LMFDB - Embedded newform 75.22.a.l.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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2 x^{7} - 13701836 x^{6} + 201589162 x^{5} + 55078074399586 x^{4} + \cdots + 23\!\cdots\!25 $ x^8 - 2x^7 - 13701836x^6 + 201589162x^5 + 55078074399586x^4 - 1921148793416750x^3 - 63775892303255475300x^2 - 5448867057840511928250*x + 2374849659307636957978125
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{10}\cdot 5^{14}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2688.59$ 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)

$f(q)$	$=$	$q-2605.59 q^{2} -59049.0 q^{3} +4.69194e6 q^{4} +1.53857e8 q^{6} -1.24132e9 q^{7} -6.76095e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-2605.59 q^{2} -59049.0 q^{3} +4.69194e6 q^{4} +1.53857e8 q^{6} -1.24132e9 q^{7} -6.76095e9 q^{8} +3.48678e9 q^{9} +6.95232e10 q^{11} -2.77054e11 q^{12} +2.02578e11 q^{13} +3.23436e12 q^{14} +7.77654e12 q^{16} -7.32903e12 q^{17} -9.08513e12 q^{18} +2.46215e13 q^{19} +7.32984e13 q^{21} -1.81149e14 q^{22} +1.68791e14 q^{23} +3.99227e14 q^{24} -5.27834e14 q^{26} -2.05891e14 q^{27} -5.82418e15 q^{28} -4.35028e15 q^{29} +4.49432e15 q^{31} -6.08373e15 q^{32} -4.10528e15 q^{33} +1.90964e16 q^{34} +1.63598e16 q^{36} -3.45204e16 q^{37} -6.41535e16 q^{38} -1.19620e16 q^{39} +6.64078e16 q^{41} -1.90986e17 q^{42} +1.34497e17 q^{43} +3.26199e17 q^{44} -4.39800e17 q^{46} +4.37113e17 q^{47} -4.59197e17 q^{48} +9.82317e17 q^{49} +4.32772e17 q^{51} +9.50482e17 q^{52} +1.75752e18 q^{53} +5.36468e17 q^{54} +8.39247e18 q^{56} -1.45387e18 q^{57} +1.13350e19 q^{58} -5.05155e18 q^{59} +7.50009e18 q^{61} -1.17104e19 q^{62} -4.32820e18 q^{63} -4.56890e17 q^{64} +1.06967e19 q^{66} -2.43348e19 q^{67} -3.43873e19 q^{68} -9.96694e18 q^{69} +8.23122e18 q^{71} -2.35740e19 q^{72} +5.21807e19 q^{73} +8.99459e19 q^{74} +1.15523e20 q^{76} -8.63002e19 q^{77} +3.11681e19 q^{78} -1.28526e20 q^{79} +1.21577e19 q^{81} -1.73032e20 q^{82} +1.15724e20 q^{83} +3.43912e20 q^{84} -3.50444e20 q^{86} +2.56880e20 q^{87} -4.70043e20 q^{88} +1.72710e20 q^{89} -2.51463e20 q^{91} +7.91958e20 q^{92} -2.65385e20 q^{93} -1.13894e21 q^{94} +3.59238e20 q^{96} -9.66495e20 q^{97} -2.55951e21 q^{98} +2.42412e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9}+O(q^{10}) $ 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9	$ 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9} - 14410165968 q^{11} - 630755749296 q^{12} - 328226481176 q^{13} + 4068036669186 q^{14} + 18553054308920 q^{16} - 5718214953936 q^{17} + 2322198411066 q^{18} + 75919698170296 q^{19} - 7914601537128 q^{21} - 426897542691372 q^{22} + 49712781537936 q^{23} - 207390009885372 q^{24} - 126056679642054 q^{26} - 16\!\cdots\!92 q^{27}+ \cdots - 50\!\cdots\!68 q^{99}+O(q^{100}) $ 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9 - 14410165968 * q^11 - 630755749296 * q^12 - 328226481176 * q^13 + 4068036669186 * q^14 + 18553054308920 * q^16 - 5718214953936 * q^17 + 2322198411066 * q^18 + 75919698170296 * q^19 - 7914601537128 * q^21 - 426897542691372 * q^22 + 49712781537936 * q^23 - 207390009885372 * q^24 - 126056679642054 * q^26 - 1647129056757192 * q^27 + 4235167126399504 * q^28 + 2111332005818352 * q^29 + 5677451410844968 * q^31 + 17528957992420632 * q^32 + 850905890244432 * q^33 + 9944754702742068 * q^34 + 37245496240179504 * q^36 - 26538401323409648 * q^37 + 85729618982552526 * q^38 + 19381445486961624 * q^39 - 9719778050982432 * q^41 - 240213497278764114 * q^42 - 20888744714565080 * q^43 - 388904820117856128 * q^44 - 156886987299808452 * q^46 + 405141072730526160 * q^47 - 1095539303887417080 * q^48 + 1172899395436368720 * q^49 + 337654874814966864 * q^51 - 2264281219115894096 * q^52 + 1890950875516930176 * q^53 - 137123493975036234 * q^54 + 19720793170524660300 * q^56 - 4482982257257808504 * q^57 + 12636224332772065092 * q^58 - 3496737905134765776 * q^59 + 17508733151770015960 * q^61 - 30404101355896200606 * q^62 + 467349306165871272 * q^63 + 27809823573720908096 * q^64 + 25207872998382825228 * q^66 - 29911607071855004216 * q^67 - 29571773973515759808 * q^68 - 2935490037033582864 * q^69 + 52204030316256846816 * q^71 + 12246172693721331228 * q^72 - 39697032757192143344 * q^73 - 32301811215943606764 * q^74 + 383913379529874070384 * q^76 - 202348742072764877136 * q^77 + 7443520876183646646 * q^78 - 274125739348498990400 * q^79 + 97261323672455430408 * q^81 - 231746572148446490256 * q^82 - 276883976086286712912 * q^83 - 250082383646764311696 * q^84 + 486708932755841131242 * q^86 - 124672043611567867248 * q^87 - 2078122304879724561672 * q^88 - 319129684977517046784 * q^89 + 45068531423660000648 * q^91 + 147706203809955886272 * q^92 - 335247828358984515432 * q^93 - 1499781052151613424236 * q^94 - 1035067440494445898968 * q^96 + 451807368184588489864 * q^97 - 2604193962467412900204 * q^98 - 50245141913043465168 * 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 $

$2$	−2605.59	−1.79925	−0.899624	−	0.436666i	$-0.856159\pi$
$2$	−2605.59	−1.79925	−0.899624	+	0.436666i	$0.856159\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	4.69194e6	2.23729
$5$	0	0
$5$	0	0
$6$	1.53857e8	1.03880
$7$	−1.24132e9	−1.66093	−0.830467	−	0.557067i	$-0.811927\pi$
$7$	−1.24132e9	−1.66093	−0.830467	+	0.557067i	$0.811927\pi$
$8$	−6.76095e9	−2.22619
$9$	3.48678e9	0.333333
$10$	0	0
$11$	6.95232e10	0.808177	0.404089	−	0.914720i	$-0.367589\pi$
$11$	6.95232e10	0.808177	0.404089	+	0.914720i	$0.367589\pi$
$12$	−2.77054e11	−1.29170
$13$	2.02578e11	0.407555	0.203778	−	0.979017i	$-0.434678\pi$
$13$	2.02578e11	0.407555	0.203778	+	0.979017i	$0.434678\pi$
$14$	3.23436e12	2.98843
$15$	0	0
$16$	7.77654e12	1.76818
$17$	−7.32903e12	−0.881724	−0.440862	−	0.897575i	$-0.645327\pi$
$17$	−7.32903e12	−0.881724	−0.440862	+	0.897575i	$0.645327\pi$
$18$	−9.08513e12	−0.599749
$19$	2.46215e13	0.921301	0.460650	−	0.887582i	$-0.347616\pi$
$19$	2.46215e13	0.921301	0.460650	+	0.887582i	$0.347616\pi$
$20$	0	0
$21$	7.32984e13	0.958941
$22$	−1.81149e14	−1.45411
$23$	1.68791e14	0.849586	0.424793	−	0.905291i	$-0.360347\pi$
$23$	1.68791e14	0.849586	0.424793	+	0.905291i	$0.360347\pi$
$24$	3.99227e14	1.28529
$25$	0	0
$26$	−5.27834e14	−0.733293
