LMFDB - Embedded newform 1083.4.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1083,4,Mod(1,1083)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1083, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1083.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1083 = 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1083.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:	$63.8990685362$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 120 x^{16} - 19 x^{15} + 5904 x^{14} + 1731 x^{13} - 153482 x^{12} - 62307 x^{11} + \cdots - 49519296 $ x^18 - 120x^16 - 19x^15 + 5904x^14 + 1731x^13 - 153482x^12 - 62307x^11 + 2264487x^10 + 1134069x^9 - 18914886x^8 - 10997190x^7 + 83975016x^6 + 53300700x^5 - 174155400x^4 - 101642904x^3 + 157450608x^2 + 62731584x - 49519296
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{3}\cdot 19^{3} $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.66881$ of defining polynomial
Character	$\chi$	$=$	1083.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.66881 q^{2} -3.00000 q^{3} +13.7978 q^{4} +4.88826 q^{5} +14.0064 q^{6} +26.8280 q^{7} -27.0688 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.66881 q^{2} -3.00000 q^{3} +13.7978 q^{4} +4.88826 q^{5} +14.0064 q^{6} +26.8280 q^{7} -27.0688 q^{8} +9.00000 q^{9} -22.8224 q^{10} +9.83125 q^{11} -41.3934 q^{12} -23.9925 q^{13} -125.255 q^{14} -14.6648 q^{15} +15.9967 q^{16} +33.1650 q^{17} -42.0193 q^{18} +67.4472 q^{20} -80.4840 q^{21} -45.9002 q^{22} +128.914 q^{23} +81.2063 q^{24} -101.105 q^{25} +112.017 q^{26} -27.0000 q^{27} +370.167 q^{28} +263.395 q^{29} +68.4671 q^{30} +187.463 q^{31} +141.865 q^{32} -29.4938 q^{33} -154.841 q^{34} +131.142 q^{35} +124.180 q^{36} +220.139 q^{37} +71.9776 q^{39} -132.319 q^{40} -206.094 q^{41} +375.764 q^{42} -466.586 q^{43} +135.650 q^{44} +43.9944 q^{45} -601.877 q^{46} +633.424 q^{47} -47.9900 q^{48} +376.741 q^{49} +472.040 q^{50} -99.4949 q^{51} -331.044 q^{52} -125.823 q^{53} +126.058 q^{54} +48.0577 q^{55} -726.201 q^{56} -1229.74 q^{58} +142.520 q^{59} -202.342 q^{60} +709.747 q^{61} -875.231 q^{62} +241.452 q^{63} -790.313 q^{64} -117.282 q^{65} +137.701 q^{66} -712.246 q^{67} +457.603 q^{68} -386.743 q^{69} -612.278 q^{70} +323.486 q^{71} -243.619 q^{72} -339.899 q^{73} -1027.79 q^{74} +303.315 q^{75} +263.753 q^{77} -336.050 q^{78} +63.7192 q^{79} +78.1959 q^{80} +81.0000 q^{81} +962.213 q^{82} +275.416 q^{83} -1110.50 q^{84} +162.119 q^{85} +2178.40 q^{86} -790.186 q^{87} -266.120 q^{88} -258.528 q^{89} -205.401 q^{90} -643.672 q^{91} +1778.73 q^{92} -562.390 q^{93} -2957.34 q^{94} -425.594 q^{96} +486.563 q^{97} -1758.93 q^{98} +88.4813 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 54 q^{3} + 96 q^{4} + 18 q^{5} + 48 q^{7} + 57 q^{8} + 162 q^{9}+O(q^{10}) $ 18 * q - 54 * q^3 + 96 * q^4 + 18 * q^5 + 48 * q^7 + 57 * q^8 + 162 * q^9	\( 18 q - 54 q^{3} + 96 q^{4} + 18 q^{5} + 48 q^{7} + 57 q^{8} + 162 q^{9} - 60 q^{10} + 108 q^{11} - 288 q^{12} - 42 q^{13} + 60 q^{14} - 54 q^{15} + 576 q^{16} + 300 q^{17} + 27 q^{20} - 144 q^{21} - 219 q^{22} + 174 q^{23} - 171 q^{24} + 1068 q^{25} - 72 q^{26} - 486 q^{27} + 867 q^{28} + 168 q^{29} + 180 q^{30} - 1032 q^{31} + 921 q^{32} - 324 q^{33} + 75 q^{34} + 1524 q^{35} + 864 q^{36} + 132 q^{37} + 126 q^{39} - 363 q^{40} + 120 q^{41} - 180 q^{42} + 420 q^{43} + 2328 q^{44} + 162 q^{45} - 2229 q^{46} + 810 q^{47} - 1728 q^{48} + 1122 q^{49} - 1503 q^{50} - 900 q^{51} + 228 q^{52} - 174 q^{53} + 2550 q^{55} + 1119 q^{56} + 756 q^{58} + 474 q^{59} - 81 q^{60} + 1488 q^{61} + 333 q^{62} + 432 q^{63} + 2679 q^{64} - 1716 q^{65} + 657 q^{66} - 3060 q^{67} + 4623 q^{68} - 522 q^{69} - 1383 q^{70} + 1464 q^{71} + 513 q^{72} + 1470 q^{73} - 135 q^{74} - 3204 q^{75} + 1014 q^{77} + 216 q^{78} - 2508 q^{79} - 2049 q^{80} + 1458 q^{81} + 1485 q^{82} + 4764 q^{83} - 2601 q^{84} + 804 q^{85} - 1068 q^{86} - 504 q^{87} - 3012 q^{88} + 1050 q^{89} - 540 q^{90} + 3408 q^{91} + 3306 q^{92} + 3096 q^{93} - 8205 q^{94} - 2763 q^{96} + 2070 q^{97} + 1767 q^{98} + 972 q^{99}+O(q^{100}) \) 18 * q - 54 * q^3 + 96 * q^4 + 18 * q^5 + 48 * q^7 + 57 * q^8 + 162 * q^9 - 60 * q^10 + 108 * q^11 - 288 * q^12 - 42 * q^13 + 60 * q^14 - 54 * q^15 + 576 * q^16 + 300 * q^17 + 27 * q^20 - 144 * q^21 - 219 * q^22 + 174 * q^23 - 171 * q^24 + 1068 * q^25 - 72 * q^26 - 486 * q^27 + 867 * q^28 + 168 * q^29 + 180 * q^30 - 1032 * q^31 + 921 * q^32 - 324 * q^33 + 75 * q^34 + 1524 * q^35 + 864 * q^36 + 132 * q^37 + 126 * q^39 - 363 * q^40 + 120 * q^41 - 180 * q^42 + 420 * q^43 + 2328 * q^44 + 162 * q^45 - 2229 * q^46 + 810 * q^47 - 1728 * q^48 + 1122 * q^49 - 1503 * q^50 - 900 * q^51 + 228 * q^52 - 174 * q^53 + 2550 * q^55 + 1119 * q^56 + 756 * q^58 + 474 * q^59 - 81 * q^60 + 1488 * q^61 + 333 * q^62 + 432 * q^63 + 2679 * q^64 - 1716 * q^65 + 657 * q^66 - 3060 * q^67 + 4623 * q^68 - 522 * q^69 - 1383 * q^70 + 1464 * q^71 + 513 * q^72 + 1470 * q^73 - 135 * q^74 - 3204 * q^75 + 1014 * q^77 + 216 * q^78 - 2508 * q^79 - 2049 * q^80 + 1458 * q^81 + 1485 * q^82 + 4764 * q^83 - 2601 * q^84 + 804 * q^85 - 1068 * q^86 - 504 * q^87 - 3012 * q^88 + 1050 * q^89 - 540 * q^90 + 3408 * q^91 + 3306 * q^92 + 3096 * q^93 - 8205 * q^94 - 2763 * q^96 + 2070 * q^97 + 1767 * q^98 + 972 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.66881	−1.65067	−0.825337	−	0.564641i	$-0.809015\pi$
$2$	−4.66881	−1.65067	−0.825337	+	0.564641i	$0.809015\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	13.7978	1.72472
$5$	4.88826	0.437220	0.218610	−	0.975812i	$-0.429848\pi$
$5$	4.88826	0.437220	0.218610	+	0.975812i	$0.429848\pi$
$6$	14.0064	0.953017
$7$	26.8280	1.44858	0.724288	−	0.689498i	$-0.242169\pi$
$7$	26.8280	1.44858	0.724288	+	0.689498i	$0.242169\pi$
$8$	−27.0688	−1.19628
$9$	9.00000	0.333333
$10$	−22.8224	−0.721707
$11$	9.83125	0.269476	0.134738	−	0.990881i	$-0.456981\pi$
$11$	9.83125	0.269476	0.134738	+	0.990881i	$0.456981\pi$
$12$	−41.3934	−0.995770
$13$	−23.9925	−0.511872	−0.255936	−	0.966694i	$-0.582384\pi$
$13$	−23.9925	−0.511872	−0.255936	+	0.966694i	$0.582384\pi$
$14$	−125.255	−2.39113
$15$	−14.6648	−0.252429
$16$	15.9967	0.249948
$17$	33.1650	0.473158	0.236579	−	0.971612i	$-0.423974\pi$
