LMFDB - Embedded newform 786.4.a.g.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [786,4,Mod(1,786)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(786, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("786.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 786 = 2 \cdot 3 \cdot 131 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	786.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:	$46.3755012645$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 436x^{6} + 1403x^{5} + 41156x^{4} - 104947x^{3} - 993314x^{2} + 1535040x + 1863168 $ x^8 - x^7 - 436x^6 + 1403x^5 + 41156x^4 - 104947x^3 - 993314x^2 + 1535040x + 1863168
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-5.44332$ of defining polynomial
Character	$\chi$	$=$	786.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +13.8760 q^{5} +6.00000 q^{6} -21.4306 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +13.8760 q^{5} +6.00000 q^{6} -21.4306 q^{7} -8.00000 q^{8} +9.00000 q^{9} -27.7519 q^{10} +30.6632 q^{11} -12.0000 q^{12} -50.3032 q^{13} +42.8611 q^{14} -41.6279 q^{15} +16.0000 q^{16} +112.266 q^{17} -18.0000 q^{18} -81.3555 q^{19} +55.5039 q^{20} +64.2917 q^{21} -61.3263 q^{22} +72.0902 q^{23} +24.0000 q^{24} +67.5423 q^{25} +100.606 q^{26} -27.0000 q^{27} -85.7222 q^{28} -239.410 q^{29} +83.2558 q^{30} -266.275 q^{31} -32.0000 q^{32} -91.9895 q^{33} -224.532 q^{34} -297.370 q^{35} +36.0000 q^{36} +133.129 q^{37} +162.711 q^{38} +150.910 q^{39} -111.008 q^{40} +269.838 q^{41} -128.583 q^{42} +21.7697 q^{43} +122.653 q^{44} +124.884 q^{45} -144.180 q^{46} -11.2740 q^{47} -48.0000 q^{48} +116.269 q^{49} -135.085 q^{50} -336.798 q^{51} -201.213 q^{52} +232.471 q^{53} +54.0000 q^{54} +425.481 q^{55} +171.444 q^{56} +244.066 q^{57} +478.819 q^{58} +38.4728 q^{59} -166.512 q^{60} +368.127 q^{61} +532.550 q^{62} -192.875 q^{63} +64.0000 q^{64} -698.005 q^{65} +183.979 q^{66} +319.527 q^{67} +449.064 q^{68} -216.271 q^{69} +594.739 q^{70} -722.844 q^{71} -72.0000 q^{72} -660.119 q^{73} -266.258 q^{74} -202.627 q^{75} -325.422 q^{76} -657.129 q^{77} -301.819 q^{78} +151.164 q^{79} +222.015 q^{80} +81.0000 q^{81} -539.677 q^{82} -750.300 q^{83} +257.167 q^{84} +1557.80 q^{85} -43.5394 q^{86} +718.229 q^{87} -245.305 q^{88} +924.897 q^{89} -249.767 q^{90} +1078.03 q^{91} +288.361 q^{92} +798.825 q^{93} +22.5479 q^{94} -1128.89 q^{95} +96.0000 q^{96} -767.380 q^{97} -232.538 q^{98} +275.969 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9	\( 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9} - 2 q^{10} - 77 q^{11} - 96 q^{12} + 29 q^{13} - 44 q^{14} - 3 q^{15} + 128 q^{16} + 59 q^{17} - 144 q^{18} - 150 q^{19} + 4 q^{20} - 66 q^{21} + 154 q^{22} - 269 q^{23} + 192 q^{24} + 273 q^{25} - 58 q^{26} - 216 q^{27} + 88 q^{28} - 123 q^{29} + 6 q^{30} - 86 q^{31} - 256 q^{32} + 231 q^{33} - 118 q^{34} - 292 q^{35} + 288 q^{36} + 412 q^{37} + 300 q^{38} - 87 q^{39} - 8 q^{40} - 114 q^{41} + 132 q^{42} + 1087 q^{43} - 308 q^{44} + 9 q^{45} + 538 q^{46} - 442 q^{47} - 384 q^{48} + 1292 q^{49} - 546 q^{50} - 177 q^{51} + 116 q^{52} + 5 q^{53} + 432 q^{54} + 456 q^{55} - 176 q^{56} + 450 q^{57} + 246 q^{58} + 252 q^{59} - 12 q^{60} + 1482 q^{61} + 172 q^{62} + 198 q^{63} + 512 q^{64} + 475 q^{65} - 462 q^{66} + 330 q^{67} + 236 q^{68} + 807 q^{69} + 584 q^{70} - 2946 q^{71} - 576 q^{72} - 214 q^{73} - 824 q^{74} - 819 q^{75} - 600 q^{76} - 960 q^{77} + 174 q^{78} - 64 q^{79} + 16 q^{80} + 648 q^{81} + 228 q^{82} - 276 q^{83} - 264 q^{84} + 80 q^{85} - 2174 q^{86} + 369 q^{87} + 616 q^{88} - 3177 q^{89} - 18 q^{90} - 781 q^{91} - 1076 q^{92} + 258 q^{93} + 884 q^{94} - 2700 q^{95} + 768 q^{96} + 200 q^{97} - 2584 q^{98} - 693 q^{99}+O(q^{100}) \) 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9 - 2 * q^10 - 77 * q^11 - 96 * q^12 + 29 * q^13 - 44 * q^14 - 3 * q^15 + 128 * q^16 + 59 * q^17 - 144 * q^18 - 150 * q^19 + 4 * q^20 - 66 * q^21 + 154 * q^22 - 269 * q^23 + 192 * q^24 + 273 * q^25 - 58 * q^26 - 216 * q^27 + 88 * q^28 - 123 * q^29 + 6 * q^30 - 86 * q^31 - 256 * q^32 + 231 * q^33 - 118 * q^34 - 292 * q^35 + 288 * q^36 + 412 * q^37 + 300 * q^38 - 87 * q^39 - 8 * q^40 - 114 * q^41 + 132 * q^42 + 1087 * q^43 - 308 * q^44 + 9 * q^45 + 538 * q^46 - 442 * q^47 - 384 * q^48 + 1292 * q^49 - 546 * q^50 - 177 * q^51 + 116 * q^52 + 5 * q^53 + 432 * q^54 + 456 * q^55 - 176 * q^56 + 450 * q^57 + 246 * q^58 + 252 * q^59 - 12 * q^60 + 1482 * q^61 + 172 * q^62 + 198 * q^63 + 512 * q^64 + 475 * q^65 - 462 * q^66 + 330 * q^67 + 236 * q^68 + 807 * q^69 + 584 * q^70 - 2946 * q^71 - 576 * q^72 - 214 * q^73 - 824 * q^74 - 819 * q^75 - 600 * q^76 - 960 * q^77 + 174 * q^78 - 64 * q^79 + 16 * q^80 + 648 * q^81 + 228 * q^82 - 276 * q^83 - 264 * q^84 + 80 * q^85 - 2174 * q^86 + 369 * q^87 + 616 * q^88 - 3177 * q^89 - 18 * q^90 - 781 * q^91 - 1076 * q^92 + 258 * q^93 + 884 * q^94 - 2700 * q^95 + 768 * q^96 + 200 * q^97 - 2584 * q^98 - 693 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	13.8760	1.24110	0.620552	−	0.784165i	$-0.286909\pi$
$5$	13.8760	1.24110	0.620552	+	0.784165i	$0.286909\pi$
$6$	6.00000	0.408248
$7$	−21.4306	−1.15714	−0.578571	−	0.815632i	$-0.696389\pi$
$7$	−21.4306	−1.15714	−0.578571	+	0.815632i	$0.696389\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−27.7519	−0.877593
$11$	30.6632	0.840481	0.420241	−	0.907413i	$-0.361946\pi$
$11$	30.6632	0.840481	0.420241	+	0.907413i	$0.361946\pi$
$12$	−12.0000	−0.288675
$13$	−50.3032	−1.07320	−0.536600	−	0.843837i	$-0.680291\pi$
$13$	−50.3032	−1.07320	−0.536600	+	0.843837i	$0.680291\pi$
$14$	42.8611	0.818223
$15$	−41.6279	−0.716552
$16$	16.0000	0.250000
$17$	112.266	1.60168	0.800839	−	0.598880i	$-0.204387\pi$
$17$	112.266	1.60168	0.800839	+	0.598880i	$0.204387\pi$
$18$	−18.0000	−0.235702
$19$	−81.3555	−0.982328	−0.491164	−	0.871067i	$-0.663428\pi$
$19$	−81.3555	−0.982328	−0.491164	+	0.871067i	$0.663428\pi$
$20$	55.5039	0.620552
$21$	64.2917	0.668076
$22$	−61.3263	−0.594310
$23$	72.0902	0.653559	0.326779	−	0.945101i	$-0.394037\pi$
$23$	72.0902	0.653559	0.326779	+	0.945101i	$0.394037\pi$
$24$	24.0000	0.204124
$25$	67.5423	0.540339
$26$	100.606	0.758867
$27$	−27.0000	−0.192450
