LMFDB - Embedded newform 1045.4.a.g.1.9

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			1.9
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)

$f(q)$	$=$	$q-1.45938 q^{2} +5.64597 q^{3} -5.87020 q^{4} -5.00000 q^{5} -8.23963 q^{6} -17.6407 q^{7} +20.2419 q^{8} +4.87695 q^{9} +O(q^{10})$	\(q-1.45938 q^{2} +5.64597 q^{3} -5.87020 q^{4} -5.00000 q^{5} -8.23963 q^{6} -17.6407 q^{7} +20.2419 q^{8} +4.87695 q^{9} +7.29692 q^{10} -11.0000 q^{11} -33.1429 q^{12} -0.138199 q^{13} +25.7446 q^{14} -28.2298 q^{15} +17.4208 q^{16} +28.7527 q^{17} -7.11734 q^{18} -19.0000 q^{19} +29.3510 q^{20} -99.5990 q^{21} +16.0532 q^{22} -148.034 q^{23} +114.285 q^{24} +25.0000 q^{25} +0.201685 q^{26} -124.906 q^{27} +103.555 q^{28} +229.633 q^{29} +41.1982 q^{30} -217.638 q^{31} -187.359 q^{32} -62.1056 q^{33} -41.9612 q^{34} +88.2036 q^{35} -28.6287 q^{36} +6.91435 q^{37} +27.7283 q^{38} -0.780267 q^{39} -101.210 q^{40} -113.298 q^{41} +145.353 q^{42} +263.713 q^{43} +64.5722 q^{44} -24.3847 q^{45} +216.039 q^{46} -448.756 q^{47} +98.3574 q^{48} -31.8047 q^{49} -36.4846 q^{50} +162.337 q^{51} +0.811256 q^{52} +411.962 q^{53} +182.286 q^{54} +55.0000 q^{55} -357.083 q^{56} -107.273 q^{57} -335.122 q^{58} +1.42587 q^{59} +165.715 q^{60} -8.19685 q^{61} +317.617 q^{62} -86.0329 q^{63} +134.063 q^{64} +0.690995 q^{65} +90.6360 q^{66} +896.045 q^{67} -168.784 q^{68} -835.795 q^{69} -128.723 q^{70} +993.114 q^{71} +98.7189 q^{72} -299.616 q^{73} -10.0907 q^{74} +141.149 q^{75} +111.534 q^{76} +194.048 q^{77} +1.13871 q^{78} +822.292 q^{79} -87.1041 q^{80} -836.893 q^{81} +165.345 q^{82} +501.460 q^{83} +584.666 q^{84} -143.763 q^{85} -384.859 q^{86} +1296.50 q^{87} -222.661 q^{88} +774.110 q^{89} +35.5867 q^{90} +2.43793 q^{91} +868.989 q^{92} -1228.77 q^{93} +654.907 q^{94} +95.0000 q^{95} -1057.82 q^{96} +767.801 q^{97} +46.4153 q^{98} -53.6464 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.45938	−0.515970	−0.257985	−	0.966149i	$-0.583059\pi$
$2$	−1.45938	−0.515970	−0.257985	+	0.966149i	$0.583059\pi$
$3$	5.64597	1.08657	0.543283	−	0.839549i	$-0.317181\pi$
$3$	5.64597	1.08657	0.543283	+	0.839549i	$0.317181\pi$
$4$	−5.87020	−0.733775
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−8.23963	−0.560636
$7$	−17.6407	−0.952510	−0.476255	−	0.879307i	$-0.658006\pi$
$7$	−17.6407	−0.952510	−0.476255	+	0.879307i	$0.658006\pi$
$8$	20.2419	0.894576
$9$	4.87695	0.180628
$10$	7.29692	0.230749
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−33.1429	−0.797295
$13$	−0.138199	−0.00294842	−0.00147421	−	0.999999i	$-0.500469\pi$
$13$	−0.138199	−0.00294842	−0.00147421	+	0.999999i	$0.500469\pi$
$14$	25.7446	0.491467
$15$	−28.2298	−0.485928
$16$	17.4208	0.272200
$17$	28.7527	0.410209	0.205104	−	0.978740i	$-0.434247\pi$
$17$	28.7527	0.410209	0.205104	+	0.978740i	$0.434247\pi$
$18$	−7.11734	−0.0931985
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	29.3510	0.328154
$21$	−99.5990	−1.03497
$22$	16.0532	0.155571
$23$	−148.034	−1.34205	−0.671027	−	0.741433i	$-0.734146\pi$
$23$	−148.034	−1.34205	−0.671027	+	0.741433i	$0.734146\pi$
$24$	114.285	0.972017
$25$	25.0000	0.200000
$26$	0.201685	0.00152130
$27$	−124.906	−0.890303
$28$	103.555	0.698928
$29$	229.633	1.47041	0.735203	−	0.677848i	$-0.237087\pi$
$29$	229.633	1.47041	0.735203	+	0.677848i	$0.237087\pi$
$30$	41.1982	0.250724
$31$	−217.638	−1.26093	−0.630465	−	0.776217i	$-0.717136\pi$
$31$	−217.638	−1.26093	−0.630465	+	0.776217i	$0.717136\pi$
$32$	−187.359	−1.03502
$33$	−62.1056	−0.327612
$34$	−41.9612	−0.211656
$35$	88.2036	0.425975
$36$	−28.6287	−0.132540
$37$	6.91435	0.0307219	0.0153610	−	0.999882i	$-0.495110\pi$
$37$	6.91435	0.0307219	0.0153610	+	0.999882i	$0.495110\pi$
$38$	27.7283	0.118372
$39$	−0.780267	−0.00320366
$40$	−101.210	−0.400067
$41$	−113.298	−0.431564	−0.215782	−	0.976442i	$-0.569230\pi$
$41$	−113.298	−0.431564	−0.215782	+	0.976442i	$0.569230\pi$
$42$	145.353	0.534011
$43$	263.713	0.935253	0.467627	−	0.883926i	$-0.345109\pi$
$43$	263.713	0.935253	0.467627	+	0.883926i	$0.345109\pi$
$44$	64.5722	0.221241
$45$	−24.3847	−0.0807792
$46$	216.039	0.692460
$47$	−448.756	−1.39272	−0.696359	−	0.717693i	$-0.745198\pi$
$47$	−448.756	−1.39272	−0.696359	+	0.717693i	$0.745198\pi$
$48$	98.3574	0.295764
$49$	−31.8047	−0.0927251
$50$	−36.4846	−0.103194
$51$	162.337	0.445720
$52$	0.811256	0.00216348
$53$	411.962	1.06769	0.533843	−	0.845583i	$-0.320747\pi$
$53$	411.962	1.06769	0.533843	+	0.845583i	$0.320747\pi$
$54$	182.286	0.459370
$55$	55.0000	0.134840
$56$	−357.083	−0.852092
$57$	−107.273	−0.249276
$58$	−335.122	−0.758685
$59$	1.42587	0.00314632	0.00157316	−	0.999999i	$-0.499499\pi$
$59$	1.42587	0.00314632	0.00157316	+	0.999999i	$0.499499\pi$
$60$	165.715	0.356561
$61$	−8.19685	−0.0172049	−0.00860245	−	0.999963i	$-0.502738\pi$
$61$	−8.19685	−0.0172049	−0.00860245	+	0.999963i	$0.502738\pi$
$62$	317.617	0.650603
$63$	−86.0329	−0.172050
$64$	134.063	0.261841
$65$	0.690995	0.00131858
$66$	90.6360	0.169038
$67$	896.045	1.63387	0.816935	−	0.576730i	$-0.195671\pi$
$67$	896.045	1.63387	0.816935	+	0.576730i	$0.195671\pi$
$68$	−168.784	−0.301001
$69$	−835.795	−1.45823
$70$	−128.723	−0.219791
$71$	993.114	1.66001	0.830006	−	0.557754i	$-0.188337\pi$
$71$	993.114	1.66001	0.830006	+	0.557754i	$0.188337\pi$
$72$	98.7189	0.161585
$73$	−299.616	−0.480375	−0.240188	−	0.970726i	$-0.577209\pi$
$73$	−299.616	−0.480375	−0.240188	+	0.970726i	$0.577209\pi$
$74$	−10.0907	−0.0158516
$75$	141.149	0.217313
$76$	111.534	0.168339
$77$	194.048	0.287193
$78$	1.13871	0.00165299
