LMFDB - Embedded newform 4002.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$1$
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	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -2.00000 q^{13} -3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} +1.00000 q^{18} -5.00000 q^{19} -1.00000 q^{20} -3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -3.00000 q^{28} +1.00000 q^{29} -1.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} +5.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} -5.00000 q^{38} -2.00000 q^{39} -1.00000 q^{40} -9.00000 q^{41} -3.00000 q^{42} +1.00000 q^{43} -1.00000 q^{45} +1.00000 q^{46} +3.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -4.00000 q^{50} +5.00000 q^{51} -2.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} -3.00000 q^{56} -5.00000 q^{57} +1.00000 q^{58} -11.0000 q^{59} -1.00000 q^{60} -6.00000 q^{61} -6.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} +5.00000 q^{68} +1.00000 q^{69} +3.00000 q^{70} -6.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} -7.00000 q^{74} -4.00000 q^{75} -5.00000 q^{76} -2.00000 q^{78} +4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} +16.0000 q^{83} -3.00000 q^{84} -5.00000 q^{85} +1.00000 q^{86} +1.00000 q^{87} -6.00000 q^{89} -1.00000 q^{90} +6.00000 q^{91} +1.00000 q^{92} -6.00000 q^{93} +3.00000 q^{94} +5.00000 q^{95} +1.00000 q^{96} +2.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	1.00000	0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−3.00000	−0.801784
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	1.00000	0.235702
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	−1.00000	−0.223607
$21$	−3.00000	−0.654654
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	−3.00000	−0.566947
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−1.00000	−0.182574
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.00000	0.857493
$35$	3.00000	0.507093
$36$	1.00000	0.166667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−5.00000	−0.811107
$39$	−2.00000	−0.320256
$40$	−1.00000	−0.158114
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	−3.00000	−0.462910
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	1.00000	0.147442
$47$	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
$49$	2.00000	0.285714
$50$	−4.00000	−0.565685
$51$	5.00000	0.700140
$52$	−2.00000	−0.277350
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−3.00000	−0.400892
$57$	−5.00000	−0.662266
$58$	1.00000	0.131306
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	−1.00000	−0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−6.00000	−0.762001
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	5.00000	0.606339
$69$	1.00000	0.120386
$70$	3.00000	0.358569
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	1.00000	0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−7.00000	−0.813733
$75$	−4.00000	−0.461880
$76$	−5.00000	−0.573539
$77$	0	0
$78$	−2.00000	−0.226455
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−9.00000	−0.993884
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	−3.00000	−0.327327
$85$	−5.00000	−0.542326
$86$	1.00000	0.107833
$87$	1.00000	0.107211
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−1.00000	−0.105409
$91$	6.00000	0.628971
$92$	1.00000	0.104257
$93$	−6.00000	−0.622171
$94$	3.00000	0.309426
$95$	5.00000	0.512989
$96$	1.00000	0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	2.00000	0.202031
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	5.00000	0.495074
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	−2.00000	−0.196116
$105$	3.00000	0.292770
$106$	10.0000	0.971286
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	1.00000	0.0962250
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	−3.00000	−0.283473
$113$	−5.00000	−0.470360	−0.235180	−	0.971952i	$-0.575568\pi$
$113$	−5.00000	−0.470360	−0.235180	+	0.971952i	$0.575568\pi$
