LMFDB - Embedded newform 1045.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7281497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 $ x^6 - 2x^5 - 5x^4 + 7x^3 + 6x^2 - 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			1.5
Root			$2.59744$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.21244 q^{2} +1.77266 q^{3} -0.529980 q^{4} -1.00000 q^{5} +2.14925 q^{6} -3.37010 q^{7} -3.06746 q^{8} +0.142317 q^{9} +O(q^{10})$	\(q+1.21244 q^{2} +1.77266 q^{3} -0.529980 q^{4} -1.00000 q^{5} +2.14925 q^{6} -3.37010 q^{7} -3.06746 q^{8} +0.142317 q^{9} -1.21244 q^{10} -1.00000 q^{11} -0.939473 q^{12} -0.439786 q^{13} -4.08605 q^{14} -1.77266 q^{15} -2.65916 q^{16} -3.08872 q^{17} +0.172551 q^{18} -1.00000 q^{19} +0.529980 q^{20} -5.97403 q^{21} -1.21244 q^{22} +1.45571 q^{23} -5.43756 q^{24} +1.00000 q^{25} -0.533216 q^{26} -5.06570 q^{27} +1.78608 q^{28} +5.85137 q^{29} -2.14925 q^{30} -6.80721 q^{31} +2.91083 q^{32} -1.77266 q^{33} -3.74490 q^{34} +3.37010 q^{35} -0.0754249 q^{36} -2.73402 q^{37} -1.21244 q^{38} -0.779590 q^{39} +3.06746 q^{40} -3.67980 q^{41} -7.24317 q^{42} +4.59330 q^{43} +0.529980 q^{44} -0.142317 q^{45} +1.76497 q^{46} +0.210971 q^{47} -4.71378 q^{48} +4.35755 q^{49} +1.21244 q^{50} -5.47525 q^{51} +0.233078 q^{52} +5.17998 q^{53} -6.14187 q^{54} +1.00000 q^{55} +10.3376 q^{56} -1.77266 q^{57} +7.09446 q^{58} -10.8732 q^{59} +0.939473 q^{60} -5.66286 q^{61} -8.25336 q^{62} -0.479621 q^{63} +8.84754 q^{64} +0.439786 q^{65} -2.14925 q^{66} +6.14764 q^{67} +1.63696 q^{68} +2.58048 q^{69} +4.08605 q^{70} -7.15175 q^{71} -0.436550 q^{72} +4.15351 q^{73} -3.31484 q^{74} +1.77266 q^{75} +0.529980 q^{76} +3.37010 q^{77} -0.945209 q^{78} +15.4881 q^{79} +2.65916 q^{80} -9.40670 q^{81} -4.46155 q^{82} +3.35537 q^{83} +3.16612 q^{84} +3.08872 q^{85} +5.56911 q^{86} +10.3725 q^{87} +3.06746 q^{88} -6.26557 q^{89} -0.172551 q^{90} +1.48212 q^{91} -0.771498 q^{92} -12.0669 q^{93} +0.255791 q^{94} +1.00000 q^{95} +5.15991 q^{96} -6.70633 q^{97} +5.28329 q^{98} -0.142317 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9}+O(q^{10}) $ 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9	\( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21} + 2 q^{22} + 4 q^{23} - q^{24} + 6 q^{25} - 14 q^{26} - 16 q^{27} + 10 q^{28} - 9 q^{29} - 21 q^{31} - q^{32} + q^{33} - 5 q^{35} - 28 q^{36} - 3 q^{37} + 2 q^{38} + 20 q^{39} + 12 q^{40} - 23 q^{41} + q^{42} + 7 q^{43} - 4 q^{44} - q^{45} - 12 q^{46} - 18 q^{47} - 3 q^{49} - 2 q^{50} - 16 q^{51} + 13 q^{52} - 17 q^{53} + q^{54} + 6 q^{55} - 2 q^{56} + q^{57} + 23 q^{58} - 29 q^{59} - q^{60} + 17 q^{61} + 2 q^{62} + 6 q^{63} - 18 q^{64} + 5 q^{65} + 8 q^{67} - q^{68} - 38 q^{69} + 8 q^{70} - 12 q^{71} + 13 q^{72} + 2 q^{73} - 37 q^{74} - q^{75} - 4 q^{76} - 5 q^{77} + q^{78} + 3 q^{79} - 4 q^{80} - 2 q^{81} + 24 q^{82} - 11 q^{83} - 3 q^{84} - q^{85} - 12 q^{86} - 12 q^{87} + 12 q^{88} - 22 q^{89} - 6 q^{90} - 18 q^{91} - 15 q^{92} + 18 q^{93} + 22 q^{94} + 6 q^{95} - 17 q^{96} - 2 q^{97} - q^{98} - q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 + 2 * q^10 - 6 * q^11 + q^12 - 5 * q^13 - 8 * q^14 + q^15 + 4 * q^16 + q^17 + 6 * q^18 - 6 * q^19 - 4 * q^20 - 21 * q^21 + 2 * q^22 + 4 * q^23 - q^24 + 6 * q^25 - 14 * q^26 - 16 * q^27 + 10 * q^28 - 9 * q^29 - 21 * q^31 - q^32 + q^33 - 5 * q^35 - 28 * q^36 - 3 * q^37 + 2 * q^38 + 20 * q^39 + 12 * q^40 - 23 * q^41 + q^42 + 7 * q^43 - 4 * q^44 - q^45 - 12 * q^46 - 18 * q^47 - 3 * q^49 - 2 * q^50 - 16 * q^51 + 13 * q^52 - 17 * q^53 + q^54 + 6 * q^55 - 2 * q^56 + q^57 + 23 * q^58 - 29 * q^59 - q^60 + 17 * q^61 + 2 * q^62 + 6 * q^63 - 18 * q^64 + 5 * q^65 + 8 * q^67 - q^68 - 38 * q^69 + 8 * q^70 - 12 * q^71 + 13 * q^72 + 2 * q^73 - 37 * q^74 - q^75 - 4 * q^76 - 5 * q^77 + q^78 + 3 * q^79 - 4 * q^80 - 2 * q^81 + 24 * q^82 - 11 * q^83 - 3 * q^84 - q^85 - 12 * q^86 - 12 * q^87 + 12 * q^88 - 22 * q^89 - 6 * q^90 - 18 * q^91 - 15 * q^92 + 18 * q^93 + 22 * q^94 + 6 * q^95 - 17 * q^96 - 2 * q^97 - q^98 - 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 $

$2$	1.21244	0.857327	0.428664	−	0.903464i	$-0.358985\pi$
$2$	1.21244	0.857327	0.428664	+	0.903464i	$0.358985\pi$
$3$	1.77266	1.02344	0.511722	−	0.859151i	$-0.329008\pi$
$3$	1.77266	1.02344	0.511722	+	0.859151i	$0.329008\pi$
$4$	−0.529980	−0.264990
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.14925	0.877427
$7$	−3.37010	−1.27378	−0.636888	−	0.770956i	$-0.719779\pi$
$7$	−3.37010	−1.27378	−0.636888	+	0.770956i	$0.719779\pi$
$8$	−3.06746	−1.08451
$9$	0.142317	0.0474389
$10$	−1.21244	−0.383408
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.939473	−0.271203
$13$	−0.439786	−0.121975	−0.0609873	−	0.998139i	$-0.519425\pi$
$13$	−0.439786	−0.121975	−0.0609873	+	0.998139i	$0.519425\pi$
$14$	−4.08605	−1.09204
$15$	−1.77266	−0.457698
$16$	−2.65916	−0.664790
$17$	−3.08872	−0.749125	−0.374563	−	0.927202i	$-0.622207\pi$
$17$	−3.08872	−0.749125	−0.374563	+	0.927202i	$0.622207\pi$
$18$	0.172551	0.0406706
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.529980	0.118507
$21$	−5.97403	−1.30364
$22$	−1.21244	−0.258494
$23$	1.45571	0.303537	0.151768	−	0.988416i	$-0.451503\pi$
$23$	1.45571	0.303537	0.151768	+	0.988416i	$0.451503\pi$
$24$	−5.43756	−1.10994
$25$	1.00000	0.200000
$26$	−0.533216	−0.104572
$27$	−5.06570	−0.974894
$28$	1.78608	0.337538
$29$	5.85137	1.08657	0.543286	−	0.839548i	$-0.317180\pi$
$29$	5.85137	1.08657	0.543286	+	0.839548i	$0.317180\pi$
$30$	−2.14925	−0.392397
$31$	−6.80721	−1.22261	−0.611306	−	0.791394i	$-0.709355\pi$
$31$	−6.80721	−1.22261	−0.611306	+	0.791394i	$0.709355\pi$
$32$	2.91083	0.514567
$33$	−1.77266	−0.308580
$34$	−3.74490	−0.642245
$35$	3.37010	0.569650
$36$	−0.0754249	−0.0125708
$37$	−2.73402	−0.449470	−0.224735	−	0.974420i	$-0.572152\pi$
$37$	−2.73402	−0.449470	−0.224735	+	0.974420i	$0.572152\pi$
$38$	−1.21244	−0.196684
$39$	−0.779590	−0.124834
$40$	3.06746	0.485008
$41$	−3.67980	−0.574687	−0.287344	−	0.957828i	$-0.592772\pi$
$41$	−3.67980	−0.574687	−0.287344	+	0.957828i	$0.592772\pi$
$42$	−7.24317	−1.11765
