	$0.605001\pi$
$468$	−1.00000	−0.0462250
$469$	−3.00000	−0.138527
$470$	−13.0000	−0.599645
$471$	−13.0000	−0.599008
$472$	5.00000	0.230144
$473$	4.00000	0.183920
$474$	4.00000	0.183726
$475$	−16.0000	−0.734130
$476$	0	0
$477$	2.00000	0.0915737
$478$	6.00000	0.274434
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	−1.00000	−0.0456435
$481$	−2.00000	−0.0911922
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−7.00000	−0.318182
$485$	14.0000	0.635707
$486$	−1.00000	−0.0453609
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	−4.00000	−0.181071
$489$	−2.00000	−0.0904431
$490$	−2.00000	−0.0903508
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−6.00000	−0.270501
$493$	0	0
$494$	4.00000	0.179969
$495$	2.00000	0.0898933
$496$	−4.00000	−0.179605
$497$	0	0
$498$	1.00000	0.0448111
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	−9.00000	−0.402492
$501$	−18.0000	−0.804181
$502$	−1.00000	−0.0446322
$503$	−37.0000	−1.64975	−0.824874	−	0.565316i	$-0.808754\pi$
$503$	−37.0000	−1.64975	−0.824874	+	0.565316i	$0.808754\pi$
$504$	−3.00000	−0.133631
$505$	14.0000	0.622992
$506$	0	0
$507$	1.00000	0.0444116
$508$	−20.0000	−0.887357
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	9.00000	0.398137
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	−13.0000	−0.573405
$515$	−1.00000	−0.0440653
$516$	2.00000	0.0880451
$517$	26.0000	1.14348
$518$	−6.00000	−0.263625
$519$	22.0000	0.965693
$520$	1.00000	0.0438529
$521$	21.0000	0.920027	0.460013	−	0.887912i	$-0.347845\pi$
$521$	21.0000	0.920027	0.460013	+	0.887912i	$0.347845\pi$
$522$	−7.00000	−0.306382
$523$	9.00000	0.393543	0.196771	−	0.980449i	$-0.436954\pi$
$523$	9.00000	0.393543	0.196771	+	0.980449i	$0.436954\pi$
$524$	2.00000	0.0873704
$525$	−12.0000	−0.523723
$526$	10.0000	0.436021
$527$	0	0
$528$	2.00000	0.0870388
$529$	−23.0000	−1.00000
$530$	−2.00000	−0.0868744
$531$	−5.00000	−0.216982
$532$	12.0000	0.520266
$533$	6.00000	0.259889
$534$	8.00000	0.346194
$535$	18.0000	0.778208
$536$	1.00000	0.0431934
$537$	−4.00000	−0.172613
$538$	−9.00000	−0.388018
$539$	4.00000	0.172292
$540$	1.00000	0.0430331
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	−30.0000	−1.28861
$543$	−6.00000	−0.257485
$544$	0	0
$545$	2.00000	0.0856706
$546$	3.00000	0.128388
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	13.0000	0.555332
$549$	4.00000	0.170716
$550$	8.00000	0.341121
$551$	28.0000	1.19284
$552$	0	0
$553$	−12.0000	−0.510292
$554$	31.0000	1.31706
$555$	2.00000	0.0848953
$556$	−4.00000	−0.169638
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	4.00000	0.169334
$559$	−2.00000	−0.0845910
$560$	3.00000	0.126773
$561$	0	0
$562$	10.0000	0.421825
$563$	−1.00000	−0.0421450	−0.0210725	−	0.999778i	$-0.506708\pi$
$563$	−1.00000	−0.0421450	−0.0210725	+	0.999778i	$0.506708\pi$
$564$	13.0000	0.547399
$565$	19.0000	0.799336
$566$	6.00000	0.252199
$567$	3.00000	0.125988
$568$	0	0
$569$	−33.0000	−1.38343	−0.691716	−	0.722170i	$-0.743145\pi$
$569$	−33.0000	−1.38343	−0.691716	+	0.722170i	$0.743145\pi$
$570$	−4.00000	−0.167542
$571$	21.0000	0.878823	0.439411	−	0.898286i	$-0.355187\pi$
$571$	21.0000	0.878823	0.439411	+	0.898286i	$0.355187\pi$
$572$	−2.00000	−0.0836242
$573$	22.0000	0.919063
$574$	18.0000	0.751305
$575$	0	0
$576$	1.00000	0.0416667
