LMFDB - Embedded newform 1045.4.a.g.1.4

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			1.4
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)

$f(q)$	$=$	$q-4.30610 q^{2} +2.30247 q^{3} +10.5425 q^{4} -5.00000 q^{5} -9.91467 q^{6} +14.9972 q^{7} -10.9483 q^{8} -21.6986 q^{9} +O(q^{10})$	\(q-4.30610 q^{2} +2.30247 q^{3} +10.5425 q^{4} -5.00000 q^{5} -9.91467 q^{6} +14.9972 q^{7} -10.9483 q^{8} -21.6986 q^{9} +21.5305 q^{10} -11.0000 q^{11} +24.2738 q^{12} +79.8622 q^{13} -64.5793 q^{14} -11.5123 q^{15} -37.1954 q^{16} +61.7616 q^{17} +93.4366 q^{18} -19.0000 q^{19} -52.7126 q^{20} +34.5305 q^{21} +47.3671 q^{22} +23.4191 q^{23} -25.2082 q^{24} +25.0000 q^{25} -343.895 q^{26} -112.127 q^{27} +158.108 q^{28} +136.220 q^{29} +49.5733 q^{30} -112.451 q^{31} +247.754 q^{32} -25.3272 q^{33} -265.952 q^{34} -74.9858 q^{35} -228.758 q^{36} -245.751 q^{37} +81.8159 q^{38} +183.880 q^{39} +54.7417 q^{40} +114.939 q^{41} -148.692 q^{42} +210.152 q^{43} -115.968 q^{44} +108.493 q^{45} -100.845 q^{46} +589.406 q^{47} -85.6414 q^{48} -118.085 q^{49} -107.653 q^{50} +142.204 q^{51} +841.949 q^{52} +409.294 q^{53} +482.831 q^{54} +55.0000 q^{55} -164.194 q^{56} -43.7469 q^{57} -586.577 q^{58} -606.990 q^{59} -121.369 q^{60} -549.777 q^{61} +484.227 q^{62} -325.418 q^{63} -769.291 q^{64} -399.311 q^{65} +109.061 q^{66} -1032.89 q^{67} +651.123 q^{68} +53.9218 q^{69} +322.896 q^{70} +995.095 q^{71} +237.564 q^{72} +176.427 q^{73} +1058.23 q^{74} +57.5617 q^{75} -200.308 q^{76} -164.969 q^{77} -791.807 q^{78} +256.162 q^{79} +185.977 q^{80} +327.694 q^{81} -494.939 q^{82} +1035.53 q^{83} +364.038 q^{84} -308.808 q^{85} -904.934 q^{86} +313.642 q^{87} +120.432 q^{88} -534.242 q^{89} -467.183 q^{90} +1197.71 q^{91} +246.896 q^{92} -258.915 q^{93} -2538.04 q^{94} +95.0000 q^{95} +570.446 q^{96} +351.842 q^{97} +508.488 q^{98} +238.685 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.30610	−1.52244	−0.761219	−	0.648495i	$-0.775399\pi$
$2$	−4.30610	−1.52244	−0.761219	+	0.648495i	$0.775399\pi$
$3$	2.30247	0.443110	0.221555	−	0.975148i	$-0.428887\pi$
$3$	2.30247	0.443110	0.221555	+	0.975148i	$0.428887\pi$
$4$	10.5425	1.31781
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−9.91467	−0.674608
$7$	14.9972	0.809770	0.404885	−	0.914368i	$-0.367312\pi$
$7$	14.9972	0.809770	0.404885	+	0.914368i	$0.367312\pi$
$8$	−10.9483	−0.483853
$9$	−21.6986	−0.803653
$10$	21.5305	0.680855
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	24.2738	0.583937
$13$	79.8622	1.70383	0.851915	−	0.523680i	$-0.175441\pi$
$13$	79.8622	1.70383	0.851915	+	0.523680i	$0.175441\pi$
$14$	−64.5793	−1.23282
$15$	−11.5123	−0.198165
$16$	−37.1954	−0.581179
$17$	61.7616	0.881140	0.440570	−	0.897718i	$-0.354776\pi$
$17$	61.7616	0.881140	0.440570	+	0.897718i	$0.354776\pi$
$18$	93.4366	1.22351
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−52.7126	−0.589345
$21$	34.5305	0.358818
$22$	47.3671	0.459032
$23$	23.4191	0.212314	0.106157	−	0.994349i	$-0.466145\pi$
$23$	23.4191	0.212314	0.106157	+	0.994349i	$0.466145\pi$
$24$	−25.2082	−0.214400
$25$	25.0000	0.200000
$26$	−343.895	−2.59397
$27$	−112.127	−0.799217
$28$	158.108	1.06713
$29$	136.220	0.872256	0.436128	−	0.899885i	$-0.356349\pi$
$29$	136.220	0.872256	0.436128	+	0.899885i	$0.356349\pi$
$30$	49.5733	0.301694
$31$	−112.451	−0.651511	−0.325755	−	0.945454i	$-0.605619\pi$
$31$	−112.451	−0.651511	−0.325755	+	0.945454i	$0.605619\pi$
$32$	247.754	1.36866
$33$	−25.3272	−0.133603
$34$	−265.952	−1.34148
$35$	−74.9858	−0.362140
$36$	−228.758	−1.05907
$37$	−245.751	−1.09193	−0.545963	−	0.837809i	$-0.683836\pi$
$37$	−245.751	−1.09193	−0.545963	+	0.837809i	$0.683836\pi$
$38$	81.8159	0.349271
$39$	183.880	0.754984
$40$	54.7417	0.216386
$41$	114.939	0.437816	0.218908	−	0.975746i	$-0.429751\pi$
$41$	114.939	0.437816	0.218908	+	0.975746i	$0.429751\pi$
$42$	−148.692	−0.546277
$43$	210.152	0.745298	0.372649	−	0.927972i	$-0.378450\pi$
$43$	210.152	0.745298	0.372649	+	0.927972i	$0.378450\pi$
$44$	−115.968	−0.397336
$45$	108.493	0.359405
$46$	−100.845	−0.323235
$47$	589.406	1.82923	0.914614	−	0.404328i	$-0.132495\pi$
$47$	589.406	1.82923	0.914614	+	0.404328i	$0.132495\pi$
$48$	−85.6414	−0.257526
$49$	−118.085	−0.344272
$50$	−107.653	−0.304487
$51$	142.204	0.390442
$52$	841.949	2.24533
$53$	409.294	1.06077	0.530386	−	0.847756i	$-0.322047\pi$
$53$	409.294	1.06077	0.530386	+	0.847756i	$0.322047\pi$
$54$	482.831	1.21676
$55$	55.0000	0.134840
$56$	−164.194	−0.391810
$57$	−43.7469	−0.101656
$58$	−586.577	−1.32796
$59$	−606.990	−1.33938	−0.669690	−	0.742641i	$-0.733573\pi$
$59$	−606.990	−1.33938	−0.669690	+	0.742641i	$0.733573\pi$
$60$	−121.369	−0.261145
$61$	−549.777	−1.15396	−0.576981	−	0.816757i	$-0.695769\pi$
$61$	−549.777	−1.15396	−0.576981	+	0.816757i	$0.695769\pi$
$62$	484.227	0.991884
$63$	−325.418	−0.650774
$64$	−769.291	−1.50252
$65$	−399.311	−0.761976
$66$	109.061	0.203402
$67$	−1032.89	−1.88340	−0.941698	−	0.336459i	$-0.890771\pi$
$67$	−1032.89	−1.88340	−0.941698	+	0.336459i	$0.890771\pi$
$68$	651.123	1.16118
$69$	53.9218	0.0940785
$70$	322.896	0.551336
$71$	995.095	1.66332	0.831662	−	0.555282i	$-0.187390\pi$
$71$	995.095	1.66332	0.831662	+	0.555282i	$0.187390\pi$
$72$	237.564	0.388850
$73$	176.427	0.282866	0.141433	−	0.989948i	$-0.454829\pi$
$73$	176.427	0.282866	0.141433	+	0.989948i	$0.454829\pi$
$74$	1058.23	1.66239
$75$	57.5617	0.0886221
$76$	−200.308	−0.302327
$77$	−164.969	−0.244155
$78$	−791.807	−1.14942