$43$	4.59330	0.700471	0.350236	−	0.936662i	$-0.386102\pi$
$43$	4.59330	0.700471	0.350236	+	0.936662i	$0.386102\pi$
$44$	0.529980	0.0798975
$45$	−0.142317	−0.0212153
$46$	1.76497	0.260230
$47$	0.210971	0.0307733	0.0153867	−	0.999882i	$-0.495102\pi$
$47$	0.210971	0.0307733	0.0153867	+	0.999882i	$0.495102\pi$
$48$	−4.71378	−0.680376
$49$	4.35755	0.622507
$50$	1.21244	0.171465
$51$	−5.47525	−0.766688
$52$	0.233078	0.0323221
$53$	5.17998	0.711525	0.355762	−	0.934576i	$-0.384221\pi$
$53$	5.17998	0.711525	0.355762	+	0.934576i	$0.384221\pi$
$54$	−6.14187	−0.835803
$55$	1.00000	0.134840
$56$	10.3376	1.38142
$57$	−1.77266	−0.234794
$58$	7.09446	0.931548
$59$	−10.8732	−1.41557	−0.707785	−	0.706428i	$-0.750305\pi$
$59$	−10.8732	−1.41557	−0.707785	+	0.706428i	$0.750305\pi$
$60$	0.939473	0.121285
$61$	−5.66286	−0.725055	−0.362527	−	0.931973i	$-0.618086\pi$
$61$	−5.66286	−0.725055	−0.362527	+	0.931973i	$0.618086\pi$
$62$	−8.25336	−1.04818
$63$	−0.479621	−0.0604265
$64$	8.84754	1.10594
$65$	0.439786	0.0545487
$66$	−2.14925	−0.264554
$67$	6.14764	0.751054	0.375527	−	0.926811i	$-0.377462\pi$
$67$	6.14764	0.751054	0.375527	+	0.926811i	$0.377462\pi$
$68$	1.63696	0.198511
$69$	2.58048	0.310653
$70$	4.08605	0.488377
$71$	−7.15175	−0.848756	−0.424378	−	0.905485i	$-0.639507\pi$
$71$	−7.15175	−0.848756	−0.424378	+	0.905485i	$0.639507\pi$
$72$	−0.436550	−0.0514479
$73$	4.15351	0.486132	0.243066	−	0.970010i	$-0.421847\pi$
$73$	4.15351	0.486132	0.243066	+	0.970010i	$0.421847\pi$
$74$	−3.31484	−0.385343
$75$	1.77266	0.204689
$76$	0.529980	0.0607929
$77$	3.37010	0.384058
$78$	−0.945209	−0.107024
$79$	15.4881	1.74254	0.871272	−	0.490800i	$-0.163295\pi$
$79$	15.4881	1.74254	0.871272	+	0.490800i	$0.163295\pi$
$80$	2.65916	0.297303
$81$	−9.40670	−1.04519
$82$	−4.46155	−0.492695
$83$	3.35537	0.368299	0.184150	−	0.982898i	$-0.441047\pi$
$83$	3.35537	0.368299	0.184150	+	0.982898i	$0.441047\pi$
$84$	3.16612	0.345452
$85$	3.08872	0.335019
$86$	5.56911	0.600533
$87$	10.3725	1.11205
$88$	3.06746	0.326992
$89$	−6.26557	−0.664149	−0.332075	−	0.943253i	$-0.607749\pi$
$89$	−6.26557	−0.664149	−0.332075	+	0.943253i	$0.607749\pi$
$90$	−0.172551	−0.0181885
$91$	1.48212	0.155368
$92$	−0.771498	−0.0804342
$93$	−12.0669	−1.25128
$94$	0.255791	0.0263828
$95$	1.00000	0.102598
$96$	5.15991	0.526631
$97$	−6.70633	−0.680924	−0.340462	−	0.940258i	$-0.610584\pi$
$97$	−6.70633	−0.680924	−0.340462	+	0.940258i	$0.610584\pi$
$98$	5.28329	0.533692
$99$	−0.142317	−0.0143034
$100$	−0.529980	−0.0529980
$101$	19.2123	1.91169	0.955846	−	0.293868i	$-0.0949426\pi$
$101$	19.2123	1.91169	0.955846	+	0.293868i	$0.0949426\pi$
$102$	−6.63843	−0.657302
$103$	−15.0086	−1.47884	−0.739418	−	0.673246i	$-0.764899\pi$
$103$	−15.0086	−1.47884	−0.739418	+	0.673246i	$0.764899\pi$
$104$	1.34902	0.132283
$105$	5.97403	0.583006
$106$	6.28043	0.610010
$107$	−7.63909	−0.738499	−0.369250	−	0.929330i	$-0.620385\pi$
$107$	−7.63909	−0.738499	−0.369250	+	0.929330i	$0.620385\pi$
$108$	2.68472	0.258337
$109$	0.0458556	0.00439217	0.00219609	−	0.999998i	$-0.499301\pi$
$109$	0.0458556	0.00439217	0.00219609	+	0.999998i	$0.499301\pi$
$110$	1.21244	0.115602
$111$	−4.84648	−0.460008
$112$	8.96163	0.846795
$113$	8.43432	0.793434	0.396717	−	0.917941i	$-0.370149\pi$
$113$	8.43432	0.793434	0.396717	+	0.917941i	$0.370149\pi$
$114$	−2.14925	−0.201296
$115$	−1.45571	−0.135746
$116$	−3.10111	−0.287931
$117$	−0.0625888	−0.00578634
$118$	−13.1831	−1.21361
$119$	10.4093	0.954218
$120$	5.43756	0.496379
$121$	1.00000	0.0909091
$122$	−6.86590	−0.621609
$123$	−6.52302	−0.588161
$124$	3.60769	0.323980
$125$	−1.00000	−0.0894427
$126$	−0.581513	−0.0518053
$127$	1.50239	0.133316	0.0666579	−	0.997776i	$-0.478766\pi$
$127$	1.50239	0.133316	0.0666579	+	0.997776i	$0.478766\pi$
$128$	4.90548	0.433588
$129$	8.14235	0.716894
$130$	0.533216	0.0467661
$131$	−0.608483	−0.0531634	−0.0265817	−	0.999647i	$-0.508462\pi$
$131$	−0.608483	−0.0531634	−0.0265817	+	0.999647i	$0.508462\pi$
$132$	0.939473	0.0817706
$133$	3.37010	0.292224
$134$	7.45367	0.643899
$135$	5.06570	0.435986
$136$	9.47453	0.812434
$137$	−21.4534	−1.83289	−0.916443	−	0.400166i	$-0.868953\pi$
$137$	−21.4534	−1.83289	−0.916443	+	0.400166i	$0.868953\pi$
$138$	3.12869	0.266331
$139$	−8.85639	−0.751189	−0.375595	−	0.926784i	$-0.622561\pi$
$139$	−8.85639	−0.751189	−0.375595	+	0.926784i	$0.622561\pi$
$140$	−1.78608	−0.150952
$141$	0.373980	0.0314948
$142$	−8.67109	−0.727662
$143$	0.439786	0.0367767
$144$	−0.378443	−0.0315369
$145$	−5.85137	−0.485930
$146$	5.03590	0.416774
$147$	7.72445	0.637102
$148$	1.44898	0.119105
$149$	−2.25215	−0.184503	−0.0922515	−	0.995736i	$-0.529406\pi$
$149$	−2.25215	−0.184503	−0.0922515	+	0.995736i	$0.529406\pi$
$150$	2.14925	0.175485
$151$	−17.3616	−1.41286	−0.706432	−	0.707781i	$-0.749696\pi$
$151$	−17.3616	−1.41286	−0.706432	+	0.707781i	$0.749696\pi$
$152$	3.06746	0.248804
$153$	−0.439576	−0.0355376
$154$	4.08605	0.329264
$155$	6.80721	0.546769
$156$	0.413167	0.0330798
$157$	−3.75886	−0.299990	−0.149995	−	0.988687i	$-0.547926\pi$
$157$	−3.75886	−0.299990	−0.149995	+	0.988687i	$0.547926\pi$
$158$	18.7784	1.49393
$159$	9.18233	0.728206
$160$	−2.91083	−0.230122
$161$	−4.90589	−0.386638
$162$	−11.4051	−0.896069
$163$	−11.8435	−0.927655	−0.463828	−	0.885925i	$-0.653524\pi$
$163$	−11.8435	−0.927655	−0.463828	+	0.885925i	$0.653524\pi$
$164$	1.95022	0.152286
$165$	1.77266	0.138001
$166$	4.06819	0.315753
$167$	6.33806	0.490454	0.245227	−	0.969466i	$-0.421137\pi$
$167$	6.33806	0.490454	0.245227	+	0.969466i	$0.421137\pi$
$168$	18.3251	1.41381
$169$	−12.8066	−0.985122
$170$	3.74490	0.287221
$171$	−0.142317	−0.0108832
$172$	−2.43436	−0.185618
$173$	−8.21796	−0.624800	−0.312400	−	0.949951i	$-0.601133\pi$
$173$	−8.21796	−0.624800	−0.312400	+	0.949951i	$0.601133\pi$
$174$	12.5760	0.953388
$175$	−3.37010	−0.254755
$176$	2.65916	0.200442
$177$	−19.2745	−1.44876
$178$	−7.59665	−0.569393
$179$	11.2448	0.840476	0.420238	−	0.907414i	$-0.361947\pi$
$179$	11.2448	0.840476	0.420238	+	0.907414i	$0.361947\pi$
$180$	0.0754249	0.00562184
$181$	8.27670	0.615202	0.307601	−	0.951515i	$-0.400474\pi$
