LMFDB - Embedded newform 1045.4.a.f.1.8

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

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.74715 q^{2} +4.30519 q^{3} -0.453153 q^{4} -5.00000 q^{5} -11.8270 q^{6} +27.4903 q^{7} +23.2221 q^{8} -8.46533 q^{9} +O(q^{10})$	\(q-2.74715 q^{2} +4.30519 q^{3} -0.453153 q^{4} -5.00000 q^{5} -11.8270 q^{6} +27.4903 q^{7} +23.2221 q^{8} -8.46533 q^{9} +13.7358 q^{10} +11.0000 q^{11} -1.95091 q^{12} -37.9897 q^{13} -75.5199 q^{14} -21.5260 q^{15} -60.1694 q^{16} +9.56072 q^{17} +23.2556 q^{18} -19.0000 q^{19} +2.26576 q^{20} +118.351 q^{21} -30.2187 q^{22} +60.7767 q^{23} +99.9756 q^{24} +25.0000 q^{25} +104.364 q^{26} -152.685 q^{27} -12.4573 q^{28} -62.6241 q^{29} +59.1351 q^{30} -270.909 q^{31} -20.4822 q^{32} +47.3571 q^{33} -26.2647 q^{34} -137.451 q^{35} +3.83609 q^{36} -389.720 q^{37} +52.1959 q^{38} -163.553 q^{39} -116.111 q^{40} +356.935 q^{41} -325.128 q^{42} +196.919 q^{43} -4.98468 q^{44} +42.3267 q^{45} -166.963 q^{46} +286.977 q^{47} -259.041 q^{48} +412.714 q^{49} -68.6788 q^{50} +41.1607 q^{51} +17.2151 q^{52} -383.733 q^{53} +419.449 q^{54} -55.0000 q^{55} +638.382 q^{56} -81.7986 q^{57} +172.038 q^{58} -75.5393 q^{59} +9.75454 q^{60} +610.371 q^{61} +744.229 q^{62} -232.714 q^{63} +537.623 q^{64} +189.949 q^{65} -130.097 q^{66} -59.3749 q^{67} -4.33246 q^{68} +261.655 q^{69} +377.600 q^{70} -882.545 q^{71} -196.583 q^{72} -771.899 q^{73} +1070.62 q^{74} +107.630 q^{75} +8.60990 q^{76} +302.393 q^{77} +449.305 q^{78} +708.113 q^{79} +300.847 q^{80} -428.774 q^{81} -980.556 q^{82} -125.274 q^{83} -53.6310 q^{84} -47.8036 q^{85} -540.968 q^{86} -269.609 q^{87} +255.443 q^{88} -1594.03 q^{89} -116.278 q^{90} -1044.35 q^{91} -27.5411 q^{92} -1166.32 q^{93} -788.369 q^{94} +95.0000 q^{95} -88.1797 q^{96} +163.509 q^{97} -1133.79 q^{98} -93.1186 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * q^98 + 1870 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.74715	−0.971265	−0.485633	−	0.874163i	$-0.661411\pi$
$2$	−2.74715	−0.971265	−0.485633	+	0.874163i	$0.661411\pi$
$3$	4.30519	0.828534	0.414267	−	0.910155i	$-0.364038\pi$
$3$	4.30519	0.828534	0.414267	+	0.910155i	$0.364038\pi$
$4$	−0.453153	−0.0566441
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−11.8270	−0.804727
$7$	27.4903	1.48433	0.742167	−	0.670215i	$-0.233798\pi$
$7$	27.4903	1.48433	0.742167	+	0.670215i	$0.233798\pi$
$8$	23.2221	1.02628
$9$	−8.46533	−0.313531
$10$	13.7358	0.434363
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−1.95091	−0.0469316
$13$	−37.9897	−0.810496	−0.405248	−	0.914207i	$-0.632815\pi$
$13$	−37.9897	−0.810496	−0.405248	+	0.914207i	$0.632815\pi$
$14$	−75.5199	−1.44168
$15$	−21.5260	−0.370532
$16$	−60.1694	−0.940147
$17$	9.56072	0.136401	0.0682004	−	0.997672i	$-0.478274\pi$
$17$	9.56072	0.136401	0.0682004	+	0.997672i	$0.478274\pi$
$18$	23.2556	0.304522
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	2.26576	0.0253320
$21$	118.351	1.22982
$22$	−30.2187	−0.292847
$23$	60.7767	0.550992	0.275496	−	0.961302i	$-0.411158\pi$
$23$	60.7767	0.550992	0.275496	+	0.961302i	$0.411158\pi$
$24$	99.9756	0.850310
$25$	25.0000	0.200000
$26$	104.364	0.787207
$27$	−152.685	−1.08831
$28$	−12.4573	−0.0840787
$29$	−62.6241	−0.401000	−0.200500	−	0.979694i	$-0.564257\pi$
$29$	−62.6241	−0.401000	−0.200500	+	0.979694i	$0.564257\pi$
$30$	59.1351	0.359885
$31$	−270.909	−1.56957	−0.784786	−	0.619766i	$-0.787227\pi$
$31$	−270.909	−1.56957	−0.784786	+	0.619766i	$0.787227\pi$
$32$	−20.4822	−0.113149
$33$	47.3571	0.249813
$34$	−26.2647	−0.132481
$35$	−137.451	−0.663815
$36$	3.83609	0.0177597
$37$	−389.720	−1.73161	−0.865804	−	0.500382i	$-0.833193\pi$
$37$	−389.720	−1.73161	−0.865804	+	0.500382i	$0.833193\pi$
$38$	52.1959	0.222824
$39$	−163.553	−0.671524
$40$	−116.111	−0.458967
$41$	356.935	1.35961	0.679804	−	0.733394i	$-0.262065\pi$
$41$	356.935	1.35961	0.679804	+	0.733394i	$0.262065\pi$
$42$	−325.128	−1.19448
$43$	196.919	0.698371	0.349185	−	0.937054i	$-0.386458\pi$
$43$	196.919	0.698371	0.349185	+	0.937054i	$0.386458\pi$
$44$	−4.98468	−0.0170788
$45$	42.3267	0.140215
$46$	−166.963	−0.535159
$47$	286.977	0.890635	0.445318	−	0.895373i	$-0.353091\pi$
$47$	286.977	0.890635	0.445318	+	0.895373i	$0.353091\pi$
$48$	−259.041	−0.778944
$49$	412.714	1.20325
$50$	−68.6788	−0.194253
$51$	41.1607	0.113013
$52$	17.2151	0.0459098
$53$	−383.733	−0.994525	−0.497262	−	0.867600i	$-0.665661\pi$
$53$	−383.733	−0.994525	−0.497262	+	0.867600i	$0.665661\pi$
$54$	419.449	1.05703
$55$	−55.0000	−0.134840
$56$	638.382	1.52334
$57$	−81.7986	−0.190079
$58$	172.038	0.389477
$59$	−75.5393	−0.166684	−0.0833422	−	0.996521i	$-0.526559\pi$
$59$	−75.5393	−0.166684	−0.0833422	+	0.996521i	$0.526559\pi$
$60$	9.75454	0.0209884
$61$	610.371	1.28115	0.640574	−	0.767897i	$-0.278697\pi$
$61$	610.371	1.28115	0.640574	+	0.767897i	$0.278697\pi$
$62$	744.229	1.52447
$63$	−232.714	−0.465385
$64$	537.623	1.05005
$65$	189.949	0.362465
$66$	−130.097	−0.242634
$67$	−59.3749	−0.108266	−0.0541328	−	0.998534i	$-0.517239\pi$
$67$	−59.3749	−0.108266	−0.0541328	+	0.998534i	$0.517239\pi$
$68$	−4.33246	−0.00772630
$69$	261.655	0.456516
$70$	377.600	0.644740
$71$	−882.545	−1.47519	−0.737597	−	0.675241i	$-0.764040\pi$
$71$	−882.545	−1.47519	−0.737597	+	0.675241i	$0.764040\pi$
$72$	−196.583	−0.321771
$73$	−771.899	−1.23759	−0.618794	−	0.785553i	$-0.712378\pi$
$73$	−771.899	−1.23759	−0.618794	+	0.785553i	$0.712378\pi$
$74$	1070.62	1.68185
$75$	107.630	0.165707
$76$	8.60990	0.0129950
$77$	302.393	0.447544