$79$	256.162	0.364816	0.182408	−	0.983223i	$-0.441611\pi$
$79$	256.162	0.364816	0.182408	+	0.983223i	$0.441611\pi$
$80$	185.977	0.259911
$81$	327.694	0.449512
$82$	−494.939	−0.666547
$83$	1035.53	1.36945	0.684726	−	0.728801i	$-0.259922\pi$
$83$	1035.53	1.36945	0.684726	+	0.728801i	$0.259922\pi$
$84$	364.038	0.472855
$85$	−308.808	−0.394058
$86$	−904.934	−1.13467
$87$	313.642	0.386506
$88$	120.432	0.145887
$89$	−534.242	−0.636287	−0.318144	−	0.948043i	$-0.603059\pi$
$89$	−534.242	−0.636287	−0.318144	+	0.948043i	$0.603059\pi$
$90$	−467.183	−0.547171
$91$	1197.71	1.37971
$92$	246.896	0.279791
$93$	−258.915	−0.288691
$94$	−2538.04	−2.78488
$95$	95.0000	0.102598
$96$	570.446	0.606468
$97$	351.842	0.368290	0.184145	−	0.982899i	$-0.441048\pi$
$97$	351.842	0.368290	0.184145	+	0.982899i	$0.441048\pi$
$98$	508.488	0.524133
$99$	238.685	0.242311
$100$	263.563	0.263563
$101$	−124.225	−0.122384	−0.0611922	−	0.998126i	$-0.519490\pi$
$101$	−124.225	−0.122384	−0.0611922	+	0.998126i	$0.519490\pi$
$102$	−612.346	−0.594424
$103$	−936.795	−0.896167	−0.448083	−	0.893992i	$-0.647893\pi$
$103$	−936.795	−0.896167	−0.448083	+	0.893992i	$0.647893\pi$
$104$	−874.359	−0.824403
$105$	−172.652	−0.160468
$106$	−1762.46	−1.61496
$107$	896.229	0.809736	0.404868	−	0.914375i	$-0.367317\pi$
$107$	896.229	0.809736	0.404868	+	0.914375i	$0.367317\pi$
$108$	−1182.10	−1.05322
$109$	430.768	0.378533	0.189266	−	0.981926i	$-0.439389\pi$
$109$	430.768	0.378533	0.189266	+	0.981926i	$0.439389\pi$
$110$	−236.836	−0.205285
$111$	−565.835	−0.483844
$112$	−557.826	−0.470621
$113$	−1308.62	−1.08942	−0.544709	−	0.838625i	$-0.683360\pi$
$113$	−1308.62	−1.08942	−0.544709	+	0.838625i	$0.683360\pi$
$114$	188.379	0.154766
$115$	−117.096	−0.0949497
$116$	1436.10	1.14947
$117$	−1732.90	−1.36929
$118$	2613.76	2.03912
$119$	926.248	0.713521
$120$	126.041	0.0958827
$121$	121.000	0.0909091
$122$	2367.40	1.75684
$123$	264.643	0.194001
$124$	−1185.52	−0.858571
$125$	−125.000	−0.0894427
$126$	1401.28	0.990763
$127$	−2085.91	−1.45744	−0.728718	−	0.684814i	$-0.759883\pi$
$127$	−2085.91	−1.45744	−0.728718	+	0.684814i	$0.759883\pi$
$128$	1330.61	0.918834
$129$	483.867	0.330249
$130$	1719.47	1.16006
$131$	−1849.34	−1.23342	−0.616709	−	0.787191i	$-0.711534\pi$
$131$	−1849.34	−1.23342	−0.616709	+	0.787191i	$0.711534\pi$
$132$	−267.012	−0.176064
$133$	−284.946	−0.185774
$134$	4447.73	2.86735
$135$	560.635	0.357421
$136$	−676.187	−0.426343
$137$	−2033.53	−1.26815	−0.634073	−	0.773274i	$-0.718618\pi$
$137$	−2033.53	−1.26815	−0.634073	+	0.773274i	$0.718618\pi$
$138$	−232.193	−0.143229
$139$	2007.48	1.22498	0.612490	−	0.790479i	$-0.290168\pi$
$139$	2007.48	1.22498	0.612490	+	0.790479i	$0.290168\pi$
$140$	−790.539	−0.477234
$141$	1357.09	0.810550
$142$	−4284.98	−2.53231
$143$	−878.484	−0.513724
$144$	807.091	0.467066
$145$	−681.100	−0.390085
$146$	−759.714	−0.430646
$147$	−271.888	−0.152551
$148$	−2590.84	−1.43896
$149$	−508.460	−0.279562	−0.139781	−	0.990182i	$-0.544640\pi$
$149$	−508.460	−0.279562	−0.139781	+	0.990182i	$0.544640\pi$
$150$	−247.867	−0.134922
$151$	−1056.35	−0.569300	−0.284650	−	0.958632i	$-0.591877\pi$
$151$	−1056.35	−0.569300	−0.284650	+	0.958632i	$0.591877\pi$
$152$	208.019	0.111004
$153$	−1340.14	−0.708131
$154$	710.372	0.371710
$155$	562.256	0.291365
$156$	1938.56	0.994930
$157$	154.886	0.0787339	0.0393669	−	0.999225i	$-0.487466\pi$
$157$	154.886	0.0787339	0.0393669	+	0.999225i	$0.487466\pi$
$158$	−1103.06	−0.555410
$159$	942.388	0.470039
$160$	−1238.77	−0.612084
$161$	351.220	0.171926
$162$	−1411.08	−0.684353
$163$	1744.07	0.838073	0.419036	−	0.907969i	$-0.362368\pi$
$163$	1744.07	0.838073	0.419036	+	0.907969i	$0.362368\pi$
$164$	1211.75	0.576960
$165$	126.636	0.0597490
$166$	−4459.11	−2.08490
$167$	1940.40	0.899118	0.449559	−	0.893251i	$-0.351581\pi$
$167$	1940.40	0.899118	0.449559	+	0.893251i	$0.351581\pi$
$168$	−378.052	−0.173615
$169$	4180.97	1.90304
$170$	1329.76	0.599929
$171$	412.274	0.184371
$172$	2215.53	0.982165
$173$	3383.21	1.48683	0.743413	−	0.668833i	$-0.233206\pi$
$173$	3383.21	1.48683	0.743413	+	0.668833i	$0.233206\pi$
$174$	−1350.58	−0.588431
$175$	374.929	0.161954
$176$	409.150	0.175232
$177$	−1397.58	−0.593493
$178$	2300.50	0.968707
$179$	3305.84	1.38039	0.690196	−	0.723622i	$-0.257524\pi$
$179$	3305.84	1.38039	0.690196	+	0.723622i	$0.257524\pi$
$180$	1143.79	0.473629
$181$	2298.57	0.943929	0.471964	−	0.881618i	$-0.343545\pi$
$181$	2298.57	0.943929	0.471964	+	0.881618i	$0.343545\pi$
$182$	−5157.44	−2.10052
$183$	−1265.84	−0.511333
$184$	−256.401	−0.102729
$185$	1228.76	0.488324
$186$	1114.92	0.439514
$187$	−679.378	−0.265674
$188$	6213.82	2.41058
$189$	−1681.59	−0.647182
$190$	−409.080	−0.156199
$191$	2798.75	1.06026	0.530132	−	0.847915i	$-0.322142\pi$
$191$	2798.75	1.06026	0.530132	+	0.847915i	$0.322142\pi$
$192$	−1771.27	−0.665783
$193$	−102.172	−0.0381062	−0.0190531	−	0.999818i	$-0.506065\pi$
$193$	−102.172	−0.0381062	−0.0190531	+	0.999818i	$0.506065\pi$
$194$	−1515.07	−0.560698
$195$	−919.401	−0.337639
$196$	−1244.92	−0.453687
$197$	5139.03	1.85858	0.929291	−	0.369348i	$-0.120419\pi$
$197$	5139.03	1.85858	0.929291	+	0.369348i	$0.120419\pi$
$198$	−1027.80	−0.368903
$199$	3576.79	1.27413	0.637065	−	0.770810i	$-0.280148\pi$
$199$	3576.79	1.27413	0.637065	+	0.770810i	$0.280148\pi$
$200$	−273.709	−0.0967706
$201$	−2378.20	−0.834552
$202$	534.925	0.186323
$203$	2042.91	0.706327
$204$	1499.19	0.514531
$205$	−574.694	−0.195797
$206$	4033.94	1.36436
$207$	−508.163	−0.170627
$208$	−2970.51	−0.990230
