LMFDB - Embedded newform 165.4.a.g.1.3

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			1.3
Root			$1.12946$ 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)

$f(q)$	$=$	$q+4.59486 q^{2} -3.00000 q^{3} +13.1127 q^{4} +5.00000 q^{5} -13.7846 q^{6} +20.6383 q^{7} +23.4921 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.59486 q^{2} -3.00000 q^{3} +13.1127 q^{4} +5.00000 q^{5} -13.7846 q^{6} +20.6383 q^{7} +23.4921 q^{8} +9.00000 q^{9} +22.9743 q^{10} +11.0000 q^{11} -39.3381 q^{12} -15.6584 q^{13} +94.8302 q^{14} -15.0000 q^{15} +3.04132 q^{16} +72.9507 q^{17} +41.3537 q^{18} +61.0513 q^{19} +65.5635 q^{20} -61.9150 q^{21} +50.5434 q^{22} -13.6605 q^{23} -70.4764 q^{24} +25.0000 q^{25} -71.9483 q^{26} -27.0000 q^{27} +270.624 q^{28} -31.4663 q^{29} -68.9228 q^{30} -243.008 q^{31} -173.963 q^{32} -33.0000 q^{33} +335.198 q^{34} +103.192 q^{35} +118.014 q^{36} -65.4018 q^{37} +280.522 q^{38} +46.9753 q^{39} +117.461 q^{40} -109.087 q^{41} -284.491 q^{42} -121.750 q^{43} +144.240 q^{44} +45.0000 q^{45} -62.7678 q^{46} -519.530 q^{47} -9.12396 q^{48} +82.9413 q^{49} +114.871 q^{50} -218.852 q^{51} -205.324 q^{52} -542.673 q^{53} -124.061 q^{54} +55.0000 q^{55} +484.839 q^{56} -183.154 q^{57} -144.583 q^{58} +109.478 q^{59} -196.691 q^{60} -89.6156 q^{61} -1116.59 q^{62} +185.745 q^{63} -823.664 q^{64} -78.2922 q^{65} -151.630 q^{66} +488.446 q^{67} +956.581 q^{68} +40.9814 q^{69} +474.151 q^{70} +837.423 q^{71} +211.429 q^{72} +351.216 q^{73} -300.512 q^{74} -75.0000 q^{75} +800.547 q^{76} +227.022 q^{77} +215.845 q^{78} -831.205 q^{79} +15.2066 q^{80} +81.0000 q^{81} -501.238 q^{82} +1389.13 q^{83} -811.873 q^{84} +364.754 q^{85} -559.423 q^{86} +94.3988 q^{87} +258.413 q^{88} +1523.70 q^{89} +206.769 q^{90} -323.164 q^{91} -179.125 q^{92} +729.025 q^{93} -2387.17 q^{94} +305.256 q^{95} +521.888 q^{96} -426.612 q^{97} +381.103 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.59486	1.62453	0.812263	−	0.583291i	$-0.198235\pi$
$2$	4.59486	1.62453	0.812263	+	0.583291i	$0.198235\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	13.1127	1.63909
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−13.7846	−0.937921
$7$	20.6383	1.11437	0.557183	−	0.830390i	$-0.311882\pi$
$7$	20.6383	1.11437	0.557183	+	0.830390i	$0.311882\pi$
$8$	23.4921	1.03822
$9$	9.00000	0.333333
$10$	22.9743	0.726511
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−39.3381	−0.946328
$13$	−15.6584	−0.334067	−0.167033	−	0.985951i	$-0.553419\pi$
$13$	−15.6584	−0.334067	−0.167033	+	0.985951i	$0.553419\pi$
$14$	94.8302	1.81032
$15$	−15.0000	−0.258199
$16$	3.04132	0.0475206
$17$	72.9507	1.04077	0.520387	−	0.853931i	$-0.325788\pi$
$17$	72.9507	1.04077	0.520387	+	0.853931i	$0.325788\pi$
$18$	41.3537	0.541509
$19$	61.0513	0.737165	0.368582	−	0.929595i	$-0.379843\pi$
$19$	61.0513	0.737165	0.368582	+	0.929595i	$0.379843\pi$
$20$	65.5635	0.733022
$21$	−61.9150	−0.643379
$22$	50.5434	0.489813
$23$	−13.6605	−0.123844	−0.0619218	−	0.998081i	$-0.519723\pi$
$23$	−13.6605	−0.123844	−0.0619218	+	0.998081i	$0.519723\pi$
$24$	−70.4764	−0.599414
$25$	25.0000	0.200000
$26$	−71.9483	−0.542701
$27$	−27.0000	−0.192450
$28$	270.624	1.82654
$29$	−31.4663	−0.201487	−0.100744	−	0.994912i	$-0.532122\pi$
$29$	−31.4663	−0.201487	−0.100744	+	0.994912i	$0.532122\pi$
$30$	−68.9228	−0.419451
$31$	−243.008	−1.40792	−0.703961	−	0.710239i	$-0.748587\pi$
$31$	−243.008	−1.40792	−0.703961	+	0.710239i	$0.748587\pi$
$32$	−173.963	−0.961016
$33$	−33.0000	−0.174078
$34$	335.198	1.69076
$35$	103.192	0.498360
$36$	118.014	0.546363
$37$	−65.4018	−0.290594	−0.145297	−	0.989388i	$-0.546414\pi$
$37$	−65.4018	−0.290594	−0.145297	+	0.989388i	$0.546414\pi$
$38$	280.522	1.19754
$39$	46.9753	0.192874
$40$	117.461	0.464304
$41$	−109.087	−0.415524	−0.207762	−	0.978179i	$-0.566618\pi$
$41$	−109.087	−0.415524	−0.207762	+	0.978179i	$0.566618\pi$
$42$	−284.491	−1.04519
$43$	−121.750	−0.431783	−0.215891	−	0.976417i	$-0.569266\pi$
$43$	−121.750	−0.431783	−0.215891	+	0.976417i	$0.569266\pi$
$44$	144.240	0.494204
$45$	45.0000	0.149071
$46$	−62.7678	−0.201187
$47$	−519.530	−1.61237	−0.806184	−	0.591665i	$-0.798471\pi$
$47$	−519.530	−1.61237	−0.806184	+	0.591665i	$0.798471\pi$
$48$	−9.12396	−0.0274361
$49$	82.9413	0.241811
$50$	114.871	0.324905
$51$	−218.852	−0.600891
$52$	−205.324	−0.547565
$53$	−542.673	−1.40645	−0.703226	−	0.710967i	$-0.748258\pi$
$53$	−542.673	−1.40645	−0.703226	+	0.710967i	$0.748258\pi$
$54$	−124.061	−0.312640
$55$	55.0000	0.134840
$56$	484.839	1.15695
$57$	−183.154	−0.425602
$58$	−144.583	−0.327322
$59$	109.478	0.241574	0.120787	−	0.992678i	$-0.461458\pi$
$59$	109.478	0.241574	0.120787	+	0.992678i	$0.461458\pi$
$60$	−196.691	−0.423211
$61$	−89.6156	−0.188100	−0.0940501	−	0.995567i	$-0.529981\pi$
$61$	−89.6156	−0.188100	−0.0940501	+	0.995567i	$0.529981\pi$
$62$	−1116.59	−2.28721
$63$	185.745	0.371455
$64$	−823.664	−1.60872
$65$	−78.2922	−0.149399
$66$	−151.630	−0.282794
$67$	488.446	0.890644	0.445322	−	0.895371i	$-0.353089\pi$
$67$	488.446	0.890644	0.445322	+	0.895371i	$0.353089\pi$
$68$	956.581	1.70592
$69$	40.9814	0.0715011
$70$	474.151	0.809599
$71$	837.423	1.39977	0.699887	−	0.714254i	$-0.253234\pi$
$71$	837.423	1.39977	0.699887	+	0.714254i	$0.253234\pi$
$72$	211.429	0.346072
$73$	351.216	0.563105	0.281553	−	0.959546i	$-0.409151\pi$
$73$	351.216	0.563105	0.281553	+	0.959546i	$0.409151\pi$
$74$	−300.512	−0.472078
$75$	−75.0000	−0.115470
$76$	800.547	1.20828
$77$	227.022	0.335994
$78$	215.845	0.313328
$79$	−831.205	−1.18377	−0.591885	−	0.806022i	$-0.701616\pi$
