LMFDB - Embedded newform 87.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [87,2,Mod(1,87)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(87, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("87.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 87 = 3 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	87.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.694698497585$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	87.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +3.23607 q^{5} -0.618034 q^{6} +0.236068 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +3.23607 q^{5} -0.618034 q^{6} +0.236068 q^{7} +2.23607 q^{8} +1.00000 q^{9} -2.00000 q^{10} -0.236068 q^{11} -1.61803 q^{12} -5.47214 q^{13} -0.145898 q^{14} +3.23607 q^{15} +1.85410 q^{16} +3.00000 q^{17} -0.618034 q^{18} -7.23607 q^{19} -5.23607 q^{20} +0.236068 q^{21} +0.145898 q^{22} -7.70820 q^{23} +2.23607 q^{24} +5.47214 q^{25} +3.38197 q^{26} +1.00000 q^{27} -0.381966 q^{28} -1.00000 q^{29} -2.00000 q^{30} +3.70820 q^{31} -5.61803 q^{32} -0.236068 q^{33} -1.85410 q^{34} +0.763932 q^{35} -1.61803 q^{36} +5.23607 q^{37} +4.47214 q^{38} -5.47214 q^{39} +7.23607 q^{40} +2.00000 q^{41} -0.145898 q^{42} +4.00000 q^{43} +0.381966 q^{44} +3.23607 q^{45} +4.76393 q^{46} +4.70820 q^{47} +1.85410 q^{48} -6.94427 q^{49} -3.38197 q^{50} +3.00000 q^{51} +8.85410 q^{52} +11.2361 q^{53} -0.618034 q^{54} -0.763932 q^{55} +0.527864 q^{56} -7.23607 q^{57} +0.618034 q^{58} +4.47214 q^{59} -5.23607 q^{60} -5.23607 q^{61} -2.29180 q^{62} +0.236068 q^{63} -0.236068 q^{64} -17.7082 q^{65} +0.145898 q^{66} -13.1803 q^{67} -4.85410 q^{68} -7.70820 q^{69} -0.472136 q^{70} -5.23607 q^{71} +2.23607 q^{72} +6.76393 q^{73} -3.23607 q^{74} +5.47214 q^{75} +11.7082 q^{76} -0.0557281 q^{77} +3.38197 q^{78} -12.7639 q^{79} +6.00000 q^{80} +1.00000 q^{81} -1.23607 q^{82} +2.94427 q^{83} -0.381966 q^{84} +9.70820 q^{85} -2.47214 q^{86} -1.00000 q^{87} -0.527864 q^{88} +5.00000 q^{89} -2.00000 q^{90} -1.29180 q^{91} +12.4721 q^{92} +3.70820 q^{93} -2.90983 q^{94} -23.4164 q^{95} -5.61803 q^{96} +18.6525 q^{97} +4.29180 q^{98} -0.236068 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 2 q^{9} - 4 q^{10} + 4 q^{11} - q^{12} - 2 q^{13} - 7 q^{14} + 2 q^{15} - 3 q^{16} + 6 q^{17} + q^{18} - 10 q^{19} - 6 q^{20} - 4 q^{21} + 7 q^{22} - 2 q^{23} + 2 q^{25} + 9 q^{26} + 2 q^{27} - 3 q^{28} - 2 q^{29} - 4 q^{30} - 6 q^{31} - 9 q^{32} + 4 q^{33} + 3 q^{34} + 6 q^{35} - q^{36} + 6 q^{37} - 2 q^{39} + 10 q^{40} + 4 q^{41} - 7 q^{42} + 8 q^{43} + 3 q^{44} + 2 q^{45} + 14 q^{46} - 4 q^{47} - 3 q^{48} + 4 q^{49} - 9 q^{50} + 6 q^{51} + 11 q^{52} + 18 q^{53} + q^{54} - 6 q^{55} + 10 q^{56} - 10 q^{57} - q^{58} - 6 q^{60} - 6 q^{61} - 18 q^{62} - 4 q^{63} + 4 q^{64} - 22 q^{65} + 7 q^{66} - 4 q^{67} - 3 q^{68} - 2 q^{69} + 8 q^{70} - 6 q^{71} + 18 q^{73} - 2 q^{74} + 2 q^{75} + 10 q^{76} - 18 q^{77} + 9 q^{78} - 30 q^{79} + 12 q^{80} + 2 q^{81} + 2 q^{82} - 12 q^{83} - 3 q^{84} + 6 q^{85} + 4 q^{86} - 2 q^{87} - 10 q^{88} + 10 q^{89} - 4 q^{90} - 16 q^{91} + 16 q^{92} - 6 q^{93} - 17 q^{94} - 20 q^{95} - 9 q^{96} + 6 q^{97} + 22 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 + 2 * q^9 - 4 * q^10 + 4 * q^11 - q^12 - 2 * q^13 - 7 * q^14 + 2 * q^15 - 3 * q^16 + 6 * q^17 + q^18 - 10 * q^19 - 6 * q^20 - 4 * q^21 + 7 * q^22 - 2 * q^23 + 2 * q^25 + 9 * q^26 + 2 * q^27 - 3 * q^28 - 2 * q^29 - 4 * q^30 - 6 * q^31 - 9 * q^32 + 4 * q^33 + 3 * q^34 + 6 * q^35 - q^36 + 6 * q^37 - 2 * q^39 + 10 * q^40 + 4 * q^41 - 7 * q^42 + 8 * q^43 + 3 * q^44 + 2 * q^45 + 14 * q^46 - 4 * q^47 - 3 * q^48 + 4 * q^49 - 9 * q^50 + 6 * q^51 + 11 * q^52 + 18 * q^53 + q^54 - 6 * q^55 + 10 * q^56 - 10 * q^57 - q^58 - 6 * q^60 - 6 * q^61 - 18 * q^62 - 4 * q^63 + 4 * q^64 - 22 * q^65 + 7 * q^66 - 4 * q^67 - 3 * q^68 - 2 * q^69 + 8 * q^70 - 6 * q^71 + 18 * q^73 - 2 * q^74 + 2 * q^75 + 10 * q^76 - 18 * q^77 + 9 * q^78 - 30 * q^79 + 12 * q^80 + 2 * q^81 + 2 * q^82 - 12 * q^83 - 3 * q^84 + 6 * q^85 + 4 * q^86 - 2 * q^87 - 10 * q^88 + 10 * q^89 - 4 * q^90 - 16 * q^91 + 16 * q^92 - 6 * q^93 - 17 * q^94 - 20 * q^95 - 9 * q^96 + 6 * q^97 + 22 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.61803	−0.809017
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	−0.618034	−0.252311
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	−1.61803	−0.467086
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	−0.145898	−0.0389929
$15$	3.23607	0.835549
$16$	1.85410	0.463525
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−0.618034	−0.145672
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	−5.23607	−1.17082
$21$	0.236068	0.0515143
$22$	0.145898	0.0311056
$23$	−7.70820	−1.60727	−0.803636	−	0.595121i	$-0.797104\pi$
$23$	−7.70820	−1.60727	−0.803636	+	0.595121i	$0.797104\pi$
$24$	2.23607	0.456435
$25$	5.47214	1.09443
$26$	3.38197	0.663258
$27$	1.00000	0.192450
$28$	−0.381966	−0.0721848
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−2.00000	−0.365148
$31$	3.70820	0.666013	0.333007	−	0.942925i	$-0.391937\pi$
$31$	3.70820	0.666013	0.333007	+	0.942925i	$0.391937\pi$
$32$	−5.61803	−0.993137
$33$	−0.236068	−0.0410942
$34$	−1.85410	−0.317976
$35$	0.763932	0.129128
$36$	−1.61803	−0.269672
$37$	5.23607	0.860804	0.430402	−	0.902637i	$-0.358372\pi$
$37$	5.23607	0.860804	0.430402	+	0.902637i	$0.358372\pi$
$38$	4.47214	0.725476
$39$	−5.47214	−0.876243
$40$	7.23607	1.14412
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−0.145898	−0.0225126
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0.381966	0.0575835
$45$	3.23607	0.482405
$46$	4.76393	0.702403
$47$	4.70820	0.686762	0.343381	−	0.939196i	$-0.388428\pi$
$47$	4.70820	0.686762	0.343381	+	0.939196i	$0.388428\pi$
$48$	1.85410	0.267617
$49$	−6.94427	−0.992039
$50$	−3.38197	−0.478282
$51$	3.00000	0.420084
$52$	8.85410	1.22784
$53$	11.2361	1.54339	0.771696	−	0.635991i	$-0.219409\pi$
$53$	11.2361	1.54339	0.771696	+	0.635991i	$0.219409\pi$