$209$	209.000	0.0691714
$210$	743.459	0.244303
$211$	2530.97	0.825777	0.412889	−	0.910782i	$-0.364520\pi$
$211$	2530.97	0.825777	0.412889	+	0.910782i	$0.364520\pi$
$212$	4314.99	1.39790
$213$	2291.18	0.737036
$214$	−3859.26	−1.23277
$215$	−1050.76	−0.333307
$216$	1227.61	0.386704
$217$	−1686.45	−0.527574
$218$	−1854.93	−0.576292
$219$	406.218	0.125341
$220$	579.839	0.177694
$221$	4932.42	1.50131
$222$	2436.54	0.736622
$223$	6179.51	1.85565	0.927826	−	0.373012i	$-0.121675\pi$
$223$	6179.51	1.85565	0.927826	+	0.373012i	$0.121675\pi$
$224$	3715.61	1.10830
$225$	−542.466	−0.160731
$226$	5635.04	1.65857
$227$	842.512	0.246341	0.123171	−	0.992386i	$-0.460694\pi$
$227$	842.512	0.246341	0.123171	+	0.992386i	$0.460694\pi$
$228$	−461.203	−0.133964
$229$	6418.51	1.85217	0.926085	−	0.377314i	$-0.123152\pi$
$229$	6418.51	1.85217	0.926085	+	0.377314i	$0.123152\pi$
$230$	504.225	0.144555
$231$	−379.835	−0.108188
$232$	−1491.38	−0.422044
$233$	−5462.91	−1.53600	−0.767998	−	0.640452i	$-0.778747\pi$
$233$	−5462.91	−1.53600	−0.767998	+	0.640452i	$0.778747\pi$
$234$	7462.05	2.08466
$235$	−2947.03	−0.818056
$236$	−6399.21	−1.76506
$237$	589.805	0.161654
$238$	−3988.52	−1.08629
$239$	6454.13	1.74679	0.873395	−	0.487012i	$-0.161913\pi$
$239$	6454.13	1.74679	0.873395	+	0.487012i	$0.161913\pi$
$240$	428.207	0.115169
$241$	348.569	0.0931671	0.0465836	−	0.998914i	$-0.485167\pi$
$241$	348.569	0.0931671	0.0465836	+	0.998914i	$0.485167\pi$
$242$	−521.038	−0.138403
$243$	3781.94	0.998401
$244$	−5796.03	−1.52071
$245$	590.427	0.153963
$246$	−1139.58	−0.295354
$247$	−1517.38	−0.390885
$248$	1231.16	0.315236
$249$	2384.28	0.606818
$250$	538.263	0.136171
$251$	−7116.11	−1.78950	−0.894751	−	0.446566i	$-0.852647\pi$
$251$	−7116.11	−1.78950	−0.894751	+	0.446566i	$0.852647\pi$
$252$	−3430.72	−0.857600
$253$	−257.610	−0.0640151
$254$	8982.13	2.21885
$255$	−711.021	−0.174611
$256$	424.571	0.103655
$257$	3296.03	0.800002	0.400001	−	0.916515i	$-0.369010\pi$
$257$	3296.03	0.800002	0.400001	+	0.916515i	$0.369010\pi$
$258$	−2083.58	−0.502784
$259$	−3685.57	−0.884210
$260$	−4209.74	−1.00414
$261$	−2955.79	−0.700991
$262$	7963.46	1.87780
$263$	−1022.98	−0.239847	−0.119923	−	0.992783i	$-0.538265\pi$
$263$	−1022.98	−0.239847	−0.119923	+	0.992783i	$0.538265\pi$
$264$	277.290	0.0646441
$265$	−2046.47	−0.474392
$266$	1227.01	0.282829
$267$	−1230.08	−0.281945
$268$	−10889.3	−2.48197
$269$	−671.463	−0.152193	−0.0760963	−	0.997100i	$-0.524246\pi$
$269$	−671.463	−0.152193	−0.0760963	+	0.997100i	$0.524246\pi$
$270$	−2414.15	−0.544151
$271$	256.036	0.0573915	0.0286957	−	0.999588i	$-0.490865\pi$
$271$	256.036	0.0573915	0.0286957	+	0.999588i	$0.490865\pi$
$272$	−2297.25	−0.512100
$273$	2757.68	0.611364
$274$	8756.57	1.93067
$275$	−275.000	−0.0603023
$276$	568.471	0.123978
$277$	4086.63	0.886432	0.443216	−	0.896415i	$-0.353837\pi$
$277$	4086.63	0.886432	0.443216	+	0.896415i	$0.353837\pi$
$278$	−8644.41	−1.86495
$279$	2440.04	0.523589
$280$	820.970	0.175223
$281$	−2476.19	−0.525683	−0.262842	−	0.964839i	$-0.584660\pi$
$281$	−2476.19	−0.525683	−0.262842	+	0.964839i	$0.584660\pi$
$282$	−5843.76	−1.23401
$283$	460.898	0.0968110	0.0484055	−	0.998828i	$-0.484586\pi$
$283$	460.898	0.0968110	0.0484055	+	0.998828i	$0.484586\pi$
$284$	10490.8	2.19195
$285$	218.735	0.0454622
$286$	3782.84	0.782112
$287$	1723.76	0.354530
$288$	−5375.93	−1.09993
$289$	−1098.51	−0.223592
$290$	2932.89	0.593880
$291$	810.105	0.163193
$292$	1859.99	0.372766
$293$	5308.92	1.05853	0.529267	−	0.848455i	$-0.322467\pi$
$293$	5308.92	1.05853	0.529267	+	0.848455i	$0.322467\pi$
$294$	1170.78	0.232249
$295$	3034.95	0.598989
$296$	2690.57	0.528332
$297$	1233.40	0.240973
$298$	2189.48	0.425615
$299$	1870.30	0.361747
$300$	606.846	0.116787
$301$	3151.67	0.603520
$302$	4548.74	0.866723
$303$	−286.024	−0.0542298
$304$	706.713	0.133332
$305$	2748.88	0.516068
$306$	5770.79	1.07809
$307$	4488.28	0.834397	0.417198	−	0.908815i	$-0.363012\pi$
$307$	4488.28	0.834397	0.417198	+	0.908815i	$0.363012\pi$
$308$	−1739.19	−0.321751
$309$	−2156.94	−0.397101
$310$	−2421.13	−0.443584
$311$	633.545	0.115515	0.0577573	−	0.998331i	$-0.481605\pi$
$311$	633.545	0.115515	0.0577573	+	0.998331i	$0.481605\pi$
$312$	−2013.18	−0.365302
$313$	3115.83	0.562674	0.281337	−	0.959609i	$-0.409222\pi$
$313$	3115.83	0.562674	0.281337	+	0.959609i	$0.409222\pi$
$314$	−666.953	−0.119867
$315$	1627.09	0.291035
$316$	2700.59	0.480760
$317$	6145.80	1.08890	0.544452	−	0.838792i	$-0.316738\pi$
$317$	6145.80	1.08890	0.544452	+	0.838792i	$0.316738\pi$
$318$	−4058.02	−0.715605
$319$	−1498.42	−0.262995
$320$	3846.46	0.671948
$321$	2063.54	0.358802
$322$	−1512.39	−0.261746
$323$	−1173.47	−0.202147
$324$	3454.72	0.592373
$325$	1996.55	0.340766
$326$	−7510.13	−1.27591
$327$	991.829	0.167732
$328$	−1258.39	−0.211838
$329$	8839.41	1.48125
$330$	−545.307	−0.0909641
$331$	−9482.07	−1.57457	−0.787284	−	0.616591i	$-0.788513\pi$
$331$	−9482.07	−1.57457	−0.787284	+	0.616591i	$0.788513\pi$
$332$	10917.1	1.80468
$333$	5332.47	0.877530
$334$	−8355.57	−1.36885
$335$	5164.45	0.842281
$336$	−1284.38	−0.208537
$337$	−3607.37	−0.583104	−0.291552	−	0.956555i	$-0.594172\pi$
$337$	−3607.37	−0.583104	−0.291552	+	0.956555i	$0.594172\pi$
$338$	−18003.7	−2.89725
$339$	−3013.05	−0.482732
$340$	−3255.61	−0.519295
$341$	1236.96	0.196438
$342$	−1775.29	−0.280693
$343$	−6914.97	−1.08855
$344$	−2300.81	−0.360615
$345$	−269.609	−0.0420732
$346$	−14568.5	−2.26360
$347$	6043.68	0.934991	0.467496	−	0.883995i	$-0.345156\pi$