$17$	33.1650	0.473158	0.236579	+	0.971612i	$0.423974\pi$
$18$	−42.0193	−0.550225
$19$	0	0
$19$	0	0
$20$	67.4472	0.754083
$21$	−80.4840	−0.836335
$22$	−45.9002	−0.444817
$23$	128.914	1.16872	0.584359	−	0.811495i	$-0.301346\pi$
$23$	128.914	1.16872	0.584359	+	0.811495i	$0.301346\pi$
$24$	81.2063	0.690674
$25$	−101.105	−0.808839
$26$	112.017	0.844934
$27$	−27.0000	−0.192450
$28$	370.167	2.49839
$29$	263.395	1.68660	0.843298	−	0.537446i	$-0.180611\pi$
$29$	263.395	1.68660	0.843298	+	0.537446i	$0.180611\pi$
$30$	68.4671	0.416678
$31$	187.463	1.08611	0.543055	−	0.839697i	$-0.317267\pi$
$31$	187.463	1.08611	0.543055	+	0.839697i	$0.317267\pi$
$32$	141.865	0.783700
$33$	−29.4938	−0.155582
$34$	−154.841	−0.781029
$35$	131.142	0.633345
$36$	124.180	0.574908
$37$	220.139	0.978126	0.489063	−	0.872249i	$-0.337339\pi$
$37$	220.139	0.978126	0.489063	+	0.872249i	$0.337339\pi$
$38$	0	0
$39$	71.9776	0.295529
$40$	−132.319	−0.523038
$41$	−206.094	−0.785035	−0.392518	−	0.919745i	$-0.628396\pi$
$41$	−206.094	−0.785035	−0.392518	+	0.919745i	$0.628396\pi$
$42$	375.764	1.38052
$43$	−466.586	−1.65474	−0.827368	−	0.561660i	$-0.810163\pi$
$43$	−466.586	−1.65474	−0.827368	+	0.561660i	$0.810163\pi$
$44$	135.650	0.464771
$45$	43.9944	0.145740
$46$	−601.877	−1.92917
$47$	633.424	1.96584	0.982919	−	0.184037i	$-0.0589167\pi$
$47$	633.424	1.96584	0.982919	+	0.184037i	$0.0589167\pi$
$48$	−47.9900	−0.144307
$49$	376.741	1.09837
$50$	472.040	1.33513
$51$	−99.4949	−0.273178
$52$	−331.044	−0.882838
$53$	−125.823	−0.326096	−0.163048	−	0.986618i	$-0.552133\pi$
$53$	−125.823	−0.326096	−0.163048	+	0.986618i	$0.552133\pi$
$54$	126.058	0.317672
$55$	48.0577	0.117820
$56$	−726.201	−1.73290
$57$	0	0
$58$	−1229.74	−2.78402
$59$	142.520	0.314483	0.157242	−	0.987560i	$-0.449740\pi$
$59$	142.520	0.314483	0.157242	+	0.987560i	$0.449740\pi$
$60$	−202.342	−0.435370
$61$	709.747	1.48973	0.744867	−	0.667213i	$-0.232513\pi$
$61$	709.747	1.48973	0.744867	+	0.667213i	$0.232513\pi$
$62$	−875.231	−1.79281
$63$	241.452	0.482858
$64$	−790.313	−1.54358
$65$	−117.282	−0.223800
$66$	137.701	0.256815
$67$	−712.246	−1.29873	−0.649363	−	0.760478i	$-0.724965\pi$
$67$	−712.246	−1.29873	−0.649363	+	0.760478i	$0.724965\pi$
$68$	457.603	0.816067
$69$	−386.743	−0.674760
$70$	−612.278	−1.04545
$71$	323.486	0.540715	0.270358	−	0.962760i	$-0.412858\pi$
$71$	323.486	0.540715	0.270358	+	0.962760i	$0.412858\pi$
$72$	−243.619	−0.398761
$73$	−339.899	−0.544962	−0.272481	−	0.962161i	$-0.587844\pi$
$73$	−339.899	−0.544962	−0.272481	+	0.962161i	$0.587844\pi$
$74$	−1027.79	−1.61457
$75$	303.315	0.466983
$76$	0	0
$77$	263.753	0.390356
$78$	−336.050	−0.487823
$79$	63.7192	0.0907465	0.0453733	−	0.998970i	$-0.485552\pi$
$79$	63.7192	0.0907465	0.0453733	+	0.998970i	$0.485552\pi$
$80$	78.1959	0.109282
$81$	81.0000	0.111111
$82$	962.213	1.29584
$83$	275.416	0.364227	0.182113	−	0.983278i	$-0.441706\pi$
$83$	275.416	0.364227	0.182113	+	0.983278i	$0.441706\pi$
$84$	−1110.50	−1.44245
$85$	162.119	0.206874
$86$	2178.40	2.73143
$87$	−790.186	−0.973757
$88$	−266.120	−0.322369
$89$	−258.528	−0.307909	−0.153954	−	0.988078i	$-0.549201\pi$
$89$	−258.528	−0.307909	−0.153954	+	0.988078i	$0.549201\pi$
$90$	−205.401	−0.240569
$91$	−643.672	−0.741485
$92$	1778.73	2.01572
$93$	−562.390	−0.627066
$94$	−2957.34	−3.24496
$95$	0	0
$96$	−425.594	−0.452469
$97$	486.563	0.509309	0.254655	−	0.967032i	$-0.418038\pi$
$97$	486.563	0.509309	0.254655	+	0.967032i	$0.418038\pi$
$98$	−1758.93	−1.81305
$99$	88.4813	0.0898253
$100$	−1395.02	−1.39502
$101$	−1899.22	−1.87109	−0.935544	−	0.353211i	$-0.885090\pi$
$101$	−1899.22	−1.87109	−0.935544	+	0.353211i	$0.885090\pi$
$102$	464.523	0.450928
$103$	−1170.57	−1.11981	−0.559903	−	0.828558i	$-0.689162\pi$
$103$	−1170.57	−1.11981	−0.559903	+	0.828558i	$0.689162\pi$
$104$	649.449	0.612343
$105$	−393.427	−0.365662
$106$	587.443	0.538279
$107$	1622.55	1.46596	0.732982	−	0.680248i	$-0.238128\pi$
$107$	1622.55	1.46596	0.732982	+	0.680248i	$0.238128\pi$
$108$	−372.540	−0.331923
$109$	713.962	0.627387	0.313693	−	0.949524i	$-0.398434\pi$
$109$	713.962	0.627387	0.313693	+	0.949524i	$0.398434\pi$
$110$	−224.372	−0.194482
$111$	−660.417	−0.564721
$112$	429.158	0.362068
$113$	−700.024	−0.582768	−0.291384	−	0.956606i	$-0.594116\pi$
$113$	−700.024	−0.582768	−0.291384	+	0.956606i	$0.594116\pi$
$114$	0	0
$115$	630.168	0.510986
$116$	3634.27	2.90891
$117$	−215.933	−0.170624
$118$	−665.399	−0.519109
$119$	889.749	0.685405
$120$	396.958	0.301976
$121$	−1234.35	−0.927383
$122$	−3313.67	−2.45906
$123$	618.281	0.453240
$124$	2586.58	1.87324
$125$	−1105.26	−0.790860
$126$	−1127.29	−0.797042
$127$	−1323.14	−0.924488	−0.462244	−	0.886753i	$-0.652956\pi$
$127$	−1323.14	−0.924488	−0.462244	+	0.886753i	$0.652956\pi$
$128$	2554.90	1.76425
$129$	1399.76	0.955362
$130$	547.567	0.369421
$131$	229.341	0.152959	0.0764795	−	0.997071i	$-0.475632\pi$
$131$	229.341	0.152959	0.0764795	+	0.997071i	$0.475632\pi$
$132$	−406.949	−0.268336
$133$	0	0
$134$	3325.34	2.14377
$135$	−131.983	−0.0841429
$136$	−897.735	−0.566030
$137$	−1337.18	−0.833891	−0.416946	−	0.908931i	$-0.636899\pi$
$137$	−1337.18	−0.833891	−0.416946	+	0.908931i	$0.636899\pi$
$138$	1805.63	1.11381
$139$	2825.65	1.72423	0.862116	−	0.506711i	$-0.169139\pi$
$139$	2825.65	1.72423	0.862116	+	0.506711i	$0.169139\pi$
$140$	1809.47	1.09235
$141$	−1900.27	−1.13498
$142$	−1510.30	−0.892544
$143$	−235.877	−0.137937
$144$	143.970	0.0833159
$145$	1287.55	0.737413
$146$	1586.93	0.899554
$147$	−1130.22	−0.634145
$148$	3037.43	1.68700
$149$	−628.501	−0.345563	−0.172781	−	0.984960i	$-0.555275\pi$
$149$	−628.501	−0.345563	−0.172781	+	0.984960i	$0.555275\pi$
$150$	−1416.12	−0.770837
$151$	1575.25	0.848952	0.424476	−	0.905439i	$-0.360458\pi$
$151$	1575.25	0.848952	0.424476	+	0.905439i	$0.360458\pi$
$152$	0	0
$153$	298.485	0.157719
$154$	−1231.41	−0.644350
$155$	916.370	0.474869
$156$	993.132	0.509707
$157$	1668.51	0.848163	0.424081	−	0.905624i	$-0.360597\pi$
$157$	1668.51	0.848163	0.424081	+	0.905624i	$0.360597\pi$
$158$	−297.493	−0.149793
$159$	377.469	0.188272
$160$	693.473	0.342649
$161$	3458.52	1.69298
$162$	−378.174	−0.183408