$78$	449.305	0.652228
$79$	708.113	1.00847	0.504234	−	0.863567i	$-0.331775\pi$
$79$	708.113	1.00847	0.504234	+	0.863567i	$0.331775\pi$
$80$	300.847	0.420447
$81$	−428.774	−0.588168
$82$	−980.556	−1.32054
$83$	−125.274	−0.165671	−0.0828353	−	0.996563i	$-0.526398\pi$
$83$	−125.274	−0.165671	−0.0828353	+	0.996563i	$0.526398\pi$
$84$	−53.6310	−0.0696621
$85$	−47.8036	−0.0610003
$86$	−540.968	−0.678303
$87$	−269.609	−0.332242
$88$	255.443	0.309436
$89$	−1594.03	−1.89850	−0.949249	−	0.314525i	$-0.898155\pi$
$89$	−1594.03	−1.89850	−0.949249	+	0.314525i	$0.898155\pi$
$90$	−116.278	−0.136186
$91$	−1044.35	−1.20305
$92$	−27.5411	−0.0312104
$93$	−1166.32	−1.30045
$94$	−788.369	−0.865043
$95$	95.0000	0.102598
$96$	−88.1797	−0.0937480
$97$	163.509	0.171153	0.0855765	−	0.996332i	$-0.472727\pi$
$97$	163.509	0.171153	0.0855765	+	0.996332i	$0.472727\pi$
$98$	−1133.79	−1.16867
$99$	−93.1186	−0.0945331
$100$	−11.3288	−0.0113288
$101$	1880.29	1.85244	0.926218	−	0.376988i	$-0.123040\pi$
$101$	1880.29	1.85244	0.926218	+	0.376988i	$0.123040\pi$
$102$	−113.075	−0.109765
$103$	404.130	0.386603	0.193302	−	0.981139i	$-0.438080\pi$
$103$	404.130	0.386603	0.193302	+	0.981139i	$0.438080\pi$
$104$	−882.201	−0.831797
$105$	−591.754	−0.549993
$106$	1054.17	0.965947
$107$	−1865.64	−1.68559	−0.842796	−	0.538232i	$-0.819092\pi$
$107$	−1865.64	−1.68559	−0.842796	+	0.538232i	$0.819092\pi$
$108$	69.1896	0.0616460
$109$	506.479	0.445064	0.222532	−	0.974925i	$-0.428568\pi$
$109$	506.479	0.445064	0.222532	+	0.974925i	$0.428568\pi$
$110$	151.093	0.130965
$111$	−1677.82	−1.43470
$112$	−1654.07	−1.39549
$113$	−925.423	−0.770411	−0.385205	−	0.922831i	$-0.625869\pi$
$113$	−925.423	−0.770411	−0.385205	+	0.922831i	$0.625869\pi$
$114$	224.713	0.184617
$115$	−303.883	−0.246411
$116$	28.3783	0.0227143
$117$	321.595	0.254116
$118$	207.518	0.161895
$119$	262.827	0.202464
$120$	−499.878	−0.380270
$121$	121.000	0.0909091
$122$	−1676.78	−1.24433
$123$	1536.67	1.12648
$124$	122.763	0.0889070
$125$	−125.000	−0.0894427
$126$	639.301	0.452012
$127$	613.234	0.428470	0.214235	−	0.976782i	$-0.431274\pi$
$127$	613.234	0.428470	0.214235	+	0.976782i	$0.431274\pi$
$128$	−1313.08	−0.906723
$129$	847.776	0.578624
$130$	−521.818	−0.352050
$131$	2318.48	1.54631	0.773156	−	0.634216i	$-0.218677\pi$
$131$	2318.48	1.54631	0.773156	+	0.634216i	$0.218677\pi$
$132$	−21.4600	−0.0141504
$133$	−522.315	−0.340530
$134$	163.112	0.105155
$135$	763.425	0.486705
$136$	222.020	0.139986
$137$	−1634.86	−1.01953	−0.509766	−	0.860313i	$-0.670268\pi$
$137$	−1634.86	−1.01953	−0.509766	+	0.860313i	$0.670268\pi$
$138$	−718.807	−0.443398
$139$	−189.234	−0.115472	−0.0577362	−	0.998332i	$-0.518388\pi$
$139$	−189.234	−0.115472	−0.0577362	+	0.998332i	$0.518388\pi$
$140$	62.2864	0.0376012
$141$	1235.49	0.737922
$142$	2424.49	1.43281
$143$	−417.887	−0.244374
$144$	509.354	0.294765
$145$	313.120	0.179333
$146$	2120.52	1.20203
$147$	1776.81	0.996933
$148$	176.602	0.0980854
$149$	−3435.55	−1.88894	−0.944468	−	0.328603i	$-0.893422\pi$
$149$	−3435.55	−1.88894	−0.944468	+	0.328603i	$0.893422\pi$
$150$	−295.675	−0.160945
$151$	−1265.12	−0.681817	−0.340908	−	0.940096i	$-0.610735\pi$
$151$	−1265.12	−0.681817	−0.340908	+	0.940096i	$0.610735\pi$
$152$	−441.220	−0.235445
$153$	−80.9346	−0.0427659
$154$	−830.719	−0.434684
$155$	1354.55	0.701934
$156$	74.1144	0.0380378
$157$	1623.83	0.825450	0.412725	−	0.910856i	$-0.364577\pi$
$157$	1623.83	0.825450	0.412725	+	0.910856i	$0.364577\pi$
$158$	−1945.29	−0.979489
$159$	−1652.04	−0.823998
$160$	102.411	0.0506018
$161$	1670.77	0.817856
$162$	1177.91	0.571267
$163$	−1333.20	−0.640640	−0.320320	−	0.947309i	$-0.603790\pi$
$163$	−1333.20	−0.640640	−0.320320	+	0.947309i	$0.603790\pi$
$164$	−161.746	−0.0770137
$165$	−236.786	−0.111720
$166$	344.148	0.160910
$167$	−2798.28	−1.29663	−0.648315	−	0.761372i	$-0.724526\pi$
$167$	−2798.28	−1.29663	−0.648315	+	0.761372i	$0.724526\pi$
$168$	2748.35	1.26214
$169$	−753.782	−0.343096
$170$	131.324	0.0592475
$171$	160.841	0.0719289
$172$	−89.2346	−0.0395586
$173$	−2140.95	−0.940885	−0.470442	−	0.882431i	$-0.655906\pi$
$173$	−2140.95	−0.940885	−0.470442	+	0.882431i	$0.655906\pi$
$174$	740.656	0.322695
$175$	687.256	0.296867
$176$	−661.864	−0.283465
$177$	−325.211	−0.138104
$178$	4379.03	1.84395
$179$	−1640.21	−0.684890	−0.342445	−	0.939538i	$-0.611255\pi$
$179$	−1640.21	−0.684890	−0.342445	+	0.939538i	$0.611255\pi$
$180$	−19.1804	−0.00794236
$181$	−1421.40	−0.583712	−0.291856	−	0.956462i	$-0.594273\pi$
$181$	−1421.40	−0.583712	−0.291856	+	0.956462i	$0.594273\pi$
$182$	2868.98	1.16848
$183$	2627.76	1.06147
$184$	1411.36	0.565473
$185$	1948.60	0.774399
$186$	3204.05	1.26308
$187$	105.168	0.0411264
$188$	−130.044	−0.0504492
$189$	−4197.35	−1.61541
$190$	−260.979	−0.0996497
$191$	−3498.70	−1.32543	−0.662715	−	0.748872i	$-0.730596\pi$
$191$	−3498.70	−1.32543	−0.662715	+	0.748872i	$0.730596\pi$
$192$	2314.57	0.869999
$193$	−2638.13	−0.983920	−0.491960	−	0.870618i	$-0.663719\pi$
$193$	−2638.13	−0.983920	−0.491960	+	0.870618i	$0.663719\pi$
$194$	−449.185	−0.166235
$195$	817.765	0.300315
$196$	−187.023	−0.0681569
$197$	−3992.22	−1.44383	−0.721914	−	0.691983i	$-0.756737\pi$
$197$	−3992.22	−1.44383	−0.721914	+	0.691983i	$0.756737\pi$
$198$	255.811	0.0918167
$199$	1678.87	0.598051	0.299025	−	0.954245i	$-0.403338\pi$
$199$	1678.87	0.598051	0.299025	+	0.954245i	$0.403338\pi$
$200$	580.553	0.205256
$201$	−255.620	−0.0897018
$202$	−5165.45	−1.79921
$203$	−1721.55	−0.595218
$204$	−18.6521	−0.00640150
