LMFDB - Embedded newform 8470.2.a.cr.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.93185$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.41421 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.41421 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.82843 q^{9} +1.00000 q^{10} -2.41421 q^{12} -5.59575 q^{13} -1.00000 q^{14} -2.41421 q^{15} +1.00000 q^{16} -0.449490 q^{17} +2.82843 q^{18} +2.27792 q^{19} +1.00000 q^{20} +2.41421 q^{21} +0.110988 q^{23} -2.41421 q^{24} +1.00000 q^{25} -5.59575 q^{26} +0.414214 q^{27} -1.00000 q^{28} +2.49938 q^{29} -2.41421 q^{30} +4.52004 q^{31} +1.00000 q^{32} -0.449490 q^{34} -1.00000 q^{35} +2.82843 q^{36} -6.42883 q^{37} +2.27792 q^{38} +13.5093 q^{39} +1.00000 q^{40} +7.67752 q^{41} +2.41421 q^{42} -1.34366 q^{43} +2.82843 q^{45} +0.110988 q^{46} +1.51399 q^{47} -2.41421 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.08516 q^{51} -5.59575 q^{52} -6.94887 q^{53} +0.414214 q^{54} -1.00000 q^{56} -5.49938 q^{57} +2.49938 q^{58} +3.97469 q^{59} -2.41421 q^{60} +4.29253 q^{61} +4.52004 q^{62} -2.82843 q^{63} +1.00000 q^{64} -5.59575 q^{65} -11.1708 q^{67} -0.449490 q^{68} -0.267949 q^{69} -1.00000 q^{70} -1.87780 q^{71} +2.82843 q^{72} +3.41421 q^{73} -6.42883 q^{74} -2.41421 q^{75} +2.27792 q^{76} +13.5093 q^{78} -11.2168 q^{79} +1.00000 q^{80} -9.48528 q^{81} +7.67752 q^{82} +16.6960 q^{83} +2.41421 q^{84} -0.449490 q^{85} -1.34366 q^{86} -6.03403 q^{87} +2.17209 q^{89} +2.82843 q^{90} +5.59575 q^{91} +0.110988 q^{92} -10.9123 q^{93} +1.51399 q^{94} +2.27792 q^{95} -2.41421 q^{96} -10.1270 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{10} - 4 q^{12} - 4 q^{14} - 4 q^{15} + 4 q^{16} + 8 q^{17} - 12 q^{19} + 4 q^{20} + 4 q^{21} - 8 q^{23} - 4 q^{24} + 4 q^{25} - 4 q^{27} - 4 q^{28} - 8 q^{29} - 4 q^{30} + 4 q^{32} + 8 q^{34} - 4 q^{35} - 16 q^{37} - 12 q^{38} + 8 q^{39} + 4 q^{40} + 8 q^{41} + 4 q^{42} - 8 q^{43} - 8 q^{46} - 16 q^{47} - 4 q^{48} + 4 q^{49} + 4 q^{50} - 8 q^{51} - 4 q^{54} - 4 q^{56} - 4 q^{57} - 8 q^{58} - 8 q^{59} - 4 q^{60} - 8 q^{61} + 4 q^{64} + 8 q^{68} - 8 q^{69} - 4 q^{70} - 8 q^{71} + 8 q^{73} - 16 q^{74} - 4 q^{75} - 12 q^{76} + 8 q^{78} - 24 q^{79} + 4 q^{80} - 4 q^{81} + 8 q^{82} + 8 q^{83} + 4 q^{84} + 8 q^{85} - 8 q^{86} + 16 q^{87} - 8 q^{92} + 16 q^{93} - 16 q^{94} - 12 q^{95} - 4 q^{96} - 8 q^{97} + 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^10 - 4 * q^12 - 4 * q^14 - 4 * q^15 + 4 * q^16 + 8 * q^17 - 12 * q^19 + 4 * q^20 + 4 * q^21 - 8 * q^23 - 4 * q^24 + 4 * q^25 - 4 * q^27 - 4 * q^28 - 8 * q^29 - 4 * q^30 + 4 * q^32 + 8 * q^34 - 4 * q^35 - 16 * q^37 - 12 * q^38 + 8 * q^39 + 4 * q^40 + 8 * q^41 + 4 * q^42 - 8 * q^43 - 8 * q^46 - 16 * q^47 - 4 * q^48 + 4 * q^49 + 4 * q^50 - 8 * q^51 - 4 * q^54 - 4 * q^56 - 4 * q^57 - 8 * q^58 - 8 * q^59 - 4 * q^60 - 8 * q^61 + 4 * q^64 + 8 * q^68 - 8 * q^69 - 4 * q^70 - 8 * q^71 + 8 * q^73 - 16 * q^74 - 4 * q^75 - 12 * q^76 + 8 * q^78 - 24 * q^79 + 4 * q^80 - 4 * q^81 + 8 * q^82 + 8 * q^83 + 4 * q^84 + 8 * q^85 - 8 * q^86 + 16 * q^87 - 8 * q^92 + 16 * q^93 - 16 * q^94 - 12 * q^95 - 4 * q^96 - 8 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.41421	−0.985599
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	2.82843	0.942809
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−2.41421	−0.696923
$13$	−5.59575	−1.55198	−0.775991	−	0.630743i	$-0.782750\pi$
$13$	−5.59575	−1.55198	−0.775991	+	0.630743i	$0.782750\pi$
$14$	−1.00000	−0.267261
$15$	−2.41421	−0.623347
$16$	1.00000	0.250000
$17$	−0.449490	−0.109017	−0.0545086	−	0.998513i	$-0.517359\pi$
$17$	−0.449490	−0.109017	−0.0545086	+	0.998513i	$0.517359\pi$
$18$	2.82843	0.666667
$19$	2.27792	0.522590	0.261295	−	0.965259i	$-0.415850\pi$
$19$	2.27792	0.522590	0.261295	+	0.965259i	$0.415850\pi$
$20$	1.00000	0.223607
$21$	2.41421	0.526825
$22$	0	0
$23$	0.110988	0.0231426	0.0115713	−	0.999933i	$-0.496317\pi$
$23$	0.110988	0.0231426	0.0115713	+	0.999933i	$0.496317\pi$
$24$	−2.41421	−0.492799
$25$	1.00000	0.200000
$26$	−5.59575	−1.09742
$27$	0.414214	0.0797154
$28$	−1.00000	−0.188982
$29$	2.49938	0.464123	0.232061	−	0.972701i	$-0.425453\pi$
$29$	2.49938	0.464123	0.232061	+	0.972701i	$0.425453\pi$
$30$	−2.41421	−0.440773
$31$	4.52004	0.811824	0.405912	−	0.913912i	$-0.366954\pi$
$31$	4.52004	0.811824	0.405912	+	0.913912i	$0.366954\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.449490	−0.0770869
$35$	−1.00000	−0.169031
$36$	2.82843	0.471405
$37$	−6.42883	−1.05689	−0.528446	−	0.848967i	$-0.677225\pi$
$37$	−6.42883	−1.05689	−0.528446	+	0.848967i	$0.677225\pi$
$38$	2.27792	0.369527
$39$	13.5093	2.16323
$40$	1.00000	0.158114
$41$	7.67752	1.19903	0.599513	−	0.800365i	$-0.295361\pi$
$41$	7.67752	1.19903	0.599513	+	0.800365i	$0.295361\pi$
$42$	2.41421	0.372521
$43$	−1.34366	−0.204906	−0.102453	−	0.994738i	$-0.532669\pi$
$43$	−1.34366	−0.204906	−0.102453	+	0.994738i	$0.532669\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0.110988	0.0163643
$47$	1.51399	0.220838	0.110419	−	0.993885i	$-0.464781\pi$
$47$	1.51399	0.220838	0.110419	+	0.993885i	$0.464781\pi$
$48$	−2.41421	−0.348462
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	1.08516	0.151953
$52$	−5.59575	−0.775991
$53$	−6.94887	−0.954500	−0.477250	−	0.878767i	$-0.658366\pi$
$53$	−6.94887	−0.954500	−0.477250	+	0.878767i	$0.658366\pi$
$54$	0.414214	0.0563673
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−5.49938	−0.728410
$58$	2.49938	0.328184
$59$	3.97469	0.517461	0.258730	−	0.965950i	$-0.416696\pi$
$59$	3.97469	0.517461	0.258730	+	0.965950i	$0.416696\pi$
$60$	−2.41421	−0.311674
$61$	4.29253	0.549602	0.274801	−	0.961501i	$-0.411388\pi$
