LMFDB - Embedded newform 471.4.a.c.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,4,Mod(1,471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("471.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	471.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:	$27.7898996127$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00169 q^{2} -3.00000 q^{3} +17.0169 q^{4} +16.1971 q^{5} -15.0051 q^{6} -23.8262 q^{7} +45.0996 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+5.00169 q^{2} -3.00000 q^{3} +17.0169 q^{4} +16.1971 q^{5} -15.0051 q^{6} -23.8262 q^{7} +45.0996 q^{8} +9.00000 q^{9} +81.0129 q^{10} +9.48462 q^{11} -51.0506 q^{12} +45.8182 q^{13} -119.171 q^{14} -48.5913 q^{15} +89.4391 q^{16} +82.9387 q^{17} +45.0152 q^{18} +28.8942 q^{19} +275.624 q^{20} +71.4785 q^{21} +47.4391 q^{22} +104.751 q^{23} -135.299 q^{24} +137.347 q^{25} +229.168 q^{26} -27.0000 q^{27} -405.447 q^{28} +35.2486 q^{29} -243.039 q^{30} -115.035 q^{31} +86.5496 q^{32} -28.4539 q^{33} +414.833 q^{34} -385.915 q^{35} +153.152 q^{36} -100.531 q^{37} +144.520 q^{38} -137.455 q^{39} +730.484 q^{40} -378.309 q^{41} +357.513 q^{42} +494.020 q^{43} +161.399 q^{44} +145.774 q^{45} +523.932 q^{46} -244.254 q^{47} -268.317 q^{48} +224.686 q^{49} +686.965 q^{50} -248.816 q^{51} +779.683 q^{52} +173.916 q^{53} -135.046 q^{54} +153.623 q^{55} -1074.55 q^{56} -86.6827 q^{57} +176.303 q^{58} +112.605 q^{59} -826.873 q^{60} +172.638 q^{61} -575.367 q^{62} -214.436 q^{63} -282.619 q^{64} +742.123 q^{65} -142.317 q^{66} -899.323 q^{67} +1411.36 q^{68} -314.253 q^{69} -1930.23 q^{70} -698.693 q^{71} +405.896 q^{72} +295.729 q^{73} -502.826 q^{74} -412.040 q^{75} +491.690 q^{76} -225.982 q^{77} -687.505 q^{78} +659.257 q^{79} +1448.66 q^{80} +81.0000 q^{81} -1892.18 q^{82} -816.298 q^{83} +1216.34 q^{84} +1343.37 q^{85} +2470.94 q^{86} -105.746 q^{87} +427.753 q^{88} +71.1185 q^{89} +729.116 q^{90} -1091.67 q^{91} +1782.54 q^{92} +345.104 q^{93} -1221.68 q^{94} +468.003 q^{95} -259.649 q^{96} -51.1002 q^{97} +1123.81 q^{98} +85.3616 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) $ 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * 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 $

$2$	5.00169	1.76836	0.884182	−	0.467143i	$-0.154717\pi$
$2$	5.00169	1.76836	0.884182	+	0.467143i	$0.154717\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	17.0169	2.12711
$5$	16.1971	1.44871	0.724357	−	0.689425i	$-0.242137\pi$
$5$	16.1971	1.44871	0.724357	+	0.689425i	$0.242137\pi$
$6$	−15.0051	−1.02097
$7$	−23.8262	−1.28649	−0.643246	−	0.765660i	$-0.722413\pi$
$7$	−23.8262	−1.28649	−0.643246	+	0.765660i	$0.722413\pi$
$8$	45.0996	1.99314
$9$	9.00000	0.333333
$10$	81.0129	2.56185
$11$	9.48462	0.259975	0.129987	−	0.991516i	$-0.458506\pi$
$11$	9.48462	0.259975	0.129987	+	0.991516i	$0.458506\pi$
$12$	−51.0506	−1.22809
$13$	45.8182	0.977514	0.488757	−	0.872420i	$-0.337450\pi$
$13$	45.8182	0.977514	0.488757	+	0.872420i	$0.337450\pi$
$14$	−119.171	−2.27499
$15$	−48.5913	−0.836415
$16$	89.4391	1.39749
$17$	82.9387	1.18327	0.591635	−	0.806206i	$-0.298483\pi$
$17$	82.9387	1.18327	0.591635	+	0.806206i	$0.298483\pi$
$18$	45.0152	0.589455
$19$	28.8942	0.348884	0.174442	−	0.984667i	$-0.444188\pi$
$19$	28.8942	0.348884	0.174442	+	0.984667i	$0.444188\pi$
$20$	275.624	3.08157
$21$	71.4785	0.742757
$22$	47.4391	0.459730
$23$	104.751	0.949657	0.474829	−	0.880078i	$-0.342510\pi$
$23$	104.751	0.949657	0.474829	+	0.880078i	$0.342510\pi$
$24$	−135.299	−1.15074
$25$	137.347	1.09877
$26$	229.168	1.72860
$27$	−27.0000	−0.192450
$28$	−405.447	−2.73651
$29$	35.2486	0.225707	0.112854	−	0.993612i	$-0.464001\pi$
$29$	35.2486	0.225707	0.112854	+	0.993612i	$0.464001\pi$
$30$	−243.039	−1.47909
$31$	−115.035	−0.666478	−0.333239	−	0.942842i	$-0.608142\pi$
$31$	−115.035	−0.666478	−0.333239	+	0.942842i	$0.608142\pi$
$32$	86.5496	0.478124
$33$	−28.4539	−0.150096
$34$	414.833	2.09245
$35$	−385.915	−1.86376
$36$	153.152	0.709037
$37$	−100.531	−0.446682	−0.223341	−	0.974740i	$-0.571696\pi$
$37$	−100.531	−0.446682	−0.223341	+	0.974740i	$0.571696\pi$
$38$	144.520	0.616953
$39$	−137.455	−0.564368
$40$	730.484	2.88749
$41$	−378.309	−1.44102	−0.720511	−	0.693443i	$-0.756093\pi$
$41$	−378.309	−1.44102	−0.720511	+	0.693443i	$0.756093\pi$
$42$	357.513	1.31346
$43$	494.020	1.75203	0.876016	−	0.482282i	$-0.160192\pi$
$43$	494.020	1.75203	0.876016	+	0.482282i	$0.160192\pi$
$44$	161.399	0.552994
$45$	145.774	0.482905
$46$	523.932	1.67934
$47$	−244.254	−0.758044	−0.379022	−	0.925388i	$-0.623740\pi$
$47$	−244.254	−0.758044	−0.379022	+	0.925388i	$0.623740\pi$
$48$	−268.317	−0.806839
$49$	224.686	0.655062
$50$	686.965	1.94303
$51$	−248.816	−0.683161
$52$	779.683	2.07928
$53$	173.916	0.450739	0.225369	−	0.974273i	$-0.427641\pi$
$53$	173.916	0.450739	0.225369	+	0.974273i	$0.427641\pi$
$54$	−135.046	−0.340322
$55$	153.623	0.376629
$56$	−1074.55	−2.56416
$57$	−86.6827	−0.201428
$58$	176.303	0.399132
$59$	112.605	0.248474	0.124237	−	0.992253i	$-0.460352\pi$
$59$	112.605	0.248474	0.124237	+	0.992253i	$0.460352\pi$
$60$	−826.873	−1.77915
$61$	172.638	0.362360	0.181180	−	0.983450i	$-0.442008\pi$
$61$	172.638	0.362360	0.181180	+	0.983450i	$0.442008\pi$
$62$	−575.367	−1.17858
$63$	−214.436	−0.428831
$64$	−282.619	−0.551990
$65$	742.123	1.41614
$66$	−142.317	−0.265425
$67$	−899.323	−1.63985	−0.819924	−	0.572473i	$-0.805984\pi$
$67$	−899.323	−1.63985	−0.819924	+	0.572473i	$0.805984\pi$
$68$	1411.36	2.51694
$69$	−314.253	−0.548285
$70$	−1930.23	−3.29580
