LMFDB - Embedded newform 8470.2.a.cp.1.3

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			1.3
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)

$f(q)$	$=$	$q-1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.414214 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.82843 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.414214 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.82843 q^{9} -1.00000 q^{10} +0.414214 q^{12} -2.13165 q^{13} -1.00000 q^{14} +0.414214 q^{15} +1.00000 q^{16} -4.44949 q^{17} +2.82843 q^{18} +8.27792 q^{19} +1.00000 q^{20} +0.414214 q^{21} -0.646887 q^{23} -0.414214 q^{24} +1.00000 q^{25} +2.13165 q^{26} -2.41421 q^{27} +1.00000 q^{28} -0.428825 q^{29} -0.414214 q^{30} -4.52004 q^{31} -1.00000 q^{32} +4.44949 q^{34} +1.00000 q^{35} -2.82843 q^{36} -8.49938 q^{37} -8.27792 q^{38} -0.882959 q^{39} -1.00000 q^{40} +10.6057 q^{41} -0.414214 q^{42} +2.65634 q^{43} -2.82843 q^{45} +0.646887 q^{46} +4.34242 q^{47} +0.414214 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.84304 q^{51} -2.13165 q^{52} +0.0206643 q^{53} +2.41421 q^{54} -1.00000 q^{56} +3.42883 q^{57} +0.428825 q^{58} -4.51059 q^{59} +0.414214 q^{60} +1.36433 q^{61} +4.52004 q^{62} -2.82843 q^{63} +1.00000 q^{64} -2.13165 q^{65} -2.68556 q^{67} -4.44949 q^{68} -0.267949 q^{69} -1.00000 q^{70} -9.05040 q^{71} +2.82843 q^{72} -0.585786 q^{73} +8.49938 q^{74} +0.414214 q^{75} +8.27792 q^{76} +0.882959 q^{78} +4.24728 q^{79} +1.00000 q^{80} +7.48528 q^{81} -10.6057 q^{82} -4.62446 q^{83} +0.414214 q^{84} -4.44949 q^{85} -2.65634 q^{86} -0.177625 q^{87} -2.17209 q^{89} +2.82843 q^{90} -2.13165 q^{91} -0.646887 q^{92} -1.87226 q^{93} -4.34242 q^{94} +8.27792 q^{95} -0.414214 q^{96} +13.0552 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.414214	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.414214	−0.169102
$7$	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
$12$	0.414214	0.119573
$13$	−2.13165	−0.591214	−0.295607	−	0.955310i	$-0.595522\pi$
$13$	−2.13165	−0.591214	−0.295607	+	0.955310i	$0.595522\pi$
$14$	−1.00000	−0.267261
$15$	0.414214	0.106949
$16$	1.00000	0.250000
$17$	−4.44949	−1.07916	−0.539580	−	0.841934i	$-0.681417\pi$
$17$	−4.44949	−1.07916	−0.539580	+	0.841934i	$0.681417\pi$
$18$	2.82843	0.666667
$19$	8.27792	1.89908	0.949542	−	0.313639i	$-0.101548\pi$
$19$	8.27792	1.89908	0.949542	+	0.313639i	$0.101548\pi$
$20$	1.00000	0.223607
$21$	0.414214	0.0903888
$22$	0	0
$23$	−0.646887	−0.134885	−0.0674426	−	0.997723i	$-0.521484\pi$
$23$	−0.646887	−0.134885	−0.0674426	+	0.997723i	$0.521484\pi$
$24$	−0.414214	−0.0845510
$25$	1.00000	0.200000
$26$	2.13165	0.418051
$27$	−2.41421	−0.464616
$28$	1.00000	0.188982
$29$	−0.428825	−0.0796309	−0.0398154	−	0.999207i	$-0.512677\pi$
$29$	−0.428825	−0.0796309	−0.0398154	+	0.999207i	$0.512677\pi$
$30$	−0.414214	−0.0756247
$31$	−4.52004	−0.811824	−0.405912	−	0.913912i	$-0.633046\pi$
$31$	−4.52004	−0.811824	−0.405912	+	0.913912i	$0.633046\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.44949	0.763081
$35$	1.00000	0.169031
$36$	−2.82843	−0.471405
$37$	−8.49938	−1.39729	−0.698645	−	0.715469i	$-0.746213\pi$
$37$	−8.49938	−1.39729	−0.698645	+	0.715469i	$0.746213\pi$
$38$	−8.27792	−1.34286
$39$	−0.882959	−0.141387
$40$	−1.00000	−0.158114
$41$	10.6057	1.65634	0.828168	−	0.560480i	$-0.189383\pi$
$41$	10.6057	1.65634	0.828168	+	0.560480i	$0.189383\pi$
$42$	−0.414214	−0.0639145
$43$	2.65634	0.405088	0.202544	−	0.979273i	$-0.435079\pi$
$43$	2.65634	0.405088	0.202544	+	0.979273i	$0.435079\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0.646887	0.0953782
$47$	4.34242	0.633407	0.316703	−	0.948525i	$-0.397424\pi$
$47$	4.34242	0.633407	0.316703	+	0.948525i	$0.397424\pi$
$48$	0.414214	0.0597866
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−1.84304	−0.258077
$52$	−2.13165	−0.295607
$53$	0.0206643	0.00283846	0.00141923	−	0.999999i	$-0.499548\pi$
$53$	0.0206643	0.00283846	0.00141923	+	0.999999i	$0.499548\pi$
$54$	2.41421	0.328533
$55$	0	0
$56$	−1.00000	−0.133631
$57$	3.42883	0.454159
$58$	0.428825	0.0563075
$59$	−4.51059	−0.587229	−0.293614	−	0.955924i	$-0.594858\pi$
$59$	−4.51059	−0.587229	−0.293614	+	0.955924i	$0.594858\pi$
$60$	0.414214	0.0534747
$61$	1.36433	0.174684	0.0873420	−	0.996178i	$-0.472163\pi$
$61$	1.36433	0.174684	0.0873420	+	0.996178i	$0.472163\pi$
$62$	4.52004	0.574046
$63$	−2.82843	−0.356348
$64$	1.00000	0.125000
$65$	−2.13165	−0.264399
$66$	0	0
$67$	−2.68556	−0.328094	−0.164047	−	0.986453i	$-0.552455\pi$
$67$	−2.68556	−0.328094	−0.164047	+	0.986453i	$0.552455\pi$
$68$	−4.44949	−0.539580
$69$	−0.267949	−0.0322573
$70$	−1.00000	−0.119523
$71$	−9.05040	−1.07409	−0.537043	−	0.843555i	$-0.680459\pi$
$71$	−9.05040	−1.07409	−0.537043	+	0.843555i	$0.680459\pi$
$72$	2.82843	0.333333
$73$	−0.585786	−0.0685611	−0.0342806	−	0.999412i	$-0.510914\pi$
$73$	−0.585786	−0.0685611	−0.0342806	+	0.999412i	$0.510914\pi$
$74$	8.49938	0.988033
$75$	0.414214	0.0478293
$76$	8.27792	0.949542
$77$	0	0
$78$	0.882959	0.0999755
$79$	4.24728	0.477857	0.238928	−	0.971037i	$-0.423204\pi$
$79$	4.24728	0.477857	0.238928	+	0.971037i	$0.423204\pi$
$80$	1.00000	0.111803
$81$	7.48528	0.831698
$82$	−10.6057	−1.17121
$83$	−4.62446	−0.507601	−0.253800	−	0.967257i	$-0.581681\pi$
$83$	−4.62446	−0.507601	−0.253800	+	0.967257i	$0.581681\pi$
