LMFDB - Embedded newform 207.8.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,8,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	207.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:	$64.6637002752$
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} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 $ x^8 - 832x^6 - 1059x^5 + 203052x^4 + 678328x^3 - 13424272x^2 - 73308944x - 37372224
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$11.0962$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.0962 q^{2} -4.87416 q^{4} -165.526 q^{5} +952.148 q^{7} +1474.40 q^{8} +O(q^{10})$	\(q-11.0962 q^{2} -4.87416 q^{4} -165.526 q^{5} +952.148 q^{7} +1474.40 q^{8} +1836.71 q^{10} +4863.13 q^{11} +12899.5 q^{13} -10565.2 q^{14} -15736.4 q^{16} +18595.8 q^{17} -8378.65 q^{19} +806.801 q^{20} -53962.3 q^{22} +12167.0 q^{23} -50726.1 q^{25} -143136. q^{26} -4640.92 q^{28} +133283. q^{29} -107642. q^{31} -14109.3 q^{32} -206343. q^{34} -157606. q^{35} +422577. q^{37} +92971.3 q^{38} -244052. q^{40} +85366.0 q^{41} +410360. q^{43} -23703.7 q^{44} -135008. q^{46} -1.11839e6 q^{47} +83043.7 q^{49} +562867. q^{50} -62874.3 q^{52} -275790. q^{53} -804976. q^{55} +1.40385e6 q^{56} -1.47893e6 q^{58} +182891. q^{59} +2.49804e6 q^{61} +1.19442e6 q^{62} +2.17081e6 q^{64} -2.13521e6 q^{65} -705106. q^{67} -90638.9 q^{68} +1.74882e6 q^{70} -4.54627e6 q^{71} -4.46832e6 q^{73} -4.68901e6 q^{74} +40838.9 q^{76} +4.63042e6 q^{77} +4.92889e6 q^{79} +2.60478e6 q^{80} -947239. q^{82} -369529. q^{83} -3.07809e6 q^{85} -4.55344e6 q^{86} +7.17020e6 q^{88} -2.70162e6 q^{89} +1.22823e7 q^{91} -59303.9 q^{92} +1.24099e7 q^{94} +1.38689e6 q^{95} +8.20102e6 q^{97} -921470. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 640 q^{4} - 444 q^{5} + 1446 q^{7} - 3177 q^{8}+O(q^{10}) $ 8 * q + 640 * q^4 - 444 * q^5 + 1446 * q^7 - 3177 * q^8	\( 8 q + 640 q^{4} - 444 q^{5} + 1446 q^{7} - 3177 q^{8} + 19502 q^{10} - 7588 q^{11} + 19862 q^{13} - 17544 q^{14} + 64336 q^{16} - 42070 q^{17} + 1050 q^{19} - 3364 q^{20} - 128220 q^{22} + 97336 q^{23} + 49496 q^{25} + 371761 q^{26} + 143050 q^{28} + 102578 q^{29} + 304172 q^{31} + 612824 q^{32} - 524530 q^{34} - 531048 q^{35} + 286472 q^{37} + 762932 q^{38} + 2105286 q^{40} - 1324414 q^{41} + 2052578 q^{43} + 867298 q^{44} - 675556 q^{47} - 55404 q^{49} - 1458528 q^{50} - 1695409 q^{52} - 203654 q^{53} - 1024444 q^{55} + 5766846 q^{56} - 5039991 q^{58} + 748892 q^{59} + 61822 q^{61} + 4939277 q^{62} + 2702267 q^{64} + 1571618 q^{65} + 3235604 q^{67} - 4914980 q^{68} + 10871764 q^{70} + 4951664 q^{71} + 11019370 q^{73} - 356954 q^{74} + 21973240 q^{76} + 5284888 q^{77} + 4202464 q^{79} - 8785886 q^{80} + 32636759 q^{82} - 518568 q^{83} + 9854220 q^{85} + 14681386 q^{86} + 20589740 q^{88} - 4203864 q^{89} + 2488406 q^{91} + 7786880 q^{92} + 12314327 q^{94} + 44485300 q^{95} + 18621134 q^{97} - 35756 q^{98}+O(q^{100}) \) 8 * q + 640 * q^4 - 444 * q^5 + 1446 * q^7 - 3177 * q^8 + 19502 * q^10 - 7588 * q^11 + 19862 * q^13 - 17544 * q^14 + 64336 * q^16 - 42070 * q^17 + 1050 * q^19 - 3364 * q^20 - 128220 * q^22 + 97336 * q^23 + 49496 * q^25 + 371761 * q^26 + 143050 * q^28 + 102578 * q^29 + 304172 * q^31 + 612824 * q^32 - 524530 * q^34 - 531048 * q^35 + 286472 * q^37 + 762932 * q^38 + 2105286 * q^40 - 1324414 * q^41 + 2052578 * q^43 + 867298 * q^44 - 675556 * q^47 - 55404 * q^49 - 1458528 * q^50 - 1695409 * q^52 - 203654 * q^53 - 1024444 * q^55 + 5766846 * q^56 - 5039991 * q^58 + 748892 * q^59 + 61822 * q^61 + 4939277 * q^62 + 2702267 * q^64 + 1571618 * q^65 + 3235604 * q^67 - 4914980 * q^68 + 10871764 * q^70 + 4951664 * q^71 + 11019370 * q^73 - 356954 * q^74 + 21973240 * q^76 + 5284888 * q^77 + 4202464 * q^79 - 8785886 * q^80 + 32636759 * q^82 - 518568 * q^83 + 9854220 * q^85 + 14681386 * q^86 + 20589740 * q^88 - 4203864 * q^89 + 2488406 * q^91 + 7786880 * q^92 + 12314327 * q^94 + 44485300 * q^95 + 18621134 * q^97 - 35756 * 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 $

$2$	−11.0962	−0.980776	−0.490388	−	0.871504i	$-0.663145\pi$
$2$	−11.0962	−0.980776	−0.490388	+	0.871504i	$0.663145\pi$
$3$	0	0
$3$	0	0
$4$	−4.87416	−0.0380794
$5$	−165.526	−0.592205	−0.296102	−	0.955156i	$-0.595687\pi$
$5$	−165.526	−0.592205	−0.296102	+	0.955156i	$0.595687\pi$
$6$	0	0
$7$	952.148	1.04921	0.524604	−	0.851346i	$-0.324213\pi$
$7$	952.148	1.04921	0.524604	+	0.851346i	$0.324213\pi$
$8$	1474.40	1.01812
$9$	0	0
$10$	1836.71	0.580820
$11$	4863.13	1.10164	0.550822	−	0.834622i	$-0.314314\pi$
$11$	4863.13	1.10164	0.550822	+	0.834622i	$0.314314\pi$
$12$	0	0
$13$	12899.5	1.62844	0.814220	−	0.580556i	$-0.197165\pi$
$13$	12899.5	1.62844	0.814220	+	0.580556i	$0.197165\pi$
$14$	−10565.2	−1.02904
$15$	0	0
$16$	−15736.4	−0.960471
$17$	18595.8	0.918002	0.459001	−	0.888436i	$-0.348207\pi$
$17$	18595.8	0.918002	0.459001	+	0.888436i	$0.348207\pi$
$18$	0	0
$19$	−8378.65	−0.280244	−0.140122	−	0.990134i	$-0.544750\pi$
$19$	−8378.65	−0.280244	−0.140122	+	0.990134i	$0.544750\pi$
$20$	806.801	0.0225508
$21$	0	0
$22$	−53962.3	−1.08047
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	0	0
$25$	−50726.1	−0.649294
$26$	−143136.	−1.59714
$27$	0	0
$28$	−4640.92	−0.0399532
$29$	133283.	1.01480	0.507400	−	0.861711i	$-0.330607\pi$
$29$	133283.	1.01480	0.507400	+	0.861711i	$0.330607\pi$
$30$	0	0
$31$	−107642.	−0.648956	−0.324478	−	0.945893i	$-0.605189\pi$
$31$	−107642.	−0.648956	−0.324478	+	0.945893i	$0.605189\pi$
$32$	−14109.3	−0.0761168
$33$	0	0
$34$	−206343.	−0.900354
$35$	−157606.	−0.621346
$36$	0	0
$37$	422577.	1.37151	0.685757	−	0.727831i	$-0.259471\pi$
$37$	422577.	1.37151	0.685757	+	0.727831i	$0.259471\pi$