$577$	−15.0000	−0.624458	−0.312229	−	0.950007i	$-0.601076\pi$
$577$	−15.0000	−0.624458	−0.312229	+	0.950007i	$0.601076\pi$
$578$	17.0000	0.707107
$579$	17.0000	0.706496
$580$	7.00000	0.290659
$581$	−3.00000	−0.124461
$582$	−14.0000	−0.580319
$583$	4.00000	0.165663
$584$	−3.00000	−0.124141
$585$	−1.00000	−0.0413449
$586$	30.0000	1.23929
$587$	41.0000	1.69225	0.846126	−	0.532984i	$-0.178929\pi$
$587$	41.0000	1.69225	0.846126	+	0.532984i	$0.178929\pi$
$588$	2.00000	0.0824786
$589$	−16.0000	−0.659269
$590$	5.00000	0.205847
$591$	−1.00000	−0.0411345
$592$	2.00000	0.0821995
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	−18.0000	−0.737309
$597$	9.00000	0.368345
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	4.00000	0.163299
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	−6.00000	−0.244542
$603$	−1.00000	−0.0407231
$604$	2.00000	0.0813788
$605$	−7.00000	−0.284590
$606$	−14.0000	−0.568711
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	−4.00000	−0.162221
$609$	21.0000	0.850963
$610$	−4.00000	−0.161955
$611$	−13.0000	−0.525924
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	28.0000	1.12999
$615$	−6.00000	−0.241943
$616$	−6.00000	−0.241747
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	1.00000	0.0402259
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	35.0000	1.40337
$623$	−24.0000	−0.961540
$624$	−1.00000	−0.0400320
$625$	11.0000	0.440000
$626$	17.0000	0.679457
$627$	8.00000	0.319489
$628$	−13.0000	−0.518756
$629$	0	0
$630$	−3.00000	−0.119523
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	4.00000	0.159111
$633$	10.0000	0.397464
$634$	−18.0000	−0.714871
$635$	−20.0000	−0.793676
$636$	2.00000	0.0793052
$637$	−2.00000	−0.0792429
$638$	−14.0000	−0.554265
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	−18.0000	−0.710403
$643$	−50.0000	−1.97181	−0.985904	−	0.167313i	$-0.946491\pi$
$643$	−50.0000	−1.97181	−0.985904	+	0.167313i	$0.946491\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	0	0
$647$	31.0000	1.21874	0.609368	−	0.792888i	$-0.291423\pi$
$647$	31.0000	1.21874	0.609368	+	0.792888i	$0.291423\pi$
$648$	−1.00000	−0.0392837
$649$	−10.0000	−0.392534
$650$	−4.00000	−0.156893
$651$	−12.0000	−0.470317
$652$	−2.00000	−0.0783260
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	−2.00000	−0.0782062
$655$	2.00000	0.0781465
$656$	−6.00000	−0.234261
$657$	3.00000	0.117041
$658$	−39.0000	−1.52038
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	2.00000	0.0778499
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	−17.0000	−0.660724
$663$	0	0
$664$	1.00000	0.0388075
$665$	12.0000	0.465340
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	−18.0000	−0.696441
$669$	−27.0000	−1.04388
$670$	1.00000	0.0386334
$671$	8.00000	0.308837
$672$	−3.00000	−0.115728
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	14.0000	0.539260
$675$	−4.00000	−0.153960
$676$	1.00000	0.0384615
$677$	13.0000	0.499631	0.249815	−	0.968294i	$-0.419630\pi$
$677$	13.0000	0.499631	0.249815	+	0.968294i	$0.419630\pi$
$678$	−19.0000	−0.729691
$679$	42.0000	1.61181
$680$	0	0
$681$	24.0000	0.919682
$682$	8.00000	0.306336
$683$	14.0000	0.535695	0.267848	−	0.963461i	$-0.413688\pi$
$683$	14.0000	0.535695	0.267848	+	0.963461i	$0.413688\pi$
$684$	4.00000	0.152944
$685$	13.0000	0.496704
