LMFDB - Embedded newform 2209.4.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2209,4,Mod(1,2209)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2209, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2209.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2209 = 47^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2209.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:	$130.335219203$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 47)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.11903$ of defining polynomial
Character	$\chi$	$=$	2209.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.60930 q^{2} +1.72833 q^{3} -5.41015 q^{4} +9.01945 q^{5} -2.78140 q^{6} -11.3182 q^{7} +21.5810 q^{8} -24.0129 q^{9} +O(q^{10})$	\(q-1.60930 q^{2} +1.72833 q^{3} -5.41015 q^{4} +9.01945 q^{5} -2.78140 q^{6} -11.3182 q^{7} +21.5810 q^{8} -24.0129 q^{9} -14.5150 q^{10} +40.6469 q^{11} -9.35051 q^{12} +12.3983 q^{13} +18.2143 q^{14} +15.5886 q^{15} +8.55095 q^{16} +59.6717 q^{17} +38.6440 q^{18} -26.4089 q^{19} -48.7966 q^{20} -19.5615 q^{21} -65.4131 q^{22} -107.097 q^{23} +37.2990 q^{24} -43.6495 q^{25} -19.9526 q^{26} -88.1670 q^{27} +61.2330 q^{28} -173.657 q^{29} -25.0867 q^{30} +332.301 q^{31} -186.409 q^{32} +70.2512 q^{33} -96.0296 q^{34} -102.084 q^{35} +129.913 q^{36} -172.339 q^{37} +42.4998 q^{38} +21.4283 q^{39} +194.648 q^{40} +178.190 q^{41} +31.4804 q^{42} -63.2928 q^{43} -219.906 q^{44} -216.583 q^{45} +172.351 q^{46} +14.7788 q^{48} -214.899 q^{49} +70.2452 q^{50} +103.132 q^{51} -67.0767 q^{52} -402.256 q^{53} +141.887 q^{54} +366.613 q^{55} -244.257 q^{56} -45.6432 q^{57} +279.466 q^{58} +305.280 q^{59} -84.3365 q^{60} -86.2464 q^{61} -534.772 q^{62} +271.782 q^{63} +231.580 q^{64} +111.826 q^{65} -113.055 q^{66} +681.333 q^{67} -322.833 q^{68} -185.098 q^{69} +164.283 q^{70} +726.348 q^{71} -518.221 q^{72} +79.8711 q^{73} +277.345 q^{74} -75.4406 q^{75} +142.876 q^{76} -460.049 q^{77} -34.4846 q^{78} +279.707 q^{79} +77.1248 q^{80} +495.966 q^{81} -286.761 q^{82} +556.598 q^{83} +105.831 q^{84} +538.206 q^{85} +101.857 q^{86} -300.135 q^{87} +877.200 q^{88} -342.069 q^{89} +348.547 q^{90} -140.326 q^{91} +579.410 q^{92} +574.325 q^{93} -238.194 q^{95} -322.175 q^{96} -1637.35 q^{97} +345.837 q^{98} -976.050 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + 6 q^{5} - 8 q^{6} - 45 q^{7} - 39 q^{8} - 42 q^{9}+O(q^{10}) $ 3 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + 6 * q^5 - 8 * q^6 - 45 * q^7 - 39 * q^8 - 42 * q^9	\( 3 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + 6 q^{5} - 8 q^{6} - 45 q^{7} - 39 q^{8} - 42 q^{9} + 40 q^{10} - 2 q^{11} + 12 q^{12} + 80 q^{13} + 162 q^{14} - 14 q^{15} + 89 q^{16} - 39 q^{17} + 181 q^{18} + 24 q^{19} - 232 q^{20} + 24 q^{21} + 14 q^{22} - 120 q^{23} + 192 q^{24} - 171 q^{25} - 316 q^{26} + 64 q^{27} - 408 q^{28} + 184 q^{29} - 116 q^{30} + 4 q^{31} - 7 q^{32} + 208 q^{33} + 218 q^{34} + 156 q^{35} - 343 q^{36} - 589 q^{37} - 42 q^{38} - 60 q^{39} + 432 q^{40} + 92 q^{41} - 54 q^{42} + 250 q^{43} - 466 q^{44} + 78 q^{45} + 816 q^{46} - 120 q^{48} + 30 q^{49} + 137 q^{50} + 317 q^{51} + 900 q^{52} + 459 q^{53} + 106 q^{54} + 448 q^{55} + 1032 q^{56} - 216 q^{57} - 684 q^{58} + 579 q^{59} + 240 q^{60} + 267 q^{61} + 244 q^{62} + 1044 q^{63} - 87 q^{64} - 424 q^{65} - 16 q^{66} + 540 q^{67} - 1334 q^{68} - 642 q^{69} - 1236 q^{70} + 749 q^{71} + 357 q^{72} + 1924 q^{73} + 950 q^{74} + 473 q^{75} + 402 q^{76} + 288 q^{77} + 152 q^{78} + 805 q^{79} - 448 q^{80} + 291 q^{81} + 938 q^{82} + 712 q^{83} + 372 q^{84} + 1038 q^{85} + 1294 q^{86} - 1216 q^{87} + 2190 q^{88} + 835 q^{89} - 764 q^{90} - 2040 q^{91} - 1596 q^{92} + 1500 q^{93} - 312 q^{95} - 1432 q^{96} - 2243 q^{97} - 2989 q^{98} - 554 q^{99}+O(q^{100}) \) 3 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + 6 * q^5 - 8 * q^6 - 45 * q^7 - 39 * q^8 - 42 * q^9 + 40 * q^10 - 2 * q^11 + 12 * q^12 + 80 * q^13 + 162 * q^14 - 14 * q^15 + 89 * q^16 - 39 * q^17 + 181 * q^18 + 24 * q^19 - 232 * q^20 + 24 * q^21 + 14 * q^22 - 120 * q^23 + 192 * q^24 - 171 * q^25 - 316 * q^26 + 64 * q^27 - 408 * q^28 + 184 * q^29 - 116 * q^30 + 4 * q^31 - 7 * q^32 + 208 * q^33 + 218 * q^34 + 156 * q^35 - 343 * q^36 - 589 * q^37 - 42 * q^38 - 60 * q^39 + 432 * q^40 + 92 * q^41 - 54 * q^42 + 250 * q^43 - 466 * q^44 + 78 * q^45 + 816 * q^46 - 120 * q^48 + 30 * q^49 + 137 * q^50 + 317 * q^51 + 900 * q^52 + 459 * q^53 + 106 * q^54 + 448 * q^55 + 1032 * q^56 - 216 * q^57 - 684 * q^58 + 579 * q^59 + 240 * q^60 + 267 * q^61 + 244 * q^62 + 1044 * q^63 - 87 * q^64 - 424 * q^65 - 16 * q^66 + 540 * q^67 - 1334 * q^68 - 642 * q^69 - 1236 * q^70 + 749 * q^71 + 357 * q^72 + 1924 * q^73 + 950 * q^74 + 473 * q^75 + 402 * q^76 + 288 * q^77 + 152 * q^78 + 805 * q^79 - 448 * q^80 + 291 * q^81 + 938 * q^82 + 712 * q^83 + 372 * q^84 + 1038 * q^85 + 1294 * q^86 - 1216 * q^87 + 2190 * q^88 + 835 * q^89 - 764 * q^90 - 2040 * q^91 - 1596 * q^92 + 1500 * q^93 - 312 * q^95 - 1432 * q^96 - 2243 * q^97 - 2989 * q^98 - 554 * 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 $

$2$	−1.60930	−0.568974	−0.284487	−	0.958680i	$-0.591823\pi$
$2$	−1.60930	−0.568974	−0.284487	+	0.958680i	$0.591823\pi$
$3$	1.72833	0.332617	0.166308	−	0.986074i	$-0.446815\pi$
$3$	1.72833	0.332617	0.166308	+	0.986074i	$0.446815\pi$
$4$	−5.41015	−0.676269
$5$	9.01945	0.806724	0.403362	−	0.915040i	$-0.367841\pi$
$5$	9.01945	0.806724	0.403362	+	0.915040i	$0.367841\pi$
$6$	−2.78140	−0.189250
$7$	−11.3182	−0.611124	−0.305562	−	0.952172i	$-0.598844\pi$
$7$	−11.3182	−0.611124	−0.305562	+	0.952172i	$0.598844\pi$
$8$	21.5810	0.953753
$9$	−24.0129	−0.889366
$10$	−14.5150	−0.459005
$11$	40.6469	1.11414	0.557069	−	0.830467i	$-0.311926\pi$
$11$	40.6469	1.11414	0.557069	+	0.830467i	$0.311926\pi$
$12$	−9.35051	−0.224938
$13$	12.3983	0.264513	0.132257	−	0.991216i	$-0.457778\pi$
$13$	12.3983	0.264513	0.132257	+	0.991216i	$0.457778\pi$
$14$	18.2143	0.347714
$15$	15.5886	0.268330
$16$	8.55095	0.133609
$17$	59.6717	0.851324	0.425662	−	0.904882i	$-0.360041\pi$
$17$	59.6717	0.851324	0.425662	+	0.904882i	$0.360041\pi$
$18$	38.6440	0.506026
$19$	−26.4089	−0.318874	−0.159437	−	0.987208i	$-0.550968\pi$
$19$	−26.4089	−0.318874	−0.159437	+	0.987208i	$0.550968\pi$