$38$	92971.3	0.274857
$39$	0	0
$40$	−244052.	−0.602937
$41$	85366.0	0.193438	0.0967189	−	0.995312i	$-0.469165\pi$
$41$	85366.0	0.193438	0.0967189	+	0.995312i	$0.469165\pi$
$42$	0	0
$43$	410360.	0.787091	0.393546	−	0.919305i	$-0.371248\pi$
$43$	410360.	0.787091	0.393546	+	0.919305i	$0.371248\pi$
$44$	−23703.7	−0.0419499
$45$	0	0
$46$	−135008.	−0.204506
$47$	−1.11839e6	−1.57127	−0.785635	−	0.618691i	$-0.787663\pi$
$47$	−1.11839e6	−1.57127	−0.785635	+	0.618691i	$0.787663\pi$
$48$	0	0
$49$	83043.7	0.100837
$50$	562867.	0.636811
$51$	0	0
$52$	−62874.3	−0.0620100
$53$	−275790.	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$53$	−275790.	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$54$	0	0
$55$	−804976.	−0.652399
$56$	1.40385e6	1.06822
$57$	0	0
$58$	−1.47893e6	−0.995291
$59$	182891.	0.115934	0.0579670	−	0.998318i	$-0.481538\pi$
$59$	182891.	0.115934	0.0579670	+	0.998318i	$0.481538\pi$
$60$	0	0
$61$	2.49804e6	1.40911	0.704556	−	0.709648i	$-0.251146\pi$
$61$	2.49804e6	1.40911	0.704556	+	0.709648i	$0.251146\pi$
$62$	1.19442e6	0.636480
$63$	0	0
$64$	2.17081e6	1.03512
$65$	−2.13521e6	−0.964370
$66$	0	0
$67$	−705106.	−0.286413	−0.143207	−	0.989693i	$-0.545741\pi$
$67$	−705106.	−0.286413	−0.143207	+	0.989693i	$0.545741\pi$
$68$	−90638.9	−0.0349569
$69$	0	0
$70$	1.74882e6	0.609401
$71$	−4.54627e6	−1.50748	−0.753740	−	0.657173i	$-0.771752\pi$
$71$	−4.54627e6	−1.50748	−0.753740	+	0.657173i	$0.771752\pi$
$72$	0	0
$73$	−4.46832e6	−1.34435	−0.672177	−	0.740390i	$-0.734641\pi$
$73$	−4.46832e6	−1.34435	−0.672177	+	0.740390i	$0.734641\pi$
$74$	−4.68901e6	−1.34515
$75$	0	0
$76$	40838.9	0.0106715
$77$	4.63042e6	1.15585
$78$	0	0
$79$	4.92889e6	1.12475	0.562373	−	0.826884i	$-0.309889\pi$
$79$	4.92889e6	1.12475	0.562373	+	0.826884i	$0.309889\pi$
$80$	2.60478e6	0.568795
$81$	0	0
$82$	−947239.	−0.189719
$83$	−369529.	−0.0709373	−0.0354687	−	0.999371i	$-0.511292\pi$
$83$	−369529.	−0.0709373	−0.0354687	+	0.999371i	$0.511292\pi$
$84$	0	0
$85$	−3.07809e6	−0.543645
$86$	−4.55344e6	−0.771960
$87$	0	0
$88$	7.17020e6	1.12161
$89$	−2.70162e6	−0.406217	−0.203109	−	0.979156i	$-0.565104\pi$
$89$	−2.70162e6	−0.406217	−0.203109	+	0.979156i	$0.565104\pi$
$90$	0	0
$91$	1.22823e7	1.70857
$92$	−59303.9	−0.00794009
$93$	0	0
$94$	1.24099e7	1.54106
$95$	1.38689e6	0.165962
$96$	0	0
$97$	8.20102e6	0.912362	0.456181	−	0.889887i	$-0.349217\pi$
$97$	8.20102e6	0.912362	0.456181	+	0.889887i	$0.349217\pi$
$98$	−921470.	−0.0988985
$99$	0	0
$100$	247247.	0.0247247
$101$	1.65975e7	1.60294	0.801471	−	0.598034i	$-0.204051\pi$
$101$	1.65975e7	1.60294	0.801471	+	0.598034i	$0.204051\pi$
$102$	0	0
$103$	−1.26773e7	−1.14313	−0.571565	−	0.820556i	$-0.693664\pi$
$103$	−1.26773e7	−1.14313	−0.571565	+	0.820556i	$0.693664\pi$
$104$	1.90191e7	1.65795
$105$	0	0
$106$	3.06023e6	0.249565
$107$	2.30763e7	1.82105	0.910526	−	0.413453i	$-0.135677\pi$
$107$	2.30763e7	1.82105	0.910526	+	0.413453i	$0.135677\pi$
$108$	0	0
$109$	2.67537e7	1.97875	0.989376	−	0.145377i	$-0.0464396\pi$
$109$	2.67537e7	1.97875	0.989376	+	0.145377i	$0.0464396\pi$
$110$	8.93218e6	0.639857
$111$	0	0
$112$	−1.49833e7	−1.00773
$113$	1.63549e6	0.106629	0.0533143	−	0.998578i	$-0.483021\pi$
$113$	1.63549e6	0.106629	0.0533143	+	0.998578i	$0.483021\pi$
$114$	0	0
$115$	−2.01396e6	−0.123483
$116$	−649640.	−0.0386429
$117$	0	0
$118$	−2.02940e6	−0.113705
$119$	1.77060e7	0.963175
$120$	0	0
$121$	4.16288e6	0.213622
$122$	−2.77188e7	−1.38202
$123$	0	0
$124$	524663.	0.0247118
$125$	2.13282e7	0.976719
$126$	0	0
$127$	−9.72199e6	−0.421155	−0.210578	−	0.977577i	$-0.567534\pi$
$127$	−9.72199e6	−0.421155	−0.210578	+	0.977577i	$0.567534\pi$
$128$	−2.22818e7	−0.939108
$129$	0	0
$130$	2.36927e7	0.945831
$131$	1.23717e7	0.480815	0.240408	−	0.970672i	$-0.422719\pi$
$131$	1.23717e7	0.480815	0.240408	+	0.970672i	$0.422719\pi$
$132$	0	0
$133$	−7.97772e6	−0.294034
$134$	7.82401e6	0.280907
$135$	0	0
$136$	2.74176e7	0.934639
$137$	−2.11350e7	−0.702231	−0.351116	−	0.936332i	$-0.614198\pi$
$137$	−2.11350e7	−0.702231	−0.351116	+	0.936332i	$0.614198\pi$
$138$	0	0
$139$	1.65004e7	0.521126	0.260563	−	0.965457i	$-0.416092\pi$
$139$	1.65004e7	0.521126	0.260563	+	0.965457i	$0.416092\pi$
$140$	768194.	0.0236604
$141$	0	0
$142$	5.04464e7	1.47850
$143$	6.27321e7	1.79396
$144$	0	0
$145$	−2.20618e7	−0.600969
$146$	4.95814e7	1.31851
$147$	0	0
$148$	−2.05971e6	−0.0522264
$149$	−7.87547e7	−1.95040	−0.975202	−	0.221315i	$-0.928965\pi$
$149$	−7.87547e7	−1.95040	−0.975202	+	0.221315i	$0.928965\pi$
$150$	0	0
$151$	−6.23367e7	−1.47341	−0.736707	−	0.676212i	$-0.763620\pi$
$151$	−6.23367e7	−1.47341	−0.736707	+	0.676212i	$0.763620\pi$
$152$	−1.23535e7	−0.285323
$153$	0	0
$154$	−5.13801e7	−1.13363
$155$	1.78175e7	0.384315
$156$	0	0
$157$	6.46758e7	1.33381	0.666903	−	0.745144i	$-0.267619\pi$
$157$	6.46758e7	1.33381	0.666903	+	0.745144i	$0.267619\pi$
$158$	−5.46920e7	−1.10312
$159$	0	0
$160$	2.33546e6	0.0450767
$161$	1.15848e7	0.218775
$162$	0	0
$163$	−5.34198e7	−0.966153	−0.483077	−	0.875578i	$-0.660481\pi$
$163$	−5.34198e7	−0.966153	−0.483077	+	0.875578i	$0.660481\pi$
$164$	−416087.	−0.00736599
$165$	0	0
$166$	4.10037e6	0.0695736
$167$	2.35239e7	0.390843	0.195421	−	0.980719i	$-0.437393\pi$
$167$	2.35239e7	0.390843	0.195421	+	0.980719i	$0.437393\pi$
$168$	0	0
$169$	1.03649e8	1.65182
$170$	3.41552e7	0.533194
$171$	0	0
$172$	−2.00016e6	−0.0299719
$173$	4.02988e7	0.591739	0.295870	−	0.955228i	$-0.404391\pi$
$173$	4.02988e7	0.591739	0.295870	+	0.955228i	$0.404391\pi$
$174$	0	0
$175$	−4.82988e7	−0.681244
$176$	−7.65279e7	−1.05810
$177$	0	0
$178$	2.99777e7	0.398408
$179$	−9.45318e7	−1.23195	−0.615974	−	0.787767i	$-0.711237\pi$
