LMFDB - Embedded newform 4998.2.a.bk.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:	$0$
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.00000 q^{11} +1.00000 q^{12} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +3.00000 q^{19} -1.00000 q^{20} -6.00000 q^{22} +7.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +7.00000 q^{37} +3.00000 q^{38} -1.00000 q^{40} -9.00000 q^{43} -6.00000 q^{44} -1.00000 q^{45} +7.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -1.00000 q^{51} +2.00000 q^{53} +1.00000 q^{54} +6.00000 q^{55} +3.00000 q^{57} +6.00000 q^{58} +11.0000 q^{59} -1.00000 q^{60} +6.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -6.00000 q^{66} +9.00000 q^{67} -1.00000 q^{68} +7.00000 q^{69} -9.00000 q^{71} +1.00000 q^{72} +12.0000 q^{73} +7.00000 q^{74} -4.00000 q^{75} +3.00000 q^{76} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{83} +1.00000 q^{85} -9.00000 q^{86} +6.00000 q^{87} -6.00000 q^{88} -5.00000 q^{89} -1.00000 q^{90} +7.00000 q^{92} +8.00000 q^{93} +6.00000 q^{94} -3.00000 q^{95} +1.00000 q^{96} +4.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
$5$	−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
$11$	−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
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	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
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−6.00000	−1.27920
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−1.00000	−0.182574
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	−6.00000	−1.04447
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	−1.00000	−0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	−6.00000	−0.904534
$45$	−1.00000	−0.149071
$46$	7.00000	1.03209
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−1.00000	−0.140028
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	1.00000	0.136083
$55$	6.00000	0.809040
$56$	0	0
$57$	3.00000	0.397360
$58$	6.00000	0.787839
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	−1.00000	−0.129099
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	9.00000	1.09952	0.549762	−	0.835321i	$-0.314718\pi$
$67$	9.00000	1.09952	0.549762	+	0.835321i	$0.314718\pi$
$68$	−1.00000	−0.121268
$69$	7.00000	0.842701
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	1.00000	0.117851
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	7.00000	0.813733
$75$	−4.00000	−0.461880
$76$	3.00000	0.344124
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	−9.00000	−0.970495
$87$	6.00000	0.643268
$88$	−6.00000	−0.639602
$89$	−5.00000	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	7.00000	0.729800
$93$	8.00000	0.829561
$94$	6.00000	0.618853
$95$	−3.00000	−0.307794
$96$	1.00000	0.102062
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	−4.00000	−0.400000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	−1.00000	−0.0990148
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	2.00000	0.194257
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	1.00000	0.0962250
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	6.00000	0.572078
$111$	7.00000	0.664411
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	3.00000	0.280976
$115$	−7.00000	−0.652753
$116$	6.00000	0.557086
$117$	0	0
$118$	11.0000	1.01263
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	25.0000	2.27273
$122$	6.00000	0.543214
$123$	0	0
$124$	8.00000	0.718421
$125$	9.00000	0.804984
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	1.00000	0.0883883
