LMFDB - Embedded newform 663.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [663,4,Mod(1,663)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(663, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("663.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 663 = 3 \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	663.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$39.1182663338$
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	$\chi$	$=$	663.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +3.00000 q^{3} +8.00000 q^{4} -10.0000 q^{5} +12.0000 q^{6} -10.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} +3.00000 q^{3} +8.00000 q^{4} -10.0000 q^{5} +12.0000 q^{6} -10.0000 q^{7} +9.00000 q^{9} -40.0000 q^{10} +18.0000 q^{11} +24.0000 q^{12} -13.0000 q^{13} -40.0000 q^{14} -30.0000 q^{15} -64.0000 q^{16} -17.0000 q^{17} +36.0000 q^{18} -74.0000 q^{19} -80.0000 q^{20} -30.0000 q^{21} +72.0000 q^{22} -132.000 q^{23} -25.0000 q^{25} -52.0000 q^{26} +27.0000 q^{27} -80.0000 q^{28} +210.000 q^{29} -120.000 q^{30} -230.000 q^{31} -256.000 q^{32} +54.0000 q^{33} -68.0000 q^{34} +100.000 q^{35} +72.0000 q^{36} -46.0000 q^{37} -296.000 q^{38} -39.0000 q^{39} -114.000 q^{41} -120.000 q^{42} +36.0000 q^{43} +144.000 q^{44} -90.0000 q^{45} -528.000 q^{46} +446.000 q^{47} -192.000 q^{48} -243.000 q^{49} -100.000 q^{50} -51.0000 q^{51} -104.000 q^{52} -754.000 q^{53} +108.000 q^{54} -180.000 q^{55} -222.000 q^{57} +840.000 q^{58} -50.0000 q^{59} -240.000 q^{60} -226.000 q^{61} -920.000 q^{62} -90.0000 q^{63} -512.000 q^{64} +130.000 q^{65} +216.000 q^{66} +582.000 q^{67} -136.000 q^{68} -396.000 q^{69} +400.000 q^{70} -370.000 q^{71} +826.000 q^{73} -184.000 q^{74} -75.0000 q^{75} -592.000 q^{76} -180.000 q^{77} -156.000 q^{78} +272.000 q^{79} +640.000 q^{80} +81.0000 q^{81} -456.000 q^{82} +162.000 q^{83} -240.000 q^{84} +170.000 q^{85} +144.000 q^{86} +630.000 q^{87} -186.000 q^{89} -360.000 q^{90} +130.000 q^{91} -1056.00 q^{92} -690.000 q^{93} +1784.00 q^{94} +740.000 q^{95} -768.000 q^{96} -790.000 q^{97} -972.000 q^{98} +162.000 q^{99} +O(q^{100})\)

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 $

$2$	4.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	4.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	8.00000	1.00000
$5$	−10.0000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−10.0000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	12.0000	0.816497
$7$	−10.0000	−0.539949	−0.269975	−	0.962867i	$-0.587015\pi$
$7$	−10.0000	−0.539949	−0.269975	+	0.962867i	$0.587015\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	−40.0000	−1.26491
$11$	18.0000	0.493382	0.246691	−	0.969094i	$-0.420657\pi$
$11$	18.0000	0.493382	0.246691	+	0.969094i	$0.420657\pi$
$12$	24.0000	0.577350
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−40.0000	−0.763604
$15$	−30.0000	−0.516398
$16$	−64.0000	−1.00000
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	36.0000	0.471405
$19$	−74.0000	−0.893514	−0.446757	−	0.894655i	$-0.647421\pi$
$19$	−74.0000	−0.893514	−0.446757	+	0.894655i	$0.647421\pi$
$20$	−80.0000	−0.894427
$21$	−30.0000	−0.311740
$22$	72.0000	0.697748
$23$	−132.000	−1.19669	−0.598346	−	0.801238i	$-0.704175\pi$
$23$	−132.000	−1.19669	−0.598346	+	0.801238i	$0.704175\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	−52.0000	−0.392232
$27$	27.0000	0.192450
$28$	−80.0000	−0.539949
$29$	210.000	1.34469	0.672345	−	0.740238i	$-0.265287\pi$
$29$	210.000	1.34469	0.672345	+	0.740238i	$0.265287\pi$
$30$	−120.000	−0.730297
$31$	−230.000	−1.33256	−0.666278	−	0.745704i	$-0.732113\pi$
$31$	−230.000	−1.33256	−0.666278	+	0.745704i	$0.732113\pi$
$32$	−256.000	−1.41421
$33$	54.0000	0.284854
$34$	−68.0000	−0.342997
$35$	100.000	0.482945
$36$	72.0000	0.333333
$37$	−46.0000	−0.204388	−0.102194	−	0.994764i	$-0.532586\pi$
$37$	−46.0000	−0.204388	−0.102194	+	0.994764i	$0.532586\pi$
$38$	−296.000	−1.26362
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−114.000	−0.434239	−0.217120	−	0.976145i	$-0.569666\pi$
$41$	−114.000	−0.434239	−0.217120	+	0.976145i	$0.569666\pi$
$42$	−120.000	−0.440867
$43$	36.0000	0.127673	0.0638366	−	0.997960i	$-0.479666\pi$
$43$	36.0000	0.127673	0.0638366	+	0.997960i	$0.479666\pi$
$44$	144.000	0.493382
$45$	−90.0000	−0.298142
$46$	−528.000	−1.69238
$47$	446.000	1.38417	0.692083	−	0.721818i	$-0.256693\pi$
$47$	446.000	1.38417	0.692083	+	0.721818i	$0.256693\pi$
$48$	−192.000	−0.577350
$49$	−243.000	−0.708455
$50$	−100.000	−0.282843
$51$	−51.0000	−0.140028
$52$	−104.000	−0.277350
$53$	−754.000	−1.95415	−0.977074	−	0.212899i	$-0.931709\pi$
$53$	−754.000	−1.95415	−0.977074	+	0.212899i	$0.931709\pi$
$54$	108.000	0.272166
$55$	−180.000	−0.441294
$56$	0	0
$57$	−222.000	−0.515870
$58$	840.000	1.90168
$59$	−50.0000	−0.110330	−0.0551648	−	0.998477i	$-0.517568\pi$
$59$	−50.0000	−0.110330	−0.0551648	+	0.998477i	$0.517568\pi$
$60$	−240.000	−0.516398
$61$	−226.000	−0.474366	−0.237183	−	0.971465i	$-0.576224\pi$
$61$	−226.000	−0.474366	−0.237183	+	0.971465i	$0.576224\pi$
$62$	−920.000	−1.88452
$63$	−90.0000	−0.179983
$64$	−512.000	−1.00000
$65$	130.000	0.248069
$66$	216.000	0.402845
$67$	582.000	1.06123	0.530617	−	0.847612i	$-0.321960\pi$