$686$	15.0000	0.572703
$687$	10.0000	0.381524
$688$	2.00000	0.0762493
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−11.0000	−0.418460	−0.209230	−	0.977866i	$-0.567096\pi$
$691$	−11.0000	−0.418460	−0.209230	+	0.977866i	$0.567096\pi$
$692$	22.0000	0.836315
$693$	6.00000	0.227921
$694$	28.0000	1.06287
$695$	−4.00000	−0.151729
$696$	−7.00000	−0.265334
$697$	0	0
$698$	−28.0000	−1.05982
$699$	6.00000	0.226941
$700$	−12.0000	−0.453557
$701$	9.00000	0.339925	0.169963	−	0.985451i	$-0.445635\pi$
$701$	9.00000	0.339925	0.169963	+	0.985451i	$0.445635\pi$
$702$	1.00000	0.0377426
$703$	8.00000	0.301726
$704$	2.00000	0.0753778
$705$	13.0000	0.489608
$706$	24.0000	0.903252
$707$	42.0000	1.57957
$708$	−5.00000	−0.187912
$709$	−11.0000	−0.413114	−0.206557	−	0.978435i	$-0.566226\pi$
$709$	−11.0000	−0.413114	−0.206557	+	0.978435i	$0.566226\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	8.00000	0.299813
$713$	0	0
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	−4.00000	−0.149487
$717$	−6.00000	−0.224074
$718$	18.0000	0.671754
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	1.00000	0.0372678
$721$	−3.00000	−0.111726
$722$	3.00000	0.111648
$723$	2.00000	0.0743808
$724$	−6.00000	−0.222988
$725$	−28.0000	−1.03989
$726$	7.00000	0.259794
$727$	−7.00000	−0.259616	−0.129808	−	0.991539i	$-0.541436\pi$
$727$	−7.00000	−0.259616	−0.129808	+	0.991539i	$0.541436\pi$
$728$	3.00000	0.111187
$729$	1.00000	0.0370370
$730$	−3.00000	−0.111035
$731$	0	0
$732$	4.00000	0.147844
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−16.0000	−0.590571
$735$	2.00000	0.0737711
$736$	0	0
$737$	−2.00000	−0.0736709
$738$	6.00000	0.220863
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	2.00000	0.0735215
$741$	−4.00000	−0.146944
$742$	−6.00000	−0.220267
$743$	15.0000	0.550297	0.275148	−	0.961402i	$-0.411273\pi$
$743$	15.0000	0.550297	0.275148	+	0.961402i	$0.411273\pi$
$744$	4.00000	0.146647
$745$	−18.0000	−0.659469
$746$	4.00000	0.146450
$747$	−1.00000	−0.0365881
$748$	0	0
$749$	54.0000	1.97312
$750$	9.00000	0.328634
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	13.0000	0.474061
$753$	1.00000	0.0364420
$754$	7.00000	0.254925
$755$	2.00000	0.0727875
$756$	3.00000	0.109109
$757$	4.00000	0.145382	0.0726912	−	0.997354i	$-0.476841\pi$
$757$	4.00000	0.145382	0.0726912	+	0.997354i	$0.476841\pi$
$758$	−1.00000	−0.0363216
$759$	0	0
$760$	−4.00000	−0.145095
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	20.0000	0.724524
$763$	6.00000	0.217215
$764$	22.0000	0.795932
$765$	0	0
$766$	−21.0000	−0.758761
$767$	5.00000	0.180540
$768$	1.00000	0.0360844
$769$	13.0000	0.468792	0.234396	−	0.972141i	$-0.424689\pi$
$769$	13.0000	0.468792	0.234396	+	0.972141i	$0.424689\pi$
$770$	−6.00000	−0.216225
$771$	13.0000	0.468184
$772$	17.0000	0.611843
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	−2.00000	−0.0718885
$775$	16.0000	0.574737
$776$	−14.0000	−0.502571
$777$	6.00000	0.215249
$778$	14.0000	0.501924
$779$	−24.0000	−0.859889
$780$	−1.00000	−0.0358057
$781$	0	0
$782$	0	0
$783$	7.00000	0.250160
$784$	2.00000	0.0714286
$785$	−13.0000	−0.463990
$786$	−2.00000	−0.0713376
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	−1.00000	−0.0356235
$789$	−10.0000	−0.356009
$790$	4.00000	0.142314
$791$	57.0000	2.02669
