LMFDB - Embedded newform 75.22.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$209.608008215$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2 x^{7} - 13701836 x^{6} + 201589162 x^{5} + 55078074399586 x^{4} + \cdots + 23\!\cdots\!25 $ x^8 - 2x^7 - 13701836x^6 + 201589162x^5 + 55078074399586x^4 - 1921148793416750x^3 - 63775892303255475300x^2 - 5448867057840511928250*x + 2374849659307636957978125
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{10}\cdot 5^{14}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2210.46$ 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)

$f(q)$	$=$	$q-2127.46 q^{2} -59049.0 q^{3} +2.42895e6 q^{4} +1.25625e8 q^{6} +1.04187e9 q^{7} -7.05898e8 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-2127.46 q^{2} -59049.0 q^{3} +2.42895e6 q^{4} +1.25625e8 q^{6} +1.04187e9 q^{7} -7.05898e8 q^{8} +3.48678e9 q^{9} +2.38707e10 q^{11} -1.43427e11 q^{12} -3.64962e11 q^{13} -2.21654e12 q^{14} -3.59211e12 q^{16} +8.45697e12 q^{17} -7.41801e12 q^{18} +5.34859e11 q^{19} -6.15215e13 q^{21} -5.07841e13 q^{22} -3.10320e14 q^{23} +4.16825e13 q^{24} +7.76443e14 q^{26} -2.05891e14 q^{27} +2.53066e15 q^{28} +1.78803e15 q^{29} +6.32789e15 q^{31} +9.12247e15 q^{32} -1.40954e15 q^{33} -1.79919e16 q^{34} +8.46924e15 q^{36} +2.11910e16 q^{37} -1.13789e15 q^{38} +2.15506e16 q^{39} -2.38647e16 q^{41} +1.30885e17 q^{42} -1.06785e17 q^{43} +5.79809e16 q^{44} +6.60194e17 q^{46} -6.65515e16 q^{47} +2.12111e17 q^{48} +5.26950e17 q^{49} -4.99375e17 q^{51} -8.86475e17 q^{52} +1.12980e18 q^{53} +4.38026e17 q^{54} -7.35454e17 q^{56} -3.15829e16 q^{57} -3.80397e18 q^{58} +3.51140e18 q^{59} +8.16179e18 q^{61} -1.34624e19 q^{62} +3.63278e18 q^{63} -1.18745e19 q^{64} +2.99875e18 q^{66} -1.34455e19 q^{67} +2.05416e19 q^{68} +1.83241e19 q^{69} +4.06302e19 q^{71} -2.46131e18 q^{72} +1.09593e19 q^{73} -4.50830e19 q^{74} +1.29915e18 q^{76} +2.48702e19 q^{77} -4.58482e19 q^{78} +3.58892e19 q^{79} +1.21577e19 q^{81} +5.07714e19 q^{82} -1.25727e20 q^{83} -1.49433e20 q^{84} +2.27181e20 q^{86} -1.05581e20 q^{87} -1.68503e19 q^{88} +4.16172e20 q^{89} -3.80243e20 q^{91} -7.53752e20 q^{92} -3.73656e20 q^{93} +1.41586e20 q^{94} -5.38673e20 q^{96} -4.65598e20 q^{97} -1.12107e21 q^{98} +8.32320e19 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9}+O(q^{10}) $ 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9	$ 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9} - 14410165968 q^{11} - 630755749296 q^{12} - 328226481176 q^{13} + 4068036669186 q^{14} + 18553054308920 q^{16} - 5718214953936 q^{17} + 2322198411066 q^{18} + 75919698170296 q^{19} - 7914601537128 q^{21} - 426897542691372 q^{22} + 49712781537936 q^{23} - 207390009885372 q^{24} - 126056679642054 q^{26} - 16\!\cdots\!92 q^{27}+ \cdots - 50\!\cdots\!68 q^{99}+O(q^{100}) $ 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9 - 14410165968 * q^11 - 630755749296 * q^12 - 328226481176 * q^13 + 4068036669186 * q^14 + 18553054308920 * q^16 - 5718214953936 * q^17 + 2322198411066 * q^18 + 75919698170296 * q^19 - 7914601537128 * q^21 - 426897542691372 * q^22 + 49712781537936 * q^23 - 207390009885372 * q^24 - 126056679642054 * q^26 - 1647129056757192 * q^27 + 4235167126399504 * q^28 + 2111332005818352 * q^29 + 5677451410844968 * q^31 + 17528957992420632 * q^32 + 850905890244432 * q^33 + 9944754702742068 * q^34 + 37245496240179504 * q^36 - 26538401323409648 * q^37 + 85729618982552526 * q^38 + 19381445486961624 * q^39 - 9719778050982432 * q^41 - 240213497278764114 * q^42 - 20888744714565080 * q^43 - 388904820117856128 * q^44 - 156886987299808452 * q^46 + 405141072730526160 * q^47 - 1095539303887417080 * q^48 + 1172899395436368720 * q^49 + 337654874814966864 * q^51 - 2264281219115894096 * q^52 + 1890950875516930176 * q^53 - 137123493975036234 * q^54 + 19720793170524660300 * q^56 - 4482982257257808504 * q^57 + 12636224332772065092 * q^58 - 3496737905134765776 * q^59 + 17508733151770015960 * q^61 - 30404101355896200606 * q^62 + 467349306165871272 * q^63 + 27809823573720908096 * q^64 + 25207872998382825228 * q^66 - 29911607071855004216 * q^67 - 29571773973515759808 * q^68 - 2935490037033582864 * q^69 + 52204030316256846816 * q^71 + 12246172693721331228 * q^72 - 39697032757192143344 * q^73 - 32301811215943606764 * q^74 + 383913379529874070384 * q^76 - 202348742072764877136 * q^77 + 7443520876183646646 * q^78 - 274125739348498990400 * q^79 + 97261323672455430408 * q^81 - 231746572148446490256 * q^82 - 276883976086286712912 * q^83 - 250082383646764311696 * q^84 + 486708932755841131242 * q^86 - 124672043611567867248 * q^87 - 2078122304879724561672 * q^88 - 319129684977517046784 * q^89 + 45068531423660000648 * q^91 + 147706203809955886272 * q^92 - 335247828358984515432 * q^93 - 1499781052151613424236 * q^94 - 1035067440494445898968 * q^96 + 451807368184588489864 * q^97 - 2604193962467412900204 * q^98 - 50245141913043465168 * 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 $

$2$	−2127.46	−1.46909	−0.734543	−	0.678562i	$-0.762604\pi$
$2$	−2127.46	−1.46909	−0.734543	+	0.678562i	$0.762604\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	2.42895e6	1.15822
$5$	0	0
$5$	0	0
$6$	1.25625e8	0.848178
$7$	1.04187e9	1.39407	0.697035	−	0.717037i	$-0.254502\pi$
$7$	1.04187e9	1.39407	0.697035	+	0.717037i	$0.254502\pi$
$8$	−7.05898e8	−0.232432
$9$	3.48678e9	0.333333
$10$	0	0
$11$	2.38707e10	0.277487	0.138743	−	0.990328i	$-0.455694\pi$
$11$	2.38707e10	0.277487	0.138743	+	0.990328i	$0.455694\pi$
$12$	−1.43427e11	−0.668696
$13$	−3.64962e11	−0.734247	−0.367123	−	0.930172i	$-0.619657\pi$
$13$	−3.64962e11	−0.734247	−0.367123	+	0.930172i	$0.619657\pi$
$14$	−2.21654e12	−2.04801
$15$	0	0
$16$	−3.59211e12	−0.816752
$17$	8.45697e12	1.01742	0.508711	−	0.860937i	$-0.330122\pi$
$17$	8.45697e12	1.01742	0.508711	+	0.860937i	$0.330122\pi$
$18$	−7.41801e12	−0.489696
$19$	5.34859e11	0.0200137	0.0100068	−	0.999950i	$-0.496815\pi$
$19$	5.34859e11	0.0200137	0.0100068	+	0.999950i	$0.496815\pi$
$20$	0	0
$21$	−6.15215e13	−0.804867
$22$	−5.07841e13	−0.407652
$23$	−3.10320e14	−1.56195	−0.780975	−	0.624562i	$-0.785277\pi$
$23$	−3.10320e14	−1.56195	−0.780975	+	0.624562i	$0.785277\pi$
$24$	4.16825e13	0.134195
$25$	0	0
$26$	7.76443e14	1.07867
