LMFDB - Embedded newform 6018.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6018,2,Mod(1,6018)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6018, 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("6018.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6018 = 2 \cdot 3 \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6018.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:	$48.0539719364$
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	$\chi$	$=$	6018.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +5.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} -6.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} -1.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +4.00000 q^{37} +5.00000 q^{38} +6.00000 q^{39} +2.00000 q^{40} +11.0000 q^{41} -1.00000 q^{42} -6.00000 q^{43} -6.00000 q^{44} +2.00000 q^{45} -1.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -6.00000 q^{52} -11.0000 q^{53} -1.00000 q^{54} -12.0000 q^{55} +1.00000 q^{56} -5.00000 q^{57} +1.00000 q^{59} -2.00000 q^{60} -14.0000 q^{61} +6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -12.0000 q^{65} +6.00000 q^{66} -2.00000 q^{67} -1.00000 q^{68} +1.00000 q^{69} +2.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} -5.00000 q^{73} +4.00000 q^{74} +1.00000 q^{75} +5.00000 q^{76} -6.00000 q^{77} +6.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +11.0000 q^{82} -15.0000 q^{83} -1.00000 q^{84} -2.00000 q^{85} -6.00000 q^{86} -6.00000 q^{88} +15.0000 q^{89} +2.00000 q^{90} -6.00000 q^{91} -1.00000 q^{92} -6.00000 q^{93} -4.00000 q^{94} +10.0000 q^{95} -1.00000 q^{96} -17.0000 q^{97} -6.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	1.00000	0.267261
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	2.00000	0.447214
$21$	−1.00000	−0.218218
$22$	−6.00000	−1.27920
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−2.00000	−0.365148
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	6.00000	1.04447
$34$	−1.00000	−0.171499
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	5.00000	0.811107
$39$	6.00000	0.960769
$40$	2.00000	0.316228
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	−1.00000	−0.154303
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−6.00000	−0.904534
$45$	2.00000	0.298142
$46$	−1.00000	−0.147442
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	−1.00000	−0.141421
$51$	1.00000	0.140028
$52$	−6.00000	−0.832050
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	−1.00000	−0.136083
$55$	−12.0000	−1.61808
$56$	1.00000	0.133631
$57$	−5.00000	−0.662266
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	−2.00000	−0.258199
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	6.00000	0.762001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	6.00000	0.738549
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−1.00000	−0.121268
$69$	1.00000	0.120386
$70$	2.00000	0.239046
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	4.00000	0.464991
$75$	1.00000	0.115470
$76$	5.00000	0.573539
$77$	−6.00000	−0.683763
$78$	6.00000	0.679366
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	11.0000	1.21475
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	−1.00000	−0.109109
$85$	−2.00000	−0.216930
$86$	−6.00000	−0.646997
$87$	0	0
$88$	−6.00000	−0.639602
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	2.00000	0.210819
$91$	−6.00000	−0.628971
$92$	−1.00000	−0.104257
$93$	−6.00000	−0.622171
$94$	−4.00000	−0.412568
$95$	10.0000	1.02598
$96$	−1.00000	−0.102062
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	−6.00000	−0.606092
$99$	−6.00000	−0.603023
$100$	−1.00000	−0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	1.00000	0.0990148
$103$	−11.0000	−1.08386	−0.541931	−	0.840423i	$-0.682307\pi$
$103$	−11.0000	−1.08386	−0.541931	+	0.840423i	$0.682307\pi$
$104$	−6.00000	−0.588348
$105$	−2.00000	−0.195180
$106$	−11.0000	−1.06841
$107$	−7.00000	−0.676716	−0.338358	−	0.941018i	$-0.609871\pi$
$107$	−7.00000	−0.676716	−0.338358	+	0.941018i	$0.609871\pi$
