LMFDB - Embedded newform 75.22.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$209.608008215$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6395796x - 2792983104 $ x^3 - x^2 - 6395796*x - 2792983104
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-451.072$ 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)

$f(q)$	$=$	$q-719.072 q^{2} -59049.0 q^{3} -1.58009e6 q^{4} +4.24605e7 q^{6} +1.45023e9 q^{7} +2.64420e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-719.072 q^{2} -59049.0 q^{3} -1.58009e6 q^{4} +4.24605e7 q^{6} +1.45023e9 q^{7} +2.64420e9 q^{8} +3.48678e9 q^{9} +5.34415e10 q^{11} +9.33026e10 q^{12} -6.66061e10 q^{13} -1.04282e12 q^{14} +1.41231e12 q^{16} -6.72914e12 q^{17} -2.50725e12 q^{18} +5.05646e13 q^{19} -8.56345e13 q^{21} -3.84283e13 q^{22} -7.33880e13 q^{23} -1.56137e14 q^{24} +4.78946e13 q^{26} -2.05891e14 q^{27} -2.29149e15 q^{28} -2.61187e15 q^{29} -3.33577e15 q^{31} -6.56084e15 q^{32} -3.15567e15 q^{33} +4.83873e15 q^{34} -5.50942e15 q^{36} +1.04258e16 q^{37} -3.63596e16 q^{38} +3.93302e15 q^{39} -1.21333e17 q^{41} +6.15773e16 q^{42} -1.69796e17 q^{43} -8.44422e16 q^{44} +5.27713e16 q^{46} -3.98939e17 q^{47} -8.33957e16 q^{48} +1.54461e18 q^{49} +3.97349e17 q^{51} +1.05243e17 q^{52} +2.16226e17 q^{53} +1.48051e17 q^{54} +3.83469e18 q^{56} -2.98579e18 q^{57} +1.87812e18 q^{58} +4.31686e18 q^{59} +1.13584e18 q^{61} +2.39866e18 q^{62} +5.05663e18 q^{63} +1.75588e18 q^{64} +2.26915e18 q^{66} +1.49163e18 q^{67} +1.06326e19 q^{68} +4.33349e18 q^{69} -2.51061e19 q^{71} +9.21975e18 q^{72} -3.19214e19 q^{73} -7.49687e18 q^{74} -7.98966e19 q^{76} +7.75023e19 q^{77} -2.82813e18 q^{78} +7.75405e19 q^{79} +1.21577e19 q^{81} +8.72468e19 q^{82} +2.62543e19 q^{83} +1.35310e20 q^{84} +1.22096e20 q^{86} +1.54228e20 q^{87} +1.41310e20 q^{88} -4.67811e20 q^{89} -9.65940e19 q^{91} +1.15960e20 q^{92} +1.96974e20 q^{93} +2.86866e20 q^{94} +3.87411e20 q^{96} -8.87155e20 q^{97} -1.11069e21 q^{98} +1.86339e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 803 q^{2} - 177147 q^{3} + 6715073 q^{4} + 47416347 q^{6} + 1577598316 q^{7} + 1424191017 q^{8} + 10460353203 q^{9}+O(q^{10}) $ 3 * q - 803 * q^2 - 177147 * q^3 + 6715073 * q^4 + 47416347 * q^6 + 1577598316 * q^7 + 1424191017 * q^8 + 10460353203 * q^9	$ 3 q - 803 q^{2} - 177147 q^{3} + 6715073 q^{4} + 47416347 q^{6} + 1577598316 q^{7} + 1424191017 q^{8} + 10460353203 q^{9} + 84497282000 q^{11} - 396518345577 q^{12} - 1065489966310 q^{13} + 1543881561348 q^{14} + 9712801855841 q^{16} - 13851876239906 q^{17} - 2799887874003 q^{18} + 26858848298644 q^{19} - 93155602961484 q^{21} + 93991312008688 q^{22} - 75776598293952 q^{23} - 84097055362833 q^{24} + 16\!\cdots\!86 q^{26}+ \cdots + 29\!\cdots\!00 q^{99}+O(q^{100}) $ 3 * q - 803 * q^2 - 177147 * q^3 + 6715073 * q^4 + 47416347 * q^6 + 1577598316 * q^7 + 1424191017 * q^8 + 10460353203 * q^9 + 84497282000 * q^11 - 396518345577 * q^12 - 1065489966310 * q^13 + 1543881561348 * q^14 + 9712801855841 * q^16 - 13851876239906 * q^17 - 2799887874003 * q^18 + 26858848298644 * q^19 - 93155602961484 * q^21 + 93991312008688 * q^22 - 75776598293952 * q^23 - 84097055362833 * q^24 + 1616971789034486 * q^26 - 617673396283947 * q^27 - 1980748081451468 * q^28 - 911172744177122 * q^29 - 14770635893644696 * q^31 - 10845641917892839 * q^32 - 4989480004818000 * q^33 - 14314214801416654 * q^34 + 23414011787976273 * q^36 + 33876604648537146 * q^37 - 187726010316740996 * q^38 + 62916117020639190 * q^39 + 4403226550677934 * q^41 - 91164662316038052 * q^42 + 76568115811359884 * q^43 + 33141017859775792 * q^44 + 260351976204717264 * q^46 - 178877159689418912 * q^47 - 573531236785555209 * q^48 + 973733149877485627 * q^49 + 817939440090209394 * q^51 - 4165878936433955826 * q^52 - 4013622553495571102 * q^53 + 165330579072003147 * q^54 + 9080922748836602580 * q^56 - 1585988133186629556 * q^57 - 12459376778387397982 * q^58 - 4476302860362619024 * q^59 + 13672812736400197042 * q^61 + 10065431538796030152 * q^62 + 5500745199272668716 * q^63 - 17161949036537140719 * q^64 - 5550092982801017712 * q^66 - 10024292550718281716 * q^67 - 17276944334927178886 * q^68 + 4474532352659571648 * q^69 + 31548512648354310016 * q^71 + 4965847022119925817 * q^72 - 80637656440064589382 * q^73 - 97250438023995188522 * q^74 - 165430884440717923764 * q^76 + 107240569572877645632 * q^77 - 95480567170697363814 * q^78 - 101660087769974505880 * q^79 + 36472996377170786403 * q^81 + 608532996859944197042 * q^82 + 98643502167172343676 * q^83 + 116961193461627733932 * q^84 - 181033595702025175180 * q^86 + 53803839370914876978 * q^87 + 397025854664543013552 * q^88 + 49757751372279077814 * q^89 + 156825279702460168744 * q^91 + 88639329782864857776 * q^92 + 872191278883825654104 * q^93 + 2329033824323004369680 * q^94 + 640424309609654250111 * q^96 - 767965200855128324166 * q^97 - 756579208914437470555 * q^98 + 294623804804498082000 * 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 $

$2$	−719.072	−0.496544	−0.248272	−	0.968690i	$-0.579863\pi$
$2$	−719.072	−0.496544	−0.248272	+	0.968690i	$0.579863\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	−1.58009e6	−0.753444
$5$	0	0
$5$	0	0
$6$	4.24605e7	0.286680
$7$	1.45023e9	1.94047	0.970234	−	0.242169i	$-0.0778587\pi$
$7$	1.45023e9	1.94047	0.970234	+	0.242169i	$0.0778587\pi$
$8$	2.64420e9	0.870662
$9$	3.48678e9	0.333333
$10$	0	0
$11$	5.34415e10	0.621234	0.310617	−	0.950535i	$-0.399464\pi$
$11$	5.34415e10	0.621234	0.310617	+	0.950535i	$0.399464\pi$
$12$	9.33026e10	0.435001
$13$	−6.66061e10	−0.134001	−0.0670006	−	0.997753i	$-0.521343\pi$
$13$	−6.66061e10	−0.134001	−0.0670006	+	0.997753i	$0.521343\pi$
$14$	−1.04282e12	−0.963527
$15$	0	0
$16$	1.41231e12	0.321123
$17$	−6.72914e12	−0.809554	−0.404777	−	0.914415i	$-0.632651\pi$
$17$	−6.72914e12	−0.809554	−0.404777	+	0.914415i	$0.632651\pi$
$18$	−2.50725e12	−0.165515
$19$	5.05646e13	1.89206	0.946028	−	0.324084i	$-0.105056\pi$
$19$	5.05646e13	1.89206	0.946028	+	0.324084i	$0.105056\pi$
$20$	0	0
$21$	−8.56345e13	−1.12033
$22$	−3.84283e13	−0.308470
$23$	−7.33880e13	−0.369388	−0.184694	−	0.982796i	$-0.559129\pi$
$23$	−7.33880e13	−0.369388	−0.184694	+	0.982796i	$0.559129\pi$
$24$	−1.56137e14	−0.502677
$25$	0	0
$26$	4.78946e13	0.0665375
$27$	−2.05891e14	−0.192450