$79$	−831.205	−1.18377	−0.591885	+	0.806022i	$0.701616\pi$
$80$	15.2066	0.0212519
$81$	81.0000	0.111111
$82$	−501.238	−0.675031
$83$	1389.13	1.83707	0.918537	−	0.395335i	$-0.129371\pi$
$83$	1389.13	1.83707	0.918537	+	0.395335i	$0.129371\pi$
$84$	−811.873	−1.05456
$85$	364.754	0.465448
$86$	−559.423	−0.701443
$87$	94.3988	0.116329
$88$	258.413	0.313034
$89$	1523.70	1.81474	0.907369	−	0.420335i	$-0.138088\pi$
$89$	1523.70	1.81474	0.907369	+	0.420335i	$0.138088\pi$
$90$	206.769	0.242170
$91$	−323.164	−0.372273
$92$	−179.125	−0.202990
$93$	729.025	0.812864
$94$	−2387.17	−2.61933
$95$	305.256	0.329670
$96$	521.888	0.554843
$97$	−426.612	−0.446555	−0.223278	−	0.974755i	$-0.571676\pi$
$97$	−426.612	−0.446555	−0.223278	+	0.974755i	$0.571676\pi$
$98$	381.103	0.392829
$99$	99.0000	0.100504
$100$	327.818	0.327818
$101$	74.1387	0.0730403	0.0365202	−	0.999333i	$-0.488373\pi$
$101$	74.1387	0.0730403	0.0365202	+	0.999333i	$0.488373\pi$
$102$	−1005.59	−0.976163
$103$	−69.3916	−0.0663821	−0.0331911	−	0.999449i	$-0.510567\pi$
$103$	−69.3916	−0.0663821	−0.0331911	+	0.999449i	$0.510567\pi$
$104$	−367.850	−0.346833
$105$	−309.575	−0.287728
$106$	−2493.51	−2.28482
$107$	1141.71	1.03152	0.515761	−	0.856733i	$-0.327509\pi$
$107$	1141.71	1.03152	0.515761	+	0.856733i	$0.327509\pi$
$108$	−354.043	−0.315443
$109$	−2226.85	−1.95682	−0.978409	−	0.206680i	$-0.933734\pi$
$109$	−2226.85	−1.95682	−0.978409	+	0.206680i	$0.933734\pi$
$110$	252.717	0.219051
$111$	196.205	0.167775
$112$	62.7678	0.0529554
$113$	−1719.76	−1.43169	−0.715847	−	0.698257i	$-0.753959\pi$
$113$	−1719.76	−1.43169	−0.715847	+	0.698257i	$0.753959\pi$
$114$	−841.566	−0.691402
$115$	−68.3023	−0.0553845
$116$	−412.608	−0.330256
$117$	−140.926	−0.111356
$118$	503.038	0.392444
$119$	1505.58	1.15980
$120$	−352.382	−0.268066
$121$	121.000	0.0909091
$122$	−411.771	−0.305574
$123$	327.260	0.239903
$124$	−3186.49	−2.30771
$125$	125.000	0.0894427
$126$	853.472	0.603439
$127$	−1601.63	−1.11907	−0.559534	−	0.828807i	$-0.689020\pi$
$127$	−1601.63	−1.11907	−0.559534	+	0.828807i	$0.689020\pi$
$128$	−2392.91	−1.65239
$129$	365.249	0.249290
$130$	−359.741	−0.242703
$131$	2004.13	1.33665	0.668327	−	0.743868i	$-0.267011\pi$
$131$	2004.13	1.33665	0.668327	+	0.743868i	$0.267011\pi$
$132$	−432.719	−0.285329
$133$	1260.00	0.821471
$134$	2244.34	1.44687
$135$	−135.000	−0.0860663
$136$	1713.77	1.08055
$137$	−1672.85	−1.04322	−0.521610	−	0.853184i	$-0.674669\pi$
$137$	−1672.85	−1.04322	−0.521610	+	0.853184i	$0.674669\pi$
$138$	188.303	0.116155
$139$	2540.38	1.55016	0.775080	−	0.631863i	$-0.217709\pi$
$139$	2540.38	1.55016	0.775080	+	0.631863i	$0.217709\pi$
$140$	1353.12	0.816855
$141$	1558.59	0.930901
$142$	3847.84	2.27397
$143$	−172.243	−0.100725
$144$	27.3719	0.0158402
$145$	−157.331	−0.0901079
$146$	1613.79	0.914780
$147$	−248.824	−0.139610
$148$	−857.594	−0.476310
$149$	3090.68	1.69932	0.849658	−	0.527334i	$-0.176808\pi$
$149$	3090.68	1.69932	0.849658	+	0.527334i	$0.176808\pi$
$150$	−344.614	−0.187584
$151$	1358.74	0.732267	0.366134	−	0.930562i	$-0.380681\pi$
$151$	1358.74	0.732267	0.366134	+	0.930562i	$0.380681\pi$
$152$	1434.22	0.765335
$153$	656.557	0.346925
$154$	1043.13	0.545831
$155$	−1215.04	−0.629642
$156$	615.973	0.316137
$157$	−1011.95	−0.514411	−0.257205	−	0.966357i	$-0.582802\pi$
$157$	−1011.95	−0.514411	−0.257205	+	0.966357i	$0.582802\pi$
$158$	−3819.27	−1.92307
$159$	1628.02	0.812015
$160$	−869.813	−0.429780
$161$	−281.929	−0.138007
$162$	372.183	0.180503
$163$	−2816.37	−1.35334	−0.676672	−	0.736285i	$-0.736578\pi$
$163$	−2816.37	−1.35334	−0.676672	+	0.736285i	$0.736578\pi$
$164$	−1430.42	−0.681081
$165$	−165.000	−0.0778499
$166$	6382.87	2.98438
$167$	3448.89	1.59810	0.799052	−	0.601262i	$-0.205335\pi$
$167$	3448.89	1.59810	0.799052	+	0.601262i	$0.205335\pi$
$168$	−1454.52	−0.667966
$169$	−1951.81	−0.888399
$170$	1675.99	0.756133
$171$	549.462	0.245722
$172$	−1596.47	−0.707730
$173$	−2287.85	−1.00545	−0.502723	−	0.864448i	$-0.667668\pi$
$173$	−2287.85	−1.00545	−0.502723	+	0.864448i	$0.667668\pi$
$174$	433.749	0.188979
$175$	515.959	0.222873
$176$	33.4545	0.0143280
$177$	−328.435	−0.139473
$178$	7001.17	2.94809
$179$	3249.06	1.35668	0.678340	−	0.734748i	$-0.262700\pi$
$179$	3249.06	1.35668	0.678340	+	0.734748i	$0.262700\pi$
$180$	590.072	0.244341
$181$	1170.45	0.480655	0.240328	−	0.970692i	$-0.422745\pi$
$181$	1170.45	0.480655	0.240328	+	0.970692i	$0.422745\pi$
$182$	−1484.89	−0.604767
$183$	268.847	0.108600
$184$	−320.913	−0.128576
$185$	−327.009	−0.129958
$186$	3349.76	1.32052
$187$	802.458	0.313805
$188$	−6812.44	−2.64281
$189$	−557.235	−0.214460
$190$	1402.61	0.535558
$191$	−2760.35	−1.04572	−0.522859	−	0.852419i	$-0.675134\pi$
$191$	−2760.35	−1.04572	−0.522859	+	0.852419i	$0.675134\pi$
$192$	2470.99	0.928794
$193$	−1250.61	−0.466430	−0.233215	−	0.972425i	$-0.574925\pi$
$193$	−1250.61	−0.466430	−0.233215	+	0.972425i	$0.574925\pi$
$194$	−1960.22	−0.725441
$195$	234.877	0.0862557
$196$	1087.58	0.396350
$197$	−143.991	−0.0520756	−0.0260378	−	0.999661i	$-0.508289\pi$
$197$	−143.991	−0.0520756	−0.0260378	+	0.999661i	$0.508289\pi$
$198$	454.891	0.163271
$199$	761.249	0.271174	0.135587	−	0.990765i	$-0.456708\pi$
$199$	761.249	0.271174	0.135587	+	0.990765i	$0.456708\pi$
$200$	587.303	0.207643
$201$	−1465.34	−0.514213
$202$	340.657	0.118656
$203$	−649.411	−0.224531
$204$	−2869.74	−0.984913
$205$	−545.434	−0.185828
$206$	−318.844	−0.107840
$207$	−122.944	−0.0412812
$208$	−47.6223	−0.0158751
$209$	671.564	0.222263
$210$	−1422.45	−0.467422
