LMFDB - Embedded newform 1045.4.a.g.1.16

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:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q+2.38912 q^{2} -1.09640 q^{3} -2.29208 q^{4} -5.00000 q^{5} -2.61943 q^{6} -1.10283 q^{7} -24.5891 q^{8} -25.7979 q^{9} +O(q^{10})$	\(q+2.38912 q^{2} -1.09640 q^{3} -2.29208 q^{4} -5.00000 q^{5} -2.61943 q^{6} -1.10283 q^{7} -24.5891 q^{8} -25.7979 q^{9} -11.9456 q^{10} -11.0000 q^{11} +2.51304 q^{12} -17.2371 q^{13} -2.63480 q^{14} +5.48199 q^{15} -40.4097 q^{16} -90.0323 q^{17} -61.6344 q^{18} -19.0000 q^{19} +11.4604 q^{20} +1.20914 q^{21} -26.2804 q^{22} -20.2343 q^{23} +26.9594 q^{24} +25.0000 q^{25} -41.1816 q^{26} +57.8875 q^{27} +2.52778 q^{28} +303.592 q^{29} +13.0972 q^{30} +158.658 q^{31} +100.169 q^{32} +12.0604 q^{33} -215.098 q^{34} +5.51415 q^{35} +59.1310 q^{36} -205.513 q^{37} -45.3934 q^{38} +18.8987 q^{39} +122.945 q^{40} +243.106 q^{41} +2.88879 q^{42} +284.494 q^{43} +25.2129 q^{44} +128.990 q^{45} -48.3422 q^{46} -423.507 q^{47} +44.3051 q^{48} -341.784 q^{49} +59.7281 q^{50} +98.7112 q^{51} +39.5089 q^{52} -392.342 q^{53} +138.300 q^{54} +55.0000 q^{55} +27.1176 q^{56} +20.8316 q^{57} +725.319 q^{58} -245.957 q^{59} -12.5652 q^{60} +140.280 q^{61} +379.053 q^{62} +28.4507 q^{63} +562.593 q^{64} +86.1855 q^{65} +28.8137 q^{66} +206.602 q^{67} +206.362 q^{68} +22.1848 q^{69} +13.1740 q^{70} +11.9429 q^{71} +634.347 q^{72} +479.940 q^{73} -490.996 q^{74} -27.4099 q^{75} +43.5496 q^{76} +12.1311 q^{77} +45.1514 q^{78} -248.417 q^{79} +202.048 q^{80} +633.076 q^{81} +580.811 q^{82} -329.347 q^{83} -2.77145 q^{84} +450.162 q^{85} +679.692 q^{86} -332.858 q^{87} +270.480 q^{88} +1272.03 q^{89} +308.172 q^{90} +19.0096 q^{91} +46.3786 q^{92} -173.952 q^{93} -1011.81 q^{94} +95.0000 q^{95} -109.825 q^{96} +266.541 q^{97} -816.564 q^{98} +283.777 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * 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 $

$2$	2.38912	0.844683	0.422342	−	0.906437i	$-0.361208\pi$
$2$	2.38912	0.844683	0.422342	+	0.906437i	$0.361208\pi$
$3$	−1.09640	−0.211002	−0.105501	−	0.994419i	$-0.533645\pi$
$3$	−1.09640	−0.211002	−0.105501	+	0.994419i	$0.533645\pi$
$4$	−2.29208	−0.286511
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−2.61943	−0.178230
$7$	−1.10283	−0.0595472	−0.0297736	−	0.999557i	$-0.509479\pi$
$7$	−1.10283	−0.0595472	−0.0297736	+	0.999557i	$0.509479\pi$
$8$	−24.5891	−1.08669
$9$	−25.7979	−0.955478
$10$	−11.9456	−0.377754
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	2.51304	0.0604543
$13$	−17.2371	−0.367747	−0.183874	−	0.982950i	$-0.558864\pi$
$13$	−17.2371	−0.367747	−0.183874	+	0.982950i	$0.558864\pi$
$14$	−2.63480	−0.0502985
$15$	5.48199	0.0943629
$16$	−40.4097	−0.631401
$17$	−90.0323	−1.28447	−0.642237	−	0.766506i	$-0.721993\pi$
$17$	−90.0323	−1.28447	−0.642237	+	0.766506i	$0.721993\pi$
$18$	−61.6344	−0.807076
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	11.4604	0.128131
$21$	1.20914	0.0125646
$22$	−26.2804	−0.254682
$23$	−20.2343	−0.183441	−0.0917203	−	0.995785i	$-0.529237\pi$
$23$	−20.2343	−0.183441	−0.0917203	+	0.995785i	$0.529237\pi$
$24$	26.9594	0.229294
$25$	25.0000	0.200000
$26$	−41.1816	−0.310630
$27$	57.8875	0.412610
$28$	2.52778	0.0170609
$29$	303.592	1.94399	0.971993	−	0.235009i	$-0.0755119\pi$
$29$	303.592	1.94399	0.971993	+	0.235009i	$0.0755119\pi$
$30$	13.0972	0.0797067
$31$	158.658	0.919217	0.459609	−	0.888122i	$-0.347990\pi$
$31$	158.658	0.919217	0.459609	+	0.888122i	$0.347990\pi$
$32$	100.169	0.553360
$33$	12.0604	0.0636195
$34$	−215.098	−1.08497
$35$	5.51415	0.0266303
$36$	59.1310	0.273755
$37$	−205.513	−0.913138	−0.456569	−	0.889688i	$-0.650922\pi$
$37$	−205.513	−0.913138	−0.456569	+	0.889688i	$0.650922\pi$
$38$	−45.3934	−0.193784
$39$	18.8987	0.0775953
$40$	122.945	0.485984
$41$	243.106	0.926020	0.463010	−	0.886353i	$-0.346769\pi$
$41$	243.106	0.926020	0.463010	+	0.886353i	$0.346769\pi$
$42$	2.88879	0.0106131
$43$	284.494	1.00895	0.504476	−	0.863426i	$-0.331686\pi$
$43$	284.494	1.00895	0.504476	+	0.863426i	$0.331686\pi$
$44$	25.2129	0.0863862
$45$	128.990	0.427303
$46$	−48.3422	−0.154949
$47$	−423.507	−1.31436	−0.657179	−	0.753735i	$-0.728250\pi$
$47$	−423.507	−1.31436	−0.657179	+	0.753735i	$0.728250\pi$
$48$	44.3051	0.133227
$49$	−341.784	−0.996454
$50$	59.7281	0.168937
$51$	98.7112	0.271026
$52$	39.5089	0.105363
$53$	−392.342	−1.01684	−0.508418	−	0.861110i	$-0.669770\pi$
$53$	−392.342	−1.01684	−0.508418	+	0.861110i	$0.669770\pi$
$54$	138.300	0.348524
$55$	55.0000	0.134840
$56$	27.1176	0.0647096
$57$	20.8316	0.0484071
$58$	725.319	1.64205
$59$	−245.957	−0.542726	−0.271363	−	0.962477i	$-0.587474\pi$
$59$	−245.957	−0.542726	−0.271363	+	0.962477i	$0.587474\pi$
$60$	−12.5652	−0.0270360
$61$	140.280	0.294442	0.147221	−	0.989104i	$-0.452967\pi$
$61$	140.280	0.294442	0.147221	+	0.989104i	$0.452967\pi$
$62$	379.053	0.776447
$63$	28.4507	0.0568961
$64$	562.593	1.09881
$65$	86.1855	0.164461
$66$	28.8137	0.0537383
$67$	206.602	0.376723	0.188362	−	0.982100i	$-0.439682\pi$
$67$	206.602	0.376723	0.188362	+	0.982100i	$0.439682\pi$
$68$	206.362	0.368015
$69$	22.1848	0.0387063
$70$	13.1740	0.0224942
$71$	11.9429	0.0199628	0.00998142	−	0.999950i	$-0.496823\pi$
$71$	11.9429	0.0199628	0.00998142	+	0.999950i	$0.496823\pi$
$72$	634.347	1.03831
$73$	479.940	0.769489	0.384745	−	0.923023i	$-0.374290\pi$
$73$	479.940	0.769489	0.384745	+	0.923023i	$0.374290\pi$
$74$	−490.996	−0.771312
$75$	−27.4099	−0.0422004
$76$	43.5496	0.0657300
$77$	12.1311	0.0179542
$78$	45.1514	0.0655434
