LMFDB - Embedded newform 570.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [570,2,Mod(1,570)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(570, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("570.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 570 = 2 \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	570.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:	$4.55147291521$
Analytic rank:	$0$
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	$\chi$	$=$	570.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} -4.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -1.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{42} +12.0000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -8.00000 q^{46} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{55} +4.00000 q^{56} -1.00000 q^{57} +6.00000 q^{58} +1.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} -8.00000 q^{69} +4.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} +1.00000 q^{75} -1.00000 q^{76} -16.0000 q^{77} -2.00000 q^{78} -12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -8.00000 q^{83} +4.00000 q^{84} -2.00000 q^{85} +12.0000 q^{86} +6.00000 q^{87} -4.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} -8.00000 q^{91} -8.00000 q^{92} +4.00000 q^{93} -1.00000 q^{95} +1.00000 q^{96} +18.0000 q^{97} +9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	4.00000	1.06904
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.00000	0.223607
$21$	4.00000	0.872872
$22$	−4.00000	−0.852803
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	1.00000	0.182574
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	−2.00000	−0.342997
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−1.00000	−0.162221
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	4.00000	0.617213
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	−4.00000	−0.603023
$45$	1.00000	0.149071
$46$	−8.00000	−1.17954
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	−2.00000	−0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	−4.00000	−0.539360
$56$	4.00000	0.534522
$57$	−1.00000	−0.132453
$58$	6.00000	0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	1.00000	0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000	0.508001
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	−4.00000	−0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−2.00000	−0.242536
$69$	−8.00000	−0.963087
$70$	4.00000	0.478091
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−10.0000	−1.16248
$75$	1.00000	0.115470
$76$	−1.00000	−0.114708
$77$	−16.0000	−1.82337
$78$	−2.00000	−0.226455
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	4.00000	0.436436
$85$	−2.00000	−0.216930
$86$	12.0000	1.29399
$87$	6.00000	0.643268
$88$	−4.00000	−0.426401
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	1.00000	0.105409
$91$	−8.00000	−0.838628
$92$	−8.00000	−0.834058
$93$	4.00000	0.414781
$94$	0	0
$95$	−1.00000	−0.102598
$96$	1.00000	0.102062
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	9.00000	0.909137
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−2.00000	−0.198030
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−2.00000	−0.196116
$105$	4.00000	0.390360
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−4.00000	−0.381385
$111$	−10.0000	−0.949158
$112$	4.00000	0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−1.00000	−0.0936586
$115$	−8.00000	−0.746004
$116$	6.00000	0.557086
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−8.00000	−0.733359
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	−2.00000	−0.180334
$124$	4.00000	0.359211
$125$	1.00000	0.0894427
$126$	4.00000	0.356348
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	12.0000	1.05654
$130$	−2.00000	−0.175412
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−4.00000	−0.348155
$133$	−4.00000	−0.346844
$134$	−4.00000	−0.345547
$135$	1.00000	0.0860663
$136$	−2.00000	−0.171499
