LMFDB - Embedded newform 8214.2.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8214,2,Mod(1,8214)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8214.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8214, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8214 = 2 \cdot 3 \cdot 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8214.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1,1,-2,-1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$65.5891202203$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 222)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8214.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +3.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -2.00000 q^{20} -3.00000 q^{21} +3.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -3.00000 q^{34} -6.00000 q^{35} +1.00000 q^{36} +1.00000 q^{38} +1.00000 q^{39} -2.00000 q^{40} -2.00000 q^{41} -3.00000 q^{42} +4.00000 q^{43} +3.00000 q^{44} -2.00000 q^{45} -1.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +3.00000 q^{51} -1.00000 q^{52} -11.0000 q^{53} -1.00000 q^{54} -6.00000 q^{55} +3.00000 q^{56} -1.00000 q^{57} -6.00000 q^{58} +6.00000 q^{59} +2.00000 q^{60} -10.0000 q^{61} -10.0000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -3.00000 q^{66} -8.00000 q^{67} -3.00000 q^{68} +1.00000 q^{69} -6.00000 q^{70} +2.00000 q^{71} +1.00000 q^{72} +1.00000 q^{73} +1.00000 q^{75} +1.00000 q^{76} +9.00000 q^{77} +1.00000 q^{78} -14.0000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -1.00000 q^{83} -3.00000 q^{84} +6.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} +3.00000 q^{88} +9.00000 q^{89} -2.00000 q^{90} -3.00000 q^{91} -1.00000 q^{92} +10.0000 q^{93} +8.00000 q^{94} -2.00000 q^{95} -1.00000 q^{96} -8.00000 q^{97} +2.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−1.00000	−0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	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
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	3.00000	0.801784
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	−2.00000	−0.447214
$21$	−3.00000	−0.654654
$22$	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
$25$	−1.00000	−0.200000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	3.00000	0.566947
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	2.00000	0.365148
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000	0.176777
$33$	−3.00000	−0.522233
$34$	−3.00000	−0.514496
$35$	−6.00000	−1.01419
$36$	1.00000	0.166667
$37$	0	0
$37$	0	0
$38$	1.00000	0.162221
$39$	1.00000	0.160128
$40$	−2.00000	−0.316228
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−3.00000	−0.462910
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	3.00000	0.452267
$45$	−2.00000	−0.298142
$46$	−1.00000	−0.147442
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	3.00000	0.420084
$52$	−1.00000	−0.138675
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	−1.00000	−0.136083
$55$	−6.00000	−0.809040
$56$	3.00000	0.400892
$57$	−1.00000	−0.132453
$58$	−6.00000	−0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	2.00000	0.258199
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−10.0000	−1.27000
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−3.00000	−0.369274
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−3.00000	−0.363803
$69$	1.00000	0.120386
$70$	−6.00000	−0.717137
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	1.00000	0.117851
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	1.00000	0.114708
$77$	9.00000	1.02565
$78$	1.00000	0.113228
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	−3.00000	−0.327327
$85$	6.00000	0.650791
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	3.00000	0.319801
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	−2.00000	−0.210819
$91$	−3.00000	−0.314485
$92$	−1.00000	−0.104257
$93$	10.0000	1.03695
$94$	8.00000	0.825137
$95$	−2.00000	−0.205196
$96$	−1.00000	−0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	2.00000	0.202031
$99$	3.00000	0.301511