$108$	−1.00000	−0.0962250
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	−12.0000	−1.14416
$111$	−4.00000	−0.379663
$112$	1.00000	0.0944911
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−5.00000	−0.468293
$115$	−2.00000	−0.186501
$116$	0	0
$117$	−6.00000	−0.554700
$118$	1.00000	0.0920575
$119$	−1.00000	−0.0916698
$120$	−2.00000	−0.182574
$121$	25.0000	2.27273
$122$	−14.0000	−1.26750
$123$	−11.0000	−0.991837
$124$	6.00000	0.538816
$125$	−12.0000	−1.07331
$126$	1.00000	0.0890871
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	1.00000	0.0883883
$129$	6.00000	0.528271
$130$	−12.0000	−1.05247
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	6.00000	0.522233
$133$	5.00000	0.433555
$134$	−2.00000	−0.172774
$135$	−2.00000	−0.172133
$136$	−1.00000	−0.0857493
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	1.00000	0.0851257
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	2.00000	0.169031
$141$	4.00000	0.336861
$142$	6.00000	0.503509
$143$	36.0000	3.01047
$144$	1.00000	0.0833333
$145$	0	0
$146$	−5.00000	−0.413803
$147$	6.00000	0.494872
$148$	4.00000	0.328798
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	1.00000	0.0816497
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	5.00000	0.405554
$153$	−1.00000	−0.0808452
$154$	−6.00000	−0.483494
$155$	12.0000	0.963863
$156$	6.00000	0.480384
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	−8.00000	−0.636446
$159$	11.0000	0.872357
$160$	2.00000	0.158114
$161$	−1.00000	−0.0788110
$162$	1.00000	0.0785674
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	11.0000	0.858956
$165$	12.0000	0.934199
$166$	−15.0000	−1.16423
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−1.00000	−0.0771517
$169$	23.0000	1.76923
$170$	−2.00000	−0.153393
$171$	5.00000	0.382360
$172$	−6.00000	−0.457496
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−6.00000	−0.452267
$177$	−1.00000	−0.0751646
$178$	15.0000	1.12430
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	2.00000	0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−6.00000	−0.444750
$183$	14.0000	1.03491
$184$	−1.00000	−0.0737210
$185$	8.00000	0.588172
$186$	−6.00000	−0.439941
$187$	6.00000	0.438763
$188$	−4.00000	−0.291730
$189$	−1.00000	−0.0727393
$190$	10.0000	0.725476
$191$	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
$193$	−12.0000	−0.863779	−0.431889	−	0.901927i	$-0.642153\pi$
$193$	−12.0000	−0.863779	−0.431889	+	0.901927i	$0.642153\pi$
$194$	−17.0000	−1.22053
$195$	12.0000	0.859338
$196$	−6.00000	−0.428571
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−6.00000	−0.426401
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	−1.00000	−0.0707107
$201$	2.00000	0.141069
$202$	−10.0000	−0.703598
$203$	0	0
$204$	1.00000	0.0700140
$205$	22.0000	1.53655
$206$	−11.0000	−0.766406
$207$	−1.00000	−0.0695048
$208$	−6.00000	−0.416025
$209$	−30.0000	−2.07514
$210$	−2.00000	−0.138013
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−11.0000	−0.755483
$213$	−6.00000	−0.411113
$214$	−7.00000	−0.478510
$215$	−12.0000	−0.818393
$216$	−1.00000	−0.0680414
$217$	6.00000	0.407307
$218$	0	0
$219$	5.00000	0.337869
$220$	−12.0000	−0.809040
$221$	6.00000	0.403604
$222$	−4.00000	−0.268462
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	1.00000	0.0668153
$225$	−1.00000	−0.0666667
$226$	2.00000	0.133038
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−5.00000	−0.331133
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	−2.00000	−0.131876
$231$	6.00000	0.394771
$232$	0	0
$233$	−30.0000	−1.96537	−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−30.0000	−1.96537	−0.982683	+	0.185296i	$0.940675\pi$
$234$	−6.00000	−0.392232
$235$	−8.00000	−0.521862
$236$	1.00000	0.0650945
$237$	8.00000	0.519656
$238$	−1.00000	−0.0648204
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	−2.00000	−0.129099
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	25.0000	1.60706
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	−12.0000	−0.766652
