LMFDB - Embedded newform 1045.4.a.c.1.20

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			1.20
Root			$-5.13031$ 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)

$f(q)$	$=$	$q+5.13031 q^{2} +1.57482 q^{3} +18.3201 q^{4} -5.00000 q^{5} +8.07930 q^{6} -16.5525 q^{7} +52.9453 q^{8} -24.5200 q^{9} +O(q^{10})$	\(q+5.13031 q^{2} +1.57482 q^{3} +18.3201 q^{4} -5.00000 q^{5} +8.07930 q^{6} -16.5525 q^{7} +52.9453 q^{8} -24.5200 q^{9} -25.6516 q^{10} -11.0000 q^{11} +28.8508 q^{12} -24.1537 q^{13} -84.9194 q^{14} -7.87409 q^{15} +125.065 q^{16} -68.6392 q^{17} -125.795 q^{18} +19.0000 q^{19} -91.6005 q^{20} -26.0671 q^{21} -56.4334 q^{22} -112.892 q^{23} +83.3792 q^{24} +25.0000 q^{25} -123.916 q^{26} -81.1345 q^{27} -303.243 q^{28} +97.6351 q^{29} -40.3965 q^{30} -33.3174 q^{31} +218.061 q^{32} -17.3230 q^{33} -352.141 q^{34} +82.7624 q^{35} -449.208 q^{36} -133.680 q^{37} +97.4759 q^{38} -38.0377 q^{39} -264.727 q^{40} -244.104 q^{41} -133.733 q^{42} +383.631 q^{43} -201.521 q^{44} +122.600 q^{45} -579.171 q^{46} -70.4164 q^{47} +196.955 q^{48} -69.0154 q^{49} +128.258 q^{50} -108.094 q^{51} -442.498 q^{52} +281.119 q^{53} -416.245 q^{54} +55.0000 q^{55} -876.377 q^{56} +29.9215 q^{57} +500.898 q^{58} +32.3374 q^{59} -144.254 q^{60} -411.232 q^{61} -170.929 q^{62} +405.866 q^{63} +118.200 q^{64} +120.769 q^{65} -88.8723 q^{66} -202.133 q^{67} -1257.48 q^{68} -177.784 q^{69} +424.597 q^{70} +616.586 q^{71} -1298.22 q^{72} +12.4199 q^{73} -685.818 q^{74} +39.3704 q^{75} +348.082 q^{76} +182.077 q^{77} -195.145 q^{78} -3.23026 q^{79} -625.326 q^{80} +534.267 q^{81} -1252.33 q^{82} -304.203 q^{83} -477.552 q^{84} +343.196 q^{85} +1968.15 q^{86} +153.757 q^{87} -582.399 q^{88} -706.546 q^{89} +628.975 q^{90} +399.804 q^{91} -2068.19 q^{92} -52.4688 q^{93} -361.258 q^{94} -95.0000 q^{95} +343.407 q^{96} +1805.02 q^{97} -354.070 q^{98} +269.719 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.13031	1.81384	0.906920	−	0.421304i	$-0.138427\pi$
$2$	5.13031	1.81384	0.906920	+	0.421304i	$0.138427\pi$
$3$	1.57482	0.303074	0.151537	−	0.988452i	$-0.451578\pi$
$3$	1.57482	0.303074	0.151537	+	0.988452i	$0.451578\pi$
$4$	18.3201	2.29001
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	8.07930	0.549727
$7$	−16.5525	−0.893750	−0.446875	−	0.894596i	$-0.647463\pi$
$7$	−16.5525	−0.893750	−0.446875	+	0.894596i	$0.647463\pi$
$8$	52.9453	2.33988
$9$	−24.5200	−0.908146
$10$	−25.6516	−0.811174
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	28.8508	0.694043
$13$	−24.1537	−0.515310	−0.257655	−	0.966237i	$-0.582950\pi$
$13$	−24.1537	−0.515310	−0.257655	+	0.966237i	$0.582950\pi$
$14$	−84.9194	−1.62112
$15$	−7.87409	−0.135539
$16$	125.065	1.95415
$17$	−68.6392	−0.979262	−0.489631	−	0.871930i	$-0.662869\pi$
$17$	−68.6392	−0.979262	−0.489631	+	0.871930i	$0.662869\pi$
$18$	−125.795	−1.64723
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−91.6005	−1.02412
$21$	−26.0671	−0.270872
$22$	−56.4334	−0.546893
$23$	−112.892	−1.02346	−0.511731	−	0.859146i	$-0.670995\pi$
$23$	−112.892	−1.02346	−0.511731	+	0.859146i	$0.670995\pi$
$24$	83.3792	0.709155
$25$	25.0000	0.200000
$26$	−123.916	−0.934690
$27$	−81.1345	−0.578309
$28$	−303.243	−2.04670
$29$	97.6351	0.625186	0.312593	−	0.949887i	$-0.398802\pi$
$29$	97.6351	0.625186	0.312593	+	0.949887i	$0.398802\pi$
$30$	−40.3965	−0.245845
$31$	−33.3174	−0.193032	−0.0965158	−	0.995331i	$-0.530770\pi$
$31$	−33.3174	−0.193032	−0.0965158	+	0.995331i	$0.530770\pi$
$32$	218.061	1.20463
$33$	−17.3230	−0.0913802
$34$	−352.141	−1.77622
$35$	82.7624	0.399697
$36$	−449.208	−2.07967
$37$	−133.680	−0.593967	−0.296983	−	0.954883i	$-0.595981\pi$
$37$	−133.680	−0.593967	−0.296983	+	0.954883i	$0.595981\pi$
$38$	97.4759	0.416123
$39$	−38.0377	−0.156177
$40$	−264.727	−1.04642
$41$	−244.104	−0.929819	−0.464910	−	0.885358i	$-0.653913\pi$
$41$	−244.104	−0.929819	−0.464910	+	0.885358i	$0.653913\pi$
$42$	−133.733	−0.491318
$43$	383.631	1.36054	0.680270	−	0.732962i	$-0.261863\pi$
$43$	383.631	1.36054	0.680270	+	0.732962i	$0.261863\pi$
$44$	−201.521	−0.690465
$45$	122.600	0.406135
$46$	−579.171	−1.85639
$47$	−70.4164	−0.218538	−0.109269	−	0.994012i	$-0.534851\pi$
$47$	−70.4164	−0.218538	−0.109269	+	0.994012i	$0.534851\pi$
$48$	196.955	0.592250
$49$	−69.0154	−0.201211
$50$	128.258	0.362768
$51$	−108.094	−0.296789
$52$	−442.498	−1.18007
$53$	281.119	0.728578	0.364289	−	0.931286i	$-0.381312\pi$
$53$	281.119	0.728578	0.364289	+	0.931286i	$0.381312\pi$
$54$	−416.245	−1.04896
$55$	55.0000	0.134840
$56$	−876.377	−2.09126
$57$	29.9215	0.0695299
$58$	500.898	1.13399
$59$	32.3374	0.0713553	0.0356777	−	0.999363i	$-0.488641\pi$
$59$	32.3374	0.0713553	0.0356777	+	0.999363i	$0.488641\pi$
$60$	−144.254	−0.310385
$61$	−411.232	−0.863162	−0.431581	−	0.902074i	$-0.642044\pi$
$61$	−411.232	−0.863162	−0.431581	+	0.902074i	$0.642044\pi$
$62$	−170.929	−0.350128
$63$	405.866	0.811656
$64$	118.200	0.230859
$65$	120.769	0.230454
$66$	−88.8723	−0.165749
$67$	−202.133	−0.368574	−0.184287	−	0.982872i	$-0.558998\pi$
$67$	−202.133	−0.368574	−0.184287	+	0.982872i	$0.558998\pi$
$68$	−1257.48	−2.24252
$69$	−177.784	−0.310184
$70$	424.597	0.724986
$71$	616.586	1.03064	0.515319	−	0.856999i	$-0.327674\pi$
$71$	616.586	1.03064	0.515319	+	0.856999i	$0.327674\pi$
$72$	−1298.22	−2.12495
$73$	12.4199	0.0199129	0.00995647	−	0.999950i	$-0.496831\pi$
$73$	12.4199	0.0199129	0.00995647	+	0.999950i	$0.496831\pi$
$74$	−685.818	−1.07736
$75$	39.3704	0.0606147
$76$	348.082	0.525365
$77$	182.077	0.269476
