LMFDB - Embedded newform 51.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [51,4,Mod(1,51)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(51, 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("51.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 51 = 3 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	51.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:	$3.00909741029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	51.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.24264 q^{2} -3.00000 q^{3} +10.0000 q^{4} +19.9706 q^{5} -12.7279 q^{6} -20.9706 q^{7} +8.48528 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.24264 q^{2} -3.00000 q^{3} +10.0000 q^{4} +19.9706 q^{5} -12.7279 q^{6} -20.9706 q^{7} +8.48528 q^{8} +9.00000 q^{9} +84.7279 q^{10} +16.0294 q^{11} -30.0000 q^{12} -34.9411 q^{13} -88.9706 q^{14} -59.9117 q^{15} -44.0000 q^{16} -17.0000 q^{17} +38.1838 q^{18} -80.8823 q^{19} +199.706 q^{20} +62.9117 q^{21} +68.0071 q^{22} -115.971 q^{23} -25.4558 q^{24} +273.823 q^{25} -148.243 q^{26} -27.0000 q^{27} -209.706 q^{28} +154.118 q^{29} -254.184 q^{30} +299.941 q^{31} -254.558 q^{32} -48.0883 q^{33} -72.1249 q^{34} -418.794 q^{35} +90.0000 q^{36} +315.529 q^{37} -343.154 q^{38} +104.823 q^{39} +169.456 q^{40} +132.265 q^{41} +266.912 q^{42} -23.1177 q^{43} +160.294 q^{44} +179.735 q^{45} -492.021 q^{46} +260.912 q^{47} +132.000 q^{48} +96.7645 q^{49} +1161.73 q^{50} +51.0000 q^{51} -349.411 q^{52} -676.087 q^{53} -114.551 q^{54} +320.117 q^{55} -177.941 q^{56} +242.647 q^{57} +653.866 q^{58} +629.294 q^{59} -599.117 q^{60} -461.852 q^{61} +1272.54 q^{62} -188.735 q^{63} -728.000 q^{64} -697.794 q^{65} -204.021 q^{66} -789.470 q^{67} -170.000 q^{68} +347.912 q^{69} -1776.79 q^{70} -686.412 q^{71} +76.3675 q^{72} +484.912 q^{73} +1338.68 q^{74} -821.470 q^{75} -808.823 q^{76} -336.146 q^{77} +444.728 q^{78} +254.000 q^{79} -878.705 q^{80} +81.0000 q^{81} +561.153 q^{82} +548.912 q^{83} +629.117 q^{84} -339.500 q^{85} -98.0803 q^{86} -462.353 q^{87} +136.014 q^{88} +925.145 q^{89} +762.551 q^{90} +732.735 q^{91} -1159.71 q^{92} -899.823 q^{93} +1106.95 q^{94} -1615.26 q^{95} +763.675 q^{96} +732.617 q^{97} +410.537 q^{98} +144.265 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 20 * q^4 + 6 * q^5 - 8 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{7} + 18 q^{9} + 144 q^{10} + 66 q^{11} - 60 q^{12} - 2 q^{13} - 144 q^{14} - 18 q^{15} - 88 q^{16} - 34 q^{17} - 26 q^{19} + 60 q^{20} + 24 q^{21} - 144 q^{22} - 198 q^{23} + 344 q^{25} - 288 q^{26} - 54 q^{27} - 80 q^{28} + 444 q^{29} - 432 q^{30} + 532 q^{31} - 198 q^{33} - 600 q^{35} + 180 q^{36} + 88 q^{37} - 576 q^{38} + 6 q^{39} + 288 q^{40} + 570 q^{41} + 432 q^{42} - 182 q^{43} + 660 q^{44} + 54 q^{45} - 144 q^{46} + 420 q^{47} + 264 q^{48} - 78 q^{49} + 864 q^{50} + 102 q^{51} - 20 q^{52} - 300 q^{53} - 378 q^{55} - 288 q^{56} + 78 q^{57} - 576 q^{58} + 444 q^{59} - 180 q^{60} + 400 q^{61} + 288 q^{62} - 72 q^{63} - 1456 q^{64} - 1158 q^{65} + 432 q^{66} - 968 q^{67} - 340 q^{68} + 594 q^{69} - 1008 q^{70} - 1848 q^{71} + 868 q^{73} + 2304 q^{74} - 1032 q^{75} - 260 q^{76} + 312 q^{77} + 864 q^{78} + 508 q^{79} - 264 q^{80} + 162 q^{81} - 1296 q^{82} + 996 q^{83} + 240 q^{84} - 102 q^{85} + 576 q^{86} - 1332 q^{87} - 288 q^{88} - 288 q^{89} + 1296 q^{90} + 1160 q^{91} - 1980 q^{92} - 1596 q^{93} + 432 q^{94} - 2382 q^{95} + 1024 q^{97} + 1152 q^{98} + 594 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 20 * q^4 + 6 * q^5 - 8 * q^7 + 18 * q^9 + 144 * q^10 + 66 * q^11 - 60 * q^12 - 2 * q^13 - 144 * q^14 - 18 * q^15 - 88 * q^16 - 34 * q^17 - 26 * q^19 + 60 * q^20 + 24 * q^21 - 144 * q^22 - 198 * q^23 + 344 * q^25 - 288 * q^26 - 54 * q^27 - 80 * q^28 + 444 * q^29 - 432 * q^30 + 532 * q^31 - 198 * q^33 - 600 * q^35 + 180 * q^36 + 88 * q^37 - 576 * q^38 + 6 * q^39 + 288 * q^40 + 570 * q^41 + 432 * q^42 - 182 * q^43 + 660 * q^44 + 54 * q^45 - 144 * q^46 + 420 * q^47 + 264 * q^48 - 78 * q^49 + 864 * q^50 + 102 * q^51 - 20 * q^52 - 300 * q^53 - 378 * q^55 - 288 * q^56 + 78 * q^57 - 576 * q^58 + 444 * q^59 - 180 * q^60 + 400 * q^61 + 288 * q^62 - 72 * q^63 - 1456 * q^64 - 1158 * q^65 + 432 * q^66 - 968 * q^67 - 340 * q^68 + 594 * q^69 - 1008 * q^70 - 1848 * q^71 + 868 * q^73 + 2304 * q^74 - 1032 * q^75 - 260 * q^76 + 312 * q^77 + 864 * q^78 + 508 * q^79 - 264 * q^80 + 162 * q^81 - 1296 * q^82 + 996 * q^83 + 240 * q^84 - 102 * q^85 + 576 * q^86 - 1332 * q^87 - 288 * q^88 - 288 * q^89 + 1296 * q^90 + 1160 * q^91 - 1980 * q^92 - 1596 * q^93 + 432 * q^94 - 2382 * q^95 + 1024 * q^97 + 1152 * q^98 + 594 * 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 $

$2$	4.24264	1.50000	0.750000	−	0.661438i	$-0.230053\pi$
$2$	4.24264	1.50000	0.750000	+	0.661438i	$0.230053\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	10.0000	1.25000
$5$	19.9706	1.78622	0.893111	−	0.449837i	$-0.148518\pi$
$5$	19.9706	1.78622	0.893111	+	0.449837i	$0.148518\pi$
$6$	−12.7279	−0.866025
$7$	−20.9706	−1.13230	−0.566152	−	0.824301i	$-0.691568\pi$
$7$	−20.9706	−1.13230	−0.566152	+	0.824301i	$0.691568\pi$
$8$	8.48528	0.375000
$9$	9.00000	0.333333
$10$	84.7279	2.67933
$11$	16.0294	0.439369	0.219684	−	0.975571i	$-0.429497\pi$
$11$	16.0294	0.439369	0.219684	+	0.975571i	$0.429497\pi$
$12$	−30.0000	−0.721688
$13$	−34.9411	−0.745456	−0.372728	−	0.927941i	$-0.621578\pi$
$13$	−34.9411	−0.745456	−0.372728	+	0.927941i	$0.621578\pi$
$14$	−88.9706	−1.69846
$15$	−59.9117	−1.03128
$16$	−44.0000	−0.687500
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	38.1838	0.500000
$19$	−80.8823	−0.976614	−0.488307	−	0.872672i	$-0.662385\pi$
$19$	−80.8823	−0.976614	−0.488307	+	0.872672i	$0.662385\pi$
$20$	199.706	2.23278
$21$	62.9117	0.653736
$22$	68.0071	0.659053
$23$	−115.971	−1.05137	−0.525686	−	0.850679i	$-0.676191\pi$
$23$	−115.971	−1.05137	−0.525686	+	0.850679i	$0.676191\pi$
$24$	−25.4558	−0.216506
$25$	273.823	2.19059
$26$	−148.243	−1.11818
$27$	−27.0000	−0.192450
$28$	−209.706	−1.41538
$29$	154.118	0.986860	0.493430	−	0.869785i	$-0.335743\pi$
$29$	154.118	0.986860	0.493430	+	0.869785i	$0.335743\pi$
$30$	−254.184	−1.54691
