LMFDB - Embedded newform 538.6.a.b.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,6,Mod(1,538)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("538.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	538.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:	$86.2864950594$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +10.6837 q^{3} +16.0000 q^{4} +72.6773 q^{5} -42.7347 q^{6} -163.422 q^{7} -64.0000 q^{8} -128.859 q^{9} -290.709 q^{10} +221.872 q^{11} +170.939 q^{12} -332.575 q^{13} +653.689 q^{14} +776.459 q^{15} +256.000 q^{16} -980.161 q^{17} +515.437 q^{18} -2433.01 q^{19} +1162.84 q^{20} -1745.95 q^{21} -887.488 q^{22} +2232.85 q^{23} -683.754 q^{24} +2156.98 q^{25} +1330.30 q^{26} -3972.82 q^{27} -2614.75 q^{28} -3079.40 q^{29} -3105.84 q^{30} +9855.95 q^{31} -1024.00 q^{32} +2370.41 q^{33} +3920.65 q^{34} -11877.1 q^{35} -2061.75 q^{36} +8882.23 q^{37} +9732.05 q^{38} -3553.12 q^{39} -4651.34 q^{40} +17703.1 q^{41} +6983.79 q^{42} +21321.4 q^{43} +3549.95 q^{44} -9365.14 q^{45} -8931.41 q^{46} +11613.1 q^{47} +2735.02 q^{48} +9899.80 q^{49} -8627.93 q^{50} -10471.7 q^{51} -5321.20 q^{52} +30085.6 q^{53} +15891.3 q^{54} +16125.0 q^{55} +10459.0 q^{56} -25993.5 q^{57} +12317.6 q^{58} +42282.4 q^{59} +12423.3 q^{60} +20201.9 q^{61} -39423.8 q^{62} +21058.5 q^{63} +4096.00 q^{64} -24170.6 q^{65} -9481.62 q^{66} -30823.7 q^{67} -15682.6 q^{68} +23855.0 q^{69} +47508.3 q^{70} -57312.7 q^{71} +8247.00 q^{72} +20408.1 q^{73} -35528.9 q^{74} +23044.5 q^{75} -38928.2 q^{76} -36258.8 q^{77} +14212.5 q^{78} -27329.0 q^{79} +18605.4 q^{80} -11131.5 q^{81} -70812.4 q^{82} -78758.0 q^{83} -27935.2 q^{84} -71235.4 q^{85} -85285.6 q^{86} -32899.3 q^{87} -14199.8 q^{88} +29455.8 q^{89} +37460.6 q^{90} +54350.1 q^{91} +35725.6 q^{92} +105298. q^{93} -46452.4 q^{94} -176825. q^{95} -10940.1 q^{96} +145868. q^{97} -39599.2 q^{98} -28590.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 108 q^{2} + 33 q^{3} + 432 q^{4} + 139 q^{5} - 132 q^{6} - 25 q^{7} - 1728 q^{8} + 2346 q^{9} - 556 q^{10} + 1241 q^{11} + 528 q^{12} - 202 q^{13} + 100 q^{14} + 1786 q^{15} + 6912 q^{16} + 1550 q^{17}+ \cdots + 335378 q^{99}+O(q^{100}) $ 27 * q - 108 * q^2 + 33 * q^3 + 432 * q^4 + 139 * q^5 - 132 * q^6 - 25 * q^7 - 1728 * q^8 + 2346 * q^9 - 556 * q^10 + 1241 * q^11 + 528 * q^12 - 202 * q^13 + 100 * q^14 + 1786 * q^15 + 6912 * q^16 + 1550 * q^17 - 9384 * q^18 - 66 * q^19 + 2224 * q^20 - 1217 * q^21 - 4964 * q^22 + 10568 * q^23 - 2112 * q^24 + 15880 * q^25 + 808 * q^26 + 8256 * q^27 - 400 * q^28 - 2122 * q^29 - 7144 * q^30 - 1621 * q^31 - 27648 * q^32 + 18369 * q^33 - 6200 * q^34 + 23123 * q^35 + 37536 * q^36 + 8542 * q^37 + 264 * q^38 + 589 * q^39 - 8896 * q^40 + 10859 * q^41 + 4868 * q^42 + 23759 * q^43 + 19856 * q^44 + 28006 * q^45 - 42272 * q^46 + 38457 * q^47 + 8448 * q^48 + 57376 * q^49 - 63520 * q^50 + 58237 * q^51 - 3232 * q^52 + 73586 * q^53 - 33024 * q^54 + 25923 * q^55 + 1600 * q^56 + 52534 * q^57 + 8488 * q^58 + 68927 * q^59 + 28576 * q^60 - 76016 * q^61 + 6484 * q^62 - 20738 * q^63 + 110592 * q^64 + 74867 * q^65 - 73476 * q^66 + 94787 * q^67 + 24800 * q^68 - 90601 * q^69 - 92492 * q^70 + 15523 * q^71 - 150144 * q^72 - 96279 * q^73 - 34168 * q^74 - 123786 * q^75 - 1056 * q^76 - 199866 * q^77 - 2356 * q^78 - 246217 * q^79 + 35584 * q^80 - 54473 * q^81 - 43436 * q^82 - 56658 * q^83 - 19472 * q^84 - 120541 * q^85 - 95036 * q^86 - 352954 * q^87 - 79424 * q^88 - 83070 * q^89 - 112024 * q^90 - 132348 * q^91 + 169088 * q^92 + 40727 * q^93 - 153828 * q^94 - 72090 * q^95 - 33792 * q^96 - 36273 * q^97 - 229504 * q^98 + 335378 * q^99

Coefficient data

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

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

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

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	10.6837	0.685357	0.342679	−	0.939453i	$-0.388666\pi$
$3$	10.6837	0.685357	0.342679	+	0.939453i	$0.388666\pi$
$4$	16.0000	0.500000
$5$	72.6773	1.30009	0.650045	−	0.759896i	$-0.274750\pi$
$5$	72.6773	1.30009	0.650045	+	0.759896i	$0.274750\pi$
$6$	−42.7347	−0.484621
$7$	−163.422	−1.26057	−0.630283	−	0.776365i	$-0.717061\pi$
$7$	−163.422	−1.26057	−0.630283	+	0.776365i	$0.717061\pi$
$8$	−64.0000	−0.353553
$9$	−128.859	−0.530285
$10$	−290.709	−0.919303
$11$	221.872	0.552867	0.276434	−	0.961033i	$-0.410847\pi$
$11$	221.872	0.552867	0.276434	+	0.961033i	$0.410847\pi$
$12$	170.939	0.342679
$13$	−332.575	−0.545797	−0.272899	−	0.962043i	$-0.587982\pi$
$13$	−332.575	−0.545797	−0.272899	+	0.962043i	$0.587982\pi$
$14$	653.689	0.891355
$15$	776.459	0.891026
$16$	256.000	0.250000
$17$	−980.161	−0.822575	−0.411287	−	0.911506i	$-0.634921\pi$
$17$	−980.161	−0.822575	−0.411287	+	0.911506i	$0.634921\pi$
$18$	515.437	0.374968
$19$	−2433.01	−1.54618	−0.773091	−	0.634295i	$-0.781290\pi$
$19$	−2433.01	−1.54618	−0.773091	+	0.634295i	$0.781290\pi$
$20$	1162.84	0.650045
$21$	−1745.95	−0.863939
$22$	−887.488	−0.390936
$23$	2232.85	0.880117	0.440058	−	0.897969i	$-0.354958\pi$
$23$	2232.85	0.880117	0.440058	+	0.897969i	$0.354958\pi$
$24$	−683.754	−0.242310
$25$	2156.98	0.690235
$26$	1330.30	0.385937
$27$	−3972.82	−1.04879
$28$	−2614.75	−0.630283
$29$	−3079.40	−0.679942	−0.339971	−	0.940436i	$-0.610417\pi$
$29$	−3079.40	−0.679942	−0.339971	+	0.940436i	$0.610417\pi$
$30$	−3105.84	−0.630051
$31$	9855.95	1.84202	0.921010	−	0.389540i	$-0.127366\pi$
$31$	9855.95	1.84202	0.921010	+	0.389540i	$0.127366\pi$
$32$	−1024.00	−0.176777
$33$	2370.41	0.378911
$34$	3920.65	0.581648
$35$	−11877.1	−1.63885
$36$	−2061.75	−0.265143
$37$	8882.23	1.06664	0.533320	−	0.845914i	$-0.320944\pi$
$37$	8882.23	1.06664	0.533320	+	0.845914i	$0.320944\pi$
$38$	9732.05	1.09332
$39$	−3553.12	−0.374066
$40$	−4651.34	−0.459651
$41$	17703.1	1.64471	0.822355	−	0.568975i	$-0.192660\pi$
$41$	17703.1	1.64471	0.822355	+	0.568975i	$0.192660\pi$
$42$	6983.79	0.610897
$43$	21321.4	1.75851	0.879254	−	0.476353i	$-0.158042\pi$
$43$	21321.4	1.75851	0.879254	+	0.476353i	$0.158042\pi$
$44$	3549.95	0.276434
$45$	−9365.14	−0.689419
$46$	−8931.41	−0.622337