$67$	582.000	1.06123	0.530617	+	0.847612i	$0.321960\pi$
$68$	−136.000	−0.242536
$69$	−396.000	−0.690910
$70$	400.000	0.682988
$71$	−370.000	−0.618464	−0.309232	−	0.950987i	$-0.600072\pi$
$71$	−370.000	−0.618464	−0.309232	+	0.950987i	$0.600072\pi$
$72$	0	0
$73$	826.000	1.32433	0.662164	−	0.749359i	$-0.269638\pi$
$73$	826.000	1.32433	0.662164	+	0.749359i	$0.269638\pi$
$74$	−184.000	−0.289048
$75$	−75.0000	−0.115470
$76$	−592.000	−0.893514
$77$	−180.000	−0.266401
$78$	−156.000	−0.226455
$79$	272.000	0.387372	0.193686	−	0.981064i	$-0.437956\pi$
$79$	272.000	0.387372	0.193686	+	0.981064i	$0.437956\pi$
$80$	640.000	0.894427
$81$	81.0000	0.111111
$82$	−456.000	−0.614107
$83$	162.000	0.214239	0.107119	−	0.994246i	$-0.465837\pi$
$83$	162.000	0.214239	0.107119	+	0.994246i	$0.465837\pi$
$84$	−240.000	−0.311740
$85$	170.000	0.216930
$86$	144.000	0.180557
$87$	630.000	0.776357
$88$	0	0
$89$	−186.000	−0.221528	−0.110764	−	0.993847i	$-0.535330\pi$
$89$	−186.000	−0.221528	−0.110764	+	0.993847i	$0.535330\pi$
$90$	−360.000	−0.421637
$91$	130.000	0.149755
$92$	−1056.00	−1.19669
$93$	−690.000	−0.769351
$94$	1784.00	1.95751
$95$	740.000	0.799183
$96$	−768.000	−0.816497
$97$	−790.000	−0.826931	−0.413466	−	0.910520i	$-0.635682\pi$
$97$	−790.000	−0.826931	−0.413466	+	0.910520i	$0.635682\pi$
$98$	−972.000	−1.00191
$99$	162.000	0.164461
$100$	−200.000	−0.200000
$101$	1518.00	1.49551	0.747756	−	0.663974i	$-0.231131\pi$
$101$	1518.00	1.49551	0.747756	+	0.663974i	$0.231131\pi$
$102$	−204.000	−0.198030
$103$	1400.00	1.33928	0.669641	−	0.742685i	$-0.266448\pi$
$103$	1400.00	1.33928	0.669641	+	0.742685i	$0.266448\pi$
$104$	0	0
$105$	300.000	0.278829
$106$	−3016.00	−2.76358
$107$	−836.000	−0.755319	−0.377660	−	0.925944i	$-0.623271\pi$
$107$	−836.000	−0.755319	−0.377660	+	0.925944i	$0.623271\pi$
$108$	216.000	0.192450
$109$	986.000	0.866437	0.433219	−	0.901289i	$-0.357378\pi$
$109$	986.000	0.866437	0.433219	+	0.901289i	$0.357378\pi$
$110$	−720.000	−0.624085
$111$	−138.000	−0.118003
$112$	640.000	0.539949
$113$	78.0000	0.0649347	0.0324674	−	0.999473i	$-0.489664\pi$
$113$	78.0000	0.0649347	0.0324674	+	0.999473i	$0.489664\pi$
$114$	−888.000	−0.729551
$115$	1320.00	1.07035
$116$	1680.00	1.34469
$117$	−117.000	−0.0924500
$118$	−200.000	−0.156030
$119$	170.000	0.130957
$120$	0	0
$121$	−1007.00	−0.756574
$122$	−904.000	−0.670855
$123$	−342.000	−0.250708
$124$	−1840.00	−1.33256
$125$	1500.00	1.07331
$126$	−360.000	−0.254535
$127$	2464.00	1.72161	0.860806	−	0.508934i	$-0.169960\pi$
$127$	2464.00	1.72161	0.860806	+	0.508934i	$0.169960\pi$
$128$	0	0
$129$	108.000	0.0737122
$130$	520.000	0.350823
$131$	−2628.00	−1.75274	−0.876372	−	0.481635i	$-0.840043\pi$
$131$	−2628.00	−1.75274	−0.876372	+	0.481635i	$0.840043\pi$
$132$	432.000	0.284854
$133$	740.000	0.482452
$134$	2328.00	1.50081
$135$	−270.000	−0.172133
$136$	0	0
$137$	−714.000	−0.445264	−0.222632	−	0.974903i	$-0.571465\pi$
$137$	−714.000	−0.445264	−0.222632	+	0.974903i	$0.571465\pi$
$138$	−1584.00	−0.977094
$139$	−356.000	−0.217234	−0.108617	−	0.994084i	$-0.534642\pi$
$139$	−356.000	−0.217234	−0.108617	+	0.994084i	$0.534642\pi$
$140$	800.000	0.482945
$141$	1338.00	0.799148
$142$	−1480.00	−0.874640
$143$	−234.000	−0.136840
$144$	−576.000	−0.333333
$145$	−2100.00	−1.20273
$146$	3304.00	1.87288
$147$	−729.000	−0.409027
$148$	−368.000	−0.204388
$149$	1446.00	0.795040	0.397520	−	0.917594i	$-0.369871\pi$
$149$	1446.00	0.795040	0.397520	+	0.917594i	$0.369871\pi$
$150$	−300.000	−0.163299
$151$	−526.000	−0.283479	−0.141739	−	0.989904i	$-0.545269\pi$
$151$	−526.000	−0.283479	−0.141739	+	0.989904i	$0.545269\pi$
$152$	0	0
$153$	−153.000	−0.0808452
$154$	−720.000	−0.376748
$155$	2300.00	1.19187
$156$	−312.000	−0.160128
$157$	−262.000	−0.133184	−0.0665920	−	0.997780i	$-0.521213\pi$
$157$	−262.000	−0.133184	−0.0665920	+	0.997780i	$0.521213\pi$
$158$	1088.00	0.547827
$159$	−2262.00	−1.12823
$160$	2560.00	1.26491
$161$	1320.00	0.646153
$162$	324.000	0.157135
$163$	−2634.00	−1.26571	−0.632855	−	0.774270i	$-0.718117\pi$
$163$	−2634.00	−1.26571	−0.632855	+	0.774270i	$0.718117\pi$
$164$	−912.000	−0.434239
$165$	−540.000	−0.254781
$166$	648.000	0.302979
$167$	2002.00	0.927661	0.463831	−	0.885924i	$-0.346475\pi$
$167$	2002.00	0.927661	0.463831	+	0.885924i	$0.346475\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	680.000	0.306786
$171$	−666.000	−0.297838
$172$	288.000	0.127673
$173$	1142.00	0.501877	0.250938	−	0.968003i	$-0.419261\pi$
$173$	1142.00	0.501877	0.250938	+	0.968003i	$0.419261\pi$
$174$	2520.00	1.09794
$175$	250.000	0.107990
$176$	−1152.00	−0.493382
$177$	−150.000	−0.0636988
$178$	−744.000	−0.313287
$179$	−2580.00	−1.07731	−0.538654	−	0.842527i	$-0.681067\pi$
$179$	−2580.00	−1.07731	−0.538654	+	0.842527i	$0.681067\pi$
$180$	−720.000	−0.298142
$181$	1874.00	0.769576	0.384788	−	0.923005i	$-0.374274\pi$
$181$	1874.00	0.769576	0.384788	+	0.923005i	$0.374274\pi$
$182$	520.000	0.211786
$183$	−678.000	−0.273875
$184$	0	0
$185$	460.000	0.182810
$186$	−2760.00	−1.08803
$187$	−306.000	−0.119663
$188$	3568.00	1.38417
$189$	−270.000	−0.103913
$190$	2960.00	1.13022
$191$	−412.000	−0.156080	−0.0780400	−	0.996950i	$-0.524866\pi$
$191$	−412.000	−0.156080	−0.0780400	+	0.996950i	$0.524866\pi$
