LMFDB - Embedded newform 625.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(1,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.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:	$4.99065012633$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -0.381966 q^{3} -1.61803 q^{4} -0.236068 q^{6} +2.85410 q^{7} -2.23607 q^{8} -2.85410 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} -0.381966 q^{3} -1.61803 q^{4} -0.236068 q^{6} +2.85410 q^{7} -2.23607 q^{8} -2.85410 q^{9} -1.61803 q^{11} +0.618034 q^{12} -3.47214 q^{13} +1.76393 q^{14} +1.85410 q^{16} -7.47214 q^{17} -1.76393 q^{18} +5.85410 q^{19} -1.09017 q^{21} -1.00000 q^{22} -4.85410 q^{23} +0.854102 q^{24} -2.14590 q^{26} +2.23607 q^{27} -4.61803 q^{28} -7.23607 q^{29} +2.00000 q^{31} +5.61803 q^{32} +0.618034 q^{33} -4.61803 q^{34} +4.61803 q^{36} -3.00000 q^{37} +3.61803 q^{38} +1.32624 q^{39} -2.47214 q^{41} -0.673762 q^{42} -8.47214 q^{43} +2.61803 q^{44} -3.00000 q^{46} -9.38197 q^{47} -0.708204 q^{48} +1.14590 q^{49} +2.85410 q^{51} +5.61803 q^{52} +0.472136 q^{53} +1.38197 q^{54} -6.38197 q^{56} -2.23607 q^{57} -4.47214 q^{58} +3.94427 q^{59} -2.14590 q^{61} +1.23607 q^{62} -8.14590 q^{63} -0.236068 q^{64} +0.381966 q^{66} +2.52786 q^{67} +12.0902 q^{68} +1.85410 q^{69} +12.3262 q^{71} +6.38197 q^{72} +6.85410 q^{73} -1.85410 q^{74} -9.47214 q^{76} -4.61803 q^{77} +0.819660 q^{78} +6.70820 q^{79} +7.70820 q^{81} -1.52786 q^{82} +4.94427 q^{83} +1.76393 q^{84} -5.23607 q^{86} +2.76393 q^{87} +3.61803 q^{88} +1.38197 q^{89} -9.90983 q^{91} +7.85410 q^{92} -0.763932 q^{93} -5.79837 q^{94} -2.14590 q^{96} +8.38197 q^{97} +0.708204 q^{98} +4.61803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 3 q^{3} - q^{4} + 4 q^{6} - q^{7} + q^{9}+O(q^{10}) $ 2 * q - q^2 - 3 * q^3 - q^4 + 4 * q^6 - q^7 + q^9	$ 2 q - q^{2} - 3 q^{3} - q^{4} + 4 q^{6} - q^{7} + q^{9} - q^{11} - q^{12} + 2 q^{13} + 8 q^{14} - 3 q^{16} - 6 q^{17} - 8 q^{18} + 5 q^{19} + 9 q^{21} - 2 q^{22} - 3 q^{23} - 5 q^{24} - 11 q^{26} - 7 q^{28} - 10 q^{29} + 4 q^{31} + 9 q^{32} - q^{33} - 7 q^{34} + 7 q^{36} - 6 q^{37} + 5 q^{38} - 13 q^{39} + 4 q^{41} - 17 q^{42} - 8 q^{43} + 3 q^{44} - 6 q^{46} - 21 q^{47} + 12 q^{48} + 9 q^{49} - q^{51} + 9 q^{52} - 8 q^{53} + 5 q^{54} - 15 q^{56} - 10 q^{59} - 11 q^{61} - 2 q^{62} - 23 q^{63} + 4 q^{64} + 3 q^{66} + 14 q^{67} + 13 q^{68} - 3 q^{69} + 9 q^{71} + 15 q^{72} + 7 q^{73} + 3 q^{74} - 10 q^{76} - 7 q^{77} + 24 q^{78} + 2 q^{81} - 12 q^{82} - 8 q^{83} + 8 q^{84} - 6 q^{86} + 10 q^{87} + 5 q^{88} + 5 q^{89} - 31 q^{91} + 9 q^{92} - 6 q^{93} + 13 q^{94} - 11 q^{96} + 19 q^{97} - 12 q^{98} + 7 q^{99}+O(q^{100}) $ 2 * q - q^2 - 3 * q^3 - q^4 + 4 * q^6 - q^7 + q^9 - q^11 - q^12 + 2 * q^13 + 8 * q^14 - 3 * q^16 - 6 * q^17 - 8 * q^18 + 5 * q^19 + 9 * q^21 - 2 * q^22 - 3 * q^23 - 5 * q^24 - 11 * q^26 - 7 * q^28 - 10 * q^29 + 4 * q^31 + 9 * q^32 - q^33 - 7 * q^34 + 7 * q^36 - 6 * q^37 + 5 * q^38 - 13 * q^39 + 4 * q^41 - 17 * q^42 - 8 * q^43 + 3 * q^44 - 6 * q^46 - 21 * q^47 + 12 * q^48 + 9 * q^49 - q^51 + 9 * q^52 - 8 * q^53 + 5 * q^54 - 15 * q^56 - 10 * q^59 - 11 * q^61 - 2 * q^62 - 23 * q^63 + 4 * q^64 + 3 * q^66 + 14 * q^67 + 13 * q^68 - 3 * q^69 + 9 * q^71 + 15 * q^72 + 7 * q^73 + 3 * q^74 - 10 * q^76 - 7 * q^77 + 24 * q^78 + 2 * q^81 - 12 * q^82 - 8 * q^83 + 8 * q^84 - 6 * q^86 + 10 * q^87 + 5 * q^88 + 5 * q^89 - 31 * q^91 + 9 * q^92 - 6 * q^93 + 13 * q^94 - 11 * q^96 + 19 * q^97 - 12 * q^98 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	−0.381966	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$3$	−0.381966	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	−0.236068	−0.0963743
$7$	2.85410	1.07875	0.539375	−	0.842066i	$-0.318661\pi$
$7$	2.85410	1.07875	0.539375	+	0.842066i	$0.318661\pi$
$8$	−2.23607	−0.790569
$9$	−2.85410	−0.951367
$10$	0	0
$11$	−1.61803	−0.487856	−0.243928	−	0.969793i	$-0.578436\pi$
$11$	−1.61803	−0.487856	−0.243928	+	0.969793i	$0.578436\pi$
$12$	0.618034	0.178411
$13$	−3.47214	−0.962997	−0.481499	−	0.876447i	$-0.659907\pi$
$13$	−3.47214	−0.962997	−0.481499	+	0.876447i	$0.659907\pi$
$14$	1.76393	0.471431
$15$	0	0
$16$	1.85410	0.463525
$17$	−7.47214	−1.81226	−0.906130	−	0.423000i	$-0.860977\pi$
$17$	−7.47214	−1.81226	−0.906130	+	0.423000i	$0.860977\pi$
$18$	−1.76393	−0.415763
$19$	5.85410	1.34302	0.671512	−	0.740994i	$-0.265645\pi$
$19$	5.85410	1.34302	0.671512	+	0.740994i	$0.265645\pi$
$20$	0	0
$21$	−1.09017	−0.237895
$22$	−1.00000	−0.213201
$23$	−4.85410	−1.01215	−0.506075	−	0.862489i	$-0.668904\pi$
$23$	−4.85410	−1.01215	−0.506075	+	0.862489i	$0.668904\pi$
$24$	0.854102	0.174343
$25$	0	0
$26$	−2.14590	−0.420845
$27$	2.23607	0.430331
$28$	−4.61803	−0.872726
$29$	−7.23607	−1.34370	−0.671852	−	0.740685i	$-0.734501\pi$
$29$	−7.23607	−1.34370	−0.671852	+	0.740685i	$0.734501\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	5.61803	0.993137
$33$	0.618034	0.107586
$34$	−4.61803	−0.791986
$35$	0	0
$36$	4.61803	0.769672
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	3.61803	0.586923
$39$	1.32624	0.212368
$40$	0	0
$41$	−2.47214	−0.386083	−0.193041	−	0.981191i	$-0.561835\pi$
$41$	−2.47214	−0.386083	−0.193041	+	0.981191i	$0.561835\pi$
$42$	−0.673762	−0.103964
$43$	−8.47214	−1.29199	−0.645994	−	0.763342i	$-0.723557\pi$
$43$	−8.47214	−1.29199	−0.645994	+	0.763342i	$0.723557\pi$
$44$	2.61803	0.394683
$45$	0	0
$46$	−3.00000	−0.442326
$47$	−9.38197	−1.36850	−0.684250	−	0.729247i	$-0.739870\pi$
$47$	−9.38197	−1.36850	−0.684250	+	0.729247i	$0.739870\pi$
$48$	−0.708204	−0.102220
$49$	1.14590	0.163700
$50$	0	0
$51$	2.85410	0.399654
$52$	5.61803	0.779081
