LMFDB - Embedded newform 1334.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1334,4,Mod(1,1334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1334, 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("1334.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1334 = 2 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1334.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:	$78.7085479477$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 327 x^{17} + 1564 x^{16} + 43869 x^{15} - 203270 x^{14} - 3103297 x^{13} + \cdots - 37580243456 $ x^19 - 5x^18 - 327x^17 + 1564x^16 + 43869x^15 - 203270x^14 - 3103297x^13 + 14176990x^12 + 123567059x^11 - 569338227x^10 - 2720149582x^9 + 13044432966x^8 + 29634597970x^7 - 156881614170x^6 - 111892601042x^5 + 797597313752x^4 - 61225108811x^3 - 836249925952x^2 - 357044713984x - 37580243456
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.55575$ of defining polynomial
Character	$\chi$	$=$	1334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -9.55575 q^{3} +4.00000 q^{4} -15.6269 q^{5} +19.1115 q^{6} +28.1388 q^{7} -8.00000 q^{8} +64.3123 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -9.55575 q^{3} +4.00000 q^{4} -15.6269 q^{5} +19.1115 q^{6} +28.1388 q^{7} -8.00000 q^{8} +64.3123 q^{9} +31.2538 q^{10} +19.4183 q^{11} -38.2230 q^{12} -64.2090 q^{13} -56.2776 q^{14} +149.327 q^{15} +16.0000 q^{16} -8.04058 q^{17} -128.625 q^{18} -110.263 q^{19} -62.5076 q^{20} -268.887 q^{21} -38.8366 q^{22} +23.0000 q^{23} +76.4460 q^{24} +119.200 q^{25} +128.418 q^{26} -356.547 q^{27} +112.555 q^{28} -29.0000 q^{29} -298.653 q^{30} -69.7435 q^{31} -32.0000 q^{32} -185.556 q^{33} +16.0812 q^{34} -439.722 q^{35} +257.249 q^{36} +182.293 q^{37} +220.527 q^{38} +613.565 q^{39} +125.015 q^{40} +234.949 q^{41} +537.774 q^{42} -348.711 q^{43} +77.6732 q^{44} -1005.00 q^{45} -46.0000 q^{46} +223.734 q^{47} -152.892 q^{48} +448.792 q^{49} -238.399 q^{50} +76.8338 q^{51} -256.836 q^{52} +519.412 q^{53} +713.094 q^{54} -303.448 q^{55} -225.110 q^{56} +1053.65 q^{57} +58.0000 q^{58} -48.2937 q^{59} +597.306 q^{60} -351.716 q^{61} +139.487 q^{62} +1809.67 q^{63} +64.0000 q^{64} +1003.39 q^{65} +371.113 q^{66} +584.297 q^{67} -32.1623 q^{68} -219.782 q^{69} +879.444 q^{70} +441.963 q^{71} -514.498 q^{72} +274.355 q^{73} -364.585 q^{74} -1139.04 q^{75} -441.054 q^{76} +546.408 q^{77} -1227.13 q^{78} -411.132 q^{79} -250.030 q^{80} +1670.64 q^{81} -469.897 q^{82} +318.762 q^{83} -1075.55 q^{84} +125.649 q^{85} +697.421 q^{86} +277.117 q^{87} -155.346 q^{88} -1599.08 q^{89} +2010.00 q^{90} -1806.76 q^{91} +92.0000 q^{92} +666.452 q^{93} -447.469 q^{94} +1723.07 q^{95} +305.784 q^{96} +1526.87 q^{97} -897.583 q^{98} +1248.84 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 38 q^{2} - 5 q^{3} + 76 q^{4} + 10 q^{6} - 13 q^{7} - 152 q^{8} + 166 q^{9}+O(q^{10}) $ 19 * q - 38 * q^2 - 5 * q^3 + 76 * q^4 + 10 * q^6 - 13 * q^7 - 152 * q^8 + 166 * q^9	\( 19 q - 38 q^{2} - 5 q^{3} + 76 q^{4} + 10 q^{6} - 13 q^{7} - 152 q^{8} + 166 q^{9} + 30 q^{11} - 20 q^{12} - 151 q^{13} + 26 q^{14} + 2 q^{15} + 304 q^{16} - 56 q^{17} - 332 q^{18} - 266 q^{19} - 239 q^{21} - 60 q^{22} + 437 q^{23} + 40 q^{24} + 395 q^{25} + 302 q^{26} - 68 q^{27} - 52 q^{28} - 551 q^{29} - 4 q^{30} - 622 q^{31} - 608 q^{32} - 291 q^{33} + 112 q^{34} + 21 q^{35} + 664 q^{36} - 183 q^{37} + 532 q^{38} - 212 q^{39} - 395 q^{41} + 478 q^{42} - 291 q^{43} + 120 q^{44} - 1018 q^{45} - 874 q^{46} - 241 q^{47} - 80 q^{48} + 1394 q^{49} - 790 q^{50} - 184 q^{51} - 604 q^{52} - 631 q^{53} + 136 q^{54} - 1765 q^{55} + 104 q^{56} + 144 q^{57} + 1102 q^{58} - 457 q^{59} + 8 q^{60} - 630 q^{61} + 1244 q^{62} + 125 q^{63} + 1216 q^{64} + 137 q^{65} + 582 q^{66} + 1421 q^{67} - 224 q^{68} - 115 q^{69} - 42 q^{70} + 983 q^{71} - 1328 q^{72} - 2532 q^{73} + 366 q^{74} - 1081 q^{75} - 1064 q^{76} - 595 q^{77} + 424 q^{78} - 4361 q^{79} - 721 q^{81} + 790 q^{82} + 1422 q^{83} - 956 q^{84} + 722 q^{85} + 582 q^{86} + 145 q^{87} - 240 q^{88} - 580 q^{89} + 2036 q^{90} - 447 q^{91} + 1748 q^{92} + 2106 q^{93} + 482 q^{94} - 310 q^{95} + 160 q^{96} - 2016 q^{97} - 2788 q^{98} - 1972 q^{99}+O(q^{100}) \) 19 * q - 38 * q^2 - 5 * q^3 + 76 * q^4 + 10 * q^6 - 13 * q^7 - 152 * q^8 + 166 * q^9 + 30 * q^11 - 20 * q^12 - 151 * q^13 + 26 * q^14 + 2 * q^15 + 304 * q^16 - 56 * q^17 - 332 * q^18 - 266 * q^19 - 239 * q^21 - 60 * q^22 + 437 * q^23 + 40 * q^24 + 395 * q^25 + 302 * q^26 - 68 * q^27 - 52 * q^28 - 551 * q^29 - 4 * q^30 - 622 * q^31 - 608 * q^32 - 291 * q^33 + 112 * q^34 + 21 * q^35 + 664 * q^36 - 183 * q^37 + 532 * q^38 - 212 * q^39 - 395 * q^41 + 478 * q^42 - 291 * q^43 + 120 * q^44 - 1018 * q^45 - 874 * q^46 - 241 * q^47 - 80 * q^48 + 1394 * q^49 - 790 * q^50 - 184 * q^51 - 604 * q^52 - 631 * q^53 + 136 * q^54 - 1765 * q^55 + 104 * q^56 + 144 * q^57 + 1102 * q^58 - 457 * q^59 + 8 * q^60 - 630 * q^61 + 1244 * q^62 + 125 * q^63 + 1216 * q^64 + 137 * q^65 + 582 * q^66 + 1421 * q^67 - 224 * q^68 - 115 * q^69 - 42 * q^70 + 983 * q^71 - 1328 * q^72 - 2532 * q^73 + 366 * q^74 - 1081 * q^75 - 1064 * q^76 - 595 * q^77 + 424 * q^78 - 4361 * q^79 - 721 * q^81 + 790 * q^82 + 1422 * q^83 - 956 * q^84 + 722 * q^85 + 582 * q^86 + 145 * q^87 - 240 * q^88 - 580 * q^89 + 2036 * q^90 - 447 * q^91 + 1748 * q^92 + 2106 * q^93 + 482 * q^94 - 310 * q^95 + 160 * q^96 - 2016 * q^97 - 2788 * q^98 - 1972 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−9.55575	−1.83900	−0.919502	−	0.393085i	$-0.871408\pi$
$3$	−9.55575	−1.83900	−0.919502	+	0.393085i	$0.871408\pi$
$4$	4.00000	0.500000
$5$	−15.6269	−1.39771	−0.698856	−	0.715263i	$-0.746307\pi$
$5$	−15.6269	−1.39771	−0.698856	+	0.715263i	$0.746307\pi$
$6$	19.1115	1.30037
$7$	28.1388	1.51935	0.759676	−	0.650302i	$-0.225358\pi$
$7$	28.1388	1.51935	0.759676	+	0.650302i	$0.225358\pi$
$8$	−8.00000	−0.353553
$9$	64.3123	2.38194
$10$	31.2538	0.988331
$11$	19.4183	0.532258	0.266129	−	0.963937i	$-0.414255\pi$
$11$	19.4183	0.532258	0.266129	+	0.963937i	$0.414255\pi$
$12$	−38.2230	−0.919502
$13$	−64.2090	−1.36988	−0.684938	−	0.728602i	$-0.740171\pi$
$13$	−64.2090	−1.36988	−0.684938	+	0.728602i	$0.740171\pi$
$14$	−56.2776	−1.07434
$15$	149.327	2.57040
$16$	16.0000	0.250000