$205$	−1784.68	−0.608035
$206$	−1110.21	−0.375494
$207$	−514.495	−0.172753
$208$	2285.82	0.761986
$209$	−209.000	−0.0691714
$210$	1625.64	0.534189
$211$	−3813.44	−1.24421	−0.622105	−	0.782934i	$-0.713722\pi$
$211$	−3813.44	−1.24421	−0.622105	+	0.782934i	$0.713722\pi$
$212$	173.890	0.0563339
$213$	−3799.52	−1.22225
$214$	5125.20	1.63716
$215$	−984.597	−0.312321
$216$	−3545.67	−1.11691
$217$	−7447.37	−2.32977
$218$	−1391.38	−0.432275
$219$	−3323.17	−1.02538
$220$	24.9234	0.00763788
$221$	−363.209	−0.110552
$222$	4609.22	1.39347
$223$	6470.54	1.94305	0.971523	−	0.236946i	$-0.0761465\pi$
$223$	6470.54	1.94305	0.971523	+	0.236946i	$0.0761465\pi$
$224$	−563.061	−0.167951
$225$	−211.633	−0.0627062
$226$	2542.28	0.748273
$227$	4259.87	1.24554	0.622770	−	0.782405i	$-0.286007\pi$
$227$	4259.87	1.24554	0.622770	+	0.782405i	$0.286007\pi$
$228$	37.0673	0.0107668
$229$	−3542.00	−1.02211	−0.511053	−	0.859549i	$-0.670744\pi$
$229$	−3542.00	−1.02211	−0.511053	+	0.859549i	$0.670744\pi$
$230$	834.814	0.239330
$231$	1301.86	0.370805
$232$	−1454.26	−0.411539
$233$	3410.80	0.959010	0.479505	−	0.877539i	$-0.340816\pi$
$233$	3410.80	0.959010	0.479505	+	0.877539i	$0.340816\pi$
$234$	−883.472	−0.246814
$235$	−1434.88	−0.398304
$236$	34.2308	0.00944168
$237$	3048.56	0.835550
$238$	−722.025	−0.196647
$239$	−4550.72	−1.23164	−0.615819	−	0.787888i	$-0.711175\pi$
$239$	−4550.72	−1.23164	−0.615819	+	0.787888i	$0.711175\pi$
$240$	1295.20	0.348355
$241$	2333.33	0.623663	0.311832	−	0.950137i	$-0.399058\pi$
$241$	2333.33	0.623663	0.311832	+	0.950137i	$0.399058\pi$
$242$	−332.405	−0.0882968
$243$	2276.54	0.600988
$244$	−276.591	−0.0725694
$245$	−2063.57	−0.538109
$246$	−4221.48	−1.09411
$247$	721.805	0.185941
$248$	−6291.08	−1.61082
$249$	−539.330	−0.137264
$250$	343.394	0.0868726
$251$	2537.43	0.638091	0.319046	−	0.947739i	$-0.396638\pi$
$251$	2537.43	0.638091	0.319046	+	0.947739i	$0.396638\pi$
$252$	105.455	0.0263613
$253$	668.543	0.166130
$254$	−1684.65	−0.416158
$255$	−205.804	−0.0505409
$256$	−693.767	−0.169377
$257$	−2224.71	−0.539974	−0.269987	−	0.962864i	$-0.587019\pi$
$257$	−2224.71	−0.539974	−0.269987	+	0.962864i	$0.587019\pi$
$258$	−2328.97	−0.561997
$259$	−10713.5	−2.57029
$260$	−86.0757	−0.0205315
$261$	530.134	0.125726
$262$	−6369.23	−1.50188
$263$	5915.90	1.38703	0.693517	−	0.720440i	$-0.256060\pi$
$263$	5915.90	1.38703	0.693517	+	0.720440i	$0.256060\pi$
$264$	1099.73	0.256378
$265$	1918.67	0.444765
$266$	1434.88	0.330745
$267$	−6862.58	−1.57297
$268$	26.9059	0.00613261
$269$	−6769.81	−1.53443	−0.767217	−	0.641388i	$-0.778359\pi$
$269$	−6769.81	−1.53443	−0.767217	+	0.641388i	$0.778359\pi$
$270$	−2097.25	−0.472720
$271$	6606.06	1.48077	0.740387	−	0.672181i	$-0.234642\pi$
$271$	6606.06	1.48077	0.740387	+	0.672181i	$0.234642\pi$
$272$	−575.263	−0.128237
$273$	−4496.11	−0.996766
$274$	4491.22	0.990236
$275$	275.000	0.0603023
$276$	−118.570	−0.0258589
$277$	−789.439	−0.171238	−0.0856188	−	0.996328i	$-0.527287\pi$
$277$	−789.439	−0.171238	−0.0856188	+	0.996328i	$0.527287\pi$
$278$	519.856	0.112154
$279$	2293.34	0.492109
$280$	−3191.91	−0.681261
$281$	−1694.99	−0.359839	−0.179919	−	0.983681i	$-0.557584\pi$
$281$	−1694.99	−0.359839	−0.179919	+	0.983681i	$0.557584\pi$
$282$	−3394.08	−0.716718
$283$	−2403.05	−0.504758	−0.252379	−	0.967628i	$-0.581213\pi$
$283$	−2403.05	−0.504758	−0.252379	+	0.967628i	$0.581213\pi$
$284$	399.927	0.0835610
$285$	408.993	0.0850058
$286$	1148.00	0.237352
$287$	9812.24	2.01811
$288$	173.388	0.0354757
$289$	−4821.59	−0.981395
$290$	−860.189	−0.174180
$291$	703.938	0.141806
$292$	349.788	0.0701020
$293$	−1697.14	−0.338389	−0.169195	−	0.985583i	$-0.554117\pi$
$293$	−1697.14	−0.338389	−0.169195	+	0.985583i	$0.554117\pi$
$294$	−4881.18	−0.968286
$295$	377.696	0.0745435
$296$	−9050.11	−1.77712
$297$	−1679.54	−0.328136
$298$	9437.99	1.83466
$299$	−2308.89	−0.446577
$300$	−48.7727	−0.00938631
$301$	5413.37	1.03662
$302$	3475.49	0.662225
$303$	8095.02	1.53481
$304$	1143.22	0.215685
$305$	−3051.86	−0.572947
$306$	222.340	0.0415370
$307$	−391.294	−0.0727438	−0.0363719	−	0.999338i	$-0.511580\pi$
$307$	−391.294	−0.0727438	−0.0363719	+	0.999338i	$0.511580\pi$
$308$	−137.030	−0.0253507
$309$	1739.86	0.320314
$310$	−3721.15	−0.681764
$311$	−6273.05	−1.14377	−0.571884	−	0.820334i	$-0.693787\pi$
$311$	−6273.05	−1.14377	−0.571884	+	0.820334i	$0.693787\pi$
$312$	−3798.04	−0.689173
$313$	−1547.69	−0.279491	−0.139745	−	0.990187i	$-0.544628\pi$
$313$	−1547.69	−0.279491	−0.139745	+	0.990187i	$0.544628\pi$
$314$	−4460.91	−0.801731
$315$	1163.57	0.208126
$316$	−320.883	−0.0571237
$317$	−4116.78	−0.729406	−0.364703	−	0.931124i	$-0.618829\pi$
$317$	−4116.78	−0.729406	−0.364703	+	0.931124i	$0.618829\pi$
$318$	4538.42	0.800320
$319$	−688.865	−0.120906
$320$	−2688.12	−0.469594
$321$	−8031.95	−1.39657
$322$	−4589.85	−0.794355
$323$	−181.654	−0.0312925
$324$	194.300	0.0333162
$325$	−949.743	−0.162099
$326$	3662.51	0.622231
$327$	2180.49	0.368750
$328$	8288.79	1.39534
$329$	7889.06	1.32200
$330$	650.486	0.108509
$331$	5099.01	0.846728	0.423364	−	0.905960i	$-0.360849\pi$
$331$	5099.01	0.846728	0.423364	+	0.905960i	$0.360849\pi$
$332$	56.7684	0.00938425
$333$	3299.11	0.542913
$334$	7687.30	1.25937
$335$	296.875	0.0484179
$336$	−7121.10	−1.15621
$337$	−11426.7	−1.84703	−0.923516	−	0.383561i	$-0.874698\pi$
$337$	−11426.7	−1.84703	−0.923516	+	0.383561i	$0.874698\pi$
$338$	2070.75	0.333237
$339$	−3984.12	−0.638312
$340$	21.6623	0.00345531
$341$	−2980.00	−0.473244
$342$	−441.856	−0.0698620
$343$	1916.46	0.301689
