LMFDB - Embedded newform 1045.4.a.c.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + \cdots + 150528 $ x^20 - x^19 - 105x^18 + 103x^17 + 4500x^16 - 4345x^15 - 101844x^14 + 95592x^13 + 1317797x^12 - 1160501x^11 - 9914845x^10 + 7570653x^9 + 42786958x^8 - 23777633x^7 - 102801526x^6 + 28436356x^5 + 122325928x^4 + 411232x^3 - 47350496x^2 - 4782848x + 150528
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$-3.59432$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.59432 q^{2} +4.02265 q^{3} +4.91912 q^{4} -5.00000 q^{5} +14.4587 q^{6} +10.5448 q^{7} -11.0737 q^{8} -10.8183 q^{9} +O(q^{10})$	\(q+3.59432 q^{2} +4.02265 q^{3} +4.91912 q^{4} -5.00000 q^{5} +14.4587 q^{6} +10.5448 q^{7} -11.0737 q^{8} -10.8183 q^{9} -17.9716 q^{10} -11.0000 q^{11} +19.7879 q^{12} -33.8888 q^{13} +37.9012 q^{14} -20.1133 q^{15} -79.1552 q^{16} +70.8746 q^{17} -38.8842 q^{18} +19.0000 q^{19} -24.5956 q^{20} +42.4180 q^{21} -39.5375 q^{22} -83.8519 q^{23} -44.5455 q^{24} +25.0000 q^{25} -121.807 q^{26} -152.130 q^{27} +51.8710 q^{28} -121.086 q^{29} -72.2935 q^{30} -46.6505 q^{31} -195.920 q^{32} -44.2492 q^{33} +254.746 q^{34} -52.7238 q^{35} -53.2163 q^{36} +180.848 q^{37} +68.2920 q^{38} -136.323 q^{39} +55.3683 q^{40} -236.659 q^{41} +152.464 q^{42} -471.447 q^{43} -54.1103 q^{44} +54.0913 q^{45} -301.390 q^{46} +137.337 q^{47} -318.414 q^{48} -231.808 q^{49} +89.8579 q^{50} +285.104 q^{51} -166.703 q^{52} -491.854 q^{53} -546.803 q^{54} +55.0000 q^{55} -116.769 q^{56} +76.4304 q^{57} -435.223 q^{58} -294.905 q^{59} -98.9396 q^{60} -303.719 q^{61} -167.677 q^{62} -114.076 q^{63} -70.9559 q^{64} +169.444 q^{65} -159.046 q^{66} -466.369 q^{67} +348.641 q^{68} -337.307 q^{69} -189.506 q^{70} -266.496 q^{71} +119.798 q^{72} +1030.88 q^{73} +650.025 q^{74} +100.566 q^{75} +93.4633 q^{76} -115.992 q^{77} -489.988 q^{78} +1174.04 q^{79} +395.776 q^{80} -319.873 q^{81} -850.627 q^{82} -360.451 q^{83} +208.659 q^{84} -354.373 q^{85} -1694.53 q^{86} -487.089 q^{87} +121.810 q^{88} +988.640 q^{89} +194.421 q^{90} -357.350 q^{91} -412.478 q^{92} -187.659 q^{93} +493.633 q^{94} -95.0000 q^{95} -788.117 q^{96} +1294.58 q^{97} -833.191 q^{98} +119.001 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) $ 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	3.59432	1.27078	0.635392	−	0.772190i	$-0.280839\pi$
$2$	3.59432	1.27078	0.635392	+	0.772190i	$0.280839\pi$
$3$	4.02265	0.774160	0.387080	−	0.922046i	$-0.373484\pi$
$3$	4.02265	0.774160	0.387080	+	0.922046i	$0.373484\pi$
$4$	4.91912	0.614890
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	14.4587	0.983790
$7$	10.5448	0.569364	0.284682	−	0.958622i	$-0.408112\pi$
$7$	10.5448	0.569364	0.284682	+	0.958622i	$0.408112\pi$
$8$	−11.0737	−0.489391
$9$	−10.8183	−0.400676
$10$	−17.9716	−0.568312
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	19.7879	0.476023
$13$	−33.8888	−0.723005	−0.361503	−	0.932371i	$-0.617736\pi$
$13$	−33.8888	−0.723005	−0.361503	+	0.932371i	$0.617736\pi$
$14$	37.9012	0.723538
$15$	−20.1133	−0.346215
$16$	−79.1552	−1.23680
$17$	70.8746	1.01115	0.505577	−	0.862782i	$-0.331280\pi$
$17$	70.8746	1.01115	0.505577	+	0.862782i	$0.331280\pi$
$18$	−38.8842	−0.509172
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−24.5956	−0.274987
$21$	42.4180	0.440779
$22$	−39.5375	−0.383156
$23$	−83.8519	−0.760189	−0.380094	−	0.924948i	$-0.624108\pi$
$23$	−83.8519	−0.760189	−0.380094	+	0.924948i	$0.624108\pi$
$24$	−44.5455	−0.378867
$25$	25.0000	0.200000
$26$	−121.807	−0.918783
$27$	−152.130	−1.08435
$28$	51.8710	0.350096
$29$	−121.086	−0.775351	−0.387676	−	0.921796i	$-0.626722\pi$
$29$	−121.086	−0.775351	−0.387676	+	0.921796i	$0.626722\pi$
$30$	−72.2935	−0.439964
$31$	−46.6505	−0.270280	−0.135140	−	0.990827i	$-0.543148\pi$
$31$	−46.6505	−0.270280	−0.135140	+	0.990827i	$0.543148\pi$
$32$	−195.920	−1.08231
$33$	−44.2492	−0.233418
$34$	254.746	1.28496
$35$	−52.7238	−0.254627
$36$	−53.2163	−0.246372
$37$	180.848	0.803547	0.401773	−	0.915739i	$-0.368394\pi$
$37$	180.848	0.803547	0.401773	+	0.915739i	$0.368394\pi$
$38$	68.2920	0.291538
$39$	−136.323	−0.559722
$40$	55.3683	0.218862
$41$	−236.659	−0.901461	−0.450730	−	0.892660i	$-0.648836\pi$
$41$	−236.659	−0.901461	−0.450730	+	0.892660i	$0.648836\pi$
$42$	152.464	0.560134
$43$	−471.447	−1.67198	−0.835988	−	0.548748i	$-0.815105\pi$
$43$	−471.447	−1.67198	−0.835988	+	0.548748i	$0.815105\pi$
$44$	−54.1103	−0.185396
$45$	54.0913	0.179188
$46$	−301.390	−0.966035
$47$	137.337	0.426227	0.213114	−	0.977027i	$-0.431640\pi$
$47$	137.337	0.426227	0.213114	+	0.977027i	$0.431640\pi$
$48$	−318.414	−0.957482
$49$	−231.808	−0.675825
$50$	89.8579	0.254157
$51$	285.104	0.782795
$52$	−166.703	−0.444569
$53$	−491.854	−1.27474	−0.637372	−	0.770556i	$-0.719978\pi$
$53$	−491.854	−1.27474	−0.637372	+	0.770556i	$0.719978\pi$
$54$	−546.803	−1.37797
$55$	55.0000	0.134840
$56$	−116.769	−0.278642
$57$	76.4304	0.177605
$58$	−435.223	−0.985303
$59$	−294.905	−0.650736	−0.325368	−	0.945588i	$-0.605488\pi$
$59$	−294.905	−0.650736	−0.325368	+	0.945588i	$0.605488\pi$
$60$	−98.9396	−0.212884
$61$	−303.719	−0.637495	−0.318747	−	0.947840i	$-0.603262\pi$
$61$	−303.719	−0.637495	−0.318747	+	0.947840i	$0.603262\pi$
$62$	−167.677	−0.343468
$63$	−114.076	−0.228131
$64$	−70.9559	−0.138586
$65$	169.444	0.323338
$66$	−159.046	−0.296624
$67$	−466.369	−0.850388	−0.425194	−	0.905102i	$-0.639794\pi$
$67$	−466.369	−0.850388	−0.425194	+	0.905102i	$0.639794\pi$
$68$	348.641	0.621748
$69$	−337.307	−0.588508
$70$	−189.506	−0.323576
$71$	−266.496	−0.445454	−0.222727	−	0.974881i	$-0.571496\pi$
