LMFDB - Embedded newform 75.22.a.h.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:	$0$
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			1.1
Root			$-1711.97$ 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)

$f(q)$	$=$	$q-1487.97 q^{2} +59049.0 q^{3} +116903. q^{4} -8.78631e7 q^{6} +1.26833e9 q^{7} +2.94655e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-1487.97 q^{2} +59049.0 q^{3} +116903. q^{4} -8.78631e7 q^{6} +1.26833e9 q^{7} +2.94655e9 q^{8} +3.48678e9 q^{9} +4.25851e10 q^{11} +6.90299e9 q^{12} -7.14056e11 q^{13} -1.88724e12 q^{14} -4.62954e12 q^{16} -9.08430e12 q^{17} -5.18823e12 q^{18} -2.62762e13 q^{19} +7.48936e13 q^{21} -6.33654e13 q^{22} +2.96171e13 q^{23} +1.73991e14 q^{24} +1.06249e15 q^{26} +2.05891e14 q^{27} +1.48271e14 q^{28} +4.45551e15 q^{29} +4.93427e14 q^{31} +7.09254e14 q^{32} +2.51461e15 q^{33} +1.35172e16 q^{34} +4.07614e14 q^{36} -8.82960e15 q^{37} +3.90982e16 q^{38} -4.21643e16 q^{39} +1.17100e17 q^{41} -1.11439e17 q^{42} +4.77156e16 q^{43} +4.97831e15 q^{44} -4.40694e16 q^{46} -3.81704e17 q^{47} -2.73370e17 q^{48} +1.05012e18 q^{49} -5.36419e17 q^{51} -8.34750e16 q^{52} -1.01485e18 q^{53} -3.06360e17 q^{54} +3.73720e18 q^{56} -1.55158e18 q^{57} -6.62967e18 q^{58} -8.26140e17 q^{59} -6.60518e18 q^{61} -7.34204e17 q^{62} +4.42239e18 q^{63} +8.65351e18 q^{64} -3.74166e18 q^{66} +2.57407e19 q^{67} -1.06198e18 q^{68} +1.74886e18 q^{69} +3.32319e19 q^{71} +1.02740e19 q^{72} +2.66123e19 q^{73} +1.31382e19 q^{74} -3.07176e18 q^{76} +5.40120e19 q^{77} +6.27392e19 q^{78} -7.85851e19 q^{79} +1.21577e19 q^{81} -1.74241e20 q^{82} -1.68959e20 q^{83} +8.75526e18 q^{84} -7.09994e19 q^{86} +2.63093e20 q^{87} +1.25479e20 q^{88} +6.58830e19 q^{89} -9.05659e20 q^{91} +3.46232e18 q^{92} +2.91363e19 q^{93} +5.67964e20 q^{94} +4.18808e19 q^{96} +1.13619e21 q^{97} -1.56254e21 q^{98} +1.48485e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1487.97	−1.02749	−0.513747	−	0.857942i	$-0.671743\pi$
$2$	−1487.97	−1.02749	−0.513747	+	0.857942i	$0.671743\pi$
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	116903.	0.0557435
$5$	0	0
$5$	0	0
$6$	−8.78631e7	−0.593224
$7$	1.26833e9	1.69708	0.848541	−	0.529129i	$-0.177481\pi$
$7$	1.26833e9	1.69708	0.848541	+	0.529129i	$0.177481\pi$
$8$	2.94655e9	0.970218
$9$	3.48678e9	0.333333
$10$	0	0
$11$	4.25851e10	0.495033	0.247517	−	0.968884i	$-0.420385\pi$
$11$	4.25851e10	0.495033	0.247517	+	0.968884i	$0.420385\pi$
$12$	6.90299e9	0.0321835
$13$	−7.14056e11	−1.43657	−0.718285	−	0.695749i	$-0.755073\pi$
$13$	−7.14056e11	−1.43657	−0.718285	+	0.695749i	$0.755073\pi$
$14$	−1.88724e12	−1.74374
$15$	0	0
$16$	−4.62954e12	−1.05264
$17$	−9.08430e12	−1.09289	−0.546447	−	0.837494i	$-0.684020\pi$
$17$	−9.08430e12	−1.09289	−0.546447	+	0.837494i	$0.684020\pi$
$18$	−5.18823e12	−0.342498
$19$	−2.62762e13	−0.983219	−0.491609	−	0.870816i	$-0.663591\pi$
$19$	−2.62762e13	−0.983219	−0.491609	+	0.870816i	$0.663591\pi$
$20$	0	0
$21$	7.48936e13	0.979811
$22$	−6.33654e13	−0.508644
$23$	2.96171e13	0.149074	0.0745368	−	0.997218i	$-0.476252\pi$
$23$	2.96171e13	0.149074	0.0745368	+	0.997218i	$0.476252\pi$
$24$	1.73991e14	0.560155
$25$	0	0
$26$	1.06249e15	1.47607
$27$	2.05891e14	0.192450
$28$	1.48271e14	0.0946014
$29$	4.45551e15	1.96661	0.983305	−	0.181963i	$-0.0582450\pi$
$29$	4.45551e15	1.96661	0.983305	+	0.181963i	$0.0582450\pi$
$30$	0	0
$31$	4.93427e14	0.108125	0.0540623	−	0.998538i	$-0.482783\pi$
$31$	4.93427e14	0.108125	0.0540623	+	0.998538i	$0.482783\pi$
$32$	7.09254e14	0.111359
$33$	2.51461e15	0.285808
$34$	1.35172e16	1.12294
$35$	0	0
$36$	4.07614e14	0.0185812
$37$	−8.82960e15	−0.301872	−0.150936	−	0.988544i	$-0.548229\pi$
$37$	−8.82960e15	−0.301872	−0.150936	+	0.988544i	$0.548229\pi$
$38$	3.90982e16	1.01025
$39$	−4.21643e16	−0.829405
$40$	0	0
$41$	1.17100e17	1.36247	0.681235	−	0.732065i	$-0.261443\pi$
$41$	1.17100e17	1.36247	0.681235	+	0.732065i	$0.261443\pi$
$42$	−1.11439e17	−1.00675
$43$	4.77156e16	0.336699	0.168350	−	0.985727i	$-0.446156\pi$
$43$	4.77156e16	0.336699	0.168350	+	0.985727i	$0.446156\pi$
$44$	4.97831e15	0.0275949
$45$	0	0
$46$	−4.40694e16	−0.153172
$47$	−3.81704e17	−1.05852	−0.529260	−	0.848460i	$-0.677530\pi$
$47$	−3.81704e17	−1.05852	−0.529260	+	0.848460i	$0.677530\pi$
$48$	−2.73370e17	−0.607740
$49$	1.05012e18	1.88009
$50$	0	0
$51$	−5.36419e17	−0.630982
$52$	−8.34750e16	−0.0800795
$53$	−1.01485e18	−0.797086	−0.398543	−	0.917150i	$-0.630484\pi$
$53$	−1.01485e18	−0.797086	−0.398543	+	0.917150i	$0.630484\pi$
$54$	−3.06360e17	−0.197741
$55$	0	0
$56$	3.73720e18	1.64654
$57$	−1.55158e18	−0.567662
$58$	−6.62967e18	−2.02068
$59$	−8.26140e17	−0.210430	−0.105215	−	0.994450i	$-0.533553\pi$
$59$	−8.26140e17	−0.210430	−0.105215	+	0.994450i	$0.533553\pi$
$60$	0	0
$61$	−6.60518e18	−1.18555	−0.592777	−	0.805367i	$-0.701969\pi$
$61$	−6.60518e18	−1.18555	−0.592777	+	0.805367i	$0.701969\pi$
$62$	−7.34204e17	−0.111097
$63$	4.42239e18	0.565694
$64$	8.65351e18	0.938215
$65$	0	0
$66$	−3.74166e18	−0.293666
$67$	2.57407e19	1.72518	0.862590	−	0.505904i	$-0.168841\pi$
$67$	2.57407e19	1.72518	0.862590	+	0.505904i	$0.168841\pi$
$68$	−1.06198e18	−0.0609218
$69$	1.74886e18	0.0860676
$70$	0	0
$71$	3.32319e19	1.21155	0.605777	−	0.795635i	$-0.292863\pi$
$71$	3.32319e19	1.21155	0.605777	+	0.795635i	$0.292863\pi$
$72$	1.02740e19	0.323406
$73$	2.66123e19	0.724757	0.362378	−	0.932031i	$-0.381965\pi$
$73$	2.66123e19	0.724757	0.362378	+	0.932031i	$0.381965\pi$
$74$	1.31382e19	0.310172
$75$	0	0
$76$	−3.07176e18	−0.0548081
$77$	5.40120e19	0.840112
$78$	6.27392e19	0.852208