$114$	−5.00000	−0.468293
$115$	−1.00000	−0.0932505
$116$	1.00000	0.0928477
$117$	−2.00000	−0.184900
$118$	−11.0000	−1.01263
$119$	−15.0000	−1.37505
$120$	−1.00000	−0.0912871
$121$	−11.0000	−1.00000
$122$	−6.00000	−0.543214
$123$	−9.00000	−0.811503
$124$	−6.00000	−0.538816
$125$	9.00000	0.804984
$126$	−3.00000	−0.267261
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	1.00000	0.0883883
$129$	1.00000	0.0880451
$130$	2.00000	0.175412
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	15.0000	1.30066
$134$	−4.00000	−0.345547
$135$	−1.00000	−0.0860663
$136$	5.00000	0.428746
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	1.00000	0.0851257
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	3.00000	0.253546
$141$	3.00000	0.252646
$142$	−6.00000	−0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	−1.00000	−0.0830455
$146$	−6.00000	−0.496564
$147$	2.00000	0.164957
$148$	−7.00000	−0.575396
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	−4.00000	−0.326599
$151$	15.0000	1.22068	0.610341	−	0.792139i	$-0.291032\pi$
$151$	15.0000	1.22068	0.610341	+	0.792139i	$0.291032\pi$
$152$	−5.00000	−0.405554
$153$	5.00000	0.404226
$154$	0	0
$155$	6.00000	0.481932
$156$	−2.00000	−0.160128
$157$	21.0000	1.67598	0.837991	−	0.545684i	$-0.183730\pi$
$157$	21.0000	1.67598	0.837991	+	0.545684i	$0.183730\pi$
$158$	4.00000	0.318223
$159$	10.0000	0.793052
$160$	−1.00000	−0.0790569
$161$	−3.00000	−0.236433
$162$	1.00000	0.0785674
$163$	−21.0000	−1.64485	−0.822423	−	0.568876i	$-0.807379\pi$
$163$	−21.0000	−1.64485	−0.822423	+	0.568876i	$0.807379\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	16.0000	1.24184
$167$	22.0000	1.70241	0.851206	−	0.524832i	$-0.175872\pi$
$167$	22.0000	1.70241	0.851206	+	0.524832i	$0.175872\pi$
$168$	−3.00000	−0.231455
$169$	−9.00000	−0.692308
$170$	−5.00000	−0.383482
$171$	−5.00000	−0.382360
$172$	1.00000	0.0762493
$173$	15.0000	1.14043	0.570214	−	0.821496i	$-0.306860\pi$
$173$	15.0000	1.14043	0.570214	+	0.821496i	$0.306860\pi$
$174$	1.00000	0.0758098
$175$	12.0000	0.907115
$176$	0	0
$177$	−11.0000	−0.826811
$178$	−6.00000	−0.449719
$179$	−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
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	6.00000	0.444750
$183$	−6.00000	−0.443533
$184$	1.00000	0.0737210
$185$	7.00000	0.514650
$186$	−6.00000	−0.439941
$187$	0	0
$188$	3.00000	0.218797
$189$	−3.00000	−0.218218
$190$	5.00000	0.362738
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	1.00000	0.0721688
$193$	−20.0000	−1.43963	−0.719816	−	0.694165i	$-0.755774\pi$
$193$	−20.0000	−1.43963	−0.719816	+	0.694165i	$0.755774\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	2.00000	0.142857
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	0	0
$199$	−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
$201$	−4.00000	−0.282138
$202$	−10.0000	−0.703598
$203$	−3.00000	−0.210559
$204$	5.00000	0.350070
$205$	9.00000	0.628587
$206$	11.0000	0.766406
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	0	0
$210$	3.00000	0.207020
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	10.0000	0.686803
$213$	−6.00000	−0.411113
$214$	−9.00000	−0.615227
$215$	−1.00000	−0.0681994
$216$	1.00000	0.0680414
$217$	18.0000	1.22192
$218$	−4.00000	−0.270914
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−10.0000	−0.672673
$222$	−7.00000	−0.469809
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−3.00000	−0.200446
$225$	−4.00000	−0.266667
$226$	−5.00000	−0.332595
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	−5.00000	−0.331133
$229$	27.0000	1.78421	0.892105	−	0.451828i	$-0.149228\pi$
$229$	27.0000	1.78421	0.892105	+	0.451828i	$0.149228\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	1.00000	0.0656532
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	−2.00000	−0.130744
$235$	−3.00000	−0.195698
$236$	−11.0000	−0.716039
$237$	4.00000	0.259828
