LMFDB - Embedded newform 75.22.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$209.608008215$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000 $ x^4 - x^3 - 3512166x^2 + 363520480x + 2321089280000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1069.36$ 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)

$f(q)$	$=$	$q+1293.36 q^{2} +59049.0 q^{3} -424379. q^{4} +7.63714e7 q^{6} -1.27960e9 q^{7} -3.26124e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q+1293.36 q^{2} +59049.0 q^{3} -424379. q^{4} +7.63714e7 q^{6} -1.27960e9 q^{7} -3.26124e9 q^{8} +3.48678e9 q^{9} -1.21178e11 q^{11} -2.50592e10 q^{12} -7.37152e11 q^{13} -1.65498e12 q^{14} -3.32796e12 q^{16} -6.05462e12 q^{17} +4.50966e12 q^{18} -3.23277e13 q^{19} -7.55589e13 q^{21} -1.56726e14 q^{22} +3.98109e13 q^{23} -1.92573e14 q^{24} -9.53401e14 q^{26} +2.05891e14 q^{27} +5.43035e14 q^{28} +1.40928e15 q^{29} -2.62721e15 q^{31} +2.53508e15 q^{32} -7.15543e15 q^{33} -7.83079e15 q^{34} -1.47972e15 q^{36} +3.44728e15 q^{37} -4.18113e16 q^{38} -4.35281e16 q^{39} +7.99269e16 q^{41} -9.77246e16 q^{42} -2.07610e17 q^{43} +5.14254e16 q^{44} +5.14897e16 q^{46} +1.95408e17 q^{47} -1.96513e17 q^{48} +1.07882e18 q^{49} -3.57519e17 q^{51} +3.12832e17 q^{52} +2.33305e17 q^{53} +2.66291e17 q^{54} +4.17307e18 q^{56} -1.90892e18 q^{57} +1.82271e18 q^{58} -3.93099e18 q^{59} +8.99462e18 q^{61} -3.39792e18 q^{62} -4.46168e18 q^{63} +1.02580e19 q^{64} -9.25453e18 q^{66} -9.21439e18 q^{67} +2.56946e18 q^{68} +2.35079e18 q^{69} -4.75444e19 q^{71} -1.13712e19 q^{72} -1.81566e19 q^{73} +4.45857e18 q^{74} +1.37192e19 q^{76} +1.55059e20 q^{77} -5.62974e19 q^{78} -1.11814e20 q^{79} +1.21577e19 q^{81} +1.03374e20 q^{82} +3.39283e19 q^{83} +3.20656e19 q^{84} -2.68514e20 q^{86} +8.32168e19 q^{87} +3.95190e20 q^{88} +3.38998e20 q^{89} +9.43257e20 q^{91} -1.68949e19 q^{92} -1.55134e20 q^{93} +2.52733e20 q^{94} +1.49694e20 q^{96} -1.12006e20 q^{97} +1.39530e21 q^{98} -4.22521e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9}+O(q^{10}) $ 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9	$ 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9} + 31491830256 q^{11} - 68681250027 q^{12} + 27017977768 q^{13} - 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17} + 3127645607697 q^{18} - 24270353300752 q^{19} + 13851567033696 q^{21} + 56303932793676 q^{22} - 10350924920928 q^{23} - 4538248036821 q^{24} + 474751622871378 q^{26} + 823564528378596 q^{27} + 18\!\cdots\!68 q^{28}+ \cdots + 10\!\cdots\!56 q^{99}+O(q^{100}) $ 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9 + 31491830256 * q^11 - 68681250027 * q^12 + 27017977768 * q^13 - 2233308126672 * q^14 - 11324681311775 * q^16 + 2946095028888 * q^17 + 3127645607697 * q^18 - 24270353300752 * q^19 + 13851567033696 * q^21 + 56303932793676 * q^22 - 10350924920928 * q^23 - 4538248036821 * q^24 + 474751622871378 * q^26 + 823564528378596 * q^27 + 1868468795909968 * q^28 + 4728924677079096 * q^29 - 1094923910405536 * q^31 - 7933327650814221 * q^32 + 1859561084786544 * q^33 + 3862795623387294 * q^34 - 4055559132844323 * q^36 - 4813435696247096 * q^37 - 21307487199674508 * q^38 + 1595384569222632 * q^39 + 289731591445930344 * q^41 - 131874611571854928 * q^42 - 451091912658458000 * q^43 + 245434444420437276 * q^44 + 645145648102459824 * q^46 - 813883435638492480 * q^47 - 668711106779001975 * q^48 + 1545180753004755300 * q^49 + 173963965360807512 * q^51 - 884645329958480822 * q^52 - 697278335404085208 * q^53 + 184684345488900153 * q^54 + 5673275878853478000 * q^56 - 1433140092056104848 * q^57 - 12277858736996428626 * q^58 + 6622888614569598192 * q^59 + 7390887218011683320 * q^61 - 11649094150323079392 * q^62 + 817921181772715104 * q^63 + 21525702776885072473 * q^64 + 3324690927533774124 * q^66 + 24188188449376788688 * q^67 - 18995582935459304346 * q^68 - 611211765655877472 * q^69 - 37390337803999713312 * q^71 - 267979008326243229 * q^72 + 37253672904265201432 * q^73 - 77875663243460100102 * q^74 - 44795472028972724668 * q^76 + 315245139112859776128 * q^77 + 28033608578931999522 * q^78 - 109751291042259570400 * q^79 + 48630661836227715204 * q^81 + 210801413757052759338 * q^82 + 149023592088638482896 * q^83 + 110331213929687700432 * q^84 - 346200583098972939204 * q^86 + 279238273256843539704 * q^87 + 116461091469581805156 * q^88 + 558502237151273959848 * q^89 + 30360075980445717184 * q^91 + 764380916144075558544 * q^92 - 64654161985536495264 * q^93 + 1096939878232058257512 * q^94 - 468455064452928935829 * q^96 + 236244923748673777528 * q^97 - 197058383612642903463 * q^98 + 109805222495560636656 * 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 $

$2$	1293.36	0.893107	0.446553	−	0.894757i	$-0.352651\pi$
$2$	1293.36	0.893107	0.446553	+	0.894757i	$0.352651\pi$
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	−424379.	−0.202360
$5$	0	0
$5$	0	0
$6$	7.63714e7	0.515636
$7$	−1.27960e9	−1.71216	−0.856079	−	0.516846i	$-0.827106\pi$
$7$	−1.27960e9	−1.71216	−0.856079	+	0.516846i	$0.827106\pi$
$8$	−3.26124e9	−1.07384
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−1.21178e11	−1.40864	−0.704320	−	0.709883i	$-0.748748\pi$
$11$	−1.21178e11	−1.40864	−0.704320	+	0.709883i	$0.748748\pi$
$12$	−2.50592e10	−0.116833
$13$	−7.37152e11	−1.48304	−0.741518	−	0.670933i	$-0.765894\pi$
$13$	−7.37152e11	−1.48304	−0.741518	+	0.670933i	$0.765894\pi$
$14$	−1.65498e12	−1.52914
$15$	0	0
$16$	−3.32796e12	−0.756691
$17$	−6.05462e12	−0.728406	−0.364203	−	0.931320i	$-0.618659\pi$
$17$	−6.05462e12	−0.728406	−0.364203	+	0.931320i	$0.618659\pi$
$18$	4.50966e12	0.297702
$19$	−3.23277e13	−1.20966	−0.604828	−	0.796356i	$-0.706758\pi$
$19$	−3.23277e13	−1.20966	−0.604828	+	0.796356i	$0.706758\pi$
$20$	0	0
$21$	−7.55589e13	−0.988515
$22$	−1.56726e14	−1.25807
$23$	3.98109e13	0.200382	0.100191	−	0.994968i	$-0.468055\pi$
$23$	3.98109e13	0.200382	0.100191	+	0.994968i	$0.468055\pi$
$24$	−1.92573e14	−0.619980
$25$	0	0
$26$	−9.53401e14	−1.32451
$27$	2.05891e14	0.192450
$28$	5.43035e14	0.346472
$29$	1.40928e15	0.622041	0.311021	−	0.950403i	$-0.399329\pi$
$29$	1.40928e15	0.622041	0.311021	+	0.950403i	$0.399329\pi$
$30$	0	0
$31$	−2.62721e15	−0.575701	−0.287850	−	0.957675i	$-0.592941\pi$
