LMFDB - Embedded newform 1849.4.a.i.1.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, 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("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.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:	$109.094531601$
Analytic rank:	$1$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.28
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618313 q^{2} +9.18025 q^{3} -7.61769 q^{4} -19.4242 q^{5} +5.67627 q^{6} -15.1594 q^{7} -9.65662 q^{8} +57.2770 q^{9} +O(q^{10})$	\(q+0.618313 q^{2} +9.18025 q^{3} -7.61769 q^{4} -19.4242 q^{5} +5.67627 q^{6} -15.1594 q^{7} -9.65662 q^{8} +57.2770 q^{9} -12.0103 q^{10} -7.19433 q^{11} -69.9323 q^{12} +36.4617 q^{13} -9.37325 q^{14} -178.319 q^{15} +54.9707 q^{16} +102.264 q^{17} +35.4151 q^{18} +51.6991 q^{19} +147.968 q^{20} -139.167 q^{21} -4.44835 q^{22} -102.545 q^{23} -88.6502 q^{24} +252.301 q^{25} +22.5447 q^{26} +277.950 q^{27} +115.480 q^{28} +96.3830 q^{29} -110.257 q^{30} -157.016 q^{31} +111.242 q^{32} -66.0458 q^{33} +63.2314 q^{34} +294.460 q^{35} -436.318 q^{36} +233.730 q^{37} +31.9662 q^{38} +334.728 q^{39} +187.572 q^{40} -39.6529 q^{41} -86.0488 q^{42} +54.8042 q^{44} -1112.56 q^{45} -63.4047 q^{46} -455.256 q^{47} +504.645 q^{48} -113.193 q^{49} +156.001 q^{50} +938.812 q^{51} -277.754 q^{52} -602.876 q^{53} +171.860 q^{54} +139.744 q^{55} +146.388 q^{56} +474.610 q^{57} +59.5949 q^{58} -124.432 q^{59} +1358.38 q^{60} -838.549 q^{61} -97.0852 q^{62} -868.284 q^{63} -370.983 q^{64} -708.241 q^{65} -40.8370 q^{66} -166.437 q^{67} -779.018 q^{68} -941.386 q^{69} +182.068 q^{70} -215.480 q^{71} -553.102 q^{72} +108.227 q^{73} +144.518 q^{74} +2316.18 q^{75} -393.827 q^{76} +109.062 q^{77} +206.966 q^{78} -494.914 q^{79} -1067.76 q^{80} +1005.17 q^{81} -24.5179 q^{82} +712.724 q^{83} +1060.13 q^{84} -1986.41 q^{85} +884.820 q^{87} +69.4729 q^{88} +431.700 q^{89} -687.911 q^{90} -552.738 q^{91} +781.154 q^{92} -1441.45 q^{93} -281.491 q^{94} -1004.21 q^{95} +1021.23 q^{96} +1299.31 q^{97} -69.9885 q^{98} -412.070 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) $ 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} - 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} - 386 q^{18} + 12 q^{19} + 108 q^{20} - 408 q^{21} + 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} + 1493 q^{26} + 10 q^{27} - 242 q^{28} + 208 q^{29} - 48 q^{30} - 932 q^{31} - 1124 q^{32} + 254 q^{33} + 765 q^{34} - 1452 q^{35} + 747 q^{36} - 90 q^{37} - 1213 q^{38} - 1610 q^{39} - 1693 q^{40} - 1354 q^{41} - 16 q^{42} - 2704 q^{44} + 4508 q^{45} + 233 q^{46} - 3484 q^{47} - 376 q^{48} + 1324 q^{49} - 408 q^{50} + 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} - 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} + 1172 q^{61} - 1546 q^{62} - 3686 q^{63} + 606 q^{64} + 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} + 136 q^{69} - 1310 q^{70} + 162 q^{71} - 5814 q^{72} + 746 q^{73} - 4332 q^{74} + 236 q^{75} + 1338 q^{76} + 2024 q^{77} - 2782 q^{78} - 2656 q^{79} - 5713 q^{80} - 86 q^{81} - 4168 q^{82} - 3514 q^{83} - 4269 q^{84} - 7558 q^{85} - 10278 q^{87} + 11692 q^{88} + 2640 q^{89} - 8286 q^{90} - 5946 q^{91} - 4271 q^{92} - 2 q^{93} + 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} - 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 - 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 - 386 * q^18 + 12 * q^19 + 108 * q^20 - 408 * q^21 + 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 + 1493 * q^26 + 10 * q^27 - 242 * q^28 + 208 * q^29 - 48 * q^30 - 932 * q^31 - 1124 * q^32 + 254 * q^33 + 765 * q^34 - 1452 * q^35 + 747 * q^36 - 90 * q^37 - 1213 * q^38 - 1610 * q^39 - 1693 * q^40 - 1354 * q^41 - 16 * q^42 - 2704 * q^44 + 4508 * q^45 + 233 * q^46 - 3484 * q^47 - 376 * q^48 + 1324 * q^49 - 408 * q^50 + 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 - 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 + 1172 * q^61 - 1546 * q^62 - 3686 * q^63 + 606 * q^64 + 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 + 136 * q^69 - 1310 * q^70 + 162 * q^71 - 5814 * q^72 + 746 * q^73 - 4332 * q^74 + 236 * q^75 + 1338 * q^76 + 2024 * q^77 - 2782 * q^78 - 2656 * q^79 - 5713 * q^80 - 86 * q^81 - 4168 * q^82 - 3514 * q^83 - 4269 * q^84 - 7558 * q^85 - 10278 * q^87 + 11692 * q^88 + 2640 * q^89 - 8286 * q^90 - 5946 * q^91 - 4271 * q^92 - 2 * q^93 + 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 - 2826 * q^98 - 8174 * 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 $

$2$	0.618313	0.218607	0.109303	−	0.994008i	$-0.465138\pi$
$2$	0.618313	0.218607	0.109303	+	0.994008i	$0.465138\pi$
$3$	9.18025	1.76674	0.883370	−	0.468676i	$-0.155269\pi$
$3$	9.18025	1.76674	0.883370	+	0.468676i	$0.155269\pi$
$4$	−7.61769	−0.952211
$5$	−19.4242	−1.73736	−0.868678	−	0.495377i	$-0.835030\pi$
$5$	−19.4242	−1.73736	−0.868678	+	0.495377i	$0.835030\pi$
$6$	5.67627	0.386221
$7$	−15.1594	−0.818531	−0.409265	−	0.912415i	$-0.634215\pi$
$7$	−15.1594	−0.818531	−0.409265	+	0.912415i	$0.634215\pi$
$8$	−9.65662	−0.426766
$9$	57.2770	2.12137
$10$	−12.0103	−0.379798
$11$	−7.19433	−0.197198	−0.0985988	−	0.995127i	$-0.531436\pi$
$11$	−7.19433	−0.197198	−0.0985988	+	0.995127i	$0.531436\pi$
$12$	−69.9323	−1.68231
$13$	36.4617	0.777897	0.388948	−	0.921260i	$-0.372838\pi$
$13$	36.4617	0.777897	0.388948	+	0.921260i	$0.372838\pi$
$14$	−9.37325	−0.178936
$15$	−178.319	−3.06946
$16$	54.9707	0.858917
$17$	102.264	1.45899	0.729493	−	0.683989i	$-0.239756\pi$
$17$	102.264	1.45899	0.729493	+	0.683989i	$0.239756\pi$
$18$	35.4151	0.463745
$19$	51.6991	0.624241	0.312121	−	0.950043i	$-0.398961\pi$
$19$	51.6991	0.624241	0.312121	+	0.950043i	$0.398961\pi$
$20$	147.968	1.65433
$21$	−139.167	−1.44613
$22$	−4.44835	−0.0431087
$23$	−102.545	−0.929655	−0.464827	−	0.885401i	$-0.653884\pi$
$23$	−102.545	−0.929655	−0.464827	+	0.885401i	$0.653884\pi$
$24$	−88.6502	−0.753985
$25$	252.301	2.01841
$26$	22.5447	0.170053
$27$	277.950	1.98117
$28$	115.480	0.779414
$29$	96.3830	0.617168	0.308584	−	0.951197i	$-0.400145\pi$
