LMFDB - Embedded newform 498.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [498,2,Mod(1,498)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(498, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("498.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 498 = 2 \cdot 3 \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	498.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:	$3.97655002066$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	498.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -3.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} -3.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +4.00000 q^{29} +1.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +4.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} +3.00000 q^{37} +3.00000 q^{38} -6.00000 q^{39} +1.00000 q^{40} +6.00000 q^{41} +4.00000 q^{42} -12.0000 q^{43} +3.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} +1.00000 q^{48} +9.00000 q^{49} +4.00000 q^{50} -4.00000 q^{51} -6.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} -3.00000 q^{55} +4.00000 q^{56} -3.00000 q^{57} -4.00000 q^{58} -7.00000 q^{59} -1.00000 q^{60} -1.00000 q^{61} +2.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -3.00000 q^{66} +7.00000 q^{67} -4.00000 q^{68} -1.00000 q^{69} -4.00000 q^{70} -1.00000 q^{72} +4.00000 q^{73} -3.00000 q^{74} -4.00000 q^{75} -3.00000 q^{76} -12.0000 q^{77} +6.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +1.00000 q^{83} -4.00000 q^{84} +4.00000 q^{85} +12.0000 q^{86} +4.00000 q^{87} -3.00000 q^{88} +5.00000 q^{89} +1.00000 q^{90} +24.0000 q^{91} -1.00000 q^{92} -2.00000 q^{93} +3.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} -9.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000	0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	4.00000	1.06904
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−1.00000	−0.235702
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	−1.00000	−0.223607
$21$	−4.00000	−0.872872
$22$	−3.00000	−0.639602
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	1.00000	0.182574
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	4.00000	0.685994
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	3.00000	0.486664
$39$	−6.00000	−0.960769
$40$	1.00000	0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	4.00000	0.617213
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	3.00000	0.452267
$45$	−1.00000	−0.149071
$46$	1.00000	0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	4.00000	0.565685
$51$	−4.00000	−0.560112
$52$	−6.00000	−0.832050
$53$	−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
$56$	4.00000	0.534522
$57$	−3.00000	−0.397360
$58$	−4.00000	−0.525226
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	−1.00000	−0.129099
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	2.00000	0.254000
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	−3.00000	−0.369274
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	−4.00000	−0.485071
$69$	−1.00000	−0.120386
$70$	−4.00000	−0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−3.00000	−0.348743
$75$	−4.00000	−0.461880
$76$	−3.00000	−0.344124
$77$	−12.0000	−1.36753
$78$	6.00000	0.679366
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	1.00000	0.109764
$83$	1.00000	0.109764
$84$	−4.00000	−0.436436
$85$	4.00000	0.433861
$86$	12.0000	1.29399
$87$	4.00000	0.428845
$88$	−3.00000	−0.319801
$89$	5.00000	0.529999	0.264999	−	0.964249i	$-0.414628\pi$
$89$	5.00000	0.529999	0.264999	+	0.964249i	$0.414628\pi$
$90$	1.00000	0.105409
$91$	24.0000	2.51588
$92$	−1.00000	−0.104257
$93$	−2.00000	−0.207390
$94$	0	0
$95$	3.00000	0.307794
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−9.00000	−0.909137
$99$	3.00000	0.301511
$100$	−4.00000	−0.400000
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	4.00000	0.396059
$103$	−19.0000	−1.87213	−0.936063	−	0.351833i	$-0.885559\pi$
$103$	−19.0000	−1.87213	−0.936063	+	0.351833i	$0.885559\pi$
$104$	6.00000	0.588348
$105$	4.00000	0.390360
$106$	9.00000	0.874157