$246$	−11.0000	−0.701334
$247$	−30.0000	−1.90885
$248$	6.00000	0.381000
$249$	15.0000	0.950586
$250$	−12.0000	−0.758947
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	1.00000	0.0629941
$253$	6.00000	0.377217
$254$	4.00000	0.250982
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	6.00000	0.373544
$259$	4.00000	0.248548
$260$	−12.0000	−0.744208
$261$	0	0
$262$	−2.00000	−0.123560
$263$	15.0000	0.924940	0.462470	−	0.886635i	$-0.346963\pi$
$263$	15.0000	0.924940	0.462470	+	0.886635i	$0.346963\pi$
$264$	6.00000	0.369274
$265$	−22.0000	−1.35145
$266$	5.00000	0.306570
$267$	−15.0000	−0.917985
$268$	−2.00000	−0.122169
$269$	−19.0000	−1.15845	−0.579225	−	0.815168i	$-0.696645\pi$
$269$	−19.0000	−1.15845	−0.579225	+	0.815168i	$0.696645\pi$
$270$	−2.00000	−0.121716
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−1.00000	−0.0606339
$273$	6.00000	0.363137
$274$	−16.0000	−0.966595
$275$	6.00000	0.361814
$276$	1.00000	0.0601929
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	−4.00000	−0.239904
$279$	6.00000	0.359211
$280$	2.00000	0.119523
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	4.00000	0.238197
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	6.00000	0.356034
$285$	−10.0000	−0.592349
$286$	36.0000	2.12872
$287$	11.0000	0.649309
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	17.0000	0.996558
$292$	−5.00000	−0.292603
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	6.00000	0.349927
$295$	2.00000	0.116445
$296$	4.00000	0.232495
$297$	6.00000	0.348155
$298$	12.0000	0.695141
$299$	6.00000	0.346989
$300$	1.00000	0.0577350
$301$	−6.00000	−0.345834
$302$	−5.00000	−0.287718
$303$	10.0000	0.574485
$304$	5.00000	0.286770
$305$	−28.0000	−1.60328
$306$	−1.00000	−0.0571662
$307$	23.0000	1.31268	0.656340	−	0.754466i	$-0.272104\pi$
$307$	23.0000	1.31268	0.656340	+	0.754466i	$0.272104\pi$
$308$	−6.00000	−0.341882
$309$	11.0000	0.625768
$310$	12.0000	0.681554
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	6.00000	0.339683
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	1.00000	0.0564333
$315$	2.00000	0.112687
$316$	−8.00000	−0.450035
$317$	16.0000	0.898650	0.449325	−	0.893368i	$-0.351665\pi$
$317$	16.0000	0.898650	0.449325	+	0.893368i	$0.351665\pi$
$318$	11.0000	0.616849
$319$	0	0
$320$	2.00000	0.111803
$321$	7.00000	0.390702
$322$	−1.00000	−0.0557278
$323$	−5.00000	−0.278207
$324$	1.00000	0.0555556
$325$	6.00000	0.332820
$326$	16.0000	0.886158
$327$	0	0
$328$	11.0000	0.607373
$329$	−4.00000	−0.220527
$330$	12.0000	0.660578
$331$	1.00000	0.0549650	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000	0.0549650	0.0274825	+	0.999622i	$0.491251\pi$
$332$	−15.0000	−0.823232
$333$	4.00000	0.219199
$334$	12.0000	0.656611
$335$	−4.00000	−0.218543
$336$	−1.00000	−0.0545545
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	23.0000	1.25104
$339$	−2.00000	−0.108625
$340$	−2.00000	−0.108465
$341$	−36.0000	−1.94951
$342$	5.00000	0.270369
$343$	−13.0000	−0.701934
$344$	−6.00000	−0.323498
$345$	2.00000	0.107676
$346$	−22.0000	−1.18273
$347$	36.0000	1.93258	0.966291	−	0.257454i	$-0.0828835\pi$
$347$	36.0000	1.93258	0.966291	+	0.257454i	$0.0828835\pi$
$348$	0	0
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	−1.00000	−0.0534522
$351$	6.00000	0.320256
$352$	−6.00000	−0.319801
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	−1.00000	−0.0531494
$355$	12.0000	0.636894
$356$	15.0000	0.794998
$357$	1.00000	0.0529256
$358$	−5.00000	−0.264258
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	2.00000	0.105409
$361$	6.00000	0.315789
$362$	−10.0000	−0.525588
$363$	−25.0000	−1.31216
$364$	−6.00000	−0.314485
$365$	−10.0000	−0.523424
$366$	14.0000	0.731792
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	−1.00000	−0.0521286
$369$	11.0000	0.572637
$370$	8.00000	0.415900
$371$	−11.0000	−0.571092
$372$	−6.00000	−0.311086
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	6.00000	0.310253
$375$	12.0000	0.619677
$376$	−4.00000	−0.206284
$377$	0	0
$378$	−1.00000	−0.0514344
