LMFDB - Embedded newform 75.22.a.l.1.6

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			1.6
Root			$1642.33$ 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)

$f(q)$	$=$	$q+1725.33 q^{2} -59049.0 q^{3} +879617. q^{4} -1.01879e8 q^{6} -5.00000e8 q^{7} -2.10065e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q+1725.33 q^{2} -59049.0 q^{3} +879617. q^{4} -1.01879e8 q^{6} -5.00000e8 q^{7} -2.10065e9 q^{8} +3.48678e9 q^{9} -1.14347e11 q^{11} -5.19405e10 q^{12} -1.56229e11 q^{13} -8.62665e11 q^{14} -5.46901e12 q^{16} +2.48725e12 q^{17} +6.01586e12 q^{18} -4.52155e13 q^{19} +2.95245e13 q^{21} -1.97286e14 q^{22} -6.05193e12 q^{23} +1.24041e14 q^{24} -2.69546e14 q^{26} -2.05891e14 q^{27} -4.39808e14 q^{28} -2.25071e15 q^{29} +2.57666e15 q^{31} -5.03047e15 q^{32} +6.75205e15 q^{33} +4.29133e15 q^{34} +3.06704e15 q^{36} -2.36303e16 q^{37} -7.80117e16 q^{38} +9.22514e15 q^{39} -7.10884e16 q^{41} +5.09395e16 q^{42} +1.38090e17 q^{43} -1.00581e17 q^{44} -1.04416e16 q^{46} -5.07457e17 q^{47} +3.22940e17 q^{48} -3.08546e17 q^{49} -1.46870e17 q^{51} -1.37421e17 q^{52} +1.50326e18 q^{53} -3.55230e17 q^{54} +1.05032e18 q^{56} +2.66993e18 q^{57} -3.88322e18 q^{58} +2.27610e18 q^{59} +5.10938e18 q^{61} +4.44559e18 q^{62} -1.74339e18 q^{63} +2.79011e18 q^{64} +1.16495e19 q^{66} -2.36689e19 q^{67} +2.18783e18 q^{68} +3.57361e17 q^{69} +1.27391e19 q^{71} -7.32452e18 q^{72} -2.08270e19 q^{73} -4.07701e19 q^{74} -3.97723e19 q^{76} +5.71732e19 q^{77} +1.59164e19 q^{78} +3.58135e19 q^{79} +1.21577e19 q^{81} -1.22651e20 q^{82} -1.35359e20 q^{83} +2.59702e19 q^{84} +2.38251e20 q^{86} +1.32902e20 q^{87} +2.40202e20 q^{88} -1.87985e20 q^{89} +7.81142e19 q^{91} -5.32338e18 q^{92} -1.52149e20 q^{93} -8.75532e20 q^{94} +2.97044e20 q^{96} +6.13650e20 q^{97} -5.32344e20 q^{98} -3.98702e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1725.33	1.19140	0.595700	−	0.803207i	$-0.296875\pi$
$2$	1725.33	1.19140	0.595700	+	0.803207i	$0.296875\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	879617.	0.419434
$5$	0	0
$5$	0	0
$6$	−1.01879e8	−0.687855
$7$	−5.00000e8	−0.669022	−0.334511	−	0.942392i	$-0.608571\pi$
$7$	−5.00000e8	−0.669022	−0.334511	+	0.942392i	$0.608571\pi$
$8$	−2.10065e9	−0.691686
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−1.14347e11	−1.32923	−0.664615	−	0.747186i	$-0.731404\pi$
$11$	−1.14347e11	−1.32923	−0.664615	+	0.747186i	$0.731404\pi$
$12$	−5.19405e10	−0.242160
$13$	−1.56229e11	−0.314308	−0.157154	−	0.987574i	$-0.550232\pi$
$13$	−1.56229e11	−0.314308	−0.157154	+	0.987574i	$0.550232\pi$
$14$	−8.62665e11	−0.797073
$15$	0	0
$16$	−5.46901e12	−1.24351
$17$	2.48725e12	0.299230	0.149615	−	0.988744i	$-0.452197\pi$
$17$	2.48725e12	0.299230	0.149615	+	0.988744i	$0.452197\pi$
$18$	6.01586e12	0.397133
$19$	−4.52155e13	−1.69190	−0.845950	−	0.533263i	$-0.820966\pi$
$19$	−4.52155e13	−1.69190	−0.845950	+	0.533263i	$0.820966\pi$
$20$	0	0
$21$	2.95245e13	0.386260
$22$	−1.97286e14	−1.58364
$23$	−6.05193e12	−0.0304615	−0.0152308	−	0.999884i	$-0.504848\pi$
$23$	−6.05193e12	−0.0304615	−0.0152308	+	0.999884i	$0.504848\pi$
$24$	1.24041e14	0.399345
$25$	0	0
$26$	−2.69546e14	−0.374466
$27$	−2.05891e14	−0.192450
$28$	−4.39808e14	−0.280611
$29$	−2.25071e15	−0.993438	−0.496719	−	0.867911i	$-0.665462\pi$
$29$	−2.25071e15	−0.993438	−0.496719	+	0.867911i	$0.665462\pi$
$30$	0	0
$31$	2.57666e15	0.564624	0.282312	−	0.959323i	$-0.408899\pi$
$31$	2.57666e15	0.564624	0.282312	+	0.959323i	$0.408899\pi$
$32$	−5.03047e15	−0.789831
$33$	6.75205e15	0.767431
$34$	4.29133e15	0.356503
$35$	0	0
$36$	3.06704e15	0.139811
$37$	−2.36303e16	−0.807889	−0.403944	−	0.914784i	$-0.632361\pi$
$37$	−2.36303e16	−0.807889	−0.403944	+	0.914784i	$0.632361\pi$
$38$	−7.80117e16	−2.01573
$39$	9.22514e15	0.181466
$40$	0	0
$41$	−7.10884e16	−0.827120	−0.413560	−	0.910477i	$-0.635715\pi$
$41$	−7.10884e16	−0.827120	−0.413560	+	0.910477i	$0.635715\pi$
$42$	5.09395e16	0.460190
$43$	1.38090e17	0.974413	0.487206	−	0.873287i	$-0.338016\pi$
$43$	1.38090e17	0.974413	0.487206	+	0.873287i	$0.338016\pi$
$44$	−1.00581e17	−0.557524
$45$	0	0
$46$	−1.04416e16	−0.0362919
$47$	−5.07457e17	−1.40725	−0.703626	−	0.710571i	$-0.748437\pi$
$47$	−5.07457e17	−1.40725	−0.703626	+	0.710571i	$0.748437\pi$
$48$	3.22940e17	0.717940
$49$	−3.08546e17	−0.552410
$50$	0	0
$51$	−1.46870e17	−0.172761
$52$	−1.37421e17	−0.131831
$53$	1.50326e18	1.18069	0.590346	−	0.807151i	$-0.298991\pi$
$53$	1.50326e18	1.18069	0.590346	+	0.807151i	$0.298991\pi$
$54$	−3.55230e17	−0.229285
$55$	0	0
$56$	1.05032e18	0.462753
$57$	2.66993e18	0.976818
$58$	−3.88322e18	−1.18358
$59$	2.27610e18	0.579756	0.289878	−	0.957064i	$-0.406385\pi$
$59$	2.27610e18	0.579756	0.289878	+	0.957064i	$0.406385\pi$
$60$	0	0
$61$	5.10938e18	0.917076	0.458538	−	0.888675i	$-0.348373\pi$
$61$	5.10938e18	0.917076	0.458538	+	0.888675i	$0.348373\pi$
$62$	4.44559e18	0.672693
$63$	−1.74339e18	−0.223007
$64$	2.79011e18	0.302505
$65$	0	0
$66$	1.16495e19	0.914317
$67$	−2.36689e19	−1.58633	−0.793164	−	0.609009i	$-0.791568\pi$
$67$	−2.36689e19	−1.58633	−0.793164	+	0.609009i	$0.791568\pi$
$68$	2.18783e18	0.125507
$69$	3.57361e17	0.0175870
$70$	0	0
$71$	1.27391e19	0.464434	0.232217	−	0.972664i	$-0.425402\pi$
$71$	1.27391e19	0.464434	0.232217	+	0.972664i	$0.425402\pi$
$72$	−7.32452e18	−0.230562
$73$	−2.08270e19	−0.567200	−0.283600	−	0.958943i	$-0.591529\pi$
$73$	−2.08270e19	−0.567200	−0.283600	+	0.958943i	$0.591529\pi$
$74$	−4.07701e19	−0.962519
$75$	0	0
$76$	−3.97723e19	−0.709640
$77$	5.71732e19	0.889283
$78$	1.59164e19	0.216198