$344$	4572.88	0.716725
$345$	−1308.28	−0.204160
$346$	5881.50	0.913849
$347$	−4389.02	−0.679006	−0.339503	−	0.940605i	$-0.610259\pi$
$347$	−4389.02	−0.679006	−0.339503	+	0.940605i	$0.610259\pi$
$348$	122.174	0.0188196
$349$	204.667	0.0313913	0.0156957	−	0.999877i	$-0.495004\pi$
$349$	204.667	0.0313913	0.0156957	+	0.999877i	$0.495004\pi$
$350$	−1888.00	−0.288336
$351$	5800.46	0.882067
$352$	−225.304	−0.0341158
$353$	−11791.4	−1.77789	−0.888943	−	0.458017i	$-0.848560\pi$
$353$	−11791.4	−1.77789	−0.888943	+	0.458017i	$0.848560\pi$
$354$	893.404	0.134135
$355$	4412.73	0.659727
$356$	722.337	0.107539
$357$	1131.52	0.167749
$358$	4505.91	0.665209
$359$	−561.064	−0.0824841	−0.0412421	−	0.999149i	$-0.513131\pi$
$359$	−561.064	−0.0824841	−0.0412421	+	0.999149i	$0.513131\pi$
$360$	982.914	0.143900
$361$	361.000	0.0526316
$362$	3904.80	0.566939
$363$	520.928	0.0753213
$364$	473.248	0.0681455
$365$	3859.49	0.553466
$366$	−7218.87	−1.03097
$367$	5801.66	0.825189	0.412594	−	0.910915i	$-0.364623\pi$
$367$	5801.66	0.825189	0.412594	+	0.910915i	$0.364623\pi$
$368$	−3656.90	−0.518013
$369$	−3021.58	−0.426279
$370$	−5353.10	−0.752147
$371$	−10548.9	−1.47621
$372$	528.519	0.0736625
$373$	2220.22	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$373$	2220.22	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$374$	−288.912	−0.0399446
$375$	−538.149	−0.0741064
$376$	6664.20	0.914042
$377$	2379.07	0.325009
$378$	11530.8	1.56899
$379$	−10541.3	−1.42868	−0.714341	−	0.699797i	$-0.753274\pi$
$379$	−10541.3	−1.42868	−0.714341	+	0.699797i	$0.753274\pi$
$380$	−43.0495	−0.00581156
$381$	2640.09	0.355002
$382$	9611.47	1.28734
$383$	13886.8	1.85269	0.926346	−	0.376674i	$-0.122932\pi$
$383$	13886.8	1.85269	0.926346	+	0.376674i	$0.122932\pi$
$384$	−5653.04	−0.751251
$385$	−1511.96	−0.200148
$386$	7247.34	0.955647
$387$	−1666.99	−0.218961
$388$	−74.0946	−0.00969481
$389$	−12086.3	−1.57532	−0.787659	−	0.616111i	$-0.788707\pi$
$389$	−12086.3	−1.57532	−0.787659	+	0.616111i	$0.788707\pi$
$390$	−2246.52	−0.291685
$391$	581.068	0.0751557
$392$	9584.09	1.23487
$393$	9981.51	1.28117
$394$	10967.2	1.40234
$395$	−3540.56	−0.451000
$396$	42.1969	0.00535474
$397$	4174.79	0.527775	0.263887	−	0.964553i	$-0.414995\pi$
$397$	4174.79	0.527775	0.263887	+	0.964553i	$0.414995\pi$
$398$	−4612.12	−0.580866
$399$	−2248.67	−0.282141
$400$	−1504.24	−0.188029
$401$	8287.21	1.03203	0.516015	−	0.856580i	$-0.327415\pi$
$401$	8287.21	1.03203	0.516015	+	0.856580i	$0.327415\pi$
$402$	702.228	0.0871243
$403$	10291.8	1.27213
$404$	−852.059	−0.104930
$405$	2143.87	0.263037
$406$	4729.37	0.578114
$407$	−4286.92	−0.522100
$408$	955.838	0.115983
$409$	−6767.38	−0.818156	−0.409078	−	0.912500i	$-0.634150\pi$
$409$	−6767.38	−0.818156	−0.409078	+	0.912500i	$0.634150\pi$
$410$	4902.78	0.590563
$411$	−7038.41	−0.844718
$412$	−183.133	−0.0218988
$413$	−2076.59	−0.247415
$414$	1413.40	0.167789
$415$	626.372	0.0740901
$416$	778.112	0.0917070
$417$	−814.690	−0.0956728
$418$	574.155	0.0671838
$419$	1636.44	0.190800	0.0954002	−	0.995439i	$-0.469587\pi$
$419$	1636.44	0.190800	0.0954002	+	0.995439i	$0.469587\pi$
$420$	268.155	0.0311538
$421$	−3643.84	−0.421829	−0.210915	−	0.977504i	$-0.567644\pi$
$421$	−3643.84	−0.421829	−0.210915	+	0.977504i	$0.567644\pi$
$422$	10476.1	1.20846
$423$	−2429.35	−0.279242
$424$	−8911.09	−1.02066
$425$	239.018	0.0272802
$426$	10437.9	1.18713
$427$	16779.3	1.90165
$428$	845.420	0.0954788
$429$	−1799.08	−0.202472
$430$	2704.84	0.303346
$431$	11834.8	1.32265	0.661324	−	0.750100i	$-0.269995\pi$
$431$	11834.8	1.32265	0.661324	+	0.750100i	$0.269995\pi$
$432$	9186.97	1.02317
$433$	8833.36	0.980379	0.490190	−	0.871616i	$-0.336928\pi$
$433$	8833.36	0.980379	0.490190	+	0.871616i	$0.336928\pi$
$434$	20459.1	2.26283
$435$	1348.04	0.148583
$436$	−229.512	−0.0252102
$437$	−1154.76	−0.126406
$438$	9129.26	0.995920
$439$	15827.6	1.72076	0.860378	−	0.509656i	$-0.170227\pi$
$439$	15827.6	1.72076	0.860378	+	0.509656i	$0.170227\pi$
$440$	−1277.22	−0.138384
$441$	−3493.76	−0.377255
$442$	997.790	0.107376
$443$	17558.8	1.88317	0.941584	−	0.336778i	$-0.109337\pi$
$443$	17558.8	1.88317	0.941584	+	0.336778i	$0.109337\pi$
$444$	760.307	0.0812671
$445$	7970.13	0.849034
$446$	−17775.6	−1.88721
$447$	−14790.7	−1.56505
$448$	14779.4	1.55862
$449$	6846.61	0.719625	0.359812	−	0.933025i	$-0.382841\pi$
$449$	6846.61	0.719625	0.359812	+	0.933025i	$0.382841\pi$
$450$	581.389	0.0609043
$451$	3926.29	0.409937
$452$	419.358	0.0436392
$453$	−5446.60	−0.564909
$454$	−11702.5	−1.20975
$455$	5221.73	0.538019
$456$	−1899.54	−0.195074
$457$	2675.16	0.273827	0.136913	−	0.990583i	$-0.456282\pi$
$457$	2675.16	0.273827	0.136913	+	0.990583i	$0.456282\pi$
$458$	9730.43	0.992736
$459$	−1459.78	−0.148446
$460$	137.705	0.0139577
$461$	−6042.97	−0.610519	−0.305259	−	0.952269i	$-0.598743\pi$
$461$	−6042.97	−0.610519	−0.305259	+	0.952269i	$0.598743\pi$
$462$	−3576.41	−0.360150
$463$	13465.9	1.35165	0.675826	−	0.737061i	$-0.263787\pi$
$463$	13465.9	1.35165	0.675826	+	0.737061i	$0.263787\pi$
$464$	3768.05	0.376999
$465$	5831.58	0.581577
$466$	−9370.00	−0.931453
$467$	14445.9	1.43143	0.715713	−	0.698395i	$-0.246102\pi$
$467$	14445.9	1.43143	0.715713	+	0.698395i	$0.246102\pi$
$468$	−145.732	−0.0143941
$469$	−1632.23	−0.160702
$470$	3941.84	0.386859
$471$	6990.89	0.683914
$472$	−1754.18	−0.171065
$473$	2166.11	0.210567
$474$	−8374.86	−0.811540
$475$	−475.000	−0.0458831
$476$	−119.101	−0.0114684
$477$	3248.43	0.311814
$478$	12501.5	1.19625
$479$	−12346.4	−1.17771	−0.588854	−	0.808240i	$-0.700421\pi$
