LMFDB - Embedded newform 1083.4.a.s.1.14

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			1.14
Root			$3.50909$ 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)

$f(q)$	$=$	$q+3.50909 q^{2} -3.00000 q^{3} +4.31372 q^{4} -14.5964 q^{5} -10.5273 q^{6} -16.0084 q^{7} -12.9355 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.50909 q^{2} -3.00000 q^{3} +4.31372 q^{4} -14.5964 q^{5} -10.5273 q^{6} -16.0084 q^{7} -12.9355 q^{8} +9.00000 q^{9} -51.2202 q^{10} -57.7554 q^{11} -12.9412 q^{12} -58.3434 q^{13} -56.1750 q^{14} +43.7893 q^{15} -79.9016 q^{16} +52.7169 q^{17} +31.5818 q^{18} -62.9649 q^{20} +48.0252 q^{21} -202.669 q^{22} -115.436 q^{23} +38.8065 q^{24} +88.0557 q^{25} -204.732 q^{26} -27.0000 q^{27} -69.0558 q^{28} +206.064 q^{29} +153.661 q^{30} +187.836 q^{31} -176.898 q^{32} +173.266 q^{33} +184.988 q^{34} +233.666 q^{35} +38.8235 q^{36} +39.8244 q^{37} +175.030 q^{39} +188.812 q^{40} +34.8820 q^{41} +168.525 q^{42} -199.246 q^{43} -249.141 q^{44} -131.368 q^{45} -405.075 q^{46} -621.811 q^{47} +239.705 q^{48} -86.7308 q^{49} +308.996 q^{50} -158.151 q^{51} -251.677 q^{52} +219.948 q^{53} -94.7455 q^{54} +843.023 q^{55} +207.077 q^{56} +723.098 q^{58} +350.205 q^{59} +188.895 q^{60} +102.916 q^{61} +659.132 q^{62} -144.076 q^{63} +18.4613 q^{64} +851.605 q^{65} +608.007 q^{66} -86.4459 q^{67} +227.406 q^{68} +346.308 q^{69} +819.954 q^{70} -968.159 q^{71} -116.419 q^{72} -101.101 q^{73} +139.747 q^{74} -264.167 q^{75} +924.572 q^{77} +614.196 q^{78} -789.590 q^{79} +1166.28 q^{80} +81.0000 q^{81} +122.404 q^{82} -373.615 q^{83} +207.167 q^{84} -769.478 q^{85} -699.172 q^{86} -618.192 q^{87} +747.094 q^{88} -160.088 q^{89} -460.982 q^{90} +933.984 q^{91} -497.959 q^{92} -563.507 q^{93} -2181.99 q^{94} +530.694 q^{96} -1023.02 q^{97} -304.346 q^{98} -519.799 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.50909	1.24065	0.620326	−	0.784344i	$-0.287000\pi$
$2$	3.50909	1.24065	0.620326	+	0.784344i	$0.287000\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.31372	0.539215
$5$	−14.5964	−1.30554	−0.652772	−	0.757554i	$-0.726394\pi$
$5$	−14.5964	−1.30554	−0.652772	+	0.757554i	$0.726394\pi$
$6$	−10.5273	−0.716290
$7$	−16.0084	−0.864373	−0.432186	−	0.901784i	$-0.642258\pi$
$7$	−16.0084	−0.864373	−0.432186	+	0.901784i	$0.642258\pi$
$8$	−12.9355	−0.571673
$9$	9.00000	0.333333
$10$	−51.2202	−1.61972
$11$	−57.7554	−1.58308	−0.791541	−	0.611115i	$-0.790721\pi$
$11$	−57.7554	−1.58308	−0.791541	+	0.611115i	$0.790721\pi$
$12$	−12.9412	−0.311316
$13$	−58.3434	−1.24473	−0.622367	−	0.782726i	$-0.713829\pi$
$13$	−58.3434	−1.24473	−0.622367	+	0.782726i	$0.713829\pi$
$14$	−56.1750	−1.07239
$15$	43.7893	0.753756
$16$	−79.9016	−1.24846
$17$	52.7169	0.752101	0.376050	−	0.926599i	$-0.377282\pi$
$17$	52.7169	0.752101	0.376050	+	0.926599i	$0.377282\pi$
$18$	31.5818	0.413550
$19$	0	0
$19$	0	0
$20$	−62.9649	−0.703969
$21$	48.0252	0.499046
$22$	−202.669	−1.96405
$23$	−115.436	−1.04652	−0.523262	−	0.852172i	$-0.675285\pi$
$23$	−115.436	−1.04652	−0.523262	+	0.852172i	$0.675285\pi$
$24$	38.8065	0.330056
$25$	88.0557	0.704446
$26$	−204.732	−1.54428
$27$	−27.0000	−0.192450
$28$	−69.0558	−0.466083
$29$	206.064	1.31949	0.659744	−	0.751491i	$-0.270665\pi$
$29$	206.064	1.31949	0.659744	+	0.751491i	$0.270665\pi$
$30$	153.661	0.935149
$31$	187.836	1.08827	0.544133	−	0.838999i	$-0.316859\pi$
$31$	187.836	1.08827	0.544133	+	0.838999i	$0.316859\pi$
$32$	−176.898	−0.977233
$33$	173.266	0.913993
$34$	184.988	0.933095
$35$	233.666	1.12848
$36$	38.8235	0.179738
$37$	39.8244	0.176948	0.0884741	−	0.996078i	$-0.471801\pi$
$37$	39.8244	0.176948	0.0884741	+	0.996078i	$0.471801\pi$
$38$	0	0
$39$	175.030	0.718647
$40$	188.812	0.746345
$41$	34.8820	0.132870	0.0664348	−	0.997791i	$-0.478838\pi$
$41$	34.8820	0.132870	0.0664348	+	0.997791i	$0.478838\pi$
$42$	168.525	0.619142
$43$	−199.246	−0.706621	−0.353311	−	0.935506i	$-0.614944\pi$
$43$	−199.246	−0.706621	−0.353311	+	0.935506i	$0.614944\pi$
$44$	−249.141	−0.853622
$45$	−131.368	−0.435181
$46$	−405.075	−1.29837
$47$	−621.811	−1.92980	−0.964899	−	0.262621i	$-0.915413\pi$
$47$	−621.811	−1.92980	−0.964899	+	0.262621i	$0.915413\pi$
$48$	239.705	0.720800
$49$	−86.7308	−0.252859
$50$	308.996	0.873971
$51$	−158.151	−0.434226
$52$	−251.677	−0.671179
$53$	219.948	0.570042	0.285021	−	0.958521i	$-0.407999\pi$
$53$	219.948	0.570042	0.285021	+	0.958521i	$0.407999\pi$
$54$	−94.7455	−0.238763
$55$	843.023	2.06678
$56$	207.077	0.494139
$57$	0	0
$58$	723.098	1.63702
$59$	350.205	0.772760	0.386380	−	0.922340i	$-0.373725\pi$
$59$	350.205	0.772760	0.386380	+	0.922340i	$0.373725\pi$
$60$	188.895	0.406437
$61$	102.916	0.216016	0.108008	−	0.994150i	$-0.465553\pi$
$61$	102.916	0.216016	0.108008	+	0.994150i	$0.465553\pi$
$62$	659.132	1.35016
$63$	−144.076	−0.288124
$64$	18.4613	0.0360573
$65$	851.605	1.62505
$66$	608.007	1.13395
$67$	−86.4459	−0.157628	−0.0788138	−	0.996889i	$-0.525113\pi$
$67$	−86.4459	−0.157628	−0.0788138	+	0.996889i	$0.525113\pi$
$68$	227.406	0.405544
$69$	346.308	0.604211
$70$	819.954	1.40005
$71$	−968.159	−1.61830	−0.809150	−	0.587602i	$-0.800072\pi$
$71$	−968.159	−1.61830	−0.809150	+	0.587602i	$0.800072\pi$
$72$	−116.419	−0.190558
$73$	−101.101	−0.162096	−0.0810478	−	0.996710i	$-0.525827\pi$
$73$	−101.101	−0.162096	−0.0810478	+	0.996710i	$0.525827\pi$
$74$	139.747	0.219531
$75$	−264.167	−0.406712
$76$	0	0
$77$	924.572	1.36837
$78$	614.196	0.891590
