LMFDB - Embedded newform 8214.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8214,2,Mod(1,8214)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 8214 = 2 \cdot 3 \cdot 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8214.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:	$65.5891202203$
Analytic rank:	$0$
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)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} +5.00000 q^{11} -1.00000 q^{12} -3.00000 q^{13} +3.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +7.00000 q^{19} +4.00000 q^{20} -3.00000 q^{21} +5.00000 q^{22} -9.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -3.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -4.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} -3.00000 q^{34} +12.0000 q^{35} +1.00000 q^{36} +7.00000 q^{38} +3.00000 q^{39} +4.00000 q^{40} +6.00000 q^{41} -3.00000 q^{42} -4.00000 q^{43} +5.00000 q^{44} +4.00000 q^{45} -9.00000 q^{46} -10.0000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +11.0000 q^{50} +3.00000 q^{51} -3.00000 q^{52} +3.00000 q^{53} -1.00000 q^{54} +20.0000 q^{55} +3.00000 q^{56} -7.00000 q^{57} +4.00000 q^{59} -4.00000 q^{60} +2.00000 q^{61} +2.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -12.0000 q^{65} -5.00000 q^{66} +6.00000 q^{67} -3.00000 q^{68} +9.00000 q^{69} +12.0000 q^{70} -12.0000 q^{71} +1.00000 q^{72} +13.0000 q^{73} -11.0000 q^{75} +7.00000 q^{76} +15.0000 q^{77} +3.00000 q^{78} +6.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +5.00000 q^{83} -3.00000 q^{84} -12.0000 q^{85} -4.00000 q^{86} +5.00000 q^{88} -11.0000 q^{89} +4.00000 q^{90} -9.00000 q^{91} -9.00000 q^{92} -2.00000 q^{93} -10.0000 q^{94} +28.0000 q^{95} -1.00000 q^{96} -6.00000 q^{97} +2.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	−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
$10$	4.00000	1.26491
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	−1.00000	−0.288675
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	3.00000	0.801784
$15$	−4.00000	−1.03280
$16$	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
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	4.00000	0.894427
$21$	−3.00000	−0.654654
$22$	5.00000	1.06600
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	−1.00000	−0.204124
$25$	11.0000	2.20000
$26$	−3.00000	−0.588348
$27$	−1.00000	−0.192450
$28$	3.00000	0.566947
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−4.00000	−0.730297
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	−5.00000	−0.870388
$34$	−3.00000	−0.514496
$35$	12.0000	2.02837
$36$	1.00000	0.166667
$37$	0	0
$37$	0	0
$38$	7.00000	1.13555
$39$	3.00000	0.480384
$40$	4.00000	0.632456
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−3.00000	−0.462910
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	5.00000	0.753778
$45$	4.00000	0.596285
$46$	−9.00000	−1.32698
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	11.0000	1.55563
$51$	3.00000	0.420084
$52$	−3.00000	−0.416025
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−1.00000	−0.136083
$55$	20.0000	2.69680
$56$	3.00000	0.400892
$57$	−7.00000	−0.927173
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−4.00000	−0.516398
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	2.00000	0.254000
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	−5.00000	−0.615457
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	−3.00000	−0.363803
$69$	9.00000	1.08347
$70$	12.0000	1.43427
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	1.00000	0.117851
$73$	13.0000	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$74$	0	0
$75$	−11.0000	−1.27017
$76$	7.00000	0.802955
$77$	15.0000	1.70941
$78$	3.00000	0.339683
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	5.00000	0.548821	0.274411	−	0.961613i	$-0.411517\pi$
$83$	5.00000	0.548821	0.274411	+	0.961613i	$0.411517\pi$
$84$	−3.00000	−0.327327
$85$	−12.0000	−1.30158
$86$	−4.00000	−0.431331
$87$	0	0
$88$	5.00000	0.533002
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	4.00000	0.421637
$91$	−9.00000	−0.943456
$92$	−9.00000	−0.938315
$93$	−2.00000	−0.207390
$94$	−10.0000	−1.03142
