LMFDB - Embedded newform 165.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,4,Mod(1,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, 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("165.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	165.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:	$9.73531515095$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1957.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 10 $ x^3 - x^2 - 9*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.04096$ of defining polynomial
Character	$\chi$	$=$	165.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.793499 q^{2} -3.00000 q^{3} -7.37036 q^{4} +5.00000 q^{5} -2.38050 q^{6} -2.90793 q^{7} -12.1964 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+0.793499 q^{2} -3.00000 q^{3} -7.37036 q^{4} +5.00000 q^{5} -2.38050 q^{6} -2.90793 q^{7} -12.1964 q^{8} +9.00000 q^{9} +3.96749 q^{10} +11.0000 q^{11} +22.1111 q^{12} +68.4882 q^{13} -2.30744 q^{14} -15.0000 q^{15} +49.2851 q^{16} -31.0515 q^{17} +7.14149 q^{18} +54.9249 q^{19} -36.8518 q^{20} +8.72380 q^{21} +8.72849 q^{22} +180.615 q^{23} +36.5891 q^{24} +25.0000 q^{25} +54.3453 q^{26} -27.0000 q^{27} +21.4325 q^{28} +67.3690 q^{29} -11.9025 q^{30} +153.367 q^{31} +136.679 q^{32} -33.0000 q^{33} -24.6393 q^{34} -14.5397 q^{35} -66.3332 q^{36} -324.485 q^{37} +43.5828 q^{38} -205.465 q^{39} -60.9818 q^{40} -25.4570 q^{41} +6.92233 q^{42} +133.864 q^{43} -81.0740 q^{44} +45.0000 q^{45} +143.318 q^{46} +113.784 q^{47} -147.855 q^{48} -334.544 q^{49} +19.8375 q^{50} +93.1546 q^{51} -504.783 q^{52} +91.6741 q^{53} -21.4245 q^{54} +55.0000 q^{55} +35.4662 q^{56} -164.775 q^{57} +53.4573 q^{58} +434.698 q^{59} +110.555 q^{60} -60.2384 q^{61} +121.696 q^{62} -26.1714 q^{63} -285.826 q^{64} +342.441 q^{65} -26.1855 q^{66} -439.825 q^{67} +228.861 q^{68} -541.845 q^{69} -11.5372 q^{70} +436.620 q^{71} -109.767 q^{72} -91.5134 q^{73} -257.478 q^{74} -75.0000 q^{75} -404.816 q^{76} -31.9873 q^{77} -163.036 q^{78} +947.462 q^{79} +246.425 q^{80} +81.0000 q^{81} -20.2001 q^{82} -944.385 q^{83} -64.2975 q^{84} -155.258 q^{85} +106.221 q^{86} -202.107 q^{87} -134.160 q^{88} +413.702 q^{89} +35.7075 q^{90} -199.159 q^{91} -1331.20 q^{92} -460.100 q^{93} +90.2872 q^{94} +274.624 q^{95} -410.036 q^{96} -1463.39 q^{97} -265.460 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q + q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 27 * q^9	\( 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9} + 5 q^{10} + 33 q^{11} - 51 q^{12} - 20 q^{13} + 144 q^{14} - 45 q^{15} + 25 q^{16} + 32 q^{17} + 9 q^{18} + 116 q^{19} + 85 q^{20} - 18 q^{21} + 11 q^{22} + 240 q^{23} + 9 q^{24} + 75 q^{25} + 302 q^{26} - 81 q^{27} + 160 q^{28} + 238 q^{29} - 15 q^{30} + 92 q^{31} + 197 q^{32} - 99 q^{33} + 354 q^{34} + 30 q^{35} + 153 q^{36} - 90 q^{37} + 324 q^{38} + 60 q^{39} - 15 q^{40} - 46 q^{41} - 432 q^{42} - 134 q^{43} + 187 q^{44} + 135 q^{45} - 240 q^{46} - 220 q^{47} - 75 q^{48} - 457 q^{49} + 25 q^{50} - 96 q^{51} - 1530 q^{52} - 798 q^{53} - 27 q^{54} + 165 q^{55} + 688 q^{56} - 348 q^{57} - 978 q^{58} + 1236 q^{59} - 255 q^{60} + 342 q^{61} - 1792 q^{62} + 54 q^{63} - 1919 q^{64} - 100 q^{65} - 33 q^{66} + 764 q^{67} + 1074 q^{68} - 720 q^{69} + 720 q^{70} + 1816 q^{71} - 27 q^{72} + 100 q^{73} - 1874 q^{74} - 225 q^{75} + 396 q^{76} + 66 q^{77} - 906 q^{78} - 96 q^{79} + 125 q^{80} + 243 q^{81} - 910 q^{82} + 858 q^{83} - 480 q^{84} + 160 q^{85} + 188 q^{86} - 714 q^{87} - 33 q^{88} + 838 q^{89} + 45 q^{90} + 332 q^{91} - 688 q^{92} - 276 q^{93} - 3112 q^{94} + 580 q^{95} - 591 q^{96} - 1322 q^{97} + 1017 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q + q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 27 * q^9 + 5 * q^10 + 33 * q^11 - 51 * q^12 - 20 * q^13 + 144 * q^14 - 45 * q^15 + 25 * q^16 + 32 * q^17 + 9 * q^18 + 116 * q^19 + 85 * q^20 - 18 * q^21 + 11 * q^22 + 240 * q^23 + 9 * q^24 + 75 * q^25 + 302 * q^26 - 81 * q^27 + 160 * q^28 + 238 * q^29 - 15 * q^30 + 92 * q^31 + 197 * q^32 - 99 * q^33 + 354 * q^34 + 30 * q^35 + 153 * q^36 - 90 * q^37 + 324 * q^38 + 60 * q^39 - 15 * q^40 - 46 * q^41 - 432 * q^42 - 134 * q^43 + 187 * q^44 + 135 * q^45 - 240 * q^46 - 220 * q^47 - 75 * q^48 - 457 * q^49 + 25 * q^50 - 96 * q^51 - 1530 * q^52 - 798 * q^53 - 27 * q^54 + 165 * q^55 + 688 * q^56 - 348 * q^57 - 978 * q^58 + 1236 * q^59 - 255 * q^60 + 342 * q^61 - 1792 * q^62 + 54 * q^63 - 1919 * q^64 - 100 * q^65 - 33 * q^66 + 764 * q^67 + 1074 * q^68 - 720 * q^69 + 720 * q^70 + 1816 * q^71 - 27 * q^72 + 100 * q^73 - 1874 * q^74 - 225 * q^75 + 396 * q^76 + 66 * q^77 - 906 * q^78 - 96 * q^79 + 125 * q^80 + 243 * q^81 - 910 * q^82 + 858 * q^83 - 480 * q^84 + 160 * q^85 + 188 * q^86 - 714 * q^87 - 33 * q^88 + 838 * q^89 + 45 * q^90 + 332 * q^91 - 688 * q^92 - 276 * q^93 - 3112 * q^94 + 580 * q^95 - 591 * q^96 - 1322 * q^97 + 1017 * q^98 + 297 * q^99

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 $

$2$	0.793499	0.280544	0.140272	−	0.990113i	$-0.455202\pi$
$2$	0.793499	0.280544	0.140272	+	0.990113i	$0.455202\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.37036	−0.921295
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−2.38050	−0.161972
$7$	−2.90793	−0.157014	−0.0785068	−	0.996914i	$-0.525015\pi$
$7$	−2.90793	−0.157014	−0.0785068	+	0.996914i	$0.525015\pi$
$8$	−12.1964	−0.539008
$9$	9.00000	0.333333
$10$	3.96749	0.125463
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	22.1111	0.531910
$13$	68.4882	1.46117	0.730585	−	0.682822i	$-0.239247\pi$
$13$	68.4882	1.46117	0.730585	+	0.682822i	$0.239247\pi$
$14$	−2.30744	−0.0440493
$15$	−15.0000	−0.258199
$16$	49.2851	0.770079
$17$	−31.0515	−0.443006	−0.221503	−	0.975160i	$-0.571096\pi$
$17$	−31.0515	−0.443006	−0.221503	+	0.975160i	$0.571096\pi$
$18$	7.14149	0.0935147
$19$	54.9249	0.663191	0.331595	−	0.943422i	$-0.392413\pi$
$19$	54.9249	0.663191	0.331595	+	0.943422i	$0.392413\pi$
$20$	−36.8518	−0.412016
$21$	8.72380	0.0906519
$22$	8.72849	0.0845873
$23$	180.615	1.63743	0.818713	−	0.574203i	$-0.194688\pi$
$23$	180.615	1.63743	0.818713	+	0.574203i	$0.194688\pi$
$24$	36.5891	0.311197
$25$	25.0000	0.200000
$26$	54.3453	0.409923
$27$	−27.0000	−0.192450
$28$	21.4325	0.144656
$29$	67.3690	0.431383	0.215692	−	0.976462i	$-0.430799\pi$