$61$	4.29253	0.549602	0.274801	+	0.961501i	$0.411388\pi$
$62$	4.52004	0.574046
$63$	−2.82843	−0.356348
$64$	1.00000	0.125000
$65$	−5.59575	−0.694068
$66$	0	0
$67$	−11.1708	−1.36474	−0.682368	−	0.731009i	$-0.739050\pi$
$67$	−11.1708	−1.36474	−0.682368	+	0.731009i	$0.739050\pi$
$68$	−0.449490	−0.0545086
$69$	−0.267949	−0.0322573
$70$	−1.00000	−0.119523
$71$	−1.87780	−0.222854	−0.111427	−	0.993773i	$-0.535542\pi$
$71$	−1.87780	−0.222854	−0.111427	+	0.993773i	$0.535542\pi$
$72$	2.82843	0.333333
$73$	3.41421	0.399603	0.199802	−	0.979836i	$-0.435970\pi$
$73$	3.41421	0.399603	0.199802	+	0.979836i	$0.435970\pi$
$74$	−6.42883	−0.747336
$75$	−2.41421	−0.278769
$76$	2.27792	0.261295
$77$	0	0
$78$	13.5093	1.52963
$79$	−11.2168	−1.26199	−0.630995	−	0.775787i	$-0.717353\pi$
$79$	−11.2168	−1.26199	−0.630995	+	0.775787i	$0.717353\pi$
$80$	1.00000	0.111803
$81$	−9.48528	−1.05392
$82$	7.67752	0.847840
$83$	16.6960	1.83263	0.916315	−	0.400459i	$-0.131149\pi$
$83$	16.6960	1.83263	0.916315	+	0.400459i	$0.131149\pi$
$84$	2.41421	0.263412
$85$	−0.449490	−0.0487540
$86$	−1.34366	−0.144891
$87$	−6.03403	−0.646916
$88$	0	0
$89$	2.17209	0.230241	0.115120	−	0.993352i	$-0.463275\pi$
$89$	2.17209	0.230241	0.115120	+	0.993352i	$0.463275\pi$
$90$	2.82843	0.298142
$91$	5.59575	0.586594
$92$	0.110988	0.0115713
$93$	−10.9123	−1.13156
$94$	1.51399	0.156156
$95$	2.27792	0.233709
$96$	−2.41421	−0.246400
$97$	−10.1270	−1.02824	−0.514121	−	0.857718i	$-0.671882\pi$
$97$	−10.1270	−1.02824	−0.514121	+	0.857718i	$0.671882\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	6.46410	0.643202	0.321601	−	0.946875i	$-0.395779\pi$
$101$	6.46410	0.643202	0.321601	+	0.946875i	$0.395779\pi$
$102$	1.08516	0.107447
$103$	−7.55051	−0.743974	−0.371987	−	0.928238i	$-0.621323\pi$
$103$	−7.55051	−0.743974	−0.371987	+	0.928238i	$0.621323\pi$
$104$	−5.59575	−0.548709
$105$	2.41421	0.235603
$106$	−6.94887	−0.674934
$107$	−6.00481	−0.580507	−0.290253	−	0.956950i	$-0.593740\pi$
$107$	−6.00481	−0.580507	−0.290253	+	0.956950i	$0.593740\pi$
$108$	0.414214	0.0398577
$109$	−1.01461	−0.0971822	−0.0485911	−	0.998819i	$-0.515473\pi$
$109$	−1.01461	−0.0971822	−0.0485911	+	0.998819i	$0.515473\pi$
$110$	0	0
$111$	15.5206	1.47315
$112$	−1.00000	−0.0944911
$113$	−5.83912	−0.549299	−0.274649	−	0.961544i	$-0.588562\pi$
$113$	−5.83912	−0.549299	−0.274649	+	0.961544i	$0.588562\pi$
$114$	−5.49938	−0.515064
$115$	0.110988	0.0103497
$116$	2.49938	0.232061
$117$	−15.8272	−1.46322
$118$	3.97469	0.365900
$119$	0.449490	0.0412047
$120$	−2.41421	−0.220387
$121$	0	0
$122$	4.29253	0.388627
$123$	−18.5352	−1.67126
$124$	4.52004	0.405912
$125$	1.00000	0.0894427
$126$	−2.82843	−0.251976
$127$	8.36257	0.742058	0.371029	−	0.928621i	$-0.379005\pi$
$127$	8.36257	0.742058	0.371029	+	0.928621i	$0.379005\pi$
$128$	1.00000	0.0883883
$129$	3.24389	0.285608
$130$	−5.59575	−0.490780
$131$	−10.5704	−0.923544	−0.461772	−	0.886999i	$-0.652786\pi$
$131$	−10.5704	−0.923544	−0.461772	+	0.886999i	$0.652786\pi$
$132$	0	0
$133$	−2.27792	−0.197520
$134$	−11.1708	−0.965014
$135$	0.414214	0.0356498
$136$	−0.449490	−0.0385434
$137$	−10.8895	−0.930355	−0.465178	−	0.885217i	$-0.654010\pi$
$137$	−10.8895	−0.930355	−0.465178	+	0.885217i	$0.654010\pi$
$138$	−0.267949	−0.0228093
$139$	−15.5493	−1.31887	−0.659436	−	0.751761i	$-0.729205\pi$
$139$	−15.5493	−1.31887	−0.659436	+	0.751761i	$0.729205\pi$
$140$	−1.00000	−0.0845154
$141$	−3.65509	−0.307814
$142$	−1.87780	−0.157581
$143$	0	0
$144$	2.82843	0.235702
$145$	2.49938	0.207562
$146$	3.41421	0.282562
$147$	−2.41421	−0.199121
$148$	−6.42883	−0.528446
$149$	19.0552	1.56106	0.780532	−	0.625116i	$-0.214948\pi$
$149$	19.0552	1.56106	0.780532	+	0.625116i	$0.214948\pi$
$150$	−2.41421	−0.197120
$151$	−23.3590	−1.90092	−0.950462	−	0.310840i	$-0.899390\pi$
$151$	−23.3590	−1.90092	−0.950462	+	0.310840i	$0.899390\pi$
$152$	2.27792	0.184763
$153$	−1.27135	−0.102782
$154$	0	0
$155$	4.52004	0.363059
$156$	13.5093	1.08161
$157$	−16.6617	−1.32975	−0.664873	−	0.746957i	$-0.731514\pi$
$157$	−16.6617	−1.32975	−0.664873	+	0.746957i	$0.731514\pi$
$158$	−11.2168	−0.892362
$159$	16.7761	1.33043
$160$	1.00000	0.0790569
$161$	−0.110988	−0.00874709
$162$	−9.48528	−0.745234
$163$	2.01994	0.158214	0.0791068	−	0.996866i	$-0.474793\pi$
$163$	2.01994	0.158214	0.0791068	+	0.996866i	$0.474793\pi$
$164$	7.67752	0.599513
$165$	0	0
$166$	16.6960	1.29586
$167$	−22.4402	−1.73647	−0.868237	−	0.496149i	$-0.834747\pi$
$167$	−22.4402	−1.73647	−0.868237	+	0.496149i	$0.834747\pi$
$168$	2.41421	0.186261
$169$	18.3125	1.40865
$170$	−0.449490	−0.0344743
$171$	6.44292	0.492703
$172$	−1.34366	−0.102453
$173$	20.2419	1.53896	0.769482	−	0.638668i	$-0.220514\pi$
$173$	20.2419	1.53896	0.769482	+	0.638668i	$0.220514\pi$
$174$	−6.03403	−0.457439
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−9.59575	−0.721261
$178$	2.17209	0.162805
$179$	−16.3132	−1.21931	−0.609653	−	0.792669i	$-0.708691\pi$
$179$	−16.3132	−1.21931	−0.609653	+	0.792669i	$0.708691\pi$
$180$	2.82843	0.210819
$181$	10.0659	0.748193	0.374097	−	0.927390i	$-0.377953\pi$
$181$	10.0659	0.748193	0.374097	+	0.927390i	$0.377953\pi$
$182$	5.59575	0.414785
$183$	−10.3631	−0.766061
$184$	0.110988	0.00818216
$185$	−6.42883	−0.472657
$186$	−10.9123	−0.800132
$187$	0	0
$188$	1.51399	0.110419
$189$	−0.414214	−0.0301296
$190$	2.27792	0.165257
$191$	−12.6211	−0.913228	−0.456614	−	0.889665i	$-0.650938\pi$
$191$	−12.6211	−0.913228	−0.456614	+	0.889665i	$0.650938\pi$
$192$	−2.41421	−0.174231
$193$	18.3984	1.32434	0.662171	−	0.749352i	$-0.269635\pi$
$193$	18.3984	1.32434	0.662171	+	0.749352i	$0.269635\pi$
$194$	−10.1270	−0.727077
$195$	13.5093	0.967424
$196$	1.00000	0.0714286