$179$	−9.45318e7	−1.23195	−0.615974	+	0.787767i	$0.711237\pi$
$180$	0	0
$181$	2.13500e7	0.267623	0.133811	−	0.991007i	$-0.457278\pi$
$181$	2.13500e7	0.267623	0.133811	+	0.991007i	$0.457278\pi$
$182$	−1.36287e8	−1.67573
$183$	0	0
$184$	1.79390e7	0.212293
$185$	−6.99476e7	−0.812217
$186$	0	0
$187$	9.04338e7	1.01131
$188$	5.45121e6	0.0598329
$189$	0	0
$190$	−1.53892e7	−0.162771
$191$	1.38843e8	1.44181	0.720903	−	0.693036i	$-0.243727\pi$
$191$	1.38843e8	1.44181	0.720903	+	0.693036i	$0.243727\pi$
$192$	0	0
$193$	−1.26393e8	−1.26553	−0.632767	−	0.774342i	$-0.718081\pi$
$193$	−1.26393e8	−1.26553	−0.632767	+	0.774342i	$0.718081\pi$
$194$	−9.10003e7	−0.894822
$195$	0	0
$196$	−404768.	−0.00383981
$197$	−1.25589e8	−1.17036	−0.585181	−	0.810903i	$-0.698977\pi$
$197$	−1.25589e8	−1.17036	−0.585181	+	0.810903i	$0.698977\pi$
$198$	0	0
$199$	−6.73412e7	−0.605752	−0.302876	−	0.953030i	$-0.597947\pi$
$199$	−6.73412e7	−0.605752	−0.302876	+	0.953030i	$0.597947\pi$
$200$	−7.47905e7	−0.661061
$201$	0	0
$202$	−1.84169e8	−1.57213
$203$	1.26905e8	1.06474
$204$	0	0
$205$	−1.41303e7	−0.114555
$206$	1.40670e8	1.12115
$207$	0	0
$208$	−2.02991e8	−1.56407
$209$	−4.07465e7	−0.308730
$210$	0	0
$211$	2.55103e8	1.86950	0.934752	−	0.355300i	$-0.115621\pi$
$211$	2.55103e8	1.86950	0.934752	+	0.355300i	$0.115621\pi$
$212$	1.34425e6	0.00968954
$213$	0	0
$214$	−2.56059e8	−1.78604
$215$	−6.79253e7	−0.466119
$216$	0	0
$217$	−1.02491e8	−0.680890
$218$	−2.96865e8	−1.94071
$219$	0	0
$220$	3.92358e6	0.0248429
$221$	2.39877e8	1.49491
$222$	0	0
$223$	−5.74660e6	−0.0347012	−0.0173506	−	0.999849i	$-0.505523\pi$
$223$	−5.74660e6	−0.0347012	−0.0173506	+	0.999849i	$0.505523\pi$
$224$	−1.34341e7	−0.0798623
$225$	0	0
$226$	−1.81478e7	−0.104579
$227$	−5.38756e7	−0.305704	−0.152852	−	0.988249i	$-0.548846\pi$
$227$	−5.38756e7	−0.305704	−0.152852	+	0.988249i	$0.548846\pi$
$228$	0	0
$229$	−1.29368e8	−0.711876	−0.355938	−	0.934510i	$-0.615838\pi$
$229$	−1.29368e8	−0.711876	−0.355938	+	0.934510i	$0.615838\pi$
$230$	2.23473e7	0.121109
$231$	0	0
$232$	1.96512e8	1.03319
$233$	−4.55108e7	−0.235705	−0.117852	−	0.993031i	$-0.537601\pi$
$233$	−4.55108e7	−0.235705	−0.117852	+	0.993031i	$0.537601\pi$
$234$	0	0
$235$	1.85123e8	0.930513
$236$	−891441.	−0.00441469
$237$	0	0
$238$	−1.96469e8	−0.944659
$239$	3.62185e8	1.71608	0.858041	−	0.513582i	$-0.171682\pi$
$239$	3.62185e8	1.71608	0.858041	+	0.513582i	$0.171682\pi$
$240$	0	0
$241$	−1.34139e8	−0.617297	−0.308648	−	0.951176i	$-0.599877\pi$
$241$	−1.34139e8	−0.617297	−0.308648	+	0.951176i	$0.599877\pi$
$242$	−4.61922e7	−0.209515
$243$	0	0
$244$	−1.21759e7	−0.0536581
$245$	−1.37459e7	−0.0597162
$246$	0	0
$247$	−1.08081e8	−0.456361
$248$	−1.58707e8	−0.660717
$249$	0	0
$250$	−2.36662e8	−0.957942
$251$	5.80152e7	0.231571	0.115785	−	0.993274i	$-0.463062\pi$
$251$	5.80152e7	0.231571	0.115785	+	0.993274i	$0.463062\pi$
$252$	0	0
$253$	5.91697e7	0.229709
$254$	1.07877e8	0.413059
$255$	0	0
$256$	−3.06205e7	−0.114070
$257$	−1.24570e8	−0.457770	−0.228885	−	0.973453i	$-0.573508\pi$
$257$	−1.24570e8	−0.457770	−0.228885	+	0.973453i	$0.573508\pi$
$258$	0	0
$259$	4.02356e8	1.43900
$260$	1.04073e7	0.0367226
$261$	0	0
$262$	−1.37278e8	−0.471572
$263$	3.79795e8	1.28737	0.643686	−	0.765289i	$-0.277404\pi$
$263$	3.79795e8	1.28737	0.643686	+	0.765289i	$0.277404\pi$
$264$	0	0
$265$	4.56505e7	0.150690
$266$	8.85225e7	0.288382
$267$	0	0
$268$	3.43680e6	0.0109064
$269$	−2.77479e8	−0.869154	−0.434577	−	0.900635i	$-0.643102\pi$
$269$	−2.77479e8	−0.869154	−0.434577	+	0.900635i	$0.643102\pi$
$270$	0	0
$271$	4.63332e8	1.41416	0.707082	−	0.707131i	$-0.250011\pi$
$271$	4.63332e8	1.41416	0.707082	+	0.707131i	$0.250011\pi$
$272$	−2.92630e8	−0.881714
$273$	0	0
$274$	2.34518e8	0.688731
$275$	−2.46688e8	−0.715291
$276$	0	0
$277$	−4.78837e8	−1.35366	−0.676828	−	0.736141i	$-0.736646\pi$
$277$	−4.78837e8	−1.35366	−0.676828	+	0.736141i	$0.736646\pi$
$278$	−1.83092e8	−0.511107
$279$	0	0
$280$	−2.32373e8	−0.632606
$281$	−7.01149e7	−0.188512	−0.0942559	−	0.995548i	$-0.530047\pi$
$281$	−7.01149e7	−0.188512	−0.0942559	+	0.995548i	$0.530047\pi$
$282$	0	0
$283$	−2.76444e8	−0.725027	−0.362513	−	0.931979i	$-0.618081\pi$
$283$	−2.76444e8	−0.725027	−0.362513	+	0.931979i	$0.618081\pi$
$284$	2.21593e7	0.0574038
$285$	0	0
$286$	−6.96088e8	−1.75948
$287$	8.12811e7	0.202957
$288$	0	0
$289$	−6.45348e7	−0.157272
$290$	2.44802e8	0.589416
$291$	0	0
$292$	2.17793e7	0.0511922
$293$	6.73610e8	1.56449	0.782243	−	0.622973i	$-0.214075\pi$
$293$	6.73610e8	1.56449	0.782243	+	0.622973i	$0.214075\pi$
$294$	0	0
$295$	−3.02733e7	−0.0686567
$296$	6.23048e8	1.39637
$297$	0	0
$298$	8.73879e8	1.91291
$299$	1.56949e8	0.339553
$300$	0	0
$301$	3.90723e8	0.825822
$302$	6.91701e8	1.44509
$303$	0	0
$304$	1.31849e8	0.269166
$305$	−4.13492e8	−0.834483
$306$	0	0
$307$	−2.02723e7	−0.0399869	−0.0199934	−	0.999800i	$-0.506365\pi$
$307$	−2.02723e7	−0.0399869	−0.0199934	+	0.999800i	$0.506365\pi$
$308$	−2.25694e7	−0.0440142
$309$	0	0
$310$	−1.97707e8	−0.376926
$311$	3.68701e8	0.695046	0.347523	−	0.937672i	$-0.387023\pi$
$311$	3.68701e8	0.695046	0.347523	+	0.937672i	$0.387023\pi$
$312$	0	0
$313$	5.51670e8	1.01689	0.508445	−	0.861094i	$-0.330220\pi$
$313$	5.51670e8	1.01689	0.508445	+	0.861094i	$0.330220\pi$
$314$	−7.17656e8	−1.30816
$315$	0	0
$316$	−2.40242e7	−0.0428296
$317$	1.59775e8	0.281709	0.140855	−	0.990030i	$-0.455015\pi$
$317$	1.59775e8	0.281709	0.140855	+	0.990030i	$0.455015\pi$
$318$	0	0
$319$	6.48171e8	1.11795
$320$	−3.59326e8	−0.613005
$321$	0	0
$322$	−1.28547e8	−0.214569
$323$	−1.55808e8	−0.257265
$324$	0	0
$325$	−6.54342e8	−1.05734
$326$	5.92758e8	0.947579
$327$	0	0