$238$	−15.0000	−0.972306
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	−1.00000	−0.0645497
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	−11.0000	−0.707107
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	−2.00000	−0.127775
$246$	−9.00000	−0.573819
$247$	10.0000	0.636285
$248$	−6.00000	−0.381000
$249$	16.0000	1.01396
$250$	9.00000	0.569210
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	−3.00000	−0.188982
$253$	0	0
$254$	−2.00000	−0.125491
$255$	−5.00000	−0.313112
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	1.00000	0.0622573
$259$	21.0000	1.30488
$260$	2.00000	0.124035
$261$	1.00000	0.0618984
$262$	−14.0000	−0.864923
$263$	1.00000	0.0616626	0.0308313	−	0.999525i	$-0.490185\pi$
$263$	1.00000	0.0616626	0.0308313	+	0.999525i	$0.490185\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	15.0000	0.919709
$267$	−6.00000	−0.367194
$268$	−4.00000	−0.244339
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	−1.00000	−0.0608581
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	5.00000	0.303170
$273$	6.00000	0.363137
$274$	−18.0000	−1.08742
$275$	0	0
$276$	1.00000	0.0601929
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	14.0000	0.839664
$279$	−6.00000	−0.359211
$280$	3.00000	0.179284
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	3.00000	0.178647
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−6.00000	−0.356034
$285$	5.00000	0.296174
$286$	0	0
$287$	27.0000	1.59376
$288$	1.00000	0.0589256
$289$	8.00000	0.470588
$290$	−1.00000	−0.0587220
$291$	0	0
$292$	−6.00000	−0.351123
$293$	−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
$295$	11.0000	0.640445
$296$	−7.00000	−0.406867
$297$	0	0
$298$	−15.0000	−0.868927
$299$	−2.00000	−0.115663
$300$	−4.00000	−0.230940
$301$	−3.00000	−0.172917
$302$	15.0000	0.863153
$303$	−10.0000	−0.574485
$304$	−5.00000	−0.286770
$305$	6.00000	0.343559
$306$	5.00000	0.285831
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	11.0000	0.625768
$310$	6.00000	0.340777
$311$	19.0000	1.07739	0.538696	−	0.842500i	$-0.318917\pi$
$311$	19.0000	1.07739	0.538696	+	0.842500i	$0.318917\pi$
$312$	−2.00000	−0.113228
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	21.0000	1.18510
$315$	3.00000	0.169031
$316$	4.00000	0.225018
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	10.0000	0.560772
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−9.00000	−0.502331
$322$	−3.00000	−0.167183
$323$	−25.0000	−1.39104
$324$	1.00000	0.0555556
$325$	8.00000	0.443760
$326$	−21.0000	−1.16308
$327$	−4.00000	−0.221201
$328$	−9.00000	−0.496942
$329$	−9.00000	−0.496186
$330$	0	0
$331$	−5.00000	−0.274825	−0.137412	−	0.990514i	$-0.543879\pi$
$331$	−5.00000	−0.274825	−0.137412	+	0.990514i	$0.543879\pi$
$332$	16.0000	0.878114
$333$	−7.00000	−0.383598
$334$	22.0000	1.20379
$335$	4.00000	0.218543
$336$	−3.00000	−0.163663
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−9.00000	−0.489535
$339$	−5.00000	−0.271563
$340$	−5.00000	−0.271163
$341$	0	0
$342$	−5.00000	−0.270369
$343$	15.0000	0.809924
$344$	1.00000	0.0539164
$345$	−1.00000	−0.0538382
$346$	15.0000	0.806405
$347$	23.0000	1.23470	0.617352	−	0.786687i	$-0.288205\pi$
$347$	23.0000	1.23470	0.617352	+	0.786687i	$0.288205\pi$
$348$	1.00000	0.0536056
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	12.0000	0.641427
$351$	−2.00000	−0.106752
$352$	0	0
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	−11.0000	−0.584643
$355$	6.00000	0.318447
$356$	−6.00000	−0.317999
$357$	−15.0000	−0.793884
$358$	−4.00000	−0.211407
$359$	−19.0000	−1.00278	−0.501391	−	0.865221i	$-0.667178\pi$
$359$	−19.0000	−1.00278	−0.501391	+	0.865221i	$0.667178\pi$
$360$	−1.00000	−0.0527046
$361$	6.00000	0.315789
$362$	2.00000	0.105118
$363$	−11.0000	−0.577350
$364$	6.00000	0.314485
$365$	6.00000	0.314054
$366$	−6.00000	−0.313625
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	1.00000	0.0521286
