LMFDB - Embedded newform 1045.4.a.d.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.65397 q^{2} +3.36009 q^{3} +5.35150 q^{4} +5.00000 q^{5} +12.2777 q^{6} -1.36266 q^{7} -9.67753 q^{8} -15.7098 q^{9} +O(q^{10})$	\(q+3.65397 q^{2} +3.36009 q^{3} +5.35150 q^{4} +5.00000 q^{5} +12.2777 q^{6} -1.36266 q^{7} -9.67753 q^{8} -15.7098 q^{9} +18.2699 q^{10} -11.0000 q^{11} +17.9815 q^{12} -71.5168 q^{13} -4.97911 q^{14} +16.8004 q^{15} -78.1734 q^{16} +55.9067 q^{17} -57.4032 q^{18} -19.0000 q^{19} +26.7575 q^{20} -4.57865 q^{21} -40.1937 q^{22} +125.854 q^{23} -32.5174 q^{24} +25.0000 q^{25} -261.320 q^{26} -143.509 q^{27} -7.29227 q^{28} -244.581 q^{29} +61.3883 q^{30} +59.2492 q^{31} -208.223 q^{32} -36.9610 q^{33} +204.282 q^{34} -6.81329 q^{35} -84.0711 q^{36} -106.327 q^{37} -69.4254 q^{38} -240.303 q^{39} -48.3877 q^{40} -523.023 q^{41} -16.7303 q^{42} +286.782 q^{43} -58.8665 q^{44} -78.5490 q^{45} +459.867 q^{46} -437.372 q^{47} -262.670 q^{48} -341.143 q^{49} +91.3493 q^{50} +187.852 q^{51} -382.723 q^{52} +235.160 q^{53} -524.377 q^{54} -55.0000 q^{55} +13.1872 q^{56} -63.8417 q^{57} -893.692 q^{58} -613.449 q^{59} +89.9076 q^{60} -126.914 q^{61} +216.495 q^{62} +21.4071 q^{63} -135.454 q^{64} -357.584 q^{65} -135.054 q^{66} +585.333 q^{67} +299.185 q^{68} +422.880 q^{69} -24.8956 q^{70} +737.432 q^{71} +152.032 q^{72} +152.003 q^{73} -388.515 q^{74} +84.0022 q^{75} -101.679 q^{76} +14.9892 q^{77} -878.060 q^{78} +457.791 q^{79} -390.867 q^{80} -58.0374 q^{81} -1911.11 q^{82} +1293.95 q^{83} -24.5027 q^{84} +279.534 q^{85} +1047.89 q^{86} -821.814 q^{87} +106.453 q^{88} -1090.77 q^{89} -287.016 q^{90} +97.4530 q^{91} +673.508 q^{92} +199.082 q^{93} -1598.15 q^{94} -95.0000 q^{95} -699.649 q^{96} -1047.37 q^{97} -1246.53 q^{98} +172.808 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10}) $ 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9 - 20 * q^10 - 242 * q^11 - 196 * q^12 - q^13 - 63 * q^14 - 105 * q^15 + 6 * q^16 + 187 * q^17 - 361 * q^18 - 418 * q^19 + 370 * q^20 - 107 * q^21 + 44 * q^22 - 361 * q^23 + 208 * q^24 + 550 * q^25 - 365 * q^26 - 1467 * q^27 - 773 * q^28 - 319 * q^29 - 45 * q^30 - 402 * q^31 - 873 * q^32 + 231 * q^33 - 717 * q^34 - 205 * q^35 + 725 * q^36 - 838 * q^37 + 76 * q^38 - 607 * q^39 - 390 * q^40 - 392 * q^41 - 1350 * q^42 - 610 * q^43 - 814 * q^44 + 1045 * q^45 - 605 * q^46 - 1866 * q^47 - 1637 * q^48 + 379 * q^49 - 100 * q^50 - 2659 * q^51 - 638 * q^52 - 1303 * q^53 + 2338 * q^54 - 1210 * q^55 + 727 * q^56 + 399 * q^57 + 44 * q^58 - 2417 * q^59 - 980 * q^60 + 918 * q^61 - 1634 * q^62 - 374 * q^63 - 1716 * q^64 - 5 * q^65 + 99 * q^66 - 2339 * q^67 + 4940 * q^68 + 127 * q^69 - 315 * q^70 - 2370 * q^71 - 3306 * q^72 + 2207 * q^73 + 2051 * q^74 - 525 * q^75 - 1406 * q^76 + 451 * q^77 + 1380 * q^78 + 586 * q^79 + 30 * q^80 + 1950 * q^81 - 1566 * q^82 - 2870 * q^83 + 3076 * q^84 + 935 * q^85 - 1246 * q^86 - 1811 * q^87 + 858 * q^88 - 1768 * q^89 - 1805 * q^90 - 2195 * q^91 - 6728 * q^92 - 2916 * q^93 + 672 * q^94 - 2090 * q^95 + 6022 * q^96 - 4022 * q^97 + 1162 * q^98 - 2299 * 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 $

$2$	3.65397	1.29187	0.645937	−	0.763391i	$-0.276467\pi$
$2$	3.65397	1.29187	0.645937	+	0.763391i	$0.276467\pi$
$3$	3.36009	0.646649	0.323325	−	0.946288i	$-0.395199\pi$
$3$	3.36009	0.646649	0.323325	+	0.946288i	$0.395199\pi$
$4$	5.35150	0.668938
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	12.2777	0.835389
$7$	−1.36266	−0.0735766	−0.0367883	−	0.999323i	$-0.511713\pi$
$7$	−1.36266	−0.0735766	−0.0367883	+	0.999323i	$0.511713\pi$
$8$	−9.67753	−0.427690
$9$	−15.7098	−0.581845
$10$	18.2699	0.577744
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	17.9815	0.432568
$13$	−71.5168	−1.52578	−0.762892	−	0.646526i	$-0.776221\pi$
$13$	−71.5168	−1.52578	−0.762892	+	0.646526i	$0.776221\pi$
$14$	−4.97911	−0.0950517
$15$	16.8004	0.289190
$16$	−78.1734	−1.22146
$17$	55.9067	0.797610	0.398805	−	0.917036i	$-0.369425\pi$
$17$	55.9067	0.797610	0.398805	+	0.917036i	$0.369425\pi$
$18$	−57.4032	−0.751670
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	26.7575	0.299158
$21$	−4.57865	−0.0475783
$22$	−40.1937	−0.389515
$23$	125.854	1.14097	0.570486	−	0.821307i	$-0.306755\pi$
$23$	125.854	1.14097	0.570486	+	0.821307i	$0.306755\pi$
$24$	−32.5174	−0.276566
$25$	25.0000	0.200000
$26$	−261.320	−1.97112
$27$	−143.509	−1.02290
$28$	−7.29227	−0.0492182
$29$	−244.581	−1.56612	−0.783062	−	0.621944i	$-0.786343\pi$
$29$	−244.581	−1.56612	−0.783062	+	0.621944i	$0.786343\pi$
$30$	61.3883	0.373597
$31$	59.2492	0.343273	0.171637	−	0.985160i	$-0.445095\pi$
$31$	59.2492	0.343273	0.171637	+	0.985160i	$0.445095\pi$
$32$	−208.223	−1.15028
$33$	−36.9610	−0.194972
$34$	204.282	1.03041
$35$	−6.81329	−0.0329045
$36$	−84.0711	−0.389218
$37$	−106.327	−0.472432	−0.236216	−	0.971701i	$-0.575907\pi$
$37$	−106.327	−0.472432	−0.236216	+	0.971701i	$0.575907\pi$
$38$	−69.4254	−0.296376
$39$	−240.303	−0.986648
$40$	−48.3877	−0.191269
$41$	−523.023	−1.99226	−0.996128	−	0.0879158i	$-0.971979\pi$
$41$	−523.023	−1.99226	−0.996128	+	0.0879158i	$0.971979\pi$
$42$	−16.7303	−0.0614651
$43$	286.782	1.01707	0.508533	−	0.861042i	$-0.330188\pi$
$43$	286.782	1.01707	0.508533	+	0.861042i	$0.330188\pi$
$44$	−58.8665	−0.201692
$45$	−78.5490	−0.260209
$46$	459.867	1.47399
$47$	−437.372	−1.35739	−0.678695	−	0.734420i	$-0.737454\pi$
$47$	−437.372	−1.35739	−0.678695	+	0.734420i	$0.737454\pi$
$48$	−262.670	−0.789856
$49$	−341.143	−0.994586
$50$	91.3493	0.258375
$51$	187.852	0.515774
$52$	−382.723	−1.02066
$53$	235.160	0.609466	0.304733	−	0.952438i	$-0.401433\pi$
$53$	235.160	0.609466	0.304733	+	0.952438i	$0.401433\pi$
$54$	−524.377	−1.32146
$55$	−55.0000	−0.134840
$56$	13.1872	0.0314680
$57$	−63.8417	−0.148352
$58$	−893.692	−2.02323
