LMFDB - Embedded newform 4998.2.a.g.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} +1.00000 q^{30} -5.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} -6.00000 q^{38} -2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +6.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} -2.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} +1.00000 q^{51} +2.00000 q^{52} +5.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} -6.00000 q^{57} -3.00000 q^{58} -3.00000 q^{59} -1.00000 q^{60} -12.0000 q^{61} +5.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -1.00000 q^{66} +16.0000 q^{67} -1.00000 q^{68} +6.00000 q^{69} -14.0000 q^{71} -1.00000 q^{72} -10.0000 q^{73} +4.00000 q^{74} +4.00000 q^{75} +6.00000 q^{76} +2.00000 q^{78} +15.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +1.00000 q^{83} -1.00000 q^{85} -6.00000 q^{86} -3.00000 q^{87} +1.00000 q^{88} -2.00000 q^{89} -1.00000 q^{90} -6.00000 q^{92} +5.00000 q^{93} +2.00000 q^{94} +6.00000 q^{95} +1.00000 q^{96} -15.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	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	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	1.00000	0.182574
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−6.00000	−0.973329
$39$	−2.00000	−0.320256
$40$	−1.00000	−0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	−1.00000	−0.150756
$45$	1.00000	0.149071
$46$	6.00000	0.884652
$47$	−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
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	1.00000	0.136083
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−6.00000	−0.794719
$58$	−3.00000	−0.393919
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	−1.00000	−0.129099
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	5.00000	0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−1.00000	−0.123091
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	−1.00000	−0.121268
$69$	6.00000	0.722315
$70$	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
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	4.00000	0.464991
$75$	4.00000	0.461880
$76$	6.00000	0.688247
$77$	0	0
$78$	2.00000	0.226455
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	−6.00000	−0.646997
$87$	−3.00000	−0.321634
$88$	1.00000	0.106600
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−6.00000	−0.625543
$93$	5.00000	0.518476
$94$	2.00000	0.206284
$95$	6.00000	0.615587
$96$	1.00000	0.102062
$97$	−15.0000	−1.52302	−0.761510	−	0.648154i	$-0.775541\pi$
$97$	−15.0000	−1.52302	−0.761510	+	0.648154i	$0.775541\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	−4.00000	−0.400000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−1.00000	−0.0990148
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−5.00000	−0.485643
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	1.00000	0.0953463
$111$	4.00000	0.379663
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	6.00000	0.561951
$115$	−6.00000	−0.559503
$116$	3.00000	0.278543
$117$	2.00000	0.184900
$118$	3.00000	0.276172
$119$	0	0
$120$	1.00000	0.0912871
$121$	−10.0000	−0.909091
$122$	12.0000	1.08643
$123$	6.00000	0.541002
$124$	−5.00000	−0.449013
$125$	−9.00000	−0.804984
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	−1.00000	−0.0883883
$129$	−6.00000	−0.528271
$130$	−2.00000	−0.175412
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−16.0000	−1.38219
$135$	−1.00000	−0.0860663
$136$	1.00000	0.0857493
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	−6.00000	−0.510754
