LMFDB - Embedded newform 4998.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4998,2,Mod(1,4998)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4998, 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("4998.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4998.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:	$39.9092309302$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 714)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4998.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} +2.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +5.00000 q^{29} -1.00000 q^{30} +3.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} -6.00000 q^{38} -2.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +10.0000 q^{43} -3.00000 q^{44} -1.00000 q^{45} +2.00000 q^{46} -2.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} -9.00000 q^{53} +1.00000 q^{54} +3.00000 q^{55} -6.00000 q^{57} +5.00000 q^{58} -13.0000 q^{59} -1.00000 q^{60} -8.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -3.00000 q^{66} +8.00000 q^{67} -1.00000 q^{68} +2.00000 q^{69} -14.0000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -8.00000 q^{74} -4.00000 q^{75} -6.00000 q^{76} -2.00000 q^{78} -9.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -9.00000 q^{83} +1.00000 q^{85} +10.0000 q^{86} +5.00000 q^{87} -3.00000 q^{88} -14.0000 q^{89} -1.00000 q^{90} +2.00000 q^{92} +3.00000 q^{93} -2.00000 q^{94} +6.00000 q^{95} +1.00000 q^{96} -3.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−3.00000	−0.639602
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	−1.00000	−0.182574
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	1.00000	0.176777
$33$	−3.00000	−0.522233
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−6.00000	−0.973329
$39$	−2.00000	−0.320256
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	−3.00000	−0.452267
$45$	−1.00000	−0.149071
$46$	2.00000	0.294884
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−1.00000	−0.140028
$52$	−2.00000	−0.277350
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	1.00000	0.136083
$55$	3.00000	0.404520
$56$	0	0
$57$	−6.00000	−0.794719
$58$	5.00000	0.656532
$59$	−13.0000	−1.69246	−0.846228	−	0.532821i	$-0.821132\pi$
$59$	−13.0000	−1.69246	−0.846228	+	0.532821i	$0.821132\pi$
$60$	−1.00000	−0.129099
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	3.00000	0.381000
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−3.00000	−0.369274
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−1.00000	−0.121268
$69$	2.00000	0.240772
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	1.00000	0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−8.00000	−0.929981
$75$	−4.00000	−0.461880
$76$	−6.00000	−0.688247
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	10.0000	1.07833
$87$	5.00000	0.536056
$88$	−3.00000	−0.319801
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	2.00000	0.208514
$93$	3.00000	0.311086
$94$	−2.00000	−0.206284
$95$	6.00000	0.615587
$96$	1.00000	0.102062
$97$	−3.00000	−0.304604	−0.152302	−	0.988334i	$-0.548669\pi$
$97$	−3.00000	−0.304604	−0.152302	+	0.988334i	$0.548669\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	−4.00000	−0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−1.00000	−0.0990148
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−9.00000	−0.874157
$107$	19.0000	1.83680	0.918400	−	0.395654i	$-0.129482\pi$
$107$	19.0000	1.83680	0.918400	+	0.395654i	$0.129482\pi$
$108$	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	$0.733911\pi$
$110$	3.00000	0.286039
$111$	−8.00000	−0.759326
$112$	0	0
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	−6.00000	−0.561951
$115$	−2.00000	−0.186501
$116$	5.00000	0.464238
$117$	−2.00000	−0.184900