$197$	−4.24921	−0.302744	−0.151372	−	0.988477i	$-0.548369\pi$
$197$	−4.24921	−0.302744	−0.151372	+	0.988477i	$0.548369\pi$
$198$	0	0
$199$	−16.8211	−1.19242	−0.596209	−	0.802829i	$-0.703327\pi$
$199$	−16.8211	−1.19242	−0.596209	+	0.802829i	$0.703327\pi$
$200$	1.00000	0.0707107
$201$	26.9688	1.90223
$202$	6.46410	0.454813
$203$	−2.49938	−0.175422
$204$	1.08516	0.0759767
$205$	7.67752	0.536221
$206$	−7.55051	−0.526069
$207$	0.313922	0.0218191
$208$	−5.59575	−0.387996
$209$	0	0
$210$	2.41421	0.166597
$211$	0.762683	0.0525052	0.0262526	−	0.999655i	$-0.491643\pi$
$211$	0.762683	0.0525052	0.0262526	+	0.999655i	$0.491643\pi$
$212$	−6.94887	−0.477250
$213$	4.53341	0.310624
$214$	−6.00481	−0.410480
$215$	−1.34366	−0.0916369
$216$	0.414214	0.0281837
$217$	−4.52004	−0.306840
$218$	−1.01461	−0.0687182
$219$	−8.24264	−0.556986
$220$	0	0
$221$	2.51523	0.169193
$222$	15.5206	1.04167
$223$	9.00532	0.603041	0.301521	−	0.953460i	$-0.402506\pi$
$223$	9.00532	0.603041	0.301521	+	0.953460i	$0.402506\pi$
$224$	−1.00000	−0.0668153
$225$	2.82843	0.188562
$226$	−5.83912	−0.388413
$227$	20.9895	1.39312	0.696560	−	0.717499i	$-0.254713\pi$
$227$	20.9895	1.39312	0.696560	+	0.717499i	$0.254713\pi$
$228$	−5.49938	−0.364205
$229$	29.9109	1.97657	0.988283	−	0.152633i	$-0.0487751\pi$
$229$	29.9109	1.97657	0.988283	+	0.152633i	$0.0487751\pi$
$230$	0.110988	0.00731834
$231$	0	0
$232$	2.49938	0.164092
$233$	−11.8917	−0.779050	−0.389525	−	0.921016i	$-0.627361\pi$
$233$	−11.8917	−0.779050	−0.389525	+	0.921016i	$0.627361\pi$
$234$	−15.8272	−1.03466
$235$	1.51399	0.0987618
$236$	3.97469	0.258730
$237$	27.0798	1.75902
$238$	0.449490	0.0291361
$239$	−22.6819	−1.46717	−0.733587	−	0.679596i	$-0.762155\pi$
$239$	−22.6819	−1.46717	−0.733587	+	0.679596i	$0.762155\pi$
$240$	−2.41421	−0.155837
$241$	−8.72135	−0.561792	−0.280896	−	0.959738i	$-0.590632\pi$
$241$	−8.72135	−0.561792	−0.280896	+	0.959738i	$0.590632\pi$
$242$	0	0
$243$	21.6569	1.38929
$244$	4.29253	0.274801
$245$	1.00000	0.0638877
$246$	−18.5352	−1.18176
$247$	−12.7467	−0.811051
$248$	4.52004	0.287023
$249$	−40.3078	−2.55440
$250$	1.00000	0.0632456
$251$	0.728651	0.0459920	0.0229960	−	0.999736i	$-0.492679\pi$
$251$	0.728651	0.0459920	0.0229960	+	0.999736i	$0.492679\pi$
$252$	−2.82843	−0.178174
$253$	0	0
$254$	8.36257	0.524714
$255$	1.08516	0.0679556
$256$	1.00000	0.0625000
$257$	10.4203	0.649998	0.324999	−	0.945714i	$-0.394636\pi$
$257$	10.4203	0.649998	0.324999	+	0.945714i	$0.394636\pi$
$258$	3.24389	0.201955
$259$	6.42883	0.399468
$260$	−5.59575	−0.347034
$261$	7.06931	0.437579
$262$	−10.5704	−0.653044
$263$	0.150909	0.00930542	0.00465271	−	0.999989i	$-0.498519\pi$
$263$	0.150909	0.00930542	0.00465271	+	0.999989i	$0.498519\pi$
$264$	0	0
$265$	−6.94887	−0.426866
$266$	−2.27792	−0.139668
$267$	−5.24389	−0.320921
$268$	−11.1708	−0.682368
$269$	1.33726	0.0815340	0.0407670	−	0.999169i	$-0.487020\pi$
$269$	1.33726	0.0815340	0.0407670	+	0.999169i	$0.487020\pi$
$270$	0.414214	0.0252082
$271$	−25.2500	−1.53383	−0.766913	−	0.641751i	$-0.778208\pi$
$271$	−25.2500	−1.53383	−0.766913	+	0.641751i	$0.778208\pi$
$272$	−0.449490	−0.0272543
$273$	−13.5093	−0.817623
$274$	−10.8895	−0.657860
$275$	0	0
$276$	−0.267949	−0.0161286
$277$	−4.14286	−0.248921	−0.124460	−	0.992225i	$-0.539720\pi$
$277$	−4.14286	−0.248921	−0.124460	+	0.992225i	$0.539720\pi$
$278$	−15.5493	−0.932583
$279$	12.7846	0.765395
$280$	−1.00000	−0.0597614
$281$	−9.33046	−0.556608	−0.278304	−	0.960493i	$-0.589772\pi$
$281$	−9.33046	−0.556608	−0.278304	+	0.960493i	$0.589772\pi$
$282$	−3.65509	−0.217658
$283$	−0.839639	−0.0499114	−0.0249557	−	0.999689i	$-0.507944\pi$
$283$	−0.839639	−0.0499114	−0.0249557	+	0.999689i	$0.507944\pi$
$284$	−1.87780	−0.111427
$285$	−5.49938	−0.325755
$286$	0	0
$287$	−7.67752	−0.453190
$288$	2.82843	0.166667
$289$	−16.7980	−0.988115
$290$	2.49938	0.146769
$291$	24.4488	1.43321
$292$	3.41421	0.199802
$293$	14.6968	0.858595	0.429297	−	0.903163i	$-0.358761\pi$
$293$	14.6968	0.858595	0.429297	+	0.903163i	$0.358761\pi$
$294$	−2.41421	−0.140800
$295$	3.97469	0.231415
$296$	−6.42883	−0.373668
$297$	0	0
$298$	19.0552	1.10384
$299$	−0.621063	−0.0359170
$300$	−2.41421	−0.139385
$301$	1.34366	0.0774473
$302$	−23.3590	−1.34416
$303$	−15.6057	−0.896525
$304$	2.27792	0.130647
$305$	4.29253	0.245789
$306$	−1.27135	−0.0726782
$307$	−21.2724	−1.21408	−0.607039	−	0.794672i	$-0.707643\pi$
$307$	−21.2724	−1.21408	−0.607039	+	0.794672i	$0.707643\pi$
$308$	0	0
$309$	18.2285	1.03699
$310$	4.52004	0.256721
$311$	−15.4623	−0.876789	−0.438394	−	0.898783i	$-0.644453\pi$
$311$	−15.4623	−0.876789	−0.438394	+	0.898783i	$0.644453\pi$
$312$	13.5093	0.764816
$313$	−10.0613	−0.568696	−0.284348	−	0.958721i	$-0.591777\pi$
$313$	−10.0613	−0.568696	−0.284348	+	0.958721i	$0.591777\pi$
$314$	−16.6617	−0.940272
$315$	−2.82843	−0.159364
$316$	−11.2168	−0.630995
$317$	12.3764	0.695131	0.347565	−	0.937656i	$-0.387008\pi$
$317$	12.3764	0.695131	0.347565	+	0.937656i	$0.387008\pi$
$318$	16.7761	0.940754
$319$	0	0
$320$	1.00000	0.0559017
$321$	14.4969	0.809137
$322$	−0.110988	−0.00618513
$323$	−1.02390	−0.0569713
$324$	−9.48528	−0.526960
$325$	−5.59575	−0.310397
$326$	2.01994	0.111874
$327$	2.44949	0.135457
$328$	7.67752	0.423920
$329$	−1.51399	−0.0834690
$330$	0	0
$331$	6.80776	0.374188	0.187094	−	0.982342i	$-0.440093\pi$
$331$	6.80776	0.374188	0.187094	+	0.982342i	$0.440093\pi$
$332$	16.6960	0.916315
$333$	−18.1835	−0.996448
$334$	−22.4402	−1.22787
$335$	−11.1708	−0.610328
$336$	2.41421	0.131706
$337$	−4.48852	−0.244505	−0.122253	−	0.992499i	$-0.539012\pi$
$337$	−4.48852	−0.244505	−0.122253	+	0.992499i	$0.539012\pi$
$338$	18.3125	0.996067
$339$	14.0969	0.765638