$79$	3.58135e19	0.425562	0.212781	−	0.977100i	$-0.431748\pi$
$79$	3.58135e19	0.425562	0.212781	+	0.977100i	$0.431748\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−1.22651e20	−0.985431
$83$	−1.35359e20	−0.957561	−0.478780	−	0.877935i	$-0.658921\pi$
$83$	−1.35359e20	−0.957561	−0.478780	+	0.877935i	$0.658921\pi$
$84$	2.59702e19	0.162011
$85$	0	0
$86$	2.38251e20	1.16092
$87$	1.32902e20	0.573562
$88$	2.40202e20	0.919409
$89$	−1.87985e20	−0.639039	−0.319520	−	0.947580i	$-0.603522\pi$
$89$	−1.87985e20	−0.639039	−0.319520	+	0.947580i	$0.603522\pi$
$90$	0	0
$91$	7.81142e19	0.210279
$92$	−5.32338e18	−0.0127766
$93$	−1.52149e20	−0.325986
$94$	−8.75532e20	−1.67660
$95$	0	0
$96$	2.97044e20	0.456009
$97$	6.13650e20	0.844924	0.422462	−	0.906381i	$-0.361166\pi$
$97$	6.13650e20	0.844924	0.422462	+	0.906381i	$0.361166\pi$
$98$	−5.32344e20	−0.658141
$99$	−3.98702e20	−0.443076
$100$	0	0
$101$	−1.02046e21	−0.919228	−0.459614	−	0.888119i	$-0.652012\pi$
$101$	−1.02046e21	−0.919228	−0.459614	+	0.888119i	$0.652012\pi$
$102$	−2.53399e20	−0.205827
$103$	−1.84787e20	−0.135482	−0.0677409	−	0.997703i	$-0.521579\pi$
$103$	−1.84787e20	−0.135482	−0.0677409	+	0.997703i	$0.521579\pi$
$104$	3.28182e20	0.217402
$105$	0	0
$106$	2.59362e21	1.40668
$107$	2.86328e21	1.40713	0.703565	−	0.710631i	$-0.251591\pi$
$107$	2.86328e21	1.40713	0.703565	+	0.710631i	$0.251591\pi$
$108$	−1.81105e20	−0.0807202
$109$	−3.33660e21	−1.34998	−0.674988	−	0.737828i	$-0.735851\pi$
$109$	−3.33660e21	−1.34998	−0.674988	+	0.737828i	$0.735851\pi$
$110$	0	0
$111$	1.39535e21	0.466435
$112$	2.73450e21	0.831935
$113$	−6.99764e20	−0.193922	−0.0969612	−	0.995288i	$-0.530912\pi$
$113$	−6.99764e20	−0.193922	−0.0969612	+	0.995288i	$0.530912\pi$
$114$	4.60651e21	1.16378
$115$	0	0
$116$	−1.97976e21	−0.416682
$117$	−5.44735e20	−0.104769
$118$	3.92703e21	0.690721
$119$	−1.24362e21	−0.200192
$120$	0	0
$121$	5.67488e21	0.766850
$122$	8.81538e21	1.09260
$123$	4.19770e21	0.477538
$124$	2.26648e21	0.236823
$125$	0	0
$126$	−3.00793e21	−0.265691
$127$	1.12891e22	0.917745	0.458872	−	0.888502i	$-0.348253\pi$
$127$	1.12891e22	0.917745	0.458872	+	0.888502i	$0.348253\pi$
$128$	1.53635e22	1.15023
$129$	−8.15407e21	−0.562577
$130$	0	0
$131$	2.83013e22	1.66133	0.830667	−	0.556769i	$-0.187959\pi$
$131$	2.83013e22	1.66133	0.830667	+	0.556769i	$0.187959\pi$
$132$	5.93922e21	0.321887
$133$	2.26077e22	1.13192
$134$	−4.08367e22	−1.88995
$135$	0	0
$136$	−5.22484e21	−0.206974
$137$	2.99272e22	1.09774	0.548871	−	0.835907i	$-0.315058\pi$
$137$	2.99272e22	1.09774	0.548871	+	0.835907i	$0.315058\pi$
$138$	6.16565e20	0.0209531
$139$	−1.34479e22	−0.423641	−0.211821	−	0.977309i	$-0.567939\pi$
$139$	−1.34479e22	−0.423641	−0.211821	+	0.977309i	$0.567939\pi$
$140$	0	0
$141$	2.99648e22	0.812477
$142$	2.19791e22	0.553327
$143$	1.78642e22	0.417787
$144$	−1.90693e22	−0.414503
$145$	0	0
$146$	−3.59335e22	−0.675763
$147$	1.82193e22	0.318934
$148$	−2.07856e22	−0.338856
$149$	7.73280e22	1.17458	0.587288	−	0.809378i	$-0.300196\pi$
$149$	7.73280e22	1.17458	0.587288	+	0.809378i	$0.300196\pi$
$150$	0	0
$151$	1.10528e23	1.45954	0.729769	−	0.683694i	$-0.239628\pi$
$151$	1.10528e23	1.45954	0.729769	+	0.683694i	$0.239628\pi$
$152$	9.49820e22	1.17026
$153$	8.67250e21	0.0997435
$154$	9.86428e22	1.05949
$155$	0	0
$156$	8.11459e21	0.0761129
$157$	7.18388e22	0.630105	0.315053	−	0.949074i	$-0.397978\pi$
$157$	7.18388e22	0.630105	0.315053	+	0.949074i	$0.397978\pi$
$158$	6.17902e22	0.507014
$159$	−8.87658e22	−0.681672
$160$	0	0
$161$	3.02596e21	0.0203794
$162$	2.09760e22	0.132378
$163$	−1.33670e23	−0.790795	−0.395398	−	0.918510i	$-0.629393\pi$
$163$	−1.33670e23	−0.790795	−0.395398	+	0.918510i	$0.629393\pi$
$164$	−6.25306e22	−0.346922
$165$	0	0
$166$	−2.33539e23	−1.14084
$167$	−1.94655e23	−0.892776	−0.446388	−	0.894840i	$-0.647290\pi$
$167$	−1.94655e23	−0.892776	−0.446388	+	0.894840i	$0.647290\pi$
$168$	−6.20206e22	−0.267171
$169$	−2.22657e23	−0.901211
$170$	0	0
$171$	−1.57657e23	−0.563966
$172$	1.21466e23	0.408702
$173$	−5.61820e22	−0.177874	−0.0889372	−	0.996037i	$-0.528347\pi$
$173$	−5.61820e22	−0.177874	−0.0889372	+	0.996037i	$0.528347\pi$
$174$	2.29300e23	0.683341
$175$	0	0
$176$	6.25363e23	1.65291
$177$	−1.34401e23	−0.334722
$178$	−3.24336e23	−0.761351
$179$	2.55847e23	0.566270	0.283135	−	0.959080i	$-0.408626\pi$
$179$	2.55847e23	0.566270	0.283135	+	0.959080i	$0.408626\pi$
$180$	0	0
$181$	5.38421e23	1.06047	0.530233	−	0.847852i	$-0.322104\pi$
$181$	5.38421e23	1.06047	0.530233	+	0.847852i	$0.322104\pi$
$182$	1.34773e23	0.250526
$183$	−3.01704e23	−0.529474
$184$	1.27130e22	0.0210698
$185$	0	0
$186$	−2.62508e23	−0.388380
$187$	−2.84408e23	−0.397746
$188$	−4.46368e23	−0.590250
$189$	1.02946e23	0.128753
$190$	0	0
$191$	−1.60279e24	−1.79484	−0.897419	−	0.441179i	$-0.854561\pi$
$191$	−1.60279e24	−1.79484	−0.897419	+	0.441179i	$0.854561\pi$
$192$	−1.64753e23	−0.174651
$193$	−3.72739e23	−0.374156	−0.187078	−	0.982345i	$-0.559902\pi$
$193$	−3.72739e23	−0.374156	−0.187078	+	0.982345i	$0.559902\pi$
$194$	1.05875e24	1.00664
$195$	0	0
$196$	−2.71403e23	−0.231700
$197$	−1.26230e24	−1.02157	−0.510785	−	0.859709i	$-0.670645\pi$
$197$	−1.26230e24	−1.02157	−0.510785	+	0.859709i	$0.670645\pi$
$198$	−6.87893e23	−0.527881
$199$	1.03027e24	0.749885	0.374943	−	0.927048i	$-0.377662\pi$
$199$	1.03027e24	0.749885	0.374943	+	0.927048i	$0.377662\pi$
$200$	0	0
$201$	1.39763e24	0.915867
$202$	−1.76064e24	−1.09517
$203$	1.12535e24	0.664632
$204$	−1.29189e23	−0.0724618
$205$	0	0
$206$	−3.18819e23	−0.161413
$207$	−2.11018e22	−0.0101538