$79$	822.292	1.17108	0.585539	−	0.810644i	$-0.300883\pi$
$79$	822.292	1.17108	0.585539	+	0.810644i	$0.300883\pi$
$80$	−87.1041	−0.121732
$81$	−836.893	−1.14800
$82$	165.345	0.222674
$83$	501.460	0.663161	0.331581	−	0.943427i	$-0.392418\pi$
$83$	501.460	0.663161	0.331581	+	0.943427i	$0.392418\pi$
$84$	584.666	0.759432
$85$	−143.763	−0.183451
$86$	−384.859	−0.482563
$87$	1296.50	1.59769
$88$	−222.661	−0.269725
$89$	774.110	0.921972	0.460986	−	0.887407i	$-0.347496\pi$
$89$	774.110	0.921972	0.460986	+	0.887407i	$0.347496\pi$
$90$	35.5867	0.0416796
$91$	2.43793	0.00280840
$92$	868.989	0.984765
$93$	−1228.77	−1.37009
$94$	654.907	0.718601
$95$	95.0000	0.102598
$96$	−1057.82	−1.12462
$97$	767.801	0.803695	0.401847	−	0.915707i	$-0.368368\pi$
$97$	767.801	0.803695	0.401847	+	0.915707i	$0.368368\pi$
$98$	46.4153	0.0478434
$99$	−53.6464	−0.0544613
$100$	−146.755	−0.146755
$101$	−435.371	−0.428921	−0.214460	−	0.976733i	$-0.568799\pi$
$101$	−435.371	−0.428921	−0.214460	+	0.976733i	$0.568799\pi$
$102$	−236.912	−0.229978
$103$	1498.83	1.43382	0.716911	−	0.697164i	$-0.245555\pi$
$103$	1498.83	1.43382	0.716911	+	0.697164i	$0.245555\pi$
$104$	−2.79742	−0.00263759
$105$	497.995	0.462851
$106$	−601.211	−0.550894
$107$	751.895	0.679331	0.339665	−	0.940546i	$-0.389686\pi$
$107$	751.895	0.679331	0.339665	+	0.940546i	$0.389686\pi$
$108$	733.223	0.653282
$109$	998.125	0.877092	0.438546	−	0.898709i	$-0.355494\pi$
$109$	998.125	0.877092	0.438546	+	0.898709i	$0.355494\pi$
$110$	−80.2661	−0.0695734
$111$	39.0382	0.0333814
$112$	−307.316	−0.259273
$113$	−2035.53	−1.69457	−0.847286	−	0.531137i	$-0.821765\pi$
$113$	−2035.53	−1.69457	−0.847286	+	0.531137i	$0.821765\pi$
$114$	156.553	0.128619
$115$	740.170	0.600185
$116$	−1347.99	−1.07895
$117$	−0.673990	−0.000532567 0
$118$	−2.08090	−0.00162341
$119$	−507.218	−0.390728
$120$	−571.427	−0.434699
$121$	121.000	0.0909091
$122$	11.9623	0.00887721
$123$	−639.675	−0.468923
$124$	1277.58	0.925239
$125$	−125.000	−0.0894427
$126$	125.555	0.0887725
$127$	738.397	0.515922	0.257961	−	0.966155i	$-0.416949\pi$
$127$	738.397	0.515922	0.257961	+	0.966155i	$0.416949\pi$
$128$	1303.23	0.899921
$129$	1488.92	1.01622
$130$	−1.00843	−0.000680346 0
$131$	377.746	0.251937	0.125969	−	0.992034i	$-0.459796\pi$
$131$	377.746	0.251937	0.125969	+	0.992034i	$0.459796\pi$
$132$	364.572	0.240394
$133$	335.174	0.218521
$134$	−1307.67	−0.843028
$135$	624.530	0.398156
$136$	582.010	0.366963
$137$	1675.97	1.04517	0.522584	−	0.852588i	$-0.324968\pi$
$137$	1675.97	1.04517	0.522584	+	0.852588i	$0.324968\pi$
$138$	1219.75	0.752404
$139$	610.161	0.372325	0.186162	−	0.982519i	$-0.440395\pi$
$139$	610.161	0.372325	0.186162	+	0.982519i	$0.440395\pi$
$140$	−517.773	−0.312570
$141$	−2533.66	−1.51328
$142$	−1449.33	−0.856517
$143$	1.52019	0.000888983 0
$144$	84.9604	0.0491669
$145$	−1148.16	−0.657585
$146$	437.255	0.247859
$147$	−179.568	−0.100752
$148$	−40.5886	−0.0225430
$149$	162.615	0.0894088	0.0447044	−	0.999000i	$-0.485765\pi$
$149$	162.615	0.0894088	0.0447044	+	0.999000i	$0.485765\pi$
$150$	−205.991	−0.112127
$151$	−2215.87	−1.19421	−0.597103	−	0.802164i	$-0.703682\pi$
$151$	−2215.87	−1.19421	−0.597103	+	0.802164i	$0.703682\pi$
$152$	−384.597	−0.205230
$153$	140.225	0.0740951
$154$	−283.191	−0.148183
$155$	1088.19	0.563905
$156$	4.58032	0.00235077
$157$	3247.09	1.65061	0.825306	−	0.564685i	$-0.191003\pi$
$157$	3247.09	1.65061	0.825306	+	0.564685i	$0.191003\pi$
$158$	−1200.04	−0.604241
$159$	2325.93	1.16011
$160$	936.796	0.462876
$161$	2611.43	1.27832
$162$	1221.35	0.592334
$163$	821.350	0.394681	0.197341	−	0.980335i	$-0.436769\pi$
$163$	821.350	0.394681	0.197341	+	0.980335i	$0.436769\pi$
$164$	665.080	0.316671
$165$	310.528	0.146513
$166$	−731.823	−0.342171
$167$	−1928.19	−0.893459	−0.446729	−	0.894669i	$-0.647411\pi$
$167$	−1928.19	−0.893459	−0.446729	+	0.894669i	$0.647411\pi$
$168$	−2016.08	−0.925856
$169$	−2196.98	−0.999991
$170$	209.806	0.0946553
$171$	−92.6620	−0.0414388
$172$	−1548.05	−0.686265
$173$	−4129.09	−1.81462	−0.907309	−	0.420465i	$-0.861867\pi$
$173$	−4129.09	−1.81462	−0.907309	+	0.420465i	$0.861867\pi$
$174$	−1892.09	−0.824362
$175$	−441.018	−0.190502
$176$	−191.629	−0.0820715
$177$	8.05043	0.00341869
$178$	−1129.72	−0.475710
$179$	310.062	0.129470	0.0647350	−	0.997902i	$-0.479380\pi$
$179$	310.062	0.129470	0.0647350	+	0.997902i	$0.479380\pi$
$180$	143.143	0.0592737
$181$	−4549.65	−1.86836	−0.934178	−	0.356807i	$-0.883865\pi$
$181$	−4549.65	−1.86836	−0.934178	+	0.356807i	$0.883865\pi$
$182$	−3.55788	−0.00144905
$183$	−46.2791	−0.0186943
$184$	−2996.50	−1.20057
$185$	−34.5717	−0.0137393
$186$	1793.25	0.706923
$187$	−316.280	−0.123683
$188$	2634.29	1.02194
$189$	2203.43	0.848022
$190$	−138.641	−0.0529374
$191$	992.128	0.375853	0.187926	−	0.982183i	$-0.439823\pi$
$191$	992.128	0.375853	0.187926	+	0.982183i	$0.439823\pi$
$192$	756.913	0.284508
$193$	2217.10	0.826893	0.413447	−	0.910528i	$-0.364325\pi$
$193$	2217.10	0.826893	0.413447	+	0.910528i	$0.364325\pi$
$194$	−1120.52	−0.414682
$195$	3.90134	0.00143272
$196$	186.700	0.0680393
$197$	−765.895	−0.276994	−0.138497	−	0.990363i	$-0.544227\pi$
$197$	−765.895	−0.276994	−0.138497	+	0.990363i	$0.544227\pi$
$198$	78.2908	0.0281004
$199$	3316.18	1.18130	0.590648	−	0.806929i	$-0.298872\pi$
$199$	3316.18	1.18130	0.590648	+	0.806929i	$0.298872\pi$
$200$	506.049	0.178915
$201$	5059.04	1.77531
$202$	635.373	0.221310
$203$	−4050.89	−1.40058
$204$	−952.949	−0.327058
$205$	566.489	0.193001
$206$	−2187.36	−0.739810
$207$	−721.955	−0.242412
$208$	−2.40754	−0.000802562 0
$209$	209.000	0.0691714