$28$	−2.29149e15	−1.46204
$29$	−2.61187e15	−1.15285	−0.576425	−	0.817150i	$-0.695553\pi$
$29$	−2.61187e15	−1.15285	−0.576425	+	0.817150i	$0.695553\pi$
$30$	0	0
$31$	−3.33577e15	−0.730968	−0.365484	−	0.930818i	$-0.619097\pi$
$31$	−3.33577e15	−0.730968	−0.365484	+	0.930818i	$0.619097\pi$
$32$	−6.56084e15	−1.03011
$33$	−3.15567e15	−0.358670
$34$	4.83873e15	0.401979
$35$	0	0
$36$	−5.50942e15	−0.251148
$37$	1.04258e16	0.356443	0.178221	−	0.983990i	$-0.442966\pi$
$37$	1.04258e16	0.356443	0.178221	+	0.983990i	$0.442966\pi$
$38$	−3.63596e16	−0.939489
$39$	3.93302e15	0.0773657
$40$	0	0
$41$	−1.21333e17	−1.41171	−0.705857	−	0.708354i	$-0.749438\pi$
$41$	−1.21333e17	−1.41171	−0.705857	+	0.708354i	$0.749438\pi$
$42$	6.15773e16	0.556293
$43$	−1.69796e17	−1.19815	−0.599073	−	0.800695i	$-0.704464\pi$
$43$	−1.69796e17	−1.19815	−0.599073	+	0.800695i	$0.704464\pi$
$44$	−8.44422e16	−0.468066
$45$	0	0
$46$	5.27713e16	0.183417
$47$	−3.98939e17	−1.10631	−0.553157	−	0.833077i	$-0.686577\pi$
$47$	−3.98939e17	−1.10631	−0.553157	+	0.833077i	$0.686577\pi$
$48$	−8.33957e16	−0.185400
$49$	1.54461e18	2.76542
$50$	0	0
$51$	3.97349e17	0.467396
$52$	1.05243e17	0.100963
$53$	2.16226e17	0.169829	0.0849146	−	0.996388i	$-0.472938\pi$
$53$	2.16226e17	0.169829	0.0849146	+	0.996388i	$0.472938\pi$
$54$	1.48051e17	0.0955599
$55$	0	0
$56$	3.83469e18	1.68949
$57$	−2.98579e18	−1.09238
$58$	1.87812e18	0.572440
$59$	4.31686e18	1.09957	0.549783	−	0.835307i	$-0.314710\pi$
$59$	4.31686e18	1.09957	0.549783	+	0.835307i	$0.314710\pi$
$60$	0	0
$61$	1.13584e18	0.203871	0.101935	−	0.994791i	$-0.467497\pi$
$61$	1.13584e18	0.203871	0.101935	+	0.994791i	$0.467497\pi$
$62$	2.39866e18	0.362958
$63$	5.05663e18	0.646823
$64$	1.75588e18	0.190373
$65$	0	0
$66$	2.26915e18	0.178095
$67$	1.49163e18	0.0999711	0.0499855	−	0.998750i	$-0.484082\pi$
$67$	1.49163e18	0.0999711	0.0499855	+	0.998750i	$0.484082\pi$
$68$	1.06326e19	0.609954
$69$	4.33349e18	0.213266
$70$	0	0
$71$	−2.51061e19	−0.915306	−0.457653	−	0.889131i	$-0.651310\pi$
$71$	−2.51061e19	−0.915306	−0.457653	+	0.889131i	$0.651310\pi$
$72$	9.21975e18	0.290221
$73$	−3.19214e19	−0.869345	−0.434672	−	0.900589i	$-0.643136\pi$
$73$	−3.19214e19	−0.869345	−0.434672	+	0.900589i	$0.643136\pi$
$74$	−7.49687e18	−0.176989
$75$	0	0
$76$	−7.98966e19	−1.42556
$77$	7.75023e19	1.20549
$78$	−2.82813e18	−0.0384154
$79$	7.75405e19	0.921391	0.460696	−	0.887558i	$-0.347600\pi$
$79$	7.75405e19	0.921391	0.460696	+	0.887558i	$0.347600\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	8.72468e19	0.700978
$83$	2.62543e19	0.185729	0.0928646	−	0.995679i	$-0.470398\pi$
$83$	2.62543e19	0.185729	0.0928646	+	0.995679i	$0.470398\pi$
$84$	1.35310e20	0.844106
$85$	0	0
$86$	1.22096e20	0.594931
$87$	1.54228e20	0.665598
$88$	1.41310e20	0.540885
$89$	−4.67811e20	−1.59029	−0.795144	−	0.606421i	$-0.792605\pi$
$89$	−4.67811e20	−1.59029	−0.795144	+	0.606421i	$0.792605\pi$
$90$	0	0
$91$	−9.65940e19	−0.260025
$92$	1.15960e20	0.278313
$93$	1.96974e20	0.422025
$94$	2.86866e20	0.549334
$95$	0	0
$96$	3.87411e20	0.594736
$97$	−8.87155e20	−1.22151	−0.610754	−	0.791820i	$-0.709134\pi$
$97$	−8.87155e20	−1.22151	−0.610754	+	0.791820i	$0.709134\pi$
$98$	−1.11069e21	−1.37315
$99$	1.86339e20	0.207078
$100$	0	0
$101$	1.60924e21	1.44960	0.724799	−	0.688960i	$-0.241933\pi$
$101$	1.60924e21	1.44960	0.724799	+	0.688960i	$0.241933\pi$
$102$	−2.85722e20	−0.232083
$103$	1.75803e21	1.28895	0.644475	−	0.764625i	$-0.277076\pi$
$103$	1.75803e21	1.28895	0.644475	+	0.764625i	$0.277076\pi$
$104$	−1.76120e20	−0.116670
$105$	0	0
$106$	−1.55482e20	−0.0843276
$107$	−6.16929e20	−0.303184	−0.151592	−	0.988443i	$-0.548440\pi$
$107$	−6.16929e20	−0.303184	−0.151592	+	0.988443i	$0.548440\pi$
$108$	3.25326e20	0.145000
$109$	1.40972e21	0.570369	0.285185	−	0.958473i	$-0.407945\pi$
$109$	1.40972e21	0.570369	0.285185	+	0.958473i	$0.407945\pi$
$110$	0	0
$111$	−6.15631e20	−0.205792
$112$	2.04818e21	0.623129
$113$	−6.89762e21	−1.91151	−0.955753	−	0.294170i	$-0.904957\pi$
$113$	−6.89762e21	−1.91151	−0.955753	+	0.294170i	$0.904957\pi$
$114$	2.14700e21	0.542414
$115$	0	0
$116$	4.12699e21	0.868608
$117$	−2.32241e20	−0.0446671
$118$	−3.10413e21	−0.545983
$119$	−9.75878e21	−1.57091
$120$	0	0
$121$	−4.54426e21	−0.614068
$122$	−8.16753e20	−0.101231
$123$	7.16457e21	0.815054
$124$	5.27081e21	0.550744
$125$	0	0
$126$	−3.63608e21	−0.321176
$127$	−1.77824e21	−0.144561	−0.0722807	−	0.997384i	$-0.523028\pi$
$127$	−1.77824e21	−0.144561	−0.0722807	+	0.997384i	$0.523028\pi$
$128$	1.24965e22	0.935585
$129$	1.00263e22	0.691750
$130$	0	0
$131$	8.48898e21	0.498318	0.249159	−	0.968463i	$-0.419846\pi$
$131$	8.48898e21	0.498318	0.249159	+	0.968463i	$0.419846\pi$
$132$	4.98623e21	0.270238
$133$	7.33302e22	3.67148
$134$	−1.07259e21	−0.0496400
$135$	0	0
$136$	−1.77932e22	−0.704848
$137$	−5.12067e22	−1.87828	−0.939140	−	0.343534i	$-0.888376\pi$
$137$	−5.12067e22	−1.87828	−0.939140	+	0.343534i	$0.888376\pi$
$138$	−3.11609e21	−0.105896
$139$	−4.53556e22	−1.42881	−0.714407	−	0.699731i	$-0.753303\pi$
$139$	−4.53556e22	−1.42881	−0.714407	+	0.699731i	$0.753303\pi$
$140$	0	0
$141$	2.35569e22	0.638731
$142$	1.80531e22	0.454489
$143$	−3.55953e21	−0.0832462
$144$	4.92443e21	0.107041
$145$	0	0
$146$	2.29538e22	0.431668
$147$	−9.12078e22	−1.59661
$148$	−1.64736e22	−0.268560
$149$	8.87592e22	1.34821	0.674105	−	0.738635i	$-0.264529\pi$
$149$	8.87592e22	1.34821	0.674105	+	0.738635i	$0.264529\pi$
$150$	0	0
$151$	2.20804e22	0.291574	0.145787	−	0.989316i	$-0.453429\pi$
$151$	2.20804e22	0.291574	0.145787	+	0.989316i	$0.453429\pi$
$152$	1.33703e23	1.64734
$153$	−2.34631e22	−0.269851
$154$	−5.57297e22	−0.598576
$155$	0	0
$156$	−6.21452e21	−0.0582907
$157$	−1.24312e22	−0.109035	−0.0545177	−	0.998513i	$-0.517362\pi$
$157$	−1.24312e22	−0.109035	−0.0545177	+	0.998513i	$0.517362\pi$
$158$	−5.57572e22	−0.457511
$159$	−1.27680e22	−0.0980509
$160$	0	0
$161$	−1.06429e23	−0.716786
$162$	−8.74224e21	−0.0551715
$163$	−1.59873e23	−0.945815	−0.472907	−	0.881112i	$-0.656796\pi$
$163$	−1.59873e23	−0.945815	−0.472907	+	0.881112i	$0.656796\pi$
$164$	1.91716e23	1.06365
$165$	0	0
$166$	−1.88787e22	−0.0922227
$167$	−2.09458e23	−0.960670	−0.480335	−	0.877085i	$-0.659485\pi$
$167$	−2.09458e23	−0.960670	−0.480335	+	0.877085i	$0.659485\pi$
$168$	−2.26435e23	−0.975428
$169$	−2.42628e23	−0.982044