$208$	8.54415e23	0.390845
$209$	5.17024e24	2.24892
$210$	0	0
$211$	−9.94277e23	−0.391329	−0.195664	−	0.980671i	$-0.562686\pi$
$211$	−9.94277e23	−0.391329	−0.195664	+	0.980671i	$0.562686\pi$
$212$	1.32229e24	0.495222
$213$	−7.52228e23	−0.268141
$214$	4.94011e24	1.67645
$215$	0	0
$216$	4.32505e23	0.133115
$217$	−1.28833e24	−0.377746
$218$	−5.75675e24	−1.60836
$219$	1.22981e24	0.327473
$220$	0	0
$221$	−3.88579e23	−0.0940504
$222$	2.40744e24	0.555710
$223$	−6.04055e24	−1.33007	−0.665037	−	0.746811i	$-0.731584\pi$
$223$	−6.04055e24	−1.33007	−0.665037	+	0.746811i	$0.731584\pi$
$224$	2.51524e24	0.528414
$225$	0	0
$226$	−1.20732e24	−0.231039
$227$	6.01325e24	1.09860	0.549298	−	0.835627i	$-0.314895\pi$
$227$	6.01325e24	1.09860	0.549298	+	0.835627i	$0.314895\pi$
$228$	2.34852e24	0.409711
$229$	4.79968e24	0.799724	0.399862	−	0.916575i	$-0.369058\pi$
$229$	4.79968e24	0.799724	0.399862	+	0.916575i	$0.369058\pi$
$230$	0	0
$231$	−3.37602e24	−0.513428
$232$	4.72796e24	0.687147
$233$	3.91324e24	0.543625	0.271812	−	0.962350i	$-0.412377\pi$
$233$	3.91324e24	0.543625	0.271812	+	0.962350i	$0.412377\pi$
$234$	−9.39849e23	−0.124822
$235$	0	0
$236$	2.00210e24	0.243170
$237$	−2.11475e24	−0.245698
$238$	−2.14566e24	−0.238508
$239$	−6.41698e24	−0.682579	−0.341289	−	0.939958i	$-0.610864\pi$
$239$	−6.41698e24	−0.682579	−0.341289	+	0.939958i	$0.610864\pi$
$240$	0	0
$241$	1.54568e25	1.50640	0.753199	−	0.657792i	$-0.228509\pi$
$241$	1.54568e25	1.50640	0.753199	+	0.657792i	$0.228509\pi$
$242$	9.79106e24	0.913625
$243$	−7.17898e23	−0.0641500
$244$	4.49430e24	0.384653
$245$	0	0
$246$	7.24243e24	0.568939
$247$	7.06395e24	0.531777
$248$	−5.41267e24	−0.390543
$249$	7.99280e24	0.552848
$250$	0	0
$251$	−1.19724e25	−0.761388	−0.380694	−	0.924701i	$-0.624315\pi$
$251$	−1.19724e25	−0.761388	−0.380694	+	0.924701i	$0.624315\pi$
$252$	−1.53352e24	−0.0935369
$253$	6.92018e23	0.0404904
$254$	1.94775e25	1.09340
$255$	0	0
$256$	2.06559e25	1.06789
$257$	2.66405e25	1.32204	0.661019	−	0.750369i	$-0.270124\pi$
$257$	2.66405e25	1.32204	0.661019	+	0.750369i	$0.270124\pi$
$258$	−1.40685e25	−0.670255
$259$	1.18152e25	0.540495
$260$	0	0
$261$	−7.84774e24	−0.331146
$262$	4.88291e25	1.97931
$263$	−2.16330e25	−0.842522	−0.421261	−	0.906940i	$-0.638412\pi$
$263$	−2.16330e25	−0.842522	−0.421261	+	0.906940i	$0.638412\pi$
$264$	−1.41837e25	−0.530821
$265$	0	0
$266$	3.90058e25	1.34857
$267$	1.11003e25	0.368949
$268$	−2.08196e25	−0.665360
$269$	2.26549e25	0.696246	0.348123	−	0.937449i	$-0.386819\pi$
$269$	2.26549e25	0.696246	0.348123	+	0.937449i	$0.386819\pi$
$270$	0	0
$271$	4.96432e25	1.41150	0.705752	−	0.708458i	$-0.250609\pi$
$271$	4.96432e25	1.41150	0.705752	+	0.708458i	$0.250609\pi$
$272$	−1.36028e25	−0.372096
$273$	−4.61257e24	−0.121404
$274$	5.16344e25	1.30785
$275$	0	0
$276$	3.14341e23	0.00737658
$277$	4.38598e25	0.990897	0.495449	−	0.868637i	$-0.335004\pi$
$277$	4.38598e25	0.990897	0.495449	+	0.868637i	$0.335004\pi$
$278$	−2.32020e25	−0.504726
$279$	8.98426e24	0.188208
$280$	0	0
$281$	8.01277e25	1.55728	0.778639	−	0.627472i	$-0.215910\pi$
$281$	8.01277e25	1.55728	0.778639	+	0.627472i	$0.215910\pi$
$282$	5.16993e25	0.967985
$283$	−5.50934e24	−0.0993898	−0.0496949	−	0.998764i	$-0.515825\pi$
$283$	−5.50934e24	−0.0993898	−0.0496949	+	0.998764i	$0.515825\pi$
$284$	1.12055e25	0.194800
$285$	0	0
$286$	3.08217e25	0.497752
$287$	3.55442e25	0.553361
$288$	−1.75402e25	−0.263277
$289$	−6.29055e25	−0.910461
$290$	0	0
$291$	−3.62354e25	−0.487817
$292$	−1.83198e25	−0.237903
$293$	−9.99971e25	−1.25279	−0.626394	−	0.779507i	$-0.715470\pi$
$293$	−9.99971e25	−1.25279	−0.626394	+	0.779507i	$0.715470\pi$
$294$	3.14344e25	0.379978
$295$	0	0
$296$	4.96391e25	0.558805
$297$	2.35429e25	0.255810
$298$	1.33416e26	1.39939
$299$	9.45484e23	0.00957430
$300$	0	0
$301$	−6.90449e25	−0.651903
$302$	1.90698e26	1.73889
$303$	6.02574e25	0.530717
$304$	2.47284e26	2.10389
$305$	0	0
$306$	1.49629e25	0.118834
$307$	−2.37110e26	−1.81969	−0.909846	−	0.414947i	$-0.863800\pi$
$307$	−2.37110e26	−1.81969	−0.909846	+	0.414947i	$0.863800\pi$
$308$	5.02906e25	0.372996
$309$	1.09115e25	0.0782205
$310$	0	0
$311$	−2.03442e26	−1.36288	−0.681438	−	0.731876i	$-0.738645\pi$
$311$	−2.03442e26	−1.36288	−0.681438	+	0.731876i	$0.738645\pi$
$312$	−1.93788e25	−0.125517
$313$	−2.21902e26	−1.38978	−0.694888	−	0.719118i	$-0.744546\pi$
$313$	−2.21902e26	−1.38978	−0.694888	+	0.719118i	$0.744546\pi$
$314$	1.23946e26	0.750707
$315$	0	0
$316$	3.15022e25	0.178495
$317$	−5.23596e25	−0.286995	−0.143497	−	0.989651i	$-0.545835\pi$
$317$	−5.23596e25	−0.286995	−0.143497	+	0.989651i	$0.545835\pi$
$318$	−1.53150e26	−0.812145
$319$	2.57361e26	1.32051
$320$	0	0
$321$	−1.69074e26	−0.812406
$322$	5.22079e24	0.0242801
$323$	−1.12462e26	−0.506268
$324$	1.06941e25	0.0466038
$325$	0	0
$326$	−2.30625e26	−0.942154
$327$	1.97023e26	0.779409
$328$	1.49332e26	0.572107
$329$	2.53728e26	0.941482
$330$	0	0
$331$	4.13812e26	1.44082	0.720409	−	0.693550i	$-0.243954\pi$
$331$	4.13812e26	1.44082	0.720409	+	0.693550i	$0.243954\pi$
$332$	−1.19064e26	−0.401634
$333$	−8.23938e25	−0.269296
$334$	−3.35844e26	−1.06365
$335$	0	0
$336$	−1.61470e26	−0.480318
$337$	−3.93698e26	−1.13514	−0.567569	−	0.823326i	$-0.692116\pi$
$337$	−3.93698e26	−1.13514	−0.567569	+	0.823326i	$0.692116\pi$
$338$	−3.84157e26	−1.07370
$339$	4.13204e25	0.111961
$340$	0	0
$341$	−2.94632e26	−0.750515
$342$	−2.72010e26	−0.671910
$343$	4.33546e26	1.03860
$344$	−2.90079e26	−0.673988
$345$	0	0
$346$	−9.69326e25	−0.211920
$347$	−8.12879e26	−1.72412	−0.862058	−	0.506810i	$-0.830824\pi$