$79$	−248.417	−0.353786	−0.176893	−	0.984230i	$-0.556605\pi$
$79$	−248.417	−0.353786	−0.176893	+	0.984230i	$0.556605\pi$
$80$	202.048	0.282371
$81$	633.076	0.868417
$82$	580.811	0.782194
$83$	−329.347	−0.435548	−0.217774	−	0.975999i	$-0.569880\pi$
$83$	−329.347	−0.435548	−0.217774	+	0.975999i	$0.569880\pi$
$84$	−2.77145	−0.00359988
$85$	450.162	0.574434
$86$	679.692	0.852245
$87$	−332.858	−0.410185
$88$	270.480	0.327650
$89$	1272.03	1.51499	0.757497	−	0.652839i	$-0.226422\pi$
$89$	1272.03	1.51499	0.757497	+	0.652839i	$0.226422\pi$
$90$	308.172	0.360935
$91$	19.0096	0.0218983
$92$	46.3786	0.0525577
$93$	−173.952	−0.193957
$94$	−1011.81	−1.11022
$95$	95.0000	0.102598
$96$	−109.825	−0.116760
$97$	266.541	0.279001	0.139501	−	0.990222i	$-0.455450\pi$
$97$	266.541	0.279001	0.139501	+	0.990222i	$0.455450\pi$
$98$	−816.564	−0.841688
$99$	283.777	0.288088
$100$	−57.3021	−0.0573021
$101$	864.295	0.851491	0.425746	−	0.904843i	$-0.360012\pi$
$101$	864.295	0.851491	0.425746	+	0.904843i	$0.360012\pi$
$102$	235.833	0.228931
$103$	1277.10	1.22171	0.610857	−	0.791741i	$-0.290825\pi$
$103$	1277.10	1.22171	0.610857	+	0.791741i	$0.290825\pi$
$104$	423.844	0.399628
$105$	−6.04570	−0.00561905
$106$	−937.354	−0.858904
$107$	1112.44	1.00508	0.502542	−	0.864553i	$-0.332398\pi$
$107$	1112.44	1.00508	0.502542	+	0.864553i	$0.332398\pi$
$108$	−132.683	−0.118217
$109$	−1651.18	−1.45096	−0.725479	−	0.688245i	$-0.758382\pi$
$109$	−1651.18	−1.45096	−0.725479	+	0.688245i	$0.758382\pi$
$110$	131.402	0.113897
$111$	225.324	0.192674
$112$	44.5650	0.0375982
$113$	−539.448	−0.449089	−0.224544	−	0.974464i	$-0.572089\pi$
$113$	−539.448	−0.449089	−0.224544	+	0.974464i	$0.572089\pi$
$114$	49.7692	0.0408887
$115$	101.171	0.0820372
$116$	−695.858	−0.556973
$117$	444.681	0.351374
$118$	−587.622	−0.458432
$119$	99.2904	0.0764868
$120$	−134.797	−0.102544
$121$	121.000	0.0909091
$122$	335.146	0.248711
$123$	−266.541	−0.195392
$124$	−363.657	−0.263365
$125$	−125.000	−0.0894427
$126$	67.9723	0.0480592
$127$	−1885.28	−1.31726	−0.658629	−	0.752468i	$-0.728863\pi$
$127$	−1885.28	−1.31726	−0.658629	+	0.752468i	$0.728863\pi$
$128$	542.755	0.374791
$129$	−311.919	−0.212891
$130$	205.908	0.138918
$131$	1475.03	0.983771	0.491885	−	0.870660i	$-0.336308\pi$
$131$	1475.03	0.983771	0.491885	+	0.870660i	$0.336308\pi$
$132$	−27.6434	−0.0182276
$133$	20.9538	0.0136611
$134$	493.598	0.318212
$135$	−289.438	−0.184525
$136$	2213.81	1.39583
$137$	2083.69	1.29943	0.649716	−	0.760177i	$-0.274888\pi$
$137$	2083.69	1.29943	0.649716	+	0.760177i	$0.274888\pi$
$138$	53.0022	0.0326946
$139$	−1595.58	−0.973635	−0.486818	−	0.873504i	$-0.661842\pi$
$139$	−1595.58	−0.973635	−0.486818	+	0.873504i	$0.661842\pi$
$140$	−12.6389	−0.00762987
$141$	464.332	0.277332
$142$	28.5331	0.0168623
$143$	189.608	0.110880
$144$	1042.49	0.603290
$145$	−1517.96	−0.869377
$146$	1146.64	0.649974
$147$	374.731	0.210254
$148$	471.053	0.261624
$149$	2328.04	1.28000	0.640001	−	0.768374i	$-0.278934\pi$
$149$	2328.04	1.28000	0.640001	+	0.768374i	$0.278934\pi$
$150$	−65.4858	−0.0356459
$151$	2772.79	1.49435	0.747175	−	0.664628i	$-0.231410\pi$
$151$	2772.79	1.49435	0.747175	+	0.664628i	$0.231410\pi$
$152$	467.192	0.249305
$153$	2322.65	1.22729
$154$	28.9828	0.0151656
$155$	−793.288	−0.411087
$156$	−43.3175	−0.0222319
$157$	1890.49	0.961006	0.480503	−	0.876993i	$-0.340454\pi$
$157$	1890.49	0.961006	0.480503	+	0.876993i	$0.340454\pi$
$158$	−593.499	−0.298837
$159$	430.163	0.214554
$160$	−500.844	−0.247470
$161$	22.3150	0.0109234
$162$	1512.50	0.733537
$163$	207.008	0.0994732	0.0497366	−	0.998762i	$-0.484162\pi$
$163$	207.008	0.0994732	0.0497366	+	0.998762i	$0.484162\pi$
$164$	−557.220	−0.265315
$165$	−60.3019	−0.0284515
$166$	−786.851	−0.367900
$167$	−3397.26	−1.57418	−0.787090	−	0.616838i	$-0.788413\pi$
$167$	−3397.26	−1.57418	−0.787090	+	0.616838i	$0.788413\pi$
$168$	−29.7316	−0.0136538
$169$	−1899.88	−0.864762
$170$	1075.49	0.485215
$171$	490.160	0.219202
$172$	−652.084	−0.289075
$173$	3208.08	1.40986	0.704931	−	0.709276i	$-0.250978\pi$
$173$	3208.08	1.40986	0.704931	+	0.709276i	$0.250978\pi$
$174$	−795.238	−0.346476
$175$	−27.5708	−0.0119094
$176$	444.506	0.190375
$177$	269.667	0.114516
$178$	3039.03	1.27969
$179$	−3199.55	−1.33601	−0.668004	−	0.744158i	$-0.732851\pi$
$179$	−3199.55	−1.33601	−0.668004	+	0.744158i	$0.732851\pi$
$180$	−295.655	−0.122427
$181$	−3517.58	−1.44453	−0.722264	−	0.691617i	$-0.756899\pi$
$181$	−3517.58	−1.44453	−0.722264	+	0.691617i	$0.756899\pi$
$182$	45.4163	0.0184971
$183$	−153.802	−0.0621279
$184$	497.542	0.199344
$185$	1027.56	0.408368
$186$	−415.593	−0.163832
$187$	990.356	0.387283
$188$	970.713	0.376577
$189$	−63.8401	−0.0245698
$190$	226.967	0.0866627
$191$	183.295	0.0694387	0.0347193	−	0.999397i	$-0.488946\pi$
$191$	183.295	0.0694387	0.0347193	+	0.999397i	$0.488946\pi$
$192$	−616.826	−0.231852
$193$	−1556.68	−0.580582	−0.290291	−	0.956938i	$-0.593752\pi$
$193$	−1556.68	−0.580582	−0.290291	+	0.956938i	$0.593752\pi$
$194$	636.800	0.235668
$195$	−94.4936	−0.0347017
$196$	783.397	0.285495
$197$	−2379.22	−0.860468	−0.430234	−	0.902717i	$-0.641569\pi$
$197$	−2379.22	−0.860468	−0.430234	+	0.902717i	$0.641569\pi$
$198$	677.979	0.243343
$199$	3675.50	1.30929	0.654647	−	0.755935i	$-0.272817\pi$
$199$	3675.50	1.30929	0.654647	+	0.755935i	$0.272817\pi$
$200$	−614.727	−0.217339
$201$	−226.518	−0.0794893
$202$	2064.91	0.719240
$203$	−334.810	−0.115759
$204$	−226.254	−0.0776519
$205$	−1215.53	−0.414129
$206$	3051.15	1.03196
$207$	522.002	0.175274
$208$	696.546	0.232196