$170$	0	0
$171$	1.76308e23	0.630686
$172$	2.68293e23	0.902736
$173$	8.91449e22	0.282236	0.141118	−	0.989993i	$-0.454930\pi$
$173$	8.91449e22	0.282236	0.141118	+	0.989993i	$0.454930\pi$
$174$	−1.10901e23	−0.330499
$175$	0	0
$176$	7.54762e22	0.199493
$177$	−2.54906e23	−0.634835
$178$	3.36390e23	0.789647
$179$	6.46845e23	1.43167	0.715836	−	0.698269i	$-0.246046\pi$
$179$	6.46845e23	1.43167	0.715836	+	0.698269i	$0.246046\pi$
$180$	0	0
$181$	−1.35874e23	−0.267615	−0.133807	−	0.991007i	$-0.542720\pi$
$181$	−1.35874e23	−0.267615	−0.133807	+	0.991007i	$0.542720\pi$
$182$	6.94580e22	0.129114
$183$	−6.70704e22	−0.117705
$184$	−1.94053e23	−0.321612
$185$	0	0
$186$	−1.41638e23	−0.209554
$187$	−3.59615e23	−0.502923
$188$	6.30358e23	0.833547
$189$	−2.98589e23	−0.373443
$190$	0	0
$191$	−5.94980e23	−0.666273	−0.333136	−	0.942879i	$-0.608107\pi$
$191$	−5.94980e23	−0.666273	−0.333136	+	0.942879i	$0.608107\pi$
$192$	−1.03683e23	−0.109912
$193$	3.65629e23	0.367019	0.183510	−	0.983018i	$-0.441254\pi$
$193$	3.65629e23	0.367019	0.183510	+	0.983018i	$0.441254\pi$
$194$	6.37928e23	0.606532
$195$	0	0
$196$	−2.44062e24	−2.08359
$197$	2.05182e24	1.66052	0.830259	−	0.557377i	$-0.188192\pi$
$197$	2.05182e24	1.66052	0.830259	+	0.557377i	$0.188192\pi$
$198$	−1.33991e23	−0.102823
$199$	−1.28088e22	−0.00932289	−0.00466145	−	0.999989i	$-0.501484\pi$
$199$	−1.28088e22	−0.00932289	−0.00466145	+	0.999989i	$0.501484\pi$
$200$	0	0
$201$	−8.80790e22	−0.0577183
$202$	−1.15716e24	−0.719789
$203$	−3.78781e24	−2.23707
$204$	−6.27846e23	−0.352157
$205$	0	0
$206$	−1.26415e24	−0.640020
$207$	−2.55888e23	−0.123129
$208$	−9.40687e22	−0.0430309
$209$	2.70225e24	1.17541
$210$	0	0
$211$	−3.15546e24	−1.24193	−0.620964	−	0.783839i	$-0.713259\pi$
$211$	−3.15546e24	−1.24193	−0.620964	+	0.783839i	$0.713259\pi$
$212$	−3.41657e23	−0.127957
$213$	1.48249e24	0.528452
$214$	4.43617e23	0.150544
$215$	0	0
$216$	−5.44417e23	−0.167559
$217$	−4.83763e24	−1.41842
$218$	−1.01369e24	−0.283213
$219$	1.88493e24	0.501916
$220$	0	0
$221$	4.48202e23	0.108481
$222$	4.42683e23	0.102185
$223$	−6.19879e24	−1.36492	−0.682458	−	0.730925i	$-0.739089\pi$
$223$	−6.19879e24	−1.36492	−0.682458	+	0.730925i	$0.739089\pi$
$224$	−9.51471e24	−1.99890
$225$	0	0
$226$	4.95989e24	0.949146
$227$	−1.08021e25	−1.97351	−0.986753	−	0.162228i	$-0.948132\pi$
$227$	−1.08021e25	−1.97351	−0.986753	+	0.162228i	$0.948132\pi$
$228$	4.71781e24	0.823047
$229$	1.04108e25	1.73465	0.867326	−	0.497740i	$-0.165837\pi$
$229$	1.04108e25	1.73465	0.867326	+	0.497740i	$0.165837\pi$
$230$	0	0
$231$	−4.57643e24	−0.695987
$232$	−6.90631e24	−1.00374
$233$	2.61385e24	0.363114	0.181557	−	0.983380i	$-0.441886\pi$
$233$	2.61385e24	0.363114	0.181557	+	0.983380i	$0.441886\pi$
$234$	1.66998e23	0.0221792
$235$	0	0
$236$	−6.82101e24	−0.828463
$237$	−4.57869e24	−0.531965
$238$	7.01726e24	0.780027
$239$	−3.17368e24	−0.337587	−0.168793	−	0.985651i	$-0.553987\pi$
$239$	−3.17368e24	−0.337587	−0.168793	+	0.985651i	$0.553987\pi$
$240$	0	0
$241$	−4.44859e24	−0.433554	−0.216777	−	0.976221i	$-0.569554\pi$
$241$	−4.44859e24	−0.433554	−0.216777	+	0.976221i	$0.569554\pi$
$242$	3.26765e24	0.304912
$243$	−7.17898e23	−0.0641500
$244$	−1.79473e24	−0.153605
$245$	0	0
$246$	−5.15184e24	−0.404710
$247$	−3.36791e24	−0.253538
$248$	−8.82045e24	−0.636426
$249$	−1.55029e24	−0.107231
$250$	0	0
$251$	−1.43489e25	−0.912525	−0.456262	−	0.889845i	$-0.650812\pi$
$251$	−1.43489e25	−0.912525	−0.456262	+	0.889845i	$0.650812\pi$
$252$	−7.98992e24	−0.487345
$253$	−3.92197e24	−0.229477
$254$	1.27869e24	0.0717810
$255$	0	0
$256$	−1.26682e25	−0.654932
$257$	6.11572e24	0.303494	0.151747	−	0.988419i	$-0.451510\pi$
$257$	6.11572e24	0.303494	0.151747	+	0.988419i	$0.451510\pi$
$258$	−7.20963e24	−0.343484
$259$	1.51197e25	0.691666
$260$	0	0
$261$	−9.10703e24	−0.384283
$262$	−6.10419e24	−0.247437
$263$	9.31741e24	0.362877	0.181439	−	0.983402i	$-0.441925\pi$
$263$	9.31741e24	0.362877	0.181439	+	0.983402i	$0.441925\pi$
$264$	−8.34421e24	−0.312280
$265$	0	0
$266$	−5.27297e25	−1.82305
$267$	2.76238e25	0.918153
$268$	−2.35690e24	−0.0753226
$269$	5.19103e25	1.59535	0.797673	−	0.603090i	$-0.206064\pi$
$269$	5.19103e25	1.59535	0.797673	+	0.603090i	$0.206064\pi$
$270$	0	0
$271$	−2.16010e24	−0.0614181	−0.0307090	−	0.999528i	$-0.509777\pi$
$271$	−2.16010e24	−0.0614181	−0.0307090	+	0.999528i	$0.509777\pi$
$272$	−9.50365e24	−0.259966
$273$	5.70378e24	0.150126
$274$	3.68213e25	0.932648
$275$	0	0
$276$	−6.84729e24	−0.160684
$277$	−7.66259e25	−1.73116	−0.865581	−	0.500769i	$-0.833051\pi$
$277$	−7.66259e25	−1.73116	−0.865581	+	0.500769i	$0.833051\pi$
$278$	3.26139e25	0.709468
$279$	−1.16311e25	−0.243656
$280$	0	0
$281$	9.52888e24	0.185193	0.0925967	−	0.995704i	$-0.470483\pi$
$281$	9.52888e24	0.185193	0.0925967	+	0.995704i	$0.470483\pi$
$282$	−1.69391e25	−0.317158
$283$	1.67669e25	0.302479	0.151240	−	0.988497i	$-0.451674\pi$
$283$	1.67669e25	0.302479	0.151240	+	0.988497i	$0.451674\pi$
$284$	3.96698e25	0.689632
$285$	0	0
$286$	2.55956e24	0.0413354
$287$	−1.75960e26	−2.73939
$288$	−2.28762e25	−0.343371
$289$	−2.38106e25	−0.344622
$290$	0	0
$291$	5.23856e25	0.705238
$292$	5.04386e25	0.655003
$293$	1.00892e26	1.26400	0.632002	−	0.774967i	$-0.282233\pi$
$293$	1.00892e26	1.26400	0.632002	+	0.774967i	$0.282233\pi$
$294$	6.55850e25	0.792789
$295$	0	0
$296$	2.75678e25	0.310341
$297$	−1.10031e25	−0.119557
$298$	−6.38243e25	−0.669445
$299$	4.88809e24	0.0494985
$300$	0	0
$301$	−2.46243e26	−2.32496
$302$	−1.58774e25	−0.144779
$303$	−9.50243e25	−0.836926
$304$	7.14131e25	0.607583
$305$	0	0
$306$	1.68716e25	0.133993
$307$	−2.64734e25	−0.203169	−0.101584	−	0.994827i	$-0.532391\pi$
$307$	−2.64734e25	−0.203169	−0.101584	+	0.994827i	$0.532391\pi$
$308$	−1.22460e26	−0.908266
$309$	−1.03810e26	−0.744176
$310$	0	0
$311$	−1.96060e26	−1.31342	−0.656711	−	0.754143i	$-0.728053\pi$
$311$	−1.96060e26	−1.31342	−0.656711	+	0.754143i	$0.728053\pi$
$312$	1.03997e25	0.0673593
$313$	1.28532e26	0.805000	0.402500	−	0.915420i	$-0.368141\pi$
$313$	1.28532e26	0.805000	0.402500	+	0.915420i	$0.368141\pi$
$314$	8.93894e24	0.0541408
$315$	0	0
$316$	−1.22521e26	−0.694217
$317$	−7.55478e25	−0.414095	−0.207047	−	0.978331i	$-0.566385\pi$
$317$	−7.55478e25	−0.414095	−0.207047	+	0.978331i	$0.566385\pi$
$318$	9.18108e24	0.0486866
$319$	−1.39582e26	−0.716190