$71$	−266.496	−0.445454	−0.222727	+	0.974881i	$0.571496\pi$
$72$	119.798	0.196087
$73$	1030.88	1.65282	0.826409	−	0.563070i	$-0.190380\pi$
$73$	1030.88	1.65282	0.826409	+	0.563070i	$0.190380\pi$
$74$	650.025	1.02113
$75$	100.566	0.154832
$76$	93.4633	0.141065
$77$	−115.992	−0.171670
$78$	−489.988	−0.711285
$79$	1174.04	1.67202	0.836012	−	0.548710i	$-0.184881\pi$
$79$	1174.04	1.67202	0.836012	+	0.548710i	$0.184881\pi$
$80$	395.776	0.553114
$81$	−319.873	−0.438783
$82$	−850.627	−1.14556
$83$	−360.451	−0.476682	−0.238341	−	0.971182i	$-0.576604\pi$
$83$	−360.451	−0.476682	−0.238341	+	0.971182i	$0.576604\pi$
$84$	208.659	0.271031
$85$	−354.373	−0.452202
$86$	−1694.53	−2.12472
$87$	−487.089	−0.600246
$88$	121.810	0.147557
$89$	988.640	1.17748	0.588739	−	0.808323i	$-0.299624\pi$
$89$	988.640	1.17748	0.588739	+	0.808323i	$0.299624\pi$
$90$	194.421	0.227709
$91$	−357.350	−0.411653
$92$	−412.478	−0.467432
$93$	−187.659	−0.209240
$94$	493.633	0.541643
$95$	−95.0000	−0.102598
$96$	−788.117	−0.837884
$97$	1294.58	1.35510	0.677550	−	0.735477i	$-0.263042\pi$
$97$	1294.58	1.35510	0.677550	+	0.735477i	$0.263042\pi$
$98$	−833.191	−0.858827
$99$	119.001	0.120808
$100$	122.978	0.122978
$101$	920.017	0.906387	0.453194	−	0.891412i	$-0.350285\pi$
$101$	920.017	0.906387	0.453194	+	0.891412i	$0.350285\pi$
$102$	1024.75	0.994763
$103$	20.3830	0.0194990	0.00974949	−	0.999952i	$-0.496897\pi$
$103$	20.3830	0.0194990	0.00974949	+	0.999952i	$0.496897\pi$
$104$	375.274	0.353833
$105$	−212.090	−0.197122
$106$	−1767.88	−1.61992
$107$	−851.740	−0.769540	−0.384770	−	0.923012i	$-0.625719\pi$
$107$	−851.740	−0.769540	−0.384770	+	0.923012i	$0.625719\pi$
$108$	−748.344	−0.666754
$109$	−2012.22	−1.76821	−0.884107	−	0.467284i	$-0.845233\pi$
$109$	−2012.22	−1.76821	−0.884107	+	0.467284i	$0.845233\pi$
$110$	197.687	0.171352
$111$	727.489	0.622074
$112$	−834.673	−0.704190
$113$	455.894	0.379530	0.189765	−	0.981830i	$-0.439227\pi$
$113$	455.894	0.379530	0.189765	+	0.981830i	$0.439227\pi$
$114$	274.715	0.225697
$115$	419.260	0.339967
$116$	−595.638	−0.476756
$117$	366.618	0.289691
$118$	−1059.98	−0.826944
$119$	747.356	0.575715
$120$	222.728	0.169435
$121$	121.000	0.0909091
$122$	−1091.66	−0.810118
$123$	−951.996	−0.697875
$124$	−229.480	−0.166193
$125$	−125.000	−0.0894427
$126$	−410.025	−0.289904
$127$	−886.810	−0.619619	−0.309810	−	0.950799i	$-0.600265\pi$
$127$	−886.810	−0.619619	−0.309810	+	0.950799i	$0.600265\pi$
$128$	1312.32	0.906201
$129$	−1896.47	−1.29438
$130$	609.036	0.410892
$131$	−66.4559	−0.0443227	−0.0221614	−	0.999754i	$-0.507055\pi$
$131$	−66.4559	−0.0443227	−0.0221614	+	0.999754i	$0.507055\pi$
$132$	−217.667	−0.143526
$133$	200.351	0.130621
$134$	−1676.28	−1.08066
$135$	760.649	0.484935
$136$	−784.842	−0.494850
$137$	2326.29	1.45072	0.725360	−	0.688370i	$-0.241673\pi$
$137$	2326.29	1.45072	0.725360	+	0.688370i	$0.241673\pi$
$138$	−1212.39	−0.747866
$139$	−1007.16	−0.614579	−0.307290	−	0.951616i	$-0.599422\pi$
$139$	−1007.16	−0.614579	−0.307290	+	0.951616i	$0.599422\pi$
$140$	−259.355	−0.156568
$141$	552.460	0.329968
$142$	−957.871	−0.566076
$143$	372.777	0.217994
$144$	856.321	0.495556
$145$	605.432	0.346748
$146$	3705.32	2.10037
$147$	−932.483	−0.523197
$148$	889.613	0.494093
$149$	330.968	0.181973	0.0909864	−	0.995852i	$-0.470998\pi$
$149$	330.968	0.181973	0.0909864	+	0.995852i	$0.470998\pi$
$150$	361.467	0.196758
$151$	1041.39	0.561242	0.280621	−	0.959819i	$-0.409460\pi$
$151$	1041.39	0.561242	0.280621	+	0.959819i	$0.409460\pi$
$152$	−210.400	−0.112274
$153$	−766.739	−0.405145
$154$	−416.914	−0.218155
$155$	233.253	0.120873
$156$	−670.589	−0.344167
$157$	−1145.75	−0.582423	−0.291212	−	0.956659i	$-0.594058\pi$
$157$	−1145.75	−0.582423	−0.291212	+	0.956659i	$0.594058\pi$
$158$	4219.88	2.12478
$159$	−1978.56	−0.986856
$160$	979.598	0.484025
$161$	−884.199	−0.432824
$162$	−1149.72	−0.557598
$163$	−1722.98	−0.827939	−0.413969	−	0.910291i	$-0.635858\pi$
$163$	−1722.98	−0.827939	−0.413969	+	0.910291i	$0.635858\pi$
$164$	−1164.15	−0.554299
$165$	221.246	0.104388
$166$	−1295.57	−0.605759
$167$	−781.510	−0.362126	−0.181063	−	0.983472i	$-0.557954\pi$
$167$	−781.510	−0.362126	−0.181063	+	0.983472i	$0.557954\pi$
$168$	−469.722	−0.215713
$169$	−1048.55	−0.477263
$170$	−1273.73	−0.574650
$171$	−205.547	−0.0919214
$172$	−2319.10	−1.02808
$173$	484.011	0.212709	0.106355	−	0.994328i	$-0.466082\pi$
$173$	484.011	0.212709	0.106355	+	0.994328i	$0.466082\pi$
$174$	−1750.75	−0.762782
$175$	263.619	0.113873
$176$	870.707	0.372909
$177$	−1186.30	−0.503774
$178$	3553.49	1.49632
$179$	−251.769	−0.105129	−0.0525646	−	0.998618i	$-0.516740\pi$
$179$	−251.769	−0.105129	−0.0525646	+	0.998618i	$0.516740\pi$
$180$	266.081	0.110181
$181$	384.762	0.158006	0.0790031	−	0.996874i	$-0.474826\pi$
$181$	384.762	0.158006	0.0790031	+	0.996874i	$0.474826\pi$
$182$	−1284.43	−0.523122
$183$	−1221.75	−0.493523
$184$	928.548	0.372030
$185$	−904.240	−0.359357
$186$	−674.506	−0.265899
$187$	−779.621	−0.304874
$188$	675.578	0.262083
$189$	−1604.17	−0.617388
$190$	−341.460	−0.130380
$191$	863.324	0.327057	0.163529	−	0.986539i	$-0.447712\pi$
$191$	863.324	0.327057	0.163529	+	0.986539i	$0.447712\pi$
$192$	−285.431	−0.107287
$193$	3430.02	1.27927	0.639633	−	0.768680i	$-0.279086\pi$
$193$	3430.02	1.27927	0.639633	+	0.768680i	$0.279086\pi$
$194$	4653.13	1.72204
$195$	681.615	0.250315
$196$	−1140.29	−0.415558
$197$	−2567.10	−0.928417	−0.464209	−	0.885726i	$-0.653661\pi$
$197$	−2567.10	−0.928417	−0.464209	+	0.885726i	$0.653661\pi$
$198$	427.727	0.153521