$78$	−195.145	−0.283280
$79$	−3.23026	−0.00460041	−0.00230020	−	0.999997i	$-0.500732\pi$
$79$	−3.23026	−0.00460041	−0.00230020	+	0.999997i	$0.500732\pi$
$80$	−625.326	−0.873920
$81$	534.267	0.732876
$82$	−1252.33	−1.68654
$83$	−304.203	−0.402296	−0.201148	−	0.979561i	$-0.564467\pi$
$83$	−304.203	−0.402296	−0.201148	+	0.979561i	$0.564467\pi$
$84$	−477.552	−0.620301
$85$	343.196	0.437939
$86$	1968.15	2.46780
$87$	153.757	0.189477
$88$	−582.399	−0.705499
$89$	−706.546	−0.841503	−0.420751	−	0.907176i	$-0.638234\pi$
$89$	−706.546	−0.841503	−0.420751	+	0.907176i	$0.638234\pi$
$90$	628.975	0.736664
$91$	399.804	0.460559
$92$	−2068.19	−2.34374
$93$	−52.4688	−0.0585028
$94$	−361.258	−0.396393
$95$	−95.0000	−0.102598
$96$	343.407	0.365092
$97$	1805.02	1.88941	0.944703	−	0.327927i	$-0.106350\pi$
$97$	1805.02	1.88941	0.944703	+	0.327927i	$0.106350\pi$
$98$	−354.070	−0.364964
$99$	269.719	0.273816
$100$	458.003	0.458003
$101$	−742.772	−0.731768	−0.365884	−	0.930661i	$-0.619233\pi$
$101$	−742.772	−0.731768	−0.365884	+	0.930661i	$0.619233\pi$
$102$	−554.557	−0.538327
$103$	−269.854	−0.258151	−0.129075	−	0.991635i	$-0.541201\pi$
$103$	−269.854	−0.258151	−0.129075	+	0.991635i	$0.541201\pi$
$104$	−1278.83	−1.20576
$105$	130.336	0.121138
$106$	1442.23	1.32152
$107$	−247.150	−0.223298	−0.111649	−	0.993748i	$-0.535613\pi$
$107$	−247.150	−0.223298	−0.111649	+	0.993748i	$0.535613\pi$
$108$	−1486.39	−1.32433
$109$	156.947	0.137916	0.0689580	−	0.997620i	$-0.478033\pi$
$109$	156.947	0.137916	0.0689580	+	0.997620i	$0.478033\pi$
$110$	282.167	0.244578
$111$	−210.521	−0.180016
$112$	−2070.14	−1.74652
$113$	267.890	0.223018	0.111509	−	0.993763i	$-0.464432\pi$
$113$	267.890	0.223018	0.111509	+	0.993763i	$0.464432\pi$
$114$	153.507	0.126116
$115$	564.460	0.457706
$116$	1788.68	1.43168
$117$	592.248	0.467977
$118$	165.901	0.129427
$119$	1136.15	0.875215
$120$	−416.896	−0.317144
$121$	121.000	0.0909091
$122$	−2109.75	−1.56564
$123$	−384.419	−0.281804
$124$	−610.378	−0.442045
$125$	−125.000	−0.0894427
$126$	2082.22	1.47221
$127$	259.022	0.180980	0.0904900	−	0.995897i	$-0.471157\pi$
$127$	259.022	0.180980	0.0904900	+	0.995897i	$0.471157\pi$
$128$	−1138.09	−0.785888
$129$	604.149	0.412344
$130$	619.580	0.418006
$131$	−1458.39	−0.972673	−0.486337	−	0.873772i	$-0.661667\pi$
$131$	−1458.39	−0.972673	−0.486337	+	0.873772i	$0.661667\pi$
$132$	−317.359	−0.209262
$133$	−314.497	−0.205040
$134$	−1037.01	−0.668535
$135$	405.673	0.258628
$136$	−3634.13	−2.29135
$137$	−938.915	−0.585525	−0.292762	−	0.956185i	$-0.594574\pi$
$137$	−938.915	−0.585525	−0.292762	+	0.956185i	$0.594574\pi$
$138$	−912.089	−0.562624
$139$	1867.10	1.13932	0.569660	−	0.821881i	$-0.307075\pi$
$139$	1867.10	1.13932	0.569660	+	0.821881i	$0.307075\pi$
$140$	1516.22	0.915311
$141$	−110.893	−0.0662332
$142$	3163.28	1.86941
$143$	265.691	0.155372
$144$	−3066.59	−1.77465
$145$	−488.175	−0.279591
$146$	63.7182	0.0361189
$147$	−108.687	−0.0609818
$148$	−2449.02	−1.36019
$149$	−1194.74	−0.656890	−0.328445	−	0.944523i	$-0.606525\pi$
$149$	−1194.74	−0.656890	−0.328445	+	0.944523i	$0.606525\pi$
$150$	201.983	0.109945
$151$	281.123	0.151506	0.0757532	−	0.997127i	$-0.475864\pi$
$151$	281.123	0.151506	0.0757532	+	0.997127i	$0.475864\pi$
$152$	1005.96	0.536804
$153$	1683.03	0.889313
$154$	934.113	0.488786
$155$	166.587	0.0863264
$156$	−696.854	−0.357647
$157$	1440.78	0.732401	0.366201	−	0.930536i	$-0.380658\pi$
$157$	1440.78	0.732401	0.366201	+	0.930536i	$0.380658\pi$
$158$	−16.5722	−0.00834440
$159$	442.711	0.220813
$160$	−1090.31	−0.538727
$161$	1868.64	0.914719
$162$	2740.95	1.32932
$163$	−101.992	−0.0490098	−0.0245049	−	0.999700i	$-0.507801\pi$
$163$	−101.992	−0.0490098	−0.0245049	+	0.999700i	$0.507801\pi$
$164$	−4472.00	−2.12930
$165$	86.6149	0.0408665
$166$	−1560.65	−0.729700
$167$	940.914	0.435989	0.217994	−	0.975950i	$-0.430049\pi$
$167$	940.914	0.435989	0.217994	+	0.975950i	$0.430049\pi$
$168$	−1380.13	−0.633807
$169$	−1613.60	−0.734455
$170$	1760.70	0.794351
$171$	−465.879	−0.208343
$172$	7028.16	3.11565
$173$	2491.05	1.09475	0.547373	−	0.836889i	$-0.315628\pi$
$173$	2491.05	1.09475	0.547373	+	0.836889i	$0.315628\pi$
$174$	788.823	0.343681
$175$	−413.812	−0.178750
$176$	−1375.72	−0.589197
$177$	50.9254	0.0216259
$178$	−3624.80	−1.52635
$179$	178.051	0.0743473	0.0371737	−	0.999309i	$-0.488165\pi$
$179$	178.051	0.0743473	0.0371737	+	0.999309i	$0.488165\pi$
$180$	2246.04	0.930055
$181$	485.104	0.199213	0.0996064	−	0.995027i	$-0.468242\pi$
$181$	485.104	0.199213	0.0996064	+	0.995027i	$0.468242\pi$
$182$	2051.12	0.835379
$183$	−647.616	−0.261602
$184$	−5977.11	−2.39477
$185$	668.398	0.265630
$186$	−269.181	−0.106115
$187$	755.031	0.295259
$188$	−1290.04	−0.500455
$189$	1342.98	0.516864
$190$	−487.380	−0.186096
$191$	524.335	0.198636	0.0993182	−	0.995056i	$-0.468334\pi$
$191$	524.335	0.198636	0.0993182	+	0.995056i	$0.468334\pi$
$192$	186.143	0.0699674
$193$	−1881.27	−0.701641	−0.350821	−	0.936443i	$-0.614097\pi$
$193$	−1881.27	−0.701641	−0.350821	+	0.936443i	$0.614097\pi$
$194$	9260.33	3.42708
$195$	190.188	0.0698445
$196$	−1264.37	−0.460776
$197$	−1618.91	−0.585495	−0.292747	−	0.956190i	$-0.594570\pi$
$197$	−1618.91	−0.585495	−0.292747	+	0.956190i	$0.594570\pi$
$198$	1383.74	0.496659
$199$	−1583.85	−0.564201	−0.282100	−	0.959385i	$-0.591031\pi$
$199$	−1583.85	−0.564201	−0.282100	+	0.959385i	$0.591031\pi$
$200$	1323.63	0.467975
$201$	−318.323	−0.111705
$202$	−3810.65	−1.32731
$203$	−1616.10	−0.558760
$204$	−1980.30	−0.679650
$205$	1220.52	0.415828
