LMFDB - Embedded newform 6078.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6078,2,Mod(1,6078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6078, 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("6078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6078 = 2 \cdot 3 \cdot 1013 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6078.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:	$48.5330743485$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6078.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +2.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +2.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.23607 q^{10} +4.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -2.61803 q^{14} -1.23607 q^{15} +1.00000 q^{16} -2.85410 q^{17} -1.00000 q^{18} +1.14590 q^{19} +1.23607 q^{20} -2.61803 q^{21} -4.00000 q^{22} +4.61803 q^{23} +1.00000 q^{24} -3.47214 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.61803 q^{28} -6.00000 q^{29} +1.23607 q^{30} +7.32624 q^{31} -1.00000 q^{32} -4.00000 q^{33} +2.85410 q^{34} +3.23607 q^{35} +1.00000 q^{36} +3.61803 q^{37} -1.14590 q^{38} +4.00000 q^{39} -1.23607 q^{40} -1.09017 q^{41} +2.61803 q^{42} -3.09017 q^{43} +4.00000 q^{44} +1.23607 q^{45} -4.61803 q^{46} +10.0000 q^{47} -1.00000 q^{48} -0.145898 q^{49} +3.47214 q^{50} +2.85410 q^{51} -4.00000 q^{52} -11.3262 q^{53} +1.00000 q^{54} +4.94427 q^{55} -2.61803 q^{56} -1.14590 q^{57} +6.00000 q^{58} +4.14590 q^{59} -1.23607 q^{60} -11.6180 q^{61} -7.32624 q^{62} +2.61803 q^{63} +1.00000 q^{64} -4.94427 q^{65} +4.00000 q^{66} +10.9443 q^{67} -2.85410 q^{68} -4.61803 q^{69} -3.23607 q^{70} +5.85410 q^{71} -1.00000 q^{72} +10.5623 q^{73} -3.61803 q^{74} +3.47214 q^{75} +1.14590 q^{76} +10.4721 q^{77} -4.00000 q^{78} +8.94427 q^{79} +1.23607 q^{80} +1.00000 q^{81} +1.09017 q^{82} +9.70820 q^{83} -2.61803 q^{84} -3.52786 q^{85} +3.09017 q^{86} +6.00000 q^{87} -4.00000 q^{88} +11.7082 q^{89} -1.23607 q^{90} -10.4721 q^{91} +4.61803 q^{92} -7.32624 q^{93} -10.0000 q^{94} +1.41641 q^{95} +1.00000 q^{96} +8.85410 q^{97} +0.145898 q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 8 q^{11} - 2 q^{12} - 8 q^{13} - 3 q^{14} + 2 q^{15} + 2 q^{16} + q^{17} - 2 q^{18} + 9 q^{19} - 2 q^{20} - 3 q^{21} - 8 q^{22} + 7 q^{23} + 2 q^{24} + 2 q^{25} + 8 q^{26} - 2 q^{27} + 3 q^{28} - 12 q^{29} - 2 q^{30} - q^{31} - 2 q^{32} - 8 q^{33} - q^{34} + 2 q^{35} + 2 q^{36} + 5 q^{37} - 9 q^{38} + 8 q^{39} + 2 q^{40} + 9 q^{41} + 3 q^{42} + 5 q^{43} + 8 q^{44} - 2 q^{45} - 7 q^{46} + 20 q^{47} - 2 q^{48} - 7 q^{49} - 2 q^{50} - q^{51} - 8 q^{52} - 7 q^{53} + 2 q^{54} - 8 q^{55} - 3 q^{56} - 9 q^{57} + 12 q^{58} + 15 q^{59} + 2 q^{60} - 21 q^{61} + q^{62} + 3 q^{63} + 2 q^{64} + 8 q^{65} + 8 q^{66} + 4 q^{67} + q^{68} - 7 q^{69} - 2 q^{70} + 5 q^{71} - 2 q^{72} + q^{73} - 5 q^{74} - 2 q^{75} + 9 q^{76} + 12 q^{77} - 8 q^{78} - 2 q^{80} + 2 q^{81} - 9 q^{82} + 6 q^{83} - 3 q^{84} - 16 q^{85} - 5 q^{86} + 12 q^{87} - 8 q^{88} + 10 q^{89} + 2 q^{90} - 12 q^{91} + 7 q^{92} + q^{93} - 20 q^{94} - 24 q^{95} + 2 q^{96} + 11 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 8 * q^11 - 2 * q^12 - 8 * q^13 - 3 * q^14 + 2 * q^15 + 2 * q^16 + q^17 - 2 * q^18 + 9 * q^19 - 2 * q^20 - 3 * q^21 - 8 * q^22 + 7 * q^23 + 2 * q^24 + 2 * q^25 + 8 * q^26 - 2 * q^27 + 3 * q^28 - 12 * q^29 - 2 * q^30 - q^31 - 2 * q^32 - 8 * q^33 - q^34 + 2 * q^35 + 2 * q^36 + 5 * q^37 - 9 * q^38 + 8 * q^39 + 2 * q^40 + 9 * q^41 + 3 * q^42 + 5 * q^43 + 8 * q^44 - 2 * q^45 - 7 * q^46 + 20 * q^47 - 2 * q^48 - 7 * q^49 - 2 * q^50 - q^51 - 8 * q^52 - 7 * q^53 + 2 * q^54 - 8 * q^55 - 3 * q^56 - 9 * q^57 + 12 * q^58 + 15 * q^59 + 2 * q^60 - 21 * q^61 + q^62 + 3 * q^63 + 2 * q^64 + 8 * q^65 + 8 * q^66 + 4 * q^67 + q^68 - 7 * q^69 - 2 * q^70 + 5 * q^71 - 2 * q^72 + q^73 - 5 * q^74 - 2 * q^75 + 9 * q^76 + 12 * q^77 - 8 * q^78 - 2 * q^80 + 2 * q^81 - 9 * q^82 + 6 * q^83 - 3 * q^84 - 16 * q^85 - 5 * q^86 + 12 * q^87 - 8 * q^88 + 10 * q^89 + 2 * q^90 - 12 * q^91 + 7 * q^92 + q^93 - 20 * q^94 - 24 * q^95 + 2 * q^96 + 11 * q^97 + 7 * q^98 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	1.00000	0.408248
$7$	2.61803	0.989524	0.494762	−	0.869029i	$-0.335255\pi$
$7$	2.61803	0.989524	0.494762	+	0.869029i	$0.335255\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.23607	−0.390879
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−1.00000	−0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−2.61803	−0.699699
$15$	−1.23607	−0.319151
$16$	1.00000	0.250000
$17$	−2.85410	−0.692221	−0.346111	−	0.938194i	$-0.612498\pi$
$17$	−2.85410	−0.692221	−0.346111	+	0.938194i	$0.612498\pi$
$18$	−1.00000	−0.235702
$19$	1.14590	0.262887	0.131444	−	0.991324i	$-0.458039\pi$
$19$	1.14590	0.262887	0.131444	+	0.991324i	$0.458039\pi$
$20$	1.23607	0.276393
$21$	−2.61803	−0.571302
$22$	−4.00000	−0.852803
$23$	4.61803	0.962927	0.481463	−	0.876466i	$-0.340105\pi$
$23$	4.61803	0.962927	0.481463	+	0.876466i	$0.340105\pi$
$24$	1.00000	0.204124
$25$	−3.47214	−0.694427
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	2.61803	0.494762
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	1.23607	0.225674
$31$	7.32624	1.31583	0.657916	−	0.753092i	$-0.271438\pi$
$31$	7.32624	1.31583	0.657916	+	0.753092i	$0.271438\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	2.85410	0.489474
$35$	3.23607	0.546995
$36$	1.00000	0.166667
$37$	3.61803	0.594801	0.297401	−	0.954753i	$-0.403880\pi$
$37$	3.61803	0.594801	0.297401	+	0.954753i	$0.403880\pi$
$38$	−1.14590	−0.185889
$39$	4.00000	0.640513
$40$	−1.23607	−0.195440
$41$	−1.09017	−0.170256	−0.0851280	−	0.996370i	$-0.527130\pi$
$41$	−1.09017	−0.170256	−0.0851280	+	0.996370i	$0.527130\pi$
$42$	2.61803	0.403971
$43$	−3.09017	−0.471246	−0.235623	−	0.971844i	$-0.575713\pi$
$43$	−3.09017	−0.471246	−0.235623	+	0.971844i	$0.575713\pi$
$44$	4.00000	0.603023
$45$	1.23607	0.184262
$46$	−4.61803	−0.680892