$28$	−85.7222	−0.578571
$29$	−239.410	−1.53301	−0.766504	−	0.642239i	$-0.778006\pi$
$29$	−239.410	−1.53301	−0.766504	+	0.642239i	$0.778006\pi$
$30$	83.2558	0.506679
$31$	−266.275	−1.54272	−0.771361	−	0.636397i	$-0.780424\pi$
$31$	−266.275	−1.54272	−0.771361	+	0.636397i	$0.780424\pi$
$32$	−32.0000	−0.176777
$33$	−91.9895	−0.485252
$34$	−224.532	−1.13256
$35$	−297.370	−1.43613
$36$	36.0000	0.166667
$37$	133.129	0.591520	0.295760	−	0.955262i	$-0.404427\pi$
$37$	133.129	0.591520	0.295760	+	0.955262i	$0.404427\pi$
$38$	162.711	0.694611
$39$	150.910	0.619612
$40$	−111.008	−0.438796
$41$	269.838	1.02785	0.513923	−	0.857836i	$-0.328192\pi$
$41$	269.838	1.02785	0.513923	+	0.857836i	$0.328192\pi$
$42$	−128.583	−0.472401
$43$	21.7697	0.0772057	0.0386028	−	0.999255i	$-0.487709\pi$
$43$	21.7697	0.0772057	0.0386028	+	0.999255i	$0.487709\pi$
$44$	122.653	0.420241
$45$	124.884	0.413701
$46$	−144.180	−0.462136
$47$	−11.2740	−0.0349888	−0.0174944	−	0.999847i	$-0.505569\pi$
$47$	−11.2740	−0.0349888	−0.0174944	+	0.999847i	$0.505569\pi$
$48$	−48.0000	−0.144338
$49$	116.269	0.338976
$50$	−135.085	−0.382077
$51$	−336.798	−0.924729
$52$	−201.213	−0.536600
$53$	232.471	0.602497	0.301249	−	0.953546i	$-0.402597\pi$
$53$	232.471	0.602497	0.301249	+	0.953546i	$0.402597\pi$
$54$	54.0000	0.136083
$55$	425.481	1.04312
$56$	171.444	0.409111
$57$	244.066	0.567147
$58$	478.819	1.08400
$59$	38.4728	0.0848937	0.0424468	−	0.999099i	$-0.486485\pi$
$59$	38.4728	0.0848937	0.0424468	+	0.999099i	$0.486485\pi$
$60$	−166.512	−0.358276
$61$	368.127	0.772687	0.386343	−	0.922355i	$-0.373738\pi$
$61$	368.127	0.772687	0.386343	+	0.922355i	$0.373738\pi$
$62$	532.550	1.09087
$63$	−192.875	−0.385714
$64$	64.0000	0.125000
$65$	−698.005	−1.33195
$66$	183.979	0.343125
$67$	319.527	0.582633	0.291317	−	0.956627i	$-0.405907\pi$
$67$	319.527	0.582633	0.291317	+	0.956627i	$0.405907\pi$
$68$	449.064	0.800839
$69$	−216.271	−0.377332
$70$	594.739	1.01550
$71$	−722.844	−1.20825	−0.604125	−	0.796889i	$-0.706477\pi$
$71$	−722.844	−1.20825	−0.604125	+	0.796889i	$0.706477\pi$
$72$	−72.0000	−0.117851
$73$	−660.119	−1.05837	−0.529185	−	0.848506i	$-0.677502\pi$
$73$	−660.119	−1.05837	−0.529185	+	0.848506i	$0.677502\pi$
$74$	−266.258	−0.418268
$75$	−202.627	−0.311965
$76$	−325.422	−0.491164
$77$	−657.129	−0.972556
$78$	−301.819	−0.438132
$79$	151.164	0.215283	0.107641	−	0.994190i	$-0.465670\pi$
$79$	151.164	0.215283	0.107641	+	0.994190i	$0.465670\pi$
$80$	222.015	0.310276
$81$	81.0000	0.111111
$82$	−539.677	−0.726797
$83$	−750.300	−0.992242	−0.496121	−	0.868253i	$-0.665243\pi$
$83$	−750.300	−0.992242	−0.496121	+	0.868253i	$0.665243\pi$
$84$	257.167	0.334038
$85$	1557.80	1.98785
$86$	−43.5394	−0.0545927
$87$	718.229	0.885083
$88$	−245.305	−0.297155
$89$	924.897	1.10156	0.550780	−	0.834650i	$-0.314330\pi$
$89$	924.897	1.10156	0.550780	+	0.834650i	$0.314330\pi$
$90$	−249.767	−0.292531
$91$	1078.03	1.24184
$92$	288.361	0.326779
$93$	798.825	0.890691
$94$	22.5479	0.0247408
$95$	−1128.89	−1.21917
$96$	96.0000	0.102062
$97$	−767.380	−0.803254	−0.401627	−	0.915803i	$-0.631555\pi$
$97$	−767.380	−0.803254	−0.401627	+	0.915803i	$0.631555\pi$
$98$	−232.538	−0.239693
$99$	275.969	0.280160
$100$	270.169	0.270169
$101$	653.294	0.643615	0.321808	−	0.946805i	$-0.395710\pi$
$101$	653.294	0.643615	0.321808	+	0.946805i	$0.395710\pi$
$102$	673.596	0.653882
$103$	−1476.95	−1.41289	−0.706446	−	0.707767i	$-0.749703\pi$
$103$	−1476.95	−1.41289	−0.706446	+	0.707767i	$0.749703\pi$
$104$	402.425	0.379433
$105$	892.109	0.829152
$106$	−464.942	−0.426030
$107$	437.930	0.395666	0.197833	−	0.980236i	$-0.436610\pi$
$107$	437.930	0.395666	0.197833	+	0.980236i	$0.436610\pi$
$108$	−108.000	−0.0962250
$109$	−1540.44	−1.35365	−0.676825	−	0.736144i	$-0.736645\pi$
$109$	−1540.44	−1.35365	−0.676825	+	0.736144i	$0.736645\pi$
$110$	−850.962	−0.737601
$111$	−399.387	−0.341514
$112$	−342.889	−0.289285
$113$	−2153.17	−1.79250	−0.896252	−	0.443545i	$-0.853721\pi$
$113$	−2153.17	−1.79250	−0.896252	+	0.443545i	$0.853721\pi$
$114$	−488.133	−0.401034
$115$	1000.32	0.811134
$116$	−957.638	−0.766504
$117$	−452.729	−0.357733
$118$	−76.9455	−0.0600289
$119$	−2405.92	−1.85337
$120$	333.023	0.253339
$121$	−390.770	−0.293591
$122$	−736.255	−0.546372
$123$	−809.515	−0.593427
$124$	−1065.10	−0.771361
$125$	−797.280	−0.570487
$126$	385.750	0.272741
$127$	−1258.00	−0.878969	−0.439485	−	0.898250i	$-0.644839\pi$
$127$	−1258.00	−0.878969	−0.439485	+	0.898250i	$0.644839\pi$
$128$	−128.000	−0.0883883
$129$	−65.3090	−0.0445747
$130$	1396.01	0.941832
$131$	131.000	0.0873704
$131$	131.000	0.0873704
$132$	−367.958	−0.242626
$133$	1743.49	1.13669
$134$	−639.054	−0.411984
$135$	−374.651	−0.238851
$136$	−898.128	−0.566279
$137$	1921.05	1.19800	0.599001	−	0.800749i	$-0.295565\pi$
$137$	1921.05	1.19800	0.599001	+	0.800749i	$0.295565\pi$
$138$	432.541	0.266814
$139$	−2640.78	−1.61142	−0.805712	−	0.592307i	$-0.798217\pi$
$139$	−2640.78	−1.61142	−0.805712	+	0.592307i	$0.798217\pi$
$140$	−1189.48	−0.718066
$141$	33.8219	0.0202008
$142$	1445.69	0.854362
$143$	−1542.46	−0.902004
$144$	144.000	0.0833333
$145$	−3322.04	−1.90262
$146$	1320.24	0.748381
$147$	−348.807	−0.195708
$148$	532.515	0.295760
$149$	−2514.10	−1.38230	−0.691151	−	0.722711i	$-0.742896\pi$
$149$	−2514.10	−1.38230	−0.691151	+	0.722711i	$0.742896\pi$
$150$	405.254	0.220592
$151$	−828.434	−0.446470	−0.223235	−	0.974765i	$-0.571662\pi$
$151$	−828.434	−0.446470	−0.223235	+	0.974765i	$0.571662\pi$
$152$	650.844	0.347305
$153$	1010.39	0.533893
$154$	1314.26	0.687701
$155$	−3694.82	−1.91468
$156$	603.638	0.309806
$157$	1944.37	0.988390	0.494195	−	0.869351i	$-0.335463\pi$
$157$	1944.37	0.988390	0.494195	+	0.869351i	$0.335463\pi$
$158$	−302.329	−0.152228
$159$	−697.413	−0.347852
$160$	−444.031	−0.219398
$161$	−1544.93	−0.756260
$162$	−162.000	−0.0785674
$163$	1527.47	0.733990	0.366995	−	0.930223i	$-0.380387\pi$
$163$	1527.47	0.733990	0.366995	+	0.930223i	$0.380387\pi$
$164$	1079.35	0.513923
$165$	−1276.44	−0.602248
$166$	1500.60	0.701621
$167$	−1241.18	−0.575123	−0.287561	−	0.957762i	$-0.592845\pi$
$167$	−1241.18	−0.575123	−0.287561	+	0.957762i	$0.592845\pi$
$168$	−514.333	−0.236201
$169$	333.410	0.151757