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	14.0000	1.17485
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	3.00000	0.249136
$146$	10.0000	0.827606
$147$	0	0
$148$	−4.00000	−0.328798
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−4.00000	−0.326599
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	−6.00000	−0.486664
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−5.00000	−0.401610
$156$	−2.00000	−0.160128
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−15.0000	−1.19334
$159$	−5.00000	−0.396526
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−6.00000	−0.468521
$165$	1.00000	0.0778499
$166$	−1.00000	−0.0776151
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	1.00000	0.0766965
$171$	6.00000	0.458831
$172$	6.00000	0.457496
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	3.00000	0.225494
$178$	2.00000	0.149906
$179$	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	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	6.00000	0.442326
$185$	−4.00000	−0.294086
$186$	−5.00000	−0.366618
$187$	1.00000	0.0731272
$188$	−2.00000	−0.145865
$189$	0	0
$190$	−6.00000	−0.435286
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−1.00000	−0.0721688
$193$	7.00000	0.503871	0.251936	−	0.967744i	$-0.418933\pi$
$193$	7.00000	0.503871	0.251936	+	0.967744i	$0.418933\pi$
$194$	15.0000	1.07694
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	1.00000	0.0710669
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	4.00000	0.282843
$201$	−16.0000	−1.12855
$202$	−10.0000	−0.703598
$203$	0	0
$204$	1.00000	0.0700140
$205$	−6.00000	−0.419058
$206$	4.00000	0.278693
$207$	−6.00000	−0.417029
$208$	2.00000	0.138675
$209$	−6.00000	−0.415029
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	5.00000	0.343401
$213$	14.0000	0.959264
$214$	15.0000	1.02538
$215$	6.00000	0.409197
$216$	1.00000	0.0680414
$217$	0	0
$218$	−10.0000	−0.677285
$219$	10.0000	0.675737
$220$	−1.00000	−0.0674200
$221$	−2.00000	−0.134535
$222$	−4.00000	−0.268462
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	16.0000	1.06430
$227$	1.00000	0.0663723	0.0331862	−	0.999449i	$-0.489435\pi$
$227$	1.00000	0.0663723	0.0331862	+	0.999449i	$0.489435\pi$
$228$	−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	$0.414850\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	−3.00000	−0.196960
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	−2.00000	−0.130744
$235$	−2.00000	−0.130466
$236$	−3.00000	−0.195283
$237$	−15.0000	−0.974355
$238$	0	0
$239$	22.0000	1.42306	0.711531	−	0.702655i	$-0.248002\pi$
$239$	22.0000	1.42306	0.711531	+	0.702655i	$0.248002\pi$
$240$	−1.00000	−0.0645497
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	10.0000	0.642824
$243$	−1.00000	−0.0641500
$244$	−12.0000	−0.768221
$245$	0	0
$246$	−6.00000	−0.382546
$247$	12.0000	0.763542
$248$	5.00000	0.317500
$249$	−1.00000	−0.0633724
$250$	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
$253$	6.00000	0.377217
$254$	−7.00000	−0.439219
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	6.00000	0.373544
$259$	0	0
$260$	2.00000	0.124035
$261$	3.00000	0.185695
$262$	−3.00000	−0.185341
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−1.00000	−0.0615457
$265$	5.00000	0.307148
$266$	0	0
$267$	2.00000	0.122398
$268$	16.0000	0.977356
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	1.00000	0.0608581
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−14.0000	−0.845771
$275$	4.00000	0.241209
$276$	6.00000	0.361158
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	−16.0000	−0.959616
