LMFDB - Embedded newform 23.4.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,4,Mod(1,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("23.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	23.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:	$1.35704393013$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.334189.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 16x^{2} - 5x + 4 $ x^4 - 2x^3 - 16x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.22031$ of defining polynomial
Character	$\chi$	$=$	23.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0323756 q^{2} +6.42170 q^{3} -7.99895 q^{4} +14.1026 q^{5} -0.207906 q^{6} -14.0109 q^{7} +0.517976 q^{8} +14.2382 q^{9} +O(q^{10})$	\(q-0.0323756 q^{2} +6.42170 q^{3} -7.99895 q^{4} +14.1026 q^{5} -0.207906 q^{6} -14.0109 q^{7} +0.517976 q^{8} +14.2382 q^{9} -0.456580 q^{10} -55.5140 q^{11} -51.3668 q^{12} -18.3149 q^{13} +0.453613 q^{14} +90.5626 q^{15} +63.9748 q^{16} +10.0273 q^{17} -0.460970 q^{18} +161.106 q^{19} -112.806 q^{20} -89.9740 q^{21} +1.79730 q^{22} -23.0000 q^{23} +3.32628 q^{24} +73.8833 q^{25} +0.592956 q^{26} -81.9525 q^{27} +112.073 q^{28} +183.185 q^{29} -2.93202 q^{30} -144.762 q^{31} -6.21503 q^{32} -356.494 q^{33} -0.324640 q^{34} -197.591 q^{35} -113.891 q^{36} +181.411 q^{37} -5.21591 q^{38} -117.613 q^{39} +7.30481 q^{40} +77.7996 q^{41} +2.91296 q^{42} +315.335 q^{43} +444.054 q^{44} +200.795 q^{45} +0.744639 q^{46} -524.190 q^{47} +410.827 q^{48} -146.693 q^{49} -2.39202 q^{50} +64.3923 q^{51} +146.500 q^{52} +73.7334 q^{53} +2.65326 q^{54} -782.892 q^{55} -7.25733 q^{56} +1034.57 q^{57} -5.93071 q^{58} -132.892 q^{59} -724.406 q^{60} +236.683 q^{61} +4.68676 q^{62} -199.490 q^{63} -511.598 q^{64} -258.288 q^{65} +11.5417 q^{66} +493.624 q^{67} -80.2079 q^{68} -147.699 q^{69} +6.39712 q^{70} -806.060 q^{71} +7.37504 q^{72} +1011.91 q^{73} -5.87328 q^{74} +474.456 q^{75} -1288.68 q^{76} +777.804 q^{77} +3.80779 q^{78} -599.386 q^{79} +902.212 q^{80} -910.705 q^{81} -2.51881 q^{82} -642.245 q^{83} +719.698 q^{84} +141.411 q^{85} -10.2092 q^{86} +1176.36 q^{87} -28.7549 q^{88} +883.399 q^{89} -6.50088 q^{90} +256.609 q^{91} +183.976 q^{92} -929.617 q^{93} +16.9710 q^{94} +2272.02 q^{95} -39.9111 q^{96} -71.2938 q^{97} +4.74929 q^{98} -790.419 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9	\( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9} - 70 q^{10} + 8 q^{11} - 67 q^{12} + 111 q^{13} - 144 q^{14} + 10 q^{15} + 64 q^{16} + 98 q^{17} + 49 q^{18} + 96 q^{19} + 140 q^{20} + 180 q^{21} + 220 q^{22} - 92 q^{23} - 188 q^{24} + 184 q^{25} - 229 q^{26} - 155 q^{27} + 282 q^{28} + 21 q^{29} - 406 q^{30} - 193 q^{31} - 432 q^{32} - 418 q^{33} + 666 q^{34} - 752 q^{35} - 629 q^{36} + 170 q^{37} + 748 q^{38} - 291 q^{39} - 26 q^{40} - 125 q^{41} + 640 q^{42} + 2 q^{43} + 830 q^{44} + 168 q^{45} - 46 q^{46} - 677 q^{47} + 551 q^{48} + 1220 q^{49} + 414 q^{50} - 340 q^{51} + 2247 q^{52} - 230 q^{53} + 641 q^{54} - 972 q^{55} - 2174 q^{56} + 1322 q^{57} - 1835 q^{58} - 1140 q^{59} - 804 q^{60} + 754 q^{61} + 443 q^{62} - 1092 q^{63} - 805 q^{64} + 1318 q^{65} - 398 q^{66} + 488 q^{67} + 284 q^{68} - 161 q^{69} - 3820 q^{70} - 401 q^{71} + 1503 q^{72} + 1509 q^{73} + 1366 q^{74} + 1401 q^{75} - 3832 q^{76} + 736 q^{77} - 1907 q^{78} - 838 q^{79} + 2846 q^{80} - 932 q^{81} - 949 q^{82} + 142 q^{83} + 2614 q^{84} + 112 q^{85} + 918 q^{86} + 2223 q^{87} - 404 q^{88} + 2342 q^{89} + 1784 q^{90} + 292 q^{91} - 460 q^{92} - 509 q^{93} + 1567 q^{94} - 956 q^{95} + 799 q^{96} + 1062 q^{97} + 2478 q^{98} - 1498 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9 - 70 * q^10 + 8 * q^11 - 67 * q^12 + 111 * q^13 - 144 * q^14 + 10 * q^15 + 64 * q^16 + 98 * q^17 + 49 * q^18 + 96 * q^19 + 140 * q^20 + 180 * q^21 + 220 * q^22 - 92 * q^23 - 188 * q^24 + 184 * q^25 - 229 * q^26 - 155 * q^27 + 282 * q^28 + 21 * q^29 - 406 * q^30 - 193 * q^31 - 432 * q^32 - 418 * q^33 + 666 * q^34 - 752 * q^35 - 629 * q^36 + 170 * q^37 + 748 * q^38 - 291 * q^39 - 26 * q^40 - 125 * q^41 + 640 * q^42 + 2 * q^43 + 830 * q^44 + 168 * q^45 - 46 * q^46 - 677 * q^47 + 551 * q^48 + 1220 * q^49 + 414 * q^50 - 340 * q^51 + 2247 * q^52 - 230 * q^53 + 641 * q^54 - 972 * q^55 - 2174 * q^56 + 1322 * q^57 - 1835 * q^58 - 1140 * q^59 - 804 * q^60 + 754 * q^61 + 443 * q^62 - 1092 * q^63 - 805 * q^64 + 1318 * q^65 - 398 * q^66 + 488 * q^67 + 284 * q^68 - 161 * q^69 - 3820 * q^70 - 401 * q^71 + 1503 * q^72 + 1509 * q^73 + 1366 * q^74 + 1401 * q^75 - 3832 * q^76 + 736 * q^77 - 1907 * q^78 - 838 * q^79 + 2846 * q^80 - 932 * q^81 - 949 * q^82 + 142 * q^83 + 2614 * q^84 + 112 * q^85 + 918 * q^86 + 2223 * q^87 - 404 * q^88 + 2342 * q^89 + 1784 * q^90 + 292 * q^91 - 460 * q^92 - 509 * q^93 + 1567 * q^94 - 956 * q^95 + 799 * q^96 + 1062 * q^97 + 2478 * q^98 - 1498 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.0323756	−0.0114465	−0.00572325	−	0.999984i	$-0.501822\pi$
$2$	−0.0323756	−0.0114465	−0.00572325	+	0.999984i	$0.501822\pi$
$3$	6.42170	1.23586	0.617928	−	0.786235i	$-0.287972\pi$
$3$	6.42170	1.23586	0.617928	+	0.786235i	$0.287972\pi$
$4$	−7.99895	−0.999869
$5$	14.1026	1.26137	0.630687	−	0.776037i	$-0.282773\pi$
$5$	14.1026	1.26137	0.630687	+	0.776037i	$0.282773\pi$
$6$	−0.207906	−0.0141462
$7$	−14.0109	−0.756520	−0.378260	−	0.925699i	$-0.623477\pi$
$7$	−14.0109	−0.756520	−0.378260	+	0.925699i	$0.623477\pi$
$8$	0.517976	0.0228915
$9$	14.2382	0.527340
$10$	−0.456580	−0.0144383
$11$	−55.5140	−1.52165	−0.760823	−	0.648959i	$-0.775204\pi$
$11$	−55.5140	−1.52165	−0.760823	+	0.648959i	$0.775204\pi$
$12$	−51.3668	−1.23569
$13$	−18.3149	−0.390742	−0.195371	−	0.980729i	$-0.562591\pi$
$13$	−18.3149	−0.390742	−0.195371	+	0.980729i	$0.562591\pi$
$14$	0.453613	0.00865951
$15$	90.5626	1.55888
$16$	63.9748	0.999607
$17$	10.0273	0.143058	0.0715288	−	0.997439i	$-0.477212\pi$
$17$	10.0273	0.143058	0.0715288	+	0.997439i	$0.477212\pi$
$18$	−0.460970	−0.00603620
$19$	161.106	1.94528	0.972639	−	0.232321i	$-0.0746318\pi$
$19$	161.106	1.94528	0.972639	+	0.232321i	$0.0746318\pi$
$20$	−112.806	−1.26121
$21$	−89.9740	−0.934950
$22$	1.79730	0.0174175
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	3.32628	0.0282906
$25$	73.8833	0.591066
$26$	0.592956	0.00447263
$27$	−81.9525	−0.584139
$28$	112.073	0.756421