$31$	−2.62721e15	−0.575701	−0.287850	+	0.957675i	$0.592941\pi$
$32$	2.53508e15	0.398030
$33$	−7.15543e15	−0.813279
$34$	−7.83079e15	−0.650544
$35$	0	0
$36$	−1.47972e15	−0.0674533
$37$	3.44728e15	0.117858	0.0589290	−	0.998262i	$-0.481231\pi$
$37$	3.44728e15	0.117858	0.0589290	+	0.998262i	$0.481231\pi$
$38$	−4.18113e16	−1.08035
$39$	−4.35281e16	−0.856232
$40$	0	0
$41$	7.99269e16	0.929956	0.464978	−	0.885322i	$-0.346062\pi$
$41$	7.99269e16	0.929956	0.464978	+	0.885322i	$0.346062\pi$
$42$	−9.77246e16	−0.882849
$43$	−2.07610e17	−1.46498	−0.732488	−	0.680780i	$-0.761641\pi$
$43$	−2.07610e17	−1.46498	−0.732488	+	0.680780i	$0.761641\pi$
$44$	5.14254e16	0.285052
$45$	0	0
$46$	5.14897e16	0.178963
$47$	1.95408e17	0.541895	0.270948	−	0.962594i	$-0.412663\pi$
$47$	1.95408e17	0.541895	0.270948	+	0.962594i	$0.412663\pi$
$48$	−1.96513e17	−0.436875
$49$	1.07882e18	1.93148
$50$	0	0
$51$	−3.57519e17	−0.420545
$52$	3.12832e17	0.300107
$53$	2.33305e17	0.183243	0.0916214	−	0.995794i	$-0.470795\pi$
$53$	2.33305e17	0.183243	0.0916214	+	0.995794i	$0.470795\pi$
$54$	2.66291e17	0.171879
$55$	0	0
$56$	4.17307e18	1.83858
$57$	−1.90892e18	−0.698395
$58$	1.82271e18	0.555550
$59$	−3.93099e18	−1.00128	−0.500640	−	0.865656i	$-0.666902\pi$
$59$	−3.93099e18	−1.00128	−0.500640	+	0.865656i	$0.666902\pi$
$60$	0	0
$61$	8.99462e18	1.61443	0.807216	−	0.590256i	$-0.200973\pi$
$61$	8.99462e18	1.61443	0.807216	+	0.590256i	$0.200973\pi$
$62$	−3.39792e18	−0.514162
$63$	−4.46168e18	−0.570719
$64$	1.02580e19	1.11217
$65$	0	0
$66$	−9.25453e18	−0.726345
$67$	−9.21439e18	−0.617563	−0.308781	−	0.951133i	$-0.599921\pi$
$67$	−9.21439e18	−0.617563	−0.308781	+	0.951133i	$0.599921\pi$
$68$	2.56946e18	0.147400
$69$	2.35079e18	0.115691
$70$	0	0
$71$	−4.75444e19	−1.73335	−0.866676	−	0.498871i	$-0.833748\pi$
$71$	−4.75444e19	−1.73335	−0.866676	+	0.498871i	$0.833748\pi$
$72$	−1.13712e19	−0.357945
$73$	−1.81566e19	−0.494474	−0.247237	−	0.968955i	$-0.579523\pi$
$73$	−1.81566e19	−0.494474	−0.247237	+	0.968955i	$0.579523\pi$
$74$	4.45857e18	0.105260
$75$	0	0
$76$	1.37192e19	0.244786
$77$	1.55059e20	2.41181
$78$	−5.62974e19	−0.764707
$79$	−1.11814e20	−1.32865	−0.664327	−	0.747442i	$-0.731282\pi$
$79$	−1.11814e20	−1.32865	−0.664327	+	0.747442i	$0.731282\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	1.03374e20	0.830550
$83$	3.39283e19	0.240017	0.120009	−	0.992773i	$-0.461708\pi$
$83$	3.39283e19	0.240017	0.120009	+	0.992773i	$0.461708\pi$
$84$	3.20656e19	0.200036
$85$	0	0
$86$	−2.68514e20	−1.30838
$87$	8.32168e19	0.359136
$88$	3.95190e20	1.51265
$89$	3.38998e20	1.15240	0.576199	−	0.817310i	$-0.304535\pi$
$89$	3.38998e20	1.15240	0.576199	+	0.817310i	$0.304535\pi$
$90$	0	0
$91$	9.43257e20	2.53919
$92$	−1.68949e19	−0.0405494
$93$	−1.55134e20	−0.332381
$94$	2.52733e20	0.483970
$95$	0	0
$96$	1.49694e20	0.229803
$97$	−1.12006e20	−0.154219	−0.0771097	−	0.997023i	$-0.524569\pi$
$97$	−1.12006e20	−0.154219	−0.0771097	+	0.997023i	$0.524569\pi$
$98$	1.39530e21	1.72502
$99$	−4.22521e20	−0.469547
$100$	0	0
$101$	2.51635e20	0.226672	0.113336	−	0.993557i	$-0.463846\pi$
$101$	2.51635e20	0.226672	0.113336	+	0.993557i	$0.463846\pi$
$102$	−4.62400e20	−0.375592
$103$	−2.29653e21	−1.68377	−0.841883	−	0.539660i	$-0.818553\pi$
$103$	−2.29653e21	−1.68377	−0.841883	+	0.539660i	$0.818553\pi$
$104$	2.40403e21	1.59254
$105$	0	0
$106$	3.01746e20	0.163655
$107$	−3.75913e21	−1.84739	−0.923694	−	0.383130i	$-0.874846\pi$
$107$	−3.75913e21	−1.84739	−0.923694	+	0.383130i	$0.874846\pi$
$108$	−8.73760e19	−0.0389442
$109$	−1.44930e21	−0.586380	−0.293190	−	0.956054i	$-0.594717\pi$
$109$	−1.44930e21	−0.586380	−0.293190	+	0.956054i	$0.594717\pi$
$110$	0	0
$111$	2.03559e20	0.0680453
$112$	4.25845e21	1.29557
$113$	5.37147e21	1.48857	0.744285	−	0.667862i	$-0.232790\pi$
$113$	5.37147e21	1.48857	0.744285	+	0.667862i	$0.232790\pi$
$114$	−2.46891e21	−0.623742
$115$	0	0
$116$	−5.98071e20	−0.125876
$117$	−2.57029e21	−0.494346
$118$	−5.08417e21	−0.894250
$119$	7.74747e21	1.24715
$120$	0	0
$121$	7.28381e21	0.984266
$122$	1.16333e22	1.44186
$123$	4.71960e21	0.536910
$124$	1.11493e21	0.116499
$125$	0	0
$126$	−5.77054e21	−0.509713
$127$	−1.99248e22	−1.61978	−0.809888	−	0.586585i	$-0.800472\pi$
$127$	−1.99248e22	−1.61978	−0.809888	+	0.586585i	$0.800472\pi$
$128$	7.95081e21	0.595260
$129$	−1.22592e22	−0.845804
$130$	0	0
$131$	1.95301e22	1.14645	0.573225	−	0.819398i	$-0.305692\pi$
$131$	1.95301e22	1.14645	0.573225	+	0.819398i	$0.305692\pi$
$132$	3.03662e21	0.164575
$133$	4.13664e22	2.07112
$134$	−1.19175e22	−0.551549
$135$	0	0
$136$	1.97456e22	0.782188
$137$	−2.54688e22	−0.934206	−0.467103	−	0.884203i	$-0.654702\pi$
$137$	−2.54688e22	−0.934206	−0.467103	+	0.884203i	$0.654702\pi$
$138$	3.04041e21	0.103324
$139$	−3.28445e22	−1.03468	−0.517341	−	0.855780i	$-0.673078\pi$
$139$	−3.28445e22	−1.03468	−0.517341	+	0.855780i	$0.673078\pi$
$140$	0	0
$141$	1.15387e22	0.312863
$142$	−6.14919e22	−1.54807
$143$	8.93265e22	2.08906
$144$	−1.16039e22	−0.252230
$145$	0	0
$146$	−2.34829e22	−0.441618
$147$	6.37033e22	1.11514
$148$	−1.46296e21	−0.0238497
$149$	−4.10815e22	−0.624009	−0.312004	−	0.950081i	$-0.601000\pi$
$149$	−4.10815e22	−0.624009	−0.312004	+	0.950081i	$0.601000\pi$
$150$	0	0
$151$	8.76777e22	1.15779	0.578897	−	0.815401i	$-0.303483\pi$
$151$	8.76777e22	1.15779	0.578897	+	0.815401i	$0.303483\pi$
$152$	1.05428e23	1.29897
$153$	−2.11112e22	−0.242802
$154$	2.00546e23	2.15401
$155$	0	0
$156$	1.84724e22	0.173267
$157$	−1.06605e23	−0.935040	−0.467520	−	0.883982i	$-0.654852\pi$
$157$	−1.06605e23	−0.935040	−0.467520	+	0.883982i	$0.654852\pi$
$158$	−1.44615e23	−1.18663
$159$	1.37764e22	0.105795
$160$	0	0
$161$	−5.09419e22	−0.343086
$162$	1.57242e22	0.0992341
$163$	−2.93390e23	−1.73571	−0.867853	−	0.496822i	$-0.834500\pi$
$163$	−2.93390e23	−1.73571	−0.867853	+	0.496822i	$0.834500\pi$
$164$	−3.39193e22	−0.188186
$165$	0	0
$166$	4.38814e22	0.214361
$167$	−1.40812e23	−0.645830	−0.322915	−	0.946428i	$-0.604663\pi$
$167$	−1.40812e23	−0.645830	−0.322915	+	0.946428i	$0.604663\pi$
$168$	2.46416e23	1.06150
$169$	2.96329e23	1.19940
$170$	0	0
$171$	−1.12720e23	−0.403219
$172$	8.81056e22	0.296452
$173$	−3.14336e21	−0.00995198	−0.00497599	−	0.999988i	$-0.501584\pi$