$118$	−13.0000	−1.19675
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−2.00000	−0.181818
$122$	−8.00000	−0.724286
$123$	−2.00000	−0.180334
$124$	3.00000	0.269408
$125$	9.00000	0.804984
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	1.00000	0.0883883
$129$	10.0000	0.880451
$130$	2.00000	0.175412
$131$	−7.00000	−0.611593	−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000	−0.611593	−0.305796	+	0.952097i	$0.598923\pi$
$132$	−3.00000	−0.261116
$133$	0	0
$134$	8.00000	0.691095
$135$	−1.00000	−0.0860663
$136$	−1.00000	−0.0857493
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	2.00000	0.170251
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	−14.0000	−1.17485
$143$	6.00000	0.501745
$144$	1.00000	0.0833333
$145$	−5.00000	−0.415227
$146$	−2.00000	−0.165521
$147$	0	0
$148$	−8.00000	−0.657596
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	−4.00000	−0.326599
$151$	13.0000	1.05792	0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000	1.05792	0.528962	+	0.848645i	$0.322581\pi$
$152$	−6.00000	−0.486664
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−3.00000	−0.240966
$156$	−2.00000	−0.160128
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−9.00000	−0.716002
$159$	−9.00000	−0.713746
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	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	$0.284443\pi$
$164$	−2.00000	−0.156174
$165$	3.00000	0.233550
$166$	−9.00000	−0.698535
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	1.00000	0.0766965
$171$	−6.00000	−0.458831
$172$	10.0000	0.762493
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	5.00000	0.379049
$175$	0	0
$176$	−3.00000	−0.226134
$177$	−13.0000	−0.977140
$178$	−14.0000	−1.04934
$179$	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
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	2.00000	0.147442
$185$	8.00000	0.588172
$186$	3.00000	0.219971
$187$	3.00000	0.219382
$188$	−2.00000	−0.145865
$189$	0	0
$190$	6.00000	0.435286
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	1.00000	0.0721688
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	−3.00000	−0.215387
$195$	2.00000	0.143223
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−3.00000	−0.213201
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−4.00000	−0.282843
$201$	8.00000	0.564276
$202$	6.00000	0.422159
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	2.00000	0.139686
$206$	4.00000	0.278693
$207$	2.00000	0.139010
$208$	−2.00000	−0.138675
$209$	18.0000	1.24509
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	−9.00000	−0.618123
$213$	−14.0000	−0.959264
$214$	19.0000	1.29881
$215$	−10.0000	−0.681994
$216$	1.00000	0.0680414
$217$	0	0
$218$	−14.0000	−0.948200
$219$	−2.00000	−0.135147
$220$	3.00000	0.202260
$221$	2.00000	0.134535
$222$	−8.00000	−0.536925
$223$	−21.0000	−1.40626	−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000	−1.40626	−0.703132	+	0.711059i	$0.748216\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	16.0000	1.06430
$227$	19.0000	1.26107	0.630537	−	0.776159i	$-0.282835\pi$
$227$	19.0000	1.26107	0.630537	+	0.776159i	$0.282835\pi$
$228$	−6.00000	−0.397360
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	5.00000	0.328266
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	−2.00000	−0.130744
$235$	2.00000	0.130466
$236$	−13.0000	−0.846228
$237$	−9.00000	−0.584613
$238$	0	0
$239$	−10.0000	−0.646846	−0.323423	−	0.946254i	$-0.604834\pi$
$239$	−10.0000	−0.646846	−0.323423	+	0.946254i	$0.604834\pi$
$240$	−1.00000	−0.0645497
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	−2.00000	−0.128565
$243$	1.00000	0.0641500
$244$	−8.00000	−0.512148
$245$	0	0
$246$	−2.00000	−0.127515
$247$	12.0000	0.763542
$248$	3.00000	0.190500
$249$	−9.00000	−0.570352