$340$	−0.449490	−0.0243770
$341$	0	0
$342$	6.44292	0.348393
$343$	−1.00000	−0.0539949
$344$	−1.34366	−0.0724454
$345$	−0.267949	−0.0144259
$346$	20.2419	1.08821
$347$	−24.7899	−1.33079	−0.665396	−	0.746490i	$-0.731737\pi$
$347$	−24.7899	−1.33079	−0.665396	+	0.746490i	$0.731737\pi$
$348$	−6.03403	−0.323458
$349$	−5.59716	−0.299609	−0.149805	−	0.988716i	$-0.547864\pi$
$349$	−5.59716	−0.299609	−0.149805	+	0.988716i	$0.547864\pi$
$350$	−1.00000	−0.0534522
$351$	−2.31784	−0.123717
$352$	0	0
$353$	−3.32956	−0.177215	−0.0886074	−	0.996067i	$-0.528242\pi$
$353$	−3.32956	−0.177215	−0.0886074	+	0.996067i	$0.528242\pi$
$354$	−9.59575	−0.510009
$355$	−1.87780	−0.0996633
$356$	2.17209	0.115120
$357$	−1.08516	−0.0574330
$358$	−16.3132	−0.862179
$359$	27.2046	1.43581	0.717903	−	0.696143i	$-0.245102\pi$
$359$	27.2046	1.43581	0.717903	+	0.696143i	$0.245102\pi$
$360$	2.82843	0.149071
$361$	−13.8111	−0.726900
$362$	10.0659	0.529052
$363$	0	0
$364$	5.59575	0.293297
$365$	3.41421	0.178708
$366$	−10.3631	−0.541687
$367$	−36.2016	−1.88971	−0.944855	−	0.327489i	$-0.893798\pi$
$367$	−36.2016	−1.88971	−0.944855	+	0.327489i	$0.893798\pi$
$368$	0.110988	0.00578566
$369$	21.7153	1.13045
$370$	−6.42883	−0.334219
$371$	6.94887	0.360767
$372$	−10.9123	−0.565779
$373$	3.02848	0.156809	0.0784043	−	0.996922i	$-0.475017\pi$
$373$	3.02848	0.156809	0.0784043	+	0.996922i	$0.475017\pi$
$374$	0	0
$375$	−2.41421	−0.124669
$376$	1.51399	0.0780781
$377$	−13.9859	−0.720311
$378$	−0.414214	−0.0213048
$379$	11.6617	0.599019	0.299510	−	0.954093i	$-0.403177\pi$
$379$	11.6617	0.599019	0.299510	+	0.954093i	$0.403177\pi$
$380$	2.27792	0.116855
$381$	−20.1890	−1.03431
$382$	−12.6211	−0.645750
$383$	−8.46943	−0.432767	−0.216384	−	0.976308i	$-0.569426\pi$
$383$	−8.46943	−0.432767	−0.216384	+	0.976308i	$0.569426\pi$
$384$	−2.41421	−0.123200
$385$	0	0
$386$	18.3984	0.936452
$387$	−3.80045	−0.193188
$388$	−10.1270	−0.514121
$389$	10.5400	0.534398	0.267199	−	0.963641i	$-0.413902\pi$
$389$	10.5400	0.534398	0.267199	+	0.963641i	$0.413902\pi$
$390$	13.5093	0.684072
$391$	−0.0498881	−0.00252295
$392$	1.00000	0.0505076
$393$	25.5193	1.28728
$394$	−4.24921	−0.214072
$395$	−11.2168	−0.564379
$396$	0	0
$397$	−25.9280	−1.30129	−0.650644	−	0.759383i	$-0.725501\pi$
$397$	−25.9280	−1.30129	−0.650644	+	0.759383i	$0.725501\pi$
$398$	−16.8211	−0.843167
$399$	5.49938	0.275313
$400$	1.00000	0.0500000
$401$	3.06251	0.152934	0.0764672	−	0.997072i	$-0.475636\pi$
$401$	3.06251	0.152934	0.0764672	+	0.997072i	$0.475636\pi$
$402$	26.9688	1.34508
$403$	−25.2930	−1.25994
$404$	6.46410	0.321601
$405$	−9.48528	−0.471327
$406$	−2.49938	−0.124042
$407$	0	0
$408$	1.08516	0.0537236
$409$	14.8635	0.734952	0.367476	−	0.930033i	$-0.380222\pi$
$409$	14.8635	0.734952	0.367476	+	0.930033i	$0.380222\pi$
$410$	7.67752	0.379166
$411$	26.2896	1.29677
$412$	−7.55051	−0.371987
$413$	−3.97469	−0.195582
$414$	0.313922	0.0154284
$415$	16.6960	0.819577
$416$	−5.59575	−0.274354
$417$	37.5392	1.83830
$418$	0	0
$419$	−30.5280	−1.49139	−0.745696	−	0.666286i	$-0.767883\pi$
$419$	−30.5280	−1.49139	−0.745696	+	0.666286i	$0.767883\pi$
$420$	2.41421	0.117802
$421$	21.5425	1.04992	0.524958	−	0.851128i	$-0.324081\pi$
$421$	21.5425	1.04992	0.524958	+	0.851128i	$0.324081\pi$
$422$	0.762683	0.0371268
$423$	4.28221	0.208208
$424$	−6.94887	−0.337467
$425$	−0.449490	−0.0218035
$426$	4.53341	0.219644
$427$	−4.29253	−0.207730
$428$	−6.00481	−0.290253
$429$	0	0
$430$	−1.34366	−0.0647971
$431$	2.31784	0.111646	0.0558231	−	0.998441i	$-0.482222\pi$
$431$	2.31784	0.111646	0.0558231	+	0.998441i	$0.482222\pi$
$432$	0.414214	0.0199289
$433$	−15.3870	−0.739451	−0.369726	−	0.929141i	$-0.620548\pi$
$433$	−15.3870	−0.739451	−0.369726	+	0.929141i	$0.620548\pi$
$434$	−4.52004	−0.216969
$435$	−6.03403	−0.289310
$436$	−1.01461	−0.0485911
$437$	0.252822	0.0120941
$438$	−8.24264	−0.393849
$439$	19.8045	0.945218	0.472609	−	0.881272i	$-0.343312\pi$
$439$	19.8045	0.945218	0.472609	+	0.881272i	$0.343312\pi$
$440$	0	0
$441$	2.82843	0.134687
$442$	2.51523	0.119637
$443$	15.4203	0.732640	0.366320	−	0.930489i	$-0.380618\pi$
$443$	15.4203	0.732640	0.366320	+	0.930489i	$0.380618\pi$
$444$	15.5206	0.736573
$445$	2.17209	0.102967
$446$	9.00532	0.426414
$447$	−46.0034	−2.17588
$448$	−1.00000	−0.0472456
$449$	−18.2046	−0.859130	−0.429565	−	0.903036i	$-0.641333\pi$
$449$	−18.2046	−0.859130	−0.429565	+	0.903036i	$0.641333\pi$
$450$	2.82843	0.133333
$451$	0	0
$452$	−5.83912	−0.274649
$453$	56.3935	2.64960
$454$	20.9895	0.985085
$455$	5.59575	0.262333
$456$	−5.49938	−0.257532
$457$	−27.6216	−1.29208	−0.646041	−	0.763302i	$-0.723577\pi$
$457$	−27.6216	−1.29208	−0.646041	+	0.763302i	$0.723577\pi$
$458$	29.9109	1.39764
$459$	−0.186185	−0.00869036
$460$	0.110988	0.00517485
$461$	−31.6027	−1.47189	−0.735943	−	0.677044i	$-0.763261\pi$
$461$	−31.6027	−1.47189	−0.735943	+	0.677044i	$0.763261\pi$
$462$	0	0
$463$	3.02990	0.140812	0.0704058	−	0.997518i	$-0.477571\pi$
$463$	3.02990	0.140812	0.0704058	+	0.997518i	$0.477571\pi$
$464$	2.49938	0.116031
$465$	−10.9123	−0.506048
$466$	−11.8917	−0.550872
$467$	−3.45679	−0.159961	−0.0799805	−	0.996796i	$-0.525486\pi$
$467$	−3.45679	−0.159961	−0.0799805	+	0.996796i	$0.525486\pi$
$468$	−15.8272	−0.731612
$469$	11.1708	0.515822
$470$	1.51399	0.0698351
$471$	40.2248	1.85346
$472$	3.97469	0.182950
$473$	0	0
$474$	27.0798	1.24382
$475$	2.27792	0.104518
$476$	0.449490	0.0206023
$477$	−19.6544	−0.899912
$478$	−22.6819	−1.03745
$479$	−2.92067	−0.133449	−0.0667245	−	0.997771i	$-0.521255\pi$
$479$	−2.92067	−0.133449	−0.0667245	+	0.997771i	$0.521255\pi$
$480$	−2.41421	−0.110193
$481$	35.9741	1.64028
$482$	−8.72135	−0.397247