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	−8.00000	−0.681005
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−8.00000	−0.671345
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	2.00000	0.165521
$147$	9.00000	0.742307
$148$	−10.0000	−0.821995
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	1.00000	0.0816497
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−1.00000	−0.0811107
$153$	−2.00000	−0.161690
$154$	−16.0000	−1.28932
$155$	4.00000	0.321288
$156$	−2.00000	−0.160128
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−12.0000	−0.954669
$159$	6.00000	0.475831
$160$	1.00000	0.0790569
$161$	−32.0000	−2.52195
$162$	1.00000	0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−2.00000	−0.156174
$165$	−4.00000	−0.311400
$166$	−8.00000	−0.620920
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	4.00000	0.308607
$169$	−9.00000	−0.692308
$170$	−2.00000	−0.153393
$171$	−1.00000	−0.0764719
$172$	12.0000	0.914991
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	6.00000	0.454859
$175$	4.00000	0.302372
$176$	−4.00000	−0.301511
$177$	0	0
$178$	6.00000	0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	1.00000	0.0745356
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−8.00000	−0.592999
$183$	−10.0000	−0.739221
$184$	−8.00000	−0.589768
$185$	−10.0000	−0.735215
$186$	4.00000	0.293294
$187$	8.00000	0.585018
$188$	0	0
$189$	4.00000	0.290957
$190$	−1.00000	−0.0725476
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	18.0000	1.29232
$195$	−2.00000	−0.143223
$196$	9.00000	0.642857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−4.00000	−0.284268
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	1.00000	0.0707107
$201$	−4.00000	−0.282138
$202$	−10.0000	−0.703598
$203$	24.0000	1.68447
$204$	−2.00000	−0.140028
$205$	−2.00000	−0.139686
$206$	16.0000	1.11477
$207$	−8.00000	−0.556038
$208$	−2.00000	−0.138675
$209$	4.00000	0.276686
$210$	4.00000	0.276026
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.00000	0.412082
$213$	−8.00000	−0.548151
$214$	−12.0000	−0.820303
$215$	12.0000	0.818393
$216$	1.00000	0.0680414
$217$	16.0000	1.08615
$218$	10.0000	0.677285
$219$	2.00000	0.135147
$220$	−4.00000	−0.269680
$221$	4.00000	0.269069
$222$	−10.0000	−0.671156
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	4.00000	0.267261
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−1.00000	−0.0662266
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	−8.00000	−0.527504
$231$	−16.0000	−1.05272
$232$	6.00000	0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	0	0
$237$	−12.0000	−0.779484
$238$	−8.00000	−0.518563
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	1.00000	0.0645497
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	9.00000	0.574989
$246$	−2.00000	−0.127515
$247$	2.00000	0.127257
$248$	4.00000	0.254000
$249$	−8.00000	−0.506979
$250$	1.00000	0.0632456
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	4.00000	0.251976
$253$	32.0000	2.01182
$254$	8.00000	0.501965
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	12.0000	0.747087
$259$	−40.0000	−2.48548
$260$	−2.00000	−0.124035
$261$	6.00000	0.371391
$262$	−12.0000	−0.741362
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	−4.00000	−0.246183
$265$	6.00000	0.368577
$266$	−4.00000	−0.245256
$267$	6.00000	0.367194
$268$	−4.00000	−0.244339
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	1.00000	0.0608581
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	−2.00000	−0.121268
$273$	−8.00000	−0.484182
$274$	14.0000	0.845771
$275$	−4.00000	−0.241209
$276$	−8.00000	−0.481543
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−12.0000	−0.719712
$279$	4.00000	0.239474
$280$	4.00000	0.239046
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−8.00000	−0.474713
$285$	−1.00000	−0.0592349
$286$	8.00000	0.473050
$287$	−8.00000	−0.472225
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	6.00000	0.352332
$291$	18.0000	1.05518
$292$	2.00000	0.117041
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	−10.0000	−0.581238