$211$	3976.58	1.29743	0.648717	−	0.761029i	$-0.275306\pi$
$211$	3976.58	1.29743	0.648717	+	0.761029i	$0.275306\pi$
$212$	−7115.91	−2.30530
$213$	−2512.27	−0.808159
$214$	5245.97	1.67573
$215$	−608.749	−0.193099
$216$	−634.287	−0.199805
$217$	−5015.29	−1.56894
$218$	−10232.0	−3.17890
$219$	−1053.65	−0.325109
$220$	721.199	0.221015
$221$	−1142.29	−0.347688
$222$	901.535	0.272555
$223$	908.084	0.272690	0.136345	−	0.990661i	$-0.456464\pi$
$223$	908.084	0.272690	0.136345	+	0.990661i	$0.456464\pi$
$224$	−3590.30	−1.07092
$225$	225.000	0.0666667
$226$	−7902.05	−2.32583
$227$	−2062.15	−0.602951	−0.301475	−	0.953474i	$-0.597479\pi$
$227$	−2062.15	−0.602951	−0.301475	+	0.953474i	$0.597479\pi$
$228$	−2401.64	−0.697599
$229$	4077.47	1.17662	0.588312	−	0.808634i	$-0.299793\pi$
$229$	4077.47	1.17662	0.588312	+	0.808634i	$0.299793\pi$
$230$	−313.839	−0.0899737
$231$	−681.065	−0.193986
$232$	−739.209	−0.209187
$233$	1682.76	0.473138	0.236569	−	0.971615i	$-0.423977\pi$
$233$	1682.76	0.473138	0.236569	+	0.971615i	$0.423977\pi$
$234$	−647.534	−0.180900
$235$	−2597.65	−0.721073
$236$	1435.56	0.395961
$237$	2493.62	0.683450
$238$	6917.93	1.88413
$239$	4024.96	1.08934	0.544672	−	0.838649i	$-0.316654\pi$
$239$	4024.96	1.08934	0.544672	+	0.838649i	$0.316654\pi$
$240$	−45.6198	−0.0122698
$241$	−2784.27	−0.744194	−0.372097	−	0.928194i	$-0.621361\pi$
$241$	−2784.27	−0.744194	−0.372097	+	0.928194i	$0.621361\pi$
$242$	555.978	0.147684
$243$	−243.000	−0.0641500
$244$	−1175.10	−0.308313
$245$	414.707	0.108141
$246$	1503.71	0.389729
$247$	−955.968	−0.246262
$248$	−5708.78	−1.46173
$249$	−4167.40	−1.06064
$250$	574.357	0.145302
$251$	−1827.60	−0.459591	−0.229796	−	0.973239i	$-0.573806\pi$
$251$	−1827.60	−0.459591	−0.229796	+	0.973239i	$0.573806\pi$
$252$	2435.62	0.608848
$253$	−150.265	−0.0373402
$254$	−7359.26	−1.81796
$255$	−1094.26	−0.268727
$256$	−4405.79	−1.07563
$257$	585.171	0.142031	0.0710155	−	0.997475i	$-0.477376\pi$
$257$	585.171	0.142031	0.0710155	+	0.997475i	$0.477376\pi$
$258$	1678.27	0.404978
$259$	−1349.78	−0.323828
$260$	−1026.62	−0.244878
$261$	−283.196	−0.0671625
$262$	9208.69	2.17143
$263$	−238.098	−0.0558241	−0.0279120	−	0.999610i	$-0.508886\pi$
$263$	−238.098	−0.0558241	−0.0279120	+	0.999610i	$0.508886\pi$
$264$	−775.240	−0.180730
$265$	−2713.37	−0.628984
$266$	5789.51	1.33450
$267$	−4571.09	−1.04774
$268$	6404.84	1.45984
$269$	4618.46	1.04681	0.523406	−	0.852083i	$-0.324661\pi$
$269$	4618.46	1.04681	0.523406	+	0.852083i	$0.324661\pi$
$270$	−620.306	−0.139817
$271$	−143.439	−0.0321525	−0.0160762	−	0.999871i	$-0.505117\pi$
$271$	−143.439	−0.0321525	−0.0160762	+	0.999871i	$0.505117\pi$
$272$	221.867	0.0494582
$273$	969.493	0.214932
$274$	−7686.51	−1.69474
$275$	275.000	0.0603023
$276$	537.376	0.117197
$277$	−8602.51	−1.86597	−0.932987	−	0.359911i	$-0.882807\pi$
$277$	−8602.51	−1.86597	−0.932987	+	0.359911i	$0.882807\pi$
$278$	11672.7	2.51828
$279$	−2187.07	−0.469307
$280$	2424.19	0.517404
$281$	−2992.81	−0.635360	−0.317680	−	0.948198i	$-0.602904\pi$
$281$	−2992.81	−0.635360	−0.317680	+	0.948198i	$0.602904\pi$
$282$	7161.50	1.51227
$283$	−6858.89	−1.44070	−0.720351	−	0.693610i	$-0.756019\pi$
$283$	−6858.89	−1.44070	−0.720351	+	0.693610i	$0.756019\pi$
$284$	10980.9	2.29435
$285$	−915.769	−0.190335
$286$	−791.431	−0.163630
$287$	−2251.37	−0.463046
$288$	−1565.66	−0.320339
$289$	408.809	0.0832096
$290$	−722.915	−0.146383
$291$	1279.84	0.257819
$292$	4605.39	0.922979
$293$	4049.70	0.807461	0.403731	−	0.914878i	$-0.367713\pi$
$293$	4049.70	0.807461	0.403731	+	0.914878i	$0.367713\pi$
$294$	−1143.31	−0.226800
$295$	547.392	0.108035
$296$	−1536.43	−0.301699
$297$	−297.000	−0.0580259
$298$	14201.2	2.76059
$299$	213.901	0.0413720
$300$	−983.453	−0.189266
$301$	−2512.71	−0.481164
$302$	6243.20	1.18959
$303$	−222.416	−0.0421699
$304$	185.677	0.0350305
$305$	−448.078	−0.0841209
$306$	3016.78	0.563588
$307$	−9572.69	−1.77962	−0.889808	−	0.456335i	$-0.849162\pi$
$307$	−9572.69	−1.77962	−0.889808	+	0.456335i	$0.849162\pi$
$308$	2976.87	0.550724
$309$	208.175	0.0383257
$310$	−5582.94	−1.02287
$311$	5396.42	0.983932	0.491966	−	0.870614i	$-0.336278\pi$
$311$	5396.42	0.983932	0.491966	+	0.870614i	$0.336278\pi$
$312$	1103.55	0.200244
$313$	9755.04	1.76162	0.880811	−	0.473469i	$-0.156998\pi$
$313$	9755.04	1.76162	0.880811	+	0.473469i	$0.156998\pi$
$314$	−4649.77	−0.835674
$315$	928.726	0.166120
$316$	−10899.3	−1.94030
$317$	−4353.75	−0.771391	−0.385695	−	0.922626i	$-0.626038\pi$
$317$	−4353.75	−0.771391	−0.385695	+	0.922626i	$0.626038\pi$
$318$	7480.52	1.31914
$319$	−346.129	−0.0607508
$320$	−4118.32	−0.719440
$321$	−3425.12	−0.595549
$322$	−1295.42	−0.224196
$323$	4453.74	0.767221
$324$	1062.13	0.182121
$325$	−391.461	−0.0668134
$326$	−12940.8	−2.19854
$327$	6680.54	1.12977
$328$	−2562.68	−0.431404
$329$	−10722.2	−1.79677
$330$	−758.151	−0.126469
$331$	5387.64	0.894656	0.447328	−	0.894370i	$-0.352376\pi$
$331$	5387.64	0.894656	0.447328	+	0.894370i	$0.352376\pi$
$332$	18215.3	3.01113
$333$	−588.616	−0.0968648
$334$	15847.2	2.59616
$335$	2442.23	0.398308
$336$	−188.303	−0.0305738
$337$	−4500.27	−0.727434	−0.363717	−	0.931509i	$-0.618492\pi$
$337$	−4500.27	−0.727434	−0.363717	+	0.931509i	$0.618492\pi$
$338$	−8968.30	−1.44323
$339$	5159.28	0.826589
$340$	4782.91	0.762910
$341$	−2673.09	−0.424504
$342$	2524.70	0.399181
$343$	−5367.18	−0.844899
$344$	−2860.16	−0.448283
$345$	204.907	0.0319763
$346$	−10512.4	−1.63337
$347$	5906.32	0.913740	0.456870	−	0.889533i	$-0.348970\pi$
$347$	5906.32	0.913740	0.456870	+	0.889533i	$0.348970\pi$
$348$	1237.82	0.190673
