LMFDB - Embedded newform 125.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [125,2,Mod(1,125)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(125, 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("125.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 125 = 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	125.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:	$0.998130025266$
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			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	125.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} -0.381966 q^{3} +0.618034 q^{4} +0.618034 q^{6} -3.00000 q^{7} +2.23607 q^{8} -2.85410 q^{9} +O(q^{10})$	\(q-1.61803 q^{2} -0.381966 q^{3} +0.618034 q^{4} +0.618034 q^{6} -3.00000 q^{7} +2.23607 q^{8} -2.85410 q^{9} -3.00000 q^{11} -0.236068 q^{12} -4.85410 q^{13} +4.85410 q^{14} -4.85410 q^{16} +4.23607 q^{17} +4.61803 q^{18} -3.61803 q^{19} +1.14590 q^{21} +4.85410 q^{22} -1.23607 q^{23} -0.854102 q^{24} +7.85410 q^{26} +2.23607 q^{27} -1.85410 q^{28} +6.70820 q^{29} +5.09017 q^{31} +3.38197 q^{32} +1.14590 q^{33} -6.85410 q^{34} -1.76393 q^{36} +3.70820 q^{37} +5.85410 q^{38} +1.85410 q^{39} -3.00000 q^{41} -1.85410 q^{42} -9.00000 q^{43} -1.85410 q^{44} +2.00000 q^{46} -8.32624 q^{47} +1.85410 q^{48} +2.00000 q^{49} -1.61803 q^{51} -3.00000 q^{52} +4.61803 q^{53} -3.61803 q^{54} -6.70820 q^{56} +1.38197 q^{57} -10.8541 q^{58} +4.14590 q^{59} -6.09017 q^{61} -8.23607 q^{62} +8.56231 q^{63} +4.23607 q^{64} -1.85410 q^{66} -13.8541 q^{67} +2.61803 q^{68} +0.472136 q^{69} -3.00000 q^{71} -6.38197 q^{72} +1.85410 q^{73} -6.00000 q^{74} -2.23607 q^{76} +9.00000 q^{77} -3.00000 q^{78} +0.527864 q^{79} +7.70820 q^{81} +4.85410 q^{82} +0.472136 q^{83} +0.708204 q^{84} +14.5623 q^{86} -2.56231 q^{87} -6.70820 q^{88} -13.4164 q^{89} +14.5623 q^{91} -0.763932 q^{92} -1.94427 q^{93} +13.4721 q^{94} -1.29180 q^{96} +7.85410 q^{97} -3.23607 q^{98} +8.56231 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9}+O(q^{10}) $ 2 * q - q^2 - 3 * q^3 - q^4 - q^6 - 6 * q^7 + q^9	$ 2 q - q^{2} - 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9} - 6 q^{11} + 4 q^{12} - 3 q^{13} + 3 q^{14} - 3 q^{16} + 4 q^{17} + 7 q^{18} - 5 q^{19} + 9 q^{21} + 3 q^{22} + 2 q^{23} + 5 q^{24} + 9 q^{26} + 3 q^{28} - q^{31} + 9 q^{32} + 9 q^{33} - 7 q^{34} - 8 q^{36} - 6 q^{37} + 5 q^{38} - 3 q^{39} - 6 q^{41} + 3 q^{42} - 18 q^{43} + 3 q^{44} + 4 q^{46} - q^{47} - 3 q^{48} + 4 q^{49} - q^{51} - 6 q^{52} + 7 q^{53} - 5 q^{54} + 5 q^{57} - 15 q^{58} + 15 q^{59} - q^{61} - 12 q^{62} - 3 q^{63} + 4 q^{64} + 3 q^{66} - 21 q^{67} + 3 q^{68} - 8 q^{69} - 6 q^{71} - 15 q^{72} - 3 q^{73} - 12 q^{74} + 18 q^{77} - 6 q^{78} + 10 q^{79} + 2 q^{81} + 3 q^{82} - 8 q^{83} - 12 q^{84} + 9 q^{86} + 15 q^{87} + 9 q^{91} - 6 q^{92} + 14 q^{93} + 18 q^{94} - 16 q^{96} + 9 q^{97} - 2 q^{98} - 3 q^{99}+O(q^{100}) $ 2 * q - q^2 - 3 * q^3 - q^4 - q^6 - 6 * q^7 + q^9 - 6 * q^11 + 4 * q^12 - 3 * q^13 + 3 * q^14 - 3 * q^16 + 4 * q^17 + 7 * q^18 - 5 * q^19 + 9 * q^21 + 3 * q^22 + 2 * q^23 + 5 * q^24 + 9 * q^26 + 3 * q^28 - q^31 + 9 * q^32 + 9 * q^33 - 7 * q^34 - 8 * q^36 - 6 * q^37 + 5 * q^38 - 3 * q^39 - 6 * q^41 + 3 * q^42 - 18 * q^43 + 3 * q^44 + 4 * q^46 - q^47 - 3 * q^48 + 4 * q^49 - q^51 - 6 * q^52 + 7 * q^53 - 5 * q^54 + 5 * q^57 - 15 * q^58 + 15 * q^59 - q^61 - 12 * q^62 - 3 * q^63 + 4 * q^64 + 3 * q^66 - 21 * q^67 + 3 * q^68 - 8 * q^69 - 6 * q^71 - 15 * q^72 - 3 * q^73 - 12 * q^74 + 18 * q^77 - 6 * q^78 + 10 * q^79 + 2 * q^81 + 3 * q^82 - 8 * q^83 - 12 * q^84 + 9 * q^86 + 15 * q^87 + 9 * q^91 - 6 * q^92 + 14 * q^93 + 18 * q^94 - 16 * q^96 + 9 * q^97 - 2 * q^98 - 3 * 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 $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	−0.381966	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$3$	−0.381966	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	2.23607	0.790569
$9$	−2.85410	−0.951367
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−0.236068	−0.0681470
$13$	−4.85410	−1.34629	−0.673143	−	0.739512i	$-0.735056\pi$
$13$	−4.85410	−1.34629	−0.673143	+	0.739512i	$0.735056\pi$
$14$	4.85410	1.29731
$15$	0	0
$16$	−4.85410	−1.21353
$17$	4.23607	1.02740	0.513699	−	0.857971i	$-0.328275\pi$
$17$	4.23607	1.02740	0.513699	+	0.857971i	$0.328275\pi$
$18$	4.61803	1.08848
$19$	−3.61803	−0.830034	−0.415017	−	0.909814i	$-0.636224\pi$
$19$	−3.61803	−0.830034	−0.415017	+	0.909814i	$0.636224\pi$
$20$	0	0
$21$	1.14590	0.250055
$22$	4.85410	1.03490
$23$	−1.23607	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$24$	−0.854102	−0.174343
$25$	0	0
$26$	7.85410	1.54032
$27$	2.23607	0.430331
$28$	−1.85410	−0.350392
$29$	6.70820	1.24568	0.622841	−	0.782348i	$-0.285978\pi$
$29$	6.70820	1.24568	0.622841	+	0.782348i	$0.285978\pi$
$30$	0	0
$31$	5.09017	0.914222	0.457111	−	0.889410i	$-0.348884\pi$
$31$	5.09017	0.914222	0.457111	+	0.889410i	$0.348884\pi$
$32$	3.38197	0.597853
$33$	1.14590	0.199475
$34$	−6.85410	−1.17547
$35$	0	0
$36$	−1.76393	−0.293989
$37$	3.70820	0.609625	0.304812	−	0.952412i	$-0.401406\pi$
$37$	3.70820	0.609625	0.304812	+	0.952412i	$0.401406\pi$
$38$	5.85410	0.949661
$39$	1.85410	0.296894
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	−1.85410	−0.286094
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	−1.85410	−0.279516
$45$	0	0
$46$	2.00000	0.294884
$47$	−8.32624	−1.21451	−0.607253	−	0.794508i	$-0.707729\pi$
$47$	−8.32624	−1.21451	−0.607253	+	0.794508i	$0.707729\pi$
$48$	1.85410	0.267617
$49$	2.00000	0.285714
$50$	0	0
$51$	−1.61803	−0.226570
$52$	−3.00000	−0.416025
$53$	4.61803	0.634336	0.317168	−	0.948369i	$-0.397268\pi$
$53$	4.61803	0.634336	0.317168	+	0.948369i	$0.397268\pi$