$71$	−698.693	−1.16788	−0.583941	−	0.811796i	$-0.698490\pi$
$71$	−698.693	−1.16788	−0.583941	+	0.811796i	$0.698490\pi$
$72$	405.896	0.664380
$73$	295.729	0.474143	0.237071	−	0.971492i	$-0.423813\pi$
$73$	295.729	0.474143	0.237071	+	0.971492i	$0.423813\pi$
$74$	−502.826	−0.789896
$75$	−412.040	−0.634377
$76$	491.690	0.742114
$77$	−225.982	−0.334455
$78$	−687.505	−0.998008
$79$	659.257	0.938888	0.469444	−	0.882962i	$-0.344454\pi$
$79$	659.257	0.938888	0.469444	+	0.882962i	$0.344454\pi$
$80$	1448.66	2.02456
$81$	81.0000	0.111111
$82$	−1892.18	−2.54825
$83$	−816.298	−1.07952	−0.539761	−	0.841818i	$-0.681485\pi$
$83$	−816.298	−1.07952	−0.539761	+	0.841818i	$0.681485\pi$
$84$	1216.34	1.57992
$85$	1343.37	1.71422
$86$	2470.94	3.09823
$87$	−105.746	−0.130312
$88$	427.753	0.518166
$89$	71.1185	0.0847027	0.0423514	−	0.999103i	$-0.486515\pi$
$89$	71.1185	0.0847027	0.0423514	+	0.999103i	$0.486515\pi$
$90$	729.116	0.853951
$91$	−1091.67	−1.25756
$92$	1782.54	2.02003
$93$	345.104	0.384791
$94$	−1221.68	−1.34050
$95$	468.003	0.505433
$96$	−259.649	−0.276045
$97$	−51.1002	−0.0534891	−0.0267445	−	0.999642i	$-0.508514\pi$
$97$	−51.1002	−0.0534891	−0.0267445	+	0.999642i	$0.508514\pi$
$98$	1123.81	1.15839
$99$	85.3616	0.0866582
$100$	2337.21	2.33721
$101$	−1130.91	−1.11415	−0.557077	−	0.830461i	$-0.688077\pi$
$101$	−1130.91	−1.11415	−0.557077	+	0.830461i	$0.688077\pi$
$102$	−1244.50	−1.20808
$103$	280.305	0.268149	0.134074	−	0.990971i	$-0.457194\pi$
$103$	280.305	0.268149	0.134074	+	0.990971i	$0.457194\pi$
$104$	2066.38	1.94832
$105$	1157.75	1.07604
$106$	869.872	0.797070
$107$	−1205.78	−1.08942	−0.544708	−	0.838626i	$-0.683359\pi$
$107$	−1205.78	−1.08942	−0.544708	+	0.838626i	$0.683359\pi$
$108$	−459.456	−0.409362
$109$	1635.35	1.43705	0.718524	−	0.695502i	$-0.244818\pi$
$109$	1635.35	1.43705	0.718524	+	0.695502i	$0.244818\pi$
$110$	768.377	0.666017
$111$	301.594	0.257892
$112$	−2130.99	−1.79785
$113$	926.499	0.771307	0.385654	−	0.922644i	$-0.373976\pi$
$113$	926.499	0.771307	0.385654	+	0.922644i	$0.373976\pi$
$114$	−433.560	−0.356198
$115$	1696.67	1.37578
$116$	599.822	0.480104
$117$	412.364	0.325838
$118$	563.217	0.439393
$119$	−1976.11	−1.52227
$120$	−2191.45	−1.66709
$121$	−1241.04	−0.932413
$122$	863.479	0.640785
$123$	1134.93	0.831975
$124$	−1957.53	−1.41767
$125$	199.979	0.143093
$126$	−1072.54	−0.758329
$127$	−2837.69	−1.98271	−0.991355	−	0.131211i	$-0.958114\pi$
$127$	−2837.69	−1.98271	−0.991355	+	0.131211i	$0.958114\pi$
$128$	−2105.97	−1.45424
$129$	−1482.06	−1.01154
$130$	3711.87	2.50425
$131$	77.9213	0.0519696	0.0259848	−	0.999662i	$-0.491728\pi$
$131$	77.9213	0.0519696	0.0259848	+	0.999662i	$0.491728\pi$
$132$	−484.196	−0.319272
$133$	−688.439	−0.448836
$134$	−4498.13	−2.89985
$135$	−437.322	−0.278805
$136$	3740.50	2.35842
$137$	553.861	0.345398	0.172699	−	0.984975i	$-0.444751\pi$
$137$	553.861	0.345398	0.172699	+	0.984975i	$0.444751\pi$
$138$	−1571.80	−0.969567
$139$	−299.028	−0.182469	−0.0912347	−	0.995829i	$-0.529081\pi$
$139$	−299.028	−0.182469	−0.0912347	+	0.995829i	$0.529081\pi$
$140$	−6567.07	−3.96442
$141$	732.761	0.437657
$142$	−3494.65	−2.06524
$143$	434.568	0.254129
$144$	804.952	0.465829
$145$	570.926	0.326985
$146$	1479.14	0.838456
$147$	−674.059	−0.378200
$148$	−1710.73	−0.950141
$149$	−1566.18	−0.861116	−0.430558	−	0.902563i	$-0.641683\pi$
$149$	−1566.18	−0.861116	−0.430558	+	0.902563i	$0.641683\pi$
$150$	−2060.89	−1.12181
$151$	−1422.89	−0.766842	−0.383421	−	0.923574i	$-0.625254\pi$
$151$	−1422.89	−0.766842	−0.383421	+	0.923574i	$0.625254\pi$
$152$	1303.12	0.695374
$153$	746.448	0.394423
$154$	−1130.29	−0.591438
$155$	−1863.23	−0.965536
$156$	−2339.05	−1.20047
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	3297.40	1.66030
$159$	−521.747	−0.260234
$160$	1401.85	0.692664
$161$	−2495.82	−1.22173
$162$	405.137	0.196485
$163$	3120.48	1.49948	0.749739	−	0.661733i	$-0.230179\pi$
$163$	3120.48	1.49948	0.749739	+	0.661733i	$0.230179\pi$
$164$	−6437.64	−3.06521
$165$	−460.870	−0.217447
$166$	−4082.87	−1.90899
$167$	−1263.55	−0.585488	−0.292744	−	0.956191i	$-0.594568\pi$
$167$	−1263.55	−0.585488	−0.292744	+	0.956191i	$0.594568\pi$
$168$	3223.65	1.48042
$169$	−97.6909	−0.0444656
$170$	6719.10	3.03136
$171$	260.048	0.116295
$172$	8406.68	3.72677
$173$	−2092.48	−0.919584	−0.459792	−	0.888027i	$-0.652076\pi$
$173$	−2092.48	−0.919584	−0.459792	+	0.888027i	$0.652076\pi$
$174$	−528.908	−0.230439
$175$	−3272.44	−1.41356
$176$	848.296	0.363311
$177$	−337.816	−0.143457
$178$	355.712	0.149785
$179$	−1882.53	−0.786071	−0.393035	−	0.919523i	$-0.628575\pi$
$179$	−1882.53	−0.786071	−0.393035	+	0.919523i	$0.628575\pi$
$180$	2480.62	1.02719
$181$	3435.46	1.41080	0.705402	−	0.708808i	$-0.250767\pi$
$181$	3435.46	1.41080	0.705402	+	0.708808i	$0.250767\pi$
$182$	−5460.20	−2.22383
$183$	−517.913	−0.209209
$184$	4724.23	1.89280
$185$	−1628.32	−0.647114
$186$	1726.10	0.680451
$187$	786.642	0.307620
$188$	−4156.44	−1.61244
$189$	643.307	0.247586
$190$	2340.81	0.893789
$191$	−1957.97	−0.741747	−0.370873	−	0.928683i	$-0.620942\pi$
$191$	−1957.97	−0.741747	−0.370873	+	0.928683i	$0.620942\pi$
$192$	847.856	0.318691
$193$	−212.424	−0.0792261	−0.0396131	−	0.999215i	$-0.512613\pi$
$193$	−212.424	−0.0792261	−0.0396131	+	0.999215i	$0.512613\pi$
$194$	−255.587	−0.0945881
$195$	−2226.37	−0.817608
$196$	3823.46	1.39339
$197$	2617.39	0.946604	0.473302	−	0.880900i	$-0.343062\pi$
$197$	2617.39	0.946604	0.473302	+	0.880900i	$0.343062\pi$
$198$	426.952	0.153243
$199$	−2423.02	−0.863133	−0.431566	−	0.902081i	$-0.642039\pi$