$29$	183.185	1.17298	0.586492	−	0.809955i	$-0.300509\pi$
$29$	183.185	1.17298	0.586492	+	0.809955i	$0.300509\pi$
$30$	−2.93202	−0.0178437
$31$	−144.762	−0.838710	−0.419355	−	0.907822i	$-0.637744\pi$
$31$	−144.762	−0.838710	−0.419355	+	0.907822i	$0.637744\pi$
$32$	−6.21503	−0.0343335
$33$	−356.494	−1.88054
$34$	−0.324640	−0.00163751
$35$	−197.591	−0.954255
$36$	−113.891	−0.527271
$37$	181.411	0.806047	0.403023	−	0.915190i	$-0.367959\pi$
$37$	181.411	0.806047	0.403023	+	0.915190i	$0.367959\pi$
$38$	−5.21591	−0.0222666
$39$	−117.613	−0.482900
$40$	7.30481	0.0288748
$41$	77.7996	0.296348	0.148174	−	0.988961i	$-0.452660\pi$
$41$	77.7996	0.296348	0.148174	+	0.988961i	$0.452660\pi$
$42$	2.91296	0.0107019
$43$	315.335	1.11833	0.559164	−	0.829057i	$-0.311122\pi$
$43$	315.335	1.11833	0.559164	+	0.829057i	$0.311122\pi$
$44$	444.054	1.52145
$45$	200.795	0.665174
$46$	0.744639	0.00238676
$47$	−524.190	−1.62683	−0.813414	−	0.581685i	$-0.802393\pi$
$47$	−524.190	−1.62683	−0.813414	+	0.581685i	$0.802393\pi$
$48$	410.827	1.23537
$49$	−146.693	−0.427678
$50$	−2.39202	−0.00676565
$51$	64.3923	0.176799
$52$	146.500	0.390690
$53$	73.7334	0.191095	0.0955477	−	0.995425i	$-0.469540\pi$
$53$	73.7334	0.191095	0.0955477	+	0.995425i	$0.469540\pi$
$54$	2.65326	0.00668636
$55$	−782.892	−1.91937
$56$	−7.25733	−0.0173179
$57$	1034.57	2.40408
$58$	−5.93071	−0.0134266
$59$	−132.892	−0.293239	−0.146619	−	0.989193i	$-0.546839\pi$
$59$	−132.892	−0.293239	−0.146619	+	0.989193i	$0.546839\pi$
$60$	−724.406	−1.55867
$61$	236.683	0.496789	0.248394	−	0.968659i	$-0.420097\pi$
$61$	236.683	0.496789	0.248394	+	0.968659i	$0.420097\pi$
$62$	4.68676	0.00960030
$63$	−199.490	−0.398943
$64$	−511.598	−0.999214
$65$	−258.288	−0.492872
$66$	11.5417	0.0215256
$67$	493.624	0.900086	0.450043	−	0.893007i	$-0.351409\pi$
$67$	493.624	0.900086	0.450043	+	0.893007i	$0.351409\pi$
$68$	−80.2079	−0.143039
$69$	−147.699	−0.257694
$70$	6.39712	0.0109229
$71$	−806.060	−1.34735	−0.673674	−	0.739029i	$-0.735285\pi$
$71$	−806.060	−1.34735	−0.673674	+	0.739029i	$0.735285\pi$
$72$	7.37504	0.0120716
$73$	1011.91	1.62241	0.811203	−	0.584764i	$-0.198813\pi$
$73$	1011.91	1.62241	0.811203	+	0.584764i	$0.198813\pi$
$74$	−5.87328	−0.00922642
$75$	474.456	0.730473
$76$	−1288.68	−1.94502
$77$	777.804	1.15116
$78$	3.80779	0.00552752
$79$	−599.386	−0.853623	−0.426811	−	0.904341i	$-0.640363\pi$
$79$	−599.386	−0.853623	−0.426811	+	0.904341i	$0.640363\pi$
$80$	902.212	1.26088
$81$	−910.705	−1.24925
$82$	−2.51881	−0.00339215
$83$	−642.245	−0.849344	−0.424672	−	0.905347i	$-0.639610\pi$
$83$	−642.245	−0.849344	−0.424672	+	0.905347i	$0.639610\pi$
$84$	719.698	0.934827
$85$	141.411	0.180449
$86$	−10.2092	−0.0128009
$87$	1176.36	1.44964
$88$	−28.7549	−0.0348328
$89$	883.399	1.05214	0.526068	−	0.850442i	$-0.323666\pi$
$89$	883.399	1.05214	0.526068	+	0.850442i	$0.323666\pi$
$90$	−6.50088	−0.00761392
$91$	256.609	0.295604
$92$	183.976	0.208487
$93$	−929.617	−1.03652
$94$	16.9710	0.0186215
$95$	2272.02	2.45373
$96$	−39.9111	−0.0424313
$97$	−71.2938	−0.0746266	−0.0373133	−	0.999304i	$-0.511880\pi$
$97$	−71.2938	−0.0746266	−0.0373133	+	0.999304i	$0.511880\pi$
$98$	4.74929	0.00489542
$99$	−790.419	−0.802425
$100$	−590.989	−0.590989
$101$	−942.689	−0.928723	−0.464361	−	0.885646i	$-0.653716\pi$
$101$	−942.689	−0.928723	−0.464361	+	0.885646i	$0.653716\pi$
$102$	−2.08474	−0.00202373
$103$	−556.624	−0.532484	−0.266242	−	0.963906i	$-0.585782\pi$
$103$	−556.624	−0.532484	−0.266242	+	0.963906i	$0.585782\pi$
$104$	−9.48668	−0.00894467
$105$	−1268.87	−1.17932
$106$	−2.38716	−0.00218738
$107$	647.477	0.584990	0.292495	−	0.956267i	$-0.405514\pi$
$107$	647.477	0.584990	0.292495	+	0.956267i	$0.405514\pi$
$108$	655.534	0.584063
$109$	−1349.13	−1.18554	−0.592768	−	0.805373i	$-0.701965\pi$
$109$	−1349.13	−1.18554	−0.592768	+	0.805373i	$0.701965\pi$
$110$	25.3466	0.0219700
$111$	1164.96	0.996158
$112$	−896.348	−0.756222
$113$	−284.549	−0.236886	−0.118443	−	0.992961i	$-0.537790\pi$
$113$	−284.549	−0.236886	−0.118443	+	0.992961i	$0.537790\pi$
$114$	−33.4950	−0.0275184
$115$	−324.360	−0.263015
$116$	−1465.28	−1.17283
$117$	−260.771	−0.206054
$118$	4.30247	0.00335656
$119$	−140.492	−0.108226
$120$	46.9093	0.0356851
$121$	1750.81	1.31541
$122$	−7.66275	−0.00568650
$123$	499.605	0.366243
$124$	1157.94	0.838600
$125$	−720.878	−0.515819
$126$	6.45863	0.00456651
$127$	−753.553	−0.526512	−0.263256	−	0.964726i	$-0.584796\pi$
$127$	−753.553	−0.526512	−0.263256	+	0.964726i	$0.584796\pi$
$128$	66.2835	0.0457710
$129$	2024.98	1.38209
$130$	8.36223	0.00564166
$131$	−769.226	−0.513035	−0.256518	−	0.966540i	$-0.582575\pi$
$131$	−769.226	−0.513035	−0.256518	+	0.966540i	$0.582575\pi$
$132$	2851.58	1.88029
$133$	−2257.25	−1.47164
$134$	−15.9814	−0.0103028
$135$	−1155.74	−0.736819
$136$	5.19390	0.00327481
$137$	−211.638	−0.131982	−0.0659908	−	0.997820i	$-0.521021\pi$
$137$	−211.638	−0.131982	−0.0659908	+	0.997820i	$0.521021\pi$
$138$	4.78185	0.00294969
$139$	1998.54	1.21953	0.609763	−	0.792584i	$-0.291265\pi$
$139$	1998.54	1.21953	0.609763	+	0.792584i	$0.291265\pi$
$140$	1580.52	0.954130
$141$	−3366.19	−2.01052
$142$	26.0967	0.0154224
$143$	1016.73	0.594570
$144$	910.886	0.527133
$145$	2583.38	1.47957
$146$	−32.7614	−0.0185709
$147$	−942.021	−0.528548
$148$	−1451.10	−0.805941
$149$	499.968	0.274893	0.137446	−	0.990509i	$-0.456111\pi$
$149$	499.968	0.274893	0.137446	+	0.990509i	$0.456111\pi$
$150$	−15.3608	−0.00836136
$151$	501.652	0.270357	0.135178	−	0.990821i	$-0.456839\pi$
$151$	501.652	0.270357	0.135178	+	0.990821i	$0.456839\pi$
$152$	83.4491	0.0445304
$153$	142.771	0.0754400
$154$	−25.1819	−0.0131767
$155$	−2041.52	−1.05793
$156$	940.779	0.482837
$157$	1686.36	0.857237	0.428619	−	0.903485i	$-0.359000\pi$
$157$	1686.36	0.857237	0.428619	+	0.903485i	$0.359000\pi$
$158$	19.4055	0.00977100
$159$	473.493	0.236166
$160$	−87.6481	−0.0433074
$161$	322.252	0.157745
$162$	29.4846	0.0142996
$163$	3183.54	1.52978	0.764890	−	0.644161i	$-0.222793\pi$
$163$	3183.54	1.52978	0.764890	+	0.644161i	$0.222793\pi$
$164$	−622.315	−0.296309
$165$	−5027.49	−2.37206
$166$	20.7931	0.00972202
$167$	−3771.37	−1.74753	−0.873764	−	0.486350i	$-0.838328\pi$
$167$	−3771.37	−1.74753	−0.873764	+	0.486350i	$0.838328\pi$
$168$	−46.6044	−0.0214024
$169$	−1861.56	−0.847321
$170$	−4.57827	−0.00206551
$171$	2293.86	1.02582
$172$	−2522.35	−1.11818