$209$	209.000	0.0691714
$210$	−14.4439	−0.00474632
$211$	3137.68	1.02373	0.511864	−	0.859066i	$-0.328955\pi$
$211$	3137.68	1.02373	0.511864	+	0.859066i	$0.328955\pi$
$212$	899.281	0.291334
$213$	−13.0942	−0.00421219
$214$	2657.77	0.848978
$215$	−1422.47	−0.451217
$216$	−1423.40	−0.448380
$217$	−174.972	−0.0547368
$218$	−3944.87	−1.22560
$219$	−526.205	−0.162364
$220$	−126.065	−0.0386331
$221$	1551.90	0.472361
$222$	538.327	0.162748
$223$	−2370.65	−0.711886	−0.355943	−	0.934508i	$-0.615840\pi$
$223$	−2370.65	−0.711886	−0.355943	+	0.934508i	$0.615840\pi$
$224$	−110.469	−0.0329510
$225$	−644.948	−0.191096
$226$	−1288.81	−0.379338
$227$	311.041	0.0909451	0.0454726	−	0.998966i	$-0.485521\pi$
$227$	311.041	0.0909451	0.0454726	+	0.998966i	$0.485521\pi$
$228$	−47.7477	−0.0138692
$229$	−4265.89	−1.23100	−0.615498	−	0.788139i	$-0.711045\pi$
$229$	−4265.89	−1.23100	−0.615498	+	0.788139i	$0.711045\pi$
$230$	241.711	0.0692954
$231$	−13.3005	−0.00378836
$232$	−7465.04	−2.11252
$233$	355.048	0.0998282	0.0499141	−	0.998754i	$-0.484105\pi$
$233$	355.048	0.0998282	0.0499141	+	0.998754i	$0.484105\pi$
$234$	1062.40	0.296800
$235$	2117.53	0.587798
$236$	563.754	0.155497
$237$	272.364	0.0746495
$238$	237.217	0.0646071
$239$	1909.17	0.516710	0.258355	−	0.966050i	$-0.416820\pi$
$239$	1909.17	0.516710	0.258355	+	0.966050i	$0.416820\pi$
$240$	−221.525	−0.0595808
$241$	2788.08	0.745212	0.372606	−	0.927990i	$-0.378464\pi$
$241$	2788.08	0.745212	0.372606	+	0.927990i	$0.378464\pi$
$242$	289.084	0.0767894
$243$	−2257.07	−0.595847
$244$	−321.533	−0.0843609
$245$	1708.92	0.445628
$246$	−636.800	−0.165044
$247$	327.505	0.0843670
$248$	−3901.24	−0.998908
$249$	361.095	0.0919015
$250$	−298.641	−0.0755508
$251$	−5173.16	−1.30091	−0.650453	−	0.759547i	$-0.725421\pi$
$251$	−5173.16	−1.30091	−0.650453	+	0.759547i	$0.725421\pi$
$252$	−65.2114	−0.0163013
$253$	222.577	0.0553094
$254$	−4504.17	−1.11266
$255$	−493.556	−0.121207
$256$	−3204.04	−0.782236
$257$	−1163.93	−0.282505	−0.141252	−	0.989974i	$-0.545113\pi$
$257$	−1163.93	−0.282505	−0.141252	+	0.989974i	$0.545113\pi$
$258$	−745.213	−0.179825
$259$	226.646	0.0543748
$260$	−197.544	−0.0471199
$261$	−7832.04	−1.85744
$262$	3524.03	0.830975
$263$	−5340.43	−1.25211	−0.626055	−	0.779779i	$-0.715331\pi$
$263$	−5340.43	−1.25211	−0.626055	+	0.779779i	$0.715331\pi$
$264$	−296.553	−0.0691349
$265$	1961.71	0.454743
$266$	50.0612	0.0115393
$267$	−1394.65	−0.319666
$268$	−473.549	−0.107935
$269$	−5936.93	−1.34565	−0.672827	−	0.739800i	$-0.734920\pi$
$269$	−5936.93	−1.34565	−0.672827	+	0.739800i	$0.734920\pi$
$270$	−691.502	−0.155865
$271$	−2866.57	−0.642552	−0.321276	−	0.946986i	$-0.604112\pi$
$271$	−2866.57	−0.642552	−0.321276	+	0.946986i	$0.604112\pi$
$272$	3638.18	0.811018
$273$	−20.8421	−0.00462059
$274$	4978.21	1.09761
$275$	−275.000	−0.0603023
$276$	−50.8494	−0.0110898
$277$	3753.35	0.814141	0.407071	−	0.913397i	$-0.366550\pi$
$277$	3753.35	0.814141	0.407071	+	0.913397i	$0.366550\pi$
$278$	−3812.04	−0.822413
$279$	−4093.03	−0.878292
$280$	−135.588	−0.0289390
$281$	7936.36	1.68485	0.842426	−	0.538812i	$-0.181127\pi$
$281$	7936.36	1.68485	0.842426	+	0.538812i	$0.181127\pi$
$282$	1109.35	0.234258
$283$	3156.12	0.662940	0.331470	−	0.943466i	$-0.392455\pi$
$283$	3156.12	0.662940	0.331470	+	0.943466i	$0.392455\pi$
$284$	−27.3741	−0.00571956
$285$	−104.158	−0.0216483
$286$	452.997	0.0936584
$287$	−268.105	−0.0551420
$288$	−2584.15	−0.528723
$289$	3192.82	0.649872
$290$	−3626.59	−0.734348
$291$	−292.235	−0.0588698
$292$	−1100.06	−0.220467
$293$	8111.05	1.61724	0.808622	−	0.588329i	$-0.200214\pi$
$293$	8111.05	1.61724	0.808622	+	0.588329i	$0.200214\pi$
$294$	895.279	0.177598
$295$	1229.78	0.242715
$296$	5053.37	0.992301
$297$	−636.763	−0.124406
$298$	5561.97	1.08120
$299$	348.780	0.0674598
$300$	62.8259	0.0120909
$301$	−313.749	−0.0600803
$302$	6624.55	1.26225
$303$	−947.612	−0.179666
$304$	767.784	0.144853
$305$	−701.399	−0.131679
$306$	5549.09	1.03667
$307$	−10112.0	−1.87987	−0.939935	−	0.341353i	$-0.889115\pi$
$307$	−10112.0	−1.87987	−0.939935	+	0.341353i	$0.889115\pi$
$308$	−27.8056	−0.00514406
$309$	−1400.21	−0.257784
$310$	−1895.26	−0.347238
$311$	3426.08	0.624678	0.312339	−	0.949971i	$-0.398887\pi$
$311$	3426.08	0.624678	0.312339	+	0.949971i	$0.398887\pi$
$312$	−464.702	−0.0843223
$313$	5642.86	1.01902	0.509510	−	0.860465i	$-0.329827\pi$
$313$	5642.86	1.01902	0.509510	+	0.860465i	$0.329827\pi$
$314$	4516.63	0.811745
$315$	−142.254	−0.0254447
$316$	569.393	0.101363
$317$	8310.38	1.47242	0.736210	−	0.676753i	$-0.236613\pi$
$317$	8310.38	1.47242	0.736210	+	0.676753i	$0.236613\pi$
$318$	1027.71	0.181230
$319$	−3339.51	−0.586134
$320$	−2812.97	−0.491405
$321$	−1219.68	−0.212075
$322$	53.3132	0.00922680
$323$	1710.61	0.294678
$324$	−1451.06	−0.248811
$325$	−430.928	−0.0735494
$326$	494.568	0.0840233
$327$	1810.35	0.306155
$328$	−5977.76	−1.00630
$329$	467.056	0.0782663
$330$	−144.069	−0.0240325
$331$	−6002.42	−0.996746	−0.498373	−	0.866963i	$-0.666069\pi$
$331$	−6002.42	−0.996746	−0.498373	+	0.866963i	$0.666069\pi$
$332$	754.891	0.124789
$333$	5301.80	0.872484
$334$	−8116.48	−1.32968
$335$	−1033.01	−0.168476
$336$	−48.8610	−0.00793329
$337$	−3454.63	−0.558415	−0.279208	−	0.960231i	$-0.590072\pi$
$337$	−3454.63	−0.558415	−0.279208	+	0.960231i	$0.590072\pi$
$338$	−4539.06	−0.730450
$339$	591.450	0.0947586
$340$	−1031.81	−0.164581
$341$	−1745.23	−0.277154
$342$	1171.05	0.185156
$343$	755.200	0.118883
$344$	−6995.45	−1.09642
$345$	−110.924	−0.0173100
$346$	7664.51	1.19089