$479$	−12346.4	−1.17771	−0.588854	+	0.808240i	$0.700421\pi$
$480$	440.899	0.0419254
$481$	14805.3	1.40346
$482$	−6410.01	−0.605742
$483$	7192.97	0.677622
$484$	−54.8315	−0.00514946
$485$	−817.546	−0.0765420
$486$	−6254.00	−0.583719
$487$	−3694.89	−0.343802	−0.171901	−	0.985114i	$-0.554991\pi$
$487$	−3694.89	−0.343802	−0.171901	+	0.985114i	$0.554991\pi$
$488$	14174.1	1.31482
$489$	−5739.68	−0.530792
$490$	5668.95	0.522647
$491$	11093.6	1.01965	0.509823	−	0.860279i	$-0.329711\pi$
$491$	11093.6	1.01965	0.509823	+	0.860279i	$0.329711\pi$
$492$	−696.348	−0.0638085
$493$	−598.731	−0.0546967
$494$	−1982.91	−0.180598
$495$	465.593	0.0422765
$496$	16300.5	1.47563
$497$	−24261.4	−2.18968
$498$	1481.62	0.133319
$499$	−2082.62	−0.186836	−0.0934178	−	0.995627i	$-0.529779\pi$
$499$	−2082.62	−0.186836	−0.0934178	+	0.995627i	$0.529779\pi$
$500$	56.6441	0.00506640
$501$	−12047.1	−1.07430
$502$	−6970.70	−0.619756
$503$	−17143.4	−1.51965	−0.759826	−	0.650126i	$-0.774716\pi$
$503$	−17143.4	−1.51965	−0.759826	+	0.650126i	$0.774716\pi$
$504$	−5404.11	−0.477616
$505$	−9401.46	−0.828435
$506$	−1836.59	−0.161357
$507$	−3245.17	−0.284267
$508$	−277.889	−0.0242703
$509$	10844.5	0.944352	0.472176	−	0.881504i	$-0.343469\pi$
$509$	10844.5	0.944352	0.472176	+	0.881504i	$0.343469\pi$
$510$	565.374	0.0490886
$511$	−21219.7	−1.83699
$512$	12410.5	1.07123
$513$	2901.02	0.249674
$514$	6111.61	0.524458
$515$	−2020.65	−0.172894
$516$	−384.172	−0.0327756
$517$	3156.74	0.268537
$518$	29431.6	2.49643
$519$	−9217.18	−0.779555
$520$	4411.00	0.371991
$521$	1458.88	0.122677	0.0613384	−	0.998117i	$-0.480463\pi$
$521$	1458.88	0.122677	0.0613384	+	0.998117i	$0.480463\pi$
$522$	−1456.36	−0.122113
$523$	11970.4	1.00082	0.500409	−	0.865789i	$-0.333183\pi$
$523$	11970.4	1.00082	0.500409	+	0.865789i	$0.333183\pi$
$524$	−1050.63	−0.0875894
$525$	2958.77	0.245964
$526$	−16251.9	−1.34718
$527$	−2590.09	−0.214091
$528$	−2849.45	−0.234861
$529$	−8473.20	−0.696408
$530$	−5270.87	−0.431985
$531$	639.465	0.0522607
$532$	236.688	0.0192890
$533$	−13559.9	−1.10196
$534$	18852.6	1.52777
$535$	9328.21	0.753820
$536$	−1378.81	−0.111111
$537$	−7061.43	−0.567455
$538$	18597.7	1.49034
$539$	4539.86	0.362793
$540$	−345.948	−0.0275689
$541$	10369.6	0.824074	0.412037	−	0.911167i	$-0.364817\pi$
$541$	10369.6	0.824074	0.412037	+	0.911167i	$0.364817\pi$
$542$	−18147.9	−1.43822
$543$	−6119.40	−0.483625
$544$	−195.824	−0.0154336
$545$	−2532.40	−0.199038
$546$	12351.5	0.968124
$547$	−16450.4	−1.28587	−0.642934	−	0.765922i	$-0.722283\pi$
$547$	−16450.4	−1.28587	−0.642934	+	0.765922i	$0.722283\pi$
$548$	740.843	0.0577505
$549$	−5166.99	−0.401679
$550$	−755.467	−0.0585695
$551$	1189.86	0.0919957
$552$	6076.18	0.468514
$553$	19466.2	1.49690
$554$	2168.71	0.166317
$555$	8389.09	0.641616
$556$	85.7520	0.00654082
$557$	−4062.43	−0.309032	−0.154516	−	0.987990i	$-0.549382\pi$
$557$	−4062.43	−0.309032	−0.154516	+	0.987990i	$0.549382\pi$
$558$	−6300.15	−0.477969
$559$	−7480.91	−0.566027
$560$	8270.37	0.624083
$561$	452.768	0.0340746
$562$	4656.40	0.349499
$563$	−16987.6	−1.27165	−0.635827	−	0.771832i	$-0.719341\pi$
$563$	−16987.6	−1.27165	−0.635827	+	0.771832i	$0.719341\pi$
$564$	−559.865	−0.0417989
$565$	4627.11	0.344538
$566$	6601.55	0.490254
$567$	−11787.1	−0.873037
$568$	−20494.5	−1.51397
$569$	11587.6	0.853736	0.426868	−	0.904314i	$-0.359617\pi$
$569$	11587.6	0.853736	0.426868	+	0.904314i	$0.359617\pi$
$570$	−1123.57	−0.0825632
$571$	3856.36	0.282633	0.141317	−	0.989964i	$-0.454866\pi$
$571$	3856.36	0.282633	0.141317	+	0.989964i	$0.454866\pi$
$572$	189.366	0.0138423
$573$	−15062.6	−1.09816
$574$	−26955.7	−1.96012
$575$	1519.42	0.110198
$576$	−4551.16	−0.329222
$577$	2493.29	0.179891	0.0899453	−	0.995947i	$-0.471331\pi$
$577$	2493.29	0.179891	0.0899453	+	0.995947i	$0.471331\pi$
$578$	13245.7	0.953195
$579$	−11357.6	−0.815212
$580$	−141.891	−0.0101581
$581$	−3443.83	−0.245910
$582$	−1933.83	−0.137731
$583$	−4221.06	−0.299860
$584$	−17925.1	−1.27011
$585$	−1607.98	−0.113644
$586$	4662.30	0.328666
$587$	−9218.28	−0.648176	−0.324088	−	0.946027i	$-0.605057\pi$
$587$	−9218.28	−0.648176	−0.324088	+	0.946027i	$0.605057\pi$
$588$	−805.168	−0.0564703
$589$	5147.28	0.360085
$590$	−1037.59	−0.0724015
$591$	−17187.3	−1.19626
$592$	23449.2	1.62797
$593$	25678.2	1.77821	0.889106	−	0.457702i	$-0.151327\pi$
$593$	25678.2	1.77821	0.889106	+	0.457702i	$0.151327\pi$
$594$	4613.94	0.318707
$595$	−1314.13	−0.0905449
$596$	1556.83	0.106997
$597$	7227.87	0.495506
$598$	6342.87	0.433744
$599$	21476.0	1.46491	0.732457	−	0.680813i	$-0.238373\pi$
$599$	21476.0	1.46491	0.732457	+	0.680813i	$0.238373\pi$
$600$	2499.39	0.170062
$601$	15432.9	1.04745	0.523726	−	0.851886i	$-0.324541\pi$
$601$	15432.9	1.04745	0.523726	+	0.851886i	$0.324541\pi$
$602$	−14871.3	−1.00683
$603$	502.628	0.0339446
$604$	573.294	0.0386209
$605$	−605.000	−0.0406558
$606$	−22238.3	−1.49070
$607$	9931.72	0.664112	0.332056	−	0.943260i	$-0.392258\pi$
$607$	9931.72	0.664112	0.332056	+	0.943260i	$0.392258\pi$
$608$	389.162	0.0259582
$609$	−7411.61	−0.493159
$610$	8383.91	0.556483
$611$	−10902.2	−0.721856
$612$	36.6757	0.00242243
$613$	4639.89	0.305715	0.152858	−	0.988248i	$-0.451152\pi$
$613$	4639.89	0.305715	0.152858	+	0.988248i	$0.451152\pi$
$614$	1074.95	0.0706535
$615$	−7683.37	−0.503778
$616$	7022.20	0.459306
$617$	16643.8	1.08598	0.542992	−	0.839738i	$-0.317291\pi$
$617$	16643.8	1.08598	0.542992	+	0.839738i	$0.317291\pi$
$618$	−4779.65	−0.311110
$619$	−9270.55	−0.601963	−0.300981	−	0.953630i	$-0.597314\pi$