$79$	−789.590	−1.12450	−0.562252	−	0.826966i	$-0.690065\pi$
$79$	−789.590	−1.12450	−0.562252	+	0.826966i	$0.690065\pi$
$80$	1166.28	1.62992
$81$	81.0000	0.111111
$82$	122.404	0.164845
$83$	−373.615	−0.494092	−0.247046	−	0.969004i	$-0.579460\pi$
$83$	−373.615	−0.494092	−0.247046	+	0.969004i	$0.579460\pi$
$84$	207.167	0.269093
$85$	−769.478	−0.981901
$86$	−699.172	−0.876670
$87$	−618.192	−0.761806
$88$	747.094	0.905006
$89$	−160.088	−0.190667	−0.0953333	−	0.995445i	$-0.530392\pi$
$89$	−160.088	−0.190667	−0.0953333	+	0.995445i	$0.530392\pi$
$90$	−460.982	−0.539908
$91$	933.984	1.07591
$92$	−497.959	−0.564302
$93$	−563.507	−0.628311
$94$	−2181.99	−2.39421
$95$	0	0
$96$	530.694	0.564206
$97$	−1023.02	−1.07084	−0.535421	−	0.844586i	$-0.679847\pi$
$97$	−1023.02	−1.07084	−0.535421	+	0.844586i	$0.679847\pi$
$98$	−304.346	−0.313710
$99$	−519.799	−0.527694
$100$	379.848	0.379848
$101$	1734.76	1.70906	0.854528	−	0.519405i	$-0.173846\pi$
$101$	1734.76	1.70906	0.854528	+	0.519405i	$0.173846\pi$
$102$	−554.965	−0.538723
$103$	1033.81	0.988974	0.494487	−	0.869185i	$-0.335356\pi$
$103$	1033.81	0.988974	0.494487	+	0.869185i	$0.335356\pi$
$104$	754.700	0.711581
$105$	−700.997	−0.651527
$106$	771.819	0.707223
$107$	1379.98	1.24680	0.623402	−	0.781901i	$-0.285750\pi$
$107$	1379.98	1.24680	0.623402	+	0.781901i	$0.285750\pi$
$108$	−116.470	−0.103772
$109$	−1474.40	−1.29561	−0.647805	−	0.761806i	$-0.724313\pi$
$109$	−1474.40	−1.29561	−0.647805	+	0.761806i	$0.724313\pi$
$110$	2958.24	2.56416
$111$	−119.473	−0.102161
$112$	1279.10	1.07914
$113$	593.199	0.493836	0.246918	−	0.969036i	$-0.420582\pi$
$113$	593.199	0.493836	0.246918	+	0.969036i	$0.420582\pi$
$114$	0	0
$115$	1684.95	1.36628
$116$	888.903	0.711488
$117$	−525.090	−0.414911
$118$	1228.90	0.958725
$119$	−843.913	−0.650096
$120$	−566.436	−0.430902
$121$	2004.69	1.50615
$122$	361.140	0.268001
$123$	−104.646	−0.0767123
$124$	810.271	0.586810
$125$	539.255	0.385859
$126$	−505.575	−0.357462
$127$	312.172	0.218116	0.109058	−	0.994035i	$-0.465217\pi$
$127$	312.172	0.218116	0.109058	+	0.994035i	$0.465217\pi$
$128$	1479.97	1.02197
$129$	597.738	0.407968
$130$	2988.36	2.01613
$131$	−1914.79	−1.27707	−0.638535	−	0.769592i	$-0.720459\pi$
$131$	−1914.79	−1.27707	−0.638535	+	0.769592i	$0.720459\pi$
$132$	747.422	0.492839
$133$	0	0
$134$	−303.347	−0.195561
$135$	394.104	0.251252
$136$	−681.918	−0.429956
$137$	−892.061	−0.556306	−0.278153	−	0.960537i	$-0.589722\pi$
$137$	−892.061	−0.556306	−0.278153	+	0.960537i	$0.589722\pi$
$138$	1215.23	0.749615
$139$	−2828.45	−1.72595	−0.862973	−	0.505251i	$-0.831400\pi$
$139$	−2828.45	−1.72595	−0.862973	+	0.505251i	$0.831400\pi$
$140$	1007.97	0.608492
$141$	1865.43	1.11417
$142$	−3397.36	−2.00775
$143$	3369.64	1.97052
$144$	−719.114	−0.416154
$145$	−3007.80	−1.72265
$146$	−354.773	−0.201104
$147$	260.192	0.145988
$148$	171.791	0.0954132
$149$	−342.736	−0.188443	−0.0942215	−	0.995551i	$-0.530036\pi$
$149$	−342.736	−0.188443	−0.0942215	+	0.995551i	$0.530036\pi$
$150$	−926.987	−0.504588
$151$	848.354	0.457206	0.228603	−	0.973520i	$-0.426584\pi$
$151$	848.354	0.457206	0.228603	+	0.973520i	$0.426584\pi$
$152$	0	0
$153$	474.452	0.250700
$154$	3244.41	1.69767
$155$	−2741.73	−1.42078
$156$	755.031	0.387505
$157$	2144.17	1.08996	0.544978	−	0.838450i	$-0.316538\pi$
$157$	2144.17	1.08996	0.544978	+	0.838450i	$0.316538\pi$
$158$	−2770.74	−1.39512
$159$	−659.845	−0.329114
$160$	2582.08	1.27582
$161$	1847.95	0.904587
$162$	284.236	0.137850
$163$	3220.66	1.54762	0.773809	−	0.633419i	$-0.218349\pi$
$163$	3220.66	1.54762	0.773809	+	0.633419i	$0.218349\pi$
$164$	150.471	0.0716453
$165$	−2529.07	−1.19326
$166$	−1311.05	−0.612995
$167$	1038.27	0.481101	0.240550	−	0.970637i	$-0.422672\pi$
$167$	1038.27	0.481101	0.240550	+	0.970637i	$0.422672\pi$
$168$	−621.230	−0.285291
$169$	1206.95	0.549361
$170$	−2700.17	−1.21820
$171$	0	0
$172$	−859.491	−0.381021
$173$	1383.78	0.608133	0.304066	−	0.952651i	$-0.401656\pi$
$173$	1383.78	0.608133	0.304066	+	0.952651i	$0.401656\pi$
$174$	−2169.29	−0.945136
$175$	−1409.63	−0.608904
$176$	4614.75	1.97642
$177$	−1050.62	−0.446153
$178$	−561.764	−0.236551
$179$	−2521.37	−1.05283	−0.526413	−	0.850229i	$-0.676463\pi$
$179$	−2521.37	−1.05283	−0.526413	+	0.850229i	$0.676463\pi$
$180$	−566.684	−0.234656
$181$	−4770.19	−1.95892	−0.979462	−	0.201629i	$-0.935377\pi$
$181$	−4770.19	−1.95892	−0.979462	+	0.201629i	$0.935377\pi$
$182$	3277.44	1.33483
$183$	−308.747	−0.124717
$184$	1493.22	0.598270
$185$	−581.294	−0.231014
$186$	−1977.40	−0.779515
$187$	−3044.68	−1.19064
$188$	−2682.32	−1.04058
$189$	432.227	0.166349
$190$	0	0
$191$	42.3261	0.0160346	0.00801731	−	0.999968i	$-0.497448\pi$
$191$	42.3261	0.0160346	0.00801731	+	0.999968i	$0.497448\pi$
$192$	−55.3839	−0.0208177
$193$	3005.14	1.12080	0.560401	−	0.828221i	$-0.310647\pi$
$193$	3005.14	1.12080	0.560401	+	0.828221i	$0.310647\pi$
$194$	−3589.86	−1.32854
$195$	−2554.81	−0.938226
$196$	−374.132	−0.136346
$197$	89.2240	0.0322688	0.0161344	−	0.999870i	$-0.494864\pi$
$197$	89.2240	0.0322688	0.0161344	+	0.999870i	$0.494864\pi$
$198$	−1824.02	−0.654685
$199$	1794.19	0.639129	0.319565	−	0.947564i	$-0.396463\pi$
$199$	1794.19	0.639129	0.319565	+	0.947564i	$0.396463\pi$
$200$	−1139.04	−0.402713
$201$	259.338	0.0910064
$202$	6087.42	2.12034
$203$	−3298.76	−1.14053
$204$	−682.217	−0.234141
$205$	−509.153	−0.173467
$206$	3627.73	1.22697
$207$	−1038.92	−0.348842
$208$	4661.73	1.55400
$209$	0	0
$210$	−2459.86	−0.808317