$173$	−3.14336e21	−0.00995198	−0.00497599	+	0.999988i	$0.501584\pi$
$174$	1.07629e23	0.320747
$175$	0	0
$176$	4.03275e23	1.06590
$177$	−2.32121e23	−0.578089
$178$	4.38446e23	1.02921
$179$	−7.66967e23	−1.69754	−0.848770	−	0.528763i	$-0.822656\pi$
$179$	−7.66967e23	−1.69754	−0.848770	+	0.528763i	$0.822656\pi$
$180$	0	0
$181$	1.51630e23	0.298649	0.149325	−	0.988788i	$-0.452290\pi$
$181$	1.51630e23	0.298649	0.149325	+	0.988788i	$0.452290\pi$
$182$	1.21997e24	2.26777
$183$	5.31123e23	0.932093
$184$	−1.29833e23	−0.215178
$185$	0	0
$186$	−2.00644e23	−0.296852
$187$	7.33686e23	1.02606
$188$	−8.29272e22	−0.109658
$189$	−2.63458e23	−0.329505
$190$	0	0
$191$	6.06790e23	0.679498	0.339749	−	0.940516i	$-0.389658\pi$
$191$	6.06790e23	0.679498	0.339749	+	0.940516i	$0.389658\pi$
$192$	6.05724e23	0.642114
$193$	−1.87596e24	−1.88309	−0.941544	−	0.336889i	$-0.890625\pi$
$193$	−1.87596e24	−1.88309	−0.941544	+	0.336889i	$0.890625\pi$
$194$	−1.44864e23	−0.137734
$195$	0	0
$196$	−4.57830e23	−0.390855
$197$	−2.26924e23	−0.183647	−0.0918237	−	0.995775i	$-0.529270\pi$
$197$	−2.26924e23	−0.183647	−0.0918237	+	0.995775i	$0.529270\pi$
$198$	−5.46470e23	−0.419355
$199$	−6.13967e23	−0.446876	−0.223438	−	0.974718i	$-0.571728\pi$
$199$	−6.13967e23	−0.446876	−0.223438	+	0.974718i	$0.571728\pi$
$200$	0	0
$201$	−5.44100e23	−0.356550
$202$	3.25454e23	0.202442
$203$	−1.80331e24	−1.06503
$204$	1.51724e23	0.0851015
$205$	0	0
$206$	−2.97024e24	−1.50378
$207$	1.38812e23	0.0667941
$208$	2.45321e24	1.12220
$209$	3.91740e24	1.70397
$210$	0	0
$211$	1.99189e24	0.783970	0.391985	−	0.919972i	$-0.371788\pi$
$211$	1.99189e24	0.783970	0.391985	+	0.919972i	$0.371788\pi$
$212$	−9.90097e22	−0.0370810
$213$	−2.80745e24	−1.00075
$214$	−4.86190e24	−1.64992
$215$	0	0
$216$	−6.71461e23	−0.206660
$217$	3.36177e24	0.985690
$218$	−1.87446e24	−0.523700
$219$	−1.07213e24	−0.285485
$220$	0	0
$221$	4.46318e24	1.08025
$222$	2.63274e23	0.0607717
$223$	−3.29187e24	−0.724839	−0.362420	−	0.932015i	$-0.618049\pi$
$223$	−3.29187e24	−0.724839	−0.362420	+	0.932015i	$0.618049\pi$
$224$	−3.24388e24	−0.681491
$225$	0	0
$226$	6.94722e24	1.32945
$227$	1.53623e24	0.280663	0.140332	−	0.990105i	$-0.455183\pi$
$227$	1.53623e24	0.280663	0.140332	+	0.990105i	$0.455183\pi$
$228$	8.10106e23	0.141327
$229$	−3.70932e24	−0.618048	−0.309024	−	0.951054i	$-0.600002\pi$
$229$	−3.70932e24	−0.618048	−0.309024	+	0.951054i	$0.600002\pi$
$230$	0	0
$231$	9.15606e24	1.39246
$232$	−4.59601e24	−0.667971
$233$	1.12524e25	1.56317	0.781587	−	0.623796i	$-0.214411\pi$
$233$	1.12524e25	1.56317	0.781587	+	0.623796i	$0.214411\pi$
$234$	−3.32430e24	−0.441504
$235$	0	0
$236$	1.66823e24	0.202619
$237$	−6.60251e24	−0.767099
$238$	1.00202e25	1.11383
$239$	1.31053e25	1.39402	0.697008	−	0.717063i	$-0.254514\pi$
$239$	1.31053e25	1.39402	0.697008	+	0.717063i	$0.254514\pi$
$240$	0	0
$241$	1.66470e24	0.162240	0.0811200	−	0.996704i	$-0.474150\pi$
$241$	1.66470e24	0.162240	0.0811200	+	0.996704i	$0.474150\pi$
$242$	9.42057e24	0.879055
$243$	7.17898e23	0.0641500
$244$	−3.81713e24	−0.326696
$245$	0	0
$246$	6.10413e24	0.479518
$247$	2.38304e25	1.79396
$248$	8.56796e24	0.618208
$249$	2.00343e24	0.138574
$250$	0	0
$251$	2.32403e25	1.47798	0.738990	−	0.673717i	$-0.235303\pi$
$251$	2.32403e25	1.47798	0.738990	+	0.673717i	$0.235303\pi$
$252$	1.89344e24	0.115491
$253$	−4.82420e24	−0.282267
$254$	−2.57699e25	−1.44663
$255$	0	0
$256$	−1.12293e25	−0.580543
$257$	2.65650e25	1.31829	0.659147	−	0.752014i	$-0.270918\pi$
$257$	2.65650e25	1.31829	0.659147	+	0.752014i	$0.270918\pi$
$258$	−1.58555e25	−0.755394
$259$	−4.41113e24	−0.201791
$260$	0	0
$261$	4.91387e24	0.207347
$262$	2.52593e25	1.02390
$263$	9.75846e24	0.380055	0.190027	−	0.981779i	$-0.439142\pi$
$263$	9.75846e24	0.380055	0.190027	+	0.981779i	$0.439142\pi$
$264$	2.33356e25	0.873328
$265$	0	0
$266$	5.35015e25	1.84973
$267$	2.00175e25	0.665337
$268$	3.91040e24	0.124970
$269$	−1.15143e25	−0.353866	−0.176933	−	0.984223i	$-0.556618\pi$
$269$	−1.15143e25	−0.353866	−0.176933	+	0.984223i	$0.556618\pi$
$270$	0	0
$271$	1.57414e25	0.447575	0.223788	−	0.974638i	$-0.428158\pi$
$271$	1.57414e25	0.447575	0.223788	+	0.974638i	$0.428158\pi$
$272$	2.01495e25	0.551178
$273$	5.56984e25	1.46600
$274$	−3.29403e25	−0.834346
$275$	0	0
$276$	−9.97628e23	−0.0234112
$277$	−1.64872e25	−0.372486	−0.186243	−	0.982504i	$-0.559631\pi$
$277$	−1.64872e25	−0.372486	−0.186243	+	0.982504i	$0.559631\pi$
$278$	−4.24796e25	−0.924081
$279$	−9.16051e24	−0.191900
$280$	0	0
$281$	6.63562e25	1.28963	0.644815	−	0.764339i	$-0.276934\pi$
$281$	6.63562e25	1.28963	0.644815	+	0.764339i	$0.276934\pi$
$282$	1.49236e25	0.279420
$283$	6.72915e25	1.21395	0.606977	−	0.794719i	$-0.292382\pi$
$283$	6.72915e25	1.21395	0.606977	+	0.794719i	$0.292382\pi$
$284$	2.01769e25	0.350761
$285$	0	0
$286$	1.15531e26	1.86576
$287$	−1.02274e26	−1.59223
$288$	8.83926e24	0.132677
$289$	−3.24335e25	−0.469425
$290$	0	0
$291$	−6.61386e24	−0.0890386
$292$	7.70527e24	0.100062
$293$	1.51857e25	0.190250	0.0951248	−	0.995465i	$-0.469675\pi$
$293$	1.51857e25	0.190250	0.0951248	+	0.995465i	$0.469675\pi$
$294$	8.23912e25	0.995941
$295$	0	0
$296$	−1.12424e25	−0.126560
$297$	−2.49494e25	−0.271093
$298$	−5.31331e25	−0.557307
$299$	−2.93467e25	−0.297174
$300$	0	0
$301$	2.65658e26	2.50827
$302$	1.13399e26	1.03403
$303$	1.48588e25	0.130869
$304$	1.07585e26	0.915335
$305$	0	0
$306$	−2.73043e25	−0.216848
$307$	2.16909e26	1.66465	0.832327	−	0.554285i	$-0.187008\pi$
$307$	2.16909e26	1.66465	0.832327	+	0.554285i	$0.187008\pi$
$308$	−6.58037e25	−0.488054
$309$	−1.35608e26	−0.972123
$310$	0	0
$311$	−9.86475e25	−0.660848	−0.330424	−	0.943833i	$-0.607192\pi$
$311$	−9.86475e25	−0.660848	−0.330424	+	0.943833i	$0.607192\pi$
$312$	1.41956e26	0.919452
$313$	−1.28416e26	−0.804273	−0.402137	−	0.915580i	$-0.631732\pi$
$313$	−1.28416e26	−0.804273	−0.402137	+	0.915580i	$0.631732\pi$
$314$	−1.37878e26	−0.835091
$315$	0	0
$316$	4.74516e25	0.268866
$317$	1.45513e26	0.797588	0.398794	−	0.917041i	$-0.369429\pi$
$317$	1.45513e26	0.797588	0.398794	+	0.917041i	$0.369429\pi$
$318$	1.78178e25	0.0944865
$319$	−1.70774e26	−0.876232
$320$	0	0
$321$	−2.21973e26	−1.06659
$322$	−6.58860e25	−0.306413
$323$	1.95732e26	0.881120
$324$	−5.15946e24	−0.0224844