$84$	0.414214	0.0451944
$85$	−4.44949	−0.482615
$86$	−2.65634	−0.286440
$87$	−0.177625	−0.0190434
$88$	0	0
$89$	−2.17209	−0.230241	−0.115120	−	0.993352i	$-0.536725\pi$
$89$	−2.17209	−0.230241	−0.115120	+	0.993352i	$0.536725\pi$
$90$	2.82843	0.298142
$91$	−2.13165	−0.223458
$92$	−0.646887	−0.0674426
$93$	−1.87226	−0.194145
$94$	−4.34242	−0.447886
$95$	8.27792	0.849296
$96$	−0.414214	−0.0422755
$97$	13.0552	1.32556	0.662778	−	0.748816i	$-0.269377\pi$
$97$	13.0552	1.32556	0.662778	+	0.748816i	$0.269377\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.46410	−0.643202	−0.321601	−	0.946875i	$-0.604221\pi$
$101$	−6.46410	−0.643202	−0.321601	+	0.946875i	$0.604221\pi$
$102$	1.84304	0.182488
$103$	−12.4495	−1.22668	−0.613342	−	0.789817i	$-0.710175\pi$
$103$	−12.4495	−1.22668	−0.613342	+	0.789817i	$0.710175\pi$
$104$	2.13165	0.209026
$105$	0.414214	0.0404231
$106$	−0.0206643	−0.00200710
$107$	−10.0048	−0.967201	−0.483601	−	0.875289i	$-0.660671\pi$
$107$	−10.0048	−0.967201	−0.483601	+	0.875289i	$0.660671\pi$
$108$	−2.41421	−0.232308
$109$	5.91359	0.566419	0.283210	−	0.959058i	$-0.408601\pi$
$109$	5.91359	0.566419	0.283210	+	0.959058i	$0.408601\pi$
$110$	0	0
$111$	−3.52056	−0.334157
$112$	1.00000	0.0944911
$113$	−3.76857	−0.354517	−0.177259	−	0.984164i	$-0.556723\pi$
$113$	−3.76857	−0.354517	−0.177259	+	0.984164i	$0.556723\pi$
$114$	−3.42883	−0.321139
$115$	−0.646887	−0.0603225
$116$	−0.428825	−0.0398154
$117$	6.02922	0.557402
$118$	4.51059	0.415233
$119$	−4.44949	−0.407884
$120$	−0.414214	−0.0378124
$121$	0	0
$122$	−1.36433	−0.123520
$123$	4.39303	0.396107
$124$	−4.52004	−0.405912
$125$	1.00000	0.0894427
$126$	2.82843	0.251976
$127$	−8.56564	−0.760077	−0.380039	−	0.924971i	$-0.624089\pi$
$127$	−8.56564	−0.760077	−0.380039	+	0.924971i	$0.624089\pi$
$128$	−1.00000	−0.0883883
$129$	1.10029	0.0968753
$130$	2.13165	0.186958
$131$	−5.64224	−0.492965	−0.246483	−	0.969147i	$-0.579275\pi$
$131$	−5.64224	−0.492965	−0.246483	+	0.969147i	$0.579275\pi$
$132$	0	0
$133$	8.27792	0.717786
$134$	2.68556	0.231997
$135$	−2.41421	−0.207782
$136$	4.44949	0.381541
$137$	−1.64637	−0.140659	−0.0703295	−	0.997524i	$-0.522405\pi$
$137$	−1.64637	−0.140659	−0.0703295	+	0.997524i	$0.522405\pi$
$138$	0.267949	0.0228093
$139$	16.3071	1.38315	0.691577	−	0.722303i	$-0.256916\pi$
$139$	16.3071	1.38315	0.691577	+	0.722303i	$0.256916\pi$
$140$	1.00000	0.0845154
$141$	1.79869	0.151477
$142$	9.05040	0.759493
$143$	0	0
$144$	−2.82843	−0.235702
$145$	−0.428825	−0.0356120
$146$	0.585786	0.0484800
$147$	0.414214	0.0341638
$148$	−8.49938	−0.698645
$149$	4.12701	0.338098	0.169049	−	0.985608i	$-0.445930\pi$
$149$	4.12701	0.338098	0.169049	+	0.985608i	$0.445930\pi$
$150$	−0.414214	−0.0338204
$151$	−11.8949	−0.967989	−0.483995	−	0.875071i	$-0.660815\pi$
$151$	−11.8949	−0.967989	−0.483995	+	0.875071i	$0.660815\pi$
$152$	−8.27792	−0.671428
$153$	12.5851	1.01744
$154$	0	0
$155$	−4.52004	−0.363059
$156$	−0.882959	−0.0706933
$157$	10.6617	0.850893	0.425447	−	0.904984i	$-0.360117\pi$
$157$	10.6617	0.850893	0.425447	+	0.904984i	$0.360117\pi$
$158$	−4.24728	−0.337896
$159$	0.00855944	0.000678807 0
$160$	−1.00000	−0.0790569
$161$	−0.646887	−0.0509818
$162$	−7.48528	−0.588099
$163$	−19.0917	−1.49538	−0.747690	−	0.664048i	$-0.768837\pi$
$163$	−19.0917	−1.49538	−0.747690	+	0.664048i	$0.768837\pi$
$164$	10.6057	0.828168
$165$	0	0
$166$	4.62446	0.358928
$167$	−13.3684	−1.03448	−0.517239	−	0.855841i	$-0.673040\pi$
$167$	−13.3684	−1.03448	−0.517239	+	0.855841i	$0.673040\pi$
$168$	−0.414214	−0.0319573
$169$	−8.45606	−0.650466
$170$	4.44949	0.341260
$171$	−23.4135	−1.79047
$172$	2.65634	0.202544
$173$	2.38551	0.181367	0.0906833	−	0.995880i	$-0.471095\pi$
$173$	2.38551	0.181367	0.0906833	+	0.995880i	$0.471095\pi$
$174$	0.177625	0.0134657
$175$	1.00000	0.0755929
$176$	0	0
$177$	−1.86835	−0.140434
$178$	2.17209	0.162805
$179$	−3.68681	−0.275565	−0.137782	−	0.990463i	$-0.543997\pi$
$179$	−3.68681	−0.275565	−0.137782	+	0.990463i	$0.543997\pi$
$180$	−2.82843	−0.210819
$181$	−9.53001	−0.708360	−0.354180	−	0.935177i	$-0.615240\pi$
$181$	−9.53001	−0.708360	−0.354180	+	0.935177i	$0.615240\pi$
$182$	2.13165	0.158009
$183$	0.565122	0.0417750
$184$	0.646887	0.0476891
$185$	−8.49938	−0.624887
$186$	1.87226	0.137281
$187$	0	0
$188$	4.34242	0.316703
$189$	−2.41421	−0.175608
$190$	−8.27792	−0.600543
$191$	−13.3789	−0.968066	−0.484033	−	0.875050i	$-0.660829\pi$
$191$	−13.3789	−0.968066	−0.484033	+	0.875050i	$0.660829\pi$
$192$	0.414214	0.0298933
$193$	−6.52985	−0.470029	−0.235014	−	0.971992i	$-0.575514\pi$
$193$	−6.52985	−0.470029	−0.235014	+	0.971992i	$0.575514\pi$
$194$	−13.0552	−0.937310
$195$	−0.882959	−0.0632300
$196$	1.00000	0.0714286
$197$	−26.1056	−1.85995	−0.929974	−	0.367625i	$-0.880171\pi$
$197$	−26.1056	−1.85995	−0.929974	+	0.367625i	$0.880171\pi$
$198$	0	0
$199$	−18.8917	−1.33920	−0.669598	−	0.742724i	$-0.733534\pi$
$199$	−18.8917	−1.33920	−0.669598	+	0.742724i	$0.733534\pi$
$200$	−1.00000	−0.0707107
$201$	−1.11240	−0.0784624
$202$	6.46410	0.454813
$203$	−0.428825	−0.0300976
$204$	−1.84304	−0.129039
$205$	10.6057	0.740736
$206$	12.4495	0.867397
$207$	1.82967	0.127171
$208$	−2.13165	−0.147804
$209$	0	0
$210$	−0.414214	−0.0285835
$211$	16.7627	1.15399	0.576995	−	0.816747i	$-0.304225\pi$
$211$	16.7627	1.15399	0.576995	+	0.816747i	$0.304225\pi$
$212$	0.0206643	0.00141923