$199$	5071.07	1.80642	0.903212	−	0.429195i	$-0.141203\pi$
$199$	5071.07	1.80642	0.903212	+	0.429195i	$0.141203\pi$
$200$	−276.842	−0.0978783
$201$	−1876.04	−0.658336
$202$	3306.83	1.15182
$203$	−1276.83	−0.441457
$204$	1402.46	0.481333
$205$	1183.29	0.403146
$206$	73.2629	0.0247790
$207$	907.132	0.304589
$208$	2682.48	0.894213
$209$	−209.000	−0.0691714
$210$	−762.318	−0.250500
$211$	−2862.49	−0.933944	−0.466972	−	0.884272i	$-0.654655\pi$
$211$	−2862.49	−0.933944	−0.466972	+	0.884272i	$0.654655\pi$
$212$	−2419.49	−0.783827
$213$	−1072.02	−0.344853
$214$	−3061.42	−0.977919
$215$	2357.23	0.747730
$216$	1684.63	0.530670
$217$	−491.919	−0.153888
$218$	−7232.54	−2.24702
$219$	4146.89	1.27955
$220$	270.552	0.0829117
$221$	−2401.86	−0.731070
$222$	2614.83	0.790521
$223$	1123.50	0.337376	0.168688	−	0.985669i	$-0.446047\pi$
$223$	1123.50	0.337376	0.168688	+	0.985669i	$0.446047\pi$
$224$	−2065.93	−0.616230
$225$	−270.456	−0.0801352
$226$	1638.63	0.482300
$227$	4297.36	1.25650	0.628250	−	0.778011i	$-0.283771\pi$
$227$	4297.36	1.25650	0.628250	+	0.778011i	$0.283771\pi$
$228$	375.970	0.109207
$229$	3859.37	1.11369	0.556843	−	0.830618i	$-0.312012\pi$
$229$	3859.37	1.11369	0.556843	+	0.830618i	$0.312012\pi$
$230$	1506.95	0.432024
$231$	−466.598	−0.132900
$232$	1340.87	0.379450
$233$	−1279.63	−0.359791	−0.179895	−	0.983686i	$-0.557576\pi$
$233$	−1279.63	−0.359791	−0.179895	+	0.983686i	$0.557576\pi$
$234$	1317.74	0.368134
$235$	−686.686	−0.190615
$236$	−1450.67	−0.400131
$237$	4722.76	1.29442
$238$	2686.24	0.731609
$239$	−181.882	−0.0492259	−0.0246130	−	0.999697i	$-0.507835\pi$
$239$	−181.882	−0.0492259	−0.0246130	+	0.999697i	$0.507835\pi$
$240$	1592.07	0.428199
$241$	1712.24	0.457655	0.228827	−	0.973467i	$-0.426511\pi$
$241$	1712.24	0.457655	0.228827	+	0.973467i	$0.426511\pi$
$242$	434.912	0.115526
$243$	2820.77	0.744660
$244$	−1494.03	−0.391989
$245$	1159.04	0.302238
$246$	−3421.78	−0.886848
$247$	−643.888	−0.165869
$248$	516.592	0.132273
$249$	−1449.97	−0.369028
$250$	−449.290	−0.113662
$251$	−392.824	−0.0987842	−0.0493921	−	0.998779i	$-0.515728\pi$
$251$	−392.824	−0.0987842	−0.0493921	+	0.998779i	$0.515728\pi$
$252$	−561.153	−0.140275
$253$	922.371	0.229205
$254$	−3187.48	−0.787402
$255$	−1425.52	−0.350077
$256$	5284.54	1.29017
$257$	−1628.06	−0.395159	−0.197579	−	0.980287i	$-0.563308\pi$
$257$	−1628.06	−0.395159	−0.197579	+	0.980287i	$0.563308\pi$
$258$	−6816.51	−1.64487
$259$	1907.00	0.457511
$260$	833.516	0.198817
$261$	1309.94	0.310665
$262$	−238.864	−0.0563246
$263$	−5121.43	−1.20076	−0.600382	−	0.799713i	$-0.704985\pi$
$263$	−5121.43	−1.20076	−0.600382	+	0.799713i	$0.704985\pi$
$264$	490.001	0.114233
$265$	2459.27	0.570083
$266$	720.124	0.165991
$267$	3976.96	0.911557
$268$	−2294.12	−0.522895
$269$	−940.184	−0.213100	−0.106550	−	0.994307i	$-0.533980\pi$
$269$	−940.184	−0.213100	−0.106550	+	0.994307i	$0.533980\pi$
$270$	2734.01	0.616247
$271$	3839.71	0.860685	0.430342	−	0.902666i	$-0.358393\pi$
$271$	3839.71	0.860685	0.430342	+	0.902666i	$0.358393\pi$
$272$	−5610.10	−1.25060
$273$	−1437.49	−0.318686
$274$	8361.44	1.84355
$275$	−275.000	−0.0603023
$276$	−1659.26	−0.361867
$277$	−5076.91	−1.10123	−0.550617	−	0.834758i	$-0.685608\pi$
$277$	−5076.91	−1.10123	−0.550617	+	0.834758i	$0.685608\pi$
$278$	−3620.07	−0.780997
$279$	504.677	0.108295
$280$	583.846	0.124612
$281$	−2830.73	−0.600951	−0.300476	−	0.953790i	$-0.597145\pi$
$281$	−2830.73	−0.600951	−0.300476	+	0.953790i	$0.597145\pi$
$282$	1985.72	0.419318
$283$	−797.293	−0.167470	−0.0837352	−	0.996488i	$-0.526685\pi$
$283$	−797.293	−0.167470	−0.0837352	+	0.996488i	$0.526685\pi$
$284$	−1310.93	−0.273905
$285$	−382.152	−0.0794272
$286$	1339.88	0.277024
$287$	−2495.51	−0.513259
$288$	2119.51	0.433657
$289$	110.210	0.0224323
$290$	2176.11	0.440641
$291$	5207.65	1.04906
$292$	5071.04	1.01630
$293$	−5690.33	−1.13458	−0.567291	−	0.823517i	$-0.692009\pi$
$293$	−5690.33	−1.13458	−0.567291	+	0.823517i	$0.692009\pi$
$294$	−3351.64	−0.664869
$295$	1474.53	0.291018
$296$	−2002.65	−0.393249
$297$	1673.43	0.326943
$298$	1189.60	0.231248
$299$	2841.64	0.549620
$300$	494.698	0.0952047
$301$	−4971.30	−0.951963
$302$	3743.10	0.713216
$303$	3700.91	0.701689
$304$	−1503.95	−0.283741
$305$	1518.59	0.285096
$306$	−2755.91	−0.514852
$307$	3156.35	0.586784	0.293392	−	0.955992i	$-0.405216\pi$
$307$	3156.35	0.586784	0.293392	+	0.955992i	$0.405216\pi$
$308$	−570.581	−0.105558
$309$	81.9937	0.0150953
$310$	838.384	0.153603
$311$	2243.87	0.409127	0.204563	−	0.978853i	$-0.434423\pi$
$311$	2243.87	0.409127	0.204563	+	0.978853i	$0.434423\pi$
$312$	1509.60	0.273923
$313$	−4984.70	−0.900165	−0.450082	−	0.892987i	$-0.648605\pi$
$313$	−4984.70	−0.900165	−0.450082	+	0.892987i	$0.648605\pi$
$314$	−4118.17	−0.740134
$315$	570.380	0.102023
$316$	5775.25	1.02811
$317$	−4774.12	−0.845872	−0.422936	−	0.906160i	$-0.639000\pi$
$317$	−4774.12	−0.845872	−0.422936	+	0.906160i	$0.639000\pi$
$318$	−7111.57	−1.25408
$319$	1331.95	0.233777
$320$	354.779	0.0619774
$321$	−3426.25	−0.595747
$322$	−3178.09	−0.550025
$323$	1346.62	0.231975
$324$	−1573.49	−0.269803
$325$	−847.221	−0.144601
$326$	−6192.93	−1.05213
$327$	−8094.45	−1.36888
$328$	2620.68	0.441167
$329$	1448.19	0.242679
$330$	795.228	0.132654
$331$	3686.50	0.612169	0.306085	−	0.952004i	$-0.400981\pi$
$331$	3686.50	0.612169	0.306085	+	0.952004i	$0.400981\pi$
$332$	−1773.10	−0.293107
$333$	−1956.46	−0.321962
$334$	−2808.99	−0.460183
$335$	2331.84	0.380305
$336$	−3357.60	−0.545156
$337$	−4919.26	−0.795160	−0.397580	−	0.917567i	$-0.630150\pi$
