LMFDB - Embedded newform 405.8.a.e.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,8,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	405.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:	$126.515935321$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 1170 x^{11} + 4622 x^{10} + 503384 x^{9} - 1392714 x^{8} - 97100172 x^{7} + \cdots + 4693741072256 $ x^13 - 5x^12 - 1170x^11 + 4622x^10 + 503384x^9 - 1392714x^8 - 97100172x^7 + 151526988x^6 + 8205832521x^5 - 4072720601x^4 - 282543812090x^3 - 118496553138x^2 + 3000776721014x + 4693741072256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{24} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$19.8324$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.8324 q^{2} +226.657 q^{4} +125.000 q^{5} +398.809 q^{7} +1857.95 q^{8} +O(q^{10})$	\(q+18.8324 q^{2} +226.657 q^{4} +125.000 q^{5} +398.809 q^{7} +1857.95 q^{8} +2354.04 q^{10} -4850.12 q^{11} -6892.16 q^{13} +7510.51 q^{14} +5977.44 q^{16} -29913.8 q^{17} -13015.7 q^{19} +28332.2 q^{20} -91339.1 q^{22} +30513.4 q^{23} +15625.0 q^{25} -129796. q^{26} +90393.0 q^{28} -97954.3 q^{29} -254406. q^{31} -125248. q^{32} -563346. q^{34} +49851.1 q^{35} +336304. q^{37} -245116. q^{38} +232244. q^{40} -24801.0 q^{41} -949803. q^{43} -1.09932e6 q^{44} +574639. q^{46} +249783. q^{47} -664494. q^{49} +294255. q^{50} -1.56216e6 q^{52} +1.16255e6 q^{53} -606265. q^{55} +740968. q^{56} -1.84471e6 q^{58} +677372. q^{59} +2.87790e6 q^{61} -4.79106e6 q^{62} -3.12384e6 q^{64} -861520. q^{65} +941082. q^{67} -6.78018e6 q^{68} +938814. q^{70} +4.06499e6 q^{71} +3.87973e6 q^{73} +6.33339e6 q^{74} -2.95010e6 q^{76} -1.93427e6 q^{77} -6.75281e6 q^{79} +747180. q^{80} -467061. q^{82} -2.65183e6 q^{83} -3.73922e6 q^{85} -1.78870e7 q^{86} -9.01128e6 q^{88} -1.76803e6 q^{89} -2.74865e6 q^{91} +6.91609e6 q^{92} +4.70400e6 q^{94} -1.62696e6 q^{95} +8.02400e6 q^{97} -1.25140e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 8 q^{2} + 704 q^{4} + 1625 q^{5} - 1455 q^{7} - 1236 q^{8}+O(q^{10}) $ 13 * q - 8 * q^2 + 704 * q^4 + 1625 * q^5 - 1455 * q^7 - 1236 * q^8	\( 13 q - 8 q^{2} + 704 q^{4} + 1625 q^{5} - 1455 q^{7} - 1236 q^{8} - 1000 q^{10} - 1658 q^{11} - 13568 q^{13} - 12351 q^{14} + 52076 q^{16} - 6944 q^{17} - 45812 q^{19} + 88000 q^{20} - 17993 q^{22} - 96441 q^{23} + 203125 q^{25} - 126146 q^{26} - 216945 q^{28} - 39043 q^{29} - 158520 q^{31} - 725206 q^{32} - 441617 q^{34} - 181875 q^{35} - 505438 q^{37} - 615041 q^{38} - 154500 q^{40} - 1578883 q^{41} - 1082090 q^{43} - 498211 q^{44} - 2312547 q^{46} - 1690139 q^{47} - 1054816 q^{49} - 125000 q^{50} - 4100644 q^{52} + 102274 q^{53} - 207250 q^{55} - 389331 q^{56} - 5780521 q^{58} + 2908966 q^{59} - 3091451 q^{61} - 5212476 q^{62} - 5659352 q^{64} - 1696000 q^{65} - 1849533 q^{67} + 563369 q^{68} - 1543875 q^{70} + 2617958 q^{71} + 5310946 q^{73} + 11485004 q^{74} - 14421739 q^{76} - 1719660 q^{77} - 3632166 q^{79} + 6509500 q^{80} - 7347658 q^{82} - 1424115 q^{83} - 868000 q^{85} + 23193379 q^{86} - 20229447 q^{88} + 2816067 q^{89} - 16929702 q^{91} + 27930183 q^{92} - 11693189 q^{94} - 5726500 q^{95} - 12062476 q^{97} + 29617805 q^{98}+O(q^{100}) \) 13 * q - 8 * q^2 + 704 * q^4 + 1625 * q^5 - 1455 * q^7 - 1236 * q^8 - 1000 * q^10 - 1658 * q^11 - 13568 * q^13 - 12351 * q^14 + 52076 * q^16 - 6944 * q^17 - 45812 * q^19 + 88000 * q^20 - 17993 * q^22 - 96441 * q^23 + 203125 * q^25 - 126146 * q^26 - 216945 * q^28 - 39043 * q^29 - 158520 * q^31 - 725206 * q^32 - 441617 * q^34 - 181875 * q^35 - 505438 * q^37 - 615041 * q^38 - 154500 * q^40 - 1578883 * q^41 - 1082090 * q^43 - 498211 * q^44 - 2312547 * q^46 - 1690139 * q^47 - 1054816 * q^49 - 125000 * q^50 - 4100644 * q^52 + 102274 * q^53 - 207250 * q^55 - 389331 * q^56 - 5780521 * q^58 + 2908966 * q^59 - 3091451 * q^61 - 5212476 * q^62 - 5659352 * q^64 - 1696000 * q^65 - 1849533 * q^67 + 563369 * q^68 - 1543875 * q^70 + 2617958 * q^71 + 5310946 * q^73 + 11485004 * q^74 - 14421739 * q^76 - 1719660 * q^77 - 3632166 * q^79 + 6509500 * q^80 - 7347658 * q^82 - 1424115 * q^83 - 868000 * q^85 + 23193379 * q^86 - 20229447 * q^88 + 2816067 * q^89 - 16929702 * q^91 + 27930183 * q^92 - 11693189 * q^94 - 5726500 * q^95 - 12062476 * q^97 + 29617805 * q^98

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 $

$2$	18.8324	1.66456	0.832280	−	0.554355i	$-0.187035\pi$
$2$	18.8324	1.66456	0.832280	+	0.554355i	$0.187035\pi$
$3$	0	0
$3$	0	0
$4$	226.657	1.77076
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	398.809	0.439462	0.219731	−	0.975560i	$-0.429482\pi$
$7$	398.809	0.439462	0.219731	+	0.975560i	$0.429482\pi$
$8$	1857.95	1.28298
$9$	0	0
$10$	2354.04	0.744414
$11$	−4850.12	−1.09870	−0.549348	−	0.835593i	$-0.685124\pi$
$11$	−4850.12	−1.09870	−0.549348	+	0.835593i	$0.685124\pi$
$12$	0	0
$13$	−6892.16	−0.870069	−0.435034	−	0.900414i	$-0.643264\pi$
$13$	−6892.16	−0.870069	−0.435034	+	0.900414i	$0.643264\pi$
$14$	7510.51	0.731512
$15$	0	0
$16$	5977.44	0.364834
$17$	−29913.8	−1.47673	−0.738363	−	0.674404i	$-0.764401\pi$
$17$	−29913.8	−1.47673	−0.738363	+	0.674404i	$0.764401\pi$
$18$	0	0
$19$	−13015.7	−0.435341	−0.217670	−	0.976022i	$-0.569846\pi$
$19$	−13015.7	−0.435341	−0.217670	+	0.976022i	$0.569846\pi$
$20$	28332.2	0.791909
$21$	0	0
$22$	−91339.1	−1.82885
$23$	30513.4	0.522930	0.261465	−	0.965213i	$-0.415794\pi$
$23$	30513.4	0.522930	0.261465	+	0.965213i	$0.415794\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−129796.	−1.44828
$27$	0	0
$28$	90393.0	0.778183
$29$	−97954.3	−0.745814	−0.372907	−	0.927869i	$-0.621639\pi$
$29$	−97954.3	−0.745814	−0.372907	+	0.927869i	$0.621639\pi$
$30$	0	0
$31$	−254406.	−1.53377	−0.766887	−	0.641782i	$-0.778195\pi$