$279$	−5.00000	−0.299342
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	−2.00000	−0.119098
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−14.0000	−0.830747
$285$	−6.00000	−0.355409
$286$	2.00000	0.118262
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	−3.00000	−0.176166
$291$	15.0000	0.879316
$292$	−10.0000	−0.585206
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	−3.00000	−0.174667
$296$	4.00000	0.232495
$297$	1.00000	0.0580259
$298$	−10.0000	−0.579284
$299$	−12.0000	−0.693978
$300$	4.00000	0.230940
$301$	0	0
$302$	7.00000	0.402805
$303$	−10.0000	−0.574485
$304$	6.00000	0.344124
$305$	−12.0000	−0.687118
$306$	1.00000	0.0571662
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	5.00000	0.283981
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	2.00000	0.113228
$313$	−9.00000	−0.508710	−0.254355	−	0.967111i	$-0.581863\pi$
$313$	−9.00000	−0.508710	−0.254355	+	0.967111i	$0.581863\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	15.0000	0.843816
$317$	−13.0000	−0.730153	−0.365076	−	0.930978i	$-0.618957\pi$
$317$	−13.0000	−0.730153	−0.365076	+	0.930978i	$0.618957\pi$
$318$	5.00000	0.280386
$319$	−3.00000	−0.167968
$320$	1.00000	0.0559017
$321$	15.0000	0.837218
$322$	0	0
$323$	−6.00000	−0.333849
$324$	1.00000	0.0555556
$325$	−8.00000	−0.443760
$326$	−4.00000	−0.221540
$327$	−10.0000	−0.553001
$328$	6.00000	0.331295
$329$	0	0
$330$	−1.00000	−0.0550482
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	1.00000	0.0548821
$333$	−4.00000	−0.219199
$334$	2.00000	0.109435
$335$	16.0000	0.874173
$336$	0	0
$337$	7.00000	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$338$	9.00000	0.489535
$339$	16.0000	0.869001
$340$	−1.00000	−0.0542326
$341$	5.00000	0.270765
$342$	−6.00000	−0.324443
$343$	0	0
$344$	−6.00000	−0.323498
$345$	6.00000	0.323029
$346$	14.0000	0.752645
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−3.00000	−0.160817
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	1.00000	0.0533002
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	−3.00000	−0.159448
$355$	−14.0000	−0.743043
$356$	−2.00000	−0.106000
$357$	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
$361$	17.0000	0.894737
$362$	12.0000	0.630706
$363$	10.0000	0.524864
$364$	0	0
$365$	−10.0000	−0.523424
$366$	−12.0000	−0.627250
$367$	−3.00000	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$368$	−6.00000	−0.312772
$369$	−6.00000	−0.312348
$370$	4.00000	0.207950
$371$	0	0
$372$	5.00000	0.259238
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	−1.00000	−0.0517088
$375$	9.00000	0.464758
$376$	2.00000	0.103142
$377$	6.00000	0.309016
$378$	0	0
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	6.00000	0.307794
$381$	−7.00000	−0.358621
$382$	6.00000	0.306987
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−7.00000	−0.356291
$387$	6.00000	0.304997
$388$	−15.0000	−0.761510
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	2.00000	0.101274
$391$	6.00000	0.303433
$392$	0	0
$393$	−3.00000	−0.151330
$394$	10.0000	0.503793
$395$	15.0000	0.754732
$396$	−1.00000	−0.0502519
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	16.0000	0.798007
$403$	−10.0000	−0.498135
$404$	10.0000	0.497519
$405$	1.00000	0.0496904
$406$	0	0
$407$	4.00000	0.198273
$408$	−1.00000	−0.0495074
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	6.00000	0.296319
$411$	−14.0000	−0.690569
$412$	−4.00000	−0.197066
$413$	0	0
$414$	6.00000	0.294884
$415$	1.00000	0.0490881
$416$	−2.00000	−0.0980581
$417$	−16.0000	−0.783523
