LMFDB - Embedded newform 75.22.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	75.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:	$209.608008215$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 $ x^6 - x^5 - 530373x^4 - 23850203x^3 + 59940607490x^2 + 2888057920800x - 684850637484000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-571.187$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2300.75 q^{2} +59049.0 q^{3} +3.19629e6 q^{4} -1.35857e8 q^{6} +2.10374e8 q^{7} -2.52885e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-2300.75 q^{2} +59049.0 q^{3} +3.19629e6 q^{4} -1.35857e8 q^{6} +2.10374e8 q^{7} -2.52885e9 q^{8} +3.48678e9 q^{9} +6.30187e10 q^{11} +1.88738e11 q^{12} +3.66488e11 q^{13} -4.84019e11 q^{14} -8.84863e11 q^{16} -5.08634e12 q^{17} -8.02222e12 q^{18} +7.36866e12 q^{19} +1.24224e13 q^{21} -1.44990e14 q^{22} -5.08468e12 q^{23} -1.49326e14 q^{24} -8.43196e14 q^{26} +2.05891e14 q^{27} +6.72418e14 q^{28} +1.15873e15 q^{29} -2.75579e15 q^{31} +7.33923e15 q^{32} +3.72119e15 q^{33} +1.17024e16 q^{34} +1.11448e16 q^{36} +1.13252e16 q^{37} -1.69534e16 q^{38} +2.16407e16 q^{39} -3.82078e16 q^{41} -2.85808e16 q^{42} -1.60836e17 q^{43} +2.01426e17 q^{44} +1.16986e16 q^{46} +7.69010e16 q^{47} -5.22503e16 q^{48} -5.14289e17 q^{49} -3.00343e17 q^{51} +1.17140e18 q^{52} -1.26003e18 q^{53} -4.73704e17 q^{54} -5.32005e17 q^{56} +4.35112e17 q^{57} -2.66595e18 q^{58} -4.45998e16 q^{59} -7.65725e18 q^{61} +6.34039e18 q^{62} +7.33530e17 q^{63} -1.50300e19 q^{64} -8.56153e18 q^{66} +4.35494e18 q^{67} -1.62574e19 q^{68} -3.00245e17 q^{69} -4.76688e19 q^{71} -8.81756e18 q^{72} +8.98402e18 q^{73} -2.60564e19 q^{74} +2.35524e19 q^{76} +1.32575e19 q^{77} -4.97899e19 q^{78} -9.52432e19 q^{79} +1.21577e19 q^{81} +8.79065e19 q^{82} -1.50799e20 q^{83} +3.97056e19 q^{84} +3.70043e20 q^{86} +6.84219e19 q^{87} -1.59365e20 q^{88} +7.17533e19 q^{89} +7.70996e19 q^{91} -1.62521e19 q^{92} -1.62727e20 q^{93} -1.76930e20 q^{94} +4.33374e20 q^{96} -2.27790e20 q^{97} +1.18325e21 q^{98} +2.19733e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 92 q^{2} + 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} - 1327143454 q^{7} + 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) $ 6 * q - 92 * q^2 + 354294 * q^3 + 4390448 * q^4 - 5432508 * q^6 - 1327143454 * q^7 + 4252271424 * q^8 + 20920706406 * q^9	$ 6 q - 92 q^{2} + 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} - 1327143454 q^{7} + 4252271424 q^{8} + 20920706406 q^{9} - 78310112516 q^{11} + 259251563952 q^{12} - 116029746338 q^{13} - 313825730148 q^{14} + 1996701612800 q^{16} - 5382900513068 q^{17} - 320784164892 q^{18} + 6969638997622 q^{19} - 78366493815246 q^{21} + 19701817271864 q^{22} - 295754895311052 q^{23} + 251092375315776 q^{24} - 274314326263628 q^{26} + 12\!\cdots\!94 q^{27}+ \cdots - 27\!\cdots\!16 q^{99}+O(q^{100}) $ 6 * q - 92 * q^2 + 354294 * q^3 + 4390448 * q^4 - 5432508 * q^6 - 1327143454 * q^7 + 4252271424 * q^8 + 20920706406 * q^9 - 78310112516 * q^11 + 259251563952 * q^12 - 116029746338 * q^13 - 313825730148 * q^14 + 1996701612800 * q^16 - 5382900513068 * q^17 - 320784164892 * q^18 + 6969638997622 * q^19 - 78366493815246 * q^21 + 19701817271864 * q^22 - 295754895311052 * q^23 + 251092375315776 * q^24 - 274314326263628 * q^26 + 1235346792567894 * q^27 - 2674107576939568 * q^28 - 3895624305077596 * q^29 + 492586859482706 * q^31 + 10862288965692416 * q^32 - 4624133833957284 * q^33 - 30773279577490744 * q^34 + 15308545599801648 * q^36 + 26101348744076316 * q^37 + 8307793564414708 * q^38 - 6851440491512562 * q^39 + 112662882011424056 * q^41 - 18531095539509252 * q^42 + 13377804412633810 * q^43 - 23820753283797856 * q^44 - 156784693648581624 * q^46 + 243206274450163780 * q^47 + 117903233534227200 * q^48 + 327097566625336780 * q^49 - 317854892396152332 * q^51 + 192375894240694512 * q^52 - 554044170675798992 * q^53 - 18941984152707708 * q^54 - 5086364147554777920 * q^56 + 411550213170581478 * q^57 - 11350712056264661624 * q^58 - 7003944869710758212 * q^59 - 10235053877143753790 * q^61 + 1855763029836316572 * q^62 - 4627463093296461054 * q^63 - 10747451596339826688 * q^64 + 1163372608086297336 * q^66 - 8975831852618567158 * q^67 - 64266312753700360864 * q^68 - 17464030813222309548 * q^69 + 18009408229245489272 * q^71 + 14826753670021257024 * q^72 - 81172357549902895652 * q^73 - 103132266830355254968 * q^74 + 111484542579179194608 * q^76 + 40817017589776296372 * q^77 - 16197986651540969772 * q^78 - 79149540279186452080 * q^79 + 72945992754341572806 * q^81 - 17339452427244467168 * q^82 - 238715465076655746756 * q^83 - 157903378310704550832 * q^84 + 28767310268029844764 * q^86 - 230032719590526966204 * q^87 - 1023055046877663881856 * q^88 - 712331293797957179808 * q^89 + 670774308716312800426 * q^91 - 880736698470324218784 * q^92 + 29086761465594306594 * q^93 + 278898069330490980248 * q^94 + 641407301135171472384 * q^96 - 1498553500583421551178 * q^97 - 2662349783988420722392 * q^98 - 273050478761343662916 * 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 $

$2$	−2300.75	−1.58875	−0.794373	−	0.607431i	$-0.792200\pi$
$2$	−2300.75	−1.58875	−0.794373	+	0.607431i	$0.792200\pi$
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	3.19629e6	1.52411
$5$	0	0
$5$	0	0
$6$	−1.35857e8	−0.917263
$7$	2.10374e8	0.281490	0.140745	−	0.990046i	$-0.455050\pi$
$7$	2.10374e8	0.281490	0.140745	+	0.990046i	$0.455050\pi$
$8$	−2.52885e9	−0.832681
$9$	3.48678e9	0.333333
$10$	0	0
$11$	6.30187e10	0.732565	0.366283	−	0.930504i	$-0.380630\pi$
$11$	6.30187e10	0.732565	0.366283	+	0.930504i	$0.380630\pi$
$12$	1.88738e11	0.879946
$13$	3.66488e11	0.737317	0.368659	−	0.929565i	$-0.379817\pi$
$13$	3.66488e11	0.737317	0.368659	+	0.929565i	$0.379817\pi$
$14$	−4.84019e11	−0.447216
$15$	0	0
$16$	−8.84863e11	−0.201195
$17$	−5.08634e12	−0.611916	−0.305958	−	0.952045i	$-0.598977\pi$
$17$	−5.08634e12	−0.611916	−0.305958	+	0.952045i	$0.598977\pi$
$18$	−8.02222e12	−0.529582
$19$	7.36866e12	0.275725	0.137862	−	0.990451i	$-0.455977\pi$
$19$	7.36866e12	0.275725	0.137862	+	0.990451i	$0.455977\pi$
$20$	0	0
$21$	1.24224e13	0.162518
$22$	−1.44990e14	−1.16386
$23$	−5.08468e12	−0.0255930	−0.0127965	−	0.999918i	$-0.504073\pi$
$23$	−5.08468e12	−0.0255930	−0.0127965	+	0.999918i	$0.504073\pi$
