LMFDB - Embedded newform 966.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,4,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("966.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	966.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:	$56.9958450655$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 356x^{2} + 245x + 16751 $ x^4 - x^3 - 356x^2 + 245x + 16751
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.86398$ of defining polynomial
Character	$\chi$	$=$	966.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.86398 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.86398 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +17.7280 q^{10} -7.07245 q^{11} -12.0000 q^{12} -31.1952 q^{13} +14.0000 q^{14} +26.5919 q^{15} +16.0000 q^{16} +95.0114 q^{17} -18.0000 q^{18} -14.6555 q^{19} -35.4559 q^{20} +21.0000 q^{21} +14.1449 q^{22} +23.0000 q^{23} +24.0000 q^{24} -46.4299 q^{25} +62.3904 q^{26} -27.0000 q^{27} -28.0000 q^{28} -1.23408 q^{29} -53.1839 q^{30} +189.709 q^{31} -32.0000 q^{32} +21.2174 q^{33} -190.023 q^{34} +62.0478 q^{35} +36.0000 q^{36} +302.496 q^{37} +29.3110 q^{38} +93.5856 q^{39} +70.9118 q^{40} -51.2951 q^{41} -42.0000 q^{42} +109.836 q^{43} -28.2898 q^{44} -79.7758 q^{45} -46.0000 q^{46} +89.2811 q^{47} -48.0000 q^{48} +49.0000 q^{49} +92.8598 q^{50} -285.034 q^{51} -124.781 q^{52} -616.455 q^{53} +54.0000 q^{54} +62.6900 q^{55} +56.0000 q^{56} +43.9665 q^{57} +2.46815 q^{58} -530.382 q^{59} +106.368 q^{60} +575.371 q^{61} -379.419 q^{62} -63.0000 q^{63} +64.0000 q^{64} +276.514 q^{65} -42.4347 q^{66} +215.120 q^{67} +380.046 q^{68} -69.0000 q^{69} -124.096 q^{70} +652.183 q^{71} -72.0000 q^{72} +780.710 q^{73} -604.992 q^{74} +139.290 q^{75} -58.6220 q^{76} +49.5072 q^{77} -187.171 q^{78} +422.179 q^{79} -141.824 q^{80} +81.0000 q^{81} +102.590 q^{82} -677.295 q^{83} +84.0000 q^{84} -842.179 q^{85} -219.672 q^{86} +3.70223 q^{87} +56.5796 q^{88} -1117.69 q^{89} +159.552 q^{90} +218.366 q^{91} +92.0000 q^{92} -569.128 q^{93} -178.562 q^{94} +129.906 q^{95} +96.0000 q^{96} +886.728 q^{97} -98.0000 q^{98} -63.6521 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 5 q^{5} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 5 * q^5 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 5 q^{5} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9} + 10 q^{10} - 41 q^{11} - 48 q^{12} + 23 q^{13} + 56 q^{14} + 15 q^{15} + 64 q^{16} + 18 q^{17} - 72 q^{18} + 15 q^{19} - 20 q^{20} + 84 q^{21} + 82 q^{22} + 92 q^{23} + 96 q^{24} + 219 q^{25} - 46 q^{26} - 108 q^{27} - 112 q^{28} - 46 q^{29} - 30 q^{30} + 142 q^{31} - 128 q^{32} + 123 q^{33} - 36 q^{34} + 35 q^{35} + 144 q^{36} - 142 q^{37} - 30 q^{38} - 69 q^{39} + 40 q^{40} - 621 q^{41} - 168 q^{42} + 185 q^{43} - 164 q^{44} - 45 q^{45} - 184 q^{46} + 669 q^{47} - 192 q^{48} + 196 q^{49} - 438 q^{50} - 54 q^{51} + 92 q^{52} + 422 q^{53} + 216 q^{54} + 1435 q^{55} + 224 q^{56} - 45 q^{57} + 92 q^{58} + 270 q^{59} + 60 q^{60} + 272 q^{61} - 284 q^{62} - 252 q^{63} + 256 q^{64} - 1992 q^{65} - 246 q^{66} - 67 q^{67} + 72 q^{68} - 276 q^{69} - 70 q^{70} + 611 q^{71} - 288 q^{72} - 236 q^{73} + 284 q^{74} - 657 q^{75} + 60 q^{76} + 287 q^{77} + 138 q^{78} + 558 q^{79} - 80 q^{80} + 324 q^{81} + 1242 q^{82} + 468 q^{83} + 336 q^{84} + 1045 q^{85} - 370 q^{86} + 138 q^{87} + 328 q^{88} - 1519 q^{89} + 90 q^{90} - 161 q^{91} + 368 q^{92} - 426 q^{93} - 1338 q^{94} + 23 q^{95} + 384 q^{96} + 600 q^{97} - 392 q^{98} - 369 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 5 * q^5 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9 + 10 * q^10 - 41 * q^11 - 48 * q^12 + 23 * q^13 + 56 * q^14 + 15 * q^15 + 64 * q^16 + 18 * q^17 - 72 * q^18 + 15 * q^19 - 20 * q^20 + 84 * q^21 + 82 * q^22 + 92 * q^23 + 96 * q^24 + 219 * q^25 - 46 * q^26 - 108 * q^27 - 112 * q^28 - 46 * q^29 - 30 * q^30 + 142 * q^31 - 128 * q^32 + 123 * q^33 - 36 * q^34 + 35 * q^35 + 144 * q^36 - 142 * q^37 - 30 * q^38 - 69 * q^39 + 40 * q^40 - 621 * q^41 - 168 * q^42 + 185 * q^43 - 164 * q^44 - 45 * q^45 - 184 * q^46 + 669 * q^47 - 192 * q^48 + 196 * q^49 - 438 * q^50 - 54 * q^51 + 92 * q^52 + 422 * q^53 + 216 * q^54 + 1435 * q^55 + 224 * q^56 - 45 * q^57 + 92 * q^58 + 270 * q^59 + 60 * q^60 + 272 * q^61 - 284 * q^62 - 252 * q^63 + 256 * q^64 - 1992 * q^65 - 246 * q^66 - 67 * q^67 + 72 * q^68 - 276 * q^69 - 70 * q^70 + 611 * q^71 - 288 * q^72 - 236 * q^73 + 284 * q^74 - 657 * q^75 + 60 * q^76 + 287 * q^77 + 138 * q^78 + 558 * q^79 - 80 * q^80 + 324 * q^81 + 1242 * q^82 + 468 * q^83 + 336 * q^84 + 1045 * q^85 - 370 * q^86 + 138 * q^87 + 328 * q^88 - 1519 * q^89 + 90 * q^90 - 161 * q^91 + 368 * q^92 - 426 * q^93 - 1338 * q^94 + 23 * q^95 + 384 * q^96 + 600 * q^97 - 392 * q^98 - 369 * 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
$5$	−8.86398	−0.792818	−0.396409	−	0.918074i	$-0.629744\pi$
$5$	−8.86398	−0.792818	−0.396409	+	0.918074i	$0.629744\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	17.7280	0.560607
$11$	−7.07245	−0.193857	−0.0969284	−	0.995291i	$-0.530902\pi$
$11$	−7.07245	−0.193857	−0.0969284	+	0.995291i	$0.530902\pi$
$12$	−12.0000	−0.288675
$13$	−31.1952	−0.665538	−0.332769	−	0.943008i	$-0.607983\pi$
$13$	−31.1952	−0.665538	−0.332769	+	0.943008i	$0.607983\pi$
$14$	14.0000	0.267261
$15$	26.5919	0.457734
$16$	16.0000	0.250000
$17$	95.0114	1.35551	0.677755	−	0.735288i	$-0.262953\pi$
$17$	95.0114	1.35551	0.677755	+	0.735288i	$0.262953\pi$
$18$	−18.0000	−0.235702
$19$	−14.6555	−0.176958	−0.0884790	−	0.996078i	$-0.528201\pi$
$19$	−14.6555	−0.176958	−0.0884790	+	0.996078i	$0.528201\pi$
$20$	−35.4559	−0.396409
$21$	21.0000	0.218218
$22$	14.1449	0.137077
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	24.0000	0.204124
$25$	−46.4299	−0.371439
$26$	62.3904	0.470606
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	−1.23408	−0.00790215	−0.00395107	−	0.999992i	$-0.501258\pi$
$29$	−1.23408	−0.00790215	−0.00395107	+	0.999992i	$0.501258\pi$
$30$	−53.1839	−0.323667
$31$	189.709	1.09912	0.549561	−	0.835453i	$-0.314795\pi$
$31$	189.709	1.09912	0.549561	+	0.835453i	$0.314795\pi$
$32$	−32.0000	−0.176777
$33$	21.2174	0.111923
$34$	−190.023	−0.958490