$250$	9.00000	0.569210
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	−13.0000	−0.815693
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	10.0000	0.622573
$259$	0	0
$260$	2.00000	0.124035
$261$	5.00000	0.309492
$262$	−7.00000	−0.432461
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	−3.00000	−0.184637
$265$	9.00000	0.552866
$266$	0	0
$267$	−14.0000	−0.856786
$268$	8.00000	0.488678
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	−1.00000	−0.0608581
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	2.00000	0.120824
$275$	12.0000	0.723627
$276$	2.00000	0.120386
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−16.0000	−0.959616
$279$	3.00000	0.179605
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	−2.00000	−0.119098
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	−14.0000	−0.830747
$285$	6.00000	0.355409
$286$	6.00000	0.354787
$287$	0	0
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	−5.00000	−0.293610
$291$	−3.00000	−0.175863
$292$	−2.00000	−0.117041
$293$	3.00000	0.175262	0.0876309	−	0.996153i	$-0.472070\pi$
$293$	3.00000	0.175262	0.0876309	+	0.996153i	$0.472070\pi$
$294$	0	0
$295$	13.0000	0.756889
$296$	−8.00000	−0.464991
$297$	−3.00000	−0.174078
$298$	22.0000	1.27443
$299$	−4.00000	−0.231326
$300$	−4.00000	−0.230940
$301$	0	0
$302$	13.0000	0.748066
$303$	6.00000	0.344691
$304$	−6.00000	−0.344124
$305$	8.00000	0.458079
$306$	−1.00000	−0.0571662
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	−3.00000	−0.170389
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	−2.00000	−0.113228
$313$	3.00000	0.169570	0.0847850	−	0.996399i	$-0.472980\pi$
$313$	3.00000	0.169570	0.0847850	+	0.996399i	$0.472980\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−9.00000	−0.506290
$317$	13.0000	0.730153	0.365076	−	0.930978i	$-0.381043\pi$
$317$	13.0000	0.730153	0.365076	+	0.930978i	$0.381043\pi$
$318$	−9.00000	−0.504695
$319$	−15.0000	−0.839839
$320$	−1.00000	−0.0559017
$321$	19.0000	1.06048
$322$	0	0
$323$	6.00000	0.333849
$324$	1.00000	0.0555556
$325$	8.00000	0.443760
$326$	16.0000	0.886158
$327$	−14.0000	−0.774202
$328$	−2.00000	−0.110432
$329$	0	0
$330$	3.00000	0.165145
$331$	6.00000	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$332$	−9.00000	−0.493939
$333$	−8.00000	−0.438397
$334$	2.00000	0.109435
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−9.00000	−0.489535
$339$	16.0000	0.869001
$340$	1.00000	0.0542326
$341$	−9.00000	−0.487377
$342$	−6.00000	−0.324443
$343$	0	0
$344$	10.0000	0.539164
$345$	−2.00000	−0.107676
$346$	−2.00000	−0.107521
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	5.00000	0.268028
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−3.00000	−0.159901
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	−13.0000	−0.690942
$355$	14.0000	0.743043
$356$	−14.0000	−0.741999
$357$	0	0
$358$	12.0000	0.634220
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	−1.00000	−0.0527046
$361$	17.0000	0.894737
$362$	12.0000	0.630706
$363$	−2.00000	−0.104973
$364$	0	0
$365$	2.00000	0.104685
$366$	−8.00000	−0.418167
$367$	−27.0000	−1.40939	−0.704694	−	0.709511i	$-0.748916\pi$
$367$	−27.0000	−1.40939	−0.704694	+	0.709511i	$0.748916\pi$
$368$	2.00000	0.104257
$369$	−2.00000	−0.104116
$370$	8.00000	0.415900
$371$	0	0
$372$	3.00000	0.155543
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	3.00000	0.155126
$375$	9.00000	0.464758
$376$	−2.00000	−0.103142
$377$	−10.0000	−0.515026
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	6.00000	0.307794
$381$	−13.0000	−0.666010
$382$	6.00000	0.306987
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	11.0000	0.559885
$387$	10.0000	0.508329
$388$	−3.00000	−0.152302
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	2.00000	0.101274
$391$	−2.00000	−0.101144