$192$	−1536.00	−0.577350
$193$	1082.00	0.403544	0.201772	−	0.979432i	$-0.435330\pi$
$193$	1082.00	0.403544	0.201772	+	0.979432i	$0.435330\pi$
$194$	−3160.00	−1.16946
$195$	390.000	0.143223
$196$	−1944.00	−0.708455
$197$	−1194.00	−0.431822	−0.215911	−	0.976413i	$-0.569272\pi$
$197$	−1194.00	−0.431822	−0.215911	+	0.976413i	$0.569272\pi$
$198$	648.000	0.232583
$199$	−2860.00	−1.01879	−0.509397	−	0.860532i	$-0.670132\pi$
$199$	−2860.00	−1.01879	−0.509397	+	0.860532i	$0.670132\pi$
$200$	0	0
$201$	1746.00	0.612703
$202$	6072.00	2.11497
$203$	−2100.00	−0.726065
$204$	−408.000	−0.140028
$205$	1140.00	0.388395
$206$	5600.00	1.89403
$207$	−1188.00	−0.398897
$208$	832.000	0.277350
$209$	−1332.00	−0.440844
$210$	1200.00	0.394323
$211$	−5972.00	−1.94848	−0.974240	−	0.225512i	$-0.927594\pi$
$211$	−5972.00	−1.94848	−0.974240	+	0.225512i	$0.927594\pi$
$212$	−6032.00	−1.95415
$213$	−1110.00	−0.357070
$214$	−3344.00	−1.06818
$215$	−360.000	−0.114194
$216$	0	0
$217$	2300.00	0.719512
$218$	3944.00	1.22533
$219$	2478.00	0.764601
$220$	−1440.00	−0.441294
$221$	221.000	0.0672673
$222$	−552.000	−0.166882
$223$	1998.00	0.599982	0.299991	−	0.953942i	$-0.403016\pi$
$223$	1998.00	0.599982	0.299991	+	0.953942i	$0.403016\pi$
$224$	2560.00	0.763604
$225$	−225.000	−0.0666667
$226$	312.000	0.0918316
$227$	−474.000	−0.138592	−0.0692962	−	0.997596i	$-0.522075\pi$
$227$	−474.000	−0.138592	−0.0692962	+	0.997596i	$0.522075\pi$
$228$	−1776.00	−0.515870
$229$	−6390.00	−1.84394	−0.921972	−	0.387257i	$-0.873423\pi$
$229$	−6390.00	−1.84394	−0.921972	+	0.387257i	$0.873423\pi$
$230$	5280.00	1.51371
$231$	−540.000	−0.153807
$232$	0	0
$233$	−6310.00	−1.77417	−0.887086	−	0.461605i	$-0.847274\pi$
$233$	−6310.00	−1.77417	−0.887086	+	0.461605i	$0.847274\pi$
$234$	−468.000	−0.130744
$235$	−4460.00	−1.23804
$236$	−400.000	−0.110330
$237$	816.000	0.223649
$238$	680.000	0.185201
$239$	−870.000	−0.235463	−0.117731	−	0.993045i	$-0.537562\pi$
$239$	−870.000	−0.235463	−0.117731	+	0.993045i	$0.537562\pi$
$240$	1920.00	0.516398
$241$	6058.00	1.61921	0.809606	−	0.586974i	$-0.199681\pi$
$241$	6058.00	1.61921	0.809606	+	0.586974i	$0.199681\pi$
$242$	−4028.00	−1.06996
$243$	243.000	0.0641500
$244$	−1808.00	−0.474366
$245$	2430.00	0.633661
$246$	−1368.00	−0.354555
$247$	962.000	0.247816
$248$	0	0
$249$	486.000	0.123691
$250$	6000.00	1.51789
$251$	436.000	0.109642	0.0548209	−	0.998496i	$-0.482541\pi$
$251$	436.000	0.109642	0.0548209	+	0.998496i	$0.482541\pi$
$252$	−720.000	−0.179983
$253$	−2376.00	−0.590426
$254$	9856.00	2.43473
$255$	510.000	0.125245
$256$	4096.00	1.00000
$257$	5402.00	1.31116	0.655579	−	0.755127i	$-0.272425\pi$
$257$	5402.00	1.31116	0.655579	+	0.755127i	$0.272425\pi$
$258$	432.000	0.104245
$259$	460.000	0.110359
$260$	1040.00	0.248069
$261$	1890.00	0.448230
$262$	−10512.0	−2.47875
$263$	6364.00	1.49210	0.746048	−	0.665893i	$-0.231949\pi$
$263$	6364.00	1.49210	0.746048	+	0.665893i	$0.231949\pi$
$264$	0	0
$265$	7540.00	1.74784
$266$	2960.00	0.682290
$267$	−558.000	−0.127899
$268$	4656.00	1.06123
$269$	−3918.00	−0.888047	−0.444024	−	0.896015i	$-0.646449\pi$
$269$	−3918.00	−0.888047	−0.444024	+	0.896015i	$0.646449\pi$
$270$	−1080.00	−0.243432
$271$	2938.00	0.658564	0.329282	−	0.944232i	$-0.393193\pi$
$271$	2938.00	0.658564	0.329282	+	0.944232i	$0.393193\pi$
$272$	1088.00	0.242536
$273$	390.000	0.0864611
$274$	−2856.00	−0.629698
$275$	−450.000	−0.0986764
$276$	−3168.00	−0.690910
$277$	−6226.00	−1.35048	−0.675242	−	0.737596i	$-0.735961\pi$
$277$	−6226.00	−1.35048	−0.675242	+	0.737596i	$0.735961\pi$
$278$	−1424.00	−0.307215
$279$	−2070.00	−0.444185
$280$	0	0
$281$	1070.00	0.227156	0.113578	−	0.993529i	$-0.463769\pi$
$281$	1070.00	0.227156	0.113578	+	0.993529i	$0.463769\pi$
$282$	5352.00	1.13017
$283$	1152.00	0.241976	0.120988	−	0.992654i	$-0.461394\pi$
$283$	1152.00	0.241976	0.120988	+	0.992654i	$0.461394\pi$
$284$	−2960.00	−0.618464
$285$	2220.00	0.461409
$286$	−936.000	−0.193520
$287$	1140.00	0.234467
$288$	−2304.00	−0.471405
$289$	289.000	0.0588235
$290$	−8400.00	−1.70091
$291$	−2370.00	−0.477429
$292$	6608.00	1.32433
$293$	−8314.00	−1.65771	−0.828855	−	0.559463i	$-0.811007\pi$
$293$	−8314.00	−1.65771	−0.828855	+	0.559463i	$0.811007\pi$
$294$	−2916.00	−0.578451
$295$	500.000	0.0986818
$296$	0	0
$297$	486.000	0.0949514
$298$	5784.00	1.12436
$299$	1716.00	0.331902
$300$	−600.000	−0.115470
$301$	−360.000	−0.0689371
$302$	−2104.00	−0.400899
$303$	4554.00	0.863434
$304$	4736.00	0.893514
$305$	2260.00	0.424286
$306$	−612.000	−0.114332
$307$	2866.00	0.532805	0.266403	−	0.963862i	$-0.414165\pi$
$307$	2866.00	0.532805	0.266403	+	0.963862i	$0.414165\pi$
$308$	−1440.00	−0.266401
$309$	4200.00	0.773235
$310$	9200.00	1.68556
$311$	−7428.00	−1.35435	−0.677176	−	0.735821i	$-0.736796\pi$
$311$	−7428.00	−1.35435	−0.677176	+	0.735821i	$0.736796\pi$
$312$	0	0
$313$	−4786.00	−0.864283	−0.432142	−	0.901806i	$-0.642242\pi$
$313$	−4786.00	−0.864283	−0.432142	+	0.901806i	$0.642242\pi$
$314$	−1048.00	−0.188351
$315$	900.000	0.160982
$316$	2176.00	0.387372
$317$	−4402.00	−0.779940	−0.389970	−	0.920828i	$-0.627515\pi$
$317$	−4402.00	−0.779940	−0.389970	+	0.920828i	$0.627515\pi$
$318$	−9048.00	−1.59556
$319$	3780.00	0.663446
$320$	5120.00	0.894427