$347$	−3217.40	−0.497749	−0.248875	−	0.968536i	$-0.580061\pi$
$347$	−3217.40	−0.497749	−0.248875	+	0.968536i	$0.580061\pi$
$348$	762.938	0.117522
$349$	9112.68	1.39768	0.698841	−	0.715277i	$-0.253700\pi$
$349$	9112.68	1.39768	0.698841	+	0.715277i	$0.253700\pi$
$350$	−65.8700	−0.0100597
$351$	−997.813	−0.151736
$352$	−1101.86	−0.166844
$353$	−12662.2	−1.90918	−0.954590	−	0.297921i	$-0.903707\pi$
$353$	−12662.2	−1.90918	−0.954590	+	0.297921i	$0.903707\pi$
$354$	644.267	0.0967299
$355$	−59.7145	−0.00892765
$356$	−2915.59	−0.434062
$357$	−108.862	−0.0161389
$358$	−7644.12	−1.12850
$359$	9707.69	1.42717	0.713583	−	0.700571i	$-0.247071\pi$
$359$	9707.69	1.42717	0.713583	+	0.700571i	$0.247071\pi$
$360$	−3171.73	−0.464347
$361$	361.000	0.0526316
$362$	−8403.94	−1.22017
$363$	−132.664	−0.0191820
$364$	−43.5716	−0.00627410
$365$	−2399.70	−0.344126
$366$	−367.453	−0.0524784
$367$	10580.8	1.50494	0.752468	−	0.658629i	$-0.228863\pi$
$367$	10580.8	1.50494	0.752468	+	0.658629i	$0.228863\pi$
$368$	817.660	0.115825
$369$	−6271.64	−0.884792
$370$	2454.98	0.344941
$371$	432.687	0.0605498
$372$	398.712	0.0555706
$373$	8166.26	1.13360	0.566800	−	0.823855i	$-0.308181\pi$
$373$	8166.26	1.13360	0.566800	+	0.823855i	$0.308181\pi$
$374$	2366.08	0.327132
$375$	137.050	0.0188726
$376$	10413.6	1.42830
$377$	−5233.05	−0.714895
$378$	−152.522	−0.0207537
$379$	−11567.0	−1.56769	−0.783847	−	0.620954i	$-0.786745\pi$
$379$	−11567.0	−1.56769	−0.783847	+	0.620954i	$0.786745\pi$
$380$	−217.748	−0.0293954
$381$	2067.02	0.277944
$382$	437.915	0.0586537
$383$	6728.83	0.897722	0.448861	−	0.893602i	$-0.351830\pi$
$383$	6728.83	0.897722	0.448861	+	0.893602i	$0.351830\pi$
$384$	−595.075	−0.0790815
$385$	−60.6557	−0.00802935
$386$	−3719.10	−0.490408
$387$	−7339.35	−0.964032
$388$	−610.935	−0.0799369
$389$	−3563.56	−0.464472	−0.232236	−	0.972660i	$-0.574604\pi$
$389$	−3563.56	−0.464472	−0.232236	+	0.972660i	$0.574604\pi$
$390$	−225.757	−0.0293119
$391$	1821.74	0.235625
$392$	8404.15	1.08284
$393$	−1617.22	−0.207577
$394$	−5684.24	−0.726823
$395$	1242.09	0.158218
$396$	−650.441	−0.0825401
$397$	−2549.57	−0.322316	−0.161158	−	0.986929i	$-0.551523\pi$
$397$	−2549.57	−0.322316	−0.161158	+	0.986929i	$0.551523\pi$
$398$	8781.23	1.10594
$399$	−22.9737	−0.00288251
$400$	−1010.24	−0.126280
$401$	12929.7	1.61017	0.805086	−	0.593158i	$-0.202119\pi$
$401$	12929.7	1.61017	0.805086	+	0.593158i	$0.202119\pi$
$402$	−541.180	−0.0671433
$403$	−2734.80	−0.338040
$404$	−1981.04	−0.243961
$405$	−3165.38	−0.388368
$406$	−799.904	−0.0977797
$407$	2260.64	0.275321
$408$	−2427.22	−0.294523
$409$	−13699.4	−1.65622	−0.828109	−	0.560568i	$-0.810583\pi$
$409$	−13699.4	−1.65622	−0.828109	+	0.560568i	$0.810583\pi$
$410$	−2904.06	−0.349808
$411$	−2284.56	−0.274182
$412$	−2927.22	−0.350034
$413$	271.249	0.0323178
$414$	1247.13	0.148051
$415$	1646.73	0.194783
$416$	−1726.62	−0.203496
$417$	1749.39	0.205439
$418$	499.327	0.0584279
$419$	8164.14	0.951896	0.475948	−	0.879473i	$-0.342105\pi$
$419$	8164.14	0.951896	0.475948	+	0.879473i	$0.342105\pi$
$420$	13.8573	0.00160992
$421$	−8612.02	−0.996970	−0.498485	−	0.866898i	$-0.666110\pi$
$421$	−8612.02	−0.996970	−0.498485	+	0.866898i	$0.666110\pi$
$422$	7496.31	0.864727
$423$	10925.6	1.25584
$424$	9647.32	1.10499
$425$	−2250.81	−0.256895
$426$	−31.2836	−0.00355797
$427$	−154.705	−0.0175332
$428$	−2549.82	−0.287967
$429$	−207.886	−0.0233959
$430$	−3398.46	−0.381135
$431$	12147.2	1.35756	0.678782	−	0.734340i	$-0.262508\pi$
$431$	12147.2	1.35756	0.678782	+	0.734340i	$0.262508\pi$
$432$	−2339.22	−0.260522
$433$	2420.61	0.268654	0.134327	−	0.990937i	$-0.457113\pi$
$433$	2420.61	0.268654	0.134327	+	0.990937i	$0.457113\pi$
$434$	−418.031	−0.0462353
$435$	1664.29	0.183440
$436$	3784.64	0.415715
$437$	384.451	0.0420842
$438$	−1257.17	−0.137146
$439$	3285.62	0.357208	0.178604	−	0.983921i	$-0.442842\pi$
$439$	3285.62	0.357208	0.178604	+	0.983921i	$0.442842\pi$
$440$	−1352.40	−0.146530
$441$	8817.31	0.952090
$442$	3707.67	0.398996
$443$	−5275.94	−0.565840	−0.282920	−	0.959143i	$-0.591303\pi$
$443$	−5275.94	−0.565840	−0.282920	+	0.959143i	$0.591303\pi$
$444$	−516.461	−0.0552031
$445$	−6360.13	−0.677526
$446$	−5663.78	−0.601318
$447$	−2552.45	−0.270083
$448$	−620.445	−0.0654314
$449$	−12236.8	−1.28617	−0.643087	−	0.765793i	$-0.722347\pi$
$449$	−12236.8	−1.28617	−0.643087	+	0.765793i	$0.722347\pi$
$450$	−1540.86	−0.161415
$451$	−2674.17	−0.279206
$452$	1236.46	0.128669
$453$	−3040.08	−0.315310
$454$	743.117	0.0768198
$455$	−95.0480	−0.00979323
$456$	−512.229	−0.0526037
$457$	304.782	0.0311971	0.0155986	−	0.999878i	$-0.495035\pi$
$457$	304.782	0.0311971	0.0155986	+	0.999878i	$0.495035\pi$
$458$	−10191.7	−1.03980
$459$	−5211.75	−0.529986
$460$	−231.893	−0.0235045
$461$	−11305.5	−1.14219	−0.571095	−	0.820884i	$-0.693481\pi$
$461$	−11305.5	−1.14219	−0.571095	+	0.820884i	$0.693481\pi$
$462$	−31.7767	−0.00319997
$463$	7269.30	0.729661	0.364830	−	0.931074i	$-0.381127\pi$
$463$	7269.30	0.729661	0.364830	+	0.931074i	$0.381127\pi$
$464$	−12268.1	−1.22744
$465$	869.759	0.0867400
$466$	848.254	0.0843232
$467$	−10411.6	−1.03168	−0.515838	−	0.856686i	$-0.672519\pi$
$467$	−10411.6	−1.03168	−0.515838	+	0.856686i	$0.672519\pi$
$468$	−1019.25	−0.100672
$469$	−227.847	−0.0224328
$470$	5059.05	0.496503
$471$	−2072.73	−0.202774
$472$	6047.85	0.589777
$473$	−3129.44	−0.304211
$474$	650.711	0.0630552
$475$	−475.000	−0.0458831
$476$	−227.582	−0.0219143
$477$	10121.6	0.971565
$478$	4561.24	0.436456
$479$	−15353.6	−1.46456	−0.732281	−	0.681003i	$-0.761544\pi$