$328$	1.25864e8	0.196944
$329$	−1.06487e9	−1.64859
$330$	0	0
$331$	−2.39933e8	−0.363658	−0.181829	−	0.983330i	$-0.558202\pi$
$331$	−2.39933e8	−0.363658	−0.181829	+	0.983330i	$0.558202\pi$
$332$	1.80114e6	0.00270125
$333$	0	0
$334$	−2.61026e8	−0.383329
$335$	1.16714e8	0.169615
$336$	0	0
$337$	1.35378e9	1.92682	0.963412	−	0.268026i	$-0.0863713\pi$
$337$	1.35378e9	1.92682	0.963412	+	0.268026i	$0.0863713\pi$
$338$	−1.15011e9	−1.62006
$339$	0	0
$340$	1.50031e7	0.0207017
$341$	−5.23476e8	−0.714919
$342$	0	0
$343$	−7.05065e8	−0.943409
$344$	6.05034e8	0.801356
$345$	0	0
$346$	−4.47164e8	−0.580364
$347$	−4.09155e7	−0.0525695	−0.0262848	−	0.999654i	$-0.508368\pi$
$347$	−4.09155e7	−0.0525695	−0.0262848	+	0.999654i	$0.508368\pi$
$348$	0	0
$349$	1.24915e9	1.57298	0.786491	−	0.617601i	$-0.211895\pi$
$349$	1.24915e9	1.57298	0.786491	+	0.617601i	$0.211895\pi$
$350$	5.35933e8	0.668148
$351$	0	0
$352$	−6.86153e7	−0.0838536
$353$	−9.60797e8	−1.16257	−0.581287	−	0.813699i	$-0.697450\pi$
$353$	−9.60797e8	−1.16257	−0.581287	+	0.813699i	$0.697450\pi$
$354$	0	0
$355$	7.52527e8	0.892736
$356$	1.31681e7	0.0154685
$357$	0	0
$358$	1.04894e9	1.20826
$359$	−7.94256e8	−0.906003	−0.453002	−	0.891510i	$-0.649647\pi$
$359$	−7.94256e8	−0.906003	−0.453002	+	0.891510i	$0.649647\pi$
$360$	0	0
$361$	−8.23670e8	−0.921463
$362$	−2.36904e8	−0.262478
$363$	0	0
$364$	−5.98657e7	−0.0650614
$365$	7.39623e8	0.796133
$366$	0	0
$367$	8.04891e8	0.849974	0.424987	−	0.905199i	$-0.360279\pi$
$367$	8.04891e8	0.849974	0.424987	+	0.905199i	$0.360279\pi$
$368$	−1.91464e8	−0.200272
$369$	0	0
$370$	7.76153e8	0.796602
$371$	−2.62593e8	−0.266978
$372$	0	0
$373$	1.19314e9	1.19045	0.595223	−	0.803561i	$-0.297064\pi$
$373$	1.19314e9	1.19045	0.595223	+	0.803561i	$0.297064\pi$
$374$	−1.00347e9	−0.991871
$375$	0	0
$376$	−1.64895e9	−1.59974
$377$	1.71928e9	1.65254
$378$	0	0
$379$	−6.14932e8	−0.580216	−0.290108	−	0.956994i	$-0.593691\pi$
$379$	−6.14932e8	−0.580216	−0.290108	+	0.956994i	$0.593691\pi$
$380$	−6.75990e6	−0.00631972
$381$	0	0
$382$	−1.54063e9	−1.41409
$383$	−1.05026e9	−0.955214	−0.477607	−	0.878574i	$-0.658496\pi$
$383$	−1.05026e9	−0.955214	−0.477607	+	0.878574i	$0.658496\pi$
$384$	0	0
$385$	−7.66456e8	−0.684502
$386$	1.40249e9	1.24121
$387$	0	0
$388$	−3.99731e7	−0.0347421
$389$	9.88396e8	0.851349	0.425674	−	0.904876i	$-0.360037\pi$
$389$	9.88396e8	0.851349	0.425674	+	0.904876i	$0.360037\pi$
$390$	0	0
$391$	2.26255e8	0.191417
$392$	1.22440e8	0.102665
$393$	0	0
$394$	1.39356e9	1.14786
$395$	−8.15861e8	−0.666080
$396$	0	0
$397$	1.79703e8	0.144141	0.0720707	−	0.997400i	$-0.477039\pi$
$397$	1.79703e8	0.144141	0.0720707	+	0.997400i	$0.477039\pi$
$398$	7.47231e8	0.594107
$399$	0	0
$400$	7.98243e8	0.623628
$401$	2.18971e9	1.69583	0.847914	−	0.530133i	$-0.177858\pi$
$401$	2.18971e9	1.69583	0.847914	+	0.530133i	$0.177858\pi$
$402$	0	0
$403$	−1.38853e9	−1.05679
$404$	−8.08988e7	−0.0610390
$405$	0	0
$406$	−1.40816e9	−1.04427
$407$	2.05505e9	1.51092
$408$	0	0
$409$	8.31329e8	0.600816	0.300408	−	0.953811i	$-0.402877\pi$
$409$	8.31329e8	0.600816	0.300408	+	0.953811i	$0.402877\pi$
$410$	1.56793e8	0.112353
$411$	0	0
$412$	6.17911e7	0.0435297
$413$	1.74140e8	0.121639
$414$	0	0
$415$	6.11667e7	0.0420094
$416$	−1.82003e8	−0.123952
$417$	0	0
$418$	4.52132e8	0.302794
$419$	1.68006e9	1.11578	0.557888	−	0.829916i	$-0.311612\pi$
$419$	1.68006e9	1.11578	0.557888	+	0.829916i	$0.311612\pi$
$420$	0	0
$421$	1.78087e8	0.116317	0.0581586	−	0.998307i	$-0.481477\pi$
$421$	1.78087e8	0.116317	0.0581586	+	0.998307i	$0.481477\pi$
$422$	−2.83067e9	−1.83356
$423$	0	0
$424$	−4.06625e8	−0.259068
$425$	−9.43292e8	−0.596053
$426$	0	0
$427$	2.37851e9	1.47845
$428$	−1.12477e8	−0.0693445
$429$	0	0
$430$	7.53713e8	0.457158
$431$	−5.82450e8	−0.350419	−0.175210	−	0.984531i	$-0.556060\pi$
$431$	−5.82450e8	−0.350419	−0.175210	+	0.984531i	$0.556060\pi$
$432$	0	0
$433$	−2.04988e9	−1.21345	−0.606723	−	0.794913i	$-0.707516\pi$
$433$	−2.04988e9	−1.21345	−0.606723	+	0.794913i	$0.707516\pi$
$434$	1.13726e9	0.667800
$435$	0	0
$436$	−1.30402e8	−0.0753496
$437$	−1.01943e8	−0.0584349
$438$	0	0
$439$	3.26179e6	0.00184005	0.000920027	−	1.00000i	$-0.499707\pi$
$439$	3.26179e6	0.00184005	0.000920027		1.00000i	$0.499707\pi$
$440$	−1.18686e9	−0.664222
$441$	0	0
$442$	−2.66173e9	−1.46617
$443$	1.85183e9	1.01202	0.506010	−	0.862528i	$-0.331120\pi$
$443$	1.85183e9	1.01202	0.506010	+	0.862528i	$0.331120\pi$
$444$	0	0
$445$	4.47188e8	0.240564
$446$	6.37655e7	0.0340340
$447$	0	0
$448$	2.06694e9	1.08606
$449$	9.26659e8	0.483123	0.241562	−	0.970385i	$-0.422340\pi$
$449$	9.26659e8	0.483123	0.241562	+	0.970385i	$0.422340\pi$
$450$	0	0
$451$	4.15146e8	0.213100
$452$	−7.97165e6	−0.00406035
$453$	0	0
$454$	5.97815e8	0.299827
$455$	−2.03304e9	−1.01182
$456$	0	0
$457$	1.41700e9	0.694483	0.347242	−	0.937776i	$-0.387118\pi$
$457$	1.41700e9	0.694483	0.347242	+	0.937776i	$0.387118\pi$
$458$	1.43550e9	0.698190
$459$	0	0
$460$	9.81634e6	0.00470216
$461$	−2.11900e9	−1.00735	−0.503673	−	0.863894i	$-0.668018\pi$
$461$	−2.11900e9	−1.00735	−0.503673	+	0.863894i	$0.668018\pi$
$462$	0	0
$463$	2.03922e9	0.954841	0.477421	−	0.878675i	$-0.341572\pi$
$463$	2.03922e9	0.954841	0.477421	+	0.878675i	$0.341572\pi$
$464$	−2.09738e9	−0.974685
$465$	0	0
$466$	5.04997e8	0.231174
$467$	1.27874e9	0.580996	0.290498	−	0.956876i	$-0.406179\pi$
$467$	1.27874e9	0.580996	0.290498	+	0.956876i	$0.406179\pi$
$468$	0	0
$469$	−6.71366e8	−0.300507
$470$	−2.05416e9	−0.912624
$471$	0	0
$472$	2.69655e8	0.118035
$473$	1.99563e9	0.867095
$474$	0	0
$475$	4.25016e8	0.181961
$476$	−8.63017e7	−0.0366771
$477$	0	0
$478$	−4.01888e9	−1.68309