$321$	−2508.00	−0.436084
$322$	5280.00	0.913798
$323$	1258.00	0.216709
$324$	648.000	0.111111
$325$	325.000	0.0554700
$326$	−10536.0	−1.78998
$327$	2958.00	0.500238
$328$	0	0
$329$	−4460.00	−0.747379
$330$	−2160.00	−0.360315
$331$	5470.00	0.908334	0.454167	−	0.890917i	$-0.349937\pi$
$331$	5470.00	0.908334	0.454167	+	0.890917i	$0.349937\pi$
$332$	1296.00	0.214239
$333$	−414.000	−0.0681293
$334$	8008.00	1.31191
$335$	−5820.00	−0.949196
$336$	1920.00	0.311740
$337$	−3634.00	−0.587408	−0.293704	−	0.955896i	$-0.594888\pi$
$337$	−3634.00	−0.587408	−0.293704	+	0.955896i	$0.594888\pi$
$338$	676.000	0.108786
$339$	234.000	0.0374901
$340$	1360.00	0.216930
$341$	−4140.00	−0.657459
$342$	−2664.00	−0.421206
$343$	5860.00	0.922479
$344$	0	0
$345$	3960.00	0.617969
$346$	4568.00	0.709761
$347$	−11588.0	−1.79273	−0.896364	−	0.443319i	$-0.853801\pi$
$347$	−11588.0	−1.79273	−0.896364	+	0.443319i	$0.853801\pi$
$348$	5040.00	0.776357
$349$	2434.00	0.373321	0.186661	−	0.982424i	$-0.440234\pi$
$349$	2434.00	0.373321	0.186661	+	0.982424i	$0.440234\pi$
$350$	1000.00	0.152721
$351$	−351.000	−0.0533761
$352$	−4608.00	−0.697748
$353$	−8258.00	−1.24512	−0.622562	−	0.782570i	$-0.713908\pi$
$353$	−8258.00	−1.24512	−0.622562	+	0.782570i	$0.713908\pi$
$354$	−600.000	−0.0900837
$355$	3700.00	0.553171
$356$	−1488.00	−0.221528
$357$	510.000	0.0756080
$358$	−10320.0	−1.52354
$359$	−7594.00	−1.11642	−0.558212	−	0.829699i	$-0.688512\pi$
$359$	−7594.00	−1.11642	−0.558212	+	0.829699i	$0.688512\pi$
$360$	0	0
$361$	−1383.00	−0.201633
$362$	7496.00	1.08835
$363$	−3021.00	−0.436808
$364$	1040.00	0.149755
$365$	−8260.00	−1.18452
$366$	−2712.00	−0.387318
$367$	−4704.00	−0.669065	−0.334532	−	0.942384i	$-0.608578\pi$
$367$	−4704.00	−0.669065	−0.334532	+	0.942384i	$0.608578\pi$
$368$	8448.00	1.19669
$369$	−1026.00	−0.144746
$370$	1840.00	0.258533
$371$	7540.00	1.05514
$372$	−5520.00	−0.769351
$373$	4178.00	0.579970	0.289985	−	0.957031i	$-0.406350\pi$
$373$	4178.00	0.579970	0.289985	+	0.957031i	$0.406350\pi$
$374$	−1224.00	−0.169229
$375$	4500.00	0.619677
$376$	0	0
$377$	−2730.00	−0.372950
$378$	−1080.00	−0.146956
$379$	7450.00	1.00971	0.504856	−	0.863204i	$-0.331546\pi$
$379$	7450.00	1.00971	0.504856	+	0.863204i	$0.331546\pi$
$380$	5920.00	0.799183
$381$	7392.00	0.993973
$382$	−1648.00	−0.220730
$383$	−7398.00	−0.986998	−0.493499	−	0.869746i	$-0.664282\pi$
$383$	−7398.00	−0.986998	−0.493499	+	0.869746i	$0.664282\pi$
$384$	0	0
$385$	1800.00	0.238277
$386$	4328.00	0.570698
$387$	324.000	0.0425577
$388$	−6320.00	−0.826931
$389$	10234.0	1.33389	0.666947	−	0.745106i	$-0.267601\pi$
$389$	10234.0	1.33389	0.666947	+	0.745106i	$0.267601\pi$
$390$	1560.00	0.202548
$391$	2244.00	0.290240
$392$	0	0
$393$	−7884.00	−1.01195
$394$	−4776.00	−0.610689
$395$	−2720.00	−0.346476
$396$	1296.00	0.164461
$397$	−1406.00	−0.177746	−0.0888729	−	0.996043i	$-0.528327\pi$
$397$	−1406.00	−0.177746	−0.0888729	+	0.996043i	$0.528327\pi$
$398$	−11440.0	−1.44079
$399$	2220.00	0.278544
$400$	1600.00	0.200000
$401$	8550.00	1.06475	0.532377	−	0.846507i	$-0.321299\pi$
$401$	8550.00	1.06475	0.532377	+	0.846507i	$0.321299\pi$
$402$	6984.00	0.866493
$403$	2990.00	0.369584
$404$	12144.0	1.49551
$405$	−810.000	−0.0993808
$406$	−8400.00	−1.02681
$407$	−828.000	−0.100841
$408$	0	0
$409$	6730.00	0.813636	0.406818	−	0.913509i	$-0.366638\pi$
$409$	6730.00	0.813636	0.406818	+	0.913509i	$0.366638\pi$
$410$	4560.00	0.549274
$411$	−2142.00	−0.257073
$412$	11200.0	1.33928
$413$	500.000	0.0595724
$414$	−4752.00	−0.564126
$415$	−1620.00	−0.191621
$416$	3328.00	0.392232
$417$	−1068.00	−0.125420
$418$	−5328.00	−0.623447
$419$	−3988.00	−0.464980	−0.232490	−	0.972599i	$-0.574687\pi$
$419$	−3988.00	−0.464980	−0.232490	+	0.972599i	$0.574687\pi$
$420$	2400.00	0.278829
$421$	14450.0	1.67280	0.836401	−	0.548118i	$-0.184655\pi$
$421$	14450.0	1.67280	0.836401	+	0.548118i	$0.184655\pi$
$422$	−23888.0	−2.75557
$423$	4014.00	0.461389
$424$	0	0
$425$	425.000	0.0485071
$426$	−4440.00	−0.504973
$427$	2260.00	0.256134
$428$	−6688.00	−0.755319
$429$	−702.000	−0.0790044
$430$	−1440.00	−0.161495
$431$	774.000	0.0865018	0.0432509	−	0.999064i	$-0.486229\pi$
$431$	774.000	0.0865018	0.0432509	+	0.999064i	$0.486229\pi$
$432$	−1728.00	−0.192450
$433$	11970.0	1.32850	0.664251	−	0.747509i	$-0.268751\pi$
$433$	11970.0	1.32850	0.664251	+	0.747509i	$0.268751\pi$
$434$	9200.00	1.01754
$435$	−6300.00	−0.694395
$436$	7888.00	0.866437
$437$	9768.00	1.06926
$438$	9912.00	1.08131
$439$	12884.0	1.40073	0.700364	−	0.713786i	$-0.253021\pi$
$439$	12884.0	1.40073	0.700364	+	0.713786i	$0.253021\pi$
$440$	0	0
$441$	−2187.00	−0.236152
$442$	884.000	0.0951303
$443$	−12928.0	−1.38652	−0.693259	−	0.720688i	$-0.743826\pi$
$443$	−12928.0	−1.38652	−0.693259	+	0.720688i	$0.743826\pi$
$444$	−1104.00	−0.118003
$445$	1860.00	0.198140
$446$	7992.00	0.848503
$447$	4338.00	0.459016
$448$	5120.00	0.539949
$449$	3814.00	0.400877	0.200439	−	0.979706i	$-0.435763\pi$
$449$	3814.00	0.400877	0.200439	+	0.979706i	$0.435763\pi$
$450$	−900.000	−0.0942809
$451$	−2052.00	−0.214246
$452$	624.000	0.0649347
$453$	−1578.00	−0.163666
$454$	−1896.00	−0.195999
$455$	−1300.00	−0.133945
$456$	0	0
$457$	−5390.00	−0.551715	−0.275857	−	0.961199i	$-0.588962\pi$