$325$	0	0
$326$	−3.79458e26	−1.55017
$327$	−8.55796e25	−0.338547
$328$	−2.60661e26	−0.998620
$329$	−2.50044e26	−0.927810
$330$	0	0
$331$	−4.40991e26	−1.53545	−0.767725	−	0.640780i	$-0.778611\pi$
$331$	−4.40991e26	−1.53545	−0.767725	+	0.640780i	$0.778611\pi$
$332$	−1.43985e25	−0.0485699
$333$	1.20199e25	0.0392860
$334$	−1.82121e26	−0.576795
$335$	0	0
$336$	2.51457e26	0.748000
$337$	−4.81798e26	−1.38915	−0.694577	−	0.719418i	$-0.744409\pi$
$337$	−4.81798e26	−1.38915	−0.694577	+	0.719418i	$0.744409\pi$
$338$	3.83259e26	1.07119
$339$	3.17180e26	0.859426
$340$	0	0
$341$	3.18359e26	0.810955
$342$	−1.45787e26	−0.360117
$343$	−6.65743e26	−1.59485
$344$	6.77068e26	1.57314
$345$	0	0
$346$	−4.06548e24	−0.00888819
$347$	3.64414e26	0.772922	0.386461	−	0.922306i	$-0.373697\pi$
$347$	3.64414e26	0.772922	0.386461	+	0.922306i	$0.373697\pi$
$348$	−3.53155e25	−0.0726747
$349$	−3.60335e26	−0.719515	−0.359758	−	0.933046i	$-0.617141\pi$
$349$	−3.60335e26	−0.719515	−0.359758	+	0.933046i	$0.617141\pi$
$350$	0	0
$351$	−1.51773e26	−0.285411
$352$	−3.07195e26	−0.560681
$353$	3.52364e26	0.624247	0.312124	−	0.950042i	$-0.398960\pi$
$353$	3.52364e26	0.624247	0.312124	+	0.950042i	$0.398960\pi$
$354$	−3.00215e26	−0.516295
$355$	0	0
$356$	−1.43864e26	−0.233199
$357$	4.57481e26	0.720040
$358$	−9.91962e26	−1.51608
$359$	1.17616e27	1.74571	0.872857	−	0.487977i	$-0.162265\pi$
$359$	1.17616e27	1.74571	0.872857	+	0.487977i	$0.162265\pi$
$360$	0	0
$361$	3.30870e26	0.463268
$362$	1.96112e26	0.266726
$363$	4.30102e26	0.568266
$364$	−4.00299e26	−0.513831
$365$	0	0
$366$	6.86932e26	0.832458
$367$	−1.52241e26	−0.179282	−0.0896411	−	0.995974i	$-0.528572\pi$
$367$	−1.52241e26	−0.179282	−0.0896411	+	0.995974i	$0.528572\pi$
$368$	−1.32489e26	−0.151627
$369$	2.78688e26	0.309985
$370$	0	0
$371$	−2.98536e26	−0.313740
$372$	6.58357e25	0.0672606
$373$	8.84632e26	0.878658	0.439329	−	0.898326i	$-0.355216\pi$
$373$	8.84632e26	0.878658	0.439329	+	0.898326i	$0.355216\pi$
$374$	9.48918e26	0.916382
$375$	0	0
$376$	−6.37273e26	−0.581907
$377$	−1.03886e27	−0.922510
$378$	−3.40745e26	−0.294283
$379$	−3.83294e26	−0.321974	−0.160987	−	0.986957i	$-0.551468\pi$
$379$	−3.83294e26	−0.321974	−0.160987	+	0.986957i	$0.551468\pi$
$380$	0	0
$381$	−1.17654e27	−0.935178
$382$	7.84796e26	0.606864
$383$	−1.86870e27	−1.40589	−0.702947	−	0.711243i	$-0.748133\pi$
$383$	−1.86870e27	−1.40589	−0.702947	+	0.711243i	$0.748133\pi$
$384$	4.69488e26	0.343674
$385$	0	0
$386$	−2.42628e27	−1.68180
$387$	−7.23893e26	−0.488325
$388$	4.75332e25	0.0312078
$389$	−7.58738e26	−0.484865	−0.242432	−	0.970168i	$-0.577945\pi$
$389$	−7.58738e26	−0.484865	−0.242432	+	0.970168i	$0.577945\pi$
$390$	0	0
$391$	−2.41040e26	−0.145960
$392$	−3.51830e27	−2.07410
$393$	1.15323e27	0.661903
$394$	−2.93494e26	−0.164017
$395$	0	0
$396$	1.79309e26	0.0950174
$397$	−5.16201e26	−0.266391	−0.133195	−	0.991090i	$-0.542524\pi$
$397$	−5.16201e26	−0.266391	−0.133195	+	0.991090i	$0.542524\pi$
$398$	−7.94078e26	−0.399108
$399$	2.44265e27	1.19576
$400$	0	0
$401$	2.28825e27	1.06289	0.531443	−	0.847094i	$-0.321650\pi$
$401$	2.28825e27	1.06289	0.531443	+	0.847094i	$0.321650\pi$
$402$	−7.03716e26	−0.318437
$403$	1.93665e27	0.853785
$404$	−1.06789e26	−0.0458692
$405$	0	0
$406$	−2.33233e27	−0.951188
$407$	−4.17734e26	−0.166019
$408$	1.16596e27	0.451597
$409$	−1.71019e27	−0.645581	−0.322791	−	0.946470i	$-0.604621\pi$
$409$	−1.71019e27	−0.645581	−0.322791	+	0.946470i	$0.604621\pi$
$410$	0	0
$411$	−1.50391e27	−0.539364
$412$	9.74601e26	0.340727
$413$	5.03008e27	1.71435
$414$	1.79533e26	0.0596543
$415$	0	0
$416$	−1.86874e27	−0.590294
$417$	−1.93943e27	−0.597374
$418$	5.06660e27	1.52183
$419$	−7.77024e25	−0.0227608	−0.0113804	−	0.999935i	$-0.503623\pi$
$419$	−7.77024e25	−0.0227608	−0.0113804	+	0.999935i	$0.503623\pi$
$420$	0	0
$421$	2.53023e27	0.705015	0.352508	−	0.935809i	$-0.385329\pi$
$421$	2.53023e27	0.705015	0.352508	+	0.935809i	$0.385329\pi$
$422$	2.57622e27	0.700169
$423$	6.81346e26	0.180632
$424$	−7.60863e26	−0.196773
$425$	0	0
$426$	−3.63104e27	−0.893778
$427$	−1.15095e28	−2.76416
$428$	1.59530e27	0.373837
$429$	5.27464e27	1.20612
$430$	0	0
$431$	−4.79626e26	−0.104446	−0.0522229	−	0.998635i	$-0.516631\pi$
$431$	−4.79626e26	−0.104446	−0.0522229	+	0.998635i	$0.516631\pi$
$432$	−6.85198e26	−0.145625
$433$	−8.00695e27	−1.66090	−0.830451	−	0.557092i	$-0.811917\pi$
$433$	−8.00695e27	−1.66090	−0.830451	+	0.557092i	$0.811917\pi$
$434$	4.34797e27	0.880327
$435$	0	0
$436$	6.15052e26	0.118660
$437$	−1.28699e27	−0.242394
$438$	−1.38664e27	−0.254968
$439$	−9.55381e27	−1.71514	−0.857569	−	0.514368i	$-0.828026\pi$
$439$	−9.55381e27	−1.71514	−0.857569	+	0.514368i	$0.828026\pi$
$440$	0	0
$441$	3.76162e27	0.643828
$442$	5.77248e27	0.964781
$443$	−5.53824e27	−0.903925	−0.451962	−	0.892037i	$-0.649276\pi$
$443$	−5.53824e27	−0.903925	−0.451962	+	0.892037i	$0.649276\pi$
$444$	−8.63861e25	−0.0137696
$445$	0	0
$446$	−4.25756e27	−0.647359
$447$	−2.42582e27	−0.360272
$448$	−1.31261e28	−1.90422
$449$	5.56138e27	0.788127	0.394063	−	0.919083i	$-0.371069\pi$
$449$	5.56138e27	0.788127	0.394063	+	0.919083i	$0.371069\pi$
$450$	0	0
$451$	−9.68536e27	−1.30997
$452$	−2.27954e27	−0.301227
$453$	5.17728e27	0.668453
$454$	1.98690e27	0.250662
$455$	0	0
$456$	6.22544e27	0.749962
$457$	−5.87968e27	−0.692203	−0.346101	−	0.938197i	$-0.612495\pi$
$457$	−5.87968e27	−0.692203	−0.346101	+	0.938197i	$0.612495\pi$
$458$	−4.79748e27	−0.551983
$459$	−1.24659e27	−0.140182
$460$	0	0
$461$	−1.13415e28	−1.21846	−0.609231	−	0.792993i	$-0.708522\pi$
$461$	−1.13415e28	−1.21846	−0.609231	+	0.792993i	$0.708522\pi$
$462$	1.18421e28	1.24362
$463$	−1.74305e28	−1.78941	−0.894705	−	0.446658i	$-0.852614\pi$
$463$	−1.74305e28	−1.78941	−0.894705	+	0.446658i	$0.852614\pi$
$464$	−4.69004e27	−0.470693
$465$	0	0
$466$	1.45533e28	1.39608
$467$	−1.60704e27	−0.150730	−0.0753651	−	0.997156i	$-0.524012\pi$
$467$	−1.60704e27	−0.150730	−0.0753651	+	0.997156i	$0.524012\pi$
$468$	1.09078e27	0.100036
$469$	1.17907e28	1.05736
$470$	0	0
$471$	−6.29490e27	−0.539846
$472$	1.28199e28	1.07521
$473$	2.51578e28	2.06362
$474$	−8.53940e27	−0.685101
$475$	0	0
$476$	−3.28787e27	−0.252372
$477$	8.13483e26	0.0610809
$478$	1.69498e28	1.24501