$35$	62.0478	0.299657
$36$	36.0000	0.166667
$37$	302.496	1.34406	0.672028	−	0.740526i	$-0.265424\pi$
$37$	302.496	1.34406	0.672028	+	0.740526i	$0.265424\pi$
$38$	29.3110	0.125128
$39$	93.5856	0.384248
$40$	70.9118	0.280304
$41$	−51.2951	−0.195389	−0.0976945	−	0.995216i	$-0.531147\pi$
$41$	−51.2951	−0.195389	−0.0976945	+	0.995216i	$0.531147\pi$
$42$	−42.0000	−0.154303
$43$	109.836	0.389531	0.194766	−	0.980850i	$-0.437605\pi$
$43$	109.836	0.389531	0.194766	+	0.980850i	$0.437605\pi$
$44$	−28.2898	−0.0969284
$45$	−79.7758	−0.264273
$46$	−46.0000	−0.147442
$47$	89.2811	0.277085	0.138542	−	0.990357i	$-0.455758\pi$
$47$	89.2811	0.277085	0.138542	+	0.990357i	$0.455758\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	92.8598	0.262647
$51$	−285.034	−0.782604
$52$	−124.781	−0.332769
$53$	−616.455	−1.59767	−0.798836	−	0.601548i	$-0.794551\pi$
$53$	−616.455	−1.59767	−0.798836	+	0.601548i	$0.794551\pi$
$54$	54.0000	0.136083
$55$	62.6900	0.153693
$56$	56.0000	0.133631
$57$	43.9665	0.102167
$58$	2.46815	0.00558766
$59$	−530.382	−1.17034	−0.585169	−	0.810912i	$-0.698972\pi$
$59$	−530.382	−1.17034	−0.585169	+	0.810912i	$0.698972\pi$
$60$	106.368	0.228867
$61$	575.371	1.20768	0.603842	−	0.797104i	$-0.293636\pi$
$61$	575.371	1.20768	0.603842	+	0.797104i	$0.293636\pi$
$62$	−379.419	−0.777197
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	276.514	0.527650
$66$	−42.4347	−0.0791417
$67$	215.120	0.392255	0.196127	−	0.980578i	$-0.437163\pi$
$67$	215.120	0.392255	0.196127	+	0.980578i	$0.437163\pi$
$68$	380.046	0.677755
$69$	−69.0000	−0.120386
$70$	−124.096	−0.211890
$71$	652.183	1.09014	0.545069	−	0.838391i	$-0.316503\pi$
$71$	652.183	1.09014	0.545069	+	0.838391i	$0.316503\pi$
$72$	−72.0000	−0.117851
$73$	780.710	1.25172	0.625858	−	0.779937i	$-0.284749\pi$
$73$	780.710	1.25172	0.625858	+	0.779937i	$0.284749\pi$
$74$	−604.992	−0.950390
$75$	139.290	0.214451
$76$	−58.6220	−0.0884790
$77$	49.5072	0.0732710
$78$	−187.171	−0.271705
$79$	422.179	0.601251	0.300625	−	0.953742i	$-0.402805\pi$
$79$	422.179	0.601251	0.300625	+	0.953742i	$0.402805\pi$
$80$	−141.824	−0.198205
$81$	81.0000	0.111111
$82$	102.590	0.138161
$83$	−677.295	−0.895696	−0.447848	−	0.894110i	$-0.647809\pi$
$83$	−677.295	−0.895696	−0.447848	+	0.894110i	$0.647809\pi$
$84$	84.0000	0.109109
$85$	−842.179	−1.07467
$86$	−219.672	−0.275440
$87$	3.70223	0.00456231
$88$	56.5796	0.0685387
$89$	−1117.69	−1.33118	−0.665591	−	0.746317i	$-0.731820\pi$
$89$	−1117.69	−1.33118	−0.665591	+	0.746317i	$0.731820\pi$
$90$	159.552	0.186869
$91$	218.366	0.251550
$92$	92.0000	0.104257
$93$	−569.128	−0.634579
$94$	−178.562	−0.195929
$95$	129.906	0.140296
$96$	96.0000	0.102062
$97$	886.728	0.928181	0.464091	−	0.885788i	$-0.346381\pi$
$97$	886.728	0.928181	0.464091	+	0.885788i	$0.346381\pi$
$98$	−98.0000	−0.101015
$99$	−63.6521	−0.0646189
$100$	−185.720	−0.185720
$101$	−291.107	−0.286794	−0.143397	−	0.989665i	$-0.545803\pi$
$101$	−291.107	−0.286794	−0.143397	+	0.989665i	$0.545803\pi$
$102$	570.069	0.553384
$103$	3.48533	0.00333417	0.00166709	−	0.999999i	$-0.499469\pi$
$103$	3.48533	0.00333417	0.00166709	+	0.999999i	$0.499469\pi$
$104$	249.562	0.235303
$105$	−186.144	−0.173007
$106$	1232.91	1.12973
$107$	−110.886	−0.100185	−0.0500923	−	0.998745i	$-0.515952\pi$
$107$	−110.886	−0.100185	−0.0500923	+	0.998745i	$0.515952\pi$
$108$	−108.000	−0.0962250
$109$	−1284.65	−1.12887	−0.564437	−	0.825476i	$-0.690907\pi$
$109$	−1284.65	−1.12887	−0.564437	+	0.825476i	$0.690907\pi$
$110$	−125.380	−0.108677
$111$	−907.488	−0.775991
$112$	−112.000	−0.0944911
$113$	−1236.10	−1.02905	−0.514526	−	0.857475i	$-0.672032\pi$
$113$	−1236.10	−1.02905	−0.514526	+	0.857475i	$0.672032\pi$
$114$	−87.9330	−0.0722428
$115$	−203.871	−0.165314
$116$	−4.93631	−0.00395107
$117$	−280.757	−0.221846
$118$	1060.76	0.827554
$119$	−665.080	−0.512334
$120$	−212.735	−0.161833
$121$	−1280.98	−0.962420
$122$	−1150.74	−0.853961
$123$	153.885	0.112808
$124$	758.837	0.549561
$125$	1519.55	1.08730
$126$	126.000	0.0890871
$127$	68.5586	0.0479023	0.0239511	−	0.999713i	$-0.492375\pi$
$127$	68.5586	0.0479023	0.0239511	+	0.999713i	$0.492375\pi$
$128$	−128.000	−0.0883883
$129$	−329.508	−0.224896
$130$	−553.027	−0.373105
$131$	−1135.59	−0.757384	−0.378692	−	0.925523i	$-0.623626\pi$
$131$	−1135.59	−0.757384	−0.378692	+	0.925523i	$0.623626\pi$
$132$	84.8694	0.0559616
$133$	102.589	0.0668839
$134$	−430.240	−0.277366
$135$	239.327	0.152578
$136$	−760.091	−0.479245
$137$	138.687	0.0864876	0.0432438	−	0.999065i	$-0.486231\pi$
$137$	138.687	0.0864876	0.0432438	+	0.999065i	$0.486231\pi$
$138$	138.000	0.0851257
$139$	267.247	0.163076	0.0815381	−	0.996670i	$-0.474017\pi$
$139$	267.247	0.163076	0.0815381	+	0.996670i	$0.474017\pi$
$140$	248.191	0.149829
$141$	−267.843	−0.159975
$142$	−1304.37	−0.770845
$143$	220.626	0.129019
$144$	144.000	0.0833333
$145$	10.9388	0.00626497
$146$	−1561.42	−0.885096
$147$	−147.000	−0.0824786
$148$	1209.98	0.672028
$149$	−656.199	−0.360792	−0.180396	−	0.983594i	$-0.557738\pi$
$149$	−656.199	−0.360792	−0.180396	+	0.983594i	$0.557738\pi$
$150$	−278.579	−0.151639
$151$	282.590	0.152297	0.0761485	−	0.997096i	$-0.475738\pi$
$151$	282.590	0.152297	0.0761485	+	0.997096i	$0.475738\pi$
$152$	117.244	0.0625641
$153$	855.103	0.451836
$154$	−99.0143	−0.0518104
$155$	−1681.58	−0.871405
$156$	374.342	0.192124
$157$	−2622.55	−1.33313	−0.666567	−	0.745445i	$-0.732237\pi$
$157$	−2622.55	−1.33313	−0.666567	+	0.745445i	$0.732237\pi$
$158$	−844.358	−0.425149
$159$	1849.37	0.922417
$160$	283.647	0.140152
$161$	−161.000	−0.0788110
$162$	−162.000	−0.0785674
$163$	−2062.03	−0.990864	−0.495432	−	0.868647i	$-0.664990\pi$
$163$	−2062.03	−0.990864	−0.495432	+	0.868647i	$0.664990\pi$
$164$	−205.180	−0.0976945
$165$	−188.070	−0.0887348
$166$	1354.59	0.633353
$167$	−1178.92	−0.546272	−0.273136	−	0.961975i	$-0.588061\pi$
$167$	−1178.92	−0.546272	−0.273136	+	0.961975i	$0.588061\pi$
$168$	−168.000	−0.0771517
$169$	−1223.86	−0.557059
$170$	1684.36	0.759908
$171$	−131.900	−0.0589860
$172$	439.344	0.194766
$173$	1292.69	0.568100	0.284050	−	0.958809i	$-0.408322\pi$
$173$	1292.69	0.568100	0.284050	+	0.958809i	$0.408322\pi$
$174$	−7.40446	−0.00322604
$175$	325.009	0.140391