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	10.0000	0.512989
$381$	−4.00000	−0.204926
$382$	22.0000	1.12562
$383$	−7.00000	−0.357683	−0.178842	−	0.983878i	$-0.557235\pi$
$383$	−7.00000	−0.357683	−0.178842	+	0.983878i	$0.557235\pi$
$384$	−1.00000	−0.0510310
$385$	−12.0000	−0.611577
$386$	−12.0000	−0.610784
$387$	−6.00000	−0.304997
$388$	−17.0000	−0.863044
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	12.0000	0.607644
$391$	1.00000	0.0505722
$392$	−6.00000	−0.303046
$393$	2.00000	0.100887
$394$	18.0000	0.906827
$395$	−16.0000	−0.805047
$396$	−6.00000	−0.301511
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	5.00000	0.250627
$399$	−5.00000	−0.250313
$400$	−1.00000	−0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	2.00000	0.0997509
$403$	−36.0000	−1.79329
$404$	−10.0000	−0.497519
$405$	2.00000	0.0993808
$406$	0	0
$407$	−24.0000	−1.18964
$408$	1.00000	0.0495074
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	22.0000	1.08650
$411$	16.0000	0.789222
$412$	−11.0000	−0.541931
$413$	1.00000	0.0492068
$414$	−1.00000	−0.0491473
$415$	−30.0000	−1.47264
$416$	−6.00000	−0.294174
$417$	4.00000	0.195881
$418$	−30.0000	−1.46735
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	−2.00000	−0.0975900
$421$	−25.0000	−1.21843	−0.609213	−	0.793007i	$-0.708514\pi$
$421$	−25.0000	−1.21843	−0.609213	+	0.793007i	$0.708514\pi$
$422$	−8.00000	−0.389434
$423$	−4.00000	−0.194487
$424$	−11.0000	−0.534207
$425$	1.00000	0.0485071
$426$	−6.00000	−0.290701
$427$	−14.0000	−0.677507
$428$	−7.00000	−0.338358
$429$	−36.0000	−1.73810
$430$	−12.0000	−0.578691
$431$	−11.0000	−0.529851	−0.264926	−	0.964269i	$-0.585347\pi$
$431$	−11.0000	−0.529851	−0.264926	+	0.964269i	$0.585347\pi$
$432$	−1.00000	−0.0481125
$433$	1.00000	0.0480569	0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000	0.0480569	0.0240285	+	0.999711i	$0.492351\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	0	0
$437$	−5.00000	−0.239182
$438$	5.00000	0.238909
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	−12.0000	−0.572078
$441$	−6.00000	−0.285714
$442$	6.00000	0.285391
$443$	15.0000	0.712672	0.356336	−	0.934358i	$-0.384026\pi$
$443$	15.0000	0.712672	0.356336	+	0.934358i	$0.384026\pi$
$444$	−4.00000	−0.189832
$445$	30.0000	1.42214
$446$	−16.0000	−0.757622
$447$	−12.0000	−0.567581
$448$	1.00000	0.0472456
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	−1.00000	−0.0471405
$451$	−66.0000	−3.10782
$452$	2.00000	0.0940721
$453$	5.00000	0.234920
$454$	−24.0000	−1.12638
$455$	−12.0000	−0.562569
$456$	−5.00000	−0.234146
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	15.0000	0.700904
$459$	1.00000	0.0466760
$460$	−2.00000	−0.0932505
$461$	27.0000	1.25752	0.628758	−	0.777601i	$-0.283564\pi$
$461$	27.0000	1.25752	0.628758	+	0.777601i	$0.283564\pi$
$462$	6.00000	0.279145
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	−30.0000	−1.38972
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	−6.00000	−0.277350
$469$	−2.00000	−0.0923514
$470$	−8.00000	−0.369012
$471$	−1.00000	−0.0460776
$472$	1.00000	0.0460287
$473$	36.0000	1.65528
$474$	8.00000	0.367452
$475$	−5.00000	−0.229416
$476$	−1.00000	−0.0458349
$477$	−11.0000	−0.503655
$478$	12.0000	0.548867
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	−2.00000	−0.0912871
$481$	−24.0000	−1.09431
$482$	−8.00000	−0.364390
$483$	1.00000	0.0455016
$484$	25.0000	1.13636
$485$	−34.0000	−1.54386
$486$	−1.00000	−0.0453609
$487$	7.00000	0.317200	0.158600	−	0.987343i	$-0.449302\pi$
$487$	7.00000	0.317200	0.158600	+	0.987343i	$0.449302\pi$
$488$	−14.0000	−0.633750
$489$	−16.0000	−0.723545
$490$	−12.0000	−0.542105
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	−11.0000	−0.495918
$493$	0	0
$494$	−30.0000	−1.34976
$495$	−12.0000	−0.539360
$496$	6.00000	0.269408
$497$	6.00000	0.269137
$498$	15.0000	0.672166
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	−12.0000	−0.536656
$501$	−12.0000	−0.536120
$502$	2.00000	0.0892644
$503$	−23.0000	−1.02552	−0.512760	−	0.858532i	$-0.671377\pi$