$54$	−3.61803	−0.492352
$55$	0	0
$56$	−6.70820	−0.896421
$57$	1.38197	0.183046
$58$	−10.8541	−1.42521
$59$	4.14590	0.539750	0.269875	−	0.962895i	$-0.413018\pi$
$59$	4.14590	0.539750	0.269875	+	0.962895i	$0.413018\pi$
$60$	0	0
$61$	−6.09017	−0.779766	−0.389883	−	0.920864i	$-0.627485\pi$
$61$	−6.09017	−0.779766	−0.389883	+	0.920864i	$0.627485\pi$
$62$	−8.23607	−1.04598
$63$	8.56231	1.07875
$64$	4.23607	0.529508
$65$	0	0
$66$	−1.85410	−0.228224
$67$	−13.8541	−1.69255	−0.846274	−	0.532748i	$-0.821159\pi$
$67$	−13.8541	−1.69255	−0.846274	+	0.532748i	$0.821159\pi$
$68$	2.61803	0.317483
$69$	0.472136	0.0568385
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	−6.38197	−0.752122
$73$	1.85410	0.217006	0.108503	−	0.994096i	$-0.465394\pi$
$73$	1.85410	0.217006	0.108503	+	0.994096i	$0.465394\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−2.23607	−0.256495
$77$	9.00000	1.02565
$78$	−3.00000	−0.339683
$79$	0.527864	0.0593893	0.0296947	−	0.999559i	$-0.490547\pi$
$79$	0.527864	0.0593893	0.0296947	+	0.999559i	$0.490547\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	4.85410	0.536046
$83$	0.472136	0.0518237	0.0259118	−	0.999664i	$-0.491751\pi$
$83$	0.472136	0.0518237	0.0259118	+	0.999664i	$0.491751\pi$
$84$	0.708204	0.0772714
$85$	0	0
$86$	14.5623	1.57029
$87$	−2.56231	−0.274708
$88$	−6.70820	−0.715097
$89$	−13.4164	−1.42214	−0.711068	−	0.703123i	$-0.751788\pi$
$89$	−13.4164	−1.42214	−0.711068	+	0.703123i	$0.751788\pi$
$90$	0	0
$91$	14.5623	1.52654
$92$	−0.763932	−0.0796454
$93$	−1.94427	−0.201612
$94$	13.4721	1.38954
$95$	0	0
$96$	−1.29180	−0.131843
$97$	7.85410	0.797463	0.398732	−	0.917068i	$-0.369451\pi$
$97$	7.85410	0.797463	0.398732	+	0.917068i	$0.369451\pi$
$98$	−3.23607	−0.326892
$99$	8.56231	0.860544
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	2.61803	0.259224
$103$	12.7082	1.25218	0.626088	−	0.779752i	$-0.284655\pi$
$103$	12.7082	1.25218	0.626088	+	0.779752i	$0.284655\pi$
$104$	−10.8541	−1.06433
$105$	0	0
$106$	−7.47214	−0.725758
$107$	0.0901699	0.00871706	0.00435853	−	0.999991i	$-0.498613\pi$
$107$	0.0901699	0.00871706	0.00435853	+	0.999991i	$0.498613\pi$
$108$	1.38197	0.132980
$109$	5.32624	0.510161	0.255081	−	0.966920i	$-0.417898\pi$
$109$	5.32624	0.510161	0.255081	+	0.966920i	$0.417898\pi$
$110$	0	0
$111$	−1.41641	−0.134439
$112$	14.5623	1.37601
$113$	2.05573	0.193387	0.0966933	−	0.995314i	$-0.469173\pi$
$113$	2.05573	0.193387	0.0966933	+	0.995314i	$0.469173\pi$
$114$	−2.23607	−0.209427
$115$	0	0
$116$	4.14590	0.384937
$117$	13.8541	1.28081
$118$	−6.70820	−0.617540
$119$	−12.7082	−1.16496
$120$	0	0
$121$	−2.00000	−0.181818
$122$	9.85410	0.892148
$123$	1.14590	0.103322
$124$	3.14590	0.282510
$125$	0	0
$126$	−13.8541	−1.23422
$127$	−9.70820	−0.861464	−0.430732	−	0.902480i	$-0.641745\pi$
$127$	−9.70820	−0.861464	−0.430732	+	0.902480i	$0.641745\pi$
$128$	−13.6180	−1.20368
$129$	3.43769	0.302672
$130$	0	0
$131$	−12.2705	−1.07208	−0.536040	−	0.844193i	$-0.680080\pi$
$131$	−12.2705	−1.07208	−0.536040	+	0.844193i	$0.680080\pi$
$132$	0.708204	0.0616412
$133$	10.8541	0.941170
$134$	22.4164	1.93648
$135$	0	0
$136$	9.47214	0.812229
$137$	5.61803	0.479981	0.239991	−	0.970775i	$-0.422856\pi$
$137$	5.61803	0.479981	0.239991	+	0.970775i	$0.422856\pi$
$138$	−0.763932	−0.0650302
$139$	−12.2361	−1.03785	−0.518925	−	0.854820i	$-0.673668\pi$
$139$	−12.2361	−1.03785	−0.518925	+	0.854820i	$0.673668\pi$
$140$	0	0
$141$	3.18034	0.267833
$142$	4.85410	0.407347
$143$	14.5623	1.21776
$144$	13.8541	1.15451
$145$	0	0
$146$	−3.00000	−0.248282
$147$	−0.763932	−0.0630081
$148$	2.29180	0.188384
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−21.0902	−1.71629	−0.858147	−	0.513404i	$-0.828384\pi$
$151$	−21.0902	−1.71629	−0.858147	+	0.513404i	$0.828384\pi$
$152$	−8.09017	−0.656199
$153$	−12.0902	−0.977432
$154$	−14.5623	−1.17346
$155$	0	0
$156$	1.14590	0.0917453
$157$	−12.2705	−0.979293	−0.489647	−	0.871921i	$-0.662874\pi$
$157$	−12.2705	−0.979293	−0.489647	+	0.871921i	$0.662874\pi$
$158$	−0.854102	−0.0679487
$159$	−1.76393	−0.139889
$160$	0	0
$161$	3.70820	0.292247
$162$	−12.4721	−0.979904
$163$	−19.8541	−1.55509	−0.777547	−	0.628825i	$-0.783536\pi$
$163$	−19.8541	−1.55509	−0.777547	+	0.628825i	$0.783536\pi$
$164$	−1.85410	−0.144781
$165$	0	0
$166$	−0.763932	−0.0592926
$167$	9.23607	0.714708	0.357354	−	0.933969i	$-0.383679\pi$
$167$	9.23607	0.714708	0.357354	+	0.933969i	$0.383679\pi$
$168$	2.56231	0.197686
$169$	10.5623	0.812485
$170$	0	0
$171$	10.3262	0.789667
$172$	−5.56231	−0.424122
$173$	−0.0557281	−0.00423693	−0.00211846	−	0.999998i	$-0.500674\pi$
$173$	−0.0557281	−0.00423693	−0.00211846	+	0.999998i	$0.500674\pi$
$174$	4.14590	0.314300
$175$	0	0
$176$	14.5623	1.09768
$177$	−1.58359	−0.119030
$178$	21.7082	1.62710
$179$	6.70820	0.501395	0.250697	−	0.968066i	$-0.419340\pi$
$179$	6.70820	0.501395	0.250697	+	0.968066i	$0.419340\pi$
$180$	0	0
$181$	18.1803	1.35133	0.675667	−	0.737207i	$-0.263856\pi$
$181$	18.1803	1.35133	0.675667	+	0.737207i	$0.263856\pi$
$182$	−23.5623	−1.74655
$183$	2.32624	0.171960
$184$	−2.76393	−0.203760
$185$	0	0
$186$	3.14590	0.230668
$187$	−12.7082	−0.929316
$188$	−5.14590	−0.375303
$189$	−6.70820	−0.487950
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.61803	−0.116772
$193$	15.2705	1.09920	0.549598	−	0.835429i	$-0.314781\pi$
$193$	15.2705	1.09920	0.549598	+	0.835429i	$0.314781\pi$
$194$	−12.7082	−0.912396
$195$	0	0
$196$	1.23607	0.0882906
$197$	−4.05573	−0.288959	−0.144479	−	0.989508i	$-0.546151\pi$
$197$	−4.05573	−0.288959	−0.144479	+	0.989508i	$0.546151\pi$
$198$	−13.8541	−0.984568
$199$	16.1803	1.14699	0.573497	−	0.819208i	$-0.305586\pi$
$199$	16.1803	1.14699	0.573497	+	0.819208i	$0.305586\pi$
$200$	0	0