$181$	8.27670	0.615202	0.307601	+	0.951515i	$0.400474\pi$
$182$	1.79699	0.133202
$183$	−10.0383	−0.742053
$184$	−4.46533	−0.329189
$185$	2.73402	0.201009
$186$	−14.6304	−1.07275
$187$	3.08872	0.225870
$188$	−0.111811	−0.00815462
$189$	17.0719	1.24180
$190$	1.21244	0.0879599
$191$	3.21777	0.232830	0.116415	−	0.993201i	$-0.462860\pi$
$191$	3.21777	0.232830	0.116415	+	0.993201i	$0.462860\pi$
$192$	15.6837	1.13187
$193$	19.0447	1.37087	0.685433	−	0.728136i	$-0.259613\pi$
$193$	19.0447	1.37087	0.685433	+	0.728136i	$0.259613\pi$
$194$	−8.13104	−0.583775
$195$	0.779590	0.0558276
$196$	−2.30941	−0.164958
$197$	2.04266	0.145534	0.0727668	−	0.997349i	$-0.476817\pi$
$197$	2.04266	0.145534	0.0727668	+	0.997349i	$0.476817\pi$
$198$	−0.172551	−0.0122627
$199$	4.61202	0.326937	0.163469	−	0.986549i	$-0.447732\pi$
$199$	4.61202	0.326937	0.163469	+	0.986549i	$0.447732\pi$
$200$	−3.06746	−0.216902
$201$	10.8977	0.768662
$202$	23.2938	1.63895
$203$	−19.7197	−1.38405
$204$	2.90177	0.203165
$205$	3.67980	0.257008
$206$	−18.1970	−1.26785
$207$	0.207172	0.0143994
$208$	1.16946	0.0810876
$209$	1.00000	0.0691714
$210$	7.24317	0.499827
$211$	−16.5102	−1.13661	−0.568303	−	0.822819i	$-0.692400\pi$
$211$	−16.5102	−1.13661	−0.568303	+	0.822819i	$0.692400\pi$
$212$	−2.74528	−0.188547
$213$	−12.6776	−0.868655
$214$	−9.26197	−0.633135
$215$	−4.59330	−0.313260
$216$	15.5388	1.05728
$217$	22.9410	1.55733
$218$	0.0555973	0.00376553
$219$	7.36276	0.497529
$220$	−0.529980	−0.0357312
$221$	1.35838	0.0913742
$222$	−5.87609	−0.394377
$223$	3.43650	0.230125	0.115062	−	0.993358i	$-0.463293\pi$
$223$	3.43650	0.230125	0.115062	+	0.993358i	$0.463293\pi$
$224$	−9.80979	−0.655444
$225$	0.142317	0.00948777
$226$	10.2261	0.680233
$227$	−9.66762	−0.641663	−0.320831	−	0.947136i	$-0.603962\pi$
$227$	−9.66762	−0.641663	−0.320831	+	0.947136i	$0.603962\pi$
$228$	0.939473	0.0622181
$229$	11.7897	0.779084	0.389542	−	0.921009i	$-0.372633\pi$
$229$	11.7897	0.779084	0.389542	+	0.921009i	$0.372633\pi$
$230$	−1.76497	−0.116379
$231$	5.97403	0.393062
$232$	−17.9488	−1.17840
$233$	2.00908	0.131619	0.0658097	−	0.997832i	$-0.479037\pi$
$233$	2.00908	0.131619	0.0658097	+	0.997832i	$0.479037\pi$
$234$	−0.0758854	−0.00496078
$235$	−0.210971	−0.0137622
$236$	5.76258	0.375112
$237$	27.4551	1.78340
$238$	12.6207	0.818077
$239$	−6.36104	−0.411461	−0.205731	−	0.978609i	$-0.565957\pi$
$239$	−6.36104	−0.411461	−0.205731	+	0.978609i	$0.565957\pi$
$240$	4.71378	0.304273
$241$	−28.3775	−1.82796	−0.913978	−	0.405763i	$-0.867006\pi$
$241$	−28.3775	−1.82796	−0.913978	+	0.405763i	$0.867006\pi$
$242$	1.21244	0.0779388
$243$	−1.47777	−0.0947989
$244$	3.00120	0.192132
$245$	−4.35755	−0.278394
$246$	−7.90879	−0.504246
$247$	0.439786	0.0279829
$248$	20.8808	1.32594
$249$	5.94792	0.376934
$250$	−1.21244	−0.0766817
$251$	−17.7932	−1.12310	−0.561550	−	0.827443i	$-0.689795\pi$
$251$	−17.7932	−1.12310	−0.561550	+	0.827443i	$0.689795\pi$
$252$	0.254189	0.0160124
$253$	−1.45571	−0.0915198
$254$	1.82157	0.114295
$255$	5.47525	0.342873
$256$	−11.7475	−0.734217
$257$	−22.5168	−1.40456	−0.702279	−	0.711902i	$-0.747834\pi$
$257$	−22.5168	−1.40456	−0.702279	+	0.711902i	$0.747834\pi$
$258$	9.87214	0.614612
$259$	9.21391	0.572524
$260$	−0.233078	−0.0144549
$261$	0.832747	0.0515457
$262$	−0.737751	−0.0455784
$263$	−5.74826	−0.354453	−0.177226	−	0.984170i	$-0.556712\pi$
$263$	−5.74826	−0.354453	−0.177226	+	0.984170i	$0.556712\pi$
$264$	5.43756	0.334658
$265$	−5.17998	−0.318204
$266$	4.08605	0.250532
$267$	−11.1067	−0.679720
$268$	−3.25813	−0.199022
$269$	−5.69265	−0.347087	−0.173544	−	0.984826i	$-0.555522\pi$
$269$	−5.69265	−0.347087	−0.173544	+	0.984826i	$0.555522\pi$
$270$	6.14187	0.373782
$271$	−16.0881	−0.977282	−0.488641	−	0.872485i	$-0.662507\pi$
$271$	−16.0881	−0.977282	−0.488641	+	0.872485i	$0.662507\pi$
$272$	8.21341	0.498011
$273$	2.62729	0.159011
$274$	−26.0110	−1.57138
$275$	−1.00000	−0.0603023
$276$	−1.36760	−0.0823200
$277$	−3.44717	−0.207120	−0.103560	−	0.994623i	$-0.533023\pi$
$277$	−3.44717	−0.207120	−0.103560	+	0.994623i	$0.533023\pi$
$278$	−10.7379	−0.644015
$279$	−0.968779	−0.0579993
$280$	−10.3376	−0.617792
$281$	−8.74423	−0.521637	−0.260819	−	0.965388i	$-0.583992\pi$
$281$	−8.74423	−0.521637	−0.260819	+	0.965388i	$0.583992\pi$
$282$	0.453430	0.0270013
$283$	30.1987	1.79513	0.897564	−	0.440885i	$-0.145335\pi$
$283$	30.1987	1.79513	0.897564	+	0.440885i	$0.145335\pi$
$284$	3.79028	0.224912
$285$	1.77266	0.105003
$286$	0.533216	0.0315297
$287$	12.4013	0.732024
$288$	0.414260	0.0244105
$289$	−7.45980	−0.438812
$290$	−7.09446	−0.416601
$291$	−11.8880	−0.696888
$292$	−2.20128	−0.128820
$293$	−11.1213	−0.649714	−0.324857	−	0.945763i	$-0.605316\pi$
$293$	−11.1213	−0.649714	−0.324857	+	0.945763i	$0.605316\pi$
$294$	9.36546	0.546205
$295$	10.8732	0.633062
$296$	8.38649	0.487455
$297$	5.06570	0.293941
$298$	−2.73060	−0.158179
$299$	−0.640201	−0.0370238
$300$	−0.939473	−0.0542405
$301$	−15.4799	−0.892244
$302$	−21.0499	−1.21129
$303$	34.0568	1.95651
$304$	2.65916	0.152513
$305$	5.66286	0.324254
$306$	−0.532961	−0.0304674
$307$	24.8969	1.42094	0.710469	−	0.703728i	$-0.248483\pi$
$307$	24.8969	1.42094	0.710469	+	0.703728i	$0.248483\pi$
$308$	−1.78608	−0.101772
$309$	−26.6050	−1.51351
$310$	8.25336	0.468760
$311$	9.24020	0.523963	0.261982	−	0.965073i	$-0.415624\pi$
$311$	9.24020	0.523963	0.261982	+	0.965073i	$0.415624\pi$
$312$	2.39136	0.135384
$313$	−14.8438	−0.839021	−0.419511	−	0.907750i	$-0.637798\pi$
$313$	−14.8438	−0.839021	−0.419511	+	0.907750i	$0.637798\pi$
$314$	−4.55740	−0.257189
$315$	0.479621	0.0270236
$316$	−8.20837	−0.461757
$317$	33.8110	1.89902	0.949509	−	0.313740i	$-0.101582\pi$
$317$	33.8110	1.89902	0.949509	+	0.313740i	$0.101582\pi$
$318$	11.1331	0.624311
$319$	−5.85137	−0.327614
$320$	−8.84754	−0.494593
$321$	−13.5415	−0.755813
$322$	−5.94811	−0.331475
$323$	3.08872	0.171861
$324$	4.98536	0.276964
$325$	−0.439786	−0.0243949
$326$	−14.3596	−0.795304
$327$	0.0812863	0.00449514
$328$	11.2876	0.623255
$329$	−0.710993	−0.0391983
$330$	2.14925	0.118312