$54$	−0.618034	−0.0841038
$55$	−0.763932	−0.103009
$56$	0.527864	0.0705388
$57$	−7.23607	−0.958441
$58$	0.618034	0.0811518
$59$	4.47214	0.582223	0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214	0.582223	0.291111	+	0.956689i	$0.405975\pi$
$60$	−5.23607	−0.675973
$61$	−5.23607	−0.670410	−0.335205	−	0.942145i	$-0.608806\pi$
$61$	−5.23607	−0.670410	−0.335205	+	0.942145i	$0.608806\pi$
$62$	−2.29180	−0.291058
$63$	0.236068	0.0297418
$64$	−0.236068	−0.0295085
$65$	−17.7082	−2.19643
$66$	0.145898	0.0179588
$67$	−13.1803	−1.61023	−0.805117	−	0.593115i	$-0.797898\pi$
$67$	−13.1803	−1.61023	−0.805117	+	0.593115i	$0.797898\pi$
$68$	−4.85410	−0.588646
$69$	−7.70820	−0.927959
$70$	−0.472136	−0.0564310
$71$	−5.23607	−0.621407	−0.310703	−	0.950507i	$-0.600565\pi$
$71$	−5.23607	−0.621407	−0.310703	+	0.950507i	$0.600565\pi$
$72$	2.23607	0.263523
$73$	6.76393	0.791658	0.395829	−	0.918324i	$-0.370457\pi$
$73$	6.76393	0.791658	0.395829	+	0.918324i	$0.370457\pi$
$74$	−3.23607	−0.376185
$75$	5.47214	0.631868
$76$	11.7082	1.34302
$77$	−0.0557281	−0.00635081
$78$	3.38197	0.382932
$79$	−12.7639	−1.43605	−0.718027	−	0.696015i	$-0.754955\pi$
$79$	−12.7639	−1.43605	−0.718027	+	0.696015i	$0.754955\pi$
$80$	6.00000	0.670820
$81$	1.00000	0.111111
$82$	−1.23607	−0.136501
$83$	2.94427	0.323176	0.161588	−	0.986858i	$-0.448338\pi$
$83$	2.94427	0.323176	0.161588	+	0.986858i	$0.448338\pi$
$84$	−0.381966	−0.0416759
$85$	9.70820	1.05300
$86$	−2.47214	−0.266577
$87$	−1.00000	−0.107211
$88$	−0.527864	−0.0562705
$89$	5.00000	0.529999	0.264999	−	0.964249i	$-0.414628\pi$
$89$	5.00000	0.529999	0.264999	+	0.964249i	$0.414628\pi$
$90$	−2.00000	−0.210819
$91$	−1.29180	−0.135417
$92$	12.4721	1.30031
$93$	3.70820	0.384523
$94$	−2.90983	−0.300126
$95$	−23.4164	−2.40247
$96$	−5.61803	−0.573388
$97$	18.6525	1.89387	0.946936	−	0.321422i	$-0.104161\pi$
$97$	18.6525	1.89387	0.946936	+	0.321422i	$0.104161\pi$
$98$	4.29180	0.433537
$99$	−0.236068	−0.0237257
$100$	−8.85410	−0.885410
$101$	−7.47214	−0.743505	−0.371753	−	0.928332i	$-0.621243\pi$
$101$	−7.47214	−0.743505	−0.371753	+	0.928332i	$0.621243\pi$
$102$	−1.85410	−0.183583
$103$	−4.94427	−0.487174	−0.243587	−	0.969879i	$-0.578324\pi$
$103$	−4.94427	−0.487174	−0.243587	+	0.969879i	$0.578324\pi$
$104$	−12.2361	−1.19985
$105$	0.763932	0.0745521
$106$	−6.94427	−0.674487
$107$	9.70820	0.938527	0.469264	−	0.883058i	$-0.344519\pi$
$107$	9.70820	0.938527	0.469264	+	0.883058i	$0.344519\pi$
$108$	−1.61803	−0.155695
$109$	9.47214	0.907266	0.453633	−	0.891189i	$-0.350128\pi$
$109$	9.47214	0.907266	0.453633	+	0.891189i	$0.350128\pi$
$110$	0.472136	0.0450164
$111$	5.23607	0.496986
$112$	0.437694	0.0413582
$113$	19.0000	1.78737	0.893685	−	0.448695i	$-0.148111\pi$
$113$	19.0000	1.78737	0.893685	+	0.448695i	$0.148111\pi$
$114$	4.47214	0.418854
$115$	−24.9443	−2.32607
$116$	1.61803	0.150231
$117$	−5.47214	−0.505899
$118$	−2.76393	−0.254441
$119$	0.708204	0.0649209
$120$	7.23607	0.660560
$121$	−10.9443	−0.994934
$122$	3.23607	0.292980
$123$	2.00000	0.180334
$124$	−6.00000	−0.538816
$125$	1.52786	0.136656
$126$	−0.145898	−0.0129976
$127$	12.4721	1.10672	0.553362	−	0.832941i	$-0.313345\pi$
$127$	12.4721	1.10672	0.553362	+	0.832941i	$0.313345\pi$
$128$	11.3820	1.00603
$129$	4.00000	0.352180
$130$	10.9443	0.959876
$131$	8.70820	0.760839	0.380420	−	0.924814i	$-0.375780\pi$
$131$	8.70820	0.760839	0.380420	+	0.924814i	$0.375780\pi$
$132$	0.381966	0.0332459
$133$	−1.70820	−0.148120
$134$	8.14590	0.703698
$135$	3.23607	0.278516
$136$	6.70820	0.575224
$137$	3.52786	0.301406	0.150703	−	0.988579i	$-0.451846\pi$
$137$	3.52786	0.301406	0.150703	+	0.988579i	$0.451846\pi$
$138$	4.76393	0.405533
$139$	1.18034	0.100115	0.0500576	−	0.998746i	$-0.484060\pi$
$139$	1.18034	0.100115	0.0500576	+	0.998746i	$0.484060\pi$
$140$	−1.23607	−0.104467
$141$	4.70820	0.396502
$142$	3.23607	0.271565
$143$	1.29180	0.108025
$144$	1.85410	0.154508
$145$	−3.23607	−0.268741
$146$	−4.18034	−0.345967
$147$	−6.94427	−0.572754
$148$	−8.47214	−0.696405
$149$	−23.4164	−1.91835	−0.959173	−	0.282819i	$-0.908731\pi$
$149$	−23.4164	−1.91835	−0.959173	+	0.282819i	$0.908731\pi$
$150$	−3.38197	−0.276136
$151$	3.05573	0.248672	0.124336	−	0.992240i	$-0.460320\pi$
$151$	3.05573	0.248672	0.124336	+	0.992240i	$0.460320\pi$
$152$	−16.1803	−1.31240
$153$	3.00000	0.242536
$154$	0.0344419	0.00277540
$155$	12.0000	0.963863
$156$	8.85410	0.708896
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	7.88854	0.627579
$159$	11.2361	0.891078
$160$	−18.1803	−1.43728
$161$	−1.81966	−0.143409
$162$	−0.618034	−0.0485573
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−3.23607	−0.252694
$165$	−0.763932	−0.0594720
$166$	−1.81966	−0.141233
$167$	−6.47214	−0.500829	−0.250414	−	0.968139i	$-0.580567\pi$
$167$	−6.47214	−0.500829	−0.250414	+	0.968139i	$0.580567\pi$
$168$	0.527864	0.0407256
$169$	16.9443	1.30341
$170$	−6.00000	−0.460179
$171$	−7.23607	−0.553356
$172$	−6.47214	−0.493496
$173$	−7.05573	−0.536437	−0.268219	−	0.963358i	$-0.586435\pi$
$173$	−7.05573	−0.536437	−0.268219	+	0.963358i	$0.586435\pi$
$174$	0.618034	0.0468530
$175$	1.29180	0.0976506
$176$	−0.437694	−0.0329924
$177$	4.47214	0.336146
$178$	−3.09017	−0.231618
$179$	17.2361	1.28828	0.644142	−	0.764906i	$-0.277215\pi$
$179$	17.2361	1.28828	0.644142	+	0.764906i	$0.277215\pi$
$180$	−5.23607	−0.390273
$181$	−3.00000	−0.222988	−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000	−0.222988	−0.111494	+	0.993765i	$0.535564\pi$
$182$	0.798374	0.0591794
$183$	−5.23607	−0.387061
$184$	−17.2361	−1.27066
$185$	16.9443	1.24577
$186$	−2.29180	−0.168043
$187$	−0.708204	−0.0517890
$188$	−7.61803	−0.555602
$189$	0.236068	0.0171714
$190$	14.4721	1.04992
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−0.236068	−0.0170367
$193$	8.47214	0.609838	0.304919	−	0.952378i	$-0.401371\pi$
$193$	8.47214	0.609838	0.304919	+	0.952378i	$0.401371\pi$
$194$	−11.5279	−0.827652
$195$	−17.7082	−1.26811
$196$	11.2361	0.802576