$27$	−2.05891e14	−0.192450
$28$	2.53066e15	1.61463
$29$	1.78803e15	0.789217	0.394608	−	0.918849i	$-0.370880\pi$
$29$	1.78803e15	0.789217	0.394608	+	0.918849i	$0.370880\pi$
$30$	0	0
$31$	6.32789e15	1.38663	0.693316	−	0.720634i	$-0.256149\pi$
$31$	6.32789e15	1.38663	0.693316	+	0.720634i	$0.256149\pi$
$32$	9.12247e15	1.43231
$33$	−1.40954e15	−0.160207
$34$	−1.79919e16	−1.49468
$35$	0	0
$36$	8.46924e15	0.386072
$37$	2.11910e16	0.724491	0.362245	−	0.932083i	$-0.382010\pi$
$37$	2.11910e16	0.724491	0.362245	+	0.932083i	$0.382010\pi$
$38$	−1.13789e15	−0.0294018
$39$	2.15506e16	0.423918
$40$	0	0
$41$	−2.38647e16	−0.277668	−0.138834	−	0.990316i	$-0.544335\pi$
$41$	−2.38647e16	−0.277668	−0.138834	+	0.990316i	$0.544335\pi$
$42$	1.30885e17	1.18242
$43$	−1.06785e17	−0.753514	−0.376757	−	0.926312i	$-0.622961\pi$
$43$	−1.06785e17	−0.753514	−0.376757	+	0.926312i	$0.622961\pi$
$44$	5.79809e16	0.321389
$45$	0	0
$46$	6.60194e17	2.29464
$47$	−6.65515e16	−0.184557	−0.0922785	−	0.995733i	$-0.529415\pi$
$47$	−6.65515e16	−0.184557	−0.0922785	+	0.995733i	$0.529415\pi$
$48$	2.12111e17	0.471552
$49$	5.26950e17	0.943431
$50$	0	0
$51$	−4.99375e17	−0.587409
$52$	−8.86475e17	−0.850416
$53$	1.12980e18	0.887370	0.443685	−	0.896183i	$-0.353671\pi$
$53$	1.12980e18	0.887370	0.443685	+	0.896183i	$0.353671\pi$
$54$	4.38026e17	0.282726
$55$	0	0
$56$	−7.35454e17	−0.324027
$57$	−3.15829e16	−0.0115549
$58$	−3.80397e18	−1.15943
$59$	3.51140e18	0.894405	0.447202	−	0.894433i	$-0.352420\pi$
$59$	3.51140e18	0.894405	0.447202	+	0.894433i	$0.352420\pi$
$60$	0	0
$61$	8.16179e18	1.46495	0.732474	−	0.680795i	$-0.238365\pi$
$61$	8.16179e18	1.46495	0.732474	+	0.680795i	$0.238365\pi$
$62$	−1.34624e19	−2.03708
$63$	3.63278e18	0.464690
$64$	−1.18745e19	−1.28744
$65$	0	0
$66$	2.99875e18	0.235358
$67$	−1.34455e19	−0.901137	−0.450569	−	0.892742i	$-0.648779\pi$
$67$	−1.34455e19	−0.901137	−0.450569	+	0.892742i	$0.648779\pi$
$68$	2.05416e19	1.17839
$69$	1.83241e19	0.901792
$70$	0	0
$71$	4.06302e19	1.48128	0.740639	−	0.671903i	$-0.234523\pi$
$71$	4.06302e19	1.48128	0.740639	+	0.671903i	$0.234523\pi$
$72$	−2.46131e18	−0.0774775
$73$	1.09593e19	0.298464	0.149232	−	0.988802i	$-0.452320\pi$
$73$	1.09593e19	0.298464	0.149232	+	0.988802i	$0.452320\pi$
$74$	−4.50830e19	−1.06434
$75$	0	0
$76$	1.29915e18	0.0231801
$77$	2.48702e19	0.386836
$78$	−4.58482e19	−0.622772
$79$	3.58892e19	0.426461	0.213231	−	0.977002i	$-0.431601\pi$
$79$	3.58892e19	0.426461	0.213231	+	0.977002i	$0.431601\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	5.07714e19	0.407919
$83$	−1.25727e20	−0.889422	−0.444711	−	0.895674i	$-0.646694\pi$
$83$	−1.25727e20	−0.889422	−0.444711	+	0.895674i	$0.646694\pi$
$84$	−1.49433e20	−0.932209
$85$	0	0
$86$	2.27181e20	1.10698
$87$	−1.05581e20	−0.455654
$88$	−1.68503e19	−0.0644969
$89$	4.16172e20	1.41474	0.707372	−	0.706841i	$-0.249880\pi$
$89$	4.16172e20	1.41474	0.707372	+	0.706841i	$0.249880\pi$
$90$	0	0
$91$	−3.80243e20	−1.02359
$92$	−7.53752e20	−1.80907
$93$	−3.73656e20	−0.800573
$94$	1.41586e20	0.271130
$95$	0	0
$96$	−5.38673e20	−0.826946
$97$	−4.65598e20	−0.641073	−0.320537	−	0.947236i	$-0.603863\pi$
$97$	−4.65598e20	−0.641073	−0.320537	+	0.947236i	$0.603863\pi$
$98$	−1.12107e21	−1.38598
$99$	8.32320e19	0.0924955
$100$	0	0
$101$	−9.68773e20	−0.872665	−0.436332	−	0.899786i	$-0.643723\pi$
$101$	−9.68773e20	−0.872665	−0.436332	+	0.899786i	$0.643723\pi$
$102$	1.06240e21	0.862954
$103$	−1.91438e21	−1.40358	−0.701790	−	0.712383i	$-0.747616\pi$
$103$	−1.91438e21	−1.40358	−0.701790	+	0.712383i	$0.747616\pi$
$104$	2.57626e20	0.170663
$105$	0	0
$106$	−2.40361e21	−1.30362
$107$	−6.22284e20	−0.305815	−0.152907	−	0.988241i	$-0.548864\pi$
$107$	−6.22284e20	−0.305815	−0.152907	+	0.988241i	$0.548864\pi$
$108$	−5.00100e20	−0.222899
$109$	−3.42222e21	−1.38462	−0.692308	−	0.721602i	$-0.743406\pi$
$109$	−3.42222e21	−1.38462	−0.692308	+	0.721602i	$0.743406\pi$
$110$	0	0
$111$	−1.25131e21	−0.418285
$112$	−3.74252e21	−1.13861
$113$	6.40614e21	1.77530	0.887651	−	0.460516i	$-0.152336\pi$
$113$	6.40614e21	1.77530	0.887651	+	0.460516i	$0.152336\pi$
$114$	6.71915e19	0.0169751
$115$	0	0
$116$	4.34305e21	0.914083
$117$	−1.27254e21	−0.244749
$118$	−7.47038e21	−1.31396
$119$	8.81107e21	1.41836
$120$	0	0
$121$	−6.83044e21	−0.923001
$122$	−1.73639e22	−2.15214
$123$	1.40919e21	0.160312
$124$	1.53702e22	1.60602
$125$	0	0
$126$	−7.72861e21	−0.682670
$127$	−3.91229e21	−0.318047	−0.159024	−	0.987275i	$-0.550835\pi$
$127$	−3.91229e21	−0.318047	−0.159024	+	0.987275i	$0.550835\pi$
$128$	6.13143e21	0.459047
$129$	6.30555e21	0.435042
$130$	0	0
$131$	−9.38194e21	−0.550737	−0.275368	−	0.961339i	$-0.588800\pi$
$131$	−9.38194e21	−0.550737	−0.275368	+	0.961339i	$0.588800\pi$
$132$	−3.42371e21	−0.185554
$133$	5.57254e20	0.0279004
$134$	2.86048e22	1.32385
$135$	0	0
$136$	−5.96975e21	−0.236482
$137$	−3.91869e22	−1.43739	−0.718696	−	0.695325i	$-0.755261\pi$
$137$	−3.91869e22	−1.43739	−0.718696	+	0.695325i	$0.755261\pi$
$138$	−3.89838e22	−1.32481
$139$	5.75024e22	1.81147	0.905733	−	0.423848i	$-0.139321\pi$
$139$	5.75024e22	1.81147	0.905733	+	0.423848i	$0.139321\pi$
$140$	0	0
$141$	3.92980e21	0.106554
$142$	−8.64394e22	−2.17613
$143$	−8.71189e21	−0.203744
$144$	−1.25249e22	−0.272251
$145$	0	0
$146$	−2.33155e22	−0.438470
$147$	−3.11159e22	−0.544690
$148$	5.14719e22	0.839116
$149$	8.30694e22	1.26178	0.630892	−	0.775871i	$-0.282689\pi$
$149$	8.30694e22	1.26178	0.630892	+	0.775871i	$0.282689\pi$
$150$	0	0
$151$	−8.13097e22	−1.07370	−0.536852	−	0.843676i	$-0.680387\pi$
$151$	−8.13097e22	−1.07370	−0.536852	+	0.843676i	$0.680387\pi$
$152$	−3.77556e20	−0.00465182
$153$	2.94876e22	0.339141
$154$	−5.29105e22	−0.568295
$155$	0	0
$156$	5.23455e22	0.490988
$157$	1.22856e23	1.07758	0.538791	−	0.842439i	$-0.318881\pi$
$157$	1.22856e23	1.07758	0.538791	+	0.842439i	$0.318881\pi$
$158$	−7.63530e22	−0.626508
$159$	−6.67136e22	−0.512323
$160$	0	0
$161$	−3.23313e23	−2.17747
$162$	−2.58650e22	−0.163232
$163$	−2.64435e23	−1.56440	−0.782202	−	0.623025i	$-0.785904\pi$
$163$	−2.64435e23	−1.56440	−0.782202	+	0.623025i	$0.785904\pi$
$164$	−5.79663e22	−0.321600
$165$	0	0
$166$	2.67479e23	1.30664
$167$	2.39814e23	1.09989	0.549947	−	0.835200i	$-0.314648\pi$
$167$	2.39814e23	1.09989	0.549947	+	0.835200i	$0.314648\pi$