$347$	−8.12879e26	−1.72412	−0.862058	+	0.506810i	$0.830824\pi$
$348$	1.16903e26	0.240571
$349$	9.01336e26	1.79978	0.899890	−	0.436116i	$-0.143646\pi$
$349$	9.01336e26	1.79978	0.899890	+	0.436116i	$0.143646\pi$
$350$	0	0
$351$	3.21661e25	0.0604886
$352$	5.75217e26	1.04987
$353$	−3.74190e26	−0.662915	−0.331458	−	0.943470i	$-0.607540\pi$
$353$	−3.74190e26	−0.662915	−0.331458	+	0.943470i	$0.607540\pi$
$354$	−2.31887e26	−0.398788
$355$	0	0
$356$	−1.65355e26	−0.268035
$357$	7.34348e25	0.115581
$358$	4.41421e26	0.674654
$359$	−7.72766e26	−1.14698	−0.573491	−	0.819212i	$-0.694411\pi$
$359$	−7.72766e26	−1.14698	−0.573491	+	0.819212i	$0.694411\pi$
$360$	0	0
$361$	1.33023e27	1.86252
$362$	9.28955e26	1.26344
$363$	−3.35096e26	−0.442741
$364$	6.87106e25	0.0881981
$365$	0	0
$366$	−5.20539e26	−0.630815
$367$	−1.47285e27	−1.73446	−0.867232	−	0.497905i	$-0.834103\pi$
$367$	−1.47285e27	−1.73446	−0.867232	+	0.497905i	$0.834103\pi$
$368$	3.30981e25	0.0378792
$369$	−2.47870e26	−0.275707
$370$	0	0
$371$	−7.51628e26	−0.789908
$372$	−1.33833e26	−0.136730
$373$	6.83099e26	0.678486	0.339243	−	0.940699i	$-0.389829\pi$
$373$	6.83099e26	0.678486	0.339243	+	0.940699i	$0.389829\pi$
$374$	−4.90699e26	−0.473874
$375$	0	0
$376$	1.06599e27	0.973377
$377$	3.51625e26	0.312245
$378$	1.77615e26	0.153397
$379$	−1.77322e27	−1.48954	−0.744768	−	0.667323i	$-0.767440\pi$
$379$	−1.77322e27	−1.48954	−0.744768	+	0.667323i	$0.767440\pi$
$380$	0	0
$381$	−6.66613e26	−0.529860
$382$	−2.76534e27	−2.13837
$383$	1.98827e27	1.49585	0.747926	−	0.663782i	$-0.231050\pi$
$383$	1.98827e27	1.49585	0.747926	+	0.663782i	$0.231050\pi$
$384$	−9.07201e26	−0.664088
$385$	0	0
$386$	−6.43098e26	−0.445770
$387$	4.81489e26	0.324804
$388$	5.39777e26	0.354390
$389$	−2.09476e27	−1.33864	−0.669321	−	0.742974i	$-0.733415\pi$
$389$	−2.09476e27	−1.33864	−0.669321	+	0.742974i	$0.733415\pi$
$390$	0	0
$391$	−1.50527e25	−0.00911502
$392$	6.48148e26	0.382094
$393$	−1.67116e27	−0.959172
$394$	−2.17789e27	−1.21710
$395$	0	0
$396$	−3.50705e26	−0.185841
$397$	5.81695e26	0.300189	0.150095	−	0.988672i	$-0.452042\pi$
$397$	5.81695e26	0.300189	0.150095	+	0.988672i	$0.452042\pi$
$398$	1.77756e27	0.893413
$399$	−1.33496e27	−0.653513
$400$	0	0
$401$	1.98794e26	0.0923395	0.0461697	−	0.998934i	$-0.485298\pi$
$401$	1.98794e26	0.0923395	0.0461697	+	0.998934i	$0.485298\pi$
$402$	2.41137e27	1.09116
$403$	−4.02548e26	−0.177466
$404$	−8.97618e26	−0.385556
$405$	0	0
$406$	1.94161e27	0.791842
$407$	2.70205e27	1.07387
$408$	3.08522e26	0.119496
$409$	2.21424e27	0.835853	0.417926	−	0.908481i	$-0.362757\pi$
$409$	2.21424e27	0.835853	0.417926	+	0.908481i	$0.362757\pi$
$410$	0	0
$411$	−1.76717e27	−0.633782
$412$	−1.62542e26	−0.0568257
$413$	−1.13805e27	−0.387870
$414$	−3.64076e25	−0.0120973
$415$	0	0
$416$	7.85903e26	0.248250
$417$	7.94083e26	0.244589
$418$	8.92037e27	2.67937
$419$	−1.77708e27	−0.520547	−0.260273	−	0.965535i	$-0.583813\pi$
$419$	−1.77708e27	−0.520547	−0.260273	+	0.965535i	$0.583813\pi$
$420$	0	0
$421$	3.92361e27	1.09326	0.546630	−	0.837374i	$-0.315910\pi$
$421$	3.92361e27	1.09326	0.546630	+	0.837374i	$0.315910\pi$
$422$	−1.71546e27	−0.466229
$423$	−1.76939e27	−0.469084
$424$	−3.15782e27	−0.816668
$425$	0	0
$426$	−1.29784e27	−0.319464
$427$	−2.55469e27	−0.613544
$428$	2.51859e27	0.590198
$429$	−1.05486e27	−0.241209
$430$	0	0
$431$	−5.87979e27	−1.28041	−0.640207	−	0.768203i	$-0.721151\pi$
$431$	−5.87979e27	−1.28041	−0.640207	+	0.768203i	$0.721151\pi$
$432$	1.12602e27	0.239313
$433$	6.98609e27	1.44914	0.724571	−	0.689200i	$-0.242038\pi$
$433$	6.98609e27	1.44914	0.724571	+	0.689200i	$0.242038\pi$
$434$	−2.22280e27	−0.450047
$435$	0	0
$436$	−2.93493e27	−0.566226
$437$	2.73641e26	0.0515378
$438$	2.12184e27	0.390152
$439$	−5.97706e27	−1.07303	−0.536513	−	0.843892i	$-0.680259\pi$
$439$	−5.97706e27	−1.07303	−0.536513	+	0.843892i	$0.680259\pi$
$440$	0	0
$441$	−1.07583e27	−0.184137
$442$	−6.70428e26	−0.112052
$443$	−4.73265e27	−0.772442	−0.386221	−	0.922406i	$-0.626220\pi$
$443$	−4.73265e27	−0.772442	−0.386221	+	0.922406i	$0.626220\pi$
$444$	1.22737e27	0.195639
$445$	0	0
$446$	−1.04220e28	−1.58465
$447$	−4.56614e27	−0.678142
$448$	−1.39506e27	−0.202382
$449$	−6.26275e27	−0.887521	−0.443760	−	0.896145i	$-0.646356\pi$
$449$	−6.26275e27	−0.887521	−0.443760	+	0.896145i	$0.646356\pi$
$450$	0	0
$451$	8.12872e27	1.09943
$452$	−6.15525e26	−0.0813377
$453$	−6.52658e27	−0.842664
$454$	1.03749e28	1.30887
$455$	0	0
$456$	−5.60859e27	−0.675652
$457$	−1.52517e28	−1.79555	−0.897776	−	0.440452i	$-0.854818\pi$
$457$	−1.52517e28	−1.79555	−0.897776	+	0.440452i	$0.854818\pi$
$458$	8.28105e27	0.952791
$459$	−5.12103e26	−0.0575869
$460$	0	0
$461$	9.27945e27	0.996925	0.498463	−	0.866911i	$-0.333898\pi$
$461$	9.27945e27	0.996925	0.498463	+	0.866911i	$0.333898\pi$
$462$	−5.82476e27	−0.611698
$463$	−3.40416e27	−0.349470	−0.174735	−	0.984616i	$-0.555907\pi$
$463$	−3.40416e27	−0.349470	−0.174735	+	0.984616i	$0.555907\pi$
$464$	1.23092e28	1.23535
$465$	0	0
$466$	6.75163e27	0.647674
$467$	−1.32611e28	−1.24381	−0.621904	−	0.783094i	$-0.713641\pi$
$467$	−1.32611e28	−1.24381	−0.621904	+	0.783094i	$0.713641\pi$
$468$	−4.79158e26	−0.0439438
$469$	1.18344e28	1.06129
$470$	0	0
$471$	−4.24201e27	−0.363791
$472$	−4.78129e27	−0.401009
$473$	−1.57901e28	−1.29522
$474$	−3.64865e27	−0.292725
$475$	0	0
$476$	−1.09391e27	−0.0839672
$477$	5.24153e27	0.393564
$478$	−1.10714e28	−0.813224
$479$	−9.11491e27	−0.654982	−0.327491	−	0.944854i	$-0.606203\pi$
$479$	−9.11491e27	−0.654982	−0.327491	+	0.944854i	$0.606203\pi$