$320$	0	0
$321$	3.64291e25	0.175043
$322$	7.65303e25	0.355916
$323$	−3.40256e26	−1.53172
$324$	−1.92102e25	−0.0837160
$325$	0	0
$326$	1.14960e26	0.469638
$327$	−8.32428e25	−0.329303
$328$	−3.20827e26	−1.22913
$329$	−5.78552e26	−2.14677
$330$	0	0
$331$	−2.16140e26	−0.752559	−0.376280	−	0.926506i	$-0.622797\pi$
$331$	−2.16140e26	−0.752559	−0.376280	+	0.926506i	$0.622797\pi$
$332$	−4.14841e25	−0.139937
$333$	3.63524e25	0.118814
$334$	1.50615e26	0.477015
$335$	0	0
$336$	−1.20943e26	−0.359764
$337$	2.57810e26	0.743337	0.371668	−	0.928366i	$-0.378786\pi$
$337$	2.57810e26	0.743337	0.371668	+	0.928366i	$0.378786\pi$
$338$	1.74467e26	0.487628
$339$	4.07298e26	1.10361
$340$	0	0
$341$	−1.78269e26	−0.454103
$342$	−1.26778e26	−0.313163
$343$	1.43002e27	3.42574
$344$	−4.48975e26	−1.04318
$345$	0	0
$346$	−6.41016e25	−0.140142
$347$	8.08697e25	0.171525	0.0857623	−	0.996316i	$-0.472667\pi$
$347$	8.08697e25	0.171525	0.0857623	+	0.996316i	$0.472667\pi$
$348$	−2.43694e26	−0.501491
$349$	3.35155e26	0.669235	0.334618	−	0.942354i	$-0.391393\pi$
$349$	3.35155e26	0.669235	0.334618	+	0.942354i	$0.391393\pi$
$350$	0	0
$351$	1.37136e25	0.0257886
$352$	−3.50621e26	−0.639942
$353$	−6.48285e26	−1.14850	−0.574251	−	0.818679i	$-0.694707\pi$
$353$	−6.48285e26	−1.14850	−0.574251	+	0.818679i	$0.694707\pi$
$354$	1.83296e26	0.315223
$355$	0	0
$356$	7.39183e26	1.19819
$357$	5.76246e26	0.906968
$358$	−4.65128e26	−0.710887
$359$	7.38689e26	1.09640	0.548201	−	0.836347i	$-0.315313\pi$
$359$	7.38689e26	1.09640	0.548201	+	0.836347i	$0.315313\pi$
$360$	0	0
$361$	1.84257e27	2.57988
$362$	9.77029e25	0.132882
$363$	2.68334e26	0.354532
$364$	1.52627e26	0.195915
$365$	0	0
$366$	4.82284e25	0.0584456
$367$	1.06086e27	1.24929	0.624647	−	0.780907i	$-0.285243\pi$
$367$	1.06086e27	1.24929	0.624647	+	0.780907i	$0.285243\pi$
$368$	−1.03647e26	−0.118619
$369$	−4.23060e26	−0.470571
$370$	0	0
$371$	3.13578e26	0.329548
$372$	−3.11236e26	−0.317972
$373$	−8.79522e26	−0.873582	−0.436791	−	0.899563i	$-0.643885\pi$
$373$	−8.79522e26	−0.873582	−0.436791	+	0.899563i	$0.643885\pi$
$374$	2.58589e26	0.249723
$375$	0	0
$376$	−1.05487e27	−0.963226
$377$	1.73967e26	0.154483
$378$	2.14707e26	0.185431
$379$	1.37283e27	1.15320	0.576600	−	0.817027i	$-0.304379\pi$
$379$	1.37283e27	1.15320	0.576600	+	0.817027i	$0.304379\pi$
$380$	0	0
$381$	1.05003e26	0.0834625
$382$	4.27834e26	0.330834
$383$	4.70426e25	0.0353919	0.0176960	−	0.999843i	$-0.494367\pi$
$383$	4.70426e25	0.0353919	0.0176960	+	0.999843i	$0.494367\pi$
$384$	−7.37905e26	−0.540160
$385$	0	0
$386$	−2.62913e26	−0.182241
$387$	−5.92043e26	−0.399382
$388$	1.40178e27	0.920338
$389$	4.91701e26	0.314218	0.157109	−	0.987581i	$-0.449783\pi$
$389$	4.91701e26	0.314218	0.157109	+	0.987581i	$0.449783\pi$
$390$	0	0
$391$	4.93838e26	0.299040
$392$	4.08426e27	2.40774
$393$	−5.01266e26	−0.287704
$394$	−1.47541e27	−0.824520
$395$	0	0
$396$	−2.94432e26	−0.156022
$397$	−4.75566e26	−0.245420	−0.122710	−	0.992443i	$-0.539159\pi$
$397$	−4.75566e26	−0.245420	−0.122710	+	0.992443i	$0.539159\pi$
$398$	9.21044e24	0.00462922
$399$	−4.33008e27	−2.11973
$400$	0	0
$401$	3.19241e27	1.48287	0.741434	−	0.671025i	$-0.234146\pi$
$401$	3.19241e27	1.48287	0.741434	+	0.671025i	$0.234146\pi$
$402$	6.33351e25	0.0286597
$403$	2.22183e26	0.0979507
$404$	−2.54275e27	−1.09219
$405$	0	0
$406$	2.72371e27	1.11080
$407$	5.57168e26	0.221434
$408$	1.05067e27	0.406944
$409$	−5.80155e26	−0.219003	−0.109501	−	0.993987i	$-0.534925\pi$
$409$	−5.80155e26	−0.219003	−0.109501	+	0.993987i	$0.534925\pi$
$410$	0	0
$411$	3.02370e27	1.08443
$412$	−2.77785e27	−0.971153
$413$	6.26042e27	2.13368
$414$	1.84002e26	0.0611391
$415$	0	0
$416$	4.36992e26	0.138036
$417$	2.67820e27	0.824926
$418$	−1.94311e27	−0.583643
$419$	−6.91834e26	−0.202654	−0.101327	−	0.994853i	$-0.532309\pi$
$419$	−6.91834e26	−0.202654	−0.101327	+	0.994853i	$0.532309\pi$
$420$	0	0
$421$	5.75790e27	1.60436	0.802181	−	0.597081i	$-0.203673\pi$
$421$	5.75790e27	1.60436	0.802181	+	0.597081i	$0.203673\pi$
$422$	2.26900e27	0.616672
$423$	−1.39101e27	−0.368772
$424$	5.71746e26	0.147864
$425$	0	0
$426$	−1.06602e27	−0.262399
$427$	1.64723e27	0.395605
$428$	9.74802e26	0.228432
$429$	2.10187e26	0.0480622
$430$	0	0
$431$	−3.85181e26	−0.0838791	−0.0419395	−	0.999120i	$-0.513354\pi$
$431$	−3.85181e26	−0.0838791	−0.0419395	+	0.999120i	$0.513354\pi$
$432$	−2.90783e26	−0.0618001
$433$	4.76715e27	0.988863	0.494432	−	0.869216i	$-0.335376\pi$
$433$	4.76715e27	0.988863	0.494432	+	0.869216i	$0.335376\pi$
$434$	3.47860e27	0.704308
$435$	0	0
$436$	−2.22749e27	−0.429742
$437$	−3.71084e27	−0.698903
$438$	−1.35540e27	−0.249223
$439$	−1.30635e27	−0.234521	−0.117260	−	0.993101i	$-0.537411\pi$
$439$	−1.30635e27	−0.234521	−0.117260	+	0.993101i	$0.537411\pi$
$440$	0	0
$441$	5.38573e27	0.921806
$442$	−3.22289e26	−0.0538657
$443$	−1.05987e28	−1.72987	−0.864936	−	0.501882i	$-0.832641\pi$
$443$	−1.05987e28	−1.72987	−0.864936	+	0.501882i	$0.832641\pi$
$444$	9.72750e26	0.155053
$445$	0	0
$446$	4.45738e27	0.677741
$447$	−5.24114e27	−0.778390
$448$	2.54643e27	0.369413
$449$	−2.62975e27	−0.372673	−0.186337	−	0.982486i	$-0.559661\pi$
$449$	−2.62975e27	−0.372673	−0.186337	+	0.982486i	$0.559661\pi$
$450$	0	0
$451$	−6.48419e27	−0.877005
$452$	1.08988e28	1.44021
$453$	−1.30382e27	−0.168340
$454$	7.76752e27	0.979932
$455$	0	0
$456$	−7.89503e27	−0.951093
$457$	5.80657e26	0.0683596	0.0341798	−	0.999416i	$-0.489118\pi$
$457$	5.80657e26	0.0683596	0.0341798	+	0.999416i	$0.489118\pi$
$458$	−7.48613e27	−0.861330
$459$	1.38547e27	0.155799
$460$	0	0
$461$	−9.20734e27	−0.989178	−0.494589	−	0.869127i	$-0.664681\pi$
$461$	−9.20734e27	−0.989178	−0.494589	+	0.869127i	$0.664681\pi$
$462$	3.29079e27	0.345588
$463$	2.81089e27	0.288564	0.144282	−	0.989537i	$-0.453913\pi$
$463$	2.81089e27	0.288564	0.144282	+	0.989537i	$0.453913\pi$
$464$	−3.68878e27	−0.370207
$465$	0	0
$466$	−1.87955e27	−0.180302
$467$	−1.43686e28	−1.34768	−0.673841	−	0.738876i	$-0.735357\pi$
$467$	−1.43686e28	−1.34768	−0.673841	+	0.738876i	$0.735357\pi$
$468$	3.66961e26	0.0336542
$469$	2.16320e27	0.193991
$470$	0	0
$471$	7.34051e26	0.0629516
$472$	1.14146e28	0.957351
$473$	−9.07417e27	−0.744329
$474$	3.29241e27	0.264144
$475$	0	0
$476$	1.54197e28	1.18360
$477$	7.53935e26	0.0566097
$478$	2.28211e27	0.167627