$197$	−19.2361	−1.37051	−0.685257	−	0.728302i	$-0.740310\pi$
$197$	−19.2361	−1.37051	−0.685257	+	0.728302i	$0.740310\pi$
$198$	0.145898	0.0103685
$199$	−15.6525	−1.10957	−0.554787	−	0.831992i	$-0.687200\pi$
$199$	−15.6525	−1.10957	−0.554787	+	0.831992i	$0.687200\pi$
$200$	12.2361	0.865221
$201$	−13.1803	−0.929669
$202$	4.61803	0.324924
$203$	−0.236068	−0.0165687
$204$	−4.85410	−0.339855
$205$	6.47214	0.452034
$206$	3.05573	0.212903
$207$	−7.70820	−0.535757
$208$	−10.1459	−0.703491
$209$	1.70820	0.118159
$210$	−0.472136	−0.0325805
$211$	10.9443	0.753435	0.376717	−	0.926328i	$-0.377053\pi$
$211$	10.9443	0.753435	0.376717	+	0.926328i	$0.377053\pi$
$212$	−18.1803	−1.24863
$213$	−5.23607	−0.358769
$214$	−6.00000	−0.410152
$215$	12.9443	0.882792
$216$	2.23607	0.152145
$217$	0.875388	0.0594252
$218$	−5.85410	−0.396490
$219$	6.76393	0.457064
$220$	1.23607	0.0833357
$221$	−16.4164	−1.10429
$222$	−3.23607	−0.217191
$223$	15.1803	1.01655	0.508275	−	0.861195i	$-0.330283\pi$
$223$	15.1803	1.01655	0.508275	+	0.861195i	$0.330283\pi$
$224$	−1.32624	−0.0886130
$225$	5.47214	0.364809
$226$	−11.7426	−0.781109
$227$	−10.9443	−0.726397	−0.363198	−	0.931712i	$-0.618315\pi$
$227$	−10.9443	−0.726397	−0.363198	+	0.931712i	$0.618315\pi$
$228$	11.7082	0.775395
$229$	−16.1803	−1.06923	−0.534613	−	0.845097i	$-0.679543\pi$
$229$	−16.1803	−1.06923	−0.534613	+	0.845097i	$0.679543\pi$
$230$	15.4164	1.01653
$231$	−0.0557281	−0.00366664
$232$	−2.23607	−0.146805
$233$	17.4164	1.14099	0.570493	−	0.821302i	$-0.306752\pi$
$233$	17.4164	1.14099	0.570493	+	0.821302i	$0.306752\pi$
$234$	3.38197	0.221086
$235$	15.2361	0.993891
$236$	−7.23607	−0.471028
$237$	−12.7639	−0.829106
$238$	−0.437694	−0.0283715
$239$	19.5967	1.26761	0.633804	−	0.773494i	$-0.281493\pi$
$239$	19.5967	1.26761	0.633804	+	0.773494i	$0.281493\pi$
$240$	6.00000	0.387298
$241$	10.4164	0.670980	0.335490	−	0.942044i	$-0.391098\pi$
$241$	10.4164	0.670980	0.335490	+	0.942044i	$0.391098\pi$
$242$	6.76393	0.434802
$243$	1.00000	0.0641500
$244$	8.47214	0.542373
$245$	−22.4721	−1.43569
$246$	−1.23607	−0.0788088
$247$	39.5967	2.51948
$248$	8.29180	0.526530
$249$	2.94427	0.186586
$250$	−0.944272	−0.0597210
$251$	−9.18034	−0.579458	−0.289729	−	0.957109i	$-0.593565\pi$
$251$	−9.18034	−0.579458	−0.289729	+	0.957109i	$0.593565\pi$
$252$	−0.381966	−0.0240616
$253$	1.81966	0.114401
$254$	−7.70820	−0.483656
$255$	9.70820	0.607951
$256$	−6.56231	−0.410144
$257$	−15.4164	−0.961649	−0.480825	−	0.876817i	$-0.659663\pi$
$257$	−15.4164	−0.961649	−0.480825	+	0.876817i	$0.659663\pi$
$258$	−2.47214	−0.153908
$259$	1.23607	0.0768055
$260$	28.6525	1.77695
$261$	−1.00000	−0.0618984
$262$	−5.38197	−0.332499
$263$	−22.8328	−1.40793	−0.703966	−	0.710234i	$-0.748589\pi$
$263$	−22.8328	−1.40793	−0.703966	+	0.710234i	$0.748589\pi$
$264$	−0.527864	−0.0324878
$265$	36.3607	2.23362
$266$	1.05573	0.0647308
$267$	5.00000	0.305995
$268$	21.3262	1.30271
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	−2.00000	−0.121716
$271$	−26.9443	−1.63675	−0.818374	−	0.574686i	$-0.805124\pi$
$271$	−26.9443	−1.63675	−0.818374	+	0.574686i	$0.805124\pi$
$272$	5.56231	0.337264
$273$	−1.29180	−0.0781831
$274$	−2.18034	−0.131719
$275$	−1.29180	−0.0778982
$276$	12.4721	0.750734
$277$	3.00000	0.180253	0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000	0.180253	0.0901263	+	0.995930i	$0.471273\pi$
$278$	−0.729490	−0.0437519
$279$	3.70820	0.222004
$280$	1.70820	0.102085
$281$	−21.4164	−1.27760	−0.638798	−	0.769375i	$-0.720568\pi$
$281$	−21.4164	−1.27760	−0.638798	+	0.769375i	$0.720568\pi$
$282$	−2.90983	−0.173278
$283$	7.41641	0.440860	0.220430	−	0.975403i	$-0.429254\pi$
$283$	7.41641	0.440860	0.220430	+	0.975403i	$0.429254\pi$
$284$	8.47214	0.502729
$285$	−23.4164	−1.38707
$286$	−0.798374	−0.0472088
$287$	0.472136	0.0278693
$288$	−5.61803	−0.331046
$289$	−8.00000	−0.470588
$290$	2.00000	0.117444
$291$	18.6525	1.09343
$292$	−10.9443	−0.640465
$293$	−19.9443	−1.16516	−0.582578	−	0.812775i	$-0.697956\pi$
$293$	−19.9443	−1.16516	−0.582578	+	0.812775i	$0.697956\pi$
$294$	4.29180	0.250303
$295$	14.4721	0.842600
$296$	11.7082	0.680526
$297$	−0.236068	−0.0136981
$298$	14.4721	0.838348
$299$	42.1803	2.43935
$300$	−8.85410	−0.511192
$301$	0.944272	0.0544269
$302$	−1.88854	−0.108673
$303$	−7.47214	−0.429263
$304$	−13.4164	−0.769484
$305$	−16.9443	−0.970226
$306$	−1.85410	−0.105992
$307$	−12.6525	−0.722115	−0.361057	−	0.932544i	$-0.617584\pi$
$307$	−12.6525	−0.722115	−0.361057	+	0.932544i	$0.617584\pi$
$308$	0.0901699	0.00513791
$309$	−4.94427	−0.281270
$310$	−7.41641	−0.421224
$311$	28.7082	1.62789	0.813946	−	0.580940i	$-0.197315\pi$
$311$	28.7082	1.62789	0.813946	+	0.580940i	$0.197315\pi$
$312$	−12.2361	−0.692731
$313$	−11.0000	−0.621757	−0.310878	−	0.950450i	$-0.600623\pi$
$313$	−11.0000	−0.621757	−0.310878	+	0.950450i	$0.600623\pi$
$314$	1.23607	0.0697554
$315$	0.763932	0.0430427
$316$	20.6525	1.16179
$317$	−1.47214	−0.0826834	−0.0413417	−	0.999145i	$-0.513163\pi$
$317$	−1.47214	−0.0826834	−0.0413417	+	0.999145i	$0.513163\pi$
$318$	−6.94427	−0.389415
$319$	0.236068	0.0132173
$320$	−0.763932	−0.0427051
$321$	9.70820	0.541859
$322$	1.12461	0.0626722
$323$	−21.7082	−1.20788
$324$	−1.61803	−0.0898908
$325$	−29.9443	−1.66101
$326$	3.70820	0.205378
$327$	9.47214	0.523810
$328$	4.47214	0.246932
$329$	1.11146	0.0612766
$330$	0.472136	0.0259902
$331$	−20.7639	−1.14129	−0.570644	−	0.821197i	$-0.693307\pi$
$331$	−20.7639	−1.14129	−0.570644	+	0.821197i	$0.693307\pi$
$332$	−4.76393	−0.261455
$333$	5.23607	0.286935
$334$	4.00000	0.218870
$335$	−42.6525	−2.33035
$336$	0.437694	0.0238782
$337$	−8.18034	−0.445612	−0.222806	−	0.974863i	$-0.571522\pi$
$337$	−8.18034	−0.445612	−0.222806	+	0.974863i	$0.571522\pi$
$338$	−10.4721	−0.569609
$339$	19.0000	1.03194
$340$	−15.7082	−0.851897
$341$	−0.875388	−0.0474049
$342$	4.47214	0.241825
$343$	−3.29180	−0.177740
$344$	8.94427	0.482243
$345$	−24.9443	−1.34295
$346$	4.36068	0.234432
$347$	25.2361	1.35474	0.677372	−	0.735641i	$-0.263119\pi$