$29$	67.3690	0.431383	0.215692	+	0.976462i	$0.430799\pi$
$30$	−11.9025	−0.0724362
$31$	153.367	0.888562	0.444281	−	0.895887i	$-0.353459\pi$
$31$	153.367	0.888562	0.444281	+	0.895887i	$0.353459\pi$
$32$	136.679	0.755050
$33$	−33.0000	−0.174078
$34$	−24.6393	−0.124283
$35$	−14.5397	−0.0702186
$36$	−66.3332	−0.307098
$37$	−324.485	−1.44176	−0.720878	−	0.693062i	$-0.756261\pi$
$37$	−324.485	−1.44176	−0.720878	+	0.693062i	$0.756261\pi$
$38$	43.5828	0.186054
$39$	−205.465	−0.843607
$40$	−60.9818	−0.241052
$41$	−25.4570	−0.0969686	−0.0484843	−	0.998824i	$-0.515439\pi$
$41$	−25.4570	−0.0969686	−0.0484843	+	0.998824i	$0.515439\pi$
$42$	6.92233	0.0254319
$43$	133.864	0.474746	0.237373	−	0.971419i	$-0.423714\pi$
$43$	133.864	0.474746	0.237373	+	0.971419i	$0.423714\pi$
$44$	−81.0740	−0.277781
$45$	45.0000	0.149071
$46$	143.318	0.459371
$47$	113.784	0.353129	0.176564	−	0.984289i	$-0.443502\pi$
$47$	113.784	0.353129	0.176564	+	0.984289i	$0.443502\pi$
$48$	−147.855	−0.444605
$49$	−334.544	−0.975347
$50$	19.8375	0.0561088
$51$	93.1546	0.255770
$52$	−504.783	−1.34617
$53$	91.6741	0.237593	0.118796	−	0.992919i	$-0.462096\pi$
$53$	91.6741	0.237593	0.118796	+	0.992919i	$0.462096\pi$
$54$	−21.4245	−0.0539908
$55$	55.0000	0.134840
$56$	35.4662	0.0846316
$57$	−164.775	−0.382893
$58$	53.4573	0.121022
$59$	434.698	0.959200	0.479600	−	0.877487i	$-0.340782\pi$
$59$	434.698	0.959200	0.479600	+	0.877487i	$0.340782\pi$
$60$	110.555	0.237877
$61$	−60.2384	−0.126438	−0.0632191	−	0.998000i	$-0.520137\pi$
$61$	−60.2384	−0.126438	−0.0632191	+	0.998000i	$0.520137\pi$
$62$	121.696	0.249281
$63$	−26.1714	−0.0523379
$64$	−285.826	−0.558254
$65$	342.441	0.653455
$66$	−26.1855	−0.0488365
$67$	−439.825	−0.801987	−0.400994	−	0.916081i	$-0.631335\pi$
$67$	−439.825	−0.801987	−0.400994	+	0.916081i	$0.631335\pi$
$68$	228.861	0.408139
$69$	−541.845	−0.945369
$70$	−11.5372	−0.0196994
$71$	436.620	0.729820	0.364910	−	0.931043i	$-0.381100\pi$
$71$	436.620	0.729820	0.364910	+	0.931043i	$0.381100\pi$
$72$	−109.767	−0.179669
$73$	−91.5134	−0.146724	−0.0733619	−	0.997305i	$-0.523373\pi$
$73$	−91.5134	−0.146724	−0.0733619	+	0.997305i	$0.523373\pi$
$74$	−257.478	−0.404477
$75$	−75.0000	−0.115470
$76$	−404.816	−0.610994
$77$	−31.9873	−0.0473414
$78$	−163.036	−0.236669
$79$	947.462	1.34934	0.674670	−	0.738120i	$-0.264286\pi$
$79$	947.462	1.34934	0.674670	+	0.738120i	$0.264286\pi$
$80$	246.425	0.344390
$81$	81.0000	0.111111
$82$	−20.2001	−0.0272040
$83$	−944.385	−1.24891	−0.624456	−	0.781060i	$-0.714679\pi$
$83$	−944.385	−1.24891	−0.624456	+	0.781060i	$0.714679\pi$
$84$	−64.2975	−0.0835171
$85$	−155.258	−0.198118
$86$	106.221	0.133187
$87$	−202.107	−0.249059
$88$	−134.160	−0.162517
$89$	413.702	0.492722	0.246361	−	0.969178i	$-0.420765\pi$
$89$	413.702	0.492722	0.246361	+	0.969178i	$0.420765\pi$
$90$	35.7075	0.0418211
$91$	−199.159	−0.229424
$92$	−1331.20	−1.50855
$93$	−460.100	−0.513012
$94$	90.2872	0.0990683
$95$	274.624	0.296588
$96$	−410.036	−0.435928
$97$	−1463.39	−1.53180	−0.765899	−	0.642960i	$-0.777706\pi$
$97$	−1463.39	−1.53180	−0.765899	+	0.642960i	$0.777706\pi$
$98$	−265.460	−0.273628
$99$	99.0000	0.100504
$100$	−184.259	−0.184259
$101$	1959.24	1.93022	0.965109	−	0.261847i	$-0.0843316\pi$
$101$	1959.24	1.93022	0.965109	+	0.261847i	$0.0843316\pi$
$102$	73.9180	0.0717547
$103$	−151.029	−0.144479	−0.0722397	−	0.997387i	$-0.523015\pi$
$103$	−151.029	−0.144479	−0.0722397	+	0.997387i	$0.523015\pi$
$104$	−835.307	−0.787583
$105$	43.6190	0.0405407
$106$	72.7433	0.0666552
$107$	54.4430	0.0491888	0.0245944	−	0.999698i	$-0.492171\pi$
$107$	54.4430	0.0491888	0.0245944	+	0.999698i	$0.492171\pi$
$108$	199.000	0.177303
$109$	619.673	0.544531	0.272266	−	0.962222i	$-0.412227\pi$
$109$	619.673	0.544531	0.272266	+	0.962222i	$0.412227\pi$
$110$	43.6424	0.0378286
$111$	973.455	0.832399
$112$	−143.318	−0.120913
$113$	171.324	0.142627	0.0713133	−	0.997454i	$-0.477281\pi$
$113$	171.324	0.142627	0.0713133	+	0.997454i	$0.477281\pi$
$114$	−130.748	−0.107419
$115$	903.074	0.732279
$116$	−496.534	−0.397431
$117$	616.394	0.487057
$118$	344.932	0.269098
$119$	90.2957	0.0695580
$120$	182.945	0.139171
$121$	121.000	0.0909091
$122$	−47.7991	−0.0354715
$123$	76.3709	0.0559848
$124$	−1130.37	−0.818628
$125$	125.000	0.0894427
$126$	−20.7670	−0.0146831
$127$	1085.23	0.758257	0.379128	−	0.925344i	$-0.376224\pi$
$127$	1085.23	0.758257	0.379128	+	0.925344i	$0.376224\pi$
$128$	−1320.23	−0.911665
$129$	−401.592	−0.274095
$130$	271.727	0.183323
$131$	2354.89	1.57059	0.785296	−	0.619120i	$-0.212511\pi$
$131$	2354.89	1.57059	0.785296	+	0.619120i	$0.212511\pi$
$132$	243.222	0.160377
$133$	−159.718	−0.104130
$134$	−349.000	−0.224993
$135$	−135.000	−0.0860663
$136$	378.716	0.238784
$137$	2408.70	1.50211	0.751056	−	0.660239i	$-0.229545\pi$
$137$	2408.70	1.50211	0.751056	+	0.660239i	$0.229545\pi$
$138$	−429.953	−0.265218
$139$	−2604.89	−1.58952	−0.794762	−	0.606921i	$-0.792405\pi$
$139$	−2604.89	−1.58952	−0.794762	+	0.606921i	$0.792405\pi$
$140$	107.163	0.0646921
$141$	−341.351	−0.203879
$142$	346.457	0.204747
$143$	753.370	0.440559
$144$	443.566	0.256693
$145$	336.845	0.192920
$146$	−72.6158	−0.0411625
$147$	1003.63	0.563117
$148$	2391.57	1.32828
$149$	−304.932	−0.167658	−0.0838288	−	0.996480i	$-0.526715\pi$
$149$	−304.932	−0.167658	−0.0838288	+	0.996480i	$0.526715\pi$
$150$	−59.5124	−0.0323945
$151$	1263.74	0.681071	0.340536	−	0.940232i	$-0.389392\pi$
$151$	1263.74	0.681071	0.340536	+	0.940232i	$0.389392\pi$
$152$	−669.883	−0.357465
$153$	−279.464	−0.147669
$154$	−25.3819	−0.0132814
$155$	766.833	0.397377
$156$	1514.35	0.777211
$157$	−3714.09	−1.88801	−0.944003	−	0.329937i	$-0.892972\pi$
$157$	−3714.09	−1.88801	−0.944003	+	0.329937i	$0.892972\pi$
$158$	751.810	0.378549
$159$	−275.022	−0.137174
$160$	683.393	0.337668
$161$	−525.216	−0.257098
$162$	64.2734	0.0311716
$163$	1862.92	0.895186	0.447593	−	0.894238i	$-0.352281\pi$
$163$	1862.92	0.895186	0.447593	+	0.894238i	$0.352281\pi$
$164$	187.627	0.0893367
$165$	−165.000	−0.0778499
$166$	−749.369	−0.350375
$167$	1092.64	0.506295	0.253147	−	0.967428i	$-0.418534\pi$
$167$	1092.64	0.506295	0.253147	+	0.967428i	$0.418534\pi$
$168$	−106.399	−0.0488621
$169$	2493.64	1.13502
$170$	−123.197	−0.0555809
$171$	494.324	0.221064
$172$	−986.627	−0.437381