$369$	−9.00000	−0.468521
$370$	7.00000	0.363913
$371$	−30.0000	−1.55752
$372$	−6.00000	−0.311086
$373$	36.0000	1.86401	0.932005	−	0.362446i	$-0.118058\pi$
$373$	36.0000	1.86401	0.932005	+	0.362446i	$0.118058\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	3.00000	0.154713
$377$	−2.00000	−0.103005
$378$	−3.00000	−0.154303
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	5.00000	0.256495
$381$	−2.00000	−0.102463
$382$	15.0000	0.767467
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−20.0000	−1.01797
$387$	1.00000	0.0508329
$388$	0	0
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	2.00000	0.101274
$391$	5.00000	0.252861
$392$	2.00000	0.101015
$393$	−14.0000	−0.706207
$394$	−15.0000	−0.755689
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	−20.0000	−1.00251
$399$	15.0000	0.750939
$400$	−4.00000	−0.200000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	−4.00000	−0.199502
$403$	12.0000	0.597763
$404$	−10.0000	−0.497519
$405$	−1.00000	−0.0496904
$406$	−3.00000	−0.148888
$407$	0	0
$408$	5.00000	0.247537
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	9.00000	0.444478
$411$	−18.0000	−0.887875
$412$	11.0000	0.541931
$413$	33.0000	1.62382
$414$	1.00000	0.0491473
$415$	−16.0000	−0.785409
$416$	−2.00000	−0.0980581
$417$	14.0000	0.685583
$418$	0	0
$419$	33.0000	1.61216	0.806078	−	0.591810i	$-0.201586\pi$
$419$	33.0000	1.61216	0.806078	+	0.591810i	$0.201586\pi$
$420$	3.00000	0.146385
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−15.0000	−0.730189
$423$	3.00000	0.145865
$424$	10.0000	0.485643
$425$	−20.0000	−0.970143
$426$	−6.00000	−0.290701
$427$	18.0000	0.871081
$428$	−9.00000	−0.435031
$429$	0	0
$430$	−1.00000	−0.0482243
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	1.00000	0.0481125
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	18.0000	0.864028
$435$	−1.00000	−0.0479463
$436$	−4.00000	−0.191565
$437$	−5.00000	−0.239182
$438$	−6.00000	−0.286691
$439$	−3.00000	−0.143182	−0.0715911	−	0.997434i	$-0.522808\pi$
$439$	−3.00000	−0.143182	−0.0715911	+	0.997434i	$0.522808\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−10.0000	−0.475651
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−7.00000	−0.332205
$445$	6.00000	0.284427
$446$	8.00000	0.378811
$447$	−15.0000	−0.709476
$448$	−3.00000	−0.141737
$449$	31.0000	1.46298	0.731490	−	0.681852i	$-0.238825\pi$
$449$	31.0000	1.46298	0.731490	+	0.681852i	$0.238825\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	−5.00000	−0.235180
$453$	15.0000	0.704761
$454$	15.0000	0.703985
$455$	−6.00000	−0.281284
$456$	−5.00000	−0.234146
$457$	37.0000	1.73079	0.865393	−	0.501093i	$-0.167069\pi$
$457$	37.0000	1.73079	0.865393	+	0.501093i	$0.167069\pi$
$458$	27.0000	1.26163
$459$	5.00000	0.233380
$460$	−1.00000	−0.0466252
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	1.00000	0.0464238
$465$	6.00000	0.278243
$466$	−4.00000	−0.185296
$467$	34.0000	1.57333	0.786666	−	0.617379i	$-0.211805\pi$
$467$	34.0000	1.57333	0.786666	+	0.617379i	$0.211805\pi$
$468$	−2.00000	−0.0924500
$469$	12.0000	0.554109
$470$	−3.00000	−0.138380
$471$	21.0000	0.967629
$472$	−11.0000	−0.506316
$473$	0	0
$474$	4.00000	0.183726
$475$	20.0000	0.917663
$476$	−15.0000	−0.687524
$477$	10.0000	0.457869
$478$	6.00000	0.274434
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	−1.00000	−0.0456435
$481$	14.0000	0.638345
$482$	19.0000	0.865426
$483$	−3.00000	−0.136505
$484$	−11.0000	−0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	−15.0000	−0.679715	−0.339857	−	0.940477i	$-0.610379\pi$
$487$	−15.0000	−0.679715	−0.339857	+	0.940477i	$0.610379\pi$
$488$	−6.00000	−0.271607
$489$	−21.0000	−0.949653
$490$	−2.00000	−0.0903508
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	−9.00000	−0.405751
$493$	5.00000	0.225189
$494$	10.0000	0.449921
$495$	0	0
$496$	−6.00000	−0.269408
$497$	18.0000	0.807410
$498$	16.0000	0.716977
