LMFDB - Embedded newform 4002.2.a.e.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:	$0$
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	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.00000 q^{11} +1.00000 q^{12} +3.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -5.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -3.00000 q^{26} +1.00000 q^{27} -1.00000 q^{29} +1.00000 q^{30} +5.00000 q^{31} -1.00000 q^{32} +5.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -3.00000 q^{37} -4.00000 q^{38} +3.00000 q^{39} +1.00000 q^{40} +9.00000 q^{41} -6.00000 q^{43} +5.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} +10.0000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +4.00000 q^{50} +4.00000 q^{51} +3.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -5.00000 q^{55} +4.00000 q^{57} +1.00000 q^{58} -7.00000 q^{59} -1.00000 q^{60} +1.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -3.00000 q^{65} -5.00000 q^{66} -13.0000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -5.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} +3.00000 q^{74} -4.00000 q^{75} +4.00000 q^{76} -3.00000 q^{78} -2.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} -6.00000 q^{83} -4.00000 q^{85} +6.00000 q^{86} -1.00000 q^{87} -5.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} -1.00000 q^{92} +5.00000 q^{93} -10.0000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +18.0000 q^{97} +7.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	1.00000	0.288675
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−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
$21$	0	0
$22$	−5.00000	−1.06600
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	−3.00000	−0.588348
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	1.00000	0.182574
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	5.00000	0.870388
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−4.00000	−0.648886
$39$	3.00000	0.480384
$40$	1.00000	0.158114
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	5.00000	0.753778
$45$	−1.00000	−0.149071
$46$	1.00000	0.147442
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	1.00000	0.144338
$49$	−7.00000	−1.00000
$50$	4.00000	0.565685
$51$	4.00000	0.560112
$52$	3.00000	0.416025
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	−5.00000	−0.674200
$56$	0	0
$57$	4.00000	0.529813
$58$	1.00000	0.131306
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	−1.00000	−0.129099
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−5.00000	−0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.00000	−0.372104
$66$	−5.00000	−0.615457
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	4.00000	0.485071
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	−1.00000	−0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	3.00000	0.348743
$75$	−4.00000	−0.461880
$76$	4.00000	0.458831
$77$	0	0
$78$	−3.00000	−0.339683
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−9.00000	−0.993884
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	6.00000	0.646997
$87$	−1.00000	−0.107211
$88$	−5.00000	−0.533002
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−1.00000	−0.104257
$93$	5.00000	0.518476
$94$	−10.0000	−1.03142
$95$	−4.00000	−0.410391
$96$	−1.00000	−0.102062
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	7.00000	0.707107
$99$	5.00000	0.502519
$100$	−4.00000	−0.400000
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	−4.00000	−0.396059
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−6.00000	−0.582772
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	1.00000	0.0962250
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	5.00000	0.476731