$457$	−5390.00	−0.551715	−0.275857	+	0.961199i	$0.588962\pi$
$458$	−25560.0	−2.60773
$459$	−459.000	−0.0466760
$460$	10560.0	1.07035
$461$	−6634.00	−0.670230	−0.335115	−	0.942177i	$-0.608775\pi$
$461$	−6634.00	−0.670230	−0.335115	+	0.942177i	$0.608775\pi$
$462$	−2160.00	−0.217516
$463$	4370.00	0.438642	0.219321	−	0.975653i	$-0.429616\pi$
$463$	4370.00	0.438642	0.219321	+	0.975653i	$0.429616\pi$
$464$	−13440.0	−1.34469
$465$	6900.00	0.688129
$466$	−25240.0	−2.50906
$467$	−15164.0	−1.50258	−0.751291	−	0.659971i	$-0.770569\pi$
$467$	−15164.0	−1.50258	−0.751291	+	0.659971i	$0.770569\pi$
$468$	−936.000	−0.0924500
$469$	−5820.00	−0.573012
$470$	−17840.0	−1.75085
$471$	−786.000	−0.0768938
$472$	0	0
$473$	648.000	0.0629917
$474$	3264.00	0.316288
$475$	1850.00	0.178703
$476$	1360.00	0.130957
$477$	−6786.00	−0.651383
$478$	−3480.00	−0.332995
$479$	10162.0	0.969340	0.484670	−	0.874697i	$-0.338940\pi$
$479$	10162.0	0.969340	0.484670	+	0.874697i	$0.338940\pi$
$480$	7680.00	0.730297
$481$	598.000	0.0566870
$482$	24232.0	2.28991
$483$	3960.00	0.373056
$484$	−8056.00	−0.756574
$485$	7900.00	0.739630
$486$	972.000	0.0907218
$487$	−17174.0	−1.59800	−0.799002	−	0.601328i	$-0.794639\pi$
$487$	−17174.0	−1.59800	−0.799002	+	0.601328i	$0.794639\pi$
$488$	0	0
$489$	−7902.00	−0.730758
$490$	9720.00	0.896132
$491$	−3828.00	−0.351844	−0.175922	−	0.984404i	$-0.556291\pi$
$491$	−3828.00	−0.351844	−0.175922	+	0.984404i	$0.556291\pi$
$492$	−2736.00	−0.250708
$493$	−3570.00	−0.326135
$494$	3848.00	0.350465
$495$	−1620.00	−0.147098
$496$	14720.0	1.33256
$497$	3700.00	0.333939
$498$	1944.00	0.174925
$499$	−10894.0	−0.977319	−0.488660	−	0.872474i	$-0.662514\pi$
$499$	−10894.0	−0.977319	−0.488660	+	0.872474i	$0.662514\pi$
$500$	12000.0	1.07331
$501$	6006.00	0.535585
$502$	1744.00	0.155057
$503$	2336.00	0.207072	0.103536	−	0.994626i	$-0.466984\pi$
$503$	2336.00	0.207072	0.103536	+	0.994626i	$0.466984\pi$
$504$	0	0
$505$	−15180.0	−1.33763
$506$	−9504.00	−0.834989
$507$	507.000	0.0444116
$508$	19712.0	1.72161
$509$	4382.00	0.381589	0.190794	−	0.981630i	$-0.438894\pi$
$509$	4382.00	0.381589	0.190794	+	0.981630i	$0.438894\pi$
$510$	2040.00	0.177123
$511$	−8260.00	−0.715070
$512$	16384.0	1.41421
$513$	−1998.00	−0.171957
$514$	21608.0	1.85426
$515$	−14000.0	−1.19789
$516$	864.000	0.0737122
$517$	8028.00	0.682923
$518$	1840.00	0.156071
$519$	3426.00	0.289759
$520$	0	0
$521$	−14634.0	−1.23057	−0.615285	−	0.788305i	$-0.710959\pi$
$521$	−14634.0	−1.23057	−0.615285	+	0.788305i	$0.710959\pi$
$522$	7560.00	0.633893
$523$	20696.0	1.73035	0.865175	−	0.501470i	$-0.167207\pi$
$523$	20696.0	1.73035	0.865175	+	0.501470i	$0.167207\pi$
$524$	−21024.0	−1.75274
$525$	750.000	0.0623480
$526$	25456.0	2.11014
$527$	3910.00	0.323192
$528$	−3456.00	−0.284854
$529$	5257.00	0.432070
$530$	30160.0	2.47182
$531$	−450.000	−0.0367765
$532$	5920.00	0.482452
$533$	1482.00	0.120436
$534$	−2232.00	−0.180877
$535$	8360.00	0.675578
$536$	0	0
$537$	−7740.00	−0.621984
$538$	−15672.0	−1.25589
$539$	−4374.00	−0.349539
$540$	−2160.00	−0.172133
$541$	18034.0	1.43316	0.716582	−	0.697502i	$-0.245705\pi$
$541$	18034.0	1.43316	0.716582	+	0.697502i	$0.245705\pi$
$542$	11752.0	0.931350
$543$	5622.00	0.444315
$544$	4352.00	0.342997
$545$	−9860.00	−0.774965
$546$	1560.00	0.122274
$547$	−10836.0	−0.847009	−0.423504	−	0.905894i	$-0.639200\pi$
$547$	−10836.0	−0.847009	−0.423504	+	0.905894i	$0.639200\pi$
$548$	−5712.00	−0.445264
$549$	−2034.00	−0.158122
$550$	−1800.00	−0.139550
$551$	−15540.0	−1.20150
$552$	0	0
$553$	−2720.00	−0.209161
$554$	−24904.0	−1.90987
$555$	1380.00	0.105545
$556$	−2848.00	−0.217234
$557$	14566.0	1.10805	0.554023	−	0.832502i	$-0.313092\pi$
$557$	14566.0	1.10805	0.554023	+	0.832502i	$0.313092\pi$
$558$	−8280.00	−0.628173
$559$	−468.000	−0.0354102
$560$	−6400.00	−0.482945
$561$	−918.000	−0.0690873
$562$	4280.00	0.321247
$563$	−13380.0	−1.00160	−0.500799	−	0.865564i	$-0.666960\pi$
$563$	−13380.0	−1.00160	−0.500799	+	0.865564i	$0.666960\pi$
$564$	10704.0	0.799148
$565$	−780.000	−0.0580794
$566$	4608.00	0.342206
$567$	−810.000	−0.0599944
$568$	0	0
$569$	−1582.00	−0.116557	−0.0582785	−	0.998300i	$-0.518561\pi$
$569$	−1582.00	−0.116557	−0.0582785	+	0.998300i	$0.518561\pi$
$570$	8880.00	0.652530
$571$	9680.00	0.709449	0.354725	−	0.934971i	$-0.384575\pi$
$571$	9680.00	0.709449	0.354725	+	0.934971i	$0.384575\pi$
$572$	−1872.00	−0.136840
$573$	−1236.00	−0.0901128
$574$	4560.00	0.331587
$575$	3300.00	0.239338
$576$	−4608.00	−0.333333
$577$	−24574.0	−1.77301	−0.886507	−	0.462715i	$-0.846875\pi$
$577$	−24574.0	−1.77301	−0.886507	+	0.462715i	$0.846875\pi$
$578$	1156.00	0.0831890
$579$	3246.00	0.232986
$580$	−16800.0	−1.20273
$581$	−1620.00	−0.115678
$582$	−9480.00	−0.675187
$583$	−13572.0	−0.964142
$584$	0	0
$585$	1170.00	0.0826898
$586$	−33256.0	−2.34436
$587$	10722.0	0.753909	0.376954	−	0.926232i	$-0.376971\pi$
$587$	10722.0	0.753909	0.376954	+	0.926232i	$0.376971\pi$
$588$	−5832.00	−0.409027
$589$	17020.0	1.19066
$590$	2000.00	0.139557
$591$	−3582.00	−0.249313
$592$	2944.00	0.204388
$593$	−8930.00	−0.618400	−0.309200	−	0.950997i	$-0.600061\pi$
$593$	−8930.00	−0.618400	−0.309200	+	0.950997i	$0.600061\pi$
$594$	1944.00	0.134282
$595$	−1700.00	−0.117131