$173$	129.941	0.0571052	0.0285526	−	0.999592i	$-0.490910\pi$
$173$	129.941	0.0571052	0.0285526	+	0.999592i	$0.490910\pi$
$174$	−38.0852	−0.0165933
$175$	−1035.17	−0.447153
$176$	−3551.50	−1.52105
$177$	−853.393	−0.362401
$178$	−28.6006	−0.0120433
$179$	810.182	0.338301	0.169150	−	0.985590i	$-0.445898\pi$
$179$	810.182	0.338301	0.169150	+	0.985590i	$0.445898\pi$
$180$	−1606.15	−0.665086
$181$	2430.33	0.998039	0.499019	−	0.866591i	$-0.333694\pi$
$181$	2430.33	0.998039	0.499019	+	0.866591i	$0.333694\pi$
$182$	−8.30788	−0.00338363
$183$	1519.90	0.613960
$184$	−11.9134	−0.00477321
$185$	2558.36	1.01673
$186$	30.0969	0.0118646
$187$	−556.656	−0.217683
$188$	4192.97	1.62661
$189$	1148.23	0.441913
$190$	−73.5579	−0.0280866
$191$	2462.87	0.933022	0.466511	−	0.884515i	$-0.345511\pi$
$191$	2462.87	0.933022	0.466511	+	0.884515i	$0.345511\pi$
$192$	−3285.32	−1.23488
$193$	4021.66	1.49992	0.749962	−	0.661481i	$-0.230072\pi$
$193$	4021.66	1.49992	0.749962	+	0.661481i	$0.230072\pi$
$194$	2.30818	0.000854214 0
$195$	−1658.65	−0.609118
$196$	1173.39	0.427622
$197$	1811.60	0.655182	0.327591	−	0.944820i	$-0.393763\pi$
$197$	1811.60	0.655182	0.327591	+	0.944820i	$0.393763\pi$
$198$	25.5903	0.00918497
$199$	723.923	0.257877	0.128939	−	0.991653i	$-0.458843\pi$
$199$	723.923	0.257877	0.128939	+	0.991653i	$0.458843\pi$
$200$	38.2698	0.0135304
$201$	3169.90	1.11238
$202$	30.5201	0.0106306
$203$	−2566.59	−0.887385
$204$	−515.071	−0.176775
$205$	1097.18	0.373805
$206$	18.0211	0.00609508
$207$	−327.478	−0.109958
$208$	−1171.69	−0.390588
$209$	−8943.65	−2.96003
$210$	41.0804	0.0134991
$211$	−583.883	−0.190503	−0.0952515	−	0.995453i	$-0.530366\pi$
$211$	−583.883	−0.190503	−0.0952515	+	0.995453i	$0.530366\pi$
$212$	−589.790	−0.191070
$213$	−5176.27	−1.66513
$214$	−20.9625	−0.00669610
$215$	4447.04	1.41063
$216$	−42.4494	−0.0133718
$217$	2028.25	0.634501
$218$	43.6790	0.0135703
$219$	6498.21	2.00506
$220$	6262.31	1.91911
$221$	−183.649	−0.0558985
$222$	−37.7164	−0.0114025
$223$	3157.57	0.948191	0.474096	−	0.880473i	$-0.342775\pi$
$223$	3157.57	0.948191	0.474096	+	0.880473i	$0.342775\pi$
$224$	87.0785	0.0259740
$225$	1051.96	0.311693
$226$	9.21245	0.00271152
$227$	−2219.80	−0.649044	−0.324522	−	0.945878i	$-0.605203\pi$
$227$	−2219.80	−0.649044	−0.324522	+	0.945878i	$0.605203\pi$
$228$	−8275.52	−2.40377
$229$	−4398.24	−1.26919	−0.634593	−	0.772846i	$-0.718832\pi$
$229$	−4398.24	−1.26919	−0.634593	+	0.772846i	$0.718832\pi$
$230$	10.5013	0.00301060
$231$	4994.82	1.42266
$232$	94.8852	0.0268514
$233$	2112.84	0.594065	0.297032	−	0.954867i	$-0.404003\pi$
$233$	2112.84	0.594065	0.297032	+	0.954867i	$0.404003\pi$
$234$	8.44262	0.00235860
$235$	−7392.43	−2.05204
$236$	1063.00	0.293200
$237$	−3849.08	−1.05496
$238$	4.54852	0.00123881
$239$	−1548.69	−0.419147	−0.209574	−	0.977793i	$-0.567208\pi$
$239$	−1548.69	−0.419147	−0.209574	+	0.977793i	$0.567208\pi$
$240$	5793.73	1.55826
$241$	−2516.92	−0.672734	−0.336367	−	0.941731i	$-0.609198\pi$
$241$	−2516.92	−0.672734	−0.336367	+	0.941731i	$0.609198\pi$
$242$	−56.6834	−0.0150568
$243$	−3635.55	−0.959757
$244$	−1893.21	−0.496724
$245$	−2068.76	−0.539462
$246$	−16.1750	−0.00419220
$247$	−2950.64	−0.760101
$248$	−74.9832	−0.0191993
$249$	−4124.30	−1.04967
$250$	23.3389	0.00590432
$251$	386.310	0.0971461	0.0485731	−	0.998820i	$-0.484533\pi$
$251$	386.310	0.0971461	0.0485731	+	0.998820i	$0.484533\pi$
$252$	1595.71	0.398891
$253$	1276.82	0.317285
$254$	24.3967	0.00602672
$255$	908.099	0.223009
$256$	4090.63	0.998690
$257$	3816.61	0.926357	0.463178	−	0.886265i	$-0.346709\pi$
$257$	3816.61	0.926357	0.463178	+	0.886265i	$0.346709\pi$
$258$	−65.5601	−0.0158201
$259$	−2541.73	−0.609790
$260$	2066.03	0.492807
$261$	2608.22	0.618561
$262$	24.9042	0.00587246
$263$	4510.11	1.05744	0.528718	−	0.848798i	$-0.322673\pi$
$263$	4510.11	1.05744	0.528718	+	0.848798i	$0.322673\pi$
$264$	−184.655	−0.0430483
$265$	1039.83	0.241043
$266$	73.0798	0.0168452
$267$	5672.92	1.30029
$268$	−3948.47	−0.899968
$269$	−550.730	−0.124827	−0.0624137	−	0.998050i	$-0.519880\pi$
$269$	−550.730	−0.124827	−0.0624137	+	0.998050i	$0.519880\pi$
$270$	37.4179	0.00843400
$271$	−897.873	−0.201262	−0.100631	−	0.994924i	$-0.532086\pi$
$271$	−897.873	−0.201262	−0.100631	+	0.994924i	$0.532086\pi$
$272$	641.495	0.143001
$273$	1647.87	0.365324
$274$	6.85192	0.00151073
$275$	−4101.56	−0.899394
$276$	1181.44	0.257660
$277$	2288.67	0.496437	0.248219	−	0.968704i	$-0.420155\pi$
$277$	2288.67	0.496437	0.248219	+	0.968704i	$0.420155\pi$
$278$	−64.7040	−0.0139593
$279$	−2061.15	−0.442285
$280$	−102.347	−0.0218443
$281$	−2587.28	−0.549268	−0.274634	−	0.961549i	$-0.588557\pi$
$281$	−2587.28	−0.549268	−0.274634	+	0.961549i	$0.588557\pi$
$282$	108.982	0.0230135
$283$	1488.48	0.312653	0.156326	−	0.987705i	$-0.450035\pi$
$283$	1488.48	0.312653	0.156326	+	0.987705i	$0.450035\pi$
$284$	6447.63	1.34717
$285$	14590.2	3.03245
$286$	−32.9174	−0.00680576
$287$	−1090.05	−0.224193
$288$	−88.4908	−0.0181055
$289$	−4812.45	−0.979535
$290$	−83.6384	−0.0169359
$291$	−457.827	−0.0922278
$292$	−8094.26	−1.62219
$293$	−2821.24	−0.562522	−0.281261	−	0.959631i	$-0.590753\pi$
$293$	−2821.24	−0.562522	−0.281261	+	0.959631i	$0.590753\pi$
$294$	30.4985	0.00605003
$295$	−1874.12	−0.369884
$296$	93.9664	0.0184516
$297$	4549.51	0.888853
$298$	−16.1868	−0.00314656
$299$	421.243	0.0814753
$300$	−3795.15	−0.730377
$301$	−4418.14	−0.846037
$302$	−16.2413	−0.00309464
$303$	−6053.66	−1.14777
$304$	10306.7	1.94451
$305$	3337.84	0.626637
$306$	−4.62229	−0.000863525 0
$307$	−4085.72	−0.759558	−0.379779	−	0.925077i	$-0.624000\pi$
$307$	−4085.72	−0.759558	−0.379779	+	0.925077i	$0.624000\pi$
$308$	−6221.61	−1.15100
$309$	−3574.47	−0.658073
$310$	66.0954	0.0121096
$311$	4044.84	0.737498	0.368749	−	0.929529i	$-0.379786\pi$
$311$	4044.84	0.737498	0.368749	+	0.929529i	$0.379786\pi$
$312$	−60.9206	−0.0110543
$313$	5111.54	0.923072	0.461536	−	0.887122i	$-0.347299\pi$
$313$	5111.54	0.923072	0.461536	+	0.887122i	$0.347299\pi$
$314$	−54.5970	−0.00981238
$315$	−2813.33	−0.503217
$316$	4794.46	0.853511
$317$	−7017.95	−1.24343	−0.621715	−	0.783244i	$-0.713564\pi$
$317$	−7017.95	−1.24343	−0.621715	+	0.783244i	$0.713564\pi$
$318$	−15.3296	−0.00270328
$319$	−10169.3	−1.78487
$320$	−7214.85	−1.26038
$321$	4157.90	0.722964
$322$	−10.4331	−0.00180563
$323$	1615.46	0.278287