$201$	5.29180	0.373255
$202$	4.85410	0.341533
$203$	−20.1246	−1.41247
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−20.5623	−1.43264
$207$	3.52786	0.245204
$208$	23.5623	1.63375
$209$	10.8541	0.750794
$210$	0	0
$211$	−4.18034	−0.287786	−0.143893	−	0.989593i	$-0.545962\pi$
$211$	−4.18034	−0.287786	−0.143893	+	0.989593i	$0.545962\pi$
$212$	2.85410	0.196021
$213$	1.14590	0.0785156
$214$	−0.145898	−0.00997338
$215$	0	0
$216$	5.00000	0.340207
$217$	−15.2705	−1.03663
$218$	−8.61803	−0.583687
$219$	−0.708204	−0.0478560
$220$	0	0
$221$	−20.5623	−1.38317
$222$	2.29180	0.153815
$223$	1.85410	0.124160	0.0620799	−	0.998071i	$-0.480227\pi$
$223$	1.85410	0.124160	0.0620799	+	0.998071i	$0.480227\pi$
$224$	−10.1459	−0.677901
$225$	0	0
$226$	−3.32624	−0.221258
$227$	−6.61803	−0.439254	−0.219627	−	0.975584i	$-0.570484\pi$
$227$	−6.61803	−0.439254	−0.219627	+	0.975584i	$0.570484\pi$
$228$	0.854102	0.0565643
$229$	−1.38197	−0.0913229	−0.0456614	−	0.998957i	$-0.514540\pi$
$229$	−1.38197	−0.0913229	−0.0456614	+	0.998957i	$0.514540\pi$
$230$	0	0
$231$	−3.43769	−0.226184
$232$	15.0000	0.984798
$233$	−26.8885	−1.76153	−0.880764	−	0.473556i	$-0.842970\pi$
$233$	−26.8885	−1.76153	−0.880764	+	0.473556i	$0.842970\pi$
$234$	−22.4164	−1.46541
$235$	0	0
$236$	2.56231	0.166792
$237$	−0.201626	−0.0130970
$238$	20.5623	1.33286
$239$	25.8541	1.67236	0.836181	−	0.548453i	$-0.184783\pi$
$239$	25.8541	1.67236	0.836181	+	0.548453i	$0.184783\pi$
$240$	0	0
$241$	−19.1803	−1.23551	−0.617757	−	0.786369i	$-0.711959\pi$
$241$	−19.1803	−1.23551	−0.617757	+	0.786369i	$0.711959\pi$
$242$	3.23607	0.208022
$243$	−9.65248	−0.619207
$244$	−3.76393	−0.240961
$245$	0	0
$246$	−1.85410	−0.118213
$247$	17.5623	1.11746
$248$	11.3820	0.722756
$249$	−0.180340	−0.0114286
$250$	0	0
$251$	6.27051	0.395791	0.197896	−	0.980223i	$-0.436589\pi$
$251$	6.27051	0.395791	0.197896	+	0.980223i	$0.436589\pi$
$252$	5.29180	0.333352
$253$	3.70820	0.233133
$254$	15.7082	0.985620
$255$	0	0
$256$	13.5623	0.847644
$257$	29.3607	1.83147	0.915734	−	0.401784i	$-0.131610\pi$
$257$	29.3607	1.83147	0.915734	+	0.401784i	$0.131610\pi$
$258$	−5.56231	−0.346294
$259$	−11.1246	−0.691250
$260$	0	0
$261$	−19.1459	−1.18510
$262$	19.8541	1.22659
$263$	16.3262	1.00672	0.503359	−	0.864077i	$-0.332097\pi$
$263$	16.3262	1.00672	0.503359	+	0.864077i	$0.332097\pi$
$264$	2.56231	0.157699
$265$	0	0
$266$	−17.5623	−1.07681
$267$	5.12461	0.313621
$268$	−8.56231	−0.523026
$269$	2.56231	0.156227	0.0781133	−	0.996944i	$-0.475110\pi$
$269$	2.56231	0.156227	0.0781133	+	0.996944i	$0.475110\pi$
$270$	0	0
$271$	20.0902	1.22039	0.610195	−	0.792251i	$-0.291091\pi$
$271$	20.0902	1.22039	0.610195	+	0.792251i	$0.291091\pi$
$272$	−20.5623	−1.24677
$273$	−5.56231	−0.336646
$274$	−9.09017	−0.549157
$275$	0	0
$276$	0.291796	0.0175641
$277$	5.29180	0.317953	0.158977	−	0.987282i	$-0.449181\pi$
$277$	5.29180	0.317953	0.158977	+	0.987282i	$0.449181\pi$
$278$	19.7984	1.18743
$279$	−14.5279	−0.869760
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	−5.14590	−0.306434
$283$	−15.7082	−0.933756	−0.466878	−	0.884322i	$-0.654621\pi$
$283$	−15.7082	−0.933756	−0.466878	+	0.884322i	$0.654621\pi$
$284$	−1.85410	−0.110021
$285$	0	0
$286$	−23.5623	−1.39327
$287$	9.00000	0.531253
$288$	−9.65248	−0.568778
$289$	0.944272	0.0555454
$290$	0	0
$291$	−3.00000	−0.175863
$292$	1.14590	0.0670586
$293$	−21.3607	−1.24790	−0.623952	−	0.781463i	$-0.714474\pi$
$293$	−21.3607	−1.24790	−0.623952	+	0.781463i	$0.714474\pi$
$294$	1.23607	0.0720889
$295$	0	0
$296$	8.29180	0.481951
$297$	−6.70820	−0.389249
$298$	24.2705	1.40595
$299$	6.00000	0.346989
$300$	0	0
$301$	27.0000	1.55625
$302$	34.1246	1.96365
$303$	1.14590	0.0658301
$304$	17.5623	1.00727
$305$	0	0
$306$	19.5623	1.11830
$307$	6.27051	0.357877	0.178938	−	0.983860i	$-0.442734\pi$
$307$	6.27051	0.357877	0.178938	+	0.983860i	$0.442734\pi$
$308$	5.56231	0.316942
$309$	−4.85410	−0.276140
$310$	0	0
$311$	21.2705	1.20614	0.603070	−	0.797688i	$-0.293944\pi$
$311$	21.2705	1.20614	0.603070	+	0.797688i	$0.293944\pi$
$312$	4.14590	0.234715
$313$	19.4164	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$313$	19.4164	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$314$	19.8541	1.12043
$315$	0	0
$316$	0.326238	0.0183523
$317$	−26.9443	−1.51334	−0.756671	−	0.653796i	$-0.773175\pi$
$317$	−26.9443	−1.51334	−0.756671	+	0.653796i	$0.773175\pi$
$318$	2.85410	0.160050
$319$	−20.1246	−1.12676
$320$	0	0
$321$	−0.0344419	−0.00192236
$322$	−6.00000	−0.334367
$323$	−15.3262	−0.852775
$324$	4.76393	0.264663
$325$	0	0
$326$	32.1246	1.77922
$327$	−2.03444	−0.112505
$328$	−6.70820	−0.370399
$329$	24.9787	1.37712
$330$	0	0
$331$	−21.0902	−1.15922	−0.579610	−	0.814894i	$-0.696795\pi$
$331$	−21.0902	−1.15922	−0.579610	+	0.814894i	$0.696795\pi$
$332$	0.291796	0.0160144
$333$	−10.5836	−0.579977
$334$	−14.9443	−0.817714
$335$	0	0
$336$	−5.56231	−0.303449
$337$	−29.8328	−1.62510	−0.812549	−	0.582894i	$-0.801920\pi$
$337$	−29.8328	−1.62510	−0.812549	+	0.582894i	$0.801920\pi$
$338$	−17.0902	−0.929583
$339$	−0.785218	−0.0426472
$340$	0	0
$341$	−15.2705	−0.826944
$342$	−16.7082	−0.903476
$343$	15.0000	0.809924
$344$	−20.1246	−1.08505
$345$	0	0
$346$	0.0901699	0.00484757
$347$	15.9443	0.855933	0.427967	−	0.903795i	$-0.359230\pi$
$347$	15.9443	0.855933	0.427967	+	0.903795i	$0.359230\pi$
$348$	−1.58359	−0.0848894
$349$	−17.3607	−0.929296	−0.464648	−	0.885496i	$-0.653819\pi$
$349$	−17.3607	−0.929296	−0.464648	+	0.885496i	$0.653819\pi$
$350$	0	0
$351$	−10.8541	−0.579349
$352$	−10.1459	−0.540778
$353$	3.88854	0.206966	0.103483	−	0.994631i	$-0.467001\pi$
$353$	3.88854	0.206966	0.103483	+	0.994631i	$0.467001\pi$
$354$	2.56231	0.136185
$355$	0	0
$356$	−8.29180	−0.439464