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	−1.00000	−0.144338
$49$	−0.145898	−0.0208426
$50$	3.47214	0.491034
$51$	2.85410	0.399654
$52$	−4.00000	−0.554700
$53$	−11.3262	−1.55578	−0.777889	−	0.628401i	$-0.783710\pi$
$53$	−11.3262	−1.55578	−0.777889	+	0.628401i	$0.783710\pi$
$54$	1.00000	0.136083
$55$	4.94427	0.666685
$56$	−2.61803	−0.349850
$57$	−1.14590	−0.151778
$58$	6.00000	0.787839
$59$	4.14590	0.539750	0.269875	−	0.962895i	$-0.413018\pi$
$59$	4.14590	0.539750	0.269875	+	0.962895i	$0.413018\pi$
$60$	−1.23607	−0.159576
$61$	−11.6180	−1.48754	−0.743768	−	0.668437i	$-0.766963\pi$
$61$	−11.6180	−1.48754	−0.743768	+	0.668437i	$0.766963\pi$
$62$	−7.32624	−0.930433
$63$	2.61803	0.329841
$64$	1.00000	0.125000
$65$	−4.94427	−0.613261
$66$	4.00000	0.492366
$67$	10.9443	1.33706	0.668528	−	0.743687i	$-0.266925\pi$
$67$	10.9443	1.33706	0.668528	+	0.743687i	$0.266925\pi$
$68$	−2.85410	−0.346111
$69$	−4.61803	−0.555946
$70$	−3.23607	−0.386784
$71$	5.85410	0.694754	0.347377	−	0.937726i	$-0.387072\pi$
$71$	5.85410	0.694754	0.347377	+	0.937726i	$0.387072\pi$
$72$	−1.00000	−0.117851
$73$	10.5623	1.23622	0.618112	−	0.786090i	$-0.287898\pi$
$73$	10.5623	1.23622	0.618112	+	0.786090i	$0.287898\pi$
$74$	−3.61803	−0.420588
$75$	3.47214	0.400928
$76$	1.14590	0.131444
$77$	10.4721	1.19341
$78$	−4.00000	−0.452911
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	1.23607	0.138197
$81$	1.00000	0.111111
$82$	1.09017	0.120389
$83$	9.70820	1.06561	0.532807	−	0.846237i	$-0.321137\pi$
$83$	9.70820	1.06561	0.532807	+	0.846237i	$0.321137\pi$
$84$	−2.61803	−0.285651
$85$	−3.52786	−0.382651
$86$	3.09017	0.333222
$87$	6.00000	0.643268
$88$	−4.00000	−0.426401
$89$	11.7082	1.24107	0.620534	−	0.784180i	$-0.286916\pi$
$89$	11.7082	1.24107	0.620534	+	0.784180i	$0.286916\pi$
$90$	−1.23607	−0.130293
$91$	−10.4721	−1.09778
$92$	4.61803	0.481463
$93$	−7.32624	−0.759695
$94$	−10.0000	−1.03142
$95$	1.41641	0.145320
$96$	1.00000	0.102062
$97$	8.85410	0.898998	0.449499	−	0.893281i	$-0.351603\pi$
$97$	8.85410	0.898998	0.449499	+	0.893281i	$0.351603\pi$
$98$	0.145898	0.0147379
$99$	4.00000	0.402015
$100$	−3.47214	−0.347214
$101$	4.18034	0.415959	0.207980	−	0.978133i	$-0.433311\pi$
$101$	4.18034	0.415959	0.207980	+	0.978133i	$0.433311\pi$
$102$	−2.85410	−0.282598
$103$	12.1459	1.19677	0.598385	−	0.801208i	$-0.295809\pi$
$103$	12.1459	1.19677	0.598385	+	0.801208i	$0.295809\pi$
$104$	4.00000	0.392232
$105$	−3.23607	−0.315808
$106$	11.3262	1.10010
$107$	6.61803	0.639789	0.319895	−	0.947453i	$-0.396352\pi$
$107$	6.61803	0.639789	0.319895	+	0.947453i	$0.396352\pi$
$108$	−1.00000	−0.0962250
$109$	−11.6180	−1.11281	−0.556403	−	0.830913i	$-0.687819\pi$
$109$	−11.6180	−1.11281	−0.556403	+	0.830913i	$0.687819\pi$
$110$	−4.94427	−0.471418
$111$	−3.61803	−0.343409
$112$	2.61803	0.247381
$113$	18.1803	1.71026	0.855131	−	0.518412i	$-0.173476\pi$
$113$	18.1803	1.71026	0.855131	+	0.518412i	$0.173476\pi$
$114$	1.14590	0.107323
$115$	5.70820	0.532293
$116$	−6.00000	−0.557086
$117$	−4.00000	−0.369800
$118$	−4.14590	−0.381661
$119$	−7.47214	−0.684970
$120$	1.23607	0.112837
$121$	5.00000	0.454545
$122$	11.6180	1.05185
$123$	1.09017	0.0982973
$124$	7.32624	0.657916
$125$	−10.4721	−0.936656
$126$	−2.61803	−0.233233
$127$	−6.94427	−0.616204	−0.308102	−	0.951353i	$-0.599694\pi$
$127$	−6.94427	−0.616204	−0.308102	+	0.951353i	$0.599694\pi$
$128$	−1.00000	−0.0883883
$129$	3.09017	0.272074
$130$	4.94427	0.433641
$131$	−16.7984	−1.46768	−0.733840	−	0.679322i	$-0.762274\pi$
$131$	−16.7984	−1.46768	−0.733840	+	0.679322i	$0.762274\pi$
$132$	−4.00000	−0.348155
$133$	3.00000	0.260133
$134$	−10.9443	−0.945441
$135$	−1.23607	−0.106384
$136$	2.85410	0.244737
$137$	1.38197	0.118069	0.0590347	−	0.998256i	$-0.481198\pi$
$137$	1.38197	0.118069	0.0590347	+	0.998256i	$0.481198\pi$
$138$	4.61803	0.393113
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	3.23607	0.273498
$141$	−10.0000	−0.842152
$142$	−5.85410	−0.491265
$143$	−16.0000	−1.33799
$144$	1.00000	0.0833333
$145$	−7.41641	−0.615899
$146$	−10.5623	−0.874143
$147$	0.145898	0.0120335
$148$	3.61803	0.297401
$149$	5.41641	0.443729	0.221865	−	0.975077i	$-0.428786\pi$
$149$	5.41641	0.443729	0.221865	+	0.975077i	$0.428786\pi$
$150$	−3.47214	−0.283499
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−1.14590	−0.0929446
$153$	−2.85410	−0.230740
$154$	−10.4721	−0.843869
$155$	9.05573	0.727374
$156$	4.00000	0.320256
$157$	−13.2361	−1.05635	−0.528177	−	0.849135i	$-0.677124\pi$
$157$	−13.2361	−1.05635	−0.528177	+	0.849135i	$0.677124\pi$
$158$	−8.94427	−0.711568
$159$	11.3262	0.898229
$160$	−1.23607	−0.0977198
$161$	12.0902	0.952839
$162$	−1.00000	−0.0785674
$163$	−12.9443	−1.01387	−0.506937	−	0.861983i	$-0.669222\pi$
$163$	−12.9443	−1.01387	−0.506937	+	0.861983i	$0.669222\pi$
$164$	−1.09017	−0.0851280
$165$	−4.94427	−0.384911
$166$	−9.70820	−0.753503
$167$	8.38197	0.648616	0.324308	−	0.945952i	$-0.394869\pi$
$167$	8.38197	0.648616	0.324308	+	0.945952i	$0.394869\pi$
$168$	2.61803	0.201986
$169$	3.00000	0.230769
$170$	3.52786	0.270575
$171$	1.14590	0.0876290
$172$	−3.09017	−0.235623
$173$	13.0344	0.990990	0.495495	−	0.868611i	$-0.334987\pi$
$173$	13.0344	0.990990	0.495495	+	0.868611i	$0.334987\pi$
$174$	−6.00000	−0.454859
$175$	−9.09017	−0.687152
$176$	4.00000	0.301511
$177$	−4.14590	−0.311625
$178$	−11.7082	−0.877567
$179$	−4.09017	−0.305714	−0.152857	−	0.988248i	$-0.548847\pi$
$179$	−4.09017	−0.305714	−0.152857	+	0.988248i	$0.548847\pi$
$180$	1.23607	0.0921311
$181$	21.2361	1.57846	0.789232	−	0.614095i	$-0.210479\pi$
$181$	21.2361	1.57846	0.789232	+	0.614095i	$0.210479\pi$
$182$	10.4721	0.776246
$183$	11.6180	0.858830
$184$	−4.61803	−0.340446
$185$	4.47214	0.328798
$186$	7.32624	0.537186
$187$	−11.4164	−0.834850
$188$	10.0000	0.729325
$189$	−2.61803	−0.190434
$190$	−1.41641	−0.102757
$191$	−23.5967	−1.70740	−0.853700	−	0.520765i	$-0.825647\pi$
$191$	−23.5967	−1.70740	−0.853700	+	0.520765i	$0.825647\pi$