$210$	−726.766	−0.238817
$211$	−5951.06	−1.94165	−0.970825	−	0.239789i	$-0.922922\pi$
$211$	−5951.06	−1.94165	−0.970825	+	0.239789i	$0.922922\pi$
$212$	−2418.30	−0.783441
$213$	5607.09	1.80372
$214$	−1097.30	−0.350514
$215$	−1318.57	−0.418258
$216$	−2528.34	−0.796444
$217$	3839.29	1.20105
$218$	−1456.65	−0.452554
$219$	−1691.62	−0.521960
$220$	−322.861	−0.0989422
$221$	−3.97359	−0.00120947
$222$	−56.9717	−0.0172238
$223$	4890.57	1.46860	0.734298	−	0.678828i	$-0.237512\pi$
$223$	4890.57	1.46860	0.734298	+	0.678828i	$0.237512\pi$
$224$	3305.15	0.985870
$225$	121.924	0.0361255
$226$	2970.62	0.874348
$227$	3781.85	1.10577	0.552886	−	0.833257i	$-0.313527\pi$
$227$	3781.85	1.10577	0.552886	+	0.833257i	$0.313527\pi$
$228$	629.716	0.182912
$229$	430.082	0.124107	0.0620537	−	0.998073i	$-0.480235\pi$
$229$	430.082	0.124107	0.0620537	+	0.998073i	$0.480235\pi$
$230$	−1080.19	−0.309677
$231$	1095.59	0.312054
$232$	4648.22	1.31539
$233$	−742.967	−0.208899	−0.104449	−	0.994530i	$-0.533308\pi$
$233$	−742.967	−0.208899	−0.104449	+	0.994530i	$0.533308\pi$
$234$	0.983610	0.000274789 0
$235$	2243.78	0.622843
$236$	−8.37016	−0.00230869
$237$	4642.64	1.27245
$238$	740.227	0.201604
$239$	3858.13	1.04419	0.522096	−	0.852887i	$-0.325150\pi$
$239$	3858.13	1.04419	0.522096	+	0.852887i	$0.325150\pi$
$240$	−491.787	−0.132270
$241$	7192.19	1.92236	0.961182	−	0.275914i	$-0.0889804\pi$
$241$	7192.19	1.92236	0.961182	+	0.275914i	$0.0889804\pi$
$242$	−176.585	−0.0469064
$243$	−1352.61	−0.357078
$244$	48.1171	0.0126245
$245$	159.024	0.0414679
$246$	933.532	0.241950
$247$	2.62578	0.000676415 0
$248$	−4405.41	−1.12800
$249$	2831.23	0.720569
$250$	182.423	0.0461498
$251$	3192.93	0.802932	0.401466	−	0.915874i	$-0.368501\pi$
$251$	3192.93	0.802932	0.401466	+	0.915874i	$0.368501\pi$
$252$	505.030	0.126246
$253$	1628.37	0.404644
$254$	−1077.60	−0.266200
$255$	−811.684	−0.199332
$256$	−2974.41	−0.726173
$257$	4097.43	0.994517	0.497259	−	0.867602i	$-0.334340\pi$
$257$	4097.43	0.994517	0.497259	+	0.867602i	$0.334340\pi$
$258$	−2172.90	−0.524337
$259$	−121.974	−0.0292630
$260$	−4.05628	−0.000967537 0
$261$	1119.91	0.265596
$262$	−551.276	−0.129992
$263$	4159.02	0.975119	0.487559	−	0.873090i	$-0.337887\pi$
$263$	4159.02	0.975119	0.487559	+	0.873090i	$0.337887\pi$
$264$	−1257.14	−0.293074
$265$	−2059.81	−0.477484
$266$	−489.147	−0.112750
$267$	4370.60	1.00178
$268$	−5259.96	−1.19889
$269$	−1251.09	−0.283570	−0.141785	−	0.989897i	$-0.545284\pi$
$269$	−1251.09	−0.283570	−0.141785	+	0.989897i	$0.545284\pi$
$270$	−911.429	−0.205436
$271$	3400.35	0.762201	0.381101	−	0.924534i	$-0.375545\pi$
$271$	3400.35	0.762201	0.381101	+	0.924534i	$0.375545\pi$
$272$	500.895	0.111659
$273$	13.7645	0.00305152
$274$	−2445.89	−0.539276
$275$	−275.000	−0.0603023
$276$	4906.29	1.07001
$277$	−3972.38	−0.861651	−0.430825	−	0.902435i	$-0.641778\pi$
$277$	−3972.38	−0.861651	−0.430825	+	0.902435i	$0.641778\pi$
$278$	−890.459	−0.192109
$279$	−1061.41	−0.227759
$280$	1785.41	0.381067
$281$	−626.430	−0.132988	−0.0664941	−	0.997787i	$-0.521181\pi$
$281$	−626.430	−0.132988	−0.0664941	+	0.997787i	$0.521181\pi$
$282$	3697.58	0.780808
$283$	5313.53	1.11610	0.558050	−	0.829807i	$-0.311550\pi$
$283$	5313.53	1.11610	0.558050	+	0.829807i	$0.311550\pi$
$284$	−5829.77	−1.21808
$285$	536.367	0.111479
$286$	−2.21854	−0.000458689 0
$287$	1998.65	0.411069
$288$	−913.741	−0.186954
$289$	−4086.28	−0.831729
$290$	1675.61	0.339294
$291$	4334.98	0.873268
$292$	1758.81	0.352487
$293$	−5773.51	−1.15117	−0.575584	−	0.817743i	$-0.695225\pi$
$293$	−5773.51	−1.15117	−0.575584	+	0.817743i	$0.695225\pi$
$294$	262.059	0.0519850
$295$	−7.12937	−0.00140708
$296$	139.960	0.0274831
$297$	1373.97	0.268436
$298$	−237.317	−0.0461323
$299$	20.4582	0.00395694
$300$	−828.574	−0.159459
$301$	−4652.09	−0.890838
$302$	3233.81	0.616175
$303$	−2458.09	−0.466051
$304$	−330.996	−0.0624470
$305$	40.9842	0.00769426
$306$	−204.643	−0.0382309
$307$	8815.71	1.63889	0.819445	−	0.573158i	$-0.194282\pi$
$307$	8815.71	1.63889	0.819445	+	0.573158i	$0.194282\pi$
$308$	−1139.10	−0.210735
$309$	8462.33	1.55794
$310$	−1588.08	−0.290958
$311$	−2571.35	−0.468836	−0.234418	−	0.972136i	$-0.575318\pi$
$311$	−2571.35	−0.468836	−0.234418	+	0.972136i	$0.575318\pi$
$312$	−15.7941	−0.00286592
$313$	3625.48	0.654710	0.327355	−	0.944901i	$-0.393843\pi$
$313$	3625.48	0.654710	0.327355	+	0.944901i	$0.393843\pi$
$314$	−4738.75	−0.851667
$315$	430.165	0.0769430
$316$	−4827.02	−0.859307
$317$	−1133.68	−0.200864	−0.100432	−	0.994944i	$-0.532023\pi$
$317$	−1133.68	−0.200864	−0.100432	+	0.994944i	$0.532023\pi$
$318$	−3394.42	−0.598584
$319$	−2525.96	−0.443344
$320$	−670.313	−0.117099
$321$	4245.17	0.738138
$322$	−3811.08	−0.659575
$323$	−546.301	−0.0941084
$324$	4912.73	0.842374
$325$	−3.45498	−0.000589685 0
$326$	−1198.66	−0.203644
$327$	5635.38	0.953020
$328$	−2293.37	−0.386067
$329$	7916.38	1.32658
$330$	−453.180	−0.0755962
$331$	1860.11	0.308885	0.154443	−	0.988002i	$-0.450642\pi$
$331$	1860.11	0.308885	0.154443	+	0.988002i	$0.450642\pi$
$332$	−2943.67	−0.486611
$333$	33.7209	0.00554924
$334$	2813.97	0.460998
$335$	−4480.22	−0.730689
$336$	−1735.10	−0.281718
$337$	−6332.27	−1.02356	−0.511782	−	0.859116i	$-0.671014\pi$
$337$	−6332.27	−1.02356	−0.511782	+	0.859116i	$0.671014\pi$
$338$	3206.24	0.515966
$339$	−11492.5	−1.84127
$340$	843.920	0.134612
$341$	2394.01	0.380185
$342$	135.229	0.0213812
$343$	6611.83	1.04083
$344$	5338.07	0.836655
$345$	4178.98	0.652141
$346$	6025.93	0.936288
$347$	−8814.43	−1.36364	−0.681821	−	0.731519i	$-0.738812\pi$