$31$	−254406.	−1.53377	−0.766887	+	0.641782i	$0.778195\pi$
$32$	−125248.	−0.675690
$33$	0	0
$34$	−563346.	−2.45810
$35$	49851.1	0.196534
$36$	0	0
$37$	336304.	1.09151	0.545753	−	0.837946i	$-0.316244\pi$
$37$	336304.	1.09151	0.545753	+	0.837946i	$0.316244\pi$
$38$	−245116.	−0.724651
$39$	0	0
$40$	232244.	0.573765
$41$	−24801.0	−0.0561986	−0.0280993	−	0.999605i	$-0.508945\pi$
$41$	−24801.0	−0.0561986	−0.0280993	+	0.999605i	$0.508945\pi$
$42$	0	0
$43$	−949803.	−1.82177	−0.910886	−	0.412659i	$-0.864600\pi$
$43$	−949803.	−1.82177	−0.910886	+	0.412659i	$0.864600\pi$
$44$	−1.09932e6	−1.94553
$45$	0	0
$46$	574639.	0.870448
$47$	249783.	0.350930	0.175465	−	0.984486i	$-0.443857\pi$
$47$	249783.	0.350930	0.175465	+	0.984486i	$0.443857\pi$
$48$	0	0
$49$	−664494.	−0.806873
$50$	294255.	0.332912
$51$	0	0
$52$	−1.56216e6	−1.54068
$53$	1.16255e6	1.07262	0.536309	−	0.844021i	$-0.319818\pi$
$53$	1.16255e6	1.07262	0.536309	+	0.844021i	$0.319818\pi$
$54$	0	0
$55$	−606265.	−0.491352
$56$	740968.	0.563821
$57$	0	0
$58$	−1.84471e6	−1.24145
$59$	677372.	0.429384	0.214692	−	0.976682i	$-0.431125\pi$
$59$	677372.	0.429384	0.214692	+	0.976682i	$0.431125\pi$
$60$	0	0
$61$	2.87790e6	1.62338	0.811692	−	0.584086i	$-0.198547\pi$
$61$	2.87790e6	1.62338	0.811692	+	0.584086i	$0.198547\pi$
$62$	−4.79106e6	−2.55306
$63$	0	0
$64$	−3.12384e6	−1.48956
$65$	−861520.	−0.389106
$66$	0	0
$67$	941082.	0.382266	0.191133	−	0.981564i	$-0.438784\pi$
$67$	941082.	0.382266	0.191133	+	0.981564i	$0.438784\pi$
$68$	−6.78018e6	−2.61493
$69$	0	0
$70$	938814.	0.327142
$71$	4.06499e6	1.34789	0.673946	−	0.738781i	$-0.264598\pi$
$71$	4.06499e6	1.34789	0.673946	+	0.738781i	$0.264598\pi$
$72$	0	0
$73$	3.87973e6	1.16727	0.583635	−	0.812016i	$-0.301630\pi$
$73$	3.87973e6	1.16727	0.583635	+	0.812016i	$0.301630\pi$
$74$	6.33339e6	1.81688
$75$	0	0
$76$	−2.95010e6	−0.770885
$77$	−1.93427e6	−0.482836
$78$	0	0
$79$	−6.75281e6	−1.54095	−0.770477	−	0.637468i	$-0.779982\pi$
$79$	−6.75281e6	−1.54095	−0.770477	+	0.637468i	$0.779982\pi$
$80$	747180.	0.163159
$81$	0	0
$82$	−467061.	−0.0935460
$83$	−2.65183e6	−0.509064	−0.254532	−	0.967064i	$-0.581921\pi$
$83$	−2.65183e6	−0.509064	−0.254532	+	0.967064i	$0.581921\pi$
$84$	0	0
$85$	−3.73922e6	−0.660412
$86$	−1.78870e7	−3.03245
$87$	0	0
$88$	−9.01128e6	−1.40960
$89$	−1.76803e6	−0.265842	−0.132921	−	0.991127i	$-0.542436\pi$
$89$	−1.76803e6	−0.265842	−0.132921	+	0.991127i	$0.542436\pi$
$90$	0	0
$91$	−2.74865e6	−0.382362
$92$	6.91609e6	0.925984
$93$	0	0
$94$	4.70400e6	0.584144
$95$	−1.62696e6	−0.194690
$96$	0	0
$97$	8.02400e6	0.892668	0.446334	−	0.894866i	$-0.352729\pi$
$97$	8.02400e6	0.892668	0.446334	+	0.894866i	$0.352729\pi$
$98$	−1.25140e7	−1.34309
$99$	0	0
$100$	3.54152e6	0.354152
$101$	−1.88099e7	−1.81661	−0.908304	−	0.418311i	$-0.862622\pi$
$101$	−1.88099e7	−1.81661	−0.908304	+	0.418311i	$0.862622\pi$
$102$	0	0
$103$	7.18915e6	0.648257	0.324129	−	0.946013i	$-0.394929\pi$
$103$	7.18915e6	0.648257	0.324129	+	0.946013i	$0.394929\pi$
$104$	−1.28053e7	−1.11628
$105$	0	0
$106$	2.18935e7	1.78544
$107$	−1.90869e7	−1.50623	−0.753117	−	0.657887i	$-0.771450\pi$
$107$	−1.90869e7	−1.50623	−0.753117	+	0.657887i	$0.771450\pi$
$108$	0	0
$109$	1.45172e7	1.07372	0.536860	−	0.843671i	$-0.319610\pi$
$109$	1.45172e7	1.07372	0.536860	+	0.843671i	$0.319610\pi$
$110$	−1.14174e7	−0.817885
$111$	0	0
$112$	2.38386e6	0.160331
$113$	6.50503e6	0.424106	0.212053	−	0.977258i	$-0.431985\pi$
$113$	6.50503e6	0.424106	0.212053	+	0.977258i	$0.431985\pi$
$114$	0	0
$115$	3.81418e6	0.233861
$116$	−2.22021e7	−1.32066
$117$	0	0
$118$	1.27565e7	0.714735
$119$	−1.19299e7	−0.648965
$120$	0	0
$121$	4.03647e6	0.207135
$122$	5.41976e7	2.70222
$123$	0	0
$124$	−5.76630e7	−2.71595
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	1.58650e7	0.687270	0.343635	−	0.939103i	$-0.388342\pi$
$127$	1.58650e7	0.687270	0.343635	+	0.939103i	$0.388342\pi$
$128$	−4.27974e7	−1.80377
$129$	0	0
$130$	−1.62244e7	−0.647691
$131$	3.42706e7	1.33190	0.665950	−	0.745996i	$-0.268026\pi$
$131$	3.42706e7	1.33190	0.665950	+	0.745996i	$0.268026\pi$
$132$	0	0
$133$	−5.19077e6	−0.191316
$134$	1.77228e7	0.636305
$135$	0	0
$136$	−5.55783e7	−1.89461
$137$	−3.18031e7	−1.05669	−0.528345	−	0.849030i	$-0.677187\pi$
$137$	−3.18031e7	−1.05669	−0.528345	+	0.849030i	$0.677187\pi$
$138$	0	0
$139$	−4.45702e7	−1.40764	−0.703822	−	0.710376i	$-0.748525\pi$
$139$	−4.45702e7	−1.40764	−0.703822	+	0.710376i	$0.748525\pi$
$140$	1.12991e7	0.348014
$141$	0	0
$142$	7.65532e7	2.24365
$143$	3.34278e7	0.955942
$144$	0	0
$145$	−1.22443e7	−0.333538
$146$	7.30644e7	1.94299
$147$	0	0
$148$	7.62258e7	1.93280
$149$	−7.76972e7	−1.92421	−0.962107	−	0.272673i	$-0.912092\pi$
$149$	−7.76972e7	−1.92421	−0.962107	+	0.272673i	$0.912092\pi$
$150$	0	0
$151$	−4.40092e7	−1.04022	−0.520108	−	0.854100i	$-0.674108\pi$
$151$	−4.40092e7	−1.04022	−0.520108	+	0.854100i	$0.674108\pi$
$152$	−2.41825e7	−0.558533
$153$	0	0
$154$	−3.64269e7	−0.803710
$155$	−3.18008e7	−0.685925
$156$	0	0
$157$	−4.93368e7	−1.01747	−0.508735	−	0.860923i	$-0.669887\pi$
$157$	−4.93368e7	−1.01747	−0.508735	+	0.860923i	$0.669887\pi$
$158$	−1.27171e8	−2.56501
$159$	0	0
$160$	−1.56561e7	−0.302178
$161$	1.21690e7	0.229808
$162$	0	0
$163$	−3.99554e7	−0.722635	−0.361317	−	0.932443i	$-0.617673\pi$
$163$	−3.99554e7	−0.722635	−0.361317	+	0.932443i	$0.617673\pi$
$164$	−5.62133e6	−0.0995144
$165$	0	0
$166$	−4.99402e7	−0.847368
$167$	2.23753e7	0.371759	0.185880	−	0.982573i	$-0.440487\pi$
$167$	2.23753e7	0.371759	0.185880	+	0.982573i	$0.440487\pi$
$168$	0	0
$169$	−1.52467e7	−0.242981
$170$	−7.04183e7	−1.09930
$171$	0	0
$172$	−2.15280e8	−3.22592
$173$	−4.21321e7	−0.618659	−0.309330	−	0.950955i	$-0.600105\pi$
