LMFDB - Embedded newform 75.18.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,18,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, 18, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 18 $
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:	$137.416565508$
Analytic rank:	$0$
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} - 37234x - 350700 $ x^3 - 37234*x - 350700
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-188.067$ 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)

$f(q)$	$=$	\(q+670.613 q^{2} +6561.00 q^{3} +318650. q^{4} +4.39989e6 q^{6} +2.70804e7 q^{7} +1.25792e8 q^{8} +4.30467e7 q^{9} +1.01644e9 q^{11} +2.09066e9 q^{12} +1.04834e9 q^{13} +1.81605e10 q^{14} +4.25918e10 q^{16} -2.00099e10 q^{17} +2.88677e10 q^{18} +1.64417e10 q^{19} +1.77675e11 q^{21} +6.81636e11 q^{22} -5.92721e11 q^{23} +8.25322e11 q^{24} +7.03032e11 q^{26} +2.82430e11 q^{27} +8.62917e12 q^{28} +4.83212e11 q^{29} -5.20802e12 q^{31} +1.20748e13 q^{32} +6.66885e12 q^{33} -1.34189e13 q^{34} +1.37168e13 q^{36} -3.82367e13 q^{37} +1.10260e13 q^{38} +6.87817e12 q^{39} -6.56402e13 q^{41} +1.19151e14 q^{42} +5.83804e13 q^{43} +3.23888e14 q^{44} -3.97486e14 q^{46} -7.43658e12 q^{47} +2.79445e14 q^{48} +5.00719e14 q^{49} -1.31285e14 q^{51} +3.34054e14 q^{52} +3.01466e14 q^{53} +1.89401e14 q^{54} +3.40650e15 q^{56} +1.07874e14 q^{57} +3.24048e14 q^{58} -3.44389e14 q^{59} -2.43237e15 q^{61} -3.49257e15 q^{62} +1.16572e15 q^{63} +2.51491e15 q^{64} +4.47222e15 q^{66} -1.19632e15 q^{67} -6.37614e15 q^{68} -3.88884e15 q^{69} +5.42254e15 q^{71} +5.41494e15 q^{72} +5.84732e15 q^{73} -2.56420e16 q^{74} +5.23914e15 q^{76} +2.75256e16 q^{77} +4.61259e15 q^{78} +2.03455e15 q^{79} +1.85302e15 q^{81} -4.40192e16 q^{82} +1.18343e16 q^{83} +5.66160e16 q^{84} +3.91507e16 q^{86} +3.17035e15 q^{87} +1.27860e17 q^{88} -2.43170e16 q^{89} +2.83895e16 q^{91} -1.88870e17 q^{92} -3.41698e16 q^{93} -4.98707e15 q^{94} +7.92225e16 q^{96} +1.06105e17 q^{97} +3.35788e17 q^{98} +4.37543e16 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 442 q^{2} + 19683 q^{3} + 298148 q^{4} + 2899962 q^{6} - 4962644 q^{7} + 108831912 q^{8} + 129140163 q^{9} + 1049849720 q^{11} + 1956149028 q^{12} + 3091742090 q^{13} + 27586028328 q^{14} + 22392797456 q^{16}+ \cdots + 45\!\cdots\!20 q^{99}+O(q^{100}) $ 3 * q + 442 * q^2 + 19683 * q^3 + 298148 * q^4 + 2899962 * q^6 - 4962644 * q^7 + 108831912 * q^8 + 129140163 * q^9 + 1049849720 * q^11 + 1956149028 * q^12 + 3091742090 * q^13 + 27586028328 * q^14 + 22392797456 * q^16 + 15119940094 * q^17 + 19026650682 * q^18 - 111896117996 * q^19 - 32559907284 * q^21 + 693818341648 * q^22 - 784535528472 * q^23 + 714046174632 * q^24 - 1092511841044 * q^26 + 847288609443 * q^27 + 7640185610512 * q^28 + 1395352153678 * q^29 - 12458197616536 * q^31 + 24074446195616 * q^32 + 6888064012920 * q^33 - 41046727271164 * q^34 + 12834293772708 * q^36 - 32093748310134 * q^37 + 68757678664024 * q^38 + 20284919852490 * q^39 - 102276917392466 * q^41 + 180991931860008 * q^42 + 141347515908644 * q^43 + 319886980653472 * q^44 - 299722933931856 * q^46 + 54182226541768 * q^47 + 146919144108816 * q^48 + 702936204707227 * q^49 + 99201926956734 * q^51 + 670193699134584 * q^52 - 345545322785582 * q^53 + 124833855124602 * q^54 + 3014407093472160 * q^56 - 734150430171756 * q^57 - 2198630240496412 * q^58 + 2075534176494536 * q^59 - 1591128188427998 * q^61 - 1555165628278608 * q^62 - 213625551690324 * q^63 + 4014461944704576 * q^64 + 4552142139552528 * q^66 + 2417737431711844 * q^67 - 1338181134241816 * q^68 - 5147337602304792 * q^69 + 5019485593673056 * q^71 + 4684856951760552 * q^72 + 10804161755859578 * q^73 - 12140475721796372 * q^74 - 3289783191661584 * q^76 + 27418121851422432 * q^77 - 7167970189089684 * q^78 - 40642131657181960 * q^79 + 5559060566555523 * q^81 - 15262994262855388 * q^82 + 21823256926416996 * q^83 + 50127257790569232 * q^84 + 8867591636371640 * q^86 + 9154905480281358 * q^87 + 125733265675831872 * q^88 - 2700230040372426 * q^89 - 46116457499478296 * q^91 - 204241599793288224 * q^92 - 81738234562092696 * q^93 - 72231791695517200 * q^94 + 157952441489436576 * q^96 + 79760576372788314 * q^97 + 128015343617626250 * q^98 + 45192587988768120 * q^99

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 $

$2$	670.613	1.85232	0.926162	−	0.377126i	$-0.123088\pi$
$2$	670.613	1.85232	0.926162	+	0.377126i	$0.123088\pi$
$3$	6561.00	0.577350
$3$	6561.00	0.577350
$4$	318650.	2.43110
$5$	0	0
$5$	0	0
$6$	4.39989e6	1.06944
$7$	2.70804e7	1.77551	0.887753	−	0.460320i	$-0.152265\pi$
$7$	2.70804e7	1.77551	0.887753	+	0.460320i	$0.152265\pi$
$8$	1.25792e8	2.65087
$9$	4.30467e7	0.333333
$10$	0	0
$11$	1.01644e9	1.42969	0.714847	−	0.699281i	$-0.246496\pi$
$11$	1.01644e9	1.42969	0.714847	+	0.699281i	$0.246496\pi$
$12$	2.09066e9	1.40360
$13$	1.04834e9	0.356438	0.178219	−	0.983991i	$-0.442966\pi$
$13$	1.04834e9	0.356438	0.178219	+	0.983991i	$0.442966\pi$
$14$	1.81605e10	3.28881
$15$	0	0
$16$	4.25918e10	2.47917
$17$	−2.00099e10	−0.695711	−0.347855	−	0.937548i	$-0.613090\pi$
$17$	−2.00099e10	−0.695711	−0.347855	+	0.937548i	$0.613090\pi$
$18$	2.88677e10	0.617441
$19$	1.64417e10	0.222096	0.111048	−	0.993815i	$-0.464579\pi$
$19$	1.64417e10	0.222096	0.111048	+	0.993815i	$0.464579\pi$
$20$	0	0
$21$	1.77675e11	1.02509
$22$	6.81636e11	2.64826
$23$	−5.92721e11	−1.57821	−0.789103	−	0.614261i	$-0.789454\pi$
$23$	−5.92721e11	−1.57821	−0.789103	+	0.614261i	$0.789454\pi$
$24$	8.25322e11	1.53048
$25$	0	0
$26$	7.03032e11	0.660239
$27$	2.82430e11	0.192450
$28$	8.62917e12	4.31644
$29$	4.83212e11	0.179372	0.0896860	−	0.995970i	$-0.471414\pi$
$29$	4.83212e11	0.179372	0.0896860	+	0.995970i	$0.471414\pi$
$30$	0	0
$31$	−5.20802e12	−1.09673	−0.548363	−	0.836240i	$-0.684749\pi$
$31$	−5.20802e12	−1.09673	−0.548363	+	0.836240i	$0.684749\pi$
$32$	1.20748e13	1.94135
$33$	6.66885e12	0.825434
$34$	−1.34189e13	−1.28868
$35$	0	0
$36$	1.37168e13	0.810368
$37$	−3.82367e13	−1.78964	−0.894820	−	0.446427i	$-0.852696\pi$
$37$	−3.82367e13	−1.78964	−0.894820	+	0.446427i	$0.852696\pi$