$95$	28.0000	2.87274
$96$	−1.00000	−0.102062
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	2.00000	0.202031
$99$	5.00000	0.502519
$100$	11.0000	1.10000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	3.00000	0.297044
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	−3.00000	−0.294174
$105$	−12.0000	−1.17108
$106$	3.00000	0.291386
$107$	−1.00000	−0.0966736	−0.0483368	−	0.998831i	$-0.515392\pi$
$107$	−1.00000	−0.0966736	−0.0483368	+	0.998831i	$0.515392\pi$
$108$	−1.00000	−0.0962250
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	20.0000	1.90693
$111$	0	0
$112$	3.00000	0.283473
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	−7.00000	−0.655610
$115$	−36.0000	−3.35702
$116$	0	0
$117$	−3.00000	−0.277350
$118$	4.00000	0.368230
$119$	−9.00000	−0.825029
$120$	−4.00000	−0.365148
$121$	14.0000	1.27273
$122$	2.00000	0.181071
$123$	−6.00000	−0.541002
$124$	2.00000	0.179605
$125$	24.0000	2.14663
$126$	3.00000	0.267261
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	−12.0000	−1.05247
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	−5.00000	−0.435194
$133$	21.0000	1.82093
$134$	6.00000	0.518321
$135$	−4.00000	−0.344265
$136$	−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	$0.671323\pi$
$138$	9.00000	0.766131
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	12.0000	1.01419
$141$	10.0000	0.842152
$142$	−12.0000	−1.00702
$143$	−15.0000	−1.25436
$144$	1.00000	0.0833333
$145$	0	0
$146$	13.0000	1.07589
$147$	−2.00000	−0.164957
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−11.0000	−0.898146
$151$	−15.0000	−1.22068	−0.610341	−	0.792139i	$-0.708968\pi$
$151$	−15.0000	−1.22068	−0.610341	+	0.792139i	$0.708968\pi$
$152$	7.00000	0.567775
$153$	−3.00000	−0.242536
$154$	15.0000	1.20873
$155$	8.00000	0.642575
$156$	3.00000	0.240192
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	6.00000	0.477334
$159$	−3.00000	−0.237915
$160$	4.00000	0.316228
$161$	−27.0000	−2.12790
$162$	1.00000	0.0785674
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	6.00000	0.468521
$165$	−20.0000	−1.55700
$166$	5.00000	0.388075
$167$	5.00000	0.386912	0.193456	−	0.981109i	$-0.438030\pi$
$167$	5.00000	0.386912	0.193456	+	0.981109i	$0.438030\pi$
$168$	−3.00000	−0.231455
$169$	−4.00000	−0.307692
$170$	−12.0000	−0.920358
$171$	7.00000	0.535303
$172$	−4.00000	−0.304997
$173$	−11.0000	−0.836315	−0.418157	−	0.908375i	$-0.637324\pi$
$173$	−11.0000	−0.836315	−0.418157	+	0.908375i	$0.637324\pi$
$174$	0	0
$175$	33.0000	2.49457
$176$	5.00000	0.376889
$177$	−4.00000	−0.300658
$178$	−11.0000	−0.824485
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	4.00000	0.298142
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	−9.00000	−0.667124
$183$	−2.00000	−0.147844
$184$	−9.00000	−0.663489
$185$	0	0
$186$	−2.00000	−0.146647
$187$	−15.0000	−1.09691
$188$	−10.0000	−0.729325
$189$	−3.00000	−0.218218
$190$	28.0000	2.03133
$191$	−13.0000	−0.940647	−0.470323	−	0.882494i	$-0.655863\pi$
$191$	−13.0000	−0.940647	−0.470323	+	0.882494i	$0.655863\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−6.00000	−0.430775
$195$	12.0000	0.859338
$196$	2.00000	0.142857
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	5.00000	0.355335
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	11.0000	0.777817
$201$	−6.00000	−0.423207
$202$	10.0000	0.703598
$203$	0	0
$204$	3.00000	0.210042
$205$	24.0000	1.67623
$206$	2.00000	0.139347
$207$	−9.00000	−0.625543
$208$	−3.00000	−0.208013
$209$	35.0000	2.42100
$210$	−12.0000	−0.828079
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	3.00000	0.206041
$213$	12.0000	0.822226
$214$	−1.00000	−0.0683586
$215$	−16.0000	−1.09119
$216$	−1.00000	−0.0680414
$217$	6.00000	0.407307
$218$	7.00000	0.474100
$219$	−13.0000	−0.878459
$220$	20.0000	1.34840
$221$	9.00000	0.605406
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	3.00000	0.200446
$225$	11.0000	0.733333
$226$	10.0000	0.665190
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−7.00000	−0.463586
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	−36.0000	−2.37377
$231$	−15.0000	−0.986928
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−3.00000	−0.196116