$24$	−1.49326e14	−0.480748
$25$	0	0
$26$	−8.43196e14	−1.17141
$27$	2.05891e14	0.192450
$28$	6.72418e14	0.429023
$29$	1.15873e15	0.511450	0.255725	−	0.966750i	$-0.417686\pi$
$29$	1.15873e15	0.511450	0.255725	+	0.966750i	$0.417686\pi$
$30$	0	0
$31$	−2.75579e15	−0.603878	−0.301939	−	0.953327i	$-0.597634\pi$
$31$	−2.75579e15	−0.603878	−0.301939	+	0.953327i	$0.597634\pi$
$32$	7.33923e15	1.15233
$33$	3.72119e15	0.422947
$34$	1.17024e16	0.972179
$35$	0	0
$36$	1.11448e16	0.508037
$37$	1.13252e16	0.387192	0.193596	−	0.981081i	$-0.437985\pi$
$37$	1.13252e16	0.387192	0.193596	+	0.981081i	$0.437985\pi$
$38$	−1.69534e16	−0.438056
$39$	2.16407e16	0.425690
$40$	0	0
$41$	−3.82078e16	−0.444551	−0.222275	−	0.974984i	$-0.571348\pi$
$41$	−3.82078e16	−0.444551	−0.222275	+	0.974984i	$0.571348\pi$
$42$	−2.85808e16	−0.258200
$43$	−1.60836e17	−1.13492	−0.567459	−	0.823402i	$-0.692073\pi$
$43$	−1.60836e17	−1.13492	−0.567459	+	0.823402i	$0.692073\pi$
$44$	2.01426e17	1.11651
$45$	0	0
$46$	1.16986e16	0.0406608
$47$	7.69010e16	0.213258	0.106629	−	0.994299i	$-0.465994\pi$
$47$	7.69010e16	0.213258	0.106629	+	0.994299i	$0.465994\pi$
$48$	−5.22503e16	−0.116160
$49$	−5.14289e17	−0.920763
$50$	0	0
$51$	−3.00343e17	−0.353290
$52$	1.17140e18	1.12375
$53$	−1.26003e18	−0.989653	−0.494826	−	0.868992i	$-0.664768\pi$
$53$	−1.26003e18	−0.989653	−0.494826	+	0.868992i	$0.664768\pi$
$54$	−4.73704e17	−0.305754
$55$	0	0
$56$	−5.32005e17	−0.234391
$57$	4.35112e17	0.159190
$58$	−2.66595e18	−0.812564
$59$	−4.45998e16	−0.0113602	−0.00568012	−	0.999984i	$-0.501808\pi$
$59$	−4.45998e16	−0.0113602	−0.00568012	+	0.999984i	$0.501808\pi$
$60$	0	0
$61$	−7.65725e18	−1.37439	−0.687195	−	0.726473i	$-0.741158\pi$
$61$	−7.65725e18	−1.37439	−0.687195	+	0.726473i	$0.741158\pi$
$62$	6.34039e18	0.959408
$63$	7.33530e17	0.0938301
$64$	−1.50300e19	−1.62956
$65$	0	0
$66$	−8.56153e18	−0.671954
$67$	4.35494e18	0.291875	0.145938	−	0.989294i	$-0.453380\pi$
$67$	4.35494e18	0.291875	0.145938	+	0.989294i	$0.453380\pi$
$68$	−1.62574e19	−0.932629
$69$	−3.00245e17	−0.0147761
$70$	0	0
$71$	−4.76688e19	−1.73789	−0.868943	−	0.494912i	$-0.835200\pi$
$71$	−4.76688e19	−1.73789	−0.868943	+	0.494912i	$0.835200\pi$
$72$	−8.81756e18	−0.277560
$73$	8.98402e18	0.244670	0.122335	−	0.992489i	$-0.460962\pi$
$73$	8.98402e18	0.244670	0.122335	+	0.992489i	$0.460962\pi$
$74$	−2.60564e19	−0.615150
$75$	0	0
$76$	2.35524e19	0.420235
$77$	1.32575e19	0.206210
$78$	−4.97899e19	−0.676313
$79$	−9.52432e19	−1.13175	−0.565874	−	0.824492i	$-0.691461\pi$
$79$	−9.52432e19	−1.13175	−0.565874	+	0.824492i	$0.691461\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	8.79065e19	0.706278
$83$	−1.50799e20	−1.06679	−0.533396	−	0.845866i	$-0.679084\pi$
$83$	−1.50799e20	−1.06679	−0.533396	+	0.845866i	$0.679084\pi$
$84$	3.97056e19	0.247696
$85$	0	0
$86$	3.70043e20	1.80310
$87$	6.84219e19	0.295286
$88$	−1.59365e20	−0.609993
$89$	7.17533e19	0.243920	0.121960	−	0.992535i	$-0.461082\pi$
$89$	7.17533e19	0.243920	0.121960	+	0.992535i	$0.461082\pi$
$90$	0	0
$91$	7.70996e19	0.207548
$92$	−1.62521e19	−0.0390066
$93$	−1.62727e20	−0.348649
$94$	−1.76930e20	−0.338812
$95$	0	0
$96$	4.33374e20	0.665297
$97$	−2.27790e20	−0.313639	−0.156820	−	0.987627i	$-0.550124\pi$
$97$	−2.27790e20	−0.313639	−0.156820	+	0.987627i	$0.550124\pi$
$98$	1.18325e21	1.46286
$99$	2.19733e20	0.244188
$100$	0	0
$101$	1.49001e21	1.34220	0.671098	−	0.741369i	$-0.265823\pi$
$101$	1.49001e21	1.34220	0.671098	+	0.741369i	$0.265823\pi$
$102$	6.91015e20	0.561288
$103$	1.97182e21	1.44569	0.722847	−	0.691008i	$-0.242833\pi$
$103$	1.97182e21	1.44569	0.722847	+	0.691008i	$0.242833\pi$
$104$	−9.26793e20	−0.613950
$105$	0	0
$106$	2.89900e21	1.57231
$107$	1.05324e21	0.517603	0.258801	−	0.965931i	$-0.416673\pi$
$107$	1.05324e21	0.517603	0.258801	+	0.965931i	$0.416673\pi$
$108$	6.58089e20	0.293315
$109$	−3.54520e21	−1.43438	−0.717188	−	0.696880i	$-0.754571\pi$
$109$	−3.54520e21	−1.43438	−0.717188	+	0.696880i	$0.754571\pi$
$110$	0	0
$111$	6.68740e20	0.223546
$112$	−1.86152e20	−0.0566343
$113$	−2.70245e20	−0.0748919	−0.0374459	−	0.999299i	$-0.511922\pi$
$113$	−2.70245e20	−0.0748919	−0.0374459	+	0.999299i	$0.511922\pi$
$114$	−1.00108e21	−0.252912
$115$	0	0
$116$	3.70364e21	0.779507
$117$	1.27786e21	0.245772
$118$	1.02613e20	0.0180485
$119$	−1.07004e21	−0.172248
$120$	0	0
$121$	−3.42889e21	−0.463348
$122$	1.76174e22	2.18355
$123$	−2.25613e21	−0.256661
$124$	−8.80833e21	−0.920377
$125$	0	0
$126$	−1.68767e21	−0.149072
$127$	2.21609e22	1.80156	0.900780	−	0.434277i	$-0.142996\pi$
$127$	2.21609e22	1.80156	0.900780	+	0.434277i	$0.142996\pi$
$128$	1.91889e22	1.43663
$129$	−9.49720e21	−0.655245
$130$	0	0
$131$	9.46139e19	0.00555400	0.00277700	−	0.999996i	$-0.499116\pi$
$131$	9.46139e19	0.00555400	0.00277700	+	0.999996i	$0.499116\pi$
$132$	1.18940e22	0.644618
$133$	1.55018e21	0.0776138
$134$	−1.00196e22	−0.463715
$135$	0	0
$136$	1.28626e22	0.509531
$137$	4.94865e22	1.81518	0.907592	−	0.419854i	$-0.137919\pi$
$137$	4.94865e22	1.81518	0.907592	+	0.419854i	$0.137919\pi$
$138$	6.90789e20	0.0234755
$139$	−1.77582e22	−0.559428	−0.279714	−	0.960083i	$-0.590240\pi$
$139$	−1.77582e22	−0.559428	−0.279714	+	0.960083i	$0.590240\pi$
$140$	0	0
$141$	4.54093e21	0.123124
$142$	1.09674e23	2.76106
$143$	2.30956e22	0.540133
$144$	−3.08533e21	−0.0670649
$145$	0	0
$146$	−2.06700e22	−0.388718
$147$	−3.03682e22	−0.531603
$148$	3.61986e22	0.590125
$149$	−3.44662e22	−0.523525	−0.261762	−	0.965132i	$-0.584304\pi$
$149$	−3.44662e22	−0.523525	−0.261762	+	0.965132i	$0.584304\pi$
$150$	0	0
$151$	6.39694e22	0.844723	0.422361	−	0.906428i	$-0.361201\pi$
$151$	6.39694e22	0.844723	0.422361	+	0.906428i	$0.361201\pi$
$152$	−1.86342e22	−0.229591
$153$	−1.77350e22	−0.203972
$154$	−3.05022e22	−0.327615
$155$	0	0
$156$	6.91702e22	0.648800
$157$	−4.97870e22	−0.436686	−0.218343	−	0.975872i	$-0.570065\pi$
$157$	−4.97870e22	−0.436686	−0.218343	+	0.975872i	$0.570065\pi$
$158$	2.19131e23	1.79806
$159$	−7.44033e22	−0.571376
$160$	0	0
$161$	−1.06969e21	−0.00720418
$162$	−2.79717e22	−0.176527
$163$	−1.16432e23	−0.688814	−0.344407	−	0.938821i	$-0.611920\pi$
$163$	−1.16432e23	−0.688814	−0.344407	+	0.938821i	$0.611920\pi$
$164$	−1.22123e23	−0.677545
$165$	0	0
$166$	3.46952e23	1.69486
$167$	5.65142e22	0.259200	0.129600	−	0.991566i	$-0.458631\pi$
$167$	5.65142e22	0.259200	0.129600	+	0.991566i	$0.458631\pi$