$211$	−5048.13	−1.64705	−0.823525	−	0.567280i	$-0.807996\pi$
$211$	−5048.13	−1.64705	−0.823525	+	0.567280i	$0.807996\pi$
$212$	948.796	0.307375
$213$	2904.48	0.934326
$214$	4842.49	1.54685
$215$	2908.28	0.922525
$216$	349.258	0.110019
$217$	−3006.95	−0.940668
$218$	−5173.79	−1.60740
$219$	303.303	0.0935859
$220$	3636.57	1.11444
$221$	−3075.68	−0.936165
$222$	−419.242	−0.126746
$223$	−5896.49	−1.77066	−0.885332	−	0.464960i	$-0.846069\pi$
$223$	−5896.49	−1.77066	−0.885332	+	0.464960i	$0.846069\pi$
$224$	2831.86	0.844694
$225$	792.501	0.234815
$226$	2081.59	0.612678
$227$	2468.81	0.721853	0.360926	−	0.932594i	$-0.382461\pi$
$227$	2468.81	0.721853	0.360926	+	0.932594i	$0.382461\pi$
$228$	0	0
$229$	−3588.34	−1.03548	−0.517739	−	0.855539i	$-0.673226\pi$
$229$	−3588.34	−1.03548	−0.517739	+	0.855539i	$0.673226\pi$
$230$	5912.65	1.69508
$231$	−2773.72	−0.790031
$232$	−2665.54	−0.754316
$233$	5896.61	1.65794	0.828969	−	0.559294i	$-0.188928\pi$
$233$	5896.61	1.65794	0.828969	+	0.559294i	$0.188928\pi$
$234$	−1842.59	−0.514760
$235$	9076.23	2.51944
$236$	1510.69	0.416684
$237$	2368.77	0.649233
$238$	−2961.37	−0.806542
$239$	−4713.59	−1.27572	−0.637859	−	0.770153i	$-0.720180\pi$
$239$	−4713.59	−1.27572	−0.637859	+	0.770153i	$0.720180\pi$
$240$	−3498.83	−0.941036
$241$	−3133.93	−0.837653	−0.418826	−	0.908066i	$-0.637558\pi$
$241$	−3133.93	−0.837653	−0.418826	+	0.908066i	$0.637558\pi$
$242$	7034.63	1.86861
$243$	−243.000	−0.0641500
$244$	443.949	0.116479
$245$	1265.96	0.330119
$246$	−367.212	−0.0951732
$247$	0	0
$248$	−2429.75	−0.622133
$249$	1120.85	0.285264
$250$	1892.29	0.478717
$251$	−677.491	−0.170370	−0.0851850	−	0.996365i	$-0.527148\pi$
$251$	−677.491	−0.170370	−0.0851850	+	0.996365i	$0.527148\pi$
$252$	−621.502	−0.155361
$253$	6667.05	1.65674
$254$	1095.44	0.270606
$255$	2308.43	0.566901
$256$	5045.65	1.23185
$257$	−4630.02	−1.12379	−0.561893	−	0.827210i	$-0.689927\pi$
$257$	−4630.02	−1.12379	−0.561893	+	0.827210i	$0.689927\pi$
$258$	2097.52	0.506146
$259$	−637.525	−0.152949
$260$	3673.58	0.876254
$261$	1854.58	0.439829
$262$	−6719.18	−1.58440
$263$	2645.92	0.620359	0.310180	−	0.950678i	$-0.399611\pi$
$263$	2645.92	0.620359	0.310180	+	0.950678i	$0.399611\pi$
$264$	−2241.28	−0.522506
$265$	−3210.46	−0.744215
$266$	0	0
$267$	480.265	0.110081
$268$	−372.904	−0.0849952
$269$	−139.886	−0.0317063	−0.0158532	−	0.999874i	$-0.505046\pi$
$269$	−139.886	−0.0317063	−0.0158532	+	0.999874i	$0.505046\pi$
$270$	1382.95	0.311716
$271$	3932.01	0.881376	0.440688	−	0.897660i	$-0.354735\pi$
$271$	3932.01	0.881376	0.440688	+	0.897660i	$0.354735\pi$
$272$	−4212.16	−0.938969
$273$	−2801.95	−0.621179
$274$	−3130.32	−0.690182
$275$	−5085.69	−1.11520
$276$	1493.88	0.325800
$277$	5619.00	1.21882	0.609409	−	0.792856i	$-0.291407\pi$
$277$	5619.00	1.21882	0.609409	+	0.792856i	$0.291407\pi$
$278$	−9925.31	−2.14130
$279$	1690.52	0.362756
$280$	−3022.58	−0.645120
$281$	−1150.40	−0.244224	−0.122112	−	0.992516i	$-0.538967\pi$
$281$	−1150.40	−0.244224	−0.122112	+	0.992516i	$0.538967\pi$
$282$	6545.98	1.38230
$283$	−1615.36	−0.339304	−0.169652	−	0.985504i	$-0.554264\pi$
$283$	−1615.36	−0.339304	−0.169652	+	0.985504i	$0.554264\pi$
$284$	−4176.37	−0.872612
$285$	0	0
$286$	11824.4	2.44472
$287$	−558.405	−0.114849
$288$	−1592.08	−0.325744
$289$	−2133.93	−0.434344
$290$	−10554.6	−2.13721
$291$	3069.05	0.618250
$292$	−436.122	−0.0874044
$293$	218.530	0.0435723	0.0217861	−	0.999763i	$-0.493065\pi$
$293$	218.530	0.0435723	0.0217861	+	0.999763i	$0.493065\pi$
$294$	913.039	0.181121
$295$	−5111.74	−1.00887
$296$	−515.148	−0.101157
$297$	1559.40	0.304664
$298$	−1202.69	−0.233792
$299$	6734.92	1.30264
$300$	−1139.54	−0.219305
$301$	3189.61	0.610784
$302$	2976.95	0.567233
$303$	−5204.27	−0.986724
$304$	0	0
$305$	−1502.20	−0.282019
$306$	1664.89	0.311032
$307$	3000.14	0.557743	0.278872	−	0.960328i	$-0.410040\pi$
$307$	3000.14	0.557743	0.278872	+	0.960328i	$0.410040\pi$
$308$	3988.35	0.737848
$309$	−3101.43	−0.570984
$310$	−9620.98	−1.76269
$311$	2473.57	0.451007	0.225503	−	0.974242i	$-0.427597\pi$
$311$	2473.57	0.451007	0.225503	+	0.974242i	$0.427597\pi$
$312$	−2264.10	−0.410831
$313$	1189.82	0.214865	0.107433	−	0.994212i	$-0.465737\pi$
$313$	1189.82	0.214865	0.107433	+	0.994212i	$0.465737\pi$
$314$	7524.07	1.35225
$315$	2102.99	0.376159
$316$	−3406.07	−0.606349
$317$	−4871.10	−0.863054	−0.431527	−	0.902100i	$-0.642025\pi$
$317$	−4871.10	−0.863054	−0.431527	+	0.902100i	$0.642025\pi$
$318$	−2315.46	−0.408315
$319$	−11901.3	−2.08886
$320$	−269.469	−0.0470743
$321$	−4139.95	−0.719843
$322$	6484.61	1.12228
$323$	0	0
$324$	349.411	0.0599128
$325$	−5137.47	−0.876847
$326$	11301.6	1.92005
$327$	4423.19	0.748021
$328$	−451.216	−0.0759580
$329$	9954.21	1.66807
$330$	−8874.73	−1.48042
$331$	6292.40	1.04490	0.522450	−	0.852670i	$-0.325018\pi$
$331$	6292.40	1.04490	0.522450	+	0.852670i	$0.325018\pi$
$332$	−1611.67	−0.266422
$333$	358.419	0.0589828
$334$	3643.39	0.596878
$335$	1261.80	0.205790
$336$	−3837.29	−0.623040
$337$	6.25398	0.00101091	0.000505454	−	1.00000i	$-0.499839\pi$
$337$	6.25398	0.00101091	0.000505454		1.00000i	$0.499839\pi$
$338$	4235.29	0.681566
$339$	−1779.60	−0.285116
$340$	−3319.31	−0.529456
$341$	−10848.5	−1.72282
$342$	0	0
$343$	6879.31	1.08294
$344$	2577.34	0.403956
$345$	−5054.86	−0.788824
$346$	4855.82	0.754481
$347$	−4352.19	−0.673308	−0.336654	−	0.941628i	$-0.609295\pi$
$347$	−4352.19	−0.673308	−0.336654	+	0.941628i	$0.609295\pi$