$20$	−48.7966	−0.545563
$21$	−19.5615	−0.203270
$22$	−65.4131	−0.633915
$23$	−107.097	−0.970923	−0.485461	−	0.874258i	$-0.661348\pi$
$23$	−107.097	−0.970923	−0.485461	+	0.874258i	$0.661348\pi$
$24$	37.2990	0.317234
$25$	−43.6495	−0.349196
$26$	−19.9526	−0.150501
$27$	−88.1670	−0.628435
$28$	61.2330	0.413284
$29$	−173.657	−1.11197	−0.555987	−	0.831191i	$-0.687659\pi$
$29$	−173.657	−1.11197	−0.555987	+	0.831191i	$0.687659\pi$
$30$	−25.0867	−0.152673
$31$	332.301	1.92526	0.962629	−	0.270823i	$-0.0872957\pi$
$31$	332.301	1.92526	0.962629	+	0.270823i	$0.0872957\pi$
$32$	−186.409	−1.02977
$33$	70.2512	0.370581
$34$	−96.0296	−0.484381
$35$	−102.084	−0.493009
$36$	129.913	0.601451
$37$	−172.339	−0.765739	−0.382869	−	0.923802i	$-0.625064\pi$
$37$	−172.339	−0.765739	−0.382869	+	0.923802i	$0.625064\pi$
$38$	42.4998	0.181431
$39$	21.4283	0.0879815
$40$	194.648	0.769416
$41$	178.190	0.678746	0.339373	−	0.940652i	$-0.389785\pi$
$41$	178.190	0.678746	0.339373	+	0.940652i	$0.389785\pi$
$42$	31.4804	0.115655
$43$	−63.2928	−0.224467	−0.112233	−	0.993682i	$-0.535800\pi$
$43$	−63.2928	−0.224467	−0.112233	+	0.993682i	$0.535800\pi$
$44$	−219.906	−0.753456
$45$	−216.583	−0.717473
$46$	172.351	0.552429
$47$	0	0
$47$	0	0
$48$	14.7788	0.0444404
$49$	−214.899	−0.626527
$50$	70.2452	0.198683
$51$	103.132	0.283165
$52$	−67.0767	−0.178882
$53$	−402.256	−1.04253	−0.521266	−	0.853395i	$-0.674540\pi$
$53$	−402.256	−1.04253	−0.521266	+	0.853395i	$0.674540\pi$
$54$	141.887	0.357563
$55$	366.613	0.898801
$56$	−244.257	−0.582861
$57$	−45.6432	−0.106063
$58$	279.466	0.632684
$59$	305.280	0.673628	0.336814	−	0.941571i	$-0.390651\pi$
$59$	305.280	0.673628	0.336814	+	0.941571i	$0.390651\pi$
$60$	−84.3365	−0.181463
$61$	−86.2464	−0.181028	−0.0905141	−	0.995895i	$-0.528851\pi$
$61$	−86.2464	−0.181028	−0.0905141	+	0.995895i	$0.528851\pi$
$62$	−534.772	−1.09542
$63$	271.782	0.543513
$64$	231.580	0.452305
$65$	111.826	0.213389
$66$	−113.055	−0.210851
$67$	681.333	1.24236	0.621179	−	0.783668i	$-0.286654\pi$
$67$	681.333	1.24236	0.621179	+	0.783668i	$0.286654\pi$
$68$	−322.833	−0.575724
$69$	−185.098	−0.322945
$70$	164.283	0.280509
$71$	726.348	1.21411	0.607054	−	0.794661i	$-0.292351\pi$
$71$	726.348	1.21411	0.607054	+	0.794661i	$0.292351\pi$
$72$	−518.221	−0.848236
$73$	79.8711	0.128058	0.0640288	−	0.997948i	$-0.479605\pi$
$73$	79.8711	0.128058	0.0640288	+	0.997948i	$0.479605\pi$
$74$	277.345	0.435685
$75$	−75.4406	−0.116148
$76$	142.876	0.215645
$77$	−460.049	−0.680876
$78$	−34.4846	−0.0500591
$79$	279.707	0.398348	0.199174	−	0.979964i	$-0.436174\pi$
$79$	279.707	0.398348	0.199174	+	0.979964i	$0.436174\pi$
$80$	77.1248	0.107785
$81$	495.966	0.680338
$82$	−286.761	−0.386189
$83$	556.598	0.736080	0.368040	−	0.929810i	$-0.380029\pi$
$83$	556.598	0.736080	0.368040	+	0.929810i	$0.380029\pi$
$84$	105.831	0.137465
$85$	538.206	0.686784
$86$	101.857	0.127716
$87$	−300.135	−0.369861
$88$	877.200	1.06261
$89$	−342.069	−0.407408	−0.203704	−	0.979033i	$-0.565298\pi$
$89$	−342.069	−0.407408	−0.203704	+	0.979033i	$0.565298\pi$
$90$	348.547	0.408223
$91$	−140.326	−0.161650
$92$	579.410	0.656605
$93$	574.325	0.640373
$94$	0	0
$95$	−238.194	−0.257244
$96$	−322.175	−0.342520
$97$	−1637.35	−1.71390	−0.856949	−	0.515402i	$-0.827643\pi$
$97$	−1637.35	−1.71390	−0.856949	+	0.515402i	$0.827643\pi$
$98$	345.837	0.356478
$99$	−976.050	−0.990876
$100$	236.150	0.236150
$101$	−1453.29	−1.43176	−0.715879	−	0.698224i	$-0.753974\pi$
$101$	−1453.29	−1.43176	−0.715879	+	0.698224i	$0.753974\pi$
$102$	−165.971	−0.161113
$103$	1363.10	1.30398	0.651992	−	0.758226i	$-0.273933\pi$
$103$	1363.10	1.30398	0.651992	+	0.758226i	$0.273933\pi$
$104$	267.567	0.252280
$105$	−176.434	−0.163983
$106$	647.351	0.593173
$107$	−274.746	−0.248231	−0.124116	−	0.992268i	$-0.539609\pi$
$107$	−274.746	−0.248231	−0.124116	+	0.992268i	$0.539609\pi$
$108$	476.997	0.424991
$109$	−1448.56	−1.27291	−0.636454	−	0.771315i	$-0.719599\pi$
$109$	−1448.56	−1.27291	−0.636454	+	0.771315i	$0.719599\pi$
$110$	−589.990	−0.511394
$111$	−297.858	−0.254698
$112$	−96.7811	−0.0816514
$113$	1591.30	1.32475	0.662376	−	0.749171i	$-0.269548\pi$
$113$	1591.30	1.32475	0.662376	+	0.749171i	$0.269548\pi$
$114$	73.4536	0.0603470
$115$	−965.954	−0.783267
$116$	939.509	0.751993
$117$	−297.719	−0.235249
$118$	−491.287	−0.383277
$119$	−675.374	−0.520264
$120$	336.416	0.255921
$121$	321.172	0.241301
$122$	138.796	0.103000
$123$	307.970	0.225762
$124$	−1797.80	−1.30199
$125$	−1521.13	−1.08843
$126$	−437.379	−0.309245
$127$	−747.668	−0.522400	−0.261200	−	0.965285i	$-0.584118\pi$
$127$	−747.668	−0.522400	−0.261200	+	0.965285i	$0.584118\pi$
$128$	1118.59	0.772423
$129$	−109.391	−0.0746614
$130$	−179.961	−0.121413
$131$	−624.528	−0.416529	−0.208264	−	0.978073i	$-0.566781\pi$
$131$	−624.528	−0.416529	−0.208264	+	0.978073i	$0.566781\pi$
$132$	−380.070	−0.250612
$133$	298.900	0.194872
$134$	−1096.47	−0.706869
$135$	−795.218	−0.506974
$136$	1287.77	0.811953
$137$	336.418	0.209796	0.104898	−	0.994483i	$-0.466548\pi$
$137$	336.418	0.209796	0.104898	+	0.994483i	$0.466548\pi$
$138$	297.879	0.183747
$139$	−1940.28	−1.18397	−0.591987	−	0.805948i	$-0.701656\pi$
$139$	−1940.28	−1.18397	−0.591987	+	0.805948i	$0.701656\pi$
$140$	552.288	0.333406
$141$	0	0
$142$	−1168.91	−0.690795
$143$	503.953	0.294704
$144$	−205.333	−0.118827
$145$	−1566.29	−0.897056
$146$	−128.537	−0.0728614
$147$	−371.416	−0.208394
$148$	932.380	0.517845
$149$	2810.71	1.54538	0.772692	−	0.634781i	$-0.218910\pi$
$149$	2810.71	1.54538	0.772692	+	0.634781i	$0.218910\pi$
$150$	121.407	0.0660854
$151$	−1710.82	−0.922014	−0.461007	−	0.887396i	$-0.652512\pi$
$151$	−1710.82	−0.922014	−0.461007	+	0.887396i	$0.652512\pi$
$152$	−569.929	−0.304127
$153$	−1432.89	−0.757138
$154$	740.357	0.387401
$155$	2997.17	1.55315
$156$	−115.930	−0.0594991
$157$	−1378.36	−0.700669	−0.350334	−	0.936625i	$-0.613932\pi$
$157$	−1378.36	−0.700669	−0.350334	+	0.936625i	$0.613932\pi$
$158$	−450.133	−0.226650
$159$	−695.231	−0.346763
$160$	−1681.30	−0.830743
$161$	1212.14	0.593354
$162$	−798.159	−0.387095
$163$	3255.46	1.56434	0.782170	−	0.623066i	$-0.214113\pi$
$163$	3255.46	1.56434	0.782170	+	0.623066i	$0.214113\pi$
$164$	−964.034	−0.459015
$165$	633.627	0.298956
$166$	−895.734	−0.418810