$163$	2359.83	1.13396	0.566982	−	0.823730i	$-0.308111\pi$
$163$	2359.83	1.13396	0.566982	+	0.823730i	$0.308111\pi$
$164$	−2843.64	−1.35397
$165$	−144.173	−0.0680235
$166$	−1285.87	−0.601220
$167$	344.365	0.159568	0.0797838	−	0.996812i	$-0.474577\pi$
$167$	344.365	0.159568	0.0797838	+	0.996812i	$0.474577\pi$
$168$	2178.60	1.00049
$169$	−1621.36	−0.737987
$170$	−756.903	−0.341481
$171$	0	0
$172$	−6437.85	−2.85396
$173$	530.051	0.232942	0.116471	−	0.993194i	$-0.462842\pi$
$173$	530.051	0.232942	0.116471	+	0.993194i	$0.462842\pi$
$174$	3689.23	1.60736
$175$	−2712.44	−1.17166
$176$	157.267	0.0673549
$177$	−427.560	−0.181567
$178$	1207.02	0.508257
$179$	623.619	0.260399	0.130200	−	0.991488i	$-0.458438\pi$
$179$	623.619	0.260399	0.130200	+	0.991488i	$0.458438\pi$
$180$	607.025	0.251361
$181$	−2738.99	−1.12479	−0.562396	−	0.826868i	$-0.690120\pi$
$181$	−2738.99	−1.12479	−0.562396	+	0.826868i	$0.690120\pi$
$182$	3005.18	1.22395
$183$	−2129.24	−0.860098
$184$	−3489.56	−1.39812
$185$	1076.10	0.427656
$186$	2625.69	1.03508
$187$	326.053	0.127505
$188$	8739.85	3.39053
$189$	−724.356	−0.278778
$190$	0	0
$191$	−3050.00	−1.15545	−0.577724	−	0.816232i	$-0.696059\pi$
$191$	−3050.00	−1.15545	−0.577724	+	0.816232i	$0.696059\pi$
$192$	2370.94	0.891187
$193$	680.303	0.253727	0.126863	−	0.991920i	$-0.459509\pi$
$193$	680.303	0.253727	0.126863	+	0.991920i	$0.459509\pi$
$194$	−2271.67	−0.840703
$195$	351.846	0.129211
$196$	5198.20	1.89439
$197$	1890.70	0.683791	0.341896	−	0.939738i	$-0.388931\pi$
$197$	1890.70	0.683791	0.341896	+	0.939738i	$0.388931\pi$
$198$	−413.102	−0.148272
$199$	−2108.11	−0.750954	−0.375477	−	0.926832i	$-0.622521\pi$
$199$	−2108.11	−0.750954	−0.375477	+	0.926832i	$0.622521\pi$
$200$	2736.79	0.967600
$201$	2136.74	0.749820
$202$	8867.12	3.08855
$203$	7066.37	2.44316
$204$	−1372.81	−0.471156
$205$	−1007.44	−0.343233
$206$	5465.19	1.84844
$207$	1160.23	0.389573
$208$	−383.801	−0.127941
$209$	0	0
$210$	1836.84	0.603589
$211$	−5577.36	−1.81972	−0.909861	−	0.414913i	$-0.863812\pi$
$211$	−5577.36	−1.81972	−0.909861	+	0.414913i	$0.863812\pi$
$212$	−1736.08	−0.562426
$213$	−970.459	−0.312182
$214$	−7575.39	−2.41983
$215$	−2280.79	−0.723483
$216$	730.857	0.230225
$217$	5029.27	1.57331
$218$	−3333.35	−1.03561
$219$	1019.70	0.314634
$220$	663.091	0.203207
$221$	−795.712	−0.242196
$222$	3083.36	0.932170
$223$	3069.74	0.921817	0.460909	−	0.887448i	$-0.347524\pi$
$223$	3069.74	0.921817	0.460909	+	0.887448i	$0.347524\pi$
$224$	3805.95	1.13525
$225$	−909.944	−0.269613
$226$	3268.28	0.961959
$227$	5391.80	1.57650	0.788252	−	0.615353i	$-0.210986\pi$
$227$	5391.80	1.57650	0.788252	+	0.615353i	$0.210986\pi$
$228$	0	0
$229$	−272.190	−0.0785450	−0.0392725	−	0.999229i	$-0.512504\pi$
$229$	−272.190	−0.0785450	−0.0392725	+	0.999229i	$0.512504\pi$
$230$	−2942.13	−0.843472
$231$	−791.258	−0.225372
$232$	−7129.79	−2.01765
$233$	2289.24	0.643661	0.321831	−	0.946797i	$-0.395702\pi$
$233$	2289.24	0.643661	0.321831	+	0.946797i	$0.395702\pi$
$234$	1008.15	0.281645
$235$	3096.34	0.859503
$236$	1966.46	0.542397
$237$	−191.158	−0.0523925
$238$	−4154.07	−1.13138
$239$	4313.10	1.16733	0.583664	−	0.811996i	$-0.301619\pi$
$239$	4313.10	1.16733	0.583664	+	0.811996i	$0.301619\pi$
$240$	−234.588	−0.0630940
$241$	575.260	0.153758	0.0768792	−	0.997040i	$-0.475504\pi$
$241$	575.260	0.153758	0.0768792	+	0.997040i	$0.475504\pi$
$242$	5762.93	1.53081
$243$	−243.000	−0.0641500
$244$	9792.94	2.56938
$245$	1841.61	0.480229
$246$	−2886.64	−0.748152
$247$	0	0
$248$	−5074.40	−1.29929
$249$	−826.248	−0.210287
$250$	5160.25	1.30545
$251$	4709.47	1.18430	0.592149	−	0.805828i	$-0.298280\pi$
$251$	4709.47	1.18430	0.592149	+	0.805828i	$0.298280\pi$
$252$	3331.50	0.832797
$253$	1267.39	0.314941
$254$	6177.50	1.52603
$255$	−486.357	−0.119439
$256$	−5605.86	−1.36862
$257$	4838.82	1.17446	0.587232	−	0.809418i	$-0.300217\pi$
$257$	4838.82	1.17446	0.587232	+	0.809418i	$0.300217\pi$
$258$	−6535.20	−1.57699
$259$	5905.89	1.41689
$260$	−1618.23	−0.385994
$261$	2370.56	0.562199
$262$	−1070.75	−0.252485
$263$	−6532.64	−1.53163	−0.765817	−	0.643058i	$-0.777665\pi$
$263$	−6532.64	−1.53163	−0.765817	+	0.643058i	$0.777665\pi$
$264$	798.360	0.186120
$265$	−615.055	−0.142576
$266$	0	0
$267$	775.583	0.177771
$268$	−9827.42	−2.23994
$269$	6378.80	1.44581	0.722904	−	0.690949i	$-0.242807\pi$
$269$	6378.80	1.44581	0.722904	+	0.690949i	$0.242807\pi$
$270$	616.204	0.138893
$271$	−2044.33	−0.458245	−0.229123	−	0.973398i	$-0.573586\pi$
$271$	−2044.33	−0.458245	−0.229123	+	0.973398i	$0.573586\pi$
$272$	530.529	0.118265
$273$	1931.02	0.428097
$274$	6243.05	1.37648
$275$	−993.988	−0.217963
$276$	−5336.20	−1.16377
$277$	3089.88	0.670226	0.335113	−	0.942178i	$-0.391225\pi$
$277$	3089.88	0.670226	0.335113	+	0.942178i	$0.391225\pi$
$278$	−13192.4	−2.84614
$279$	1687.17	0.362037
$280$	−3549.86	−0.757660
$281$	−2621.51	−0.556535	−0.278268	−	0.960504i	$-0.589760\pi$
$281$	−2621.51	−0.556535	−0.278268	+	0.960504i	$0.589760\pi$
$282$	8872.01	1.87348
$283$	4698.62	0.986939	0.493469	−	0.869763i	$-0.335728\pi$
$283$	4698.62	0.986939	0.493469	+	0.869763i	$0.335728\pi$
$284$	4463.40	0.932584
$285$	0	0
$286$	1101.26	0.227689
$287$	−5529.08	−1.13718
$288$	1276.78	0.261233
$289$	−3813.09	−0.776122
$290$	−6011.31	−1.21723
$291$	−1459.69	−0.294050
$292$	−4689.86	−0.939909
$293$	−7416.22	−1.47870	−0.739352	−	0.673319i	$-0.764868\pi$
$293$	−7416.22	−1.47870	−0.739352	+	0.673319i	$0.764868\pi$
$294$	5276.80	1.04677
$295$	696.675	0.137498
$296$	−5958.90	−1.17011
$297$	−265.444	−0.0518606
$298$	2934.35	0.570411
$299$	−3092.99	−0.598234
$300$	4185.07	0.805417
$301$	−12517.6	−2.39701
$302$	−7354.53	−1.40134
$303$	5697.67	1.08027
$304$	0	0
$305$	3469.43	0.651341
$306$	−1393.57	−0.260343
$307$	5048.59	0.938561	0.469280	−	0.883049i	$-0.344513\pi$
$307$	5048.59	0.938561	0.469280	+	0.883049i	$0.344513\pi$
$308$	3639.20	0.673256
$309$	3511.72	0.646521
$310$	−4278.36	−0.783853
$311$	3697.30	0.674132	0.337066	−	0.941481i	$-0.390566\pi$
$311$	3697.30	0.674132	0.337066	+	0.941481i	$0.390566\pi$
$312$	−1948.35	−0.353537
$313$	−1334.48	−0.240989	−0.120494	−	0.992714i	$-0.538448\pi$
$313$	−1334.48	−0.240989	−0.120494	+	0.992714i	$0.538448\pi$
$314$	−7789.96	−1.40004