$349$	3636.26	0.557721	0.278860	−	0.960332i	$-0.410043\pi$
$349$	3636.26	0.557721	0.278860	+	0.960332i	$0.410043\pi$
$350$	2370.76	0.362063
$351$	422.778	0.0642912
$352$	−1913.59	−0.289757
$353$	210.408	0.0317248	0.0158624	−	0.999874i	$-0.494951\pi$
$353$	210.408	0.0317248	0.0158624	+	0.999874i	$0.494951\pi$
$354$	−1509.11	−0.226577
$355$	4187.12	0.625998
$356$	19979.8	2.97451
$357$	−4516.75	−0.669612
$358$	14928.9	2.20396
$359$	2499.68	0.367488	0.183744	−	0.982974i	$-0.441178\pi$
$359$	2499.68	0.367488	0.183744	+	0.982974i	$0.441178\pi$
$360$	1057.15	0.154768
$361$	−3131.74	−0.456588
$362$	5378.03	0.780837
$363$	−363.000	−0.0524864
$364$	−4237.56	−0.610188
$365$	1756.08	0.251828
$366$	1235.31	0.176423
$367$	5748.70	0.817656	0.408828	−	0.912612i	$-0.365938\pi$
$367$	5748.70	0.817656	0.408828	+	0.912612i	$0.365938\pi$
$368$	−41.5458	−0.00588512
$369$	−981.781	−0.138508
$370$	−1502.56	−0.211120
$371$	−11199.9	−1.56730
$372$	9559.48	1.33236
$373$	4467.78	0.620196	0.310098	−	0.950705i	$-0.399638\pi$
$373$	4467.78	0.620196	0.310098	+	0.950705i	$0.399638\pi$
$374$	3687.18	0.509785
$375$	−375.000	−0.0516398
$376$	−12204.9	−1.67398
$377$	492.712	0.0673103
$378$	−2560.42	−0.348396
$379$	7804.08	1.05770	0.528851	−	0.848715i	$-0.322623\pi$
$379$	7804.08	1.05770	0.528851	+	0.848715i	$0.322623\pi$
$380$	4002.74	0.540358
$381$	4804.89	0.646094
$382$	−12683.4	−1.69880
$383$	−11161.1	−1.48904	−0.744522	−	0.667597i	$-0.767323\pi$
$383$	−11161.1	−1.48904	−0.744522	+	0.667597i	$0.767323\pi$
$384$	7178.74	0.954007
$385$	1135.11	0.150261
$386$	−5746.38	−0.757728
$387$	−1095.75	−0.143928
$388$	−5594.04	−0.731944
$389$	8490.24	1.10661	0.553306	−	0.832978i	$-0.313366\pi$
$389$	8490.24	1.10661	0.553306	+	0.832978i	$0.313366\pi$
$390$	1079.22	0.140125
$391$	−996.540	−0.128893
$392$	1948.47	0.251052
$393$	−6012.39	−0.771717
$394$	−661.616	−0.0845983
$395$	−4156.03	−0.529398
$396$	1298.16	0.164735
$397$	6019.74	0.761013	0.380507	−	0.924778i	$-0.375750\pi$
$397$	6019.74	0.761013	0.380507	+	0.924778i	$0.375750\pi$
$398$	3497.83	0.440529
$399$	−3779.99	−0.474277
$400$	76.0330	0.00950413
$401$	−10398.8	−1.29499	−0.647495	−	0.762069i	$-0.724184\pi$
$401$	−10398.8	−1.29499	−0.647495	+	0.762069i	$0.724184\pi$
$402$	−6733.01	−0.835354
$403$	3805.13	0.470340
$404$	972.158	0.119720
$405$	405.000	0.0496904
$406$	−2983.95	−0.364756
$407$	−719.420	−0.0876175
$408$	−5141.30	−0.623854
$409$	−4733.68	−0.572287	−0.286144	−	0.958187i	$-0.592373\pi$
$409$	−4733.68	−0.572287	−0.286144	+	0.958187i	$0.592373\pi$
$410$	−2506.19	−0.301883
$411$	5018.55	0.602304
$412$	−909.911	−0.108806
$413$	2259.45	0.269202
$414$	−564.910	−0.0670624
$415$	6945.67	0.821565
$416$	2723.98	0.321044
$417$	−7621.14	−0.894985
$418$	3085.74	0.361073
$419$	−8117.57	−0.946466	−0.473233	−	0.880937i	$-0.656913\pi$
$419$	−8117.57	−0.946466	−0.473233	+	0.880937i	$0.656913\pi$
$420$	−4059.37	−0.471611
$421$	−9484.27	−1.09795	−0.548973	−	0.835840i	$-0.684981\pi$
$421$	−9484.27	−1.09795	−0.548973	+	0.835840i	$0.684981\pi$
$422$	18271.8	2.10772
$423$	−4675.77	−0.537456
$424$	−12748.5	−1.46020
$425$	1823.77	0.208155
$426$	−11543.5	−1.31288
$427$	−1849.52	−0.209612
$428$	14970.8	1.69075
$429$	516.728	0.0581536
$430$	−2797.11	−0.313695
$431$	9335.16	1.04329	0.521646	−	0.853162i	$-0.325318\pi$
$431$	9335.16	1.04329	0.521646	+	0.853162i	$0.325318\pi$
$432$	−82.1157	−0.00914535
$433$	−2983.02	−0.331074	−0.165537	−	0.986204i	$-0.552936\pi$
$433$	−2983.02	−0.331074	−0.165537	+	0.986204i	$0.552936\pi$
$434$	−23044.5	−2.54878
$435$	471.994	0.0520238
$436$	−29200.0	−3.20739
$437$	−833.988	−0.0912931
$438$	−4841.36	−0.528148
$439$	−5232.32	−0.568850	−0.284425	−	0.958698i	$-0.591803\pi$
$439$	−5232.32	−0.568850	−0.284425	+	0.958698i	$0.591803\pi$
$440$	1292.07	0.139993
$441$	746.472	0.0806038
$442$	−5248.68	−0.564828
$443$	7517.71	0.806269	0.403135	−	0.915141i	$-0.367921\pi$
$443$	7517.71	0.806269	0.403135	+	0.915141i	$0.367921\pi$
$444$	2572.78	0.274997
$445$	7618.49	0.811575
$446$	4172.52	0.442992
$447$	−9272.03	−0.981101
$448$	−16999.1	−1.79270
$449$	16070.9	1.68916	0.844581	−	0.535428i	$-0.179850\pi$
$449$	16070.9	1.68916	0.844581	+	0.535428i	$0.179850\pi$
$450$	1033.84	0.108302
$451$	−1199.96	−0.125285
$452$	−22550.7	−2.34667
$453$	−4076.21	−0.422775
$454$	−9475.29	−0.979510
$455$	−1615.82	−0.166485
$456$	−4302.67	−0.441867
$457$	9718.51	0.994776	0.497388	−	0.867528i	$-0.334293\pi$
$457$	9718.51	0.994776	0.497388	+	0.867528i	$0.334293\pi$
$458$	18735.4	1.91146
$459$	−1969.67	−0.200297
$460$	−895.627	−0.0907801
$461$	−14538.0	−1.46877	−0.734385	−	0.678733i	$-0.762529\pi$
$461$	−14538.0	−1.46877	−0.734385	+	0.678733i	$0.762529\pi$
$462$	−3129.40	−0.315136
$463$	−9978.17	−1.00157	−0.500783	−	0.865573i	$-0.666955\pi$
$463$	−9978.17	−1.00157	−0.500783	+	0.865573i	$0.666955\pi$
$464$	−95.6990	−0.00957481
$465$	3645.12	0.363524
$466$	7732.03	0.768625
$467$	15188.2	1.50498	0.752489	−	0.658605i	$-0.228853\pi$
$467$	15188.2	1.50498	0.752489	+	0.658605i	$0.228853\pi$
$468$	−1847.92	−0.182522
$469$	10080.7	0.992503
$470$	−11935.8	−1.17140
$471$	3035.85	0.296995
$472$	2571.88	0.250806
$473$	−1339.25	−0.130187
$474$	11457.8	1.11028
$475$	1526.28	0.147433
$476$	19742.3	1.90102
$477$	−4884.06	−0.468817
$478$	18494.1	1.76967
$479$	11330.8	1.08083	0.540415	−	0.841399i	$-0.318267\pi$
$479$	11330.8	1.08083	0.540415	+	0.841399i	$0.318267\pi$
$480$	2609.44	0.248133
$481$	1024.09	0.0970779
$482$	−12793.3	−1.20896
$483$	845.788	0.0796784
$484$	1586.64	0.149008
$485$	−2133.06	−0.199706
$486$	−1116.55	−0.104213