$47$	11613.1	0.766837	0.383418	−	0.923575i	$-0.374747\pi$
$47$	11613.1	0.766837	0.383418	+	0.923575i	$0.374747\pi$
$48$	2735.02	0.171339
$49$	9899.80	0.589028
$50$	−8627.93	−0.488070
$51$	−10471.7	−0.563758
$52$	−5321.20	−0.272899
$53$	30085.6	1.47119	0.735596	−	0.677420i	$-0.236902\pi$
$53$	30085.6	1.47119	0.735596	+	0.677420i	$0.236902\pi$
$54$	15891.3	0.741608
$55$	16125.0	0.718777
$56$	10459.0	0.445678
$57$	−25993.5	−1.05969
$58$	12317.6	0.480791
$59$	42282.4	1.58136	0.790678	−	0.612232i	$-0.209728\pi$
$59$	42282.4	1.58136	0.790678	+	0.612232i	$0.209728\pi$
$60$	12423.3	0.445513
$61$	20201.9	0.695134	0.347567	−	0.937655i	$-0.387008\pi$
$61$	20201.9	0.695134	0.347567	+	0.937655i	$0.387008\pi$
$62$	−39423.8	−1.30250
$63$	21058.5	0.668460
$64$	4096.00	0.125000
$65$	−24170.6	−0.709585
$66$	−9481.62	−0.267931
$67$	−30823.7	−0.838876	−0.419438	−	0.907784i	$-0.637773\pi$
$67$	−30823.7	−0.838876	−0.419438	+	0.907784i	$0.637773\pi$
$68$	−15682.6	−0.411287
$69$	23855.0	0.603195
$70$	47508.3	1.15884
$71$	−57312.7	−1.34929	−0.674645	−	0.738143i	$-0.735703\pi$
$71$	−57312.7	−1.34929	−0.674645	+	0.738143i	$0.735703\pi$
$72$	8247.00	0.187484
$73$	20408.1	0.448224	0.224112	−	0.974563i	$-0.428052\pi$
$73$	20408.1	0.448224	0.224112	+	0.974563i	$0.428052\pi$
$74$	−35528.9	−0.754228
$75$	23044.5	0.473057
$76$	−38928.2	−0.773091
$77$	−36258.8	−0.696926
$78$	14212.5	0.264505
$79$	−27329.0	−0.492670	−0.246335	−	0.969185i	$-0.579226\pi$
$79$	−27329.0	−0.492670	−0.246335	+	0.969185i	$0.579226\pi$
$80$	18605.4	0.325023
$81$	−11131.5	−0.188512
$82$	−70812.4	−1.16299
$83$	−78758.0	−1.25487	−0.627436	−	0.778668i	$-0.715896\pi$
$83$	−78758.0	−1.25487	−0.627436	+	0.778668i	$0.715896\pi$
$84$	−27935.2	−0.431969
$85$	−71235.4	−1.06942
$86$	−85285.6	−1.24345
$87$	−32899.3	−0.466003
$88$	−14199.8	−0.195468
$89$	29455.8	0.394181	0.197091	−	0.980385i	$-0.436851\pi$
$89$	29455.8	0.394181	0.197091	+	0.980385i	$0.436851\pi$
$90$	37460.6	0.487493
$91$	54350.1	0.688014
$92$	35725.6	0.440058
$93$	105298.	1.26244
$94$	−46452.4	−0.542236
$95$	−176825.	−2.01018
$96$	−10940.1	−0.121155
$97$	145868.	1.57410	0.787048	−	0.616892i	$-0.211608\pi$
$97$	145868.	1.57410	0.787048	+	0.616892i	$0.211608\pi$
$98$	−39599.2	−0.416506
$99$	−28590.3	−0.293177
$100$	34511.7	0.345117
$101$	118582.	1.15668	0.578342	−	0.815794i	$-0.303700\pi$
$101$	118582.	1.15668	0.578342	+	0.815794i	$0.303700\pi$
$102$	41886.9	0.398637
$103$	145364.	1.35010	0.675049	−	0.737773i	$-0.264123\pi$
$103$	145364.	1.35010	0.675049	+	0.737773i	$0.264123\pi$
$104$	21284.8	0.192968
$105$	−126891.	−1.12320
$106$	−120343.	−1.04029
$107$	−143266.	−1.20972	−0.604858	−	0.796334i	$-0.706770\pi$
$107$	−143266.	−1.20972	−0.604858	+	0.796334i	$0.706770\pi$
$108$	−63565.1	−0.524396
$109$	173115.	1.39563	0.697814	−	0.716279i	$-0.254156\pi$
$109$	173115.	1.39563	0.697814	+	0.716279i	$0.254156\pi$
$110$	−64500.2	−0.508252
$111$	94894.8	0.731029
$112$	−41836.1	−0.315142
$113$	−130507.	−0.961475	−0.480737	−	0.876865i	$-0.659631\pi$
$113$	−130507.	−0.961475	−0.480737	+	0.876865i	$0.659631\pi$
$114$	103974.	0.749312
$115$	162278.	1.14423
$116$	−49270.5	−0.339971
$117$	42855.4	0.289428
$118$	−169130.	−1.11819
$119$	160180.	1.03691
$120$	−49693.4	−0.315025
$121$	−111824.	−0.694338
$122$	−80807.7	−0.491534
$123$	189134.	1.12721
$124$	157695.	0.921010
$125$	−70352.8	−0.402723
$126$	−84233.9	−0.472673
$127$	92885.7	0.511022	0.255511	−	0.966806i	$-0.417756\pi$
$127$	92885.7	0.511022	0.255511	+	0.966806i	$0.417756\pi$
$128$	−16384.0	−0.0883883
$129$	227791.	1.20521
$130$	96682.5	0.501753
$131$	64776.6	0.329792	0.164896	−	0.986311i	$-0.447271\pi$
$131$	64776.6	0.329792	0.164896	+	0.986311i	$0.447271\pi$
$132$	37926.5	0.189456
$133$	397608.	1.94907
$134$	123295.	0.593175
$135$	−288734.	−1.36352
$136$	62730.3	0.290824
$137$	−216586.	−0.985890	−0.492945	−	0.870060i	$-0.664080\pi$
$137$	−216586.	−0.985890	−0.492945	+	0.870060i	$0.664080\pi$
$138$	−95420.2	−0.426523
$139$	9409.80	0.0413089	0.0206544	−	0.999787i	$-0.493425\pi$
$139$	9409.80	0.0413089	0.0206544	+	0.999787i	$0.493425\pi$
$140$	−190033.	−0.819425
$141$	124070.	0.525557
$142$	229251.	0.954091
$143$	−73789.1	−0.301753
$144$	−32988.0	−0.132571
$145$	−223803.	−0.883986
$146$	−81632.3	−0.316942
$147$	105766.	0.403695
$148$	142116.	0.533320
$149$	−461499.	−1.70296	−0.851482	−	0.524384i	$-0.824295\pi$
$149$	−461499.	−1.70296	−0.851482	+	0.524384i	$0.824295\pi$
$150$	−92177.9	−0.334502
$151$	73810.3	0.263436	0.131718	−	0.991287i	$-0.457951\pi$
$151$	73810.3	0.263436	0.131718	+	0.991287i	$0.457951\pi$
$152$	155713.	0.546658
$153$	126303.	0.436199
$154$	145035.	0.492801
$155$	716303.	2.39479
$156$	−56849.9	−0.187033
$157$	386739.	1.25219	0.626093	−	0.779749i	$-0.284653\pi$
$157$	386739.	1.25219	0.626093	+	0.779749i	$0.284653\pi$
$158$	109316.	0.348370
$159$	321425.	1.00829
$160$	−74421.5	−0.229826
$161$	−364898.	−1.10945
$162$	44525.8	0.133298
$163$	−84131.1	−0.248020	−0.124010	−	0.992281i	$-0.539576\pi$
$163$	−84131.1	−0.248020	−0.124010	+	0.992281i	$0.539576\pi$
$164$	283249.	0.822355
$165$	172275.	0.492619
$166$	315032.	0.887329
$167$	379986.	1.05433	0.527164	−	0.849763i	$-0.323255\pi$
$167$	379986.	1.05433	0.527164	+	0.849763i	$0.323255\pi$
$168$	111741.	0.305448
$169$	−260687.	−0.702106
$170$	284942.	0.756195
$171$	313516.	0.819917
$172$	341142.	0.879254
$173$	−56864.2	−0.144452	−0.0722260	−	0.997388i	$-0.523010\pi$
$173$	−56864.2	−0.144452	−0.0722260	+	0.997388i	$0.523010\pi$
$174$	131597.	0.329514
$175$	−352499.	−0.870087
$176$	56799.2	0.138217
$177$	451731.	1.08379
$178$	−117823.	−0.278728
$179$	552368.	1.28853	0.644267	−	0.764801i	$-0.277163\pi$
$179$	552368.	1.28853	0.644267	+	0.764801i	$0.277163\pi$
$180$	−149842.	−0.344709
$181$	57601.9	0.130689	0.0653447	−	0.997863i	$-0.479185\pi$
$181$	57601.9	0.130689	0.0653447	+	0.997863i	$0.479185\pi$
$182$	−217400.	−0.486499
$183$	215831.	0.476415
$184$	−142903.	−0.311168
$185$	645536.	1.38673
$186$	−421190.	−0.892681
$187$	−217470.	−0.454774
$188$	185809.	0.383418
$189$	649247.	1.32207
$190$	707299.	1.42141
$191$	−464050.	−0.920409	−0.460205	−	0.887813i	$-0.652224\pi$