$38$	1.10260e13	0.411393
$39$	6.87817e12	0.205790
$40$	0	0
$41$	−6.56402e13	−1.28383	−0.641915	−	0.766776i	$-0.721860\pi$
$41$	−6.56402e13	−1.28383	−0.641915	+	0.766776i	$0.721860\pi$
$42$	1.19151e14	1.89880
$43$	5.83804e13	0.761703	0.380851	−	0.924636i	$-0.375631\pi$
$43$	5.83804e13	0.761703	0.380851	+	0.924636i	$0.375631\pi$
$44$	3.23888e14	3.47573
$45$	0	0
$46$	−3.97486e14	−2.92335
$47$	−7.43658e12	−0.0455556	−0.0227778	−	0.999741i	$-0.507251\pi$
$47$	−7.43658e12	−0.0455556	−0.0227778	+	0.999741i	$0.507251\pi$
$48$	2.79445e14	1.43135
$49$	5.00719e14	2.15242
$50$	0	0
$51$	−1.31285e14	−0.401669
$52$	3.34054e14	0.866539
$53$	3.01466e14	0.665109	0.332555	−	0.943084i	$-0.392089\pi$
$53$	3.01466e14	0.665109	0.332555	+	0.943084i	$0.392089\pi$
$54$	1.89401e14	0.356480
$55$	0	0
$56$	3.40650e15	4.70664
$57$	1.07874e14	0.128227
$58$	3.24048e14	0.332255
$59$	−3.44389e14	−0.305357	−0.152678	−	0.988276i	$-0.548790\pi$
$59$	−3.44389e14	−0.305357	−0.152678	+	0.988276i	$0.548790\pi$
$60$	0	0
$61$	−2.43237e15	−1.62452	−0.812261	−	0.583294i	$-0.801763\pi$
$61$	−2.43237e15	−1.62452	−0.812261	+	0.583294i	$0.801763\pi$
$62$	−3.49257e15	−2.03149
$63$	1.16572e15	0.591835
$64$	2.51491e15	1.11684
$65$	0	0
$66$	4.47222e15	1.52897
$67$	−1.19632e15	−0.359924	−0.179962	−	0.983674i	$-0.557597\pi$
$67$	−1.19632e15	−0.359924	−0.179962	+	0.983674i	$0.557597\pi$
$68$	−6.37614e15	−1.69135
$69$	−3.88884e15	−0.911177
$70$	0	0
$71$	5.42254e15	0.996567	0.498283	−	0.867014i	$-0.333964\pi$
$71$	5.42254e15	0.996567	0.498283	+	0.867014i	$0.333964\pi$
$72$	5.41494e15	0.883623
$73$	5.84732e15	0.848619	0.424309	−	0.905517i	$-0.360517\pi$
$73$	5.84732e15	0.848619	0.424309	+	0.905517i	$0.360517\pi$
$74$	−2.56420e16	−3.31499
$75$	0	0
$76$	5.23914e15	0.539938
$77$	2.75256e16	2.53843
$78$	4.61259e15	0.381189
$79$	2.03455e15	0.150882	0.0754411	−	0.997150i	$-0.475964\pi$
$79$	2.03455e15	0.150882	0.0754411	+	0.997150i	$0.475964\pi$
$80$	0	0
$81$	1.85302e15	0.111111
$82$	−4.40192e16	−2.37807
$83$	1.18343e16	0.576740	0.288370	−	0.957519i	$-0.406887\pi$
$83$	1.18343e16	0.576740	0.288370	+	0.957519i	$0.406887\pi$
$84$	5.66160e16	2.49210
$85$	0	0
$86$	3.91507e16	1.41092
$87$	3.17035e15	0.103560
$88$	1.27860e17	3.78993
$89$	−2.43170e16	−0.654780	−0.327390	−	0.944889i	$-0.606169\pi$
$89$	−2.43170e16	−0.654780	−0.327390	+	0.944889i	$0.606169\pi$
$90$	0	0
$91$	2.83895e16	0.632858
$92$	−1.88870e17	−3.83678
$93$	−3.41698e16	−0.633195
$94$	−4.98707e15	−0.0843837
$95$	0	0
$96$	7.92225e16	1.12084
$97$	1.06105e17	1.37459	0.687296	−	0.726377i	$-0.258797\pi$
$97$	1.06105e17	1.37459	0.687296	+	0.726377i	$0.258797\pi$
$98$	3.35788e17	3.98698
$99$	4.37543e16	0.476564
$100$	0	0
$101$	6.36534e16	0.584911	0.292455	−	0.956279i	$-0.405528\pi$
$101$	6.36534e16	0.584911	0.292455	+	0.956279i	$0.405528\pi$
$102$	−8.80413e16	−0.744021
$103$	3.76414e16	0.292785	0.146393	−	0.989227i	$-0.453234\pi$
$103$	3.76414e16	0.292785	0.146393	+	0.989227i	$0.453234\pi$
$104$	1.31873e17	0.944872
$105$	0	0
$106$	2.02167e17	1.23200
$107$	−1.97603e17	−1.11181	−0.555906	−	0.831245i	$-0.687629\pi$
$107$	−1.97603e17	−1.11181	−0.555906	+	0.831245i	$0.687629\pi$
$108$	8.99961e16	0.467866
$109$	2.14472e17	1.03097	0.515484	−	0.856899i	$-0.327612\pi$
$109$	2.14472e17	1.03097	0.515484	+	0.856899i	$0.327612\pi$
$110$	0	0
$111$	−2.50871e17	−1.03325
$112$	1.15340e18	4.40177
$113$	−3.29313e17	−1.16531	−0.582656	−	0.812719i	$-0.697986\pi$
$113$	−3.29313e17	−1.16531	−0.582656	+	0.812719i	$0.697986\pi$
$114$	7.23416e16	0.237518
$115$	0	0
$116$	1.53975e17	0.436072
$117$	4.51277e16	0.118813
$118$	−2.30952e17	−0.565620
$119$	−5.41876e17	−1.23524
$120$	0	0
$121$	5.27698e17	1.04402
$122$	−1.63118e18	−3.00914
$123$	−4.30666e17	−0.741219
$124$	−1.65953e18	−2.66626
$125$	0	0
$126$	7.81749e17	1.09627
$127$	−4.24273e17	−0.556306	−0.278153	−	0.960537i	$-0.589722\pi$
$127$	−4.24273e17	−0.556306	−0.278153	+	0.960537i	$0.589722\pi$
$128$	1.03865e17	0.127405
$129$	3.83034e17	0.439769
$130$	0	0
$131$	−1.01448e18	−1.02197	−0.510985	−	0.859590i	$-0.670719\pi$
$131$	−1.01448e18	−1.02197	−0.510985	+	0.859590i	$0.670719\pi$
$132$	2.12503e18	2.00672
$133$	4.45247e17	0.394332
$134$	−8.02266e17	−0.666696
$135$	0	0
$136$	−2.51708e18	−1.84424
$137$	1.60921e18	1.10787	0.553934	−	0.832560i	$-0.313126\pi$
$137$	1.60921e18	1.10787	0.553934	+	0.832560i	$0.313126\pi$
$138$	−2.60791e18	−1.68780
$139$	−1.37382e18	−0.836188	−0.418094	−	0.908404i	$-0.637302\pi$
$139$	−1.37382e18	−0.836188	−0.418094	+	0.908404i	$0.637302\pi$
$140$	0	0
$141$	−4.87914e16	−0.0263015
$142$	3.63642e18	1.84596
$143$	1.06557e18	0.509597
$144$	1.83344e18	0.826389
$145$	0	0
$146$	3.92129e18	1.57192
$147$	3.28521e18	1.24270
$148$	−1.21841e19	−4.35080
$149$	−3.59822e18	−1.21340	−0.606701	−	0.794930i	$-0.707507\pi$
$149$	−3.59822e18	−1.21340	−0.606701	+	0.794930i	$0.707507\pi$
$150$	0	0
$151$	1.83062e18	0.551182	0.275591	−	0.961275i	$-0.411127\pi$
$151$	1.83062e18	0.551182	0.275591	+	0.961275i	$0.411127\pi$
$152$	2.06823e18	0.588747
$153$	−8.61360e17	−0.231904
$154$	1.84590e19	4.70199
$155$	0	0
$156$	2.19173e18	0.500296
$157$	−2.03936e18	−0.440907	−0.220453	−	0.975398i	$-0.570754\pi$
$157$	−2.03936e18	−0.440907	−0.220453	+	0.975398i	$0.570754\pi$
$158$	1.36440e18	0.279483
$159$	1.97792e18	0.384001
$160$	0	0
$161$	−1.60511e19	−2.80211
$162$	1.24266e18	0.205814
$163$	1.26018e19	1.98078	0.990390	−	0.138302i	$-0.0441646\pi$
$163$	1.26018e19	1.98078	0.990390	+	0.138302i	$0.0441646\pi$
$164$	−2.09162e19	−3.12112
$165$	0	0
$166$	7.93626e18	1.06831
$167$	−9.46306e18	−1.21044	−0.605218	−	0.796060i	$-0.706914\pi$
$167$	−9.46306e18	−1.21044	−0.605218	+	0.796060i	$0.706914\pi$
$168$	2.23501e19	2.71738
$169$	−7.55140e18	−0.872952
$170$	0	0
$171$	7.07760e17	0.0740319
$172$	1.86029e19	1.85178
$173$	5.05953e18	0.479422	0.239711	−	0.970844i	$-0.422947\pi$
$173$	5.05953e18	0.479422	0.239711	+	0.970844i	$0.422947\pi$
$174$	2.12608e18	0.191828
$175$	0	0
$176$	4.32919e19	3.54445
$177$	−2.25954e18	−0.176298
$178$	−1.63073e19	−1.21287