$27$	−2.05891e14	−0.192450
$28$	−5.82418e15	−3.71600
$29$	−4.35028e15	−1.92016	−0.960082	−	0.279720i	$-0.909758\pi$
$29$	−4.35028e15	−1.92016	−0.960082	+	0.279720i	$0.909758\pi$
$30$	0	0
$31$	4.49432e15	0.984841	0.492421	−	0.870357i	$-0.336112\pi$
$31$	4.49432e15	0.984841	0.492421	+	0.870357i	$0.336112\pi$
$32$	−6.08373e15	−0.955202
$33$	−4.10528e15	−0.466601
$34$	1.90964e16	1.58644
$35$	0	0
$36$	1.63598e16	0.745764
$37$	−3.45204e16	−1.18021	−0.590103	−	0.807328i	$-0.700913\pi$
$37$	−3.45204e16	−1.18021	−0.590103	+	0.807328i	$0.700913\pi$
$38$	−6.41535e16	−1.65765
$39$	−1.19620e16	−0.235302
$40$	0	0
$41$	6.64078e16	0.772661	0.386330	−	0.922360i	$-0.373742\pi$
$41$	6.64078e16	0.772661	0.386330	+	0.922360i	$0.373742\pi$
$42$	−1.90986e17	−1.72537
$43$	1.34497e17	0.949062	0.474531	−	0.880239i	$-0.342618\pi$
$43$	1.34497e17	0.949062	0.474531	+	0.880239i	$0.342618\pi$
$44$	3.26199e17	1.80813
$45$	0	0
$46$	−4.39800e17	−1.52861
$47$	4.37113e17	1.21218	0.606088	−	0.795397i	$-0.292738\pi$
$47$	4.37113e17	1.21218	0.606088	+	0.795397i	$0.292738\pi$
$48$	−4.59197e17	−1.02086
$49$	9.82317e17	1.75870
$50$	0	0
$51$	4.32772e17	0.509064
$52$	9.50482e17	0.911820
$53$	1.75752e18	1.38040	0.690198	−	0.723621i	$-0.257524\pi$
$53$	1.75752e18	1.38040	0.690198	+	0.723621i	$0.257524\pi$
$54$	5.36468e17	0.346265
$55$	0	0
$56$	8.39247e18	3.69756
$57$	−1.45387e18	−0.531913
$58$	1.13350e19	3.45485
$59$	−5.05155e18	−1.28670	−0.643352	−	0.765571i	$-0.722457\pi$
$59$	−5.05155e18	−1.28670	−0.643352	+	0.765571i	$0.722457\pi$
$60$	0	0
$61$	7.50009e18	1.34618	0.673090	−	0.739561i	$-0.264967\pi$
$61$	7.50009e18	1.34618	0.673090	+	0.739561i	$0.264967\pi$
$62$	−1.17104e19	−1.77197
$63$	−4.32820e18	−0.553645
$64$	−4.56890e17	−0.0495362
$65$	0	0
$66$	1.06967e19	0.839531
$67$	−2.43348e19	−1.63096	−0.815479	−	0.578786i	$-0.803527\pi$
$67$	−2.43348e19	−1.63096	−0.815479	+	0.578786i	$0.803527\pi$
$68$	−3.43873e19	−1.97267
$69$	−9.96694e18	−0.490509
$70$	0	0
$71$	8.23122e18	0.300090	0.150045	−	0.988679i	$-0.452058\pi$
$71$	8.23122e18	0.300090	0.150045	+	0.988679i	$0.452058\pi$
$72$	−2.35740e19	−0.742064
$73$	5.21807e19	1.42108	0.710542	−	0.703655i	$-0.248450\pi$
$73$	5.21807e19	1.42108	0.710542	+	0.703655i	$0.248450\pi$
$74$	8.99459e19	2.12348
$75$	0	0
$76$	1.15523e20	2.06122
$77$	−8.63002e19	−1.34233
$78$	3.11681e19	0.423367
$79$	−1.28526e20	−1.52724	−0.763621	−	0.645665i	$-0.776580\pi$
$79$	−1.28526e20	−1.52724	−0.763621	+	0.645665i	$0.776580\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−1.73032e20	−1.39021
$83$	1.15724e20	0.818660	0.409330	−	0.912386i	$-0.365762\pi$
$83$	1.15724e20	0.818660	0.409330	+	0.912386i	$0.365762\pi$
$84$	3.43912e20	2.14543
$85$	0	0
$86$	−3.50444e20	−1.70760
$87$	2.56880e20	1.10861
$88$	−4.70043e20	−1.79916
$89$	1.72710e20	0.587115	0.293558	−	0.955941i	$-0.405161\pi$
$89$	1.72710e20	0.587115	0.293558	+	0.955941i	$0.405161\pi$
$90$	0	0
$91$	−2.51463e20	−0.676922
$92$	7.91958e20	1.90077
$93$	−2.65385e20	−0.568598
$94$	−1.13894e21	−2.18101
$95$	0	0
$96$	3.59238e20	0.551486
$97$	−9.66495e20	−1.33075	−0.665375	−	0.746509i	$-0.731728\pi$
$97$	−9.66495e20	−1.33075	−0.665375	+	0.746509i	$0.731728\pi$
$98$	−2.55951e21	−3.16434
$99$	2.42412e20	0.269392
$100$	0	0
$101$	−4.14795e20	−0.373645	−0.186822	−	0.982394i	$-0.559819\pi$
$101$	−4.14795e20	−0.373645	−0.186822	+	0.982394i	$0.559819\pi$
$102$	−1.12762e21	−0.915931
$103$	2.38519e20	0.174877	0.0874385	−	0.996170i	$-0.472132\pi$
$103$	2.38519e20	0.174877	0.0874385	+	0.996170i	$0.472132\pi$
$104$	−1.36962e21	−0.907296
$105$	0	0
$106$	−4.57937e21	−2.48367
$107$	1.90939e21	0.938352	0.469176	−	0.883105i	$-0.344551\pi$
$107$	1.90939e21	0.938352	0.469176	+	0.883105i	$0.344551\pi$
$108$	−9.66029e20	−0.430567
$109$	6.07725e20	0.245883	0.122942	−	0.992414i	$-0.460767\pi$
$109$	6.07725e20	0.245883	0.122942	+	0.992414i	$0.460767\pi$
$110$	0	0
$111$	2.03839e21	0.681392
$112$	−9.65314e21	−2.93683
$113$	−6.84045e21	−1.89566	−0.947831	−	0.318773i	$-0.896729\pi$
$113$	−6.84045e21	−1.89566	−0.947831	+	0.318773i	$0.896729\pi$
$114$	3.78820e21	0.957043
$115$	0	0
$116$	−2.04112e22	−4.29596
$117$	7.06345e20	0.135852
$118$	1.31623e22	2.31510
$119$	9.09763e21	1.46449
$120$	0	0
$121$	−2.56677e21	−0.346850
$122$	−1.95421e22	−2.42211
$123$	−3.92132e21	−0.446096
$124$	2.10871e22	2.20338
$125$	0	0
$126$	1.12775e22	0.996144
$127$	−1.99009e22	−1.61783	−0.808917	−	0.587922i	$-0.799946\pi$
$127$	−1.99009e22	−1.61783	−0.808917	+	0.587922i	$0.799946\pi$
$128$	1.39490e22	1.04433
$129$	−7.94192e21	−0.547941
$130$	0	0
$131$	1.86818e22	1.09665	0.548327	−	0.836264i	$-0.315265\pi$
$131$	1.86818e22	1.09665	0.548327	+	0.836264i	$0.315265\pi$
$132$	−1.92617e22	−1.04392
$133$	−3.05630e22	−1.53022
$134$	6.34066e22	2.93450
$135$	0	0
$136$	4.95512e22	1.96289
$137$	3.01207e21	0.110484	0.0552420	−	0.998473i	$-0.482407\pi$
$137$	3.01207e21	0.110484	0.0552420	+	0.998473i	$0.482407\pi$
$138$	2.59698e22	0.882546
$139$	1.76332e21	0.0555489	0.0277745	−	0.999614i	$-0.491158\pi$
$139$	1.76332e21	0.0555489	0.0277745	+	0.999614i	$0.491158\pi$
$140$	0	0
$141$	−2.58111e22	−0.699851
$142$	−2.14472e22	−0.539936
$143$	1.40838e22	0.329377
$144$	2.71151e22	0.589394
$145$	0	0
$146$	−1.35961e23	−2.55688
$147$	−5.80048e22	−1.01539
$148$	−1.61968e23	−2.64046
$149$	−7.46366e22	−1.13369	−0.566847	−	0.823823i	$-0.691837\pi$
$149$	−7.46366e22	−1.13369	−0.566847	+	0.823823i	$0.691837\pi$
$150$	0	0
$151$	−4.26848e22	−0.563657	−0.281829	−	0.959465i	$-0.590941\pi$
$151$	−4.26848e22	−0.563657	−0.281829	+	0.959465i	$0.590941\pi$
$152$	−1.66465e23	−2.05099
$153$	−2.55547e22	−0.293908
$154$	2.24863e23	2.41518
$155$	0	0
$156$	−5.61250e22	−0.526439
$157$	−5.91618e22	−0.518913	−0.259457	−	0.965755i	$-0.583543\pi$
$157$	−5.91618e22	−0.518913	−0.259457	+	0.965755i	$0.583543\pi$
$158$	3.34887e23	2.74789
$159$	−1.03780e23	−0.796972
$160$	0	0
$161$	−2.09523e23	−1.41111
$162$	−3.16779e22	−0.199916
$163$	−1.00577e23	−0.595017	−0.297508	−	0.954719i	$-0.596156\pi$
$163$	−1.00577e23	−0.595017	−0.297508	+	0.954719i	$0.596156\pi$
$164$	3.11582e23	1.72867
$165$	0	0
$166$	−3.01529e23	−1.47297
$167$	2.07849e23	0.953289	0.476645	−	0.879096i	$-0.341853\pi$
$167$	2.07849e23	0.953289	0.476645	+	0.879096i	$0.341853\pi$