$191$	−464050.	−0.920409	−0.460205	+	0.887813i	$0.652224\pi$
$192$	43760.3	0.0856697
$193$	−957625.	−1.85056	−0.925278	−	0.379289i	$-0.876168\pi$
$193$	−957625.	−1.85056	−0.925278	+	0.379289i	$0.876168\pi$
$194$	−583473.	−1.11305
$195$	−258231.	−0.486320
$196$	158397.	0.294514
$197$	−24571.3	−0.0451089	−0.0225545	−	0.999746i	$-0.507180\pi$
$197$	−24571.3	−0.0451089	−0.0225545	+	0.999746i	$0.507180\pi$
$198$	114361.	0.207308
$199$	18997.5	0.0340066	0.0170033	−	0.999855i	$-0.494587\pi$
$199$	18997.5	0.0340066	0.0170033	+	0.999855i	$0.494587\pi$
$200$	−138047.	−0.244035
$201$	−329310.	−0.574930
$202$	−474327.	−0.817899
$203$	503243.	0.857112
$204$	−167547.	−0.281879
$205$	1.28661e6	2.13827
$206$	−581458.	−0.954663
$207$	−287724.	−0.466713
$208$	−85139.2	−0.136449
$209$	−539817.	−0.854833
$210$	507563.	0.794221
$211$	−584038.	−0.903099	−0.451549	−	0.892246i	$-0.649129\pi$
$211$	−584038.	−0.903099	−0.451549	+	0.892246i	$0.649129\pi$
$212$	481370.	0.735596
$213$	−612310.	−0.924745
$214$	573063.	0.855398
$215$	1.54958e6	2.28622
$216$	254260.	0.370804
$217$	−1.61068e6	−2.32199
$218$	−692462.	−0.986858
$219$	218033.	0.307193
$220$	258001.	0.359389
$221$	325977.	0.448959
$222$	−379579.	−0.516916
$223$	851726.	1.14693	0.573466	−	0.819229i	$-0.305598\pi$
$223$	851726.	1.14693	0.573466	+	0.819229i	$0.305598\pi$
$224$	167344.	0.222839
$225$	−277947.	−0.366021
$226$	522028.	0.679865
$227$	821810.	1.05854	0.529270	−	0.848454i	$-0.322466\pi$
$227$	821810.	1.05854	0.529270	+	0.848454i	$0.322466\pi$
$228$	−415896.	−0.529843
$229$	963746.	1.21443	0.607217	−	0.794536i	$-0.292286\pi$
$229$	963746.	1.21443	0.607217	+	0.794536i	$0.292286\pi$
$230$	−649110.	−0.809094
$231$	−387377.	−0.477643
$232$	197082.	0.240396
$233$	904381.	1.09134	0.545672	−	0.837999i	$-0.316275\pi$
$233$	904381.	1.09134	0.545672	+	0.837999i	$0.316275\pi$
$234$	−171422.	−0.204657
$235$	844007.	0.996957
$236$	676518.	0.790678
$237$	−291974.	−0.337655
$238$	−640720.	−0.733206
$239$	304218.	0.344501	0.172250	−	0.985053i	$-0.444896\pi$
$239$	304218.	0.344501	0.172250	+	0.985053i	$0.444896\pi$
$240$	198774.	0.222757
$241$	−1.10700e6	−1.22774	−0.613868	−	0.789409i	$-0.710387\pi$
$241$	−1.10700e6	−1.22774	−0.613868	+	0.789409i	$0.710387\pi$
$242$	447295.	0.490971
$243$	846471.	0.919594
$244$	323231.	0.347567
$245$	719490.	0.765790
$246$	−756535.	−0.797061
$247$	809159.	0.843901
$248$	−630781.	−0.651252
$249$	−841424.	−0.860036
$250$	281411.	0.284768
$251$	514112.	0.515078	0.257539	−	0.966268i	$-0.417088\pi$
$251$	514112.	0.515078	0.257539	+	0.966268i	$0.417088\pi$
$252$	336936.	0.334230
$253$	495407.	0.486588
$254$	−371543.	−0.361347
$255$	−761055.	−0.732936
$256$	65536.0	0.0625000
$257$	−430131.	−0.406226	−0.203113	−	0.979155i	$-0.565106\pi$
$257$	−430131.	−0.406226	−0.203113	+	0.979155i	$0.565106\pi$
$258$	−911162.	−0.852210
$259$	−1.45155e6	−1.34457
$260$	−386730.	−0.354793
$261$	396810.	0.360563
$262$	−259106.	−0.233198
$263$	−901743.	−0.803884	−0.401942	−	0.915665i	$-0.631665\pi$
$263$	−901743.	−0.803884	−0.401942	+	0.915665i	$0.631665\pi$
$264$	−151706.	−0.133965
$265$	2.18654e6	1.91268
$266$	−1.59043e6	−1.37820
$267$	314696.	0.270155
$268$	−493179.	−0.419438
$269$	72361.0	0.0609711
$269$	72361.0	0.0609711
$270$	1.15493e6	0.964157
$271$	787231.	0.651147	0.325573	−	0.945517i	$-0.394443\pi$
$271$	787231.	0.651147	0.325573	+	0.945517i	$0.394443\pi$
$272$	−250921.	−0.205644
$273$	580658.	0.471535
$274$	866343.	0.697130
$275$	478574.	0.381608
$276$	381681.	0.301597
$277$	−52784.1	−0.0413337	−0.0206668	−	0.999786i	$-0.506579\pi$
$277$	−52784.1	−0.0413337	−0.0206668	+	0.999786i	$0.506579\pi$
$278$	−37639.2	−0.0292098
$279$	−1.27003e6	−0.976796
$280$	760133.	0.579421
$281$	−856015.	−0.646719	−0.323359	−	0.946276i	$-0.604812\pi$
$281$	−856015.	−0.646719	−0.323359	+	0.946276i	$0.604812\pi$
$282$	−496281.	−0.371625
$283$	2.66614e6	1.97887	0.989434	−	0.144984i	$-0.0463132\pi$
$283$	2.66614e6	1.97887	0.989434	+	0.144984i	$0.0463132\pi$
$284$	−917003.	−0.674645
$285$	−1.88914e6	−1.37769
$286$	295156.	0.213372
$287$	−2.89308e6	−2.07327
$288$	131952.	0.0937421
$289$	−459140.	−0.323371
$290$	895210.	0.625072
$291$	1.55841e6	1.07882
$292$	326529.	0.224112
$293$	1.71356e6	1.16608	0.583041	−	0.812443i	$-0.301863\pi$
$293$	1.71356e6	1.16608	0.583041	+	0.812443i	$0.301863\pi$
$294$	−423065.	−0.285455
$295$	3.07297e6	2.05591
$296$	−568463.	−0.377114
$297$	−881457.	−0.579843
$298$	1.84600e6	1.20418
$299$	−742591.	−0.480365
$300$	368712.	0.236529
$301$	−3.48439e6	−2.21672
$302$	−295241.	−0.186277
$303$	1.26689e6	0.792742
$304$	−622851.	−0.386545
$305$	1.46822e6	0.903737
$306$	−505212.	−0.308439
$307$	371856.	0.225179	0.112590	−	0.993642i	$-0.464085\pi$
$307$	371856.	0.225179	0.112590	+	0.993642i	$0.464085\pi$
$308$	−580141.	−0.348463
$309$	1.55303e6	0.925299
$310$	−2.86521e6	−1.69337
$311$	2.09992e6	1.23112	0.615561	−	0.788089i	$-0.288929\pi$
$311$	2.09992e6	1.23112	0.615561	+	0.788089i	$0.288929\pi$
$312$	227400.	0.132252
$313$	2.23415e6	1.28900	0.644499	−	0.764606i	$-0.277066\pi$
$313$	2.23415e6	1.28900	0.644499	+	0.764606i	$0.277066\pi$
$314$	−1.54695e6	−0.885429
$315$	1.53047e6	0.869058
$316$	−437264.	−0.246335
$317$	−2.74743e6	−1.53560	−0.767800	−	0.640689i	$-0.778649\pi$
$317$	−2.74743e6	−1.53560	−0.767800	+	0.640689i	$0.778649\pi$
$318$	−1.28570e6	−0.712971
$319$	−683233.	−0.375917
$320$	297686.	0.162511
$321$	−1.53060e6	−0.829087
$322$	1.45959e6	0.784497
$323$	2.38475e6	1.27185
$324$	−178103.	−0.0942561
$325$	−717358.	−0.376728
$326$	336524.	0.175377
$327$	1.84951e6	0.956504
$328$	−1.13300e6	−0.581493
$329$	−1.89784e6	−0.966649
$330$	−689098.	−0.348334
$331$	−1.68448e6	−0.845076	−0.422538	−	0.906345i	$-0.638861\pi$
$331$	−1.68448e6	−0.845076	−0.422538	+	0.906345i	$0.638861\pi$
$332$	−1.26013e6	−0.627436
$333$	−1.14456e6	−0.565623
$334$	−1.51994e6	−0.745523
$335$	−2.24018e6	−1.09061
$336$	−446963.	−0.215985
$337$	−2.36863e6	−1.13611	−0.568057	−	0.822989i	$-0.692305\pi$
$337$	−2.36863e6	−1.13611	−0.568057	+	0.822989i	$0.692305\pi$
$338$	1.04275e6	0.496464
$339$	−1.39429e6	−0.658954