$79$	−7.85851e19	−0.933804	−0.466902	−	0.884309i	$-0.654630\pi$
$79$	−7.85851e19	−0.933804	−0.466902	+	0.884309i	$0.654630\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−1.74241e20	−1.39993
$83$	−1.68959e20	−1.19526	−0.597630	−	0.801772i	$-0.703891\pi$
$83$	−1.68959e20	−1.19526	−0.597630	+	0.801772i	$0.703891\pi$
$84$	8.75526e18	0.0546181
$85$	0	0
$86$	−7.09994e19	−0.345956
$87$	2.63093e20	1.13542
$88$	1.25479e20	0.480290
$89$	6.58830e19	0.223964	0.111982	−	0.993710i	$-0.464280\pi$
$89$	6.58830e19	0.223964	0.111982	+	0.993710i	$0.464280\pi$
$90$	0	0
$91$	−9.05659e20	−2.43798
$92$	3.46232e18	0.00830989
$93$	2.91363e19	0.0624258
$94$	5.67964e20	1.08762
$95$	0	0
$96$	4.18808e19	0.0642934
$97$	1.13619e21	1.56440	0.782201	−	0.623026i	$-0.214097\pi$
$97$	1.13619e21	1.56440	0.782201	+	0.623026i	$0.214097\pi$
$98$	−1.56254e21	−1.93178
$99$	1.48485e20	0.165011
$100$	0	0
$101$	6.08234e20	0.547894	0.273947	−	0.961745i	$-0.411671\pi$
$101$	6.08234e20	0.547894	0.273947	+	0.961745i	$0.411671\pi$
$102$	7.98175e20	0.648331
$103$	−1.93831e21	−1.42113	−0.710564	−	0.703633i	$-0.751560\pi$
$103$	−1.93831e21	−1.42113	−0.710564	+	0.703633i	$0.751560\pi$
$104$	−2.10400e21	−1.39379
$105$	0	0
$106$	1.51007e21	0.819001
$107$	1.21945e20	0.0599286	0.0299643	−	0.999551i	$-0.490461\pi$
$107$	1.21945e20	0.0599286	0.0299643	+	0.999551i	$0.490461\pi$
$108$	2.40692e19	0.0107278
$109$	4.65871e21	1.88490	0.942448	−	0.334351i	$-0.108517\pi$
$109$	4.65871e21	1.88490	0.942448	+	0.334351i	$0.108517\pi$
$110$	0	0
$111$	−5.21379e20	−0.174286
$112$	−5.87179e21	−1.78641
$113$	−1.27385e21	−0.353017	−0.176509	−	0.984299i	$-0.556480\pi$
$113$	−1.27385e21	−0.353017	−0.176509	+	0.984299i	$0.556480\pi$
$114$	2.30871e21	0.583269
$115$	0	0
$116$	5.20861e20	0.109626
$117$	−2.48976e21	−0.478857
$118$	1.22927e21	0.216215
$119$	−1.15219e22	−1.85473
$120$	0	0
$121$	−5.58676e21	−0.754942
$122$	9.82831e21	1.21815
$123$	6.91464e21	0.786622
$124$	5.76829e19	0.00602725
$125$	0	0
$126$	−6.58039e21	−0.581247
$127$	1.08103e22	0.878821	0.439411	−	0.898286i	$-0.355187\pi$
$127$	1.08103e22	0.878821	0.439411	+	0.898286i	$0.355187\pi$
$128$	−1.43636e22	−1.07537
$129$	2.81756e21	0.194393
$130$	0	0
$131$	−2.63526e22	−1.54694	−0.773472	−	0.633830i	$-0.781482\pi$
$131$	−2.63526e22	−1.54694	−0.773472	+	0.633830i	$0.781482\pi$
$132$	2.93964e20	0.0159319
$133$	−3.33269e22	−1.66860
$134$	−3.83013e22	−1.77261
$135$	0	0
$136$	−2.67674e22	−1.06034
$137$	1.96734e22	0.721629	0.360815	−	0.932638i	$-0.382499\pi$
$137$	1.96734e22	0.721629	0.360815	+	0.932638i	$0.382499\pi$
$138$	−2.60225e21	−0.0884340
$139$	1.37361e22	0.432722	0.216361	−	0.976313i	$-0.430581\pi$
$139$	1.37361e22	0.432722	0.216361	+	0.976313i	$0.430581\pi$
$140$	0	0
$141$	−2.25392e22	−0.611137
$142$	−4.94481e22	−1.24486
$143$	−3.04081e22	−0.711151
$144$	−1.61422e22	−0.350879
$145$	0	0
$146$	−3.95983e22	−0.744683
$147$	6.20083e22	1.08547
$148$	−1.03220e21	−0.0168274
$149$	9.14642e22	1.38930	0.694649	−	0.719349i	$-0.255560\pi$
$149$	9.14642e22	1.38930	0.694649	+	0.719349i	$0.255560\pi$
$150$	0	0
$151$	1.22303e23	1.61502	0.807511	−	0.589853i	$-0.200814\pi$
$151$	1.22303e23	1.61502	0.807511	+	0.589853i	$0.200814\pi$
$152$	−7.74243e22	−0.953936
$153$	−3.16750e22	−0.364298
$154$	−8.03682e22	−0.863210
$155$	0	0
$156$	−4.92912e21	−0.0462339
$157$	1.10353e23	0.967918	0.483959	−	0.875091i	$-0.339198\pi$
$157$	1.10353e23	0.967918	0.483959	+	0.875091i	$0.339198\pi$
$158$	1.16932e23	0.959477
$159$	−5.99259e22	−0.460198
$160$	0	0
$161$	3.75643e22	0.252990
$162$	−1.80902e22	−0.114166
$163$	−3.12086e23	−1.84631	−0.923155	−	0.384429i	$-0.874398\pi$
$163$	−3.12086e23	−1.84631	−0.923155	+	0.384429i	$0.874398\pi$
$164$	1.36893e22	0.0759488
$165$	0	0
$166$	2.51406e23	1.22812
$167$	−5.58649e22	−0.256222	−0.128111	−	0.991760i	$-0.540891\pi$
$167$	−5.58649e22	−0.256222	−0.128111	+	0.991760i	$0.540891\pi$
$168$	2.20678e23	0.950630
$169$	2.62811e23	1.06374
$170$	0	0
$171$	−9.16195e22	−0.327740
$172$	5.57809e21	0.0187688
$173$	1.34399e23	0.425512	0.212756	−	0.977105i	$-0.431756\pi$
$173$	1.34399e23	0.425512	0.212756	+	0.977105i	$0.431756\pi$
$174$	−3.91475e23	−1.16664
$175$	0	0
$176$	−1.97150e23	−0.521090
$177$	−4.87827e22	−0.121492
$178$	−9.80319e22	−0.230122
$179$	5.77287e23	1.27772	0.638859	−	0.769324i	$-0.279407\pi$
$179$	5.77287e23	1.27772	0.638859	+	0.769324i	$0.279407\pi$
$180$	0	0
$181$	2.15236e23	0.423925	0.211963	−	0.977278i	$-0.432014\pi$
$181$	2.15236e23	0.423925	0.211963	+	0.977278i	$0.432014\pi$
$182$	1.34759e24	2.50501
$183$	−3.90029e23	−0.684480
$184$	8.72684e22	0.144634
$185$	0	0
$186$	−4.33540e22	−0.0641421
$187$	−3.86856e23	−0.541019
$188$	−4.46222e22	−0.0590057
$189$	2.61138e23	0.326604
$190$	0	0
$191$	4.41173e23	0.494036	0.247018	−	0.969011i	$-0.420549\pi$
$191$	4.41173e23	0.494036	0.247018	+	0.969011i	$0.420549\pi$
$192$	5.10981e23	0.541679
$193$	7.07886e23	0.710578	0.355289	−	0.934757i	$-0.384383\pi$
$193$	7.07886e23	0.710578	0.355289	+	0.934757i	$0.384383\pi$
$194$	−1.69062e24	−1.60741
$195$	0	0
$196$	1.22761e23	0.104803
$197$	−2.25891e23	−0.182812	−0.0914058	−	0.995814i	$-0.529136\pi$
$197$	−2.25891e23	−0.182812	−0.0914058	+	0.995814i	$0.529136\pi$
$198$	−2.20941e23	−0.169548
$199$	−5.29674e22	−0.0385524	−0.0192762	−	0.999814i	$-0.506136\pi$
$199$	−5.29674e22	−0.0385524	−0.0192762	+	0.999814i	$0.506136\pi$
$200$	0	0
$201$	1.51996e24	0.996033
$202$	−9.05034e23	−0.562957
$203$	5.65106e24	3.33750
$204$	−6.27088e22	−0.0351732
$205$	0	0
$206$	2.88415e24	1.46020
$207$	1.03269e23	0.0496912