$331$	11.5116	0.632737	0.316369	−	0.948636i	$-0.397536\pi$
$331$	11.5116	0.632737	0.316369	+	0.948636i	$0.397536\pi$
$332$	−1.77828	−0.0975956
$333$	−0.389096	−0.0213223
$334$	7.68455	0.420480
$335$	−6.14764	−0.335882
$336$	15.8859	0.866647
$337$	25.3591	1.38140	0.690698	−	0.723143i	$-0.257303\pi$
$337$	25.3591	1.38140	0.690698	+	0.723143i	$0.257303\pi$
$338$	−15.5273	−0.844572
$339$	14.9512	0.812036
$340$	−1.63696	−0.0887766
$341$	6.80721	0.368631
$342$	−0.172551	−0.00933048
$343$	8.90531	0.480841
$344$	−14.0897	−0.759668
$345$	−2.58048	−0.138928
$346$	−9.96382	−0.535658
$347$	−9.66755	−0.518982	−0.259491	−	0.965746i	$-0.583555\pi$
$347$	−9.66755	−0.518982	−0.259491	+	0.965746i	$0.583555\pi$
$348$	−5.49721	−0.294681
$349$	30.9349	1.65591	0.827953	−	0.560797i	$-0.189505\pi$
$349$	30.9349	1.65591	0.827953	+	0.560797i	$0.189505\pi$
$350$	−4.08605	−0.218409
$351$	2.22782	0.118912
$352$	−2.91083	−0.155148
$353$	10.9128	0.580830	0.290415	−	0.956901i	$-0.406207\pi$
$353$	10.9128	0.580830	0.290415	+	0.956901i	$0.406207\pi$
$354$	−23.3692	−1.24206
$355$	7.15175	0.379575
$356$	3.32063	0.175993
$357$	18.4521	0.976589
$358$	13.6337	0.720563
$359$	−29.0146	−1.53133	−0.765667	−	0.643237i	$-0.777591\pi$
$359$	−29.0146	−1.53133	−0.765667	+	0.643237i	$0.777591\pi$
$360$	0.436550	0.0230082
$361$	1.00000	0.0526316
$362$	10.0350	0.527429
$363$	1.77266	0.0930404
$364$	−0.785494	−0.0411711
$365$	−4.15351	−0.217405
$366$	−12.1709	−0.636182
$367$	21.1172	1.10231	0.551154	−	0.834404i	$-0.314188\pi$
$367$	21.1172	1.10231	0.551154	+	0.834404i	$0.314188\pi$
$368$	−3.87097	−0.201788
$369$	−0.523696	−0.0272625
$370$	3.31484	0.172331
$371$	−17.4570	−0.906324
$372$	6.39519	0.331575
$373$	24.3430	1.26043	0.630215	−	0.776421i	$-0.282967\pi$
$373$	24.3430	1.26043	0.630215	+	0.776421i	$0.282967\pi$
$374$	3.74490	0.193644
$375$	−1.77266	−0.0915397
$376$	−0.647146	−0.0333740
$377$	−2.57335	−0.132534
$378$	20.6987	1.06463
$379$	12.5458	0.644433	0.322216	−	0.946666i	$-0.395572\pi$
$379$	12.5458	0.644433	0.322216	+	0.946666i	$0.395572\pi$
$380$	−0.529980	−0.0271874
$381$	2.66323	0.136441
$382$	3.90137	0.199611
$383$	8.27908	0.423041	0.211521	−	0.977374i	$-0.432158\pi$
$383$	8.27908	0.423041	0.211521	+	0.977374i	$0.432158\pi$
$384$	8.69575	0.443753
$385$	−3.37010	−0.171756
$386$	23.0906	1.17528
$387$	0.653702	0.0332296
$388$	3.55422	0.180438
$389$	−7.91740	−0.401428	−0.200714	−	0.979650i	$-0.564326\pi$
$389$	−7.91740	−0.401428	−0.200714	+	0.979650i	$0.564326\pi$
$390$	0.945209	0.0478625
$391$	−4.49629	−0.227387
$392$	−13.3666	−0.675116
$393$	−1.07863	−0.0544098
$394$	2.47661	0.124770
$395$	−15.4881	−0.779290
$396$	0.0754249	0.00379024
$397$	−24.3775	−1.22347	−0.611737	−	0.791062i	$-0.709529\pi$
$397$	−24.3775	−1.22347	−0.611737	+	0.791062i	$0.709529\pi$
$398$	5.59181	0.280292
$399$	5.97403	0.299076
$400$	−2.65916	−0.132958
$401$	32.6120	1.62857	0.814283	−	0.580469i	$-0.197131\pi$
$401$	32.6120	1.62857	0.814283	+	0.580469i	$0.197131\pi$
$402$	13.2128	0.658995
$403$	2.99372	0.149128
$404$	−10.1821	−0.506579
$405$	9.40670	0.467422
$406$	−23.9090	−1.18658
$407$	2.73402	0.135520
$408$	16.7951	0.831481
$409$	−36.7429	−1.81682	−0.908409	−	0.418083i	$-0.862702\pi$
$409$	−36.7429	−1.81682	−0.908409	+	0.418083i	$0.862702\pi$
$410$	4.46155	0.220340
$411$	−38.0295	−1.87586
$412$	7.95423	0.391877
$413$	36.6437	1.80312
$414$	0.251184	0.0123450
$415$	−3.35537	−0.164708
$416$	−1.28014	−0.0627642
$417$	−15.6994	−0.768801
$418$	1.21244	0.0593026
$419$	−20.7229	−1.01238	−0.506191	−	0.862422i	$-0.668947\pi$
$419$	−20.7229	−1.01238	−0.506191	+	0.862422i	$0.668947\pi$
$420$	−3.16612	−0.154491
$421$	−23.3391	−1.13748	−0.568738	−	0.822518i	$-0.692568\pi$
$421$	−23.3391	−1.13748	−0.568738	+	0.822518i	$0.692568\pi$
$422$	−20.0176	−0.974443
$423$	0.0300247	0.00145985
$424$	−15.8894	−0.771656
$425$	−3.08872	−0.149825
$426$	−15.3709	−0.744722
$427$	19.0844	0.923558
$428$	4.04857	0.195695
$429$	0.779590	0.0376390
$430$	−5.56911	−0.268567
$431$	2.35724	0.113544	0.0567721	−	0.998387i	$-0.481919\pi$
$431$	2.35724	0.113544	0.0567721	+	0.998387i	$0.481919\pi$
$432$	13.4705	0.648100
$433$	37.3199	1.79348	0.896741	−	0.442555i	$-0.145928\pi$
$433$	37.3199	1.79348	0.896741	+	0.442555i	$0.145928\pi$
$434$	27.8146	1.33515
$435$	−10.3725	−0.497322
$436$	−0.0243026	−0.00116388
$437$	−1.45571	−0.0696361
$438$	8.92693	0.426545
$439$	26.3625	1.25821	0.629106	−	0.777320i	$-0.283421\pi$
$439$	26.3625	1.25821	0.629106	+	0.777320i	$0.283421\pi$
$440$	−3.06746	−0.146235
$441$	0.620152	0.0295310
$442$	1.64695	0.0783376
$443$	−7.31633	−0.347609	−0.173805	−	0.984780i	$-0.555606\pi$
$443$	−7.31633	−0.347609	−0.173805	+	0.984780i	$0.555606\pi$
$444$	2.56854	0.121897
$445$	6.26557	0.297017
$446$	4.16656	0.197292
$447$	−3.99228	−0.188829
$448$	−29.8171	−1.40872
$449$	−0.872582	−0.0411797	−0.0205898	−	0.999788i	$-0.506554\pi$
$449$	−0.872582	−0.0411797	−0.0205898	+	0.999788i	$0.506554\pi$
$450$	0.172551	0.00813412
$451$	3.67980	0.173275
$452$	−4.47002	−0.210252
$453$	−30.7761	−1.44599
$454$	−11.7215	−0.550115
$455$	−1.48212	−0.0694829
$456$	5.43756	0.254637
$457$	−2.04749	−0.0957774	−0.0478887	−	0.998853i	$-0.515249\pi$
$457$	−2.04749	−0.0957774	−0.0478887	+	0.998853i	$0.515249\pi$
$458$	14.2943	0.667930
$459$	15.6465	0.730317
$460$	0.771498	0.0359713
$461$	26.6538	1.24139	0.620695	−	0.784052i	$-0.286851\pi$
$461$	26.6538	1.24139	0.620695	+	0.784052i	$0.286851\pi$
$462$	7.24317	0.336983
$463$	−19.5335	−0.907798	−0.453899	−	0.891053i	$-0.649967\pi$
$463$	−19.5335	−0.907798	−0.453899	+	0.891053i	$0.649967\pi$
$464$	−15.5597	−0.722343
$465$	12.0669	0.559587
$466$	2.43590	0.112841
$467$	−29.3029	−1.35598	−0.677988	−	0.735073i	$-0.737148\pi$
$467$	−29.3029	−1.35598	−0.677988	+	0.735073i	$0.737148\pi$
$468$	0.0331708	0.00153332
$469$	−20.7182	−0.956675
$470$	−0.255791	−0.0117987
$471$	−6.66317	−0.307023
$472$	33.3531	1.53520
$473$	−4.59330	−0.211200
$474$	33.2877	1.52896
$475$	−1.00000	−0.0458831
$476$	−5.51671	−0.252858
$477$	0.737197	0.0337539
$478$	−7.71240	−0.352757
$479$	−0.965639	−0.0441212	−0.0220606	−	0.999757i	$-0.507023\pi$