$479$	−5.39349e27	−0.387567	−0.193783	−	0.981044i	$-0.562076\pi$
$479$	−5.39349e27	−0.387567	−0.193783	+	0.981044i	$0.562076\pi$
$480$	0	0
$481$	−2.54117e27	−0.174788
$482$	2.15306e27	0.144898
$483$	−3.00807e27	−0.198081
$484$	−3.09110e27	−0.199176
$485$	0	0
$486$	9.28498e26	0.0572928
$487$	−1.39994e28	−0.845386	−0.422693	−	0.906273i	$-0.638915\pi$
$487$	−1.39994e28	−0.845386	−0.422693	+	0.906273i	$0.638915\pi$
$488$	−2.93336e28	−1.73363
$489$	−1.73244e28	−1.00211
$490$	0	0
$491$	1.66296e27	0.0921567	0.0460783	−	0.998938i	$-0.485328\pi$
$491$	1.66296e27	0.0921567	0.0460783	+	0.998938i	$0.485328\pi$
$492$	−2.00290e27	−0.108649
$493$	−8.53268e27	−0.453099
$494$	3.08213e28	1.60220
$495$	0	0
$496$	8.74325e27	0.435627
$497$	6.08377e28	2.96777
$498$	2.59116e27	0.123761
$499$	−2.24968e28	−1.05212	−0.526059	−	0.850448i	$-0.676331\pi$
$499$	−2.24968e28	−1.05212	−0.526059	+	0.850448i	$0.676331\pi$
$500$	0	0
$501$	−8.31483e27	−0.372870
$502$	3.00581e28	1.31999
$503$	2.16964e28	0.933090	0.466545	−	0.884497i	$-0.345499\pi$
$503$	2.16964e28	0.933090	0.466545	+	0.884497i	$0.345499\pi$
$504$	1.45506e28	0.612859
$505$	0	0
$506$	−6.23941e27	−0.252094
$507$	1.74979e28	0.692473
$508$	8.45567e27	0.327778
$509$	−1.86305e28	−0.707436	−0.353718	−	0.935352i	$-0.615083\pi$
$509$	−1.86305e28	−0.707436	−0.353718	+	0.935352i	$0.615083\pi$
$510$	0	0
$511$	2.32331e28	0.846618
$512$	−3.11976e28	−1.11375
$513$	−6.65599e27	−0.232798
$514$	3.43580e28	1.17738
$515$	0	0
$516$	5.20255e27	0.171157
$517$	−2.36791e28	−0.763335
$518$	−5.70517e27	−0.180221
$519$	−1.85612e26	−0.00574578
$520$	0	0
$521$	6.39183e28	1.90033	0.950165	−	0.311747i	$-0.100914\pi$
$521$	6.39183e28	1.90033	0.950165	+	0.311747i	$0.100914\pi$
$522$	6.35538e27	0.185183
$523$	9.84531e27	0.281165	0.140583	−	0.990069i	$-0.455102\pi$
$523$	9.84531e27	0.281165	0.140583	+	0.990069i	$0.455102\pi$
$524$	−8.28815e27	−0.231995
$525$	0	0
$526$	1.26212e28	0.339429
$527$	1.59068e28	0.419344
$528$	2.38130e28	0.615400
$529$	−3.78867e28	−0.959847
$530$	0	0
$531$	−1.37065e28	−0.333760
$532$	−1.75551e28	−0.419112
$533$	−5.89183e28	−1.37916
$534$	2.58898e28	0.594217
$535$	0	0
$536$	3.00503e28	0.663161
$537$	−4.52886e28	−0.980075
$538$	−1.48921e28	−0.316040
$539$	−1.30729e29	−2.72076
$540$	0	0
$541$	6.36576e28	1.27432	0.637161	−	0.770731i	$-0.280109\pi$
$541$	6.36576e28	1.27432	0.637161	+	0.770731i	$0.280109\pi$
$542$	2.03593e28	0.399732
$543$	8.95362e27	0.172425
$544$	−1.53489e28	−0.289928
$545$	0	0
$546$	7.20379e28	1.30930
$547$	7.99629e28	1.42568	0.712839	−	0.701328i	$-0.247409\pi$
$547$	7.99629e28	1.42568	0.712839	+	0.701328i	$0.247409\pi$
$548$	1.08084e28	0.189046
$549$	3.13623e28	0.538144
$550$	0	0
$551$	−4.55589e28	−0.752456
$552$	−7.66650e27	−0.124233
$553$	1.43077e29	2.27486
$554$	−2.13238e28	−0.332669
$555$	0	0
$556$	1.39385e28	0.209378
$557$	−6.15885e28	−0.907862	−0.453931	−	0.891037i	$-0.649979\pi$
$557$	−6.15885e28	−0.907862	−0.453931	+	0.891037i	$0.649979\pi$
$558$	−1.18478e28	−0.171387
$559$	1.53040e29	2.17261
$560$	0	0
$561$	4.33234e28	0.592397
$562$	8.58222e28	1.15178
$563$	−2.00390e28	−0.263960	−0.131980	−	0.991252i	$-0.542134\pi$
$563$	−2.00390e28	−0.263960	−0.131980	+	0.991252i	$0.542134\pi$
$564$	−4.89677e27	−0.0633110
$565$	0	0
$566$	8.70319e28	1.08419
$567$	−1.55569e28	−0.190240
$568$	1.55054e29	1.86134
$569$	6.14459e28	0.724126	0.362063	−	0.932154i	$-0.382073\pi$
$569$	6.14459e28	0.724126	0.362063	+	0.932154i	$0.382073\pi$
$570$	0	0
$571$	−4.65211e28	−0.528410	−0.264205	−	0.964467i	$-0.585110\pi$
$571$	−4.65211e28	−0.528410	−0.264205	+	0.964467i	$0.585110\pi$
$572$	−3.79083e28	−0.422743
$573$	3.58304e28	0.392308
$574$	−1.32277e29	−1.42203
$575$	0	0
$576$	3.57674e28	0.370725
$577$	−1.04152e28	−0.106004	−0.0530020	−	0.998594i	$-0.516879\pi$
$577$	−1.04152e28	−0.106004	−0.0530020	+	0.998594i	$0.516879\pi$
$578$	−4.19481e28	−0.419247
$579$	−1.10773e29	−1.08720
$580$	0	0
$581$	−4.34146e28	−0.410947
$582$	−8.55408e27	−0.0795210
$583$	−2.82714e28	−0.258123
$584$	5.92129e28	0.530984
$585$	0	0
$586$	1.96405e28	0.169913
$587$	4.54423e28	0.386154	0.193077	−	0.981184i	$-0.438153\pi$
$587$	4.54423e28	0.386154	0.193077	+	0.981184i	$0.438153\pi$
$588$	−2.70344e28	−0.225660
$589$	8.49316e28	0.696400
$590$	0	0
$591$	−1.33996e28	−0.106029
$592$	−1.14724e28	−0.0891820
$593$	−3.29422e28	−0.251581	−0.125791	−	0.992057i	$-0.540147\pi$
$593$	−3.29422e28	−0.251581	−0.125791	+	0.992057i	$0.540147\pi$
$594$	−3.22685e28	−0.242115
$595$	0	0
$596$	1.74342e28	0.126274
$597$	−3.62541e28	−0.258004
$598$	−3.79557e28	−0.265409
$599$	−1.44151e29	−0.990458	−0.495229	−	0.868763i	$-0.664916\pi$
$599$	−1.44151e29	−0.990458	−0.495229	+	0.868763i	$0.664916\pi$
$600$	0	0
$601$	1.75962e28	0.116745	0.0583723	−	0.998295i	$-0.481409\pi$
$601$	1.75962e28	0.116745	0.0583723	+	0.998295i	$0.481409\pi$
$602$	3.43590e29	2.24015
$603$	−3.21286e28	−0.205854
$604$	−3.72086e28	−0.234291
$605$	0	0
$606$	1.92177e28	0.116880
$607$	7.40539e28	0.442656	0.221328	−	0.975199i	$-0.428961\pi$
$607$	7.40539e28	0.442656	0.221328	+	0.975199i	$0.428961\pi$
$608$	−8.19532e28	−0.481480
$609$	−1.06484e29	−0.614897
$610$	0	0
$611$	−1.44046e29	−0.803650
$612$	8.95914e27	0.0491334
$613$	2.04497e29	1.10243	0.551216	−	0.834363i	$-0.314164\pi$
$613$	2.04497e29	1.10243	0.551216	+	0.834363i	$0.314164\pi$
$614$	2.80540e29	1.48671
$615$	0	0
$616$	−5.05684e29	−2.58989
$617$	−3.28969e29	−1.65638	−0.828192	−	0.560445i	$-0.810630\pi$
$617$	−3.28969e29	−1.65638	−0.828192	+	0.560445i	$0.810630\pi$
$618$	−1.75389e29	−0.868210
$619$	1.51436e29	0.737018	0.368509	−	0.929624i	$-0.379868\pi$
$619$	1.51436e29	0.737018	0.368509	+	0.929624i	$0.379868\pi$
$620$	0	0
$621$	8.19671e27	0.0385636
$622$	−1.27586e29	−0.590208
$623$	−4.33781e29	−1.97309
$624$	1.44860e29	0.647902
$625$	0	0
$626$	−1.66088e29	−0.718302
$627$	2.31319e29	0.983787
$628$	4.52409e28	0.189215
$629$	−2.08720e28	−0.0858484
$630$	0	0
$631$	1.00288e29	0.398969	0.199485	−	0.979901i	$-0.436073\pi$
$631$	1.00288e29	0.398969	0.199485	+	0.979901i	$0.436073\pi$
$632$	3.64652e29	1.42676
$633$	1.17619e29	0.452625
$634$	1.88200e29	0.712331
$635$	0	0
$636$	−5.84643e27	−0.0214087
$637$	−7.95256e29	−2.86446
$638$	−2.20872e29	−0.782569
$639$	−1.65777e29	−0.577784
$640$	0	0