$348$	−2666.71	−0.410778
$349$	−6422.60	−0.985083	−0.492542	−	0.870289i	$-0.663932\pi$
$349$	−6422.60	−0.985083	−0.492542	+	0.870289i	$0.663932\pi$
$350$	−4946.53	−0.755437
$351$	1575.27	0.239549
$352$	10216.8	1.54704
$353$	−8456.04	−1.27498	−0.637492	−	0.770457i	$-0.720028\pi$
$353$	−8456.04	−1.27498	−0.637492	+	0.770457i	$0.720028\pi$
$354$	−3686.70	−0.553520
$355$	14131.7	2.11276
$356$	−690.576	−0.102810
$357$	2531.74	0.375333
$358$	−8847.70	−1.30619
$359$	−10785.1	−1.58556	−0.792782	−	0.609506i	$-0.791368\pi$
$359$	−10785.1	−1.58556	−0.792782	+	0.609506i	$0.791368\pi$
$360$	1699.31	0.248782
$361$	0	0
$362$	−16739.0	−2.43034
$363$	−6014.06	−0.869577
$364$	4028.95	0.580149
$365$	1475.71	0.211623
$366$	−1083.42	−0.154730
$367$	−1194.77	−0.169936	−0.0849681	−	0.996384i	$-0.527079\pi$
$367$	−1194.77	−0.169936	−0.0849681	+	0.996384i	$0.527079\pi$
$368$	9223.52	1.30655
$369$	313.938	0.0442899
$370$	−2039.81	−0.286608
$371$	−3521.02	−0.492729
$372$	−2430.81	−0.338795
$373$	8719.04	1.21033	0.605167	−	0.796099i	$-0.293106\pi$
$373$	8719.04	1.21033	0.605167	+	0.796099i	$0.293106\pi$
$374$	−10684.1	−1.47717
$375$	−1617.76	−0.222776
$376$	8043.43	1.10321
$377$	−12022.5	−1.64241
$378$	1516.72	0.206381
$379$	7409.59	1.00423	0.502117	−	0.864800i	$-0.332555\pi$
$379$	7409.59	1.00423	0.502117	+	0.864800i	$0.332555\pi$
$380$	0	0
$381$	−936.516	−0.125930
$382$	148.526	0.0198934
$383$	−3929.92	−0.524307	−0.262153	−	0.965026i	$-0.584433\pi$
$383$	−3929.92	−0.524307	−0.262153	+	0.965026i	$0.584433\pi$
$384$	−4439.90	−0.590033
$385$	−13495.5	−1.78647
$386$	10545.3	1.39052
$387$	−1793.21	−0.235540
$388$	−4413.01	−0.577414
$389$	8448.91	1.10123	0.550613	−	0.834761i	$-0.314394\pi$
$389$	8448.91	1.10123	0.550613	+	0.834761i	$0.314394\pi$
$390$	−8965.07	−1.16401
$391$	−6085.42	−0.787092
$392$	1121.91	0.144553
$393$	5744.38	0.737317
$394$	313.095	0.0400343
$395$	11525.2	1.46809
$396$	−2242.27	−0.284541
$397$	5377.90	0.679872	0.339936	−	0.940449i	$-0.389595\pi$
$397$	5377.90	0.679872	0.339936	+	0.940449i	$0.389595\pi$
$398$	6295.98	0.792937
$399$	0	0
$400$	−7035.79	−0.879474
$401$	481.387	0.0599484	0.0299742	−	0.999551i	$-0.490457\pi$
$401$	481.387	0.0599484	0.0299742	+	0.999551i	$0.490457\pi$
$402$	910.040	0.112907
$403$	−10959.0	−1.35460
$404$	7483.25	0.921549
$405$	−1182.31	−0.145060
$406$	−11575.6	−1.41500
$407$	−2300.07	−0.280124
$408$	2045.75	0.248235
$409$	−10349.8	−1.25125	−0.625627	−	0.780123i	$-0.715157\pi$
$409$	−10349.8	−1.25125	−0.625627	+	0.780123i	$0.715157\pi$
$410$	−1786.66	−0.215212
$411$	2676.18	0.321183
$412$	4459.57	0.533270
$413$	−5606.23	−0.667952
$414$	−3645.68	−0.432791
$415$	5453.45	0.645059
$416$	10320.8	1.21639
$417$	8485.36	0.996475
$418$	0	0
$419$	15955.2	1.86030	0.930148	−	0.367184i	$-0.119678\pi$
$419$	15955.2	1.86030	0.930148	+	0.367184i	$0.119678\pi$
$420$	−3023.91	−0.351313
$421$	11656.9	1.34946	0.674732	−	0.738062i	$-0.264259\pi$
$421$	11656.9	1.34946	0.674732	+	0.738062i	$0.264259\pi$
$422$	−17714.4	−2.04342
$423$	−5596.30	−0.643266
$424$	−2845.14	−0.325878
$425$	4642.02	0.529814
$426$	10192.1	1.15917
$427$	−1647.52	−0.186719
$428$	5952.86	0.672296
$429$	−10108.9	−1.13768
$430$	10205.4	1.14453
$431$	3971.13	0.443812	0.221906	−	0.975068i	$-0.428772\pi$
$431$	3971.13	0.443812	0.221906	+	0.975068i	$0.428772\pi$
$432$	2157.34	0.240267
$433$	−9040.87	−1.00341	−0.501705	−	0.865039i	$-0.667294\pi$
$433$	−9040.87	−1.00341	−0.501705	+	0.865039i	$0.667294\pi$
$434$	−10551.7	−1.16704
$435$	9023.40	0.994572
$436$	−6360.13	−0.698613
$437$	0	0
$438$	1064.32	0.116107
$439$	1192.36	0.129631	0.0648157	−	0.997897i	$-0.479354\pi$
$439$	1192.36	0.129631	0.0648157	+	0.997897i	$0.479354\pi$
$440$	−10904.9	−1.18153
$441$	−780.577	−0.0842865
$442$	−10792.8	−1.16145
$443$	4271.42	0.458106	0.229053	−	0.973414i	$-0.426437\pi$
$443$	4271.42	0.458106	0.229053	+	0.973414i	$0.426437\pi$
$444$	−515.374	−0.0550868
$445$	2336.72	0.248924
$446$	−20691.3	−2.19678
$447$	1028.21	0.108798
$448$	−295.536	−0.0311669
$449$	15857.8	1.66676	0.833379	−	0.552702i	$-0.186403\pi$
$449$	15857.8	1.66676	0.833379	+	0.552702i	$0.186403\pi$
$450$	2780.96	0.291324
$451$	−2014.62	−0.210344
$452$	2558.89	0.266284
$453$	−2545.06	−0.263968
$454$	8663.28	0.895568
$455$	−13632.8	−1.40465
$456$	0	0
$457$	−7007.40	−0.717270	−0.358635	−	0.933478i	$-0.616758\pi$
$457$	−7007.40	−0.717270	−0.358635	+	0.933478i	$0.616758\pi$
$458$	−12591.8	−1.28467
$459$	−1423.36	−0.144742
$460$	7268.42	0.736721
$461$	8180.84	0.826507	0.413254	−	0.910616i	$-0.364392\pi$
$461$	8180.84	0.826507	0.413254	+	0.910616i	$0.364392\pi$
$462$	−9733.23	−0.980153
$463$	4354.50	0.437086	0.218543	−	0.975827i	$-0.429870\pi$
$463$	4354.50	0.437086	0.218543	+	0.975827i	$0.429870\pi$
$464$	−16464.8	−1.64733
$465$	8225.19	0.820288
$466$	20691.7	2.05692
$467$	−9347.11	−0.926195	−0.463097	−	0.886307i	$-0.653262\pi$
$467$	−9347.11	−0.926195	−0.463097	+	0.886307i	$0.653262\pi$
$468$	−2265.09	−0.223726
$469$	1383.86	0.136249
$470$	31849.3	3.12574
$471$	−6432.50	−0.629286
$472$	−4530.07	−0.441766
$473$	11507.5	1.11864
$474$	8312.23	0.805471
$475$	0	0
$476$	−3640.41	−0.350541
$477$	1979.53	0.190014
$478$	−16540.4	−1.58272
$479$	1557.64	0.148581	0.0742907	−	0.997237i	$-0.476331\pi$
$479$	1557.64	0.148581	0.0742907	+	0.997237i	$0.476331\pi$
$480$	−7746.24	−0.736595
$481$	−2323.49	−0.220253
$482$	−10997.3	−1.03923
$483$	−5543.84	−0.522264
$484$	8647.67	0.812140
$485$	14932.4	1.39803
$486$	−852.709	−0.0795878