$347$	25.2361	1.35474	0.677372	+	0.735641i	$0.263119\pi$
$348$	1.61803	0.0867357
$349$	−13.4164	−0.718164	−0.359082	−	0.933306i	$-0.616910\pi$
$349$	−13.4164	−0.718164	−0.359082	+	0.933306i	$0.616910\pi$
$350$	−0.798374	−0.0426749
$351$	−5.47214	−0.292081
$352$	1.32624	0.0706887
$353$	−3.23607	−0.172239	−0.0861193	−	0.996285i	$-0.527447\pi$
$353$	−3.23607	−0.172239	−0.0861193	+	0.996285i	$0.527447\pi$
$354$	−2.76393	−0.146901
$355$	−16.9443	−0.899309
$356$	−8.09017	−0.428778
$357$	0.708204	0.0374821
$358$	−10.6525	−0.563001
$359$	−14.4721	−0.763810	−0.381905	−	0.924202i	$-0.624732\pi$
$359$	−14.4721	−0.763810	−0.381905	+	0.924202i	$0.624732\pi$
$360$	7.23607	0.381374
$361$	33.3607	1.75583
$362$	1.85410	0.0974494
$363$	−10.9443	−0.574425
$364$	2.09017	0.109555
$365$	21.8885	1.14570
$366$	3.23607	0.169152
$367$	13.5279	0.706149	0.353074	−	0.935595i	$-0.385136\pi$
$367$	13.5279	0.706149	0.353074	+	0.935595i	$0.385136\pi$
$368$	−14.2918	−0.745011
$369$	2.00000	0.104116
$370$	−10.4721	−0.544420
$371$	2.65248	0.137710
$372$	−6.00000	−0.311086
$373$	−11.5279	−0.596890	−0.298445	−	0.954427i	$-0.596468\pi$
$373$	−11.5279	−0.596890	−0.298445	+	0.954427i	$0.596468\pi$
$374$	0.437694	0.0226326
$375$	1.52786	0.0788986
$376$	10.5279	0.542933
$377$	5.47214	0.281829
$378$	−0.145898	−0.00750419
$379$	1.70820	0.0877445	0.0438723	−	0.999037i	$-0.486031\pi$
$379$	1.70820	0.0877445	0.0438723	+	0.999037i	$0.486031\pi$
$380$	37.8885	1.94364
$381$	12.4721	0.638967
$382$	−7.41641	−0.379456
$383$	11.2361	0.574136	0.287068	−	0.957910i	$-0.407319\pi$
$383$	11.2361	0.574136	0.287068	+	0.957910i	$0.407319\pi$
$384$	11.3820	0.580834
$385$	−0.180340	−0.00919097
$386$	−5.23607	−0.266509
$387$	4.00000	0.203331
$388$	−30.1803	−1.53217
$389$	17.3607	0.880221	0.440111	−	0.897944i	$-0.354939\pi$
$389$	17.3607	0.880221	0.440111	+	0.897944i	$0.354939\pi$
$390$	10.9443	0.554185
$391$	−23.1246	−1.16946
$392$	−15.5279	−0.784276
$393$	8.70820	0.439271
$394$	11.8885	0.598936
$395$	−41.3050	−2.07828
$396$	0.381966	0.0191945
$397$	6.94427	0.348523	0.174262	−	0.984699i	$-0.444246\pi$
$397$	6.94427	0.348523	0.174262	+	0.984699i	$0.444246\pi$
$398$	9.67376	0.484902
$399$	−1.70820	−0.0855172
$400$	10.1459	0.507295
$401$	28.1803	1.40726	0.703630	−	0.710567i	$-0.251561\pi$
$401$	28.1803	1.40726	0.703630	+	0.710567i	$0.251561\pi$
$402$	8.14590	0.406280
$403$	−20.2918	−1.01081
$404$	12.0902	0.601508
$405$	3.23607	0.160802
$406$	0.145898	0.00724080
$407$	−1.23607	−0.0612696
$408$	6.70820	0.332106
$409$	−4.47214	−0.221133	−0.110566	−	0.993869i	$-0.535266\pi$
$409$	−4.47214	−0.221133	−0.110566	+	0.993869i	$0.535266\pi$
$410$	−4.00000	−0.197546
$411$	3.52786	0.174017
$412$	8.00000	0.394132
$413$	1.05573	0.0519490
$414$	4.76393	0.234134
$415$	9.52786	0.467704
$416$	30.7426	1.50728
$417$	1.18034	0.0578015
$418$	−1.05573	−0.0516373
$419$	27.2361	1.33057	0.665284	−	0.746590i	$-0.268310\pi$
$419$	27.2361	1.33057	0.665284	+	0.746590i	$0.268310\pi$
$420$	−1.23607	−0.0603139
$421$	−35.8885	−1.74910	−0.874550	−	0.484935i	$-0.838843\pi$
$421$	−35.8885	−1.74910	−0.874550	+	0.484935i	$0.838843\pi$
$422$	−6.76393	−0.329263
$423$	4.70820	0.228921
$424$	25.1246	1.22016
$425$	16.4164	0.796313
$426$	3.23607	0.156788
$427$	−1.23607	−0.0598175
$428$	−15.7082	−0.759285
$429$	1.29180	0.0623685
$430$	−8.00000	−0.385794
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	1.85410	0.0892055
$433$	−6.65248	−0.319698	−0.159849	−	0.987142i	$-0.551101\pi$
$433$	−6.65248	−0.319698	−0.159849	+	0.987142i	$0.551101\pi$
$434$	−0.541020	−0.0259698
$435$	−3.23607	−0.155158
$436$	−15.3262	−0.733994
$437$	55.7771	2.66818
$438$	−4.18034	−0.199744
$439$	13.2918	0.634383	0.317191	−	0.948362i	$-0.397260\pi$
$439$	13.2918	0.634383	0.317191	+	0.948362i	$0.397260\pi$
$440$	−1.70820	−0.0814354
$441$	−6.94427	−0.330680
$442$	10.1459	0.482591
$443$	−3.76393	−0.178830	−0.0894149	−	0.995994i	$-0.528500\pi$
$443$	−3.76393	−0.178830	−0.0894149	+	0.995994i	$0.528500\pi$
$444$	−8.47214	−0.402070
$445$	16.1803	0.767022
$446$	−9.38197	−0.444249
$447$	−23.4164	−1.10756
$448$	−0.0557281	−0.00263290
$449$	−19.4721	−0.918947	−0.459473	−	0.888192i	$-0.651962\pi$
$449$	−19.4721	−0.918947	−0.459473	+	0.888192i	$0.651962\pi$
$450$	−3.38197	−0.159427
$451$	−0.472136	−0.0222320
$452$	−30.7426	−1.44601
$453$	3.05573	0.143571
$454$	6.76393	0.317447
$455$	−4.18034	−0.195977
$456$	−16.1803	−0.757714
$457$	−1.47214	−0.0688636	−0.0344318	−	0.999407i	$-0.510962\pi$
$457$	−1.47214	−0.0688636	−0.0344318	+	0.999407i	$0.510962\pi$
$458$	10.0000	0.467269
$459$	3.00000	0.140028
$460$	40.3607	1.88183
$461$	−15.8885	−0.740003	−0.370002	−	0.929031i	$-0.620643\pi$
$461$	−15.8885	−0.740003	−0.370002	+	0.929031i	$0.620643\pi$
$462$	0.0344419	0.00160238
$463$	15.1803	0.705490	0.352745	−	0.935719i	$-0.385248\pi$
$463$	15.1803	0.705490	0.352745	+	0.935719i	$0.385248\pi$
$464$	−1.85410	−0.0860745
$465$	12.0000	0.556487
$466$	−10.7639	−0.498630
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	8.85410	0.409281
$469$	−3.11146	−0.143674
$470$	−9.41641	−0.434347
$471$	−2.00000	−0.0921551
$472$	10.0000	0.460287
$473$	−0.944272	−0.0434177
$474$	7.88854	0.362333
$475$	−39.5967	−1.81682
$476$	−1.14590	−0.0525222
$477$	11.2361	0.514464
$478$	−12.1115	−0.553965
$479$	14.4721	0.661249	0.330624	−	0.943762i	$-0.392741\pi$
$479$	14.4721	0.661249	0.330624	+	0.943762i	$0.392741\pi$
$480$	−18.1803	−0.829815
$481$	−28.6525	−1.30644
$482$	−6.43769	−0.293229
$483$	−1.81966	−0.0827974
$484$	17.7082	0.804918
$485$	60.3607	2.74084
$486$	−0.618034	−0.0280346
$487$	36.9443	1.67410	0.837052	−	0.547123i	$-0.184277\pi$
$487$	36.9443	1.67410	0.837052	+	0.547123i	$0.184277\pi$
$488$	−11.7082	−0.530005
$489$	−6.00000	−0.271329
$490$	13.8885	0.627420
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	−3.23607	−0.145893
$493$	−3.00000	−0.135113
$494$	−24.4721	−1.10105
$495$	−0.763932	−0.0343362
$496$	6.87539	0.308714
$497$	−1.23607	−0.0554452
$498$	−1.81966	−0.0815409