$347$	−8814.43	−1.36364	−0.681821	+	0.731519i	$0.738812\pi$
$348$	−7610.71	−1.17235
$349$	−10792.1	−1.65527	−0.827636	−	0.561265i	$-0.810315\pi$
$349$	−10792.1	−1.65527	−0.827636	+	0.561265i	$0.810315\pi$
$350$	643.615	0.0982933
$351$	17.2619	0.00262499
$352$	2060.95	0.312071
$353$	5793.00	0.873457	0.436728	−	0.899593i	$-0.356137\pi$
$353$	5793.00	0.873457	0.436728	+	0.899593i	$0.356137\pi$
$354$	−11.7487	−0.00176394
$355$	−4965.57	−0.742380
$356$	−4544.18	−0.676520
$357$	−2863.74	−0.424552
$358$	−452.500	−0.0668026
$359$	10330.2	1.51868	0.759339	−	0.650695i	$-0.225522\pi$
$359$	10330.2	1.51868	0.759339	+	0.650695i	$0.225522\pi$
$360$	−493.595	−0.0722631
$361$	361.000	0.0526316
$362$	6639.68	0.964016
$363$	683.162	0.0987788
$364$	−14.3111	−0.00206074
$365$	1498.08	0.214830
$366$	67.5390	0.00964569
$367$	−10099.4	−1.43647	−0.718236	−	0.695799i	$-0.755050\pi$
$367$	−10099.4	−1.43647	−0.718236	+	0.695799i	$0.755050\pi$
$368$	−2578.87	−0.365307
$369$	−552.547	−0.0779525
$370$	50.4534	0.00708905
$371$	−7267.32	−1.01698
$372$	7213.15	1.00533
$373$	−12501.0	−1.73533	−0.867665	−	0.497150i	$-0.834380\pi$
$373$	−12501.0	−1.73533	−0.867665	+	0.497150i	$0.834380\pi$
$374$	461.573	0.0638166
$375$	−705.746	−0.0971855
$376$	−9083.69	−1.24589
$377$	−31.7350	−0.00433538
$378$	−3215.66	−0.437554
$379$	8936.83	1.21122	0.605612	−	0.795760i	$-0.292928\pi$
$379$	8936.83	1.21122	0.605612	+	0.795760i	$0.292928\pi$
$380$	−557.669	−0.0752837
$381$	4168.96	0.560584
$382$	−1447.90	−0.193929
$383$	−11108.6	−1.48205	−0.741023	−	0.671479i	$-0.765659\pi$
$383$	−11108.6	−1.48205	−0.741023	+	0.671479i	$0.765659\pi$
$384$	7357.97	0.977825
$385$	−970.240	−0.128436
$386$	−3235.60	−0.426652
$387$	1286.12	0.168933
$388$	−4507.14	−0.589731
$389$	8100.60	1.05583	0.527914	−	0.849298i	$-0.322974\pi$
$389$	8100.60	1.05583	0.527914	+	0.849298i	$0.322974\pi$
$390$	−5.69355	−0.000739241 0
$391$	−4256.38	−0.550522
$392$	−643.789	−0.0829496
$393$	2132.74	0.273747
$394$	1117.73	0.142920
$395$	−4111.46	−0.523722
$396$	314.915	0.0399623
$397$	−12483.4	−1.57814	−0.789070	−	0.614303i	$-0.789437\pi$
$397$	−12483.4	−1.57814	−0.789070	+	0.614303i	$0.789437\pi$
$398$	−4839.58	−0.609513
$399$	1892.38	0.237437
$400$	435.520	0.0544401
$401$	−3260.74	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$401$	−3260.74	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$402$	−7383.08	−0.916007
$403$	30.0773	0.00371776
$404$	2555.71	0.314731
$405$	4184.46	0.513402
$406$	5911.80	0.722655
$407$	−76.0578	−0.00926301
$408$	3286.01	0.398730
$409$	7831.69	0.946827	0.473414	−	0.880840i	$-0.343022\pi$
$409$	7831.69	0.946827	0.473414	+	0.880840i	$0.343022\pi$
$410$	−826.724	−0.0995830
$411$	9462.49	1.13565
$412$	−8798.41	−1.05210
$413$	−25.1534	−0.00299690
$414$	1053.61	0.125077
$415$	−2507.30	−0.296575
$416$	25.8929	0.00305169
$417$	3444.95	0.404556
$418$	−305.011	−0.0356904
$419$	−7557.99	−0.881222	−0.440611	−	0.897698i	$-0.645238\pi$
$419$	−7557.99	−0.881222	−0.440611	+	0.897698i	$0.645238\pi$
$420$	−2923.33	−0.339628
$421$	8085.07	0.935968	0.467984	−	0.883737i	$-0.344981\pi$
$421$	8085.07	0.935968	0.467984	+	0.883737i	$0.344981\pi$
$422$	8684.89	1.00183
$423$	−2188.56	−0.251564
$424$	8338.92	0.955127
$425$	718.817	0.0820418
$426$	−8182.89	−0.930663
$427$	144.598	0.0163878
$428$	−4413.77	−0.498476
$429$	8.58294	0.000965940 0
$430$	1924.29	0.215809
$431$	−7080.33	−0.791293	−0.395647	−	0.918403i	$-0.629479\pi$
$431$	−7080.33	−0.791293	−0.395647	+	0.918403i	$0.629479\pi$
$432$	−2175.96	−0.242341
$433$	−15523.6	−1.72290	−0.861450	−	0.507842i	$-0.830443\pi$
$433$	−15523.6	−1.72290	−0.861450	+	0.507842i	$0.830443\pi$
$434$	−5602.99	−0.619706
$435$	−6482.50	−0.714510
$436$	−5859.19	−0.643588
$437$	2812.65	0.307888
$438$	2468.73	0.269316
$439$	1475.24	0.160386	0.0801928	−	0.996779i	$-0.474446\pi$
$439$	1475.24	0.160386	0.0801928	+	0.996779i	$0.474446\pi$
$440$	1113.31	0.120625
$441$	−155.110	−0.0167487
$442$	5.79900	0.000624050 0
$443$	−9868.63	−1.05840	−0.529202	−	0.848496i	$-0.677509\pi$
$443$	−9868.63	−1.05840	−0.529202	+	0.848496i	$0.677509\pi$
$444$	−229.162	−0.0244945
$445$	−3870.55	−0.412319
$446$	−7137.22	−0.757751
$447$	918.118	0.0971487
$448$	−2364.96	−0.249406
$449$	11452.2	1.20370	0.601852	−	0.798608i	$-0.294430\pi$
$449$	11452.2	1.20370	0.601852	+	0.798608i	$0.294430\pi$
$450$	−177.934	−0.0186397
$451$	1246.28	0.130122
$452$	11949.0	1.24343
$453$	−12510.7	−1.29759
$454$	−5519.17	−0.570545
$455$	−12.1897	−0.00125596
$456$	−2171.42	−0.222996
$457$	1538.54	0.157483	0.0787414	−	0.996895i	$-0.474910\pi$
$457$	1538.54	0.157483	0.0787414	+	0.996895i	$0.474910\pi$
$458$	−627.654	−0.0640357
$459$	−3591.38	−0.365210
$460$	−4344.95	−0.440400
$461$	790.685	0.0798826	0.0399413	−	0.999202i	$-0.487283\pi$
$461$	790.685	0.0798826	0.0399413	+	0.999202i	$0.487283\pi$
$462$	−1598.88	−0.161010
$463$	−3442.25	−0.345518	−0.172759	−	0.984964i	$-0.555268\pi$
$463$	−3442.25	−0.345518	−0.172759	+	0.984964i	$0.555268\pi$
$464$	4000.39	0.400245
$465$	6143.87	0.612721
$466$	1084.27	0.107785
$467$	19733.6	1.95538	0.977692	−	0.210044i	$-0.0673607\pi$
$467$	19733.6	1.95538	0.977692	+	0.210044i	$0.0673607\pi$
$468$	3.95645	0.000390784 0
$469$	−15806.9	−1.55628
$470$	−3274.54	−0.321368
$471$	18333.0	1.79350
$472$	28.8625	0.00281462
$473$	−2900.85	−0.281990
$474$	−6775.39	−0.656548
$475$	−475.000	−0.0458831
$476$	2977.47	0.286706
$477$	2009.12	0.192854
$478$	−5630.49	−0.538772
$479$	−2858.27	−0.272647	−0.136323	−	0.990664i	$-0.543529\pi$
$479$	−2858.27	−0.272647	−0.136323	+	0.990664i	$0.543529\pi$
$480$	5289.12	0.502946