$315$	1180.28	0.211115
$316$	879.185	0.156513
$317$	5872.13	1.04041	0.520207	−	0.854040i	$-0.325855\pi$
$317$	5872.13	1.04041	0.520207	+	0.854040i	$0.325855\pi$
$318$	−1762.33	−0.310775
$319$	2589.51	0.454497
$320$	−3863.26	−0.674884
$321$	−4867.66	−0.846375
$322$	−16147.2	−2.79455
$323$	0	0
$324$	1117.62	0.191636
$325$	2425.76	0.414022
$326$	−11017.6	−1.87181
$327$	−2141.89	−0.362222
$328$	5578.71	0.939123
$329$	16993.5	2.84767
$330$	673.117	0.112285
$331$	−4744.53	−0.787864	−0.393932	−	0.919140i	$-0.628885\pi$
$331$	−4744.53	−0.787864	−0.393932	+	0.919140i	$0.628885\pi$
$332$	3800.13	0.628191
$333$	1981.25	0.326042
$334$	−1607.78	−0.263394
$335$	−3481.65	−0.567829
$336$	−1287.47	−0.209040
$337$	8527.00	1.37832	0.689162	−	0.724607i	$-0.257979\pi$
$337$	8527.00	1.37832	0.689162	+	0.724607i	$0.257979\pi$
$338$	7569.81	1.21818
$339$	2100.07	0.336461
$340$	2236.88	0.356800
$341$	1843.00	0.292680
$342$	0	0
$343$	905.209	0.142498
$344$	12629.9	1.97953
$345$	−1890.50	−0.295018
$346$	−2474.71	−0.384512
$347$	5251.44	0.812427	0.406213	−	0.913778i	$-0.366849\pi$
$347$	5251.44	0.812427	0.406213	+	0.913778i	$0.366849\pi$
$348$	−10902.8	−1.67946
$349$	−5227.69	−0.801810	−0.400905	−	0.916120i	$-0.631304\pi$
$349$	−5227.69	−0.801810	−0.400905	+	0.916120i	$0.631304\pi$
$350$	12663.9	1.93404
$351$	647.799	0.0985098
$352$	1394.71	0.211188
$353$	−5199.57	−0.783981	−0.391990	−	0.919969i	$-0.628213\pi$
$353$	−5199.57	−0.783981	−0.391990	+	0.919969i	$0.628213\pi$
$354$	1996.20	0.299708
$355$	1581.29	0.236411
$356$	−3567.11	−0.531057
$357$	−2669.25	−0.395719
$358$	−2911.56	−0.429834
$359$	−11784.8	−1.73253	−0.866265	−	0.499585i	$-0.833486\pi$
$359$	−11784.8	−1.73253	−0.866265	+	0.499585i	$0.833486\pi$
$360$	−1190.87	−0.174346
$361$	0	0
$362$	12787.8	1.85666
$363$	3703.04	0.535425
$364$	−8881.25	−1.27886
$365$	−1661.52	−0.238268
$366$	9941.02	1.41974
$367$	−9203.56	−1.30905	−0.654526	−	0.756040i	$-0.727132\pi$
$367$	−9203.56	−1.30905	−0.654526	+	0.756040i	$0.727132\pi$
$368$	2062.20	0.292119
$369$	−1854.84	−0.261678
$370$	−5024.10	−0.705920
$371$	−3375.58	−0.472375
$372$	−7759.74	−1.08152
$373$	−2074.85	−0.288020	−0.144010	−	0.989576i	$-0.546000\pi$
$373$	−2074.85	−0.288020	−0.144010	+	0.989576i	$0.546000\pi$
$374$	−1522.28	−0.210469
$375$	3315.78	0.456603
$376$	−17146.0	−2.35170
$377$	−6319.53	−0.863322
$378$	3381.88	0.460172
$379$	−857.501	−0.116219	−0.0581093	−	0.998310i	$-0.518507\pi$
$379$	−857.501	−0.116219	−0.0581093	+	0.998310i	$0.518507\pi$
$380$	0	0
$381$	3969.43	0.533753
$382$	14239.9	1.90727
$383$	1763.17	0.235232	0.117616	−	0.993059i	$-0.462475\pi$
$383$	1763.17	0.235232	0.117616	+	0.993059i	$0.462475\pi$
$384$	−7664.71	−1.01859
$385$	1289.29	0.170671
$386$	−3176.21	−0.418820
$387$	−4199.27	−0.551579
$388$	6713.49	0.878417
$389$	2153.13	0.280638	0.140319	−	0.990106i	$-0.455187\pi$
$389$	2153.13	0.280638	0.140319	+	0.990106i	$0.455187\pi$
$390$	−1642.70	−0.213286
$391$	4275.44	0.552988
$392$	−10197.9	−1.31396
$393$	−688.023	−0.0883109
$394$	−8827.33	−1.12872
$395$	311.476	0.0396761
$396$	1220.85	0.154924
$397$	8081.11	1.02161	0.510805	−	0.859697i	$-0.329347\pi$
$397$	8081.11	1.02161	0.510805	+	0.859697i	$0.329347\pi$
$398$	9842.36	1.23958
$399$	0	0
$400$	−1617.34	−0.202168
$401$	15583.2	1.94061	0.970307	−	0.241875i	$-0.0777626\pi$
$401$	15583.2	1.94061	0.970307	+	0.241875i	$0.0777626\pi$
$402$	−9976.02	−1.23771
$403$	−4497.72	−0.555949
$404$	−26205.1	−3.22711
$405$	395.949	0.0485799
$406$	−32991.5	−4.03286
$407$	2164.24	0.263581
$408$	2693.20	0.326798
$409$	−5100.52	−0.616636	−0.308318	−	0.951283i	$-0.599766\pi$
$409$	−5100.52	−0.616636	−0.308318	+	0.951283i	$0.599766\pi$
$410$	4703.55	0.566565
$411$	4011.54	0.481447
$412$	−16151.3	−1.93136
$413$	3823.52	0.455553
$414$	−5416.89	−0.643057
$415$	1346.31	0.159247
$416$	−3403.70	−0.401154
$417$	−8476.94	−0.995486
$418$	0	0
$419$	−11238.5	−1.31035	−0.655176	−	0.755476i	$-0.727406\pi$
$419$	−11238.5	−1.31035	−0.655176	+	0.755476i	$0.727406\pi$
$420$	−5428.42	−0.630666
$421$	5920.36	0.685370	0.342685	−	0.939450i	$-0.388664\pi$
$421$	5920.36	0.685370	0.342685	+	0.939450i	$0.388664\pi$
$422$	26039.6	3.00377
$423$	5700.82	0.655280
$424$	3405.87	0.390103
$425$	−3353.14	−0.382709
$426$	4530.89	0.515311
$427$	19041.1	2.15799
$428$	22387.6	2.52838
$429$	707.630	0.0796380
$430$	10648.6	1.19423
$431$	−14100.2	−1.57583	−0.787917	−	0.615782i	$-0.788840\pi$
$431$	−14100.2	−1.57583	−0.787917	+	0.615782i	$0.788840\pi$
$432$	−431.910	−0.0481025
$433$	1087.21	0.120665	0.0603327	−	0.998178i	$-0.480784\pi$
$433$	1087.21	0.120665	0.0603327	+	0.998178i	$0.480784\pi$
$434$	−23480.7	−2.59703
$435$	−3862.64	−0.425746
$436$	9851.10	1.08207
$437$	0	0
$438$	−4760.78	−0.519358
$439$	−9351.49	−1.01668	−0.508340	−	0.861157i	$-0.669741\pi$
$439$	−9351.49	−1.01668	−0.508340	+	0.861157i	$0.669741\pi$
$440$	−1300.86	−0.140946
$441$	3390.67	0.366124
$442$	3715.03	0.399787
$443$	6986.37	0.749283	0.374641	−	0.927170i	$-0.377766\pi$
$443$	6986.37	0.749283	0.374641	+	0.927170i	$0.377766\pi$
$444$	−9112.30	−0.973988
$445$	−1263.75	−0.134624
$446$	−14332.0	−1.52162
$447$	1885.50	0.199511
$448$	−21202.5	−2.23599
$449$	6317.26	0.663986	0.331993	−	0.943282i	$-0.392279\pi$
$449$	6317.26	0.663986	0.331993	+	0.943282i	$0.392279\pi$
$450$	4248.36	0.445043
$451$	−2026.16	−0.211548
$452$	−9658.79	−1.00511
$453$	−4725.74	−0.490143
$454$	−25173.3	−2.60229
$455$	−3146.44	−0.324192
$456$	0	0
$457$	7474.38	0.765069	0.382534	−	0.923941i	$-0.375051\pi$
$457$	7474.38	0.765069	0.382534	+	0.923941i	$0.375051\pi$
$458$	1270.80	0.129652
$459$	−895.454	−0.0910593
$460$	8694.92	0.881310
$461$	9530.54	0.962866	0.481433	−	0.876483i	$-0.340116\pi$
$461$	9530.54	0.962866	0.481433	+	0.876483i	$0.340116\pi$
$462$	3694.23	0.372016
$463$	13174.0	1.32235	0.661173	−	0.750234i	$-0.270059\pi$
$463$	13174.0	1.32235	0.661173	+	0.750234i	$0.270059\pi$
$464$	4213.45	0.421561
$465$	−2749.11	−0.274166
$466$	−10688.0	−1.06247
$467$	6607.44	0.654724	0.327362	−	0.944899i	$-0.393840\pi$
$467$	6607.44	0.654724	0.327362	+	0.944899i	$0.393840\pi$
$468$	−2979.40	−0.294279
$469$	−19108.1	−1.88130
$470$	−14456.2	−1.41876
$471$	−5005.53	−0.489687