$168$	−4.95567e23	−2.13479
$169$	−2.06027e23	−0.833899
$170$	0	0
$171$	8.58498e22	0.307100
$172$	6.31053e23	2.12333
$173$	−3.66342e23	−1.15985	−0.579927	−	0.814669i	$-0.696919\pi$
$173$	−3.66342e23	−1.15985	−0.579927	+	0.814669i	$0.696919\pi$
$174$	−6.69323e23	−1.99466
$175$	0	0
$176$	5.40650e23	1.42900
$177$	2.98289e23	0.742879
$178$	−4.50012e23	−1.05637
$179$	1.83321e23	0.405747	0.202874	−	0.979205i	$-0.434972\pi$
$179$	1.83321e23	0.405747	0.202874	+	0.979205i	$0.434972\pi$
$180$	0	0
$181$	1.25118e23	0.246431	0.123215	−	0.992380i	$-0.460679\pi$
$181$	1.25118e23	0.246431	0.123215	+	0.992380i	$0.460679\pi$
$182$	6.55208e23	1.21795
$183$	−4.42873e23	−0.777217
$184$	−1.14119e24	−1.89134
$185$	0	0
$186$	6.91485e23	1.02305
$187$	−5.09537e23	−0.712589
$188$	2.05091e24	2.71199
$189$	2.55576e23	0.319647
$190$	0	0
$191$	5.46709e23	0.612218	0.306109	−	0.951996i	$-0.400973\pi$
$191$	5.46709e23	0.612218	0.306109	+	0.951996i	$0.400973\pi$
$192$	2.69789e22	0.0285997
$193$	−8.98545e23	−0.901962	−0.450981	−	0.892534i	$-0.648926\pi$
$193$	−8.98545e23	−0.901962	−0.450981	+	0.892534i	$0.648926\pi$
$194$	2.51829e24	2.39435
$195$	0	0
$196$	4.60897e24	3.93473
$197$	1.82118e24	1.47386	0.736931	−	0.675968i	$-0.236274\pi$
$197$	1.82118e24	1.47386	0.736931	+	0.675968i	$0.236274\pi$
$198$	−6.31627e23	−0.484704
$199$	−8.12616e23	−0.591464	−0.295732	−	0.955271i	$-0.595564\pi$
$199$	−8.12616e23	−0.591464	−0.295732	+	0.955271i	$0.595564\pi$
$200$	0	0
$201$	1.43695e24	0.941634
$202$	1.08078e24	0.672280
$203$	5.40007e24	3.18927
$204$	2.03054e24	1.13892
$205$	0	0
$206$	−6.21483e23	−0.314647
$207$	5.88538e23	0.283195
$208$	1.57535e24	0.720631
$209$	1.71176e24	0.744574
$210$	0	0
$211$	−4.16409e24	−1.63891	−0.819454	−	0.573144i	$-0.805723\pi$
$211$	−4.16409e24	−1.63891	−0.819454	+	0.573144i	$0.805723\pi$
$212$	8.24617e24	3.08835
$213$	−4.86045e23	−0.173257
$214$	−4.97509e24	−1.68833
$215$	0	0
$216$	1.39202e24	0.428431
$217$	−5.57887e24	−1.63576
$218$	−1.58348e24	−0.442405
$219$	−3.08122e24	−0.820463
$220$	0	0
$221$	−1.48470e24	−0.359351
$222$	−5.31122e24	−1.22599
$223$	−1.98298e24	−0.436633	−0.218317	−	0.975878i	$-0.570057\pi$
$223$	−1.98298e24	−0.436633	−0.218317	+	0.975878i	$0.570057\pi$
$224$	7.55183e24	1.58653
$225$	0	0
$226$	1.78234e25	3.41077
$227$	3.11369e24	0.568858	0.284429	−	0.958697i	$-0.408196\pi$
$227$	3.11369e24	0.568858	0.284429	+	0.958697i	$0.408196\pi$
$228$	−6.82149e24	−1.19004
$229$	6.11135e24	1.01827	0.509137	−	0.860685i	$-0.329965\pi$
$229$	6.11135e24	1.01827	0.509137	+	0.860685i	$0.329965\pi$
$230$	0	0
$231$	5.09594e24	0.774994
$232$	2.94120e25	4.27465
$233$	2.97322e24	0.413037	0.206519	−	0.978443i	$-0.433787\pi$
$233$	2.97322e24	0.413037	0.206519	+	0.978443i	$0.433787\pi$
$234$	−1.84044e24	−0.244431
$235$	0	0
$236$	−2.37016e25	−2.87873
$237$	7.58935e24	0.881753
$238$	−2.37047e25	−2.63497
$239$	−1.14700e25	−1.22007	−0.610037	−	0.792373i	$-0.708845\pi$
$239$	−1.14700e25	−1.22007	−0.610037	+	0.792373i	$0.708845\pi$
$240$	0	0
$241$	−1.74978e25	−1.70531	−0.852655	−	0.522474i	$-0.825009\pi$
$241$	−1.74978e25	−1.70531	−0.852655	+	0.522474i	$0.825009\pi$
$242$	6.68796e24	0.624068
$243$	−7.17898e23	−0.0641500
$244$	3.51899e25	3.01180
$245$	0	0
$246$	1.02173e25	0.802637
$247$	4.98776e24	0.375481
$248$	−3.03859e25	−2.19245
$249$	−6.83339e24	−0.472654
$250$	0	0
$251$	−2.88875e24	−0.183711	−0.0918557	−	0.995772i	$-0.529280\pi$
$251$	−2.88875e24	−0.183711	−0.0918557	+	0.995772i	$0.529280\pi$
$252$	−2.03076e25	−1.23867
$253$	1.17349e25	0.686616
$254$	5.18536e25	2.91089
$255$	0	0
$256$	−3.53871e25	−1.82947
$257$	−1.99717e25	−0.991100	−0.495550	−	0.868579i	$-0.665033\pi$
$257$	−1.99717e25	−0.991100	−0.495550	+	0.868579i	$0.665033\pi$
$258$	2.06934e25	0.985881
$259$	4.28507e25	1.96024
$260$	0	0
$261$	−1.51685e25	−0.640054
$262$	−4.86770e25	−1.97315
$263$	−2.81379e23	−0.0109586	−0.00547931	−	0.999985i	$-0.501744\pi$
$263$	−2.81379e23	−0.0109586	−0.00547931	+	0.999985i	$0.501744\pi$
$264$	2.77556e25	1.03874
$265$	0	0
$266$	7.96346e25	2.75325
$267$	−1.01984e25	−0.338971
$268$	−1.14178e26	−3.64893
$269$	1.56428e25	0.480746	0.240373	−	0.970681i	$-0.422730\pi$
$269$	1.56428e25	0.480746	0.240373	+	0.970681i	$0.422730\pi$
$270$	0	0
$271$	4.66463e25	1.32629	0.663147	−	0.748489i	$-0.269220\pi$
$271$	4.66463e25	1.32629	0.663147	+	0.748489i	$0.269220\pi$
$272$	−5.69945e25	−1.55905
$273$	1.48486e25	0.390821
$274$	−7.84822e24	−0.198788
$275$	0	0
$276$	−4.67643e25	−1.09741
$277$	−4.63819e25	−1.04788	−0.523939	−	0.851756i	$-0.675538\pi$
$277$	−4.63819e25	−1.04788	−0.523939	+	0.851756i	$0.675538\pi$
$278$	−4.59449e24	−0.0999463
$279$	1.56707e25	0.328280
$280$	0	0
$281$	6.82634e25	1.32670	0.663348	−	0.748311i	$-0.269135\pi$
$281$	6.82634e25	1.32670	0.663348	+	0.748311i	$0.269135\pi$
$282$	6.72530e25	1.25920
$283$	−2.75714e25	−0.497395	−0.248697	−	0.968581i	$-0.580002\pi$
$283$	−2.75714e25	−0.497395	−0.248697	+	0.968581i	$0.580002\pi$
$284$	3.86204e25	0.671389
$285$	0	0
$286$	−3.66967e25	−0.592630
$287$	−8.24331e25	−1.28334
$288$	−2.12127e25	−0.318401
$289$	−1.53773e25	−0.222563
$290$	0	0
$291$	5.70706e25	0.768309
$292$	2.44829e26	3.17938
$293$	−8.09822e25	−1.01456	−0.507282	−	0.861780i	$-0.669350\pi$
$293$	−8.09822e25	−1.01456	−0.507282	+	0.861780i	$0.669350\pi$
$294$	1.51137e26	1.82694
$295$	0	0
$296$	2.33391e26	2.62737
$297$	−1.43142e25	−0.155534
$298$	1.94472e26	2.03980
$299$	3.41933e25	0.346253
$300$	0	0
$301$	−1.66953e26	−1.57633
$302$	1.11219e26	1.01416
$303$	2.44932e25	0.215724
$304$	1.91470e26	1.62903
$305$	0	0
$306$	6.65851e25	0.528813
$307$	9.02694e25	0.692768	0.346384	−	0.938093i	$-0.387409\pi$
$307$	9.02694e25	0.692768	0.346384	+	0.938093i	$0.387409\pi$
$308$	−4.04915e26	−3.00318
$309$	−1.40843e25	−0.100965
$310$	0	0
$311$	−5.41968e25	−0.363069	−0.181535	−	0.983385i	$-0.558106\pi$
$311$	−5.41968e25	−0.363069	−0.181535	+	0.983385i	$0.558106\pi$
$312$	8.08745e25	0.523828
$313$	2.33160e26	1.46029	0.730143	−	0.683294i	$-0.239453\pi$
$313$	2.33160e26	1.46029	0.730143	+	0.683294i	$0.239453\pi$
$314$	1.54151e26	0.933654
$315$	0	0
$316$	−6.03038e26	−3.41688
$317$	−4.78391e25	−0.262217	−0.131109	−	0.991368i	$-0.541854\pi$
$317$	−4.78391e25	−0.262217	−0.131109	+	0.991368i	$0.541854\pi$