$31$	299.941	1.73777	0.868887	−	0.495010i	$-0.164836\pi$
$31$	299.941	1.73777	0.868887	+	0.495010i	$0.164836\pi$
$32$	−254.558	−1.40625
$33$	−48.0883	−0.253670
$34$	−72.1249	−0.363803
$35$	−418.794	−2.02255
$36$	90.0000	0.416667
$37$	315.529	1.40196	0.700982	−	0.713179i	$-0.252745\pi$
$37$	315.529	1.40196	0.700982	+	0.713179i	$0.252745\pi$
$38$	−343.154	−1.46492
$39$	104.823	0.430389
$40$	169.456	0.669833
$41$	132.265	0.503813	0.251906	−	0.967752i	$-0.418943\pi$
$41$	132.265	0.503813	0.251906	+	0.967752i	$0.418943\pi$
$42$	266.912	0.980604
$43$	−23.1177	−0.0819866	−0.0409933	−	0.999159i	$-0.513052\pi$
$43$	−23.1177	−0.0819866	−0.0409933	+	0.999159i	$0.513052\pi$
$44$	160.294	0.549211
$45$	179.735	0.595407
$46$	−492.021	−1.57706
$47$	260.912	0.809742	0.404871	−	0.914374i	$-0.367316\pi$
$47$	260.912	0.809742	0.404871	+	0.914374i	$0.367316\pi$
$48$	132.000	0.396928
$49$	96.7645	0.282112
$50$	1161.73	3.28588
$51$	51.0000	0.140028
$52$	−349.411	−0.931820
$53$	−676.087	−1.75222	−0.876111	−	0.482110i	$-0.839871\pi$
$53$	−676.087	−1.75222	−0.876111	+	0.482110i	$0.839871\pi$
$54$	−114.551	−0.288675
$55$	320.117	0.784810
$56$	−177.941	−0.424614
$57$	242.647	0.563848
$58$	653.866	1.48029
$59$	629.294	1.38859	0.694297	−	0.719689i	$-0.255715\pi$
$59$	629.294	1.38859	0.694297	+	0.719689i	$0.255715\pi$
$60$	−599.117	−1.28909
$61$	−461.852	−0.969411	−0.484706	−	0.874677i	$-0.661073\pi$
$61$	−461.852	−0.969411	−0.484706	+	0.874677i	$0.661073\pi$
$62$	1272.54	2.60666
$63$	−188.735	−0.377435
$64$	−728.000	−1.42188
$65$	−697.794	−1.33155
$66$	−204.021	−0.380505
$67$	−789.470	−1.43954	−0.719770	−	0.694213i	$-0.755753\pi$
$67$	−789.470	−1.43954	−0.719770	+	0.694213i	$0.755753\pi$
$68$	−170.000	−0.303170
$69$	347.912	0.607009
$70$	−1776.79	−3.03382
$71$	−686.412	−1.14735	−0.573677	−	0.819082i	$-0.694484\pi$
$71$	−686.412	−1.14735	−0.573677	+	0.819082i	$0.694484\pi$
$72$	76.3675	0.125000
$73$	484.912	0.777461	0.388730	−	0.921352i	$-0.372914\pi$
$73$	484.912	0.777461	0.388730	+	0.921352i	$0.372914\pi$
$74$	1338.68	2.10295
$75$	−821.470	−1.26474
$76$	−808.823	−1.22077
$77$	−336.146	−0.497499
$78$	444.728	0.645584
$79$	254.000	0.361737	0.180869	−	0.983507i	$-0.442109\pi$
$79$	254.000	0.361737	0.180869	+	0.983507i	$0.442109\pi$
$80$	−878.705	−1.22803
$81$	81.0000	0.111111
$82$	561.153	0.755719
$83$	548.912	0.725914	0.362957	−	0.931806i	$-0.381767\pi$
$83$	548.912	0.725914	0.362957	+	0.931806i	$0.381767\pi$
$84$	629.117	0.817170
$85$	−339.500	−0.433222
$86$	−98.0803	−0.122980
$87$	−462.353	−0.569764
$88$	136.014	0.164763
$89$	925.145	1.10186	0.550928	−	0.834553i	$-0.314274\pi$
$89$	925.145	1.10186	0.550928	+	0.834553i	$0.314274\pi$
$90$	762.551	0.893111
$91$	732.735	0.844082
$92$	−1159.71	−1.31421
$93$	−899.823	−1.00330
$94$	1106.95	1.21461
$95$	−1615.26	−1.74445
$96$	763.675	0.811899
$97$	732.617	0.766866	0.383433	−	0.923569i	$-0.374742\pi$
$97$	732.617	0.766866	0.383433	+	0.923569i	$0.374742\pi$
$98$	410.537	0.423168
$99$	144.265	0.146456
$100$	2738.23	2.73823
$101$	−128.528	−0.126624	−0.0633120	−	0.997994i	$-0.520166\pi$
$101$	−128.528	−0.126624	−0.0633120	+	0.997994i	$0.520166\pi$
$102$	216.375	0.210042
$103$	5.35325	0.00512108	0.00256054	−	0.999997i	$-0.499185\pi$
$103$	5.35325	0.00512108	0.00256054	+	0.999997i	$0.499185\pi$
$104$	−296.485	−0.279546
$105$	1256.38	1.16772
$106$	−2868.40	−2.62833
$107$	473.382	0.427697	0.213848	−	0.976867i	$-0.431400\pi$
$107$	473.382	0.427697	0.213848	+	0.976867i	$0.431400\pi$
$108$	−270.000	−0.240563
$109$	−352.353	−0.309627	−0.154813	−	0.987944i	$-0.549478\pi$
$109$	−352.353	−0.309627	−0.154813	+	0.987944i	$0.549478\pi$
$110$	1358.14	1.17722
$111$	−946.587	−0.809424
$112$	922.705	0.778459
$113$	−733.617	−0.610734	−0.305367	−	0.952235i	$-0.598779\pi$
$113$	−733.617	−0.610734	−0.305367	+	0.952235i	$0.598779\pi$
$114$	1029.46	0.845772
$115$	−2316.00	−1.87798
$116$	1541.18	1.23358
$117$	−314.470	−0.248485
$118$	2669.87	2.08289
$119$	356.500	0.274624
$120$	−508.368	−0.386728
$121$	−1074.06	−0.806955
$122$	−1959.47	−1.45412
$123$	−396.795	−0.290876
$124$	2999.41	2.17222
$125$	2972.09	2.12665
$126$	−800.735	−0.566152
$127$	340.410	0.237846	0.118923	−	0.992903i	$-0.462056\pi$
$127$	340.410	0.237846	0.118923	+	0.992903i	$0.462056\pi$
$128$	−1052.17	−0.726562
$129$	69.3532	0.0473350
$130$	−2960.49	−1.99732
$131$	−2133.85	−1.42317	−0.711586	−	0.702599i	$-0.752023\pi$
$131$	−2133.85	−1.42317	−0.711586	+	0.702599i	$0.752023\pi$
$132$	−480.883	−0.317087
$133$	1696.15	1.10582
$134$	−3349.44	−2.15931
$135$	−539.205	−0.343758
$136$	−144.250	−0.0909509
$137$	55.1455	0.0343897	0.0171949	−	0.999852i	$-0.494526\pi$
$137$	55.1455	0.0343897	0.0171949	+	0.999852i	$0.494526\pi$
$138$	1476.06	0.910514
$139$	256.266	0.156375	0.0781877	−	0.996939i	$-0.475087\pi$
$139$	256.266	0.156375	0.0781877	+	0.996939i	$0.475087\pi$
$140$	−4187.94	−2.52818
$141$	−782.735	−0.467505
$142$	−2912.20	−1.72103
$143$	−560.087	−0.327530
$144$	−396.000	−0.229167
$145$	3077.82	1.76275
$146$	2057.31	1.16619
$147$	−290.294	−0.162878
$148$	3155.29	1.75245
$149$	−84.3836	−0.0463958	−0.0231979	−	0.999731i	$-0.507385\pi$
$149$	−84.3836	−0.0463958	−0.0231979	+	0.999731i	$0.507385\pi$
$150$	−3485.20	−1.89710
$151$	1051.65	0.566767	0.283383	−	0.959007i	$-0.408543\pi$
$151$	1051.65	0.566767	0.283383	+	0.959007i	$0.408543\pi$
$152$	−686.309	−0.366230
$153$	−153.000	−0.0808452
$154$	−1426.15	−0.746249
$155$	5989.99	3.10405
$156$	1048.23	0.537986
$157$	1587.47	0.806968	0.403484	−	0.914987i	$-0.367799\pi$
$157$	1587.47	0.806968	0.403484	+	0.914987i	$0.367799\pi$
$158$	1077.63	0.542606
$159$	2028.26	1.01165
$160$	−5083.68	−2.51187
$161$	2431.97	1.19047
$162$	343.654	0.166667
$163$	1749.59	0.840727	0.420363	−	0.907356i	$-0.361903\pi$
$163$	1749.59	0.840727	0.420363	+	0.907356i	$0.361903\pi$
$164$	1322.65	0.629766
$165$	−960.351	−0.453110
$166$	2328.84	1.08887
$167$	−1624.21	−0.752604	−0.376302	−	0.926497i	$-0.622804\pi$
$167$	−1624.21	−0.752604	−0.376302	+	0.926497i	$0.622804\pi$
$168$	533.823	0.245151
$169$	−976.118	−0.444296
$170$	−1440.37	−0.649833
$171$	−727.940	−0.325538
$172$	−231.177	−0.102483
$173$	−3734.50	−1.64120	−0.820602	−	0.571500i	$-0.806362\pi$