$176$	−113.159	−0.0484642
$177$	1591.15	0.675695
$178$	2235.38	0.941287
$179$	−2273.50	−0.949328	−0.474664	−	0.880167i	$-0.657430\pi$
$179$	−2273.50	−0.949328	−0.474664	+	0.880167i	$0.657430\pi$
$180$	−319.103	−0.132136
$181$	1554.61	0.638415	0.319208	−	0.947685i	$-0.396583\pi$
$181$	1554.61	0.638415	0.319208	+	0.947685i	$0.396583\pi$
$182$	−436.733	−0.177872
$183$	−1726.11	−0.697257
$184$	−184.000	−0.0737210
$185$	−2681.32	−1.06559
$186$	1138.26	0.448715
$187$	−671.964	−0.262775
$188$	357.124	0.138542
$189$	189.000	0.0727393
$190$	−259.812	−0.0992039
$191$	−2184.46	−0.827549	−0.413775	−	0.910379i	$-0.635790\pi$
$191$	−2184.46	−0.827549	−0.413775	+	0.910379i	$0.635790\pi$
$192$	−192.000	−0.0721688
$193$	−1717.33	−0.640496	−0.320248	−	0.947334i	$-0.603766\pi$
$193$	−1717.33	−0.640496	−0.320248	+	0.947334i	$0.603766\pi$
$194$	−1773.46	−0.656323
$195$	−829.541	−0.304639
$196$	196.000	0.0714286
$197$	−748.677	−0.270767	−0.135383	−	0.990793i	$-0.543227\pi$
$197$	−748.677	−0.270767	−0.135383	+	0.990793i	$0.543227\pi$
$198$	127.304	0.0456925
$199$	−908.709	−0.323702	−0.161851	−	0.986815i	$-0.551746\pi$
$199$	−908.709	−0.323702	−0.161851	+	0.986815i	$0.551746\pi$
$200$	371.439	0.131324
$201$	−645.360	−0.226468
$202$	582.214	0.202794
$203$	8.63854	0.00298673
$204$	−1140.14	−0.391302
$205$	454.679	0.154908
$206$	−6.97066	−0.00235762
$207$	207.000	0.0695048
$208$	−499.123	−0.166384
$209$	103.650	0.0343045
$210$	372.287	0.122335
$211$	369.632	0.120600	0.0602999	−	0.998180i	$-0.480794\pi$
$211$	369.632	0.120600	0.0602999	+	0.998180i	$0.480794\pi$
$212$	−2465.82	−0.798836
$213$	−1956.55	−0.629392
$214$	221.772	0.0708413
$215$	−973.585	−0.308828
$216$	216.000	0.0680414
$217$	−1327.97	−0.415429
$218$	2569.30	0.798235
$219$	−2342.13	−0.722678
$220$	250.760	0.0768466
$221$	−2963.90	−0.902142
$222$	1814.98	0.548708
$223$	−1128.24	−0.338801	−0.169400	−	0.985547i	$-0.554183\pi$
$223$	−1128.24	−0.338801	−0.169400	+	0.985547i	$0.554183\pi$
$224$	224.000	0.0668153
$225$	−417.869	−0.123813
$226$	2472.21	0.727649
$227$	−2270.87	−0.663979	−0.331989	−	0.943283i	$-0.607720\pi$
$227$	−2270.87	−0.663979	−0.331989	+	0.943283i	$0.607720\pi$
$228$	175.866	0.0510834
$229$	−4859.77	−1.40237	−0.701184	−	0.712980i	$-0.747345\pi$
$229$	−4859.77	−1.40237	−0.701184	+	0.712980i	$0.747345\pi$
$230$	407.743	0.116895
$231$	−148.521	−0.0423030
$232$	9.87261	0.00279383
$233$	−1976.11	−0.555620	−0.277810	−	0.960636i	$-0.589609\pi$
$233$	−1976.11	−0.555620	−0.277810	+	0.960636i	$0.589609\pi$
$234$	561.514	0.156869
$235$	−791.386	−0.219678
$236$	−2121.53	−0.585169
$237$	−1266.54	−0.347132
$238$	1330.16	0.362275
$239$	−6215.45	−1.68219	−0.841097	−	0.540885i	$-0.818089\pi$
$239$	−6215.45	−1.68219	−0.841097	+	0.540885i	$0.818089\pi$
$240$	425.471	0.114433
$241$	−3069.22	−0.820355	−0.410178	−	0.912006i	$-0.634533\pi$
$241$	−3069.22	−0.820355	−0.410178	+	0.912006i	$0.634533\pi$
$242$	2561.96	0.680533
$243$	−243.000	−0.0641500
$244$	2301.48	0.603842
$245$	−434.335	−0.113260
$246$	−307.771	−0.0797672
$247$	457.181	0.117772
$248$	−1517.67	−0.388599
$249$	2031.88	0.517130
$250$	−3039.10	−0.768839
$251$	6490.49	1.63217	0.816087	−	0.577929i	$-0.196139\pi$
$251$	6490.49	1.63217	0.816087	+	0.577929i	$0.196139\pi$
$252$	−252.000	−0.0629941
$253$	−162.666	−0.0404219
$254$	−137.117	−0.0338720
$255$	2526.54	0.620462
$256$	256.000	0.0625000
$257$	−860.232	−0.208793	−0.104396	−	0.994536i	$-0.533291\pi$
$257$	−860.232	−0.208793	−0.104396	+	0.994536i	$0.533291\pi$
$258$	659.017	0.159025
$259$	−2117.47	−0.508005
$260$	1106.05	0.263825
$261$	−11.1067	−0.00263405
$262$	2271.19	0.535552
$263$	−2716.25	−0.636849	−0.318424	−	0.947948i	$-0.603154\pi$
$263$	−2716.25	−0.636849	−0.318424	+	0.947948i	$0.603154\pi$
$264$	−169.739	−0.0395708
$265$	5464.25	1.26666
$266$	−205.177	−0.0472940
$267$	3353.08	0.768558
$268$	860.480	0.196127
$269$	−6723.81	−1.52401	−0.762004	−	0.647573i	$-0.775784\pi$
$269$	−6723.81	−1.52401	−0.762004	+	0.647573i	$0.775784\pi$
$270$	−478.655	−0.107889
$271$	1258.57	0.282113	0.141056	−	0.990002i	$-0.454950\pi$
$271$	1258.57	0.282113	0.141056	+	0.990002i	$0.454950\pi$
$272$	1520.18	0.338877
$273$	−655.099	−0.145232
$274$	−277.373	−0.0611559
$275$	328.373	0.0720060
$276$	−276.000	−0.0601929
$277$	−1429.20	−0.310009	−0.155004	−	0.987914i	$-0.549539\pi$
$277$	−1429.20	−0.310009	−0.155004	+	0.987914i	$0.549539\pi$
$278$	−534.494	−0.115312
$279$	1707.38	0.366374
$280$	−496.383	−0.105945
$281$	3502.03	0.743465	0.371732	−	0.928340i	$-0.378764\pi$
$281$	3502.03	0.743465	0.371732	+	0.928340i	$0.378764\pi$
$282$	535.687	0.113119
$283$	5303.80	1.11406	0.557028	−	0.830493i	$-0.311941\pi$
$283$	5303.80	1.11406	0.557028	+	0.830493i	$0.311941\pi$
$284$	2608.73	0.545069
$285$	−389.718	−0.0809997
$286$	−441.253	−0.0912302
$287$	359.066	0.0738501
$288$	−288.000	−0.0589256
$289$	4114.17	0.837405
$290$	−21.8777	−0.00443000
$291$	−2660.18	−0.535886
$292$	3122.84	0.625858
$293$	4694.89	0.936103	0.468052	−	0.883701i	$-0.344956\pi$
$293$	4694.89	0.936103	0.468052	+	0.883701i	$0.344956\pi$
$294$	294.000	0.0583212
$295$	4701.30	0.927865
$296$	−2419.97	−0.475195
$297$	190.956	0.0373077
$298$	1312.40	0.255118
$299$	−717.490	−0.138774
$300$	557.159	0.107225
$301$	−768.853	−0.147229
$302$	−565.180	−0.107690
$303$	873.321	0.165581
$304$	−234.488	−0.0442395
$305$	−5100.08	−0.957474
$306$	−1710.21	−0.319497
$307$	3888.70	0.722931	0.361466	−	0.932385i	$-0.382276\pi$
$307$	3888.70	0.722931	0.361466	+	0.932385i	$0.382276\pi$
$308$	198.029	0.0366355
$309$	−10.4560	−0.00192499
$310$	3363.16	0.616176
$311$	−3039.42	−0.554179	−0.277090	−	0.960844i	$-0.589370\pi$
$311$	−3039.42	−0.554179	−0.277090	+	0.960844i	$0.589370\pi$
$312$	−748.685	−0.135852
$313$	7273.35	1.31346	0.656732	−	0.754124i	$-0.271938\pi$
$313$	7273.35	1.31346	0.656732	+	0.754124i	$0.271938\pi$
$314$	5245.09	0.942668
$315$	558.431	0.0998857
$316$	1688.72	0.300625
$317$	−1075.86	−0.190619	−0.0953094	−	0.995448i	$-0.530384\pi$
$317$	−1075.86	−0.190619	−0.0953094	+	0.995448i	$0.530384\pi$
$318$	−3698.73	−0.652247
$319$	8.72795	0.00153188
$320$	−567.295	−0.0991023
$321$	332.658	0.0578417
$322$	322.000	0.0557278
$323$	−1392.44	−0.239868
$324$	324.000	0.0555556