$168$	−3.14144e22	−0.135326
$169$	−1.12751e23	−0.456363
$170$	0	0
$171$	2.56929e22	0.0919082
$172$	−5.14079e23	−1.72974
$173$	3.02320e23	0.957157	0.478579	−	0.878045i	$-0.341152\pi$
$173$	3.02320e23	0.957157	0.478579	+	0.878045i	$0.341152\pi$
$174$	−1.57422e23	−0.469134
$175$	0	0
$176$	−5.57629e22	−0.147388
$177$	−2.63358e21	−0.00655883
$178$	−1.65086e23	−0.387526
$179$	8.48756e23	1.87856	0.939282	−	0.343147i	$-0.111493\pi$
$179$	8.48756e23	1.87856	0.939282	+	0.343147i	$0.111493\pi$
$180$	0	0
$181$	−3.06671e23	−0.604016	−0.302008	−	0.953305i	$-0.597657\pi$
$181$	−3.06671e23	−0.604016	−0.302008	+	0.953305i	$0.597657\pi$
$182$	−1.77387e23	−0.329740
$183$	−4.52153e23	−0.793504
$184$	1.28584e22	0.0213108
$185$	0	0
$186$	3.74394e23	0.553914
$187$	−3.20535e23	−0.448268
$188$	2.45798e23	0.325028
$189$	4.33142e22	0.0541728
$190$	0	0
$191$	−1.59483e24	−1.78593	−0.892964	−	0.450128i	$-0.851378\pi$
$191$	−1.59483e24	−1.78593	−0.892964	+	0.450128i	$0.851378\pi$
$192$	−8.87509e23	−0.940827
$193$	−1.20399e24	−1.20857	−0.604283	−	0.796770i	$-0.706540\pi$
$193$	−1.20399e24	−1.20857	−0.604283	+	0.796770i	$0.706540\pi$
$194$	5.24087e23	0.498293
$195$	0	0
$196$	−1.64382e24	−1.40335
$197$	2.54274e23	0.205782	0.102891	−	0.994693i	$-0.467191\pi$
$197$	2.54274e23	0.205782	0.102891	+	0.994693i	$0.467191\pi$
$198$	−5.05550e23	−0.387953
$199$	−1.83859e24	−1.33822	−0.669111	−	0.743162i	$-0.733325\pi$
$199$	−1.83859e24	−1.33822	−0.669111	+	0.743162i	$0.733325\pi$
$200$	0	0
$201$	2.57155e23	0.168514
$202$	−3.42815e24	−2.13241
$203$	2.43767e23	0.143968
$204$	−9.59986e23	−0.538453
$205$	0	0
$206$	−4.53667e24	−2.29684
$207$	−1.77292e22	−0.00853100
$208$	−3.24292e23	−0.148344
$209$	4.64363e23	0.201986
$210$	0	0
$211$	−2.45445e24	−0.966025	−0.483013	−	0.875613i	$-0.660457\pi$
$211$	−2.45445e24	−0.966025	−0.483013	+	0.875613i	$0.660457\pi$
$212$	−4.02741e24	−1.50834
$213$	−2.81479e24	−1.00337
$214$	−2.42323e24	−0.822339
$215$	0	0
$216$	−5.20668e23	−0.160249
$217$	−5.79748e23	−0.169986
$218$	8.15662e24	2.27886
$219$	5.30497e23	0.141260
$220$	0	0
$221$	−1.86408e24	−0.451176
$222$	−1.53860e24	−0.355157
$223$	2.05271e24	0.451988	0.225994	−	0.974129i	$-0.427437\pi$
$223$	2.05271e24	0.451988	0.225994	+	0.974129i	$0.427437\pi$
$224$	1.54399e24	0.324369
$225$	0	0
$226$	6.21767e23	0.118984
$227$	−5.12767e24	−0.936804	−0.468402	−	0.883516i	$-0.655170\pi$
$227$	−5.12767e24	−0.936804	−0.468402	+	0.883516i	$0.655170\pi$
$228$	1.39075e24	0.242623
$229$	2.29982e24	0.383196	0.191598	−	0.981473i	$-0.438633\pi$
$229$	2.29982e24	0.383196	0.191598	+	0.981473i	$0.438633\pi$
$230$	0	0
$231$	7.82843e23	0.119055
$232$	−2.93026e24	−0.425875
$233$	1.35634e25	1.88422	0.942109	−	0.335308i	$-0.108840\pi$
$233$	1.35634e25	1.88422	0.942109	+	0.335308i	$0.108840\pi$
$234$	−2.94004e24	−0.390470
$235$	0	0
$236$	−1.42554e23	−0.0173143
$237$	−5.62402e24	−0.653415
$238$	2.46188e24	0.273659
$239$	1.88596e24	0.200610	0.100305	−	0.994957i	$-0.468018\pi$
$239$	1.88596e24	0.200610	0.100305	+	0.994957i	$0.468018\pi$
$240$	0	0
$241$	1.04662e25	1.02002	0.510010	−	0.860168i	$-0.329642\pi$
$241$	1.04662e25	1.02002	0.510010	+	0.860168i	$0.329642\pi$
$242$	7.88903e24	0.736143
$243$	7.17898e23	0.0641500
$244$	−2.44748e25	−2.09472
$245$	0	0
$246$	5.19079e24	0.407770
$247$	2.70052e24	0.203297
$248$	6.96899e24	0.502837
$249$	−8.90456e24	−0.615913
$250$	0	0
$251$	1.72531e25	1.09722	0.548610	−	0.836078i	$-0.315157\pi$
$251$	1.72531e25	1.09722	0.548610	+	0.836078i	$0.315157\pi$
$252$	2.34458e24	0.143008
$253$	−3.20430e23	−0.0187485
$254$	−5.09867e25	−2.86222
$255$	0	0
$256$	−1.26285e25	−0.652878
$257$	−8.63659e24	−0.428593	−0.214296	−	0.976769i	$-0.568746\pi$
$257$	−8.63659e24	−0.428593	−0.214296	+	0.976769i	$0.568746\pi$
$258$	2.18507e25	1.04102
$259$	2.38253e24	0.108991
$260$	0	0
$261$	4.04024e24	0.170483
$262$	−2.17683e23	−0.00882390
$263$	−6.94089e24	−0.270321	−0.135161	−	0.990824i	$-0.543155\pi$
$263$	−6.94089e24	−0.270321	−0.135161	+	0.990824i	$0.543155\pi$
$264$	−9.41034e24	−0.352179
$265$	0	0
$266$	−3.56657e24	−0.123309
$267$	4.23696e24	0.140827
$268$	1.39197e25	0.444850
$269$	−2.37580e25	−0.730149	−0.365075	−	0.930978i	$-0.618957\pi$
$269$	−2.37580e25	−0.730149	−0.365075	+	0.930978i	$0.618957\pi$
$270$	0	0
$271$	−4.58603e25	−1.30394	−0.651972	−	0.758243i	$-0.726058\pi$
$271$	−4.58603e25	−1.30394	−0.651972	+	0.758243i	$0.726058\pi$
$272$	4.50072e24	0.123114
$273$	4.55266e24	0.119828
$274$	−1.13856e26	−2.88386
$275$	0	0
$276$	−9.59672e23	−0.0225205
$277$	6.09487e25	1.37698	0.688489	−	0.725247i	$-0.258274\pi$
$277$	6.09487e25	1.37698	0.688489	+	0.725247i	$0.258274\pi$
$278$	4.08572e25	0.888788
$279$	−9.60886e24	−0.201293
$280$	0	0
$281$	3.41139e25	0.663003	0.331501	−	0.943455i	$-0.392445\pi$
$281$	3.41139e25	0.663003	0.331501	+	0.943455i	$0.392445\pi$
$282$	−1.04475e25	−0.195613
$283$	4.80459e25	0.866760	0.433380	−	0.901211i	$-0.357321\pi$
$283$	4.80459e25	0.866760	0.433380	+	0.901211i	$0.357321\pi$
$284$	−1.52363e26	−2.64873
$285$	0	0
$286$	−5.31371e25	−0.858133
$287$	−8.03794e24	−0.125137
$288$	2.55903e25	0.384109
$289$	−4.32211e25	−0.625559
$290$	0	0
$291$	−1.34507e25	−0.181080
$292$	2.87156e25	0.372904
$293$	−1.15610e26	−1.44840	−0.724198	−	0.689593i	$-0.757790\pi$
$293$	−1.15610e26	−1.44840	−0.724198	+	0.689593i	$0.757790\pi$
$294$	6.98697e25	0.844582
$295$	0	0
$296$	−2.86397e25	−0.322408
$297$	1.29750e25	0.140982
$298$	7.92980e25	0.831748
$299$	−1.86347e24	−0.0188702
$300$	0	0
$301$	−3.38358e25	−0.319468
$302$	−1.47177e26	−1.34205
$303$	8.79838e25	0.774917
$304$	−6.52025e24	−0.0554743
$305$	0	0
$306$	4.08037e25	0.324060
$307$	−1.37622e26	−1.05618	−0.528088	−	0.849190i	$-0.677091\pi$
$307$	−1.37622e26	−1.05618	−0.528088	+	0.849190i	$0.677091\pi$
$308$	4.23749e25	0.314287
$309$	1.16434e26	0.834672
$310$	0	0
$311$	1.64151e26	1.09967	0.549833	−	0.835275i	$-0.314691\pi$
$311$	1.64151e26	1.09967	0.549833	+	0.835275i	$0.314691\pi$
$312$	−5.47262e25	−0.354464
$313$	−8.22532e25	−0.515154	−0.257577	−	0.966258i	$-0.582924\pi$
$313$	−8.22532e25	−0.515154	−0.257577	+	0.966258i	$0.582924\pi$
$314$	1.14547e26	0.693783
$315$	0	0
$316$	−3.04425e26	−1.72491
$317$	−2.64449e26	−1.44950	−0.724752	−	0.689010i	$-0.758046\pi$
$317$	−2.64449e26	−1.44950	−0.724752	+	0.689010i	$0.758046\pi$
$318$	1.71183e26	0.907772