$483$	0.267949	0.0121921
$484$	0	0
$485$	−10.1270	−0.459844
$486$	21.6569	0.982375
$487$	−4.24248	−0.192245	−0.0961225	−	0.995370i	$-0.530644\pi$
$487$	−4.24248	−0.192245	−0.0961225	+	0.995370i	$0.530644\pi$
$488$	4.29253	0.194314
$489$	−4.87656	−0.220525
$490$	1.00000	0.0451754
$491$	−30.8597	−1.39268	−0.696340	−	0.717712i	$-0.745190\pi$
$491$	−30.8597	−1.39268	−0.696340	+	0.717712i	$0.745190\pi$
$492$	−18.5352	−0.835630
$493$	−1.12344	−0.0505974
$494$	−12.7467	−0.573499
$495$	0	0
$496$	4.52004	0.202956
$497$	1.87780	0.0842308
$498$	−40.3078	−1.80624
$499$	−38.3475	−1.71667	−0.858336	−	0.513089i	$-0.828501\pi$
$499$	−38.3475	−1.71667	−0.858336	+	0.513089i	$0.828501\pi$
$500$	1.00000	0.0447214
$501$	54.1754	2.42038
$502$	0.728651	0.0325213
$503$	21.7360	0.969159	0.484579	−	0.874747i	$-0.338973\pi$
$503$	21.7360	0.969159	0.484579	+	0.874747i	$0.338973\pi$
$504$	−2.82843	−0.125988
$505$	6.46410	0.287649
$506$	0	0
$507$	−44.2102	−1.96344
$508$	8.36257	0.371029
$509$	−25.7306	−1.14049	−0.570244	−	0.821475i	$-0.693151\pi$
$509$	−25.7306	−1.14049	−0.570244	+	0.821475i	$0.693151\pi$
$510$	1.08516	0.0480519
$511$	−3.41421	−0.151036
$512$	1.00000	0.0441942
$513$	0.943544	0.0416585
$514$	10.4203	0.459618
$515$	−7.55051	−0.332715
$516$	3.24389	0.142804
$517$	0	0
$518$	6.42883	0.282466
$519$	−48.8683	−2.14508
$520$	−5.59575	−0.245390
$521$	−6.74378	−0.295450	−0.147725	−	0.989028i	$-0.547195\pi$
$521$	−6.74378	−0.295450	−0.147725	+	0.989028i	$0.547195\pi$
$522$	7.06931	0.309415
$523$	−2.58595	−0.113076	−0.0565379	−	0.998400i	$-0.518006\pi$
$523$	−2.58595	−0.113076	−0.0565379	+	0.998400i	$0.518006\pi$
$524$	−10.5704	−0.461772
$525$	2.41421	0.105365
$526$	0.150909	0.00657993
$527$	−2.03171	−0.0885028
$528$	0	0
$529$	−22.9877	−0.999464
$530$	−6.94887	−0.301840
$531$	11.2421	0.487867
$532$	−2.27792	−0.0987602
$533$	−42.9615	−1.86087
$534$	−5.24389	−0.226925
$535$	−6.00481	−0.259610
$536$	−11.1708	−0.482507
$537$	39.3835	1.69952
$538$	1.33726	0.0576532
$539$	0	0
$540$	0.414214	0.0178249
$541$	42.4707	1.82596	0.912979	−	0.408007i	$-0.133776\pi$
$541$	42.4707	1.82596	0.912979	+	0.408007i	$0.133776\pi$
$542$	−25.2500	−1.08458
$543$	−24.3013	−1.04287
$544$	−0.449490	−0.0192717
$545$	−1.01461	−0.0434612
$546$	−13.5093	−0.578147
$547$	−29.6629	−1.26830	−0.634148	−	0.773212i	$-0.718649\pi$
$547$	−29.6629	−1.26830	−0.634148	+	0.773212i	$0.718649\pi$
$548$	−10.8895	−0.465178
$549$	12.1411	0.518170
$550$	0	0
$551$	5.69337	0.242546
$552$	−0.267949	−0.0114047
$553$	11.2168	0.476988
$554$	−4.14286	−0.176013
$555$	15.5206	0.658811
$556$	−15.5493	−0.659436
$557$	−1.64299	−0.0696157	−0.0348079	−	0.999394i	$-0.511082\pi$
$557$	−1.64299	−0.0696157	−0.0348079	+	0.999394i	$0.511082\pi$
$558$	12.7846	0.541216
$559$	7.51880	0.318011
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−9.33046	−0.393582
$563$	6.29591	0.265341	0.132670	−	0.991160i	$-0.457645\pi$
$563$	6.29591	0.265341	0.132670	+	0.991160i	$0.457645\pi$
$564$	−3.65509	−0.153907
$565$	−5.83912	−0.245654
$566$	−0.839639	−0.0352927
$567$	9.48528	0.398344
$568$	−1.87780	−0.0787907
$569$	19.7613	0.828436	0.414218	−	0.910178i	$-0.364055\pi$
$569$	19.7613	0.828436	0.414218	+	0.910178i	$0.364055\pi$
$570$	−5.49938	−0.230344
$571$	−15.9882	−0.669086	−0.334543	−	0.942380i	$-0.608582\pi$
$571$	−15.9882	−0.669086	−0.334543	+	0.942380i	$0.608582\pi$
$572$	0	0
$573$	30.4699	1.27290
$574$	−7.67752	−0.320453
$575$	0.110988	0.00462853
$576$	2.82843	0.117851
$577$	−35.5977	−1.48195	−0.740975	−	0.671532i	$-0.765636\pi$
$577$	−35.5977	−1.48195	−0.740975	+	0.671532i	$0.765636\pi$
$578$	−16.7980	−0.698703
$579$	−44.4176	−1.84593
$580$	2.49938	0.103781
$581$	−16.6960	−0.692669
$582$	24.4488	1.01343
$583$	0	0
$584$	3.41421	0.141281
$585$	−15.8272	−0.654373
$586$	14.6968	0.607118
$587$	−39.2344	−1.61938	−0.809688	−	0.586860i	$-0.800364\pi$
$587$	−39.2344	−1.61938	−0.809688	+	0.586860i	$0.800364\pi$
$588$	−2.41421	−0.0995605
$589$	10.2963	0.424251
$590$	3.97469	0.163635
$591$	10.2585	0.421978
$592$	−6.42883	−0.264223
$593$	24.7544	1.01654	0.508270	−	0.861198i	$-0.330285\pi$
$593$	24.7544	1.01654	0.508270	+	0.861198i	$0.330285\pi$
$594$	0	0
$595$	0.449490	0.0184273
$596$	19.0552	0.780532
$597$	40.6098	1.66205
$598$	−0.621063	−0.0253971
$599$	−35.5118	−1.45097	−0.725486	−	0.688237i	$-0.758385\pi$
$599$	−35.5118	−1.45097	−0.725486	+	0.688237i	$0.758385\pi$
$600$	−2.41421	−0.0985599
$601$	−31.5173	−1.28562	−0.642810	−	0.766026i	$-0.722231\pi$
$601$	−31.5173	−1.28562	−0.642810	+	0.766026i	$0.722231\pi$
$602$	1.34366	0.0547635
$603$	−31.5959	−1.28669
$604$	−23.3590	−0.950462
$605$	0	0
$606$	−15.6057	−0.633939
$607$	−11.0849	−0.449924	−0.224962	−	0.974368i	$-0.572226\pi$
$607$	−11.0849	−0.449924	−0.224962	+	0.974368i	$0.572226\pi$
$608$	2.27792	0.0923817
$609$	6.03403	0.244511
$610$	4.29253	0.173799
$611$	−8.47191	−0.342737
$612$	−1.27135	−0.0513912
$613$	−19.3616	−0.782008	−0.391004	−	0.920389i	$-0.627872\pi$
$613$	−19.3616	−0.782008	−0.391004	+	0.920389i	$0.627872\pi$
$614$	−21.2724	−0.858483
$615$	−18.5352	−0.747410
$616$	0	0
$617$	39.7449	1.60007	0.800034	−	0.599955i	$-0.204815\pi$
$617$	39.7449	1.60007	0.800034	+	0.599955i	$0.204815\pi$
$618$	18.2285	0.733260
$619$	−35.5001	−1.42687	−0.713434	−	0.700723i	$-0.752861\pi$
$619$	−35.5001	−1.42687	−0.713434	+	0.700723i	$0.752861\pi$
$620$	4.52004	0.181529
$621$	0.0459728	0.00184483
$622$	−15.4623	−0.619983
$623$	−2.17209	−0.0870229
$624$	13.5093	0.540807
$625$	1.00000	0.0400000
$626$	−10.0613	−0.402129
$627$	0	0
$628$	−16.6617	−0.664873
$629$	2.88969	0.115220
$630$	−2.82843	−0.112687
$631$	31.7687	1.26469	0.632347	−	0.774686i	$-0.282092\pi$