$324$	7284.69	1.24909
$325$	−1353.17	−0.230954
$326$	−103.069	−0.0175106
$327$	−8663.72	−1.46515
$328$	40.2983	0.00678385
$329$	7344.39	1.23073
$330$	162.768	0.0271518
$331$	−6537.02	−1.08552	−0.542760	−	0.839888i	$-0.682621\pi$
$331$	−6537.02	−1.08552	−0.542760	+	0.839888i	$0.682621\pi$
$332$	5137.28	0.849232
$333$	2582.96	0.425061
$334$	122.100	0.0200031
$335$	6961.38	1.13535
$336$	−5756.07	−0.934582
$337$	−838.254	−0.135497	−0.0677486	−	0.997702i	$-0.521582\pi$
$337$	−838.254	−0.135497	−0.0677486	+	0.997702i	$0.521582\pi$
$338$	60.2693	0.00969887
$339$	−1827.29	−0.292757
$340$	−1131.14	−0.180426
$341$	8036.32	1.27622
$342$	−74.2651	−0.0117421
$343$	6861.07	1.08007
$344$	163.336	0.0256002
$345$	−2082.94	−0.325048
$346$	−4.20691	−0.000653655 0
$347$	7101.86	1.09870	0.549348	−	0.835593i	$-0.314876\pi$
$347$	7101.86	1.09870	0.549348	+	0.835593i	$0.314876\pi$
$348$	−9409.61	−1.44945
$349$	−2675.10	−0.410300	−0.205150	−	0.978731i	$-0.565768\pi$
$349$	−2675.10	−0.410300	−0.205150	+	0.978731i	$0.565768\pi$
$350$	33.5144	0.00511835
$351$	1500.95	0.228248
$352$	345.021	0.0522435
$353$	−8032.84	−1.21118	−0.605588	−	0.795778i	$-0.707062\pi$
$353$	−8032.84	−1.21118	−0.605588	+	0.795778i	$0.707062\pi$
$354$	27.6291	0.00414823
$355$	−11367.5	−1.69951
$356$	−7066.27	−1.05200
$357$	−902.197	−0.133752
$358$	−26.2301	−0.00387236
$359$	4828.72	0.709889	0.354944	−	0.934887i	$-0.384500\pi$
$359$	4828.72	0.709889	0.354944	+	0.934887i	$0.384500\pi$
$360$	104.007	0.0152268
$361$	19096.2	2.78411
$362$	−78.6834	−0.0114241
$363$	11243.1	1.62565
$364$	−2052.60	−0.295565
$365$	14270.6	2.04646
$366$	−49.2079	−0.00702769
$367$	89.8597	0.0127810	0.00639052	−	0.999980i	$-0.497966\pi$
$367$	89.8597	0.0127810	0.00639052	+	0.999980i	$0.497966\pi$
$368$	−1471.42	−0.208432
$369$	1107.72	0.156276
$370$	−82.8285	−0.0116380
$371$	−1033.07	−0.144567
$372$	7435.96	1.03639
$373$	5196.16	0.721306	0.360653	−	0.932700i	$-0.382554\pi$
$373$	5196.16	0.721306	0.360653	+	0.932700i	$0.382554\pi$
$374$	18.0221	0.00249171
$375$	−4629.26	−0.637478
$376$	−271.518	−0.0372406
$377$	−3355.01	−0.458333
$378$	−37.1747	−0.00505836
$379$	4900.54	0.664179	0.332090	−	0.943248i	$-0.392246\pi$
$379$	4900.54	0.664179	0.332090	+	0.943248i	$0.392246\pi$
$380$	−18173.7	−2.45340
$381$	−4839.09	−0.650693
$382$	−79.7370	−0.0106798
$383$	9973.73	1.33064	0.665318	−	0.746560i	$-0.268296\pi$
$383$	9973.73	1.33064	0.665318	+	0.746560i	$0.268296\pi$
$384$	425.653	0.0565664
$385$	10969.1	1.45204
$386$	−130.204	−0.0171689
$387$	4489.79	0.589739
$388$	570.275	0.0746169
$389$	−6277.96	−0.818266	−0.409133	−	0.912475i	$-0.634169\pi$
$389$	−6277.96	−0.818266	−0.409133	+	0.912475i	$0.634169\pi$
$390$	53.6997	0.00697228
$391$	−230.628	−0.0298296
$392$	−75.9837	−0.00979019
$393$	−4939.74	−0.634038
$394$	−58.6515	−0.00749955
$395$	−8452.90	−1.07674
$396$	6322.52	0.802320
$397$	−11309.9	−1.42979	−0.714895	−	0.699232i	$-0.753526\pi$
$397$	−11309.9	−1.42979	−0.714895	+	0.699232i	$0.753526\pi$
$398$	−23.4375	−0.00295179
$399$	−14495.4	−1.81874
$400$	4726.67	0.590834
$401$	−14306.5	−1.78163	−0.890813	−	0.454371i	$-0.849864\pi$
$401$	−14306.5	−1.78163	−0.890813	+	0.454371i	$0.849864\pi$
$402$	−102.628	−0.0127328
$403$	2651.30	0.327719
$404$	7540.52	0.928601
$405$	−12843.3	−1.57578
$406$	83.0949	0.0101575
$407$	−10070.8	−1.22652
$408$	33.3537	0.00404719
$409$	4363.34	0.527514	0.263757	−	0.964589i	$-0.415038\pi$
$409$	4363.34	0.527514	0.263757	+	0.964589i	$0.415038\pi$
$410$	−35.5218	−0.00427877
$411$	−1359.08	−0.163110
$412$	4452.41	0.532414
$413$	1861.94	0.221841
$414$	10.6023	0.00125864
$415$	−9057.32	−1.07134
$416$	113.828	0.0134155
$417$	12834.0	1.50716
$418$	289.556	0.0338820
$419$	12304.9	1.43468	0.717341	−	0.696722i	$-0.245359\pi$
$419$	12304.9	1.43468	0.717341	+	0.696722i	$0.245359\pi$
$420$	10149.6	1.17917
$421$	−8353.05	−0.966990	−0.483495	−	0.875347i	$-0.660633\pi$
$421$	−8353.05	−0.966990	−0.483495	+	0.875347i	$0.660633\pi$
$422$	18.9036	0.00218059
$423$	−7463.51	−0.857892
$424$	38.1921	0.00437446
$425$	740.850	0.0845565
$426$	167.585	0.0190599
$427$	−3316.15	−0.375831
$428$	−5179.14	−0.584914
$429$	6529.16	0.734803
$430$	−143.976	−0.0161468
$431$	−15959.9	−1.78367	−0.891835	−	0.452361i	$-0.850582\pi$
$431$	−15959.9	−1.78367	−0.891835	+	0.452361i	$0.850582\pi$
$432$	−5242.90	−0.583910
$433$	5779.60	0.641454	0.320727	−	0.947172i	$-0.396073\pi$
$433$	5779.60	0.641454	0.320727	+	0.947172i	$0.396073\pi$
$434$	−65.6659	−0.00726282
$435$	16589.7	1.82854
$436$	10791.6	1.18538
$437$	−3705.44	−0.405619
$438$	−210.384	−0.0229509
$439$	795.488	0.0864842	0.0432421	−	0.999065i	$-0.486231\pi$
$439$	795.488	0.0864842	0.0432421	+	0.999065i	$0.486231\pi$
$440$	−405.519	−0.0439372
$441$	−2088.65	−0.225532
$442$	5.94575	0.000639843 0
$443$	7357.81	0.789120	0.394560	−	0.918870i	$-0.370897\pi$
$443$	7357.81	0.789120	0.394560	+	0.918870i	$0.370897\pi$
$444$	−9318.49	−0.996027
$445$	12458.2	1.32714
$446$	−102.228	−0.0108535
$447$	3210.65	0.339728
$448$	7167.96	0.755925
$449$	−6672.34	−0.701308	−0.350654	−	0.936505i	$-0.614041\pi$
$449$	−6672.34	−0.701308	−0.350654	+	0.936505i	$0.614041\pi$
$450$	−34.0580	−0.00356780
$451$	−4318.97	−0.450936
$452$	2276.09	0.236855
$453$	3221.46	0.334122
$454$	71.8673	0.00742929
$455$	3618.86	0.372867
$456$	535.885	0.0550331
$457$	−6324.72	−0.647392	−0.323696	−	0.946161i	$-0.604925\pi$
$457$	−6324.72	−0.647392	−0.323696	+	0.946161i	$0.604925\pi$
$458$	142.396	0.0145278
$459$	−821.763	−0.0835656
$460$	2594.54	0.262980
$461$	−641.556	−0.0648162	−0.0324081	−	0.999475i	$-0.510318\pi$
$461$	−641.556	−0.0648162	−0.0324081	+	0.999475i	$0.510318\pi$
$462$	−161.710	−0.0162845
$463$	714.350	0.0717034	0.0358517	−	0.999357i	$-0.488586\pi$
$463$	714.350	0.0717034	0.0358517	+	0.999357i	$0.488586\pi$
$464$	11719.2	1.17252
$465$	−13110.0	−1.30745
$466$	−68.4047	−0.00679997
$467$	3937.64	0.390176	0.195088	−	0.980786i	$-0.437501\pi$
$467$	3937.64	0.390176	0.195088	+	0.980786i	$0.437501\pi$
$468$	2085.89	0.206027
$469$	−6916.13	−0.680933
$470$	239.335	0.0234887
$471$	10829.3	1.05942
$472$	−68.8349	−0.00671268
$473$	−17505.5	−1.70170
$474$	124.616	0.0120756
$475$	11903.1	1.14979
$476$	1123.79	0.108212
$477$	1049.83	0.100772
$478$	50.1397	0.00479777
$479$	−2252.09	−0.214824	−0.107412	−	0.994215i	$-0.534256\pi$
$479$	−2252.09	−0.214824	−0.107412	+	0.994215i	$0.534256\pi$