$392$	0	0
$393$	−7.00000	−0.353103
$394$	−6.00000	−0.302276
$395$	9.00000	0.452839
$396$	−3.00000	−0.150756
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	8.00000	0.399004
$403$	−6.00000	−0.298881
$404$	6.00000	0.298511
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	24.0000	1.18964
$408$	−1.00000	−0.0495074
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	2.00000	0.0987730
$411$	2.00000	0.0986527
$412$	4.00000	0.197066
$413$	0	0
$414$	2.00000	0.0982946
$415$	9.00000	0.441793
$416$	−2.00000	−0.0980581
$417$	−16.0000	−0.783523
$418$	18.0000	0.880409
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	12.0000	0.584844	0.292422	−	0.956289i	$-0.405539\pi$
$421$	12.0000	0.584844	0.292422	+	0.956289i	$0.405539\pi$
$422$	−10.0000	−0.486792
$423$	−2.00000	−0.0972433
$424$	−9.00000	−0.437079
$425$	4.00000	0.194029
$426$	−14.0000	−0.678302
$427$	0	0
$428$	19.0000	0.918400
$429$	6.00000	0.289683
$430$	−10.0000	−0.482243
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	1.00000	0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	−5.00000	−0.239732
$436$	−14.0000	−0.670478
$437$	−12.0000	−0.574038
$438$	−2.00000	−0.0955637
$439$	−25.0000	−1.19318	−0.596592	−	0.802544i	$-0.703479\pi$
$439$	−25.0000	−1.19318	−0.596592	+	0.802544i	$0.703479\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	2.00000	0.0951303
$443$	−1.00000	−0.0475114	−0.0237557	−	0.999718i	$-0.507562\pi$
$443$	−1.00000	−0.0475114	−0.0237557	+	0.999718i	$0.507562\pi$
$444$	−8.00000	−0.379663
$445$	14.0000	0.663664
$446$	−21.0000	−0.994379
$447$	22.0000	1.04056
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−4.00000	−0.188562
$451$	6.00000	0.282529
$452$	16.0000	0.752577
$453$	13.0000	0.610793
$454$	19.0000	0.891714
$455$	0	0
$456$	−6.00000	−0.280976
$457$	7.00000	0.327446	0.163723	−	0.986506i	$-0.447650\pi$
$457$	7.00000	0.327446	0.163723	+	0.986506i	$0.447650\pi$
$458$	8.00000	0.373815
$459$	−1.00000	−0.0466760
$460$	−2.00000	−0.0932505
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	5.00000	0.232119
$465$	−3.00000	−0.139122
$466$	4.00000	0.185296
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	2.00000	0.0922531
$471$	10.0000	0.460776
$472$	−13.0000	−0.598374
$473$	−30.0000	−1.37940
$474$	−9.00000	−0.413384
$475$	24.0000	1.10120
$476$	0	0
$477$	−9.00000	−0.412082
$478$	−10.0000	−0.457389
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	−1.00000	−0.0456435
$481$	16.0000	0.729537
$482$	−11.0000	−0.501036
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	3.00000	0.136223
$486$	1.00000	0.0453609
$487$	13.0000	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$487$	13.0000	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$488$	−8.00000	−0.362143
$489$	16.0000	0.723545
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	−2.00000	−0.0901670
$493$	−5.00000	−0.225189
$494$	12.0000	0.539906
$495$	3.00000	0.134840
$496$	3.00000	0.134704
$497$	0	0
$498$	−9.00000	−0.403300
$499$	−42.0000	−1.88018	−0.940089	−	0.340929i	$-0.889258\pi$
$499$	−42.0000	−1.88018	−0.940089	+	0.340929i	$0.889258\pi$
$500$	9.00000	0.402492
$501$	2.00000	0.0893534
$502$	−9.00000	−0.401690
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	−6.00000	−0.266733
$507$	−9.00000	−0.399704
$508$	−13.0000	−0.576782
$509$	−37.0000	−1.64000	−0.819998	−	0.572366i	$-0.806026\pi$
$509$	−37.0000	−1.64000	−0.819998	+	0.572366i	$0.806026\pi$
$510$	1.00000	0.0442807
$511$	0	0
$512$	1.00000	0.0441942
$513$	−6.00000	−0.264906
$514$	12.0000	0.529297
$515$	−4.00000	−0.176261
$516$	10.0000	0.440225
$517$	6.00000	0.263880
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	2.00000	0.0877058
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	5.00000	0.218844
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−7.00000	−0.305796