$631$	31.7687	1.26469	0.632347	+	0.774686i	$0.282092\pi$
$632$	−11.2168	−0.446181
$633$	−1.84128	−0.0731843
$634$	12.3764	0.491532
$635$	8.36257	0.331858
$636$	16.7761	0.665214
$637$	−5.59575	−0.221712
$638$	0	0
$639$	−5.31122	−0.210109
$640$	1.00000	0.0395285
$641$	48.4441	1.91343	0.956713	−	0.291034i	$-0.0939992\pi$
$641$	48.4441	1.91343	0.956713	+	0.291034i	$0.0939992\pi$
$642$	14.4969	0.572147
$643$	−5.25526	−0.207247	−0.103624	−	0.994617i	$-0.533044\pi$
$643$	−5.25526	−0.207247	−0.103624	+	0.994617i	$0.533044\pi$
$644$	−0.110988	−0.00437355
$645$	3.24389	0.127728
$646$	−1.02390	−0.0402848
$647$	49.1639	1.93283	0.966416	−	0.256984i	$-0.0827287\pi$
$647$	49.1639	1.93283	0.966416	+	0.256984i	$0.0827287\pi$
$648$	−9.48528	−0.372617
$649$	0	0
$650$	−5.59575	−0.219484
$651$	10.9123	0.427689
$652$	2.01994	0.0791068
$653$	−35.4550	−1.38746	−0.693732	−	0.720234i	$-0.744035\pi$
$653$	−35.4550	−1.38746	−0.693732	+	0.720234i	$0.744035\pi$
$654$	2.44949	0.0957826
$655$	−10.5704	−0.413021
$656$	7.67752	0.299757
$657$	9.65685	0.376750
$658$	−1.51399	−0.0590215
$659$	15.5831	0.607031	0.303515	−	0.952827i	$-0.401840\pi$
$659$	15.5831	0.607031	0.303515	+	0.952827i	$0.401840\pi$
$660$	0	0
$661$	−22.5419	−0.876778	−0.438389	−	0.898785i	$-0.644451\pi$
$661$	−22.5419	−0.876778	−0.438389	+	0.898785i	$0.644451\pi$
$662$	6.80776	0.264591
$663$	−6.07231	−0.235829
$664$	16.6960	0.647932
$665$	−2.27792	−0.0883338
$666$	−18.1835	−0.704595
$667$	0.277401	0.0107410
$668$	−22.4402	−0.868237
$669$	−21.7408	−0.840547
$670$	−11.1708	−0.431567
$671$	0	0
$672$	2.41421	0.0931303
$673$	−42.2397	−1.62822	−0.814111	−	0.580710i	$-0.802775\pi$
$673$	−42.2397	−1.62822	−0.814111	+	0.580710i	$0.802775\pi$
$674$	−4.48852	−0.172891
$675$	0.414214	0.0159431
$676$	18.3125	0.704326
$677$	−27.2287	−1.04648	−0.523242	−	0.852184i	$-0.675278\pi$
$677$	−27.2287	−1.04648	−0.523242	+	0.852184i	$0.675278\pi$
$678$	14.0969	0.541388
$679$	10.1270	0.388639
$680$	−0.449490	−0.0172371
$681$	−50.6731	−1.94180
$682$	0	0
$683$	−33.2266	−1.27138	−0.635690	−	0.771945i	$-0.719284\pi$
$683$	−33.2266	−1.27138	−0.635690	+	0.771945i	$0.719284\pi$
$684$	6.44292	0.246351
$685$	−10.8895	−0.416067
$686$	−1.00000	−0.0381802
$687$	−72.2112	−2.75503
$688$	−1.34366	−0.0512266
$689$	38.8842	1.48137
$690$	−0.267949	−0.0102007
$691$	3.12921	0.119041	0.0595204	−	0.998227i	$-0.481043\pi$
$691$	3.12921	0.119041	0.0595204	+	0.998227i	$0.481043\pi$
$692$	20.2419	0.769482
$693$	0	0
$694$	−24.7899	−0.941012
$695$	−15.5493	−0.589817
$696$	−6.03403	−0.228719
$697$	−3.45097	−0.130715
$698$	−5.59716	−0.211856
$699$	28.7091	1.08588
$700$	−1.00000	−0.0377964
$701$	8.48402	0.320437	0.160218	−	0.987082i	$-0.448780\pi$
$701$	8.48402	0.320437	0.160218	+	0.987082i	$0.448780\pi$
$702$	−2.31784	−0.0874811
$703$	−14.6443	−0.552321
$704$	0	0
$705$	−3.65509	−0.137659
$706$	−3.32956	−0.125310
$707$	−6.46410	−0.243108
$708$	−9.59575	−0.360631
$709$	1.62962	0.0612017	0.0306009	−	0.999532i	$-0.490258\pi$
$709$	1.62962	0.0612017	0.0306009	+	0.999532i	$0.490258\pi$
$710$	−1.87780	−0.0704726
$711$	−31.7259	−1.18982
$712$	2.17209	0.0814025
$713$	0.501671	0.0187877
$714$	−1.08516	−0.0406113
$715$	0	0
$716$	−16.3132	−0.609653
$717$	54.7591	2.04501
$718$	27.2046	1.01527
$719$	−23.4644	−0.875076	−0.437538	−	0.899200i	$-0.644149\pi$
$719$	−23.4644	−0.875076	−0.437538	+	0.899200i	$0.644149\pi$
$720$	2.82843	0.105409
$721$	7.55051	0.281196
$722$	−13.8111	−0.513996
$723$	21.0552	0.783052
$724$	10.0659	0.374097
$725$	2.49938	0.0928246
$726$	0	0
$727$	−6.87175	−0.254859	−0.127429	−	0.991848i	$-0.540673\pi$
$727$	−6.87175	−0.254859	−0.127429	+	0.991848i	$0.540673\pi$
$728$	5.59575	0.207392
$729$	−23.8284	−0.882534
$730$	3.41421	0.126366
$731$	0.603962	0.0223383
$732$	−10.3631	−0.383030
$733$	33.9730	1.25482	0.627411	−	0.778688i	$-0.284115\pi$
$733$	33.9730	1.25482	0.627411	+	0.778688i	$0.284115\pi$
$734$	−36.2016	−1.33623
$735$	−2.41421	−0.0890496
$736$	0.110988	0.00409108
$737$	0	0
$738$	21.7153	0.799351
$739$	29.0301	1.06789	0.533944	−	0.845520i	$-0.320709\pi$
$739$	29.0301	1.06789	0.533944	+	0.845520i	$0.320709\pi$
$740$	−6.42883	−0.236328
$741$	30.7732	1.13048
$742$	6.94887	0.255101
$743$	47.1515	1.72982	0.864911	−	0.501925i	$-0.167375\pi$
$743$	47.1515	1.72982	0.864911	+	0.501925i	$0.167375\pi$
$744$	−10.9123	−0.400066
$745$	19.0552	0.698129
$746$	3.02848	0.110880
$747$	47.2235	1.72782
$748$	0	0
$749$	6.00481	0.219411
$750$	−2.41421	−0.0881546
$751$	−21.1160	−0.770532	−0.385266	−	0.922805i	$-0.625890\pi$
$751$	−21.1160	−0.770532	−0.385266	+	0.922805i	$0.625890\pi$
$752$	1.51399	0.0552095
$753$	−1.75912	−0.0641059
$754$	−13.9859	−0.509337
$755$	−23.3590	−0.850119
$756$	−0.414214	−0.0150648
$757$	11.2590	0.409216	0.204608	−	0.978844i	$-0.434408\pi$
$757$	11.2590	0.409216	0.204608	+	0.978844i	$0.434408\pi$
$758$	11.6617	0.423571
$759$	0	0
$760$	2.27792	0.0826287
$761$	10.2708	0.372317	0.186159	−	0.982520i	$-0.440396\pi$
$761$	10.2708	0.372317	0.186159	+	0.982520i	$0.440396\pi$
$762$	−20.1890	−0.731371
$763$	1.01461	0.0367314
$764$	−12.6211	−0.456614
$765$	−1.27135	−0.0459657
$766$	−8.46943	−0.306013
$767$	−22.2414	−0.803090
$768$	−2.41421	−0.0871154
$769$	−11.3560	−0.409508	−0.204754	−	0.978813i	$-0.565639\pi$
$769$	−11.3560	−0.409508	−0.204754	+	0.978813i	$0.565639\pi$
$770$	0	0
$771$	−25.1567	−0.905998
$772$	18.3984	0.662171
$773$	−41.0116	−1.47508	−0.737542	−	0.675301i	$-0.764014\pi$
$773$	−41.0116	−1.47508	−0.737542	+	0.675301i	$0.764014\pi$
$774$	−3.80045	−0.136604
$775$	4.52004	0.162365
$776$	−10.1270	−0.363538
$777$	−15.5206	−0.556797
$778$	10.5400	0.377876
$779$	17.4887	0.626599
$780$	13.5093	0.483712
$781$	0	0
$782$	−0.0498881	−0.00178399