$168$	4.34278e22	0.187077
$169$	−1.13867e23	−0.460882
$170$	0	0
$171$	1.86494e21	0.00667122
$172$	−2.59376e23	−0.872732
$173$	3.35267e23	1.06147	0.530734	−	0.847538i	$-0.321916\pi$
$173$	3.35267e23	1.06147	0.530734	+	0.847538i	$0.321916\pi$
$174$	2.24621e23	0.669396
$175$	0	0
$176$	−8.57463e22	−0.226638
$177$	−2.07345e23	−0.516385
$178$	−8.85392e23	−2.07838
$179$	2.45865e22	0.0544177	0.0272088	−	0.999630i	$-0.491338\pi$
$179$	2.45865e22	0.0544177	0.0272088	+	0.999630i	$0.491338\pi$
$180$	0	0
$181$	3.11848e23	0.614212	0.307106	−	0.951675i	$-0.400639\pi$
$181$	3.11848e23	0.614212	0.307106	+	0.951675i	$0.400639\pi$
$182$	8.08954e23	1.50374
$183$	−4.81946e23	−0.845788
$184$	2.19054e23	0.363048
$185$	0	0
$186$	7.94939e23	1.17611
$187$	2.01874e23	0.282321
$188$	−1.61651e23	−0.213757
$189$	−2.14512e23	−0.268289
$190$	0	0
$191$	−3.19817e23	−0.358139	−0.179070	−	0.983836i	$-0.557309\pi$
$191$	−3.19817e23	−0.358139	−0.179070	+	0.983836i	$0.557309\pi$
$192$	7.01179e23	0.743303
$193$	1.38926e24	1.39454	0.697269	−	0.716810i	$-0.254398\pi$
$193$	1.38926e24	1.39454	0.697269	+	0.716810i	$0.254398\pi$
$194$	9.90543e23	0.941792
$195$	0	0
$196$	1.27994e24	1.09270
$197$	1.35954e24	1.10026	0.550132	−	0.835078i	$-0.314577\pi$
$197$	1.35954e24	1.10026	0.550132	+	0.835078i	$0.314577\pi$
$198$	−1.77073e23	−0.135884
$199$	−8.32615e23	−0.606020	−0.303010	−	0.952987i	$-0.597992\pi$
$199$	−8.32615e23	−0.606020	−0.303010	+	0.952987i	$0.597992\pi$
$200$	0	0
$201$	7.93942e23	0.520272
$202$	2.06103e24	1.28202
$203$	1.86290e24	1.10022
$204$	−1.21296e24	−0.680346
$205$	0	0
$206$	4.07278e24	2.06198
$207$	−1.08202e24	−0.520650
$208$	1.31098e24	0.599698
$209$	1.27675e22	0.00555352
$210$	0	0
$211$	4.90320e24	1.92981	0.964904	−	0.262603i	$-0.0845809\pi$
$211$	4.90320e24	1.92981	0.964904	+	0.262603i	$0.0845809\pi$
$212$	2.74423e24	1.02777
$213$	−2.39917e24	−0.855216
$214$	1.32389e24	0.449269
$215$	0	0
$216$	1.45338e23	0.0447317
$217$	6.59285e24	1.93306
$218$	7.28065e24	2.03412
$219$	−6.47135e23	−0.172318
$220$	0	0
$221$	−3.08647e24	−0.747039
$222$	2.66211e24	0.614497
$223$	−7.50133e24	−1.65172	−0.825862	−	0.563872i	$-0.809311\pi$
$223$	−7.50133e24	−1.65172	−0.825862	+	0.563872i	$0.809311\pi$
$224$	9.50444e24	1.99674
$225$	0	0
$226$	−1.36288e25	−2.60807
$227$	−3.57793e23	−0.0653673	−0.0326836	−	0.999466i	$-0.510405\pi$
$227$	−3.57793e23	−0.0653673	−0.0326836	+	0.999466i	$0.510405\pi$
$228$	−7.67134e22	−0.0133831
$229$	−4.08201e24	−0.680145	−0.340072	−	0.940399i	$-0.610452\pi$
$229$	−4.08201e24	−0.680145	−0.340072	+	0.940399i	$0.610452\pi$
$230$	0	0
$231$	−1.46856e24	−0.223340
$232$	−1.26217e24	−0.183440
$233$	−6.11617e24	−0.849655	−0.424827	−	0.905274i	$-0.639665\pi$
$233$	−6.11617e24	−0.849655	−0.424827	+	0.905274i	$0.639665\pi$
$234$	2.70729e24	0.359557
$235$	0	0
$236$	8.52903e24	1.03591
$237$	−2.11922e24	−0.246217
$238$	−1.87452e25	−2.08369
$239$	1.68371e25	1.79098	0.895488	−	0.445085i	$-0.146827\pi$
$239$	1.68371e25	1.79098	0.895488	+	0.445085i	$0.146827\pi$
$240$	0	0
$241$	7.91923e24	0.771799	0.385899	−	0.922541i	$-0.373891\pi$
$241$	7.91923e24	0.771799	0.385899	+	0.922541i	$0.373891\pi$
$242$	1.45315e25	1.35597
$243$	−7.17898e23	−0.0641500
$244$	1.98246e25	1.69673
$245$	0	0
$246$	−2.99800e24	−0.235512
$247$	−1.95203e23	−0.0146950
$248$	−4.46684e24	−0.322298
$249$	7.42404e24	0.513508
$250$	0	0
$251$	−1.00598e25	−0.639758	−0.319879	−	0.947458i	$-0.603642\pi$
$251$	−1.00598e25	−0.639758	−0.319879	+	0.947458i	$0.603642\pi$
$252$	8.82386e24	0.538211
$253$	−7.40755e24	−0.433420
$254$	8.32326e24	0.467239
$255$	0	0
$256$	1.18583e25	0.613059
$257$	−2.24368e25	−1.11343	−0.556716	−	0.830703i	$-0.687939\pi$
$257$	−2.24368e25	−1.11343	−0.556716	+	0.830703i	$0.687939\pi$
$258$	−1.34148e25	−0.639114
$259$	2.20783e25	1.00999
$260$	0	0
$261$	6.23448e24	0.263072
$262$	1.99597e25	0.809080
$263$	−1.25135e25	−0.487353	−0.243676	−	0.969857i	$-0.578353\pi$
$263$	−1.25135e25	−0.487353	−0.243676	+	0.969857i	$0.578353\pi$
$264$	9.94992e23	0.0372373
$265$	0	0
$266$	−1.18554e24	−0.0409882
$267$	−2.45746e25	−0.816803
$268$	−3.26585e25	−1.04371
$269$	−8.81802e24	−0.271002	−0.135501	−	0.990777i	$-0.543264\pi$
$269$	−8.81802e24	−0.271002	−0.135501	+	0.990777i	$0.543264\pi$
$270$	0	0
$271$	−4.96008e25	−1.41030	−0.705150	−	0.709058i	$-0.749120\pi$
$271$	−4.96008e25	−1.41030	−0.705150	+	0.709058i	$0.749120\pi$
$272$	−3.03784e25	−0.830981
$273$	2.24530e25	0.590971
$274$	8.33688e25	2.11165
$275$	0	0
$276$	4.45083e25	1.04447
$277$	−3.36814e25	−0.760943	−0.380471	−	0.924793i	$-0.624238\pi$
$277$	−3.36814e25	−0.760943	−0.380471	+	0.924793i	$0.624238\pi$
$278$	−1.22334e26	−2.66120
$279$	2.20640e25	0.462211
$280$	0	0
$281$	8.37255e25	1.62720	0.813601	−	0.581423i	$-0.197504\pi$
$281$	8.37255e25	1.62720	0.813601	+	0.581423i	$0.197504\pi$
$282$	−8.36051e24	−0.156537
$283$	7.42339e25	1.33920	0.669599	−	0.742723i	$-0.266466\pi$
$283$	7.42339e25	1.33920	0.669599	+	0.742723i	$0.266466\pi$
$284$	9.86889e25	1.71564
$285$	0	0
$286$	1.85342e25	0.299317
$287$	−2.48640e25	−0.387089
$288$	3.18081e25	0.477437
$289$	2.42835e24	0.0351466
$290$	0	0
$291$	2.74931e25	0.370124
$292$	2.66196e25	0.345686
$293$	−8.24760e25	−1.03328	−0.516639	−	0.856203i	$-0.672817\pi$
$293$	−8.24760e25	−1.03328	−0.516639	+	0.856203i	$0.672817\pi$
$294$	6.61979e25	0.800197
$295$	0	0
$296$	−1.49587e25	−0.168395
$297$	−4.91477e24	−0.0534023
$298$	−1.76727e26	−1.85367
$299$	1.13255e26	1.14686
$300$	0	0
$301$	−1.11256e26	−1.05045
$302$	1.72984e26	1.57736
$303$	5.72051e25	0.503833
$304$	−1.92127e24	−0.0163462
$305$	0	0
$306$	−6.27339e25	−0.498227
$307$	1.43666e26	1.10256	0.551279	−	0.834321i	$-0.314140\pi$
$307$	1.43666e26	1.10256	0.551279	+	0.834321i	$0.314140\pi$
$308$	6.04086e25	0.448039
$309$	1.13042e26	0.810358
$310$	0	0
$311$	1.34915e25	0.0903811	0.0451906	−	0.998978i	$-0.485610\pi$
$311$	1.34915e25	0.0903811	0.0451906	+	0.998978i	$0.485610\pi$
$312$	−1.52125e25	−0.0985322
$313$	−3.83034e25	−0.239896	−0.119948	−	0.992780i	$-0.538273\pi$
$313$	−3.83034e25	−0.239896	−0.119948	+	0.992780i	$0.538273\pi$
$314$	−2.61372e26	−1.58306
$315$	0	0
$316$	8.71733e25	0.493934
$317$	2.08246e26	1.14144	0.570721	−	0.821144i	$-0.306664\pi$