$641$	−2.60873e29	−0.879871	−0.439936	−	0.898029i	$-0.644999\pi$
$641$	−2.60873e29	−0.879871	−0.439936	+	0.898029i	$0.644999\pi$
$642$	−2.87091e29	−0.952579
$643$	−5.30533e28	−0.173180	−0.0865898	−	0.996244i	$-0.527597\pi$
$643$	−5.30533e28	−0.173180	−0.0865898	+	0.996244i	$0.527597\pi$
$644$	2.16187e28	0.0694269
$645$	0	0
$646$	2.53151e29	0.786935
$647$	7.13885e27	0.0218340	0.0109170	−	0.999940i	$-0.496525\pi$
$647$	7.13885e27	0.0218340	0.0109170	+	0.999940i	$0.496525\pi$
$648$	−3.96491e28	−0.119315
$649$	4.76348e29	1.41044
$650$	0	0
$651$	1.98509e29	0.569089
$652$	1.24509e29	0.351237
$653$	8.18691e28	0.227264	0.113632	−	0.993523i	$-0.463751\pi$
$653$	8.18691e28	0.227264	0.113632	+	0.993523i	$0.463751\pi$
$654$	−1.10685e29	−0.302359
$655$	0	0
$656$	−2.65993e29	−0.703689
$657$	−6.33080e28	−0.164825
$658$	−3.23396e29	−0.828633
$659$	−3.25306e28	−0.0820342	−0.0410171	−	0.999158i	$-0.513060\pi$
$659$	−3.25306e28	−0.0820342	−0.0410171	+	0.999158i	$0.513060\pi$
$660$	0	0
$661$	8.22623e28	0.200949	0.100474	−	0.994940i	$-0.467964\pi$
$661$	8.22623e28	0.200949	0.100474	+	0.994940i	$0.467964\pi$
$662$	−5.70359e29	−1.37132
$663$	2.63546e29	0.623684
$664$	−1.10648e29	−0.257739
$665$	0	0
$666$	1.55461e28	0.0350866
$667$	5.61048e28	0.124646
$668$	5.97579e28	0.130690
$669$	−1.94382e29	−0.418486
$670$	0	0
$671$	−1.08995e30	−2.27415
$672$	−1.91548e29	−0.393459
$673$	−1.16937e29	−0.236479	−0.118239	−	0.992985i	$-0.537725\pi$
$673$	−1.16937e29	−0.236479	−0.118239	+	0.992985i	$0.537725\pi$
$674$	−6.23136e29	−1.24066
$675$	0	0
$676$	−1.25756e29	−0.242710
$677$	−4.28660e28	−0.0814577	−0.0407289	−	0.999170i	$-0.512968\pi$
$677$	−4.28660e28	−0.0814577	−0.0407289	+	0.999170i	$0.512968\pi$
$678$	4.10227e29	0.767560
$679$	1.43323e29	0.264048
$680$	0	0
$681$	9.07130e28	0.162041
$682$	4.11752e29	0.724270
$683$	−5.45367e29	−0.944651	−0.472325	−	0.881424i	$-0.656585\pi$
$683$	−5.45367e29	−0.944651	−0.472325	+	0.881424i	$0.656585\pi$
$684$	4.78359e28	0.0815953
$685$	0	0
$686$	−8.61044e29	−1.42437
$687$	−2.19032e29	−0.356830
$688$	6.90919e29	1.10853
$689$	−1.71981e29	−0.271756
$690$	0	0
$691$	−9.02075e29	−1.38269	−0.691343	−	0.722527i	$-0.742980\pi$
$691$	−9.02075e29	−1.38269	−0.691343	+	0.722527i	$0.742980\pi$
$692$	1.33398e27	0.00201388
$693$	5.40656e29	0.803938
$694$	4.71318e29	0.690302
$695$	0	0
$696$	−2.71390e29	−0.385653
$697$	−4.83927e29	−0.677385
$698$	−4.66042e29	−0.642604
$699$	6.64442e29	0.902499
$700$	0	0
$701$	−1.36429e29	−0.179833	−0.0899164	−	0.995949i	$-0.528660\pi$
$701$	−1.36429e29	−0.179833	−0.0899164	+	0.995949i	$0.528660\pi$
$702$	−1.96297e29	−0.254902
$703$	−1.11443e29	−0.142568
$704$	−1.24304e30	−1.56665
$705$	0	0
$706$	4.55732e29	0.557520
$707$	−3.21992e29	−0.388097
$708$	9.85073e28	0.116982
$709$	−1.96075e29	−0.229423	−0.114712	−	0.993399i	$-0.536594\pi$
$709$	−1.96075e29	−0.229423	−0.114712	+	0.993399i	$0.536594\pi$
$710$	0	0
$711$	−3.89871e29	−0.442885
$712$	−1.10555e30	−1.23749
$713$	−1.04591e29	−0.115360
$714$	5.91686e29	0.643072
$715$	0	0
$716$	3.25485e29	0.343514
$717$	7.73853e29	0.804836
$718$	1.52119e30	1.55911
$719$	6.61361e29	0.668013	0.334007	−	0.942571i	$-0.391599\pi$
$719$	6.61361e29	0.668013	0.334007	+	0.942571i	$0.391599\pi$
$720$	0	0
$721$	2.93864e30	2.88287
$722$	4.27933e29	0.413748
$723$	9.82991e28	0.0936693
$724$	−6.43488e28	−0.0604346
$725$	0	0
$726$	5.56275e29	0.507523
$727$	5.62124e29	0.505500	0.252750	−	0.967532i	$-0.418665\pi$
$727$	5.62124e29	0.505500	0.252750	+	0.967532i	$0.418665\pi$
$728$	−3.07619e30	−2.72668
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	1.25700e30	1.06710
$732$	−2.25398e29	−0.188618
$733$	1.44462e30	1.19169	0.595843	−	0.803101i	$-0.296818\pi$
$733$	1.44462e30	1.19169	0.595843	+	0.803101i	$0.296818\pi$
$734$	−1.96902e29	−0.160118
$735$	0	0
$736$	1.00924e29	0.0797583
$737$	1.11658e30	0.869923
$738$	3.60443e29	0.276850
$739$	4.75852e29	0.360334	0.180167	−	0.983636i	$-0.442336\pi$
$739$	4.75852e29	0.360334	0.180167	+	0.983636i	$0.442336\pi$
$740$	0	0
$741$	1.40716e30	1.03575
$742$	−3.86114e29	−0.280204
$743$	1.64704e30	1.17848	0.589239	−	0.807959i	$-0.299428\pi$
$743$	1.64704e30	1.17848	0.589239	+	0.807959i	$0.299428\pi$
$744$	5.05929e29	0.356923
$745$	0	0
$746$	1.14415e30	0.784736
$747$	1.18301e29	0.0800058
$748$	−3.11361e29	−0.207634
$749$	4.81018e30	3.16302
$750$	0	0
$751$	−1.15249e30	−0.736914	−0.368457	−	0.929645i	$-0.620114\pi$
$751$	−1.15249e30	−0.736914	−0.368457	+	0.929645i	$0.620114\pi$
$752$	−6.50311e29	−0.410047
$753$	1.37232e30	0.853312
$754$	−1.34361e30	−0.823900
$755$	0	0
$756$	1.11806e29	0.0666786
$757$	−6.06438e29	−0.356681	−0.178340	−	0.983969i	$-0.557073\pi$
$757$	−6.06438e29	−0.356681	−0.178340	+	0.983969i	$0.557073\pi$
$758$	−4.95736e29	−0.287557
$759$	−2.84864e29	−0.162967
$760$	0	0
$761$	−3.30053e30	−1.83673	−0.918364	−	0.395737i	$-0.870489\pi$
$761$	−3.30053e30	−1.83673	−0.918364	+	0.395737i	$0.870489\pi$
$762$	−1.52169e30	−0.835214
$763$	1.85452e30	1.00398
$764$	−2.57509e29	−0.137503
$765$	0	0
$766$	−2.41690e30	−1.25561
$767$	2.89773e30	1.48493
$768$	−6.63081e29	−0.335177
$769$	−2.35995e30	−1.17673	−0.588363	−	0.808597i	$-0.700228\pi$
$769$	−2.35995e30	−1.17673	−0.588363	+	0.808597i	$0.700228\pi$
$770$	0	0
$771$	1.56864e30	0.761117
$772$	7.96117e29	0.381062
$773$	−2.75286e30	−1.29987	−0.649935	−	0.759990i	$-0.725204\pi$
$773$	−2.75286e30	−1.29987	−0.649935	+	0.759990i	$0.725204\pi$
$774$	−9.36252e29	−0.436127
$775$	0	0
$776$	3.65279e29	0.165606
$777$	−2.60473e29	−0.116504
$778$	−9.81319e29	−0.433036
$779$	−2.58385e30	−1.12493
$780$	0	0
$781$	5.76133e30	2.44167
$782$	−3.11751e29	−0.130358
$783$	2.90159e29	0.119712
$784$	−3.59028e30	−1.46153
$785$	0	0
$786$	1.49154e30	0.591150
$787$	1.96178e30	0.767213	0.383606	−	0.923497i	$-0.374682\pi$
$787$	1.96178e30	0.767213	0.383606	+	0.923497i	$0.374682\pi$
$788$	9.63019e28	0.0371629
$789$	5.76227e29	0.219425
$790$	0	0
$791$	−6.87331e30	−2.54867
$792$	1.37794e30	0.504216
$793$	−6.63040e30	−2.39426
$794$	−6.67633e29	−0.237915
$795$	0	0
$796$	2.60555e29	0.0904299
$797$	3.68558e30	1.26239	0.631195	−	0.775624i	$-0.282565\pi$
$797$	3.68558e30	1.26239	0.631195	+	0.775624i	$0.282565\pi$
$798$	3.15921e30	1.06794
$799$	−1.18312e30	−0.394720