$487$	−11150.6	−1.03754	−0.518769	−	0.854914i	$-0.673610\pi$
$487$	−11150.6	−1.03754	−0.518769	+	0.854914i	$0.673610\pi$
$488$	−1331.26	−0.123491
$489$	−9661.99	−0.893518
$490$	4442.37	0.409563
$491$	6378.44	0.586262	0.293131	−	0.956072i	$-0.405303\pi$
$491$	6378.44	0.586262	0.293131	+	0.956072i	$0.405303\pi$
$492$	−451.414	−0.0413644
$493$	10863.0	0.992388
$494$	0	0
$495$	7587.20	0.688928
$496$	−15008.4	−1.35866
$497$	15498.7	1.39882
$498$	3933.15	0.353913
$499$	349.260	0.0313328	0.0156664	−	0.999877i	$-0.495013\pi$
$499$	349.260	0.0313328	0.0156664	+	0.999877i	$0.495013\pi$
$500$	2326.19	0.208061
$501$	−3114.81	−0.277764
$502$	−2377.38	−0.211370
$503$	19474.5	1.72629	0.863144	−	0.504958i	$-0.168492\pi$
$503$	19474.5	1.72629	0.863144	+	0.504958i	$0.168492\pi$
$504$	1863.69	0.164713
$505$	−25321.2	−2.23125
$506$	23395.3	2.05543
$507$	−3620.84	−0.317174
$508$	1346.62	0.117612
$509$	5217.28	0.454326	0.227163	−	0.973857i	$-0.427055\pi$
$509$	5217.28	0.454326	0.227163	+	0.973857i	$0.427055\pi$
$510$	8100.50	0.703326
$511$	1618.47	0.140111
$512$	5865.90	0.506326
$513$	0	0
$514$	−16247.2	−1.39423
$515$	−15089.9	−1.29115
$516$	2578.47	0.219983
$517$	35913.0	3.05503
$518$	−2237.13	−0.189757
$519$	−4151.34	−0.351106
$520$	−11015.9	−0.929000
$521$	16307.6	1.37130	0.685652	−	0.727930i	$-0.259517\pi$
$521$	16307.6	1.37130	0.685652	+	0.727930i	$0.259517\pi$
$522$	6507.88	0.545675
$523$	−794.339	−0.0664130	−0.0332065	−	0.999449i	$-0.510572\pi$
$523$	−794.339	−0.0664130	−0.0332065	+	0.999449i	$0.510572\pi$
$524$	−8259.88	−0.688616
$525$	4228.90	0.351551
$526$	9284.78	0.769649
$527$	9902.10	0.818486
$528$	−13844.2	−1.14109
$529$	1158.46	0.0952137
$530$	−11265.8	−0.923311
$531$	3151.85	0.257587
$532$	0	0
$533$	−2035.13	−0.165387
$534$	1685.29	0.136573
$535$	−20142.8	−1.62776
$536$	1118.22	0.0901115
$537$	7564.10	0.607849
$538$	−490.873	−0.0393365
$539$	5009.17	0.400298
$540$	1700.05	0.135479
$541$	−11968.3	−0.951119	−0.475560	−	0.879683i	$-0.657754\pi$
$541$	−11968.3	−0.951119	−0.475560	+	0.879683i	$0.657754\pi$
$542$	13797.8	1.09348
$543$	14310.6	1.13099
$544$	−9325.51	−0.734978
$545$	21520.9	1.69148
$546$	−9832.31	−0.770667
$547$	−1123.46	−0.0878168	−0.0439084	−	0.999036i	$-0.513981\pi$
$547$	−1123.46	−0.0878168	−0.0439084	+	0.999036i	$0.513981\pi$
$548$	−3848.10	−0.299969
$549$	926.241	0.0720055
$550$	−17846.2	−1.38357
$551$	0	0
$552$	−4479.66	−0.345411
$553$	12640.1	0.971991
$554$	19717.6	1.51213
$555$	1743.88	0.133376
$556$	−12201.2	−0.930656
$557$	17284.9	1.31488	0.657438	−	0.753509i	$-0.271640\pi$
$557$	17284.9	1.31488	0.657438	+	0.753509i	$0.271640\pi$
$558$	5932.19	0.450053
$559$	11624.7	0.879555
$560$	−18670.3	−1.40886
$561$	9134.05	0.687415
$562$	−4036.85	−0.302997
$563$	−7658.42	−0.573293	−0.286646	−	0.958036i	$-0.592540\pi$
$563$	−7658.42	−0.573293	−0.286646	+	0.958036i	$0.592540\pi$
$564$	8046.96	0.600777
$565$	−8658.58	−0.644724
$566$	−5668.43	−0.420958
$567$	−1296.68	−0.0960414
$568$	12523.6	0.925139
$569$	4395.54	0.323850	0.161925	−	0.986803i	$-0.448230\pi$
$569$	4395.54	0.323850	0.161925	+	0.986803i	$0.448230\pi$
$570$	0	0
$571$	4380.71	0.321063	0.160532	−	0.987031i	$-0.448679\pi$
$571$	4380.71	0.321063	0.160532	+	0.987031i	$0.448679\pi$
$572$	14535.7	1.06253
$573$	−126.978	−0.00925759
$574$	−1959.50	−0.142487
$575$	−10164.8	−0.737220
$576$	166.152	0.0120191
$577$	−13071.8	−0.943127	−0.471563	−	0.881832i	$-0.656310\pi$
$577$	−13071.8	−0.943127	−0.471563	+	0.881832i	$0.656310\pi$
$578$	−7488.17	−0.538870
$579$	−9015.42	−0.647095
$580$	−12974.8	−0.928879
$581$	5980.99	0.427079
$582$	10769.6	0.767033
$583$	−12703.2	−0.902424
$584$	1307.79	0.0926657
$585$	7664.44	0.541685
$586$	766.843	0.0540580
$587$	7363.09	0.517729	0.258865	−	0.965914i	$-0.416652\pi$
$587$	7363.09	0.517729	0.258865	+	0.965914i	$0.416652\pi$
$588$	1122.40	0.0787192
$589$	0	0
$590$	−17937.6	−1.25166
$591$	−267.672	−0.0186304
$592$	−3182.03	−0.220913
$593$	919.662	0.0636863	0.0318432	−	0.999493i	$-0.489862\pi$
$593$	919.662	0.0636863	0.0318432	+	0.999493i	$0.489862\pi$
$594$	5472.06	0.377982
$595$	12318.1	0.848729
$596$	−1478.47	−0.101611
$597$	−5382.57	−0.369002
$598$	23633.5	1.61613
$599$	6139.91	0.418815	0.209407	−	0.977828i	$-0.432847\pi$
$599$	6139.91	0.418815	0.209407	+	0.977828i	$0.432847\pi$
$600$	3417.13	0.232506
$601$	−26989.6	−1.83183	−0.915915	−	0.401373i	$-0.868533\pi$
$601$	−26989.6	−1.83183	−0.915915	+	0.401373i	$0.868533\pi$
$602$	11192.6	0.757770
$603$	−778.013	−0.0525426
$604$	3659.56	0.246532
$605$	−29261.3	−1.96635
$606$	−18262.3	−1.22418
$607$	−4200.27	−0.280863	−0.140431	−	0.990090i	$-0.544849\pi$
$607$	−4200.27	−0.280863	−0.140431	+	0.990090i	$0.544849\pi$
$608$	0	0
$609$	9896.27	0.658485
$610$	−5271.36	−0.349887
$611$	36278.6	2.40208
$612$	2046.65	0.135181
$613$	−23443.4	−1.54465	−0.772325	−	0.635228i	$-0.780906\pi$
$613$	−23443.4	−1.54465	−0.772325	+	0.635228i	$0.780906\pi$
$614$	10527.8	0.691965
$615$	1527.46	0.100151
$616$	−11959.8	−0.782263
$617$	−1589.53	−0.103715	−0.0518573	−	0.998655i	$-0.516514\pi$
$617$	−1589.53	−0.103715	−0.0518573	+	0.998655i	$0.516514\pi$
$618$	−10883.2	−0.708392
$619$	−22899.3	−1.48692	−0.743458	−	0.668783i	$-0.766816\pi$
$619$	−22899.3	−1.48692	−0.743458	+	0.668783i	$0.766816\pi$
$620$	−11827.1	−0.766106
$621$	3116.77	0.201404
$622$	8679.98	0.559542
$623$	2562.76	0.164807
$624$	−13985.2	−0.897204
$625$	−18878.2	−1.20820
$626$	4175.20	0.266573
$627$	0	0
$628$	9249.33	0.587721