$783$	1.03528	0.0369978
$784$	1.00000	0.0357143
$785$	−16.6617	−0.594680
$786$	25.5193	0.910244
$787$	45.1477	1.60934	0.804671	−	0.593721i	$-0.202342\pi$
$787$	45.1477	1.60934	0.804671	+	0.593721i	$0.202342\pi$
$788$	−4.24921	−0.151372
$789$	−0.364326	−0.0129703
$790$	−11.2168	−0.399076
$791$	5.83912	0.207615
$792$	0	0
$793$	−24.0199	−0.852973
$794$	−25.9280	−0.920150
$795$	16.7761	0.594985
$796$	−16.8211	−0.596209
$797$	−34.8398	−1.23409	−0.617045	−	0.786928i	$-0.711670\pi$
$797$	−34.8398	−1.23409	−0.617045	+	0.786928i	$0.711670\pi$
$798$	5.49938	0.194676
$799$	−0.680523	−0.0240752
$800$	1.00000	0.0353553
$801$	6.14359	0.217073
$802$	3.06251	0.108141
$803$	0	0
$804$	26.9688	0.951116
$805$	−0.110988	−0.00391182
$806$	−25.2930	−0.890909
$807$	−3.22842	−0.113646
$808$	6.46410	0.227406
$809$	23.5770	0.828924	0.414462	−	0.910067i	$-0.363970\pi$
$809$	23.5770	0.828924	0.414462	+	0.910067i	$0.363970\pi$
$810$	−9.48528	−0.333279
$811$	14.9977	0.526641	0.263321	−	0.964708i	$-0.415182\pi$
$811$	14.9977	0.526641	0.263321	+	0.964708i	$0.415182\pi$
$812$	−2.49938	−0.0877110
$813$	60.9588	2.13792
$814$	0	0
$815$	2.01994	0.0707552
$816$	1.08516	0.0379883
$817$	−3.06075	−0.107082
$818$	14.8635	0.519690
$819$	15.8272	0.553047
$820$	7.67752	0.268111
$821$	38.2540	1.33507	0.667537	−	0.744576i	$-0.267349\pi$
$821$	38.2540	1.33507	0.667537	+	0.744576i	$0.267349\pi$
$822$	26.2896	0.916957
$823$	−25.3006	−0.881923	−0.440961	−	0.897526i	$-0.645362\pi$
$823$	−25.3006	−0.881923	−0.440961	+	0.897526i	$0.645362\pi$
$824$	−7.55051	−0.263034
$825$	0	0
$826$	−3.97469	−0.138297
$827$	29.9973	1.04311	0.521554	−	0.853218i	$-0.325353\pi$
$827$	29.9973	1.04311	0.521554	+	0.853218i	$0.325353\pi$
$828$	0.313922	0.0109095
$829$	44.3053	1.53879	0.769394	−	0.638775i	$-0.220558\pi$
$829$	44.3053	1.53879	0.769394	+	0.638775i	$0.220558\pi$
$830$	16.6960	0.579528
$831$	10.0018	0.346957
$832$	−5.59575	−0.193998
$833$	−0.449490	−0.0155739
$834$	37.5392	1.29988
$835$	−22.4402	−0.776575
$836$	0	0
$837$	1.87226	0.0647149
$838$	−30.5280	−1.05457
$839$	33.3319	1.15074	0.575372	−	0.817892i	$-0.304857\pi$
$839$	33.3319	1.15074	0.575372	+	0.817892i	$0.304857\pi$
$840$	2.41421	0.0832983
$841$	−22.7531	−0.784590
$842$	21.5425	0.742403
$843$	22.5257	0.775827
$844$	0.762683	0.0262526
$845$	18.3125	0.629968
$846$	4.28221	0.147225
$847$	0	0
$848$	−6.94887	−0.238625
$849$	2.02707	0.0695688
$850$	−0.449490	−0.0154174
$851$	−0.713524	−0.0244593
$852$	4.53341	0.155312
$853$	11.3834	0.389760	0.194880	−	0.980827i	$-0.437568\pi$
$853$	11.3834	0.389760	0.194880	+	0.980827i	$0.437568\pi$
$854$	−4.29253	−0.146887
$855$	6.44292	0.220343
$856$	−6.00481	−0.205240
$857$	37.8048	1.29139	0.645693	−	0.763597i	$-0.276569\pi$
$857$	37.8048	1.29139	0.645693	+	0.763597i	$0.276569\pi$
$858$	0	0
$859$	−57.8457	−1.97367	−0.986834	−	0.161735i	$-0.948291\pi$
$859$	−57.8457	−1.97367	−0.986834	+	0.161735i	$0.948291\pi$
$860$	−1.34366	−0.0458185
$861$	18.5352	0.631677
$862$	2.31784	0.0789458
$863$	45.1248	1.53607	0.768033	−	0.640410i	$-0.221236\pi$
$863$	45.1248	1.53607	0.768033	+	0.640410i	$0.221236\pi$
$864$	0.414214	0.0140918
$865$	20.2419	0.688246
$866$	−15.3870	−0.522871
$867$	40.5539	1.37728
$868$	−4.52004	−0.153420
$869$	0	0
$870$	−6.03403	−0.204573
$871$	62.5093	2.11805
$872$	−1.01461	−0.0343591
$873$	−28.6435	−0.969436
$874$	0.252822	0.00855183
$875$	−1.00000	−0.0338062
$876$	−8.24264	−0.278493
$877$	−20.5249	−0.693075	−0.346537	−	0.938036i	$-0.612643\pi$
$877$	−20.5249	−0.693075	−0.346537	+	0.938036i	$0.612643\pi$
$878$	19.8045	0.668370
$879$	−35.4812	−1.19675
$880$	0	0
$881$	−8.43131	−0.284058	−0.142029	−	0.989862i	$-0.545363\pi$
$881$	−8.43131	−0.284058	−0.142029	+	0.989862i	$0.545363\pi$
$882$	2.82843	0.0952381
$883$	51.5287	1.73408	0.867040	−	0.498239i	$-0.166020\pi$
$883$	51.5287	1.73408	0.867040	+	0.498239i	$0.166020\pi$
$884$	2.51523	0.0845965
$885$	−9.59575	−0.322558
$886$	15.4203	0.518055
$887$	28.9499	0.972042	0.486021	−	0.873947i	$-0.338448\pi$
$887$	28.9499	0.972042	0.486021	+	0.873947i	$0.338448\pi$
$888$	15.5206	0.520836
$889$	−8.36257	−0.280471
$890$	2.17209	0.0728086
$891$	0	0
$892$	9.00532	0.301521
$893$	3.44874	0.115408
$894$	−46.0034	−1.53858
$895$	−16.3132	−0.545290
$896$	−1.00000	−0.0334077
$897$	1.49938	0.0500628
$898$	−18.2046	−0.607497
$899$	11.2973	0.376786
$900$	2.82843	0.0942809
$901$	3.12344	0.104057
$902$	0	0
$903$	−3.24389	−0.107950
$904$	−5.83912	−0.194206
$905$	10.0659	0.334602
$906$	56.3935	1.87355
$907$	−5.81255	−0.193003	−0.0965013	−	0.995333i	$-0.530765\pi$
$907$	−5.81255	−0.193003	−0.0965013	+	0.995333i	$0.530765\pi$
$908$	20.9895	0.696560
$909$	18.2832	0.606417
$910$	5.59575	0.185497
$911$	21.8126	0.722683	0.361341	−	0.932434i	$-0.382319\pi$
$911$	21.8126	0.722683	0.361341	+	0.932434i	$0.382319\pi$
$912$	−5.49938	−0.182103
$913$	0	0
$914$	−27.6216	−0.913641
$915$	−10.3631	−0.342593
$916$	29.9109	0.988283
$917$	10.5704	0.349067
$918$	−0.186185	−0.00614501
$919$	−48.0456	−1.58488	−0.792440	−	0.609950i	$-0.791189\pi$
$919$	−48.0456	−1.58488	−0.792440	+	0.609950i	$0.791189\pi$
$920$	0.110988	0.00365917
$921$	51.3561	1.69224
$922$	−31.6027	−1.04078
$923$	10.5077	0.345865
$924$	0	0
$925$	−6.42883	−0.211378
$926$	3.02990	0.0995688
$927$	−21.3561	−0.701425
$928$	2.49938	0.0820461
$929$	28.8259	0.945748	0.472874	−	0.881130i	$-0.343217\pi$
$929$	28.8259	0.945748	0.472874	+	0.881130i	$0.343217\pi$
$930$	−10.9123	−0.357830
$931$	2.27792	0.0746557
$932$	−11.8917	−0.389525
$933$	37.3294	1.22211
$934$	−3.45679	−0.113110
$935$	0	0
$936$	−15.8272	−0.517328
$937$	−12.7627	−0.416939	−0.208469	−	0.978029i	$-0.566848\pi$