$100$	−1.00000	−0.100000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	3.00000	0.297044
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−1.00000	−0.0980581
$105$	6.00000	0.585540
$106$	−11.0000	−1.06841
$107$	13.0000	1.25676	0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000	1.25676	0.628379	+	0.777908i	$0.283719\pi$
$108$	−1.00000	−0.0962250
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	−6.00000	−0.572078
$111$	0	0
$112$	3.00000	0.283473
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−1.00000	−0.0936586
$115$	2.00000	0.186501
$116$	−6.00000	−0.557086
$117$	−1.00000	−0.0924500
$118$	6.00000	0.552345
$119$	−9.00000	−0.825029
$120$	2.00000	0.182574
$121$	−2.00000	−0.181818
$122$	−10.0000	−0.905357
$123$	2.00000	0.180334
$124$	−10.0000	−0.898027
$125$	12.0000	1.07331
$126$	3.00000	0.267261
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	2.00000	0.175412
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	−3.00000	−0.261116
$133$	3.00000	0.260133
$134$	−8.00000	−0.691095
$135$	2.00000	0.172133
$136$	−3.00000	−0.257248
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	1.00000	0.0851257
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−6.00000	−0.507093
$141$	−8.00000	−0.673722
$142$	2.00000	0.167836
$143$	−3.00000	−0.250873
$144$	1.00000	0.0833333
$145$	12.0000	0.996546
$146$	1.00000	0.0827606
$147$	−2.00000	−0.164957
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	1.00000	0.0816497
$151$	13.0000	1.05792	0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000	1.05792	0.528962	+	0.848645i	$0.322581\pi$
$152$	1.00000	0.0811107
$153$	−3.00000	−0.242536
$154$	9.00000	0.725241
$155$	20.0000	1.60644
$156$	1.00000	0.0800641
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	−14.0000	−1.11378
$159$	11.0000	0.872357
$160$	−2.00000	−0.158114
$161$	−3.00000	−0.236433
$162$	1.00000	0.0785674
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	−2.00000	−0.156174
$165$	6.00000	0.467099
$166$	−1.00000	−0.0776151
$167$	17.0000	1.31550	0.657750	−	0.753237i	$-0.271508\pi$
$167$	17.0000	1.31550	0.657750	+	0.753237i	$0.271508\pi$
$168$	−3.00000	−0.231455
$169$	−12.0000	−0.923077
$170$	6.00000	0.460179
$171$	1.00000	0.0764719
$172$	4.00000	0.304997
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	6.00000	0.454859
$175$	−3.00000	−0.226779
$176$	3.00000	0.226134
$177$	−6.00000	−0.450988
$178$	9.00000	0.674579
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	−2.00000	−0.149071
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−3.00000	−0.222375
$183$	10.0000	0.739221
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	10.0000	0.733236
$187$	−9.00000	−0.658145
$188$	8.00000	0.583460
$189$	−3.00000	−0.218218
$190$	−2.00000	−0.145095
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	−1.00000	−0.0721688
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−8.00000	−0.574367
$195$	−2.00000	−0.143223
$196$	2.00000	0.142857
$197$	−17.0000	−1.21120	−0.605600	−	0.795769i	$-0.707067\pi$
$197$	−17.0000	−1.21120	−0.605600	+	0.795769i	$0.707067\pi$
$198$	3.00000	0.213201
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	−1.00000	−0.0707107
$201$	8.00000	0.564276
$202$	18.0000	1.26648
$203$	−18.0000	−1.26335
$204$	3.00000	0.210042
$205$	4.00000	0.279372
$206$	16.0000	1.11477
$207$	−1.00000	−0.0695048
$208$	−1.00000	−0.0693375
$209$	3.00000	0.207514
$210$	6.00000	0.414039
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	−11.0000	−0.755483
$213$	−2.00000	−0.137038
$214$	13.0000	0.888662
$215$	−8.00000	−0.545595
$216$	−1.00000	−0.0680414
$217$	−30.0000	−2.03653
$218$	−19.0000	−1.28684
$219$	−1.00000	−0.0675737
$220$	−6.00000	−0.404520
$221$	3.00000	0.201802
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	3.00000	0.200446
$225$	−1.00000	−0.0666667
$226$	−14.0000	−0.931266
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	−1.00000	−0.0662266
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	2.00000	0.131876
$231$	−9.00000	−0.592157
$232$	−6.00000	−0.393919