$357$	4.85410	0.256906
$358$	−10.8541	−0.573657
$359$	−10.8541	−0.572858	−0.286429	−	0.958102i	$-0.592468\pi$
$359$	−10.8541	−0.572858	−0.286429	+	0.958102i	$0.592468\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	−29.4164	−1.54609
$363$	0.763932	0.0400960
$364$	9.00000	0.471728
$365$	0	0
$366$	−3.76393	−0.196744
$367$	1.14590	0.0598154	0.0299077	−	0.999553i	$-0.490479\pi$
$367$	1.14590	0.0598154	0.0299077	+	0.999553i	$0.490479\pi$
$368$	6.00000	0.312772
$369$	8.56231	0.445736
$370$	0	0
$371$	−13.8541	−0.719269
$372$	−1.20163	−0.0623014
$373$	1.85410	0.0960018	0.0480009	−	0.998847i	$-0.484715\pi$
$373$	1.85410	0.0960018	0.0480009	+	0.998847i	$0.484715\pi$
$374$	20.5623	1.06325
$375$	0	0
$376$	−18.6180	−0.960152
$377$	−32.5623	−1.67704
$378$	10.8541	0.558275
$379$	−7.56231	−0.388450	−0.194225	−	0.980957i	$-0.562219\pi$
$379$	−7.56231	−0.388450	−0.194225	+	0.980957i	$0.562219\pi$
$380$	0	0
$381$	3.70820	0.189977
$382$	−19.4164	−0.993430
$383$	35.4721	1.81254	0.906271	−	0.422698i	$-0.138917\pi$
$383$	35.4721	1.81254	0.906271	+	0.422698i	$0.138917\pi$
$384$	5.20163	0.265444
$385$	0	0
$386$	−24.7082	−1.25761
$387$	25.6869	1.30574
$388$	4.85410	0.246430
$389$	1.58359	0.0802913	0.0401457	−	0.999194i	$-0.487218\pi$
$389$	1.58359	0.0802913	0.0401457	+	0.999194i	$0.487218\pi$
$390$	0	0
$391$	−5.23607	−0.264799
$392$	4.47214	0.225877
$393$	4.68692	0.236424
$394$	6.56231	0.330604
$395$	0	0
$396$	5.29180	0.265923
$397$	−16.4164	−0.823916	−0.411958	−	0.911203i	$-0.635155\pi$
$397$	−16.4164	−0.823916	−0.411958	+	0.911203i	$0.635155\pi$
$398$	−26.1803	−1.31230
$399$	−4.14590	−0.207555
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	−8.56231	−0.427049
$403$	−24.7082	−1.23080
$404$	−1.85410	−0.0922450
$405$	0	0
$406$	32.5623	1.61604
$407$	−11.1246	−0.551427
$408$	−3.61803	−0.179119
$409$	−22.7639	−1.12560	−0.562802	−	0.826592i	$-0.690277\pi$
$409$	−22.7639	−1.12560	−0.562802	+	0.826592i	$0.690277\pi$
$410$	0	0
$411$	−2.14590	−0.105849
$412$	7.85410	0.386944
$413$	−12.4377	−0.612019
$414$	−5.70820	−0.280543
$415$	0	0
$416$	−16.4164	−0.804881
$417$	4.67376	0.228875
$418$	−17.5623	−0.859000
$419$	−6.70820	−0.327717	−0.163859	−	0.986484i	$-0.552394\pi$
$419$	−6.70820	−0.327717	−0.163859	+	0.986484i	$0.552394\pi$
$420$	0	0
$421$	8.90983	0.434239	0.217119	−	0.976145i	$-0.430334\pi$
$421$	8.90983	0.434239	0.217119	+	0.976145i	$0.430334\pi$
$422$	6.76393	0.329263
$423$	23.7639	1.15544
$424$	10.3262	0.501486
$425$	0	0
$426$	−1.85410	−0.0898315
$427$	18.2705	0.884172
$428$	0.0557281	0.00269372
$429$	−5.56231	−0.268551
$430$	0	0
$431$	−12.2705	−0.591050	−0.295525	−	0.955335i	$-0.595495\pi$
$431$	−12.2705	−0.591050	−0.295525	+	0.955335i	$0.595495\pi$
$432$	−10.8541	−0.522218
$433$	14.2918	0.686820	0.343410	−	0.939186i	$-0.388418\pi$
$433$	14.2918	0.686820	0.343410	+	0.939186i	$0.388418\pi$
$434$	24.7082	1.18603
$435$	0	0
$436$	3.29180	0.157648
$437$	4.47214	0.213931
$438$	1.14590	0.0547531
$439$	30.1246	1.43777	0.718885	−	0.695129i	$-0.244653\pi$
$439$	30.1246	1.43777	0.718885	+	0.695129i	$0.244653\pi$
$440$	0	0
$441$	−5.70820	−0.271819
$442$	33.2705	1.58252
$443$	7.18034	0.341148	0.170574	−	0.985345i	$-0.445438\pi$
$443$	7.18034	0.341148	0.170574	+	0.985345i	$0.445438\pi$
$444$	−0.875388	−0.0415441
$445$	0	0
$446$	−3.00000	−0.142054
$447$	5.72949	0.270996
$448$	−12.7082	−0.600406
$449$	−39.2705	−1.85329	−0.926645	−	0.375938i	$-0.877321\pi$
$449$	−39.2705	−1.85329	−0.926645	+	0.375938i	$0.877321\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	1.27051	0.0597598
$453$	8.05573	0.378491
$454$	10.7082	0.502561
$455$	0	0
$456$	3.09017	0.144710
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	2.23607	0.104485
$459$	9.47214	0.442121
$460$	0	0
$461$	36.2705	1.68929	0.844643	−	0.535330i	$-0.179813\pi$
$461$	36.2705	1.68929	0.844643	+	0.535330i	$0.179813\pi$
$462$	5.56231	0.258782
$463$	−14.1246	−0.656426	−0.328213	−	0.944604i	$-0.606446\pi$
$463$	−14.1246	−0.656426	−0.328213	+	0.944604i	$0.606446\pi$
$464$	−32.5623	−1.51167
$465$	0	0
$466$	43.5066	2.01540
$467$	39.2361	1.81563	0.907814	−	0.419372i	$-0.137750\pi$
$467$	39.2361	1.81563	0.907814	+	0.419372i	$0.137750\pi$
$468$	8.56231	0.395793
$469$	41.5623	1.91917
$470$	0	0
$471$	4.68692	0.215962
$472$	9.27051	0.426710
$473$	27.0000	1.24146
$474$	0.326238	0.0149846
$475$	0	0
$476$	−7.85410	−0.359992
$477$	−13.1803	−0.603486
$478$	−41.8328	−1.91339
$479$	−32.5623	−1.48781	−0.743905	−	0.668286i	$-0.767028\pi$
$479$	−32.5623	−1.48781	−0.743905	+	0.668286i	$0.767028\pi$
$480$	0	0
$481$	−18.0000	−0.820729
$482$	31.0344	1.41358
$483$	−1.41641	−0.0644488
$484$	−1.23607	−0.0561849
$485$	0	0
$486$	15.6180	0.708448
$487$	3.70820	0.168035	0.0840174	−	0.996464i	$-0.473225\pi$
$487$	3.70820	0.168035	0.0840174	+	0.996464i	$0.473225\pi$
$488$	−13.6180	−0.616459
$489$	7.58359	0.342942
$490$	0	0
$491$	36.2705	1.63687	0.818433	−	0.574603i	$-0.194843\pi$
$491$	36.2705	1.63687	0.818433	+	0.574603i	$0.194843\pi$
$492$	0.708204	0.0319283
$493$	28.4164	1.27981
$494$	−28.4164	−1.27851
$495$	0	0
$496$	−24.7082	−1.10943
$497$	9.00000	0.403705
$498$	0.291796	0.0130757
$499$	42.3607	1.89632	0.948162	−	0.317787i	$-0.102940\pi$
$499$	42.3607	1.89632	0.948162	+	0.317787i	$0.102940\pi$
$500$	0	0
$501$	−3.52786	−0.157613
$502$	−10.1459	−0.452834
$503$	25.7984	1.15029	0.575146	−	0.818051i	$-0.304945\pi$
$503$	25.7984	1.15029	0.575146	+	0.818051i	$0.304945\pi$
$504$	19.1459	0.852826
$505$	0	0
$506$	−6.00000	−0.266733
$507$	−4.03444	−0.179176
$508$	−6.00000	−0.266207
$509$	13.4164	0.594672	0.297336	−	0.954773i	$-0.403902\pi$
$509$	13.4164	0.594672	0.297336	+	0.954773i	$0.403902\pi$
$510$	0	0
$511$	−5.56231	−0.246062
$512$	5.29180	0.233867
$513$	−8.09017	−0.357190
