LMFDB - Embedded newform 75.22.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$209.608008215$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 125326x + 2416960 $ x^3 - x^2 - 125326*x + 2416960
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-362.802$ 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)

$f(q)$	$=$	$q-999.805 q^{2} -59049.0 q^{3} -1.09754e6 q^{4} +5.90375e7 q^{6} -9.82313e7 q^{7} +3.19407e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-999.805 q^{2} -59049.0 q^{3} -1.09754e6 q^{4} +5.90375e7 q^{6} -9.82313e7 q^{7} +3.19407e9 q^{8} +3.48678e9 q^{9} -9.60815e10 q^{11} +6.48088e10 q^{12} -6.22052e11 q^{13} +9.82121e10 q^{14} -8.91733e11 q^{16} +1.42559e13 q^{17} -3.48610e12 q^{18} -8.28760e12 q^{19} +5.80046e12 q^{21} +9.60627e13 q^{22} +1.17504e14 q^{23} -1.88607e14 q^{24} +6.21930e14 q^{26} -2.05891e14 q^{27} +1.07813e14 q^{28} -1.98798e15 q^{29} -5.31766e15 q^{31} -5.80689e15 q^{32} +5.67352e15 q^{33} -1.42532e16 q^{34} -3.82689e15 q^{36} -1.99479e16 q^{37} +8.28598e15 q^{38} +3.67315e16 q^{39} -2.99615e16 q^{41} -5.79933e15 q^{42} +3.44307e16 q^{43} +1.05454e17 q^{44} -1.17481e17 q^{46} +5.23092e17 q^{47} +5.26560e16 q^{48} -5.48896e17 q^{49} -8.41799e17 q^{51} +6.82728e17 q^{52} +1.54033e18 q^{53} +2.05851e17 q^{54} -3.13758e17 q^{56} +4.89374e17 q^{57} +1.98759e18 q^{58} -6.81914e17 q^{59} -3.28598e18 q^{61} +5.31662e18 q^{62} -3.42511e17 q^{63} +7.67586e18 q^{64} -5.67241e18 q^{66} -1.07261e19 q^{67} -1.56465e19 q^{68} -6.93850e18 q^{69} +3.71838e19 q^{71} +1.11370e19 q^{72} +6.79746e19 q^{73} +1.99440e19 q^{74} +9.09599e18 q^{76} +9.43821e18 q^{77} -3.67244e19 q^{78} +1.28759e20 q^{79} +1.21577e19 q^{81} +2.99556e19 q^{82} -1.24075e20 q^{83} -6.36625e18 q^{84} -3.44240e19 q^{86} +1.17388e20 q^{87} -3.06891e20 q^{88} +4.94950e20 q^{89} +6.11049e19 q^{91} -1.28966e20 q^{92} +3.14002e20 q^{93} -5.22990e20 q^{94} +3.42891e20 q^{96} -4.69497e20 q^{97} +5.48789e20 q^{98} -3.35015e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2300 q^{2} - 177147 q^{3} + 1264400 q^{4} - 135812700 q^{6} - 465666872 q^{7} + 3839876544 q^{8} + 10460353203 q^{9}+O(q^{10}) $ 3 * q + 2300 * q^2 - 177147 * q^3 + 1264400 * q^4 - 135812700 * q^6 - 465666872 * q^7 + 3839876544 * q^8 + 10460353203 * q^9	$ 3 q + 2300 q^{2} - 177147 q^{3} + 1264400 q^{4} - 135812700 q^{6} - 465666872 q^{7} + 3839876544 q^{8} + 10460353203 q^{9} - 167336332556 q^{11} - 74661555600 q^{12} + 545571033878 q^{13} - 1568858902656 q^{14} + 255267954944 q^{16} + 8104424487194 q^{17} + 8019604122300 q^{18} + 3937700740828 q^{19} + 27497163124728 q^{21} - 114198109969712 q^{22} + 156235274730744 q^{23} - 226740870046656 q^{24} + 29\!\cdots\!56 q^{26}+ \cdots - 58\!\cdots\!56 q^{99}+O(q^{100}) $ 3 * q + 2300 * q^2 - 177147 * q^3 + 1264400 * q^4 - 135812700 * q^6 - 465666872 * q^7 + 3839876544 * q^8 + 10460353203 * q^9 - 167336332556 * q^11 - 74661555600 * q^12 + 545571033878 * q^13 - 1568858902656 * q^14 + 255267954944 * q^16 + 8104424487194 * q^17 + 8019604122300 * q^18 + 3937700740828 * q^19 + 27497163124728 * q^21 - 114198109969712 * q^22 + 156235274730744 * q^23 - 226740870046656 * q^24 + 2972936042645656 * q^26 - 617673396283947 * q^27 - 3826676848454144 * q^28 - 930273612785494 * q^29 - 6257709152718928 * q^31 - 4297283007276032 * q^32 + 9881043101099244 * q^33 - 20587982883981400 * q^34 + 4408690196624400 * q^36 + 22246337613227118 * q^37 + 82202958592557968 * q^38 - 32215423979462022 * q^39 - 186265908060974338 * q^41 + 92639549342934144 * q^42 + 268609288174096316 * q^43 - 284579440966881088 * q^44 + 375824751838355712 * q^46 + 900034127817222032 * q^47 - 15073317471488256 * q^48 - 586354193149814837 * q^49 - 478558161544318506 * q^51 + 3462564631549962144 * q^52 + 1269623243180583374 * q^53 - 473549603817692700 * q^54 - 5236064873787955200 * q^56 - 232517291045152572 * q^57 + 1091617398834773576 * q^58 - 8551487099411338268 * q^59 - 7181148471323735222 * q^61 + 3050004306382316256 * q^62 - 1623679985352063672 * q^63 - 557099647389855744 * q^64 + 6743284195601523888 * q^66 - 2946635148405656396 * q^67 - 10323921738190858400 * q^68 - 9225536737575702456 * q^69 - 37849731561832987624 * q^71 + 13388821635384990144 * q^72 + 7149058835824826594 * q^73 + 50872537645745842424 * q^74 + 200886445168773700032 * q^76 + 110970090560628236256 * q^77 - 175548900382183341144 * q^78 + 309682369826312504000 * q^79 + 36472996377170786403 * q^81 - 265773545783653420264 * q^82 - 199780825758268631580 * q^83 + 225961441224368749056 * q^84 + 267144820042737104944 * q^86 + 54931726561370635206 * q^87 - 749649166937994283776 * q^88 + 162380735751340085838 * q^89 - 558457704043459369328 * q^91 + 1333721831719821170688 * q^92 + 369511467758899979472 * q^93 - 149583353667227862112 * q^94 + 253750264296642413568 * q^96 + 1208943423358924299354 * q^97 + 876777377902285855132 * q^98 - 583465714076809258956 * 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 $

$2$	−999.805	−0.690399	−0.345200	−	0.938529i	$-0.612189\pi$
$2$	−999.805	−0.690399	−0.345200	+	0.938529i	$0.612189\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	−1.09754e6	−0.523349
$5$	0	0
$5$	0	0
$6$	5.90375e7	0.398602
$7$	−9.82313e7	−0.131438	−0.0657189	−	0.997838i	$-0.520934\pi$
$7$	−9.82313e7	−0.131438	−0.0657189	+	0.997838i	$0.520934\pi$
$8$	3.19407e9	1.05172
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−9.60815e10	−1.11691	−0.558453	−	0.829536i	$-0.688605\pi$
$11$	−9.60815e10	−1.11691	−0.558453	+	0.829536i	$0.688605\pi$
$12$	6.48088e10	0.302156
$13$	−6.22052e11	−1.25147	−0.625736	−	0.780035i	$-0.715201\pi$
$13$	−6.22052e11	−1.25147	−0.625736	+	0.780035i	$0.715201\pi$
$14$	9.82121e10	0.0907446
$15$	0	0
$16$	−8.91733e11	−0.202757
$17$	1.42559e13	1.71507	0.857536	−	0.514425i	$-0.171994\pi$
$17$	1.42559e13	1.71507	0.857536	+	0.514425i	$0.171994\pi$
$18$	−3.48610e12	−0.230133
$19$	−8.28760e12	−0.310110	−0.155055	−	0.987906i	$-0.549555\pi$
$19$	−8.28760e12	−0.310110	−0.155055	+	0.987906i	$0.549555\pi$
$20$	0	0
$21$	5.80046e12	0.0758857
$22$	9.60627e13	0.771111
$23$	1.17504e14	0.591440	0.295720	−	0.955275i	$-0.404440\pi$
$23$	1.17504e14	0.591440	0.295720	+	0.955275i	$0.404440\pi$
$24$	−1.88607e14	−0.607210
$25$	0	0
$26$	6.21930e14	0.864015
$27$	−2.05891e14	−0.192450