$340$	−1.13977e6	−0.534711
$341$	2.18676e6	1.01839
$342$	−1.25407e6	−0.579769
$343$	1.12879e6	0.518057
$344$	−1.36457e6	−0.621727
$345$	1.73372e6	0.784207
$346$	227457.	0.102143
$347$	−444350.	−0.198108	−0.0990539	−	0.995082i	$-0.531582\pi$
$347$	−444350.	−0.198108	−0.0990539	+	0.995082i	$0.531582\pi$
$348$	−526389.	−0.233002
$349$	−213616.	−0.0938793	−0.0469397	−	0.998898i	$-0.514947\pi$
$349$	−213616.	−0.0938793	−0.0469397	+	0.998898i	$0.514947\pi$
$350$	1.41000e6	0.615244
$351$	1.32126e6	0.572428
$352$	−227197.	−0.0977340
$353$	119070.	0.0508587	0.0254293	−	0.999677i	$-0.491905\pi$
$353$	119070.	0.0508587	0.0254293	+	0.999677i	$0.491905\pi$
$354$	−1.80692e6	−0.766358
$355$	−4.16533e6	−1.75420
$356$	471293.	0.197091
$357$	1.71131e6	0.710654
$358$	−2.20947e6	−0.911131
$359$	4.01362e6	1.64361	0.821807	−	0.569766i	$-0.192966\pi$
$359$	4.01362e6	1.64361	0.821807	+	0.569766i	$0.192966\pi$
$360$	599369.	0.243746
$361$	3.44346e6	1.39068
$362$	−230408.	−0.0924114
$363$	−1.19469e6	−0.475870
$364$	869602.	0.344007
$365$	1.48320e6	0.582731
$366$	−863323.	−0.336876
$367$	−68084.7	−0.0263867	−0.0131933	−	0.999913i	$-0.504200\pi$
$367$	−68084.7	−0.0263867	−0.0131933	+	0.999913i	$0.504200\pi$
$368$	571610.	0.220029
$369$	−2.28121e6	−0.872166
$370$	−2.58214e6	−0.980565
$371$	−4.91666e6	−1.85454
$372$	1.68476e6	0.631221
$373$	−3.15537e6	−1.17430	−0.587150	−	0.809478i	$-0.699750\pi$
$373$	−3.15537e6	−1.17430	−0.587150	+	0.809478i	$0.699750\pi$
$374$	869881.	0.321574
$375$	−751626.	−0.276009
$376$	−743238.	−0.271118
$377$	1.02413e6	0.371110
$378$	−2.59699e6	−0.934847
$379$	−1.99619e6	−0.713844	−0.356922	−	0.934134i	$-0.616174\pi$
$379$	−1.99619e6	−0.713844	−0.356922	+	0.934134i	$0.616174\pi$
$380$	−2.82920e6	−1.00509
$381$	992359.	0.350233
$382$	1.85620e6	0.650828
$383$	−5.51741e6	−1.92193	−0.960966	−	0.276665i	$-0.910771\pi$
$383$	−5.51741e6	−1.92193	−0.960966	+	0.276665i	$0.910771\pi$
$384$	−175041.	−0.0605776
$385$	−2.63519e6	−0.906066
$386$	3.83050e6	1.30854
$387$	−2.74746e6	−0.932511
$388$	2.33389e6	0.787048
$389$	1.33600e6	0.447643	0.223821	−	0.974630i	$-0.428147\pi$
$389$	1.33600e6	0.447643	0.223821	+	0.974630i	$0.428147\pi$
$390$	1.03292e6	0.343880
$391$	−2.18856e6	−0.723962
$392$	−633587.	−0.208253
$393$	692051.	0.226025
$394$	98285.2	0.0318968
$395$	−1.98620e6	−0.640516
$396$	−457444.	−0.146589
$397$	2.23100e6	0.710433	0.355216	−	0.934784i	$-0.384407\pi$
$397$	2.23100e6	0.710433	0.355216	+	0.934784i	$0.384407\pi$
$398$	−75989.8	−0.0240463
$399$	4.24791e6	1.33581
$400$	552188.	0.172559
$401$	3.51596e6	1.09190	0.545950	−	0.837818i	$-0.316169\pi$
$401$	3.51596e6	1.09190	0.545950	+	0.837818i	$0.316169\pi$
$402$	1.31724e6	0.406537
$403$	−3.27784e6	−1.00537
$404$	1.89731e6	0.578342
$405$	−809003.	−0.245083
$406$	−2.01297e6	−0.606070
$407$	1.97072e6	0.589710
$408$	670190.	0.199318
$409$	−4.75138e6	−1.40447	−0.702233	−	0.711947i	$-0.747814\pi$
$409$	−4.75138e6	−1.40447	−0.702233	+	0.711947i	$0.747814\pi$
$410$	−5.14645e6	−1.51199
$411$	−2.31393e6	−0.675687
$412$	2.32583e6	0.675049
$413$	−6.90988e6	−1.99340
$414$	1.15090e6	0.330016
$415$	−5.72392e6	−1.63145
$416$	340557.	0.0964842
$417$	100531.	0.0283114
$418$	2.15927e6	0.604458
$419$	−6.18829e6	−1.72201	−0.861005	−	0.508596i	$-0.830165\pi$
$419$	−6.18829e6	−1.72201	−0.861005	+	0.508596i	$0.830165\pi$
$420$	−2.03025e6	−0.561599
$421$	−6.47076e6	−1.77930	−0.889652	−	0.456639i	$-0.849053\pi$
$421$	−6.47076e6	−1.77930	−0.889652	+	0.456639i	$0.849053\pi$
$422$	2.33615e6	0.638587
$423$	−1.49645e6	−0.406642
$424$	−1.92548e6	−0.520145
$425$	−2.11419e6	−0.567769
$426$	2.44924e6	0.653894
$427$	−3.30144e6	−0.876262
$428$	−2.29225e6	−0.604858
$429$	−788337.	−0.206809
$430$	−6.19832e6	−1.61660
$431$	1.02200e6	0.265008	0.132504	−	0.991182i	$-0.457698\pi$
$431$	1.02200e6	0.265008	0.132504	+	0.991182i	$0.457698\pi$
$432$	−1.01704e6	−0.262198
$433$	1.88514e6	0.483196	0.241598	−	0.970376i	$-0.422328\pi$
$433$	1.88514e6	0.483196	0.241598	+	0.970376i	$0.422328\pi$
$434$	6.44272e6	1.64189
$435$	−2.39103e6	−0.605846
$436$	2.76985e6	0.697814
$437$	−5.43256e6	−1.36082
$438$	−872132.	−0.217219
$439$	2.65719e6	0.658054	0.329027	−	0.944320i	$-0.393279\pi$
$439$	2.65719e6	0.658054	0.329027	+	0.944320i	$0.393279\pi$
$440$	−1.03200e6	−0.254126
$441$	−1.27568e6	−0.312353
$442$	−1.30391e6	−0.317462
$443$	6.37750e6	1.54398	0.771989	−	0.635636i	$-0.219262\pi$
$443$	6.37750e6	1.54398	0.771989	+	0.635636i	$0.219262\pi$
$444$	1.51832e6	0.365515
$445$	2.14077e6	0.512471
$446$	−3.40691e6	−0.811004
$447$	−4.93050e6	−1.16714
$448$	−669377.	−0.157571
$449$	6.85016e6	1.60356	0.801779	−	0.597620i	$-0.203887\pi$
$449$	6.85016e6	1.60356	0.801779	+	0.597620i	$0.203887\pi$
$450$	1.11179e6	0.258816
$451$	3.92782e6	0.909306
$452$	−2.08811e6	−0.480737
$453$	788565.	0.180548
$454$	−3.28724e6	−0.748500
$455$	3.95002e6	0.894480
$456$	1.66358e6	0.374656
$457$	1.99282e6	0.446352	0.223176	−	0.974778i	$-0.428358\pi$
$457$	1.99282e6	0.446352	0.223176	+	0.974778i	$0.428358\pi$
$458$	−3.85499e6	−0.858735
$459$	3.89401e6	0.862710
$460$	2.59644e6	0.572116
$461$	4.94978e6	1.08476	0.542380	−	0.840134i	$-0.317523\pi$
$461$	4.94978e6	1.08476	0.542380	+	0.840134i	$0.317523\pi$
$462$	1.54951e6	0.337745
$463$	−2.37016e6	−0.513837	−0.256919	−	0.966433i	$-0.582707\pi$
$463$	−2.37016e6	−0.513837	−0.256919	+	0.966433i	$0.582707\pi$
$464$	−788327.	−0.169985
$465$	7.65274e6	1.64129
$466$	−3.61752e6	−0.771696
$467$	4.83745e6	1.02642	0.513209	−	0.858264i	$-0.328456\pi$
$467$	4.83745e6	1.02642	0.513209	+	0.858264i	$0.328456\pi$
$468$	685686.	0.144714
$469$	5.03727e6	1.05746
$470$	−3.37603e6	−0.704955
$471$	4.13179e6	0.858194
$472$	−2.70607e6	−0.559094
$473$	4.73062e6	0.972221
$474$	1.16790e6	0.238758
$475$	−5.24797e6	−1.06723
$476$	2.56288e6	0.518455
$477$	−3.87682e6	−0.780152
$478$	−1.21687e6	−0.243599
$479$	2.56529e6	0.510856	0.255428	−	0.966828i	$-0.417784\pi$
$479$	2.56529e6	0.510856	0.255428	+	0.966828i	$0.417784\pi$
$480$	−795094.	−0.157513
$481$	−2.95401e6	−0.582169
$482$	4.42800e6	0.868140
$483$	−3.89844e6	−0.760367
$484$	−1.78918e6	−0.347169
$485$	1.06013e7	2.04647