$208$	3.30575e24	1.51219
$209$	−1.11898e24	−0.486726
$210$	0	0
$211$	2.91715e24	1.14813	0.574067	−	0.818808i	$-0.305365\pi$
$211$	2.91715e24	1.14813	0.574067	+	0.818808i	$0.305365\pi$
$212$	−1.18639e23	−0.0444324
$213$	1.96231e24	0.699490
$214$	−1.81450e23	−0.0615763
$215$	0	0
$216$	6.06669e23	0.186718
$217$	6.25828e23	0.183496
$218$	−6.93202e24	−1.93672
$219$	1.57143e24	0.418439
$220$	0	0
$221$	6.48670e24	1.57002
$222$	7.75796e23	0.179078
$223$	−4.10996e24	−0.904975	−0.452487	−	0.891771i	$-0.649463\pi$
$223$	−4.10996e24	−0.904975	−0.452487	+	0.891771i	$0.649463\pi$
$224$	8.99569e23	0.188986
$225$	0	0
$226$	1.89546e24	0.362723
$227$	−4.85462e24	−0.886919	−0.443460	−	0.896294i	$-0.646249\pi$
$227$	−4.85462e24	−0.886919	−0.443460	+	0.896294i	$0.646249\pi$
$228$	−1.81384e23	−0.0316435
$229$	−3.91845e24	−0.652893	−0.326446	−	0.945216i	$-0.605851\pi$
$229$	−3.91845e24	−0.652893	−0.326446	+	0.945216i	$0.605851\pi$
$230$	0	0
$231$	3.18935e24	0.485039
$232$	1.31284e25	1.90804
$233$	−1.12801e25	−1.56702	−0.783511	−	0.621378i	$-0.786573\pi$
$233$	−1.12801e25	−1.56702	−0.783511	+	0.621378i	$0.786573\pi$
$234$	3.70469e24	0.492023
$235$	0	0
$236$	−9.65779e22	−0.0117301
$237$	−4.64037e24	−0.539132
$238$	1.71442e25	1.90572
$239$	−9.29369e24	−0.988577	−0.494288	−	0.869298i	$-0.664571\pi$
$239$	−9.29369e24	−0.988577	−0.494288	+	0.869298i	$0.664571\pi$
$240$	0	0
$241$	−2.88908e24	−0.281567	−0.140783	−	0.990040i	$-0.544962\pi$
$241$	−2.88908e24	−0.281567	−0.140783	+	0.990040i	$0.544962\pi$
$242$	8.31293e24	0.775698
$243$	7.17898e23	0.0641500
$244$	−7.72163e23	−0.0660870
$245$	0	0
$246$	−1.02888e25	−0.808249
$247$	1.87627e25	1.41246
$248$	1.45391e24	0.104904
$249$	−9.97688e24	−0.690084
$250$	0	0
$251$	1.09500e25	0.696370	0.348185	−	0.937426i	$-0.386798\pi$
$251$	1.09500e25	0.696370	0.348185	+	0.937426i	$0.386798\pi$
$252$	5.16990e23	0.0315338
$253$	1.26125e24	0.0737964
$254$	−1.60855e25	−0.902983
$255$	0	0
$256$	3.22485e24	0.166721
$257$	1.13821e25	0.564840	0.282420	−	0.959291i	$-0.408863\pi$
$257$	1.13821e25	0.564840	0.282420	+	0.959291i	$0.408863\pi$
$258$	−4.19245e24	−0.199738
$259$	−1.11988e25	−0.512302
$260$	0	0
$261$	1.55354e25	0.655537
$262$	3.92119e25	1.58948
$263$	−3.72385e23	−0.0145030	−0.00725148	−	0.999974i	$-0.502308\pi$
$263$	−3.72385e23	−0.0145030	−0.00725148	+	0.999974i	$0.502308\pi$
$264$	7.40942e24	0.277296
$265$	0	0
$266$	4.95895e25	1.71448
$267$	3.89032e24	0.129306
$268$	3.00915e24	0.0961676
$269$	3.72748e25	1.14556	0.572779	−	0.819710i	$-0.305865\pi$
$269$	3.72748e25	1.14556	0.572779	+	0.819710i	$0.305865\pi$
$270$	0	0
$271$	2.21667e25	0.630267	0.315133	−	0.949047i	$-0.397951\pi$
$271$	2.21667e25	0.630267	0.315133	+	0.949047i	$0.397951\pi$
$272$	4.20562e25	1.15042
$273$	−5.34782e25	−1.40757
$274$	−2.92735e25	−0.741470
$275$	0	0
$276$	2.04447e23	0.00479771
$277$	−1.52006e25	−0.343419	−0.171709	−	0.985148i	$-0.554929\pi$
$277$	−1.52006e25	−0.343419	−0.171709	+	0.985148i	$0.554929\pi$
$278$	−2.04389e25	−0.444619
$279$	1.72047e24	0.0360415
$280$	0	0
$281$	−4.82436e25	−0.937612	−0.468806	−	0.883301i	$-0.655316\pi$
$281$	−4.82436e25	−0.937612	−0.468806	+	0.883301i	$0.655316\pi$
$282$	3.35377e25	0.627939
$283$	−3.95762e25	−0.713965	−0.356983	−	0.934111i	$-0.616194\pi$
$283$	−3.95762e25	−0.713965	−0.356983	+	0.934111i	$0.616194\pi$
$284$	3.88490e24	0.0675362
$285$	0	0
$286$	4.52464e25	0.730703
$287$	1.48522e26	2.31222
$288$	2.47302e24	0.0371198
$289$	1.34326e25	0.194416
$290$	0	0
$291$	6.70910e25	0.903208
$292$	3.11105e24	0.0404005
$293$	6.72372e25	0.842364	0.421182	−	0.906976i	$-0.361615\pi$
$293$	6.72372e25	0.842364	0.421182	+	0.906976i	$0.361615\pi$
$294$	−9.22664e25	−1.11531
$295$	0	0
$296$	−2.60169e25	−0.292882
$297$	8.76789e24	0.0952692
$298$	−1.36096e26	−1.42749
$299$	−2.11483e25	−0.214155
$300$	0	0
$301$	6.05192e25	0.571406
$302$	−1.81983e26	−1.65942
$303$	3.59156e25	0.316327
$304$	1.21647e26	1.03497
$305$	0	0
$306$	4.71315e25	0.374314
$307$	1.82823e25	0.140307	0.0701533	−	0.997536i	$-0.477651\pi$
$307$	1.82823e25	0.140307	0.0701533	+	0.997536i	$0.477651\pi$
$308$	6.31414e24	0.0468308
$309$	−1.14455e26	−0.820488
$310$	0	0
$311$	1.19682e26	0.801762	0.400881	−	0.916130i	$-0.368704\pi$
$311$	1.19682e26	0.801762	0.400881	+	0.916130i	$0.368704\pi$
$312$	−1.24239e26	−0.804703
$313$	2.72575e26	1.70715	0.853573	−	0.520974i	$-0.174431\pi$
$313$	2.72575e26	1.70715	0.853573	+	0.520974i	$0.174431\pi$
$314$	−1.64202e26	−0.994530
$315$	0	0
$316$	−9.18680e24	−0.0520535
$317$	6.50489e25	0.356548	0.178274	−	0.983981i	$-0.442949\pi$
$317$	6.50489e25	0.356548	0.178274	+	0.983981i	$0.442949\pi$
$318$	8.91679e25	0.472851
$319$	1.89738e26	0.973538
$320$	0	0
$321$	7.20073e24	0.0345998
$322$	−5.58945e25	−0.259946
$323$	2.38701e26	1.07455
$324$	1.42126e24	0.00619373
$325$	0	0
$326$	4.64375e26	1.89707
$327$	2.75092e26	1.08825
$328$	3.45041e26	1.32189
$329$	−4.84127e26	−1.79640
$330$	0	0
$331$	2.10948e26	0.734482	0.367241	−	0.930126i	$-0.380302\pi$
$331$	2.10948e26	0.734482	0.367241	+	0.930126i	$0.380302\pi$
$332$	−1.97518e25	−0.0666280
$333$	−3.07869e25	−0.100624
$334$	8.31254e25	0.263267
$335$	0	0
$336$	−3.46723e26	−1.03138
$337$	−1.58978e26	−0.458377	−0.229189	−	0.973382i	$-0.573607\pi$
$337$	−1.58978e26	−0.458377	−0.229189	+	0.973382i	$0.573607\pi$
$338$	−3.91056e26	−1.09298
$339$	−7.52198e25	−0.203815
$340$	0	0
$341$	2.10126e25	0.0535253
$342$	1.36327e26	0.336750
$343$	6.23473e26	1.49358
$344$	1.40597e26	0.326671
$345$	0	0
$346$	−1.99982e26	−0.437211
$347$	3.27291e26	0.694184	0.347092	−	0.937831i	$-0.387169\pi$