$111$	−3.00000	−0.284747
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−4.00000	−0.374634
$115$	1.00000	0.0932505
$116$	−1.00000	−0.0928477
$117$	3.00000	0.277350
$118$	7.00000	0.644402
$119$	0	0
$120$	1.00000	0.0912871
$121$	14.0000	1.27273
$122$	−1.00000	−0.0905357
$123$	9.00000	0.811503
$124$	5.00000	0.449013
$125$	9.00000	0.804984
$126$	0	0
$127$	9.00000	0.798621	0.399310	−	0.916816i	$-0.369250\pi$
$127$	9.00000	0.798621	0.399310	+	0.916816i	$0.369250\pi$
$128$	−1.00000	−0.0883883
$129$	−6.00000	−0.528271
$130$	3.00000	0.263117
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	5.00000	0.435194
$133$	0	0
$134$	13.0000	1.12303
$135$	−1.00000	−0.0860663
$136$	−4.00000	−0.342997
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	1.00000	0.0851257
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	5.00000	0.419591
$143$	15.0000	1.25436
$144$	1.00000	0.0833333
$145$	1.00000	0.0830455
$146$	4.00000	0.331042
$147$	−7.00000	−0.577350
$148$	−3.00000	−0.246598
$149$	−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
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	−4.00000	−0.324443
$153$	4.00000	0.323381
$154$	0	0
$155$	−5.00000	−0.401610
$156$	3.00000	0.240192
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	2.00000	0.159111
$159$	6.00000	0.475831
$160$	1.00000	0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	13.0000	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$163$	13.0000	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$164$	9.00000	0.702782
$165$	−5.00000	−0.389249
$166$	6.00000	0.465690
$167$	13.0000	1.00597	0.502985	−	0.864295i	$-0.332235\pi$
$167$	13.0000	1.00597	0.502985	+	0.864295i	$0.332235\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	4.00000	0.306786
$171$	4.00000	0.305888
$172$	−6.00000	−0.457496
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	5.00000	0.376889
$177$	−7.00000	−0.526152
$178$	10.0000	0.749532
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−1.00000	−0.0745356
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	1.00000	0.0737210
$185$	3.00000	0.220564
$186$	−5.00000	−0.366618
$187$	20.0000	1.46254
$188$	10.0000	0.729325
$189$	0	0
$190$	4.00000	0.290191
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	1.00000	0.0721688
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−18.0000	−1.29232
$195$	−3.00000	−0.214834
$196$	−7.00000	−0.500000
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−5.00000	−0.355335
$199$	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
$201$	−13.0000	−0.916949
$202$	−15.0000	−1.05540
$203$	0	0
$204$	4.00000	0.280056
$205$	−9.00000	−0.628587
$206$	−13.0000	−0.905753
$207$	−1.00000	−0.0695048
$208$	3.00000	0.208013
$209$	20.0000	1.38343
$210$	0	0
$211$	−17.0000	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$211$	−17.0000	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$212$	6.00000	0.412082
$213$	−5.00000	−0.342594
$214$	−4.00000	−0.273434
$215$	6.00000	0.409197
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	14.0000	0.948200
$219$	−4.00000	−0.270295
$220$	−5.00000	−0.337100
$221$	12.0000	0.807207
$222$	3.00000	0.201347
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	−6.00000	−0.399114
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	4.00000	0.264906
$229$	29.0000	1.91637	0.958187	−	0.286143i	$-0.0923732\pi$
$229$	29.0000	1.91637	0.958187	+	0.286143i	$0.0923732\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	1.00000	0.0656532
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	−3.00000	−0.196116
$235$	−10.0000	−0.652328
$236$	−7.00000	−0.455661
$237$	−2.00000	−0.129914
$238$	0	0