$619$	−9270.55	−0.601963	−0.300981	+	0.953630i	$0.597314\pi$
$620$	−613.816	−0.0397604
$621$	−9279.69	−0.599647
$622$	17233.0	1.11090
$623$	−43820.2	−2.81801
$624$	9840.89	0.631331
$625$	625.000	0.0400000
$626$	4251.74	0.271460
$627$	−899.785	−0.0573109
$628$	−735.842	−0.0467568
$629$	−3726.00	−0.236193
$630$	−3196.51	−0.202146
$631$	20030.9	1.26374	0.631870	−	0.775075i	$-0.282288\pi$
$631$	20030.9	1.26374	0.631870	+	0.775075i	$0.282288\pi$
$632$	16443.9	1.03497
$633$	−16417.6	−1.03087
$634$	11309.4	0.708447
$635$	−3066.17	−0.191618
$636$	748.628	0.0466746
$637$	−15678.9	−0.975228
$638$	1892.42	0.117432
$639$	7471.04	0.462519
$640$	6565.38	0.405499
$641$	5649.68	0.348126	0.174063	−	0.984734i	$-0.444310\pi$
$641$	5649.68	0.348126	0.174063	+	0.984734i	$0.444310\pi$
$642$	22065.0	1.35644
$643$	15630.4	0.958635	0.479318	−	0.877642i	$-0.340884\pi$
$643$	15630.4	0.958635	0.479318	+	0.877642i	$0.340884\pi$
$644$	−757.112	−0.0463267
$645$	−4238.88	−0.258769
$646$	499.030	0.0303933
$647$	1180.45	0.0717286	0.0358643	−	0.999357i	$-0.488582\pi$
$647$	1180.45	0.0717286	0.0358643	+	0.999357i	$0.488582\pi$
$648$	−9957.04	−0.603626
$649$	−830.932	−0.0502572
$650$	2609.09	0.157441
$651$	−32062.3	−1.93030
$652$	604.143	0.0362885
$653$	−19107.3	−1.14507	−0.572533	−	0.819882i	$-0.694039\pi$
$653$	−19107.3	−1.14507	−0.572533	+	0.819882i	$0.694039\pi$
$654$	−5990.14	−0.358154
$655$	−11592.4	−0.691531
$656$	−21476.6	−1.27823
$657$	6534.38	0.388022
$658$	−21672.5	−1.28401
$659$	20908.3	1.23592	0.617959	−	0.786210i	$-0.287960\pi$
$659$	20908.3	1.23592	0.617959	+	0.786210i	$0.287960\pi$
$660$	107.300	0.00632825
$661$	2012.79	0.118440	0.0592198	−	0.998245i	$-0.481139\pi$
$661$	2012.79	0.118440	0.0592198	+	0.998245i	$0.481139\pi$
$662$	−14007.8	−0.822398
$663$	−1563.68	−0.0915964
$664$	−2909.13	−0.170025
$665$	2611.57	0.152289
$666$	−9063.15	−0.527312
$667$	−3806.08	−0.220948
$668$	1268.05	0.0734464
$669$	27856.9	1.60988
$670$	−815.560	−0.0470266
$671$	6714.08	0.386281
$672$	−2424.08	−0.139153
$673$	−10418.9	−0.596759	−0.298379	−	0.954447i	$-0.596446\pi$
$673$	−10418.9	−0.596759	−0.298379	+	0.954447i	$0.596446\pi$
$674$	31390.8	1.79396
$675$	−3817.13	−0.217661
$676$	341.578	0.0194343
$677$	−6113.99	−0.347090	−0.173545	−	0.984826i	$-0.555522\pi$
$677$	−6113.99	−0.347090	−0.173545	+	0.984826i	$0.555522\pi$
$678$	10945.0	0.619970
$679$	4494.91	0.254048
$680$	−1110.10	−0.0626035
$681$	18339.5	1.03197
$682$	8186.52	0.459645
$683$	−16526.6	−0.925875	−0.462938	−	0.886391i	$-0.653205\pi$
$683$	−16526.6	−0.925875	−0.462938	+	0.886391i	$0.653205\pi$
$684$	−72.8856	−0.00407435
$685$	8174.32	0.455949
$686$	−5264.82	−0.293020
$687$	−15249.0	−0.846850
$688$	−11848.5	−0.656571
$689$	14577.9	0.806058
$690$	3594.03	0.198293
$691$	29284.1	1.61218	0.806092	−	0.591790i	$-0.201578\pi$
$691$	29284.1	1.61218	0.806092	+	0.591790i	$0.201578\pi$
$692$	970.175	0.0532955
$693$	−2559.86	−0.140319
$694$	12057.3	0.659495
$695$	946.172	0.0516408
$696$	−6260.88	−0.340974
$697$	3412.56	0.185452
$698$	−562.252	−0.0304893
$699$	14684.2	0.794572
$700$	−311.432	−0.0168157
$701$	−20707.4	−1.11570	−0.557852	−	0.829941i	$-0.688374\pi$
$701$	−20707.4	−1.11570	−0.557852	+	0.829941i	$0.688374\pi$
$702$	−15934.7	−0.856721
$703$	7404.67	0.397258
$704$	5913.85	0.316601
$705$	−6177.45	−0.330009
$706$	32392.8	1.72680
$707$	51689.7	2.74964
$708$	147.370	0.00782276
$709$	−12560.9	−0.665352	−0.332676	−	0.943041i	$-0.607952\pi$
$709$	−12560.9	−0.665352	−0.332676	+	0.943041i	$0.607952\pi$
$710$	−12122.4	−0.640770
$711$	−5994.41	−0.316186
$712$	−37016.6	−1.94839
$713$	−16465.0	−0.864822
$714$	−3108.45	−0.162929
$715$	2089.43	0.109287
$716$	743.267	0.0387949
$717$	−19591.7	−1.02045
$718$	1541.33	0.0801139
$719$	7945.42	0.412120	0.206060	−	0.978539i	$-0.433936\pi$
$719$	7945.42	0.412120	0.206060	+	0.978539i	$0.433936\pi$
$720$	−2546.77	−0.131823
$721$	11109.6	0.573848
$722$	−991.722	−0.0511192
$723$	10045.4	0.516727
$724$	644.111	0.0330638
$725$	−1565.60	−0.0802000
$726$	−1431.07	−0.0731570
$727$	−25978.8	−1.32531	−0.662655	−	0.748925i	$-0.730571\pi$
$727$	−25978.8	−1.32531	−0.662655	+	0.748925i	$0.730571\pi$
$728$	−24251.9	−1.23467
$729$	21377.8	1.08611
$730$	−10602.6	−0.537562
$731$	1882.69	0.0952583
$732$	−1190.78	−0.0601263
$733$	−11699.7	−0.589549	−0.294774	−	0.955567i	$-0.595244\pi$
$733$	−11699.7	−0.589549	−0.294774	+	0.955567i	$0.595244\pi$
$734$	−15938.0	−0.801477
$735$	−8884.07	−0.445842
$736$	−1244.84	−0.0623443
$737$	−653.124	−0.0326433
$738$	8300.73	0.414030
$739$	28184.7	1.40297	0.701483	−	0.712686i	$-0.252522\pi$
$739$	28184.7	1.40297	0.701483	+	0.712686i	$0.252522\pi$
$740$	−883.012	−0.0438651
$741$	3107.51	0.154058
$742$	28979.5	1.43379
$743$	5917.10	0.292163	0.146082	−	0.989273i	$-0.453334\pi$
$743$	5917.10	0.292163	0.146082	+	0.989273i	$0.453334\pi$
$744$	−27084.3	−1.33462
$745$	17177.8	0.844758
$746$	−6099.27	−0.299343
$747$	1060.49	0.0519428
$748$	−47.6571	−0.00232957
$749$	−51287.0	−2.50198
$750$	1478.38	0.0719769
$751$	−16368.5	−0.795333	−0.397667	−	0.917530i	$-0.630180\pi$
$751$	−16368.5	−0.795333	−0.397667	+	0.917530i	$0.630180\pi$
$752$	−17267.2	−0.837328
$753$	10924.1	0.528681
$754$	−6535.67	−0.315670
$755$	6325.62	0.304918
$756$	1902.04	0.0915033
$757$	3588.53	0.172295	0.0861475	−	0.996282i	$-0.472544\pi$
$757$	3588.53	0.172295	0.0861475	+	0.996282i	$0.472544\pi$
$758$	28958.6	1.38763
$759$	2878.21	0.137645
$760$	2206.10	0.105294
$761$	−24369.2	−1.16082	−0.580410	−	0.814325i	$-0.697108\pi$