$337$	−4919.26	−0.795160	−0.397580	+	0.917567i	$0.630150\pi$
$338$	−3768.81	−0.606498
$339$	1833.90	0.293817
$340$	−1743.20	−0.278054
$341$	513.156	0.0814925
$342$	−738.801	−0.116812
$343$	−6061.22	−0.954154
$344$	5220.64	0.818251
$345$	1686.54	0.263189
$346$	1739.69	0.270307
$347$	−2358.41	−0.364859	−0.182430	−	0.983219i	$-0.558396\pi$
$347$	−2358.41	−0.364859	−0.182430	+	0.983219i	$0.558396\pi$
$348$	−2396.05	−0.369085
$349$	−1502.56	−0.230459	−0.115229	−	0.993339i	$-0.536760\pi$
$349$	−1502.56	−0.230459	−0.115229	+	0.993339i	$0.536760\pi$
$350$	947.531	0.144708
$351$	5155.50	0.783989
$352$	2155.12	0.326330
$353$	−6471.49	−0.975759	−0.487879	−	0.872911i	$-0.662229\pi$
$353$	−6471.49	−0.975759	−0.487879	+	0.872911i	$0.662229\pi$
$354$	−4263.95	−0.640187
$355$	1332.48	0.199213
$356$	4863.24	0.724020
$357$	3006.36	0.445695
$358$	−904.939	−0.133596
$359$	5482.69	0.806032	0.403016	−	0.915193i	$-0.367962\pi$
$359$	5482.69	0.806032	0.403016	+	0.915193i	$0.367962\pi$
$360$	−598.989	−0.0876930
$361$	361.000	0.0526316
$362$	1382.96	0.200792
$363$	486.741	0.0703782
$364$	−1757.85	−0.253121
$365$	−5154.42	−0.739163
$366$	−4391.37	−0.627161
$367$	1049.99	0.149343	0.0746714	−	0.997208i	$-0.476209\pi$
$367$	1049.99	0.149343	0.0746714	+	0.997208i	$0.476209\pi$
$368$	6637.32	0.940201
$369$	2560.23	0.361194
$370$	−3250.13	−0.456665
$371$	−5186.49	−0.725793
$372$	−923.117	−0.128660
$373$	11106.9	1.54180	0.770902	−	0.636953i	$-0.219806\pi$
$373$	11106.9	1.54180	0.770902	+	0.636953i	$0.219806\pi$
$374$	−2802.20	−0.387429
$375$	−502.832	−0.0692430
$376$	−1520.83	−0.208592
$377$	4103.48	0.560583
$378$	−5765.91	−0.784567
$379$	−1390.60	−0.188471	−0.0942353	−	0.995550i	$-0.530041\pi$
$379$	−1390.60	−0.188471	−0.0942353	+	0.995550i	$0.530041\pi$
$380$	−467.316	−0.0630864
$381$	−3567.33	−0.479685
$382$	3103.06	0.415619
$383$	1683.03	0.224540	0.112270	−	0.993678i	$-0.464188\pi$
$383$	1683.03	0.224540	0.112270	+	0.993678i	$0.464188\pi$
$384$	5279.01	0.701545
$385$	579.962	0.0767730
$386$	12328.6	1.62567
$387$	5100.23	0.669921
$388$	6368.20	0.833238
$389$	−4021.88	−0.524210	−0.262105	−	0.965039i	$-0.584417\pi$
$389$	−4021.88	−0.524210	−0.262105	+	0.965039i	$0.584417\pi$
$390$	2449.94	0.318096
$391$	−5942.97	−0.768668
$392$	2566.96	0.330743
$393$	−267.329	−0.0343129
$394$	−9226.97	−1.17982
$395$	−5870.21	−0.747752
$396$	585.379	0.0742839
$397$	3946.62	0.498930	0.249465	−	0.968384i	$-0.419745\pi$
$397$	3946.62	0.498930	0.249465	+	0.968384i	$0.419745\pi$
$398$	18227.0	2.29557
$399$	805.941	0.101122
$400$	−1978.88	−0.247360
$401$	4261.66	0.530716	0.265358	−	0.964150i	$-0.414510\pi$
$401$	4261.66	0.530716	0.265358	+	0.964150i	$0.414510\pi$
$402$	−6743.08	−0.836603
$403$	1580.93	0.195414
$404$	4525.67	0.557328
$405$	1599.36	0.196230
$406$	−4589.33	−0.560996
$407$	−1989.33	−0.242279
$408$	−3157.15	−0.383093
$409$	10343.7	1.25052	0.625260	−	0.780416i	$-0.284993\pi$
$409$	10343.7	1.25052	0.625260	+	0.780416i	$0.284993\pi$
$410$	4253.13	0.512311
$411$	9357.88	1.12309
$412$	100.266	0.0119897
$413$	−3109.71	−0.370505
$414$	3260.52	0.387067
$415$	1802.25	0.213179
$416$	6639.49	0.782519
$417$	−4051.47	−0.475783
$418$	−751.212	−0.0879019
$419$	−9746.66	−1.13641	−0.568205	−	0.822887i	$-0.692362\pi$
$419$	−9746.66	−1.13641	−0.568205	+	0.822887i	$0.692362\pi$
$420$	−1043.29	−0.121209
$421$	4864.35	0.563121	0.281561	−	0.959543i	$-0.409148\pi$
$421$	4864.35	0.563121	0.281561	+	0.959543i	$0.409148\pi$
$422$	−10288.7	−1.18684
$423$	−1485.75	−0.170779
$424$	5446.63	0.623849
$425$	1771.87	0.202231
$426$	−3853.18	−0.438233
$427$	−3202.64	−0.362967
$428$	−4189.81	−0.473182
$429$	1499.55	0.168763
$430$	8472.65	0.950203
$431$	−12532.0	−1.40057	−0.700285	−	0.713863i	$-0.746944\pi$
$431$	−12532.0	−1.40057	−0.700285	+	0.713863i	$0.746944\pi$
$432$	12041.9	1.34112
$433$	−12184.0	−1.35225	−0.676126	−	0.736786i	$-0.736342\pi$
$433$	−12184.0	−1.35225	−0.676126	+	0.736786i	$0.736342\pi$
$434$	−1768.11	−0.195558
$435$	2435.44	0.268438
$436$	−9898.33	−1.08726
$437$	−1593.19	−0.174399
$438$	14905.2	1.62603
$439$	15031.0	1.63414	0.817072	−	0.576536i	$-0.195596\pi$
$439$	15031.0	1.63414	0.817072	+	0.576536i	$0.195596\pi$
$440$	−609.051	−0.0659895
$441$	2507.76	0.270787
$442$	−8633.04	−0.929031
$443$	6360.31	0.682138	0.341069	−	0.940038i	$-0.389211\pi$
$443$	6360.31	0.682138	0.341069	+	0.940038i	$0.389211\pi$
$444$	3578.61	0.382507
$445$	−4943.20	−0.526585
$446$	4038.20	0.428732
$447$	1331.37	0.140876
$448$	−748.213	−0.0789057
$449$	4578.96	0.481279	0.240640	−	0.970615i	$-0.422643\pi$
$449$	4578.96	0.481279	0.240640	+	0.970615i	$0.422643\pi$
$450$	−972.106	−0.101834
$451$	2603.25	0.271801
$452$	2242.59	0.233369
$453$	4189.17	0.434491
$454$	15446.1	1.59674
$455$	1786.75	0.184097
$456$	−846.365	−0.0869181
$457$	−19236.5	−1.96903	−0.984513	−	0.175313i	$-0.943906\pi$
$457$	−19236.5	−1.96903	−0.984513	+	0.175313i	$0.943906\pi$
$458$	13871.8	1.41525
$459$	−10782.1	−1.09644
$460$	2062.39	0.209042
$461$	−465.907	−0.0470704	−0.0235352	−	0.999723i	$-0.507492\pi$
$461$	−465.907	−0.0470704	−0.0235352	+	0.999723i	$0.507492\pi$
$462$	−1677.10	−0.168887
$463$	−14416.8	−1.44710	−0.723551	−	0.690271i	$-0.757491\pi$
$463$	−14416.8	−1.44710	−0.723551	+	0.690271i	$0.757491\pi$
$464$	9584.62	0.958954
$465$	938.295	0.0935750
$466$	−4599.39	−0.457216
$467$	5194.12	0.514679	0.257340	−	0.966321i	$-0.417154\pi$
$467$	5194.12	0.514679	0.257340	+	0.966321i	$0.417154\pi$
$468$	1803.44	0.178128
$469$	−4917.75	−0.484180
$470$	−2468.17	−0.242230
$471$	−4608.94	−0.450889
$472$	3265.68	0.318464
$473$	5185.91	0.504120