$167$	2337.08	1.08293	0.541463	−	0.840725i	$-0.317871\pi$
$167$	2337.08	1.08293	0.541463	+	0.840725i	$0.317871\pi$
$168$	−422.156	−0.193869
$169$	−2043.28	−0.930033
$170$	−866.135	−0.390762
$171$	634.153	0.283596
$172$	342.424	0.151800
$173$	−3269.43	−1.43682	−0.718412	−	0.695618i	$-0.755130\pi$
$173$	−3269.43	−1.43682	−0.718412	+	0.695618i	$0.755130\pi$
$174$	483.008	0.210441
$175$	494.033	0.213402
$176$	347.570	0.148858
$177$	527.624	0.224060
$178$	550.492	0.231804
$179$	2203.96	0.920290	0.460145	−	0.887844i	$-0.347797\pi$
$179$	2203.96	0.920290	0.460145	+	0.887844i	$0.347797\pi$
$180$	1171.75	0.485205
$181$	−1522.63	−0.625282	−0.312641	−	0.949871i	$-0.601214\pi$
$181$	−1522.63	−0.625282	−0.312641	+	0.949871i	$0.601214\pi$
$182$	225.827	0.0919748
$183$	−149.062	−0.0602130
$184$	−2311.25	−0.926020
$185$	−1554.40	−0.617740
$186$	−924.261	−0.364356
$187$	2425.47	0.948491
$188$	0	0
$189$	997.889	0.384052
$190$	383.325	0.146365
$191$	−2019.27	−0.764968	−0.382484	−	0.923962i	$-0.624931\pi$
$191$	−2019.27	−0.764968	−0.382484	+	0.923962i	$0.624931\pi$
$192$	400.246	0.150444
$193$	4447.95	1.65891	0.829456	−	0.558571i	$-0.188650\pi$
$193$	4447.95	1.65891	0.829456	+	0.558571i	$0.188650\pi$
$194$	2634.99	0.975163
$195$	193.272	0.0709768
$196$	1162.64	0.423701
$197$	1790.04	0.647388	0.323694	−	0.946162i	$-0.395075\pi$
$197$	1790.04	0.647388	0.323694	+	0.946162i	$0.395075\pi$
$198$	1570.76	0.563782
$199$	−3481.42	−1.24016	−0.620079	−	0.784540i	$-0.712899\pi$
$199$	−3481.42	−1.24016	−0.620079	+	0.784540i	$0.712899\pi$
$200$	−941.998	−0.333047
$201$	1177.57	0.413229
$202$	2338.78	0.814633
$203$	1965.48	0.679554
$204$	−557.961	−0.191495
$205$	1607.18	0.547561
$206$	−2193.64	−0.741933
$207$	2571.70	0.863506
$208$	106.017	0.0353412
$209$	−1073.44	−0.355270
$210$	283.936	0.0933020
$211$	−5216.72	−1.70206	−0.851028	−	0.525120i	$-0.824021\pi$
$211$	−5216.72	−1.70206	−0.851028	+	0.525120i	$0.824021\pi$
$212$	2176.27	0.705031
$213$	1255.37	0.403833
$214$	442.150	0.141237
$215$	−570.867	−0.181083
$216$	−1902.73	−0.599372
$217$	−3761.04	−1.17657
$218$	2331.17	0.724251
$219$	138.043	0.0425941
$220$	−1983.43	−0.607831
$221$	739.827	0.225186
$222$	479.343	0.144916
$223$	−5075.44	−1.52411	−0.762055	−	0.647512i	$-0.775810\pi$
$223$	−5075.44	−1.52411	−0.762055	+	0.647512i	$0.775810\pi$
$224$	2109.81	0.629319
$225$	1048.15	0.310563
$226$	−2560.88	−0.753749
$227$	−1752.66	−0.512460	−0.256230	−	0.966616i	$-0.582480\pi$
$227$	−1752.66	−0.512460	−0.256230	+	0.966616i	$0.582480\pi$
$228$	246.936	0.0717270
$229$	4676.77	1.34956	0.674781	−	0.738018i	$-0.264238\pi$
$229$	4676.77	1.34956	0.674781	+	0.738018i	$0.264238\pi$
$230$	1554.51	0.445658
$231$	−795.115	−0.226471
$232$	−3747.68	−1.06055
$233$	−4261.89	−1.19831	−0.599154	−	0.800634i	$-0.704496\pi$
$233$	−4261.89	−1.19831	−0.599154	+	0.800634i	$0.704496\pi$
$234$	479.119	0.133850
$235$	0	0
$236$	−1651.61	−0.455554
$237$	483.426	0.132497
$238$	1086.88	0.296017
$239$	−3227.38	−0.873479	−0.436740	−	0.899588i	$-0.643867\pi$
$239$	−3227.38	−0.873479	−0.436740	+	0.899588i	$0.643867\pi$
$240$	133.297	0.0358512
$241$	−1310.37	−0.350242	−0.175121	−	0.984547i	$-0.556032\pi$
$241$	−1310.37	−0.350242	−0.175121	+	0.984547i	$0.556032\pi$
$242$	−516.863	−0.137294
$243$	3237.70	0.854727
$244$	466.606	0.122424
$245$	−1938.27	−0.505435
$246$	−495.617	−0.128453
$247$	−327.425	−0.0843464
$248$	7171.38	1.83622
$249$	961.984	0.244832
$250$	2447.95	0.619288
$251$	4906.61	1.23387	0.616937	−	0.787013i	$-0.288373\pi$
$251$	4906.61	1.23387	0.616937	+	0.787013i	$0.288373\pi$
$252$	−1470.38	−0.367561
$253$	−4353.15	−1.08174
$254$	1203.22	0.297232
$255$	930.196	0.228436
$256$	−3652.79	−0.891793
$257$	−2103.21	−0.510485	−0.255243	−	0.966877i	$-0.582155\pi$
$257$	−2103.21	−0.510485	−0.255243	+	0.966877i	$0.582155\pi$
$258$	176.043	0.0424804
$259$	1950.56	0.467962
$260$	−604.995	−0.144308
$261$	4170.00	0.988951
$262$	1005.05	0.236994
$263$	−6993.90	−1.63978	−0.819890	−	0.572521i	$-0.805965\pi$
$263$	−6993.90	−1.63978	−0.819890	+	0.572521i	$0.805965\pi$
$264$	1516.09	0.353442
$265$	−3628.13	−0.841035
$266$	−481.020	−0.110877
$267$	−591.208	−0.135511
$268$	−3686.11	−0.840168
$269$	6466.63	1.46571	0.732857	−	0.680382i	$-0.238186\pi$
$269$	6466.63	1.46571	0.732857	+	0.680382i	$0.238186\pi$
$270$	1279.74	0.288455
$271$	−8154.65	−1.82790	−0.913948	−	0.405831i	$-0.866982\pi$
$271$	−8154.65	−1.82790	−0.913948	+	0.405831i	$0.866982\pi$
$272$	510.249	0.113744
$273$	−242.530	−0.0537676
$274$	−541.397	−0.119369
$275$	−1774.22	−0.389052
$276$	1001.41	0.218398
$277$	3722.71	0.807495	0.403748	−	0.914870i	$-0.367707\pi$
$277$	3722.71	0.807495	0.403748	+	0.914870i	$0.367707\pi$
$278$	3122.49	0.673650
$279$	−7979.50	−1.71226
$280$	−2203.07	−0.470208
$281$	6046.33	1.28361	0.641804	−	0.766869i	$-0.278186\pi$
$281$	6046.33	1.28361	0.641804	+	0.766869i	$0.278186\pi$
$282$	0	0
$283$	268.000	0.0562931	0.0281466	−	0.999604i	$-0.491039\pi$
$283$	268.000	0.0562931	0.0281466	+	0.999604i	$0.491039\pi$
$284$	−3929.65	−0.821063
$285$	−411.676	−0.0855635
$286$	−811.011	−0.167679
$287$	−2016.78	−0.414798
$288$	4476.21	0.915845
$289$	−1352.29	−0.275248
$290$	2520.63	0.510401
$291$	−2829.88	−0.570071
$292$	−432.115	−0.0866014
$293$	−2731.03	−0.544535	−0.272267	−	0.962222i	$-0.587774\pi$
$293$	−2731.03	−0.544535	−0.272267	+	0.962222i	$0.587774\pi$
$294$	597.719	0.118570
$295$	2753.46	0.543432
$296$	−3719.24	−0.730326
$297$	−3583.72	−0.700163
$298$	−4523.28	−0.879283
$299$	−1327.82	−0.256822
$300$	408.145	0.0785476
$301$	716.360	0.137177
$302$	2753.22	0.524602
$303$	−2511.76	−0.476227
$304$	−225.821	−0.0426043
$305$	−777.896	−0.146040
$306$	2305.95	0.430792
$307$	−7009.57	−1.30312	−0.651559	−	0.758598i	$-0.725885\pi$
$307$	−7009.57	−1.30312	−0.651559	+	0.758598i	$0.725885\pi$
$308$	2488.93	0.460455
$309$	2355.89	0.433727
$310$	−4823.35	−0.883703
$311$	867.418	0.158157	0.0790784	−	0.996868i	$-0.474802\pi$
$311$	867.418	0.158157	0.0790784	+	0.996868i	$0.474802\pi$
$312$	462.444	0.0839126
$313$	−5724.10	−1.03369	−0.516845	−	0.856079i	$-0.672894\pi$
$313$	−5724.10	−1.03369	−0.516845	+	0.856079i	$0.672894\pi$
$314$	2218.19	0.398662
$315$	2451.33	0.438465
$316$	−1513.26	−0.269391
$317$	−5821.24	−1.03140	−0.515700	−	0.856769i	$-0.672468\pi$
$317$	−5821.24	−1.03140	−0.515700	+	0.856769i	$0.672468\pi$