$179$	1.38935e18	0.0985284	0.0492642	−	0.998786i	$-0.484312\pi$
$179$	1.38935e18	0.0985284	0.0492642	+	0.998786i	$0.484312\pi$
$180$	0	0
$181$	6.56442e18	0.423573	0.211787	−	0.977316i	$-0.432072\pi$
$181$	6.56442e18	0.423573	0.211787	+	0.977316i	$0.432072\pi$
$182$	1.90384e19	1.17226
$183$	−1.59588e19	−0.937918
$184$	−7.45596e19	−4.18362
$185$	0	0
$186$	−2.29147e19	−1.17288
$187$	−2.03388e19	−0.994653
$188$	−2.36966e18	−0.110750
$189$	7.64831e18	0.341696
$190$	0	0
$191$	1.88960e19	0.771943	0.385972	−	0.922511i	$-0.373866\pi$
$191$	1.88960e19	0.771943	0.385972	+	0.922511i	$0.373866\pi$
$192$	1.65003e19	0.644809
$193$	2.57702e18	0.0963563	0.0481781	−	0.998839i	$-0.484658\pi$
$193$	2.57702e18	0.0963563	0.0481781	+	0.998839i	$0.484658\pi$
$194$	7.11551e19	2.54619
$195$	0	0
$196$	1.59554e20	5.23276
$197$	2.58028e19	0.810410	0.405205	−	0.914226i	$-0.367200\pi$
$197$	2.58028e19	0.810410	0.405205	+	0.914226i	$0.367200\pi$
$198$	2.93422e19	0.882752
$199$	5.04896e18	0.145529	0.0727647	−	0.997349i	$-0.476818\pi$
$199$	5.04896e18	0.145529	0.0727647	+	0.997349i	$0.476818\pi$
$200$	0	0
$201$	−7.84904e18	−0.207802
$202$	4.26868e19	1.08344
$203$	1.30856e19	0.318476
$204$	−4.18339e19	−0.976499
$205$	0	0
$206$	2.52428e19	0.542333
$207$	−2.55147e19	−0.526069
$208$	4.46507e19	0.883670
$209$	1.67119e19	0.317529
$210$	0	0
$211$	4.88902e19	0.856685	0.428342	−	0.903616i	$-0.359098\pi$
$211$	4.88902e19	0.856685	0.428342	+	0.903616i	$0.359098\pi$
$212$	9.60620e19	1.61695
$213$	3.55773e19	0.575368
$214$	−1.32515e20	−2.05944
$215$	0	0
$216$	3.55274e19	0.510160
$217$	−1.41035e20	−1.94724
$218$	1.43828e20	1.90969
$219$	3.83643e19	0.489950
$220$	0	0
$221$	−2.09772e19	−0.247978
$222$	−1.68237e20	−1.91391
$223$	−6.03668e18	−0.0661008	−0.0330504	−	0.999454i	$-0.510522\pi$
$223$	−6.03668e18	−0.0661008	−0.0330504	+	0.999454i	$0.510522\pi$
$224$	3.26990e20	3.44688
$225$	0	0
$226$	−2.20842e20	−2.15854
$227$	1.85243e20	1.74390	0.871952	−	0.489591i	$-0.162854\pi$
$227$	1.85243e20	1.74390	0.871952	+	0.489591i	$0.162854\pi$
$228$	3.43740e19	0.311733
$229$	1.07933e20	0.943092	0.471546	−	0.881841i	$-0.343696\pi$
$229$	1.07933e20	0.943092	0.471546	+	0.881841i	$0.343696\pi$
$230$	0	0
$231$	1.80595e20	1.46556
$232$	6.07842e19	0.475492
$233$	1.20686e20	0.910190	0.455095	−	0.890443i	$-0.349605\pi$
$233$	1.20686e20	0.910190	0.455095	+	0.890443i	$0.349605\pi$
$234$	3.02632e19	0.220080
$235$	0	0
$236$	−1.09740e20	−0.742355
$237$	1.33487e19	0.0871119
$238$	−3.63389e20	−2.28806
$239$	1.11522e20	0.677607	0.338804	−	0.940857i	$-0.389978\pi$
$239$	1.11522e20	0.677607	0.338804	+	0.940857i	$0.389978\pi$
$240$	0	0
$241$	−1.89825e20	−1.07451	−0.537253	−	0.843421i	$-0.680538\pi$
$241$	−1.89825e20	−1.07451	−0.537253	+	0.843421i	$0.680538\pi$
$242$	3.53881e20	1.93387
$243$	1.21577e19	0.0641500
$244$	−7.75073e20	−3.94938
$245$	0	0
$246$	−2.88810e20	−1.37298
$247$	1.72365e19	0.0791634
$248$	−6.55128e20	−2.90728
$249$	7.76451e19	0.332981
$250$	0	0
$251$	−1.47560e20	−0.591210	−0.295605	−	0.955310i	$-0.595521\pi$
$251$	−1.47560e20	−0.591210	−0.295605	+	0.955310i	$0.595521\pi$
$252$	3.71457e20	1.43881
$253$	−6.02464e20	−2.25635
$254$	−2.84523e20	−1.03046
$255$	0	0
$256$	−2.59980e20	−0.880848
$257$	−5.88705e20	−1.92959	−0.964797	−	0.262994i	$-0.915290\pi$
$257$	−5.88705e20	−1.92959	−0.964797	+	0.262994i	$0.915290\pi$
$258$	2.56868e20	0.814595
$259$	−1.03547e21	−3.17752
$260$	0	0
$261$	2.08007e19	0.0597906
$262$	−6.80324e20	−1.89302
$263$	2.53714e20	0.683472	0.341736	−	0.939796i	$-0.388985\pi$
$263$	2.53714e20	0.683472	0.341736	+	0.939796i	$0.388985\pi$
$264$	8.38888e20	2.18812
$265$	0	0
$266$	2.98589e20	0.730431
$267$	−1.59544e20	−0.378038
$268$	−3.81206e20	−0.875013
$269$	−3.54749e20	−0.788909	−0.394455	−	0.918915i	$-0.629067\pi$
$269$	−3.54749e20	−0.788909	−0.394455	+	0.918915i	$0.629067\pi$
$270$	0	0
$271$	6.20629e20	1.29596	0.647982	−	0.761656i	$-0.275613\pi$
$271$	6.20629e20	1.29596	0.647982	+	0.761656i	$0.275613\pi$
$272$	−8.52256e20	−1.72478
$273$	1.86264e20	0.365381
$274$	1.07916e21	2.05213
$275$	0	0
$276$	−1.23918e21	−2.21517
$277$	5.37902e20	0.932449	0.466225	−	0.884666i	$-0.345614\pi$
$277$	5.37902e20	0.932449	0.466225	+	0.884666i	$0.345614\pi$
$278$	−9.21301e20	−1.54889
$279$	−2.24188e20	−0.365575
$280$	0	0
$281$	−8.40846e20	−1.29037	−0.645183	−	0.764028i	$-0.723219\pi$
$281$	−8.40846e20	−1.29037	−0.645183	+	0.764028i	$0.723219\pi$
$282$	−3.27201e19	−0.0487189
$283$	4.49542e20	0.649510	0.324755	−	0.945798i	$-0.394718\pi$
$283$	4.49542e20	0.649510	0.324755	+	0.945798i	$0.394718\pi$
$284$	1.72789e21	2.42276
$285$	0	0
$286$	7.14588e20	0.943940
$287$	−1.77756e21	−2.27945
$288$	5.19779e20	0.647117
$289$	−4.26845e20	−0.515987
$290$	0	0
$291$	6.96152e20	0.793622
$292$	1.86325e21	2.06308
$293$	8.24381e20	0.886652	0.443326	−	0.896360i	$-0.353798\pi$
$293$	8.24381e20	0.886652	0.443326	+	0.896360i	$0.353798\pi$
$294$	2.20311e21	2.30188
$295$	0	0
$296$	−4.80987e21	−4.74410
$297$	2.87072e20	0.275145
$298$	−2.41301e21	−2.24761
$299$	−6.21374e20	−0.562533
$300$	0	0
$301$	1.58097e21	1.35241
$302$	1.22764e21	1.02097
$303$	4.17630e20	0.337698
$304$	7.00280e20	0.550612
$305$	0	0
$306$	−5.77639e20	−0.429560
$307$	−2.06003e20	−0.149004	−0.0745021	−	0.997221i	$-0.523737\pi$
$307$	−2.06003e20	−0.149004	−0.0745021	+	0.997221i	$0.523737\pi$
$308$	8.77101e21	6.17119
$309$	2.46965e20	0.169040
$310$	0	0
$311$	9.73195e20	0.630575	0.315287	−	0.948996i	$-0.397899\pi$
$311$	9.73195e20	0.630575	0.315287	+	0.948996i	$0.397899\pi$
$312$	8.65219e20	0.545522
$313$	1.65984e21	1.01845	0.509226	−	0.860633i	$-0.329932\pi$
$313$	1.65984e21	1.01845	0.509226	+	0.860633i	$0.329932\pi$
$314$	−1.36762e21	−0.816702
$315$	0	0
$316$	6.48309e20	0.366811
$317$	−3.08017e21	−1.69657	−0.848283	−	0.529543i	$-0.822364\pi$
$317$	−3.08017e21	−1.69657	−0.848283	+	0.529543i	$0.822364\pi$
$318$	1.32642e21	0.711294
$319$	4.91155e20	0.256447
$320$	0	0
$321$	−1.29647e21	−0.641905
$322$	−1.07641e22	−5.19042
$323$	−3.28996e20	−0.154514
$324$	5.90465e20	0.270123
$325$	0	0
$326$	8.45090e21	3.66905
$327$	1.40715e21	0.595229