$479$	−4.76473e8	−0.198091	−0.0990453	−	0.995083i	$-0.531579\pi$
$479$	−4.76473e8	−0.198091	−0.0990453	+	0.995083i	$0.531579\pi$
$480$	0	0
$481$	5.45105e9	2.23343
$482$	1.48843e9	0.605430
$483$	0	0
$484$	−2.02905e7	−0.00813457
$485$	−1.35748e9	−0.540305
$486$	0	0
$487$	−9.49824e8	−0.372642	−0.186321	−	0.982489i	$-0.559656\pi$
$487$	−9.49824e8	−0.372642	−0.186321	+	0.982489i	$0.559656\pi$
$488$	3.68311e9	1.43465
$489$	0	0
$490$	1.52527e8	0.0585682
$491$	1.10186e8	0.0420089	0.0210044	−	0.999779i	$-0.493314\pi$
$491$	1.10186e8	0.0420089	0.0210044	+	0.999779i	$0.493314\pi$
$492$	0	0
$493$	2.47850e9	0.931589
$494$	1.19929e9	0.447588
$495$	0	0
$496$	1.69389e9	0.623303
$497$	−4.32873e9	−1.58166
$498$	0	0
$499$	−5.66434e8	−0.204079	−0.102039	−	0.994780i	$-0.532537\pi$
$499$	−5.66434e8	−0.204079	−0.102039	+	0.994780i	$0.532537\pi$
$500$	−1.03957e8	−0.0371928
$501$	0	0
$502$	−6.43748e8	−0.227119
$503$	3.11228e8	0.109041	0.0545207	−	0.998513i	$-0.482637\pi$
$503$	3.11228e8	0.109041	0.0545207	+	0.998513i	$0.482637\pi$
$504$	0	0
$505$	−2.74732e9	−0.949269
$506$	−6.56560e8	−0.225293
$507$	0	0
$508$	4.73865e7	0.0160373
$509$	−5.06352e8	−0.170193	−0.0850963	−	0.996373i	$-0.527120\pi$
$509$	−5.06352e8	−0.170193	−0.0850963	+	0.996373i	$0.527120\pi$
$510$	0	0
$511$	−4.25450e9	−1.41051
$512$	3.19184e9	1.05098
$513$	0	0
$514$	1.38225e9	0.448970
$515$	2.09842e9	0.676967
$516$	0	0
$517$	−5.43888e9	−1.73098
$518$	−4.46463e9	−1.41134
$519$	0	0
$520$	−3.14815e9	−0.981847
$521$	3.18059e9	0.985317	0.492658	−	0.870223i	$-0.336025\pi$
$521$	3.18059e9	0.985317	0.492658	+	0.870223i	$0.336025\pi$
$522$	0	0
$523$	−4.24695e9	−1.29814	−0.649069	−	0.760730i	$-0.724841\pi$
$523$	−4.24695e9	−1.29814	−0.649069	+	0.760730i	$0.724841\pi$
$524$	−6.03014e7	−0.0183091
$525$	0	0
$526$	−4.21429e9	−1.26262
$527$	−2.00169e9	−0.595743
$528$	0	0
$529$	1.48036e8	0.0434783
$530$	−5.06548e8	−0.147793
$531$	0	0
$532$	3.88847e7	0.0111966
$533$	1.10118e9	0.315002
$534$	0	0
$535$	−3.81973e9	−1.07843
$536$	−1.03961e9	−0.291604
$537$	0	0
$538$	3.07896e9	0.852445
$539$	4.03852e8	0.111087
$540$	0	0
$541$	1.47556e9	0.400651	0.200325	−	0.979729i	$-0.435800\pi$
$541$	1.47556e9	0.400651	0.200325	+	0.979729i	$0.435800\pi$
$542$	−5.14123e9	−1.38698
$543$	0	0
$544$	−2.62374e8	−0.0698754
$545$	−4.42844e9	−1.17183
$546$	0	0
$547$	−3.24569e9	−0.847914	−0.423957	−	0.905682i	$-0.639359\pi$
$547$	−3.24569e9	−0.847914	−0.423957	+	0.905682i	$0.639359\pi$
$548$	1.03015e8	0.0267405
$549$	0	0
$550$	2.73730e9	0.701540
$551$	−1.11673e9	−0.284392
$552$	0	0
$553$	4.69304e9	1.18009
$554$	5.31327e9	1.32763
$555$	0	0
$556$	−8.04255e7	−0.0198441
$557$	−5.12858e9	−1.25749	−0.628744	−	0.777612i	$-0.716431\pi$
$557$	−5.12858e9	−1.25749	−0.628744	+	0.777612i	$0.716431\pi$
$558$	0	0
$559$	5.29345e9	1.28173
$560$	2.48014e9	0.596784
$561$	0	0
$562$	7.78010e8	0.184888
$563$	1.95702e9	0.462185	0.231092	−	0.972932i	$-0.425770\pi$
$563$	1.95702e9	0.462185	0.231092	+	0.972932i	$0.425770\pi$
$564$	0	0
$565$	−2.70717e8	−0.0631460
$566$	3.06748e9	0.711088
$567$	0	0
$568$	−6.70302e9	−1.53480
$569$	−8.49926e8	−0.193414	−0.0967070	−	0.995313i	$-0.530831\pi$
$569$	−8.49926e8	−0.193414	−0.0967070	+	0.995313i	$0.530831\pi$
$570$	0	0
$571$	3.26079e9	0.732988	0.366494	−	0.930421i	$-0.380558\pi$
$571$	3.26079e9	0.732988	0.366494	+	0.930421i	$0.380558\pi$
$572$	−3.05766e8	−0.0683130
$573$	0	0
$574$	−9.01912e8	−0.199055
$575$	−6.17184e8	−0.135387
$576$	0	0
$577$	2.86087e9	0.619988	0.309994	−	0.950739i	$-0.399673\pi$
$577$	2.86087e9	0.619988	0.309994	+	0.950739i	$0.399673\pi$
$578$	7.16091e8	0.154248
$579$	0	0
$580$	1.07532e8	0.0228845
$581$	−3.51846e8	−0.0744280
$582$	0	0
$583$	−1.34121e9	−0.280321
$584$	−6.58808e9	−1.36872
$585$	0	0
$586$	−7.47452e9	−1.53441
$587$	8.05972e9	1.64470	0.822349	−	0.568983i	$-0.192663\pi$
$587$	8.05972e9	1.64470	0.822349	+	0.568983i	$0.192663\pi$
$588$	0	0
$589$	9.01893e8	0.181866
$590$	3.35919e8	0.0673368
$591$	0	0
$592$	−6.64982e9	−1.31730
$593$	−2.06892e9	−0.407429	−0.203714	−	0.979030i	$-0.565301\pi$
$593$	−2.06892e9	−0.407429	−0.203714	+	0.979030i	$0.565301\pi$
$594$	0	0
$595$	−2.93080e9	−0.570397
$596$	3.83863e8	0.0742702
$597$	0	0
$598$	−1.74153e9	−0.333026
$599$	2.27549e9	0.432595	0.216298	−	0.976328i	$-0.430602\pi$
$599$	2.27549e9	0.432595	0.216298	+	0.976328i	$0.430602\pi$
$600$	0	0
$601$	−4.39655e9	−0.826136	−0.413068	−	0.910700i	$-0.635543\pi$
$601$	−4.39655e9	−0.826136	−0.413068	+	0.910700i	$0.635543\pi$
$602$	−4.33555e9	−0.809946
$603$	0	0
$604$	3.03839e8	0.0561066
$605$	−6.89066e8	−0.126508
$606$	0	0
$607$	−1.55818e9	−0.282785	−0.141393	−	0.989954i	$-0.545158\pi$
$607$	−1.55818e9	−0.282785	−0.141393	+	0.989954i	$0.545158\pi$
$608$	1.18217e8	0.0213313
$609$	0	0
$610$	4.58819e9	0.818440
$611$	−1.44267e10	−2.55872
$612$	0	0
$613$	2.08002e9	0.364717	0.182359	−	0.983232i	$-0.441627\pi$
$613$	2.08002e9	0.364717	0.182359	+	0.983232i	$0.441627\pi$
$614$	2.24945e8	0.0392182
$615$	0	0
$616$	6.82709e9	1.17680
$617$	−1.48880e9	−0.255175	−0.127588	−	0.991827i	$-0.540723\pi$
$617$	−1.48880e9	−0.255175	−0.127588	+	0.991827i	$0.540723\pi$
$618$	0	0
$619$	−4.92878e9	−0.835261	−0.417630	−	0.908617i	$-0.637139\pi$
$619$	−4.92878e9	−0.835261	−0.417630	+	0.908617i	$0.637139\pi$
$620$	−8.68455e7	−0.0146345
$621$	0	0
$622$	−4.09119e9	−0.681684
$623$	−2.57234e9	−0.426206
$624$	0	0
$625$	4.32594e8	0.0708763
$626$	−6.12145e9	−0.997342
$627$	0	0
$628$	−3.15240e8	−0.0507905
$629$	7.85816e9	1.25905
$630$	0	0
$631$	−3.18924e9	−0.505340	−0.252670	−	0.967552i	$-0.581309\pi$
$631$	−3.18924e9	−0.505340	−0.252670	+	0.967552i	$0.581309\pi$
$632$	7.26715e9	1.14513
$633$	0	0
$634$	−1.77290e9	−0.276294