$318$	1118.84	0.197299
$319$	−7058.61	−1.23889
$320$	2088.73	0.364886
$321$	−474.852	−0.0825659
$322$	−1950.70	−0.337603
$323$	−1575.86	−0.271465
$324$	−2683.25	−0.460092
$325$	−541.179	−0.0923669
$326$	−5239.01	−0.890068
$327$	−2503.59	−0.423390
$328$	3845.51	0.647356
$329$	0	0
$330$	−1019.70	−0.170098
$331$	−2300.44	−0.382005	−0.191002	−	0.981590i	$-0.561174\pi$
$331$	−2300.44	−0.382005	−0.191002	+	0.981590i	$0.561174\pi$
$332$	−3011.28	−0.497788
$333$	4138.35	0.681022
$334$	−3761.06	−0.616156
$335$	6145.25	1.00224
$336$	−167.269	−0.0271586
$337$	−6399.32	−1.03440	−0.517201	−	0.855864i	$-0.673026\pi$
$337$	−6399.32	−1.03440	−0.517201	+	0.855864i	$0.673026\pi$
$338$	3288.26	0.529164
$339$	2750.29	0.440635
$340$	−2911.77	−0.464450
$341$	13507.0	2.14500
$342$	−1020.54	−0.161359
$343$	6314.40	0.994010
$344$	−1365.92	−0.214086
$345$	−1669.48	−0.260528
$346$	5261.50	0.817515
$347$	−9188.96	−1.42158	−0.710791	−	0.703403i	$-0.751663\pi$
$347$	−9188.96	−1.42158	−0.710791	+	0.703403i	$0.751663\pi$
$348$	1623.78	0.250125
$349$	−6327.24	−0.970457	−0.485229	−	0.874387i	$-0.661264\pi$
$349$	−6327.24	−0.970457	−0.485229	+	0.874387i	$0.661264\pi$
$350$	−795.047	−0.121420
$351$	−1093.12	−0.166229
$352$	−7576.94	−1.14731
$353$	8212.76	1.23830	0.619152	−	0.785271i	$-0.287477\pi$
$353$	8212.76	1.23830	0.619152	+	0.785271i	$0.287477\pi$
$354$	−849.105	−0.127484
$355$	6551.26	0.979450
$356$	1850.65	0.275517
$357$	−1167.27	−0.173049
$358$	−3546.84	−0.523621
$359$	5614.24	0.825371	0.412685	−	0.910874i	$-0.364591\pi$
$359$	5614.24	0.825371	0.412685	+	0.910874i	$0.364591\pi$
$360$	−4674.07	−0.684292
$361$	−6161.57	−0.898319
$362$	2450.37	0.355769
$363$	555.091	0.0802609
$364$	759.186	0.109319
$365$	720.394	0.103307
$366$	239.886	0.0342596
$367$	3282.91	0.466938	0.233469	−	0.972364i	$-0.424992\pi$
$367$	3282.91	0.466938	0.233469	+	0.972364i	$0.424992\pi$
$368$	−915.778	−0.129724
$369$	−4278.85	−0.603654
$370$	2501.50	0.351478
$371$	4552.81	0.637116
$372$	−3107.18	−0.433065
$373$	9749.67	1.35340	0.676701	−	0.736258i	$-0.263409\pi$
$373$	9749.67	1.35340	0.676701	+	0.736258i	$0.263409\pi$
$374$	−3903.31	−0.539667
$375$	−2629.00	−0.362030
$376$	0	0
$377$	−2153.05	−0.294131
$378$	−1605.90	−0.218515
$379$	−7875.26	−1.06735	−0.533674	−	0.845690i	$-0.679189\pi$
$379$	−7875.26	−1.06735	−0.533674	+	0.845690i	$0.679189\pi$
$380$	1288.66	0.173966
$381$	−1292.21	−0.173759
$382$	3249.61	0.435247
$383$	−1599.65	−0.213416	−0.106708	−	0.994290i	$-0.534031\pi$
$383$	−1599.65	−0.213416	−0.106708	+	0.994290i	$0.534031\pi$
$384$	1933.29	0.256921
$385$	−4149.39	−0.549279
$386$	−7158.08	−0.943878
$387$	1519.84	0.199633
$388$	8858.33	1.15906
$389$	5101.31	0.664901	0.332451	−	0.943121i	$-0.392124\pi$
$389$	5101.31	0.664901	0.332451	+	0.943121i	$0.392124\pi$
$390$	−311.032	−0.0403839
$391$	−6390.64	−0.826569
$392$	−4637.73	−0.597552
$393$	−1079.39	−0.138544
$394$	−2880.72	−0.368347
$395$	2522.81	0.321357
$396$	5280.58	0.670098
$397$	−8835.01	−1.11692	−0.558459	−	0.829532i	$-0.688607\pi$
$397$	−8835.01	−1.11692	−0.558459	+	0.829532i	$0.688607\pi$
$398$	5602.65	0.705617
$399$	516.597	0.0648176
$400$	−373.244	−0.0466555
$401$	−15444.1	−1.92330	−0.961649	−	0.274284i	$-0.911559\pi$
$401$	−15444.1	−1.92330	−0.961649	+	0.274284i	$0.911559\pi$
$402$	−1895.06	−0.235117
$403$	4119.97	0.509256
$404$	7862.51	0.968253
$405$	4473.35	0.548845
$406$	−3163.04	−0.386648
$407$	−7005.05	−0.853138
$408$	2225.69	0.270069
$409$	13309.7	1.60910	0.804552	−	0.593882i	$-0.202405\pi$
$409$	13309.7	1.60910	0.804552	+	0.593882i	$0.202405\pi$
$410$	−2586.43	−0.311548
$411$	581.440	0.0697818
$412$	−7374.59	−0.881844
$413$	−3455.21	−0.411671
$414$	−4138.64	−0.491312
$415$	5020.21	0.593813
$416$	−2311.15	−0.272388
$417$	−3353.44	−0.393809
$418$	1727.49	0.202139
$419$	2074.44	0.241869	0.120934	−	0.992661i	$-0.461411\pi$
$419$	2074.44	0.241869	0.120934	+	0.992661i	$0.461411\pi$
$420$	954.535	0.110897
$421$	−14903.2	−1.72526	−0.862631	−	0.505833i	$-0.831185\pi$
$421$	−14903.2	−1.72526	−0.862631	+	0.505833i	$0.831185\pi$
$422$	8395.27	0.968426
$423$	0	0
$424$	−8681.08	−0.994317
$425$	−2604.64	−0.297279
$426$	−2020.26	−0.229770
$427$	976.152	0.110631
$428$	1486.42	0.167871
$429$	870.995	0.0980234
$430$	918.696	0.103031
$431$	7434.74	0.830902	0.415451	−	0.909616i	$-0.363624\pi$
$431$	7434.74	0.830902	0.415451	+	0.909616i	$0.363624\pi$
$432$	−753.911	−0.0839642
$433$	−4385.47	−0.486725	−0.243363	−	0.969935i	$-0.578251\pi$
$433$	−4385.47	−0.486725	−0.243363	+	0.969935i	$0.578251\pi$
$434$	6052.65	0.669438
$435$	−2707.06	−0.298376
$436$	7836.93	0.860828
$437$	2828.30	0.309602
$438$	−222.153	−0.0242349
$439$	−10300.7	−1.11988	−0.559939	−	0.828534i	$-0.689175\pi$
$439$	−10300.7	−1.11988	−0.559939	+	0.828534i	$0.689175\pi$
$440$	7911.86	0.857235
$441$	5160.34	0.557212
$442$	−1190.60	−0.128125
$443$	−5700.48	−0.611372	−0.305686	−	0.952132i	$-0.598886\pi$
$443$	−5700.48	−0.611372	−0.305686	+	0.952132i	$0.598886\pi$
$444$	1611.46	0.172244
$445$	−3085.28	−0.328666
$446$	8167.91	0.867179
$447$	4857.83	0.514021
$448$	−2621.07	−0.276415
$449$	8946.30	0.940317	0.470158	−	0.882582i	$-0.344197\pi$
$449$	8946.30	0.940317	0.470158	+	0.882582i	$0.344197\pi$
$450$	−1686.79	−0.176702
$451$	7242.87	0.756216
$452$	−8609.18	−0.895889
$453$	−2956.85	−0.306677
$454$	2820.56	0.291576
$455$	−1265.66	−0.130407
$456$	−985.024	−0.101158
$457$	−16498.3	−1.68875	−0.844374	−	0.535755i	$-0.820027\pi$
$457$	−16498.3	−1.68875	−0.844374	+	0.535755i	$0.820027\pi$
$458$	−7526.33	−0.767865
$459$	−5261.07	−0.535001
$460$	5225.96	0.529699
$461$	−9016.08	−0.910891	−0.455445	−	0.890264i	$-0.650520\pi$
$461$	−9016.08	−0.910891	−0.455445	+	0.890264i	$0.650520\pi$
$462$	1279.58	0.128856
$463$	−3241.41	−0.325359	−0.162679	−	0.986679i	$-0.552014\pi$
$463$	−3241.41	−0.325359	−0.162679	+	0.986679i	$0.552014\pi$
$464$	−1484.93	−0.148569
$465$	5180.10	0.516605
$466$	6858.67	0.681806
$467$	−12186.1	−1.20751	−0.603755	−	0.797170i	$-0.706329\pi$
$467$	−12186.1	−1.20751	−0.603755	+	0.797170i	$0.706329\pi$
$468$	1610.70	0.159092
$469$	−7711.44	−0.759235
$470$	0	0
$471$	−2382.25	−0.233054
$472$	6588.24	0.642475
$473$	−2572.66	−0.250087
$474$	−777.977	−0.0753875
$475$	1152.73	0.111350
$476$	3653.88	0.351839
$477$	9659.33	0.927192