$479$	−0.965639	−0.0441212	−0.0220606	+	0.999757i	$0.507023\pi$
$480$	−5.15991	−0.235517
$481$	1.20238	0.0548239
$482$	−34.4061	−1.56716
$483$	−8.69646	−0.395703
$484$	−0.529980	−0.0240900
$485$	6.70633	0.304519
$486$	−1.79171	−0.0812737
$487$	−3.36125	−0.152313	−0.0761564	−	0.997096i	$-0.524265\pi$
$487$	−3.36125	−0.152313	−0.0761564	+	0.997096i	$0.524265\pi$
$488$	17.3706	0.786329
$489$	−20.9945	−0.949404
$490$	−5.28329	−0.238675
$491$	−32.5832	−1.47046	−0.735230	−	0.677818i	$-0.762926\pi$
$491$	−32.5832	−1.47046	−0.735230	+	0.677818i	$0.762926\pi$
$492$	3.45707	0.155857
$493$	−18.0733	−0.813978
$494$	0.533216	0.0239905
$495$	0.142317	0.00639665
$496$	18.1015	0.812780
$497$	24.1021	1.08113
$498$	7.21152	0.323156
$499$	7.53868	0.337477	0.168739	−	0.985661i	$-0.446031\pi$
$499$	7.53868	0.337477	0.168739	+	0.985661i	$0.446031\pi$
$500$	0.529980	0.0237014
$501$	11.2352	0.501953
$502$	−21.5733	−0.962864
$503$	−9.02729	−0.402507	−0.201254	−	0.979539i	$-0.564502\pi$
$503$	−9.02729	−0.402507	−0.201254	+	0.979539i	$0.564502\pi$
$504$	1.47122	0.0655332
$505$	−19.2123	−0.854935
$506$	−1.76497	−0.0784624
$507$	−22.7017	−1.00822
$508$	−0.796238	−0.0353273
$509$	−20.8922	−0.926031	−0.463015	−	0.886350i	$-0.653233\pi$
$509$	−20.8922	−0.926031	−0.463015	+	0.886350i	$0.653233\pi$
$510$	6.63843	0.293955
$511$	−13.9977	−0.619223
$512$	−24.0541	−1.06305
$513$	5.06570	0.223656
$514$	−27.3003	−1.20417
$515$	15.0086	0.661356
$516$	−4.31528	−0.189970
$517$	−0.210971	−0.00927850
$518$	11.1713	0.490841
$519$	−14.5676	−0.639448
$520$	−1.34902	−0.0591586
$521$	−43.6640	−1.91296	−0.956478	−	0.291803i	$-0.905745\pi$
$521$	−43.6640	−1.91296	−0.956478	+	0.291803i	$0.905745\pi$
$522$	1.00966	0.0441916
$523$	26.4677	1.15735	0.578675	−	0.815558i	$-0.303570\pi$
$523$	26.4677	1.15735	0.578675	+	0.815558i	$0.303570\pi$
$524$	0.322484	0.0140878
$525$	−5.97403	−0.260728
$526$	−6.96944	−0.303882
$527$	21.0256	0.915889
$528$	4.71378	0.205141
$529$	−20.8809	−0.907865
$530$	−6.28043	−0.272805
$531$	−1.54744	−0.0671530
$532$	−1.78608	−0.0774365
$533$	1.61832	0.0700973
$534$	−13.4663	−0.582742
$535$	7.63909	0.330267
$536$	−18.8576	−0.814526
$537$	19.9332	0.860180
$538$	−6.90202	−0.297567
$539$	−4.35755	−0.187693
$540$	−2.68472	−0.115532
$541$	−1.04162	−0.0447829	−0.0223915	−	0.999749i	$-0.507128\pi$
$541$	−1.04162	−0.0447829	−0.0223915	+	0.999749i	$0.507128\pi$
$542$	−19.5059	−0.837851
$543$	14.6718	0.629625
$544$	−8.99075	−0.385475
$545$	−0.0458556	−0.00196424
$546$	3.18545	0.136324
$547$	−5.27609	−0.225589	−0.112795	−	0.993618i	$-0.535980\pi$
$547$	−5.27609	−0.225589	−0.112795	+	0.993618i	$0.535980\pi$
$548$	11.3699	0.485696
$549$	−0.805918	−0.0343958
$550$	−1.21244	−0.0516988
$551$	−5.85137	−0.249277
$552$	−7.91551	−0.336907
$553$	−52.1963	−2.21961
$554$	−4.17949	−0.177570
$555$	4.84648	0.205722
$556$	4.69371	0.199058
$557$	−12.9219	−0.547517	−0.273759	−	0.961798i	$-0.588267\pi$
$557$	−12.9219	−0.547517	−0.273759	+	0.961798i	$0.588267\pi$
$558$	−1.17459	−0.0497244
$559$	−2.02007	−0.0854397
$560$	−8.96163	−0.378698
$561$	5.47525	0.231165
$562$	−10.6019	−0.447214
$563$	−8.76657	−0.369467	−0.184733	−	0.982789i	$-0.559142\pi$
$563$	−8.76657	−0.369467	−0.184733	+	0.982789i	$0.559142\pi$
$564$	−0.198202	−0.00834580
$565$	−8.43432	−0.354834
$566$	36.6142	1.53901
$567$	31.7015	1.33134
$568$	21.9377	0.920485
$569$	−14.7828	−0.619726	−0.309863	−	0.950781i	$-0.600283\pi$
$569$	−14.7828	−0.619726	−0.309863	+	0.950781i	$0.600283\pi$
$570$	2.14925	0.0900221
$571$	22.6278	0.946943	0.473472	−	0.880809i	$-0.343001\pi$
$571$	22.6278	0.946943	0.473472	+	0.880809i	$0.343001\pi$
$572$	−0.233078	−0.00974547
$573$	5.70401	0.238288
$574$	15.0358	0.627584
$575$	1.45571	0.0607074
$576$	1.25915	0.0524647
$577$	3.73371	0.155436	0.0777182	−	0.996975i	$-0.475237\pi$
$577$	3.73371	0.155436	0.0777182	+	0.996975i	$0.475237\pi$
$578$	−9.04459	−0.376205
$579$	33.7597	1.40301
$580$	3.10111	0.128767
$581$	−11.3079	−0.469131
$582$	−14.4136	−0.597461
$583$	−5.17998	−0.214533
$584$	−12.7407	−0.527215
$585$	0.0625888	0.00258773
$586$	−13.4840	−0.557018
$587$	−19.6389	−0.810584	−0.405292	−	0.914187i	$-0.632830\pi$
$587$	−19.6389	−0.810584	−0.405292	+	0.914187i	$0.632830\pi$
$588$	−4.09380	−0.168826
$589$	6.80721	0.280486
$590$	13.1831	0.542741
$591$	3.62094	0.148946
$592$	7.27020	0.298803
$593$	−32.8110	−1.34739	−0.673693	−	0.739011i	$-0.735293\pi$
$593$	−32.8110	−1.34739	−0.673693	+	0.739011i	$0.735293\pi$
$594$	6.14187	0.252004
$595$	−10.4093	−0.426739
$596$	1.19359	0.0488914
$597$	8.17553	0.334602
$598$	−0.776208	−0.0317415
$599$	−41.5943	−1.69950	−0.849750	−	0.527187i	$-0.823247\pi$
$599$	−41.5943	−1.69950	−0.849750	+	0.527187i	$0.823247\pi$
$600$	−5.43756	−0.221987
$601$	−16.1727	−0.659698	−0.329849	−	0.944034i	$-0.606998\pi$
$601$	−16.1727	−0.659698	−0.329849	+	0.944034i	$0.606998\pi$
$602$	−18.7685	−0.764945
$603$	0.874911	0.0356291
$604$	9.20127	0.374395
$605$	−1.00000	−0.0406558
$606$	41.2919	1.67737
$607$	−14.9877	−0.608330	−0.304165	−	0.952619i	$-0.598377\pi$
$607$	−14.9877	−0.608330	−0.304165	+	0.952619i	$0.598377\pi$
$608$	−2.91083	−0.118050
$609$	−34.9563	−1.41650
$610$	6.86590	0.277992
$611$	−0.0927821	−0.00375356
$612$	0.232967	0.00941712
$613$	19.1145	0.772027	0.386014	−	0.922493i	$-0.373852\pi$
$613$	19.1145	0.772027	0.386014	+	0.922493i	$0.373852\pi$
$614$	30.1860	1.21821
$615$	6.52302	0.263033
$616$	−10.3376	−0.416515
$617$	−1.20030	−0.0483223	−0.0241612	−	0.999708i	$-0.507691\pi$
$617$	−1.20030	−0.0483223	−0.0241612	+	0.999708i	$0.507691\pi$
$618$	−32.2571	−1.29757
$619$	−44.2783	−1.77969	−0.889847	−	0.456258i	$-0.849189\pi$
$619$	−44.2783	−1.77969	−0.889847	+	0.456258i	$0.849189\pi$
$620$	−3.60769	−0.144888
$621$	−7.37419	−0.295916
$622$	11.2032	0.449208
$623$	21.1156	0.845978
$624$	2.07306	0.0829886
$625$	1.00000	0.0400000
$626$	−17.9973	−0.719316
$627$	1.77266	0.0707931
$628$	1.99212	0.0794942
$629$	8.44462	0.336709
$630$	0.581513	0.0231680
$631$	7.76811	0.309244	0.154622	−	0.987974i	$-0.450584\pi$
$631$	7.76811	0.309244	0.154622	+	0.987974i	$0.450584\pi$