$480$	−562.850	−0.0535218
$481$	−3322.52	−0.314956
$482$	81.4868	0.00770046
$483$	2069.40	0.194950
$484$	−14004.6	−1.31523
$485$	−1005.43	−0.0941322
$486$	117.703	0.0109859
$487$	11821.8	1.09999	0.549995	−	0.835168i	$-0.314630\pi$
$487$	11821.8	1.09999	0.549995	+	0.835168i	$0.314630\pi$
$488$	122.596	0.0113723
$489$	20443.7	1.89059
$490$	66.9773	0.00617496
$491$	20198.1	1.85647	0.928235	−	0.371993i	$-0.121326\pi$
$491$	20198.1	1.85647	0.928235	+	0.371993i	$0.121326\pi$
$492$	−3996.32	−0.366195
$493$	1836.85	0.167804
$494$	95.5289	0.00870051
$495$	−11147.0	−1.01216
$496$	−9261.12	−0.838380
$497$	11293.7	1.01930
$498$	133.527	0.0120150
$499$	−633.684	−0.0568488	−0.0284244	−	0.999596i	$-0.509049\pi$
$499$	−633.684	−0.0568488	−0.0284244	+	0.999596i	$0.509049\pi$
$500$	5766.27	0.515751
$501$	−24218.6	−2.15969
$502$	−12.5070	−0.00111198
$503$	10068.6	0.892518	0.446259	−	0.894904i	$-0.352756\pi$
$503$	10068.6	0.892518	0.446259	+	0.894904i	$0.352756\pi$
$504$	−103.331	−0.00913242
$505$	−13294.4	−1.17147
$506$	−41.3379	−0.00363181
$507$	−11954.4	−1.04717
$508$	6027.63	0.526443
$509$	−4287.40	−0.373351	−0.186675	−	0.982422i	$-0.559771\pi$
$509$	−4287.40	−0.373351	−0.186675	+	0.982422i	$0.559771\pi$
$510$	−29.4003	−0.00255268
$511$	−14177.9	−1.22738
$512$	−662.705	−0.0572026
$513$	−13203.1	−1.13631
$514$	−123.565	−0.0106035
$515$	−7849.85	−0.671662
$516$	−16197.7	−1.38191
$517$	29099.9	2.47546
$518$	82.2902	0.00697997
$519$	834.439	0.0705738
$520$	−133.787	−0.0112826
$521$	−19842.9	−1.66858	−0.834291	−	0.551324i	$-0.814123\pi$
$521$	−19842.9	−1.66858	−0.834291	+	0.551324i	$0.814123\pi$
$522$	−84.4426	−0.00708037
$523$	−10894.9	−0.910897	−0.455448	−	0.890262i	$-0.650521\pi$
$523$	−10894.9	−0.910897	−0.455448	+	0.890262i	$0.650521\pi$
$524$	6153.00	0.512968
$525$	−6647.58	−0.552617
$526$	−146.018	−0.0121039
$527$	−1451.57	−0.119984
$528$	−22806.7	−1.87980
$529$	529.000	0.0434783
$530$	−33.6652	−0.00275910
$531$	−1892.14	−0.154637
$532$	18055.6	1.47145
$533$	−1424.89	−0.115795
$534$	−183.664	−0.0148838
$535$	9131.11	0.737892
$536$	255.685	0.0206043
$537$	5202.74	0.418091
$538$	17.8302	0.00142884
$539$	8143.54	0.650774
$540$	9244.73	0.736722
$541$	4405.04	0.350069	0.175035	−	0.984562i	$-0.443996\pi$
$541$	4405.04	0.350069	0.175035	+	0.984562i	$0.443996\pi$
$542$	29.0692	0.00230374
$543$	15606.8	1.23343
$544$	−62.3200	−0.00491167
$545$	−19026.3	−1.49541
$546$	−53.3507	−0.00418168
$547$	10031.2	0.784099	0.392050	−	0.919944i	$-0.371766\pi$
$547$	10031.2	0.784099	0.392050	+	0.919944i	$0.371766\pi$
$548$	1692.88	0.131964
$549$	3369.93	0.261977
$550$	132.790	0.0102949
$551$	29512.2	2.28178
$552$	−76.5045	−0.00589900
$553$	8397.97	0.645783
$554$	−74.0972	−0.00568247
$555$	16429.0	1.25653
$556$	−15986.2	−1.21937
$557$	23278.3	1.77079	0.885396	−	0.464837i	$-0.153887\pi$
$557$	23278.3	1.77079	0.885396	+	0.464837i	$0.153887\pi$
$558$	66.7309	0.00506262
$559$	−5775.32	−0.436977
$560$	−12640.8	−0.953880
$561$	−3574.68	−0.269025
$562$	83.7648	0.00628720
$563$	−20989.8	−1.57125	−0.785625	−	0.618702i	$-0.787659\pi$
$563$	−20989.8	−1.57125	−0.785625	+	0.618702i	$0.787659\pi$
$564$	26926.0	2.01026
$565$	−4012.88	−0.298802
$566$	−48.1904	−0.00357878
$567$	12759.8	0.945084
$568$	−417.520	−0.0308428
$569$	−11942.2	−0.879868	−0.439934	−	0.898030i	$-0.644998\pi$
$569$	−11942.2	−0.879868	−0.439934	+	0.898030i	$0.644998\pi$
$570$	−472.367	−0.0347110
$571$	−13981.9	−1.02473	−0.512367	−	0.858767i	$-0.671231\pi$
$571$	−13981.9	−1.02473	−0.512367	+	0.858767i	$0.671231\pi$
$572$	−8132.81	−0.594492
$573$	15815.8	1.15308
$574$	35.2909	0.00256623
$575$	−1699.32	−0.123246
$576$	−7284.22	−0.526926
$577$	−5024.27	−0.362501	−0.181251	−	0.983437i	$-0.558015\pi$
$577$	−5024.27	−0.362501	−0.181251	+	0.983437i	$0.558015\pi$
$578$	155.806	0.0112123
$579$	25825.9	1.85369
$580$	−20664.3	−1.47938
$581$	8998.45	0.642545
$582$	14.8224	0.00105569
$583$	−4093.24	−0.290780
$584$	524.148	0.0371393
$585$	−3677.55	−0.259911
$586$	91.3395	0.00643891
$587$	−22464.4	−1.57957	−0.789784	−	0.613385i	$-0.789807\pi$
$587$	−22464.4	−1.57957	−0.789784	+	0.613385i	$0.789807\pi$
$588$	7535.18	0.528479
$589$	−23322.0	−1.63152
$590$	60.6760	0.00423388
$591$	11633.5	0.809710
$592$	11605.7	0.805730
$593$	14073.1	0.974561	0.487281	−	0.873245i	$-0.337989\pi$
$593$	14073.1	0.974561	0.487281	+	0.873245i	$0.337989\pi$
$594$	−147.293	−0.0101743
$595$	−1981.30	−0.136513
$596$	−3999.22	−0.274857
$597$	4648.82	0.318699
$598$	−13.6380	−0.000932607 0
$599$	−8952.63	−0.610675	−0.305338	−	0.952244i	$-0.598769\pi$
$599$	−8952.63	−0.610675	−0.305338	+	0.952244i	$0.598769\pi$
$600$	245.757	0.0167216
$601$	−20522.5	−1.39290	−0.696449	−	0.717607i	$-0.745238\pi$
$601$	−20522.5	−1.39290	−0.696449	+	0.717607i	$0.745238\pi$
$602$	143.040	0.00968417
$603$	7028.31	0.474651
$604$	−4012.69	−0.270321
$605$	24690.9	1.65922
$606$	195.991	0.0131379
$607$	23022.8	1.53949	0.769743	−	0.638353i	$-0.220384\pi$
$607$	23022.8	1.53949	0.769743	+	0.638353i	$0.220384\pi$
$608$	−1001.28	−0.0667883
$609$	−16481.8	−1.09668
$610$	−108.065	−0.00717281
$611$	9600.48	0.635669
$612$	−1142.02	−0.0754301
$613$	−6159.74	−0.405856	−0.202928	−	0.979194i	$-0.565046\pi$
$613$	−6159.74	−0.405856	−0.202928	+	0.979194i	$0.565046\pi$
$614$	132.278	0.00869429
$615$	7045.73	0.461970
$616$	402.884	0.0263517
$617$	26889.1	1.75448	0.877241	−	0.480050i	$-0.159381\pi$
$617$	26889.1	1.75448	0.877241	+	0.480050i	$0.159381\pi$
$618$	115.726	0.00753264
$619$	−6478.47	−0.420665	−0.210333	−	0.977630i	$-0.567455\pi$
$619$	−6478.47	−0.420665	−0.210333	+	0.977630i	$0.567455\pi$
$620$	16330.0	1.05779
$621$	1884.91	0.121801
$622$	−130.954	−0.00844178
$623$	−12377.3	−0.795962
$624$	−7524.26	−0.482711
$625$	−19401.7	−1.24171
$626$	−165.489	−0.0105659
$627$	−57433.4	−3.65817
$628$	−13489.1	−0.857125
$629$	1819.06	0.115311
$630$	91.0834	0.00576008
$631$	2956.54	0.186526	0.0932630	−	0.995642i	$-0.470270\pi$
$631$	2956.54	0.186526	0.0932630	+	0.995642i	$0.470270\pi$
$632$	−310.468	−0.0195407
$633$	−3749.52	−0.235434
$634$	227.210	0.0142329
$635$	−10627.1	−0.664129
$636$	−3787.45	−0.236135
$637$	2686.68	0.167111
$638$	329.238	0.0204305
$639$	−11476.8	−0.710511
$640$	934.770	0.0577344
$641$	20604.4	1.26961	0.634807	−	0.772670i	$-0.281079\pi$
$641$	20604.4	1.26961	0.634807	+	0.772670i	$0.281079\pi$