$486$	−3.38588e6	−0.650251
$487$	8.19420e6	1.56561	0.782806	−	0.622266i	$-0.213788\pi$
$487$	8.19420e6	1.56561	0.782806	+	0.622266i	$0.213788\pi$
$488$	−1.29292e6	−0.245767
$489$	−898828.	−0.169983
$490$	−2.87796e6	−0.541495
$491$	−1.98990e6	−0.372501	−0.186250	−	0.982502i	$-0.559634\pi$
$491$	−1.98990e6	−0.372501	−0.186250	+	0.982502i	$0.559634\pi$
$492$	3.02614e6	0.563607
$493$	3.01831e6	0.559303
$494$	−3.23664e6	−0.596728
$495$	−2.07786e6	−0.381157
$496$	2.52312e6	0.460505
$497$	9.36617e6	1.70087
$498$	3.36570e6	0.608137
$499$	−7.29203e6	−1.31098	−0.655491	−	0.755203i	$-0.727538\pi$
$499$	−7.29203e6	−1.31098	−0.655491	+	0.755203i	$0.727538\pi$
$500$	−1.12565e6	−0.201362
$501$	4.05964e6	0.722592
$502$	−2.05645e6	−0.364215
$503$	6.39026e6	1.12616	0.563078	−	0.826404i	$-0.309617\pi$
$503$	6.39026e6	1.12616	0.563078	+	0.826404i	$0.309617\pi$
$504$	−1.34774e6	−0.236336
$505$	8.61820e6	1.50379
$506$	−1.98163e6	−0.344069
$507$	−2.78509e6	−0.481193
$508$	1.48617e6	0.255511
$509$	−7.01692e6	−1.20047	−0.600236	−	0.799823i	$-0.704927\pi$
$509$	−7.01692e6	−1.20047	−0.600236	+	0.799823i	$0.704927\pi$
$510$	3.04422e6	0.518264
$511$	−3.33513e6	−0.565016
$512$	−262144.	−0.0441942
$513$	9.66592e6	1.62162
$514$	1.72052e6	0.287245
$515$	1.05647e7	1.75525
$516$	3.64465e6	0.602603
$517$	2.57662e6	0.423959
$518$	5.80621e6	0.950755
$519$	−607518.	−0.0990013
$520$	1.54692e6	0.250876
$521$	−7.81316e6	−1.26105	−0.630525	−	0.776169i	$-0.717160\pi$
$521$	−7.81316e6	−1.26105	−0.630525	+	0.776169i	$0.717160\pi$
$522$	−1.58724e6	−0.254957
$523$	−474750.	−0.0758945	−0.0379473	−	0.999280i	$-0.512082\pi$
$523$	−474750.	−0.0758945	−0.0379473	+	0.999280i	$0.512082\pi$
$524$	1.03643e6	0.164896
$525$	−3.76598e6	−0.596320
$526$	3.60697e6	0.568432
$527$	−9.66042e6	−1.51520
$528$	606824.	0.0947279
$529$	−1.45071e6	−0.225394
$530$	−8.74617e6	−1.35247
$531$	−5.44848e6	−0.838570
$532$	6.36173e6	0.974533
$533$	−5.88760e6	−0.897678
$534$	−1.25878e6	−0.191028
$535$	−1.04122e7	−1.57274
$536$	1.97272e6	0.296587
$537$	5.90131e6	0.883106
$538$	−289444.	−0.0431131
$539$	2.19649e6	0.325654
$540$	−4.61974e6	−0.681762
$541$	−1.01655e7	−1.49326	−0.746632	−	0.665237i	$-0.768330\pi$
$541$	−1.01655e7	−1.49326	−0.746632	+	0.665237i	$0.768330\pi$
$542$	−3.14892e6	−0.460430
$543$	615400.	0.0895690
$544$	1.00369e6	0.145412
$545$	1.25816e7	1.81444
$546$	−2.32263e6	−0.333426
$547$	8.37810e6	1.19723	0.598614	−	0.801037i	$-0.295718\pi$
$547$	8.37810e6	1.19723	0.598614	+	0.801037i	$0.295718\pi$
$548$	−3.46537e6	−0.492945
$549$	−2.60321e6	−0.368619
$550$	−1.91430e6	−0.269838
$551$	7.49223e6	1.05131
$552$	−1.52672e6	−0.213262
$553$	4.46617e6	0.621044
$554$	211137.	0.0292273
$555$	6.89669e6	0.950404
$556$	150557.	0.0206544
$557$	8.54939e6	1.16761	0.583804	−	0.811895i	$-0.301564\pi$
$557$	8.54939e6	1.16761	0.583804	+	0.811895i	$0.301564\pi$
$558$	5.08012e6	0.690699
$559$	−7.09096e6	−0.959789
$560$	−3.04053e6	−0.409713
$561$	−2.32338e6	−0.311683
$562$	3.42406e6	0.457299
$563$	25029.6	0.00332799	0.00166400	−	0.999999i	$-0.499470\pi$
$563$	25029.6	0.00332799	0.00166400	+	0.999999i	$0.499470\pi$
$564$	1.98513e6	0.262779
$565$	−9.48490e6	−1.25000
$566$	−1.06646e7	−1.39927
$567$	1.81913e6	0.237632
$568$	3.66801e6	0.477046
$569$	−675924.	−0.0875220	−0.0437610	−	0.999042i	$-0.513934\pi$
$569$	−675924.	−0.0875220	−0.0437610	+	0.999042i	$0.513934\pi$
$570$	7.55654e6	0.974173
$571$	1.29887e7	1.66715	0.833574	−	0.552407i	$-0.186291\pi$
$571$	1.29887e7	1.66715	0.833574	+	0.552407i	$0.186291\pi$
$572$	−1.18062e6	−0.150877
$573$	−4.95775e6	−0.630809
$574$	1.15723e7	1.46602
$575$	4.81622e6	0.607487
$576$	−527808.	−0.0662857
$577$	−3.02708e6	−0.378516	−0.189258	−	0.981927i	$-0.560608\pi$
$577$	−3.02708e6	−0.378516	−0.189258	+	0.981927i	$0.560608\pi$
$578$	1.83656e6	0.228658
$579$	−1.02309e7	−1.26829
$580$	−3.58084e6	−0.441993
$581$	1.28708e7	1.58185
$582$	−6.23363e6	−0.762840
$583$	6.67516e6	0.813374
$584$	−1.30612e6	−0.158471
$585$	3.11461e6	0.376283
$586$	−6.85422e6	−0.824544
$587$	−1.27023e7	−1.52155	−0.760775	−	0.649015i	$-0.775181\pi$
$587$	−1.27023e7	−1.52155	−0.760775	+	0.649015i	$0.775181\pi$
$588$	1.69226e6	0.201847
$589$	−2.39797e7	−2.84810
$590$	−1.22919e7	−1.45374
$591$	−262512.	−0.0309157
$592$	2.27385e6	0.266660
$593$	−5.95457e6	−0.695366	−0.347683	−	0.937612i	$-0.613031\pi$
$593$	−5.95457e6	−0.695366	−0.347683	+	0.937612i	$0.613031\pi$
$594$	3.52583e6	0.410011
$595$	1.16414e7	1.34808
$596$	−7.38399e6	−0.851482
$597$	202962.	0.0233066
$598$	2.97036e6	0.339670
$599$	93939.8	0.0106975	0.00534875	−	0.999986i	$-0.498297\pi$
$599$	93939.8	0.0106975	0.00534875	+	0.999986i	$0.498297\pi$
$600$	−1.47485e6	−0.167251
$601$	1.16122e7	1.31138	0.655691	−	0.755029i	$-0.272377\pi$
$601$	1.16122e7	1.31138	0.655691	+	0.755029i	$0.272377\pi$
$602$	1.39376e7	1.56746
$603$	3.97192e6	0.444843
$604$	1.18097e6	0.131718
$605$	−8.12705e6	−0.902702
$606$	−5.06755e6	−0.560553
$607$	−3.66792e6	−0.404063	−0.202031	−	0.979379i	$-0.564754\pi$
$607$	−3.66792e6	−0.404063	−0.202031	+	0.979379i	$0.564754\pi$
$608$	2.49141e6	0.273329
$609$	5.37648e6	0.587428
$610$	−5.87288e6	−0.639038
$611$	−3.86222e6	−0.418537
$612$	2.02085e6	0.218100
$613$	7.87305e6	0.846237	0.423118	−	0.906074i	$-0.360935\pi$
$613$	7.87305e6	0.846237	0.423118	+	0.906074i	$0.360935\pi$
$614$	−1.48742e6	−0.159226
$615$	1.37457e7	1.46548
$616$	2.32056e6	0.246400
$617$	6.17776e6	0.653308	0.326654	−	0.945144i	$-0.394079\pi$
$617$	6.17776e6	0.653308	0.326654	+	0.945144i	$0.394079\pi$
$618$	−6.21210e6	−0.654285
$619$	6.01288e6	0.630748	0.315374	−	0.948967i	$-0.397870\pi$
$619$	6.01288e6	0.630748	0.315374	+	0.948967i	$0.397870\pi$
$620$	1.14609e7	1.19740
$621$	−8.87072e6	−0.923060
$622$	−8.39967e6	−0.870535
$623$	−4.81373e6	−0.496891
$624$	−909598.	−0.0935165
$625$	−1.18536e7	−1.21381
$626$	−8.93661e6	−0.911459
$627$	−5.76723e6	−0.585866
$628$	6.18782e6	0.626093
$629$	−8.70602e6	−0.877391
$630$	−6.12189e6	−0.614517
$631$	−1.47485e7	−1.47460	−0.737300	−	0.675566i	$-0.763899\pi$
$631$	−1.47485e7	−1.47460	−0.737300	+	0.675566i	$0.763899\pi$
$632$	1.74906e6	0.174185
$633$	−6.23967e6	−0.618945