$173$	−4.21321e7	−0.618659	−0.309330	+	0.950955i	$0.600105\pi$
$174$	0	0
$175$	6.23139e6	0.0878925
$176$	−2.89913e7	−0.400842
$177$	0	0
$178$	−3.32961e7	−0.442511
$179$	7.23126e7	0.942385	0.471193	−	0.882030i	$-0.343824\pi$
$179$	7.23126e7	0.942385	0.471193	+	0.882030i	$0.343824\pi$
$180$	0	0
$181$	−7.39713e6	−0.0927232	−0.0463616	−	0.998925i	$-0.514763\pi$
$181$	−7.39713e6	−0.0927232	−0.0463616	+	0.998925i	$0.514763\pi$
$182$	−5.17636e7	−0.636465
$183$	0	0
$184$	5.66924e7	0.670908
$185$	4.20380e7	0.488136
$186$	0	0
$187$	1.45085e8	1.62247
$188$	5.66152e7	0.621413
$189$	0	0
$190$	−3.06395e7	−0.324074
$191$	5.78761e7	0.601012	0.300506	−	0.953780i	$-0.402845\pi$
$191$	5.78761e7	0.601012	0.300506	+	0.953780i	$0.402845\pi$
$192$	0	0
$193$	−1.00257e8	−1.00384	−0.501918	−	0.864915i	$-0.667372\pi$
$193$	−1.00257e8	−1.00384	−0.501918	+	0.864915i	$0.667372\pi$
$194$	1.51111e8	1.48590
$195$	0	0
$196$	−1.50613e8	−1.42878
$197$	1.17718e8	1.09702	0.548508	−	0.836146i	$-0.315196\pi$
$197$	1.17718e8	1.09702	0.548508	+	0.836146i	$0.315196\pi$
$198$	0	0
$199$	−2.95137e7	−0.265484	−0.132742	−	0.991151i	$-0.542378\pi$
$199$	−2.95137e7	−0.265484	−0.132742	+	0.991151i	$0.542378\pi$
$200$	2.90305e7	0.256596
$201$	0	0
$202$	−3.54234e8	−3.02385
$203$	−3.90651e7	−0.327757
$204$	0	0
$205$	−3.10013e6	−0.0251328
$206$	1.35389e8	1.07906
$207$	0	0
$208$	−4.11975e7	−0.317431
$209$	6.31276e7	0.478308
$210$	0	0
$211$	1.73747e8	1.27330	0.636648	−	0.771155i	$-0.280320\pi$
$211$	1.73747e8	1.27330	0.636648	+	0.771155i	$0.280320\pi$
$212$	2.63500e8	1.89935
$213$	0	0
$214$	−3.59451e8	−2.50722
$215$	−1.18725e8	−0.814721
$216$	0	0
$217$	−1.01459e8	−0.674036
$218$	2.73394e8	1.78727
$219$	0	0
$220$	−1.37414e8	−0.870067
$221$	2.06170e8	1.28485
$222$	0	0
$223$	−1.24308e7	−0.0750642	−0.0375321	−	0.999295i	$-0.511950\pi$
$223$	−1.24308e7	−0.0750642	−0.0375321	+	0.999295i	$0.511950\pi$
$224$	−4.99502e7	−0.296940
$225$	0	0
$226$	1.22505e8	0.705951
$227$	2.14129e8	1.21502	0.607511	−	0.794311i	$-0.292168\pi$
$227$	2.14129e8	1.21502	0.607511	+	0.794311i	$0.292168\pi$
$228$	0	0
$229$	2.00173e8	1.10149	0.550745	−	0.834674i	$-0.314344\pi$
$229$	2.00173e8	1.10149	0.550745	+	0.834674i	$0.314344\pi$
$230$	7.18299e7	0.389276
$231$	0	0
$232$	−1.81994e8	−0.956864
$233$	−4.83075e7	−0.250189	−0.125095	−	0.992145i	$-0.539923\pi$
$233$	−4.83075e7	−0.250189	−0.125095	+	0.992145i	$0.539923\pi$
$234$	0	0
$235$	3.12229e7	0.156941
$236$	1.53531e8	0.760336
$237$	0	0
$238$	−2.24668e8	−1.08024
$239$	−1.06760e8	−0.505843	−0.252921	−	0.967487i	$-0.581391\pi$
$239$	−1.06760e8	−0.505843	−0.252921	+	0.967487i	$0.581391\pi$
$240$	0	0
$241$	−3.58266e8	−1.64872	−0.824358	−	0.566069i	$-0.808464\pi$
$241$	−3.58266e8	−1.64872	−0.824358	+	0.566069i	$0.808464\pi$
$242$	7.60162e7	0.344788
$243$	0	0
$244$	6.52297e8	2.87462
$245$	−8.30618e7	−0.360844
$246$	0	0
$247$	8.97061e7	0.378776
$248$	−4.72674e8	−1.96780
$249$	0	0
$250$	3.67819e7	0.148883
$251$	3.11746e8	1.24435	0.622175	−	0.782878i	$-0.286249\pi$
$251$	3.11746e8	1.24435	0.622175	+	0.782878i	$0.286249\pi$
$252$	0	0
$253$	−1.47994e8	−0.574541
$254$	2.98775e8	1.14400
$255$	0	0
$256$	−4.06124e8	−1.51293
$257$	3.23370e8	1.18832	0.594161	−	0.804346i	$-0.297484\pi$
$257$	3.23370e8	1.18832	0.594161	+	0.804346i	$0.297484\pi$
$258$	0	0
$259$	1.34121e8	0.479676
$260$	−1.95270e8	−0.689015
$261$	0	0
$262$	6.45395e8	2.21703
$263$	−1.23948e8	−0.420139	−0.210070	−	0.977686i	$-0.567369\pi$
$263$	−1.23948e8	−0.420139	−0.210070	+	0.977686i	$0.567369\pi$
$264$	0	0
$265$	1.45319e8	0.479690
$266$	−9.77544e7	−0.318457
$267$	0	0
$268$	2.13303e8	0.676901
$269$	2.43144e8	0.761605	0.380802	−	0.924656i	$-0.375648\pi$
$269$	2.43144e8	0.761605	0.380802	+	0.924656i	$0.375648\pi$
$270$	0	0
$271$	−3.02969e8	−0.924711	−0.462356	−	0.886695i	$-0.652996\pi$
$271$	−3.02969e8	−0.924711	−0.462356	+	0.886695i	$0.652996\pi$
$272$	−1.78808e8	−0.538760
$273$	0	0
$274$	−5.98927e8	−1.75892
$275$	−7.57831e7	−0.219739
$276$	0	0
$277$	5.40744e7	0.152867	0.0764333	−	0.997075i	$-0.475647\pi$
$277$	5.40744e7	0.152867	0.0764333	+	0.997075i	$0.475647\pi$
$278$	−8.39362e8	−2.34311
$279$	0	0
$280$	9.26210e7	0.252148
$281$	4.38152e8	1.17802	0.589010	−	0.808126i	$-0.299518\pi$
$281$	4.38152e8	1.17802	0.589010	+	0.808126i	$0.299518\pi$
$282$	0	0
$283$	5.82094e8	1.52665	0.763326	−	0.646013i	$-0.223565\pi$
$283$	5.82094e8	1.52665	0.763326	+	0.646013i	$0.223565\pi$
$284$	9.21359e8	2.38679
$285$	0	0
$286$	6.29524e8	1.59122
$287$	−9.89086e6	−0.0246972
$288$	0	0
$289$	4.84494e8	1.18072
$290$	−2.30589e8	−0.555195
$291$	0	0
$292$	8.79369e8	2.06696
$293$	−7.53377e8	−1.74975	−0.874874	−	0.484350i	$-0.839056\pi$
$293$	−7.53377e8	−1.74975	−0.874874	+	0.484350i	$0.839056\pi$
$294$	0	0
$295$	8.46715e7	0.192026
$296$	6.24836e8	1.40038
$297$	0	0
$298$	−1.46322e9	−3.20297
$299$	−2.10303e8	−0.454985
$300$	0	0
$301$	−3.78790e8	−0.800600
$302$	−8.28796e8	−1.73150
$303$	0	0
$304$	−7.78005e7	−0.158827
$305$	3.59737e8	0.725999
$306$	0	0
$307$	−3.72439e7	−0.0734634	−0.0367317	−	0.999325i	$-0.511695\pi$
$307$	−3.72439e7	−0.0734634	−0.0367317	+	0.999325i	$0.511695\pi$
$308$	−4.38417e8	−0.854987
$309$	0	0
$310$	−5.98883e8	−1.14176
$311$	3.49898e8	0.659600	0.329800	−	0.944051i	$-0.393019\pi$
$311$	3.49898e8	0.659600	0.329800	+	0.944051i	$0.393019\pi$
$312$	0	0
$313$	−1.84947e8	−0.340911	−0.170456	−	0.985365i	$-0.554524\pi$
$313$	−1.84947e8	−0.340911	−0.170456	+	0.985365i	$0.554524\pi$
$314$	−9.29127e8	−1.69364
$315$	0	0
$316$	−1.53057e9	−2.72866
$317$	7.68992e8	1.35586	0.677930	−	0.735127i	$-0.262877\pi$
$317$	7.68992e8	1.35586	0.677930	+	0.735127i	$0.262877\pi$
$318$	0	0
$319$	4.75090e8	0.819424
$320$	−3.90480e8	−0.666152