$642$	−134.615	−0.00827541
$643$	−23773.4	−1.45806	−0.729030	−	0.684482i	$-0.760028\pi$
$643$	−23773.4	−1.45806	−0.729030	+	0.684482i	$0.760028\pi$
$644$	−2577.68	−0.157725
$645$	28557.5	1.74334
$646$	−52.3015	−0.00318541
$647$	−24907.8	−1.51349	−0.756744	−	0.653712i	$-0.773211\pi$
$647$	−24907.8	−1.51349	−0.756744	+	0.653712i	$0.773211\pi$
$648$	−471.723	−0.0285973
$649$	7377.38	0.446206
$650$	43.8096	0.00264362
$651$	13024.8	0.784152
$652$	−25465.0	−1.52958
$653$	−18523.9	−1.11010	−0.555050	−	0.831817i	$-0.687301\pi$
$653$	−18523.9	−1.11010	−0.555050	+	0.831817i	$0.687301\pi$
$654$	280.493	0.0167709
$655$	−10848.1	−0.647130
$656$	4977.22	0.296231
$657$	14407.8	0.855560
$658$	−237.779	−0.0140875
$659$	24408.5	1.44282	0.721411	−	0.692507i	$-0.243494\pi$
$659$	24408.5	1.44282	0.721411	+	0.692507i	$0.243494\pi$
$660$	40214.7	2.37175
$661$	20341.7	1.19697	0.598487	−	0.801133i	$-0.295769\pi$
$661$	20341.7	1.19697	0.598487	+	0.801133i	$0.295769\pi$
$662$	211.640	0.0124254
$663$	−1179.34	−0.0690826
$664$	−332.667	−0.0194428
$665$	−31833.1	−1.85629
$666$	−83.6249	−0.00486546
$667$	−4213.24	−0.244584
$668$	30167.0	1.74730
$669$	20277.0	1.17183
$670$	−225.379	−0.0129957
$671$	−13139.2	−0.755937
$672$	559.192	0.0321001
$673$	12282.3	0.703490	0.351745	−	0.936096i	$-0.385588\pi$
$673$	12282.3	0.703490	0.351745	+	0.936096i	$0.385588\pi$
$674$	27.1390	0.00155097
$675$	−6054.92	−0.345265
$676$	14890.6	0.847210
$677$	21868.0	1.24144	0.620721	−	0.784031i	$-0.286840\pi$
$677$	21868.0	1.24144	0.620721	+	0.784031i	$0.286840\pi$
$678$	59.1596	0.00335105
$679$	998.893	0.0564565
$680$	73.2475	0.00413076
$681$	−14254.9	−0.802125
$682$	−260.181	−0.0146083
$683$	−22877.0	−1.28165	−0.640823	−	0.767688i	$-0.721407\pi$
$683$	−22877.0	−1.28165	−0.640823	+	0.767688i	$0.721407\pi$
$684$	−18348.5	−1.02569
$685$	−2984.65	−0.166478
$686$	−222.131	−0.0123630
$687$	−28244.2	−1.56853
$688$	20173.5	1.11789
$689$	−1350.42	−0.0746689
$690$	67.4365	0.00372067
$691$	24499.1	1.34875	0.674376	−	0.738388i	$-0.264413\pi$
$691$	24499.1	1.34875	0.674376	+	0.738388i	$0.264413\pi$
$692$	−1039.39	−0.0570977
$693$	11074.5	0.607051
$694$	−229.927	−0.0125762
$695$	28184.6	1.53828
$696$	609.324	0.0331844
$697$	780.120	0.0423948
$698$	86.6080	0.00469650
$699$	13568.0	0.734178
$700$	8280.31	0.447095
$701$	−25020.5	−1.34809	−0.674044	−	0.738691i	$-0.735444\pi$
$701$	−25020.5	−1.34809	−0.674044	+	0.738691i	$0.735444\pi$
$702$	−48.5943	−0.00261264
$703$	29226.4	1.56799
$704$	28400.8	1.52045
$705$	−47472.0	−2.53603
$706$	260.068	0.0138637
$707$	13208.0	0.702597
$708$	6826.25	0.362353
$709$	−15069.4	−0.798225	−0.399112	−	0.916902i	$-0.630682\pi$
$709$	−15069.4	−0.798225	−0.399112	+	0.916902i	$0.630682\pi$
$710$	368.031	0.0194535
$711$	−8534.17	−0.450150
$712$	457.579	0.0240850
$713$	3329.52	0.174883
$714$	29.2092	0.00153099
$715$	14338.6	0.749976
$716$	−6480.61	−0.338256
$717$	−9945.19	−0.518006
$718$	−156.333	−0.00812575
$719$	−23894.6	−1.23938	−0.619692	−	0.784845i	$-0.712743\pi$
$719$	−23894.6	−1.23938	−0.619692	+	0.784845i	$0.712743\pi$
$720$	12845.9	0.664912
$721$	7798.83	0.402835
$722$	−618.251	−0.0318683
$723$	−16162.9	−0.831403
$724$	−19440.1	−0.997908
$725$	13534.3	0.693311
$726$	−364.004	−0.0186081
$727$	32641.3	1.66520	0.832598	−	0.553877i	$-0.186852\pi$
$727$	32641.3	1.66520	0.832598	+	0.553877i	$0.186852\pi$
$728$	132.917	0.00676682
$729$	1242.61	0.0631312
$730$	−462.020	−0.0234249
$731$	3161.96	0.159985
$732$	−12157.6	−0.613879
$733$	−628.010	−0.0316454	−0.0158227	−	0.999875i	$-0.505037\pi$
$733$	−628.010	−0.0316454	−0.0158227	+	0.999875i	$0.505037\pi$
$734$	−2.90926	−0.000146298 0
$735$	−13284.9	−0.666697
$736$	142.946	0.00715904
$737$	−27403.0	−1.36961
$738$	−35.8633	−0.00178881
$739$	−20111.4	−1.00109	−0.500547	−	0.865709i	$-0.666868\pi$
$739$	−20111.4	−1.00109	−0.500547	+	0.865709i	$0.666868\pi$
$740$	−20464.2	−1.01659
$741$	−18948.1	−0.939376
$742$	33.4464	0.00165479
$743$	19508.4	0.963251	0.481625	−	0.876377i	$-0.340047\pi$
$743$	19508.4	0.963251	0.481625	+	0.876377i	$0.340047\pi$
$744$	−481.519	−0.0237276
$745$	7050.85	0.346743
$746$	−168.229	−0.00825643
$747$	−9144.40	−0.447893
$748$	4452.66	0.217654
$749$	−9071.77	−0.442557
$750$	149.875	0.00729689
$751$	5494.75	0.266986	0.133493	−	0.991050i	$-0.457381\pi$
$751$	5494.75	0.266986	0.133493	+	0.991050i	$0.457381\pi$
$752$	−33534.9	−1.62619
$753$	2480.77	0.120059
$754$	108.620	0.00524632
$755$	7074.60	0.341021
$756$	−9184.65	−0.441855
$757$	−3411.29	−0.163785	−0.0818926	−	0.996641i	$-0.526096\pi$
$757$	−3411.29	−0.163785	−0.0818926	+	0.996641i	$0.526096\pi$
$758$	−158.658	−0.00760253
$759$	8199.37	0.392119
$760$	1176.85	0.0561695
$761$	−15927.5	−0.758700	−0.379350	−	0.925253i	$-0.623852\pi$
$761$	−15927.5	−0.758700	−0.379350	+	0.925253i	$0.623852\pi$
$762$	156.668	0.00744816
$763$	18902.6	0.896882
$764$	−19700.4	−0.932899
$765$	2013.44	0.0951581
$766$	−322.906	−0.0152311
$767$	2433.91	0.114581
$768$	26268.8	1.23424
$769$	20621.3	0.967000	0.483500	−	0.875344i	$-0.339365\pi$
$769$	20621.3	0.967000	0.483500	+	0.875344i	$0.339365\pi$
$770$	−355.130	−0.0166208
$771$	24509.1	1.14484
$772$	−32169.1	−1.49973
$773$	18163.8	0.845156	0.422578	−	0.906326i	$-0.361125\pi$
$773$	18163.8	0.845156	0.422578	+	0.906326i	$0.361125\pi$
$774$	−145.360	−0.00675045
$775$	−10695.5	−0.495733
$776$	−36.9284	−0.00170832
$777$	−16322.2	−0.753613
$778$	203.253	0.00936628
$779$	12534.0	0.576479
$780$	13267.4	0.609039
$781$	44747.6	2.05019
$782$	7.46672	0.000341444 0
$783$	−15012.4	−0.685186
$784$	−9384.69	−0.427510
$785$	23782.1	1.08130
$786$	159.927	0.00725752
$787$	−30750.8	−1.39282	−0.696408	−	0.717646i	$-0.745220\pi$
$787$	−30750.8	−1.39282	−0.696408	+	0.717646i	$0.745220\pi$
$788$	−14490.9	−0.655096
$789$	28962.6	1.30684
$790$	273.668	0.0123249
$791$	3986.80	0.179209
$792$	−409.418	−0.0183687
$793$	−4334.82	−0.194116
$794$	366.164	0.0163661
$795$	6677.49	0.297894
$796$	−5790.63	−0.257843
$797$	−32871.8	−1.46095	−0.730475	−	0.682939i	$-0.760701\pi$
$797$	−32871.8	−1.46095	−0.730475	+	0.682939i	$0.760701\pi$
$798$	469.297	0.0208182
$799$	−5256.21	−0.232730
$800$	−459.187	−0.0202934
$801$	12578.0	0.554834
$802$	463.181	0.0203934
$803$	−56175.5	−2.46873
$804$	−25355.9	−1.11223
$805$	4544.59	0.198976
$806$	−85.8375	−0.00375124
$807$	−3536.62	−0.154269
$808$	−488.290	−0.0212599