$325$	1448.39	0.247207
$326$	4124.06	0.700646
$327$	3853.96	0.651756
$328$	410.361	0.0690805
$329$	−624.968	−0.104728
$330$	376.140	0.0627450
$331$	−5865.92	−0.974079	−0.487040	−	0.873380i	$-0.661923\pi$
$331$	−5865.92	−0.974079	−0.487040	+	0.873380i	$0.661923\pi$
$332$	−2709.18	−0.447848
$333$	2722.46	0.448018
$334$	2357.84	0.386273
$335$	−1906.82	−0.310987
$336$	336.000	0.0545545
$337$	1257.03	0.203189	0.101594	−	0.994826i	$-0.467606\pi$
$337$	1257.03	0.203189	0.101594	+	0.994826i	$0.467606\pi$
$338$	2447.72	0.393901
$339$	3708.31	0.594123
$340$	−3368.72	−0.537336
$341$	−1341.71	−0.213072
$342$	263.799	0.0417094
$343$	−343.000	−0.0539949
$344$	−878.689	−0.137720
$345$	611.614	0.0954441
$346$	−2585.38	−0.401707
$347$	1428.32	0.220969	0.110485	−	0.993878i	$-0.464760\pi$
$347$	1428.32	0.220969	0.110485	+	0.993878i	$0.464760\pi$
$348$	14.8089	0.00228115
$349$	−2120.19	−0.325189	−0.162595	−	0.986693i	$-0.551986\pi$
$349$	−2120.19	−0.325189	−0.162595	+	0.986693i	$0.551986\pi$
$350$	−650.019	−0.0992713
$351$	842.270	0.128083
$352$	226.318	0.0342694
$353$	12152.4	1.83231	0.916157	−	0.400820i	$-0.131275\pi$
$353$	12152.4	1.83231	0.916157	+	0.400820i	$0.131275\pi$
$354$	−3182.29	−0.477788
$355$	−5780.93	−0.864282
$356$	−4470.77	−0.665591
$357$	1995.24	0.295796
$358$	4547.01	0.671276
$359$	8099.32	1.19071	0.595356	−	0.803462i	$-0.297011\pi$
$359$	8099.32	1.19071	0.595356	+	0.803462i	$0.297011\pi$
$360$	638.206	0.0934345
$361$	−6644.22	−0.968686
$362$	−3109.22	−0.451428
$363$	3842.94	0.555653
$364$	873.466	0.125775
$365$	−6920.20	−0.992383
$366$	3452.23	0.493035
$367$	7850.76	1.11664	0.558319	−	0.829626i	$-0.311446\pi$
$367$	7850.76	1.11664	0.558319	+	0.829626i	$0.311446\pi$
$368$	368.000	0.0521286
$369$	−461.656	−0.0651297
$370$	5362.64	0.753487
$371$	4315.19	0.603864
$372$	−2276.51	−0.317289
$373$	−8188.64	−1.13671	−0.568354	−	0.822784i	$-0.692419\pi$
$373$	−8188.64	−1.13671	−0.568354	+	0.822784i	$0.692419\pi$
$374$	1343.93	0.185810
$375$	−4558.65	−0.627754
$376$	−714.249	−0.0979643
$377$	38.4973	0.00525918
$378$	−378.000	−0.0514344
$379$	−4916.15	−0.666294	−0.333147	−	0.942875i	$-0.608111\pi$
$379$	−4916.15	−0.666294	−0.333147	+	0.942875i	$0.608111\pi$
$380$	519.624	0.0701478
$381$	−205.676	−0.0276564
$382$	4368.92	0.585166
$383$	2170.62	0.289592	0.144796	−	0.989462i	$-0.453747\pi$
$383$	2170.62	0.289592	0.144796	+	0.989462i	$0.453747\pi$
$384$	384.000	0.0510310
$385$	−438.830	−0.0580905
$386$	3434.65	0.452899
$387$	988.525	0.129844
$388$	3546.91	0.464091
$389$	−14404.1	−1.87742	−0.938708	−	0.344714i	$-0.887976\pi$
$389$	−14404.1	−1.87742	−0.938708	+	0.344714i	$0.887976\pi$
$390$	1659.08	0.215412
$391$	2185.26	0.282643
$392$	−392.000	−0.0505076
$393$	3406.78	0.437276
$394$	1497.35	0.191461
$395$	−3742.18	−0.476683
$396$	−254.608	−0.0323095
$397$	822.247	0.103948	0.0519740	−	0.998648i	$-0.483449\pi$
$397$	822.247	0.103948	0.0519740	+	0.998648i	$0.483449\pi$
$398$	1817.42	0.228892
$399$	−307.766	−0.0386154
$400$	−742.879	−0.0928598
$401$	6178.43	0.769416	0.384708	−	0.923038i	$-0.374302\pi$
$401$	6178.43	0.769416	0.384708	+	0.923038i	$0.374302\pi$
$402$	1290.72	0.160137
$403$	−5918.02	−0.731508
$404$	−1164.43	−0.143397
$405$	−717.982	−0.0880909
$406$	−17.2771	−0.00211194
$407$	−2139.39	−0.260554
$408$	2280.27	0.276692
$409$	7026.01	0.849423	0.424712	−	0.905329i	$-0.360376\pi$
$409$	7026.01	0.849423	0.424712	+	0.905329i	$0.360376\pi$
$410$	−909.357	−0.109536
$411$	−416.060	−0.0499336
$412$	13.9413	0.00166709
$413$	3712.68	0.442346
$414$	−414.000	−0.0491473
$415$	6003.52	0.710124
$416$	998.246	0.117652
$417$	−801.741	−0.0941521
$418$	−207.301	−0.0242570
$419$	16533.0	1.92766	0.963832	−	0.266511i	$-0.0858707\pi$
$419$	16533.0	1.92766	0.963832	+	0.266511i	$0.0858707\pi$
$420$	−744.574	−0.0865036
$421$	−1647.69	−0.190745	−0.0953725	−	0.995442i	$-0.530404\pi$
$421$	−1647.69	−0.190745	−0.0953725	+	0.995442i	$0.530404\pi$
$422$	−739.265	−0.0852769
$423$	803.530	0.0923616
$424$	4931.64	0.564863
$425$	−4411.37	−0.503489
$426$	3913.10	0.445047
$427$	−4027.60	−0.456462
$428$	−443.544	−0.0500923
$429$	−661.879	−0.0744891
$430$	1947.17	0.218374
$431$	−485.957	−0.0543103	−0.0271552	−	0.999631i	$-0.508645\pi$
$431$	−485.957	−0.0543103	−0.0271552	+	0.999631i	$0.508645\pi$
$432$	−432.000	−0.0481125
$433$	5600.73	0.621603	0.310802	−	0.950475i	$-0.399403\pi$
$433$	5600.73	0.621603	0.310802	+	0.950475i	$0.399403\pi$
$434$	2655.93	0.293753
$435$	−32.8165	−0.00361708
$436$	−5138.61	−0.564437
$437$	−337.077	−0.0368983
$438$	4684.26	0.511011
$439$	11011.8	1.19718	0.598591	−	0.801055i	$-0.295727\pi$
$439$	11011.8	1.19718	0.598591	+	0.801055i	$0.295727\pi$
$440$	−501.520	−0.0543387
$441$	441.000	0.0476190
$442$	5927.80	0.637911
$443$	8842.62	0.948365	0.474182	−	0.880427i	$-0.342744\pi$
$443$	8842.62	0.948365	0.474182	+	0.880427i	$0.342744\pi$
$444$	−3629.95	−0.387995
$445$	9907.20	1.05538
$446$	2256.48	0.239568
$447$	1968.60	0.208303
$448$	−448.000	−0.0472456
$449$	4573.14	0.480668	0.240334	−	0.970690i	$-0.422743\pi$
$449$	4573.14	0.480668	0.240334	+	0.970690i	$0.422743\pi$
$450$	835.738	0.0875491
$451$	362.782	0.0378775
$452$	−4944.41	−0.514526
$453$	−847.770	−0.0879287
$454$	4541.75	0.469504
$455$	−1935.59	−0.199433
$456$	−351.732	−0.0361214
$457$	5984.15	0.612531	0.306265	−	0.951946i	$-0.400920\pi$
$457$	5984.15	0.612531	0.306265	+	0.951946i	$0.400920\pi$
$458$	9719.53	0.991624
$459$	−2565.31	−0.260868
$460$	−815.486	−0.0826570
$461$	−6032.72	−0.609484	−0.304742	−	0.952435i	$-0.598570\pi$
$461$	−6032.72	−0.609484	−0.304742	+	0.952435i	$0.598570\pi$
$462$	297.043	0.0299127
$463$	14063.6	1.41164	0.705820	−	0.708391i	$-0.250579\pi$
$463$	14063.6	1.41164	0.705820	+	0.708391i	$0.250579\pi$
$464$	−19.7452	−0.00197554
$465$	5044.74	0.503106
$466$	3952.23	0.392883
$467$	6123.56	0.606776	0.303388	−	0.952867i	$-0.401882\pi$
$467$	6123.56	0.606776	0.303388	+	0.952867i	$0.401882\pi$
$468$	−1123.03	−0.110923
$469$	−1505.84	−0.148258
$470$	1582.77	0.155336
$471$	7867.64	0.769685
$472$	4243.06	0.413777
$473$	−776.810	−0.0755133
$474$	2533.07	0.245460
$475$	680.454	0.0657292
$476$	−2660.32	−0.256167
$477$	−5548.10	−0.532558
$478$	12430.9	1.18949