$297$	−4.00000	−0.232104
$298$	22.0000	1.27443
$299$	16.0000	0.925304
$300$	1.00000	0.0577350
$301$	48.0000	2.76667
$302$	4.00000	0.230174
$303$	−10.0000	−0.574485
$304$	−1.00000	−0.0573539
$305$	−10.0000	−0.572598
$306$	−2.00000	−0.114332
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	−16.0000	−0.911685
$309$	16.0000	0.910208
$310$	4.00000	0.227185
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	−2.00000	−0.113228
$313$	−30.0000	−1.69570	−0.847850	−	0.530236i	$-0.822103\pi$
$313$	−30.0000	−1.69570	−0.847850	+	0.530236i	$0.822103\pi$
$314$	−6.00000	−0.338600
$315$	4.00000	0.225374
$316$	−12.0000	−0.675053
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	6.00000	0.336463
$319$	−24.0000	−1.34374
$320$	1.00000	0.0559017
$321$	−12.0000	−0.669775
$322$	−32.0000	−1.78329
$323$	2.00000	0.111283
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−4.00000	−0.221540
$327$	10.0000	0.553001
$328$	−2.00000	−0.110432
$329$	0	0
$330$	−4.00000	−0.220193
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−8.00000	−0.439057
$333$	−10.0000	−0.547997
$334$	−16.0000	−0.875481
$335$	−4.00000	−0.218543
$336$	4.00000	0.218218
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	−9.00000	−0.489535
$339$	−6.00000	−0.325875
$340$	−2.00000	−0.108465
$341$	−16.0000	−0.866449
$342$	−1.00000	−0.0540738
$343$	8.00000	0.431959
$344$	12.0000	0.646997
$345$	−8.00000	−0.430706
$346$	14.0000	0.752645
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	6.00000	0.321634
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	4.00000	0.213809
$351$	−2.00000	−0.106752
$352$	−4.00000	−0.213201
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	6.00000	0.317999
$357$	−8.00000	−0.423405
$358$	0	0
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	1.00000	0.0527046
$361$	1.00000	0.0526316
$362$	2.00000	0.105118
$363$	5.00000	0.262432
$364$	−8.00000	−0.419314
$365$	2.00000	0.104685
$366$	−10.0000	−0.522708
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	−8.00000	−0.417029
$369$	−2.00000	−0.104116
$370$	−10.0000	−0.519875
$371$	24.0000	1.24602
$372$	4.00000	0.207390
$373$	30.0000	1.55334	0.776671	−	0.629907i	$-0.216907\pi$
$373$	30.0000	1.55334	0.776671	+	0.629907i	$0.216907\pi$
$374$	8.00000	0.413670
$375$	1.00000	0.0516398
$376$	0	0
$377$	−12.0000	−0.618031
$378$	4.00000	0.205738
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−1.00000	−0.0512989
$381$	8.00000	0.409852
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	0.0510310
$385$	−16.0000	−0.815436
$386$	18.0000	0.916176
$387$	12.0000	0.609994
$388$	18.0000	0.913812
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−2.00000	−0.101274
$391$	16.0000	0.809155
$392$	9.00000	0.454569
$393$	−12.0000	−0.605320
$394$	−18.0000	−0.906827
$395$	−12.0000	−0.603786
$396$	−4.00000	−0.201008
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	16.0000	0.802008
$399$	−4.00000	−0.200250
$400$	1.00000	0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−4.00000	−0.199502
$403$	−8.00000	−0.398508
$404$	−10.0000	−0.497519
$405$	1.00000	0.0496904
$406$	24.0000	1.19110
$407$	40.0000	1.98273
$408$	−2.00000	−0.0990148
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	−2.00000	−0.0987730
$411$	14.0000	0.690569
$412$	16.0000	0.788263
$413$	0	0
$414$	−8.00000	−0.393179
$415$	−8.00000	−0.392705
$416$	−2.00000	−0.0980581
$417$	−12.0000	−0.587643
$418$	4.00000	0.195646
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	4.00000	0.195180
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	6.00000	0.291386
$425$	−2.00000	−0.0970143
$426$	−8.00000	−0.387601
$427$	−40.0000	−1.93574
$428$	−12.0000	−0.580042
$429$	8.00000	0.386244
$430$	12.0000	0.578691
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	16.0000	0.768025
$435$	6.00000	0.287678
$436$	10.0000	0.478913
$437$	8.00000	0.382692
$438$	2.00000	0.0955637
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	−4.00000	−0.190693
$441$	9.00000	0.428571