$596$	11568.0	0.795040
$597$	−8580.00	−0.588201
$598$	6864.00	0.469381
$599$	−20284.0	−1.38361	−0.691804	−	0.722085i	$-0.743184\pi$
$599$	−20284.0	−1.38361	−0.691804	+	0.722085i	$0.743184\pi$
$600$	0	0
$601$	7790.00	0.528720	0.264360	−	0.964424i	$-0.414839\pi$
$601$	7790.00	0.528720	0.264360	+	0.964424i	$0.414839\pi$
$602$	−1440.00	−0.0974917
$603$	5238.00	0.353744
$604$	−4208.00	−0.283479
$605$	10070.0	0.676700
$606$	18216.0	1.22108
$607$	13008.0	0.869816	0.434908	−	0.900475i	$-0.356781\pi$
$607$	13008.0	0.869816	0.434908	+	0.900475i	$0.356781\pi$
$608$	18944.0	1.26362
$609$	−6300.00	−0.419194
$610$	9040.00	0.600031
$611$	−5798.00	−0.383898
$612$	−1224.00	−0.0808452
$613$	−26478.0	−1.74459	−0.872297	−	0.488976i	$-0.837371\pi$
$613$	−26478.0	−1.74459	−0.872297	+	0.488976i	$0.837371\pi$
$614$	11464.0	0.753501
$615$	3420.00	0.224240
$616$	0	0
$617$	−18290.0	−1.19340	−0.596700	−	0.802464i	$-0.703522\pi$
$617$	−18290.0	−1.19340	−0.596700	+	0.802464i	$0.703522\pi$
$618$	16800.0	1.09352
$619$	10614.0	0.689197	0.344598	−	0.938750i	$-0.388015\pi$
$619$	10614.0	0.689197	0.344598	+	0.938750i	$0.388015\pi$
$620$	18400.0	1.19187
$621$	−3564.00	−0.230303
$622$	−29712.0	−1.91534
$623$	1860.00	0.119614
$624$	2496.00	0.160128
$625$	−11875.0	−0.760000
$626$	−19144.0	−1.22228
$627$	−3996.00	−0.254521
$628$	−2096.00	−0.133184
$629$	782.000	0.0495714
$630$	3600.00	0.227663
$631$	−20062.0	−1.26570	−0.632849	−	0.774275i	$-0.718115\pi$
$631$	−20062.0	−1.26570	−0.632849	+	0.774275i	$0.718115\pi$
$632$	0	0
$633$	−17916.0	−1.12496
$634$	−17608.0	−1.10300
$635$	−24640.0	−1.53986
$636$	−18096.0	−1.12823
$637$	3159.00	0.196490
$638$	15120.0	0.938255
$639$	−3330.00	−0.206155
$640$	0	0
$641$	−12962.0	−0.798702	−0.399351	−	0.916798i	$-0.630765\pi$
$641$	−12962.0	−0.798702	−0.399351	+	0.916798i	$0.630765\pi$
$642$	−10032.0	−0.616716
$643$	−21238.0	−1.30256	−0.651279	−	0.758838i	$-0.725767\pi$
$643$	−21238.0	−1.30256	−0.651279	+	0.758838i	$0.725767\pi$
$644$	10560.0	0.646153
$645$	−1080.00	−0.0659302
$646$	5032.00	0.306473
$647$	−20480.0	−1.24444	−0.622219	−	0.782843i	$-0.713769\pi$
$647$	−20480.0	−1.24444	−0.622219	+	0.782843i	$0.713769\pi$
$648$	0	0
$649$	−900.000	−0.0544347
$650$	1300.00	0.0784465
$651$	6900.00	0.415411
$652$	−21072.0	−1.26571
$653$	−14558.0	−0.872433	−0.436216	−	0.899842i	$-0.643682\pi$
$653$	−14558.0	−0.872433	−0.436216	+	0.899842i	$0.643682\pi$
$654$	11832.0	0.707443
$655$	26280.0	1.56770
$656$	7296.00	0.434239
$657$	7434.00	0.441443
$658$	−17840.0	−1.05695
$659$	888.000	0.0524910	0.0262455	−	0.999656i	$-0.491645\pi$
$659$	888.000	0.0524910	0.0262455	+	0.999656i	$0.491645\pi$
$660$	−4320.00	−0.254781
$661$	−31142.0	−1.83250	−0.916251	−	0.400605i	$-0.868800\pi$
$661$	−31142.0	−1.83250	−0.916251	+	0.400605i	$0.868800\pi$
$662$	21880.0	1.28458
$663$	663.000	0.0388368
$664$	0	0
$665$	−7400.00	−0.431518
$666$	−1656.00	−0.0963494
$667$	−27720.0	−1.60918
$668$	16016.0	0.927661
$669$	5994.00	0.346400
$670$	−23280.0	−1.34237
$671$	−4068.00	−0.234044
$672$	7680.00	0.440867
$673$	−15666.0	−0.897296	−0.448648	−	0.893709i	$-0.648094\pi$
$673$	−15666.0	−0.897296	−0.448648	+	0.893709i	$0.648094\pi$
$674$	−14536.0	−0.830721
$675$	−675.000	−0.0384900
$676$	1352.00	0.0769231
$677$	−14098.0	−0.800340	−0.400170	−	0.916441i	$-0.631049\pi$
$677$	−14098.0	−0.800340	−0.400170	+	0.916441i	$0.631049\pi$
$678$	936.000	0.0530190
$679$	7900.00	0.446501
$680$	0	0
$681$	−1422.00	−0.0800164
$682$	−16560.0	−0.929788
$683$	666.000	0.0373115	0.0186558	−	0.999826i	$-0.494061\pi$
$683$	666.000	0.0373115	0.0186558	+	0.999826i	$0.494061\pi$
$684$	−5328.00	−0.297838
$685$	7140.00	0.398256
$686$	23440.0	1.30458
$687$	−19170.0	−1.06460
$688$	−2304.00	−0.127673
$689$	9802.00	0.541983
$690$	15840.0	0.873940
$691$	−4254.00	−0.234197	−0.117098	−	0.993120i	$-0.537359\pi$
$691$	−4254.00	−0.234197	−0.117098	+	0.993120i	$0.537359\pi$
$692$	9136.00	0.501877
$693$	−1620.00	−0.0888004
$694$	−46352.0	−2.53530
$695$	3560.00	0.194300
$696$	0	0
$697$	1938.00	0.105318
$698$	9736.00	0.527956
$699$	−18930.0	−1.02432
$700$	2000.00	0.107990
$701$	−9230.00	−0.497307	−0.248654	−	0.968592i	$-0.579988\pi$
$701$	−9230.00	−0.497307	−0.248654	+	0.968592i	$0.579988\pi$
$702$	−1404.00	−0.0754851
$703$	3404.00	0.182623
$704$	−9216.00	−0.493382
$705$	−13380.0	−0.714780
$706$	−33032.0	−1.76087
$707$	−15180.0	−0.807500
$708$	−1200.00	−0.0636988
$709$	1994.00	0.105622	0.0528112	−	0.998605i	$-0.483182\pi$
$709$	1994.00	0.105622	0.0528112	+	0.998605i	$0.483182\pi$
$710$	14800.0	0.782302
$711$	2448.00	0.129124
$712$	0	0
$713$	30360.0	1.59466
$714$	2040.00	0.106926
$715$	2340.00	0.122393
$716$	−20640.0	−1.07731
$717$	−2610.00	−0.135945
$718$	−30376.0	−1.57886
$719$	2260.00	0.117224	0.0586118	−	0.998281i	$-0.481333\pi$
$719$	2260.00	0.117224	0.0586118	+	0.998281i	$0.481333\pi$
$720$	5760.00	0.298142
$721$	−14000.0	−0.723145
$722$	−5532.00	−0.285152
$723$	18174.0	0.934852
$724$	14992.0	0.769576
$725$	−5250.00	−0.268938
$726$	−12084.0	−0.617740
$727$	1136.00	0.0579531	0.0289766	−	0.999580i	$-0.490775\pi$
$727$	1136.00	0.0579531	0.0289766	+	0.999580i	$0.490775\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−33040.0	−1.67516
$731$	−612.000	−0.0309653
$732$	−5424.00	−0.273875