$321$	0	0
$322$	2.29171e8	0.382529
$323$	3.89348e8	0.642879
$324$	0	0
$325$	−1.07690e8	−0.174014
$326$	−7.52454e8	−1.20287
$327$	0	0
$328$	−4.60791e7	−0.0721017
$329$	9.96157e7	0.154221
$330$	0	0
$331$	−5.91249e8	−0.896133	−0.448066	−	0.894000i	$-0.647887\pi$
$331$	−5.91249e8	−0.896133	−0.448066	+	0.894000i	$0.647887\pi$
$332$	−6.01057e8	−0.901431
$333$	0	0
$334$	4.21380e8	0.618816
$335$	1.17635e8	0.170954
$336$	0	0
$337$	8.89725e8	1.26634	0.633171	−	0.774012i	$-0.281753\pi$
$337$	8.89725e8	1.26634	0.633171	+	0.774012i	$0.281753\pi$
$338$	−2.87131e8	−0.404456
$339$	0	0
$340$	−8.47522e8	−1.16943
$341$	1.23390e9	1.68515
$342$	0	0
$343$	−5.93443e8	−0.794053
$344$	−1.76469e9	−2.33729
$345$	0	0
$346$	−7.93446e8	−1.02980
$347$	−8.50243e8	−1.09242	−0.546210	−	0.837648i	$-0.683930\pi$
$347$	−8.50243e8	−1.09242	−0.546210	+	0.837648i	$0.683930\pi$
$348$	0	0
$349$	−1.14362e8	−0.144011	−0.0720053	−	0.997404i	$-0.522940\pi$
$349$	−1.14362e8	−0.144011	−0.0720053	+	0.997404i	$0.522940\pi$
$350$	1.17352e8	0.146302
$351$	0	0
$352$	6.07470e8	0.742379
$353$	−7.72593e8	−0.934844	−0.467422	−	0.884034i	$-0.654817\pi$
$353$	−7.72593e8	−0.934844	−0.467422	+	0.884034i	$0.654817\pi$
$354$	0	0
$355$	5.08123e8	0.602795
$356$	−4.00737e8	−0.470743
$357$	0	0
$358$	1.36182e9	1.56866
$359$	−6.50789e8	−0.742351	−0.371176	−	0.928563i	$-0.621045\pi$
$359$	−6.50789e8	−0.742351	−0.371176	+	0.928563i	$0.621045\pi$
$360$	0	0
$361$	−7.24464e8	−0.810478
$362$	−1.39305e8	−0.154343
$363$	0	0
$364$	−6.23003e8	−0.677072
$365$	4.84966e8	0.522019
$366$	0	0
$367$	−1.42424e9	−1.50401	−0.752006	−	0.659156i	$-0.770914\pi$
$367$	−1.42424e9	−1.50401	−0.752006	+	0.659156i	$0.770914\pi$
$368$	1.82392e8	0.190783
$369$	0	0
$370$	7.91674e8	0.812532
$371$	4.63635e8	0.471376
$372$	0	0
$373$	−9.15841e8	−0.913774	−0.456887	−	0.889525i	$-0.651036\pi$
$373$	−9.15841e8	−0.913774	−0.456887	+	0.889525i	$0.651036\pi$
$374$	2.73230e9	2.70071
$375$	0	0
$376$	4.64085e8	0.450236
$377$	6.75117e8	0.648910
$378$	0	0
$379$	−5.95667e8	−0.562039	−0.281019	−	0.959702i	$-0.590673\pi$
$379$	−5.95667e8	−0.562039	−0.281019	+	0.959702i	$0.590673\pi$
$380$	−3.68763e8	−0.344750
$381$	0	0
$382$	1.08994e9	1.00042
$383$	9.13969e8	0.831258	0.415629	−	0.909534i	$-0.363562\pi$
$383$	9.13969e8	0.831258	0.415629	+	0.909534i	$0.363562\pi$
$384$	0	0
$385$	−2.41784e8	−0.215931
$386$	−1.88807e9	−1.67095
$387$	0	0
$388$	1.81870e9	1.58070
$389$	4.27769e8	0.368456	0.184228	−	0.982883i	$-0.441021\pi$
$389$	4.27769e8	0.368456	0.184228	+	0.982883i	$0.441021\pi$
$390$	0	0
$391$	−9.12771e8	−0.772224
$392$	−1.23460e9	−1.03520
$393$	0	0
$394$	2.21691e9	1.82605
$395$	−8.44101e8	−0.689136
$396$	0	0
$397$	−3.07353e8	−0.246530	−0.123265	−	0.992374i	$-0.539337\pi$
$397$	−3.07353e8	−0.246530	−0.123265	+	0.992374i	$0.539337\pi$
$398$	−5.55813e8	−0.441914
$399$	0	0
$400$	9.33975e7	0.0729668
$401$	−3.17027e8	−0.245522	−0.122761	−	0.992436i	$-0.539175\pi$
$401$	−3.17027e8	−0.245522	−0.122761	+	0.992436i	$0.539175\pi$
$402$	0	0
$403$	1.75341e9	1.33449
$404$	−4.26340e9	−3.21678
$405$	0	0
$406$	−7.35687e8	−0.545572
$407$	−1.63111e9	−1.19923
$408$	0	0
$409$	−2.05291e9	−1.48367	−0.741837	−	0.670581i	$-0.766045\pi$
$409$	−2.05291e9	−1.48367	−0.741837	+	0.670581i	$0.766045\pi$
$410$	−5.83827e7	−0.0418351
$411$	0	0
$412$	1.62947e9	1.14791
$413$	2.70142e8	0.188698
$414$	0	0
$415$	−3.31479e8	−0.227660
$416$	8.63232e8	0.587897
$417$	0	0
$418$	1.18884e9	0.796172
$419$	2.01263e9	1.33664	0.668320	−	0.743874i	$-0.267014\pi$
$419$	2.01263e9	1.33664	0.668320	+	0.743874i	$0.267014\pi$
$420$	0	0
$421$	−1.37757e9	−0.899758	−0.449879	−	0.893090i	$-0.648533\pi$
$421$	−1.37757e9	−0.899758	−0.449879	+	0.893090i	$0.648533\pi$
$422$	3.27207e9	2.11948
$423$	0	0
$424$	2.15996e9	1.37615
$425$	−4.67402e8	−0.295345
$426$	0	0
$427$	1.14773e9	0.713416
$428$	−4.32619e9	−2.66718
$429$	0	0
$430$	−2.23588e9	−1.35615
$431$	−1.27517e9	−0.767178	−0.383589	−	0.923504i	$-0.625312\pi$
$431$	−1.27517e9	−0.767178	−0.383589	+	0.923504i	$0.625312\pi$
$432$	0	0
$433$	1.81133e9	1.07224	0.536119	−	0.844143i	$-0.319890\pi$
$433$	1.81133e9	1.07224	0.536119	+	0.844143i	$0.319890\pi$
$434$	−1.91072e9	−1.12197
$435$	0	0
$436$	3.29044e9	1.90130
$437$	−3.97153e8	−0.227653
$438$	0	0
$439$	1.31726e8	0.0743099	0.0371550	−	0.999310i	$-0.488170\pi$
$439$	1.31726e8	0.0743099	0.0371550	+	0.999310i	$0.488170\pi$
$440$	−1.12641e9	−0.630394
$441$	0	0
$442$	3.88267e9	2.13871
$443$	5.63961e8	0.308203	0.154101	−	0.988055i	$-0.450752\pi$
$443$	5.63961e8	0.308203	0.154101	+	0.988055i	$0.450752\pi$
$444$	0	0
$445$	−2.21004e8	−0.118888
$446$	−2.34102e8	−0.124949
$447$	0	0
$448$	−1.24581e9	−0.654606
$449$	3.29292e9	1.71679	0.858397	−	0.512985i	$-0.171460\pi$
$449$	3.29292e9	1.71679	0.858397	+	0.512985i	$0.171460\pi$
$450$	0	0
$451$	1.20288e8	0.0617453
$452$	1.47441e9	0.750991
$453$	0	0
$454$	4.03255e9	2.02248
$455$	−3.43582e8	−0.170998
$456$	0	0
$457$	2.65558e9	1.30152	0.650762	−	0.759282i	$-0.274450\pi$
$457$	2.65558e9	1.30152	0.650762	+	0.759282i	$0.274450\pi$
$458$	3.76972e9	1.83350
$459$	0	0
$460$	8.64511e8	0.414112
$461$	2.70258e9	1.28477	0.642385	−	0.766382i	$-0.277945\pi$
$461$	2.70258e9	1.28477	0.642385	+	0.766382i	$0.277945\pi$
$462$	0	0
$463$	3.07783e9	1.44116	0.720578	−	0.693374i	$-0.243877\pi$
$463$	3.07783e9	1.44116	0.720578	+	0.693374i	$0.243877\pi$
$464$	−5.85516e8	−0.272099
$465$	0	0
$466$	−9.09743e8	−0.416455
$467$	−1.76625e9	−0.802497	−0.401249	−	0.915969i	$-0.631424\pi$
$467$	−1.76625e9	−0.802497	−0.401249	+	0.915969i	$0.631424\pi$
$468$	0	0
$469$	3.75312e8	0.167991
$470$	5.88001e8	0.261237
$471$	0	0
$472$	1.25852e9	0.550890
$473$	4.60666e9	2.00157
$474$	0	0
$475$	−2.03370e8	−0.0870682