$481$	−0.955556	−9.05813e−5 0
$482$	−10496.2	−0.991883
$483$	14744.0	1.38898
$484$	−710.294	−0.0667068
$485$	−3839.00	−0.359423
$486$	1973.97	0.184241
$487$	−13190.9	−1.22739	−0.613695	−	0.789543i	$-0.710318\pi$
$487$	−13190.9	−1.22739	−0.613695	+	0.789543i	$0.710318\pi$
$488$	−165.920	−0.0153911
$489$	4637.31	0.428848
$490$	−232.076	−0.0213962
$491$	13680.3	1.25740	0.628700	−	0.777648i	$-0.283587\pi$
$491$	13680.3	1.25740	0.628700	+	0.777648i	$0.283587\pi$
$492$	3755.02	0.344084
$493$	6602.56	0.603173
$494$	−3.83202	−0.000349010 0
$495$	268.232	0.0243558
$496$	−3791.42	−0.343226
$497$	−17519.2	−1.58118
$498$	−4131.85	−0.371792
$499$	−8293.19	−0.743996	−0.371998	−	0.928233i	$-0.621327\pi$
$499$	−8293.19	−0.743996	−0.371998	+	0.928233i	$0.621327\pi$
$500$	733.775	0.0656308
$501$	−10886.5	−0.970803
$502$	−4659.71	−0.414289
$503$	6581.96	0.583449	0.291725	−	0.956502i	$-0.405771\pi$
$503$	6581.96	0.583449	0.291725	+	0.956502i	$0.405771\pi$
$504$	−1741.47	−0.153912
$505$	2176.85	0.191819
$506$	−2376.42	−0.208784
$507$	−12404.1	−1.08656
$508$	−4334.53	−0.378571
$509$	−788.106	−0.0686291	−0.0343145	−	0.999411i	$-0.510925\pi$
$509$	−788.106	−0.0686291	−0.0343145	+	0.999411i	$0.510925\pi$
$510$	1184.56	0.102849
$511$	5285.44	0.457562
$512$	−6085.00	−0.525237
$513$	2373.21	0.204249
$514$	−5979.73	−0.513141
$515$	−7494.13	−0.641225
$516$	−8740.24	−0.745673
$517$	4936.31	0.419920
$518$	178.007	0.0150988
$519$	−23312.7	−1.97170
$520$	13.9871	0.00117957
$521$	15659.5	1.31680	0.658401	−	0.752667i	$-0.271233\pi$
$521$	15659.5	1.31680	0.658401	+	0.752667i	$0.271233\pi$
$522$	−1634.38	−0.137040
$523$	−9996.68	−0.835801	−0.417901	−	0.908493i	$-0.637234\pi$
$523$	−9996.68	−0.835801	−0.417901	+	0.908493i	$0.637234\pi$
$524$	−2217.44	−0.184865
$525$	−2489.97	−0.206993
$526$	−6069.61	−0.503132
$527$	−6257.67	−0.517245
$528$	−1081.93	−0.0891761
$529$	9747.08	0.801108
$530$	3006.06	0.246367
$531$	6.95391	0.000568313 0
$532$	−1967.54	−0.160345
$533$	15.6576	0.00127243
$534$	−6378.39	−0.516891
$535$	−3759.47	−0.303806
$536$	18137.7	1.46162
$537$	1750.60	0.140678
$538$	1825.82	0.146314
$539$	349.852	0.0279577
$540$	−3666.12	−0.292156
$541$	−11374.5	−0.903930	−0.451965	−	0.892036i	$-0.649277\pi$
$541$	−11374.5	−0.903930	−0.451965	+	0.892036i	$0.649277\pi$
$542$	−4962.42	−0.393273
$543$	−25687.1	−2.03009
$544$	−5387.08	−0.424576
$545$	−4990.63	−0.392248
$546$	−20.0877	−0.00157449
$547$	−4484.08	−0.350503	−0.175252	−	0.984524i	$-0.556074\pi$
$547$	−4484.08	−0.350503	−0.175252	+	0.984524i	$0.556074\pi$
$548$	−9838.30	−0.766918
$549$	−39.9756	−0.00310768
$550$	401.331	0.0311142
$551$	−4363.02	−0.337334
$552$	−16918.1	−1.30450
$553$	−14505.8	−1.11546
$554$	5797.23	0.444586
$555$	−195.191	−0.0149286
$556$	−3581.77	−0.273203
$557$	−4533.57	−0.344872	−0.172436	−	0.985021i	$-0.555164\pi$
$557$	−4533.57	−0.344872	−0.172436	+	0.985021i	$0.555164\pi$
$558$	1549.00	0.117517
$559$	−36.4449	−0.00275752
$560$	1536.58	0.115951
$561$	−1785.70	−0.134389
$562$	914.202	0.0686180
$563$	11021.6	0.825054	0.412527	−	0.910945i	$-0.364646\pi$
$563$	11021.6	0.825054	0.412527	+	0.910945i	$0.364646\pi$
$564$	14873.1	1.11041
$565$	10177.7	0.757835
$566$	−7754.48	−0.575875
$567$	14763.4	1.09348
$568$	20102.6	1.48501
$569$	−13604.4	−1.00233	−0.501164	−	0.865353i	$-0.667094\pi$
$569$	−13604.4	−1.00233	−0.501164	+	0.865353i	$0.667094\pi$
$570$	−782.765	−0.0575201
$571$	−18414.1	−1.34958	−0.674788	−	0.738012i	$-0.735765\pi$
$571$	−18414.1	−1.34958	−0.674788	+	0.738012i	$0.735765\pi$
$572$	−8.92381	−0.000652314 0
$573$	5601.52	0.408389
$574$	−2916.80	−0.212099
$575$	−3700.85	−0.268411
$576$	653.816	0.0472957
$577$	18037.2	1.30138	0.650690	−	0.759343i	$-0.274480\pi$
$577$	18037.2	1.30138	0.650690	+	0.759343i	$0.274480\pi$
$578$	5963.46	0.429147
$579$	12517.7	0.898475
$580$	6739.95	0.482519
$581$	−8846.12	−0.631668
$582$	−6326.40	−0.450580
$583$	−4531.59	−0.321920
$584$	−6064.81	−0.429732
$585$	3.36995	0.000238171 0
$586$	8425.77	0.593968
$587$	−15317.0	−1.07700	−0.538501	−	0.842625i	$-0.681009\pi$
$587$	−15317.0	−1.07700	−0.538501	+	0.842625i	$0.681009\pi$
$588$	1054.10	0.0739293
$589$	4135.11	0.289277
$590$	10.4045	0.000726010 0
$591$	−4324.22	−0.300972
$592$	120.454	0.00836252
$593$	−8156.38	−0.564827	−0.282413	−	0.959293i	$-0.591135\pi$
$593$	−8156.38	−0.564827	−0.282413	+	0.959293i	$0.591135\pi$
$594$	−2005.14	−0.138505
$595$	2536.09	0.174739
$596$	−954.581	−0.0656060
$597$	18723.1	1.28356
$598$	−29.8563	−0.00204166
$599$	22501.6	1.53487	0.767437	−	0.641124i	$-0.221532\pi$
$599$	22501.6	1.53487	0.767437	+	0.641124i	$0.221532\pi$
$600$	2857.13	0.194403
$601$	−17675.8	−1.19969	−0.599844	−	0.800117i	$-0.704771\pi$
$601$	−17675.8	−1.19969	−0.599844	+	0.800117i	$0.704771\pi$
$602$	6789.19	0.459646
$603$	4369.96	0.295122
$604$	13007.6	0.876279
$605$	−605.000	−0.0406558
$606$	3587.30	0.240468
$607$	−15181.6	−1.01516	−0.507582	−	0.861604i	$-0.669460\pi$
$607$	−15181.6	−1.01516	−0.507582	+	0.861604i	$0.669460\pi$
$608$	3559.83	0.237451
$609$	−22871.2	−1.52182
$610$	−59.8117	−0.00397001
$611$	62.0176	0.00410632
$612$	−823.151	−0.0543691
$613$	−19843.6	−1.30747	−0.653733	−	0.756725i	$-0.726798\pi$
$613$	−19843.6	−1.30747	−0.653733	+	0.756725i	$0.726798\pi$
$614$	−12865.5	−0.845618
$615$	3198.38	0.209709
$616$	3927.91	0.256916
$617$	6624.69	0.432253	0.216127	−	0.976365i	$-0.430658\pi$
$617$	6624.69	0.432253	0.216127	+	0.976365i	$0.430658\pi$
$618$	−12349.8	−0.803853
$619$	26260.2	1.70515	0.852573	−	0.522609i	$-0.175041\pi$
$619$	26260.2	1.70515	0.852573	+	0.522609i	$0.175041\pi$