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	1.00000	0.0962250
$109$	15.0000	1.43674	0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000	1.43674	0.718370	+	0.695662i	$0.244889\pi$
$110$	3.00000	0.286039
$111$	3.00000	0.284747
$112$	−4.00000	−0.377964
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	3.00000	0.280976
$115$	1.00000	0.0932505
$116$	4.00000	0.371391
$117$	−6.00000	−0.554700
$118$	7.00000	0.644402
$119$	16.0000	1.46672
$120$	1.00000	0.0912871
$121$	−2.00000	−0.181818
$122$	1.00000	0.0905357
$123$	6.00000	0.541002
$124$	−2.00000	−0.179605
$125$	9.00000	0.804984
$126$	4.00000	0.356348
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	−1.00000	−0.0883883
$129$	−12.0000	−1.05654
$130$	−6.00000	−0.526235
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	3.00000	0.261116
$133$	12.0000	1.04053
$134$	−7.00000	−0.604708
$135$	−1.00000	−0.0860663
$136$	4.00000	0.342997
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	1.00000	0.0851257
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	0	0
$143$	−18.0000	−1.50524
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	−4.00000	−0.331042
$147$	9.00000	0.742307
$148$	3.00000	0.246598
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	4.00000	0.326599
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	3.00000	0.243332
$153$	−4.00000	−0.323381
$154$	12.0000	0.966988
$155$	2.00000	0.160644
$156$	−6.00000	−0.480384
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	4.00000	0.318223
$159$	−9.00000	−0.713746
$160$	1.00000	0.0790569
$161$	4.00000	0.315244
$162$	−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	$0.284443\pi$
$164$	6.00000	0.468521
$165$	−3.00000	−0.233550
$166$	−1.00000	−0.0776151
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	4.00000	0.308607
$169$	23.0000	1.76923
$170$	−4.00000	−0.306786
$171$	−3.00000	−0.229416
$172$	−12.0000	−0.914991
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	−4.00000	−0.303239
$175$	16.0000	1.20949
$176$	3.00000	0.226134
$177$	−7.00000	−0.526152
$178$	−5.00000	−0.374766
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	−1.00000	−0.0745356
$181$	24.0000	1.78391	0.891953	−	0.452128i	$-0.149335\pi$
$181$	24.0000	1.78391	0.891953	+	0.452128i	$0.149335\pi$
$182$	−24.0000	−1.77900
$183$	−1.00000	−0.0739221
$184$	1.00000	0.0737210
$185$	−3.00000	−0.220564
$186$	2.00000	0.146647
$187$	−12.0000	−0.877527
$188$	0	0
$189$	−4.00000	−0.290957
$190$	−3.00000	−0.217643
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	−3.00000	−0.215945	−0.107972	−	0.994154i	$-0.534436\pi$
$193$	−3.00000	−0.215945	−0.107972	+	0.994154i	$0.534436\pi$
$194$	−10.0000	−0.717958
$195$	6.00000	0.429669
$196$	9.00000	0.642857
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	−3.00000	−0.213201
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	4.00000	0.282843
$201$	7.00000	0.493742
$202$	9.00000	0.633238
$203$	−16.0000	−1.12298
$204$	−4.00000	−0.280056
$205$	−6.00000	−0.419058
$206$	19.0000	1.32379
$207$	−1.00000	−0.0695048
$208$	−6.00000	−0.416025
$209$	−9.00000	−0.622543
$210$	−4.00000	−0.276026
$211$	−17.0000	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$211$	−17.0000	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$212$	−9.00000	−0.618123
$213$	0	0
$214$	−18.0000	−1.23045
$215$	12.0000	0.818393
$216$	−1.00000	−0.0680414
$217$	8.00000	0.543075
$218$	−15.0000	−1.01593
$219$	4.00000	0.270295
$220$	−3.00000	−0.202260
$221$	24.0000	1.61441
$222$	−3.00000	−0.201347
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	4.00000	0.267261
$225$	−4.00000	−0.266667
$226$	−12.0000	−0.798228
$227$	9.00000	0.597351	0.298675	−	0.954355i	$-0.403455\pi$
$227$	9.00000	0.597351	0.298675	+	0.954355i	$0.403455\pi$
$228$	−3.00000	−0.198680
$229$	−23.0000	−1.51988	−0.759941	−	0.649992i	$-0.774772\pi$
$229$	−23.0000	−1.51988	−0.759941	+	0.649992i	$0.774772\pi$
$230$	−1.00000	−0.0659380
$231$	−12.0000	−0.789542
$232$	−4.00000	−0.262613
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−7.00000	−0.455661
$237$	−4.00000	−0.259828