$28$	1.07813e14	0.0687879
$29$	−1.98798e15	−0.877471	−0.438735	−	0.898616i	$-0.644573\pi$
$29$	−1.98798e15	−0.877471	−0.438735	+	0.898616i	$0.644573\pi$
$30$	0	0
$31$	−5.31766e15	−1.16526	−0.582629	−	0.812738i	$-0.697976\pi$
$31$	−5.31766e15	−1.16526	−0.582629	+	0.812738i	$0.697976\pi$
$32$	−5.80689e15	−0.911736
$33$	5.67352e15	0.644846
$34$	−1.42532e16	−1.18408
$35$	0	0
$36$	−3.82689e15	−0.174450
$37$	−1.99479e16	−0.681992	−0.340996	−	0.940065i	$-0.610764\pi$
$37$	−1.99479e16	−0.681992	−0.340996	+	0.940065i	$0.610764\pi$
$38$	8.28598e15	0.214100
$39$	3.67315e16	0.722538
$40$	0	0
$41$	−2.99615e16	−0.348604	−0.174302	−	0.984692i	$-0.555767\pi$
$41$	−2.99615e16	−0.348604	−0.174302	+	0.984692i	$0.555767\pi$
$42$	−5.79933e15	−0.0523914
$43$	3.44307e16	0.242956	0.121478	−	0.992594i	$-0.461237\pi$
$43$	3.44307e16	0.242956	0.121478	+	0.992594i	$0.461237\pi$
$44$	1.05454e17	0.584532
$45$	0	0
$46$	−1.17481e17	−0.408330
$47$	5.23092e17	1.45061	0.725305	−	0.688428i	$-0.241699\pi$
$47$	5.23092e17	1.45061	0.725305	+	0.688428i	$0.241699\pi$
$48$	5.26560e16	0.117062
$49$	−5.48896e17	−0.982724
$50$	0	0
$51$	−8.41799e17	−0.990197
$52$	6.82728e17	0.654957
$53$	1.54033e18	1.20981	0.604904	−	0.796298i	$-0.293211\pi$
$53$	1.54033e18	1.20981	0.604904	+	0.796298i	$0.293211\pi$
$54$	2.05851e17	0.132867
$55$	0	0
$56$	−3.13758e17	−0.138236
$57$	4.89374e17	0.179042
$58$	1.98759e18	0.605805
$59$	−6.81914e17	−0.173693	−0.0868467	−	0.996222i	$-0.527679\pi$
$59$	−6.81914e17	−0.173693	−0.0868467	+	0.996222i	$0.527679\pi$
$60$	0	0
$61$	−3.28598e18	−0.589796	−0.294898	−	0.955529i	$-0.595286\pi$
$61$	−3.28598e18	−0.589796	−0.294898	+	0.955529i	$0.595286\pi$
$62$	5.31662e18	0.804494
$63$	−3.42511e17	−0.0438126
$64$	7.67586e18	0.832218
$65$	0	0
$66$	−5.67241e18	−0.445201
$67$	−1.07261e19	−0.718882	−0.359441	−	0.933168i	$-0.617033\pi$
$67$	−1.07261e19	−0.718882	−0.359441	+	0.933168i	$0.617033\pi$
$68$	−1.56465e19	−0.897581
$69$	−6.93850e18	−0.341468
$70$	0	0
$71$	3.71838e19	1.35563	0.677815	−	0.735232i	$-0.262927\pi$
$71$	3.71838e19	1.35563	0.677815	+	0.735232i	$0.262927\pi$
$72$	1.11370e19	0.350573
$73$	6.79746e19	1.85121	0.925607	−	0.378487i	$-0.123555\pi$
$73$	6.79746e19	1.85121	0.925607	+	0.378487i	$0.123555\pi$
$74$	1.99440e19	0.470847
$75$	0	0
$76$	9.09599e18	0.162296
$77$	9.43821e18	0.146804
$78$	−3.67244e19	−0.498839
$79$	1.28759e20	1.53001	0.765004	−	0.644026i	$-0.222737\pi$
$79$	1.28759e20	1.53001	0.765004	+	0.644026i	$0.222737\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	2.99556e19	0.240676
$83$	−1.24075e20	−0.877738	−0.438869	−	0.898551i	$-0.644621\pi$
$83$	−1.24075e20	−0.877738	−0.438869	+	0.898551i	$0.644621\pi$
$84$	−6.36625e18	−0.0397147
$85$	0	0
$86$	−3.44240e19	−0.167736
$87$	1.17388e20	0.506608
$88$	−3.06891e20	−1.17467
$89$	4.94950e20	1.68254	0.841271	−	0.540613i	$-0.181808\pi$
$89$	4.94950e20	1.68254	0.841271	+	0.540613i	$0.181808\pi$
$90$	0	0
$91$	6.11049e19	0.164491
$92$	−1.28966e20	−0.309530
$93$	3.14002e20	0.672763
$94$	−5.22990e20	−1.00150
$95$	0	0
$96$	3.42891e20	0.526391
$97$	−4.69497e20	−0.646443	−0.323221	−	0.946323i	$-0.604766\pi$
$97$	−4.69497e20	−0.646443	−0.323221	+	0.946323i	$0.604766\pi$
$98$	5.48789e20	0.678472
$99$	−3.35015e20	−0.372302
$100$	0	0
$101$	5.69917e20	0.513378	0.256689	−	0.966494i	$-0.417368\pi$
$101$	5.69917e20	0.513378	0.256689	+	0.966494i	$0.417368\pi$
$102$	8.41635e20	0.683631
$103$	−6.53842e19	−0.0479382	−0.0239691	−	0.999713i	$-0.507630\pi$
$103$	−6.53842e19	−0.0479382	−0.0239691	+	0.999713i	$0.507630\pi$
$104$	−1.98688e21	−1.31620
$105$	0	0
$106$	−1.54003e21	−0.835250
$107$	1.83068e21	0.899669	0.449834	−	0.893112i	$-0.351483\pi$
$107$	1.83068e21	0.899669	0.449834	+	0.893112i	$0.351483\pi$
$108$	2.25974e20	0.100719
$109$	2.30722e21	0.933491	0.466745	−	0.884392i	$-0.345426\pi$
$109$	2.30722e21	0.933491	0.466745	+	0.884392i	$0.345426\pi$
$110$	0	0
$111$	1.17790e21	0.393748
$112$	8.75961e19	0.0266499
$113$	4.46490e21	1.23734	0.618668	−	0.785652i	$-0.287673\pi$
$113$	4.46490e21	1.23734	0.618668	+	0.785652i	$0.287673\pi$
$114$	−4.89279e20	−0.123611
$115$	0	0
$116$	2.18189e21	0.459223
$117$	−2.16896e21	−0.417157
$118$	6.81781e20	0.119918
$119$	−1.40038e21	−0.225425
$120$	0	0
$121$	1.83140e21	0.247479
$122$	3.28534e21	0.407195
$123$	1.76919e21	0.201267
$124$	5.83635e21	0.609837
$125$	0	0
$126$	3.42445e20	0.0302482
$127$	−2.21160e22	−1.79791	−0.898955	−	0.438040i	$-0.855673\pi$
$127$	−2.21160e22	−1.79791	−0.898955	+	0.438040i	$0.855673\pi$
$128$	4.50358e21	0.337173
$129$	−2.03310e21	−0.140271
$130$	0	0
$131$	7.93939e21	0.466057	0.233028	−	0.972470i	$-0.425137\pi$
$131$	7.93939e21	0.466057	0.233028	+	0.972470i	$0.425137\pi$
$132$	−6.22693e21	−0.337480
$133$	8.14102e20	0.0407602
$134$	1.07240e22	0.496315
$135$	0	0
$136$	4.55345e22	1.80377
$137$	−1.87691e22	−0.688457	−0.344229	−	0.938886i	$-0.611860\pi$
$137$	−1.87691e22	−0.688457	−0.344229	+	0.938886i	$0.611860\pi$
$138$	6.93715e21	0.235749
$139$	6.17628e21	0.194568	0.0972841	−	0.995257i	$-0.468984\pi$
$139$	6.17628e21	0.194568	0.0972841	+	0.995257i	$0.468984\pi$
$140$	0	0
$141$	−3.08880e22	−0.837510
$142$	−3.71766e22	−0.935926
$143$	5.97676e22	1.39778
$144$	−3.10928e21	−0.0675856
$145$	0	0
$146$	−6.79613e22	−1.27808
$147$	3.24118e22	0.567376
$148$	2.18937e22	0.356920
$149$	−1.87086e22	−0.284175	−0.142088	−	0.989854i	$-0.545381\pi$
$149$	−1.87086e22	−0.284175	−0.142088	+	0.989854i	$0.545381\pi$
$150$	0	0
$151$	9.39155e22	1.24016	0.620082	−	0.784537i	$-0.287099\pi$
$151$	9.39155e22	1.24016	0.620082	+	0.784537i	$0.287099\pi$
$152$	−2.64712e22	−0.326149
$153$	4.97074e22	0.571690
$154$	−9.43637e21	−0.101353
$155$	0	0
$156$	−4.03144e22	−0.378140
$157$	5.10719e22	0.447956	0.223978	−	0.974594i	$-0.428096\pi$
$157$	5.10719e22	0.447956	0.223978	+	0.974594i	$0.428096\pi$
$158$	−1.28734e23	−1.05632
$159$	−9.09548e22	−0.698483
$160$	0	0
$161$	−1.15426e22	−0.0777377
$162$	−1.21553e22	−0.0767110
$163$	1.61660e22	0.0956386	0.0478193	−	0.998856i	$-0.484773\pi$
$163$	1.61660e22	0.0956386	0.0478193	+	0.998856i	$0.484773\pi$
$164$	3.28840e22	0.182442
$165$	0	0
$166$	1.24051e23	0.605989
$167$	2.32624e23	1.06692	0.533459	−	0.845826i	$-0.320892\pi$
$167$	2.32624e23	1.06692	0.533459	+	0.845826i	$0.320892\pi$
$168$	1.85271e22	0.0798104
$169$	1.39884e23	0.566183