$317$	2.08246e26	1.14144	0.570721	+	0.821144i	$0.306664\pi$
$318$	1.41931e26	0.752648
$319$	4.26816e25	0.218997
$320$	0	0
$321$	3.67452e25	0.176562
$322$	6.87837e26	3.19889
$323$	4.52328e24	0.0203623
$324$	2.95304e25	0.128691
$325$	0	0
$326$	5.62576e26	2.29825
$327$	2.02079e26	0.799409
$328$	1.68461e25	0.0645391
$329$	−6.93381e25	−0.257285
$330$	0	0
$331$	−4.16316e26	−1.44954	−0.724768	−	0.688992i	$-0.758053\pi$
$331$	−4.16316e26	−1.44954	−0.724768	+	0.688992i	$0.758053\pi$
$332$	−3.05385e26	−1.03014
$333$	7.38883e25	0.241497
$334$	−5.10195e26	−1.61584
$335$	0	0
$336$	2.20992e26	0.657377
$337$	1.00308e26	0.289216	0.144608	−	0.989489i	$-0.453808\pi$
$337$	1.00308e26	0.289216	0.144608	+	0.989489i	$0.453808\pi$
$338$	2.42249e26	0.677075
$339$	−3.78276e26	−1.02497
$340$	0	0
$341$	1.51051e26	0.384772
$342$	−3.96759e24	−0.00980060
$343$	−3.29192e25	−0.0788607
$344$	7.53793e25	0.175141
$345$	0	0
$346$	−7.13269e26	−1.55939
$347$	−2.62880e26	−0.557568	−0.278784	−	0.960354i	$-0.589931\pi$
$347$	−2.62880e26	−0.557568	−0.278784	+	0.960354i	$0.589931\pi$
$348$	−2.56453e26	−0.527746
$349$	8.84994e25	0.176715	0.0883575	−	0.996089i	$-0.471838\pi$
$349$	8.84994e25	0.176715	0.0883575	+	0.996089i	$0.471838\pi$
$350$	0	0
$351$	7.51424e25	0.141306
$352$	2.17760e26	0.397447
$353$	4.85017e26	0.859255	0.429628	−	0.903006i	$-0.358645\pi$
$353$	4.85017e26	0.859255	0.429628	+	0.903006i	$0.358645\pi$
$354$	4.41118e26	0.758614
$355$	0	0
$356$	1.01086e27	1.63858
$357$	−5.20285e26	−0.818889
$358$	−5.23069e25	−0.0799443
$359$	1.60735e26	0.238571	0.119285	−	0.992860i	$-0.461940\pi$
$359$	1.60735e26	0.238571	0.119285	+	0.992860i	$0.461940\pi$
$360$	0	0
$361$	−7.13923e26	−0.999599
$362$	−6.63446e26	−0.902330
$363$	4.03331e26	0.532895
$364$	−9.23593e26	−1.18554
$365$	0	0
$366$	1.02532e27	1.24254
$367$	8.97540e25	0.105696	0.0528482	−	0.998603i	$-0.483170\pi$
$367$	8.97540e25	0.105696	0.0528482	+	0.998603i	$0.483170\pi$
$368$	1.11470e27	1.27573
$369$	−8.32112e25	−0.0925560
$370$	0	0
$371$	1.17711e27	1.23706
$372$	−9.07593e26	−0.927236
$373$	−1.71990e27	−1.70828	−0.854142	−	0.520040i	$-0.825917\pi$
$373$	−1.71990e27	−1.70828	−0.854142	+	0.520040i	$0.825917\pi$
$374$	−4.29479e26	−0.414754
$375$	0	0
$376$	4.69786e25	0.0428970
$377$	−6.52563e26	−0.579480
$378$	4.56367e26	0.394140
$379$	−6.22737e26	−0.523110	−0.261555	−	0.965189i	$-0.584235\pi$
$379$	−6.22737e26	−0.523110	−0.261555	+	0.965189i	$0.584235\pi$
$380$	0	0
$381$	2.31017e26	0.183625
$382$	6.80400e26	0.526137
$383$	3.54226e26	0.266498	0.133249	−	0.991083i	$-0.457459\pi$
$383$	3.54226e26	0.266498	0.133249	+	0.991083i	$0.457459\pi$
$384$	−3.62055e26	−0.265031
$385$	0	0
$386$	−2.95559e27	−2.04870
$387$	−3.72336e26	−0.251171
$388$	−1.13092e27	−0.742501
$389$	1.41987e27	0.907356	0.453678	−	0.891166i	$-0.350112\pi$
$389$	1.41987e27	0.907356	0.453678	+	0.891166i	$0.350112\pi$
$390$	0	0
$391$	−2.62436e27	−1.58916
$392$	−3.71973e26	−0.219284
$393$	5.53994e26	0.317968
$394$	−2.89238e27	−1.61638
$395$	0	0
$396$	2.02167e26	0.107130
$397$	−3.01335e27	−1.55507	−0.777534	−	0.628841i	$-0.783529\pi$
$397$	−3.01335e27	−1.55507	−0.777534	+	0.628841i	$0.783529\pi$
$398$	1.77136e27	0.890295
$399$	−3.29053e25	−0.0161083
$400$	0	0
$401$	−2.47480e27	−1.14954	−0.574769	−	0.818316i	$-0.694908\pi$
$401$	−2.47480e27	−1.14954	−0.574769	+	0.818316i	$0.694908\pi$
$402$	−1.68908e27	−0.764324
$403$	−2.30944e27	−1.01813
$404$	−2.35310e27	−1.01073
$405$	0	0
$406$	−3.96325e27	−1.61632
$407$	5.05843e26	0.201036
$408$	3.52508e26	0.136533
$409$	−2.52746e27	−0.954090	−0.477045	−	0.878879i	$-0.658292\pi$
$409$	−2.52746e27	−0.954090	−0.477045	+	0.878879i	$0.658292\pi$
$410$	0	0
$411$	2.31395e27	0.829878
$412$	−4.64994e27	−1.62565
$413$	3.65843e27	1.24686
$414$	2.30195e27	0.764880
$415$	0	0
$416$	−3.32935e27	−1.05167
$417$	−3.39546e27	−1.04585
$418$	−2.71623e25	−0.00815860
$419$	5.89009e27	1.72534	0.862670	−	0.505768i	$-0.168791\pi$
$419$	5.89009e27	1.72534	0.862670	+	0.505768i	$0.168791\pi$
$420$	0	0
$421$	−6.40338e27	−1.78421	−0.892107	−	0.451823i	$-0.850774\pi$
$421$	−6.40338e27	−1.78421	−0.892107	+	0.451823i	$0.850774\pi$
$422$	−1.04314e28	−2.83506
$423$	−2.32051e26	−0.0615190
$424$	−7.97523e26	−0.206254
$425$	0	0
$426$	5.10416e27	1.25639
$427$	8.50354e27	2.04224
$428$	−1.51150e27	−0.354200
$429$	5.14429e26	0.117631
$430$	0	0
$431$	−2.23137e27	−0.485914	−0.242957	−	0.970037i	$-0.578117\pi$
$431$	−2.23137e27	−0.485914	−0.242957	+	0.970037i	$0.578117\pi$
$432$	7.39584e26	0.157184
$433$	5.24943e27	1.08890	0.544451	−	0.838793i	$-0.316738\pi$
$433$	5.24943e27	1.08890	0.544451	+	0.838793i	$0.316738\pi$
$434$	−1.40261e28	−2.83984
$435$	0	0
$436$	−8.31241e27	−1.60369
$437$	−1.65977e26	−0.0312603
$438$	1.37676e27	0.253150
$439$	−4.22260e26	−0.0758058	−0.0379029	−	0.999281i	$-0.512068\pi$
$439$	−4.22260e26	−0.0758058	−0.0379029	+	0.999281i	$0.512068\pi$
$440$	0	0
$441$	1.83736e27	0.314477
$442$	6.56635e27	1.09746
$443$	6.71148e27	1.09542	0.547708	−	0.836670i	$-0.315501\pi$
$443$	6.71148e27	1.09542	0.547708	+	0.836670i	$0.315501\pi$
$444$	−3.03936e27	−0.484464
$445$	0	0
$446$	1.59588e28	2.42653
$447$	−4.90516e27	−0.728491
$448$	−1.23717e28	−1.79478
$449$	1.36311e28	1.93171	0.965857	−	0.259074i	$-0.0834173\pi$
$449$	1.36311e28	1.93171	0.965857	+	0.259074i	$0.0834173\pi$
$450$	0	0
$451$	−5.69668e26	−0.0770492
$452$	1.55602e28	2.05618
$453$	4.80126e27	0.619903
$454$	7.61192e26	0.0960302
$455$	0	0
$456$	2.22943e25	0.00268573
$457$	−6.75530e26	−0.0795288	−0.0397644	−	0.999209i	$-0.512661\pi$
$457$	−6.75530e26	−0.0795288	−0.0397644	+	0.999209i	$0.512661\pi$
$458$	8.68433e27	0.999191
$459$	−1.74121e27	−0.195803
$460$	0	0
$461$	1.77372e28	1.90557	0.952784	−	0.303648i	$-0.0982046\pi$
$461$	1.77372e28	1.90557	0.952784	+	0.303648i	$0.0982046\pi$
$462$	3.12431e27	0.328105
$463$	1.71011e28	1.75559	0.877795	−	0.479036i	$-0.159014\pi$
$463$	1.71011e28	1.75559	0.877795	+	0.479036i	$0.159014\pi$
$464$	−6.42281e27	−0.644594
$465$	0	0
$466$	1.30119e28	1.24822
$467$	−4.22031e27	−0.395838	−0.197919	−	0.980218i	$-0.563418\pi$
$467$	−4.22031e27	−0.395838	−0.197919	+	0.980218i	$0.563418\pi$
$468$	−3.09095e27	−0.283472
$469$	−1.40085e28	−1.25625
$470$	0	0
$471$	−7.25453e27	−0.622143
$472$	−2.47869e27	−0.207889
$473$	−2.54903e27	−0.209090