$170$	−3115.60	−1.40562
$171$	−732.199	−0.327443
$172$	87.0787	0.0386028
$173$	−2143.13	−0.941843	−0.470922	−	0.882175i	$-0.656079\pi$
$173$	−2143.13	−0.941843	−0.470922	+	0.882175i	$0.656079\pi$
$174$	−1436.46	−0.625848
$175$	−1447.47	−0.625248
$176$	490.611	0.210120
$177$	−115.418	−0.0490134
$178$	−1849.79	−0.778921
$179$	−2832.68	−1.18282	−0.591409	−	0.806372i	$-0.701428\pi$
$179$	−2832.68	−1.18282	−0.591409	+	0.806372i	$0.701428\pi$
$180$	499.535	0.206851
$181$	−1719.03	−0.705935	−0.352967	−	0.935636i	$-0.614827\pi$
$181$	−1719.03	−0.705935	−0.352967	+	0.935636i	$0.614827\pi$
$182$	−2156.05	−0.878116
$183$	−1104.38	−0.446111
$184$	−576.722	−0.231068
$185$	1847.29	0.734138
$186$	−1597.65	−0.629814
$187$	3442.43	1.34618
$188$	−45.0958	−0.0174944
$189$	578.625	0.222692
$190$	2257.77	0.862084
$191$	−1629.56	−0.617334	−0.308667	−	0.951170i	$-0.599883\pi$
$191$	−1629.56	−0.617334	−0.308667	+	0.951170i	$0.599883\pi$
$192$	−192.000	−0.0721688
$193$	1900.81	0.708931	0.354465	−	0.935069i	$-0.384663\pi$
$193$	1900.81	0.708931	0.354465	+	0.935069i	$0.384663\pi$
$194$	1534.76	0.567986
$195$	2094.02	0.769003
$196$	465.076	0.169488
$197$	634.976	0.229646	0.114823	−	0.993386i	$-0.463370\pi$
$197$	634.976	0.229646	0.114823	+	0.993386i	$0.463370\pi$
$198$	−551.937	−0.198103
$199$	−1889.63	−0.673128	−0.336564	−	0.941661i	$-0.609265\pi$
$199$	−1889.63	−0.673128	−0.336564	+	0.941661i	$0.609265\pi$
$200$	−540.339	−0.191039
$201$	−958.581	−0.336383
$202$	−1306.59	−0.455105
$203$	5130.68	1.77391
$204$	−1347.19	−0.462364
$205$	3744.27	1.27566
$206$	2953.89	0.999066
$207$	648.812	0.217853
$208$	−804.851	−0.268300
$209$	−2494.62	−0.825628
$210$	−1784.22	−0.586299
$211$	−131.852	−0.0430193	−0.0215096	−	0.999769i	$-0.506847\pi$
$211$	−131.852	−0.0430193	−0.0215096	+	0.999769i	$0.506847\pi$
$212$	929.884	0.301249
$213$	2168.53	0.697584
$214$	−875.859	−0.279778
$215$	302.075	0.0958203
$216$	216.000	0.0680414
$217$	5706.42	1.78515
$218$	3080.89	0.957175
$219$	1980.36	0.611050
$220$	1701.92	0.521562
$221$	−5647.34	−1.71892
$222$	798.773	0.241487
$223$	−826.117	−0.248076	−0.124038	−	0.992277i	$-0.539584\pi$
$223$	−826.117	−0.248076	−0.124038	+	0.992277i	$0.539584\pi$
$224$	685.778	0.204556
$225$	607.881	0.180113
$226$	4306.33	1.26749
$227$	−3748.28	−1.09596	−0.547978	−	0.836493i	$-0.684602\pi$
$227$	−3748.28	−1.09596	−0.547978	+	0.836493i	$0.684602\pi$
$228$	976.266	0.283574
$229$	−4686.48	−1.35236	−0.676181	−	0.736735i	$-0.736366\pi$
$229$	−4686.48	−1.35236	−0.676181	+	0.736735i	$0.736366\pi$
$230$	−2000.64	−0.573558
$231$	1971.39	0.561505
$232$	1915.28	0.542000
$233$	5860.84	1.64788	0.823941	−	0.566676i	$-0.191771\pi$
$233$	5860.84	1.64788	0.823941	+	0.566676i	$0.191771\pi$
$234$	905.457	0.252956
$235$	−156.437	−0.0434248
$236$	153.891	0.0424468
$237$	−453.493	−0.124294
$238$	4811.85	1.31053
$239$	−3668.86	−0.992966	−0.496483	−	0.868046i	$-0.665376\pi$
$239$	−3668.86	−0.992966	−0.496483	+	0.868046i	$0.665376\pi$
$240$	−666.046	−0.179138
$241$	5691.73	1.52131	0.760657	−	0.649154i	$-0.224877\pi$
$241$	5691.73	1.52131	0.760657	+	0.649154i	$0.224877\pi$
$242$	781.540	0.207600
$243$	−243.000	−0.0641500
$244$	1472.51	0.386343
$245$	1613.34	0.420705
$246$	1619.03	0.419616
$247$	4092.44	1.05423
$248$	2130.20	0.545435
$249$	2250.90	0.572871
$250$	1594.56	0.403395
$251$	−1705.37	−0.428852	−0.214426	−	0.976740i	$-0.568788\pi$
$251$	−1705.37	−0.428852	−0.214426	+	0.976740i	$0.568788\pi$
$252$	−771.500	−0.192857
$253$	2210.51	0.549304
$254$	2515.99	0.621525
$255$	−4673.40	−1.14768
$256$	256.000	0.0625000
$257$	3107.34	0.754204	0.377102	−	0.926172i	$-0.376921\pi$
$257$	3107.34	0.754204	0.377102	+	0.926172i	$0.376921\pi$
$258$	130.618	0.0315191
$259$	−2853.03	−0.684473
$260$	−2792.02	−0.665976
$261$	−2154.69	−0.511003
$262$	−262.000	−0.0617802
$263$	1079.24	0.253037	0.126518	−	0.991964i	$-0.459620\pi$
$263$	1079.24	0.253037	0.126518	+	0.991964i	$0.459620\pi$
$264$	735.916	0.171563
$265$	3225.76	0.747762
$266$	−3486.99	−0.803763
$267$	−2774.69	−0.635986
$268$	1278.11	0.291317
$269$	6415.42	1.45411	0.727054	−	0.686580i	$-0.240889\pi$
$269$	6415.42	1.45411	0.727054	+	0.686580i	$0.240889\pi$
$270$	749.302	0.168893
$271$	−5291.55	−1.18612	−0.593061	−	0.805158i	$-0.702081\pi$
$271$	−5291.55	−1.18612	−0.593061	+	0.805158i	$0.702081\pi$
$272$	1796.26	0.400419
$273$	−3234.08	−0.716979
$274$	−3842.10	−0.847115
$275$	2071.06	0.454145
$276$	−865.082	−0.188666
$277$	−334.278	−0.0725083	−0.0362542	−	0.999343i	$-0.511543\pi$
$277$	−334.278	−0.0725083	−0.0362542	+	0.999343i	$0.511543\pi$
$278$	5281.56	1.13945
$279$	−2396.47	−0.514241
$280$	2378.96	0.507750
$281$	−6441.28	−1.36745	−0.683727	−	0.729738i	$-0.739642\pi$
$281$	−6441.28	−1.36745	−0.683727	+	0.729738i	$0.739642\pi$
$282$	−67.6437	−0.0142841
$283$	−5194.26	−1.09105	−0.545524	−	0.838095i	$-0.683669\pi$
$283$	−5194.26	−1.09105	−0.545524	+	0.838095i	$0.683669\pi$
$284$	−2891.38	−0.604125
$285$	3386.66	0.703889
$286$	3084.91	0.637813
$287$	−5782.79	−1.18936
$288$	−288.000	−0.0589256
$289$	7690.67	1.56537
$290$	6644.08	1.34536
$291$	2302.14	0.463759
$292$	−2640.47	−0.529185
$293$	−6962.12	−1.38816	−0.694081	−	0.719897i	$-0.744189\pi$
$293$	−6962.12	−1.38816	−0.694081	+	0.719897i	$0.744189\pi$
$294$	697.614	0.138387
$295$	533.847	0.105362
$296$	−1065.03	−0.209134
$297$	−827.906	−0.161751
$298$	5028.20	0.977435
$299$	−3626.37	−0.701399
$300$	−810.508	−0.155982
$301$	−466.536	−0.0893379
$302$	1656.87	0.315702
$303$	−1959.88	−0.371592
$304$	−1301.69	−0.245582
$305$	5108.12	0.958984
$306$	−2020.79	−0.377519
$307$	2155.52	0.400723	0.200361	−	0.979722i	$-0.435788\pi$
$307$	2155.52	0.400723	0.200361	+	0.979722i	$0.435788\pi$
$308$	−2628.52	−0.486278
$309$	4430.84	0.815734
$310$	7389.64	1.35388
$311$	2478.70	0.451943	0.225972	−	0.974134i	$-0.427444\pi$
$311$	2478.70	0.451943	0.225972	+	0.974134i	$0.427444\pi$
$312$	−1207.28	−0.219066
$313$	2592.32	0.468136	0.234068	−	0.972220i	$-0.424796\pi$
$313$	2592.32	0.468136	0.234068	+	0.972220i	$0.424796\pi$
$314$	−3888.73	−0.698897
$315$	−2676.33	−0.478711
$316$	604.658	0.107641
$317$	5830.34	1.03301	0.516506	−	0.856284i	$-0.327232\pi$
$317$	5830.34	1.03301	0.516506	+	0.856284i	$0.327232\pi$