$479$	−15353.6	−1.46456	−0.732281	+	0.681003i	$0.761544\pi$
$480$	549.124	0.0522166
$481$	3542.45	0.335804
$482$	6661.07	0.629468
$483$	−24.4661	−0.00230485
$484$	−277.342	−0.0260464
$485$	−1332.71	−0.124773
$486$	−5392.41	−0.503302
$487$	11606.6	1.07997	0.539986	−	0.841674i	$-0.318429\pi$
$487$	11606.6	1.07997	0.539986	+	0.841674i	$0.318429\pi$
$488$	−3449.35	−0.319969
$489$	−226.963	−0.0209890
$490$	4082.82	0.376414
$491$	16895.9	1.55296	0.776479	−	0.630143i	$-0.217004\pi$
$491$	16895.9	1.55296	0.776479	+	0.630143i	$0.217004\pi$
$492$	610.935	0.0559819
$493$	−27333.1	−2.49700
$494$	782.450	0.0712633
$495$	−1418.89	−0.128837
$496$	−6411.30	−0.580395
$497$	−13.1710	−0.00118873
$498$	862.701	0.0776277
$499$	20973.1	1.88154	0.940768	−	0.339050i	$-0.110106\pi$
$499$	20973.1	1.88154	0.940768	+	0.339050i	$0.110106\pi$
$500$	286.511	0.0256263
$501$	3724.75	0.332155
$502$	−12359.3	−1.09885
$503$	8817.17	0.781587	0.390794	−	0.920478i	$-0.372201\pi$
$503$	8817.17	0.781587	0.390794	+	0.920478i	$0.372201\pi$
$504$	−699.577	−0.0618286
$505$	−4321.48	−0.380798
$506$	531.764	0.0467189
$507$	2083.03	0.182466
$508$	4321.22	0.377408
$509$	11799.5	1.02751	0.513755	−	0.857937i	$-0.328254\pi$
$509$	11799.5	1.02751	0.513755	+	0.857937i	$0.328254\pi$
$510$	−1179.17	−0.102381
$511$	−529.292	−0.0458209
$512$	−11996.9	−1.03553
$513$	−1099.86	−0.0946591
$514$	−2780.77	−0.238627
$515$	−6385.51	−0.546367
$516$	714.944	0.0609955
$517$	4658.57	0.396294
$518$	541.485	0.0459295
$519$	−3517.34	−0.297483
$520$	−2119.22	−0.178719
$521$	20886.7	1.75636	0.878178	−	0.478333i	$-0.158759\pi$
$521$	20886.7	1.75636	0.878178	+	0.478333i	$0.158759\pi$
$522$	−18711.7	−1.56895
$523$	−6384.88	−0.533827	−0.266913	−	0.963721i	$-0.586004\pi$
$523$	−6384.88	−0.533827	−0.266913	+	0.963721i	$0.586004\pi$
$524$	−3380.89	−0.281861
$525$	30.2285	0.00251292
$526$	−12758.9	−1.05764
$527$	−14284.3	−1.18071
$528$	−487.356	−0.0401694
$529$	−11757.6	−0.966350
$530$	4686.77	0.384114
$531$	6345.17	0.518563
$532$	−48.0278	−0.00391404
$533$	−4190.45	−0.340541
$534$	−3331.98	−0.270017
$535$	−5562.22	−0.449487
$536$	−5080.15	−0.409383
$537$	3507.98	0.281900
$538$	−14184.1	−1.13665
$539$	3759.62	0.300442
$540$	663.415	0.0528682
$541$	20815.7	1.65423	0.827114	−	0.562035i	$-0.189981\pi$
$541$	20815.7	1.65423	0.827114	+	0.562035i	$0.189981\pi$
$542$	−6848.58	−0.542753
$543$	3856.67	0.304798
$544$	−9018.43	−0.710776
$545$	8255.90	0.648888
$546$	−49.7943	−0.00390293
$547$	23129.8	1.80797	0.903983	−	0.427568i	$-0.140630\pi$
$547$	23129.8	1.80797	0.903983	+	0.427568i	$0.140630\pi$
$548$	−4776.00	−0.372301
$549$	−3618.93	−0.281333
$550$	−657.009	−0.0509363
$551$	−5768.25	−0.445981
$552$	−545.504	−0.0420619
$553$	273.962	0.0210670
$554$	8967.23	0.687691
$555$	−1126.62	−0.0861664
$556$	3657.20	0.278957
$557$	15139.5	1.15167	0.575835	−	0.817566i	$-0.304677\pi$
$557$	15139.5	1.15167	0.575835	+	0.817566i	$0.304677\pi$
$558$	−9778.77	−0.741879
$559$	−4903.85	−0.371039
$560$	−222.825	−0.0168144
$561$	−1085.82	−0.0817175
$562$	18960.9	1.42317
$563$	−22374.5	−1.67491	−0.837453	−	0.546509i	$-0.815957\pi$
$563$	−22374.5	−1.67491	−0.837453	+	0.546509i	$0.815957\pi$
$564$	−1064.29	−0.0794585
$565$	2697.24	0.200839
$566$	7540.37	0.559974
$567$	−698.175	−0.0517118
$568$	−293.665	−0.0216935
$569$	−6970.59	−0.513572	−0.256786	−	0.966468i	$-0.582664\pi$
$569$	−6970.59	−0.513572	−0.256786	+	0.966468i	$0.582664\pi$
$570$	−248.846	−0.0182860
$571$	8871.05	0.650161	0.325080	−	0.945686i	$-0.394609\pi$
$571$	8871.05	0.650161	0.325080	+	0.945686i	$0.394609\pi$
$572$	−434.598	−0.0317683
$573$	−200.965	−0.0146517
$574$	−640.536	−0.0465775
$575$	−505.857	−0.0366881
$576$	−14513.7	−1.04989
$577$	−5112.36	−0.368857	−0.184429	−	0.982846i	$-0.559043\pi$
$577$	−5112.36	−0.368857	−0.184429	+	0.982846i	$0.559043\pi$
$578$	7628.04	0.548936
$579$	1706.74	0.122504
$580$	3479.29	0.249086
$581$	363.214	0.0259357
$582$	−698.186	−0.0497264
$583$	4315.76	0.306588
$584$	−11801.3	−0.836199
$585$	−2223.41	−0.157139
$586$	19378.3	1.36606
$587$	−4921.57	−0.346056	−0.173028	−	0.984917i	$-0.555355\pi$
$587$	−4921.57	−0.346056	−0.173028	+	0.984917i	$0.555355\pi$
$588$	−858.915	−0.0602399
$589$	−3014.49	−0.210883
$590$	2938.11	0.205017
$591$	2608.57	0.181560
$592$	8304.71	0.576556
$593$	15409.6	1.06711	0.533556	−	0.845765i	$-0.320855\pi$
$593$	15409.6	1.06711	0.533556	+	0.845765i	$0.320855\pi$
$594$	−1521.31	−0.105084
$595$	−496.452	−0.0342059
$596$	−5336.05	−0.366734
$597$	−4029.81	−0.276263
$598$	833.279	0.0569821
$599$	−2793.53	−0.190552	−0.0952760	−	0.995451i	$-0.530373\pi$
$599$	−2793.53	−0.190552	−0.0952760	+	0.995451i	$0.530373\pi$
$600$	673.985	0.0458589
$601$	9994.27	0.678328	0.339164	−	0.940727i	$-0.389856\pi$
$601$	9994.27	0.678328	0.339164	+	0.940727i	$0.389856\pi$
$602$	−749.585	−0.0507488
$603$	−5329.90	−0.359951
$604$	−6355.48	−0.428147
$605$	−605.000	−0.0406558
$606$	−2263.96	−0.151761
$607$	17682.5	1.18239	0.591193	−	0.806530i	$-0.298657\pi$
$607$	17682.5	1.18239	0.591193	+	0.806530i	$0.298657\pi$
$608$	−1903.21	−0.126949
$609$	367.085	0.0244254
$610$	−1675.73	−0.111227
$611$	7300.03	0.483351
$612$	−5323.70	−0.351630
$613$	−22829.9	−1.50423	−0.752114	−	0.659033i	$-0.770966\pi$
$613$	−22829.9	−1.50423	−0.752114	+	0.659033i	$0.770966\pi$
$614$	−24158.7	−1.58789
$615$	1332.71	0.0873820
$616$	−298.293	−0.0195107
$617$	3147.71	0.205384	0.102692	−	0.994713i	$-0.467254\pi$
$617$	3147.71	0.205384	0.102692	+	0.994713i	$0.467254\pi$
$618$	−3345.28	−0.217746
$619$	11409.1	0.740823	0.370412	−	0.928868i	$-0.379217\pi$