$53$	0.472136	0.0648529	0.0324264	−	0.999474i	$-0.489677\pi$
$53$	0.472136	0.0648529	0.0324264	+	0.999474i	$0.489677\pi$
$54$	1.38197	0.188062
$55$	0	0
$56$	−6.38197	−0.852826
$57$	−2.23607	−0.296174
$58$	−4.47214	−0.587220
$59$	3.94427	0.513500	0.256750	−	0.966478i	$-0.417348\pi$
$59$	3.94427	0.513500	0.256750	+	0.966478i	$0.417348\pi$
$60$	0	0
$61$	−2.14590	−0.274754	−0.137377	−	0.990519i	$-0.543867\pi$
$61$	−2.14590	−0.274754	−0.137377	+	0.990519i	$0.543867\pi$
$62$	1.23607	0.156981
$63$	−8.14590	−1.02629
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0.381966	0.0470168
$67$	2.52786	0.308828	0.154414	−	0.988006i	$-0.450651\pi$
$67$	2.52786	0.308828	0.154414	+	0.988006i	$0.450651\pi$
$68$	12.0902	1.46615
$69$	1.85410	0.223208
$70$	0	0
$71$	12.3262	1.46286	0.731428	−	0.681919i	$-0.238854\pi$
$71$	12.3262	1.46286	0.731428	+	0.681919i	$0.238854\pi$
$72$	6.38197	0.752122
$73$	6.85410	0.802212	0.401106	−	0.916032i	$-0.368626\pi$
$73$	6.85410	0.802212	0.401106	+	0.916032i	$0.368626\pi$
$74$	−1.85410	−0.215535
$75$	0	0
$76$	−9.47214	−1.08653
$77$	−4.61803	−0.526274
$78$	0.819660	0.0928082
$79$	6.70820	0.754732	0.377366	−	0.926064i	$-0.376830\pi$
$79$	6.70820	0.754732	0.377366	+	0.926064i	$0.376830\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	−1.52786	−0.168724
$83$	4.94427	0.542704	0.271352	−	0.962480i	$-0.412529\pi$
$83$	4.94427	0.542704	0.271352	+	0.962480i	$0.412529\pi$
$84$	1.76393	0.192461
$85$	0	0
$86$	−5.23607	−0.564620
$87$	2.76393	0.296325
$88$	3.61803	0.385684
$89$	1.38197	0.146488	0.0732441	−	0.997314i	$-0.476665\pi$
$89$	1.38197	0.146488	0.0732441	+	0.997314i	$0.476665\pi$
$90$	0	0
$91$	−9.90983	−1.03883
$92$	7.85410	0.818847
$93$	−0.763932	−0.0792161
$94$	−5.79837	−0.598057
$95$	0	0
$96$	−2.14590	−0.219015
$97$	8.38197	0.851060	0.425530	−	0.904944i	$-0.360088\pi$
$97$	8.38197	0.851060	0.425530	+	0.904944i	$0.360088\pi$
$98$	0.708204	0.0715394
$99$	4.61803	0.464130
$100$	0	0
$101$	14.2361	1.41654	0.708271	−	0.705941i	$-0.249476\pi$
$101$	14.2361	1.41654	0.708271	+	0.705941i	$0.249476\pi$
$102$	1.76393	0.174655
$103$	11.8541	1.16802	0.584010	−	0.811747i	$-0.301483\pi$
$103$	11.8541	1.16802	0.584010	+	0.811747i	$0.301483\pi$
$104$	7.76393	0.761316
$105$	0	0
$106$	0.291796	0.0283417
$107$	−12.7984	−1.23727	−0.618633	−	0.785680i	$-0.712313\pi$
$107$	−12.7984	−1.23727	−0.618633	+	0.785680i	$0.712313\pi$
$108$	−3.61803	−0.348145
$109$	9.14590	0.876018	0.438009	−	0.898971i	$-0.355684\pi$
$109$	9.14590	0.876018	0.438009	+	0.898971i	$0.355684\pi$
$110$	0	0
$111$	1.14590	0.108764
$112$	5.29180	0.500028
$113$	12.7082	1.19549	0.597744	−	0.801687i	$-0.296064\pi$
$113$	12.7082	1.19549	0.597744	+	0.801687i	$0.296064\pi$
$114$	−1.38197	−0.129433
$115$	0	0
$116$	11.7082	1.08708
$117$	9.90983	0.916164
$118$	2.43769	0.224408
$119$	−21.3262	−1.95497
$120$	0	0
$121$	−8.38197	−0.761997
$122$	−1.32624	−0.120072
$123$	0.944272	0.0851421
$124$	−3.23607	−0.290607
$125$	0	0
$126$	−5.03444	−0.448504
$127$	−13.8541	−1.22935	−0.614676	−	0.788779i	$-0.710713\pi$
$127$	−13.8541	−1.22935	−0.614676	+	0.788779i	$0.710713\pi$
$128$	−11.3820	−1.00603
$129$	3.23607	0.284920
$130$	0	0
$131$	−5.23607	−0.457477	−0.228739	−	0.973488i	$-0.573460\pi$
$131$	−5.23607	−0.457477	−0.228739	+	0.973488i	$0.573460\pi$
$132$	−1.00000	−0.0870388
$133$	16.7082	1.44879
$134$	1.56231	0.134963
$135$	0	0
$136$	16.7082	1.43272
$137$	−15.7639	−1.34680	−0.673402	−	0.739277i	$-0.735168\pi$
$137$	−15.7639	−1.34680	−0.673402	+	0.739277i	$0.735168\pi$
$138$	1.14590	0.0975453
$139$	−18.4164	−1.56206	−0.781030	−	0.624494i	$-0.785305\pi$
$139$	−18.4164	−1.56206	−0.781030	+	0.624494i	$0.785305\pi$
$140$	0	0
$141$	3.58359	0.301793
$142$	7.61803	0.639291
$143$	5.61803	0.469804
$144$	−5.29180	−0.440983
$145$	0	0
$146$	4.23607	0.350579
$147$	−0.437694	−0.0361004
$148$	4.85410	0.399005
$149$	−11.3820	−0.932447	−0.466223	−	0.884667i	$-0.654386\pi$
$149$	−11.3820	−0.932447	−0.466223	+	0.884667i	$0.654386\pi$
$150$	0	0
$151$	−1.94427	−0.158223	−0.0791113	−	0.996866i	$-0.525208\pi$
$151$	−1.94427	−0.158223	−0.0791113	+	0.996866i	$0.525208\pi$
$152$	−13.0902	−1.06175
$153$	21.3262	1.72412
$154$	−2.85410	−0.229990
$155$	0	0
$156$	−2.14590	−0.171809
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	4.14590	0.329830
$159$	−0.180340	−0.0143019
$160$	0	0
$161$	−13.8541	−1.09186
$162$	4.76393	0.374290
$163$	−4.85410	−0.380203	−0.190101	−	0.981764i	$-0.560882\pi$
$163$	−4.85410	−0.380203	−0.190101	+	0.981764i	$0.560882\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	3.05573	0.237170
$167$	−8.52786	−0.659906	−0.329953	−	0.943997i	$-0.607033\pi$
$167$	−8.52786	−0.659906	−0.329953	+	0.943997i	$0.607033\pi$
$168$	2.43769	0.188072
$169$	−0.944272	−0.0726363
$170$	0	0
$171$	−16.7082	−1.27771
$172$	13.7082	1.04524
$173$	−2.29180	−0.174242	−0.0871210	−	0.996198i	$-0.527767\pi$
$173$	−2.29180	−0.174242	−0.0871210	+	0.996198i	$0.527767\pi$
$174$	1.70820	0.129499
$175$	0	0
$176$	−3.00000	−0.226134
$177$	−1.50658	−0.113241
$178$	0.854102	0.0640176
$179$	20.1246	1.50418	0.752092	−	0.659058i	$-0.229045\pi$
$179$	20.1246	1.50418	0.752092	+	0.659058i	$0.229045\pi$
$180$	0	0
$181$	2.32624	0.172908	0.0864540	−	0.996256i	$-0.472446\pi$
$181$	2.32624	0.172908	0.0864540	+	0.996256i	$0.472446\pi$
$182$	−6.12461	−0.453986
$183$	0.819660	0.0605910
$184$	10.8541	0.800175
$185$	0	0
$186$	−0.472136	−0.0346187
$187$	12.0902	0.884121
$188$	15.1803	1.10714
$189$	6.38197	0.464220
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0.0901699	0.00650746
$193$	−5.05573	−0.363919	−0.181960	−	0.983306i	$-0.558244\pi$
$193$	−5.05573	−0.363919	−0.181960	+	0.983306i	$0.558244\pi$