$29$	96.3830	0.617168	0.308584	+	0.951197i	$0.400145\pi$
$30$	−110.257	−0.671003
$31$	−157.016	−0.909709	−0.454854	−	0.890566i	$-0.650309\pi$
$31$	−157.016	−0.909709	−0.454854	+	0.890566i	$0.650309\pi$
$32$	111.242	0.614531
$33$	−66.0458	−0.348397
$34$	63.2314	0.318944
$35$	294.460	1.42208
$36$	−436.318	−2.01999
$37$	233.730	1.03851	0.519257	−	0.854618i	$-0.326209\pi$
$37$	233.730	1.03851	0.519257	+	0.854618i	$0.326209\pi$
$38$	31.9662	0.136463
$39$	334.728	1.37434
$40$	187.572	0.741445
$41$	−39.6529	−0.151043	−0.0755213	−	0.997144i	$-0.524062\pi$
$41$	−39.6529	−0.151043	−0.0755213	+	0.997144i	$0.524062\pi$
$42$	−86.0488	−0.316134
$43$	0	0
$43$	0	0
$44$	54.8042	0.187774
$45$	−1112.56	−3.68557
$46$	−63.4047	−0.203229
$47$	−455.256	−1.41289	−0.706446	−	0.707767i	$-0.749703\pi$
$47$	−455.256	−1.41289	−0.706446	+	0.707767i	$0.749703\pi$
$48$	504.645	1.51748
$49$	−113.193	−0.330008
$50$	156.001	0.441237
$51$	938.812	2.57765
$52$	−277.754	−0.740722
$53$	−602.876	−1.56248	−0.781240	−	0.624231i	$-0.785413\pi$
$53$	−602.876	−1.56248	−0.781240	+	0.624231i	$0.785413\pi$
$54$	171.860	0.433096
$55$	139.744	0.342602
$56$	146.388	0.349321
$57$	474.610	1.10287
$58$	59.5949	0.134917
$59$	−124.432	−0.274571	−0.137285	−	0.990532i	$-0.543838\pi$
$59$	−124.432	−0.274571	−0.137285	+	0.990532i	$0.543838\pi$
$60$	1358.38	2.92277
$61$	−838.549	−1.76008	−0.880042	−	0.474896i	$-0.842486\pi$
$61$	−838.549	−1.76008	−0.880042	+	0.474896i	$0.842486\pi$
$62$	−97.0852	−0.198868
$63$	−868.284	−1.73641
$64$	−370.983	−0.724577
$65$	−708.241	−1.35148
$66$	−40.8370	−0.0761618
$67$	−166.437	−0.303486	−0.151743	−	0.988420i	$-0.548489\pi$
$67$	−166.437	−0.303486	−0.151743	+	0.988420i	$0.548489\pi$
$68$	−779.018	−1.38926
$69$	−941.386	−1.64246
$70$	182.068	0.310876
$71$	−215.480	−0.360180	−0.180090	−	0.983650i	$-0.557639\pi$
$71$	−215.480	−0.360180	−0.180090	+	0.983650i	$0.557639\pi$
$72$	−553.102	−0.905329
$73$	108.227	0.173521	0.0867606	−	0.996229i	$-0.472348\pi$
$73$	108.227	0.173521	0.0867606	+	0.996229i	$0.472348\pi$
$74$	144.518	0.227026
$75$	2316.18	3.56600
$76$	−393.827	−0.594409
$77$	109.062	0.161412
$78$	206.966	0.300440
$79$	−494.914	−0.704837	−0.352419	−	0.935843i	$-0.614641\pi$
$79$	−494.914	−0.704837	−0.352419	+	0.935843i	$0.614641\pi$
$80$	−1067.76	−1.49225
$81$	1005.17	1.37884
$82$	−24.5179	−0.0330189
$83$	712.724	0.942549	0.471275	−	0.881987i	$-0.343794\pi$
$83$	712.724	0.942549	0.471275	+	0.881987i	$0.343794\pi$
$84$	1060.13	1.37702
$85$	−1986.41	−2.53478
$86$	0	0
$87$	884.820	1.09038
$88$	69.4729	0.0841573
$89$	431.700	0.514158	0.257079	−	0.966390i	$-0.417240\pi$
$89$	431.700	0.514158	0.257079	+	0.966390i	$0.417240\pi$
$90$	−687.911	−0.805691
$91$	−552.738	−0.636732
$92$	781.154	0.885227
$93$	−1441.45	−1.60722
$94$	−281.491	−0.308868
$95$	−1004.21	−1.08453
$96$	1021.23	1.08572
$97$	1299.31	1.36005	0.680025	−	0.733189i	$-0.261969\pi$
$97$	1299.31	1.36005	0.680025	+	0.733189i	$0.261969\pi$
$98$	−69.9885	−0.0721419
$99$	−412.070	−0.418329
$100$	−1921.95	−1.92195
$101$	−1475.52	−1.45366	−0.726831	−	0.686817i	$-0.759007\pi$
$101$	−1475.52	−1.45366	−0.726831	+	0.686817i	$0.759007\pi$
$102$	580.480	0.563491
$103$	−291.930	−0.279269	−0.139635	−	0.990203i	$-0.544593\pi$
$103$	−291.930	−0.279269	−0.139635	+	0.990203i	$0.544593\pi$
$104$	−352.097	−0.331980
$105$	2703.21	2.51244
$106$	−372.766	−0.341568
$107$	−1552.74	−1.40289	−0.701444	−	0.712725i	$-0.747461\pi$
$107$	−1552.74	−1.40289	−0.701444	+	0.712725i	$0.747461\pi$
$108$	−2117.34	−1.88649
$109$	1809.11	1.58973	0.794867	−	0.606784i	$-0.207541\pi$
$109$	1809.11	1.58973	0.794867	+	0.606784i	$0.207541\pi$
$110$	86.4058	0.0748952
$111$	2145.70	1.83478
$112$	−833.323	−0.703050
$113$	677.063	0.563652	0.281826	−	0.959466i	$-0.409060\pi$
$113$	677.063	0.563652	0.281826	+	0.959466i	$0.409060\pi$
$114$	293.458	0.241095
$115$	1991.85	1.61514
$116$	−734.216	−0.587675
$117$	2088.42	1.65021
$118$	−76.9380	−0.0600230
$119$	−1550.27	−1.19422
$120$	1721.96	1.30994
$121$	−1279.24	−0.961113
$122$	−518.485	−0.384766
$123$	−364.024	−0.266853
$124$	1196.10	0.866235
$125$	−2472.72	−1.76934
$126$	−536.871	−0.379590
$127$	−10.4671	−0.00731340	−0.00365670	−	0.999993i	$-0.501164\pi$
$127$	−10.4671	−0.00731340	−0.00365670	+	0.999993i	$0.501164\pi$
$128$	−1119.32	−0.772928
$129$	0	0
$130$	−437.914	−0.295443
$131$	26.6364	0.0177652	0.00888258	−	0.999961i	$-0.497173\pi$
$131$	26.6364	0.0177652	0.00888258	+	0.999961i	$0.497173\pi$
$132$	503.116	0.331747
$133$	−783.727	−0.510960
$134$	−102.910	−0.0663439
$135$	−5398.97	−3.44199
$136$	−987.528	−0.622646
$137$	−748.264	−0.466632	−0.233316	−	0.972401i	$-0.574958\pi$
$137$	−748.264	−0.466632	−0.233316	+	0.972401i	$0.574958\pi$
$138$	−582.071	−0.359052
$139$	−1029.41	−0.628152	−0.314076	−	0.949398i	$-0.601695\pi$
$139$	−1029.41	−0.628152	−0.314076	+	0.949398i	$0.601695\pi$
$140$	−2243.10	−1.35412
$141$	−4179.37	−2.49621
$142$	−133.234	−0.0787378
$143$	−262.318	−0.153399
$144$	3148.56	1.82208
$145$	−1872.17	−1.07224
$146$	66.9183	0.0379329
$147$	−1039.14	−0.583038
$148$	−1780.48	−0.988885
$149$	1048.80	0.576649	0.288324	−	0.957533i	$-0.406902\pi$
$149$	1048.80	0.576649	0.288324	+	0.957533i	$0.406902\pi$
$150$	1432.13	0.779551
$151$	2902.66	1.56434	0.782168	−	0.623068i	$-0.214114\pi$
$151$	2902.66	1.56434	0.782168	+	0.623068i	$0.214114\pi$
$152$	−499.238	−0.266405
$153$	5857.39	3.09505
$154$	67.4343	0.0352858
$155$	3049.92	1.58049
$156$	−2549.85	−1.30866
$157$	524.942	0.266847	0.133423	−	0.991059i	$-0.457403\pi$
$157$	524.942	0.266847	0.133423	+	0.991059i	$0.457403\pi$
$158$	−306.011	−0.154082
$159$	−5534.56	−2.76050
$160$	−2160.79	−1.06766
$161$	1554.52	0.760951
$162$	621.512	0.301423
$163$	40.9398	0.0196727	0.00983635	−	0.999952i	$-0.496869\pi$
$163$	40.9398	0.0196727	0.00983635	+	0.999952i	$0.496869\pi$
$164$	302.064	0.143824
$165$	1282.89	0.605289
$166$	440.686	0.206047
$167$	−1168.95	−0.541652	−0.270826	−	0.962628i	$-0.587297\pi$
$167$	−1168.95	−0.541652	−0.270826	+	0.962628i	$0.587297\pi$
$168$	1343.88	0.617160
$169$	−867.544	−0.394877
$170$	−1228.22	−0.554119
$171$	2961.17	1.32425