$632$	−47.5090	−1.88981
$633$	−29.2669	−1.16325
$634$	40.9940	1.62808
$635$	−1.50239	−0.0596206
$636$	−4.86645	−0.192967
$637$	−1.91639	−0.0759301
$638$	−7.09446	−0.280872
$639$	−1.01781	−0.0402640
$640$	−4.90548	−0.193906
$641$	27.0513	1.06846	0.534231	−	0.845339i	$-0.320601\pi$
$641$	27.0513	1.06846	0.534231	+	0.845339i	$0.320601\pi$
$642$	−16.4183	−0.647979
$643$	−15.1507	−0.597485	−0.298742	−	0.954334i	$-0.596567\pi$
$643$	−15.1507	−0.597485	−0.298742	+	0.954334i	$0.596567\pi$
$644$	2.60002	0.102455
$645$	−8.14235	−0.320605
$646$	3.74490	0.147341
$647$	−22.7874	−0.895866	−0.447933	−	0.894067i	$-0.647840\pi$
$647$	−22.7874	−0.895866	−0.447933	+	0.894067i	$0.647840\pi$
$648$	28.8546	1.13352
$649$	10.8732	0.426810
$650$	−0.533216	−0.0209144
$651$	40.6665	1.59385
$652$	6.27682	0.245819
$653$	36.1144	1.41327	0.706633	−	0.707580i	$-0.250213\pi$
$653$	36.1144	1.41327	0.706633	+	0.707580i	$0.250213\pi$
$654$	0.0985551	0.00385381
$655$	0.608483	0.0237754
$656$	9.78517	0.382047
$657$	0.591113	0.0230615
$658$	−0.862040	−0.0336058
$659$	−5.96009	−0.232172	−0.116086	−	0.993239i	$-0.537035\pi$
$659$	−5.96009	−0.232172	−0.116086	+	0.993239i	$0.537035\pi$
$660$	−0.939473	−0.0365689
$661$	47.5667	1.85013	0.925065	−	0.379808i	$-0.124010\pi$
$661$	47.5667	1.85013	0.925065	+	0.379808i	$0.124010\pi$
$662$	13.9572	0.542463
$663$	2.40794	0.0935165
$664$	−10.2924	−0.399425
$665$	−3.37010	−0.130687
$666$	−0.471757	−0.0182802
$667$	8.51791	0.329815
$668$	−3.35905	−0.129965
$669$	6.09174	0.235520
$670$	−7.45367	−0.287960
$671$	5.66286	0.218612
$672$	−17.3894	−0.670811
$673$	23.3646	0.900639	0.450319	−	0.892868i	$-0.351310\pi$
$673$	23.3646	0.900639	0.450319	+	0.892868i	$0.351310\pi$
$674$	30.7465	1.18431
$675$	−5.06570	−0.194979
$676$	6.78723	0.261047
$677$	−22.7258	−0.873423	−0.436712	−	0.899602i	$-0.643857\pi$
$677$	−22.7258	−0.873423	−0.436712	+	0.899602i	$0.643857\pi$
$678$	18.1274	0.696180
$679$	22.6010	0.867346
$680$	−9.47453	−0.363331
$681$	−17.1374	−0.656706
$682$	8.25336	0.316038
$683$	−6.08405	−0.232800	−0.116400	−	0.993202i	$-0.537135\pi$
$683$	−6.08405	−0.232800	−0.116400	+	0.993202i	$0.537135\pi$
$684$	0.0754249	0.00288394
$685$	21.4534	0.819691
$686$	10.7972	0.412239
$687$	20.8991	0.797350
$688$	−12.2143	−0.465667
$689$	−2.27808	−0.0867880
$690$	−3.12869	−0.119107
$691$	10.2313	0.389215	0.194608	−	0.980881i	$-0.437657\pi$
$691$	10.2313	0.389215	0.194608	+	0.980881i	$0.437657\pi$
$692$	4.35535	0.165566
$693$	0.479621	0.0182193
$694$	−11.7214	−0.444937
$695$	8.85639	0.335942
$696$	−31.8171	−1.20603
$697$	11.3659	0.430513
$698$	37.5068	1.41965
$699$	3.56141	0.134705
$700$	1.78608	0.0675076
$701$	20.1154	0.759749	0.379875	−	0.925038i	$-0.375967\pi$
$701$	20.1154	0.759749	0.379875	+	0.925038i	$0.375967\pi$
$702$	2.70111	0.101947
$703$	2.73402	0.103115
$704$	−8.84754	−0.333454
$705$	−0.373980	−0.0140849
$706$	13.2312	0.497962
$707$	−64.7472	−2.43507
$708$	10.2151	0.383906
$709$	−18.6469	−0.700301	−0.350150	−	0.936694i	$-0.613870\pi$
$709$	−18.6469	−0.700301	−0.350150	+	0.936694i	$0.613870\pi$
$710$	8.67109	0.325420
$711$	2.20421	0.0826643
$712$	19.2194	0.720277
$713$	−9.90934	−0.371108
$714$	22.3721	0.837257
$715$	−0.439786	−0.0164471
$716$	−5.95952	−0.222718
$717$	−11.2759	−0.421108
$718$	−35.1786	−1.31285
$719$	−25.8776	−0.965073	−0.482537	−	0.875876i	$-0.660284\pi$
$719$	−25.8776	−0.965073	−0.482537	+	0.875876i	$0.660284\pi$
$720$	0.378443	0.0141037
$721$	50.5803	1.88371
$722$	1.21244	0.0451225
$723$	−50.3036	−1.87081
$724$	−4.38648	−0.163022
$725$	5.85137	0.217314
$726$	2.14925	0.0797661
$727$	48.1800	1.78690	0.893450	−	0.449164i	$-0.148278\pi$
$727$	48.1800	1.78690	0.893450	+	0.449164i	$0.148278\pi$
$728$	−4.54634	−0.168499
$729$	25.6005	0.948167
$730$	−5.03590	−0.186387
$731$	−14.1874	−0.524741
$732$	5.32010	0.196637
$733$	5.47157	0.202097	0.101049	−	0.994881i	$-0.467780\pi$
$733$	5.47157	0.202097	0.101049	+	0.994881i	$0.467780\pi$
$734$	25.6034	0.945038
$735$	−7.72445	−0.284921
$736$	4.23733	0.156190
$737$	−6.14764	−0.226451
$738$	−0.634952	−0.0233729
$739$	11.0454	0.406311	0.203156	−	0.979146i	$-0.434880\pi$
$739$	11.0454	0.406311	0.203156	+	0.979146i	$0.434880\pi$
$740$	−1.44898	−0.0532654
$741$	0.779590	0.0286389
$742$	−21.1657	−0.777016
$743$	−16.8303	−0.617443	−0.308721	−	0.951153i	$-0.599901\pi$
$743$	−16.8303	−0.617443	−0.308721	+	0.951153i	$0.599901\pi$
$744$	37.0146	1.35702
$745$	2.25215	0.0825122
$746$	29.5145	1.08060
$747$	0.477524	0.0174717
$748$	−1.63696	−0.0598532
$749$	25.7445	0.940683
$750$	−2.14925	−0.0784795
$751$	−8.34710	−0.304590	−0.152295	−	0.988335i	$-0.548666\pi$
$751$	−8.34710	−0.304590	−0.152295	+	0.988335i	$0.548666\pi$
$752$	−0.561007	−0.0204578
$753$	−31.5413	−1.14943
$754$	−3.12004	−0.113625
$755$	17.3616	0.631852
$756$	−9.04776	−0.329064
$757$	2.18852	0.0795431	0.0397716	−	0.999209i	$-0.487337\pi$
$757$	2.18852	0.0795431	0.0397716	+	0.999209i	$0.487337\pi$
$758$	15.2110	0.552490
$759$	−2.58048	−0.0936654
$760$	−3.06746	−0.111268
$761$	33.4322	1.21192	0.605958	−	0.795496i	$-0.292790\pi$
$761$	33.4322	1.21192	0.605958	+	0.795496i	$0.292790\pi$
$762$	3.22901	0.116975
$763$	−0.154538	−0.00559464
$764$	−1.70535	−0.0616975
$765$	0.439576	0.0158929
$766$	10.0379	0.362685
$767$	4.78188	0.172664
$768$	−20.8242	−0.751430
$769$	35.7102	1.28774	0.643871	−	0.765134i	$-0.277327\pi$
$769$	35.7102	1.28774	0.643871	+	0.765134i	$0.277327\pi$
$770$	−4.08605	−0.147251
$771$	−39.9145	−1.43749
$772$	−10.0933	−0.363266
$773$	12.1707	0.437749	0.218875	−	0.975753i	$-0.429761\pi$
$773$	12.1707	0.437749	0.218875	+	0.975753i	$0.429761\pi$
$774$	0.792577	0.0284886
$775$	−6.80721	−0.244522
$776$	20.5714	0.738469
$777$	16.3331	0.585947
$778$	−9.59940	−0.344155
$779$	3.67980	0.131842
$780$	−0.413167	−0.0147938
$781$	7.15175	0.255910
$782$	−5.45150	−0.194945
$783$	−29.6413	−1.05929
$784$	−11.5874	−0.413837
$785$	3.75886	0.134159
$786$	−1.30778	−0.0466470
$787$	47.9347	1.70869	0.854344	−	0.519707i	$-0.173959\pi$
$787$	47.9347	1.70869	0.854344	+	0.519707i	$0.173959\pi$
$788$	−1.08257	−0.0385649
$789$	−10.1897	−0.362763