$480$	0	0
$481$	3.69173e27	0.253926
$482$	2.66681e28	1.79472
$483$	−1.78680e26	−0.0117661
$484$	4.99173e27	0.321643
$485$	0	0
$486$	−1.23861e27	−0.0764284
$487$	−2.44964e27	−0.147927	−0.0739636	−	0.997261i	$-0.523565\pi$
$487$	−2.44964e27	−0.147927	−0.0739636	+	0.997261i	$0.523565\pi$
$488$	−1.07330e28	−0.634328
$489$	7.89308e27	0.456566
$490$	0	0
$491$	−1.18523e28	−0.656819	−0.328409	−	0.944535i	$-0.606513\pi$
$491$	−1.18523e28	−0.656819	−0.328409	+	0.944535i	$0.606513\pi$
$492$	3.69237e27	0.200296
$493$	−5.59808e27	−0.297267
$494$	1.21877e28	0.633559
$495$	0	0
$496$	−1.40918e28	−0.702115
$497$	−6.36952e27	−0.310717
$498$	1.37902e28	0.658663
$499$	1.95470e28	0.914166	0.457083	−	0.889424i	$-0.348894\pi$
$499$	1.95470e28	0.914166	0.457083	+	0.889424i	$0.348894\pi$
$500$	0	0
$501$	1.14942e28	0.515444
$502$	−2.06563e28	−0.907118
$503$	−3.70053e28	−1.59148	−0.795739	−	0.605640i	$-0.792917\pi$
$503$	−3.70053e28	−1.59148	−0.795739	+	0.605640i	$0.792917\pi$
$504$	3.66226e27	0.154251
$505$	0	0
$506$	1.19396e27	0.0482402
$507$	1.31477e28	0.520314
$508$	9.93013e27	0.384934
$509$	−1.86616e28	−0.708620	−0.354310	−	0.935128i	$-0.615284\pi$
$509$	−1.86616e28	−0.708620	−0.354310	+	0.935128i	$0.615284\pi$
$510$	0	0
$511$	1.04135e28	0.379470
$512$	3.41862e27	0.122044
$513$	9.30947e27	0.325606
$514$	4.59636e28	1.57508
$515$	0	0
$516$	−7.17246e27	−0.235964
$517$	5.80260e28	1.87056
$518$	2.03851e28	0.643946
$519$	3.31749e27	0.102696
$520$	0	0
$521$	3.75470e28	1.11630	0.558148	−	0.829741i	$-0.311512\pi$
$521$	3.75470e28	1.11630	0.558148	+	0.829741i	$0.311512\pi$
$522$	−1.35400e28	−0.394527
$523$	−4.77722e28	−1.36429	−0.682147	−	0.731215i	$-0.738953\pi$
$523$	−4.77722e28	−1.36429	−0.682147	+	0.731215i	$0.738953\pi$
$524$	2.48943e28	0.696821
$525$	0	0
$526$	−3.73241e28	−1.00378
$527$	6.40880e27	0.168953
$528$	−3.69270e28	−0.954307
$529$	−3.94350e28	−0.999072
$530$	0	0
$531$	7.93627e27	0.193252
$532$	1.98862e28	0.474765
$533$	1.11060e28	0.259970
$534$	1.91517e28	0.439566
$535$	0	0
$536$	4.97201e28	1.09724
$537$	−1.51075e28	−0.326936
$538$	3.90872e28	0.829508
$539$	3.52812e28	0.734279
$540$	0	0
$541$	3.28704e27	0.0658012	0.0329006	−	0.999459i	$-0.489526\pi$
$541$	3.28704e27	0.0658012	0.0329006	+	0.999459i	$0.489526\pi$
$542$	8.56510e28	1.68167
$543$	−3.17932e28	−0.612261
$544$	−1.25120e28	−0.236341
$545$	0	0
$546$	−7.95821e27	−0.144641
$547$	2.43473e28	0.434093	0.217047	−	0.976161i	$-0.430358\pi$
$547$	2.43473e28	0.434093	0.217047	+	0.976161i	$0.430358\pi$
$548$	2.63245e28	0.460431
$549$	1.78153e28	0.305692
$550$	0	0
$551$	1.01767e29	1.68080
$552$	−7.50690e26	−0.0121647
$553$	−1.79067e28	−0.284710
$554$	7.56726e28	1.18056
$555$	0	0
$556$	−1.18290e28	−0.177690
$557$	6.34849e28	0.935818	0.467909	−	0.883777i	$-0.345008\pi$
$557$	6.34849e28	0.935818	0.467909	+	0.883777i	$0.345008\pi$
$558$	1.55008e28	0.224231
$559$	−2.15736e28	−0.306265
$560$	0	0
$561$	1.67940e28	0.229639
$562$	1.38247e29	1.85534
$563$	1.37075e29	1.80559	0.902794	−	0.430072i	$-0.141512\pi$
$563$	1.37075e29	1.80559	0.902794	+	0.430072i	$0.141512\pi$
$564$	2.63576e28	0.340781
$565$	0	0
$566$	−9.50544e27	−0.118413
$567$	−6.07883e27	−0.0743358
$568$	−2.67603e28	−0.321243
$569$	−1.06962e29	−1.26053	−0.630263	−	0.776382i	$-0.717053\pi$
$569$	−1.06962e29	−1.26053	−0.630263	+	0.776382i	$0.717053\pi$
$570$	0	0
$571$	7.63012e28	0.866667	0.433334	−	0.901234i	$-0.357337\pi$
$571$	7.63012e28	0.866667	0.433334	+	0.901234i	$0.357337\pi$
$572$	1.57137e28	0.175234
$573$	9.46430e28	1.03625
$574$	6.13255e28	0.659275
$575$	0	0
$576$	9.72852e27	0.100835
$577$	−2.87183e28	−0.292289	−0.146144	−	0.989263i	$-0.546686\pi$
$577$	−2.87183e28	−0.292289	−0.146144	+	0.989263i	$0.546686\pi$
$578$	−1.08533e29	−1.08472
$579$	2.20099e28	0.216019
$580$	0	0
$581$	6.76793e28	0.640629
$582$	−6.25181e28	−0.581185
$583$	−1.71892e29	−1.56941
$584$	4.37502e28	0.392325
$585$	0	0
$586$	−1.72528e29	−1.49257
$587$	8.10915e28	0.689089	0.344545	−	0.938770i	$-0.388033\pi$
$587$	8.10915e28	0.689089	0.344545	+	0.938770i	$0.388033\pi$
$588$	1.60261e28	0.133772
$589$	−1.16505e29	−0.955287
$590$	0	0
$591$	7.45376e28	0.589803
$592$	1.29234e29	1.00462
$593$	−7.78815e28	−0.594785	−0.297393	−	0.954755i	$-0.596117\pi$
$593$	−7.78815e28	−0.594785	−0.297393	+	0.954755i	$0.596117\pi$
$594$	4.06194e28	0.304772
$595$	0	0
$596$	6.80191e28	0.492657
$597$	−6.08366e28	−0.432947
$598$	1.63127e27	0.0114068
$599$	8.84089e28	0.607455	0.303728	−	0.952759i	$-0.401769\pi$
$599$	8.84089e28	0.607455	0.303728	+	0.952759i	$0.401769\pi$
$600$	0	0
$601$	−1.61606e29	−1.07220	−0.536098	−	0.844156i	$-0.680102\pi$
$601$	−1.61606e29	−1.07220	−0.536098	+	0.844156i	$0.680102\pi$
$602$	−1.19125e29	−0.776678
$603$	−8.25284e28	−0.528776
$604$	9.72225e28	0.612180
$605$	0	0
$606$	1.03964e29	0.632296
$607$	−2.23740e29	−1.33740	−0.668702	−	0.743531i	$-0.733150\pi$
$607$	−2.23740e29	−1.33740	−0.668702	+	0.743531i	$0.733150\pi$
$608$	2.27455e29	1.33631
$609$	−6.64511e28	−0.383725
$610$	0	0
$611$	7.92793e28	0.442310
$612$	7.62848e27	0.0418358
$613$	−3.07543e29	−1.65795	−0.828974	−	0.559286i	$-0.811075\pi$
$613$	−3.07543e29	−1.65795	−0.828974	+	0.559286i	$0.811075\pi$
$614$	−4.09094e29	−2.16798
$615$	0	0
$616$	−1.20101e29	−0.615105
$617$	−4.19694e28	−0.211319	−0.105660	−	0.994402i	$-0.533695\pi$
$617$	−4.19694e28	−0.211319	−0.105660	+	0.994402i	$0.533695\pi$
$618$	1.88260e28	0.0931919
$619$	1.57471e28	0.0766390	0.0383195	−	0.999266i	$-0.487800\pi$
$619$	1.57471e28	0.0766390	0.0383195	+	0.999266i	$0.487800\pi$
$620$	0	0
$621$	1.24604e27	0.00586232