$199$	−2423.02	−0.863133	−0.431566	+	0.902081i	$0.642039\pi$
$200$	6194.28	2.19001
$201$	2697.97	0.946766
$202$	−5656.45	−1.97023
$203$	−839.840	−0.290371
$204$	−4234.07	−1.45316
$205$	−6127.51	−2.08763
$206$	1402.00	0.474184
$207$	942.760	0.316552
$208$	4097.94	1.36606
$209$	274.051	0.0907009
$210$	5790.68	1.90283
$211$	−1567.73	−0.511502	−0.255751	−	0.966743i	$-0.582323\pi$
$211$	−1567.73	−0.511502	−0.255751	+	0.966743i	$0.582323\pi$
$212$	2959.50	0.958771
$213$	2096.08	0.674277
$214$	−6030.95	−1.92648
$215$	8001.70	2.53819
$216$	−1217.69	−0.383580
$217$	2740.83	0.857418
$218$	8179.51	2.54122
$219$	−887.186	−0.273746
$220$	2614.19	0.801131
$221$	3800.10	1.15666
$222$	1508.48	0.456047
$223$	−4758.67	−1.42899	−0.714493	−	0.699643i	$-0.753343\pi$
$223$	−4758.67	−1.42899	−0.714493	+	0.699643i	$0.753343\pi$
$224$	−2062.15	−0.615102
$225$	1236.12	0.366257
$226$	4634.06	1.36395
$227$	6348.50	1.85623	0.928117	−	0.372290i	$-0.121427\pi$
$227$	6348.50	1.85623	0.928117	+	0.372290i	$0.121427\pi$
$228$	−1475.07	−0.428460
$229$	3257.31	0.939952	0.469976	−	0.882679i	$-0.344263\pi$
$229$	3257.31	0.939952	0.469976	+	0.882679i	$0.344263\pi$
$230$	8486.19	2.43288
$231$	677.946	0.193098
$232$	1589.70	0.449866
$233$	−2277.38	−0.640326	−0.320163	−	0.947363i	$-0.603738\pi$
$233$	−2277.38	−0.640326	−0.320163	+	0.947363i	$0.603738\pi$
$234$	2062.52	0.576200
$235$	−3956.21	−1.09819
$236$	1916.19	0.528532
$237$	−1977.77	−0.542068
$238$	−9883.89	−2.69192
$239$	−6922.33	−1.87351	−0.936753	−	0.349990i	$-0.886185\pi$
$239$	−6922.33	−1.87351	−0.936753	+	0.349990i	$0.886185\pi$
$240$	−4345.97	−1.16888
$241$	146.451	0.0391441	0.0195721	−	0.999808i	$-0.493770\pi$
$241$	146.451	0.0391441	0.0195721	+	0.999808i	$0.493770\pi$
$242$	−6207.30	−1.64885
$243$	−243.000	−0.0641500
$244$	2937.75	0.770780
$245$	3639.27	0.948997
$246$	5676.55	1.47123
$247$	1323.88	0.341039
$248$	−5188.01	−1.32838
$249$	2448.89	0.623263
$250$	1000.23	0.253040
$251$	7233.09	1.81892	0.909459	−	0.415794i	$-0.136496\pi$
$251$	7233.09	1.81892	0.909459	+	0.415794i	$0.136496\pi$
$252$	−3649.02	−0.912170
$253$	993.525	0.246887
$254$	−14193.2	−3.50615
$255$	−4030.10	−0.989705
$256$	−8272.44	−2.01964
$257$	7456.92	1.80992	0.904960	−	0.425496i	$-0.139900\pi$
$257$	7456.92	1.80992	0.904960	+	0.425496i	$0.139900\pi$
$258$	−7412.81	−1.78876
$259$	2395.27	0.574653
$260$	12628.6	3.01228
$261$	317.238	0.0752357
$262$	389.738	0.0919011
$263$	579.171	0.135792	0.0678958	−	0.997692i	$-0.478371\pi$
$263$	579.171	0.135792	0.0678958	+	0.997692i	$0.478371\pi$
$264$	−1283.26	−0.299163
$265$	2816.93	0.652992
$266$	−3443.36	−0.793705
$267$	−213.355	−0.0489031
$268$	−15303.7	−3.48813
$269$	8164.20	1.85048	0.925242	−	0.379378i	$-0.123862\pi$
$269$	8164.20	1.85048	0.925242	+	0.379378i	$0.123862\pi$
$270$	−2187.35	−0.493029
$271$	5186.40	1.16255	0.581276	−	0.813706i	$-0.302554\pi$
$271$	5186.40	1.16255	0.581276	+	0.813706i	$0.302554\pi$
$272$	7417.96	1.65360
$273$	3275.02	0.726055
$274$	2770.24	0.610790
$275$	1302.68	0.285653
$276$	−5347.61	−1.16626
$277$	5325.42	1.15514	0.577569	−	0.816342i	$-0.304001\pi$
$277$	5325.42	1.15514	0.577569	+	0.816342i	$0.304001\pi$
$278$	−1495.65	−0.322672
$279$	−1035.31	−0.222159
$280$	−17404.6	−3.71473
$281$	−4121.14	−0.874898	−0.437449	−	0.899243i	$-0.644118\pi$
$281$	−4121.14	−0.874898	−0.437449	+	0.899243i	$0.644118\pi$
$282$	3665.04	0.773937
$283$	−6766.41	−1.42128	−0.710638	−	0.703557i	$-0.751594\pi$
$283$	−6766.41	−1.42128	−0.710638	+	0.703557i	$0.751594\pi$
$284$	−11889.6	−2.48421
$285$	−1404.01	−0.291812
$286$	2173.58	0.449392
$287$	9013.65	1.85386
$288$	778.947	0.159375
$289$	1965.82	0.400127
$290$	2855.60	0.578229
$291$	153.301	0.0308819
$292$	5032.38	1.00855
$293$	6576.57	1.31129	0.655644	−	0.755070i	$-0.272397\pi$
$293$	6576.57	1.31129	0.655644	+	0.755070i	$0.272397\pi$
$294$	−3371.43	−0.668795
$295$	1823.88	0.359968
$296$	−4533.92	−0.890299
$297$	−256.085	−0.0500321
$298$	−7833.53	−1.52277
$299$	4799.51	0.928304
$300$	−7011.63	−1.34939
$301$	−11770.6	−2.25398
$302$	−7116.85	−1.35606
$303$	3392.73	0.643258
$304$	2584.27	0.487560
$305$	2796.23	0.524956
$306$	3733.50	0.697484
$307$	−8317.89	−1.54634	−0.773171	−	0.634198i	$-0.781331\pi$
$307$	−8317.89	−1.54634	−0.773171	+	0.634198i	$0.781331\pi$
$308$	−3845.51	−0.711423
$309$	−840.916	−0.154816
$310$	−9319.28	−1.70742
$311$	8042.00	1.46630	0.733151	−	0.680066i	$-0.238049\pi$
$311$	8042.00	1.46630	0.733151	+	0.680066i	$0.238049\pi$
$312$	−6199.15	−1.12486
$313$	−9051.96	−1.63465	−0.817327	−	0.576174i	$-0.804545\pi$
$313$	−9051.96	−1.63465	−0.817327	+	0.576174i	$0.804545\pi$
$314$	−785.265	−0.141131
$315$	−3473.24	−0.621253
$316$	11218.5	1.99712
$317$	−8979.31	−1.59094	−0.795470	−	0.605993i	$-0.792776\pi$
$317$	−8979.31	−1.59094	−0.795470	+	0.605993i	$0.792776\pi$
$318$	−2609.62	−0.460189
$319$	334.320	0.0586781
$320$	−4577.61	−0.799675
$321$	3617.35	0.628974
$322$	−12483.3	−2.16046
$323$	2396.45	0.412823
$324$	1378.37	0.236346
$325$	6292.97	1.07407
$326$	15607.7	2.65162
$327$	−4906.05	−0.829680
$328$	−17061.6	−2.87216
$329$	5819.63	0.975218
$330$	−2305.13	−0.384525
$331$	4156.62	0.690238	0.345119	−	0.938559i	$-0.387839\pi$
$331$	4156.62	0.690238	0.345119	+	0.938559i	$0.387839\pi$
$332$	−13890.8	−2.29626
$333$	−904.781	−0.148894
$334$	−6319.89	−1.03536
$335$	−14566.4	−2.37567
$336$	6392.97	1.03799
$337$	5887.11	0.951607	0.475803	−	0.879552i	$-0.342157\pi$
$337$	5887.11	0.951607	0.475803	+	0.879552i	$0.342157\pi$
$338$	−488.619	−0.0786313
$339$	−2779.50	−0.445315