$418$	6.00000	0.293470
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	−10.0000	−0.486792
$423$	−2.00000	−0.0972433
$424$	−5.00000	−0.242821
$425$	4.00000	0.194029
$426$	−14.0000	−0.678302
$427$	0	0
$428$	−15.0000	−0.725052
$429$	2.00000	0.0965609
$430$	−6.00000	−0.289346
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	−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
$435$	−3.00000	−0.143839
$436$	10.0000	0.478913
$437$	−36.0000	−1.72211
$438$	−10.0000	−0.477818
$439$	31.0000	1.47955	0.739775	−	0.672855i	$-0.234932\pi$
$439$	31.0000	1.47955	0.739775	+	0.672855i	$0.234932\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	2.00000	0.0951303
$443$	9.00000	0.427603	0.213801	−	0.976877i	$-0.431415\pi$
$443$	9.00000	0.427603	0.213801	+	0.976877i	$0.431415\pi$
$444$	4.00000	0.189832
$445$	−2.00000	−0.0948091
$446$	9.00000	0.426162
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	4.00000	0.188562
$451$	6.00000	0.282529
$452$	−16.0000	−0.752577
$453$	7.00000	0.328889
$454$	−1.00000	−0.0469323
$455$	0	0
$456$	6.00000	0.280976
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	−8.00000	−0.373815
$459$	1.00000	0.0466760
$460$	−6.00000	−0.279751
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	3.00000	0.139272
$465$	5.00000	0.231869
$466$	−8.00000	−0.370593
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	2.00000	0.0922531
$471$	14.0000	0.645086
$472$	3.00000	0.138086
$473$	−6.00000	−0.275880
$474$	15.0000	0.688973
$475$	−24.0000	−1.10120
$476$	0	0
$477$	5.00000	0.228934
$478$	−22.0000	−1.00626
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	1.00000	0.0456435
$481$	−8.00000	−0.364769
$482$	−25.0000	−1.13872
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−15.0000	−0.681115
$486$	1.00000	0.0453609
$487$	−35.0000	−1.58600	−0.793001	−	0.609221i	$-0.791482\pi$
$487$	−35.0000	−1.58600	−0.793001	+	0.609221i	$0.791482\pi$
$488$	12.0000	0.543214
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−7.00000	−0.315906	−0.157953	−	0.987447i	$-0.550489\pi$
$491$	−7.00000	−0.315906	−0.157953	+	0.987447i	$0.550489\pi$
$492$	6.00000	0.270501
$493$	−3.00000	−0.135113
$494$	−12.0000	−0.539906
$495$	−1.00000	−0.0449467
$496$	−5.00000	−0.224507
$497$	0	0
$498$	1.00000	0.0448111
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	−9.00000	−0.402492
$501$	2.00000	0.0893534
$502$	−17.0000	−0.758747
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	−6.00000	−0.266733
$507$	9.00000	0.399704
$508$	7.00000	0.310575
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	−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
$516$	−6.00000	−0.264135
$517$	2.00000	0.0879599
$518$	0	0
$519$	14.0000	0.614532
$520$	−2.00000	−0.0877058
$521$	−40.0000	−1.75243	−0.876216	−	0.481919i	$-0.839940\pi$
$521$	−40.0000	−1.75243	−0.876216	+	0.481919i	$0.839940\pi$
$522$	−3.00000	−0.131306
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	24.0000	1.04645
$527$	5.00000	0.217803
$528$	1.00000	0.0435194
$529$	13.0000	0.565217
$530$	−5.00000	−0.217186
$531$	−3.00000	−0.130189
$532$	0	0
$533$	−12.0000	−0.519778
$534$	−2.00000	−0.0865485
$535$	−15.0000	−0.648507
$536$	−16.0000	−0.691095
$537$	−12.0000	−0.517838
$538$	9.00000	0.388018
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	15.0000	0.644305
$543$	12.0000	0.514969
$544$	1.00000	0.0428746
$545$	10.0000	0.428353
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	14.0000	0.598050
$549$	−12.0000	−0.512148
$550$	−4.00000	−0.170561
$551$	18.0000	0.766826
$552$	−6.00000	−0.255377
$553$	0	0