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	9.00000	0.402492
$501$	22.0000	0.982888
$502$	4.00000	0.178529
$503$	−41.0000	−1.82810	−0.914050	−	0.405602i	$-0.867062\pi$
$503$	−41.0000	−1.82810	−0.914050	+	0.405602i	$0.867062\pi$
$504$	−3.00000	−0.133631
$505$	10.0000	0.444994
$506$	0	0
$507$	−9.00000	−0.399704
$508$	−2.00000	−0.0887357
$509$	39.0000	1.72864	0.864322	−	0.502938i	$-0.167748\pi$
$509$	39.0000	1.72864	0.864322	+	0.502938i	$0.167748\pi$
$510$	−5.00000	−0.221404
$511$	18.0000	0.796273
$512$	1.00000	0.0441942
$513$	−5.00000	−0.220755
$514$	−6.00000	−0.264649
$515$	−11.0000	−0.484718
$516$	1.00000	0.0440225
$517$	0	0
$518$	21.0000	0.922687
$519$	15.0000	0.658427
$520$	2.00000	0.0877058
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	1.00000	0.0437688
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	−14.0000	−0.611593
$525$	12.0000	0.523723
$526$	1.00000	0.0436021
$527$	−30.0000	−1.30682
$528$	0	0
$529$	1.00000	0.0434783
$530$	−10.0000	−0.434372
$531$	−11.0000	−0.477359
$532$	15.0000	0.650332
$533$	18.0000	0.779667
$534$	−6.00000	−0.259645
$535$	9.00000	0.389104
$536$	−4.00000	−0.172774
$537$	−4.00000	−0.172613
$538$	0	0
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	35.0000	1.50477	0.752384	−	0.658725i	$-0.228904\pi$
$541$	35.0000	1.50477	0.752384	+	0.658725i	$0.228904\pi$
$542$	20.0000	0.859074
$543$	2.00000	0.0858282
$544$	5.00000	0.214373
$545$	4.00000	0.171341
$546$	6.00000	0.256776
$547$	−38.0000	−1.62476	−0.812381	−	0.583127i	$-0.801829\pi$
$547$	−38.0000	−1.62476	−0.812381	+	0.583127i	$0.801829\pi$
$548$	−18.0000	−0.768922
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−5.00000	−0.213007
$552$	1.00000	0.0425628
$553$	−12.0000	−0.510292
$554$	8.00000	0.339887
$555$	7.00000	0.297133
$556$	14.0000	0.593732
$557$	−41.0000	−1.73723	−0.868613	−	0.495491i	$-0.834988\pi$
$557$	−41.0000	−1.73723	−0.868613	+	0.495491i	$0.834988\pi$
$558$	−6.00000	−0.254000
$559$	−2.00000	−0.0845910
$560$	3.00000	0.126773
$561$	0	0
$562$	−6.00000	−0.253095
$563$	−42.0000	−1.77009	−0.885044	−	0.465506i	$-0.845872\pi$
$563$	−42.0000	−1.77009	−0.885044	+	0.465506i	$0.845872\pi$
$564$	3.00000	0.126323
$565$	5.00000	0.210352
$566$	16.0000	0.672530
$567$	−3.00000	−0.125988
$568$	−6.00000	−0.251754
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	5.00000	0.209427
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	15.0000	0.626634
$574$	27.0000	1.12696
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	8.00000	0.332756
$579$	−20.0000	−0.831172
$580$	−1.00000	−0.0415227
$581$	−48.0000	−1.99138
$582$	0	0
$583$	0	0
$584$	−6.00000	−0.248282
$585$	2.00000	0.0826898
$586$	−30.0000	−1.23929
$587$	−13.0000	−0.536567	−0.268284	−	0.963340i	$-0.586456\pi$
$587$	−13.0000	−0.536567	−0.268284	+	0.963340i	$0.586456\pi$
$588$	2.00000	0.0824786
$589$	30.0000	1.23613
$590$	11.0000	0.452863
$591$	−15.0000	−0.617018
$592$	−7.00000	−0.287698
$593$	8.00000	0.328521	0.164260	−	0.986417i	$-0.447476\pi$
$593$	8.00000	0.328521	0.164260	+	0.986417i	$0.447476\pi$
$594$	0	0
$595$	15.0000	0.614940
$596$	−15.0000	−0.614424
$597$	−20.0000	−0.818546
$598$	−2.00000	−0.0817861
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	−4.00000	−0.163299
$601$	12.0000	0.489490	0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000	0.489490	0.244745	+	0.969587i	$0.421296\pi$
$602$	−3.00000	−0.122271
$603$	−4.00000	−0.162893
$604$	15.0000	0.610341
$605$	11.0000	0.447214
$606$	−10.0000	−0.406222
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	−5.00000	−0.202777
$609$	−3.00000	−0.121566
$610$	6.00000	0.242933
$611$	−6.00000	−0.242734
$612$	5.00000	0.202113
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	−12.0000	−0.484281
$615$	9.00000	0.362915
$616$	0	0
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	11.0000	0.442485
$619$	−39.0000	−1.56754	−0.783771	−	0.621050i	$-0.786706\pi$
$619$	−39.0000	−1.56754	−0.783771	+	0.621050i	$0.786706\pi$