$213$	−3.74880	−0.256864
$214$	10.0048	0.683915
$215$	2.65634	0.181161
$216$	2.41421	0.164266
$217$	−4.52004	−0.306840
$218$	−5.91359	−0.400519
$219$	−0.242641	−0.0163961
$220$	0	0
$221$	9.48477	0.638014
$222$	3.52056	0.236284
$223$	−17.0053	−1.13876	−0.569380	−	0.822074i	$-0.692817\pi$
$223$	−17.0053	−1.13876	−0.569380	+	0.822074i	$0.692817\pi$
$224$	−1.00000	−0.0668153
$225$	−2.82843	−0.188562
$226$	3.76857	0.250682
$227$	14.0613	0.933279	0.466639	−	0.884448i	$-0.345465\pi$
$227$	14.0613	0.933279	0.466639	+	0.884448i	$0.345465\pi$
$228$	3.42883	0.227080
$229$	−27.7673	−1.83491	−0.917457	−	0.397836i	$-0.869761\pi$
$229$	−27.7673	−1.83491	−0.917457	+	0.397836i	$0.869761\pi$
$230$	0.646887	0.0426544
$231$	0	0
$232$	0.428825	0.0281538
$233$	9.82113	0.643404	0.321702	−	0.946841i	$-0.395745\pi$
$233$	9.82113	0.643404	0.321702	+	0.946841i	$0.395745\pi$
$234$	−6.02922	−0.394143
$235$	4.34242	0.283268
$236$	−4.51059	−0.293614
$237$	1.75928	0.114278
$238$	4.44949	0.288418
$239$	−4.28964	−0.277474	−0.138737	−	0.990329i	$-0.544304\pi$
$239$	−4.28964	−0.277474	−0.138737	+	0.990329i	$0.544304\pi$
$240$	0.414214	0.0267374
$241$	5.13505	0.330778	0.165389	−	0.986228i	$-0.447112\pi$
$241$	5.13505	0.330778	0.165389	+	0.986228i	$0.447112\pi$
$242$	0	0
$243$	10.3431	0.663513
$244$	1.36433	0.0873420
$245$	1.00000	0.0638877
$246$	−4.39303	−0.280090
$247$	−17.6456	−1.12277
$248$	4.52004	0.287023
$249$	−1.91552	−0.121391
$250$	−1.00000	−0.0632456
$251$	−10.5851	−0.668123	−0.334061	−	0.942551i	$-0.608419\pi$
$251$	−10.5851	−0.668123	−0.334061	+	0.942551i	$0.608419\pi$
$252$	−2.82843	−0.178174
$253$	0	0
$254$	8.56564	0.537456
$255$	−1.84304	−0.115416
$256$	1.00000	0.0625000
$257$	−4.27667	−0.266771	−0.133386	−	0.991064i	$-0.542585\pi$
$257$	−4.27667	−0.266771	−0.133386	+	0.991064i	$0.542585\pi$
$258$	−1.10029	−0.0685012
$259$	−8.49938	−0.528126
$260$	−2.13165	−0.132199
$261$	1.21290	0.0750767
$262$	5.64224	0.348579
$263$	−12.7773	−0.787882	−0.393941	−	0.919136i	$-0.628889\pi$
$263$	−12.7773	−0.787882	−0.393941	+	0.919136i	$0.628889\pi$
$264$	0	0
$265$	0.0206643	0.00126940
$266$	−8.27792	−0.507552
$267$	−0.899708	−0.0550613
$268$	−2.68556	−0.164047
$269$	−6.94495	−0.423441	−0.211721	−	0.977330i	$-0.567907\pi$
$269$	−6.94495	−0.423441	−0.211721	+	0.977330i	$0.567907\pi$
$270$	2.41421	0.146924
$271$	29.3911	1.78538	0.892690	−	0.450671i	$-0.148815\pi$
$271$	29.3911	1.78538	0.892690	+	0.450671i	$0.148815\pi$
$272$	−4.44949	−0.269790
$273$	−0.882959	−0.0534391
$274$	1.64637	0.0994609
$275$	0	0
$276$	−0.267949	−0.0161286
$277$	−9.99927	−0.600798	−0.300399	−	0.953814i	$-0.597120\pi$
$277$	−9.99927	−0.600798	−0.300399	+	0.953814i	$0.597120\pi$
$278$	−16.3071	−0.978037
$279$	12.7846	0.765395
$280$	−1.00000	−0.0597614
$281$	−21.5792	−1.28731	−0.643653	−	0.765318i	$-0.722582\pi$
$281$	−21.5792	−1.28731	−0.643653	+	0.765318i	$0.722582\pi$
$282$	−1.79869	−0.107110
$283$	−11.2319	−0.667670	−0.333835	−	0.942632i	$-0.608343\pi$
$283$	−11.2319	−0.667670	−0.333835	+	0.942632i	$0.608343\pi$
$284$	−9.05040	−0.537043
$285$	3.42883	0.203106
$286$	0	0
$287$	10.6057	0.626036
$288$	2.82843	0.166667
$289$	2.79796	0.164586
$290$	0.428825	0.0251815
$291$	5.40765	0.317002
$292$	−0.585786	−0.0342806
$293$	−16.7673	−0.979558	−0.489779	−	0.871847i	$-0.662922\pi$
$293$	−16.7673	−0.979558	−0.489779	+	0.871847i	$0.662922\pi$
$294$	−0.414214	−0.0241574
$295$	−4.51059	−0.262617
$296$	8.49938	0.494016
$297$	0	0
$298$	−4.12701	−0.239071
$299$	1.37894	0.0797460
$300$	0.414214	0.0239146
$301$	2.65634	0.153109
$302$	11.8949	0.684472
$303$	−2.67752	−0.153819
$304$	8.27792	0.474771
$305$	1.36433	0.0781211
$306$	−12.5851	−0.719440
$307$	12.5840	0.718208	0.359104	−	0.933297i	$-0.383082\pi$
$307$	12.5840	0.718208	0.359104	+	0.933297i	$0.383082\pi$
$308$	0	0
$309$	−5.15675	−0.293357
$310$	4.52004	0.256721
$311$	−21.3223	−1.20907	−0.604537	−	0.796577i	$-0.706642\pi$
$311$	−21.3223	−1.20907	−0.604537	+	0.796577i	$0.706642\pi$
$312$	0.882959	0.0499877
$313$	24.9895	1.41249	0.706244	−	0.707968i	$-0.250388\pi$
$313$	24.9895	1.41249	0.706244	+	0.707968i	$0.250388\pi$
$314$	−10.6617	−0.601672
$315$	−2.82843	−0.159364
$316$	4.24728	0.238928
$317$	3.33636	0.187389	0.0936944	−	0.995601i	$-0.470132\pi$
$317$	3.33636	0.187389	0.0936944	+	0.995601i	$0.470132\pi$
$318$	−0.00855944	−0.000479989 0
$319$	0	0
$320$	1.00000	0.0559017
$321$	−4.14413	−0.231303
$322$	0.646887	0.0360496
$323$	−36.8325	−2.04942
$324$	7.48528	0.415849
$325$	−2.13165	−0.118243
$326$	19.0917	1.05739
$327$	2.44949	0.135457
$328$	−10.6057	−0.585603
$329$	4.34242	0.239405
$330$	0	0
$331$	8.12044	0.446340	0.223170	−	0.974780i	$-0.428360\pi$
$331$	8.12044	0.446340	0.223170	+	0.974780i	$0.428360\pi$
$332$	−4.62446	−0.253800
$333$	24.0399	1.31738
$334$	13.3684	0.731487
$335$	−2.68556	−0.146728
$336$	0.414214	0.0225972
$337$	30.2961	1.65033	0.825167	−	0.564889i	$-0.191081\pi$
$337$	30.2961	1.65033	0.825167	+	0.564889i	$0.191081\pi$
$338$	8.45606	0.459949
$339$	−1.56099	−0.0847815
$340$	−4.44949	−0.241307
$341$	0	0
$342$	23.4135	1.26606
$343$	1.00000	0.0539949
$344$	−2.65634	−0.143220
$345$	−0.267949	−0.0144259
$346$	−2.38551	−0.128246
$347$	−21.5745	−1.15818	−0.579091	−	0.815263i	$-0.696592\pi$
$347$	−21.5745	−1.15818	−0.579091	+	0.815263i	$0.696592\pi$
$348$	−0.177625	−0.00952172