$554$	−12.0000	−0.509831
$555$	4.00000	0.169791
$556$	16.0000	0.678551
$557$	−33.0000	−1.39825	−0.699127	−	0.714997i	$-0.746428\pi$
$557$	−33.0000	−1.39825	−0.699127	+	0.714997i	$0.746428\pi$
$558$	5.00000	0.211667
$559$	12.0000	0.507546
$560$	0	0
$561$	−1.00000	−0.0422200
$562$	24.0000	1.01238
$563$	−31.0000	−1.30649	−0.653247	−	0.757145i	$-0.726594\pi$
$563$	−31.0000	−1.30649	−0.653247	+	0.757145i	$0.726594\pi$
$564$	2.00000	0.0842152
$565$	−16.0000	−0.673125
$566$	4.00000	0.168133
$567$	0	0
$568$	14.0000	0.587427
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	6.00000	0.251312
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	−2.00000	−0.0836242
$573$	6.00000	0.250654
$574$	0	0
$575$	24.0000	1.00087
$576$	1.00000	0.0416667
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	−1.00000	−0.0415945
$579$	−7.00000	−0.290910
$580$	3.00000	0.124568
$581$	0	0
$582$	−15.0000	−0.621770
$583$	−5.00000	−0.207079
$584$	10.0000	0.413803
$585$	2.00000	0.0826898
$586$	−9.00000	−0.371787
$587$	−33.0000	−1.36206	−0.681028	−	0.732257i	$-0.738467\pi$
$587$	−33.0000	−1.36206	−0.681028	+	0.732257i	$0.738467\pi$
$588$	0	0
$589$	−30.0000	−1.23613
$590$	3.00000	0.123508
$591$	10.0000	0.411345
$592$	−4.00000	−0.164399
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	10.0000	0.409616
$597$	8.00000	0.327418
$598$	12.0000	0.490716
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	−4.00000	−0.163299
$601$	−43.0000	−1.75401	−0.877003	−	0.480484i	$-0.840461\pi$
$601$	−43.0000	−1.75401	−0.877003	+	0.480484i	$0.840461\pi$
$602$	0	0
$603$	16.0000	0.651570
$604$	−7.00000	−0.284826
$605$	−10.0000	−0.406558
$606$	10.0000	0.406222
$607$	25.0000	1.01472	0.507359	−	0.861735i	$-0.330622\pi$
$607$	25.0000	1.01472	0.507359	+	0.861735i	$0.330622\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	12.0000	0.485866
$611$	−4.00000	−0.161823
$612$	−1.00000	−0.0404226
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	2.00000	0.0807134
$615$	6.00000	0.241943
$616$	0	0
$617$	−28.0000	−1.12724	−0.563619	−	0.826035i	$-0.690591\pi$
$617$	−28.0000	−1.12724	−0.563619	+	0.826035i	$0.690591\pi$
$618$	−4.00000	−0.160904
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	−5.00000	−0.200805
$621$	6.00000	0.240772
$622$	28.0000	1.12270
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	11.0000	0.440000
$626$	9.00000	0.359712
$627$	6.00000	0.239617
$628$	−14.0000	−0.558661
$629$	4.00000	0.159490
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	−15.0000	−0.596668
$633$	−10.0000	−0.397464
$634$	13.0000	0.516296
$635$	7.00000	0.277787
$636$	−5.00000	−0.198263
$637$	0	0
$638$	3.00000	0.118771
$639$	−14.0000	−0.553831
$640$	−1.00000	−0.0395285
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	−15.0000	−0.592003
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	−6.00000	−0.236250
$646$	6.00000	0.236067
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	−1.00000	−0.0392837
$649$	3.00000	0.117760
$650$	8.00000	0.313786
$651$	0	0
$652$	4.00000	0.156652
$653$	−7.00000	−0.273931	−0.136966	−	0.990576i	$-0.543735\pi$
$653$	−7.00000	−0.273931	−0.136966	+	0.990576i	$0.543735\pi$
$654$	10.0000	0.391031
$655$	3.00000	0.117220
$656$	−6.00000	−0.234261
$657$	−10.0000	−0.390137
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	1.00000	0.0389249
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	18.0000	0.699590
$663$	2.00000	0.0776736
$664$	−1.00000	−0.0388075
$665$	0	0
$666$	4.00000	0.154997
$667$	−18.0000	−0.696963
$668$	−2.00000	−0.0773823
$669$	9.00000	0.347960