$238$	−16.0000	−1.03713
$239$	−26.0000	−1.68180	−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000	−1.68180	−0.840900	+	0.541190i	$0.817974\pi$
$240$	−1.00000	−0.0645497
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	2.00000	0.128565
$243$	1.00000	0.0641500
$244$	−1.00000	−0.0640184
$245$	−9.00000	−0.574989
$246$	−6.00000	−0.382546
$247$	18.0000	1.14531
$248$	2.00000	0.127000
$249$	1.00000	0.0633724
$250$	−9.00000	−0.569210
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	−4.00000	−0.251976
$253$	−3.00000	−0.188608
$254$	−10.0000	−0.627456
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	15.0000	0.935674	0.467837	−	0.883815i	$-0.345033\pi$
$257$	15.0000	0.935674	0.467837	+	0.883815i	$0.345033\pi$
$258$	12.0000	0.747087
$259$	−12.0000	−0.745644
$260$	6.00000	0.372104
$261$	4.00000	0.247594
$262$	0	0
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	−3.00000	−0.184637
$265$	9.00000	0.552866
$266$	−12.0000	−0.735767
$267$	5.00000	0.305995
$268$	7.00000	0.427593
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	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	$0.650573\pi$
$272$	−4.00000	−0.242536
$273$	24.0000	1.45255
$274$	−10.0000	−0.604122
$275$	−12.0000	−0.723627
$276$	−1.00000	−0.0601929
$277$	3.00000	0.180253	0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000	0.180253	0.0901263	+	0.995930i	$0.471273\pi$
$278$	13.0000	0.779688
$279$	−2.00000	−0.119737
$280$	−4.00000	−0.239046
$281$	7.00000	0.417585	0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000	0.417585	0.208792	+	0.977960i	$0.433047\pi$
$282$	0	0
$283$	−21.0000	−1.24832	−0.624160	−	0.781296i	$-0.714559\pi$
$283$	−21.0000	−1.24832	−0.624160	+	0.781296i	$0.714559\pi$
$284$	0	0
$285$	3.00000	0.177705
$286$	18.0000	1.06436
$287$	−24.0000	−1.41668
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	4.00000	0.234888
$291$	10.0000	0.586210
$292$	4.00000	0.234082
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−9.00000	−0.524891
$295$	7.00000	0.407556
$296$	−3.00000	−0.174371
$297$	3.00000	0.174078
$298$	14.0000	0.810998
$299$	6.00000	0.346989
$300$	−4.00000	−0.230940
$301$	48.0000	2.76667
$302$	22.0000	1.26596
$303$	−9.00000	−0.517036
$304$	−3.00000	−0.172062
$305$	1.00000	0.0572598
$306$	4.00000	0.228665
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	−12.0000	−0.683763
$309$	−19.0000	−1.08087
$310$	−2.00000	−0.113592
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	6.00000	0.339683
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−2.00000	−0.112867
$315$	4.00000	0.225374
$316$	−4.00000	−0.225018
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	9.00000	0.504695
$319$	12.0000	0.671871
$320$	−1.00000	−0.0559017
$321$	18.0000	1.00466
$322$	−4.00000	−0.222911
$323$	12.0000	0.667698
$324$	1.00000	0.0555556
$325$	24.0000	1.33128
$326$	−16.0000	−0.886158
$327$	15.0000	0.829502
$328$	−6.00000	−0.331295
$329$	0	0
$330$	3.00000	0.165145
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	1.00000	0.0548821
$333$	3.00000	0.164399
$334$	3.00000	0.164153
$335$	−7.00000	−0.382451
$336$	−4.00000	−0.218218
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−23.0000	−1.25104
$339$	12.0000	0.651751
$340$	4.00000	0.216930
$341$	−6.00000	−0.324918
$342$	3.00000	0.162221
$343$	−8.00000	−0.431959
$344$	12.0000	0.646997
$345$	1.00000	0.0538382
$346$	24.0000	1.29025
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	4.00000	0.214423
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	−16.0000	−0.855236
$351$	−6.00000	−0.320256
$352$	−3.00000	−0.159901
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	7.00000	0.372046
$355$	0	0
$356$	5.00000	0.264999
$357$	16.0000	0.846810
$358$	−6.00000	−0.317110
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	1.00000	0.0527046
$361$	−10.0000	−0.526316
$362$	−24.0000	−1.26141
$363$	−2.00000	−0.104973
$364$	24.0000	1.25794
$365$	−4.00000	−0.209370
$366$	1.00000	0.0522708
$367$	31.0000	1.61819	0.809093	−	0.587680i	$-0.199959\pi$
$367$	31.0000	1.61819	0.809093	+	0.587680i	$0.199959\pi$