$170$	0	0
$171$	−2.88971e22	−0.103370
$172$	−3.77892e22	−0.127151
$173$	1.84508e23	0.584158	0.292079	−	0.956394i	$-0.405653\pi$
$173$	1.84508e23	0.584158	0.292079	+	0.956394i	$0.405653\pi$
$174$	−1.17365e23	−0.349762
$175$	0	0
$176$	8.56791e22	0.226460
$177$	4.02663e22	0.100282
$178$	−4.94853e23	−1.16163
$179$	4.64644e22	0.102840	0.0514202	−	0.998677i	$-0.483625\pi$
$179$	4.64644e22	0.102840	0.0514202	+	0.998677i	$0.483625\pi$
$180$	0	0
$181$	−9.44640e23	−1.86055	−0.930275	−	0.366863i	$-0.880432\pi$
$181$	−9.44640e23	−1.86055	−0.930275	+	0.366863i	$0.880432\pi$
$182$	−6.10930e22	−0.113564
$183$	1.94034e23	0.340519
$184$	3.75317e23	0.622029
$185$	0	0
$186$	−3.13941e23	−0.464475
$187$	−1.36973e24	−1.91557
$188$	−5.74116e23	−0.759175
$189$	2.02250e22	0.0252952
$190$	0	0
$191$	−8.05978e23	−0.902554	−0.451277	−	0.892384i	$-0.649031\pi$
$191$	−8.05978e23	−0.902554	−0.451277	+	0.892384i	$0.649031\pi$
$192$	−4.53252e23	−0.480481
$193$	−1.40323e24	−1.40857	−0.704283	−	0.709919i	$-0.748731\pi$
$193$	−1.40323e24	−1.40857	−0.704283	+	0.709919i	$0.748731\pi$
$194$	4.69406e23	0.446303
$195$	0	0
$196$	6.02437e23	0.514308
$197$	1.78218e24	1.44230	0.721152	−	0.692777i	$-0.243613\pi$
$197$	1.78218e24	1.44230	0.721152	+	0.692777i	$0.243613\pi$
$198$	3.34950e23	0.257037
$199$	1.01185e24	0.736478	0.368239	−	0.929731i	$-0.379961\pi$
$199$	1.01185e24	0.736478	0.368239	+	0.929731i	$0.379961\pi$
$200$	0	0
$201$	6.33367e23	0.415047
$202$	−5.69806e23	−0.354436
$203$	1.95282e23	0.115333
$204$	9.23910e23	0.518219
$205$	0	0
$206$	6.53714e22	0.0330965
$207$	4.09712e23	0.197147
$208$	5.54704e23	0.253744
$209$	7.96285e23	0.346364
$210$	0	0
$211$	−4.73938e24	−1.86533	−0.932664	−	0.360745i	$-0.882522\pi$
$211$	−4.73938e24	−1.86533	−0.932664	+	0.360745i	$0.882522\pi$
$212$	−1.69057e24	−0.633152
$213$	−2.19567e24	−0.782674
$214$	−1.83032e24	−0.621130
$215$	0	0
$216$	−6.57631e23	−0.202403
$217$	5.22360e23	0.153159
$218$	−2.30677e24	−0.644481
$219$	−4.01383e24	−1.06880
$220$	0	0
$221$	−8.86793e24	−2.14636
$222$	−1.17767e24	−0.271843
$223$	7.74634e24	1.70567	0.852836	−	0.522179i	$-0.174881\pi$
$223$	7.74634e24	1.70567	0.852836	+	0.522179i	$0.174881\pi$
$224$	5.70419e23	0.119837
$225$	0	0
$226$	−4.46402e24	−0.854256
$227$	−6.99617e24	−1.27817	−0.639085	−	0.769136i	$-0.720687\pi$
$227$	−6.99617e24	−1.27817	−0.639085	+	0.769136i	$0.720687\pi$
$228$	−5.37109e23	−0.0937015
$229$	−4.76320e24	−0.793645	−0.396822	−	0.917895i	$-0.629887\pi$
$229$	−4.76320e24	−0.793645	−0.396822	+	0.917895i	$0.629887\pi$
$230$	0	0
$231$	−5.57317e23	−0.0847572
$232$	−6.34974e24	−0.922852
$233$	−2.30237e24	−0.319843	−0.159922	−	0.987130i	$-0.551124\pi$
$233$	−2.30237e24	−0.319843	−0.159922	+	0.987130i	$0.551124\pi$
$234$	2.16854e24	0.288005
$235$	0	0
$236$	7.48429e23	0.0909023
$237$	−7.60310e24	−0.883350
$238$	1.40011e24	0.155633
$239$	4.40816e24	0.468899	0.234450	−	0.972128i	$-0.424671\pi$
$239$	4.40816e24	0.468899	0.234450	+	0.972128i	$0.424671\pi$
$240$	0	0
$241$	−2.88573e24	−0.281240	−0.140620	−	0.990064i	$-0.544910\pi$
$241$	−2.88573e24	−0.281240	−0.140620	+	0.990064i	$0.544910\pi$
$242$	−1.83105e24	−0.170859
$243$	−7.17898e23	−0.0641500
$244$	3.60650e24	0.308669
$245$	0	0
$246$	−1.76885e24	−0.138954
$247$	5.15531e24	0.388094
$248$	−1.69850e25	−1.22552
$249$	7.32651e24	0.506762
$250$	0	0
$251$	6.06385e24	0.385633	0.192817	−	0.981235i	$-0.438238\pi$
$251$	6.06385e24	0.385633	0.192817	+	0.981235i	$0.438238\pi$
$252$	3.75921e23	0.0229293
$253$	−1.12900e25	−0.660583
$254$	2.21117e25	1.24128
$255$	0	0
$256$	−2.06001e25	−1.06500
$257$	−3.42813e25	−1.70122	−0.850608	−	0.525800i	$-0.823766\pi$
$257$	−3.42813e25	−1.70122	−0.850608	+	0.525800i	$0.823766\pi$
$258$	2.03270e24	0.0968426
$259$	1.95951e24	0.0896396
$260$	0	0
$261$	−6.93165e24	−0.292490
$262$	−7.93784e24	−0.321765
$263$	3.93877e25	1.53400	0.767000	−	0.641647i	$-0.221748\pi$
$263$	3.93877e25	1.53400	0.767000	+	0.641647i	$0.221748\pi$
$264$	1.81216e25	0.678197
$265$	0	0
$266$	−8.13943e23	−0.0281408
$267$	−2.92263e25	−0.971416
$268$	1.17724e25	0.376226
$269$	1.18456e25	0.364049	0.182024	−	0.983294i	$-0.441735\pi$
$269$	1.18456e25	0.364049	0.182024	+	0.983294i	$0.441735\pi$
$270$	0	0
$271$	−1.63162e25	−0.463918	−0.231959	−	0.972726i	$-0.574514\pi$
$271$	−1.63162e25	−0.463918	−0.231959	+	0.972726i	$0.574514\pi$
$272$	−1.27125e25	−0.347742
$273$	−3.60819e24	−0.0949688
$274$	1.87654e25	0.475310
$275$	0	0
$276$	7.61530e24	0.178707
$277$	−2.06084e24	−0.0465594	−0.0232797	−	0.999729i	$-0.507411\pi$
$277$	−2.06084e24	−0.0465594	−0.0232797	+	0.999729i	$0.507411\pi$
$278$	−6.17508e24	−0.134330
$279$	−1.85415e25	−0.388420
$280$	0	0
$281$	−8.23966e25	−1.60138	−0.800688	−	0.599082i	$-0.795532\pi$
$281$	−8.23966e25	−1.60138	−0.800688	+	0.599082i	$0.795532\pi$
$282$	3.08820e25	0.578216
$283$	6.15303e25	1.11002	0.555011	−	0.831843i	$-0.312714\pi$
$283$	6.15303e25	1.11002	0.555011	+	0.831843i	$0.312714\pi$
$284$	−4.08108e25	−0.709468
$285$	0	0
$286$	−5.97560e25	−0.965024
$287$	2.94315e24	0.0458198
$288$	−2.02474e25	−0.303912
$289$	1.34140e26	1.94147
$290$	0	0
$291$	2.77234e25	0.373224
$292$	−7.46050e25	−0.968831
$293$	−2.98062e24	−0.0373419	−0.0186709	−	0.999826i	$-0.505943\pi$
$293$	−2.98062e24	−0.0373419	−0.0186709	+	0.999826i	$0.505943\pi$
$294$	−3.24055e25	−0.391716
$295$	0	0
$296$	−6.37150e25	−0.717264
$297$	1.97823e25	0.214949
$298$	1.87050e25	0.196194
$299$	−7.30936e25	−0.740171
$300$	0	0
$301$	−3.38217e24	−0.0319336
$302$	−9.38972e25	−0.856208
$303$	−3.36530e25	−0.296399
$304$	7.39033e24	0.0628769
$305$	0	0
$306$	−4.96977e25	−0.394695
$307$	−1.83958e26	−1.41177	−0.705886	−	0.708325i	$-0.749451\pi$
$307$	−1.83958e26	−1.41177	−0.705886	+	0.708325i	$0.749451\pi$
$308$	−1.03588e25	−0.0768296
$309$	3.86087e24	0.0276771
$310$	0	0
$311$	−1.01647e26	−0.680944	−0.340472	−	0.940255i	$-0.610587\pi$
$311$	−1.01647e26	−0.680944	−0.340472	+	0.940255i	$0.610587\pi$
$312$	1.17323e26	0.759907
$313$	−2.15822e26	−1.35170	−0.675850	−	0.737040i	$-0.736223\pi$
$313$	−2.15822e26	−1.35170	−0.675850	+	0.737040i	$0.736223\pi$
$314$	−5.10619e25	−0.309269
$315$	0	0
$316$	−1.41319e26	−0.800728
$317$	−1.10445e26	−0.605376	−0.302688	−	0.953090i	$-0.597884\pi$
$317$	−1.10445e26	−0.605376	−0.302688	+	0.953090i	$0.597884\pi$
$318$	9.09370e25	0.482232
$319$	1.91008e26	0.980052