$670$	−16.0000	−0.618134
$671$	12.0000	0.463255
$672$	0	0
$673$	−33.0000	−1.27206	−0.636028	−	0.771666i	$-0.719424\pi$
$673$	−33.0000	−1.27206	−0.636028	+	0.771666i	$0.719424\pi$
$674$	−7.00000	−0.269630
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	37.0000	1.42203	0.711013	−	0.703179i	$-0.248237\pi$
$677$	37.0000	1.42203	0.711013	+	0.703179i	$0.248237\pi$
$678$	−16.0000	−0.614476
$679$	0	0
$680$	1.00000	0.0383482
$681$	−1.00000	−0.0383201
$682$	−5.00000	−0.191460
$683$	29.0000	1.10965	0.554827	−	0.831966i	$-0.312784\pi$
$683$	29.0000	1.10965	0.554827	+	0.831966i	$0.312784\pi$
$684$	6.00000	0.229416
$685$	14.0000	0.534913
$686$	0	0
$687$	−8.00000	−0.305219
$688$	6.00000	0.228748
$689$	10.0000	0.380970
$690$	−6.00000	−0.228416
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	12.0000	0.455514
$695$	16.0000	0.606915
$696$	3.00000	0.113715
$697$	6.00000	0.227266
$698$	16.0000	0.605609
$699$	−8.00000	−0.302588
$700$	0	0
$701$	21.0000	0.793159	0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000	0.793159	0.396580	+	0.918000i	$0.370197\pi$
$702$	2.00000	0.0754851
$703$	−24.0000	−0.905177
$704$	−1.00000	−0.0376889
$705$	2.00000	0.0753244
$706$	34.0000	1.27961
$707$	0	0
$708$	3.00000	0.112747
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	14.0000	0.525411
$711$	15.0000	0.562544
$712$	2.00000	0.0749532
$713$	30.0000	1.12351
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	12.0000	0.448461
$717$	−22.0000	−0.821605
$718$	24.0000	0.895672
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−17.0000	−0.632674
$723$	−25.0000	−0.929760
$724$	−12.0000	−0.445976
$725$	−12.0000	−0.445669
$726$	−10.0000	−0.371135
$727$	−27.0000	−1.00137	−0.500687	−	0.865628i	$-0.666919\pi$
$727$	−27.0000	−1.00137	−0.500687	+	0.865628i	$0.666919\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	−6.00000	−0.221918
$732$	12.0000	0.443533
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	3.00000	0.110732
$735$	0	0
$736$	6.00000	0.221163
$737$	−16.0000	−0.589368
$738$	6.00000	0.220863
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	−4.00000	−0.147043
$741$	−12.0000	−0.440831
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	−5.00000	−0.183309
$745$	10.0000	0.366372
$746$	22.0000	0.805477
$747$	1.00000	0.0365881
$748$	1.00000	0.0365636
$749$	0	0
$750$	−9.00000	−0.328634
$751$	−31.0000	−1.13121	−0.565603	−	0.824678i	$-0.691357\pi$
$751$	−31.0000	−1.13121	−0.565603	+	0.824678i	$0.691357\pi$
$752$	−2.00000	−0.0729325
$753$	−17.0000	−0.619514
$754$	−6.00000	−0.218507
$755$	−7.00000	−0.254756
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	−26.0000	−0.944363
$759$	−6.00000	−0.217786
$760$	−6.00000	−0.217643
$761$	−40.0000	−1.45000	−0.724999	−	0.688749i	$-0.758160\pi$
$761$	−40.0000	−1.45000	−0.724999	+	0.688749i	$0.758160\pi$
$762$	7.00000	0.253583
$763$	0	0
$764$	−6.00000	−0.217072
$765$	−1.00000	−0.0361551
$766$	20.0000	0.722629
$767$	−6.00000	−0.216647
$768$	−1.00000	−0.0360844
$769$	−47.0000	−1.69486	−0.847432	−	0.530904i	$-0.821852\pi$
$769$	−47.0000	−1.69486	−0.847432	+	0.530904i	$0.821852\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	7.00000	0.251936
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	−6.00000	−0.215666
$775$	20.0000	0.718421
$776$	15.0000	0.538469
$777$	0	0
$778$	6.00000	0.215110
$779$	−36.0000	−1.28983
$780$	−2.00000	−0.0716115
$781$	14.0000	0.500959
$782$	−6.00000	−0.214560
$783$	−3.00000	−0.107211
$784$	0	0
$785$	−14.0000	−0.499681
$786$	3.00000	0.107006
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	−10.0000	−0.356235