$235$	−40.0000	−2.60931
$236$	4.00000	0.260378
$237$	−6.00000	−0.389742
$238$	−9.00000	−0.583383
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	−4.00000	−0.258199
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	14.0000	0.899954
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	8.00000	0.511101
$246$	−6.00000	−0.382546
$247$	−21.0000	−1.33620
$248$	2.00000	0.127000
$249$	−5.00000	−0.316862
$250$	24.0000	1.51789
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	3.00000	0.188982
$253$	−45.0000	−2.82913
$254$	1.00000	0.0627456
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	−19.0000	−1.18519	−0.592594	−	0.805502i	$-0.701896\pi$
$257$	−19.0000	−1.18519	−0.592594	+	0.805502i	$0.701896\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	−12.0000	−0.744208
$261$	0	0
$262$	−14.0000	−0.864923
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	−5.00000	−0.307729
$265$	12.0000	0.737154
$266$	21.0000	1.28759
$267$	11.0000	0.673189
$268$	6.00000	0.366508
$269$	−19.0000	−1.15845	−0.579225	−	0.815168i	$-0.696645\pi$
$269$	−19.0000	−1.15845	−0.579225	+	0.815168i	$0.696645\pi$
$270$	−4.00000	−0.243432
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−3.00000	−0.181902
$273$	9.00000	0.544705
$274$	−12.0000	−0.724947
$275$	55.0000	3.31662
$276$	9.00000	0.541736
$277$	−21.0000	−1.26177	−0.630884	−	0.775877i	$-0.717308\pi$
$277$	−21.0000	−1.26177	−0.630884	+	0.775877i	$0.717308\pi$
$278$	2.00000	0.119952
$279$	2.00000	0.119737
$280$	12.0000	0.717137
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	10.0000	0.595491
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	−12.0000	−0.712069
$285$	−28.0000	−1.65858
$286$	−15.0000	−0.886969
$287$	18.0000	1.06251
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	6.00000	0.351726
$292$	13.0000	0.760767
$293$	19.0000	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$294$	−2.00000	−0.116642
$295$	16.0000	0.931556
$296$	0	0
$297$	−5.00000	−0.290129
$298$	−6.00000	−0.347571
$299$	27.0000	1.56145
$300$	−11.0000	−0.635085
$301$	−12.0000	−0.691669
$302$	−15.0000	−0.863153
$303$	−10.0000	−0.574485
$304$	7.00000	0.401478
$305$	8.00000	0.458079
$306$	−3.00000	−0.171499
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	15.0000	0.854704
$309$	−2.00000	−0.113776
$310$	8.00000	0.454369
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	3.00000	0.169842
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	4.00000	0.225733
$315$	12.0000	0.676123
$316$	6.00000	0.337526
$317$	34.0000	1.90963	0.954815	−	0.297200i	$-0.0960529\pi$
$317$	34.0000	1.90963	0.954815	+	0.297200i	$0.0960529\pi$
$318$	−3.00000	−0.168232
$319$	0	0
$320$	4.00000	0.223607
$321$	1.00000	0.0558146
$322$	−27.0000	−1.50465
$323$	−21.0000	−1.16847
$324$	1.00000	0.0555556
$325$	−33.0000	−1.83051
$326$	5.00000	0.276924
$327$	−7.00000	−0.387101
$328$	6.00000	0.331295
$329$	−30.0000	−1.65395
$330$	−20.0000	−1.10096
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	5.00000	0.274411
$333$	0	0
$334$	5.00000	0.273588
$335$	24.0000	1.31126
$336$	−3.00000	−0.163663
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	−4.00000	−0.217571
$339$	−10.0000	−0.543125
$340$	−12.0000	−0.650791
$341$	10.0000	0.541530
$342$	7.00000	0.378517
$343$	−15.0000	−0.809924
$344$	−4.00000	−0.215666
$345$	36.0000	1.93817
$346$	−11.0000	−0.591364
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	33.0000	1.76392
$351$	3.00000	0.160128
$352$	5.00000	0.266501
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−4.00000	−0.212598
$355$	−48.0000	−2.54758
$356$	−11.0000	−0.582999
$357$	9.00000	0.476331
$358$	24.0000	1.26844
$359$	−26.0000	−1.37223	−0.686114	−	0.727494i	$-0.740685\pi$
$359$	−26.0000	−1.37223	−0.686114	+	0.727494i	$0.740685\pi$
$360$	4.00000	0.210819
$361$	30.0000	1.57895
$362$	−16.0000	−0.840941
$363$	−14.0000	−0.734809
$364$	−9.00000	−0.471728
$365$	52.0000	2.72180
$366$	−2.00000	−0.104542
$367$	−13.0000	−0.678594	−0.339297	−	0.940679i	$-0.610189\pi$
$367$	−13.0000	−0.678594	−0.339297	+	0.940679i	$0.610189\pi$
$368$	−9.00000	−0.469157
$369$	6.00000	0.312348
$370$	0	0
$371$	9.00000	0.467257
$372$	−2.00000	−0.103695