$619$	11409.1	0.740823	0.370412	+	0.928868i	$0.379217\pi$
$620$	1818.28	0.117781
$621$	−1171.31	−0.0756894
$622$	8185.32	0.527655
$623$	−1402.83	−0.0902137
$624$	−763.691	−0.0489938
$625$	625.000	0.0400000
$626$	13481.5	0.860749
$627$	−229.147	−0.0145953
$628$	−4333.17	−0.275338
$629$	18502.8	1.17290
$630$	−339.861	−0.0214927
$631$	21031.6	1.32687	0.663436	−	0.748233i	$-0.269097\pi$
$631$	21031.6	1.32687	0.663436	+	0.748233i	$0.269097\pi$
$632$	6108.34	0.384457
$633$	−3440.15	−0.216009
$634$	19854.5	1.24373
$635$	9426.41	0.589095
$636$	−985.970	−0.0614721
$637$	5891.36	0.366443
$638$	−7978.51	−0.495097
$639$	−308.102	−0.0190740
$640$	−2713.77	−0.167611
$641$	−28627.1	−1.76396	−0.881982	−	0.471284i	$-0.843791\pi$
$641$	−28627.1	−1.76396	−0.881982	+	0.471284i	$0.843791\pi$
$642$	−2913.97	−0.179136
$643$	−5927.03	−0.363514	−0.181757	−	0.983344i	$-0.558178\pi$
$643$	−5927.03	−0.363514	−0.181757	+	0.983344i	$0.558178\pi$
$644$	−51.1477	−0.00312966
$645$	1559.59	0.0952077
$646$	4086.87	0.248910
$647$	−18027.6	−1.09542	−0.547710	−	0.836668i	$-0.684500\pi$
$647$	−18027.6	−1.09542	−0.547710	+	0.836668i	$0.684500\pi$
$648$	−15566.7	−0.943703
$649$	2705.53	0.163638
$650$	−1029.54	−0.0621259
$651$	191.839	0.0115496
$652$	−474.480	−0.0285001
$653$	20204.2	1.21080	0.605400	−	0.795921i	$-0.293013\pi$
$653$	20204.2	1.21080	0.605400	+	0.795921i	$0.293013\pi$
$654$	4325.15	0.258604
$655$	−7375.15	−0.439956
$656$	−9823.85	−0.584690
$657$	−12381.4	−0.735230
$658$	1115.85	0.0661102
$659$	7317.01	0.432519	0.216260	−	0.976336i	$-0.430614\pi$
$659$	7317.01	0.432519	0.216260	+	0.976336i	$0.430614\pi$
$660$	138.217	0.00815165
$661$	11597.3	0.682423	0.341211	−	0.939987i	$-0.389163\pi$
$661$	11597.3	0.682423	0.341211	+	0.939987i	$0.389163\pi$
$662$	−14340.5	−0.841934
$663$	−1701.50	−0.0996691
$664$	8098.33	0.473308
$665$	−104.769	−0.00610942
$666$	12666.7	0.736972
$667$	−6142.96	−0.356606
$668$	7786.81	0.451019
$669$	2599.18	0.150209
$670$	−2467.99	−0.142309
$671$	−1543.08	−0.0887777
$672$	121.118	0.00695273
$673$	−6366.84	−0.364671	−0.182336	−	0.983236i	$-0.558366\pi$
$673$	−6366.84	−0.364671	−0.182336	+	0.983236i	$0.558366\pi$
$674$	−8253.55	−0.471684
$675$	1447.19	0.0825219
$676$	4354.69	0.247763
$677$	5775.20	0.327856	0.163928	−	0.986472i	$-0.447583\pi$
$677$	5775.20	0.327856	0.163928	+	0.986472i	$0.447583\pi$
$678$	1413.05	0.0800410
$679$	−293.950	−0.0166138
$680$	−11069.1	−0.624234
$681$	−341.025	−0.0191896
$682$	−4169.58	−0.234108
$683$	−13218.5	−0.740544	−0.370272	−	0.928923i	$-0.620736\pi$
$683$	−13218.5	−0.740544	−0.370272	+	0.928923i	$0.620736\pi$
$684$	−1123.49	−0.0628036
$685$	−10418.5	−0.581123
$686$	1804.27	0.100419
$687$	4677.11	0.259742
$688$	−11496.3	−0.637054
$689$	6762.84	0.373939
$690$	−265.011	−0.0146215
$691$	10897.0	0.599917	0.299958	−	0.953952i	$-0.403027\pi$
$691$	10897.0	0.599917	0.299958	+	0.953952i	$0.403027\pi$
$692$	−7353.20	−0.403940
$693$	−312.958	−0.0171548
$694$	−7686.77	−0.420440
$695$	7977.90	0.435423
$696$	8184.66	0.445745
$697$	−21887.4	−1.18945
$698$	21771.3	1.18060
$699$	−389.274	−0.0210639
$700$	63.1945	0.00341218
$701$	−13271.7	−0.715069	−0.357535	−	0.933900i	$-0.616383\pi$
$701$	−13271.7	−0.715069	−0.357535	+	0.933900i	$0.616383\pi$
$702$	−2383.90	−0.128169
$703$	3904.74	0.209488
$704$	−6188.53	−0.331305
$705$	−2321.66	−0.124027
$706$	−30251.6	−1.61265
$707$	−953.171	−0.0507039
$708$	−618.098	−0.0328101
$709$	3745.25	0.198386	0.0991932	−	0.995068i	$-0.468374\pi$
$709$	3745.25	0.198386	0.0991932	+	0.995068i	$0.468374\pi$
$710$	−142.665	−0.00754103
$711$	6408.64	0.338035
$712$	−31277.9	−1.64633
$713$	−3210.32	−0.168622
$714$	−260.084	−0.0136322
$715$	−948.041	−0.0495870
$716$	7333.63	0.382780
$717$	−2093.21	−0.109027
$718$	23192.9	1.20550
$719$	31438.8	1.63069	0.815346	−	0.578974i	$-0.196547\pi$
$719$	31438.8	1.63069	0.815346	+	0.578974i	$0.196547\pi$
$720$	−5212.43	−0.269800
$721$	−1408.43	−0.0727497
$722$	862.474	0.0444570
$723$	−3056.84	−0.157241
$724$	8062.59	0.413873
$725$	7589.80	0.388797
$726$	−316.951	−0.0162027
$727$	13375.4	0.682349	0.341174	−	0.940000i	$-0.389175\pi$
$727$	13375.4	0.682349	0.341174	+	0.940000i	$0.389175\pi$
$728$	−467.428	−0.0237968
$729$	−14618.4	−0.742692
$730$	−5733.18	−0.290677
$731$	−25613.7	−1.29597
$732$	352.528	0.0178003
$733$	10725.4	0.540450	0.270225	−	0.962797i	$-0.412902\pi$
$733$	10725.4	0.540450	0.270225	+	0.962797i	$0.412902\pi$
$734$	25278.8	1.27119
$735$	−1873.65	−0.0940283
$736$	−2026.84	−0.101509
$737$	−2272.62	−0.113586
$738$	−14983.7	−0.747369
$739$	6852.48	0.341099	0.170550	−	0.985349i	$-0.445446\pi$
$739$	6852.48	0.341099	0.170550	+	0.985349i	$0.445446\pi$
$740$	−2355.26	−0.117002
$741$	−359.076	−0.0178016
$742$	1033.74	0.0511454
$743$	−35123.8	−1.73428	−0.867138	−	0.498068i	$-0.834043\pi$
$743$	−35123.8	−1.73428	−0.867138	+	0.498068i	$0.834043\pi$
$744$	4277.31	0.210771
$745$	−11640.2	−0.572434
$746$	19510.2	0.957533
$747$	8496.46	0.416157
$748$	−2269.98	−0.110961
$749$	−1226.84	−0.0598500
$750$	327.429	0.0159413
$751$	−16946.8	−0.823433	−0.411717	−	0.911312i	$-0.635071\pi$
$751$	−16946.8	−0.823433	−0.411717	+	0.911312i	$0.635071\pi$
$752$	17113.8	0.829887
$753$	5671.85	0.274493
$754$	−12502.4	−0.603860
$755$	−13864.0	−0.668293
$756$	146.327	0.00703949
$757$	−21051.8	−1.01076	−0.505378	−	0.862898i	$-0.668647\pi$
$757$	−21051.8	−1.01076	−0.505378	+	0.862898i	$0.668647\pi$
$758$	−27635.0	−1.32420
$759$	−244.033	−0.0116704
$760$	−2335.96	−0.111492
$761$	18352.7	0.874224	0.437112	−	0.899407i	$-0.356001\pi$