$17$	−8.04058	−0.114713	−0.0573567	−	0.998354i	$-0.518267\pi$
$17$	−8.04058	−0.114713	−0.0573567	+	0.998354i	$0.518267\pi$
$18$	−128.625	−1.68428
$19$	−110.263	−1.33138	−0.665689	−	0.746230i	$-0.731862\pi$
$19$	−110.263	−1.33138	−0.665689	+	0.746230i	$0.731862\pi$
$20$	−62.5076	−0.698856
$21$	−268.887	−2.79409
$22$	−38.8366	−0.376363
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	76.4460	0.650186
$25$	119.200	0.953597
$26$	128.418	0.968648
$27$	−356.547	−2.54139
$28$	112.555	0.759676
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	−298.653	−1.81755
$31$	−69.7435	−0.404075	−0.202037	−	0.979378i	$-0.564756\pi$
$31$	−69.7435	−0.404075	−0.202037	+	0.979378i	$0.564756\pi$
$32$	−32.0000	−0.176777
$33$	−185.556	−0.978825
$34$	16.0812	0.0811146
$35$	−439.722	−2.12362
$36$	257.249	1.19097
$37$	182.293	0.809966	0.404983	−	0.914324i	$-0.367278\pi$
$37$	182.293	0.809966	0.404983	+	0.914324i	$0.367278\pi$
$38$	220.527	0.941426
$39$	613.565	2.51921
$40$	125.015	0.494166
$41$	234.949	0.894947	0.447473	−	0.894297i	$-0.352324\pi$
$41$	234.949	0.894947	0.447473	+	0.894297i	$0.352324\pi$
$42$	537.774	1.97572
$43$	−348.711	−1.23670	−0.618348	−	0.785905i	$-0.712198\pi$
$43$	−348.711	−1.23670	−0.618348	+	0.785905i	$0.712198\pi$
$44$	77.6732	0.266129
$45$	−1005.00	−3.32926
$46$	−46.0000	−0.147442
$47$	223.734	0.694362	0.347181	−	0.937798i	$-0.387139\pi$
$47$	223.734	0.694362	0.347181	+	0.937798i	$0.387139\pi$
$48$	−152.892	−0.459751
$49$	448.792	1.30843
$50$	−238.399	−0.674295
$51$	76.8338	0.210958
$52$	−256.836	−0.684938
$53$	519.412	1.34616	0.673082	−	0.739568i	$-0.264970\pi$
$53$	519.412	1.34616	0.673082	+	0.739568i	$0.264970\pi$
$54$	713.094	1.79703
$55$	−303.448	−0.743943
$56$	−225.110	−0.537172
$57$	1053.65	2.44841
$58$	58.0000	0.131306
$59$	−48.2937	−0.106565	−0.0532823	−	0.998579i	$-0.516968\pi$
$59$	−48.2937	−0.106565	−0.0532823	+	0.998579i	$0.516968\pi$
$60$	597.306	1.28520
$61$	−351.716	−0.738241	−0.369120	−	0.929382i	$-0.620341\pi$
$61$	−351.716	−0.738241	−0.369120	+	0.929382i	$0.620341\pi$
$62$	139.487	0.285724
$63$	1809.67	3.61900
$64$	64.0000	0.125000
$65$	1003.39	1.91469
$66$	371.113	0.692134
$67$	584.297	1.06542	0.532710	−	0.846298i	$-0.321174\pi$
$67$	584.297	1.06542	0.532710	+	0.846298i	$0.321174\pi$
$68$	−32.1623	−0.0573567
$69$	−219.782	−0.383459
$70$	879.444	1.50162
$71$	441.963	0.738751	0.369375	−	0.929280i	$-0.379572\pi$
$71$	441.963	0.738751	0.369375	+	0.929280i	$0.379572\pi$
$72$	−514.498	−0.842142
$73$	274.355	0.439874	0.219937	−	0.975514i	$-0.429415\pi$
$73$	274.355	0.439874	0.219937	+	0.975514i	$0.429415\pi$
$74$	−364.585	−0.572732
$75$	−1139.04	−1.75367
$76$	−441.054	−0.665689
$77$	546.408	0.808688
$78$	−1227.13	−1.78135
$79$	−411.132	−0.585519	−0.292760	−	0.956186i	$-0.594574\pi$
$79$	−411.132	−0.585519	−0.292760	+	0.956186i	$0.594574\pi$
$80$	−250.030	−0.349428
$81$	1670.64	2.29169
$82$	−469.897	−0.632823
$83$	318.762	0.421551	0.210775	−	0.977535i	$-0.432401\pi$
$83$	318.762	0.421551	0.210775	+	0.977535i	$0.432401\pi$
$84$	−1075.55	−1.39705
$85$	125.649	0.160336
$86$	697.421	0.874475
$87$	277.117	0.341495
$88$	−155.346	−0.188182
$89$	−1599.08	−1.90452	−0.952258	−	0.305296i	$-0.901245\pi$
$89$	−1599.08	−1.90452	−0.952258	+	0.305296i	$0.901245\pi$
$90$	2010.00	2.35414
$91$	−1806.76	−2.08132
$92$	92.0000	0.104257
$93$	666.452	0.743095
$94$	−447.469	−0.490988
$95$	1723.07	1.86088
$96$	305.784	0.325093
$97$	1526.87	1.59825	0.799123	−	0.601168i	$-0.205298\pi$
$97$	1526.87	1.59825	0.799123	+	0.601168i	$0.205298\pi$
$98$	−897.583	−0.925200
$99$	1248.84	1.26781
$100$	476.799	0.476799
$101$	624.578	0.615325	0.307663	−	0.951496i	$-0.400453\pi$
$101$	624.578	0.615325	0.307663	+	0.951496i	$0.400453\pi$
$102$	−153.668	−0.149170
$103$	730.618	0.698931	0.349465	−	0.936949i	$-0.386363\pi$
$103$	730.618	0.698931	0.349465	+	0.936949i	$0.386363\pi$
$104$	513.672	0.484324
$105$	4201.87	3.90534
$106$	−1038.82	−0.951881
$107$	43.2396	0.0390666	0.0195333	−	0.999809i	$-0.493782\pi$
$107$	43.2396	0.0390666	0.0195333	+	0.999809i	$0.493782\pi$
$108$	−1426.19	−1.27069
$109$	1026.05	0.901634	0.450817	−	0.892617i	$-0.351133\pi$
$109$	1026.05	0.901634	0.450817	+	0.892617i	$0.351133\pi$
$110$	606.896	0.526047
$111$	−1741.94	−1.48953
$112$	450.221	0.379838
$113$	−2154.47	−1.79359	−0.896794	−	0.442449i	$-0.854110\pi$
$113$	−2154.47	−1.79359	−0.896794	+	0.442449i	$0.854110\pi$
$114$	−2107.30	−1.73129
$115$	−359.418	−0.291443
$116$	−116.000	−0.0928477
$117$	−4129.43	−3.26296
$118$	96.5875	0.0753525
$119$	−226.252	−0.174290
$120$	−1194.61	−0.908773
$121$	−953.929	−0.716701
$122$	703.433	0.522015
$123$	−2245.11	−1.64581
$124$	−278.974	−0.202037
$125$	90.6416	0.0648578
$126$	−3619.34	−2.55902
$127$	−889.841	−0.621737	−0.310868	−	0.950453i	$-0.600620\pi$
$127$	−889.841	−0.621737	−0.310868	+	0.950453i	$0.600620\pi$
$128$	−128.000	−0.0883883
$129$	3332.19	2.27429
$130$	−2006.78	−1.35389
$131$	1986.97	1.32521	0.662603	−	0.748971i	$-0.269452\pi$
$131$	1986.97	1.32521	0.662603	+	0.748971i	$0.269452\pi$
$132$	−742.226	−0.489413
$133$	−3102.68	−2.02283
$134$	−1168.59	−0.753366
$135$	5571.72	3.55213
$136$	64.3247	0.0405573
$137$	1262.56	0.787356	0.393678	−	0.919248i	$-0.371203\pi$
$137$	1262.56	0.787356	0.393678	+	0.919248i	$0.371203\pi$
$138$	439.564	0.271146
$139$	−975.494	−0.595254	−0.297627	−	0.954682i	$-0.596195\pi$
$139$	−975.494	−0.595254	−0.297627	+	0.954682i	$0.596195\pi$
$140$	−1758.89	−1.06181
$141$	−2137.95	−1.27693
$142$	−883.925	−0.522376
$143$	−1246.83	−0.729128
$144$	1029.00	0.595484
$145$	453.180	0.259548
$146$	−548.710	−0.311038
$147$	−4288.54	−2.40621
$148$	729.171	0.404983
$149$	128.981	0.0709164	0.0354582	−	0.999371i	$-0.488711\pi$
$149$	128.981	0.0709164	0.0354582	+	0.999371i	$0.488711\pi$
$150$	2278.08	1.24003
$151$	−169.875	−0.0915510	−0.0457755	−	0.998952i	$-0.514576\pi$
$151$	−169.875	−0.0915510	−0.0457755	+	0.998952i	$0.514576\pi$
$152$	882.108	0.470713
$153$	−517.108	−0.273240
$154$	−1092.82	−0.571829
$155$	1089.87	0.564780
$156$	2454.26	1.25960
$157$	3118.67	1.58533	0.792665	−	0.609658i	$-0.208693\pi$
$157$	3118.67	1.58533	0.792665	+	0.609658i	$0.208693\pi$
$158$	822.265	0.414025
$159$	−4963.37	−2.47560
$160$	500.060	0.247083
$161$	647.192	0.316807
$162$	−3341.28	−1.62047