$809$	−2591.22	−0.112611	−0.0563055	−	0.998414i	$-0.517932\pi$
$809$	−2591.22	−0.112611	−0.0563055	+	0.998414i	$0.517932\pi$
$810$	415.810	0.0180371
$811$	37339.5	1.61673	0.808364	−	0.588683i	$-0.200353\pi$
$811$	37339.5	1.61673	0.808364	+	0.588683i	$0.200353\pi$
$812$	20530.0	0.887269
$813$	−5765.87	−0.248730
$814$	326.050	0.0140393
$815$	44896.2	1.92963
$816$	4119.49	0.176729
$817$	50802.4	2.17546
$818$	−141.266	−0.00603819
$819$	3653.65	0.155884
$820$	−8776.26	−0.373756
$821$	20494.9	0.871226	0.435613	−	0.900134i	$-0.356532\pi$
$821$	20494.9	0.871226	0.435613	+	0.900134i	$0.356532\pi$
$822$	44.0009	0.00186704
$823$	−13568.2	−0.574675	−0.287338	−	0.957829i	$-0.592770\pi$
$823$	−13568.2	−0.574675	−0.287338	+	0.957829i	$0.592770\pi$
$824$	−288.318	−0.0121894
$825$	−26339.0	−1.11152
$826$	−60.2816	−0.00253930
$827$	−39430.4	−1.65796	−0.828978	−	0.559281i	$-0.811077\pi$
$827$	−39430.4	−1.65796	−0.828978	+	0.559281i	$0.811077\pi$
$828$	2619.48	0.109944
$829$	14439.0	0.604931	0.302465	−	0.953160i	$-0.402190\pi$
$829$	14439.0	0.604931	0.302465	+	0.953160i	$0.402190\pi$
$830$	293.236	0.0122631
$831$	14697.2	0.613525
$832$	9369.86	0.390434
$833$	−1470.94	−0.0611825
$834$	−415.510	−0.0172517
$835$	−53186.1	−2.20429
$836$	71539.8	2.95964
$837$	11863.6	0.489924
$838$	−398.378	−0.0164221
$839$	33140.6	1.36369	0.681847	−	0.731495i	$-0.261177\pi$
$839$	33140.6	1.36369	0.681847	+	0.731495i	$0.261177\pi$
$840$	−657.243	−0.0269965
$841$	9167.56	0.375889
$842$	270.435	0.0110687
$843$	−16614.7	−0.678816
$844$	4670.45	0.190478
$845$	−26252.9	−1.06879
$846$	241.636	0.00981986
$847$	−24530.4	−0.995131
$848$	4717.08	0.191020
$849$	9558.55	0.386394
$850$	−23.9855	−0.000967877 0
$851$	−4172.45	−0.168072
$852$	41404.7	1.66491
$853$	27379.7	1.09902	0.549510	−	0.835487i	$-0.314815\pi$
$853$	27379.7	1.09902	0.549510	+	0.835487i	$0.314815\pi$
$854$	107.362	0.00430195
$855$	32349.4	1.29395
$856$	335.378	0.0133913
$857$	20497.8	0.817025	0.408513	−	0.912753i	$-0.366048\pi$
$857$	20497.8	0.817025	0.408513	+	0.912753i	$0.366048\pi$
$858$	−211.385	−0.00841093
$859$	−26376.7	−1.04768	−0.523842	−	0.851816i	$-0.675502\pi$
$859$	−26376.7	−1.04768	−0.523842	+	0.851816i	$0.675502\pi$
$860$	−35571.6	−1.41045
$861$	−6999.94	−0.277070
$862$	516.712	0.0204168
$863$	−32543.2	−1.28364	−0.641821	−	0.766854i	$-0.721821\pi$
$863$	−32543.2	−1.28364	−0.641821	+	0.766854i	$0.721821\pi$
$864$	509.337	0.0200556
$865$	1832.50	0.0720311
$866$	−187.118	−0.00734241
$867$	−30904.1	−1.21056
$868$	−16223.9	−0.634418
$869$	33274.3	1.29891
$870$	−537.101	−0.0209304
$871$	−9040.67	−0.351701
$872$	−698.818	−0.0271387
$873$	−1015.09	−0.0393536
$874$	119.966	0.00464292
$875$	10100.2	0.390227
$876$	−51978.9	−2.00480
$877$	6671.86	0.256890	0.128445	−	0.991717i	$-0.459001\pi$
$877$	6671.86	0.256890	0.128445	+	0.991717i	$0.459001\pi$
$878$	−25.7544	−0.000989942 0
$879$	−18117.2	−0.695196
$880$	−50085.4	−1.91861
$881$	−25432.9	−0.972595	−0.486298	−	0.873793i	$-0.661653\pi$
$881$	−25432.9	−0.972595	−0.486298	+	0.873793i	$0.661653\pi$
$882$	67.6213	0.00258155
$883$	16292.4	0.620933	0.310467	−	0.950584i	$-0.399515\pi$
$883$	16292.4	0.620933	0.310467	+	0.950584i	$0.399515\pi$
$884$	1469.00	0.0558912
$885$	−12035.1	−0.457123
$886$	−238.214	−0.00903267
$887$	8139.80	0.308126	0.154063	−	0.988061i	$-0.450764\pi$
$887$	8139.80	0.308126	0.154063	+	0.988061i	$0.450764\pi$
$888$	603.423	0.0228036
$889$	10558.0	0.398317
$890$	−403.343	−0.0151911
$891$	50556.9	1.90092
$892$	−25257.3	−0.948067
$893$	−84450.2	−3.16463
$894$	−103.947	−0.00388870
$895$	11425.7	0.426724
$896$	−928.695	−0.0346267
$897$	2705.09	0.100692
$898$	216.021	0.00802753
$899$	−26518.1	−0.983793
$900$	−8414.61	−0.311652
$901$	739.347	0.0273376
$902$	139.829	0.00516164
$903$	−28371.9	−1.04558
$904$	−147.390	−0.00542268
$905$	34274.0	1.25890
$906$	−104.297	−0.00382453
$907$	16087.6	0.588954	0.294477	−	0.955659i	$-0.404855\pi$
$907$	16087.6	0.588954	0.294477	+	0.955659i	$0.404855\pi$
$908$	17756.0	0.648959
$909$	−13422.2	−0.489753
$910$	−117.163	−0.00426803
$911$	−36379.0	−1.32304	−0.661521	−	0.749927i	$-0.730089\pi$
$911$	−36379.0	−1.32304	−0.661521	+	0.749927i	$0.730089\pi$
$912$	66186.8	2.40314
$913$	35653.6	1.29240
$914$	204.767	0.00741037
$915$	21434.6	0.774433
$916$	35181.3	1.26902
$917$	10777.6	0.388121
$918$	26.6051	0.000956534 0
$919$	−10077.9	−0.361739	−0.180869	−	0.983507i	$-0.557891\pi$
$919$	−10077.9	−0.361739	−0.180869	+	0.983507i	$0.557891\pi$
$920$	−168.011	−0.00602081
$921$	−26237.2	−0.938704
$922$	20.7708	0.000741919 0
$923$	14762.9	0.526465
$924$	−39953.3	−1.42248
$925$	13403.2	0.476427
$926$	−23.1275	−0.000820753 0
$927$	−7925.32	−0.280800
$928$	−1138.50	−0.0402726
$929$	−7768.50	−0.274355	−0.137178	−	0.990546i	$-0.543803\pi$
$929$	−7768.50	−0.274355	−0.137178	+	0.990546i	$0.543803\pi$
$930$	424.445	0.0149657
$931$	−23633.2	−0.831952
$932$	−16900.5	−0.593987
$933$	25974.8	0.911442
$934$	−127.484	−0.00446615
$935$	−7850.30	−0.274580
$936$	−135.073	−0.00471688
$937$	−1611.36	−0.0561804	−0.0280902	−	0.999605i	$-0.508943\pi$
$937$	−1611.36	−0.0561804	−0.0280902	+	0.999605i	$0.508943\pi$
$938$	223.914	0.00779430
$939$	32824.8	1.14078
$940$	59131.7	2.05177
$941$	−1974.16	−0.0683909	−0.0341954	−	0.999415i	$-0.510887\pi$
$941$	−1974.16	−0.0683909	−0.0341954	+	0.999415i	$0.510887\pi$
$942$	−350.605	−0.0121267
$943$	−1789.39	−0.0617927
$944$	−8501.76	−0.293124
$945$	16193.1	0.557418
$946$	566.751	0.0194785
$947$	57242.2	1.96422	0.982112	−	0.188297i	$-0.0602969\pi$
$947$	57242.2	1.96422	0.982112	+	0.188297i	$0.0602969\pi$
$948$	30788.6	1.05482
$949$	−18533.1	−0.633942
$950$	−385.369	−0.0131611
$951$	−45067.1	−1.53670
$952$	−72.7715	−0.00247746
$953$	−14723.4	−0.500458	−0.250229	−	0.968187i	$-0.580506\pi$
$953$	−14723.4	−0.500458	−0.250229	+	0.968187i	$0.580506\pi$
$954$	−33.9889	−0.00115349
$955$	34732.9	1.17689
$956$	12387.9	0.419092
$957$	−65304.2	−2.20584
$958$	72.9128	0.00245898
$959$	2965.25	0.0998467
$960$	−46331.6	−1.55765
$961$	−8834.99	−0.296566
$962$	107.569	0.00360515
$963$	9218.90	0.308489
$964$	20132.7	0.672646
$965$	56715.9	1.89197
$966$	−66.9982	−0.00223150
$967$	−46140.9	−1.53443	−0.767214	−	0.641391i	$-0.778358\pi$
$967$	−46140.9	−1.53443	−0.767214	+	0.641391i	$0.778358\pi$
$968$	906.875	0.0301117
$969$	10374.0	0.343922
$970$	32.5513	0.00107748