$319$	7.30217e25	0.374671
$320$	0	0
$321$	6.21926e25	0.298838
$322$	2.46108e24	0.0114456
$323$	−3.74795e25	−0.168720
$324$	3.88595e25	0.169346
$325$	0	0
$326$	2.67880e26	1.09435
$327$	−2.09341e26	−0.828137
$328$	9.66218e25	0.370169
$329$	1.61780e25	0.0600299
$330$	0	0
$331$	−5.13496e26	−1.78790	−0.893949	−	0.448168i	$-0.852077\pi$
$331$	−5.13496e26	−1.78790	−0.893949	+	0.448168i	$0.852077\pi$
$332$	−4.81999e26	−1.62591
$333$	3.94884e25	0.129064
$334$	−1.30025e26	−0.411802
$335$	0	0
$336$	−1.09921e25	−0.0326978
$337$	−7.57301e25	−0.218351	−0.109175	−	0.994023i	$-0.534821\pi$
$337$	−7.57301e25	−0.218351	−0.109175	+	0.994023i	$0.534821\pi$
$338$	2.59412e26	0.725045
$339$	−1.59577e25	−0.0432388
$340$	0	0
$341$	−1.73667e26	−0.442380
$342$	−5.91130e25	−0.146019
$343$	−2.25697e26	−0.540676
$344$	4.06730e26	0.945024
$345$	0	0
$346$	−6.95563e26	−1.52068
$347$	−2.39671e26	−0.508343	−0.254171	−	0.967159i	$-0.581803\pi$
$347$	−2.39671e26	−0.508343	−0.254171	+	0.967159i	$0.581803\pi$
$348$	2.18696e26	0.450049
$349$	2.95784e26	0.590619	0.295310	−	0.955402i	$-0.404577\pi$
$349$	2.95784e26	0.590619	0.295310	+	0.955402i	$0.404577\pi$
$350$	0	0
$351$	7.54566e25	0.141897
$352$	4.62509e26	0.844155
$353$	−4.45964e26	−0.790069	−0.395034	−	0.918666i	$-0.629267\pi$
$353$	−4.45964e26	−0.790069	−0.395034	+	0.918666i	$0.629267\pi$
$354$	6.05920e24	0.0104203
$355$	0	0
$356$	2.29345e26	0.371761
$357$	−6.31846e25	−0.0994477
$358$	−1.95277e27	−2.98456
$359$	−1.82800e26	−0.271321	−0.135661	−	0.990755i	$-0.543316\pi$
$359$	−1.82800e26	−0.271321	−0.135661	+	0.990755i	$0.543316\pi$
$360$	0	0
$361$	−6.59912e26	−0.923976
$362$	7.05574e26	0.959627
$363$	−2.02473e26	−0.267514
$364$	2.46433e26	0.316326
$365$	0	0
$366$	1.04029e27	1.26068
$367$	7.38077e26	0.869176	0.434588	−	0.900629i	$-0.356894\pi$
$367$	7.38077e26	0.869176	0.434588	+	0.900629i	$0.356894\pi$
$368$	4.49925e24	0.00514917
$369$	−1.33222e26	−0.148184
$370$	0	0
$371$	−2.65077e26	−0.278578
$372$	−5.20123e26	−0.531380
$373$	−1.76554e27	−1.75362	−0.876808	−	0.480841i	$-0.840332\pi$
$373$	−1.76554e27	−1.75362	−0.876808	+	0.480841i	$0.840332\pi$
$374$	7.37470e26	0.712184
$375$	0	0
$376$	−1.94471e26	−0.177575
$377$	4.24660e26	0.377101
$378$	−9.96551e25	−0.0860668
$379$	−9.86553e26	−0.828722	−0.414361	−	0.910113i	$-0.635995\pi$
$379$	−9.86553e26	−0.828722	−0.414361	+	0.910113i	$0.635995\pi$
$380$	0	0
$381$	1.30858e27	1.04013
$382$	3.66930e27	2.83739
$383$	1.49870e27	1.12753	0.563766	−	0.825934i	$-0.309352\pi$
$383$	1.49870e27	1.12753	0.563766	+	0.825934i	$0.309352\pi$
$384$	1.13308e27	0.829438
$385$	0	0
$386$	2.77008e27	1.92010
$387$	−5.60800e26	−0.378306
$388$	−7.28083e26	−0.478022
$389$	1.22249e27	0.781224	0.390612	−	0.920556i	$-0.372264\pi$
$389$	1.22249e27	0.781224	0.390612	+	0.920556i	$0.372264\pi$
$390$	0	0
$391$	2.58624e25	0.0156608
$392$	1.30056e27	0.766702
$393$	5.58685e24	0.00320661
$394$	−5.85021e26	−0.326935
$395$	0	0
$396$	7.02330e26	0.372170
$397$	3.53943e26	0.182656	0.0913279	−	0.995821i	$-0.470889\pi$
$397$	3.53943e26	0.182656	0.0913279	+	0.995821i	$0.470889\pi$
$398$	4.23014e27	2.12610
$399$	9.15364e25	0.0448103
$400$	0	0
$401$	1.38817e27	0.644804	0.322402	−	0.946603i	$-0.395510\pi$
$401$	1.38817e27	0.644804	0.322402	+	0.946603i	$0.395510\pi$
$402$	−5.91649e26	−0.267726
$403$	−1.00996e27	−0.445249
$404$	4.76252e27	2.04566
$405$	0	0
$406$	−5.60847e26	−0.228729
$407$	7.13698e26	0.283644
$408$	7.59524e26	0.294178
$409$	−4.10310e27	−1.54888	−0.774439	−	0.632648i	$-0.781968\pi$
$409$	−4.10310e27	−1.54888	−0.774439	+	0.632648i	$0.781968\pi$
$410$	0	0
$411$	2.92213e27	1.04800
$412$	6.30252e27	2.20340
$413$	−9.38266e24	−0.00319779
$414$	4.07904e25	0.0135536
$415$	0	0
$416$	2.68974e27	0.849631
$417$	−1.04860e27	−0.322986
$418$	−1.06838e27	−0.320905
$419$	1.74062e27	0.509867	0.254934	−	0.966959i	$-0.417946\pi$
$419$	1.74062e27	0.509867	0.254934	+	0.966959i	$0.417946\pi$
$420$	0	0
$421$	3.18977e26	0.0888786	0.0444393	−	0.999012i	$-0.485850\pi$
$421$	3.18977e26	0.0888786	0.0444393	+	0.999012i	$0.485850\pi$
$422$	5.64707e27	1.53477
$423$	2.68137e26	0.0710859
$424$	3.18642e27	0.824065
$425$	0	0
$426$	6.47613e27	1.59410
$427$	−1.61089e27	−0.386877
$428$	3.36646e27	0.788884
$429$	1.36377e27	0.311846
$430$	0	0
$431$	4.01990e27	0.875395	0.437697	−	0.899122i	$-0.355794\pi$
$431$	4.01990e27	0.875395	0.437697	+	0.899122i	$0.355794\pi$
$432$	−1.82185e26	−0.0387199
$433$	−5.98127e27	−1.24071	−0.620355	−	0.784321i	$-0.713012\pi$
$433$	−5.98127e27	−1.24071	−0.620355	+	0.784321i	$0.713012\pi$
$434$	1.33386e27	0.270064
$435$	0	0
$436$	−1.13315e28	−2.18615
$437$	−3.74673e25	−0.00705662
$438$	−1.22054e27	−0.224427
$439$	−2.93253e27	−0.526460	−0.263230	−	0.964733i	$-0.584788\pi$
$439$	−2.93253e27	−0.526460	−0.263230	+	0.964733i	$0.584788\pi$
$440$	0	0
$441$	−1.79321e27	−0.306921
$442$	4.28879e27	0.716804
$443$	−2.72621e27	−0.444959	−0.222480	−	0.974937i	$-0.571415\pi$
$443$	−2.72621e27	−0.444959	−0.222480	+	0.974937i	$0.571415\pi$
$444$	2.13749e27	0.340709
$445$	0	0
$446$	−4.72277e27	−0.718094
$447$	−2.03519e27	−0.302257
$448$	−3.16194e27	−0.458705
$449$	5.45985e27	0.773739	0.386869	−	0.922135i	$-0.373556\pi$
$449$	5.45985e27	0.773739	0.386869	+	0.922135i	$0.373556\pi$
$450$	0	0
$451$	−2.40780e27	−0.325662
$452$	−8.63784e26	−0.114144
$453$	3.77733e27	0.487701
$454$	1.17975e28	1.48834
$455$	0	0
$456$	−1.10033e27	−0.132554
$457$	3.99132e27	0.469891	0.234945	−	0.972009i	$-0.424509\pi$
$457$	3.99132e27	0.469891	0.234945	+	0.972009i	$0.424509\pi$
$458$	−5.29131e27	−0.608801
$459$	−1.04723e27	−0.117763
$460$	0	0
$461$	4.71199e27	0.506226	0.253113	−	0.967437i	$-0.418546\pi$
$461$	4.71199e27	0.506226	0.253113	+	0.967437i	$0.418546\pi$
$462$	−1.80113e27	−0.189149
$463$	−1.54367e28	−1.58473	−0.792364	−	0.610049i	$-0.791150\pi$
$463$	−1.54367e28	−1.58473	−0.792364	+	0.610049i	$0.791150\pi$
$464$	−1.02532e27	−0.102901
$465$	0	0
$466$	−3.12060e28	−2.99354
$467$	−1.83848e28	−1.72438	−0.862188	−	0.506589i	$-0.830906\pi$
$467$	−1.83848e28	−1.72438	−0.862188	+	0.506589i	$0.830906\pi$
$468$	4.08443e27	0.374585
$469$	9.16168e26	0.0821600
$470$	0	0
$471$	−2.93987e27	−0.252121
$472$	1.12786e26	0.00945944
$473$	−1.01357e28	−0.831401
$474$	1.29395e28	1.03811
$475$	0	0
$476$	−3.42015e27	−0.262526
$477$	−4.39344e27	−0.329884
$478$	−4.33911e27	−0.318719