$173$	−3734.50	−1.64120	−0.820602	+	0.571500i	$0.806362\pi$
$174$	−1961.60	−0.854646
$175$	−5742.23	−2.48041
$176$	−705.295	−0.302066
$177$	−1887.88	−0.801705
$178$	3925.06	1.65278
$179$	−1434.65	−0.599053	−0.299526	−	0.954088i	$-0.596829\pi$
$179$	−1434.65	−0.599053	−0.299526	+	0.954088i	$0.596829\pi$
$180$	1797.35	0.744259
$181$	−219.263	−0.0900426	−0.0450213	−	0.998986i	$-0.514336\pi$
$181$	−219.263	−0.0900426	−0.0450213	+	0.998986i	$0.514336\pi$
$182$	3108.73	1.26612
$183$	1385.56	0.559690
$184$	−984.043	−0.394264
$185$	6301.29	2.50422
$186$	−3817.63	−1.50496
$187$	−272.500	−0.106563
$188$	2609.12	1.01218
$189$	566.205	0.217912
$190$	−6852.99	−2.61667
$191$	−3116.44	−1.18062	−0.590308	−	0.807178i	$-0.700994\pi$
$191$	−3116.44	−1.18062	−0.590308	+	0.807178i	$0.700994\pi$
$192$	2184.00	0.820920
$193$	−3921.82	−1.46269	−0.731344	−	0.682009i	$-0.761106\pi$
$193$	−3921.82	−1.46269	−0.731344	+	0.682009i	$0.761106\pi$
$194$	3108.23	1.15030
$195$	2093.38	0.768770
$196$	967.645	0.352640
$197$	3141.68	1.13622	0.568110	−	0.822953i	$-0.307675\pi$
$197$	3141.68	1.13622	0.568110	+	0.822953i	$0.307675\pi$
$198$	612.064	0.219684
$199$	1832.79	0.652880	0.326440	−	0.945218i	$-0.394151\pi$
$199$	1832.79	0.652880	0.326440	+	0.945218i	$0.394151\pi$
$200$	2323.47	0.821470
$201$	2368.41	0.831118
$202$	−545.299	−0.189936
$203$	−3231.94	−1.11743
$204$	510.000	0.175035
$205$	2641.41	0.899921
$206$	22.7119	0.00768162
$207$	−1043.74	−0.350457
$208$	1537.41	0.512501
$209$	−1296.50	−0.429094
$210$	5330.38	1.75158
$211$	4928.41	1.60799	0.803994	−	0.594637i	$-0.202704\pi$
$211$	4928.41	1.60799	0.803994	+	0.594637i	$0.202704\pi$
$212$	−6760.87	−2.19028
$213$	2059.24	0.662425
$214$	2008.39	0.641545
$215$	−461.674	−0.146446
$216$	−229.103	−0.0721688
$217$	−6289.93	−1.96769
$218$	−1494.91	−0.464440
$219$	−1454.74	−0.448867
$220$	3201.17	0.981013
$221$	593.999	0.180800
$222$	−4016.03	−1.21414
$223$	−75.4727	−0.0226638	−0.0113319	−	0.999936i	$-0.503607\pi$
$223$	−75.4727	−0.0226638	−0.0113319	+	0.999936i	$0.503607\pi$
$224$	5338.23	1.59230
$225$	2464.41	0.730196
$226$	−3112.47	−0.916101
$227$	−1629.97	−0.476587	−0.238293	−	0.971193i	$-0.576588\pi$
$227$	−1629.97	−0.476587	−0.238293	+	0.971193i	$0.576588\pi$
$228$	2426.47	0.704810
$229$	2000.35	0.577235	0.288617	−	0.957445i	$-0.406804\pi$
$229$	2000.35	0.577235	0.288617	+	0.957445i	$0.406804\pi$
$230$	−9825.94	−2.81697
$231$	1008.44	0.287231
$232$	1307.73	0.370073
$233$	−4225.26	−1.18801	−0.594004	−	0.804462i	$-0.702454\pi$
$233$	−4225.26	−1.18801	−0.594004	+	0.804462i	$0.702454\pi$
$234$	−1334.18	−0.372728
$235$	5210.55	1.44638
$236$	6292.94	1.73574
$237$	−762.000	−0.208849
$238$	1512.50	0.411936
$239$	−1614.97	−0.437086	−0.218543	−	0.975827i	$-0.570130\pi$
$239$	−1614.97	−0.437086	−0.218543	+	0.975827i	$0.570130\pi$
$240$	2636.11	0.709002
$241$	−6761.67	−1.80729	−0.903646	−	0.428280i	$-0.859120\pi$
$241$	−6761.67	−1.80729	−0.903646	+	0.428280i	$0.859120\pi$
$242$	−4556.84	−1.21043
$243$	−243.000	−0.0641500
$244$	−4618.52	−1.21176
$245$	1932.44	0.503915
$246$	−1683.46	−0.436314
$247$	2826.12	0.728022
$248$	2545.08	0.651666
$249$	−1646.74	−0.419107
$250$	12609.5	3.18998
$251$	2880.29	0.724313	0.362156	−	0.932117i	$-0.382041\pi$
$251$	2880.29	0.724313	0.362156	+	0.932117i	$0.382041\pi$
$252$	−1887.35	−0.471793
$253$	−1858.94	−0.461940
$254$	1444.24	0.356769
$255$	1018.50	0.250121
$256$	1360.00	0.332031
$257$	−1460.44	−0.354474	−0.177237	−	0.984168i	$-0.556716\pi$
$257$	−1460.44	−0.354474	−0.177237	+	0.984168i	$0.556716\pi$
$258$	294.241	0.0710025
$259$	−6616.82	−1.58745
$260$	−6977.94	−1.66444
$261$	1387.06	0.328953
$262$	−9053.17	−2.13476
$263$	−7601.29	−1.78219	−0.891094	−	0.453819i	$-0.850061\pi$
$263$	−7601.29	−1.78219	−0.891094	+	0.453819i	$0.850061\pi$
$264$	−408.043	−0.0951261
$265$	−13501.8	−3.12986
$266$	7196.14	1.65874
$267$	−2775.44	−0.636157
$268$	−7894.70	−1.79942
$269$	6896.26	1.56309	0.781547	−	0.623846i	$-0.214431\pi$
$269$	6896.26	1.56309	0.781547	+	0.623846i	$0.214431\pi$
$270$	−2287.65	−0.515638
$271$	849.234	0.190359	0.0951795	−	0.995460i	$-0.469658\pi$
$271$	849.234	0.190359	0.0951795	+	0.995460i	$0.469658\pi$
$272$	748.000	0.166743
$273$	−2198.21	−0.487331
$274$	233.962	0.0515846
$275$	4389.23	0.962476
$276$	3479.12	0.758762
$277$	4080.09	0.885013	0.442507	−	0.896765i	$-0.354089\pi$
$277$	4080.09	0.885013	0.442507	+	0.896765i	$0.354089\pi$
$278$	1087.24	0.234563
$279$	2699.47	0.579258
$280$	−3553.58	−0.758455
$281$	4967.90	1.05466	0.527331	−	0.849660i	$-0.323193\pi$
$281$	4967.90	1.05466	0.527331	+	0.849660i	$0.323193\pi$
$282$	−3320.86	−0.701257
$283$	−1475.83	−0.309996	−0.154998	−	0.987915i	$-0.549537\pi$
$283$	−1475.83	−0.309996	−0.154998	+	0.987915i	$0.549537\pi$
$284$	−6864.12	−1.43419
$285$	4845.79	1.00716
$286$	−2376.25	−0.491295
$287$	−2773.67	−0.570469
$288$	−2291.03	−0.468750
$289$	289.000	0.0588235
$290$	13058.1	2.64413
$291$	−2197.85	−0.442750
$292$	4849.12	0.971826
$293$	521.056	0.103892	0.0519461	−	0.998650i	$-0.483458\pi$
$293$	521.056	0.103892	0.0519461	+	0.998650i	$0.483458\pi$
$294$	−1231.61	−0.244316
$295$	12567.3	2.48034
$296$	2677.35	0.525736
$297$	−432.795	−0.0845566
$298$	−358.009	−0.0695937
$299$	4052.14	0.783751
$300$	−8214.70	−1.58092
$301$	484.792	0.0928337
$302$	4461.76	0.850150
$303$	385.584	0.0731064
$304$	3558.82	0.671422
$305$	−9223.44	−1.73158
$306$	−649.124	−0.121268
$307$	−1718.23	−0.319429	−0.159715	−	0.987163i	$-0.551057\pi$
$307$	−1718.23	−0.319429	−0.159715	+	0.987163i	$0.551057\pi$
$308$	−3361.46	−0.621874
$309$	−16.0597	−0.00295666
$310$	25413.4	4.65608
$311$	4916.05	0.896347	0.448173	−	0.893947i	$-0.352075\pi$
$311$	4916.05	0.896347	0.448173	+	0.893947i	$0.352075\pi$
$312$	889.456	0.161396
$313$	7375.84	1.33197	0.665986	−	0.745964i	$-0.268011\pi$
$313$	7375.84	1.33197	0.665986	+	0.745964i	$0.268011\pi$
$314$	6735.07	1.21045
$315$	−3769.15	−0.674182
$316$	2540.00	0.452171
$317$	522.706	0.0926124	0.0463062	−	0.998927i	$-0.485255\pi$
$317$	522.706	0.0926124	0.0463062	+	0.998927i	$0.485255\pi$
$318$	8605.19	1.51747
$319$	2470.42	0.433596
$320$	−14538.6	−2.53978
$321$	−1420.15	−0.246931
$322$	10318.0	1.78571
$323$	1375.00	0.236864
$324$	810.000	0.138889