$192$	−1.00000	−0.0721688
$193$	−16.6180	−1.19619	−0.598096	−	0.801424i	$-0.704076\pi$
$193$	−16.6180	−1.19619	−0.598096	+	0.801424i	$0.704076\pi$
$194$	−8.85410	−0.635687
$195$	4.94427	0.354067
$196$	−0.145898	−0.0104213
$197$	22.5066	1.60353	0.801764	−	0.597641i	$-0.203895\pi$
$197$	22.5066	1.60353	0.801764	+	0.597641i	$0.203895\pi$
$198$	−4.00000	−0.284268
$199$	−13.4164	−0.951064	−0.475532	−	0.879698i	$-0.657744\pi$
$199$	−13.4164	−0.951064	−0.475532	+	0.879698i	$0.657744\pi$
$200$	3.47214	0.245517
$201$	−10.9443	−0.771949
$202$	−4.18034	−0.294128
$203$	−15.7082	−1.10250
$204$	2.85410	0.199827
$205$	−1.34752	−0.0941152
$206$	−12.1459	−0.846245
$207$	4.61803	0.320976
$208$	−4.00000	−0.277350
$209$	4.58359	0.317054
$210$	3.23607	0.223310
$211$	−4.61803	−0.317919	−0.158959	−	0.987285i	$-0.550814\pi$
$211$	−4.61803	−0.317919	−0.158959	+	0.987285i	$0.550814\pi$
$212$	−11.3262	−0.777889
$213$	−5.85410	−0.401116
$214$	−6.61803	−0.452399
$215$	−3.81966	−0.260499
$216$	1.00000	0.0680414
$217$	19.1803	1.30205
$218$	11.6180	0.786873
$219$	−10.5623	−0.713734
$220$	4.94427	0.333343
$221$	11.4164	0.767951
$222$	3.61803	0.242827
$223$	2.29180	0.153470	0.0767350	−	0.997052i	$-0.475550\pi$
$223$	2.29180	0.153470	0.0767350	+	0.997052i	$0.475550\pi$
$224$	−2.61803	−0.174925
$225$	−3.47214	−0.231476
$226$	−18.1803	−1.20934
$227$	4.43769	0.294540	0.147270	−	0.989096i	$-0.452951\pi$
$227$	4.43769	0.294540	0.147270	+	0.989096i	$0.452951\pi$
$228$	−1.14590	−0.0758890
$229$	2.29180	0.151446	0.0757231	−	0.997129i	$-0.475874\pi$
$229$	2.29180	0.151446	0.0757231	+	0.997129i	$0.475874\pi$
$230$	−5.70820	−0.376388
$231$	−10.4721	−0.689016
$232$	6.00000	0.393919
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	4.00000	0.261488
$235$	12.3607	0.806322
$236$	4.14590	0.269875
$237$	−8.94427	−0.580993
$238$	7.47214	0.484347
$239$	22.3607	1.44639	0.723196	−	0.690643i	$-0.242672\pi$
$239$	22.3607	1.44639	0.723196	+	0.690643i	$0.242672\pi$
$240$	−1.23607	−0.0797878
$241$	−27.0902	−1.74503	−0.872516	−	0.488586i	$-0.837513\pi$
$241$	−27.0902	−1.74503	−0.872516	+	0.488586i	$0.837513\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−11.6180	−0.743768
$245$	−0.180340	−0.0115215
$246$	−1.09017	−0.0695067
$247$	−4.58359	−0.291647
$248$	−7.32624	−0.465217
$249$	−9.70820	−0.615232
$250$	10.4721	0.662316
$251$	10.1803	0.642577	0.321289	−	0.946981i	$-0.395884\pi$
$251$	10.1803	0.642577	0.321289	+	0.946981i	$0.395884\pi$
$252$	2.61803	0.164921
$253$	18.4721	1.16133
$254$	6.94427	0.435722
$255$	3.52786	0.220923
$256$	1.00000	0.0625000
$257$	29.7082	1.85315	0.926573	−	0.376114i	$-0.122740\pi$
$257$	29.7082	1.85315	0.926573	+	0.376114i	$0.122740\pi$
$258$	−3.09017	−0.192386
$259$	9.47214	0.588570
$260$	−4.94427	−0.306631
$261$	−6.00000	−0.371391
$262$	16.7984	1.03781
$263$	30.4721	1.87899	0.939496	−	0.342559i	$-0.111294\pi$
$263$	30.4721	1.87899	0.939496	+	0.342559i	$0.111294\pi$
$264$	4.00000	0.246183
$265$	−14.0000	−0.860013
$266$	−3.00000	−0.183942
$267$	−11.7082	−0.716530
$268$	10.9443	0.668528
$269$	0.583592	0.0355822	0.0177911	−	0.999842i	$-0.494337\pi$
$269$	0.583592	0.0355822	0.0177911	+	0.999842i	$0.494337\pi$
$270$	1.23607	0.0752247
$271$	−18.9443	−1.15078	−0.575391	−	0.817878i	$-0.695150\pi$
$271$	−18.9443	−1.15078	−0.575391	+	0.817878i	$0.695150\pi$
$272$	−2.85410	−0.173055
$273$	10.4721	0.633803
$274$	−1.38197	−0.0834876
$275$	−13.8885	−0.837511
$276$	−4.61803	−0.277973
$277$	2.85410	0.171486	0.0857432	−	0.996317i	$-0.472674\pi$
$277$	2.85410	0.171486	0.0857432	+	0.996317i	$0.472674\pi$
$278$	12.0000	0.719712
$279$	7.32624	0.438610
$280$	−3.23607	−0.193392
$281$	−0.763932	−0.0455724	−0.0227862	−	0.999740i	$-0.507254\pi$
$281$	−0.763932	−0.0455724	−0.0227862	+	0.999740i	$0.507254\pi$
$282$	10.0000	0.595491
$283$	−9.85410	−0.585766	−0.292883	−	0.956148i	$-0.594615\pi$
$283$	−9.85410	−0.585766	−0.292883	+	0.956148i	$0.594615\pi$
$284$	5.85410	0.347377
$285$	−1.41641	−0.0839008
$286$	16.0000	0.946100
$287$	−2.85410	−0.168472
$288$	−1.00000	−0.0589256
$289$	−8.85410	−0.520830
$290$	7.41641	0.435506
$291$	−8.85410	−0.519037
$292$	10.5623	0.618112
$293$	−32.0689	−1.87348	−0.936742	−	0.350020i	$-0.886175\pi$
$293$	−32.0689	−1.87348	−0.936742	+	0.350020i	$0.886175\pi$
$294$	−0.145898	−0.00850895
$295$	5.12461	0.298366
$296$	−3.61803	−0.210294
$297$	−4.00000	−0.232104
$298$	−5.41641	−0.313764
$299$	−18.4721	−1.06827
$300$	3.47214	0.200464
$301$	−8.09017	−0.466310
$302$	−4.00000	−0.230174
$303$	−4.18034	−0.240154
$304$	1.14590	0.0657218
$305$	−14.3607	−0.822290
$306$	2.85410	0.163158
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	10.4721	0.596705
$309$	−12.1459	−0.690956
$310$	−9.05573	−0.514331
$311$	−9.14590	−0.518616	−0.259308	−	0.965795i	$-0.583495\pi$
$311$	−9.14590	−0.518616	−0.259308	+	0.965795i	$0.583495\pi$
$312$	−4.00000	−0.226455
$313$	1.23607	0.0698667	0.0349333	−	0.999390i	$-0.488878\pi$
$313$	1.23607	0.0698667	0.0349333	+	0.999390i	$0.488878\pi$
$314$	13.2361	0.746955
$315$	3.23607	0.182332
$316$	8.94427	0.503155
$317$	−10.9443	−0.614692	−0.307346	−	0.951598i	$-0.599441\pi$
$317$	−10.9443	−0.614692	−0.307346	+	0.951598i	$0.599441\pi$
$318$	−11.3262	−0.635144
$319$	−24.0000	−1.34374
$320$	1.23607	0.0690983
$321$	−6.61803	−0.369383
$322$	−12.0902	−0.673759
$323$	−3.27051	−0.181976
$324$	1.00000	0.0555556
$325$	13.8885	0.770398
$326$	12.9443	0.716917
$327$	11.6180	0.642479
$328$	1.09017	0.0601946
$329$	26.1803	1.44337
$330$	4.94427	0.272173
$331$	24.3262	1.33709	0.668545	−	0.743671i	$-0.266917\pi$
$331$	24.3262	1.33709	0.668545	+	0.743671i	$0.266917\pi$
$332$	9.70820	0.532807
$333$	3.61803	0.198267
$334$	−8.38197	−0.458641
$335$	13.5279	0.739106
$336$	−2.61803	−0.142825
$337$	25.4164	1.38452	0.692260	−	0.721648i	$-0.256615\pi$
$337$	25.4164	1.38452	0.692260	+	0.721648i	$0.256615\pi$
$338$	−3.00000	−0.163178
$339$	−18.1803	−0.987421
$340$	−3.52786	−0.191325