$59$	−613.449	−1.35363	−0.676816	−	0.736152i	$-0.736641\pi$
$59$	−613.449	−1.35363	−0.676816	+	0.736152i	$0.736641\pi$
$60$	89.9076	0.193450
$61$	−126.914	−0.266388	−0.133194	−	0.991090i	$-0.542523\pi$
$61$	−126.914	−0.266388	−0.133194	+	0.991090i	$0.542523\pi$
$62$	216.495	0.443465
$63$	21.4071	0.0428102
$64$	−135.454	−0.264559
$65$	−357.584	−0.682352
$66$	−135.054	−0.251879
$67$	585.333	1.06731	0.533655	−	0.845702i	$-0.320818\pi$
$67$	585.333	1.06731	0.533655	+	0.845702i	$0.320818\pi$
$68$	299.185	0.533552
$69$	422.880	0.737809
$70$	−24.8956	−0.0425084
$71$	737.432	1.23264	0.616318	−	0.787498i	$-0.288624\pi$
$71$	737.432	1.23264	0.616318	+	0.787498i	$0.288624\pi$
$72$	152.032	0.248849
$73$	152.003	0.243707	0.121854	−	0.992548i	$-0.461116\pi$
$73$	152.003	0.243707	0.121854	+	0.992548i	$0.461116\pi$
$74$	−388.515	−0.610323
$75$	84.0022	0.129330
$76$	−101.679	−0.153465
$77$	14.9892	0.0221842
$78$	−878.060	−1.27462
$79$	457.791	0.651969	0.325984	−	0.945375i	$-0.394304\pi$
$79$	457.791	0.651969	0.325984	+	0.945375i	$0.394304\pi$
$80$	−390.867	−0.546254
$81$	−58.0374	−0.0796123
$82$	−1911.11	−2.57374
$83$	1293.95	1.71119	0.855596	−	0.517644i	$-0.173191\pi$
$83$	1293.95	1.71119	0.855596	+	0.517644i	$0.173191\pi$
$84$	−24.5027	−0.0318269
$85$	279.534	0.356702
$86$	1047.89	1.31392
$87$	−821.814	−1.01273
$88$	106.453	0.128954
$89$	−1090.77	−1.29912	−0.649558	−	0.760312i	$-0.725046\pi$
$89$	−1090.77	−1.29912	−0.649558	+	0.760312i	$0.725046\pi$
$90$	−287.016	−0.336157
$91$	97.4530	0.112262
$92$	673.508	0.763239
$93$	199.082	0.221977
$94$	−1598.15	−1.75358
$95$	−95.0000	−0.102598
$96$	−699.649	−0.743829
$97$	−1047.37	−1.09633	−0.548167	−	0.836369i	$-0.684674\pi$
$97$	−1047.37	−1.09633	−0.548167	+	0.836369i	$0.684674\pi$
$98$	−1246.53	−1.28488
$99$	172.808	0.175433
$100$	133.788	0.133788
$101$	1536.06	1.51330	0.756650	−	0.653820i	$-0.226835\pi$
$101$	1536.06	1.51330	0.756650	+	0.653820i	$0.226835\pi$
$102$	686.404	0.666315
$103$	−948.160	−0.907039	−0.453520	−	0.891246i	$-0.649832\pi$
$103$	−948.160	−0.907039	−0.453520	+	0.891246i	$0.649832\pi$
$104$	692.106	0.652563
$105$	−22.8933	−0.0212777
$106$	859.267	0.787353
$107$	1612.43	1.45682	0.728411	−	0.685141i	$-0.240259\pi$
$107$	1612.43	1.45682	0.728411	+	0.685141i	$0.240259\pi$
$108$	−767.987	−0.684256
$109$	−682.057	−0.599351	−0.299675	−	0.954041i	$-0.596878\pi$
$109$	−682.057	−0.599351	−0.299675	+	0.954041i	$0.596878\pi$
$110$	−200.968	−0.174196
$111$	−357.267	−0.305498
$112$	106.524	0.0898709
$113$	7.28207	0.00606230	0.00303115	−	0.999995i	$-0.499035\pi$
$113$	7.28207	0.00606230	0.00303115	+	0.999995i	$0.499035\pi$
$114$	−233.276	−0.191651
$115$	629.270	0.510258
$116$	−1308.88	−1.04764
$117$	1123.52	0.887769
$118$	−2241.53	−1.74872
$119$	−76.1817	−0.0586855
$120$	−162.587	−0.123684
$121$	121.000	0.0909091
$122$	−463.740	−0.344140
$123$	−1757.40	−1.28829
$124$	317.072	0.229628
$125$	125.000	0.0894427
$126$	78.2209	0.0553053
$127$	−1785.00	−1.24719	−0.623596	−	0.781747i	$-0.714329\pi$
$127$	−1785.00	−1.24719	−0.623596	+	0.781747i	$0.714329\pi$
$128$	1170.84	0.808505
$129$	963.613	0.657685
$130$	−1306.60	−0.881512
$131$	−1291.33	−0.861254	−0.430627	−	0.902530i	$-0.641707\pi$
$131$	−1291.33	−0.861254	−0.430627	+	0.902530i	$0.641707\pi$
$132$	−197.797	−0.130424
$133$	25.8905	0.0168796
$134$	2138.79	1.37883
$135$	−717.544	−0.457454
$136$	−541.039	−0.341130
$137$	−434.352	−0.270870	−0.135435	−	0.990786i	$-0.543243\pi$
$137$	−434.352	−0.270870	−0.135435	+	0.990786i	$0.543243\pi$
$138$	1545.19	0.953156
$139$	1899.56	1.15913	0.579565	−	0.814926i	$-0.303223\pi$
$139$	1899.56	1.15913	0.579565	+	0.814926i	$0.303223\pi$
$140$	−36.4613	−0.0220110
$141$	−1469.61	−0.877755
$142$	2694.56	1.59241
$143$	786.685	0.460041
$144$	1228.09	0.710700
$145$	−1222.91	−0.700392
$146$	555.415	0.314839
$147$	−1146.27	−0.643149
$148$	−569.008	−0.316028
$149$	486.320	0.267389	0.133694	−	0.991023i	$-0.457316\pi$
$149$	486.320	0.267389	0.133694	+	0.991023i	$0.457316\pi$
$150$	306.942	0.167078
$151$	−1846.25	−0.995005	−0.497503	−	0.867462i	$-0.665750\pi$
$151$	−1846.25	−0.995005	−0.497503	+	0.867462i	$0.665750\pi$
$152$	183.873	0.0981189
$153$	−878.284	−0.464085
$154$	54.7702	0.0286592
$155$	296.246	0.153516
$156$	−1285.98	−0.660006
$157$	−1380.31	−0.701660	−0.350830	−	0.936439i	$-0.614100\pi$
$157$	−1380.31	−0.701660	−0.350830	+	0.936439i	$0.614100\pi$
$158$	1672.76	0.842261
$159$	790.158	0.394111
$160$	−1041.12	−0.514422
$161$	−171.496	−0.0839489
$162$	−212.067	−0.102849
$163$	3201.15	1.53824	0.769121	−	0.639103i	$-0.220694\pi$
$163$	3201.15	1.53824	0.769121	+	0.639103i	$0.220694\pi$
$164$	−2798.96	−1.33270
$165$	−184.805	−0.0871942
$166$	4728.04	2.21064
$167$	1727.66	0.800541	0.400270	−	0.916397i	$-0.368916\pi$
$167$	1727.66	0.800541	0.400270	+	0.916397i	$0.368916\pi$
$168$	44.3100	0.0203488
$169$	2917.66	1.32802
$170$	1021.41	0.460814
$171$	298.486	0.133484
$172$	1534.71	0.680354
$173$	−1366.74	−0.600645	−0.300322	−	0.953838i	$-0.597094\pi$
$173$	−1366.74	−0.600645	−0.300322	+	0.953838i	$0.597094\pi$
$174$	−3002.89	−1.30832
$175$	−34.0665	−0.0147153
$176$	859.908	0.368284
$177$	−2061.24	−0.875325
$178$	−3985.64	−1.67829
$179$	−4087.07	−1.70660	−0.853301	−	0.521418i	$-0.825403\pi$
$179$	−4087.07	−1.70660	−0.853301	+	0.521418i	$0.825403\pi$
$180$	−420.355	−0.174064
$181$	−3149.66	−1.29344	−0.646720	−	0.762728i	$-0.723860\pi$
$181$	−3149.66	−1.29344	−0.646720	+	0.762728i	$0.723860\pi$
$182$	356.090	0.145028
$183$	−426.443	−0.172260
$184$	−1217.96	−0.487983
$185$	−531.633	−0.211278
$186$	727.441	0.286767
$187$	−614.974	−0.240488
$188$	−2340.60	−0.908009
$189$	195.553	0.0752614
$190$	−347.127	−0.132543
$191$	341.566	0.129397	0.0646985	−	0.997905i	$-0.479391\pi$
$191$	341.566	0.129397	0.0646985	+	0.997905i	$0.479391\pi$
$192$	−455.138	−0.171077