$320$	0	0
$321$	−1.08100e26	−0.519424
$322$	1.15403e25	0.0536700
$323$	−1.18147e26	−0.531861
$324$	−1.33436e25	−0.0581499
$325$	0	0
$326$	−1.61629e25	−0.0660288
$327$	−1.36239e26	−0.538951
$328$	−9.56990e25	−0.366634
$329$	−5.13840e25	−0.190665
$330$	0	0
$331$	2.13586e26	0.743668	0.371834	−	0.928299i	$-0.378729\pi$
$331$	2.13586e26	0.743668	0.371834	+	0.928299i	$0.378729\pi$
$332$	1.36178e26	0.459363
$333$	−6.95540e25	−0.227331
$334$	−2.32578e26	−0.736599
$335$	0	0
$336$	−5.17246e24	−0.0153863
$337$	−1.22428e26	−0.352994	−0.176497	−	0.984301i	$-0.556477\pi$
$337$	−1.22428e26	−0.352994	−0.176497	+	0.984301i	$0.556477\pi$
$338$	−1.39856e26	−0.390892
$339$	−2.63648e26	−0.714376
$340$	0	0
$341$	5.10928e26	1.30148
$342$	2.88914e25	0.0713666
$343$	1.08786e26	0.260605
$344$	1.09974e26	0.255521
$345$	0	0
$346$	−1.84472e26	−0.403302
$347$	−3.11301e26	−0.660270	−0.330135	−	0.943934i	$-0.607094\pi$
$347$	−3.11301e26	−0.660270	−0.330135	+	0.943934i	$0.607094\pi$
$348$	−1.28838e26	−0.265133
$349$	7.85942e26	1.56936	0.784681	−	0.619899i	$-0.212827\pi$
$349$	7.85942e26	1.56936	0.784681	+	0.619899i	$0.212827\pi$
$350$	0	0
$351$	1.28075e26	0.240846
$352$	5.57935e26	1.01832
$353$	1.02640e26	0.181837	0.0909185	−	0.995858i	$-0.471020\pi$
$353$	1.02640e26	0.181837	0.0909185	+	0.995858i	$0.471020\pi$
$354$	−4.02585e25	−0.0692346
$355$	0	0
$356$	−5.43228e26	−0.880557
$357$	8.26910e25	0.130149
$358$	−4.64553e25	−0.0710009
$359$	−9.30102e26	−1.38051	−0.690254	−	0.723568i	$-0.742501\pi$
$359$	−9.30102e26	−1.38051	−0.690254	+	0.723568i	$0.742501\pi$
$360$	0	0
$361$	−6.45525e26	−0.903832
$362$	9.44455e26	1.28452
$363$	−1.08143e26	−0.142882
$364$	−6.70653e25	−0.0860861
$365$	0	0
$366$	−1.93996e26	−0.235094
$367$	1.27277e27	1.49884	0.749422	−	0.662093i	$-0.230332\pi$
$367$	1.27277e27	1.49884	0.749422	+	0.662093i	$0.230332\pi$
$368$	−1.04782e26	−0.119918
$369$	−1.04469e26	−0.116201
$370$	0	0
$371$	−1.51308e26	−0.159015
$372$	−3.44631e26	−0.352090
$373$	1.92075e27	1.90778	0.953889	−	0.300159i	$-0.0970397\pi$
$373$	1.92075e27	1.90778	0.953889	+	0.300159i	$0.0970397\pi$
$374$	1.36946e27	1.32251
$375$	0	0
$376$	1.67079e27	1.52563
$377$	1.23662e27	1.09813
$378$	−2.02210e25	−0.0174638
$379$	4.90168e26	0.411750	0.205875	−	0.978578i	$-0.433996\pi$
$379$	4.90168e26	0.411750	0.205875	+	0.978578i	$0.433996\pi$
$380$	0	0
$381$	1.30593e27	1.03802
$382$	8.05821e26	0.623122
$383$	−2.45414e26	−0.184634	−0.0923170	−	0.995730i	$-0.529427\pi$
$383$	−2.45414e26	−0.184634	−0.0923170	+	0.995730i	$0.529427\pi$
$384$	−2.65932e26	−0.194667
$385$	0	0
$386$	1.40296e27	0.972473
$387$	1.20052e26	0.0809852
$388$	5.15293e26	0.338315
$389$	−1.02936e27	−0.657802	−0.328901	−	0.944364i	$-0.606678\pi$
$389$	−1.02936e27	−0.657802	−0.328901	+	0.944364i	$0.606678\pi$
$390$	0	0
$391$	1.67513e27	1.01436
$392$	−1.75321e27	−1.03355
$393$	−4.68813e26	−0.269078
$394$	−1.78184e27	−0.995766
$395$	0	0
$396$	3.67694e26	0.194844
$397$	−1.36083e27	−0.702270	−0.351135	−	0.936325i	$-0.614204\pi$
$397$	−1.36083e27	−0.702270	−0.351135	+	0.936325i	$0.614204\pi$
$398$	−1.01165e27	−0.508464
$399$	−4.80719e25	−0.0235329
$400$	0	0
$401$	3.27496e27	1.52121	0.760606	−	0.649214i	$-0.224902\pi$
$401$	3.27496e27	1.52121	0.760606	+	0.649214i	$0.224902\pi$
$402$	−6.33243e26	−0.286548
$403$	3.30786e27	1.45829
$404$	−6.25508e26	−0.268676
$405$	0	0
$406$	−1.95244e26	−0.0796257
$407$	1.91662e27	0.761721
$408$	−2.68877e27	−1.04141
$409$	−6.11720e26	−0.230918	−0.115459	−	0.993312i	$-0.536834\pi$
$409$	−6.11720e26	−0.230918	−0.115459	+	0.993312i	$0.536834\pi$
$410$	0	0
$411$	1.10830e27	0.397481
$412$	7.17619e25	0.0250884
$413$	6.69853e25	0.0228299
$414$	−4.09632e26	−0.136110
$415$	0	0
$416$	3.61219e27	1.14101
$417$	−3.64703e26	−0.112334
$418$	−7.96129e26	−0.239129
$419$	−1.85944e27	−0.544673	−0.272336	−	0.962202i	$-0.587796\pi$
$419$	−1.85944e27	−0.544673	−0.272336	+	0.962202i	$0.587796\pi$
$420$	0	0
$421$	−2.61982e27	−0.729976	−0.364988	−	0.931012i	$-0.618927\pi$
$421$	−2.61982e27	−0.729976	−0.364988	+	0.931012i	$0.618927\pi$
$422$	4.73845e27	1.28782
$423$	1.82391e27	0.483536
$424$	4.91991e27	1.27238
$425$	0	0
$426$	2.19524e27	0.540357
$427$	3.22786e26	0.0775215
$428$	−2.00925e27	−0.470841
$429$	−3.52922e27	−0.807007
$430$	0	0
$431$	8.46324e27	1.84300	0.921500	−	0.388378i	$-0.126965\pi$
$431$	8.46324e27	1.84300	0.921500	+	0.388378i	$0.126965\pi$
$432$	1.83600e26	0.0390205
$433$	7.14150e27	1.48138	0.740690	−	0.671847i	$-0.234499\pi$
$433$	7.14150e27	1.48138	0.740690	+	0.671847i	$0.234499\pi$
$434$	−5.22258e26	−0.105741
$435$	0	0
$436$	−2.53227e27	−0.488542
$437$	−9.73827e26	−0.183412
$438$	4.01305e27	0.737898
$439$	−2.98385e27	−0.535672	−0.267836	−	0.963464i	$-0.586309\pi$
$439$	−2.98385e27	−0.535672	−0.267836	+	0.963464i	$0.586309\pi$
$440$	0	0
$441$	−1.91388e27	−0.327575
$442$	8.86620e27	1.48185
$443$	−1.36610e27	−0.222969	−0.111484	−	0.993766i	$-0.535560\pi$
$443$	−1.36610e27	−0.222969	−0.111484	+	0.993766i	$0.535560\pi$
$444$	−1.29280e27	−0.206068
$445$	0	0
$446$	−7.74483e27	−1.17759
$447$	1.10473e27	0.164069
$448$	−7.54010e26	−0.109385
$449$	−5.56569e27	−0.788737	−0.394369	−	0.918952i	$-0.629037\pi$
$449$	−5.56569e27	−0.788737	−0.394369	+	0.918952i	$0.629037\pi$
$450$	0	0
$451$	2.87874e27	0.389358
$452$	−4.90041e27	−0.647559
$453$	−5.54562e27	−0.716009
$454$	6.99480e27	0.882448
$455$	0	0
$456$	1.56310e27	0.188302
$457$	−3.92206e27	−0.461736	−0.230868	−	0.972985i	$-0.574157\pi$
$457$	−3.92206e27	−0.461736	−0.230868	+	0.972985i	$0.574157\pi$
$458$	4.76227e27	0.547932
$459$	−2.93517e27	−0.330066
$460$	0	0
$461$	−1.51086e28	−1.62317	−0.811585	−	0.584235i	$-0.801395\pi$
$461$	−1.51086e28	−1.62317	−0.811585	+	0.584235i	$0.801395\pi$
$462$	5.57208e26	0.0585163
$463$	−6.41081e27	−0.658131	−0.329066	−	0.944307i	$-0.606734\pi$
$463$	−6.41081e27	−0.658131	−0.329066	+	0.944307i	$0.606734\pi$
$464$	1.77275e27	0.177913
$465$	0	0
$466$	2.30192e27	0.220820
$467$	2.82682e27	0.265137	0.132569	−	0.991174i	$-0.457678\pi$
$467$	2.82682e27	0.265137	0.132569	+	0.991174i	$0.457678\pi$
$468$	2.38053e27	0.218319
$469$	1.05364e27	0.0944883
$470$	0	0
$471$	−3.01574e27	−0.258628
$472$	−2.17808e27	−0.182677
$473$	−3.30815e27	−0.271359
$474$	7.60161e27	0.609864
$475$	0	0
$476$	1.53698e27	0.117976
$477$	5.37079e27	0.403269