$172$	0	0
$173$	−2210.67	−0.971528	−0.485764	−	0.874090i	$-0.661459\pi$
$173$	−2210.67	−0.971528	−0.485764	+	0.874090i	$0.661459\pi$
$174$	547.096	0.238363
$175$	−3824.73	−1.65213
$176$	−395.478	−0.169376
$177$	−1142.32	−0.485095
$178$	266.925	0.112398
$179$	2397.80	1.00123	0.500615	−	0.865670i	$-0.333107\pi$
$179$	2397.80	1.00123	0.500615	+	0.865670i	$0.333107\pi$
$180$	8475.15	3.50945
$181$	68.3076	0.0280512	0.0140256	−	0.999902i	$-0.495535\pi$
$181$	68.3076	0.0280512	0.0140256	+	0.999902i	$0.495535\pi$
$182$	−341.765	−0.139194
$183$	−7698.08	−3.10961
$184$	990.235	0.396745
$185$	−4540.03	−1.80427
$186$	−891.267	−0.351349
$187$	−735.724	−0.287708
$188$	3468.00	1.34537
$189$	−4213.56	−1.62165
$190$	−620.919	−0.237085
$191$	−585.643	−0.221862	−0.110931	−	0.993828i	$-0.535383\pi$
$191$	−585.643	−0.221862	−0.110931	+	0.993828i	$0.535383\pi$
$192$	−3405.72	−1.28014
$193$	−4104.42	−1.53079	−0.765395	−	0.643561i	$-0.777456\pi$
$193$	−4104.42	−1.53079	−0.765395	+	0.643561i	$0.777456\pi$
$194$	803.380	0.297316
$195$	−6501.83	−2.38772
$196$	862.267	0.314237
$197$	787.155	0.284683	0.142341	−	0.989818i	$-0.454537\pi$
$197$	787.155	0.284683	0.142341	+	0.989818i	$0.454537\pi$
$198$	−254.788	−0.0914495
$199$	−4673.04	−1.66464	−0.832319	−	0.554297i	$-0.812987\pi$
$199$	−4673.04	−1.66464	−0.832319	+	0.554297i	$0.812987\pi$
$200$	−2436.37	−0.861388
$201$	−1527.93	−0.536180
$202$	−912.333	−0.317780
$203$	−1461.11	−0.505171
$204$	−7151.58	−2.45446
$205$	770.227	0.262415
$206$	−180.504	−0.0610501
$207$	−5873.45	−1.97214
$208$	2004.33	0.668149
$209$	−371.940	−0.123099
$210$	1671.43	0.549237
$211$	−1152.23	−0.375939	−0.187969	−	0.982175i	$-0.560191\pi$
$211$	−1152.23	−0.375939	−0.187969	+	0.982175i	$0.560191\pi$
$212$	4592.53	1.48781
$213$	−1978.16	−0.636345
$214$	−960.079	−0.306680
$215$	0	0
$216$	−2684.06	−0.845496
$217$	2380.27	0.744624
$218$	1118.59	0.347526
$219$	993.554	0.306567
$220$	−1064.53	−0.326230
$221$	3728.73	1.13494
$222$	1326.72	0.401096
$223$	−2655.51	−0.797426	−0.398713	−	0.917076i	$-0.630543\pi$
$223$	−2655.51	−0.797426	−0.398713	+	0.917076i	$0.630543\pi$
$224$	−1686.36	−0.503013
$225$	14451.0	4.28179
$226$	418.636	0.123218
$227$	−3747.98	−1.09587	−0.547934	−	0.836521i	$-0.684586\pi$
$227$	−3747.98	−1.09587	−0.547934	+	0.836521i	$0.684586\pi$
$228$	−3615.43	−1.05017
$229$	1183.25	0.341446	0.170723	−	0.985319i	$-0.445390\pi$
$229$	1183.25	0.341446	0.170723	+	0.985319i	$0.445390\pi$
$230$	1231.59	0.353080
$231$	1001.21	0.285173
$232$	−930.734	−0.263387
$233$	2800.12	0.787304	0.393652	−	0.919259i	$-0.371211\pi$
$233$	2800.12	0.787304	0.393652	+	0.919259i	$0.371211\pi$
$234$	1291.29	0.360746
$235$	8843.00	2.45470
$236$	947.885	0.261449
$237$	−4543.43	−1.24526
$238$	−958.549	−0.261065
$239$	−2888.20	−0.781682	−0.390841	−	0.920458i	$-0.627816\pi$
$239$	−2888.20	−0.781682	−0.390841	+	0.920458i	$0.627816\pi$
$240$	−9802.34	−2.63641
$241$	−48.4148	−0.0129405	−0.00647027	−	0.999979i	$-0.502060\pi$
$241$	−48.4148	−0.0129405	−0.00647027	+	0.999979i	$0.502060\pi$
$242$	−790.971	−0.210106
$243$	1723.09	0.454882
$244$	6387.80	1.67597
$245$	2198.68	0.573341
$246$	−225.080	−0.0583358
$247$	1885.04	0.485595
$248$	1516.25	0.388233
$249$	6542.98	1.66524
$250$	−1528.92	−0.386788
$251$	−4735.73	−1.19090	−0.595452	−	0.803391i	$-0.703027\pi$
$251$	−4735.73	−1.19090	−0.595452	+	0.803391i	$0.703027\pi$
$252$	6614.32	1.65342
$253$	737.741	0.183326
$254$	−6.47192	−0.00159876
$255$	−18235.7	−4.47829
$256$	2275.78	0.555609
$257$	−642.806	−0.156020	−0.0780100	−	0.996953i	$-0.524857\pi$
$257$	−642.806	−0.156020	−0.0780100	+	0.996953i	$0.524857\pi$
$258$	0	0
$259$	−3543.21	−0.850055
$260$	5395.16	1.28690
$261$	5520.53	1.30924
$262$	16.4696	0.00388358
$263$	−5427.16	−1.27245	−0.636223	−	0.771505i	$-0.719504\pi$
$263$	−5427.16	−1.27245	−0.636223	+	0.771505i	$0.719504\pi$
$264$	637.779	0.148684
$265$	11710.4	2.71458
$266$	−484.588	−0.111699
$267$	3963.11	0.908383
$268$	1267.87	0.288982
$269$	4408.39	0.999199	0.499599	−	0.866257i	$-0.333481\pi$
$269$	4408.39	0.999199	0.499599	+	0.866257i	$0.333481\pi$
$270$	−3338.25	−0.752443
$271$	5005.29	1.12196	0.560978	−	0.827831i	$-0.310425\pi$
$271$	5005.29	1.12196	0.560978	+	0.827831i	$0.310425\pi$
$272$	5621.54	1.25315
$273$	−5074.27	−1.12494
$274$	−462.661	−0.102009
$275$	−1815.14	−0.398025
$276$	7171.19	1.56397
$277$	8864.55	1.92281	0.961407	−	0.275132i	$-0.0887215\pi$
$277$	8864.55	1.92281	0.961407	+	0.275132i	$0.0887215\pi$
$278$	−636.495	−0.137318
$279$	−8993.42	−1.92983
$280$	−2843.48	−0.606895
$281$	3796.45	0.805968	0.402984	−	0.915207i	$-0.367973\pi$
$281$	3796.45	0.805968	0.402984	+	0.915207i	$0.367973\pi$
$282$	−2584.16	−0.545689
$283$	−4707.79	−0.988867	−0.494433	−	0.869216i	$-0.664624\pi$
$283$	−4707.79	−0.988867	−0.494433	+	0.869216i	$0.664624\pi$
$284$	1641.46	0.342968
$285$	−9218.94	−1.91608
$286$	−162.194	−0.0335341
$287$	601.114	0.123633
$288$	6371.61	1.30365
$289$	5545.00	1.12864
$290$	−1157.58	−0.234399
$291$	11928.0	2.40286
$292$	−824.442	−0.165229
$293$	−6455.57	−1.28716	−0.643581	−	0.765378i	$-0.722552\pi$
$293$	−6455.57	−1.28716	−0.643581	+	0.765378i	$0.722552\pi$
$294$	−642.512	−0.127456
$295$	2417.00	0.477027
$296$	−2257.04	−0.443203
$297$	−1999.67	−0.390682
$298$	648.484	0.126059
$299$	−3738.96	−0.723175
$300$	−17644.0	−3.39558
$301$	0	0
$302$	1794.75	0.341974
$303$	−13545.6	−2.56824
$304$	2841.93	0.536171
$305$	16288.2	3.05789
$306$	3621.70	0.676598
$307$	373.270	0.0693930	0.0346965	−	0.999398i	$-0.488954\pi$
$307$	373.270	0.0693930	0.0346965	+	0.999398i	$0.488954\pi$
$308$	−830.799	−0.153699
$309$	−2679.99	−0.493396
$310$	1885.81	0.345505
$311$	2560.85	0.466921	0.233461	−	0.972366i	$-0.424995\pi$
$311$	2560.85	0.466921	0.233461	+	0.972366i	$0.424995\pi$
$312$	−3232.34	−0.586522
$313$	9039.94	1.63249	0.816243	−	0.577709i	$-0.196053\pi$
$313$	9039.94	1.63249	0.816243	+	0.577709i	$0.196053\pi$
$314$	324.578	0.0583344
$315$	16865.8	3.01676
$316$	3770.10	0.671154
$317$	2482.03	0.439762	0.219881	−	0.975527i	$-0.429433\pi$
$317$	2482.03	0.439762	0.219881	+	0.975527i	$0.429433\pi$