$479$	−1.66001e28	−1.19285	−0.596426	−	0.802668i	$-0.703413\pi$
$479$	−1.66001e28	−1.19285	−0.596426	+	0.802668i	$0.703413\pi$
$480$	0	0
$481$	−6.94419e26	−0.0477638
$482$	3.19885e27	0.215278
$483$	6.28454e27	0.413837
$484$	7.18032e27	0.462666
$485$	0	0
$486$	5.16220e26	0.0318533
$487$	2.39148e28	1.44415	0.722076	−	0.691814i	$-0.243188\pi$
$487$	2.39148e28	1.44415	0.722076	+	0.691814i	$0.243188\pi$
$488$	3.00340e27	0.177502
$489$	9.44036e27	0.546066
$490$	0	0
$491$	−2.52047e28	−1.39677	−0.698386	−	0.715721i	$-0.746098\pi$
$491$	−2.52047e28	−1.39677	−0.698386	+	0.715721i	$0.746098\pi$
$492$	−1.13206e28	−0.614098
$493$	1.75756e28	0.933294
$494$	2.42177e27	0.125893
$495$	0	0
$496$	−4.71116e27	−0.234731
$497$	−3.64095e28	−1.77612
$498$	1.11477e27	0.0532448
$499$	−4.60542e27	−0.215384	−0.107692	−	0.994184i	$-0.534346\pi$
$499$	−4.60542e27	−0.215384	−0.107692	+	0.994184i	$0.534346\pi$
$500$	0	0
$501$	1.23683e28	0.554643
$502$	1.03179e28	0.453108
$503$	−1.46909e28	−0.631806	−0.315903	−	0.948791i	$-0.602307\pi$
$503$	−1.46909e28	−0.631806	−0.315903	+	0.948791i	$0.602307\pi$
$504$	1.33707e28	0.563164
$505$	0	0
$506$	2.82018e27	0.113945
$507$	1.43269e28	0.566983
$508$	2.80978e27	0.108919
$509$	4.03387e27	0.153174	0.0765871	−	0.997063i	$-0.475598\pi$
$509$	4.03387e27	0.153174	0.0765871	+	0.997063i	$0.475598\pi$
$510$	0	0
$511$	−4.62933e28	−1.68694
$512$	−1.70977e28	−0.610382
$513$	−1.04108e28	−0.364126
$514$	−4.39764e27	−0.150698
$515$	0	0
$516$	−1.58424e28	−0.521195
$517$	−2.13199e28	−0.687281
$518$	−1.08722e28	−0.343442
$519$	−5.26392e27	−0.162949
$520$	0	0
$521$	−3.75449e28	−1.11623	−0.558117	−	0.829762i	$-0.688476\pi$
$521$	−3.75449e28	−1.11623	−0.558117	+	0.829762i	$0.688476\pi$
$522$	6.54861e27	0.190813
$523$	−5.27634e28	−1.50683	−0.753417	−	0.657543i	$-0.771596\pi$
$523$	−5.27634e28	−1.50683	−0.753417	+	0.657543i	$0.771596\pi$
$524$	−1.34133e28	−0.375455
$525$	0	0
$526$	−6.69989e27	−0.180184
$527$	2.24469e28	0.591758
$528$	−4.45679e27	−0.115177
$529$	−3.40858e28	−0.863552
$530$	0	0
$531$	1.50520e28	0.366522
$532$	−1.15868e29	−2.76625
$533$	8.08149e27	0.189172
$534$	−1.98635e28	−0.455903
$535$	0	0
$536$	3.94416e27	0.0870410
$537$	−3.81955e28	−0.826576
$538$	−3.73273e28	−0.792159
$539$	8.25464e28	1.71797
$540$	0	0
$541$	1.40670e28	0.281598	0.140799	−	0.990038i	$-0.455033\pi$
$541$	1.40670e28	0.281598	0.140799	+	0.990038i	$0.455033\pi$
$542$	1.55327e27	0.0304968
$543$	8.02320e27	0.154508
$544$	4.41488e28	0.833932
$545$	0	0
$546$	−4.10143e27	−0.0745439
$547$	−6.71332e28	−1.19693	−0.598467	−	0.801147i	$-0.704223\pi$
$547$	−6.71332e28	−1.19693	−0.598467	+	0.801147i	$0.704223\pi$
$548$	8.09110e28	1.41518
$549$	3.96044e27	0.0679569
$550$	0	0
$551$	−1.32068e29	−2.18126
$552$	1.14586e28	0.185683
$553$	1.12451e29	1.78793
$554$	5.50995e28	0.859598
$555$	0	0
$556$	7.16658e28	1.07653
$557$	−4.16746e28	−0.614316	−0.307158	−	0.951658i	$-0.599378\pi$
$557$	−4.16746e28	−0.614316	−0.307158	+	0.951658i	$0.599378\pi$
$558$	8.36361e27	0.120986
$559$	1.13095e28	0.160553
$560$	0	0
$561$	2.12349e28	0.290363
$562$	−6.85195e27	−0.0919566
$563$	6.01432e27	0.0792225	0.0396112	−	0.999215i	$-0.487388\pi$
$563$	6.01432e27	0.0792225	0.0396112	+	0.999215i	$0.487388\pi$
$564$	−3.72220e28	−0.481248
$565$	0	0
$566$	−1.20566e28	−0.150194
$567$	1.76314e28	0.215608
$568$	−6.63855e28	−0.796922
$569$	−1.58661e28	−0.186979	−0.0934893	−	0.995620i	$-0.529802\pi$
$569$	−1.58661e28	−0.186979	−0.0934893	+	0.995620i	$0.529802\pi$
$570$	0	0
$571$	3.59799e28	0.408677	0.204339	−	0.978900i	$-0.434496\pi$
$571$	3.59799e28	0.408677	0.204339	+	0.978900i	$0.434496\pi$
$572$	5.62437e27	0.0627214
$573$	3.51330e28	0.384673
$574$	1.26528e29	1.36023
$575$	0	0
$576$	6.12239e27	0.0634577
$577$	4.38317e28	0.446111	0.223055	−	0.974806i	$-0.428397\pi$
$577$	4.38317e28	0.446111	0.223055	+	0.974806i	$0.428397\pi$
$578$	1.71216e28	0.171120
$579$	−2.15900e28	−0.211899
$580$	0	0
$581$	3.80747e28	0.360402
$582$	−3.76690e28	−0.350181
$583$	1.15555e28	0.105504
$584$	−8.44066e28	−0.756905
$585$	0	0
$586$	−7.25489e28	−0.627633
$587$	1.20595e29	1.02478	0.512389	−	0.858754i	$-0.328761\pi$
$587$	1.20595e29	1.02478	0.512389	+	0.858754i	$0.328761\pi$
$588$	1.44116e29	1.20296
$589$	−1.68672e29	−1.38303
$590$	0	0
$591$	−1.21158e29	−0.958701
$592$	1.47244e28	0.114462
$593$	7.53007e27	0.0575075	0.0287538	−	0.999587i	$-0.490846\pi$
$593$	7.53007e27	0.0575075	0.0287538	+	0.999587i	$0.490846\pi$
$594$	7.91204e27	0.0593651
$595$	0	0
$596$	−1.40247e29	−1.01580
$597$	7.56346e26	0.00538257
$598$	−3.51489e27	−0.0245782
$599$	2.68676e28	0.184607	0.0923033	−	0.995731i	$-0.470577\pi$
$599$	2.68676e28	0.184607	0.0923033	+	0.995731i	$0.470577\pi$
$600$	0	0
$601$	6.40947e28	0.425246	0.212623	−	0.977134i	$-0.431799\pi$
$601$	6.40947e28	0.425246	0.212623	+	0.977134i	$0.431799\pi$
$602$	1.77067e29	1.15445
$603$	5.20098e27	0.0333237
$604$	−3.48889e28	−0.219685
$605$	0	0
$606$	6.83293e28	0.415570
$607$	−4.48774e28	−0.268254	−0.134127	−	0.990964i	$-0.542823\pi$
$607$	−4.48774e28	−0.268254	−0.134127	+	0.990964i	$0.542823\pi$
$608$	−3.31747e29	−1.94903
$609$	2.23666e29	1.29157
$610$	0	0
$611$	2.65718e28	0.148248
$612$	3.70737e28	0.203318
$613$	−1.67946e29	−0.905390	−0.452695	−	0.891665i	$-0.649537\pi$
$613$	−1.67946e29	−0.905390	−0.452695	+	0.891665i	$0.649537\pi$
$614$	1.90363e28	0.100882
$615$	0	0
$616$	2.04932e29	1.04957
$617$	−1.45325e29	−0.731723	−0.365862	−	0.930669i	$-0.619226\pi$
$617$	−1.45325e29	−0.731723	−0.365862	+	0.930669i	$0.619226\pi$
$618$	7.46469e28	0.369516
$619$	−2.90308e29	−1.41289	−0.706443	−	0.707770i	$-0.749701\pi$
$619$	−2.90308e29	−1.41289	−0.706443	+	0.707770i	$0.749701\pi$
$620$	0	0
$621$	1.51099e28	0.0710888
$622$	1.40981e29	0.652171
$623$	−6.78432e29	−3.08590
$624$	5.55466e27	0.0248439
$625$	0	0
$626$	−9.24238e28	−0.399718
$627$	−1.59565e29	−0.678624
$628$	1.96424e28	0.0821521
$629$	−7.01564e28	−0.288559
$630$	0	0
$631$	−5.05402e28	−0.201062	−0.100531	−	0.994934i	$-0.532054\pi$
$631$	−5.05402e28	−0.201062	−0.100531	+	0.994934i	$0.532054\pi$
$632$	2.05033e29	0.802220
$633$	1.86327e29	0.717028
$634$	5.43243e28	0.205616
$635$	0	0
$636$	2.01745e28	0.0738759
$637$	−1.02881e29	−0.370569
$638$	1.00370e29	0.355620
$639$	−8.75395e28	−0.305102
$640$	0	0
$641$	4.86202e29	1.63986	0.819931	−	0.572462i	$-0.194012\pi$