$474$	16975.1	1.64492
$475$	475.000	0.0458831
$476$	3676.34	0.354001
$477$	5321.01	0.510759
$478$	−653.743	−0.0625554
$479$	9212.89	0.878805	0.439403	−	0.898290i	$-0.355190\pi$
$479$	9212.89	0.878805	0.439403	+	0.898290i	$0.355190\pi$
$480$	3940.59	0.374713
$481$	−6128.73	−0.580969
$482$	6154.32	0.581580
$483$	−3556.83	−0.335075
$484$	595.213	0.0558991
$485$	−6472.90	−0.606019
$486$	10138.7	0.946301
$487$	−13265.2	−1.23430	−0.617152	−	0.786844i	$-0.711714\pi$
$487$	−13265.2	−1.23430	−0.617152	+	0.786844i	$0.711714\pi$
$488$	3363.28	0.311984
$489$	−6930.94	−0.640957
$490$	4165.96	0.384079
$491$	−10607.8	−0.974995	−0.487498	−	0.873124i	$-0.662090\pi$
$491$	−10607.8	−0.974995	−0.487498	+	0.873124i	$0.662090\pi$
$492$	−4682.98	−0.429116
$493$	−8581.95	−0.783999
$494$	−2314.34	−0.210783
$495$	−595.004	−0.0540271
$496$	3692.63	0.334283
$497$	−2810.14	−0.253626
$498$	−5211.64	−0.468955
$499$	15115.8	1.35607	0.678034	−	0.735030i	$-0.262832\pi$
$499$	15115.8	1.35607	0.678034	+	0.735030i	$0.262832\pi$
$500$	−614.890	−0.0549974
$501$	−3143.74	−0.280343
$502$	−1411.94	−0.125533
$503$	8485.80	0.752213	0.376106	−	0.926577i	$-0.377263\pi$
$503$	8485.80	0.752213	0.376106	+	0.926577i	$0.377263\pi$
$504$	1263.24	0.111645
$505$	−4600.08	−0.405349
$506$	3315.30	0.291270
$507$	−4217.94	−0.369478
$508$	−4362.32	−0.380998
$509$	9601.10	0.836073	0.418036	−	0.908430i	$-0.362718\pi$
$509$	9601.10	0.836073	0.418036	+	0.908430i	$0.362718\pi$
$510$	−5123.77	−0.444871
$511$	10870.4	0.941055
$512$	8495.76	0.733326
$513$	−2890.47	−0.248766
$514$	−5851.78	−0.502161
$515$	−101.915	−0.00872021
$516$	−9328.95	−0.795899
$517$	−1510.71	−0.128512
$518$	6854.37	0.581397
$519$	1947.01	0.164671
$520$	−1876.37	−0.158239
$521$	−3937.13	−0.331073	−0.165536	−	0.986204i	$-0.552936\pi$
$521$	−3937.13	−0.331073	−0.165536	+	0.986204i	$0.552936\pi$
$522$	4708.35	0.394787
$523$	−3942.45	−0.329620	−0.164810	−	0.986325i	$-0.552701\pi$
$523$	−3942.45	−0.329620	−0.164810	+	0.986325i	$0.552701\pi$
$524$	−326.905	−0.0272536
$525$	1060.45	0.0881558
$526$	−18408.1	−1.52591
$527$	−3306.34	−0.273295
$528$	3502.55	0.288692
$529$	−5135.85	−0.422113
$530$	8839.41	0.724451
$531$	3190.36	0.260734
$532$	985.549	0.0803176
$533$	8020.09	0.651761
$534$	14294.4	1.15839
$535$	4258.70	0.344149
$536$	5164.41	0.416173
$537$	−1012.78	−0.0813868
$538$	−3379.32	−0.270804
$539$	2549.89	0.203769
$540$	3741.72	0.298182
$541$	6122.28	0.486539	0.243269	−	0.969959i	$-0.421780\pi$
$541$	6122.28	0.486539	0.243269	+	0.969959i	$0.421780\pi$
$542$	13801.1	1.09374
$543$	1547.76	0.122322
$544$	−13885.7	−1.09439
$545$	10061.1	0.790770
$546$	−5166.81	−0.404980
$547$	11279.7	0.881689	0.440844	−	0.897584i	$-0.354679\pi$
$547$	11279.7	0.881689	0.440844	+	0.897584i	$0.354679\pi$
$548$	11443.3	0.892033
$549$	3285.70	0.255429
$550$	−988.437	−0.0766311
$551$	−2300.64	−0.177878
$552$	3735.23	0.288011
$553$	12380.0	0.951991
$554$	−18248.0	−1.39943
$555$	−3637.45	−0.278200
$556$	−4954.36	−0.377899
$557$	−24386.0	−1.85506	−0.927529	−	0.373752i	$-0.878071\pi$
$557$	−24386.0	−1.85506	−0.927529	+	0.373752i	$0.878071\pi$
$558$	1813.97	0.137619
$559$	15976.8	1.20885
$560$	4173.37	0.314923
$561$	−3136.14	−0.236022
$562$	−10174.5	−0.763678
$563$	−11983.0	−0.897022	−0.448511	−	0.893777i	$-0.648046\pi$
$563$	−11983.0	−0.897022	−0.448511	+	0.893777i	$0.648046\pi$
$564$	2717.62	0.202894
$565$	−2279.47	−0.169731
$566$	−2865.72	−0.212819
$567$	−3372.98	−0.249827
$568$	2951.09	0.218001
$569$	−8467.22	−0.623839	−0.311919	−	0.950109i	$-0.600972\pi$
$569$	−8467.22	−0.623839	−0.311919	+	0.950109i	$0.600972\pi$
$570$	−1373.58	−0.100935
$571$	−26265.3	−1.92499	−0.962493	−	0.271308i	$-0.912544\pi$
$571$	−26265.3	−1.92499	−0.962493	+	0.271308i	$0.912544\pi$
$572$	1833.74	0.134043
$573$	3472.86	0.253195
$574$	−8969.66	−0.652241
$575$	−2096.30	−0.152038
$576$	767.618	0.0555280
$577$	7838.60	0.565555	0.282777	−	0.959186i	$-0.408744\pi$
$577$	7838.60	0.565555	0.282777	+	0.959186i	$0.408744\pi$
$578$	396.130	0.0285066
$579$	13797.8	0.990357
$580$	2978.19	0.213212
$581$	−3800.87	−0.271405
$582$	18717.9	1.33313
$583$	5410.40	0.384350
$584$	−11415.7	−0.808875
$585$	−1833.09	−0.129554
$586$	−20452.9	−1.44181
$587$	−21907.4	−1.54040	−0.770199	−	0.637804i	$-0.779843\pi$
$587$	−21907.4	−1.54040	−0.770199	+	0.637804i	$0.779843\pi$
$588$	−4586.99	−0.321708
$589$	−886.360	−0.0620065
$590$	5299.92	0.369820
$591$	−10326.6	−0.718744
$592$	−14315.1	−0.993827
$593$	19208.3	1.33017	0.665086	−	0.746767i	$-0.268395\pi$
$593$	19208.3	1.33017	0.665086	+	0.746767i	$0.268395\pi$
$594$	6014.83	0.415474
$595$	−3736.78	−0.257467
$596$	1628.07	0.111893
$597$	20399.1	1.39846
$598$	10213.8	0.698448
$599$	29179.6	1.99040	0.995198	−	0.0978853i	$-0.0312078\pi$
$599$	29179.6	1.99040	0.995198	+	0.0978853i	$0.0312078\pi$
$600$	−1113.64	−0.0757735
$601$	−26008.0	−1.76521	−0.882603	−	0.470118i	$-0.844211\pi$
$601$	−26008.0	−1.76521	−0.882603	+	0.470118i	$0.844211\pi$
$602$	−17868.4	−1.20974
$603$	5045.29	0.340730
$604$	5122.74	0.345102
$605$	−605.000	−0.0406558
$606$	13302.2	0.891694
$607$	−184.750	−0.0123538	−0.00617690	−	0.999981i	$-0.501966\pi$
$607$	−184.750	−0.0123538	−0.00617690	+	0.999981i	$0.501966\pi$
$608$	−3722.47	−0.248300
$609$	−5136.24	−0.341758
$610$	5458.31	0.362296
$611$	−4654.20	−0.308165
$612$	−3771.68	−0.249120
$613$	26.0343	0.00171536	0.000857681	−	1.00000i	$-0.499727\pi$
$613$	26.0343	0.00171536	0.000857681		1.00000i	$0.499727\pi$
$614$	11344.9	0.745675
$615$	4759.98	0.312099
$616$	1284.46	0.0840137
$617$	−8708.33	−0.568208	−0.284104	−	0.958794i	$-0.591696\pi$