$620$	−6387.88	−0.413780
$621$	18490.3	1.19483
$622$	3752.59	0.241905
$623$	−13655.9	−0.878188
$624$	−13.5929	−0.000872037 0
$625$	625.000	0.0400000
$626$	−5290.97	−0.337811
$627$	1180.01	0.0751594
$628$	−19061.1	−1.21118
$629$	198.806	0.0126024
$630$	−627.775	−0.0397003
$631$	18500.7	1.16719	0.583597	−	0.812043i	$-0.301645\pi$
$631$	18500.7	1.16719	0.583597	+	0.812043i	$0.301645\pi$
$632$	16644.8	1.04762
$633$	−33599.5	−2.10973
$634$	1654.48	0.103640
$635$	−3691.98	−0.230727
$636$	−13653.6	−0.851262
$637$	4.39538	0.000273393 0
$638$	3686.35	0.228752
$639$	4843.37	0.299844
$640$	−6516.13	−0.402457
$641$	−2626.04	−0.161813	−0.0809067	−	0.996722i	$-0.525782\pi$
$641$	−2626.04	−0.161813	−0.0809067	+	0.996722i	$0.525782\pi$
$642$	−6195.34	−0.380857
$643$	12981.0	0.796146	0.398073	−	0.917354i	$-0.369679\pi$
$643$	12981.0	0.796146	0.398073	+	0.917354i	$0.369679\pi$
$644$	−15329.6	−0.937998
$645$	−7444.58	−0.454465
$646$	797.263	0.0485571
$647$	12840.1	0.780210	0.390105	−	0.920770i	$-0.372439\pi$
$647$	12840.1	0.780210	0.390105	+	0.920770i	$0.372439\pi$
$648$	−16940.3	−1.02697
$649$	−15.6846	−0.000948651 0
$650$	5.04214	0.000304260 0
$651$	21676.5	1.30502
$652$	−4821.49	−0.289607
$653$	3401.90	0.203869	0.101935	−	0.994791i	$-0.467497\pi$
$653$	3401.90	0.203869	0.101935	+	0.994791i	$0.467497\pi$
$654$	−8224.19	−0.491730
$655$	−1888.73	−0.112670
$656$	−1973.74	−0.117472
$657$	−1461.21	−0.0867691
$658$	−11553.0	−0.684475
$659$	8993.52	0.531620	0.265810	−	0.964025i	$-0.414361\pi$
$659$	8993.52	0.531620	0.265810	+	0.964025i	$0.414361\pi$
$660$	−1822.86	−0.107507
$661$	21425.1	1.26073	0.630363	−	0.776301i	$-0.282906\pi$
$661$	21425.1	1.26073	0.630363	+	0.776301i	$0.282906\pi$
$662$	−2714.62	−0.159376
$663$	−22.4348	−0.00131417
$664$	10150.5	0.593248
$665$	−1675.87	−0.0977254
$666$	−49.2118	−0.00286324
$667$	−33993.5	−1.97336
$668$	11318.8	0.655598
$669$	27612.0	1.59573
$670$	6538.37	0.377014
$671$	90.1653	0.00518747
$672$	18660.8	1.07121
$673$	−27126.6	−1.55372	−0.776860	−	0.629673i	$-0.783189\pi$
$673$	−27126.6	−1.55372	−0.776860	+	0.629673i	$0.783189\pi$
$674$	9241.21	0.528128
$675$	−3122.65	−0.178061
$676$	12896.7	0.733768
$677$	7717.80	0.438138	0.219069	−	0.975709i	$-0.429698\pi$
$677$	7717.80	0.438138	0.219069	+	0.975709i	$0.429698\pi$
$678$	16772.0	0.950038
$679$	−13544.6	−0.765527
$680$	−2910.05	−0.164111
$681$	21352.2	1.20149
$682$	−3493.78	−0.196164
$683$	−29.4877	−0.00165200	−0.000825999	−	1.00000i	$-0.500263\pi$
$683$	−29.4877	−0.00165200	−0.000825999		1.00000i	$0.500263\pi$
$684$	543.945	0.0304068
$685$	−8379.87	−0.467413
$686$	−9649.20	−0.537038
$687$	2428.23	0.134851
$688$	4594.10	0.254576
$689$	−56.9328	−0.00314799
$690$	−6098.73	−0.336485
$691$	14183.6	0.780852	0.390426	−	0.920634i	$-0.372328\pi$
$691$	14183.6	0.780852	0.390426	+	0.920634i	$0.372328\pi$
$692$	24238.6	1.33152
$693$	946.362	0.0518749
$694$	12863.6	0.703598
$695$	−3050.80	−0.166509
$696$	26243.7	1.42926
$697$	−3257.62	−0.177032
$698$	15749.9	0.854071
$699$	−4194.77	−0.226982
$700$	2588.86	0.139786
$701$	24841.6	1.33845	0.669226	−	0.743059i	$-0.266626\pi$
$701$	24841.6	1.33845	0.669226	+	0.743059i	$0.266626\pi$
$702$	−25.1917	−0.00135442
$703$	−131.373	−0.00704810
$704$	−1474.69	−0.0789480
$705$	12668.3	0.676760
$706$	−8454.21	−0.450678
$707$	7680.26	0.408551
$708$	−47.2576	−0.00250855
$709$	−2813.39	−0.149026	−0.0745128	−	0.997220i	$-0.523740\pi$
$709$	−2813.39	−0.149026	−0.0745128	+	0.997220i	$0.523740\pi$
$710$	7246.67	0.383046
$711$	4010.28	0.211529
$712$	15669.5	0.824774
$713$	32217.8	1.69224
$714$	4179.29	0.219056
$715$	−7.60095	−0.000397565 0
$716$	−1820.13	−0.0950018
$717$	21782.9	1.13458
$718$	−15075.7	−0.783593
$719$	15293.6	0.793260	0.396630	−	0.917979i	$-0.370180\pi$
$719$	15293.6	0.793260	0.396630	+	0.917979i	$0.370180\pi$
$720$	−424.802	−0.0219881
$721$	−26440.4	−1.36573
$722$	−526.838	−0.0271563
$723$	40606.9	2.08878
$724$	26707.3	1.37095
$725$	5740.82	0.294081
$726$	−996.996	−0.0509669
$727$	−23014.8	−1.17410	−0.587051	−	0.809550i	$-0.699711\pi$
$727$	−23014.8	−1.17410	−0.587051	+	0.809550i	$0.699711\pi$
$728$	49.3485	0.00251233
$729$	14959.3	0.760013
$730$	−2186.27	−0.110846
$731$	7582.47	0.383649
$732$	271.668	0.0137174
$733$	−3839.09	−0.193452	−0.0967259	−	0.995311i	$-0.530837\pi$
$733$	−3839.09	−0.193452	−0.0967259	+	0.995311i	$0.530837\pi$
$734$	14738.9	0.741177
$735$	897.842	0.0450577
$736$	27735.5	1.38906
$737$	−9856.49	−0.492630
$738$	806.379	0.0402211
$739$	17221.4	0.857238	0.428619	−	0.903485i	$-0.359000\pi$
$739$	17221.4	0.857238	0.428619	+	0.903485i	$0.359000\pi$
$740$	202.943	0.0100815
$741$	14.8251	0.000734970 0
$742$	10605.8	0.524732
$743$	−11212.5	−0.553628	−0.276814	−	0.960924i	$-0.589279\pi$
$743$	−11212.5	−0.553628	−0.276814	+	0.960924i	$0.589279\pi$
$744$	−24872.8	−1.22565
$745$	−813.074	−0.0399849
$746$	18243.8	0.895378
$747$	2445.59	0.119785
$748$	1856.62	0.0907552
$749$	−13264.0	−0.647069
$750$	1029.95	0.0501448
$751$	−11965.5	−0.581395	−0.290698	−	0.956815i	$-0.593887\pi$
$751$	−11965.5	−0.581395	−0.290698	+	0.956815i	$0.593887\pi$
$752$	−7817.69	−0.379098
$753$	18027.2	0.872439
$754$	46.3136	0.00223693
$755$	11079.4	0.534065
$756$	−12934.6	−0.622257
$757$	−20199.9	−0.969854	−0.484927	−	0.874555i	$-0.661154\pi$
$757$	−20199.9	−0.969854	−0.484927	+	0.874555i	$0.661154\pi$
$758$	−13042.3	−0.624956
$759$	9193.75	0.439673
$760$	1922.98	0.0917816
$761$	7132.06	0.339733	0.169867	−	0.985467i	$-0.445666\pi$
$761$	7132.06	0.339733	0.169867	+	0.985467i	$0.445666\pi$
$762$	−6084.12	−0.289245