$193$	−2888.68	−1.07737	−0.538683	−	0.842509i	$-0.681078\pi$
$193$	−2888.68	−1.07737	−0.538683	+	0.842509i	$0.681078\pi$
$194$	−3827.06	−1.41633
$195$	−1201.51	−0.441242
$196$	−1825.63	−0.665317
$197$	1031.47	0.373041	0.186521	−	0.982451i	$-0.440279\pi$
$197$	1031.47	0.373041	0.186521	+	0.982451i	$0.440279\pi$
$198$	631.435	0.226637
$199$	−1249.43	−0.445075	−0.222537	−	0.974924i	$-0.571434\pi$
$199$	−1249.43	−0.445075	−0.222537	+	0.974924i	$0.571434\pi$
$200$	−241.938	−0.0855381
$201$	1966.77	0.690176
$202$	5612.70	1.95499
$203$	333.280	0.115230
$204$	1005.29	0.345021
$205$	−2615.12	−0.890964
$206$	−3464.55	−1.17178
$207$	−1977.14	−0.663868
$208$	5590.72	1.86368
$209$	209.000	0.0691714
$210$	−83.6513	−0.0274880
$211$	1117.74	0.364686	0.182343	−	0.983235i	$-0.441632\pi$
$211$	1117.74	0.364686	0.182343	+	0.983235i	$0.441632\pi$
$212$	1258.46	0.407695
$213$	2477.84	0.797083
$214$	5891.79	1.88203
$215$	1433.91	0.454846
$216$	1388.81	0.437484
$217$	−80.7363	−0.0252569
$218$	−2492.22	−0.774285
$219$	510.744	0.157593
$220$	−294.333	−0.0901996
$221$	−3998.27	−1.21698
$222$	−1305.44	−0.394665
$223$	−670.642	−0.201388	−0.100694	−	0.994917i	$-0.532106\pi$
$223$	−670.642	−0.201388	−0.100694	+	0.994917i	$0.532106\pi$
$224$	283.737	0.0846338
$225$	−392.745	−0.116369
$226$	26.6085	0.00783172
$227$	3463.91	1.01281	0.506405	−	0.862296i	$-0.330974\pi$
$227$	3463.91	1.01281	0.506405	+	0.862296i	$0.330974\pi$
$228$	−341.649	−0.0992380
$229$	3244.29	0.936196	0.468098	−	0.883677i	$-0.344939\pi$
$229$	3244.29	0.936196	0.468098	+	0.883677i	$0.344939\pi$
$230$	2299.33	0.659189
$231$	50.3652	0.0143454
$232$	2366.94	0.669816
$233$	−2702.97	−0.759989	−0.379995	−	0.924989i	$-0.624074\pi$
$233$	−2702.97	−0.759989	−0.379995	+	0.924989i	$0.624074\pi$
$234$	4105.29	1.14689
$235$	−2186.86	−0.607043
$236$	−3282.88	−0.905496
$237$	1538.22	0.421595
$238$	−278.366	−0.0758142
$239$	−707.108	−0.191377	−0.0956883	−	0.995411i	$-0.530505\pi$
$239$	−707.108	−0.191377	−0.0956883	+	0.995411i	$0.530505\pi$
$240$	−1313.35	−0.353234
$241$	4070.10	1.08788	0.543938	−	0.839125i	$-0.316933\pi$
$241$	4070.10	1.08788	0.543938	+	0.839125i	$0.316933\pi$
$242$	442.130	0.117443
$243$	3679.73	0.971418
$244$	−679.181	−0.178197
$245$	−1705.72	−0.444793
$246$	−6421.50	−1.66431
$247$	1358.82	0.350039
$248$	−573.386	−0.146815
$249$	4347.77	1.10654
$250$	456.746	0.115549
$251$	1491.47	0.375063	0.187532	−	0.982259i	$-0.439951\pi$
$251$	1491.47	0.375063	0.187532	+	0.982259i	$0.439951\pi$
$252$	114.560	0.0286373
$253$	−1384.39	−0.344016
$254$	−6522.35	−1.61121
$255$	939.258	0.230661
$256$	5361.85	1.30905
$257$	−74.6875	−0.0181279	−0.00906397	−	0.999959i	$-0.502885\pi$
$257$	−74.6875	−0.0181279	−0.00906397	+	0.999959i	$0.502885\pi$
$258$	3521.01	0.849646
$259$	144.887	0.0347600
$260$	−1913.61	−0.456451
$261$	3842.32	0.911240
$262$	−4718.49	−1.11263
$263$	2587.90	0.606757	0.303378	−	0.952870i	$-0.401885\pi$
$263$	2587.90	0.606757	0.303378	+	0.952870i	$0.401885\pi$
$264$	357.691	0.0833877
$265$	1175.80	0.272561
$266$	94.6031	0.0218064
$267$	−3665.08	−0.840073
$268$	3132.41	0.713965
$269$	−1584.76	−0.359198	−0.179599	−	0.983740i	$-0.557480\pi$
$269$	−1584.76	−0.359198	−0.179599	+	0.983740i	$0.557480\pi$
$270$	−2621.88	−0.590973
$271$	342.084	0.0766794	0.0383397	−	0.999265i	$-0.487793\pi$
$271$	342.084	0.0766794	0.0383397	+	0.999265i	$0.487793\pi$
$272$	−4370.42	−0.974249
$273$	327.451	0.0725942
$274$	−1587.11	−0.349930
$275$	−275.000	−0.0603023
$276$	2263.05	0.493548
$277$	5532.84	1.20013	0.600065	−	0.799951i	$-0.295141\pi$
$277$	5532.84	1.20013	0.600065	+	0.799951i	$0.295141\pi$
$278$	6940.95	1.49745
$279$	−930.793	−0.199732
$280$	65.9358	0.0140729
$281$	1455.87	0.309075	0.154538	−	0.987987i	$-0.450611\pi$
$281$	1455.87	0.309075	0.154538	+	0.987987i	$0.450611\pi$
$282$	−5369.91	−1.13395
$283$	1808.30	0.379831	0.189915	−	0.981800i	$-0.439179\pi$
$283$	1808.30	0.379831	0.189915	+	0.981800i	$0.439179\pi$
$284$	3946.37	0.824556
$285$	−319.208	−0.0663448
$286$	2874.52	0.594315
$287$	712.702	0.146583
$288$	3271.15	0.669285
$289$	−1787.44	−0.363818
$290$	−4468.46	−0.904817
$291$	−3519.26	−0.708944
$292$	813.446	0.163025
$293$	−235.695	−0.0469947	−0.0234973	−	0.999724i	$-0.507480\pi$
$293$	−235.695	−0.0469947	−0.0234973	+	0.999724i	$0.507480\pi$
$294$	−4188.44	−0.830867
$295$	−3067.25	−0.605363
$296$	1028.98	0.202055
$297$	1578.60	0.308416
$298$	1777.00	0.345433
$299$	−9000.67	−1.74088
$300$	449.538	0.0865137
$301$	−390.786	−0.0748323
$302$	−6746.15	−1.28542
$303$	5161.28	0.978574
$304$	1485.30	0.280222
$305$	−634.571	−0.119132
$306$	−3209.22	−0.599539
$307$	−2540.46	−0.472285	−0.236142	−	0.971718i	$-0.575883\pi$
$307$	−2540.46	−0.472285	−0.236142	+	0.971718i	$0.575883\pi$
$308$	80.2150	0.0148398
$309$	−3185.90	−0.586536
$310$	1082.47	0.198324
$311$	7147.77	1.30326	0.651628	−	0.758538i	$-0.274086\pi$
$311$	7147.77	1.30326	0.651628	+	0.758538i	$0.274086\pi$
$312$	2325.54	0.421980
$313$	4302.99	0.777058	0.388529	−	0.921436i	$-0.372983\pi$
$313$	4302.99	0.777058	0.388529	+	0.921436i	$0.372983\pi$
$314$	−5043.60	−0.906456
$315$	107.035	0.0191453
$316$	2449.87	0.436126
$317$	4858.80	0.860875	0.430438	−	0.902620i	$-0.358359\pi$
$317$	4858.80	0.860875	0.430438	+	0.902620i	$0.358359\pi$
$318$	2887.21	0.509141
$319$	2690.39	0.472204
$320$	−677.270	−0.118314
$321$	5417.92	0.942053
$322$	−626.641	−0.108451
$323$	−1062.23	−0.182984
$324$	−310.587	−0.0532557
$325$	−1787.92	−0.305157
$326$	11696.9	1.98721
$327$	−2291.77	−0.387570
$328$	5061.57	0.852069
$329$	595.989	0.0998722
$330$	−675.272	−0.112644
$331$	−3944.03	−0.654935	−0.327468	−	0.944862i	$-0.606195\pi$
$331$	−3944.03	−0.654935	−0.327468	+	0.944862i	$0.606195\pi$
$332$	6924.55	1.14468
$333$	1670.37	0.274882
$334$	6312.82	1.03420
$335$	2926.67	0.477316
$336$	357.929	0.0581150