$479$	−9506.66	−0.906828	−0.453414	−	0.891300i	$-0.649794\pi$
$479$	−9506.66	−0.906828	−0.453414	+	0.891300i	$0.649794\pi$
$480$	−850.942	−0.0809167
$481$	−9436.42	−0.894519
$482$	6138.43	0.580079
$483$	483.000	0.0455016
$484$	−5123.92	−0.481210
$485$	−7859.94	−0.735879
$486$	486.000	0.0453609
$487$	−8709.17	−0.810370	−0.405185	−	0.914235i	$-0.632793\pi$
$487$	−8709.17	−0.810370	−0.405185	+	0.914235i	$0.632793\pi$
$488$	−4602.97	−0.426981
$489$	6186.09	0.572075
$490$	868.670	0.0800867
$491$	10337.2	0.950122	0.475061	−	0.879953i	$-0.342426\pi$
$491$	10337.2	0.950122	0.475061	+	0.879953i	$0.342426\pi$
$492$	615.541	0.0564040
$493$	−117.251	−0.0107114
$494$	−914.363	−0.0832776
$495$	564.210	0.0512310
$496$	3035.35	0.274781
$497$	−4565.28	−0.412034
$498$	−4063.77	−0.365666
$499$	−3483.64	−0.312524	−0.156262	−	0.987716i	$-0.549944\pi$
$499$	−3483.64	−0.312524	−0.156262	+	0.987716i	$0.549944\pi$
$500$	6078.20	0.543651
$501$	3536.76	0.315390
$502$	−12981.0	−1.15412
$503$	−4253.23	−0.377022	−0.188511	−	0.982071i	$-0.560366\pi$
$503$	−4253.23	−0.377022	−0.188511	+	0.982071i	$0.560366\pi$
$504$	504.000	0.0445435
$505$	2580.36	0.227376
$506$	325.333	0.0285826
$507$	3671.58	0.321618
$508$	274.234	0.0239511
$509$	−21427.2	−1.86590	−0.932951	−	0.360003i	$-0.882776\pi$
$509$	−21427.2	−1.86590	−0.932951	+	0.360003i	$0.882776\pi$
$510$	−5053.07	−0.438733
$511$	−5464.97	−0.473104
$512$	−512.000	−0.0441942
$513$	395.699	0.0340556
$514$	1720.46	0.147639
$515$	−30.8939	−0.00264339
$516$	−1318.03	−0.112448
$517$	−631.436	−0.0537148
$518$	4234.94	0.359214
$519$	−3878.07	−0.327993
$520$	−2212.11	−0.186553
$521$	−6521.15	−0.548362	−0.274181	−	0.961678i	$-0.588407\pi$
$521$	−6521.15	−0.548362	−0.274181	+	0.961678i	$0.588407\pi$
$522$	22.2134	0.00186255
$523$	−7822.43	−0.654017	−0.327009	−	0.945021i	$-0.606041\pi$
$523$	−7822.43	−0.654017	−0.327009	+	0.945021i	$0.606041\pi$
$524$	−4542.38	−0.378692
$525$	−975.028	−0.0810547
$526$	5432.50	0.450320
$527$	18024.6	1.48987
$528$	339.478	0.0279808
$529$	529.000	0.0434783
$530$	−10928.5	−0.895667
$531$	−4773.44	−0.390113
$532$	410.354	0.0334419
$533$	1600.16	0.130039
$534$	−6706.15	−0.543453
$535$	982.892	0.0794283
$536$	−1720.96	−0.138683
$537$	6820.51	0.548095
$538$	13447.6	1.07764
$539$	−346.550	−0.0276938
$540$	957.309	0.0762890
$541$	3831.80	0.304513	0.152257	−	0.988341i	$-0.451346\pi$
$541$	3831.80	0.304513	0.152257	+	0.988341i	$0.451346\pi$
$542$	−2517.13	−0.199484
$543$	−4663.82	−0.368589
$544$	−3040.37	−0.239622
$545$	11387.1	0.894993
$546$	1310.20	0.102695
$547$	8892.76	0.695113	0.347557	−	0.937659i	$-0.387011\pi$
$547$	8892.76	0.695113	0.347557	+	0.937659i	$0.387011\pi$
$548$	554.746	0.0432438
$549$	5178.34	0.402561
$550$	−656.747	−0.0509159
$551$	18.0860	0.00139835
$552$	552.000	0.0425628
$553$	−2955.25	−0.227252
$554$	2858.40	0.219209
$555$	8043.95	0.615219
$556$	1068.99	0.0815381
$557$	−18345.4	−1.39554	−0.697772	−	0.716319i	$-0.745825\pi$
$557$	−18345.4	−1.39554	−0.697772	+	0.716319i	$0.745825\pi$
$558$	−3414.77	−0.259066
$559$	−3426.36	−0.259248
$560$	992.765	0.0749143
$561$	2015.89	0.151713
$562$	−7004.06	−0.525709
$563$	−9236.77	−0.691445	−0.345722	−	0.938337i	$-0.612366\pi$
$563$	−9236.77	−0.691445	−0.345722	+	0.938337i	$0.612366\pi$
$564$	−1071.37	−0.0799875
$565$	10956.8	0.815850
$566$	−10607.6	−0.787757
$567$	−567.000	−0.0419961
$568$	−5217.46	−0.385422
$569$	−4585.46	−0.337843	−0.168921	−	0.985630i	$-0.554028\pi$
$569$	−4585.46	−0.337843	−0.168921	+	0.985630i	$0.554028\pi$
$570$	779.436	0.0572754
$571$	5223.28	0.382815	0.191408	−	0.981511i	$-0.438695\pi$
$571$	5223.28	0.382815	0.191408	+	0.981511i	$0.438695\pi$
$572$	882.506	0.0645095
$573$	6553.38	0.477786
$574$	−718.132	−0.0522199
$575$	−1067.89	−0.0774505
$576$	576.000	0.0416667
$577$	15089.5	1.08871	0.544353	−	0.838856i	$-0.316775\pi$
$577$	15089.5	1.08871	0.544353	+	0.838856i	$0.316775\pi$
$578$	−8228.34	−0.592135
$579$	5151.98	0.369791
$580$	43.7553	0.00313248
$581$	4741.06	0.338541
$582$	5320.37	0.378928
$583$	4359.85	0.309720
$584$	−6245.68	−0.442548
$585$	2488.62	0.175883
$586$	−9389.77	−0.661925
$587$	99.5023	0.00699642	0.00349821	−	0.999994i	$-0.498886\pi$
$587$	99.5023	0.00699642	0.00349821	+	0.999994i	$0.498886\pi$
$588$	−588.000	−0.0412393
$589$	−2780.29	−0.194499
$590$	−9402.60	−0.656100
$591$	2246.03	0.156327
$592$	4839.94	0.336014
$593$	−20653.8	−1.43027	−0.715136	−	0.698986i	$-0.753635\pi$
$593$	−20653.8	−1.43027	−0.715136	+	0.698986i	$0.753635\pi$
$594$	−381.912	−0.0263806
$595$	5895.25	0.406188
$596$	−2624.80	−0.180396
$597$	2726.13	0.186889
$598$	1434.98	0.0981282
$599$	−24466.0	−1.66887	−0.834435	−	0.551106i	$-0.814206\pi$
$599$	−24466.0	−1.66887	−0.834435	+	0.551106i	$0.814206\pi$
$600$	−1114.32	−0.0758197
$601$	−9107.53	−0.618143	−0.309071	−	0.951039i	$-0.600018\pi$
$601$	−9107.53	−0.618143	−0.309071	+	0.951039i	$0.600018\pi$
$602$	1537.71	0.104107
$603$	1936.08	0.130752
$604$	1130.36	0.0761485
$605$	11354.6	0.763024
$606$	−1746.64	−0.117083
$607$	−11822.0	−0.790509	−0.395254	−	0.918572i	$-0.629344\pi$
$607$	−11822.0	−0.790509	−0.395254	+	0.918572i	$0.629344\pi$
$608$	468.976	0.0312821
$609$	−25.9156	−0.00172439
$610$	10200.2	0.677036
$611$	−2785.14	−0.184410
$612$	3420.41	0.225918
$613$	10183.0	0.670943	0.335471	−	0.942050i	$-0.391104\pi$
$613$	10183.0	0.670943	0.335471	+	0.942050i	$0.391104\pi$
$614$	−7777.40	−0.511190
$615$	−1364.04	−0.0894362
$616$	−396.057	−0.0259052
$617$	−16358.6	−1.06738	−0.533688	−	0.845682i	$-0.679194\pi$
$617$	−16358.6	−1.06738	−0.533688	+	0.845682i	$0.679194\pi$
$618$	20.9120	0.00136117
$619$	1415.05	0.0918829	0.0459415	−	0.998944i	$-0.485371\pi$
$619$	1415.05	0.0918829	0.0459415	+	0.998944i	$0.485371\pi$
$620$	−6726.32	−0.435702
$621$	−621.000	−0.0401286
$622$	6078.84	0.391864
$623$	7823.85	0.503139
$624$	1497.37	0.0960621
$625$	−7665.52	−0.490593
$626$	−14546.7	−0.928759
$627$	−310.951	−0.0198057
$628$	−10490.2	−0.666567
$629$	28740.6	1.82188
$630$	−1116.86	−0.0706299
$631$	−22696.5	−1.43190	−0.715952	−	0.698149i	$-0.754007\pi$
$631$	−22696.5	−1.43190	−0.715952	+	0.698149i	$0.754007\pi$
$632$	−3377.43	−0.212574
$633$	−1108.90	−0.0696283
$634$	2151.71	0.134788
$635$	−607.702	−0.0379778
$636$	7397.47	0.461208