$478$	−4.40730e27	−0.323728
$479$	8.65614e27	0.622016	0.311008	−	0.950407i	$-0.399333\pi$
$479$	8.65614e27	0.622016	0.311008	+	0.950407i	$0.399333\pi$
$480$	0	0
$481$	1.24086e28	0.853494
$482$	2.88517e27	0.194168
$483$	6.81578e26	0.0448819
$484$	−2.01004e27	−0.129518
$485$	0	0
$486$	7.17758e26	0.0442891
$487$	3.37782e27	0.203978	0.101989	−	0.994786i	$-0.467479\pi$
$487$	3.37782e27	0.203978	0.101989	+	0.994786i	$0.467479\pi$
$488$	−1.04957e28	−0.620299
$489$	−9.54588e26	−0.0552170
$490$	0	0
$491$	−2.94209e28	−1.63042	−0.815212	−	0.579163i	$-0.803379\pi$
$491$	−2.94209e28	−1.63042	−0.815212	+	0.579163i	$0.803379\pi$
$492$	−1.94177e27	−0.105333
$493$	−2.83405e28	−1.50492
$494$	−5.15431e27	−0.267940
$495$	0	0
$496$	4.74193e27	0.236264
$497$	−3.65262e27	−0.178181
$498$	−7.32508e27	−0.349868
$499$	3.12544e28	1.46169	0.730846	−	0.682543i	$-0.239126\pi$
$499$	3.12544e28	1.46169	0.730846	+	0.682543i	$0.239126\pi$
$500$	0	0
$501$	−1.37362e28	−0.615985
$502$	−6.06267e27	−0.266241
$503$	−1.52567e28	−0.656140	−0.328070	−	0.944653i	$-0.606398\pi$
$503$	−1.52567e28	−0.656140	−0.328070	+	0.944653i	$0.606398\pi$
$504$	−1.09401e27	−0.0460786
$505$	0	0
$506$	1.12878e28	0.456066
$507$	−8.25999e27	−0.326886
$508$	2.42733e28	0.940935
$509$	−2.89297e28	−1.09852	−0.549259	−	0.835652i	$-0.685090\pi$
$509$	−2.89297e28	−1.09852	−0.549259	+	0.835652i	$0.685090\pi$
$510$	0	0
$511$	−6.67723e27	−0.243320
$512$	1.11514e28	0.398104
$513$	1.70634e27	0.0596807
$514$	3.42746e28	1.17452
$515$	0	0
$516$	2.23141e27	0.0734104
$517$	−5.02594e28	−1.62019
$518$	−1.95913e27	−0.0618871
$519$	−1.08950e28	−0.337264
$520$	0	0
$521$	−1.88105e28	−0.559247	−0.279623	−	0.960110i	$-0.590210\pi$
$521$	−1.88105e28	−0.559247	−0.279623	+	0.960110i	$0.590210\pi$
$522$	6.93030e27	0.201935
$523$	2.81334e28	0.803443	0.401722	−	0.915762i	$-0.368412\pi$
$523$	2.81334e28	0.803443	0.401722	+	0.915762i	$0.368412\pi$
$524$	−8.71382e27	−0.243910
$525$	0	0
$526$	−3.93800e28	−1.05907
$527$	−7.58082e28	−1.99850
$528$	−5.05926e27	−0.130747
$529$	−2.56644e28	−0.650198
$530$	0	0
$531$	−2.37769e27	−0.0578978
$532$	−8.93511e26	−0.0213318
$533$	1.86376e28	0.436268
$534$	2.92206e28	0.670665
$535$	0	0
$536$	−3.42600e28	−0.756061
$537$	−2.74368e27	−0.0593749
$538$	−1.18433e28	−0.251339
$539$	5.27388e28	1.09761
$540$	0	0
$541$	−8.42394e28	−1.68634	−0.843168	−	0.537650i	$-0.819312\pi$
$541$	−8.42394e28	−1.68634	−0.843168	+	0.537650i	$0.819312\pi$
$542$	1.63130e28	0.320289
$543$	5.57800e28	1.07419
$544$	−8.27827e28	−1.56369
$545$	0	0
$546$	3.60748e27	0.0655664
$547$	−5.96331e28	−1.06321	−0.531607	−	0.846991i	$-0.678412\pi$
$547$	−5.96331e28	−1.06321	−0.531607	+	0.846991i	$0.678412\pi$
$548$	2.05999e28	0.360304
$549$	−1.14575e28	−0.196599
$550$	0	0
$551$	1.64756e28	0.272112
$552$	−2.21621e28	−0.359129
$553$	−1.26482e28	−0.201101
$554$	2.06044e27	0.0321445
$555$	0	0
$556$	−6.77873e27	−0.101827
$557$	−2.67043e28	−0.393642	−0.196821	−	0.980439i	$-0.563062\pi$
$557$	−2.67043e28	−0.393642	−0.196821	+	0.980439i	$0.563062\pi$
$558$	1.85379e28	0.268165
$559$	−2.14177e28	−0.304052
$560$	0	0
$561$	8.08813e28	1.10596
$562$	8.23805e28	1.10559
$563$	2.15210e28	0.283482	0.141741	−	0.989904i	$-0.454730\pi$
$563$	2.15210e28	0.283482	0.141741	+	0.989904i	$0.454730\pi$
$564$	3.39009e28	0.438310
$565$	0	0
$566$	−6.15183e28	−0.766358
$567$	−1.19426e27	−0.0146042
$568$	1.18768e29	1.42574
$569$	6.95667e28	0.819827	0.409914	−	0.912124i	$-0.365559\pi$
$569$	6.95667e28	0.819827	0.409914	+	0.912124i	$0.365559\pi$
$570$	0	0
$571$	5.52815e28	0.627915	0.313957	−	0.949437i	$-0.398345\pi$
$571$	5.52815e28	0.627915	0.313957	+	0.949437i	$0.398345\pi$
$572$	−6.55975e28	−0.731525
$573$	4.75922e28	0.521090
$574$	−2.94258e27	−0.0316339
$575$	0	0
$576$	2.67641e28	0.277406
$577$	−4.36967e28	−0.444736	−0.222368	−	0.974963i	$-0.571379\pi$
$577$	−4.36967e28	−0.444736	−0.222368	+	0.974963i	$0.571379\pi$
$578$	−1.34114e29	−1.34039
$579$	8.28594e28	0.813236
$580$	0	0
$581$	1.21881e28	0.115368
$582$	−2.77179e28	−0.257673
$583$	−1.47997e29	−1.35124
$584$	2.17116e29	1.94696
$585$	0	0
$586$	2.98003e27	0.0257808
$587$	8.04739e28	0.683841	0.341921	−	0.939729i	$-0.388923\pi$
$587$	8.04739e28	0.683841	0.341921	+	0.939729i	$0.388923\pi$
$588$	−3.55733e28	−0.296936
$589$	4.40706e28	0.361359
$590$	0	0
$591$	−1.05236e29	−0.832715
$592$	1.77882e28	0.138278
$593$	−1.36633e29	−1.04348	−0.521738	−	0.853106i	$-0.674716\pi$
$593$	−1.36633e29	−1.04348	−0.521738	+	0.853106i	$0.674716\pi$
$594$	−1.97785e28	−0.148400
$595$	0	0
$596$	2.05335e28	0.148723
$597$	−5.97489e28	−0.425206
$598$	7.30794e28	0.511013
$599$	−2.62483e29	−1.80351	−0.901757	−	0.432243i	$-0.857722\pi$
$599$	−2.62483e29	−1.80351	−0.901757	+	0.432243i	$0.857722\pi$
$600$	0	0
$601$	2.24428e29	1.48900	0.744501	−	0.667621i	$-0.232687\pi$
$601$	2.24428e29	1.48900	0.744501	+	0.667621i	$0.232687\pi$
$602$	3.38151e27	0.0220469
$603$	−3.73997e28	−0.239627
$604$	−1.03076e29	−0.649039
$605$	0	0
$606$	3.36465e28	0.204634
$607$	−1.15112e29	−0.688081	−0.344041	−	0.938955i	$-0.611796\pi$
$607$	−1.15112e29	−0.688081	−0.344041	+	0.938955i	$0.611796\pi$
$608$	4.81252e28	0.282738
$609$	−1.15312e28	−0.0665875
$610$	0	0
$611$	−3.25390e29	−1.81540
$612$	−5.45560e28	−0.299194
$613$	1.38014e29	0.744028	0.372014	−	0.928227i	$-0.378667\pi$
$613$	1.38014e29	0.744028	0.372014	+	0.928227i	$0.378667\pi$
$614$	1.83922e29	0.974686
$615$	0	0
$616$	3.01463e28	0.154396
$617$	3.29841e29	1.66077	0.830387	−	0.557188i	$-0.188120\pi$
$617$	3.29841e29	1.66077	0.830387	+	0.557188i	$0.188120\pi$
$618$	−3.86012e27	−0.0191083
$619$	−3.27323e29	−1.59304	−0.796518	−	0.604615i	$-0.793327\pi$
$619$	−3.27323e29	−1.59304	−0.796518	+	0.604615i	$0.793327\pi$
$620$	0	0
$621$	−2.41931e28	−0.113823
$622$	1.01627e29	0.470123
$623$	−4.86196e28	−0.221150
$624$	−3.27547e28	−0.146499
$625$	0	0
$626$	2.15780e29	0.933212
$627$	−4.70198e28	−0.199973
$628$	−5.60536e28	−0.234437
$629$	−2.84376e29	−1.16966
$630$	0	0
$631$	1.40426e29	0.558649	0.279325	−	0.960197i	$-0.409889\pi$
$631$	1.40426e29	0.558649	0.279325	+	0.960197i	$0.409889\pi$
$632$	4.11266e29	1.60914
$633$	2.79855e29	1.07695
$634$	1.10424e29	0.417951
$635$	0	0
$636$	9.98268e28	0.365550
$637$	3.41442e29	1.22985
$638$	−1.90971e29	−0.676627
$639$	1.29652e29	0.451877
$640$	0	0
