LMFDB - Embedded newform 690.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,4,Mod(1,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, 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("690.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	690.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:	$40.7113179040$
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	$\chi$	$=$	690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +6.00000 q^{6} -19.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +6.00000 q^{6} -19.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -24.0000 q^{11} -12.0000 q^{12} +44.0000 q^{13} +38.0000 q^{14} -15.0000 q^{15} +16.0000 q^{16} +75.0000 q^{17} -18.0000 q^{18} -16.0000 q^{19} +20.0000 q^{20} +57.0000 q^{21} +48.0000 q^{22} -23.0000 q^{23} +24.0000 q^{24} +25.0000 q^{25} -88.0000 q^{26} -27.0000 q^{27} -76.0000 q^{28} -123.000 q^{29} +30.0000 q^{30} -43.0000 q^{31} -32.0000 q^{32} +72.0000 q^{33} -150.000 q^{34} -95.0000 q^{35} +36.0000 q^{36} -43.0000 q^{37} +32.0000 q^{38} -132.000 q^{39} -40.0000 q^{40} +207.000 q^{41} -114.000 q^{42} +236.000 q^{43} -96.0000 q^{44} +45.0000 q^{45} +46.0000 q^{46} -30.0000 q^{47} -48.0000 q^{48} +18.0000 q^{49} -50.0000 q^{50} -225.000 q^{51} +176.000 q^{52} +519.000 q^{53} +54.0000 q^{54} -120.000 q^{55} +152.000 q^{56} +48.0000 q^{57} +246.000 q^{58} +39.0000 q^{59} -60.0000 q^{60} -190.000 q^{61} +86.0000 q^{62} -171.000 q^{63} +64.0000 q^{64} +220.000 q^{65} -144.000 q^{66} -295.000 q^{67} +300.000 q^{68} +69.0000 q^{69} +190.000 q^{70} -603.000 q^{71} -72.0000 q^{72} +668.000 q^{73} +86.0000 q^{74} -75.0000 q^{75} -64.0000 q^{76} +456.000 q^{77} +264.000 q^{78} -1276.00 q^{79} +80.0000 q^{80} +81.0000 q^{81} -414.000 q^{82} -573.000 q^{83} +228.000 q^{84} +375.000 q^{85} -472.000 q^{86} +369.000 q^{87} +192.000 q^{88} +456.000 q^{89} -90.0000 q^{90} -836.000 q^{91} -92.0000 q^{92} +129.000 q^{93} +60.0000 q^{94} -80.0000 q^{95} +96.0000 q^{96} -1186.00 q^{97} -36.0000 q^{98} -216.000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	6.00000	0.408248
$7$	−19.0000	−1.02590	−0.512952	−	0.858417i	$-0.671448\pi$
$7$	−19.0000	−1.02590	−0.512952	+	0.858417i	$0.671448\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	−24.0000	−0.657843	−0.328921	−	0.944357i	$-0.606685\pi$
$11$	−24.0000	−0.657843	−0.328921	+	0.944357i	$0.606685\pi$
$12$	−12.0000	−0.288675
$13$	44.0000	0.938723	0.469362	−	0.883006i	$-0.344484\pi$
$13$	44.0000	0.938723	0.469362	+	0.883006i	$0.344484\pi$
$14$	38.0000	0.725423
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	75.0000	1.07001	0.535005	−	0.844849i	$-0.320310\pi$
$17$	75.0000	1.07001	0.535005	+	0.844849i	$0.320310\pi$
$18$	−18.0000	−0.235702
$19$	−16.0000	−0.193192	−0.0965961	−	0.995324i	$-0.530796\pi$
$19$	−16.0000	−0.193192	−0.0965961	+	0.995324i	$0.530796\pi$
$20$	20.0000	0.223607
$21$	57.0000	0.592306
$22$	48.0000	0.465165
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	−88.0000	−0.663778
$27$	−27.0000	−0.192450
$28$	−76.0000	−0.512952
$29$	−123.000	−0.787604	−0.393802	−	0.919195i	$-0.628841\pi$
$29$	−123.000	−0.787604	−0.393802	+	0.919195i	$0.628841\pi$
$30$	30.0000	0.182574
$31$	−43.0000	−0.249130	−0.124565	−	0.992211i	$-0.539754\pi$
$31$	−43.0000	−0.249130	−0.124565	+	0.992211i	$0.539754\pi$
$32$	−32.0000	−0.176777
$33$	72.0000	0.379806
$34$	−150.000	−0.756611
$35$	−95.0000	−0.458798
$36$	36.0000	0.166667
$37$	−43.0000	−0.191058	−0.0955291	−	0.995427i	$-0.530454\pi$
$37$	−43.0000	−0.191058	−0.0955291	+	0.995427i	$0.530454\pi$
$38$	32.0000	0.136608
$39$	−132.000	−0.541972
$40$	−40.0000	−0.158114
$41$	207.000	0.788487	0.394244	−	0.919006i	$-0.371007\pi$
$41$	207.000	0.788487	0.394244	+	0.919006i	$0.371007\pi$
$42$	−114.000	−0.418823
$43$	236.000	0.836969	0.418484	−	0.908224i	$-0.362561\pi$
$43$	236.000	0.836969	0.418484	+	0.908224i	$0.362561\pi$
$44$	−96.0000	−0.328921
$45$	45.0000	0.149071
$46$	46.0000	0.147442
$47$	−30.0000	−0.0931053	−0.0465527	−	0.998916i	$-0.514824\pi$
$47$	−30.0000	−0.0931053	−0.0465527	+	0.998916i	$0.514824\pi$
$48$	−48.0000	−0.144338
$49$	18.0000	0.0524781
$50$	−50.0000	−0.141421
$51$	−225.000	−0.617771
$52$	176.000	0.469362
$53$	519.000	1.34510	0.672548	−	0.740053i	$-0.265200\pi$
$53$	519.000	1.34510	0.672548	+	0.740053i	$0.265200\pi$
$54$	54.0000	0.136083
$55$	−120.000	−0.294196
$56$	152.000	0.362712
$57$	48.0000	0.111540
$58$	246.000	0.556920
$59$	39.0000	0.0860571	0.0430285	−	0.999074i	$-0.486299\pi$
$59$	39.0000	0.0860571	0.0430285	+	0.999074i	$0.486299\pi$
$60$	−60.0000	−0.129099
$61$	−190.000	−0.398803	−0.199402	−	0.979918i	$-0.563900\pi$
$61$	−190.000	−0.398803	−0.199402	+	0.979918i	$0.563900\pi$
$62$	86.0000	0.176161
$63$	−171.000	−0.341968
$64$	64.0000	0.125000
$65$	220.000	0.419810
$66$	−144.000	−0.268563
$67$	−295.000	−0.537910	−0.268955	−	0.963153i	$-0.586678\pi$
$67$	−295.000	−0.537910	−0.268955	+	0.963153i	$0.586678\pi$
$68$	300.000	0.535005
$69$	69.0000	0.120386
$70$	190.000	0.324419
$71$	−603.000	−1.00793	−0.503964	−	0.863724i	$-0.668126\pi$
$71$	−603.000	−1.00793	−0.503964	+	0.863724i	$0.668126\pi$
$72$	−72.0000	−0.117851
$73$	668.000	1.07101	0.535503	−	0.844533i	$-0.320122\pi$
$73$	668.000	1.07101	0.535503	+	0.844533i	$0.320122\pi$
$74$	86.0000	0.135099
$75$	−75.0000	−0.115470