$525$	0	0
$526$	4.00000	0.174408
$527$	−3.00000	−0.130682
$528$	−3.00000	−0.130558
$529$	−19.0000	−0.826087
$530$	9.00000	0.390935
$531$	−13.0000	−0.564152
$532$	0	0
$533$	4.00000	0.173259
$534$	−14.0000	−0.605839
$535$	−19.0000	−0.821442
$536$	8.00000	0.345547
$537$	12.0000	0.517838
$538$	9.00000	0.388018
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	−3.00000	−0.128861
$543$	12.0000	0.514969
$544$	−1.00000	−0.0428746
$545$	14.0000	0.599694
$546$	0	0
$547$	−10.0000	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$547$	−10.0000	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$548$	2.00000	0.0854358
$549$	−8.00000	−0.341432
$550$	12.0000	0.511682
$551$	−30.0000	−1.27804
$552$	2.00000	0.0851257
$553$	0	0
$554$	−28.0000	−1.18961
$555$	8.00000	0.339581
$556$	−16.0000	−0.678551
$557$	5.00000	0.211857	0.105928	−	0.994374i	$-0.466219\pi$
$557$	5.00000	0.211857	0.105928	+	0.994374i	$0.466219\pi$
$558$	3.00000	0.127000
$559$	−20.0000	−0.845910
$560$	0	0
$561$	3.00000	0.126660
$562$	−16.0000	−0.674919
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	−2.00000	−0.0842152
$565$	−16.0000	−0.673125
$566$	−24.0000	−1.00880
$567$	0	0
$568$	−14.0000	−0.587427
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	6.00000	0.251312
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	6.00000	0.250873
$573$	6.00000	0.250654
$574$	0	0
$575$	−8.00000	−0.333623
$576$	1.00000	0.0416667
$577$	31.0000	1.29055	0.645273	−	0.763952i	$-0.276743\pi$
$577$	31.0000	1.29055	0.645273	+	0.763952i	$0.276743\pi$
$578$	1.00000	0.0415945
$579$	11.0000	0.457144
$580$	−5.00000	−0.207614
$581$	0	0
$582$	−3.00000	−0.124354
$583$	27.0000	1.11823
$584$	−2.00000	−0.0827606
$585$	2.00000	0.0826898
$586$	3.00000	0.123929
$587$	−23.0000	−0.949312	−0.474656	−	0.880172i	$-0.657427\pi$
$587$	−23.0000	−0.949312	−0.474656	+	0.880172i	$0.657427\pi$
$588$	0	0
$589$	−18.0000	−0.741677
$590$	13.0000	0.535202
$591$	−6.00000	−0.246807
$592$	−8.00000	−0.328798
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	22.0000	0.901155
$597$	−16.0000	−0.654836
$598$	−4.00000	−0.163572
$599$	26.0000	1.06233	0.531166	−	0.847268i	$-0.321754\pi$
$599$	26.0000	1.06233	0.531166	+	0.847268i	$0.321754\pi$
$600$	−4.00000	−0.163299
$601$	1.00000	0.0407909	0.0203954	−	0.999792i	$-0.493507\pi$
$601$	1.00000	0.0407909	0.0203954	+	0.999792i	$0.493507\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	13.0000	0.528962
$605$	2.00000	0.0813116
$606$	6.00000	0.243733
$607$	1.00000	0.0405887	0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000	0.0405887	0.0202944	+	0.999794i	$0.493540\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	8.00000	0.323911
$611$	4.00000	0.161823
$612$	−1.00000	−0.0404226
$613$	−4.00000	−0.161558	−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000	−0.161558	−0.0807792	+	0.996732i	$0.525741\pi$
$614$	10.0000	0.403567
$615$	2.00000	0.0806478
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	4.00000	0.160904
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	−3.00000	−0.120483
$621$	2.00000	0.0802572
$622$	20.0000	0.801927
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	11.0000	0.440000
$626$	3.00000	0.119904
$627$	18.0000	0.718851
$628$	10.0000	0.399043
$629$	8.00000	0.318981
$630$	0	0
$631$	−45.0000	−1.79142	−0.895711	−	0.444637i	$-0.853333\pi$
$631$	−45.0000	−1.79142	−0.895711	+	0.444637i	$0.853333\pi$
$632$	−9.00000	−0.358001
$633$	−10.0000	−0.397464
$634$	13.0000	0.516296
$635$	13.0000	0.515889
$636$	−9.00000	−0.356873
$637$	0	0
$638$	−15.0000	−0.593856
$639$	−14.0000	−0.553831
$640$	−1.00000	−0.0395285
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	19.0000	0.749870
$643$	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