$337$	−4567.58	−0.738315	−0.369157	−	0.929367i	$-0.620354\pi$
$337$	−4567.58	−0.738315	−0.369157	+	0.929367i	$0.620354\pi$
$338$	10661.0	1.71563
$339$	24.4684	0.00392018
$340$	1495.92	0.238612
$341$	−651.741	−0.103501
$342$	1090.66	0.172445
$343$	932.253	0.146755
$344$	−2775.34	−0.434989
$345$	2114.40	0.329958
$346$	−4994.04	−0.775957
$347$	3717.49	0.575117	0.287558	−	0.957763i	$-0.407156\pi$
$347$	3717.49	0.575117	0.287558	+	0.957763i	$0.407156\pi$
$348$	−4397.94	−0.677455
$349$	−11402.2	−1.74884	−0.874418	−	0.485173i	$-0.838757\pi$
$349$	−11402.2	−1.74884	−0.874418	+	0.485173i	$0.838757\pi$
$350$	−124.478	−0.0190103
$351$	10263.3	1.56072
$352$	2290.46	0.346823
$353$	−5994.36	−0.903817	−0.451909	−	0.892064i	$-0.649257\pi$
$353$	−5994.36	−0.903817	−0.451909	+	0.892064i	$0.649257\pi$
$354$	−7531.72	−1.13081
$355$	3687.16	0.551251
$356$	−5837.26	−0.869028
$357$	−255.977	−0.0379489
$358$	−14934.0	−2.20472
$359$	2837.37	0.417132	0.208566	−	0.978008i	$-0.433120\pi$
$359$	2837.37	0.417132	0.208566	+	0.978008i	$0.433120\pi$
$360$	760.161	0.111289
$361$	361.000	0.0526316
$362$	−11508.8	−1.67096
$363$	406.571	0.0587863
$364$	521.520	0.0750964
$365$	760.016	0.108989
$366$	−1558.21	−0.222538
$367$	−8558.98	−1.21737	−0.608685	−	0.793412i	$-0.708303\pi$
$367$	−8558.98	−1.21737	−0.608685	+	0.793412i	$0.708303\pi$
$368$	−9838.43	−1.39365
$369$	8216.59	1.15918
$370$	−1942.57	−0.272945
$371$	−320.443	−0.0448424
$372$	1065.39	0.148489
$373$	−2255.02	−0.313031	−0.156515	−	0.987676i	$-0.550026\pi$
$373$	−2255.02	−0.313031	−0.156515	+	0.987676i	$0.550026\pi$
$374$	−2247.10	−0.310681
$375$	420.011	0.0578381
$376$	4232.68	0.580543
$377$	17491.7	2.38957
$378$	714.546	0.0972283
$379$	−9184.01	−1.24472	−0.622362	−	0.782729i	$-0.713827\pi$
$379$	−9184.01	−1.24472	−0.622362	+	0.782729i	$0.713827\pi$
$380$	−508.393	−0.0686316
$381$	−5997.77	−0.806496
$382$	1248.07	0.167165
$383$	6361.02	0.848649	0.424325	−	0.905510i	$-0.360512\pi$
$383$	6361.02	0.848649	0.424325	+	0.905510i	$0.360512\pi$
$384$	3934.13	0.522819
$385$	74.9462	0.00992107
$386$	−10555.1	−1.39182
$387$	−4505.29	−0.591774
$388$	−5605.01	−0.733380
$389$	2334.60	0.304291	0.152145	−	0.988358i	$-0.451382\pi$
$389$	2334.60	0.304291	0.152145	+	0.988358i	$0.451382\pi$
$390$	−4390.30	−0.570029
$391$	7036.08	0.910051
$392$	3301.42	0.425375
$393$	−4338.99	−0.556929
$394$	3768.96	0.481922
$395$	2288.96	0.291569
$396$	924.782	0.117354
$397$	−2294.44	−0.290062	−0.145031	−	0.989427i	$-0.546328\pi$
$397$	−2294.44	−0.290062	−0.145031	+	0.989427i	$0.546328\pi$
$398$	−4565.39	−0.574981
$399$	86.9944	0.0109152
$400$	−1954.34	−0.244292
$401$	−12131.7	−1.51080	−0.755399	−	0.655265i	$-0.772557\pi$
$401$	−12131.7	−1.51080	−0.755399	+	0.655265i	$0.772557\pi$
$402$	7186.52	0.891620
$403$	−4237.31	−0.523761
$404$	8220.21	1.01230
$405$	−290.187	−0.0356037
$406$	1217.80	0.148863
$407$	1169.59	0.142444
$408$	−1817.94	−0.220592
$409$	−12437.7	−1.50368	−0.751841	−	0.659344i	$-0.770834\pi$
$409$	−12437.7	−1.50368	−0.751841	+	0.659344i	$0.770834\pi$
$410$	−9555.56	−1.15101
$411$	−1459.46	−0.175158
$412$	−5074.08	−0.606753
$413$	835.921	0.0995957
$414$	−7224.41	−0.857634
$415$	6469.73	0.765269
$416$	14891.5	1.75508
$417$	6382.71	0.749550
$418$	763.680	0.0893608
$419$	3450.77	0.402341	0.201171	−	0.979556i	$-0.435525\pi$
$419$	3450.77	0.402341	0.201171	+	0.979556i	$0.435525\pi$
$420$	−122.513	−0.0142334
$421$	−13085.9	−1.51489	−0.757445	−	0.652900i	$-0.773552\pi$
$421$	−13085.9	−1.51489	−0.757445	+	0.652900i	$0.773552\pi$
$422$	4084.20	0.471128
$423$	6871.03	0.789790
$424$	−2275.77	−0.260663
$425$	1397.67	0.159522
$426$	9053.95	1.02973
$427$	172.941	0.0196000
$428$	8628.95	0.974523
$429$	2643.33	0.297485
$430$	5239.46	0.587603
$431$	10086.0	1.12721	0.563603	−	0.826046i	$-0.309415\pi$
$431$	10086.0	1.12721	0.563603	+	0.826046i	$0.309415\pi$
$432$	11218.6	1.24943
$433$	−4373.45	−0.485391	−0.242696	−	0.970102i	$-0.578032\pi$
$433$	−4373.45	−0.485391	−0.242696	+	0.970102i	$0.578032\pi$
$434$	−295.008	−0.0326287
$435$	−4109.07	−0.452908
$436$	−3650.03	−0.400928
$437$	−2391.22	−0.261757
$438$	1866.24	0.203590
$439$	−15007.0	−1.63153	−0.815766	−	0.578382i	$-0.803684\pi$
$439$	−15007.0	−1.63153	−0.815766	+	0.578382i	$0.803684\pi$
$440$	532.264	0.0576698
$441$	5359.29	0.578695
$442$	−14609.6	−1.57219
$443$	7440.33	0.797970	0.398985	−	0.916957i	$-0.369362\pi$
$443$	7440.33	0.797970	0.398985	+	0.916957i	$0.369362\pi$
$444$	−1911.92	−0.204359
$445$	−5453.85	−0.580983
$446$	−2450.51	−0.260168
$447$	1634.08	0.172907
$448$	184.578	0.0194653
$449$	−11130.0	−1.16984	−0.584919	−	0.811092i	$-0.698874\pi$
$449$	−11130.0	−1.16984	−0.584919	+	0.811092i	$0.698874\pi$
$450$	−1435.08	−0.150334
$451$	5753.25	0.600688
$452$	38.9700	0.00405530
$453$	−6203.57	−0.643420
$454$	12657.0	1.30842
$455$	487.265	0.0502051
$456$	617.830	0.0634485
$457$	14831.3	1.51812	0.759060	−	0.651021i	$-0.225659\pi$
$457$	14831.3	1.51812	0.759060	+	0.651021i	$0.225659\pi$
$458$	11854.5	1.20945
$459$	−8023.10	−0.815874
$460$	3367.54	0.341331
$461$	4929.97	0.498073	0.249037	−	0.968494i	$-0.419886\pi$
$461$	4929.97	0.498073	0.249037	+	0.968494i	$0.419886\pi$
$462$	184.033	0.0185324
$463$	15879.4	1.59390	0.796951	−	0.604044i	$-0.206445\pi$
$463$	15879.4	1.59390	0.796951	+	0.604044i	$0.206445\pi$
$464$	19119.7	1.91296
$465$	995.412	0.0992713
$466$	−9876.57	−0.981810
$467$	−2221.50	−0.220126	−0.110063	−	0.993925i	$-0.535105\pi$
$467$	−2221.50	−0.220126	−0.110063	+	0.993925i	$0.535105\pi$
$468$	6012.50	0.593863
$469$	−797.609	−0.0785291
$470$	−7990.73	−0.784223
$471$	−4637.96	−0.453728
$472$	5936.67	0.578935
$473$	−3154.60	−0.306657
$474$	5620.61	0.544648
$475$	−475.000	−0.0458831
$476$	−407.687	−0.0392569
$477$	−3694.32	−0.354614
$478$	−2583.75	−0.247234
$479$	−16649.9	−1.58821	−0.794105	−	0.607780i	$-0.792060\pi$