$347$	6043.68	0.934991	0.467496	+	0.883995i	$0.345156\pi$
$348$	3306.58	0.509343
$349$	4979.91	0.763807	0.381903	−	0.924202i	$-0.375269\pi$
$349$	4979.91	0.763807	0.381903	+	0.924202i	$0.375269\pi$
$350$	−1614.48	−0.246565
$351$	−8954.71	−1.36173
$352$	−2725.30	−0.412667
$353$	−2487.04	−0.374990	−0.187495	−	0.982266i	$-0.560037\pi$
$353$	−2487.04	−0.374990	−0.187495	+	0.982266i	$0.560037\pi$
$354$	6018.11	0.903556
$355$	−4975.47	−0.743861
$356$	−5632.26	−0.838509
$357$	2132.66	0.316169
$358$	−14235.3	−2.10156
$359$	−6380.25	−0.937985	−0.468992	−	0.883202i	$-0.655383\pi$
$359$	−6380.25	−0.937985	−0.468992	+	0.883202i	$0.655383\pi$
$360$	−1187.82	−0.173899
$361$	361.000	0.0526316
$362$	−9897.87	−1.43707
$363$	278.599	0.0402828
$364$	12626.8	1.81820
$365$	−882.136	−0.126502
$366$	5450.85	0.778472
$367$	−1445.05	−0.205534	−0.102767	−	0.994705i	$-0.532770\pi$
$367$	−1445.05	−0.205534	−0.102767	+	0.994705i	$0.532770\pi$
$368$	−871.084	−0.123392
$369$	−2494.02	−0.351852
$370$	−5291.15	−0.743443
$371$	6138.25	0.858981
$372$	−2729.62	−0.380442
$373$	12497.4	1.73483	0.867414	−	0.497587i	$-0.165781\pi$
$373$	12497.4	1.73483	0.867414	+	0.497587i	$0.165781\pi$
$374$	2925.47	0.404472
$375$	−287.809	−0.0396330
$376$	−6453.02	−0.885078
$377$	10878.8	1.48618
$378$	7241.09	0.985294
$379$	−10656.2	−1.44425	−0.722126	−	0.691762i	$-0.756835\pi$
$379$	−10656.2	−1.44425	−0.722126	+	0.691762i	$0.756835\pi$
$380$	1001.54	0.135205
$381$	−4802.74	−0.645805
$382$	−12051.7	−1.61418
$383$	4806.76	0.641290	0.320645	−	0.947199i	$-0.396100\pi$
$383$	4806.76	0.641290	0.320645	+	0.947199i	$0.396100\pi$
$384$	3063.70	0.407145
$385$	824.843	0.109189
$386$	439.963	0.0580143
$387$	−4560.00	−0.598961
$388$	3709.30	0.485338
$389$	−5108.16	−0.665794	−0.332897	−	0.942963i	$-0.608026\pi$
$389$	−5108.16	−0.665794	−0.332897	+	0.942963i	$0.608026\pi$
$390$	3959.03	0.514035
$391$	1446.40	0.187078
$392$	1292.84	0.166577
$393$	−4258.05	−0.546540
$394$	−22129.2	−2.82958
$395$	−1280.81	−0.163151
$396$	2516.34	0.319320
$397$	−4307.53	−0.544557	−0.272278	−	0.962219i	$-0.587777\pi$
$397$	−4307.53	−0.544557	−0.272278	+	0.962219i	$0.587777\pi$
$398$	−15402.0	−1.93978
$399$	−656.079	−0.0823184
$400$	−929.886	−0.116236
$401$	−2653.46	−0.330443	−0.165222	−	0.986256i	$-0.552834\pi$
$401$	−2653.46	−0.330443	−0.165222	+	0.986256i	$0.552834\pi$
$402$	10240.8	1.27055
$403$	−8980.60	−1.11006
$404$	−1309.64	−0.161280
$405$	−1638.47	−0.201028
$406$	−8796.99	−1.07534
$407$	2703.27	0.329228
$408$	−1556.90	−0.188917
$409$	7337.68	0.887103	0.443551	−	0.896249i	$-0.353718\pi$
$409$	7337.68	0.887103	0.443551	+	0.896249i	$0.353718\pi$
$410$	2474.69	0.298089
$411$	−4682.13	−0.561928
$412$	−9876.18	−1.18098
$413$	−9103.13	−1.08459
$414$	2188.20	0.259769
$415$	−5177.67	−0.612438
$416$	19786.2	2.33197
$417$	4622.16	0.542801
$418$	−899.975	−0.105309
$419$	9685.87	1.12932	0.564661	−	0.825323i	$-0.309007\pi$
$419$	9685.87	1.12932	0.564661	+	0.825323i	$0.309007\pi$
$420$	−1820.19	−0.211467
$421$	−4616.71	−0.534453	−0.267227	−	0.963634i	$-0.586107\pi$
$421$	−4616.71	−0.534453	−0.267227	+	0.963634i	$0.586107\pi$
$422$	−10898.6	−1.25719
$423$	−12789.3	−1.47006
$424$	−4481.10	−0.513258
$425$	1544.04	0.176228
$426$	−9866.03	−1.12209
$427$	−8245.09	−0.934444
$428$	9448.52	1.06708
$429$	−2022.68	−0.227636
$430$	4524.67	0.507439
$431$	−9384.03	−1.04875	−0.524377	−	0.851486i	$-0.675702\pi$
$431$	−9384.03	−1.04875	−0.524377	+	0.851486i	$0.675702\pi$
$432$	4170.62	0.464488
$433$	−999.927	−0.110978	−0.0554889	−	0.998459i	$-0.517672\pi$
$433$	−999.927	−0.110978	−0.0554889	+	0.998459i	$0.517672\pi$
$434$	7262.02	0.803198
$435$	−1568.21	−0.172851
$436$	4541.37	0.498836
$437$	−444.963	−0.0487082
$438$	−1749.22	−0.190824
$439$	15988.7	1.73827	0.869133	−	0.494578i	$-0.164677\pi$
$439$	15988.7	1.73827	0.869133	+	0.494578i	$0.164677\pi$
$440$	−602.159	−0.0652427
$441$	2562.29	0.276676
$442$	−21239.5	−2.28565
$443$	−15165.9	−1.62653	−0.813267	−	0.581891i	$-0.802313\pi$
$443$	−15165.9	−1.62653	−0.813267	+	0.581891i	$0.802313\pi$
$444$	−5965.33	−0.637617
$445$	2671.21	0.284556
$446$	−26609.6	−2.82511
$447$	−1170.71	−0.123877
$448$	−11537.2	−1.21670
$449$	17244.3	1.81249	0.906244	−	0.422756i	$-0.138937\pi$
$449$	17244.3	1.81249	0.906244	+	0.422756i	$0.138937\pi$
$450$	2335.91	0.244702
$451$	−1264.33	−0.132006
$452$	−13796.1	−1.43565
$453$	−2432.21	−0.252263
$454$	−3627.94	−0.375039
$455$	−5988.53	−0.617025
$456$	478.956	0.0491868
$457$	777.446	0.0795785	0.0397892	−	0.999208i	$-0.487331\pi$
$457$	777.446	0.0795785	0.0397892	+	0.999208i	$0.487331\pi$
$458$	−27638.8	−2.81981
$459$	−6925.15	−0.704223
$460$	−1234.48	−0.125126
$461$	491.737	0.0496800	0.0248400	−	0.999691i	$-0.492092\pi$
$461$	491.737	0.0496800	0.0248400	+	0.999691i	$0.492092\pi$
$462$	1635.61	0.164709
$463$	−5086.07	−0.510518	−0.255259	−	0.966873i	$-0.582161\pi$
$463$	−5086.07	−0.510518	−0.255259	+	0.966873i	$0.582161\pi$
$464$	−5066.77	−0.506937
$465$	1294.58	0.129107
$466$	23523.8	2.33846
$467$	523.873	0.0519099	0.0259550	−	0.999663i	$-0.491737\pi$
$467$	523.873	0.0519099	0.0259550	+	0.999663i	$0.491737\pi$
$468$	−18269.1	−1.80447
$469$	−15490.4	−1.52512
$470$	12690.2	1.24544
$471$	356.619	0.0348878
$472$	6645.54	0.648063
$473$	−2311.67	−0.224716
$474$	−2539.76	−0.246108
$475$	−475.000	−0.0458831
$476$	9764.99	0.940289
$477$	−8881.13	−0.852493
$478$	−27792.1	−2.65938
$479$	−13196.6	−1.25880	−0.629402	−	0.777080i	$-0.716700\pi$
$479$	−13196.6	−1.25880	−0.629402	+	0.777080i	$0.716700\pi$