$129$	−9.00000	−0.792406
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	9.00000	0.777482
$135$	−1.00000	−0.0860663
$136$	−1.00000	−0.0857493
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	7.00000	0.595880
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	−9.00000	−0.755263
$143$	0	0
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	12.0000	0.993127
$147$	0	0
$148$	7.00000	0.575396
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−4.00000	−0.326599
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	3.00000	0.243332
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−8.00000	−0.636446
$159$	2.00000	0.158610
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	6.00000	0.467099
$166$	−4.00000	−0.310460
$167$	−19.0000	−1.47026	−0.735132	−	0.677924i	$-0.762880\pi$
$167$	−19.0000	−1.47026	−0.735132	+	0.677924i	$0.762880\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	1.00000	0.0766965
$171$	3.00000	0.229416
$172$	−9.00000	−0.686244
$173$	11.0000	0.836315	0.418157	−	0.908375i	$-0.362676\pi$
$173$	11.0000	0.836315	0.418157	+	0.908375i	$0.362676\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−6.00000	−0.452267
$177$	11.0000	0.826811
$178$	−5.00000	−0.374766
$179$	1.00000	0.0747435	0.0373718	−	0.999301i	$-0.488101\pi$
$179$	1.00000	0.0747435	0.0373718	+	0.999301i	$0.488101\pi$
$180$	−1.00000	−0.0745356
$181$	−11.0000	−0.817624	−0.408812	−	0.912619i	$-0.634057\pi$
$181$	−11.0000	−0.817624	−0.408812	+	0.912619i	$0.634057\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	7.00000	0.516047
$185$	−7.00000	−0.514650
$186$	8.00000	0.586588
$187$	6.00000	0.438763
$188$	6.00000	0.437595
$189$	0	0
$190$	−3.00000	−0.217643
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	1.00000	0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	0	0
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	−6.00000	−0.426401
$199$	15.0000	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$200$	−4.00000	−0.282843
$201$	9.00000	0.634811
$202$	12.0000	0.844317
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	14.0000	0.975426
$207$	7.00000	0.486534
$208$	0	0
$209$	−18.0000	−1.24509
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	2.00000	0.137361
$213$	−9.00000	−0.616670
$214$	−8.00000	−0.546869
$215$	9.00000	0.613795
$216$	1.00000	0.0680414
$217$	0	0
$218$	9.00000	0.609557
$219$	12.0000	0.810885
$220$	6.00000	0.404520
$221$	0	0
$222$	7.00000	0.469809
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	−6.00000	−0.399114
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	3.00000	0.198680
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	−7.00000	−0.461566
$231$	0	0
$232$	6.00000	0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	11.0000	0.716039
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	−1.00000	−0.0645497
$241$	−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
$244$	6.00000	0.384111
$245$	0	0
$246$	0	0
$247$	0	0
$248$	8.00000	0.508001
$249$	−4.00000	−0.253490
$250$	9.00000	0.569210
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−42.0000	−2.64052
$254$	20.0000	1.25491
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	−9.00000	−0.560316
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	10.0000	0.617802
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	−6.00000	−0.369274
$265$	−2.00000	−0.122859
$266$	0	0
$267$	−5.00000	−0.305995
$268$	9.00000	0.549762
$269$	−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