$478$	5193.82	0.496987
$479$	−9110.87	−0.869074	−0.434537	−	0.900654i	$-0.643088\pi$
$479$	−9110.87	−0.869074	−0.434537	+	0.900654i	$0.643088\pi$
$480$	−2905.85	−0.276319
$481$	−2136.71	−0.202548
$482$	2108.78	0.199279
$483$	2094.97	0.197360
$484$	−1737.59	−0.163185
$485$	−14768.0	−1.38264
$486$	−5210.43	−0.486317
$487$	535.218	0.0498009	0.0249004	−	0.999690i	$-0.492073\pi$
$487$	535.218	0.0498009	0.0249004	+	0.999690i	$0.492073\pi$
$488$	−1861.28	−0.172656
$489$	5626.50	0.520325
$490$	3119.26	0.287579
$491$	10810.8	0.993658	0.496829	−	0.867848i	$-0.334498\pi$
$491$	10810.8	0.993658	0.496829	+	0.867848i	$0.334498\pi$
$492$	−1666.17	−0.152676
$493$	−10362.4	−0.946649
$494$	526.925	0.0479909
$495$	−8803.43	−0.799364
$496$	2841.49	0.257231
$497$	−8220.93	−0.741970
$498$	−1548.12	−0.139303
$499$	−16736.4	−1.50146	−0.750728	−	0.660612i	$-0.770297\pi$
$499$	−16736.4	−1.50146	−0.750728	+	0.660612i	$0.770297\pi$
$500$	8229.52	0.736071
$501$	4039.24	0.360199
$502$	−7896.21	−0.702042
$503$	−6612.65	−0.586170	−0.293085	−	0.956086i	$-0.594682\pi$
$503$	−6612.65	−0.586170	−0.293085	+	0.956086i	$0.594682\pi$
$504$	5865.32	0.518377
$505$	−13107.9	−1.15503
$506$	7005.53	0.615482
$507$	−3531.46	−0.309345
$508$	4045.00	0.353283
$509$	−14953.5	−1.30216	−0.651081	−	0.759008i	$-0.725684\pi$
$509$	−14953.5	−1.30216	−0.651081	+	0.759008i	$0.725684\pi$
$510$	−1496.96	−0.129974
$511$	−903.995	−0.0782591
$512$	−3070.27	−0.265016
$513$	2328.39	0.200392
$514$	3384.70	0.290453
$515$	12294.4	1.05196
$516$	591.820	0.0504912
$517$	0	0
$518$	−3139.04	−0.266258
$519$	−5650.65	−0.477911
$520$	2413.31	0.203520
$521$	822.169	0.0691361	0.0345680	−	0.999402i	$-0.488994\pi$
$521$	822.169	0.0691361	0.0345680	+	0.999402i	$0.488994\pi$
$522$	−6710.78	−0.562687
$523$	97.2265	0.00812890	0.00406445	−	0.999992i	$-0.498706\pi$
$523$	97.2265	0.00812890	0.00406445	+	0.999992i	$0.498706\pi$
$524$	3378.79	0.281685
$525$	853.850	0.0709811
$526$	11255.3	0.932992
$527$	19828.9	1.63902
$528$	600.714	0.0495127
$529$	−697.284	−0.0573095
$530$	5838.75	0.478527
$531$	−7330.65	−0.599102
$532$	−1617.10	−0.131786
$533$	2209.25	0.179537
$534$	951.431	0.0771020
$535$	−2478.06	−0.200254
$536$	14703.8	1.18490
$537$	3809.17	0.306104
$538$	−10406.7	−0.833953
$539$	−8734.98	−0.698037
$540$	4302.25	0.342851
$541$	−3856.31	−0.306462	−0.153231	−	0.988190i	$-0.548968\pi$
$541$	−3856.31	−0.306462	−0.153231	+	0.988190i	$0.548968\pi$
$542$	13123.3	1.04003
$543$	−2631.60	−0.207979
$544$	−11123.3	−0.876670
$545$	−13065.2	−1.02689
$546$	390.303	0.0305923
$547$	4147.40	0.324186	0.162093	−	0.986775i	$-0.448175\pi$
$547$	4147.40	0.324186	0.162093	+	0.986775i	$0.448175\pi$
$548$	−1820.07	−0.141879
$549$	2071.03	0.161000
$550$	2855.25	0.221360
$551$	4586.07	0.354580
$552$	−3994.60	−0.308010
$553$	−3165.78	−0.243440
$554$	−5990.96	−0.459443
$555$	−2686.52	−0.205471
$556$	10497.2	0.800684
$557$	−85.2280	−0.00648335	−0.00324168	−	0.999995i	$-0.501032\pi$
$557$	−85.2280	−0.00648335	−0.00324168	+	0.999995i	$0.501032\pi$
$558$	12841.4	0.974231
$559$	−784.724	−0.0593744
$560$	−872.913	−0.0658702
$561$	4192.00	0.315484
$562$	−9730.37	−0.730340
$563$	−12824.3	−0.960003	−0.480001	−	0.877268i	$-0.659364\pi$
$563$	−12824.3	−0.960003	−0.480001	+	0.877268i	$0.659364\pi$
$564$	0	0
$565$	14352.7	1.06871
$566$	−431.293	−0.0320293
$567$	−5613.44	−0.415771
$568$	15675.3	1.15796
$569$	12096.8	0.891253	0.445626	−	0.895219i	$-0.352981\pi$
$569$	12096.8	0.891253	0.445626	+	0.895219i	$0.352981\pi$
$570$	662.511	0.0486834
$571$	5688.49	0.416910	0.208455	−	0.978032i	$-0.433156\pi$
$571$	5688.49	0.416910	0.208455	+	0.978032i	$0.433156\pi$
$572$	−2726.46	−0.199299
$573$	−3489.95	−0.254441
$574$	3245.61	0.236009
$575$	4674.72	0.339042
$576$	−5560.91	−0.402265
$577$	9878.78	0.712754	0.356377	−	0.934342i	$-0.384012\pi$
$577$	9878.78	0.712754	0.356377	+	0.934342i	$0.384012\pi$
$578$	2176.25	0.156609
$579$	7687.51	0.551782
$580$	8473.85	0.606651
$581$	−6299.68	−0.449836
$582$	4554.13	0.324355
$583$	−16350.5	−1.16152
$584$	1723.70	0.122135
$585$	−2685.26	−0.189781
$586$	4395.05	0.309826
$587$	17342.9	1.21945	0.609726	−	0.792613i	$-0.291280\pi$
$587$	17342.9	1.21945	0.609726	+	0.792613i	$0.291280\pi$
$588$	2009.41	0.140930
$589$	−8775.69	−0.613915
$590$	−4431.14	−0.309199
$591$	3093.78	0.215332
$592$	−1473.66	−0.102309
$593$	−698.313	−0.0483580	−0.0241790	−	0.999708i	$-0.507697\pi$
$593$	−698.313	−0.0483580	−0.0241790	+	0.999708i	$0.507697\pi$
$594$	5767.28	0.398374
$595$	−6091.51	−0.419710
$596$	−15206.4	−1.04510
$597$	−6017.03	−0.412497
$598$	2136.86	0.146125
$599$	−9549.46	−0.651386	−0.325693	−	0.945476i	$-0.605598\pi$
$599$	−9549.46	−0.651386	−0.325693	+	0.945476i	$0.605598\pi$
$600$	−1628.08	−0.110777
$601$	27369.5	1.85761	0.928805	−	0.370568i	$-0.120837\pi$
$601$	27369.5	1.85761	0.928805	+	0.370568i	$0.120837\pi$
$602$	−1152.84	−0.0780501
$603$	−16360.8	−1.10491
$604$	9255.77	0.623530
$605$	2896.80	0.194664
$606$	4042.17	0.270960
$607$	−18179.8	−1.21564	−0.607821	−	0.794074i	$-0.707956\pi$
$607$	−18179.8	−1.21564	−0.607821	+	0.794074i	$0.707956\pi$
$608$	4922.84	0.328368
$609$	3396.99	0.226031
$610$	1251.87	0.0830929
$611$	0	0
$612$	7752.14	0.512029
$613$	−3148.69	−0.207462	−0.103731	−	0.994605i	$-0.533078\pi$
$613$	−3148.69	−0.207462	−0.103731	+	0.994605i	$0.533078\pi$
$614$	11280.5	0.741440
$615$	2777.72	0.182128
$616$	−9928.30	−0.649387
$617$	−24002.0	−1.56610	−0.783050	−	0.621959i	$-0.786337\pi$
$617$	−24002.0	−1.56610	−0.783050	+	0.621959i	$0.786337\pi$
$618$	−3791.33	−0.246779
$619$	17997.2	1.16861	0.584304	−	0.811535i	$-0.301367\pi$
$619$	17997.2	1.16861	0.584304	+	0.811535i	$0.301367\pi$
$620$	−16215.2	−1.05035
$621$	9442.40	0.610162
$622$	−1395.94	−0.0899871
$623$	3871.60	0.248977
$624$	183.232	0.0117551
$625$	−8263.54	−0.528866
$626$	9211.80	0.588143
$627$	−1855.25	−0.118169
$628$	7457.13	0.473840
$629$	−10283.7	−0.651892
$630$	−3944.92	−0.249475
$631$	−5508.64	−0.347536	−0.173768	−	0.984787i	$-0.555594\pi$
$631$	−5508.64	−0.347536	−0.173768	+	0.984787i	$0.555594\pi$
$632$	6036.35	0.379926
$633$	−9016.20	−0.566133
$634$	9368.13	0.586839
$635$	−6743.55	−0.421433
$636$	3761.30	0.234505
$637$	−2664.38	−0.165725
$638$	11359.4	0.704896
$639$	−17441.7	−1.07979
$640$	10089.1	0.623132