$499$	3.29180	0.147361	0.0736805	−	0.997282i	$-0.476525\pi$
$499$	3.29180	0.147361	0.0736805	+	0.997282i	$0.476525\pi$
$500$	−2.47214	−0.110557
$501$	−6.47214	−0.289154
$502$	5.67376	0.253232
$503$	27.5410	1.22799	0.613997	−	0.789309i	$-0.289561\pi$
$503$	27.5410	1.22799	0.613997	+	0.789309i	$0.289561\pi$
$504$	0.527864	0.0235129
$505$	−24.1803	−1.07601
$506$	−1.12461	−0.0499951
$507$	16.9443	0.752522
$508$	−20.1803	−0.895358
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	−6.00000	−0.265684
$511$	1.59675	0.0706360
$512$	−18.7082	−0.826794
$513$	−7.23607	−0.319480
$514$	9.52786	0.420256
$515$	−16.0000	−0.705044
$516$	−6.47214	−0.284920
$517$	−1.11146	−0.0488818
$518$	−0.763932	−0.0335652
$519$	−7.05573	−0.309712
$520$	−39.5967	−1.73643
$521$	−31.4164	−1.37638	−0.688189	−	0.725532i	$-0.741594\pi$
$521$	−31.4164	−1.37638	−0.688189	+	0.725532i	$0.741594\pi$
$522$	0.618034	0.0270506
$523$	14.1246	0.617626	0.308813	−	0.951123i	$-0.400068\pi$
$523$	14.1246	0.617626	0.308813	+	0.951123i	$0.400068\pi$
$524$	−14.0902	−0.615532
$525$	1.29180	0.0563786
$526$	14.1115	0.615289
$527$	11.1246	0.484596
$528$	−0.437694	−0.0190482
$529$	36.4164	1.58332
$530$	−22.4721	−0.976127
$531$	4.47214	0.194074
$532$	2.76393	0.119832
$533$	−10.9443	−0.474049
$534$	−3.09017	−0.133725
$535$	31.4164	1.35825
$536$	−29.4721	−1.27300
$537$	17.2361	0.743791
$538$	3.09017	0.133227
$539$	1.63932	0.0706105
$540$	−5.23607	−0.225324
$541$	8.18034	0.351700	0.175850	−	0.984417i	$-0.443733\pi$
$541$	8.18034	0.351700	0.175850	+	0.984417i	$0.443733\pi$
$542$	16.6525	0.715285
$543$	−3.00000	−0.128742
$544$	−16.8541	−0.722614
$545$	30.6525	1.31301
$546$	0.798374	0.0341672
$547$	3.65248	0.156169	0.0780843	−	0.996947i	$-0.475120\pi$
$547$	3.65248	0.156169	0.0780843	+	0.996947i	$0.475120\pi$
$548$	−5.70820	−0.243842
$549$	−5.23607	−0.223470
$550$	0.798374	0.0340428
$551$	7.23607	0.308267
$552$	−17.2361	−0.733616
$553$	−3.01316	−0.128132
$554$	−1.85410	−0.0787732
$555$	16.9443	0.719244
$556$	−1.90983	−0.0809948
$557$	−25.4164	−1.07693	−0.538464	−	0.842649i	$-0.680995\pi$
$557$	−25.4164	−1.07693	−0.538464	+	0.842649i	$0.680995\pi$
$558$	−2.29180	−0.0970195
$559$	−21.8885	−0.925787
$560$	1.41641	0.0598542
$561$	−0.708204	−0.0299004
$562$	13.2361	0.558330
$563$	−45.0689	−1.89943	−0.949713	−	0.313120i	$-0.898626\pi$
$563$	−45.0689	−1.89943	−0.949713	+	0.313120i	$0.898626\pi$
$564$	−7.61803	−0.320777
$565$	61.4853	2.58671
$566$	−4.58359	−0.192663
$567$	0.236068	0.00991392
$568$	−11.7082	−0.491265
$569$	35.2492	1.47772	0.738862	−	0.673857i	$-0.235363\pi$
$569$	35.2492	1.47772	0.738862	+	0.673857i	$0.235363\pi$
$570$	14.4721	0.606171
$571$	15.4164	0.645157	0.322578	−	0.946543i	$-0.395450\pi$
$571$	15.4164	0.645157	0.322578	+	0.946543i	$0.395450\pi$
$572$	−2.09017	−0.0873944
$573$	12.0000	0.501307
$574$	−0.291796	−0.0121793
$575$	−42.1803	−1.75904
$576$	−0.236068	−0.00983617
$577$	−40.9443	−1.70453	−0.852266	−	0.523108i	$-0.824772\pi$
$577$	−40.9443	−1.70453	−0.852266	+	0.523108i	$0.824772\pi$
$578$	4.94427	0.205655
$579$	8.47214	0.352090
$580$	5.23607	0.217416
$581$	0.695048	0.0288355
$582$	−11.5279	−0.477845
$583$	−2.65248	−0.109854
$584$	15.1246	0.625861
$585$	−17.7082	−0.732144
$586$	12.3262	0.509192
$587$	−5.41641	−0.223559	−0.111780	−	0.993733i	$-0.535655\pi$
$587$	−5.41641	−0.223559	−0.111780	+	0.993733i	$0.535655\pi$
$588$	11.2361	0.463368
$589$	−26.8328	−1.10563
$590$	−8.94427	−0.368230
$591$	−19.2361	−0.791266
$592$	9.70820	0.399005
$593$	−15.5967	−0.640482	−0.320241	−	0.947336i	$-0.603764\pi$
$593$	−15.5967	−0.640482	−0.320241	+	0.947336i	$0.603764\pi$
$594$	0.145898	0.00598627
$595$	2.29180	0.0939545
$596$	37.8885	1.55198
$597$	−15.6525	−0.640613
$598$	−26.0689	−1.06604
$599$	33.2918	1.36027	0.680133	−	0.733089i	$-0.261922\pi$
$599$	33.2918	1.36027	0.680133	+	0.733089i	$0.261922\pi$
$600$	12.2361	0.499535
$601$	8.18034	0.333683	0.166842	−	0.985984i	$-0.446643\pi$
$601$	8.18034	0.333683	0.166842	+	0.985984i	$0.446643\pi$
$602$	−0.583592	−0.0237854
$603$	−13.1803	−0.536745
$604$	−4.94427	−0.201180
$605$	−35.4164	−1.43988
$606$	4.61803	0.187595
$607$	1.41641	0.0574902	0.0287451	−	0.999587i	$-0.490849\pi$
$607$	1.41641	0.0574902	0.0287451	+	0.999587i	$0.490849\pi$
$608$	40.6525	1.64868
$609$	−0.236068	−0.00956596
$610$	10.4721	0.424004
$611$	−25.7639	−1.04230
$612$	−4.85410	−0.196215
$613$	5.58359	0.225519	0.112760	−	0.993622i	$-0.464031\pi$
$613$	5.58359	0.225519	0.112760	+	0.993622i	$0.464031\pi$
$614$	7.81966	0.315576
$615$	6.47214	0.260982
$616$	−0.124612	−0.00502075
$617$	12.4721	0.502109	0.251055	−	0.967973i	$-0.419223\pi$
$617$	12.4721	0.502109	0.251055	+	0.967973i	$0.419223\pi$
$618$	3.05573	0.122919
$619$	−1.70820	−0.0686585	−0.0343293	−	0.999411i	$-0.510929\pi$
$619$	−1.70820	−0.0686585	−0.0343293	+	0.999411i	$0.510929\pi$
$620$	−19.4164	−0.779782
$621$	−7.70820	−0.309320
$622$	−17.7426	−0.711415
$623$	1.18034	0.0472893
$624$	−10.1459	−0.406161
$625$	−22.4164	−0.896656
$626$	6.79837	0.271718
$627$	1.70820	0.0682191
$628$	3.23607	0.129133
$629$	15.7082	0.626327
$630$	−0.472136	−0.0188103
$631$	−23.6525	−0.941590	−0.470795	−	0.882243i	$-0.656033\pi$
$631$	−23.6525	−0.941590	−0.470795	+	0.882243i	$0.656033\pi$
$632$	−28.5410	−1.13530
$633$	10.9443	0.434996
$634$	0.909830	0.0361340
$635$	40.3607	1.60166
$636$	−18.1803	−0.720897
$637$	38.0000	1.50561
$638$	−0.145898	−0.00577616
$639$	−5.23607	−0.207136
$640$	36.8328	1.45594
$641$	38.3050	1.51295	0.756477	−	0.654020i	$-0.226919\pi$
$641$	38.3050	1.51295	0.756477	+	0.654020i	$0.226919\pi$
$642$	−6.00000	−0.236801
$643$	0.708204	0.0279288	0.0139644	−	0.999902i	$-0.495555\pi$
$643$	0.708204	0.0279288	0.0139644	+	0.999902i	$0.495555\pi$
$644$	2.94427	0.116021
$645$	12.9443	0.509680
$646$	13.4164	0.527862
$647$	14.1803	0.557487	0.278743	−	0.960366i	$-0.410082\pi$
$647$	14.1803	0.557487	0.278743	+	0.960366i	$0.410082\pi$
$648$	2.23607	0.0878410
$649$	−1.05573	−0.0414410
$650$	18.5066	0.725888
$651$	0.875388	0.0343092