$325$	−9567.70	−1.63299
$326$	7422.88	1.26109
$327$	1057.06	0.178763
$328$	1122.31	0.188930
$329$	−5471.46	−0.916874
$330$	−4074.42	−0.679665
$331$	5016.46	0.833020	0.416510	−	0.909131i	$-0.363253\pi$
$331$	5016.46	0.833020	0.416510	+	0.909131i	$0.363253\pi$
$332$	5489.12	0.907393
$333$	2839.76	0.467321
$334$	−6890.92	−1.12891
$335$	−15766.2	−2.57134
$336$	−2768.11	−0.449443
$337$	−10810.8	−1.74748	−0.873741	−	0.486392i	$-0.838313\pi$
$337$	−10810.8	−1.74748	−0.873741	+	0.486392i	$0.838313\pi$
$338$	−4141.32	−0.666444
$339$	2200.85	0.352607
$340$	−3395.00	−0.541528
$341$	4807.89	0.763524
$342$	−3088.39	−0.488307
$343$	5163.70	0.812867
$344$	−196.161	−0.0307450
$345$	6947.99	1.08425
$346$	−15844.1	−2.46181
$347$	−4137.35	−0.640070	−0.320035	−	0.947406i	$-0.603695\pi$
$347$	−4137.35	−0.640070	−0.320035	+	0.947406i	$0.603695\pi$
$348$	−4623.53	−0.712205
$349$	7531.11	1.15510	0.577552	−	0.816354i	$-0.304008\pi$
$349$	7531.11	1.15510	0.577552	+	0.816354i	$0.304008\pi$
$350$	−24362.2	−3.72062
$351$	943.410	0.143463
$352$	−4080.43	−0.617862
$353$	−4872.29	−0.734634	−0.367317	−	0.930096i	$-0.619724\pi$
$353$	−4872.29	−0.734634	−0.367317	+	0.930096i	$0.619724\pi$
$354$	−8009.60	−1.20256
$355$	−13708.0	−2.04943
$356$	9251.45	1.37732
$357$	−1069.50	−0.158554
$358$	−6086.69	−0.898579
$359$	−31.9429	−0.00469604	−0.00234802	−	0.999997i	$-0.500747\pi$
$359$	−31.9429	−0.00469604	−0.00234802	+	0.999997i	$0.500747\pi$
$360$	1525.10	0.223278
$361$	−317.061	−0.0462256
$362$	−930.255	−0.135064
$363$	3222.17	0.465896
$364$	7327.35	1.05510
$365$	9683.96	1.38872
$366$	5878.42	0.839535
$367$	2262.42	0.321791	0.160895	−	0.986971i	$-0.448562\pi$
$367$	2262.42	0.321791	0.160895	+	0.986971i	$0.448562\pi$
$368$	5102.70	0.722818
$369$	1190.38	0.167938
$370$	26734.1	3.75633
$371$	14177.9	1.98405
$372$	−8998.23	−1.25413
$373$	−5788.70	−0.803559	−0.401780	−	0.915736i	$-0.631608\pi$
$373$	−5788.70	−0.803559	−0.401780	+	0.915736i	$0.631608\pi$
$374$	−1156.12	−0.159844
$375$	−8916.26	−1.22782
$376$	2213.91	0.303653
$377$	−5385.05	−0.735661
$378$	2402.21	0.326868
$379$	10940.9	1.48284	0.741420	−	0.671041i	$-0.234153\pi$
$379$	10940.9	1.48284	0.741420	+	0.671041i	$0.234153\pi$
$380$	−16152.6	−2.18056
$381$	−1021.23	−0.137321
$382$	−13221.9	−1.77092
$383$	2414.03	0.322066	0.161033	−	0.986949i	$-0.448517\pi$
$383$	2414.03	0.322066	0.161033	+	0.986949i	$0.448517\pi$
$384$	3156.52	0.419481
$385$	−6713.03	−0.888644
$386$	−16638.9	−2.19403
$387$	−208.060	−0.0273289
$388$	7326.17	0.958583
$389$	−4479.41	−0.583844	−0.291922	−	0.956442i	$-0.594295\pi$
$389$	−4479.41	−0.583844	−0.291922	+	0.956442i	$0.594295\pi$
$390$	8881.47	1.15316
$391$	1971.50	0.254995
$392$	821.074	0.105792
$393$	6401.56	0.821669
$394$	13329.0	1.70433
$395$	5072.52	0.646143
$396$	1442.65	0.183070
$397$	−1780.62	−0.225105	−0.112553	−	0.993646i	$-0.535903\pi$
$397$	−1780.62	−0.225105	−0.112553	+	0.993646i	$0.535903\pi$
$398$	7775.88	0.979321
$399$	−5088.44	−0.638448
$400$	−12048.2	−1.50603
$401$	−4067.97	−0.506595	−0.253297	−	0.967388i	$-0.581515\pi$
$401$	−4067.97	−0.506595	−0.253297	+	0.967388i	$0.581515\pi$
$402$	10048.3	1.24668
$403$	−10480.3	−1.29543
$404$	−1285.28	−0.158280
$405$	1617.62	0.198469
$406$	−13711.9	−1.67614
$407$	5057.75	0.615979
$408$	432.749	0.0525105
$409$	−632.302	−0.0764434	−0.0382217	−	0.999269i	$-0.512169\pi$
$409$	−632.302	−0.0764434	−0.0382217	+	0.999269i	$0.512169\pi$
$410$	11206.5	1.34988
$411$	−165.436	−0.0198549
$412$	53.5325	0.00640135
$413$	−13196.6	−1.57231
$414$	−4428.19	−0.525686
$415$	10962.1	1.29664
$416$	8894.56	1.04830
$417$	−768.797	−0.0902834
$418$	−5500.57	−0.643640
$419$	−1107.06	−0.129077	−0.0645386	−	0.997915i	$-0.520558\pi$
$419$	−1107.06	−0.129077	−0.0645386	+	0.997915i	$0.520558\pi$
$420$	12563.8	1.45965
$421$	−15977.3	−1.84961	−0.924804	−	0.380444i	$-0.875771\pi$
$421$	−15977.3	−1.84961	−0.924804	+	0.380444i	$0.875771\pi$
$422$	20909.5	2.41198
$423$	2348.21	0.269914
$424$	−5736.79	−0.657083
$425$	−4655.00	−0.531295
$426$	8736.60	0.993638
$427$	9685.30	1.09767
$428$	4733.82	0.534621
$429$	1680.26	0.189100
$430$	−1958.72	−0.219669
$431$	10283.0	1.14922	0.574610	−	0.818427i	$-0.305154\pi$
$431$	10283.0	1.14922	0.574610	+	0.818427i	$0.305154\pi$
$432$	1188.00	0.132309
$433$	−7084.29	−0.786257	−0.393128	−	0.919484i	$-0.628607\pi$
$433$	−7084.29	−0.786257	−0.393128	+	0.919484i	$0.628607\pi$
$434$	−26685.9	−2.95153
$435$	−9233.45	−1.01772
$436$	−3523.53	−0.387033
$437$	9379.96	1.02678
$438$	−6171.92	−0.673301
$439$	−5308.51	−0.577133	−0.288567	−	0.957460i	$-0.593179\pi$
$439$	−5308.51	−0.577133	−0.288567	+	0.957460i	$0.593179\pi$
$440$	2716.28	0.294304
$441$	870.881	0.0940374
$442$	2520.12	0.271199
$443$	3533.26	0.378939	0.189470	−	0.981887i	$-0.439323\pi$
$443$	3533.26	0.378939	0.189470	+	0.981887i	$0.439323\pi$
$444$	−9465.87	−1.01178
$445$	18475.7	1.96816
$446$	−320.204	−0.0339957
$447$	253.151	0.0267866
$448$	15266.6	1.60999
$449$	4787.18	0.503165	0.251582	−	0.967836i	$-0.419049\pi$
$449$	4787.18	0.503165	0.251582	+	0.967836i	$0.419049\pi$
$450$	10455.6	1.09529
$451$	2120.13	0.221360
$452$	−7336.17	−0.763417
$453$	−3154.94	−0.327223
$454$	−6915.39	−0.714880
$455$	14633.1	1.50772
$456$	2058.93	0.211443
$457$	13168.8	1.34795	0.673973	−	0.738756i	$-0.264586\pi$
$457$	13168.8	1.34795	0.673973	+	0.738756i	$0.264586\pi$
$458$	8486.76	0.865852
$459$	459.000	0.0466760
$460$	−23160.0	−2.34748
$461$	−7145.81	−0.721938	−0.360969	−	0.932578i	$-0.617554\pi$
$461$	−7145.81	−0.721938	−0.360969	+	0.932578i	$0.617554\pi$
$462$	4278.44	0.430847
$463$	9462.71	0.949826	0.474913	−	0.880033i	$-0.342480\pi$
$463$	9462.71	0.949826	0.474913	+	0.880033i	$0.342480\pi$
$464$	−6781.18	−0.678466
$465$	−17970.0	−1.79212
$466$	−17926.3	−1.78201
$467$	−5306.09	−0.525774	−0.262887	−	0.964827i	$-0.584675\pi$
$467$	−5306.09	−0.525774	−0.262887	+	0.964827i	$0.584675\pi$
$468$	−3144.70	−0.310607
$469$	16555.6	1.63000
$470$	22106.5	2.16957
$471$	−4762.41	−0.465903
$472$	5339.73	0.520723
$473$	−370.565	−0.0360224
$474$	−3232.89	−0.313274
$475$	−22147.5	−2.13936
$476$	3565.00	0.343280
$477$	−6084.79	−0.584074
$478$	−6851.73	−0.655630
$479$	11731.4	1.11904	0.559522	−	0.828815i	$-0.310985\pi$