$318$	−3422.09	−0.603463
$319$	−693.412	−0.121704
$320$	7206.07	1.25885
$321$	−14254.5	−2.47854
$322$	961.177	0.166349
$323$	5286.97	0.910758
$324$	−7657.10	−1.31295
$325$	9199.32	1.57011
$326$	25.3136	0.00430058
$327$	16608.0	2.80864
$328$	382.913	0.0644599
$329$	6901.41	1.15650
$330$	793.227	0.132320
$331$	−4063.14	−0.674715	−0.337357	−	0.941377i	$-0.609533\pi$
$331$	−4063.14	−0.674715	−0.337357	+	0.941377i	$0.609533\pi$
$332$	−5429.31	−0.897506
$333$	13387.4	2.20307
$334$	−722.775	−0.118409
$335$	3232.91	0.527263
$336$	−7650.11	−1.24211
$337$	−985.260	−0.159260	−0.0796299	−	0.996824i	$-0.525374\pi$
$337$	−985.260	−0.159260	−0.0796299	+	0.996824i	$0.525374\pi$
$338$	−536.413	−0.0863226
$339$	6215.60	0.995827
$340$	15131.8	2.41364
$341$	1129.63	0.179392
$342$	1830.93	0.289489
$343$	6915.61	1.08865
$344$	0	0
$345$	18285.7	2.85353
$346$	−1366.89	−0.212383
$347$	1072.53	0.165927	0.0829633	−	0.996553i	$-0.473562\pi$
$347$	1072.53	0.165927	0.0829633	+	0.996553i	$0.473562\pi$
$348$	−6740.29	−1.03827
$349$	−293.309	−0.0449870	−0.0224935	−	0.999747i	$-0.507161\pi$
$349$	−293.309	−0.0449870	−0.0224935	+	0.999747i	$0.507161\pi$
$350$	−2364.88	−0.361166
$351$	10134.5	1.54114
$352$	−800.312	−0.121184
$353$	−1149.23	−0.173279	−0.0866396	−	0.996240i	$-0.527613\pi$
$353$	−1149.23	−0.173279	−0.0866396	+	0.996240i	$0.527613\pi$
$354$	−706.310	−0.106045
$355$	4185.54	0.625762
$356$	−3288.55	−0.489587
$357$	−14231.8	−2.10988
$358$	1482.59	0.218875
$359$	8685.57	1.27690	0.638450	−	0.769663i	$-0.279576\pi$
$359$	8685.57	1.27690	0.638450	+	0.769663i	$0.279576\pi$
$360$	10743.6	1.57288
$361$	−4186.21	−0.610323
$362$	42.2354	0.00613217
$363$	−11743.8	−1.69804
$364$	4210.58	0.606304
$365$	−2102.23	−0.301468
$366$	−4759.82	−0.679781
$367$	−12096.1	−1.72047	−0.860234	−	0.509899i	$-0.829683\pi$
$367$	−12096.1	−1.72047	−0.860234	+	0.509899i	$0.829683\pi$
$368$	−5636.96	−0.798496
$369$	−2271.20	−0.320417
$370$	−2807.16	−0.394425
$371$	9139.24	1.27894
$372$	10980.5	1.53041
$373$	−6887.99	−0.956157	−0.478079	−	0.878317i	$-0.658667\pi$
$373$	−6887.99	−0.956157	−0.478079	+	0.878317i	$0.658667\pi$
$374$	−454.908	−0.0628949
$375$	−22700.2	−3.12595
$376$	4396.23	0.602975
$377$	3514.29	0.480093
$378$	−2605.30	−0.354503
$379$	−2089.36	−0.283175	−0.141588	−	0.989926i	$-0.545221\pi$
$379$	−2089.36	−0.283175	−0.141588	+	0.989926i	$0.545221\pi$
$380$	7649.80	1.03270
$381$	−96.0903	−0.0129209
$382$	−362.110	−0.0485005
$383$	−3454.11	−0.460827	−0.230413	−	0.973093i	$-0.574008\pi$
$383$	−3454.11	−0.460827	−0.230413	+	0.973093i	$0.574008\pi$
$384$	−10275.6	−1.36556
$385$	−2118.44	−0.280431
$386$	−2537.81	−0.334641
$387$	0	0
$388$	−9897.74	−1.29506
$389$	−15107.6	−1.96911	−0.984555	−	0.175073i	$-0.943984\pi$
$389$	−15107.6	−1.96911	−0.984555	+	0.175073i	$0.943984\pi$
$390$	−4020.16	−0.521971
$391$	−10486.7	−1.35635
$392$	1093.06	0.140836
$393$	244.529	0.0313864
$394$	486.708	0.0622335
$395$	9613.32	1.22455
$396$	3139.02	0.398338
$397$	1850.76	0.233972	0.116986	−	0.993134i	$-0.462677\pi$
$397$	1850.76	0.233972	0.116986	+	0.993134i	$0.462677\pi$
$398$	−2889.40	−0.363901
$399$	−7194.81	−0.902734
$400$	13869.2	1.73364
$401$	−4757.23	−0.592430	−0.296215	−	0.955121i	$-0.595725\pi$
$401$	−4757.23	−0.592430	−0.296215	+	0.955121i	$0.595725\pi$
$402$	−944.741	−0.117212
$403$	−5725.09	−0.707660
$404$	11240.1	1.38419
$405$	−19524.7	−2.39553
$406$	−903.422	−0.110434
$407$	−1681.53	−0.204792
$408$	−9065.75	−1.10005
$409$	−1964.94	−0.237555	−0.118778	−	0.992921i	$-0.537898\pi$
$409$	−1964.94	−0.237555	−0.118778	+	0.992921i	$0.537898\pi$
$410$	476.241	0.0573656
$411$	−6869.25	−0.824417
$412$	2223.83	0.265923
$413$	1886.32	0.224745
$414$	−3631.63	−0.431123
$415$	−13844.1	−1.63754
$416$	4056.07	0.478042
$417$	−9450.21	−1.10978
$418$	−229.976	−0.0269102
$419$	−12920.8	−1.50650	−0.753249	−	0.657735i	$-0.771515\pi$
$419$	−12920.8	−1.50650	−0.753249	+	0.657735i	$0.771515\pi$
$420$	−20592.2	−2.39238
$421$	−1779.75	−0.206033	−0.103016	−	0.994680i	$-0.532849\pi$
$421$	−1779.75	−0.206033	−0.103016	+	0.994680i	$0.532849\pi$
$422$	−712.441	−0.0821827
$423$	−26075.7	−2.99727
$424$	5821.75	0.666814
$425$	25801.4	2.94483
$426$	−1223.12	−0.139109
$427$	12711.9	1.44068
$428$	11828.3	1.33585
$429$	−2408.14	−0.271017
$430$	0	0
$431$	−10329.4	−1.15441	−0.577205	−	0.816599i	$-0.695857\pi$
$431$	−10329.4	−1.15441	−0.577205	+	0.816599i	$0.695857\pi$
$432$	15279.1	1.70166
$433$	35.7668	0.00396961	0.00198481	−	0.999998i	$-0.499368\pi$
$433$	35.7668	0.00396961	0.00198481	+	0.999998i	$0.499368\pi$
$434$	1471.75	0.162780
$435$	−17187.0	−1.89437
$436$	−13781.2	−1.51376
$437$	−5301.47	−0.580329
$438$	614.327	0.0670175
$439$	5776.48	0.628010	0.314005	−	0.949421i	$-0.398329\pi$
$439$	5776.48	0.628010	0.314005	+	0.949421i	$0.398329\pi$
$440$	−1349.46	−0.146211
$441$	−6483.33	−0.700068
$442$	2305.52	0.248105
$443$	−12268.5	−1.31578	−0.657892	−	0.753112i	$-0.728552\pi$
$443$	−12268.5	−1.31578	−0.657892	+	0.753112i	$0.728552\pi$
$444$	−16345.3	−1.74710
$445$	−8385.43	−0.893276
$446$	−1641.93	−0.174323
$447$	9628.20	1.01879
$448$	5623.88	0.593088
$449$	−11796.6	−1.23990	−0.619952	−	0.784640i	$-0.712848\pi$
$449$	−11796.6	−1.23990	−0.619952	+	0.784640i	$0.712848\pi$
$450$	8935.26	0.936027
$451$	285.276	0.0297852
$452$	−5157.65	−0.536716
$453$	26647.1	2.76377
$454$	−2317.42	−0.239564
$455$	10736.5	1.10623
$456$	−4583.13	−0.470668
$457$	−8101.46	−0.829256	−0.414628	−	0.909991i	$-0.636088\pi$
$457$	−8101.46	−0.829256	−0.414628	+	0.909991i	$0.636088\pi$
$458$	731.616	0.0746423
$459$	28424.4	2.89050
$460$	−15173.3	−1.53796
$461$	11117.9	1.12324	0.561618	−	0.827396i	$-0.310179\pi$
$461$	11117.9	1.12324	0.561618	+	0.827396i	$0.310179\pi$
$462$	619.064	0.0623408
$463$	16742.2	1.68051	0.840255	−	0.542191i	$-0.182405\pi$
$463$	16742.2	1.68051	0.840255	+	0.542191i	$0.182405\pi$
$464$	5298.24	0.530096
$465$	27999.1	2.79231
$466$	1731.35	0.172110
$467$	−16116.5	−1.59697	−0.798485	−	0.602015i	$-0.794365\pi$
$467$	−16116.5	−1.59697	−0.798485	+	0.602015i	$0.794365\pi$
$468$	−15908.9	−1.57135
$469$	2523.09	0.248412
$470$	5467.74	0.536613