$347$	3.27291e26	0.694184	0.347092	+	0.937831i	$0.387169\pi$
$348$	3.07563e25	0.0632925
$349$	6.28483e26	1.25495	0.627475	−	0.778637i	$-0.284088\pi$
$349$	6.28483e26	1.25495	0.627475	+	0.778637i	$0.284088\pi$
$350$	0	0
$351$	−1.47018e26	−0.276468
$352$	3.02037e25	0.0551267
$353$	−5.78370e26	−1.02464	−0.512320	−	0.858795i	$-0.671214\pi$
$353$	−5.78370e26	−1.02464	−0.512320	+	0.858795i	$0.671214\pi$
$354$	7.25872e25	0.124832
$355$	0	0
$356$	7.70189e24	0.0124845
$357$	−6.80356e26	−1.07083
$358$	−8.58985e26	−1.31285
$359$	−5.85883e25	−0.0869599	−0.0434799	−	0.999054i	$-0.513844\pi$
$359$	−5.85883e25	−0.0869599	−0.0434799	+	0.999054i	$0.513844\pi$
$360$	0	0
$361$	−2.37694e25	−0.0332807
$362$	−3.20264e26	−0.435581
$363$	−3.29893e26	−0.435866
$364$	−1.05874e26	−0.135902
$365$	0	0
$366$	5.80352e26	0.703299
$367$	6.27704e25	0.0739198	0.0369599	−	0.999317i	$-0.488233\pi$
$367$	6.27704e25	0.0739198	0.0369599	+	0.999317i	$0.488233\pi$
$368$	−1.37114e26	−0.156920
$369$	4.08303e26	0.454156
$370$	0	0
$371$	−1.28717e27	−1.35272
$372$	3.40612e24	0.00347983
$373$	1.70679e27	1.69526	0.847631	−	0.530587i	$-0.178028\pi$
$373$	1.70679e27	1.69526	0.847631	+	0.530587i	$0.178028\pi$
$374$	5.75630e26	0.555893
$375$	0	0
$376$	−1.12471e27	−1.02700
$377$	−3.18148e27	−2.82518
$378$	−3.88565e26	−0.335583
$379$	2.35215e27	1.97585	0.987923	−	0.154946i	$-0.0495204\pi$
$379$	2.35215e27	1.97585	0.987923	+	0.154946i	$0.0495204\pi$
$380$	0	0
$381$	6.38340e26	0.507388
$382$	−6.56452e26	−0.507619
$383$	−9.18944e26	−0.691356	−0.345678	−	0.938353i	$-0.612351\pi$
$383$	−9.18944e26	−0.691356	−0.345678	+	0.938353i	$0.612351\pi$
$384$	−8.48155e26	−0.620865
$385$	0	0
$386$	−1.05331e27	−0.730114
$387$	1.66374e26	0.112233
$388$	1.32824e26	0.0872053
$389$	−1.24941e27	−0.798428	−0.399214	−	0.916858i	$-0.630717\pi$
$389$	−1.24941e27	−0.798428	−0.399214	+	0.916858i	$0.630717\pi$
$390$	0	0
$391$	−2.69051e26	−0.162922
$392$	3.09422e27	1.82409
$393$	−1.55609e27	−0.893129
$394$	3.36119e26	0.187838
$395$	0	0
$396$	1.73583e25	0.00919830
$397$	3.17780e27	1.63993	0.819967	−	0.572410i	$-0.193992\pi$
$397$	3.17780e27	1.63993	0.819967	+	0.572410i	$0.193992\pi$
$398$	7.88139e25	0.0396123
$399$	−1.96792e27	−0.963369
$400$	0	0
$401$	−2.00476e27	−0.931208	−0.465604	−	0.884993i	$-0.654163\pi$
$401$	−2.00476e27	−0.931208	−0.465604	+	0.884993i	$0.654163\pi$
$402$	−2.26166e27	−1.02342
$403$	−3.52334e26	−0.155329
$404$	7.11042e25	0.0305415
$405$	0	0
$406$	−8.40860e27	−3.42926
$407$	−3.76009e26	−0.149437
$408$	−1.58059e27	−0.612190
$409$	1.65077e27	0.623150	0.311575	−	0.950222i	$-0.399144\pi$
$409$	1.65077e27	0.623150	0.311575	+	0.950222i	$0.399144\pi$
$410$	0	0
$411$	1.16170e27	0.416633
$412$	−2.26594e26	−0.0792187
$413$	−1.04782e27	−0.357117
$414$	−1.53660e26	−0.0510574
$415$	0	0
$416$	−5.06447e26	−0.159976
$417$	8.11105e26	0.249832
$418$	1.66500e27	0.500108
$419$	2.41867e27	0.708484	0.354242	−	0.935154i	$-0.384739\pi$
$419$	2.41867e27	0.708484	0.354242	+	0.935154i	$0.384739\pi$
$420$	0	0
$421$	−5.33902e27	−1.48764	−0.743822	−	0.668377i	$-0.766989\pi$
$421$	−5.33902e27	−1.48764	−0.743822	+	0.668377i	$0.766989\pi$
$422$	−4.34063e27	−1.17970
$423$	−1.33092e27	−0.352840
$424$	−2.99031e27	−0.773347
$425$	0	0
$426$	−2.91986e27	−0.718722
$427$	−8.37755e27	−2.01198
$428$	1.42557e25	0.00334063
$429$	−1.79557e27	−0.410583
$430$	0	0
$431$	1.47687e27	0.321610	0.160805	−	0.986986i	$-0.448591\pi$
$431$	1.47687e27	0.321610	0.160805	+	0.986986i	$0.448591\pi$
$432$	−9.53182e26	−0.202580
$433$	−1.41696e27	−0.293924	−0.146962	−	0.989142i	$-0.546949\pi$
$433$	−1.41696e27	−0.293924	−0.146962	+	0.989142i	$0.546949\pi$
$434$	−9.31213e26	−0.188541
$435$	0	0
$436$	5.44616e26	0.105071
$437$	−7.78226e26	−0.146572
$438$	−2.33824e27	−0.429943
$439$	1.76504e27	0.316867	0.158433	−	0.987370i	$-0.449356\pi$
$439$	1.76504e27	0.316867	0.158433	+	0.987370i	$0.449356\pi$
$440$	0	0
$441$	3.66153e27	0.626696
$442$	−9.65202e27	−1.61318
$443$	−1.08549e28	−1.77169	−0.885843	−	0.463984i	$-0.846419\pi$
$443$	−1.08549e28	−1.77169	−0.885843	+	0.463984i	$0.846419\pi$
$444$	−6.09506e25	−0.00971531
$445$	0	0
$446$	6.11550e27	0.929856
$447$	5.40087e27	0.802111
$448$	1.09755e28	1.59223
$449$	7.34487e27	1.04087	0.520436	−	0.853900i	$-0.325769\pi$
$449$	7.34487e27	1.04087	0.520436	+	0.853900i	$0.325769\pi$
$450$	0	0
$451$	4.98672e27	0.674468
$452$	−1.48917e26	−0.0196784
$453$	7.22186e27	0.932433
$454$	7.22353e27	0.911304
$455$	0	0
$456$	−4.57183e27	−0.550755
$457$	1.52263e28	1.79256	0.896282	−	0.443485i	$-0.146258\pi$
$457$	1.52263e28	1.79256	0.896282	+	0.443485i	$0.146258\pi$
$458$	5.83054e27	0.670843
$459$	−1.87038e27	−0.210327
$460$	0	0
$461$	−1.36988e28	−1.47171	−0.735855	−	0.677140i	$-0.763219\pi$
$461$	−1.36988e28	−1.47171	−0.735855	+	0.677140i	$0.763219\pi$
$462$	−4.74566e27	−0.498375
$463$	8.82461e27	0.905931	0.452965	−	0.891528i	$-0.350366\pi$
$463$	8.82461e27	0.905931	0.452965	+	0.891528i	$0.350366\pi$
$464$	−2.06270e28	−2.07013
$465$	0	0
$466$	1.67844e28	1.61011
$467$	−1.04648e28	−0.981532	−0.490766	−	0.871291i	$-0.663283\pi$
$467$	−1.04648e28	−0.981532	−0.490766	+	0.871291i	$0.663283\pi$
$468$	−2.91059e26	−0.0266932
$469$	3.26477e28	2.92777
$470$	0	0
$471$	6.51625e27	0.558828
$472$	−2.43426e27	−0.204163
$473$	2.03198e27	0.166677
$474$	6.90473e27	0.553955
$475$	0	0
$476$	−1.34694e27	−0.103389
$477$	−3.53856e27	−0.265695
$478$	1.38287e28	1.01576
$479$	−9.99111e27	−0.717944	−0.358972	−	0.933348i	$-0.616873\pi$
$479$	−9.99111e27	−0.717944	−0.358972	+	0.933348i	$0.616873\pi$
$480$	0	0
$481$	6.30483e27	0.433661