$479$	11731.4	1.11904	0.559522	+	0.828815i	$0.310985\pi$
$480$	15251.0	1.45023
$481$	−11024.9	−1.04510
$482$	−28687.3	−2.71094
$483$	−7295.90	−0.687319
$484$	−10740.6	−1.00869
$485$	14630.8	1.36979
$486$	−1030.96	−0.0962250
$487$	11075.5	1.03055	0.515274	−	0.857026i	$-0.327690\pi$
$487$	11075.5	1.03055	0.515274	+	0.857026i	$0.327690\pi$
$488$	−3918.94	−0.363529
$489$	−5248.77	−0.485394
$490$	8198.66	0.755872
$491$	2009.14	0.184666	0.0923331	−	0.995728i	$-0.470568\pi$
$491$	2009.14	0.184666	0.0923331	+	0.995728i	$0.470568\pi$
$492$	−3967.95	−0.363595
$493$	−2620.00	−0.239349
$494$	11990.2	1.09203
$495$	2881.05	0.261603
$496$	−13197.4	−1.19472
$497$	14394.4	1.29915
$498$	−6986.51	−0.628660
$499$	9956.24	0.893192	0.446596	−	0.894736i	$-0.352636\pi$
$499$	9956.24	0.893192	0.446596	+	0.894736i	$0.352636\pi$
$500$	29720.9	2.65832
$501$	4872.62	0.434516
$502$	12220.0	1.08647
$503$	11760.9	1.04253	0.521266	−	0.853394i	$-0.325460\pi$
$503$	11760.9	1.04253	0.521266	+	0.853394i	$0.325460\pi$
$504$	−1601.47	−0.141538
$505$	−2566.78	−0.226179
$506$	−7886.83	−0.692910
$507$	2928.35	0.256514
$508$	3404.10	0.297308
$509$	−948.293	−0.0825783	−0.0412891	−	0.999147i	$-0.513146\pi$
$509$	−948.293	−0.0825783	−0.0412891	+	0.999147i	$0.513146\pi$
$510$	4321.12	0.375182
$511$	−10168.9	−0.880322
$512$	14187.4	1.22461
$513$	2183.82	0.187949
$514$	−6196.13	−0.531711
$515$	106.907	0.00914738
$516$	693.532	0.0591687
$517$	4182.27	0.355775
$518$	−28072.8	−2.38117
$519$	11203.5	0.947550
$520$	−5920.98	−0.499331
$521$	−9796.08	−0.823751	−0.411875	−	0.911240i	$-0.635126\pi$
$521$	−9796.08	−0.823751	−0.411875	+	0.911240i	$0.635126\pi$
$522$	5884.80	0.493430
$523$	−4501.89	−0.376394	−0.188197	−	0.982131i	$-0.560264\pi$
$523$	−4501.89	−0.376394	−0.188197	+	0.982131i	$0.560264\pi$
$524$	−21338.5	−1.77897
$525$	17226.7	1.43207
$526$	−32249.5	−2.67328
$527$	−5099.00	−0.421472
$528$	2115.89	0.174398
$529$	1282.17	0.105381
$530$	−57283.5	−4.69478
$531$	5663.64	0.462865
$532$	16961.5	1.38228
$533$	−4621.49	−0.375570
$534$	−11775.2	−0.954236
$535$	9453.70	0.763961
$536$	−6698.88	−0.539827
$537$	4303.94	0.345863
$538$	29258.3	2.34464
$539$	1551.08	0.123951
$540$	−5392.05	−0.429698
$541$	−8419.68	−0.669114	−0.334557	−	0.942376i	$-0.608587\pi$
$541$	−8419.68	−0.669114	−0.334557	+	0.942376i	$0.608587\pi$
$542$	3602.99	0.285538
$543$	657.790	0.0519861
$544$	4327.49	0.341066
$545$	−7036.69	−0.553062
$546$	−9326.19	−0.730997
$547$	4655.26	0.363884	0.181942	−	0.983309i	$-0.441762\pi$
$547$	4655.26	0.363884	0.181942	+	0.983309i	$0.441762\pi$
$548$	551.455	0.0429872
$549$	−4156.67	−0.323137
$550$	18621.9	1.44371
$551$	−12465.4	−0.963781
$552$	2952.13	0.227629
$553$	−5326.52	−0.409596
$554$	17310.3	1.32752
$555$	−18903.9	−1.44581
$556$	2562.66	0.195469
$557$	17855.9	1.35831	0.679157	−	0.733993i	$-0.262346\pi$
$557$	17855.9	1.35831	0.679157	+	0.733993i	$0.262346\pi$
$558$	11452.9	0.868887
$559$	807.760	0.0611174
$560$	18426.9	1.39050
$561$	817.501	0.0615239
$562$	21077.0	1.58199
$563$	−21434.5	−1.60454	−0.802269	−	0.596962i	$-0.796374\pi$
$563$	−21434.5	−1.60454	−0.802269	+	0.596962i	$0.796374\pi$
$564$	−7827.35	−0.584381
$565$	−14650.8	−1.09091
$566$	−6261.40	−0.464994
$567$	−1698.62	−0.125812
$568$	−5824.40	−0.430258
$569$	−14412.8	−1.06189	−0.530945	−	0.847406i	$-0.678163\pi$
$569$	−14412.8	−1.06189	−0.530945	+	0.847406i	$0.678163\pi$
$570$	20559.0	1.51074
$571$	4492.48	0.329255	0.164627	−	0.986356i	$-0.447358\pi$
$571$	4492.48	0.329255	0.164627	+	0.986356i	$0.447358\pi$
$572$	−5600.87	−0.409413
$573$	9349.31	0.681628
$574$	−11767.7	−0.855703
$575$	−31755.5	−2.30312
$576$	−6552.00	−0.473958
$577$	9544.29	0.688621	0.344310	−	0.938856i	$-0.388113\pi$
$577$	9544.29	0.688621	0.344310	+	0.938856i	$0.388113\pi$
$578$	1226.12	0.0882353
$579$	11765.5	0.844483
$580$	30778.2	2.20344
$581$	−11511.0	−0.821956
$582$	−9324.70	−0.664126
$583$	−10837.3	−0.769872
$584$	4114.61	0.291548
$585$	−6280.15	−0.443850
$586$	2210.65	0.155838
$587$	−15671.9	−1.10195	−0.550977	−	0.834520i	$-0.685745\pi$
$587$	−15671.9	−1.10195	−0.550977	+	0.834520i	$0.685745\pi$
$588$	−2902.94	−0.203597
$589$	−24259.9	−1.69713
$590$	53318.7	3.72050
$591$	−9425.03	−0.655996
$592$	−13883.3	−0.963850
$593$	6737.03	0.466537	0.233269	−	0.972412i	$-0.425058\pi$
$593$	6737.03	0.466537	0.233269	+	0.972412i	$0.425058\pi$
$594$	−1836.19	−0.126835
$595$	7119.50	0.490539
$596$	−843.836	−0.0579947
$597$	−5498.38	−0.376941
$598$	17191.8	1.17563
$599$	−25077.1	−1.71056	−0.855278	−	0.518169i	$-0.826614\pi$
$599$	−25077.1	−1.71056	−0.855278	+	0.518169i	$0.826614\pi$
$600$	−6970.41	−0.474276
$601$	18123.5	1.23007	0.615037	−	0.788499i	$-0.289141\pi$
$601$	18123.5	1.23007	0.615037	+	0.788499i	$0.289141\pi$
$602$	2056.80	0.139251
$603$	−7105.23	−0.479846
$604$	10516.5	0.708459
$605$	−21449.5	−1.44140
$606$	1635.90	0.109660
$607$	18377.1	1.22884	0.614419	−	0.788980i	$-0.289390\pi$
$607$	18377.1	1.22884	0.614419	+	0.788980i	$0.289390\pi$
$608$	20589.3	1.37336
$609$	9695.81	0.645146
$610$	−39131.8	−2.59737
$611$	−9116.55	−0.603627
$612$	−1530.00	−0.101057
$613$	19642.4	1.29421	0.647105	−	0.762401i	$-0.275979\pi$
$613$	19642.4	1.29421	0.647105	+	0.762401i	$0.275979\pi$
$614$	−7289.85	−0.479144
$615$	−7924.22	−0.519569
$616$	−2852.30	−0.186562
$617$	8738.58	0.570181	0.285091	−	0.958501i	$-0.407976\pi$
$617$	8738.58	0.570181	0.285091	+	0.958501i	$0.407976\pi$
$618$	−68.1357	−0.00443498
$619$	18996.2	1.23348	0.616739	−	0.787168i	$-0.288453\pi$
$619$	18996.2	1.23348	0.616739	+	0.787168i	$0.288453\pi$
$620$	59899.9	3.88006
$621$	3131.21	0.202336
$622$	20857.0	1.34452
$623$	−19400.8	−1.24764
$624$	−4612.23	−0.295892
$625$	25126.3	1.60808
$626$	31293.1	1.99796
$627$	3889.49	0.247737
$628$	15874.7	1.00871
$629$	−5363.99	−0.340026
$630$	−15991.1	−1.01127
$631$	18454.2	1.16426	0.582130	−	0.813096i	$-0.302219\pi$
$631$	18454.2	1.16426	0.582130	+	0.813096i	$0.302219\pi$
$632$	2155.26	0.135651
$633$	−14785.2	−0.928373
$634$	2217.66	0.138919
$635$	6798.17	0.424846
$636$	20282.6	1.26456
$637$	−3381.06	−0.210302
$638$	10481.1	0.650393
$639$	−6177.71	−0.382451
$640$	−21012.5	−1.29780
$641$	−2208.09	−0.136060	−0.0680300	−	0.997683i	$-0.521671\pi$