$340$	22859.9	3.64633
$341$	−1091.06	−0.173267
$342$	1300.68	0.205651
$343$	2818.96	0.443760
$344$	22280.1	3.49205
$345$	−5090.00	−0.794308
$346$	−10465.9	−1.62616
$347$	5338.60	0.825911	0.412955	−	0.910751i	$-0.364497\pi$
$347$	5338.60	0.825911	0.412955	+	0.910751i	$0.364497\pi$
$348$	−1799.47	−0.277188
$349$	10699.7	1.64110	0.820551	−	0.571574i	$-0.193667\pi$
$349$	10699.7	1.64110	0.820551	+	0.571574i	$0.193667\pi$
$350$	−16367.7	−2.49969
$351$	−1237.09	−0.188123
$352$	820.890	0.124300
$353$	12698.3	1.91463	0.957313	−	0.289052i	$-0.0933400\pi$
$353$	12698.3	1.91463	0.957313	+	0.289052i	$0.0933400\pi$
$354$	−1689.65	−0.253683
$355$	−11316.8	−1.69193
$356$	1210.21	0.180172
$357$	5928.33	0.878881
$358$	−9415.81	−1.39006
$359$	8171.68	1.20135	0.600675	−	0.799493i	$-0.294898\pi$
$359$	8171.68	1.20135	0.600675	+	0.799493i	$0.294898\pi$
$360$	6574.35	0.962497
$361$	−6024.12	−0.878280
$362$	17183.1	2.49481
$363$	3723.13	0.538329
$364$	−18576.9	−2.67498
$365$	4789.95	0.686897
$366$	−2590.44	−0.369957
$367$	11772.4	1.67443	0.837216	−	0.546873i	$-0.184182\pi$
$367$	11772.4	1.67443	0.837216	+	0.546873i	$0.184182\pi$
$368$	9368.85	1.32713
$369$	−3404.78	−0.480341
$370$	−8144.33	−1.14433
$371$	−4143.75	−0.579872
$372$	5872.59	0.818493
$373$	−7366.37	−1.02256	−0.511282	−	0.859413i	$-0.670829\pi$
$373$	−7366.37	−1.02256	−0.511282	+	0.859413i	$0.670829\pi$
$374$	3934.54	0.543984
$375$	−599.936	−0.0826148
$376$	−11015.7	−1.51089
$377$	1615.03	0.220632
$378$	3217.62	0.437821
$379$	10276.9	1.39285	0.696424	−	0.717631i	$-0.254773\pi$
$379$	10276.9	1.39285	0.696424	+	0.717631i	$0.254773\pi$
$380$	7963.95	1.07511
$381$	8513.06	1.14472
$382$	−9793.15	−1.31168
$383$	−13216.0	−1.76320	−0.881601	−	0.471996i	$-0.843534\pi$
$383$	−13216.0	−1.76320	−0.881601	+	0.471996i	$0.843534\pi$
$384$	6317.90	0.839607
$385$	−3660.26	−0.484530
$386$	−1062.48	−0.140101
$387$	4446.18	0.584011
$388$	−869.566	−0.113777
$389$	−8815.13	−1.14896	−0.574479	−	0.818519i	$-0.694795\pi$
$389$	−8815.13	−1.14896	−0.574479	+	0.818519i	$0.694795\pi$
$390$	−11135.6	−1.44583
$391$	8687.92	1.12370
$392$	10133.3	1.30563
$393$	−233.764	−0.0300047
$394$	13091.3	1.67394
$395$	10678.1	1.36018
$396$	1452.59	0.184331
$397$	−36.1013	−0.00456391	−0.00228196	−	0.999997i	$-0.500726\pi$
$397$	−36.1013	−0.00456391	−0.00228196	+	0.999997i	$0.500726\pi$
$398$	−12119.2	−1.52633
$399$	2065.32	0.259136
$400$	12284.2	1.53552
$401$	4961.67	0.617890	0.308945	−	0.951080i	$-0.400024\pi$
$401$	4961.67	0.617890	0.308945	+	0.951080i	$0.400024\pi$
$402$	13494.4	1.67423
$403$	−5270.68	−0.651492
$404$	−19244.5	−2.36993
$405$	1311.97	0.160968
$406$	−4200.62	−0.513481
$407$	−953.500	−0.116126
$408$	−11221.5	−1.36164
$409$	−6733.92	−0.814110	−0.407055	−	0.913404i	$-0.633444\pi$
$409$	−6733.92	−0.814110	−0.407055	+	0.913404i	$0.633444\pi$
$410$	−30647.9	−3.69169
$411$	−1661.58	−0.199416
$412$	4769.92	0.570381
$413$	−2682.96	−0.319660
$414$	4715.39	0.559780
$415$	−13221.7	−1.56392
$416$	3965.55	0.467373
$417$	897.085	0.105349
$418$	1370.72	0.160392
$419$	2414.86	0.281560	0.140780	−	0.990041i	$-0.455039\pi$
$419$	2414.86	0.281560	0.140780	+	0.990041i	$0.455039\pi$
$420$	19701.2	2.28886
$421$	−2899.83	−0.335699	−0.167849	−	0.985813i	$-0.553682\pi$
$421$	−2899.83	−0.335699	−0.167849	+	0.985813i	$0.553682\pi$
$422$	−7841.29	−0.904521
$423$	−2198.28	−0.252681
$424$	7843.53	0.898386
$425$	11391.3	1.30014
$426$	10483.9	1.19237
$427$	−4113.29	−0.466174
$428$	−20518.7	−2.31731
$429$	−1303.71	−0.146721
$430$	40022.0	4.48845
$431$	5129.95	0.573320	0.286660	−	0.958032i	$-0.407455\pi$
$431$	5129.95	0.573320	0.286660	+	0.958032i	$0.407455\pi$
$432$	−2414.86	−0.268946
$433$	9933.58	1.10249	0.551244	−	0.834344i	$-0.314153\pi$
$433$	9933.58	1.10249	0.551244	+	0.834344i	$0.314153\pi$
$434$	13708.8	1.51623
$435$	−1712.78	−0.188785
$436$	27828.6	3.05676
$437$	3026.70	0.331320
$438$	−4437.42	−0.484083
$439$	−11846.1	−1.28789	−0.643946	−	0.765071i	$-0.722704\pi$
$439$	−11846.1	−1.28789	−0.643946	+	0.765071i	$0.722704\pi$
$440$	6928.36	0.750674
$441$	2022.18	0.218354
$442$	19006.9	2.04540
$443$	1448.59	0.155361	0.0776803	−	0.996978i	$-0.475249\pi$
$443$	1448.59	0.155361	0.0776803	+	0.996978i	$0.475249\pi$
$444$	5132.18	0.548564
$445$	1151.91	0.122710
$446$	−23801.4	−2.52697
$447$	4698.53	0.497165
$448$	6733.72	0.710130
$449$	−2026.67	−0.213017	−0.106508	−	0.994312i	$-0.533967\pi$
$449$	−2026.67	−0.213017	−0.106508	+	0.994312i	$0.533967\pi$
$450$	6182.68	0.647676
$451$	−3588.12	−0.374629
$452$	15766.1	1.64066
$453$	4268.67	0.442736
$454$	31753.2	3.28250
$455$	−17681.9	−1.82185
$456$	−3909.35	−0.401474
$457$	−16789.6	−1.71856	−0.859281	−	0.511503i	$-0.829089\pi$
$457$	−16789.6	−1.71856	−0.859281	+	0.511503i	$0.829089\pi$
$458$	16292.0	1.66218
$459$	−2239.34	−0.227720
$460$	28872.0	2.92644
$461$	−882.815	−0.0891904	−0.0445952	−	0.999005i	$-0.514200\pi$
$461$	−882.815	−0.0891904	−0.0445952	+	0.999005i	$0.514200\pi$
$462$	3390.88	0.341467
$463$	2507.00	0.251642	0.125821	−	0.992053i	$-0.459843\pi$
$463$	2507.00	0.251642	0.125821	+	0.992053i	$0.459843\pi$
$464$	3152.61	0.315423
$465$	5589.68	0.557452
$466$	−11390.7	−1.13233
$467$	8734.02	0.865443	0.432722	−	0.901528i	$-0.357553\pi$
$467$	8734.02	0.865443	0.432722	+	0.901528i	$0.357553\pi$
$468$	7017.15	0.693093
$469$	21427.4	2.10965
$470$	−19787.7	−1.94200
$471$	471.000	0.0460776
$472$	5078.46	0.495244
$473$	4685.59	0.455484
$474$	−9892.19	−0.958572
$475$	3968.52	0.383344
$476$	−33627.2	−3.23803
$477$	1565.24	0.150246