$341$	29.3050	1.58695
$342$	−1.14590	−0.0619631
$343$	−18.7082	−1.01015
$344$	3.09017	0.166611
$345$	−5.70820	−0.307319
$346$	−13.0344	−0.700736
$347$	−17.8885	−0.960307	−0.480154	−	0.877184i	$-0.659419\pi$
$347$	−17.8885	−0.960307	−0.480154	+	0.877184i	$0.659419\pi$
$348$	6.00000	0.321634
$349$	16.5066	0.883577	0.441788	−	0.897119i	$-0.354344\pi$
$349$	16.5066	0.883577	0.441788	+	0.897119i	$0.354344\pi$
$350$	9.09017	0.485890
$351$	4.00000	0.213504
$352$	−4.00000	−0.213201
$353$	−13.2361	−0.704485	−0.352242	−	0.935909i	$-0.614581\pi$
$353$	−13.2361	−0.704485	−0.352242	+	0.935909i	$0.614581\pi$
$354$	4.14590	0.220352
$355$	7.23607	0.384051
$356$	11.7082	0.620534
$357$	7.47214	0.395467
$358$	4.09017	0.216172
$359$	19.8885	1.04968	0.524839	−	0.851202i	$-0.324126\pi$
$359$	19.8885	1.04968	0.524839	+	0.851202i	$0.324126\pi$
$360$	−1.23607	−0.0651465
$361$	−17.6869	−0.930890
$362$	−21.2361	−1.11614
$363$	−5.00000	−0.262432
$364$	−10.4721	−0.548889
$365$	13.0557	0.683368
$366$	−11.6180	−0.607284
$367$	0.111456	0.00581797	0.00290898	−	0.999996i	$-0.499074\pi$
$367$	0.111456	0.00581797	0.00290898	+	0.999996i	$0.499074\pi$
$368$	4.61803	0.240732
$369$	−1.09017	−0.0567520
$370$	−4.47214	−0.232495
$371$	−29.6525	−1.53948
$372$	−7.32624	−0.379848
$373$	21.8885	1.13335	0.566673	−	0.823943i	$-0.308230\pi$
$373$	21.8885	1.13335	0.566673	+	0.823943i	$0.308230\pi$
$374$	11.4164	0.590328
$375$	10.4721	0.540779
$376$	−10.0000	−0.515711
$377$	24.0000	1.23606
$378$	2.61803	0.134657
$379$	27.0344	1.38867	0.694333	−	0.719654i	$-0.255700\pi$
$379$	27.0344	1.38867	0.694333	+	0.719654i	$0.255700\pi$
$380$	1.41641	0.0726602
$381$	6.94427	0.355766
$382$	23.5967	1.20731
$383$	33.8885	1.73162	0.865812	−	0.500370i	$-0.166803\pi$
$383$	33.8885	1.73162	0.865812	+	0.500370i	$0.166803\pi$
$384$	1.00000	0.0510310
$385$	12.9443	0.659701
$386$	16.6180	0.845836
$387$	−3.09017	−0.157082
$388$	8.85410	0.449499
$389$	7.81966	0.396473	0.198236	−	0.980154i	$-0.436479\pi$
$389$	7.81966	0.396473	0.198236	+	0.980154i	$0.436479\pi$
$390$	−4.94427	−0.250363
$391$	−13.1803	−0.666558
$392$	0.145898	0.00736896
$393$	16.7984	0.847366
$394$	−22.5066	−1.13387
$395$	11.0557	0.556274
$396$	4.00000	0.201008
$397$	4.00000	0.200754	0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000	0.200754	0.100377	+	0.994949i	$0.467995\pi$
$398$	13.4164	0.672504
$399$	−3.00000	−0.150188
$400$	−3.47214	−0.173607
$401$	23.2148	1.15929	0.579645	−	0.814869i	$-0.303191\pi$
$401$	23.2148	1.15929	0.579645	+	0.814869i	$0.303191\pi$
$402$	10.9443	0.545851
$403$	−29.3050	−1.45978
$404$	4.18034	0.207980
$405$	1.23607	0.0614207
$406$	15.7082	0.779585
$407$	14.4721	0.717357
$408$	−2.85410	−0.141299
$409$	−31.3050	−1.54793	−0.773965	−	0.633228i	$-0.781729\pi$
$409$	−31.3050	−1.54793	−0.773965	+	0.633228i	$0.781729\pi$
$410$	1.34752	0.0665495
$411$	−1.38197	−0.0681674
$412$	12.1459	0.598385
$413$	10.8541	0.534095
$414$	−4.61803	−0.226964
$415$	12.0000	0.589057
$416$	4.00000	0.196116
$417$	12.0000	0.587643
$418$	−4.58359	−0.224191
$419$	13.2361	0.646624	0.323312	−	0.946292i	$-0.395204\pi$
$419$	13.2361	0.646624	0.323312	+	0.946292i	$0.395204\pi$
$420$	−3.23607	−0.157904
$421$	−15.5623	−0.758460	−0.379230	−	0.925302i	$-0.623811\pi$
$421$	−15.5623	−0.758460	−0.379230	+	0.925302i	$0.623811\pi$
$422$	4.61803	0.224802
$423$	10.0000	0.486217
$424$	11.3262	0.550051
$425$	9.90983	0.480697
$426$	5.85410	0.283632
$427$	−30.4164	−1.47195
$428$	6.61803	0.319895
$429$	16.0000	0.772487
$430$	3.81966	0.184200
$431$	−3.38197	−0.162904	−0.0814518	−	0.996677i	$-0.525956\pi$
$431$	−3.38197	−0.162904	−0.0814518	+	0.996677i	$0.525956\pi$
$432$	−1.00000	−0.0481125
$433$	1.20163	0.0577465	0.0288732	−	0.999583i	$-0.490808\pi$
$433$	1.20163	0.0577465	0.0288732	+	0.999583i	$0.490808\pi$
$434$	−19.1803	−0.920686
$435$	7.41641	0.355590
$436$	−11.6180	−0.556403
$437$	5.29180	0.253141
$438$	10.5623	0.504686
$439$	−7.41641	−0.353966	−0.176983	−	0.984214i	$-0.556634\pi$
$439$	−7.41641	−0.353966	−0.176983	+	0.984214i	$0.556634\pi$
$440$	−4.94427	−0.235709
$441$	−0.145898	−0.00694753
$442$	−11.4164	−0.543023
$443$	34.7426	1.65067	0.825336	−	0.564641i	$-0.190985\pi$
$443$	34.7426	1.65067	0.825336	+	0.564641i	$0.190985\pi$
$444$	−3.61803	−0.171704
$445$	14.4721	0.686045
$446$	−2.29180	−0.108520
$447$	−5.41641	−0.256187
$448$	2.61803	0.123690
$449$	24.3607	1.14965	0.574826	−	0.818276i	$-0.305070\pi$
$449$	24.3607	1.14965	0.574826	+	0.818276i	$0.305070\pi$
$450$	3.47214	0.163678
$451$	−4.36068	−0.205336
$452$	18.1803	0.855131
$453$	−4.00000	−0.187936
$454$	−4.43769	−0.208271
$455$	−12.9443	−0.606837
$456$	1.14590	0.0536616
$457$	−17.4164	−0.814705	−0.407353	−	0.913271i	$-0.633548\pi$
$457$	−17.4164	−0.814705	−0.407353	+	0.913271i	$0.633548\pi$
$458$	−2.29180	−0.107089
$459$	2.85410	0.133218
$460$	5.70820	0.266146
$461$	−28.6869	−1.33608	−0.668041	−	0.744124i	$-0.732867\pi$
$461$	−28.6869	−1.33608	−0.668041	+	0.744124i	$0.732867\pi$
$462$	10.4721	0.487208
$463$	−10.3262	−0.479901	−0.239950	−	0.970785i	$-0.577131\pi$
$463$	−10.3262	−0.479901	−0.239950	+	0.970785i	$0.577131\pi$
$464$	−6.00000	−0.278543
$465$	−9.05573	−0.419949
$466$	−14.0000	−0.648537
$467$	8.72949	0.403953	0.201976	−	0.979390i	$-0.435264\pi$
$467$	8.72949	0.403953	0.201976	+	0.979390i	$0.435264\pi$
$468$	−4.00000	−0.184900
$469$	28.6525	1.32305
$470$	−12.3607	−0.570156
$471$	13.2361	0.609886
$472$	−4.14590	−0.190830
$473$	−12.3607	−0.568345
$474$	8.94427	0.410824
$475$	−3.97871	−0.182556
$476$	−7.47214	−0.342485
$477$	−11.3262	−0.518593
$478$	−22.3607	−1.02275
$479$	−19.7984	−0.904611	−0.452305	−	0.891863i	$-0.649398\pi$
$479$	−19.7984	−0.904611	−0.452305	+	0.891863i	$0.649398\pi$
$480$	1.23607	0.0564185
$481$	−14.4721	−0.659873
$482$	27.0902	1.23392
$483$	−12.0902	−0.550122
$484$	5.00000	0.227273
$485$	10.9443	0.496954
$486$	1.00000	0.0453609
$487$	25.2361	1.14356	0.571778	−	0.820409i	$-0.306254\pi$