$239$	19.0000	1.22901	0.614504	−	0.788914i	$-0.289356\pi$
$239$	19.0000	1.22901	0.614504	+	0.788914i	$0.289356\pi$
$240$	−1.00000	−0.0645497
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	−14.0000	−0.899954
$243$	1.00000	0.0641500
$244$	1.00000	0.0640184
$245$	7.00000	0.447214
$246$	−9.00000	−0.573819
$247$	12.0000	0.763542
$248$	−5.00000	−0.317500
$249$	−6.00000	−0.380235
$250$	−9.00000	−0.569210
$251$	17.0000	1.07303	0.536515	−	0.843891i	$-0.319740\pi$
$251$	17.0000	1.07303	0.536515	+	0.843891i	$0.319740\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	−9.00000	−0.564710
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	6.00000	0.373544
$259$	0	0
$260$	−3.00000	−0.186052
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	−5.00000	−0.307729
$265$	−6.00000	−0.368577
$266$	0	0
$267$	−10.0000	−0.611990
$268$	−13.0000	−0.794101
$269$	1.00000	0.0609711	0.0304855	−	0.999535i	$-0.490295\pi$
$269$	1.00000	0.0609711	0.0304855	+	0.999535i	$0.490295\pi$
$270$	1.00000	0.0608581
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−8.00000	−0.483298
$275$	−20.0000	−1.20605
$276$	−1.00000	−0.0601929
$277$	7.00000	0.420589	0.210295	−	0.977638i	$-0.432558\pi$
$277$	7.00000	0.420589	0.210295	+	0.977638i	$0.432558\pi$
$278$	2.00000	0.119952
$279$	5.00000	0.299342
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−10.0000	−0.595491
$283$	−29.0000	−1.72387	−0.861936	−	0.507018i	$-0.830748\pi$
$283$	−29.0000	−1.72387	−0.861936	+	0.507018i	$0.830748\pi$
$284$	−5.00000	−0.296695
$285$	−4.00000	−0.236940
$286$	−15.0000	−0.886969
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	−1.00000	−0.0587220
$291$	18.0000	1.05518
$292$	−4.00000	−0.234082
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	7.00000	0.408248
$295$	7.00000	0.407556
$296$	3.00000	0.174371
$297$	5.00000	0.290129
$298$	15.0000	0.868927
$299$	−3.00000	−0.173494
$300$	−4.00000	−0.230940
$301$	0	0
$302$	20.0000	1.15087
$303$	15.0000	0.861727
$304$	4.00000	0.229416
$305$	−1.00000	−0.0572598
$306$	−4.00000	−0.228665
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	0	0
$309$	13.0000	0.739544
$310$	5.00000	0.283981
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	−3.00000	−0.169842
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−2.00000	−0.112509
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	−6.00000	−0.336463
$319$	−5.00000	−0.279946
$320$	−1.00000	−0.0559017
$321$	4.00000	0.223258
$322$	0	0
$323$	16.0000	0.890264
$324$	1.00000	0.0555556
$325$	−12.0000	−0.665640
$326$	−13.0000	−0.720003
$327$	−14.0000	−0.774202
$328$	−9.00000	−0.496942
$329$	0	0
$330$	5.00000	0.275241
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−6.00000	−0.329293
$333$	−3.00000	−0.164399
$334$	−13.0000	−0.711328
$335$	13.0000	0.710266
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	4.00000	0.217571
$339$	6.00000	0.325875
$340$	−4.00000	−0.216930
$341$	25.0000	1.35383
$342$	−4.00000	−0.216295
$343$	0	0
$344$	6.00000	0.323498
$345$	1.00000	0.0538382
$346$	6.00000	0.322562
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	−1.00000	−0.0536056
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	3.00000	0.160128
$352$	−5.00000	−0.266501
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	7.00000	0.372046
$355$	5.00000	0.265372
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	14.0000	0.734809
$364$	0	0
$365$	4.00000	0.209370
$366$	−1.00000	−0.0522708
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	−1.00000	−0.0521286
$369$	9.00000	0.468521
$370$	−3.00000	−0.155963
$371$	0	0
$372$	5.00000	0.259238
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	−20.0000	−1.03418
$375$	9.00000	0.464758