$937$	−12.7627	−0.416939	−0.208469	+	0.978029i	$0.566848\pi$
$938$	11.1708	0.364741
$939$	24.2900	0.792675
$940$	1.51399	0.0493809
$941$	20.1593	0.657173	0.328586	−	0.944474i	$-0.393428\pi$
$941$	20.1593	0.657173	0.328586	+	0.944474i	$0.393428\pi$
$942$	40.2248	1.31060
$943$	0.852114	0.0277486
$944$	3.97469	0.129365
$945$	−0.414214	−0.0134744
$946$	0	0
$947$	1.46868	0.0477256	0.0238628	−	0.999715i	$-0.492404\pi$
$947$	1.46868	0.0477256	0.0238628	+	0.999715i	$0.492404\pi$
$948$	27.0798	0.879511
$949$	−19.1051	−0.620178
$950$	2.27792	0.0739054
$951$	−29.8794	−0.968906
$952$	0.449490	0.0145680
$953$	38.5687	1.24936	0.624681	−	0.780880i	$-0.285229\pi$
$953$	38.5687	1.24936	0.624681	+	0.780880i	$0.285229\pi$
$954$	−19.6544	−0.636334
$955$	−12.6211	−0.408408
$956$	−22.6819	−0.733587
$957$	0	0
$958$	−2.92067	−0.0943627
$959$	10.8895	0.351641
$960$	−2.41421	−0.0779184
$961$	−10.5692	−0.340943
$962$	35.9741	1.15985
$963$	−16.9842	−0.547307
$964$	−8.72135	−0.280896
$965$	18.3984	0.592264
$966$	0.267949	0.00862112
$967$	44.4040	1.42794	0.713968	−	0.700178i	$-0.246896\pi$
$967$	44.4040	1.42794	0.713968	+	0.700178i	$0.246896\pi$
$968$	0	0
$969$	2.47191	0.0794093
$970$	−10.1270	−0.325159
$971$	−17.4270	−0.559260	−0.279630	−	0.960108i	$-0.590212\pi$
$971$	−17.4270	−0.559260	−0.279630	+	0.960108i	$0.590212\pi$
$972$	21.6569	0.694644
$973$	15.5493	0.498487
$974$	−4.24248	−0.135938
$975$	13.5093	0.432645
$976$	4.29253	0.137400
$977$	−1.26471	−0.0404618	−0.0202309	−	0.999795i	$-0.506440\pi$
$977$	−1.26471	−0.0404618	−0.0202309	+	0.999795i	$0.506440\pi$
$978$	−4.87656	−0.155935
$979$	0	0
$980$	1.00000	0.0319438
$981$	−2.86976	−0.0916242
$982$	−30.8597	−0.984774
$983$	−30.6312	−0.976983	−0.488492	−	0.872569i	$-0.662453\pi$
$983$	−30.6312	−0.976983	−0.488492	+	0.872569i	$0.662453\pi$
$984$	−18.5352	−0.590880
$985$	−4.24921	−0.135391
$986$	−1.12344	−0.0357778
$987$	3.65509	0.116343
$988$	−12.7467	−0.405525
$989$	−0.149131	−0.00474207
$990$	0	0
$991$	26.3840	0.838116	0.419058	−	0.907959i	$-0.362360\pi$
$991$	26.3840	0.838116	0.419058	+	0.907959i	$0.362360\pi$
$992$	4.52004	0.143511
$993$	−16.4354	−0.521561
$994$	1.87780	0.0595602
$995$	−16.8211	−0.533266
$996$	−40.3078	−1.27720
$997$	50.2403	1.59113	0.795563	−	0.605871i	$-0.207175\pi$
$997$	50.2403	1.59113	0.795563	+	0.605871i	$0.207175\pi$
$998$	−38.3475	−1.21387
$999$	−2.66291	−0.0842506

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cr.1.1	yes	4
11.10	odd	2		8470.2.a.cp.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.cp.1.2	✓	4	11.10	odd	2
8470.2.a.cr.1.1	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.41421 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.41421 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.82843 q^{9} +1.00000 q^{10} -2.41421 q^{12} -5.59575 q^{13} -1.00000 q^{14} -2.41421 q^{15} +1.00000 q^{16} -0.449490 q^{17} +2.82843 q^{18} +2.27792 q^{19} +1.00000 q^{20} +2.41421 q^{21} +0.110988 q^{23} -2.41421 q^{24} +1.00000 q^{25} -5.59575 q^{26} +0.414214 q^{27} -1.00000 q^{28} +2.49938 q^{29} -2.41421 q^{30} +4.52004 q^{31} +1.00000 q^{32} -0.449490 q^{34} -1.00000 q^{35} +2.82843 q^{36} -6.42883 q^{37} +2.27792 q^{38} +13.5093 q^{39} +1.00000 q^{40} +7.67752 q^{41} +2.41421 q^{42} -1.34366 q^{43} +2.82843 q^{45} +0.110988 q^{46} +1.51399 q^{47} -2.41421 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.08516 q^{51} -5.59575 q^{52} -6.94887 q^{53} +0.414214 q^{54} -1.00000 q^{56} -5.49938 q^{57} +2.49938 q^{58} +3.97469 q^{59} -2.41421 q^{60} +4.29253 q^{61} +4.52004 q^{62} -2.82843 q^{63} +1.00000 q^{64} -5.59575 q^{65} -11.1708 q^{67} -0.449490 q^{68} -0.267949 q^{69} -1.00000 q^{70} -1.87780 q^{71} +2.82843 q^{72} +3.41421 q^{73} -6.42883 q^{74} -2.41421 q^{75} +2.27792 q^{76} +13.5093 q^{78} -11.2168 q^{79} +1.00000 q^{80} -9.48528 q^{81} +7.67752 q^{82} +16.6960 q^{83} +2.41421 q^{84} -0.449490 q^{85} -1.34366 q^{86} -6.03403 q^{87} +2.17209 q^{89} +2.82843 q^{90} +5.59575 q^{91} +0.110988 q^{92} -10.9123 q^{93} +1.51399 q^{94} +2.27792 q^{95} -2.41421 q^{96} -10.1270 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{10} - 4 q^{12} - 4 q^{14} - 4 q^{15} + 4 q^{16} + 8 q^{17} - 12 q^{19} + 4 q^{20} + 4 q^{21} - 8 q^{23} - 4 q^{24} + 4 q^{25} - 4 q^{27} - 4 q^{28} - 8 q^{29} - 4 q^{30} + 4 q^{32} + 8 q^{34} - 4 q^{35} - 16 q^{37} - 12 q^{38} + 8 q^{39} + 4 q^{40} + 8 q^{41} + 4 q^{42} - 8 q^{43} - 8 q^{46} - 16 q^{47} - 4 q^{48} + 4 q^{49} + 4 q^{50} - 8 q^{51} - 4 q^{54} - 4 q^{56} - 4 q^{57} - 8 q^{58} - 8 q^{59} - 4 q^{60} - 8 q^{61} + 4 q^{64} + 8 q^{68} - 8 q^{69} - 4 q^{70} - 8 q^{71} + 8 q^{73} - 16 q^{74} - 4 q^{75} - 12 q^{76} + 8 q^{78} - 24 q^{79} + 4 q^{80} - 4 q^{81} + 8 q^{82} + 8 q^{83} + 4 q^{84} + 8 q^{85} - 8 q^{86} + 16 q^{87} - 8 q^{92} + 16 q^{93} - 16 q^{94} - 12 q^{95} - 4 q^{96} - 8 q^{97} + 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^10 - 4 * q^12 - 4 * q^14 - 4 * q^15 + 4 * q^16 + 8 * q^17 - 12 * q^19 + 4 * q^20 + 4 * q^21 - 8 * q^23 - 4 * q^24 + 4 * q^25 - 4 * q^27 - 4 * q^28 - 8 * q^29 - 4 * q^30 + 4 * q^32 + 8 * q^34 - 4 * q^35 - 16 * q^37 - 12 * q^38 + 8 * q^39 + 4 * q^40 + 8 * q^41 + 4 * q^42 - 8 * q^43 - 8 * q^46 - 16 * q^47 - 4 * q^48 + 4 * q^49 + 4 * q^50 - 8 * q^51 - 4 * q^54 - 4 * q^56 - 4 * q^57 - 8 * q^58 - 8 * q^59 - 4 * q^60 - 8 * q^61 + 4 * q^64 + 8 * q^68 - 8 * q^69 - 4 * q^70 - 8 * q^71 + 8 * q^73 - 16 * q^74 - 4 * q^75 - 12 * q^76 + 8 * q^78 - 24 * q^79 + 4 * q^80 - 4 * q^81 + 8 * q^82 + 8 * q^83 + 4 * q^84 + 8 * q^85 - 8 * q^86 + 16 * q^87 - 8 * q^92 + 16 * q^93 - 16 * q^94 - 12 * q^95 - 4 * q^96 - 8 * q^97 + 4 * q^98

Introduction

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