$487$	19086.9	1.77599	0.887997	−	0.459850i	$-0.152097\pi$
$487$	19086.9	1.77599	0.887997	+	0.459850i	$0.152097\pi$
$488$	−2105.26	−0.195288
$489$	8449.11	0.781353
$490$	1905.52	0.175679
$491$	−8112.85	−0.745677	−0.372839	−	0.927896i	$-0.621616\pi$
$491$	−8112.85	−0.745677	−0.372839	+	0.927896i	$0.621616\pi$
$492$	4291.27	0.393222
$493$	−2295.49	−0.209703
$494$	−4392.53	−0.400060
$495$	495.000	0.0449467
$496$	−739.066	−0.0669053
$497$	17283.0	1.55986
$498$	−19148.6	−1.72303
$499$	18329.1	1.64433	0.822167	−	0.569246i	$-0.192765\pi$
$499$	18329.1	1.64433	0.822167	+	0.569246i	$0.192765\pi$
$500$	1639.09	0.146604
$501$	−10346.7	−0.922666
$502$	−8397.58	−0.746618
$503$	7739.57	0.686064	0.343032	−	0.939324i	$-0.388546\pi$
$503$	7739.57	0.686064	0.343032	+	0.939324i	$0.388546\pi$
$504$	4363.55	0.385651
$505$	370.693	0.0326646
$506$	−690.446	−0.0606602
$507$	5855.44	0.512918
$508$	−21001.7	−1.83425
$509$	15914.9	1.38589	0.692943	−	0.720993i	$-0.256314\pi$
$509$	15914.9	1.38589	0.692943	+	0.720993i	$0.256314\pi$
$510$	−5027.97	−0.436554
$511$	7248.51	0.627505
$512$	−1100.65	−0.0950048
$513$	−1648.38	−0.141867
$514$	2688.78	0.230733
$515$	−346.958	−0.0296870
$516$	4789.40	0.408608
$517$	−5714.83	−0.486147
$518$	−6202.07	−0.526068
$519$	6863.56	0.580495
$520$	−1839.25	−0.155109
$521$	2274.50	0.191262	0.0956312	−	0.995417i	$-0.469513\pi$
$521$	2274.50	0.191262	0.0956312	+	0.995417i	$0.469513\pi$
$522$	−1301.25	−0.109107
$523$	10971.1	0.917274	0.458637	−	0.888624i	$-0.348338\pi$
$523$	10971.1	0.917274	0.458637	+	0.888624i	$0.348338\pi$
$524$	26279.6	2.19089
$525$	−1547.88	−0.128676
$526$	−1094.02	−0.0906877
$527$	−17727.6	−1.46533
$528$	−100.364	−0.00827228
$529$	−11980.4	−0.984663
$530$	−12467.5	−1.02180
$531$	985.306	0.0805247
$532$	16522.0	1.34646
$533$	1708.13	0.138813
$534$	−21003.5	−1.70208
$535$	5708.53	0.461310
$536$	11474.6	0.924680
$537$	−9747.17	−0.783280
$538$	21221.2	1.70058
$539$	912.355	0.0729089
$540$	−1770.21	−0.141070
$541$	5313.05	0.422229	0.211115	−	0.977461i	$-0.432291\pi$
$541$	5313.05	0.422229	0.211115	+	0.977461i	$0.432291\pi$
$542$	−659.084	−0.0522326
$543$	−3511.34	−0.277506
$544$	−12690.7	−1.00020
$545$	−11134.2	−0.875115
$546$	4454.68	0.349162
$547$	20685.1	1.61688	0.808439	−	0.588581i	$-0.200313\pi$
$547$	20685.1	1.61688	0.808439	+	0.588581i	$0.200313\pi$
$548$	−21935.6	−1.70993
$549$	−806.541	−0.0627000
$550$	1263.59	0.0979627
$551$	−1921.06	−0.148529
$552$	962.739	0.0742335
$553$	−17154.7	−1.31915
$554$	−39527.3	−3.03132
$555$	981.027	0.0750311
$556$	33311.2	2.54085
$557$	−10853.8	−0.825659	−0.412830	−	0.910808i	$-0.635460\pi$
$557$	−10853.8	−0.825659	−0.412830	+	0.910808i	$0.635460\pi$
$558$	−10049.3	−0.762402
$559$	1906.41	0.144244
$560$	313.839	0.0236824
$561$	−2407.37	−0.181175
$562$	−13751.5	−1.03216
$563$	−15381.2	−1.15141	−0.575704	−	0.817658i	$-0.695272\pi$
$563$	−15381.2	−1.15141	−0.575704	+	0.817658i	$0.695272\pi$
$564$	20437.3	1.52583
$565$	−8598.80	−0.640273
$566$	−31515.6	−2.34046
$567$	1671.71	0.123818
$568$	19672.9	1.45327
$569$	−1348.88	−0.0993814	−0.0496907	−	0.998765i	$-0.515824\pi$
$569$	−1348.88	−0.0993814	−0.0496907	+	0.998765i	$0.515824\pi$
$570$	−4207.83	−0.309204
$571$	3463.51	0.253841	0.126920	−	0.991913i	$-0.459491\pi$
$571$	3463.51	0.253841	0.126920	+	0.991913i	$0.459491\pi$
$572$	−2258.57	−0.165097
$573$	8281.05	0.603745
$574$	−10344.7	−0.752231
$575$	−341.511	−0.0247687
$576$	−7412.97	−0.536239
$577$	12052.6	0.869598	0.434799	−	0.900528i	$-0.356819\pi$
$577$	12052.6	0.869598	0.434799	+	0.900528i	$0.356819\pi$
$578$	1878.42	0.135176
$579$	3751.83	0.269293
$580$	−2063.04	−0.147695
$581$	28669.4	2.04717
$582$	5880.66	0.418834
$583$	−5969.41	−0.424061
$584$	8250.80	0.584624
$585$	−704.630	−0.0497997
$586$	18607.8	1.31174
$587$	−11133.1	−0.782813	−0.391407	−	0.920218i	$-0.628011\pi$
$587$	−11133.1	−0.782813	−0.391407	+	0.920218i	$0.628011\pi$
$588$	−3262.75	−0.228833
$589$	−14836.0	−1.03787
$590$	2515.19	0.175506
$591$	431.972	0.0300659
$592$	−198.908	−0.0138092
$593$	−7939.69	−0.549821	−0.274911	−	0.961470i	$-0.588648\pi$
$593$	−7939.69	−0.549821	−0.274911	+	0.961470i	$0.588648\pi$
$594$	−1364.67	−0.0942646
$595$	7527.91	0.518680
$596$	40527.1	2.78533
$597$	−2283.75	−0.156562
$598$	982.846	0.0672100
$599$	−19474.7	−1.32840	−0.664202	−	0.747553i	$-0.731229\pi$
$599$	−19474.7	−1.32840	−0.664202	+	0.747553i	$0.731229\pi$
$600$	−1761.91	−0.119883
$601$	−19946.1	−1.35377	−0.676887	−	0.736087i	$-0.736671\pi$
$601$	−19946.1	−1.35377	−0.676887	+	0.736087i	$0.736671\pi$
$602$	−11545.6	−0.781664
$603$	4396.01	0.296881
$604$	17816.7	1.20025
$605$	605.000	0.0406558
$606$	−1021.97	−0.0685061
$607$	1427.44	0.0954496	0.0477248	−	0.998861i	$-0.484803\pi$
$607$	1427.44	0.0954496	0.0477248	+	0.998861i	$0.484803\pi$
$608$	−10620.6	−0.708427
$609$	1948.23	0.129633
$610$	−2058.85	−0.136657
$611$	8135.03	0.538638
$612$	8609.23	0.568640
$613$	−8029.40	−0.529045	−0.264522	−	0.964380i	$-0.585214\pi$
$613$	−8029.40	−0.529045	−0.264522	+	0.964380i	$0.585214\pi$
$614$	−43985.1	−2.89103
$615$	1636.30	0.107288
$616$	5333.22	0.348834
$617$	20795.5	1.35688	0.678440	−	0.734655i	$-0.262656\pi$
$617$	20795.5	1.35688	0.678440	+	0.734655i	$0.262656\pi$
$618$	956.533	0.0622612
$619$	1677.43	0.108920	0.0544602	−	0.998516i	$-0.482656\pi$
$619$	1677.43	0.108920	0.0544602	+	0.998516i	$0.482656\pi$
$620$	−15932.5	−1.03204
$621$	368.832	0.0238337
$622$	24795.8	1.59842
$623$	31446.6	2.02228
$624$	142.867	0.00916547
$625$	625.000	0.0400000
$626$	44823.0	2.86180
$627$	−2014.69	−0.128324
$628$	−13269.4	−0.843165