$194$	5.18034	0.371927
$195$	0	0
$196$	−1.85410	−0.132436
$197$	7.52786	0.536338	0.268169	−	0.963372i	$-0.413581\pi$
$197$	7.52786	0.536338	0.268169	+	0.963372i	$0.413581\pi$
$198$	2.85410	0.202832
$199$	−4.14590	−0.293895	−0.146947	−	0.989144i	$-0.546945\pi$
$199$	−4.14590	−0.293895	−0.146947	+	0.989144i	$0.546945\pi$
$200$	0	0
$201$	−0.965558	−0.0681052
$202$	8.79837	0.619051
$203$	−20.6525	−1.44952
$204$	−4.61803	−0.323327
$205$	0	0
$206$	7.32624	0.510443
$207$	13.8541	0.962927
$208$	−6.43769	−0.446374
$209$	−9.47214	−0.655201
$210$	0	0
$211$	−9.18034	−0.632001	−0.316000	−	0.948759i	$-0.602340\pi$
$211$	−9.18034	−0.632001	−0.316000	+	0.948759i	$0.602340\pi$
$212$	−0.763932	−0.0524671
$213$	−4.70820	−0.322601
$214$	−7.90983	−0.540705
$215$	0	0
$216$	−5.00000	−0.340207
$217$	5.70820	0.387498
$218$	5.65248	0.382834
$219$	−2.61803	−0.176910
$220$	0	0
$221$	25.9443	1.74520
$222$	0.708204	0.0475315
$223$	6.85410	0.458985	0.229492	−	0.973310i	$-0.426293\pi$
$223$	6.85410	0.458985	0.229492	+	0.973310i	$0.426293\pi$
$224$	16.0344	1.07135
$225$	0	0
$226$	7.85410	0.522447
$227$	−5.23607	−0.347530	−0.173765	−	0.984787i	$-0.555593\pi$
$227$	−5.23607	−0.347530	−0.173765	+	0.984787i	$0.555593\pi$
$228$	3.61803	0.239610
$229$	−16.7082	−1.10411	−0.552055	−	0.833808i	$-0.686156\pi$
$229$	−16.7082	−1.10411	−0.552055	+	0.833808i	$0.686156\pi$
$230$	0	0
$231$	1.76393	0.116058
$232$	16.1803	1.06229
$233$	−10.9098	−0.714727	−0.357363	−	0.933965i	$-0.616324\pi$
$233$	−10.9098	−0.714727	−0.357363	+	0.933965i	$0.616324\pi$
$234$	6.12461	0.400378
$235$	0	0
$236$	−6.38197	−0.415431
$237$	−2.56231	−0.166440
$238$	−13.1803	−0.854355
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−7.14590	−0.460308	−0.230154	−	0.973154i	$-0.573923\pi$
$241$	−7.14590	−0.460308	−0.230154	+	0.973154i	$0.573923\pi$
$242$	−5.18034	−0.333005
$243$	−9.65248	−0.619207
$244$	3.47214	0.222281
$245$	0	0
$246$	0.583592	0.0372085
$247$	−20.3262	−1.29333
$248$	−4.47214	−0.283981
$249$	−1.88854	−0.119682
$250$	0	0
$251$	13.1803	0.831936	0.415968	−	0.909379i	$-0.363443\pi$
$251$	13.1803	0.831936	0.415968	+	0.909379i	$0.363443\pi$
$252$	13.1803	0.830283
$253$	7.85410	0.493783
$254$	−8.56231	−0.537247
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−9.70820	−0.605581	−0.302791	−	0.953057i	$-0.597918\pi$
$257$	−9.70820	−0.605581	−0.302791	+	0.953057i	$0.597918\pi$
$258$	2.00000	0.124515
$259$	−8.56231	−0.532036
$260$	0	0
$261$	20.6525	1.27836
$262$	−3.23607	−0.199925
$263$	−3.67376	−0.226534	−0.113267	−	0.993565i	$-0.536132\pi$
$263$	−3.67376	−0.226534	−0.113267	+	0.993565i	$0.536132\pi$
$264$	−1.38197	−0.0850541
$265$	0	0
$266$	10.3262	0.633142
$267$	−0.527864	−0.0323048
$268$	−4.09017	−0.249847
$269$	26.1803	1.59624	0.798122	−	0.602496i	$-0.205827\pi$
$269$	26.1803	1.59624	0.798122	+	0.602496i	$0.205827\pi$
$270$	0	0
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	−13.8541	−0.840028
$273$	3.78522	0.229092
$274$	−9.74265	−0.588575
$275$	0	0
$276$	−3.00000	−0.180579
$277$	28.7082	1.72491	0.862454	−	0.506135i	$-0.168926\pi$
$277$	28.7082	1.72491	0.862454	+	0.506135i	$0.168926\pi$
$278$	−11.3820	−0.682645
$279$	−5.70820	−0.341741
$280$	0	0
$281$	−25.3607	−1.51289	−0.756446	−	0.654057i	$-0.773066\pi$
$281$	−25.3607	−1.51289	−0.756446	+	0.654057i	$0.773066\pi$
$282$	2.21478	0.131888
$283$	−21.3607	−1.26976	−0.634880	−	0.772611i	$-0.718951\pi$
$283$	−21.3607	−1.26976	−0.634880	+	0.772611i	$0.718951\pi$
$284$	−19.9443	−1.18347
$285$	0	0
$286$	3.47214	0.205312
$287$	−7.05573	−0.416486
$288$	−16.0344	−0.944839
$289$	38.8328	2.28428
$290$	0	0
$291$	−3.20163	−0.187683
$292$	−11.0902	−0.649003
$293$	1.32624	0.0774796	0.0387398	−	0.999249i	$-0.487666\pi$
$293$	1.32624	0.0774796	0.0387398	+	0.999249i	$0.487666\pi$
$294$	−0.270510	−0.0157765
$295$	0	0
$296$	6.70820	0.389906
$297$	−3.61803	−0.209940
$298$	−7.03444	−0.407494
$299$	16.8541	0.974698
$300$	0	0
$301$	−24.1803	−1.39373
$302$	−1.20163	−0.0691458
$303$	−5.43769	−0.312387
$304$	10.8541	0.622525
$305$	0	0
$306$	13.1803	0.753470
$307$	32.9787	1.88219	0.941097	−	0.338136i	$-0.109796\pi$
$307$	32.9787	1.88219	0.941097	+	0.338136i	$0.109796\pi$
$308$	7.47214	0.425764
$309$	−4.52786	−0.257581
$310$	0	0
$311$	−5.56231	−0.315409	−0.157705	−	0.987486i	$-0.550409\pi$
$311$	−5.56231	−0.315409	−0.157705	+	0.987486i	$0.550409\pi$
$312$	−2.96556	−0.167892
$313$	8.03444	0.454134	0.227067	−	0.973879i	$-0.427086\pi$
$313$	8.03444	0.454134	0.227067	+	0.973879i	$0.427086\pi$
$314$	−8.03444	−0.453410
$315$	0	0
$316$	−10.8541	−0.610591
$317$	−17.7984	−0.999656	−0.499828	−	0.866125i	$-0.666604\pi$
$317$	−17.7984	−0.999656	−0.499828	+	0.866125i	$0.666604\pi$
$318$	−0.111456	−0.00625015
$319$	11.7082	0.655534
$320$	0	0
$321$	4.88854	0.272852
$322$	−8.56231	−0.477159
$323$	−43.7426	−2.43391
$324$	−12.4721	−0.692896
$325$	0	0
$326$	−3.00000	−0.166155
$327$	−3.49342	−0.193187
$328$	5.52786	0.305225
$329$	−26.7771	−1.47627
$330$	0	0
$331$	28.7082	1.57795	0.788973	−	0.614428i	$-0.210613\pi$
$331$	28.7082	1.57795	0.788973	+	0.614428i	$0.210613\pi$
$332$	−8.00000	−0.439057
$333$	8.56231	0.469211
$334$	−5.27051	−0.288389
$335$	0	0
$336$	−2.02129	−0.110270
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−0.583592	−0.0317432
$339$	−4.85410	−0.263639
$340$	0	0
$341$	−3.23607	−0.175243
$342$	−10.3262	−0.558379
$343$	−16.7082	−0.902158
$344$	18.9443	1.02141
$345$	0	0
$346$	−1.41641	−0.0761466
$347$	−10.2361	−0.549501	−0.274750	−	0.961516i	$-0.588595\pi$
$347$	−10.2361	−0.549501	−0.274750	+	0.961516i	$0.588595\pi$
$348$	−4.47214	−0.239732
$349$	20.1246	1.07725	0.538623	−	0.842547i	$-0.318945\pi$