$442$	4.00000	0.190261
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−10.0000	−0.474579
$445$	6.00000	0.284427
$446$	16.0000	0.757622
$447$	22.0000	1.04056
$448$	4.00000	0.188982
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	1.00000	0.0471405
$451$	8.00000	0.376705
$452$	−6.00000	−0.282216
$453$	4.00000	0.187936
$454$	−12.0000	−0.563188
$455$	−8.00000	−0.375046
$456$	−1.00000	−0.0468293
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	22.0000	1.02799
$459$	−2.00000	−0.0933520
$460$	−8.00000	−0.373002
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	−16.0000	−0.744387
$463$	−36.0000	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$463$	−36.0000	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$464$	6.00000	0.278543
$465$	4.00000	0.185496
$466$	6.00000	0.277945
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	−2.00000	−0.0924500
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	−48.0000	−2.20704
$474$	−12.0000	−0.551178
$475$	−1.00000	−0.0458831
$476$	−8.00000	−0.366679
$477$	6.00000	0.274721
$478$	−8.00000	−0.365911
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	1.00000	0.0456435
$481$	20.0000	0.911922
$482$	2.00000	0.0910975
$483$	−32.0000	−1.45605
$484$	5.00000	0.227273
$485$	18.0000	0.817338
$486$	1.00000	0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−10.0000	−0.452679
$489$	−4.00000	−0.180886
$490$	9.00000	0.406579
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	−2.00000	−0.0901670
$493$	−12.0000	−0.540453
$494$	2.00000	0.0899843
$495$	−4.00000	−0.179787
$496$	4.00000	0.179605
$497$	−32.0000	−1.43540
$498$	−8.00000	−0.358489
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	1.00000	0.0447214
$501$	−16.0000	−0.714827
$502$	−28.0000	−1.24970
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	4.00000	0.178174
$505$	−10.0000	−0.444994
$506$	32.0000	1.42257
$507$	−9.00000	−0.399704
$508$	8.00000	0.354943
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	−2.00000	−0.0885615
$511$	8.00000	0.353899
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	18.0000	0.793946
$515$	16.0000	0.705044
$516$	12.0000	0.528271
$517$	0	0
$518$	−40.0000	−1.75750
$519$	14.0000	0.614532
$520$	−2.00000	−0.0877058
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	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	$0.472127\pi$
$524$	−12.0000	−0.524222
$525$	4.00000	0.174574
$526$	24.0000	1.04645
$527$	−8.00000	−0.348485
$528$	−4.00000	−0.174078
$529$	41.0000	1.78261
$530$	6.00000	0.260623
$531$	0	0
$532$	−4.00000	−0.173422
$533$	4.00000	0.173259
$534$	6.00000	0.259645
$535$	−12.0000	−0.518805
$536$	−4.00000	−0.172774
$537$	0	0
$538$	6.00000	0.258678
$539$	−36.0000	−1.55063
$540$	1.00000	0.0430331
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	−24.0000	−1.03089
$543$	2.00000	0.0858282
$544$	−2.00000	−0.0857493
$545$	10.0000	0.428353
$546$	−8.00000	−0.342368
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	14.0000	0.598050
$549$	−10.0000	−0.426790
$550$	−4.00000	−0.170561
$551$	−6.00000	−0.255609
$552$	−8.00000	−0.340503
$553$	−48.0000	−2.04117
$554$	18.0000	0.764747
$555$	−10.0000	−0.424476
$556$	−12.0000	−0.508913
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	4.00000	0.169334
$559$	−24.0000	−1.01509
$560$	4.00000	0.169031
$561$	8.00000	0.337760
$562$	30.0000	1.26547
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	−4.00000	−0.168133
$567$	4.00000	0.167984
$568$	−8.00000	−0.335673
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	−1.00000	−0.0418854
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	8.00000	0.334497
$573$	0	0
$574$	−8.00000	−0.333914
$575$	−8.00000	−0.333623
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−13.0000	−0.540729
$579$	18.0000	0.748054
$580$	6.00000	0.249136
$581$	−32.0000	−1.32758
$582$	18.0000	0.746124
$583$	−24.0000	−0.993978
$584$	2.00000	0.0827606
$585$	−2.00000	−0.0826898
$586$	−10.0000	−0.413096
$587$	−40.0000	−1.65098	−0.825488	−	0.564419i	$-0.809100\pi$
