LMFDB - Embedded newform 23.4.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,4,Mod(1,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("23.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	23.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:	$1.35704393013$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.334189.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 16x^{2} - 5x + 4 $ x^4 - 2x^3 - 16x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.83969$ of defining polynomial
Character	$\chi$	$=$	23.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.86845 q^{2} +3.43737 q^{3} +0.228032 q^{4} -17.9704 q^{5} +9.85995 q^{6} +32.7301 q^{7} -22.2935 q^{8} -15.1845 q^{9} +O(q^{10})$	\(q+2.86845 q^{2} +3.43737 q^{3} +0.228032 q^{4} -17.9704 q^{5} +9.85995 q^{6} +32.7301 q^{7} -22.2935 q^{8} -15.1845 q^{9} -51.5473 q^{10} +26.7049 q^{11} +0.783832 q^{12} -14.4956 q^{13} +93.8848 q^{14} -61.7710 q^{15} -65.7723 q^{16} +24.7016 q^{17} -43.5560 q^{18} +94.6224 q^{19} -4.09784 q^{20} +112.505 q^{21} +76.6018 q^{22} -23.0000 q^{23} -76.6312 q^{24} +197.935 q^{25} -41.5801 q^{26} -145.004 q^{27} +7.46352 q^{28} -57.5965 q^{29} -177.187 q^{30} +88.8691 q^{31} -10.3165 q^{32} +91.7948 q^{33} +70.8553 q^{34} -588.173 q^{35} -3.46255 q^{36} -305.467 q^{37} +271.420 q^{38} -49.8269 q^{39} +400.624 q^{40} -179.205 q^{41} +322.717 q^{42} -96.5826 q^{43} +6.08959 q^{44} +272.871 q^{45} -65.9745 q^{46} +218.484 q^{47} -226.084 q^{48} +728.258 q^{49} +567.769 q^{50} +84.9085 q^{51} -3.30548 q^{52} -519.174 q^{53} -415.937 q^{54} -479.898 q^{55} -729.669 q^{56} +325.252 q^{57} -165.213 q^{58} -37.2884 q^{59} -14.0858 q^{60} +96.3052 q^{61} +254.917 q^{62} -496.989 q^{63} +496.586 q^{64} +260.493 q^{65} +263.309 q^{66} +497.552 q^{67} +5.63276 q^{68} -79.0596 q^{69} -1687.15 q^{70} -19.6235 q^{71} +338.515 q^{72} -208.235 q^{73} -876.219 q^{74} +680.378 q^{75} +21.5770 q^{76} +874.054 q^{77} -142.926 q^{78} -446.200 q^{79} +1181.95 q^{80} -88.4513 q^{81} -514.042 q^{82} +501.151 q^{83} +25.6549 q^{84} -443.897 q^{85} -277.043 q^{86} -197.981 q^{87} -595.347 q^{88} +1102.82 q^{89} +782.718 q^{90} -474.444 q^{91} -5.24475 q^{92} +305.476 q^{93} +626.711 q^{94} -1700.40 q^{95} -35.4615 q^{96} +1814.37 q^{97} +2088.98 q^{98} -405.500 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9	\( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9} - 70 q^{10} + 8 q^{11} - 67 q^{12} + 111 q^{13} - 144 q^{14} + 10 q^{15} + 64 q^{16} + 98 q^{17} + 49 q^{18} + 96 q^{19} + 140 q^{20} + 180 q^{21} + 220 q^{22} - 92 q^{23} - 188 q^{24} + 184 q^{25} - 229 q^{26} - 155 q^{27} + 282 q^{28} + 21 q^{29} - 406 q^{30} - 193 q^{31} - 432 q^{32} - 418 q^{33} + 666 q^{34} - 752 q^{35} - 629 q^{36} + 170 q^{37} + 748 q^{38} - 291 q^{39} - 26 q^{40} - 125 q^{41} + 640 q^{42} + 2 q^{43} + 830 q^{44} + 168 q^{45} - 46 q^{46} - 677 q^{47} + 551 q^{48} + 