$514$	−47.5066	−2.09543
$515$	0	0
$516$	2.12461	0.0935308
$517$	24.9787	1.09856
$518$	18.0000	0.790875
$519$	0.0212862	0.000934362 0
$520$	0	0
$521$	6.27051	0.274716	0.137358	−	0.990521i	$-0.456139\pi$
$521$	6.27051	0.274716	0.137358	+	0.990521i	$0.456139\pi$
$522$	30.9787	1.35590
$523$	−37.4164	−1.63611	−0.818053	−	0.575143i	$-0.804946\pi$
$523$	−37.4164	−1.63611	−0.818053	+	0.575143i	$0.804946\pi$
$524$	−7.58359	−0.331291
$525$	0	0
$526$	−26.4164	−1.15181
$527$	21.5623	0.939269
$528$	−5.56231	−0.242068
$529$	−21.4721	−0.933571
$530$	0	0
$531$	−11.8328	−0.513500
$532$	6.70820	0.290838
$533$	14.5623	0.630763
$534$	−8.29180	−0.358821
$535$	0	0
$536$	−30.9787	−1.33808
$537$	−2.56231	−0.110572
$538$	−4.14590	−0.178742
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−34.1803	−1.46953	−0.734764	−	0.678323i	$-0.762707\pi$
$541$	−34.1803	−1.46953	−0.734764	+	0.678323i	$0.762707\pi$
$542$	−32.5066	−1.39628
$543$	−6.94427	−0.298007
$544$	14.3262	0.614232
$545$	0	0
$546$	9.00000	0.385164
$547$	−22.1459	−0.946890	−0.473445	−	0.880823i	$-0.656990\pi$
$547$	−22.1459	−0.946890	−0.473445	+	0.880823i	$0.656990\pi$
$548$	3.47214	0.148322
$549$	17.3820	0.741844
$550$	0	0
$551$	−24.2705	−1.03396
$552$	1.05573	0.0449348
$553$	−1.58359	−0.0673412
$554$	−8.56231	−0.363778
$555$	0	0
$556$	−7.56231	−0.320713
$557$	−40.3607	−1.71014	−0.855068	−	0.518515i	$-0.826485\pi$
$557$	−40.3607	−1.71014	−0.855068	+	0.518515i	$0.826485\pi$
$558$	23.5066	0.995113
$559$	43.6869	1.84776
$560$	0	0
$561$	4.85410	0.204940
$562$	−19.4164	−0.819032
$563$	7.05573	0.297363	0.148682	−	0.988885i	$-0.452497\pi$
$563$	7.05573	0.297363	0.148682	+	0.988885i	$0.452497\pi$
$564$	1.96556	0.0827649
$565$	0	0
$566$	25.4164	1.06833
$567$	−23.1246	−0.971142
$568$	−6.70820	−0.281470
$569$	26.8328	1.12489	0.562445	−	0.826835i	$-0.309861\pi$
$569$	26.8328	1.12489	0.562445	+	0.826835i	$0.309861\pi$
$570$	0	0
$571$	−13.0000	−0.544033	−0.272017	−	0.962293i	$-0.587691\pi$
$571$	−13.0000	−0.544033	−0.272017	+	0.962293i	$0.587691\pi$
$572$	9.00000	0.376309
$573$	−4.58359	−0.191482
$574$	−14.5623	−0.607819
$575$	0	0
$576$	−12.0902	−0.503757
$577$	29.5623	1.23069	0.615347	−	0.788256i	$-0.289016\pi$
$577$	29.5623	1.23069	0.615347	+	0.788256i	$0.289016\pi$
$578$	−1.52786	−0.0635508
$579$	−5.83282	−0.242404
$580$	0	0
$581$	−1.41641	−0.0587625
$582$	4.85410	0.201209
$583$	−13.8541	−0.573778
$584$	4.14590	0.171558
$585$	0	0
$586$	34.5623	1.42776
$587$	−2.47214	−0.102036	−0.0510180	−	0.998698i	$-0.516247\pi$
$587$	−2.47214	−0.102036	−0.0510180	+	0.998698i	$0.516247\pi$
$588$	−0.472136	−0.0194706
$589$	−18.4164	−0.758835
$590$	0	0
$591$	1.54915	0.0637235
$592$	−18.0000	−0.739795
$593$	−41.3607	−1.69848	−0.849240	−	0.528007i	$-0.822939\pi$
$593$	−41.3607	−1.69848	−0.849240	+	0.528007i	$0.822939\pi$
$594$	10.8541	0.445349
$595$	0	0
$596$	−9.27051	−0.379735
$597$	−6.18034	−0.252944
$598$	−9.70820	−0.396998
$599$	−24.2705	−0.991666	−0.495833	−	0.868418i	$-0.665137\pi$
$599$	−24.2705	−0.991666	−0.495833	+	0.868418i	$0.665137\pi$
$600$	0	0
$601$	−0.639320	−0.0260784	−0.0130392	−	0.999915i	$-0.504151\pi$
$601$	−0.639320	−0.0260784	−0.0130392	+	0.999915i	$0.504151\pi$
$602$	−43.6869	−1.78055
$603$	39.5410	1.61023
$604$	−13.0344	−0.530364
$605$	0	0
$606$	−1.85410	−0.0753177
$607$	30.5410	1.23962	0.619811	−	0.784751i	$-0.287209\pi$
$607$	30.5410	1.23962	0.619811	+	0.784751i	$0.287209\pi$
$608$	−12.2361	−0.496238
$609$	7.68692	0.311490
$610$	0	0
$611$	40.4164	1.63507
$612$	−7.47214	−0.302043
$613$	4.41641	0.178377	0.0891885	−	0.996015i	$-0.471573\pi$
$613$	4.41641	0.178377	0.0891885	+	0.996015i	$0.471573\pi$
$614$	−10.1459	−0.409455
$615$	0	0
$616$	20.1246	0.810844
$617$	−25.7639	−1.03722	−0.518608	−	0.855012i	$-0.673550\pi$
$617$	−25.7639	−1.03722	−0.518608	+	0.855012i	$0.673550\pi$
$618$	7.85410	0.315938
$619$	−20.5279	−0.825085	−0.412542	−	0.910938i	$-0.635359\pi$
$619$	−20.5279	−0.825085	−0.412542	+	0.910938i	$0.635359\pi$
$620$	0	0
$621$	−2.76393	−0.110913
$622$	−34.4164	−1.37997
$623$	40.2492	1.61255
$624$	−9.00000	−0.360288
$625$	0	0
$626$	−31.4164	−1.25565
$627$	−4.14590	−0.165571
$628$	−7.58359	−0.302618
$629$	15.7082	0.626327
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	1.18034	0.0469514
$633$	1.59675	0.0634650
$634$	43.5967	1.73145
$635$	0	0
$636$	−1.09017	−0.0432281
$637$	−9.70820	−0.384653
$638$	32.5623	1.28915
$639$	8.56231	0.338720
$640$	0	0
$641$	−8.72949	−0.344794	−0.172397	−	0.985028i	$-0.555151\pi$
$641$	−8.72949	−0.344794	−0.172397	+	0.985028i	$0.555151\pi$
$642$	0.0557281	0.00219941
$643$	−18.2705	−0.720519	−0.360259	−	0.932852i	$-0.617312\pi$
$643$	−18.2705	−0.720519	−0.360259	+	0.932852i	$0.617312\pi$
$644$	2.29180	0.0903094
$645$	0	0
$646$	24.7984	0.975679
$647$	−20.2361	−0.795562	−0.397781	−	0.917480i	$-0.630220\pi$
$647$	−20.2361	−0.795562	−0.397781	+	0.917480i	$0.630220\pi$
$648$	17.2361	0.677097
$649$	−12.4377	−0.488222
$650$	0	0
$651$	5.83282	0.228606
$652$	−12.2705	−0.480550
$653$	−4.65248	−0.182065	−0.0910327	−	0.995848i	$-0.529017\pi$
$653$	−4.65248	−0.182065	−0.0910327	+	0.995848i	$0.529017\pi$
$654$	3.29180	0.128719
$655$	0	0
$656$	14.5623	0.568563
$657$	−5.29180	−0.206453
$658$	−40.4164	−1.57560
$659$	−25.8541	−1.00713	−0.503566	−	0.863957i	$-0.667979\pi$
$659$	−25.8541	−1.00713	−0.503566	+	0.863957i	$0.667979\pi$
$660$	0	0
$661$	14.3607	0.558566	0.279283	−	0.960209i	$-0.409903\pi$
$661$	14.3607	0.558566	0.279283	+	0.960209i	$0.409903\pi$
$662$	34.1246	1.32629
$663$	7.85410	0.305028
$664$	1.05573	0.0409702
$665$	0	0
$666$	17.1246	0.663565
$667$	−8.29180	−0.321060
$668$	5.70820	0.220857
$669$	−0.708204	−0.0273807
$670$	0	0
$671$	18.2705	0.705325
$672$	3.87539	0.149496