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	−1.00000	−0.0653720
$235$	−16.0000	−1.04372
$236$	6.00000	0.390567
$237$	14.0000	0.909398
$238$	−9.00000	−0.583383
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	2.00000	0.129099
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−2.00000	−0.128565
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	−4.00000	−0.255551
$246$	2.00000	0.127515
$247$	−1.00000	−0.0636285
$248$	−10.0000	−0.635001
$249$	1.00000	0.0633724
$250$	12.0000	0.758947
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	3.00000	0.188982
$253$	−3.00000	−0.188608
$254$	−7.00000	−0.439219
$255$	−6.00000	−0.375735
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	2.00000	0.124035
$261$	−6.00000	−0.371391
$262$	−10.0000	−0.617802
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−3.00000	−0.184637
$265$	22.0000	1.35145
$266$	3.00000	0.183942
$267$	−9.00000	−0.550791
$268$	−8.00000	−0.488678
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	2.00000	0.121716
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	−3.00000	−0.181902
$273$	3.00000	0.181568
$274$	−12.0000	−0.724947
$275$	−3.00000	−0.180907
$276$	1.00000	0.0601929
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	−6.00000	−0.358569
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	−8.00000	−0.476393
$283$	−29.0000	−1.72387	−0.861936	−	0.507018i	$-0.830748\pi$
$283$	−29.0000	−1.72387	−0.861936	+	0.507018i	$0.830748\pi$
$284$	2.00000	0.118678
$285$	2.00000	0.118470
$286$	−3.00000	−0.177394
$287$	−6.00000	−0.354169
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	12.0000	0.704664
$291$	8.00000	0.468968
$292$	1.00000	0.0585206
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	−2.00000	−0.116642
$295$	−12.0000	−0.698667
$296$	0	0
$297$	−3.00000	−0.174078
$298$	−10.0000	−0.579284
$299$	1.00000	0.0578315
$300$	1.00000	0.0577350
$301$	12.0000	0.691669
$302$	13.0000	0.748066
$303$	−18.0000	−1.03407
$304$	1.00000	0.0573539
$305$	20.0000	1.14520
$306$	−3.00000	−0.171499
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	9.00000	0.512823
$309$	−16.0000	−0.910208
$310$	20.0000	1.13592
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	1.00000	0.0566139
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	8.00000	0.451466
$315$	−6.00000	−0.338062
$316$	−14.0000	−0.787562
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	11.0000	0.616849
$319$	−18.0000	−1.00781
$320$	−2.00000	−0.111803
$321$	−13.0000	−0.725589
$322$	−3.00000	−0.167183
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	1.00000	0.0554700
$326$	−9.00000	−0.498464
$327$	19.0000	1.05070
$328$	−2.00000	−0.110432
$329$	24.0000	1.32316
$330$	6.00000	0.330289
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−1.00000	−0.0548821
$333$	0	0
$334$	17.0000	0.930199
$335$	16.0000	0.874173
$336$	−3.00000	−0.163663
$337$	−3.00000	−0.163420	−0.0817102	−	0.996656i	$-0.526038\pi$
$337$	−3.00000	−0.163420	−0.0817102	+	0.996656i	$0.526038\pi$
$338$	−12.0000	−0.652714
$339$	14.0000	0.760376
$340$	6.00000	0.325396
$341$	−30.0000	−1.62459
$342$	1.00000	0.0540738
$343$	−15.0000	−0.809924
$344$	4.00000	0.215666
$345$	−2.00000	−0.107676
$346$	−9.00000	−0.483843
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	6.00000	0.321634
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	−3.00000	−0.160357
$351$	1.00000	0.0533761
$352$	3.00000	0.159901
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	−6.00000	−0.318896
$355$	−4.00000	−0.212298
$356$	9.00000	0.476999
$357$	9.00000	0.476331
$358$	24.0000	1.26844
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	−2.00000	−0.105409
$361$	−18.0000	−0.947368
$362$	2.00000	0.105118
$363$	2.00000	0.104973
$364$	−3.00000	−0.157243
$365$	−2.00000	−0.104685
$366$	10.0000	0.522708
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	−1.00000	−0.0521286
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−33.0000	−1.71327
$372$	10.0000	0.518476
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	−9.00000	−0.465379
$375$	−12.0000	−0.619677