$368$	−1.00000	−0.0521286
$369$	6.00000	0.312348
$370$	3.00000	0.155963
$371$	36.0000	1.86903
$372$	−2.00000	−0.103695
$373$	23.0000	1.19089	0.595447	−	0.803394i	$-0.296975\pi$
$373$	23.0000	1.19089	0.595447	+	0.803394i	$0.296975\pi$
$374$	12.0000	0.620505
$375$	9.00000	0.464758
$376$	0	0
$377$	−24.0000	−1.23606
$378$	4.00000	0.205738
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	3.00000	0.153897
$381$	10.0000	0.512316
$382$	0	0
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	−1.00000	−0.0510310
$385$	12.0000	0.611577
$386$	3.00000	0.152696
$387$	−12.0000	−0.609994
$388$	10.0000	0.507673
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	−6.00000	−0.303822
$391$	4.00000	0.202289
$392$	−9.00000	−0.454569
$393$	0	0
$394$	−24.0000	−1.20910
$395$	4.00000	0.201262
$396$	3.00000	0.150756
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	12.0000	0.601506
$399$	12.0000	0.600751
$400$	−4.00000	−0.200000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	−7.00000	−0.349128
$403$	12.0000	0.597763
$404$	−9.00000	−0.447767
$405$	−1.00000	−0.0496904
$406$	16.0000	0.794067
$407$	9.00000	0.446113
$408$	4.00000	0.198030
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	6.00000	0.296319
$411$	10.0000	0.493264
$412$	−19.0000	−0.936063
$413$	28.0000	1.37779
$414$	1.00000	0.0491473
$415$	−1.00000	−0.0490881
$416$	6.00000	0.294174
$417$	−13.0000	−0.636613
$418$	9.00000	0.440204
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	4.00000	0.195180
$421$	−24.0000	−1.16969	−0.584844	−	0.811146i	$-0.698844\pi$
$421$	−24.0000	−1.16969	−0.584844	+	0.811146i	$0.698844\pi$
$422$	17.0000	0.827547
$423$	0	0
$424$	9.00000	0.437079
$425$	16.0000	0.776114
$426$	0	0
$427$	4.00000	0.193574
$428$	18.0000	0.870063
$429$	−18.0000	−0.869048
$430$	−12.0000	−0.578691
$431$	39.0000	1.87856	0.939282	−	0.343146i	$-0.111493\pi$
$431$	39.0000	1.87856	0.939282	+	0.343146i	$0.111493\pi$
$432$	1.00000	0.0481125
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−8.00000	−0.384012
$435$	−4.00000	−0.191785
$436$	15.0000	0.718370
$437$	3.00000	0.143509
$438$	−4.00000	−0.191127
$439$	−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
$441$	9.00000	0.428571
$442$	−24.0000	−1.14156
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	3.00000	0.142374
$445$	−5.00000	−0.237023
$446$	−8.00000	−0.378811
$447$	−14.0000	−0.662177
$448$	−4.00000	−0.188982
$449$	−23.0000	−1.08544	−0.542719	−	0.839915i	$-0.682605\pi$
$449$	−23.0000	−1.08544	−0.542719	+	0.839915i	$0.682605\pi$
$450$	4.00000	0.188562
$451$	18.0000	0.847587
$452$	12.0000	0.564433
$453$	−22.0000	−1.03365
$454$	−9.00000	−0.422391
$455$	−24.0000	−1.12514
$456$	3.00000	0.140488
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	23.0000	1.07472
$459$	−4.00000	−0.186704
$460$	1.00000	0.0466252
$461$	27.0000	1.25752	0.628758	−	0.777601i	$-0.283564\pi$
$461$	27.0000	1.25752	0.628758	+	0.777601i	$0.283564\pi$
$462$	12.0000	0.558291
$463$	−18.0000	−0.836531	−0.418265	−	0.908325i	$-0.637362\pi$
$463$	−18.0000	−0.836531	−0.418265	+	0.908325i	$0.637362\pi$
$464$	4.00000	0.185695
$465$	2.00000	0.0927478
$466$	1.00000	0.0463241
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	−6.00000	−0.277350
$469$	−28.0000	−1.29292
$470$	0	0
$471$	2.00000	0.0921551
$472$	7.00000	0.322201
$473$	−36.0000	−1.65528
$474$	4.00000	0.183726
$475$	12.0000	0.550598
$476$	16.0000	0.733359
$477$	−9.00000	−0.412082
$478$	26.0000	1.18921
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	1.00000	0.0456435
$481$	−18.0000	−0.820729
$482$	−1.00000	−0.0455488
$483$	4.00000	0.182006
$484$	−2.00000	−0.0909091
$485$	−10.0000	−0.454077
$486$	−1.00000	−0.0453609
$487$	−13.0000	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$487$	−13.0000	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$488$	1.00000	0.0452679
$489$	16.0000	0.723545
$490$	9.00000	0.406579
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	6.00000	0.270501
$493$	−16.0000	−0.720604
$494$	−18.0000	−0.809858
$495$	−3.00000	−0.134840