$487$	25.2361	1.14356	0.571778	+	0.820409i	$0.306254\pi$
$488$	11.6180	0.525924
$489$	12.9443	0.585360
$490$	0.180340	0.00814693
$491$	2.94427	0.132873	0.0664366	−	0.997791i	$-0.478837\pi$
$491$	2.94427	0.132873	0.0664366	+	0.997791i	$0.478837\pi$
$492$	1.09017	0.0491487
$493$	17.1246	0.771254
$494$	4.58359	0.206226
$495$	4.94427	0.222228
$496$	7.32624	0.328958
$497$	15.3262	0.687476
$498$	9.70820	0.435035
$499$	33.7771	1.51207	0.756035	−	0.654531i	$-0.227134\pi$
$499$	33.7771	1.51207	0.756035	+	0.654531i	$0.227134\pi$
$500$	−10.4721	−0.468328
$501$	−8.38197	−0.374479
$502$	−10.1803	−0.454371
$503$	9.85410	0.439373	0.219686	−	0.975571i	$-0.429497\pi$
$503$	9.85410	0.439373	0.219686	+	0.975571i	$0.429497\pi$
$504$	−2.61803	−0.116617
$505$	5.16718	0.229937
$506$	−18.4721	−0.821187
$507$	−3.00000	−0.133235
$508$	−6.94427	−0.308102
$509$	23.6869	1.04990	0.524952	−	0.851132i	$-0.324083\pi$
$509$	23.6869	1.04990	0.524952	+	0.851132i	$0.324083\pi$
$510$	−3.52786	−0.156216
$511$	27.6525	1.22327
$512$	−1.00000	−0.0441942
$513$	−1.14590	−0.0505926
$514$	−29.7082	−1.31037
$515$	15.0132	0.661559
$516$	3.09017	0.136037
$517$	40.0000	1.75920
$518$	−9.47214	−0.416182
$519$	−13.0344	−0.572148
$520$	4.94427	0.216821
$521$	−40.8328	−1.78892	−0.894459	−	0.447150i	$-0.852439\pi$
$521$	−40.8328	−1.78892	−0.894459	+	0.447150i	$0.852439\pi$
$522$	6.00000	0.262613
$523$	12.9443	0.566013	0.283007	−	0.959118i	$-0.408668\pi$
$523$	12.9443	0.566013	0.283007	+	0.959118i	$0.408668\pi$
$524$	−16.7984	−0.733840
$525$	9.09017	0.396728
$526$	−30.4721	−1.32865
$527$	−20.9098	−0.910847
$528$	−4.00000	−0.174078
$529$	−1.67376	−0.0727723
$530$	14.0000	0.608121
$531$	4.14590	0.179917
$532$	3.00000	0.130066
$533$	4.36068	0.188882
$534$	11.7082	0.506664
$535$	8.18034	0.353667
$536$	−10.9443	−0.472721
$537$	4.09017	0.176504
$538$	−0.583592	−0.0251604
$539$	−0.583592	−0.0251371
$540$	−1.23607	−0.0531919
$541$	13.1246	0.564271	0.282136	−	0.959375i	$-0.408957\pi$
$541$	13.1246	0.564271	0.282136	+	0.959375i	$0.408957\pi$
$542$	18.9443	0.813726
$543$	−21.2361	−0.911327
$544$	2.85410	0.122369
$545$	−14.3607	−0.615144
$546$	−10.4721	−0.448166
$547$	14.7639	0.631260	0.315630	−	0.948882i	$-0.397784\pi$
$547$	14.7639	0.631260	0.315630	+	0.948882i	$0.397784\pi$
$548$	1.38197	0.0590347
$549$	−11.6180	−0.495846
$550$	13.8885	0.592209
$551$	−6.87539	−0.292901
$552$	4.61803	0.196557
$553$	23.4164	0.995767
$554$	−2.85410	−0.121259
$555$	−4.47214	−0.189832
$556$	−12.0000	−0.508913
$557$	−41.6312	−1.76397	−0.881985	−	0.471277i	$-0.843793\pi$
$557$	−41.6312	−1.76397	−0.881985	+	0.471277i	$0.843793\pi$
$558$	−7.32624	−0.310144
$559$	12.3607	0.522801
$560$	3.23607	0.136749
$561$	11.4164	0.482001
$562$	0.763932	0.0322245
$563$	24.4508	1.03048	0.515240	−	0.857046i	$-0.327703\pi$
$563$	24.4508	1.03048	0.515240	+	0.857046i	$0.327703\pi$
$564$	−10.0000	−0.421076
$565$	22.4721	0.945410
$566$	9.85410	0.414199
$567$	2.61803	0.109947
$568$	−5.85410	−0.245633
$569$	−14.8328	−0.621824	−0.310912	−	0.950439i	$-0.600634\pi$
$569$	−14.8328	−0.621824	−0.310912	+	0.950439i	$0.600634\pi$
$570$	1.41641	0.0593268
$571$	−6.43769	−0.269409	−0.134705	−	0.990886i	$-0.543009\pi$
$571$	−6.43769	−0.269409	−0.134705	+	0.990886i	$0.543009\pi$
$572$	−16.0000	−0.668994
$573$	23.5967	0.985768
$574$	2.85410	0.119128
$575$	−16.0344	−0.668682
$576$	1.00000	0.0416667
$577$	19.5279	0.812956	0.406478	−	0.913661i	$-0.366757\pi$
$577$	19.5279	0.812956	0.406478	+	0.913661i	$0.366757\pi$
$578$	8.85410	0.368282
$579$	16.6180	0.690622
$580$	−7.41641	−0.307950
$581$	25.4164	1.05445
$582$	8.85410	0.367014
$583$	−45.3050	−1.87634
$584$	−10.5623	−0.437071
$585$	−4.94427	−0.204420
$586$	32.0689	1.32475
$587$	7.41641	0.306108	0.153054	−	0.988218i	$-0.451089\pi$
$587$	7.41641	0.306108	0.153054	+	0.988218i	$0.451089\pi$
$588$	0.145898	0.00601673
$589$	8.39512	0.345915
$590$	−5.12461	−0.210977
$591$	−22.5066	−0.925797
$592$	3.61803	0.148700
$593$	−25.1459	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$593$	−25.1459	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$594$	4.00000	0.164122
$595$	−9.23607	−0.378642
$596$	5.41641	0.221865
$597$	13.4164	0.549097
$598$	18.4721	0.755382
$599$	26.1803	1.06970	0.534850	−	0.844947i	$-0.320368\pi$
$599$	26.1803	1.06970	0.534850	+	0.844947i	$0.320368\pi$
$600$	−3.47214	−0.141749
$601$	2.11146	0.0861281	0.0430640	−	0.999072i	$-0.486288\pi$
$601$	2.11146	0.0861281	0.0430640	+	0.999072i	$0.486288\pi$
$602$	8.09017	0.329731
$603$	10.9443	0.445685
$604$	4.00000	0.162758
$605$	6.18034	0.251267
$606$	4.18034	0.169815
$607$	4.21478	0.171073	0.0855364	−	0.996335i	$-0.472740\pi$
$607$	4.21478	0.171073	0.0855364	+	0.996335i	$0.472740\pi$
$608$	−1.14590	−0.0464723
$609$	15.7082	0.636529
$610$	14.3607	0.581447
$611$	−40.0000	−1.61823
$612$	−2.85410	−0.115370
$613$	38.8328	1.56844	0.784221	−	0.620481i	$-0.213063\pi$
$613$	38.8328	1.56844	0.784221	+	0.620481i	$0.213063\pi$
$614$	−22.0000	−0.887848
$615$	1.34752	0.0543374
$616$	−10.4721	−0.421934
$617$	−5.41641	−0.218056	−0.109028	−	0.994039i	$-0.534774\pi$
$617$	−5.41641	−0.218056	−0.109028	+	0.994039i	$0.534774\pi$
$618$	12.1459	0.488580
$619$	−16.6525	−0.669320	−0.334660	−	0.942339i	$-0.608621\pi$
$619$	−16.6525	−0.669320	−0.334660	+	0.942339i	$0.608621\pi$
$620$	9.05573	0.363687
$621$	−4.61803	−0.185315
$622$	9.14590	0.366717
$623$	30.6525	1.22807
$624$	4.00000	0.160128
$625$	4.41641	0.176656
$626$	−1.23607	−0.0494032
$627$	−4.58359	−0.183051
$628$	−13.2361	−0.528177
$629$	−10.3262	−0.411734
$630$	−3.23607	−0.128928
$631$	−13.3050	−0.529662	−0.264831	−	0.964295i	$-0.585316\pi$
$631$	−13.3050	−0.529662	−0.264831	+	0.964295i	$0.585316\pi$
$632$	−8.94427	−0.355784
$633$	4.61803	0.183550
$634$	10.9443	0.434653
$635$	−8.58359	−0.340629
$636$	11.3262	0.449115
$637$	0.583592	0.0231228
$638$	24.0000	0.950169
$639$	5.85410	0.231585
$640$	−1.23607	−0.0488599
$641$	18.9443	0.748254	0.374127	−	0.927378i	$-0.377942\pi$