$376$	8.00000	0.412568
$377$	6.00000	0.309016
$378$	−3.00000	−0.154303
$379$	−30.0000	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$380$	−2.00000	−0.102598
$381$	7.00000	0.358621
$382$	15.0000	0.767467
$383$	−11.0000	−0.562074	−0.281037	−	0.959697i	$-0.590678\pi$
$383$	−11.0000	−0.562074	−0.281037	+	0.959697i	$0.590678\pi$
$384$	−1.00000	−0.0510310
$385$	−18.0000	−0.917365
$386$	−16.0000	−0.814379
$387$	4.00000	0.203331
$388$	−8.00000	−0.406138
$389$	−4.00000	−0.202808	−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000	−0.202808	−0.101404	+	0.994845i	$0.532333\pi$
$390$	−2.00000	−0.101274
$391$	3.00000	0.151717
$392$	2.00000	0.101015
$393$	10.0000	0.504433
$394$	−17.0000	−0.856448
$395$	28.0000	1.40883
$396$	3.00000	0.150756
$397$	32.0000	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$398$	14.0000	0.701757
$399$	−3.00000	−0.150188
$400$	−1.00000	−0.0500000
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	8.00000	0.399004
$403$	10.0000	0.498135
$404$	18.0000	0.895533
$405$	−2.00000	−0.0993808
$406$	−18.0000	−0.893325
$407$	0	0
$408$	3.00000	0.148522
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	4.00000	0.197546
$411$	12.0000	0.591916
$412$	16.0000	0.788263
$413$	18.0000	0.885722
$414$	−1.00000	−0.0491473
$415$	2.00000	0.0981761
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	3.00000	0.146735
$419$	−35.0000	−1.70986	−0.854931	−	0.518742i	$-0.826401\pi$
$419$	−35.0000	−1.70986	−0.854931	+	0.518742i	$0.826401\pi$
$420$	6.00000	0.292770
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	22.0000	1.07094
$423$	8.00000	0.388973
$424$	−11.0000	−0.534207
$425$	3.00000	0.145521
$426$	−2.00000	−0.0969003
$427$	−30.0000	−1.45180
$428$	13.0000	0.628379
$429$	3.00000	0.144841
$430$	−8.00000	−0.385794
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	−1.00000	−0.0481125
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	−30.0000	−1.44005
$435$	−12.0000	−0.575356
$436$	−19.0000	−0.909935
$437$	−1.00000	−0.0478365
$438$	−1.00000	−0.0477818
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	−6.00000	−0.286039
$441$	2.00000	0.0952381
$442$	3.00000	0.142695
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	4.00000	0.189405
$447$	10.0000	0.472984
$448$	3.00000	0.141737
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	−1.00000	−0.0471405
$451$	−6.00000	−0.282529
$452$	−14.0000	−0.658505
$453$	−13.0000	−0.610793
$454$	−8.00000	−0.375459
$455$	6.00000	0.281284
$456$	−1.00000	−0.0468293
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	−20.0000	−0.934539
$459$	3.00000	0.140028
$460$	2.00000	0.0932505
$461$	40.0000	1.86299	0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000	1.86299	0.931493	+	0.363760i	$0.118507\pi$
$462$	−9.00000	−0.418718
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	−6.00000	−0.278543
$465$	−20.0000	−0.927478
$466$	−24.0000	−1.11178
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	−1.00000	−0.0462250
$469$	−24.0000	−1.10822
$470$	−16.0000	−0.738025
$471$	−8.00000	−0.368621
$472$	6.00000	0.276172
$473$	12.0000	0.551761
$474$	14.0000	0.643041
$475$	−1.00000	−0.0458831
$476$	−9.00000	−0.412514
$477$	−11.0000	−0.503655
$478$	16.0000	0.731823
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	2.00000	0.0912871
$481$	0	0
$482$	0	0
$483$	3.00000	0.136505
$484$	−2.00000	−0.0909091
$485$	16.0000	0.726523
$486$	−1.00000	−0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−10.0000	−0.452679
$489$	9.00000	0.406994
$490$	−4.00000	−0.180702
$491$	27.0000	1.21849	0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000	1.21849	0.609246	+	0.792981i	$0.291472\pi$
$492$	2.00000	0.0901670
$493$	18.0000	0.810679
$494$	−1.00000	−0.0449921
$495$	−6.00000	−0.269680
$496$	−10.0000	−0.449013
$497$	6.00000	0.269137
$498$	1.00000	0.0448111
$499$	39.0000	1.74588	0.872940	−	0.487828i	$-0.162211\pi$
$499$	39.0000	1.74588	0.872940	+	0.487828i	$0.162211\pi$
$500$	12.0000	0.536656
$501$	−17.0000	−0.759504
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	3.00000	0.133631