$318$	2.70407e26	1.43395
$319$	−3.02445e26	−1.55183
$320$	0	0
$321$	−1.12748e26	−0.541757
$322$	5.45930e26	2.53893
$323$	−1.80451e26	−0.812333
$324$	5.70430e25	0.248588
$325$	0	0
$326$	2.62063e26	1.07058
$327$	−3.58855e25	−0.141961
$328$	−4.48980e26	−1.72009
$329$	−5.42595e26	−2.01335
$330$	0	0
$331$	−1.10032e26	−0.383112	−0.191556	−	0.981482i	$-0.561353\pi$
$331$	−1.10032e26	−0.383112	−0.191556	+	0.981482i	$0.561353\pi$
$332$	5.42970e26	1.83158
$333$	−1.20365e26	−0.393402
$334$	−5.41569e26	−1.71520
$335$	0	0
$336$	5.70008e26	1.69558
$337$	4.09738e26	1.18139	0.590693	−	0.806896i	$-0.298854\pi$
$337$	4.09738e26	1.18139	0.590693	+	0.806896i	$0.298854\pi$
$338$	5.36821e26	1.50039
$339$	4.03922e26	1.09446
$340$	0	0
$341$	3.12460e26	0.795926
$342$	−2.23689e26	−0.552549
$343$	−5.26034e26	−1.26016
$344$	−9.09329e26	−2.11279
$345$	0	0
$346$	9.54538e26	2.08686
$347$	−5.16092e25	−0.109463	−0.0547315	−	0.998501i	$-0.517430\pi$
$347$	−5.16092e25	−0.109463	−0.0547315	+	0.998501i	$0.517430\pi$
$348$	1.20526e27	2.48028
$349$	8.78050e26	1.75328	0.876642	−	0.481144i	$-0.159779\pi$
$349$	8.78050e26	1.75328	0.876642	+	0.481144i	$0.159779\pi$
$350$	0	0
$351$	−4.17089e25	−0.0784340
$352$	−4.22961e26	−0.771973
$353$	9.97990e26	1.76804	0.884019	−	0.467450i	$-0.154827\pi$
$353$	9.97990e26	1.76804	0.884019	+	0.467450i	$0.154827\pi$
$354$	−7.77218e26	−1.33662
$355$	0	0
$356$	8.10347e26	1.31355
$357$	−5.37206e26	−0.845521
$358$	−4.77659e26	−0.730040
$359$	5.62411e26	0.834761	0.417380	−	0.908732i	$-0.362948\pi$
$359$	5.62411e26	0.834761	0.417380	+	0.908732i	$0.362948\pi$
$360$	0	0
$361$	−1.07992e26	−0.151205
$362$	−3.26006e26	−0.443390
$363$	1.51565e26	0.200254
$364$	−1.17985e27	−1.51447
$365$	0	0
$366$	1.15394e27	1.39841
$367$	1.20110e27	1.41445	0.707223	−	0.706990i	$-0.249948\pi$
$367$	1.20110e27	1.41445	0.707223	+	0.706990i	$0.249948\pi$
$368$	1.31261e27	1.50222
$369$	2.31550e26	0.257554
$370$	0	0
$371$	−2.18163e27	−2.29275
$372$	−1.24517e27	−1.27212
$373$	2.05161e26	0.203776	0.101888	−	0.994796i	$-0.467512\pi$
$373$	2.05161e26	0.203776	0.101888	+	0.994796i	$0.467512\pi$
$374$	1.32764e27	1.28212
$375$	0	0
$376$	−2.95530e27	−2.69854
$377$	−8.81269e26	−0.782572
$378$	−6.65925e26	−0.575124
$379$	4.88468e26	0.410321	0.205161	−	0.978728i	$-0.434228\pi$
$379$	4.88468e26	0.410321	0.205161	+	0.978728i	$0.434228\pi$
$380$	0	0
$381$	1.17513e27	0.934057
$382$	−1.42450e27	−1.10153
$383$	2.62713e26	0.197649	0.0988245	−	0.995105i	$-0.468492\pi$
$383$	2.62713e26	0.197649	0.0988245	+	0.995105i	$0.468492\pi$
$384$	−8.23673e26	−0.602944
$385$	0	0
$386$	2.34124e27	1.62285
$387$	4.68963e26	0.316354
$388$	−4.53474e27	−2.97727
$389$	1.55416e27	0.993171	0.496586	−	0.867988i	$-0.334587\pi$
$389$	1.55416e27	0.993171	0.496586	+	0.867988i	$0.334587\pi$
$390$	0	0
$391$	−1.23707e27	−0.749100
$392$	−6.64140e27	−3.91522
$393$	−1.10314e27	−0.633154
$394$	−4.74524e27	−2.65184
$395$	0	0
$396$	1.13738e27	0.602709
$397$	1.47347e27	0.760400	0.380200	−	0.924904i	$-0.375855\pi$
$397$	1.47347e27	0.760400	0.380200	+	0.924904i	$0.375855\pi$
$398$	2.11734e27	1.06419
$399$	1.80472e27	0.883473
$400$	0	0
$401$	−2.04288e27	−0.948914	−0.474457	−	0.880279i	$-0.657356\pi$
$401$	−2.04288e27	−0.948914	−0.474457	+	0.880279i	$0.657356\pi$
$402$	−3.74409e27	−1.69423
$403$	9.10449e26	0.401377
$404$	−1.94619e27	−0.835952
$405$	0	0
$406$	−1.40704e28	−5.73828
$407$	−2.39997e27	−0.953815
$408$	−2.92595e27	−1.13327
$409$	−3.56939e27	−1.34741	−0.673705	−	0.739000i	$-0.735298\pi$
$409$	−3.56939e27	−1.34741	−0.673705	+	0.739000i	$0.735298\pi$
$410$	0	0
$411$	−1.77860e26	−0.0637880
$412$	1.11912e27	0.391251
$413$	6.27056e27	2.13713
$414$	−1.53349e27	−0.509538
$415$	0	0
$416$	−1.23243e27	−0.389298
$417$	−1.04122e26	−0.0320712
$418$	−4.46015e27	−1.33967
$419$	2.84960e27	0.834711	0.417356	−	0.908743i	$-0.362957\pi$
$419$	2.84960e27	0.834711	0.417356	+	0.908743i	$0.362957\pi$
$420$	0	0
$421$	−4.28342e27	−1.19352	−0.596759	−	0.802421i	$-0.703545\pi$
$421$	−4.28342e27	−1.19352	−0.596759	+	0.802421i	$0.703545\pi$
$422$	1.08499e28	2.94880
$423$	1.52412e27	0.404059
$424$	−1.18825e28	−3.07303
$425$	0	0
$426$	1.26643e27	0.311732
$427$	−9.30997e27	−2.23592
$428$	8.95876e27	2.09937
$429$	−8.31637e26	−0.190166
$430$	0	0
$431$	1.26102e27	0.274606	0.137303	−	0.990529i	$-0.456157\pi$
$431$	1.26102e27	0.274606	0.137303	+	0.990529i	$0.456157\pi$
$432$	−1.60112e27	−0.340287
$433$	−2.23591e27	−0.463801	−0.231901	−	0.972739i	$-0.574494\pi$
$433$	−2.23591e27	−0.463801	−0.231901	+	0.972739i	$0.574494\pi$
$434$	1.45362e28	2.94313
$435$	0	0
$436$	2.85141e27	0.550112
$437$	4.15589e27	0.782724
$438$	8.02838e27	1.47622
$439$	5.77197e27	1.03621	0.518104	−	0.855318i	$-0.326638\pi$
$439$	5.77197e27	1.03621	0.518104	+	0.855318i	$0.326638\pi$
$440$	0	0
$441$	3.42513e27	0.586235
$442$	3.86851e27	0.646561
$443$	5.18891e27	0.846909	0.423455	−	0.905917i	$-0.360817\pi$
$443$	5.18891e27	0.846909	0.423455	+	0.905917i	$0.360817\pi$
$444$	9.56403e27	1.52447
$445$	0	0
$446$	5.16682e27	0.785611
$447$	4.40721e27	0.654539
$448$	5.67145e26	0.0822763
$449$	−9.43896e27	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$449$	−9.43896e27	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$450$	0	0
$451$	4.61689e27	0.624447
$452$	−3.20950e28	−4.24115
$453$	2.52049e27	0.325428
$454$	−8.11299e27	−1.02352
$455$	0	0
$456$	9.82957e27	1.18414
$457$	6.36474e27	0.749309	0.374654	−	0.927165i	$-0.377761\pi$
$457$	6.36474e27	0.749309	0.374654	+	0.927165i	$0.377761\pi$
$458$	−1.59237e28	−1.83213
$459$	1.50898e27	0.169688
$460$	0	0
$461$	5.92334e27	0.636366	0.318183	−	0.948029i	$-0.396927\pi$
$461$	5.92334e27	0.636366	0.318183	+	0.948029i	$0.396927\pi$
$462$	−1.32779e28	−1.39441
$463$	−1.45586e28	−1.49458	−0.747291	−	0.664497i	$-0.768646\pi$
$463$	−1.45586e28	−1.49458	−0.747291	+	0.664497i	$0.768646\pi$
$464$	−3.38301e28	−3.39520
$465$	0	0
$466$	−7.74698e27	−0.743156
$467$	3.43275e27	0.321970	0.160985	−	0.986957i	$-0.448533\pi$
$467$	3.43275e27	0.321970	0.160985	+	0.986957i	$0.448533\pi$
$468$	3.31413e27	0.303940
$469$	3.02072e28	2.70892
$470$	0	0
$471$	3.49344e27	0.299595
$472$	3.41533e28	2.86445
$473$	9.35068e27	0.767010
$474$	−1.97747e28	−1.58649