$349$	−29.4536	−1.57661	−0.788307	−	0.615282i	$-0.789042\pi$
$349$	−29.4536	−1.57661	−0.788307	+	0.615282i	$0.789042\pi$
$350$	−1.00000	−0.0534522
$351$	5.14626	0.274687
$352$	0	0
$353$	10.2578	0.545966	0.272983	−	0.962019i	$-0.411990\pi$
$353$	10.2578	0.545966	0.272983	+	0.962019i	$0.411990\pi$
$354$	1.86835	0.0993015
$355$	−9.05040	−0.480346
$356$	−2.17209	−0.115120
$357$	−1.84304	−0.0975440
$358$	3.68681	0.194854
$359$	31.9893	1.68833	0.844164	−	0.536085i	$-0.180097\pi$
$359$	31.9893	1.68833	0.844164	+	0.536085i	$0.180097\pi$
$360$	2.82843	0.149071
$361$	49.5239	2.60652
$362$	9.53001	0.500886
$363$	0	0
$364$	−2.13165	−0.111729
$365$	−0.585786	−0.0306615
$366$	−0.565122	−0.0295394
$367$	21.2734	1.11046	0.555232	−	0.831695i	$-0.312629\pi$
$367$	21.2734	1.11046	0.555232	+	0.831695i	$0.312629\pi$
$368$	−0.646887	−0.0337213
$369$	−29.9975	−1.56161
$370$	8.49938	0.441862
$371$	0.0206643	0.00107284
$372$	−1.87226	−0.0970723
$373$	21.6695	1.12200	0.561002	−	0.827815i	$-0.310416\pi$
$373$	21.6695	1.12200	0.561002	+	0.827815i	$0.310416\pi$
$374$	0	0
$375$	0.414214	0.0213899
$376$	−4.34242	−0.223943
$377$	0.914107	0.0470789
$378$	2.41421	0.124174
$379$	−15.6617	−0.804486	−0.402243	−	0.915533i	$-0.631769\pi$
$379$	−15.6617	−0.804486	−0.402243	+	0.915533i	$0.631769\pi$
$380$	8.27792	0.424648
$381$	−3.54800	−0.181770
$382$	13.3789	0.684526
$383$	17.5412	0.896315	0.448157	−	0.893955i	$-0.352080\pi$
$383$	17.5412	0.896315	0.448157	+	0.893955i	$0.352080\pi$
$384$	−0.414214	−0.0211377
$385$	0	0
$386$	6.52985	0.332360
$387$	−7.51326	−0.381921
$388$	13.0552	0.662778
$389$	−19.6118	−0.994356	−0.497178	−	0.867648i	$-0.665630\pi$
$389$	−19.6118	−0.994356	−0.497178	+	0.867648i	$0.665630\pi$
$390$	0.882959	0.0447104
$391$	2.87832	0.145563
$392$	−1.00000	−0.0505076
$393$	−2.33709	−0.117891
$394$	26.1056	1.31518
$395$	4.24728	0.213704
$396$	0	0
$397$	−1.78482	−0.0895777	−0.0447889	−	0.998996i	$-0.514262\pi$
$397$	−1.78482	−0.0895777	−0.0447889	+	0.998996i	$0.514262\pi$
$398$	18.8917	0.946954
$399$	3.42883	0.171656
$400$	1.00000	0.0500000
$401$	−27.8471	−1.39062	−0.695309	−	0.718711i	$-0.744733\pi$
$401$	−27.8471	−1.39062	−0.695309	+	0.718711i	$0.744733\pi$
$402$	1.11240	0.0554813
$403$	9.63516	0.479961
$404$	−6.46410	−0.321601
$405$	7.48528	0.371947
$406$	0.428825	0.0212822
$407$	0	0
$408$	1.84304	0.0912440
$409$	17.0071	0.840946	0.420473	−	0.907305i	$-0.361864\pi$
$409$	17.0071	0.840946	0.420473	+	0.907305i	$0.361864\pi$
$410$	−10.6057	−0.523779
$411$	−0.681949	−0.0336381
$412$	−12.4495	−0.613342
$413$	−4.51059	−0.221952
$414$	−1.82967	−0.0899235
$415$	−4.62446	−0.227006
$416$	2.13165	0.104513
$417$	6.75464	0.330776
$418$	0	0
$419$	7.35112	0.359126	0.179563	−	0.983746i	$-0.442532\pi$
$419$	7.35112	0.359126	0.179563	+	0.983746i	$0.442532\pi$
$420$	0.414214	0.0202116
$421$	20.0267	0.976043	0.488022	−	0.872832i	$-0.337719\pi$
$421$	20.0267	0.976043	0.488022	+	0.872832i	$0.337719\pi$
$422$	−16.7627	−0.815995
$423$	−12.2822	−0.597181
$424$	−0.0206643	−0.00100355
$425$	−4.44949	−0.215832
$426$	3.74880	0.181630
$427$	1.36433	0.0660243
$428$	−10.0048	−0.483601
$429$	0	0
$430$	−2.65634	−0.128100
$431$	−5.14626	−0.247887	−0.123943	−	0.992289i	$-0.539554\pi$
$431$	−5.14626	−0.247887	−0.123943	+	0.992289i	$0.539554\pi$
$432$	−2.41421	−0.116154
$433$	−41.3976	−1.98944	−0.994722	−	0.102611i	$-0.967280\pi$
$433$	−41.3976	−1.98944	−0.994722	+	0.102611i	$0.967280\pi$
$434$	4.52004	0.216969
$435$	−0.177625	−0.00851648
$436$	5.91359	0.283210
$437$	−5.35487	−0.256158
$438$	0.242641	0.0115938
$439$	21.6609	1.03382	0.516910	−	0.856040i	$-0.327082\pi$
$439$	21.6609	1.03382	0.516910	+	0.856040i	$0.327082\pi$
$440$	0	0
$441$	−2.82843	−0.134687
$442$	−9.48477	−0.451144
$443$	21.0771	1.00141	0.500703	−	0.865619i	$-0.333075\pi$
$443$	21.0771	1.00141	0.500703	+	0.865619i	$0.333075\pi$
$444$	−3.52056	−0.167078
$445$	−2.17209	−0.102967
$446$	17.0053	0.805225
$447$	1.70946	0.0808548
$448$	1.00000	0.0472456
$449$	40.9893	1.93440	0.967201	−	0.254011i	$-0.0817501\pi$
$449$	40.9893	1.93440	0.967201	+	0.254011i	$0.0817501\pi$
$450$	2.82843	0.133333
$451$	0	0
$452$	−3.76857	−0.177259
$453$	−4.92701	−0.231491
$454$	−14.0613	−0.659928
$455$	−2.13165	−0.0999334
$456$	−3.42883	−0.160569
$457$	18.3784	0.859706	0.429853	−	0.902899i	$-0.358565\pi$
$457$	18.3784	0.859706	0.429853	+	0.902899i	$0.358565\pi$
$458$	27.7673	1.29748
$459$	10.7420	0.501394
$460$	−0.646887	−0.0301612
$461$	15.0383	0.700403	0.350202	−	0.936674i	$-0.386113\pi$
$461$	15.0383	0.700403	0.350202	+	0.936674i	$0.386113\pi$
$462$	0	0
$463$	−28.6376	−1.33090	−0.665451	−	0.746442i	$-0.731761\pi$
$463$	−28.6376	−1.33090	−0.665451	+	0.746442i	$0.731761\pi$
$464$	−0.428825	−0.0199077
$465$	−1.87226	−0.0868241
$466$	−9.82113	−0.454955
$467$	9.16960	0.424318	0.212159	−	0.977235i	$-0.431950\pi$
$467$	9.16960	0.424318	0.212159	+	0.977235i	$0.431950\pi$
$468$	6.02922	0.278701
$469$	−2.68556	−0.124008
$470$	−4.34242	−0.200301
$471$	4.41621	0.203488
$472$	4.51059	0.207617
$473$	0	0
$474$	−1.75928	−0.0808065
$475$	8.27792	0.379817
$476$	−4.44949	−0.203942
$477$	−0.0584475	−0.00267613
$478$	4.28964	0.196204
$479$	−33.8489	−1.54659	−0.773297	−	0.634044i	$-0.781394\pi$
$479$	−33.8489	−1.54659	−0.773297	+	0.634044i	$0.781394\pi$
$480$	−0.414214	−0.0189062
$481$	18.1177	0.826097
$482$	−5.13505	−0.233895