$474$	4.50857e27	0.361715
$475$	0	0
$476$	2.14017e28	1.64276
$477$	3.93937e27	0.295790
$478$	−3.58204e28	−2.63110
$479$	−1.35397e28	−0.972940	−0.486470	−	0.873697i	$-0.661716\pi$
$479$	−1.35397e28	−0.972940	−0.486470	+	0.873697i	$0.661716\pi$
$480$	0	0
$481$	−7.73389e27	−0.531955
$482$	−1.68479e28	−1.13384
$483$	1.90913e28	1.25716
$484$	−1.65908e28	−1.06903
$485$	0	0
$486$	1.52730e27	0.0942420
$487$	3.09803e28	1.87082	0.935409	−	0.353566i	$-0.115031\pi$
$487$	3.09803e28	1.87082	0.935409	+	0.353566i	$0.115031\pi$
$488$	−5.76139e27	−0.340502
$489$	1.56146e28	0.903209
$490$	0	0
$491$	−9.91039e27	−0.549206	−0.274603	−	0.961558i	$-0.588546\pi$
$491$	−9.91039e27	−0.549206	−0.274603	+	0.961558i	$0.588546\pi$
$492$	3.42285e27	0.185676
$493$	1.51213e28	0.802966
$494$	4.15288e26	0.0215882
$495$	0	0
$496$	−2.27305e28	−1.13253
$497$	4.23315e28	2.06501
$498$	−1.57944e28	−0.754388
$499$	1.73803e28	0.812833	0.406416	−	0.913688i	$-0.366778\pi$
$499$	1.73803e28	0.812833	0.406416	+	0.913688i	$0.366778\pi$
$500$	0	0
$501$	−1.41607e28	−0.635024
$502$	2.14019e28	0.939860
$503$	−2.38152e28	−1.02421	−0.512106	−	0.858922i	$-0.671135\pi$
$503$	−2.38152e28	−1.02421	−0.512106	+	0.858922i	$0.671135\pi$
$504$	−2.56437e27	−0.108009
$505$	0	0
$506$	1.57593e28	0.636732
$507$	6.72376e27	0.266090
$508$	−9.50277e27	−0.368367
$509$	1.77287e28	0.673195	0.336598	−	0.941649i	$-0.390724\pi$
$509$	1.77287e28	0.673195	0.336598	+	0.941649i	$0.390724\pi$
$510$	0	0
$511$	1.14182e28	0.416080
$512$	−3.80866e28	−1.35968
$513$	−1.10123e26	−0.00385163
$514$	4.77336e28	1.63573
$515$	0	0
$516$	1.53159e28	0.503872
$517$	−1.58863e27	−0.0512121
$518$	−4.69707e28	−1.48376
$519$	−1.97972e28	−0.612839
$520$	0	0
$521$	−1.06630e28	−0.317017	−0.158509	−	0.987358i	$-0.550669\pi$
$521$	−1.06630e28	−0.317017	−0.158509	+	0.987358i	$0.550669\pi$
$522$	−1.32636e28	−0.386476
$523$	4.82119e27	0.137685	0.0688424	−	0.997628i	$-0.478069\pi$
$523$	4.82119e27	0.137685	0.0688424	+	0.997628i	$0.478069\pi$
$524$	−2.27883e28	−0.637872
$525$	0	0
$526$	2.66220e28	0.715963
$527$	5.35148e28	1.41079
$528$	5.06323e27	0.130849
$529$	5.68267e28	1.43969
$530$	0	0
$531$	1.22435e28	0.298135
$532$	1.35354e27	0.0323147
$533$	8.70971e27	0.203877
$534$	5.22815e28	1.19995
$535$	0	0
$536$	9.49113e27	0.209454
$537$	−1.45181e27	−0.0314181
$538$	1.87600e28	0.398125
$539$	1.25787e28	0.261790
$540$	0	0
$541$	−5.70287e28	−1.14162	−0.570810	−	0.821082i	$-0.693371\pi$
$541$	−5.70287e28	−1.14162	−0.570810	+	0.821082i	$0.693371\pi$
$542$	1.05524e29	2.07185
$543$	−1.84143e28	−0.354615
$544$	7.71484e28	1.45727
$545$	0	0
$546$	−4.77679e28	−0.868187
$547$	−7.81143e28	−1.39272	−0.696360	−	0.717693i	$-0.745198\pi$
$547$	−7.81143e28	−1.39272	−0.696360	+	0.717693i	$0.745198\pi$
$548$	−9.51832e28	−1.66481
$549$	2.84584e28	0.488316
$550$	0	0
$551$	9.56345e26	0.0157951
$552$	−1.29349e28	−0.209606
$553$	3.73919e28	0.594517
$554$	7.16559e28	1.11789
$555$	0	0
$556$	1.39671e29	2.09807
$557$	−8.52376e28	−1.25647	−0.628234	−	0.778024i	$-0.716222\pi$
$557$	−8.52376e28	−1.25647	−0.628234	+	0.778024i	$0.716222\pi$
$558$	−4.69404e28	−0.679028
$559$	3.89724e28	0.553265
$560$	0	0
$561$	−1.19204e28	−0.162998
$562$	−1.78123e29	−2.39050
$563$	7.73536e28	1.01893	0.509463	−	0.860493i	$-0.329844\pi$
$563$	7.73536e28	1.01893	0.509463	+	0.860493i	$0.329844\pi$
$564$	9.54531e27	0.123413
$565$	0	0
$566$	−1.57930e29	−1.96740
$567$	1.26667e28	0.154897
$568$	−2.86808e28	−0.344297
$569$	1.06186e29	1.25137	0.625687	−	0.780074i	$-0.284819\pi$
$569$	1.06186e29	1.25137	0.625687	+	0.780074i	$0.284819\pi$
$570$	0	0
$571$	6.33038e28	0.719036	0.359518	−	0.933138i	$-0.382941\pi$
$571$	6.33038e28	0.719036	0.359518	+	0.933138i	$0.382941\pi$
$572$	−2.11608e28	−0.235979
$573$	1.88849e28	0.206772
$574$	5.28972e28	0.568667
$575$	0	0
$576$	−4.14039e28	−0.429146
$577$	−1.30204e29	−1.32519	−0.662596	−	0.748977i	$-0.730545\pi$
$577$	−1.30204e29	−1.32519	−0.662596	+	0.748977i	$0.730545\pi$
$578$	−5.16623e27	−0.0516335
$579$	−8.20341e28	−0.805137
$580$	0	0
$581$	−1.30991e29	−1.23992
$582$	−5.84905e28	−0.543744
$583$	2.69691e28	0.246233
$584$	−7.73613e27	−0.0693727
$585$	0	0
$586$	1.75465e29	1.51798
$587$	2.01309e29	1.71066	0.855328	−	0.518086i	$-0.173355\pi$
$587$	2.01309e29	1.71066	0.855328	+	0.518086i	$0.173355\pi$
$588$	−7.55790e28	−0.630869
$589$	3.38453e27	0.0277516
$590$	0	0
$591$	−8.02796e28	−0.635238
$592$	−7.61204e28	−0.591729
$593$	5.38839e28	0.411514	0.205757	−	0.978603i	$-0.434034\pi$
$593$	5.38839e28	0.411514	0.205757	+	0.978603i	$0.434034\pi$
$594$	1.04560e28	0.0784526
$595$	0	0
$596$	2.01772e29	1.46142
$597$	4.91651e28	0.349886
$598$	−2.40946e29	−1.68483
$599$	−1.93023e29	−1.32626	−0.663128	−	0.748506i	$-0.730772\pi$
$599$	−1.93023e29	−1.32626	−0.663128	+	0.748506i	$0.730772\pi$
$600$	0	0
$601$	−2.42672e28	−0.161005	−0.0805023	−	0.996754i	$-0.525652\pi$
$601$	−2.42672e28	−0.161005	−0.0805023	+	0.996754i	$0.525652\pi$
$602$	2.36694e29	1.54320
$603$	−4.68815e28	−0.300379
$604$	−1.97498e29	−1.24358
$605$	0	0
$606$	−1.21702e29	−0.740175
$607$	3.24649e29	1.94058	0.970292	−	0.241936i	$-0.0777823\pi$
$607$	3.24649e29	1.94058	0.970292	+	0.241936i	$0.0777823\pi$
$608$	4.87923e27	0.0286658
$609$	−1.10002e29	−0.635214
$610$	0	0
$611$	2.42888e28	0.135510
$612$	7.16241e28	0.392798
$613$	1.28641e29	0.693495	0.346747	−	0.937959i	$-0.387286\pi$
$613$	1.28641e29	0.693495	0.346747	+	0.937959i	$0.387286\pi$
$614$	−3.05644e29	−1.61975
$615$	0	0
$616$	−1.75558e28	−0.0899132
$617$	1.63967e28	0.0825584	0.0412792	−	0.999148i	$-0.486857\pi$
$617$	1.63967e28	0.0825584	0.0412792	+	0.999148i	$0.486857\pi$
$618$	−2.40493e29	−1.19049
$619$	2.86407e29	1.39390	0.696949	−	0.717120i	$-0.254540\pi$
$619$	2.86407e29	1.39390	0.696949	+	0.717120i	$0.254540\pi$
$620$	0	0
$621$	6.38921e28	0.300597
$622$	−2.87028e28	−0.132778
$623$	4.33598e29	1.97225
$624$	−7.74123e28	−0.346236
$625$	0	0
$626$	8.14892e28	0.352427
$627$	−7.53906e26	−0.00320633
$628$	2.98412e29	1.24807
$629$	1.79211e29	0.737112
$630$	0	0
$631$	3.17949e29	1.26488	0.632439	−	0.774610i	$-0.282054\pi$
$631$	3.17949e29	1.26488	0.632439	+	0.774610i	$0.282054\pi$
$632$	−2.53341e28	−0.0991234
$633$	−2.89529e29	−1.11418
$634$	−4.43035e29	−1.67688
$635$	0	0
$636$	−1.62044e29	−0.593381