$76$	−64.0000	−0.0965961
$77$	456.000	0.674883
$78$	264.000	0.383232
$79$	−1276.00	−1.81723	−0.908615	−	0.417634i	$-0.862859\pi$
$79$	−1276.00	−1.81723	−0.908615	+	0.417634i	$0.862859\pi$
$80$	80.0000	0.111803
$81$	81.0000	0.111111
$82$	−414.000	−0.557545
$83$	−573.000	−0.757770	−0.378885	−	0.925444i	$-0.623692\pi$
$83$	−573.000	−0.757770	−0.378885	+	0.925444i	$0.623692\pi$
$84$	228.000	0.296153
$85$	375.000	0.478523
$86$	−472.000	−0.591826
$87$	369.000	0.454724
$88$	192.000	0.232583
$89$	456.000	0.543100	0.271550	−	0.962424i	$-0.412464\pi$
$89$	456.000	0.543100	0.271550	+	0.962424i	$0.412464\pi$
$90$	−90.0000	−0.105409
$91$	−836.000	−0.963040
$92$	−92.0000	−0.104257
$93$	129.000	0.143835
$94$	60.0000	0.0658354
$95$	−80.0000	−0.0863982
$96$	96.0000	0.102062
$97$	−1186.00	−1.24144	−0.620722	−	0.784031i	$-0.713160\pi$
$97$	−1186.00	−1.24144	−0.620722	+	0.784031i	$0.713160\pi$
$98$	−36.0000	−0.0371076
$99$	−216.000	−0.219281
$100$	100.000	0.100000
$101$	−1323.00	−1.30340	−0.651700	−	0.758477i	$-0.725944\pi$
$101$	−1323.00	−1.30340	−0.651700	+	0.758477i	$0.725944\pi$
$102$	450.000	0.436830
$103$	−40.0000	−0.0382652	−0.0191326	−	0.999817i	$-0.506090\pi$
$103$	−40.0000	−0.0382652	−0.0191326	+	0.999817i	$0.506090\pi$
$104$	−352.000	−0.331889
$105$	285.000	0.264887
$106$	−1038.00	−0.951127
$107$	543.000	0.490596	0.245298	−	0.969448i	$-0.421114\pi$
$107$	543.000	0.490596	0.245298	+	0.969448i	$0.421114\pi$
$108$	−108.000	−0.0962250
$109$	−1852.00	−1.62743	−0.813713	−	0.581267i	$-0.802557\pi$
$109$	−1852.00	−1.62743	−0.813713	+	0.581267i	$0.802557\pi$
$110$	240.000	0.208028
$111$	129.000	0.110308
$112$	−304.000	−0.256476
$113$	−1377.00	−1.14635	−0.573174	−	0.819434i	$-0.694288\pi$
$113$	−1377.00	−1.14635	−0.573174	+	0.819434i	$0.694288\pi$
$114$	−96.0000	−0.0788704
$115$	−115.000	−0.0932505
$116$	−492.000	−0.393802
$117$	396.000	0.312908
$118$	−78.0000	−0.0608515
$119$	−1425.00	−1.09773
$120$	120.000	0.0912871
$121$	−755.000	−0.567243
$122$	380.000	0.281997
$123$	−621.000	−0.455233
$124$	−172.000	−0.124565
$125$	125.000	0.0894427
$126$	342.000	0.241808
$127$	−1534.00	−1.07181	−0.535907	−	0.844277i	$-0.680030\pi$
$127$	−1534.00	−1.07181	−0.535907	+	0.844277i	$0.680030\pi$
$128$	−128.000	−0.0883883
$129$	−708.000	−0.483224
$130$	−440.000	−0.296850
$131$	468.000	0.312132	0.156066	−	0.987747i	$-0.450119\pi$
$131$	468.000	0.312132	0.156066	+	0.987747i	$0.450119\pi$
$132$	288.000	0.189903
$133$	304.000	0.198197
$134$	590.000	0.380360
$135$	−135.000	−0.0860663
$136$	−600.000	−0.378306
$137$	990.000	0.617383	0.308691	−	0.951162i	$-0.400109\pi$
$137$	990.000	0.617383	0.308691	+	0.951162i	$0.400109\pi$
$138$	−138.000	−0.0851257
$139$	−2041.00	−1.24543	−0.622717	−	0.782447i	$-0.713971\pi$
$139$	−2041.00	−1.24543	−0.622717	+	0.782447i	$0.713971\pi$
$140$	−380.000	−0.229399
$141$	90.0000	0.0537544
$142$	1206.00	0.712713
$143$	−1056.00	−0.617533
$144$	144.000	0.0833333
$145$	−615.000	−0.352227
$146$	−1336.00	−0.757316
$147$	−54.0000	−0.0302983
$148$	−172.000	−0.0955291
$149$	1470.00	0.808236	0.404118	−	0.914707i	$-0.367579\pi$
$149$	1470.00	0.808236	0.404118	+	0.914707i	$0.367579\pi$
$150$	150.000	0.0816497
$151$	−2200.00	−1.18565	−0.592826	−	0.805331i	$-0.701988\pi$
$151$	−2200.00	−1.18565	−0.592826	+	0.805331i	$0.701988\pi$
$152$	128.000	0.0683038
$153$	675.000	0.356670
$154$	−912.000	−0.477215
$155$	−215.000	−0.111414
$156$	−528.000	−0.270986
$157$	−1033.00	−0.525111	−0.262555	−	0.964917i	$-0.584565\pi$
$157$	−1033.00	−0.525111	−0.262555	+	0.964917i	$0.584565\pi$
$158$	2552.00	1.28498
$159$	−1557.00	−0.776592
$160$	−160.000	−0.0790569
$161$	437.000	0.213916
$162$	−162.000	−0.0785674
$163$	−2494.00	−1.19844	−0.599218	−	0.800586i	$-0.704522\pi$
$163$	−2494.00	−1.19844	−0.599218	+	0.800586i	$0.704522\pi$
$164$	828.000	0.394244
$165$	360.000	0.169854
$166$	1146.00	0.535824
$167$	3810.00	1.76543	0.882715	−	0.469910i	$-0.155714\pi$
$167$	3810.00	1.76543	0.882715	+	0.469910i	$0.155714\pi$
$168$	−456.000	−0.209412
$169$	−261.000	−0.118798
$170$	−750.000	−0.338367
$171$	−144.000	−0.0643974
$172$	944.000	0.418484
$173$	−432.000	−0.189852	−0.0949259	−	0.995484i	$-0.530261\pi$
$173$	−432.000	−0.189852	−0.0949259	+	0.995484i	$0.530261\pi$
$174$	−738.000	−0.321538
$175$	−475.000	−0.205181
$176$	−384.000	−0.164461
$177$	−117.000	−0.0496851
$178$	−912.000	−0.384030
$179$	2640.00	1.10236	0.551181	−	0.834386i	$-0.314177\pi$
$179$	2640.00	1.10236	0.551181	+	0.834386i	$0.314177\pi$
$180$	180.000	0.0745356
$181$	−3616.00	−1.48495	−0.742473	−	0.669876i	$-0.766347\pi$
$181$	−3616.00	−1.48495	−0.742473	+	0.669876i	$0.766347\pi$
$182$	1672.00	0.680972
$183$	570.000	0.230249
$184$	184.000	0.0737210
$185$	−215.000	−0.0854439
$186$	−258.000	−0.101707
$187$	−1800.00	−0.703899
$188$	−120.000	−0.0465527
$189$	513.000	0.197435
$190$	160.000	0.0610927
$191$	−942.000	−0.356862	−0.178431	−	0.983952i	$-0.557102\pi$
$191$	−942.000	−0.356862	−0.178431	+	0.983952i	$0.557102\pi$
$192$	−192.000	−0.0721688
$193$	−1864.00	−0.695200	−0.347600	−	0.937643i	$-0.613003\pi$
$193$	−1864.00	−0.695200	−0.347600	+	0.937643i	$0.613003\pi$
$194$	2372.00	0.877833
$195$	−660.000	−0.242377
$196$	72.0000	0.0262391
$197$	−3774.00	−1.36491	−0.682453	−	0.730930i	$-0.739087\pi$
$197$	−3774.00	−1.36491	−0.682453	+	0.730930i	$0.739087\pi$
$198$	432.000	0.155055