$483$	−0.267949	−0.0121921
$484$	0	0
$485$	13.0552	0.592807
$486$	−10.3431	−0.469175
$487$	−27.2216	−1.23353	−0.616765	−	0.787148i	$-0.711557\pi$
$487$	−27.2216	−1.23353	−0.616765	+	0.787148i	$0.711557\pi$
$488$	−1.36433	−0.0617601
$489$	−7.90805	−0.357615
$490$	−1.00000	−0.0451754
$491$	−33.7879	−1.52483	−0.762414	−	0.647089i	$-0.775986\pi$
$491$	−33.7879	−1.52483	−0.762414	+	0.647089i	$0.775986\pi$
$492$	4.39303	0.198053
$493$	1.90805	0.0859344
$494$	17.6456	0.793915
$495$	0	0
$496$	−4.52004	−0.202956
$497$	−9.05040	−0.405966
$498$	1.91552	0.0858363
$499$	14.6347	0.655140	0.327570	−	0.944827i	$-0.393770\pi$
$499$	14.6347	0.655140	0.327570	+	0.944827i	$0.393770\pi$
$500$	1.00000	0.0447214
$501$	−5.53737	−0.247392
$502$	10.5851	0.472434
$503$	−23.0486	−1.02769	−0.513844	−	0.857884i	$-0.671779\pi$
$503$	−23.0486	−1.02769	−0.513844	+	0.857884i	$0.671779\pi$
$504$	2.82843	0.125988
$505$	−6.46410	−0.287649
$506$	0	0
$507$	−3.50261	−0.155557
$508$	−8.56564	−0.380039
$509$	2.55368	0.113190	0.0565949	−	0.998397i	$-0.481976\pi$
$509$	2.55368	0.113190	0.0565949	+	0.998397i	$0.481976\pi$
$510$	1.84304	0.0816111
$511$	−0.585786	−0.0259137
$512$	−1.00000	−0.0441942
$513$	−19.9847	−0.882344
$514$	4.27667	0.188636
$515$	−12.4495	−0.548590
$516$	1.10029	0.0484376
$517$	0	0
$518$	8.49938	0.373441
$519$	0.988109	0.0433732
$520$	2.13165	0.0934791
$521$	9.67198	0.423737	0.211869	−	0.977298i	$-0.432045\pi$
$521$	9.67198	0.423737	0.211869	+	0.977298i	$0.432045\pi$
$522$	−1.21290	−0.0530873
$523$	−26.0501	−1.13909	−0.569545	−	0.821960i	$-0.692880\pi$
$523$	−26.0501	−1.13909	−0.569545	+	0.821960i	$0.692880\pi$
$524$	−5.64224	−0.246483
$525$	0.414214	0.0180778
$526$	12.7773	0.557117
$527$	20.1119	0.876087
$528$	0	0
$529$	−22.5815	−0.981806
$530$	−0.0206643	−0.000897600 0
$531$	12.7579	0.553645
$532$	8.27792	0.358893
$533$	−22.6077	−0.979249
$534$	0.899708	0.0389342
$535$	−10.0048	−0.432546
$536$	2.68556	0.115999
$537$	−1.52713	−0.0659003
$538$	6.94495	0.299418
$539$	0	0
$540$	−2.41421	−0.103891
$541$	−40.9549	−1.76079	−0.880395	−	0.474241i	$-0.842723\pi$
$541$	−40.9549	−1.76079	−0.880395	+	0.474241i	$0.842723\pi$
$542$	−29.3911	−1.26245
$543$	−3.94746	−0.169402
$544$	4.44949	0.190770
$545$	5.91359	0.253310
$546$	0.882959	0.0377872
$547$	26.8345	1.14736	0.573680	−	0.819079i	$-0.305515\pi$
$547$	26.8345	1.14736	0.573680	+	0.819079i	$0.305515\pi$
$548$	−1.64637	−0.0703295
$549$	−3.85890	−0.164694
$550$	0	0
$551$	−3.54978	−0.151226
$552$	0.267949	0.0114047
$553$	4.24728	0.180613
$554$	9.99927	0.424828
$555$	−3.52056	−0.149439
$556$	16.3071	0.691577
$557$	19.9262	0.844301	0.422151	−	0.906526i	$-0.361275\pi$
$557$	19.9262	0.844301	0.422151	+	0.906526i	$0.361275\pi$
$558$	−12.7846	−0.541216
$559$	−5.66239	−0.239494
$560$	1.00000	0.0422577
$561$	0	0
$562$	21.5792	0.910262
$563$	8.40103	0.354061	0.177031	−	0.984205i	$-0.443351\pi$
$563$	8.40103	0.354061	0.177031	+	0.984205i	$0.443351\pi$
$564$	1.79869	0.0757384
$565$	−3.76857	−0.158545
$566$	11.2319	0.472114
$567$	7.48528	0.314352
$568$	9.05040	0.379746
$569$	−29.5592	−1.23919	−0.619594	−	0.784923i	$-0.712703\pi$
$569$	−29.5592	−1.23919	−0.619594	+	0.784923i	$0.712703\pi$
$570$	−3.42883	−0.143618
$571$	−23.2036	−0.971041	−0.485520	−	0.874225i	$-0.661370\pi$
$571$	−23.2036	−0.971041	−0.485520	+	0.874225i	$0.661370\pi$
$572$	0	0
$573$	−5.54174	−0.231509
$574$	−10.6057	−0.442674
$575$	−0.646887	−0.0269770
$576$	−2.82843	−0.117851
$577$	9.45408	0.393579	0.196789	−	0.980446i	$-0.436948\pi$
$577$	9.45408	0.393579	0.196789	+	0.980446i	$0.436948\pi$
$578$	−2.79796	−0.116380
$579$	−2.70475	−0.112406
$580$	−0.428825	−0.0178060
$581$	−4.62446	−0.191855
$582$	−5.40765	−0.224154
$583$	0	0
$584$	0.585786	0.0242400
$585$	6.02922	0.249278
$586$	16.7673	0.692652
$587$	20.1626	0.832199	0.416100	−	0.909319i	$-0.363397\pi$
$587$	20.1626	0.832199	0.416100	+	0.909319i	$0.363397\pi$
$588$	0.414214	0.0170819
$589$	−37.4165	−1.54172
$590$	4.51059	0.185698
$591$	−10.8133	−0.444800
$592$	−8.49938	−0.349322
$593$	−34.9585	−1.43557	−0.717786	−	0.696264i	$-0.754844\pi$
$593$	−34.9585	−1.43557	−0.717786	+	0.696264i	$0.754844\pi$
$594$	0	0
$595$	−4.44949	−0.182411
$596$	4.12701	0.169049
$597$	−7.82519	−0.320264
$598$	−1.37894	−0.0563889
$599$	24.4400	0.998591	0.499295	−	0.866432i	$-0.333592\pi$
$599$	24.4400	0.998591	0.499295	+	0.866432i	$0.333592\pi$
$600$	−0.414214	−0.0169102
$601$	−9.94812	−0.405792	−0.202896	−	0.979200i	$-0.565035\pi$
$601$	−9.94812	−0.405792	−0.202896	+	0.979200i	$0.565035\pi$
$602$	−2.65634	−0.108264
$603$	7.59592	0.309330
$604$	−11.8949	−0.483995
$605$	0	0
$606$	2.67752	0.108767
$607$	−32.6542	−1.32539	−0.662696	−	0.748889i	$-0.730588\pi$
$607$	−32.6542	−1.32539	−0.662696	+	0.748889i	$0.730588\pi$
$608$	−8.27792	−0.335714
$609$	−0.177625	−0.00719774
$610$	−1.36433	−0.0552399
$611$	−9.25652	−0.374479
$612$	12.5851	0.508721
$613$	−39.0744	−1.57820	−0.789100	−	0.614264i	$-0.789453\pi$
$613$	−39.0744	−1.57820	−0.789100	+	0.614264i	$0.789453\pi$
$614$	−12.5840	−0.507850
$615$	4.39303	0.177144
$616$	0	0
$617$	−5.10384	−0.205473	−0.102736	−	0.994709i	$-0.532760\pi$
$617$	−5.10384	−0.205473	−0.102736	+	0.994709i	$0.532760\pi$
$618$	5.15675	0.207435
$619$	8.03596	0.322992	0.161496	−	0.986873i	$-0.448368\pi$
$619$	8.03596	0.322992	0.161496	+	0.986873i	$0.448368\pi$
$620$	−4.52004	−0.181529