$349$	20.1246	1.07725	0.538623	+	0.842547i	$0.318945\pi$
$350$	0	0
$351$	−7.76393	−0.414408
$352$	−9.09017	−0.484508
$353$	−24.6525	−1.31212	−0.656059	−	0.754709i	$-0.727778\pi$
$353$	−24.6525	−1.31212	−0.656059	+	0.754709i	$0.727778\pi$
$354$	−0.931116	−0.0494883
$355$	0	0
$356$	−2.23607	−0.118511
$357$	8.14590	0.431127
$358$	12.4377	0.657353
$359$	5.65248	0.298326	0.149163	−	0.988813i	$-0.452342\pi$
$359$	5.65248	0.298326	0.149163	+	0.988813i	$0.452342\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	1.43769	0.0755635
$363$	3.20163	0.168042
$364$	16.0344	0.840433
$365$	0	0
$366$	0.506578	0.0264792
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	−9.00000	−0.469157
$369$	7.05573	0.367307
$370$	0	0
$371$	1.34752	0.0699600
$372$	1.23607	0.0640871
$373$	11.8541	0.613782	0.306891	−	0.951745i	$-0.400711\pi$
$373$	11.8541	0.613782	0.306891	+	0.951745i	$0.400711\pi$
$374$	7.47214	0.386375
$375$	0	0
$376$	20.9787	1.08189
$377$	25.1246	1.29398
$378$	3.94427	0.202871
$379$	7.23607	0.371692	0.185846	−	0.982579i	$-0.440497\pi$
$379$	7.23607	0.371692	0.185846	+	0.982579i	$0.440497\pi$
$380$	0	0
$381$	5.29180	0.271107
$382$	−1.85410	−0.0948641
$383$	−27.6180	−1.41122	−0.705608	−	0.708603i	$-0.749326\pi$
$383$	−27.6180	−1.41122	−0.705608	+	0.708603i	$0.749326\pi$
$384$	4.34752	0.221859
$385$	0	0
$386$	−3.12461	−0.159039
$387$	24.1803	1.22916
$388$	−13.5623	−0.688522
$389$	−5.32624	−0.270051	−0.135025	−	0.990842i	$-0.543112\pi$
$389$	−5.32624	−0.270051	−0.135025	+	0.990842i	$0.543112\pi$
$390$	0	0
$391$	36.2705	1.83428
$392$	−2.56231	−0.129416
$393$	2.00000	0.100887
$394$	4.65248	0.234388
$395$	0	0
$396$	−7.47214	−0.375489
$397$	15.4164	0.773727	0.386864	−	0.922137i	$-0.373558\pi$
$397$	15.4164	0.773727	0.386864	+	0.922137i	$0.373558\pi$
$398$	−2.56231	−0.128437
$399$	−6.38197	−0.319498
$400$	0	0
$401$	−5.56231	−0.277768	−0.138884	−	0.990309i	$-0.544352\pi$
$401$	−5.56231	−0.277768	−0.138884	+	0.990309i	$0.544352\pi$
$402$	−0.596748	−0.0297631
$403$	−6.94427	−0.345919
$404$	−23.0344	−1.14601
$405$	0	0
$406$	−12.7639	−0.633463
$407$	4.85410	0.240609
$408$	−6.38197	−0.315954
$409$	−18.2918	−0.904471	−0.452236	−	0.891899i	$-0.649373\pi$
$409$	−18.2918	−0.904471	−0.452236	+	0.891899i	$0.649373\pi$
$410$	0	0
$411$	6.02129	0.297008
$412$	−19.1803	−0.944948
$413$	11.2574	0.553938
$414$	8.56231	0.420814
$415$	0	0
$416$	−19.5066	−0.956389
$417$	7.03444	0.344478
$418$	−5.85410	−0.286333
$419$	−27.7639	−1.35636	−0.678178	−	0.734897i	$-0.737230\pi$
$419$	−27.7639	−1.35636	−0.678178	+	0.734897i	$0.737230\pi$
$420$	0	0
$421$	−9.18034	−0.447422	−0.223711	−	0.974655i	$-0.571817\pi$
$421$	−9.18034	−0.447422	−0.223711	+	0.974655i	$0.571817\pi$
$422$	−5.67376	−0.276194
$423$	26.7771	1.30195
$424$	−1.05573	−0.0512707
$425$	0	0
$426$	−2.90983	−0.140982
$427$	−6.12461	−0.296391
$428$	20.7082	1.00097
$429$	−2.14590	−0.103605
$430$	0	0
$431$	−31.4164	−1.51328	−0.756638	−	0.653835i	$-0.773159\pi$
$431$	−31.4164	−1.51328	−0.756638	+	0.653835i	$0.773159\pi$
$432$	4.14590	0.199470
$433$	6.85410	0.329387	0.164694	−	0.986345i	$-0.447336\pi$
$433$	6.85410	0.329387	0.164694	+	0.986345i	$0.447336\pi$
$434$	3.52786	0.169343
$435$	0	0
$436$	−14.7984	−0.708714
$437$	−28.4164	−1.35934
$438$	−1.61803	−0.0773127
$439$	−24.2705	−1.15837	−0.579184	−	0.815197i	$-0.696629\pi$
$439$	−24.2705	−1.15837	−0.579184	+	0.815197i	$0.696629\pi$
$440$	0	0
$441$	−3.27051	−0.155739
$442$	16.0344	0.762681
$443$	14.2918	0.679024	0.339512	−	0.940602i	$-0.389738\pi$
$443$	14.2918	0.679024	0.339512	+	0.940602i	$0.389738\pi$
$444$	−1.85410	−0.0879918
$445$	0	0
$446$	4.23607	0.200584
$447$	4.34752	0.205631
$448$	−0.673762	−0.0318323
$449$	−29.0689	−1.37185	−0.685923	−	0.727674i	$-0.740601\pi$
$449$	−29.0689	−1.37185	−0.685923	+	0.727674i	$0.740601\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	−20.5623	−0.967170
$453$	0.742646	0.0348925
$454$	−3.23607	−0.151876
$455$	0	0
$456$	5.00000	0.234146
$457$	−0.888544	−0.0415643	−0.0207822	−	0.999784i	$-0.506616\pi$
$457$	−0.888544	−0.0415643	−0.0207822	+	0.999784i	$0.506616\pi$
$458$	−10.3262	−0.482513
$459$	−16.7082	−0.779872
$460$	0	0
$461$	−26.0902	−1.21514	−0.607570	−	0.794266i	$-0.707856\pi$
$461$	−26.0902	−1.21514	−0.607570	+	0.794266i	$0.707856\pi$
$462$	1.09017	0.0507193
$463$	−19.3262	−0.898166	−0.449083	−	0.893490i	$-0.648249\pi$
$463$	−19.3262	−0.898166	−0.449083	+	0.893490i	$0.648249\pi$
$464$	−13.4164	−0.622841
$465$	0	0
$466$	−6.74265	−0.312347
$467$	12.3262	0.570390	0.285195	−	0.958469i	$-0.407942\pi$
$467$	12.3262	0.570390	0.285195	+	0.958469i	$0.407942\pi$
$468$	−16.0344	−0.741192
$469$	7.21478	0.333148
$470$	0	0
$471$	4.96556	0.228801
$472$	−8.81966	−0.405958
$473$	13.7082	0.630304
$474$	−1.58359	−0.0727368
$475$	0	0
$476$	34.5066	1.58161
$477$	−1.34752	−0.0616989
$478$	0	0
$479$	27.2361	1.24445	0.622224	−	0.782839i	$-0.286229\pi$
$479$	27.2361	1.24445	0.622224	+	0.782839i	$0.286229\pi$
$480$	0	0
$481$	10.4164	0.474947
$482$	−4.41641	−0.201162
$483$	5.29180	0.240785
$484$	13.5623	0.616468
$485$	0	0
$486$	−5.96556	−0.270603
$487$	39.3607	1.78360	0.891801	−	0.452427i	$-0.149442\pi$
$487$	39.3607	1.78360	0.891801	+	0.452427i	$0.149442\pi$
$488$	4.79837	0.217212
$489$	1.85410	0.0838454
$490$	0	0
$491$	13.0557	0.589197	0.294598	−	0.955621i	$-0.404814\pi$
$491$	13.0557	0.589197	0.294598	+	0.955621i	$0.404814\pi$
$492$	−1.52786	−0.0688814
$493$	54.0689	2.43514
$494$	−12.5623	−0.565205
$495$	0	0
$496$	3.70820	0.166503
$497$	35.1803	1.57805
$498$	−1.16718	−0.0523028
$499$	−1.58359	−0.0708913	−0.0354457	−	0.999372i	$-0.511285\pi$
$499$	−1.58359	−0.0708913	−0.0354457	+	0.999372i	$0.511285\pi$