$199$	422.000	0.150326	0.0751628	−	0.997171i	$-0.476052\pi$
$199$	422.000	0.150326	0.0751628	+	0.997171i	$0.476052\pi$
$200$	−200.000	−0.0707107
$201$	885.000	0.310563
$202$	2646.00	0.921643
$203$	2337.00	0.808006
$204$	−900.000	−0.308885
$205$	1035.00	0.352622
$206$	80.0000	0.0270576
$207$	−207.000	−0.0695048
$208$	704.000	0.234681
$209$	384.000	0.127090
$210$	−570.000	−0.187304
$211$	2093.00	0.682882	0.341441	−	0.939903i	$-0.389085\pi$
$211$	2093.00	0.682882	0.341441	+	0.939903i	$0.389085\pi$
$212$	2076.00	0.672548
$213$	1809.00	0.581928
$214$	−1086.00	−0.346904
$215$	1180.00	0.374304
$216$	216.000	0.0680414
$217$	817.000	0.255583
$218$	3704.00	1.15076
$219$	−2004.00	−0.618346
$220$	−480.000	−0.147098
$221$	3300.00	1.00444
$222$	−258.000	−0.0779992
$223$	3446.00	1.03480	0.517402	−	0.855743i	$-0.326899\pi$
$223$	3446.00	1.03480	0.517402	+	0.855743i	$0.326899\pi$
$224$	608.000	0.181356
$225$	225.000	0.0666667
$226$	2754.00	0.810590
$227$	−1980.00	−0.578930	−0.289465	−	0.957189i	$-0.593477\pi$
$227$	−1980.00	−0.578930	−0.289465	+	0.957189i	$0.593477\pi$
$228$	192.000	0.0557698
$229$	4256.00	1.22814	0.614071	−	0.789251i	$-0.289531\pi$
$229$	4256.00	1.22814	0.614071	+	0.789251i	$0.289531\pi$
$230$	230.000	0.0659380
$231$	−1368.00	−0.389644
$232$	984.000	0.278460
$233$	−2964.00	−0.833382	−0.416691	−	0.909048i	$-0.636810\pi$
$233$	−2964.00	−0.833382	−0.416691	+	0.909048i	$0.636810\pi$
$234$	−792.000	−0.221259
$235$	−150.000	−0.0416380
$236$	156.000	0.0430285
$237$	3828.00	1.04918
$238$	2850.00	0.776210
$239$	−165.000	−0.0446567	−0.0223284	−	0.999751i	$-0.507108\pi$
$239$	−165.000	−0.0446567	−0.0223284	+	0.999751i	$0.507108\pi$
$240$	−240.000	−0.0645497
$241$	−1072.00	−0.286529	−0.143265	−	0.989684i	$-0.545760\pi$
$241$	−1072.00	−0.286529	−0.143265	+	0.989684i	$0.545760\pi$
$242$	1510.00	0.401101
$243$	−243.000	−0.0641500
$244$	−760.000	−0.199402
$245$	90.0000	0.0234689
$246$	1242.00	0.321898
$247$	−704.000	−0.181354
$248$	344.000	0.0880807
$249$	1719.00	0.437499
$250$	−250.000	−0.0632456
$251$	−1398.00	−0.351558	−0.175779	−	0.984430i	$-0.556244\pi$
$251$	−1398.00	−0.351558	−0.175779	+	0.984430i	$0.556244\pi$
$252$	−684.000	−0.170984
$253$	552.000	0.137170
$254$	3068.00	0.757888
$255$	−1125.00	−0.276275
$256$	256.000	0.0625000
$257$	378.000	0.0917471	0.0458735	−	0.998947i	$-0.485393\pi$
$257$	378.000	0.0917471	0.0458735	+	0.998947i	$0.485393\pi$
$258$	1416.00	0.341691
$259$	817.000	0.196007
$260$	880.000	0.209905
$261$	−1107.00	−0.262535
$262$	−936.000	−0.220711
$263$	−2823.00	−0.661877	−0.330938	−	0.943652i	$-0.607365\pi$
$263$	−2823.00	−0.661877	−0.330938	+	0.943652i	$0.607365\pi$
$264$	−576.000	−0.134282
$265$	2595.00	0.601546
$266$	−608.000	−0.140146
$267$	−1368.00	−0.313559
$268$	−1180.00	−0.268955
$269$	1755.00	0.397785	0.198893	−	0.980021i	$-0.436265\pi$
$269$	1755.00	0.397785	0.198893	+	0.980021i	$0.436265\pi$
$270$	270.000	0.0608581
$271$	−769.000	−0.172374	−0.0861871	−	0.996279i	$-0.527468\pi$
$271$	−769.000	−0.172374	−0.0861871	+	0.996279i	$0.527468\pi$
$272$	1200.00	0.267503
$273$	2508.00	0.556011
$274$	−1980.00	−0.436555
$275$	−600.000	−0.131569
$276$	276.000	0.0601929
$277$	−3730.00	−0.809076	−0.404538	−	0.914521i	$-0.632568\pi$
$277$	−3730.00	−0.809076	−0.404538	+	0.914521i	$0.632568\pi$
$278$	4082.00	0.880655
$279$	−387.000	−0.0830433
$280$	760.000	0.162210
$281$	444.000	0.0942591	0.0471296	−	0.998889i	$-0.484993\pi$
$281$	444.000	0.0942591	0.0471296	+	0.998889i	$0.484993\pi$
$282$	−180.000	−0.0380101
$283$	−2161.00	−0.453916	−0.226958	−	0.973905i	$-0.572878\pi$
$283$	−2161.00	−0.453916	−0.226958	+	0.973905i	$0.572878\pi$
$284$	−2412.00	−0.503964
$285$	240.000	0.0498820
$286$	2112.00	0.436661
$287$	−3933.00	−0.808912
$288$	−288.000	−0.0589256
$289$	712.000	0.144922
$290$	1230.00	0.249062
$291$	3558.00	0.716748
$292$	2672.00	0.535503
$293$	2481.00	0.494681	0.247341	−	0.968929i	$-0.420443\pi$
$293$	2481.00	0.494681	0.247341	+	0.968929i	$0.420443\pi$
$294$	108.000	0.0214241
$295$	195.000	0.0384859
$296$	344.000	0.0675493
$297$	648.000	0.126602
$298$	−2940.00	−0.571509
$299$	−1012.00	−0.195737
$300$	−300.000	−0.0577350
$301$	−4484.00	−0.858649
$302$	4400.00	0.838383
$303$	3969.00	0.752518
$304$	−256.000	−0.0482980
$305$	−950.000	−0.178350
$306$	−1350.00	−0.252204
$307$	3056.00	0.568127	0.284064	−	0.958805i	$-0.408317\pi$
$307$	3056.00	0.568127	0.284064	+	0.958805i	$0.408317\pi$
$308$	1824.00	0.337442
$309$	120.000	0.0220924
$310$	430.000	0.0787818
$311$	−5352.00	−0.975833	−0.487917	−	0.872890i	$-0.662243\pi$
$311$	−5352.00	−0.975833	−0.487917	+	0.872890i	$0.662243\pi$
$312$	1056.00	0.191616
$313$	4751.00	0.857963	0.428981	−	0.903313i	$-0.358873\pi$
$313$	4751.00	0.857963	0.428981	+	0.903313i	$0.358873\pi$
$314$	2066.00	0.371309
$315$	−855.000	−0.152933
$316$	−5104.00	−0.908615
$317$	−4530.00	−0.802619	−0.401309	−	0.915943i	$-0.631445\pi$
$317$	−4530.00	−0.802619	−0.401309	+	0.915943i	$0.631445\pi$
$318$	3114.00	0.549133
$319$	2952.00	0.518120
$320$	320.000	0.0559017
$321$	−1629.00	−0.283246
$322$	−874.000	−0.151261
$323$	−1200.00	−0.206718
$324$	324.000	0.0555556
$325$	1100.00	0.187745
$326$	4988.00	0.847423
$327$	5556.00	0.939595
$328$	−1656.00	−0.278772
$329$	570.000	0.0955171
$330$	−720.000	−0.120105
$331$	−11155.0	−1.85237	−0.926185	−	0.377070i	$-0.876932\pi$