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	−1.00000	−0.0448111
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	9.00000	0.402492
$501$	−3.00000	−0.134030
$502$	30.0000	1.33897
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	4.00000	0.178174
$505$	9.00000	0.400495
$506$	3.00000	0.133366
$507$	23.0000	1.02147
$508$	10.0000	0.443678
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	−4.00000	−0.177123
$511$	−16.0000	−0.707798
$512$	−1.00000	−0.0441942
$513$	−3.00000	−0.132453
$514$	−15.0000	−0.661622
$515$	19.0000	0.837240
$516$	−12.0000	−0.528271
$517$	0	0
$518$	12.0000	0.527250
$519$	−24.0000	−1.05348
$520$	−6.00000	−0.263117
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−4.00000	−0.175075
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	0	0
$525$	16.0000	0.698297
$526$	28.0000	1.22086
$527$	8.00000	0.348485
$528$	3.00000	0.130558
$529$	−22.0000	−0.956522
$530$	−9.00000	−0.390935
$531$	−7.00000	−0.303774
$532$	12.0000	0.520266
$533$	−36.0000	−1.55933
$534$	−5.00000	−0.216371
$535$	−18.0000	−0.778208
$536$	−7.00000	−0.302354
$537$	6.00000	0.258919
$538$	10.0000	0.431131
$539$	27.0000	1.16297
$540$	−1.00000	−0.0430331
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	15.0000	0.644305
$543$	24.0000	1.02994
$544$	4.00000	0.171499
$545$	−15.0000	−0.642529
$546$	−24.0000	−1.02711
$547$	−14.0000	−0.598597	−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000	−0.598597	−0.299298	+	0.954160i	$0.596753\pi$
$548$	10.0000	0.427179
$549$	−1.00000	−0.0426790
$550$	12.0000	0.511682
$551$	−12.0000	−0.511217
$552$	1.00000	0.0425628
$553$	16.0000	0.680389
$554$	−3.00000	−0.127458
$555$	−3.00000	−0.127343
$556$	−13.0000	−0.551323
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	2.00000	0.0846668
$559$	72.0000	3.04528
$560$	4.00000	0.169031
$561$	−12.0000	−0.506640
$562$	−7.00000	−0.295277
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	21.0000	0.882696
$567$	−4.00000	−0.167984
$568$	0	0
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	−3.00000	−0.125656
$571$	41.0000	1.71580	0.857898	−	0.513820i	$-0.171770\pi$
$571$	41.0000	1.71580	0.857898	+	0.513820i	$0.171770\pi$
$572$	−18.0000	−0.752618
$573$	0	0
$574$	24.0000	1.00174
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	1.00000	0.0415945
$579$	−3.00000	−0.124676
$580$	−4.00000	−0.166091
$581$	−4.00000	−0.165948
$582$	−10.0000	−0.414513
$583$	−27.0000	−1.11823
$584$	−4.00000	−0.165521
$585$	6.00000	0.248069
$586$	−6.00000	−0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	9.00000	0.371154
$589$	6.00000	0.247226
$590$	−7.00000	−0.288185
$591$	24.0000	0.987228
$592$	3.00000	0.123299
$593$	−20.0000	−0.821302	−0.410651	−	0.911793i	$-0.634698\pi$
$593$	−20.0000	−0.821302	−0.410651	+	0.911793i	$0.634698\pi$
$594$	−3.00000	−0.123091
$595$	−16.0000	−0.655936
$596$	−14.0000	−0.573462
$597$	−12.0000	−0.491127
$598$	−6.00000	−0.245358
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	4.00000	0.163299
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	−48.0000	−1.95633
$603$	7.00000	0.285062
$604$	−22.0000	−0.895167
$605$	2.00000	0.0813116
$606$	9.00000	0.365600
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	3.00000	0.121666
$609$	−16.0000	−0.648353
$610$	−1.00000	−0.0404888
$611$	0	0
$612$	−4.00000	−0.161690
$613$	46.0000	1.85792	0.928961	−	0.370177i	$-0.120703\pi$
$613$	46.0000	1.85792	0.928961	+	0.370177i	$0.120703\pi$
$614$	7.00000	0.282497
$615$	−6.00000	−0.241943
$616$	12.0000	0.483494
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	19.0000	0.764292
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	2.00000	0.0803219
$621$	−1.00000	−0.0401286
$622$	6.00000	0.240578
$623$	−20.0000	−0.801283
$624$	−6.00000	−0.240192
$625$	11.0000	0.440000
$626$	6.00000	0.239808
$627$	−9.00000	−0.359425
$628$	2.00000	0.0798087
$629$	−12.0000	−0.478471
$630$	−4.00000	−0.159364
$631$	−49.0000	−1.95066	−0.975330	−	0.220754i	$-0.929148\pi$
$631$	−49.0000	−1.95066	−0.975330	+	0.220754i	$0.929148\pi$
$632$	4.00000	0.159111
$633$	−17.0000	−0.675689
$634$	28.0000	1.11202