$479$	−16649.9	−1.58821	−0.794105	+	0.607780i	$0.792060\pi$
$480$	−3498.24	−0.332650
$481$	7604.15	0.720830
$482$	14872.0	1.40540
$483$	−576.241	−0.0542855
$484$	647.532	0.0608125
$485$	−5236.86	−0.490296
$486$	13445.6	1.25495
$487$	14791.8	1.37634	0.688171	−	0.725548i	$-0.258414\pi$
$487$	14791.8	1.37634	0.688171	+	0.725548i	$0.258414\pi$
$488$	1228.22	0.113932
$489$	10756.1	0.994703
$490$	−6232.64	−0.574616
$491$	−16308.1	−1.49893	−0.749463	−	0.662046i	$-0.769688\pi$
$491$	−16308.1	−1.49893	−0.749463	+	0.662046i	$0.769688\pi$
$492$	−9404.75	−0.861787
$493$	−13673.7	−1.24916
$494$	4965.09	0.452206
$495$	864.039	0.0784559
$496$	−4631.71	−0.419294
$497$	−1004.87	−0.0906931
$498$	15886.6	1.42951
$499$	−15279.2	−1.37073	−0.685363	−	0.728201i	$-0.740357\pi$
$499$	−15279.2	−1.37073	−0.685363	+	0.728201i	$0.740357\pi$
$500$	668.938	0.0598316
$501$	5805.09	0.517669
$502$	5449.79	0.484534
$503$	−11335.4	−1.00481	−0.502406	−	0.864632i	$-0.667552\pi$
$503$	−11335.4	−1.00481	−0.502406	+	0.864632i	$0.667552\pi$
$504$	−207.168	−0.0183095
$505$	7680.28	0.676768
$506$	−5058.53	−0.444425
$507$	9803.59	0.858762
$508$	−9552.45	−0.834294
$509$	3200.91	0.278739	0.139369	−	0.990240i	$-0.455492\pi$
$509$	3200.91	0.278739	0.139369	+	0.990240i	$0.455492\pi$
$510$	3432.02	0.297985
$511$	−207.128	−0.0179312
$512$	10225.3	0.882616
$513$	2726.67	0.234669
$514$	−272.906	−0.0234190
$515$	−4740.80	−0.405640
$516$	5156.78	0.439951
$517$	4811.10	0.409268
$518$	529.413	0.0449055
$519$	−4592.38	−0.388407
$520$	3460.53	0.291835
$521$	−10054.9	−0.845515	−0.422757	−	0.906243i	$-0.638938\pi$
$521$	−10054.9	−0.845515	−0.422757	+	0.906243i	$0.638938\pi$
$522$	14039.7	1.17721
$523$	−796.139	−0.0665635	−0.0332818	−	0.999446i	$-0.510596\pi$
$523$	−796.139	−0.0665635	−0.0332818	+	0.999446i	$0.510596\pi$
$524$	−6910.57	−0.576125
$525$	−114.466	−0.00951566
$526$	9456.13	0.783853
$527$	3312.43	0.273798
$528$	2889.37	0.238151
$529$	3672.21	0.301817
$530$	4296.34	0.352115
$531$	9637.17	0.787603
$532$	138.553	0.0112914
$533$	37405.0	3.03975
$534$	−13392.1	−1.08527
$535$	8062.17	0.651510
$536$	−5664.58	−0.456479
$537$	−13732.9	−1.10357
$538$	−5790.65	−0.464039
$539$	3752.57	0.299879
$540$	−3839.94	−0.306008
$541$	4488.02	0.356664	0.178332	−	0.983970i	$-0.442930\pi$
$541$	4488.02	0.356664	0.178332	+	0.983970i	$0.442930\pi$
$542$	1249.96	0.0990600
$543$	−10583.1	−0.836402
$544$	−11641.1	−0.917476
$545$	−3410.29	−0.268038
$546$	1196.50	0.0937825
$547$	−6865.40	−0.536642	−0.268321	−	0.963330i	$-0.586469\pi$
$547$	−6865.40	−0.536642	−0.268321	+	0.963330i	$0.586469\pi$
$548$	−2324.44	−0.181195
$549$	1993.80	0.154997
$550$	−1004.84	−0.0779029
$551$	4647.04	0.359293
$552$	−4092.44	−0.315554
$553$	−623.813	−0.0479696
$554$	20216.8	1.55042
$555$	−1786.34	−0.136623
$556$	10165.5	0.775385
$557$	−4135.45	−0.314587	−0.157293	−	0.987552i	$-0.550277\pi$
$557$	−4135.45	−0.314587	−0.157293	+	0.987552i	$0.550277\pi$
$558$	−3401.09	−0.258028
$559$	−20509.7	−1.55182
$560$	532.618	0.0401915
$561$	−2066.37	−0.155512
$562$	5319.72	0.399286
$563$	−13134.9	−0.983252	−0.491626	−	0.870806i	$-0.663597\pi$
$563$	−13134.9	−0.983252	−0.491626	+	0.870806i	$0.663597\pi$
$564$	−7864.62	−0.587164
$565$	36.4104	0.00271114
$566$	6607.46	0.490693
$567$	79.0851	0.00585760
$568$	−7136.52	−0.527186
$569$	10883.7	0.801881	0.400940	−	0.916104i	$-0.368683\pi$
$569$	10883.7	0.801881	0.400940	+	0.916104i	$0.368683\pi$
$570$	−1166.38	−0.0857091
$571$	5834.56	0.427616	0.213808	−	0.976876i	$-0.431413\pi$
$571$	5834.56	0.427616	0.213808	+	0.976876i	$0.431413\pi$
$572$	4209.95	0.307739
$573$	1147.69	0.0836745
$574$	2604.19	0.189367
$575$	3146.35	0.228194
$576$	2127.96	0.153932
$577$	−5810.86	−0.419254	−0.209627	−	0.977781i	$-0.567225\pi$
$577$	−5810.86	−0.419254	−0.209627	+	0.977781i	$0.567225\pi$
$578$	−6531.25	−0.470007
$579$	−9706.21	−0.696678
$580$	−6544.38	−0.468518
$581$	−1763.21	−0.125904
$582$	−12859.3	−0.915866
$583$	−2586.76	−0.183761
$584$	−1471.02	−0.104231
$585$	5617.58	0.397023
$586$	−861.222	−0.0607112
$587$	−10274.2	−0.722419	−0.361210	−	0.932485i	$-0.617636\pi$
$587$	−10274.2	−0.722419	−0.361210	+	0.932485i	$0.617636\pi$
$588$	−6134.27	−0.430227
$589$	−1125.73	−0.0787522
$590$	−11207.6	−0.782052
$591$	3465.83	0.241227
$592$	8311.92	0.577057
$593$	10956.6	0.758739	0.379369	−	0.925245i	$-0.376141\pi$
$593$	10956.6	0.758739	0.379369	+	0.925245i	$0.376141\pi$
$594$	5768.14	0.398434
$595$	−380.909	−0.0262449
$596$	2602.55	0.178866
$597$	−4198.20	−0.287807
$598$	−32888.2	−2.24899
$599$	12782.0	0.871882	0.435941	−	0.899975i	$-0.356416\pi$
$599$	12782.0	0.871882	0.435941	+	0.899975i	$0.356416\pi$
$600$	−812.934	−0.0553132
$601$	17715.9	1.20240	0.601202	−	0.799097i	$-0.294689\pi$
$601$	17715.9	1.20240	0.601202	+	0.799097i	$0.294689\pi$
$602$	−1427.92	−0.0966739
$603$	−9195.47	−0.621009
$604$	−9880.22	−0.665597
$605$	605.000	0.0406558
$606$	18859.2	1.26419
$607$	15691.8	1.04927	0.524637	−	0.851326i	$-0.324201\pi$
$607$	15691.8	1.04927	0.524637	+	0.851326i	$0.324201\pi$
$608$	3956.24	0.263893
$609$	1119.85	0.0745134
$610$	−2318.70	−0.153904
$611$	31279.5	2.07108
$612$	−4700.14	−0.310444
$613$	−25998.7	−1.71301	−0.856507	−	0.516136i	$-0.827370\pi$
$613$	−25998.7	−1.71301	−0.856507	+	0.516136i	$0.827370\pi$
$614$	−9282.75	−0.610133
$615$	−8787.02	−0.576141
$616$	−145.059	−0.00948796
$617$	−24067.6	−1.57038	−0.785191	−	0.619253i	$-0.787435\pi$
$617$	−24067.6	−1.57038	−0.785191	+	0.619253i	$0.787435\pi$
$618$	−11641.2	−0.757731
$619$	30.1848	0.00195998	0.000979991	−	1.00000i	$-0.499688\pi$
$619$	30.1848	0.00195998	0.000979991		1.00000i	$0.499688\pi$
$620$	1585.36	0.102693
$621$	−18061.1	−1.16710
$622$	26117.7	1.68364
$623$	1486.35	0.0955846
$624$	18785.3	1.20515
$625$	625.000	0.0400000
$626$	15723.0	1.00386
$627$	702.259	0.0447297
$628$	−7386.72	−0.469367