$641$	−2208.09	−0.136060	−0.0680300	+	0.997683i	$0.521671\pi$
$642$	−6025.17	−0.370396
$643$	−14048.7	−0.861625	−0.430813	−	0.902441i	$-0.641773\pi$
$643$	−14048.7	−0.861625	−0.430813	+	0.902441i	$0.641773\pi$
$644$	24319.7	1.48809
$645$	1385.02	0.0845508
$646$	5833.62	0.355295
$647$	7959.60	0.483654	0.241827	−	0.970319i	$-0.422253\pi$
$647$	7959.60	0.483654	0.241827	+	0.970319i	$0.422253\pi$
$648$	687.308	0.0416667
$649$	10087.2	0.610105
$650$	−40592.3	−2.44948
$651$	18869.8	1.13605
$652$	17495.9	1.05091
$653$	8533.37	0.511388	0.255694	−	0.966758i	$-0.417696\pi$
$653$	8533.37	0.511388	0.255694	+	0.966758i	$0.417696\pi$
$654$	4484.72	0.268145
$655$	−42614.2	−2.54210
$656$	−5819.66	−0.346371
$657$	4364.21	0.259154
$658$	−23213.5	−1.37531
$659$	8532.32	0.504358	0.252179	−	0.967681i	$-0.418853\pi$
$659$	8532.32	0.504358	0.252179	+	0.967681i	$0.418853\pi$
$660$	−9603.51	−0.566388
$661$	24194.0	1.42366	0.711828	−	0.702353i	$-0.247867\pi$
$661$	24194.0	1.42366	0.711828	+	0.702353i	$0.247867\pi$
$662$	21283.1	1.24953
$663$	−1782.00	−0.104385
$664$	4657.67	0.272218
$665$	33873.0	1.97525
$666$	12048.1	0.700982
$667$	−17873.1	−1.03756
$668$	−16242.1	−0.940755
$669$	226.418	0.0130850
$670$	−66890.2	−3.85700
$671$	−7403.23	−0.425929
$672$	−16014.7	−0.919316
$673$	−10069.6	−0.576750	−0.288375	−	0.957518i	$-0.593115\pi$
$673$	−10069.6	−0.576750	−0.288375	+	0.957518i	$0.593115\pi$
$674$	−45866.3	−2.62122
$675$	−7393.23	−0.421579
$676$	−9761.18	−0.555370
$677$	13155.9	0.746856	0.373428	−	0.927659i	$-0.378182\pi$
$677$	13155.9	0.746856	0.373428	+	0.927659i	$0.378182\pi$
$678$	9337.42	0.528911
$679$	−15363.4	−0.868326
$680$	−2880.75	−0.162458
$681$	4889.92	0.275157
$682$	20398.1	1.14529
$683$	−21814.1	−1.22210	−0.611049	−	0.791593i	$-0.709252\pi$
$683$	−21814.1	−1.22210	−0.611049	+	0.791593i	$0.709252\pi$
$684$	−7279.40	−0.406922
$685$	1101.29	0.0614277
$686$	21907.7	1.21930
$687$	−6001.04	−0.333267
$688$	1017.18	0.0563658
$689$	23623.3	1.30620
$690$	29477.8	1.62638
$691$	−8237.48	−0.453500	−0.226750	−	0.973953i	$-0.572810\pi$
$691$	−8237.48	−0.453500	−0.226750	+	0.973953i	$0.572810\pi$
$692$	−37345.0	−2.05151
$693$	−3025.32	−0.165833
$694$	−17553.3	−0.960105
$695$	5117.77	0.279321
$696$	−3923.20	−0.213662
$697$	−2248.50	−0.122192
$698$	31951.8	1.73265
$699$	12675.8	0.685897
$700$	−57422.3	−3.10051
$701$	3684.13	0.198499	0.0992495	−	0.995063i	$-0.468356\pi$
$701$	3684.13	0.198499	0.0992495	+	0.995063i	$0.468356\pi$
$702$	4002.55	0.215195
$703$	−25520.7	−1.36918
$704$	−11669.4	−0.624728
$705$	−15631.7	−0.835067
$706$	−20671.4	−1.10195
$707$	2695.31	0.143377
$708$	−18878.8	−1.00213
$709$	26765.8	1.41779	0.708894	−	0.705315i	$-0.249194\pi$
$709$	26765.8	1.41779	0.708894	+	0.705315i	$0.249194\pi$
$710$	−58158.3	−3.07414
$711$	2286.00	0.120579
$712$	7850.12	0.413196
$713$	−34784.3	−1.82705
$714$	−4537.50	−0.237831
$715$	−11185.2	−0.585041
$716$	−14346.5	−0.748816
$717$	4844.91	0.252352
$718$	−135.522	−0.00704407
$719$	20163.3	1.04585	0.522924	−	0.852379i	$-0.324841\pi$
$719$	20163.3	1.04585	0.522924	+	0.852379i	$0.324841\pi$
$720$	−7908.34	−0.409342
$721$	−112.261	−0.00579862
$722$	−1345.18	−0.0693384
$723$	20285.0	1.04344
$724$	−2192.63	−0.112553
$725$	42201.0	2.16180
$726$	13670.5	0.698844
$727$	23230.6	1.18511	0.592556	−	0.805529i	$-0.298119\pi$
$727$	23230.6	1.18511	0.592556	+	0.805529i	$0.298119\pi$
$728$	6217.46	0.316531
$729$	729.000	0.0370370
$730$	41085.6	2.08308
$731$	393.002	0.0198847
$732$	13855.6	0.699612
$733$	−28594.4	−1.44087	−0.720436	−	0.693521i	$-0.756058\pi$
$733$	−28594.4	−1.44087	−0.720436	+	0.693521i	$0.756058\pi$
$734$	9598.62	0.482686
$735$	−5797.32	−0.290935
$736$	29521.3	1.47849
$737$	−12654.8	−0.632489
$738$	5050.37	0.251906
$739$	8294.73	0.412891	0.206446	−	0.978458i	$-0.433810\pi$
$739$	8294.73	0.412891	0.206446	+	0.978458i	$0.433810\pi$
$740$	63012.9	3.13027
$741$	−8478.35	−0.420324
$742$	60151.9	2.97607
$743$	−5881.61	−0.290411	−0.145205	−	0.989402i	$-0.546384\pi$
$743$	−5881.61	−0.290411	−0.145205	+	0.989402i	$0.546384\pi$
$744$	−7635.25	−0.376239
$745$	−1685.19	−0.0828731
$746$	−24559.4	−1.20534
$747$	4940.21	0.241971
$748$	−2725.00	−0.133203
$749$	−9927.08	−0.484283
$750$	−37828.5	−1.84173
$751$	1255.39	0.0609984	0.0304992	−	0.999535i	$-0.490290\pi$
$751$	1255.39	0.0609984	0.0304992	+	0.999535i	$0.490290\pi$
$752$	−11480.1	−0.556698
$753$	−8640.88	−0.418182
$754$	−22846.8	−1.10349
$755$	21002.0	1.01237
$756$	5662.05	0.272390
$757$	−38652.8	−1.85582	−0.927912	−	0.372799i	$-0.878398\pi$
$757$	−38652.8	−1.85582	−0.927912	+	0.372799i	$0.878398\pi$
$758$	46418.3	2.22426
$759$	5576.83	0.266701
$760$	−13706.0	−0.654168
$761$	−3286.48	−0.156550	−0.0782752	−	0.996932i	$-0.524941\pi$
$761$	−3286.48	−0.156550	−0.0782752	+	0.996932i	$0.524941\pi$
$762$	−4332.71	−0.205981
$763$	7389.05	0.350592
$764$	−31164.4	−1.47577
$765$	−3055.50	−0.144407
$766$	10241.9	0.483099
$767$	−21988.2	−1.03514
$768$	−4080.00	−0.191698
$769$	−29939.6	−1.40396	−0.701982	−	0.712195i	$-0.747701\pi$
$769$	−29939.6	−1.40396	−0.701982	+	0.712195i	$0.747701\pi$
$770$	−28481.0	−1.33297
$771$	4381.33	0.204656
$772$	−39218.2	−1.82836
$773$	40700.7	1.89379	0.946896	−	0.321541i	$-0.104201\pi$
$773$	40700.7	1.89379	0.946896	+	0.321541i	$0.104201\pi$
$774$	−882.723	−0.0409933
$775$	82130.9	3.80675
$776$	6216.46	0.287575
$777$	19850.5	0.916514
$778$	−19004.5	−0.875765
$779$	−10697.9	−0.492030
$780$	20933.8	0.960963
$781$	−11002.8	−0.504112
$782$	8364.36	0.382492
$783$	−4161.18	−0.189921
$784$	−4257.64	−0.193952
$785$	31702.7	1.44142
$786$	27159.5	1.23250
$787$	−8126.31	−0.368071	−0.184035	−	0.982920i	$-0.558916\pi$
$787$	−8126.31	−0.368071	−0.184035	+	0.982920i	$0.558916\pi$
$788$	31416.8	1.42027
$789$	22803.9	1.02895
$790$	21520.9	0.969214
$791$	15384.4	0.691536
$792$	1224.13	0.0549211
$793$	16137.6	0.722653
$794$	−7554.53	−0.337658
$795$	40505.5	1.80702
$796$	18327.9	0.816100
$797$	11987.8	0.532787	0.266393	−	0.963864i	$-0.414168\pi$
$797$	11987.8	0.532787	0.266393	+	0.963864i	$0.414168\pi$
$798$	−21588.4	−0.957671
$799$	−4435.50	−0.196391
$800$	−69704.1	−3.08051
$801$	8326.31	0.367285
$802$	−17258.9	−0.759892
$803$	7772.86	0.341592