$163$	−2675.07	−1.28545	−0.642724	−	0.766098i	$-0.722196\pi$
$163$	−2675.07	−1.28545	−0.642724	+	0.766098i	$0.722196\pi$
$164$	939.795	0.447473
$165$	2899.67	1.36812
$166$	−637.525	−0.298082
$167$	605.289	0.280471	0.140235	−	0.990118i	$-0.455214\pi$
$167$	605.289	0.280471	0.140235	+	0.990118i	$0.455214\pi$
$168$	2151.10	0.987862
$169$	1925.80	0.876559
$170$	−251.299	−0.113375
$171$	−7091.30	−3.17126
$172$	−1394.84	−0.618348
$173$	−989.545	−0.434877	−0.217438	−	0.976074i	$-0.569770\pi$
$173$	−989.545	−0.434877	−0.217438	+	0.976074i	$0.569770\pi$
$174$	−554.233	−0.241473
$175$	3354.13	1.44885
$176$	310.693	0.133065
$177$	461.483	0.195973
$178$	3198.15	1.34670
$179$	−4088.86	−1.70735	−0.853676	−	0.520804i	$-0.825632\pi$
$179$	−4088.86	−1.70735	−0.853676	+	0.520804i	$0.825632\pi$
$180$	−4020.00	−1.66463
$181$	−1317.81	−0.541173	−0.270586	−	0.962696i	$-0.587218\pi$
$181$	−1317.81	−0.541173	−0.270586	+	0.962696i	$0.587218\pi$
$182$	3613.53	1.47172
$183$	3360.91	1.35763
$184$	−184.000	−0.0737210
$185$	−2848.67	−1.13210
$186$	−1332.90	−0.525447
$187$	−156.135	−0.0610571
$188$	894.937	0.347181
$189$	−10032.8	−3.86126
$190$	−3446.15	−1.31584
$191$	4817.83	1.82516	0.912582	−	0.408895i	$-0.134086\pi$
$191$	4817.83	1.82516	0.912582	+	0.408895i	$0.134086\pi$
$192$	−611.568	−0.229876
$193$	3937.26	1.46845	0.734223	−	0.678908i	$-0.237546\pi$
$193$	3937.26	1.46845	0.734223	+	0.678908i	$0.237546\pi$
$194$	−3053.73	−1.13013
$195$	−9588.12	−3.52112
$196$	1795.17	0.654215
$197$	4808.93	1.73920	0.869599	−	0.493759i	$-0.164378\pi$
$197$	4808.93	1.73920	0.869599	+	0.493759i	$0.164378\pi$
$198$	−2497.67	−0.896474
$199$	−2299.36	−0.819082	−0.409541	−	0.912292i	$-0.634311\pi$
$199$	−2299.36	−0.819082	−0.409541	+	0.912292i	$0.634311\pi$
$200$	−953.597	−0.337147
$201$	−5583.39	−1.95931
$202$	−1249.16	−0.435101
$203$	−816.025	−0.282137
$204$	307.335	0.105479
$205$	−3671.52	−1.25088
$206$	−1461.24	−0.494219
$207$	1479.18	0.496668
$208$	−1027.34	−0.342469
$209$	−2141.13	−0.708637
$210$	−8403.74	−2.76149
$211$	−1678.05	−0.547497	−0.273749	−	0.961801i	$-0.588264\pi$
$211$	−1678.05	−0.547497	−0.273749	+	0.961801i	$0.588264\pi$
$212$	2077.65	0.673082
$213$	−4223.28	−1.35857
$214$	−86.4792	−0.0276243
$215$	5449.26	1.72854
$216$	2852.37	0.898516
$217$	−1962.50	−0.613931
$218$	−2052.11	−0.637551
$219$	−2621.67	−0.808931
$220$	−1213.79	−0.371972
$221$	516.278	0.157143
$222$	3483.88	1.05326
$223$	−989.387	−0.297104	−0.148552	−	0.988905i	$-0.547461\pi$
$223$	−989.387	−0.297104	−0.148552	+	0.988905i	$0.547461\pi$
$224$	−900.441	−0.268586
$225$	7666.00	2.27141
$226$	4308.94	1.26826
$227$	800.361	0.234017	0.117008	−	0.993131i	$-0.462670\pi$
$227$	800.361	0.234017	0.117008	+	0.993131i	$0.462670\pi$
$228$	4214.60	1.22420
$229$	2845.33	0.821068	0.410534	−	0.911845i	$-0.365342\pi$
$229$	2845.33	0.821068	0.410534	+	0.911845i	$0.365342\pi$
$230$	718.837	0.206081
$231$	−5221.34	−1.48718
$232$	232.000	0.0656532
$233$	4142.46	1.16473	0.582364	−	0.812928i	$-0.302128\pi$
$233$	4142.46	1.16473	0.582364	+	0.812928i	$0.302128\pi$
$234$	8258.86	2.30726
$235$	−3496.27	−0.970517
$236$	−193.175	−0.0532823
$237$	3928.68	1.07677
$238$	452.505	0.123242
$239$	5091.05	1.37788	0.688939	−	0.724819i	$-0.258077\pi$
$239$	5091.05	1.37788	0.688939	+	0.724819i	$0.258077\pi$
$240$	2389.23	0.642599
$241$	−6701.11	−1.79110	−0.895552	−	0.444957i	$-0.853219\pi$
$241$	−6701.11	−1.79110	−0.895552	+	0.444957i	$0.853219\pi$
$242$	1907.86	0.506784
$243$	−6337.44	−1.67303
$244$	−1406.87	−0.369120
$245$	−7013.22	−1.82881
$246$	4490.22	1.16376
$247$	7079.91	1.82382
$248$	557.948	0.142862
$249$	−3046.01	−0.775234
$250$	−181.283	−0.0458614
$251$	1778.58	0.447262	0.223631	−	0.974674i	$-0.428209\pi$
$251$	1778.58	0.447262	0.223631	+	0.974674i	$0.428209\pi$
$252$	7238.68	1.80950
$253$	446.621	0.110984
$254$	1779.68	0.439634
$255$	−1200.67	−0.294859
$256$	256.000	0.0625000
$257$	−1120.02	−0.271848	−0.135924	−	0.990719i	$-0.543400\pi$
$257$	−1120.02	−0.271848	−0.135924	+	0.990719i	$0.543400\pi$
$258$	−6664.38	−1.60816
$259$	5129.50	1.23062
$260$	4013.55	0.957345
$261$	−1865.06	−0.442315
$262$	−3973.93	−0.937062
$263$	5868.75	1.37598	0.687990	−	0.725720i	$-0.258493\pi$
$263$	5868.75	1.37598	0.687990	+	0.725720i	$0.258493\pi$
$264$	1484.45	0.346067
$265$	−8116.79	−1.88155
$266$	6205.36	1.43036
$267$	15280.4	3.50241
$268$	2337.19	0.532710
$269$	726.228	0.164606	0.0823028	−	0.996607i	$-0.473773\pi$
$269$	726.228	0.164606	0.0823028	+	0.996607i	$0.473773\pi$
$270$	−11143.4	−2.51173
$271$	−623.860	−0.139841	−0.0699203	−	0.997553i	$-0.522274\pi$
$271$	−623.860	−0.139841	−0.0699203	+	0.997553i	$0.522274\pi$
$272$	−128.649	−0.0286783
$273$	17265.0	3.82756
$274$	−2525.12	−0.556744
$275$	2314.66	0.507560
$276$	−879.129	−0.191729
$277$	−8426.76	−1.82785	−0.913925	−	0.405882i	$-0.866964\pi$
$277$	−8426.76	−1.82785	−0.913925	+	0.405882i	$0.866964\pi$
$278$	1950.99	0.420908
$279$	−4485.37	−0.962480
$280$	3517.77	0.750812
$281$	1571.53	0.333629	0.166814	−	0.985988i	$-0.446652\pi$
$281$	1571.53	0.333629	0.166814	+	0.985988i	$0.446652\pi$
$282$	4275.90	0.902929
$283$	3868.47	0.812567	0.406284	−	0.913747i	$-0.366824\pi$
$283$	3868.47	0.812567	0.406284	+	0.913747i	$0.366824\pi$
$284$	1767.85	0.369375
$285$	−16465.3	−3.42217
$286$	2493.66	0.515571
$287$	6611.17	1.35974
$288$	−2057.99	−0.421071
$289$	−4848.35	−0.986841
$290$	−906.360	−0.183528
$291$	−14590.4	−2.93918
$292$	1097.42	0.219937
$293$	−6533.12	−1.30262	−0.651312	−	0.758810i	$-0.725781\pi$
$293$	−6533.12	−1.30262	−0.651312	+	0.758810i	$0.725781\pi$
$294$	8577.08	1.70145
$295$	754.681	0.148947
$296$	−1458.34	−0.286366
$297$	−6923.54	−1.35267
$298$	−257.962	−0.0501454
$299$	−1476.81	−0.285639
$300$	−4556.17	−0.876835
$301$	−9812.30	−1.87898
$302$	339.749	0.0647364
$303$	−5968.31	−1.13159
$304$	−1764.22	−0.332844
$305$	5496.23	1.03185
$306$	1034.22	0.193210
$307$	−1542.53	−0.286764	−0.143382	−	0.989667i	$-0.545798\pi$
$307$	−1542.53	−0.286764	−0.143382	+	0.989667i	$0.545798\pi$
$308$	2185.63	0.404344
$309$	−6981.60	−1.28534
$310$	−2179.75	−0.399359
$311$	−10077.9	−1.83751	−0.918757	−	0.394824i	$-0.870806\pi$
$311$	−10077.9	−1.83751	−0.918757	+	0.394824i	$0.870806\pi$
$312$	−4908.52	−0.890674
$313$	−10506.0	−1.89723	−0.948617	−	0.316428i	$-0.897517\pi$