$482$	4.29887e27	0.289308
$483$	2.21813e27	0.146064
$484$	−6.53107e26	−0.0420831
$485$	0	0
$486$	−1.06821e27	−0.0659138
$487$	2.33380e28	1.40932	0.704661	−	0.709544i	$-0.251099\pi$
$487$	2.33380e28	1.40932	0.704661	+	0.709544i	$0.251099\pi$
$488$	−1.94625e28	−1.15025
$489$	−1.84284e28	−1.06597
$490$	0	0
$491$	2.29620e28	1.27249	0.636245	−	0.771487i	$-0.280487\pi$
$491$	2.29620e28	1.27249	0.636245	+	0.771487i	$0.280487\pi$
$492$	8.08340e26	0.0438491
$493$	−4.04752e28	−2.14930
$494$	−2.79183e28	−1.45130
$495$	0	0
$496$	−2.28434e27	−0.113816
$497$	4.21490e28	2.05611
$498$	1.48453e28	0.709057
$499$	−1.58930e28	−0.743276	−0.371638	−	0.928378i	$-0.621204\pi$
$499$	−1.58930e28	−0.743276	−0.371638	+	0.928378i	$0.621204\pi$
$500$	0	0
$501$	−3.29877e27	−0.147930
$502$	−1.62933e28	−0.715516
$503$	4.31616e28	1.85624	0.928120	−	0.372282i	$-0.121425\pi$
$503$	4.31616e28	1.85624	0.928120	+	0.372282i	$0.121425\pi$
$504$	1.30308e28	0.548846
$505$	0	0
$506$	−1.87670e27	−0.0758253
$507$	1.55188e28	0.614148
$508$	1.26376e27	0.0489886
$509$	3.30343e28	1.25438	0.627189	−	0.778867i	$-0.284206\pi$
$509$	3.30343e28	1.25438	0.627189	+	0.778867i	$0.284206\pi$
$510$	0	0
$511$	3.37532e28	1.22997
$512$	2.53241e28	0.904065
$513$	−5.41004e27	−0.189221
$514$	−1.69362e28	−0.580369
$515$	0	0
$516$	3.29380e26	0.0108362
$517$	−1.62549e28	−0.524003
$518$	1.66635e28	0.526387
$519$	7.93613e27	0.245670
$520$	0	0
$521$	4.42108e27	0.131442	0.0657208	−	0.997838i	$-0.479065\pi$
$521$	4.42108e27	0.131442	0.0657208	+	0.997838i	$0.479065\pi$
$522$	−2.31162e28	−0.673560
$523$	6.75490e28	1.92909	0.964543	−	0.263927i	$-0.0850179\pi$
$523$	6.75490e28	1.92909	0.964543	+	0.263927i	$0.0850179\pi$
$524$	−3.08069e27	−0.0862321
$525$	0	0
$526$	5.54097e26	0.0149017
$527$	−4.48244e27	−0.118169
$528$	−1.16415e28	−0.300851
$529$	−3.85944e28	−0.977777
$530$	0	0
$531$	−2.88057e27	−0.0701433
$532$	−3.89601e27	−0.0930138
$533$	−8.36160e28	−1.95728
$534$	−5.78868e27	−0.132861
$535$	0	0
$536$	7.58462e28	1.67380
$537$	3.40882e28	0.737690
$538$	−5.54638e28	−1.17705
$539$	4.47193e28	0.930706
$540$	0	0
$541$	9.90502e27	0.198282	0.0991412	−	0.995073i	$-0.468390\pi$
$541$	9.90502e27	0.198282	0.0991412	+	0.995073i	$0.468390\pi$
$542$	−3.29834e28	−0.647595
$543$	1.27095e28	0.244753
$544$	−6.44308e27	−0.121704
$545$	0	0
$546$	7.95740e28	1.44627
$547$	1.52806e28	0.272442	0.136221	−	0.990678i	$-0.456504\pi$
$547$	1.52806e28	0.272442	0.136221	+	0.990678i	$0.456504\pi$
$548$	2.29988e27	0.0402262
$549$	−2.30308e28	−0.395185
$550$	0	0
$551$	−1.17074e29	−1.93361
$552$	5.15311e27	0.0835043
$553$	−9.96718e28	−1.58474
$554$	2.26181e28	0.352861
$555$	0	0
$556$	1.60579e27	0.0241214
$557$	6.67397e28	0.983795	0.491897	−	0.870653i	$-0.336304\pi$
$557$	6.67397e28	0.983795	0.491897	+	0.870653i	$0.336304\pi$
$558$	−2.56001e27	−0.0370325
$559$	−3.40716e28	−0.483692
$560$	0	0
$561$	−2.28435e28	−0.312357
$562$	7.17850e28	0.963390
$563$	−4.32189e28	−0.569293	−0.284646	−	0.958633i	$-0.591876\pi$
$563$	−4.32189e28	−0.569293	−0.284646	+	0.958633i	$0.591876\pi$
$564$	−2.63490e27	−0.0340669
$565$	0	0
$566$	5.88883e28	0.733595
$567$	1.54199e28	0.188565
$568$	9.79195e28	1.17547
$569$	−2.28226e28	−0.268960	−0.134480	−	0.990916i	$-0.542936\pi$
$569$	−2.28226e28	−0.268960	−0.134480	+	0.990916i	$0.542936\pi$
$570$	0	0
$571$	1.97421e28	0.224240	0.112120	−	0.993695i	$-0.464236\pi$
$571$	1.97421e28	0.224240	0.112120	+	0.993695i	$0.464236\pi$
$572$	−3.55479e27	−0.0396420
$573$	2.60508e28	0.285232
$574$	−2.20996e29	−2.37579
$575$	0	0
$576$	3.01729e28	0.312738
$577$	1.13098e29	1.15109	0.575546	−	0.817770i	$-0.304790\pi$
$577$	1.13098e29	1.15109	0.575546	+	0.817770i	$0.304790\pi$
$578$	−1.99873e28	−0.199762
$579$	4.18000e28	0.410252
$580$	0	0
$581$	−2.14296e29	−2.02845
$582$	−9.98294e28	−0.928040
$583$	−4.32175e28	−0.394584
$584$	7.84145e28	0.703172
$585$	0	0
$586$	−1.00047e29	−0.865523
$587$	6.65281e28	0.565334	0.282667	−	0.959218i	$-0.408781\pi$
$587$	6.65281e28	0.565334	0.282667	+	0.959218i	$0.408781\pi$
$588$	7.24893e27	0.0605079
$589$	−1.29654e28	−0.106310
$590$	0	0
$591$	−1.33386e28	−0.105546
$592$	4.08770e28	0.317761
$593$	3.42927e28	0.261895	0.130948	−	0.991389i	$-0.458198\pi$
$593$	3.42927e28	0.261895	0.130948	+	0.991389i	$0.458198\pi$
$594$	−1.30464e28	−0.0978885
$595$	0	0
$596$	1.06924e28	0.0774443
$597$	−3.12767e27	−0.0222582
$598$	3.14680e28	0.220043
$599$	−7.57014e28	−0.520143	−0.260071	−	0.965589i	$-0.583746\pi$
$599$	−7.57014e28	−0.520143	−0.260071	+	0.965589i	$0.583746\pi$
$600$	0	0
$601$	2.14743e29	1.42475	0.712374	−	0.701800i	$-0.247620\pi$
$601$	2.14743e29	1.42475	0.712374	+	0.701800i	$0.247620\pi$
$602$	−9.00507e28	−0.587116
$603$	8.97521e28	0.575060
$604$	1.42975e28	0.0900270
$605$	0	0
$606$	−5.34413e28	−0.325024
$607$	−3.33523e28	−0.199363	−0.0996815	−	0.995019i	$-0.531782\pi$
$607$	−3.33523e28	−0.199363	−0.0996815	+	0.995019i	$0.531782\pi$
$608$	−1.86365e28	−0.109491
$609$	3.33689e29	1.92691
$610$	0	0
$611$	2.72558e29	1.52064
$612$	−3.70289e27	−0.0203073
$613$	−6.67007e28	−0.359580	−0.179790	−	0.983705i	$-0.557542\pi$
$613$	−6.67007e28	−0.359580	−0.179790	+	0.983705i	$0.557542\pi$
$614$	−2.72035e28	−0.144164
$615$	0	0
$616$	1.59149e29	0.815092
$617$	2.22057e29	1.11807	0.559037	−	0.829143i	$-0.311171\pi$
$617$	2.22057e29	1.11807	0.559037	+	0.829143i	$0.311171\pi$
$618$	1.70306e29	0.843047
$619$	1.99692e29	0.971869	0.485935	−	0.873995i	$-0.338479\pi$
$619$	1.99692e29	0.971869	0.485935	+	0.873995i	$0.338479\pi$
$620$	0	0
$621$	6.09790e27	0.0286892
$622$	−1.78084e29	−0.823806