$790$	−18.7784	−0.668106
$791$	−28.4245	−1.01066
$792$	0.436550	0.0155121
$793$	2.49044	0.0884383
$794$	−29.5564	−1.04892
$795$	−9.18233	−0.325664
$796$	−2.44428	−0.0866351
$797$	50.1321	1.77577	0.887885	−	0.460066i	$-0.152174\pi$
$797$	50.1321	1.77577	0.887885	+	0.460066i	$0.152174\pi$
$798$	7.24317	0.256406
$799$	−0.651631	−0.0230531
$800$	2.91083	0.102913
$801$	−0.891695	−0.0315065
$802$	39.5402	1.39621
$803$	−4.15351	−0.146574
$804$	−5.77555	−0.203688
$805$	4.90589	0.172910
$806$	3.62971	0.127851
$807$	−10.0911	−0.355225
$808$	−58.9328	−2.07325
$809$	−39.8704	−1.40177	−0.700884	−	0.713275i	$-0.747211\pi$
$809$	−39.8704	−1.40177	−0.700884	+	0.713275i	$0.747211\pi$
$810$	11.4051	0.400734
$811$	−54.0597	−1.89829	−0.949147	−	0.314835i	$-0.898051\pi$
$811$	−54.0597	−1.89829	−0.949147	+	0.314835i	$0.898051\pi$
$812$	10.4510	0.366759
$813$	−28.5187	−1.00019
$814$	3.31484	0.116185
$815$	11.8435	0.414860
$816$	14.5596	0.509687
$817$	−4.59330	−0.160699
$818$	−44.5487	−1.55761
$819$	0.210930	0.00737050
$820$	−1.95022	−0.0681046
$821$	28.2714	0.986677	0.493339	−	0.869837i	$-0.335776\pi$
$821$	28.2714	0.986677	0.493339	+	0.869837i	$0.335776\pi$
$822$	−46.1086	−1.60822
$823$	−28.9439	−1.00892	−0.504461	−	0.863434i	$-0.668309\pi$
$823$	−28.9439	−1.00892	−0.504461	+	0.863434i	$0.668309\pi$
$824$	46.0381	1.60381
$825$	−1.77266	−0.0617160
$826$	44.4285	1.54586
$827$	−6.79361	−0.236237	−0.118118	−	0.993000i	$-0.537686\pi$
$827$	−6.79361	−0.236237	−0.118118	+	0.993000i	$0.537686\pi$
$828$	−0.109797	−0.00381571
$829$	4.79089	0.166394	0.0831972	−	0.996533i	$-0.473487\pi$
$829$	4.79089	0.166394	0.0831972	+	0.996533i	$0.473487\pi$
$830$	−4.06819	−0.141209
$831$	−6.11065	−0.211976
$832$	−3.89102	−0.134897
$833$	−13.4593	−0.466336
$834$	−19.0346	−0.659114
$835$	−6.33806	−0.219338
$836$	−0.529980	−0.0183297
$837$	34.4833	1.19192
$838$	−25.1254	−0.867942
$839$	−46.8222	−1.61648	−0.808241	−	0.588852i	$-0.799580\pi$
$839$	−46.8222	−1.61648	−0.808241	+	0.588852i	$0.799580\pi$
$840$	−18.3251	−0.632276
$841$	5.23853	0.180639
$842$	−28.2973	−0.975190
$843$	−15.5005	−0.533867
$844$	8.75005	0.301189
$845$	12.8066	0.440560
$846$	0.0364033	0.00125157
$847$	−3.37010	−0.115798
$848$	−13.7744	−0.473015
$849$	53.5320	1.83721
$850$	−3.74490	−0.128449
$851$	−3.97994	−0.136431
$852$	6.71888	0.230185
$853$	4.99555	0.171044	0.0855222	−	0.996336i	$-0.472744\pi$
$853$	4.99555	0.171044	0.0855222	+	0.996336i	$0.472744\pi$
$854$	23.1387	0.791791
$855$	0.142317	0.00486712
$856$	23.4326	0.800910
$857$	3.32386	0.113541	0.0567705	−	0.998387i	$-0.481920\pi$
$857$	3.32386	0.113541	0.0567705	+	0.998387i	$0.481920\pi$
$858$	0.945209	0.0322689
$859$	−32.5496	−1.11058	−0.555288	−	0.831658i	$-0.687392\pi$
$859$	−32.5496	−1.11058	−0.555288	+	0.831658i	$0.687392\pi$
$860$	2.43436	0.0830108
$861$	21.9832	0.749186
$862$	2.85802	0.0973446
$863$	7.39803	0.251832	0.125916	−	0.992041i	$-0.459813\pi$
$863$	7.39803	0.251832	0.125916	+	0.992041i	$0.459813\pi$
$864$	−14.7454	−0.501649
$865$	8.21796	0.279419
$866$	45.2483	1.53760
$867$	−13.2237	−0.449099
$868$	−12.1583	−0.412678
$869$	−15.4881	−0.525397
$870$	−12.5760	−0.426368
$871$	−2.70365	−0.0916096
$872$	−0.140660	−0.00476335
$873$	−0.954421	−0.0323023
$874$	−1.76497	−0.0597009
$875$	3.37010	0.113930
$876$	−3.90211	−0.131840
$877$	40.5912	1.37067	0.685333	−	0.728230i	$-0.259657\pi$
$877$	40.5912	1.37067	0.685333	+	0.728230i	$0.259657\pi$
$878$	31.9630	1.07870
$879$	−19.7143	−0.664946
$880$	−2.65916	−0.0896403
$881$	16.4133	0.552978	0.276489	−	0.961017i	$-0.410829\pi$
$881$	16.4133	0.552978	0.276489	+	0.961017i	$0.410829\pi$
$882$	0.751899	0.0253178
$883$	41.3719	1.39227	0.696137	−	0.717909i	$-0.254900\pi$
$883$	41.3719	1.39227	0.696137	+	0.717909i	$0.254900\pi$
$884$	−0.719912	−0.0242133
$885$	19.2745	0.647904
$886$	−8.87064	−0.298015
$887$	11.9068	0.399791	0.199896	−	0.979817i	$-0.435940\pi$
$887$	11.9068	0.399791	0.199896	+	0.979817i	$0.435940\pi$
$888$	14.8664	0.498883
$889$	−5.06321	−0.169814
$890$	7.59665	0.254640
$891$	9.40670	0.315136
$892$	−1.82128	−0.0609808
$893$	−0.210971	−0.00705988
$894$	−4.84042	−0.161888
$895$	−11.2448	−0.375872
$896$	−16.5320	−0.552294
$897$	−1.13486	−0.0378918
$898$	−1.05796	−0.0353045
$899$	−39.8315	−1.32846
$900$	−0.0754249	−0.00251416
$901$	−15.9995	−0.533021
$902$	4.46155	0.148553
$903$	−27.4405	−0.913162
$904$	−25.8719	−0.860487
$905$	−8.27670	−0.275127
$906$	−37.3143	−1.23968
$907$	−18.1787	−0.603614	−0.301807	−	0.953369i	$-0.597590\pi$
$907$	−18.1787	−0.603614	−0.301807	+	0.953369i	$0.597590\pi$
$908$	5.12365	0.170034
$909$	2.73422	0.0906885
$910$	−1.79699	−0.0595696
$911$	32.5463	1.07831	0.539154	−	0.842207i	$-0.318744\pi$
$911$	32.5463	1.07831	0.539154	+	0.842207i	$0.318744\pi$
$912$	4.71378	0.156089
$913$	−3.35537	−0.111046
$914$	−2.48246	−0.0821125
$915$	10.0383	0.331856
$916$	−6.24830	−0.206450
$917$	2.05065	0.0677183
$918$	18.9705	0.626121
$919$	−22.8963	−0.755280	−0.377640	−	0.925952i	$-0.623264\pi$
$919$	−22.8963	−0.755280	−0.377640	+	0.925952i	$0.623264\pi$
$920$	4.46533	0.147218
$921$	44.1336	1.45425
$922$	32.3162	1.06428
$923$	3.14524	0.103527
$924$	−3.16612	−0.104158
$925$	−2.73402	−0.0898940
$926$	−23.6832	−0.778280
$927$	−2.13597	−0.0701543
$928$	17.0324	0.559115
$929$	−39.9363	−1.31027	−0.655135	−	0.755512i	$-0.727388\pi$
$929$	−39.9363	−1.31027	−0.655135	+	0.755512i	$0.727388\pi$
$930$	14.6304	0.479749
$931$	−4.35755	−0.142813
$932$	−1.06477	−0.0348778
$933$	16.3797	0.536248
$934$	−35.5281	−1.16252
$935$	−3.08872	−0.101012
$936$	0.191989	0.00627534
$937$	51.4680	1.68139	0.840693	−	0.541511i	$-0.182148\pi$
$937$	51.4680	1.68139	0.840693	+	0.541511i	$0.182148\pi$
$938$	−25.1196	−0.820184
$939$	−26.3130	−0.858692
$940$	0.111811	0.00364686
$941$	56.3979	1.83852	0.919259	−	0.393652i	$-0.128789\pi$
$941$	56.3979	1.83852	0.919259	+	0.393652i	$0.128789\pi$
$942$	−8.07872	−0.263219
$943$	−5.35672	−0.174439
$944$	28.9136	0.941057
$945$	−17.0719	−0.555348
$946$	−5.56911	−0.181068
$947$	26.6837	0.867103	0.433552	−	0.901129i	$-0.357260\pi$