$479$	−1.87722e28	−1.34894	−0.674469	−	0.738303i	$-0.735628\pi$
$479$	−1.87722e28	−1.34894	−0.674469	+	0.738303i	$0.735628\pi$
$480$	0	0
$481$	4.15054e27	0.285484
$482$	−2.40800e28	−1.62055
$483$	−6.31639e25	−0.00415934
$484$	−1.09598e28	−0.706195
$485$	0	0
$486$	−1.65170e27	−0.101918
$487$	−7.96543e27	−0.481011	−0.240506	−	0.970648i	$-0.577313\pi$
$487$	−7.96543e27	−0.481011	−0.240506	+	0.970648i	$0.577313\pi$
$488$	1.93641e28	1.14443
$489$	−6.87518e27	−0.397687
$490$	0	0
$491$	2.38795e28	1.32333	0.661666	−	0.749799i	$-0.269850\pi$
$491$	2.38795e28	1.32333	0.661666	+	0.749799i	$0.269850\pi$
$492$	−7.21126e27	−0.391181
$493$	−5.89370e27	−0.312965
$494$	−6.21323e27	−0.322986
$495$	0	0
$496$	2.43850e27	0.121497
$497$	−1.00283e28	−0.489198
$498$	2.04872e28	0.978528
$499$	−1.85870e27	−0.0869270	−0.0434635	−	0.999055i	$-0.513839\pi$
$499$	−1.85870e27	−0.0869270	−0.0434635	+	0.999055i	$0.513839\pi$
$500$	0	0
$501$	3.33711e27	0.149649
$502$	−3.96951e28	−1.74320
$503$	−3.05087e28	−1.31208	−0.656039	−	0.754727i	$-0.727769\pi$
$503$	−3.05087e28	−1.31208	−0.656039	+	0.754727i	$0.727769\pi$
$504$	−1.85499e27	−0.0781305
$505$	0	0
$506$	7.37229e26	0.0297867
$507$	−6.65785e27	−0.263482
$508$	7.08328e28	2.74578
$509$	−2.93108e28	−1.11299	−0.556494	−	0.830851i	$-0.687854\pi$
$509$	−2.93108e28	−1.11299	−0.556494	+	0.830851i	$0.687854\pi$
$510$	0	0
$511$	1.89001e27	0.0688722
$512$	−1.11870e28	−0.399373
$513$	1.51714e27	0.0530632
$514$	1.98706e28	0.680924
$515$	0	0
$516$	−3.03559e28	−0.998667
$517$	4.84620e27	0.156225
$518$	−5.48160e27	−0.173159
$519$	1.78517e28	0.552615
$520$	0	0
$521$	1.39401e28	0.414448	0.207224	−	0.978293i	$-0.433557\pi$
$521$	1.39401e28	0.414448	0.207224	+	0.978293i	$0.433557\pi$
$522$	−9.29558e27	−0.270855
$523$	1.92103e28	0.548613	0.274307	−	0.961642i	$-0.411552\pi$
$523$	1.92103e28	0.548613	0.274307	+	0.961642i	$0.411552\pi$
$524$	3.02414e26	0.00846492
$525$	0	0
$526$	1.59693e28	0.429472
$527$	1.40169e28	0.369522
$528$	−3.29274e27	−0.0850946
$529$	−3.94457e28	−0.999345
$530$	0	0
$531$	−1.55510e26	−0.00378674
$532$	4.95482e27	0.118292
$533$	−1.40027e28	−0.327775
$534$	−9.74819e27	−0.223739
$535$	0	0
$536$	−1.10130e28	−0.243039
$537$	5.01182e28	1.08459
$538$	5.46613e28	1.16002
$539$	−3.24098e28	−0.674519
$540$	0	0
$541$	2.24623e27	0.0449659	0.0224830	−	0.999747i	$-0.492843\pi$
$541$	2.24623e27	0.0449659	0.0224830	+	0.999747i	$0.492843\pi$
$542$	1.05513e29	2.07164
$543$	−1.81086e28	−0.348729
$544$	−3.73299e28	−0.705128
$545$	0	0
$546$	−1.04745e28	−0.190376
$547$	1.02940e28	0.183535	0.0917675	−	0.995780i	$-0.470748\pi$
$547$	1.02940e28	0.183535	0.0917675	+	0.995780i	$0.470748\pi$
$548$	1.58173e29	2.76654
$549$	−2.66992e28	−0.458130
$550$	0	0
$551$	8.53829e27	0.141019
$552$	7.59276e26	0.0123038
$553$	−2.00367e28	−0.318576
$554$	−1.40228e29	−2.18767
$555$	0	0
$556$	−5.67605e28	−0.852630
$557$	7.27319e28	1.07213	0.536063	−	0.844178i	$-0.319911\pi$
$557$	7.27319e28	1.07213	0.536063	+	0.844178i	$0.319911\pi$
$558$	2.21076e28	0.319803
$559$	−5.89444e28	−0.836794
$560$	0	0
$561$	−1.89273e28	−0.258808
$562$	−7.84876e28	−1.05334
$563$	−9.50226e28	−1.25167	−0.625834	−	0.779957i	$-0.715241\pi$
$563$	−9.50226e28	−1.25167	−0.625834	+	0.779957i	$0.715241\pi$
$564$	1.45141e28	0.187655
$565$	0	0
$566$	−1.10542e29	−1.37706
$567$	2.55766e27	0.0312767
$568$	1.20547e29	1.44710
$569$	−9.95783e28	−1.17351	−0.586754	−	0.809765i	$-0.699594\pi$
$569$	−9.95783e28	−1.17351	−0.586754	+	0.809765i	$0.699594\pi$
$570$	0	0
$571$	9.60678e28	1.09119	0.545593	−	0.838050i	$-0.316304\pi$
$571$	9.60678e28	1.09119	0.545593	+	0.838050i	$0.316304\pi$
$572$	7.38203e28	0.823223
$573$	−9.41731e28	−1.03111
$574$	1.84933e28	0.198810
$575$	0	0
$576$	−5.24065e28	−0.543187
$577$	−9.48715e28	−0.965583	−0.482792	−	0.875735i	$-0.660377\pi$
$577$	−9.48715e28	−0.965583	−0.482792	+	0.875735i	$0.660377\pi$
$578$	9.94408e28	0.993853
$579$	−7.10943e28	−0.697766
$580$	0	0
$581$	−3.17243e28	−0.300292
$582$	3.09468e28	0.287690
$583$	−7.94052e28	−0.724985
$584$	−2.27192e28	−0.203732
$585$	0	0
$586$	2.65991e29	2.30113
$587$	7.25220e28	0.616268	0.308134	−	0.951343i	$-0.400295\pi$
$587$	7.25220e28	0.616268	0.308134	+	0.951343i	$0.400295\pi$
$588$	−9.70658e28	−0.810222
$589$	−2.03065e28	−0.166504
$590$	0	0
$591$	1.50146e28	0.118808
$592$	−1.00212e28	−0.0779010
$593$	5.96548e28	0.455587	0.227793	−	0.973710i	$-0.426849\pi$
$593$	5.96548e28	0.455587	0.227793	+	0.973710i	$0.426849\pi$
$594$	−2.98522e28	−0.223985
$595$	0	0
$596$	−1.10164e29	−0.797910
$597$	−1.08567e29	−0.772623
$598$	4.28738e27	0.0299799
$599$	1.36596e29	0.938548	0.469274	−	0.883053i	$-0.344516\pi$
$599$	1.36596e29	0.938548	0.469274	+	0.883053i	$0.344516\pi$
$600$	0	0
$601$	2.91002e29	1.93069	0.965347	−	0.260968i	$-0.0840418\pi$
$601$	2.91002e29	1.93069	0.965347	+	0.260968i	$0.0840418\pi$
$602$	7.78476e28	0.507554
$603$	1.51847e28	0.0972917
$604$	2.04465e29	1.28745
$605$	0	0
$606$	−2.02429e29	−1.23115
$607$	−1.53630e29	−0.918322	−0.459161	−	0.888353i	$-0.651850\pi$
$607$	−1.53630e29	−0.918322	−0.459161	+	0.888353i	$0.651850\pi$
$608$	5.40803e28	0.317725
$609$	1.43942e28	0.0831201
$610$	0	0
$611$	2.81833e28	0.157238
$612$	−5.66862e28	−0.310876
$613$	6.03634e28	0.325416	0.162708	−	0.986674i	$-0.447977\pi$
$613$	6.03634e28	0.325416	0.162708	+	0.986674i	$0.447977\pi$
$614$	3.16634e29	1.67799
$615$	0	0
$616$	−3.35263e28	−0.171707
$617$	−6.56975e28	−0.330791	−0.165396	−	0.986227i	$-0.552890\pi$
$617$	−6.56975e28	−0.330791	−0.165396	+	0.986227i	$0.552890\pi$
$618$	−2.67886e29	−1.32608
$619$	2.24352e29	1.09189	0.545945	−	0.837821i	$-0.316171\pi$
$619$	2.24352e29	1.09189	0.545945	+	0.837821i	$0.316171\pi$
$620$	0	0
$621$	−1.04689e27	−0.00492538
$622$	−3.77671e29	−1.74709
$623$	1.50951e28	0.0686610
$624$	−1.91491e28	−0.0856466
$625$	0	0
$626$	1.89244e29	0.818449
$627$	2.74202e28	0.116617
$628$	−1.59134e29	−0.665558
$629$	−5.76037e28	−0.236929
$630$	0	0
$631$	1.40053e29	0.557166	0.278583	−	0.960412i	$-0.410135\pi$
$631$	1.40053e29	0.557166	0.278583	+	0.960412i	$0.410135\pi$
$632$	2.40856e29	0.942384
$633$	−1.44933e29	−0.557735
$634$	6.08430e29	2.30289
$635$	0	0
$636$	−2.37815e29	−0.870842
$637$	−1.88480e29	−0.678894
$638$	−1.68005e29	−0.595256
$639$	−1.66211e29	−0.579295
$640$	0	0
$641$	−2.34559e29	−0.791121	−0.395560	−	0.918440i	$-0.629450\pi$