$641$	4.86202e29	1.63986	0.819931	+	0.572462i	$0.194012\pi$
$642$	−2.61951e28	−0.0869166
$643$	3.08867e29	1.00822	0.504111	−	0.863639i	$-0.331820\pi$
$643$	3.08867e29	1.00822	0.504111	+	0.863639i	$0.331820\pi$
$644$	1.68168e29	0.540058
$645$	0	0
$646$	2.44669e29	0.760567
$647$	4.70559e29	1.43919	0.719596	−	0.694393i	$-0.244327\pi$
$647$	4.70559e29	1.43919	0.719596	+	0.694393i	$0.244327\pi$
$648$	3.21473e28	0.0967402
$649$	2.30699e29	0.683089
$650$	0	0
$651$	2.85657e29	0.818926
$652$	2.52614e29	0.712619
$653$	1.04545e29	0.290211	0.145106	−	0.989416i	$-0.453648\pi$
$653$	1.04545e29	0.290211	0.145106	+	0.989416i	$0.453648\pi$
$654$	5.98576e28	0.163513
$655$	0	0
$656$	−1.71360e29	−0.453334
$657$	−1.11303e29	−0.289782
$658$	4.16020e29	1.06596
$659$	−3.23730e29	−0.816367	−0.408184	−	0.912900i	$-0.633838\pi$
$659$	−3.23730e29	−0.816367	−0.408184	+	0.912900i	$0.633838\pi$
$660$	0	0
$661$	2.98605e29	0.729428	0.364714	−	0.931120i	$-0.381167\pi$
$661$	2.98605e29	0.729428	0.364714	+	0.931120i	$0.381167\pi$
$662$	1.55420e29	0.373678
$663$	−2.64659e28	−0.0626317
$664$	6.94216e28	0.161707
$665$	0	0
$666$	−2.61400e28	−0.0589964
$667$	1.91680e29	0.425849
$668$	3.30962e29	0.723812
$669$	3.66033e29	0.788035
$670$	0	0
$671$	6.07011e28	0.126652
$672$	5.61834e29	1.15407
$673$	2.49272e29	0.504099	0.252049	−	0.967714i	$-0.418895\pi$
$673$	2.49272e29	0.504099	0.252049	+	0.967714i	$0.418895\pi$
$674$	−1.85384e29	−0.369099
$675$	0	0
$676$	3.83374e29	0.739915
$677$	−6.42171e29	−1.22031	−0.610154	−	0.792283i	$-0.708892\pi$
$677$	−6.42171e29	−1.22031	−0.610154	+	0.792283i	$0.708892\pi$
$678$	−2.92876e29	−0.547990
$679$	−1.28658e30	−2.37030
$680$	0	0
$681$	6.37856e29	1.13940
$682$	1.28188e29	0.225482
$683$	−5.44201e29	−0.942632	−0.471316	−	0.881964i	$-0.656221\pi$
$683$	−5.44201e29	−0.942632	−0.471316	+	0.881964i	$0.656221\pi$
$684$	−2.78582e29	−0.475187
$685$	0	0
$686$	−1.02829e30	−1.70103
$687$	−6.14749e29	−1.00150
$688$	−2.39806e29	−0.384752
$689$	−1.44020e28	−0.0227573
$690$	0	0
$691$	−1.13440e30	−1.73879	−0.869397	−	0.494114i	$-0.835492\pi$
$691$	−1.13440e30	−1.73879	−0.869397	+	0.494114i	$0.835492\pi$
$692$	−1.40857e29	−0.212649
$693$	2.70234e29	0.401829
$694$	−5.81511e28	−0.0851694
$695$	0	0
$696$	4.07811e29	0.579511
$697$	8.16463e29	1.14286
$698$	−2.41001e29	−0.332305
$699$	−1.54345e29	−0.209644
$700$	0	0
$701$	4.56419e28	0.0601623	0.0300812	−	0.999547i	$-0.490423\pi$
$701$	4.56419e28	0.0601623	0.0300812	+	0.999547i	$0.490423\pi$
$702$	−9.86107e27	−0.0128051
$703$	5.27175e29	0.674410
$704$	9.38370e28	0.118266
$705$	0	0
$706$	4.66164e29	0.570281
$707$	2.33377e30	2.81290
$708$	4.02774e29	0.478313
$709$	4.21173e29	0.492805	0.246403	−	0.969168i	$-0.420751\pi$
$709$	4.21173e29	0.492805	0.246403	+	0.969168i	$0.420751\pi$
$710$	0	0
$711$	2.70367e29	0.307130
$712$	−1.23699e30	−1.38460
$713$	2.44806e29	0.270011
$714$	−4.14362e29	−0.450349
$715$	0	0
$716$	−1.02207e30	−1.07868
$717$	1.87403e29	0.194906
$718$	−5.31171e29	−0.544412
$719$	9.88968e29	0.998916	0.499458	−	0.866338i	$-0.333533\pi$
$719$	9.88968e29	0.998916	0.499458	+	0.866338i	$0.333533\pi$
$720$	0	0
$721$	2.54955e30	2.50117
$722$	−1.32494e30	−1.28102
$723$	2.62685e29	0.250312
$724$	2.14692e29	0.201633
$725$	0	0
$726$	−1.92951e29	−0.176041
$727$	−2.23750e28	−0.0201211	−0.0100605	−	0.999949i	$-0.503202\pi$
$727$	−2.23750e28	−0.0201211	−0.0100605	+	0.999949i	$0.503202\pi$
$728$	−2.55414e29	−0.226394
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	1.14258e30	0.969963
$732$	1.05977e29	0.0886841
$733$	4.89534e29	0.403823	0.201912	−	0.979404i	$-0.435285\pi$
$733$	4.89534e29	0.403823	0.201912	+	0.979404i	$0.435285\pi$
$734$	−7.62836e29	−0.620329
$735$	0	0
$736$	4.81487e29	0.380512
$737$	7.97147e28	0.0621055
$738$	3.04211e29	0.233659
$739$	1.30296e30	0.986652	0.493326	−	0.869845i	$-0.335781\pi$
$739$	1.30296e30	0.986652	0.493326	+	0.869845i	$0.335781\pi$
$740$	0	0
$741$	1.98872e29	0.146380
$742$	−2.25485e29	−0.163635
$743$	−1.91846e30	−1.37268	−0.686341	−	0.727280i	$-0.740784\pi$
$743$	−1.91846e30	−1.37268	−0.686341	+	0.727280i	$0.740784\pi$
$744$	5.20839e29	0.367441
$745$	0	0
$746$	6.32439e29	0.433772
$747$	9.15430e28	0.0619097
$748$	5.68224e29	0.378924
$749$	−8.94688e29	−0.588318
$750$	0	0
$751$	−2.65366e29	−0.169678	−0.0848390	−	0.996395i	$-0.527038\pi$
$751$	−2.65366e29	−0.169678	−0.0848390	+	0.996395i	$0.527038\pi$
$752$	−5.63427e29	−0.355263
$753$	8.47288e29	0.526846
$754$	−1.25095e29	−0.0767077
$755$	0	0
$756$	4.71797e29	0.281369
$757$	−1.26747e30	−0.745471	−0.372736	−	0.927938i	$-0.621580\pi$
$757$	−1.26747e30	−0.745471	−0.372736	+	0.927938i	$0.621580\pi$
$758$	−9.87162e29	−0.572614
$759$	2.31588e29	0.132488
$760$	0	0
$761$	−2.69492e30	−1.49971	−0.749854	−	0.661603i	$-0.769876\pi$
$761$	−2.69492e30	−1.49971	−0.749854	+	0.661603i	$0.769876\pi$
$762$	−7.55051e28	−0.0414428
$763$	2.04442e30	1.10678
$764$	9.40121e29	0.502000
$765$	0	0
$766$	−3.38270e28	−0.0175736
$767$	−2.87529e29	−0.147343
$768$	7.48046e29	0.378125
$769$	2.90682e30	1.44941	0.724707	−	0.689058i	$-0.241975\pi$
$769$	2.90682e30	1.44941	0.724707	+	0.689058i	$0.241975\pi$
$770$	0	0
$771$	−3.61127e29	−0.175222
$772$	−5.77726e29	−0.276529
$773$	−3.54314e30	−1.67303	−0.836516	−	0.547943i	$-0.815411\pi$
$773$	−3.54314e30	−1.67303	−0.836516	+	0.547943i	$0.815411\pi$
$774$	4.25722e29	0.198310
$775$	0	0
$776$	−2.34582e30	−1.06352
$777$	−8.92804e29	−0.399333
$778$	−3.53569e29	−0.156023
$779$	−6.13514e30	−2.67104
$780$	0	0
$781$	−1.34171e30	−0.568619
$782$	−3.55105e29	−0.148486
$783$	5.37761e29	0.221866
$784$	2.18148e30	0.888039
$785$	0	0
$786$	3.60446e29	0.142858
$787$	2.52723e30	0.988347	0.494174	−	0.869363i	$-0.335471\pi$
$787$	2.52723e30	0.988347	0.494174	+	0.869363i	$0.335471\pi$
$788$	−3.24205e30	−1.25111
$789$	−5.50184e29	−0.209507
$790$	0	0
$791$	−1.00031e31	−3.70922
$792$	4.92718e29	0.180295
$793$	−7.56541e28	−0.0273189
$794$	3.41966e29	0.121862
$795$	0	0
$796$	2.02390e28	0.00702428
$797$	7.95340e29	0.272421	0.136211	−	0.990680i	$-0.456508\pi$
$797$	7.95340e29	0.272421	0.136211	+	0.990680i	$0.456508\pi$
$798$	3.11364e30	1.05254
$799$	2.68451e30	0.895621
$800$	0	0
$801$	−1.63116e30	−0.530096
$802$	−2.29557e30	−0.736309
$803$	−1.70593e30	−0.540067
$804$	1.39173e29	0.0434876
$805$	0	0
$806$	−1.59765e29	−0.0486368