$804$	23684.1	1.03890
$805$	48567.8	2.12645
$806$	−44464.1	−1.94315
$807$	−20688.8	−0.902453
$808$	−1090.60	−0.0474840
$809$	−36146.3	−1.57087	−0.785436	−	0.618943i	$-0.787561\pi$
$809$	−36146.3	−1.57087	−0.785436	+	0.618943i	$0.787561\pi$
$810$	6862.96	0.297704
$811$	−6244.38	−0.270370	−0.135185	−	0.990820i	$-0.543163\pi$
$811$	−6244.38	−0.270370	−0.135185	+	0.990820i	$0.543163\pi$
$812$	−32319.4	−1.39678
$813$	−2547.70	−0.109904
$814$	21458.2	0.923969
$815$	34940.3	1.50172
$816$	−2244.00	−0.0962693
$817$	1869.82	0.0800692
$818$	−2682.63	−0.114665
$819$	6594.62	0.281361
$820$	26414.1	1.12490
$821$	28413.7	1.20785	0.603924	−	0.797042i	$-0.293603\pi$
$821$	28413.7	1.20785	0.603924	+	0.797042i	$0.293603\pi$
$822$	−701.887	−0.0297824
$823$	−18803.1	−0.796399	−0.398199	−	0.917299i	$-0.630365\pi$
$823$	−18803.1	−0.796399	−0.398199	+	0.917299i	$0.630365\pi$
$824$	45.4238	0.00192040
$825$	−13167.7	−0.555686
$826$	−55988.6	−2.35847
$827$	33860.3	1.42375	0.711873	−	0.702309i	$-0.247847\pi$
$827$	33860.3	1.42375	0.711873	+	0.702309i	$0.247847\pi$
$828$	−10437.4	−0.438071
$829$	−19746.0	−0.827270	−0.413635	−	0.910443i	$-0.635741\pi$
$829$	−19746.0	−0.827270	−0.413635	+	0.910443i	$0.635741\pi$
$830$	46508.1	1.94497
$831$	−12240.3	−0.510963
$832$	25437.1	1.05994
$833$	−1645.00	−0.0684223
$834$	−3261.73	−0.135425
$835$	−32436.3	−1.34432
$836$	−12965.0	−0.536367
$837$	−8098.41	−0.334435
$838$	−4696.85	−0.193616
$839$	−31045.8	−1.27750	−0.638748	−	0.769416i	$-0.720547\pi$
$839$	−31045.8	−1.27750	−0.638748	+	0.769416i	$0.720547\pi$
$840$	10660.8	0.437894
$841$	−636.719	−0.0261068
$842$	−67785.9	−2.77441
$843$	−14903.7	−0.608910
$844$	49284.1	2.00999
$845$	−19493.6	−0.793611
$846$	9962.59	0.404871
$847$	22523.6	0.913718
$848$	29747.8	1.20465
$849$	4427.48	0.178976
$850$	−19749.5	−0.796943
$851$	−36592.1	−1.47398
$852$	20592.4	0.828031
$853$	36020.9	1.44587	0.722937	−	0.690914i	$-0.242792\pi$
$853$	36020.9	1.44587	0.722937	+	0.690914i	$0.242792\pi$
$854$	41091.2	1.64650
$855$	−14537.4	−0.581483
$856$	4016.78	0.160386
$857$	−35927.0	−1.43202	−0.716010	−	0.698090i	$-0.754034\pi$
$857$	−35927.0	−1.43202	−0.716010	+	0.698090i	$0.754034\pi$
$858$	7128.74	0.283649
$859$	−20914.3	−0.830718	−0.415359	−	0.909658i	$-0.636344\pi$
$859$	−20914.3	−0.830718	−0.415359	+	0.909658i	$0.636344\pi$
$860$	−4616.74	−0.183058
$861$	8321.01	0.329360
$862$	43627.0	1.72383
$863$	−5395.57	−0.212824	−0.106412	−	0.994322i	$-0.533936\pi$
$863$	−5395.57	−0.212824	−0.106412	+	0.994322i	$0.533936\pi$
$864$	6873.08	0.270633
$865$	−74580.0	−2.93156
$866$	−30056.1	−1.17939
$867$	−867.000	−0.0339618
$868$	−62899.3	−2.45961
$869$	4071.48	0.158936
$870$	−39174.2	−1.52659
$871$	27585.0	1.07311
$872$	−2989.82	−0.116110
$873$	6593.56	0.255622
$874$	39795.8	1.54018
$875$	−62326.3	−2.40802
$876$	−14547.4	−0.561084
$877$	−31305.2	−1.20536	−0.602681	−	0.797983i	$-0.705901\pi$
$877$	−31305.2	−1.20536	−0.602681	+	0.797983i	$0.705901\pi$
$878$	−22522.1	−0.865700
$879$	−1563.17	−0.0599822
$880$	−14085.1	−0.539557
$881$	19975.4	0.763891	0.381945	−	0.924185i	$-0.375254\pi$
$881$	19975.4	0.763891	0.381945	+	0.924185i	$0.375254\pi$
$882$	3694.83	0.141056
$883$	32032.5	1.22081	0.610407	−	0.792088i	$-0.291006\pi$
$883$	32032.5	1.22081	0.610407	+	0.792088i	$0.291006\pi$
$884$	5939.99	0.225999
$885$	−37702.0	−1.43202
$886$	14990.3	0.568409
$887$	32668.4	1.23664	0.618319	−	0.785927i	$-0.287814\pi$
$887$	32668.4	1.23664	0.618319	+	0.785927i	$0.287814\pi$
$888$	−8032.06	−0.303534
$889$	−7138.58	−0.269314
$890$	78385.7	2.95224
$891$	1298.38	0.0488188
$892$	−754.727	−0.0283298
$893$	−21103.1	−0.790805
$894$	1074.03	0.0401799
$895$	−28650.7	−1.07004
$896$	22064.7	0.822690
$897$	−12156.4	−0.452499
$898$	20310.3	0.754747
$899$	46226.3	1.71494
$900$	24644.1	0.912745
$901$	11493.5	0.424976
$902$	8994.96	0.332039
$903$	−1454.38	−0.0535976
$904$	−6224.95	−0.229025
$905$	−4378.81	−0.160836
$906$	−13385.3	−0.490835
$907$	−39521.1	−1.44683	−0.723417	−	0.690411i	$-0.757429\pi$
$907$	−39521.1	−1.44683	−0.723417	+	0.690411i	$0.757429\pi$
$908$	−16299.7	−0.595733
$909$	−1156.75	−0.0422080
$910$	62083.1	2.26158
$911$	47835.9	1.73971	0.869854	−	0.493310i	$-0.164213\pi$
$911$	47835.9	1.73971	0.869854	+	0.493310i	$0.164213\pi$
$912$	−10676.5	−0.387646
$913$	8798.75	0.318944
$914$	55870.6	2.02192
$915$	27670.3	0.999730
$916$	20003.5	0.721543
$917$	44748.1	1.61146
$918$	1947.37	0.0700140
$919$	−39689.7	−1.42464	−0.712319	−	0.701856i	$-0.752355\pi$
$919$	−39689.7	−1.42464	−0.712319	+	0.701856i	$0.752355\pi$
$920$	−19651.9	−0.704243
$921$	5154.70	0.184423
$922$	−30317.1	−1.08291
$923$	23984.0	0.855302
$924$	10084.4	0.359039
$925$	86399.2	3.07112
$926$	40146.9	1.42474
$927$	48.1792	0.00170703
$928$	−39232.0	−1.38777
$929$	−4579.27	−0.161723	−0.0808617	−	0.996725i	$-0.525767\pi$
$929$	−4579.27	−0.161723	−0.0808617	+	0.996725i	$0.525767\pi$
$930$	−76240.2	−2.68819
$931$	−7826.53	−0.275515
$932$	−42252.6	−1.48501
$933$	−14748.2	−0.517506
$934$	−22511.8	−0.788661
$935$	−5441.99	−0.190344
$936$	−2668.37	−0.0931820
$937$	−38046.5	−1.32649	−0.663246	−	0.748401i	$-0.730822\pi$
$937$	−38046.5	−1.32649	−0.663246	+	0.748401i	$0.730822\pi$
$938$	70239.6	2.44499
$939$	−22127.5	−0.769015
$940$	52105.5	1.80797
$941$	−9081.31	−0.314604	−0.157302	−	0.987551i	$-0.550280\pi$
$941$	−9081.31	−0.314604	−0.157302	+	0.987551i	$0.550280\pi$
$942$	−20205.2	−0.698855
$943$	−15338.8	−0.529694
$944$	−27688.9	−0.954658
$945$	11307.4	0.389239
$946$	−1572.17	−0.0540335
$947$	39575.9	1.35802	0.679009	−	0.734130i	$-0.262410\pi$
$947$	39575.9	1.35802	0.679009	+	0.734130i	$0.262410\pi$
$948$	−7620.00	−0.261061
$949$	−16943.4	−0.579562
$950$	−93963.7	−3.20904
$951$	−1568.12	−0.0534698
$952$	3025.00	0.102984
$953$	−28601.6	−0.972189	−0.486095	−	0.873906i	$-0.661579\pi$
$953$	−28601.6	−0.972189	−0.486095	+	0.873906i	$0.661579\pi$
$954$	−25815.6	−0.876111
$955$	−62237.0	−2.10884
$956$	−16149.7	−0.546358
$957$	−7411.26	−0.250337
$958$	49772.2	1.67857
$959$	−1156.43	−0.0389396
$960$	43615.7	1.46634
$961$	60173.7	2.01986
$962$	−46774.9	−1.56765
$963$	4260.44	0.142566
$964$	−67616.7	−2.25912
$965$	−78320.9	−2.61268
$966$	−30953.9	−1.03098