$733$	17434.0	0.878499	0.439249	−	0.898365i	$-0.355244\pi$
$733$	17434.0	0.878499	0.439249	+	0.898365i	$0.355244\pi$
$734$	−18816.0	−0.946201
$735$	7290.00	0.365844
$736$	33792.0	1.69238
$737$	10476.0	0.523594
$738$	−4104.00	−0.204702
$739$	25234.0	1.25609	0.628043	−	0.778179i	$-0.283856\pi$
$739$	25234.0	1.25609	0.628043	+	0.778179i	$0.283856\pi$
$740$	3680.00	0.182810
$741$	2886.00	0.143077
$742$	30160.0	1.49219
$743$	−33546.0	−1.65637	−0.828185	−	0.560454i	$-0.810627\pi$
$743$	−33546.0	−1.65637	−0.828185	+	0.560454i	$0.810627\pi$
$744$	0	0
$745$	−14460.0	−0.711105
$746$	16712.0	0.820201
$747$	1458.00	0.0714129
$748$	−2448.00	−0.119663
$749$	8360.00	0.407834
$750$	18000.0	0.876356
$751$	−4636.00	−0.225260	−0.112630	−	0.993637i	$-0.535927\pi$
$751$	−4636.00	−0.225260	−0.112630	+	0.993637i	$0.535927\pi$
$752$	−28544.0	−1.38417
$753$	1308.00	0.0633017
$754$	−10920.0	−0.527431
$755$	5260.00	0.253551
$756$	−2160.00	−0.103913
$757$	−11010.0	−0.528620	−0.264310	−	0.964438i	$-0.585144\pi$
$757$	−11010.0	−0.528620	−0.264310	+	0.964438i	$0.585144\pi$
$758$	29800.0	1.42795
$759$	−7128.00	−0.340883
$760$	0	0
$761$	27686.0	1.31881	0.659407	−	0.751787i	$-0.270808\pi$
$761$	27686.0	1.31881	0.659407	+	0.751787i	$0.270808\pi$
$762$	29568.0	1.40569
$763$	−9860.00	−0.467832
$764$	−3296.00	−0.156080
$765$	1530.00	0.0723102
$766$	−29592.0	−1.39583
$767$	650.000	0.0305999
$768$	12288.0	0.577350
$769$	−21934.0	−1.02856	−0.514278	−	0.857623i	$-0.671940\pi$
$769$	−21934.0	−1.02856	−0.514278	+	0.857623i	$0.671940\pi$
$770$	7200.00	0.336974
$771$	16206.0	0.756997
$772$	8656.00	0.403544
$773$	27630.0	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$773$	27630.0	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$774$	1296.00	0.0601857
$775$	5750.00	0.266511
$776$	0	0
$777$	1380.00	0.0637159
$778$	40936.0	1.88641
$779$	8436.00	0.387999
$780$	3120.00	0.143223
$781$	−6660.00	−0.305139
$782$	8976.00	0.410462
$783$	5670.00	0.258786
$784$	15552.0	0.708455
$785$	2620.00	0.119123
$786$	−31536.0	−1.43111
$787$	16886.0	0.764830	0.382415	−	0.923991i	$-0.375093\pi$
$787$	16886.0	0.764830	0.382415	+	0.923991i	$0.375093\pi$
$788$	−9552.00	−0.431822
$789$	19092.0	0.861462
$790$	−10880.0	−0.489991
$791$	−780.000	−0.0350615
$792$	0	0
$793$	2938.00	0.131565
$794$	−5624.00	−0.251371
$795$	22620.0	1.00912
$796$	−22880.0	−1.01879
$797$	3282.00	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$797$	3282.00	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$798$	8880.00	0.393921
$799$	−7582.00	−0.335709
$800$	6400.00	0.282843
$801$	−1674.00	−0.0738425
$802$	34200.0	1.50579
$803$	14868.0	0.653400
$804$	13968.0	0.612703
$805$	−13200.0	−0.577936
$806$	11960.0	0.522671
$807$	−11754.0	−0.512714
$808$	0	0
$809$	9830.00	0.427199	0.213600	−	0.976921i	$-0.431481\pi$
$809$	9830.00	0.427199	0.213600	+	0.976921i	$0.431481\pi$
$810$	−3240.00	−0.140546
$811$	−582.000	−0.0251995	−0.0125997	−	0.999921i	$-0.504011\pi$
$811$	−582.000	−0.0251995	−0.0125997	+	0.999921i	$0.504011\pi$
$812$	−16800.0	−0.726065
$813$	8814.00	0.380222
$814$	−3312.00	−0.142611
$815$	26340.0	1.13209
$816$	3264.00	0.140028
$817$	−2664.00	−0.114078
$818$	26920.0	1.15065
$819$	1170.00	0.0499183
$820$	9120.00	0.388395
$821$	−32650.0	−1.38793	−0.693966	−	0.720007i	$-0.744138\pi$
$821$	−32650.0	−1.38793	−0.693966	+	0.720007i	$0.744138\pi$
$822$	−8568.00	−0.363556
$823$	30084.0	1.27419	0.637097	−	0.770783i	$-0.280135\pi$
$823$	30084.0	1.27419	0.637097	+	0.770783i	$0.280135\pi$
$824$	0	0
$825$	−1350.00	−0.0569709
$826$	2000.00	0.0842481
$827$	16842.0	0.708167	0.354083	−	0.935214i	$-0.384793\pi$
$827$	16842.0	0.708167	0.354083	+	0.935214i	$0.384793\pi$
$828$	−9504.00	−0.398897
$829$	22430.0	0.939718	0.469859	−	0.882742i	$-0.344305\pi$
$829$	22430.0	0.939718	0.469859	+	0.882742i	$0.344305\pi$
$830$	−6480.00	−0.270993
$831$	−18678.0	−0.779702
$832$	6656.00	0.277350
$833$	4131.00	0.171826
$834$	−4272.00	−0.177371
$835$	−20020.0	−0.829725
$836$	−10656.0	−0.440844
$837$	−6210.00	−0.256450
$838$	−15952.0	−0.657581
$839$	13074.0	0.537979	0.268990	−	0.963143i	$-0.413310\pi$
$839$	13074.0	0.537979	0.268990	+	0.963143i	$0.413310\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	57800.0	2.36570
$843$	3210.00	0.131149
$844$	−47776.0	−1.94848
$845$	−1690.00	−0.0688021
$846$	16056.0	0.652502
$847$	10070.0	0.408512
$848$	48256.0	1.95415
$849$	3456.00	0.139705
$850$	1700.00	0.0685994
$851$	6072.00	0.244589
$852$	−8880.00	−0.357070
$853$	47938.0	1.92423	0.962114	−	0.272649i	$-0.0878997\pi$
$853$	47938.0	1.92423	0.962114	+	0.272649i	$0.0878997\pi$
$854$	9040.00	0.362228
$855$	6660.00	0.266394
$856$	0	0
$857$	−38.0000	−0.00151465	−0.000757325	−	1.00000i	$-0.500241\pi$
$857$	−38.0000	−0.00151465	−0.000757325		1.00000i	$0.500241\pi$
$858$	−2808.00	−0.111729
$859$	18256.0	0.725130	0.362565	−	0.931958i	$-0.381901\pi$
$859$	18256.0	0.725130	0.362565	+	0.931958i	$0.381901\pi$
$860$	−2880.00	−0.114194
$861$	3420.00	0.135370
$862$	3096.00	0.122332
$863$	14994.0	0.591427	0.295714	−	0.955277i	$-0.404443\pi$
$863$	14994.0	0.591427	0.295714	+	0.955277i	$0.404443\pi$
$864$	−6912.00	−0.272166
$865$	−11420.0	−0.448892
$866$	47880.0	1.87879
$867$	867.000	0.0339618
$868$	18400.0	0.719512