$637$	−1.92316e29	−0.692712
$638$	−9.08036e28	−0.321726
$639$	1.41669e29	0.493759
$640$	0	0
$641$	−1.00981e29	−0.340589	−0.170294	−	0.985393i	$-0.554472\pi$
$641$	−1.00981e29	−0.340589	−0.170294	+	0.985393i	$0.554472\pi$
$642$	−7.81742e28	−0.259385
$643$	4.70649e29	1.53632	0.768161	−	0.640257i	$-0.221172\pi$
$643$	4.70649e29	1.53632	0.768161	+	0.640257i	$0.221172\pi$
$644$	−7.85313e29	−2.52198
$645$	0	0
$646$	−9.62313e27	−0.0299140
$647$	4.78801e29	1.46440	0.732201	−	0.681089i	$-0.238493\pi$
$647$	4.78801e29	1.46440	0.732201	+	0.681089i	$0.238493\pi$
$648$	−8.58207e27	−0.0258258
$649$	8.38196e28	0.248185
$650$	0	0
$651$	−3.89301e29	−1.11605
$652$	−6.42300e29	−1.81192
$653$	1.11323e29	0.309027	0.154514	−	0.987991i	$-0.450619\pi$
$653$	1.11323e29	0.309027	0.154514	+	0.987991i	$0.450619\pi$
$654$	−4.29915e29	−1.17440
$655$	0	0
$656$	8.57248e28	0.226786
$657$	3.82126e28	0.0994880
$658$	1.47514e29	0.377975
$659$	−3.13403e29	−0.790327	−0.395163	−	0.918611i	$-0.629312\pi$
$659$	−3.13403e29	−0.790327	−0.395163	+	0.918611i	$0.629312\pi$
$660$	0	0
$661$	1.94234e29	0.474471	0.237235	−	0.971452i	$-0.423759\pi$
$661$	1.94234e29	0.474471	0.237235	+	0.971452i	$0.423759\pi$
$662$	8.85698e29	2.12950
$663$	1.82253e29	0.431303
$664$	8.87503e28	0.206731
$665$	0	0
$666$	−1.57195e29	−0.354780
$667$	−5.54861e29	−1.23272
$668$	5.82496e29	1.27391
$669$	4.42946e29	0.953623
$670$	0	0
$671$	1.94828e29	0.406504
$672$	−5.61228e29	−1.15282
$673$	−2.68841e28	−0.0543673	−0.0271836	−	0.999630i	$-0.508654\pi$
$673$	−2.68841e28	−0.0543673	−0.0271836	+	0.999630i	$0.508654\pi$
$674$	−2.13402e29	−0.424884
$675$	0	0
$676$	−2.76579e29	−0.533800
$677$	−8.12214e29	−1.54344	−0.771720	−	0.635963i	$-0.780603\pi$
$677$	−8.12214e29	−1.54344	−0.771720	+	0.635963i	$0.780603\pi$
$678$	8.04769e29	1.50577
$679$	−4.85093e29	−0.893701
$680$	0	0
$681$	2.11273e28	0.0377398
$682$	−3.21356e29	−0.565263
$683$	2.74550e29	0.475559	0.237779	−	0.971319i	$-0.423580\pi$
$683$	2.74550e29	0.475559	0.237779	+	0.971319i	$0.423580\pi$
$684$	4.52985e27	0.00772671
$685$	0	0
$686$	7.00343e28	0.115853
$687$	2.41038e29	0.392682
$688$	3.83584e29	0.615434
$689$	−4.12334e29	−0.651549
$690$	0	0
$691$	7.16099e29	1.09762	0.548812	−	0.835946i	$-0.315080\pi$
$691$	7.16099e29	1.09762	0.548812	+	0.835946i	$0.315080\pi$
$692$	8.14349e29	1.22941
$693$	8.67170e28	0.128945
$694$	5.59268e29	0.819116
$695$	0	0
$696$	7.45297e28	0.105909
$697$	−2.01823e29	−0.282506
$698$	−1.88279e29	−0.259610
$699$	3.61154e29	0.490548
$700$	0	0
$701$	−1.03091e29	−0.135888	−0.0679438	−	0.997689i	$-0.521644\pi$
$701$	−1.03091e29	−0.135888	−0.0679438	+	0.997689i	$0.521644\pi$
$702$	−1.59863e29	−0.207591
$703$	1.13342e28	0.0144997
$704$	−2.83453e29	−0.357247
$705$	0	0
$706$	−1.03186e30	−1.26232
$707$	−1.00934e30	−1.21656
$708$	−5.03631e29	−0.598085
$709$	−5.75181e28	−0.0673006	−0.0336503	−	0.999434i	$-0.510713\pi$
$709$	−5.75181e28	−0.0673006	−0.0336503	+	0.999434i	$0.510713\pi$
$710$	0	0
$711$	1.25138e29	0.142154
$712$	−2.93775e29	−0.328833
$713$	−1.96367e30	−2.16585
$714$	1.10689e30	1.20302
$715$	0	0
$716$	5.97195e28	0.0630274
$717$	−9.94215e29	−1.03402
$718$	−3.41957e29	−0.350481
$719$	−1.09295e30	−1.10395	−0.551974	−	0.833861i	$-0.686125\pi$
$719$	−1.09295e30	−1.10395	−0.551974	+	0.833861i	$0.686125\pi$
$720$	0	0
$721$	−1.99454e30	−1.95669
$722$	1.51885e30	1.46850
$723$	−4.67623e29	−0.445598
$724$	7.57465e29	0.711389
$725$	0	0
$726$	−8.58072e29	−0.782869
$727$	1.41110e29	0.126896	0.0634478	−	0.997985i	$-0.479790\pi$
$727$	1.41110e29	0.126896	0.0634478	+	0.997985i	$0.479790\pi$
$728$	2.68413e29	0.237916
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−9.03077e29	−0.766642
$732$	−1.17062e30	−0.979605
$733$	−1.08687e30	−0.896575	−0.448287	−	0.893890i	$-0.647966\pi$
$733$	−1.08687e30	−0.896575	−0.448287	+	0.893890i	$0.647966\pi$
$734$	−1.90949e29	−0.155277
$735$	0	0
$736$	−2.83088e30	−2.23720
$737$	−3.20953e29	−0.250054
$738$	1.77029e29	0.135973
$739$	1.32212e30	1.00116	0.500582	−	0.865689i	$-0.333119\pi$
$739$	1.32212e30	1.00116	0.500582	+	0.865689i	$0.333119\pi$
$740$	0	0
$741$	1.15265e28	0.00848414
$742$	−2.50425e30	−1.81734
$743$	−1.32336e30	−0.946884	−0.473442	−	0.880825i	$-0.656989\pi$
$743$	−1.32336e30	−0.946884	−0.473442	+	0.880825i	$0.656989\pi$
$744$	2.63763e29	0.186079
$745$	0	0
$746$	3.65902e30	2.50962
$747$	−4.38382e29	−0.296474
$748$	4.90342e29	0.326988
$749$	−6.48339e29	−0.426327
$750$	0	0
$751$	−1.81366e29	−0.115968	−0.0579839	−	0.998318i	$-0.518467\pi$
$751$	−1.81366e29	−0.115968	−0.0579839	+	0.998318i	$0.518467\pi$
$752$	2.39061e29	0.150737
$753$	5.94022e29	0.369364
$754$	1.38831e30	0.851306
$755$	0	0
$756$	−5.21040e29	−0.310736
$757$	1.12853e30	0.663753	0.331877	−	0.943323i	$-0.392318\pi$
$757$	1.12853e30	0.663753	0.331877	+	0.943323i	$0.392318\pi$
$758$	1.32485e30	0.768494
$759$	4.37408e29	0.250235
$760$	0	0
$761$	2.77844e30	1.54619	0.773093	−	0.634293i	$-0.218709\pi$
$761$	2.77844e30	1.54619	0.773093	+	0.634293i	$0.218709\pi$
$762$	−4.91480e29	−0.269761
$763$	−3.56551e30	−1.93025
$764$	−7.76822e29	−0.414802
$765$	0	0
$766$	−7.53604e29	−0.391509
$767$	−1.28153e30	−0.656714
$768$	−7.00220e29	−0.353950
$769$	−2.88419e29	−0.143813	−0.0719064	−	0.997411i	$-0.522908\pi$
$769$	−2.88419e29	−0.143813	−0.0719064	+	0.997411i	$0.522908\pi$
$770$	0	0
$771$	1.32487e30	0.642840
$772$	3.37444e30	1.61518
$773$	3.73885e30	1.76544	0.882720	−	0.469899i	$-0.155710\pi$
$773$	3.73885e30	1.76544	0.882720	+	0.469899i	$0.155710\pi$
$774$	7.92132e29	0.368993
$775$	0	0
$776$	3.28664e29	0.149006
$777$	−1.30370e30	−0.583118
$778$	−3.02072e30	−1.33298
$779$	−1.27643e28	−0.00555715
$780$	0	0
$781$	9.69872e29	0.411035
$782$	5.58324e30	2.33462
$783$	−3.68140e29	−0.151885
$784$	−1.89286e30	−0.770550
$785$	0	0
$786$	−1.17860e30	−0.467123
$787$	−7.64436e29	−0.298955	−0.149478	−	0.988765i	$-0.547759\pi$
$787$	−7.64436e29	−0.298955	−0.149478	+	0.988765i	$0.547759\pi$
$788$	3.30226e30	1.27434
$789$	7.38909e29	0.281373
$790$	0	0
$791$	6.67437e30	2.47490
$792$	−5.87533e28	−0.0214990
$793$	−2.97874e30	−1.07563
$794$	6.41079e30	2.28453
$795$	0	0
$796$	−2.02238e30	−0.701901
$797$	4.75836e30	1.62984	0.814920	−	0.579573i	$-0.196781\pi$
$797$	4.75836e30	1.62984	0.814920	+	0.579573i	$0.196781\pi$