$641$	18.9443	0.748254	0.374127	+	0.927378i	$0.377942\pi$
$642$	6.61803	0.261193
$643$	−8.76393	−0.345616	−0.172808	−	0.984956i	$-0.555284\pi$
$643$	−8.76393	−0.345616	−0.172808	+	0.984956i	$0.555284\pi$
$644$	12.0902	0.476419
$645$	3.81966	0.150399
$646$	3.27051	0.128676
$647$	42.8328	1.68393	0.841966	−	0.539531i	$-0.181398\pi$
$647$	42.8328	1.68393	0.841966	+	0.539531i	$0.181398\pi$
$648$	−1.00000	−0.0392837
$649$	16.5836	0.650963
$650$	−13.8885	−0.544754
$651$	−19.1803	−0.751737
$652$	−12.9443	−0.506937
$653$	26.1459	1.02317	0.511584	−	0.859233i	$-0.329059\pi$
$653$	26.1459	1.02317	0.511584	+	0.859233i	$0.329059\pi$
$654$	−11.6180	−0.454301
$655$	−20.7639	−0.811314
$656$	−1.09017	−0.0425640
$657$	10.5623	0.412075
$658$	−26.1803	−1.02062
$659$	−14.8328	−0.577804	−0.288902	−	0.957359i	$-0.593290\pi$
$659$	−14.8328	−0.577804	−0.288902	+	0.957359i	$0.593290\pi$
$660$	−4.94427	−0.192456
$661$	−6.85410	−0.266594	−0.133297	−	0.991076i	$-0.542556\pi$
$661$	−6.85410	−0.266594	−0.133297	+	0.991076i	$0.542556\pi$
$662$	−24.3262	−0.945466
$663$	−11.4164	−0.443377
$664$	−9.70820	−0.376751
$665$	3.70820	0.143798
$666$	−3.61803	−0.140196
$667$	−27.7082	−1.07287
$668$	8.38197	0.324308
$669$	−2.29180	−0.0886060
$670$	−13.5279	−0.522627
$671$	−46.4721	−1.79404
$672$	2.61803	0.100993
$673$	−48.2705	−1.86069	−0.930346	−	0.366684i	$-0.880493\pi$
$673$	−48.2705	−1.86069	−0.930346	+	0.366684i	$0.880493\pi$
$674$	−25.4164	−0.979003
$675$	3.47214	0.133643
$676$	3.00000	0.115385
$677$	−1.03444	−0.0397568	−0.0198784	−	0.999802i	$-0.506328\pi$
$677$	−1.03444	−0.0397568	−0.0198784	+	0.999802i	$0.506328\pi$
$678$	18.1803	0.698212
$679$	23.1803	0.889580
$680$	3.52786	0.135287
$681$	−4.43769	−0.170053
$682$	−29.3050	−1.12214
$683$	15.9098	0.608773	0.304386	−	0.952549i	$-0.401549\pi$
$683$	15.9098	0.608773	0.304386	+	0.952549i	$0.401549\pi$
$684$	1.14590	0.0438145
$685$	1.70820	0.0652671
$686$	18.7082	0.714283
$687$	−2.29180	−0.0874375
$688$	−3.09017	−0.117812
$689$	45.3050	1.72598
$690$	5.70820	0.217308
$691$	−28.0344	−1.06648	−0.533240	−	0.845964i	$-0.679026\pi$
$691$	−28.0344	−1.06648	−0.533240	+	0.845964i	$0.679026\pi$
$692$	13.0344	0.495495
$693$	10.4721	0.397804
$694$	17.8885	0.679040
$695$	−14.8328	−0.562641
$696$	−6.00000	−0.227429
$697$	3.11146	0.117855
$698$	−16.5066	−0.624783
$699$	−14.0000	−0.529529
$700$	−9.09017	−0.343576
$701$	−37.2705	−1.40769	−0.703844	−	0.710355i	$-0.748535\pi$
$701$	−37.2705	−1.40769	−0.703844	+	0.710355i	$0.748535\pi$
$702$	−4.00000	−0.150970
$703$	4.14590	0.156366
$704$	4.00000	0.150756
$705$	−12.3607	−0.465530
$706$	13.2361	0.498146
$707$	10.9443	0.411602
$708$	−4.14590	−0.155812
$709$	12.1803	0.457442	0.228721	−	0.973492i	$-0.426546\pi$
$709$	12.1803	0.457442	0.228721	+	0.973492i	$0.426546\pi$
$710$	−7.23607	−0.271565
$711$	8.94427	0.335436
$712$	−11.7082	−0.438783
$713$	33.8328	1.26705
$714$	−7.47214	−0.279638
$715$	−19.7771	−0.739621
$716$	−4.09017	−0.152857
$717$	−22.3607	−0.835075
$718$	−19.8885	−0.742234
$719$	−12.3607	−0.460976	−0.230488	−	0.973075i	$-0.574032\pi$
$719$	−12.3607	−0.460976	−0.230488	+	0.973075i	$0.574032\pi$
$720$	1.23607	0.0460655
$721$	31.7984	1.18423
$722$	17.6869	0.658239
$723$	27.0902	1.00749
$724$	21.2361	0.789232
$725$	20.8328	0.773711
$726$	5.00000	0.185567
$727$	29.5279	1.09513	0.547564	−	0.836764i	$-0.315555\pi$
$727$	29.5279	1.09513	0.547564	+	0.836764i	$0.315555\pi$
$728$	10.4721	0.388123
$729$	1.00000	0.0370370
$730$	−13.0557	−0.483214
$731$	8.81966	0.326207
$732$	11.6180	0.429415
$733$	−44.2148	−1.63311	−0.816555	−	0.577267i	$-0.804119\pi$
$733$	−44.2148	−1.63311	−0.816555	+	0.577267i	$0.804119\pi$
$734$	−0.111456	−0.00411392
$735$	0.180340	0.00665194
$736$	−4.61803	−0.170223
$737$	43.7771	1.61255
$738$	1.09017	0.0401297
$739$	19.4377	0.715027	0.357514	−	0.933908i	$-0.383625\pi$
$739$	19.4377	0.715027	0.357514	+	0.933908i	$0.383625\pi$
$740$	4.47214	0.164399
$741$	4.58359	0.168382
$742$	29.6525	1.08858
$743$	−36.3607	−1.33394	−0.666972	−	0.745083i	$-0.732410\pi$
$743$	−36.3607	−1.33394	−0.666972	+	0.745083i	$0.732410\pi$
$744$	7.32624	0.268593
$745$	6.69505	0.245288
$746$	−21.8885	−0.801397
$747$	9.70820	0.355205
$748$	−11.4164	−0.417425
$749$	17.3262	0.633087
$750$	−10.4721	−0.382388
$751$	6.47214	0.236172	0.118086	−	0.993003i	$-0.462324\pi$
$751$	6.47214	0.236172	0.118086	+	0.993003i	$0.462324\pi$
$752$	10.0000	0.364662
$753$	−10.1803	−0.370992
$754$	−24.0000	−0.874028
$755$	4.94427	0.179940
$756$	−2.61803	−0.0952170
$757$	14.8328	0.539108	0.269554	−	0.962985i	$-0.413124\pi$
$757$	14.8328	0.539108	0.269554	+	0.962985i	$0.413124\pi$
$758$	−27.0344	−0.981935
$759$	−18.4721	−0.670496
$760$	−1.41641	−0.0513785
$761$	50.0902	1.81577	0.907884	−	0.419222i	$-0.137697\pi$
$761$	50.0902	1.81577	0.907884	+	0.419222i	$0.137697\pi$
$762$	−6.94427	−0.251564
$763$	−30.4164	−1.10115
$764$	−23.5967	−0.853700
$765$	−3.52786	−0.127550
$766$	−33.8885	−1.22444
$767$	−16.5836	−0.598799
$768$	−1.00000	−0.0360844
$769$	29.2361	1.05428	0.527140	−	0.849779i	$-0.323264\pi$
$769$	29.2361	1.05428	0.527140	+	0.849779i	$0.323264\pi$
$770$	−12.9443	−0.466479
$771$	−29.7082	−1.06991
$772$	−16.6180	−0.598096
$773$	−28.3262	−1.01882	−0.509412	−	0.860523i	$-0.670137\pi$
$773$	−28.3262	−1.01882	−0.509412	+	0.860523i	$0.670137\pi$
$774$	3.09017	0.111074
$775$	−25.4377	−0.913749
$776$	−8.85410	−0.317844
$777$	−9.47214	−0.339811
$778$	−7.81966	−0.280348
$779$	−1.24922	−0.0447581
$780$	4.94427	0.177033
$781$	23.4164	0.837905
$782$	13.1803	0.471328
$783$	6.00000	0.214423
$784$	−0.145898	−0.00521064
$785$	−16.3607	−0.583938
$786$	−16.7984	−0.599178
$787$	32.6525	1.16394	0.581968	−	0.813212i	$-0.302283\pi$
$787$	32.6525	1.16394	0.581968	+	0.813212i	$0.302283\pi$
$788$	22.5066	0.801764
$789$	−30.4721	−1.08484
$790$	−11.0557	−0.393345
$791$	47.5967	1.69235
$792$	−4.00000	−0.142134
$793$	46.4721	1.65027
$794$	−4.00000	−0.141955