$476$	−2.70399e9	−1.14916
$477$	0	0
$478$	−2.01054e9	−0.842006
$479$	1.22117e9	0.507694	0.253847	−	0.967244i	$-0.418304\pi$
$479$	1.22117e9	0.507694	0.253847	+	0.967244i	$0.418304\pi$
$480$	0	0
$481$	−2.31786e9	−0.949685
$482$	−6.74699e9	−2.74439
$483$	0	0
$484$	9.14896e8	0.366786
$485$	1.00300e9	0.399213
$486$	0	0
$487$	1.95261e9	0.766062	0.383031	−	0.923735i	$-0.374880\pi$
$487$	1.95261e9	0.766062	0.383031	+	0.923735i	$0.374880\pi$
$488$	5.34700e9	2.08277
$489$	0	0
$490$	−1.56425e9	−0.600647
$491$	−2.60079e9	−0.991563	−0.495782	−	0.868447i	$-0.665118\pi$
$491$	−2.60079e9	−0.991563	−0.495782	+	0.868447i	$0.665118\pi$
$492$	0	0
$493$	2.93018e9	1.10136
$494$	1.68938e9	0.630496
$495$	0	0
$496$	−1.52070e9	−0.559573
$497$	1.62115e9	0.592347
$498$	0	0
$499$	−3.39974e9	−1.22488	−0.612440	−	0.790517i	$-0.709812\pi$
$499$	−3.39974e9	−1.22488	−0.612440	+	0.790517i	$0.709812\pi$
$500$	4.42690e8	0.158382
$501$	0	0
$502$	5.87090e9	2.07130
$503$	4.17304e9	1.46206	0.731029	−	0.682346i	$-0.239040\pi$
$503$	4.17304e9	1.46206	0.731029	+	0.682346i	$0.239040\pi$
$504$	0	0
$505$	−2.35123e9	−0.812412
$506$	−2.78707e9	−0.956359
$507$	0	0
$508$	3.59592e9	1.21699
$509$	−8.83002e8	−0.296790	−0.148395	−	0.988928i	$-0.547411\pi$
$509$	−8.83002e8	−0.296790	−0.148395	+	0.988928i	$0.547411\pi$
$510$	0	0
$511$	1.54727e9	0.512971
$512$	−2.17021e9	−0.714589
$513$	0	0
$514$	6.08982e9	1.97803
$515$	8.98644e8	0.289909
$516$	0	0
$517$	−1.21148e9	−0.385566
$518$	2.52581e9	0.798449
$519$	0	0
$520$	−1.60066e9	−0.499215
$521$	−1.18572e9	−0.367325	−0.183663	−	0.982989i	$-0.558795\pi$
$521$	−1.18572e9	−0.367325	−0.183663	+	0.982989i	$0.558795\pi$
$522$	0	0
$523$	−2.59301e9	−0.792590	−0.396295	−	0.918123i	$-0.629704\pi$
$523$	−2.59301e9	−0.792590	−0.396295	+	0.918123i	$0.629704\pi$
$524$	7.76768e9	2.35848
$525$	0	0
$526$	−2.33423e9	−0.699347
$527$	7.61024e9	2.26496
$528$	0	0
$529$	−2.47376e9	−0.726544
$530$	2.73669e9	0.798473
$531$	0	0
$532$	−1.17653e9	−0.338775
$533$	1.70932e8	0.0488967
$534$	0	0
$535$	−2.38586e9	−0.673608
$536$	1.74848e9	0.490439
$537$	0	0
$538$	4.57896e9	1.26774
$539$	3.22288e9	0.886509
$540$	0	0
$541$	2.75676e9	0.748529	0.374265	−	0.927322i	$-0.377895\pi$
$541$	2.75676e9	0.748529	0.374265	+	0.927322i	$0.377895\pi$
$542$	−5.70563e9	−1.53924
$543$	0	0
$544$	3.74665e9	0.997809
$545$	1.81465e9	0.480182
$546$	0	0
$547$	−1.84785e9	−0.482737	−0.241369	−	0.970433i	$-0.577596\pi$
$547$	−1.84785e9	−0.482737	−0.241369	+	0.970433i	$0.577596\pi$
$548$	−7.20840e9	−1.87114
$549$	0	0
$550$	−1.42717e9	−0.365769
$551$	1.27494e9	0.324683
$552$	0	0
$553$	−2.69308e9	−0.677191
$554$	1.01835e9	0.254456
$555$	0	0
$556$	−1.01022e10	−2.49260
$557$	−4.36605e9	−1.07052	−0.535261	−	0.844686i	$-0.679787\pi$
$557$	−4.36605e9	−1.07052	−0.535261	+	0.844686i	$0.679787\pi$
$558$	0	0
$559$	6.54619e9	1.58507
$560$	2.97982e8	0.0717021
$561$	0	0
$562$	8.25143e9	1.96089
$563$	1.42782e9	0.337206	0.168603	−	0.985684i	$-0.446074\pi$
$563$	1.42782e9	0.337206	0.168603	+	0.985684i	$0.446074\pi$
$564$	0	0
$565$	8.13129e8	0.189666
$566$	1.09622e10	2.54121
$567$	0	0
$568$	7.55255e9	1.72932
$569$	−2.68109e9	−0.610124	−0.305062	−	0.952332i	$-0.598677\pi$
$569$	−2.68109e9	−0.610124	−0.305062	+	0.952332i	$0.598677\pi$
$570$	0	0
$571$	−7.98475e9	−1.79488	−0.897439	−	0.441140i	$-0.854574\pi$
$571$	−7.98475e9	−1.79488	−0.897439	+	0.441140i	$0.854574\pi$
$572$	7.57665e9	1.69274
$573$	0	0
$574$	−1.86268e8	−0.0411100
$575$	4.76772e8	0.104586
$576$	0	0
$577$	1.99867e9	0.433138	0.216569	−	0.976267i	$-0.430513\pi$
$577$	1.99867e9	0.433138	0.216569	+	0.976267i	$0.430513\pi$
$578$	9.12416e9	1.96538
$579$	0	0
$580$	−2.77526e9	−0.590617
$581$	−1.05757e9	−0.223715
$582$	0	0
$583$	−5.63850e9	−1.17848
$584$	7.20835e9	1.49758
$585$	0	0
$586$	−1.41879e10	−2.91256
$587$	1.15087e9	0.234851	0.117425	−	0.993082i	$-0.462536\pi$
$587$	1.15087e9	0.234851	0.117425	+	0.993082i	$0.462536\pi$
$588$	0	0
$589$	3.31127e9	0.667715
$590$	1.59456e9	0.319639
$591$	0	0
$592$	2.01024e9	0.398218
$593$	3.62465e9	0.713796	0.356898	−	0.934143i	$-0.383834\pi$
$593$	3.62465e9	0.713796	0.356898	+	0.934143i	$0.383834\pi$
$594$	0	0
$595$	−1.49123e9	−0.290226
$596$	−1.76106e10	−3.40732
$597$	0	0
$598$	−3.96050e9	−0.757349
$599$	−5.13590e9	−0.976388	−0.488194	−	0.872735i	$-0.662344\pi$
$599$	−5.13590e9	−0.976388	−0.488194	+	0.872735i	$0.662344\pi$
$600$	0	0
$601$	−5.38693e9	−1.01223	−0.506116	−	0.862465i	$-0.668919\pi$
$601$	−5.38693e9	−1.01223	−0.506116	+	0.862465i	$0.668919\pi$
$602$	−7.13350e9	−1.33265
$603$	0	0
$604$	−9.97500e9	−1.84198
$605$	5.04559e8	0.0926335
$606$	0	0
$607$	5.98039e8	0.108535	0.0542674	−	0.998526i	$-0.482718\pi$
$607$	5.98039e8	0.108535	0.0542674	+	0.998526i	$0.482718\pi$
$608$	1.63019e9	0.294156
$609$	0	0
$610$	6.77470e9	1.20847
$611$	−1.72154e9	−0.305333
$612$	0	0
$613$	−6.93260e9	−1.21558	−0.607791	−	0.794097i	$-0.707944\pi$
$613$	−6.93260e9	−1.21558	−0.607791	+	0.794097i	$0.707944\pi$
$614$	−7.01390e8	−0.122284
$615$	0	0
$616$	−3.59378e9	−0.619468
$617$	4.31707e9	0.739930	0.369965	−	0.929046i	$-0.379370\pi$
$617$	4.31707e9	0.739930	0.369965	+	0.929046i	$0.379370\pi$
$618$	0	0
$619$	−1.84194e9	−0.312147	−0.156074	−	0.987745i	$-0.549884\pi$
$619$	−1.84194e9	−0.312147	−0.156074	+	0.987745i	$0.549884\pi$
$620$	−7.20788e9	−1.21461
$621$	0	0
$622$	6.58941e9	1.09794
$623$	−7.05105e8	−0.116828
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−3.48298e9	−0.567467
$627$	0	0
$628$	−1.11825e10	−1.80170
$629$	−1.00601e10	−1.61185
$630$	0	0
$631$	−1.84379e9	−0.292152	−0.146076	−	0.989273i	$-0.546664\pi$
$631$	−1.84379e9	−0.292152	−0.146076	+	0.989273i	$0.546664\pi$
$632$	−1.25464e10	−1.97701
$633$	0	0
$634$	1.44819e10	2.25691