$623$	8.35614e28	0.380085
$624$	1.95201e29	0.873061
$625$	0	0
$626$	−4.05583e29	−1.75408
$627$	−6.60744e28	−0.281011
$628$	1.29006e28	0.0539552
$629$	8.02107e28	0.329914
$630$	0	0
$631$	1.07700e29	0.428456	0.214228	−	0.976784i	$-0.431276\pi$
$631$	1.07700e29	0.428456	0.214228	+	0.976784i	$0.431276\pi$
$632$	−2.31555e29	−0.905993
$633$	1.72255e29	0.662875
$634$	−9.67908e28	−0.366351
$635$	0	0
$636$	−7.00550e27	−0.0256531
$637$	−7.49841e29	−2.70088
$638$	−2.82325e29	−1.00030
$639$	1.15872e29	0.403851
$640$	0	0
$641$	−4.82362e28	−0.162691	−0.0813455	−	0.996686i	$-0.525922\pi$
$641$	−4.82362e28	−0.162691	−0.0813455	+	0.996686i	$0.525922\pi$
$642$	−1.07145e28	−0.0355511
$643$	−1.12338e29	−0.366699	−0.183350	−	0.983048i	$-0.558694\pi$
$643$	−1.12338e29	−0.366699	−0.183350	+	0.983048i	$0.558694\pi$
$644$	4.39136e27	0.0141026
$645$	0	0
$646$	−3.55180e29	−1.10410
$647$	2.58352e28	0.0790162	0.0395081	−	0.999219i	$-0.487421\pi$
$647$	2.58352e28	0.0790162	0.0395081	+	0.999219i	$0.487421\pi$
$648$	3.58232e28	0.107802
$649$	−3.51812e28	−0.104170
$650$	0	0
$651$	3.69545e28	0.105942
$652$	−3.64837e28	−0.102920
$653$	−5.85458e29	−1.62520	−0.812601	−	0.582820i	$-0.801949\pi$
$653$	−5.85458e29	−1.62520	−0.812601	+	0.582820i	$0.801949\pi$
$654$	−4.09329e29	−1.11817
$655$	0	0
$656$	−5.42120e29	−1.43418
$657$	9.27913e28	0.241586
$658$	7.20366e29	1.84579
$659$	−1.22801e29	−0.309673	−0.154837	−	0.987940i	$-0.549485\pi$
$659$	−1.22801e29	−0.309673	−0.154837	+	0.987940i	$0.549485\pi$
$660$	0	0
$661$	4.50136e29	1.09958	0.549792	−	0.835301i	$-0.314707\pi$
$661$	4.50136e29	1.09958	0.549792	+	0.835301i	$0.314707\pi$
$662$	−3.13884e29	−0.754676
$663$	3.83033e29	0.906451
$664$	−4.97848e29	−1.15966
$665$	0	0
$666$	4.58100e28	0.103391
$667$	1.31959e29	0.293170
$668$	−6.53076e27	−0.0142827
$669$	−2.42689e29	−0.522487
$670$	0	0
$671$	−2.81282e29	−0.586889
$672$	5.31186e28	0.109111
$673$	−6.19299e29	−1.25240	−0.626199	−	0.779663i	$-0.715390\pi$
$673$	−6.19299e29	−1.25240	−0.626199	+	0.779663i	$0.715390\pi$
$674$	2.36555e29	0.470980
$675$	0	0
$676$	3.07234e28	0.0592964
$677$	−9.15907e28	−0.174049	−0.0870243	−	0.996206i	$-0.527736\pi$
$677$	−9.15907e28	−0.174049	−0.0870243	+	0.996206i	$0.527736\pi$
$678$	1.11925e29	0.209418
$679$	1.44107e30	2.65492
$680$	0	0
$681$	−2.86661e29	−0.512063
$682$	−3.12661e28	−0.0549969
$683$	1.08753e30	1.88375	0.941876	−	0.335961i	$-0.109061\pi$
$683$	1.08753e30	1.88375	0.941876	+	0.335961i	$0.109061\pi$
$684$	−1.07106e28	−0.0182694
$685$	0	0
$686$	−9.27708e29	−1.53465
$687$	−2.31381e29	−0.376948
$688$	−2.20902e29	−0.354422
$689$	7.24660e29	1.14507
$690$	0	0
$691$	−4.53419e29	−0.694992	−0.347496	−	0.937681i	$-0.612968\pi$
$691$	−4.53419e29	−0.694992	−0.347496	+	0.937681i	$0.612968\pi$
$692$	1.57116e28	0.0237196
$693$	1.88328e29	0.280037
$694$	−4.86999e29	−0.713269
$695$	0	0
$696$	7.75218e29	1.10161
$697$	−1.06377e30	−1.48903
$698$	−9.35163e29	−1.28945
$699$	−6.66078e29	−0.904721
$700$	0	0
$701$	−9.03929e29	−1.19150	−0.595752	−	0.803169i	$-0.703146\pi$
$701$	−9.03929e29	−1.19150	−0.595752	+	0.803169i	$0.703146\pi$
$702$	2.18758e29	0.284069
$703$	2.32009e29	0.296806
$704$	3.68510e29	0.464448
$705$	0	0
$706$	8.60597e29	1.05281
$707$	7.71441e29	0.929821
$708$	−5.70283e27	−0.00677238
$709$	−9.52975e29	−1.11505	−0.557527	−	0.830159i	$-0.688250\pi$
$709$	−9.52975e29	−1.11505	−0.557527	+	0.830159i	$0.688250\pi$
$710$	0	0
$711$	−2.74009e29	−0.311268
$712$	1.94128e29	0.217294
$713$	1.46139e28	0.0161185
$714$	1.01235e30	1.10027
$715$	0	0
$716$	6.74864e28	0.0712245
$717$	−5.48783e29	−0.570755
$718$	8.71776e28	0.0893507
$719$	−1.28953e30	−1.30250	−0.651249	−	0.758864i	$-0.725755\pi$
$719$	−1.28953e30	−1.30250	−0.651249	+	0.758864i	$0.725755\pi$
$720$	0	0
$721$	−2.45842e30	−2.41177
$722$	3.53681e28	0.0341957
$723$	−1.70597e29	−0.162563
$724$	2.51616e28	0.0236311
$725$	0	0
$726$	4.90870e29	0.447850
$727$	−1.09976e30	−0.988981	−0.494490	−	0.869183i	$-0.664645\pi$
$727$	−1.09976e30	−0.988981	−0.494490	+	0.869183i	$0.664645\pi$
$728$	−2.66857e30	−2.36537
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−4.33463e29	−0.367976
$732$	−4.55955e28	−0.0381553
$733$	−5.44126e29	−0.448856	−0.224428	−	0.974491i	$-0.572051\pi$
$733$	−5.44126e29	−0.448856	−0.224428	+	0.974491i	$0.572051\pi$
$734$	−9.34004e28	−0.0759522
$735$	0	0
$736$	2.10061e28	0.0166008
$737$	1.09617e30	0.854021
$738$	−6.07542e29	−0.466643
$739$	−3.50055e29	−0.265076	−0.132538	−	0.991178i	$-0.542313\pi$
$739$	−3.50055e29	−0.265076	−0.132538	+	0.991178i	$0.542313\pi$
$740$	0	0
$741$	1.10792e30	0.815486
$742$	1.91526e30	1.38991
$743$	−1.31687e30	−0.942239	−0.471120	−	0.882069i	$-0.656150\pi$
$743$	−1.31687e30	−0.942239	−0.471120	+	0.882069i	$0.656150\pi$
$744$	8.58517e28	0.0605666
$745$	0	0
$746$	−2.53965e30	−1.74187
$747$	−5.89125e29	−0.398420
$748$	−4.52245e28	−0.0301583
$749$	1.54666e29	0.101704
$750$	0	0
$751$	2.67518e30	1.71054	0.855271	−	0.518181i	$-0.173391\pi$
$751$	2.67518e30	1.71054	0.855271	+	0.518181i	$0.173391\pi$
$752$	1.76712e30	1.11424
$753$	6.46586e29	0.402049
$754$	4.73395e30	2.90285
$755$	0	0
$756$	3.05277e28	0.0182060
$757$	1.57579e30	0.926812	0.463406	−	0.886146i	$-0.346627\pi$
$757$	1.57579e30	0.926812	0.463406	+	0.886146i	$0.346627\pi$
$758$	−3.49993e30	−2.03017
$759$	7.44754e28	0.0426064
$760$	0	0
$761$	1.14447e30	0.636893	0.318447	−	0.947941i	$-0.396839\pi$
$761$	1.14447e30	0.636893	0.318447	+	0.947941i	$0.396839\pi$
$762$	−9.49831e29	−0.521338
$763$	5.90878e30	3.19883
$764$	5.15743e28	0.0275393
$765$	0	0
$766$	1.36736e30	0.710364