$328$	−8.25702e21	−3.40327
$329$	−2.01386e20	−0.0808841
$330$	0	0
$331$	−3.68783e21	−1.40680	−0.703401	−	0.710794i	$-0.748336\pi$
$331$	−3.68783e21	−1.40680	−0.703401	+	0.710794i	$0.748336\pi$
$332$	3.77101e21	1.40212
$333$	−1.64596e21	−0.596547
$334$	−6.34605e21	−2.24212
$335$	0	0
$336$	7.56747e21	2.54137
$337$	−2.06121e20	−0.0674946	−0.0337473	−	0.999430i	$-0.510744\pi$
$337$	−2.06121e20	−0.0674946	−0.0337473	+	0.999430i	$0.510744\pi$
$338$	−5.06406e21	−1.61699
$339$	−2.16062e21	−0.672793
$340$	0	0
$341$	−5.29363e21	−1.56798
$342$	4.74633e20	0.137131
$343$	7.25994e21	2.04613
$344$	7.34380e21	2.01918
$345$	0	0
$346$	3.39299e21	0.888045
$347$	−1.45695e21	−0.372087	−0.186044	−	0.982541i	$-0.559567\pi$
$347$	−1.45695e21	−0.372087	−0.186044	+	0.982541i	$0.559567\pi$
$348$	1.01023e21	0.251766
$349$	−6.15182e21	−1.49619	−0.748096	−	0.663590i	$-0.769032\pi$
$349$	−6.15182e21	−1.49619	−0.748096	+	0.663590i	$0.769032\pi$
$350$	0	0
$351$	2.96083e20	0.0685966
$352$	1.22732e22	2.77553
$353$	−4.05523e21	−0.895222	−0.447611	−	0.894228i	$-0.647725\pi$
$353$	−4.05523e21	−0.895222	−0.447611	+	0.894228i	$0.647725\pi$
$354$	−1.51528e21	−0.326561
$355$	0	0
$356$	−7.74862e21	−1.59184
$357$	−3.55525e21	−0.713165
$358$	9.31717e20	0.182507
$359$	2.30064e21	0.440094	0.220047	−	0.975489i	$-0.429379\pi$
$359$	2.30064e21	0.440094	0.220047	+	0.975489i	$0.429379\pi$
$360$	0	0
$361$	−5.21006e21	−0.950673
$362$	4.40219e21	0.784595
$363$	3.46223e21	0.602767
$364$	9.04632e21	1.53854
$365$	0	0
$366$	−1.07022e22	−1.73733
$367$	−3.92558e21	−0.622648	−0.311324	−	0.950304i	$-0.600772\pi$
$367$	−3.92558e21	−0.622648	−0.311324	+	0.950304i	$0.600772\pi$
$368$	−2.52450e22	−3.91263
$369$	−2.82560e21	−0.427943
$370$	0	0
$371$	8.16382e21	1.18091
$372$	−1.08882e22	−1.53936
$373$	−1.72627e20	−0.0238553	−0.0119276	−	0.999929i	$-0.503797\pi$
$373$	−1.72627e20	−0.0238553	−0.0119276	+	0.999929i	$0.503797\pi$
$374$	−1.36395e22	−1.84242
$375$	0	0
$376$	−9.35463e20	−0.120762
$377$	5.06571e20	0.0639350
$378$	5.12906e21	0.632932
$379$	−2.57296e21	−0.310455	−0.155228	−	0.987879i	$-0.549611\pi$
$379$	−2.57296e21	−0.310455	−0.155228	+	0.987879i	$0.549611\pi$
$380$	0	0
$381$	−2.78365e21	−0.321183
$382$	1.26719e22	1.42989
$383$	−1.66889e21	−0.184178	−0.0920890	−	0.995751i	$-0.529354\pi$
$383$	−1.66889e21	−0.184178	−0.0920890	+	0.995751i	$0.529354\pi$
$384$	6.81460e20	0.0735572
$385$	0	0
$386$	1.72818e21	0.178483
$387$	2.51309e21	0.253901
$388$	3.38102e22	3.34178
$389$	5.43023e21	0.525106	0.262553	−	0.964918i	$-0.415436\pi$
$389$	5.43023e21	0.525106	0.262553	+	0.964918i	$0.415436\pi$
$390$	0	0
$391$	1.18603e22	1.09797
$392$	6.29865e22	5.70579
$393$	−6.65601e21	−0.590034
$394$	1.73037e22	1.50114
$395$	0	0
$396$	1.39423e22	1.15858
$397$	1.04340e22	0.848659	0.424330	−	0.905508i	$-0.360510\pi$
$397$	1.04340e22	0.848659	0.424330	+	0.905508i	$0.360510\pi$
$398$	3.38590e21	0.269568
$399$	2.92127e21	0.227668
$400$	0	0
$401$	−1.07284e22	−0.801326	−0.400663	−	0.916225i	$-0.631220\pi$
$401$	−1.07284e22	−0.801326	−0.400663	+	0.916225i	$0.631220\pi$
$402$	−5.26367e21	−0.384917
$403$	−5.45979e21	−0.390915
$404$	2.02831e22	1.42198
$405$	0	0
$406$	8.77536e21	0.589921
$407$	−3.88652e22	−2.55864
$408$	−1.65146e22	−1.06477
$409$	−3.09142e22	−1.95213	−0.976066	−	0.217474i	$-0.930218\pi$
$409$	−3.09142e22	−1.95213	−0.976066	+	0.217474i	$0.930218\pi$
$410$	0	0
$411$	1.05580e22	0.639628
$412$	1.19944e22	0.711791
$413$	−9.32621e21	−0.542163
$414$	−1.71105e22	−0.974450
$415$	0	0
$416$	1.26585e22	0.691971
$417$	−9.01363e21	−0.482773
$418$	1.12072e22	0.588166
$419$	−1.50465e22	−0.773780	−0.386890	−	0.922126i	$-0.626451\pi$
$419$	−1.50465e22	−0.773780	−0.386890	+	0.922126i	$0.626451\pi$
$420$	0	0
$421$	−2.30163e22	−1.13668	−0.568339	−	0.822795i	$-0.692414\pi$
$421$	−2.30163e22	−1.13668	−0.568339	+	0.822795i	$0.692414\pi$
$422$	3.27864e22	1.58686
$423$	−3.20120e20	−0.0151852
$424$	3.79220e22	1.76312
$425$	0	0
$426$	2.38586e22	1.06577
$427$	−6.58695e22	−2.88435
$428$	−6.29662e22	−2.70293
$429$	6.99123e21	0.294216
$430$	0	0
$431$	1.28296e22	0.518985	0.259493	−	0.965745i	$-0.416445\pi$
$431$	1.28296e22	0.518985	0.259493	+	0.965745i	$0.416445\pi$
$432$	1.20292e22	0.477116
$433$	−7.37153e21	−0.286689	−0.143344	−	0.989673i	$-0.545786\pi$
$433$	−7.37153e21	−0.286689	−0.143344	+	0.989673i	$0.545786\pi$
$434$	−9.45802e22	−3.60693
$435$	0	0
$436$	6.83414e22	2.50639
$437$	−9.74532e21	−0.350513
$438$	2.57276e22	0.907547
$439$	3.49450e22	1.20903	0.604514	−	0.796595i	$-0.293367\pi$
$439$	3.49450e22	1.20903	0.604514	+	0.796595i	$0.293367\pi$
$440$	0	0
$441$	2.15543e22	0.717473
$442$	−1.40676e22	−0.459335
$443$	3.18362e22	1.01974	0.509870	−	0.860251i	$-0.329693\pi$
$443$	3.18362e22	1.01974	0.509870	+	0.860251i	$0.329693\pi$
$444$	−7.99400e22	−2.51194
$445$	0	0
$446$	−4.04828e21	−0.122440
$447$	−2.36079e22	−0.700558
$448$	6.81047e22	1.98296
$449$	−1.60496e22	−0.458532	−0.229266	−	0.973364i	$-0.573633\pi$
$449$	−1.60496e22	−0.458532	−0.229266	+	0.973364i	$0.573633\pi$
$450$	0	0
$451$	−6.67192e22	−1.83548
$452$	−1.04936e23	−2.83300
$453$	1.20107e22	0.318225
$454$	1.24227e23	3.23028
$455$	0	0
$456$	1.35697e22	0.339913
$457$	6.60625e22	1.62430	0.812152	−	0.583446i	$-0.198296\pi$
$457$	6.60625e22	1.62430	0.812152	+	0.583446i	$0.198296\pi$
$458$	7.23815e22	1.74691
$459$	−5.65138e21	−0.133890
$460$	0	0
$461$	2.90192e22	0.662563	0.331282	−	0.943532i	$-0.392519\pi$
$461$	2.90192e22	0.662563	0.331282	+	0.943532i	$0.392519\pi$
$462$	1.21109e23	2.71470
$463$	8.02536e22	1.76614	0.883072	−	0.469237i	$-0.155471\pi$
$463$	8.02536e22	1.76614	0.883072	+	0.469237i	$0.155471\pi$
$464$	2.05808e22	0.444693
$465$	0	0
$466$	8.09337e22	1.68597
$467$	3.77237e22	0.771651	0.385826	−	0.922572i	$-0.373917\pi$
$467$	3.77237e22	0.771651	0.385826	+	0.922572i	$0.373917\pi$
$468$	1.43799e22	0.288846
$469$	−3.23968e22	−0.639047
$470$	0	0
$471$	−1.33802e22	−0.254558
$472$	−4.33215e22	−0.809462
$473$	5.93401e22	1.08900
$474$	8.95180e21	0.161359
$475$	0	0
$476$	−1.72669e23	−3.00299
$477$	1.29771e22	0.221703