$173$	−2645.18	−1.16248	−0.581241	−	0.813732i	$-0.697433\pi$
$173$	−2645.18	−1.16248	−0.581241	+	0.813732i	$0.697433\pi$
$174$	−160.372	−0.0698721
$175$	−72.6983	−0.0314027
$176$	542.136	0.232188
$177$	−1304.09	−0.553794
$178$	328.272	0.138230
$179$	658.228	0.274851	0.137425	−	0.990512i	$-0.456117\pi$
$179$	658.228	0.274851	0.137425	+	0.990512i	$0.456117\pi$
$180$	−331.666	−0.137339
$181$	3986.11	1.63694	0.818468	−	0.574552i	$-0.194824\pi$
$181$	3986.11	1.63694	0.818468	+	0.574552i	$0.194824\pi$
$182$	−158.033	−0.0643635
$183$	180.715	0.0729992
$184$	−2202.84	−0.882586
$185$	−1622.43	−0.644773
$186$	−365.088	−0.143923
$187$	−341.567	−0.133571
$188$	−838.627	−0.325336
$189$	78.5142	0.0302173
$190$	217.914	0.0832060
$191$	2104.50	0.797260	0.398630	−	0.917112i	$-0.369486\pi$
$191$	2104.50	0.797260	0.398630	+	0.917112i	$0.369486\pi$
$192$	857.479	0.322308
$193$	−433.649	−0.161734	−0.0808672	−	0.996725i	$-0.525769\pi$
$193$	−433.649	−0.161734	−0.0808672	+	0.996725i	$0.525769\pi$
$194$	−1161.20	−0.429737
$195$	−1027.32	−0.377273
$196$	2465.71	0.898582
$197$	−4898.20	−1.77148	−0.885742	−	0.464177i	$-0.846350\pi$
$197$	−4898.20	−1.77148	−0.885742	+	0.464177i	$0.846350\pi$
$198$	78.5564	0.0281958
$199$	1866.38	0.664844	0.332422	−	0.943131i	$-0.392134\pi$
$199$	1866.38	0.664844	0.332422	+	0.943131i	$0.392134\pi$
$200$	−304.909	−0.107802
$201$	1319.47	0.463027
$202$	1554.66	0.541512
$203$	−195.905	−0.0677331
$204$	−686.583	−0.235639
$205$	−127.285	−0.0433657
$206$	−119.842	−0.0405328
$207$	1625.53	0.545809
$208$	3375.45	1.12522
$209$	604.173	0.199960
$210$	34.6116	0.0113735
$211$	−4165.49	−1.35907	−0.679537	−	0.733642i	$-0.737819\pi$
$211$	−4165.49	−1.35907	−0.679537	+	0.733642i	$0.737819\pi$
$212$	−675.671	−0.218893
$213$	−1309.86	−0.421362
$214$	43.2004	0.0137996
$215$	669.320	0.212313
$216$	329.302	0.103732
$217$	−445.980	−0.139516
$218$	491.710	0.152765
$219$	274.540	0.0847110
$220$	−405.370	−0.124227
$221$	−2126.66	−0.647307
$222$	772.435	0.233525
$223$	−3151.18	−0.946271	−0.473136	−	0.880990i	$-0.656878\pi$
$223$	−3151.18	−0.946271	−0.473136	+	0.880990i	$0.656878\pi$
$224$	−397.452	−0.118553
$225$	225.000	0.0666667
$226$	135.945	0.0400131
$227$	−3430.20	−1.00295	−0.501477	−	0.865171i	$-0.667210\pi$
$227$	−3430.20	−1.00295	−0.501477	+	0.865171i	$0.667210\pi$
$228$	1214.45	0.352758
$229$	−4041.79	−1.16633	−0.583164	−	0.812354i	$-0.698186\pi$
$229$	−4041.79	−1.16633	−0.583164	+	0.812354i	$0.698186\pi$
$230$	716.589	0.205437
$231$	95.9618	0.0273326
$232$	−821.657	−0.232519
$233$	−2565.83	−0.721431	−0.360715	−	0.932676i	$-0.617467\pi$
$233$	−2565.83	−0.721431	−0.360715	+	0.932676i	$0.617467\pi$
$234$	489.108	0.136641
$235$	568.918	0.157924
$236$	−3203.88	−0.883706
$237$	−2842.39	−0.779042
$238$	71.6496	0.0195141
$239$	−5540.15	−1.49942	−0.749712	−	0.661764i	$-0.769808\pi$
$239$	−5540.15	−1.49942	−0.749712	+	0.661764i	$0.769808\pi$
$240$	−739.276	−0.198834
$241$	−5980.22	−1.59842	−0.799211	−	0.601050i	$-0.794749\pi$
$241$	−5980.22	−1.59842	−0.799211	+	0.601050i	$0.794749\pi$
$242$	96.0134	0.0255040
$243$	−243.000	−0.0641500
$244$	443.979	0.116487
$245$	−1672.72	−0.436188
$246$	60.6003	0.0157062
$247$	3761.71	0.969035
$248$	−1870.51	−0.478942
$249$	2833.16	0.721060
$250$	99.1874	0.0250926
$251$	1527.19	0.384044	0.192022	−	0.981391i	$-0.438495\pi$
$251$	1527.19	0.384044	0.192022	+	0.981391i	$0.438495\pi$
$252$	192.893	0.0482186
$253$	1986.76	0.493703
$254$	861.129	0.212725
$255$	465.773	0.114384
$256$	1239.01	0.302492
$257$	−1193.28	−0.289629	−0.144814	−	0.989459i	$-0.546259\pi$
$257$	−1193.28	−0.289629	−0.144814	+	0.989459i	$0.546259\pi$
$258$	−318.663	−0.0768957
$259$	943.581	0.226376
$260$	−2523.91	−0.602025
$261$	606.321	0.143794
$262$	1868.60	0.440621
$263$	−6793.71	−1.59285	−0.796423	−	0.604740i	$-0.793277\pi$
$263$	−6793.71	−1.59285	−0.796423	+	0.604740i	$0.793277\pi$
$264$	402.480	0.0938293
$265$	458.370	0.106255
$266$	−126.736	−0.0292131
$267$	−1241.11	−0.284473
$268$	3241.67	0.738867
$269$	7443.13	1.68705	0.843524	−	0.537091i	$-0.180477\pi$
$269$	7443.13	1.68705	0.843524	+	0.537091i	$0.180477\pi$
$270$	−107.122	−0.0241454
$271$	−7912.17	−1.77354	−0.886771	−	0.462209i	$-0.847057\pi$
$271$	−7912.17	−1.77354	−0.886771	+	0.462209i	$0.847057\pi$
$272$	−1530.38	−0.341150
$273$	597.478	0.132458
$274$	1911.30	0.421409
$275$	275.000	0.0603023
$276$	3993.59	0.870963
$277$	3462.64	0.751083	0.375541	−	0.926806i	$-0.377457\pi$
$277$	3462.64	0.751083	0.375541	+	0.926806i	$0.377457\pi$
$278$	−2066.98	−0.445932
$279$	1380.30	0.296187
$280$	177.331	0.0378484
$281$	−7546.11	−1.60200	−0.801002	−	0.598662i	$-0.795699\pi$
$281$	−7546.11	−1.60200	−0.801002	+	0.598662i	$0.795699\pi$
$282$	−270.862	−0.0571971
$283$	3070.04	0.644859	0.322430	−	0.946593i	$-0.395500\pi$
$283$	3070.04	0.644859	0.322430	+	0.946593i	$0.395500\pi$
$284$	−3218.05	−0.672380
$285$	−823.873	−0.171235
$286$	597.799	0.123596
$287$	74.0272	0.0152254
$288$	1230.11	0.251683
$289$	−3948.80	−0.803746
$290$	267.286	0.0541227
$291$	4390.16	0.884384
$292$	674.487	0.135176
$293$	3760.43	0.749785	0.374892	−	0.927068i	$-0.377680\pi$
$293$	3760.43	0.749785	0.374892	+	0.927068i	$0.377680\pi$
$294$	796.381	0.157979
$295$	2173.49	0.428967
$296$	3957.54	0.777119
$297$	−297.000	−0.0580259
$298$	−241.963	−0.0470354
$299$	12370.0	2.39256
$300$	552.777	0.106382
$301$	−389.268	−0.0745416
$302$	1002.78	0.191071
$303$	−5877.73	−1.11441
$304$	2706.98	0.510709
$305$	−301.192	−0.0565449
$306$	−221.754	−0.0414276
$307$	2132.91	0.396521	0.198260	−	0.980149i	$-0.436471\pi$
$307$	2132.91	0.396521	0.198260	+	0.980149i	$0.436471\pi$
$308$	235.758	0.0436154
$309$	453.088	0.0834152
$310$	608.481	0.111482
$311$	354.565	0.0646481	0.0323241	−	0.999477i	$-0.489709\pi$
$311$	354.565	0.0646481	0.0323241	+	0.999477i	$0.489709\pi$
$312$	2505.92	0.454711
$313$	8700.26	1.57114	0.785572	−	0.618771i	$-0.212369\pi$
$313$	8700.26	1.57114	0.785572	+	0.618771i	$0.212369\pi$
$314$	−2947.13	−0.529669
$315$	−130.857	−0.0234062
$316$	−6983.14	−1.24314
$317$	8115.01	1.43781	0.718903	−	0.695111i	$-0.244645\pi$
$317$	8115.01	1.43781	0.718903	+	0.695111i	$0.244645\pi$
$318$	−218.230	−0.0384834
$319$	741.059	0.130067
$320$	−1429.13	−0.249659
$321$	−163.329	−0.0283992
$322$	−416.758	−0.0721274
$323$	−1705.50	−0.293797