$635$	1.60924e9	0.249410
$636$	0	0
$637$	1.07122e9	0.164207
$638$	−7.19224e9	−1.09646
$639$	0	0
$640$	3.68822e9	0.556144
$641$	−3.87554e9	−0.581204	−0.290602	−	0.956844i	$-0.593856\pi$
$641$	−3.87554e9	−0.581204	−0.290602	+	0.956844i	$0.593856\pi$
$642$	0	0
$643$	−9.82602e9	−1.45760	−0.728801	−	0.684725i	$-0.759922\pi$
$643$	−9.82602e9	−1.45760	−0.728801	+	0.684725i	$0.759922\pi$
$644$	−5.64661e7	−0.00833081
$645$	0	0
$646$	1.72888e9	0.252319
$647$	3.65079e9	0.529934	0.264967	−	0.964258i	$-0.414639\pi$
$647$	3.65079e9	0.529934	0.264967	+	0.964258i	$0.414639\pi$
$648$	0	0
$649$	8.89425e8	0.127718
$650$	7.26072e9	1.03701
$651$	0	0
$652$	2.60377e8	0.0367905
$653$	−1.36014e9	−0.191156	−0.0955779	−	0.995422i	$-0.530470\pi$
$653$	−1.36014e9	−0.191156	−0.0955779	+	0.995422i	$0.530470\pi$
$654$	0	0
$655$	−2.04783e9	−0.284741
$656$	−1.34335e9	−0.185791
$657$	0	0
$658$	1.18161e10	1.61689
$659$	4.55486e9	0.619978	0.309989	−	0.950740i	$-0.399675\pi$
$659$	4.55486e9	0.619978	0.309989	+	0.950740i	$0.399675\pi$
$660$	0	0
$661$	7.93234e9	1.06831	0.534154	−	0.845387i	$-0.320630\pi$
$661$	7.93234e9	1.06831	0.534154	+	0.845387i	$0.320630\pi$
$662$	2.66235e9	0.356667
$663$	0	0
$664$	−5.44833e8	−0.0722229
$665$	1.32052e9	0.174128
$666$	0	0
$667$	1.62165e9	0.211600
$668$	−1.14659e8	−0.0148830
$669$	0	0
$670$	−1.29508e9	−0.166354
$671$	1.21483e10	1.55234
$672$	0	0
$673$	−1.94604e9	−0.246093	−0.123046	−	0.992401i	$-0.539266\pi$
$673$	−1.94604e9	−0.246093	−0.123046	+	0.992401i	$0.539266\pi$
$674$	−1.50218e10	−1.88978
$675$	0	0
$676$	−5.05203e8	−0.0629002
$677$	1.37845e10	1.70738	0.853692	−	0.520778i	$-0.174358\pi$
$677$	1.37845e10	1.70738	0.853692	+	0.520778i	$0.174358\pi$
$678$	0	0
$679$	7.80859e9	0.957257
$680$	−4.53834e9	−0.553497
$681$	0	0
$682$	5.80860e9	0.701175
$683$	−7.27273e9	−0.873424	−0.436712	−	0.899601i	$-0.643857\pi$
$683$	−7.27273e9	−0.873424	−0.436712	+	0.899601i	$0.643857\pi$
$684$	0	0
$685$	3.49839e9	0.415864
$686$	7.82355e9	0.925272
$687$	0	0
$688$	−6.45756e9	−0.755978
$689$	−3.55757e9	−0.414368
$690$	0	0
$691$	1.50396e10	1.73405	0.867027	−	0.498261i	$-0.166028\pi$
$691$	1.50396e10	1.73405	0.867027	+	0.498261i	$0.166028\pi$
$692$	−1.96423e8	−0.0225331
$693$	0	0
$694$	4.54006e8	0.0515589
$695$	−2.73125e9	−0.308613
$696$	0	0
$697$	1.58745e9	0.177576
$698$	−1.38608e10	−1.54274
$699$	0	0
$700$	2.35416e8	0.0259413
$701$	−8.30205e7	−0.00910274	−0.00455137	−	0.999990i	$-0.501449\pi$
$701$	−8.30205e7	−0.00910274	−0.00455137	+	0.999990i	$0.501449\pi$
$702$	0	0
$703$	−3.54063e9	−0.384359
$704$	1.05569e10	1.14034
$705$	0	0
$706$	1.06612e10	1.14022
$707$	1.58033e10	1.68182
$708$	0	0
$709$	5.57061e9	0.587004	0.293502	−	0.955959i	$-0.405179\pi$
$709$	5.57061e9	0.587004	0.293502	+	0.955959i	$0.405179\pi$
$710$	−8.35020e9	−0.875574
$711$	0	0
$712$	−3.98326e9	−0.413579
$713$	−1.30968e9	−0.135317
$714$	0	0
$715$	−1.03838e10	−1.06239
$716$	4.60763e8	0.0469118
$717$	0	0
$718$	8.81323e9	0.888586
$719$	−6.00718e9	−0.602726	−0.301363	−	0.953510i	$-0.597442\pi$
$719$	−6.00718e9	−0.602726	−0.301363	+	0.953510i	$0.597442\pi$
$720$	0	0
$721$	−1.20707e10	−1.19938
$722$	9.13961e9	0.903749
$723$	0	0
$724$	−1.04063e8	−0.0101909
$725$	−6.76090e9	−0.658903
$726$	0	0
$727$	1.41144e10	1.36236	0.681182	−	0.732114i	$-0.261466\pi$
$727$	1.41144e10	1.36236	0.681182	+	0.732114i	$0.261466\pi$
$728$	1.81090e10	1.73954
$729$	0	0
$730$	−8.20702e9	−0.780828
$731$	7.63097e9	0.722552
$732$	0	0
$733$	1.56763e10	1.47021	0.735104	−	0.677954i	$-0.237133\pi$
$733$	1.56763e10	1.47021	0.735104	+	0.677954i	$0.237133\pi$
$734$	−8.93124e9	−0.833634
$735$	0	0
$736$	−1.71668e8	−0.0158714
$737$	−3.42903e9	−0.315526
$738$	0	0
$739$	5.76779e9	0.525720	0.262860	−	0.964834i	$-0.415334\pi$
$739$	5.76779e9	0.525720	0.262860	+	0.964834i	$0.415334\pi$
$740$	3.40936e8	0.0309287
$741$	0	0
$742$	2.91379e9	0.261845
$743$	−1.11257e10	−0.995096	−0.497548	−	0.867436i	$-0.665766\pi$
$743$	−1.11257e10	−0.995096	−0.497548	+	0.867436i	$0.665766\pi$
$744$	0	0
$745$	1.30360e10	1.15504
$746$	−1.32393e10	−1.16756
$747$	0	0
$748$	−4.40789e8	−0.0385101
$749$	2.19720e10	1.91066
$750$	0	0
$751$	2.96634e9	0.255553	0.127777	−	0.991803i	$-0.459216\pi$
$751$	2.96634e9	0.255553	0.127777	+	0.991803i	$0.459216\pi$
$752$	1.75994e10	1.50916
$753$	0	0
$754$	−1.90775e10	−1.62077
$755$	1.03184e10	0.872562
$756$	0	0
$757$	1.45781e10	1.22142	0.610710	−	0.791854i	$-0.290884\pi$
$757$	1.45781e10	1.22142	0.610710	+	0.791854i	$0.290884\pi$
$758$	6.82341e9	0.569062
$759$	0	0
$760$	2.04482e9	0.168970
$761$	1.30095e10	1.07007	0.535036	−	0.844829i	$-0.320298\pi$
$761$	1.30095e10	1.07007	0.535036	+	0.844829i	$0.320298\pi$
$762$	0	0
$763$	2.54735e10	2.07612
$764$	−6.76742e8	−0.0549031
$765$	0	0
$766$	1.16539e10	0.936850
$767$	2.35921e9	0.188792
$768$	0	0
$769$	1.68664e9	0.133746	0.0668728	−	0.997762i	$-0.478698\pi$
$769$	1.68664e9	0.133746	0.0668728	+	0.997762i	$0.478698\pi$
$770$	8.50476e9	0.671343
$771$	0	0
$772$	6.16062e8	0.0481907
$773$	1.63352e10	1.27203	0.636013	−	0.771678i	$-0.280582\pi$
$773$	1.63352e10	1.27203	0.636013	+	0.771678i	$0.280582\pi$
$774$	0	0
$775$	5.46025e9	0.421363
$776$	1.20916e10	0.928896
$777$	0	0
$778$	−1.09674e10	−0.834982
$779$	−7.15252e8	−0.0542098
$780$	0	0
$781$	−2.21091e10	−1.66071
$782$	−2.51057e9	−0.187737
$783$	0	0
$784$	−1.30680e9	−0.0968510
$785$	−1.07055e10	−0.789886
$786$	0	0
$787$	−1.13744e10	−0.831795	−0.415898	−	0.909411i	$-0.636533\pi$
$787$	−1.13744e10	−0.831795	−0.415898	+	0.909411i	$0.636533\pi$
$788$	6.12141e8	0.0445666
$789$	0	0
$790$	9.05296e9	0.653275
$791$	1.55723e9	0.111876
$792$	0	0
$793$	3.22236e10	2.29466
$794$	−1.99402e9	−0.141370