$475$	0	0
$476$	4.26855e28	3.27648
$477$	6.12809e27	0.460132
$478$	2.98861e28	2.19521
$479$	−1.84344e27	−0.132466	−0.0662331	−	0.997804i	$-0.521098\pi$
$479$	−1.84344e27	−0.132466	−0.0662331	+	0.997804i	$0.521098\pi$
$480$	0	0
$481$	−6.99306e27	−0.480999
$482$	4.55920e28	3.06828
$483$	1.23721e28	0.814703
$484$	−1.20431e28	−0.776004
$485$	0	0
$486$	1.87055e27	0.115422
$487$	2.26903e28	1.37021	0.685103	−	0.728446i	$-0.259757\pi$
$487$	2.26903e28	1.37021	0.685103	+	0.728446i	$0.259757\pi$
$488$	−5.07077e28	−2.99686
$489$	5.93898e27	0.343533
$490$	0	0
$491$	−8.90333e27	−0.493398	−0.246699	−	0.969092i	$-0.579346\pi$
$491$	−8.90333e27	−0.493398	−0.246699	+	0.969092i	$0.579346\pi$
$492$	−1.83986e28	−0.998047
$493$	3.18833e28	1.69305
$494$	−1.29961e28	−0.675583
$495$	0	0
$496$	3.49503e28	1.74138
$497$	−1.02175e28	−0.498430
$498$	1.78050e28	0.850421
$499$	−3.12044e27	−0.145935	−0.0729675	−	0.997334i	$-0.523247\pi$
$499$	−3.12044e27	−0.145935	−0.0729675	+	0.997334i	$0.523247\pi$
$500$	0	0
$501$	−1.22733e28	−0.550382
$502$	7.52690e27	0.330542
$503$	2.96443e28	1.27490	0.637452	−	0.770490i	$-0.279988\pi$
$503$	2.96443e28	1.27490	0.637452	+	0.770490i	$0.279988\pi$
$504$	2.92627e28	1.23252
$505$	0	0
$506$	−3.05763e28	−1.23539
$507$	1.21657e28	0.481452
$508$	−9.33739e28	−3.61957
$509$	−4.68441e28	−1.77876	−0.889382	−	0.457164i	$-0.848865\pi$
$509$	−4.68441e28	−1.77876	−0.889382	+	0.457164i	$0.848865\pi$
$510$	0	0
$511$	−6.47727e28	−2.36033
$512$	6.29512e28	2.24734
$513$	−5.06934e27	−0.177304
$514$	5.20381e28	1.78323
$515$	0	0
$516$	−3.72630e28	−1.22590
$517$	3.03895e28	0.979654
$518$	−1.11651e29	−3.52696
$519$	2.16322e28	0.669642
$520$	0	0
$521$	−1.67929e27	−0.0499262	−0.0249631	−	0.999688i	$-0.507947\pi$
$521$	−1.67929e27	−0.0499262	−0.0249631	+	0.999688i	$0.507947\pi$
$522$	3.95228e28	1.15162
$523$	2.57265e28	0.734705	0.367353	−	0.930082i	$-0.380264\pi$
$523$	2.57265e28	0.734705	0.367353	+	0.930082i	$0.380264\pi$
$524$	8.76538e28	2.45353
$525$	0	0
$526$	7.33158e26	0.0197173
$527$	−3.29390e28	−0.868358
$528$	−3.19249e28	−0.825036
$529$	−1.09812e28	−0.278204
$530$	0	0
$531$	−1.76137e28	−0.428901
$532$	−1.43400e29	−3.42355
$533$	1.34527e28	0.314902
$534$	2.65728e28	0.609893
$535$	0	0
$536$	1.64527e29	3.63083
$537$	−1.08249e28	−0.234258
$538$	−4.07587e28	−0.864981
$539$	6.82938e28	1.42134
$540$	0	0
$541$	6.95896e28	1.39307	0.696535	−	0.717523i	$-0.254724\pi$
$541$	6.95896e28	1.39307	0.696535	+	0.717523i	$0.254724\pi$
$542$	−1.21541e29	−2.38633
$543$	−7.38809e27	−0.142277
$544$	4.45878e28	0.842224
$545$	0	0
$546$	−3.86894e28	−0.703184
$547$	−9.44414e27	−0.168382	−0.0841910	−	0.996450i	$-0.526831\pi$
$547$	−9.44414e27	−0.168382	−0.0841910	+	0.996450i	$0.526831\pi$
$548$	1.41325e28	0.247185
$549$	2.61512e28	0.448726
$550$	0	0
$551$	−1.07110e29	−1.76905
$552$	6.73860e28	1.09197
$553$	1.59542e29	2.53665
$554$	1.20852e29	1.88539
$555$	0	0
$556$	8.27339e27	0.124279
$557$	1.10162e29	1.62388	0.811939	−	0.583743i	$-0.198412\pi$
$557$	1.10162e29	1.62388	0.811939	+	0.583743i	$0.198412\pi$
$558$	−4.08315e28	−0.590658
$559$	2.72461e28	0.386795
$560$	0	0
$561$	3.00877e28	0.411414
$562$	−1.77866e29	−2.38705
$563$	4.71492e28	0.621064	0.310532	−	0.950563i	$-0.399493\pi$
$563$	4.71492e28	0.621064	0.310532	+	0.950563i	$0.399493\pi$
$564$	−1.21104e29	−1.56577
$565$	0	0
$566$	7.18397e28	0.894937
$567$	−1.50915e28	−0.184548
$568$	−5.56509e28	−0.668058
$569$	9.07959e28	1.07001	0.535005	−	0.844849i	$-0.320310\pi$
$569$	9.07959e28	1.07001	0.535005	+	0.844849i	$0.320310\pi$
$570$	0	0
$571$	−5.62345e28	−0.638740	−0.319370	−	0.947630i	$-0.603471\pi$
$571$	−5.62345e28	−0.638740	−0.319370	+	0.947630i	$0.603471\pi$
$572$	6.60806e28	0.736912
$573$	−3.22826e28	−0.353464
$574$	2.14787e29	2.30904
$575$	0	0
$576$	−1.59308e27	−0.0165121
$577$	1.22469e29	1.24646	0.623232	−	0.782037i	$-0.285819\pi$
$577$	1.22469e29	1.24646	0.623232	+	0.782037i	$0.285819\pi$
$578$	4.00669e28	0.400446
$579$	5.30582e28	0.520748
$580$	0	0
$581$	−1.43650e29	−1.35974
$582$	−1.48702e29	−1.38238
$583$	1.22188e29	1.11560
$584$	−3.52791e29	−3.16361
$585$	0	0
$586$	2.11006e29	1.82545
$587$	−2.66134e28	−0.226152	−0.113076	−	0.993586i	$-0.536070\pi$
$587$	−2.66134e28	−0.226152	−0.113076	+	0.993586i	$0.536070\pi$
$588$	−2.72155e29	−2.27172
$589$	1.10657e29	0.907335
$590$	0	0
$591$	−1.07539e29	−0.850934
$592$	−2.68449e29	−2.08682
$593$	1.86093e29	1.42120	0.710600	−	0.703597i	$-0.248424\pi$
$593$	1.86093e29	1.42120	0.710600	+	0.703597i	$0.248424\pi$
$594$	3.72969e28	0.279844
$595$	0	0
$596$	−3.50190e29	−2.53640
$597$	4.79842e28	0.341482
$598$	−8.90937e28	−0.622995
$599$	1.93452e28	0.132920	0.0664602	−	0.997789i	$-0.478829\pi$
$599$	1.93452e28	0.132920	0.0664602	+	0.997789i	$0.478829\pi$
$600$	0	0
$601$	2.71373e29	1.80046	0.900231	−	0.435412i	$-0.143397\pi$
$601$	2.71373e29	1.80046	0.900231	+	0.435412i	$0.143397\pi$
$602$	4.35012e29	2.83621
$603$	−8.48503e28	−0.543653
$604$	−2.00274e29	−1.26107
$605$	0	0
$606$	−6.38193e28	−0.388141
$607$	−1.50527e29	−0.899772	−0.449886	−	0.893086i	$-0.648535\pi$
$607$	−1.50527e29	−0.899772	−0.449886	+	0.893086i	$0.648535\pi$
$608$	−1.49790e29	−0.880028
$609$	−3.18869e29	−1.84132
$610$	0	0
$611$	8.85493e28	0.494029
$612$	−1.19901e29	−0.657558
$613$	−2.75303e29	−1.48414	−0.742072	−	0.670320i	$-0.766157\pi$
$613$	−2.75303e29	−1.48414	−0.742072	+	0.670320i	$0.766157\pi$
$614$	−2.35205e29	−1.24646
$615$	0	0
$616$	5.83471e29	2.98829
$617$	2.18187e29	1.09859	0.549294	−	0.835629i	$-0.314897\pi$
$617$	2.18187e29	1.09859	0.549294	+	0.835629i	$0.314897\pi$
$618$	3.66979e28	0.181661
$619$	−6.66145e28	−0.324203	−0.162102	−	0.986774i	$-0.551827\pi$
$619$	−6.66145e28	−0.324203	−0.162102	+	0.986774i	$0.551827\pi$
$620$	0	0
$621$	−3.47526e28	−0.163503
$622$	1.41214e29	0.653251
$623$	−2.14388e29	−0.975160
$624$	−9.30231e28	−0.416057
$625$	0	0
$626$	−6.07518e29	−2.62742
$627$	−1.01078e29	−0.429880
$628$	−2.77584e29	−1.16096
$629$	2.53001e29	1.04062
$630$	0	0
$631$	−1.69347e29	−0.673703	−0.336852	−	0.941558i	$-0.609362\pi$
$631$	−1.69347e29	−0.673703	−0.336852	+	0.941558i	$0.609362\pi$
$632$	8.68960e29	3.39993
$633$	2.45886e29	0.946224
$634$	1.24649e29	0.471793
$635$	0	0
$636$	−4.86928e29	−1.78306