$622$	−3.51005e29	−1.62373
$623$	9.39923e28	0.427531
$624$	−5.04524e28	−0.225654
$625$	0	0
$626$	−3.82854e29	−1.65578
$627$	−3.05297e29	−1.29842
$628$	6.31907e28	0.264288
$629$	−5.87745e28	−0.241745
$630$	0	0
$631$	−4.85526e29	−1.93154	−0.965771	−	0.259395i	$-0.916477\pi$
$631$	−4.85526e29	−1.93154	−0.965771	+	0.259395i	$0.916477\pi$
$632$	−7.52317e28	−0.294355
$633$	5.87111e28	0.225934
$634$	−9.03376e28	−0.341925
$635$	0	0
$636$	−7.80799e28	−0.285917
$637$	4.82037e28	0.173627
$638$	4.44033e29	1.57325
$639$	4.44183e28	0.154811
$640$	0	0
$641$	3.13906e29	1.05874	0.529371	−	0.848391i	$-0.322428\pi$
$641$	3.13906e29	1.05874	0.529371	+	0.848391i	$0.322428\pi$
$642$	−2.91708e29	−0.967901
$643$	−3.57264e29	−1.16620	−0.583101	−	0.812399i	$-0.698161\pi$
$643$	−3.57264e29	−1.16620	−0.583101	+	0.812399i	$0.698161\pi$
$644$	2.66169e27	0.00854783
$645$	0	0
$646$	−1.94035e29	−0.603167
$647$	2.26919e28	0.0694026	0.0347013	−	0.999398i	$-0.488952\pi$
$647$	2.26919e28	0.0694026	0.0347013	+	0.999398i	$0.488952\pi$
$648$	−2.55390e28	−0.0768540
$649$	−2.60264e29	−0.770629
$650$	0	0
$651$	7.60746e28	0.218092
$652$	−1.17578e29	−0.331687
$653$	−4.18665e28	−0.116219	−0.0581096	−	0.998310i	$-0.518507\pi$
$653$	−4.18665e28	−0.116219	−0.0581096	+	0.998310i	$0.518507\pi$
$654$	3.39930e29	0.928589
$655$	0	0
$656$	3.88783e29	1.02853
$657$	−7.26192e28	−0.189067
$658$	4.37766e29	1.12168
$659$	8.56521e27	0.0215994	0.0107997	−	0.999942i	$-0.496562\pi$
$659$	8.56521e27	0.0215994	0.0107997	+	0.999942i	$0.496562\pi$
$660$	0	0
$661$	−8.93249e28	−0.218201	−0.109101	−	0.994031i	$-0.534797\pi$
$661$	−8.93249e28	−0.218201	−0.109101	+	0.994031i	$0.534797\pi$
$662$	7.13963e29	1.71659
$663$	2.29452e28	0.0543000
$664$	2.84342e29	0.662332
$665$	0	0
$666$	−1.42157e29	−0.320840
$667$	1.36211e28	0.0302616
$668$	−1.71222e29	−0.374461
$669$	3.56689e29	0.767918
$670$	0	0
$671$	−5.84240e29	−1.21900
$672$	−1.48522e29	−0.305080
$673$	7.65352e29	1.54776	0.773879	−	0.633334i	$-0.218314\pi$
$673$	7.65352e29	1.54776	0.773879	+	0.633334i	$0.218314\pi$
$674$	−6.79259e29	−1.35240
$675$	0	0
$676$	−1.95853e29	−0.377999
$677$	−7.75800e29	−1.47424	−0.737121	−	0.675761i	$-0.763815\pi$
$677$	−7.75800e29	−1.47424	−0.737121	+	0.675761i	$0.763815\pi$
$678$	7.12913e28	0.133391
$679$	−3.06825e29	−0.565272
$680$	0	0
$681$	−3.55076e29	−0.634274
$682$	−5.08338e29	−0.894164
$683$	1.31829e29	0.228345	0.114173	−	0.993461i	$-0.463578\pi$
$683$	1.31829e29	0.228345	0.114173	+	0.993461i	$0.463578\pi$
$684$	−1.38678e29	−0.236547
$685$	0	0
$686$	7.48010e29	1.23738
$687$	−2.83417e29	−0.461721
$688$	−7.55215e29	−1.21169
$689$	−2.34851e29	−0.371100
$690$	0	0
$691$	−7.37987e28	−0.113117	−0.0565587	−	0.998399i	$-0.518013\pi$
$691$	−7.37987e28	−0.113117	−0.0565587	+	0.998399i	$0.518013\pi$
$692$	−4.94187e28	−0.0746066
$693$	1.99351e29	0.296428
$694$	−1.40249e30	−2.05411
$695$	0	0
$696$	−2.79181e29	−0.396725
$697$	−1.76815e29	−0.247499
$698$	1.55510e30	2.14426
$699$	−2.31073e29	−0.313862
$700$	0	0
$701$	1.08205e30	1.42629	0.713146	−	0.701016i	$-0.247270\pi$
$701$	1.08205e30	1.42629	0.713146	+	0.701016i	$0.247270\pi$
$702$	5.54971e28	0.0720661
$703$	1.06846e30	1.36687
$704$	−3.19040e29	−0.402098
$705$	0	0
$706$	−6.45602e29	−0.789797
$707$	5.10232e29	0.614984
$708$	−1.18222e29	−0.140394
$709$	3.32002e29	0.388468	0.194234	−	0.980955i	$-0.437778\pi$
$709$	3.32002e29	0.388468	0.194234	+	0.980955i	$0.437778\pi$
$710$	0	0
$711$	1.24874e29	0.141854
$712$	3.94890e29	0.442015
$713$	−1.55938e28	−0.0171993
$714$	1.26699e29	0.137703
$715$	0	0
$716$	2.25047e29	0.237513
$717$	3.78916e29	0.394087
$718$	−1.33328e30	−1.36651
$719$	−6.85117e29	−0.692008	−0.346004	−	0.938233i	$-0.612462\pi$
$719$	−6.85117e29	−0.692008	−0.346004	+	0.938233i	$0.612462\pi$
$720$	0	0
$721$	9.23935e28	0.0906403
$722$	2.29509e30	2.21901
$723$	−9.12707e29	−0.869720
$724$	4.73605e29	0.444796
$725$	0	0
$726$	−5.78152e29	−0.527482
$727$	−4.54942e29	−0.409115	−0.204557	−	0.978855i	$-0.565576\pi$
$727$	−4.54942e29	−0.409115	−0.204557	+	0.978855i	$0.565576\pi$
$728$	−1.64091e29	−0.145447
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	3.43464e29	0.291574
$732$	−2.65384e29	−0.222079
$733$	8.27007e29	0.682209	0.341104	−	0.940025i	$-0.389199\pi$
$733$	8.27007e29	0.682209	0.341104	+	0.940025i	$0.389199\pi$
$734$	−2.54116e30	−2.06644
$735$	0	0
$736$	3.04441e28	0.0240595
$737$	2.70646e30	2.10859
$738$	−4.27658e29	−0.328477
$739$	1.90443e30	1.44211	0.721054	−	0.692879i	$-0.243658\pi$
$739$	1.90443e30	1.44211	0.721054	+	0.692879i	$0.243658\pi$
$740$	0	0
$741$	−4.17119e29	−0.307022
$742$	−1.29681e30	−0.941097
$743$	9.44840e29	0.676045	0.338023	−	0.941138i	$-0.390242\pi$
$743$	9.44840e29	0.676045	0.338023	+	0.941138i	$0.390242\pi$
$744$	3.19613e29	0.225480
$745$	0	0
$746$	1.17857e30	0.808348
$747$	−4.71967e29	−0.319187
$748$	−2.50171e29	−0.166828
$749$	−1.43164e30	−0.941400
$750$	0	0
$751$	1.71619e30	1.09735	0.548677	−	0.836034i	$-0.315132\pi$
$751$	1.71619e30	1.09735	0.548677	+	0.836034i	$0.315132\pi$
$752$	2.77529e30	1.74993
$753$	7.06957e29	0.439588
$754$	6.06670e29	0.372009
$755$	0	0
$756$	9.05527e28	0.0540036
$757$	−2.11711e29	−0.124520	−0.0622598	−	0.998060i	$-0.519831\pi$
$757$	−2.11711e29	−0.124520	−0.0622598	+	0.998060i	$0.519831\pi$
$758$	−3.05939e30	−1.77463
$759$	−4.08629e28	−0.0233771
$760$	0	0
$761$	1.32222e30	0.735810	0.367905	−	0.929863i	$-0.380075\pi$
$761$	1.32222e30	0.735810	0.367905	+	0.929863i	$0.380075\pi$
$762$	−1.15013e30	−0.631276
$763$	1.66830e30	0.903164
$764$	−1.40984e30	−0.752817
$765$	0	0