$478$	7.47881e22	1.25515
$479$	6.60372e21	0.108877	0.0544385	−	0.998517i	$-0.482663\pi$
$479$	6.60372e21	0.108877	0.0544385	+	0.998517i	$0.482663\pi$
$480$	0	0
$481$	−4.00851e22	−0.637896
$482$	−1.27299e23	−1.99033
$483$	−1.05311e23	−1.61780
$484$	1.68151e23	2.53813
$485$	0	0
$486$	8.15309e21	0.118827
$487$	5.47261e22	0.783788	0.391894	−	0.920010i	$-0.371820\pi$
$487$	5.47261e22	0.783788	0.391894	+	0.920010i	$0.371820\pi$
$488$	−3.05973e23	−4.30640
$489$	8.26801e22	1.14360
$490$	0	0
$491$	8.78896e22	1.17421	0.587103	−	0.809512i	$-0.300268\pi$
$491$	8.78896e22	1.17421	0.587103	+	0.809512i	$0.300268\pi$
$492$	−1.37231e23	−1.80198
$493$	−9.66901e21	−0.124791
$494$	1.15590e22	0.146636
$495$	0	0
$496$	−2.21819e23	−2.71897
$497$	1.46845e23	1.76941
$498$	5.20698e22	0.616789
$499$	5.68475e22	0.661998	0.330999	−	0.943631i	$-0.392614\pi$
$499$	5.68475e22	0.661998	0.330999	+	0.943631i	$0.392614\pi$
$500$	0	0
$501$	−6.20871e22	−0.698845
$502$	−9.89556e22	−1.09511
$503$	9.35523e22	1.01795	0.508975	−	0.860781i	$-0.330024\pi$
$503$	9.35523e22	1.01795	0.508975	+	0.860781i	$0.330024\pi$
$504$	1.46639e23	1.56888
$505$	0	0
$506$	−4.04020e23	−4.17949
$507$	−4.95447e22	−0.503999
$508$	−1.35194e23	−1.35244
$509$	−1.72895e23	−1.70091	−0.850454	−	0.526050i	$-0.823673\pi$
$509$	−1.72895e23	−1.70091	−0.850454	+	0.526050i	$0.823673\pi$
$510$	0	0
$511$	1.58348e23	1.50673
$512$	−1.87960e23	−1.75902
$513$	4.64361e21	0.0427424
$514$	−3.94793e23	−3.57423
$515$	0	0
$516$	1.22054e23	1.06913
$517$	−7.55882e21	−0.0651305
$518$	−6.94397e23	−5.88579
$519$	3.31956e22	0.276795
$520$	0	0
$521$	−1.19149e23	−0.961542	−0.480771	−	0.876846i	$-0.659643\pi$
$521$	−1.19149e23	−0.961542	−0.480771	+	0.876846i	$0.659643\pi$
$522$	1.39492e22	0.110752
$523$	−1.24654e22	−0.0973735	−0.0486868	−	0.998814i	$-0.515504\pi$
$523$	−1.24654e22	−0.0973735	−0.0486868	+	0.998814i	$0.515504\pi$
$524$	−3.23264e23	−2.48452
$525$	0	0
$526$	1.70144e23	1.26601
$527$	1.04212e23	0.763004
$528$	2.84038e23	2.04639
$529$	2.10268e23	1.49073
$530$	0	0
$531$	−1.48248e22	−0.101786
$532$	1.41878e23	0.958663
$533$	−6.88134e22	−0.457606
$534$	−1.06992e23	−0.700248
$535$	0	0
$536$	−1.50487e23	−0.954112
$537$	9.11554e21	0.0568854
$538$	−2.37899e23	−1.46132
$539$	5.08949e23	3.07730
$540$	0	0
$541$	−7.61271e22	−0.446028	−0.223014	−	0.974815i	$-0.571590\pi$
$541$	−7.61271e22	−0.446028	−0.223014	+	0.974815i	$0.571590\pi$
$542$	4.16202e23	2.40054
$543$	4.30692e22	0.244550
$544$	−2.41614e23	−1.35062
$545$	0	0
$546$	1.24911e23	0.676804
$547$	1.74228e23	0.929447	0.464723	−	0.885456i	$-0.346154\pi$
$547$	1.74228e23	0.929447	0.464723	+	0.885456i	$0.346154\pi$
$548$	5.12775e23	2.69335
$549$	−1.04705e23	−0.541507
$550$	0	0
$551$	7.94481e21	0.0398378
$552$	−4.89186e23	−2.41541
$553$	5.50965e22	0.267892
$554$	3.60724e23	1.72720
$555$	0	0
$556$	−4.37767e23	−2.03286
$557$	−4.06323e22	−0.185824	−0.0929122	−	0.995674i	$-0.529618\pi$
$557$	−4.06323e22	−0.185824	−0.0929122	+	0.995674i	$0.529618\pi$
$558$	−1.50344e23	−0.677164
$559$	6.12027e22	0.271500
$560$	0	0
$561$	−1.33443e23	−0.574263
$562$	−5.63882e23	−2.39017
$563$	−3.27464e23	−1.36723	−0.683615	−	0.729843i	$-0.739593\pi$
$563$	−3.27464e23	−1.36723	−0.683615	+	0.729843i	$0.739593\pi$
$564$	−1.55474e22	−0.0639417
$565$	0	0
$566$	3.01469e23	1.20310
$567$	5.01806e22	0.197278
$568$	6.82112e23	2.64177
$569$	3.74009e23	1.42701	0.713506	−	0.700649i	$-0.247106\pi$
$569$	3.74009e23	1.42701	0.713506	+	0.700649i	$0.247106\pi$
$570$	0	0
$571$	2.13597e23	0.791021	0.395511	−	0.918461i	$-0.370568\pi$
$571$	2.13597e23	0.791021	0.395511	+	0.918461i	$0.370568\pi$
$572$	3.39545e23	1.23888
$573$	1.23976e23	0.445682
$574$	−1.19206e24	−4.22227
$575$	0	0
$576$	1.08258e23	0.372281
$577$	−8.63899e22	−0.292731	−0.146366	−	0.989231i	$-0.546758\pi$
$577$	−8.63899e22	−0.292731	−0.146366	+	0.989231i	$0.546758\pi$
$578$	−2.86248e23	−0.955775
$579$	1.69078e22	0.0556313
$580$	0	0
$581$	3.20479e23	1.02400
$582$	4.66848e23	1.47004
$583$	3.06421e23	0.950902
$584$	7.35547e23	2.24958
$585$	0	0
$586$	5.52841e23	1.64237
$587$	−1.46869e23	−0.430037	−0.215018	−	0.976610i	$-0.568981\pi$
$587$	−1.46869e23	−0.430037	−0.215018	+	0.976610i	$0.568981\pi$
$588$	1.04683e24	3.02114
$589$	−8.56286e22	−0.243578
$590$	0	0
$591$	1.69292e23	0.467890
$592$	−1.62857e24	−4.43682
$593$	6.72164e23	1.80514	0.902569	−	0.430545i	$-0.141679\pi$
$593$	6.72164e23	1.80514	0.902569	+	0.430545i	$0.141679\pi$
$594$	1.92514e23	0.509657
$595$	0	0
$596$	−1.14657e24	−2.94991
$597$	3.31262e22	0.0840214
$598$	−4.16701e23	−1.04199
$599$	−2.15383e23	−0.530986	−0.265493	−	0.964113i	$-0.585535\pi$
$599$	−2.15383e23	−0.530986	−0.265493	+	0.964113i	$0.585535\pi$
$600$	0	0
$601$	−3.86809e23	−0.926964	−0.463482	−	0.886106i	$-0.653400\pi$
$601$	−3.86809e23	−0.926964	−0.463482	+	0.886106i	$0.653400\pi$
$602$	1.06022e24	2.50510
$603$	−5.14976e22	−0.119975
$604$	5.83328e23	1.33998
$605$	0	0
$606$	2.80068e23	0.625527
$607$	5.57542e23	1.22793	0.613965	−	0.789333i	$-0.289574\pi$
$607$	5.57542e23	1.22793	0.613965	+	0.789333i	$0.289574\pi$
$608$	1.98529e23	0.431166
$609$	8.58545e22	0.183872
$610$	0	0
$611$	−7.79607e21	−0.0162377
$612$	−2.74472e23	−0.563782
$613$	−3.77874e23	−0.765478	−0.382739	−	0.923857i	$-0.625019\pi$
$613$	−3.77874e23	−0.765478	−0.382739	+	0.923857i	$0.625019\pi$
$614$	−1.38149e23	−0.276004
$615$	0	0
$616$	3.46250e24	6.72904
$617$	2.68278e23	0.514235	0.257118	−	0.966380i	$-0.417227\pi$
$617$	2.68278e23	0.514235	0.257118	+	0.966380i	$0.417227\pi$
$618$	1.65618e23	0.313116
$619$	3.02103e23	0.563358	0.281679	−	0.959509i	$-0.409109\pi$
$619$	3.02103e23	0.563358	0.281679	+	0.959509i	$0.409109\pi$
$620$	0	0
$621$	−1.67402e23	−0.303726
$622$	6.52637e23	1.16803
$623$	−6.58516e23	−1.16257
$624$	2.92953e23	0.510187
$625$	0	0
$626$	1.11311e24	1.88650
$627$	1.09647e23	0.183325
$628$	−6.49842e23	−1.07189
$629$	7.65112e23	1.24507
$630$	0	0
$631$	4.97599e23	0.788189	0.394094	−	0.919070i	$-0.371058\pi$
$631$	4.97599e23	0.788189	0.394094	+	0.919070i	$0.371058\pi$
$632$	2.55930e23	0.399969
$633$	3.20769e23	0.494607