$318$	1394.83	0.245968
$319$	−7341.06	−1.28847
$320$	888.062	0.155138
$321$	−1313.79	−0.228438
$322$	3089.87	0.534756
$323$	−9133.46	−1.57337
$324$	324.000	0.0555556
$325$	−3397.59	−0.579891
$326$	−3054.93	−0.519009
$327$	4621.33	0.781530
$328$	−2158.71	−0.363398
$329$	241.607	0.0404870
$330$	2552.89	0.425854
$331$	−2663.45	−0.442286	−0.221143	−	0.975241i	$-0.570979\pi$
$331$	−2663.45	−0.442286	−0.221143	+	0.975241i	$0.570979\pi$
$332$	−3001.20	−0.496121
$333$	1198.16	0.197173
$334$	2482.36	0.406673
$335$	4433.74	0.723108
$336$	1028.67	0.167019
$337$	−7724.50	−1.24861	−0.624303	−	0.781182i	$-0.714617\pi$
$337$	−7724.50	−1.24861	−0.624303	+	0.781182i	$0.714617\pi$
$338$	−666.820	−0.107308
$339$	6459.50	1.03490
$340$	6231.20	0.993924
$341$	−8164.84	−1.29663
$342$	1464.40	0.231537
$343$	4858.97	0.764898
$344$	−174.157	−0.0272963
$345$	−3000.96	−0.468308
$346$	4286.25	0.665984
$347$	6562.99	1.01533	0.507665	−	0.861555i	$-0.330509\pi$
$347$	6562.99	1.01533	0.507665	+	0.861555i	$0.330509\pi$
$348$	2872.92	0.442541
$349$	−7873.08	−1.20755	−0.603777	−	0.797153i	$-0.706338\pi$
$349$	−7873.08	−1.20755	−0.603777	+	0.797153i	$0.706338\pi$
$350$	2894.94	0.442117
$351$	1358.19	0.206537
$352$	−981.222	−0.148578
$353$	2736.70	0.412634	0.206317	−	0.978485i	$-0.433852\pi$
$353$	2736.70	0.412634	0.206317	+	0.978485i	$0.433852\pi$
$354$	230.837	0.0346577
$355$	−10030.2	−1.49956
$356$	3699.59	0.550780
$357$	7217.77	1.07004
$358$	5665.36	0.836378
$359$	6308.75	0.927474	0.463737	−	0.885973i	$-0.346508\pi$
$359$	6308.75	0.927474	0.463737	+	0.885973i	$0.346508\pi$
$360$	−999.069	−0.146265
$361$	−240.283	−0.0350317
$362$	3438.05	0.499171
$363$	1172.31	0.169505
$364$	4312.10	0.620922
$365$	−9159.78	−1.31355
$366$	2208.76	0.315448
$367$	−12007.4	−1.70785	−0.853924	−	0.520397i	$-0.825784\pi$
$367$	−12007.4	−1.70785	−0.853924	+	0.520397i	$0.825784\pi$
$368$	1153.44	0.163390
$369$	2428.55	0.342615
$370$	−3694.58	−0.519114
$371$	−4981.98	−0.697174
$372$	3195.30	0.445346
$373$	10622.3	1.47453	0.737264	−	0.675604i	$-0.236117\pi$
$373$	10622.3	1.47453	0.737264	+	0.675604i	$0.236117\pi$
$374$	−6884.87	−0.951893
$375$	2391.84	0.329371
$376$	90.1916	0.0123704
$377$	12043.1	1.64522
$378$	−1157.25	−0.157467
$379$	8337.59	1.13001	0.565004	−	0.825088i	$-0.308874\pi$
$379$	8337.59	1.13001	0.565004	+	0.825088i	$0.308874\pi$
$380$	−4515.54	−0.609586
$381$	3773.99	0.507473
$382$	3259.12	0.436521
$383$	1256.81	0.167677	0.0838383	−	0.996479i	$-0.473282\pi$
$383$	1256.81	0.167677	0.0838383	+	0.996479i	$0.473282\pi$
$384$	384.000	0.0510310
$385$	−9118.30	−1.20704
$386$	−3801.63	−0.501290
$387$	195.927	0.0257352
$388$	−3069.52	−0.401627
$389$	2590.66	0.337665	0.168833	−	0.985645i	$-0.446000\pi$
$389$	2590.66	0.337665	0.168833	+	0.985645i	$0.446000\pi$
$390$	−4188.03	−0.543767
$391$	8093.28	1.04679
$392$	−930.151	−0.119846
$393$	−393.000	−0.0504433
$394$	−1269.95	−0.162384
$395$	2097.55	0.267188
$396$	1103.87	0.140080
$397$	9430.62	1.19222	0.596108	−	0.802905i	$-0.296713\pi$
$397$	9430.62	1.19222	0.596108	+	0.802905i	$0.296713\pi$
$398$	3779.26	0.475973
$399$	−5230.48	−0.656270
$400$	1080.68	0.135085
$401$	−5089.65	−0.633828	−0.316914	−	0.948454i	$-0.602647\pi$
$401$	−5089.65	−0.633828	−0.316914	+	0.948454i	$0.602647\pi$
$402$	1917.16	0.237859
$403$	13394.5	1.65565
$404$	2613.17	0.321808
$405$	1123.95	0.137900
$406$	−10261.4	−1.25434
$407$	4082.15	0.497162
$408$	2694.39	0.326941
$409$	2609.06	0.315427	0.157713	−	0.987485i	$-0.449588\pi$
$409$	2609.06	0.315427	0.157713	+	0.987485i	$0.449588\pi$
$410$	−7488.53	−0.902030
$411$	−5763.14	−0.691666
$412$	−5907.79	−0.706446
$413$	−824.493	−0.0982340
$414$	−1297.62	−0.154045
$415$	−10411.1	−1.23148
$416$	1609.70	0.189717
$417$	7922.34	0.930356
$418$	4989.24	0.583807
$419$	−8740.94	−1.01915	−0.509574	−	0.860427i	$-0.670197\pi$
$419$	−8740.94	−1.01915	−0.509574	+	0.860427i	$0.670197\pi$
$420$	3568.44	0.414576
$421$	−13772.8	−1.59441	−0.797204	−	0.603710i	$-0.793689\pi$
$421$	−13772.8	−1.59441	−0.797204	+	0.603710i	$0.793689\pi$
$422$	263.704	0.0304192
$423$	−101.466	−0.0116629
$424$	−1859.77	−0.213015
$425$	7582.71	0.865448
$426$	−4337.06	−0.493266
$427$	−7889.18	−0.894108
$428$	1751.72	0.197833
$429$	4627.37	0.520772
$430$	−604.151	−0.0677552
$431$	8783.70	0.981661	0.490831	−	0.871255i	$-0.336693\pi$
$431$	8783.70	0.981661	0.490831	+	0.871255i	$0.336693\pi$
$432$	−432.000	−0.0481125
$433$	10620.7	1.17875	0.589376	−	0.807859i	$-0.299374\pi$
$433$	10620.7	1.17875	0.589376	+	0.807859i	$0.299374\pi$
$434$	−11412.8	−1.26229
$435$	9966.12	1.09848
$436$	−6161.78	−0.676825
$437$	−5864.93	−0.642009
$438$	−3960.71	−0.432078
$439$	1474.42	0.160297	0.0801484	−	0.996783i	$-0.474461\pi$
$439$	1474.42	0.160297	0.0801484	+	0.996783i	$0.474461\pi$
$440$	−3403.85	−0.368800
$441$	1046.42	0.112992
$442$	11294.7	1.21546
$443$	16473.4	1.76676	0.883381	−	0.468655i	$-0.155261\pi$
$443$	16473.4	1.76676	0.883381	+	0.468655i	$0.155261\pi$
$444$	−1597.55	−0.170757
$445$	12833.8	1.36715
$446$	1652.23	0.175416
$447$	7542.29	0.798072
$448$	−1371.56	−0.144643
$449$	2519.34	0.264800	0.132400	−	0.991196i	$-0.457732\pi$
$449$	2519.34	0.264800	0.132400	+	0.991196i	$0.457732\pi$
$450$	−1215.76	−0.127359
$451$	8274.10	0.863885
$452$	−8612.67	−0.896252
$453$	2485.30	0.257770
$454$	7496.55	0.774957
$455$	14958.6	1.54126
$456$	−1952.53	−0.200517
$457$	−11593.7	−1.18672	−0.593358	−	0.804939i	$-0.702198\pi$
$457$	−11593.7	−1.18672	−0.593358	+	0.804939i	$0.702198\pi$
$458$	9372.95	0.956265
$459$	−3031.18	−0.308243
$460$	4001.28	0.405567
$461$	−3153.71	−0.318618	−0.159309	−	0.987229i	$-0.550927\pi$
$461$	−3153.71	−0.318618	−0.159309	+	0.987229i	$0.550927\pi$
$462$	−3942.77	−0.397044
$463$	1965.03	0.197241	0.0986205	−	0.995125i	$-0.468557\pi$
$463$	1965.03	0.197241	0.0986205	+	0.995125i	$0.468557\pi$
$464$	−3830.55	−0.383252
$465$	11084.5	1.10544
$466$	−11721.7	−1.16523
$467$	19193.2	1.90184	0.950918	−	0.309443i	$-0.100143\pi$
$467$	19193.2	1.90184	0.950918	+	0.309443i	$0.100143\pi$
$468$	−1810.91	−0.178867
$469$	−6847.64	−0.674189
$470$	312.874	0.0307060
$471$	−5833.10	−0.570647
$472$	−307.782	−0.0300144
$473$	667.527	0.0648899
$474$	906.987	0.0878888