$480$	−2852.23	−0.271221
$481$	−19626.2	−1.86046
$482$	−1500.97	−0.141841
$483$	808.673	0.0761820
$484$	1275.64	0.119801
$485$	−1759.21	−0.164704
$486$	−16285.4	−1.52000
$487$	18265.4	1.69955	0.849777	−	0.527142i	$-0.176736\pi$
$487$	18265.4	1.69955	0.849777	+	0.527142i	$0.176736\pi$
$488$	6019.15	0.558348
$489$	4015.66	0.371359
$490$	−2542.44	−0.234399
$491$	−6305.34	−0.579544	−0.289772	−	0.957096i	$-0.593579\pi$
$491$	−6305.34	−0.579544	−0.289772	+	0.957096i	$0.593579\pi$
$492$	2790.01	0.255657
$493$	8413.17	0.768580
$494$	6534.00	0.595098
$495$	−1193.43	−0.108365
$496$	4182.67	0.378644
$497$	14923.6	1.34691
$498$	−10267.0	−0.923843
$499$	10004.1	0.897485	0.448742	−	0.893661i	$-0.351872\pi$
$499$	10004.1	0.897485	0.448742	+	0.893661i	$0.351872\pi$
$500$	−1317.81	−0.117869
$501$	4467.71	0.398409
$502$	30642.7	2.72440
$503$	3871.90	0.343220	0.171610	−	0.985165i	$-0.445103\pi$
$503$	3871.90	0.343220	0.171610	+	0.985165i	$0.445103\pi$
$504$	3562.79	0.314879
$505$	621.124	0.0547320
$506$	1109.30	0.0974589
$507$	9626.55	0.843255
$508$	−21990.7	−1.92063
$509$	629.785	0.0548423	0.0274211	−	0.999624i	$-0.491270\pi$
$509$	629.785	0.0548423	0.0274211	+	0.999624i	$0.491270\pi$
$510$	3061.73	0.265835
$511$	2645.91	0.229057
$512$	−12473.2	−1.07664
$513$	2130.41	0.183353
$514$	−14193.0	−1.21795
$515$	4683.98	0.400778
$516$	5101.18	0.435207
$517$	−6483.47	−0.551533
$518$	15870.4	1.34615
$519$	7789.74	0.658828
$520$	4371.79	0.368684
$521$	13896.5	1.16855	0.584275	−	0.811556i	$-0.301379\pi$
$521$	13896.5	1.16855	0.584275	+	0.811556i	$0.301379\pi$
$522$	12727.9	1.06722
$523$	−17457.6	−1.45959	−0.729796	−	0.683665i	$-0.760385\pi$
$523$	−17457.6	−1.45959	−0.729796	+	0.683665i	$0.760385\pi$
$524$	−19496.7	−1.62542
$525$	863.262	0.0717635
$526$	4405.06	0.365152
$527$	−6945.17	−0.574073
$528$	942.055	0.0776471
$529$	−11618.5	−0.954923
$530$	8812.32	0.722231
$531$	13170.9	1.07640
$532$	−3004.05	−0.244816
$533$	9179.27	0.745963
$534$	5296.83	0.429244
$535$	−4481.15	−0.362125
$536$	11308.4	0.911287
$537$	7611.60	0.611666
$538$	2891.39	0.231704
$539$	1298.94	0.103802
$540$	5910.51	0.471015
$541$	18212.9	1.44738	0.723692	−	0.690124i	$-0.242444\pi$
$541$	18212.9	1.44738	0.723692	+	0.690124i	$0.242444\pi$
$542$	−1102.52	−0.0873749
$543$	5292.38	0.418265
$544$	15301.7	1.20598
$545$	−2153.84	−0.169285
$546$	−11874.8	−0.930763
$547$	12903.8	1.00864	0.504322	−	0.863515i	$-0.331742\pi$
$547$	12903.8	1.00864	0.504322	+	0.863515i	$0.331742\pi$
$548$	−21438.5	−1.67118
$549$	11929.4	0.927386
$550$	1184.18	0.0918064
$551$	−2588.18	−0.200109
$552$	−590.354	−0.0455202
$553$	3841.70	0.295417
$554$	−17597.4	−1.34954
$555$	2829.17	0.216382
$556$	21163.9	1.61430
$557$	−23644.7	−1.79867	−0.899334	−	0.437263i	$-0.855948\pi$
$557$	−23644.7	−1.79867	−0.899334	+	0.437263i	$0.855948\pi$
$558$	−10507.1	−0.797131
$559$	16783.2	1.26986
$560$	2789.13	0.210468
$561$	−1564.25	−0.117723
$562$	10662.7	0.800320
$563$	−5371.87	−0.402127	−0.201063	−	0.979578i	$-0.564440\pi$
$563$	−5371.87	−0.402127	−0.201063	+	0.979578i	$0.564440\pi$
$564$	14307.1	1.06815
$565$	6543.08	0.487203
$566$	−1984.67	−0.147389
$567$	4914.48	0.364001
$568$	−10894.6	−0.804805
$569$	−10868.4	−0.800753	−0.400377	−	0.916351i	$-0.631121\pi$
$569$	−10868.4	−0.800753	−0.400377	+	0.916351i	$0.631121\pi$
$570$	−941.893	−0.0692133
$571$	13359.7	0.979135	0.489567	−	0.871966i	$-0.337155\pi$
$571$	13359.7	0.979135	0.489567	+	0.871966i	$0.337155\pi$
$572$	−9261.43	−0.676993
$573$	6444.03	0.469814
$574$	−7422.67	−0.539750
$575$	585.478	0.0424628
$576$	16692.6	1.20751
$577$	−23006.3	−1.65991	−0.829953	−	0.557833i	$-0.811633\pi$
$577$	−23006.3	−1.65991	−0.829953	+	0.557833i	$0.811633\pi$
$578$	4730.28	0.340404
$579$	−235.248	−0.0168853
$580$	−7180.51	−0.514060
$581$	15530.0	1.10894
$582$	−3488.39	−0.248451
$583$	−4502.24	−0.319835
$584$	−1931.59	−0.136866
$585$	8664.50	0.612364
$586$	−22860.8	−1.61155
$587$	−5987.29	−0.420992	−0.210496	−	0.977595i	$-0.567508\pi$
$587$	−5987.29	−0.420992	−0.210496	+	0.977595i	$0.567508\pi$
$588$	−2866.38	−0.201033
$589$	2136.57	0.149467
$590$	−13068.8	−0.911923
$591$	11832.5	0.823557
$592$	9140.83	0.634605
$593$	10977.6	0.760197	0.380099	−	0.924946i	$-0.375890\pi$
$593$	10977.6	0.760197	0.380099	+	0.924946i	$0.375890\pi$
$594$	−5311.14	−0.366866
$595$	−4631.24	−0.319096
$596$	−5360.45	−0.368410
$597$	8235.45	0.564580
$598$	−8053.71	−0.550737
$599$	24542.1	1.67406	0.837030	−	0.547157i	$-0.184290\pi$
$599$	24542.1	1.67406	0.837030	+	0.547157i	$0.184290\pi$
$600$	−630.206	−0.0428801
$601$	−12637.2	−0.857710	−0.428855	−	0.903373i	$-0.641083\pi$
$601$	−12637.2	−0.857710	−0.428855	+	0.903373i	$0.641083\pi$
$602$	−13571.4	−0.918821
$603$	22412.3	1.51360
$604$	−11136.6	−0.750232
$605$	−605.000	−0.0406558
$606$	1231.65	0.0825615
$607$	15970.2	1.06789	0.533946	−	0.845518i	$-0.320708\pi$
$607$	15970.2	1.06789	0.533946	+	0.845518i	$0.320708\pi$
$608$	−4707.33	−0.313992
$609$	4703.74	0.312981
$610$	−11837.0	−0.785681
$611$	47071.3	3.11669
$612$	−14128.5	−0.933186
$613$	−10457.2	−0.689012	−0.344506	−	0.938784i	$-0.611953\pi$
$613$	−10457.2	−0.689012	−0.344506	+	0.938784i	$0.611953\pi$
$614$	−19327.0	−1.27032
$615$	−1323.22	−0.0867597
$616$	1806.13	0.118135
$617$	19175.7	1.25119	0.625597	−	0.780147i	$-0.284856\pi$
$617$	19175.7	1.25119	0.625597	+	0.780147i	$0.284856\pi$
$618$	9288.01	0.604561
$619$	−6936.56	−0.450410	−0.225205	−	0.974311i	$-0.572305\pi$
$619$	−6936.56	−0.450410	−0.225205	+	0.974311i	$0.572305\pi$