$645$	−10.0000	−0.393750
$646$	6.00000	0.236067
$647$	40.0000	1.57256	0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000	1.57256	0.786281	+	0.617869i	$0.212004\pi$
$648$	1.00000	0.0392837
$649$	39.0000	1.53088
$650$	8.00000	0.313786
$651$	0	0
$652$	16.0000	0.626608
$653$	−1.00000	−0.0391330	−0.0195665	−	0.999809i	$-0.506229\pi$
$653$	−1.00000	−0.0391330	−0.0195665	+	0.999809i	$0.506229\pi$
$654$	−14.0000	−0.547443
$655$	7.00000	0.273513
$656$	−2.00000	−0.0780869
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	3.00000	0.116775
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	6.00000	0.233197
$663$	2.00000	0.0776736
$664$	−9.00000	−0.349268
$665$	0	0
$666$	−8.00000	−0.309994
$667$	10.0000	0.387202
$668$	2.00000	0.0773823
$669$	−21.0000	−0.811907
$670$	−8.00000	−0.309067
$671$	24.0000	0.926510
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	−13.0000	−0.500741
$675$	−4.00000	−0.153960
$676$	−9.00000	−0.346154
$677$	43.0000	1.65262	0.826312	−	0.563212i	$-0.190435\pi$
$677$	43.0000	1.65262	0.826312	+	0.563212i	$0.190435\pi$
$678$	16.0000	0.614476
$679$	0	0
$680$	1.00000	0.0383482
$681$	19.0000	0.728082
$682$	−9.00000	−0.344628
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	−6.00000	−0.229416
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	8.00000	0.305219
$688$	10.0000	0.381246
$689$	18.0000	0.685745
$690$	−2.00000	−0.0761387
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	−12.0000	−0.455514
$695$	16.0000	0.606915
$696$	5.00000	0.189525
$697$	2.00000	0.0757554
$698$	16.0000	0.605609
$699$	4.00000	0.151294
$700$	0	0
$701$	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
$703$	48.0000	1.81035
$704$	−3.00000	−0.113067
$705$	2.00000	0.0753244
$706$	−30.0000	−1.12906
$707$	0	0
$708$	−13.0000	−0.488570
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	14.0000	0.525411
$711$	−9.00000	−0.337526
$712$	−14.0000	−0.524672
$713$	6.00000	0.224702
$714$	0	0
$715$	−6.00000	−0.224387
$716$	12.0000	0.448461
$717$	−10.0000	−0.373457
$718$	−8.00000	−0.298557
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	17.0000	0.632674
$723$	−11.0000	−0.409094
$724$	12.0000	0.445976
$725$	−20.0000	−0.742781
$726$	−2.00000	−0.0742270
$727$	33.0000	1.22390	0.611951	−	0.790896i	$-0.290385\pi$
$727$	33.0000	1.22390	0.611951	+	0.790896i	$0.290385\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	2.00000	0.0740233
$731$	−10.0000	−0.369863
$732$	−8.00000	−0.295689
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	−27.0000	−0.996588
$735$	0	0
$736$	2.00000	0.0737210
$737$	−24.0000	−0.884051
$738$	−2.00000	−0.0736210
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	8.00000	0.294086
$741$	12.0000	0.440831
$742$	0	0
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	3.00000	0.109985
$745$	−22.0000	−0.806018
$746$	10.0000	0.366126
$747$	−9.00000	−0.329293
$748$	3.00000	0.109691
$749$	0	0
$750$	9.00000	0.328634
$751$	−39.0000	−1.42313	−0.711565	−	0.702620i	$-0.752013\pi$
$751$	−39.0000	−1.42313	−0.711565	+	0.702620i	$0.752013\pi$
$752$	−2.00000	−0.0729325
$753$	−9.00000	−0.327978
$754$	−10.0000	−0.364179
$755$	−13.0000	−0.473118
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	10.0000	0.363216
$759$	−6.00000	−0.217786
$760$	6.00000	0.217643
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−13.0000	−0.470940
$763$	0	0
$764$	6.00000	0.217072
$765$	1.00000	0.0361551
$766$	−4.00000	−0.144526
$767$	26.0000	0.938806
$768$	1.00000	0.0360844
$769$	−23.0000	−0.829401	−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000	−0.829401	−0.414701	+	0.909958i	$0.636114\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	11.0000	0.395899
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	10.0000	0.359443
$775$	−12.0000	−0.431053
$776$	−3.00000	−0.107694