$621$	1.56172	0.0626698
$622$	21.3223	0.854945
$623$	−2.17209	−0.0870229
$624$	−0.882959	−0.0353467
$625$	1.00000	0.0400000
$626$	−24.9895	−0.998780
$627$	0	0
$628$	10.6617	0.425447
$629$	37.8179	1.50790
$630$	2.82843	0.112687
$631$	2.37486	0.0945416	0.0472708	−	0.998882i	$-0.484948\pi$
$631$	2.37486	0.0945416	0.0472708	+	0.998882i	$0.484948\pi$
$632$	−4.24728	−0.168948
$633$	6.94333	0.275973
$634$	−3.33636	−0.132504
$635$	−8.56564	−0.339917
$636$	0.00855944	0.000339404 0
$637$	−2.13165	−0.0844591
$638$	0	0
$639$	25.5984	1.01266
$640$	−1.00000	−0.0395285
$641$	−41.6595	−1.64545	−0.822725	−	0.568440i	$-0.807547\pi$
$641$	−41.6595	−1.64545	−0.822725	+	0.568440i	$0.807547\pi$
$642$	4.14413	0.163556
$643$	36.9681	1.45788	0.728939	−	0.684578i	$-0.240014\pi$
$643$	36.9681	1.45788	0.728939	+	0.684578i	$0.240014\pi$
$644$	−0.646887	−0.0254909
$645$	1.10029	0.0433239
$646$	36.8325	1.44916
$647$	−14.5229	−0.570952	−0.285476	−	0.958386i	$-0.592152\pi$
$647$	−14.5229	−0.570952	−0.285476	+	0.958386i	$0.592152\pi$
$648$	−7.48528	−0.294050
$649$	0	0
$650$	2.13165	0.0836103
$651$	−1.87226	−0.0733798
$652$	−19.0917	−0.747690
$653$	−49.0424	−1.91918	−0.959588	−	0.281408i	$-0.909199\pi$
$653$	−49.0424	−1.91918	−0.959588	+	0.281408i	$0.909199\pi$
$654$	−2.44949	−0.0957826
$655$	−5.64224	−0.220461
$656$	10.6057	0.414084
$657$	1.65685	0.0646400
$658$	−4.34242	−0.169285
$659$	14.0139	0.545903	0.272951	−	0.962028i	$-0.412000\pi$
$659$	14.0139	0.545903	0.272951	+	0.962028i	$0.412000\pi$
$660$	0	0
$661$	44.9342	1.74774	0.873869	−	0.486161i	$-0.161603\pi$
$661$	44.9342	1.74774	0.873869	+	0.486161i	$0.161603\pi$
$662$	−8.12044	−0.315610
$663$	3.92872	0.152579
$664$	4.62446	0.179464
$665$	8.27792	0.321004
$666$	−24.0399	−0.931526
$667$	0.277401	0.0107410
$668$	−13.3684	−0.517239
$669$	−7.04384	−0.272330
$670$	2.68556	0.103752
$671$	0	0
$672$	−0.414214	−0.0159786
$673$	−43.1679	−1.66400	−0.832000	−	0.554775i	$-0.812804\pi$
$673$	−43.1679	−1.66400	−0.832000	+	0.554775i	$0.812804\pi$
$674$	−30.2961	−1.16696
$675$	−2.41421	−0.0929231
$676$	−8.45606	−0.325233
$677$	−7.26720	−0.279301	−0.139651	−	0.990201i	$-0.544598\pi$
$677$	−7.26720	−0.279301	−0.139651	+	0.990201i	$0.544598\pi$
$678$	1.56099	0.0599496
$679$	13.0552	0.501013
$680$	4.44949	0.170630
$681$	5.82437	0.223190
$682$	0	0
$683$	8.44196	0.323023	0.161511	−	0.986871i	$-0.448363\pi$
$683$	8.44196	0.323023	0.161511	+	0.986871i	$0.448363\pi$
$684$	−23.4135	−0.895237
$685$	−1.64637	−0.0629046
$686$	−1.00000	−0.0381802
$687$	−11.5016	−0.438813
$688$	2.65634	0.101272
$689$	−0.0440491	−0.00167814
$690$	0.267949	0.0102007
$691$	42.7272	1.62542	0.812710	−	0.582669i	$-0.197992\pi$
$691$	42.7272	1.62542	0.812710	+	0.582669i	$0.197992\pi$
$692$	2.38551	0.0906833
$693$	0	0
$694$	21.5745	0.818958
$695$	16.3071	0.618565
$696$	0.177625	0.00673287
$697$	−47.1901	−1.78745
$698$	29.4536	1.11483
$699$	4.06805	0.153868
$700$	1.00000	0.0377964
$701$	32.9814	1.24569	0.622846	−	0.782345i	$-0.285976\pi$
$701$	32.9814	1.24569	0.622846	+	0.782345i	$0.285976\pi$
$702$	−5.14626	−0.194233
$703$	−70.3571	−2.65357
$704$	0	0
$705$	1.79869	0.0677425
$706$	−10.2578	−0.386056
$707$	−6.46410	−0.243108
$708$	−1.86835	−0.0702168
$709$	19.1550	0.719381	0.359690	−	0.933072i	$-0.382882\pi$
$709$	19.1550	0.719381	0.359690	+	0.933072i	$0.382882\pi$
$710$	9.05040	0.339656
$711$	−12.0131	−0.450528
$712$	2.17209	0.0814025
$713$	2.92395	0.109503
$714$	1.84304	0.0689740
$715$	0	0
$716$	−3.68681	−0.137782
$717$	−1.77683	−0.0663569
$718$	−31.9893	−1.19383
$719$	39.4644	1.47177	0.735887	−	0.677104i	$-0.236765\pi$
$719$	39.4644	1.47177	0.735887	+	0.677104i	$0.236765\pi$
$720$	−2.82843	−0.105409
$721$	−12.4495	−0.463643
$722$	−49.5239	−1.84309
$723$	2.12701	0.0791043
$724$	−9.53001	−0.354180
$725$	−0.428825	−0.0159262
$726$	0	0
$727$	−25.9129	−0.961055	−0.480527	−	0.876980i	$-0.659555\pi$
$727$	−25.9129	−0.961055	−0.480527	+	0.876980i	$0.659555\pi$
$728$	2.13165	0.0790043
$729$	−18.1716	−0.673021
$730$	0.585786	0.0216809
$731$	−11.8194	−0.437155
$732$	0.565122	0.0208875
$733$	47.2935	1.74683	0.873413	−	0.486981i	$-0.161902\pi$
$733$	47.2935	1.74683	0.873413	+	0.486981i	$0.161902\pi$
$734$	−21.2734	−0.785217
$735$	0.414214	0.0152785
$736$	0.646887	0.0238446
$737$	0	0
$738$	29.9975	1.10422
$739$	34.1019	1.25446	0.627229	−	0.778835i	$-0.284189\pi$
$739$	34.1019	1.25446	0.627229	+	0.778835i	$0.284189\pi$
$740$	−8.49938	−0.312443
$741$	−7.30906	−0.268505
$742$	−0.0206643	−0.000758611 0
$743$	37.2951	1.36823	0.684113	−	0.729376i	$-0.260190\pi$
$743$	37.2951	1.36823	0.684113	+	0.729376i	$0.260190\pi$
$744$	1.87226	0.0686405
$745$	4.12701	0.151202
$746$	−21.6695	−0.793376
$747$	13.0800	0.478571
$748$	0	0
$749$	−10.0048	−0.365568
$750$	−0.414214	−0.0151249
$751$	27.1160	0.989475	0.494738	−	0.869042i	$-0.335264\pi$
$751$	27.1160	0.989475	0.494738	+	0.869042i	$0.335264\pi$
$752$	4.34242	0.158352
$753$	−4.38447	−0.159779
$754$	−0.914107	−0.0332898
$755$	−11.8949	−0.432898
$756$	−2.41421	−0.0878041
$757$	1.81278	0.0658868	0.0329434	−	0.999457i	$-0.489512\pi$
$757$	1.81278	0.0658868	0.0329434	+	0.999457i	$0.489512\pi$
$758$	15.6617	0.568857
$759$	0	0
$760$	−8.27792	−0.300272
$761$	−31.5856	−1.14498	−0.572488	−	0.819913i	$-0.694022\pi$
$761$	−31.5856	−1.14498	−0.572488	+	0.819913i	$0.694022\pi$
$762$	3.54800	0.128531