$271$	18.0000	1.09342	0.546711	−	0.837321i	$-0.315880\pi$
$271$	18.0000	1.09342	0.546711	+	0.837321i	$0.315880\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−13.0000	−0.785359
$275$	24.0000	1.44725
$276$	7.00000	0.421350
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	−8.00000	−0.479808
$279$	8.00000	0.478947
$280$	0	0
$281$	−31.0000	−1.84930	−0.924652	−	0.380812i	$-0.875644\pi$
$281$	−31.0000	−1.84930	−0.924652	+	0.380812i	$0.875644\pi$
$282$	6.00000	0.357295
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	−9.00000	−0.534052
$285$	−3.00000	−0.177705
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	−6.00000	−0.352332
$291$	4.00000	0.234484
$292$	12.0000	0.702247
$293$	32.0000	1.86946	0.934730	−	0.355359i	$-0.115641\pi$
$293$	32.0000	1.86946	0.934730	+	0.355359i	$0.115641\pi$
$294$	0	0
$295$	−11.0000	−0.640445
$296$	7.00000	0.406867
$297$	−6.00000	−0.348155
$298$	−18.0000	−1.04271
$299$	0	0
$300$	−4.00000	−0.230940
$301$	0	0
$302$	8.00000	0.460348
$303$	12.0000	0.689382
$304$	3.00000	0.172062
$305$	−6.00000	−0.343559
$306$	−1.00000	−0.0571662
$307$	5.00000	0.285365	0.142683	−	0.989769i	$-0.454427\pi$
$307$	5.00000	0.285365	0.142683	+	0.989769i	$0.454427\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	−8.00000	−0.454369
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	2.00000	0.112154
$319$	−36.0000	−2.01561
$320$	−1.00000	−0.0559017
$321$	−8.00000	−0.446516
$322$	0	0
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	9.00000	0.497701
$328$	0	0
$329$	0	0
$330$	6.00000	0.330289
$331$	−15.0000	−0.824475	−0.412237	−	0.911077i	$-0.635253\pi$
$331$	−15.0000	−0.824475	−0.412237	+	0.911077i	$0.635253\pi$
$332$	−4.00000	−0.219529
$333$	7.00000	0.383598
$334$	−19.0000	−1.03963
$335$	−9.00000	−0.491723
$336$	0	0
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	−13.0000	−0.707107
$339$	−6.00000	−0.325875
$340$	1.00000	0.0542326
$341$	−48.0000	−2.59935
$342$	3.00000	0.162221
$343$	0	0
$344$	−9.00000	−0.485247
$345$	−7.00000	−0.376867
$346$	11.0000	0.591364
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	6.00000	0.321634
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	0	0
$351$	0	0
$352$	−6.00000	−0.319801
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	11.0000	0.584643
$355$	9.00000	0.477670
$356$	−5.00000	−0.264999
$357$	0	0
$358$	1.00000	0.0528516
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	−1.00000	−0.0527046
$361$	−10.0000	−0.526316
$362$	−11.0000	−0.578147
$363$	25.0000	1.31216
$364$	0	0
$365$	−12.0000	−0.628109
$366$	6.00000	0.313625
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	7.00000	0.364900
$369$	0	0
$370$	−7.00000	−0.363913
$371$	0	0
$372$	8.00000	0.414781
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	6.00000	0.310253
$375$	9.00000	0.464758
$376$	6.00000	0.309426
$377$	0	0
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	−3.00000	−0.153897
$381$	20.0000	1.02463
$382$	−20.0000	−1.02329
$383$	34.0000	1.73732	0.868659	−	0.495410i	$-0.164982\pi$
$383$	34.0000	1.73732	0.868659	+	0.495410i	$0.164982\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	−9.00000	−0.457496
$388$	4.00000	0.203069
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	−7.00000	−0.354005
$392$	0	0
$393$	10.0000	0.504433
$394$	−9.00000	−0.453413
$395$	8.00000	0.402524
$396$	−6.00000	−0.301511
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	15.0000	0.751882
$399$	0	0
$400$	−4.00000	−0.200000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	9.00000	0.448879