$635$	1.98313e9	0.307356
$636$	0	0
$637$	4.57980e9	0.702035
$638$	8.94706e9	1.36398
$639$	0	0
$640$	−5.34967e9	−0.806672
$641$	6.21243e8	0.0931662	0.0465831	−	0.998914i	$-0.485167\pi$
$641$	6.21243e8	0.0931662	0.0465831	+	0.998914i	$0.485167\pi$
$642$	0	0
$643$	−2.42221e8	−0.0359314	−0.0179657	−	0.999839i	$-0.505719\pi$
$643$	−2.42221e8	−0.0359314	−0.0179657	+	0.999839i	$0.505719\pi$
$644$	2.75820e9	0.406935
$645$	0	0
$646$	7.33234e9	1.07011
$647$	−1.45140e9	−0.210679	−0.105340	−	0.994436i	$-0.533593\pi$
$647$	−1.45140e9	−0.210679	−0.105340	+	0.994436i	$0.533593\pi$
$648$	0	0
$649$	−3.28534e9	−0.471762
$650$	−2.02806e9	−0.289656
$651$	0	0
$652$	−9.05619e9	−1.27961
$653$	9.06794e9	1.27442	0.637210	−	0.770690i	$-0.280088\pi$
$653$	9.06794e9	1.27442	0.637210	+	0.770690i	$0.280088\pi$
$654$	0	0
$655$	4.28382e9	0.595644
$656$	−1.48247e8	−0.0205032
$657$	0	0
$658$	1.87600e9	0.256709
$659$	4.78730e9	0.651616	0.325808	−	0.945436i	$-0.394364\pi$
$659$	4.78730e9	0.651616	0.325808	+	0.945436i	$0.394364\pi$
$660$	0	0
$661$	−3.44227e9	−0.463596	−0.231798	−	0.972764i	$-0.574461\pi$
$661$	−3.44227e9	−0.463596	−0.231798	+	0.972764i	$0.574461\pi$
$662$	−1.11346e10	−1.49167
$663$	0	0
$664$	−4.92697e9	−0.653118
$665$	−6.48846e8	−0.0855591
$666$	0	0
$667$	−2.98892e9	−0.390008
$668$	5.07153e9	0.658297
$669$	0	0
$670$	2.21535e9	0.284564
$671$	−1.39581e10	−1.78361
$672$	0	0
$673$	5.75857e9	0.728219	0.364110	−	0.931356i	$-0.381373\pi$
$673$	5.75857e9	0.728219	0.364110	+	0.931356i	$0.381373\pi$
$674$	1.67556e10	2.10790
$675$	0	0
$676$	−3.45577e9	−0.430261
$677$	6.63522e8	0.0821855	0.0410927	−	0.999155i	$-0.486916\pi$
$677$	6.63522e8	0.0821855	0.0410927	+	0.999155i	$0.486916\pi$
$678$	0	0
$679$	3.20004e9	0.392294
$680$	−6.94729e9	−0.847294
$681$	0	0
$682$	2.32372e10	2.80504
$683$	3.31284e9	0.397858	0.198929	−	0.980014i	$-0.436254\pi$
$683$	3.31284e9	0.397858	0.198929	+	0.980014i	$0.436254\pi$
$684$	0	0
$685$	−3.97539e9	−0.472566
$686$	−1.11759e10	−1.32175
$687$	0	0
$688$	−5.67739e9	−0.664644
$689$	−8.01247e9	−0.933252
$690$	0	0
$691$	−1.44895e10	−1.67063	−0.835316	−	0.549770i	$-0.814715\pi$
$691$	−1.44895e10	−1.67063	−0.835316	+	0.549770i	$0.814715\pi$
$692$	−9.54955e9	−1.09550
$693$	0	0
$694$	−1.60121e10	−1.81840
$695$	−5.57128e9	−0.629518
$696$	0	0
$697$	7.41891e8	0.0829900
$698$	−2.15371e9	−0.239714
$699$	0	0
$700$	1.41239e9	0.155637
$701$	−1.18802e10	−1.30260	−0.651298	−	0.758822i	$-0.725775\pi$
$701$	−1.18802e10	−1.30260	−0.651298	+	0.758822i	$0.725775\pi$
$702$	0	0
$703$	−4.37723e9	−0.475177
$704$	1.51510e10	1.63658
$705$	0	0
$706$	−1.45497e10	−1.55610
$707$	−7.50155e9	−0.798331
$708$	0	0
$709$	1.18144e10	1.24494	0.622470	−	0.782643i	$-0.286129\pi$
$709$	1.18144e10	1.24494	0.622470	+	0.782643i	$0.286129\pi$
$710$	9.56916e9	1.00339
$711$	0	0
$712$	−3.28491e9	−0.341070
$713$	−7.76280e9	−0.802056
$714$	0	0
$715$	4.17847e9	0.427510
$716$	1.63902e10	1.66874
$717$	0	0
$718$	−1.22559e10	−1.23569
$719$	−1.45067e9	−0.145552	−0.0727761	−	0.997348i	$-0.523186\pi$
$719$	−1.45067e9	−0.145552	−0.0727761	+	0.997348i	$0.523186\pi$
$720$	0	0
$721$	2.86710e9	0.284885
$722$	−1.36434e10	−1.34909
$723$	0	0
$724$	−1.67661e9	−0.164191
$725$	−1.53054e9	−0.149163
$726$	0	0
$727$	−8.41861e9	−0.812587	−0.406293	−	0.913743i	$-0.633179\pi$
$727$	−8.41861e9	−0.812587	−0.406293	+	0.913743i	$0.633179\pi$
$728$	−5.10687e9	−0.490563
$729$	0	0
$730$	9.13305e9	0.868932
$731$	2.84122e10	2.69026
$732$	0	0
$733$	−5.33890e8	−0.0500712	−0.0250356	−	0.999687i	$-0.507970\pi$
$733$	−5.33890e8	−0.0500712	−0.0250356	+	0.999687i	$0.507970\pi$
$734$	−2.68218e10	−2.50352
$735$	0	0
$736$	−3.82176e9	−0.353338
$737$	−4.56436e9	−0.419994
$738$	0	0
$739$	1.27798e10	1.16484	0.582421	−	0.812887i	$-0.302105\pi$
$739$	1.27798e10	1.16484	0.582421	+	0.812887i	$0.302105\pi$
$740$	9.52822e9	0.864373
$741$	0	0
$742$	8.73133e9	0.784633
$743$	−1.01093e10	−0.904195	−0.452097	−	0.891969i	$-0.649324\pi$
$743$	−1.01093e10	−0.904195	−0.452097	+	0.891969i	$0.649324\pi$
$744$	0	0
$745$	−9.71215e9	−0.860534
$746$	−1.72474e10	−1.52103
$747$	0	0
$748$	3.28846e10	2.87301
$749$	−7.61203e9	−0.661933
$750$	0	0
$751$	1.74614e10	1.50432	0.752159	−	0.658982i	$-0.229013\pi$
$751$	1.74614e10	1.50432	0.752159	+	0.658982i	$0.229013\pi$
$752$	1.49306e9	0.128031
$753$	0	0
$754$	1.27140e10	1.08015
$755$	−5.50115e9	−0.465199
$756$	0	0
$757$	5.15460e9	0.431876	0.215938	−	0.976407i	$-0.430719\pi$
$757$	5.15460e9	0.431876	0.215938	+	0.976407i	$0.430719\pi$
$758$	−1.12178e10	−0.935548
$759$	0	0
$760$	−3.02281e9	−0.249784
$761$	4.17524e9	0.343428	0.171714	−	0.985147i	$-0.445070\pi$
$761$	4.17524e9	0.343428	0.171714	+	0.985147i	$0.445070\pi$
$762$	0	0
$763$	5.78960e9	0.471859
$764$	1.31181e10	1.06425
$765$	0	0
$766$	1.72122e10	1.38368
$767$	−4.66856e9	−0.373593
$768$	0	0
$769$	−6.79752e9	−0.539024	−0.269512	−	0.962997i	$-0.586862\pi$
$769$	−6.79752e9	−0.539024	−0.269512	+	0.962997i	$0.586862\pi$
$770$	−4.55336e9	−0.359430
$771$	0	0
$772$	−2.27239e10	−1.77755
$773$	−6.48363e9	−0.504882	−0.252441	−	0.967612i	$-0.581233\pi$
$773$	−6.48363e9	−0.504882	−0.252441	+	0.967612i	$0.581233\pi$
$774$	0	0
$775$	−3.97509e9	−0.306755
$776$	1.49082e10	1.14527
$777$	0	0
$778$	8.05590e9	0.613318
$779$	3.22802e8	0.0244656
$780$	0	0
$781$	−1.97157e10	−1.48092
$782$	−1.71896e10	−1.28541
$783$	0	0
$784$	−3.97198e9	−0.294375
$785$	−6.16709e9	−0.455027
$786$	0	0
$787$	−2.39900e10	−1.75436	−0.877178	−	0.480165i	$-0.840577\pi$
$787$	−2.39900e10	−1.75436	−0.877178	+	0.480165i	$0.840577\pi$
$788$	2.66818e10	1.94255
$789$	0	0
$790$	−1.58964e10	−1.14711
$791$	2.59426e9	0.186379
$792$	0	0
$793$	−1.98349e10	−1.41245
$794$	−5.78817e9	−0.410364
$795$	0	0