$313$	−10506.0	−1.89723	−0.948617	+	0.316428i	$0.897517\pi$
$314$	−6237.33	−1.12100
$315$	−28279.5	−5.05832
$316$	−1644.53	−0.292760
$317$	−2419.88	−0.428751	−0.214376	−	0.976751i	$-0.568772\pi$
$317$	−2419.88	−0.428751	−0.214376	+	0.976751i	$0.568772\pi$
$318$	9926.73	1.75051
$319$	−563.131	−0.0988379
$320$	−1000.12	−0.174714
$321$	−413.187	−0.0718437
$322$	−1294.38	−0.224016
$323$	886.582	0.152727
$324$	6682.56	1.14584
$325$	−7653.69	−1.30631
$326$	5350.15	0.908949
$327$	−9804.70	−1.65811
$328$	−1879.59	−0.316411
$329$	6295.61	1.05498
$330$	−5799.34	−0.967404
$331$	−7301.37	−1.21245	−0.606223	−	0.795295i	$-0.707316\pi$
$331$	−7301.37	−1.21245	−0.606223	+	0.795295i	$0.707316\pi$
$332$	1275.05	0.210775
$333$	11723.7	1.92929
$334$	−1210.58	−0.198323
$335$	−9130.74	−1.48915
$336$	−4302.20	−0.698524
$337$	−11528.9	−1.86355	−0.931777	−	0.363031i	$-0.881742\pi$
$337$	−11528.9	−1.86355	−0.931777	+	0.363031i	$0.881742\pi$
$338$	−3851.60	−0.619821
$339$	20587.6	3.29842
$340$	502.597	0.0801681
$341$	−1354.30	−0.215072
$342$	14182.6	2.24242
$343$	2976.85	0.468615
$344$	2789.69	0.437238
$345$	3434.51	0.535965
$346$	1979.09	0.307504
$347$	4035.97	0.624388	0.312194	−	0.950018i	$-0.398936\pi$
$347$	4035.97	0.624388	0.312194	+	0.950018i	$0.398936\pi$
$348$	1108.47	0.170747
$349$	4918.84	0.754440	0.377220	−	0.926124i	$-0.376880\pi$
$349$	4918.84	0.754440	0.377220	+	0.926124i	$0.376880\pi$
$350$	−6708.27	−1.02449
$351$	22893.5	3.48139
$352$	−621.386	−0.0940909
$353$	−4180.28	−0.630295	−0.315147	−	0.949043i	$-0.602054\pi$
$353$	−4180.28	−0.630295	−0.315147	+	0.949043i	$0.602054\pi$
$354$	−922.965	−0.138574
$355$	−6906.50	−1.03256
$356$	−6396.31	−0.952258
$357$	2162.01	0.320520
$358$	8177.73	1.20728
$359$	−4825.99	−0.709488	−0.354744	−	0.934963i	$-0.615432\pi$
$359$	−4825.99	−0.709488	−0.354744	+	0.934963i	$0.615432\pi$
$360$	8040.01	1.17707
$361$	5299.03	0.772566
$362$	2635.63	0.382667
$363$	9115.51	1.31802
$364$	−7227.06	−1.04066
$365$	−4287.32	−0.614817
$366$	−6721.83	−0.959988
$367$	−11093.6	−1.57788	−0.788942	−	0.614468i	$-0.789371\pi$
$367$	−11093.6	−1.57788	−0.788942	+	0.614468i	$0.789371\pi$
$368$	368.000	0.0521286
$369$	15110.1	2.13171
$370$	5697.33	0.800514
$371$	14615.6	2.04530
$372$	2665.81	0.371547
$373$	−10407.5	−1.44472	−0.722359	−	0.691518i	$-0.756942\pi$
$373$	−10407.5	−1.44472	−0.722359	+	0.691518i	$0.756942\pi$
$374$	312.269	0.0431739
$375$	−866.148	−0.119274
$376$	−1789.87	−0.245494
$377$	1862.06	0.254380
$378$	20065.6	2.73033
$379$	12930.0	1.75243	0.876213	−	0.481924i	$-0.160062\pi$
$379$	12930.0	1.75243	0.876213	+	0.481924i	$0.160062\pi$
$380$	6892.30	0.930441
$381$	8503.09	1.14338
$382$	−9635.67	−1.29059
$383$	12038.2	1.60607	0.803033	−	0.595935i	$-0.203218\pi$
$383$	12038.2	1.60607	0.803033	+	0.595935i	$0.203218\pi$
$384$	1223.14	0.162547
$385$	−8538.65	−1.13031
$386$	−7874.52	−1.03835
$387$	−22426.4	−2.94573
$388$	6107.47	0.799123
$389$	−7881.10	−1.02722	−0.513609	−	0.858024i	$-0.671692\pi$
$389$	−7881.10	−1.02722	−0.513609	+	0.858024i	$0.671692\pi$
$390$	19176.2	2.48981
$391$	−184.933	−0.0239194
$392$	−3590.33	−0.462600
$393$	−18986.9	−2.43706
$394$	−9617.86	−1.22980
$395$	6424.72	0.818387
$396$	4995.34	0.633903
$397$	1012.20	0.127961	0.0639807	−	0.997951i	$-0.479620\pi$
$397$	1012.20	0.127961	0.0639807	+	0.997951i	$0.479620\pi$
$398$	4598.72	0.579178
$399$	29648.4	3.72000
$400$	1907.19	0.238399
$401$	7397.41	0.921220	0.460610	−	0.887603i	$-0.347631\pi$
$401$	7397.41	0.921220	0.460610	+	0.887603i	$0.347631\pi$
$402$	11166.8	1.38544
$403$	4478.17	0.553532
$404$	2498.31	0.307663
$405$	−26106.9	−3.20312
$406$	1632.05	0.199501
$407$	3539.82	0.431111
$408$	−614.670	−0.0745851
$409$	−8989.42	−1.08679	−0.543396	−	0.839476i	$-0.682862\pi$
$409$	−8989.42	−1.08679	−0.543396	+	0.839476i	$0.682862\pi$
$410$	7343.03	0.884504
$411$	−12064.7	−1.44795
$412$	2922.47	0.349465
$413$	−1358.93	−0.161909
$414$	−2958.37	−0.351197
$415$	−4981.27	−0.589207
$416$	2054.69	0.242162
$417$	9321.57	1.09467
$418$	4282.26	0.501082
$419$	−3442.26	−0.401349	−0.200675	−	0.979658i	$-0.564313\pi$
$419$	−3442.26	−0.401349	−0.200675	+	0.979658i	$0.564313\pi$
$420$	16807.5	1.95267
$421$	−10118.0	−1.17131	−0.585654	−	0.810561i	$-0.699162\pi$
$421$	−10118.0	−1.17131	−0.585654	+	0.810561i	$0.699162\pi$
$422$	3356.11	0.387139
$423$	14388.9	1.65393
$424$	−4155.29	−0.475941
$425$	−958.434	−0.109390
$426$	8446.56	0.960651
$427$	−9896.88	−1.12165
$428$	172.958	0.0195333
$429$	11914.4	1.34087
$430$	−10898.5	−1.22226
$431$	7077.68	0.790997	0.395499	−	0.918467i	$-0.370572\pi$
$431$	7077.68	0.790997	0.395499	+	0.918467i	$0.370572\pi$
$432$	−5704.75	−0.635347
$433$	−15322.7	−1.70061	−0.850303	−	0.526294i	$-0.823581\pi$
$433$	−15322.7	−1.70061	−0.850303	+	0.526294i	$0.823581\pi$
$434$	3925.00	0.434115
$435$	−4330.47	−0.477311
$436$	4104.21	0.450817
$437$	−2536.06	−0.277611
$438$	5243.33	0.572000
$439$	13309.1	1.44694	0.723472	−	0.690353i	$-0.242545\pi$
$439$	13309.1	1.44694	0.723472	+	0.690353i	$0.242545\pi$
$440$	2427.58	0.263024
$441$	28862.8	3.11660
$442$	−1032.56	−0.111117
$443$	10316.1	1.10639	0.553196	−	0.833051i	$-0.313408\pi$
$443$	10316.1	1.10639	0.553196	+	0.833051i	$0.313408\pi$
$444$	−6967.77	−0.744765
$445$	24988.6	2.66196
$446$	1978.77	0.210084
$447$	−1232.51	−0.130416
$448$	1800.88	0.189919
$449$	4214.41	0.442963	0.221482	−	0.975165i	$-0.428911\pi$
$449$	4214.41	0.442963	0.221482	+	0.975165i	$0.428911\pi$
$450$	−15332.0	−1.60613
$451$	4562.31	0.476343
$452$	−8617.88	−0.896794
$453$	1623.28	0.168363
$454$	−1600.72	−0.165475
$455$	28234.1	2.90909
$456$	−8429.20	−0.865643
$457$	13573.4	1.38936	0.694679	−	0.719320i	$-0.255546\pi$
$457$	13573.4	1.38936	0.694679	+	0.719320i	$0.255546\pi$
$458$	−5690.65	−0.580583
$459$	2866.84	0.291531
$460$	−1437.67	−0.145721
$461$	−4400.71	−0.444602	−0.222301	−	0.974978i	$-0.571357\pi$
$461$	−4400.71	−0.444602	−0.222301	+	0.974978i	$0.571357\pi$
$462$	10442.7	1.05160
$463$	10047.6	1.00854	0.504269	−	0.863547i	$-0.331762\pi$
$463$	10047.6	1.00854	0.504269	+	0.863547i	$0.331762\pi$
$464$	−464.000	−0.0464238
$465$	−10414.6	−1.03863
$466$	−8284.92	−0.823587
$467$	9244.66	0.916043	0.458021	−	0.888941i	$-0.348558\pi$
$467$	9244.66	0.916043	0.458021	+	0.888941i	$0.348558\pi$