$641$	−2.34559e29	−0.791121	−0.395560	+	0.918440i	$0.629450\pi$
$642$	−1.43090e29	−0.474777
$643$	−1.42258e29	−0.464368	−0.232184	−	0.972672i	$-0.574587\pi$
$643$	−1.42258e29	−0.464368	−0.232184	+	0.972672i	$0.574587\pi$
$644$	−3.41903e27	−0.0109800
$645$	0	0
$646$	8.62310e28	0.268054
$647$	−1.58157e29	−0.483720	−0.241860	−	0.970311i	$-0.577757\pi$
$647$	−1.58157e29	−0.483720	−0.241860	+	0.970311i	$0.577757\pi$
$648$	−3.07449e28	−0.0925201
$649$	−2.81062e27	−0.00832211
$650$	0	0
$651$	−3.42336e28	−0.0981413
$652$	−3.72150e29	−1.04983
$653$	−4.13878e29	−1.14890	−0.574452	−	0.818539i	$-0.694785\pi$
$653$	−4.13878e29	−1.14890	−0.574452	+	0.818539i	$0.694785\pi$
$654$	4.81640e29	1.31570
$655$	0	0
$656$	3.38087e28	0.0894412
$657$	3.13253e28	0.0815566
$658$	−3.72215e28	−0.0953723
$659$	−7.78856e28	−0.196409	−0.0982043	−	0.995166i	$-0.531310\pi$
$659$	−7.78856e28	−0.196409	−0.0982043	+	0.995166i	$0.531310\pi$
$660$	0	0
$661$	−9.22514e28	−0.225350	−0.112675	−	0.993632i	$-0.535942\pi$
$661$	−9.22514e28	−0.225350	−0.112675	+	0.993632i	$0.535942\pi$
$662$	1.18142e30	2.84052
$663$	−1.10072e29	−0.260487
$664$	3.81349e29	0.888297
$665$	0	0
$666$	−9.08530e28	−0.205050
$667$	−5.89177e27	−0.0130895
$668$	1.80636e29	0.395049
$669$	1.21211e29	0.260955
$670$	0	0
$671$	−4.82550e29	−1.00683
$672$	9.11709e28	0.187274
$673$	−3.42657e27	−0.00692949	−0.00346475	−	0.999994i	$-0.501103\pi$
$673$	−3.42657e27	−0.00692949	−0.00346475	+	0.999994i	$0.501103\pi$
$674$	1.74236e29	0.346904
$675$	0	0
$676$	−3.60386e29	−0.695549
$677$	1.60981e29	0.305910	0.152955	−	0.988233i	$-0.451121\pi$
$677$	1.60981e29	0.305910	0.152955	+	0.988233i	$0.451121\pi$
$678$	3.67147e28	0.0686955
$679$	−4.79211e28	−0.0882864
$680$	0	0
$681$	−3.02784e29	−0.540864
$682$	3.99563e29	0.702829
$683$	−2.93386e28	−0.0508186	−0.0254093	−	0.999677i	$-0.508089\pi$
$683$	−2.93386e28	−0.0508186	−0.0254093	+	0.999677i	$0.508089\pi$
$684$	8.21221e28	0.140078
$685$	0	0
$686$	5.19272e29	0.858997
$687$	1.35802e29	0.221238
$688$	1.42318e29	0.228339
$689$	−4.61784e29	−0.729688
$690$	0	0
$691$	−7.95216e29	−1.21889	−0.609446	−	0.792827i	$-0.708608\pi$
$691$	−7.95216e29	−1.21889	−0.609446	+	0.792827i	$0.708608\pi$
$692$	9.66305e29	1.45881
$693$	4.62261e28	0.0687366
$694$	5.51424e29	0.807627
$695$	0	0
$696$	−1.73029e29	−0.245879
$697$	1.94338e29	0.272028
$698$	−6.80525e29	−0.938344
$699$	8.00904e29	1.08785
$700$	0	0
$701$	1.10069e30	1.45086	0.725430	−	0.688296i	$-0.241641\pi$
$701$	1.10069e30	1.45086	0.725430	+	0.688296i	$0.241641\pi$
$702$	−1.73607e29	−0.225438
$703$	8.34514e28	0.106759
$704$	−9.47174e29	−1.19376
$705$	0	0
$706$	1.02605e30	1.25522
$707$	3.13461e29	0.377815
$708$	−8.41769e27	−0.00999640
$709$	9.95317e29	1.16460	0.582299	−	0.812975i	$-0.302153\pi$
$709$	9.95317e29	1.16460	0.582299	+	0.812975i	$0.302153\pi$
$710$	0	0
$711$	−3.32093e29	−0.377249
$712$	−1.81454e29	−0.203107
$713$	1.40123e28	0.0154550
$714$	1.45372e29	0.157997
$715$	0	0
$716$	2.71287e30	2.86314
$717$	1.11364e29	0.115822
$718$	4.20577e29	0.431061
$719$	−1.52067e29	−0.153597	−0.0767983	−	0.997047i	$-0.524470\pi$
$719$	−1.52067e29	−0.153597	−0.0767983	+	0.997047i	$0.524470\pi$
$720$	0	0
$721$	4.14820e29	0.406949
$722$	1.51829e30	1.46796
$723$	6.18017e29	0.588909
$724$	−9.80212e29	−0.920587
$725$	0	0
$726$	4.65839e29	0.425012
$727$	8.18896e29	0.736406	0.368203	−	0.929745i	$-0.379973\pi$
$727$	8.18896e29	0.736406	0.368203	+	0.929745i	$0.379973\pi$
$728$	−1.94973e29	−0.172821
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	8.18067e29	0.694475
$732$	−1.44521e30	−1.20939
$733$	4.95962e29	0.409125	0.204563	−	0.978853i	$-0.434423\pi$
$733$	4.95962e29	0.409125	0.204563	+	0.978853i	$0.434423\pi$
$734$	−1.69813e30	−1.38090
$735$	0	0
$736$	−3.73177e28	−0.0294915
$737$	2.74443e29	0.213818
$738$	3.06511e29	0.235426
$739$	−7.82517e29	−0.592553	−0.296277	−	0.955102i	$-0.595745\pi$
$739$	−7.82517e29	−0.592553	−0.296277	+	0.955102i	$0.595745\pi$
$740$	0	0
$741$	1.59463e29	0.117373
$742$	6.09876e29	0.442589
$743$	3.93104e29	0.281271	0.140636	−	0.990061i	$-0.455085\pi$
$743$	3.93104e29	0.281271	0.140636	+	0.990061i	$0.455085\pi$
$744$	4.11512e29	0.290313
$745$	0	0
$746$	4.06206e30	2.78605
$747$	−5.25805e29	−0.355597
$748$	−1.02452e30	−0.683211
$749$	2.21574e29	0.145700
$750$	0	0
$751$	−1.84532e30	−1.17992	−0.589959	−	0.807433i	$-0.700856\pi$
$751$	−1.84532e30	−1.17992	−0.589959	+	0.807433i	$0.700856\pi$
$752$	−6.80469e28	−0.0429063
$753$	1.01878e30	0.633480
$754$	−9.77037e29	−0.599117
$755$	0	0
$756$	1.38445e29	0.0825654
$757$	1.84934e30	1.08770	0.543852	−	0.839181i	$-0.316965\pi$
$757$	1.84934e30	1.08770	0.543852	+	0.839181i	$0.316965\pi$
$758$	2.26981e30	1.31663
$759$	−1.89211e28	−0.0108245
$760$	0	0
$761$	−2.72378e30	−1.51577	−0.757885	−	0.652389i	$-0.773767\pi$
$761$	−2.72378e30	−1.51577	−0.757885	+	0.652389i	$0.773767\pi$
$762$	−3.01071e30	−1.65250
$763$	−7.45820e29	−0.403763
$764$	−5.09755e30	−2.72195
$765$	0	0
$766$	−3.44814e30	−1.79136
$767$	−1.63453e28	−0.00837609
$768$	−7.45700e29	−0.376939
$769$	−3.33728e29	−0.166405	−0.0832026	−	0.996533i	$-0.526515\pi$
$769$	−3.33728e29	−0.166405	−0.0832026	+	0.996533i	$0.526515\pi$
$770$	0	0
$771$	−5.09982e29	−0.247448
$772$	−3.84830e30	−1.84199
$773$	3.35597e30	1.58465	0.792325	−	0.610100i	$-0.208871\pi$
$773$	3.35597e30	1.58465	0.792325	+	0.610100i	$0.208871\pi$
$774$	1.29026e30	0.601032
$775$	0	0
$776$	5.76046e29	0.261161
$777$	1.40686e29	0.0629259
$778$	−2.81265e30	−1.24117
$779$	−2.81540e29	−0.122574
$780$	0	0
$781$	−3.00402e30	−1.27311
$782$	−5.95029e28	−0.0248810
$783$	2.38572e29	0.0984286
$784$	4.55075e29	0.185253
$785$	0	0
$786$	−1.28539e28	−0.00509448
$787$	−1.64029e29	−0.0641484	−0.0320742	−	0.999485i	$-0.510211\pi$
$787$	−1.64029e29	−0.0641484	−0.0320742	+	0.999485i	$0.510211\pi$
$788$	8.12735e29	0.313634
$789$	−4.09853e29	−0.156070
$790$	0	0
$791$	−5.68527e28	−0.0210813
$792$	−5.55671e29	−0.203331
$793$	−2.80629e30	−1.01336
$794$	−8.14335e29	−0.290194
$795$	0	0
$796$	−5.87669e30	−2.03960
$797$	1.33943e30	0.458784	0.229392	−	0.973334i	$-0.426326\pi$
$797$	1.33943e30	0.458784	0.229392	+	0.973334i	$0.426326\pi$
$798$	−2.10602e29	−0.0711922
$799$	−3.91145e29	−0.130496
$800$	0	0
$801$	2.50188e29	0.0813066
$802$	−3.19384e30	−1.02443
$803$	5.66161e29	0.179237
$804$	8.21943e29	0.256835
$805$	0	0