$331$	−11155.0	−1.85237	−0.926185	+	0.377070i	$0.876932\pi$
$332$	−2292.00	−0.378885
$333$	−387.000	−0.0636861
$334$	−7620.00	−1.24835
$335$	−1475.00	−0.240561
$336$	912.000	0.148076
$337$	7214.00	1.16609	0.583044	−	0.812441i	$-0.301862\pi$
$337$	7214.00	1.16609	0.583044	+	0.812441i	$0.301862\pi$
$338$	522.000	0.0840031
$339$	4131.00	0.661844
$340$	1500.00	0.239262
$341$	1032.00	0.163888
$342$	288.000	0.0455358
$343$	6175.00	0.972066
$344$	−1888.00	−0.295913
$345$	345.000	0.0538382
$346$	864.000	0.134245
$347$	4476.00	0.692462	0.346231	−	0.938149i	$-0.387461\pi$
$347$	4476.00	0.692462	0.346231	+	0.938149i	$0.387461\pi$
$348$	1476.00	0.227362
$349$	4115.00	0.631149	0.315574	−	0.948901i	$-0.397803\pi$
$349$	4115.00	0.631149	0.315574	+	0.948901i	$0.397803\pi$
$350$	950.000	0.145085
$351$	−1188.00	−0.180657
$352$	768.000	0.116291
$353$	3198.00	0.482188	0.241094	−	0.970502i	$-0.422494\pi$
$353$	3198.00	0.482188	0.241094	+	0.970502i	$0.422494\pi$
$354$	234.000	0.0351327
$355$	−3015.00	−0.450759
$356$	1824.00	0.271550
$357$	4275.00	0.633773
$358$	−5280.00	−0.779488
$359$	10818.0	1.59040	0.795198	−	0.606350i	$-0.207367\pi$
$359$	10818.0	1.59040	0.795198	+	0.606350i	$0.207367\pi$
$360$	−360.000	−0.0527046
$361$	−6603.00	−0.962677
$362$	7232.00	1.05002
$363$	2265.00	0.327498
$364$	−3344.00	−0.481520
$365$	3340.00	0.478969
$366$	−1140.00	−0.162811
$367$	5159.00	0.733781	0.366890	−	0.930264i	$-0.380422\pi$
$367$	5159.00	0.733781	0.366890	+	0.930264i	$0.380422\pi$
$368$	−368.000	−0.0521286
$369$	1863.00	0.262829
$370$	430.000	0.0604179
$371$	−9861.00	−1.37994
$372$	516.000	0.0719176
$373$	−5362.00	−0.744327	−0.372163	−	0.928167i	$-0.621384\pi$
$373$	−5362.00	−0.744327	−0.372163	+	0.928167i	$0.621384\pi$
$374$	3600.00	0.497731
$375$	−375.000	−0.0516398
$376$	240.000	0.0329177
$377$	−5412.00	−0.739343
$378$	−1026.00	−0.139608
$379$	1016.00	0.137700	0.0688502	−	0.997627i	$-0.478067\pi$
$379$	1016.00	0.137700	0.0688502	+	0.997627i	$0.478067\pi$
$380$	−320.000	−0.0431991
$381$	4602.00	0.618813
$382$	1884.00	0.252340
$383$	5169.00	0.689618	0.344809	−	0.938673i	$-0.387944\pi$
$383$	5169.00	0.689618	0.344809	+	0.938673i	$0.387944\pi$
$384$	384.000	0.0510310
$385$	2280.00	0.301817
$386$	3728.00	0.491581
$387$	2124.00	0.278990
$388$	−4744.00	−0.620722
$389$	5010.00	0.653000	0.326500	−	0.945197i	$-0.394131\pi$
$389$	5010.00	0.653000	0.326500	+	0.945197i	$0.394131\pi$
$390$	1320.00	0.171387
$391$	−1725.00	−0.223113
$392$	−144.000	−0.0185538
$393$	−1404.00	−0.180210
$394$	7548.00	0.965134
$395$	−6380.00	−0.812690
$396$	−864.000	−0.109640
$397$	8108.00	1.02501	0.512505	−	0.858684i	$-0.328718\pi$
$397$	8108.00	1.02501	0.512505	+	0.858684i	$0.328718\pi$
$398$	−844.000	−0.106296
$399$	−912.000	−0.114429
$400$	400.000	0.0500000
$401$	6558.00	0.816686	0.408343	−	0.912829i	$-0.366107\pi$
$401$	6558.00	0.816686	0.408343	+	0.912829i	$0.366107\pi$
$402$	−1770.00	−0.219601
$403$	−1892.00	−0.233864
$404$	−5292.00	−0.651700
$405$	405.000	0.0496904
$406$	−4674.00	−0.571347
$407$	1032.00	0.125686
$408$	1800.00	0.218415
$409$	16025.0	1.93737	0.968686	−	0.248289i	$-0.0798681\pi$
$409$	16025.0	1.93737	0.968686	+	0.248289i	$0.0798681\pi$
$410$	−2070.00	−0.249341
$411$	−2970.00	−0.356446
$412$	−160.000	−0.0191326
$413$	−741.000	−0.0882863
$414$	414.000	0.0491473
$415$	−2865.00	−0.338885
$416$	−1408.00	−0.165944
$417$	6123.00	0.719052
$418$	−768.000	−0.0898663
$419$	7824.00	0.912237	0.456119	−	0.889919i	$-0.349239\pi$
$419$	7824.00	0.912237	0.456119	+	0.889919i	$0.349239\pi$
$420$	1140.00	0.132444
$421$	−14956.0	−1.73138	−0.865690	−	0.500581i	$-0.833120\pi$
$421$	−14956.0	−1.73138	−0.865690	+	0.500581i	$0.833120\pi$
$422$	−4186.00	−0.482870
$423$	−270.000	−0.0310351
$424$	−4152.00	−0.475564
$425$	1875.00	0.214002
$426$	−3618.00	−0.411485
$427$	3610.00	0.409134
$428$	2172.00	0.245298
$429$	3168.00	0.356533
$430$	−2360.00	−0.264673
$431$	4944.00	0.552539	0.276269	−	0.961080i	$-0.410902\pi$
$431$	4944.00	0.552539	0.276269	+	0.961080i	$0.410902\pi$
$432$	−432.000	−0.0481125
$433$	3197.00	0.354822	0.177411	−	0.984137i	$-0.443228\pi$
$433$	3197.00	0.354822	0.177411	+	0.984137i	$0.443228\pi$
$434$	−1634.00	−0.180725
$435$	1845.00	0.203359
$436$	−7408.00	−0.813713
$437$	368.000	0.0402834
$438$	4008.00	0.437237
$439$	−10840.0	−1.17851	−0.589254	−	0.807948i	$-0.700578\pi$
$439$	−10840.0	−1.17851	−0.589254	+	0.807948i	$0.700578\pi$
$440$	960.000	0.104014
$441$	162.000	0.0174927
$442$	−6600.00	−0.710249
$443$	8922.00	0.956878	0.478439	−	0.878121i	$-0.341203\pi$
$443$	8922.00	0.956878	0.478439	+	0.878121i	$0.341203\pi$
$444$	516.000	0.0551538
$445$	2280.00	0.242882
$446$	−6892.00	−0.731717
$447$	−4410.00	−0.466635
$448$	−1216.00	−0.128238
$449$	−13875.0	−1.45836	−0.729178	−	0.684324i	$-0.760097\pi$
$449$	−13875.0	−1.45836	−0.729178	+	0.684324i	$0.760097\pi$
$450$	−450.000	−0.0471405
$451$	−4968.00	−0.518701
$452$	−5508.00	−0.573174
$453$	6600.00	0.684537
$454$	3960.00	0.409366
$455$	−4180.00	−0.430684
$456$	−384.000	−0.0394352
$457$	−7429.00	−0.760424	−0.380212	−	0.924899i	$-0.624149\pi$
$457$	−7429.00	−0.760424	−0.380212	+	0.924899i	$0.624149\pi$
$458$	−8512.00	−0.868427
$459$	−2025.00	−0.205924
$460$	−460.000	−0.0466252
$461$	−12054.0	−1.21781	−0.608905	−	0.793243i	$-0.708391\pi$
$461$	−12054.0	−1.21781	−0.608905	+	0.793243i	$0.708391\pi$