$324$	−596.999	−0.102366
$325$	1712.21	0.292234
$326$	1478.23	0.251139
$327$	−1859.02	−0.314385
$328$	310.483	0.0522669
$329$	−330.875	−0.0554461
$330$	−130.927	−0.0218403
$331$	−6738.63	−1.11900	−0.559499	−	0.828831i	$-0.689006\pi$
$331$	−6738.63	−1.11900	−0.559499	+	0.828831i	$0.689006\pi$
$332$	6960.46	1.15062
$333$	−2920.37	−0.480586
$334$	867.011	0.142038
$335$	−2199.12	−0.358660
$336$	429.953	0.0698091
$337$	−10836.8	−1.75169	−0.875843	−	0.482595i	$-0.839694\pi$
$337$	−10836.8	−1.75169	−0.875843	+	0.482595i	$0.839694\pi$
$338$	1978.70	0.318423
$339$	−513.972	−0.0823455
$340$	1144.30	0.182525
$341$	1687.03	0.267912
$342$	392.245	0.0620181
$343$	1970.25	0.310156
$344$	−1632.66	−0.255892
$345$	−2709.22	−0.422782
$346$	−2098.95	−0.326128
$347$	4745.54	0.734161	0.367081	−	0.930189i	$-0.380357\pi$
$347$	4745.54	0.734161	0.367081	+	0.930189i	$0.380357\pi$
$348$	1489.60	0.229457
$349$	−744.830	−0.114240	−0.0571201	−	0.998367i	$-0.518192\pi$
$349$	−744.830	−0.114240	−0.0571201	+	0.998367i	$0.518192\pi$
$350$	−57.6861	−0.00880985
$351$	−1849.18	−0.281202
$352$	1503.46	0.227656
$353$	−5863.06	−0.884021	−0.442011	−	0.897010i	$-0.645735\pi$
$353$	−5863.06	−0.884021	−0.442011	+	0.897010i	$0.645735\pi$
$354$	−1034.80	−0.155364
$355$	2183.10	0.326386
$356$	−3049.13	−0.453943
$357$	−270.887	−0.0401593
$358$	522.303	0.0771078
$359$	12398.9	1.82281	0.911403	−	0.411515i	$-0.135000\pi$
$359$	12398.9	1.82281	0.911403	+	0.411515i	$0.135000\pi$
$360$	−548.836	−0.0803506
$361$	−3842.26	−0.560178
$362$	3162.98	0.459233
$363$	−363.000	−0.0524864
$364$	1467.87	0.211367
$365$	−457.567	−0.0656169
$366$	143.397	0.0204795
$367$	10059.5	1.43080	0.715399	−	0.698717i	$-0.246245\pi$
$367$	10059.5	1.43080	0.715399	+	0.698717i	$0.246245\pi$
$368$	8901.62	1.26095
$369$	−229.113	−0.0323229
$370$	−1287.39	−0.180887
$371$	−266.582	−0.0373053
$372$	3391.10	0.472635
$373$	−10527.6	−1.46139	−0.730694	−	0.682705i	$-0.760803\pi$
$373$	−10527.6	−1.46139	−0.730694	+	0.682705i	$0.760803\pi$
$374$	−271.033	−0.0374727
$375$	−375.000	−0.0516398
$376$	−1387.75	−0.190339
$377$	4613.98	0.630324
$378$	62.3009	0.00847729
$379$	−3772.28	−0.511264	−0.255632	−	0.966774i	$-0.582284\pi$
$379$	−3772.28	−0.511264	−0.255632	+	0.966774i	$0.582284\pi$
$380$	−2024.08	−0.273245
$381$	−3255.69	−0.437780
$382$	1669.92	0.223667
$383$	6667.97	0.889602	0.444801	−	0.895629i	$-0.353274\pi$
$383$	6667.97	0.889602	0.444801	+	0.895629i	$0.353274\pi$
$384$	3960.69	0.526350
$385$	−159.936	−0.0211717
$386$	−344.100	−0.0453736
$387$	1204.78	0.158249
$388$	10785.7	1.41124
$389$	3460.19	0.450999	0.225500	−	0.974243i	$-0.427599\pi$
$389$	3460.19	0.450999	0.225500	+	0.974243i	$0.427599\pi$
$390$	−815.180	−0.105842
$391$	−5608.37	−0.725389
$392$	4080.22	0.525720
$393$	−7064.67	−0.906782
$394$	−3886.72	−0.496980
$395$	4737.31	0.603443
$396$	−729.666	−0.0925936
$397$	−3250.11	−0.410878	−0.205439	−	0.978670i	$-0.565862\pi$
$397$	−3250.11	−0.410878	−0.205439	+	0.978670i	$0.565862\pi$
$398$	1480.97	0.186518
$399$	479.153	0.0601195
$400$	1232.13	0.154016
$401$	−2870.79	−0.357507	−0.178753	−	0.983894i	$-0.557206\pi$
$401$	−2870.79	−0.357507	−0.178753	+	0.983894i	$0.557206\pi$
$402$	1047.00	0.129900
$403$	10503.8	1.29834
$404$	−14440.3	−1.77830
$405$	405.000	0.0496904
$406$	−155.450	−0.0190021
$407$	−3569.34	−0.434706
$408$	−1136.15	−0.137862
$409$	−3166.04	−0.382765	−0.191382	−	0.981516i	$-0.561297\pi$
$409$	−3166.04	−0.382765	−0.191382	+	0.981516i	$0.561297\pi$
$410$	−101.000	−0.0121660
$411$	−7226.11	−0.867245
$412$	1113.14	0.133108
$413$	−1264.07	−0.150607
$414$	1289.86	0.153124
$415$	−4721.93	−0.558531
$416$	9360.87	1.10326
$417$	7814.67	0.917712
$418$	479.411	0.0560975
$419$	2984.75	0.348007	0.174003	−	0.984745i	$-0.444330\pi$
$419$	2984.75	0.348007	0.174003	+	0.984745i	$0.444330\pi$
$420$	−321.488	−0.0373500
$421$	1280.15	0.148197	0.0740985	−	0.997251i	$-0.476392\pi$
$421$	1280.15	0.148197	0.0740985	+	0.997251i	$0.476392\pi$
$422$	−3305.32	−0.381280
$423$	1024.05	0.117710
$424$	−1118.09	−0.128064
$425$	−776.288	−0.0886012
$426$	−1039.37	−0.118211
$427$	175.169	0.0198525
$428$	−401.264	−0.0453174
$429$	−2260.11	−0.254357
$430$	531.105	0.0595632
$431$	5952.92	0.665294	0.332647	−	0.943051i	$-0.392058\pi$
$431$	5952.92	0.665294	0.332647	+	0.943051i	$0.392058\pi$
$432$	−1330.70	−0.148202
$433$	4178.14	0.463715	0.231858	−	0.972750i	$-0.425520\pi$
$433$	4178.14	0.463715	0.231858	+	0.972750i	$0.425520\pi$
$434$	−353.884	−0.0391405
$435$	−1010.54	−0.111383
$436$	−4567.21	−0.501674
$437$	9920.25	1.08593
$438$	217.847	0.0237652
$439$	389.487	0.0423445	0.0211722	−	0.999776i	$-0.493260\pi$
$439$	389.487	0.0423445	0.0211722	+	0.999776i	$0.493260\pi$
$440$	−670.800	−0.0726799
$441$	−3010.90	−0.325116
$442$	−1687.50	−0.181598
$443$	−4395.08	−0.471369	−0.235685	−	0.971830i	$-0.575733\pi$
$443$	−4395.08	−0.471369	−0.235685	+	0.971830i	$0.575733\pi$
$444$	−7174.71	−0.766885
$445$	2068.51	0.220352
$446$	−2500.46	−0.265471
$447$	914.795	0.0967971
$448$	831.164	0.0876536
$449$	−5354.13	−0.562755	−0.281378	−	0.959597i	$-0.590791\pi$
$449$	−5354.13	−0.562755	−0.281378	+	0.959597i	$0.590791\pi$
$450$	178.537	0.0187029
$451$	−280.027	−0.0292371
$452$	−1262.72	−0.131401
$453$	−3791.22	−0.393217
$454$	−2721.86	−0.281373
$455$	−995.796	−0.102601
$456$	2009.65	0.206383
$457$	−3780.82	−0.387001	−0.193501	−	0.981100i	$-0.561984\pi$
$457$	−3780.82	−0.387001	−0.193501	+	0.981100i	$0.561984\pi$
$458$	−3207.16	−0.327207
$459$	838.391	0.0852565
$460$	−6655.98	−0.674645
$461$	3549.14	0.358568	0.179284	−	0.983797i	$-0.442622\pi$
$461$	3549.14	0.358568	0.179284	+	0.983797i	$0.442622\pi$
$462$	76.1456	0.00766799
$463$	3119.32	0.313104	0.156552	−	0.987670i	$-0.449962\pi$
$463$	3119.32	0.313104	0.156552	+	0.987670i	$0.449962\pi$
$464$	3320.29	0.332199
$465$	−2300.50	−0.229426
$466$	−2035.99	−0.202393
$467$	524.225	0.0519448	0.0259724	−	0.999663i	$-0.491732\pi$
$467$	524.225	0.0519448	0.0259724	+	0.999663i	$0.491732\pi$
$468$	−4543.05	−0.448723
$469$	1278.98	0.125923
$470$	451.436	0.0443047
$471$	11142.3	1.09004
$472$	−5301.73	−0.517017
$473$	1472.51	0.143141
$474$	−2255.43	−0.218556
$475$	1373.12	0.132638
$476$	−665.512	−0.0640834
$477$	825.067	0.0791975
$478$	−4396.10	−0.420655
$479$	−4648.94	−0.443456	−0.221728	−	0.975109i	$-0.571170\pi$