$947$	26.6837	0.867103	0.433552	+	0.901129i	$0.357260\pi$
$948$	−14.5506	−0.472583
$949$	−1.82666	−0.0592957
$950$	−1.21244	−0.0393369
$951$	59.9354	1.94354
$952$	−31.9301	−1.03486
$953$	−14.9629	−0.484695	−0.242348	−	0.970190i	$-0.577917\pi$
$953$	−14.9629	−0.484695	−0.242348	+	0.970190i	$0.577917\pi$
$954$	0.893810	0.0289382
$955$	−3.21777	−0.104125
$956$	3.37122	0.109033
$957$	−10.3725	−0.335295
$958$	−1.17078	−0.0378263
$959$	72.2999	2.33469
$960$	−15.6837	−0.506188
$961$	15.3382	0.494779
$962$	1.45782	0.0470021
$963$	−1.08717	−0.0350336
$964$	15.0395	0.484390
$965$	−19.0447	−0.613070
$966$	−10.5440	−0.339247
$967$	19.7493	0.635094	0.317547	−	0.948242i	$-0.397141\pi$
$967$	19.7493	0.635094	0.317547	+	0.948242i	$0.397141\pi$
$968$	−3.06746	−0.0985919
$969$	5.47525	0.175890
$970$	8.13104	0.261072
$971$	−53.2159	−1.70778	−0.853889	−	0.520455i	$-0.825762\pi$
$971$	−53.2159	−1.70778	−0.853889	+	0.520455i	$0.825762\pi$
$972$	0.783187	0.0251207
$973$	29.8469	0.956847
$974$	−4.07533	−0.130582
$975$	−0.779590	−0.0249669
$976$	15.0585	0.482009
$977$	−49.9728	−1.59877	−0.799385	−	0.600819i	$-0.794841\pi$
$977$	−49.9728	−1.59877	−0.799385	+	0.600819i	$0.794841\pi$
$978$	−25.4546	−0.813950
$979$	6.26557	0.200249
$980$	2.30941	0.0737715
$981$	0.00652601	0.000208360 0
$982$	−39.5053	−1.26067
$983$	29.2283	0.932239	0.466119	−	0.884722i	$-0.345652\pi$
$983$	29.2283	0.932239	0.466119	+	0.884722i	$0.345652\pi$
$984$	20.0091	0.637866
$985$	−2.04266	−0.0650846
$986$	−21.9128	−0.697846
$987$	−1.26035	−0.0401173
$988$	−0.233078	−0.00741519
$989$	6.68651	0.212619
$990$	0.172551	0.00548403
$991$	−7.92503	−0.251747	−0.125873	−	0.992046i	$-0.540173\pi$
$991$	−7.92503	−0.251747	−0.125873	+	0.992046i	$0.540173\pi$
$992$	−19.8147	−0.629116
$993$	20.4062	0.647571
$994$	29.2224	0.926879
$995$	−4.61202	−0.146211
$996$	−3.15228	−0.0998837
$997$	11.9229	0.377601	0.188800	−	0.982015i	$-0.439540\pi$
$997$	11.9229	0.377601	0.188800	+	0.982015i	$0.439540\pi$
$998$	9.14022	0.289329
$999$	13.8497	0.438185

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.f.1.5	✓	6
3.2	odd	2		9405.2.a.z.1.2		6
5.4	even	2		5225.2.a.l.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.f.1.5	✓	6	1.1	even	1	trivial
5225.2.a.l.1.2		6	5.4	even	2
9405.2.a.z.1.2		6	3.2	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+1.21244 q^{2} +1.77266 q^{3} -0.529980 q^{4} -1.00000 q^{5} +2.14925 q^{6} -3.37010 q^{7} -3.06746 q^{8} +0.142317 q^{9} +O(q^{10})\)	\(q+1.21244 q^{2} +1.77266 q^{3} -0.529980 q^{4} -1.00000 q^{5} +2.14925 q^{6} -3.37010 q^{7} -3.06746 q^{8} +0.142317 q^{9} -1.21244 q^{10} -1.00000 q^{11} -0.939473 q^{12} -0.439786 q^{13} -4.08605 q^{14} -1.77266 q^{15} -2.65916 q^{16} -3.08872 q^{17} +0.172551 q^{18} -1.00000 q^{19} +0.529980 q^{20} -5.97403 q^{21} -1.21244 q^{22} +1.45571 q^{23} -5.43756 q^{24} +1.00000 q^{25} -0.533216 q^{26} -5.06570 q^{27} +1.78608 q^{28} +5.85137 q^{29} -2.14925 q^{30} -6.80721 q^{31} +2.91083 q^{32} -1.77266 q^{33} -3.74490 q^{34} +3.37010 q^{35} -0.0754249 q^{36} -2.73402 q^{37} -1.21244 q^{38} -0.779590 q^{39} +3.06746 q^{40} -3.67980 q^{41} -7.24317 q^{42} +4.59330 q^{43} +0.529980 q^{44} -0.142317 q^{45} +1.76497 q^{46} +0.210971 q^{47} -4.71378 q^{48} +4.35755 q^{49} +1.21244 q^{50} -5.47525 q^{51} +0.233078 q^{52} +5.17998 q^{53} -6.14187 q^{54} +1.00000 q^{55} +10.3376 q^{56} -1.77266 q^{57} +7.09446 q^{58} -10.8732 q^{59} +0.939473 q^{60} -5.66286 q^{61} -8.25336 q^{62} -0.479621 q^{63} +8.84754 q^{64} +0.439786 q^{65} -2.14925 q^{66} +6.14764 q^{67} +1.63696 q^{68} +2.58048 q^{69} +4.08605 q^{70} -7.15175 q^{71} -0.436550 q^{72} +4.15351 q^{73} -3.31484 q^{74} +1.77266 q^{75} +0.529980 q^{76} +3.37010 q^{77} -0.945209 q^{78} +15.4881 q^{79} +2.65916 q^{80} -9.40670 q^{81} -4.46155 q^{82} +3.35537 q^{83} +3.16612 q^{84} +3.08872 q^{85} +5.56911 q^{86} +10.3725 q^{87} +3.06746 q^{88} -6.26557 q^{89} -0.172551 q^{90} +1.48212 q^{91} -0.771498 q^{92} -12.0669 q^{93} +0.255791 q^{94} +1.00000 q^{95} +5.15991 q^{96} -6.70633 q^{97} +5.28329 q^{98} -0.142317 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9}+O(q^{10}) \) 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9	\( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21} + 2 q^{22} + 4 q^{23} - q^{24} + 6 q^{25} - 14 q^{26} - 16 q^{27} + 10 q^{28} - 9 q^{29} - 21 q^{31} - q^{32} + q^{33} - 5 q^{35} - 28 q^{36} - 3 q^{37} + 2 q^{38} + 20 q^{39} + 12 q^{40} - 23 q^{41} + q^{42} + 7 q^{43} - 4 q^{44} - q^{45} - 12 q^{46} - 18 q^{47} - 3 q^{49} - 2 q^{50} - 16 q^{51} + 13 q^{52} - 17 q^{53} + q^{54} + 6 q^{55} - 2 q^{56} + q^{57} + 23 q^{58} - 29 q^{59} - q^{60} + 17 q^{61} + 2 q^{62} + 6 q^{63} - 18 q^{64} + 5 q^{65} + 8 q^{67} - q^{68} - 38 q^{69} + 8 q^{70} - 12 q^{71} + 13 q^{72} + 2 q^{73} - 37 q^{74} - q^{75} - 4 q^{76} - 5 q^{77} + q^{78} + 3 q^{79} - 4 q^{80} - 2 q^{81} + 24 q^{82} - 11 q^{83} - 3 q^{84} - q^{85} - 12 q^{86} - 12 q^{87} + 12 q^{88} - 22 q^{89} - 6 q^{90} - 18 q^{91} - 15 q^{92} + 18 q^{93} + 22 q^{94} + 6 q^{95} - 17 q^{96} - 2 q^{97} - q^{98} - q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 + 2 * q^10 - 6 * q^11 + q^12 - 5 * q^13 - 8 * q^14 + q^15 + 4 * q^16 + q^17 + 6 * q^18 - 6 * q^19 - 4 * q^20 - 21 * q^21 + 2 * q^22 + 4 * q^23 - q^24 + 6 * q^25 - 14 * q^26 - 16 * q^27 + 10 * q^28 - 9 * q^29 - 21 * q^31 - q^32 + q^33 - 5 * q^35 - 28 * q^36 - 3 * q^37 + 2 * q^38 + 20 * q^39 + 12 * q^40 - 23 * q^41 + q^42 + 7 * q^43 - 4 * q^44 - q^45 - 12 * q^46 - 18 * q^47 - 3 * q^49 - 2 * q^50 - 16 * q^51 + 13 * q^52 - 17 * q^53 + q^54 + 6 * q^55 - 2 * q^56 + q^57 + 23 * q^58 - 29 * q^59 - q^60 + 17 * q^61 + 2 * q^62 + 6 * q^63 - 18 * q^64 + 5 * q^65 + 8 * q^67 - q^68 - 38 * q^69 + 8 * q^70 - 12 * q^71 + 13 * q^72 + 2 * q^73 - 37 * q^74 - q^75 - 4 * q^76 - 5 * q^77 + q^78 + 3 * q^79 - 4 * q^80 - 2 * q^81 + 24 * q^82 - 11 * q^83 - 3 * q^84 - q^85 - 12 * q^86 - 12 * q^87 + 12 * q^88 - 22 * q^89 - 6 * q^90 - 18 * q^91 - 15 * q^92 + 18 * q^93 + 22 * q^94 + 6 * q^95 - 17 * q^96 - 2 * q^97 - q^98 - q^99

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