$500$	0	0
$501$	3.25735	0.145528
$502$	8.14590	0.363569
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	18.2148	0.811351
$505$	0	0
$506$	4.85410	0.215791
$507$	0.360680	0.0160184
$508$	22.4164	0.994567
$509$	33.6180	1.49009	0.745047	−	0.667012i	$-0.232427\pi$
$509$	33.6180	1.49009	0.745047	+	0.667012i	$0.232427\pi$
$510$	0	0
$511$	19.5623	0.865385
$512$	18.7082	0.826794
$513$	13.0902	0.577945
$514$	−6.00000	−0.264649
$515$	0	0
$516$	−5.23607	−0.230505
$517$	15.1803	0.667631
$518$	−5.29180	−0.232508
$519$	0.875388	0.0384253
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	12.7639	0.558662
$523$	−19.1246	−0.836261	−0.418130	−	0.908387i	$-0.637315\pi$
$523$	−19.1246	−0.836261	−0.418130	+	0.908387i	$0.637315\pi$
$524$	8.47214	0.370107
$525$	0	0
$526$	−2.27051	−0.0989989
$527$	−14.9443	−0.650983
$528$	1.14590	0.0498688
$529$	0.562306	0.0244481
$530$	0	0
$531$	−11.2574	−0.488528
$532$	−27.0344	−1.17209
$533$	8.58359	0.371797
$534$	−0.326238	−0.0141177
$535$	0	0
$536$	−5.65248	−0.244150
$537$	−7.68692	−0.331715
$538$	16.1803	0.697584
$539$	−1.85410	−0.0798618
$540$	0	0
$541$	10.2918	0.442479	0.221239	−	0.975220i	$-0.428990\pi$
$541$	10.2918	0.442479	0.221239	+	0.975220i	$0.428990\pi$
$542$	−8.03444	−0.345109
$543$	−0.888544	−0.0381311
$544$	−41.9787	−1.79982
$545$	0	0
$546$	2.33939	0.100117
$547$	18.7082	0.799905	0.399953	−	0.916536i	$-0.369027\pi$
$547$	18.7082	0.799905	0.399953	+	0.916536i	$0.369027\pi$
$548$	25.5066	1.08959
$549$	6.12461	0.261392
$550$	0	0
$551$	−42.3607	−1.80463
$552$	−4.14590	−0.176461
$553$	19.1459	0.814166
$554$	17.7426	0.753813
$555$	0	0
$556$	29.7984	1.26373
$557$	−16.6180	−0.704129	−0.352064	−	0.935976i	$-0.614520\pi$
$557$	−16.6180	−0.704129	−0.352064	+	0.935976i	$0.614520\pi$
$558$	−3.52786	−0.149346
$559$	29.4164	1.24418
$560$	0	0
$561$	−4.61803	−0.194974
$562$	−15.6738	−0.661158
$563$	17.8328	0.751564	0.375782	−	0.926708i	$-0.377374\pi$
$563$	17.8328	0.751564	0.375782	+	0.926708i	$0.377374\pi$
$564$	−5.79837	−0.244156
$565$	0	0
$566$	−13.2016	−0.554906
$567$	22.0000	0.923913
$568$	−27.5623	−1.15649
$569$	−7.76393	−0.325481	−0.162740	−	0.986669i	$-0.552033\pi$
$569$	−7.76393	−0.325481	−0.162740	+	0.986669i	$0.552033\pi$
$570$	0	0
$571$	28.7082	1.20140	0.600700	−	0.799474i	$-0.294888\pi$
$571$	28.7082	1.20140	0.600700	+	0.799474i	$0.294888\pi$
$572$	−9.09017	−0.380079
$573$	1.14590	0.0478706
$574$	−4.36068	−0.182011
$575$	0	0
$576$	0.673762	0.0280734
$577$	−32.5967	−1.35702	−0.678510	−	0.734591i	$-0.737374\pi$
$577$	−32.5967	−1.35702	−0.678510	+	0.734591i	$0.737374\pi$
$578$	24.0000	0.998268
$579$	1.93112	0.0802545
$580$	0	0
$581$	14.1115	0.585442
$582$	−1.97871	−0.0820203
$583$	−0.763932	−0.0316388
$584$	−15.3262	−0.634204
$585$	0	0
$586$	0.819660	0.0338598
$587$	7.85410	0.324173	0.162087	−	0.986777i	$-0.448178\pi$
$587$	7.85410	0.324173	0.162087	+	0.986777i	$0.448178\pi$
$588$	0.708204	0.0292058
$589$	11.7082	0.482428
$590$	0	0
$591$	−2.87539	−0.118278
$592$	−5.56231	−0.228609
$593$	21.0000	0.862367	0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000	0.862367	0.431183	+	0.902264i	$0.358096\pi$
$594$	−2.23607	−0.0917470
$595$	0	0
$596$	18.4164	0.754365
$597$	1.58359	0.0648121
$598$	10.4164	0.425959
$599$	2.76393	0.112931	0.0564656	−	0.998405i	$-0.482017\pi$
$599$	2.76393	0.112931	0.0564656	+	0.998405i	$0.482017\pi$
$600$	0	0
$601$	−1.81966	−0.0742255	−0.0371127	−	0.999311i	$-0.511816\pi$
$601$	−1.81966	−0.0742255	−0.0371127	+	0.999311i	$0.511816\pi$
$602$	−14.9443	−0.609083
$603$	−7.21478	−0.293809
$604$	3.14590	0.128005
$605$	0	0
$606$	−3.36068	−0.136518
$607$	−31.4164	−1.27515	−0.637576	−	0.770387i	$-0.720063\pi$
$607$	−31.4164	−1.27515	−0.637576	+	0.770387i	$0.720063\pi$
$608$	32.8885	1.33381
$609$	7.88854	0.319660
$610$	0	0
$611$	32.5755	1.31786
$612$	−34.5066	−1.39485
$613$	14.2918	0.577240	0.288620	−	0.957444i	$-0.406804\pi$
$613$	14.2918	0.577240	0.288620	+	0.957444i	$0.406804\pi$
$614$	20.3820	0.822549
$615$	0	0
$616$	10.3262	0.416056
$617$	26.6738	1.07385	0.536923	−	0.843631i	$-0.319587\pi$
$617$	26.6738	1.07385	0.536923	+	0.843631i	$0.319587\pi$
$618$	−2.79837	−0.112567
$619$	−36.3050	−1.45922	−0.729610	−	0.683864i	$-0.760298\pi$
$619$	−36.3050	−1.45922	−0.729610	+	0.683864i	$0.760298\pi$
$620$	0	0
$621$	−10.8541	−0.435560
$622$	−3.43769	−0.137839
$623$	3.94427	0.158024
$624$	2.45898	0.0984380
$625$	0	0
$626$	4.96556	0.198464
$627$	3.61803	0.144490
$628$	21.0344	0.839366
$629$	22.4164	0.893801
$630$	0	0
$631$	−38.5279	−1.53377	−0.766885	−	0.641785i	$-0.778194\pi$
$631$	−38.5279	−1.53377	−0.766885	+	0.641785i	$0.778194\pi$
$632$	−15.0000	−0.596668
$633$	3.50658	0.139374
$634$	−11.0000	−0.436866
$635$	0	0
$636$	0.291796	0.0115705
$637$	−3.97871	−0.157642
$638$	7.23607	0.286479
$639$	−35.1803	−1.39171
$640$	0	0
$641$	30.5410	1.20630	0.603149	−	0.797629i	$-0.293912\pi$
$641$	30.5410	1.20630	0.603149	+	0.797629i	$0.293912\pi$
$642$	3.02129	0.119241
$643$	13.7639	0.542796	0.271398	−	0.962467i	$-0.412514\pi$
$643$	13.7639	0.542796	0.271398	+	0.962467i	$0.412514\pi$
$644$	22.4164	0.883330
$645$	0	0
$646$	−27.0344	−1.06366
$647$	−32.5967	−1.28151	−0.640755	−	0.767745i	$-0.721379\pi$
$647$	−32.5967	−1.28151	−0.640755	+	0.767745i	$0.721379\pi$
$648$	−17.2361	−0.677097
$649$	−6.38197	−0.250514
$650$	0	0
$651$	−2.18034	−0.0854543
$652$	7.85410	0.307590
$653$	−7.74265	−0.302993	−0.151497	−	0.988458i	$-0.548409\pi$
$653$	−7.74265	−0.302993	−0.151497	+	0.988458i	$0.548409\pi$
$654$	−2.15905	−0.0844257
$655$	0	0
$656$	−4.58359	−0.178959
$657$	−19.5623	−0.763198
$658$	−16.5492	−0.645153
$659$	35.4508	1.38097	0.690485	−	0.723347i	$-0.257397\pi$