$967$	2591.39	0.0861773	0.0430887	−	0.999071i	$-0.486280\pi$
$967$	2591.39	0.0861773	0.0430887	+	0.999071i	$0.486280\pi$
$968$	−9113.68	−0.302608
$969$	−4124.99	−0.136753
$970$	62073.1	2.05469
$971$	−49692.4	−1.64233	−0.821166	−	0.570689i	$-0.806676\pi$
$971$	−49692.4	−1.64233	−0.821166	+	0.570689i	$0.806676\pi$
$972$	−2430.00	−0.0801875
$973$	−5374.04	−0.177064
$974$	46989.2	1.54582
$975$	28703.1	0.942805
$976$	20321.5	0.666470
$977$	−44489.8	−1.45686	−0.728432	−	0.685119i	$-0.759750\pi$
$977$	−44489.8	−1.45686	−0.728432	+	0.685119i	$0.759750\pi$
$978$	−22268.6	−0.728091
$979$	14829.6	0.484121
$980$	19324.4	0.629894
$981$	−3171.18	−0.103209
$982$	8524.05	0.276999
$983$	−37606.9	−1.22022	−0.610109	−	0.792318i	$-0.708874\pi$
$983$	−37606.9	−1.22022	−0.610109	+	0.792318i	$0.708874\pi$
$984$	−3366.92	−0.109079
$985$	62741.0	2.02954
$986$	−11115.7	−0.359023
$987$	16414.4	0.529358
$988$	28261.2	0.910028
$989$	2680.98	0.0861983
$990$	12223.3	0.392405
$991$	−55446.9	−1.77732	−0.888662	−	0.458563i	$-0.848364\pi$
$991$	−55446.9	−1.77732	−0.888662	+	0.458563i	$0.848364\pi$
$992$	−76352.5	−2.44375
$993$	−15049.4	−0.480945
$994$	61070.5	1.94873
$995$	36601.9	1.16619
$996$	−16467.4	−0.523884
$997$	−39314.6	−1.24885	−0.624426	−	0.781084i	$-0.714667\pi$
$997$	−39314.6	−1.24885	−0.624426	+	0.781084i	$0.714667\pi$
$998$	42240.8	1.33979
$999$	−8519.28	−0.269808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.4.a.d.1.2	✓	2
3.2	odd	2		153.4.a.e.1.1		2
4.3	odd	2		816.4.a.o.1.2		2
5.4	even	2		1275.4.a.m.1.1		2
7.6	odd	2		2499.4.a.l.1.2		2
12.11	even	2		2448.4.a.v.1.1		2
17.16	even	2		867.4.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.d.1.2	✓	2	1.1	even	1	trivial
153.4.a.e.1.1		2	3.2	odd	2
816.4.a.o.1.2		2	4.3	odd	2
867.4.a.j.1.2		2	17.16	even	2
1275.4.a.m.1.1		2	5.4	even	2
2448.4.a.v.1.1		2	12.11	even	2
2499.4.a.l.1.2		2	7.6	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:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	51.a (trivial)

\(f(q)\)	\(=\)	\(q+4.24264 q^{2} -3.00000 q^{3} +10.0000 q^{4} +19.9706 q^{5} -12.7279 q^{6} -20.9706 q^{7} +8.48528 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.24264 q^{2} -3.00000 q^{3} +10.0000 q^{4} +19.9706 q^{5} -12.7279 q^{6} -20.9706 q^{7} +8.48528 q^{8} +9.00000 q^{9} +84.7279 q^{10} +16.0294 q^{11} -30.0000 q^{12} -34.9411 q^{13} -88.9706 q^{14} -59.9117 q^{15} -44.0000 q^{16} -17.0000 q^{17} +38.1838 q^{18} -80.8823 q^{19} +199.706 q^{20} +62.9117 q^{21} +68.0071 q^{22} -115.971 q^{23} -25.4558 q^{24} +273.823 q^{25} -148.243 q^{26} -27.0000 q^{27} -209.706 q^{28} +154.118 q^{29} -254.184 q^{30} +299.941 q^{31} -254.558 q^{32} -48.0883 q^{33} -72.1249 q^{34} -418.794 q^{35} +90.0000 q^{36} +315.529 q^{37} -343.154 q^{38} +104.823 q^{39} +169.456 q^{40} +132.265 q^{41} +266.912 q^{42} -23.1177 q^{43} +160.294 q^{44} +179.735 q^{45} -492.021 q^{46} +260.912 q^{47} +132.000 q^{48} +96.7645 q^{49} +1161.73 q^{50} +51.0000 q^{51} -349.411 q^{52} -676.087 q^{53} -114.551 q^{54} +320.117 q^{55} -177.941 q^{56} +242.647 q^{57} +653.866 q^{58} +629.294 q^{59} -599.117 q^{60} -461.852 q^{61} +1272.54 q^{62} -188.735 q^{63} -728.000 q^{64} -697.794 q^{65} -204.021 q^{66} -789.470 q^{67} -170.000 q^{68} +347.912 q^{69} -1776.79 q^{70} -686.412 q^{71} +76.3675 q^{72} +484.912 q^{73} +1338.68 q^{74} -821.470 q^{75} -808.823 q^{76} -336.146 q^{77} +444.728 q^{78} +254.000 q^{79} -878.705 q^{80} +81.0000 q^{81} +561.153 q^{82} +548.912 q^{83} +629.117 q^{84} -339.500 q^{85} -98.0803 q^{86} -462.353 q^{87} +136.014 q^{88} +925.145 q^{89} +762.551 q^{90} +732.735 q^{91} -1159.71 q^{92} -899.823 q^{93} +1106.95 q^{94} -1615.26 q^{95} +763.675 q^{96} +732.617 q^{97} +410.537 q^{98} +144.265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 20 * q^4 + 6 * q^5 - 8 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{7} + 18 q^{9} + 144 q^{10} + 66 q^{11} - 60 q^{12} - 2 q^{13} - 144 q^{14} - 18 q^{15} - 88 q^{16} - 34 q^{17} - 26 q^{19} + 60 q^{20} + 24 q^{21} - 144 q^{22} - 198 q^{23} + 344 q^{25} - 288 q^{26} - 54 q^{27} - 80 q^{28} + 444 q^{29} - 432 q^{30} + 532 q^{31} - 198 q^{33} - 600 q^{35} + 180 q^{36} + 88 q^{37} - 576 q^{38} + 6 q^{39} + 288 q^{40} + 570 q^{41} + 432 q^{42} - 182 q^{43} + 660 q^{44} + 54 q^{45} - 144 q^{46} + 420 q^{47} + 264 q^{48} - 78 q^{49} + 864 q^{50} + 102 q^{51} - 20 q^{52} - 300 q^{53} - 378 q^{55} - 288 q^{56} + 78 q^{57} - 576 q^{58} + 444 q^{59} - 180 q^{60} + 400 q^{61} + 288 q^{62} - 72 q^{63} - 1456 q^{64} - 1158 q^{65} + 432 q^{66} - 968 q^{67} - 340 q^{68} + 594 q^{69} - 1008 q^{70} - 1848 q^{71} + 868 q^{73} + 2304 q^{74} - 1032 q^{75} - 260 q^{76} + 312 q^{77} + 864 q^{78} + 508 q^{79} - 264 q^{80} + 162 q^{81} - 1296 q^{82} + 996 q^{83} + 240 q^{84} - 102 q^{85} + 576 q^{86} - 1332 q^{87} - 288 q^{88} - 288 q^{89} + 1296 q^{90} + 1160 q^{91} - 1980 q^{92} - 1596 q^{93} + 432 q^{94} - 2382 q^{95} + 1024 q^{97} + 1152 q^{98} + 594 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 20 * q^4 + 6 * q^5 - 8 * q^7 + 18 * q^9 + 144 * q^10 + 66 * q^11 - 60 * q^12 - 2 * q^13 - 144 * q^14 - 18 * q^15 - 88 * q^16 - 34 * q^17 - 26 * q^19 + 60 * q^20 + 24 * q^21 - 144 * q^22 - 198 * q^23 + 344 * q^25 - 288 * q^26 - 54 * q^27 - 80 * q^28 + 444 * q^29 - 432 * q^30 + 532 * q^31 - 198 * q^33 - 600 * q^35 + 180 * q^36 + 88 * q^37 - 576 * q^38 + 6 * q^39 + 288 * q^40 + 570 * q^41 + 432 * q^42 - 182 * q^43 + 660 * q^44 + 54 * q^45 - 144 * q^46 + 420 * q^47 + 264 * q^48 - 78 * q^49 + 864 * q^50 + 102 * q^51 - 20 * q^52 - 300 * q^53 - 378 * q^55 - 288 * q^56 + 78 * q^57 - 576 * q^58 + 444 * q^59 - 180 * q^60 + 400 * q^61 + 288 * q^62 - 72 * q^63 - 1456 * q^64 - 1158 * q^65 + 432 * q^66 - 968 * q^67 - 340 * q^68 + 594 * q^69 - 1008 * q^70 - 1848 * q^71 + 868 * q^73 + 2304 * q^74 - 1032 * q^75 - 260 * q^76 + 312 * q^77 + 864 * q^78 + 508 * q^79 - 264 * q^80 + 162 * q^81 - 1296 * q^82 + 996 * q^83 + 240 * q^84 - 102 * q^85 + 576 * q^86 - 1332 * q^87 - 288 * q^88 - 288 * q^89 + 1296 * q^90 + 1160 * q^91 - 1980 * q^92 - 1596 * q^93 + 432 * q^94 - 2382 * q^95 + 1024 * q^97 + 1152 * q^98 + 594 * q^99

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