$652$	9.70820	0.380203
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	−5.85410	−0.228914
$655$	28.1803	1.10110
$656$	3.70820	0.144781
$657$	6.76393	0.263886
$658$	−0.686918	−0.0267788
$659$	24.5967	0.958153	0.479077	−	0.877773i	$-0.340972\pi$
$659$	24.5967	0.958153	0.479077	+	0.877773i	$0.340972\pi$
$660$	1.23607	0.0481139
$661$	−23.0000	−0.894596	−0.447298	−	0.894385i	$-0.647614\pi$
$661$	−23.0000	−0.894596	−0.447298	+	0.894385i	$0.647614\pi$
$662$	12.8328	0.498762
$663$	−16.4164	−0.637560
$664$	6.58359	0.255493
$665$	−5.52786	−0.214361
$666$	−3.23607	−0.125395
$667$	7.70820	0.298463
$668$	10.4721	0.405179
$669$	15.1803	0.586906
$670$	26.3607	1.01840
$671$	1.23607	0.0477179
$672$	−1.32624	−0.0511607
$673$	30.3050	1.16817	0.584085	−	0.811692i	$-0.301453\pi$
$673$	30.3050	1.16817	0.584085	+	0.811692i	$0.301453\pi$
$674$	5.05573	0.194739
$675$	5.47214	0.210623
$676$	−27.4164	−1.05448
$677$	−0.416408	−0.0160039	−0.00800193	−	0.999968i	$-0.502547\pi$
$677$	−0.416408	−0.0160039	−0.00800193	+	0.999968i	$0.502547\pi$
$678$	−11.7426	−0.450974
$679$	4.40325	0.168981
$680$	21.7082	0.832472
$681$	−10.9443	−0.419385
$682$	0.541020	0.0207167
$683$	10.8328	0.414506	0.207253	−	0.978287i	$-0.433548\pi$
$683$	10.8328	0.414506	0.207253	+	0.978287i	$0.433548\pi$
$684$	11.7082	0.447674
$685$	11.4164	0.436199
$686$	2.03444	0.0776754
$687$	−16.1803	−0.617318
$688$	7.41641	0.282748
$689$	−61.4853	−2.34240
$690$	15.4164	0.586893
$691$	−32.5967	−1.24004	−0.620019	−	0.784587i	$-0.712875\pi$
$691$	−32.5967	−1.24004	−0.620019	+	0.784587i	$0.712875\pi$
$692$	11.4164	0.433987
$693$	−0.0557281	−0.00211694
$694$	−15.5967	−0.592044
$695$	3.81966	0.144888
$696$	−2.23607	−0.0847579
$697$	6.00000	0.227266
$698$	8.29180	0.313849
$699$	17.4164	0.658749
$700$	−2.09017	−0.0790010
$701$	−26.5410	−1.00244	−0.501220	−	0.865320i	$-0.667115\pi$
$701$	−26.5410	−1.00244	−0.501220	+	0.865320i	$0.667115\pi$
$702$	3.38197	0.127644
$703$	−37.8885	−1.42899
$704$	0.0557281	0.00210033
$705$	15.2361	0.573824
$706$	2.00000	0.0752710
$707$	−1.76393	−0.0663395
$708$	−7.23607	−0.271948
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	10.4721	0.393012
$711$	−12.7639	−0.478685
$712$	11.1803	0.419001
$713$	−28.5836	−1.07046
$714$	−0.437694	−0.0163803
$715$	4.18034	0.156336
$716$	−27.8885	−1.04224
$717$	19.5967	0.731854
$718$	8.94427	0.333797
$719$	−50.2492	−1.87398	−0.936990	−	0.349356i	$-0.886400\pi$
$719$	−50.2492	−1.87398	−0.936990	+	0.349356i	$0.886400\pi$
$720$	6.00000	0.223607
$721$	−1.16718	−0.0434682
$722$	−20.6180	−0.767324
$723$	10.4164	0.387390
$724$	4.85410	0.180401
$725$	−5.47214	−0.203230
$726$	6.76393	0.251033
$727$	−14.7639	−0.547564	−0.273782	−	0.961792i	$-0.588275\pi$
$727$	−14.7639	−0.547564	−0.273782	+	0.961792i	$0.588275\pi$
$728$	−2.88854	−0.107057
$729$	1.00000	0.0370370
$730$	−13.5279	−0.500689
$731$	12.0000	0.443836
$732$	8.47214	0.313139
$733$	18.4721	0.682284	0.341142	−	0.940012i	$-0.389186\pi$
$733$	18.4721	0.682284	0.341142	+	0.940012i	$0.389186\pi$
$734$	−8.36068	−0.308598
$735$	−22.4721	−0.828897
$736$	43.3050	1.59624
$737$	3.11146	0.114612
$738$	−1.23607	−0.0455003
$739$	6.18034	0.227347	0.113674	−	0.993518i	$-0.463738\pi$
$739$	6.18034	0.227347	0.113674	+	0.993518i	$0.463738\pi$
$740$	−27.4164	−1.00785
$741$	39.5967	1.45462
$742$	−1.63932	−0.0601813
$743$	−10.3475	−0.379614	−0.189807	−	0.981821i	$-0.560786\pi$
$743$	−10.3475	−0.379614	−0.189807	+	0.981821i	$0.560786\pi$
$744$	8.29180	0.303992
$745$	−75.7771	−2.77626
$746$	7.12461	0.260851
$747$	2.94427	0.107725
$748$	1.14590	0.0418982
$749$	2.29180	0.0837404
$750$	−0.944272	−0.0344799
$751$	−43.7771	−1.59745	−0.798724	−	0.601697i	$-0.794491\pi$
$751$	−43.7771	−1.59745	−0.798724	+	0.601697i	$0.794491\pi$
$752$	8.72949	0.318332
$753$	−9.18034	−0.334550
$754$	−3.38197	−0.123164
$755$	9.88854	0.359881
$756$	−0.381966	−0.0138920
$757$	−20.9443	−0.761233	−0.380616	−	0.924733i	$-0.624288\pi$
$757$	−20.9443	−0.761233	−0.380616	+	0.924733i	$0.624288\pi$
$758$	−1.05573	−0.0383458
$759$	1.81966	0.0660495
$760$	−52.3607	−1.89932
$761$	28.1803	1.02154	0.510768	−	0.859718i	$-0.329361\pi$
$761$	28.1803	1.02154	0.510768	+	0.859718i	$0.329361\pi$
$762$	−7.70820	−0.279239
$763$	2.23607	0.0809511
$764$	−19.4164	−0.702461
$765$	9.70820	0.351001
$766$	−6.94427	−0.250907
$767$	−24.4721	−0.883638
$768$	−6.56231	−0.236797
$769$	17.2361	0.621549	0.310774	−	0.950484i	$-0.399412\pi$
$769$	17.2361	0.621549	0.310774	+	0.950484i	$0.399412\pi$
$770$	0.111456	0.00401660
$771$	−15.4164	−0.555208
$772$	−13.7082	−0.493369
$773$	28.4721	1.02407	0.512036	−	0.858964i	$-0.328892\pi$
$773$	28.4721	1.02407	0.512036	+	0.858964i	$0.328892\pi$
$774$	−2.47214	−0.0888591
$775$	20.2918	0.728903
$776$	41.7082	1.49724
$777$	1.23607	0.0443437
$778$	−10.7295	−0.384671
$779$	−14.4721	−0.518518
$780$	28.6525	1.02592
$781$	1.23607	0.0442300
$782$	14.2918	0.511074
$783$	−1.00000	−0.0357371
$784$	−12.8754	−0.459835
$785$	−6.47214	−0.231000
$786$	−5.38197	−0.191968
$787$	−46.4721	−1.65655	−0.828276	−	0.560320i	$-0.810678\pi$
$787$	−46.4721	−1.65655	−0.828276	+	0.560320i	$0.810678\pi$
$788$	31.1246	1.10877
$789$	−22.8328	−0.812870
$790$	25.5279	0.908241
$791$	4.48529	0.159479
$792$	−0.527864	−0.0187568
$793$	28.6525	1.01748
$794$	−4.29180	−0.152310
$795$	36.3607	1.28958
$796$	25.3262	0.897665
$797$	9.05573	0.320770	0.160385	−	0.987055i	$-0.448726\pi$
$797$	9.05573	0.320770	0.160385	+	0.987055i	$0.448726\pi$
$798$	1.05573	0.0373724
$799$	14.1246	0.499693
$800$	−30.7426	−1.08692
$801$	5.00000	0.176666
$802$	−17.4164	−0.614995
$803$	−1.59675	−0.0563480
$804$	21.3262	0.752118
$805$	−5.88854	−0.207544
$806$	12.5410	0.441739
$807$	−5.00000	−0.176008
$808$	−16.7082	−0.587793
$809$	36.3050	1.27641	0.638207	−	0.769865i	$-0.279676\pi$
$809$	36.3050	1.27641	0.638207	+	0.769865i	$0.279676\pi$
$810$	−2.00000	−0.0702728
$811$	−23.6525	−0.830551	−0.415275	−	0.909696i	$-0.636315\pi$
$811$	−23.6525	−0.830551	−0.415275	+	0.909696i	$0.636315\pi$