$659$	35.4508	1.38097	0.690485	+	0.723347i	$0.257397\pi$
$660$	0	0
$661$	33.7082	1.31110	0.655549	−	0.755153i	$-0.272437\pi$
$661$	33.7082	1.31110	0.655549	+	0.755153i	$0.272437\pi$
$662$	17.7426	0.689588
$663$	−9.90983	−0.384866
$664$	−11.0557	−0.429045
$665$	0	0
$666$	5.29180	0.205053
$667$	35.1246	1.36003
$668$	13.7984	0.533875
$669$	−2.61803	−0.101219
$670$	0	0
$671$	3.47214	0.134040
$672$	−6.12461	−0.236262
$673$	−12.4164	−0.478617	−0.239309	−	0.970944i	$-0.576921\pi$
$673$	−12.4164	−0.478617	−0.239309	+	0.970944i	$0.576921\pi$
$674$	1.23607	0.0476116
$675$	0	0
$676$	1.52786	0.0587640
$677$	23.5066	0.903431	0.451716	−	0.892162i	$-0.350812\pi$
$677$	23.5066	0.903431	0.451716	+	0.892162i	$0.350812\pi$
$678$	−3.00000	−0.115214
$679$	23.9230	0.918080
$680$	0	0
$681$	2.00000	0.0766402
$682$	−2.00000	−0.0765840
$683$	26.3262	1.00735	0.503673	−	0.863895i	$-0.331982\pi$
$683$	26.3262	1.00735	0.503673	+	0.863895i	$0.331982\pi$
$684$	27.0344	1.03369
$685$	0	0
$686$	−10.3262	−0.394258
$687$	6.38197	0.243487
$688$	−15.7082	−0.598870
$689$	−1.63932	−0.0624531
$690$	0	0
$691$	−51.5410	−1.96071	−0.980356	−	0.197234i	$-0.936804\pi$
$691$	−51.5410	−1.96071	−0.980356	+	0.197234i	$0.936804\pi$
$692$	3.70820	0.140965
$693$	13.1803	0.500680
$694$	−6.32624	−0.240141
$695$	0	0
$696$	−6.18034	−0.234265
$697$	18.4721	0.699682
$698$	12.4377	0.470774
$699$	4.16718	0.157617
$700$	0	0
$701$	5.94427	0.224512	0.112256	−	0.993679i	$-0.464192\pi$
$701$	5.94427	0.224512	0.112256	+	0.993679i	$0.464192\pi$
$702$	−4.79837	−0.181103
$703$	−17.5623	−0.662375
$704$	0.381966	0.0143959
$705$	0	0
$706$	−15.2361	−0.573417
$707$	40.6312	1.52809
$708$	2.43769	0.0916142
$709$	−40.1246	−1.50691	−0.753456	−	0.657499i	$-0.771615\pi$
$709$	−40.1246	−1.50691	−0.753456	+	0.657499i	$0.771615\pi$
$710$	0	0
$711$	−19.1459	−0.718027
$712$	−3.09017	−0.115809
$713$	−9.70820	−0.363575
$714$	5.03444	0.188409
$715$	0	0
$716$	−32.5623	−1.21691
$717$	0	0
$718$	3.49342	0.130373
$719$	−8.21478	−0.306360	−0.153180	−	0.988198i	$-0.548951\pi$
$719$	−8.21478	−0.306360	−0.153180	+	0.988198i	$0.548951\pi$
$720$	0	0
$721$	33.8328	1.26000
$722$	9.43769	0.351235
$723$	2.72949	0.101511
$724$	−3.76393	−0.139885
$725$	0	0
$726$	1.97871	0.0734370
$727$	17.9787	0.666794	0.333397	−	0.942787i	$-0.391805\pi$
$727$	17.9787	0.666794	0.333397	+	0.942787i	$0.391805\pi$
$728$	22.1591	0.821269
$729$	−19.4377	−0.719915
$730$	0	0
$731$	63.3050	2.34142
$732$	−1.32624	−0.0490192
$733$	−25.8328	−0.954157	−0.477078	−	0.878861i	$-0.658304\pi$
$733$	−25.8328	−0.954157	−0.477078	+	0.878861i	$0.658304\pi$
$734$	−11.1246	−0.410617
$735$	0	0
$736$	−27.2705	−1.00520
$737$	−4.09017	−0.150663
$738$	4.36068	0.160519
$739$	20.5279	0.755130	0.377565	−	0.925983i	$-0.376762\pi$
$739$	20.5279	0.755130	0.377565	+	0.925983i	$0.376762\pi$
$740$	0	0
$741$	7.76393	0.285215
$742$	0.832816	0.0305736
$743$	−37.0902	−1.36071	−0.680353	−	0.732884i	$-0.738174\pi$
$743$	−37.0902	−1.36071	−0.680353	+	0.732884i	$0.738174\pi$
$744$	1.70820	0.0626258
$745$	0	0
$746$	7.32624	0.268233
$747$	−14.1115	−0.516311
$748$	−19.5623	−0.715269
$749$	−36.5279	−1.33470
$750$	0	0
$751$	9.03444	0.329671	0.164836	−	0.986321i	$-0.447291\pi$
$751$	9.03444	0.329671	0.164836	+	0.986321i	$0.447291\pi$
$752$	−17.3951	−0.634335
$753$	−5.03444	−0.183465
$754$	15.5279	0.565491
$755$	0	0
$756$	−10.3262	−0.375562
$757$	0.167184	0.00607642	0.00303821	−	0.999995i	$-0.499033\pi$
$757$	0.167184	0.00607642	0.00303821	+	0.999995i	$0.499033\pi$
$758$	4.47214	0.162435
$759$	−3.00000	−0.108893
$760$	0	0
$761$	6.27051	0.227306	0.113653	−	0.993521i	$-0.463745\pi$
$761$	6.27051	0.227306	0.113653	+	0.993521i	$0.463745\pi$
$762$	3.27051	0.118478
$763$	26.1033	0.945004
$764$	4.85410	0.175615
$765$	0	0
$766$	−17.0689	−0.616724
$767$	−13.6950	−0.494500
$768$	2.50658	0.0904483
$769$	33.5410	1.20952	0.604760	−	0.796408i	$-0.293269\pi$
$769$	33.5410	1.20952	0.604760	+	0.796408i	$0.293269\pi$
$770$	0	0
$771$	3.70820	0.133548
$772$	8.18034	0.294417
$773$	45.7214	1.64448	0.822242	−	0.569139i	$-0.192723\pi$
$773$	45.7214	1.64448	0.822242	+	0.569139i	$0.192723\pi$
$774$	14.9443	0.537161
$775$	0	0
$776$	−18.7426	−0.672822
$777$	3.27051	0.117329
$778$	−3.29180	−0.118017
$779$	−14.4721	−0.518518
$780$	0	0
$781$	−19.9443	−0.713662
$782$	22.4164	0.801609
$783$	−16.1803	−0.578238
$784$	2.12461	0.0758790
$785$	0	0
$786$	1.23607	0.0440891
$787$	4.88854	0.174258	0.0871289	−	0.996197i	$-0.472231\pi$
$787$	4.88854	0.174258	0.0871289	+	0.996197i	$0.472231\pi$
$788$	−12.1803	−0.433907
$789$	1.40325	0.0499571
$790$	0	0
$791$	36.2705	1.28963
$792$	−10.3262	−0.366927
$793$	7.45085	0.264587
$794$	9.52786	0.338131
$795$	0	0
$796$	6.70820	0.237766
$797$	19.6869	0.697346	0.348673	−	0.937244i	$-0.386632\pi$
$797$	19.6869	0.697346	0.348673	+	0.937244i	$0.386632\pi$
$798$	−3.94427	−0.139626
$799$	70.1033	2.48008
$800$	0	0
$801$	−3.94427	−0.139364
$802$	−3.43769	−0.121389
$803$	−11.0902	−0.391364
$804$	1.56231	0.0550983
$805$	0	0
$806$	−4.29180	−0.151172
$807$	−10.0000	−0.352017
$808$	−31.8328	−1.11987
$809$	20.4508	0.719014	0.359507	−	0.933142i	$-0.382945\pi$
$809$	20.4508	0.719014	0.359507	+	0.933142i	$0.382945\pi$
$810$	0	0
$811$	−38.1246	−1.33874	−0.669368	−	0.742931i	$-0.733435\pi$
$811$	−38.1246	−1.33874	−0.669368	+	0.742931i	$0.733435\pi$
$812$	33.4164	1.17269
$813$	4.96556	0.174150
$814$	3.00000	0.105150
$815$	0	0
$816$	5.29180	0.185250
$817$	−49.5967	−1.73517
$818$	−11.3050	−0.395268
$819$	28.2837	0.988311
$820$	0	0
$821$	−13.5279	−0.472126	−0.236063	−	0.971738i	$-0.575857\pi$
$821$	−13.5279	−0.472126	−0.236063	+	0.971738i	$0.575857\pi$