$475$	−5494.94	−0.530790
$476$	−9623.70	−0.926684
$477$	2092.24	0.200832
$478$	7337.72	0.702133
$479$	15590.0	1.48711	0.743553	−	0.668677i	$-0.233139\pi$
$479$	15590.0	1.48711	0.743553	+	0.668677i	$0.233139\pi$
$480$	1332.09	0.126670
$481$	−6696.80	−0.634819
$482$	−11383.5	−1.07573
$483$	4634.80	0.436627
$484$	−1563.08	−0.146796
$485$	−10648.1	−0.996921
$486$	486.000	0.0453609
$487$	12877.4	1.19821	0.599106	−	0.800669i	$-0.295523\pi$
$487$	12877.4	1.19821	0.599106	+	0.800669i	$0.295523\pi$
$488$	−2945.02	−0.273186
$489$	−4582.40	−0.423769
$490$	−3226.69	−0.297483
$491$	−1758.90	−0.161666	−0.0808331	−	0.996728i	$-0.525758\pi$
$491$	−1758.90	−0.161666	−0.0808331	+	0.996728i	$0.525758\pi$
$492$	−3238.06	−0.296713
$493$	−26877.6	−2.45539
$494$	−8184.88	−0.745456
$495$	3829.33	0.347708
$496$	−4260.40	−0.385681
$497$	15491.0	1.39812
$498$	−4501.80	−0.405081
$499$	1623.48	0.145645	0.0728224	−	0.997345i	$-0.476799\pi$
$499$	1623.48	0.145645	0.0728224	+	0.997345i	$0.476799\pi$
$500$	−3189.12	−0.285244
$501$	3723.54	0.332047
$502$	3410.74	0.303244
$503$	−6983.06	−0.619004	−0.309502	−	0.950899i	$-0.600162\pi$
$503$	−6983.06	−0.619004	−0.309502	+	0.950899i	$0.600162\pi$
$504$	1543.00	0.136370
$505$	9065.08	0.798794
$506$	−4421.03	−0.388416
$507$	−1000.23	−0.0876169
$508$	−5031.98	−0.439485
$509$	4731.55	0.412028	0.206014	−	0.978549i	$-0.433951\pi$
$509$	4731.55	0.412028	0.206014	+	0.978549i	$0.433951\pi$
$510$	9346.80	0.811536
$511$	14146.7	1.22468
$512$	−512.000	−0.0441942
$513$	2196.60	0.189049
$514$	−6214.68	−0.533303
$515$	−20494.1	−1.75355
$516$	−261.236	−0.0222874
$517$	−345.695	−0.0294075
$518$	5706.05	0.483995
$519$	6429.38	0.543774
$520$	5584.04	0.470916
$521$	3983.02	0.334931	0.167466	−	0.985878i	$-0.446442\pi$
$521$	3983.02	0.334931	0.167466	+	0.985878i	$0.446442\pi$
$522$	4309.37	0.361334
$523$	11545.8	0.965317	0.482658	−	0.875809i	$-0.339671\pi$
$523$	11545.8	0.965317	0.482658	+	0.875809i	$0.339671\pi$
$524$	524.000	0.0436852
$525$	4342.41	0.360987
$526$	−2158.48	−0.178924
$527$	−29893.6	−2.47094
$528$	−1471.83	−0.121313
$529$	−6970.00	−0.572861
$530$	−6451.52	−0.528747
$531$	346.255	0.0282979
$532$	6973.98	0.568346
$533$	−13573.7	−1.10308
$534$	5549.38	0.449710
$535$	6076.70	0.491063
$536$	−2556.21	−0.205992
$537$	8498.04	0.682900
$538$	−12830.8	−1.02821
$539$	3565.17	0.284903
$540$	−1498.60	−0.119425
$541$	−19783.7	−1.57221	−0.786106	−	0.618092i	$-0.787906\pi$
$541$	−19783.7	−1.57221	−0.786106	+	0.618092i	$0.787906\pi$
$542$	10583.1	0.838714
$543$	5157.08	0.407572
$544$	−3592.51	−0.283139
$545$	−21375.2	−1.68002
$546$	6468.15	0.506980
$547$	13870.4	1.08420	0.542098	−	0.840315i	$-0.317630\pi$
$547$	13870.4	1.08420	0.542098	+	0.840315i	$0.317630\pi$
$548$	7684.19	0.599001
$549$	3313.15	0.257562
$550$	−4142.13	−0.321129
$551$	19477.3	1.50592
$552$	1730.16	0.133407
$553$	−3239.54	−0.249113
$554$	668.556	0.0512711
$555$	−5541.87	−0.423855
$556$	−10563.1	−0.805712
$557$	−10447.3	−0.794733	−0.397367	−	0.917660i	$-0.630076\pi$
$557$	−10447.3	−0.794733	−0.397367	+	0.917660i	$0.630076\pi$
$558$	4792.95	0.363623
$559$	−1095.08	−0.0828571
$560$	−4757.91	−0.359033
$561$	−10327.3	−0.777217
$562$	12882.6	0.966935
$563$	−21165.5	−1.58441	−0.792204	−	0.610257i	$-0.791066\pi$
$563$	−21165.5	−1.58441	−0.792204	+	0.610257i	$0.791066\pi$
$564$	135.287	0.0101004
$565$	−29877.3	−2.22468
$566$	10388.5	0.771488
$567$	−1735.88	−0.128571
$568$	5782.75	0.427181
$569$	4360.57	0.321273	0.160637	−	0.987014i	$-0.448645\pi$
$569$	4360.57	0.321273	0.160637	+	0.987014i	$0.448645\pi$
$570$	−6773.32	−0.497724
$571$	−8569.23	−0.628041	−0.314020	−	0.949416i	$-0.601676\pi$
$571$	−8569.23	−0.628041	−0.314020	+	0.949416i	$0.601676\pi$
$572$	−6169.82	−0.451002
$573$	4888.68	0.356418
$574$	11565.6	0.841007
$575$	4869.14	0.353143
$576$	576.000	0.0416667
$577$	17921.3	1.29302	0.646512	−	0.762904i	$-0.276227\pi$
$577$	17921.3	1.29302	0.646512	+	0.762904i	$0.276227\pi$
$578$	−15381.3	−1.10688
$579$	−5702.44	−0.409301
$580$	−13288.2	−0.951311
$581$	16079.3	1.14816
$582$	−4604.28	−0.327927
$583$	7128.30	0.506388
$584$	5280.95	0.374190
$585$	−6282.05	−0.443984
$586$	13924.2	0.981579
$587$	−17247.5	−1.21274	−0.606371	−	0.795182i	$-0.707375\pi$
$587$	−17247.5	−1.21274	−0.606371	+	0.795182i	$0.707375\pi$
$588$	−1395.23	−0.0978541
$589$	21662.9	1.51546
$590$	−1067.69	−0.0745021
$591$	−1904.93	−0.132586
$592$	2130.06	0.147880
$593$	−19004.0	−1.31602	−0.658012	−	0.753007i	$-0.728602\pi$
$593$	−19004.0	−1.31602	−0.658012	+	0.753007i	$0.728602\pi$
$594$	1655.81	0.114375
$595$	−33384.5	−2.30022
$596$	−10056.4	−0.691151
$597$	5668.89	0.388630
$598$	7252.73	0.495964
$599$	−26955.2	−1.83866	−0.919332	−	0.393482i	$-0.871270\pi$
$599$	−26955.2	−1.83866	−0.919332	+	0.393482i	$0.871270\pi$
$600$	1621.02	0.110296
$601$	−6087.09	−0.413140	−0.206570	−	0.978432i	$-0.566230\pi$
$601$	−6087.09	−0.413140	−0.206570	+	0.978432i	$0.566230\pi$
$602$	933.073	0.0631714
$603$	2875.74	0.194211
$604$	−3313.73	−0.223235
$605$	−5422.31	−0.364377
$606$	3919.76	0.262755
$607$	−15722.0	−1.05129	−0.525646	−	0.850703i	$-0.676176\pi$
$607$	−15722.0	−1.05129	−0.525646	+	0.850703i	$0.676176\pi$
$608$	2603.38	0.173653
$609$	−15392.0	−1.02417
$610$	−10216.2	−0.678104
$611$	567.116	0.0375500
$612$	4041.58	0.266946
$613$	−386.090	−0.0254389	−0.0127194	−	0.999919i	$-0.504049\pi$
$613$	−386.090	−0.0254389	−0.0127194	+	0.999919i	$0.504049\pi$
$614$	−4311.03	−0.283354
$615$	−11232.8	−0.736505
$616$	5257.03	0.343850
$617$	14520.6	0.947453	0.473726	−	0.880672i	$-0.342909\pi$
$617$	14520.6	0.947453	0.473726	+	0.880672i	$0.342909\pi$
$618$	−8861.68	−0.576811
$619$	29576.2	1.92047	0.960234	−	0.279197i	$-0.0900684\pi$
$619$	29576.2	1.92047	0.960234	+	0.279197i	$0.0900684\pi$
$620$	−14779.3	−0.957339
$621$	−1946.44	−0.125777
$622$	−4957.41	−0.319572
$623$	−19821.1	−1.27466
$624$	2414.55	0.154903
$625$	−19505.8	−1.24837
$626$	−5184.64	−0.331022
$627$	7483.85	0.476677
$628$	7777.46	0.494195
$629$	14945.9	0.947425
$630$	5352.65	0.338500
$631$	4052.28	0.255656	0.127828	−	0.991796i	$-0.459200\pi$
$631$	4052.28	0.255656	0.127828	+	0.991796i	$0.459200\pi$
$632$	−1209.32	−0.0761139
$633$	395.556	0.0248372