$673$	−31.6869	−1.22144	−0.610720	−	0.791846i	$-0.709120\pi$
$673$	−31.6869	−1.22144	−0.610720	+	0.791846i	$0.709120\pi$
$674$	48.2705	1.85931
$675$	0	0
$676$	6.52786	0.251072
$677$	4.11146	0.158016	0.0790080	−	0.996874i	$-0.474825\pi$
$677$	4.11146	0.158016	0.0790080	+	0.996874i	$0.474825\pi$
$678$	1.27051	0.0487936
$679$	−23.5623	−0.904238
$680$	0	0
$681$	2.52786	0.0968680
$682$	24.7082	0.946126
$683$	−43.0689	−1.64799	−0.823993	−	0.566601i	$-0.808258\pi$
$683$	−43.0689	−1.64799	−0.823993	+	0.566601i	$0.808258\pi$
$684$	6.38197	0.244021
$685$	0	0
$686$	−24.2705	−0.926652
$687$	0.527864	0.0201393
$688$	43.6869	1.66555
$689$	−22.4164	−0.853997
$690$	0	0
$691$	25.8197	0.982226	0.491113	−	0.871096i	$-0.336590\pi$
$691$	25.8197	0.982226	0.491113	+	0.871096i	$0.336590\pi$
$692$	−0.0344419	−0.00130928
$693$	−25.6869	−0.975765
$694$	−25.7984	−0.979293
$695$	0	0
$696$	−5.72949	−0.217176
$697$	−12.7082	−0.481358
$698$	28.0902	1.06323
$699$	10.2705	0.388466
$700$	0	0
$701$	2.72949	0.103091	0.0515457	−	0.998671i	$-0.483585\pi$
$701$	2.72949	0.103091	0.0515457	+	0.998671i	$0.483585\pi$
$702$	17.5623	0.662847
$703$	−13.4164	−0.506009
$704$	−12.7082	−0.478958
$705$	0	0
$706$	−6.29180	−0.236795
$707$	9.00000	0.338480
$708$	−0.978714	−0.0367823
$709$	−17.0344	−0.639742	−0.319871	−	0.947461i	$-0.603640\pi$
$709$	−17.0344	−0.639742	−0.319871	+	0.947461i	$0.603640\pi$
$710$	0	0
$711$	−1.50658	−0.0565011
$712$	−30.0000	−1.12430
$713$	−6.29180	−0.235630
$714$	−7.85410	−0.293932
$715$	0	0
$716$	4.14590	0.154939
$717$	−9.87539	−0.368803
$718$	17.5623	0.655419
$719$	47.5623	1.77377	0.886887	−	0.461986i	$-0.152863\pi$
$719$	47.5623	1.77377	0.886887	+	0.461986i	$0.152863\pi$
$720$	0	0
$721$	−38.1246	−1.41983
$722$	9.56231	0.355872
$723$	7.32624	0.272466
$724$	11.2361	0.417585
$725$	0	0
$726$	−1.23607	−0.0458748
$727$	38.8328	1.44023	0.720115	−	0.693855i	$-0.244089\pi$
$727$	38.8328	1.44023	0.720115	+	0.693855i	$0.244089\pi$
$728$	32.5623	1.20684
$729$	−19.4377	−0.719915
$730$	0	0
$731$	−38.1246	−1.41009
$732$	1.43769	0.0531387
$733$	23.5623	0.870294	0.435147	−	0.900360i	$-0.356696\pi$
$733$	23.5623	0.870294	0.435147	+	0.900360i	$0.356696\pi$
$734$	−1.85410	−0.0684362
$735$	0	0
$736$	−4.18034	−0.154089
$737$	41.5623	1.53097
$738$	−13.8541	−0.509977
$739$	17.7639	0.653457	0.326728	−	0.945118i	$-0.394054\pi$
$739$	17.7639	0.653457	0.326728	+	0.945118i	$0.394054\pi$
$740$	0	0
$741$	−6.70820	−0.246432
$742$	22.4164	0.822932
$743$	−25.9098	−0.950539	−0.475270	−	0.879840i	$-0.657650\pi$
$743$	−25.9098	−0.950539	−0.475270	+	0.879840i	$0.657650\pi$
$744$	−4.34752	−0.159388
$745$	0	0
$746$	−3.00000	−0.109838
$747$	−1.34752	−0.0493033
$748$	−7.85410	−0.287174
$749$	−0.270510	−0.00988421
$750$	0	0
$751$	−15.3607	−0.560519	−0.280260	−	0.959924i	$-0.590421\pi$
$751$	−15.3607	−0.560519	−0.280260	+	0.959924i	$0.590421\pi$
$752$	40.4164	1.47383
$753$	−2.39512	−0.0872831
$754$	52.6869	1.91874
$755$	0	0
$756$	−4.14590	−0.150785
$757$	27.0000	0.981332	0.490666	−	0.871348i	$-0.336754\pi$
$757$	27.0000	0.981332	0.490666	+	0.871348i	$0.336754\pi$
$758$	12.2361	0.444434
$759$	−1.41641	−0.0514123
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−6.00000	−0.217357
$763$	−15.9787	−0.578468
$764$	7.41641	0.268316
$765$	0	0
$766$	−57.3951	−2.07377
$767$	−20.1246	−0.726658
$768$	−5.18034	−0.186929
$769$	−12.8885	−0.464773	−0.232386	−	0.972624i	$-0.574653\pi$
$769$	−12.8885	−0.464773	−0.232386	+	0.972624i	$0.574653\pi$
$770$	0	0
$771$	−11.2148	−0.403890
$772$	9.43769	0.339670
$773$	−21.2361	−0.763808	−0.381904	−	0.924202i	$-0.624732\pi$
$773$	−21.2361	−0.763808	−0.381904	+	0.924202i	$0.624732\pi$
$774$	−41.5623	−1.49393
$775$	0	0
$776$	17.5623	0.630450
$777$	4.24922	0.152440
$778$	−2.56231	−0.0918631
$779$	10.8541	0.388889
$780$	0	0
$781$	9.00000	0.322045
$782$	8.47214	0.302963
$783$	15.0000	0.536056
$784$	−9.70820	−0.346722
$785$	0	0
$786$	−7.58359	−0.270498
$787$	−11.2918	−0.402509	−0.201255	−	0.979539i	$-0.564502\pi$
$787$	−11.2918	−0.402509	−0.201255	+	0.979539i	$0.564502\pi$
$788$	−2.50658	−0.0892931
$789$	−6.23607	−0.222010
$790$	0	0
$791$	−6.16718	−0.219280
$792$	19.1459	0.680320
$793$	29.5623	1.04979
$794$	26.5623	0.942661
$795$	0	0
$796$	10.0000	0.354441
$797$	35.2148	1.24737	0.623686	−	0.781675i	$-0.285634\pi$
$797$	35.2148	1.24737	0.623686	+	0.781675i	$0.285634\pi$
$798$	6.70820	0.237468
$799$	−35.2705	−1.24778
$800$	0	0
$801$	38.2918	1.35297
$802$	−19.4164	−0.685617
$803$	−5.56231	−0.196290
$804$	3.27051	0.115342
$805$	0	0
$806$	39.9787	1.40819
$807$	−0.978714	−0.0344524
$808$	−6.70820	−0.235994
$809$	−5.12461	−0.180172	−0.0900859	−	0.995934i	$-0.528714\pi$
$809$	−5.12461	−0.180172	−0.0900859	+	0.995934i	$0.528714\pi$
$810$	0	0
$811$	−11.8197	−0.415044	−0.207522	−	0.978230i	$-0.566540\pi$
$811$	−11.8197	−0.415044	−0.207522	+	0.978230i	$0.566540\pi$
$812$	−12.4377	−0.436477
$813$	−7.67376	−0.269131
$814$	18.0000	0.630900
$815$	0	0
$816$	7.85410	0.274949
$817$	32.5623	1.13921
$818$	36.8328	1.28783
$819$	−41.5623	−1.45230
$820$	0	0
$821$	21.2705	0.742346	0.371173	−	0.928564i	$-0.378956\pi$
$821$	21.2705	0.742346	0.371173	+	0.928564i	$0.378956\pi$
$822$	3.47214	0.121105
$823$	−22.4164	−0.781387	−0.390693	−	0.920521i	$-0.627765\pi$
$823$	−22.4164	−0.781387	−0.390693	+	0.920521i	$0.627765\pi$
$824$	28.4164	0.989932
$825$	0	0
$826$	20.1246	0.700225
$827$	−31.6180	−1.09947	−0.549733	−	0.835340i	$-0.685271\pi$
$827$	−31.6180	−1.09947	−0.549733	+	0.835340i	$0.685271\pi$
$828$	2.18034	0.0757720
$829$	6.25735	0.217327	0.108663	−	0.994079i	$-0.465343\pi$
$829$	6.25735	0.217327	0.108663	+	0.994079i	$0.465343\pi$
$830$	0	0
$831$	−2.02129	−0.0701176
$832$	−20.5623	−0.712870
$833$	8.47214	0.293542
$834$	−7.56231	−0.261861