$761$	18352.7	0.874224	0.437112	+	0.899407i	$0.356001\pi$
$762$	4938.36	0.234774
$763$	1820.97	0.0864005
$764$	−420.128	−0.0198949
$765$	−11613.2	−0.548859
$766$	16076.0	0.758290
$767$	4239.58	0.199586
$768$	3512.90	0.165053
$769$	−15415.8	−0.722899	−0.361449	−	0.932392i	$-0.617718\pi$
$769$	−15415.8	−0.722899	−0.361449	+	0.932392i	$0.617718\pi$
$770$	−144.914	−0.00678225
$771$	1276.13	0.0596091
$772$	3568.04	0.166343
$773$	735.833	0.0342381	0.0171191	−	0.999853i	$-0.494551\pi$
$773$	735.833	0.0342381	0.0171191	+	0.999853i	$0.494551\pi$
$774$	−17534.6	−0.814301
$775$	3966.44	0.183843
$776$	−6554.00	−0.303189
$777$	−248.494	−0.0114732
$778$	−8513.78	−0.392331
$779$	−4619.02	−0.212444
$780$	216.587	0.00994240
$781$	−131.372	−0.00601902
$782$	4352.36	0.199028
$783$	17574.2	0.802107
$784$	13811.4	0.629162
$785$	−9452.47	−0.429775
$786$	−3863.74	−0.175337
$787$	−11292.9	−0.511499	−0.255750	−	0.966743i	$-0.582322\pi$
$787$	−11292.9	−0.511499	−0.255750	+	0.966743i	$0.582322\pi$
$788$	5453.36	0.246533
$789$	5855.23	0.264197
$790$	2967.50	0.133644
$791$	594.920	0.0267420
$792$	−6977.81	−0.313063
$793$	−2418.02	−0.108280
$794$	−6091.25	−0.272255
$795$	−2150.81	−0.0959516
$796$	−8424.56	−0.375126
$797$	3981.57	0.176956	0.0884782	−	0.996078i	$-0.471800\pi$
$797$	3981.57	0.176956	0.0884782	+	0.996078i	$0.471800\pi$
$798$	−54.8870	−0.00243481
$799$	38129.3	1.68826
$800$	2504.22	0.110672
$801$	−32815.6	−1.44754
$802$	30890.7	1.36009
$803$	−5279.34	−0.232010
$804$	519.199	0.0227745
$805$	−111.575	−0.00488508
$806$	−6533.77	−0.285536
$807$	6509.23	0.283935
$808$	−21252.2	−0.925310
$809$	27324.8	1.18750	0.593750	−	0.804650i	$-0.297647\pi$
$809$	27324.8	1.18750	0.593750	+	0.804650i	$0.297647\pi$
$810$	−7562.49	−0.328048
$811$	20334.5	0.880443	0.440222	−	0.897889i	$-0.354900\pi$
$811$	20334.5	0.880443	0.440222	+	0.897889i	$0.354900\pi$
$812$	767.413	0.0331662
$813$	3142.90	0.135580
$814$	5400.95	0.232559
$815$	−1035.04	−0.0444858
$816$	−3988.89	−0.171126
$817$	−5405.39	−0.231470
$818$	−32729.6	−1.39898
$819$	−490.408	−0.0209234
$820$	2786.10	0.118652
$821$	2747.86	0.116810	0.0584050	−	0.998293i	$-0.481399\pi$
$821$	2747.86	0.116810	0.0584050	+	0.998293i	$0.481399\pi$
$822$	−5458.09	−0.231597
$823$	1888.16	0.0799721	0.0399861	−	0.999200i	$-0.487269\pi$
$823$	1888.16	0.0799721	0.0399861	+	0.999200i	$0.487269\pi$
$824$	−31402.7	−1.32763
$825$	301.509	0.0127239
$826$	648.047	0.0272983
$827$	22204.6	0.933653	0.466827	−	0.884349i	$-0.345397\pi$
$827$	22204.6	0.933653	0.466827	+	0.884349i	$0.345397\pi$
$828$	−1196.47	−0.0502177
$829$	−21477.0	−0.899790	−0.449895	−	0.893082i	$-0.648539\pi$
$829$	−21477.0	−0.899790	−0.449895	+	0.893082i	$0.648539\pi$
$830$	3934.25	0.164530
$831$	−4115.17	−0.171785
$832$	−9697.48	−0.404086
$833$	30771.6	1.27992
$834$	4179.51	0.173531
$835$	16986.3	0.703995
$836$	−479.046	−0.0198183
$837$	9184.29	0.379278
$838$	19505.2	0.804050
$839$	41745.0	1.71775	0.858877	−	0.512181i	$-0.171162\pi$
$839$	41745.0	1.71775	0.858877	+	0.512181i	$0.171162\pi$
$840$	148.658	0.00610618
$841$	67779.1	2.77908
$842$	−20575.2	−0.842123
$843$	−8701.40	−0.355507
$844$	−7191.83	−0.293309
$845$	9499.41	0.386733
$846$	26102.6	1.06079
$847$	−133.442	−0.00541338
$848$	15854.4	0.642032
$849$	−3460.37	−0.139882
$850$	−5377.46	−0.216995
$851$	4158.40	0.167507
$852$	30.0129	0.00120684
$853$	16564.2	0.664884	0.332442	−	0.943124i	$-0.392127\pi$
$853$	16564.2	0.664884	0.332442	+	0.943124i	$0.392127\pi$
$854$	−369.609	−0.0148100
$855$	−2450.80	−0.0980300
$856$	−27354.0	−1.09222
$857$	2280.82	0.0909117	0.0454558	−	0.998966i	$-0.485526\pi$
$857$	2280.82	0.0909117	0.0454558	+	0.998966i	$0.485526\pi$
$858$	−496.665	−0.0197621
$859$	3383.49	0.134393	0.0671963	−	0.997740i	$-0.478595\pi$
$859$	3383.49	0.134393	0.0671963	+	0.997740i	$0.478595\pi$
$860$	3260.42	0.129278
$861$	293.950	0.0116351
$862$	29021.2	1.14671
$863$	98.4376	0.00388280	0.00194140	−	0.999998i	$-0.499382\pi$
$863$	98.4376	0.00388280	0.00194140	+	0.999998i	$0.499382\pi$
$864$	5798.52	0.228322
$865$	−16040.4	−0.630509
$866$	5783.15	0.226928
$867$	−3500.60	−0.137124
$868$	401.051	0.0156827
$869$	2732.59	0.106671
$870$	3976.19	0.154949
$871$	−3561.22	−0.138539
$872$	40601.0	1.57675
$873$	−6876.20	−0.266580
$874$	918.501	0.0355478
$875$	137.854	0.00532607
$876$	1206.11	0.0465189
$877$	5410.16	0.208310	0.104155	−	0.994561i	$-0.466786\pi$
$877$	5410.16	0.208310	0.104155	+	0.994561i	$0.466786\pi$
$878$	7849.76	0.301727
$879$	−8892.93	−0.341241
$880$	−2222.53	−0.0851381
$881$	−48258.2	−1.84547	−0.922735	−	0.385436i	$-0.874051\pi$
$881$	−48258.2	−1.84547	−0.922735	+	0.385436i	$0.874051\pi$
$882$	21065.6	0.804214
$883$	−25265.6	−0.962917	−0.481459	−	0.876469i	$-0.659893\pi$
$883$	−25265.6	−0.962917	−0.481459	+	0.876469i	$0.659893\pi$
$884$	−3557.08	−0.135336
$885$	−1348.33	−0.0512132
$886$	−12604.9	−0.477956
$887$	12786.2	0.484011	0.242005	−	0.970275i	$-0.422195\pi$
$887$	12786.2	0.484011	0.242005	+	0.970275i	$0.422195\pi$
$888$	−5540.50	−0.209377
$889$	2079.14	0.0784390
$890$	−15195.1	−0.572295
$891$	−6963.83	−0.261838
$892$	5433.73	0.203963
$893$	8046.63	0.301534
$894$	−6098.13	−0.228134
$895$	15997.7	0.597481
$896$	−598.566	−0.0223177
$897$	−382.402	−0.0142341
$898$	−29235.3	−1.08641
$899$	48167.2	1.78695
$900$	1478.27	0.0547509
$901$	35323.5	1.30610
$902$	−6388.93	−0.235840
$903$	343.993	0.0126771
$904$	13264.5	0.488022
$905$	17587.9	0.646013
$906$	−7263.14	−0.266337
$907$	23860.6	0.873517	0.436758	−	0.899579i	$-0.356126\pi$
$907$	23860.6	0.873517	0.436758	+	0.899579i	$0.356126\pi$