$800$	0	0
$801$	1.18201e30	0.384133
$802$	2.95952e30	0.949271
$803$	2.20017e30	0.696536
$804$	2.30905e29	0.0721514
$805$	0	0
$806$	2.50478e30	0.762522
$807$	−6.79908e29	−0.204305
$808$	−8.20643e29	−0.243408
$809$	−1.59133e30	−0.465909	−0.232954	−	0.972488i	$-0.574839\pi$
$809$	−1.59133e30	−0.465909	−0.232954	+	0.972488i	$0.574839\pi$
$810$	0	0
$811$	2.03075e30	0.579345	0.289673	−	0.957126i	$-0.406454\pi$
$811$	2.03075e30	0.579345	0.289673	+	0.957126i	$0.406454\pi$
$812$	7.65290e29	0.215520
$813$	9.29514e29	0.258408
$814$	−5.40280e29	−0.148273
$815$	0	0
$816$	1.18981e30	0.318223
$817$	6.71157e30	1.77212
$818$	−2.21189e30	−0.576573
$819$	3.28894e30	0.846397
$820$	0	0
$821$	−1.16660e30	−0.292631	−0.146315	−	0.989238i	$-0.546741\pi$
$821$	−1.16660e30	−0.292631	−0.146315	+	0.989238i	$0.546741\pi$
$822$	−1.94509e30	−0.481710
$823$	4.40906e30	1.07807	0.539036	−	0.842283i	$-0.318789\pi$
$823$	4.40906e30	1.07807	0.539036	+	0.842283i	$0.318789\pi$
$824$	7.48954e30	1.80809
$825$	0	0
$826$	6.50568e30	1.53110
$827$	2.06510e30	0.479881	0.239941	−	0.970788i	$-0.422872\pi$
$827$	2.06510e30	0.479881	0.239941	+	0.970788i	$0.422872\pi$
$828$	−5.89089e28	−0.0135165
$829$	−3.07768e30	−0.697271	−0.348635	−	0.937258i	$-0.613355\pi$
$829$	−3.07768e30	−0.697271	−0.348635	+	0.937258i	$0.613355\pi$
$830$	0	0
$831$	−9.73553e29	−0.215055
$832$	−7.56170e30	−1.64940
$833$	−6.53186e30	−1.40690
$834$	−2.50838e30	−0.533519
$835$	0	0
$836$	−1.66246e30	−0.344815
$837$	−5.40919e29	−0.110794
$838$	−1.00497e29	−0.0203278
$839$	−4.30426e30	−0.859800	−0.429900	−	0.902876i	$-0.641451\pi$
$839$	−4.30426e30	−0.859800	−0.429900	+	0.902876i	$0.641451\pi$
$840$	0	0
$841$	−3.14676e30	−0.613064
$842$	3.27249e30	0.629654
$843$	3.91827e30	0.744568
$844$	−8.45317e29	−0.158644
$845$	0	0
$846$	8.81224e29	0.161323
$847$	−9.32035e30	−1.68522
$848$	−7.76429e29	−0.138658
$849$	3.97349e30	0.700877
$850$	0	0
$851$	1.37239e29	0.0236167
$852$	1.19142e30	0.202512
$853$	5.29542e30	0.889069	0.444534	−	0.895762i	$-0.353369\pi$
$853$	5.29542e30	0.889069	0.444534	+	0.895762i	$0.353369\pi$
$854$	−1.48859e31	−2.46869
$855$	0	0
$856$	1.22594e31	1.98379
$857$	−8.07412e29	−0.129062	−0.0645308	−	0.997916i	$-0.520555\pi$
$857$	−8.07412e29	−0.129062	−0.0645308	+	0.997916i	$0.520555\pi$
$858$	6.82199e30	1.07720
$859$	4.76822e30	0.743751	0.371876	−	0.928283i	$-0.378715\pi$
$859$	4.76822e30	0.743751	0.371876	+	0.928283i	$0.378715\pi$
$860$	0	0
$861$	−6.03919e30	−0.919275
$862$	−6.20328e29	−0.0932814
$863$	−1.13640e31	−1.68817	−0.844085	−	0.536209i	$-0.819856\pi$
$863$	−1.13640e31	−1.68817	−0.844085	+	0.536209i	$0.819856\pi$
$864$	5.21950e29	0.0766010
$865$	0	0
$866$	−1.03558e31	−1.48336
$867$	−1.91517e30	−0.271023
$868$	−1.42667e30	−0.199464
$869$	1.35494e31	1.87159
$870$	0	0
$871$	6.79240e30	0.915868
$872$	4.72651e30	0.629676
$873$	−3.90542e29	−0.0514065
$874$	−1.66454e30	−0.216484
$875$	0	0
$876$	4.54989e29	0.0577707
$877$	−6.52492e30	−0.818615	−0.409307	−	0.912397i	$-0.634230\pi$
$877$	−6.52492e30	−0.818615	−0.409307	+	0.912397i	$0.634230\pi$
$878$	−1.23565e31	−1.53180
$879$	8.96699e29	0.109841
$880$	0	0
$881$	6.24200e30	0.746581	0.373290	−	0.927715i	$-0.378230\pi$
$881$	6.24200e30	0.746581	0.373290	+	0.927715i	$0.378230\pi$
$882$	4.86512e30	0.575007
$883$	5.35499e30	0.625419	0.312709	−	0.949849i	$-0.398763\pi$
$883$	5.35499e30	0.625419	0.312709	+	0.949849i	$0.398763\pi$
$884$	−1.89408e30	−0.218600
$885$	0	0
$886$	−7.16292e30	−0.807302
$887$	1.58739e31	1.76802	0.884009	−	0.467470i	$-0.154834\pi$
$887$	1.58739e31	1.76802	0.884009	+	0.467470i	$0.154834\pi$
$888$	−6.63854e29	−0.0730695
$889$	2.54957e31	2.77331
$890$	0	0
$891$	−1.47324e30	−0.156516
$892$	1.39700e30	0.146678
$893$	−6.31710e30	−0.655507
$894$	−3.13745e30	−0.321761
$895$	0	0
$896$	−1.01738e31	−1.01918
$897$	−1.73289e30	−0.171574
$898$	7.19285e30	0.703881
$899$	−3.70248e30	−0.358110
$900$	0	0
$901$	−1.41257e30	−0.133475
$902$	−1.25266e31	−1.16995
$903$	1.56868e31	1.44815
$904$	−1.75176e31	−1.59848
$905$	0	0
$906$	6.69607e30	0.597000
$907$	2.03878e31	1.79677	0.898386	−	0.439206i	$-0.144740\pi$
$907$	2.03878e31	1.79677	0.898386	+	0.439206i	$0.144740\pi$
$908$	−6.51945e29	−0.0567950
$909$	8.77398e29	0.0755572
$910$	0	0
$911$	1.16077e31	0.976792	0.488396	−	0.872622i	$-0.337582\pi$
$911$	1.16077e31	0.976792	0.488396	+	0.872622i	$0.337582\pi$
$912$	6.35280e30	0.528469
$913$	−4.11136e30	−0.338098
$914$	−7.60452e30	−0.618211
$915$	0	0
$916$	1.57416e30	0.125068
$917$	−2.49906e31	−1.96290
$918$	−1.61229e30	−0.125197
$919$	1.11455e31	0.855631	0.427816	−	0.903866i	$-0.359283\pi$
$919$	1.11455e31	0.855631	0.427816	+	0.903866i	$0.359283\pi$
$920$	0	0
$921$	1.28082e31	0.961088
$922$	−1.46687e31	−1.08822
$923$	3.50475e31	2.57062
$924$	−3.88565e30	−0.281778
$925$	0	0
$926$	−2.25439e31	−1.59813
$927$	−8.00751e30	−0.561255
$928$	3.57264e30	0.247591
$929$	−1.58810e30	−0.108821	−0.0544105	−	0.998519i	$-0.517328\pi$
$929$	−1.58810e30	−0.108821	−0.0544105	+	0.998519i	$0.517328\pi$
$930$	0	0
$931$	−3.48758e31	−2.33643
$932$	−4.77528e30	−0.316324
$933$	−5.82503e30	−0.381541
$934$	−2.07848e30	−0.134618
$935$	0	0
$936$	8.38234e30	0.530846
$937$	3.78856e29	0.0237251	0.0118626	−	0.999930i	$-0.496224\pi$
$937$	3.78856e29	0.0237251	0.0118626	+	0.999930i	$0.496224\pi$
$938$	1.52496e31	0.944339
$939$	−7.58284e30	−0.464347
$940$	0	0
$941$	−2.33225e31	−1.39664	−0.698320	−	0.715786i	$-0.746069\pi$
$941$	−2.33225e31	−1.39664	−0.698320	+	0.715786i	$0.746069\pi$
$942$	−8.14156e30	−0.482140
$943$	3.18196e30	0.186347
$944$	1.30822e31	0.757659
$945$	0	0
$946$	3.25380e31	1.84304
$947$	1.89428e31	1.06113	0.530566	−	0.847643i	$-0.321979\pi$
$947$	1.89428e31	1.06113	0.530566	+	0.847643i	$0.321979\pi$
$948$	2.80197e30	0.155230
$949$	1.33841e31	0.733323
$950$	0	0
$951$	8.59238e30	0.460488
$952$	−2.52664e31	−1.33923
$953$	−2.74511e31	−1.43908	−0.719538	−	0.694453i	$-0.755646\pi$
$953$	−2.74511e31	−1.43908	−0.719538	+	0.694453i	$0.755646\pi$
$954$	1.05212e30	0.0545518
$955$	0	0
$956$	−5.56161e30	−0.282093
$957$	−1.00840e31	−0.505893
$958$	−6.97570e30	−0.346139
$959$	3.25898e31	1.59951
$960$	0	0
$961$	−1.39233e31	−0.668569
$962$	−3.28664e30	−0.156104
$963$	−1.31073e31	−0.615796