$641$	−3.83660e29	−1.29401	−0.647004	−	0.762487i	$-0.723978\pi$
$641$	−3.83660e29	−1.29401	−0.647004	+	0.762487i	$0.723978\pi$
$642$	1.08079e29	0.358610
$643$	−5.53770e29	−1.80765	−0.903824	−	0.427903i	$-0.859252\pi$
$643$	−5.53770e29	−1.80765	−0.903824	+	0.427903i	$0.859252\pi$
$644$	1.26685e28	0.0406839
$645$	0	0
$646$	1.18124e29	0.367196
$647$	6.57835e28	0.201197	0.100599	−	0.994927i	$-0.467924\pi$
$647$	6.57835e28	0.201197	0.100599	+	0.994927i	$0.467924\pi$
$648$	3.88324e28	0.116858
$649$	6.55193e28	0.193999
$650$	0	0
$651$	−3.08449e28	−0.0884265
$652$	−1.77429e28	−0.0500524
$653$	−1.12463e29	−0.312193	−0.156096	−	0.987742i	$-0.549891\pi$
$653$	−1.12463e29	−0.312193	−0.156096	+	0.987742i	$0.549891\pi$
$654$	1.36212e29	0.372091
$655$	0	0
$656$	2.67176e28	0.0706818
$657$	2.37013e29	0.617071
$658$	5.13740e28	0.131635
$659$	2.28737e29	0.576820	0.288410	−	0.957507i	$-0.406873\pi$
$659$	2.28737e29	0.576820	0.288410	+	0.957507i	$0.406873\pi$
$660$	0	0
$661$	5.05209e29	1.23412	0.617058	−	0.786918i	$-0.288325\pi$
$661$	5.05209e29	1.23412	0.617058	+	0.786918i	$0.288325\pi$
$662$	−2.13544e29	−0.513427
$663$	5.23642e29	1.23920
$664$	−3.96305e29	−0.923133
$665$	0	0
$666$	6.95405e28	0.156949
$667$	−2.33596e29	−0.518971
$668$	−2.55314e29	−0.558370
$669$	−4.57414e29	−0.984770
$670$	0	0
$671$	3.15722e29	0.658747
$672$	−3.36827e28	−0.0691877
$673$	3.47080e29	0.701894	0.350947	−	0.936395i	$-0.385860\pi$
$673$	3.47080e29	0.701894	0.350947	+	0.936395i	$0.385860\pi$
$674$	1.22404e29	0.243707
$675$	0	0
$676$	−1.53528e29	−0.296311
$677$	4.53881e29	0.862504	0.431252	−	0.902232i	$-0.358072\pi$
$677$	4.53881e29	0.862504	0.431252	+	0.902232i	$0.358072\pi$
$678$	2.63596e29	0.493205
$679$	4.61194e28	0.0849671
$680$	0	0
$681$	4.13117e29	0.737952
$682$	−5.10829e29	−0.898544
$683$	3.49762e29	0.605836	0.302918	−	0.953017i	$-0.402039\pi$
$683$	3.49762e29	0.605836	0.302918	+	0.953017i	$0.402039\pi$
$684$	3.17158e28	0.0540986
$685$	0	0
$686$	−1.08764e29	−0.179921
$687$	2.81262e29	0.458211
$688$	−3.07030e28	−0.0492609
$689$	−9.58163e29	−1.51404
$690$	0	0
$691$	1.19978e30	1.83900	0.919502	−	0.393085i	$-0.128592\pi$
$691$	1.19978e30	1.83900	0.919502	+	0.393085i	$0.128592\pi$
$692$	−2.02505e29	−0.305719
$693$	3.29090e28	0.0489346
$694$	3.11240e29	0.455850
$695$	0	0
$696$	3.74946e29	0.532809
$697$	−4.27129e29	−0.597881
$698$	−7.85788e29	−1.08349
$699$	1.35952e29	0.184662
$700$	0	0
$701$	−4.17850e29	−0.550784	−0.275392	−	0.961332i	$-0.588808\pi$
$701$	−4.17850e29	−0.550784	−0.275392	+	0.961332i	$0.588808\pi$
$702$	−1.28050e29	−0.166280
$703$	1.65320e29	0.211493
$704$	−7.37508e29	−0.929509
$705$	0	0
$706$	−1.02620e29	−0.125540
$707$	−5.59837e28	−0.0674773
$708$	−4.41940e28	−0.0524825
$709$	9.29367e29	1.08743	0.543716	−	0.839270i	$-0.317017\pi$
$709$	9.29367e29	1.08743	0.543716	+	0.839270i	$0.317017\pi$
$710$	0	0
$711$	4.48955e29	0.510002
$712$	1.58090e30	1.76956
$713$	−6.24847e29	−0.689181
$714$	−8.26749e28	−0.0898550
$715$	0	0
$716$	−5.09967e28	−0.0538214
$717$	−2.60298e29	−0.270719
$718$	9.29920e29	0.953101
$719$	9.87158e29	0.997088	0.498544	−	0.866864i	$-0.333868\pi$
$719$	9.87158e29	0.997088	0.498544	+	0.866864i	$0.333868\pi$
$720$	0	0
$721$	6.42278e27	0.00630090
$722$	6.45399e29	0.624005
$723$	1.70400e29	0.162374
$724$	1.03678e30	0.973717
$725$	0	0
$726$	1.08121e29	0.0986455
$727$	−1.00668e30	−0.905271	−0.452636	−	0.891696i	$-0.649516\pi$
$727$	−1.00668e30	−0.905271	−0.452636	+	0.891696i	$0.649516\pi$
$728$	1.95174e29	0.172998
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	4.90842e29	0.416686
$732$	−2.12960e29	−0.178210
$733$	1.72342e30	1.42167	0.710837	−	0.703356i	$-0.248316\pi$
$733$	1.72342e30	1.42167	0.710837	+	0.703356i	$0.248316\pi$
$734$	−1.27252e30	−1.03480
$735$	0	0
$736$	−6.82334e29	−0.539237
$737$	1.03058e30	0.802923
$738$	1.04449e29	0.0802253
$739$	−1.92102e30	−1.45467	−0.727337	−	0.686280i	$-0.759242\pi$
$739$	−1.92102e30	−1.45467	−0.727337	+	0.686280i	$0.759242\pi$
$740$	0	0
$741$	−3.04416e29	−0.224066
$742$	1.51279e29	0.109784
$743$	−1.99512e30	−1.42753	−0.713766	−	0.700384i	$-0.753012\pi$
$743$	−1.99512e30	−1.42753	−0.713766	+	0.700384i	$0.753012\pi$
$744$	1.00295e30	0.707557
$745$	0	0
$746$	−1.92037e30	−1.31713
$747$	−4.32623e29	−0.292579
$748$	1.50334e30	1.00251
$749$	−1.79830e29	−0.118251
$750$	0	0
$751$	2.59166e30	1.65714	0.828570	−	0.559885i	$-0.189155\pi$
$751$	2.59166e30	1.65714	0.828570	+	0.559885i	$0.189155\pi$
$752$	−4.66458e29	−0.294121
$753$	−3.58065e29	−0.222646
$754$	−1.23638e30	−0.758148
$755$	0	0
$756$	−2.21978e28	−0.0132382
$757$	7.37403e29	0.433709	0.216854	−	0.976204i	$-0.430420\pi$
$757$	7.37403e29	0.433709	0.216854	+	0.976204i	$0.430420\pi$
$758$	−4.90072e29	−0.284272
$759$	6.66662e29	0.381388
$760$	0	0
$761$	−2.89068e30	−1.60865	−0.804325	−	0.594189i	$-0.797473\pi$
$761$	−2.89068e30	−1.60865	−0.804325	+	0.594189i	$0.797473\pi$
$762$	−1.30567e30	−0.716651
$763$	−2.26641e29	−0.122696
$764$	8.84596e29	0.472351
$765$	0	0
$766$	2.45366e29	0.127471
$767$	4.24186e29	0.217372
$768$	1.21642e30	0.614879
$769$	1.48232e30	0.739119	0.369559	−	0.929207i	$-0.379509\pi$
$769$	1.48232e30	0.739119	0.369559	+	0.929207i	$0.379509\pi$
$770$	0	0
$771$	2.02428e30	0.982198
$772$	1.54011e30	0.737172
$773$	−2.57666e30	−1.21667	−0.608335	−	0.793680i	$-0.708162\pi$
$773$	−2.57666e30	−1.21667	−0.608335	+	0.793680i	$0.708162\pi$
$774$	−1.20029e29	−0.0559121
$775$	0	0
$776$	−1.49961e30	−0.679876
$777$	−1.15707e29	−0.0517534
$778$	1.02916e30	0.454146
$779$	2.48309e29	0.108106
$780$	0	0
$781$	−3.57268e30	−1.51411
$782$	−1.67480e30	−0.700315
$783$	4.09307e29	0.168869
$784$	4.89469e29	0.199254
$785$	0	0
$786$	4.68722e29	0.185771
$787$	−2.25891e30	−0.883413	−0.441707	−	0.897160i	$-0.645627\pi$
$787$	−2.25891e30	−0.883413	−0.441707	+	0.897160i	$0.645627\pi$
$788$	−1.95602e30	−0.754829
$789$	−2.32581e30	−0.885656
$790$	0	0
$791$	−4.38593e29	−0.162633
$792$	−1.07006e30	−0.391557
$793$	2.04405e30	0.738113
$794$	1.36057e30	0.484847
$795$	0	0
$796$	−1.11055e30	−0.385435
$797$	−4.64460e30	−1.59087	−0.795437	−	0.606036i	$-0.792759\pi$
$797$	−4.64460e30	−1.59087	−0.795437	+	0.606036i	$0.792759\pi$
$798$	4.80625e28	0.0162471
$799$	7.45717e30	2.48790
$800$	0	0
$801$	1.72578e30	0.560847
$802$	−3.27432e30	−1.05024
$803$	−6.53110e30	−2.06763
$804$	−6.95147e29	−0.217214