$478$	−34623.3	−3.31304
$479$	−2950.66	−0.281460	−0.140730	−	0.990048i	$-0.544945\pi$
$479$	−2950.66	−0.281460	−0.140730	+	0.990048i	$0.544945\pi$
$480$	−4205.56	−0.399910
$481$	−4606.16	−0.436638
$482$	732.502	0.0692210
$483$	7487.45	0.705364
$484$	−21118.7	−1.98335
$485$	−827.676	−0.0774903
$486$	−1215.41	−0.113441
$487$	−15499.4	−1.44218	−0.721091	−	0.692840i	$-0.756359\pi$
$487$	−15499.4	−1.44218	−0.721091	+	0.692840i	$0.756359\pi$
$488$	7785.89	0.722235
$489$	−9361.45	−0.865724
$490$	18202.5	1.67817
$491$	13651.4	1.25474	0.627371	−	0.778721i	$-0.284131\pi$
$491$	13651.4	1.25474	0.627371	+	0.778721i	$0.284131\pi$
$492$	19312.9	1.76970
$493$	2923.48	0.267072
$494$	6621.64	0.603081
$495$	1382.61	0.125543
$496$	−10288.6	−0.931393
$497$	16647.2	1.50247
$498$	12248.6	1.10215
$499$	14507.7	1.30151	0.650753	−	0.759289i	$-0.274453\pi$
$499$	14507.7	1.30151	0.650753	+	0.759289i	$0.274453\pi$
$500$	3403.01	0.304375
$501$	3790.65	0.338032
$502$	36177.6	3.21651
$503$	−9798.69	−0.868592	−0.434296	−	0.900770i	$-0.643003\pi$
$503$	−9798.69	−0.868592	−0.434296	+	0.900770i	$0.643003\pi$
$504$	−9670.96	−0.854720
$505$	−18317.5	−1.61409
$506$	4969.30	0.436586
$507$	293.073	0.0256722
$508$	−48288.6	−4.21744
$509$	7967.73	0.693838	0.346919	−	0.937895i	$-0.387228\pi$
$509$	7967.73	0.693838	0.346919	+	0.937895i	$0.387228\pi$
$510$	−20157.3	−1.75016
$511$	−7046.08	−0.609981
$512$	−24528.4	−2.11721
$513$	−780.144	−0.0671427
$514$	37297.2	3.20060
$515$	4540.14	0.388471
$516$	−25220.1	−2.15165
$517$	−2316.65	−0.197072
$518$	11980.4	1.01619
$519$	6277.43	0.530922
$520$	33469.5	2.82256
$521$	2755.65	0.231722	0.115861	−	0.993265i	$-0.463037\pi$
$521$	2755.65	0.231722	0.115861	+	0.993265i	$0.463037\pi$
$522$	1586.72	0.133044
$523$	8873.52	0.741897	0.370948	−	0.928653i	$-0.379033\pi$
$523$	8873.52	0.741897	0.370948	+	0.928653i	$0.379033\pi$
$524$	1325.98	0.110545
$525$	9817.33	0.816120
$526$	2896.83	0.240129
$527$	−9540.81	−0.788623
$528$	−2544.89	−0.209758
$529$	−1194.20	−0.0981508
$530$	14089.4	1.15473
$531$	1013.45	0.0828247
$532$	−11715.1	−0.954724
$533$	−17333.4	−1.40862
$534$	−1067.14	−0.0864785
$535$	−19530.2	−1.57825
$536$	−40559.1	−3.26844
$537$	5647.58	0.453838
$538$	40834.8	3.27233
$539$	2131.06	0.170299
$540$	−7441.86	−0.593049
$541$	10367.5	0.823909	0.411954	−	0.911204i	$-0.364846\pi$
$541$	10367.5	0.823909	0.411954	+	0.911204i	$0.364846\pi$
$542$	25940.8	2.05581
$543$	−10306.4	−0.814528
$544$	7178.31	0.565749
$545$	26488.0	2.08187
$546$	16380.6	1.28393
$547$	−9661.54	−0.755206	−0.377603	−	0.925968i	$-0.623252\pi$
$547$	−9661.54	−0.755206	−0.377603	+	0.925968i	$0.623252\pi$
$548$	9424.99	0.734700
$549$	1553.74	0.120787
$550$	6515.60	0.505138
$551$	1018.48	0.0787456
$552$	−14172.7	−1.09281
$553$	−15707.6	−1.20787
$554$	26636.1	2.04270
$555$	4884.95	0.373612
$556$	−5088.53	−0.388132
$557$	−10008.4	−0.761349	−0.380674	−	0.924709i	$-0.624308\pi$
$557$	−10008.4	−0.761349	−0.380674	+	0.924709i	$0.624308\pi$
$558$	−5178.30	−0.392858
$559$	22635.1	1.71264
$560$	−34515.9	−2.60458
$561$	−2359.93	−0.177605
$562$	−20612.6	−1.54714
$563$	16684.8	1.24899	0.624493	−	0.781030i	$-0.285306\pi$
$563$	16684.8	1.24899	0.624493	+	0.781030i	$0.285306\pi$
$564$	12469.3	0.930944
$565$	15006.6	1.11740
$566$	−33843.5	−2.51333
$567$	−1929.92	−0.142944
$568$	−31510.8	−2.32775
$569$	−7884.34	−0.580894	−0.290447	−	0.956891i	$-0.593804\pi$
$569$	−7884.34	−0.580894	−0.290447	+	0.956891i	$0.593804\pi$
$570$	−7022.42	−0.516029
$571$	12591.2	0.922814	0.461407	−	0.887189i	$-0.347345\pi$
$571$	12591.2	0.922814	0.461407	+	0.887189i	$0.347345\pi$
$572$	7395.00	0.540560
$573$	5873.91	0.428248
$574$	45083.5	3.27831
$575$	14387.2	1.04346
$576$	−2543.57	−0.183997
$577$	−11241.1	−0.811046	−0.405523	−	0.914085i	$-0.632911\pi$
$577$	−11241.1	−0.811046	−0.405523	+	0.914085i	$0.632911\pi$
$578$	9832.44	0.707570
$579$	637.273	0.0457412
$580$	9715.38	0.695533
$581$	19449.3	1.38880
$582$	766.762	0.0546105
$583$	1649.52	0.117181
$584$	13337.2	0.945032
$585$	6679.11	0.472046
$586$	32894.0	2.31884
$587$	1015.94	0.0714347	0.0357173	−	0.999362i	$-0.488628\pi$
$587$	1015.94	0.0714347	0.0357173	+	0.999362i	$0.488628\pi$
$588$	−11470.4	−0.804473
$589$	−3323.83	−0.232523
$590$	9122.49	0.636554
$591$	−7852.16	−0.546522
$592$	−8991.42	−0.624232
$593$	−26028.1	−1.80244	−0.901220	−	0.433362i	$-0.857327\pi$
$593$	−26028.1	−1.80244	−0.901220	+	0.433362i	$0.857327\pi$
$594$	−1280.86	−0.0884750
$595$	−32007.3	−2.20533
$596$	−26651.5	−1.83169
$597$	7269.07	0.498330
$598$	24005.6	1.64158
$599$	4134.12	0.281996	0.140998	−	0.990010i	$-0.454969\pi$
$599$	4134.12	0.281996	0.140998	+	0.990010i	$0.454969\pi$
$600$	−18582.8	−1.26440
$601$	−9808.71	−0.665733	−0.332866	−	0.942974i	$-0.608016\pi$
$601$	−9808.71	−0.665733	−0.332866	+	0.942974i	$0.608016\pi$
$602$	−58872.9	−3.98585
$603$	−8093.90	−0.546616
$604$	−24213.1	−1.63116
$605$	−20101.3	−1.35080
$606$	16969.4	1.13751
$607$	−23419.8	−1.56603	−0.783015	−	0.622003i	$-0.786319\pi$
$607$	−23419.8	−1.56603	−0.783015	+	0.622003i	$0.786319\pi$
$608$	2500.78	0.166810
$609$	2519.52	0.167645
$610$	13985.9	0.928314
$611$	−11191.3	−0.740999
$612$	12702.2	0.838981
$613$	−1908.75	−0.125765	−0.0628823	−	0.998021i	$-0.520029\pi$
$613$	−1908.75	−0.125765	−0.0628823	+	0.998021i	$0.520029\pi$
$614$	−41603.5	−2.73449
$615$	18382.5	1.20529
$616$	−10191.7	−0.666616
$617$	−20370.0	−1.32912	−0.664559	−	0.747236i	$-0.731380\pi$
$617$	−20370.0	−1.32912	−0.664559	+	0.747236i	$0.731380\pi$
$618$	−4206.00	−0.273770