$403$	0	0
$404$	12.0000	0.597022
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−42.0000	−2.08186
$408$	−1.00000	−0.0495074
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	−13.0000	−0.641243
$412$	14.0000	0.689730
$413$	0	0
$414$	7.00000	0.344031
$415$	4.00000	0.196352
$416$	0	0
$417$	−8.00000	−0.391762
$418$	−18.0000	−0.880409
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	2.00000	0.0973585
$423$	6.00000	0.291730
$424$	2.00000	0.0971286
$425$	4.00000	0.194029
$426$	−9.00000	−0.436051
$427$	0	0
$428$	−8.00000	−0.386695
$429$	0	0
$430$	9.00000	0.434019
$431$	23.0000	1.10787	0.553936	−	0.832560i	$-0.313125\pi$
$431$	23.0000	1.10787	0.553936	+	0.832560i	$0.313125\pi$
$432$	1.00000	0.0481125
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	9.00000	0.431022
$437$	21.0000	1.00457
$438$	12.0000	0.573382
$439$	29.0000	1.38409	0.692047	−	0.721852i	$-0.256709\pi$
$439$	29.0000	1.38409	0.692047	+	0.721852i	$0.256709\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	0	0
$443$	−5.00000	−0.237557	−0.118779	−	0.992921i	$-0.537898\pi$
$443$	−5.00000	−0.237557	−0.118779	+	0.992921i	$0.537898\pi$
$444$	7.00000	0.332205
$445$	5.00000	0.237023
$446$	−4.00000	−0.189405
$447$	−18.0000	−0.851371
$448$	0	0
$449$	16.0000	0.755087	0.377543	−	0.925992i	$-0.376769\pi$
$449$	16.0000	0.755087	0.377543	+	0.925992i	$0.376769\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	−6.00000	−0.282216
$453$	8.00000	0.375873
$454$	−14.0000	−0.657053
$455$	0	0
$456$	3.00000	0.140488
$457$	−35.0000	−1.63723	−0.818615	−	0.574342i	$-0.805258\pi$
$457$	−35.0000	−1.63723	−0.818615	+	0.574342i	$0.805258\pi$
$458$	20.0000	0.934539
$459$	−1.00000	−0.0466760
$460$	−7.00000	−0.326377
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−28.0000	−1.30127	−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000	−1.30127	−0.650635	+	0.759390i	$0.725497\pi$
$464$	6.00000	0.278543
$465$	−8.00000	−0.370991
$466$	18.0000	0.833834
$467$	−35.0000	−1.61961	−0.809803	−	0.586701i	$-0.800426\pi$
$467$	−35.0000	−1.61961	−0.809803	+	0.586701i	$0.800426\pi$
$468$	0	0
$469$	0	0
$470$	−6.00000	−0.276759
$471$	−18.0000	−0.829396
$472$	11.0000	0.506316
$473$	54.0000	2.48292
$474$	−8.00000	−0.367452
$475$	−12.0000	−0.550598
$476$	0	0
$477$	2.00000	0.0915737
$478$	−12.0000	−0.548867
$479$	17.0000	0.776750	0.388375	−	0.921501i	$-0.373037\pi$
$479$	17.0000	0.776750	0.388375	+	0.921501i	$0.373037\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−8.00000	−0.364390
$483$	0	0
$484$	25.0000	1.13636
$485$	−4.00000	−0.181631
$486$	1.00000	0.0453609
$487$	5.00000	0.226572	0.113286	−	0.993562i	$-0.463862\pi$
$487$	5.00000	0.226572	0.113286	+	0.993562i	$0.463862\pi$
$488$	6.00000	0.271607
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−35.0000	−1.57953	−0.789764	−	0.613411i	$-0.789797\pi$
$491$	−35.0000	−1.57953	−0.789764	+	0.613411i	$0.789797\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	6.00000	0.269680
$496$	8.00000	0.359211
$497$	0	0
$498$	−4.00000	−0.179244
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	9.00000	0.402492
$501$	−19.0000	−0.848857
$502$	12.0000	0.535586
$503$	33.0000	1.47140	0.735699	−	0.677309i	$-0.236854\pi$
$503$	33.0000	1.47140	0.735699	+	0.677309i	$0.236854\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	−42.0000	−1.86713
$507$	−13.0000	−0.577350
$508$	20.0000	0.887357
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	1.00000	0.0442807
$511$	0	0
$512$	1.00000	0.0441942
$513$	3.00000	0.132453