$479$	−4648.94	−0.443456	−0.221728	+	0.975109i	$0.571170\pi$
$480$	−2050.18	−0.194953
$481$	−22223.4	−2.10665
$482$	−4745.30	−0.448428
$483$	1575.65	0.148436
$484$	−891.813	−0.0837541
$485$	−7316.94	−0.685041
$486$	−192.820	−0.0179969
$487$	−17883.3	−1.66400	−0.832002	−	0.554773i	$-0.812805\pi$
$487$	−17883.3	−1.66400	−0.832002	+	0.554773i	$0.812805\pi$
$488$	734.689	0.0681513
$489$	−5588.76	−0.516836
$490$	−1327.30	−0.122370
$491$	−10702.7	−0.983715	−0.491857	−	0.870676i	$-0.663682\pi$
$491$	−10702.7	−0.983715	−0.491857	+	0.870676i	$0.663682\pi$
$492$	−562.881	−0.0515785
$493$	−2091.91	−0.191105
$494$	2984.91	0.271857
$495$	495.000	0.0449467
$496$	7558.68	0.684264
$497$	−1269.66	−0.114592
$498$	2248.11	0.202289
$499$	4092.39	0.367135	0.183567	−	0.983007i	$-0.441235\pi$
$499$	4092.39	0.367135	0.183567	+	0.983007i	$0.441235\pi$
$500$	−921.295	−0.0824031
$501$	−3277.93	−0.292309
$502$	1211.82	0.107741
$503$	945.979	0.0838551	0.0419275	−	0.999121i	$-0.486650\pi$
$503$	945.979	0.0838551	0.0419275	+	0.999121i	$0.486650\pi$
$504$	319.196	0.0282105
$505$	9796.22	0.863220
$506$	1576.49	0.138505
$507$	−7480.91	−0.655303
$508$	−7998.54	−0.698578
$509$	3674.38	0.319968	0.159984	−	0.987120i	$-0.448856\pi$
$509$	3674.38	0.319968	0.159984	+	0.987120i	$0.448856\pi$
$510$	369.590	0.0320897
$511$	266.115	0.0230376
$512$	11545.0	0.996527
$513$	−1482.97	−0.127631
$514$	−946.864	−0.0812537
$515$	−755.147	−0.0646131
$516$	2959.88	0.252522
$517$	1251.62	0.106472
$518$	748.730	0.0635083
$519$	7935.54	0.671159
$520$	−4176.54	−0.352218
$521$	−545.436	−0.0458656	−0.0229328	−	0.999737i	$-0.507300\pi$
$521$	−545.436	−0.0458656	−0.0229328	+	0.999737i	$0.507300\pi$
$522$	481.115	0.0403407
$523$	9060.79	0.757554	0.378777	−	0.925488i	$-0.376345\pi$
$523$	9060.79	0.757554	0.378777	+	0.925488i	$0.376345\pi$
$524$	−17356.4	−1.44698
$525$	218.095	0.0181304
$526$	−5390.80	−0.446864
$527$	−4762.26	−0.393638
$528$	−1626.41	−0.134054
$529$	20454.7	1.68117
$530$	363.716	0.0298091
$531$	3912.28	0.319733
$532$	1177.18	0.0959344
$533$	−1743.50	−0.141688
$534$	−984.816	−0.0798074
$535$	272.215	0.0219979
$536$	5364.26	0.432278
$537$	−1974.68	−0.158685
$538$	5906.12	0.473292
$539$	−3679.98	−0.294078
$540$	994.999	0.0792924
$541$	−15842.3	−1.25899	−0.629495	−	0.777005i	$-0.716738\pi$
$541$	−15842.3	−1.25899	−0.629495	+	0.777005i	$0.716738\pi$
$542$	−6278.30	−0.497557
$543$	−11958.3	−0.945085
$544$	−4244.08	−0.334491
$545$	3098.36	0.243522
$546$	474.098	0.0371603
$547$	19567.5	1.52952	0.764759	−	0.644316i	$-0.222858\pi$
$547$	19567.5	1.52952	0.764759	+	0.644316i	$0.222858\pi$
$548$	−17753.0	−1.38389
$549$	−542.145	−0.0421461
$550$	218.212	0.0169175
$551$	3700.23	0.286089
$552$	6608.53	0.509561
$553$	−2755.16	−0.211865
$554$	2747.60	0.210712
$555$	4867.28	0.372260
$556$	19199.0	1.46442
$557$	24262.5	1.84567	0.922833	−	0.385200i	$-0.125867\pi$
$557$	24262.5	1.84567	0.922833	+	0.385200i	$0.125867\pi$
$558$	1095.27	0.0830937
$559$	9168.11	0.693685
$560$	−716.589	−0.0540739
$561$	1024.70	0.0771174
$562$	−5987.83	−0.449433
$563$	9322.42	0.697857	0.348928	−	0.937149i	$-0.386546\pi$
$563$	9322.42	0.697857	0.348928	+	0.937149i	$0.386546\pi$
$564$	2515.88	0.187833
$565$	856.620	0.0637845
$566$	2436.08	0.180912
$567$	−235.543	−0.0174460
$568$	−5325.18	−0.393379
$569$	12441.6	0.916662	0.458331	−	0.888782i	$-0.348447\pi$
$569$	12441.6	0.916662	0.458331	+	0.888782i	$0.348447\pi$
$570$	−653.742	−0.0480390
$571$	−2994.01	−0.219432	−0.109716	−	0.993963i	$-0.534994\pi$
$571$	−2994.01	−0.219432	−0.109716	+	0.993963i	$0.534994\pi$
$572$	−5552.61	−0.405885
$573$	−6313.51	−0.460298
$574$	58.7405	0.00427140
$575$	4515.37	0.327485
$576$	−2572.44	−0.186085
$577$	5436.07	0.392212	0.196106	−	0.980583i	$-0.437170\pi$
$577$	5436.07	0.392212	0.196106	+	0.980583i	$0.437170\pi$
$578$	−3133.37	−0.225486
$579$	1300.95	0.0933774
$580$	−2482.67	−0.177737
$581$	2746.21	0.196096
$582$	3483.59	0.248109
$583$	1008.41	0.0716368
$584$	1116.13	0.0790853
$585$	3081.97	0.217818
$586$	2983.90	0.210348
$587$	7342.27	0.516265	0.258133	−	0.966109i	$-0.416893\pi$
$587$	7342.27	0.516265	0.258133	+	0.966109i	$0.416893\pi$
$588$	−7397.13	−0.518797
$589$	8423.63	0.589286
$590$	1724.66	0.120344
$591$	14694.6	1.02277
$592$	−15992.3	−1.11027
$593$	3362.02	0.232819	0.116409	−	0.993201i	$-0.462862\pi$
$593$	3362.02	0.232819	0.116409	+	0.993201i	$0.462862\pi$
$594$	−235.669	−0.0162788
$595$	451.479	0.0311073
$596$	2247.46	0.154462
$597$	−5599.13	−0.383848
$598$	9815.58	0.671219
$599$	−2197.77	−0.149914	−0.0749569	−	0.997187i	$-0.523882\pi$
$599$	−2197.77	−0.149914	−0.0749569	+	0.997187i	$0.523882\pi$
$600$	914.727	0.0622393
$601$	−10659.0	−0.723442	−0.361721	−	0.932286i	$-0.617811\pi$
$601$	−10659.0	−0.723442	−0.361721	+	0.932286i	$0.617811\pi$
$602$	−308.884	−0.0209122
$603$	−3958.42	−0.267329
$604$	−9314.22	−0.627467
$605$	605.000	0.0406558
$606$	−4663.97	−0.312642
$607$	−18305.0	−1.22402	−0.612008	−	0.790851i	$-0.709638\pi$
$607$	−18305.0	−1.22402	−0.612008	+	0.790851i	$0.709638\pi$
$608$	7507.05	0.500742
$609$	587.714	0.0391057
$610$	−238.995	−0.0158633
$611$	7792.84	0.515981
$612$	2059.75	0.136046
$613$	2721.13	0.179291	0.0896455	−	0.995974i	$-0.471427\pi$
$613$	2721.13	0.179291	0.0896455	+	0.995974i	$0.471427\pi$
$614$	1692.47	0.111242
$615$	381.855	0.0250372
$616$	390.128	0.0255174
$617$	−21160.0	−1.38066	−0.690331	−	0.723493i	$-0.742535\pi$
$617$	−21160.0	−1.38066	−0.690331	+	0.723493i	$0.742535\pi$
$618$	359.525	0.0234017
$619$	15215.3	0.987971	0.493986	−	0.869470i	$-0.335539\pi$
$619$	15215.3	0.987971	0.493986	+	0.869470i	$0.335539\pi$
$620$	−5651.83	−0.366102
$621$	−4876.60	−0.315123
$622$	281.347	0.0181367
$623$	−1203.02	−0.0773641
$624$	−10126.3	−0.649644
$625$	625.000	0.0400000
$626$	6903.65	0.440775
$627$	−1812.52	−0.115447
$628$	27374.2	1.73941
$629$	10075.8	0.638707
$630$	−103.835	−0.00656648
$631$	27192.9	1.71558	0.857790	−	0.514001i	$-0.171837\pi$
$631$	27192.9	1.71558	0.857790	+	0.514001i	$0.171837\pi$
$632$	−11555.6	−0.727305
$633$	12496.5	0.784661
$634$	6439.25	0.403368
$635$	5426.15	0.339103
$636$	2027.01	0.126378
$637$	−22912.3	−1.42515
$638$	588.030	0.0364895
$639$	3929.58	0.243273
$640$	−6601.16	−0.407709
$641$	19887.7	1.22546	0.612728	−	0.790294i	$-0.290072\pi$