$468$	−16517.7	−1.63148
$469$	16441.4	1.61875
$470$	6992.54	0.686259
$471$	−29801.2	−2.91543
$472$	386.350	0.0376763
$473$	−6771.37	−0.658241
$474$	−7857.35	−0.761393
$475$	−13143.4	−1.26960
$476$	−905.009	−0.0871450
$477$	33404.6	3.20648
$478$	−10182.1	−0.974307
$479$	−14953.7	−1.42641	−0.713205	−	0.700955i	$-0.752757\pi$
$479$	−14953.7	−1.42641	−0.713205	+	0.700955i	$0.752757\pi$
$480$	−4778.45	−0.454386
$481$	−11704.8	−1.10955
$482$	13402.2	1.26650
$483$	−6184.41	−0.582609
$484$	−3815.72	−0.358351
$485$	−23860.2	−2.23389
$486$	12674.9	1.18301
$487$	−2963.44	−0.275742	−0.137871	−	0.990450i	$-0.544026\pi$
$487$	−2963.44	−0.275742	−0.137871	+	0.990450i	$0.544026\pi$
$488$	2813.73	0.261008
$489$	25562.3	2.36394
$490$	14026.4	1.29316
$491$	−5219.48	−0.479739	−0.239869	−	0.970805i	$-0.577105\pi$
$491$	−5219.48	−0.479739	−0.239869	+	0.970805i	$0.577105\pi$
$492$	−8980.44	−0.822905
$493$	233.177	0.0213017
$494$	−14159.8	−1.28964
$495$	−19515.4	−1.77203
$496$	−1115.90	−0.101019
$497$	12436.3	1.12242
$498$	6092.03	0.548173
$499$	−2050.75	−0.183976	−0.0919881	−	0.995760i	$-0.529322\pi$
$499$	−2050.75	−0.183976	−0.0919881	+	0.995760i	$0.529322\pi$
$500$	362.566	0.0324289
$501$	−5783.99	−0.515787
$502$	−3557.16	−0.316262
$503$	−20030.0	−1.77553	−0.887767	−	0.460292i	$-0.847745\pi$
$503$	−20030.0	−1.77553	−0.887767	+	0.460292i	$0.847745\pi$
$504$	−14477.4	−1.27951
$505$	−9760.21	−0.860047
$506$	−893.242	−0.0784772
$507$	−18402.5	−1.61200
$508$	−3559.36	−0.310868
$509$	−993.546	−0.0865190	−0.0432595	−	0.999064i	$-0.513774\pi$
$509$	−993.546	−0.0865190	−0.0432595	+	0.999064i	$0.513774\pi$
$510$	2401.35	0.208497
$511$	7720.02	0.668324
$512$	−512.000	−0.0441942
$513$	39314.1	3.38355
$514$	2240.04	0.192225
$515$	−11417.3	−0.976904
$516$	13328.8	1.13714
$517$	4344.54	0.369580
$518$	−10259.0	−0.870182
$519$	9455.84	0.799740
$520$	−8027.10	−0.676945
$521$	−13763.7	−1.15739	−0.578695	−	0.815544i	$-0.696438\pi$
$521$	−13763.7	−1.15739	−0.578695	+	0.815544i	$0.696438\pi$
$522$	3730.11	0.312764
$523$	14278.1	1.19376	0.596881	−	0.802330i	$-0.296407\pi$
$523$	14278.1	1.19376	0.596881	+	0.802330i	$0.296407\pi$
$524$	7947.86	0.662603
$525$	−32051.3	−2.66444
$526$	−11737.5	−0.972964
$527$	560.779	0.0463528
$528$	−2968.90	−0.244706
$529$	529.000	0.0434783
$530$	16233.6	1.33046
$531$	−3105.88	−0.253830
$532$	−12410.7	−1.01142
$533$	−15085.8	−1.22597
$534$	−30560.8	−2.47658
$535$	−675.700	−0.0546039
$536$	−4674.37	−0.376683
$537$	39072.1	3.13983
$538$	−1452.46	−0.116394
$539$	8714.78	0.696423
$540$	22286.9	1.77606
$541$	−5874.55	−0.466851	−0.233426	−	0.972375i	$-0.574994\pi$
$541$	−5874.55	−0.466851	−0.233426	+	0.972375i	$0.574994\pi$
$542$	1247.72	0.0988822
$543$	12592.7	0.995219
$544$	257.299	0.0202787
$545$	−16034.0	−1.26022
$546$	−34530.0	−2.70650
$547$	−1890.11	−0.147742	−0.0738712	−	0.997268i	$-0.523535\pi$
$547$	−1890.11	−0.147742	−0.0738712	+	0.997268i	$0.523535\pi$
$548$	5050.24	0.393678
$549$	−22619.7	−1.75844
$550$	−4629.31	−0.358899
$551$	3197.64	0.247231
$552$	1758.26	0.135573
$553$	−11568.8	−0.889610
$554$	16853.5	1.29249
$555$	27221.1	2.08193
$556$	−3901.98	−0.297627
$557$	10146.8	0.771875	0.385938	−	0.922525i	$-0.373878\pi$
$557$	10146.8	0.771875	0.385938	+	0.922525i	$0.373878\pi$
$558$	8970.74	0.680576
$559$	22390.4	1.69412
$560$	−7035.55	−0.530904
$561$	1491.98	0.112284
$562$	−3143.06	−0.235911
$563$	−20886.8	−1.56354	−0.781769	−	0.623568i	$-0.785682\pi$
$563$	−20886.8	−1.56354	−0.781769	+	0.623568i	$0.785682\pi$
$564$	−8551.79	−0.638467
$565$	33667.6	2.50692
$566$	−7736.94	−0.574572
$567$	47009.8	3.48188
$568$	−3535.70	−0.261188
$569$	−8462.05	−0.623458	−0.311729	−	0.950171i	$-0.600908\pi$
$569$	−8462.05	−0.623458	−0.311729	+	0.950171i	$0.600908\pi$
$570$	32930.5	2.41984
$571$	22126.3	1.62164	0.810819	−	0.585297i	$-0.199022\pi$
$571$	22126.3	1.62164	0.810819	+	0.585297i	$0.199022\pi$
$572$	−4987.32	−0.364564
$573$	−46038.0	−3.35648
$574$	−13222.3	−0.961481
$575$	2741.59	0.198839
$576$	4115.99	0.297742
$577$	21785.4	1.57181	0.785907	−	0.618345i	$-0.212196\pi$
$577$	21785.4	1.57181	0.785907	+	0.618345i	$0.212196\pi$
$578$	9696.70	0.697802
$579$	−37623.5	−2.70048
$580$	1812.72	0.129774
$581$	8969.59	0.640484
$582$	29180.7	2.07831
$583$	10086.1	0.716507
$584$	−2194.84	−0.155519
$585$	64530.2	4.56067
$586$	13066.2	0.921094
$587$	−1023.12	−0.0719397	−0.0359698	−	0.999353i	$-0.511452\pi$
$587$	−1023.12	−0.0719397	−0.0359698	+	0.999353i	$0.511452\pi$
$588$	−17154.2	−1.20310
$589$	7690.16	0.537976
$590$	−1509.36	−0.105321
$591$	−45952.9	−3.19839
$592$	2916.68	0.202491
$593$	5629.01	0.389807	0.194904	−	0.980822i	$-0.437561\pi$
$593$	5629.01	0.389807	0.194904	+	0.980822i	$0.437561\pi$
$594$	13847.1	0.956486
$595$	3535.62	0.243607
$596$	515.924	0.0354582
$597$	21972.1	1.50629
$598$	2953.62	0.201977
$599$	7753.89	0.528907	0.264454	−	0.964398i	$-0.414808\pi$
$599$	7753.89	0.528907	0.264454	+	0.964398i	$0.414808\pi$
$600$	9112.33	0.620016
$601$	−27849.0	−1.89016	−0.945079	−	0.326842i	$-0.894015\pi$
$601$	−27849.0	−1.89016	−0.945079	+	0.326842i	$0.894015\pi$
$602$	19624.6	1.32864
$603$	37577.5	2.53776
$604$	−679.499	−0.0457755
$605$	14906.9	1.00174
$606$	11936.6	0.800152
$607$	−11828.4	−0.790938	−0.395469	−	0.918479i	$-0.629418\pi$
$607$	−11828.4	−0.790938	−0.395469	+	0.918479i	$0.629418\pi$
$608$	3528.43	0.235357
$609$	7797.73	0.518850
$610$	−10992.5	−0.729626
$611$	−14365.8	−0.951189
$612$	−2068.43	−0.136620
$613$	−5121.76	−0.337465	−0.168732	−	0.985662i	$-0.553967\pi$
$613$	−5121.76	−0.337465	−0.168732	+	0.985662i	$0.553967\pi$
$614$	3085.05	0.202773
$615$	35084.1	2.30037
$616$	−4371.26	−0.285914
$617$	9030.84	0.589251	0.294625	−	0.955613i	$-0.404805\pi$
$617$	9030.84	0.589251	0.294625	+	0.955613i	$0.404805\pi$
$618$	13963.2	0.908870
$619$	−12798.1	−0.831014	−0.415507	−	0.909590i	$-0.636396\pi$
$619$	−12798.1	−0.831014	−0.415507	+	0.909590i	$0.636396\pi$
$620$	4359.50	0.282390
$621$	−8200.58	−0.529916
$622$	20155.9	1.29932
$623$	−44996.1	−2.89363
$624$	9817.05	0.629802
$625$	−16316.4	−1.04425
$626$	21012.0	1.34155
$627$	20460.1	1.30319
$628$	12474.7	0.792665
$629$	−1465.74	−0.0929139
$630$	56559.0	3.57677
$631$	−27159.8	−1.71349	−0.856745	−	0.515739i	$-0.827517\pi$
$631$	−27159.8	−1.71349	−0.856745	+	0.515739i	$0.827517\pi$