$822$	3.72136	0.129797
$823$	12.7082	0.442980	0.221490	−	0.975163i	$-0.428908\pi$
$823$	12.7082	0.442980	0.221490	+	0.975163i	$0.428908\pi$
$824$	−26.5066	−0.923400
$825$	0	0
$826$	6.95743	0.242080
$827$	47.1246	1.63868	0.819342	−	0.573306i	$-0.194339\pi$
$827$	47.1246	1.63868	0.819342	+	0.573306i	$0.194339\pi$
$828$	−22.4164	−0.779024
$829$	−55.1246	−1.91456	−0.957278	−	0.289168i	$-0.906621\pi$
$829$	−55.1246	−1.91456	−0.957278	+	0.289168i	$0.906621\pi$
$830$	0	0
$831$	−10.9656	−0.380391
$832$	0.819660	0.0284166
$833$	−8.56231	−0.296666
$834$	4.34752	0.150542
$835$	0	0
$836$	15.3262	0.530069
$837$	4.47214	0.154580
$838$	−17.1591	−0.592750
$839$	11.5066	0.397251	0.198626	−	0.980075i	$-0.436352\pi$
$839$	11.5066	0.397251	0.198626	+	0.980075i	$0.436352\pi$
$840$	0	0
$841$	23.3607	0.805541
$842$	−5.67376	−0.195531
$843$	9.68692	0.333635
$844$	14.8541	0.511299
$845$	0	0
$846$	16.5492	0.568972
$847$	−23.9230	−0.822004
$848$	0.875388	0.0300610
$849$	8.15905	0.280018
$850$	0	0
$851$	14.5623	0.499189
$852$	7.61803	0.260990
$853$	8.68692	0.297434	0.148717	−	0.988880i	$-0.452486\pi$
$853$	8.68692	0.297434	0.148717	+	0.988880i	$0.452486\pi$
$854$	−3.78522	−0.129528
$855$	0	0
$856$	28.6180	0.978144
$857$	36.0689	1.23209	0.616045	−	0.787711i	$-0.288734\pi$
$857$	36.0689	1.23209	0.616045	+	0.787711i	$0.288734\pi$
$858$	−1.32624	−0.0452770
$859$	−12.2361	−0.417489	−0.208745	−	0.977970i	$-0.566938\pi$
$859$	−12.2361	−0.417489	−0.208745	+	0.977970i	$0.566938\pi$
$860$	0	0
$861$	2.69505	0.0918470
$862$	−19.4164	−0.661325
$863$	−49.2492	−1.67646	−0.838232	−	0.545314i	$-0.816410\pi$
$863$	−49.2492	−1.67646	−0.838232	+	0.545314i	$0.816410\pi$
$864$	12.5623	0.427378
$865$	0	0
$866$	4.23607	0.143947
$867$	−14.8328	−0.503749
$868$	−9.23607	−0.313493
$869$	−10.8541	−0.368200
$870$	0	0
$871$	−8.77709	−0.297400
$872$	−20.4508	−0.692553
$873$	−23.9230	−0.809670
$874$	−17.5623	−0.594054
$875$	0	0
$876$	4.23607	0.143123
$877$	−28.9787	−0.978542	−0.489271	−	0.872132i	$-0.662737\pi$
$877$	−28.9787	−0.978542	−0.489271	+	0.872132i	$0.662737\pi$
$878$	−15.0000	−0.506225
$879$	−0.506578	−0.0170864
$880$	0	0
$881$	−23.4508	−0.790079	−0.395040	−	0.918664i	$-0.629269\pi$
$881$	−23.4508	−0.790079	−0.395040	+	0.918664i	$0.629269\pi$
$882$	−2.02129	−0.0680602
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	−41.9787	−1.41190
$885$	0	0
$886$	8.83282	0.296744
$887$	−42.0689	−1.41253	−0.706267	−	0.707945i	$-0.749622\pi$
$887$	−42.0689	−1.41253	−0.706267	+	0.707945i	$0.749622\pi$
$888$	−2.56231	−0.0859854
$889$	−39.5410	−1.32616
$890$	0	0
$891$	−12.4721	−0.417832
$892$	−11.0902	−0.371326
$893$	−54.9230	−1.83793
$894$	2.68692	0.0898640
$895$	0	0
$896$	−32.4853	−1.08526
$897$	−6.43769	−0.214948
$898$	−17.9656	−0.599518
$899$	−14.4721	−0.482673
$900$	0	0
$901$	−3.52786	−0.117530
$902$	2.47214	0.0823131
$903$	9.23607	0.307357
$904$	−28.4164	−0.945116
$905$	0	0
$906$	0.458980	0.0152486
$907$	46.1459	1.53225	0.766125	−	0.642692i	$-0.222182\pi$
$907$	46.1459	1.53225	0.766125	+	0.642692i	$0.222182\pi$
$908$	8.47214	0.281158
$909$	−40.6312	−1.34765
$910$	0	0
$911$	41.4721	1.37403	0.687017	−	0.726642i	$-0.258920\pi$
$911$	41.4721	1.37403	0.687017	+	0.726642i	$0.258920\pi$
$912$	−4.14590	−0.137284
$913$	−8.00000	−0.264761
$914$	−0.549150	−0.0181643
$915$	0	0
$916$	27.0344	0.893243
$917$	−14.9443	−0.493503
$918$	−10.3262	−0.340817
$919$	12.6393	0.416933	0.208466	−	0.978030i	$-0.433153\pi$
$919$	12.6393	0.416933	0.208466	+	0.978030i	$0.433153\pi$
$920$	0	0
$921$	−12.5967	−0.415077
$922$	−16.1246	−0.531036
$923$	−42.7984	−1.40873
$924$	−2.85410	−0.0938931
$925$	0	0
$926$	−11.9443	−0.392513
$927$	−33.8328	−1.11122
$928$	−40.6525	−1.33448
$929$	−15.6525	−0.513541	−0.256771	−	0.966472i	$-0.582658\pi$
$929$	−15.6525	−0.513541	−0.256771	+	0.966472i	$0.582658\pi$
$930$	0	0
$931$	6.70820	0.219853
$932$	17.6525	0.578226
$933$	2.12461	0.0695567
$934$	7.61803	0.249270
$935$	0	0
$936$	−22.1591	−0.724291
$937$	8.70820	0.284485	0.142242	−	0.989832i	$-0.454569\pi$
$937$	8.70820	0.284485	0.142242	+	0.989832i	$0.454569\pi$
$938$	4.45898	0.145591
$939$	−3.06888	−0.100149
$940$	0	0
$941$	−57.0689	−1.86039	−0.930196	−	0.367063i	$-0.880363\pi$
$941$	−57.0689	−1.86039	−0.930196	+	0.367063i	$0.880363\pi$
$942$	3.06888	0.0999896
$943$	12.0000	0.390774
$944$	7.31308	0.238021
$945$	0	0
$946$	8.47214	0.275453
$947$	−3.00000	−0.0974869	−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000	−0.0974869	−0.0487435	+	0.998811i	$0.515522\pi$
$948$	4.14590	0.134653
$949$	−23.7984	−0.772528
$950$	0	0
$951$	6.79837	0.220452
$952$	47.6869	1.54554
$953$	20.9230	0.677762	0.338881	−	0.940829i	$-0.389952\pi$
$953$	20.9230	0.677762	0.338881	+	0.940829i	$0.389952\pi$
$954$	−0.832816	−0.0269634
$955$	0	0
$956$	0	0
$957$	−4.47214	−0.144564
$958$	16.8328	0.543844
$959$	−44.9919	−1.45286
$960$	0	0
$961$	−27.0000	−0.870968
$962$	6.43769	0.207560
$963$	36.5279	1.17709
$964$	11.5623	0.372397
$965$	0	0
$966$	3.27051	0.105227
$967$	−27.0689	−0.870477	−0.435238	−	0.900315i	$-0.643336\pi$
$967$	−27.0689	−0.870477	−0.435238	+	0.900315i	$0.643336\pi$
$968$	18.7426	0.602411
$969$	16.7082	0.536745
$970$	0	0
$971$	−55.6869	−1.78708	−0.893539	−	0.448985i	$-0.851786\pi$
$971$	−55.6869	−1.78708	−0.893539	+	0.448985i	$0.851786\pi$
$972$	15.6180	0.500949
$973$	−52.5623	−1.68507
$974$	24.3262	0.779463
$975$	0	0
$976$	−3.97871	−0.127356
$977$	31.0689	0.993982	0.496991	−	0.867756i	$-0.334438\pi$
$977$	31.0689	0.993982	0.496991	+	0.867756i	$0.334438\pi$
$978$	1.14590	0.0366418
$979$	−2.23607	−0.0714650
$980$	0	0
$981$	−26.1033	−0.833415
$982$	8.06888	0.257488