$634$	1.09897e7	1.08583
$635$	6.75068e6	0.664374
$636$	5.14280e6	0.504146
$637$	−3.29243e6	−0.321490
$638$	2.73293e6	0.265814
$639$	7.38528e6	0.715508
$640$	−1.19074e6	−0.114913
$641$	1.98558e6	0.190872	0.0954360	−	0.995436i	$-0.469575\pi$
$641$	1.98558e6	0.190872	0.0954360	+	0.995436i	$0.469575\pi$
$642$	6.12241e6	0.586253
$643$	7.27451e6	0.693867	0.346934	−	0.937890i	$-0.387223\pi$
$643$	7.27451e6	0.693867	0.346934	+	0.937890i	$0.387223\pi$
$644$	−5.83836e6	−0.554723
$645$	1.65552e7	1.56688
$646$	−9.53898e6	−0.899334
$647$	7.90667e6	0.742563	0.371281	−	0.928520i	$-0.378919\pi$
$647$	7.90667e6	0.742563	0.371281	+	0.928520i	$0.378919\pi$
$648$	712413.	0.0666491
$649$	9.38128e6	0.874280
$650$	2.86943e6	0.266387
$651$	−1.72080e7	−1.59139
$652$	−1.34610e6	−0.124010
$653$	1.67178e7	1.53425	0.767124	−	0.641499i	$-0.221687\pi$
$653$	1.67178e7	1.53425	0.767124	+	0.641499i	$0.221687\pi$
$654$	−7.39803e6	−0.676350
$655$	4.70778e6	0.428759
$656$	4.53199e6	0.411178
$657$	−2.62977e6	−0.237686
$658$	7.59134e6	0.683524
$659$	6.78003e6	0.608160	0.304080	−	0.952646i	$-0.401651\pi$
$659$	6.78003e6	0.608160	0.304080	+	0.952646i	$0.401651\pi$
$660$	2.75639e6	0.246310
$661$	−8.57343e6	−0.763222	−0.381611	−	0.924323i	$-0.624631\pi$
$661$	−8.57343e6	−0.763222	−0.381611	+	0.924323i	$0.624631\pi$
$662$	6.73792e6	0.597559
$663$	3.48263e6	0.307697
$664$	5.04051e6	0.443664
$665$	2.88971e7	2.53396
$666$	4.57823e6	0.399956
$667$	−6.87585e6	−0.598428
$668$	6.07977e6	0.527164
$669$	9.09956e6	0.786059
$670$	8.96072e6	0.771181
$671$	4.48224e6	0.384317
$672$	1.78785e6	0.152724
$673$	6.56839e6	0.559012	0.279506	−	0.960144i	$-0.409829\pi$
$673$	6.56839e6	0.559012	0.279506	+	0.960144i	$0.409829\pi$
$674$	9.47451e6	0.803354
$675$	−8.56930e6	−0.723913
$676$	−4.17099e6	−0.351053
$677$	−7.98281e6	−0.669397	−0.334699	−	0.942325i	$-0.608635\pi$
$677$	−7.98281e6	−0.669397	−0.334699	+	0.942325i	$0.608635\pi$
$678$	5.57717e6	0.465951
$679$	−2.38381e7	−1.98425
$680$	4.55907e6	0.378098
$681$	8.77994e6	0.725478
$682$	−8.74703e6	−0.720112
$683$	−1.79637e7	−1.47348	−0.736740	−	0.676176i	$-0.763636\pi$
$683$	−1.79637e7	−1.47348	−0.736740	+	0.676176i	$0.763636\pi$
$684$	5.01626e6	0.409959
$685$	−1.57409e7	−1.28175
$686$	−4.51516e6	−0.366322
$687$	1.02963e7	0.832321
$688$	5.45828e6	0.439627
$689$	−1.00057e7	−0.802973
$690$	−6.93488e6	−0.554518
$691$	1.11833e7	0.890996	0.445498	−	0.895283i	$-0.353027\pi$
$691$	1.11833e7	0.890996	0.445498	+	0.895283i	$0.353027\pi$
$692$	−909828.	−0.0722260
$693$	4.67228e6	0.369570
$694$	1.77740e6	0.140083
$695$	683879.	0.0537053
$696$	2.10556e6	0.164757
$697$	−1.73519e7	−1.35290
$698$	854464.	0.0663827
$699$	9.66210e6	0.747960
$700$	−5.63998e6	−0.435043
$701$	−1.08396e7	−0.833144	−0.416572	−	0.909103i	$-0.636769\pi$
$701$	−1.08396e7	−0.833144	−0.416572	+	0.909103i	$0.636769\pi$
$702$	−5.28504e6	−0.404768
$703$	−2.16106e7	−1.64922
$704$	908788.	0.0691084
$705$	9.01709e6	0.683272
$706$	−476279.	−0.0359625
$707$	−1.93789e7	−1.45808
$708$	7.22770e6	0.541897
$709$	2.46360e6	0.184058	0.0920292	−	0.995756i	$-0.470665\pi$
$709$	2.46360e6	0.184058	0.0920292	+	0.995756i	$0.470665\pi$
$710$	1.66613e7	1.24040
$711$	3.52160e6	0.261256
$712$	−1.88517e6	−0.139364
$713$	2.20069e7	1.62119
$714$	−6.84524e6	−0.502508
$715$	−5.36279e6	−0.392306
$716$	8.83789e6	0.644267
$717$	3.25016e6	0.236106
$718$	−1.60545e7	−1.16221
$719$	−6.46308e6	−0.466248	−0.233124	−	0.972447i	$-0.574895\pi$
$719$	−6.46308e6	−0.466248	−0.233124	+	0.972447i	$0.574895\pi$
$720$	−2.39748e6	−0.172355
$721$	−2.37558e7	−1.70189
$722$	−1.37738e7	−0.983358
$723$	−1.18268e7	−0.841437
$724$	921631.	0.0653447
$725$	−6.64222e6	−0.469319
$726$	4.77875e6	0.336491
$727$	−5.09622e6	−0.357612	−0.178806	−	0.983884i	$-0.557223\pi$
$727$	−5.09622e6	−0.357612	−0.178806	+	0.983884i	$0.557223\pi$
$728$	−3.47841e6	−0.243250
$729$	1.17483e7	0.818763
$730$	−5.93281e6	−0.412053
$731$	−2.08984e7	−1.44650
$732$	3.45329e6	0.238208
$733$	−1.92779e7	−1.32525	−0.662626	−	0.748950i	$-0.730558\pi$
$733$	−1.92779e7	−1.32525	−0.662626	+	0.748950i	$0.730558\pi$
$734$	272339.	0.0186582
$735$	7.68679e6	0.524840
$736$	−2.28644e6	−0.155584
$737$	−6.83891e6	−0.463787
$738$	9.12483e6	0.616714
$739$	2.48384e7	1.67307	0.836533	−	0.547917i	$-0.184579\pi$
$739$	2.48384e7	1.67307	0.836533	+	0.547917i	$0.184579\pi$
$740$	1.03286e7	0.693364
$741$	8.64479e6	0.578374
$742$	1.96666e7	1.31136
$743$	1.86619e7	1.24018	0.620090	−	0.784531i	$-0.287096\pi$
$743$	1.86619e7	1.24018	0.620090	+	0.784531i	$0.287096\pi$
$744$	−6.73905e6	−0.446340
$745$	−3.35405e7	−2.21401
$746$	1.26215e7	0.830355
$747$	1.01487e7	0.665441
$748$	−3.47953e6	−0.227387
$749$	2.34128e7	1.52493
$750$	3.00650e6	0.195168
$751$	1.82226e7	1.17899	0.589495	−	0.807772i	$-0.299327\pi$
$751$	1.82226e7	1.17899	0.589495	+	0.807772i	$0.299327\pi$
$752$	2.97295e6	0.191709
$753$	5.49260e6	0.353013
$754$	−4.09653e6	−0.262415
$755$	5.36433e6	0.342490
$756$	1.03879e7	0.661036
$757$	−1.13560e7	−0.720254	−0.360127	−	0.932903i	$-0.617267\pi$
$757$	−1.13560e7	−0.720254	−0.360127	+	0.932903i	$0.617267\pi$
$758$	7.98475e6	0.504764
$759$	5.29277e6	0.333486
$760$	1.13168e7	0.710704
$761$	−2.06212e6	−0.129078	−0.0645389	−	0.997915i	$-0.520558\pi$
$761$	−2.06212e6	−0.129078	−0.0645389	+	0.997915i	$0.520558\pi$
$762$	−3.96944e6	−0.247652
$763$	−2.82909e7	−1.75928
$764$	−7.42479e6	−0.460205
$765$	9.17935e6	0.567098
$766$	2.20696e7	1.35901
$767$	−1.40621e7	−0.863099
$768$	700165.	0.0428348
$769$	−777529.	−0.0474134	−0.0237067	−	0.999719i	$-0.507547\pi$
$769$	−777529.	−0.0474134	−0.0237067	+	0.999719i	$0.507547\pi$
$770$	1.05408e7	0.640686
$771$	−4.59537e6	−0.278410
$772$	−1.53220e7	−0.925278
$773$	2.74816e7	1.65422	0.827111	−	0.562038i	$-0.189983\pi$
$773$	2.74816e7	1.65422	0.827111	+	0.562038i	$0.189983\pi$
$774$	1.09898e7	0.659385
$775$	2.12591e7	1.27143
$776$	−9.33556e6	−0.556527
$777$	−1.55079e7	−0.921511
$778$	−5.34399e6	−0.316531
$779$	−4.30719e7	−2.54302
$780$	−4.13169e6	−0.243160
$781$	−1.27161e7	−0.745978
$782$	8.75422e6	0.511918
$783$	1.22339e7	0.713118
$784$	2.53435e6	0.147257
$785$	2.81071e7	1.62795