$634$	−11660.7	−0.730450
$635$	−17455.9	−1.09089
$636$	−2789.65	−0.173926
$637$	−5848.70	−0.363789
$638$	14682.1	0.911082
$639$	−6505.60	−0.402750
$640$	−1776.12	−0.109699
$641$	−1625.42	−0.100156	−0.0500782	−	0.998745i	$-0.515947\pi$
$641$	−1625.42	−0.100156	−0.0500782	+	0.998745i	$0.515947\pi$
$642$	2627.58	0.161530
$643$	−2755.04	−0.168971	−0.0844854	−	0.996425i	$-0.526925\pi$
$643$	−2755.04	−0.168971	−0.0844854	+	0.996425i	$0.526925\pi$
$644$	−6179.73	−0.378130
$645$	−906.226	−0.0553219
$646$	18266.9	1.11254
$647$	3765.78	0.228822	0.114411	−	0.993433i	$-0.463502\pi$
$647$	3765.78	0.228822	0.114411	+	0.993433i	$0.463502\pi$
$648$	−648.000	−0.0392837
$649$	1179.70	0.0713515
$650$	6795.19	0.410045
$651$	−17119.3	−1.03066
$652$	6109.87	0.366995
$653$	12488.4	0.748408	0.374204	−	0.927346i	$-0.377916\pi$
$653$	12488.4	0.748408	0.374204	+	0.927346i	$0.377916\pi$
$654$	−9242.67	−0.552625
$655$	1817.75	0.108436
$656$	4317.41	0.256961
$657$	−5941.07	−0.352790
$658$	−483.214	−0.0286287
$659$	13586.3	0.803108	0.401554	−	0.915835i	$-0.368470\pi$
$659$	13586.3	0.803108	0.401554	+	0.915835i	$0.368470\pi$
$660$	−5105.77	−0.301124
$661$	−10535.7	−0.619959	−0.309979	−	0.950743i	$-0.600322\pi$
$661$	−10535.7	−0.619959	−0.309979	+	0.950743i	$0.600322\pi$
$662$	5326.91	0.312743
$663$	16942.0	0.992419
$664$	6002.40	0.350811
$665$	24192.7	1.41075
$666$	−2396.32	−0.139423
$667$	−17259.1	−1.00191
$668$	−4964.72	−0.287561
$669$	2478.35	0.143227
$670$	−8867.49	−0.511315
$671$	11288.0	0.649429
$672$	−2057.33	−0.118100
$673$	−5717.61	−0.327486	−0.163743	−	0.986503i	$-0.552357\pi$
$673$	−5717.61	−0.327486	−0.163743	+	0.986503i	$0.552357\pi$
$674$	15449.0	0.882898
$675$	−1823.64	−0.103988
$676$	1333.64	0.0758784
$677$	13127.3	0.745234	0.372617	−	0.927985i	$-0.378460\pi$
$677$	13127.3	0.745234	0.372617	+	0.927985i	$0.378460\pi$
$678$	−12919.0	−0.731787
$679$	16445.4	0.929478
$680$	−12462.4	−0.702810
$681$	11244.8	0.632750
$682$	16329.7	0.916856
$683$	29695.1	1.66362	0.831810	−	0.555060i	$-0.187305\pi$
$683$	29695.1	1.66362	0.831810	+	0.555060i	$0.187305\pi$
$684$	−2928.80	−0.163721
$685$	26656.4	1.48684
$686$	−9717.95	−0.540864
$687$	14059.4	0.780787
$688$	348.315	0.0193014
$689$	−11694.0	−0.646600
$690$	6001.93	0.331144
$691$	−26461.5	−1.45679	−0.728395	−	0.685157i	$-0.759734\pi$
$691$	−26461.5	−1.45679	−0.728395	+	0.685157i	$0.759734\pi$
$692$	−8572.51	−0.470922
$693$	−5914.16	−0.324185
$694$	−13126.0	−0.717947
$695$	−36643.4	−1.99994
$696$	−5745.83	−0.312924
$697$	30293.7	1.64628
$698$	15746.2	0.853870
$699$	−17582.5	−0.951405
$700$	−5789.88	−0.312624
$701$	30856.5	1.66253	0.831266	−	0.555874i	$-0.187616\pi$
$701$	30856.5	1.66253	0.831266	+	0.555874i	$0.187616\pi$
$702$	−2716.37	−0.146044
$703$	−10830.8	−0.581067
$704$	1962.44	0.105060
$705$	469.311	0.0250713
$706$	−5473.40	−0.291776
$707$	−14000.5	−0.744754
$708$	−461.673	−0.0245067
$709$	19016.6	1.00731	0.503655	−	0.863905i	$-0.331988\pi$
$709$	19016.6	1.00731	0.503655	+	0.863905i	$0.331988\pi$
$710$	20060.3	1.06035
$711$	1360.48	0.0717609
$712$	−7399.17	−0.389460
$713$	−19195.8	−1.00826
$714$	−14435.5	−0.756634
$715$	−21403.0	−1.11948
$716$	−11330.7	−0.591409
$717$	11006.6	0.573289
$718$	−12617.5	−0.655823
$719$	19518.6	1.01241	0.506204	−	0.862414i	$-0.331048\pi$
$719$	19518.6	1.01241	0.506204	+	0.862414i	$0.331048\pi$
$720$	1998.14	0.103425
$721$	31651.8	1.63492
$722$	480.565	0.0247712
$723$	−17075.2	−0.878331
$724$	−6876.10	−0.352967
$725$	−16170.3	−0.828344
$726$	−2344.62	−0.119858
$727$	−12838.8	−0.654970	−0.327485	−	0.944856i	$-0.606201\pi$
$727$	−12838.8	−0.654970	−0.327485	+	0.944856i	$0.606201\pi$
$728$	−8624.20	−0.439058
$729$	729.000	0.0370370
$730$	18319.6	0.928819
$731$	2444.00	0.123659
$732$	−4417.53	−0.223055
$733$	−7823.61	−0.394232	−0.197116	−	0.980380i	$-0.563157\pi$
$733$	−7823.61	−0.394232	−0.197116	+	0.980380i	$0.563157\pi$
$734$	24014.8	1.20763
$735$	−4840.03	−0.242894
$736$	−2306.89	−0.115534
$737$	9797.71	0.489692
$738$	−4857.09	−0.242266
$739$	−12749.2	−0.634622	−0.317311	−	0.948322i	$-0.602780\pi$
$739$	−12749.2	−0.634622	−0.317311	+	0.948322i	$0.602780\pi$
$740$	7389.16	0.367069
$741$	−12277.3	−0.608662
$742$	9963.97	0.492977
$743$	−6940.66	−0.342703	−0.171351	−	0.985210i	$-0.554813\pi$
$743$	−6940.66	−0.342703	−0.171351	+	0.985210i	$0.554813\pi$
$744$	−6390.60	−0.314907
$745$	−34885.5	−1.71558
$746$	−21244.5	−1.04265
$747$	−6752.70	−0.330747
$748$	13769.7	0.673090
$749$	−9385.08	−0.457841
$750$	−4783.68	−0.232900
$751$	−20782.4	−1.00980	−0.504902	−	0.863177i	$-0.668471\pi$
$751$	−20782.4	−1.00980	−0.504902	+	0.863177i	$0.668471\pi$
$752$	−180.383	−0.00874721
$753$	5116.11	0.247598
$754$	−24086.1	−1.16335
$755$	−11495.3	−0.554116
$756$	2314.50	0.111346
$757$	2775.80	0.133274	0.0666369	−	0.997777i	$-0.478773\pi$
$757$	2775.80	0.133274	0.0666369	+	0.997777i	$0.478773\pi$
$758$	−16675.2	−0.799037
$759$	−6631.54	−0.317141
$760$	9031.09	0.431042
$761$	26484.5	1.26158	0.630791	−	0.775953i	$-0.282731\pi$
$761$	26484.5	1.26158	0.630791	+	0.775953i	$0.282731\pi$
$762$	−7547.98	−0.358838
$763$	33012.6	1.56636
$764$	−6518.24	−0.308667
$765$	14020.2	0.662616
$766$	−2513.63	−0.118565
$767$	−1935.30	−0.0911078
$768$	−768.000	−0.0360844
$769$	28936.2	1.35691	0.678457	−	0.734640i	$-0.262649\pi$
$769$	28936.2	1.35691	0.678457	+	0.734640i	$0.262649\pi$
$770$	18236.6	0.853508
$771$	−9322.01	−0.435440
$772$	7603.26	0.354465
$773$	−13307.8	−0.619208	−0.309604	−	0.950866i	$-0.600197\pi$
$773$	−13307.8	−0.619208	−0.309604	+	0.950866i	$0.600197\pi$
$774$	−391.854	−0.0181976
$775$	−17984.8	−0.833593
$776$	6139.04	0.283993
$777$	8559.08	0.395180
$778$	−5181.33	−0.238765
$779$	−21952.8	−1.00968
$780$	8376.06	0.384501
$781$	−22164.7	−1.01551
$782$	−16186.6	−0.740192
$783$	6464.06	0.295028
$784$	1860.30	0.0847441
$785$	26979.9	1.22669
$786$	786.000	0.0356688
$787$	−6906.85	−0.312837	−0.156419	−	0.987691i	$-0.549995\pi$
$787$	−6906.85	−0.312837	−0.156419	+	0.987691i	$0.549995\pi$
$788$	2539.91	0.114823
$789$	−3237.72	−0.146091
$790$	−4195.11	−0.188931
$791$	46143.6	2.07418
$792$	−2207.75	−0.0990517
$793$	−18518.0	−0.829247
$794$	−18861.2	−0.843023
$795$	−9677.28	−0.431720