$908$	−712.933	−0.0260567
$909$	−22297.0	−0.813581
$910$	−227.081	−0.00827217
$911$	51077.0	1.85758	0.928791	−	0.370603i	$-0.120849\pi$
$911$	51077.0	1.85758	0.928791	+	0.370603i	$0.120849\pi$
$912$	−841.797	−0.0305643
$913$	3622.82	0.131323
$914$	728.162	0.0263517
$915$	769.012	0.0277844
$916$	9777.78	0.352693
$917$	−1626.71	−0.0585808
$918$	−12451.5	−0.447670
$919$	−26441.5	−0.949102	−0.474551	−	0.880228i	$-0.657390\pi$
$919$	−26441.5	−0.949102	−0.474551	+	0.880228i	$0.657390\pi$
$920$	−2487.71	−0.0891493
$921$	11086.7	0.396656
$922$	−27010.2	−0.964788
$923$	−205.861	−0.00734127
$924$	30.4860	0.00108541
$925$	−5137.82	−0.182628
$926$	17367.3	0.616332
$927$	−32946.5	−1.16732
$928$	30410.4	1.07572
$929$	47556.4	1.67952	0.839761	−	0.542956i	$-0.182695\pi$
$929$	47556.4	1.67952	0.839761	+	0.542956i	$0.182695\pi$
$930$	2077.96	0.0732678
$931$	6493.89	0.228602
$932$	−813.800	−0.0286018
$933$	−3756.34	−0.131808
$934$	−24874.7	−0.871440
$935$	−4951.78	−0.173198
$936$	−10934.3	−0.381836
$937$	27254.0	0.950214	0.475107	−	0.879928i	$-0.342409\pi$
$937$	27254.0	0.950214	0.475107	+	0.879928i	$0.342409\pi$
$938$	−544.355	−0.0189486
$939$	−6186.82	−0.215015
$940$	−4853.56	−0.168410
$941$	23110.5	0.800618	0.400309	−	0.916380i	$-0.368903\pi$
$941$	23110.5	0.800618	0.400309	+	0.916380i	$0.368903\pi$
$942$	−4952.02	−0.171280
$943$	−4919.08	−0.169870
$944$	9939.04	0.342678
$945$	319.200	0.0109879
$946$	−7476.61	−0.256961
$947$	46536.9	1.59688	0.798440	−	0.602074i	$-0.205659\pi$
$947$	46536.9	1.59688	0.798440	+	0.602074i	$0.205659\pi$
$948$	−624.281	−0.0213879
$949$	−8272.77	−0.282977
$950$	−1134.83	−0.0387567
$951$	−9111.48	−0.310683
$952$	−2441.46	−0.0831177
$953$	−22841.6	−0.776404	−0.388202	−	0.921574i	$-0.626904\pi$
$953$	−22841.6	−0.776404	−0.388202	+	0.921574i	$0.626904\pi$
$954$	24181.8	0.820664
$955$	−916.477	−0.0310539
$956$	−4375.97	−0.148043
$957$	3661.43	0.123675
$958$	−36681.7	−1.23709
$959$	−2297.96	−0.0773775
$960$	3084.13	0.103687
$961$	−4618.78	−0.155039
$962$	8463.35	0.283648
$963$	−28698.7	−0.960336
$964$	−6390.51	−0.213511
$965$	7783.40	0.259644
$966$	−58.4525	−0.00194687
$967$	−25859.4	−0.859961	−0.429980	−	0.902838i	$-0.641480\pi$
$967$	−25859.4	−0.859961	−0.429980	+	0.902838i	$0.641480\pi$
$968$	−2975.28	−0.0987903
$969$	−1875.51	−0.0621777
$970$	−3184.00	−0.105394
$971$	−7224.28	−0.238762	−0.119381	−	0.992848i	$-0.538091\pi$
$971$	−7224.28	−0.238762	−0.119381	+	0.992848i	$0.538091\pi$
$972$	5173.39	0.170716
$973$	1759.65	0.0579773
$974$	27729.7	0.912235
$975$	472.468	0.0155191
$976$	−5668.66	−0.185911
$977$	−33307.2	−1.09068	−0.545338	−	0.838216i	$-0.683599\pi$
$977$	−33307.2	−1.09068	−0.545338	+	0.838216i	$0.683599\pi$
$978$	−542.244	−0.0177291
$979$	−13992.3	−0.456788
$980$	−3916.99	−0.127677
$981$	42597.0	1.38636
$982$	40366.5	1.31176
$983$	−45211.9	−1.46697	−0.733487	−	0.679704i	$-0.762108\pi$
$983$	−45211.9	−1.46697	−0.733487	+	0.679704i	$0.762108\pi$
$984$	6554.00	0.212331
$985$	11896.1	0.384813
$986$	−65302.2	−2.10917
$987$	−512.079	−0.0165143
$988$	−750.669	−0.0241720
$989$	−5756.53	−0.185083
$990$	−3389.89	−0.108826
$991$	−16097.7	−0.516004	−0.258002	−	0.966144i	$-0.583064\pi$
$991$	−16097.7	−0.516004	−0.258002	+	0.966144i	$0.583064\pi$
$992$	15892.5	0.508658
$993$	6581.04	0.210315
$994$	−31.4671	−0.00100410
$995$	−18377.5	−0.585534
$996$	−827.661	−0.0263308
$997$	36604.3	1.16276	0.581379	−	0.813633i	$-0.302513\pi$
$997$	36604.3	1.16276	0.581379	+	0.813633i	$0.302513\pi$
$998$	50107.5	1.58930
$999$	−11896.6	−0.376769

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.g.1.16	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.g.1.16	✓	23	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)

\(f(q)\)	\(=\)	\(q+2.38912 q^{2} -1.09640 q^{3} -2.29208 q^{4} -5.00000 q^{5} -2.61943 q^{6} -1.10283 q^{7} -24.5891 q^{8} -25.7979 q^{9} +O(q^{10})\)	\(q+2.38912 q^{2} -1.09640 q^{3} -2.29208 q^{4} -5.00000 q^{5} -2.61943 q^{6} -1.10283 q^{7} -24.5891 q^{8} -25.7979 q^{9} -11.9456 q^{10} -11.0000 q^{11} +2.51304 q^{12} -17.2371 q^{13} -2.63480 q^{14} +5.48199 q^{15} -40.4097 q^{16} -90.0323 q^{17} -61.6344 q^{18} -19.0000 q^{19} +11.4604 q^{20} +1.20914 q^{21} -26.2804 q^{22} -20.2343 q^{23} +26.9594 q^{24} +25.0000 q^{25} -41.1816 q^{26} +57.8875 q^{27} +2.52778 q^{28} +303.592 q^{29} +13.0972 q^{30} +158.658 q^{31} +100.169 q^{32} +12.0604 q^{33} -215.098 q^{34} +5.51415 q^{35} +59.1310 q^{36} -205.513 q^{37} -45.3934 q^{38} +18.8987 q^{39} +122.945 q^{40} +243.106 q^{41} +2.88879 q^{42} +284.494 q^{43} +25.2129 q^{44} +128.990 q^{45} -48.3422 q^{46} -423.507 q^{47} +44.3051 q^{48} -341.784 q^{49} +59.7281 q^{50} +98.7112 q^{51} +39.5089 q^{52} -392.342 q^{53} +138.300 q^{54} +55.0000 q^{55} +27.1176 q^{56} +20.8316 q^{57} +725.319 q^{58} -245.957 q^{59} -12.5652 q^{60} +140.280 q^{61} +379.053 q^{62} +28.4507 q^{63} +562.593 q^{64} +86.1855 q^{65} +28.8137 q^{66} +206.602 q^{67} +206.362 q^{68} +22.1848 q^{69} +13.1740 q^{70} +11.9429 q^{71} +634.347 q^{72} +479.940 q^{73} -490.996 q^{74} -27.4099 q^{75} +43.5496 q^{76} +12.1311 q^{77} +45.1514 q^{78} -248.417 q^{79} +202.048 q^{80} +633.076 q^{81} +580.811 q^{82} -329.347 q^{83} -2.77145 q^{84} +450.162 q^{85} +679.692 q^{86} -332.858 q^{87} +270.480 q^{88} +1272.03 q^{89} +308.172 q^{90} +19.0096 q^{91} +46.3786 q^{92} -173.952 q^{93} -1011.81 q^{94} +95.0000 q^{95} -109.825 q^{96} +266.541 q^{97} -816.564 q^{98} +283.777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * q^99

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