$373$	−28.0000	−1.44979	−0.724893	−	0.688862i	$-0.758111\pi$
$373$	−28.0000	−1.44979	−0.724893	+	0.688862i	$0.758111\pi$
$374$	−15.0000	−0.775632
$375$	−24.0000	−1.23935
$376$	−10.0000	−0.515711
$377$	0	0
$378$	−3.00000	−0.154303
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	28.0000	1.43637
$381$	−1.00000	−0.0512316
$382$	−13.0000	−0.665138
$383$	1.00000	0.0510976	0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000	0.0510976	0.0255488	+	0.999674i	$0.491867\pi$
$384$	−1.00000	−0.0510310
$385$	60.0000	3.05788
$386$	−14.0000	−0.712581
$387$	−4.00000	−0.203331
$388$	−6.00000	−0.304604
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	12.0000	0.607644
$391$	27.0000	1.36545
$392$	2.00000	0.101015
$393$	14.0000	0.706207
$394$	−15.0000	−0.755689
$395$	24.0000	1.20757
$396$	5.00000	0.251259
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	−4.00000	−0.200502
$399$	−21.0000	−1.05131
$400$	11.0000	0.550000
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	−6.00000	−0.299253
$403$	−6.00000	−0.298881
$404$	10.0000	0.497519
$405$	4.00000	0.198762
$406$	0	0
$407$	0	0
$408$	3.00000	0.148522
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	24.0000	1.18528
$411$	12.0000	0.591916
$412$	2.00000	0.0985329
$413$	12.0000	0.590481
$414$	−9.00000	−0.442326
$415$	20.0000	0.981761
$416$	−3.00000	−0.147087
$417$	−2.00000	−0.0979404
$418$	35.0000	1.71191
$419$	−25.0000	−1.22133	−0.610665	−	0.791889i	$-0.709098\pi$
$419$	−25.0000	−1.22133	−0.610665	+	0.791889i	$0.709098\pi$
$420$	−12.0000	−0.585540
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	12.0000	0.584151
$423$	−10.0000	−0.486217
$424$	3.00000	0.145693
$425$	−33.0000	−1.60074
$426$	12.0000	0.581402
$427$	6.00000	0.290360
$428$	−1.00000	−0.0483368
$429$	15.0000	0.724207
$430$	−16.0000	−0.771589
$431$	29.0000	1.39688	0.698440	−	0.715668i	$-0.253878\pi$
$431$	29.0000	1.39688	0.698440	+	0.715668i	$0.253878\pi$
$432$	−1.00000	−0.0481125
$433$	39.0000	1.87422	0.937110	−	0.349034i	$-0.113490\pi$
$433$	39.0000	1.87422	0.937110	+	0.349034i	$0.113490\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	7.00000	0.335239
$437$	−63.0000	−3.01370
$438$	−13.0000	−0.621164
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	20.0000	0.953463
$441$	2.00000	0.0952381
$442$	9.00000	0.428086
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−44.0000	−2.08580
$446$	−16.0000	−0.757622
$447$	6.00000	0.283790
$448$	3.00000	0.141737
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	11.0000	0.518545
$451$	30.0000	1.41264
$452$	10.0000	0.470360
$453$	15.0000	0.704761
$454$	12.0000	0.563188
$455$	−36.0000	−1.68771
$456$	−7.00000	−0.327805
$457$	24.0000	1.12267	0.561336	−	0.827588i	$-0.310287\pi$
$457$	24.0000	1.12267	0.561336	+	0.827588i	$0.310287\pi$
$458$	−8.00000	−0.373815
$459$	3.00000	0.140028
$460$	−36.0000	−1.67851
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	−15.0000	−0.697863
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	−10.0000	−0.463241
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	−3.00000	−0.138675
$469$	18.0000	0.831163
$470$	−40.0000	−1.84506
$471$	−4.00000	−0.184310
$472$	4.00000	0.184115
$473$	−20.0000	−0.919601
$474$	−6.00000	−0.275589
$475$	77.0000	3.53300
$476$	−9.00000	−0.412514
$477$	3.00000	0.137361
$478$	8.00000	0.365911
$479$	−11.0000	−0.502603	−0.251301	−	0.967909i	$-0.580859\pi$
$479$	−11.0000	−0.502603	−0.251301	+	0.967909i	$0.580859\pi$
$480$	−4.00000	−0.182574
$481$	0	0
$482$	−8.00000	−0.364390
$483$	27.0000	1.22854
$484$	14.0000	0.636364
$485$	−24.0000	−1.08978
$486$	−1.00000	−0.0453609
$487$	−10.0000	−0.453143	−0.226572	−	0.973995i	$-0.572752\pi$
$487$	−10.0000	−0.453143	−0.226572	+	0.973995i	$0.572752\pi$
$488$	2.00000	0.0905357
$489$	−5.00000	−0.226108
$490$	8.00000	0.361403
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	−6.00000	−0.270501
$493$	0	0
$494$	−21.0000	−0.944835
$495$	20.0000	0.898933
$496$	2.00000	0.0898027
$497$	−36.0000	−1.61482
$498$	−5.00000	−0.224055
$499$	21.0000	0.940089	0.470045	−	0.882643i	$-0.344238\pi$
$499$	21.0000	0.940089	0.470045	+	0.882643i	$0.344238\pi$
$500$	24.0000	1.07331
$501$	−5.00000	−0.223384
$502$	10.0000	0.446322