$805$	0	0
$806$	−3.30721e30	−1.00680
$807$	−6.99473e29	−0.210184
$808$	1.82036e30	0.539929
$809$	2.89685e30	0.848139	0.424069	−	0.905630i	$-0.360601\pi$
$809$	2.89685e30	0.848139	0.424069	+	0.905630i	$0.360601\pi$
$810$	0	0
$811$	−5.60886e30	−1.60013	−0.800066	−	0.599912i	$-0.795202\pi$
$811$	−5.60886e30	−1.60013	−0.800066	+	0.599912i	$0.795202\pi$
$812$	−2.14330e29	−0.0603594
$813$	9.63455e29	0.267843
$814$	−1.91625e30	−0.525891
$815$	0	0
$816$	7.50660e29	0.200769
$817$	−2.85348e29	−0.0753430
$818$	6.11600e29	0.159426
$819$	2.13060e29	0.0548303
$820$	0	0
$821$	−5.41255e30	−1.35768	−0.678842	−	0.734284i	$-0.737518\pi$
$821$	−5.41255e30	−1.35768	−0.678842	+	0.734284i	$0.737518\pi$
$822$	−1.10808e30	−0.274421
$823$	−2.61047e30	−0.638293	−0.319147	−	0.947705i	$-0.603396\pi$
$823$	−2.61047e30	−0.638293	−0.319147	+	0.947705i	$0.603396\pi$
$824$	−2.08842e29	−0.0504175
$825$	0	0
$826$	−6.69722e28	−0.0157617
$827$	7.22685e30	1.67935	0.839675	−	0.543090i	$-0.182746\pi$
$827$	7.22685e30	1.67935	0.839675	+	0.543090i	$0.182746\pi$
$828$	−4.49676e29	−0.103177
$829$	−3.91702e30	−0.887429	−0.443715	−	0.896168i	$-0.646340\pi$
$829$	−3.91702e30	−0.887429	−0.443715	+	0.896168i	$0.646340\pi$
$830$	0	0
$831$	1.21691e29	0.0268811
$832$	−4.77478e30	−1.04150
$833$	−7.82503e30	−1.68544
$834$	3.64632e29	0.0775553
$835$	0	0
$836$	−8.73956e29	−0.181269
$837$	1.09486e30	0.224254
$838$	1.85908e30	0.376041
$839$	−8.21745e30	−1.64148	−0.820742	−	0.571299i	$-0.806440\pi$
$839$	−8.21745e30	−1.64148	−0.820742	+	0.571299i	$0.806440\pi$
$840$	0	0
$841$	−1.18079e30	−0.230045
$842$	2.61930e30	0.503975
$843$	4.86544e30	0.924554
$844$	5.20167e30	0.976218
$845$	0	0
$846$	−1.82355e30	−0.333833
$847$	−1.79901e29	−0.0325281
$848$	−1.37356e30	−0.245297
$849$	−3.63330e30	−0.640871
$850$	0	0
$851$	−2.34396e30	−0.403357
$852$	2.40984e30	0.409612
$853$	8.59334e30	1.44277	0.721385	−	0.692534i	$-0.243506\pi$
$853$	8.59334e30	1.44277	0.721385	+	0.692534i	$0.243506\pi$
$854$	−3.22723e29	−0.0535208
$855$	0	0
$856$	5.84732e30	0.946198
$857$	3.26828e30	0.522421	0.261211	−	0.965282i	$-0.415878\pi$
$857$	3.26828e30	0.522421	0.261211	+	0.965282i	$0.415878\pi$
$858$	3.52853e30	0.557157
$859$	9.63038e29	0.150216	0.0751078	−	0.997175i	$-0.476070\pi$
$859$	9.63038e29	0.150216	0.0751078	+	0.997175i	$0.476070\pi$
$860$	0	0
$861$	−1.73790e29	−0.0264541
$862$	−8.46159e30	−1.27241
$863$	−3.97633e30	−0.590702	−0.295351	−	0.955389i	$-0.595437\pi$
$863$	−3.97633e30	−0.590702	−0.295351	+	0.955389i	$0.595437\pi$
$864$	1.19559e30	0.175464
$865$	0	0
$866$	−7.14011e30	−1.02274
$867$	−7.92083e30	−1.12091
$868$	−5.73313e29	−0.0801557
$869$	−1.23714e31	−1.70887
$870$	0	0
$871$	6.67220e30	0.899661
$872$	7.36941e30	0.981770
$873$	−1.63704e30	−0.215481
$874$	9.73637e29	0.126627
$875$	0	0
$876$	4.40535e30	0.559355
$877$	5.52079e29	0.0692636	0.0346318	−	0.999400i	$-0.488974\pi$
$877$	5.52079e29	0.0692636	0.0346318	+	0.999400i	$0.488974\pi$
$878$	2.98326e30	0.369828
$879$	1.76002e29	0.0215593
$880$	0	0
$881$	−3.98610e30	−0.476761	−0.238380	−	0.971172i	$-0.576617\pi$
$881$	−3.98610e30	−0.476761	−0.238380	+	0.971172i	$0.576617\pi$
$882$	1.91351e30	0.226157
$883$	−1.11066e31	−1.29717	−0.648583	−	0.761144i	$-0.724638\pi$
$883$	−1.11066e31	−1.29717	−0.648583	+	0.761144i	$0.724638\pi$
$884$	9.73293e30	1.12330
$885$	0	0
$886$	1.36583e30	0.153937
$887$	2.93164e30	0.326522	0.163261	−	0.986583i	$-0.447799\pi$
$887$	2.93164e30	0.326522	0.163261	+	0.986583i	$0.447799\pi$
$888$	3.76231e30	0.414112
$889$	2.17249e30	0.236314
$890$	0	0
$891$	−1.16813e30	−0.124101
$892$	−8.50194e30	−0.892662
$893$	−4.33517e30	−0.449848
$894$	−1.10451e30	−0.113273
$895$	0	0
$896$	−4.42392e29	−0.0443173
$897$	4.31611e30	0.427338
$898$	5.56460e30	0.544544
$899$	1.05714e31	1.02248
$900$	0	0
$901$	2.19588e31	2.07491
$902$	−2.87818e30	−0.268812
$903$	1.99714e29	0.0184369
$904$	1.42612e31	1.30133
$905$	0	0
$906$	5.54453e30	0.494332
$907$	1.45433e31	1.28170	0.640852	−	0.767665i	$-0.278581\pi$
$907$	1.45433e31	1.28170	0.640852	+	0.767665i	$0.278581\pi$
$908$	7.67859e30	0.668929
$909$	1.98718e30	0.171126
$910$	0	0
$911$	6.40205e30	0.538736	0.269368	−	0.963037i	$-0.413185\pi$
$911$	6.40205e30	0.538736	0.269368	+	0.963037i	$0.413185\pi$
$912$	−4.36391e29	−0.0363020
$913$	1.19213e31	0.980350
$914$	3.92129e30	0.318782
$915$	0	0
$916$	5.22781e30	0.415353
$917$	−7.79897e29	−0.0612575
$918$	2.93460e30	0.227877
$919$	−5.93034e30	−0.455268	−0.227634	−	0.973747i	$-0.573099\pi$
$919$	−5.93034e30	−0.455268	−0.227634	+	0.973747i	$0.573099\pi$
$920$	0	0
$921$	1.08625e31	0.815087
$922$	1.51056e31	1.12063
$923$	−2.31303e31	−1.69653
$924$	6.11679e29	0.0443576
$925$	0	0
$926$	6.40956e30	0.454373
$927$	−2.27981e29	−0.0159794
$928$	1.15440e31	0.800021
$929$	−1.75288e31	−1.20112	−0.600561	−	0.799579i	$-0.705056\pi$
$929$	−1.75288e31	−1.20112	−0.600561	+	0.799579i	$0.705056\pi$
$930$	0	0
$931$	4.54903e30	0.304753
$932$	2.52695e30	0.167390
$933$	6.00217e30	0.393143
$934$	−2.82627e30	−0.183051
$935$	0	0
$936$	−6.92781e30	−0.438732
$937$	1.86505e31	1.16795	0.583977	−	0.811770i	$-0.301496\pi$
$937$	1.86505e31	1.16795	0.583977	+	0.811770i	$0.301496\pi$
$938$	−1.05344e30	−0.0652346
$939$	1.27441e31	0.780404
$940$	0	0
$941$	−7.81711e30	−0.468118	−0.234059	−	0.972222i	$-0.575201\pi$
$941$	−7.81711e30	−0.468118	−0.234059	+	0.972222i	$0.575201\pi$
$942$	3.01515e30	0.178556
$943$	−3.52060e30	−0.206179
$944$	6.08085e29	0.0352175
$945$	0	0
$946$	3.30751e30	0.187346
$947$	−1.46277e31	−0.819411	−0.409705	−	0.912218i	$-0.634368\pi$
$947$	−1.46277e31	−0.819411	−0.409705	+	0.912218i	$0.634368\pi$
$948$	8.34472e30	0.462301
$949$	−4.22837e31	−2.31674
$950$	0	0
$951$	6.52169e30	0.349514
$952$	−4.47291e30	−0.237084
$953$	1.47934e31	0.775521	0.387760	−	0.921760i	$-0.373249\pi$
$953$	1.47934e31	0.775521	0.387760	+	0.921760i	$0.373249\pi$
$954$	−5.36974e30	−0.278417
$955$	0	0
$956$	−4.83815e30	−0.245398
$957$	−1.12788e31	−0.565833
$958$	−8.65445e30	−0.429439
$959$	1.84371e30	0.0904894
$960$	0	0
$961$	7.45196e30	0.357828
$962$	−1.24062e31	−0.589251
$963$	6.38318e30	0.299890
$964$	3.16722e30	0.147187
$965$	0	0
$966$	−6.81445e29	−0.0309864
$967$	−1.70219e31	−0.765649	−0.382824	−	0.923821i	$-0.625049\pi$
$967$	−1.70219e31	−0.765649	−0.382824	+	0.923821i	$0.625049\pi$
$968$	5.84963e30	0.260278