$514$	−27.0000	−1.19092
$515$	−14.0000	−0.616914
$516$	−9.00000	−0.396203
$517$	−36.0000	−1.58328
$518$	0	0
$519$	11.0000	0.482846
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	6.00000	0.262613
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	10.0000	0.436852
$525$	0	0
$526$	14.0000	0.610429
$527$	−8.00000	−0.348485
$528$	−6.00000	−0.261116
$529$	26.0000	1.13043
$530$	−2.00000	−0.0868744
$531$	11.0000	0.477359
$532$	0	0
$533$	0	0
$534$	−5.00000	−0.216371
$535$	8.00000	0.345870
$536$	9.00000	0.388741
$537$	1.00000	0.0431532
$538$	−9.00000	−0.388018
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	18.0000	0.773166
$543$	−11.0000	−0.472055
$544$	−1.00000	−0.0428746
$545$	−9.00000	−0.385518
$546$	0	0
$547$	10.0000	0.427569	0.213785	−	0.976881i	$-0.431421\pi$
$547$	10.0000	0.427569	0.213785	+	0.976881i	$0.431421\pi$
$548$	−13.0000	−0.555332
$549$	6.00000	0.256074
$550$	24.0000	1.02336
$551$	18.0000	0.766826
$552$	7.00000	0.297940
$553$	0	0
$554$	6.00000	0.254916
$555$	−7.00000	−0.297133
$556$	−8.00000	−0.339276
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	8.00000	0.338667
$559$	0	0
$560$	0	0
$561$	6.00000	0.253320
$562$	−31.0000	−1.30766
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	6.00000	0.252646
$565$	6.00000	0.252422
$566$	−32.0000	−1.34506
$567$	0	0
$568$	−9.00000	−0.377632
$569$	−9.00000	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$570$	−3.00000	−0.125656
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	0	0
$573$	−20.0000	−0.835512
$574$	0	0
$575$	−28.0000	−1.16768
$576$	1.00000	0.0416667
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	1.00000	0.0415945
$579$	−6.00000	−0.249351
$580$	−6.00000	−0.249136
$581$	0	0
$582$	4.00000	0.165805
$583$	−12.0000	−0.496989
$584$	12.0000	0.496564
$585$	0	0
$586$	32.0000	1.32191
$587$	−13.0000	−0.536567	−0.268284	−	0.963340i	$-0.586456\pi$
$587$	−13.0000	−0.536567	−0.268284	+	0.963340i	$0.586456\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	−11.0000	−0.452863
$591$	−9.00000	−0.370211
$592$	7.00000	0.287698
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	−6.00000	−0.246183
$595$	0	0
$596$	−18.0000	−0.737309
$597$	15.0000	0.613909
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	−4.00000	−0.163299
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	9.00000	0.366508
$604$	8.00000	0.325515
$605$	−25.0000	−1.01639
$606$	12.0000	0.487467
$607$	27.0000	1.09590	0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	3.00000	0.121666
$609$	0	0
$610$	−6.00000	−0.242933
$611$	0	0
$612$	−1.00000	−0.0404226
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	5.00000	0.201784
$615$	0	0
$616$	0	0
$617$	46.0000	1.85189	0.925945	−	0.377658i	$-0.123271\pi$
$617$	46.0000	1.85189	0.925945	+	0.377658i	$0.123271\pi$
$618$	14.0000	0.563163
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	−8.00000	−0.321288
$621$	7.00000	0.280900
$622$	21.0000	0.842023
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	10.0000	0.399680
$627$	−18.0000	−0.718851
$628$	−18.0000	−0.718278
$629$	−7.00000	−0.279108
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	−8.00000	−0.318223
$633$	2.00000	0.0794929
$634$	3.00000	0.119145
$635$	−20.0000	−0.793676
$636$	2.00000	0.0793052
$637$	0	0
$638$	−36.0000	−1.42525
$639$	−9.00000	−0.356034
$640$	−1.00000	−0.0395285
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	−8.00000	−0.315735
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	0	0
$645$	9.00000	0.354375
$646$	−3.00000	−0.118033