$777$	0	0
$778$	30.0000	1.07555
$779$	12.0000	0.429945
$780$	2.00000	0.0716115
$781$	42.0000	1.50288
$782$	−2.00000	−0.0715199
$783$	5.00000	0.178685
$784$	0	0
$785$	−10.0000	−0.356915
$786$	−7.00000	−0.249682
$787$	−50.0000	−1.78231	−0.891154	−	0.453701i	$-0.850103\pi$
$787$	−50.0000	−1.78231	−0.891154	+	0.453701i	$0.850103\pi$
$788$	−6.00000	−0.213741
$789$	4.00000	0.142404
$790$	9.00000	0.320206
$791$	0	0
$792$	−3.00000	−0.106600
$793$	16.0000	0.568177
$794$	2.00000	0.0709773
$795$	9.00000	0.319197
$796$	−16.0000	−0.567105
$797$	−3.00000	−0.106265	−0.0531327	−	0.998587i	$-0.516921\pi$
$797$	−3.00000	−0.106265	−0.0531327	+	0.998587i	$0.516921\pi$
$798$	0	0
$799$	2.00000	0.0707549
$800$	−4.00000	−0.141421
$801$	−14.0000	−0.494666
$802$	−18.0000	−0.635602
$803$	6.00000	0.211735
$804$	8.00000	0.282138
$805$	0	0
$806$	−6.00000	−0.211341
$807$	9.00000	0.316815
$808$	6.00000	0.211079
$809$	−44.0000	−1.54696	−0.773479	−	0.633822i	$-0.781485\pi$
$809$	−44.0000	−1.54696	−0.773479	+	0.633822i	$0.781485\pi$
$810$	−1.00000	−0.0351364
$811$	−50.0000	−1.75574	−0.877869	−	0.478901i	$-0.841035\pi$
$811$	−50.0000	−1.75574	−0.877869	+	0.478901i	$0.841035\pi$
$812$	0	0
$813$	−3.00000	−0.105215
$814$	24.0000	0.841200
$815$	−16.0000	−0.560456
$816$	−1.00000	−0.0350070
$817$	−60.0000	−2.09913
$818$	23.0000	0.804176
$819$	0	0
$820$	2.00000	0.0698430
$821$	−11.0000	−0.383903	−0.191951	−	0.981404i	$-0.561482\pi$
$821$	−11.0000	−0.383903	−0.191951	+	0.981404i	$0.561482\pi$
$822$	2.00000	0.0697580
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	4.00000	0.139347
$825$	12.0000	0.417786
$826$	0	0
$827$	−15.0000	−0.521601	−0.260801	−	0.965393i	$-0.583986\pi$
$827$	−15.0000	−0.521601	−0.260801	+	0.965393i	$0.583986\pi$
$828$	2.00000	0.0695048
$829$	44.0000	1.52818	0.764092	−	0.645108i	$-0.223188\pi$
$829$	44.0000	1.52818	0.764092	+	0.645108i	$0.223188\pi$
$830$	9.00000	0.312395
$831$	−28.0000	−0.971309
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	−16.0000	−0.554035
$835$	−2.00000	−0.0692129
$836$	18.0000	0.622543
$837$	3.00000	0.103695
$838$	4.00000	0.138178
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	12.0000	0.413547
$843$	−16.0000	−0.551069
$844$	−10.0000	−0.344214
$845$	9.00000	0.309609
$846$	−2.00000	−0.0687614
$847$	0	0
$848$	−9.00000	−0.309061
$849$	−24.0000	−0.823678
$850$	4.00000	0.137199
$851$	−16.0000	−0.548473
$852$	−14.0000	−0.479632
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	19.0000	0.649407
$857$	−32.0000	−1.09310	−0.546550	−	0.837427i	$-0.684059\pi$
$857$	−32.0000	−1.09310	−0.546550	+	0.837427i	$0.684059\pi$
$858$	6.00000	0.204837
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	−10.0000	−0.340997
$861$	0	0
$862$	30.0000	1.02180
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	1.00000	0.0340207
$865$	2.00000	0.0680020
$866$	−34.0000	−1.15537
$867$	1.00000	0.0339618
$868$	0	0
$869$	27.0000	0.915912
$870$	−5.00000	−0.169516
$871$	−16.0000	−0.542139
$872$	−14.0000	−0.474100
$873$	−3.00000	−0.101535
$874$	−12.0000	−0.405906
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	−56.0000	−1.89099	−0.945493	−	0.325643i	$-0.894419\pi$
$877$	−56.0000	−1.89099	−0.945493	+	0.325643i	$0.894419\pi$
$878$	−25.0000	−0.843709
$879$	3.00000	0.101187
$880$	3.00000	0.101130
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−46.0000	−1.54802	−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000	−1.54802	−0.774012	+	0.633171i	$0.781753\pi$
$884$	2.00000	0.0672673
$885$	13.0000	0.436990
$886$	−1.00000	−0.0335957
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	−8.00000	−0.268462
$889$	0	0
$890$	14.0000	0.469281
$891$	−3.00000	−0.100504
$892$	−21.0000	−0.703132
$893$	12.0000	0.401565
$894$	22.0000	0.735790
$895$	−12.0000	−0.401116
$896$	0	0