$767$	5.89910e29	0.302297
$768$	1.90424e29	0.0962562
$769$	−1.55372e29	−0.0774724	−0.0387362	−	0.999249i	$-0.512333\pi$
$769$	−1.55372e29	−0.0774724	−0.0387362	+	0.999249i	$0.512333\pi$
$770$	0	0
$771$	6.72103e29	0.326110
$772$	8.27538e28	0.0396101
$773$	−2.20959e30	−1.04334	−0.521671	−	0.853147i	$-0.674691\pi$
$773$	−2.20959e30	−1.04334	−0.521671	+	0.853147i	$0.674691\pi$
$774$	−2.47560e29	−0.115319
$775$	0	0
$776$	3.34785e30	1.51781
$777$	−6.61281e29	−0.295778
$778$	1.85909e30	0.820380
$779$	−3.07695e30	−1.33961
$780$	0	0
$781$	1.41518e30	0.599759
$782$	4.00340e29	0.167401
$783$	9.17350e29	0.378474
$784$	−4.86155e30	−1.97905
$785$	0	0
$786$	2.31542e30	0.917684
$787$	2.75620e30	1.07789	0.538947	−	0.842340i	$-0.318822\pi$
$787$	2.75620e30	1.07789	0.538947	+	0.842340i	$0.318822\pi$
$788$	−2.64073e28	−0.0101906
$789$	−2.19889e28	−0.00837329
$790$	0	0
$791$	−1.61567e30	−0.599099
$792$	4.37519e29	0.160097
$793$	4.71647e30	1.70313
$794$	−4.72847e30	−1.68502
$795$	0	0
$796$	−6.19203e27	−0.00214905
$797$	−4.23039e30	−1.44900	−0.724500	−	0.689275i	$-0.757929\pi$
$797$	−4.23039e30	−1.44900	−0.724500	+	0.689275i	$0.757929\pi$
$798$	2.92821e30	0.989855
$799$	3.46751e30	1.15685
$800$	0	0
$801$	2.29720e29	0.0746546
$802$	2.98302e30	0.956810
$803$	1.13329e30	0.358779
$804$	1.77687e29	0.0555224
$805$	0	0
$806$	5.24263e29	0.159599
$807$	2.20104e30	0.661388
$808$	1.79219e30	0.531576
$809$	3.93165e30	1.15111	0.575553	−	0.817764i	$-0.304787\pi$
$809$	3.93165e30	1.15111	0.575553	+	0.817764i	$0.304787\pi$
$810$	0	0
$811$	−1.50931e30	−0.430585	−0.215293	−	0.976550i	$-0.569071\pi$
$811$	−1.50931e30	−0.430585	−0.215293	+	0.976550i	$0.569071\pi$
$812$	6.60624e29	0.186044
$813$	1.30892e30	0.363885
$814$	5.59491e29	0.153545
$815$	0	0
$816$	2.48337e30	0.664195
$817$	−1.25379e30	−0.331049
$818$	−2.45630e30	−0.640282
$819$	−3.15784e30	−0.812660
$820$	0	0
$821$	−1.87901e29	−0.0471332	−0.0235666	−	0.999722i	$-0.507502\pi$
$821$	−1.87901e29	−0.0471332	−0.0235666	+	0.999722i	$0.507502\pi$
$822$	−1.72857e30	−0.428088
$823$	5.28445e30	1.29212	0.646058	−	0.763289i	$-0.276417\pi$
$823$	5.28445e30	1.29212	0.646058	+	0.763289i	$0.276417\pi$
$824$	−5.71134e30	−1.37880
$825$	0	0
$826$	1.55912e30	0.366935
$827$	2.20857e30	0.513220	0.256610	−	0.966515i	$-0.417394\pi$
$827$	2.20857e30	0.513220	0.256610	+	0.966515i	$0.417394\pi$
$828$	1.20724e28	0.00276996
$829$	5.77035e30	1.30731	0.653657	−	0.756791i	$-0.273234\pi$
$829$	5.77035e30	1.30731	0.653657	+	0.756791i	$0.273234\pi$
$830$	0	0
$831$	−8.97582e29	−0.198273
$832$	−6.17909e30	−1.34781
$833$	−9.53957e30	−2.05474
$834$	−1.20690e30	−0.256701
$835$	0	0
$836$	−1.30811e29	−0.0271318
$837$	1.01592e29	0.0208086
$838$	−3.59892e30	−0.727963
$839$	−4.79329e30	−0.957488	−0.478744	−	0.877955i	$-0.658908\pi$
$839$	−4.79329e30	−0.957488	−0.478744	+	0.877955i	$0.658908\pi$
$840$	0	0
$841$	1.47187e31	2.86756
$842$	7.94430e30	1.52855
$843$	−2.84873e30	−0.541330
$844$	3.41022e29	0.0640010
$845$	0	0
$846$	1.98037e30	0.362541
$847$	−7.08586e30	−1.28120
$848$	4.69829e30	0.839042
$849$	−2.33694e30	−0.412208
$850$	0	0
$851$	−2.61507e29	−0.0450011
$852$	2.29399e29	0.0389921
$853$	2.24317e30	0.376615	0.188308	−	0.982110i	$-0.439700\pi$
$853$	2.24317e30	0.376615	0.188308	+	0.982110i	$0.439700\pi$
$854$	1.24655e31	2.06730
$855$	0	0
$856$	3.59317e29	0.0581438
$857$	1.67730e30	0.268109	0.134055	−	0.990974i	$-0.457200\pi$
$857$	1.67730e30	0.268109	0.134055	+	0.990974i	$0.457200\pi$
$858$	2.67176e30	0.421871
$859$	−2.75580e30	−0.429853	−0.214926	−	0.976630i	$-0.568951\pi$
$859$	−2.75580e30	−0.429853	−0.214926	+	0.976630i	$0.568951\pi$
$860$	0	0
$861$	8.77005e30	1.33496
$862$	−2.19753e30	−0.330453
$863$	−4.97847e30	−0.739576	−0.369788	−	0.929116i	$-0.620570\pi$
$863$	−4.97847e30	−0.739576	−0.369788	+	0.929116i	$0.620570\pi$
$864$	1.46029e29	0.0214311
$865$	0	0
$866$	2.10839e30	0.302005
$867$	7.93182e29	0.112246
$868$	7.31609e28	0.0102287
$869$	−3.34655e30	−0.462264
$870$	0	0
$871$	−1.83803e31	−2.47834
$872$	1.37271e31	1.82876
$873$	3.96165e30	0.521467
$874$	1.15798e30	0.150602
$875$	0	0
$876$	1.83704e29	0.0233252
$877$	5.93482e29	0.0744581	0.0372291	−	0.999307i	$-0.488147\pi$
$877$	5.93482e29	0.0744581	0.0372291	+	0.999307i	$0.488147\pi$
$878$	−2.62632e30	−0.325579
$879$	3.97029e30	0.486339
$880$	0	0
$881$	−1.27954e31	−1.53040	−0.765202	−	0.643791i	$-0.777361\pi$
$881$	−1.27954e31	−1.53040	−0.765202	+	0.643791i	$0.777361\pi$
$882$	−5.44824e30	−0.643926
$883$	−1.12787e30	−0.131726	−0.0658632	−	0.997829i	$-0.520980\pi$
$883$	−1.12787e30	−0.131726	−0.0658632	+	0.997829i	$0.520980\pi$
$884$	7.58313e29	0.0875184
$885$	0	0
$886$	1.61518e31	1.82040
$887$	6.34061e30	0.706209	0.353104	−	0.935584i	$-0.385126\pi$
$887$	6.34061e30	0.706209	0.353104	+	0.935584i	$0.385126\pi$
$888$	−1.53627e30	−0.169095
$889$	1.37111e31	1.49143
$890$	0	0
$891$	5.17735e29	0.0550037
$892$	−4.80465e29	−0.0504465
$893$	1.00297e31	1.04076
$894$	−8.03633e30	−0.824164
$895$	0	0
$896$	−1.82178e31	−1.82499
$897$	−1.24878e30	−0.123642
$898$	−1.09289e31	−1.06949
$899$	2.19847e30	0.212639
$900$	0	0
$901$	9.21921e30	0.871130
$902$	−7.42009e30	−0.693011
$903$	3.57360e30	0.329901
$904$	−3.75347e30	−0.342503
$905$	0	0
$906$	−1.07459e31	−0.958069
$907$	4.56592e30	0.402394	0.201197	−	0.979551i	$-0.435517\pi$
$907$	4.56592e30	0.402394	0.201197	+	0.979551i	$0.435517\pi$
$908$	−5.67518e29	−0.0494400
$909$	2.12078e30	0.182631
$910$	0	0
$911$	−2.06100e31	−1.73434	−0.867172	−	0.498009i	$-0.834065\pi$
$911$	−2.06100e31	−1.73434	−0.867172	+	0.498009i	$0.834065\pi$