$637$	1.98995e29	0.716769
$638$	7.88048e29	2.79213
$639$	2.87005e28	0.100030
$640$	0	0
$641$	−6.07884e28	−0.205027	−0.102514	−	0.994732i	$-0.532688\pi$
$641$	−6.07884e28	−0.205027	−0.102514	+	0.994732i	$0.532688\pi$
$642$	2.93774e29	0.974756
$643$	5.51584e29	1.80051	0.900256	−	0.435360i	$-0.143379\pi$
$643$	5.51584e29	1.80051	0.900256	+	0.435360i	$0.143379\pi$
$644$	−9.83069e29	−3.15706
$645$	0	0
$646$	4.70182e29	1.46159
$647$	−4.87598e29	−1.49131	−0.745653	−	0.666334i	$-0.767863\pi$
$647$	−4.87598e29	−1.49131	−0.745653	+	0.666334i	$0.767863\pi$
$648$	−8.21974e28	−0.247355
$649$	−3.51200e29	−1.03988
$650$	0	0
$651$	3.29427e29	0.944405
$652$	−4.71902e29	−1.33123
$653$	2.06458e29	0.573118	0.286559	−	0.958063i	$-0.407488\pi$
$653$	2.06458e29	0.573118	0.286559	+	0.958063i	$0.407488\pi$
$654$	9.35030e28	0.255422
$655$	0	0
$656$	5.16423e29	1.36620
$657$	1.81943e29	0.473695
$658$	1.41378e30	3.62251
$659$	−1.48007e29	−0.373238	−0.186619	−	0.982432i	$-0.559753\pi$
$659$	−1.48007e29	−0.373238	−0.186619	+	0.982432i	$0.559753\pi$
$660$	0	0
$661$	3.33446e29	0.814536	0.407268	−	0.913309i	$-0.366481\pi$
$661$	3.33446e29	0.814536	0.407268	+	0.913309i	$0.366481\pi$
$662$	2.86698e29	0.689313
$663$	8.76699e28	0.207471
$664$	−7.82404e29	−1.82250
$665$	0	0
$666$	3.13622e29	0.707827
$667$	−7.34288e29	−1.63134
$668$	9.75215e29	2.13279
$669$	1.17093e29	0.252090
$670$	0	0
$671$	5.21430e29	1.08795
$672$	−4.45928e29	−0.915983
$673$	2.34439e29	0.474101	0.237051	−	0.971497i	$-0.423819\pi$
$673$	2.34439e29	0.474101	0.237051	+	0.971497i	$0.423819\pi$
$674$	−1.06761e30	−2.12561
$675$	0	0
$676$	−9.66665e29	−1.86567
$677$	6.72580e29	1.27809	0.639047	−	0.769168i	$-0.279329\pi$
$677$	6.72580e29	1.27809	0.639047	+	0.769168i	$0.279329\pi$
$678$	−1.05245e30	−1.96921
$679$	1.19972e30	2.21029
$680$	0	0
$681$	−1.83860e29	−0.328430
$682$	−8.14141e29	−1.43207
$683$	−8.49477e29	−1.47141	−0.735706	−	0.677302i	$-0.763149\pi$
$683$	−8.49477e29	−1.47141	−0.735706	+	0.677302i	$0.763149\pi$
$684$	4.02802e29	0.687073
$685$	0	0
$686$	1.37063e30	2.26734
$687$	−3.60869e29	−0.587901
$688$	1.04592e30	1.67811
$689$	3.56034e29	0.562587
$690$	0	0
$691$	−5.41967e29	−0.830718	−0.415359	−	0.909658i	$-0.636344\pi$
$691$	−5.41967e29	−0.830718	−0.415359	+	0.909658i	$0.636344\pi$
$692$	−1.71886e30	−2.59493
$693$	−3.00910e29	−0.447443
$694$	1.34472e29	0.196951
$695$	0	0
$696$	−1.73675e30	−2.46797
$697$	−4.86705e29	−0.681273
$698$	−2.28784e30	−3.15459
$699$	−1.75565e29	−0.238467
$700$	0	0
$701$	6.70904e29	0.884344	0.442172	−	0.896930i	$-0.354208\pi$
$701$	6.70904e29	0.884344	0.442172	+	0.896930i	$0.354208\pi$
$702$	1.08676e29	0.141122
$703$	−8.49943e29	−1.08732
$704$	−3.17645e28	−0.0400340
$705$	0	0
$706$	−2.60035e30	−3.18114
$707$	5.14891e29	0.620600
$708$	1.39955e30	1.66204
$709$	−3.20580e29	−0.375103	−0.187551	−	0.982255i	$-0.560055\pi$
$709$	−3.20580e29	−0.375103	−0.187551	+	0.982255i	$0.560055\pi$
$710$	0	0
$711$	−4.48144e29	−0.509080
$712$	−1.16769e30	−1.30703
$713$	7.58601e29	0.836707
$714$	1.39974e30	1.52130
$715$	0	0
$716$	8.60131e29	0.907775
$717$	6.77293e29	0.704410
$718$	−1.46541e30	−1.50194
$719$	8.76340e29	0.885155	0.442578	−	0.896730i	$-0.354064\pi$
$719$	8.76340e29	0.885155	0.442578	+	0.896730i	$0.354064\pi$
$720$	0	0
$721$	−2.96077e29	−0.290459
$722$	2.81383e29	0.272055
$723$	1.03323e30	0.984562
$724$	5.87046e29	0.551337
$725$	0	0
$726$	−3.94917e29	−0.360306
$727$	9.36998e28	0.0842612	0.0421306	−	0.999112i	$-0.486585\pi$
$727$	9.36998e28	0.0842612	0.0421306	+	0.999112i	$0.486585\pi$
$728$	1.70013e30	1.50696
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−9.85733e29	−0.836810
$732$	−2.07793e30	−1.73886
$733$	−1.06565e30	−0.879070	−0.439535	−	0.898226i	$-0.644857\pi$
$733$	−1.06565e30	−0.879070	−0.439535	+	0.898226i	$0.644857\pi$
$734$	−3.12958e30	−2.54494
$735$	0	0
$736$	−1.02688e30	−0.811526
$737$	−1.69184e30	−1.31810
$738$	−6.03324e29	−0.463403
$739$	−3.80508e29	−0.288136	−0.144068	−	0.989568i	$-0.546018\pi$
$739$	−3.80508e29	−0.288136	−0.144068	+	0.989568i	$0.546018\pi$
$740$	0	0
$741$	−2.94522e29	−0.216784
$742$	5.68444e30	4.12522
$743$	−1.93395e30	−1.38377	−0.691885	−	0.722008i	$-0.743219\pi$
$743$	−1.93395e30	−1.38377	−0.691885	+	0.722008i	$0.743219\pi$
$744$	1.79426e30	1.26581
$745$	0	0
$746$	−5.34566e29	−0.366643
$747$	4.03505e29	0.272887
$748$	−2.39072e30	−1.59427
$749$	−2.37016e30	−1.55854
$750$	0	0
$751$	−8.10437e29	−0.518203	−0.259101	−	0.965850i	$-0.583426\pi$
$751$	−8.10437e29	−0.518203	−0.259101	+	0.965850i	$0.583426\pi$
$752$	3.39923e30	2.14335
$753$	1.70578e29	0.106066
$754$	2.29623e30	1.40804
$755$	0	0
$756$	1.19915e30	0.715144
$757$	−2.01389e30	−1.18448	−0.592241	−	0.805761i	$-0.701757\pi$
$757$	−2.01389e30	−1.18448	−0.592241	+	0.805761i	$0.701757\pi$
$758$	−1.27275e30	−0.738270
$759$	−6.92934e29	−0.396418
$760$	0	0
$761$	−7.97241e29	−0.443661	−0.221830	−	0.975085i	$-0.571203\pi$
$761$	−7.97241e29	−0.443661	−0.221830	+	0.975085i	$0.571203\pi$
$762$	−3.06190e30	−1.68060
$763$	−7.54378e29	−0.408396
$764$	2.56513e30	1.36971
$765$	0	0
$766$	−6.84522e29	−0.355620
$767$	−1.02333e30	−0.524403
$768$	2.08957e30	1.05625
$769$	−1.60279e30	−0.799191	−0.399596	−	0.916692i	$-0.630849\pi$
$769$	−1.60279e30	−0.799191	−0.399596	+	0.916692i	$0.630849\pi$
$770$	0	0
$771$	1.17931e30	0.572212
$772$	−4.21592e30	−2.01795
$773$	−1.80800e30	−0.853716	−0.426858	−	0.904319i	$-0.640380\pi$
$773$	−1.80800e30	−0.853716	−0.426858	+	0.904319i	$0.640380\pi$
$774$	−1.22192e30	−0.569199
$775$	0	0
$776$	6.53442e30	2.96251
$777$	−2.53029e30	−1.13175
$778$	−4.04949e30	−1.78696
$779$	1.63506e30	0.711853
$780$	0	0
$781$	5.72261e29	0.242526
$782$	3.22331e30	1.34782
$783$	8.95684e29	0.369536
$784$	7.63903e30	3.10971
$785$	0	0
$786$	2.87433e30	1.13920
$787$	2.20871e30	0.863783	0.431892	−	0.901926i	$-0.357846\pi$
$787$	2.20871e30	0.863783	0.431892	+	0.901926i	$0.357846\pi$
$788$	8.54485e30	3.29746
$789$	1.66151e28	0.00632697
$790$	0	0
$791$	8.49115e30	3.14857
$792$	−1.63894e30	−0.599720
$793$	1.51935e30	0.548642
$794$	−3.83927e30	−1.36815
$795$	0	0
$796$	−3.81275e30	−1.32328
$797$	−1.05607e30	−0.361725	−0.180863	−	0.983508i	$-0.557889\pi$
$797$	−1.05607e30	−0.361725	−0.180863	+	0.983508i	$0.557889\pi$