$763$	−17607.7	−0.835439
$764$	−5823.99	−0.275791
$765$	−701.127	−0.0331363
$766$	16211.7	0.764692
$767$	−0.197054	−9.27669e−6 0
$768$	−16793.4	−0.789036
$769$	32993.7	1.54718	0.773591	−	0.633685i	$-0.218458\pi$
$769$	32993.7	1.54718	0.773591	+	0.633685i	$0.218458\pi$
$770$	1415.95	0.0662693
$771$	23134.0	1.08061
$772$	−13014.8	−0.606754
$773$	34621.0	1.61091	0.805454	−	0.592658i	$-0.201922\pi$
$773$	34621.0	1.61091	0.805454	+	0.592658i	$0.201922\pi$
$774$	−1876.94	−0.0871642
$775$	−5440.94	−0.252186
$776$	15541.8	0.718966
$777$	−688.662	−0.0317962
$778$	−11821.9	−0.544775
$779$	2152.66	0.0990076
$780$	−22.9016	−0.00105129
$781$	−10924.3	−0.500513
$782$	6211.69	0.284053
$783$	−28682.5	−1.30911
$784$	−554.064	−0.0252398
$785$	−16235.5	−0.738176
$786$	−3112.49	−0.141245
$787$	−24295.2	−1.10042	−0.550209	−	0.835027i	$-0.685452\pi$
$787$	−24295.2	−1.10042	−0.550209	+	0.835027i	$0.685452\pi$
$788$	4495.96	0.203251
$789$	23481.7	1.05953
$790$	6000.20	0.270225
$791$	35908.2	1.61410
$792$	−1085.91	−0.0487198
$793$	1.13280	5.07273e−5 0
$794$	18218.0	0.814273
$795$	−11629.6	−0.518818
$796$	−19466.6	−0.866805
$797$	21058.5	0.935924	0.467962	−	0.883749i	$-0.344988\pi$
$797$	21058.5	0.935924	0.467962	+	0.883749i	$0.344988\pi$
$798$	−2761.71	−0.122511
$799$	−12902.9	−0.571306
$800$	−4683.98	−0.207005
$801$	3775.30	0.166534
$802$	4758.67	0.209519
$803$	3295.78	0.144839
$804$	−29697.6	−1.30268
$805$	−13057.1	−0.571682
$806$	−43.8943	−0.00191825
$807$	−7063.62	−0.308118
$808$	−8812.75	−0.383702
$809$	−6993.30	−0.303920	−0.151960	−	0.988387i	$-0.548559\pi$
$809$	−6993.30	−0.303920	−0.151960	+	0.988387i	$0.548559\pi$
$810$	−6106.74	−0.264900
$811$	11766.2	0.509453	0.254726	−	0.967013i	$-0.418015\pi$
$811$	11766.2	0.509453	0.254726	+	0.967013i	$0.418015\pi$
$812$	23779.5	1.02771
$813$	19198.3	0.828183
$814$	110.998	0.00477944
$815$	−4106.75	−0.176507
$816$	2828.04	0.121325
$817$	−5010.55	−0.214562
$818$	−11429.4	−0.488535
$819$	11.8897	0.000507275 0
$820$	−3325.40	−0.141620
$821$	22373.7	0.951091	0.475546	−	0.879691i	$-0.342251\pi$
$821$	22373.7	0.951091	0.475546	+	0.879691i	$0.342251\pi$
$822$	−13809.4	−0.585959
$823$	6124.00	0.259379	0.129690	−	0.991555i	$-0.458602\pi$
$823$	6124.00	0.259379	0.129690	+	0.991555i	$0.458602\pi$
$824$	30339.2	1.28266
$825$	−1552.64	−0.0655225
$826$	36.7085	0.00154631
$827$	16394.7	0.689358	0.344679	−	0.938721i	$-0.387988\pi$
$827$	16394.7	0.689358	0.344679	+	0.938721i	$0.387988\pi$
$828$	4238.02	0.177876
$829$	−24772.1	−1.03784	−0.518922	−	0.854822i	$-0.673666\pi$
$829$	−24772.1	−1.03784	−0.518922	+	0.854822i	$0.673666\pi$
$830$	3659.11	0.153024
$831$	−22427.9	−0.936241
$832$	−18.5273	−0.000772018 0
$833$	−914.471	−0.0380367
$834$	−5027.50	−0.208739
$835$	9640.94	0.399567
$836$	−1226.87	−0.0507563
$837$	27184.2	1.12261
$838$	11030.0	0.454684
$839$	−46348.2	−1.90717	−0.953587	−	0.301118i	$-0.902640\pi$
$839$	−46348.2	−1.90717	−0.953587	+	0.301118i	$0.902640\pi$
$840$	10080.4	0.414055
$841$	28342.2	1.16209
$842$	−11799.2	−0.482931
$843$	−3536.81	−0.144501
$844$	34933.9	1.42473
$845$	10984.9	0.447210
$846$	3193.95	0.129799
$847$	−2134.53	−0.0865918
$848$	7176.72	0.290625
$849$	30000.0	1.21272
$850$	−1049.03	−0.0423311
$851$	−1023.56	−0.0412305
$852$	−32914.7	−1.32352
$853$	35433.3	1.42229	0.711145	−	0.703046i	$-0.248177\pi$
$853$	35433.3	1.42229	0.711145	+	0.703046i	$0.248177\pi$
$854$	−211.024	−0.00845563
$855$	463.310	0.0185320
$856$	15219.8	0.607713
$857$	19433.5	0.774606	0.387303	−	0.921953i	$-0.373407\pi$
$857$	19433.5	0.774606	0.387303	+	0.921953i	$0.373407\pi$
$858$	−12.5258	−0.000498396 0
$859$	11893.4	0.472408	0.236204	−	0.971703i	$-0.424097\pi$
$859$	11893.4	0.472408	0.236204	+	0.971703i	$0.424097\pi$
$860$	7740.25	0.306907
$861$	11284.3	0.446654
$862$	10332.9	0.408284
$863$	−17641.2	−0.695845	−0.347923	−	0.937523i	$-0.613113\pi$
$863$	−17641.2	−0.695845	−0.347923	+	0.937523i	$0.613113\pi$
$864$	23402.3	0.921484
$865$	20645.4	0.811522
$866$	22654.9	0.888965
$867$	−23071.0	−0.903729
$868$	−22537.4	−0.881300
$869$	−9045.22	−0.353093
$870$	9460.45	0.368666
$871$	−123.833	−0.00481734
$872$	20204.0	0.784626
$873$	3744.53	0.145170
$874$	−4104.73	−0.158861
$875$	2205.09	0.0851951
$876$	9930.16	0.383001
$877$	−48524.8	−1.86838	−0.934188	−	0.356780i	$-0.883875\pi$
$877$	−48524.8	−1.86838	−0.934188	+	0.356780i	$0.883875\pi$
$878$	−2152.94	−0.0827542
$879$	−32597.1	−1.25082
$880$	958.145	0.0367035
$881$	49613.7	1.89731	0.948653	−	0.316319i	$-0.102447\pi$
$881$	49613.7	1.89731	0.948653	+	0.316319i	$0.102447\pi$
$882$	226.365	0.00864184
$883$	5285.72	0.201448	0.100724	−	0.994914i	$-0.467884\pi$
$883$	5285.72	0.201448	0.100724	+	0.994914i	$0.467884\pi$
$884$	23.3258	0.000887479 0
$885$	−40.2522	−0.00152888
$886$	14402.1	0.546105
$887$	45362.2	1.71715	0.858576	−	0.512687i	$-0.171350\pi$
$887$	45362.2	1.71715	0.858576	+	0.512687i	$0.171350\pi$
$888$	790.209	0.0298622
$889$	−13025.9	−0.491421
$890$	5648.62	0.212744
$891$	9205.82	0.346135
$892$	−28708.6	−1.07762
$893$	8526.36	0.319512
$894$	−1339.89	−0.0501258
$895$	−1550.31	−0.0579007
$896$	−22989.8	−0.857184
$897$	115.506	0.00429948
$898$	−16713.2	−0.621075
$899$	−49976.7	−1.85408
$900$	−715.716	−0.0265080
$901$	11845.0	0.437975
$902$	−1818.79	−0.0671388
$903$	−26265.6	−0.967955
$904$	−41203.1	−1.51592
$905$	22748.2	0.835554
$906$	18258.0	0.669515
$907$	15625.6	0.572041	0.286020	−	0.958224i	$-0.407668\pi$
$907$	15625.6	0.572041	0.286020	+	0.958224i	$0.407668\pi$
$908$	−22200.2	−0.811387
$909$	−2123.28	−0.0774750
$910$	17.7894	0.000648036 0