$206$	−1384.44	−0.468244
$207$	2768.11	0.929453
$208$	−3020.79	−1.00699
$209$	−209.000	−0.0691714
$210$	668.663	0.219724
$211$	−51.8030	−0.0169017	−0.00845087	−	0.999964i	$-0.502690\pi$
$211$	−51.8030	−0.0169017	−0.00845087	+	0.999964i	$0.502690\pi$
$212$	5150.12	1.66845
$213$	971.010	0.312359
$214$	−1267.96	−0.405027
$215$	−1918.16	−0.608452
$216$	−4295.69	−1.35317
$217$	551.486	0.172522
$218$	805.189	0.250157
$219$	19.5591	0.00603509
$220$	1007.61	0.308785
$221$	1657.89	0.504624
$222$	−1080.04	−0.326520
$223$	−1396.62	−0.419394	−0.209697	−	0.977766i	$-0.567248\pi$
$223$	−1396.62	−0.419394	−0.209697	+	0.977766i	$0.567248\pi$
$224$	−3609.45	−1.07664
$225$	−612.999	−0.181629
$226$	1374.36	0.404518
$227$	4233.99	1.23797	0.618986	−	0.785402i	$-0.287544\pi$
$227$	4233.99	1.23797	0.618986	+	0.785402i	$0.287544\pi$
$228$	548.165	0.159224
$229$	−3565.12	−1.02878	−0.514389	−	0.857557i	$-0.671981\pi$
$229$	−3565.12	−1.02878	−0.514389	+	0.857557i	$0.671981\pi$
$230$	2895.86	0.830205
$231$	286.738	0.0816710
$232$	5169.32	1.46286
$233$	−2337.94	−0.657354	−0.328677	−	0.944442i	$-0.606603\pi$
$233$	−2337.94	−0.657354	−0.328677	+	0.944442i	$0.606603\pi$
$234$	3038.42	0.848835
$235$	352.082	0.0977332
$236$	592.424	0.163405
$237$	−5.08706	−0.00139426
$238$	5828.80	1.58750
$239$	−417.181	−0.112909	−0.0564544	−	0.998405i	$-0.517980\pi$
$239$	−417.181	−0.112909	−0.0564544	+	0.998405i	$0.517980\pi$
$240$	−984.775	−0.264862
$241$	−6350.39	−1.69736	−0.848682	−	0.528904i	$-0.822603\pi$
$241$	−6350.39	−1.69736	−0.848682	+	0.528904i	$0.822603\pi$
$242$	620.768	0.164894
$243$	3032.00	0.800424
$244$	−7533.82	−1.97665
$245$	345.077	0.0899843
$246$	−1972.19	−0.511147
$247$	−458.921	−0.118220
$248$	−1764.00	−0.451670
$249$	−479.064	−0.121925
$250$	−641.289	−0.162235
$251$	3486.77	0.876825	0.438413	−	0.898774i	$-0.355541\pi$
$251$	3486.77	0.876825	0.438413	+	0.898774i	$0.355541\pi$
$252$	7435.51	1.85870
$253$	1241.81	0.308585
$254$	1328.86	0.328269
$255$	540.471	0.132728
$256$	−6784.34	−1.65633
$257$	−4517.62	−1.09650	−0.548252	−	0.836313i	$-0.684707\pi$
$257$	−4517.62	−1.09650	−0.548252	+	0.836313i	$0.684707\pi$
$258$	3099.47	0.747925
$259$	2212.73	0.530858
$260$	2212.49	0.527742
$261$	−2394.01	−0.567760
$262$	−7482.00	−1.76427
$263$	−3540.36	−0.830069	−0.415034	−	0.909806i	$-0.636230\pi$
$263$	−3540.36	−0.830069	−0.415034	+	0.909806i	$0.636230\pi$
$264$	−917.172	−0.213818
$265$	−1405.59	−0.325830
$266$	−1613.47	−0.371910
$267$	−1112.68	−0.255037
$268$	−3703.10	−0.844040
$269$	6630.47	1.50285	0.751426	−	0.659818i	$-0.229367\pi$
$269$	6630.47	1.50285	0.751426	+	0.659818i	$0.229367\pi$
$270$	2081.23	0.469109
$271$	−7297.91	−1.63585	−0.817927	−	0.575322i	$-0.804877\pi$
$271$	−7297.91	−1.63585	−0.817927	+	0.575322i	$0.804877\pi$
$272$	−8584.38	−1.91362
$273$	629.618	0.139583
$274$	−4816.92	−1.06205
$275$	−275.000	−0.0603023
$276$	−3257.03	−0.710326
$277$	3980.14	0.863333	0.431667	−	0.902033i	$-0.357926\pi$
$277$	3980.14	0.863333	0.431667	+	0.902033i	$0.357926\pi$
$278$	9578.81	2.06654
$279$	816.941	0.175301
$280$	4381.88	0.935242
$281$	−4876.20	−1.03520	−0.517598	−	0.855624i	$-0.673174\pi$
$281$	−4876.20	−1.03520	−0.517598	+	0.855624i	$0.673174\pi$
$282$	−568.916	−0.120136
$283$	332.292	0.0697976	0.0348988	−	0.999391i	$-0.488889\pi$
$283$	332.292	0.0697976	0.0348988	+	0.999391i	$0.488889\pi$
$284$	11295.9	2.36017
$285$	−149.608	−0.0310947
$286$	1363.08	0.281820
$287$	4040.52	0.831026
$288$	−5346.85	−1.09398
$289$	−201.658	−0.0410459
$290$	−2504.49	−0.507134
$291$	2842.58	0.572629
$292$	227.535	0.0456009
$293$	−6492.31	−1.29449	−0.647244	−	0.762283i	$-0.724078\pi$
$293$	−6492.31	−1.29449	−0.647244	+	0.762283i	$0.724078\pi$
$294$	−557.596	−0.110611
$295$	−161.687	−0.0319111
$296$	−7077.71	−1.38981
$297$	892.480	0.174367
$298$	−6129.37	−1.19149
$299$	2726.76	0.527400
$300$	721.270	0.138809
$301$	−6350.05	−1.21598
$302$	1442.25	0.274808
$303$	−1169.73	−0.221780
$304$	2376.24	0.448312
$305$	2056.16	0.386018
$306$	8634.47	1.61307
$307$	4893.26	0.909684	0.454842	−	0.890572i	$-0.349696\pi$
$307$	4893.26	0.909684	0.454842	+	0.890572i	$0.349696\pi$
$308$	3335.67	0.617103
$309$	−424.971	−0.0782386
$310$	854.643	0.156582
$311$	−4299.13	−0.783862	−0.391931	−	0.919995i	$-0.628193\pi$
$311$	−4299.13	−0.783862	−0.391931	+	0.919995i	$0.628193\pi$
$312$	−2013.92	−0.365435
$313$	6583.52	1.18889	0.594445	−	0.804137i	$-0.297372\pi$
$313$	6583.52	1.18889	0.594445	+	0.804137i	$0.297372\pi$
$314$	7391.66	1.32846
$315$	−2029.33	−0.362983
$316$	−59.1786	−0.0105350
$317$	−1790.86	−0.317301	−0.158651	−	0.987335i	$-0.550714\pi$
$317$	−1790.86	−0.317301	−0.158651	+	0.987335i	$0.550714\pi$
$318$	2271.24	0.400519
$319$	−1073.99	−0.188501
$320$	−591.000	−0.103243
$321$	−389.217	−0.0676759
$322$	9586.72	1.65915
$323$	−1304.15	−0.224658
$324$	9787.82	1.67830
$325$	−603.843	−0.103062
$326$	−523.249	−0.0888959
$327$	247.164	0.0417987
$328$	−12924.2	−2.17566
$329$	1165.57	0.195318
$330$	444.362	0.0741252
$331$	−433.131	−0.0719245	−0.0359623	−	0.999353i	$-0.511450\pi$
$331$	−433.131	−0.0719245	−0.0359623	+	0.999353i	$0.511450\pi$
$332$	−5573.02	−0.921263
$333$	3277.81	0.539409
$334$	4827.18	0.790814
$335$	1010.67	0.164831
$336$	−3260.09	−0.529323
$337$	−2245.64	−0.362990	−0.181495	−	0.983392i	$-0.558094\pi$
$337$	−2245.64	−0.362990	−0.181495	+	0.983392i	$0.558094\pi$
$338$	−8278.26	−1.33218
$339$	421.878	0.0675908
$340$	6287.39	1.00289
$341$	366.491	0.0582012
$342$	−2390.10	−0.377901
$343$	6819.88	1.07358
$344$	20311.5	3.18349