$869$	4896.00	0.191122
$870$	−25200.0	−0.982023
$871$	−7566.00	−0.294333
$872$	0	0
$873$	−7110.00	−0.275644
$874$	39072.0	1.51216
$875$	−15000.0	−0.579534
$876$	19824.0	0.764601
$877$	−43478.0	−1.67406	−0.837028	−	0.547160i	$-0.815709\pi$
$877$	−43478.0	−1.67406	−0.837028	+	0.547160i	$0.815709\pi$
$878$	51536.0	1.98093
$879$	−24942.0	−0.957079
$880$	11520.0	0.441294
$881$	−36670.0	−1.40232	−0.701160	−	0.713004i	$-0.747334\pi$
$881$	−36670.0	−1.40232	−0.701160	+	0.713004i	$0.747334\pi$
$882$	−8748.00	−0.333969
$883$	22772.0	0.867881	0.433940	−	0.900942i	$-0.357123\pi$
$883$	22772.0	0.867881	0.433940	+	0.900942i	$0.357123\pi$
$884$	1768.00	0.0672673
$885$	1500.00	0.0569740
$886$	−51712.0	−1.96083
$887$	36736.0	1.39061	0.695307	−	0.718713i	$-0.255269\pi$
$887$	36736.0	1.39061	0.695307	+	0.718713i	$0.255269\pi$
$888$	0	0
$889$	−24640.0	−0.929583
$890$	7440.00	0.280213
$891$	1458.00	0.0548202
$892$	15984.0	0.599982
$893$	−33004.0	−1.23677
$894$	17352.0	0.649147
$895$	25800.0	0.963574
$896$	0	0
$897$	5148.00	0.191624
$898$	15256.0	0.566926
$899$	−48300.0	−1.79187
$900$	−1800.00	−0.0666667
$901$	12818.0	0.473951
$902$	−8208.00	−0.302989
$903$	−1080.00	−0.0398008
$904$	0	0
$905$	−18740.0	−0.688330
$906$	−6312.00	−0.231459
$907$	4912.00	0.179824	0.0899120	−	0.995950i	$-0.471341\pi$
$907$	4912.00	0.179824	0.0899120	+	0.995950i	$0.471341\pi$
$908$	−3792.00	−0.138592
$909$	13662.0	0.498504
$910$	−5200.00	−0.189427
$911$	−3312.00	−0.120452	−0.0602258	−	0.998185i	$-0.519182\pi$
$911$	−3312.00	−0.120452	−0.0602258	+	0.998185i	$0.519182\pi$
$912$	14208.0	0.515870
$913$	2916.00	0.105702
$914$	−21560.0	−0.780242
$915$	6780.00	0.244962
$916$	−51120.0	−1.84394
$917$	26280.0	0.946393
$918$	−1836.00	−0.0660098
$919$	−17508.0	−0.628439	−0.314220	−	0.949350i	$-0.601743\pi$
$919$	−17508.0	−0.628439	−0.314220	+	0.949350i	$0.601743\pi$
$920$	0	0
$921$	8598.00	0.307615
$922$	−26536.0	−0.947849
$923$	4810.00	0.171531
$924$	−4320.00	−0.153807
$925$	1150.00	0.0408776
$926$	17480.0	0.620333
$927$	12600.0	0.446428
$928$	−53760.0	−1.90168
$929$	−27898.0	−0.985257	−0.492628	−	0.870240i	$-0.663964\pi$
$929$	−27898.0	−0.985257	−0.492628	+	0.870240i	$0.663964\pi$
$930$	27600.0	0.973161
$931$	17982.0	0.633014
$932$	−50480.0	−1.77417
$933$	−22284.0	−0.781935
$934$	−60656.0	−2.12497
$935$	3060.00	0.107030
$936$	0	0
$937$	13354.0	0.465588	0.232794	−	0.972526i	$-0.425213\pi$
$937$	13354.0	0.465588	0.232794	+	0.972526i	$0.425213\pi$
$938$	−23280.0	−0.810361
$939$	−14358.0	−0.498994
$940$	−35680.0	−1.23804
$941$	39742.0	1.37678	0.688391	−	0.725340i	$-0.258317\pi$
$941$	39742.0	1.37678	0.688391	+	0.725340i	$0.258317\pi$
$942$	−3144.00	−0.108744
$943$	15048.0	0.519650
$944$	3200.00	0.110330
$945$	2700.00	0.0929429
$946$	2592.00	0.0890837
$947$	2474.00	0.0848936	0.0424468	−	0.999099i	$-0.486485\pi$
$947$	2474.00	0.0848936	0.0424468	+	0.999099i	$0.486485\pi$
$948$	6528.00	0.223649
$949$	−10738.0	−0.367303
$950$	7400.00	0.252724
$951$	−13206.0	−0.450299
$952$	0	0
$953$	20882.0	0.709795	0.354897	−	0.934905i	$-0.384516\pi$
$953$	20882.0	0.709795	0.354897	+	0.934905i	$0.384516\pi$
$954$	−27144.0	−0.921194
$955$	4120.00	0.139602
$956$	−6960.00	−0.235463
$957$	11340.0	0.383041
$958$	40648.0	1.37085
$959$	7140.00	0.240420
$960$	15360.0	0.516398
$961$	23109.0	0.775704
$962$	2392.00	0.0801675
$963$	−7524.00	−0.251773
$964$	48464.0	1.61921
$965$	−10820.0	−0.360941
$966$	15840.0	0.527581
$967$	47942.0	1.59432	0.797162	−	0.603766i	$-0.206334\pi$
$967$	47942.0	1.59432	0.797162	+	0.603766i	$0.206334\pi$
$968$	0	0
$969$	3774.00	0.125117
$970$	31600.0	1.04599
$971$	56208.0	1.85767	0.928837	−	0.370490i	$-0.120810\pi$
$971$	56208.0	1.85767	0.928837	+	0.370490i	$0.120810\pi$
$972$	1944.00	0.0641500
$973$	3560.00	0.117295
$974$	−68696.0	−2.25992
$975$	975.000	0.0320256
$976$	14464.0	0.474366
$977$	23718.0	0.776669	0.388335	−	0.921518i	$-0.373050\pi$
$977$	23718.0	0.776669	0.388335	+	0.921518i	$0.373050\pi$
$978$	−31608.0	−1.03345
$979$	−3348.00	−0.109298
$980$	19440.0	0.633661
$981$	8874.00	0.288812
$982$	−15312.0	−0.497582
$983$	−48974.0	−1.58904	−0.794521	−	0.607237i	$-0.792278\pi$
$983$	−48974.0	−1.58904	−0.794521	+	0.607237i	$0.792278\pi$
$984$	0	0
$985$	11940.0	0.386234
$986$	−14280.0	−0.461225
$987$	−13380.0	−0.431500
$988$	7696.00	0.247816
$989$	−4752.00	−0.152785
$990$	−6480.00	−0.208028
$991$	−36760.0	−1.17832	−0.589162	−	0.808015i	$-0.700542\pi$
$991$	−36760.0	−1.17832	−0.589162	+	0.808015i	$0.700542\pi$
$992$	58880.0	1.88452
$993$	16410.0	0.524427
$994$	14800.0	0.472261
$995$	28600.0	0.911237
$996$	3888.00	0.123691
$997$	−6778.00	−0.215307	−0.107654	−	0.994188i	$-0.534334\pi$
$997$	−6778.00	−0.215307	−0.107654	+	0.994188i	$0.534334\pi$
$998$	−43576.0	−1.38214
$999$	−1242.00	−0.0393345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	663.4.a.a.1.1	✓	1
3.2	odd	2		1989.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
663.4.a.a.1.1	✓	1	1.1	even	1	trivial
1989.4.a.b.1.1		1	3.2	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:	\( N \)	\(=\)	\( 663 = 3 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	663.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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