$619$	6964.74	0.452240	0.226120	−	0.974099i	$-0.427396\pi$
$619$	6964.74	0.452240	0.226120	+	0.974099i	$0.427396\pi$
$620$	−31706.3	−2.05380
$621$	−2828.28	−0.182762
$622$	40223.6	2.59296
$623$	−1694.48	−0.108969
$624$	−12293.8	−0.788697
$625$	−13929.2	−0.891472
$626$	−45275.1	−2.89066
$627$	−822.152	−0.0523662
$628$	−2671.65	−0.169762
$629$	−8337.93	−0.528545
$630$	−17372.0	−1.09860
$631$	−14074.6	−0.887958	−0.443979	−	0.896037i	$-0.646433\pi$
$631$	−14074.6	−0.887958	−0.443979	+	0.896037i	$0.646433\pi$
$632$	29732.2	1.87134
$633$	4703.19	0.295316
$634$	−44911.7	−2.81336
$635$	−45962.4	−2.87238
$636$	−8878.51	−0.553547
$637$	10294.7	0.640333
$638$	1672.16	0.103764
$639$	−6288.24	−0.389294
$640$	−34110.6	−2.10678
$641$	−3764.07	−0.231937	−0.115969	−	0.993253i	$-0.536997\pi$
$641$	−3764.07	−0.231937	−0.115969	+	0.993253i	$0.536997\pi$
$642$	18092.9	1.11226
$643$	16905.1	1.03681	0.518407	−	0.855134i	$-0.326525\pi$
$643$	16905.1	1.03681	0.518407	+	0.855134i	$0.326525\pi$
$644$	−42471.0	−2.59875
$645$	−24005.1	−1.46543
$646$	11986.3	0.730022
$647$	18899.6	1.14841	0.574205	−	0.818712i	$-0.305311\pi$
$647$	18899.6	1.14841	0.574205	+	0.818712i	$0.305311\pi$
$648$	3653.07	0.221460
$649$	1068.02	0.0645970
$650$	31475.5	1.89934
$651$	−8222.50	−0.495031
$652$	53100.9	3.18956
$653$	−8626.95	−0.516996	−0.258498	−	0.966012i	$-0.583228\pi$
$653$	−8626.95	−0.516996	−0.258498	+	0.966012i	$0.583228\pi$
$654$	−24538.5	−1.46718
$655$	1262.10	0.0752891
$656$	−33835.6	−2.01381
$657$	2661.56	0.158048
$658$	29108.0	1.72454
$659$	11366.3	0.671877	0.335938	−	0.941884i	$-0.390947\pi$
$659$	11366.3	0.671877	0.335938	+	0.941884i	$0.390947\pi$
$660$	−7842.58	−0.462533
$661$	−6697.04	−0.394077	−0.197038	−	0.980396i	$-0.563132\pi$
$661$	−6697.04	−0.394077	−0.197038	+	0.980396i	$0.563132\pi$
$662$	20790.1	1.22059
$663$	−11400.3	−0.667800
$664$	−36814.7	−2.15164
$665$	−11150.7	−0.650235
$666$	−4525.43	−0.263299
$667$	3692.34	0.214345
$668$	−21501.7	−1.24540
$669$	14276.0	0.825025
$670$	−72856.8	−4.20105
$671$	1637.40	0.0942045
$672$	6186.44	0.355129
$673$	−3181.76	−0.182241	−0.0911204	−	0.995840i	$-0.529045\pi$
$673$	−3181.76	−0.182241	−0.0911204	+	0.995840i	$0.529045\pi$
$674$	29445.5	1.68279
$675$	−3708.36	−0.211459
$676$	−1662.39	−0.0945832
$677$	−5000.81	−0.283895	−0.141947	−	0.989874i	$-0.545336\pi$
$677$	−5000.81	−0.283895	−0.141947	+	0.989874i	$0.545336\pi$
$678$	−13902.2	−0.787478
$679$	1217.52	0.0688133
$680$	60585.3	3.41668
$681$	−19045.5	−1.07170
$682$	−5457.13	−0.306400
$683$	−21253.5	−1.19069	−0.595347	−	0.803468i	$-0.702986\pi$
$683$	−21253.5	−1.19069	−0.595347	+	0.803468i	$0.702986\pi$
$684$	4425.21	0.247371
$685$	8970.95	0.500383
$686$	14099.6	0.784729
$687$	−9771.92	−0.542681
$688$	44184.7	2.44844
$689$	7968.51	0.440604
$690$	−25458.6	−1.40463
$691$	25550.2	1.40662	0.703311	−	0.710882i	$-0.251704\pi$
$691$	25550.2	1.40662	0.703311	+	0.710882i	$0.251704\pi$
$692$	−35607.4	−1.95606
$693$	−2033.84	−0.111485
$694$	26702.0	1.46051
$695$	−4843.39	−0.264346
$696$	−4769.10	−0.259730
$697$	−31376.4	−1.70512
$698$	53516.8	2.90206
$699$	6832.13	0.369692
$700$	−55686.7	−3.00680
$701$	7764.46	0.418345	0.209172	−	0.977879i	$-0.432923\pi$
$701$	7764.46	0.418345	0.209172	+	0.977879i	$0.432923\pi$
$702$	−6187.55	−0.332669
$703$	−2904.77	−0.155840
$704$	−2680.53	−0.143503
$705$	11868.6	0.634040
$706$	63513.0	3.38576
$707$	26945.2	1.43335
$708$	−5748.58	−0.305148
$709$	6895.79	0.365271	0.182635	−	0.983181i	$-0.441537\pi$
$709$	6895.79	0.365271	0.182635	+	0.983181i	$0.441537\pi$
$710$	−56603.2	−2.99194
$711$	5933.31	0.312963
$712$	3207.41	0.168824
$713$	−12050.0	−0.632926
$714$	29651.7	1.55418
$715$	7038.75	0.368160
$716$	−32034.7	−1.67206
$717$	20767.0	1.08167
$718$	40872.2	2.12443
$719$	−10522.0	−0.545765	−0.272883	−	0.962047i	$-0.587977\pi$
$719$	−10522.0	−0.545765	−0.272883	+	0.962047i	$0.587977\pi$
$720$	13037.9	0.674853
$721$	−6678.60	−0.344971
$722$	−30130.8	−1.55312
$723$	−439.353	−0.0225999
$724$	58460.7	3.00093
$725$	4841.28	0.248001
$726$	18621.9	0.951961
$727$	28804.1	1.46944	0.734722	−	0.678368i	$-0.237312\pi$
$727$	28804.1	1.46944	0.734722	+	0.678368i	$0.237312\pi$
$728$	−49234.0	−2.50650
$729$	729.000	0.0370370
$730$	23957.8	1.21468
$731$	40973.4	2.07313
$732$	−8813.26	−0.445010
$733$	−32906.9	−1.65818	−0.829088	−	0.559118i	$-0.811140\pi$
$733$	−32906.9	−1.65818	−0.829088	+	0.559118i	$0.811140\pi$
$734$	58882.1	2.96100
$735$	−10917.8	−0.547904
$736$	9066.17	0.454054
$737$	−8529.73	−0.426319
$738$	−17029.6	−0.849417
$739$	20779.1	1.03433	0.517166	−	0.855885i	$-0.326987\pi$
$739$	20779.1	1.03433	0.517166	+	0.855885i	$0.326987\pi$
$740$	−27708.9	−1.37648
$741$	−3971.65	−0.196899
$742$	−20725.7	−1.02542
$743$	24019.1	1.18597	0.592985	−	0.805213i	$-0.297949\pi$
$743$	24019.1	1.18597	0.592985	+	0.805213i	$0.297949\pi$
$744$	15564.0	0.766942
$745$	−25367.6	−1.24751
$746$	−36844.3	−1.80826
$747$	−7346.68	−0.359841
$748$	13386.2	0.654342
$749$	28729.2	1.40152
$750$	−3000.69	−0.146093
$751$	30342.3	1.47431	0.737155	−	0.675723i	$-0.236169\pi$
$751$	30342.3	1.47431	0.737155	+	0.675723i	$0.236169\pi$
$752$	−21845.8	−1.05936
$753$	−21699.3	−1.05015
$754$	8077.88	0.390158
$755$	−23046.7	−1.11093
$756$	10947.1	0.526642
$757$	2043.35	0.0981068	0.0490534	−	0.998796i	$-0.484380\pi$
$757$	2043.35	0.0981068	0.0490534	+	0.998796i	$0.484380\pi$
$758$	51401.9	2.46306
$759$	−2980.57	−0.142540
$760$	21106.8	1.00740
$761$	25175.2	1.19921	0.599606	−	0.800295i	$-0.295324\pi$
$761$	25175.2	1.19921	0.599606	+	0.800295i	$0.295324\pi$