$632$	3289.06	0.207012
$633$	16035.0	1.00685
$634$	4839.77	0.303173
$635$	13905.4	0.869009
$636$	−19853.5	−1.23780
$637$	−28816.5	−1.79239
$638$	1126.26	0.0698889
$639$	28423.6	1.75966
$640$	2000.24	0.123541
$641$	−21820.7	−1.34457	−0.672283	−	0.740294i	$-0.734686\pi$
$641$	−21820.7	−1.34457	−0.672283	+	0.740294i	$0.734686\pi$
$642$	826.373	0.0508012
$643$	177.352	0.0108773	0.00543863	−	0.999985i	$-0.498269\pi$
$643$	177.352	0.0108773	0.00543863	+	0.999985i	$0.498269\pi$
$644$	2588.77	0.158403
$645$	−52071.8	−3.17880
$646$	−1773.16	−0.107994
$647$	8902.41	0.540943	0.270471	−	0.962728i	$-0.412820\pi$
$647$	8902.41	0.540943	0.270471	+	0.962728i	$0.412820\pi$
$648$	−13365.1	−0.810234
$649$	−937.783	−0.0567199
$650$	15307.4	0.923700
$651$	18753.1	1.12902
$652$	−10700.3	−0.642724
$653$	−30133.7	−1.80585	−0.902927	−	0.429794i	$-0.858586\pi$
$653$	−30133.7	−1.80585	−0.902927	+	0.429794i	$0.858586\pi$
$654$	19609.4	1.17246
$655$	−31050.1	−1.85226
$656$	3759.18	0.223737
$657$	17644.4	1.04775
$658$	−12591.2	−0.745983
$659$	−7797.80	−0.460939	−0.230470	−	0.973080i	$-0.574026\pi$
$659$	−7797.80	−0.460939	−0.230470	+	0.973080i	$0.574026\pi$
$660$	11598.7	0.684058
$661$	25462.3	1.49829	0.749143	−	0.662409i	$-0.230466\pi$
$661$	25462.3	1.49829	0.749143	+	0.662409i	$0.230466\pi$
$662$	14602.7	0.857329
$663$	−4933.42	−0.288987
$664$	−2550.10	−0.149041
$665$	48485.2	2.82733
$666$	−23447.3	−1.36421
$667$	−667.000	−0.0387202
$668$	2421.15	0.140235
$669$	9454.33	0.546376
$670$	18261.5	1.05299
$671$	−6829.74	−0.392935
$672$	8604.39	0.493931
$673$	4828.65	0.276569	0.138284	−	0.990393i	$-0.455841\pi$
$673$	4828.65	0.276569	0.138284	+	0.990393i	$0.455841\pi$
$674$	23057.7	1.31773
$675$	−42500.3	−2.42346
$676$	7703.20	0.438280
$677$	−5301.71	−0.300977	−0.150489	−	0.988612i	$-0.548085\pi$
$677$	−5301.71	−0.300977	−0.150489	+	0.988612i	$0.548085\pi$
$678$	−41175.1	−2.33233
$679$	42964.2	2.42830
$680$	−1005.19	−0.0566874
$681$	−7648.04	−0.430358
$682$	2708.60	0.152079
$683$	−310.363	−0.0173876	−0.00869379	−	0.999962i	$-0.502767\pi$
$683$	−310.363	−0.0173876	−0.00869379	+	0.999962i	$0.502767\pi$
$684$	−28365.2	−1.58563
$685$	−19729.9	−1.10050
$686$	−5953.70	−0.331361
$687$	−27189.2	−1.50995
$688$	−5579.37	−0.309174
$689$	−33350.9	−1.84408
$690$	−6869.02	−0.378984
$691$	−4497.51	−0.247603	−0.123801	−	0.992307i	$-0.539509\pi$
$691$	−4497.51	−0.247603	−0.123801	+	0.992307i	$0.539509\pi$
$692$	−3958.18	−0.217438
$693$	35140.7	1.92624
$694$	−8071.95	−0.441509
$695$	15243.9	0.831993
$696$	−2216.93	−0.120737
$697$	−1889.12	−0.102662
$698$	−9837.68	−0.533469
$699$	−39584.3	−2.14194
$700$	13416.5	0.724425
$701$	−11985.6	−0.645779	−0.322889	−	0.946437i	$-0.604654\pi$
$701$	−11985.6	−0.645779	−0.322889	+	0.946437i	$0.604654\pi$
$702$	−45787.1	−2.46171
$703$	−20100.2	−1.07837
$704$	1242.77	0.0665323
$705$	33409.5	1.78479
$706$	8360.57	0.445686
$707$	17574.9	0.934896
$708$	1845.93	0.0979863
$709$	−9604.85	−0.508770	−0.254385	−	0.967103i	$-0.581873\pi$
$709$	−9604.85	−0.508770	−0.254385	+	0.967103i	$0.581873\pi$
$710$	13813.0	0.730130
$711$	−26440.9	−1.39467
$712$	12792.6	0.673348
$713$	−1604.10	−0.0842554
$714$	−4324.02	−0.226642
$715$	19484.1	1.01911
$716$	−16355.5	−0.853676
$717$	−48648.8	−2.53392
$718$	9651.99	0.501684
$719$	8872.50	0.460207	0.230103	−	0.973166i	$-0.426094\pi$
$719$	8872.50	0.460207	0.230103	+	0.973166i	$0.426094\pi$
$720$	−16080.0	−0.832315
$721$	20558.7	1.06192
$722$	−10598.1	−0.546287
$723$	64034.1	3.29385
$724$	−5271.25	−0.270586
$725$	−3456.79	−0.177079
$726$	−18231.0	−0.931978
$727$	2567.03	0.130957	0.0654786	−	0.997854i	$-0.479143\pi$
$727$	2567.03	0.130957	0.0654786	+	0.997854i	$0.479143\pi$
$728$	14454.1	0.735859
$729$	15451.7	0.785030
$730$	8574.63	0.434742
$731$	2803.84	0.141865
$732$	13443.7	0.678814
$733$	−33641.1	−1.69517	−0.847586	−	0.530657i	$-0.821945\pi$
$733$	−33641.1	−1.69517	−0.847586	+	0.530657i	$0.821945\pi$
$734$	22187.3	1.11573
$735$	67016.5	3.36319
$736$	−736.000	−0.0368605
$737$	11346.1	0.567079
$738$	−30220.2	−1.50734
$739$	−11006.5	−0.547876	−0.273938	−	0.961747i	$-0.588326\pi$
$739$	−11006.5	−0.547876	−0.273938	+	0.961747i	$0.588326\pi$
$740$	−11394.7	−0.566049
$741$	−67653.8	−3.35402
$742$	−29231.2	−1.44624
$743$	16585.7	0.818939	0.409469	−	0.912324i	$-0.365714\pi$
$743$	16585.7	0.818939	0.409469	+	0.912324i	$0.365714\pi$
$744$	−5331.61	−0.262724
$745$	−2015.57	−0.0991206
$746$	20815.0	1.02157
$747$	20500.3	1.00411
$748$	−624.538	−0.0305286
$749$	1216.71	0.0593560
$750$	1732.30	0.0843393
$751$	18770.2	0.912029	0.456014	−	0.889972i	$-0.349277\pi$
$751$	18770.2	0.912029	0.456014	+	0.889972i	$0.349277\pi$
$752$	3579.75	0.173590
$753$	−16995.6	−0.822518
$754$	−3724.12	−0.179873
$755$	2654.61	0.127962
$756$	−40131.2	−1.93063
$757$	8050.49	0.386526	0.193263	−	0.981147i	$-0.438093\pi$
$757$	8050.49	0.386526	0.193263	+	0.981147i	$0.438093\pi$
$758$	−25860.0	−1.23915
$759$	−4267.80	−0.204099
$760$	−13784.6	−0.657921
$761$	19543.4	0.930941	0.465471	−	0.885063i	$-0.345885\pi$
$761$	19543.4	0.930941	0.465471	+	0.885063i	$0.345885\pi$
$762$	−17006.2	−0.808490
$763$	28871.9	1.36990
$764$	19271.3	0.912582
$765$	8080.79	0.381911
$766$	−24076.4	−1.13566
$767$	3100.89	0.145980
$768$	−2446.27	−0.114938
$769$	−2640.65	−0.123829	−0.0619143	−	0.998081i	$-0.519721\pi$
$769$	−2640.65	−0.123829	−0.0619143	+	0.998081i	$0.519721\pi$
$770$	17077.3	0.799251
$771$	10702.6	0.499929
$772$	15749.0	0.734223
$773$	9596.75	0.446534	0.223267	−	0.974757i	$-0.428328\pi$
$773$	9596.75	0.446534	0.223267	+	0.974757i	$0.428328\pi$
$774$	44852.8	2.08295
$775$	−8313.41	−0.385324
$776$	−12214.9	−0.565065
$777$	−49016.2	−2.26312
$778$	15762.2	0.726353
$779$	−25906.2	−1.19151
$780$	−38352.5	−1.76056
$781$	8582.17	0.393206
$782$	369.867	0.0169136
$783$	10339.9	0.471924
$784$	7180.67	0.327108
$785$	−48735.1	−2.21583
$786$	37973.9	1.72326
$787$	19399.4	0.878673	0.439336	−	0.898323i	$-0.355214\pi$
$787$	19399.4	0.878673	0.439336	+	0.898323i	$0.355214\pi$
$788$	19235.7	0.869599
$789$	−56080.3	−2.53043
$790$	−12849.4	−0.578687
$791$	−60624.2	−2.72509
$792$	−9990.69	−0.448237
$793$	22583.4	1.01130
$794$	−2024.39	−0.0904824
$795$	77562.0	3.46017
$796$	−9197.43	−0.409541
$797$	−32601.5	−1.44894	−0.724469	−	0.689308i	$-0.757915\pi$