$629$	−4771.11	−0.302443
$630$	4267.36	0.269866
$631$	−25225.2	−1.59144	−0.795719	−	0.605666i	$-0.792907\pi$
$631$	−25225.2	−1.59144	−0.795719	+	0.605666i	$0.792907\pi$
$632$	−19526.8	−1.22901
$633$	−11929.7	−0.749074
$634$	−20004.8	−1.25315
$635$	−8008.15	−0.500462
$636$	21347.7	1.33096
$637$	−1298.73	−0.0807812
$638$	−1590.41	−0.0986912
$639$	7536.81	0.466591
$640$	−11964.6	−0.738971
$641$	15165.3	0.934468	0.467234	−	0.884134i	$-0.345251\pi$
$641$	15165.3	0.934468	0.467234	+	0.884134i	$0.345251\pi$
$642$	−15737.9	−0.967486
$643$	27156.1	1.66553	0.832763	−	0.553630i	$-0.186758\pi$
$643$	27156.1	1.66553	0.832763	+	0.553630i	$0.186758\pi$
$644$	−3696.85	−0.226206
$645$	1826.25	0.111486
$646$	20464.3	1.24637
$647$	29154.9	1.77156	0.885778	−	0.464110i	$-0.153626\pi$
$647$	29154.9	1.77156	0.885778	+	0.464110i	$0.153626\pi$
$648$	1902.86	0.115357
$649$	1204.26	0.0728374
$650$	−1798.71	−0.108540
$651$	15045.9	0.905828
$652$	−36930.2	−2.21825
$653$	−19141.7	−1.14713	−0.573564	−	0.819161i	$-0.694440\pi$
$653$	−19141.7	−1.14713	−0.573564	+	0.819161i	$0.694440\pi$
$654$	30696.1	1.83534
$655$	10020.7	0.597770
$656$	−331.768	−0.0197460
$657$	3160.94	0.187702
$658$	−49267.2	−2.91890
$659$	−24939.6	−1.47422	−0.737110	−	0.675773i	$-0.763810\pi$
$659$	−24939.6	−1.47422	−0.737110	+	0.675773i	$0.763810\pi$
$660$	−2163.60	−0.127603
$661$	22617.7	1.33090	0.665452	−	0.746440i	$-0.268239\pi$
$661$	22617.7	1.33090	0.665452	+	0.746440i	$0.268239\pi$
$662$	24755.4	1.45339
$663$	3426.88	0.200738
$664$	32633.7	1.90728
$665$	6299.99	0.367373
$666$	−2704.61	−0.157359
$667$	429.843	0.0249529
$668$	45224.3	2.61943
$669$	−2724.25	−0.157438
$670$	11221.7	0.647062
$671$	−985.772	−0.0567143
$672$	10770.9	0.618298
$673$	−13855.8	−0.793615	−0.396807	−	0.917902i	$-0.629882\pi$
$673$	−13855.8	−0.793615	−0.396807	+	0.917902i	$0.629882\pi$
$674$	−20678.1	−1.18174
$675$	−675.000	−0.0384900
$676$	−25593.5	−1.45616
$677$	−24992.8	−1.41884	−0.709419	−	0.704787i	$-0.751043\pi$
$677$	−24992.8	−1.41884	−0.709419	+	0.704787i	$0.751043\pi$
$678$	23706.1	1.34282
$679$	−8804.57	−0.497626
$680$	8568.84	0.483235
$681$	6186.46	0.348114
$682$	−12282.5	−0.689619
$683$	−14420.5	−0.807887	−0.403943	−	0.914784i	$-0.632361\pi$
$683$	−14420.5	−0.807887	−0.403943	+	0.914784i	$0.632361\pi$
$684$	7204.93	0.402759
$685$	−8364.25	−0.466543
$686$	−24661.4	−1.37256
$687$	−12232.4	−0.679324
$688$	−370.280	−0.0205186
$689$	8497.42	0.469849
$690$	941.517	0.0519463
$691$	30552.4	1.68201	0.841005	−	0.541027i	$-0.181964\pi$
$691$	30552.4	1.68201	0.841005	+	0.541027i	$0.181964\pi$
$692$	−29999.9	−1.64801
$693$	2043.20	0.111998
$694$	27138.7	1.48440
$695$	12701.9	0.693253
$696$	2217.63	0.120774
$697$	−7957.96	−0.432467
$698$	16708.1	0.906032
$699$	−5048.27	−0.273166
$700$	6765.61	0.365309
$701$	−9151.47	−0.493076	−0.246538	−	0.969133i	$-0.579293\pi$
$701$	−9151.47	−0.493076	−0.246538	+	0.969133i	$0.579293\pi$
$702$	1942.60	0.104443
$703$	−3992.86	−0.214216
$704$	−9060.30	−0.485047
$705$	7792.95	0.416311
$706$	966.793	0.0515378
$707$	1530.10	0.0813937
$708$	−4306.67	−0.228608
$709$	−6261.96	−0.331697	−0.165848	−	0.986151i	$-0.553036\pi$
$709$	−6261.96	−0.331697	−0.165848	+	0.986151i	$0.553036\pi$
$710$	19239.2	1.01695
$711$	−7480.85	−0.394590
$712$	35794.9	1.88409
$713$	3319.60	0.174362
$714$	−20753.8	−1.08780
$715$	−861.214	−0.0450456
$716$	42603.9	2.22372
$717$	−12074.9	−0.628933
$718$	11485.7	0.596995
$719$	−18228.7	−0.945500	−0.472750	−	0.881197i	$-0.656739\pi$
$719$	−18228.7	−0.945500	−0.472750	+	0.881197i	$0.656739\pi$
$720$	136.859	0.00708396
$721$	−1432.13	−0.0739740
$722$	−14389.9	−0.741740
$723$	8352.81	0.429660
$724$	15347.7	0.787836
$725$	−786.656	−0.0402975
$726$	−1667.93	−0.0852655
$727$	−7233.66	−0.369026	−0.184513	−	0.982830i	$-0.559071\pi$
$727$	−7233.66	−0.369026	−0.184513	+	0.982830i	$0.559071\pi$
$728$	−7591.81	−0.386499
$729$	729.000	0.0370370
$730$	8068.93	0.409102
$731$	−8881.73	−0.449388
$732$	3525.31	0.178004
$733$	−13444.8	−0.677485	−0.338743	−	0.940879i	$-0.610002\pi$
$733$	−13444.8	−0.677485	−0.338743	+	0.940879i	$0.610002\pi$
$734$	26414.4	1.32830
$735$	−1244.12	−0.0624355
$736$	2376.41	0.119016
$737$	5372.90	0.268539
$738$	−4511.14	−0.225010
$739$	18490.9	0.920432	0.460216	−	0.887807i	$-0.347772\pi$
$739$	18490.9	0.920432	0.460216	+	0.887807i	$0.347772\pi$
$740$	−4287.97	−0.213012
$741$	2867.90	0.142180
$742$	−51461.8	−2.54612
$743$	−25160.9	−1.24235	−0.621173	−	0.783674i	$-0.713343\pi$
$743$	−25160.9	−1.24235	−0.621173	+	0.783674i	$0.713343\pi$
$744$	17126.3	0.843927
$745$	15453.4	0.759957
$746$	20528.8	1.00752
$747$	12502.2	0.612358
$748$	10522.4	0.514354
$749$	23562.9	1.14949
$750$	−1723.07	−0.0838902
$751$	−13419.5	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$751$	−13419.5	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$752$	−1580.06	−0.0766207
$753$	5482.81	0.265345
$754$	2263.94	0.109347
$755$	6793.68	0.327480
$756$	−7306.86	−0.351518
$757$	−7014.90	−0.336804	−0.168402	−	0.985718i	$-0.553861\pi$
$757$	−7014.90	−0.336804	−0.168402	+	0.985718i	$0.553861\pi$
$758$	35858.6	1.71826
$759$	450.795	0.0215584
$760$	7171.12	0.342268
$761$	−30156.9	−1.43651	−0.718256	−	0.695779i	$-0.755059\pi$
$761$	−30156.9	−1.43651	−0.718256	+	0.695779i	$0.755059\pi$
$762$	22077.8	1.04960
$763$	−45958.4	−2.18061
$764$	−36195.7	−1.71402
$765$	3282.78	0.155149
$766$	−51283.5	−2.41899
$767$	−1714.26	−0.0807019
$768$	13217.4	0.621017
$769$	11292.2	0.529530	0.264765	−	0.964313i	$-0.414706\pi$
$769$	11292.2	0.529530	0.264765	+	0.964313i	$0.414706\pi$