$587$	−40.0000	−1.65098	−0.825488	+	0.564419i	$0.809100\pi$
$588$	9.00000	0.371154
$589$	−4.00000	−0.164817
$590$	0	0
$591$	−18.0000	−0.740421
$592$	−10.0000	−0.410997
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	−4.00000	−0.164122
$595$	−8.00000	−0.327968
$596$	22.0000	0.901155
$597$	16.0000	0.654836
$598$	16.0000	0.654289
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	1.00000	0.0408248
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	48.0000	1.95633
$603$	−4.00000	−0.162893
$604$	4.00000	0.162758
$605$	5.00000	0.203279
$606$	−10.0000	−0.406222
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	−1.00000	−0.0405554
$609$	24.0000	0.972529
$610$	−10.0000	−0.404888
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	28.0000	1.12999
$615$	−2.00000	−0.0806478
$616$	−16.0000	−0.644658
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	16.0000	0.643614
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	4.00000	0.160644
$621$	−8.00000	−0.321029
$622$	24.0000	0.962312
$623$	24.0000	0.961540
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−30.0000	−1.19904
$627$	4.00000	0.159745
$628$	−6.00000	−0.239426
$629$	20.0000	0.797452
$630$	4.00000	0.159364
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−12.0000	−0.477334
$633$	−12.0000	−0.476957
$634$	30.0000	1.19145
$635$	8.00000	0.317470
$636$	6.00000	0.237915
$637$	−18.0000	−0.713186
$638$	−24.0000	−0.950169
$639$	−8.00000	−0.316475
$640$	1.00000	0.0395285
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	−12.0000	−0.473602
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	−32.0000	−1.26098
$645$	12.0000	0.472500
$646$	2.00000	0.0786889
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	16.0000	0.627089
$652$	−4.00000	−0.156652
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	10.0000	0.391031
$655$	−12.0000	−0.468879
$656$	−2.00000	−0.0780869
$657$	2.00000	0.0780274
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	−4.00000	−0.155700
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	−12.0000	−0.466393
$663$	4.00000	0.155347
$664$	−8.00000	−0.310460
$665$	−4.00000	−0.155113
$666$	−10.0000	−0.387492
$667$	−48.0000	−1.85857
$668$	−16.0000	−0.619059
$669$	16.0000	0.618596
$670$	−4.00000	−0.154533
$671$	40.0000	1.54418
$672$	4.00000	0.154303
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	34.0000	1.30963
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−6.00000	−0.230429
$679$	72.0000	2.76311
$680$	−2.00000	−0.0766965
$681$	−12.0000	−0.459841
$682$	−16.0000	−0.612672
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−1.00000	−0.0382360
$685$	14.0000	0.534913
$686$	8.00000	0.305441
$687$	22.0000	0.839352
$688$	12.0000	0.457496
$689$	−12.0000	−0.457164
$690$	−8.00000	−0.304555
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	14.0000	0.532200
$693$	−16.0000	−0.607790
$694$	24.0000	0.911028
$695$	−12.0000	−0.455186
$696$	6.00000	0.227429
$697$	4.00000	0.151511
$698$	−34.0000	−1.28692
$699$	6.00000	0.226941
$700$	4.00000	0.151186
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	−2.00000	−0.0754851
$703$	10.0000	0.377157
$704$	−4.00000	−0.150756
$705$	0	0
$706$	6.00000	0.225813
$707$	−40.0000	−1.50435
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	−8.00000	−0.300235
$711$	−12.0000	−0.450035
$712$	6.00000	0.224860
$713$	−32.0000	−1.19841
$714$	−8.00000	−0.299392
$715$	8.00000	0.299183
$716$	0	0
$717$	−8.00000	−0.298765
$718$	32.0000	1.19423
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	1.00000	0.0372678
$721$	64.0000	2.38348
$722$	1.00000	0.0372161
$723$	2.00000	0.0743808
$724$	2.00000	0.0743294
$725$	6.00000	0.222834
$726$	5.00000	0.185567
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	2.00000	0.0740233
$731$	−24.0000	−0.887672
$732$	−10.0000	−0.369611
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−12.0000	−0.442928
$735$	9.00000	0.331970
$736$	−8.00000	−0.294884
$737$	16.0000	0.589368