$641$	−11605.9	−0.715142	−0.357571	−	0.933886i	$-0.616395\pi$
$641$	−11605.9	−0.715142	−0.357571	+	0.933886i	$0.616395\pi$
$642$	764.179	0.0469778
$643$	7998.88	0.490583	0.245292	−	0.969449i	$-0.421116\pi$
$643$	7998.88	0.490583	0.245292	+	0.969449i	$0.421116\pi$
$644$	−6557.86	−0.401267
$645$	−986.645	−0.0602312
$646$	2536.03	0.154457
$647$	−5421.64	−0.329438	−0.164719	−	0.986341i	$-0.552672\pi$
$647$	−5421.64	−0.329438	−0.164719	+	0.986341i	$0.552672\pi$
$648$	10703.4	0.648875
$649$	12408.7	0.750514
$650$	870.920	0.0525543
$651$	−6500.31	−0.391348
$652$	−17612.5	−1.05791
$653$	−21419.1	−1.28361	−0.641804	−	0.766869i	$-0.721814\pi$
$653$	−21419.1	−1.28361	−0.641804	+	0.766869i	$0.721814\pi$
$654$	4029.02	0.240898
$655$	−5632.90	−0.336024
$656$	1523.69	0.0906862
$657$	−1917.94	−0.113890
$658$	0	0
$659$	−11435.7	−0.675982	−0.337991	−	0.941149i	$-0.609747\pi$
$659$	−11435.7	−0.675982	−0.337991	+	0.941149i	$0.609747\pi$
$660$	−3428.02	−0.202175
$661$	27123.2	1.59602	0.798011	−	0.602643i	$-0.205886\pi$
$661$	27123.2	1.59602	0.798011	+	0.602643i	$0.205886\pi$
$662$	3702.10	0.217351
$663$	1278.66	0.0749007
$664$	12011.9	0.702038
$665$	2695.92	0.157208
$666$	−6659.86	−0.387484
$667$	18598.1	1.07964
$668$	−12644.0	−0.732349
$669$	−8772.02	−0.506945
$670$	−9889.55	−0.570249
$671$	−3505.65	−0.201690
$672$	3646.44	0.209322
$673$	11097.2	0.635612	0.317806	−	0.948156i	$-0.397054\pi$
$673$	11097.2	0.635612	0.317806	+	0.948156i	$0.397054\pi$
$674$	10298.4	0.588547
$675$	3848.44	0.219447
$676$	11054.5	0.628952
$677$	1647.83	0.0935468	0.0467734	−	0.998906i	$-0.485106\pi$
$677$	1647.83	0.0935468	0.0467734	+	0.998906i	$0.485106\pi$
$678$	−4426.04	−0.250710
$679$	18531.9	1.04740
$680$	11615.0	0.655022
$681$	−3029.18	−0.170453
$682$	−21736.8	−1.22045
$683$	9256.41	0.518575	0.259287	−	0.965800i	$-0.416512\pi$
$683$	9256.41	0.518575	0.259287	+	0.965800i	$0.416512\pi$
$684$	−3430.86	−0.191787
$685$	3034.30	0.169248
$686$	−10161.8	−0.565566
$687$	8082.99	0.448887
$688$	−541.214	−0.0299907
$689$	−4987.29	−0.275763
$690$	2686.70	0.148233
$691$	2404.88	0.132397	0.0661983	−	0.997806i	$-0.478913\pi$
$691$	2404.88	0.132397	0.0661983	+	0.997806i	$0.478913\pi$
$692$	17688.1	0.971679
$693$	11047.1	0.605548
$694$	14787.8	0.808843
$695$	−17500.3	−0.955140
$696$	−6477.21	−0.352756
$697$	10632.9	0.577833
$698$	10182.4	0.552165
$699$	−7365.94	−0.398577
$700$	−2672.79	−0.144317
$701$	−1158.57	−0.0624232	−0.0312116	−	0.999513i	$-0.509937\pi$
$701$	−1158.57	−0.0624232	−0.0312116	+	0.999513i	$0.509937\pi$
$702$	1759.16	0.0945801
$703$	4551.28	0.244174
$704$	9413.02	0.503930
$705$	0	0
$706$	−13216.8	−0.704562
$707$	16448.6	0.874982
$708$	−2854.52	−0.151525
$709$	−26868.0	−1.42320	−0.711600	−	0.702585i	$-0.752029\pi$
$709$	−26868.0	−1.42320	−0.711600	+	0.702585i	$0.752029\pi$
$710$	−10542.9	−0.557281
$711$	−6716.58	−0.354278
$712$	−7382.19	−0.388566
$713$	−35588.4	−1.86928
$714$	1878.49	0.0984601
$715$	4545.38	0.237745
$716$	−11923.8	−0.622363
$717$	−5577.96	−0.290534
$718$	−9035.00	−0.469614
$719$	12313.1	0.638666	0.319333	−	0.947643i	$-0.396541\pi$
$719$	12313.1	0.638666	0.319333	+	0.947643i	$0.396541\pi$
$720$	−1851.99	−0.0958605
$721$	−15427.8	−0.796896
$722$	9915.82	0.511120
$723$	−2264.75	−0.116496
$724$	8237.65	0.422859
$725$	7580.02	0.388296
$726$	−893.308	−0.0456663
$727$	−4104.02	−0.209367	−0.104683	−	0.994506i	$-0.533383\pi$
$727$	−4104.02	−0.209367	−0.104683	+	0.994506i	$0.533383\pi$
$728$	−3028.37	−0.154174
$729$	−7795.29	−0.396042
$730$	−1159.33	−0.0587791
$731$	−3776.79	−0.191094
$732$	806.448	0.0407202
$733$	36113.9	1.81978	0.909888	−	0.414854i	$-0.136167\pi$
$733$	36113.9	1.81978	0.909888	+	0.414854i	$0.136167\pi$
$734$	−5283.18	−0.265675
$735$	−3349.97	−0.168116
$736$	19963.8	0.999829
$737$	27694.1	1.38416
$738$	6885.96	0.343463
$739$	−4185.77	−0.208357	−0.104179	−	0.994559i	$-0.533221\pi$
$739$	−4185.77	−0.208357	−0.104179	+	0.994559i	$0.533221\pi$
$740$	8409.55	0.417759
$741$	−565.898	−0.0280550
$742$	−7326.84	−0.362502
$743$	24708.4	1.22001	0.610003	−	0.792399i	$-0.291168\pi$
$743$	24708.4	1.22001	0.610003	+	0.792399i	$0.291168\pi$
$744$	12394.5	0.610758
$745$	25351.1	1.24670
$746$	−15690.2	−0.770050
$747$	−13365.5	−0.654644
$748$	−13122.2	−0.641435
$749$	3109.63	0.151700
$750$	4230.86	0.205985
$751$	−24154.5	−1.17365	−0.586824	−	0.809714i	$-0.699622\pi$
$751$	−24154.5	−1.17365	−0.586824	+	0.809714i	$0.699622\pi$
$752$	0	0
$753$	8480.22	0.410407
$754$	3464.90	0.167353
$755$	−15430.6	−0.743811
$756$	−5398.73	−0.259722
$757$	29215.5	1.40272	0.701358	−	0.712809i	$-0.252577\pi$
$757$	29215.5	1.40272	0.701358	+	0.712809i	$0.252577\pi$
$758$	12673.7	0.607293
$759$	−7523.67	−0.359805
$760$	−5140.45	−0.245347
$761$	29582.7	1.40916	0.704581	−	0.709624i	$-0.251135\pi$
$761$	29582.7	1.40916	0.704581	+	0.709624i	$0.251135\pi$
$762$	2079.56	0.0988643
$763$	16395.1	0.777905
$764$	10924.5	0.517324
$765$	−12923.9	−0.610802
$766$	2574.32	0.121428
$767$	3784.95	0.178183
$768$	−6313.21	−0.296625
$769$	−30737.1	−1.44136	−0.720682	−	0.693265i	$-0.756171\pi$
$769$	−30737.1	−1.44136	−0.720682	+	0.693265i	$0.756171\pi$
$770$	6677.62	0.312525
$771$	−3635.04	−0.169796
$772$	−24064.1	−1.12187
$773$	−13581.6	−0.631947	−0.315973	−	0.948768i	$-0.602331\pi$
$773$	−13581.6	−0.631947	−0.315973	+	0.948768i	$0.602331\pi$
$774$	−2445.89	−0.113586
$775$	−14504.8	−0.672292
$776$	−35335.7	−1.63463
$777$	3371.21	0.155652
$778$	−8209.54	−0.378311
$779$	−4705.79	−0.216435
$780$	−1045.63	−0.0479994
$781$	29523.8	1.35268
$782$	10284.5	0.470296
$783$	15310.8	0.698803
$784$	−1837.59	−0.0837094
$785$	−12432.0	−0.565246
$786$	1737.06	0.0788281
$787$	37723.8	1.70865	0.854325	−	0.519739i	$-0.173971\pi$
$787$	37723.8	1.70865	0.854325	+	0.519739i	$0.173971\pi$
$788$	−9684.41	−0.437808
$789$	−12087.7	−0.545418
$790$	−4059.95	−0.182844
$791$	−18010.6	−0.809588
$792$	−21064.1	−0.945051
$793$	−1069.31	−0.0478843
$794$	14218.2	0.635497
$795$	−6270.60	−0.279742
$796$	18835.0	0.838680
$797$	6349.61	0.282202	0.141101	−	0.989995i	$-0.454936\pi$
$797$	6349.61	0.282202	0.141101	+	0.989995i	$0.454936\pi$
$798$	−831.361	−0.0368795
$799$	0	0
$800$	8136.65	0.359592
$801$	8214.07	0.362334
$802$	24854.2	1.09431
$803$	3246.51	0.142674
$804$	−6370.81	−0.279454
$805$	10932.8	0.478673
$806$	−6630.27	−0.289753