$505$	−36.0000	−1.60198
$506$	−3.00000	−0.133366
$507$	12.0000	0.532939
$508$	−7.00000	−0.310575
$509$	5.00000	0.221621	0.110811	−	0.993842i	$-0.464655\pi$
$509$	5.00000	0.221621	0.110811	+	0.993842i	$0.464655\pi$
$510$	−6.00000	−0.265684
$511$	3.00000	0.132712
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−3.00000	−0.132324
$515$	−32.0000	−1.41009
$516$	−4.00000	−0.176090
$517$	24.0000	1.05552
$518$	0	0
$519$	9.00000	0.395056
$520$	2.00000	0.0877058
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−6.00000	−0.262613
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−10.0000	−0.436852
$525$	3.00000	0.130931
$526$	−24.0000	−1.04645
$527$	30.0000	1.30682
$528$	−3.00000	−0.130558
$529$	−22.0000	−0.956522
$530$	22.0000	0.955619
$531$	6.00000	0.260378
$532$	3.00000	0.130066
$533$	2.00000	0.0866296
$534$	−9.00000	−0.389468
$535$	−26.0000	−1.12408
$536$	−8.00000	−0.345547
$537$	−24.0000	−1.03568
$538$	−5.00000	−0.215565
$539$	6.00000	0.258438
$540$	2.00000	0.0860663
$541$	−45.0000	−1.93470	−0.967351	−	0.253442i	$-0.918437\pi$
$541$	−45.0000	−1.93470	−0.967351	+	0.253442i	$0.918437\pi$
$542$	−28.0000	−1.20270
$543$	−2.00000	−0.0858282
$544$	−3.00000	−0.128624
$545$	38.0000	1.62774
$546$	3.00000	0.128388
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	−12.0000	−0.512615
$549$	−10.0000	−0.426790
$550$	−3.00000	−0.127920
$551$	−6.00000	−0.255609
$552$	1.00000	0.0425628
$553$	−42.0000	−1.78602
$554$	−7.00000	−0.297402
$555$	0	0
$556$	0	0
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	−10.0000	−0.423334
$559$	−4.00000	−0.169182
$560$	−6.00000	−0.253546
$561$	9.00000	0.379980
$562$	−15.0000	−0.632737
$563$	14.0000	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$564$	−8.00000	−0.336861
$565$	28.0000	1.17797
$566$	−29.0000	−1.21896
$567$	3.00000	0.125988
$568$	2.00000	0.0839181
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	2.00000	0.0837708
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−3.00000	−0.125436
$573$	−15.0000	−0.626634
$574$	−6.00000	−0.250435
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	−8.00000	−0.332756
$579$	16.0000	0.664937
$580$	12.0000	0.498273
$581$	−3.00000	−0.124461
$582$	8.00000	0.331611
$583$	−33.0000	−1.36672
$584$	1.00000	0.0413803
$585$	2.00000	0.0826898
$586$	9.00000	0.371787
$587$	−2.00000	−0.0825488	−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000	−0.0825488	−0.0412744	+	0.999148i	$0.513142\pi$
$588$	−2.00000	−0.0824786
$589$	−10.0000	−0.412043
$590$	−12.0000	−0.494032
$591$	17.0000	0.699287
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	−3.00000	−0.123091
$595$	18.0000	0.737928
$596$	−10.0000	−0.409616
$597$	−14.0000	−0.572982
$598$	1.00000	0.0408930
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	1.00000	0.0408248
$601$	7.00000	0.285536	0.142768	−	0.989756i	$-0.454400\pi$
$601$	7.00000	0.285536	0.142768	+	0.989756i	$0.454400\pi$
$602$	12.0000	0.489083
$603$	−8.00000	−0.325785
$604$	13.0000	0.528962
$605$	4.00000	0.162623
$606$	−18.0000	−0.731200
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	1.00000	0.0405554
$609$	18.0000	0.729397
$610$	20.0000	0.809776
$611$	−8.00000	−0.323645
$612$	−3.00000	−0.121268
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	12.0000	0.484281
$615$	−4.00000	−0.161296
$616$	9.00000	0.362620
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−16.0000	−0.643614
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	20.0000	0.803219
$621$	1.00000	0.0401286
$622$	20.0000	0.801927
$623$	27.0000	1.08173
$624$	1.00000	0.0400320
$625$	−19.0000	−0.760000
$626$	−6.00000	−0.239808
$627$	−3.00000	−0.119808
$628$	8.00000	0.319235
$629$	0	0
$630$	−6.00000	−0.239046
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−14.0000	−0.556890
$633$	−22.0000	−0.874421
$634$	2.00000	0.0794301
$635$	14.0000	0.555573
$636$	11.0000	0.436178
$637$	−2.00000	−0.0792429
$638$	−18.0000	−0.712627
$639$	2.00000	0.0791188
$640$	−2.00000	−0.0790569
$641$	−8.00000	−0.315981	−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000	−0.315981	−0.157991	+	0.987441i	$0.550502\pi$