$472$	−3857.84	−0.376211
$473$	−4587.12	−0.445911
$474$	892.479	0.0864830
$475$	0	0
$476$	12276.6	1.18213
$477$	−1132.41	−0.108699
$478$	−20137.0	−1.92688
$479$	12346.3	1.17769	0.588847	−	0.808245i	$-0.299582\pi$
$479$	12346.3	1.17769	0.588847	+	0.808245i	$0.299582\pi$
$480$	−2080.42	−0.197828
$481$	−5281.70	−0.500675
$482$	−2685.78	−0.253805
$483$	−10375.5	−0.977440
$484$	−17031.3	−1.59948
$485$	2378.45	0.222680
$486$	1134.52	0.105891
$487$	1579.60	0.146979	0.0734894	−	0.997296i	$-0.476586\pi$
$487$	1579.60	0.146979	0.0734894	+	0.997296i	$0.476586\pi$
$488$	−19212.0	−1.78214
$489$	−7079.49	−0.654695
$490$	−8598.13	−0.792702
$491$	1631.70	0.149975	0.0749874	−	0.997184i	$-0.476108\pi$
$491$	1631.70	0.149975	0.0749874	+	0.997184i	$0.476108\pi$
$492$	8530.92	0.781714
$493$	8735.50	0.798027
$494$	0	0
$495$	432.520	0.0392734
$496$	2998.79	0.271471
$497$	8678.49	0.783267
$498$	3857.60	0.347114
$499$	−7996.96	−0.717421	−0.358710	−	0.933449i	$-0.616783\pi$
$499$	−7996.96	−0.717421	−0.358710	+	0.933449i	$0.616783\pi$
$500$	−15250.1	−1.36401
$501$	−1033.10	−0.0921264
$502$	−21987.6	−1.95489
$503$	7400.10	0.655972	0.327986	−	0.944683i	$-0.393630\pi$
$503$	7400.10	0.655972	0.327986	+	0.944683i	$0.393630\pi$
$504$	−6535.81	−0.577635
$505$	−9283.91	−0.818076
$506$	−5917.20	−0.519865
$507$	4864.07	0.426077
$508$	−18256.4	−1.59449
$509$	−5639.49	−0.491093	−0.245546	−	0.969385i	$-0.578967\pi$
$509$	−5639.49	−0.491093	−0.245546	+	0.969385i	$0.578967\pi$
$510$	2270.71	0.197154
$511$	−9118.82	−0.789419
$512$	5733.44	0.494892
$513$	0	0
$514$	−22591.5	−1.93866
$515$	−5722.07	−0.489601
$516$	19313.5	1.64774
$517$	6227.35	0.529746
$518$	−27573.5	−2.33882
$519$	−1590.15	−0.134489
$520$	3174.68	0.267728
$521$	2331.39	0.196046	0.0980230	−	0.995184i	$-0.468748\pi$
$521$	2331.39	0.196046	0.0980230	+	0.995184i	$0.468748\pi$
$522$	−11067.7	−0.928007
$523$	−750.014	−0.0627071	−0.0313536	−	0.999508i	$-0.509982\pi$
$523$	−750.014	−0.0627071	−0.0313536	+	0.999508i	$0.509982\pi$
$524$	3164.40	0.263812
$525$	8137.32	0.676461
$526$	30499.7	2.52823
$527$	6217.22	0.513902
$528$	−471.802	−0.0388874
$529$	4451.93	0.365902
$530$	2871.58	0.235346
$531$	1282.68	0.104828
$532$	0	0
$533$	4944.71	0.401837
$534$	−3621.05	−0.293442
$535$	7931.47	0.640948
$536$	19279.6	1.55364
$537$	−1870.86	−0.150342
$538$	−29781.4	−2.38656
$539$	3703.84	0.295984
$540$	−1821.07	−0.145123
$541$	9551.37	0.759049	0.379524	−	0.925182i	$-0.376088\pi$
$541$	9551.37	0.759049	0.379524	+	0.925182i	$0.376088\pi$
$542$	9544.61	0.756413
$543$	8216.96	0.649399
$544$	4704.94	0.370814
$545$	3490.03	0.274306
$546$	−9015.55	−0.706648
$547$	−5663.60	−0.442702	−0.221351	−	0.975194i	$-0.571047\pi$
$547$	−5663.60	−0.442702	−0.221351	+	0.975194i	$0.571047\pi$
$548$	−18450.1	−1.43823
$549$	6387.72	0.496578
$550$	4640.74	0.359785
$551$	0	0
$552$	10468.7	0.807203
$553$	1709.46	0.131453
$554$	−14426.0	−1.10632
$555$	−3228.29	−0.246907
$556$	38987.7	2.97382
$557$	−18637.0	−1.41773	−0.708864	−	0.705345i	$-0.750792\pi$
$557$	−18637.0	−1.41773	−0.708864	+	0.705345i	$0.750792\pi$
$558$	−7877.08	−0.597605
$559$	11194.6	0.847013
$560$	2097.84	0.158303
$561$	−978.159	−0.0736148
$562$	12239.3	0.918658
$563$	−2318.92	−0.173589	−0.0867946	−	0.996226i	$-0.527662\pi$
$563$	−2318.92	−0.173589	−0.0867946	+	0.996226i	$0.527662\pi$
$564$	−26219.6	−1.95752
$565$	−3421.90	−0.254797
$566$	−21936.9	−1.62911
$567$	2173.07	0.160953
$568$	−8756.38	−0.646848
$569$	13160.0	0.969592	0.484796	−	0.874627i	$-0.338894\pi$
$569$	13160.0	0.969592	0.484796	+	0.874627i	$0.338894\pi$
$570$	0	0
$571$	24111.1	1.76711	0.883553	−	0.468331i	$-0.155145\pi$
$571$	24111.1	1.76711	0.883553	+	0.468331i	$0.155145\pi$
$572$	−3254.58	−0.237903
$573$	9150.01	0.667098
$574$	25814.2	1.87712
$575$	−13033.9	−0.945305
$576$	−7112.82	−0.514527
$577$	23096.8	1.66643	0.833217	−	0.552947i	$-0.186497\pi$
$577$	23096.8	1.66643	0.833217	+	0.552947i	$0.186497\pi$
$578$	17802.6	1.28112
$579$	−2040.91	−0.146489
$580$	17765.3	1.27183
$581$	7388.86	0.527610
$582$	6815.01	0.485380
$583$	−1237.00	−0.0878751
$584$	9200.66	0.651928
$585$	−1055.54	−0.0746001
$586$	34624.9	2.44086
$587$	8460.54	0.594896	0.297448	−	0.954738i	$-0.403865\pi$
$587$	8460.54	0.594896	0.297448	+	0.954738i	$0.403865\pi$
$588$	−15594.6	−1.09372
$589$	0	0
$590$	−3252.64	−0.226965
$591$	−5672.11	−0.394787
$592$	3521.49	0.244480
$593$	−8947.97	−0.619644	−0.309822	−	0.950795i	$-0.600270\pi$
$593$	−8947.97	−0.619644	−0.309822	+	0.950795i	$0.600270\pi$
$594$	1239.31	0.0856050
$595$	4349.33	0.299672
$596$	−8671.93	−0.596000
$597$	6324.33	0.433564
$598$	14440.6	0.987489
$599$	−176.090	−0.0120114	−0.00600570	−	0.999982i	$-0.501912\pi$
$599$	−176.090	−0.0120114	−0.00600570	+	0.999982i	$0.501912\pi$
$600$	−8210.36	−0.558644
$601$	−10178.0	−0.690798	−0.345399	−	0.938456i	$-0.612256\pi$
$601$	−10178.0	−0.690798	−0.345399	+	0.938456i	$0.612256\pi$
$602$	58442.1	3.95668
$603$	−6410.21	−0.432909
$604$	21734.9	1.46421
$605$	−6033.81	−0.405470
$606$	−26601.3	−1.78318
$607$	19016.5	1.27159	0.635796	−	0.771857i	$-0.280672\pi$
$607$	19016.5	1.27159	0.635796	+	0.771857i	$0.280672\pi$
$608$	0	0
$609$	−21199.1	−1.41056
$610$	−16198.1	−1.07515
$611$	−15197.5	−1.00626
$612$	4118.43	0.272022
$613$	4042.66	0.266364	0.133182	−	0.991092i	$-0.457480\pi$
$613$	4042.66	0.266364	0.133182	+	0.991092i	$0.457480\pi$
$614$	−23570.9	−1.54926
$615$	3022.32	0.198165
$616$	−7139.46	−0.466976
$617$	9137.37	0.596202	0.298101	−	0.954534i	$-0.403647\pi$
$617$	9137.37	0.596202	0.298101	+	0.954534i	$0.403647\pi$
$618$	−16395.6	−1.06719
$619$	−3770.73	−0.244844	−0.122422	−	0.992478i	$-0.539066\pi$
$619$	−3770.73	−0.244844	−0.122422	+	0.992478i	$0.539066\pi$
$620$	12643.9	0.819017
$621$	−3480.69	−0.224920
$622$	−17262.0	−1.11277
$623$	−6935.77	−0.446029
$624$	1151.40	0.0738669
$625$	7235.31	0.463060
$626$	6230.45	0.397794
$627$	0	0
$628$	23021.7	1.46285
$629$	7300.91	0.462808
$630$	−5510.51	−0.348482
$631$	23833.2	1.50362	0.751809	−	0.659381i	$-0.229182\pi$
$631$	23833.2	1.50362	0.751809	+	0.659381i	$0.229182\pi$
$632$	−1724.80	−0.108558
$633$	16732.1	1.05062
$634$	−27415.8	−1.71739
$635$	−6467.87	−0.404204
$636$	5208.23	0.324717
$637$	−9038.98	−0.562225