$738$	−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	$0.380092\pi$
$740$	−10.0000	−0.367607
$741$	2.00000	0.0734718
$742$	24.0000	0.881068
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	4.00000	0.146647
$745$	22.0000	0.806018
$746$	30.0000	1.09838
$747$	−8.00000	−0.292705
$748$	8.00000	0.292509
$749$	−48.0000	−1.75388
$750$	1.00000	0.0365148
$751$	52.0000	1.89751	0.948753	−	0.316017i	$-0.102346\pi$
$751$	52.0000	1.89751	0.948753	+	0.316017i	$0.102346\pi$
$752$	0	0
$753$	−28.0000	−1.02038
$754$	−12.0000	−0.437014
$755$	4.00000	0.145575
$756$	4.00000	0.145479
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	−20.0000	−0.726433
$759$	32.0000	1.16153
$760$	−1.00000	−0.0362738
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	8.00000	0.289809
$763$	40.0000	1.44810
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	0	0
$767$	0	0
$768$	1.00000	0.0360844
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	−16.0000	−0.576600
$771$	18.0000	0.648254
$772$	18.0000	0.647834
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	12.0000	0.431331
$775$	4.00000	0.143684
$776$	18.0000	0.646162
$777$	−40.0000	−1.43499
$778$	6.00000	0.215110
$779$	2.00000	0.0716574
$780$	−2.00000	−0.0716115
$781$	32.0000	1.14505
$782$	16.0000	0.572159
$783$	6.00000	0.214423
$784$	9.00000	0.321429
$785$	−6.00000	−0.214149
$786$	−12.0000	−0.428026
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	−18.0000	−0.641223
$789$	24.0000	0.854423
$790$	−12.0000	−0.426941
$791$	−24.0000	−0.853342
$792$	−4.00000	−0.142134
$793$	20.0000	0.710221
$794$	2.00000	0.0709773
$795$	6.00000	0.212798
$796$	16.0000	0.567105
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	−4.00000	−0.141598
$799$	0	0
$800$	1.00000	0.0353553
$801$	6.00000	0.212000
$802$	−18.0000	−0.635602
$803$	−8.00000	−0.282314
$804$	−4.00000	−0.141069
$805$	−32.0000	−1.12785
$806$	−8.00000	−0.281788
$807$	6.00000	0.211210
$808$	−10.0000	−0.351799
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	1.00000	0.0351364
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	24.0000	0.842235
$813$	−24.0000	−0.841717
$814$	40.0000	1.40200
$815$	−4.00000	−0.140114
$816$	−2.00000	−0.0700140
$817$	−12.0000	−0.419827
$818$	−6.00000	−0.209785
$819$	−8.00000	−0.279543
$820$	−2.00000	−0.0698430
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	14.0000	0.488306
$823$	52.0000	1.81261	0.906303	−	0.422628i	$-0.138892\pi$
$823$	52.0000	1.81261	0.906303	+	0.422628i	$0.138892\pi$
$824$	16.0000	0.557386
$825$	−4.00000	−0.139262
$826$	0	0
$827$	52.0000	1.80822	0.904109	−	0.427303i	$-0.140536\pi$
$827$	52.0000	1.80822	0.904109	+	0.427303i	$0.140536\pi$
$828$	−8.00000	−0.278019
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	−8.00000	−0.277684
$831$	18.0000	0.624413
$832$	−2.00000	−0.0693375
$833$	−18.0000	−0.623663
$834$	−12.0000	−0.415526
$835$	−16.0000	−0.553703
$836$	4.00000	0.138343
$837$	4.00000	0.138260
$838$	12.0000	0.414533
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	4.00000	0.138013
$841$	7.00000	0.241379
$842$	−14.0000	−0.482472
$843$	30.0000	1.03325
$844$	−12.0000	−0.413057
$845$	−9.00000	−0.309609
$846$	0	0
$847$	20.0000	0.687208
$848$	6.00000	0.206041
$849$	−4.00000	−0.137280
$850$	−2.00000	−0.0685994
$851$	80.0000	2.74236
$852$	−8.00000	−0.274075
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	−40.0000	−1.36877
$855$	−1.00000	−0.0341993
$856$	−12.0000	−0.410152
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	8.00000	0.273115
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	12.0000	0.409197
$861$	−8.00000	−0.272639
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	1.00000	0.0340207
$865$	14.0000	0.476014
$866$	2.00000	0.0679628
$867$	−13.0000	−0.441503
$868$	16.0000	0.543075
$869$	48.0000	1.62829
$870$	6.00000	0.203419
$871$	8.00000	0.271070
$872$	10.0000	0.338643
$873$	18.0000	0.609208
$874$	8.00000	0.270604
$875$	4.00000	0.135225
$876$	2.00000	0.0675737
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	−4.00000	−0.134993