$798$	−4.70235e30	−1.58959
$799$	−3.20361e30	−1.06881
$800$	0	0
$801$	6.02204e29	0.195705
$802$	5.32291e30	1.70733
$803$	3.62777e30	1.14849
$804$	6.74207e30	2.10671
$805$	0	0
$806$	−2.37226e30	−0.722177
$807$	−9.23692e29	−0.277559
$808$	2.80441e30	0.831806
$809$	3.73525e30	1.09360	0.546802	−	0.837262i	$-0.315845\pi$
$809$	3.73525e30	1.09360	0.546802	+	0.837262i	$0.315845\pi$
$810$	0	0
$811$	6.00447e30	1.71299	0.856497	−	0.516153i	$-0.172636\pi$
$811$	6.00447e30	1.71299	0.856497	+	0.516153i	$0.172636\pi$
$812$	2.53368e31	7.13532
$813$	−2.75442e30	−0.765736
$814$	6.25333e30	1.71615
$815$	0	0
$816$	3.36547e30	0.900116
$817$	3.31152e30	0.874371
$818$	9.30037e30	2.42432
$819$	−8.76796e29	−0.225641
$820$	0	0
$821$	1.49122e28	0.00374058	0.00187029	−	0.999998i	$-0.499405\pi$
$821$	1.49122e28	0.00374058	0.00187029	+	0.999998i	$0.499405\pi$
$822$	4.63429e29	0.114770
$823$	−5.03035e30	−1.22999	−0.614993	−	0.788533i	$-0.710841\pi$
$823$	−5.03035e30	−1.22999	−0.614993	+	0.788533i	$0.710841\pi$
$824$	−1.61262e30	−0.389310
$825$	0	0
$826$	−1.63385e31	−3.84523
$827$	−5.60022e30	−1.30136	−0.650679	−	0.759353i	$-0.725516\pi$
$827$	−5.60022e30	−1.30136	−0.650679	+	0.759353i	$0.725516\pi$
$828$	2.76139e30	0.633590
$829$	−4.33088e30	−0.981192	−0.490596	−	0.871387i	$-0.663221\pi$
$829$	−4.33088e30	−0.981192	−0.490596	+	0.871387i	$0.663221\pi$
$830$	0	0
$831$	2.73880e30	0.604993
$832$	−9.25558e28	−0.0201887
$833$	−7.19943e30	−1.55069
$834$	2.71300e29	0.0577040
$835$	0	0
$836$	8.03150e30	1.66583
$837$	−9.25341e29	−0.189533
$838$	−7.42488e30	−1.50185
$839$	−1.83104e29	−0.0365761	−0.0182880	−	0.999833i	$-0.505822\pi$
$839$	−1.83104e29	−0.0365761	−0.0182880	+	0.999833i	$0.505822\pi$
$840$	0	0
$841$	1.37921e31	2.68703
$842$	1.11608e31	2.14743
$843$	−4.03088e30	−0.765968
$844$	−1.95377e31	−3.66672
$845$	0	0
$846$	−3.97122e30	−0.727002
$847$	3.18618e30	0.576095
$848$	1.36674e31	2.44079
$849$	1.62806e30	0.287171
$850$	0	0
$851$	−5.82673e30	−1.00269
$852$	−2.28050e30	−0.387626
$853$	1.14361e30	0.192004	0.0960022	−	0.995381i	$-0.469394\pi$
$853$	1.14361e30	0.192004	0.0960022	+	0.995381i	$0.469394\pi$
$854$	2.42579e31	4.02297
$855$	0	0
$856$	−1.29093e31	−2.08895
$857$	−1.60507e30	−0.256564	−0.128282	−	0.991738i	$-0.540946\pi$
$857$	−1.60507e30	−0.256564	−0.128282	+	0.991738i	$0.540946\pi$
$858$	2.16690e30	0.342155
$859$	−9.89432e30	−1.54333	−0.771663	−	0.636031i	$-0.780575\pi$
$859$	−9.89432e30	−1.54333	−0.771663	+	0.636031i	$0.780575\pi$
$860$	0	0
$861$	4.86759e30	0.740936
$862$	−3.28569e30	−0.494083
$863$	5.54727e30	0.824073	0.412036	−	0.911167i	$-0.364818\pi$
$863$	5.54727e30	0.824073	0.412036	+	0.911167i	$0.364818\pi$
$864$	1.25259e30	0.183829
$865$	0	0
$866$	5.82587e30	0.834493
$867$	9.08015e29	0.128497
$868$	−2.61757e31	−3.65967
$869$	−8.93556e30	−1.23428
$870$	0	0
$871$	−4.92969e30	−0.664705
$872$	−4.10880e30	−0.547383
$873$	−3.36996e30	−0.443583
$874$	−1.08285e31	−1.40831
$875$	0	0
$876$	−1.44569e31	−1.83562
$877$	−7.95169e30	−0.997617	−0.498809	−	0.866712i	$-0.666229\pi$
$877$	−7.95169e30	−0.997617	−0.498809	+	0.866712i	$0.666229\pi$
$878$	−1.50394e31	−1.86439
$879$	4.78192e30	0.585759
$880$	0	0
$881$	6.82498e30	0.816308	0.408154	−	0.912913i	$-0.366173\pi$
$881$	6.82498e30	0.816308	0.408154	+	0.912913i	$0.366173\pi$
$882$	−8.92447e30	−1.05478
$883$	−7.69636e30	−0.898873	−0.449436	−	0.893312i	$-0.648375\pi$
$883$	−7.69636e30	−0.898873	−0.449436	+	0.893312i	$0.648375\pi$
$884$	−6.96611e30	−0.803973
$885$	0	0
$886$	−1.35202e31	−1.52380
$887$	2.64624e30	0.294735	0.147367	−	0.989082i	$-0.452920\pi$
$887$	2.64624e30	0.294735	0.147367	+	0.989082i	$0.452920\pi$
$888$	−1.37815e31	−1.51691
$889$	2.47033e31	2.68712
$890$	0	0
$891$	8.45240e29	0.0897975
$892$	−9.30401e30	−0.976875
$893$	1.07624e31	1.11678
$894$	−1.14834e31	−1.17768
$895$	0	0
$896$	−1.73151e31	−1.73456
$897$	−2.01908e30	−0.199909
$898$	2.45940e31	2.40674
$899$	−1.95516e31	−1.89106
$900$	0	0
$901$	−1.28809e31	−1.21713
$902$	−1.20297e31	−1.12353
$903$	9.85843e30	0.910094
$904$	4.62479e31	4.22011
$905$	0	0
$906$	−6.56737e30	−0.585525
$907$	−3.13450e30	−0.276244	−0.138122	−	0.990415i	$-0.544107\pi$
$907$	−3.13450e30	−0.276244	−0.138122	+	0.990415i	$0.544107\pi$
$908$	1.46092e31	1.27270
$909$	−1.44630e30	−0.124548
$910$	0	0
$911$	1.31558e31	1.10707	0.553534	−	0.832827i	$-0.313279\pi$
$911$	1.31558e31	1.10707	0.553534	+	0.832827i	$0.313279\pi$
$912$	−1.13061e31	−0.940519
$913$	8.04551e30	0.661622
$914$	−1.65839e31	−1.34819
$915$	0	0
$916$	2.86741e31	2.27818
$917$	−2.31900e31	−1.82147
$918$	−3.93178e30	−0.305310
$919$	2.35377e31	1.80697	0.903485	−	0.428620i	$-0.141000\pi$
$919$	2.35377e31	1.80697	0.903485	+	0.428620i	$0.141000\pi$
$920$	0	0
$921$	−5.33032e30	−0.399970
$922$	−1.54338e31	−1.14498
$923$	1.66746e30	0.122303
$924$	2.39098e31	1.73389
$925$	0	0
$926$	3.79337e31	2.68912
$927$	8.31665e29	0.0582923
$928$	2.64659e31	1.83414
$929$	1.87776e30	0.128669	0.0643347	−	0.997928i	$-0.479507\pi$
$929$	1.87776e30	0.128669	0.0643347	+	0.997928i	$0.479507\pi$
$930$	0	0
$931$	2.41861e31	1.62030
$932$	1.39501e31	0.924085
$933$	3.20027e30	0.209618
$934$	−8.94434e30	−0.579303
$935$	0	0
$936$	−4.77556e30	−0.302432
$937$	−1.60028e30	−0.100214	−0.0501072	−	0.998744i	$-0.515956\pi$
$937$	−1.60028e30	−0.100214	−0.0501072	+	0.998744i	$0.515956\pi$
$938$	−7.87075e31	−4.87401
$939$	−1.37679e31	−0.843097
$940$	0	0
$941$	−1.04985e31	−0.628687	−0.314343	−	0.949309i	$-0.601784\pi$
$941$	−1.04985e31	−0.628687	−0.314343	+	0.949309i	$0.601784\pi$
$942$	−9.10248e30	−0.539045
$943$	1.12091e31	0.656442
$944$	−3.92836e31	−2.27512
$945$	0	0
$946$	−2.43640e31	−1.38004
$947$	6.46124e30	0.361944	0.180972	−	0.983488i	$-0.442076\pi$
$947$	6.46124e30	0.361944	0.180972	+	0.983488i	$0.442076\pi$
$948$	3.56088e31	1.97274
$949$	1.05706e31	0.579170
$950$	0	0
$951$	2.82485e30	0.151391
$952$	−6.15086e31	−3.26023
$953$	9.31770e30	0.488465	0.244232	−	0.969717i	$-0.421464\pi$
$953$	9.31770e30	0.488465	0.244232	+	0.969717i	$0.421464\pi$
$954$	−1.59673e31	−0.827891
$955$	0	0
$956$	−5.38166e31	−2.72966
$957$	1.78591e31	0.895951
$958$	4.80324e30	0.238340
$959$	−3.73893e30	−0.183507
$960$	0	0
$961$	−6.26586e29	−0.0300874
$962$	1.82210e31	0.865436
$963$	6.65764e30	0.312784