$911$	15420.5	0.560818	0.280409	−	0.959881i	$-0.409530\pi$
$911$	15420.5	0.560818	0.280409	+	0.959881i	$0.409530\pi$
$912$	−1868.79	−0.0678529
$913$	−5516.06	−0.199951
$914$	−2245.31	−0.0812564
$915$	231.396	0.00836033
$916$	−2524.66	−0.0910669
$917$	−6663.71	−0.239973
$918$	5241.21	0.188438
$919$	−2905.89	−0.104305	−0.0521525	−	0.998639i	$-0.516608\pi$
$919$	−2905.89	−0.104305	−0.0521525	+	0.998639i	$0.516608\pi$
$920$	14982.5	0.536911
$921$	49773.2	1.78076
$922$	−1153.91	−0.0412170
$923$	−137.247	−0.00489442
$924$	−6431.32	−0.228977
$925$	172.859	0.00614439
$926$	5023.56	0.178277
$927$	7309.70	0.258988
$928$	−43023.8	−1.52190
$929$	43243.4	1.52720	0.763601	−	0.645689i	$-0.223430\pi$
$929$	43243.4	1.52720	0.763601	+	0.645689i	$0.223430\pi$
$930$	−8966.27	−0.316146
$931$	604.289	0.0212726
$932$	4361.36	0.153284
$933$	−14517.8	−0.509422
$934$	−28799.0	−1.00892
$935$	1581.40	0.0553126
$936$	−13.6429	−0.000476422 0
$937$	25998.7	0.906447	0.453224	−	0.891397i	$-0.350274\pi$
$937$	25998.7	0.906447	0.453224	+	0.891397i	$0.350274\pi$
$938$	23068.3	0.802993
$939$	20469.3	0.711386
$940$	−13171.4	−0.457026
$941$	2696.34	0.0934092	0.0467046	−	0.998909i	$-0.485128\pi$
$941$	2696.34	0.0934092	0.0467046	+	0.998909i	$0.485128\pi$
$942$	−26754.9	−0.925393
$943$	16771.9	0.579182
$944$	24.8399	0.000856429 0
$945$	−11017.2	−0.379247
$946$	4233.45	0.145498
$947$	−53595.8	−1.83910	−0.919550	−	0.392972i	$-0.871447\pi$
$947$	−53595.8	−1.83910	−0.919550	+	0.392972i	$0.871447\pi$
$948$	−27253.2	−0.933695
$949$	41.4066	0.00141635
$950$	693.207	0.0236743
$951$	−6400.73	−0.218252
$952$	−10267.1	−0.349536
$953$	−12656.9	−0.430219	−0.215109	−	0.976590i	$-0.569011\pi$
$953$	−12656.9	−0.430219	−0.215109	+	0.976590i	$0.569011\pi$
$954$	−2932.08	−0.0995068
$955$	−4960.64	−0.168086
$956$	−22648.0	−0.766201
$957$	−14261.5	−0.481723
$958$	4171.31	0.140677
$959$	−29565.4	−0.995533
$960$	−3784.56	−0.127236
$961$	17575.1	0.589947
$962$	1.39452	4.67373e−5 0
$963$	3666.95	0.122706
$964$	−42219.6	−1.41058
$965$	−11085.5	−0.369798
$966$	−21517.2	−0.716672
$967$	40675.5	1.35267	0.676337	−	0.736592i	$-0.263566\pi$
$967$	40675.5	1.35267	0.676337	+	0.736592i	$0.263566\pi$
$968$	2449.28	0.0813251
$969$	−3084.40	−0.102255
$970$	5602.58	0.185452
$971$	−931.871	−0.0307983	−0.0153992	−	0.999881i	$-0.504902\pi$
$971$	−931.871	−0.0307983	−0.0153992	+	0.999881i	$0.504902\pi$
$972$	7940.08	0.262014
$973$	−10763.7	−0.354643
$974$	19250.7	0.633297
$975$	−19.5067	−0.000640732 0
$976$	−142.796	−0.00468318
$977$	−194.453	−0.00636757	−0.00318379	−	0.999995i	$-0.501013\pi$
$977$	−194.453	−0.00636757	−0.00318379	+	0.999995i	$0.501013\pi$
$978$	−6767.62	−0.221273
$979$	−8515.21	−0.277985
$980$	−933.500	−0.0304281
$981$	4867.81	0.158427
$982$	−19964.8	−0.648781
$983$	34446.5	1.11767	0.558837	−	0.829277i	$-0.311248\pi$
$983$	34446.5	1.11767	0.558837	+	0.829277i	$0.311248\pi$
$984$	−12948.3	−0.419488
$985$	3829.47	0.123875
$986$	−9635.67	−0.311219
$987$	44695.6	1.44142
$988$	−15.4139	−0.000496336 0
$989$	−39038.5	−1.25516
$990$	−391.454	−0.0125669
$991$	−12553.1	−0.402385	−0.201192	−	0.979552i	$-0.564482\pi$
$991$	−12553.1	−0.402385	−0.201192	+	0.979552i	$0.564482\pi$
$992$	40776.4	1.30509
$993$	10502.1	0.335625
$994$	25567.3	0.815841
$995$	−16580.9	−0.528292
$996$	−16619.9	−0.528735
$997$	−23436.3	−0.744469	−0.372234	−	0.928139i	$-0.621408\pi$
$997$	−23436.3	−0.744469	−0.372234	+	0.928139i	$0.621408\pi$
$998$	12102.9	0.383880
$999$	−863.644	−0.0273518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.g.1.9	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.g.1.9	✓	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)

\(f(q)\)	\(=\)	\(q-1.45938 q^{2} +5.64597 q^{3} -5.87020 q^{4} -5.00000 q^{5} -8.23963 q^{6} -17.6407 q^{7} +20.2419 q^{8} +4.87695 q^{9} +O(q^{10})\)	\(q-1.45938 q^{2} +5.64597 q^{3} -5.87020 q^{4} -5.00000 q^{5} -8.23963 q^{6} -17.6407 q^{7} +20.2419 q^{8} +4.87695 q^{9} +7.29692 q^{10} -11.0000 q^{11} -33.1429 q^{12} -0.138199 q^{13} +25.7446 q^{14} -28.2298 q^{15} +17.4208 q^{16} +28.7527 q^{17} -7.11734 q^{18} -19.0000 q^{19} +29.3510 q^{20} -99.5990 q^{21} +16.0532 q^{22} -148.034 q^{23} +114.285 q^{24} +25.0000 q^{25} +0.201685 q^{26} -124.906 q^{27} +103.555 q^{28} +229.633 q^{29} +41.1982 q^{30} -217.638 q^{31} -187.359 q^{32} -62.1056 q^{33} -41.9612 q^{34} +88.2036 q^{35} -28.6287 q^{36} +6.91435 q^{37} +27.7283 q^{38} -0.780267 q^{39} -101.210 q^{40} -113.298 q^{41} +145.353 q^{42} +263.713 q^{43} +64.5722 q^{44} -24.3847 q^{45} +216.039 q^{46} -448.756 q^{47} +98.3574 q^{48} -31.8047 q^{49} -36.4846 q^{50} +162.337 q^{51} +0.811256 q^{52} +411.962 q^{53} +182.286 q^{54} +55.0000 q^{55} -357.083 q^{56} -107.273 q^{57} -335.122 q^{58} +1.42587 q^{59} +165.715 q^{60} -8.19685 q^{61} +317.617 q^{62} -86.0329 q^{63} +134.063 q^{64} +0.690995 q^{65} +90.6360 q^{66} +896.045 q^{67} -168.784 q^{68} -835.795 q^{69} -128.723 q^{70} +993.114 q^{71} +98.7189 q^{72} -299.616 q^{73} -10.0907 q^{74} +141.149 q^{75} +111.534 q^{76} +194.048 q^{77} +1.13871 q^{78} +822.292 q^{79} -87.1041 q^{80} -836.893 q^{81} +165.345 q^{82} +501.460 q^{83} +584.666 q^{84} -143.763 q^{85} -384.859 q^{86} +1296.50 q^{87} -222.661 q^{88} +774.110 q^{89} +35.5867 q^{90} +2.43793 q^{91} +868.989 q^{92} -1228.77 q^{93} +654.907 q^{94} +95.0000 q^{95} -1057.82 q^{96} +767.801 q^{97} +46.4153 q^{98} -53.6464 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) \) 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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