$807$	−3.06525e30	−0.921073
$808$	4.25516e30	1.26211
$809$	4.89784e29	0.143399	0.0716993	−	0.997426i	$-0.477158\pi$
$809$	4.89784e29	0.143399	0.0716993	+	0.997426i	$0.477158\pi$
$810$	0	0
$811$	−1.26964e30	−0.362211	−0.181106	−	0.983464i	$-0.557968\pi$
$811$	−1.26964e30	−0.362211	−0.181106	+	0.983464i	$0.557968\pi$
$812$	5.98507e30	1.68551
$813$	1.27552e29	0.0354598
$814$	−4.00644e29	−0.109952
$815$	0	0
$816$	5.61181e29	0.150092
$817$	−8.58569e30	−2.26696
$818$	4.17173e29	0.108744
$819$	−3.36802e29	−0.0866751
$820$	0	0
$821$	3.19408e30	0.801203	0.400601	−	0.916252i	$-0.368801\pi$
$821$	3.19408e30	0.801203	0.400601	+	0.916252i	$0.368801\pi$
$822$	−2.17426e30	−0.538465
$823$	4.59957e30	1.12465	0.562327	−	0.826915i	$-0.309906\pi$
$823$	4.59957e30	1.12465	0.562327	+	0.826915i	$0.309906\pi$
$824$	4.64859e30	1.12224
$825$	0	0
$826$	−4.50170e30	−1.05946
$827$	7.96735e29	0.185142	0.0925712	−	0.995706i	$-0.470491\pi$
$827$	7.96735e29	0.185142	0.0925712	+	0.995706i	$0.470491\pi$
$828$	4.04326e29	0.0927712
$829$	−5.78171e30	−1.30989	−0.654944	−	0.755678i	$-0.727308\pi$
$829$	−5.78171e30	−1.30989	−0.654944	+	0.755678i	$0.727308\pi$
$830$	0	0
$831$	4.52468e30	0.999487
$832$	−1.16953e29	−0.0255103
$833$	−1.03939e31	−2.23875
$834$	−1.92582e30	−0.409612
$835$	0	0
$836$	−4.26979e30	−0.885607
$837$	6.86806e29	0.140675
$838$	4.97478e29	0.100626
$839$	3.18907e30	0.637036	0.318518	−	0.947917i	$-0.396815\pi$
$839$	3.18907e30	0.637036	0.318518	+	0.947917i	$0.396815\pi$
$840$	0	0
$841$	1.68903e30	0.329063
$842$	−4.14035e30	−0.796636
$843$	−5.62671e29	−0.106922
$844$	4.98590e30	0.935724
$845$	0	0
$846$	1.00024e30	0.183111
$847$	−6.59020e30	−1.19158
$848$	3.05380e29	0.0545360
$849$	−9.90069e29	−0.174636
$850$	0	0
$851$	−7.65126e29	−0.131666
$852$	−2.34246e30	−0.398159
$853$	−7.60896e30	−1.27750	−0.638750	−	0.769415i	$-0.720548\pi$
$853$	−7.60896e30	−1.27750	−0.638750	+	0.769415i	$0.720548\pi$
$854$	−1.18448e30	−0.196435
$855$	0	0
$856$	−1.63128e30	−0.263970
$857$	−3.72716e30	−0.595771	−0.297886	−	0.954602i	$-0.596281\pi$
$857$	−3.72716e30	−0.595771	−0.297886	+	0.954602i	$0.596281\pi$
$858$	−1.51139e29	−0.0238650
$859$	−7.44888e30	−1.16188	−0.580942	−	0.813945i	$-0.697316\pi$
$859$	−7.44888e30	−1.16188	−0.580942	+	0.813945i	$0.697316\pi$
$860$	0	0
$861$	1.03902e31	1.58159
$862$	2.76973e29	0.0416496
$863$	−2.89953e30	−0.430740	−0.215370	−	0.976533i	$-0.569096\pi$
$863$	−2.89953e30	−0.430740	−0.215370	+	0.976533i	$0.569096\pi$
$864$	1.35082e30	0.198245
$865$	0	0
$866$	−3.42793e30	−0.491014
$867$	1.40599e30	0.198968
$868$	7.64387e30	1.06870
$869$	4.14388e30	0.572400
$870$	0	0
$871$	−9.93514e28	−0.0133963
$872$	3.72759e30	0.496599
$873$	−3.09332e30	−0.407169
$874$	2.66836e30	0.347036
$875$	0	0
$876$	−2.97835e30	−0.378166
$877$	1.35915e31	1.70518	0.852592	−	0.522578i	$-0.175030\pi$
$877$	1.35915e31	1.70518	0.852592	+	0.522578i	$0.175030\pi$
$878$	9.39358e29	0.116450
$879$	−5.95760e30	−0.729773
$880$	0	0
$881$	−1.10116e31	−1.31705	−0.658525	−	0.752559i	$-0.728819\pi$
$881$	−1.10116e31	−1.31705	−0.658525	+	0.752559i	$0.728819\pi$
$882$	−3.87273e30	−0.457717
$883$	4.42112e30	0.516351	0.258175	−	0.966098i	$-0.416879\pi$
$883$	4.42112e30	0.516351	0.258175	+	0.966098i	$0.416879\pi$
$884$	−7.08198e29	−0.0817346
$885$	0	0
$886$	7.62124e30	0.858957
$887$	5.76372e30	0.641955	0.320978	−	0.947087i	$-0.395989\pi$
$887$	5.76372e30	0.641955	0.320978	+	0.947087i	$0.395989\pi$
$888$	−1.62785e30	−0.179175
$889$	−2.57886e30	−0.280517
$890$	0	0
$891$	6.49724e29	0.0690260
$892$	9.79464e30	1.02839
$893$	−2.01722e31	−2.09321
$894$	3.76876e30	0.386505
$895$	0	0
$896$	1.81227e31	1.81547
$897$	−2.88637e29	−0.0285780
$898$	1.89098e30	0.185048
$899$	8.71261e30	0.842697
$900$	0	0
$901$	−1.45502e30	−0.137486
$902$	4.66260e30	0.435471
$903$	1.45404e31	1.34232
$904$	−1.82387e31	−1.66428
$905$	0	0
$906$	9.37543e29	0.0835882
$907$	−2.23060e29	−0.0196583	−0.00982913	−	0.999952i	$-0.503129\pi$
$907$	−2.23060e29	−0.0196583	−0.00982913	+	0.999952i	$0.503129\pi$
$908$	1.70683e31	1.48693
$909$	5.61109e30	0.483199
$910$	0	0
$911$	−1.26374e31	−1.06344	−0.531722	−	0.846919i	$-0.678455\pi$
$911$	−1.26374e31	−1.06344	−0.531722	+	0.846919i	$0.678455\pi$
$912$	−4.21687e30	−0.350788
$913$	1.40307e30	0.115381
$914$	−4.17534e29	−0.0339435
$915$	0	0
$916$	−1.64500e31	−1.30696
$917$	1.23109e31	0.966971
$918$	−9.96253e29	−0.0773609
$919$	1.57231e31	1.20705	0.603526	−	0.797343i	$-0.293762\pi$
$919$	1.57231e31	1.20705	0.603526	+	0.797343i	$0.293762\pi$
$920$	0	0
$921$	1.56323e30	0.117300
$922$	6.62074e30	0.491170
$923$	1.67222e30	0.122652
$924$	7.23117e30	0.524388
$925$	0	0
$926$	−2.02123e30	−0.143285
$927$	6.12988e30	0.429650
$928$	1.71361e31	1.18757
$929$	−1.16955e31	−0.801410	−0.400705	−	0.916207i	$-0.631235\pi$
$929$	−1.16955e31	−0.801410	−0.400705	+	0.916207i	$0.631235\pi$
$930$	0	0
$931$	7.81028e31	5.23233
$932$	−4.13011e30	−0.273587
$933$	1.15771e31	0.758304
$934$	1.03321e31	0.669183
$935$	0	0
$936$	−6.14092e29	−0.0388899
$937$	−3.53331e30	−0.221267	−0.110633	−	0.993861i	$-0.535288\pi$
$937$	−3.53331e30	−0.221267	−0.110633	+	0.993861i	$0.535288\pi$
$938$	−1.55549e30	−0.0963248
$939$	−7.58969e30	−0.464767
$940$	0	0
$941$	−4.65830e30	−0.278956	−0.139478	−	0.990225i	$-0.544543\pi$
$941$	−4.65830e30	−0.278956	−0.139478	+	0.990225i	$0.544543\pi$
$942$	−5.27836e29	−0.0312582
$943$	8.90436e30	0.521471
$944$	6.09676e30	0.353096
$945$	0	0
$946$	6.52498e30	0.369592
$947$	−3.24251e30	−0.181638	−0.0908189	−	0.995867i	$-0.528948\pi$
$947$	−3.24251e30	−0.181638	−0.0908189	+	0.995867i	$0.528948\pi$
$948$	7.23473e30	0.400806
$949$	2.12616e30	0.116493
$950$	0	0
$951$	4.46102e30	0.239078
$952$	−2.58042e31	−1.36773
$953$	−9.11533e30	−0.477856	−0.238928	−	0.971037i	$-0.576796\pi$
$953$	−9.11533e30	−0.477856	−0.238928	+	0.971037i	$0.576796\pi$
$954$	−5.42134e29	−0.0281092
$955$	0	0
$956$	5.01470e30	0.254353
$957$	8.24220e30	0.413492
$958$	1.19366e31	0.592303
$959$	−7.42613e31	−3.64474
$960$	0	0
$961$	−9.69813e30	−0.465685
$962$	4.99337e29	0.0237168
$963$	−2.15110e30	−0.101061
$964$	7.02915e30	0.326659
$965$	0	0
$966$	−4.51904e30	−0.205488
$967$	3.05091e31	1.37231	0.686153	−	0.727458i	$-0.259298\pi$
$967$	3.05091e31	1.37231	0.686153	+	0.727458i	$0.259298\pi$
$968$	−1.20159e31	−0.534645
$969$	2.00918e31	0.884340