$792$	−2.00000	−0.0710669
$793$	−4.00000	−0.142044
$794$	4.00000	0.141955
$795$	2.00000	0.0709327
$796$	9.00000	0.318997
$797$	−7.00000	−0.247953	−0.123976	−	0.992285i	$-0.539565\pi$
$797$	−7.00000	−0.247953	−0.123976	+	0.992285i	$0.539565\pi$
$798$	−12.0000	−0.424795
$799$	0	0
$800$	4.00000	0.141421
$801$	−8.00000	−0.282666
$802$	−33.0000	−1.16527
$803$	6.00000	0.211735
$804$	−1.00000	−0.0352673
$805$	0	0
$806$	−4.00000	−0.140894
$807$	9.00000	0.316815
$808$	−14.0000	−0.492518
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	−1.00000	−0.0351364
$811$	19.0000	0.667180	0.333590	−	0.942718i	$-0.391740\pi$
$811$	19.0000	0.667180	0.333590	+	0.942718i	$0.391740\pi$
$812$	21.0000	0.736956
$813$	30.0000	1.05215
$814$	−4.00000	−0.140200
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	8.00000	0.279885
$818$	34.0000	1.18878
$819$	−3.00000	−0.104828
$820$	−6.00000	−0.209529
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	−13.0000	−0.453427
$823$	−5.00000	−0.174289	−0.0871445	−	0.996196i	$-0.527774\pi$
$823$	−5.00000	−0.174289	−0.0871445	+	0.996196i	$0.527774\pi$
$824$	1.00000	0.0348367
$825$	−8.00000	−0.278524
$826$	15.0000	0.521917
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	23.0000	0.798823	0.399412	−	0.916772i	$-0.369214\pi$
$829$	23.0000	0.798823	0.399412	+	0.916772i	$0.369214\pi$
$830$	1.00000	0.0347105
$831$	−31.0000	−1.07538
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	4.00000	0.138509
$835$	−18.0000	−0.622916
$836$	8.00000	0.276686
$837$	−4.00000	−0.138260
$838$	−10.0000	−0.345444
$839$	14.0000	0.483334	0.241667	−	0.970359i	$-0.422306\pi$
$839$	14.0000	0.483334	0.241667	+	0.970359i	$0.422306\pi$
$840$	−3.00000	−0.103510
$841$	20.0000	0.689655
$842$	−37.0000	−1.27510
$843$	−10.0000	−0.344418
$844$	10.0000	0.344214
$845$	1.00000	0.0344010
$846$	−13.0000	−0.446949
$847$	−21.0000	−0.721569
$848$	2.00000	0.0686803
$849$	−6.00000	−0.205919
$850$	0	0
$851$	0	0
$852$	0	0
$853$	15.0000	0.513590	0.256795	−	0.966466i	$-0.417333\pi$
$853$	15.0000	0.513590	0.256795	+	0.966466i	$0.417333\pi$
$854$	−12.0000	−0.410632
$855$	4.00000	0.136797
$856$	−18.0000	−0.615227
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	2.00000	0.0682789
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	2.00000	0.0681994
$861$	−18.0000	−0.613438
$862$	10.0000	0.340601
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	−1.00000	−0.0340207
$865$	22.0000	0.748022
$866$	−26.0000	−0.883516
$867$	−17.0000	−0.577350
$868$	−12.0000	−0.407307
$869$	−8.00000	−0.271381
$870$	−7.00000	−0.237322
$871$	1.00000	0.0338837
$872$	−2.00000	−0.0677285
$873$	14.0000	0.473828
$874$	0	0
$875$	−27.0000	−0.912767
$876$	3.00000	0.101361
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	−13.0000	−0.438729
$879$	−30.0000	−1.01187
$880$	2.00000	0.0674200
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	−2.00000	−0.0673435
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	−5.00000	−0.168073
$886$	−33.0000	−1.10866
$887$	−29.0000	−0.973725	−0.486862	−	0.873479i	$-0.661859\pi$
$887$	−29.0000	−0.973725	−0.486862	+	0.873479i	$0.661859\pi$
$888$	−2.00000	−0.0671156
$889$	−60.0000	−2.01234
$890$	8.00000	0.268161
$891$	2.00000	0.0670025
$892$	−27.0000	−0.904027
$893$	52.0000	1.74011
$894$	18.0000	0.602010