$835$	0	0
$836$	6.70820	0.232008
$837$	11.3820	0.393418
$838$	10.8541	0.374949
$839$	29.3951	1.01483	0.507416	−	0.861701i	$-0.330601\pi$
$839$	29.3951	1.01483	0.507416	+	0.861701i	$0.330601\pi$
$840$	0	0
$841$	16.0000	0.551724
$842$	−14.4164	−0.496822
$843$	−4.58359	−0.157867
$844$	−2.58359	−0.0889309
$845$	0	0
$846$	−38.4508	−1.32197
$847$	6.00000	0.206162
$848$	−22.4164	−0.769783
$849$	6.00000	0.205919
$850$	0	0
$851$	−4.58359	−0.157124
$852$	0.708204	0.0242627
$853$	−5.83282	−0.199712	−0.0998559	−	0.995002i	$-0.531838\pi$
$853$	−5.83282	−0.199712	−0.0998559	+	0.995002i	$0.531838\pi$
$854$	−29.5623	−1.01160
$855$	0	0
$856$	0.201626	0.00689144
$857$	−34.1803	−1.16758	−0.583789	−	0.811905i	$-0.698431\pi$
$857$	−34.1803	−1.16758	−0.583789	+	0.811905i	$0.698431\pi$
$858$	9.00000	0.307255
$859$	−25.1246	−0.857241	−0.428620	−	0.903485i	$-0.641000\pi$
$859$	−25.1246	−0.857241	−0.428620	+	0.903485i	$0.641000\pi$
$860$	0	0
$861$	−3.43769	−0.117156
$862$	19.8541	0.676233
$863$	−6.76393	−0.230247	−0.115123	−	0.993351i	$-0.536726\pi$
$863$	−6.76393	−0.230247	−0.115123	+	0.993351i	$0.536726\pi$
$864$	7.56231	0.257275
$865$	0	0
$866$	−23.1246	−0.785806
$867$	−0.360680	−0.0122493
$868$	−9.43769	−0.320336
$869$	−1.58359	−0.0537197
$870$	0	0
$871$	67.2492	2.27865
$872$	11.9098	0.403318
$873$	−22.4164	−0.758680
$874$	−7.23607	−0.244764
$875$	0	0
$876$	−0.437694	−0.0147883
$877$	−18.9787	−0.640866	−0.320433	−	0.947271i	$-0.603828\pi$
$877$	−18.9787	−0.640866	−0.320433	+	0.947271i	$0.603828\pi$
$878$	−48.7426	−1.64498
$879$	8.15905	0.275198
$880$	0	0
$881$	45.5410	1.53432	0.767158	−	0.641458i	$-0.221670\pi$
$881$	45.5410	1.53432	0.767158	+	0.641458i	$0.221670\pi$
$882$	9.23607	0.310995
$883$	17.8328	0.600122	0.300061	−	0.953920i	$-0.402993\pi$
$883$	17.8328	0.600122	0.300061	+	0.953920i	$0.402993\pi$
$884$	−12.7082	−0.427423
$885$	0	0
$886$	−11.6180	−0.390315
$887$	−42.4721	−1.42607	−0.713037	−	0.701126i	$-0.752681\pi$
$887$	−42.4721	−1.42607	−0.713037	+	0.701126i	$0.752681\pi$
$888$	−3.16718	−0.106284
$889$	29.1246	0.976808
$890$	0	0
$891$	−23.1246	−0.774704
$892$	1.14590	0.0383675
$893$	30.1246	1.00808
$894$	−9.27051	−0.310052
$895$	0	0
$896$	40.8541	1.36484
$897$	−2.29180	−0.0765208
$898$	63.5410	2.12039
$899$	34.1459	1.13883
$900$	0	0
$901$	19.5623	0.651715
$902$	−14.5623	−0.484872
$903$	−10.3131	−0.343198
$904$	4.59675	0.152886
$905$	0	0
$906$	−13.0344	−0.433040
$907$	−42.2705	−1.40357	−0.701785	−	0.712389i	$-0.747613\pi$
$907$	−42.2705	−1.40357	−0.701785	+	0.712389i	$0.747613\pi$
$908$	−4.09017	−0.135737
$909$	8.56231	0.283994
$910$	0	0
$911$	−36.5410	−1.21066	−0.605329	−	0.795975i	$-0.706958\pi$
$911$	−36.5410	−1.21066	−0.605329	+	0.795975i	$0.706958\pi$
$912$	−6.70820	−0.222131
$913$	−1.41641	−0.0468763
$914$	29.1246	0.963357
$915$	0	0
$916$	−0.854102	−0.0282203
$917$	36.8115	1.21562
$918$	−15.3262	−0.505841
$919$	−24.0689	−0.793959	−0.396980	−	0.917827i	$-0.629942\pi$
$919$	−24.0689	−0.793959	−0.396980	+	0.917827i	$0.629942\pi$
$920$	0	0
$921$	−2.39512	−0.0789219
$922$	−58.6869	−1.93275
$923$	14.5623	0.479324
$924$	−2.12461	−0.0698946
$925$	0	0
$926$	22.8541	0.751032
$927$	−36.2705	−1.19128
$928$	22.6869	0.744735
$929$	36.7082	1.20436	0.602179	−	0.798361i	$-0.294300\pi$
$929$	36.7082	1.20436	0.602179	+	0.798361i	$0.294300\pi$
$930$	0	0
$931$	−7.23607	−0.237153
$932$	−16.6180	−0.544342
$933$	−8.12461	−0.265988
$934$	−63.4853	−2.07730
$935$	0	0
$936$	30.9787	1.01257
$937$	9.43769	0.308316	0.154158	−	0.988046i	$-0.450734\pi$
$937$	9.43769	0.308316	0.154158	+	0.988046i	$0.450734\pi$
$938$	−67.2492	−2.19576
$939$	−7.41641	−0.242025
$940$	0	0
$941$	−27.2705	−0.888993	−0.444497	−	0.895781i	$-0.646617\pi$
$941$	−27.2705	−0.888993	−0.444497	+	0.895781i	$0.646617\pi$
$942$	−7.58359	−0.247087
$943$	3.70820	0.120756
$944$	−20.1246	−0.655000
$945$	0	0
$946$	−43.6869	−1.42038
$947$	−22.5967	−0.734296	−0.367148	−	0.930163i	$-0.619666\pi$
$947$	−22.5967	−0.734296	−0.367148	+	0.930163i	$0.619666\pi$
$948$	−0.124612	−0.00404720
$949$	−9.00000	−0.292152
$950$	0	0
$951$	10.2918	0.333734
$952$	−28.4164	−0.920981
$953$	13.1591	0.426264	0.213132	−	0.977023i	$-0.431634\pi$
$953$	13.1591	0.426264	0.213132	+	0.977023i	$0.431634\pi$
$954$	21.3262	0.690462
$955$	0	0
$956$	15.9787	0.516789
$957$	7.68692	0.248483
$958$	52.6869	1.70224
$959$	−16.8541	−0.544247
$960$	0	0
$961$	−5.09017	−0.164199
$962$	29.1246	0.939015
$963$	−0.257354	−0.00829312
$964$	−11.8541	−0.381795
$965$	0	0
$966$	2.29180	0.0737373
$967$	16.1459	0.519217	0.259609	−	0.965714i	$-0.416406\pi$
$967$	16.1459	0.519217	0.259609	+	0.965714i	$0.416406\pi$
$968$	−4.47214	−0.143740
$969$	5.85410	0.188061
$970$	0	0
$971$	−51.5410	−1.65403	−0.827015	−	0.562180i	$-0.809963\pi$
$971$	−51.5410	−1.65403	−0.827015	+	0.562180i	$0.809963\pi$
$972$	−5.96556	−0.191345
$973$	36.7082	1.17681
$974$	−6.00000	−0.192252
$975$	0	0
$976$	29.5623	0.946266
$977$	23.3820	0.748055	0.374028	−	0.927418i	$-0.377976\pi$
$977$	23.3820	0.748055	0.374028	+	0.927418i	$0.377976\pi$
$978$	−12.2705	−0.392368
$979$	40.2492	1.28637
$980$	0	0
$981$	−15.2016	−0.485351
$982$	−58.6869	−1.87277
$983$	−5.25735	−0.167684	−0.0838418	−	0.996479i	$-0.526719\pi$
$983$	−5.25735	−0.167684	−0.0838418	+	0.996479i	$0.526719\pi$
$984$	2.56231	0.0816833
$985$	0	0
$986$	−45.9787	−1.46426
$987$	−9.54102	−0.303694
$988$	10.8541	0.345315
$989$	11.1246	0.353742
$990$	0	0
$991$	−26.8197	−0.851955	−0.425977	−	0.904734i	$-0.640070\pi$
$991$	−26.8197	−0.851955	−0.425977	+	0.904734i	$0.640070\pi$
$992$	17.2148	0.546570
$993$	8.05573	0.255641
$994$	−14.5623	−0.461888
$995$	0	0
$996$	−0.111456	−0.00353162
$997$	52.8541	1.67391	0.836953	−	0.547275i	$-0.184335\pi$