$345$	888.922	0.138719
$346$	12779.9	1.98569
$347$	5195.43	0.803762	0.401881	−	0.915692i	$-0.368357\pi$
$347$	5195.43	0.803762	0.401881	+	0.915692i	$0.368357\pi$
$348$	2816.85	0.433905
$349$	8840.40	1.35592	0.677959	−	0.735099i	$-0.262865\pi$
$349$	8840.40	1.35592	0.677959	+	0.735099i	$0.262865\pi$
$350$	−2122.98	−0.324224
$351$	1959.70	0.298009
$352$	−2398.67	−0.363210
$353$	−7751.30	−1.16873	−0.584363	−	0.811492i	$-0.698656\pi$
$353$	−7751.30	−1.16873	−0.584363	+	0.811492i	$0.698656\pi$
$354$	261.263	0.0392260
$355$	−3082.93	−0.460915
$356$	−12944.0	−1.92705
$357$	1789.23	0.265255
$358$	913.458	0.134854
$359$	2681.06	0.394153	0.197076	−	0.980388i	$-0.436855\pi$
$359$	2681.06	0.394153	0.197076	+	0.980388i	$0.436855\pi$
$360$	6491.09	0.950306
$361$	361.000	0.0526316
$362$	2488.74	0.361340
$363$	190.553	0.0275522
$364$	7324.45	1.05468
$365$	−62.0997	−0.00890534
$366$	−3322.47	−0.474504
$367$	−1716.59	−0.244156	−0.122078	−	0.992520i	$-0.538956\pi$
$367$	−1716.59	−0.244156	−0.122078	+	0.992520i	$0.538956\pi$
$368$	−14118.9	−1.99999
$369$	5985.41	0.844412
$370$	3429.09	0.481810
$371$	−4653.21	−0.651166
$372$	−961.234	−0.133972
$373$	2432.53	0.337672	0.168836	−	0.985644i	$-0.445999\pi$
$373$	2432.53	0.337672	0.168836	+	0.985644i	$0.445999\pi$
$374$	3873.55	0.535552
$375$	−196.852	−0.0271077
$376$	−3728.22	−0.511352
$377$	−2358.25	−0.322165
$378$	6889.89	0.937508
$379$	2227.21	0.301858	0.150929	−	0.988545i	$-0.451773\pi$
$379$	2227.21	0.301858	0.150929	+	0.988545i	$0.451773\pi$
$380$	−1740.41	−0.234950
$381$	407.912	0.0548503
$382$	2690.00	0.360294
$383$	−11162.2	−1.48919	−0.744595	−	0.667516i	$-0.767357\pi$
$383$	−11162.2	−1.48919	−0.744595	+	0.667516i	$0.767357\pi$
$384$	−1792.28	−0.238182
$385$	−910.386	−0.120513
$386$	−9651.50	−1.27266
$387$	−9406.62	−1.23557
$388$	33068.2	4.32676
$389$	5746.87	0.749044	0.374522	−	0.927218i	$-0.377807\pi$
$389$	5746.87	0.749044	0.374522	+	0.927218i	$0.377807\pi$
$390$	975.726	0.126687
$391$	7748.82	1.00224
$392$	−3654.04	−0.470809
$393$	−2296.70	−0.294792
$394$	−8305.51	−1.06199
$395$	16.1513	0.00205736
$396$	4941.29	0.627043
$397$	878.605	0.111073	0.0555364	−	0.998457i	$-0.482313\pi$
$397$	878.605	0.111073	0.0555364	+	0.998457i	$0.482313\pi$
$398$	−8125.63	−1.02337
$399$	−495.275	−0.0621423
$400$	3126.63	0.390829
$401$	8408.75	1.04716	0.523582	−	0.851975i	$-0.324595\pi$
$401$	8408.75	1.04716	0.523582	+	0.851975i	$0.324595\pi$
$402$	−1633.09	−0.202615
$403$	804.739	0.0994712
$404$	−13607.6	−1.67576
$405$	−2671.33	−0.327752
$406$	−8291.11	−1.01350
$407$	1470.47	0.179088
$408$	−5723.08	−0.694448
$409$	1611.21	0.194790	0.0973949	−	0.995246i	$-0.468949\pi$
$409$	1611.21	0.194790	0.0973949	+	0.995246i	$0.468949\pi$
$410$	6261.64	0.754245
$411$	−1478.62	−0.177457
$412$	−4943.75	−0.591168
$413$	−535.263	−0.0637738
$414$	14201.3	1.68588
$415$	1521.01	0.179912
$416$	−5266.99	−0.620758
$417$	2940.34	0.345298
$418$	−1072.24	−0.125466
$419$	4640.88	0.541102	0.270551	−	0.962706i	$-0.412794\pi$
$419$	4640.88	0.541102	0.270551	+	0.962706i	$0.412794\pi$
$420$	2387.76	0.277407
$421$	−2451.11	−0.283752	−0.141876	−	0.989884i	$-0.545313\pi$
$421$	−2451.11	−0.283752	−0.141876	+	0.989884i	$0.545313\pi$
$422$	−265.766	−0.0306570
$423$	1726.61	0.198465
$424$	14883.9	1.70478
$425$	−1715.98	−0.195852
$426$	4981.58	0.566569
$427$	6806.92	0.771451
$428$	−4527.82	−0.511356
$429$	418.414	0.0470891
$430$	−9840.74	−1.10363
$431$	−1207.29	−0.134926	−0.0674632	−	0.997722i	$-0.521491\pi$
$431$	−1207.29	−0.134926	−0.0674632	+	0.997722i	$0.521491\pi$
$432$	−10147.1	−1.13010
$433$	−17772.6	−1.97251	−0.986255	−	0.165232i	$-0.947163\pi$
$433$	−17772.6	−1.97251	−0.986255	+	0.165232i	$0.947163\pi$
$434$	2829.29	0.312927
$435$	−768.787	−0.0847368
$436$	2875.29	0.315829
$437$	−2144.95	−0.234798
$438$	100.345	0.0109467
$439$	−3011.06	−0.327358	−0.163679	−	0.986514i	$-0.552336\pi$
$439$	−3011.06	−0.327358	−0.163679	+	0.986514i	$0.552336\pi$
$440$	2911.99	0.315509
$441$	1692.25	0.182729
$442$	8505.50	0.915306
$443$	15797.6	1.69429	0.847143	−	0.531365i	$-0.178321\pi$
$443$	15797.6	1.69429	0.847143	+	0.531365i	$0.178321\pi$
$444$	−3856.76	−0.412238
$445$	3532.73	0.376331
$446$	−7165.11	−0.760713
$447$	−1881.49	−0.199086
$448$	−1956.50	−0.206331
$449$	−5092.05	−0.535208	−0.267604	−	0.963529i	$-0.586232\pi$
$449$	−5092.05	−0.535208	−0.267604	+	0.963529i	$0.586232\pi$
$450$	−3144.87	−0.329446
$451$	2685.14	0.280351
$452$	4907.77	0.510713
$453$	442.718	0.0459176
$454$	21721.7	2.24548
$455$	−1999.02	−0.205968
$456$	1584.21	0.162691
$457$	15484.7	1.58500	0.792499	−	0.609873i	$-0.208779\pi$
$457$	15484.7	1.58500	0.792499	+	0.609873i	$0.208779\pi$
$458$	−18290.2	−1.86604
$459$	5569.01	0.566316
$460$	10341.0	1.04815
$461$	13179.1	1.33148	0.665742	−	0.746182i	$-0.268115\pi$
$461$	13179.1	1.33148	0.665742	+	0.746182i	$0.268115\pi$
$462$	1471.06	0.148138
$463$	3486.24	0.349933	0.174967	−	0.984574i	$-0.444018\pi$
$463$	3486.24	0.349933	0.174967	+	0.984574i	$0.444018\pi$
$464$	12210.8	1.22170
$465$	262.344	0.0261633
$466$	−11994.4	−1.19234
$467$	−4048.42	−0.401154	−0.200577	−	0.979678i	$-0.564282\pi$
$467$	−4048.42	−0.401154	−0.200577	+	0.979678i	$0.564282\pi$
$468$	10850.0	1.07167
$469$	3345.80	0.329413
$470$	1806.29	0.177272
$471$	2268.97	0.221972
$472$	1712.11	0.166963
$473$	−4219.94	−0.410218
$474$	−26.0982	−0.00252897
$475$	475.000	0.0458831
$476$	20814.4	2.00425
$477$	−6893.02	−0.661655
$478$	−2140.27	−0.204798
$479$	−3853.57	−0.367587	−0.183794	−	0.982965i	$-0.558838\pi$