$969$	6.97649e30	0.307070
$970$	0	0
$971$	2.36139e31	1.01711	0.508553	−	0.861031i	$-0.330181\pi$
$971$	2.36139e31	1.01711	0.508553	+	0.861031i	$0.330181\pi$
$972$	7.87924e29	0.0335729
$973$	−6.06704e29	−0.0255736
$974$	−3.37716e30	−0.140826
$975$	0	0
$976$	2.93022e30	0.119585
$977$	4.50176e31	1.81756	0.908781	−	0.417273i	$-0.137014\pi$
$977$	4.50176e31	1.81756	0.908781	+	0.417273i	$0.137014\pi$
$978$	9.54401e29	0.0381218
$979$	−4.75555e31	−1.87924
$980$	0	0
$981$	8.04477e30	0.311164
$982$	2.94152e31	1.12564
$983$	1.95413e31	0.739845	0.369922	−	0.929063i	$-0.379384\pi$
$983$	1.95413e31	0.739845	0.369922	+	0.929063i	$0.379384\pi$
$984$	5.65093e30	0.211676
$985$	0	0
$986$	2.83350e31	1.03900
$987$	3.03417e30	0.110080
$988$	−5.65818e30	−0.203109
$989$	4.04575e30	0.143694
$990$	0	0
$991$	−3.06722e31	−1.06653	−0.533263	−	0.845950i	$-0.679034\pi$
$991$	−3.06722e31	−1.06653	−0.533263	+	0.845950i	$0.679034\pi$
$992$	3.08791e31	1.06241
$993$	−1.26120e31	−0.429357
$994$	3.65190e30	0.123016
$995$	0	0
$996$	−8.04116e30	−0.265213
$997$	8.72776e30	0.284842	0.142421	−	0.989806i	$-0.454511\pi$
$997$	8.72776e30	0.284842	0.142421	+	0.989806i	$0.454511\pi$
$998$	−3.12483e31	−1.00915
$999$	4.10710e30	0.131249

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.g.1.1		3
5.2	odd	4		75.22.b.g.49.2		6
5.3	odd	4		75.22.b.g.49.5		6
5.4	even	2		15.22.a.b.1.3	✓	3
15.14	odd	2		45.22.a.e.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.a.b.1.3	✓	3	5.4	even	2
45.22.a.e.1.1		3	15.14	odd	2
75.22.a.g.1.1		3	1.1	even	1	trivial
75.22.b.g.49.2		6	5.2	odd	4
75.22.b.g.49.5		6	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-999.805 q^{2} -59049.0 q^{3} -1.09754e6 q^{4} +5.90375e7 q^{6} -9.82313e7 q^{7} +3.19407e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-999.805 q^{2} -59049.0 q^{3} -1.09754e6 q^{4} +5.90375e7 q^{6} -9.82313e7 q^{7} +3.19407e9 q^{8} +3.48678e9 q^{9} -9.60815e10 q^{11} +6.48088e10 q^{12} -6.22052e11 q^{13} +9.82121e10 q^{14} -8.91733e11 q^{16} +1.42559e13 q^{17} -3.48610e12 q^{18} -8.28760e12 q^{19} +5.80046e12 q^{21} +9.60627e13 q^{22} +1.17504e14 q^{23} -1.88607e14 q^{24} +6.21930e14 q^{26} -2.05891e14 q^{27} +1.07813e14 q^{28} -1.98798e15 q^{29} -5.31766e15 q^{31} -5.80689e15 q^{32} +5.67352e15 q^{33} -1.42532e16 q^{34} -3.82689e15 q^{36} -1.99479e16 q^{37} +8.28598e15 q^{38} +3.67315e16 q^{39} -2.99615e16 q^{41} -5.79933e15 q^{42} +3.44307e16 q^{43} +1.05454e17 q^{44} -1.17481e17 q^{46} +5.23092e17 q^{47} +5.26560e16 q^{48} -5.48896e17 q^{49} -8.41799e17 q^{51} +6.82728e17 q^{52} +1.54033e18 q^{53} +2.05851e17 q^{54} -3.13758e17 q^{56} +4.89374e17 q^{57} +1.98759e18 q^{58} -6.81914e17 q^{59} -3.28598e18 q^{61} +5.31662e18 q^{62} -3.42511e17 q^{63} +7.67586e18 q^{64} -5.67241e18 q^{66} -1.07261e19 q^{67} -1.56465e19 q^{68} -6.93850e18 q^{69} +3.71838e19 q^{71} +1.11370e19 q^{72} +6.79746e19 q^{73} +1.99440e19 q^{74} +9.09599e18 q^{76} +9.43821e18 q^{77} -3.67244e19 q^{78} +1.28759e20 q^{79} +1.21577e19 q^{81} +2.99556e19 q^{82} -1.24075e20 q^{83} -6.36625e18 q^{84} -3.44240e19 q^{86} +1.17388e20 q^{87} -3.06891e20 q^{88} +4.94950e20 q^{89} +6.11049e19 q^{91} -1.28966e20 q^{92} +3.14002e20 q^{93} -5.22990e20 q^{94} +3.42891e20 q^{96} -4.69497e20 q^{97} +5.48789e20 q^{98} -3.35015e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2300 q^{2} - 177147 q^{3} + 1264400 q^{4} - 135812700 q^{6} - 465666872 q^{7} + 3839876544 q^{8} + 10460353203 q^{9}+O(q^{10}) \) 3 * q + 2300 * q^2 - 177147 * q^3 + 1264400 * q^4 - 135812700 * q^6 - 465666872 * q^7 + 3839876544 * q^8 + 10460353203 * q^9	\( 3 q + 2300 q^{2} - 177147 q^{3} + 1264400 q^{4} - 135812700 q^{6} - 465666872 q^{7} + 3839876544 q^{8} + 10460353203 q^{9} - 167336332556 q^{11} - 74661555600 q^{12} + 545571033878 q^{13} - 1568858902656 q^{14} + 255267954944 q^{16} + 8104424487194 q^{17} + 8019604122300 q^{18} + 3937700740828 q^{19} + 27497163124728 q^{21} - 114198109969712 q^{22} + 156235274730744 q^{23} - 226740870046656 q^{24} + 29\!\cdots\!56 q^{26}+ \cdots - 58\!\cdots\!56 q^{99}+O(q^{100}) \) 3 * q + 2300 * q^2 - 177147 * q^3 + 1264400 * q^4 - 135812700 * q^6 - 465666872 * q^7 + 3839876544 * q^8 + 10460353203 * q^9 - 167336332556 * q^11 - 74661555600 * q^12 + 545571033878 * q^13 - 1568858902656 * q^14 + 255267954944 * q^16 + 8104424487194 * q^17 + 8019604122300 * q^18 + 3937700740828 * q^19 + 27497163124728 * q^21 - 114198109969712 * q^22 + 156235274730744 * q^23 - 226740870046656 * q^24 + 2972936042645656 * q^26 - 617673396283947 * q^27 - 3826676848454144 * q^28 - 930273612785494 * q^29 - 6257709152718928 * q^31 - 4297283007276032 * q^32 + 9881043101099244 * q^33 - 20587982883981400 * q^34 + 4408690196624400 * q^36 + 22246337613227118 * q^37 + 82202958592557968 * q^38 - 32215423979462022 * q^39 - 186265908060974338 * q^41 + 92639549342934144 * q^42 + 268609288174096316 * q^43 - 284579440966881088 * q^44 + 375824751838355712 * q^46 + 900034127817222032 * q^47 - 15073317471488256 * q^48 - 586354193149814837 * q^49 - 478558161544318506 * q^51 + 3462564631549962144 * q^52 + 1269623243180583374 * q^53 - 473549603817692700 * q^54 - 5236064873787955200 * q^56 - 232517291045152572 * q^57 + 1091617398834773576 * q^58 - 8551487099411338268 * q^59 - 7181148471323735222 * q^61 + 3050004306382316256 * q^62 - 1623679985352063672 * q^63 - 557099647389855744 * q^64 + 6743284195601523888 * q^66 - 2946635148405656396 * q^67 - 10323921738190858400 * q^68 - 9225536737575702456 * q^69 - 37849731561832987624 * q^71 + 13388821635384990144 * q^72 + 7149058835824826594 * q^73 + 50872537645745842424 * q^74 + 200886445168773700032 * q^76 + 110970090560628236256 * q^77 - 175548900382183341144 * q^78 + 309682369826312504000 * q^79 + 36472996377170786403 * q^81 - 265773545783653420264 * q^82 - 199780825758268631580 * q^83 + 225961441224368749056 * q^84 + 267144820042737104944 * q^86 + 54931726561370635206 * q^87 - 749649166937994283776 * q^88 + 162380735751340085838 * q^89 - 558457704043459369328 * q^91 + 1333721831719821170688 * q^92 + 369511467758899979472 * q^93 - 149583353667227862112 * q^94 + 253750264296642413568 * q^96 + 1208943423358924299354 * q^97 + 876777377902285855132 * q^98 - 583465714076809258956 * q^99

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