$762$	42579.7	2.02428
$763$	−38964.1	−1.84875
$764$	−33318.5	−1.57778
$765$	12090.3	0.571406
$766$	−66102.3	−3.11798
$767$	5159.38	0.242887
$768$	24817.3	1.16604
$769$	31282.8	1.46695	0.733477	−	0.679714i	$-0.237896\pi$
$769$	31282.8	1.46695	0.733477	+	0.679714i	$0.237896\pi$
$770$	−18307.5	−0.856825
$771$	−22370.7	−1.04496
$772$	−3614.80	−0.168523
$773$	4063.82	0.189088	0.0945442	−	0.995521i	$-0.469861\pi$
$773$	4063.82	0.189088	0.0945442	+	0.995521i	$0.469861\pi$
$774$	22238.4	1.03274
$775$	−15799.6	−0.732307
$776$	−2304.60	−0.106611
$777$	−7185.82	−0.331776
$778$	−44090.5	−2.03178
$779$	−10930.9	−0.502749
$780$	−37885.8	−1.73914
$781$	−6626.84	−0.303620
$782$	43454.3	1.98711
$783$	−951.713	−0.0434374
$784$	20095.7	0.915440
$785$	−2542.95	−0.115620
$786$	−1169.21	−0.0530591
$787$	752.031	0.0340623	0.0170311	−	0.999855i	$-0.494579\pi$
$787$	752.031	0.0340623	0.0170311	+	0.999855i	$0.494579\pi$
$788$	44539.7	2.01353
$789$	−1737.51	−0.0783994
$790$	53408.3	2.40529
$791$	−22074.9	−0.992281
$792$	3849.77	0.172722
$793$	7909.95	0.354212
$794$	−180.567	−0.00807065
$795$	−8450.80	−0.377005
$796$	−41232.3	−1.83598
$797$	−20204.5	−0.897966	−0.448983	−	0.893540i	$-0.648214\pi$
$797$	−20204.5	−0.897966	−0.448983	+	0.893540i	$0.648214\pi$
$798$	10330.1	0.458246
$799$	−20258.1	−0.896971
$800$	11887.3	0.525349
$801$	640.066	0.0282342
$802$	24816.7	1.09265
$803$	2804.87	0.123265
$804$	45911.0	2.01388
$805$	−40425.1	−1.76993
$806$	−26362.3	−1.15207
$807$	−24492.6	−1.06838
$808$	−51003.5	−2.22067
$809$	−195.683	−0.00850412	−0.00425206	−	0.999991i	$-0.501353\pi$
$809$	−195.683	−0.00850412	−0.00425206	+	0.999991i	$0.501353\pi$
$810$	6562.05	0.284650
$811$	−27172.3	−1.17651	−0.588255	−	0.808676i	$-0.700185\pi$
$811$	−27172.3	−1.17651	−0.588255	+	0.808676i	$0.700185\pi$
$812$	−14291.5	−0.617650
$813$	−15559.2	−0.671200
$814$	−4769.11	−0.205353
$815$	50542.8	2.17232
$816$	−22253.9	−0.954708
$817$	14274.3	0.611256
$818$	−33681.0	−1.43964
$819$	−9825.05	−0.419188
$820$	−104271.	−4.44062
$821$	27846.4	1.18374	0.591868	−	0.806035i	$-0.298391\pi$
$821$	27846.4	1.18374	0.591868	+	0.806035i	$0.298391\pi$
$822$	−8310.72	−0.352640
$823$	−16107.1	−0.682208	−0.341104	−	0.940026i	$-0.610801\pi$
$823$	−16107.1	−0.682208	−0.341104	+	0.940026i	$0.610801\pi$
$824$	12641.7	0.534458
$825$	−3908.04	−0.164922
$826$	−13419.3	−0.565275
$827$	−9949.86	−0.418368	−0.209184	−	0.977876i	$-0.567081\pi$
$827$	−9949.86	−0.418368	−0.209184	+	0.977876i	$0.567081\pi$
$828$	16042.8	0.673342
$829$	38491.2	1.61261	0.806305	−	0.591500i	$-0.201464\pi$
$829$	38491.2	1.61261	0.806305	+	0.591500i	$0.201464\pi$
$830$	−66130.7	−2.76558
$831$	−15976.3	−0.666919
$832$	−12949.1	−0.539578
$833$	18635.2	0.775115
$834$	4486.94	0.186295
$835$	−20465.9	−0.848205
$836$	4663.49	0.192931
$837$	3105.93	0.128264
$838$	12078.4	0.497900
$839$	18053.8	0.742890	0.371445	−	0.928455i	$-0.378862\pi$
$839$	18053.8	0.742890	0.371445	+	0.928455i	$0.378862\pi$
$840$	52213.9	2.14470
$841$	−23146.5	−0.949056
$842$	−14504.0	−0.593637
$843$	12363.4	0.505123
$844$	−26677.8	−1.08802
$845$	−1582.31	−0.0644179
$846$	−10995.1	−0.446832
$847$	29569.3	1.19954
$848$	15554.9	0.629901
$849$	20299.2	0.820575
$850$	56975.9	2.29913
$851$	−10530.8	−0.424195
$852$	35668.7	1.43426
$853$	14463.7	0.580573	0.290287	−	0.956940i	$-0.406249\pi$
$853$	14463.7	0.580573	0.290287	+	0.956940i	$0.406249\pi$
$854$	−20573.4	−0.824365
$855$	4212.03	0.168478
$856$	−54380.3	−2.17136
$857$	385.951	0.0153837	0.00769185	−	0.999970i	$-0.497552\pi$
$857$	385.951	0.0153837	0.00769185	+	0.999970i	$0.497552\pi$
$858$	−6520.73	−0.259457
$859$	16804.0	0.667456	0.333728	−	0.942669i	$-0.391693\pi$
$859$	16804.0	0.667456	0.333728	+	0.942669i	$0.391693\pi$
$860$	136164.	5.39902
$861$	−27041.0	−1.07033
$862$	25658.4	1.01384
$863$	17785.6	0.701540	0.350770	−	0.936462i	$-0.385920\pi$
$863$	17785.6	0.701540	0.350770	+	0.936462i	$0.385920\pi$
$864$	−2336.84	−0.0920149
$865$	−33892.1	−1.33221
$866$	49684.7	1.94960
$867$	−5897.47	−0.231013
$868$	46640.4	1.82382
$869$	6252.80	0.244087
$870$	−8566.79	−0.333840
$871$	−41205.4	−1.60297
$872$	73753.7	2.86424
$873$	−459.902	−0.0178297
$874$	15138.6	0.585894
$875$	−4764.72	−0.184088
$876$	−15097.1	−0.582288
$877$	8352.15	0.321587	0.160794	−	0.986988i	$-0.448595\pi$
$877$	8352.15	0.321587	0.160794	+	0.986988i	$0.448595\pi$
$878$	−59250.6	−2.27746
$879$	−19729.7	−0.757073
$880$	13739.9	0.526334
$881$	−31141.9	−1.19092	−0.595459	−	0.803386i	$-0.703030\pi$
$881$	−31141.9	−1.19092	−0.595459	+	0.803386i	$0.703030\pi$
$882$	10114.3	0.386129
$883$	−16318.1	−0.621912	−0.310956	−	0.950424i	$-0.600649\pi$
$883$	−16318.1	−0.621912	−0.310956	+	0.950424i	$0.600649\pi$
$884$	64665.9	2.46035
$885$	−5471.65	−0.207828
$886$	7245.41	0.274734
$887$	12127.9	0.459094	0.229547	−	0.973298i	$-0.426276\pi$
$887$	12127.9	0.459094	0.229547	+	0.973298i	$0.426276\pi$
$888$	13601.8	0.514015
$889$	67611.2	2.55074
$890$	5761.51	0.216996
$891$	768.254	0.0288861
$892$	−80977.6	−3.03961
$893$	−7057.52	−0.264469
$894$	23500.6	0.879169
$895$	−30491.5	−1.13879
$896$	50177.1	1.87087
$897$	−14398.5	−0.535956
$898$	−10136.8	−0.376691
$899$	−4054.81	−0.150429
$900$	21034.9	0.779070
$901$	14424.3	0.533346
$902$	−17946.6	−0.662481
$903$	35311.8	1.30133
$904$	41784.8	1.53732
$905$	55644.5	2.04385
$906$	21350.5	0.782919
$907$	−8075.93	−0.295652	−0.147826	−	0.989013i	$-0.547228\pi$
$907$	−8075.93	−0.295652	−0.147826	+	0.989013i	$0.547228\pi$
$908$	108032.	3.94841
$909$	−10178.2	−0.371385
$910$	−88439.6	−3.22170