$629$	2099.42	0.133083
$630$	7379.59	0.466682
$631$	−23356.1	−1.47352	−0.736762	−	0.676152i	$-0.763646\pi$
$631$	−23356.1	−1.47352	−0.736762	+	0.676152i	$0.763646\pi$
$632$	10213.7	0.642849
$633$	15144.4	0.950925
$634$	−17093.1	−1.07075
$635$	−4556.60	−0.284761
$636$	−2846.39	−0.177463
$637$	5060.17	0.314743
$638$	−41762.8	−2.59154
$639$	−8713.43	−0.539434
$640$	−21602.2	−1.33422
$641$	3614.07	0.222694	0.111347	−	0.993782i	$-0.464483\pi$
$641$	3614.07	0.222694	0.111347	+	0.993782i	$0.464483\pi$
$642$	−14527.5	−0.893074
$643$	13757.1	0.843746	0.421873	−	0.906655i	$-0.361373\pi$
$643$	13757.1	0.843746	0.421873	+	0.906655i	$0.361373\pi$
$644$	7971.53	0.487767
$645$	−8724.84	−0.532620
$646$	0	0
$647$	5629.45	0.342066	0.171033	−	0.985265i	$-0.445290\pi$
$647$	5629.45	0.342066	0.171033	+	0.985265i	$0.445290\pi$
$648$	−1047.77	−0.0635192
$649$	−20226.2	−1.22334
$650$	−18027.8	−1.08786
$651$	9020.85	0.543095
$652$	13893.0	0.834499
$653$	17164.4	1.02863	0.514316	−	0.857601i	$-0.328046\pi$
$653$	17164.4	1.02863	0.514316	+	0.857601i	$0.328046\pi$
$654$	15521.4	0.928033
$655$	27949.1	1.66727
$656$	−2787.13	−0.165883
$657$	−909.909	−0.0540319
$658$	34930.2	2.06949
$659$	−9456.53	−0.558990	−0.279495	−	0.960147i	$-0.590167\pi$
$659$	−9456.53	−0.558990	−0.279495	+	0.960147i	$0.590167\pi$
$660$	−10909.7	−0.643423
$661$	−4435.93	−0.261025	−0.130513	−	0.991447i	$-0.541662\pi$
$661$	−4435.93	−0.261025	−0.130513	+	0.991447i	$0.541662\pi$
$662$	22080.6	1.29636
$663$	9227.03	0.540495
$664$	4832.90	0.282459
$665$	0	0
$666$	1257.73	0.0731770
$667$	−23787.2	−1.38088
$668$	4478.81	0.259417
$669$	17689.5	1.02229
$670$	4427.78	0.255313
$671$	−5943.94	−0.341972
$672$	−8495.57	−0.487684
$673$	8060.65	0.461687	0.230843	−	0.972991i	$-0.425851\pi$
$673$	8060.65	0.461687	0.230843	+	0.972991i	$0.425851\pi$
$674$	21.9458	0.00125418
$675$	−2377.50	−0.135571
$676$	5206.43	0.296224
$677$	−12585.3	−0.714464	−0.357232	−	0.934016i	$-0.616280\pi$
$677$	−12585.3	−0.714464	−0.357232	+	0.934016i	$0.616280\pi$
$678$	−6244.77	−0.353730
$679$	16376.9	0.925606
$680$	9953.57	0.561326
$681$	−7406.43	−0.416762
$682$	−38068.5	−2.13741
$683$	13116.8	0.734848	0.367424	−	0.930054i	$-0.380240\pi$
$683$	13116.8	0.734848	0.367424	+	0.930054i	$0.380240\pi$
$684$	0	0
$685$	13020.9	0.726282
$686$	24140.1	1.34355
$687$	10765.0	0.597833
$688$	15920.1	0.882190
$689$	−12832.5	−0.709550
$690$	−17738.0	−0.978656
$691$	23651.4	1.30209	0.651044	−	0.759040i	$-0.274332\pi$
$691$	23651.4	1.30209	0.651044	+	0.759040i	$0.274332\pi$
$692$	5969.25	0.327914
$693$	8321.15	0.456125
$694$	−15272.2	−0.835340
$695$	41285.3	2.25330
$696$	7996.62	0.435504
$697$	1838.87	0.0999313
$698$	−22537.5	−1.22214
$699$	−17689.8	−0.957211
$700$	−6080.76	−0.328330
$701$	940.958	0.0506983	0.0253492	−	0.999679i	$-0.491930\pi$
$701$	940.958	0.0506983	0.0253492	+	0.999679i	$0.491930\pi$
$702$	5527.77	0.297197
$703$	0	0
$704$	−1066.24	−0.0570816
$705$	−27228.7	−1.45460
$706$	−29673.0	−1.58181
$707$	−27770.7	−1.47726
$708$	−4532.06	−0.240572
$709$	−4558.63	−0.241471	−0.120735	−	0.992685i	$-0.538525\pi$
$709$	−4558.63	−0.241471	−0.120735	+	0.992685i	$0.538525\pi$
$710$	49589.3	2.62120
$711$	−7106.31	−0.374835
$712$	2070.82	0.108999
$713$	−21683.0	−1.13890
$714$	8884.10	0.465657
$715$	−49184.8	−2.57260
$716$	−10876.5	−0.567699
$717$	14140.8	0.736537
$718$	−37846.0	−1.96713
$719$	18434.9	0.956197	0.478098	−	0.878306i	$-0.341326\pi$
$719$	18434.9	0.956197	0.478098	+	0.878306i	$0.341326\pi$
$720$	10496.5	0.543308
$721$	−16549.7	−0.854842
$722$	0	0
$723$	9401.80	0.483619
$724$	−20577.3	−1.05628
$725$	18145.1	0.929507
$726$	−21103.9	−1.07884
$727$	−10535.5	−0.537467	−0.268734	−	0.963215i	$-0.586605\pi$
$727$	−10535.5	−0.537467	−0.268734	+	0.963215i	$0.586605\pi$
$728$	−12081.5	−0.615071
$729$	729.000	0.0370370
$730$	5178.41	0.262550
$731$	−10503.6	−0.531450
$732$	−1331.85	−0.0672494
$733$	27672.6	1.39442	0.697212	−	0.716865i	$-0.254424\pi$
$733$	27672.6	1.39442	0.697212	+	0.716865i	$0.254424\pi$
$734$	−4192.56	−0.210831
$735$	−3797.88	−0.190594
$736$	20420.4	1.02270
$737$	4992.72	0.249538
$738$	1101.64	0.0549483
$739$	−17280.9	−0.860201	−0.430100	−	0.902781i	$-0.641522\pi$
$739$	−17280.9	−0.860201	−0.430100	+	0.902781i	$0.641522\pi$
$740$	−2507.54	−0.124566
$741$	0	0
$742$	−12355.6	−0.611305
$743$	5049.24	0.249312	0.124656	−	0.992200i	$-0.460217\pi$
$743$	5049.24	0.249312	0.124656	+	0.992200i	$0.460217\pi$
$744$	7289.24	0.359189
$745$	5002.72	0.246021
$746$	30595.9	1.50160
$747$	−3362.54	−0.164697
$748$	−13133.9	−0.642010
$749$	−22091.3	−1.07770
$750$	−5676.88	−0.276387
$751$	27755.0	1.34859	0.674297	−	0.738460i	$-0.264447\pi$
$751$	27755.0	1.34859	0.674297	+	0.738460i	$0.264447\pi$
$752$	49683.7	2.40928
$753$	2032.47	0.0983632
$754$	−42187.9	−2.03766
$755$	−12382.9	−0.596902
$756$	1864.51	0.0896977
$757$	18889.5	0.906937	0.453469	−	0.891272i	$-0.350186\pi$
$757$	18889.5	0.906937	0.453469	+	0.891272i	$0.350186\pi$
$758$	26000.9	1.24590
$759$	−20001.2	−0.956517
$760$	0	0
$761$	16730.6	0.796957	0.398478	−	0.917178i	$-0.369538\pi$
$761$	16730.6	0.796957	0.398478	+	0.917178i	$0.369538\pi$
$762$	−3286.32	−0.156235
$763$	23602.7	1.11989
$764$	182.583	0.00864611
$765$	−6925.30	−0.327300
$766$	−13790.4	−0.650482
$767$	−20432.1	−0.961880
$768$	−15136.9	−0.711208
$769$	−36917.1	−1.73116	−0.865582	−	0.500767i	$-0.833051\pi$
$769$	−36917.1	−1.73116	−0.865582	+	0.500767i	$0.833051\pi$
$770$	−47356.8	−2.21639