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	1.00000	0.0392837
$649$	−66.0000	−2.59073
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−39.0000	−1.52619	−0.763094	−	0.646288i	$-0.776321\pi$
$653$	−39.0000	−1.52619	−0.763094	+	0.646288i	$0.776321\pi$
$654$	9.00000	0.351928
$655$	−10.0000	−0.390732
$656$	0	0
$657$	12.0000	0.468165
$658$	0	0
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	6.00000	0.233550
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−15.0000	−0.582992
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	7.00000	0.271244
$667$	42.0000	1.62625
$668$	−19.0000	−0.735132
$669$	−4.00000	−0.154649
$670$	−9.00000	−0.347700
$671$	−36.0000	−1.38976
$672$	0	0
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	20.0000	0.770371
$675$	−4.00000	−0.153960
$676$	−13.0000	−0.500000
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	1.00000	0.0383482
$681$	−14.0000	−0.536481
$682$	−48.0000	−1.83801
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	3.00000	0.114708
$685$	13.0000	0.496704
$686$	0	0
$687$	20.0000	0.763048
$688$	−9.00000	−0.343122
$689$	0	0
$690$	−7.00000	−0.266485
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	11.0000	0.418157
$693$	0	0
$694$	12.0000	0.455514
$695$	8.00000	0.303457
$696$	6.00000	0.227429
$697$	0	0
$698$	−8.00000	−0.302804
$699$	18.0000	0.680823
$700$	0	0
$701$	−16.0000	−0.604312	−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000	−0.604312	−0.302156	+	0.953259i	$0.597706\pi$
$702$	0	0
$703$	21.0000	0.792030
$704$	−6.00000	−0.226134
$705$	−6.00000	−0.225973
$706$	9.00000	0.338719
$707$	0	0
$708$	11.0000	0.413405
$709$	5.00000	0.187779	0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000	0.187779	0.0938895	+	0.995583i	$0.470070\pi$
$710$	9.00000	0.337764
$711$	−8.00000	−0.300023
$712$	−5.00000	−0.187383
$713$	56.0000	2.09722
$714$	0	0
$715$	0	0
$716$	1.00000	0.0373718
$717$	−12.0000	−0.448148
$718$	10.0000	0.373197
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−10.0000	−0.372161
$723$	−8.00000	−0.297523
$724$	−11.0000	−0.408812
$725$	−24.0000	−0.891338
$726$	25.0000	0.927837
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−12.0000	−0.444140
$731$	9.00000	0.332877
$732$	6.00000	0.221766
$733$	−16.0000	−0.590973	−0.295487	−	0.955347i	$-0.595482\pi$
$733$	−16.0000	−0.590973	−0.295487	+	0.955347i	$0.595482\pi$
$734$	−17.0000	−0.627481
$735$	0	0
$736$	7.00000	0.258023
$737$	−54.0000	−1.98912
$738$	0	0
$739$	−19.0000	−0.698926	−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000	−0.698926	−0.349463	+	0.936950i	$0.613636\pi$
$740$	−7.00000	−0.257325
$741$	0	0
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	8.00000	0.293294
$745$	18.0000	0.659469
$746$	34.0000	1.24483
$747$	−4.00000	−0.146352
$748$	6.00000	0.219382
$749$	0	0
$750$	9.00000	0.328634
$751$	1.00000	0.0364905	0.0182453	−	0.999834i	$-0.494192\pi$
$751$	1.00000	0.0364905	0.0182453	+	0.999834i	$0.494192\pi$
$752$	6.00000	0.218797
$753$	12.0000	0.437304
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	12.0000	0.436147	0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000	0.436147	0.218074	+	0.975932i	$0.430023\pi$
$758$	16.0000	0.581146
$759$	−42.0000	−1.52450
$760$	−3.00000	−0.108821
$761$	25.0000	0.906249	0.453125	−	0.891447i	$-0.350309\pi$
$761$	25.0000	0.906249	0.453125	+	0.891447i	$0.350309\pi$
$762$	20.0000	0.724524
$763$	0	0
$764$	−20.0000	−0.723575
$765$	1.00000	0.0361551
$766$	34.0000	1.22847
$767$	0	0
$768$	1.00000	0.0360844