$795$	0	0
$796$	3.28231e8	0.0230666
$797$	−1.05659e10	−0.739267	−0.369633	−	0.929178i	$-0.620517\pi$
$797$	−1.05659e10	−0.739267	−0.369633	+	0.929178i	$0.620517\pi$
$798$	0	0
$799$	−2.07974e10	−1.44243
$800$	7.15709e8	0.0494221
$801$	0	0
$802$	−2.42975e10	−1.66323
$803$	−2.17300e10	−1.48100
$804$	0	0
$805$	−1.91759e9	−0.129560
$806$	1.54074e10	1.03647
$807$	0	0
$808$	2.44713e10	1.63199
$809$	−1.96297e10	−1.30345	−0.651724	−	0.758457i	$-0.725954\pi$
$809$	−1.96297e10	−1.30345	−0.651724	+	0.758457i	$0.725954\pi$
$810$	0	0
$811$	−2.53973e10	−1.67191	−0.835957	−	0.548795i	$-0.815087\pi$
$811$	−2.53973e10	−1.67191	−0.835957	+	0.548795i	$0.815087\pi$
$812$	−6.18554e8	−0.0405445
$813$	0	0
$814$	−2.28033e10	−1.48187
$815$	8.84238e9	0.572160
$816$	0	0
$817$	−3.43826e9	−0.220578
$818$	−9.22461e9	−0.589266
$819$	0	0
$820$	6.88734e7	0.00436217
$821$	−3.97732e9	−0.250836	−0.125418	−	0.992104i	$-0.540027\pi$
$821$	−3.97732e9	−0.250836	−0.125418	+	0.992104i	$0.540027\pi$
$822$	0	0
$823$	−9.82551e9	−0.614407	−0.307203	−	0.951644i	$-0.599393\pi$
$823$	−9.82551e9	−0.614407	−0.307203	+	0.951644i	$0.599393\pi$
$824$	−1.86914e10	−1.16385
$825$	0	0
$826$	−1.93229e9	−0.119300
$827$	−1.33644e10	−0.821636	−0.410818	−	0.911717i	$-0.634757\pi$
$827$	−1.33644e10	−0.821636	−0.410818	+	0.911717i	$0.634757\pi$
$828$	0	0
$829$	−1.05176e10	−0.641173	−0.320586	−	0.947219i	$-0.603880\pi$
$829$	−1.05176e10	−0.641173	−0.320586	+	0.947219i	$0.603880\pi$
$830$	−6.78718e8	−0.0412018
$831$	0	0
$832$	2.80025e10	1.68564
$833$	1.54426e9	0.0925687
$834$	0	0
$835$	−3.89383e9	−0.231459
$836$	1.98605e8	0.0117562
$837$	0	0
$838$	−1.86423e10	−1.09433
$839$	−1.44529e10	−0.844865	−0.422433	−	0.906394i	$-0.638824\pi$
$839$	−1.44529e10	−0.844865	−0.422433	+	0.906394i	$0.638824\pi$
$840$	0	0
$841$	5.14370e8	0.0298188
$842$	−1.97609e9	−0.114081
$843$	0	0
$844$	−1.24341e9	−0.0711895
$845$	−1.71567e10	−0.978215
$846$	0	0
$847$	3.96368e9	0.224133
$848$	4.33994e9	0.244398
$849$	0	0
$850$	1.04670e10	0.584594
$851$	5.14150e9	0.285980
$852$	0	0
$853$	−2.49192e10	−1.37471	−0.687356	−	0.726321i	$-0.741229\pi$
$853$	−2.49192e10	−1.37471	−0.687356	+	0.726321i	$0.741229\pi$
$854$	−2.63924e10	−1.45003
$855$	0	0
$856$	3.40236e10	1.85405
$857$	−1.98477e10	−1.07715	−0.538577	−	0.842576i	$-0.681038\pi$
$857$	−1.98477e10	−1.07715	−0.538577	+	0.842576i	$0.681038\pi$
$858$	0	0
$859$	−2.59097e10	−1.39472	−0.697359	−	0.716722i	$-0.745642\pi$
$859$	−2.59097e10	−1.39472	−0.697359	+	0.716722i	$0.745642\pi$
$860$	3.31078e8	0.0177495
$861$	0	0
$862$	6.46298e9	0.343683
$863$	2.02220e9	0.107099	0.0535496	−	0.998565i	$-0.482946\pi$
$863$	2.02220e9	0.107099	0.0535496	+	0.998565i	$0.482946\pi$
$864$	0	0
$865$	−6.67050e9	−0.350431
$866$	2.27459e10	1.19012
$867$	0	0
$868$	4.99557e8	0.0259278
$869$	2.39698e10	1.23907
$870$	0	0
$871$	−9.09554e9	−0.466407
$872$	3.94457e10	2.01461
$873$	0	0
$874$	1.13118e9	0.0573116
$875$	2.03076e10	1.02478
$876$	0	0
$877$	9.24606e9	0.462869	0.231435	−	0.972850i	$-0.425658\pi$
$877$	9.24606e9	0.462869	0.231435	+	0.972850i	$0.425658\pi$
$878$	−3.61935e7	−0.00180468
$879$	0	0
$880$	1.26674e10	0.626610
$881$	1.84268e10	0.907893	0.453946	−	0.891029i	$-0.350016\pi$
$881$	1.84268e10	0.907893	0.453946	+	0.891029i	$0.350016\pi$
$882$	0	0
$883$	−1.07940e10	−0.527618	−0.263809	−	0.964575i	$-0.584979\pi$
$883$	−1.07940e10	−0.527618	−0.263809	+	0.964575i	$0.584979\pi$
$884$	−1.16920e9	−0.0569253
$885$	0	0
$886$	−2.05483e10	−0.992564
$887$	1.27893e10	0.615339	0.307670	−	0.951493i	$-0.400451\pi$
$887$	1.27893e10	0.615339	0.307670	+	0.951493i	$0.400451\pi$
$888$	0	0
$889$	−9.25677e9	−0.441879
$890$	−4.96209e9	−0.235939
$891$	0	0
$892$	2.80098e7	0.00132140
$893$	9.37060e9	0.440339
$894$	0	0
$895$	1.56475e10	0.729565
$896$	−2.12156e10	−0.985319
$897$	0	0
$898$	−1.02824e10	−0.473835
$899$	−1.43468e10	−0.658560
$900$	0	0
$901$	−5.12855e9	−0.233592
$902$	−4.60655e9	−0.209003
$903$	0	0
$904$	2.41137e9	0.108561
$905$	−3.53398e9	−0.158487
$906$	0	0
$907$	1.90825e10	0.849199	0.424599	−	0.905381i	$-0.360415\pi$
$907$	1.90825e10	0.849199	0.424599	+	0.905381i	$0.360415\pi$
$908$	2.62598e8	0.0116410
$909$	0	0
$910$	2.25590e10	0.992373
$911$	−5.20460e9	−0.228072	−0.114036	−	0.993477i	$-0.536378\pi$
$911$	−5.20460e9	−0.228072	−0.114036	+	0.993477i	$0.536378\pi$
$912$	0	0
$913$	−1.79707e9	−0.0781477
$914$	−1.57233e10	−0.681132
$915$	0	0
$916$	6.30562e8	0.0271078
$917$	1.17797e10	0.504475
$918$	0	0
$919$	−1.65425e10	−0.703066	−0.351533	−	0.936176i	$-0.614339\pi$
$919$	−1.65425e10	−0.703066	−0.351533	+	0.936176i	$0.614339\pi$
$920$	−2.96938e9	−0.125721
$921$	0	0
$922$	2.35129e10	0.987980
$923$	−5.86448e10	−2.45484
$924$	0	0
$925$	−2.14357e10	−0.890515
$926$	−2.26276e10	−0.936485
$927$	0	0
$928$	−1.88052e9	−0.0772433
$929$	−2.36746e10	−0.968786	−0.484393	−	0.874850i	$-0.660960\pi$
$929$	−2.36746e10	−0.968786	−0.484393	+	0.874850i	$0.660960\pi$
$930$	0	0
$931$	−6.95794e8	−0.0282590
$932$	2.21827e8	0.00897549
$933$	0	0
$934$	−1.41892e10	−0.569826
$935$	−1.49692e10	−0.598904
$936$	0	0
$937$	3.19714e10	1.26962	0.634809	−	0.772669i	$-0.281079\pi$
$937$	3.19714e10	1.26962	0.634809	+	0.772669i	$0.281079\pi$
$938$	7.44962e9	0.294730
$939$	0	0
$940$	−9.02317e8	−0.0354333
$941$	−3.81681e10	−1.49326	−0.746632	−	0.665237i	$-0.768331\pi$
$941$	−3.81681e10	−1.49326	−0.746632	+	0.665237i	$0.768331\pi$
$942$	0	0
$943$	1.03865e9	0.0403346
$944$	−2.87804e9	−0.111351
$945$	0	0
$946$	−2.21440e10	−0.850426
$947$	3.14336e10	1.20273	0.601366	−	0.798974i	$-0.294623\pi$
$947$	3.14336e10	1.20273	0.601366	+	0.798974i	$0.294623\pi$
$948$	0	0
$949$	−5.76392e10	−2.18920
$950$	−4.71607e9	−0.178463