$770$	5215.66	0.244103
$771$	−1755.51	−0.0820016
$772$	−16398.9	−0.764519
$773$	8524.10	0.396624	0.198312	−	0.980139i	$-0.436454\pi$
$773$	8524.10	0.396624	0.198312	+	0.980139i	$0.436454\pi$
$774$	−5034.80	−0.233814
$775$	−6075.21	−0.281584
$776$	−10022.0	−0.463621
$777$	4049.35	0.186962
$778$	39011.4	1.79772
$779$	−6659.89	−0.306310
$780$	3079.87	0.141381
$781$	9211.66	0.422047
$782$	−4578.96	−0.209390
$783$	849.589	0.0387763
$784$	252.251	0.0114910
$785$	−5059.76	−0.230052
$786$	−27626.1	−1.25368
$787$	−14983.9	−0.678676	−0.339338	−	0.940665i	$-0.610203\pi$
$787$	−14983.9	−0.678676	−0.339338	+	0.940665i	$0.610203\pi$
$788$	−1888.10	−0.0853565
$789$	714.293	0.0322300
$790$	−19096.3	−0.860022
$791$	−35493.0	−1.59543
$792$	2325.72	0.104345
$793$	1403.24	0.0628380
$794$	27659.9	1.23629
$795$	8140.10	0.363144
$796$	9982.04	0.444477
$797$	37172.3	1.65208	0.826041	−	0.563610i	$-0.190588\pi$
$797$	37172.3	1.65208	0.826041	+	0.563610i	$0.190588\pi$
$798$	−17368.5	−0.770475
$799$	−37900.1	−1.67811
$800$	−4349.06	−0.192203
$801$	13713.3	0.604913
$802$	−47781.0	−2.10375
$803$	3863.37	0.169783
$804$	−19214.5	−0.842841
$805$	−1409.65	−0.0617186
$806$	17484.0	0.764080
$807$	−13855.4	−0.604378
$808$	1741.68	0.0758316
$809$	23797.1	1.03419	0.517096	−	0.855928i	$-0.327013\pi$
$809$	23797.1	1.03419	0.517096	+	0.855928i	$0.327013\pi$
$810$	1860.92	0.0807234
$811$	8988.35	0.389178	0.194589	−	0.980885i	$-0.437663\pi$
$811$	8988.35	0.389178	0.194589	+	0.980885i	$0.437663\pi$
$812$	−8515.54	−0.368026
$813$	430.318	0.0185633
$814$	−3305.63	−0.142337
$815$	−14081.8	−0.605234
$816$	−665.600	−0.0285547
$817$	−7432.98	−0.318295
$818$	−21750.6	−0.929696
$819$	−2908.48	−0.124091
$820$	−7152.11	−0.304589
$821$	−25156.8	−1.06940	−0.534702	−	0.845041i	$-0.679576\pi$
$821$	−25156.8	−1.06940	−0.534702	+	0.845041i	$0.679576\pi$
$822$	23059.5	0.978459
$823$	1318.51	0.0558447	0.0279224	−	0.999610i	$-0.491111\pi$
$823$	1318.51	0.0558447	0.0279224	+	0.999610i	$0.491111\pi$
$824$	−1630.16	−0.0689189
$825$	−825.000	−0.0348155
$826$	10381.9	0.437326
$827$	124.982	0.00525519	0.00262760	−	0.999997i	$-0.499164\pi$
$827$	124.982	0.00525519	0.00262760	+	0.999997i	$0.499164\pi$
$828$	−1612.13	−0.0676635
$829$	−8886.80	−0.372318	−0.186159	−	0.982520i	$-0.559604\pi$
$829$	−8886.80	−0.372318	−0.186159	+	0.982520i	$0.559604\pi$
$830$	31914.3	1.33465
$831$	25807.5	1.07732
$832$	12897.3	0.537419
$833$	6050.63	0.251671
$834$	−35018.0	−1.45393
$835$	17244.5	0.714694
$836$	8806.02	0.364309
$837$	6561.22	0.270955
$838$	−37299.1	−1.53756
$839$	2995.21	0.123249	0.0616247	−	0.998099i	$-0.480372\pi$
$839$	2995.21	0.123249	0.0616247	+	0.998099i	$0.480372\pi$
$840$	−7272.58	−0.298724
$841$	−23398.9	−0.959403
$842$	−43578.8	−1.78364
$843$	8978.43	0.366825
$844$	52143.6	2.12661
$845$	−9759.07	−0.397304
$846$	−21484.5	−0.873111
$847$	2497.24	0.101306
$848$	−1650.44	−0.0668355
$849$	20576.7	0.831789
$850$	8379.95	0.338153
$851$	893.418	0.0359882
$852$	−32942.7	−1.32464
$853$	18130.5	0.727757	0.363878	−	0.931446i	$-0.381452\pi$
$853$	18130.5	0.727757	0.363878	+	0.931446i	$0.381452\pi$
$854$	−8498.27	−0.340521
$855$	2747.31	0.109890
$856$	26821.1	1.07094
$857$	26394.1	1.05205	0.526024	−	0.850470i	$-0.323682\pi$
$857$	26394.1	1.05205	0.526024	+	0.850470i	$0.323682\pi$
$858$	2374.29	0.0944720
$859$	−29456.2	−1.17000	−0.585002	−	0.811032i	$-0.698906\pi$
$859$	−29456.2	−1.17000	−0.585002	+	0.811032i	$0.698906\pi$
$860$	−7982.34	−0.316506
$861$	6754.12	0.267340
$862$	42893.7	1.69486
$863$	−762.616	−0.0300808	−0.0150404	−	0.999887i	$-0.504788\pi$
$863$	−762.616	−0.0300808	−0.0150404	+	0.999887i	$0.504788\pi$
$864$	4696.99	0.184948
$865$	−11439.3	−0.449649
$866$	−13706.6	−0.537838
$867$	−1226.43	−0.0480411
$868$	−65764.0	−2.57163
$869$	−9143.26	−0.356920
$870$	2168.74	0.0845141
$871$	−7648.29	−0.297535
$872$	−52313.3	−2.03160
$873$	−3839.51	−0.148852
$874$	−3832.06	−0.148308
$875$	2579.79	0.0996719
$876$	−13816.2	−0.532882
$877$	44767.2	1.72369	0.861847	−	0.507168i	$-0.169308\pi$
$877$	44767.2	1.72369	0.861847	+	0.507168i	$0.169308\pi$
$878$	−24041.8	−0.924112
$879$	−12149.1	−0.466188
$880$	167.273	0.00640768
$881$	−32057.9	−1.22595	−0.612973	−	0.790104i	$-0.710027\pi$
$881$	−32057.9	−1.22595	−0.612973	+	0.790104i	$0.710027\pi$
$882$	3429.93	0.130943
$883$	7078.95	0.269791	0.134896	−	0.990860i	$-0.456930\pi$
$883$	7078.95	0.269791	0.134896	+	0.990860i	$0.456930\pi$
$884$	−14978.6	−0.569891
$885$	−1642.18	−0.0623742
$886$	34542.8	1.30981
$887$	−25148.1	−0.951964	−0.475982	−	0.879455i	$-0.657907\pi$
$887$	−25148.1	−0.951964	−0.475982	+	0.879455i	$0.657907\pi$
$888$	4609.28	0.174186
$889$	−33055.0	−1.24705
$890$	35005.9	1.31843
$891$	891.000	0.0335013
$892$	11907.4	0.446963
$893$	−31718.0	−1.18858
$894$	−42603.6	−1.59382
$895$	16245.3	0.606726
$896$	−49385.8	−1.84137
$897$	−641.704	−0.0238862
$898$	73843.6	2.74409
$899$	7646.56	0.283679
$900$	2950.36	0.109273
$901$	−39588.4	−1.46380
$902$	−5513.62	−0.203529
$903$	7538.14	0.277800
$904$	−40400.8	−1.48641
$905$	5852.23	0.214955
$906$	−18729.6	−0.686809
$907$	1269.76	0.0464848	0.0232424	−	0.999730i	$-0.492601\pi$
$907$	1269.76	0.0464848	0.0232424	+	0.999730i	$0.492601\pi$
$908$	−27040.4	−0.988289
$909$	667.248	0.0243468
$910$	−7424.47	−0.270460
$911$	−33783.1	−1.22863	−0.614316	−	0.789060i	$-0.710568\pi$
$911$	−33783.1	−1.22863	−0.614316	+	0.789060i	$0.710568\pi$
$912$	−557.030	−0.0202249
$913$	15280.5	0.553899
$914$	44655.1	1.61604
$915$	1344.23	0.0485672
$916$	53466.6	1.92859
$917$	41361.9	1.48952
$918$	−9050.35	−0.325388