$637$	−1528.56	−0.0950768
$638$	−17.4559	−0.00108321
$639$	5869.65	0.363380
$640$	1134.59	0.0700759
$641$	−3899.34	−0.240273	−0.120136	−	0.992757i	$-0.538333\pi$
$641$	−3899.34	−0.240273	−0.120136	+	0.992757i	$0.538333\pi$
$642$	−665.317	−0.0409002
$643$	−9407.00	−0.576945	−0.288473	−	0.957488i	$-0.593147\pi$
$643$	−9407.00	−0.576945	−0.288473	+	0.957488i	$0.593147\pi$
$644$	−644.000	−0.0394055
$645$	2920.75	0.178302
$646$	2784.88	0.169612
$647$	17260.3	1.04880	0.524399	−	0.851472i	$-0.324290\pi$
$647$	17260.3	1.04880	0.524399	+	0.851472i	$0.324290\pi$
$648$	−648.000	−0.0392837
$649$	3751.10	0.226878
$650$	−2896.78	−0.174802
$651$	3983.90	0.239848
$652$	−8248.13	−0.495432
$653$	−10536.3	−0.631419	−0.315710	−	0.948856i	$-0.602243\pi$
$653$	−10536.3	−0.631419	−0.315710	+	0.948856i	$0.602243\pi$
$654$	−7707.91	−0.460861
$655$	10065.9	0.600468
$656$	−820.722	−0.0488473
$657$	7026.39	0.417238
$658$	1249.94	0.0740540
$659$	1603.93	0.0948105	0.0474053	−	0.998876i	$-0.484905\pi$
$659$	1603.93	0.0948105	0.0474053	+	0.998876i	$0.484905\pi$
$660$	−752.280	−0.0443674
$661$	−30640.2	−1.80297	−0.901487	−	0.432807i	$-0.857523\pi$
$661$	−30640.2	−1.80297	−0.901487	+	0.432807i	$0.857523\pi$
$662$	11731.8	0.688778
$663$	8891.70	0.520852
$664$	5418.36	0.316676
$665$	−909.342	−0.0530267
$666$	−5444.93	−0.316797
$667$	−28.3838	−0.00164771
$668$	−4715.68	−0.273136
$669$	3384.72	0.195607
$670$	3813.64	0.219901
$671$	−4069.28	−0.234118
$672$	−672.000	−0.0385758
$673$	2008.11	0.115018	0.0575088	−	0.998345i	$-0.481684\pi$
$673$	2008.11	0.115018	0.0575088	+	0.998345i	$0.481684\pi$
$674$	−2514.05	−0.143676
$675$	1253.61	0.0714835
$676$	−4895.44	−0.278530
$677$	−27391.3	−1.55500	−0.777498	−	0.628885i	$-0.783512\pi$
$677$	−27391.3	−1.55500	−0.777498	+	0.628885i	$0.783512\pi$
$678$	−7416.62	−0.420108
$679$	−6207.10	−0.350820
$680$	6737.43	0.379954
$681$	6812.62	0.383348
$682$	2683.42	0.150665
$683$	14923.3	0.836054	0.418027	−	0.908435i	$-0.362722\pi$
$683$	14923.3	0.836054	0.418027	+	0.908435i	$0.362722\pi$
$684$	−527.598	−0.0294930
$685$	−1229.31	−0.0685689
$686$	686.000	0.0381802
$687$	14579.3	0.809658
$688$	1757.38	0.0973828
$689$	19230.4	1.06331
$690$	−1223.23	−0.0674892
$691$	18561.2	1.02185	0.510926	−	0.859624i	$-0.329302\pi$
$691$	18561.2	1.02185	0.510926	+	0.859624i	$0.329302\pi$
$692$	5170.76	0.284050
$693$	445.564	0.0244237
$694$	−2856.64	−0.156249
$695$	−2368.87	−0.129290
$696$	−29.6178	−0.00161302
$697$	−4873.62	−0.264852
$698$	4240.37	0.229943
$699$	5928.34	0.320787
$700$	1300.04	0.0701954
$701$	−30458.8	−1.64110	−0.820552	−	0.571571i	$-0.806334\pi$
$701$	−30458.8	−1.64110	−0.820552	+	0.571571i	$0.806334\pi$
$702$	−1684.54	−0.0905682
$703$	−4433.23	−0.237841
$704$	−452.637	−0.0242321
$705$	2374.16	0.126831
$706$	−24304.8	−1.29564
$707$	2037.75	0.108398
$708$	6364.59	0.337847
$709$	−11544.9	−0.611535	−0.305767	−	0.952106i	$-0.598913\pi$
$709$	−11544.9	−0.611535	−0.305767	+	0.952106i	$0.598913\pi$
$710$	11561.9	0.611140
$711$	3799.61	0.200417
$712$	8941.54	0.470644
$713$	4363.32	0.229183
$714$	−3990.48	−0.209160
$715$	−1955.63	−0.102289
$716$	−9094.02	−0.474664
$717$	18646.4	0.971215
$718$	−16198.6	−0.841960
$719$	27470.5	1.42486	0.712431	−	0.701743i	$-0.247594\pi$
$719$	27470.5	1.42486	0.712431	+	0.701743i	$0.247594\pi$
$720$	−1276.41	−0.0660682
$721$	−24.3973	−0.00126020
$722$	13288.4	0.684964
$723$	9207.65	0.473632
$724$	6218.43	0.319208
$725$	57.2981	0.00293517
$726$	−7685.88	−0.392906
$727$	9393.33	0.479201	0.239601	−	0.970872i	$-0.422983\pi$
$727$	9393.33	0.479201	0.239601	+	0.970872i	$0.422983\pi$
$728$	−1746.93	−0.0889362
$729$	729.000	0.0370370
$730$	13840.4	0.701720
$731$	10435.7	0.528013
$732$	−6904.45	−0.348628
$733$	−26141.4	−1.31727	−0.658633	−	0.752464i	$-0.728865\pi$
$733$	−26141.4	−1.31727	−0.658633	+	0.752464i	$0.728865\pi$
$734$	−15701.5	−0.789583
$735$	1303.00	0.0653905
$736$	−736.000	−0.0368605
$737$	−1521.42	−0.0760413
$738$	923.312	0.0460536
$739$	−27158.3	−1.35188	−0.675938	−	0.736959i	$-0.736261\pi$
$739$	−27158.3	−1.35188	−0.675938	+	0.736959i	$0.736261\pi$
$740$	−10725.3	−0.532796
$741$	−1371.54	−0.0679959
$742$	−8630.38	−0.426996
$743$	−4914.95	−0.242681	−0.121340	−	0.992611i	$-0.538719\pi$
$743$	−4914.95	−0.242681	−0.121340	+	0.992611i	$0.538719\pi$
$744$	4553.02	0.224358
$745$	5816.53	0.286042
$746$	16377.3	0.803773
$747$	−6095.65	−0.298565
$748$	−2687.85	−0.131387
$749$	776.203	0.0378663
$750$	9117.30	0.443889
$751$	1998.90	0.0971252	0.0485626	−	0.998820i	$-0.484536\pi$
$751$	1998.90	0.0971252	0.0485626	+	0.998820i	$0.484536\pi$
$752$	1428.50	0.0692712
$753$	−19471.5	−0.942336
$754$	−76.9945	−0.00371880
$755$	−2504.87	−0.120744
$756$	756.000	0.0363696
$757$	35871.0	1.72226	0.861131	−	0.508382i	$-0.169756\pi$
$757$	35871.0	1.72226	0.861131	+	0.508382i	$0.169756\pi$
$758$	9832.29	0.471141
$759$	487.999	0.0233376
$760$	−1039.25	−0.0496020
$761$	−12082.7	−0.575557	−0.287778	−	0.957697i	$-0.592917\pi$
$761$	−12082.7	−0.575557	−0.287778	+	0.957697i	$0.592917\pi$
$762$	411.352	0.0195560
$763$	8992.57	0.426675
$764$	−8737.84	−0.413775
$765$	−7579.61	−0.358224
$766$	−4341.25	−0.204772
$767$	16545.4	0.778904
$768$	−768.000	−0.0360844
$769$	−24222.2	−1.13586	−0.567929	−	0.823077i	$-0.692255\pi$
$769$	−24222.2	−1.13586	−0.567929	+	0.823077i	$0.692255\pi$
$770$	877.660	0.0410762
$771$	2580.70	0.120547
$772$	−6869.30	−0.320248
$773$	−26693.5	−1.24204	−0.621020	−	0.783794i	$-0.713282\pi$
$773$	−26693.5	−1.24204	−0.621020	+	0.783794i	$0.713282\pi$
$774$	−1977.05	−0.0918134
$775$	−8808.19	−0.408257
$776$	−7093.82	−0.328162
$777$	6352.42	0.293297
$778$	28808.1	1.32753
$779$	751.756	0.0345757
$780$	−3318.16	−0.152320
$781$	−4612.53	−0.211331
$782$	−4370.53	−0.199859
$783$	33.3201	0.00152077
$784$	784.000	0.0357143
$785$	23246.2	1.05693
$786$	−6813.57	−0.309201
$787$	25632.9	1.16101	0.580504	−	0.814257i	$-0.302855\pi$
$787$	25632.9	1.16101	0.580504	+	0.814257i	$0.302855\pi$
$788$	−2994.71	−0.135383
$789$	8148.75	0.367685
$790$	7484.37	0.337066
$791$	8652.72	0.388945
$792$	509.216	0.0228462
$793$	−17948.8	−0.803759
$794$	−1644.49	−0.0735024
$795$	−16392.7	−0.731309
$796$	−3634.84	−0.161851
$797$	−15592.2	−0.692977	−0.346488	−	0.938054i	$-0.612626\pi$