$503$	−23.0000	−1.02552	−0.512760	+	0.858532i	$0.671377\pi$
$504$	1.00000	0.0445435
$505$	−20.0000	−0.889988
$506$	6.00000	0.266733
$507$	−23.0000	−1.02147
$508$	4.00000	0.177471
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	2.00000	0.0885615
$511$	−5.00000	−0.221187
$512$	1.00000	0.0441942
$513$	−5.00000	−0.220755
$514$	22.0000	0.970378
$515$	−22.0000	−0.969436
$516$	6.00000	0.264135
$517$	24.0000	1.05552
$518$	4.00000	0.175750
$519$	22.0000	0.965693
$520$	−12.0000	−0.526235
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−2.00000	−0.0873704
$525$	1.00000	0.0436436
$526$	15.0000	0.654031
$527$	−6.00000	−0.261364
$528$	6.00000	0.261116
$529$	−22.0000	−0.956522
$530$	−22.0000	−0.955619
$531$	1.00000	0.0433963
$532$	5.00000	0.216777
$533$	−66.0000	−2.85878
$534$	−15.0000	−0.649113
$535$	−14.0000	−0.605273
$536$	−2.00000	−0.0863868
$537$	5.00000	0.215766
$538$	−19.0000	−0.819148
$539$	36.0000	1.55063
$540$	−2.00000	−0.0860663
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	8.00000	0.343629
$543$	10.0000	0.429141
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	6.00000	0.256776
$547$	18.0000	0.769624	0.384812	−	0.922995i	$-0.374266\pi$
$547$	18.0000	0.769624	0.384812	+	0.922995i	$0.374266\pi$
$548$	−16.0000	−0.683486
$549$	−14.0000	−0.597505
$550$	6.00000	0.255841
$551$	0	0
$552$	1.00000	0.0425628
$553$	−8.00000	−0.340195
$554$	−19.0000	−0.807233
$555$	−8.00000	−0.339581
$556$	−4.00000	−0.169638
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	6.00000	0.254000
$559$	36.0000	1.52264
$560$	2.00000	0.0845154
$561$	−6.00000	−0.253320
$562$	−6.00000	−0.253095
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	4.00000	0.168430
$565$	4.00000	0.168281
$566$	−13.0000	−0.546431
$567$	1.00000	0.0419961
$568$	6.00000	0.251754
$569$	−5.00000	−0.209611	−0.104805	−	0.994493i	$-0.533422\pi$
$569$	−5.00000	−0.209611	−0.104805	+	0.994493i	$0.533422\pi$
$570$	−10.0000	−0.418854
$571$	35.0000	1.46470	0.732352	−	0.680926i	$-0.238422\pi$
$571$	35.0000	1.46470	0.732352	+	0.680926i	$0.238422\pi$
$572$	36.0000	1.50524
$573$	−22.0000	−0.919063
$574$	11.0000	0.459131
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	1.00000	0.0415945
$579$	12.0000	0.498703
$580$	0	0
$581$	−15.0000	−0.622305
$582$	17.0000	0.704673
$583$	66.0000	2.73344
$584$	−5.00000	−0.206901
$585$	−12.0000	−0.496139
$586$	−10.0000	−0.413096
$587$	−5.00000	−0.206372	−0.103186	−	0.994662i	$-0.532904\pi$
$587$	−5.00000	−0.206372	−0.103186	+	0.994662i	$0.532904\pi$
$588$	6.00000	0.247436
$589$	30.0000	1.23613
$590$	2.00000	0.0823387
$591$	−18.0000	−0.740421
$592$	4.00000	0.164399
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	6.00000	0.246183
$595$	−2.00000	−0.0819920
$596$	12.0000	0.491539
$597$	−5.00000	−0.204636
$598$	6.00000	0.245358
$599$	5.00000	0.204294	0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000	0.204294	0.102147	+	0.994769i	$0.467429\pi$
$600$	1.00000	0.0408248
$601$	−21.0000	−0.856608	−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000	−0.856608	−0.428304	+	0.903635i	$0.640889\pi$
$602$	−6.00000	−0.244542
$603$	−2.00000	−0.0814463
$604$	−5.00000	−0.203447
$605$	50.0000	2.03279
$606$	10.0000	0.406222
$607$	7.00000	0.284121	0.142061	−	0.989858i	$-0.454627\pi$
$607$	7.00000	0.284121	0.142061	+	0.989858i	$0.454627\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	−28.0000	−1.13369
$611$	24.0000	0.970936
$612$	−1.00000	−0.0404226
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	23.0000	0.928204
$615$	−22.0000	−0.887126
$616$	−6.00000	−0.241747
$617$	−21.0000	−0.845428	−0.422714	−	0.906263i	$-0.638923\pi$
$617$	−21.0000	−0.845428	−0.422714	+	0.906263i	$0.638923\pi$
$618$	11.0000	0.442485
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	12.0000	0.481932
$621$	1.00000	0.0401286
$622$	6.00000	0.240578
$623$	15.0000	0.600962
$624$	6.00000	0.240192
$625$	−19.0000	−0.760000
$626$	26.0000	1.03917
$627$	30.0000	1.19808
$628$	1.00000	0.0399043
$629$	−4.00000	−0.159490