$479$	−3853.57	−0.367587	−0.183794	+	0.982965i	$0.558838\pi$
$480$	−1717.03	−0.163274
$481$	3228.86	0.306077
$482$	−32579.5	−3.07874
$483$	2942.77	0.277227
$484$	2216.73	0.208183
$485$	−9025.12	−0.844968
$486$	15555.1	1.45184
$487$	−2706.20	−0.251806	−0.125903	−	0.992043i	$-0.540183\pi$
$487$	−2706.20	−0.251806	−0.125903	+	0.992043i	$0.540183\pi$
$488$	−21772.8	−2.01969
$489$	−160.618	−0.0148536
$490$	1770.35	0.163217
$491$	−17369.4	−1.59648	−0.798239	−	0.602341i	$-0.794235\pi$
$491$	−17369.4	−1.59648	−0.798239	+	0.602341i	$0.794235\pi$
$492$	−7042.59	−0.645334
$493$	−6701.60	−0.612220
$494$	−2354.41	−0.214433
$495$	−1348.60	−0.122454
$496$	−4166.85	−0.377212
$497$	−10206.0	−0.921132
$498$	−2457.75	−0.221153
$499$	−18396.5	−1.65038	−0.825191	−	0.564853i	$-0.808933\pi$
$499$	−18396.5	−1.65038	−0.825191	+	0.564853i	$0.808933\pi$
$500$	−2290.01	−0.204825
$501$	1481.77	0.132137
$502$	17888.2	1.59042
$503$	−13138.6	−1.16466	−0.582328	−	0.812954i	$-0.697858\pi$
$503$	−13138.6	−1.16466	−0.582328	+	0.812954i	$0.697858\pi$
$504$	21488.7	1.89917
$505$	3713.86	0.327256
$506$	6370.89	0.559724
$507$	−2541.12	−0.222594
$508$	4745.31	0.414447
$509$	−1946.41	−0.169496	−0.0847478	−	0.996402i	$-0.527008\pi$
$509$	−1946.41	−0.169496	−0.0847478	+	0.996402i	$0.527008\pi$
$510$	2772.79	0.240747
$511$	−205.581	−0.0177972
$512$	−25701.1	−2.21844
$513$	−1541.56	−0.132673
$514$	−23176.8	−1.98888
$515$	1349.27	0.115448
$516$	11068.1	0.944273
$517$	774.581	0.0658917
$518$	11352.0	0.962891
$519$	3922.95	0.331789
$520$	6394.13	0.539233
$521$	−14230.1	−1.19661	−0.598304	−	0.801270i	$-0.704158\pi$
$521$	−14230.1	−1.19661	−0.598304	+	0.801270i	$0.704158\pi$
$522$	−12282.0	−1.02983
$523$	−18893.4	−1.57964	−0.789821	−	0.613338i	$-0.789827\pi$
$523$	−18893.4	−1.57964	−0.789821	+	0.613338i	$0.789827\pi$
$524$	−26717.9	−2.22743
$525$	−651.678	−0.0541744
$526$	−18163.2	−1.50561
$527$	2286.88	0.189029
$528$	−2166.50	−0.178570
$529$	577.612	0.0474737
$530$	−7211.13	−0.591003
$531$	−792.910	−0.0648011
$532$	−5761.62	−0.469545
$533$	5896.01	0.479145
$534$	−5708.40	−0.462597
$535$	1235.75	0.0998621
$536$	−10702.0	−0.862418
$537$	280.398	0.0225327
$538$	34016.4	2.72593
$539$	759.169	0.0606674
$540$	7431.96	0.592261
$541$	−10517.2	−0.835803	−0.417902	−	0.908492i	$-0.637234\pi$
$541$	−10517.2	−0.835803	−0.417902	+	0.908492i	$0.637234\pi$
$542$	−37440.6	−2.96718
$543$	763.951	0.0603762
$544$	−14967.6	−1.17965
$545$	−784.737	−0.0616779
$546$	3230.14	0.253181
$547$	−18490.2	−1.44531	−0.722655	−	0.691208i	$-0.757079\pi$
$547$	−18490.2	−1.44531	−0.722655	+	0.691208i	$0.757079\pi$
$548$	−17201.0	−1.34086
$549$	10083.4	0.783878
$550$	−1410.84	−0.109379
$551$	1855.07	0.143427
$552$	−9412.85	−0.725793
$553$	53.4687	0.00411161
$554$	20419.4	1.56595
$555$	1052.60	0.0805055
$556$	34205.5	2.60906
$557$	23883.3	1.81682	0.908410	−	0.418081i	$-0.137297\pi$
$557$	23883.3	1.81682	0.908410	+	0.418081i	$0.137297\pi$
$558$	4191.16	0.317968
$559$	−9266.12	−0.701100
$560$	10350.7	0.781066
$561$	1189.04	0.0894851
$562$	−25016.4	−1.87768
$563$	−11555.9	−0.865047	−0.432524	−	0.901623i	$-0.642377\pi$
$563$	−11555.9	−0.865047	−0.432524	+	0.901623i	$0.642377\pi$
$564$	−2031.57	−0.151675
$565$	−1339.45	−0.0997365
$566$	1704.76	0.126602
$567$	−8843.44	−0.655008
$568$	32645.3	2.41156
$569$	−1849.29	−0.136250	−0.0681251	−	0.997677i	$-0.521702\pi$
$569$	−1849.29	−0.136250	−0.0681251	+	0.997677i	$0.521702\pi$
$570$	−767.534	−0.0564008
$571$	16678.3	1.22236	0.611179	−	0.791493i	$-0.290696\pi$
$571$	16678.3	1.22236	0.611179	+	0.791493i	$0.290696\pi$
$572$	4867.48	0.355804
$573$	825.732	0.0602015
$574$	20729.1	1.50735
$575$	−2822.30	−0.204692
$576$	−2898.26	−0.209654
$577$	−15927.5	−1.14917	−0.574583	−	0.818446i	$-0.694836\pi$
$577$	−15927.5	−1.14917	−0.574583	+	0.818446i	$0.694836\pi$
$578$	−1034.57	−0.0744506
$579$	−2962.66	−0.212649
$580$	−8943.42	−0.640268
$581$	5035.31	0.359552
$582$	14583.3	1.03866
$583$	−3092.31	−0.219674
$584$	657.578	0.0465938
$585$	−2961.24	−0.209286
$586$	−33307.6	−2.34799
$587$	−5608.46	−0.394354	−0.197177	−	0.980368i	$-0.563177\pi$
$587$	−5608.46	−0.394354	−0.197177	+	0.980368i	$0.563177\pi$
$588$	−1991.15	−0.139649
$589$	−633.031	−0.0442845
$590$	−829.504	−0.0578816
$591$	−2549.49	−0.177448
$592$	−16718.7	−1.16070
$593$	−11134.0	−0.771023	−0.385512	−	0.922703i	$-0.625975\pi$
$593$	−11134.0	−0.771023	−0.385512	+	0.922703i	$0.625975\pi$
$594$	4578.70	0.316273
$595$	−5680.75	−0.391408
$596$	−21887.7	−1.50429
$597$	−2494.27	−0.170994
$598$	13989.1	0.956619
$599$	13068.0	0.891390	0.445695	−	0.895185i	$-0.352957\pi$
$599$	13068.0	0.891390	0.445695	+	0.895185i	$0.352957\pi$
$600$	2084.48	0.141831
$601$	15824.0	1.07400	0.536999	−	0.843583i	$-0.319558\pi$
$601$	15824.0	1.07400	0.536999	+	0.843583i	$0.319558\pi$
$602$	−32577.7	−2.20560
$603$	4956.29	0.334719
$604$	5150.21	0.346952
$605$	−605.000	−0.0406558
$606$	−6001.08	−0.402272
$607$	−8292.36	−0.554492	−0.277246	−	0.960799i	$-0.589422\pi$
$607$	−8292.36	−0.554492	−0.277246	+	0.960799i	$0.589422\pi$
$608$	4143.16	0.276361
$609$	−2545.07	−0.169345
$610$	10548.8	0.700175
$611$	1700.82	0.112615
$612$	30833.3	2.03654
$613$	21835.7	1.43872	0.719360	−	0.694637i	$-0.244435\pi$
$613$	21835.7	1.43872	0.719360	+	0.694637i	$0.244435\pi$
$614$	25104.0	1.65002
$615$	1922.09	0.126026
$616$	9640.14	0.630540
$617$	18162.0	1.18505	0.592524	−	0.805553i	$-0.298131\pi$
$617$	18162.0	1.18505	0.592524	+	0.805553i	$0.298131\pi$
$618$	−2180.23	−0.141912
$619$	21313.4	1.38394	0.691969	−	0.721927i	$-0.256743\pi$