$376$	−10.0000	−0.515711
$377$	−3.00000	−0.154508
$378$	0	0
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	−4.00000	−0.205196
$381$	9.00000	0.461084
$382$	3.00000	0.153493
$383$	−10.0000	−0.510976	−0.255488	−	0.966812i	$-0.582236\pi$
$383$	−10.0000	−0.510976	−0.255488	+	0.966812i	$0.582236\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	16.0000	0.814379
$387$	−6.00000	−0.304997
$388$	18.0000	0.913812
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	3.00000	0.151911
$391$	−4.00000	−0.202289
$392$	7.00000	0.353553
$393$	0	0
$394$	−18.0000	−0.906827
$395$	2.00000	0.100631
$396$	5.00000	0.251259
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	−9.00000	−0.451129
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−23.0000	−1.14857	−0.574283	−	0.818657i	$-0.694719\pi$
$401$	−23.0000	−1.14857	−0.574283	+	0.818657i	$0.694719\pi$
$402$	13.0000	0.648381
$403$	15.0000	0.747203
$404$	15.0000	0.746278
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−15.0000	−0.743522
$408$	−4.00000	−0.198030
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	9.00000	0.444478
$411$	8.00000	0.394611
$412$	13.0000	0.640464
$413$	0	0
$414$	1.00000	0.0491473
$415$	6.00000	0.294528
$416$	−3.00000	−0.147087
$417$	−2.00000	−0.0979404
$418$	−20.0000	−0.978232
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	15.0000	0.731055	0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000	0.731055	0.365528	+	0.930800i	$0.380889\pi$
$422$	17.0000	0.827547
$423$	10.0000	0.486217
$424$	−6.00000	−0.291386
$425$	−16.0000	−0.776114
$426$	5.00000	0.242251
$427$	0	0
$428$	4.00000	0.193347
$429$	15.0000	0.724207
$430$	−6.00000	−0.289346
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	1.00000	0.0481125
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	1.00000	0.0479463
$436$	−14.0000	−0.670478
$437$	−4.00000	−0.191346
$438$	4.00000	0.191127
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	5.00000	0.238366
$441$	−7.00000	−0.333333
$442$	−12.0000	−0.570782
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	−3.00000	−0.142374
$445$	10.0000	0.474045
$446$	12.0000	0.568216
$447$	−15.0000	−0.709476
$448$	0	0
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	4.00000	0.188562
$451$	45.0000	2.11897
$452$	6.00000	0.282216
$453$	−20.0000	−0.939682
$454$	−14.0000	−0.657053
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	−29.0000	−1.35508
$459$	4.00000	0.186704
$460$	1.00000	0.0466252
$461$	23.0000	1.07122	0.535608	−	0.844466i	$-0.320082\pi$
$461$	23.0000	1.07122	0.535608	+	0.844466i	$0.320082\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	−1.00000	−0.0464238
$465$	−5.00000	−0.231869
$466$	−12.0000	−0.555889
$467$	−23.0000	−1.06431	−0.532157	−	0.846646i	$-0.678618\pi$
$467$	−23.0000	−1.06431	−0.532157	+	0.846646i	$0.678618\pi$
$468$	3.00000	0.138675
$469$	0	0
$470$	10.0000	0.461266
$471$	14.0000	0.645086
$472$	7.00000	0.322201
$473$	−30.0000	−1.37940
$474$	2.00000	0.0918630
$475$	−16.0000	−0.734130
$476$	0	0
$477$	6.00000	0.274721
$478$	−19.0000	−0.869040
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	1.00000	0.0456435
$481$	−9.00000	−0.410365
$482$	4.00000	0.182195
$483$	0	0
$484$	14.0000	0.636364
$485$	−18.0000	−0.817338
$486$	−1.00000	−0.0453609
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	−1.00000	−0.0452679
$489$	13.0000	0.587880
$490$	−7.00000	−0.316228
$491$	−10.0000	−0.451294	−0.225647	−	0.974209i	$-0.572450\pi$
$491$	−10.0000	−0.451294	−0.225647	+	0.974209i	$0.572450\pi$
$492$	9.00000	0.405751
$493$	−4.00000	−0.180151
$494$	−12.0000	−0.539906
$495$	−5.00000	−0.224733
$496$	5.00000	0.224507
$497$	0	0
$498$	6.00000	0.268866