$806$	2.32368e30	0.707388
$807$	−1.40289e30	−0.421552
$808$	−3.76802e30	−1.11762
$809$	5.46468e30	1.59995	0.799973	−	0.600036i	$-0.204847\pi$
$809$	5.46468e30	1.59995	0.799973	+	0.600036i	$0.204847\pi$
$810$	0	0
$811$	−6.64567e30	−1.89592	−0.947959	−	0.318393i	$-0.896857\pi$
$811$	−6.64567e30	−1.89592	−0.947959	+	0.318393i	$0.896857\pi$
$812$	7.79151e29	0.219424
$813$	−2.70800e30	−0.752833
$814$	−1.64204e30	−0.450638
$815$	0	0
$816$	2.65763e29	0.0710800
$817$	−1.18515e30	−0.312925
$818$	9.44020e30	2.46077
$819$	2.68830e29	0.0691825
$820$	0	0
$821$	−3.94037e30	−0.988403	−0.494201	−	0.869347i	$-0.664539\pi$
$821$	−3.94037e30	−0.988403	−0.494201	+	0.869347i	$0.664539\pi$
$822$	−6.72308e30	−1.66500
$823$	2.57770e29	0.0630280	0.0315140	−	0.999503i	$-0.489967\pi$
$823$	2.57770e29	0.0630280	0.0315140	+	0.999503i	$0.489967\pi$
$824$	−4.98644e30	−1.20380
$825$	0	0
$826$	2.15872e28	0.00508048
$827$	9.53726e29	0.221624	0.110812	−	0.993841i	$-0.464655\pi$
$827$	9.53726e29	0.221624	0.110812	+	0.993841i	$0.464655\pi$
$828$	−5.66677e28	−0.0130022
$829$	−2.35934e29	−0.0534526	−0.0267263	−	0.999643i	$-0.508508\pi$
$829$	−2.35934e29	−0.0534526	−0.0267263	+	0.999643i	$0.508508\pi$
$830$	0	0
$831$	3.59896e30	0.794998
$832$	−5.50833e30	−1.20150
$833$	2.61585e30	0.563430
$834$	2.41258e30	0.513142
$835$	0	0
$836$	1.48424e30	0.307850
$837$	−5.67394e29	−0.116216
$838$	−4.00473e30	−0.810049
$839$	1.50684e30	0.301000	0.150500	−	0.988610i	$-0.451912\pi$
$839$	1.50684e30	0.301000	0.150500	+	0.988610i	$0.451912\pi$
$840$	0	0
$841$	−3.79019e30	−0.738419
$842$	−7.33886e29	−0.141206
$843$	2.01439e30	0.382785
$844$	−7.84514e30	−1.47233
$845$	0	0
$846$	−6.16916e29	−0.112937
$847$	−7.21351e29	−0.130428
$848$	1.11495e30	0.199113
$849$	2.83706e30	0.500424
$850$	0	0
$851$	−5.75849e28	−0.00990942
$852$	−8.99691e30	−1.52925
$853$	8.74672e30	1.46852	0.734261	−	0.678867i	$-0.237529\pi$
$853$	8.74672e30	1.46852	0.734261	+	0.678867i	$0.237529\pi$
$854$	3.70625e30	0.614649
$855$	0	0
$856$	−2.66348e30	−0.430998
$857$	1.79796e30	0.287397	0.143699	−	0.989622i	$-0.454100\pi$
$857$	1.79796e30	0.287397	0.143699	+	0.989622i	$0.454100\pi$
$858$	−3.13769e30	−0.495444
$859$	2.30872e30	0.360116	0.180058	−	0.983656i	$-0.442371\pi$
$859$	2.30872e30	0.360116	0.180058	+	0.983656i	$0.442371\pi$
$860$	0	0
$861$	−4.74632e29	−0.0722477
$862$	−9.24878e30	−1.39078
$863$	7.08890e30	1.05309	0.526545	−	0.850147i	$-0.323487\pi$
$863$	7.08890e30	1.05309	0.526545	+	0.850147i	$0.323487\pi$
$864$	1.51108e30	0.221766
$865$	0	0
$866$	1.37614e31	1.97117
$867$	−2.55216e30	−0.361166
$868$	−1.85305e30	−0.259077
$869$	−6.00210e30	−0.829079
$870$	0	0
$871$	1.59603e30	0.215205
$872$	8.96529e30	1.19438
$873$	−7.94253e29	−0.104546
$874$	8.62028e28	0.0112112
$875$	0	0
$876$	1.69563e30	0.215296
$877$	6.15937e30	0.772752	0.386376	−	0.922341i	$-0.373727\pi$
$877$	6.15937e30	0.772752	0.386376	+	0.922341i	$0.373727\pi$
$878$	6.74702e30	0.836411
$879$	−6.82668e30	−0.836231
$880$	0	0
$881$	4.62731e30	0.553454	0.276727	−	0.960949i	$-0.410750\pi$
$881$	4.62731e30	0.553454	0.276727	+	0.960949i	$0.410750\pi$
$882$	4.12573e30	0.487619
$883$	−1.04142e30	−0.121629	−0.0608147	−	0.998149i	$-0.519370\pi$
$883$	−1.04142e30	−0.121629	−0.0608147	+	0.998149i	$0.519370\pi$
$884$	−5.95816e30	−0.687643
$885$	0	0
$886$	6.27233e30	0.706927
$887$	1.43153e31	1.59441	0.797207	−	0.603705i	$-0.206310\pi$
$887$	1.43153e31	1.59441	0.797207	+	0.603705i	$0.206310\pi$
$888$	−1.69114e30	−0.186142
$889$	4.66209e30	0.507121
$890$	0	0
$891$	7.66160e29	0.0813961
$892$	6.56107e30	0.688880
$893$	5.66657e29	0.0588004
$894$	4.68247e30	0.480210
$895$	0	0
$896$	4.03685e30	0.404397
$897$	−1.10036e29	−0.0108947
$898$	−1.25618e31	−1.22927
$899$	−3.19322e30	−0.308853
$900$	0	0
$901$	6.40892e30	0.605585
$902$	5.53975e30	0.517395
$903$	−1.99797e30	−0.184445
$904$	6.83410e29	0.0623610
$905$	0	0
$906$	−8.69068e30	−0.774833
$907$	1.50854e31	1.32947	0.664736	−	0.747079i	$-0.268544\pi$
$907$	1.50854e31	1.32947	0.664736	+	0.747079i	$0.268544\pi$
$908$	−1.63895e31	−1.42779
$909$	5.19536e30	0.447399
$910$	0	0
$911$	−4.70909e30	−0.396272	−0.198136	−	0.980175i	$-0.563489\pi$
$911$	−4.70909e30	−0.396272	−0.198136	+	0.980175i	$0.563489\pi$
$912$	−3.85014e29	−0.0320281
$913$	−9.50318e30	−0.781495
$914$	−9.18303e30	−0.746536
$915$	0	0
$916$	7.35090e30	0.584034
$917$	1.99043e28	0.00156340
$918$	2.40942e30	0.187096
$919$	−3.23144e30	−0.248075	−0.124038	−	0.992278i	$-0.539584\pi$
$919$	−3.23144e30	−0.248075	−0.124038	+	0.992278i	$0.539584\pi$
$920$	0	0
$921$	−8.12646e30	−0.609783
$922$	−1.08411e31	−0.804264
$923$	−1.74700e31	−1.28137
$924$	2.50220e30	0.181454
$925$	0	0
$926$	3.55160e31	2.51773
$927$	6.87531e30	0.481898
$928$	8.50419e30	0.589358
$929$	−2.63040e31	−1.80242	−0.901210	−	0.433382i	$-0.857320\pi$
$929$	−2.63040e31	−1.80242	−0.901210	+	0.433382i	$0.857320\pi$
$930$	0	0
$931$	−3.78962e30	−0.253877
$932$	4.33526e31	2.87176
$933$	9.69298e30	0.634892
$934$	4.22988e31	2.73959
$935$	0	0
$936$	−3.23153e30	−0.204650
$937$	2.02013e31	1.26507	0.632535	−	0.774532i	$-0.282014\pi$
$937$	2.02013e31	1.26507	0.632535	+	0.774532i	$0.282014\pi$
$938$	−2.10787e30	−0.130531
$939$	−4.85697e30	−0.297424
$940$	0	0
$941$	−3.09975e31	−1.85625	−0.928123	−	0.372275i	$-0.878578\pi$
$941$	−3.09975e31	−1.85625	−0.928123	+	0.372275i	$0.878578\pi$
$942$	6.76390e30	0.400556
$943$	1.94274e29	0.0113774
$944$	3.94648e28	0.00228562
$945$	0	0
$946$	2.33196e31	1.32088
$947$	2.18440e31	1.22365	0.611826	−	0.790992i	$-0.290435\pi$
$947$	2.18440e31	1.22365	0.611826	+	0.790992i	$0.290435\pi$
$948$	−1.79760e31	−0.995877
$949$	3.29253e30	0.180399
$950$	0	0
$951$	−1.56154e31	−0.836871
$952$	2.70596e30	0.143428
$953$	−1.26936e31	−0.665439	−0.332719	−	0.943026i	$-0.607966\pi$
$953$	−1.26936e31	−0.665439	−0.332719	+	0.943026i	$0.607966\pi$
$954$	1.01082e31	0.524102
$955$	0	0
$956$	6.02807e30	0.305753
$957$	4.31186e30	0.216316
$958$	4.31901e31	2.14312
$959$	1.04107e31	0.510956
$960$	0	0
$961$	−1.32311e31	−0.635332
$962$	−9.54935e30	−0.453561
$963$	3.67241e30	0.172534
$964$	3.34530e31	1.55463
$965$	0	0
$966$	1.45324e29	0.00660813
$967$	−1.90794e31	−0.858195	−0.429097	−	0.903258i	$-0.641168\pi$
$967$	−1.90794e31	−0.858195	−0.429097	+	0.903258i	$0.641168\pi$
$968$	8.67116e30	0.385821
$969$	−2.21313e30	−0.0974108
$970$	0	0