$641$	19887.7	1.22546	0.612728	+	0.790294i	$0.290072\pi$
$642$	−129.601	−0.00796722
$643$	−920.348	−0.0564463	−0.0282232	−	0.999602i	$-0.508985\pi$
$643$	−920.348	−0.0564463	−0.0282232	+	0.999602i	$0.508985\pi$
$644$	3871.03	0.236863
$645$	−2007.96	−0.122579
$646$	−1353.31	−0.0824232
$647$	−21378.9	−1.29906	−0.649528	−	0.760337i	$-0.725034\pi$
$647$	−21378.9	−1.29906	−0.649528	+	0.760337i	$0.725034\pi$
$648$	−987.905	−0.0598898
$649$	4781.67	0.289210
$650$	1358.63	0.0819846
$651$	1337.94	0.0805499
$652$	−13730.4	−0.824730
$653$	−16874.7	−1.01127	−0.505633	−	0.862749i	$-0.668741\pi$
$653$	−16874.7	−1.01127	−0.505633	+	0.862749i	$0.668741\pi$
$654$	−1475.13	−0.0881990
$655$	11774.4	0.702390
$656$	−1254.65	−0.0746735
$657$	−823.621	−0.0489079
$658$	−262.549	−0.0155551
$659$	−19891.2	−1.17580	−0.587898	−	0.808935i	$-0.700044\pi$
$659$	−19891.2	−1.17580	−0.587898	+	0.808935i	$0.700044\pi$
$660$	1216.11	0.0717227
$661$	29758.4	1.75109	0.875543	−	0.483140i	$-0.160504\pi$
$661$	29758.4	1.75109	0.875543	+	0.483140i	$0.160504\pi$
$662$	−5347.09	−0.313929
$663$	6379.99	0.373723
$664$	11518.1	0.673174
$665$	−798.589	−0.0465684
$666$	−2317.31	−0.134826
$667$	12167.8	0.706358
$668$	−8053.17	−0.466447
$669$	9453.53	0.546330
$670$	−1745.00	−0.100620
$671$	−662.622	−0.0381226
$672$	1192.36	0.0684467
$673$	−26748.0	−1.53203	−0.766017	−	0.642820i	$-0.777764\pi$
$673$	−26748.0	−1.53203	−0.766017	+	0.642820i	$0.777764\pi$
$674$	−8599.00	−0.491426
$675$	−675.000	−0.0384900
$676$	−18379.0	−1.04569
$677$	−2938.53	−0.166820	−0.0834098	−	0.996515i	$-0.526581\pi$
$677$	−2938.53	−0.166820	−0.0834098	+	0.996515i	$0.526581\pi$
$678$	−407.836	−0.0231016
$679$	4255.43	0.240513
$680$	1893.58	0.106787
$681$	10290.6	0.579056
$682$	1338.66	0.0751611
$683$	−17208.6	−0.964083	−0.482042	−	0.876148i	$-0.660105\pi$
$683$	−17208.6	−0.964083	−0.482042	+	0.876148i	$0.660105\pi$
$684$	−3643.34	−0.203665
$685$	12043.5	0.671765
$686$	1563.39	0.0870126
$687$	12125.4	0.673380
$688$	6597.50	0.365592
$689$	6278.59	0.347163
$690$	−2149.77	−0.118609
$691$	19023.6	1.04731	0.523655	−	0.851930i	$-0.324568\pi$
$691$	19023.6	1.04731	0.523655	+	0.851930i	$0.324568\pi$
$692$	19495.9	1.07099
$693$	−287.885	−0.0157805
$694$	3765.58	0.205965
$695$	−13024.5	−0.710857
$696$	2464.97	0.134245
$697$	790.478	0.0429576
$698$	−591.022	−0.0320494
$699$	7697.50	0.416518
$700$	535.813	0.0289312
$701$	−8531.68	−0.459682	−0.229841	−	0.973228i	$-0.573821\pi$
$701$	−8531.68	−0.459682	−0.229841	+	0.973228i	$0.573821\pi$
$702$	−1467.32	−0.0788897
$703$	−17822.3	−0.956160
$704$	−3144.09	−0.168320
$705$	−1706.76	−0.0911775
$706$	−4652.34	−0.248007
$707$	−5697.35	−0.303071
$708$	9611.63	0.510208
$709$	−11511.0	−0.609736	−0.304868	−	0.952395i	$-0.598612\pi$
$709$	−11511.0	−0.609736	−0.304868	+	0.952395i	$0.598612\pi$
$710$	1732.29	0.0915656
$711$	8527.16	0.449780
$712$	−5045.66	−0.265581
$713$	27700.3	1.45496
$714$	−214.949	−0.0112665
$715$	3766.85	0.197024
$716$	−4851.38	−0.253218
$717$	16620.4	0.865693
$718$	9838.49	0.511378
$719$	−23498.9	−1.21886	−0.609430	−	0.792840i	$-0.708602\pi$
$719$	−23498.9	−1.21886	−0.609430	+	0.792840i	$0.708602\pi$
$720$	2217.83	0.114797
$721$	439.184	0.0226852
$722$	−3048.83	−0.157155
$723$	17940.7	0.922850
$724$	−29379.1	−1.50810
$725$	1684.23	0.0862767
$726$	−288.040	−0.0147248
$727$	19718.4	1.00593	0.502967	−	0.864306i	$-0.332242\pi$
$727$	19718.4	1.00593	0.502967	+	0.864306i	$0.332242\pi$
$728$	2429.02	0.123661
$729$	729.000	0.0370370
$730$	−363.079	−0.0184084
$731$	−4156.68	−0.210315
$732$	−1331.94	−0.0672538
$733$	−3618.29	−0.182325	−0.0911627	−	0.995836i	$-0.529058\pi$
$733$	−3618.29	−0.182325	−0.0911627	+	0.995836i	$0.529058\pi$
$734$	7982.22	0.401402
$735$	5018.16	0.251833
$736$	24686.2	1.23634
$737$	−4838.07	−0.241808
$738$	−181.801	−0.00906799
$739$	−29651.3	−1.47597	−0.737985	−	0.674817i	$-0.764223\pi$
$739$	−29651.3	−1.47597	−0.737985	+	0.674817i	$0.764223\pi$
$740$	11957.9	0.594026
$741$	−11285.1	−0.559472
$742$	−211.533	−0.0104658
$743$	−28501.9	−1.40731	−0.703657	−	0.710540i	$-0.748451\pi$
$743$	−28501.9	−1.40731	−0.703657	+	0.710540i	$0.748451\pi$
$744$	5611.54	0.276518
$745$	−1524.66	−0.0749787
$746$	−8353.63	−0.409984
$747$	−8499.47	−0.416304
$748$	2517.47	0.123059
$749$	−158.317	−0.00772331
$750$	−297.562	−0.0144872
$751$	33121.8	1.60936	0.804681	−	0.593707i	$-0.202336\pi$
$751$	33121.8	1.60936	0.804681	+	0.593707i	$0.202336\pi$
$752$	5607.84	0.271937
$753$	−4581.56	−0.221728
$754$	3661.19	0.176834
$755$	6318.70	0.304584
$756$	−578.678	−0.0278390
$757$	−7189.40	−0.345182	−0.172591	−	0.984994i	$-0.555214\pi$
$757$	−7189.40	−0.345182	−0.172591	+	0.984994i	$0.555214\pi$
$758$	−2993.30	−0.143432
$759$	−5960.29	−0.285039
$760$	−3349.42	−0.159863
$761$	−12289.2	−0.585392	−0.292696	−	0.956205i	$-0.594552\pi$
$761$	−12289.2	−0.585392	−0.292696	+	0.956205i	$0.594552\pi$
$762$	−2583.39	−0.122817
$763$	−1801.97	−0.0854988
$764$	−15511.0	−0.734511
$765$	−1397.32	−0.0660394
$766$	5291.03	0.249573
$767$	29771.7	1.40155
$768$	−3717.02	−0.174644
$769$	−38979.9	−1.82790	−0.913948	−	0.405832i	$-0.866982\pi$
$769$	−38979.9	−1.82790	−0.913948	+	0.405832i	$0.866982\pi$
$770$	−126.909	−0.00593960
$771$	3579.83	0.167217
$772$	3196.15	0.149005
$773$	−6846.25	−0.318554	−0.159277	−	0.987234i	$-0.550916\pi$
$773$	−6846.25	−0.318554	−0.159277	+	0.987234i	$0.550916\pi$
$774$	955.989	0.0443958
$775$	3834.16	0.177712
$776$	17848.0	0.825652
$777$	−2830.74	−0.130698
$778$	2745.66	0.126525
$779$	−1398.22	−0.0643087
$780$	7571.74	0.347579
$781$	4802.82	0.220049
$782$	−4450.23	−0.203504
$783$	−1818.96	−0.0830197
$784$	−16488.0	−0.751094
$785$	−18570.5	−0.844342
$786$	−5605.80	−0.254392
$787$	17561.3	0.795416	0.397708	−	0.917512i	$-0.369806\pi$
$787$	17561.3	0.795416	0.397708	+	0.917512i	$0.369806\pi$
$788$	36101.5	1.63206
$789$	20381.1	0.919630
$790$	3759.05	0.169292
$791$	−498.199	−0.0223943
$792$	−1207.44	−0.0541724
$793$	−4125.62	−0.184748
$794$	−2578.96	−0.115269
$795$	−1375.11	−0.0613461
$796$	−13755.9	−0.612517
$797$	−40618.8	−1.80526	−0.902629	−	0.430419i	$-0.858366\pi$
$797$	−40618.8	−1.80526	−0.902629	+	0.430419i	$0.858366\pi$
$798$	380.208	0.0168662
$799$	−3533.16	−0.156438
$800$	3416.96	0.151010
$801$	3723.32	0.164241
$802$	−2277.97	−0.100296
$803$	−1006.65	−0.0442389