$619$	21313.4	1.38394	0.691969	+	0.721927i	$0.256743\pi$
$620$	3051.89	0.197688
$621$	9159.44	0.591877
$622$	−22055.9	−1.42180
$623$	11695.1	0.752093
$624$	−4757.19	−0.305193
$625$	625.000	0.0400000
$626$	33775.5	2.15645
$627$	−329.137	−0.0209640
$628$	26395.3	1.67721
$629$	9175.66	0.581649
$630$	−10411.1	−0.658394
$631$	−10398.9	−0.656061	−0.328031	−	0.944667i	$-0.606385\pi$
$631$	−10398.9	−0.656061	−0.328031	+	0.944667i	$0.606385\pi$
$632$	−171.027	−0.0107644
$633$	−81.5803	−0.00512247
$634$	−9187.65	−0.575533
$635$	−1295.11	−0.0809367
$636$	8110.50	0.505664
$637$	1666.98	0.103686
$638$	−5509.88	−0.341910
$639$	−15118.7	−0.935970
$640$	5690.44	0.351460
$641$	−20049.2	−1.23541	−0.617705	−	0.786410i	$-0.711937\pi$
$641$	−20049.2	−1.23541	−0.617705	+	0.786410i	$0.711937\pi$
$642$	−1996.80	−0.122753
$643$	8290.23	0.508452	0.254226	−	0.967145i	$-0.418179\pi$
$643$	8290.23	0.508452	0.254226	+	0.967145i	$0.418179\pi$
$644$	34233.7	2.09472
$645$	−3020.75	−0.184406
$646$	−6690.67	−0.407494
$647$	4584.11	0.278547	0.139274	−	0.990254i	$-0.455523\pi$
$647$	4584.11	0.278547	0.139274	+	0.990254i	$0.455523\pi$
$648$	28286.9	1.71484
$649$	−355.711	−0.0215144
$650$	−3097.90	−0.186938
$651$	868.489	0.0522869
$652$	−1868.50	−0.112233
$653$	7801.20	0.467511	0.233755	−	0.972295i	$-0.424899\pi$
$653$	7801.20	0.467511	0.233755	+	0.972295i	$0.424899\pi$
$654$	1268.03	0.0758161
$655$	7291.95	0.434993
$656$	−30528.9	−1.81700
$657$	−304.536	−0.0180839
$658$	5979.72	0.354276
$659$	19923.7	1.17772	0.588859	−	0.808236i	$-0.299577\pi$
$659$	19923.7	1.17772	0.588859	+	0.808236i	$0.299577\pi$
$660$	1586.79	0.0935847
$661$	−27052.2	−1.59185	−0.795923	−	0.605398i	$-0.793014\pi$
$661$	−27052.2	−1.59185	−0.795923	+	0.605398i	$0.793014\pi$
$662$	−2222.10	−0.130460
$663$	2610.88	0.152938
$664$	−16106.1	−0.941323
$665$	1572.49	0.0916968
$666$	16816.2	0.978401
$667$	−11022.2	−0.639853
$668$	17237.6	0.998420
$669$	−2199.43	−0.127107
$670$	5185.03	0.298978
$671$	4523.56	0.260253
$672$	−5684.23	−0.326301
$673$	28710.2	1.64442	0.822211	−	0.569183i	$-0.192740\pi$
$673$	28710.2	1.64442	0.822211	+	0.569183i	$0.192740\pi$
$674$	−11520.8	−0.658406
$675$	−2028.36	−0.115662
$676$	−29561.3	−1.68191
$677$	−15113.1	−0.857968	−0.428984	−	0.903312i	$-0.641128\pi$
$677$	−15113.1	−0.857968	−0.428984	+	0.903312i	$0.641128\pi$
$678$	2164.37	0.122599
$679$	−29877.6	−1.68866
$680$	18170.6	1.02472
$681$	6667.76	0.375197
$682$	1880.21	0.105568
$683$	11707.0	0.655864	0.327932	−	0.944701i	$-0.393648\pi$
$683$	11707.0	0.655864	0.327932	+	0.944701i	$0.393648\pi$
$684$	−8534.95	−0.477108
$685$	4694.57	0.261855
$686$	34988.1	1.94731
$687$	−5614.42	−0.311795
$688$	47979.0	2.65869
$689$	−6790.06	−0.375444
$690$	4560.45	0.251613
$691$	−30706.3	−1.69048	−0.845240	−	0.534388i	$-0.820542\pi$
$691$	−30706.3	−1.69048	−0.845240	+	0.534388i	$0.820542\pi$
$692$	45636.3	2.50698
$693$	−4464.53	−0.244723
$694$	26654.2	1.45789
$695$	−9335.50	−0.509519
$696$	8140.74	0.443353
$697$	16755.1	0.910537
$698$	45354.0	2.45942
$699$	−3681.83	−0.199227
$700$	−7581.08	−0.409340
$701$	26493.8	1.42747	0.713735	−	0.700416i	$-0.247002\pi$
$701$	26493.8	1.42747	0.713735	+	0.700416i	$0.247002\pi$
$702$	10053.9	0.540540
$703$	−2539.91	−0.136265
$704$	−1300.20	−0.0696067
$705$	554.465	0.0296204
$706$	−39766.6	−2.11988
$707$	12294.7	0.654017
$708$	932.959	0.0495236
$709$	14262.7	0.755497	0.377748	−	0.925908i	$-0.376698\pi$
$709$	14262.7	0.755497	0.377748	+	0.925908i	$0.376698\pi$
$710$	−15816.4	−0.836026
$711$	79.2057	0.00417784
$712$	−37408.3	−1.96901
$713$	3761.27	0.197560
$714$	9179.29	0.481130
$715$	−1328.45	−0.0694844
$716$	3261.92	0.170256
$717$	−656.984	−0.0342197
$718$	13754.7	0.714930
$719$	11233.7	0.582679	0.291339	−	0.956620i	$-0.405899\pi$
$719$	11233.7	0.582679	0.291339	+	0.956620i	$0.405899\pi$
$720$	15333.0	0.793648
$721$	4466.75	0.230722
$722$	1852.04	0.0954652
$723$	−10000.7	−0.514426
$724$	8887.16	0.456200
$725$	2440.88	0.125037
$726$	977.596	0.0499752
$727$	17492.5	0.892383	0.446191	−	0.894938i	$-0.352780\pi$
$727$	17492.5	0.892383	0.446191	+	0.894938i	$0.352780\pi$
$728$	21167.7	1.07765
$729$	−9650.35	−0.490288
$730$	−318.591	−0.0161529
$731$	−26332.1	−1.33232
$732$	−11864.4	−0.599072
$733$	−37962.7	−1.91294	−0.956469	−	0.291833i	$-0.905735\pi$
$733$	−37962.7	−1.91294	−0.956469	+	0.291833i	$0.905735\pi$
$734$	−8806.66	−0.442861
$735$	543.433	0.0272719
$736$	−24617.4	−1.23289
$737$	2223.46	0.111129
$738$	30707.0	1.53163
$739$	26691.3	1.32863	0.664313	−	0.747455i	$-0.268724\pi$
$739$	26691.3	1.32863	0.664313	+	0.747455i	$0.268724\pi$
$740$	12245.1	0.608296
$741$	−722.716	−0.0358295
$742$	−23872.4	−1.18111
$743$	8658.74	0.427535	0.213767	−	0.976885i	$-0.431427\pi$
$743$	8658.74	0.427535	0.213767	+	0.976885i	$0.431427\pi$
$744$	−2777.98	−0.136889
$745$	5973.68	0.293770
$746$	12479.6	0.612483
$747$	7459.03	0.365344
$748$	13832.3	0.676146
$749$	4090.95	0.199573
$750$	−1009.91	−0.0491691
$751$	4928.25	0.239460	0.119730	−	0.992806i	$-0.461797\pi$
$751$	4928.25	0.239460	0.119730	+	0.992806i	$0.461797\pi$
$752$	−8806.65	−0.427055
$753$	5491.03	0.265743
$754$	−12098.6	−0.584355
$755$	−1405.62	−0.0677558
$756$	24603.5	1.18362
$757$	10701.4	0.513804	0.256902	−	0.966437i	$-0.417298\pi$
$757$	10701.4	0.513804	0.256902	+	0.966437i	$0.417298\pi$
$758$	11426.3	0.547522
$759$	1955.63	0.0935241
$760$	−5029.81	−0.240066
$761$	−15221.4	−0.725068	−0.362534	−	0.931970i	$-0.618088\pi$
$761$	−15221.4	−0.725068	−0.362534	+	0.931970i	$0.618088\pi$
$762$	2092.72	0.0994896