$812$	0.381966	0.0134044
$813$	−26.9443	−0.944977
$814$	0.763932	0.0267758
$815$	−19.4164	−0.680127
$816$	5.56231	0.194720
$817$	−28.9443	−1.01263
$818$	2.76393	0.0966386
$819$	−1.29180	−0.0451390
$820$	−10.4721	−0.365703
$821$	25.4164	0.887039	0.443519	−	0.896265i	$-0.353730\pi$
$821$	25.4164	0.887039	0.443519	+	0.896265i	$0.353730\pi$
$822$	−2.18034	−0.0760481
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−11.0557	−0.385145
$825$	−1.29180	−0.0449746
$826$	−0.652476	−0.0227025
$827$	1.16718	0.0405870	0.0202935	−	0.999794i	$-0.493540\pi$
$827$	1.16718	0.0405870	0.0202935	+	0.999794i	$0.493540\pi$
$828$	12.4721	0.433437
$829$	30.2492	1.05060	0.525299	−	0.850917i	$-0.323953\pi$
$829$	30.2492	1.05060	0.525299	+	0.850917i	$0.323953\pi$
$830$	−5.88854	−0.204394
$831$	3.00000	0.104069
$832$	1.29180	0.0447850
$833$	−20.8328	−0.721814
$834$	−0.729490	−0.0252602
$835$	−20.9443	−0.724806
$836$	−2.76393	−0.0955926
$837$	3.70820	0.128174
$838$	−16.8328	−0.581480
$839$	−35.6525	−1.23086	−0.615430	−	0.788191i	$-0.711018\pi$
$839$	−35.6525	−1.23086	−0.615430	+	0.788191i	$0.711018\pi$
$840$	1.70820	0.0589386
$841$	1.00000	0.0344828
$842$	22.1803	0.764385
$843$	−21.4164	−0.737620
$844$	−17.7082	−0.609542
$845$	54.8328	1.88631
$846$	−2.90983	−0.100042
$847$	−2.58359	−0.0887733
$848$	20.8328	0.715402
$849$	7.41641	0.254530
$850$	−10.1459	−0.348001
$851$	−40.3607	−1.38355
$852$	8.47214	0.290251
$853$	7.41641	0.253933	0.126966	−	0.991907i	$-0.459476\pi$
$853$	7.41641	0.253933	0.126966	+	0.991907i	$0.459476\pi$
$854$	0.763932	0.0261412
$855$	−23.4164	−0.800824
$856$	21.7082	0.741971
$857$	−31.5967	−1.07932	−0.539662	−	0.841882i	$-0.681448\pi$
$857$	−31.5967	−1.07932	−0.539662	+	0.841882i	$0.681448\pi$
$858$	−0.798374	−0.0272560
$859$	−36.8328	−1.25672	−0.628360	−	0.777923i	$-0.716273\pi$
$859$	−36.8328	−1.25672	−0.628360	+	0.777923i	$0.716273\pi$
$860$	−20.9443	−0.714194
$861$	0.472136	0.0160904
$862$	4.94427	0.168403
$863$	37.0132	1.25994	0.629971	−	0.776618i	$-0.283067\pi$
$863$	37.0132	1.25994	0.629971	+	0.776618i	$0.283067\pi$
$864$	−5.61803	−0.191129
$865$	−22.8328	−0.776339
$866$	4.11146	0.139713
$867$	−8.00000	−0.271694
$868$	−1.41641	−0.0480760
$869$	3.01316	0.102214
$870$	2.00000	0.0678064
$871$	72.1246	2.44385
$872$	21.1803	0.717257
$873$	18.6525	0.631291
$874$	−34.4721	−1.16604
$875$	0.360680	0.0121932
$876$	−10.9443	−0.369773
$877$	21.4164	0.723181	0.361590	−	0.932337i	$-0.382234\pi$
$877$	21.4164	0.723181	0.361590	+	0.932337i	$0.382234\pi$
$878$	−8.21478	−0.277235
$879$	−19.9443	−0.672704
$880$	−1.41641	−0.0477471
$881$	22.5279	0.758983	0.379492	−	0.925195i	$-0.376099\pi$
$881$	22.5279	0.758983	0.379492	+	0.925195i	$0.376099\pi$
$882$	4.29180	0.144512
$883$	27.4164	0.922636	0.461318	−	0.887235i	$-0.347377\pi$
$883$	27.4164	0.922636	0.461318	+	0.887235i	$0.347377\pi$
$884$	26.5623	0.893387
$885$	14.4721	0.486476
$886$	2.32624	0.0781515
$887$	16.8197	0.564749	0.282374	−	0.959304i	$-0.408878\pi$
$887$	16.8197	0.564749	0.282374	+	0.959304i	$0.408878\pi$
$888$	11.7082	0.392902
$889$	2.94427	0.0987477
$890$	−10.0000	−0.335201
$891$	−0.236068	−0.00790857
$892$	−24.5623	−0.822407
$893$	−34.0689	−1.14007
$894$	14.4721	0.484021
$895$	55.7771	1.86442
$896$	2.68692	0.0897636
$897$	42.1803	1.40836
$898$	12.0344	0.401595
$899$	−3.70820	−0.123676
$900$	−8.85410	−0.295137
$901$	33.7082	1.12298
$902$	0.291796	0.00971575
$903$	0.944272	0.0314234
$904$	42.4853	1.41304
$905$	−9.70820	−0.322712
$906$	−1.88854	−0.0627427
$907$	−17.7771	−0.590279	−0.295139	−	0.955454i	$-0.595366\pi$
$907$	−17.7771	−0.590279	−0.295139	+	0.955454i	$0.595366\pi$
$908$	17.7082	0.587667
$909$	−7.47214	−0.247835
$910$	2.58359	0.0856452
$911$	10.8197	0.358471	0.179236	−	0.983806i	$-0.442638\pi$
$911$	10.8197	0.358471	0.179236	+	0.983806i	$0.442638\pi$
$912$	−13.4164	−0.444262
$913$	−0.695048	−0.0230027
$914$	0.909830	0.0300945
$915$	−16.9443	−0.560160
$916$	26.1803	0.865023
$917$	2.05573	0.0678861
$918$	−1.85410	−0.0611945
$919$	58.0132	1.91368	0.956839	−	0.290619i	$-0.0938614\pi$
$919$	58.0132	1.91368	0.956839	+	0.290619i	$0.0938614\pi$
$920$	−55.7771	−1.83892
$921$	−12.6525	−0.416913
$922$	9.81966	0.323393
$923$	28.6525	0.943108
$924$	0.0901699	0.00296637
$925$	28.6525	0.942088
$926$	−9.38197	−0.308311
$927$	−4.94427	−0.162391
$928$	5.61803	0.184421
$929$	−35.5279	−1.16563	−0.582816	−	0.812604i	$-0.698049\pi$
$929$	−35.5279	−1.16563	−0.582816	+	0.812604i	$0.698049\pi$
$930$	−7.41641	−0.243194
$931$	50.2492	1.64685
$932$	−28.1803	−0.923078
$933$	28.7082	0.939864
$934$	7.41641	0.242672
$935$	−2.29180	−0.0749497
$936$	−12.2361	−0.399948
$937$	35.3607	1.15518	0.577592	−	0.816326i	$-0.303993\pi$
$937$	35.3607	1.15518	0.577592	+	0.816326i	$0.303993\pi$
$938$	1.92299	0.0627877
$939$	−11.0000	−0.358971
$940$	−24.6525	−0.804075
$941$	−10.1115	−0.329624	−0.164812	−	0.986325i	$-0.552702\pi$
$941$	−10.1115	−0.329624	−0.164812	+	0.986325i	$0.552702\pi$
$942$	1.23607	0.0402733
$943$	−15.4164	−0.502027
$944$	8.29180	0.269875
$945$	0.763932	0.0248507
$946$	0.583592	0.0189742
$947$	−2.12461	−0.0690406	−0.0345203	−	0.999404i	$-0.510990\pi$
$947$	−2.12461	−0.0690406	−0.0345203	+	0.999404i	$0.510990\pi$
$948$	20.6525	0.670761
$949$	−37.0132	−1.20150
$950$	24.4721	0.793981
$951$	−1.47214	−0.0477373
$952$	1.58359	0.0513245
$953$	41.8885	1.35690	0.678452	−	0.734645i	$-0.262651\pi$
$953$	41.8885	1.35690	0.678452	+	0.734645i	$0.262651\pi$
$954$	−6.94427	−0.224829
$955$	38.8328	1.25660
$956$	−31.7082	−1.02552
$957$	0.236068	0.00763099
$958$	−8.94427	−0.288976
$959$	0.832816	0.0268930
$960$	−0.763932	−0.0246558
$961$	−17.2492	−0.556427
$962$	17.7082	0.570935
$963$	9.70820	0.312842
$964$	−16.8541	−0.542834
$965$	27.4164	0.882565
$966$	1.12461	0.0361838
$967$	0.763932	0.0245664	0.0122832	−	0.999925i	$-0.496090\pi$
$967$	0.763932	0.0245664	0.0122832	+	0.999925i	$0.496090\pi$
$968$	−24.4721	−0.786564
$969$	−21.7082	−0.697368
$970$	−37.3050	−1.19779