$462$	2736.00	0.275520
$463$	−1324.00	−0.132897	−0.0664487	−	0.997790i	$-0.521167\pi$
$463$	−1324.00	−0.132897	−0.0664487	+	0.997790i	$0.521167\pi$
$464$	−1968.00	−0.196901
$465$	645.000	0.0643251
$466$	5928.00	0.589290
$467$	5895.00	0.584129	0.292064	−	0.956399i	$-0.405658\pi$
$467$	5895.00	0.584129	0.292064	+	0.956399i	$0.405658\pi$
$468$	1584.00	0.156454
$469$	5605.00	0.551844
$470$	300.000	0.0294425
$471$	3099.00	0.303173
$472$	−312.000	−0.0304258
$473$	−5664.00	−0.550594
$474$	−7656.00	−0.741881
$475$	−400.000	−0.0386384
$476$	−5700.00	−0.548864
$477$	4671.00	0.448366
$478$	330.000	0.0315771
$479$	−7110.00	−0.678213	−0.339107	−	0.940748i	$-0.610125\pi$
$479$	−7110.00	−0.678213	−0.339107	+	0.940748i	$0.610125\pi$
$480$	480.000	0.0456435
$481$	−1892.00	−0.179351
$482$	2144.00	0.202607
$483$	−1311.00	−0.123504
$484$	−3020.00	−0.283621
$485$	−5930.00	−0.555191
$486$	486.000	0.0453609
$487$	−514.000	−0.0478266	−0.0239133	−	0.999714i	$-0.507613\pi$
$487$	−514.000	−0.0478266	−0.0239133	+	0.999714i	$0.507613\pi$
$488$	1520.00	0.140998
$489$	7482.00	0.691918
$490$	−180.000	−0.0165950
$491$	−5607.00	−0.515357	−0.257679	−	0.966231i	$-0.582958\pi$
$491$	−5607.00	−0.515357	−0.257679	+	0.966231i	$0.582958\pi$
$492$	−2484.00	−0.227617
$493$	−9225.00	−0.842745
$494$	1408.00	0.128237
$495$	−1080.00	−0.0980654
$496$	−688.000	−0.0622825
$497$	11457.0	1.03404
$498$	−3438.00	−0.309358
$499$	4199.00	0.376700	0.188350	−	0.982102i	$-0.439686\pi$
$499$	4199.00	0.376700	0.188350	+	0.982102i	$0.439686\pi$
$500$	500.000	0.0447214
$501$	−11430.0	−1.01927
$502$	2796.00	0.248589
$503$	−2451.00	−0.217266	−0.108633	−	0.994082i	$-0.534647\pi$
$503$	−2451.00	−0.217266	−0.108633	+	0.994082i	$0.534647\pi$
$504$	1368.00	0.120904
$505$	−6615.00	−0.582898
$506$	−1104.00	−0.0969936
$507$	783.000	0.0685883
$508$	−6136.00	−0.535907
$509$	15990.0	1.39242	0.696212	−	0.717836i	$-0.254867\pi$
$509$	15990.0	1.39242	0.696212	+	0.717836i	$0.254867\pi$
$510$	2250.00	0.195356
$511$	−12692.0	−1.09875
$512$	−512.000	−0.0441942
$513$	432.000	0.0371799
$514$	−756.000	−0.0648750
$515$	−200.000	−0.0171127
$516$	−2832.00	−0.241612
$517$	720.000	0.0612487
$518$	−1634.00	−0.138598
$519$	1296.00	0.109611
$520$	−1760.00	−0.148425
$521$	−16884.0	−1.41977	−0.709886	−	0.704316i	$-0.751254\pi$
$521$	−16884.0	−1.41977	−0.709886	+	0.704316i	$0.751254\pi$
$522$	2214.00	0.185640
$523$	7940.00	0.663847	0.331923	−	0.943306i	$-0.392302\pi$
$523$	7940.00	0.663847	0.331923	+	0.943306i	$0.392302\pi$
$524$	1872.00	0.156066
$525$	1425.00	0.118461
$526$	5646.00	0.468018
$527$	−3225.00	−0.266572
$528$	1152.00	0.0949514
$529$	529.000	0.0434783
$530$	−5190.00	−0.425357
$531$	351.000	0.0286857
$532$	1216.00	0.0990983
$533$	9108.00	0.740171
$534$	2736.00	0.221720
$535$	2715.00	0.219401
$536$	2360.00	0.190180
$537$	−7920.00	−0.636449
$538$	−3510.00	−0.281277
$539$	−432.000	−0.0345224
$540$	−540.000	−0.0430331
$541$	11054.0	0.878463	0.439232	−	0.898374i	$-0.355251\pi$
$541$	11054.0	0.878463	0.439232	+	0.898374i	$0.355251\pi$
$542$	1538.00	0.121887
$543$	10848.0	0.857334
$544$	−2400.00	−0.189153
$545$	−9260.00	−0.727807
$546$	−5016.00	−0.393159
$547$	−14524.0	−1.13529	−0.567643	−	0.823275i	$-0.692145\pi$
$547$	−14524.0	−1.13529	−0.567643	+	0.823275i	$0.692145\pi$
$548$	3960.00	0.308691
$549$	−1710.00	−0.132934
$550$	1200.00	0.0930330
$551$	1968.00	0.152159
$552$	−552.000	−0.0425628
$553$	24244.0	1.86430
$554$	7460.00	0.572103
$555$	645.000	0.0493310
$556$	−8164.00	−0.622717
$557$	−4185.00	−0.318356	−0.159178	−	0.987250i	$-0.550884\pi$
$557$	−4185.00	−0.318356	−0.159178	+	0.987250i	$0.550884\pi$
$558$	774.000	0.0587205
$559$	10384.0	0.785682
$560$	−1520.00	−0.114700
$561$	5400.00	0.406396
$562$	−888.000	−0.0666513
$563$	−9711.00	−0.726945	−0.363472	−	0.931605i	$-0.618409\pi$
$563$	−9711.00	−0.726945	−0.363472	+	0.931605i	$0.618409\pi$
$564$	360.000	0.0268772
$565$	−6885.00	−0.512662
$566$	4322.00	0.320967
$567$	−1539.00	−0.113989
$568$	4824.00	0.356357
$569$	6606.00	0.486710	0.243355	−	0.969937i	$-0.421752\pi$
$569$	6606.00	0.486710	0.243355	+	0.969937i	$0.421752\pi$
$570$	−480.000	−0.0352719
$571$	−21802.0	−1.59787	−0.798936	−	0.601416i	$-0.794604\pi$
$571$	−21802.0	−1.59787	−0.798936	+	0.601416i	$0.794604\pi$
$572$	−4224.00	−0.308766
$573$	2826.00	0.206035
$574$	7866.00	0.571987
$575$	−575.000	−0.0417029
$576$	576.000	0.0416667
$577$	452.000	0.0326118	0.0163059	−	0.999867i	$-0.494809\pi$
$577$	452.000	0.0326118	0.0163059	+	0.999867i	$0.494809\pi$
$578$	−1424.00	−0.102475
$579$	5592.00	0.401374
$580$	−2460.00	−0.176114
$581$	10887.0	0.777399
$582$	−7116.00	−0.506817
$583$	−12456.0	−0.884862
$584$	−5344.00	−0.378658
$585$	1980.00	0.139937
$586$	−4962.00	−0.349792
$587$	7938.00	0.558154	0.279077	−	0.960269i	$-0.409972\pi$
$587$	7938.00	0.558154	0.279077	+	0.960269i	$0.409972\pi$
$588$	−216.000	−0.0151491
$589$	688.000	0.0481300
$590$	−390.000	−0.0272136
$591$	11322.0	0.788029
$592$	−688.000	−0.0477646
$593$	462.000	0.0319934	0.0159967	−	0.999872i	$-0.494908\pi$
$593$	462.000	0.0319934	0.0159967	+	0.999872i	$0.494908\pi$
$594$	−1296.00	−0.0895211
$595$	−7125.00	−0.490919
$596$	5880.00	0.404118
$597$	−1266.00	−0.0867905
$598$	2024.00	0.138407
$599$	−15276.0	−1.04200	−0.521002	−	0.853555i	$-0.674442\pi$
$599$	−15276.0	−1.04200	−0.521002	+	0.853555i	$0.674442\pi$