$620$	6.00000	0.240966
$621$	1.00000	0.0401286
$622$	19.0000	0.761831
$623$	18.0000	0.721155
$624$	−2.00000	−0.0800641
$625$	11.0000	0.440000
$626$	1.00000	0.0399680
$627$	0	0
$628$	21.0000	0.837991
$629$	−35.0000	−1.39554
$630$	3.00000	0.119523
$631$	19.0000	0.756378	0.378189	−	0.925728i	$-0.376547\pi$
$631$	19.0000	0.756378	0.378189	+	0.925728i	$0.376547\pi$
$632$	4.00000	0.159111
$633$	−15.0000	−0.596196
$634$	2.00000	0.0794301
$635$	2.00000	0.0793676
$636$	10.0000	0.396526
$637$	−4.00000	−0.158486
$638$	0	0
$639$	−6.00000	−0.237356
$640$	−1.00000	−0.0395285
$641$	−21.0000	−0.829450	−0.414725	−	0.909947i	$-0.636122\pi$
$641$	−21.0000	−0.829450	−0.414725	+	0.909947i	$0.636122\pi$
$642$	−9.00000	−0.355202
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	−3.00000	−0.118217
$645$	−1.00000	−0.0393750
$646$	−25.0000	−0.983612
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	8.00000	0.313786
$651$	18.0000	0.705476
$652$	−21.0000	−0.822423
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−4.00000	−0.156412
$655$	14.0000	0.547025
$656$	−9.00000	−0.351391
$657$	−6.00000	−0.234082
$658$	−9.00000	−0.350857
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	−5.00000	−0.194331
$663$	−10.0000	−0.388368
$664$	16.0000	0.620920
$665$	−15.0000	−0.581675
$666$	−7.00000	−0.271244
$667$	1.00000	0.0387202
$668$	22.0000	0.851206
$669$	8.00000	0.309298
$670$	4.00000	0.154533
$671$	0	0
$672$	−3.00000	−0.115728
$673$	−21.0000	−0.809491	−0.404745	−	0.914429i	$-0.632640\pi$
$673$	−21.0000	−0.809491	−0.404745	+	0.914429i	$0.632640\pi$
$674$	22.0000	0.847408
$675$	−4.00000	−0.153960
$676$	−9.00000	−0.346154
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	−5.00000	−0.192024
$679$	0	0
$680$	−5.00000	−0.191741
$681$	15.0000	0.574801
$682$	0	0
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	−5.00000	−0.191180
$685$	18.0000	0.687745
$686$	15.0000	0.572703
$687$	27.0000	1.03011
$688$	1.00000	0.0381246
$689$	−20.0000	−0.761939
$690$	−1.00000	−0.0380693
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	15.0000	0.570214
$693$	0	0
$694$	23.0000	0.873068
$695$	−14.0000	−0.531050
$696$	1.00000	0.0379049
$697$	−45.0000	−1.70450
$698$	−2.00000	−0.0757011
$699$	−4.00000	−0.151294
$700$	12.0000	0.453557
$701$	7.00000	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$701$	7.00000	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$702$	−2.00000	−0.0754851
$703$	35.0000	1.32005
$704$	0	0
$705$	−3.00000	−0.112987
$706$	34.0000	1.27961
$707$	30.0000	1.12827
$708$	−11.0000	−0.413405
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	6.00000	0.225176
$711$	4.00000	0.150012
$712$	−6.00000	−0.224860
$713$	−6.00000	−0.224702
$714$	−15.0000	−0.561361
$715$	0	0
$716$	−4.00000	−0.149487
$717$	6.00000	0.224074
$718$	−19.0000	−0.709074
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	−1.00000	−0.0372678
$721$	−33.0000	−1.22898
$722$	6.00000	0.223297
$723$	19.0000	0.706618
$724$	2.00000	0.0743294
$725$	−4.00000	−0.148556
$726$	−11.0000	−0.408248
$727$	38.0000	1.40934	0.704671	−	0.709534i	$-0.251095\pi$
$727$	38.0000	1.40934	0.704671	+	0.709534i	$0.251095\pi$
$728$	6.00000	0.222375
$729$	1.00000	0.0370370
$730$	6.00000	0.222070
$731$	5.00000	0.184932
$732$	−6.00000	−0.221766
$733$	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
$736$	1.00000	0.0368605
$737$	0	0
$738$	−9.00000	−0.331295
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	7.00000	0.257325
$741$	10.0000	0.367359
$742$	−30.0000	−1.10133
$743$	39.0000	1.43077	0.715386	−	0.698730i	$-0.246251\pi$
$743$	39.0000	1.43077	0.715386	+	0.698730i	$0.246251\pi$
$744$	−6.00000	−0.219971
$745$	15.0000	0.549557
$746$	36.0000	1.31805
$747$	16.0000	0.585409
$748$	0	0
$749$	27.0000	0.986559
$750$	9.00000	0.328634
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	3.00000	0.109399
$753$	4.00000	0.145768
$754$	−2.00000	−0.0728357