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	9.00000	0.402492
$501$	13.0000	0.580797
$502$	−17.0000	−0.758747
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	−15.0000	−0.667491
$506$	5.00000	0.222277
$507$	−4.00000	−0.177646
$508$	9.00000	0.399310
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	14.0000	0.617514
$515$	−13.0000	−0.572848
$516$	−6.00000	−0.264135
$517$	50.0000	2.19900
$518$	0	0
$519$	−6.00000	−0.263371
$520$	3.00000	0.131559
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	1.00000	0.0437688
$523$	−9.00000	−0.393543	−0.196771	−	0.980449i	$-0.563046\pi$
$523$	−9.00000	−0.393543	−0.196771	+	0.980449i	$0.563046\pi$
$524$	0	0
$525$	0	0
$526$	−16.0000	−0.697633
$527$	20.0000	0.871214
$528$	5.00000	0.217597
$529$	1.00000	0.0434783
$530$	6.00000	0.260623
$531$	−7.00000	−0.303774
$532$	0	0
$533$	27.0000	1.16950
$534$	10.0000	0.432742
$535$	−4.00000	−0.172935
$536$	13.0000	0.561514
$537$	12.0000	0.517838
$538$	−1.00000	−0.0431131
$539$	−35.0000	−1.50756
$540$	−1.00000	−0.0430331
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	13.0000	0.558398
$543$	−2.00000	−0.0858282
$544$	−4.00000	−0.171499
$545$	14.0000	0.599694
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	8.00000	0.341743
$549$	1.00000	0.0426790
$550$	20.0000	0.852803
$551$	−4.00000	−0.170406
$552$	1.00000	0.0425628
$553$	0	0
$554$	−7.00000	−0.297402
$555$	3.00000	0.127343
$556$	−2.00000	−0.0848189
$557$	25.0000	1.05928	0.529642	−	0.848221i	$-0.322326\pi$
$557$	25.0000	1.05928	0.529642	+	0.848221i	$0.322326\pi$
$558$	−5.00000	−0.211667
$559$	−18.0000	−0.761319
$560$	0	0
$561$	20.0000	0.844401
$562$	−6.00000	−0.253095
$563$	11.0000	0.463595	0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000	0.463595	0.231797	+	0.972764i	$0.425539\pi$
$564$	10.0000	0.421076
$565$	−6.00000	−0.252422
$566$	29.0000	1.21896
$567$	0	0
$568$	5.00000	0.209795
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	4.00000	0.167542
$571$	−11.0000	−0.460336	−0.230168	−	0.973151i	$-0.573928\pi$
$571$	−11.0000	−0.460336	−0.230168	+	0.973151i	$0.573928\pi$
$572$	15.0000	0.627182
$573$	−3.00000	−0.125327
$574$	0	0
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	1.00000	0.0415945
$579$	−16.0000	−0.664937
$580$	1.00000	0.0415227
$581$	0	0
$582$	−18.0000	−0.746124
$583$	30.0000	1.24247
$584$	4.00000	0.165521
$585$	−3.00000	−0.124035
$586$	24.0000	0.991431
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	−7.00000	−0.288675
$589$	20.0000	0.824086
$590$	−7.00000	−0.288185
$591$	18.0000	0.740421
$592$	−3.00000	−0.123299
$593$	4.00000	0.164260	0.0821302	−	0.996622i	$-0.473828\pi$
$593$	4.00000	0.164260	0.0821302	+	0.996622i	$0.473828\pi$
$594$	−5.00000	−0.205152
$595$	0	0
$596$	−15.0000	−0.614424
$597$	9.00000	0.368345
$598$	3.00000	0.122679
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	4.00000	0.163299
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	−13.0000	−0.529401
$604$	−20.0000	−0.813788
$605$	−14.0000	−0.569181
$606$	−15.0000	−0.609333
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	1.00000	0.0404888
$611$	30.0000	1.21367
$612$	4.00000	0.161690
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	−7.00000	−0.282497
$615$	−9.00000	−0.362915
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	−13.0000	−0.522937
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	−5.00000	−0.200805
$621$	−1.00000	−0.0401286
$622$	−6.00000	−0.240578
$623$	0	0
$624$	3.00000	0.120096
$625$	11.0000	0.440000
$626$	10.0000	0.399680
$627$	20.0000	0.798723
$628$	14.0000	0.558661
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−37.0000	−1.47295	−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000	−1.47295	−0.736473	+	0.676467i	$0.763510\pi$