$763$	5.91359	0.214086
$764$	−13.3789	−0.484033
$765$	12.5851	0.455014
$766$	−17.5412	−0.633790
$767$	9.61501	0.347178
$768$	0.414214	0.0149466
$769$	33.4286	1.20547	0.602733	−	0.797943i	$-0.294078\pi$
$769$	33.4286	1.20547	0.602733	+	0.797943i	$0.294078\pi$
$770$	0	0
$771$	−1.77146	−0.0637974
$772$	−6.52985	−0.235014
$773$	−47.6294	−1.71311	−0.856555	−	0.516055i	$-0.827400\pi$
$773$	−47.6294	−1.71311	−0.856555	+	0.516055i	$0.827400\pi$
$774$	7.51326	0.270059
$775$	−4.52004	−0.162365
$776$	−13.0552	−0.468655
$777$	−3.52056	−0.126299
$778$	19.6118	0.703116
$779$	87.7933	3.14552
$780$	−0.882959	−0.0316150
$781$	0	0
$782$	−2.87832	−0.102928
$783$	1.03528	0.0369978
$784$	1.00000	0.0357143
$785$	10.6617	0.380531
$786$	2.33709	0.0833614
$787$	−35.3497	−1.26008	−0.630041	−	0.776562i	$-0.716962\pi$
$787$	−35.3497	−1.26008	−0.630041	+	0.776562i	$0.716962\pi$
$788$	−26.1056	−0.929974
$789$	−5.29253	−0.188419
$790$	−4.24728	−0.151112
$791$	−3.76857	−0.133995
$792$	0	0
$793$	−2.90827	−0.103276
$794$	1.78482	0.0633410
$795$	0.00855944	0.000303572 0
$796$	−18.8917	−0.669598
$797$	−11.6576	−0.412933	−0.206467	−	0.978454i	$-0.566197\pi$
$797$	−11.6576	−0.412933	−0.206467	+	0.978454i	$0.566197\pi$
$798$	−3.42883	−0.121379
$799$	−19.3215	−0.683547
$800$	−1.00000	−0.0353553
$801$	6.14359	0.217073
$802$	27.8471	0.983316
$803$	0	0
$804$	−1.11240	−0.0392312
$805$	−0.646887	−0.0227998
$806$	−9.63516	−0.339384
$807$	−2.87669	−0.101264
$808$	6.46410	0.227406
$809$	14.5052	0.509976	0.254988	−	0.966944i	$-0.417928\pi$
$809$	14.5052	0.509976	0.254988	+	0.966944i	$0.417928\pi$
$810$	−7.48528	−0.263006
$811$	−30.8587	−1.08359	−0.541797	−	0.840509i	$-0.682256\pi$
$811$	−30.8587	−1.08359	−0.541797	+	0.840509i	$0.682256\pi$
$812$	−0.428825	−0.0150488
$813$	12.1742	0.426967
$814$	0	0
$815$	−19.0917	−0.668754
$816$	−1.84304	−0.0645193
$817$	21.9890	0.769296
$818$	−17.0071	−0.594639
$819$	6.02922	0.210678
$820$	10.6057	0.370368
$821$	8.11042	0.283056	0.141528	−	0.989934i	$-0.454799\pi$
$821$	8.11042	0.283056	0.141528	+	0.989934i	$0.454799\pi$
$822$	0.681949	0.0237857
$823$	−46.4122	−1.61783	−0.808914	−	0.587926i	$-0.799944\pi$
$823$	−46.4122	−1.61783	−0.808914	+	0.587926i	$0.799944\pi$
$824$	12.4495	0.433699
$825$	0	0
$826$	4.51059	0.156943
$827$	22.7819	0.792204	0.396102	−	0.918207i	$-0.370363\pi$
$827$	22.7819	0.792204	0.396102	+	0.918207i	$0.370363\pi$
$828$	1.82967	0.0635855
$829$	−6.20021	−0.215342	−0.107671	−	0.994187i	$-0.534339\pi$
$829$	−6.20021	−0.215342	−0.107671	+	0.994187i	$0.534339\pi$
$830$	4.62446	0.160517
$831$	−4.14183	−0.143679
$832$	−2.13165	−0.0739018
$833$	−4.44949	−0.154166
$834$	−6.75464	−0.233894
$835$	−13.3684	−0.462633
$836$	0	0
$837$	10.9123	0.377186
$838$	−7.35112	−0.253940
$839$	−4.54727	−0.156989	−0.0784947	−	0.996915i	$-0.525011\pi$
$839$	−4.54727	−0.156989	−0.0784947	+	0.996915i	$0.525011\pi$
$840$	−0.414214	−0.0142917
$841$	−28.8161	−0.993659
$842$	−20.0267	−0.690167
$843$	−8.93838	−0.307854
$844$	16.7627	0.576995
$845$	−8.45606	−0.290897
$846$	12.2822	0.422271
$847$	0	0
$848$	0.0206643	0.000709615 0
$849$	−4.65242	−0.159671
$850$	4.44949	0.152616
$851$	5.49813	0.188474
$852$	−3.74880	−0.128432
$853$	26.8475	0.919241	0.459620	−	0.888115i	$-0.347985\pi$
$853$	26.8475	0.919241	0.459620	+	0.888115i	$0.347985\pi$
$854$	−1.36433	−0.0466863
$855$	−23.4135	−0.800724
$856$	10.0048	0.341957
$857$	−40.8363	−1.39494	−0.697470	−	0.716614i	$-0.745691\pi$
$857$	−40.8363	−1.39494	−0.697470	+	0.716614i	$0.745691\pi$
$858$	0	0
$859$	1.34824	0.0460013	0.0230006	−	0.999735i	$-0.492678\pi$
$859$	1.34824	0.0460013	0.0230006	+	0.999735i	$0.492678\pi$
$860$	2.65634	0.0905804
$861$	4.39303	0.149714
$862$	5.14626	0.175282
$863$	−40.8376	−1.39013	−0.695064	−	0.718948i	$-0.744624\pi$
$863$	−40.8376	−1.39013	−0.695064	+	0.718948i	$0.744624\pi$
$864$	2.41421	0.0821332
$865$	2.38551	0.0811096
$866$	41.3976	1.40675
$867$	1.15895	0.0393601
$868$	−4.52004	−0.153420
$869$	0	0
$870$	0.177625	0.00602206
$871$	5.72469	0.193974
$872$	−5.91359	−0.200259
$873$	−36.9257	−1.24975
$874$	5.35487	0.181131
$875$	1.00000	0.0338062
$876$	−0.242641	−0.00819807
$877$	−4.52485	−0.152793	−0.0763967	−	0.997078i	$-0.524342\pi$
$877$	−4.52485	−0.152793	−0.0763967	+	0.997078i	$0.524342\pi$
$878$	−21.6609	−0.731021
$879$	−6.94525	−0.234258
$880$	0	0
$881$	−18.7841	−0.632852	−0.316426	−	0.948617i	$-0.602483\pi$
$881$	−18.7841	−0.632852	−0.316426	+	0.948617i	$0.602483\pi$
$882$	2.82843	0.0952381
$883$	−27.8159	−0.936080	−0.468040	−	0.883707i	$-0.655040\pi$
$883$	−27.8159	−0.936080	−0.468040	+	0.883707i	$0.655040\pi$
$884$	9.48477	0.319007
$885$	−1.86835	−0.0628038
$886$	−21.0771	−0.708100
$887$	−1.97830	−0.0664250	−0.0332125	−	0.999448i	$-0.510574\pi$
$887$	−1.97830	−0.0664250	−0.0332125	+	0.999448i	$0.510574\pi$
$888$	3.52056	0.118142
$889$	−8.56564	−0.287282
$890$	2.17209	0.0728086
$891$	0	0
$892$	−17.0053	−0.569380
$893$	35.9462	1.20289
$894$	−1.70946	−0.0571730
$895$	−3.68681	−0.123236
$896$	−1.00000	−0.0334077
$897$	0.571175	0.0190710
$898$	−40.9893	−1.36783
$899$	1.93831	0.0646462
$900$	−2.82843	−0.0942809
$901$	−0.0919456	−0.00306315
$902$	0	0
$903$	1.10029	0.0366154
$904$	3.76857	0.125341
$905$	−9.53001	−0.316788
$906$	4.92701	0.163689
$907$	29.2382	0.970838	0.485419	−	0.874282i	$-0.338667\pi$
$907$	29.2382	0.970838	0.485419	+	0.874282i	$0.338667\pi$