$951$	0	0
$952$	2.61057e10	0.980631
$953$	1.41899e10	0.531072	0.265536	−	0.964101i	$-0.414451\pi$
$953$	1.41899e10	0.531072	0.265536	+	0.964101i	$0.414451\pi$
$954$	0	0
$955$	−2.29821e10	−0.853844
$956$	−1.76535e9	−0.0653473
$957$	0	0
$958$	5.28704e9	0.194282
$959$	−2.01236e10	−0.736786
$960$	0	0
$961$	−1.59259e10	−0.578856
$962$	−6.04860e10	−2.19049
$963$	0	0
$964$	6.53813e8	0.0235063
$965$	2.09214e10	0.749455
$966$	0	0
$967$	−4.79184e10	−1.70416	−0.852078	−	0.523415i	$-0.824658\pi$
$967$	−4.79184e10	−1.70416	−0.852078	+	0.523415i	$0.824658\pi$
$968$	6.13775e9	0.217493
$969$	0	0
$970$	1.50629e10	0.529918
$971$	−3.34597e10	−1.17288	−0.586442	−	0.809991i	$-0.699472\pi$
$971$	−3.34597e10	−1.17288	−0.586442	+	0.809991i	$0.699472\pi$
$972$	0	0
$973$	1.57108e10	0.546769
$974$	1.05394e10	0.365478
$975$	0	0
$976$	−3.93101e10	−1.35341
$977$	6.62901e9	0.227414	0.113707	−	0.993514i	$-0.463727\pi$
$977$	6.62901e9	0.227414	0.113707	+	0.993514i	$0.463727\pi$
$978$	0	0
$979$	−1.31383e10	−0.447507
$980$	6.69997e7	0.00227395
$981$	0	0
$982$	−1.22265e9	−0.0412013
$983$	5.89365e10	1.97900	0.989502	−	0.144519i	$-0.0461635\pi$
$983$	5.89365e10	1.97900	0.989502	+	0.144519i	$0.0461635\pi$
$984$	0	0
$985$	2.07883e10	0.693094
$986$	−2.75019e10	−0.913679
$987$	0	0
$988$	5.26802e8	0.0173779
$989$	4.99285e9	0.164120
$990$	0	0
$991$	4.22516e10	1.37907	0.689533	−	0.724254i	$-0.257816\pi$
$991$	4.22516e10	1.37907	0.689533	+	0.724254i	$0.257816\pi$
$992$	1.51875e9	0.0493964
$993$	0	0
$994$	4.80325e10	1.55125
$995$	1.11467e10	0.358729
$996$	0	0
$997$	−4.76627e10	−1.52316	−0.761581	−	0.648070i	$-0.775576\pi$
$997$	−4.76627e10	−1.52316	−0.761581	+	0.648070i	$0.775576\pi$
$998$	6.28527e9	0.200155
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.8.a.f.1.3		8
3.2	odd	2		23.8.a.b.1.6	✓	8
12.11	even	2		368.8.a.h.1.3		8
15.14	odd	2		575.8.a.b.1.3		8
69.68	even	2		529.8.a.c.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.a.b.1.6	✓	8	3.2	odd	2
207.8.a.f.1.3		8	1.1	even	1	trivial
368.8.a.h.1.3		8	12.11	even	2
529.8.a.c.1.6		8	69.68	even	2
575.8.a.b.1.3		8	15.14	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-11.0962 q^{2} -4.87416 q^{4} -165.526 q^{5} +952.148 q^{7} +1474.40 q^{8} +O(q^{10})\)	\(q-11.0962 q^{2} -4.87416 q^{4} -165.526 q^{5} +952.148 q^{7} +1474.40 q^{8} +1836.71 q^{10} +4863.13 q^{11} +12899.5 q^{13} -10565.2 q^{14} -15736.4 q^{16} +18595.8 q^{17} -8378.65 q^{19} +806.801 q^{20} -53962.3 q^{22} +12167.0 q^{23} -50726.1 q^{25} -143136. q^{26} -4640.92 q^{28} +133283. q^{29} -107642. q^{31} -14109.3 q^{32} -206343. q^{34} -157606. q^{35} +422577. q^{37} +92971.3 q^{38} -244052. q^{40} +85366.0 q^{41} +410360. q^{43} -23703.7 q^{44} -135008. q^{46} -1.11839e6 q^{47} +83043.7 q^{49} +562867. q^{50} -62874.3 q^{52} -275790. q^{53} -804976. q^{55} +1.40385e6 q^{56} -1.47893e6 q^{58} +182891. q^{59} +2.49804e6 q^{61} +1.19442e6 q^{62} +2.17081e6 q^{64} -2.13521e6 q^{65} -705106. q^{67} -90638.9 q^{68} +1.74882e6 q^{70} -4.54627e6 q^{71} -4.46832e6 q^{73} -4.68901e6 q^{74} +40838.9 q^{76} +4.63042e6 q^{77} +4.92889e6 q^{79} +2.60478e6 q^{80} -947239. q^{82} -369529. q^{83} -3.07809e6 q^{85} -4.55344e6 q^{86} +7.17020e6 q^{88} -2.70162e6 q^{89} +1.22823e7 q^{91} -59303.9 q^{92} +1.24099e7 q^{94} +1.38689e6 q^{95} +8.20102e6 q^{97} -921470. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 640 q^{4} - 444 q^{5} + 1446 q^{7} - 3177 q^{8}+O(q^{10}) \) 8 * q + 640 * q^4 - 444 * q^5 + 1446 * q^7 - 3177 * q^8	\( 8 q + 640 q^{4} - 444 q^{5} + 1446 q^{7} - 3177 q^{8} + 19502 q^{10} - 7588 q^{11} + 19862 q^{13} - 17544 q^{14} + 64336 q^{16} - 42070 q^{17} + 1050 q^{19} - 3364 q^{20} - 128220 q^{22} + 97336 q^{23} + 49496 q^{25} + 371761 q^{26} + 143050 q^{28} + 102578 q^{29} + 304172 q^{31} + 612824 q^{32} - 524530 q^{34} - 531048 q^{35} + 286472 q^{37} + 762932 q^{38} + 2105286 q^{40} - 1324414 q^{41} + 2052578 q^{43} + 867298 q^{44} - 675556 q^{47} - 55404 q^{49} - 1458528 q^{50} - 1695409 q^{52} - 203654 q^{53} - 1024444 q^{55} + 5766846 q^{56} - 5039991 q^{58} + 748892 q^{59} + 61822 q^{61} + 4939277 q^{62} + 2702267 q^{64} + 1571618 q^{65} + 3235604 q^{67} - 4914980 q^{68} + 10871764 q^{70} + 4951664 q^{71} + 11019370 q^{73} - 356954 q^{74} + 21973240 q^{76} + 5284888 q^{77} + 4202464 q^{79} - 8785886 q^{80} + 32636759 q^{82} - 518568 q^{83} + 9854220 q^{85} + 14681386 q^{86} + 20589740 q^{88} - 4203864 q^{89} + 2488406 q^{91} + 7786880 q^{92} + 12314327 q^{94} + 44485300 q^{95} + 18621134 q^{97} - 35756 q^{98}+O(q^{100}) \) 8 * q + 640 * q^4 - 444 * q^5 + 1446 * q^7 - 3177 * q^8 + 19502 * q^10 - 7588 * q^11 + 19862 * q^13 - 17544 * q^14 + 64336 * q^16 - 42070 * q^17 + 1050 * q^19 - 3364 * q^20 - 128220 * q^22 + 97336 * q^23 + 49496 * q^25 + 371761 * q^26 + 143050 * q^28 + 102578 * q^29 + 304172 * q^31 + 612824 * q^32 - 524530 * q^34 - 531048 * q^35 + 286472 * q^37 + 762932 * q^38 + 2105286 * q^40 - 1324414 * q^41 + 2052578 * q^43 + 867298 * q^44 - 675556 * q^47 - 55404 * q^49 - 1458528 * q^50 - 1695409 * q^52 - 203654 * q^53 - 1024444 * q^55 + 5766846 * q^56 - 5039991 * q^58 + 748892 * q^59 + 61822 * q^61 + 4939277 * q^62 + 2702267 * q^64 + 1571618 * q^65 + 3235604 * q^67 - 4914980 * q^68 + 10871764 * q^70 + 4951664 * q^71 + 11019370 * q^73 - 356954 * q^74 + 21973240 * q^76 + 5284888 * q^77 + 4202464 * q^79 - 8785886 * q^80 + 32636759 * q^82 - 518568 * q^83 + 9854220 * q^85 + 14681386 * q^86 + 20589740 * q^88 - 4203864 * q^89 + 2488406 * q^91 + 7786880 * q^92 + 12314327 * q^94 + 44485300 * q^95 + 18621134 * q^97 - 35756 * q^98

Introduction

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