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	3.00000	0.133631
$505$	40.0000	1.77998
$506$	−45.0000	−2.00049
$507$	4.00000	0.177646
$508$	1.00000	0.0443678
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	12.0000	0.531369
$511$	39.0000	1.72526
$512$	1.00000	0.0441942
$513$	−7.00000	−0.309058
$514$	−19.0000	−0.838054
$515$	8.00000	0.352522
$516$	4.00000	0.176090
$517$	−50.0000	−2.19900
$518$	0	0
$519$	11.0000	0.482846
$520$	−12.0000	−0.526235
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	−14.0000	−0.611593
$525$	−33.0000	−1.44024
$526$	6.00000	0.261612
$527$	−6.00000	−0.261364
$528$	−5.00000	−0.217597
$529$	58.0000	2.52174
$530$	12.0000	0.521247
$531$	4.00000	0.173585
$532$	21.0000	0.910465
$533$	−18.0000	−0.779667
$534$	11.0000	0.476017
$535$	−4.00000	−0.172935
$536$	6.00000	0.259161
$537$	−24.0000	−1.03568
$538$	−19.0000	−0.819148
$539$	10.0000	0.430730
$540$	−4.00000	−0.172133
$541$	33.0000	1.41878	0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000	1.41878	0.709390	+	0.704816i	$0.248970\pi$
$542$	0	0
$543$	16.0000	0.686626
$544$	−3.00000	−0.128624
$545$	28.0000	1.19939
$546$	9.00000	0.385164
$547$	−3.00000	−0.128271	−0.0641354	−	0.997941i	$-0.520429\pi$
$547$	−3.00000	−0.128271	−0.0641354	+	0.997941i	$0.520429\pi$
$548$	−12.0000	−0.512615
$549$	2.00000	0.0853579
$550$	55.0000	2.34521
$551$	0	0
$552$	9.00000	0.383065
$553$	18.0000	0.765438
$554$	−21.0000	−0.892205
$555$	0	0
$556$	2.00000	0.0848189
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	2.00000	0.0846668
$559$	12.0000	0.507546
$560$	12.0000	0.507093
$561$	15.0000	0.633300
$562$	−15.0000	−0.632737
$563$	8.00000	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$563$	8.00000	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$564$	10.0000	0.421076
$565$	40.0000	1.68281
$566$	17.0000	0.714563
$567$	3.00000	0.125988
$568$	−12.0000	−0.503509
$569$	−5.00000	−0.209611	−0.104805	−	0.994493i	$-0.533422\pi$
$569$	−5.00000	−0.209611	−0.104805	+	0.994493i	$0.533422\pi$
$570$	−28.0000	−1.17279
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−15.0000	−0.627182
$573$	13.0000	0.543083
$574$	18.0000	0.751305
$575$	−99.0000	−4.12859
$576$	1.00000	0.0416667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	−8.00000	−0.332756
$579$	14.0000	0.581820
$580$	0	0
$581$	15.0000	0.622305
$582$	6.00000	0.248708
$583$	15.0000	0.621237
$584$	13.0000	0.537944
$585$	−12.0000	−0.496139
$586$	19.0000	0.784883
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	−2.00000	−0.0824786
$589$	14.0000	0.576860
$590$	16.0000	0.658710
$591$	15.0000	0.617018
$592$	0	0
$593$	−44.0000	−1.80686	−0.903432	−	0.428732i	$-0.858960\pi$
$593$	−44.0000	−1.80686	−0.903432	+	0.428732i	$0.858960\pi$
$594$	−5.00000	−0.205152
$595$	−36.0000	−1.47586
$596$	−6.00000	−0.245770
$597$	4.00000	0.163709
$598$	27.0000	1.10411
$599$	−14.0000	−0.572024	−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000	−0.572024	−0.286012	+	0.958226i	$0.592330\pi$
$600$	−11.0000	−0.449073
$601$	−17.0000	−0.693444	−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000	−0.693444	−0.346722	+	0.937968i	$0.612705\pi$
$602$	−12.0000	−0.489083
$603$	6.00000	0.244339
$604$	−15.0000	−0.610341
$605$	56.0000	2.27672
$606$	−10.0000	−0.406222
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	7.00000	0.283887
$609$	0	0
$610$	8.00000	0.323911
$611$	30.0000	1.21367
$612$	−3.00000	−0.121268
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	−4.00000	−0.161427
$615$	−24.0000	−0.967773
$616$	15.0000	0.604367
$617$	−4.00000	−0.161034	−0.0805170	−	0.996753i	$-0.525657\pi$
$617$	−4.00000	−0.161034	−0.0805170	+	0.996753i	$0.525657\pi$
$618$	−2.00000	−0.0804518
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	8.00000	0.321288
$621$	9.00000	0.361158
$622$	24.0000	0.962312
$623$	−33.0000	−1.32212
$624$	3.00000	0.120096
$625$	41.0000	1.64000
$626$	−10.0000	−0.399680
$627$	−35.0000	−1.39777
$628$	4.00000	0.159617
$629$	0	0
$630$	12.0000	0.478091
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	6.00000	0.238667
$633$	−12.0000	−0.476957
$634$	34.0000	1.35031
$635$	4.00000	0.158735
$636$	−3.00000	−0.118958
$637$	−6.00000	−0.237729
$638$	0	0