$642$	−13.0000	−0.513069
$643$	21.0000	0.828159	0.414080	−	0.910241i	$-0.364104\pi$
$643$	21.0000	0.828159	0.414080	+	0.910241i	$0.364104\pi$
$644$	−3.00000	−0.118217
$645$	8.00000	0.315000
$646$	−3.00000	−0.118033
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	1.00000	0.0392837
$649$	18.0000	0.706562
$650$	1.00000	0.0392232
$651$	30.0000	1.17579
$652$	−9.00000	−0.352467
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	19.0000	0.742959
$655$	20.0000	0.781465
$656$	−2.00000	−0.0780869
$657$	1.00000	0.0390137
$658$	24.0000	0.935617
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	6.00000	0.233550
$661$	−25.0000	−0.972387	−0.486194	−	0.873851i	$-0.661615\pi$
$661$	−25.0000	−0.972387	−0.486194	+	0.873851i	$0.661615\pi$
$662$	−20.0000	−0.777322
$663$	−3.00000	−0.116510
$664$	−1.00000	−0.0388075
$665$	−6.00000	−0.232670
$666$	0	0
$667$	6.00000	0.232321
$668$	17.0000	0.657750
$669$	−4.00000	−0.154649
$670$	16.0000	0.618134
$671$	−30.0000	−1.15814
$672$	−3.00000	−0.115728
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	−3.00000	−0.115556
$675$	1.00000	0.0384900
$676$	−12.0000	−0.461538
$677$	7.00000	0.269032	0.134516	−	0.990911i	$-0.457052\pi$
$677$	7.00000	0.269032	0.134516	+	0.990911i	$0.457052\pi$
$678$	14.0000	0.537667
$679$	−24.0000	−0.921035
$680$	6.00000	0.230089
$681$	8.00000	0.306561
$682$	−30.0000	−1.14876
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	1.00000	0.0382360
$685$	24.0000	0.916993
$686$	−15.0000	−0.572703
$687$	20.0000	0.763048
$688$	4.00000	0.152499
$689$	11.0000	0.419067
$690$	−2.00000	−0.0761387
$691$	−2.00000	−0.0760836	−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000	−0.0760836	−0.0380418	+	0.999276i	$0.512112\pi$
$692$	−9.00000	−0.342129
$693$	9.00000	0.341882
$694$	18.0000	0.683271
$695$	0	0
$696$	6.00000	0.227429
$697$	6.00000	0.227266
$698$	30.0000	1.13552
$699$	24.0000	0.907763
$700$	−3.00000	−0.113389
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	1.00000	0.0377426
$703$	0	0
$704$	3.00000	0.113067
$705$	16.0000	0.602595
$706$	−34.0000	−1.27961
$707$	54.0000	2.03088
$708$	−6.00000	−0.225494
$709$	−49.0000	−1.84023	−0.920117	−	0.391644i	$-0.871906\pi$
$709$	−49.0000	−1.84023	−0.920117	+	0.391644i	$0.871906\pi$
$710$	−4.00000	−0.150117
$711$	−14.0000	−0.525041
$712$	9.00000	0.337289
$713$	10.0000	0.374503
$714$	9.00000	0.336817
$715$	6.00000	0.224387
$716$	24.0000	0.896922
$717$	−16.0000	−0.597531
$718$	10.0000	0.373197
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	−2.00000	−0.0745356
$721$	48.0000	1.78761
$722$	−18.0000	−0.669891
$723$	0	0
$724$	2.00000	0.0743294
$725$	6.00000	0.222834
$726$	2.00000	0.0742270
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	−3.00000	−0.111187
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	−12.0000	−0.443836
$732$	10.0000	0.369611
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	−17.0000	−0.627481
$735$	4.00000	0.147542
$736$	−1.00000	−0.0368605
$737$	−24.0000	−0.884051
$738$	−2.00000	−0.0736210
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	1.00000	0.0367359
$742$	−33.0000	−1.21147
$743$	−34.0000	−1.24734	−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000	−1.24734	−0.623670	+	0.781688i	$0.714359\pi$
$744$	10.0000	0.366618
$745$	20.0000	0.732743
$746$	−4.00000	−0.146450
$747$	−1.00000	−0.0365881
$748$	−9.00000	−0.329073
$749$	39.0000	1.42503
$750$	−12.0000	−0.438178
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	6.00000	0.218507
$755$	−26.0000	−0.946237
$756$	−3.00000	−0.109109
$757$	17.0000	0.617876	0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000	0.617876	0.308938	+	0.951082i	$0.400027\pi$
$758$	−30.0000	−1.08965
$759$	3.00000	0.108893
$760$	−2.00000	−0.0725476
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	7.00000	0.253583
$763$	−57.0000	−2.06354
$764$	15.0000	0.542681
$765$	6.00000	0.216930
$766$	−11.0000	−0.397446
$767$	−6.00000	−0.216647
$768$	−1.00000	−0.0360844
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	−18.0000	−0.648675