$632$	2.00000	0.0795557
$633$	−17.0000	−0.675689
$634$	−3.00000	−0.119145
$635$	−9.00000	−0.357154
$636$	6.00000	0.237915
$637$	−21.0000	−0.832050
$638$	5.00000	0.197952
$639$	−5.00000	−0.197797
$640$	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	$0.657180\pi$
$642$	−4.00000	−0.157867
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	−16.0000	−0.629512
$647$	23.0000	0.904223	0.452112	−	0.891961i	$-0.350671\pi$
$647$	23.0000	0.904223	0.452112	+	0.891961i	$0.350671\pi$
$648$	−1.00000	−0.0392837
$649$	−35.0000	−1.37387
$650$	12.0000	0.470679
$651$	0	0
$652$	13.0000	0.509119
$653$	−43.0000	−1.68272	−0.841360	−	0.540475i	$-0.818245\pi$
$653$	−43.0000	−1.68272	−0.841360	+	0.540475i	$0.818245\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	9.00000	0.351391
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	−5.00000	−0.194625
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	12.0000	0.466393
$663$	12.0000	0.466041
$664$	6.00000	0.232845
$665$	0	0
$666$	3.00000	0.116248
$667$	1.00000	0.0387202
$668$	13.0000	0.502985
$669$	−12.0000	−0.463947
$670$	−13.0000	−0.502234
$671$	5.00000	0.193023
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−13.0000	−0.500741
$675$	−4.00000	−0.153960
$676$	−4.00000	−0.153846
$677$	28.0000	1.07613	0.538064	−	0.842904i	$-0.319156\pi$
$677$	28.0000	1.07613	0.538064	+	0.842904i	$0.319156\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	4.00000	0.153393
$681$	14.0000	0.536481
$682$	−25.0000	−0.957299
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	4.00000	0.152944
$685$	−8.00000	−0.305664
$686$	0	0
$687$	29.0000	1.10642
$688$	−6.00000	−0.228748
$689$	18.0000	0.685745
$690$	−1.00000	−0.0380693
$691$	−42.0000	−1.59776	−0.798878	−	0.601494i	$-0.794573\pi$
$691$	−42.0000	−1.59776	−0.798878	+	0.601494i	$0.794573\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	20.0000	0.759190
$695$	2.00000	0.0758643
$696$	1.00000	0.0379049
$697$	36.0000	1.36360
$698$	−1.00000	−0.0378506
$699$	12.0000	0.453882
$700$	0	0
$701$	9.00000	0.339925	0.169963	−	0.985451i	$-0.445635\pi$
$701$	9.00000	0.339925	0.169963	+	0.985451i	$0.445635\pi$
$702$	−3.00000	−0.113228
$703$	−12.0000	−0.452589
$704$	5.00000	0.188445
$705$	−10.0000	−0.376622
$706$	6.00000	0.225813
$707$	0	0
$708$	−7.00000	−0.263076
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	−5.00000	−0.187647
$711$	−2.00000	−0.0750059
$712$	10.0000	0.374766
$713$	−5.00000	−0.187251
$714$	0	0
$715$	−15.0000	−0.560968
$716$	12.0000	0.448461
$717$	19.0000	0.709568
$718$	24.0000	0.895672
$719$	1.00000	0.0372937	0.0186469	−	0.999826i	$-0.494064\pi$
$719$	1.00000	0.0372937	0.0186469	+	0.999826i	$0.494064\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	3.00000	0.111648
$723$	−4.00000	−0.148762
$724$	−2.00000	−0.0743294
$725$	4.00000	0.148556
$726$	−14.0000	−0.519589
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	−24.0000	−0.887672
$732$	1.00000	0.0369611
$733$	−23.0000	−0.849524	−0.424762	−	0.905305i	$-0.639642\pi$
$733$	−23.0000	−0.849524	−0.424762	+	0.905305i	$0.639642\pi$
$734$	−14.0000	−0.516749
$735$	7.00000	0.258199
$736$	1.00000	0.0368605
$737$	−65.0000	−2.39431
$738$	−9.00000	−0.331295
$739$	19.0000	0.698926	0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000	0.698926	0.349463	+	0.936950i	$0.386364\pi$
$740$	3.00000	0.110282
$741$	12.0000	0.440831
$742$	0	0
$743$	−31.0000	−1.13728	−0.568640	−	0.822587i	$-0.692530\pi$
$743$	−31.0000	−1.13728	−0.568640	+	0.822587i	$0.692530\pi$
$744$	−5.00000	−0.183309
$745$	15.0000	0.549557
$746$	−2.00000	−0.0732252
$747$	−6.00000	−0.219529
$748$	20.0000	0.731272
$749$	0	0
$750$	−9.00000	−0.328634