$617$	−8708.33	−0.568208	−0.284104	+	0.958794i	$0.591696\pi$
$618$	294.711	0.0191829
$619$	9979.81	0.648017	0.324008	−	0.946054i	$-0.394969\pi$
$619$	9979.81	0.648017	0.324008	+	0.946054i	$0.394969\pi$
$620$	1147.40	0.0743236
$621$	12756.4	0.824309
$622$	8065.20	0.519911
$623$	10425.0	0.670414
$624$	10790.7	0.692264
$625$	625.000	0.0400000
$626$	−17916.6	−1.14391
$627$	−840.735	−0.0535498
$628$	−5636.06	−0.358126
$629$	12817.5	0.812510
$630$	2050.13	0.129649
$631$	−2728.74	−0.172154	−0.0860771	−	0.996288i	$-0.527433\pi$
$631$	−2728.74	−0.172154	−0.0860771	+	0.996288i	$0.527433\pi$
$632$	−13000.9	−0.818275
$633$	−11514.8	−0.723022
$634$	−17159.7	−1.07492
$635$	4434.05	0.277102
$636$	−9732.78	−0.606808
$637$	7855.70	0.488625
$638$	4787.45	0.297080
$639$	2883.02	0.178483
$640$	−6561.60	−0.405266
$641$	9365.09	0.577065	0.288533	−	0.957470i	$-0.406833\pi$
$641$	9365.09	0.577065	0.288533	+	0.957470i	$0.406833\pi$
$642$	−12315.0	−0.757066
$643$	8773.54	0.538094	0.269047	−	0.963127i	$-0.413291\pi$
$643$	8773.54	0.538094	0.269047	+	0.963127i	$0.413291\pi$
$644$	−4349.48	−0.266139
$645$	9482.34	0.578863
$646$	4840.17	0.294789
$647$	−15751.9	−0.957140	−0.478570	−	0.878049i	$-0.658845\pi$
$647$	−15751.9	−0.957140	−0.478570	+	0.878049i	$0.658845\pi$
$648$	3542.16	0.214736
$649$	3243.96	0.196204
$650$	−3045.18	−0.183757
$651$	−1978.82	−0.119134
$652$	−8475.53	−0.509091
$653$	−29366.4	−1.75987	−0.879935	−	0.475094i	$-0.842414\pi$
$653$	−29366.4	−1.75987	−0.879935	+	0.475094i	$0.842414\pi$
$654$	−29094.0	−1.73955
$655$	332.280	0.0198217
$656$	18732.8	1.11493
$657$	−11152.4	−0.662245
$658$	5205.25	0.308392
$659$	−7085.46	−0.418832	−0.209416	−	0.977827i	$-0.567156\pi$
$659$	−7085.46	−0.418832	−0.209416	+	0.977827i	$0.567156\pi$
$660$	1088.34	0.0641870
$661$	−6020.01	−0.354238	−0.177119	−	0.984189i	$-0.556678\pi$
$661$	−6020.01	−0.354238	−0.177119	+	0.984189i	$0.556678\pi$
$662$	13250.4	0.777935
$663$	−9661.84	−0.565965
$664$	3991.51	0.233284
$665$	−1001.75	−0.0584155
$666$	−7032.14	−0.409144
$667$	10153.3	0.589413
$668$	−3844.34	−0.222668
$669$	4519.44	0.261183
$670$	8381.38	0.483285
$671$	3340.90	0.192212
$672$	−8310.51	−0.477061
$673$	9514.09	0.544935	0.272468	−	0.962165i	$-0.412160\pi$
$673$	9514.09	0.544935	0.272468	+	0.962165i	$0.412160\pi$
$674$	−17681.4	−1.01048
$675$	−3803.24	−0.216870
$676$	−5157.93	−0.293464
$677$	25255.2	1.43373	0.716865	−	0.697212i	$-0.245576\pi$
$677$	25255.2	1.43373	0.716865	+	0.697212i	$0.245576\pi$
$678$	6591.63	0.373377
$679$	13651.1	0.771545
$680$	3924.21	0.221304
$681$	17286.8	0.972732
$682$	1844.45	0.103559
$683$	−5594.79	−0.313439	−0.156719	−	0.987643i	$-0.550092\pi$
$683$	−5594.79	−0.313439	−0.156719	+	0.987643i	$0.550092\pi$
$684$	−1011.11	−0.0565215
$685$	−11631.5	−0.648782
$686$	−21785.9	−1.21252
$687$	15524.9	0.862172
$688$	37317.5	2.06790
$689$	16668.4	0.921647
$690$	6061.95	0.334456
$691$	8966.72	0.493647	0.246824	−	0.969060i	$-0.420613\pi$
$691$	8966.72	0.493647	0.246824	+	0.969060i	$0.420613\pi$
$692$	2380.91	0.130793
$693$	1254.84	0.0687839
$694$	−8476.88	−0.463657
$695$	5035.82	0.274848
$696$	5393.86	0.293755
$697$	−16773.1	−0.911516
$698$	−5400.68	−0.292863
$699$	−5147.50	−0.278536
$700$	1296.77	0.0700192
$701$	−9175.65	−0.494379	−0.247189	−	0.968967i	$-0.579507\pi$
$701$	−9175.65	−0.494379	−0.247189	+	0.968967i	$0.579507\pi$
$702$	18530.5	0.996280
$703$	3436.11	0.184346
$704$	780.514	0.0417851
$705$	−2762.30	−0.147566
$706$	−23260.6	−1.23998
$707$	9701.37	0.516064
$708$	−5835.56	−0.309765
$709$	−18524.6	−0.981247	−0.490624	−	0.871372i	$-0.663231\pi$
$709$	−18524.6	−0.981247	−0.490624	+	0.871372i	$0.663231\pi$
$710$	4789.36	0.253157
$711$	−12701.1	−0.669940
$712$	−10947.9	−0.576248
$713$	3911.74	0.205464
$714$	10805.8	0.566382
$715$	−1863.89	−0.0974900
$716$	−1238.48	−0.0646429
$717$	−731.650	−0.0381087
$718$	19706.5	1.02429
$719$	−376.118	−0.0195088	−0.00975441	−	0.999952i	$-0.503105\pi$
$719$	−376.118	−0.0195088	−0.00975441	+	0.999952i	$0.503105\pi$
$720$	−4281.61	−0.221620
$721$	214.934	0.0111020
$722$	1297.55	0.0668833
$723$	6887.73	0.354298
$724$	1892.69	0.0971564
$725$	−3027.16	−0.155070
$726$	1749.50	0.0894354
$727$	2761.55	0.140880	0.0704402	−	0.997516i	$-0.477560\pi$
$727$	2761.55	0.140880	0.0704402	+	0.997516i	$0.477560\pi$
$728$	3957.17	0.201460
$729$	19983.5	1.01527
$730$	−18526.6	−0.939316
$731$	−33413.6	−1.69063
$732$	−6009.96	−0.303462
$733$	−30604.4	−1.54215	−0.771077	−	0.636742i	$-0.780282\pi$
$733$	−30604.4	−1.54215	−0.771077	+	0.636742i	$0.780282\pi$
$734$	3773.98	0.189782
$735$	4662.41	0.233981
$736$	16428.2	0.822762
$737$	5130.05	0.256402
$738$	9202.30	0.458999
$739$	−4943.77	−0.246088	−0.123044	−	0.992401i	$-0.539266\pi$
$739$	−4943.77	−0.246088	−0.123044	+	0.992401i	$0.539266\pi$
$740$	−4448.07	−0.220965
$741$	−2590.14	−0.128409
$742$	−18641.9	−0.922326
$743$	40198.0	1.98482	0.992410	−	0.122974i	$-0.0392430\pi$
$743$	40198.0	1.98482	0.992410	+	0.122974i	$0.0392430\pi$
$744$	2078.07	0.102400
$745$	−1654.84	−0.0813807
$746$	39921.7	1.95930
$747$	3899.45	0.190995
$748$	−3835.05	−0.187464
$749$	−8981.40	−0.438148
$750$	−1807.34	−0.0879928
$751$	−29166.1	−1.41716	−0.708579	−	0.705632i	$-0.750663\pi$
$751$	−29166.1	−1.41716	−0.708579	+	0.705632i	$0.750663\pi$
$752$	−10871.0	−0.527158
$753$	−1580.20	−0.0764748
$754$	14749.2	0.712379
$755$	−5206.97	−0.250995
$756$	−7891.12	−0.379626
$757$	21648.2	1.03939	0.519695	−	0.854352i	$-0.326046\pi$
$757$	21648.2	1.03939	0.519695	+	0.854352i	$0.326046\pi$
$758$	−4998.26	−0.239505
$759$	3710.38	0.177442