$769$	13.0000	0.468792	0.234396	−	0.972141i	$-0.424689\pi$
$769$	13.0000	0.468792	0.234396	+	0.972141i	$0.424689\pi$
$770$	0	0
$771$	−27.0000	−0.972381
$772$	−6.00000	−0.215945
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	−9.00000	−0.323498
$775$	−32.0000	−1.14947
$776$	4.00000	0.143592
$777$	0	0
$778$	−24.0000	−0.860442
$779$	0	0
$780$	0	0
$781$	54.0000	1.93227
$782$	−7.00000	−0.250319
$783$	6.00000	0.214423
$784$	0	0
$785$	18.0000	0.642448
$786$	10.0000	0.356688
$787$	−26.0000	−0.926800	−0.463400	−	0.886149i	$-0.653371\pi$
$787$	−26.0000	−0.926800	−0.463400	+	0.886149i	$0.653371\pi$
$788$	−9.00000	−0.320612
$789$	14.0000	0.498413
$790$	8.00000	0.284627
$791$	0	0
$792$	−6.00000	−0.213201
$793$	0	0
$794$	−3.00000	−0.106466
$795$	−2.00000	−0.0709327
$796$	15.0000	0.531661
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	−4.00000	−0.141421
$801$	−5.00000	−0.176666
$802$	2.00000	0.0706225
$803$	−72.0000	−2.54082
$804$	9.00000	0.317406
$805$	0	0
$806$	0	0
$807$	−9.00000	−0.316815
$808$	12.0000	0.422159
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	−1.00000	−0.0351364
$811$	−30.0000	−1.05344	−0.526721	−	0.850038i	$-0.676579\pi$
$811$	−30.0000	−1.05344	−0.526721	+	0.850038i	$0.676579\pi$
$812$	0	0
$813$	18.0000	0.631288
$814$	−42.0000	−1.47210
$815$	4.00000	0.140114
$816$	−1.00000	−0.0350070
$817$	−27.0000	−0.944610
$818$	14.0000	0.489499
$819$	0	0
$820$	0	0
$821$	−25.0000	−0.872506	−0.436253	−	0.899824i	$-0.643695\pi$
$821$	−25.0000	−0.872506	−0.436253	+	0.899824i	$0.643695\pi$
$822$	−13.0000	−0.453427
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	14.0000	0.487713
$825$	24.0000	0.835573
$826$	0	0
$827$	−2.00000	−0.0695468	−0.0347734	−	0.999395i	$-0.511071\pi$
$827$	−2.00000	−0.0695468	−0.0347734	+	0.999395i	$0.511071\pi$
$828$	7.00000	0.243267
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	4.00000	0.138842
$831$	6.00000	0.208138
$832$	0	0
$833$	0	0
$834$	−8.00000	−0.277017
$835$	19.0000	0.657522
$836$	−18.0000	−0.622543
$837$	8.00000	0.276520
$838$	0	0
$839$	−49.0000	−1.69167	−0.845834	−	0.533446i	$-0.820897\pi$
$839$	−49.0000	−1.69167	−0.845834	+	0.533446i	$0.820897\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	8.00000	0.275698
$843$	−31.0000	−1.06770
$844$	2.00000	0.0688428
$845$	13.0000	0.447214
$846$	6.00000	0.206284
$847$	0	0
$848$	2.00000	0.0686803
$849$	−32.0000	−1.09824
$850$	4.00000	0.137199
$851$	49.0000	1.67970
$852$	−9.00000	−0.308335
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	0	0
$855$	−3.00000	−0.102598
$856$	−8.00000	−0.273434
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	23.0000	0.784750	0.392375	−	0.919805i	$-0.371654\pi$
$859$	23.0000	0.784750	0.392375	+	0.919805i	$0.371654\pi$
$860$	9.00000	0.306897
$861$	0	0
$862$	23.0000	0.783383
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	1.00000	0.0340207
$865$	−11.0000	−0.374011
$866$	19.0000	0.645646
$867$	1.00000	0.0339618
$868$	0	0
$869$	48.0000	1.62829
$870$	−6.00000	−0.203419
$871$	0	0
$872$	9.00000	0.304778
$873$	4.00000	0.135379
$874$	21.0000	0.710336
$875$	0	0
$876$	12.0000	0.405442
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	29.0000	0.978703
$879$	32.0000	1.07933
$880$	6.00000	0.202260
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	0	0
$883$	−40.0000	−1.34611	−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000	−1.34611	−0.673054	+	0.739594i	$0.735018\pi$
$884$	0	0
$885$	−11.0000	−0.369761
$886$	−5.00000	−0.167978
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	7.00000	0.234905