$797$	−15592.2	−0.692977	−0.346488	+	0.938054i	$0.612626\pi$
$798$	615.531	0.0273052
$799$	8482.72	0.375591
$800$	1485.76	0.0656618
$801$	−10059.2	−0.443727
$802$	−12356.9	−0.544059
$803$	−5521.53	−0.242653
$804$	−2581.44	−0.113234
$805$	1427.10	0.0624828
$806$	11836.0	0.517254
$807$	20171.4	0.879886
$808$	2328.86	0.101397
$809$	−2033.82	−0.0883874	−0.0441937	−	0.999023i	$-0.514072\pi$
$809$	−2033.82	−0.0883874	−0.0441937	+	0.999023i	$0.514072\pi$
$810$	1435.96	0.0622897
$811$	8404.94	0.363918	0.181959	−	0.983306i	$-0.441756\pi$
$811$	8404.94	0.363918	0.181959	+	0.983306i	$0.441756\pi$
$812$	34.5541	0.00149337
$813$	−3775.70	−0.162878
$814$	4278.78	0.184240
$815$	18277.8	0.785575
$816$	−4560.55	−0.195651
$817$	−1609.70	−0.0689307
$818$	−14052.0	−0.600633
$819$	1965.30	0.0838499
$820$	1818.71	0.0774540
$821$	−6266.03	−0.266365	−0.133183	−	0.991092i	$-0.542520\pi$
$821$	−6266.03	−0.266365	−0.133183	+	0.991092i	$0.542520\pi$
$822$	832.120	0.0353084
$823$	−38871.7	−1.64639	−0.823197	−	0.567757i	$-0.807812\pi$
$823$	−38871.7	−1.64639	−0.823197	+	0.567757i	$0.807812\pi$
$824$	−27.8826	−0.00117881
$825$	−985.120	−0.0415727
$826$	−7425.35	−0.312786
$827$	36494.1	1.53449	0.767245	−	0.641354i	$-0.221627\pi$
$827$	36494.1	1.53449	0.767245	+	0.641354i	$0.221627\pi$
$828$	828.000	0.0347524
$829$	−31297.3	−1.31122	−0.655609	−	0.755101i	$-0.727588\pi$
$829$	−31297.3	−1.31122	−0.655609	+	0.755101i	$0.727588\pi$
$830$	−12007.0	−0.502133
$831$	4287.60	0.178984
$832$	−1996.49	−0.0831922
$833$	4655.56	0.193644
$834$	1603.48	0.0665756
$835$	10449.9	0.433095
$836$	414.601	0.0171523
$837$	−5122.15	−0.211526
$838$	−33066.1	−1.36306
$839$	−8949.88	−0.368277	−0.184138	−	0.982900i	$-0.558949\pi$
$839$	−8949.88	−0.368277	−0.184138	+	0.982900i	$0.558949\pi$
$840$	1489.15	0.0611673
$841$	−24387.5	−0.999938
$842$	3295.39	0.134877
$843$	−10506.1	−0.429240
$844$	1478.53	0.0602999
$845$	10848.3	0.441647
$846$	−1607.06	−0.0653095
$847$	8966.86	0.363760
$848$	−9863.29	−0.399418
$849$	−15911.4	−0.643201
$850$	8822.74	0.356021
$851$	6957.41	0.280255
$852$	−7826.19	−0.314696
$853$	42079.5	1.68907	0.844533	−	0.535503i	$-0.179878\pi$
$853$	42079.5	1.68907	0.844533	+	0.535503i	$0.179878\pi$
$854$	8055.19	0.322767
$855$	1169.15	0.0467652
$856$	887.089	0.0354206
$857$	−40504.6	−1.61448	−0.807241	−	0.590222i	$-0.799040\pi$
$857$	−40504.6	−1.61448	−0.807241	+	0.590222i	$0.799040\pi$
$858$	1323.76	0.0526718
$859$	−39288.9	−1.56056	−0.780279	−	0.625431i	$-0.784923\pi$
$859$	−39288.9	−1.56056	−0.780279	+	0.625431i	$0.784923\pi$
$860$	−3894.34	−0.154414
$861$	−1077.20	−0.0426374
$862$	971.915	0.0384032
$863$	31531.3	1.24373	0.621865	−	0.783124i	$-0.286375\pi$
$863$	31531.3	1.24373	0.621865	+	0.783124i	$0.286375\pi$
$864$	864.000	0.0340207
$865$	−11458.4	−0.450400
$866$	−11201.5	−0.439540
$867$	−12342.5	−0.483476
$868$	−5311.86	−0.207715
$869$	−2985.84	−0.116557
$870$	65.6330	0.00255766
$871$	−6710.71	−0.261060
$872$	10277.2	0.399118
$873$	7980.55	0.309394
$874$	674.153	0.0260910
$875$	−10636.9	−0.410962
$876$	−9368.52	−0.361339
$877$	17490.8	0.673460	0.336730	−	0.941601i	$-0.390679\pi$
$877$	17490.8	0.673460	0.336730	+	0.941601i	$0.390679\pi$
$878$	−22023.5	−0.846536
$879$	−14084.7	−0.540459
$880$	1003.04	0.0384233
$881$	13868.5	0.530355	0.265177	−	0.964200i	$-0.414570\pi$
$881$	13868.5	0.530355	0.265177	+	0.964200i	$0.414570\pi$
$882$	−882.000	−0.0336718
$883$	38927.4	1.48359	0.741796	−	0.670626i	$-0.233974\pi$
$883$	38927.4	1.48359	0.741796	+	0.670626i	$0.233974\pi$
$884$	−11855.6	−0.451071
$885$	−14103.9	−0.535703
$886$	−17685.2	−0.670595
$887$	26936.5	1.01966	0.509830	−	0.860275i	$-0.329708\pi$
$887$	26936.5	1.01966	0.509830	+	0.860275i	$0.329708\pi$
$888$	7259.90	0.274354
$889$	−479.910	−0.0181054
$890$	−19814.4	−0.746270
$891$	−572.868	−0.0215396
$892$	−4512.96	−0.169400
$893$	−1308.46	−0.0490324
$894$	−3937.20	−0.147293
$895$	20152.3	0.752644
$896$	896.000	0.0334077
$897$	2152.47	0.0801213
$898$	−9146.28	−0.339884
$899$	−234.116	−0.00868543
$900$	−1671.48	−0.0619066
$901$	−58570.3	−2.16566
$902$	−725.564	−0.0267834
$903$	2306.56	0.0850027
$904$	9888.82	0.363825
$905$	−13780.0	−0.506147
$906$	1695.54	0.0621750
$907$	−34604.4	−1.26684	−0.633418	−	0.773810i	$-0.718349\pi$
$907$	−34604.4	−1.26684	−0.633418	+	0.773810i	$0.718349\pi$
$908$	−9083.50	−0.331989
$909$	−2619.96	−0.0955981
$910$	3871.19	0.141021
$911$	16749.6	0.609155	0.304578	−	0.952488i	$-0.401485\pi$
$911$	16749.6	0.609155	0.304578	+	0.952488i	$0.401485\pi$
$912$	703.464	0.0255417
$913$	4790.13	0.173637
$914$	−11968.3	−0.433125
$915$	15300.2	0.552798
$916$	−19439.1	−0.701184
$917$	7949.16	0.286264
$918$	5130.62	0.184461
$919$	−44592.5	−1.60062	−0.800311	−	0.599586i	$-0.795332\pi$
$919$	−44592.5	−1.60062	−0.800311	+	0.599586i	$0.795332\pi$
$920$	1630.97	0.0584473
$921$	−11666.1	−0.417385
$922$	12065.4	0.430970
$923$	−20345.0	−0.725529
$924$	−594.086	−0.0211515
$925$	−14044.9	−0.499235
$926$	−28127.1	−0.998181
$927$	31.3680	0.00111139
$928$	39.4905	0.00139692
$929$	−7759.49	−0.274037	−0.137019	−	0.990568i	$-0.543752\pi$
$929$	−7759.49	−0.274037	−0.137019	+	0.990568i	$0.543752\pi$
$930$	−10089.5	−0.355749
$931$	−718.120	−0.0252797
$932$	−7904.45	−0.277810
$933$	9118.27	0.319956
$934$	−12247.1	−0.429056
$935$	5956.27	0.208332
$936$	2246.05	0.0784344
$937$	−24652.7	−0.859517	−0.429758	−	0.902944i	$-0.641401\pi$
$937$	−24652.7	−0.859517	−0.429758	+	0.902944i	$0.641401\pi$
$938$	3011.68	0.104835
$939$	−21820.0	−0.758328
$940$	−3165.54	−0.109839
$941$	18584.8	0.643832	0.321916	−	0.946768i	$-0.395673\pi$
$941$	18584.8	0.643832	0.321916	+	0.946768i	$0.395673\pi$
$942$	−15735.3	−0.544250
$943$	−1179.79	−0.0407414
$944$	−8486.12	−0.292584
$945$	−1675.29	−0.0576690
$946$	1553.62	0.0533959
$947$	31989.7	1.09770	0.548852	−	0.835920i	$-0.315065\pi$
$947$	31989.7	1.09770	0.548852	+	0.835920i	$0.315065\pi$
$948$	−5066.15	−0.173566
$949$	−24354.4	−0.833064
$950$	−1360.91	−0.0464775
$951$	3227.57	0.110054
$952$	5320.64	0.181138
$953$	−24046.7	−0.817366	−0.408683	−	0.912676i	$-0.634012\pi$
$953$	−24046.7	−0.817366	−0.408683	+	0.912676i	$0.634012\pi$
$954$	11096.2	0.376575
$955$	19363.0	0.656096