$761$	−24369.2	−1.16082	−0.580410	+	0.814325i	$0.697108\pi$
$762$	−7252.73	−0.344801
$763$	13923.2	0.660623
$764$	1585.45	0.0750778
$765$	404.673	0.0191255
$766$	−38149.1	−1.79945
$767$	2869.72	0.135097
$768$	−2986.80	−0.140334
$769$	−39526.9	−1.85354	−0.926772	−	0.375624i	$-0.877428\pi$
$769$	−39526.9	−1.85354	−0.926772	+	0.375624i	$0.877428\pi$
$770$	4153.60	0.194396
$771$	−9577.78	−0.447387
$772$	1195.47	0.0557332
$773$	−9890.63	−0.460209	−0.230104	−	0.973166i	$-0.573907\pi$
$773$	−9890.63	−0.460209	−0.230104	+	0.973166i	$0.573907\pi$
$774$	4579.47	0.212669
$775$	−6772.73	−0.313915
$776$	3797.03	0.175651
$777$	−46123.6	−2.12957
$778$	33202.9	1.53005
$779$	−6781.77	−0.311915
$780$	−370.572	−0.0170110
$781$	−9708.00	−0.444788
$782$	−1596.28	−0.0729961
$783$	9561.76	0.436410
$784$	−24832.8	−1.13123
$785$	−8119.14	−0.369152
$786$	−27420.7	−1.24436
$787$	27596.5	1.24995	0.624975	−	0.780645i	$-0.285109\pi$
$787$	27596.5	1.24995	0.624975	+	0.780645i	$0.285109\pi$
$788$	1809.08	0.0817842
$789$	25469.1	1.14921
$790$	9726.47	0.438041
$791$	−25440.1	−1.14355
$792$	−2162.41	−0.0970176
$793$	−23187.8	−1.03837
$794$	−11468.8	−0.512609
$795$	8260.22	0.368503
$796$	−760.785	−0.0338760
$797$	17537.3	0.779427	0.389713	−	0.920936i	$-0.372574\pi$
$797$	17537.3	0.779427	0.389713	+	0.920936i	$0.372574\pi$
$798$	6177.43	0.274033
$799$	2743.70	0.121483
$800$	−512.055	−0.0226298
$801$	13494.0	0.595238
$802$	−22766.2	−1.00237
$803$	−8490.89	−0.373147
$804$	115.835	0.00508108
$805$	−8353.83	−0.365756
$806$	−28273.1	−1.23558
$807$	−29145.3	−1.27133
$808$	43664.3	1.90112
$809$	−31400.0	−1.36461	−0.682303	−	0.731069i	$-0.739022\pi$
$809$	−31400.0	−1.36461	−0.682303	+	0.731069i	$0.739022\pi$
$810$	−5889.54	−0.255478
$811$	−45228.6	−1.95831	−0.979156	−	0.203111i	$-0.934895\pi$
$811$	−45228.6	−1.95831	−0.979156	+	0.203111i	$0.934895\pi$
$812$	780.126	0.0337156
$813$	28440.4	1.22687
$814$	11776.8	0.507097
$815$	6666.00	0.286503
$816$	−2476.62	−0.106249
$817$	−3741.47	−0.160217
$818$	18591.0	0.794646
$819$	8840.74	0.377192
$820$	808.731	0.0344416
$821$	−38342.8	−1.62993	−0.814966	−	0.579509i	$-0.803244\pi$
$821$	−38342.8	−1.62993	−0.814966	+	0.579509i	$0.803244\pi$
$822$	19335.6	0.820445
$823$	−12047.6	−0.510273	−0.255136	−	0.966905i	$-0.582120\pi$
$823$	−12047.6	−0.510273	−0.255136	+	0.966905i	$0.582120\pi$
$824$	9384.75	0.396764
$825$	1183.93	0.0499625
$826$	5704.72	0.240306
$827$	−41994.5	−1.76577	−0.882885	−	0.469589i	$-0.844402\pi$
$827$	−41994.5	−1.76577	−0.882885	+	0.469589i	$0.844402\pi$
$828$	233.145	0.00978543
$829$	2254.93	0.0944715	0.0472357	−	0.998884i	$-0.484959\pi$
$829$	2254.93	0.0944715	0.0472357	+	0.998884i	$0.484959\pi$
$830$	−1720.74	−0.0719611
$831$	−3398.69	−0.141876
$832$	−20424.1	−0.851058
$833$	3945.84	0.164124
$834$	2238.08	0.0929236
$835$	13991.4	0.579871
$836$	94.7089	0.00391815
$837$	41363.8	1.70817
$838$	−4495.55	−0.185318
$839$	11859.2	0.487993	0.243997	−	0.969776i	$-0.421541\pi$
$839$	11859.2	0.487993	0.243997	+	0.969776i	$0.421541\pi$
$840$	−13741.8	−0.564448
$841$	−20467.2	−0.839199
$842$	10010.2	0.409708
$843$	−7297.26	−0.298139
$844$	1728.07	0.0704771
$845$	3768.91	0.153437
$846$	6673.80	0.271218
$847$	3326.32	0.134939
$848$	23089.0	0.935000
$849$	−10345.6	−0.418210
$850$	−656.619	−0.0264963
$851$	−23685.9	−0.954102
$852$	1721.76	0.0692332
$853$	3783.53	0.151871	0.0759353	−	0.997113i	$-0.475806\pi$
$853$	3783.53	0.151871	0.0759353	+	0.997113i	$0.475806\pi$
$854$	−46095.2	−1.84701
$855$	−804.206	−0.0321676
$856$	−43324.1	−1.72989
$857$	3394.56	0.135304	0.0676522	−	0.997709i	$-0.478449\pi$
$857$	3394.56	0.135304	0.0676522	+	0.997709i	$0.478449\pi$
$858$	4942.35	0.196654
$859$	−9695.49	−0.385106	−0.192553	−	0.981287i	$-0.561677\pi$
$859$	−9695.49	−0.385106	−0.192553	+	0.981287i	$0.561677\pi$
$860$	446.173	0.0176911
$861$	42243.6	1.67208
$862$	−32512.0	−1.28464
$863$	−9995.39	−0.394261	−0.197130	−	0.980377i	$-0.563162\pi$
$863$	−9995.39	−0.394261	−0.197130	+	0.980377i	$0.563162\pi$
$864$	3127.32	0.123141
$865$	10704.7	0.420777
$866$	−24266.6	−0.952208
$867$	−20757.9	−0.813119
$868$	3374.79	0.131968
$869$	7789.24	0.304064
$870$	−3703.28	−0.144314
$871$	2255.64	0.0877489
$872$	11761.5	0.456760
$873$	−1384.16	−0.0536618
$874$	3172.29	0.122774
$875$	−3436.28	−0.132763
$876$	1505.90	0.0580819
$877$	21339.5	0.821647	0.410824	−	0.911715i	$-0.365241\pi$
$877$	21339.5	0.821647	0.410824	+	0.911715i	$0.365241\pi$
$878$	−43480.9	−1.67131
$879$	−7306.51	−0.280367
$880$	3309.32	0.126769
$881$	22374.7	0.855645	0.427823	−	0.903863i	$-0.359281\pi$
$881$	22374.7	0.855645	0.427823	+	0.903863i	$0.359281\pi$
$882$	9597.90	0.366415
$883$	−35909.3	−1.36856	−0.684282	−	0.729217i	$-0.739884\pi$
$883$	−35909.3	−1.36856	−0.684282	+	0.729217i	$0.739884\pi$
$884$	164.589	0.00626214
$885$	1626.06	0.0617619
$886$	−48236.7	−1.82906
$887$	5013.76	0.189792	0.0948960	−	0.995487i	$-0.469748\pi$
$887$	5013.76	0.189792	0.0948960	+	0.995487i	$0.469748\pi$
$888$	−38962.5	−1.47240
$889$	16858.0	0.635993
$890$	−21895.2	−0.824637
$891$	−4716.52	−0.177339
$892$	−2932.14	−0.110062
$893$	−5452.56	−0.204326
$894$	40632.3	1.52008
$895$	8201.06	0.306292
$896$	−36096.8	−1.34588
$897$	−9940.20	−0.370004
$898$	−18808.7	−0.698946
$899$	16965.4	0.629399
$900$	95.9022	0.00355193
$901$	−3668.76	−0.135654
$902$	−10786.1	−0.398158
$903$	23305.6	0.858872
$904$	−21490.3	−0.790659
$905$	7107.00	0.261044
$906$	14962.6	0.548676
$907$	47723.1	1.74710	0.873550	−	0.486734i	$-0.161812\pi$
$907$	47723.1	1.74710	0.873550	+	0.486734i	$0.161812\pi$