$629$	−5944.38	−0.376817
$630$	391.104	0.0247333
$631$	−4316.76	−0.272342	−0.136171	−	0.990685i	$-0.543480\pi$
$631$	−4316.76	−0.272342	−0.136171	+	0.990685i	$0.543480\pi$
$632$	−4430.29	−0.278841
$633$	3755.72	0.235824
$634$	17753.9	1.11214
$635$	−8925.01	−0.557761
$636$	4228.53	0.263636
$637$	24397.5	1.51752
$638$	9830.61	0.610028
$639$	−11584.9	−0.717202
$640$	5854.20	0.361574
$641$	2087.70	0.128641	0.0643207	−	0.997929i	$-0.479512\pi$
$641$	2087.70	0.128641	0.0643207	+	0.997929i	$0.479512\pi$
$642$	19796.9	1.21701
$643$	−11141.9	−0.683350	−0.341675	−	0.939818i	$-0.610994\pi$
$643$	−11141.9	−0.683350	−0.341675	+	0.939818i	$0.610994\pi$
$644$	−917.761	−0.0561566
$645$	4818.06	0.294126
$646$	−3881.35	−0.236393
$647$	3757.99	0.228349	0.114175	−	0.993461i	$-0.463578\pi$
$647$	3757.99	0.228349	0.114175	+	0.993461i	$0.463578\pi$
$648$	561.658	0.0340494
$649$	6747.94	0.408135
$650$	−6533.01	−0.394224
$651$	−271.281	−0.0163323
$652$	17131.0	1.02899
$653$	11744.3	0.703814	0.351907	−	0.936035i	$-0.385533\pi$
$653$	11744.3	0.703814	0.351907	+	0.936035i	$0.385533\pi$
$654$	−8374.07	−0.500691
$655$	−6456.66	−0.385164
$656$	40886.5	2.43346
$657$	−2387.94	−0.141800
$658$	2177.73	0.129022
$659$	15110.3	0.893194	0.446597	−	0.894735i	$-0.352636\pi$
$659$	15110.3	0.893194	0.446597	+	0.894735i	$0.352636\pi$
$660$	−988.984	−0.0583275
$661$	−23918.2	−1.40743	−0.703714	−	0.710483i	$-0.748476\pi$
$661$	−23918.2	−1.40743	−0.703714	+	0.710483i	$0.748476\pi$
$662$	−14411.4	−0.846094
$663$	−13434.5	−0.786960
$664$	−12522.2	−0.731861
$665$	129.453	0.00754880
$666$	6103.49	0.355113
$667$	−30781.5	−1.78690
$668$	9245.58	0.535512
$669$	−2253.42	−0.130227
$670$	10693.9	0.616632
$671$	1396.06	0.0803191
$672$	953.382	0.0547284
$673$	19268.0	1.10360	0.551802	−	0.833975i	$-0.313940\pi$
$673$	19268.0	1.10360	0.551802	+	0.833975i	$0.313940\pi$
$674$	−16689.8	−0.953810
$675$	−3587.72	−0.204580
$676$	15613.8	0.888362
$677$	19433.3	1.10323	0.551613	−	0.834100i	$-0.314013\pi$
$677$	19433.3	1.10323	0.551613	+	0.834100i	$0.314013\pi$
$678$	89.4069	0.00506438
$679$	1427.21	0.0806646
$680$	−2705.19	−0.152558
$681$	11639.1	0.654933
$682$	−2381.44	−0.133710
$683$	−30727.0	−1.72143	−0.860714	−	0.509089i	$-0.829982\pi$
$683$	−30727.0	−1.72143	−0.860714	+	0.509089i	$0.829982\pi$
$684$	1597.35	0.0892927
$685$	−2171.76	−0.121137
$686$	3406.43	0.189589
$687$	10901.1	0.605390
$688$	−22418.7	−1.24231
$689$	−16817.9	−0.929914
$690$	7725.96	0.426264
$691$	29310.7	1.61365	0.806823	−	0.590793i	$-0.201185\pi$
$691$	29310.7	1.61365	0.806823	+	0.590793i	$0.201185\pi$
$692$	−7314.13	−0.401794
$693$	−235.478	−0.0129077
$694$	13583.6	0.742978
$695$	9497.82	0.518378
$696$	7953.13	0.433136
$697$	−29240.5	−1.58904
$698$	−41663.2	−2.25928
$699$	−9082.22	−0.491446
$700$	−182.307	−0.00984364
$701$	−10535.9	−0.567671	−0.283835	−	0.958873i	$-0.591607\pi$
$701$	−10535.9	−0.567671	−0.283835	+	0.958873i	$0.591607\pi$
$702$	37501.8	2.01626
$703$	2020.21	0.108383
$704$	1490.00	0.0797675
$705$	−7348.05	−0.392544
$706$	−21903.2	−1.16762
$707$	−2093.12	−0.111343
$708$	−11030.8	−0.585538
$709$	14295.1	0.757211	0.378605	−	0.925558i	$-0.376404\pi$
$709$	14295.1	0.757211	0.378605	+	0.925558i	$0.376404\pi$
$710$	13472.8	0.712147
$711$	−7191.81	−0.379344
$712$	10556.0	0.555620
$713$	7456.74	0.391665
$714$	−935.334	−0.0490252
$715$	3933.43	0.205737
$716$	−21872.0	−1.14161
$717$	−2375.94	−0.123753
$718$	10367.7	0.538882
$719$	−13160.0	−0.682595	−0.341298	−	0.939955i	$-0.610866\pi$
$719$	−13160.0	−0.682595	−0.341298	+	0.939955i	$0.610866\pi$
$720$	6140.45	0.317835
$721$	1292.02	0.0667369
$722$	1319.08	0.0679934
$723$	13675.9	0.703475
$724$	−16855.4	−0.865231
$725$	−6114.53	−0.313225
$726$	1485.60	0.0759445
$727$	−9955.26	−0.507868	−0.253934	−	0.967222i	$-0.581725\pi$
$727$	−9955.26	−0.507868	−0.253934	+	0.967222i	$0.581725\pi$
$728$	−943.104	−0.0480134
$729$	13931.2	0.707779
$730$	2777.08	0.140800
$731$	16033.0	0.811222
$732$	−2282.11	−0.115231
$733$	21132.8	1.06488	0.532440	−	0.846468i	$-0.321275\pi$
$733$	21132.8	1.06488	0.532440	+	0.846468i	$0.321275\pi$
$734$	−31274.3	−1.57269
$735$	−5731.36	−0.287625
$736$	−26205.7	−1.31244
$737$	−6438.66	−0.321806
$738$	30023.2	1.49752
$739$	−30716.6	−1.52900	−0.764499	−	0.644624i	$-0.777014\pi$
$739$	−30716.6	−1.52900	−0.764499	+	0.644624i	$0.777014\pi$
$740$	−2845.04	−0.141332
$741$	4565.75	0.226352
$742$	−1170.89	−0.0579308
$743$	−4970.27	−0.245413	−0.122706	−	0.992443i	$-0.539157\pi$
$743$	−4970.27	−0.245413	−0.122706	+	0.992443i	$0.539157\pi$
$744$	−1926.63	−0.0949376
$745$	2431.60	0.119580
$746$	−8239.78	−0.404396
$747$	−20327.6	−0.995648
$748$	−3291.03	−0.160872
$749$	−2197.20	−0.107188
$750$	1534.71	0.0747195
$751$	15535.1	0.754841	0.377420	−	0.926042i	$-0.376811\pi$
$751$	15535.1	0.754841	0.377420	+	0.926042i	$0.376811\pi$
$752$	34190.9	1.65800
$753$	5011.48	0.242534
$754$	63914.0	3.08702
$755$	−9231.26	−0.444980
$756$	1046.50	0.0503452
$757$	8452.81	0.405842	0.202921	−	0.979195i	$-0.434956\pi$
$757$	8452.81	0.405842	0.202921	+	0.979195i	$0.434956\pi$
$758$	−33558.1	−1.60803
$759$	−4651.68	−0.222458
$760$	919.365	0.0438801
$761$	20960.9	0.998463	0.499232	−	0.866469i	$-0.333616\pi$
$761$	20960.9	0.998463	0.499232	+	0.866469i	$0.333616\pi$
$762$	−21915.7	−1.04189
$763$	929.411	0.0440982
$764$	1827.89	0.0865586
$765$	−4391.42	−0.207545
$766$	23243.0	1.09635
$767$	43871.9	2.06535
$768$	18016.3	0.846493
$769$	32084.5	1.50455	0.752274	−	0.658850i	$-0.228957\pi$
$769$	32084.5	1.50455	0.752274	+	0.658850i	$0.228957\pi$
$770$	273.851	0.0128168
$771$	−250.957	−0.0117224
$772$	−15458.8	−0.720690
$773$	−12955.7	−0.602827	−0.301413	−	0.953494i	$-0.597458\pi$
$773$	−12955.7	−0.602827	−0.301413	+	0.953494i	$0.597458\pi$
$774$	−16462.2	−0.764498
$775$	1481.23	0.0686546
$776$	10136.0	0.468892