$600$	600.000	0.0408248
$601$	19847.0	1.34705	0.673524	−	0.739165i	$-0.264780\pi$
$601$	19847.0	1.34705	0.673524	+	0.739165i	$0.264780\pi$
$602$	8968.00	0.607157
$603$	−2655.00	−0.179303
$604$	−8800.00	−0.592826
$605$	−3775.00	−0.253679
$606$	−7938.00	−0.532111
$607$	−9682.00	−0.647414	−0.323707	−	0.946157i	$-0.604929\pi$
$607$	−9682.00	−0.647414	−0.323707	+	0.946157i	$0.604929\pi$
$608$	512.000	0.0341519
$609$	−7011.00	−0.466503
$610$	1900.00	0.126113
$611$	−1320.00	−0.0874001
$612$	2700.00	0.178335
$613$	−6358.00	−0.418919	−0.209459	−	0.977817i	$-0.567170\pi$
$613$	−6358.00	−0.418919	−0.209459	+	0.977817i	$0.567170\pi$
$614$	−6112.00	−0.401727
$615$	−3105.00	−0.203586
$616$	−3648.00	−0.238607
$617$	−4215.00	−0.275024	−0.137512	−	0.990500i	$-0.543910\pi$
$617$	−4215.00	−0.275024	−0.137512	+	0.990500i	$0.543910\pi$
$618$	−240.000	−0.0156217
$619$	18356.0	1.19191	0.595953	−	0.803019i	$-0.296774\pi$
$619$	18356.0	1.19191	0.595953	+	0.803019i	$0.296774\pi$
$620$	−860.000	−0.0557071
$621$	621.000	0.0401286
$622$	10704.0	0.690018
$623$	−8664.00	−0.557168
$624$	−2112.00	−0.135493
$625$	625.000	0.0400000
$626$	−9502.00	−0.606671
$627$	−1152.00	−0.0733755
$628$	−4132.00	−0.262555
$629$	−3225.00	−0.204434
$630$	1710.00	0.108140
$631$	−7114.00	−0.448818	−0.224409	−	0.974495i	$-0.572045\pi$
$631$	−7114.00	−0.448818	−0.224409	+	0.974495i	$0.572045\pi$
$632$	10208.0	0.642488
$633$	−6279.00	−0.394262
$634$	9060.00	0.567537
$635$	−7670.00	−0.479330
$636$	−6228.00	−0.388296
$637$	792.000	0.0492625
$638$	−5904.00	−0.366366
$639$	−5427.00	−0.335976
$640$	−640.000	−0.0395285
$641$	−21144.0	−1.30287	−0.651434	−	0.758706i	$-0.725832\pi$
$641$	−21144.0	−1.30287	−0.651434	+	0.758706i	$0.725832\pi$
$642$	3258.00	0.200285
$643$	−12823.0	−0.786454	−0.393227	−	0.919441i	$-0.628641\pi$
$643$	−12823.0	−0.786454	−0.393227	+	0.919441i	$0.628641\pi$
$644$	1748.00	0.106958
$645$	−3540.00	−0.216104
$646$	2400.00	0.146171
$647$	−30792.0	−1.87103	−0.935517	−	0.353283i	$-0.885065\pi$
$647$	−30792.0	−1.87103	−0.935517	+	0.353283i	$0.885065\pi$
$648$	−648.000	−0.0392837
$649$	−936.000	−0.0566120
$650$	−2200.00	−0.132756
$651$	−2451.00	−0.147561
$652$	−9976.00	−0.599218
$653$	−8808.00	−0.527846	−0.263923	−	0.964544i	$-0.585017\pi$
$653$	−8808.00	−0.527846	−0.263923	+	0.964544i	$0.585017\pi$
$654$	−11112.0	−0.664394
$655$	2340.00	0.139590
$656$	3312.00	0.197122
$657$	6012.00	0.357002
$658$	−1140.00	−0.0675408
$659$	−9960.00	−0.588750	−0.294375	−	0.955690i	$-0.595112\pi$
$659$	−9960.00	−0.588750	−0.294375	+	0.955690i	$0.595112\pi$
$660$	1440.00	0.0849272
$661$	−16810.0	−0.989158	−0.494579	−	0.869133i	$-0.664678\pi$
$661$	−16810.0	−0.989158	−0.494579	+	0.869133i	$0.664678\pi$
$662$	22310.0	1.30982
$663$	−9900.00	−0.579916
$664$	4584.00	0.267912
$665$	1520.00	0.0886362
$666$	774.000	0.0450329
$667$	2829.00	0.164227
$668$	15240.0	0.882715
$669$	−10338.0	−0.597444
$670$	2950.00	0.170102
$671$	4560.00	0.262350
$672$	−1824.00	−0.104706
$673$	530.000	0.0303566	0.0151783	−	0.999885i	$-0.495168\pi$
$673$	530.000	0.0303566	0.0151783	+	0.999885i	$0.495168\pi$
$674$	−14428.0	−0.824549
$675$	−675.000	−0.0384900
$676$	−1044.00	−0.0593992
$677$	−15039.0	−0.853760	−0.426880	−	0.904308i	$-0.640387\pi$
$677$	−15039.0	−0.853760	−0.426880	+	0.904308i	$0.640387\pi$
$678$	−8262.00	−0.467994
$679$	22534.0	1.27360
$680$	−3000.00	−0.169183
$681$	5940.00	0.334246
$682$	−2064.00	−0.115887
$683$	−29262.0	−1.63935	−0.819677	−	0.572825i	$-0.805847\pi$
$683$	−29262.0	−1.63935	−0.819677	+	0.572825i	$0.805847\pi$
$684$	−576.000	−0.0321987
$685$	4950.00	0.276102
$686$	−12350.0	−0.687355
$687$	−12768.0	−0.709068
$688$	3776.00	0.209242
$689$	22836.0	1.26267
$690$	−690.000	−0.0380693
$691$	7640.00	0.420607	0.210303	−	0.977636i	$-0.432555\pi$
$691$	7640.00	0.420607	0.210303	+	0.977636i	$0.432555\pi$
$692$	−1728.00	−0.0949259
$693$	4104.00	0.224961
$694$	−8952.00	−0.489644
$695$	−10205.0	−0.556975
$696$	−2952.00	−0.160769
$697$	15525.0	0.843689
$698$	−8230.00	−0.446290
$699$	8892.00	0.481154
$700$	−1900.00	−0.102590
$701$	28866.0	1.55528	0.777642	−	0.628708i	$-0.216416\pi$
$701$	28866.0	1.55528	0.777642	+	0.628708i	$0.216416\pi$
$702$	2376.00	0.127744
$703$	688.000	0.0369110
$704$	−1536.00	−0.0822304
$705$	450.000	0.0240397
$706$	−6396.00	−0.340958
$707$	25137.0	1.33716
$708$	−468.000	−0.0248425
$709$	15986.0	0.846780	0.423390	−	0.905948i	$-0.360840\pi$
$709$	15986.0	0.846780	0.423390	+	0.905948i	$0.360840\pi$
$710$	6030.00	0.318735
$711$	−11484.0	−0.605744
$712$	−3648.00	−0.192015
$713$	989.000	0.0519472
$714$	−8550.00	−0.448145
$715$	−5280.00	−0.276169
$716$	10560.0	0.551181
$717$	495.000	0.0257826
$718$	−21636.0	−1.12458
$719$	11103.0	0.575900	0.287950	−	0.957645i	$-0.407026\pi$
$719$	11103.0	0.575900	0.287950	+	0.957645i	$0.407026\pi$
$720$	720.000	0.0372678
$721$	760.000	0.0392564
$722$	13206.0	0.680715
$723$	3216.00	0.165428
$724$	−14464.0	−0.742473
$725$	−3075.00	−0.157521
$726$	−4530.00	−0.231576
$727$	6491.00	0.331139	0.165569	−	0.986198i	$-0.447054\pi$
$727$	6491.00	0.331139	0.165569	+	0.986198i	$0.447054\pi$
$728$	6688.00	0.340486
$729$	729.000	0.0370370
$730$	−6680.00	−0.338682
$731$	17700.0	0.895565
$732$	2280.00	0.115125
$733$	−11839.0	−0.596567	−0.298283	−	0.954477i	$-0.596414\pi$