$760$	1052.00	0.0502105
$761$	29779.1	1.41852	0.709259	−	0.704948i	$-0.249030\pi$
$761$	29779.1	1.41852	0.709259	+	0.704948i	$0.249030\pi$
$762$	−12822.1	−0.609575
$763$	−21218.4	−1.00676
$764$	4246.80	0.201104
$765$	3833.70	0.181186
$766$	6049.34	0.285342
$767$	9993.99	0.470485
$768$	21257.9	0.998799
$769$	−7847.97	−0.368017	−0.184008	−	0.982925i	$-0.558907\pi$
$769$	−7847.97	−0.368017	−0.184008	+	0.982925i	$0.558907\pi$
$770$	2084.57	0.0975619
$771$	−6549.13	−0.305916
$772$	16872.7	0.786608
$773$	−4331.18	−0.201529	−0.100764	−	0.994910i	$-0.532129\pi$
$773$	−4331.18	−0.201529	−0.100764	+	0.994910i	$0.532129\pi$
$774$	18331.9	0.851324
$775$	−1166.26	−0.0540560
$776$	−14335.7	−0.663174
$777$	7671.21	0.354187
$778$	−14455.9	−0.666157
$779$	−4496.52	−0.206809
$780$	3352.95	0.153916
$781$	2931.46	0.134310
$782$	−21360.9	−0.976810
$783$	18420.8	0.840750
$784$	18348.8	0.835860
$785$	5728.73	0.260468
$786$	−960.866	−0.0436043
$787$	−2772.13	−0.125560	−0.0627801	−	0.998027i	$-0.519997\pi$
$787$	−2772.13	−0.125560	−0.0627801	+	0.998027i	$0.519997\pi$
$788$	−12627.9	−0.570874
$789$	−20601.8	−0.929584
$790$	−21099.4	−0.950231
$791$	4807.29	0.216091
$792$	−1317.77	−0.0591226
$793$	10292.7	0.460912
$794$	14185.4	0.634032
$795$	9892.80	0.441335
$796$	24945.2	1.11075
$797$	27943.2	1.24190	0.620952	−	0.783848i	$-0.286746\pi$
$797$	27943.2	1.24190	0.620952	+	0.783848i	$0.286746\pi$
$798$	2896.81	0.128504
$799$	9733.72	0.430981
$800$	−4897.99	−0.216463
$801$	−10695.4	−0.471788
$802$	15317.8	0.674425
$803$	−11339.7	−0.498344
$804$	−9228.46	−0.404804
$805$	4421.00	0.193565
$806$	5682.37	0.248329
$807$	−3782.03	−0.164974
$808$	−10188.0	−0.443578
$809$	7822.12	0.339940	0.169970	−	0.985449i	$-0.445633\pi$
$809$	7822.12	0.339940	0.169970	+	0.985449i	$0.445633\pi$
$810$	5748.62	0.249365
$811$	−26969.6	−1.16773	−0.583867	−	0.811850i	$-0.698461\pi$
$811$	−26969.6	−1.16773	−0.583867	+	0.811850i	$0.698461\pi$
$812$	−6280.87	−0.271447
$813$	15445.8	0.666308
$814$	−7150.28	−0.307883
$815$	8614.89	0.370265
$816$	−22567.5	−0.968161
$817$	−8957.49	−0.383578
$818$	37178.5	1.58914
$819$	3865.90	0.164940
$820$	5820.76	0.247890
$821$	−17650.2	−0.750299	−0.375150	−	0.926964i	$-0.622409\pi$
$821$	−17650.2	−0.750299	−0.375150	+	0.926964i	$0.622409\pi$
$822$	33635.2	1.42720
$823$	39919.7	1.69078	0.845391	−	0.534148i	$-0.179368\pi$
$823$	39919.7	1.69078	0.845391	+	0.534148i	$0.179368\pi$
$824$	−225.714	−0.00954263
$825$	−1106.23	−0.0466836
$826$	−11177.3	−0.470832
$827$	−32658.1	−1.37320	−0.686598	−	0.727037i	$-0.740897\pi$
$827$	−32658.1	−1.37320	−0.686598	+	0.727037i	$0.740897\pi$
$828$	4462.29	0.187289
$829$	−446.740	−0.0187164	−0.00935821	−	0.999956i	$-0.502979\pi$
$829$	−446.740	−0.0187164	−0.00935821	+	0.999956i	$0.502979\pi$
$830$	6477.87	0.270904
$831$	−20422.7	−0.852532
$832$	2404.61	0.100198
$833$	−16429.3	−0.683363
$834$	−14562.3	−0.604617
$835$	3907.55	0.161948
$836$	−1028.10	−0.0425328
$837$	7096.94	0.293078
$838$	−35032.6	−1.44413
$839$	14975.6	0.616228	0.308114	−	0.951349i	$-0.400302\pi$
$839$	14975.6	0.616228	0.308114	+	0.951349i	$0.400302\pi$
$840$	2348.61	0.0964700
$841$	−9727.08	−0.398831
$842$	17484.0	0.715605
$843$	−11387.1	−0.465232
$844$	−14080.9	−0.574273
$845$	5242.74	0.213439
$846$	−5340.25	−0.217023
$847$	1275.92	0.0517604
$848$	38932.8	1.57660
$849$	−3207.23	−0.129649
$850$	6368.65	0.256991
$851$	−15164.5	−0.610847
$852$	−5273.40	−0.212047
$853$	48986.0	1.96629	0.983146	−	0.182822i	$-0.0585233\pi$
$853$	48986.0	1.96629	0.983146	+	0.182822i	$0.0585233\pi$
$854$	−11511.3	−0.461252
$855$	1027.73	0.0411085
$856$	9431.88	0.376606
$857$	−19067.3	−0.760008	−0.380004	−	0.924985i	$-0.624077\pi$
$857$	−19067.3	−0.760008	−0.380004	+	0.924985i	$0.624077\pi$
$858$	5389.87	0.214461
$859$	−19880.1	−0.789638	−0.394819	−	0.918759i	$-0.629193\pi$
$859$	−19880.1	−0.789638	−0.394819	+	0.918759i	$0.629193\pi$
$860$	11595.5	0.459772
$861$	−10038.6	−0.397345
$862$	−45044.0	−1.77982
$863$	6346.71	0.250341	0.125171	−	0.992135i	$-0.460052\pi$
$863$	6346.71	0.250341	0.125171	+	0.992135i	$0.460052\pi$
$864$	29805.2	1.17360
$865$	−2420.06	−0.0951265
$866$	−43793.1	−1.71842
$867$	443.337	0.0173662
$868$	−2419.81	−0.0946241
$869$	−12914.5	−0.504134
$870$	8753.76	0.341127
$871$	15804.7	0.614835
$872$	22282.6	0.865349
$873$	−14005.1	−0.542956
$874$	−5726.42	−0.221624
$875$	−1318.10	−0.0509255
$876$	20399.0	0.786780
$877$	20298.2	0.781551	0.390776	−	0.920486i	$-0.372207\pi$
$877$	20298.2	0.781551	0.390776	+	0.920486i	$0.372207\pi$
$878$	54026.1	2.07664
$879$	−22890.2	−0.878348
$880$	−4353.54	−0.166770
$881$	−40855.2	−1.56237	−0.781185	−	0.624299i	$-0.785385\pi$
$881$	−40855.2	−1.56237	−0.781185	+	0.624299i	$0.785385\pi$
$882$	9013.67	0.344111
$883$	16389.3	0.624623	0.312312	−	0.949980i	$-0.398897\pi$
$883$	16389.3	0.624623	0.312312	+	0.949980i	$0.398897\pi$
$884$	−11815.0	−0.449527
$885$	5931.51	0.225294
$886$	22861.0	0.866850
$887$	−30745.8	−1.16386	−0.581929	−	0.813240i	$-0.697702\pi$
$887$	−30745.8	−1.16386	−0.581929	+	0.813240i	$0.697702\pi$
$888$	−8055.97	−0.304438
$889$	−9351.20	−0.352789
$890$	−17767.4	−0.669175
$891$	3518.60	0.132298
$892$	5526.61	0.207449
$893$	2609.41	0.0977833
$894$	4785.36	0.179023
$895$	1258.85	0.0470152
$896$	13838.1	0.515958
$897$	11431.0	0.425494
$898$	16458.2	0.611602
$899$	5648.75	0.209562
$900$	−1330.41	−0.0492743
$901$	−34860.0	−1.28896
$902$	9356.89	0.345400
$903$	−19997.8	−0.736972
$904$	−5048.41	−0.185739
$905$	−1923.81	−0.0706625
$906$	15057.2	0.552144
$907$	−40209.6	−1.47204	−0.736019	−	0.676961i	$-0.763297\pi$