$471$	4819.10	0.471448
$472$	1201.59	0.117178
$473$	0	0
$474$	−2809.26	−0.272223
$475$	13043.7	1.25997
$476$	11809.4	1.13715
$477$	−34530.9	−3.31460
$478$	−1785.81	−0.170881
$479$	2987.24	0.284949	0.142474	−	0.989798i	$-0.454494\pi$
$479$	2987.24	0.284949	0.142474	+	0.989798i	$0.454494\pi$
$480$	−19836.6	−1.88628
$481$	8522.21	0.807857
$482$	−29.9355	−0.00282889
$483$	14270.8	1.34440
$484$	9744.86	0.915183
$485$	−25238.1	−2.36289
$486$	1065.41	0.0994402
$487$	−19072.7	−1.77468	−0.887338	−	0.461119i	$-0.847448\pi$
$487$	−19072.7	−1.77468	−0.887338	+	0.461119i	$0.847448\pi$
$488$	8097.54	0.751144
$489$	375.837	0.0347566
$490$	1359.47	0.125336
$491$	8855.63	0.813949	0.406975	−	0.913439i	$-0.366584\pi$
$491$	8855.63	0.813949	0.406975	+	0.913439i	$0.366584\pi$
$492$	2773.02	0.254100
$493$	9856.55	0.900439
$494$	1165.54	0.106154
$495$	8004.14	0.726786
$496$	−8631.30	−0.781365
$497$	3266.55	0.294819
$498$	4045.61	0.364032
$499$	−12489.5	−1.12046	−0.560229	−	0.828338i	$-0.689287\pi$
$499$	−12489.5	−1.12046	−0.560229	+	0.828338i	$0.689287\pi$
$500$	18836.4	1.68478
$501$	−10731.2	−0.956958
$502$	−2928.17	−0.260339
$503$	−1022.04	−0.0905977	−0.0452989	−	0.998973i	$-0.514424\pi$
$503$	−1022.04	−0.0905977	−0.0452989	+	0.998973i	$0.514424\pi$
$504$	8384.69	0.741039
$505$	28660.9	2.52553
$506$	456.155	0.0400762
$507$	−7964.27	−0.697644
$508$	79.7349	0.00696390
$509$	3665.05	0.319156	0.159578	−	0.987185i	$-0.448987\pi$
$509$	3665.05	0.319156	0.159578	+	0.987185i	$0.448987\pi$
$510$	−11275.4	−0.978984
$511$	−1640.66	−0.142032
$512$	10361.7	0.894388
$513$	14369.8	1.23673
$514$	−397.455	−0.0341070
$515$	5670.52	0.485190
$516$	0	0
$517$	3275.27	0.278619
$518$	−2190.81	−0.185828
$519$	−20294.5	−1.71644
$520$	6839.21	0.576768
$521$	5845.70	0.491564	0.245782	−	0.969325i	$-0.420955\pi$
$521$	5845.70	0.491564	0.245782	+	0.969325i	$0.420955\pi$
$522$	3413.41	0.286209
$523$	−16090.3	−1.34528	−0.672639	−	0.739971i	$-0.734839\pi$
$523$	−16090.3	−1.34528	−0.672639	+	0.739971i	$0.734839\pi$
$524$	−202.908	−0.0169162
$525$	−35112.0	−2.91888
$526$	−3355.69	−0.278165
$527$	−16057.2	−1.32725
$528$	−3630.58	−0.299244
$529$	−1651.58	−0.135743
$530$	7240.70	0.593426
$531$	−7127.09	−0.582466
$532$	5970.19	0.486542
$533$	−1445.81	−0.117496
$534$	2450.44	0.198579
$535$	30160.8	2.43732
$536$	1607.22	0.129517
$537$	22012.4	1.76891
$538$	2725.77	0.218431
$539$	814.346	0.0650767
$540$	41127.7	3.27751
$541$	2502.60	0.198882	0.0994410	−	0.995043i	$-0.468295\pi$
$541$	2502.60	0.198882	0.0994410	+	0.995043i	$0.468295\pi$
$542$	3094.84	0.245267
$543$	627.080	0.0495591
$544$	11376.1	0.896592
$545$	−35140.5	−2.76193
$546$	−3137.49	−0.245919
$547$	25182.5	1.96842	0.984210	−	0.177004i	$-0.0566405\pi$
$547$	25182.5	1.96842	0.984210	+	0.177004i	$0.0566405\pi$
$548$	5700.04	0.444332
$549$	−48029.5	−3.73379
$550$	−1122.32	−0.0870109
$551$	4982.91	0.385262
$552$	9090.61	0.700945
$553$	7502.59	0.576931
$554$	5481.07	0.420340
$555$	−41678.6	−3.18767
$556$	7841.70	0.598133
$557$	3635.57	0.276560	0.138280	−	0.990393i	$-0.455843\pi$
$557$	3635.57	0.276560	0.138280	+	0.990393i	$0.455843\pi$
$558$	−5560.75	−0.421873
$559$	0	0
$560$	16186.7	1.22145
$561$	−6754.13	−0.508306
$562$	2347.39	0.176190
$563$	12158.4	0.910153	0.455077	−	0.890452i	$-0.349612\pi$
$563$	12158.4	0.910153	0.455077	+	0.890452i	$0.349612\pi$
$564$	31837.1	2.37692
$565$	−13151.4	−0.979264
$566$	−2910.89	−0.216173
$567$	−15237.8	−1.12862
$568$	2080.81	0.153713
$569$	−16684.3	−1.22925	−0.614625	−	0.788819i	$-0.710693\pi$
$569$	−16684.3	−1.22925	−0.614625	+	0.788819i	$0.710693\pi$
$570$	−5700.19	−0.418868
$571$	20545.1	1.50576	0.752879	−	0.658159i	$-0.228665\pi$
$571$	20545.1	1.50576	0.752879	+	0.658159i	$0.228665\pi$
$572$	1998.26	0.146069
$573$	−5376.35	−0.391972
$574$	371.677	0.0270270
$575$	−25872.1	−1.87642
$576$	−21248.8	−1.53709
$577$	−25550.0	−1.84344	−0.921718	−	0.387861i	$-0.873214\pi$
$577$	−25550.0	−1.84344	−0.921718	+	0.387861i	$0.873214\pi$
$578$	3428.54	0.246728
$579$	−37679.6	−2.70451
$580$	14261.6	1.02100
$581$	−10804.5	−0.771505
$582$	7375.23	0.525280
$583$	4337.30	0.308117
$584$	−1045.11	−0.0740530
$585$	−40565.9	−2.86700
$586$	−3991.56	−0.281382
$587$	−21850.9	−1.53643	−0.768214	−	0.640193i	$-0.778854\pi$
$587$	−21850.9	−1.53643	−0.768214	+	0.640193i	$0.778854\pi$
$588$	7915.82	0.555175
$589$	−8117.60	−0.567878
$590$	1494.46	0.104281
$591$	7226.28	0.502960
$592$	12848.3	0.891998
$593$	15140.1	1.04845	0.524224	−	0.851580i	$-0.324355\pi$
$593$	15140.1	1.04845	0.524224	+	0.851580i	$0.324355\pi$
$594$	−1236.42	−0.0854056
$595$	30112.7	2.07479
$596$	−7989.40	−0.549091
$597$	−42899.7	−2.94098
$598$	−2311.84	−0.158091
$599$	−7836.05	−0.534511	−0.267256	−	0.963626i	$-0.586117\pi$
$599$	−7836.05	−0.534511	−0.267256	+	0.963626i	$0.586117\pi$
$600$	−22366.5	−1.52185
$601$	11113.3	0.754279	0.377140	−	0.926156i	$-0.376908\pi$
$601$	11113.3	0.754279	0.377140	+	0.926156i	$0.376908\pi$
$602$	0	0
$603$	−9533.02	−0.643805
$604$	−22111.5	−1.48958
$605$	24848.3	1.66980
$606$	−8375.45	−0.561434
$607$	−17388.4	−1.16272	−0.581361	−	0.813646i	$-0.697480\pi$
$607$	−17388.4	−1.16272	−0.581361	+	0.813646i	$0.697480\pi$
$608$	5751.11	0.383616
$609$	−13413.3	−0.892506
$610$	10071.2	0.668476
$611$	−16599.4	−1.09908
$612$	−44619.8	−2.94714
$613$	16616.6	1.09484	0.547421	−	0.836857i	$-0.315610\pi$
$613$	16616.6	1.09484	0.547421	+	0.836857i	$0.315610\pi$
$614$	230.798	0.0151698
$615$	7070.88	0.463619
$616$	−1053.17	−0.0688853
$617$	4402.51	0.287258	0.143629	−	0.989632i	$-0.454123\pi$
$617$	4402.51	0.287258	0.143629	+	0.989632i	$0.454123\pi$
$618$	−1657.07	−0.107860
$619$	−16295.5	−1.05811	−0.529055	−	0.848587i	$-0.677454\pi$
$619$	−16295.5	−1.05811	−0.529055	+	0.848587i	$0.677454\pi$
$620$	−23233.4	−1.50496
$621$	−28502.3	−1.84180
$622$	1583.41	0.102072
$623$	−6544.30	−0.420854
$624$	18400.2	1.18045
$625$	16493.1	1.05556
$626$	5589.51	0.356872
$627$	−3414.51	−0.217484
$628$	−3998.84	−0.254094
$629$	23902.3	1.51518
$630$	10428.3	0.659483
$631$	−10777.3	−0.679935	−0.339967	−	0.940437i	$-0.610416\pi$