$804$	−9725.00	−0.426585
$805$	−2626.08	−0.114978
$806$	8334.75	0.364242
$807$	−22329.4	−0.974018
$808$	−23895.7	−1.04040
$809$	−33200.7	−1.44286	−0.721430	−	0.692488i	$-0.756515\pi$
$809$	−33200.7	−1.44286	−0.721430	+	0.692488i	$0.756515\pi$
$810$	321.367	0.0139404
$811$	37275.9	1.61397	0.806987	−	0.590570i	$-0.201097\pi$
$811$	37275.9	1.61397	0.806987	+	0.590570i	$0.201097\pi$
$812$	1443.89	0.0624021
$813$	23736.5	1.02396
$814$	−2832.26	−0.121954
$815$	9314.61	0.400339
$816$	4591.13	0.196963
$817$	7352.47	0.314847
$818$	−2512.25	−0.107382
$819$	−1792.43	−0.0764746
$820$	938.135	0.0399526
$821$	−26000.2	−1.10525	−0.552626	−	0.833429i	$-0.686374\pi$
$821$	−26000.2	−1.10525	−0.552626	+	0.833429i	$0.686374\pi$
$822$	−5733.91	−0.243301
$823$	−43390.1	−1.83777	−0.918884	−	0.394527i	$-0.870908\pi$
$823$	−43390.1	−1.83777	−0.918884	+	0.394527i	$0.870908\pi$
$824$	1842.01	0.0778756
$825$	−825.000	−0.0348155
$826$	−1003.04	−0.0422521
$827$	−33496.3	−1.40844	−0.704221	−	0.709981i	$-0.748703\pi$
$827$	−33496.3	−1.40844	−0.704221	+	0.709981i	$0.748703\pi$
$828$	−11980.8	−0.502851
$829$	14895.4	0.624051	0.312025	−	0.950074i	$-0.398993\pi$
$829$	14895.4	0.624051	0.312025	+	0.950074i	$0.398993\pi$
$830$	−3746.84	−0.156693
$831$	−10387.9	−0.433638
$832$	−19575.7	−0.815705
$833$	10388.1	0.432084
$834$	6200.93	0.257459
$835$	5463.21	0.226422
$836$	−4452.97	−0.184222
$837$	−4140.90	−0.171004
$838$	2368.40	0.0976313
$839$	−20519.9	−0.844370	−0.422185	−	0.906510i	$-0.638737\pi$
$839$	−20519.9	−0.844370	−0.422185	+	0.906510i	$0.638737\pi$
$840$	−531.993	−0.0218518
$841$	−19850.4	−0.813908
$842$	1015.80	0.0415758
$843$	22638.3	0.924917
$844$	30701.2	1.25211
$845$	12468.2	0.507596
$846$	812.585	0.0330228
$847$	−351.860	−0.0142740
$848$	4518.16	0.182965
$849$	−9210.13	−0.372310
$850$	−615.984	−0.0248565
$851$	−58606.8	−2.36077
$852$	9654.14	0.388199
$853$	27624.2	1.10883	0.554416	−	0.832240i	$-0.312942\pi$
$853$	27624.2	1.10883	0.554416	+	0.832240i	$0.312942\pi$
$854$	138.997	0.00556951
$855$	2471.62	0.0988626
$856$	−664.006	−0.0265132
$857$	−31086.1	−1.23907	−0.619533	−	0.784970i	$-0.712678\pi$
$857$	−31086.1	−1.23907	−0.619533	+	0.784970i	$0.712678\pi$
$858$	−1793.40	−0.0713584
$859$	−32657.7	−1.29717	−0.648583	−	0.761144i	$-0.724638\pi$
$859$	−32657.7	−1.29717	−0.648583	+	0.761144i	$0.724638\pi$
$860$	−4933.13	−0.195603
$861$	−222.082	−0.00879038
$862$	4723.63	0.186644
$863$	−10499.4	−0.414141	−0.207070	−	0.978326i	$-0.566393\pi$
$863$	−10499.4	−0.414141	−0.207070	+	0.978326i	$0.566393\pi$
$864$	−3690.32	−0.145309
$865$	−13225.9	−0.519878
$866$	3315.35	0.130093
$867$	11846.4	0.464043
$868$	3287.03	0.128536
$869$	10422.1	0.406841
$870$	−801.859	−0.0312478
$871$	−30122.8	−1.17184
$872$	−7557.76	−0.293507
$873$	−13170.5	−0.510600
$874$	7871.70	0.304650
$875$	−363.492	−0.0140437
$876$	−2023.46	−0.0780438
$877$	6736.30	0.259371	0.129686	−	0.991555i	$-0.458603\pi$
$877$	6736.30	0.259371	0.129686	+	0.991555i	$0.458603\pi$
$878$	309.058	0.0118795
$879$	−11281.3	−0.432888
$880$	2710.68	0.103837
$881$	38860.1	1.48607	0.743037	−	0.669251i	$-0.233385\pi$
$881$	38860.1	1.48607	0.743037	+	0.669251i	$0.233385\pi$
$882$	−2389.14	−0.0912093
$883$	−13526.6	−0.515524	−0.257762	−	0.966208i	$-0.582985\pi$
$883$	−13526.6	−0.515524	−0.257762	+	0.966208i	$0.582985\pi$
$884$	15674.3	0.596361
$885$	−6520.46	−0.247664
$886$	−3487.49	−0.132240
$887$	2089.69	0.0791036	0.0395518	−	0.999218i	$-0.487407\pi$
$887$	2089.69	0.0791036	0.0395518	+	0.999218i	$0.487407\pi$
$888$	−11872.6	−0.448670
$889$	−3155.78	−0.119057
$890$	1641.36	0.0618185
$891$	891.000	0.0335013
$892$	23225.3	0.871795
$893$	6249.55	0.234192
$894$	725.889	0.0271559
$895$	3291.14	0.122917
$896$	3839.14	0.143144
$897$	−37110.0	−1.38134
$898$	−4248.50	−0.157878
$899$	10332.2	0.383311
$900$	−1658.33	−0.0614197
$901$	−2846.62	−0.105255
$902$	−222.201	−0.00820231
$903$	1167.80	0.0430366
$904$	−2089.53	−0.0768769
$905$	19930.6	0.732060
$906$	−3008.33	−0.110315
$907$	20510.3	0.750862	0.375431	−	0.926850i	$-0.377495\pi$
$907$	20510.3	0.750862	0.375431	+	0.926850i	$0.377495\pi$
$908$	25281.8	0.924016
$909$	17633.2	0.643406
$910$	−790.163	−0.0287842
$911$	34898.0	1.26918	0.634589	−	0.772850i	$-0.281169\pi$
$911$	34898.0	1.26918	0.634589	+	0.772850i	$0.281169\pi$
$912$	−8120.93	−0.294858
$913$	−10388.2	−0.376561
$914$	−3000.08	−0.108571
$915$	903.576	0.0326462
$916$	29789.5	1.07453
$917$	−6847.86	−0.246604
$918$	665.262	0.0239182
$919$	−680.620	−0.0244304	−0.0122152	−	0.999925i	$-0.503888\pi$
$919$	−680.620	−0.0244304	−0.0122152	+	0.999925i	$0.503888\pi$
$920$	−11014.2	−0.394705
$921$	−6398.74	−0.228931
$922$	2816.24	0.100594
$923$	29903.3	1.06639
$924$	−707.273	−0.0251814
$925$	−8112.13	−0.288351
$926$	2475.18	0.0878396
$927$	−1359.26	−0.0481598
$928$	9207.90	0.325716
$929$	46344.7	1.63673	0.818364	−	0.574700i	$-0.194881\pi$
$929$	46344.7	1.63673	0.818364	+	0.574700i	$0.194881\pi$
$930$	−1825.44	−0.0643641
$931$	−18374.8	−0.646841
$932$	18911.1	0.664651
$933$	−1063.70	−0.0373246
$934$	415.972	0.0145728
$935$	−1707.83	−0.0597349
$936$	−7517.77	−0.262528
$937$	50003.7	1.74338	0.871691	−	0.490056i	$-0.163024\pi$
$937$	50003.7	1.74338	0.871691	+	0.490056i	$0.163024\pi$
$938$	1014.87	0.0353269
$939$	−26100.8	−0.907100
$940$	−4193.13	−0.145495
$941$	10202.4	0.353442	0.176721	−	0.984261i	$-0.443451\pi$
$941$	10202.4	0.353442	0.176721	+	0.984261i	$0.443451\pi$
$942$	8841.39	0.305805
$943$	−4597.91	−0.158779
$944$	21424.1	0.738660
$945$	392.571	0.0135136
$946$	1168.43	0.0401575
$947$	1386.09	0.0475625	0.0237813	−	0.999717i	$-0.492429\pi$
$947$	1386.09	0.0475625	0.0237813	+	0.999717i	$0.492429\pi$
$948$	20949.4	0.717727
$949$	−6267.59	−0.214388
$950$	1089.57	0.0372109
$951$	−24345.0	−0.830117
$952$	−1101.28	−0.0374923
$953$	16825.6	0.571915	0.285958	−	0.958242i	$-0.407688\pi$
$953$	16825.6	0.571915	0.285958	+	0.958242i	$0.407688\pi$
$954$	654.690	0.0222184
$955$	10522.5	0.356545
$956$	40832.9	1.38141
$957$	−2223.18	−0.0750942
$958$	−3688.93	−0.124409
$959$	−7004.35	−0.235852
$960$	4287.39	0.144141
$961$	−6269.72	−0.210457
$962$	−17634.2	−0.591009
$963$	489.987	0.0163963
$964$	44076.4	1.47262
$965$	−2168.24	−0.0723298
$966$	1250.28	0.0416428
$967$	−39600.8	−1.31694	−0.658468	−	0.752609i	$-0.728795\pi$