$639$	−12.0000	−0.474713
$640$	4.00000	0.158114
$641$	16.0000	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$641$	16.0000	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$642$	1.00000	0.0394669
$643$	43.0000	1.69575	0.847877	−	0.530193i	$-0.177880\pi$
$643$	43.0000	1.69575	0.847877	+	0.530193i	$0.177880\pi$
$644$	−27.0000	−1.06395
$645$	16.0000	0.629999
$646$	−21.0000	−0.826234
$647$	−25.0000	−0.982851	−0.491426	−	0.870919i	$-0.663524\pi$
$647$	−25.0000	−0.982851	−0.491426	+	0.870919i	$0.663524\pi$
$648$	1.00000	0.0392837
$649$	20.0000	0.785069
$650$	−33.0000	−1.29437
$651$	−6.00000	−0.235159
$652$	5.00000	0.195815
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	−7.00000	−0.273722
$655$	−56.0000	−2.18810
$656$	6.00000	0.234261
$657$	13.0000	0.507178
$658$	−30.0000	−1.16952
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	−20.0000	−0.778499
$661$	−51.0000	−1.98367	−0.991835	−	0.127527i	$-0.959296\pi$
$661$	−51.0000	−1.98367	−0.991835	+	0.127527i	$0.959296\pi$
$662$	20.0000	0.777322
$663$	−9.00000	−0.349531
$664$	5.00000	0.194038
$665$	84.0000	3.25738
$666$	0	0
$667$	0	0
$668$	5.00000	0.193456
$669$	16.0000	0.618596
$670$	24.0000	0.927201
$671$	10.0000	0.386046
$672$	−3.00000	−0.115728
$673$	37.0000	1.42625	0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000	1.42625	0.713123	+	0.701039i	$0.247280\pi$
$674$	−27.0000	−1.04000
$675$	−11.0000	−0.423390
$676$	−4.00000	−0.153846
$677$	−7.00000	−0.269032	−0.134516	−	0.990911i	$-0.542948\pi$
$677$	−7.00000	−0.269032	−0.134516	+	0.990911i	$0.542948\pi$
$678$	−10.0000	−0.384048
$679$	−18.0000	−0.690777
$680$	−12.0000	−0.460179
$681$	−12.0000	−0.459841
$682$	10.0000	0.382920
$683$	42.0000	1.60709	0.803543	−	0.595247i	$-0.202946\pi$
$683$	42.0000	1.60709	0.803543	+	0.595247i	$0.202946\pi$
$684$	7.00000	0.267652
$685$	−48.0000	−1.83399
$686$	−15.0000	−0.572703
$687$	8.00000	0.305219
$688$	−4.00000	−0.152499
$689$	−9.00000	−0.342873
$690$	36.0000	1.37050
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	−11.0000	−0.418157
$693$	15.0000	0.569803
$694$	−18.0000	−0.683271
$695$	8.00000	0.303457
$696$	0	0
$697$	−18.0000	−0.681799
$698$	34.0000	1.28692
$699$	10.0000	0.378235
$700$	33.0000	1.24728
$701$	−16.0000	−0.604312	−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000	−0.604312	−0.302156	+	0.953259i	$0.597706\pi$
$702$	3.00000	0.113228
$703$	0	0
$704$	5.00000	0.188445
$705$	40.0000	1.50649
$706$	18.0000	0.677439
$707$	30.0000	1.12827
$708$	−4.00000	−0.150329
$709$	−3.00000	−0.112667	−0.0563337	−	0.998412i	$-0.517941\pi$
$709$	−3.00000	−0.112667	−0.0563337	+	0.998412i	$0.517941\pi$
$710$	−48.0000	−1.80141
$711$	6.00000	0.225018
$712$	−11.0000	−0.412242
$713$	−18.0000	−0.674105
$714$	9.00000	0.336817
$715$	−60.0000	−2.24387
$716$	24.0000	0.896922
$717$	−8.00000	−0.298765
$718$	−26.0000	−0.970311
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	4.00000	0.149071
$721$	6.00000	0.223452
$722$	30.0000	1.11648
$723$	8.00000	0.297523
$724$	−16.0000	−0.594635
$725$	0	0
$726$	−14.0000	−0.519589
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	−9.00000	−0.333562
$729$	1.00000	0.0370370
$730$	52.0000	1.92461
$731$	12.0000	0.443836
$732$	−2.00000	−0.0739221
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−13.0000	−0.479839
$735$	−8.00000	−0.295084
$736$	−9.00000	−0.331744
$737$	30.0000	1.10506
$738$	6.00000	0.220863
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	21.0000	0.771454
$742$	9.00000	0.330400
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	−2.00000	−0.0733236
$745$	−24.0000	−0.879292
$746$	−28.0000	−1.02515
$747$	5.00000	0.182940
$748$	−15.0000	−0.548454
$749$	−3.00000	−0.109618
$750$	−24.0000	−0.876356
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−10.0000	−0.364662
$753$	−10.0000	−0.364420
$754$	0	0
$755$	−60.0000	−2.18362
$756$	−3.00000	−0.109109
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	16.0000	0.581146
$759$	45.0000	1.63340
$760$	28.0000	1.01567
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	−1.00000	−0.0362262
$763$	21.0000	0.760251
$764$	−13.0000	−0.470323
$765$	−12.0000	−0.433861
$766$	1.00000	0.0361315
$767$	−12.0000	−0.433295