$795$	14.0000	0.496529
$796$	−13.4164	−0.475532
$797$	23.3050	0.825504	0.412752	−	0.910844i	$-0.364568\pi$
$797$	23.3050	0.825504	0.412752	+	0.910844i	$0.364568\pi$
$798$	3.00000	0.106199
$799$	−28.5410	−1.00971
$800$	3.47214	0.122759
$801$	11.7082	0.413689
$802$	−23.2148	−0.819742
$803$	42.2492	1.49094
$804$	−10.9443	−0.385975
$805$	14.9443	0.526716
$806$	29.3050	1.03222
$807$	−0.583592	−0.0205434
$808$	−4.18034	−0.147064
$809$	14.6525	0.515154	0.257577	−	0.966258i	$-0.417076\pi$
$809$	14.6525	0.515154	0.257577	+	0.966258i	$0.417076\pi$
$810$	−1.23607	−0.0434310
$811$	31.4164	1.10318	0.551590	−	0.834116i	$-0.314021\pi$
$811$	31.4164	1.10318	0.551590	+	0.834116i	$0.314021\pi$
$812$	−15.7082	−0.551250
$813$	18.9443	0.664405
$814$	−14.4721	−0.507248
$815$	−16.0000	−0.560456
$816$	2.85410	0.0999136
$817$	−3.54102	−0.123885
$818$	31.3050	1.09455
$819$	−10.4721	−0.365926
$820$	−1.34752	−0.0470576
$821$	−8.06888	−0.281606	−0.140803	−	0.990038i	$-0.544968\pi$
$821$	−8.06888	−0.281606	−0.140803	+	0.990038i	$0.544968\pi$
$822$	1.38197	0.0482016
$823$	−14.7639	−0.514638	−0.257319	−	0.966326i	$-0.582839\pi$
$823$	−14.7639	−0.514638	−0.257319	+	0.966326i	$0.582839\pi$
$824$	−12.1459	−0.423122
$825$	13.8885	0.483537
$826$	−10.8541	−0.377663
$827$	−44.7984	−1.55779	−0.778896	−	0.627153i	$-0.784220\pi$
$827$	−44.7984	−1.55779	−0.778896	+	0.627153i	$0.784220\pi$
$828$	4.61803	0.160488
$829$	13.4508	0.467167	0.233584	−	0.972337i	$-0.424955\pi$
$829$	13.4508	0.467167	0.233584	+	0.972337i	$0.424955\pi$
$830$	−12.0000	−0.416526
$831$	−2.85410	−0.0990077
$832$	−4.00000	−0.138675
$833$	0.416408	0.0144277
$834$	−12.0000	−0.415526
$835$	10.3607	0.358546
$836$	4.58359	0.158527
$837$	−7.32624	−0.253232
$838$	−13.2361	−0.457232
$839$	19.8197	0.684251	0.342125	−	0.939654i	$-0.388853\pi$
$839$	19.8197	0.684251	0.342125	+	0.939654i	$0.388853\pi$
$840$	3.23607	0.111655
$841$	7.00000	0.241379
$842$	15.5623	0.536312
$843$	0.763932	0.0263112
$844$	−4.61803	−0.158959
$845$	3.70820	0.127566
$846$	−10.0000	−0.343807
$847$	13.0902	0.449784
$848$	−11.3262	−0.388945
$849$	9.85410	0.338192
$850$	−9.90983	−0.339904
$851$	16.7082	0.572750
$852$	−5.85410	−0.200558
$853$	−51.2361	−1.75429	−0.877145	−	0.480226i	$-0.840555\pi$
$853$	−51.2361	−1.75429	−0.877145	+	0.480226i	$0.840555\pi$
$854$	30.4164	1.04083
$855$	1.41641	0.0484401
$856$	−6.61803	−0.226200
$857$	−29.7771	−1.01717	−0.508583	−	0.861013i	$-0.669830\pi$
$857$	−29.7771	−1.01717	−0.508583	+	0.861013i	$0.669830\pi$
$858$	−16.0000	−0.546231
$859$	30.8541	1.05273	0.526364	−	0.850259i	$-0.323555\pi$
$859$	30.8541	1.05273	0.526364	+	0.850259i	$0.323555\pi$
$860$	−3.81966	−0.130249
$861$	2.85410	0.0972675
$862$	3.38197	0.115190
$863$	51.4164	1.75023	0.875117	−	0.483911i	$-0.160784\pi$
$863$	51.4164	1.75023	0.875117	+	0.483911i	$0.160784\pi$
$864$	1.00000	0.0340207
$865$	16.1115	0.547806
$866$	−1.20163	−0.0408329
$867$	8.85410	0.300701
$868$	19.1803	0.651023
$869$	35.7771	1.21365
$870$	−7.41641	−0.251440
$871$	−43.7771	−1.48333
$872$	11.6180	0.393436
$873$	8.85410	0.299666
$874$	−5.29180	−0.178998
$875$	−27.4164	−0.926844
$876$	−10.5623	−0.356867
$877$	−53.7771	−1.81592	−0.907962	−	0.419053i	$-0.862362\pi$
$877$	−53.7771	−1.81592	−0.907962	+	0.419053i	$0.862362\pi$
$878$	7.41641	0.250292
$879$	32.0689	1.08166
$880$	4.94427	0.166671
$881$	−11.6180	−0.391422	−0.195711	−	0.980662i	$-0.562701\pi$
$881$	−11.6180	−0.391422	−0.195711	+	0.980662i	$0.562701\pi$
$882$	0.145898	0.00491264
$883$	−33.3262	−1.12152	−0.560759	−	0.827979i	$-0.689490\pi$
$883$	−33.3262	−1.12152	−0.560759	+	0.827979i	$0.689490\pi$
$884$	11.4164	0.383975
$885$	−5.12461	−0.172262
$886$	−34.7426	−1.16720
$887$	−19.3820	−0.650783	−0.325391	−	0.945579i	$-0.605496\pi$
$887$	−19.3820	−0.650783	−0.325391	+	0.945579i	$0.605496\pi$
$888$	3.61803	0.121413
$889$	−18.1803	−0.609749
$890$	−14.4721	−0.485107
$891$	4.00000	0.134005
$892$	2.29180	0.0767350
$893$	11.4590	0.383460
$894$	5.41641	0.181152
$895$	−5.05573	−0.168994
$896$	−2.61803	−0.0874624
$897$	18.4721	0.616767
$898$	−24.3607	−0.812926
$899$	−43.9574	−1.46606
$900$	−3.47214	−0.115738
$901$	32.3262	1.07694
$902$	4.36068	0.145195
$903$	8.09017	0.269224
$904$	−18.1803	−0.604669
$905$	26.2492	0.872554
$906$	4.00000	0.132891
$907$	−12.5836	−0.417831	−0.208916	−	0.977934i	$-0.566993\pi$
$907$	−12.5836	−0.417831	−0.208916	+	0.977934i	$0.566993\pi$
$908$	4.43769	0.147270
$909$	4.18034	0.138653
$910$	12.9443	0.429098
$911$	46.4296	1.53828	0.769140	−	0.639080i	$-0.220685\pi$
$911$	46.4296	1.53828	0.769140	+	0.639080i	$0.220685\pi$
$912$	−1.14590	−0.0379445
$913$	38.8328	1.28518
$914$	17.4164	0.576084
$915$	14.3607	0.474749
$916$	2.29180	0.0757231
$917$	−43.9787	−1.45231
$918$	−2.85410	−0.0941994
$919$	−24.6525	−0.813210	−0.406605	−	0.913604i	$-0.633287\pi$
$919$	−24.6525	−0.813210	−0.406605	+	0.913604i	$0.633287\pi$
$920$	−5.70820	−0.188194
$921$	−22.0000	−0.724925
$922$	28.6869	0.944753
$923$	−23.4164	−0.770760
$924$	−10.4721	−0.344508
$925$	−12.5623	−0.413046
$926$	10.3262	0.339341
$927$	12.1459	0.398924
$928$	6.00000	0.196960
$929$	−26.6525	−0.874439	−0.437220	−	0.899355i	$-0.644037\pi$
$929$	−26.6525	−0.874439	−0.437220	+	0.899355i	$0.644037\pi$
$930$	9.05573	0.296949
$931$	−0.167184	−0.00547924
$932$	14.0000	0.458585
$933$	9.14590	0.299423
$934$	−8.72949	−0.285638
$935$	−14.1115	−0.461494
$936$	4.00000	0.130744
$937$	37.0344	1.20986	0.604931	−	0.796278i	$-0.293201\pi$
$937$	37.0344	1.20986	0.604931	+	0.796278i	$0.293201\pi$
$938$	−28.6525	−0.935536
$939$	−1.23607	−0.0403376
$940$	12.3607	0.403161
$941$	27.2361	0.887870	0.443935	−	0.896059i	$-0.353582\pi$
$941$	27.2361	0.887870	0.443935	+	0.896059i	$0.353582\pi$
$942$	−13.2361	−0.431254
$943$	−5.03444	−0.163944
$944$	4.14590	0.134937
$945$	−3.23607	−0.105269
$946$	12.3607	0.401880
$947$	−22.1115	−0.718526	−0.359263	−	0.933236i	$-0.616972\pi$
$947$	−22.1115	−0.718526	−0.359263	+	0.933236i	$0.616972\pi$