$796$	−7558.52	−0.336564
$797$	31142.2	1.38408	0.692042	−	0.721858i	$-0.256712\pi$
$797$	31142.2	1.38408	0.692042	+	0.721858i	$0.256712\pi$
$798$	10461.0	0.464053
$799$	−1265.68	−0.0560408
$800$	−2161.36	−0.0955193
$801$	8324.07	0.367187
$802$	10179.3	0.448184
$803$	−20241.3	−0.889541
$804$	−3834.32	−0.168192
$805$	−21437.4	−0.938597
$806$	−26789.0	−1.17072
$807$	−19246.3	−0.839530
$808$	−5226.35	−0.227552
$809$	18428.2	0.800866	0.400433	−	0.916326i	$-0.368860\pi$
$809$	18428.2	0.800866	0.400433	+	0.916326i	$0.368860\pi$
$810$	−2247.91	−0.0975103
$811$	−23031.8	−0.997233	−0.498616	−	0.866823i	$-0.666158\pi$
$811$	−23031.8	−0.997233	−0.498616	+	0.866823i	$0.666158\pi$
$812$	20522.7	0.886954
$813$	15874.7	0.684807
$814$	−8164.31	−0.351546
$815$	21195.1	0.910958
$816$	−5388.77	−0.231182
$817$	−1771.08	−0.0758413
$818$	−5218.11	−0.223040
$819$	9702.23	0.413948
$820$	14977.1	0.637832
$821$	−25369.8	−1.07846	−0.539228	−	0.842160i	$-0.681284\pi$
$821$	−25369.8	−1.07846	−0.539228	+	0.842160i	$0.681284\pi$
$822$	11526.3	0.489082
$823$	30115.4	1.27553	0.637763	−	0.770233i	$-0.279860\pi$
$823$	30115.4	1.27553	0.637763	+	0.770233i	$0.279860\pi$
$824$	11815.6	0.499533
$825$	−6213.19	−0.262201
$826$	1648.99	0.0694619
$827$	−35039.0	−1.47331	−0.736655	−	0.676269i	$-0.763596\pi$
$827$	−35039.0	−1.47331	−0.736655	+	0.676269i	$0.763596\pi$
$828$	2595.25	0.108926
$829$	3364.45	0.140955	0.0704777	−	0.997513i	$-0.477548\pi$
$829$	3364.45	0.140955	0.0704777	+	0.997513i	$0.477548\pi$
$830$	20822.3	0.870785
$831$	1002.83	0.0418627
$832$	−3219.40	−0.134150
$833$	13053.1	0.542931
$834$	−15844.7	−0.657861
$835$	−17222.6	−0.713787
$836$	−9978.47	−0.412814
$837$	7189.42	0.296897
$838$	17481.9	0.720646
$839$	−8767.57	−0.360775	−0.180388	−	0.983596i	$-0.557735\pi$
$839$	−8767.57	−0.360775	−0.180388	+	0.983596i	$0.557735\pi$
$840$	−7136.87	−0.293149
$841$	32928.0	1.35012
$842$	27545.6	1.12742
$843$	19323.8	0.789499
$844$	−527.408	−0.0215096
$845$	4626.38	0.188346
$846$	202.931	0.00824695
$847$	8374.42	0.339726
$848$	3719.54	0.150624
$849$	15582.8	0.629917
$850$	−15165.4	−0.611964
$851$	9597.29	0.386593
$852$	8674.13	0.348792
$853$	−23161.4	−0.929698	−0.464849	−	0.885390i	$-0.653891\pi$
$853$	−23161.4	−0.929698	−0.464849	+	0.885390i	$0.653891\pi$
$854$	15778.4	0.632230
$855$	−10160.0	−0.406390
$856$	−3503.44	−0.139889
$857$	−3541.01	−0.141142	−0.0705709	−	0.997507i	$-0.522482\pi$
$857$	−3541.01	−0.141142	−0.0705709	+	0.997507i	$0.522482\pi$
$858$	−9254.73	−0.368242
$859$	−7044.22	−0.279797	−0.139898	−	0.990166i	$-0.544678\pi$
$859$	−7044.22	−0.279797	−0.139898	+	0.990166i	$0.544678\pi$
$860$	1208.30	0.0479101
$861$	17348.4	0.686679
$862$	−17567.4	−0.694139
$863$	−27910.7	−1.10092	−0.550459	−	0.834862i	$-0.685547\pi$
$863$	−27910.7	−1.10092	−0.550459	+	0.834862i	$0.685547\pi$
$864$	864.000	0.0340207
$865$	−29737.9	−1.16893
$866$	−21241.4	−0.833503
$867$	−23072.0	−0.903767
$868$	22825.7	0.892574
$869$	4635.18	0.180941
$870$	−19932.2	−0.776743
$871$	−16073.2	−0.625281
$872$	12323.6	0.478587
$873$	−6906.42	−0.267751
$874$	11729.9	0.453969
$875$	17086.2	0.660135
$876$	7921.42	0.305525
$877$	14906.5	0.573954	0.286977	−	0.957937i	$-0.407350\pi$
$877$	14906.5	0.573954	0.286977	+	0.957937i	$0.407350\pi$
$878$	−2948.84	−0.113347
$879$	20886.4	0.801456
$880$	6807.70	0.260781
$881$	−26549.2	−1.01528	−0.507642	−	0.861568i	$-0.669483\pi$
$881$	−26549.2	−1.01528	−0.507642	+	0.861568i	$0.669483\pi$
$882$	−2092.84	−0.0798975
$883$	39047.0	1.48815	0.744074	−	0.668097i	$-0.232891\pi$
$883$	39047.0	1.48815	0.744074	+	0.668097i	$0.232891\pi$
$884$	−22589.4	−0.859460
$885$	−1601.54	−0.0608307
$886$	−32946.8	−1.24929
$887$	36691.6	1.38893	0.694466	−	0.719526i	$-0.255641\pi$
$887$	36691.6	1.38893	0.694466	+	0.719526i	$0.255641\pi$
$888$	3195.09	0.120744
$889$	26959.6	1.01709
$890$	−25667.7	−0.966721
$891$	2483.72	0.0933868
$892$	−3304.47	−0.124038
$893$	917.198	0.0343705
$894$	−15084.6	−0.564322
$895$	−39306.2	−1.46800
$896$	2743.11	0.102278
$897$	10879.1	0.404953
$898$	−5038.68	−0.187242
$899$	63748.8	2.36501
$900$	2431.52	0.0900565
$901$	26098.6	0.965006
$902$	−16548.2	−0.610859
$903$	1399.61	0.0515793
$904$	17225.3	0.633746
$905$	−23853.1	−0.876138
$906$	−4970.60	−0.182271
$907$	−52602.2	−1.92572	−0.962860	−	0.269999i	$-0.912976\pi$
$907$	−52602.2	−1.92572	−0.962860	+	0.269999i	$0.912976\pi$
$908$	−14993.1	−0.547978
$909$	5879.64	0.214538
$910$	−29917.3	−1.08983
$911$	21642.4	0.787097	0.393549	−	0.919304i	$-0.371247\pi$
$911$	21642.4	0.787097	0.393549	+	0.919304i	$0.371247\pi$
$912$	3905.06	0.141787
$913$	−23006.6	−0.833961
$914$	23187.3	0.839135
$915$	−15324.4	−0.553670
$916$	−18745.9	−0.676181
$917$	−2807.40	−0.101100
$918$	6062.37	0.217961
$919$	−23554.2	−0.845464	−0.422732	−	0.906255i	$-0.638929\pi$
$919$	−23554.2	−0.845464	−0.422732	+	0.906255i	$0.638929\pi$
$920$	−8002.57	−0.286779
$921$	−6466.55	−0.231357
$922$	6307.42	0.225297
$923$	36361.3	1.29669
$924$	7885.55	0.280753
$925$	8991.84	0.319621
$926$	−3930.05	−0.139470
$927$	−13292.5	−0.470964
$928$	7661.11	0.271000
$929$	−53343.1	−1.88389	−0.941943	−	0.335773i	$-0.891002\pi$
$929$	−53343.1	−1.88389	−0.941943	+	0.335773i	$0.891002\pi$
$930$	−22168.9	−0.781664
$931$	−9459.12	−0.332986
$932$	23443.4	0.823941
$933$	−7436.11	−0.260930
$934$	−38386.5	−1.34480
$935$	47767.1	1.67075
$936$	3621.83	0.126478
$937$	3593.16	0.125276	0.0626379	−	0.998036i	$-0.480049\pi$
$937$	3593.16	0.125276	0.0626379	+	0.998036i	$0.480049\pi$
$938$	13695.3	0.476724
$939$	−7776.96	−0.270279
$940$	−625.748	−0.0217124
$941$	41411.8	1.43463	0.717315	−	0.696749i	$-0.245371\pi$
$941$	41411.8	1.43463	0.717315	+	0.696749i	$0.245371\pi$
$942$	11666.2	0.403509
$943$	19452.7	0.671757
$944$	615.564	0.0212234
$945$	8028.98	0.276384
$946$	−1335.05	−0.0458841
$947$	52685.8	1.80788	0.903939	−	0.427662i	$-0.140663\pi$
$947$	52685.8	1.80788	0.903939	+	0.427662i	$0.140663\pi$
$948$	−1813.97	−0.0621468
$949$	33206.1	1.13584
$950$	10989.9	0.375325
$951$	−17491.0	−0.596410
$952$	19247.4	0.655264
$953$	51976.1	1.76671	0.883353	−	0.468708i	$-0.155280\pi$
$953$	51976.1	1.76671	0.883353	+	0.468708i	$0.155280\pi$
$954$	−4184.48	−0.142010
$955$	−22611.7	−0.766176