$797$	−32601.5	−1.44894	−0.724469	+	0.689308i	$0.757915\pi$
$798$	−59296.9	−2.63043
$799$	−1798.95	−0.0796526
$800$	−3814.39	−0.168574
$801$	−102840.	−4.53643
$802$	−14794.8	−0.651401
$803$	5327.51	0.234127
$804$	−22333.6	−0.979657
$805$	−10113.6	−0.442804
$806$	−8956.33	−0.391406
$807$	−6939.65	−0.302710
$808$	−4996.63	−0.217550
$809$	19702.9	0.856262	0.428131	−	0.903717i	$-0.359172\pi$
$809$	19702.9	0.856262	0.428131	+	0.903717i	$0.359172\pi$
$810$	52213.8	2.26495
$811$	−22750.1	−0.985036	−0.492518	−	0.870302i	$-0.663923\pi$
$811$	−22750.1	−0.985036	−0.492518	+	0.870302i	$0.663923\pi$
$812$	−3264.10	−0.141068
$813$	5961.45	0.257167
$814$	−7079.63	−0.304841
$815$	41803.1	1.79668
$816$	1229.34	0.0527396
$817$	38450.0	1.64651
$818$	17978.8	0.768478
$819$	−116197.	−4.95758
$820$	−14686.1	−0.625439
$821$	−33257.5	−1.41376	−0.706880	−	0.707334i	$-0.749898\pi$
$821$	−33257.5	−1.41376	−0.706880	+	0.707334i	$0.749898\pi$
$822$	24129.4	1.02386
$823$	−14478.2	−0.613217	−0.306608	−	0.951836i	$-0.599194\pi$
$823$	−14478.2	−0.613217	−0.306608	+	0.951836i	$0.599194\pi$
$824$	−5844.94	−0.247109
$825$	−22118.3	−0.933405
$826$	2717.86	0.114487
$827$	43047.2	1.81004	0.905018	−	0.425374i	$-0.139858\pi$
$827$	43047.2	1.81004	0.905018	+	0.425374i	$0.139858\pi$
$828$	5916.73	0.248334
$829$	42326.9	1.77331	0.886655	−	0.462432i	$-0.153023\pi$
$829$	42326.9	1.77331	0.886655	+	0.462432i	$0.153023\pi$
$830$	9962.53	0.416632
$831$	80523.9	3.36143
$832$	−4109.38	−0.171234
$833$	−3608.55	−0.150094
$834$	−18643.1	−0.774052
$835$	−9458.78	−0.392017
$836$	−8564.52	−0.354318
$837$	24866.8	1.02691
$838$	6884.52	0.283797
$839$	6983.54	0.287364	0.143682	−	0.989624i	$-0.454106\pi$
$839$	6983.54	0.287364	0.143682	+	0.989624i	$0.454106\pi$
$840$	−33615.0	−1.38075
$841$	841.000	0.0344828
$842$	20236.0	0.828240
$843$	−15017.2	−0.613545
$844$	−6712.21	−0.273749
$845$	−30094.3	−1.22518
$846$	−28777.7	−1.16950
$847$	−26842.4	−1.08892
$848$	8310.58	0.336541
$849$	−36966.1	−1.49431
$850$	1916.87	0.0773507
$851$	4192.73	0.168890
$852$	−16893.1	−0.679283
$853$	7856.22	0.315348	0.157674	−	0.987491i	$-0.449601\pi$
$853$	7856.22	0.315348	0.157674	+	0.987491i	$0.449601\pi$
$854$	19793.8	0.793125
$855$	110815.	4.43250
$856$	−345.917	−0.0138121
$857$	−29936.6	−1.19325	−0.596624	−	0.802521i	$-0.703492\pi$
$857$	−29936.6	−1.19325	−0.596624	+	0.802521i	$0.703492\pi$
$858$	−23828.8	−0.948138
$859$	−13096.8	−0.520208	−0.260104	−	0.965581i	$-0.583757\pi$
$859$	−13096.8	−0.520208	−0.260104	+	0.965581i	$0.583757\pi$
$860$	21797.1	0.864271
$861$	−63174.7	−2.50057
$862$	−14155.4	−0.559320
$863$	−30850.2	−1.21686	−0.608431	−	0.793606i	$-0.708201\pi$
$863$	−30850.2	−1.21686	−0.608431	+	0.793606i	$0.708201\pi$
$864$	11409.5	0.449258
$865$	15463.5	0.607832
$866$	30645.4	1.20251
$867$	46329.6	1.81480
$868$	−7850.00	−0.306966
$869$	−7983.50	−0.311647
$870$	8660.94	0.337510
$871$	−37517.1	−1.45949
$872$	−8208.42	−0.318776
$873$	98196.3	3.80692
$874$	5072.12	0.196301
$875$	2550.54	0.0985419
$876$	−10486.7	−0.404465
$877$	−23167.2	−0.892019	−0.446010	−	0.895028i	$-0.647155\pi$
$877$	−23167.2	−0.892019	−0.446010	+	0.895028i	$0.647155\pi$
$878$	−26618.2	−1.02314
$879$	62428.8	2.39553
$880$	−4855.16	−0.185986
$881$	−41244.2	−1.57725	−0.788623	−	0.614877i	$-0.789206\pi$
$881$	−41244.2	−1.57725	−0.788623	+	0.614877i	$0.789206\pi$
$882$	−57725.7	−2.20377
$883$	10931.7	0.416625	0.208312	−	0.978062i	$-0.433203\pi$
$883$	10931.7	0.416625	0.208312	+	0.978062i	$0.433203\pi$
$884$	2065.11	0.0785715
$885$	−7211.54	−0.273913
$886$	−20632.1	−0.782337
$887$	−13828.2	−0.523455	−0.261728	−	0.965142i	$-0.584292\pi$
$887$	−13828.2	−0.523455	−0.261728	+	0.965142i	$0.584292\pi$
$888$	13935.5	0.526628
$889$	−25039.0	−0.944637
$890$	−49977.2	−1.88229
$891$	32441.0	1.21977
$892$	−3957.55	−0.148552
$893$	−24669.7	−0.924457
$894$	2465.02	0.0922177
$895$	63896.2	2.38639
$896$	−3601.77	−0.134293
$897$	14112.0	0.525291
$898$	−8428.83	−0.313222
$899$	2022.56	0.0750348
$900$	30664.0	1.13570
$901$	−4176.37	−0.154423
$902$	−9124.61	−0.336825
$903$	93763.8	3.45544
$904$	17235.8	0.634129
$905$	20593.3	0.756403
$906$	−3246.56	−0.119050
$907$	53559.0	1.96075	0.980374	−	0.197147i	$-0.0631675\pi$
$907$	53559.0	1.96075	0.980374	+	0.197147i	$0.0631675\pi$
$908$	3201.44	0.117008
$909$	40168.1	1.46567
$910$	−56468.2	−2.05704
$911$	−28393.3	−1.03261	−0.516307	−	0.856404i	$-0.672694\pi$
$911$	−28393.3	−1.03261	−0.516307	+	0.856404i	$0.672694\pi$
$912$	16858.4	0.612102
$913$	6189.83	0.224374
$914$	−27146.8	−0.982425
$915$	−52520.6	−1.89757
$916$	11381.3	0.410534
$917$	55910.8	2.01345
$918$	−5733.69	−0.206144
$919$	−8855.37	−0.317858	−0.158929	−	0.987290i	$-0.550804\pi$
$919$	−8855.37	−0.317858	−0.158929	+	0.987290i	$0.550804\pi$
$920$	2875.35	0.103041
$921$	14740.0	0.527361
$922$	8801.41	0.314381
$923$	−28378.0	−1.01200
$924$	−20885.3	−0.743590
$925$	21729.2	0.772381
$926$	−20095.3	−0.713144
$927$	46987.7	1.66481
$928$	928.000	0.0328266
$929$	2354.18	0.0831413	0.0415706	−	0.999136i	$-0.486764\pi$
$929$	2354.18	0.0831413	0.0415706	+	0.999136i	$0.486764\pi$
$930$	20829.1	0.734424
$931$	−49485.3	−1.74202
$932$	16569.8	0.582364
$933$	96302.1	3.37920
$934$	−18489.3	−0.647740
$935$	2439.90	0.0853403
$936$	33035.4	1.15363
$937$	42449.0	1.47999	0.739994	−	0.672614i	$-0.234828\pi$
$937$	42449.0	1.47999	0.739994	+	0.672614i	$0.234828\pi$
$938$	−32882.8	−1.14463
$939$	100393.	3.48902
$940$	−13985.1	−0.485259
$941$	−35120.2	−1.21667	−0.608334	−	0.793681i	$-0.708162\pi$
$941$	−35120.2	−1.21667	−0.608334	+	0.793681i	$0.708162\pi$
$942$	59602.4	2.06152
$943$	5403.82	0.186609
$944$	−772.700	−0.0266411
$945$	156781.	5.39693
$946$	13542.7	0.465447
$947$	−30948.3	−1.06197	−0.530985	−	0.847381i	$-0.678178\pi$
$947$	−30948.3	−1.06197	−0.530985	+	0.847381i	$0.678178\pi$
$948$	15714.7	0.538386
$949$	−17616.1	−0.602573
$950$	26286.7	0.897741
$951$	23123.8	0.788476
$952$	1810.02	0.0616208
$953$	−389.436	−0.0132372	−0.00661861	−	0.999978i	$-0.502107\pi$
$953$	−389.436	−0.0132372	−0.00661861	+	0.999978i	$0.502107\pi$
$954$	−66809.1	−2.26732
$955$	−75287.7	−2.55105
$956$	20364.2	0.688939
$957$	5381.14	0.181763
$958$	29907.3	1.00862
$959$	35526.9	1.19627
$960$	9556.90	0.321300
$961$	−24926.8	−0.836724