$635$	−10.0000	−0.396838
$636$	−9.00000	−0.356873
$637$	−54.0000	−2.13956
$638$	−12.0000	−0.475085
$639$	0	0
$640$	1.00000	0.0395285
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	−18.0000	−0.710403
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	4.00000	0.157622
$645$	12.0000	0.472500
$646$	−12.0000	−0.472134
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	−1.00000	−0.0392837
$649$	−21.0000	−0.824322
$650$	−24.0000	−0.941357
$651$	8.00000	0.313545
$652$	16.0000	0.626608
$653$	43.0000	1.68272	0.841360	−	0.540475i	$-0.181755\pi$
$653$	43.0000	1.68272	0.841360	+	0.540475i	$0.181755\pi$
$654$	−15.0000	−0.586546
$655$	0	0
$656$	6.00000	0.234261
$657$	4.00000	0.156055
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	−3.00000	−0.116775
$661$	34.0000	1.32245	0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000	1.32245	0.661223	+	0.750189i	$0.270038\pi$
$662$	13.0000	0.505259
$663$	24.0000	0.932083
$664$	−1.00000	−0.0388075
$665$	−12.0000	−0.465340
$666$	−3.00000	−0.116248
$667$	−4.00000	−0.154881
$668$	−3.00000	−0.116073
$669$	8.00000	0.309298
$670$	7.00000	0.270434
$671$	−3.00000	−0.115814
$672$	4.00000	0.154303
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	18.0000	0.693334
$675$	−4.00000	−0.153960
$676$	23.0000	0.884615
$677$	−1.00000	−0.0384331	−0.0192166	−	0.999815i	$-0.506117\pi$
$677$	−1.00000	−0.0384331	−0.0192166	+	0.999815i	$0.506117\pi$
$678$	−12.0000	−0.460857
$679$	−40.0000	−1.53506
$680$	−4.00000	−0.153393
$681$	9.00000	0.344881
$682$	6.00000	0.229752
$683$	26.0000	0.994862	0.497431	−	0.867503i	$-0.334277\pi$
$683$	26.0000	0.994862	0.497431	+	0.867503i	$0.334277\pi$
$684$	−3.00000	−0.114708
$685$	−10.0000	−0.382080
$686$	8.00000	0.305441
$687$	−23.0000	−0.877505
$688$	−12.0000	−0.457496
$689$	54.0000	2.05724
$690$	−1.00000	−0.0380693
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−24.0000	−0.912343
$693$	−12.0000	−0.455842
$694$	32.0000	1.21470
$695$	13.0000	0.493118
$696$	−4.00000	−0.151620
$697$	−24.0000	−0.909065
$698$	−6.00000	−0.227103
$699$	−1.00000	−0.0378235
$700$	16.0000	0.604743
$701$	40.0000	1.51078	0.755390	−	0.655276i	$-0.227448\pi$
$701$	40.0000	1.51078	0.755390	+	0.655276i	$0.227448\pi$
$702$	6.00000	0.226455
$703$	−9.00000	−0.339441
$704$	3.00000	0.113067
$705$	0	0
$706$	0	0
$707$	36.0000	1.35392
$708$	−7.00000	−0.263076
$709$	−24.0000	−0.901339	−0.450669	−	0.892691i	$-0.648815\pi$
$709$	−24.0000	−0.901339	−0.450669	+	0.892691i	$0.648815\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	−5.00000	−0.187383
$713$	2.00000	0.0749006
$714$	−16.0000	−0.598785
$715$	18.0000	0.673162
$716$	6.00000	0.224231
$717$	−26.0000	−0.970988
$718$	−32.0000	−1.19423
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−1.00000	−0.0372678
$721$	76.0000	2.83039
$722$	10.0000	0.372161
$723$	1.00000	0.0371904
$724$	24.0000	0.891953
$725$	−16.0000	−0.594225
$726$	2.00000	0.0742270
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	−24.0000	−0.889499
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	48.0000	1.77534
$732$	−1.00000	−0.0369611
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−31.0000	−1.14423
$735$	−9.00000	−0.331970
$736$	1.00000	0.0368605
$737$	21.0000	0.773545
$738$	−6.00000	−0.220863
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	−3.00000	−0.110282
$741$	18.0000	0.661247
$742$	−36.0000	−1.32160
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	2.00000	0.0733236
$745$	14.0000	0.512920
$746$	−23.0000	−0.842090
$747$	1.00000	0.0365881
$748$	−12.0000	−0.438763
$749$	−72.0000	−2.63082
$750$	−9.00000	−0.328634
$751$	6.00000	0.218943	0.109472	−	0.993990i	$-0.465084\pi$
$751$	6.00000	0.218943	0.109472	+	0.993990i	$0.465084\pi$
$752$	0	0
$753$	−30.0000	−1.09326
$754$	24.0000	0.874028
$755$	22.0000	0.800662
$756$	−4.00000	−0.145479
$757$	−9.00000	−0.327111	−0.163555	−	0.986534i	$-0.552296\pi$
$757$	−9.00000	−0.327111	−0.163555	+	0.986534i	$0.552296\pi$
$758$	20.0000	0.726433
$759$	−3.00000	−0.108893
$760$	−3.00000	−0.108821
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	−10.0000	−0.362262