$634$	−2.06560e24	−3.14259
$635$	0	0
$636$	6.30263e23	0.933547
$637$	5.24924e23	0.767205
$638$	3.29375e23	0.475023
$639$	2.33422e23	0.332189
$640$	0	0
$641$	2.45883e23	0.340750	0.170375	−	0.985379i	$-0.445502\pi$
$641$	2.45883e23	0.340750	0.170375	+	0.985379i	$0.445502\pi$
$642$	−8.69432e23	−1.18902
$643$	−1.29703e24	−1.75048	−0.875239	−	0.483691i	$-0.839296\pi$
$643$	−1.29703e24	−1.75048	−0.875239	+	0.483691i	$0.839296\pi$
$644$	−5.11469e24	−6.81223
$645$	0	0
$646$	−2.20629e23	−0.286211
$647$	−6.72178e23	−0.860593	−0.430297	−	0.902688i	$-0.641591\pi$
$647$	−6.72178e23	−0.860593	−0.430297	+	0.902688i	$0.641591\pi$
$648$	2.33095e23	0.294541
$649$	−3.50050e23	−0.436567
$650$	0	0
$651$	−9.25333e23	−1.12424
$652$	4.01555e24	4.81548
$653$	9.05901e23	1.07231	0.536153	−	0.844121i	$-0.319877\pi$
$653$	9.05901e23	1.07231	0.536153	+	0.844121i	$0.319877\pi$
$654$	9.43653e23	1.10256
$655$	0	0
$656$	−2.79573e24	−3.18283
$657$	2.51708e23	0.282873
$658$	−1.35052e23	−0.149824
$659$	9.59082e23	1.05034	0.525170	−	0.850997i	$-0.324002\pi$
$659$	9.59082e23	1.05034	0.525170	+	0.850997i	$0.324002\pi$
$660$	0	0
$661$	−9.10262e23	−0.971525	−0.485762	−	0.874091i	$-0.661458\pi$
$661$	−9.10262e23	−0.971525	−0.485762	+	0.874091i	$0.661458\pi$
$662$	−2.47311e24	−2.60585
$663$	−1.37631e23	−0.143170
$664$	1.48867e24	1.52886
$665$	0	0
$666$	−1.10381e24	−1.10500
$667$	−2.86410e23	−0.283086
$668$	−3.01540e24	−2.94270
$669$	−3.96067e22	−0.0381633
$670$	0	0
$671$	−2.47235e24	−2.32257
$672$	2.14538e24	1.99006
$673$	8.50468e23	0.778987	0.389493	−	0.921029i	$-0.372650\pi$
$673$	8.50468e23	0.778987	0.389493	+	0.921029i	$0.372650\pi$
$674$	−1.38228e23	−0.125022
$675$	0	0
$676$	−2.40625e24	−2.12224
$677$	−4.30263e23	−0.374741	−0.187370	−	0.982289i	$-0.559996\pi$
$677$	−4.30263e23	−0.374741	−0.187370	+	0.982289i	$0.559996\pi$
$678$	−1.44894e24	−1.24623
$679$	2.87335e24	2.44060
$680$	0	0
$681$	1.21538e24	1.00684
$682$	−3.54998e24	−2.90441
$683$	1.92309e23	0.155390	0.0776950	−	0.996977i	$-0.475244\pi$
$683$	1.92309e23	0.155390	0.0776950	+	0.996977i	$0.475244\pi$
$684$	2.25528e23	0.179979
$685$	0	0
$686$	4.86861e24	3.79009
$687$	7.08151e23	0.544495
$688$	2.48653e24	1.88839
$689$	3.16039e23	0.237070
$690$	0	0
$691$	9.14440e23	0.669255	0.334627	−	0.942350i	$-0.391390\pi$
$691$	9.14440e23	0.669255	0.334627	+	0.942350i	$0.391390\pi$
$692$	1.61222e24	1.16553
$693$	1.18488e24	0.846143
$694$	−9.77051e23	−0.689226
$695$	0	0
$696$	3.98805e23	0.274525
$697$	1.31345e24	0.893174
$698$	−4.12549e24	−2.77143
$699$	7.91822e23	0.525499
$700$	0	0
$701$	−1.18189e24	−0.765548	−0.382774	−	0.923842i	$-0.625031\pi$
$701$	−1.18189e24	−0.765548	−0.382774	+	0.923842i	$0.625031\pi$
$702$	1.98557e23	0.127063
$703$	−6.28675e23	−0.397472
$704$	2.55624e24	1.59674
$705$	0	0
$706$	−2.71949e24	−1.65824
$707$	1.72376e24	1.03851
$708$	−7.20002e23	−0.428599
$709$	3.06841e23	0.180476	0.0902381	−	0.995920i	$-0.471237\pi$
$709$	3.06841e23	0.180476	0.0902381	+	0.995920i	$0.471237\pi$
$710$	0	0
$711$	8.75807e22	0.0502941
$712$	−3.05889e24	−1.73574
$713$	3.08690e24	1.73086
$714$	−2.38420e24	−1.32101
$715$	0	0
$716$	4.42717e23	0.239533
$717$	7.31695e23	0.391217
$718$	1.54284e24	0.815197
$719$	1.71139e24	0.893621	0.446811	−	0.894629i	$-0.352560\pi$
$719$	1.71139e24	0.893621	0.446811	+	0.894629i	$0.352560\pi$
$720$	0	0
$721$	1.01934e24	0.519841
$722$	−3.49393e24	−1.76096
$723$	−1.24544e24	−0.620366
$724$	2.09175e24	1.02975
$725$	0	0
$726$	2.32182e24	1.11652
$727$	−1.28685e24	−0.611623	−0.305812	−	0.952092i	$-0.598928\pi$
$727$	−1.28685e24	−0.611623	−0.305812	+	0.952092i	$0.598928\pi$
$728$	3.57118e24	1.67763
$729$	7.97664e22	0.0370370
$730$	0	0
$731$	−1.16819e24	−0.529925
$732$	−5.08526e24	−2.28018
$733$	3.55442e24	1.57538	0.787690	−	0.616072i	$-0.211277\pi$
$733$	3.55442e24	1.57538	0.787690	+	0.616072i	$0.211277\pi$
$734$	−2.63255e24	−1.15335
$735$	0	0
$736$	−7.15696e24	−3.06385
$737$	−1.21598e24	−0.514581
$738$	−1.89488e24	−0.792689
$739$	−3.45741e24	−1.42979	−0.714895	−	0.699231i	$-0.753526\pi$
$739$	−3.45741e24	−1.42979	−0.714895	+	0.699231i	$0.753526\pi$
$740$	0	0
$741$	1.13089e23	0.0457050
$742$	5.47476e24	2.18742
$743$	−1.33858e24	−0.528737	−0.264369	−	0.964422i	$-0.585164\pi$
$743$	−1.33858e24	−0.528737	−0.264369	+	0.964422i	$0.585164\pi$
$744$	−4.29829e24	−1.67852
$745$	0	0
$746$	−1.15766e23	−0.0441877
$747$	5.09429e23	0.192247
$748$	−6.48095e24	−2.41810
$749$	−5.35117e24	−1.97403
$750$	0	0
$751$	1.02936e24	0.371217	0.185609	−	0.982624i	$-0.440574\pi$
$751$	1.02936e24	0.371217	0.185609	+	0.982624i	$0.440574\pi$
$752$	−3.16737e23	−0.112940
$753$	−9.68141e23	−0.341335
$754$	3.39713e23	0.118428
$755$	0	0
$756$	2.43713e24	0.830699
$757$	−2.63626e24	−0.888531	−0.444265	−	0.895895i	$-0.646535\pi$
$757$	−2.63626e24	−0.888531	−0.444265	+	0.895895i	$0.646535\pi$
$758$	−1.72546e24	−0.575064
$759$	−3.95276e24	−1.30270
$760$	0	0
$761$	−6.77132e23	−0.218225	−0.109112	−	0.994029i	$-0.534801\pi$
$761$	−6.77132e23	−0.218225	−0.109112	+	0.994029i	$0.534801\pi$
$762$	−1.86675e24	−0.594935
$763$	5.80799e24	1.83049
$764$	6.02119e24	1.87667
$765$	0	0
$766$	−1.11918e24	−0.341157
$767$	−3.61038e23	−0.108841
$768$	−1.70573e24	−0.508558
$769$	2.15350e24	0.634996	0.317498	−	0.948259i	$-0.397157\pi$
$769$	2.15350e24	0.634996	0.317498	+	0.948259i	$0.397157\pi$
$770$	0	0
$771$	−3.86249e24	−1.11405
$772$	8.21166e23	0.234252
$773$	8.52370e23	0.240493	0.120246	−	0.992744i	$-0.461632\pi$
$773$	8.52370e23	0.240493	0.120246	+	0.992744i	$0.461632\pi$
$774$	1.68531e24	0.470307
$775$	0	0
$776$	1.33471e25	3.64387
$777$	−6.79369e24	−1.83454
$778$	3.64158e24	0.972666
$779$	−1.07924e24	−0.285133
$780$	0	0
$781$	5.51167e24	1.42478
$782$	7.95365e24	2.03380
$783$	1.36473e23	0.0345201
$784$	2.13265e25	5.33621
$785$	0	0
$786$	−4.46361e24	−1.09294
$787$	−5.38452e24	−1.30425	−0.652126	−	0.758110i	$-0.726123\pi$
$787$	−5.38452e24	−1.30425	−0.652126	+	0.758110i	$0.726123\pi$
$788$	8.22207e24	1.97019
$789$	1.66462e24	0.394603
$790$	0	0
$791$	−8.91794e24	−2.06902
$792$	5.50395e24	1.26331