$638$	−12089.9	−0.750226
$639$	2911.38	0.180238
$640$	12489.0	0.771364
$641$	−2135.31	−0.131575	−0.0657876	−	0.997834i	$-0.520956\pi$
$641$	−2135.31	−0.131575	−0.0657876	+	0.997834i	$0.520956\pi$
$642$	22726.2	1.39709
$643$	−2471.39	−0.151574	−0.0757869	−	0.997124i	$-0.524147\pi$
$643$	−2471.39	−0.151574	−0.0757869	+	0.997124i	$0.524147\pi$
$644$	47719.9	2.91992
$645$	6842.38	0.417703
$646$	0	0
$647$	−19184.9	−1.16574	−0.582871	−	0.812565i	$-0.698071\pi$
$647$	−19184.9	−1.16574	−0.582871	+	0.812565i	$0.698071\pi$
$648$	−2192.57	−0.132920
$649$	1401.15	0.0847456
$650$	−11325.4	−0.683415
$651$	−15087.8	−0.908352
$652$	32560.4	1.95578
$653$	4300.69	0.257732	0.128866	−	0.991662i	$-0.458866\pi$
$653$	4300.69	0.257732	0.128866	+	0.991662i	$0.458866\pi$
$654$	10000.1	0.597910
$655$	1121.08	0.0668766
$656$	−3296.81	−0.196218
$657$	−3059.10	−0.181654
$658$	−79339.4	−4.70057
$659$	−25671.5	−1.51748	−0.758741	−	0.651393i	$-0.774185\pi$
$659$	−25671.5	−1.51748	−0.758741	+	0.651393i	$0.774185\pi$
$660$	−1989.27	−0.117322
$661$	19697.5	1.15907	0.579533	−	0.814949i	$-0.303235\pi$
$661$	19697.5	1.15907	0.579533	+	0.814949i	$0.303235\pi$
$662$	22151.3	1.30051
$663$	2387.14	0.139832
$664$	−7455.17	−0.435718
$665$	0	0
$666$	−9250.09	−0.538189
$667$	33955.5	1.97116
$668$	4751.48	0.275210
$669$	−9209.23	−0.532211
$670$	16255.1	0.937300
$671$	6977.70	0.401447
$672$	−11417.8	−0.655436
$673$	9755.79	0.558779	0.279389	−	0.960178i	$-0.409868\pi$
$673$	9755.79	0.558779	0.279389	+	0.960178i	$0.409868\pi$
$674$	−39811.0	−2.27516
$675$	2729.83	0.155661
$676$	−22371.1	−1.27282
$677$	16060.3	0.911739	0.455870	−	0.890047i	$-0.349328\pi$
$677$	16060.3	0.911739	0.455870	+	0.890047i	$0.349328\pi$
$678$	−9804.84	−0.555387
$679$	13053.5	0.737773
$680$	−4388.36	−0.247480
$681$	−16175.4	−0.910195
$682$	−8604.62	−0.483120
$683$	13918.2	0.779741	0.389871	−	0.920870i	$-0.372520\pi$
$683$	13918.2	0.779741	0.389871	+	0.920870i	$0.372520\pi$
$684$	0	0
$685$	−6536.49	−0.364594
$686$	−4226.25	−0.235217
$687$	816.569	0.0453480
$688$	−7463.81	−0.413598
$689$	3018.81	0.166920
$690$	8826.40	0.486979
$691$	21265.4	1.17073	0.585366	−	0.810769i	$-0.300951\pi$
$691$	21265.4	1.17073	0.585366	+	0.810769i	$0.300951\pi$
$692$	7313.53	0.401761
$693$	2373.77	0.130119
$694$	−24518.0	−1.34105
$695$	13812.5	0.753868
$696$	21389.4	1.16489
$697$	−6835.09	−0.371446
$698$	24407.1	1.32353
$699$	−6867.72	−0.371618
$700$	−37425.7	−2.02080
$701$	−5426.60	−0.292382	−0.146191	−	0.989256i	$-0.546701\pi$
$701$	−5426.60	−0.292382	−0.146191	+	0.989256i	$0.546701\pi$
$702$	−3024.45	−0.162608
$703$	0	0
$704$	−7769.77	−0.415958
$705$	−9289.03	−0.496234
$706$	24275.8	1.29410
$707$	−50952.4	−2.71041
$708$	−5899.38	−0.313153
$709$	23370.9	1.23796	0.618978	−	0.785408i	$-0.287547\pi$
$709$	23370.9	1.23796	0.618978	+	0.785408i	$0.287547\pi$
$710$	−7382.73	−0.390238
$711$	573.473	0.0302488
$712$	6998.02	0.368345
$713$	24166.7	1.26936
$714$	12462.2	0.653203
$715$	−1153.03	−0.0603088
$716$	8604.57	0.449117
$717$	−12939.3	−0.673957
$718$	55021.0	2.85984
$719$	−11136.7	−0.577646	−0.288823	−	0.957382i	$-0.593264\pi$
$719$	−11136.7	−0.577646	−0.288823	+	0.957382i	$0.593264\pi$
$720$	703.763	0.0364274
$721$	−31404.1	−1.62212
$722$	0	0
$723$	−1725.78	−0.0887724
$724$	−37792.0	−1.93996
$725$	−26630.6	−1.36419
$726$	−17288.8	−0.883811
$727$	13633.2	0.695497	0.347748	−	0.937588i	$-0.386946\pi$
$727$	13633.2	0.695497	0.347748	+	0.937588i	$0.386946\pi$
$728$	17423.4	0.887025
$729$	729.000	0.0370370
$730$	7757.31	0.393303
$731$	−15474.3	−0.782951
$732$	−29378.8	−1.48343
$733$	31.3799	0.00158123	0.000790617	−	1.00000i	$-0.499748\pi$
$733$	31.3799	0.00158123	0.000790617		1.00000i	$0.499748\pi$
$734$	42969.7	2.16082
$735$	−5524.83	−0.277260
$736$	18288.4	0.915924
$737$	−7002.27	−0.349975
$738$	8659.91	0.431946
$739$	22363.4	1.11320	0.556598	−	0.830782i	$-0.312107\pi$
$739$	22363.4	1.11320	0.556598	+	0.830782i	$0.312107\pi$
$740$	14847.8	0.737588
$741$	0	0
$742$	15759.9	0.779737
$743$	21022.6	1.03801	0.519007	−	0.854770i	$-0.326302\pi$
$743$	21022.6	1.03801	0.519007	+	0.854770i	$0.326302\pi$
$744$	15223.2	0.750148
$745$	−3072.28	−0.151087
$746$	9687.07	0.475427
$747$	2478.74	0.121409
$748$	4498.81	0.219910
$749$	43529.8	2.12356
$750$	−15480.7	−0.753703
$751$	−37616.6	−1.82776	−0.913881	−	0.405983i	$-0.866929\pi$
$751$	−37616.6	−1.82776	−0.913881	+	0.405983i	$0.866929\pi$
$752$	10132.7	0.491357
$753$	−14128.4	−0.683755
$754$	29504.7	1.42506
$755$	7700.22	0.371178
$756$	−9994.51	−0.480816
$757$	31136.7	1.49496	0.747478	−	0.664287i	$-0.231265\pi$
$757$	31136.7	1.49496	0.747478	+	0.664287i	$0.231265\pi$
$758$	4003.51	0.191839
$759$	−3802.17	−0.181831
$760$	0	0
$761$	11216.4	0.534289	0.267145	−	0.963656i	$-0.413920\pi$
$761$	11216.4	0.534289	0.267145	+	0.963656i	$0.413920\pi$
$762$	−18532.5	−0.881053
$763$	19154.2	0.908817
$764$	−42083.3	−1.99283
$765$	1459.07	0.0689580
$766$	−8231.92	−0.388291
$767$	−3419.42	−0.160975
$768$	16817.6	0.790171
$769$	3498.50	0.164056	0.0820280	−	0.996630i	$-0.473860\pi$
$769$	3498.50	0.164056	0.0820280	+	0.996630i	$0.473860\pi$
$770$	−6019.46	−0.281723
$771$	−14516.5	−0.678078
$772$	9386.68	0.437609
$773$	29915.7	1.39197	0.695985	−	0.718057i	$-0.254968\pi$
$773$	29915.7	1.39197	0.695985	+	0.718057i	$0.254968\pi$
$774$	19605.6	0.910476
$775$	−18953.5	−0.878488
$776$	−13170.7	−0.609277
$777$	−17717.7	−0.818041
$778$	−10052.6	−0.463241
$779$	0	0
$780$	4854.69	0.222854
$781$	3180.28	0.145710
$782$	−19961.2	−0.912803
$783$	−7111.68	−0.324586
$784$	6026.60	0.274535
$785$	8156.11	0.370833
$786$	3212.25	0.145772
$787$	−16611.1	−0.752378	−0.376189	−	0.926543i	$-0.622766\pi$
$787$	−16611.1	−0.752378	−0.376189	+	0.926543i	$0.622766\pi$
$788$	26087.5	1.17935
$789$	19597.9	0.884290
$790$	−1454.22	−0.0654924
$791$	−18780.2	−0.844183
$792$	−2395.08	−0.107456
$793$	−17028.6	−0.762553
$794$	−37729.2	−1.68634
$795$	1845.17	0.0823161
$796$	−29087.2	−1.29519
$797$	−33959.3	−1.50928	−0.754641	−	0.656137i	$-0.772189\pi$
$797$	−33959.3	−1.50928	−0.754641	+	0.656137i	$0.772189\pi$
$798$	0	0
$799$	21007.5	0.930152
$800$	−14343.2	−0.633887
$801$	−2326.75	−0.102636
$802$	−72754.9	−3.20332