$911$	−9127.05	−0.331935	−0.165967	−	0.986131i	$-0.553075\pi$
$911$	−9127.05	−0.331935	−0.165967	+	0.986131i	$0.553075\pi$
$912$	−7752.82	−0.281493
$913$	−7742.28	−0.280648
$914$	−83976.2	−3.03904
$915$	−8388.69	−0.303084
$916$	55429.2	1.99938
$917$	−1856.57	−0.0668585
$918$	−11200.5	−0.402692
$919$	21530.1	0.772809	0.386405	−	0.922329i	$-0.373717\pi$
$919$	21530.1	0.772809	0.386405	+	0.922329i	$0.373717\pi$
$920$	76519.0	2.74213
$921$	24953.7	0.892781
$922$	−4415.56	−0.157721
$923$	−32012.9	−1.14162
$924$	11536.5	0.410740
$925$	−13807.6	−0.490802
$926$	12539.2	0.444995
$927$	2522.75	0.0893829
$928$	3050.76	0.107916
$929$	−43537.1	−1.53757	−0.768786	−	0.639506i	$-0.779139\pi$
$929$	−43537.1	−1.53757	−0.768786	+	0.639506i	$0.779139\pi$
$930$	27957.8	0.985778
$931$	6492.14	0.228540
$932$	−38753.8	−1.36204
$933$	−24126.0	−0.846570
$934$	43684.8	1.53042
$935$	12741.3	0.445653
$936$	18597.5	0.649441
$937$	24820.3	0.865361	0.432680	−	0.901547i	$-0.357568\pi$
$937$	24820.3	0.865361	0.432680	+	0.901547i	$0.357568\pi$
$938$	107173.	3.73063
$939$	27155.9	0.943768
$940$	−67322.3	−2.33597
$941$	27826.7	0.964000	0.482000	−	0.876171i	$-0.339911\pi$
$941$	27826.7	0.964000	0.482000	+	0.876171i	$0.339911\pi$
$942$	2355.79	0.0814819
$943$	−39628.3	−1.36848
$944$	10071.3	0.347239
$945$	10419.7	0.358681
$946$	23435.9	0.805461
$947$	−21997.2	−0.754817	−0.377408	−	0.926047i	$-0.623185\pi$
$947$	−21997.2	−0.754817	−0.377408	+	0.926047i	$0.623185\pi$
$948$	−33655.5	−1.15304
$949$	13549.8	0.463481
$950$	19849.3	0.677891
$951$	26937.9	0.918530
$952$	−89121.8	−3.03409
$953$	40457.6	1.37518	0.687592	−	0.726098i	$-0.258668\pi$
$953$	40457.6	1.37518	0.687592	+	0.726098i	$0.258668\pi$
$954$	7828.85	0.265690
$955$	−31713.4	−1.07458
$956$	−117796.	−3.98515
$957$	−1002.96	−0.0338778
$958$	−14758.3	−0.497723
$959$	−13196.4	−0.444352
$960$	13732.8	0.461693
$961$	−16558.1	−0.555807
$962$	−23038.6	−0.772135
$963$	−10852.0	−0.363139
$964$	2492.14	0.0832638
$965$	−3440.66	−0.114776
$966$	37449.9	1.24734
$967$	−8685.53	−0.288840	−0.144420	−	0.989517i	$-0.546132\pi$
$967$	−8685.53	−0.288840	−0.144420	+	0.989517i	$0.546132\pi$
$968$	−55970.5	−1.85843
$969$	−7189.35	−0.238344
$970$	−4139.78	−0.137031
$971$	7553.69	0.249649	0.124825	−	0.992179i	$-0.460163\pi$
$971$	7553.69	0.249649	0.124825	+	0.992179i	$0.460163\pi$
$972$	−4135.10	−0.136454
$973$	7124.70	0.234745
$974$	−77522.9	−2.55030
$975$	−18878.9	−0.620112
$976$	15440.6	0.506393
$977$	−5300.45	−0.173568	−0.0867842	−	0.996227i	$-0.527659\pi$
$977$	−5300.45	−0.173568	−0.0867842	+	0.996227i	$0.527659\pi$
$978$	−46823.0	−1.53092
$979$	674.532	0.0220206
$980$	61929.0	2.01862
$981$	14718.2	0.479016
$982$	68279.9	2.21884
$983$	−35337.5	−1.14658	−0.573292	−	0.819351i	$-0.694334\pi$
$983$	−35337.5	−1.14658	−0.573292	+	0.819351i	$0.694334\pi$
$984$	51184.7	1.65824
$985$	42394.1	1.37136
$986$	14622.3	0.472281
$987$	−17458.9	−0.563042
$988$	22528.3	0.725427
$989$	51749.2	1.66383
$990$	6915.39	0.222006
$991$	6600.35	0.211571	0.105786	−	0.994389i	$-0.466264\pi$
$991$	6600.35	0.211571	0.105786	+	0.994389i	$0.466264\pi$
$992$	−9956.19	−0.318659
$993$	−12469.9	−0.398509
$994$	83264.0	2.65692
$995$	−39246.0	−1.25043
$996$	41672.5	1.32575
$997$	−9156.85	−0.290873	−0.145437	−	0.989368i	$-0.546459\pi$
$997$	−9156.85	−0.290873	−0.145437	+	0.989368i	$0.546459\pi$
$998$	72562.8	2.30154
$999$	2714.34	0.0859640

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.20	✓	22
3.2	odd	2		1413.4.a.e.1.3		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.20	✓	22	1.1	even	1	trivial
1413.4.a.e.1.3		22	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	471.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00169 q^{2} -3.00000 q^{3} +17.0169 q^{4} +16.1971 q^{5} -15.0051 q^{6} -23.8262 q^{7} +45.0996 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+5.00169 q^{2} -3.00000 q^{3} +17.0169 q^{4} +16.1971 q^{5} -15.0051 q^{6} -23.8262 q^{7} +45.0996 q^{8} +9.00000 q^{9} +81.0129 q^{10} +9.48462 q^{11} -51.0506 q^{12} +45.8182 q^{13} -119.171 q^{14} -48.5913 q^{15} +89.4391 q^{16} +82.9387 q^{17} +45.0152 q^{18} +28.8942 q^{19} +275.624 q^{20} +71.4785 q^{21} +47.4391 q^{22} +104.751 q^{23} -135.299 q^{24} +137.347 q^{25} +229.168 q^{26} -27.0000 q^{27} -405.447 q^{28} +35.2486 q^{29} -243.039 q^{30} -115.035 q^{31} +86.5496 q^{32} -28.4539 q^{33} +414.833 q^{34} -385.915 q^{35} +153.152 q^{36} -100.531 q^{37} +144.520 q^{38} -137.455 q^{39} +730.484 q^{40} -378.309 q^{41} +357.513 q^{42} +494.020 q^{43} +161.399 q^{44} +145.774 q^{45} +523.932 q^{46} -244.254 q^{47} -268.317 q^{48} +224.686 q^{49} +686.965 q^{50} -248.816 q^{51} +779.683 q^{52} +173.916 q^{53} -135.046 q^{54} +153.623 q^{55} -1074.55 q^{56} -86.6827 q^{57} +176.303 q^{58} +112.605 q^{59} -826.873 q^{60} +172.638 q^{61} -575.367 q^{62} -214.436 q^{63} -282.619 q^{64} +742.123 q^{65} -142.317 q^{66} -899.323 q^{67} +1411.36 q^{68} -314.253 q^{69} -1930.23 q^{70} -698.693 q^{71} +405.896 q^{72} +295.729 q^{73} -502.826 q^{74} -412.040 q^{75} +491.690 q^{76} -225.982 q^{77} -687.505 q^{78} +659.257 q^{79} +1448.66 q^{80} +81.0000 q^{81} -1892.18 q^{82} -816.298 q^{83} +1216.34 q^{84} +1343.37 q^{85} +2470.94 q^{86} -105.746 q^{87} +427.753 q^{88} +71.1185 q^{89} +729.116 q^{90} -1091.67 q^{91} +1782.54 q^{92} +345.104 q^{93} -1221.68 q^{94} +468.003 q^{95} -259.649 q^{96} -51.1002 q^{97} +1123.81 q^{98} +85.3616 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * q^99

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