$755$	−15.0000	−0.545906
$756$	−3.00000	−0.109109
$757$	−29.0000	−1.05402	−0.527011	−	0.849858i	$-0.676688\pi$
$757$	−29.0000	−1.05402	−0.527011	+	0.849858i	$0.676688\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	5.00000	0.181369
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	−2.00000	−0.0724524
$763$	12.0000	0.434429
$764$	15.0000	0.542681
$765$	−5.00000	−0.180775
$766$	−8.00000	−0.289052
$767$	22.0000	0.794374
$768$	1.00000	0.0360844
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	−20.0000	−0.719816
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	1.00000	0.0359443
$775$	24.0000	0.862105
$776$	0	0
$777$	21.0000	0.753371
$778$	−20.0000	−0.717035
$779$	45.0000	1.61229
$780$	2.00000	0.0716115
$781$	0	0
$782$	5.00000	0.178800
$783$	1.00000	0.0357371
$784$	2.00000	0.0714286
$785$	−21.0000	−0.749522
$786$	−14.0000	−0.499363
$787$	−2.00000	−0.0712923	−0.0356462	−	0.999364i	$-0.511349\pi$
$787$	−2.00000	−0.0712923	−0.0356462	+	0.999364i	$0.511349\pi$
$788$	−15.0000	−0.534353
$789$	1.00000	0.0356009
$790$	−4.00000	−0.142314
$791$	15.0000	0.533339
$792$	0	0
$793$	12.0000	0.426132
$794$	−20.0000	−0.709773
$795$	−10.0000	−0.354663
$796$	−20.0000	−0.708881
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	15.0000	0.530994
$799$	15.0000	0.530662
$800$	−4.00000	−0.141421
$801$	−6.00000	−0.212000
$802$	−2.00000	−0.0706225
$803$	0	0
$804$	−4.00000	−0.141069
$805$	3.00000	0.105736
$806$	12.0000	0.422682
$807$	0	0
$808$	−10.0000	−0.351799
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	−1.00000	−0.0351364
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−3.00000	−0.105279
$813$	20.0000	0.701431
$814$	0	0
$815$	21.0000	0.735598
$816$	5.00000	0.175035
$817$	−5.00000	−0.174928
$818$	−38.0000	−1.32864
$819$	6.00000	0.209657
$820$	9.00000	0.314294
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	−18.0000	−0.627822
$823$	−52.0000	−1.81261	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−52.0000	−1.81261	−0.906303	+	0.422628i	$0.861108\pi$
$824$	11.0000	0.383203
$825$	0	0
$826$	33.0000	1.14822
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	1.00000	0.0347524
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	−16.0000	−0.555368
$831$	8.00000	0.277517
$832$	−2.00000	−0.0693375
$833$	10.0000	0.346479
$834$	14.0000	0.484780
$835$	−22.0000	−0.761341
$836$	0	0
$837$	−6.00000	−0.207390
$838$	33.0000	1.13997
$839$	1.00000	0.0345238	0.0172619	−	0.999851i	$-0.494505\pi$
$839$	1.00000	0.0345238	0.0172619	+	0.999851i	$0.494505\pi$
$840$	3.00000	0.103510
$841$	1.00000	0.0344828
$842$	10.0000	0.344623
$843$	−6.00000	−0.206651
$844$	−15.0000	−0.516321
$845$	9.00000	0.309609
$846$	3.00000	0.103142
$847$	33.0000	1.13389
$848$	10.0000	0.343401
$849$	16.0000	0.549119
$850$	−20.0000	−0.685994
$851$	−7.00000	−0.239957
$852$	−6.00000	−0.205557
$853$	13.0000	0.445112	0.222556	−	0.974920i	$-0.428560\pi$
$853$	13.0000	0.445112	0.222556	+	0.974920i	$0.428560\pi$
$854$	18.0000	0.615947
$855$	5.00000	0.170996
$856$	−9.00000	−0.307614
$857$	−52.0000	−1.77629	−0.888143	−	0.459567i	$-0.848005\pi$
$857$	−52.0000	−1.77629	−0.888143	+	0.459567i	$0.848005\pi$
$858$	0	0
$859$	35.0000	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$859$	35.0000	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$860$	−1.00000	−0.0340997
$861$	27.0000	0.920158
$862$	6.00000	0.204361
$863$	−22.0000	−0.748889	−0.374444	−	0.927249i	$-0.622167\pi$
$863$	−22.0000	−0.748889	−0.374444	+	0.927249i	$0.622167\pi$
$864$	1.00000	0.0340207
$865$	−15.0000	−0.510015
$866$	−4.00000	−0.135926
$867$	8.00000	0.271694
$868$	18.0000	0.610960
$869$	0	0
$870$	−1.00000	−0.0339032
$871$	8.00000	0.271070
$872$	−4.00000	−0.135457
$873$	0	0
$874$	−5.00000	−0.169128
$875$	−27.0000	−0.912767
$876$	−6.00000	−0.202721
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−3.00000	−0.101245
$879$	−30.0000	−1.01187