$631$	−10777.3	−0.679935	−0.339967	+	0.940437i	$0.610416\pi$
$632$	4779.19	0.300801
$633$	−10577.8	−0.664186
$634$	1534.67	0.0961350
$635$	203.315	0.0127060
$636$	42160.5	2.62858
$637$	−4127.20	−0.256712
$638$	−428.745	−0.0266053
$639$	−12342.1	−0.764076
$640$	21741.9	1.34285
$641$	18764.1	1.15622	0.578109	−	0.815959i	$-0.303791\pi$
$641$	18764.1	1.15622	0.578109	+	0.815959i	$0.303791\pi$
$642$	−8813.76	−0.541825
$643$	−7517.75	−0.461075	−0.230537	−	0.973063i	$-0.574048\pi$
$643$	−7517.75	−0.461075	−0.230537	+	0.973063i	$0.574048\pi$
$644$	−11841.8	−0.724586
$645$	0	0
$646$	3269.00	0.199098
$647$	20711.1	1.25848	0.629241	−	0.777210i	$-0.283366\pi$
$647$	20711.1	1.25848	0.629241	+	0.777210i	$0.283366\pi$
$648$	−9706.58	−0.588442
$649$	895.206	0.0541447
$650$	5688.06	0.343237
$651$	21851.5	1.31556
$652$	−311.867	−0.0187326
$653$	6658.64	0.399039	0.199520	−	0.979894i	$-0.436062\pi$
$653$	6658.64	0.399039	0.199520	+	0.979894i	$0.436062\pi$
$654$	10269.0	0.613988
$655$	−517.392	−0.0308644
$656$	−2179.75	−0.129733
$657$	6198.93	0.368103
$658$	4267.23	0.252818
$659$	−11926.7	−0.705004	−0.352502	−	0.935811i	$-0.614669\pi$
$659$	−11926.7	−0.705004	−0.352502	+	0.935811i	$0.614669\pi$
$660$	−9772.65	−0.576363
$661$	−31886.9	−1.87634	−0.938168	−	0.346180i	$-0.887479\pi$
$661$	−31886.9	−1.87634	−0.938168	+	0.346180i	$0.887479\pi$
$662$	−2512.29	−0.147497
$663$	34230.7	2.00514
$664$	−6882.50	−0.402248
$665$	15223.3	0.887720
$666$	8277.58	0.481606
$667$	−9883.57	−0.573753
$668$	8904.67	0.515767
$669$	−24378.2	−1.40884
$670$	1998.95	0.115263
$671$	6032.80	0.347084
$672$	−15481.2	−0.888692
$673$	21903.5	1.25456	0.627279	−	0.778794i	$-0.284168\pi$
$673$	21903.5	1.25456	0.627279	+	0.778794i	$0.284168\pi$
$674$	−609.199	−0.0348152
$675$	70127.1	3.99880
$676$	6608.68	0.376006
$677$	−12230.4	−0.694317	−0.347159	−	0.937806i	$-0.612853\pi$
$677$	−12230.4	−0.694317	−0.347159	+	0.937806i	$0.612853\pi$
$678$	3843.19	0.217694
$679$	−19696.8	−1.11324
$680$	19182.0	1.08176
$681$	−34407.4	−1.93611
$682$	698.464	0.0392164
$683$	−17621.4	−0.987207	−0.493604	−	0.869687i	$-0.664321\pi$
$683$	−17621.4	−0.987207	−0.493604	+	0.869687i	$0.664321\pi$
$684$	−22557.2	−1.26096
$685$	14534.5	0.810705
$686$	4276.01	0.237986
$687$	10862.5	0.603246
$688$	0	0
$689$	−21981.9	−1.21545
$690$	11306.3	0.623801
$691$	−25629.9	−1.41101	−0.705504	−	0.708706i	$-0.749279\pi$
$691$	−25629.9	−1.41101	−0.705504	+	0.708706i	$0.749279\pi$
$692$	16840.2	0.925100
$693$	6246.73	0.342415
$694$	663.160	0.0362726
$695$	19995.4	1.09132
$696$	−8544.37	−0.465336
$697$	−4055.08	−0.220369
$698$	−181.357	−0.00983445
$699$	25705.8	1.39096
$700$	29135.6	1.57317
$701$	6140.99	0.330873	0.165437	−	0.986220i	$-0.447097\pi$
$701$	6140.99	0.330873	0.165437	+	0.986220i	$0.447097\pi$
$702$	6266.32	0.336904
$703$	12083.6	0.648283
$704$	2668.98	0.142885
$705$	81181.0	4.33681
$706$	−710.587	−0.0378800
$707$	22368.0	1.18987
$708$	8701.82	0.461913
$709$	−36700.3	−1.94402	−0.972009	−	0.234943i	$-0.924510\pi$
$709$	−36700.3	−1.94402	−0.972009	+	0.234943i	$0.924510\pi$
$710$	2587.97	0.136796
$711$	−28347.2	−1.49522
$712$	−4168.76	−0.219425
$713$	16101.2	0.845715
$714$	−8799.72	−0.461234
$715$	5095.32	0.266509
$716$	−18265.7	−0.953382
$717$	−26514.4	−1.38103
$718$	5370.40	0.279139
$719$	19467.1	1.00973	0.504867	−	0.863197i	$-0.331542\pi$
$719$	19467.1	1.00973	0.504867	+	0.863197i	$0.331542\pi$
$720$	−61158.3	−3.16560
$721$	4425.49	0.228590
$722$	−2588.38	−0.133421
$723$	−444.460	−0.0228626
$724$	−520.346	−0.0267106
$725$	24317.5	1.24570
$726$	−7261.31	−0.371202
$727$	11021.6	0.562269	0.281135	−	0.959668i	$-0.409289\pi$
$727$	11021.6	0.562269	0.281135	+	0.959668i	$0.409289\pi$
$728$	5337.57	0.271736
$729$	−11321.3	−0.575181
$730$	−1299.84	−0.0659029
$731$	0	0
$732$	58641.6	2.96101
$733$	29262.9	1.47455	0.737277	−	0.675590i	$-0.236111\pi$
$733$	29262.9	1.47455	0.737277	+	0.675590i	$0.236111\pi$
$734$	−7479.18	−0.376106
$735$	20184.4	1.01294
$736$	−11407.3	−0.571302
$737$	1197.40	0.0598466
$738$	−1404.31	−0.0700453
$739$	−14025.7	−0.698167	−0.349083	−	0.937092i	$-0.613507\pi$
$739$	−14025.7	−0.698167	−0.349083	+	0.937092i	$0.613507\pi$
$740$	34584.5	1.71804
$741$	17305.1	0.857920
$742$	5650.91	0.279584
$743$	2161.22	0.106712	0.0533562	−	0.998576i	$-0.483008\pi$
$743$	2161.22	0.106712	0.0533562	+	0.998576i	$0.483008\pi$
$744$	13919.5	0.685907
$745$	−20372.0	−1.00184
$746$	−4258.93	−0.209022
$747$	40822.7	1.99950
$748$	5604.52	0.273959
$749$	23538.6	1.14831
$750$	−14035.8	−0.683354
$751$	29152.5	1.41650	0.708250	−	0.705962i	$-0.249485\pi$
$751$	29152.5	1.41650	0.708250	+	0.705962i	$0.249485\pi$
$752$	−25025.8	−1.21356
$753$	−43475.2	−2.10402
$754$	2172.93	0.104952
$755$	−56381.9	−2.71781
$756$	32097.6	1.54415
$757$	26927.3	1.29285	0.646427	−	0.762976i	$-0.276263\pi$
$757$	26927.3	1.29285	0.646427	+	0.762976i	$0.276263\pi$
$758$	−1291.88	−0.0619040
$759$	6772.65	0.323889
$760$	9697.32	0.462840
$761$	−1001.36	−0.0476996	−0.0238498	−	0.999716i	$-0.507592\pi$
$761$	−1001.36	−0.0476996	−0.0238498	+	0.999716i	$0.507592\pi$
$762$	−59.4139	−0.00282459
$763$	−27425.0	−1.30125
$764$	4461.24	0.211259
$765$	−113775.	−5.37720
$766$	−2135.72	−0.100740
$767$	−4537.01	−0.213588
$768$	20892.2	0.981617
$769$	34146.0	1.60122	0.800609	−	0.599187i	$-0.204510\pi$
$769$	34146.0	1.60122	0.800609	+	0.599187i	$0.204510\pi$
$770$	−1309.86	−0.0613040
$771$	−5901.12	−0.275647
$772$	31266.2	1.45764
$773$	12617.5	0.587090	0.293545	−	0.955945i	$-0.405165\pi$
$773$	12617.5	0.587090	0.293545	+	0.955945i	$0.405165\pi$
$774$	0	0
$775$	−39615.4	−1.83616
$776$	−12546.9	−0.580424
$777$	−32527.6	−1.50183
$778$	−9341.20	−0.430461
$779$	−2050.02	−0.0942870
$780$	49528.9	2.27361
$781$	1550.24	0.0710267
$782$	−6484.04	−0.296508
$783$	26789.7	1.22271
$784$	−6222.28	−0.283449
$785$	−10196.6	−0.463608
$786$	151.195	0.00686128
$787$	−3944.55	−0.178663	−0.0893317	−	0.996002i	$-0.528473\pi$
$787$	−3944.55	−0.178663	−0.0893317	+	0.996002i	$0.528473\pi$
$788$	−5996.30	−0.271078
$789$	−49822.7	−2.24808
$790$	5944.04	0.267695
$791$	−10263.9	−0.461366
$792$	3979.20	0.178529