$796$	−6.68951e9	−0.470109
$797$	−2.05721e10	−1.43938	−0.719688	−	0.694298i	$-0.755715\pi$
$797$	−2.05721e10	−1.43938	−0.719688	+	0.694298i	$0.755715\pi$
$798$	0	0
$799$	−7.47195e9	−0.518227
$800$	−1.95701e9	−0.135138
$801$	0	0
$802$	−5.97035e9	−0.408686
$803$	−1.88171e10	−1.28248
$804$	0	0
$805$	1.52113e9	0.102773
$806$	3.30208e10	2.22134
$807$	0	0
$808$	−3.49478e10	−2.33067
$809$	−6.02726e9	−0.400221	−0.200111	−	0.979773i	$-0.564130\pi$
$809$	−6.02726e9	−0.400221	−0.200111	+	0.979773i	$0.564130\pi$
$810$	0	0
$811$	6.06003e9	0.398935	0.199467	−	0.979904i	$-0.436079\pi$
$811$	6.06003e9	0.398935	0.199467	+	0.979904i	$0.436079\pi$
$812$	−8.85439e9	−0.580380
$813$	0	0
$814$	−3.07177e10	−1.99620
$815$	−4.99443e9	−0.323172
$816$	0	0
$817$	1.23623e10	0.793092
$818$	−3.86611e10	−2.46966
$819$	0	0
$820$	−7.02667e8	−0.0445042
$821$	−3.58783e9	−0.226272	−0.113136	−	0.993580i	$-0.536090\pi$
$821$	−3.58783e9	−0.226272	−0.113136	+	0.993580i	$0.536090\pi$
$822$	0	0
$823$	−1.61521e10	−1.01002	−0.505011	−	0.863113i	$-0.668511\pi$
$823$	−1.61521e10	−1.01002	−0.505011	+	0.863113i	$0.668511\pi$
$824$	1.33571e10	0.831700
$825$	0	0
$826$	5.08741e9	0.314099
$827$	1.88121e10	1.15656	0.578279	−	0.815839i	$-0.303724\pi$
$827$	1.88121e10	1.15656	0.578279	+	0.815839i	$0.303724\pi$
$828$	0	0
$829$	−1.52364e10	−0.928844	−0.464422	−	0.885614i	$-0.653738\pi$
$829$	−1.52364e10	−0.928844	−0.464422	+	0.885614i	$0.653738\pi$
$830$	−6.24253e9	−0.378954
$831$	0	0
$832$	2.15300e10	1.29602
$833$	1.98775e10	1.19153
$834$	0	0
$835$	2.79692e9	0.166256
$836$	1.43083e10	0.846969
$837$	0	0
$838$	3.79025e10	2.22492
$839$	6.10761e9	0.357030	0.178515	−	0.983937i	$-0.442871\pi$
$839$	6.10761e9	0.357030	0.178515	+	0.983937i	$0.442871\pi$
$840$	0	0
$841$	−7.65482e9	−0.443761
$842$	−2.59428e10	−1.49770
$843$	0	0
$844$	3.93811e10	2.25470
$845$	−1.90584e9	−0.108664
$846$	0	0
$847$	1.60978e9	0.0910279
$848$	6.94907e9	0.391328
$849$	0	0
$850$	−8.80229e9	−0.491620
$851$	1.02618e10	0.570781
$852$	0	0
$853$	−7.65016e9	−0.422036	−0.211018	−	0.977482i	$-0.567678\pi$
$853$	−7.65016e9	−0.422036	−0.211018	+	0.977482i	$0.567678\pi$
$854$	2.16145e10	1.18752
$855$	0	0
$856$	−3.54625e10	−1.93247
$857$	3.44703e10	1.87073	0.935367	−	0.353679i	$-0.115070\pi$
$857$	3.44703e10	1.87073	0.935367	+	0.353679i	$0.115070\pi$
$858$	0	0
$859$	5.63231e9	0.303187	0.151593	−	0.988443i	$-0.451560\pi$
$859$	5.63231e9	0.303187	0.151593	+	0.988443i	$0.451560\pi$
$860$	−2.69100e10	−1.44268
$861$	0	0
$862$	−2.40144e10	−1.27701
$863$	−5.40030e9	−0.286009	−0.143005	−	0.989722i	$-0.545676\pi$
$863$	−5.40030e9	−0.286009	−0.143005	+	0.989722i	$0.545676\pi$
$864$	0	0
$865$	−5.26651e9	−0.276673
$866$	3.41117e10	1.78480
$867$	0	0
$868$	−2.29965e10	−1.19356
$869$	3.27519e10	1.69304
$870$	0	0
$871$	−6.48608e9	−0.332597
$872$	2.69723e10	1.37756
$873$	0	0
$874$	−7.47932e9	−0.378942
$875$	7.78924e8	0.0393067
$876$	0	0
$877$	−2.19659e10	−1.09964	−0.549821	−	0.835283i	$-0.685304\pi$
$877$	−2.19659e10	−1.09964	−0.549821	+	0.835283i	$0.685304\pi$
$878$	2.48072e9	0.123693
$879$	0	0
$880$	−3.62391e9	−0.179262
$881$	−2.04368e10	−1.00692	−0.503462	−	0.864018i	$-0.667940\pi$
$881$	−2.04368e10	−1.00692	−0.503462	+	0.864018i	$0.667940\pi$
$882$	0	0
$883$	−2.23485e10	−1.09241	−0.546206	−	0.837651i	$-0.683928\pi$
$883$	−2.23485e10	−1.09241	−0.546206	+	0.837651i	$0.683928\pi$
$884$	4.67300e10	2.27517
$885$	0	0
$886$	1.06207e10	0.513022
$887$	−2.17190e10	−1.04498	−0.522490	−	0.852645i	$-0.674997\pi$
$887$	−2.17190e10	−1.04498	−0.522490	+	0.852645i	$0.674997\pi$
$888$	0	0
$889$	6.32711e9	0.302029
$890$	−4.16202e9	−0.197897
$891$	0	0
$892$	−2.81754e9	−0.132921
$893$	−3.25110e9	−0.152774
$894$	0	0
$895$	9.03908e9	0.421448
$896$	−1.70680e10	−0.792691
$897$	0	0
$898$	6.20133e10	2.85771
$899$	2.49202e10	1.14391
$900$	0	0
$901$	−3.47762e10	−1.58396
$902$	2.26530e9	0.102779
$903$	0	0
$904$	1.20860e10	0.544119
$905$	−9.24641e8	−0.0414671
$906$	0	0
$907$	−3.90600e10	−1.73823	−0.869113	−	0.494614i	$-0.835310\pi$
$907$	−3.90600e10	−1.73823	−0.869113	+	0.494614i	$0.835310\pi$
$908$	4.85339e10	2.15152
$909$	0	0
$910$	−6.47045e9	−0.284636
$911$	−2.41491e10	−1.05825	−0.529123	−	0.848545i	$-0.677479\pi$
$911$	−2.41491e10	−1.05825	−0.529123	+	0.848545i	$0.677479\pi$
$912$	0	0
$913$	1.28617e10	0.559307
$914$	5.00107e10	2.16646
$915$	0	0
$916$	4.53706e10	1.95048
$917$	1.36674e10	0.585320
$918$	0	0
$919$	−3.27364e10	−1.39132	−0.695659	−	0.718372i	$-0.744888\pi$
$919$	−3.27364e10	−1.39132	−0.695659	+	0.718372i	$0.744888\pi$
$920$	7.08655e9	0.300039
$921$	0	0
$922$	5.08960e10	2.13858
$923$	−2.80165e10	−1.17276
$924$	0	0
$925$	5.25475e9	0.218301
$926$	5.79627e10	2.39889
$927$	0	0
$928$	1.22686e10	0.503939
$929$	−1.36588e10	−0.558929	−0.279464	−	0.960156i	$-0.590157\pi$
$929$	−1.36588e10	−0.558929	−0.279464	+	0.960156i	$0.590157\pi$
$930$	0	0
$931$	8.64885e9	0.351265
$932$	−1.09492e10	−0.443025
$933$	0	0
$934$	−3.32627e10	−1.33580
$935$	1.81357e10	0.725592
$936$	0	0
$937$	2.29174e10	0.910075	0.455038	−	0.890472i	$-0.349626\pi$
$937$	2.29174e10	0.910075	0.455038	+	0.890472i	$0.349626\pi$
$938$	7.06800e9	0.279632
$939$	0	0
$940$	7.07690e9	0.277905
$941$	−3.48492e9	−0.136342	−0.0681708	−	0.997674i	$-0.521716\pi$
$941$	−3.48492e9	−0.136342	−0.0681708	+	0.997674i	$0.521716\pi$
$942$	0	0
$943$	−7.56763e8	−0.0293879
$944$	4.04895e9	0.156654
$945$	0	0
$946$	8.67542e10	3.33174
$947$	−2.74452e10	−1.05013	−0.525063	−	0.851064i	$-0.675958\pi$
$947$	−2.74452e10	−1.05013	−0.525063	+	0.851064i	$0.675958\pi$
$948$	0	0
$949$	−2.67397e10	−1.01560
$950$	−3.82994e9	−0.144930
$951$	0	0
$952$	−2.21651e10	−0.832609
$953$	8.20112e9	0.306936	0.153468	−	0.988154i	$-0.450956\pi$