$798$	7.00049e28	0.0236645
$799$	−5.62824e29	−0.187772
$800$	0	0
$801$	1.45110e30	0.471582
$802$	5.26504e30	1.68877
$803$	2.61606e29	0.0828198
$804$	1.92845e30	0.602587
$805$	0	0
$806$	4.91325e30	1.49572
$807$	5.20695e29	0.156463
$808$	6.83854e29	0.202836
$809$	5.90538e27	0.00172897	0.000864487	−	1.00000i	$-0.499725\pi$
$809$	5.90538e27	0.00172897	0.000864487		1.00000i	$0.499725\pi$
$810$	0	0
$811$	6.54875e29	0.186827	0.0934134	−	0.995627i	$-0.470222\pi$
$811$	6.54875e29	0.186827	0.0934134	+	0.995627i	$0.470222\pi$
$812$	4.52490e30	1.27430
$813$	2.92888e30	0.814237
$814$	−1.07616e30	−0.295340
$815$	0	0
$816$	1.79381e30	0.479767
$817$	−5.71149e28	−0.0150806
$818$	5.37708e30	1.40164
$819$	−1.32583e30	−0.341197
$820$	0	0
$821$	4.27671e30	1.07277	0.536385	−	0.843973i	$-0.319789\pi$
$821$	4.27671e30	1.07277	0.536385	+	0.843973i	$0.319789\pi$
$822$	−4.92284e30	−1.21916
$823$	1.27831e30	0.312564	0.156282	−	0.987712i	$-0.450049\pi$
$823$	1.27831e30	0.312564	0.156282	+	0.987712i	$0.450049\pi$
$824$	1.35136e30	0.326238
$825$	0	0
$826$	−7.78317e30	−1.83175
$827$	1.57854e30	0.366815	0.183408	−	0.983037i	$-0.441287\pi$
$827$	1.57854e30	0.366815	0.183408	+	0.983037i	$0.441287\pi$
$828$	−2.62817e30	−0.603025
$829$	1.72347e30	0.390463	0.195232	−	0.980757i	$-0.437454\pi$
$829$	1.72347e30	0.390463	0.195232	+	0.980757i	$0.437454\pi$
$830$	0	0
$831$	1.98885e30	0.439331
$832$	4.33375e30	0.945298
$833$	4.45640e30	0.959867
$834$	7.22372e30	1.53645
$835$	0	0
$836$	3.10116e28	0.00643218
$837$	−1.30286e30	−0.266858
$838$	−1.25310e31	−2.53467
$839$	9.09359e29	0.181650	0.0908249	−	0.995867i	$-0.471050\pi$
$839$	9.09359e29	0.181650	0.0908249	+	0.995867i	$0.471050\pi$
$840$	0	0
$841$	−1.93578e30	−0.377137
$842$	1.36230e31	2.62117
$843$	−4.94391e30	−0.939466
$844$	1.19097e31	2.23513
$845$	0	0
$846$	4.93680e29	0.0903767
$847$	−7.11644e30	−1.28673
$848$	−4.05837e30	−0.724762
$849$	−4.38344e30	−0.773187
$850$	0	0
$851$	−6.57597e30	−1.13162
$852$	−5.82748e30	−0.990525
$853$	−3.54786e30	−0.595664	−0.297832	−	0.954618i	$-0.596264\pi$
$853$	−3.54786e30	−0.595664	−0.297832	+	0.954618i	$0.596264\pi$
$854$	−1.80910e31	−3.00023
$855$	0	0
$856$	4.39268e29	0.0710813
$857$	3.53257e29	0.0564667	0.0282334	−	0.999601i	$-0.491012\pi$
$857$	3.53257e29	0.0564667	0.0282334	+	0.999601i	$0.491012\pi$
$858$	−1.09443e30	−0.172811
$859$	−6.94778e30	−1.08372	−0.541861	−	0.840468i	$-0.682280\pi$
$859$	−6.94778e30	−1.08372	−0.541861	+	0.840468i	$0.682280\pi$
$860$	0	0
$861$	1.46819e30	0.223486
$862$	4.74716e30	0.713850
$863$	4.91327e29	0.0729890	0.0364945	−	0.999334i	$-0.488381\pi$
$863$	4.91327e29	0.0729890	0.0364945	+	0.999334i	$0.488381\pi$
$864$	−1.87824e30	−0.275649
$865$	0	0
$866$	−1.11680e31	−1.59969
$867$	−1.43392e29	−0.0202919
$868$	1.60137e31	2.23890
$869$	8.56701e29	0.118337
$870$	0	0
$871$	4.90709e30	0.661657
$872$	2.41574e30	0.321830
$873$	−1.62344e30	−0.213691
$874$	3.53111e29	0.0459241
$875$	0	0
$876$	−1.57186e30	−0.199582
$877$	1.31164e30	0.164558	0.0822790	−	0.996609i	$-0.473780\pi$
$877$	1.31164e30	0.164558	0.0822790	+	0.996609i	$0.473780\pi$
$878$	8.98344e29	0.111365
$879$	4.87012e30	0.596564
$880$	0	0
$881$	1.32982e31	1.59055	0.795273	−	0.606252i	$-0.207328\pi$
$881$	1.32982e31	1.59055	0.795273	+	0.606252i	$0.207328\pi$
$882$	−3.90892e30	−0.461994
$883$	4.84651e30	0.566034	0.283017	−	0.959115i	$-0.408665\pi$
$883$	4.84651e30	0.566034	0.283017	+	0.959115i	$0.408665\pi$
$884$	−7.49689e30	−0.865232
$885$	0	0
$886$	−1.42784e31	−1.60926
$887$	3.16217e30	0.352198	0.176099	−	0.984372i	$-0.443652\pi$
$887$	3.16217e30	0.352198	0.176099	+	0.984372i	$0.443652\pi$
$888$	8.83294e29	0.0972230
$889$	−4.07610e30	−0.443380
$890$	0	0
$891$	2.90212e29	0.0308318
$892$	−1.82204e31	−1.91305
$893$	−3.55957e28	−0.00369366
$894$	1.04356e31	1.07022
$895$	0	0
$896$	6.38816e30	0.639943
$897$	−6.68758e30	−0.662138
$898$	−2.89996e31	−2.83786
$899$	1.13145e31	1.09435
$900$	0	0
$901$	9.55468e30	0.902830
$902$	1.21195e30	0.113192
$903$	6.56957e30	0.606479
$904$	−4.52208e30	−0.412638
$905$	0	0
$906$	−1.02145e31	−0.910692
$907$	−1.04066e31	−0.917137	−0.458569	−	0.888659i	$-0.651638\pi$
$907$	−1.04066e31	−0.917137	−0.458569	+	0.888659i	$0.651638\pi$
$908$	−8.69063e29	−0.0757094
$909$	−3.37790e30	−0.290888
$910$	0	0
$911$	1.24395e31	1.04679	0.523394	−	0.852091i	$-0.324666\pi$
$911$	1.24395e31	1.04679	0.523394	+	0.852091i	$0.324666\pi$
$912$	1.13449e29	0.00943748
$913$	−3.00119e30	−0.246803
$914$	1.43717e30	0.116835
$915$	0	0
$916$	−9.91501e30	−0.787754
$917$	−9.77477e30	−0.767766
$918$	3.70437e30	0.287651
$919$	1.38734e31	1.06505	0.532526	−	0.846413i	$-0.321243\pi$
$919$	1.38734e31	1.06505	0.532526	+	0.846413i	$0.321243\pi$
$920$	0	0
$921$	−8.48334e30	−0.636562
$922$	−3.77352e31	−2.79945
$923$	−1.48285e31	−1.08762
$924$	−3.56707e30	−0.258676
$925$	0	0
$926$	−3.63820e31	−2.57911
$927$	−6.67503e30	−0.467860
$928$	1.63113e31	1.13040
$929$	−5.06678e30	−0.347190	−0.173595	−	0.984817i	$-0.555538\pi$
$929$	−5.06678e30	−0.347190	−0.173595	+	0.984817i	$0.555538\pi$
$930$	0	0
$931$	2.81844e29	0.0188815
$932$	−1.48559e31	−0.984083
$933$	−7.96662e29	−0.0521816
$934$	8.97856e30	0.581520
$935$	0	0
$936$	8.98285e29	0.0568876
$937$	2.26921e30	0.142105	0.0710523	−	0.997473i	$-0.477364\pi$
$937$	2.26921e30	0.142105	0.0710523	+	0.997473i	$0.477364\pi$
$938$	2.98025e31	1.84554
$939$	2.26178e30	0.138504
$940$	0	0
$941$	9.10781e30	0.545410	0.272705	−	0.962098i	$-0.412082\pi$
$941$	9.10781e30	0.545410	0.272705	+	0.962098i	$0.412082\pi$
$942$	1.54338e31	0.913981
$943$	7.40570e30	0.433704
$944$	−1.26133e31	−0.730507
$945$	0	0
$946$	5.42298e30	0.307172
$947$	−2.33280e31	−1.30678	−0.653389	−	0.757022i	$-0.726653\pi$
$947$	−2.33280e31	−1.30678	−0.653389	+	0.757022i	$0.726653\pi$
$948$	−5.14749e30	−0.285173
$949$	−3.99972e30	−0.219146
$950$	0	0
$951$	−1.22967e31	−0.659012
$952$	−6.21971e30	−0.329672
$953$	−1.16285e31	−0.609606	−0.304803	−	0.952415i	$-0.598591\pi$
$953$	−1.16285e31	−0.609606	−0.304803	+	0.952415i	$0.598591\pi$
$954$	−8.38087e30	−0.434541
$955$	0	0
$956$	4.08966e31	2.07434
$957$	−2.52030e30	−0.126438
$958$	2.88052e31	1.42933
$959$	−4.08277e31	−2.00382
$960$	0	0
$961$	1.92167e31	0.922749
$962$	1.64536e31	0.781488