$948$	−8.94427	−0.290496
$949$	−42.2492	−1.37147
$950$	3.97871	0.129087
$951$	10.9443	0.354892
$952$	7.47214	0.242173
$953$	−12.8328	−0.415696	−0.207848	−	0.978161i	$-0.566646\pi$
$953$	−12.8328	−0.415696	−0.207848	+	0.978161i	$0.566646\pi$
$954$	11.3262	0.366700
$955$	−29.1672	−0.943828
$956$	22.3607	0.723196
$957$	24.0000	0.775810
$958$	19.7984	0.639656
$959$	3.61803	0.116832
$960$	−1.23607	−0.0398939
$961$	22.6738	0.731412
$962$	14.4721	0.466600
$963$	6.61803	0.213263
$964$	−27.0902	−0.872516
$965$	−20.5410	−0.661239
$966$	12.0902	0.388995
$967$	−17.6180	−0.566558	−0.283279	−	0.959038i	$-0.591422\pi$
$967$	−17.6180	−0.566558	−0.283279	+	0.959038i	$0.591422\pi$
$968$	−5.00000	−0.160706
$969$	3.27051	0.105064
$970$	−10.9443	−0.351399
$971$	49.8885	1.60100	0.800500	−	0.599333i	$-0.204567\pi$
$971$	49.8885	1.60100	0.800500	+	0.599333i	$0.204567\pi$
$972$	−1.00000	−0.0320750
$973$	−31.4164	−1.00716
$974$	−25.2361	−0.808616
$975$	−13.8885	−0.444789
$976$	−11.6180	−0.371884
$977$	−40.5410	−1.29702	−0.648511	−	0.761205i	$-0.724608\pi$
$977$	−40.5410	−1.29702	−0.648511	+	0.761205i	$0.724608\pi$
$978$	−12.9443	−0.413912
$979$	46.8328	1.49678
$980$	−0.180340	−0.00576075
$981$	−11.6180	−0.370935
$982$	−2.94427	−0.0939555
$983$	−15.2361	−0.485955	−0.242978	−	0.970032i	$-0.578124\pi$
$983$	−15.2361	−0.485955	−0.242978	+	0.970032i	$0.578124\pi$
$984$	−1.09017	−0.0347533
$985$	27.8197	0.886408
$986$	−17.1246	−0.545359
$987$	−26.1803	−0.833329
$988$	−4.58359	−0.145823
$989$	−14.2705	−0.453776
$990$	−4.94427	−0.157139
$991$	−16.5836	−0.526795	−0.263398	−	0.964687i	$-0.584843\pi$
$991$	−16.5836	−0.526795	−0.263398	+	0.964687i	$0.584843\pi$
$992$	−7.32624	−0.232608
$993$	−24.3262	−0.771970
$994$	−15.3262	−0.486119
$995$	−16.5836	−0.525735
$996$	−9.70820	−0.307616
$997$	29.5279	0.935157	0.467578	−	0.883952i	$-0.345127\pi$
$997$	29.5279	0.935157	0.467578	+	0.883952i	$0.345127\pi$
$998$	−33.7771	−1.06920
$999$	−3.61803	−0.114470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6078.2.a.c.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6078.2.a.c.1.2	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6078 = 2 \cdot 3 \cdot 1013 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6078.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +2.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +2.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.23607 q^{10} +4.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -2.61803 q^{14} -1.23607 q^{15} +1.00000 q^{16} -2.85410 q^{17} -1.00000 q^{18} +1.14590 q^{19} +1.23607 q^{20} -2.61803 q^{21} -4.00000 q^{22} +4.61803 q^{23} +1.00000 q^{24} -3.47214 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.61803 q^{28} -6.00000 q^{29} +1.23607 q^{30} +7.32624 q^{31} -1.00000 q^{32} -4.00000 q^{33} +2.85410 q^{34} +3.23607 q^{35} +1.00000 q^{36} +3.61803 q^{37} -1.14590 q^{38} +4.00000 q^{39} -1.23607 q^{40} -1.09017 q^{41} +2.61803 q^{42} -3.09017 q^{43} +4.00000 q^{44} +1.23607 q^{45} -4.61803 q^{46} +10.0000 q^{47} -1.00000 q^{48} -0.145898 q^{49} +3.47214 q^{50} +2.85410 q^{51} -4.00000 q^{52} -11.3262 q^{53} +1.00000 q^{54} +4.94427 q^{55} -2.61803 q^{56} -1.14590 q^{57} +6.00000 q^{58} +4.14590 q^{59} -1.23607 q^{60} -11.6180 q^{61} -7.32624 q^{62} +2.61803 q^{63} +1.00000 q^{64} -4.94427 q^{65} +4.00000 q^{66} +10.9443 q^{67} -2.85410 q^{68} -4.61803 q^{69} -3.23607 q^{70} +5.85410 q^{71} -1.00000 q^{72} +10.5623 q^{73} -3.61803 q^{74} +3.47214 q^{75} +1.14590 q^{76} +10.4721 q^{77} -4.00000 q^{78} +8.94427 q^{79} +1.23607 q^{80} +1.00000 q^{81} +1.09017 q^{82} +9.70820 q^{83} -2.61803 q^{84} -3.52786 q^{85} +3.09017 q^{86} +6.00000 q^{87} -4.00000 q^{88} +11.7082 q^{89} -1.23607 q^{90} -10.4721 q^{91} +4.61803 q^{92} -7.32624 q^{93} -10.0000 q^{94} +1.41641 q^{95} +1.00000 q^{96} +8.85410 q^{97} +0.145898 q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 8 q^{11} - 2 q^{12} - 8 q^{13} - 3 q^{14} + 2 q^{15} + 2 q^{16} + q^{17} - 2 q^{18} + 9 q^{19} - 2 q^{20} - 3 q^{21} - 8 q^{22} + 7 q^{23} + 2 q^{24} + 2 q^{25} + 8 q^{26} - 2 q^{27} + 3 q^{28} - 12 q^{29} - 2 q^{30} - q^{31} - 2 q^{32} - 8 q^{33} - q^{34} + 2 q^{35} + 2 q^{36} + 5 q^{37} - 9 q^{38} + 8 q^{39} + 2 q^{40} + 9 q^{41} + 3 q^{42} + 5 q^{43} + 8 q^{44} - 2 q^{45} - 7 q^{46} + 20 q^{47} - 2 q^{48} - 7 q^{49} - 2 q^{50} - q^{51} - 8 q^{52} - 7 q^{53} + 2 q^{54} - 8 q^{55} - 3 q^{56} - 9 q^{57} + 12 q^{58} + 15 q^{59} + 2 q^{60} - 21 q^{61} + q^{62} + 3 q^{63} + 2 q^{64} + 8 q^{65} + 8 q^{66} + 4 q^{67} + q^{68} - 7 q^{69} - 2 q^{70} + 5 q^{71} - 2 q^{72} + q^{73} - 5 q^{74} - 2 q^{75} + 9 q^{76} + 12 q^{77} - 8 q^{78} - 2 q^{80} + 2 q^{81} - 9 q^{82} + 6 q^{83} - 3 q^{84} - 16 q^{85} - 5 q^{86} + 12 q^{87} - 8 q^{88} + 10 q^{89} + 2 q^{90} - 12 q^{91} + 7 q^{92} + q^{93} - 20 q^{94} - 24 q^{95} + 2 q^{96} + 11 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 8 * q^11 - 2 * q^12 - 8 * q^13 - 3 * q^14 + 2 * q^15 + 2 * q^16 + q^17 - 2 * q^18 + 9 * q^19 - 2 * q^20 - 3 * q^21 - 8 * q^22 + 7 * q^23 + 2 * q^24 + 2 * q^25 + 8 * q^26 - 2 * q^27 + 3 * q^28 - 12 * q^29 - 2 * q^30 - q^31 - 2 * q^32 - 8 * q^33 - q^34 + 2 * q^35 + 2 * q^36 + 5 * q^37 - 9 * q^38 + 8 * q^39 + 2 * q^40 + 9 * q^41 + 3 * q^42 + 5 * q^43 + 8 * q^44 - 2 * q^45 - 7 * q^46 + 20 * q^47 - 2 * q^48 - 7 * q^49 - 2 * q^50 - q^51 - 8 * q^52 - 7 * q^53 + 2 * q^54 - 8 * q^55 - 3 * q^56 - 9 * q^57 + 12 * q^58 + 15 * q^59 + 2 * q^60 - 21 * q^61 + q^62 + 3 * q^63 + 2 * q^64 + 8 * q^65 + 8 * q^66 + 4 * q^67 + q^68 - 7 * q^69 - 2 * q^70 + 5 * q^71 - 2 * q^72 + q^73 - 5 * q^74 - 2 * q^75 + 9 * q^76 + 12 * q^77 - 8 * q^78 - 2 * q^80 + 2 * q^81 - 9 * q^82 + 6 * q^83 - 3 * q^84 - 16 * q^85 - 5 * q^86 + 12 * q^87 - 8 * q^88 + 10 * q^89 + 2 * q^90 - 12 * q^91 + 7 * q^92 + q^93 - 20 * q^94 - 24 * q^95 + 2 * q^96 + 11 * q^97 + 7 * q^98 + 8 * q^99

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