$983$	−52.9443	−1.68866	−0.844330	−	0.535824i	$-0.820001\pi$
$983$	−52.9443	−1.68866	−0.844330	+	0.535824i	$0.820001\pi$
$984$	−2.11146	−0.0673108
$985$	0	0
$986$	33.4164	1.06420
$987$	10.2279	0.325559
$988$	32.8885	1.04632
$989$	41.1246	1.30769
$990$	0	0
$991$	30.5410	0.970167	0.485084	−	0.874468i	$-0.338789\pi$
$991$	30.5410	0.970167	0.485084	+	0.874468i	$0.338789\pi$
$992$	11.2361	0.356746
$993$	−10.9656	−0.347981
$994$	21.7426	0.689635
$995$	0	0
$996$	3.05573	0.0968244
$997$	−25.5623	−0.809566	−0.404783	−	0.914413i	$-0.632653\pi$
$997$	−25.5623	−0.809566	−0.404783	+	0.914413i	$0.632653\pi$
$998$	−0.978714	−0.0309806
$999$	−6.70820	−0.212238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.a.1.2	✓	2
3.2	odd	2		5625.2.a.e.1.1		2
4.3	odd	2		10000.2.a.m.1.1		2
5.2	odd	4		625.2.b.b.624.3		4
5.3	odd	4		625.2.b.b.624.2		4
5.4	even	2		625.2.a.d.1.1	yes	2
15.14	odd	2		5625.2.a.c.1.2		2
20.19	odd	2		10000.2.a.b.1.2		2
25.2	odd	20		625.2.e.e.124.2		8
25.3	odd	20		625.2.e.f.249.1		8
25.4	even	10		625.2.d.f.376.1		4
25.6	even	5		625.2.d.e.251.1		4
25.8	odd	20		625.2.e.f.374.2		8
25.9	even	10		625.2.d.c.126.1		4
25.11	even	5		625.2.d.i.501.1		4
25.12	odd	20		625.2.e.e.499.1		8
25.13	odd	20		625.2.e.e.499.2		8
25.14	even	10		625.2.d.c.501.1		4
25.16	even	5		625.2.d.i.126.1		4
25.17	odd	20		625.2.e.f.374.1		8
25.19	even	10		625.2.d.f.251.1		4
25.21	even	5		625.2.d.e.376.1		4
25.22	odd	20		625.2.e.f.249.2		8
25.23	odd	20		625.2.e.e.124.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.a.1.2	✓	2	1.1	even	1	trivial
625.2.a.d.1.1	yes	2	5.4	even	2
625.2.b.b.624.2		4	5.3	odd	4
625.2.b.b.624.3		4	5.2	odd	4
625.2.d.c.126.1		4	25.9	even	10
625.2.d.c.501.1		4	25.14	even	10
625.2.d.e.251.1		4	25.6	even	5
625.2.d.e.376.1		4	25.21	even	5
625.2.d.f.251.1		4	25.19	even	10
625.2.d.f.376.1		4	25.4	even	10
625.2.d.i.126.1		4	25.16	even	5
625.2.d.i.501.1		4	25.11	even	5
625.2.e.e.124.1		8	25.23	odd	20
625.2.e.e.124.2		8	25.2	odd	20
625.2.e.e.499.1		8	25.12	odd	20
625.2.e.e.499.2		8	25.13	odd	20
625.2.e.f.249.1		8	25.3	odd	20
625.2.e.f.249.2		8	25.22	odd	20
625.2.e.f.374.1		8	25.17	odd	20
625.2.e.f.374.2		8	25.8	odd	20
5625.2.a.c.1.2		2	15.14	odd	2
5625.2.a.e.1.1		2	3.2	odd	2
10000.2.a.b.1.2		2	20.19	odd	2
10000.2.a.m.1.1		2	4.3	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:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -0.381966 q^{3} -1.61803 q^{4} -0.236068 q^{6} +2.85410 q^{7} -2.23607 q^{8} -2.85410 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} -0.381966 q^{3} -1.61803 q^{4} -0.236068 q^{6} +2.85410 q^{7} -2.23607 q^{8} -2.85410 q^{9} -1.61803 q^{11} +0.618034 q^{12} -3.47214 q^{13} +1.76393 q^{14} +1.85410 q^{16} -7.47214 q^{17} -1.76393 q^{18} +5.85410 q^{19} -1.09017 q^{21} -1.00000 q^{22} -4.85410 q^{23} +0.854102 q^{24} -2.14590 q^{26} +2.23607 q^{27} -4.61803 q^{28} -7.23607 q^{29} +2.00000 q^{31} +5.61803 q^{32} +0.618034 q^{33} -4.61803 q^{34} +4.61803 q^{36} -3.00000 q^{37} +3.61803 q^{38} +1.32624 q^{39} -2.47214 q^{41} -0.673762 q^{42} -8.47214 q^{43} +2.61803 q^{44} -3.00000 q^{46} -9.38197 q^{47} -0.708204 q^{48} +1.14590 q^{49} +2.85410 q^{51} +5.61803 q^{52} +0.472136 q^{53} +1.38197 q^{54} -6.38197 q^{56} -2.23607 q^{57} -4.47214 q^{58} +3.94427 q^{59} -2.14590 q^{61} +1.23607 q^{62} -8.14590 q^{63} -0.236068 q^{64} +0.381966 q^{66} +2.52786 q^{67} +12.0902 q^{68} +1.85410 q^{69} +12.3262 q^{71} +6.38197 q^{72} +6.85410 q^{73} -1.85410 q^{74} -9.47214 q^{76} -4.61803 q^{77} +0.819660 q^{78} +6.70820 q^{79} +7.70820 q^{81} -1.52786 q^{82} +4.94427 q^{83} +1.76393 q^{84} -5.23607 q^{86} +2.76393 q^{87} +3.61803 q^{88} +1.38197 q^{89} -9.90983 q^{91} +7.85410 q^{92} -0.763932 q^{93} -5.79837 q^{94} -2.14590 q^{96} +8.38197 q^{97} +0.708204 q^{98} +4.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 3 q^{3} - q^{4} + 4 q^{6} - q^{7} + q^{9}+O(q^{10}) \) 2 * q - q^2 - 3 * q^3 - q^4 + 4 * q^6 - q^7 + q^9	\( 2 q - q^{2} - 3 q^{3} - q^{4} + 4 q^{6} - q^{7} + q^{9} - q^{11} - q^{12} + 2 q^{13} + 8 q^{14} - 3 q^{16} - 6 q^{17} - 8 q^{18} + 5 q^{19} + 9 q^{21} - 2 q^{22} - 3 q^{23} - 5 q^{24} - 11 q^{26} - 7 q^{28} - 10 q^{29} + 4 q^{31} + 9 q^{32} - q^{33} - 7 q^{34} + 7 q^{36} - 6 q^{37} + 5 q^{38} - 13 q^{39} + 4 q^{41} - 17 q^{42} - 8 q^{43} + 3 q^{44} - 6 q^{46} - 21 q^{47} + 12 q^{48} + 9 q^{49} - q^{51} + 9 q^{52} - 8 q^{53} + 5 q^{54} - 15 q^{56} - 10 q^{59} - 11 q^{61} - 2 q^{62} - 23 q^{63} + 4 q^{64} + 3 q^{66} + 14 q^{67} + 13 q^{68} - 3 q^{69} + 9 q^{71} + 15 q^{72} + 7 q^{73} + 3 q^{74} - 10 q^{76} - 7 q^{77} + 24 q^{78} + 2 q^{81} - 12 q^{82} - 8 q^{83} + 8 q^{84} - 6 q^{86} + 10 q^{87} + 5 q^{88} + 5 q^{89} - 31 q^{91} + 9 q^{92} - 6 q^{93} + 13 q^{94} - 11 q^{96} + 19 q^{97} - 12 q^{98} + 7 q^{99}+O(q^{100}) \) 2 * q - q^2 - 3 * q^3 - q^4 + 4 * q^6 - q^7 + q^9 - q^11 - q^12 + 2 * q^13 + 8 * q^14 - 3 * q^16 - 6 * q^17 - 8 * q^18 + 5 * q^19 + 9 * q^21 - 2 * q^22 - 3 * q^23 - 5 * q^24 - 11 * q^26 - 7 * q^28 - 10 * q^29 + 4 * q^31 + 9 * q^32 - q^33 - 7 * q^34 + 7 * q^36 - 6 * q^37 + 5 * q^38 - 13 * q^39 + 4 * q^41 - 17 * q^42 - 8 * q^43 + 3 * q^44 - 6 * q^46 - 21 * q^47 + 12 * q^48 + 9 * q^49 - q^51 + 9 * q^52 - 8 * q^53 + 5 * q^54 - 15 * q^56 - 10 * q^59 - 11 * q^61 - 2 * q^62 - 23 * q^63 + 4 * q^64 + 3 * q^66 + 14 * q^67 + 13 * q^68 - 3 * q^69 + 9 * q^71 + 15 * q^72 + 7 * q^73 + 3 * q^74 - 10 * q^76 - 7 * q^77 + 24 * q^78 + 2 * q^81 - 12 * q^82 - 8 * q^83 + 8 * q^84 - 6 * q^86 + 10 * q^87 + 5 * q^88 + 5 * q^89 - 31 * q^91 + 9 * q^92 - 6 * q^93 + 13 * q^94 - 11 * q^96 + 19 * q^97 - 12 * q^98 + 7 * q^99

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