$970$	0	0
$971$	3.72130e31	1.60285	0.801426	−	0.598094i	$-0.204075\pi$
$971$	3.72130e31	1.60285	0.801426	+	0.598094i	$0.204075\pi$
$972$	1.13434e30	0.0483335
$973$	−6.57759e31	−2.77257
$974$	−1.71965e31	−0.717084
$975$	0	0
$976$	1.60417e30	0.0654676
$977$	−1.12682e31	−0.454949	−0.227475	−	0.973784i	$-0.573047\pi$
$977$	−1.12682e31	−0.454949	−0.227475	+	0.973784i	$0.573047\pi$
$978$	−6.78830e30	−0.271146
$979$	−2.50005e31	−0.987941
$980$	0	0
$981$	4.91541e30	0.190123
$982$	1.81240e31	0.693558
$983$	−1.28263e31	−0.485613	−0.242807	−	0.970075i	$-0.578068\pi$
$983$	−1.28263e31	−0.485613	−0.242807	+	0.970075i	$0.578068\pi$
$984$	1.89445e31	0.709636
$985$	0	0
$986$	−1.26382e31	−0.463421
$987$	3.41629e31	1.23944
$988$	5.32160e30	0.191027
$989$	1.24610e31	0.442581
$990$	0	0
$991$	−3.03312e31	−1.05467	−0.527334	−	0.849658i	$-0.676808\pi$
$991$	−3.03312e31	−1.05467	−0.527334	+	0.849658i	$0.676808\pi$
$992$	2.18855e31	0.752980
$993$	1.27628e31	0.434490
$994$	2.61811e31	0.881922
$995$	0	0
$996$	2.44959e30	0.0807925
$997$	−5.62554e31	−1.83597	−0.917984	−	0.396618i	$-0.870184\pi$
$997$	−5.62554e31	−1.83597	−0.917984	+	0.396618i	$0.870184\pi$
$998$	3.31163e30	0.106948
$999$	−2.14657e30	−0.0685974

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.e.1.2		3
5.2	odd	4		75.22.b.e.49.3		6
5.3	odd	4		75.22.b.e.49.4		6
5.4	even	2		15.22.a.d.1.2	✓	3
15.14	odd	2		45.22.a.b.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.a.d.1.2	✓	3	5.4	even	2
45.22.a.b.1.2		3	15.14	odd	2
75.22.a.e.1.2		3	1.1	even	1	trivial
75.22.b.e.49.3		6	5.2	odd	4
75.22.b.e.49.4		6	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)

\(f(q)\)	\(=\)	\(q-719.072 q^{2} -59049.0 q^{3} -1.58009e6 q^{4} +4.24605e7 q^{6} +1.45023e9 q^{7} +2.64420e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-719.072 q^{2} -59049.0 q^{3} -1.58009e6 q^{4} +4.24605e7 q^{6} +1.45023e9 q^{7} +2.64420e9 q^{8} +3.48678e9 q^{9} +5.34415e10 q^{11} +9.33026e10 q^{12} -6.66061e10 q^{13} -1.04282e12 q^{14} +1.41231e12 q^{16} -6.72914e12 q^{17} -2.50725e12 q^{18} +5.05646e13 q^{19} -8.56345e13 q^{21} -3.84283e13 q^{22} -7.33880e13 q^{23} -1.56137e14 q^{24} +4.78946e13 q^{26} -2.05891e14 q^{27} -2.29149e15 q^{28} -2.61187e15 q^{29} -3.33577e15 q^{31} -6.56084e15 q^{32} -3.15567e15 q^{33} +4.83873e15 q^{34} -5.50942e15 q^{36} +1.04258e16 q^{37} -3.63596e16 q^{38} +3.93302e15 q^{39} -1.21333e17 q^{41} +6.15773e16 q^{42} -1.69796e17 q^{43} -8.44422e16 q^{44} +5.27713e16 q^{46} -3.98939e17 q^{47} -8.33957e16 q^{48} +1.54461e18 q^{49} +3.97349e17 q^{51} +1.05243e17 q^{52} +2.16226e17 q^{53} +1.48051e17 q^{54} +3.83469e18 q^{56} -2.98579e18 q^{57} +1.87812e18 q^{58} +4.31686e18 q^{59} +1.13584e18 q^{61} +2.39866e18 q^{62} +5.05663e18 q^{63} +1.75588e18 q^{64} +2.26915e18 q^{66} +1.49163e18 q^{67} +1.06326e19 q^{68} +4.33349e18 q^{69} -2.51061e19 q^{71} +9.21975e18 q^{72} -3.19214e19 q^{73} -7.49687e18 q^{74} -7.98966e19 q^{76} +7.75023e19 q^{77} -2.82813e18 q^{78} +7.75405e19 q^{79} +1.21577e19 q^{81} +8.72468e19 q^{82} +2.62543e19 q^{83} +1.35310e20 q^{84} +1.22096e20 q^{86} +1.54228e20 q^{87} +1.41310e20 q^{88} -4.67811e20 q^{89} -9.65940e19 q^{91} +1.15960e20 q^{92} +1.96974e20 q^{93} +2.86866e20 q^{94} +3.87411e20 q^{96} -8.87155e20 q^{97} -1.11069e21 q^{98} +1.86339e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 803 q^{2} - 177147 q^{3} + 6715073 q^{4} + 47416347 q^{6} + 1577598316 q^{7} + 1424191017 q^{8} + 10460353203 q^{9}+O(q^{10}) \) 3 * q - 803 * q^2 - 177147 * q^3 + 6715073 * q^4 + 47416347 * q^6 + 1577598316 * q^7 + 1424191017 * q^8 + 10460353203 * q^9	\( 3 q - 803 q^{2} - 177147 q^{3} + 6715073 q^{4} + 47416347 q^{6} + 1577598316 q^{7} + 1424191017 q^{8} + 10460353203 q^{9} + 84497282000 q^{11} - 396518345577 q^{12} - 1065489966310 q^{13} + 1543881561348 q^{14} + 9712801855841 q^{16} - 13851876239906 q^{17} - 2799887874003 q^{18} + 26858848298644 q^{19} - 93155602961484 q^{21} + 93991312008688 q^{22} - 75776598293952 q^{23} - 84097055362833 q^{24} + 16\!\cdots\!86 q^{26}+ \cdots + 29\!\cdots\!00 q^{99}+O(q^{100}) \) 3 * q - 803 * q^2 - 177147 * q^3 + 6715073 * q^4 + 47416347 * q^6 + 1577598316 * q^7 + 1424191017 * q^8 + 10460353203 * q^9 + 84497282000 * q^11 - 396518345577 * q^12 - 1065489966310 * q^13 + 1543881561348 * q^14 + 9712801855841 * q^16 - 13851876239906 * q^17 - 2799887874003 * q^18 + 26858848298644 * q^19 - 93155602961484 * q^21 + 93991312008688 * q^22 - 75776598293952 * q^23 - 84097055362833 * q^24 + 1616971789034486 * q^26 - 617673396283947 * q^27 - 1980748081451468 * q^28 - 911172744177122 * q^29 - 14770635893644696 * q^31 - 10845641917892839 * q^32 - 4989480004818000 * q^33 - 14314214801416654 * q^34 + 23414011787976273 * q^36 + 33876604648537146 * q^37 - 187726010316740996 * q^38 + 62916117020639190 * q^39 + 4403226550677934 * q^41 - 91164662316038052 * q^42 + 76568115811359884 * q^43 + 33141017859775792 * q^44 + 260351976204717264 * q^46 - 178877159689418912 * q^47 - 573531236785555209 * q^48 + 973733149877485627 * q^49 + 817939440090209394 * q^51 - 4165878936433955826 * q^52 - 4013622553495571102 * q^53 + 165330579072003147 * q^54 + 9080922748836602580 * q^56 - 1585988133186629556 * q^57 - 12459376778387397982 * q^58 - 4476302860362619024 * q^59 + 13672812736400197042 * q^61 + 10065431538796030152 * q^62 + 5500745199272668716 * q^63 - 17161949036537140719 * q^64 - 5550092982801017712 * q^66 - 10024292550718281716 * q^67 - 17276944334927178886 * q^68 + 4474532352659571648 * q^69 + 31548512648354310016 * q^71 + 4965847022119925817 * q^72 - 80637656440064589382 * q^73 - 97250438023995188522 * q^74 - 165430884440717923764 * q^76 + 107240569572877645632 * q^77 - 95480567170697363814 * q^78 - 101660087769974505880 * q^79 + 36472996377170786403 * q^81 + 608532996859944197042 * q^82 + 98643502167172343676 * q^83 + 116961193461627733932 * q^84 - 181033595702025175180 * q^86 + 53803839370914876978 * q^87 + 397025854664543013552 * q^88 + 49757751372279077814 * q^89 + 156825279702460168744 * q^91 + 88639329782864857776 * q^92 + 872191278883825654104 * q^93 + 2329033824323004369680 * q^94 + 640424309609654250111 * q^96 - 767965200855128324166 * q^97 - 756579208914437470555 * q^98 + 294623804804498082000 * q^99

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