$771$	3.00000	0.108042
$772$	−16.0000	−0.575853
$773$	39.0000	1.40273	0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000	1.40273	0.701366	+	0.712801i	$0.252574\pi$
$774$	4.00000	0.143777
$775$	10.0000	0.359211
$776$	−8.00000	−0.287183
$777$	0	0
$778$	−4.00000	−0.143407
$779$	−2.00000	−0.0716574
$780$	−2.00000	−0.0716115
$781$	6.00000	0.214697
$782$	3.00000	0.107280
$783$	6.00000	0.214423
$784$	2.00000	0.0714286
$785$	−16.0000	−0.571064
$786$	10.0000	0.356688
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	−17.0000	−0.605600
$789$	24.0000	0.854423
$790$	28.0000	0.996195
$791$	−42.0000	−1.49335
$792$	3.00000	0.106600
$793$	10.0000	0.355110
$794$	32.0000	1.13564
$795$	−22.0000	−0.780260
$796$	14.0000	0.496217
$797$	8.00000	0.283375	0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000	0.283375	0.141687	+	0.989911i	$0.454747\pi$
$798$	−3.00000	−0.106199
$799$	−24.0000	−0.849059
$800$	−1.00000	−0.0353553
$801$	9.00000	0.317999
$802$	−15.0000	−0.529668
$803$	3.00000	0.105868
$804$	8.00000	0.282138
$805$	6.00000	0.211472
$806$	10.0000	0.352235
$807$	5.00000	0.176008
$808$	18.0000	0.633238
$809$	−31.0000	−1.08990	−0.544951	−	0.838468i	$-0.683452\pi$
$809$	−31.0000	−1.08990	−0.544951	+	0.838468i	$0.683452\pi$
$810$	−2.00000	−0.0702728
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	−18.0000	−0.631676
$813$	28.0000	0.982003
$814$	0	0
$815$	18.0000	0.630512
$816$	3.00000	0.105021
$817$	4.00000	0.139942
$818$	24.0000	0.839140
$819$	−3.00000	−0.104828
$820$	4.00000	0.139686
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	12.0000	0.418548
$823$	29.0000	1.01088	0.505438	−	0.862863i	$-0.331331\pi$
$823$	29.0000	1.01088	0.505438	+	0.862863i	$0.331331\pi$
$824$	16.0000	0.557386
$825$	3.00000	0.104447
$826$	18.0000	0.626300
$827$	2.00000	0.0695468	0.0347734	−	0.999395i	$-0.488929\pi$
$827$	2.00000	0.0695468	0.0347734	+	0.999395i	$0.488929\pi$
$828$	−1.00000	−0.0347524
$829$	−11.0000	−0.382046	−0.191023	−	0.981586i	$-0.561180\pi$
$829$	−11.0000	−0.382046	−0.191023	+	0.981586i	$0.561180\pi$
$830$	2.00000	0.0694210
$831$	7.00000	0.242827
$832$	−1.00000	−0.0346688
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−34.0000	−1.17662
$836$	3.00000	0.103757
$837$	10.0000	0.345651
$838$	−35.0000	−1.20905
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	6.00000	0.207020
$841$	7.00000	0.241379
$842$	10.0000	0.344623
$843$	15.0000	0.516627
$844$	22.0000	0.757271
$845$	24.0000	0.825625
$846$	8.00000	0.275046
$847$	−6.00000	−0.206162
$848$	−11.0000	−0.377742
$849$	29.0000	0.995277
$850$	3.00000	0.102899
$851$	0	0
$852$	−2.00000	−0.0685189
$853$	−49.0000	−1.67773	−0.838864	−	0.544341i	$-0.816780\pi$
$853$	−49.0000	−1.67773	−0.838864	+	0.544341i	$0.816780\pi$
$854$	−30.0000	−1.02658
$855$	−2.00000	−0.0683986
$856$	13.0000	0.444331
$857$	−13.0000	−0.444072	−0.222036	−	0.975039i	$-0.571270\pi$
$857$	−13.0000	−0.444072	−0.222036	+	0.975039i	$0.571270\pi$
$858$	3.00000	0.102418
$859$	31.0000	1.05771	0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000	1.05771	0.528853	+	0.848713i	$0.322622\pi$
$860$	−8.00000	−0.272798
$861$	6.00000	0.204479
$862$	−15.0000	−0.510902
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	−1.00000	−0.0340207
$865$	18.0000	0.612018
$866$	−1.00000	−0.0339814
$867$	8.00000	0.271694
$868$	−30.0000	−1.01827
$869$	−42.0000	−1.42475
$870$	−12.0000	−0.406838
$871$	8.00000	0.271070
$872$	−19.0000	−0.643421
$873$	−8.00000	−0.270759
$874$	−1.00000	−0.0338255
$875$	36.0000	1.21702
$876$	−1.00000	−0.0337869
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	−26.0000	−0.877457
$879$	−9.00000	−0.303562
$880$	−6.00000	−0.202260
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	2.00000	0.0673435
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	3.00000	0.100901
$885$	12.0000	0.403376
$886$	−4.00000	−0.134383
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	0	0
$889$	−21.0000	−0.704317
$890$	−18.0000	−0.603361
$891$	3.00000	0.100504
$892$	4.00000	0.133930
$893$	8.00000	0.267710
$894$	10.0000	0.334450