$907$	−40209.6	−1.47204	−0.736019	+	0.676961i	$0.763297\pi$
$908$	21139.2	0.772609
$909$	−9952.98	−0.363168
$910$	6422.14	0.233947
$911$	−40716.8	−1.48080	−0.740400	−	0.672167i	$-0.765364\pi$
$911$	−40716.8	−1.48080	−0.740400	+	0.672167i	$0.765364\pi$
$912$	−6049.87	−0.219661
$913$	3964.96	0.143725
$914$	−69142.0	−2.50220
$915$	6108.77	0.220710
$916$	18984.7	0.684795
$917$	−700.762	−0.0252358
$918$	−38754.4	−1.39334
$919$	−51957.8	−1.86500	−0.932498	−	0.361176i	$-0.882375\pi$
$919$	−51957.8	−1.86500	−0.932498	+	0.361176i	$0.882375\pi$
$920$	−4642.74	−0.166377
$921$	12696.9	0.454265
$922$	−1674.62	−0.0598162
$923$	9031.24	0.322066
$924$	−2295.25	−0.0817188
$925$	4521.20	0.160709
$926$	−51818.7	−1.83895
$927$	−220.508	−0.00781278
$928$	23723.2	0.839173
$929$	−38224.0	−1.34993	−0.674966	−	0.737848i	$-0.735842\pi$
$929$	−38224.0	−1.34993	−0.674966	+	0.737848i	$0.735842\pi$
$930$	3372.53	0.118914
$931$	−4404.35	−0.155045
$932$	−6294.65	−0.221232
$933$	9026.33	0.316730
$934$	18669.3	0.654046
$935$	3898.10	0.136344
$936$	−4059.80	−0.141772
$937$	7129.07	0.248555	0.124278	−	0.992247i	$-0.460339\pi$
$937$	7129.07	0.248555	0.124278	+	0.992247i	$0.460339\pi$
$938$	−17676.0	−0.615288
$939$	−20051.7	−0.696872
$940$	−3377.89	−0.117207
$941$	12566.0	0.435323	0.217661	−	0.976024i	$-0.430157\pi$
$941$	12566.0	0.435323	0.217661	+	0.976024i	$0.430157\pi$
$942$	−16566.0	−0.572982
$943$	19844.3	0.685280
$944$	23343.3	0.804830
$945$	8020.87	0.276105
$946$	18639.8	0.640627
$947$	31374.5	1.07659	0.538297	−	0.842755i	$-0.319068\pi$
$947$	31374.5	1.07659	0.538297	+	0.842755i	$0.319068\pi$
$948$	23231.8	0.795923
$949$	−34935.4	−1.19500
$950$	1707.30	0.0583075
$951$	−19204.6	−0.654840
$952$	−8275.97	−0.281750
$953$	−26876.9	−0.913567	−0.456783	−	0.889578i	$-0.650999\pi$
$953$	−26876.9	−0.913567	−0.456783	+	0.889578i	$0.650999\pi$
$954$	19125.4	0.649064
$955$	−4316.62	−0.146265
$956$	−894.701	−0.0302685
$957$	5357.98	0.180981
$958$	33114.1	1.11677
$959$	24530.2	0.825988
$960$	1427.15	0.0479804
$961$	−27614.7	−0.926949
$962$	−22028.6	−0.738285
$963$	9214.34	0.308336
$964$	8422.69	0.281407
$965$	−17150.1	−0.572105
$966$	−12784.4	−0.425808
$967$	24688.0	0.821007	0.410504	−	0.911859i	$-0.365353\pi$
$967$	24688.0	0.821007	0.410504	+	0.911859i	$0.365353\pi$
$968$	−1339.91	−0.0444901
$969$	5416.98	0.179586
$970$	−23265.7	−0.770119
$971$	22539.0	0.744914	0.372457	−	0.928050i	$-0.378515\pi$
$971$	22539.0	0.744914	0.372457	+	0.928050i	$0.378515\pi$
$972$	13875.7	0.457884
$973$	−10620.3	−0.349919
$974$	−47679.5	−1.56853
$975$	−3408.08	−0.111944
$976$	24040.9	0.788454
$977$	34992.1	1.14585	0.572925	−	0.819607i	$-0.305808\pi$
$977$	34992.1	1.14585	0.572925	+	0.819607i	$0.305808\pi$
$978$	−24912.0	−0.814518
$979$	−10875.0	−0.355023
$980$	5701.45	0.185843
$981$	21768.7	0.708481
$982$	−38127.7	−1.23901
$983$	−5752.54	−0.186651	−0.0933253	−	0.995636i	$-0.529750\pi$
$983$	−5752.54	−0.186651	−0.0933253	+	0.995636i	$0.529750\pi$
$984$	10542.1	0.341534
$985$	12835.5	0.415201
$986$	−30846.3	−0.996293
$987$	5825.56	0.187872
$988$	−3167.36	−0.101991
$989$	39531.7	1.27102
$990$	−2138.63	−0.0686568
$991$	−28229.1	−0.904871	−0.452436	−	0.891797i	$-0.649445\pi$
$991$	−28229.1	−0.904871	−0.452436	+	0.891797i	$0.649445\pi$
$992$	9139.76	0.292528
$993$	14829.5	0.473917
$994$	−10100.5	−0.322303
$995$	−25355.3	−0.807857
$996$	−7132.56	−0.226912
$997$	−50127.9	−1.59234	−0.796171	−	0.605071i	$-0.793145\pi$
$997$	−50127.9	−1.59234	−0.796171	+	0.605071i	$0.793145\pi$
$998$	54331.2	1.72327
$999$	−27512.4	−0.871324

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.c.1.17	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.c.1.17	✓	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+3.59432 q^{2} +4.02265 q^{3} +4.91912 q^{4} -5.00000 q^{5} +14.4587 q^{6} +10.5448 q^{7} -11.0737 q^{8} -10.8183 q^{9} +O(q^{10})\)	\(q+3.59432 q^{2} +4.02265 q^{3} +4.91912 q^{4} -5.00000 q^{5} +14.4587 q^{6} +10.5448 q^{7} -11.0737 q^{8} -10.8183 q^{9} -17.9716 q^{10} -11.0000 q^{11} +19.7879 q^{12} -33.8888 q^{13} +37.9012 q^{14} -20.1133 q^{15} -79.1552 q^{16} +70.8746 q^{17} -38.8842 q^{18} +19.0000 q^{19} -24.5956 q^{20} +42.4180 q^{21} -39.5375 q^{22} -83.8519 q^{23} -44.5455 q^{24} +25.0000 q^{25} -121.807 q^{26} -152.130 q^{27} +51.8710 q^{28} -121.086 q^{29} -72.2935 q^{30} -46.6505 q^{31} -195.920 q^{32} -44.2492 q^{33} +254.746 q^{34} -52.7238 q^{35} -53.2163 q^{36} +180.848 q^{37} +68.2920 q^{38} -136.323 q^{39} +55.3683 q^{40} -236.659 q^{41} +152.464 q^{42} -471.447 q^{43} -54.1103 q^{44} +54.0913 q^{45} -301.390 q^{46} +137.337 q^{47} -318.414 q^{48} -231.808 q^{49} +89.8579 q^{50} +285.104 q^{51} -166.703 q^{52} -491.854 q^{53} -546.803 q^{54} +55.0000 q^{55} -116.769 q^{56} +76.4304 q^{57} -435.223 q^{58} -294.905 q^{59} -98.9396 q^{60} -303.719 q^{61} -167.677 q^{62} -114.076 q^{63} -70.9559 q^{64} +169.444 q^{65} -159.046 q^{66} -466.369 q^{67} +348.641 q^{68} -337.307 q^{69} -189.506 q^{70} -266.496 q^{71} +119.798 q^{72} +1030.88 q^{73} +650.025 q^{74} +100.566 q^{75} +93.4633 q^{76} -115.992 q^{77} -489.988 q^{78} +1174.04 q^{79} +395.776 q^{80} -319.873 q^{81} -850.627 q^{82} -360.451 q^{83} +208.659 q^{84} -354.373 q^{85} -1694.53 q^{86} -487.089 q^{87} +121.810 q^{88} +988.640 q^{89} +194.421 q^{90} -357.350 q^{91} -412.478 q^{92} -187.659 q^{93} +493.633 q^{94} -95.0000 q^{95} -788.117 q^{96} +1294.58 q^{97} -833.191 q^{98} +119.001 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * q^99

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