$967$	−39600.8	−1.31694	−0.658468	+	0.752609i	$0.728795\pi$
$968$	−1475.76	−0.0490007
$969$	5116.50	0.169624
$970$	−5805.98	−0.192184
$971$	36947.4	1.22111	0.610555	−	0.791974i	$-0.290946\pi$
$971$	36947.4	1.22111	0.610555	+	0.791974i	$0.290946\pi$
$972$	1791.00	0.0591011
$973$	7574.85	0.249577
$974$	−14190.4	−0.466827
$975$	−5136.62	−0.168721
$976$	−2968.85	−0.0973675
$977$	20284.0	0.664220	0.332110	−	0.943241i	$-0.392240\pi$
$977$	20284.0	0.664220	0.332110	+	0.943241i	$0.392240\pi$
$978$	−4434.68	−0.144995
$979$	4550.72	0.148561
$980$	12328.5	0.401858
$981$	5577.06	0.181510
$982$	−8492.54	−0.275976
$983$	−30081.3	−0.976038	−0.488019	−	0.872833i	$-0.662280\pi$
$983$	−30081.3	−0.976038	−0.488019	+	0.872833i	$0.662280\pi$
$984$	−931.448	−0.0301763
$985$	−24491.0	−0.792232
$986$	−1659.93	−0.0536135
$987$	992.626	0.0320118
$988$	−27725.1	−0.892767
$989$	24177.8	0.777362
$990$	392.782	0.0126095
$991$	−41404.2	−1.32719	−0.663596	−	0.748091i	$-0.730971\pi$
$991$	−41404.2	−1.32719	−0.663596	+	0.748091i	$0.730971\pi$
$992$	20961.9	0.670909
$993$	20215.9	0.646054
$994$	−1007.48	−0.0321481
$995$	9331.88	0.297327
$996$	−20881.4	−0.664309
$997$	23882.5	0.758642	0.379321	−	0.925265i	$-0.376158\pi$
$997$	23882.5	0.758642	0.379321	+	0.925265i	$0.376158\pi$
$998$	3247.30	0.102998
$999$	8761.10	0.277466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.4.a.g.1.2	✓	3
3.2	odd	2		495.4.a.i.1.2		3
5.2	odd	4		825.4.c.m.199.4		6
5.3	odd	4		825.4.c.m.199.3		6
5.4	even	2		825.4.a.p.1.2		3
11.10	odd	2		1815.4.a.q.1.2		3
15.14	odd	2		2475.4.a.z.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.g.1.2	✓	3	1.1	even	1	trivial
495.4.a.i.1.2		3	3.2	odd	2
825.4.a.p.1.2		3	5.4	even	2
825.4.c.m.199.3		6	5.3	odd	4
825.4.c.m.199.4		6	5.2	odd	4
1815.4.a.q.1.2		3	11.10	odd	2
2475.4.a.z.1.2		3	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:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)

\(f(q)\)	\(=\)	\(q+0.793499 q^{2} -3.00000 q^{3} -7.37036 q^{4} +5.00000 q^{5} -2.38050 q^{6} -2.90793 q^{7} -12.1964 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+0.793499 q^{2} -3.00000 q^{3} -7.37036 q^{4} +5.00000 q^{5} -2.38050 q^{6} -2.90793 q^{7} -12.1964 q^{8} +9.00000 q^{9} +3.96749 q^{10} +11.0000 q^{11} +22.1111 q^{12} +68.4882 q^{13} -2.30744 q^{14} -15.0000 q^{15} +49.2851 q^{16} -31.0515 q^{17} +7.14149 q^{18} +54.9249 q^{19} -36.8518 q^{20} +8.72380 q^{21} +8.72849 q^{22} +180.615 q^{23} +36.5891 q^{24} +25.0000 q^{25} +54.3453 q^{26} -27.0000 q^{27} +21.4325 q^{28} +67.3690 q^{29} -11.9025 q^{30} +153.367 q^{31} +136.679 q^{32} -33.0000 q^{33} -24.6393 q^{34} -14.5397 q^{35} -66.3332 q^{36} -324.485 q^{37} +43.5828 q^{38} -205.465 q^{39} -60.9818 q^{40} -25.4570 q^{41} +6.92233 q^{42} +133.864 q^{43} -81.0740 q^{44} +45.0000 q^{45} +143.318 q^{46} +113.784 q^{47} -147.855 q^{48} -334.544 q^{49} +19.8375 q^{50} +93.1546 q^{51} -504.783 q^{52} +91.6741 q^{53} -21.4245 q^{54} +55.0000 q^{55} +35.4662 q^{56} -164.775 q^{57} +53.4573 q^{58} +434.698 q^{59} +110.555 q^{60} -60.2384 q^{61} +121.696 q^{62} -26.1714 q^{63} -285.826 q^{64} +342.441 q^{65} -26.1855 q^{66} -439.825 q^{67} +228.861 q^{68} -541.845 q^{69} -11.5372 q^{70} +436.620 q^{71} -109.767 q^{72} -91.5134 q^{73} -257.478 q^{74} -75.0000 q^{75} -404.816 q^{76} -31.9873 q^{77} -163.036 q^{78} +947.462 q^{79} +246.425 q^{80} +81.0000 q^{81} -20.2001 q^{82} -944.385 q^{83} -64.2975 q^{84} -155.258 q^{85} +106.221 q^{86} -202.107 q^{87} -134.160 q^{88} +413.702 q^{89} +35.7075 q^{90} -199.159 q^{91} -1331.20 q^{92} -460.100 q^{93} +90.2872 q^{94} +274.624 q^{95} -410.036 q^{96} -1463.39 q^{97} -265.460 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q + q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 27 * q^9	\( 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9} + 5 q^{10} + 33 q^{11} - 51 q^{12} - 20 q^{13} + 144 q^{14} - 45 q^{15} + 25 q^{16} + 32 q^{17} + 9 q^{18} + 116 q^{19} + 85 q^{20} - 18 q^{21} + 11 q^{22} + 240 q^{23} + 9 q^{24} + 75 q^{25} + 302 q^{26} - 81 q^{27} + 160 q^{28} + 238 q^{29} - 15 q^{30} + 92 q^{31} + 197 q^{32} - 99 q^{33} + 354 q^{34} + 30 q^{35} + 153 q^{36} - 90 q^{37} + 324 q^{38} + 60 q^{39} - 15 q^{40} - 46 q^{41} - 432 q^{42} - 134 q^{43} + 187 q^{44} + 135 q^{45} - 240 q^{46} - 220 q^{47} - 75 q^{48} - 457 q^{49} + 25 q^{50} - 96 q^{51} - 1530 q^{52} - 798 q^{53} - 27 q^{54} + 165 q^{55} + 688 q^{56} - 348 q^{57} - 978 q^{58} + 1236 q^{59} - 255 q^{60} + 342 q^{61} - 1792 q^{62} + 54 q^{63} - 1919 q^{64} - 100 q^{65} - 33 q^{66} + 764 q^{67} + 1074 q^{68} - 720 q^{69} + 720 q^{70} + 1816 q^{71} - 27 q^{72} + 100 q^{73} - 1874 q^{74} - 225 q^{75} + 396 q^{76} + 66 q^{77} - 906 q^{78} - 96 q^{79} + 125 q^{80} + 243 q^{81} - 910 q^{82} + 858 q^{83} - 480 q^{84} + 160 q^{85} + 188 q^{86} - 714 q^{87} - 33 q^{88} + 838 q^{89} + 45 q^{90} + 332 q^{91} - 688 q^{92} - 276 q^{93} - 3112 q^{94} + 580 q^{95} - 591 q^{96} - 1322 q^{97} + 1017 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q + q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 27 * q^9 + 5 * q^10 + 33 * q^11 - 51 * q^12 - 20 * q^13 + 144 * q^14 - 45 * q^15 + 25 * q^16 + 32 * q^17 + 9 * q^18 + 116 * q^19 + 85 * q^20 - 18 * q^21 + 11 * q^22 + 240 * q^23 + 9 * q^24 + 75 * q^25 + 302 * q^26 - 81 * q^27 + 160 * q^28 + 238 * q^29 - 15 * q^30 + 92 * q^31 + 197 * q^32 - 99 * q^33 + 354 * q^34 + 30 * q^35 + 153 * q^36 - 90 * q^37 + 324 * q^38 + 60 * q^39 - 15 * q^40 - 46 * q^41 - 432 * q^42 - 134 * q^43 + 187 * q^44 + 135 * q^45 - 240 * q^46 - 220 * q^47 - 75 * q^48 - 457 * q^49 + 25 * q^50 - 96 * q^51 - 1530 * q^52 - 798 * q^53 - 27 * q^54 + 165 * q^55 + 688 * q^56 - 348 * q^57 - 978 * q^58 + 1236 * q^59 - 255 * q^60 + 342 * q^61 - 1792 * q^62 + 54 * q^63 - 1919 * q^64 - 100 * q^65 - 33 * q^66 + 764 * q^67 + 1074 * q^68 - 720 * q^69 + 720 * q^70 + 1816 * q^71 - 27 * q^72 + 100 * q^73 - 1874 * q^74 - 225 * q^75 + 396 * q^76 + 66 * q^77 - 906 * q^78 - 96 * q^79 + 125 * q^80 + 243 * q^81 - 910 * q^82 + 858 * q^83 - 480 * q^84 + 160 * q^85 + 188 * q^86 - 714 * q^87 - 33 * q^88 + 838 * q^89 + 45 * q^90 + 332 * q^91 - 688 * q^92 - 276 * q^93 - 3112 * q^94 + 580 * q^95 - 591 * q^96 - 1322 * q^97 + 1017 * q^98 + 297 * q^99

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