$912$	7.18313e30	0.597541
$913$	−7.19515e30	−0.591694
$914$	−2.26563e31	−1.84185
$915$	0	0
$916$	−4.58077e29	−0.0363945
$917$	−3.34238e31	−2.62529
$918$	2.78307e30	0.216110
$919$	−1.86231e31	−1.42968	−0.714839	−	0.699289i	$-0.753500\pi$
$919$	−1.86231e31	−1.42968	−0.714839	+	0.699289i	$0.753500\pi$
$920$	0	0
$921$	1.07955e30	0.0810061
$922$	2.03834e31	1.51217
$923$	−2.37294e31	−1.74048
$924$	3.72844e29	0.0270378
$925$	0	0
$926$	−1.31308e31	−0.930838
$927$	−6.75848e30	−0.473709
$928$	3.16009e30	0.219001
$929$	2.63294e31	1.80417	0.902083	−	0.431562i	$-0.142037\pi$
$929$	2.63294e31	1.80417	0.902083	+	0.431562i	$0.142037\pi$
$930$	0	0
$931$	−2.75931e31	−1.84854
$932$	−1.31867e30	−0.0873513
$933$	7.06712e30	0.462898
$934$	1.55713e31	1.00852
$935$	0	0
$936$	−7.33620e30	−0.464596
$937$	−4.07588e30	−0.255244	−0.127622	−	0.991823i	$-0.540734\pi$
$937$	−4.07588e30	−0.255244	−0.127622	+	0.991823i	$0.540734\pi$
$938$	−4.85787e31	−3.00827
$939$	1.60953e31	0.985621
$940$	0	0
$941$	−3.48852e30	−0.208906	−0.104453	−	0.994530i	$-0.533309\pi$
$941$	−3.48852e30	−0.208906	−0.104453	+	0.994530i	$0.533309\pi$
$942$	−9.69598e30	−0.574192
$943$	3.46817e30	0.203108
$944$	3.82465e30	0.221506
$945$	0	0
$946$	−3.02352e30	−0.171260
$947$	3.11529e31	1.74511	0.872557	−	0.488513i	$-0.162460\pi$
$947$	3.11529e31	1.74511	0.872557	+	0.488513i	$0.162460\pi$
$948$	−5.42472e29	−0.0300531
$949$	−1.90027e31	−1.04116
$950$	0	0
$951$	3.84107e30	0.205853
$952$	−3.39499e31	−1.79949
$953$	1.98167e31	1.03886	0.519429	−	0.854514i	$-0.326145\pi$
$953$	1.98167e31	1.03886	0.519429	+	0.854514i	$0.326145\pi$
$954$	5.26528e30	0.273000
$955$	0	0
$956$	−1.08646e30	−0.0551068
$957$	1.12039e31	0.562072
$958$	1.48665e31	0.737683
$959$	2.49524e31	1.22466
$960$	0	0
$961$	−2.05820e31	−0.988309
$962$	−9.38140e30	−0.445584
$963$	4.25196e29	0.0199762
$964$	−3.37742e29	−0.0156955
$965$	0	0
$966$	−3.30052e30	−0.150080
$967$	8.26115e30	0.371589	0.185794	−	0.982589i	$-0.440514\pi$
$967$	8.26115e30	0.371589	0.185794	+	0.982589i	$0.440514\pi$
$968$	−1.64617e31	−0.732458
$969$	1.40951e31	0.620394
$970$	0	0
$971$	5.95729e30	0.256594	0.128297	−	0.991736i	$-0.459049\pi$
$971$	5.95729e30	0.256594	0.128297	+	0.991736i	$0.459049\pi$
$972$	8.39242e28	0.00357595
$973$	1.74219e31	0.734365
$974$	−3.47263e31	−1.44807
$975$	0	0
$976$	3.05790e31	1.24796
$977$	−3.99242e31	−1.61192	−0.805960	−	0.591970i	$-0.798351\pi$
$977$	−3.99242e31	−1.61192	−0.805960	+	0.591970i	$0.798351\pi$
$978$	2.74209e31	1.09527
$979$	2.80563e30	0.110870
$980$	0	0
$981$	1.62439e31	0.628299
$982$	−3.41668e31	−1.30747
$983$	−2.59366e30	−0.0981977	−0.0490988	−	0.998794i	$-0.515635\pi$
$983$	−2.59366e30	−0.0981977	−0.0490988	+	0.998794i	$0.515635\pi$
$984$	2.03743e31	0.763194
$985$	0	0
$986$	6.02259e31	2.20839
$987$	−2.85872e31	−1.03715
$988$	2.19341e30	0.0787357
$989$	1.41320e30	0.0501929
$990$	0	0
$991$	6.46664e30	0.224856	0.112428	−	0.993660i	$-0.464137\pi$
$991$	6.46664e30	0.224856	0.112428	+	0.993660i	$0.464137\pi$
$992$	3.49965e29	0.0120407
$993$	1.24563e31	0.424053
$994$	−6.27165e31	−2.11264
$995$	0	0
$996$	−1.16632e30	−0.0384677
$997$	−3.67859e31	−1.20056	−0.600278	−	0.799791i	$-0.704943\pi$
$997$	−3.67859e31	−1.20056	−0.600278	+	0.799791i	$0.704943\pi$
$998$	2.36483e31	0.763712
$999$	−1.81794e30	−0.0580953

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.h.1.1		4
5.2	odd	4		75.22.b.h.49.2		8
5.3	odd	4		75.22.b.h.49.7		8
5.4	even	2		15.22.a.e.1.4	✓	4
15.14	odd	2		45.22.a.g.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.a.e.1.4	✓	4	5.4	even	2
45.22.a.g.1.1		4	15.14	odd	2
75.22.a.h.1.1		4	1.1	even	1	trivial
75.22.b.h.49.2		8	5.2	odd	4
75.22.b.h.49.7		8	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)

\(f(q)\)	\(=\)	\(q-1487.97 q^{2} +59049.0 q^{3} +116903. q^{4} -8.78631e7 q^{6} +1.26833e9 q^{7} +2.94655e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-1487.97 q^{2} +59049.0 q^{3} +116903. q^{4} -8.78631e7 q^{6} +1.26833e9 q^{7} +2.94655e9 q^{8} +3.48678e9 q^{9} +4.25851e10 q^{11} +6.90299e9 q^{12} -7.14056e11 q^{13} -1.88724e12 q^{14} -4.62954e12 q^{16} -9.08430e12 q^{17} -5.18823e12 q^{18} -2.62762e13 q^{19} +7.48936e13 q^{21} -6.33654e13 q^{22} +2.96171e13 q^{23} +1.73991e14 q^{24} +1.06249e15 q^{26} +2.05891e14 q^{27} +1.48271e14 q^{28} +4.45551e15 q^{29} +4.93427e14 q^{31} +7.09254e14 q^{32} +2.51461e15 q^{33} +1.35172e16 q^{34} +4.07614e14 q^{36} -8.82960e15 q^{37} +3.90982e16 q^{38} -4.21643e16 q^{39} +1.17100e17 q^{41} -1.11439e17 q^{42} +4.77156e16 q^{43} +4.97831e15 q^{44} -4.40694e16 q^{46} -3.81704e17 q^{47} -2.73370e17 q^{48} +1.05012e18 q^{49} -5.36419e17 q^{51} -8.34750e16 q^{52} -1.01485e18 q^{53} -3.06360e17 q^{54} +3.73720e18 q^{56} -1.55158e18 q^{57} -6.62967e18 q^{58} -8.26140e17 q^{59} -6.60518e18 q^{61} -7.34204e17 q^{62} +4.42239e18 q^{63} +8.65351e18 q^{64} -3.74166e18 q^{66} +2.57407e19 q^{67} -1.06198e18 q^{68} +1.74886e18 q^{69} +3.32319e19 q^{71} +1.02740e19 q^{72} +2.66123e19 q^{73} +1.31382e19 q^{74} -3.07176e18 q^{76} +5.40120e19 q^{77} +6.27392e19 q^{78} -7.85851e19 q^{79} +1.21577e19 q^{81} -1.74241e20 q^{82} -1.68959e20 q^{83} +8.75526e18 q^{84} -7.09994e19 q^{86} +2.63093e20 q^{87} +1.25479e20 q^{88} +6.58830e19 q^{89} -9.05659e20 q^{91} +3.46232e18 q^{92} +2.91363e19 q^{93} +5.67964e20 q^{94} +4.18808e19 q^{96} +1.13619e21 q^{97} -1.56254e21 q^{98} +1.48485e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9}+O(q^{10}) \) 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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