$895$	−48.0000	−1.60446
$896$	3.00000	0.100223
$897$	−1.00000	−0.0333890
$898$	−34.0000	−1.13459
$899$	60.0000	2.00111
$900$	−1.00000	−0.0333333
$901$	33.0000	1.09939
$902$	−6.00000	−0.199778
$903$	−12.0000	−0.399335
$904$	−14.0000	−0.465633
$905$	−4.00000	−0.132964
$906$	−13.0000	−0.431896
$907$	7.00000	0.232431	0.116216	−	0.993224i	$-0.462924\pi$
$907$	7.00000	0.232431	0.116216	+	0.993224i	$0.462924\pi$
$908$	−8.00000	−0.265489
$909$	18.0000	0.597022
$910$	6.00000	0.198898
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−1.00000	−0.0331133
$913$	−3.00000	−0.0992855
$914$	2.00000	0.0661541
$915$	−20.0000	−0.661180
$916$	−20.0000	−0.660819
$917$	−30.0000	−0.990687
$918$	3.00000	0.0990148
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	2.00000	0.0659380
$921$	−12.0000	−0.395413
$922$	40.0000	1.31733
$923$	−2.00000	−0.0658308
$924$	−9.00000	−0.296078
$925$	0	0
$926$	34.0000	1.11731
$927$	16.0000	0.525509
$928$	−6.00000	−0.196960
$929$	60.0000	1.96854	0.984268	−	0.176682i	$-0.0565363\pi$
$929$	60.0000	1.96854	0.984268	+	0.176682i	$0.0565363\pi$
$930$	−20.0000	−0.655826
$931$	2.00000	0.0655474
$932$	−24.0000	−0.786146
$933$	−20.0000	−0.654771
$934$	−8.00000	−0.261768
$935$	18.0000	0.588663
$936$	−1.00000	−0.0326860
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	−24.0000	−0.783628
$939$	6.00000	0.195803
$940$	−16.0000	−0.521862
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	−8.00000	−0.260654
$943$	2.00000	0.0651290
$944$	6.00000	0.195283
$945$	6.00000	0.195180
$946$	12.0000	0.390154
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	14.0000	0.454699
$949$	−1.00000	−0.0324614
$950$	−1.00000	−0.0324443
$951$	−2.00000	−0.0648544
$952$	−9.00000	−0.291692
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	−11.0000	−0.356138
$955$	−30.0000	−0.970777
$956$	16.0000	0.517477
$957$	18.0000	0.581857
$958$	21.0000	0.678479
$959$	−36.0000	−1.16250
$960$	2.00000	0.0645497
$961$	69.0000	2.22581
$962$	0	0
$963$	13.0000	0.418919
$964$	0	0
$965$	32.0000	1.03012
$966$	3.00000	0.0965234
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	−2.00000	−0.0642824
$969$	3.00000	0.0963739
$970$	16.0000	0.513729
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−8.00000	−0.256337
$975$	−1.00000	−0.0320256
$976$	−10.0000	−0.320092
$977$	53.0000	1.69562	0.847810	−	0.530300i	$-0.177921\pi$
$977$	53.0000	1.69562	0.847810	+	0.530300i	$0.177921\pi$
$978$	9.00000	0.287788
$979$	27.0000	0.862924
$980$	−4.00000	−0.127775
$981$	−19.0000	−0.606623
$982$	27.0000	0.861605
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	2.00000	0.0637577
$985$	34.0000	1.08333
$986$	18.0000	0.573237
$987$	−24.0000	−0.763928
$988$	−1.00000	−0.0318142
$989$	−4.00000	−0.127193
$990$	−6.00000	−0.190693
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	−10.0000	−0.317500
$993$	20.0000	0.634681
$994$	6.00000	0.190308
$995$	−28.0000	−0.887660
$996$	1.00000	0.0316862
$997$	17.0000	0.538395	0.269198	−	0.963085i	$-0.413241\pi$
$997$	17.0000	0.538395	0.269198	+	0.963085i	$0.413241\pi$
$998$	39.0000	1.23452
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8214.2.a.g.1.1		1
37.6	odd	4		222.2.c.a.73.1	✓	2
37.31	odd	4		222.2.c.a.73.2	yes	2
37.36	even	2		8214.2.a.c.1.1		1
111.68	even	4		666.2.c.a.73.1		2
111.80	even	4		666.2.c.a.73.2		2
148.31	even	4		1776.2.h.a.961.2		2
148.43	even	4		1776.2.h.a.961.1		2
444.179	odd	4		5328.2.h.b.2737.1		2
444.191	odd	4		5328.2.h.b.2737.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.2.c.a.73.1	✓	2	37.6	odd	4
222.2.c.a.73.2	yes	2	37.31	odd	4
666.2.c.a.73.1		2	111.68	even	4
666.2.c.a.73.2		2	111.80	even	4
1776.2.h.a.961.1		2	148.43	even	4
1776.2.h.a.961.2		2	148.31	even	4
5328.2.h.b.2737.1		2	444.179	odd	4
5328.2.h.b.2737.2		2	444.191	odd	4
8214.2.a.c.1.1		1	37.36	even	2
8214.2.a.g.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8214 = 2 \cdot 3 \cdot 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8214.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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