$620$	5927.60	0.383964
$621$	−2625.92	−0.169685
$622$	−2728.11	−0.175864
$623$	−8012.11	−0.515246
$624$	−6839.51	−0.438781
$625$	625.000	0.0400000
$626$	−13417.1	−0.856636
$627$	481.216	0.0306506
$628$	1632.88	0.103757
$629$	−15178.0	−0.962141
$630$	−7006.41	−0.443083
$631$	−11143.9	−0.703064	−0.351532	−	0.936176i	$-0.614339\pi$
$631$	−11143.9	−0.703064	−0.351532	+	0.936176i	$0.614339\pi$
$632$	−2804.55	−0.176517
$633$	5827.47	0.365910
$634$	−26464.4	−1.65779
$635$	10429.5	0.651785
$636$	9935.14	0.619424
$637$	−9430.56	−0.586581
$638$	6452.35	0.400394
$639$	−21592.2	−1.33674
$640$	−6653.07	−0.410915
$641$	4464.97	0.275126	0.137563	−	0.990493i	$-0.456073\pi$
$641$	4464.97	0.275126	0.137563	+	0.990493i	$0.456073\pi$
$642$	−8885.82	−0.546254
$643$	−22187.5	−1.36079	−0.680396	−	0.732845i	$-0.738192\pi$
$643$	−22187.5	−1.36079	−0.680396	+	0.732845i	$0.738192\pi$
$644$	3702.74	0.226566
$645$	−2419.34	−0.147692
$646$	5053.08	0.307757
$647$	−10909.6	−0.662905	−0.331452	−	0.943472i	$-0.607539\pi$
$647$	−10909.6	−0.662905	−0.331452	+	0.943472i	$0.607539\pi$
$648$	−3587.71	−0.217498
$649$	6676.90	0.403838
$650$	−8597.37	−0.518795
$651$	−3882.99	−0.233774
$652$	18386.9	1.10442
$653$	20154.4	1.20781	0.603906	−	0.797056i	$-0.293610\pi$
$653$	20154.4	1.20781	0.603906	+	0.797056i	$0.293610\pi$
$654$	−4270.92	−0.255361
$655$	9246.71	0.551601
$656$	−4275.20	−0.254449
$657$	−3828.23	−0.227326
$658$	−38063.4	−2.25512
$659$	−17525.4	−1.03595	−0.517976	−	0.855395i	$-0.673314\pi$
$659$	−17525.4	−1.03595	−0.517976	+	0.855395i	$0.673314\pi$
$660$	1335.06	0.0787381
$661$	−16438.0	−0.967267	−0.483634	−	0.875271i	$-0.660683\pi$
$661$	−16438.0	−0.967267	−0.483634	+	0.875271i	$0.660683\pi$
$662$	40830.8	2.39718
$663$	11356.7	0.665247
$664$	−11337.4	−0.662614
$665$	1424.73	0.0830807
$666$	−22962.2	−1.33598
$667$	3190.15	0.185192
$668$	20456.7	1.18487
$669$	14228.1	0.822259
$670$	−22238.6	−1.28232
$671$	6047.55	0.347933
$672$	8555.07	0.491100
$673$	8485.75	0.486035	0.243018	−	0.970022i	$-0.421863\pi$
$673$	8485.75	0.486035	0.243018	+	0.970022i	$0.421863\pi$
$674$	15533.7	0.887740
$675$	−2803.18	−0.159843
$676$	44077.9	2.50785
$677$	5956.31	0.338138	0.169069	−	0.985604i	$-0.445924\pi$
$677$	5956.31	0.338138	0.169069	+	0.985604i	$0.445924\pi$
$678$	12974.5	0.734930
$679$	5276.63	0.298230
$680$	3380.94	0.190666
$681$	1939.86	0.109156
$682$	−5326.49	−0.299064
$683$	−25058.0	−1.40383	−0.701915	−	0.712261i	$-0.747671\pi$
$683$	−25058.0	−1.40383	−0.701915	+	0.712261i	$0.747671\pi$
$684$	4346.41	0.242966
$685$	10167.6	0.567132
$686$	29776.6	1.65725
$687$	14778.4	0.820716
$688$	−7816.68	−0.433151
$689$	32687.1	1.80737
$690$	1160.96	0.0640538
$691$	−19072.5	−1.05001	−0.525003	−	0.851101i	$-0.675936\pi$
$691$	−19072.5	−1.05001	−0.525003	+	0.851101i	$0.675936\pi$
$692$	35667.6	1.95936
$693$	3579.60	0.196216
$694$	−26024.7	−1.42347
$695$	−10037.4	−0.547827
$696$	−3433.87	−0.187012
$697$	7098.81	0.385777
$698$	−21444.0	−1.16285
$699$	−12578.2	−0.680616
$700$	3952.69	0.213425
$701$	19908.1	1.07264	0.536319	−	0.844015i	$-0.319814\pi$
$701$	19908.1	1.07264	0.536319	+	0.844015i	$0.319814\pi$
$702$	38559.9	2.07315
$703$	4669.28	0.250505
$704$	8462.20	0.453027
$705$	−6785.45	−0.362489
$706$	10709.4	0.570899
$707$	−1863.02	−0.0991033
$708$	−14734.0	−0.782114
$709$	−1257.61	−0.0666155	−0.0333078	−	0.999445i	$-0.510604\pi$
$709$	−1257.61	−0.0666155	−0.0333078	+	0.999445i	$0.510604\pi$
$710$	21424.9	1.13248
$711$	−5558.37	−0.293186
$712$	5849.07	0.307870
$713$	−2633.51	−0.138325
$714$	−9183.44	−0.481347
$715$	4392.42	0.229744
$716$	34851.9	1.81910
$717$	14860.4	0.774021
$718$	27474.0	1.42802
$719$	8778.00	0.455305	0.227652	−	0.973742i	$-0.426895\pi$
$719$	8778.00	0.455305	0.227652	+	0.973742i	$0.426895\pi$
$720$	−4035.45	−0.208878
$721$	−14049.3	−0.725689
$722$	−1554.50	−0.0801283
$723$	802.568	0.0412833
$724$	24232.7	1.24392
$725$	3405.50	0.174451
$726$	−1199.67	−0.0613280
$727$	23111.2	1.17902	0.589510	−	0.807761i	$-0.299321\pi$
$727$	23111.2	1.17902	0.589510	+	0.807761i	$0.299321\pi$
$728$	−13112.9	−0.667577
$729$	−139.948	−0.00711011
$730$	3798.57	0.192591
$731$	12979.3	0.656712
$732$	−13345.2	−0.673842
$733$	−27998.0	−1.41082	−0.705408	−	0.708801i	$-0.749236\pi$
$733$	−27998.0	−1.41082	−0.705408	+	0.708801i	$0.749236\pi$
$734$	6222.54	0.312913
$735$	1359.44	0.0682227
$736$	5802.18	0.290586
$737$	11361.8	0.567865
$738$	10739.5	0.535672
$739$	−5545.81	−0.276056	−0.138028	−	0.990428i	$-0.544076\pi$
$739$	−5545.81	−0.276056	−0.138028	+	0.990428i	$0.544076\pi$
$740$	12954.2	0.643521
$741$	−3493.72	−0.173205
$742$	−26431.9	−1.30775
$743$	38354.9	1.89381	0.946907	−	0.321507i	$-0.104189\pi$
$743$	38354.9	1.89381	0.946907	+	0.321507i	$0.104189\pi$
$744$	2834.70	0.139684
$745$	2542.30	0.125024
$746$	−53815.1	−2.64117
$747$	−22469.7	−1.10056
$748$	−7162.35	−0.350109
$749$	13440.9	0.655700
$750$	1239.33	0.0603387
$751$	−6309.91	−0.306594	−0.153297	−	0.988180i	$-0.548989\pi$
$751$	−6309.91	−0.306594	−0.153297	+	0.988180i	$0.548989\pi$
$752$	−21923.2	−1.06311
$753$	−16384.6	−0.792946
$754$	−46845.4	−2.26261
$755$	5281.73	0.254599
$756$	−17728.2	−0.852867
$757$	−13314.5	−0.639267	−0.319633	−	0.947541i	$-0.603560\pi$
$757$	−13314.5	−0.639267	−0.319633	+	0.947541i	$0.603560\pi$
$758$	45886.6	2.19878
$759$	−593.139	−0.0283657
$760$	−1040.09	−0.0496423
$761$	−26537.8	−1.26412	−0.632059	−	0.774920i	$-0.717790\pi$
$761$	−26537.8	−1.26412	−0.632059	+	0.774920i	$0.717790\pi$
$762$	20681.1	0.983197