$763$	−2597.87	−0.123262
$764$	9605.87	0.454880
$765$	−8415.15	−0.397713
$766$	−57265.4	−2.70115
$767$	−781.067	−0.0367701
$768$	−10684.1	−0.501991
$769$	4564.96	0.214066	0.107033	−	0.994255i	$-0.465865\pi$
$769$	4564.96	0.214066	0.107033	+	0.994255i	$0.465865\pi$
$770$	−4670.57	−0.218592
$771$	−7114.43	−0.332322
$772$	−34465.1	−1.60677
$773$	−30237.6	−1.40695	−0.703474	−	0.710721i	$-0.748369\pi$
$773$	−30237.6	−1.40695	−0.703474	+	0.710721i	$0.748369\pi$
$774$	−48258.9	−2.24112
$775$	−832.935	−0.0386063
$776$	95567.6	4.42097
$777$	3484.64	0.160889
$778$	29483.2	1.35864
$779$	−4637.97	−0.213315
$780$	3484.27	0.159945
$781$	−6782.44	−0.310749
$782$	39753.9	1.81790
$783$	−7921.57	−0.361550
$784$	−8631.43	−0.393196
$785$	−7203.91	−0.327540
$786$	−11782.8	−0.534705
$787$	−14594.9	−0.661057	−0.330528	−	0.943796i	$-0.607227\pi$
$787$	−14594.9	−0.661057	−0.330528	+	0.943796i	$0.607227\pi$
$788$	−29658.6	−1.34079
$789$	−5575.42	−0.251572
$790$	82.8611	0.00373173
$791$	−4434.25	−0.199322
$792$	14280.4	0.640696
$793$	9932.79	0.444797
$794$	4507.52	0.201468
$795$	−2213.55	−0.0987505
$796$	−29016.2	−1.29203
$797$	−32107.5	−1.42699	−0.713493	−	0.700663i	$-0.752888\pi$
$797$	−32107.5	−1.42699	−0.713493	+	0.700663i	$0.752888\pi$
$798$	−2540.92	−0.112716
$799$	4833.33	0.214006
$800$	5451.53	0.240926
$801$	17324.5	0.764208
$802$	43139.5	1.89939
$803$	−136.619	−0.00600398
$804$	−5831.70	−0.255806
$805$	−9343.22	−0.409075
$806$	4128.56	0.180425
$807$	10441.8	0.455475
$808$	−39326.3	−1.71225
$809$	−25767.0	−1.11980	−0.559900	−	0.828560i	$-0.689160\pi$
$809$	−25767.0	−1.11980	−0.559900	+	0.828560i	$0.689160\pi$
$810$	−13704.8	−0.594490
$811$	15778.6	0.683181	0.341591	−	0.939849i	$-0.389034\pi$
$811$	15778.6	0.683181	0.341591	+	0.939849i	$0.389034\pi$
$812$	−29607.2	−1.27957
$813$	−11492.9	−0.495785
$814$	7543.99	0.324836
$815$	509.958	0.0219179
$816$	−13518.8	−0.579968
$817$	7288.99	0.312129
$818$	8265.99	0.353317
$819$	−9803.17	−0.418255
$820$	22360.0	0.952251
$821$	−40567.2	−1.72449	−0.862244	−	0.506493i	$-0.830942\pi$
$821$	−40567.2	−1.72449	−0.862244	+	0.506493i	$0.830942\pi$
$822$	−7585.78	−0.321879
$823$	−20025.9	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$823$	−20025.9	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$824$	−14287.5	−0.604040
$825$	−433.075	−0.0182760
$826$	−2746.07	−0.115675
$827$	−26838.3	−1.12849	−0.564243	−	0.825609i	$-0.690832\pi$
$827$	−26838.3	−1.12849	−0.564243	+	0.825609i	$0.690832\pi$
$828$	50712.0	2.12846
$829$	−29420.4	−1.23258	−0.616292	−	0.787518i	$-0.711366\pi$
$829$	−29420.4	−1.23258	−0.616292	+	0.787518i	$0.711366\pi$
$830$	7803.27	0.326332
$831$	6267.99	0.261654
$832$	−2854.97	−0.118964
$833$	4737.16	0.197038
$834$	15084.9	0.626314
$835$	−4704.57	−0.194980
$836$	−3828.90	−0.158403
$837$	2703.19	0.111632
$838$	23809.2	0.981472
$839$	28114.6	1.15688	0.578441	−	0.815724i	$-0.303661\pi$
$839$	28114.6	1.15688	0.578441	+	0.815724i	$0.303661\pi$
$840$	6900.67	0.283447
$841$	−14856.4	−0.609143
$842$	−12575.0	−0.514681
$843$	−7679.13	−0.313740
$844$	−949.037	−0.0387052
$845$	8067.99	0.328458
$846$	8858.04	0.359983
$847$	−2002.85	−0.0812500
$848$	35158.2	1.42375
$849$	523.300	0.0211538
$850$	−8803.51	−0.355245
$851$	15091.4	0.607902
$852$	17789.0	0.715306
$853$	31820.0	1.27725	0.638625	−	0.769518i	$-0.279503\pi$
$853$	31820.0	1.27725	0.638625	+	0.769518i	$0.279503\pi$
$854$	34921.6	1.39929
$855$	2329.40	0.0931738
$856$	−13085.5	−0.522491
$857$	11510.5	0.458798	0.229399	−	0.973332i	$-0.426324\pi$
$857$	11510.5	0.458798	0.229399	+	0.973332i	$0.426324\pi$
$858$	2146.60	0.0854121
$859$	−27269.7	−1.08316	−0.541578	−	0.840650i	$-0.682173\pi$
$859$	−27269.7	−1.08316	−0.541578	+	0.840650i	$0.682173\pi$
$860$	−35140.8	−1.39336
$861$	6363.08	0.251862
$862$	−6193.79	−0.244735
$863$	−22353.3	−0.881711	−0.440855	−	0.897578i	$-0.645325\pi$
$863$	−22353.3	−0.881711	−0.440855	+	0.897578i	$0.645325\pi$
$864$	−17692.3	−0.696648
$865$	−12455.3	−0.489585
$866$	−91179.0	−3.57781
$867$	−317.575	−0.0124399
$868$	10103.3	0.395078
$869$	35.5328	0.00138707
$870$	−3944.12	−0.153699
$871$	4882.26	0.189930
$872$	8309.64	0.322706
$873$	−44259.1	−1.71586
$874$	−11004.3	−0.425886
$875$	2069.06	0.0799394
$876$	358.326	0.0138204
$877$	14159.9	0.545208	0.272604	−	0.962126i	$-0.412115\pi$
$877$	14159.9	0.545208	0.272604	+	0.962126i	$0.412115\pi$
$878$	−15447.7	−0.593774
$879$	−10224.2	−0.392325
$880$	6878.59	0.263497
$881$	−20102.6	−0.768757	−0.384379	−	0.923176i	$-0.625584\pi$
$881$	−20102.6	−0.768757	−0.384379	+	0.923176i	$0.625584\pi$
$882$	8681.79	0.331441
$883$	26425.2	1.00711	0.503555	−	0.863963i	$-0.332025\pi$
$883$	26425.2	1.00711	0.503555	+	0.863963i	$0.332025\pi$
$884$	30372.7	1.15559
$885$	−254.627	−0.00967141
$886$	81046.9	3.07316
$887$	9766.07	0.369687	0.184844	−	0.982768i	$-0.440822\pi$
$887$	9766.07	0.369687	0.184844	+	0.982768i	$0.440822\pi$
$888$	−11146.1	−0.421214
$889$	−4287.45	−0.161751
$890$	18124.0	0.682605
$891$	−5876.93	−0.220970
$892$	−25586.3	−0.960417
$893$	−1337.91	−0.0501361
$894$	−9652.63	−0.361110
$895$	−890.256	−0.0332491
$896$	18838.2	0.702387
$897$	4294.15	0.159841
$898$	−26123.8	−0.970782
$899$	−3252.95	−0.120681
$900$	−11230.2	−0.415933
$901$	−19295.8	−0.713469
$902$	13775.6	0.508512
$903$	−10000.2	−0.368532
$904$	14183.5	0.521833
$905$	−2425.52	−0.0890907
$906$	2271.28	0.0832872
$907$	34965.6	1.28006	0.640029	−	0.768350i	$-0.278922\pi$
$907$	34965.6	1.28006	0.640029	+	0.768350i	$0.278922\pi$
$908$	77567.1	2.83497
$909$	18212.7	0.664552