$971$	−0.360680	−0.0115748	−0.00578738	−	0.999983i	$-0.501842\pi$
$971$	−0.360680	−0.0115748	−0.00578738	+	0.999983i	$0.501842\pi$
$972$	−1.61803	−0.0518985
$973$	0.278640	0.00893280
$974$	−22.8328	−0.731611
$975$	−29.9443	−0.958984
$976$	−9.70820	−0.310752
$977$	39.7082	1.27038	0.635189	−	0.772357i	$-0.280922\pi$
$977$	39.7082	1.27038	0.635189	+	0.772357i	$0.280922\pi$
$978$	3.70820	0.118575
$979$	−1.18034	−0.0377238
$980$	36.3607	1.16150
$981$	9.47214	0.302422
$982$	4.94427	0.157778
$983$	−4.94427	−0.157698	−0.0788489	−	0.996887i	$-0.525124\pi$
$983$	−4.94427	−0.157698	−0.0788489	+	0.996887i	$0.525124\pi$
$984$	4.47214	0.142566
$985$	−62.2492	−1.98343
$986$	1.85410	0.0590466
$987$	1.11146	0.0353780
$988$	−64.0689	−2.03830
$989$	−30.8328	−0.980427
$990$	0.472136	0.0150055
$991$	−26.0132	−0.826335	−0.413168	−	0.910655i	$-0.635578\pi$
$991$	−26.0132	−0.826335	−0.413168	+	0.910655i	$0.635578\pi$
$992$	−20.8328	−0.661443
$993$	−20.7639	−0.658923
$994$	0.763932	0.0242305
$995$	−50.6525	−1.60579
$996$	−4.76393	−0.150951
$997$	−60.1378	−1.90458	−0.952291	−	0.305191i	$-0.901280\pi$
$997$	−60.1378	−1.90458	−0.952291	+	0.305191i	$0.901280\pi$
$998$	−2.03444	−0.0643991
$999$	5.23607	0.165662

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.2.a.a.1.1	✓	2
3.2	odd	2		261.2.a.b.1.2		2
4.3	odd	2		1392.2.a.q.1.2		2
5.2	odd	4		2175.2.c.k.349.2		4
5.3	odd	4		2175.2.c.k.349.3		4
5.4	even	2		2175.2.a.l.1.2		2
7.6	odd	2		4263.2.a.j.1.1		2
8.3	odd	2		5568.2.a.bs.1.1		2
8.5	even	2		5568.2.a.bl.1.1		2
12.11	even	2		4176.2.a.bn.1.1		2
15.14	odd	2		6525.2.a.ba.1.1		2
29.28	even	2		2523.2.a.c.1.2		2
87.86	odd	2		7569.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.a.a.1.1	✓	2	1.1	even	1	trivial
261.2.a.b.1.2		2	3.2	odd	2
1392.2.a.q.1.2		2	4.3	odd	2
2175.2.a.l.1.2		2	5.4	even	2
2175.2.c.k.349.2		4	5.2	odd	4
2175.2.c.k.349.3		4	5.3	odd	4
2523.2.a.c.1.2		2	29.28	even	2
4176.2.a.bn.1.1		2	12.11	even	2
4263.2.a.j.1.1		2	7.6	odd	2
5568.2.a.bl.1.1		2	8.5	even	2
5568.2.a.bs.1.1		2	8.3	odd	2
6525.2.a.ba.1.1		2	15.14	odd	2
7569.2.a.k.1.1		2	87.86	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:	\( N \)	\(=\)	\( 87 = 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	87.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +3.23607 q^{5} -0.618034 q^{6} +0.236068 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +3.23607 q^{5} -0.618034 q^{6} +0.236068 q^{7} +2.23607 q^{8} +1.00000 q^{9} -2.00000 q^{10} -0.236068 q^{11} -1.61803 q^{12} -5.47214 q^{13} -0.145898 q^{14} +3.23607 q^{15} +1.85410 q^{16} +3.00000 q^{17} -0.618034 q^{18} -7.23607 q^{19} -5.23607 q^{20} +0.236068 q^{21} +0.145898 q^{22} -7.70820 q^{23} +2.23607 q^{24} +5.47214 q^{25} +3.38197 q^{26} +1.00000 q^{27} -0.381966 q^{28} -1.00000 q^{29} -2.00000 q^{30} +3.70820 q^{31} -5.61803 q^{32} -0.236068 q^{33} -1.85410 q^{34} +0.763932 q^{35} -1.61803 q^{36} +5.23607 q^{37} +4.47214 q^{38} -5.47214 q^{39} +7.23607 q^{40} +2.00000 q^{41} -0.145898 q^{42} +4.00000 q^{43} +0.381966 q^{44} +3.23607 q^{45} +4.76393 q^{46} +4.70820 q^{47} +1.85410 q^{48} -6.94427 q^{49} -3.38197 q^{50} +3.00000 q^{51} +8.85410 q^{52} +11.2361 q^{53} -0.618034 q^{54} -0.763932 q^{55} +0.527864 q^{56} -7.23607 q^{57} +0.618034 q^{58} +4.47214 q^{59} -5.23607 q^{60} -5.23607 q^{61} -2.29180 q^{62} +0.236068 q^{63} -0.236068 q^{64} -17.7082 q^{65} +0.145898 q^{66} -13.1803 q^{67} -4.85410 q^{68} -7.70820 q^{69} -0.472136 q^{70} -5.23607 q^{71} +2.23607 q^{72} +6.76393 q^{73} -3.23607 q^{74} +5.47214 q^{75} +11.7082 q^{76} -0.0557281 q^{77} +3.38197 q^{78} -12.7639 q^{79} +6.00000 q^{80} +1.00000 q^{81} -1.23607 q^{82} +2.94427 q^{83} -0.381966 q^{84} +9.70820 q^{85} -2.47214 q^{86} -1.00000 q^{87} -0.527864 q^{88} +5.00000 q^{89} -2.00000 q^{90} -1.29180 q^{91} +12.4721 q^{92} +3.70820 q^{93} -2.90983 q^{94} -23.4164 q^{95} -5.61803 q^{96} +18.6525 q^{97} +4.29180 q^{98} -0.236068 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 2 q^{9} - 4 q^{10} + 4 q^{11} - q^{12} - 2 q^{13} - 7 q^{14} + 2 q^{15} - 3 q^{16} + 6 q^{17} + q^{18} - 10 q^{19} - 6 q^{20} - 4 q^{21} + 7 q^{22} - 2 q^{23} + 2 q^{25} + 9 q^{26} + 2 q^{27} - 3 q^{28} - 2 q^{29} - 4 q^{30} - 6 q^{31} - 9 q^{32} + 4 q^{33} + 3 q^{34} + 6 q^{35} - q^{36} + 6 q^{37} - 2 q^{39} + 10 q^{40} + 4 q^{41} - 7 q^{42} + 8 q^{43} + 3 q^{44} + 2 q^{45} + 14 q^{46} - 4 q^{47} - 3 q^{48} + 4 q^{49} - 9 q^{50} + 6 q^{51} + 11 q^{52} + 18 q^{53} + q^{54} - 6 q^{55} + 10 q^{56} - 10 q^{57} - q^{58} - 6 q^{60} - 6 q^{61} - 18 q^{62} - 4 q^{63} + 4 q^{64} - 22 q^{65} + 7 q^{66} - 4 q^{67} - 3 q^{68} - 2 q^{69} + 8 q^{70} - 6 q^{71} + 18 q^{73} - 2 q^{74} + 2 q^{75} + 10 q^{76} - 18 q^{77} + 9 q^{78} - 30 q^{79} + 12 q^{80} + 2 q^{81} + 2 q^{82} - 12 q^{83} - 3 q^{84} + 6 q^{85} + 4 q^{86} - 2 q^{87} - 10 q^{88} + 10 q^{89} - 4 q^{90} - 16 q^{91} + 16 q^{92} - 6 q^{93} - 17 q^{94} - 20 q^{95} - 9 q^{96} + 6 q^{97} + 22 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 + 2 * q^9 - 4 * q^10 + 4 * q^11 - q^12 - 2 * q^13 - 7 * q^14 + 2 * q^15 - 3 * q^16 + 6 * q^17 + q^18 - 10 * q^19 - 6 * q^20 - 4 * q^21 + 7 * q^22 - 2 * q^23 + 2 * q^25 + 9 * q^26 + 2 * q^27 - 3 * q^28 - 2 * q^29 - 4 * q^30 - 6 * q^31 - 9 * q^32 + 4 * q^33 + 3 * q^34 + 6 * q^35 - q^36 + 6 * q^37 - 2 * q^39 + 10 * q^40 + 4 * q^41 - 7 * q^42 + 8 * q^43 + 3 * q^44 + 2 * q^45 + 14 * q^46 - 4 * q^47 - 3 * q^48 + 4 * q^49 - 9 * q^50 + 6 * q^51 + 11 * q^52 + 18 * q^53 + q^54 - 6 * q^55 + 10 * q^56 - 10 * q^57 - q^58 - 6 * q^60 - 6 * q^61 - 18 * q^62 - 4 * q^63 + 4 * q^64 - 22 * q^65 + 7 * q^66 - 4 * q^67 - 3 * q^68 - 2 * q^69 + 8 * q^70 - 6 * q^71 + 18 * q^73 - 2 * q^74 + 2 * q^75 + 10 * q^76 - 18 * q^77 + 9 * q^78 - 30 * q^79 + 12 * q^80 + 2 * q^81 + 2 * q^82 - 12 * q^83 - 3 * q^84 + 6 * q^85 + 4 * q^86 - 2 * q^87 - 10 * q^88 + 10 * q^89 - 4 * q^90 - 16 * q^91 + 16 * q^92 - 6 * q^93 - 17 * q^94 - 20 * q^95 - 9 * q^96 + 6 * q^97 + 22 * q^98 + 4 * q^99

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