$786$	−2.76820e6	−0.159824
$787$	−1.93429e7	−1.11323	−0.556615	−	0.830771i	$-0.687900\pi$
$787$	−1.93429e7	−1.11323	−0.556615	+	0.830771i	$0.687900\pi$
$788$	−393141.	−0.0225545
$789$	−9.63391e6	−0.550948
$790$	7.94479e6	0.452913
$791$	2.13277e7	1.21200
$792$	1.82978e6	0.103654
$793$	−6.71866e6	−0.379402
$794$	−8.92399e6	−0.502352
$795$	2.33603e7	1.31087
$796$	303959.	0.0170033
$797$	1.27455e7	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$797$	1.27455e7	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$798$	−1.69917e7	−0.944558
$799$	−1.13827e7	−0.630781
$800$	−2.20875e6	−0.122017
$801$	−3.79565e6	−0.209028
$802$	−1.40638e7	−0.772090
$803$	4.52798e6	0.247808
$804$	−5.26896e6	−0.287465
$805$	−2.65197e7	−1.44238
$806$	1.31114e7	0.710903
$807$	773081.	0.0417870
$808$	−7.58924e6	−0.408950
$809$	−1.25900e7	−0.676322	−0.338161	−	0.941088i	$-0.609805\pi$
$809$	−1.25900e7	−0.676322	−0.338161	+	0.941088i	$0.609805\pi$
$810$	3.23601e6	0.173300
$811$	2.21540e7	1.18277	0.591386	−	0.806389i	$-0.298581\pi$
$811$	2.21540e7	1.18277	0.591386	+	0.806389i	$0.298581\pi$
$812$	8.05189e6	0.428556
$813$	8.41051e6	0.446268
$814$	−7.88287e6	−0.416988
$815$	−6.11441e6	−0.322449
$816$	−2.68076e6	−0.140939
$817$	−5.18752e7	−2.71897
$818$	1.90055e7	0.993108
$819$	−7.00352e6	−0.364843
$820$	2.05858e7	1.06914
$821$	−3.15446e7	−1.63330	−0.816651	−	0.577131i	$-0.804172\pi$
$821$	−3.15446e7	−1.63330	−0.816651	+	0.577131i	$0.804172\pi$
$822$	9.25572e6	0.477783
$823$	3.54094e7	1.82230	0.911148	−	0.412079i	$-0.135197\pi$
$823$	3.54094e7	1.82230	0.911148	+	0.412079i	$0.135197\pi$
$824$	−9.30333e6	−0.477332
$825$	5.11292e6	0.261538
$826$	2.76395e7	1.40955
$827$	−6.84017e6	−0.347779	−0.173889	−	0.984765i	$-0.555634\pi$
$827$	−6.84017e6	−0.347779	−0.173889	+	0.984765i	$0.555634\pi$
$828$	−4.60358e6	−0.233357
$829$	−1.68528e6	−0.0851697	−0.0425848	−	0.999093i	$-0.513559\pi$
$829$	−1.68528e6	−0.0851697	−0.0425848	+	0.999093i	$0.513559\pi$
$830$	2.28957e7	1.15361
$831$	−563928.	−0.0283283
$832$	−1.36223e6	−0.0682246
$833$	−9.70340e6	−0.484520
$834$	−402125.	−0.0200192
$835$	2.76163e7	1.37072
$836$	−8.63708e6	−0.427416
$837$	−3.91559e7	−1.93190
$838$	2.47532e7	1.21765
$839$	2.08944e7	1.02477	0.512384	−	0.858757i	$-0.328763\pi$
$839$	2.08944e7	1.02477	0.512384	+	0.858757i	$0.328763\pi$
$840$	8.12100e6	0.397111
$841$	−1.10284e7	−0.537679
$842$	2.58830e7	1.25816
$843$	−9.14537e6	−0.443233
$844$	−9.34461e6	−0.451549
$845$	−1.89460e7	−0.912801
$846$	5.98582e6	0.287540
$847$	1.82745e7	0.875259
$848$	7.70192e6	0.367798
$849$	2.84841e7	1.35623
$850$	8.45677e6	0.401474
$851$	1.98327e7	0.938768
$852$	−9.79695e6	−0.462373
$853$	1.86077e7	0.875628	0.437814	−	0.899066i	$-0.355753\pi$
$853$	1.86077e7	0.875628	0.437814	+	0.899066i	$0.355753\pi$
$854$	1.32058e7	0.619611
$855$	2.27855e7	1.06597
$856$	9.16901e6	0.427699
$857$	−9.57315e6	−0.445249	−0.222624	−	0.974904i	$-0.571462\pi$
$857$	−9.57315e6	−0.445249	−0.222624	+	0.974904i	$0.571462\pi$
$858$	3.15335e6	0.146236
$859$	1.29912e7	0.600713	0.300356	−	0.953827i	$-0.402894\pi$
$859$	1.29912e7	0.600713	0.300356	+	0.953827i	$0.402894\pi$
$860$	2.47933e7	1.14311
$861$	−3.09087e7	−1.42093
$862$	−4.08801e6	−0.187389
$863$	2.11178e7	0.965211	0.482606	−	0.875838i	$-0.339690\pi$
$863$	2.11178e7	0.965211	0.482606	+	0.875838i	$0.339690\pi$
$864$	4.06817e6	0.185402
$865$	−4.13274e6	−0.187801
$866$	−7.54055e6	−0.341671
$867$	−4.90530e6	−0.221625
$868$	−2.57709e7	−1.16099
$869$	−6.06354e6	−0.272381
$870$	9.56413e6	0.428398
$871$	1.02512e7	0.457856
$872$	−1.10794e7	−0.493429
$873$	−1.87965e7	−0.834720
$874$	2.17302e7	0.962246
$875$	1.14972e7	0.507659
$876$	3.48853e6	0.153597
$877$	−2.06495e7	−0.906589	−0.453295	−	0.891361i	$-0.649751\pi$
$877$	−2.06495e7	−0.906589	−0.453295	+	0.891361i	$0.649751\pi$
$878$	−1.06288e7	−0.465315
$879$	1.83070e7	0.799183
$880$	4.12801e6	0.179694
$881$	2.77202e7	1.20325	0.601627	−	0.798777i	$-0.294519\pi$
$881$	2.77202e7	1.20325	0.601627	+	0.798777i	$0.294519\pi$
$882$	5.10273e6	0.220867
$883$	−3.16009e7	−1.36395	−0.681974	−	0.731376i	$-0.738878\pi$
$883$	−3.16009e7	−1.36395	−0.681974	+	0.731376i	$0.738878\pi$
$884$	5.21563e6	0.224479
$885$	3.28306e7	1.40903
$886$	−2.55100e7	−1.09176
$887$	−3.99404e7	−1.70452	−0.852262	−	0.523115i	$-0.824770\pi$
$887$	−3.99404e7	−1.70452	−0.852262	+	0.523115i	$0.824770\pi$
$888$	−6.07326e6	−0.258458
$889$	−1.51796e7	−0.644177
$890$	−8.56306e6	−0.362372
$891$	−2.46976e6	−0.104222
$892$	1.36276e7	0.573466
$893$	−2.82548e7	−1.18567
$894$	1.97220e7	0.825291
$895$	4.01446e7	1.67521
$896$	2.67751e6	0.111419
$897$	−7.93359e6	−0.329222
$898$	−2.74006e7	−1.13389
$899$	−3.03504e7	−1.25247
$900$	−4.44716e6	−0.183011
$901$	−2.94888e7	−1.21017
$902$	−1.57113e7	−0.642976
$903$	−3.72260e7	−1.51924
$904$	8.35245e6	0.339933
$905$	4.18635e6	0.169908
$906$	−3.15426e6	−0.127666
$907$	2.22380e7	0.897589	0.448795	−	0.893635i	$-0.351853\pi$
$907$	2.22380e7	0.897589	0.448795	+	0.893635i	$0.351853\pi$
$908$	1.31490e7	0.529270
$909$	−1.52804e7	−0.613373
$910$	−1.58001e7	−0.632493
$911$	−7.75511e6	−0.309594	−0.154797	−	0.987946i	$-0.549472\pi$
$911$	−7.75511e6	−0.309594	−0.154797	+	0.987946i	$0.549472\pi$
$912$	−6.65434e6	−0.264922
$913$	−1.74742e7	−0.693778
$914$	−7.97127e6	−0.315618
$915$	1.56860e7	0.619383
$916$	1.54199e7	0.607217
$917$	−1.05859e7	−0.415724
$918$	−1.55760e7	−0.610028
$919$	−2.30155e7	−0.898940	−0.449470	−	0.893295i	$-0.648387\pi$
$919$	−2.30155e7	−0.898940	−0.449470	+	0.893295i	$0.648387\pi$
$920$	−1.03858e7	−0.404547
$921$	3.97278e6	0.154328
$922$	−1.97991e7	−0.767041
$923$	1.90608e7	0.736438
$924$	−6.19803e6	−0.238822
$925$	1.91588e7	0.736231
$926$	9.48064e6	0.363338
$927$	−1.87316e7	−0.715937
$928$	3.15331e6	0.120198
$929$	1.62634e7	0.618261	0.309130	−	0.951020i	$-0.399962\pi$
$929$	1.62634e7	0.618261	0.309130	+	0.951020i	$0.399962\pi$
$930$	−3.06110e7	−1.16057
$931$	−2.40863e7	−0.910745
$932$	1.44701e7	0.545672
$933$	2.24348e7	0.843759
$934$	−1.93498e7	−0.725788
$935$	−1.58051e7	−0.591248
$936$	−2.74274e6	−0.102328
$937$	1.21689e7	0.452796	0.226398	−	0.974035i	$-0.427305\pi$