$733$	−11839.0	−0.596567	−0.298283	+	0.954477i	$0.596414\pi$
$734$	−10318.0	−0.518861
$735$	−270.000	−0.0135498
$736$	736.000	0.0368605
$737$	7080.00	0.353860
$738$	−3726.00	−0.185848
$739$	9473.00	0.471543	0.235771	−	0.971809i	$-0.424238\pi$
$739$	9473.00	0.471543	0.235771	+	0.971809i	$0.424238\pi$
$740$	−860.000	−0.0427219
$741$	2112.00	0.104705
$742$	19722.0	0.975765
$743$	−8304.00	−0.410019	−0.205010	−	0.978760i	$-0.565723\pi$
$743$	−8304.00	−0.410019	−0.205010	+	0.978760i	$0.565723\pi$
$744$	−1032.00	−0.0508534
$745$	7350.00	0.361454
$746$	10724.0	0.526318
$747$	−5157.00	−0.252590
$748$	−7200.00	−0.351949
$749$	−10317.0	−0.503304
$750$	750.000	0.0365148
$751$	14996.0	0.728644	0.364322	−	0.931273i	$-0.381301\pi$
$751$	14996.0	0.728644	0.364322	+	0.931273i	$0.381301\pi$
$752$	−480.000	−0.0232763
$753$	4194.00	0.202972
$754$	10824.0	0.522794
$755$	−11000.0	−0.530240
$756$	2052.00	0.0987176
$757$	28055.0	1.34700	0.673498	−	0.739189i	$-0.264791\pi$
$757$	28055.0	1.34700	0.673498	+	0.739189i	$0.264791\pi$
$758$	−2032.00	−0.0973688
$759$	−1656.00	−0.0791950
$760$	640.000	0.0305464
$761$	−17625.0	−0.839561	−0.419780	−	0.907626i	$-0.637893\pi$
$761$	−17625.0	−0.839561	−0.419780	+	0.907626i	$0.637893\pi$
$762$	−9204.00	−0.437567
$763$	35188.0	1.66958
$764$	−3768.00	−0.178431
$765$	3375.00	0.159508
$766$	−10338.0	−0.487633
$767$	1716.00	0.0807838
$768$	−768.000	−0.0360844
$769$	20762.0	0.973598	0.486799	−	0.873514i	$-0.338164\pi$
$769$	20762.0	0.973598	0.486799	+	0.873514i	$0.338164\pi$
$770$	−4560.00	−0.213417
$771$	−1134.00	−0.0529702
$772$	−7456.00	−0.347600
$773$	7854.00	0.365445	0.182722	−	0.983165i	$-0.441509\pi$
$773$	7854.00	0.365445	0.182722	+	0.983165i	$0.441509\pi$
$774$	−4248.00	−0.197275
$775$	−1075.00	−0.0498260
$776$	9488.00	0.438917
$777$	−2451.00	−0.113165
$778$	−10020.0	−0.461741
$779$	−3312.00	−0.152330
$780$	−2640.00	−0.121189
$781$	14472.0	0.663059
$782$	3450.00	0.157764
$783$	3321.00	0.151575
$784$	288.000	0.0131195
$785$	−5165.00	−0.234837
$786$	2808.00	0.127428
$787$	11801.0	0.534511	0.267256	−	0.963626i	$-0.413883\pi$
$787$	11801.0	0.534511	0.267256	+	0.963626i	$0.413883\pi$
$788$	−15096.0	−0.682453
$789$	8469.00	0.382135
$790$	12760.0	0.574659
$791$	26163.0	1.17604
$792$	1728.00	0.0775275
$793$	−8360.00	−0.374366
$794$	−16216.0	−0.724791
$795$	−7785.00	−0.347303
$796$	1688.00	0.0751628
$797$	−15735.0	−0.699325	−0.349663	−	0.936876i	$-0.613704\pi$
$797$	−15735.0	−0.699325	−0.349663	+	0.936876i	$0.613704\pi$
$798$	1824.00	0.0809134
$799$	−2250.00	−0.0996236
$800$	−800.000	−0.0353553
$801$	4104.00	0.181033
$802$	−13116.0	−0.577484
$803$	−16032.0	−0.704554
$804$	3540.00	0.155281
$805$	2185.00	0.0956660
$806$	3784.00	0.165367
$807$	−5265.00	−0.229661
$808$	10584.0	0.460822
$809$	−38415.0	−1.66947	−0.834734	−	0.550654i	$-0.814379\pi$
$809$	−38415.0	−1.66947	−0.834734	+	0.550654i	$0.814379\pi$
$810$	−810.000	−0.0351364
$811$	2381.00	0.103093	0.0515464	−	0.998671i	$-0.483585\pi$
$811$	2381.00	0.103093	0.0515464	+	0.998671i	$0.483585\pi$
$812$	9348.00	0.404003
$813$	2307.00	0.0995203
$814$	−2064.00	−0.0888737
$815$	−12470.0	−0.535957
$816$	−3600.00	−0.154443
$817$	−3776.00	−0.161696
$818$	−32050.0	−1.36993
$819$	−7524.00	−0.321013
$820$	4140.00	0.176311
$821$	37038.0	1.57446	0.787232	−	0.616657i	$-0.211513\pi$
$821$	37038.0	1.57446	0.787232	+	0.616657i	$0.211513\pi$
$822$	5940.00	0.252045
$823$	24104.0	1.02091	0.510457	−	0.859903i	$-0.329476\pi$
$823$	24104.0	1.02091	0.510457	+	0.859903i	$0.329476\pi$
$824$	320.000	0.0135288
$825$	1800.00	0.0759612
$826$	1482.00	0.0624278
$827$	25185.0	1.05897	0.529485	−	0.848319i	$-0.322385\pi$
$827$	25185.0	1.05897	0.529485	+	0.848319i	$0.322385\pi$
$828$	−828.000	−0.0347524
$829$	8489.00	0.355652	0.177826	−	0.984062i	$-0.443094\pi$
$829$	8489.00	0.355652	0.177826	+	0.984062i	$0.443094\pi$
$830$	5730.00	0.239628
$831$	11190.0	0.467120
$832$	2816.00	0.117340
$833$	1350.00	0.0561521
$834$	−12246.0	−0.508446
$835$	19050.0	0.789524
$836$	1536.00	0.0635451
$837$	1161.00	0.0479451
$838$	−15648.0	−0.645049
$839$	−32286.0	−1.32853	−0.664265	−	0.747497i	$-0.731255\pi$
$839$	−32286.0	−1.32853	−0.664265	+	0.747497i	$0.731255\pi$
$840$	−2280.00	−0.0936518
$841$	−9260.00	−0.379679
$842$	29912.0	1.22427
$843$	−1332.00	−0.0544205
$844$	8372.00	0.341441
$845$	−1305.00	−0.0531282
$846$	540.000	0.0219451
$847$	14345.0	0.581936
$848$	8304.00	0.336274
$849$	6483.00	0.262068
$850$	−3750.00	−0.151322
$851$	989.000	0.0398384
$852$	7236.00	0.290964
$853$	−2914.00	−0.116968	−0.0584839	−	0.998288i	$-0.518627\pi$
$853$	−2914.00	−0.116968	−0.0584839	+	0.998288i	$0.518627\pi$
$854$	−7220.00	−0.289301
$855$	−720.000	−0.0287994
$856$	−4344.00	−0.173452
$857$	46590.0	1.85704	0.928520	−	0.371281i	$-0.121081\pi$
$857$	46590.0	1.85704	0.928520	+	0.371281i	$0.121081\pi$
$858$	−6336.00	−0.252107
$859$	−27079.0	−1.07558	−0.537790	−	0.843079i	$-0.680741\pi$
$859$	−27079.0	−1.07558	−0.537790	+	0.843079i	$0.680741\pi$
$860$	4720.00	0.187152
$861$	11799.0	0.467025
$862$	−9888.00	−0.390704
$863$	27966.0	1.10310	0.551549	−	0.834142i	$-0.314037\pi$
$863$	27966.0	1.10310	0.551549	+	0.834142i	$0.314037\pi$
$864$	864.000	0.0340207
$865$	−2160.00	−0.0849043
$866$	−6394.00	−0.250897
$867$	−2136.00	−0.0836705
$868$	3268.00	0.127792
$869$	30624.0	1.19545
$870$	−3690.00	−0.143796
$871$	−12980.0	−0.504949