$766$	3.43043e30	1.78216
$767$	−3.55592e29	−0.182222
$768$	−1.21971e30	−0.616544
$769$	3.70600e30	1.84790	0.923950	−	0.382514i	$-0.124942\pi$
$769$	3.70600e30	1.84790	0.923950	+	0.382514i	$0.124942\pi$
$770$	0	0
$771$	−1.57309e30	−0.763279
$772$	−3.27868e29	−0.156934
$773$	−5.92668e29	−0.279851	−0.139926	−	0.990162i	$-0.544686\pi$
$773$	−5.92668e29	−0.279851	−0.139926	+	0.990162i	$0.544686\pi$
$774$	8.30729e29	0.386972
$775$	0	0
$776$	−1.28906e30	−0.584422
$777$	−6.97673e29	−0.312055
$778$	−3.61416e30	−1.59486
$779$	3.21430e30	1.39940
$780$	0	0
$781$	−1.45667e30	−0.617340
$782$	−2.59708e28	−0.0108596
$783$	4.63401e29	0.191187
$784$	1.68744e30	0.686927
$785$	0	0
$786$	−2.88331e30	−1.14276
$787$	−3.21544e30	−1.25749	−0.628747	−	0.777610i	$-0.716432\pi$
$787$	−3.21544e30	−1.25749	−0.628747	+	0.777610i	$0.716432\pi$
$788$	−1.11034e30	−0.428481
$789$	1.27741e30	0.486430
$790$	0	0
$791$	3.49882e29	0.129738
$792$	8.37533e29	0.306470
$793$	−7.98231e29	−0.288244
$794$	1.00362e30	0.357645
$795$	0	0
$796$	9.06246e29	0.314528
$797$	4.64788e30	1.59200	0.795999	−	0.605299i	$-0.206946\pi$
$797$	4.64788e30	1.59200	0.795999	+	0.605299i	$0.206946\pi$
$798$	−2.30326e30	−0.778595
$799$	−1.26217e30	−0.421093
$800$	0	0
$801$	−6.55462e29	−0.213013
$802$	3.42986e29	0.110013
$803$	2.38149e30	0.753939
$804$	1.22938e30	0.384146
$805$	0	0
$806$	−6.94529e29	−0.211433
$807$	−1.33775e30	−0.401978
$808$	2.14364e30	0.635817
$809$	4.17202e30	1.22148	0.610741	−	0.791831i	$-0.290872\pi$
$809$	4.17202e30	1.22148	0.610741	+	0.791831i	$0.290872\pi$
$810$	0	0
$811$	4.32133e30	1.23282	0.616408	−	0.787427i	$-0.288587\pi$
$811$	4.32133e30	1.23282	0.616408	+	0.787427i	$0.288587\pi$
$812$	9.89882e29	0.278769
$813$	−2.93138e30	−0.814933
$814$	4.66192e30	1.27941
$815$	0	0
$816$	8.03231e29	0.214830
$817$	−6.24380e30	−1.64861
$818$	3.82030e30	0.995835
$819$	2.72367e29	0.0700929
$820$	0	0
$821$	−4.90060e30	−1.22927	−0.614633	−	0.788813i	$-0.710696\pi$
$821$	−4.90060e30	−1.22927	−0.614633	+	0.788813i	$0.710696\pi$
$822$	−3.04896e30	−0.755088
$823$	2.48030e30	0.606465	0.303233	−	0.952917i	$-0.401934\pi$
$823$	2.48030e30	0.606465	0.303233	+	0.952917i	$0.401934\pi$
$824$	3.88173e29	0.0937109
$825$	0	0
$826$	−1.96351e30	−0.462108
$827$	5.50933e30	1.28024	0.640119	−	0.768275i	$-0.278885\pi$
$827$	5.50933e30	1.28024	0.640119	+	0.768275i	$0.278885\pi$
$828$	−1.85615e28	−0.00425887
$829$	7.39777e30	1.67602	0.838009	−	0.545657i	$-0.183720\pi$
$829$	7.39777e30	1.67602	0.838009	+	0.545657i	$0.183720\pi$
$830$	0	0
$831$	−2.58987e30	−0.572095
$832$	−4.35895e29	−0.0950795
$833$	−7.67431e29	−0.165298
$834$	1.37006e30	0.291404
$835$	0	0
$836$	4.54783e30	0.943275
$837$	−5.30512e29	−0.108662
$838$	−3.06605e30	−0.620179
$839$	−4.38789e30	−0.876507	−0.438253	−	0.898851i	$-0.644403\pi$
$839$	−4.38789e30	−0.876507	−0.438253	+	0.898851i	$0.644403\pi$
$840$	0	0
$841$	−6.71437e28	−0.0130812
$842$	6.76952e30	1.30251
$843$	−4.73146e30	−0.899095
$844$	−8.74583e29	−0.164137
$845$	0	0
$846$	−3.05279e30	−0.558867
$847$	−2.83744e30	−0.513040
$848$	−8.22132e30	−1.46820
$849$	3.25321e29	0.0573827
$850$	0	0
$851$	1.43009e29	0.0246095
$852$	−6.61673e29	−0.112468
$853$	−5.89309e29	−0.0989414	−0.0494707	−	0.998776i	$-0.515753\pi$
$853$	−5.89309e29	−0.0989414	−0.0494707	+	0.998776i	$0.515753\pi$
$854$	−4.40769e30	−0.730976
$855$	0	0
$856$	−6.01475e30	−0.973292
$857$	4.97398e30	0.795069	0.397535	−	0.917587i	$-0.369866\pi$
$857$	4.97398e30	0.795069	0.397535	+	0.917587i	$0.369866\pi$
$858$	−1.81999e30	−0.287377
$859$	5.32465e30	0.830545	0.415272	−	0.909697i	$-0.363686\pi$
$859$	5.32465e30	0.830545	0.415272	+	0.909697i	$0.363686\pi$
$860$	0	0
$861$	−2.09885e30	−0.319483
$862$	−1.01446e31	−1.52548
$863$	9.22156e30	1.36991	0.684954	−	0.728587i	$-0.259822\pi$
$863$	9.22156e30	1.36991	0.684954	+	0.728587i	$0.259822\pi$
$864$	1.03573e30	0.152003
$865$	0	0
$866$	1.20533e31	1.72651
$867$	3.71451e30	0.525655
$868$	−1.13324e30	−0.158440
$869$	−4.09515e30	−0.565669
$870$	0	0
$871$	3.69776e30	0.498595
$872$	7.00904e30	0.933760
$873$	2.13967e30	0.281641
$874$	4.72122e29	0.0614022
$875$	0	0
$876$	1.08177e30	0.137354
$877$	3.49398e30	0.438354	0.219177	−	0.975685i	$-0.429663\pi$
$877$	3.49398e30	0.438354	0.219177	+	0.975685i	$0.429663\pi$
$878$	−1.03124e31	−1.27840
$879$	5.90473e30	0.723297
$880$	0	0
$881$	1.35804e31	1.62430	0.812148	−	0.583452i	$-0.198298\pi$
$881$	1.35804e31	1.62430	0.812148	+	0.583452i	$0.198298\pi$
$882$	−1.85617e30	−0.219380
$883$	−3.71189e30	−0.433519	−0.216759	−	0.976225i	$-0.569549\pi$
$883$	−3.71189e30	−0.433519	−0.216759	+	0.976225i	$0.569549\pi$
$884$	−3.41801e29	−0.0394480
$885$	0	0
$886$	−8.16540e30	−0.920287
$887$	−2.33212e30	−0.259748	−0.129874	−	0.991531i	$-0.541457\pi$
$887$	−2.33212e30	−0.259748	−0.129874	+	0.991531i	$0.541457\pi$
$888$	−2.93114e30	−0.322626
$889$	−5.64457e30	−0.613991
$890$	0	0
$891$	−1.39019e30	−0.147692
$892$	−5.31337e30	−0.557878
$893$	2.29449e31	2.38093
$894$	−7.87811e30	−0.807938
$895$	0	0
$896$	−7.68176e30	−0.769532
$897$	−5.58299e28	−0.00552772
$898$	−1.08053e31	−1.05739
$899$	−5.79932e30	−0.560919
$900$	0	0
$901$	3.73897e30	0.353299
$902$	1.40247e31	1.30986
$903$	4.07703e30	0.376377
$904$	1.46996e30	0.134133
$905$	0	0
$906$	−1.12605e31	−1.00395
$907$	1.21247e31	1.06855	0.534273	−	0.845312i	$-0.320585\pi$
$907$	1.21247e31	1.06855	0.534273	+	0.845312i	$0.320585\pi$
$908$	5.28936e30	0.460789
$909$	−3.55814e30	−0.306409
$910$	0	0
$911$	5.96633e30	0.502070	0.251035	−	0.967978i	$-0.419229\pi$
$911$	5.96633e30	0.502070	0.251035	+	0.967978i	$0.419229\pi$