$956$	−14675.4	−0.496483
$957$	22023.2	0.743896
$958$	−31179.9	−1.05154
$959$	−41169.1	−1.38626
$960$	−2664.18	−0.0895690
$961$	41111.4	1.37999
$962$	13393.6	0.448885
$963$	3941.37	0.131889
$964$	22766.9	0.760657
$965$	26375.6	0.879856
$966$	−9269.60	−0.308742
$967$	10020.1	0.333222	0.166611	−	0.986023i	$-0.446718\pi$
$967$	10020.1	0.333222	0.166611	+	0.986023i	$0.446718\pi$
$968$	3126.16	0.103800
$969$	27400.4	0.908387
$970$	21296.3	0.704930
$971$	−22597.6	−0.746851	−0.373426	−	0.927660i	$-0.621817\pi$
$971$	−22597.6	−0.746851	−0.373426	+	0.927660i	$0.621817\pi$
$972$	−972.000	−0.0320750
$973$	56593.4	1.86465
$974$	−25754.8	−0.847265
$975$	10192.8	0.334800
$976$	5890.04	0.193172
$977$	−48709.3	−1.59503	−0.797517	−	0.603297i	$-0.793854\pi$
$977$	−48709.3	−1.59503	−0.797517	+	0.603297i	$0.793854\pi$
$978$	9164.80	0.299650
$979$	28360.3	0.925841
$980$	6453.37	0.210352
$981$	−13864.0	−0.451217
$982$	3517.80	0.114315
$983$	−7942.83	−0.257718	−0.128859	−	0.991663i	$-0.541132\pi$
$983$	−7942.83	−0.257718	−0.128859	+	0.991663i	$0.541132\pi$
$984$	6476.12	0.209808
$985$	8810.91	0.285014
$986$	53755.1	1.73622
$987$	−724.821	−0.0233752
$988$	16369.8	0.527117
$989$	1569.38	0.0504584
$990$	−7658.66	−0.245867
$991$	40660.5	1.30335	0.651677	−	0.758497i	$-0.274066\pi$
$991$	40660.5	1.30335	0.651677	+	0.758497i	$0.274066\pi$
$992$	8520.80	0.272717
$993$	7990.36	0.255354
$994$	−30981.9	−0.988618
$995$	−26220.4	−0.835421
$996$	9003.60	0.286436
$997$	51703.0	1.64238	0.821188	−	0.570657i	$-0.193311\pi$
$997$	51703.0	1.64238	0.821188	+	0.570657i	$0.193311\pi$
$998$	−3246.95	−0.102986
$999$	−3594.48	−0.113838

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	786.4.a.g.1.7	✓	8
3.2	odd	2		2358.4.a.i.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.4.a.g.1.7	✓	8	1.1	even	1	trivial
2358.4.a.i.1.2		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 786 = 2 \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	786.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +13.8760 q^{5} +6.00000 q^{6} -21.4306 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +13.8760 q^{5} +6.00000 q^{6} -21.4306 q^{7} -8.00000 q^{8} +9.00000 q^{9} -27.7519 q^{10} +30.6632 q^{11} -12.0000 q^{12} -50.3032 q^{13} +42.8611 q^{14} -41.6279 q^{15} +16.0000 q^{16} +112.266 q^{17} -18.0000 q^{18} -81.3555 q^{19} +55.5039 q^{20} +64.2917 q^{21} -61.3263 q^{22} +72.0902 q^{23} +24.0000 q^{24} +67.5423 q^{25} +100.606 q^{26} -27.0000 q^{27} -85.7222 q^{28} -239.410 q^{29} +83.2558 q^{30} -266.275 q^{31} -32.0000 q^{32} -91.9895 q^{33} -224.532 q^{34} -297.370 q^{35} +36.0000 q^{36} +133.129 q^{37} +162.711 q^{38} +150.910 q^{39} -111.008 q^{40} +269.838 q^{41} -128.583 q^{42} +21.7697 q^{43} +122.653 q^{44} +124.884 q^{45} -144.180 q^{46} -11.2740 q^{47} -48.0000 q^{48} +116.269 q^{49} -135.085 q^{50} -336.798 q^{51} -201.213 q^{52} +232.471 q^{53} +54.0000 q^{54} +425.481 q^{55} +171.444 q^{56} +244.066 q^{57} +478.819 q^{58} +38.4728 q^{59} -166.512 q^{60} +368.127 q^{61} +532.550 q^{62} -192.875 q^{63} +64.0000 q^{64} -698.005 q^{65} +183.979 q^{66} +319.527 q^{67} +449.064 q^{68} -216.271 q^{69} +594.739 q^{70} -722.844 q^{71} -72.0000 q^{72} -660.119 q^{73} -266.258 q^{74} -202.627 q^{75} -325.422 q^{76} -657.129 q^{77} -301.819 q^{78} +151.164 q^{79} +222.015 q^{80} +81.0000 q^{81} -539.677 q^{82} -750.300 q^{83} +257.167 q^{84} +1557.80 q^{85} -43.5394 q^{86} +718.229 q^{87} -245.305 q^{88} +924.897 q^{89} -249.767 q^{90} +1078.03 q^{91} +288.361 q^{92} +798.825 q^{93} +22.5479 q^{94} -1128.89 q^{95} +96.0000 q^{96} -767.380 q^{97} -232.538 q^{98} +275.969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9	\( 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9} - 2 q^{10} - 77 q^{11} - 96 q^{12} + 29 q^{13} - 44 q^{14} - 3 q^{15} + 128 q^{16} + 59 q^{17} - 144 q^{18} - 150 q^{19} + 4 q^{20} - 66 q^{21} + 154 q^{22} - 269 q^{23} + 192 q^{24} + 273 q^{25} - 58 q^{26} - 216 q^{27} + 88 q^{28} - 123 q^{29} + 6 q^{30} - 86 q^{31} - 256 q^{32} + 231 q^{33} - 118 q^{34} - 292 q^{35} + 288 q^{36} + 412 q^{37} + 300 q^{38} - 87 q^{39} - 8 q^{40} - 114 q^{41} + 132 q^{42} + 1087 q^{43} - 308 q^{44} + 9 q^{45} + 538 q^{46} - 442 q^{47} - 384 q^{48} + 1292 q^{49} - 546 q^{50} - 177 q^{51} + 116 q^{52} + 5 q^{53} + 432 q^{54} + 456 q^{55} - 176 q^{56} + 450 q^{57} + 246 q^{58} + 252 q^{59} - 12 q^{60} + 1482 q^{61} + 172 q^{62} + 198 q^{63} + 512 q^{64} + 475 q^{65} - 462 q^{66} + 330 q^{67} + 236 q^{68} + 807 q^{69} + 584 q^{70} - 2946 q^{71} - 576 q^{72} - 214 q^{73} - 824 q^{74} - 819 q^{75} - 600 q^{76} - 960 q^{77} + 174 q^{78} - 64 q^{79} + 16 q^{80} + 648 q^{81} + 228 q^{82} - 276 q^{83} - 264 q^{84} + 80 q^{85} - 2174 q^{86} + 369 q^{87} + 616 q^{88} - 3177 q^{89} - 18 q^{90} - 781 q^{91} - 1076 q^{92} + 258 q^{93} + 884 q^{94} - 2700 q^{95} + 768 q^{96} + 200 q^{97} - 2584 q^{98} - 693 q^{99}+O(q^{100}) \) 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9 - 2 * q^10 - 77 * q^11 - 96 * q^12 + 29 * q^13 - 44 * q^14 - 3 * q^15 + 128 * q^16 + 59 * q^17 - 144 * q^18 - 150 * q^19 + 4 * q^20 - 66 * q^21 + 154 * q^22 - 269 * q^23 + 192 * q^24 + 273 * q^25 - 58 * q^26 - 216 * q^27 + 88 * q^28 - 123 * q^29 + 6 * q^30 - 86 * q^31 - 256 * q^32 + 231 * q^33 - 118 * q^34 - 292 * q^35 + 288 * q^36 + 412 * q^37 + 300 * q^38 - 87 * q^39 - 8 * q^40 - 114 * q^41 + 132 * q^42 + 1087 * q^43 - 308 * q^44 + 9 * q^45 + 538 * q^46 - 442 * q^47 - 384 * q^48 + 1292 * q^49 - 546 * q^50 - 177 * q^51 + 116 * q^52 + 5 * q^53 + 432 * q^54 + 456 * q^55 - 176 * q^56 + 450 * q^57 + 246 * q^58 + 252 * q^59 - 12 * q^60 + 1482 * q^61 + 172 * q^62 + 198 * q^63 + 512 * q^64 + 475 * q^65 - 462 * q^66 + 330 * q^67 + 236 * q^68 + 807 * q^69 + 584 * q^70 - 2946 * q^71 - 576 * q^72 - 214 * q^73 - 824 * q^74 - 819 * q^75 - 600 * q^76 - 960 * q^77 + 174 * q^78 - 64 * q^79 + 16 * q^80 + 648 * q^81 + 228 * q^82 - 276 * q^83 - 264 * q^84 + 80 * q^85 - 2174 * q^86 + 369 * q^87 + 616 * q^88 - 3177 * q^89 - 18 * q^90 - 781 * q^91 - 1076 * q^92 + 258 * q^93 + 884 * q^94 - 2700 * q^95 + 768 * q^96 + 200 * q^97 - 2584 * q^98 - 693 * q^99

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