$771$	13890.1	0.648818
$772$	12963.3	0.604353
$773$	35503.1	1.65195	0.825975	−	0.563706i	$-0.190625\pi$
$773$	35503.1	1.65195	0.825975	+	0.563706i	$0.190625\pi$
$774$	−6292.55	−0.292223
$775$	16540.0	0.766625
$776$	13233.2	0.612171
$777$	1912.57	0.0883053
$778$	29648.0	1.36624
$779$	0	0
$780$	−11020.8	−0.505906
$781$	55916.4	2.56190
$782$	−21354.3	−0.976507
$783$	−5563.73	−0.253935
$784$	6929.93	0.315685
$785$	−31297.2	−1.42299
$786$	20157.6	0.914753
$787$	3651.70	0.165399	0.0826994	−	0.996575i	$-0.473646\pi$
$787$	3651.70	0.165399	0.0826994	+	0.996575i	$0.473646\pi$
$788$	384.887	0.0173998
$789$	−7937.76	−0.358165
$790$	40442.9	1.82139
$791$	−9496.17	−0.426858
$792$	6723.85	0.301669
$793$	−6004.44	−0.268883
$794$	18871.6	0.843484
$795$	9631.38	0.429673
$796$	7739.64	0.344628
$797$	−6942.94	−0.308571	−0.154286	−	0.988026i	$-0.549308\pi$
$797$	−6942.94	−0.308571	−0.154286	+	0.988026i	$0.549308\pi$
$798$	0	0
$799$	−32779.9	−1.45140
$800$	−15576.9	−0.688407
$801$	−1440.79	−0.0635555
$802$	1689.23	0.0743751
$803$	5839.13	0.256611
$804$	1118.71	0.0490720
$805$	−26973.4	−1.18098
$806$	−38456.0	−1.68059
$807$	419.658	0.0183057
$808$	−22439.9	−0.977022
$809$	5290.05	0.229899	0.114949	−	0.993371i	$-0.463329\pi$
$809$	5290.05	0.229899	0.114949	+	0.993371i	$0.463329\pi$
$810$	−4148.84	−0.179969
$811$	−554.384	−0.0240038	−0.0120019	−	0.999928i	$-0.503820\pi$
$811$	−554.384	−0.0240038	−0.0120019	+	0.999928i	$0.503820\pi$
$812$	−14229.9	−0.614991
$813$	−11796.0	−0.508862
$814$	−8071.17	−0.347536
$815$	−47010.2	−2.02048
$816$	12636.5	0.542114
$817$	0	0
$818$	−36318.2	−1.55237
$819$	8405.86	0.358638
$820$	−2196.34	−0.0935361
$821$	−15775.7	−0.670618	−0.335309	−	0.942108i	$-0.608841\pi$
$821$	−15775.7	−0.670618	−0.335309	+	0.942108i	$0.608841\pi$
$822$	9390.97	0.398477
$823$	−28230.4	−1.19569	−0.597843	−	0.801613i	$-0.703975\pi$
$823$	−28230.4	−1.19569	−0.597843	+	0.801613i	$0.703975\pi$
$824$	−13372.8	−0.565370
$825$	15257.1	0.643859
$826$	−19672.8	−0.828696
$827$	33314.4	1.40079	0.700397	−	0.713754i	$-0.253006\pi$
$827$	33314.4	1.40079	0.700397	+	0.713754i	$0.253006\pi$
$828$	−4481.63	−0.188101
$829$	−11597.0	−0.485861	−0.242931	−	0.970044i	$-0.578109\pi$
$829$	−11597.0	−0.485861	−0.242931	+	0.970044i	$0.578109\pi$
$830$	19136.6	0.800293
$831$	−16857.0	−0.703685
$832$	−1077.09	−0.0448817
$833$	−4572.17	−0.190176
$834$	29775.9	1.23628
$835$	−15155.0	−0.628098
$836$	0	0
$837$	−5071.56	−0.209437
$838$	55988.4	2.30798
$839$	−19005.2	−0.782039	−0.391019	−	0.920382i	$-0.627878\pi$
$839$	−19005.2	−0.782039	−0.391019	+	0.920382i	$0.627878\pi$
$840$	9067.74	0.372460
$841$	18073.4	0.741047
$842$	40905.3	1.67422
$843$	3451.19	0.141003
$844$	−21776.2	−0.888115
$845$	−17617.1	−0.717215
$846$	−19637.9	−0.798069
$847$	−32091.9	−1.30188
$848$	−17574.2	−0.711676
$849$	4846.07	0.195897
$850$	16289.3	0.657315
$851$	−4597.17	−0.185181
$852$	12529.1	0.503803
$853$	−25223.9	−1.01248	−0.506242	−	0.862391i	$-0.668966\pi$
$853$	−25223.9	−1.01248	−0.506242	+	0.862391i	$0.668966\pi$
$854$	−5781.28	−0.231653
$855$	0	0
$856$	−17850.8	−0.712764
$857$	−15349.4	−0.611814	−0.305907	−	0.952061i	$-0.598960\pi$
$857$	−15349.4	−0.611814	−0.305907	+	0.952061i	$0.598960\pi$
$858$	−35473.2	−1.41146
$859$	3658.79	0.145328	0.0726638	−	0.997356i	$-0.476850\pi$
$859$	3658.79	0.145328	0.0726638	+	0.997356i	$0.476850\pi$
$860$	12545.5	0.497440
$861$	1675.22	0.0663080
$862$	13935.1	0.550615
$863$	34115.5	1.34566	0.672831	−	0.739797i	$-0.265078\pi$
$863$	34115.5	1.34566	0.672831	+	0.739797i	$0.265078\pi$
$864$	4776.25	0.188069
$865$	−20198.3	−0.793944
$866$	−31725.2	−1.24488
$867$	6401.80	0.250769
$868$	−12971.1	−0.507223
$869$	45603.1	1.78018
$870$	31663.9	1.23392
$871$	5043.55	0.196204
$872$	19072.0	0.740666
$873$	−9207.15	−0.356947
$874$	0	0
$875$	−8632.61	−0.333526
$876$	1308.36	0.0504630
$877$	43811.7	1.68690	0.843452	−	0.537204i	$-0.180520\pi$
$877$	43811.7	1.68690	0.843452	+	0.537204i	$0.180520\pi$
$878$	4184.10	0.160827
$879$	−655.591	−0.0251565
$880$	−67358.8	−2.58030
$881$	31668.0	1.21103	0.605517	−	0.795833i	$-0.292966\pi$
$881$	31668.0	1.21103	0.605517	+	0.795833i	$0.292966\pi$
$882$	−2739.12	−0.104570
$883$	19784.2	0.754011	0.377005	−	0.926211i	$-0.376954\pi$
$883$	19784.2	0.754011	0.377005	+	0.926211i	$0.376954\pi$
$884$	−13267.6	−0.504794
$885$	15335.2	0.582472
$886$	14988.8	0.568350
$887$	−37256.7	−1.41032	−0.705161	−	0.709047i	$-0.749125\pi$
$887$	−37256.7	−1.41032	−0.705161	+	0.709047i	$0.749125\pi$
$888$	1545.44	0.0584028
$889$	−4997.38	−0.188534
$890$	8199.75	0.308827
$891$	−4678.19	−0.175898
$892$	−25435.8	−0.954769
$893$	0	0
$894$	3608.07	0.134980
$895$	36802.9	1.37451
$896$	−23691.9	−0.883361
$897$	−20204.8	−0.752082
$898$	55646.3	2.06786
$899$	38706.2	1.43595
$900$	3418.63	0.126616
$901$	11595.0	0.428729
$902$	−7069.50	−0.260963
$903$	−9568.83	−0.352636
$904$	−7673.32	−0.282313
$905$	69627.7	2.55746
$906$	−8930.85	−0.327492
$907$	−28676.5	−1.04982	−0.524911	−	0.851157i	$-0.675901\pi$
$907$	−28676.5	−1.04982	−0.524911	+	0.851157i	$0.675901\pi$
$908$	10649.8	0.389234
$909$	15612.8	0.569685
$910$	−47838.9	−1.74268
$911$	−1420.43	−0.0516584	−0.0258292	−	0.999666i	$-0.508223\pi$
$911$	−1420.43	−0.0516584	−0.0258292	+	0.999666i	$0.508223\pi$
$912$	0	0
$913$	21578.3	0.782188
$914$	−24589.6	−0.889881
$915$	4506.60	0.162824
$916$	−15479.1	−0.558345
$917$	30652.8	1.10387
$918$	−4994.68	−0.179574
$919$	−1881.00	−0.0675175	−0.0337588	−	0.999430i	$-0.510748\pi$