$908$	−1930.37	−0.0705524
$909$	−15917.3	−0.580796
$910$	−14344.9	−0.522559
$911$	12755.5	0.463894	0.231947	−	0.972728i	$-0.425490\pi$
$911$	12755.5	0.463894	0.231947	+	0.972728i	$0.425490\pi$
$912$	4921.78	0.178702
$913$	−1378.02	−0.0499515
$914$	−7349.09	−0.265959
$915$	−13138.8	−0.474706
$916$	1605.07	0.0578962
$917$	63735.7	2.29524
$918$	4010.23	0.144180
$919$	8907.69	0.319736	0.159868	−	0.987138i	$-0.448893\pi$
$919$	8907.69	0.319736	0.159868	+	0.987138i	$0.448893\pi$
$920$	−7056.81	−0.252887
$921$	−1684.60	−0.0602707
$922$	16601.0	0.592976
$923$	33527.6	1.19564
$924$	−589.941	−0.0210039
$925$	−9742.99	−0.346322
$926$	−36993.0	−1.31281
$927$	−3421.09	−0.121212
$928$	1282.68	0.0453728
$929$	1112.88	0.0393030	0.0196515	−	0.999807i	$-0.493744\pi$
$929$	1112.88	0.0393030	0.0196515	+	0.999807i	$0.493744\pi$
$930$	−16020.2	−0.564865
$931$	−7841.57	−0.276044
$932$	−1545.61	−0.0543222
$933$	−27006.7	−0.947652
$934$	−39685.0	−1.39029
$935$	−525.839	−0.0183923
$936$	7468.12	0.260794
$937$	45773.2	1.59588	0.797942	−	0.602734i	$-0.205922\pi$
$937$	45773.2	1.59588	0.797942	+	0.602734i	$0.205922\pi$
$938$	4483.99	0.156085
$939$	−6663.11	−0.231568
$940$	650.221	0.0225616
$941$	−11654.1	−0.403734	−0.201867	−	0.979413i	$-0.564701\pi$
$941$	−11654.1	−0.403734	−0.201867	+	0.979413i	$0.564701\pi$
$942$	−19205.1	−0.664262
$943$	21693.3	0.749133
$944$	4545.16	0.156708
$945$	20986.8	0.722433
$946$	−5950.65	−0.204516
$947$	−7971.18	−0.273525	−0.136763	−	0.990604i	$-0.543670\pi$
$947$	−7971.18	−0.273525	−0.136763	+	0.990604i	$0.543670\pi$
$948$	−1381.46	−0.0473289
$949$	29324.2	1.00306
$950$	1304.90	0.0445647
$951$	−17723.5	−0.604338
$952$	6103.38	0.207786
$953$	11868.7	0.403426	0.201713	−	0.979445i	$-0.435349\pi$
$953$	11868.7	0.403426	0.201713	+	0.979445i	$0.435349\pi$
$954$	−8923.93	−0.302854
$955$	17493.5	0.592750
$956$	2062.17	0.0697650
$957$	−2965.69	−0.100175
$958$	33917.5	1.14387
$959$	−44942.9	−1.51333
$960$	−11572.9	−0.389075
$961$	43600.9	1.46356
$962$	−40672.5	−1.36313
$963$	15793.3	0.528485
$964$	−1057.35	−0.0353268
$965$	13190.6	0.440022
$966$	−19760.2	−0.658150
$967$	−32740.0	−1.08878	−0.544389	−	0.838833i	$-0.683238\pi$
$967$	−32740.0	−1.08878	−0.544389	+	0.838833i	$0.683238\pi$
$968$	2809.87	0.0932983
$969$	−782.053	−0.0259269
$970$	2245.92	0.0743426
$971$	20025.2	0.661832	0.330916	−	0.943660i	$-0.392642\pi$
$971$	20025.2	0.661832	0.330916	+	0.943660i	$0.392642\pi$
$972$	−1031.62	−0.0340424
$973$	−5202.10	−0.171400
$974$	10150.4	0.333923
$975$	−4088.82	−0.134305
$976$	−36725.7	−1.20447
$977$	−7552.83	−0.247325	−0.123662	−	0.992324i	$-0.539464\pi$
$977$	−7552.83	−0.247325	−0.123662	+	0.992324i	$0.539464\pi$
$978$	15767.8	0.515540
$979$	−17534.3	−0.572419
$980$	935.113	0.0304807
$981$	−4287.52	−0.139541
$982$	−30475.8	−0.990347
$983$	−45072.9	−1.46246	−0.731232	−	0.682129i	$-0.761054\pi$
$983$	−45072.9	−1.46246	−0.731232	+	0.682129i	$0.761054\pi$
$984$	35684.8	1.15609
$985$	19961.1	0.645699
$986$	1644.81	0.0531250
$987$	33963.9	1.09532
$988$	−327.088	−0.0105324
$989$	11968.1	0.384796
$990$	−1279.06	−0.0410617
$991$	27225.2	0.872693	0.436346	−	0.899779i	$-0.356272\pi$
$991$	27225.2	0.872693	0.436346	+	0.899779i	$0.356272\pi$
$992$	5548.82	0.177596
$993$	21952.2	0.701544
$994$	66649.7	2.12676
$995$	−8394.36	−0.267456
$996$	244.399	0.00777517
$997$	−42138.4	−1.33855	−0.669276	−	0.743014i	$-0.733396\pi$
$997$	−42138.4	−1.33855	−0.669276	+	0.743014i	$0.733396\pi$
$998$	5721.28	0.181467
$999$	59504.4	1.88452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.f.1.8	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.f.1.8	✓	23	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.74715 q^{2} +4.30519 q^{3} -0.453153 q^{4} -5.00000 q^{5} -11.8270 q^{6} +27.4903 q^{7} +23.2221 q^{8} -8.46533 q^{9} +O(q^{10})\)	\(q-2.74715 q^{2} +4.30519 q^{3} -0.453153 q^{4} -5.00000 q^{5} -11.8270 q^{6} +27.4903 q^{7} +23.2221 q^{8} -8.46533 q^{9} +13.7358 q^{10} +11.0000 q^{11} -1.95091 q^{12} -37.9897 q^{13} -75.5199 q^{14} -21.5260 q^{15} -60.1694 q^{16} +9.56072 q^{17} +23.2556 q^{18} -19.0000 q^{19} +2.26576 q^{20} +118.351 q^{21} -30.2187 q^{22} +60.7767 q^{23} +99.9756 q^{24} +25.0000 q^{25} +104.364 q^{26} -152.685 q^{27} -12.4573 q^{28} -62.6241 q^{29} +59.1351 q^{30} -270.909 q^{31} -20.4822 q^{32} +47.3571 q^{33} -26.2647 q^{34} -137.451 q^{35} +3.83609 q^{36} -389.720 q^{37} +52.1959 q^{38} -163.553 q^{39} -116.111 q^{40} +356.935 q^{41} -325.128 q^{42} +196.919 q^{43} -4.98468 q^{44} +42.3267 q^{45} -166.963 q^{46} +286.977 q^{47} -259.041 q^{48} +412.714 q^{49} -68.6788 q^{50} +41.1607 q^{51} +17.2151 q^{52} -383.733 q^{53} +419.449 q^{54} -55.0000 q^{55} +638.382 q^{56} -81.7986 q^{57} +172.038 q^{58} -75.5393 q^{59} +9.75454 q^{60} +610.371 q^{61} +744.229 q^{62} -232.714 q^{63} +537.623 q^{64} +189.949 q^{65} -130.097 q^{66} -59.3749 q^{67} -4.33246 q^{68} +261.655 q^{69} +377.600 q^{70} -882.545 q^{71} -196.583 q^{72} -771.899 q^{73} +1070.62 q^{74} +107.630 q^{75} +8.60990 q^{76} +302.393 q^{77} +449.305 q^{78} +708.113 q^{79} +300.847 q^{80} -428.774 q^{81} -980.556 q^{82} -125.274 q^{83} -53.6310 q^{84} -47.8036 q^{85} -540.968 q^{86} -269.609 q^{87} +255.443 q^{88} -1594.03 q^{89} -116.278 q^{90} -1044.35 q^{91} -27.5411 q^{92} -1166.32 q^{93} -788.369 q^{94} +95.0000 q^{95} -88.1797 q^{96} +163.509 q^{97} -1133.79 q^{98} -93.1186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * q^98 + 1870 * q^99

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