$908$	14.0613	0.466639
$909$	18.2832	0.606417
$910$	2.13165	0.0706636
$911$	7.11563	0.235751	0.117876	−	0.993028i	$-0.462392\pi$
$911$	7.11563	0.235751	0.117876	+	0.993028i	$0.462392\pi$
$912$	3.42883	0.113540
$913$	0	0
$914$	−18.3784	−0.607904
$915$	0.565122	0.0186824
$916$	−27.7673	−0.917457
$917$	−5.64224	−0.186323
$918$	−10.7420	−0.354539
$919$	48.4518	1.59828	0.799138	−	0.601147i	$-0.205289\pi$
$919$	48.4518	1.59828	0.799138	+	0.601147i	$0.205289\pi$
$920$	0.646887	0.0213272
$921$	5.21247	0.171757
$922$	−15.0383	−0.495260
$923$	19.2923	0.635014
$924$	0	0
$925$	−8.49938	−0.279458
$926$	28.6376	0.941090
$927$	35.2125	1.15653
$928$	0.428825	0.0140769
$929$	14.8869	0.488423	0.244211	−	0.969722i	$-0.421471\pi$
$929$	14.8869	0.488423	0.244211	+	0.969722i	$0.421471\pi$
$930$	1.87226	0.0613939
$931$	8.27792	0.271298
$932$	9.82113	0.321702
$933$	−8.83197	−0.289146
$934$	−9.16960	−0.300038
$935$	0	0
$936$	−6.02922	−0.197071
$937$	−4.76268	−0.155590	−0.0777950	−	0.996969i	$-0.524788\pi$
$937$	−4.76268	−0.155590	−0.0777950	+	0.996969i	$0.524788\pi$
$938$	2.68556	0.0876867
$939$	10.3510	0.337791
$940$	4.34242	0.141634
$941$	−25.4100	−0.828341	−0.414171	−	0.910199i	$-0.635928\pi$
$941$	−25.4100	−0.828341	−0.414171	+	0.910199i	$0.635928\pi$
$942$	−4.41621	−0.143888
$943$	−6.86070	−0.223415
$944$	−4.51059	−0.146807
$945$	−2.41421	−0.0785344
$946$	0	0
$947$	−59.0379	−1.91847	−0.959237	−	0.282603i	$-0.908802\pi$
$947$	−59.0379	−1.91847	−0.959237	+	0.282603i	$0.908802\pi$
$948$	1.75928	0.0571389
$949$	1.24869	0.0405343
$950$	−8.27792	−0.268571
$951$	1.38197	0.0448134
$952$	4.44949	0.144209
$953$	−28.2159	−0.914003	−0.457002	−	0.889466i	$-0.651077\pi$
$953$	−28.2159	−0.914003	−0.457002	+	0.889466i	$0.651077\pi$
$954$	0.0584475	0.00189231
$955$	−13.3789	−0.432932
$956$	−4.28964	−0.138737
$957$	0	0
$958$	33.8489	1.09361
$959$	−1.64637	−0.0531641
$960$	0.414214	0.0133687
$961$	−10.5692	−0.340943
$962$	−18.1177	−0.584139
$963$	28.2979	0.911886
$964$	5.13505	0.165389
$965$	−6.52985	−0.210203
$966$	0.267949	0.00862112
$967$	27.6194	0.888179	0.444090	−	0.895982i	$-0.353527\pi$
$967$	27.6194	0.888179	0.444090	+	0.895982i	$0.353527\pi$
$968$	0	0
$969$	−15.2565	−0.490110
$970$	−13.0552	−0.419178
$971$	30.2501	0.970772	0.485386	−	0.874300i	$-0.338679\pi$
$971$	30.2501	0.970772	0.485386	+	0.874300i	$0.338679\pi$
$972$	10.3431	0.331757
$973$	16.3071	0.522783
$974$	27.2216	0.872237
$975$	−0.882959	−0.0282773
$976$	1.36433	0.0436710
$977$	41.5134	1.32813	0.664066	−	0.747674i	$-0.268829\pi$
$977$	41.5134	1.32813	0.664066	+	0.747674i	$0.268829\pi$
$978$	7.90805	0.252872
$979$	0	0
$980$	1.00000	0.0319438
$981$	−16.7262	−0.534025
$982$	33.7879	1.07822
$983$	10.6312	0.339082	0.169541	−	0.985523i	$-0.445771\pi$
$983$	10.6312	0.339082	0.169541	+	0.985523i	$0.445771\pi$
$984$	−4.39303	−0.140045
$985$	−26.1056	−0.831794
$986$	−1.90805	−0.0607648
$987$	1.79869	0.0572529
$988$	−17.6456	−0.561383
$989$	−1.71835	−0.0546403
$990$	0	0
$991$	−44.8815	−1.42571	−0.712853	−	0.701313i	$-0.752597\pi$
$991$	−44.8815	−1.42571	−0.712853	+	0.701313i	$0.752597\pi$
$992$	4.52004	0.143511
$993$	3.36360	0.106741
$994$	9.05040	0.287061
$995$	−18.8917	−0.598907
$996$	−1.91552	−0.0606954
$997$	−30.7930	−0.975225	−0.487613	−	0.873060i	$-0.662132\pi$
$997$	−30.7930	−0.975225	−0.487613	+	0.873060i	$0.662132\pi$
$998$	−14.6347	−0.463254
$999$	20.5193	0.649202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cp.1.3	✓	4
11.10	odd	2		8470.2.a.cr.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.cp.1.3	✓	4	1.1	even	1	trivial
8470.2.a.cr.1.4	yes	4	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.414214 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.82843 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.414214 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.82843 q^{9} -1.00000 q^{10} +0.414214 q^{12} -2.13165 q^{13} -1.00000 q^{14} +0.414214 q^{15} +1.00000 q^{16} -4.44949 q^{17} +2.82843 q^{18} +8.27792 q^{19} +1.00000 q^{20} +0.414214 q^{21} -0.646887 q^{23} -0.414214 q^{24} +1.00000 q^{25} +2.13165 q^{26} -2.41421 q^{27} +1.00000 q^{28} -0.428825 q^{29} -0.414214 q^{30} -4.52004 q^{31} -1.00000 q^{32} +4.44949 q^{34} +1.00000 q^{35} -2.82843 q^{36} -8.49938 q^{37} -8.27792 q^{38} -0.882959 q^{39} -1.00000 q^{40} +10.6057 q^{41} -0.414214 q^{42} +2.65634 q^{43} -2.82843 q^{45} +0.646887 q^{46} +4.34242 q^{47} +0.414214 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.84304 q^{51} -2.13165 q^{52} +0.0206643 q^{53} +2.41421 q^{54} -1.00000 q^{56} +3.42883 q^{57} +0.428825 q^{58} -4.51059 q^{59} +0.414214 q^{60} +1.36433 q^{61} +4.52004 q^{62} -2.82843 q^{63} +1.00000 q^{64} -2.13165 q^{65} -2.68556 q^{67} -4.44949 q^{68} -0.267949 q^{69} -1.00000 q^{70} -9.05040 q^{71} +2.82843 q^{72} -0.585786 q^{73} +8.49938 q^{74} +0.414214 q^{75} +8.27792 q^{76} +0.882959 q^{78} +4.24728 q^{79} +1.00000 q^{80} +7.48528 q^{81} -10.6057 q^{82} -4.62446 q^{83} +0.414214 q^{84} -4.44949 q^{85} -2.65634 q^{86} -0.177625 q^{87} -2.17209 q^{89} +2.82843 q^{90} -2.13165 q^{91} -0.646887 q^{92} -1.87226 q^{93} -4.34242 q^{94} +8.27792 q^{95} -0.414214 q^{96} +13.0552 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) 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

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