$919$	−1881.00	−0.0675175	−0.0337588	+	0.999430i	$0.510748\pi$
$920$	−21795.7	−0.781068
$921$	−9000.43	−0.322013
$922$	28707.3	1.02541
$923$	56485.7	2.01435
$924$	−11965.0	−0.425997
$925$	3506.76	0.124650
$926$	15280.3	0.542271
$927$	9304.29	0.329658
$928$	−36452.3	−1.28945
$929$	−14215.4	−0.502038	−0.251019	−	0.967982i	$-0.580766\pi$
$929$	−14215.4	−0.502038	−0.251019	+	0.967982i	$0.580766\pi$
$930$	28862.9	1.01769
$931$	0	0
$932$	25436.3	0.893986
$933$	−7420.70	−0.260389
$934$	−32799.9	−1.14908
$935$	44441.5	1.55443
$936$	6792.30	0.237194
$937$	−13908.4	−0.484918	−0.242459	−	0.970162i	$-0.577954\pi$
$937$	−13908.4	−0.484918	−0.242459	+	0.970162i	$0.577954\pi$
$938$	4856.10	0.169038
$939$	−3569.47	−0.124053
$940$	39152.3	1.35852
$941$	−27749.9	−0.961341	−0.480671	−	0.876901i	$-0.659607\pi$
$941$	−27749.9	−0.961341	−0.480671	+	0.876901i	$0.659607\pi$
$942$	−22572.2	−0.780725
$943$	−4026.64	−0.139051
$944$	−27981.9	−0.964761
$945$	−6308.97	−0.217176
$946$	40381.0	1.38784
$947$	6010.50	0.206246	0.103123	−	0.994669i	$-0.467116\pi$
$947$	6010.50	0.206246	0.103123	+	0.994669i	$0.467116\pi$
$948$	10218.2	0.350076
$949$	5898.57	0.201766
$950$	0	0
$951$	14613.3	0.498285
$952$	10916.4	0.371642
$953$	12736.9	0.432938	0.216469	−	0.976289i	$-0.430546\pi$
$953$	12736.9	0.432938	0.216469	+	0.976289i	$0.430546\pi$
$954$	6946.37	0.235741
$955$	−617.811	−0.0209339
$956$	−20333.1	−0.687887
$957$	35703.9	1.20600
$958$	5465.91	0.184338
$959$	14280.5	0.480856
$960$	808.408	0.0271784
$961$	5491.22	0.184325
$962$	−8153.33	−0.273258
$963$	12419.9	0.415601
$964$	−13518.9	−0.451675
$965$	−43864.3	−1.46326
$966$	−19453.8	−0.647947
$967$	12005.7	0.399252	0.199626	−	0.979872i	$-0.436027\pi$
$967$	12005.7	0.399252	0.199626	+	0.979872i	$0.436027\pi$
$968$	−25931.6	−0.861027
$969$	0	0
$970$	52399.1	1.73447
$971$	14498.4	0.479171	0.239585	−	0.970875i	$-0.422988\pi$
$971$	14498.4	0.479171	0.239585	+	0.970875i	$0.422988\pi$
$972$	−1048.23	−0.0345907
$973$	45279.1	1.49186
$974$	−39128.4	−1.28722
$975$	15412.4	0.506248
$976$	−8223.12	−0.269688
$977$	19718.0	0.645686	0.322843	−	0.946452i	$-0.395361\pi$
$977$	19718.0	0.645686	0.322843	+	0.946452i	$0.395361\pi$
$978$	−33904.8	−1.10854
$979$	9245.96	0.301841
$980$	5461.00	0.178005
$981$	−13269.6	−0.431870
$982$	22382.5	0.727347
$983$	−27805.5	−0.902194	−0.451097	−	0.892475i	$-0.648967\pi$
$983$	−27805.5	−0.902194	−0.451097	+	0.892475i	$0.648967\pi$
$984$	1353.65	0.0438544
$985$	−1302.35	−0.0421283
$986$	38119.4	1.23121
$987$	−29862.6	−0.963058
$988$	0	0
$989$	23000.1	0.739497
$990$	26624.2	0.854720
$991$	−54235.9	−1.73851	−0.869253	−	0.494367i	$-0.835400\pi$
$991$	−54235.9	−1.73851	−0.869253	+	0.494367i	$0.835400\pi$
$992$	−33227.7	−1.06349
$993$	−18877.2	−0.603273
$994$	54386.3	1.73544
$995$	−26188.8	−0.834412
$996$	4835.02	0.153819
$997$	4303.36	0.136699	0.0683495	−	0.997661i	$-0.478227\pi$
$997$	4303.36	0.136699	0.0683495	+	0.997661i	$0.478227\pi$
$998$	1225.59	0.0388730
$999$	−1075.26	−0.0340537

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1083.4.a.s.1.14		18
19.9	even	9		57.4.i.b.43.2	yes	36
19.17	even	9		57.4.i.b.4.2	✓	36
19.18	odd	2		1083.4.a.t.1.5		18
57.17	odd	18		171.4.u.c.118.5		36
57.47	odd	18		171.4.u.c.100.5		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.4.i.b.4.2	✓	36	19.17	even	9
57.4.i.b.43.2	yes	36	19.9	even	9
171.4.u.c.100.5		36	57.47	odd	18
171.4.u.c.118.5		36	57.17	odd	18
1083.4.a.s.1.14		18	1.1	even	1	trivial
1083.4.a.t.1.5		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)

\(f(q)\)	\(=\)	\(q+3.50909 q^{2} -3.00000 q^{3} +4.31372 q^{4} -14.5964 q^{5} -10.5273 q^{6} -16.0084 q^{7} -12.9355 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.50909 q^{2} -3.00000 q^{3} +4.31372 q^{4} -14.5964 q^{5} -10.5273 q^{6} -16.0084 q^{7} -12.9355 q^{8} +9.00000 q^{9} -51.2202 q^{10} -57.7554 q^{11} -12.9412 q^{12} -58.3434 q^{13} -56.1750 q^{14} +43.7893 q^{15} -79.9016 q^{16} +52.7169 q^{17} +31.5818 q^{18} -62.9649 q^{20} +48.0252 q^{21} -202.669 q^{22} -115.436 q^{23} +38.8065 q^{24} +88.0557 q^{25} -204.732 q^{26} -27.0000 q^{27} -69.0558 q^{28} +206.064 q^{29} +153.661 q^{30} +187.836 q^{31} -176.898 q^{32} +173.266 q^{33} +184.988 q^{34} +233.666 q^{35} +38.8235 q^{36} +39.8244 q^{37} +175.030 q^{39} +188.812 q^{40} +34.8820 q^{41} +168.525 q^{42} -199.246 q^{43} -249.141 q^{44} -131.368 q^{45} -405.075 q^{46} -621.811 q^{47} +239.705 q^{48} -86.7308 q^{49} +308.996 q^{50} -158.151 q^{51} -251.677 q^{52} +219.948 q^{53} -94.7455 q^{54} +843.023 q^{55} +207.077 q^{56} +723.098 q^{58} +350.205 q^{59} +188.895 q^{60} +102.916 q^{61} +659.132 q^{62} -144.076 q^{63} +18.4613 q^{64} +851.605 q^{65} +608.007 q^{66} -86.4459 q^{67} +227.406 q^{68} +346.308 q^{69} +819.954 q^{70} -968.159 q^{71} -116.419 q^{72} -101.101 q^{73} +139.747 q^{74} -264.167 q^{75} +924.572 q^{77} +614.196 q^{78} -789.590 q^{79} +1166.28 q^{80} +81.0000 q^{81} +122.404 q^{82} -373.615 q^{83} +207.167 q^{84} -769.478 q^{85} -699.172 q^{86} -618.192 q^{87} +747.094 q^{88} -160.088 q^{89} -460.982 q^{90} +933.984 q^{91} -497.959 q^{92} -563.507 q^{93} -2181.99 q^{94} +530.694 q^{96} -1023.02 q^{97} -304.346 q^{98} -519.799 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 54 q^{3} + 96 q^{4} + 18 q^{5} + 48 q^{7} + 57 q^{8} + 162 q^{9}+O(q^{10}) \) 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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