$768$	−1.00000	−0.0360844
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	60.0000	2.16225
$771$	19.0000	0.684268
$772$	−14.0000	−0.503871
$773$	−7.00000	−0.251773	−0.125886	−	0.992045i	$-0.540177\pi$
$773$	−7.00000	−0.251773	−0.125886	+	0.992045i	$0.540177\pi$
$774$	−4.00000	−0.143777
$775$	22.0000	0.790263
$776$	−6.00000	−0.215387
$777$	0	0
$778$	−14.0000	−0.501924
$779$	42.0000	1.50481
$780$	12.0000	0.429669
$781$	−60.0000	−2.14697
$782$	27.0000	0.965518
$783$	0	0
$784$	2.00000	0.0714286
$785$	16.0000	0.571064
$786$	14.0000	0.499363
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	−15.0000	−0.534353
$789$	−6.00000	−0.213606
$790$	24.0000	0.853882
$791$	30.0000	1.06668
$792$	5.00000	0.177667
$793$	−6.00000	−0.213066
$794$	−4.00000	−0.141955
$795$	−12.0000	−0.425596
$796$	−4.00000	−0.141776
$797$	−32.0000	−1.13350	−0.566749	−	0.823890i	$-0.691799\pi$
$797$	−32.0000	−1.13350	−0.566749	+	0.823890i	$0.691799\pi$
$798$	−21.0000	−0.743392
$799$	30.0000	1.06132
$800$	11.0000	0.388909
$801$	−11.0000	−0.388666
$802$	−27.0000	−0.953403
$803$	65.0000	2.29380
$804$	−6.00000	−0.211604
$805$	−108.000	−3.80650
$806$	−6.00000	−0.211341
$807$	19.0000	0.668832
$808$	10.0000	0.351799
$809$	−51.0000	−1.79306	−0.896532	−	0.442978i	$-0.853922\pi$
$809$	−51.0000	−1.79306	−0.896532	+	0.442978i	$0.853922\pi$
$810$	4.00000	0.140546
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.0000	0.700569
$816$	3.00000	0.105021
$817$	−28.0000	−0.979596
$818$	6.00000	0.209785
$819$	−9.00000	−0.314485
$820$	24.0000	0.838116
$821$	−1.00000	−0.0349002	−0.0174501	−	0.999848i	$-0.505555\pi$
$821$	−1.00000	−0.0349002	−0.0174501	+	0.999848i	$0.505555\pi$
$822$	12.0000	0.418548
$823$	−27.0000	−0.941161	−0.470580	−	0.882357i	$-0.655955\pi$
$823$	−27.0000	−0.941161	−0.470580	+	0.882357i	$0.655955\pi$
$824$	2.00000	0.0696733
$825$	−55.0000	−1.91485
$826$	12.0000	0.417533
$827$	34.0000	1.18230	0.591148	−	0.806563i	$-0.298675\pi$
$827$	34.0000	1.18230	0.591148	+	0.806563i	$0.298675\pi$
$828$	−9.00000	−0.312772
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	20.0000	0.694210
$831$	21.0000	0.728482
$832$	−3.00000	−0.104006
$833$	−6.00000	−0.207888
$834$	−2.00000	−0.0692543
$835$	20.0000	0.692129
$836$	35.0000	1.21050
$837$	−2.00000	−0.0691301
$838$	−25.0000	−0.863611
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	−12.0000	−0.414039
$841$	−29.0000	−1.00000
$842$	10.0000	0.344623
$843$	15.0000	0.516627
$844$	12.0000	0.413057
$845$	−16.0000	−0.550417
$846$	−10.0000	−0.343807
$847$	42.0000	1.44314
$848$	3.00000	0.103020
$849$	−17.0000	−0.583438
$850$	−33.0000	−1.13189
$851$	0	0
$852$	12.0000	0.411113
$853$	49.0000	1.67773	0.838864	−	0.544341i	$-0.183220\pi$
$853$	49.0000	1.67773	0.838864	+	0.544341i	$0.183220\pi$
$854$	6.00000	0.205316
$855$	28.0000	0.957580
$856$	−1.00000	−0.0341793
$857$	11.0000	0.375753	0.187876	−	0.982193i	$-0.439840\pi$
$857$	11.0000	0.375753	0.187876	+	0.982193i	$0.439840\pi$
$858$	15.0000	0.512092
$859$	−19.0000	−0.648272	−0.324136	−	0.946011i	$-0.605073\pi$
$859$	−19.0000	−0.648272	−0.324136	+	0.946011i	$0.605073\pi$
$860$	−16.0000	−0.545595
$861$	−18.0000	−0.613438
$862$	29.0000	0.987744
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	−1.00000	−0.0340207
$865$	−44.0000	−1.49604
$866$	39.0000	1.32527
$867$	8.00000	0.271694
$868$	6.00000	0.203653
$869$	30.0000	1.01768
$870$	0	0
$871$	−18.0000	−0.609907
$872$	7.00000	0.237050
$873$	−6.00000	−0.203069
$874$	−63.0000	−2.13101
$875$	72.0000	2.43404
$876$	−13.0000	−0.439229
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−22.0000	−0.742464
$879$	−19.0000	−0.640854
$880$	20.0000	0.674200
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	2.00000	0.0673435
$883$	−31.0000	−1.04323	−0.521617	−	0.853180i	$-0.674671\pi$
$883$	−31.0000	−1.04323	−0.521617	+	0.853180i	$0.674671\pi$
$884$	9.00000	0.302703
$885$	−16.0000	−0.537834
$886$	12.0000	0.403148
$887$	−50.0000	−1.67884	−0.839418	−	0.543487i	$-0.817104\pi$
$887$	−50.0000	−1.67884	−0.839418	+	0.543487i	$0.817104\pi$
$888$	0	0
$889$	3.00000	0.100617
$890$	−44.0000	−1.47488
$891$	5.00000	0.167506
$892$	−16.0000	−0.535720
$893$	−70.0000	−2.34246