$963$	−2.16977e30	−0.101938
$964$	1.92354e31	0.893909
$965$	0	0
$966$	−4.06161e31	−1.84688
$967$	−9.97749e30	−0.448790	−0.224395	−	0.974498i	$-0.572041\pi$
$967$	−9.97749e30	−0.448790	−0.224395	+	0.974498i	$0.572041\pi$
$968$	4.82159e30	0.214535
$969$	−2.67095e29	−0.0117562
$970$	0	0
$971$	3.69422e30	0.159119	0.0795593	−	0.996830i	$-0.474649\pi$
$971$	3.69422e30	0.159119	0.0795593	+	0.996830i	$0.474649\pi$
$972$	−1.74374e30	−0.0742996
$973$	5.99101e31	2.52531
$974$	−6.59095e31	−2.74840
$975$	0	0
$976$	−2.93181e31	−1.19650
$977$	2.05704e31	0.830518	0.415259	−	0.909703i	$-0.363691\pi$
$977$	2.05704e31	0.830518	0.415259	+	0.909703i	$0.363691\pi$
$978$	−3.32195e31	−1.32689
$979$	9.93433e30	0.392573
$980$	0	0
$981$	−1.19325e31	−0.461539
$982$	2.10840e31	0.806831
$983$	7.27829e30	0.275561	0.137780	−	0.990463i	$-0.456003\pi$
$983$	7.27829e30	0.275561	0.137780	+	0.990463i	$0.456003\pi$
$984$	−9.94743e29	−0.0372617
$985$	0	0
$986$	−3.21701e31	−1.17963
$987$	4.09435e30	0.148544
$988$	−4.74139e29	−0.0170199
$989$	3.31375e31	1.17695
$990$	0	0
$991$	−1.68737e31	−0.586728	−0.293364	−	0.956001i	$-0.594775\pi$
$991$	−1.68737e31	−0.586728	−0.293364	+	0.956001i	$0.594775\pi$
$992$	5.77260e31	1.98609
$993$	2.45831e31	0.836891
$994$	−9.00587e31	−3.03367
$995$	0	0
$996$	1.80327e31	0.594753
$997$	1.86625e31	0.609075	0.304538	−	0.952500i	$-0.401498\pi$
$997$	1.86625e31	0.609075	0.304538	+	0.952500i	$0.401498\pi$
$998$	−3.69760e31	−1.19412
$999$	−4.36303e30	−0.139428

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.l.1.2	yes	8
5.2	odd	4		75.22.b.j.49.3		16
5.3	odd	4		75.22.b.j.49.14		16
5.4	even	2		75.22.a.k.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.22.a.k.1.7	✓	8	5.4	even	2
75.22.a.l.1.2	yes	8	1.1	even	1	trivial
75.22.b.j.49.3		16	5.2	odd	4
75.22.b.j.49.14		16	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)

\(f(q)\)	\(=\)	\(q-2127.46 q^{2} -59049.0 q^{3} +2.42895e6 q^{4} +1.25625e8 q^{6} +1.04187e9 q^{7} -7.05898e8 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-2127.46 q^{2} -59049.0 q^{3} +2.42895e6 q^{4} +1.25625e8 q^{6} +1.04187e9 q^{7} -7.05898e8 q^{8} +3.48678e9 q^{9} +2.38707e10 q^{11} -1.43427e11 q^{12} -3.64962e11 q^{13} -2.21654e12 q^{14} -3.59211e12 q^{16} +8.45697e12 q^{17} -7.41801e12 q^{18} +5.34859e11 q^{19} -6.15215e13 q^{21} -5.07841e13 q^{22} -3.10320e14 q^{23} +4.16825e13 q^{24} +7.76443e14 q^{26} -2.05891e14 q^{27} +2.53066e15 q^{28} +1.78803e15 q^{29} +6.32789e15 q^{31} +9.12247e15 q^{32} -1.40954e15 q^{33} -1.79919e16 q^{34} +8.46924e15 q^{36} +2.11910e16 q^{37} -1.13789e15 q^{38} +2.15506e16 q^{39} -2.38647e16 q^{41} +1.30885e17 q^{42} -1.06785e17 q^{43} +5.79809e16 q^{44} +6.60194e17 q^{46} -6.65515e16 q^{47} +2.12111e17 q^{48} +5.26950e17 q^{49} -4.99375e17 q^{51} -8.86475e17 q^{52} +1.12980e18 q^{53} +4.38026e17 q^{54} -7.35454e17 q^{56} -3.15829e16 q^{57} -3.80397e18 q^{58} +3.51140e18 q^{59} +8.16179e18 q^{61} -1.34624e19 q^{62} +3.63278e18 q^{63} -1.18745e19 q^{64} +2.99875e18 q^{66} -1.34455e19 q^{67} +2.05416e19 q^{68} +1.83241e19 q^{69} +4.06302e19 q^{71} -2.46131e18 q^{72} +1.09593e19 q^{73} -4.50830e19 q^{74} +1.29915e18 q^{76} +2.48702e19 q^{77} -4.58482e19 q^{78} +3.58892e19 q^{79} +1.21577e19 q^{81} +5.07714e19 q^{82} -1.25727e20 q^{83} -1.49433e20 q^{84} +2.27181e20 q^{86} -1.05581e20 q^{87} -1.68503e19 q^{88} +4.16172e20 q^{89} -3.80243e20 q^{91} -7.53752e20 q^{92} -3.73656e20 q^{93} +1.41586e20 q^{94} -5.38673e20 q^{96} -4.65598e20 q^{97} -1.12107e21 q^{98} +8.32320e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9}+O(q^{10}) \) 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9	\( 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9} - 14410165968 q^{11} - 630755749296 q^{12} - 328226481176 q^{13} + 4068036669186 q^{14} + 18553054308920 q^{16} - 5718214953936 q^{17} + 2322198411066 q^{18} + 75919698170296 q^{19} - 7914601537128 q^{21} - 426897542691372 q^{22} + 49712781537936 q^{23} - 207390009885372 q^{24} - 126056679642054 q^{26} - 16\!\cdots\!92 q^{27}+ \cdots - 50\!\cdots\!68 q^{99}+O(q^{100}) \) 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9 - 14410165968 * q^11 - 630755749296 * q^12 - 328226481176 * q^13 + 4068036669186 * q^14 + 18553054308920 * q^16 - 5718214953936 * q^17 + 2322198411066 * q^18 + 75919698170296 * q^19 - 7914601537128 * q^21 - 426897542691372 * q^22 + 49712781537936 * q^23 - 207390009885372 * q^24 - 126056679642054 * q^26 - 1647129056757192 * q^27 + 4235167126399504 * q^28 + 2111332005818352 * q^29 + 5677451410844968 * q^31 + 17528957992420632 * q^32 + 850905890244432 * q^33 + 9944754702742068 * q^34 + 37245496240179504 * q^36 - 26538401323409648 * q^37 + 85729618982552526 * q^38 + 19381445486961624 * q^39 - 9719778050982432 * q^41 - 240213497278764114 * q^42 - 20888744714565080 * q^43 - 388904820117856128 * q^44 - 156886987299808452 * q^46 + 405141072730526160 * q^47 - 1095539303887417080 * q^48 + 1172899395436368720 * q^49 + 337654874814966864 * q^51 - 2264281219115894096 * q^52 + 1890950875516930176 * q^53 - 137123493975036234 * q^54 + 19720793170524660300 * q^56 - 4482982257257808504 * q^57 + 12636224332772065092 * q^58 - 3496737905134765776 * q^59 + 17508733151770015960 * q^61 - 30404101355896200606 * q^62 + 467349306165871272 * q^63 + 27809823573720908096 * q^64 + 25207872998382825228 * q^66 - 29911607071855004216 * q^67 - 29571773973515759808 * q^68 - 2935490037033582864 * q^69 + 52204030316256846816 * q^71 + 12246172693721331228 * q^72 - 39697032757192143344 * q^73 - 32301811215943606764 * q^74 + 383913379529874070384 * q^76 - 202348742072764877136 * q^77 + 7443520876183646646 * q^78 - 274125739348498990400 * q^79 + 97261323672455430408 * q^81 - 231746572148446490256 * q^82 - 276883976086286712912 * q^83 - 250082383646764311696 * q^84 + 486708932755841131242 * q^86 - 124672043611567867248 * q^87 - 2078122304879724561672 * q^88 - 319129684977517046784 * q^89 + 45068531423660000648 * q^91 + 147706203809955886272 * q^92 - 335247828358984515432 * q^93 - 1499781052151613424236 * q^94 - 1035067440494445898968 * q^96 + 451807368184588489864 * q^97 - 2604193962467412900204 * q^98 - 50245141913043465168 * q^99

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