$763$	6460.29	0.306524
$764$	29505.9	1.39723
$765$	6700.71	0.316686
$766$	−20698.4	−0.976324
$767$	−48475.6	−2.28208
$768$	977.561	0.0459306
$769$	16188.3	0.759123	0.379561	−	0.925167i	$-0.376075\pi$
$769$	16188.3	0.759123	0.379561	+	0.925167i	$0.376075\pi$
$770$	−3551.86	−0.166234
$771$	7589.00	0.354489
$772$	−1077.15	−0.0502169
$773$	23554.3	1.09598	0.547989	−	0.836486i	$-0.315394\pi$
$773$	23554.3	1.09598	0.547989	+	0.836486i	$0.315394\pi$
$774$	19635.8	0.911881
$775$	−2811.28	−0.130302
$776$	−3852.09	−0.178198
$777$	−8485.91	−0.391802
$778$	21996.3	1.01363
$779$	−2183.84	−0.100442
$780$	−9692.80	−0.444946
$781$	−10946.0	−0.501511
$782$	−6228.35	−0.284815
$783$	−15274.0	−0.697122
$784$	4392.24	0.200084
$785$	−774.428	−0.0352109
$786$	18335.6	0.832073
$787$	−20550.9	−0.930826	−0.465413	−	0.885094i	$-0.654094\pi$
$787$	−20550.9	−0.930826	−0.465413	+	0.885094i	$0.654094\pi$
$788$	54178.3	2.44927
$789$	−2355.38	−0.106279
$790$	5515.30	0.248387
$791$	−19625.5	−0.882178
$792$	−2613.21	−0.117243
$793$	−43906.4	−1.96616
$794$	18548.7	0.829053
$795$	−4711.94	−0.210208
$796$	37708.4	1.67907
$797$	−42045.3	−1.86866	−0.934330	−	0.356409i	$-0.884001\pi$
$797$	−42045.3	−1.86866	−0.934330	+	0.356409i	$0.884001\pi$
$798$	2825.14	0.125325
$799$	36402.7	1.61181
$800$	6193.85	0.273732
$801$	11592.3	0.511354
$802$	11426.1	0.503079
$803$	−1940.70	−0.0852874
$804$	−25072.2	−1.09979
$805$	−1756.10	−0.0768874
$806$	38671.4	1.69000
$807$	−1546.02	−0.0674382
$808$	1360.06	0.0592161
$809$	27647.2	1.20151	0.600756	−	0.799433i	$-0.294866\pi$
$809$	27647.2	1.20151	0.600756	+	0.799433i	$0.294866\pi$
$810$	7055.42	0.306052
$811$	−29480.5	−1.27645	−0.638225	−	0.769850i	$-0.720331\pi$
$811$	−29480.5	−1.27645	−0.638225	+	0.769850i	$0.720331\pi$
$812$	21537.4	0.930808
$813$	589.515	0.0254308
$814$	−11640.5	−0.501229
$815$	−8720.34	−0.374798
$816$	−5289.35	−0.226917
$817$	−3992.88	−0.170983
$818$	−31596.8	−1.35056
$819$	−25988.6	−1.10881
$820$	−6058.73	−0.258024
$821$	−5272.93	−0.224149	−0.112075	−	0.993700i	$-0.535750\pi$
$821$	−5272.93	−0.224149	−0.112075	+	0.993700i	$0.535750\pi$
$822$	20161.7	0.855500
$823$	16764.5	0.710054	0.355027	−	0.934856i	$-0.384472\pi$
$823$	16764.5	0.710054	0.355027	+	0.934856i	$0.384472\pi$
$824$	10256.4	0.433613
$825$	−633.179	−0.0267206
$826$	39199.0	1.65122
$827$	45522.0	1.91409	0.957047	−	0.289934i	$-0.0936335\pi$
$827$	45522.0	1.91409	0.957047	+	0.289934i	$0.0936335\pi$
$828$	−5357.32	−0.224855
$829$	−20950.2	−0.877720	−0.438860	−	0.898555i	$-0.644618\pi$
$829$	−20950.2	−0.877720	−0.438860	+	0.898555i	$0.644618\pi$
$830$	22295.6	0.932398
$831$	9409.34	0.392787
$832$	−61437.3	−2.56004
$833$	−7293.14	−0.303352
$834$	−19903.5	−0.826380
$835$	−9702.00	−0.402098
$836$	2203.39	0.0911552
$837$	12608.8	0.520699
$838$	−41708.4	−1.71932
$839$	−23334.5	−0.960188	−0.480094	−	0.877217i	$-0.659397\pi$
$839$	−23334.5	−0.960188	−0.480094	+	0.877217i	$0.659397\pi$
$840$	1890.26	0.0776430
$841$	−5833.10	−0.239169
$842$	19880.0	0.813671
$843$	−5701.35	−0.232936
$844$	26682.8	1.08822
$845$	−20904.8	−0.851063
$846$	55072.1	2.23808
$847$	1814.66	0.0736155
$848$	−15223.9	−0.616498
$849$	1061.20	0.0428980
$850$	−6648.79	−0.268296
$851$	−5755.28	−0.231831
$852$	24154.8	0.971277
$853$	37486.9	1.50472	0.752361	−	0.658751i	$-0.228915\pi$
$853$	37486.9	1.50472	0.752361	+	0.658751i	$0.228915\pi$
$854$	35504.2	1.42263
$855$	−2061.37	−0.0824531
$856$	−9812.23	−0.391793
$857$	16801.4	0.669690	0.334845	−	0.942273i	$-0.391316\pi$
$857$	16801.4	0.669690	0.334845	+	0.942273i	$0.391316\pi$
$858$	8709.88	0.346562
$859$	−23156.0	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$859$	−23156.0	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$860$	−11077.6	−0.439237
$861$	3968.89	0.157096
$862$	40408.6	1.59666
$863$	24112.9	0.951114	0.475557	−	0.879685i	$-0.342247\pi$
$863$	24112.9	0.951114	0.475557	+	0.879685i	$0.342247\pi$
$864$	−27780.0	−1.09386
$865$	−16916.1	−0.664929
$866$	4305.79	0.168957
$867$	−2529.27	−0.0990757
$868$	−17779.4	−0.695245
$869$	−2817.78	−0.109996
$870$	6752.88	0.263154
$871$	−82488.8	−3.20899
$872$	−4716.19	−0.183154
$873$	−7634.49	−0.295977
$874$	1916.06	0.0741551
$875$	−1874.64	−0.0724280
$876$	4282.56	0.165176
$877$	34902.4	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$877$	34902.4	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$878$	−68849.0	−2.64640
$879$	12223.6	0.469048
$880$	−2045.75	−0.0783661
$881$	−36269.9	−1.38702	−0.693510	−	0.720447i	$-0.743936\pi$
$881$	−36269.9	−1.38702	−0.693510	+	0.720447i	$0.743936\pi$
$882$	−11033.5	−0.421221
$883$	23593.1	0.899173	0.449586	−	0.893237i	$-0.351571\pi$
$883$	23593.1	0.899173	0.449586	+	0.893237i	$0.351571\pi$
$884$	52000.1	1.97845
$885$	6987.88	0.265418
$886$	65306.0	2.47630
$887$	−18755.6	−0.709979	−0.354989	−	0.934870i	$-0.615516\pi$
$887$	−18755.6	−0.709979	−0.354989	+	0.934870i	$0.615516\pi$
$888$	6194.96	0.234109
$889$	−31282.7	−1.18019
$890$	−11502.5	−0.433219
$891$	−3604.63	−0.135533
$892$	65147.6	2.44541
$893$	−11198.7	−0.419654
$894$	5041.21	0.188594
$895$	−16529.2	−0.617330
$896$	19955.4	0.744044
$897$	4306.31	0.160294
$898$	−74255.5	−2.75940
$899$	−15318.1	−0.568284
$900$	−5718.96	−0.211813
$901$	25278.7	0.934689
$902$	5444.32	0.200971
$903$	7256.63	0.267426
$904$	14327.2	0.527118
$905$	−11492.8	−0.422138
$906$	10473.3	0.384054
$907$	15746.2	0.576454	0.288227	−	0.957562i	$-0.406934\pi$
$907$	15746.2	0.576454	0.288227	+	0.957562i	$0.406934\pi$
$908$	8882.20	0.324632