$763$	−60.0000	−2.17215
$764$	0	0
$765$	4.00000	0.144620
$766$	36.0000	1.30073
$767$	42.0000	1.51653
$768$	1.00000	0.0360844
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	−12.0000	−0.432450
$771$	15.0000	0.540212
$772$	−3.00000	−0.107972
$773$	−16.0000	−0.575480	−0.287740	−	0.957709i	$-0.592904\pi$
$773$	−16.0000	−0.575480	−0.287740	+	0.957709i	$0.592904\pi$
$774$	12.0000	0.431331
$775$	8.00000	0.287368
$776$	−10.0000	−0.358979
$777$	−12.0000	−0.430498
$778$	−30.0000	−1.07555
$779$	−18.0000	−0.644917
$780$	6.00000	0.214834
$781$	0	0
$782$	−4.00000	−0.143040
$783$	4.00000	0.142948
$784$	9.00000	0.321429
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	24.0000	0.854965
$789$	−28.0000	−0.996826
$790$	−4.00000	−0.142314
$791$	−48.0000	−1.70668
$792$	−3.00000	−0.106600
$793$	6.00000	0.213066
$794$	−13.0000	−0.461353
$795$	9.00000	0.319197
$796$	−12.0000	−0.425329
$797$	35.0000	1.23976	0.619882	−	0.784695i	$-0.287181\pi$
$797$	35.0000	1.23976	0.619882	+	0.784695i	$0.287181\pi$
$798$	−12.0000	−0.424795
$799$	0	0
$800$	4.00000	0.141421
$801$	5.00000	0.176666
$802$	30.0000	1.05934
$803$	12.0000	0.423471
$804$	7.00000	0.246871
$805$	−4.00000	−0.140981
$806$	−12.0000	−0.422682
$807$	−10.0000	−0.352017
$808$	9.00000	0.316619
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	1.00000	0.0351364
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−16.0000	−0.561490
$813$	−15.0000	−0.526073
$814$	−9.00000	−0.315450
$815$	−16.0000	−0.560456
$816$	−4.00000	−0.140028
$817$	36.0000	1.25948
$818$	7.00000	0.244749
$819$	24.0000	0.838628
$820$	−6.00000	−0.209529
$821$	−7.00000	−0.244302	−0.122151	−	0.992512i	$-0.538979\pi$
$821$	−7.00000	−0.244302	−0.122151	+	0.992512i	$0.538979\pi$
$822$	−10.0000	−0.348790
$823$	−39.0000	−1.35945	−0.679727	−	0.733465i	$-0.737902\pi$
$823$	−39.0000	−1.35945	−0.679727	+	0.733465i	$0.737902\pi$
$824$	19.0000	0.661896
$825$	−12.0000	−0.417786
$826$	−28.0000	−0.974245
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	−1.00000	−0.0347524
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	1.00000	0.0347105
$831$	3.00000	0.104069
$832$	−6.00000	−0.208013
$833$	−36.0000	−1.24733
$834$	13.0000	0.450153
$835$	3.00000	0.103819
$836$	−9.00000	−0.311272
$837$	−2.00000	−0.0691301
$838$	−9.00000	−0.310900
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	−4.00000	−0.138013
$841$	−13.0000	−0.448276
$842$	24.0000	0.827095
$843$	7.00000	0.241093
$844$	−17.0000	−0.585164
$845$	−23.0000	−0.791224
$846$	0	0
$847$	8.00000	0.274883
$848$	−9.00000	−0.309061
$849$	−21.0000	−0.720718
$850$	−16.0000	−0.548795
$851$	−3.00000	−0.102839
$852$	0	0
$853$	31.0000	1.06142	0.530710	−	0.847554i	$-0.321925\pi$
$853$	31.0000	1.06142	0.530710	+	0.847554i	$0.321925\pi$
$854$	−4.00000	−0.136877
$855$	3.00000	0.102598
$856$	−18.0000	−0.615227
$857$	−4.00000	−0.136637	−0.0683187	−	0.997664i	$-0.521763\pi$
$857$	−4.00000	−0.136637	−0.0683187	+	0.997664i	$0.521763\pi$
$858$	18.0000	0.614510
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	12.0000	0.409197
$861$	−24.0000	−0.817918
$862$	−39.0000	−1.32835
$863$	45.0000	1.53182	0.765909	−	0.642949i	$-0.222289\pi$
$863$	45.0000	1.53182	0.765909	+	0.642949i	$0.222289\pi$
$864$	−1.00000	−0.0340207
$865$	24.0000	0.816024
$866$	−6.00000	−0.203888
$867$	−1.00000	−0.0339618
$868$	8.00000	0.271538
$869$	−12.0000	−0.407072
$870$	4.00000	0.135613
$871$	−42.0000	−1.42312
$872$	−15.0000	−0.507964
$873$	10.0000	0.338449
$874$	−3.00000	−0.101477
$875$	−36.0000	−1.21702
$876$	4.00000	0.135147
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	25.0000	0.843709
$879$	6.00000	0.202375
$880$	−3.00000	−0.101130
$881$	34.0000	1.14549	0.572745	−	0.819734i	$-0.305879\pi$
$881$	34.0000	1.14549	0.572745	+	0.819734i	$0.305879\pi$
$882$	−9.00000	−0.303046
$883$	13.0000	0.437485	0.218742	−	0.975783i	$-0.429805\pi$
$883$	13.0000	0.437485	0.218742	+	0.975783i	$0.429805\pi$
$884$	24.0000	0.807207
$885$	7.00000	0.235302
$886$	−39.0000	−1.31023
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	−3.00000	−0.100673