$953$	8.20112e9	0.306936	0.153468	+	0.988154i	$0.450956\pi$
$954$	0	0
$955$	7.23452e9	0.268781
$956$	−2.41979e10	−0.895727
$957$	0	0
$958$	2.29975e10	0.845087
$959$	−1.26834e10	−0.464375
$960$	0	0
$961$	3.72098e10	1.35246
$962$	−4.36508e10	−1.58081
$963$	0	0
$964$	−8.12036e10	−2.91948
$965$	−1.25321e10	−0.448929
$966$	0	0
$967$	9.17790e9	0.326400	0.163200	−	0.986593i	$-0.447818\pi$
$967$	9.17790e9	0.326400	0.163200	+	0.986593i	$0.447818\pi$
$968$	7.49957e9	0.265749
$969$	0	0
$970$	1.88889e10	0.664515
$971$	1.94352e10	0.681273	0.340637	−	0.940195i	$-0.389357\pi$
$971$	1.94352e10	0.681273	0.340637	+	0.940195i	$0.389357\pi$
$972$	0	0
$973$	−1.77750e10	−0.618607
$974$	3.67722e10	1.27516
$975$	0	0
$976$	1.72025e10	0.592266
$977$	4.02767e10	1.38173	0.690864	−	0.722984i	$-0.257230\pi$
$977$	4.02767e10	1.38173	0.690864	+	0.722984i	$0.257230\pi$
$978$	0	0
$979$	8.57514e9	0.292080
$980$	−1.88266e10	−0.638969
$981$	0	0
$982$	−4.89790e10	−1.65052
$983$	−5.79736e10	−1.94667	−0.973337	−	0.229381i	$-0.926330\pi$
$983$	−5.79736e10	−1.94667	−0.973337	+	0.229381i	$0.926330\pi$
$984$	0	0
$985$	1.47148e10	0.490600
$986$	5.51822e10	1.83329
$987$	0	0
$988$	2.03326e10	0.670723
$989$	−2.89817e10	−0.952658
$990$	0	0
$991$	4.07224e10	1.32916	0.664578	−	0.747219i	$-0.268612\pi$
$991$	4.07224e10	1.32916	0.664578	+	0.747219i	$0.268612\pi$
$992$	3.18640e10	1.03636
$993$	0	0
$994$	3.05301e10	0.985998
$995$	−3.68922e9	−0.118728
$996$	0	0
$997$	1.15141e10	0.367956	0.183978	−	0.982930i	$-0.441103\pi$
$997$	1.15141e10	0.367956	0.183978	+	0.982930i	$0.441103\pi$
$998$	−6.40251e10	−2.03889
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.e.1.13		13
3.2	odd	2		405.8.a.f.1.1		13
9.2	odd	6		135.8.e.a.91.13		26
9.4	even	3		45.8.e.a.16.1	✓	26
9.5	odd	6		135.8.e.a.46.13		26
9.7	even	3		45.8.e.a.31.1	yes	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.8.e.a.16.1	✓	26	9.4	even	3
45.8.e.a.31.1	yes	26	9.7	even	3
135.8.e.a.46.13		26	9.5	odd	6
135.8.e.a.91.13		26	9.2	odd	6
405.8.a.e.1.13		13	1.1	even	1	trivial
405.8.a.f.1.1		13	3.2	odd	2

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+18.8324 q^{2} +226.657 q^{4} +125.000 q^{5} +398.809 q^{7} +1857.95 q^{8} +O(q^{10})\)	\(q+18.8324 q^{2} +226.657 q^{4} +125.000 q^{5} +398.809 q^{7} +1857.95 q^{8} +2354.04 q^{10} -4850.12 q^{11} -6892.16 q^{13} +7510.51 q^{14} +5977.44 q^{16} -29913.8 q^{17} -13015.7 q^{19} +28332.2 q^{20} -91339.1 q^{22} +30513.4 q^{23} +15625.0 q^{25} -129796. q^{26} +90393.0 q^{28} -97954.3 q^{29} -254406. q^{31} -125248. q^{32} -563346. q^{34} +49851.1 q^{35} +336304. q^{37} -245116. q^{38} +232244. q^{40} -24801.0 q^{41} -949803. q^{43} -1.09932e6 q^{44} +574639. q^{46} +249783. q^{47} -664494. q^{49} +294255. q^{50} -1.56216e6 q^{52} +1.16255e6 q^{53} -606265. q^{55} +740968. q^{56} -1.84471e6 q^{58} +677372. q^{59} +2.87790e6 q^{61} -4.79106e6 q^{62} -3.12384e6 q^{64} -861520. q^{65} +941082. q^{67} -6.78018e6 q^{68} +938814. q^{70} +4.06499e6 q^{71} +3.87973e6 q^{73} +6.33339e6 q^{74} -2.95010e6 q^{76} -1.93427e6 q^{77} -6.75281e6 q^{79} +747180. q^{80} -467061. q^{82} -2.65183e6 q^{83} -3.73922e6 q^{85} -1.78870e7 q^{86} -9.01128e6 q^{88} -1.76803e6 q^{89} -2.74865e6 q^{91} +6.91609e6 q^{92} +4.70400e6 q^{94} -1.62696e6 q^{95} +8.02400e6 q^{97} -1.25140e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 8 q^{2} + 704 q^{4} + 1625 q^{5} - 1455 q^{7} - 1236 q^{8}+O(q^{10}) \) 13 * q - 8 * q^2 + 704 * q^4 + 1625 * q^5 - 1455 * q^7 - 1236 * q^8	\( 13 q - 8 q^{2} + 704 q^{4} + 1625 q^{5} - 1455 q^{7} - 1236 q^{8} - 1000 q^{10} - 1658 q^{11} - 13568 q^{13} - 12351 q^{14} + 52076 q^{16} - 6944 q^{17} - 45812 q^{19} + 88000 q^{20} - 17993 q^{22} - 96441 q^{23} + 203125 q^{25} - 126146 q^{26} - 216945 q^{28} - 39043 q^{29} - 158520 q^{31} - 725206 q^{32} - 441617 q^{34} - 181875 q^{35} - 505438 q^{37} - 615041 q^{38} - 154500 q^{40} - 1578883 q^{41} - 1082090 q^{43} - 498211 q^{44} - 2312547 q^{46} - 1690139 q^{47} - 1054816 q^{49} - 125000 q^{50} - 4100644 q^{52} + 102274 q^{53} - 207250 q^{55} - 389331 q^{56} - 5780521 q^{58} + 2908966 q^{59} - 3091451 q^{61} - 5212476 q^{62} - 5659352 q^{64} - 1696000 q^{65} - 1849533 q^{67} + 563369 q^{68} - 1543875 q^{70} + 2617958 q^{71} + 5310946 q^{73} + 11485004 q^{74} - 14421739 q^{76} - 1719660 q^{77} - 3632166 q^{79} + 6509500 q^{80} - 7347658 q^{82} - 1424115 q^{83} - 868000 q^{85} + 23193379 q^{86} - 20229447 q^{88} + 2816067 q^{89} - 16929702 q^{91} + 27930183 q^{92} - 11693189 q^{94} - 5726500 q^{95} - 12062476 q^{97} + 29617805 q^{98}+O(q^{100}) \) 13 * q - 8 * q^2 + 704 * q^4 + 1625 * q^5 - 1455 * q^7 - 1236 * q^8 - 1000 * q^10 - 1658 * q^11 - 13568 * q^13 - 12351 * q^14 + 52076 * q^16 - 6944 * q^17 - 45812 * q^19 + 88000 * q^20 - 17993 * q^22 - 96441 * q^23 + 203125 * q^25 - 126146 * q^26 - 216945 * q^28 - 39043 * q^29 - 158520 * q^31 - 725206 * q^32 - 441617 * q^34 - 181875 * q^35 - 505438 * q^37 - 615041 * q^38 - 154500 * q^40 - 1578883 * q^41 - 1082090 * q^43 - 498211 * q^44 - 2312547 * q^46 - 1690139 * q^47 - 1054816 * q^49 - 125000 * q^50 - 4100644 * q^52 + 102274 * q^53 - 207250 * q^55 - 389331 * q^56 - 5780521 * q^58 + 2908966 * q^59 - 3091451 * q^61 - 5212476 * q^62 - 5659352 * q^64 - 1696000 * q^65 - 1849533 * q^67 + 563369 * q^68 - 1543875 * q^70 + 2617958 * q^71 + 5310946 * q^73 + 11485004 * q^74 - 14421739 * q^76 - 1719660 * q^77 - 3632166 * q^79 + 6509500 * q^80 - 7347658 * q^82 - 1424115 * q^83 - 868000 * q^85 + 23193379 * q^86 - 20229447 * q^88 + 2816067 * q^89 - 16929702 * q^91 + 27930183 * q^92 - 11693189 * q^94 - 5726500 * q^95 - 12062476 * q^97 + 29617805 * q^98

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