$807$	11176.4	0.487521
$808$	−31363.4	−1.36554
$809$	−32315.3	−1.40438	−0.702191	−	0.711988i	$-0.747795\pi$
$809$	−32315.3	−1.40438	−0.702191	+	0.711988i	$0.747795\pi$
$810$	−7198.96	−0.312279
$811$	−27000.2	−1.16906	−0.584528	−	0.811373i	$-0.698720\pi$
$811$	−27000.2	−1.16906	−0.584528	+	0.811373i	$0.698720\pi$
$812$	−10633.5	−0.459561
$813$	−14093.9	−0.607989
$814$	11273.2	0.485413
$815$	29362.5	1.26199
$816$	881.877	0.0378332
$817$	1671.49	0.0715766
$818$	−21419.4	−0.915538
$819$	3369.64	0.143766
$820$	−8695.06	−0.370298
$821$	8734.23	0.371287	0.185644	−	0.982617i	$-0.440563\pi$
$821$	8734.23	0.371287	0.185644	+	0.982617i	$0.440563\pi$
$822$	−935.712	−0.0397040
$823$	37165.8	1.57414	0.787070	−	0.616864i	$-0.211597\pi$
$823$	37165.8	1.57414	0.787070	+	0.616864i	$0.211597\pi$
$824$	29417.1	1.24368
$825$	−3066.43	−0.129405
$826$	5560.48	0.234230
$827$	44504.9	1.87133	0.935664	−	0.352892i	$-0.114802\pi$
$827$	44504.9	1.87133	0.935664	+	0.352892i	$0.114802\pi$
$828$	−13913.3	−0.583962
$829$	−8435.36	−0.353404	−0.176702	−	0.984264i	$-0.556543\pi$
$829$	−8435.36	−0.353404	−0.176702	+	0.984264i	$0.556543\pi$
$830$	−8079.03	−0.337864
$831$	6434.07	0.268586
$832$	2871.20	0.119641
$833$	−12823.4	−0.533378
$834$	5396.69	0.224067
$835$	21079.2	0.873623
$836$	5807.47	0.240258
$837$	−29298.0	−1.20990
$838$	−3338.40	−0.137617
$839$	15830.0	0.651384	0.325692	−	0.945476i	$-0.394403\pi$
$839$	15830.0	0.651384	0.325692	+	0.945476i	$0.394403\pi$
$840$	−3807.62	−0.156399
$841$	5767.62	0.236484
$842$	23983.7	0.981629
$843$	10450.0	0.426950
$844$	28223.3	1.15105
$845$	−18429.3	−0.750280
$846$	0	0
$847$	−3635.08	−0.147465
$848$	−3439.67	−0.139291
$849$	463.192	0.0187240
$850$	4191.64	0.169144
$851$	18456.9	0.743473
$852$	−6791.73	−0.273099
$853$	−25517.9	−1.02429	−0.512144	−	0.858900i	$-0.671148\pi$
$853$	−25517.9	−1.02429	−0.512144	+	0.858900i	$0.671148\pi$
$854$	−1570.92	−0.0629460
$855$	5719.71	0.228784
$856$	−5929.29	−0.236751
$857$	−1353.59	−0.0539530	−0.0269765	−	0.999636i	$-0.508588\pi$
$857$	−1353.59	−0.0539530	−0.0269765	+	0.999636i	$0.508588\pi$
$858$	−1401.69	−0.0557728
$859$	−44829.9	−1.78065	−0.890325	−	0.455326i	$-0.849523\pi$
$859$	−44829.9	−1.78065	−0.890325	+	0.455326i	$0.849523\pi$
$860$	3088.48	0.122461
$861$	−3485.66	−0.137969
$862$	−11964.7	−0.472761
$863$	−22467.5	−0.886216	−0.443108	−	0.896468i	$-0.646124\pi$
$863$	−22467.5	−0.886216	−0.443108	+	0.896468i	$0.646124\pi$
$864$	16435.1	0.647145
$865$	−29488.5	−1.15912
$866$	7057.53	0.276934
$867$	−2337.21	−0.0915521
$868$	20347.8	0.795679
$869$	11369.2	0.443815
$870$	4356.47	0.169768
$871$	8447.37	0.328620
$872$	−31261.3	−1.21404
$873$	39317.6	1.52428
$874$	−4551.59	−0.176155
$875$	17216.4	0.665165
$876$	−746.836	−0.0288051
$877$	−408.941	−0.0157457	−0.00787283	−	0.999969i	$-0.502506\pi$
$877$	−408.941	−0.0157457	−0.00787283	+	0.999969i	$0.502506\pi$
$878$	16576.9	0.637181
$879$	−4720.12	−0.181121
$880$	3134.89	0.120088
$881$	20458.5	0.782366	0.391183	−	0.920313i	$-0.372066\pi$
$881$	20458.5	0.782366	0.391183	+	0.920313i	$0.372066\pi$
$882$	−8304.54	−0.317039
$883$	3746.69	0.142793	0.0713965	−	0.997448i	$-0.477254\pi$
$883$	3746.69	0.142793	0.0713965	+	0.997448i	$0.477254\pi$
$884$	−4002.58	−0.152286
$885$	4758.88	0.180755
$886$	9173.78	0.347855
$887$	−5289.38	−0.200225	−0.100113	−	0.994976i	$-0.531920\pi$
$887$	−5289.38	−0.200225	−0.100113	+	0.994976i	$0.531920\pi$
$888$	−6428.06	−0.242919
$889$	8462.24	0.319251
$890$	4965.14	0.187002
$891$	20159.5	0.757990
$892$	27458.9	1.03071
$893$	0	0
$894$	−7817.70	−0.292464
$895$	19878.5	0.742420
$896$	−12660.4	−0.472046
$897$	−2294.90	−0.0854232
$898$	−14397.3	−0.535015
$899$	−57706.3	−2.14084
$900$	−5670.65	−0.210024
$901$	−24003.3	−0.887531
$902$	−11656.0	−0.430267
$903$	1238.10	0.0456274
$904$	34341.8	1.26349
$905$	−13733.3	−0.504430
$906$	4758.46	0.174491
$907$	−10720.8	−0.392480	−0.196240	−	0.980556i	$-0.562873\pi$
$907$	−10720.8	−0.392480	−0.196240	+	0.980556i	$0.562873\pi$
$908$	9482.18	0.346561
$909$	34897.6	1.27336
$910$	2036.84	0.0741983
$911$	3405.66	0.123858	0.0619290	−	0.998081i	$-0.480275\pi$
$911$	3405.66	0.123858	0.0619290	+	0.998081i	$0.480275\pi$
$912$	−390.292	−0.0141709
$913$	22624.0	0.820094
$914$	26550.7	0.960853
$915$	−1344.46	−0.0485753
$916$	−25302.0	−0.912666
$917$	7068.52	0.254551
$918$	8466.64	0.304402
$919$	−3184.34	−0.114300	−0.0571500	−	0.998366i	$-0.518201\pi$
$919$	−3184.34	−0.114300	−0.0571500	+	0.998366i	$0.518201\pi$
$920$	−20846.2	−0.747043
$921$	−12114.8	−0.433439
$922$	14509.6	0.518273
$923$	9005.48	0.321147
$924$	4301.69	0.153155
$925$	7522.51	0.267393
$926$	5216.40	0.185121
$927$	−32732.0	−1.15972
$928$	32371.1	1.14508
$929$	24847.1	0.877510	0.438755	−	0.898607i	$-0.355420\pi$
$929$	24847.1	0.877510	0.438755	+	0.898607i	$0.355420\pi$
$930$	−8336.33	−0.293934
$931$	5675.24	0.199783
$932$	23057.5	0.810379
$933$	1499.18	0.0526056
$934$	19611.2	0.687041
$935$	21876.4	0.765171
$936$	−6425.06	−0.224369
$937$	−16072.8	−0.560381	−0.280191	−	0.959944i	$-0.590398\pi$
$937$	−16072.8	−0.560381	−0.280191	+	0.959944i	$0.590398\pi$
$938$	12410.0	0.431985
$939$	−9893.12	−0.343823
$940$	0	0
$941$	49359.1	1.70995	0.854974	−	0.518672i	$-0.173573\pi$
$941$	49359.1	1.70995	0.854974	+	0.518672i	$0.173573\pi$
$942$	3833.76	0.132602
$943$	−19083.6	−0.659010
$944$	2610.43	0.0900025
$945$	9000.41	0.309824
$946$	4140.18	0.142293
$947$	19624.5	0.673400	0.336700	−	0.941612i	$-0.390689\pi$
$947$	19624.5	0.673400	0.336700	+	0.941612i	$0.390689\pi$
$948$	−2615.41	−0.0896038
$949$	990.266	0.0338729
$950$	−1855.09	−0.0633550
$951$	−10061.0	−0.343061
$952$	−14575.2	−0.496204
$953$	−39172.6	−1.33151	−0.665753	−	0.746172i	$-0.731890\pi$
$953$	−39172.6	−1.33151	−0.665753	+	0.746172i	$0.731890\pi$
$954$	−15544.8	−0.527548
$955$	−18212.7	−0.617118
$956$	17460.6	0.590707
$957$	−12199.6	−0.412076
$958$	14662.1	0.494480
$959$	−3807.64	−0.128212
$960$	3610.00	0.121367
$961$	80632.9	2.70662
$962$	3438.61	0.115244
$963$	6597.46	0.220768
$964$	7089.30	0.236858
$965$	40118.0	1.33829
$966$	−3371.44	−0.112292
$967$	−50809.7	−1.68969	−0.844845	−	0.535010i	$-0.820308\pi$
$967$	−50809.7	−1.68969	−0.844845	+	0.535010i	$0.820308\pi$
$968$	6931.21	0.230142
$969$	−2723.60	−0.0902939
$970$	23766.2	0.786687