$937$	1.21689e7	0.452796	0.226398	+	0.974035i	$0.427305\pi$
$938$	−2.01491e7	−0.747736
$939$	2.38689e7	0.883424
$940$	1.35041e7	0.498479
$941$	4.43110e7	1.63131	0.815656	−	0.578537i	$-0.196376\pi$
$941$	4.43110e7	1.63131	0.815656	+	0.578537i	$0.196376\pi$
$942$	−1.65271e7	−0.606835
$943$	3.95284e7	1.44754
$944$	1.08243e7	0.395339
$945$	4.71855e7	1.71881
$946$	−1.89225e7	−0.687464
$947$	2.88978e7	1.04711	0.523553	−	0.851993i	$-0.324606\pi$
$947$	2.88978e7	1.04711	0.523553	+	0.851993i	$0.324606\pi$
$948$	−4.67158e6	−0.168828
$949$	−6.78721e6	−0.244639
$950$	2.09919e7	0.754644
$951$	−2.93526e7	−1.05244
$952$	−1.02515e7	−0.366603
$953$	−3.89881e7	−1.39059	−0.695297	−	0.718723i	$-0.744727\pi$
$953$	−3.89881e7	−1.39059	−0.695297	+	0.718723i	$0.744727\pi$
$954$	1.55073e7	0.551651
$955$	−3.37258e7	−1.19662
$956$	4.86749e6	0.172250
$957$	−7.29944e6	−0.257638
$958$	−1.02612e7	−0.361230
$959$	3.53949e7	1.24278
$960$	3.18038e6	0.111378
$961$	6.85105e7	2.39303
$962$	1.18160e7	0.411655
$963$	1.84611e7	0.641494
$964$	−1.77120e7	−0.613868
$965$	−6.95976e7	−2.40589
$966$	1.55938e7	0.537661
$967$	−4.16060e7	−1.43084	−0.715418	−	0.698697i	$-0.753764\pi$
$967$	−4.16060e7	−1.43084	−0.715418	+	0.698697i	$0.753764\pi$
$968$	7.15673e6	0.245486
$969$	2.54778e7	0.871672
$970$	−4.24052e7	−1.44707
$971$	3.84903e7	1.31010	0.655049	−	0.755587i	$-0.272648\pi$
$971$	3.84903e7	1.31010	0.655049	+	0.755587i	$0.272648\pi$
$972$	1.35435e7	0.459797
$973$	−1.53777e6	−0.0520726
$974$	−3.27768e7	−1.10706
$975$	−7.66402e6	−0.258193
$976$	5.17170e6	0.173783
$977$	−4.02738e7	−1.34985	−0.674926	−	0.737885i	$-0.735825\pi$
$977$	−4.02738e7	−1.34985	−0.674926	+	0.737885i	$0.735825\pi$
$978$	3.59531e6	0.120196
$979$	6.53541e6	0.217930
$980$	1.15118e7	0.382895
$981$	−2.23075e7	−0.740081
$982$	7.95959e6	0.263398
$983$	5.19848e7	1.71590	0.857951	−	0.513731i	$-0.171737\pi$
$983$	5.19848e7	1.71590	0.857951	+	0.513731i	$0.171737\pi$
$984$	−1.21046e7	−0.398530
$985$	−1.78577e6	−0.0586457
$986$	−1.20733e7	−0.395487
$987$	−2.02758e7	−0.662500
$988$	1.29465e7	0.421951
$989$	4.76075e7	1.54769
$990$	8.31145e6	0.269519
$991$	2.23612e6	0.0723287	0.0361644	−	0.999346i	$-0.488486\pi$
$991$	2.23612e6	0.0723287	0.0361644	+	0.999346i	$0.488486\pi$
$992$	−1.00925e7	−0.325626
$993$	−1.79964e7	−0.579179
$994$	−3.74647e7	−1.20270
$995$	1.38068e6	0.0442116
$996$	−1.34628e7	−0.430018
$997$	4.05121e7	1.29077	0.645383	−	0.763859i	$-0.276698\pi$
$997$	4.05121e7	1.29077	0.645383	+	0.763859i	$0.276698\pi$
$998$	2.91681e7	0.927005
$999$	−3.52875e7	−1.11868

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.6.a.b.1.18	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.6.a.b.1.18	✓	27	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:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +10.6837 q^{3} +16.0000 q^{4} +72.6773 q^{5} -42.7347 q^{6} -163.422 q^{7} -64.0000 q^{8} -128.859 q^{9} -290.709 q^{10} +221.872 q^{11} +170.939 q^{12} -332.575 q^{13} +653.689 q^{14} +776.459 q^{15} +256.000 q^{16} -980.161 q^{17} +515.437 q^{18} -2433.01 q^{19} +1162.84 q^{20} -1745.95 q^{21} -887.488 q^{22} +2232.85 q^{23} -683.754 q^{24} +2156.98 q^{25} +1330.30 q^{26} -3972.82 q^{27} -2614.75 q^{28} -3079.40 q^{29} -3105.84 q^{30} +9855.95 q^{31} -1024.00 q^{32} +2370.41 q^{33} +3920.65 q^{34} -11877.1 q^{35} -2061.75 q^{36} +8882.23 q^{37} +9732.05 q^{38} -3553.12 q^{39} -4651.34 q^{40} +17703.1 q^{41} +6983.79 q^{42} +21321.4 q^{43} +3549.95 q^{44} -9365.14 q^{45} -8931.41 q^{46} +11613.1 q^{47} +2735.02 q^{48} +9899.80 q^{49} -8627.93 q^{50} -10471.7 q^{51} -5321.20 q^{52} +30085.6 q^{53} +15891.3 q^{54} +16125.0 q^{55} +10459.0 q^{56} -25993.5 q^{57} +12317.6 q^{58} +42282.4 q^{59} +12423.3 q^{60} +20201.9 q^{61} -39423.8 q^{62} +21058.5 q^{63} +4096.00 q^{64} -24170.6 q^{65} -9481.62 q^{66} -30823.7 q^{67} -15682.6 q^{68} +23855.0 q^{69} +47508.3 q^{70} -57312.7 q^{71} +8247.00 q^{72} +20408.1 q^{73} -35528.9 q^{74} +23044.5 q^{75} -38928.2 q^{76} -36258.8 q^{77} +14212.5 q^{78} -27329.0 q^{79} +18605.4 q^{80} -11131.5 q^{81} -70812.4 q^{82} -78758.0 q^{83} -27935.2 q^{84} -71235.4 q^{85} -85285.6 q^{86} -32899.3 q^{87} -14199.8 q^{88} +29455.8 q^{89} +37460.6 q^{90} +54350.1 q^{91} +35725.6 q^{92} +105298. q^{93} -46452.4 q^{94} -176825. q^{95} -10940.1 q^{96} +145868. q^{97} -39599.2 q^{98} -28590.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 108 q^{2} + 33 q^{3} + 432 q^{4} + 139 q^{5} - 132 q^{6} - 25 q^{7} - 1728 q^{8} + 2346 q^{9} - 556 q^{10} + 1241 q^{11} + 528 q^{12} - 202 q^{13} + 100 q^{14} + 1786 q^{15} + 6912 q^{16} + 1550 q^{17}+ \cdots + 335378 q^{99}+O(q^{100}) \) 27 * q - 108 * q^2 + 33 * q^3 + 432 * q^4 + 139 * q^5 - 132 * q^6 - 25 * q^7 - 1728 * q^8 + 2346 * q^9 - 556 * q^10 + 1241 * q^11 + 528 * q^12 - 202 * q^13 + 100 * q^14 + 1786 * q^15 + 6912 * q^16 + 1550 * q^17 - 9384 * q^18 - 66 * q^19 + 2224 * q^20 - 1217 * q^21 - 4964 * q^22 + 10568 * q^23 - 2112 * q^24 + 15880 * q^25 + 808 * q^26 + 8256 * q^27 - 400 * q^28 - 2122 * q^29 - 7144 * q^30 - 1621 * q^31 - 27648 * q^32 + 18369 * q^33 - 6200 * q^34 + 23123 * q^35 + 37536 * q^36 + 8542 * q^37 + 264 * q^38 + 589 * q^39 - 8896 * q^40 + 10859 * q^41 + 4868 * q^42 + 23759 * q^43 + 19856 * q^44 + 28006 * q^45 - 42272 * q^46 + 38457 * q^47 + 8448 * q^48 + 57376 * q^49 - 63520 * q^50 + 58237 * q^51 - 3232 * q^52 + 73586 * q^53 - 33024 * q^54 + 25923 * q^55 + 1600 * q^56 + 52534 * q^57 + 8488 * q^58 + 68927 * q^59 + 28576 * q^60 - 76016 * q^61 + 6484 * q^62 - 20738 * q^63 + 110592 * q^64 + 74867 * q^65 - 73476 * q^66 + 94787 * q^67 + 24800 * q^68 - 90601 * q^69 - 92492 * q^70 + 15523 * q^71 - 150144 * q^72 - 96279 * q^73 - 34168 * q^74 - 123786 * q^75 - 1056 * q^76 - 199866 * q^77 - 2356 * q^78 - 246217 * q^79 + 35584 * q^80 - 54473 * q^81 - 43436 * q^82 - 56658 * q^83 - 19472 * q^84 - 120541 * q^85 - 95036 * q^86 - 352954 * q^87 - 79424 * q^88 - 83070 * q^89 - 112024 * q^90 - 132348 * q^91 + 169088 * q^92 + 40727 * q^93 - 153828 * q^94 - 72090 * q^95 - 33792 * q^96 - 36273 * q^97 - 229504 * q^98 + 335378 * q^99

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