$803$	−3341.64	−0.146854
$804$	29482.3	1.29323
$805$	16906.1	0.740202
$806$	20999.0	0.917691
$807$	−19136.4	−0.834737
$808$	51409.7	2.23835
$809$	8680.58	0.377247	0.188624	−	0.982049i	$-0.439597\pi$
$809$	8680.58	0.377247	0.188624	+	0.982049i	$0.439597\pi$
$810$	−1848.61	−0.0801896
$811$	−40693.4	−1.76195	−0.880974	−	0.473165i	$-0.843111\pi$
$811$	−40693.4	−1.76195	−0.880974	+	0.473165i	$0.843111\pi$
$812$	97500.3	4.21378
$813$	6133.00	0.264568
$814$	−10104.4	−0.435087
$815$	11535.5	0.495791
$816$	−1591.59	−0.0682802
$817$	0	0
$818$	23813.3	1.01787
$819$	−5793.05	−0.247162
$820$	−13900.5	−0.591981
$821$	−32533.2	−1.38297	−0.691484	−	0.722392i	$-0.743043\pi$
$821$	−32533.2	−1.38297	−0.691484	+	0.722392i	$0.743043\pi$
$822$	−18729.1	−0.794713
$823$	−28630.2	−1.21262	−0.606310	−	0.795228i	$-0.707351\pi$
$823$	−28630.2	−1.21262	−0.606310	+	0.795228i	$0.707351\pi$
$824$	31686.0	1.33960
$825$	2981.96	0.125841
$826$	−17851.3	−0.751969
$827$	14235.1	0.598554	0.299277	−	0.954166i	$-0.403255\pi$
$827$	14235.1	0.598554	0.299277	+	0.954166i	$0.403255\pi$
$828$	16008.6	0.671905
$829$	−3929.50	−0.164629	−0.0823144	−	0.996606i	$-0.526231\pi$
$829$	−3929.50	−0.164629	−0.0823144	+	0.996606i	$0.526231\pi$
$830$	−6285.65	−0.262865
$831$	−9269.63	−0.386955
$832$	18961.6	0.790116
$833$	12494.6	0.519703
$834$	39577.2	1.64322
$835$	1683.35	0.0697660
$836$	0	0
$837$	−5061.51	−0.209022
$838$	52470.5	2.16296
$839$	22085.5	0.908793	0.454397	−	0.890799i	$-0.349855\pi$
$839$	22085.5	0.908793	0.454397	+	0.890799i	$0.349855\pi$
$840$	10649.6	0.437435
$841$	44988.2	1.84461
$842$	−27641.1	−1.13132
$843$	7864.54	0.321316
$844$	−76955.3	−3.13852
$845$	−7925.62	−0.322662
$846$	−26616.0	−1.08165
$847$	−33115.0	−1.34338
$848$	−2012.75	−0.0815071
$849$	−14095.8	−0.569809
$850$	15655.2	0.631727
$851$	28379.1	1.14315
$852$	−13390.2	−0.538428
$853$	13041.7	0.523493	0.261746	−	0.965137i	$-0.415702\pi$
$853$	13041.7	0.523493	0.261746	+	0.965137i	$0.415702\pi$
$854$	−88899.2	−3.56214
$855$	0	0
$856$	−43920.5	−1.75371
$857$	1122.98	0.0447612	0.0223806	−	0.999750i	$-0.492875\pi$
$857$	1122.98	0.0447612	0.0223806	+	0.999750i	$0.492875\pi$
$858$	−3303.79	−0.131456
$859$	−21262.4	−0.844543	−0.422271	−	0.906469i	$-0.638767\pi$
$859$	−21262.4	−0.844543	−0.422271	+	0.906469i	$0.638767\pi$
$860$	−31469.9	−1.24781
$861$	16587.2	0.656553
$862$	65831.3	2.60119
$863$	−8303.17	−0.327512	−0.163756	−	0.986501i	$-0.552361\pi$
$863$	−8303.17	−0.327512	−0.163756	+	0.986501i	$0.552361\pi$
$864$	−3830.35	−0.150823
$865$	2591.03	0.101847
$866$	−5075.99	−0.199179
$867$	11439.3	0.448094
$868$	69392.8	2.71353
$869$	626.440	0.0244540
$870$	18033.9	0.702767
$871$	17088.6	0.664782
$872$	−19326.1	−0.750532
$873$	4379.07	0.169770
$874$	0	0
$875$	−29651.9	−1.14562
$876$	14069.6	0.542657
$877$	49224.5	1.89532	0.947659	−	0.319284i	$-0.103442\pi$
$877$	49224.5	1.89532	0.947659	+	0.319284i	$0.103442\pi$
$878$	43660.3	1.67821
$879$	22248.7	0.853730
$880$	768.763	0.0294489
$881$	−32900.4	−1.25816	−0.629082	−	0.777339i	$-0.716569\pi$
$881$	−32900.4	−1.25816	−0.629082	+	0.777339i	$0.716569\pi$
$882$	−15830.4	−0.604351
$883$	−2014.34	−0.0767700	−0.0383850	−	0.999263i	$-0.512221\pi$
$883$	−2014.34	−0.0767700	−0.0383850	+	0.999263i	$0.512221\pi$
$884$	−10979.1	−0.417722
$885$	−2090.02	−0.0793846
$886$	−32618.0	−1.23682
$887$	−27490.2	−1.04062	−0.520310	−	0.853978i	$-0.674184\pi$
$887$	−27490.2	−1.04062	−0.520310	+	0.853978i	$0.674184\pi$
$888$	17876.7	0.675566
$889$	−35497.3	−1.33919
$890$	5900.21	0.222220
$891$	796.331	0.0299418
$892$	42355.7	1.58988
$893$	0	0
$894$	−8803.06	−0.329327
$895$	3048.41	0.113852
$896$	68542.9	2.55565
$897$	9278.96	0.345391
$898$	−29494.1	−1.09602
$899$	49377.0	1.83183
$900$	−12555.2	−0.465008
$901$	−4172.91	−0.154295
$902$	9459.76	0.349197
$903$	37552.7	1.38391
$904$	18948.8	0.697155
$905$	−13388.9	−0.491781
$906$	22063.6	0.809065
$907$	25452.7	0.931800	0.465900	−	0.884837i	$-0.345731\pi$
$907$	25452.7	0.931800	0.465900	+	0.884837i	$0.345731\pi$
$908$	74394.9	2.71903
$909$	−17093.0	−0.623696
$910$	14690.1	0.535135
$911$	34009.0	1.23685	0.618424	−	0.785845i	$-0.287772\pi$
$911$	34009.0	1.23685	0.618424	+	0.785845i	$0.287772\pi$
$912$	0	0
$913$	2707.68	0.0981504
$914$	−34896.4	−1.26288
$915$	−10408.3	−0.376052
$916$	−3755.62	−0.135468
$917$	6152.76	0.221573
$918$	4180.71	0.150309
$919$	−32908.8	−1.18124	−0.590621	−	0.806949i	$-0.701117\pi$
$919$	−32908.8	−1.18124	−0.590621	+	0.806949i	$0.701117\pi$
$920$	−17057.9	−0.611284
$921$	−15145.8	−0.541878
$922$	−44496.3	−1.58938
$923$	−7761.26	−0.276777
$924$	−10917.6	−0.388705
$925$	−22257.1	−0.791146
$926$	−61506.7	−2.18276
$927$	−10535.2	−0.373269
$928$	37366.5	1.32179
$929$	40426.1	1.42770	0.713851	−	0.700297i	$-0.246949\pi$
$929$	40426.1	1.42770	0.713851	+	0.700297i	$0.246949\pi$
$930$	12835.1	0.452558
$931$	0	0
$932$	31586.4	1.11014
$933$	−11091.9	−0.389210
$934$	−30848.9	−1.08074
$935$	1593.83	0.0557475
$936$	5845.04	0.204114
$937$	1687.30	0.0588279	0.0294139	−	0.999567i	$-0.490636\pi$
$937$	1687.30	0.0588279	0.0294139	+	0.999567i	$0.490636\pi$
$938$	89212.2	3.10542
$939$	4003.45	0.139135
$940$	42722.7	1.48241
$941$	3950.04	0.136841	0.0684207	−	0.997657i	$-0.478204\pi$
$941$	3950.04	0.136841	0.0684207	+	0.997657i	$0.478204\pi$
$942$	23369.9	0.808314
$943$	−26568.5	−0.917485
$944$	2279.84	0.0786044
$945$	−3540.84	−0.121887
$946$	21416.4	0.736054
$947$	30460.3	1.04522	0.522612	−	0.852570i	$-0.324958\pi$
$947$	30460.3	1.04522	0.522612	+	0.852570i	$0.324958\pi$
$948$	−2637.55	−0.0903626
$949$	8155.05	0.278951
$950$	0	0
$951$	−17616.4	−0.600684
$952$	−24084.4	−0.819938
$953$	−49900.4	−1.69615	−0.848076	−	0.529874i	$-0.822239\pi$
$953$	−49900.4	−1.69615	−0.848076	+	0.529874i	$0.822239\pi$
$954$	5286.99	0.179426
$955$	−14909.2	−0.505184
$956$	59511.2	2.01332
$957$	−7768.52	−0.262404
$958$	−57642.4	−1.94399
$959$	−35873.9	−1.20795
$960$	11589.8	0.389644
$961$	5351.52	0.179636
$962$	24659.2	0.826451
$963$	14603.0	0.488655
$964$	7937.32	0.265191
$965$	3325.50	0.110934
$966$	48441.5	1.61344
$967$	−18205.1	−0.605416	−0.302708	−	0.953083i	$-0.597891\pi$
$967$	−18205.1	−0.605416	−0.302708	+	0.953083i	$0.597891\pi$
$968$	33412.2	1.10941