$889$	−40.0000	−1.34156
$890$	5.00000	0.167600
$891$	3.00000	0.100504
$892$	8.00000	0.267860
$893$	0	0
$894$	14.0000	0.468230
$895$	−6.00000	−0.200558
$896$	4.00000	0.133631
$897$	6.00000	0.200334
$898$	23.0000	0.767520
$899$	−8.00000	−0.266815
$900$	−4.00000	−0.133333
$901$	36.0000	1.19933
$902$	−18.0000	−0.599334
$903$	48.0000	1.59734
$904$	−12.0000	−0.399114
$905$	−24.0000	−0.797787
$906$	22.0000	0.730901
$907$	14.0000	0.464862	0.232431	−	0.972613i	$-0.425332\pi$
$907$	14.0000	0.464862	0.232431	+	0.972613i	$0.425332\pi$
$908$	9.00000	0.298675
$909$	−9.00000	−0.298511
$910$	24.0000	0.795592
$911$	−3.00000	−0.0993944	−0.0496972	−	0.998764i	$-0.515826\pi$
$911$	−3.00000	−0.0993944	−0.0496972	+	0.998764i	$0.515826\pi$
$912$	−3.00000	−0.0993399
$913$	3.00000	0.0992855
$914$	32.0000	1.05847
$915$	1.00000	0.0330590
$916$	−23.0000	−0.759941
$917$	0	0
$918$	4.00000	0.132020
$919$	5.00000	0.164935	0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000	0.164935	0.0824674	+	0.996594i	$0.473720\pi$
$920$	−1.00000	−0.0329690
$921$	−7.00000	−0.230658
$922$	−27.0000	−0.889198
$923$	0	0
$924$	−12.0000	−0.394771
$925$	−12.0000	−0.394558
$926$	18.0000	0.591517
$927$	−19.0000	−0.624042
$928$	−4.00000	−0.131306
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	−2.00000	−0.0655826
$931$	−27.0000	−0.884889
$932$	−1.00000	−0.0327561
$933$	−6.00000	−0.196431
$934$	28.0000	0.916188
$935$	12.0000	0.392442
$936$	6.00000	0.196116
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	28.0000	0.914232
$939$	−6.00000	−0.195803
$940$	0	0
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	−2.00000	−0.0651635
$943$	−6.00000	−0.195387
$944$	−7.00000	−0.227831
$945$	4.00000	0.130120
$946$	36.0000	1.17046
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	−4.00000	−0.129914
$949$	−24.0000	−0.779073
$950$	−12.0000	−0.389331
$951$	−28.0000	−0.907962
$952$	−16.0000	−0.518563
$953$	−48.0000	−1.55487	−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000	−1.55487	−0.777436	+	0.628962i	$0.783480\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	−26.0000	−0.840900
$957$	12.0000	0.387905
$958$	−4.00000	−0.129234
$959$	−40.0000	−1.29167
$960$	−1.00000	−0.0322749
$961$	−27.0000	−0.870968
$962$	18.0000	0.580343
$963$	18.0000	0.580042
$964$	1.00000	0.0322078
$965$	3.00000	0.0965734
$966$	−4.00000	−0.128698
$967$	−1.00000	−0.0321578	−0.0160789	−	0.999871i	$-0.505118\pi$
$967$	−1.00000	−0.0321578	−0.0160789	+	0.999871i	$0.505118\pi$
$968$	2.00000	0.0642824
$969$	12.0000	0.385496
$970$	10.0000	0.321081
$971$	22.0000	0.706014	0.353007	−	0.935621i	$-0.385159\pi$
$971$	22.0000	0.706014	0.353007	+	0.935621i	$0.385159\pi$
$972$	1.00000	0.0320750
$973$	52.0000	1.66704
$974$	13.0000	0.416547
$975$	24.0000	0.768615
$976$	−1.00000	−0.0320092
$977$	52.0000	1.66363	0.831814	−	0.555055i	$-0.187303\pi$
$977$	52.0000	1.66363	0.831814	+	0.555055i	$0.187303\pi$
$978$	−16.0000	−0.511624
$979$	15.0000	0.479402
$980$	−9.00000	−0.287494
$981$	15.0000	0.478913
$982$	−36.0000	−1.14881
$983$	1.00000	0.0318950	0.0159475	−	0.999873i	$-0.494924\pi$
$983$	1.00000	0.0318950	0.0159475	+	0.999873i	$0.494924\pi$
$984$	−6.00000	−0.191273
$985$	−24.0000	−0.764704
$986$	16.0000	0.509544
$987$	0	0
$988$	18.0000	0.572656
$989$	12.0000	0.381578
$990$	3.00000	0.0953463
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	2.00000	0.0635001
$993$	−13.0000	−0.412543
$994$	0	0
$995$	12.0000	0.380426
$996$	1.00000	0.0316862
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	−40.0000	−1.26618
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	498.2.a.a.1.1	✓	1
3.2	odd	2		1494.2.a.d.1.1		1
4.3	odd	2		3984.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
498.2.a.a.1.1	✓	1	1.1	even	1	trivial
1494.2.a.d.1.1		1	3.2	odd	2
3984.2.a.d.1.1		1	4.3	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:	\( N \)	\(=\)	\( 498 = 2 \cdot 3 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	498.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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