$971$	15349.8	0.507310	0.253655	−	0.967295i	$-0.418367\pi$
$971$	15349.8	0.507310	0.253655	+	0.967295i	$0.418367\pi$
$972$	29080.6	0.959631
$973$	−28001.5	−0.922596
$974$	−382.737	−0.0125910
$975$	−8689.62	−0.285426
$976$	15141.7	0.496594
$977$	−1517.18	−0.0496815	−0.0248407	−	0.999691i	$-0.507908\pi$
$977$	−1517.18	−0.0496815	−0.0248407	+	0.999691i	$0.507908\pi$
$978$	−661.879	−0.0216406
$979$	−49041.0	−1.60098
$980$	16547.9	0.539391
$981$	−19209.2	−0.625181
$982$	−653.926	−0.0212501
$983$	22648.4	0.734866	0.367433	−	0.930050i	$-0.380237\pi$
$983$	22648.4	0.734866	0.367433	+	0.930050i	$0.380237\pi$
$984$	258.783	0.00838386
$985$	25548.2	0.826430
$986$	−59.4691	−0.00192077
$987$	47163.4	1.52100
$988$	23602.1	0.760002
$989$	−7252.70	−0.233187
$990$	360.890	0.0115857
$991$	12346.4	0.395760	0.197880	−	0.980226i	$-0.436594\pi$
$991$	12346.4	0.395760	0.197880	+	0.980226i	$0.436594\pi$
$992$	899.700	0.0287959
$993$	−41978.8	−1.34155
$994$	−365.639	−0.0116674
$995$	10209.2	0.325280
$996$	32990.1	1.04953
$997$	−43566.0	−1.38390	−0.691950	−	0.721945i	$-0.743248\pi$
$997$	−43566.0	−1.38390	−0.691950	+	0.721945i	$0.743248\pi$
$998$	20.5159	0.000650721 0
$999$	−14867.1	−0.470844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.4.a.b.1.2	✓	4
3.2	odd	2		207.4.a.e.1.3		4
4.3	odd	2		368.4.a.l.1.1		4
5.2	odd	4		575.4.b.g.24.4		8
5.3	odd	4		575.4.b.g.24.5		8
5.4	even	2		575.4.a.i.1.3		4
7.6	odd	2		1127.4.a.c.1.2		4
8.3	odd	2		1472.4.a.bf.1.4		4
8.5	even	2		1472.4.a.y.1.1		4
23.22	odd	2		529.4.a.g.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.a.b.1.2	✓	4	1.1	even	1	trivial
207.4.a.e.1.3		4	3.2	odd	2
368.4.a.l.1.1		4	4.3	odd	2
529.4.a.g.1.2		4	23.22	odd	2
575.4.a.i.1.3		4	5.4	even	2
575.4.b.g.24.4		8	5.2	odd	4
575.4.b.g.24.5		8	5.3	odd	4
1127.4.a.c.1.2		4	7.6	odd	2
1472.4.a.y.1.1		4	8.5	even	2
1472.4.a.bf.1.4		4	8.3	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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	23.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0323756 q^{2} +6.42170 q^{3} -7.99895 q^{4} +14.1026 q^{5} -0.207906 q^{6} -14.0109 q^{7} +0.517976 q^{8} +14.2382 q^{9} +O(q^{10})\)	\(q-0.0323756 q^{2} +6.42170 q^{3} -7.99895 q^{4} +14.1026 q^{5} -0.207906 q^{6} -14.0109 q^{7} +0.517976 q^{8} +14.2382 q^{9} -0.456580 q^{10} -55.5140 q^{11} -51.3668 q^{12} -18.3149 q^{13} +0.453613 q^{14} +90.5626 q^{15} +63.9748 q^{16} +10.0273 q^{17} -0.460970 q^{18} +161.106 q^{19} -112.806 q^{20} -89.9740 q^{21} +1.79730 q^{22} -23.0000 q^{23} +3.32628 q^{24} +73.8833 q^{25} +0.592956 q^{26} -81.9525 q^{27} +112.073 q^{28} +183.185 q^{29} -2.93202 q^{30} -144.762 q^{31} -6.21503 q^{32} -356.494 q^{33} -0.324640 q^{34} -197.591 q^{35} -113.891 q^{36} +181.411 q^{37} -5.21591 q^{38} -117.613 q^{39} +7.30481 q^{40} +77.7996 q^{41} +2.91296 q^{42} +315.335 q^{43} +444.054 q^{44} +200.795 q^{45} +0.744639 q^{46} -524.190 q^{47} +410.827 q^{48} -146.693 q^{49} -2.39202 q^{50} +64.3923 q^{51} +146.500 q^{52} +73.7334 q^{53} +2.65326 q^{54} -782.892 q^{55} -7.25733 q^{56} +1034.57 q^{57} -5.93071 q^{58} -132.892 q^{59} -724.406 q^{60} +236.683 q^{61} +4.68676 q^{62} -199.490 q^{63} -511.598 q^{64} -258.288 q^{65} +11.5417 q^{66} +493.624 q^{67} -80.2079 q^{68} -147.699 q^{69} +6.39712 q^{70} -806.060 q^{71} +7.37504 q^{72} +1011.91 q^{73} -5.87328 q^{74} +474.456 q^{75} -1288.68 q^{76} +777.804 q^{77} +3.80779 q^{78} -599.386 q^{79} +902.212 q^{80} -910.705 q^{81} -2.51881 q^{82} -642.245 q^{83} +719.698 q^{84} +141.411 q^{85} -10.2092 q^{86} +1176.36 q^{87} -28.7549 q^{88} +883.399 q^{89} -6.50088 q^{90} +256.609 q^{91} +183.976 q^{92} -929.617 q^{93} +16.9710 q^{94} +2272.02 q^{95} -39.9111 q^{96} -71.2938 q^{97} +4.74929 q^{98} -790.419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9	\( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9} - 70 q^{10} + 8 q^{11} - 67 q^{12} + 111 q^{13} - 144 q^{14} + 10 q^{15} + 64 q^{16} + 98 q^{17} + 49 q^{18} + 96 q^{19} + 140 q^{20} + 180 q^{21} + 220 q^{22} - 92 q^{23} - 188 q^{24} + 184 q^{25} - 229 q^{26} - 155 q^{27} + 282 q^{28} + 21 q^{29} - 406 q^{30} - 193 q^{31} - 432 q^{32} - 418 q^{33} + 666 q^{34} - 752 q^{35} - 629 q^{36} + 170 q^{37} + 748 q^{38} - 291 q^{39} - 26 q^{40} - 125 q^{41} + 640 q^{42} + 2 q^{43} + 830 q^{44} + 168 q^{45} - 46 q^{46} - 677 q^{47} + 551 q^{48} + 1220 q^{49} + 414 q^{50} - 340 q^{51} + 2247 q^{52} - 230 q^{53} + 641 q^{54} - 972 q^{55} - 2174 q^{56} + 1322 q^{57} - 1835 q^{58} - 1140 q^{59} - 804 q^{60} + 754 q^{61} + 443 q^{62} - 1092 q^{63} - 805 q^{64} + 1318 q^{65} - 398 q^{66} + 488 q^{67} + 284 q^{68} - 161 q^{69} - 3820 q^{70} - 401 q^{71} + 1503 q^{72} + 1509 q^{73} + 1366 q^{74} + 1401 q^{75} - 3832 q^{76} + 736 q^{77} - 1907 q^{78} - 838 q^{79} + 2846 q^{80} - 932 q^{81} - 949 q^{82} + 142 q^{83} + 2614 q^{84} + 112 q^{85} + 918 q^{86} + 2223 q^{87} - 404 q^{88} + 2342 q^{89} + 1784 q^{90} + 292 q^{91} - 460 q^{92} - 509 q^{93} + 1567 q^{94} - 956 q^{95} + 799 q^{96} + 1062 q^{97} + 2478 q^{98} - 1498 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9 - 70 * q^10 + 8 * q^11 - 67 * q^12 + 111 * q^13 - 144 * q^14 + 10 * q^15 + 64 * q^16 + 98 * q^17 + 49 * q^18 + 96 * q^19 + 140 * q^20 + 180 * q^21 + 220 * q^22 - 92 * q^23 - 188 * q^24 + 184 * q^25 - 229 * q^26 - 155 * q^27 + 282 * q^28 + 21 * q^29 - 406 * q^30 - 193 * q^31 - 432 * q^32 - 418 * q^33 + 666 * q^34 - 752 * q^35 - 629 * q^36 + 170 * q^37 + 748 * q^38 - 291 * q^39 - 26 * q^40 - 125 * q^41 + 640 * q^42 + 2 * q^43 + 830 * q^44 + 168 * q^45 - 46 * q^46 - 677 * q^47 + 551 * q^48 + 1220 * q^49 + 414 * q^50 - 340 * q^51 + 2247 * q^52 - 230 * q^53 + 641 * q^54 - 972 * q^55 - 2174 * q^56 + 1322 * q^57 - 1835 * q^58 - 1140 * q^59 - 804 * q^60 + 754 * q^61 + 443 * q^62 - 1092 * q^63 - 805 * q^64 + 1318 * q^65 - 398 * q^66 + 488 * q^67 + 284 * q^68 - 161 * q^69 - 3820 * q^70 - 401 * q^71 + 1503 * q^72 + 1509 * q^73 + 1366 * q^74 + 1401 * q^75 - 3832 * q^76 + 736 * q^77 - 1907 * q^78 - 838 * q^79 + 2846 * q^80 - 932 * q^81 - 949 * q^82 + 142 * q^83 + 2614 * q^84 + 112 * q^85 + 918 * q^86 + 2223 * q^87 - 404 * q^88 + 2342 * q^89 + 1784 * q^90 + 292 * q^91 - 460 * q^92 - 509 * q^93 + 1567 * q^94 - 956 * q^95 + 799 * q^96 + 1062 * q^97 + 2478 * q^98 - 1498 * q^99

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