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	2.00000	0.0673435
$883$	54.0000	1.81724	0.908622	−	0.417619i	$-0.137135\pi$
$883$	54.0000	1.81724	0.908622	+	0.417619i	$0.137135\pi$
$884$	−10.0000	−0.336336
$885$	11.0000	0.369761
$886$	−24.0000	−0.806296
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	−7.00000	−0.234905
$889$	6.00000	0.201234
$890$	6.00000	0.201120
$891$	0	0
$892$	8.00000	0.267860
$893$	−15.0000	−0.501956
$894$	−15.0000	−0.501675
$895$	4.00000	0.133705
$896$	−3.00000	−0.100223
$897$	−2.00000	−0.0667781
$898$	31.0000	1.03448
$899$	−6.00000	−0.200111
$900$	−4.00000	−0.133333
$901$	50.0000	1.66574
$902$	0	0
$903$	−3.00000	−0.0998337
$904$	−5.00000	−0.166298
$905$	−2.00000	−0.0664822
$906$	15.0000	0.498342
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	15.0000	0.497792
$909$	−10.0000	−0.331679
$910$	−6.00000	−0.198898
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	−5.00000	−0.165567
$913$	0	0
$914$	37.0000	1.22385
$915$	6.00000	0.198354
$916$	27.0000	0.892105
$917$	42.0000	1.38696
$918$	5.00000	0.165025
$919$	25.0000	0.824674	0.412337	−	0.911031i	$-0.364713\pi$
$919$	25.0000	0.824674	0.412337	+	0.911031i	$0.364713\pi$
$920$	−1.00000	−0.0329690
$921$	−12.0000	−0.395413
$922$	−26.0000	−0.856264
$923$	12.0000	0.394985
$924$	0	0
$925$	28.0000	0.920634
$926$	24.0000	0.788689
$927$	11.0000	0.361287
$928$	1.00000	0.0328266
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	6.00000	0.196748
$931$	−10.0000	−0.327737
$932$	−4.00000	−0.131024
$933$	19.0000	0.622032
$934$	34.0000	1.11251
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−3.00000	−0.0980057	−0.0490029	−	0.998799i	$-0.515604\pi$
$937$	−3.00000	−0.0980057	−0.0490029	+	0.998799i	$0.515604\pi$
$938$	12.0000	0.391814
$939$	1.00000	0.0326338
$940$	−3.00000	−0.0978492
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	21.0000	0.684217
$943$	−9.00000	−0.293080
$944$	−11.0000	−0.358020
$945$	3.00000	0.0975900
$946$	0	0
$947$	−54.0000	−1.75476	−0.877382	−	0.479792i	$-0.840712\pi$
$947$	−54.0000	−1.75476	−0.877382	+	0.479792i	$0.840712\pi$
$948$	4.00000	0.129914
$949$	12.0000	0.389536
$950$	20.0000	0.648886
$951$	2.00000	0.0648544
$952$	−15.0000	−0.486153
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	10.0000	0.323762
$955$	−15.0000	−0.485389
$956$	6.00000	0.194054
$957$	0	0
$958$	24.0000	0.775405
$959$	54.0000	1.74375
$960$	−1.00000	−0.0322749
$961$	5.00000	0.161290
$962$	14.0000	0.451378
$963$	−9.00000	−0.290021
$964$	19.0000	0.611949
$965$	20.0000	0.643823
$966$	−3.00000	−0.0965234
$967$	10.0000	0.321578	0.160789	−	0.986989i	$-0.448596\pi$
$967$	10.0000	0.321578	0.160789	+	0.986989i	$0.448596\pi$
$968$	−11.0000	−0.353553
$969$	−25.0000	−0.803116
$970$	0	0
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	1.00000	0.0320750
$973$	−42.0000	−1.34646
$974$	−15.0000	−0.480631
$975$	8.00000	0.256205
$976$	−6.00000	−0.192055
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	−21.0000	−0.671506
$979$	0	0
$980$	−2.00000	−0.0638877
$981$	−4.00000	−0.127710
$982$	16.0000	0.510581
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	−9.00000	−0.286910
$985$	15.0000	0.477940
$986$	5.00000	0.159232
$987$	−9.00000	−0.286473
$988$	10.0000	0.318142
$989$	1.00000	0.0317982
$990$	0	0
$991$	−35.0000	−1.11181	−0.555906	−	0.831245i	$-0.687628\pi$
$991$	−35.0000	−1.11181	−0.555906	+	0.831245i	$0.687628\pi$
$992$	−6.00000	−0.190500
$993$	−5.00000	−0.158670
$994$	18.0000	0.570925
$995$	20.0000	0.634043
$996$	16.0000	0.506979
$997$	−3.00000	−0.0950110	−0.0475055	−	0.998871i	$-0.515127\pi$
$997$	−3.00000	−0.0950110	−0.0475055	+	0.998871i	$0.515127\pi$
$998$	−24.0000	−0.759707
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.o.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.o.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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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