$894$	6.00000	0.200670
$895$	96.0000	3.20893
$896$	3.00000	0.100223
$897$	−27.0000	−0.901504
$898$	22.0000	0.734150
$899$	0	0
$900$	11.0000	0.366667
$901$	−9.00000	−0.299833
$902$	30.0000	0.998891
$903$	12.0000	0.399335
$904$	10.0000	0.332595
$905$	−64.0000	−2.12743
$906$	15.0000	0.498342
$907$	1.00000	0.0332045	0.0166022	−	0.999862i	$-0.494715\pi$
$907$	1.00000	0.0332045	0.0166022	+	0.999862i	$0.494715\pi$
$908$	12.0000	0.398234
$909$	10.0000	0.331679
$910$	−36.0000	−1.19339
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	−7.00000	−0.231793
$913$	25.0000	0.827379
$914$	24.0000	0.793849
$915$	−8.00000	−0.264472
$916$	−8.00000	−0.264327
$917$	−42.0000	−1.38696
$918$	3.00000	0.0990148
$919$	50.0000	1.64935	0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000	1.64935	0.824674	+	0.565608i	$0.191359\pi$
$920$	−36.0000	−1.18688
$921$	4.00000	0.131804
$922$	10.0000	0.329332
$923$	36.0000	1.18495
$924$	−15.0000	−0.493464
$925$	0	0
$926$	−34.0000	−1.11731
$927$	2.00000	0.0656886
$928$	0	0
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	−8.00000	−0.262330
$931$	14.0000	0.458831
$932$	−10.0000	−0.327561
$933$	−24.0000	−0.785725
$934$	−18.0000	−0.588978
$935$	−60.0000	−1.96221
$936$	−3.00000	−0.0980581
$937$	−58.0000	−1.89478	−0.947389	−	0.320085i	$-0.896288\pi$
$937$	−58.0000	−1.89478	−0.947389	+	0.320085i	$0.896288\pi$
$938$	18.0000	0.587721
$939$	10.0000	0.326338
$940$	−40.0000	−1.30466
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	−4.00000	−0.130327
$943$	−54.0000	−1.75848
$944$	4.00000	0.130189
$945$	−12.0000	−0.390360
$946$	−20.0000	−0.650256
$947$	6.00000	0.194974	0.0974869	−	0.995237i	$-0.468920\pi$
$947$	6.00000	0.194974	0.0974869	+	0.995237i	$0.468920\pi$
$948$	−6.00000	−0.194871
$949$	−39.0000	−1.26599
$950$	77.0000	2.49821
$951$	−34.0000	−1.10253
$952$	−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	$0.752505\pi$
$954$	3.00000	0.0971286
$955$	−52.0000	−1.68268
$956$	8.00000	0.258738
$957$	0	0
$958$	−11.0000	−0.355394
$959$	−36.0000	−1.16250
$960$	−4.00000	−0.129099
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−1.00000	−0.0322245
$964$	−8.00000	−0.257663
$965$	−56.0000	−1.80270
$966$	27.0000	0.868711
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	14.0000	0.449977
$969$	21.0000	0.674617
$970$	−24.0000	−0.770594
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	−1.00000	−0.0320750
$973$	6.00000	0.192351
$974$	−10.0000	−0.320421
$975$	33.0000	1.05685
$976$	2.00000	0.0640184
$977$	45.0000	1.43968	0.719839	−	0.694141i	$-0.244216\pi$
$977$	45.0000	1.43968	0.719839	+	0.694141i	$0.244216\pi$
$978$	−5.00000	−0.159882
$979$	−55.0000	−1.75781
$980$	8.00000	0.255551
$981$	7.00000	0.223493
$982$	33.0000	1.05307
$983$	−30.0000	−0.956851	−0.478426	−	0.878128i	$-0.658792\pi$
$983$	−30.0000	−0.956851	−0.478426	+	0.878128i	$0.658792\pi$
$984$	−6.00000	−0.191273
$985$	−60.0000	−1.91176
$986$	0	0
$987$	30.0000	0.954911
$988$	−21.0000	−0.668099
$989$	36.0000	1.14473
$990$	20.0000	0.635642
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	2.00000	0.0635001
$993$	−20.0000	−0.634681
$994$	−36.0000	−1.14185
$995$	−16.0000	−0.507234
$996$	−5.00000	−0.158431
$997$	39.0000	1.23514	0.617571	−	0.786515i	$-0.288117\pi$
$997$	39.0000	1.23514	0.617571	+	0.786515i	$0.288117\pi$
$998$	21.0000	0.664743
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8214.2.a.i.1.1		1
37.36	even	2		222.2.a.a.1.1	✓	1
111.110	odd	2		666.2.a.g.1.1		1
148.147	odd	2		1776.2.a.f.1.1		1
185.184	even	2		5550.2.a.bh.1.1		1
296.147	odd	2		7104.2.a.m.1.1		1
296.221	even	2		7104.2.a.bb.1.1		1
444.443	even	2		5328.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.2.a.a.1.1	✓	1	37.36	even	2
666.2.a.g.1.1		1	111.110	odd	2
1776.2.a.f.1.1		1	148.147	odd	2
5328.2.a.v.1.1		1	444.443	even	2
5550.2.a.bh.1.1		1	185.184	even	2
7104.2.a.m.1.1		1	296.147	odd	2
7104.2.a.bb.1.1		1	296.221	even	2
8214.2.a.i.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:	\( N \)	\(=\)	\( 8214 = 2 \cdot 3 \cdot 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8214.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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