$793$	−30574.9	−1.36916
$794$	1144.35	0.0511479
$795$	107505.	4.79596
$796$	35597.7	1.58509
$797$	−15295.0	−0.679768	−0.339884	−	0.940467i	$-0.610388\pi$
$797$	−15295.0	−0.679768	−0.339884	+	0.940467i	$0.610388\pi$
$798$	−4448.64	−0.197344
$799$	−46556.5	−2.06139
$800$	28066.5	1.24037
$801$	24726.4	1.09072
$802$	−2941.45	−0.129509
$803$	−778.624	−0.0342180
$804$	11639.3	0.510557
$805$	−30195.3	−1.32204
$806$	−3539.89	−0.154699
$807$	40470.1	1.76532
$808$	14248.5	0.620374
$809$	−10980.7	−0.477208	−0.238604	−	0.971117i	$-0.576690\pi$
$809$	−10980.7	−0.477208	−0.238604	+	0.971117i	$0.576690\pi$
$810$	−12072.4	−0.523680
$811$	40654.8	1.76028	0.880138	−	0.474719i	$-0.157450\pi$
$811$	40654.8	1.76028	0.880138	+	0.474719i	$0.157450\pi$
$812$	11130.3	0.481030
$813$	45949.8	1.98220
$814$	−1039.71	−0.0447690
$815$	−795.224	−0.0341785
$816$	51607.2	2.21399
$817$	0	0
$818$	−1214.95	−0.0519312
$819$	−31659.1	−1.35074
$820$	−5867.35	−0.249874
$821$	21851.2	0.928880	0.464440	−	0.885604i	$-0.346256\pi$
$821$	21851.2	0.928880	0.464440	+	0.885604i	$0.346256\pi$
$822$	−4247.35	−0.180223
$823$	−8135.35	−0.344569	−0.172285	−	0.985047i	$-0.555115\pi$
$823$	−8135.35	−0.344569	−0.172285	+	0.985047i	$0.555115\pi$
$824$	2819.06	0.119183
$825$	−16663.4	−0.703206
$826$	1166.33	0.0491307
$827$	−39430.9	−1.65798	−0.828988	−	0.559266i	$-0.811083\pi$
$827$	−39430.9	−1.65798	−0.828988	+	0.559266i	$0.811083\pi$
$828$	44742.1	1.87789
$829$	−7493.89	−0.313961	−0.156980	−	0.987602i	$-0.550176\pi$
$829$	−7493.89	−0.313961	−0.156980	+	0.987602i	$0.550176\pi$
$830$	−8559.99	−0.357978
$831$	81378.8	3.39711
$832$	−13526.7	−0.563646
$833$	−11575.6	−0.481476
$834$	−5843.18	−0.242605
$835$	22705.9	0.941042
$836$	2833.33	0.117216
$837$	−43642.7	−1.80229
$838$	−7989.10	−0.329330
$839$	31112.0	1.28022	0.640110	−	0.768283i	$-0.278889\pi$
$839$	31112.0	1.28022	0.640110	+	0.768283i	$0.278889\pi$
$840$	−26103.9	−1.07223
$841$	−15099.3	−0.619103
$842$	−1100.44	−0.0450401
$843$	34852.3	1.42394
$844$	8777.36	0.357973
$845$	16851.4	0.686041
$846$	−16122.9	−0.655222
$847$	19392.5	0.786700
$848$	−33140.5	−1.34204
$849$	−43218.7	−1.74707
$850$	15953.3	0.643758
$851$	−23967.8	−0.965459
$852$	15069.0	0.605935
$853$	34197.2	1.37267	0.686336	−	0.727284i	$-0.259218\pi$
$853$	34197.2	1.37267	0.686336	+	0.727284i	$0.259218\pi$
$854$	7859.92	0.314943
$855$	−57518.4	−2.30069
$856$	14994.2	0.598705
$857$	48162.3	1.91971	0.959856	−	0.280493i	$-0.0904981\pi$
$857$	48162.3	1.91971	0.959856	+	0.280493i	$0.0904981\pi$
$858$	−1488.99	−0.0592461
$859$	−16199.5	−0.643447	−0.321723	−	0.946834i	$-0.604262\pi$
$859$	−16199.5	−0.643447	−0.321723	+	0.946834i	$0.604262\pi$
$860$	0	0
$861$	5518.38	0.218427
$862$	−6386.81	−0.252362
$863$	33115.5	1.30622	0.653108	−	0.757264i	$-0.273465\pi$
$863$	33115.5	1.30622	0.653108	+	0.757264i	$0.273465\pi$
$864$	30919.7	1.21749
$865$	42940.6	1.68789
$866$	22.1151	0.000867784 0
$867$	50904.5	1.99401
$868$	−18132.2	−0.709040
$869$	3560.57	0.138992
$870$	−10626.9	−0.414122
$871$	−6068.58	−0.236080
$872$	−17469.8	−0.678444
$873$	74420.5	2.88517
$874$	−3277.96	−0.126864
$875$	37485.0	1.44825
$876$	−7568.58	−0.291916
$877$	−1323.92	−0.0509756	−0.0254878	−	0.999675i	$-0.508114\pi$
$877$	−1323.92	−0.0509756	−0.0254878	+	0.999675i	$0.508114\pi$
$878$	3571.67	0.137287
$879$	−59263.8	−2.27408
$880$	7681.85	0.294267
$881$	−27090.2	−1.03597	−0.517987	−	0.855388i	$-0.673318\pi$
$881$	−27090.2	−1.03597	−0.517987	+	0.855388i	$0.673318\pi$
$882$	−4008.73	−0.153040
$883$	42910.9	1.63541	0.817705	−	0.575637i	$-0.195246\pi$
$883$	42910.9	1.63541	0.817705	+	0.575637i	$0.195246\pi$
$884$	−28404.3	−1.08070
$885$	22188.6	0.842783
$886$	−7585.75	−0.287639
$887$	−42010.3	−1.59027	−0.795133	−	0.606435i	$-0.792599\pi$
$887$	−42010.3	−1.59027	−0.795133	+	0.606435i	$0.792599\pi$
$888$	−20720.2	−0.783024
$889$	158.674	0.00598624
$890$	−5184.82	−0.195276
$891$	−7231.56	−0.271904
$892$	20228.8	0.759318
$893$	−23536.3	−0.881985
$894$	5953.24	0.222714
$895$	−46575.4	−1.73949
$896$	16968.2	0.632666
$897$	−34324.5	−1.27766
$898$	−7294.00	−0.271051
$899$	−15133.7	−0.561443
$900$	−110083.	−4.07716
$901$	−61652.8	−2.27964
$902$	176.390	0.00651125
$903$	0	0
$904$	−6538.13	−0.240548
$905$	−1326.82	−0.0487349
$906$	16476.2	0.604179
$907$	11550.7	0.422861	0.211431	−	0.977393i	$-0.432188\pi$
$907$	11550.7	0.422861	0.211431	+	0.977393i	$0.432188\pi$
$908$	28550.9	1.04350
$909$	−84513.4	−3.08375
$910$	6638.52	0.241829
$911$	−5375.85	−0.195510	−0.0977552	−	0.995210i	$-0.531166\pi$
$911$	−5375.85	−0.195510	−0.0977552	+	0.995210i	$0.531166\pi$
$912$	26089.7	0.947275
$913$	−5127.57	−0.185868
$914$	−5009.23	−0.181281
$915$	149529.	5.40250
$916$	−9013.60	−0.325129
$917$	−403.792	−0.0145413
$918$	17575.2	0.631881
$919$	1541.87	0.0553447	0.0276723	−	0.999617i	$-0.491190\pi$
$919$	1541.87	0.0553447	0.0276723	+	0.999617i	$0.491190\pi$
$920$	−19234.6	−0.689288
$921$	3426.72	0.122599
$922$	6874.34	0.245547
$923$	−7856.78	−0.280183
$924$	−7626.94	−0.271545
$925$	58970.3	2.09614
$926$	10351.9	0.367371
$927$	−16720.9	−0.592433
$928$	10721.8	0.379269
$929$	−20669.5	−0.729973	−0.364987	−	0.931013i	$-0.618926\pi$
$929$	−20669.5	−0.729973	−0.364987	+	0.931013i	$0.618926\pi$
$930$	17312.2	0.610418
$931$	−5851.96	−0.206004
$932$	−21330.4	−0.749680
$933$	23509.3	0.824929
$934$	−9965.07	−0.349108
$935$	14290.9	0.499852
$936$	−20167.0	−0.704252
$937$	31002.3	1.08090	0.540449	−	0.841377i	$-0.318255\pi$
$937$	31002.3	1.08090	0.540449	+	0.841377i	$0.318255\pi$
$938$	1560.06	0.0543045
$939$	82988.9	2.88418
$940$	−67363.2	−2.33739
$941$	−31759.6	−1.10025	−0.550124	−	0.835083i	$-0.685420\pi$
$941$	−31759.6	−1.10025	−0.550124	+	0.835083i	$0.685420\pi$
$942$	2979.71	0.103062
$943$	4066.20	0.140417
$944$	−6840.12	−0.235834
$945$	81845.1	2.81738
$946$	0	0
$947$	31056.3	1.06567	0.532837	−	0.846218i	$-0.321126\pi$
$947$	31056.3	1.06567	0.532837	+	0.846218i	$0.321126\pi$
$948$	34610.4	1.18575
$949$	3946.15	0.134982
$950$	8065.10	0.275438
$951$	22785.7	0.776946
$952$	14970.3	0.509654
$953$	22084.7	0.750676	0.375338	−	0.926888i	$-0.377527\pi$