$919$	39262.5	1.40930	0.704652	−	0.709553i	$-0.251103\pi$
$919$	39262.5	1.40930	0.704652	+	0.709553i	$0.251103\pi$
$920$	−1604.57	−0.0575010
$921$	28718.1	1.02746
$922$	−66800.1	−2.38606
$923$	−13112.7	−0.467618
$924$	−8930.61	−0.317960
$925$	−1635.04	−0.0581189
$926$	−45848.3	−1.62707
$927$	−624.525	−0.0221274
$928$	5473.95	0.193633
$929$	−21175.0	−0.747825	−0.373913	−	0.927464i	$-0.621984\pi$
$929$	−21175.0	−0.747825	−0.373913	+	0.927464i	$0.621984\pi$
$930$	16748.8	0.590554
$931$	5063.68	0.178255
$932$	22065.5	0.775515
$933$	−16189.3	−0.568073
$934$	69787.5	2.44488
$935$	4012.29	0.140338
$936$	−3310.65	−0.115611
$937$	−5135.11	−0.179036	−0.0895180	−	0.995985i	$-0.528533\pi$
$937$	−5135.11	−0.179036	−0.0895180	+	0.995985i	$0.528533\pi$
$938$	46319.4	1.61235
$939$	−29265.1	−1.01707
$940$	−34062.2	−1.18190
$941$	−9702.77	−0.336133	−0.168067	−	0.985776i	$-0.553752\pi$
$941$	−9702.77	−0.336133	−0.168067	+	0.985776i	$0.553752\pi$
$942$	13949.3	0.482477
$943$	1490.18	0.0514600
$944$	332.959	0.0114798
$945$	−2786.18	−0.0959094
$946$	−6153.65	−0.211493
$947$	−699.579	−0.0240055	−0.0120028	−	0.999928i	$-0.503821\pi$
$947$	−699.579	−0.0240055	−0.0120028	+	0.999928i	$0.503821\pi$
$948$	32698.0	1.12024
$949$	−5499.49	−0.188115
$950$	7013.05	0.239509
$951$	13061.2	0.445363
$952$	35369.3	1.20412
$953$	42039.3	1.42895	0.714473	−	0.699663i	$-0.246667\pi$
$953$	42039.3	1.42895	0.714473	+	0.699663i	$0.246667\pi$
$954$	−22441.6	−0.761606
$955$	−13801.8	−0.467659
$956$	52778.1	1.78553
$957$	1038.39	0.0350745
$958$	52063.4	1.75584
$959$	−34524.9	−1.16253
$960$	12355.0	0.415369
$961$	29262.0	0.982243
$962$	4705.55	0.157706
$963$	10275.3	0.343841
$964$	−36509.3	−1.21980
$965$	−6253.05	−0.208594
$966$	3886.27	0.129440
$967$	−32794.8	−1.09060	−0.545299	−	0.838242i	$-0.683584\pi$
$967$	−32794.8	−1.09060	−0.545299	+	0.838242i	$0.683584\pi$
$968$	2842.55	0.0943832
$969$	−13361.2	−0.442955
$970$	−9801.10	−0.324427
$971$	−3322.53	−0.109810	−0.0549048	−	0.998492i	$-0.517486\pi$
$971$	−3322.53	−0.109810	−0.0549048	+	0.998492i	$0.517486\pi$
$972$	−3186.39	−0.105148
$973$	52429.3	1.72745
$974$	87701.4	2.88515
$975$	1174.38	0.0385747
$976$	−272.550	−0.00893864
$977$	22192.5	0.726716	0.363358	−	0.931650i	$-0.381630\pi$
$977$	22192.5	0.726716	0.363358	+	0.931650i	$0.381630\pi$
$978$	38822.4	1.26933
$979$	16760.7	0.547164
$980$	5437.92	0.177253
$981$	−20041.6	−0.652272
$982$	−37277.4	−1.21137
$983$	7383.09	0.239556	0.119778	−	0.992801i	$-0.461782\pi$
$983$	7383.09	0.239556	0.119778	+	0.992801i	$0.461782\pi$
$984$	7688.04	0.249071
$985$	−719.953	−0.0232889
$986$	−10547.4	−0.340668
$987$	32166.7	1.03736
$988$	−12535.3	−0.403645
$989$	1663.16	0.0534735
$990$	2274.45	0.0730171
$991$	46260.8	1.48287	0.741434	−	0.671026i	$-0.234146\pi$
$991$	46260.8	1.48287	0.741434	+	0.671026i	$0.234146\pi$
$992$	42274.3	1.35304
$993$	−16162.9	−0.516530
$994$	79413.1	2.53403
$995$	3806.25	0.121273
$996$	−54645.9	−1.73847
$997$	41196.8	1.30864	0.654320	−	0.756217i	$-0.272955\pi$
$997$	41196.8	1.30864	0.654320	+	0.756217i	$0.272955\pi$
$998$	84219.5	2.67127
$999$	1765.85	0.0559249

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.4.a.g.1.3	✓	3
3.2	odd	2		495.4.a.i.1.1		3
5.2	odd	4		825.4.c.m.199.6		6
5.3	odd	4		825.4.c.m.199.1		6
5.4	even	2		825.4.a.p.1.1		3
11.10	odd	2		1815.4.a.q.1.1		3
15.14	odd	2		2475.4.a.z.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.g.1.3	✓	3	1.1	even	1	trivial
495.4.a.i.1.1		3	3.2	odd	2
825.4.a.p.1.1		3	5.4	even	2
825.4.c.m.199.1		6	5.3	odd	4
825.4.c.m.199.6		6	5.2	odd	4
1815.4.a.q.1.1		3	11.10	odd	2
2475.4.a.z.1.3		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)

\(f(q)\)	\(=\)	\(q+4.59486 q^{2} -3.00000 q^{3} +13.1127 q^{4} +5.00000 q^{5} -13.7846 q^{6} +20.6383 q^{7} +23.4921 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.59486 q^{2} -3.00000 q^{3} +13.1127 q^{4} +5.00000 q^{5} -13.7846 q^{6} +20.6383 q^{7} +23.4921 q^{8} +9.00000 q^{9} +22.9743 q^{10} +11.0000 q^{11} -39.3381 q^{12} -15.6584 q^{13} +94.8302 q^{14} -15.0000 q^{15} +3.04132 q^{16} +72.9507 q^{17} +41.3537 q^{18} +61.0513 q^{19} +65.5635 q^{20} -61.9150 q^{21} +50.5434 q^{22} -13.6605 q^{23} -70.4764 q^{24} +25.0000 q^{25} -71.9483 q^{26} -27.0000 q^{27} +270.624 q^{28} -31.4663 q^{29} -68.9228 q^{30} -243.008 q^{31} -173.963 q^{32} -33.0000 q^{33} +335.198 q^{34} +103.192 q^{35} +118.014 q^{36} -65.4018 q^{37} +280.522 q^{38} +46.9753 q^{39} +117.461 q^{40} -109.087 q^{41} -284.491 q^{42} -121.750 q^{43} +144.240 q^{44} +45.0000 q^{45} -62.7678 q^{46} -519.530 q^{47} -9.12396 q^{48} +82.9413 q^{49} +114.871 q^{50} -218.852 q^{51} -205.324 q^{52} -542.673 q^{53} -124.061 q^{54} +55.0000 q^{55} +484.839 q^{56} -183.154 q^{57} -144.583 q^{58} +109.478 q^{59} -196.691 q^{60} -89.6156 q^{61} -1116.59 q^{62} +185.745 q^{63} -823.664 q^{64} -78.2922 q^{65} -151.630 q^{66} +488.446 q^{67} +956.581 q^{68} +40.9814 q^{69} +474.151 q^{70} +837.423 q^{71} +211.429 q^{72} +351.216 q^{73} -300.512 q^{74} -75.0000 q^{75} +800.547 q^{76} +227.022 q^{77} +215.845 q^{78} -831.205 q^{79} +15.2066 q^{80} +81.0000 q^{81} -501.238 q^{82} +1389.13 q^{83} -811.873 q^{84} +364.754 q^{85} -559.423 q^{86} +94.3988 q^{87} +258.413 q^{88} +1523.70 q^{89} +206.769 q^{90} -323.164 q^{91} -179.125 q^{92} +729.025 q^{93} -2387.17 q^{94} +305.256 q^{95} +521.888 q^{96} -426.612 q^{97} +381.103 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9}+O(q^{10}) \) 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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