$889$	0	0
$890$	5.00000	0.167600
$891$	−6.00000	−0.201008
$892$	−4.00000	−0.133930
$893$	18.0000	0.602347
$894$	−18.0000	−0.602010
$895$	−1.00000	−0.0334263
$896$	0	0
$897$	0	0
$898$	16.0000	0.533927
$899$	48.0000	1.60089
$900$	−4.00000	−0.133333
$901$	−2.00000	−0.0666297
$902$	0	0
$903$	0	0
$904$	−6.00000	−0.199557
$905$	11.0000	0.365652
$906$	8.00000	0.265782
$907$	−54.0000	−1.79304	−0.896520	−	0.443003i	$-0.853913\pi$
$907$	−54.0000	−1.79304	−0.896520	+	0.443003i	$0.853913\pi$
$908$	−14.0000	−0.464606
$909$	12.0000	0.398015
$910$	0	0
$911$	19.0000	0.629498	0.314749	−	0.949175i	$-0.398080\pi$
$911$	19.0000	0.629498	0.314749	+	0.949175i	$0.398080\pi$
$912$	3.00000	0.0993399
$913$	24.0000	0.794284
$914$	−35.0000	−1.15770
$915$	−6.00000	−0.198354
$916$	20.0000	0.660819
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	−7.00000	−0.230783
$921$	5.00000	0.164756
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	−28.0000	−0.920634
$926$	−28.0000	−0.920137
$927$	14.0000	0.459820
$928$	6.00000	0.196960
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	−8.00000	−0.262330
$931$	0	0
$932$	18.0000	0.589610
$933$	21.0000	0.687509
$934$	−35.0000	−1.14523
$935$	−6.00000	−0.196221
$936$	0	0
$937$	31.0000	1.01273	0.506363	−	0.862320i	$-0.330990\pi$
$937$	31.0000	1.01273	0.506363	+	0.862320i	$0.330990\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	−6.00000	−0.195698
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	−18.0000	−0.586472
$943$	0	0
$944$	11.0000	0.358020
$945$	0	0
$946$	54.0000	1.75569
$947$	34.0000	1.10485	0.552426	−	0.833562i	$-0.313702\pi$
$947$	34.0000	1.10485	0.552426	+	0.833562i	$0.313702\pi$
$948$	−8.00000	−0.259828
$949$	0	0
$950$	−12.0000	−0.389331
$951$	3.00000	0.0972817
$952$	0	0
$953$	−9.00000	−0.291539	−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000	−0.291539	−0.145769	+	0.989319i	$0.546566\pi$
$954$	2.00000	0.0647524
$955$	20.0000	0.647185
$956$	−12.0000	−0.388108
$957$	−36.0000	−1.16371
$958$	17.0000	0.549245
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	33.0000	1.06452
$962$	0	0
$963$	−8.00000	−0.257796
$964$	−8.00000	−0.257663
$965$	6.00000	0.193147
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	25.0000	0.803530
$969$	−3.00000	−0.0963739
$970$	−4.00000	−0.128432
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	5.00000	0.160210
$975$	0	0
$976$	6.00000	0.192055
$977$	3.00000	0.0959785	0.0479893	−	0.998848i	$-0.484719\pi$
$977$	3.00000	0.0959785	0.0479893	+	0.998848i	$0.484719\pi$
$978$	−4.00000	−0.127906
$979$	30.0000	0.958804
$980$	0	0
$981$	9.00000	0.287348
$982$	−35.0000	−1.11689
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	−6.00000	−0.191079
$987$	0	0
$988$	0	0
$989$	−63.0000	−2.00328
$990$	6.00000	0.190693
$991$	25.0000	0.794151	0.397076	−	0.917786i	$-0.370025\pi$
$991$	25.0000	0.794151	0.397076	+	0.917786i	$0.370025\pi$
$992$	8.00000	0.254000
$993$	−15.0000	−0.476011
$994$	0	0
$995$	−15.0000	−0.475532
$996$	−4.00000	−0.126745
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	32.0000	1.01294
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4998.2.a.bk.1.1		1
7.3	odd	6		714.2.i.e.205.1	✓	2
7.5	odd	6		714.2.i.e.613.1	yes	2
7.6	odd	2		4998.2.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.i.e.205.1	✓	2	7.3	odd	6
714.2.i.e.613.1	yes	2	7.5	odd	6
4998.2.a.ba.1.1		1	7.6	odd	2
4998.2.a.bk.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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