$793$	−2.54995e24	−0.579042
$794$	6.99721e24	1.57199
$795$	0	0
$796$	1.60885e24	0.353797
$797$	−1.12702e24	−0.245208	−0.122604	−	0.992456i	$-0.539124\pi$
$797$	−1.12702e24	−0.245208	−0.122604	+	0.992456i	$0.539124\pi$
$798$	1.95904e24	0.421715
$799$	1.48805e23	0.0316935
$800$	0	0
$801$	−1.04677e24	−0.218260
$802$	−7.19463e24	−1.48432
$803$	5.94344e24	1.21326
$804$	−2.50109e24	−0.505189
$805$	0	0
$806$	−3.66140e24	−0.724102
$807$	−2.32751e24	−0.455477
$808$	8.00709e24	1.55052
$809$	−1.34479e24	−0.257686	−0.128843	−	0.991665i	$-0.541126\pi$
$809$	−1.34479e24	−0.257686	−0.128843	+	0.991665i	$0.541126\pi$
$810$	0	0
$811$	4.40358e24	0.826283	0.413141	−	0.910667i	$-0.364432\pi$
$811$	4.40358e24	0.826283	0.413141	+	0.910667i	$0.364432\pi$
$812$	4.16972e24	0.774248
$813$	4.07194e24	0.748225
$814$	−2.60635e25	−4.73942
$815$	0	0
$816$	−5.59165e24	−0.995803
$817$	9.59872e23	0.169171
$818$	−2.07314e25	−3.61598
$819$	1.22208e24	0.210953
$820$	0	0
$821$	1.72971e24	0.292453	0.146227	−	0.989251i	$-0.453287\pi$
$821$	1.72971e24	0.292453	0.146227	+	0.989251i	$0.453287\pi$
$822$	7.08036e24	1.18480
$823$	1.17051e25	1.93855	0.969274	−	0.245986i	$-0.0791116\pi$
$823$	1.17051e25	1.93855	0.969274	+	0.245986i	$0.0791116\pi$
$824$	4.73499e24	0.776135
$825$	0	0
$826$	−6.25428e24	−1.00426
$827$	−6.19900e24	−0.985202	−0.492601	−	0.870255i	$-0.663954\pi$
$827$	−6.19900e24	−0.985202	−0.492601	+	0.870255i	$0.663954\pi$
$828$	−8.13025e24	−1.27893
$829$	5.61442e24	0.874161	0.437081	−	0.899422i	$-0.356012\pi$
$829$	5.61442e24	0.874161	0.437081	+	0.899422i	$0.356012\pi$
$830$	0	0
$831$	3.52918e24	0.538350
$832$	2.63648e24	0.398085
$833$	−1.00193e25	−1.49746
$834$	−6.04465e24	−0.894252
$835$	0	0
$836$	5.32525e24	0.771946
$837$	−1.47090e24	−0.211065
$838$	−1.00904e25	−1.43329
$839$	8.30559e24	1.16787	0.583934	−	0.811801i	$-0.301513\pi$
$839$	8.30559e24	1.16787	0.583934	+	0.811801i	$0.301513\pi$
$840$	0	0
$841$	−7.02365e24	−0.967826
$842$	−1.54350e25	−2.10550
$843$	−5.51679e24	−0.744993
$844$	1.55789e25	2.08269
$845$	0	0
$846$	−2.14677e23	−0.0281279
$847$	1.42903e25	1.85367
$848$	1.28400e25	1.64892
$849$	2.94945e24	0.374995
$850$	0	0
$851$	2.26637e25	2.82442
$852$	1.13367e25	1.39878
$853$	−9.95169e24	−1.21571	−0.607855	−	0.794048i	$-0.707970\pi$
$853$	−9.95169e24	−1.21571	−0.607855	+	0.794048i	$0.707970\pi$
$854$	−4.41730e25	−5.34275
$855$	0	0
$856$	−2.48569e25	−2.94727
$857$	−2.47493e24	−0.290553	−0.145276	−	0.989391i	$-0.546407\pi$
$857$	−2.47493e24	−0.290553	−0.145276	+	0.989391i	$0.546407\pi$
$858$	4.68841e24	0.544984
$859$	3.92176e24	0.451377	0.225689	−	0.974200i	$-0.427537\pi$
$859$	3.92176e24	0.451377	0.225689	+	0.974200i	$0.427537\pi$
$860$	0	0
$861$	−1.16626e25	−1.31604
$862$	8.60367e24	0.961329
$863$	1.05637e25	1.16876	0.584381	−	0.811480i	$-0.301337\pi$
$863$	1.05637e25	1.16876	0.584381	+	0.811480i	$0.301337\pi$
$864$	3.41027e24	0.373613
$865$	0	0
$866$	−4.94344e24	−0.531040
$867$	−2.80053e24	−0.297905
$868$	−4.49409e25	−4.73395
$869$	2.06799e24	0.215715
$870$	0	0
$871$	−1.25415e24	−0.128291
$872$	2.69789e25	2.73296
$873$	4.56745e24	0.458198
$874$	−6.53534e24	−0.649263
$875$	0	0
$876$	1.22248e25	1.19112
$877$	8.71589e23	0.0841037	0.0420518	−	0.999115i	$-0.486611\pi$
$877$	8.71589e23	0.0841037	0.0420518	+	0.999115i	$0.486611\pi$
$878$	2.34346e25	2.23951
$879$	5.40876e24	0.511909
$880$	0	0
$881$	3.51372e24	0.326191	0.163096	−	0.986610i	$-0.447852\pi$
$881$	3.51372e24	0.326191	0.163096	+	0.986610i	$0.447852\pi$
$882$	1.44546e25	1.32899
$883$	−1.29043e25	−1.17508	−0.587540	−	0.809195i	$-0.699904\pi$
$883$	−1.29043e25	−1.17508	−0.587540	+	0.809195i	$0.699904\pi$
$884$	−6.68438e24	−0.602860
$885$	0	0
$886$	2.13498e25	1.88889
$887$	3.42654e24	0.300265	0.150133	−	0.988666i	$-0.452030\pi$
$887$	3.42654e24	0.300265	0.150133	+	0.988666i	$0.452030\pi$
$888$	−3.15576e25	−2.73901
$889$	−1.14895e25	−0.987724
$890$	0	0
$891$	1.88348e24	0.158855
$892$	−1.92359e24	−0.160698
$893$	−1.22270e23	−0.0101177
$894$	−1.58318e25	−1.29766
$895$	0	0
$896$	2.81272e24	0.226208
$897$	−4.07683e24	−0.324779
$898$	−1.07630e25	−0.849349
$899$	−2.51658e24	−0.196722
$900$	0	0
$901$	−6.03229e24	−0.462724
$902$	−4.47428e25	−3.39991
$903$	1.03727e25	0.780813
$904$	−4.14250e25	−3.08909
$905$	0	0
$906$	8.05454e24	0.589456
$907$	1.11742e25	0.810126	0.405063	−	0.914289i	$-0.367250\pi$
$907$	1.11742e25	0.810126	0.405063	+	0.914289i	$0.367250\pi$
$908$	5.90277e25	4.23961
$909$	2.74007e24	0.194970
$910$	0	0
$911$	−2.26295e25	−1.58041	−0.790204	−	0.612843i	$-0.790026\pi$
$911$	−2.26295e25	−1.58041	−0.790204	+	0.612843i	$0.790026\pi$
$912$	4.59453e24	0.317896
$913$	1.20289e25	0.824561
$914$	4.43024e25	3.00874
$915$	0	0
$916$	3.43930e25	2.29276
$917$	−2.74726e25	−1.81451
$918$	−3.78989e24	−0.248007
$919$	−2.06020e25	−1.33576	−0.667879	−	0.744270i	$-0.732798\pi$
$919$	−2.06020e25	−1.33576	−0.667879	+	0.744270i	$0.732798\pi$
$920$	0	0
$921$	−1.35159e24	−0.0860276
$922$	1.94606e25	1.22728
$923$	5.68467e24	0.355215
$924$	5.75466e25	3.56294
$925$	0	0
$926$	5.38191e25	3.27147
$927$	1.62034e24	0.0975950
$928$	5.83467e24	0.348224
$929$	1.38098e25	0.816681	0.408341	−	0.912830i	$-0.366108\pi$
$929$	1.38098e25	0.816681	0.408341	+	0.912830i	$0.366108\pi$
$930$	0	0
$931$	8.23265e24	0.478043
$932$	3.84566e25	2.21277
$933$	6.38513e24	0.364063
$934$	2.52980e25	1.42935
$935$	0	0
$936$	5.67670e24	0.314957
$937$	1.59511e25	0.877011	0.438506	−	0.898728i	$-0.355508\pi$
$937$	1.59511e25	0.877011	0.438506	+	0.898728i	$0.355508\pi$
$938$	−2.17257e25	−1.18372
$939$	1.08902e25	0.588003
$940$	0	0
$941$	−2.23853e25	−1.18700	−0.593500	−	0.804834i	$-0.702254\pi$
$941$	−2.23853e25	−1.18700	−0.593500	+	0.804834i	$0.702254\pi$
$942$	−8.97296e24	−0.471523
$943$	3.89063e25	2.02615
$944$	−1.46682e25	−0.757031
$945$	0	0
$946$	3.97942e25	2.01718
$947$	2.89242e25	1.45307	0.726534	−	0.687130i	$-0.241130\pi$
$947$	2.89242e25	1.45307	0.726534	+	0.687130i	$0.241130\pi$
$948$	4.25356e24	0.211778
$949$	6.12999e24	0.302480
$950$	0	0
$951$	−2.02090e25	−0.979513
$952$	−6.81637e25	−3.27446