$872$	14816.0	0.575382
$873$	−10674.0	−0.413815
$874$	−736.000	−0.0284846
$875$	−2375.00	−0.0917596
$876$	−8016.00	−0.309173
$877$	13178.0	0.507400	0.253700	−	0.967283i	$-0.418352\pi$
$877$	13178.0	0.507400	0.253700	+	0.967283i	$0.418352\pi$
$878$	21680.0	0.833331
$879$	−7443.00	−0.285604
$880$	−1920.00	−0.0735491
$881$	28470.0	1.08874	0.544369	−	0.838846i	$-0.316769\pi$
$881$	28470.0	1.08874	0.544369	+	0.838846i	$0.316769\pi$
$882$	−324.000	−0.0123692
$883$	−16270.0	−0.620078	−0.310039	−	0.950724i	$-0.600342\pi$
$883$	−16270.0	−0.620078	−0.310039	+	0.950724i	$0.600342\pi$
$884$	13200.0	0.502222
$885$	−585.000	−0.0222198
$886$	−17844.0	−0.676615
$887$	−8730.00	−0.330468	−0.165234	−	0.986254i	$-0.552838\pi$
$887$	−8730.00	−0.330468	−0.165234	+	0.986254i	$0.552838\pi$
$888$	−1032.00	−0.0389996
$889$	29146.0	1.09958
$890$	−4560.00	−0.171743
$891$	−1944.00	−0.0730937
$892$	13784.0	0.517402
$893$	480.000	0.0179872
$894$	8820.00	0.329961
$895$	13200.0	0.492991
$896$	2432.00	0.0906779
$897$	3036.00	0.113009
$898$	27750.0	1.03121
$899$	5289.00	0.196216
$900$	900.000	0.0333333
$901$	38925.0	1.43927
$902$	9936.00	0.366777
$903$	13452.0	0.495741
$904$	11016.0	0.405295
$905$	−18080.0	−0.664088
$906$	−13200.0	−0.484040
$907$	−1423.00	−0.0520948	−0.0260474	−	0.999661i	$-0.508292\pi$
$907$	−1423.00	−0.0520948	−0.0260474	+	0.999661i	$0.508292\pi$
$908$	−7920.00	−0.289465
$909$	−11907.0	−0.434467
$910$	8360.00	0.304540
$911$	−12972.0	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$911$	−12972.0	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$912$	768.000	0.0278849
$913$	13752.0	0.498494
$914$	14858.0	0.537701
$915$	2850.00	0.102971
$916$	17024.0	0.614071
$917$	−8892.00	−0.320218
$918$	4050.00	0.145610
$919$	−3100.00	−0.111273	−0.0556363	−	0.998451i	$-0.517719\pi$
$919$	−3100.00	−0.111273	−0.0556363	+	0.998451i	$0.517719\pi$
$920$	920.000	0.0329690
$921$	−9168.00	−0.328009
$922$	24108.0	0.861122
$923$	−26532.0	−0.946166
$924$	−5472.00	−0.194822
$925$	−1075.00	−0.0382117
$926$	2648.00	0.0939727
$927$	−360.000	−0.0127551
$928$	3936.00	0.139230
$929$	13953.0	0.492770	0.246385	−	0.969172i	$-0.420757\pi$
$929$	13953.0	0.492770	0.246385	+	0.969172i	$0.420757\pi$
$930$	−1290.00	−0.0454847
$931$	−288.000	−0.0101384
$932$	−11856.0	−0.416691
$933$	16056.0	0.563397
$934$	−11790.0	−0.413041
$935$	−9000.00	−0.314793
$936$	−3168.00	−0.110630
$937$	−33826.0	−1.17935	−0.589673	−	0.807642i	$-0.700743\pi$
$937$	−33826.0	−1.17935	−0.589673	+	0.807642i	$0.700743\pi$
$938$	−11210.0	−0.390213
$939$	−14253.0	−0.495345
$940$	−600.000	−0.0208190
$941$	2010.00	0.0696324	0.0348162	−	0.999394i	$-0.488915\pi$
$941$	2010.00	0.0696324	0.0348162	+	0.999394i	$0.488915\pi$
$942$	−6198.00	−0.214376
$943$	−4761.00	−0.164411
$944$	624.000	0.0215143
$945$	2565.00	0.0882957
$946$	11328.0	0.389329
$947$	−4068.00	−0.139591	−0.0697953	−	0.997561i	$-0.522235\pi$
$947$	−4068.00	−0.139591	−0.0697953	+	0.997561i	$0.522235\pi$
$948$	15312.0	0.524589
$949$	29392.0	1.00538
$950$	800.000	0.0273215
$951$	13590.0	0.463392
$952$	11400.0	0.388105
$953$	19302.0	0.656089	0.328045	−	0.944662i	$-0.393610\pi$
$953$	19302.0	0.656089	0.328045	+	0.944662i	$0.393610\pi$
$954$	−9342.00	−0.317042
$955$	−4710.00	−0.159594
$956$	−660.000	−0.0223284
$957$	−8856.00	−0.299137
$958$	14220.0	0.479569
$959$	−18810.0	−0.633375
$960$	−960.000	−0.0322749
$961$	−27942.0	−0.937934
$962$	3784.00	0.126820
$963$	4887.00	0.163532
$964$	−4288.00	−0.143265
$965$	−9320.00	−0.310903
$966$	2622.00	0.0873307
$967$	21614.0	0.718779	0.359390	−	0.933188i	$-0.382985\pi$
$967$	21614.0	0.718779	0.359390	+	0.933188i	$0.382985\pi$
$968$	6040.00	0.200551
$969$	3600.00	0.119348
$970$	11860.0	0.392579
$971$	−14520.0	−0.479886	−0.239943	−	0.970787i	$-0.577129\pi$
$971$	−14520.0	−0.479886	−0.239943	+	0.970787i	$0.577129\pi$
$972$	−972.000	−0.0320750
$973$	38779.0	1.27770
$974$	1028.00	0.0338185
$975$	−3300.00	−0.108394
$976$	−3040.00	−0.0997008
$977$	−24639.0	−0.806829	−0.403414	−	0.915017i	$-0.632177\pi$
$977$	−24639.0	−0.806829	−0.403414	+	0.915017i	$0.632177\pi$
$978$	−14964.0	−0.489260
$979$	−10944.0	−0.357275
$980$	360.000	0.0117345
$981$	−16668.0	−0.542475
$982$	11214.0	0.364413
$983$	15279.0	0.495752	0.247876	−	0.968792i	$-0.420267\pi$
$983$	15279.0	0.495752	0.247876	+	0.968792i	$0.420267\pi$
$984$	4968.00	0.160949
$985$	−18870.0	−0.610404
$986$	18450.0	0.595910
$987$	−1710.00	−0.0551468
$988$	−2816.00	−0.0906770
$989$	−5428.00	−0.174520
$990$	2160.00	0.0693427
$991$	−52639.0	−1.68732	−0.843659	−	0.536879i	$-0.819603\pi$
$991$	−52639.0	−1.68732	−0.843659	+	0.536879i	$0.819603\pi$
$992$	1376.00	0.0440404
$993$	33465.0	1.06947
$994$	−22914.0	−0.731175
$995$	2110.00	0.0672276
$996$	6876.00	0.218749
$997$	−50992.0	−1.61979	−0.809896	−	0.586573i	$-0.800477\pi$
$997$	−50992.0	−1.61979	−0.809896	+	0.586573i	$0.800477\pi$
$998$	−8398.00	−0.266367
$999$	1161.00	0.0367692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.4.a.a.1.1	✓	1
3.2	odd	2		2070.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.4.a.a.1.1	✓	1	1.1	even	1	trivial
2070.4.a.g.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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	690.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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