$964$	−7.06466e29	−0.0328309
$965$	0	0
$966$	−3.89050e30	−0.176907
$967$	−3.03551e31	−1.36538	−0.682690	−	0.730708i	$-0.739190\pi$
$967$	−3.03551e31	−1.36538	−0.682690	+	0.730708i	$0.739190\pi$
$968$	−2.37543e31	−1.05694
$969$	1.15578e31	0.508715
$970$	0	0
$971$	3.36946e31	1.45130	0.725651	−	0.688063i	$-0.241539\pi$
$971$	3.36946e31	1.45130	0.725651	+	0.688063i	$0.241539\pi$
$972$	−3.04661e29	−0.0129814
$973$	4.20277e31	1.77154
$974$	−1.81062e31	−0.755020
$975$	0	0
$976$	−2.99337e31	−1.22163
$977$	−1.47911e31	−0.597184	−0.298592	−	0.954381i	$-0.596517\pi$
$977$	−1.47911e31	−0.597184	−0.298592	+	0.954381i	$0.596517\pi$
$978$	−2.24066e31	−0.894991
$979$	−4.10791e31	−1.62331
$980$	0	0
$981$	−5.05339e30	−0.195460
$982$	2.15080e30	0.0823057
$983$	−3.37648e31	−1.27836	−0.639180	−	0.769058i	$-0.720726\pi$
$983$	−3.37648e31	−1.27836	−0.639180	+	0.769058i	$0.720726\pi$
$984$	−1.53918e31	−0.576554
$985$	0	0
$986$	−1.10358e31	−0.404665
$987$	−1.47648e31	−0.535671
$988$	−1.01131e31	−0.363026
$989$	−8.26515e30	−0.293555
$990$	0	0
$991$	−5.48368e31	−1.90677	−0.953387	−	0.301751i	$-0.902429\pi$
$991$	−5.48368e31	−1.90677	−0.953387	+	0.301751i	$0.902429\pi$
$992$	−6.66017e30	−0.229146
$993$	−2.60401e31	−0.886492
$994$	7.86848e31	2.65054
$995$	0	0
$996$	−8.50216e29	−0.0280418
$997$	2.82839e31	0.923082	0.461541	−	0.887119i	$-0.347297\pi$
$997$	2.82839e31	0.923082	0.461541	+	0.887119i	$0.347297\pi$
$998$	−2.90964e31	−0.939654
$999$	7.09765e29	0.0226818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.h.1.3		4
5.2	odd	4		75.22.b.h.49.6		8
5.3	odd	4		75.22.b.h.49.3		8
5.4	even	2		15.22.a.e.1.2	✓	4
15.14	odd	2		45.22.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.a.e.1.2	✓	4	5.4	even	2
45.22.a.g.1.3		4	15.14	odd	2
75.22.a.h.1.3		4	1.1	even	1	trivial
75.22.b.h.49.3		8	5.3	odd	4
75.22.b.h.49.6		8	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1293.36 q^{2} +59049.0 q^{3} -424379. q^{4} +7.63714e7 q^{6} -1.27960e9 q^{7} -3.26124e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q+1293.36 q^{2} +59049.0 q^{3} -424379. q^{4} +7.63714e7 q^{6} -1.27960e9 q^{7} -3.26124e9 q^{8} +3.48678e9 q^{9} -1.21178e11 q^{11} -2.50592e10 q^{12} -7.37152e11 q^{13} -1.65498e12 q^{14} -3.32796e12 q^{16} -6.05462e12 q^{17} +4.50966e12 q^{18} -3.23277e13 q^{19} -7.55589e13 q^{21} -1.56726e14 q^{22} +3.98109e13 q^{23} -1.92573e14 q^{24} -9.53401e14 q^{26} +2.05891e14 q^{27} +5.43035e14 q^{28} +1.40928e15 q^{29} -2.62721e15 q^{31} +2.53508e15 q^{32} -7.15543e15 q^{33} -7.83079e15 q^{34} -1.47972e15 q^{36} +3.44728e15 q^{37} -4.18113e16 q^{38} -4.35281e16 q^{39} +7.99269e16 q^{41} -9.77246e16 q^{42} -2.07610e17 q^{43} +5.14254e16 q^{44} +5.14897e16 q^{46} +1.95408e17 q^{47} -1.96513e17 q^{48} +1.07882e18 q^{49} -3.57519e17 q^{51} +3.12832e17 q^{52} +2.33305e17 q^{53} +2.66291e17 q^{54} +4.17307e18 q^{56} -1.90892e18 q^{57} +1.82271e18 q^{58} -3.93099e18 q^{59} +8.99462e18 q^{61} -3.39792e18 q^{62} -4.46168e18 q^{63} +1.02580e19 q^{64} -9.25453e18 q^{66} -9.21439e18 q^{67} +2.56946e18 q^{68} +2.35079e18 q^{69} -4.75444e19 q^{71} -1.13712e19 q^{72} -1.81566e19 q^{73} +4.45857e18 q^{74} +1.37192e19 q^{76} +1.55059e20 q^{77} -5.62974e19 q^{78} -1.11814e20 q^{79} +1.21577e19 q^{81} +1.03374e20 q^{82} +3.39283e19 q^{83} +3.20656e19 q^{84} -2.68514e20 q^{86} +8.32168e19 q^{87} +3.95190e20 q^{88} +3.38998e20 q^{89} +9.43257e20 q^{91} -1.68949e19 q^{92} -1.55134e20 q^{93} +2.52733e20 q^{94} +1.49694e20 q^{96} -1.12006e20 q^{97} +1.39530e21 q^{98} -4.22521e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9}+O(q^{10}) \) 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9	\( 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9} + 31491830256 q^{11} - 68681250027 q^{12} + 27017977768 q^{13} - 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17} + 3127645607697 q^{18} - 24270353300752 q^{19} + 13851567033696 q^{21} + 56303932793676 q^{22} - 10350924920928 q^{23} - 4538248036821 q^{24} + 474751622871378 q^{26} + 823564528378596 q^{27} + 18\!\cdots\!68 q^{28}+ \cdots + 10\!\cdots\!56 q^{99}+O(q^{100}) \) 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9 + 31491830256 * q^11 - 68681250027 * q^12 + 27017977768 * q^13 - 2233308126672 * q^14 - 11324681311775 * q^16 + 2946095028888 * q^17 + 3127645607697 * q^18 - 24270353300752 * q^19 + 13851567033696 * q^21 + 56303932793676 * q^22 - 10350924920928 * q^23 - 4538248036821 * q^24 + 474751622871378 * q^26 + 823564528378596 * q^27 + 1868468795909968 * q^28 + 4728924677079096 * q^29 - 1094923910405536 * q^31 - 7933327650814221 * q^32 + 1859561084786544 * q^33 + 3862795623387294 * q^34 - 4055559132844323 * q^36 - 4813435696247096 * q^37 - 21307487199674508 * q^38 + 1595384569222632 * q^39 + 289731591445930344 * q^41 - 131874611571854928 * q^42 - 451091912658458000 * q^43 + 245434444420437276 * q^44 + 645145648102459824 * q^46 - 813883435638492480 * q^47 - 668711106779001975 * q^48 + 1545180753004755300 * q^49 + 173963965360807512 * q^51 - 884645329958480822 * q^52 - 697278335404085208 * q^53 + 184684345488900153 * q^54 + 5673275878853478000 * q^56 - 1433140092056104848 * q^57 - 12277858736996428626 * q^58 + 6622888614569598192 * q^59 + 7390887218011683320 * q^61 - 11649094150323079392 * q^62 + 817921181772715104 * q^63 + 21525702776885072473 * q^64 + 3324690927533774124 * q^66 + 24188188449376788688 * q^67 - 18995582935459304346 * q^68 - 611211765655877472 * q^69 - 37390337803999713312 * q^71 - 267979008326243229 * q^72 + 37253672904265201432 * q^73 - 77875663243460100102 * q^74 - 44795472028972724668 * q^76 + 315245139112859776128 * q^77 + 28033608578931999522 * q^78 - 109751291042259570400 * q^79 + 48630661836227715204 * q^81 + 210801413757052759338 * q^82 + 149023592088638482896 * q^83 + 110331213929687700432 * q^84 - 346200583098972939204 * q^86 + 279238273256843539704 * q^87 + 116461091469581805156 * q^88 + 558502237151273959848 * q^89 + 30360075980445717184 * q^91 + 764380916144075558544 * q^92 - 64654161985536495264 * q^93 + 1096939878232058257512 * q^94 - 468455064452928935829 * q^96 + 236244923748673777528 * q^97 - 197058383612642903463 * q^98 + 109805222495560636656 * q^99

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