$630$	2.00000	0.0796819
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	−8.00000	−0.318223
$633$	8.00000	0.317971
$634$	16.0000	0.635441
$635$	8.00000	0.317470
$636$	11.0000	0.436178
$637$	36.0000	1.42637
$638$	0	0
$639$	6.00000	0.237356
$640$	2.00000	0.0790569
$641$	−17.0000	−0.671460	−0.335730	−	0.941958i	$-0.608983\pi$
$641$	−17.0000	−0.671460	−0.335730	+	0.941958i	$0.608983\pi$
$642$	7.00000	0.276268
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	−1.00000	−0.0394055
$645$	12.0000	0.472500
$646$	−5.00000	−0.196722
$647$	43.0000	1.69050	0.845252	−	0.534368i	$-0.179450\pi$
$647$	43.0000	1.69050	0.845252	+	0.534368i	$0.179450\pi$
$648$	1.00000	0.0392837
$649$	−6.00000	−0.235521
$650$	6.00000	0.235339
$651$	−6.00000	−0.235159
$652$	16.0000	0.626608
$653$	−46.0000	−1.80012	−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000	−1.80012	−0.900060	+	0.435767i	$0.856477\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	11.0000	0.429478
$657$	−5.00000	−0.195069
$658$	−4.00000	−0.155936
$659$	41.0000	1.59713	0.798567	−	0.601906i	$-0.205592\pi$
$659$	41.0000	1.59713	0.798567	+	0.601906i	$0.205592\pi$
$660$	12.0000	0.467099
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	1.00000	0.0388661
$663$	−6.00000	−0.233021
$664$	−15.0000	−0.582113
$665$	10.0000	0.387783
$666$	4.00000	0.154997
$667$	0	0
$668$	12.0000	0.464294
$669$	16.0000	0.618596
$670$	−4.00000	−0.154533
$671$	84.0000	3.24278
$672$	−1.00000	−0.0385758
$673$	11.0000	0.424019	0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000	0.424019	0.212009	+	0.977268i	$0.431999\pi$
$674$	−27.0000	−1.04000
$675$	1.00000	0.0384900
$676$	23.0000	0.884615
$677$	−28.0000	−1.07613	−0.538064	−	0.842904i	$-0.680844\pi$
$677$	−28.0000	−1.07613	−0.538064	+	0.842904i	$0.680844\pi$
$678$	−2.00000	−0.0768095
$679$	−17.0000	−0.652400
$680$	−2.00000	−0.0766965
$681$	24.0000	0.919682
$682$	−36.0000	−1.37851
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	5.00000	0.191180
$685$	−32.0000	−1.22266
$686$	−13.0000	−0.496342
$687$	−15.0000	−0.572286
$688$	−6.00000	−0.228748
$689$	66.0000	2.51440
$690$	2.00000	0.0761387
$691$	1.00000	0.0380418	0.0190209	−	0.999819i	$-0.493945\pi$
$691$	1.00000	0.0380418	0.0190209	+	0.999819i	$0.493945\pi$
$692$	−22.0000	−0.836315
$693$	−6.00000	−0.227921
$694$	36.0000	1.36654
$695$	−8.00000	−0.303457
$696$	0	0
$697$	−11.0000	−0.416655
$698$	−19.0000	−0.719161
$699$	30.0000	1.13470
$700$	−1.00000	−0.0377964
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	6.00000	0.226455
$703$	20.0000	0.754314
$704$	−6.00000	−0.226134
$705$	8.00000	0.301297
$706$	−3.00000	−0.112906
$707$	−10.0000	−0.376089
$708$	−1.00000	−0.0375823
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	12.0000	0.450352
$711$	−8.00000	−0.300023
$712$	15.0000	0.562149
$713$	−6.00000	−0.224702
$714$	1.00000	0.0374241
$715$	72.0000	2.69265
$716$	−5.00000	−0.186859
$717$	−12.0000	−0.448148
$718$	3.00000	0.111959
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	2.00000	0.0745356
$721$	−11.0000	−0.409661
$722$	6.00000	0.223297
$723$	8.00000	0.297523
$724$	−10.0000	−0.371647
$725$	0	0
$726$	−25.0000	−0.927837
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	−10.0000	−0.370117
$731$	6.00000	0.221918
$732$	14.0000	0.517455
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	16.0000	0.590571
$735$	12.0000	0.442627
$736$	−1.00000	−0.0368605
$737$	12.0000	0.442026
$738$	11.0000	0.404916
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	8.00000	0.294086
$741$	30.0000	1.10208
$742$	−11.0000	−0.403823
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−6.00000	−0.219971
$745$	24.0000	0.879292
$746$	12.0000	0.439351
$747$	−15.0000	−0.548821
$748$	6.00000	0.219382
$749$	−7.00000	−0.255774
$750$	12.0000	0.438178
$751$	50.0000	1.82453	0.912263	−	0.409605i	$-0.134333\pi$
$751$	50.0000	1.82453	0.912263	+	0.409605i	$0.134333\pi$
$752$	−4.00000	−0.145865
$753$	−2.00000	−0.0728841
$754$	0	0
$755$	−10.0000	−0.363937
$756$	−1.00000	−0.0363696