$897$	−4.00000	−0.133556
$898$	6.00000	0.200223
$899$	15.0000	0.500278
$900$	−4.00000	−0.133333
$901$	9.00000	0.299833
$902$	6.00000	0.199778
$903$	0	0
$904$	16.0000	0.532152
$905$	−12.0000	−0.398893
$906$	13.0000	0.431896
$907$	−22.0000	−0.730498	−0.365249	−	0.930910i	$-0.619016\pi$
$907$	−22.0000	−0.730498	−0.365249	+	0.930910i	$0.619016\pi$
$908$	19.0000	0.630537
$909$	6.00000	0.199007
$910$	0	0
$911$	−4.00000	−0.132526	−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000	−0.132526	−0.0662630	+	0.997802i	$0.521108\pi$
$912$	−6.00000	−0.198680
$913$	27.0000	0.893570
$914$	7.00000	0.231539
$915$	8.00000	0.264472
$916$	8.00000	0.264327
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	−2.00000	−0.0659380
$921$	10.0000	0.329511
$922$	18.0000	0.592798
$923$	28.0000	0.921631
$924$	0	0
$925$	32.0000	1.05215
$926$	−16.0000	−0.525793
$927$	4.00000	0.131377
$928$	5.00000	0.164133
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	−3.00000	−0.0983739
$931$	0	0
$932$	4.00000	0.131024
$933$	20.0000	0.654771
$934$	8.00000	0.261768
$935$	−3.00000	−0.0981105
$936$	−2.00000	−0.0653720
$937$	−13.0000	−0.424691	−0.212346	−	0.977195i	$-0.568110\pi$
$937$	−13.0000	−0.424691	−0.212346	+	0.977195i	$0.568110\pi$
$938$	0	0
$939$	3.00000	0.0979013
$940$	2.00000	0.0652328
$941$	31.0000	1.01057	0.505286	−	0.862952i	$-0.331387\pi$
$941$	31.0000	1.01057	0.505286	+	0.862952i	$0.331387\pi$
$942$	10.0000	0.325818
$943$	−4.00000	−0.130258
$944$	−13.0000	−0.423114
$945$	0	0
$946$	−30.0000	−0.975384
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−9.00000	−0.292306
$949$	4.00000	0.129845
$950$	24.0000	0.778663
$951$	13.0000	0.421554
$952$	0	0
$953$	52.0000	1.68445	0.842223	−	0.539130i	$-0.181247\pi$
$953$	52.0000	1.68445	0.842223	+	0.539130i	$0.181247\pi$
$954$	−9.00000	−0.291386
$955$	−6.00000	−0.194155
$956$	−10.0000	−0.323423
$957$	−15.0000	−0.484881
$958$	32.0000	1.03387
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	−22.0000	−0.709677
$962$	16.0000	0.515861
$963$	19.0000	0.612266
$964$	−11.0000	−0.354286
$965$	−11.0000	−0.354103
$966$	0	0
$967$	−27.0000	−0.868261	−0.434131	−	0.900850i	$-0.642944\pi$
$967$	−27.0000	−0.868261	−0.434131	+	0.900850i	$0.642944\pi$
$968$	−2.00000	−0.0642824
$969$	6.00000	0.192748
$970$	3.00000	0.0963242
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	13.0000	0.416547
$975$	8.00000	0.256205
$976$	−8.00000	−0.256074
$977$	−8.00000	−0.255943	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000	−0.255943	−0.127971	+	0.991778i	$0.540847\pi$
$978$	16.0000	0.511624
$979$	42.0000	1.34233
$980$	0	0
$981$	−14.0000	−0.446986
$982$	15.0000	0.478669
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	−2.00000	−0.0637577
$985$	6.00000	0.191176
$986$	−5.00000	−0.159232
$987$	0	0
$988$	12.0000	0.381771
$989$	20.0000	0.635963
$990$	3.00000	0.0953463
$991$	−45.0000	−1.42947	−0.714736	−	0.699394i	$-0.753453\pi$
$991$	−45.0000	−1.42947	−0.714736	+	0.699394i	$0.753453\pi$
$992$	3.00000	0.0952501
$993$	6.00000	0.190404
$994$	0	0
$995$	16.0000	0.507234
$996$	−9.00000	−0.285176
$997$	32.0000	1.01345	0.506725	−	0.862108i	$-0.330856\pi$
$997$	32.0000	1.01345	0.506725	+	0.862108i	$0.330856\pi$
$998$	−42.0000	−1.32949
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4998.2.a.bl.1.1		1
7.2	even	3		714.2.i.b.613.1	yes	2
7.4	even	3		714.2.i.b.205.1	✓	2
7.6	odd	2		4998.2.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.i.b.205.1	✓	2	7.4	even	3
714.2.i.b.613.1	yes	2	7.2	even	3
4998.2.a.bb.1.1		1	7.6	odd	2
4998.2.a.bl.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:	\( N \)	\(=\)	\( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4998.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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