$879$	−10.0000	−0.337292
$880$	−4.00000	−0.134840
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	9.00000	0.303046
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−24.0000	−0.806296
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	−10.0000	−0.335578
$889$	32.0000	1.07325
$890$	6.00000	0.201120
$891$	−4.00000	−0.134005
$892$	16.0000	0.535720
$893$	0	0
$894$	22.0000	0.735790
$895$	0	0
$896$	4.00000	0.133631
$897$	16.0000	0.534224
$898$	6.00000	0.200223
$899$	24.0000	0.800445
$900$	1.00000	0.0333333
$901$	−12.0000	−0.399778
$902$	8.00000	0.266371
$903$	48.0000	1.59734
$904$	−6.00000	−0.199557
$905$	2.00000	0.0664822
$906$	4.00000	0.132891
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	−12.0000	−0.398234
$909$	−10.0000	−0.331679
$910$	−8.00000	−0.265197
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	−1.00000	−0.0331133
$913$	32.0000	1.05905
$914$	−6.00000	−0.198462
$915$	−10.0000	−0.330590
$916$	22.0000	0.726900
$917$	−48.0000	−1.58510
$918$	−2.00000	−0.0660098
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	−8.00000	−0.263752
$921$	28.0000	0.922631
$922$	14.0000	0.461065
$923$	16.0000	0.526646
$924$	−16.0000	−0.526361
$925$	−10.0000	−0.328798
$926$	−36.0000	−1.18303
$927$	16.0000	0.525509
$928$	6.00000	0.196960
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	4.00000	0.131165
$931$	−9.00000	−0.294963
$932$	6.00000	0.196537
$933$	24.0000	0.785725
$934$	8.00000	0.261768
$935$	8.00000	0.261628
$936$	−2.00000	−0.0653720
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	−16.0000	−0.522419
$939$	−30.0000	−0.979013
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	−6.00000	−0.195491
$943$	16.0000	0.521032
$944$	0	0
$945$	4.00000	0.130120
$946$	−48.0000	−1.56061
$947$	−40.0000	−1.29983	−0.649913	−	0.760009i	$-0.725195\pi$
$947$	−40.0000	−1.29983	−0.649913	+	0.760009i	$0.725195\pi$
$948$	−12.0000	−0.389742
$949$	−4.00000	−0.129845
$950$	−1.00000	−0.0324443
$951$	30.0000	0.972817
$952$	−8.00000	−0.259281
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−8.00000	−0.258738
$957$	−24.0000	−0.775810
$958$	−16.0000	−0.516937
$959$	56.0000	1.80833
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	20.0000	0.644826
$963$	−12.0000	−0.386695
$964$	2.00000	0.0644157
$965$	18.0000	0.579441
$966$	−32.0000	−1.02958
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	5.00000	0.160706
$969$	2.00000	0.0642493
$970$	18.0000	0.577945
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	1.00000	0.0320750
$973$	−48.0000	−1.53881
$974$	−32.0000	−1.02535
$975$	−2.00000	−0.0640513
$976$	−10.0000	−0.320092
$977$	50.0000	1.59964	0.799821	−	0.600239i	$-0.204928\pi$
$977$	50.0000	1.59964	0.799821	+	0.600239i	$0.204928\pi$
$978$	−4.00000	−0.127906
$979$	−24.0000	−0.767043
$980$	9.00000	0.287494
$981$	10.0000	0.319275
$982$	28.0000	0.893516
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	−2.00000	−0.0637577
$985$	−18.0000	−0.573528
$986$	−12.0000	−0.382158
$987$	0	0
$988$	2.00000	0.0636285
$989$	−96.0000	−3.05262
$990$	−4.00000	−0.127128
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	4.00000	0.127000
$993$	−12.0000	−0.380808
$994$	−32.0000	−1.01498
$995$	16.0000	0.507234
$996$	−8.00000	−0.253490
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−4.00000	−0.126618
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	570.2.a.m.1.1	✓	1
3.2	odd	2		1710.2.a.f.1.1		1
4.3	odd	2		4560.2.a.k.1.1		1
5.2	odd	4		2850.2.d.k.799.2		2
5.3	odd	4		2850.2.d.k.799.1		2
5.4	even	2		2850.2.a.a.1.1		1
15.14	odd	2		8550.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.m.1.1	✓	1	1.1	even	1	trivial
1710.2.a.f.1.1		1	3.2	odd	2
2850.2.a.a.1.1		1	5.4	even	2
2850.2.d.k.799.1		2	5.3	odd	4
2850.2.d.k.799.2		2	5.2	odd	4
4560.2.a.k.1.1		1	4.3	odd	2
8550.2.a.t.1.1		1	15.14	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:	\( N \)	\(=\)	\( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	570.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more