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	22.0000	0.799076
$759$	−6.00000	−0.217786
$760$	10.0000	0.362738
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	−4.00000	−0.144905
$763$	0	0
$764$	22.0000	0.795932
$765$	−2.00000	−0.0723102
$766$	−7.00000	−0.252920
$767$	−6.00000	−0.216647
$768$	−1.00000	−0.0360844
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	−12.0000	−0.432450
$771$	−22.0000	−0.792311
$772$	−12.0000	−0.431889
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	−6.00000	−0.215666
$775$	−6.00000	−0.215526
$776$	−17.0000	−0.610264
$777$	−4.00000	−0.143499
$778$	30.0000	1.07555
$779$	55.0000	1.97058
$780$	12.0000	0.429669
$781$	−36.0000	−1.28818
$782$	1.00000	0.0357599
$783$	0	0
$784$	−6.00000	−0.214286
$785$	2.00000	0.0713831
$786$	2.00000	0.0713376
$787$	−52.0000	−1.85360	−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000	−1.85360	−0.926800	+	0.375555i	$0.877452\pi$
$788$	18.0000	0.641223
$789$	−15.0000	−0.534014
$790$	−16.0000	−0.569254
$791$	2.00000	0.0711118
$792$	−6.00000	−0.213201
$793$	84.0000	2.98293
$794$	6.00000	0.212932
$795$	22.0000	0.780260
$796$	5.00000	0.177220
$797$	36.0000	1.27519	0.637593	−	0.770374i	$-0.279930\pi$
$797$	36.0000	1.27519	0.637593	+	0.770374i	$0.279930\pi$
$798$	−5.00000	−0.176998
$799$	4.00000	0.141510
$800$	−1.00000	−0.0353553
$801$	15.0000	0.529999
$802$	−6.00000	−0.211867
$803$	30.0000	1.05868
$804$	2.00000	0.0705346
$805$	−2.00000	−0.0704907
$806$	−36.0000	−1.26805
$807$	19.0000	0.668832
$808$	−10.0000	−0.351799
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	2.00000	0.0702728
$811$	−49.0000	−1.72062	−0.860311	−	0.509769i	$-0.829731\pi$
$811$	−49.0000	−1.72062	−0.860311	+	0.509769i	$0.829731\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	−24.0000	−0.841200
$815$	32.0000	1.12091
$816$	1.00000	0.0350070
$817$	−30.0000	−1.04957
$818$	2.00000	0.0699284
$819$	−6.00000	−0.209657
$820$	22.0000	0.768273
$821$	−21.0000	−0.732905	−0.366453	−	0.930437i	$-0.619428\pi$
$821$	−21.0000	−0.732905	−0.366453	+	0.930437i	$0.619428\pi$
$822$	16.0000	0.558064
$823$	6.00000	0.209147	0.104573	−	0.994517i	$-0.466652\pi$
$823$	6.00000	0.209147	0.104573	+	0.994517i	$0.466652\pi$
$824$	−11.0000	−0.383203
$825$	−6.00000	−0.208893
$826$	1.00000	0.0347945
$827$	51.0000	1.77344	0.886722	−	0.462303i	$-0.152977\pi$
$827$	51.0000	1.77344	0.886722	+	0.462303i	$0.152977\pi$
$828$	−1.00000	−0.0347524
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	−30.0000	−1.04132
$831$	19.0000	0.659103
$832$	−6.00000	−0.208013
$833$	6.00000	0.207888
$834$	4.00000	0.138509
$835$	24.0000	0.830554
$836$	−30.0000	−1.03757
$837$	−6.00000	−0.207390
$838$	6.00000	0.207267
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	−2.00000	−0.0690066
$841$	−29.0000	−1.00000
$842$	−25.0000	−0.861557
$843$	6.00000	0.206651
$844$	−8.00000	−0.275371
$845$	46.0000	1.58245
$846$	−4.00000	−0.137523
$847$	25.0000	0.859010
$848$	−11.0000	−0.377742
$849$	13.0000	0.446159
$850$	1.00000	0.0342997
$851$	−4.00000	−0.137118
$852$	−6.00000	−0.205557
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	−14.0000	−0.479070
$855$	10.0000	0.341993
$856$	−7.00000	−0.239255
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	−36.0000	−1.22902
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	−12.0000	−0.409197
$861$	−11.0000	−0.374879
$862$	−11.0000	−0.374661
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	−1.00000	−0.0340207
$865$	−44.0000	−1.49604
$866$	1.00000	0.0339814
$867$	−1.00000	−0.0339618
$868$	6.00000	0.203653
$869$	48.0000	1.62829
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	−17.0000	−0.575363
$874$	−5.00000	−0.169128
$875$	−12.0000	−0.405674
$876$	5.00000	0.168934
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	0	0
$879$	10.0000	0.337292
$880$	−12.0000	−0.404520
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	−6.00000	−0.202031
$883$	−37.0000	−1.24515	−0.622575	−	0.782560i	$-0.713913\pi$
$883$	−37.0000	−1.24515	−0.622575	+	0.782560i	$0.713913\pi$