$751$	−44.0000	−1.60558	−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000	−1.60558	−0.802791	+	0.596260i	$0.796653\pi$
$752$	10.0000	0.364662
$753$	17.0000	0.619514
$754$	3.00000	0.109254
$755$	20.0000	0.727875
$756$	0	0
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	−6.00000	−0.217930
$759$	−5.00000	−0.181489
$760$	4.00000	0.145095
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−9.00000	−0.326036
$763$	0	0
$764$	−3.00000	−0.108536
$765$	−4.00000	−0.144620
$766$	10.0000	0.361315
$767$	−21.0000	−0.758266
$768$	1.00000	0.0360844
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	−16.0000	−0.575853
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	6.00000	0.215666
$775$	−20.0000	−0.718421
$776$	−18.0000	−0.646162
$777$	0	0
$778$	−8.00000	−0.286814
$779$	36.0000	1.28983
$780$	−3.00000	−0.107417
$781$	−25.0000	−0.894570
$782$	4.00000	0.143040
$783$	−1.00000	−0.0357371
$784$	−7.00000	−0.250000
$785$	−14.0000	−0.499681
$786$	0	0
$787$	45.0000	1.60408	0.802038	−	0.597272i	$-0.203749\pi$
$787$	45.0000	1.60408	0.802038	+	0.597272i	$0.203749\pi$
$788$	18.0000	0.641223
$789$	16.0000	0.569615
$790$	−2.00000	−0.0711568
$791$	0	0
$792$	−5.00000	−0.177667
$793$	3.00000	0.106533
$794$	−38.0000	−1.34857
$795$	−6.00000	−0.212798
$796$	9.00000	0.318997
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	4.00000	0.141421
$801$	−10.0000	−0.353333
$802$	23.0000	0.812158
$803$	−20.0000	−0.705785
$804$	−13.0000	−0.458475
$805$	0	0
$806$	−15.0000	−0.528352
$807$	1.00000	0.0352017
$808$	−15.0000	−0.527698
$809$	−53.0000	−1.86338	−0.931690	−	0.363253i	$-0.881666\pi$
$809$	−53.0000	−1.86338	−0.931690	+	0.363253i	$0.881666\pi$
$810$	1.00000	0.0351364
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	−13.0000	−0.455930
$814$	15.0000	0.525750
$815$	−13.0000	−0.455370
$816$	4.00000	0.140028
$817$	−24.0000	−0.839654
$818$	−14.0000	−0.489499
$819$	0	0
$820$	−9.00000	−0.314294
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	−8.00000	−0.279032
$823$	−19.0000	−0.662298	−0.331149	−	0.943578i	$-0.607436\pi$
$823$	−19.0000	−0.662298	−0.331149	+	0.943578i	$0.607436\pi$
$824$	−13.0000	−0.452876
$825$	−20.0000	−0.696311
$826$	0	0
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	−1.00000	−0.0347524
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	−6.00000	−0.208263
$831$	7.00000	0.242827
$832$	3.00000	0.104006
$833$	−28.0000	−0.970143
$834$	2.00000	0.0692543
$835$	−13.0000	−0.449884
$836$	20.0000	0.691714
$837$	5.00000	0.172825
$838$	−6.00000	−0.207267
$839$	−41.0000	−1.41548	−0.707739	−	0.706474i	$-0.750285\pi$
$839$	−41.0000	−1.41548	−0.707739	+	0.706474i	$0.750285\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−15.0000	−0.516934
$843$	6.00000	0.206651
$844$	−17.0000	−0.585164
$845$	4.00000	0.137604
$846$	−10.0000	−0.343807
$847$	0	0
$848$	6.00000	0.206041
$849$	−29.0000	−0.995277
$850$	16.0000	0.548795
$851$	3.00000	0.102839
$852$	−5.00000	−0.171297
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	−4.00000	−0.136717
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	−15.0000	−0.512092
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	6.00000	0.204598
$861$	0	0
$862$	−18.0000	−0.613082
$863$	13.0000	0.442525	0.221263	−	0.975214i	$-0.428982\pi$
$863$	13.0000	0.442525	0.221263	+	0.975214i	$0.428982\pi$
$864$	−1.00000	−0.0340207
$865$	6.00000	0.204006
$866$	−14.0000	−0.475739
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−10.0000	−0.339227
$870$	−1.00000	−0.0339032
$871$	−39.0000	−1.32146
$872$	14.0000	0.474100
$873$	18.0000	0.609208
$874$	4.00000	0.135302
$875$	0	0
$876$	−4.00000	−0.135147
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	10.0000	0.337484