$895$	−4.00000	−0.133705
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−42.0000	−1.40156
$899$	−28.0000	−0.933852
$900$	−4.00000	−0.133333
$901$	0	0
$902$	12.0000	0.399556
$903$	6.00000	0.199667
$904$	−19.0000	−0.631931
$905$	−6.00000	−0.199447
$906$	−2.00000	−0.0664455
$907$	−27.0000	−0.896520	−0.448260	−	0.893903i	$-0.647956\pi$
$907$	−27.0000	−0.896520	−0.448260	+	0.893903i	$0.647956\pi$
$908$	24.0000	0.796468
$909$	14.0000	0.464351
$910$	3.00000	0.0994490
$911$	38.0000	1.25900	0.629498	−	0.777002i	$-0.283261\pi$
$911$	38.0000	1.25900	0.629498	+	0.777002i	$0.283261\pi$
$912$	4.00000	0.132453
$913$	−2.00000	−0.0661903
$914$	−9.00000	−0.297694
$915$	4.00000	0.132236
$916$	10.0000	0.330409
$917$	6.00000	0.198137
$918$	0	0
$919$	−43.0000	−1.41844	−0.709220	−	0.704988i	$-0.750953\pi$
$919$	−43.0000	−1.41844	−0.709220	+	0.704988i	$0.750953\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	−24.0000	−0.790398
$923$	0	0
$924$	6.00000	0.197386
$925$	−8.00000	−0.263038
$926$	34.0000	1.11731
$927$	−1.00000	−0.0328443
$928$	−7.00000	−0.229786
$929$	−51.0000	−1.67326	−0.836628	−	0.547772i	$-0.815476\pi$
$929$	−51.0000	−1.67326	−0.836628	+	0.547772i	$0.815476\pi$
$930$	4.00000	0.131165
$931$	8.00000	0.262189
$932$	6.00000	0.196537
$933$	−35.0000	−1.14585
$934$	14.0000	0.458094
$935$	0	0
$936$	1.00000	0.0326860
$937$	−58.0000	−1.89478	−0.947389	−	0.320085i	$-0.896288\pi$
$937$	−58.0000	−1.89478	−0.947389	+	0.320085i	$0.896288\pi$
$938$	3.00000	0.0979535
$939$	−17.0000	−0.554774
$940$	13.0000	0.424013
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	13.0000	0.423563
$943$	0	0
$944$	−5.00000	−0.162736
$945$	3.00000	0.0975900
$946$	−4.00000	−0.130051
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	−4.00000	−0.129914
$949$	−3.00000	−0.0973841
$950$	16.0000	0.519109
$951$	18.0000	0.583690
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	−2.00000	−0.0647524
$955$	22.0000	0.711903
$956$	−6.00000	−0.194054
$957$	14.0000	0.452556
$958$	15.0000	0.484628
$959$	39.0000	1.25938
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	2.00000	0.0644826
$963$	18.0000	0.580042
$964$	2.00000	0.0644157
$965$	17.0000	0.547249
$966$	0	0
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	−14.0000	−0.449513
$971$	−51.0000	−1.63667	−0.818334	−	0.574743i	$-0.805102\pi$
$971$	−51.0000	−1.63667	−0.818334	+	0.574743i	$0.805102\pi$
$972$	1.00000	0.0320750
$973$	−12.0000	−0.384702
$974$	−12.0000	−0.384505
$975$	4.00000	0.128103
$976$	4.00000	0.128037
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	2.00000	0.0639529
$979$	−16.0000	−0.511362
$980$	2.00000	0.0638877
$981$	2.00000	0.0638551
$982$	−12.0000	−0.382935
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	6.00000	0.191273
$985$	−1.00000	−0.0318626
$986$	0	0
$987$	39.0000	1.24138
$988$	−4.00000	−0.127257
$989$	0	0
$990$	−2.00000	−0.0635642
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	4.00000	0.127000
$993$	17.0000	0.539479
$994$	0	0
$995$	9.00000	0.285319
$996$	−1.00000	−0.0316862
$997$	−23.0000	−0.728417	−0.364209	−	0.931317i	$-0.618661\pi$
$997$	−23.0000	−0.728417	−0.364209	+	0.931317i	$0.618661\pi$
$998$	16.0000	0.506471
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.c.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more