$953$	−2.32682e25	−1.10783	−0.553915	−	0.832573i	$-0.686867\pi$
$953$	−2.32682e25	−1.10783	−0.553915	+	0.832573i	$0.686867\pi$
$954$	8.70262e24	0.410666
$955$	0	0
$956$	3.55364e25	1.64733
$957$	3.22246e24	0.148060
$958$	4.42854e24	0.201676
$959$	4.35781e25	1.96703
$960$	0	0
$961$	4.57337e24	0.202809
$962$	−2.68816e25	−1.18159
$963$	−8.50616e24	−0.370604
$964$	−6.04877e25	−2.61224
$965$	0	0
$966$	−7.06232e25	−2.99669
$967$	3.76920e25	1.58535	0.792674	−	0.609646i	$-0.208689\pi$
$967$	3.76920e25	1.58535	0.792674	+	0.609646i	$0.208689\pi$
$968$	6.63803e25	2.76757
$969$	−2.15854e24	−0.0892089
$970$	0	0
$971$	−1.07235e25	−0.435484	−0.217742	−	0.976006i	$-0.569869\pi$
$971$	−1.07235e25	−0.435484	−0.217742	+	0.976006i	$0.569869\pi$
$972$	3.87404e24	0.155955
$973$	−3.72036e25	−1.48466
$974$	3.67000e25	1.45183
$975$	0	0
$976$	−1.03599e26	−4.02746
$977$	−1.87551e25	−0.722795	−0.361398	−	0.932412i	$-0.617700\pi$
$977$	−1.87551e25	−0.722795	−0.361398	+	0.932412i	$0.617700\pi$
$978$	5.54463e25	2.11833
$979$	−2.47168e25	−0.936135
$980$	0	0
$981$	9.23231e24	0.343656
$982$	5.89399e25	2.17501
$983$	1.72208e24	0.0630013	0.0315006	−	0.999504i	$-0.489971\pi$
$983$	1.72208e24	0.0630013	0.0315006	+	0.999504i	$0.489971\pi$
$984$	−5.41743e25	−1.96488
$985$	0	0
$986$	−6.48416e24	−0.231153
$987$	−1.32129e24	−0.0466985
$988$	5.49240e24	0.192455
$989$	−3.46033e25	−1.20212
$990$	0	0
$991$	5.69965e25	1.94636	0.973178	−	0.230053i	$-0.0738901\pi$
$991$	5.69965e25	1.94636	0.973178	+	0.230053i	$0.0738901\pi$
$992$	−6.28856e25	−2.12913
$993$	−2.41958e25	−0.812217
$994$	9.84759e25	3.27752
$995$	0	0
$996$	2.47416e25	0.809512
$997$	−1.93741e24	−0.0628512	−0.0314256	−	0.999506i	$-0.510005\pi$
$997$	−1.93741e24	−0.0628512	−0.0314256	+	0.999506i	$0.510005\pi$
$998$	3.81227e25	1.22623
$999$	−1.07992e25	−0.344416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.18.a.e.1.3		3
5.2	odd	4		75.18.b.d.49.6		6
5.3	odd	4		75.18.b.d.49.1		6
5.4	even	2		15.18.a.b.1.1	✓	3
15.14	odd	2		45.18.a.e.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.18.a.b.1.1	✓	3	5.4	even	2
45.18.a.e.1.3		3	15.14	odd	2
75.18.a.e.1.3		3	1.1	even	1	trivial
75.18.b.d.49.1		6	5.3	odd	4
75.18.b.d.49.6		6	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:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+670.613 q^{2} +6561.00 q^{3} +318650. q^{4} +4.39989e6 q^{6} +2.70804e7 q^{7} +1.25792e8 q^{8} +4.30467e7 q^{9} +1.01644e9 q^{11} +2.09066e9 q^{12} +1.04834e9 q^{13} +1.81605e10 q^{14} +4.25918e10 q^{16} -2.00099e10 q^{17} +2.88677e10 q^{18} +1.64417e10 q^{19} +1.77675e11 q^{21} +6.81636e11 q^{22} -5.92721e11 q^{23} +8.25322e11 q^{24} +7.03032e11 q^{26} +2.82430e11 q^{27} +8.62917e12 q^{28} +4.83212e11 q^{29} -5.20802e12 q^{31} +1.20748e13 q^{32} +6.66885e12 q^{33} -1.34189e13 q^{34} +1.37168e13 q^{36} -3.82367e13 q^{37} +1.10260e13 q^{38} +6.87817e12 q^{39} -6.56402e13 q^{41} +1.19151e14 q^{42} +5.83804e13 q^{43} +3.23888e14 q^{44} -3.97486e14 q^{46} -7.43658e12 q^{47} +2.79445e14 q^{48} +5.00719e14 q^{49} -1.31285e14 q^{51} +3.34054e14 q^{52} +3.01466e14 q^{53} +1.89401e14 q^{54} +3.40650e15 q^{56} +1.07874e14 q^{57} +3.24048e14 q^{58} -3.44389e14 q^{59} -2.43237e15 q^{61} -3.49257e15 q^{62} +1.16572e15 q^{63} +2.51491e15 q^{64} +4.47222e15 q^{66} -1.19632e15 q^{67} -6.37614e15 q^{68} -3.88884e15 q^{69} +5.42254e15 q^{71} +5.41494e15 q^{72} +5.84732e15 q^{73} -2.56420e16 q^{74} +5.23914e15 q^{76} +2.75256e16 q^{77} +4.61259e15 q^{78} +2.03455e15 q^{79} +1.85302e15 q^{81} -4.40192e16 q^{82} +1.18343e16 q^{83} +5.66160e16 q^{84} +3.91507e16 q^{86} +3.17035e15 q^{87} +1.27860e17 q^{88} -2.43170e16 q^{89} +2.83895e16 q^{91} -1.88870e17 q^{92} -3.41698e16 q^{93} -4.98707e15 q^{94} +7.92225e16 q^{96} +1.06105e17 q^{97} +3.35788e17 q^{98} +4.37543e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 442 q^{2} + 19683 q^{3} + 298148 q^{4} + 2899962 q^{6} - 4962644 q^{7} + 108831912 q^{8} + 129140163 q^{9} + 1049849720 q^{11} + 1956149028 q^{12} + 3091742090 q^{13} + 27586028328 q^{14} + 22392797456 q^{16}+ \cdots + 45\!\cdots\!20 q^{99}+O(q^{100}) \) 3 * q + 442 * q^2 + 19683 * q^3 + 298148 * q^4 + 2899962 * q^6 - 4962644 * q^7 + 108831912 * q^8 + 129140163 * q^9 + 1049849720 * q^11 + 1956149028 * q^12 + 3091742090 * q^13 + 27586028328 * q^14 + 22392797456 * q^16 + 15119940094 * q^17 + 19026650682 * q^18 - 111896117996 * q^19 - 32559907284 * q^21 + 693818341648 * q^22 - 784535528472 * q^23 + 714046174632 * q^24 - 1092511841044 * q^26 + 847288609443 * q^27 + 7640185610512 * q^28 + 1395352153678 * q^29 - 12458197616536 * q^31 + 24074446195616 * q^32 + 6888064012920 * q^33 - 41046727271164 * q^34 + 12834293772708 * q^36 - 32093748310134 * q^37 + 68757678664024 * q^38 + 20284919852490 * q^39 - 102276917392466 * q^41 + 180991931860008 * q^42 + 141347515908644 * q^43 + 319886980653472 * q^44 - 299722933931856 * q^46 + 54182226541768 * q^47 + 146919144108816 * q^48 + 702936204707227 * q^49 + 99201926956734 * q^51 + 670193699134584 * q^52 - 345545322785582 * q^53 + 124833855124602 * q^54 + 3014407093472160 * q^56 - 734150430171756 * q^57 - 2198630240496412 * q^58 + 2075534176494536 * q^59 - 1591128188427998 * q^61 - 1555165628278608 * q^62 - 213625551690324 * q^63 + 4014461944704576 * q^64 + 4552142139552528 * q^66 + 2417737431711844 * q^67 - 1338181134241816 * q^68 - 5147337602304792 * q^69 + 5019485593673056 * q^71 + 4684856951760552 * q^72 + 10804161755859578 * q^73 - 12140475721796372 * q^74 - 3289783191661584 * q^76 + 27418121851422432 * q^77 - 7167970189089684 * q^78 - 40642131657181960 * q^79 + 5559060566555523 * q^81 - 15262994262855388 * q^82 + 21823256926416996 * q^83 + 50127257790569232 * q^84 + 8867591636371640 * q^86 + 9154905480281358 * q^87 + 125733265675831872 * q^88 - 2700230040372426 * q^89 - 46116457499478296 * q^91 - 204241599793288224 * q^92 - 81738234562092696 * q^93 - 72231791695517200 * q^94 + 157952441489436576 * q^96 + 79760576372788314 * q^97 + 128015343617626250 * q^98 + 45192587988768120 * q^99

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