$997$	52.8541	1.67391	0.836953	+	0.547275i	$0.184335\pi$
$998$	−68.5410	−2.16963
$999$	8.29180	0.262341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	125.2.a.a.1.1	✓	2
3.2	odd	2		1125.2.a.d.1.2		2
4.3	odd	2		2000.2.a.l.1.1		2
5.2	odd	4		125.2.b.b.124.1		4
5.3	odd	4		125.2.b.b.124.4		4
5.4	even	2		125.2.a.b.1.2	yes	2
7.6	odd	2		6125.2.a.d.1.1		2
8.3	odd	2		8000.2.a.c.1.2		2
8.5	even	2		8000.2.a.v.1.1		2
15.2	even	4		1125.2.b.f.874.4		4
15.8	even	4		1125.2.b.f.874.1		4
15.14	odd	2		1125.2.a.c.1.1		2
20.3	even	4		2000.2.c.e.1249.3		4
20.7	even	4		2000.2.c.e.1249.2		4
20.19	odd	2		2000.2.a.a.1.2		2
25.2	odd	20		625.2.e.g.124.1		8
25.3	odd	20		625.2.e.d.249.2		8
25.4	even	10		625.2.d.a.376.1		4
25.6	even	5		625.2.d.j.251.1		4
25.8	odd	20		625.2.e.d.374.1		8
25.9	even	10		625.2.d.g.126.1		4
25.11	even	5		625.2.d.d.501.1		4
25.12	odd	20		625.2.e.g.499.2		8
25.13	odd	20		625.2.e.g.499.1		8
25.14	even	10		625.2.d.g.501.1		4
25.16	even	5		625.2.d.d.126.1		4
25.17	odd	20		625.2.e.d.374.2		8
25.19	even	10		625.2.d.a.251.1		4
25.21	even	5		625.2.d.j.376.1		4
25.22	odd	20		625.2.e.d.249.1		8
25.23	odd	20		625.2.e.g.124.2		8
35.34	odd	2		6125.2.a.g.1.2		2
40.19	odd	2		8000.2.a.u.1.1		2
40.29	even	2		8000.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
125.2.a.a.1.1	✓	2	1.1	even	1	trivial
125.2.a.b.1.2	yes	2	5.4	even	2
125.2.b.b.124.1		4	5.2	odd	4
125.2.b.b.124.4		4	5.3	odd	4
625.2.d.a.251.1		4	25.19	even	10
625.2.d.a.376.1		4	25.4	even	10
625.2.d.d.126.1		4	25.16	even	5
625.2.d.d.501.1		4	25.11	even	5
625.2.d.g.126.1		4	25.9	even	10
625.2.d.g.501.1		4	25.14	even	10
625.2.d.j.251.1		4	25.6	even	5
625.2.d.j.376.1		4	25.21	even	5
625.2.e.d.249.1		8	25.22	odd	20
625.2.e.d.249.2		8	25.3	odd	20
625.2.e.d.374.1		8	25.8	odd	20
625.2.e.d.374.2		8	25.17	odd	20
625.2.e.g.124.1		8	25.2	odd	20
625.2.e.g.124.2		8	25.23	odd	20
625.2.e.g.499.1		8	25.13	odd	20
625.2.e.g.499.2		8	25.12	odd	20
1125.2.a.c.1.1		2	15.14	odd	2
1125.2.a.d.1.2		2	3.2	odd	2
1125.2.b.f.874.1		4	15.8	even	4
1125.2.b.f.874.4		4	15.2	even	4
2000.2.a.a.1.2		2	20.19	odd	2
2000.2.a.l.1.1		2	4.3	odd	2
2000.2.c.e.1249.2		4	20.7	even	4
2000.2.c.e.1249.3		4	20.3	even	4
6125.2.a.d.1.1		2	7.6	odd	2
6125.2.a.g.1.2		2	35.34	odd	2
8000.2.a.c.1.2		2	8.3	odd	2
8000.2.a.d.1.2		2	40.29	even	2
8000.2.a.u.1.1		2	40.19	odd	2
8000.2.a.v.1.1		2	8.5	even	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:	\( N \)	\(=\)	\( 125 = 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	125.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} -0.381966 q^{3} +0.618034 q^{4} +0.618034 q^{6} -3.00000 q^{7} +2.23607 q^{8} -2.85410 q^{9} +O(q^{10})\)	\(q-1.61803 q^{2} -0.381966 q^{3} +0.618034 q^{4} +0.618034 q^{6} -3.00000 q^{7} +2.23607 q^{8} -2.85410 q^{9} -3.00000 q^{11} -0.236068 q^{12} -4.85410 q^{13} +4.85410 q^{14} -4.85410 q^{16} +4.23607 q^{17} +4.61803 q^{18} -3.61803 q^{19} +1.14590 q^{21} +4.85410 q^{22} -1.23607 q^{23} -0.854102 q^{24} +7.85410 q^{26} +2.23607 q^{27} -1.85410 q^{28} +6.70820 q^{29} +5.09017 q^{31} +3.38197 q^{32} +1.14590 q^{33} -6.85410 q^{34} -1.76393 q^{36} +3.70820 q^{37} +5.85410 q^{38} +1.85410 q^{39} -3.00000 q^{41} -1.85410 q^{42} -9.00000 q^{43} -1.85410 q^{44} +2.00000 q^{46} -8.32624 q^{47} +1.85410 q^{48} +2.00000 q^{49} -1.61803 q^{51} -3.00000 q^{52} +4.61803 q^{53} -3.61803 q^{54} -6.70820 q^{56} +1.38197 q^{57} -10.8541 q^{58} +4.14590 q^{59} -6.09017 q^{61} -8.23607 q^{62} +8.56231 q^{63} +4.23607 q^{64} -1.85410 q^{66} -13.8541 q^{67} +2.61803 q^{68} +0.472136 q^{69} -3.00000 q^{71} -6.38197 q^{72} +1.85410 q^{73} -6.00000 q^{74} -2.23607 q^{76} +9.00000 q^{77} -3.00000 q^{78} +0.527864 q^{79} +7.70820 q^{81} +4.85410 q^{82} +0.472136 q^{83} +0.708204 q^{84} +14.5623 q^{86} -2.56231 q^{87} -6.70820 q^{88} -13.4164 q^{89} +14.5623 q^{91} -0.763932 q^{92} -1.94427 q^{93} +13.4721 q^{94} -1.29180 q^{96} +7.85410 q^{97} -3.23607 q^{98} +8.56231 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9}+O(q^{10}) \) 2 * q - q^2 - 3 * q^3 - q^4 - q^6 - 6 * q^7 + q^9	\( 2 q - q^{2} - 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9} - 6 q^{11} + 4 q^{12} - 3 q^{13} + 3 q^{14} - 3 q^{16} + 4 q^{17} + 7 q^{18} - 5 q^{19} + 9 q^{21} + 3 q^{22} + 2 q^{23} + 5 q^{24} + 9 q^{26} + 3 q^{28} - q^{31} + 9 q^{32} + 9 q^{33} - 7 q^{34} - 8 q^{36} - 6 q^{37} + 5 q^{38} - 3 q^{39} - 6 q^{41} + 3 q^{42} - 18 q^{43} + 3 q^{44} + 4 q^{46} - q^{47} - 3 q^{48} + 4 q^{49} - q^{51} - 6 q^{52} + 7 q^{53} - 5 q^{54} + 5 q^{57} - 15 q^{58} + 15 q^{59} - q^{61} - 12 q^{62} - 3 q^{63} + 4 q^{64} + 3 q^{66} - 21 q^{67} + 3 q^{68} - 8 q^{69} - 6 q^{71} - 15 q^{72} - 3 q^{73} - 12 q^{74} + 18 q^{77} - 6 q^{78} + 10 q^{79} + 2 q^{81} + 3 q^{82} - 8 q^{83} - 12 q^{84} + 9 q^{86} + 15 q^{87} + 9 q^{91} - 6 q^{92} + 14 q^{93} + 18 q^{94} - 16 q^{96} + 9 q^{97} - 2 q^{98} - 3 q^{99}+O(q^{100}) \) 2 * q - q^2 - 3 * q^3 - q^4 - q^6 - 6 * q^7 + q^9 - 6 * q^11 + 4 * q^12 - 3 * q^13 + 3 * q^14 - 3 * q^16 + 4 * q^17 + 7 * q^18 - 5 * q^19 + 9 * q^21 + 3 * q^22 + 2 * q^23 + 5 * q^24 + 9 * q^26 + 3 * q^28 - q^31 + 9 * q^32 + 9 * q^33 - 7 * q^34 - 8 * q^36 - 6 * q^37 + 5 * q^38 - 3 * q^39 - 6 * q^41 + 3 * q^42 - 18 * q^43 + 3 * q^44 + 4 * q^46 - q^47 - 3 * q^48 + 4 * q^49 - q^51 - 6 * q^52 + 7 * q^53 - 5 * q^54 + 5 * q^57 - 15 * q^58 + 15 * q^59 - q^61 - 12 * q^62 - 3 * q^63 + 4 * q^64 + 3 * q^66 - 21 * q^67 + 3 * q^68 - 8 * q^69 - 6 * q^71 - 15 * q^72 - 3 * q^73 - 12 * q^74 + 18 * q^77 - 6 * q^78 + 10 * q^79 + 2 * q^81 + 3 * q^82 - 8 * q^83 - 12 * q^84 + 9 * q^86 + 15 * q^87 + 9 * q^91 - 6 * q^92 + 14 * q^93 + 18 * q^94 - 16 * q^96 + 9 * q^97 - 2 * q^98 - 3 * q^99

Introduction

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