$777$	486.833	0.0224775
$778$	8530.58	0.393105
$779$	9937.44	0.457055
$780$	−6429.91	−0.295164
$781$	−8111.75	−0.371653
$782$	25709.6	1.17567
$783$	35099.5	1.60199
$784$	26668.3	1.21485
$785$	−6901.54	−0.313792
$786$	−15854.5	−0.719482
$787$	15113.9	0.684565	0.342282	−	0.939597i	$-0.388800\pi$
$787$	15113.9	0.684565	0.342282	+	0.939597i	$0.388800\pi$
$788$	5519.91	0.249541
$789$	8695.59	0.392359
$790$	8363.78	0.376671
$791$	−9.92297	−0.000446043 0
$792$	−1672.35	−0.0750309
$793$	9076.49	0.406451
$794$	−8383.81	−0.374723
$795$	3950.79	0.176252
$796$	−6686.34	−0.297727
$797$	−11456.2	−0.509159	−0.254580	−	0.967052i	$-0.581937\pi$
$797$	−11456.2	−0.509159	−0.254580	+	0.967052i	$0.581937\pi$
$798$	317.875	0.0141011
$799$	−24452.1	−1.08267
$800$	−5205.58	−0.230056
$801$	17135.8	0.755884
$802$	−44329.0	−1.95176
$803$	−1672.04	−0.0734805
$804$	10525.2	0.461685
$805$	−857.479	−0.0375431
$806$	−15483.0	−0.676633
$807$	−5324.92	−0.232275
$808$	−14865.2	−0.647224
$809$	11358.0	0.493603	0.246802	−	0.969066i	$-0.420620\pi$
$809$	11358.0	0.493603	0.246802	+	0.969066i	$0.420620\pi$
$810$	−1060.33	−0.0459955
$811$	31036.9	1.34384	0.671919	−	0.740625i	$-0.265470\pi$
$811$	31036.9	1.34384	0.671919	+	0.740625i	$0.265470\pi$
$812$	1783.55	0.0770817
$813$	1149.43	0.0495847
$814$	4273.66	0.184019
$815$	16005.8	0.687923
$816$	−14685.0	−0.629997
$817$	−5448.86	−0.233331
$818$	−45447.1	−1.94257
$819$	−1530.97	−0.0653191
$820$	−13994.8	−0.596000
$821$	−8530.36	−0.362621	−0.181310	−	0.983426i	$-0.558034\pi$
$821$	−8530.36	−0.362621	−0.181310	+	0.983426i	$0.558034\pi$
$822$	−5332.83	−0.226282
$823$	−25062.9	−1.06153	−0.530765	−	0.847519i	$-0.678095\pi$
$823$	−25062.9	−1.06153	−0.530765	+	0.847519i	$0.678095\pi$
$824$	9175.85	0.387932
$825$	−924.024	−0.0389944
$826$	3054.43	0.128665
$827$	−40794.4	−1.71531	−0.857654	−	0.514228i	$-0.828079\pi$
$827$	−40794.4	−1.71531	−0.857654	+	0.514228i	$0.828079\pi$
$828$	−10580.7	−0.444087
$829$	−46684.1	−1.95586	−0.977928	−	0.208942i	$-0.932998\pi$
$829$	−46684.1	−1.95586	−0.977928	+	0.208942i	$0.932998\pi$
$830$	23640.2	0.988630
$831$	18590.8	0.776063
$832$	9687.25	0.403660
$833$	−19072.2	−0.793292
$834$	23322.2	0.968324
$835$	8638.30	0.358013
$836$	1118.46	0.0462714
$837$	−8502.77	−0.351134
$838$	12609.0	0.519774
$839$	−21987.6	−0.904763	−0.452382	−	0.891824i	$-0.649426\pi$
$839$	−21987.6	−0.904763	−0.452382	+	0.891824i	$0.649426\pi$
$840$	221.550	0.00910025
$841$	35430.9	1.45274
$842$	−47815.5	−1.95705
$843$	4891.86	0.199863
$844$	5981.61	0.243952
$845$	14588.3	0.593908
$846$	25106.6	1.02031
$847$	−164.882	−0.00668878
$848$	−18383.3	−0.744438
$849$	6076.04	0.245617
$850$	5107.04	0.206082
$851$	−13381.6	−0.539032
$852$	13260.2	0.533199
$853$	−29157.5	−1.17038	−0.585191	−	0.810896i	$-0.698980\pi$
$853$	−29157.5	−1.17038	−0.585191	+	0.810896i	$0.698980\pi$
$854$	631.920	0.0253207
$855$	1492.43	0.0596960
$856$	−15604.4	−0.623069
$857$	1866.81	0.0744094	0.0372047	−	0.999308i	$-0.488155\pi$
$857$	1866.81	0.0744094	0.0372047	+	0.999308i	$0.488155\pi$
$858$	9658.66	0.384314
$859$	3494.52	0.138803	0.0694013	−	0.997589i	$-0.477891\pi$
$859$	3494.52	0.138803	0.0694013	+	0.997589i	$0.477891\pi$
$860$	7673.57	0.304264
$861$	2394.74	0.0947881
$862$	36854.0	1.45621
$863$	−5051.57	−0.199255	−0.0996276	−	0.995025i	$-0.531765\pi$
$863$	−5051.57	−0.199255	−0.0996276	+	0.995025i	$0.531765\pi$
$864$	29881.9	1.17662
$865$	−6833.72	−0.268617
$866$	−15980.4	−0.627064
$867$	−6005.95	−0.235263
$868$	−432.061	−0.0168953
$869$	−5035.70	−0.196576
$870$	−15014.4	−0.585100
$871$	−41861.2	−1.62849
$872$	6600.63	0.256337
$873$	16454.0	0.637896
$874$	−8737.47	−0.338157
$875$	−170.332	−0.00658089
$876$	2733.25	0.105420
$877$	−13637.9	−0.525107	−0.262554	−	0.964917i	$-0.584565\pi$
$877$	−13637.9	−0.525107	−0.262554	+	0.964917i	$0.584565\pi$
$878$	−54835.0	−2.10773
$879$	−791.955	−0.0303891
$880$	4299.54	0.164702
$881$	−2545.94	−0.0973610	−0.0486805	−	0.998814i	$-0.515502\pi$
$881$	−2545.94	−0.0973610	−0.0486805	+	0.998814i	$0.515502\pi$
$882$	19582.7	0.747601
$883$	24253.3	0.924334	0.462167	−	0.886793i	$-0.347072\pi$
$883$	24253.3	0.924334	0.462167	+	0.886793i	$0.347072\pi$
$884$	−21396.8	−0.814085
$885$	−10306.2	−0.391457
$886$	27186.8	1.03088
$887$	−23666.3	−0.895870	−0.447935	−	0.894066i	$-0.647840\pi$
$887$	−23666.3	−0.895870	−0.447935	+	0.894066i	$0.647840\pi$
$888$	3457.46	0.130659
$889$	2432.35	0.0917642
$890$	−19928.2	−0.750556
$891$	638.411	0.0240040
$892$	−3588.94	−0.134716
$893$	8310.08	0.311407
$894$	5970.88	0.223374
$895$	−20435.3	−0.763216
$896$	−1595.46	−0.0594871
$897$	−30243.1	−1.12574
$898$	−40668.7	−1.51128
$899$	−14491.2	−0.537608
$900$	−2101.78	−0.0778436
$901$	13147.0	0.486116
$902$	21022.2	0.776013
$903$	−1313.07	−0.0483903
$904$	−70.4725	−0.00259279
$905$	−15748.3	−0.578444
$906$	−22667.7	−0.831217
$907$	−20165.2	−0.738231	−0.369116	−	0.929383i	$-0.620339\pi$
$907$	−20165.2	−0.738231	−0.369116	+	0.929383i	$0.620339\pi$
$908$	18537.1	0.677508
$909$	−24131.1	−0.880505
$910$	1780.45	0.0648587
$911$	44617.5	1.62266	0.811330	−	0.584589i	$-0.198744\pi$
$911$	44617.5	1.62266	0.811330	+	0.584589i	$0.198744\pi$
$912$	4990.72	0.181205
$913$	−14233.4	−0.515944
$914$	54193.3	1.96122
$915$	−2132.21	−0.0770369
$916$	17361.8	0.626257
$917$	1759.64	0.0633681
$918$	−29316.2	−1.05401
$919$	38351.8	1.37662	0.688308	−	0.725419i	$-0.258354\pi$
$919$	38351.8	1.37662	0.688308	+	0.725419i	$0.258354\pi$
$920$	−6089.78	−0.218233
$921$	−8536.16	−0.305403
$922$	18014.0	0.643447
$923$	−52738.8	−1.88074
$924$	269.529	0.00959618
$925$	−2658.17	−0.0944865
$926$	58022.7	2.05912
$927$	14895.4	0.527756
$928$	50927.5	1.80148
$929$	−28949.1	−1.02238	−0.511190	−	0.859468i	$-0.670795\pi$
$929$	−28949.1	−1.02238	−0.511190	+	0.859468i	$0.670795\pi$
$930$	3637.21	0.128246
$931$	6481.72	0.228174