$910$	−10255.6	−0.373593
$911$	21896.2	0.796327	0.398163	−	0.917314i	$-0.369648\pi$
$911$	21896.2	0.796327	0.398163	+	0.917314i	$0.369648\pi$
$912$	3742.14	0.135871
$913$	3346.23	0.121297
$914$	79441.4	2.87493
$915$	3238.08	0.116992
$916$	−65313.4	−2.35591
$917$	24140.0	0.869326
$918$	28570.8	1.02721
$919$	14541.5	0.521960	0.260980	−	0.965344i	$-0.415954\pi$
$919$	14541.5	0.521960	0.260980	+	0.965344i	$0.415954\pi$
$920$	29885.5	1.07097
$921$	7705.99	0.275701
$922$	67613.1	2.41510
$923$	−14892.8	−0.531098
$924$	5253.08	0.187028
$925$	−3341.99	−0.118793
$926$	17885.5	0.634723
$927$	6616.81	0.234438
$928$	21290.4	0.753117
$929$	−11454.3	−0.404526	−0.202263	−	0.979331i	$-0.564830\pi$
$929$	−11454.3	−0.404526	−0.202263	+	0.979331i	$0.564830\pi$
$930$	1345.91	0.0474559
$931$	−1311.29	−0.0461610
$932$	−42831.3	−1.50535
$933$	−6770.34	−0.237568
$934$	−20769.7	−0.727628
$935$	−3775.16	−0.132044
$936$	31356.8	1.09501
$937$	−27082.3	−0.944226	−0.472113	−	0.881538i	$-0.656509\pi$
$937$	−27082.3	−0.944226	−0.472113	+	0.881538i	$0.656509\pi$
$938$	17165.0	0.597503
$939$	10367.8	0.360321
$940$	6450.18	0.223810
$941$	16283.2	0.564098	0.282049	−	0.959400i	$-0.408986\pi$
$941$	16283.2	0.564098	0.282049	+	0.959400i	$0.408986\pi$
$942$	11640.5	0.402621
$943$	27557.4	0.951634
$944$	4044.28	0.139439
$945$	−6714.89	−0.231148
$946$	−21649.6	−0.744070
$947$	28228.3	0.968636	0.484318	−	0.874892i	$-0.339068\pi$
$947$	28228.3	0.968636	0.484318	+	0.874892i	$0.339068\pi$
$948$	−93.1955	−0.00319288
$949$	−299.988	−0.0102613
$950$	2436.90	0.0832247
$951$	−2820.27	−0.0961657
$952$	60153.8	2.04790
$953$	−3740.73	−0.127150	−0.0635750	−	0.997977i	$-0.520250\pi$
$953$	−3740.73	−0.127150	−0.0635750	+	0.997977i	$0.520250\pi$
$954$	−35363.3	−1.20014
$955$	−2621.67	−0.0888329
$956$	−7642.80	−0.258563
$957$	−1691.33	−0.0571296
$958$	−19770.0	−0.666744
$959$	15541.4	0.523313
$960$	−930.717	−0.0312904
$961$	−28681.0	−0.962739
$962$	16565.0	0.555175
$963$	6060.12	0.202788
$964$	−116340.	−3.88698
$965$	9406.35	0.313783
$966$	15097.3	0.502846
$967$	−38351.9	−1.27540	−0.637701	−	0.770284i	$-0.720115\pi$
$967$	−38351.9	−1.27540	−0.637701	+	0.770284i	$0.720115\pi$
$968$	6406.39	0.212716
$969$	−2053.79	−0.0680880
$970$	−46301.7	−1.53264
$971$	−13694.5	−0.452603	−0.226302	−	0.974057i	$-0.572663\pi$
$971$	−13694.5	−0.452603	−0.226302	+	0.974057i	$0.572663\pi$
$972$	55546.6	1.83298
$973$	−30905.1	−1.01827
$974$	−13883.7	−0.456736
$975$	−950.942	−0.0312354
$976$	−51430.9	−1.68674
$977$	16127.9	0.528126	0.264063	−	0.964505i	$-0.414937\pi$
$977$	16127.9	0.528126	0.264063	+	0.964505i	$0.414937\pi$
$978$	−824.021	−0.0269420
$979$	7772.01	0.253723
$980$	6321.84	0.206065
$981$	−3848.34	−0.125248
$982$	−89110.5	−2.89576
$983$	−30946.2	−1.00410	−0.502051	−	0.864838i	$-0.667421\pi$
$983$	−30946.2	−1.00410	−0.502051	+	0.864838i	$0.667421\pi$
$984$	−20353.2	−0.659386
$985$	8094.54	0.261841
$986$	−34381.3	−1.11047
$987$	1835.55	0.0591959
$988$	−8407.47	−0.270726
$989$	−43308.9	−1.39246
$990$	−6918.72	−0.222113
$991$	−57301.2	−1.83676	−0.918382	−	0.395695i	$-0.870504\pi$
$991$	−57301.2	−1.83676	−0.918382	+	0.395695i	$0.870504\pi$
$992$	−7265.23	−0.232532
$993$	−682.102	−0.0217984
$994$	−52360.1	−1.67079
$995$	7919.24	0.252318
$996$	−8776.49	−0.279211
$997$	2780.57	0.0883266	0.0441633	−	0.999024i	$-0.485938\pi$
$997$	2780.57	0.0883266	0.0441633	+	0.999024i	$0.485938\pi$
$998$	−94379.8	−2.99353
$999$	10846.0	0.343496

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.c.1.20	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.c.1.20	✓	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)

\(f(q)\)	\(=\)	\(q+5.13031 q^{2} +1.57482 q^{3} +18.3201 q^{4} -5.00000 q^{5} +8.07930 q^{6} -16.5525 q^{7} +52.9453 q^{8} -24.5200 q^{9} +O(q^{10})\)	\(q+5.13031 q^{2} +1.57482 q^{3} +18.3201 q^{4} -5.00000 q^{5} +8.07930 q^{6} -16.5525 q^{7} +52.9453 q^{8} -24.5200 q^{9} -25.6516 q^{10} -11.0000 q^{11} +28.8508 q^{12} -24.1537 q^{13} -84.9194 q^{14} -7.87409 q^{15} +125.065 q^{16} -68.6392 q^{17} -125.795 q^{18} +19.0000 q^{19} -91.6005 q^{20} -26.0671 q^{21} -56.4334 q^{22} -112.892 q^{23} +83.3792 q^{24} +25.0000 q^{25} -123.916 q^{26} -81.1345 q^{27} -303.243 q^{28} +97.6351 q^{29} -40.3965 q^{30} -33.3174 q^{31} +218.061 q^{32} -17.3230 q^{33} -352.141 q^{34} +82.7624 q^{35} -449.208 q^{36} -133.680 q^{37} +97.4759 q^{38} -38.0377 q^{39} -264.727 q^{40} -244.104 q^{41} -133.733 q^{42} +383.631 q^{43} -201.521 q^{44} +122.600 q^{45} -579.171 q^{46} -70.4164 q^{47} +196.955 q^{48} -69.0154 q^{49} +128.258 q^{50} -108.094 q^{51} -442.498 q^{52} +281.119 q^{53} -416.245 q^{54} +55.0000 q^{55} -876.377 q^{56} +29.9215 q^{57} +500.898 q^{58} +32.3374 q^{59} -144.254 q^{60} -411.232 q^{61} -170.929 q^{62} +405.866 q^{63} +118.200 q^{64} +120.769 q^{65} -88.8723 q^{66} -202.133 q^{67} -1257.48 q^{68} -177.784 q^{69} +424.597 q^{70} +616.586 q^{71} -1298.22 q^{72} +12.4199 q^{73} -685.818 q^{74} +39.3704 q^{75} +348.082 q^{76} +182.077 q^{77} -195.145 q^{78} -3.23026 q^{79} -625.326 q^{80} +534.267 q^{81} -1252.33 q^{82} -304.203 q^{83} -477.552 q^{84} +343.196 q^{85} +1968.15 q^{86} +153.757 q^{87} -582.399 q^{88} -706.546 q^{89} +628.975 q^{90} +399.804 q^{91} -2068.19 q^{92} -52.4688 q^{93} -361.258 q^{94} -95.0000 q^{95} +343.407 q^{96} +1805.02 q^{97} -354.070 q^{98} +269.719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) \) 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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