$956$	−24861.8	−0.841097
$957$	−26.1838	−0.000884434 0
$958$	19013.3	0.641224
$959$	−970.806	−0.0326892
$960$	1701.88	0.0572167
$961$	6198.65	0.208071
$962$	18872.8	0.632521
$963$	−997.975	−0.0333949
$964$	−12276.9	−0.410178
$965$	15222.3	0.507797
$966$	−966.000	−0.0321745
$967$	−24812.6	−0.825149	−0.412574	−	0.910924i	$-0.635370\pi$
$967$	−24812.6	−0.825149	−0.412574	+	0.910924i	$0.635370\pi$
$968$	10247.8	0.340267
$969$	4177.32	0.138488
$970$	15719.9	0.520345
$971$	16357.7	0.540622	0.270311	−	0.962773i	$-0.412873\pi$
$971$	16357.7	0.540622	0.270311	+	0.962773i	$0.412873\pi$
$972$	−972.000	−0.0320750
$973$	−1870.73	−0.0616370
$974$	17418.3	0.573018
$975$	−4345.17	−0.142725
$976$	9205.94	0.301921
$977$	−8475.06	−0.277524	−0.138762	−	0.990326i	$-0.544312\pi$
$977$	−8475.06	−0.277524	−0.138762	+	0.990326i	$0.544312\pi$
$978$	−12372.2	−0.404518
$979$	7904.82	0.258058
$980$	−1737.34	−0.0566299
$981$	−11561.9	−0.376292
$982$	−20674.3	−0.671838
$983$	−24951.8	−0.809601	−0.404801	−	0.914405i	$-0.632659\pi$
$983$	−24951.8	−0.809601	−0.404801	+	0.914405i	$0.632659\pi$
$984$	−1231.08	−0.0398836
$985$	6636.26	0.214669
$986$	234.503	0.00757413
$987$	1874.90	0.0604649
$988$	1828.73	0.0588861
$989$	2526.23	0.0812229
$990$	−1128.42	−0.0362258
$991$	−11445.6	−0.366885	−0.183442	−	0.983030i	$-0.558724\pi$
$991$	−11445.6	−0.366885	−0.183442	+	0.983030i	$0.558724\pi$
$992$	−6070.70	−0.194299
$993$	17597.8	0.562385
$994$	9130.56	0.291352
$995$	8054.77	0.256637
$996$	8127.54	0.258565
$997$	−16234.6	−0.515702	−0.257851	−	0.966185i	$-0.583014\pi$
$997$	−16234.6	−0.515702	−0.257851	+	0.966185i	$0.583014\pi$
$998$	6967.29	0.220988
$999$	−8167.39	−0.258664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.4.a.h.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.4.a.h.1.2	✓	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	966.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.86398 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.86398 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +17.7280 q^{10} -7.07245 q^{11} -12.0000 q^{12} -31.1952 q^{13} +14.0000 q^{14} +26.5919 q^{15} +16.0000 q^{16} +95.0114 q^{17} -18.0000 q^{18} -14.6555 q^{19} -35.4559 q^{20} +21.0000 q^{21} +14.1449 q^{22} +23.0000 q^{23} +24.0000 q^{24} -46.4299 q^{25} +62.3904 q^{26} -27.0000 q^{27} -28.0000 q^{28} -1.23408 q^{29} -53.1839 q^{30} +189.709 q^{31} -32.0000 q^{32} +21.2174 q^{33} -190.023 q^{34} +62.0478 q^{35} +36.0000 q^{36} +302.496 q^{37} +29.3110 q^{38} +93.5856 q^{39} +70.9118 q^{40} -51.2951 q^{41} -42.0000 q^{42} +109.836 q^{43} -28.2898 q^{44} -79.7758 q^{45} -46.0000 q^{46} +89.2811 q^{47} -48.0000 q^{48} +49.0000 q^{49} +92.8598 q^{50} -285.034 q^{51} -124.781 q^{52} -616.455 q^{53} +54.0000 q^{54} +62.6900 q^{55} +56.0000 q^{56} +43.9665 q^{57} +2.46815 q^{58} -530.382 q^{59} +106.368 q^{60} +575.371 q^{61} -379.419 q^{62} -63.0000 q^{63} +64.0000 q^{64} +276.514 q^{65} -42.4347 q^{66} +215.120 q^{67} +380.046 q^{68} -69.0000 q^{69} -124.096 q^{70} +652.183 q^{71} -72.0000 q^{72} +780.710 q^{73} -604.992 q^{74} +139.290 q^{75} -58.6220 q^{76} +49.5072 q^{77} -187.171 q^{78} +422.179 q^{79} -141.824 q^{80} +81.0000 q^{81} +102.590 q^{82} -677.295 q^{83} +84.0000 q^{84} -842.179 q^{85} -219.672 q^{86} +3.70223 q^{87} +56.5796 q^{88} -1117.69 q^{89} +159.552 q^{90} +218.366 q^{91} +92.0000 q^{92} -569.128 q^{93} -178.562 q^{94} +129.906 q^{95} +96.0000 q^{96} +886.728 q^{97} -98.0000 q^{98} -63.6521 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 5 q^{5} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 5 * q^5 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 5 q^{5} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9} + 10 q^{10} - 41 q^{11} - 48 q^{12} + 23 q^{13} + 56 q^{14} + 15 q^{15} + 64 q^{16} + 18 q^{17} - 72 q^{18} + 15 q^{19} - 20 q^{20} + 84 q^{21} + 82 q^{22} + 92 q^{23} + 96 q^{24} + 219 q^{25} - 46 q^{26} - 108 q^{27} - 112 q^{28} - 46 q^{29} - 30 q^{30} + 142 q^{31} - 128 q^{32} + 123 q^{33} - 36 q^{34} + 35 q^{35} + 144 q^{36} - 142 q^{37} - 30 q^{38} - 69 q^{39} + 40 q^{40} - 621 q^{41} - 168 q^{42} + 185 q^{43} - 164 q^{44} - 45 q^{45} - 184 q^{46} + 669 q^{47} - 192 q^{48} + 196 q^{49} - 438 q^{50} - 54 q^{51} + 92 q^{52} + 422 q^{53} + 216 q^{54} + 1435 q^{55} + 224 q^{56} - 45 q^{57} + 92 q^{58} + 270 q^{59} + 60 q^{60} + 272 q^{61} - 284 q^{62} - 252 q^{63} + 256 q^{64} - 1992 q^{65} - 246 q^{66} - 67 q^{67} + 72 q^{68} - 276 q^{69} - 70 q^{70} + 611 q^{71} - 288 q^{72} - 236 q^{73} + 284 q^{74} - 657 q^{75} + 60 q^{76} + 287 q^{77} + 138 q^{78} + 558 q^{79} - 80 q^{80} + 324 q^{81} + 1242 q^{82} + 468 q^{83} + 336 q^{84} + 1045 q^{85} - 370 q^{86} + 138 q^{87} + 328 q^{88} - 1519 q^{89} + 90 q^{90} - 161 q^{91} + 368 q^{92} - 426 q^{93} - 1338 q^{94} + 23 q^{95} + 384 q^{96} + 600 q^{97} - 392 q^{98} - 369 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 5 * q^5 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9 + 10 * q^10 - 41 * q^11 - 48 * q^12 + 23 * q^13 + 56 * q^14 + 15 * q^15 + 64 * q^16 + 18 * q^17 - 72 * q^18 + 15 * q^19 - 20 * q^20 + 84 * q^21 + 82 * q^22 + 92 * q^23 + 96 * q^24 + 219 * q^25 - 46 * q^26 - 108 * q^27 - 112 * q^28 - 46 * q^29 - 30 * q^30 + 142 * q^31 - 128 * q^32 + 123 * q^33 - 36 * q^34 + 35 * q^35 + 144 * q^36 - 142 * q^37 - 30 * q^38 - 69 * q^39 + 40 * q^40 - 621 * q^41 - 168 * q^42 + 185 * q^43 - 164 * q^44 - 45 * q^45 - 184 * q^46 + 669 * q^47 - 192 * q^48 + 196 * q^49 - 438 * q^50 - 54 * q^51 + 92 * q^52 + 422 * q^53 + 216 * q^54 + 1435 * q^55 + 224 * q^56 - 45 * q^57 + 92 * q^58 + 270 * q^59 + 60 * q^60 + 272 * q^61 - 284 * q^62 - 252 * q^63 + 256 * q^64 - 1992 * q^65 - 246 * q^66 - 67 * q^67 + 72 * q^68 - 276 * q^69 - 70 * q^70 + 611 * q^71 - 288 * q^72 - 236 * q^73 + 284 * q^74 - 657 * q^75 + 60 * q^76 + 287 * q^77 + 138 * q^78 + 558 * q^79 - 80 * q^80 + 324 * q^81 + 1242 * q^82 + 468 * q^83 + 336 * q^84 + 1045 * q^85 - 370 * q^86 + 138 * q^87 + 328 * q^88 - 1519 * q^89 + 90 * q^90 - 161 * q^91 + 368 * q^92 - 426 * q^93 - 1338 * q^94 + 23 * q^95 + 384 * q^96 + 600 * q^97 - 392 * q^98 - 369 * q^99

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