$953$	22084.7	0.750676	0.375338	+	0.926888i	$0.377527\pi$
$954$	−21350.9	−0.724593
$955$	11375.7	0.385453
$956$	22001.4	0.744327
$957$	−6365.69	−0.215019
$958$	1847.05	0.0622917
$959$	11343.2	0.381952
$960$	66153.5	2.22406
$961$	−5136.86	−0.172430
$962$	5269.39	0.176603
$963$	−88936.2	−2.97604
$964$	368.809	0.0123221
$965$	79725.2	2.65953
$966$	8823.85	0.293895
$967$	45620.1	1.51711	0.758554	−	0.651610i	$-0.225906\pi$
$967$	45620.1	1.51711	0.758554	+	0.651610i	$0.225906\pi$
$968$	12353.1	0.410171
$969$	48535.7	1.60907
$970$	−15605.0	−0.516544
$971$	−39645.2	−1.31027	−0.655137	−	0.755510i	$-0.727389\pi$
$971$	−39645.2	−1.31027	−0.655137	+	0.755510i	$0.727389\pi$
$972$	−13126.0	−0.433144
$973$	15605.2	0.514162
$974$	−11792.9	−0.387956
$975$	84452.0	2.77398
$976$	−46095.6	−1.51177
$977$	−58085.8	−1.90208	−0.951038	−	0.309074i	$-0.899981\pi$
$977$	−58085.8	−1.90208	−0.951038	+	0.309074i	$0.899981\pi$
$978$	232.385	0.00759801
$979$	−3105.79	−0.101391
$980$	−16748.9	−0.545942
$981$	103620.	3.37241
$982$	5475.55	0.177935
$983$	−8705.96	−0.282479	−0.141240	−	0.989975i	$-0.545109\pi$
$983$	−8705.96	−0.282479	−0.141240	+	0.989975i	$0.545109\pi$
$984$	3515.24	0.113884
$985$	−15289.9	−0.494595
$986$	6094.43	0.196842
$987$	63356.7	2.04323
$988$	−14359.6	−0.462389
$989$	0	0
$990$	4949.06	0.158880
$991$	−234.516	−0.00751732	−0.00375866	−	0.999993i	$-0.501196\pi$
$991$	−234.516	−0.00751732	−0.00375866	+	0.999993i	$0.501196\pi$
$992$	−17466.8	−0.559044
$993$	−37300.7	−1.19205
$994$	2019.75	0.0644493
$995$	90770.2	2.89207
$996$	−49842.4	−1.58566
$997$	−36443.3	−1.15764	−0.578822	−	0.815454i	$-0.696488\pi$
$997$	−36443.3	−1.15764	−0.578822	+	0.815454i	$0.696488\pi$
$998$	−7722.44	−0.244939
$999$	64965.4	2.05747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.28	✓	50
43.42	odd	2		1849.4.a.j.1.23	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.28	✓	50	1.1	even	1	trivial
1849.4.a.j.1.23	yes	50	43.42	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618313 q^{2} +9.18025 q^{3} -7.61769 q^{4} -19.4242 q^{5} +5.67627 q^{6} -15.1594 q^{7} -9.65662 q^{8} +57.2770 q^{9} +O(q^{10})\)	\(q+0.618313 q^{2} +9.18025 q^{3} -7.61769 q^{4} -19.4242 q^{5} +5.67627 q^{6} -15.1594 q^{7} -9.65662 q^{8} +57.2770 q^{9} -12.0103 q^{10} -7.19433 q^{11} -69.9323 q^{12} +36.4617 q^{13} -9.37325 q^{14} -178.319 q^{15} +54.9707 q^{16} +102.264 q^{17} +35.4151 q^{18} +51.6991 q^{19} +147.968 q^{20} -139.167 q^{21} -4.44835 q^{22} -102.545 q^{23} -88.6502 q^{24} +252.301 q^{25} +22.5447 q^{26} +277.950 q^{27} +115.480 q^{28} +96.3830 q^{29} -110.257 q^{30} -157.016 q^{31} +111.242 q^{32} -66.0458 q^{33} +63.2314 q^{34} +294.460 q^{35} -436.318 q^{36} +233.730 q^{37} +31.9662 q^{38} +334.728 q^{39} +187.572 q^{40} -39.6529 q^{41} -86.0488 q^{42} +54.8042 q^{44} -1112.56 q^{45} -63.4047 q^{46} -455.256 q^{47} +504.645 q^{48} -113.193 q^{49} +156.001 q^{50} +938.812 q^{51} -277.754 q^{52} -602.876 q^{53} +171.860 q^{54} +139.744 q^{55} +146.388 q^{56} +474.610 q^{57} +59.5949 q^{58} -124.432 q^{59} +1358.38 q^{60} -838.549 q^{61} -97.0852 q^{62} -868.284 q^{63} -370.983 q^{64} -708.241 q^{65} -40.8370 q^{66} -166.437 q^{67} -779.018 q^{68} -941.386 q^{69} +182.068 q^{70} -215.480 q^{71} -553.102 q^{72} +108.227 q^{73} +144.518 q^{74} +2316.18 q^{75} -393.827 q^{76} +109.062 q^{77} +206.966 q^{78} -494.914 q^{79} -1067.76 q^{80} +1005.17 q^{81} -24.5179 q^{82} +712.724 q^{83} +1060.13 q^{84} -1986.41 q^{85} +884.820 q^{87} +69.4729 q^{88} +431.700 q^{89} -687.911 q^{90} -552.738 q^{91} +781.154 q^{92} -1441.45 q^{93} -281.491 q^{94} -1004.21 q^{95} +1021.23 q^{96} +1299.31 q^{97} -69.9885 q^{98} -412.070 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} - 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} - 386 q^{18} + 12 q^{19} + 108 q^{20} - 408 q^{21} + 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} + 1493 q^{26} + 10 q^{27} - 242 q^{28} + 208 q^{29} - 48 q^{30} - 932 q^{31} - 1124 q^{32} + 254 q^{33} + 765 q^{34} - 1452 q^{35} + 747 q^{36} - 90 q^{37} - 1213 q^{38} - 1610 q^{39} - 1693 q^{40} - 1354 q^{41} - 16 q^{42} - 2704 q^{44} + 4508 q^{45} + 233 q^{46} - 3484 q^{47} - 376 q^{48} + 1324 q^{49} - 408 q^{50} + 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} - 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} + 1172 q^{61} - 1546 q^{62} - 3686 q^{63} + 606 q^{64} + 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} + 136 q^{69} - 1310 q^{70} + 162 q^{71} - 5814 q^{72} + 746 q^{73} - 4332 q^{74} + 236 q^{75} + 1338 q^{76} + 2024 q^{77} - 2782 q^{78} - 2656 q^{79} - 5713 q^{80} - 86 q^{81} - 4168 q^{82} - 3514 q^{83} - 4269 q^{84} - 7558 q^{85} - 10278 q^{87} + 11692 q^{88} + 2640 q^{89} - 8286 q^{90} - 5946 q^{91} - 4271 q^{92} - 2 q^{93} + 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} - 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 - 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 - 386 * q^18 + 12 * q^19 + 108 * q^20 - 408 * q^21 + 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 + 1493 * q^26 + 10 * q^27 - 242 * q^28 + 208 * q^29 - 48 * q^30 - 932 * q^31 - 1124 * q^32 + 254 * q^33 + 765 * q^34 - 1452 * q^35 + 747 * q^36 - 90 * q^37 - 1213 * q^38 - 1610 * q^39 - 1693 * q^40 - 1354 * q^41 - 16 * q^42 - 2704 * q^44 + 4508 * q^45 + 233 * q^46 - 3484 * q^47 - 376 * q^48 + 1324 * q^49 - 408 * q^50 + 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 - 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 + 1172 * q^61 - 1546 * q^62 - 3686 * q^63 + 606 * q^64 + 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 + 136 * q^69 - 1310 * q^70 + 162 * q^71 - 5814 * q^72 + 746 * q^73 - 4332 * q^74 + 236 * q^75 + 1338 * q^76 + 2024 * q^77 - 2782 * q^78 - 2656 * q^79 - 5713 * q^80 - 86 * q^81 - 4168 * q^82 - 3514 * q^83 - 4269 * q^84 - 7558 * q^85 - 10278 * q^87 + 11692 * q^88 + 2640 * q^89 - 8286 * q^90 - 5946 * q^91 - 4271 * q^92 - 2 * q^93 + 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 - 2826 * q^98 - 8174 * q^99

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