$879$	−24.0000	−0.809500
$880$	−5.00000	−0.168550
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	7.00000	0.235702
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	12.0000	0.403604
$885$	7.00000	0.235302
$886$	−18.0000	−0.604722
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	3.00000	0.100673
$889$	0	0
$890$	−10.0000	−0.335201
$891$	5.00000	0.167506
$892$	−12.0000	−0.401790
$893$	40.0000	1.33855
$894$	15.0000	0.501675
$895$	−12.0000	−0.401116
$896$	0	0
$897$	−3.00000	−0.100167
$898$	27.0000	0.901002
$899$	−5.00000	−0.166759
$900$	−4.00000	−0.133333
$901$	24.0000	0.799556
$902$	−45.0000	−1.49834
$903$	0	0
$904$	−6.00000	−0.199557
$905$	2.00000	0.0664822
$906$	20.0000	0.664455
$907$	54.0000	1.79304	0.896520	−	0.443003i	$-0.146087\pi$
$907$	54.0000	1.79304	0.896520	+	0.443003i	$0.146087\pi$
$908$	14.0000	0.464606
$909$	15.0000	0.497519
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	4.00000	0.132453
$913$	−30.0000	−0.992855
$914$	28.0000	0.926158
$915$	−1.00000	−0.0330590
$916$	29.0000	0.958187
$917$	0	0
$918$	−4.00000	−0.132020
$919$	−27.0000	−0.890648	−0.445324	−	0.895370i	$-0.646911\pi$
$919$	−27.0000	−0.890648	−0.445324	+	0.895370i	$0.646911\pi$
$920$	−1.00000	−0.0329690
$921$	7.00000	0.230658
$922$	−23.0000	−0.757465
$923$	−15.0000	−0.493731
$924$	0	0
$925$	12.0000	0.394558
$926$	−20.0000	−0.657241
$927$	13.0000	0.426976
$928$	1.00000	0.0328266
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	5.00000	0.163956
$931$	−28.0000	−0.917663
$932$	12.0000	0.393073
$933$	6.00000	0.196431
$934$	23.0000	0.752583
$935$	−20.0000	−0.654070
$936$	−3.00000	−0.0980581
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	−10.0000	−0.326164
$941$	−33.0000	−1.07577	−0.537885	−	0.843018i	$-0.680776\pi$
$941$	−33.0000	−1.07577	−0.537885	+	0.843018i	$0.680776\pi$
$942$	−14.0000	−0.456145
$943$	−9.00000	−0.293080
$944$	−7.00000	−0.227831
$945$	0	0
$946$	30.0000	0.975384
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−2.00000	−0.0649570
$949$	−12.0000	−0.389536
$950$	16.0000	0.519109
$951$	3.00000	0.0972817
$952$	0	0
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	−6.00000	−0.194257
$955$	3.00000	0.0970777
$956$	19.0000	0.614504
$957$	−5.00000	−0.161627
$958$	−15.0000	−0.484628
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	−6.00000	−0.193548
$962$	9.00000	0.290172
$963$	4.00000	0.128898
$964$	−4.00000	−0.128831
$965$	16.0000	0.515058
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−14.0000	−0.449977
$969$	16.0000	0.513994
$970$	18.0000	0.577945
$971$	7.00000	0.224641	0.112320	−	0.993672i	$-0.464172\pi$
$971$	7.00000	0.224641	0.112320	+	0.993672i	$0.464172\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	40.0000	1.28168
$975$	−12.0000	−0.384308
$976$	1.00000	0.0320092
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	−13.0000	−0.415694
$979$	−50.0000	−1.59801
$980$	7.00000	0.223607
$981$	−14.0000	−0.446986
$982$	10.0000	0.319113
$983$	17.0000	0.542216	0.271108	−	0.962549i	$-0.412610\pi$
$983$	17.0000	0.542216	0.271108	+	0.962549i	$0.412610\pi$
$984$	−9.00000	−0.286910
$985$	−18.0000	−0.573528
$986$	4.00000	0.127386
$987$	0	0
$988$	12.0000	0.381771
$989$	6.00000	0.190789
$990$	5.00000	0.158910
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−5.00000	−0.158750
$993$	−12.0000	−0.380808
$994$	0	0
$995$	−9.00000	−0.285319
$996$	−6.00000	−0.190117
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	38.0000	1.20287
$999$	−3.00000	−0.0949158

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.e.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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