$932$	−14465.0	−0.508385
$933$	24017.1	0.842750
$934$	−8117.31	−0.284375
$935$	−3074.87	−0.107550
$936$	−10872.9	−0.379691
$937$	37002.9	1.29011	0.645054	−	0.764137i	$-0.276835\pi$
$937$	37002.9	1.29011	0.645054	+	0.764137i	$0.276835\pi$
$938$	−2914.44	−0.101450
$939$	14458.4	0.502484
$940$	−11703.0	−0.406074
$941$	−43236.2	−1.49783	−0.748916	−	0.662665i	$-0.769425\pi$
$941$	−43236.2	−1.49783	−0.748916	+	0.662665i	$0.769425\pi$
$942$	−16947.0	−0.586159
$943$	−65824.5	−2.27311
$944$	47955.4	1.65341
$945$	977.767	0.0336579
$946$	−11526.8	−0.396162
$947$	−13217.9	−0.453562	−0.226781	−	0.973946i	$-0.572820\pi$
$947$	−13217.9	−0.453562	−0.226781	+	0.973946i	$0.572820\pi$
$948$	8231.78	0.282021
$949$	−10870.8	−0.371845
$950$	−1735.64	−0.0592752
$951$	16326.0	0.556684
$952$	737.251	0.0250992
$953$	−56255.5	−1.91217	−0.956083	−	0.293095i	$-0.905315\pi$
$953$	−56255.5	−1.91217	−0.956083	+	0.293095i	$0.905315\pi$
$954$	−13498.9	−0.458117
$955$	1707.83	0.0578681
$956$	−3784.09	−0.128019
$957$	9039.96	0.305350
$958$	−60838.2	−2.05177
$959$	591.873	0.0199297
$960$	−2275.69	−0.0765079
$961$	−26280.5	−0.882164
$962$	27785.3	0.931221
$963$	−25331.0	−0.847644
$964$	21781.2	0.727722
$965$	−14443.4	−0.481812
$966$	−2105.57	−0.0701300
$967$	−4156.89	−0.138239	−0.0691193	−	0.997608i	$-0.522019\pi$
$967$	−4156.89	−0.138239	−0.0691193	+	0.997608i	$0.522019\pi$
$968$	−1170.98	−0.0388810
$969$	−3569.18	−0.118327
$970$	−19135.3	−0.633400
$971$	37468.6	1.23834	0.619169	−	0.785258i	$-0.287470\pi$
$971$	37468.6	1.23834	0.619169	+	0.785258i	$0.287470\pi$
$972$	19692.1	0.649818
$973$	−2588.46	−0.0852848
$974$	54048.7	1.77806
$975$	−6007.57	−0.197330
$976$	9921.31	0.325383
$977$	19520.5	0.639217	0.319609	−	0.947550i	$-0.396449\pi$
$977$	19520.5	0.639217	0.319609	+	0.947550i	$0.396449\pi$
$978$	39302.7	1.28503
$979$	11998.5	0.391698
$980$	−9128.14	−0.297539
$981$	10715.0	0.348729
$982$	−59589.2	−1.93642
$983$	22929.6	0.743988	0.371994	−	0.928235i	$-0.378674\pi$
$983$	22929.6	0.743988	0.371994	+	0.928235i	$0.378674\pi$
$984$	17007.3	0.550990
$985$	5157.35	0.166829
$986$	−49963.4	−1.61375
$987$	2002.58	0.0645823
$988$	7271.73	0.234154
$989$	36092.6	1.16044
$990$	3157.17	0.101355
$991$	38360.4	1.22962	0.614812	−	0.788673i	$-0.289232\pi$
$991$	38360.4	1.22962	0.614812	+	0.788673i	$0.289232\pi$
$992$	−12337.1	−0.394861
$993$	−13252.3	−0.423513
$994$	−3671.76	−0.117164
$995$	−6247.16	−0.199044
$996$	23267.1	0.740208
$997$	−43460.8	−1.38056	−0.690280	−	0.723542i	$-0.742513\pi$
$997$	−43460.8	−1.38056	−0.690280	+	0.723542i	$0.742513\pi$
$998$	−55829.9	−1.77081
$999$	15258.8	0.483251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.d.1.17	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.d.1.17	✓	22	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+3.65397 q^{2} +3.36009 q^{3} +5.35150 q^{4} +5.00000 q^{5} +12.2777 q^{6} -1.36266 q^{7} -9.67753 q^{8} -15.7098 q^{9} +O(q^{10})\)	\(q+3.65397 q^{2} +3.36009 q^{3} +5.35150 q^{4} +5.00000 q^{5} +12.2777 q^{6} -1.36266 q^{7} -9.67753 q^{8} -15.7098 q^{9} +18.2699 q^{10} -11.0000 q^{11} +17.9815 q^{12} -71.5168 q^{13} -4.97911 q^{14} +16.8004 q^{15} -78.1734 q^{16} +55.9067 q^{17} -57.4032 q^{18} -19.0000 q^{19} +26.7575 q^{20} -4.57865 q^{21} -40.1937 q^{22} +125.854 q^{23} -32.5174 q^{24} +25.0000 q^{25} -261.320 q^{26} -143.509 q^{27} -7.29227 q^{28} -244.581 q^{29} +61.3883 q^{30} +59.2492 q^{31} -208.223 q^{32} -36.9610 q^{33} +204.282 q^{34} -6.81329 q^{35} -84.0711 q^{36} -106.327 q^{37} -69.4254 q^{38} -240.303 q^{39} -48.3877 q^{40} -523.023 q^{41} -16.7303 q^{42} +286.782 q^{43} -58.8665 q^{44} -78.5490 q^{45} +459.867 q^{46} -437.372 q^{47} -262.670 q^{48} -341.143 q^{49} +91.3493 q^{50} +187.852 q^{51} -382.723 q^{52} +235.160 q^{53} -524.377 q^{54} -55.0000 q^{55} +13.1872 q^{56} -63.8417 q^{57} -893.692 q^{58} -613.449 q^{59} +89.9076 q^{60} -126.914 q^{61} +216.495 q^{62} +21.4071 q^{63} -135.454 q^{64} -357.584 q^{65} -135.054 q^{66} +585.333 q^{67} +299.185 q^{68} +422.880 q^{69} -24.8956 q^{70} +737.432 q^{71} +152.032 q^{72} +152.003 q^{73} -388.515 q^{74} +84.0022 q^{75} -101.679 q^{76} +14.9892 q^{77} -878.060 q^{78} +457.791 q^{79} -390.867 q^{80} -58.0374 q^{81} -1911.11 q^{82} +1293.95 q^{83} -24.5027 q^{84} +279.534 q^{85} +1047.89 q^{86} -821.814 q^{87} +106.453 q^{88} -1090.77 q^{89} -287.016 q^{90} +97.4530 q^{91} +673.508 q^{92} +199.082 q^{93} -1598.15 q^{94} -95.0000 q^{95} -699.649 q^{96} -1047.37 q^{97} -1246.53 q^{98} +172.808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9 - 20 * q^10 - 242 * q^11 - 196 * q^12 - q^13 - 63 * q^14 - 105 * q^15 + 6 * q^16 + 187 * q^17 - 361 * q^18 - 418 * q^19 + 370 * q^20 - 107 * q^21 + 44 * q^22 - 361 * q^23 + 208 * q^24 + 550 * q^25 - 365 * q^26 - 1467 * q^27 - 773 * q^28 - 319 * q^29 - 45 * q^30 - 402 * q^31 - 873 * q^32 + 231 * q^33 - 717 * q^34 - 205 * q^35 + 725 * q^36 - 838 * q^37 + 76 * q^38 - 607 * q^39 - 390 * q^40 - 392 * q^41 - 1350 * q^42 - 610 * q^43 - 814 * q^44 + 1045 * q^45 - 605 * q^46 - 1866 * q^47 - 1637 * q^48 + 379 * q^49 - 100 * q^50 - 2659 * q^51 - 638 * q^52 - 1303 * q^53 + 2338 * q^54 - 1210 * q^55 + 727 * q^56 + 399 * q^57 + 44 * q^58 - 2417 * q^59 - 980 * q^60 + 918 * q^61 - 1634 * q^62 - 374 * q^63 - 1716 * q^64 - 5 * q^65 + 99 * q^66 - 2339 * q^67 + 4940 * q^68 + 127 * q^69 - 315 * q^70 - 2370 * q^71 - 3306 * q^72 + 2207 * q^73 + 2051 * q^74 - 525 * q^75 - 1406 * q^76 + 451 * q^77 + 1380 * q^78 + 586 * q^79 + 30 * q^80 + 1950 * q^81 - 1566 * q^82 - 2870 * q^83 + 3076 * q^84 + 935 * q^85 - 1246 * q^86 - 1811 * q^87 + 858 * q^88 - 1768 * q^89 - 1805 * q^90 - 2195 * q^91 - 6728 * q^92 - 2916 * q^93 + 672 * q^94 - 2090 * q^95 + 6022 * q^96 - 4022 * q^97 + 1162 * q^98 - 2299 * q^99

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