$964$	−8.20984e31	−3.81528
$965$	0	0
$966$	−3.22366e31	−1.46585
$967$	−4.91970e30	−0.221289	−0.110645	−	0.993860i	$-0.535292\pi$
$967$	−4.91970e30	−0.221289	−0.110645	+	0.993860i	$0.535292\pi$
$968$	1.73538e31	0.772154
$969$	1.06555e31	0.469001
$970$	0	0
$971$	1.52597e31	0.657272	0.328636	−	0.944457i	$-0.393411\pi$
$971$	1.52597e31	0.657272	0.328636	+	0.944457i	$0.393411\pi$
$972$	−3.36833e30	−0.143522
$973$	−2.18884e30	−0.0922632
$974$	−5.91216e31	−2.46534
$975$	0	0
$976$	5.83247e31	2.38029
$977$	8.17943e30	0.330240	0.165120	−	0.986273i	$-0.447199\pi$
$977$	8.17943e30	0.330240	0.165120	+	0.986273i	$0.447199\pi$
$978$	−1.54745e31	−0.618101
$979$	1.20074e31	0.474493
$980$	0	0
$981$	2.11901e30	0.0819611
$982$	2.31984e31	0.887744
$983$	−1.88214e30	−0.0712590	−0.0356295	−	0.999365i	$-0.511344\pi$
$983$	−1.88214e30	−0.0712590	−0.0356295	+	0.999365i	$0.511344\pi$
$984$	2.65118e31	0.993096
$985$	0	0
$986$	−8.30748e31	−3.04622
$987$	3.20397e31	1.16241
$988$	2.34023e31	0.840060
$989$	2.27019e31	0.806309
$990$	0	0
$991$	1.22743e31	0.426798	0.213399	−	0.976965i	$-0.431547\pi$
$991$	1.22743e31	0.426798	0.213399	+	0.976965i	$0.431547\pi$
$992$	−2.73422e31	−0.940723
$993$	6.49729e30	0.221190
$994$	2.66227e31	0.896799
$995$	0	0
$996$	−3.20618e31	−1.05746
$997$	−1.43738e31	−0.469107	−0.234554	−	0.972103i	$-0.575363\pi$
$997$	−1.43738e31	−0.469107	−0.234554	+	0.972103i	$0.575363\pi$
$998$	8.13058e30	0.262573
$999$	7.10744e30	0.227131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.l.1.1	yes	8
5.2	odd	4		75.22.b.j.49.2		16
5.3	odd	4		75.22.b.j.49.15		16
5.4	even	2		75.22.a.k.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.22.a.k.1.8	✓	8	5.4	even	2
75.22.a.l.1.1	yes	8	1.1	even	1	trivial
75.22.b.j.49.2		16	5.2	odd	4
75.22.b.j.49.15		16	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)

\(f(q)\)	\(=\)	\(q-2605.59 q^{2} -59049.0 q^{3} +4.69194e6 q^{4} +1.53857e8 q^{6} -1.24132e9 q^{7} -6.76095e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-2605.59 q^{2} -59049.0 q^{3} +4.69194e6 q^{4} +1.53857e8 q^{6} -1.24132e9 q^{7} -6.76095e9 q^{8} +3.48678e9 q^{9} +6.95232e10 q^{11} -2.77054e11 q^{12} +2.02578e11 q^{13} +3.23436e12 q^{14} +7.77654e12 q^{16} -7.32903e12 q^{17} -9.08513e12 q^{18} +2.46215e13 q^{19} +7.32984e13 q^{21} -1.81149e14 q^{22} +1.68791e14 q^{23} +3.99227e14 q^{24} -5.27834e14 q^{26} -2.05891e14 q^{27} -5.82418e15 q^{28} -4.35028e15 q^{29} +4.49432e15 q^{31} -6.08373e15 q^{32} -4.10528e15 q^{33} +1.90964e16 q^{34} +1.63598e16 q^{36} -3.45204e16 q^{37} -6.41535e16 q^{38} -1.19620e16 q^{39} +6.64078e16 q^{41} -1.90986e17 q^{42} +1.34497e17 q^{43} +3.26199e17 q^{44} -4.39800e17 q^{46} +4.37113e17 q^{47} -4.59197e17 q^{48} +9.82317e17 q^{49} +4.32772e17 q^{51} +9.50482e17 q^{52} +1.75752e18 q^{53} +5.36468e17 q^{54} +8.39247e18 q^{56} -1.45387e18 q^{57} +1.13350e19 q^{58} -5.05155e18 q^{59} +7.50009e18 q^{61} -1.17104e19 q^{62} -4.32820e18 q^{63} -4.56890e17 q^{64} +1.06967e19 q^{66} -2.43348e19 q^{67} -3.43873e19 q^{68} -9.96694e18 q^{69} +8.23122e18 q^{71} -2.35740e19 q^{72} +5.21807e19 q^{73} +8.99459e19 q^{74} +1.15523e20 q^{76} -8.63002e19 q^{77} +3.11681e19 q^{78} -1.28526e20 q^{79} +1.21577e19 q^{81} -1.73032e20 q^{82} +1.15724e20 q^{83} +3.43912e20 q^{84} -3.50444e20 q^{86} +2.56880e20 q^{87} -4.70043e20 q^{88} +1.72710e20 q^{89} -2.51463e20 q^{91} +7.91958e20 q^{92} -2.65385e20 q^{93} -1.13894e21 q^{94} +3.59238e20 q^{96} -9.66495e20 q^{97} -2.55951e21 q^{98} +2.42412e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9}+O(q^{10}) \) 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9	\( 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9} - 14410165968 q^{11} - 630755749296 q^{12} - 328226481176 q^{13} + 4068036669186 q^{14} + 18553054308920 q^{16} - 5718214953936 q^{17} + 2322198411066 q^{18} + 75919698170296 q^{19} - 7914601537128 q^{21} - 426897542691372 q^{22} + 49712781537936 q^{23} - 207390009885372 q^{24} - 126056679642054 q^{26} - 16\!\cdots\!92 q^{27}+ \cdots - 50\!\cdots\!68 q^{99}+O(q^{100}) \) 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9 - 14410165968 * q^11 - 630755749296 * q^12 - 328226481176 * q^13 + 4068036669186 * q^14 + 18553054308920 * q^16 - 5718214953936 * q^17 + 2322198411066 * q^18 + 75919698170296 * q^19 - 7914601537128 * q^21 - 426897542691372 * q^22 + 49712781537936 * q^23 - 207390009885372 * q^24 - 126056679642054 * q^26 - 1647129056757192 * q^27 + 4235167126399504 * q^28 + 2111332005818352 * q^29 + 5677451410844968 * q^31 + 17528957992420632 * q^32 + 850905890244432 * q^33 + 9944754702742068 * q^34 + 37245496240179504 * q^36 - 26538401323409648 * q^37 + 85729618982552526 * q^38 + 19381445486961624 * q^39 - 9719778050982432 * q^41 - 240213497278764114 * q^42 - 20888744714565080 * q^43 - 388904820117856128 * q^44 - 156886987299808452 * q^46 + 405141072730526160 * q^47 - 1095539303887417080 * q^48 + 1172899395436368720 * q^49 + 337654874814966864 * q^51 - 2264281219115894096 * q^52 + 1890950875516930176 * q^53 - 137123493975036234 * q^54 + 19720793170524660300 * q^56 - 4482982257257808504 * q^57 + 12636224332772065092 * q^58 - 3496737905134765776 * q^59 + 17508733151770015960 * q^61 - 30404101355896200606 * q^62 + 467349306165871272 * q^63 + 27809823573720908096 * q^64 + 25207872998382825228 * q^66 - 29911607071855004216 * q^67 - 29571773973515759808 * q^68 - 2935490037033582864 * q^69 + 52204030316256846816 * q^71 + 12246172693721331228 * q^72 - 39697032757192143344 * q^73 - 32301811215943606764 * q^74 + 383913379529874070384 * q^76 - 202348742072764877136 * q^77 + 7443520876183646646 * q^78 - 274125739348498990400 * q^79 + 97261323672455430408 * q^81 - 231746572148446490256 * q^82 - 276883976086286712912 * q^83 - 250082383646764311696 * q^84 + 486708932755841131242 * q^86 - 124672043611567867248 * q^87 - 2078122304879724561672 * q^88 - 319129684977517046784 * q^89 + 45068531423660000648 * q^91 + 147706203809955886272 * q^92 - 335247828358984515432 * q^93 - 1499781052151613424236 * q^94 - 1035067440494445898968 * q^96 + 451807368184588489864 * q^97 - 2604193962467412900204 * q^98 - 50245141913043465168 * q^99

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