$789$	24.0000	0.854423
$790$	−15.0000	−0.533676
$791$	0	0
$792$	1.00000	0.0355335
$793$	−24.0000	−0.852265
$794$	22.0000	0.780751
$795$	−5.00000	−0.177332
$796$	−8.00000	−0.283552
$797$	−1.00000	−0.0354218	−0.0177109	−	0.999843i	$-0.505638\pi$
$797$	−1.00000	−0.0354218	−0.0177109	+	0.999843i	$0.505638\pi$
$798$	0	0
$799$	2.00000	0.0707549
$800$	4.00000	0.141421
$801$	−2.00000	−0.0706665
$802$	6.00000	0.211867
$803$	10.0000	0.352892
$804$	−16.0000	−0.564276
$805$	0	0
$806$	10.0000	0.352235
$807$	9.00000	0.316815
$808$	−10.0000	−0.351799
$809$	52.0000	1.82822	0.914111	−	0.405463i	$-0.132890\pi$
$809$	52.0000	1.82822	0.914111	+	0.405463i	$0.132890\pi$
$810$	−1.00000	−0.0351364
$811$	−26.0000	−0.912983	−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000	−0.912983	−0.456492	+	0.889728i	$0.650894\pi$
$812$	0	0
$813$	15.0000	0.526073
$814$	−4.00000	−0.140200
$815$	4.00000	0.140114
$816$	1.00000	0.0350070
$817$	36.0000	1.25948
$818$	−7.00000	−0.244749
$819$	0	0
$820$	−6.00000	−0.209529
$821$	51.0000	1.77991	0.889956	−	0.456046i	$-0.150735\pi$
$821$	51.0000	1.77991	0.889956	+	0.456046i	$0.150735\pi$
$822$	14.0000	0.488306
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	4.00000	0.139347
$825$	−4.00000	−0.139262
$826$	0	0
$827$	19.0000	0.660695	0.330347	−	0.943859i	$-0.392834\pi$
$827$	19.0000	0.660695	0.330347	+	0.943859i	$0.392834\pi$
$828$	−6.00000	−0.208514
$829$	−8.00000	−0.277851	−0.138926	−	0.990303i	$-0.544365\pi$
$829$	−8.00000	−0.277851	−0.138926	+	0.990303i	$0.544365\pi$
$830$	−1.00000	−0.0347105
$831$	−12.0000	−0.416275
$832$	2.00000	0.0693375
$833$	0	0
$834$	16.0000	0.554035
$835$	−2.00000	−0.0692129
$836$	−6.00000	−0.207514
$837$	5.00000	0.172825
$838$	12.0000	0.414533
$839$	28.0000	0.966667	0.483334	−	0.875436i	$-0.339426\pi$
$839$	28.0000	0.966667	0.483334	+	0.875436i	$0.339426\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	24.0000	0.826604
$844$	10.0000	0.344214
$845$	−9.00000	−0.309609
$846$	2.00000	0.0687614
$847$	0	0
$848$	5.00000	0.171701
$849$	4.00000	0.137280
$850$	−4.00000	−0.137199
$851$	24.0000	0.822709
$852$	14.0000	0.479632
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	15.0000	0.512689
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	−2.00000	−0.0682789
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	6.00000	0.204598
$861$	0	0
$862$	−10.0000	−0.340601
$863$	52.0000	1.77010	0.885050	−	0.465495i	$-0.154124\pi$
$863$	52.0000	1.77010	0.885050	+	0.465495i	$0.154124\pi$
$864$	1.00000	0.0340207
$865$	−14.0000	−0.476014
$866$	−14.0000	−0.475739
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−15.0000	−0.508840
$870$	3.00000	0.101710
$871$	32.0000	1.08428
$872$	−10.0000	−0.338643
$873$	−15.0000	−0.507673
$874$	36.0000	1.21772
$875$	0	0
$876$	10.0000	0.337869
$877$	40.0000	1.35070	0.675352	−	0.737496i	$-0.263992\pi$
$877$	40.0000	1.35070	0.675352	+	0.737496i	$0.263992\pi$
$878$	−31.0000	−1.04620
$879$	−9.00000	−0.303562
$880$	−1.00000	−0.0337100
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	−2.00000	−0.0672673
$885$	3.00000	0.100844
$886$	−9.00000	−0.302361
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	−4.00000	−0.134231
$889$	0	0
$890$	2.00000	0.0670402
$891$	−1.00000	−0.0335013
$892$	−9.00000	−0.301342
$893$	−12.0000	−0.401565
$894$	10.0000	0.334450
$895$	12.0000	0.401116
$896$	0	0
$897$	12.0000	0.400668
$898$	26.0000	0.867631
$899$	−15.0000	−0.500278
$900$	−4.00000	−0.133333
$901$	−5.00000	−0.166574
$902$	−6.00000	−0.199778
$903$	0	0
$904$	16.0000	0.532152