$971$	49995.3	1.65235	0.826173	−	0.563417i	$-0.190514\pi$
$971$	49995.3	1.65235	0.826173	+	0.563417i	$0.190514\pi$
$972$	−17516.4	−0.578025
$973$	21960.4	0.723555
$974$	−861.326	−0.0283354
$975$	−935.335	−0.0307228
$976$	−737.488	−0.0241869
$977$	−6507.03	−0.213079	−0.106540	−	0.994308i	$-0.533977\pi$
$977$	−6507.03	−0.213079	−0.106540	+	0.994308i	$0.533977\pi$
$978$	−9054.73	−0.296052
$979$	−13904.1	−0.453908
$980$	10486.3	0.341810
$981$	34784.1	1.13208
$982$	−17397.9	−0.565365
$983$	37044.4	1.20197	0.600983	−	0.799262i	$-0.294776\pi$
$983$	37044.4	1.20197	0.600983	+	0.799262i	$0.294776\pi$
$984$	6646.30	0.215321
$985$	16145.2	0.522263
$986$	16676.2	0.538618
$987$	0	0
$988$	1771.42	0.0570408
$989$	6778.46	0.217940
$990$	14167.4	0.454817
$991$	−17372.8	−0.556877	−0.278439	−	0.960454i	$-0.589817\pi$
$991$	−17372.8	−0.556877	−0.278439	+	0.960454i	$0.589817\pi$
$992$	−61943.8	−1.98258
$993$	−3975.91	−0.127061
$994$	13230.0	0.422162
$995$	−31400.5	−1.00046
$996$	−5204.48	−0.165573
$997$	45015.4	1.42994	0.714971	−	0.699154i	$-0.246440\pi$
$997$	45015.4	1.42994	0.714971	+	0.699154i	$0.246440\pi$
$998$	26934.0	0.854289
$999$	15194.6	0.481217

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2209.4.a.a.1.2		3
47.46	odd	2		47.4.a.a.1.2	✓	3
141.140	even	2		423.4.a.b.1.2		3
188.187	even	2		752.4.a.c.1.1		3
235.234	odd	2		1175.4.a.a.1.2		3
329.328	even	2		2303.4.a.a.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
47.4.a.a.1.2	✓	3	47.46	odd	2
423.4.a.b.1.2		3	141.140	even	2
752.4.a.c.1.1		3	188.187	even	2
1175.4.a.a.1.2		3	235.234	odd	2
2209.4.a.a.1.2		3	1.1	even	1	trivial
2303.4.a.a.1.2		3	329.328	even	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:	\( N \)	\(=\)	\( 2209 = 47^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2209.a (trivial)

\(f(q)\)	\(=\)	\(q-1.60930 q^{2} +1.72833 q^{3} -5.41015 q^{4} +9.01945 q^{5} -2.78140 q^{6} -11.3182 q^{7} +21.5810 q^{8} -24.0129 q^{9} +O(q^{10})\)	\(q-1.60930 q^{2} +1.72833 q^{3} -5.41015 q^{4} +9.01945 q^{5} -2.78140 q^{6} -11.3182 q^{7} +21.5810 q^{8} -24.0129 q^{9} -14.5150 q^{10} +40.6469 q^{11} -9.35051 q^{12} +12.3983 q^{13} +18.2143 q^{14} +15.5886 q^{15} +8.55095 q^{16} +59.6717 q^{17} +38.6440 q^{18} -26.4089 q^{19} -48.7966 q^{20} -19.5615 q^{21} -65.4131 q^{22} -107.097 q^{23} +37.2990 q^{24} -43.6495 q^{25} -19.9526 q^{26} -88.1670 q^{27} +61.2330 q^{28} -173.657 q^{29} -25.0867 q^{30} +332.301 q^{31} -186.409 q^{32} +70.2512 q^{33} -96.0296 q^{34} -102.084 q^{35} +129.913 q^{36} -172.339 q^{37} +42.4998 q^{38} +21.4283 q^{39} +194.648 q^{40} +178.190 q^{41} +31.4804 q^{42} -63.2928 q^{43} -219.906 q^{44} -216.583 q^{45} +172.351 q^{46} +14.7788 q^{48} -214.899 q^{49} +70.2452 q^{50} +103.132 q^{51} -67.0767 q^{52} -402.256 q^{53} +141.887 q^{54} +366.613 q^{55} -244.257 q^{56} -45.6432 q^{57} +279.466 q^{58} +305.280 q^{59} -84.3365 q^{60} -86.2464 q^{61} -534.772 q^{62} +271.782 q^{63} +231.580 q^{64} +111.826 q^{65} -113.055 q^{66} +681.333 q^{67} -322.833 q^{68} -185.098 q^{69} +164.283 q^{70} +726.348 q^{71} -518.221 q^{72} +79.8711 q^{73} +277.345 q^{74} -75.4406 q^{75} +142.876 q^{76} -460.049 q^{77} -34.4846 q^{78} +279.707 q^{79} +77.1248 q^{80} +495.966 q^{81} -286.761 q^{82} +556.598 q^{83} +105.831 q^{84} +538.206 q^{85} +101.857 q^{86} -300.135 q^{87} +877.200 q^{88} -342.069 q^{89} +348.547 q^{90} -140.326 q^{91} +579.410 q^{92} +574.325 q^{93} -238.194 q^{95} -322.175 q^{96} -1637.35 q^{97} +345.837 q^{98} -976.050 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + 6 q^{5} - 8 q^{6} - 45 q^{7} - 39 q^{8} - 42 q^{9}+O(q^{10}) \) 3 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + 6 * q^5 - 8 * q^6 - 45 * q^7 - 39 * q^8 - 42 * q^9	\( 3 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + 6 q^{5} - 8 q^{6} - 45 q^{7} - 39 q^{8} - 42 q^{9} + 40 q^{10} - 2 q^{11} + 12 q^{12} + 80 q^{13} + 162 q^{14} - 14 q^{15} + 89 q^{16} - 39 q^{17} + 181 q^{18} + 24 q^{19} - 232 q^{20} + 24 q^{21} + 14 q^{22} - 120 q^{23} + 192 q^{24} - 171 q^{25} - 316 q^{26} + 64 q^{27} - 408 q^{28} + 184 q^{29} - 116 q^{30} + 4 q^{31} - 7 q^{32} + 208 q^{33} + 218 q^{34} + 156 q^{35} - 343 q^{36} - 589 q^{37} - 42 q^{38} - 60 q^{39} + 432 q^{40} + 92 q^{41} - 54 q^{42} + 250 q^{43} - 466 q^{44} + 78 q^{45} + 816 q^{46} - 120 q^{48} + 30 q^{49} + 137 q^{50} + 317 q^{51} + 900 q^{52} + 459 q^{53} + 106 q^{54} + 448 q^{55} + 1032 q^{56} - 216 q^{57} - 684 q^{58} + 579 q^{59} + 240 q^{60} + 267 q^{61} + 244 q^{62} + 1044 q^{63} - 87 q^{64} - 424 q^{65} - 16 q^{66} + 540 q^{67} - 1334 q^{68} - 642 q^{69} - 1236 q^{70} + 749 q^{71} + 357 q^{72} + 1924 q^{73} + 950 q^{74} + 473 q^{75} + 402 q^{76} + 288 q^{77} + 152 q^{78} + 805 q^{79} - 448 q^{80} + 291 q^{81} + 938 q^{82} + 712 q^{83} + 372 q^{84} + 1038 q^{85} + 1294 q^{86} - 1216 q^{87} + 2190 q^{88} + 835 q^{89} - 764 q^{90} - 2040 q^{91} - 1596 q^{92} + 1500 q^{93} - 312 q^{95} - 1432 q^{96} - 2243 q^{97} - 2989 q^{98} - 554 q^{99}+O(q^{100}) \) 3 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + 6 * q^5 - 8 * q^6 - 45 * q^7 - 39 * q^8 - 42 * q^9 + 40 * q^10 - 2 * q^11 + 12 * q^12 + 80 * q^13 + 162 * q^14 - 14 * q^15 + 89 * q^16 - 39 * q^17 + 181 * q^18 + 24 * q^19 - 232 * q^20 + 24 * q^21 + 14 * q^22 - 120 * q^23 + 192 * q^24 - 171 * q^25 - 316 * q^26 + 64 * q^27 - 408 * q^28 + 184 * q^29 - 116 * q^30 + 4 * q^31 - 7 * q^32 + 208 * q^33 + 218 * q^34 + 156 * q^35 - 343 * q^36 - 589 * q^37 - 42 * q^38 - 60 * q^39 + 432 * q^40 + 92 * q^41 - 54 * q^42 + 250 * q^43 - 466 * q^44 + 78 * q^45 + 816 * q^46 - 120 * q^48 + 30 * q^49 + 137 * q^50 + 317 * q^51 + 900 * q^52 + 459 * q^53 + 106 * q^54 + 448 * q^55 + 1032 * q^56 - 216 * q^57 - 684 * q^58 + 579 * q^59 + 240 * q^60 + 267 * q^61 + 244 * q^62 + 1044 * q^63 - 87 * q^64 - 424 * q^65 - 16 * q^66 + 540 * q^67 - 1334 * q^68 - 642 * q^69 - 1236 * q^70 + 749 * q^71 + 357 * q^72 + 1924 * q^73 + 950 * q^74 + 473 * q^75 + 402 * q^76 + 288 * q^77 + 152 * q^78 + 805 * q^79 - 448 * q^80 + 291 * q^81 + 938 * q^82 + 712 * q^83 + 372 * q^84 + 1038 * q^85 + 1294 * q^86 - 1216 * q^87 + 2190 * q^88 + 835 * q^89 - 764 * q^90 - 2040 * q^91 - 1596 * q^92 + 1500 * q^93 - 312 * q^95 - 1432 * q^96 - 2243 * q^97 - 2989 * q^98 - 554 * q^99

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