$909$	2695.51	0.0983547
$910$	25787.2	0.939382
$911$	27387.4	0.996033	0.498016	−	0.867168i	$-0.334062\pi$
$911$	27387.4	0.996033	0.498016	+	0.867168i	$0.334062\pi$
$912$	1627.19	0.0590806
$913$	−11390.9	−0.412905
$914$	−3347.76	−0.121153
$915$	6329.22	0.228675
$916$	67667.3	2.44082
$917$	−27734.9	−0.998785
$918$	29820.4	1.07213
$919$	17595.7	0.631587	0.315794	−	0.948828i	$-0.397729\pi$
$919$	17595.7	0.631587	0.315794	+	0.948828i	$0.397729\pi$
$920$	1282.00	0.0459417
$921$	10334.1	0.369730
$922$	−2117.47	−0.0756347
$923$	79470.5	2.83402
$924$	−4004.42	−0.142571
$925$	−6143.79	−0.218385
$926$	21901.1	0.777231
$927$	20327.2	0.720207
$928$	33749.1	1.19382
$929$	20479.8	0.723272	0.361636	−	0.932319i	$-0.382218\pi$
$929$	20479.8	0.723272	0.361636	+	0.932319i	$0.382218\pi$
$930$	−5574.58	−0.196557
$931$	2243.62	0.0789815
$932$	−57592.8	−2.02416
$933$	1458.72	0.0511857
$934$	−2255.85	−0.0790296
$935$	3396.89	0.118813
$936$	18972.4	0.662534
$937$	−18896.6	−0.658831	−0.329416	−	0.944185i	$-0.606852\pi$
$937$	−18896.6	−0.658831	−0.329416	+	0.944185i	$0.606852\pi$
$938$	66703.3	2.32190
$939$	7174.10	0.249327
$940$	−31069.1	−1.07805
$941$	−8904.08	−0.308464	−0.154232	−	0.988035i	$-0.549290\pi$
$941$	−8904.08	−0.308464	−0.154232	+	0.988035i	$0.549290\pi$
$942$	−1535.64	−0.0531145
$943$	2691.77	0.0929544
$944$	22577.3	0.778419
$945$	8407.94	0.289429
$946$	9954.27	0.342116
$947$	23986.0	0.823062	0.411531	−	0.911396i	$-0.364994\pi$
$947$	23986.0	0.823062	0.411531	+	0.911396i	$0.364994\pi$
$948$	6218.03	0.213030
$949$	14089.9	0.481956
$950$	2045.40	0.0698542
$951$	14150.5	0.482504
$952$	−10140.9	−0.345239
$953$	14331.4	0.487136	0.243568	−	0.969884i	$-0.421682\pi$
$953$	14331.4	0.487136	0.243568	+	0.969884i	$0.421682\pi$
$954$	38243.1	1.29787
$955$	−13993.7	−0.474164
$956$	68042.8	2.30195
$957$	−3450.07	−0.116536
$958$	56825.8	1.91645
$959$	−30497.1	−1.02691
$960$	8856.35	0.297747
$961$	−17145.7	−0.575534
$962$	84512.6	2.83243
$963$	−19447.0	−0.650747
$964$	3674.79	0.122777
$965$	510.860	0.0170416
$966$	−3482.23	−0.115982
$967$	24200.5	0.804794	0.402397	−	0.915465i	$-0.368177\pi$
$967$	24200.5	0.804794	0.402397	+	0.915465i	$0.368177\pi$
$968$	−1324.75	−0.0439866
$969$	−2701.88	−0.0895736
$970$	7575.33	0.250752
$971$	−9969.91	−0.329505	−0.164753	−	0.986335i	$-0.552683\pi$
$971$	−9969.91	−0.329505	−0.164753	+	0.986335i	$0.552683\pi$
$972$	39871.1	1.31571
$973$	30106.5	0.991952
$974$	−78652.6	−2.58746
$975$	4597.00	0.150997
$976$	20449.2	0.670659
$977$	−45181.8	−1.47952	−0.739761	−	0.672870i	$-0.765062\pi$
$977$	−45181.8	−1.47952	−0.739761	+	0.672870i	$0.765062\pi$
$978$	−17291.8	−0.565370
$979$	5876.66	0.191848
$980$	6224.59	0.202895
$981$	−9347.07	−0.304209
$982$	27151.4	0.882319
$983$	9305.63	0.301937	0.150968	−	0.988539i	$-0.451761\pi$
$983$	9305.63	0.301937	0.150968	+	0.988539i	$0.451761\pi$
$984$	−2897.41	−0.0938678
$985$	−25695.2	−0.831183
$986$	−36228.0	−1.17011
$987$	20352.5	0.656359
$988$	−15997.0	−0.515114
$989$	4921.56	0.158237
$990$	5139.01	0.164978
$991$	−25846.6	−0.828500	−0.414250	−	0.910163i	$-0.635956\pi$
$991$	−25846.6	−0.828500	−0.414250	+	0.910163i	$0.635956\pi$
$992$	−27860.3	−0.891698
$993$	−21832.2	−0.697707
$994$	−64262.5	−2.05059
$995$	−17884.0	−0.569808
$996$	25136.3	0.799674
$997$	−18031.5	−0.572781	−0.286390	−	0.958113i	$-0.592455\pi$
$997$	−18031.5	−0.572781	−0.286390	+	0.958113i	$0.592455\pi$
$998$	−43078.7	−1.36636
$999$	27555.4	0.872687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.g.1.4	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.g.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-4.30610 q^{2} +2.30247 q^{3} +10.5425 q^{4} -5.00000 q^{5} -9.91467 q^{6} +14.9972 q^{7} -10.9483 q^{8} -21.6986 q^{9} +O(q^{10})\)	\(q-4.30610 q^{2} +2.30247 q^{3} +10.5425 q^{4} -5.00000 q^{5} -9.91467 q^{6} +14.9972 q^{7} -10.9483 q^{8} -21.6986 q^{9} +21.5305 q^{10} -11.0000 q^{11} +24.2738 q^{12} +79.8622 q^{13} -64.5793 q^{14} -11.5123 q^{15} -37.1954 q^{16} +61.7616 q^{17} +93.4366 q^{18} -19.0000 q^{19} -52.7126 q^{20} +34.5305 q^{21} +47.3671 q^{22} +23.4191 q^{23} -25.2082 q^{24} +25.0000 q^{25} -343.895 q^{26} -112.127 q^{27} +158.108 q^{28} +136.220 q^{29} +49.5733 q^{30} -112.451 q^{31} +247.754 q^{32} -25.3272 q^{33} -265.952 q^{34} -74.9858 q^{35} -228.758 q^{36} -245.751 q^{37} +81.8159 q^{38} +183.880 q^{39} +54.7417 q^{40} +114.939 q^{41} -148.692 q^{42} +210.152 q^{43} -115.968 q^{44} +108.493 q^{45} -100.845 q^{46} +589.406 q^{47} -85.6414 q^{48} -118.085 q^{49} -107.653 q^{50} +142.204 q^{51} +841.949 q^{52} +409.294 q^{53} +482.831 q^{54} +55.0000 q^{55} -164.194 q^{56} -43.7469 q^{57} -586.577 q^{58} -606.990 q^{59} -121.369 q^{60} -549.777 q^{61} +484.227 q^{62} -325.418 q^{63} -769.291 q^{64} -399.311 q^{65} +109.061 q^{66} -1032.89 q^{67} +651.123 q^{68} +53.9218 q^{69} +322.896 q^{70} +995.095 q^{71} +237.564 q^{72} +176.427 q^{73} +1058.23 q^{74} +57.5617 q^{75} -200.308 q^{76} -164.969 q^{77} -791.807 q^{78} +256.162 q^{79} +185.977 q^{80} +327.694 q^{81} -494.939 q^{82} +1035.53 q^{83} +364.038 q^{84} -308.808 q^{85} -904.934 q^{86} +313.642 q^{87} +120.432 q^{88} -534.242 q^{89} -467.183 q^{90} +1197.71 q^{91} +246.896 q^{92} -258.915 q^{93} -2538.04 q^{94} +95.0000 q^{95} +570.446 q^{96} +351.842 q^{97} +508.488 q^{98} +238.685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) \) 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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