$971$	2.37666e31	1.02368	0.511841	−	0.859080i	$-0.328964\pi$
$971$	2.37666e31	1.02368	0.511841	+	0.859080i	$0.328964\pi$
$972$	2.29461e30	0.0977718
$973$	−3.73587e30	−0.157473
$974$	1.83265e31	0.764204
$975$	0	0
$976$	6.77562e30	0.276520
$977$	−2.16504e31	−0.874123	−0.437062	−	0.899432i	$-0.643981\pi$
$977$	−2.16504e31	−0.874123	−0.437062	+	0.899432i	$0.643981\pi$
$978$	1.58181e31	0.631823
$979$	4.52180e30	0.178687
$980$	0	0
$981$	−1.23614e31	−0.478125
$982$	−5.49406e31	−2.10244
$983$	−3.62066e31	−1.37080	−0.685402	−	0.728165i	$-0.740373\pi$
$983$	−3.62066e31	−1.37080	−0.685402	+	0.728165i	$0.740373\pi$
$984$	5.70542e30	0.213717
$985$	0	0
$986$	1.35599e31	0.497221
$987$	9.55294e29	0.0346583
$988$	8.63167e30	0.309847
$989$	8.17799e29	0.0290460
$990$	0	0
$991$	2.36223e30	0.0821387	0.0410694	−	0.999156i	$-0.486924\pi$
$991$	2.36223e30	0.0821387	0.0410694	+	0.999156i	$0.486924\pi$
$992$	−2.02254e31	−0.695865
$993$	−3.03214e31	−1.03224
$994$	2.30726e31	0.777211
$995$	0	0
$996$	−2.84616e31	−0.938720
$997$	−2.58169e31	−0.842570	−0.421285	−	0.906928i	$-0.638421\pi$
$997$	−2.58169e31	−0.842570	−0.421285	+	0.906928i	$0.638421\pi$
$998$	4.27641e30	0.138105
$999$	2.33175e30	0.0745152

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.i.1.1	✓	6
5.2	odd	4		75.22.b.i.49.2		12
5.3	odd	4		75.22.b.i.49.11		12
5.4	even	2		75.22.a.j.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.22.a.i.1.1	✓	6	1.1	even	1	trivial
75.22.a.j.1.6	yes	6	5.4	even	2
75.22.b.i.49.2		12	5.2	odd	4
75.22.b.i.49.11		12	5.3	odd	4

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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-2300.75 q^{2} +59049.0 q^{3} +3.19629e6 q^{4} -1.35857e8 q^{6} +2.10374e8 q^{7} -2.52885e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-2300.75 q^{2} +59049.0 q^{3} +3.19629e6 q^{4} -1.35857e8 q^{6} +2.10374e8 q^{7} -2.52885e9 q^{8} +3.48678e9 q^{9} +6.30187e10 q^{11} +1.88738e11 q^{12} +3.66488e11 q^{13} -4.84019e11 q^{14} -8.84863e11 q^{16} -5.08634e12 q^{17} -8.02222e12 q^{18} +7.36866e12 q^{19} +1.24224e13 q^{21} -1.44990e14 q^{22} -5.08468e12 q^{23} -1.49326e14 q^{24} -8.43196e14 q^{26} +2.05891e14 q^{27} +6.72418e14 q^{28} +1.15873e15 q^{29} -2.75579e15 q^{31} +7.33923e15 q^{32} +3.72119e15 q^{33} +1.17024e16 q^{34} +1.11448e16 q^{36} +1.13252e16 q^{37} -1.69534e16 q^{38} +2.16407e16 q^{39} -3.82078e16 q^{41} -2.85808e16 q^{42} -1.60836e17 q^{43} +2.01426e17 q^{44} +1.16986e16 q^{46} +7.69010e16 q^{47} -5.22503e16 q^{48} -5.14289e17 q^{49} -3.00343e17 q^{51} +1.17140e18 q^{52} -1.26003e18 q^{53} -4.73704e17 q^{54} -5.32005e17 q^{56} +4.35112e17 q^{57} -2.66595e18 q^{58} -4.45998e16 q^{59} -7.65725e18 q^{61} +6.34039e18 q^{62} +7.33530e17 q^{63} -1.50300e19 q^{64} -8.56153e18 q^{66} +4.35494e18 q^{67} -1.62574e19 q^{68} -3.00245e17 q^{69} -4.76688e19 q^{71} -8.81756e18 q^{72} +8.98402e18 q^{73} -2.60564e19 q^{74} +2.35524e19 q^{76} +1.32575e19 q^{77} -4.97899e19 q^{78} -9.52432e19 q^{79} +1.21577e19 q^{81} +8.79065e19 q^{82} -1.50799e20 q^{83} +3.97056e19 q^{84} +3.70043e20 q^{86} +6.84219e19 q^{87} -1.59365e20 q^{88} +7.17533e19 q^{89} +7.70996e19 q^{91} -1.62521e19 q^{92} -1.62727e20 q^{93} -1.76930e20 q^{94} +4.33374e20 q^{96} -2.27790e20 q^{97} +1.18325e21 q^{98} +2.19733e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 92 q^{2} + 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} - 1327143454 q^{7} + 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) \) 6 * q - 92 * q^2 + 354294 * q^3 + 4390448 * q^4 - 5432508 * q^6 - 1327143454 * q^7 + 4252271424 * q^8 + 20920706406 * q^9	\( 6 q - 92 q^{2} + 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} - 1327143454 q^{7} + 4252271424 q^{8} + 20920706406 q^{9} - 78310112516 q^{11} + 259251563952 q^{12} - 116029746338 q^{13} - 313825730148 q^{14} + 1996701612800 q^{16} - 5382900513068 q^{17} - 320784164892 q^{18} + 6969638997622 q^{19} - 78366493815246 q^{21} + 19701817271864 q^{22} - 295754895311052 q^{23} + 251092375315776 q^{24} - 274314326263628 q^{26} + 12\!\cdots\!94 q^{27}+ \cdots - 27\!\cdots\!16 q^{99}+O(q^{100}) \) 6 * q - 92 * q^2 + 354294 * q^3 + 4390448 * q^4 - 5432508 * q^6 - 1327143454 * q^7 + 4252271424 * q^8 + 20920706406 * q^9 - 78310112516 * q^11 + 259251563952 * q^12 - 116029746338 * q^13 - 313825730148 * q^14 + 1996701612800 * q^16 - 5382900513068 * q^17 - 320784164892 * q^18 + 6969638997622 * q^19 - 78366493815246 * q^21 + 19701817271864 * q^22 - 295754895311052 * q^23 + 251092375315776 * q^24 - 274314326263628 * q^26 + 1235346792567894 * q^27 - 2674107576939568 * q^28 - 3895624305077596 * q^29 + 492586859482706 * q^31 + 10862288965692416 * q^32 - 4624133833957284 * q^33 - 30773279577490744 * q^34 + 15308545599801648 * q^36 + 26101348744076316 * q^37 + 8307793564414708 * q^38 - 6851440491512562 * q^39 + 112662882011424056 * q^41 - 18531095539509252 * q^42 + 13377804412633810 * q^43 - 23820753283797856 * q^44 - 156784693648581624 * q^46 + 243206274450163780 * q^47 + 117903233534227200 * q^48 + 327097566625336780 * q^49 - 317854892396152332 * q^51 + 192375894240694512 * q^52 - 554044170675798992 * q^53 - 18941984152707708 * q^54 - 5086364147554777920 * q^56 + 411550213170581478 * q^57 - 11350712056264661624 * q^58 - 7003944869710758212 * q^59 - 10235053877143753790 * q^61 + 1855763029836316572 * q^62 - 4627463093296461054 * q^63 - 10747451596339826688 * q^64 + 1163372608086297336 * q^66 - 8975831852618567158 * q^67 - 64266312753700360864 * q^68 - 17464030813222309548 * q^69 + 18009408229245489272 * q^71 + 14826753670021257024 * q^72 - 81172357549902895652 * q^73 - 103132266830355254968 * q^74 + 111484542579179194608 * q^76 + 40817017589776296372 * q^77 - 16197986651540969772 * q^78 - 79149540279186452080 * q^79 + 72945992754341572806 * q^81 - 17339452427244467168 * q^82 - 238715465076655746756 * q^83 - 157903378310704550832 * q^84 + 28767310268029844764 * q^86 - 230032719590526966204 * q^87 - 1023055046877663881856 * q^88 - 712331293797957179808 * q^89 + 670774308716312800426 * q^91 - 880736698470324218784 * q^92 + 29086761465594306594 * q^93 + 278898069330490980248 * q^94 + 641407301135171472384 * q^96 - 1498553500583421551178 * q^97 - 2662349783988420722392 * q^98 - 273050478761343662916 * q^99

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