$884$	6.00000	0.201802
$885$	−2.00000	−0.0672293
$886$	15.0000	0.503935
$887$	−7.00000	−0.235037	−0.117518	−	0.993071i	$-0.537494\pi$
$887$	−7.00000	−0.235037	−0.117518	+	0.993071i	$0.537494\pi$
$888$	−4.00000	−0.134231
$889$	4.00000	0.134156
$890$	30.0000	1.00560
$891$	−6.00000	−0.201008
$892$	−16.0000	−0.535720
$893$	−20.0000	−0.669274
$894$	−12.0000	−0.401340
$895$	−10.0000	−0.334263
$896$	1.00000	0.0334077
$897$	−6.00000	−0.200334
$898$	−15.0000	−0.500556
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	11.0000	0.366463
$902$	−66.0000	−2.19756
$903$	6.00000	0.199667
$904$	2.00000	0.0665190
$905$	−20.0000	−0.664822
$906$	5.00000	0.166114
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−24.0000	−0.796468
$909$	−10.0000	−0.331679
$910$	−12.0000	−0.397796
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−5.00000	−0.165567
$913$	90.0000	2.97857
$914$	34.0000	1.12462
$915$	28.0000	0.925651
$916$	15.0000	0.495614
$917$	−2.00000	−0.0660458
$918$	1.00000	0.0330049
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	−2.00000	−0.0659380
$921$	−23.0000	−0.757876
$922$	27.0000	0.889198
$923$	−36.0000	−1.18495
$924$	6.00000	0.197386
$925$	−4.00000	−0.131519
$926$	−24.0000	−0.788689
$927$	−11.0000	−0.361287
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	−12.0000	−0.393496
$931$	−30.0000	−0.983210
$932$	−30.0000	−0.982683
$933$	−6.00000	−0.196431
$934$	24.0000	0.785304
$935$	12.0000	0.392442
$936$	−6.00000	−0.196116
$937$	16.0000	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$938$	−2.00000	−0.0653023
$939$	−26.0000	−0.848478
$940$	−8.00000	−0.260931
$941$	49.0000	1.59735	0.798677	−	0.601760i	$-0.205534\pi$
$941$	49.0000	1.59735	0.798677	+	0.601760i	$0.205534\pi$
$942$	−1.00000	−0.0325818
$943$	−11.0000	−0.358209
$944$	1.00000	0.0325472
$945$	−2.00000	−0.0650600
$946$	36.0000	1.17046
$947$	−13.0000	−0.422443	−0.211222	−	0.977438i	$-0.567744\pi$
$947$	−13.0000	−0.422443	−0.211222	+	0.977438i	$0.567744\pi$
$948$	8.00000	0.259828
$949$	30.0000	0.973841
$950$	−5.00000	−0.162221
$951$	−16.0000	−0.518836
$952$	−1.00000	−0.0324102
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	−11.0000	−0.356138
$955$	44.0000	1.42381
$956$	12.0000	0.388108
$957$	0	0
$958$	30.0000	0.969256
$959$	−16.0000	−0.516667
$960$	−2.00000	−0.0645497
$961$	5.00000	0.161290
$962$	−24.0000	−0.773791
$963$	−7.00000	−0.225572
$964$	−8.00000	−0.257663
$965$	−24.0000	−0.772587
$966$	1.00000	0.0321745
$967$	−3.00000	−0.0964735	−0.0482367	−	0.998836i	$-0.515360\pi$
$967$	−3.00000	−0.0964735	−0.0482367	+	0.998836i	$0.515360\pi$
$968$	25.0000	0.803530
$969$	5.00000	0.160623
$970$	−34.0000	−1.09167
$971$	26.0000	0.834380	0.417190	−	0.908819i	$-0.363015\pi$
$971$	26.0000	0.834380	0.417190	+	0.908819i	$0.363015\pi$
$972$	−1.00000	−0.0320750
$973$	−4.00000	−0.128234
$974$	7.00000	0.224294
$975$	−6.00000	−0.192154
$976$	−14.0000	−0.448129
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	−16.0000	−0.511624
$979$	−90.0000	−2.87641
$980$	−12.0000	−0.383326
$981$	0	0
$982$	−6.00000	−0.191468
$983$	31.0000	0.988746	0.494373	−	0.869250i	$-0.335398\pi$
$983$	31.0000	0.988746	0.494373	+	0.869250i	$0.335398\pi$
$984$	−11.0000	−0.350667
$985$	36.0000	1.14706
$986$	0	0
$987$	4.00000	0.127321
$988$	−30.0000	−0.954427
$989$	6.00000	0.190789
$990$	−12.0000	−0.381385
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	6.00000	0.190500
$993$	−1.00000	−0.0317340
$994$	6.00000	0.190308
$995$	10.0000	0.317021
$996$	15.0000	0.475293
$997$	−33.0000	−1.04512	−0.522560	−	0.852602i	$-0.675023\pi$
$997$	−33.0000	−1.04512	−0.522560	+	0.852602i	$0.675023\pi$
$998$	34.0000	1.07625
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6018.2.a.i.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6018.2.a.i.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:	\( N \)	\(=\)	\( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6018.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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