$969$	0	0
$970$	−11104.5	−0.367572
$971$	−7539.03	−0.249165	−0.124582	−	0.992209i	$-0.539759\pi$
$971$	−7539.03	−0.249165	−0.124582	+	0.992209i	$0.539759\pi$
$972$	−3352.86	−0.110641
$973$	75806.4	2.49768
$974$	−7374.87	−0.242614
$975$	−7277.29	−0.239036
$976$	11353.6	0.372356
$977$	−8821.91	−0.288882	−0.144441	−	0.989513i	$-0.546138\pi$
$977$	−8821.91	−0.288882	−0.144441	+	0.989513i	$0.546138\pi$
$978$	33052.8	1.08069
$979$	−2541.65	−0.0829739
$980$	25410.1	0.828263
$981$	6425.66	0.209129
$982$	−7618.10	−0.247559
$983$	−33885.4	−1.09947	−0.549733	−	0.835340i	$-0.685271\pi$
$983$	−33885.4	−1.09947	−0.549733	+	0.835340i	$0.685271\pi$
$984$	−16736.1	−0.542203
$985$	9242.25	0.298967
$986$	−40784.4	−1.31728
$987$	−50980.5	−1.64410
$988$	0	0
$989$	−60149.6	−1.93392
$990$	−2019.35	−0.0648275
$991$	−48884.1	−1.56696	−0.783478	−	0.621420i	$-0.786556\pi$
$991$	−48884.1	−1.56696	−0.783478	+	0.621420i	$0.786556\pi$
$992$	26594.5	0.851184
$993$	14233.6	0.454874
$994$	−40518.2	−1.29292
$995$	−10305.0	−0.328332
$996$	−11400.4	−0.362686
$997$	59232.5	1.88156	0.940779	−	0.339021i	$-0.110096\pi$
$997$	59232.5	1.88156	0.940779	+	0.339021i	$0.110096\pi$
$998$	37336.3	1.18423
$999$	−5943.76	−0.188240

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1083.4.a.s.1.3		18
19.6	even	9		57.4.i.b.55.1	yes	36
19.16	even	9		57.4.i.b.28.1	✓	36
19.18	odd	2		1083.4.a.t.1.16		18
57.35	odd	18		171.4.u.c.28.6		36
57.44	odd	18		171.4.u.c.55.6		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.4.i.b.28.1	✓	36	19.16	even	9
57.4.i.b.55.1	yes	36	19.6	even	9
171.4.u.c.28.6		36	57.35	odd	18
171.4.u.c.55.6		36	57.44	odd	18
1083.4.a.s.1.3		18	1.1	even	1	trivial
1083.4.a.t.1.16		18	19.18	odd	2

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:	\( N \)	\(=\)	\( 1083 = 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1083.a (trivial)

\(f(q)\)	\(=\)	\(q-4.66881 q^{2} -3.00000 q^{3} +13.7978 q^{4} +4.88826 q^{5} +14.0064 q^{6} +26.8280 q^{7} -27.0688 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.66881 q^{2} -3.00000 q^{3} +13.7978 q^{4} +4.88826 q^{5} +14.0064 q^{6} +26.8280 q^{7} -27.0688 q^{8} +9.00000 q^{9} -22.8224 q^{10} +9.83125 q^{11} -41.3934 q^{12} -23.9925 q^{13} -125.255 q^{14} -14.6648 q^{15} +15.9967 q^{16} +33.1650 q^{17} -42.0193 q^{18} +67.4472 q^{20} -80.4840 q^{21} -45.9002 q^{22} +128.914 q^{23} +81.2063 q^{24} -101.105 q^{25} +112.017 q^{26} -27.0000 q^{27} +370.167 q^{28} +263.395 q^{29} +68.4671 q^{30} +187.463 q^{31} +141.865 q^{32} -29.4938 q^{33} -154.841 q^{34} +131.142 q^{35} +124.180 q^{36} +220.139 q^{37} +71.9776 q^{39} -132.319 q^{40} -206.094 q^{41} +375.764 q^{42} -466.586 q^{43} +135.650 q^{44} +43.9944 q^{45} -601.877 q^{46} +633.424 q^{47} -47.9900 q^{48} +376.741 q^{49} +472.040 q^{50} -99.4949 q^{51} -331.044 q^{52} -125.823 q^{53} +126.058 q^{54} +48.0577 q^{55} -726.201 q^{56} -1229.74 q^{58} +142.520 q^{59} -202.342 q^{60} +709.747 q^{61} -875.231 q^{62} +241.452 q^{63} -790.313 q^{64} -117.282 q^{65} +137.701 q^{66} -712.246 q^{67} +457.603 q^{68} -386.743 q^{69} -612.278 q^{70} +323.486 q^{71} -243.619 q^{72} -339.899 q^{73} -1027.79 q^{74} +303.315 q^{75} +263.753 q^{77} -336.050 q^{78} +63.7192 q^{79} +78.1959 q^{80} +81.0000 q^{81} +962.213 q^{82} +275.416 q^{83} -1110.50 q^{84} +162.119 q^{85} +2178.40 q^{86} -790.186 q^{87} -266.120 q^{88} -258.528 q^{89} -205.401 q^{90} -643.672 q^{91} +1778.73 q^{92} -562.390 q^{93} -2957.34 q^{94} -425.594 q^{96} +486.563 q^{97} -1758.93 q^{98} +88.4813 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 54 q^{3} + 96 q^{4} + 18 q^{5} + 48 q^{7} + 57 q^{8} + 162 q^{9}+O(q^{10}) \) 18 * q - 54 * q^3 + 96 * q^4 + 18 * q^5 + 48 * q^7 + 57 * q^8 + 162 * q^9	\( 18 q - 54 q^{3} + 96 q^{4} + 18 q^{5} + 48 q^{7} + 57 q^{8} + 162 q^{9} - 60 q^{10} + 108 q^{11} - 288 q^{12} - 42 q^{13} + 60 q^{14} - 54 q^{15} + 576 q^{16} + 300 q^{17} + 27 q^{20} - 144 q^{21} - 219 q^{22} + 174 q^{23} - 171 q^{24} + 1068 q^{25} - 72 q^{26} - 486 q^{27} + 867 q^{28} + 168 q^{29} + 180 q^{30} - 1032 q^{31} + 921 q^{32} - 324 q^{33} + 75 q^{34} + 1524 q^{35} + 864 q^{36} + 132 q^{37} + 126 q^{39} - 363 q^{40} + 120 q^{41} - 180 q^{42} + 420 q^{43} + 2328 q^{44} + 162 q^{45} - 2229 q^{46} + 810 q^{47} - 1728 q^{48} + 1122 q^{49} - 1503 q^{50} - 900 q^{51} + 228 q^{52} - 174 q^{53} + 2550 q^{55} + 1119 q^{56} + 756 q^{58} + 474 q^{59} - 81 q^{60} + 1488 q^{61} + 333 q^{62} + 432 q^{63} + 2679 q^{64} - 1716 q^{65} + 657 q^{66} - 3060 q^{67} + 4623 q^{68} - 522 q^{69} - 1383 q^{70} + 1464 q^{71} + 513 q^{72} + 1470 q^{73} - 135 q^{74} - 3204 q^{75} + 1014 q^{77} + 216 q^{78} - 2508 q^{79} - 2049 q^{80} + 1458 q^{81} + 1485 q^{82} + 4764 q^{83} - 2601 q^{84} + 804 q^{85} - 1068 q^{86} - 504 q^{87} - 3012 q^{88} + 1050 q^{89} - 540 q^{90} + 3408 q^{91} + 3306 q^{92} + 3096 q^{93} - 8205 q^{94} - 2763 q^{96} + 2070 q^{97} + 1767 q^{98} + 972 q^{99}+O(q^{100}) \) 18 * q - 54 * q^3 + 96 * q^4 + 18 * q^5 + 48 * q^7 + 57 * q^8 + 162 * q^9 - 60 * q^10 + 108 * q^11 - 288 * q^12 - 42 * q^13 + 60 * q^14 - 54 * q^15 + 576 * q^16 + 300 * q^17 + 27 * q^20 - 144 * q^21 - 219 * q^22 + 174 * q^23 - 171 * q^24 + 1068 * q^25 - 72 * q^26 - 486 * q^27 + 867 * q^28 + 168 * q^29 + 180 * q^30 - 1032 * q^31 + 921 * q^32 - 324 * q^33 + 75 * q^34 + 1524 * q^35 + 864 * q^36 + 132 * q^37 + 126 * q^39 - 363 * q^40 + 120 * q^41 - 180 * q^42 + 420 * q^43 + 2328 * q^44 + 162 * q^45 - 2229 * q^46 + 810 * q^47 - 1728 * q^48 + 1122 * q^49 - 1503 * q^50 - 900 * q^51 + 228 * q^52 - 174 * q^53 + 2550 * q^55 + 1119 * q^56 + 756 * q^58 + 474 * q^59 - 81 * q^60 + 1488 * q^61 + 333 * q^62 + 432 * q^63 + 2679 * q^64 - 1716 * q^65 + 657 * q^66 - 3060 * q^67 + 4623 * q^68 - 522 * q^69 - 1383 * q^70 + 1464 * q^71 + 513 * q^72 + 1470 * q^73 - 135 * q^74 - 3204 * q^75 + 1014 * q^77 + 216 * q^78 - 2508 * q^79 - 2049 * q^80 + 1458 * q^81 + 1485 * q^82 + 4764 * q^83 - 2601 * q^84 + 804 * q^85 - 1068 * q^86 - 504 * q^87 - 3012 * q^88 + 1050 * q^89 - 540 * q^90 + 3408 * q^91 + 3306 * q^92 + 3096 * q^93 - 8205 * q^94 - 2763 * q^96 + 2070 * q^97 + 1767 * q^98 + 972 * q^99

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