$905$	−12.0000	−0.398893
$906$	−7.00000	−0.232559
$907$	−2.00000	−0.0664089	−0.0332045	−	0.999449i	$-0.510571\pi$
$907$	−2.00000	−0.0664089	−0.0332045	+	0.999449i	$0.510571\pi$
$908$	1.00000	0.0331862
$909$	10.0000	0.331679
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	−6.00000	−0.198680
$913$	−1.00000	−0.0330952
$914$	17.0000	0.562310
$915$	12.0000	0.396708
$916$	8.00000	0.264327
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	6.00000	0.197814
$921$	2.00000	0.0659022
$922$	2.00000	0.0658665
$923$	−28.0000	−0.921631
$924$	0	0
$925$	16.0000	0.526077
$926$	−8.00000	−0.262896
$927$	−4.00000	−0.131377
$928$	−3.00000	−0.0984798
$929$	−38.0000	−1.24674	−0.623370	−	0.781927i	$-0.714237\pi$
$929$	−38.0000	−1.24674	−0.623370	+	0.781927i	$0.714237\pi$
$930$	−5.00000	−0.163956
$931$	0	0
$932$	8.00000	0.262049
$933$	28.0000	0.916679
$934$	24.0000	0.785304
$935$	1.00000	0.0327035
$936$	−2.00000	−0.0653720
$937$	19.0000	0.620703	0.310351	−	0.950622i	$-0.399553\pi$
$937$	19.0000	0.620703	0.310351	+	0.950622i	$0.399553\pi$
$938$	0	0
$939$	9.00000	0.293704
$940$	−2.00000	−0.0652328
$941$	57.0000	1.85815	0.929073	−	0.369895i	$-0.120606\pi$
$941$	57.0000	1.85815	0.929073	+	0.369895i	$0.120606\pi$
$942$	−14.0000	−0.456145
$943$	36.0000	1.17232
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	6.00000	0.195077
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	−15.0000	−0.487177
$949$	−20.0000	−0.649227
$950$	24.0000	0.778663
$951$	13.0000	0.421554
$952$	0	0
$953$	56.0000	1.81402	0.907009	−	0.421111i	$-0.138360\pi$
$953$	56.0000	1.81402	0.907009	+	0.421111i	$0.138360\pi$
$954$	−5.00000	−0.161881
$955$	−6.00000	−0.194155
$956$	22.0000	0.711531
$957$	3.00000	0.0969762
$958$	20.0000	0.646171
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	−6.00000	−0.193548
$962$	8.00000	0.257930
$963$	−15.0000	−0.483368
$964$	25.0000	0.805196
$965$	7.00000	0.225338
$966$	0	0
$967$	41.0000	1.31847	0.659236	−	0.751936i	$-0.270880\pi$
$967$	41.0000	1.31847	0.659236	+	0.751936i	$0.270880\pi$
$968$	10.0000	0.321412
$969$	6.00000	0.192748
$970$	15.0000	0.481621
$971$	43.0000	1.37994	0.689968	−	0.723840i	$-0.257625\pi$
$971$	43.0000	1.37994	0.689968	+	0.723840i	$0.257625\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	35.0000	1.12147
$975$	8.00000	0.256205
$976$	−12.0000	−0.384111
$977$	−60.0000	−1.91957	−0.959785	−	0.280736i	$-0.909421\pi$
$977$	−60.0000	−1.91957	−0.959785	+	0.280736i	$0.909421\pi$
$978$	4.00000	0.127906
$979$	2.00000	0.0639203
$980$	0	0
$981$	10.0000	0.319275
$982$	7.00000	0.223379
$983$	−40.0000	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$983$	−40.0000	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$984$	−6.00000	−0.191273
$985$	−10.0000	−0.318626
$986$	3.00000	0.0955395
$987$	0	0
$988$	12.0000	0.381771
$989$	−36.0000	−1.14473
$990$	1.00000	0.0317821
$991$	27.0000	0.857683	0.428842	−	0.903380i	$-0.358922\pi$
$991$	27.0000	0.857683	0.428842	+	0.903380i	$0.358922\pi$
$992$	5.00000	0.158750
$993$	18.0000	0.571213
$994$	0	0
$995$	−8.00000	−0.253617
$996$	−1.00000	−0.0316862
$997$	32.0000	1.01345	0.506725	−	0.862108i	$-0.330856\pi$
$997$	32.0000	1.01345	0.506725	+	0.862108i	$0.330856\pi$
$998$	−6.00000	−0.189927
$999$	4.00000	0.126554

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.i.k.205.1	✓	2	7.4	even	3
714.2.i.k.613.1	yes	2	7.2	even	3
4998.2.a.g.1.1		1	1.1	even	1	trivial
4998.2.a.n.1.1		1	7.6	odd	2

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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