$962$	23409.7	0.784572
$963$	2780.84	0.0930542
$964$	−26804.4	−0.895552
$965$	−61527.1	−2.05246
$966$	12368.8	0.411967
$967$	−40683.2	−1.35293	−0.676465	−	0.736474i	$-0.736489\pi$
$967$	−40683.2	−1.35293	−0.676465	+	0.736474i	$0.736489\pi$
$968$	7631.43	0.253392
$969$	−8471.96	−0.280865
$970$	47720.4	1.57960
$971$	45051.2	1.48894	0.744470	−	0.667656i	$-0.232702\pi$
$971$	45051.2	1.48894	0.744470	+	0.667656i	$0.232702\pi$
$972$	−25349.8	−0.836517
$973$	−27449.2	−0.904400
$974$	5926.88	0.194979
$975$	73136.8	2.40231
$976$	−5627.46	−0.184560
$977$	−27823.4	−0.911103	−0.455552	−	0.890209i	$-0.650558\pi$
$977$	−27823.4	−0.911103	−0.455552	+	0.890209i	$0.650558\pi$
$978$	−51124.7	−1.67156
$979$	−31051.4	−1.01369
$980$	−28052.9	−0.914404
$981$	65987.8	2.14763
$982$	10439.0	0.339227
$983$	27893.1	0.905036	0.452518	−	0.891755i	$-0.350526\pi$
$983$	27893.1	0.905036	0.452518	+	0.891755i	$0.350526\pi$
$984$	17960.9	0.581882
$985$	−75148.6	−2.43090
$986$	−466.354	−0.0150626
$987$	−60159.3	−1.94011
$988$	28319.6	0.911911
$989$	−8020.35	−0.257869
$990$	39030.8	1.25301
$991$	−31864.2	−1.02139	−0.510696	−	0.859762i	$-0.670612\pi$
$991$	−31864.2	−1.02139	−0.510696	+	0.859762i	$0.670612\pi$
$992$	2231.79	0.0714310
$993$	69770.0	2.22969
$994$	−24872.6	−0.793672
$995$	35931.8	1.14484
$996$	−12184.1	−0.387617
$997$	−30296.6	−0.962391	−0.481196	−	0.876613i	$-0.659797\pi$
$997$	−30296.6	−0.962391	−0.481196	+	0.876613i	$0.659797\pi$
$998$	4101.50	0.130091
$999$	−64995.9	−2.05844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1334.4.a.d.1.1	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1334.4.a.d.1.1	✓	19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1334 = 2 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1334.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -9.55575 q^{3} +4.00000 q^{4} -15.6269 q^{5} +19.1115 q^{6} +28.1388 q^{7} -8.00000 q^{8} +64.3123 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -9.55575 q^{3} +4.00000 q^{4} -15.6269 q^{5} +19.1115 q^{6} +28.1388 q^{7} -8.00000 q^{8} +64.3123 q^{9} +31.2538 q^{10} +19.4183 q^{11} -38.2230 q^{12} -64.2090 q^{13} -56.2776 q^{14} +149.327 q^{15} +16.0000 q^{16} -8.04058 q^{17} -128.625 q^{18} -110.263 q^{19} -62.5076 q^{20} -268.887 q^{21} -38.8366 q^{22} +23.0000 q^{23} +76.4460 q^{24} +119.200 q^{25} +128.418 q^{26} -356.547 q^{27} +112.555 q^{28} -29.0000 q^{29} -298.653 q^{30} -69.7435 q^{31} -32.0000 q^{32} -185.556 q^{33} +16.0812 q^{34} -439.722 q^{35} +257.249 q^{36} +182.293 q^{37} +220.527 q^{38} +613.565 q^{39} +125.015 q^{40} +234.949 q^{41} +537.774 q^{42} -348.711 q^{43} +77.6732 q^{44} -1005.00 q^{45} -46.0000 q^{46} +223.734 q^{47} -152.892 q^{48} +448.792 q^{49} -238.399 q^{50} +76.8338 q^{51} -256.836 q^{52} +519.412 q^{53} +713.094 q^{54} -303.448 q^{55} -225.110 q^{56} +1053.65 q^{57} +58.0000 q^{58} -48.2937 q^{59} +597.306 q^{60} -351.716 q^{61} +139.487 q^{62} +1809.67 q^{63} +64.0000 q^{64} +1003.39 q^{65} +371.113 q^{66} +584.297 q^{67} -32.1623 q^{68} -219.782 q^{69} +879.444 q^{70} +441.963 q^{71} -514.498 q^{72} +274.355 q^{73} -364.585 q^{74} -1139.04 q^{75} -441.054 q^{76} +546.408 q^{77} -1227.13 q^{78} -411.132 q^{79} -250.030 q^{80} +1670.64 q^{81} -469.897 q^{82} +318.762 q^{83} -1075.55 q^{84} +125.649 q^{85} +697.421 q^{86} +277.117 q^{87} -155.346 q^{88} -1599.08 q^{89} +2010.00 q^{90} -1806.76 q^{91} +92.0000 q^{92} +666.452 q^{93} -447.469 q^{94} +1723.07 q^{95} +305.784 q^{96} +1526.87 q^{97} -897.583 q^{98} +1248.84 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 38 q^{2} - 5 q^{3} + 76 q^{4} + 10 q^{6} - 13 q^{7} - 152 q^{8} + 166 q^{9}+O(q^{10}) \) 19 * q - 38 * q^2 - 5 * q^3 + 76 * q^4 + 10 * q^6 - 13 * q^7 - 152 * q^8 + 166 * q^9	\( 19 q - 38 q^{2} - 5 q^{3} + 76 q^{4} + 10 q^{6} - 13 q^{7} - 152 q^{8} + 166 q^{9} + 30 q^{11} - 20 q^{12} - 151 q^{13} + 26 q^{14} + 2 q^{15} + 304 q^{16} - 56 q^{17} - 332 q^{18} - 266 q^{19} - 239 q^{21} - 60 q^{22} + 437 q^{23} + 40 q^{24} + 395 q^{25} + 302 q^{26} - 68 q^{27} - 52 q^{28} - 551 q^{29} - 4 q^{30} - 622 q^{31} - 608 q^{32} - 291 q^{33} + 112 q^{34} + 21 q^{35} + 664 q^{36} - 183 q^{37} + 532 q^{38} - 212 q^{39} - 395 q^{41} + 478 q^{42} - 291 q^{43} + 120 q^{44} - 1018 q^{45} - 874 q^{46} - 241 q^{47} - 80 q^{48} + 1394 q^{49} - 790 q^{50} - 184 q^{51} - 604 q^{52} - 631 q^{53} + 136 q^{54} - 1765 q^{55} + 104 q^{56} + 144 q^{57} + 1102 q^{58} - 457 q^{59} + 8 q^{60} - 630 q^{61} + 1244 q^{62} + 125 q^{63} + 1216 q^{64} + 137 q^{65} + 582 q^{66} + 1421 q^{67} - 224 q^{68} - 115 q^{69} - 42 q^{70} + 983 q^{71} - 1328 q^{72} - 2532 q^{73} + 366 q^{74} - 1081 q^{75} - 1064 q^{76} - 595 q^{77} + 424 q^{78} - 4361 q^{79} - 721 q^{81} + 790 q^{82} + 1422 q^{83} - 956 q^{84} + 722 q^{85} + 582 q^{86} + 145 q^{87} - 240 q^{88} - 580 q^{89} + 2036 q^{90} - 447 q^{91} + 1748 q^{92} + 2106 q^{93} + 482 q^{94} - 310 q^{95} + 160 q^{96} - 2016 q^{97} - 2788 q^{98} - 1972 q^{99}+O(q^{100}) \) 19 * q - 38 * q^2 - 5 * q^3 + 76 * q^4 + 10 * q^6 - 13 * q^7 - 152 * q^8 + 166 * q^9 + 30 * q^11 - 20 * q^12 - 151 * q^13 + 26 * q^14 + 2 * q^15 + 304 * q^16 - 56 * q^17 - 332 * q^18 - 266 * q^19 - 239 * q^21 - 60 * q^22 + 437 * q^23 + 40 * q^24 + 395 * q^25 + 302 * q^26 - 68 * q^27 - 52 * q^28 - 551 * q^29 - 4 * q^30 - 622 * q^31 - 608 * q^32 - 291 * q^33 + 112 * q^34 + 21 * q^35 + 664 * q^36 - 183 * q^37 + 532 * q^38 - 212 * q^39 - 395 * q^41 + 478 * q^42 - 291 * q^43 + 120 * q^44 - 1018 * q^45 - 874 * q^46 - 241 * q^47 - 80 * q^48 + 1394 * q^49 - 790 * q^50 - 184 * q^51 - 604 * q^52 - 631 * q^53 + 136 * q^54 - 1765 * q^55 + 104 * q^56 + 144 * q^57 + 1102 * q^58 - 457 * q^59 + 8 * q^60 - 630 * q^61 + 1244 * q^62 + 125 * q^63 + 1216 * q^64 + 137 * q^65 + 582 * q^66 + 1421 * q^67 - 224 * q^68 - 115 * q^69 - 42 * q^70 + 983 * q^71 - 1328 * q^72 - 2532 * q^73 + 366 * q^74 - 1081 * q^75 - 1064 * q^76 - 595 * q^77 + 424 * q^78 - 4361 * q^79 - 721 * q^81 + 790 * q^82 + 1422 * q^83 - 956 * q^84 + 722 * q^85 + 582 * q^86 + 145 * q^87 - 240 * q^88 - 580 * q^89 + 2036 * q^90 - 447 * q^91 + 1748 * q^92 + 2106 * q^93 + 482 * q^94 - 310 * q^95 + 160 * q^96 - 2016 * q^97 - 2788 * q^98 - 1972 * q^99

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