$912$	−1.46019e31	−1.21468
$913$	1.54778e31	1.27282
$914$	−2.63142e31	−2.13922
$915$	0	0
$916$	4.22189e30	0.335432
$917$	−1.41506e31	−1.11147
$918$	−8.83547e29	−0.0686091
$919$	−2.27380e31	−1.74558	−0.872791	−	0.488094i	$-0.837692\pi$
$919$	−2.27380e31	−1.74558	−0.872791	+	0.488094i	$0.837692\pi$
$920$	0	0
$921$	1.40011e31	1.05060
$922$	1.60101e31	1.18774
$923$	−1.99020e30	−0.145975
$924$	−2.96961e30	−0.215349
$925$	0	0
$926$	−5.87330e30	−0.416358
$927$	−6.44313e29	−0.0451606
$928$	1.13221e31	0.784648
$929$	1.61203e30	0.110461	0.0552305	−	0.998474i	$-0.482411\pi$
$929$	1.61203e30	0.110461	0.0552305	+	0.998474i	$0.482411\pi$
$930$	0	0
$931$	1.39511e31	0.934622
$932$	3.44215e30	0.228015
$933$	1.20130e31	0.786857
$934$	−2.28798e31	−1.48187
$935$	0	0
$936$	1.14430e30	0.0724674
$937$	−2.22522e31	−1.39350	−0.696751	−	0.717313i	$-0.745372\pi$
$937$	−2.22522e31	−1.39350	−0.696751	+	0.717313i	$0.745372\pi$
$938$	2.04183e31	1.26442
$939$	1.31031e31	0.802387
$940$	0	0
$941$	1.29802e31	0.777305	0.388652	−	0.921384i	$-0.372941\pi$
$941$	1.29802e31	0.777305	0.388652	+	0.921384i	$0.372941\pi$
$942$	−7.31888e30	−0.433421
$943$	4.30222e29	0.0251953
$944$	−1.24480e31	−0.720932
$945$	0	0
$946$	−2.72431e31	−1.54312
$947$	1.16388e31	0.651981	0.325990	−	0.945373i	$-0.394302\pi$
$947$	1.16388e31	0.651981	0.325990	+	0.945373i	$0.394302\pi$
$948$	−1.86017e30	−0.103054
$949$	3.25377e30	0.178276
$950$	0	0
$951$	3.09178e30	0.165696
$952$	2.61242e30	0.138470
$953$	−1.96051e30	−0.102776	−0.0513881	−	0.998679i	$-0.516365\pi$
$953$	−1.96051e30	−0.102776	−0.0513881	+	0.998679i	$0.516365\pi$
$954$	9.04338e30	0.468892
$955$	0	0
$956$	−5.64449e30	−0.286297
$957$	−1.51969e31	−0.762395
$958$	−1.57262e31	−0.780346
$959$	−1.49636e31	−0.734414
$960$	0	0
$961$	−1.41863e31	−0.681199
$962$	6.36946e30	0.302527
$963$	9.98364e30	0.469043
$964$	1.35960e31	0.631835
$965$	0	0
$966$	−3.08283e29	−0.0140181
$967$	2.56034e31	1.15165	0.575823	−	0.817574i	$-0.304682\pi$
$967$	2.56034e31	1.15165	0.575823	+	0.817574i	$0.304682\pi$
$968$	−1.19209e31	−0.530420
$969$	6.64078e30	0.292294
$970$	0	0
$971$	2.52049e31	1.08563	0.542816	−	0.839852i	$-0.317358\pi$
$971$	2.52049e31	1.08563	0.542816	+	0.839852i	$0.317358\pi$
$972$	−6.31476e29	−0.0269067
$973$	6.72393e30	0.283425
$974$	−4.22644e30	−0.176240
$975$	0	0
$976$	−2.79433e31	−1.14039
$977$	−1.51336e31	−0.611010	−0.305505	−	0.952190i	$-0.598825\pi$
$977$	−1.51336e31	−0.611010	−0.305505	+	0.952190i	$0.598825\pi$
$978$	1.36182e31	0.543953
$979$	2.14954e31	0.849430
$980$	0	0
$981$	−1.16340e31	−0.449992
$982$	−2.04491e31	−0.782534
$983$	−1.11411e31	−0.421810	−0.210905	−	0.977507i	$-0.567641\pi$
$983$	−1.11411e31	−0.421810	−0.210905	+	0.977507i	$0.567641\pi$
$984$	−8.81790e30	−0.330306
$985$	0	0
$986$	−9.65854e30	−0.354164
$987$	−1.49824e31	−0.543565
$988$	6.21357e30	0.223045
$989$	−8.35710e29	−0.0296821
$990$	0	0
$991$	−5.26732e31	−1.83154	−0.915770	−	0.401704i	$-0.868418\pi$
$991$	−5.26732e31	−1.83154	−0.915770	+	0.401704i	$0.868418\pi$
$992$	−1.29618e31	−0.445958
$993$	−2.44352e31	−0.831856
$994$	−1.09895e31	−0.370188
$995$	0	0
$996$	7.03061e30	0.231883
$997$	−4.55662e31	−1.48711	−0.743556	−	0.668673i	$-0.766863\pi$
$997$	−4.55662e31	−1.48711	−0.743556	+	0.668673i	$0.766863\pi$
$998$	3.37251e31	1.08914
$999$	4.86527e30	0.155478

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.l.1.6	yes	8
5.2	odd	4		75.22.b.j.49.12		16
5.3	odd	4		75.22.b.j.49.5		16
5.4	even	2		75.22.a.k.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.22.a.k.1.3	✓	8	5.4	even	2
75.22.a.l.1.6	yes	8	1.1	even	1	trivial
75.22.b.j.49.5		16	5.3	odd	4
75.22.b.j.49.12		16	5.2	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)

\(f(q)\)	\(=\)	\(q+1725.33 q^{2} -59049.0 q^{3} +879617. q^{4} -1.01879e8 q^{6} -5.00000e8 q^{7} -2.10065e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q+1725.33 q^{2} -59049.0 q^{3} +879617. q^{4} -1.01879e8 q^{6} -5.00000e8 q^{7} -2.10065e9 q^{8} +3.48678e9 q^{9} -1.14347e11 q^{11} -5.19405e10 q^{12} -1.56229e11 q^{13} -8.62665e11 q^{14} -5.46901e12 q^{16} +2.48725e12 q^{17} +6.01586e12 q^{18} -4.52155e13 q^{19} +2.95245e13 q^{21} -1.97286e14 q^{22} -6.05193e12 q^{23} +1.24041e14 q^{24} -2.69546e14 q^{26} -2.05891e14 q^{27} -4.39808e14 q^{28} -2.25071e15 q^{29} +2.57666e15 q^{31} -5.03047e15 q^{32} +6.75205e15 q^{33} +4.29133e15 q^{34} +3.06704e15 q^{36} -2.36303e16 q^{37} -7.80117e16 q^{38} +9.22514e15 q^{39} -7.10884e16 q^{41} +5.09395e16 q^{42} +1.38090e17 q^{43} -1.00581e17 q^{44} -1.04416e16 q^{46} -5.07457e17 q^{47} +3.22940e17 q^{48} -3.08546e17 q^{49} -1.46870e17 q^{51} -1.37421e17 q^{52} +1.50326e18 q^{53} -3.55230e17 q^{54} +1.05032e18 q^{56} +2.66993e18 q^{57} -3.88322e18 q^{58} +2.27610e18 q^{59} +5.10938e18 q^{61} +4.44559e18 q^{62} -1.74339e18 q^{63} +2.79011e18 q^{64} +1.16495e19 q^{66} -2.36689e19 q^{67} +2.18783e18 q^{68} +3.57361e17 q^{69} +1.27391e19 q^{71} -7.32452e18 q^{72} -2.08270e19 q^{73} -4.07701e19 q^{74} -3.97723e19 q^{76} +5.71732e19 q^{77} +1.59164e19 q^{78} +3.58135e19 q^{79} +1.21577e19 q^{81} -1.22651e20 q^{82} -1.35359e20 q^{83} +2.59702e19 q^{84} +2.38251e20 q^{86} +1.32902e20 q^{87} +2.40202e20 q^{88} -1.87985e20 q^{89} +7.81142e19 q^{91} -5.32338e18 q^{92} -1.52149e20 q^{93} -8.75532e20 q^{94} +2.97044e20 q^{96} +6.13650e20 q^{97} -5.32344e20 q^{98} -3.98702e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9}+O(q^{10}) \) 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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