1220 q^{49} + 414 q^{50} - 340 q^{51} + 2247 q^{52} - 230 q^{53} + 641 q^{54} - 972 q^{55} - 2174 q^{56} + 1322 q^{57} - 1835 q^{58} - 1140 q^{59} - 804 q^{60} + 754 q^{61} + 443 q^{62} - 1092 q^{63} - 805 q^{64} + 1318 q^{65} - 398 q^{66} + 488 q^{67} + 284 q^{68} - 161 q^{69} - 3820 q^{70} - 401 q^{71} + 1503 q^{72} + 1509 q^{73} + 1366 q^{74} + 1401 q^{75} - 3832 q^{76} + 736 q^{77} - 1907 q^{78} - 838 q^{79} + 2846 q^{80} - 932 q^{81} - 949 q^{82} + 142 q^{83} + 2614 q^{84} + 112 q^{85} + 918 q^{86} + 2223 q^{87} - 404 q^{88} + 2342 q^{89} + 1784 q^{90} + 292 q^{91} - 460 q^{92} - 509 q^{93} + 1567 q^{94} - 956 q^{95} + 799 q^{96} + 1062 q^{97} + 2478 q^{98} - 1498 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9 - 70 * q^10 + 8 * q^11 - 67 * q^12 + 111 * q^13 - 144 * q^14 + 10 * q^15 + 64 * q^16 + 98 * q^17 + 49 * q^18 + 96 * q^19 + 140 * q^20 + 180 * q^21 + 220 * q^22 - 92 * q^23 - 188 * q^24 + 184 * q^25 - 229 * q^26 - 155 * q^27 + 282 * q^28 + 21 * q^29 - 406 * q^30 - 193 * q^31 - 432 * q^32 - 418 * q^33 + 666 * q^34 - 752 * q^35 - 629 * q^36 + 170 * q^37 + 748 * q^38 - 291 * q^39 - 26 * q^40 - 125 * q^41 + 640 * q^42 + 2 * q^43 + 830 * q^44 + 168 * q^45 - 46 * q^46 - 677 * q^47 + 551 * q^48 + 1220 * q^49 + 414 * q^50 - 340 * q^51 + 2247 * q^52 - 230 * q^53 + 641 * q^54 - 972 * q^55 - 2174 * q^56 + 1322 * q^57 - 1835 * q^58 - 1140 * q^59 - 804 * q^60 + 754 * q^61 + 443 * q^62 - 1092 * q^63 - 805 * q^64 + 1318 * q^65 - 398 * q^66 + 488 * q^67 + 284 * q^68 - 161 * q^69 - 3820 * q^70 - 401 * q^71 + 1503 * q^72 + 1509 * q^73 + 1366 * q^74 + 1401 * q^75 - 3832 * q^76 + 736 * q^77 - 1907 * q^78 - 838 * q^79 + 2846 * q^80 - 932 * q^81 - 949 * q^82 + 142 * q^83 + 2614 * q^84 + 112 * q^85 + 918 * q^86 + 2223 * q^87 - 404 * q^88 + 2342 * q^89 + 1784 * q^90 + 292 * q^91 - 460 * q^92 - 509 * q^93 + 1567 * q^94 - 956 * q^95 + 799 * q^96 + 1062 * q^97 + 2478 * q^98 - 1498 * 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 $

$2$	2.86845	1.01415	0.507076	−	0.861901i	$-0.330726\pi$
$2$	2.86845	1.01415	0.507076	+	0.861901i	$0.330726\pi$
$3$	3.43737	0.661523	0.330761	−	0.943714i	$-0.392694\pi$
$3$	3.43737	0.661523	0.330761	+	0.943714i	$0.392694\pi$
$4$	0.228032	0.0285041
$5$	−17.9704	−1.60732	−0.803661	−	0.595087i	$-0.797117\pi$
$5$	−17.9704	−1.60732	−0.803661	+	0.595087i	$0.797117\pi$
$6$	9.85995	0.670885
$7$	32.7301	1.76726	0.883629	−	0.468187i	$-0.155093\pi$
$7$	32.7301	1.76726	0.883629	+	0.468187i	$0.155093\pi$
$8$	−22.2935	−0.985244
$9$	−15.1845	−0.562388
$10$	−51.5473	−1.63007
$11$	26.7049	0.731985	0.365993	−	0.930618i	$-0.380730\pi$
$11$	26.7049	0.731985	0.365993	+	0.930618i	$0.380730\pi$
$12$	0.783832	0.0188561
$13$	−14.4956	−0.309259	−0.154630	−	0.987973i	$-0.549418\pi$
$13$	−14.4956	−0.309259	−0.154630	+	0.987973i	$0.549418\pi$
$14$	93.8848	1.79227
$15$	−61.7710	−1.06328
$16$	−65.7723	−1.02769
$17$	24.7016	0.352412	0.176206	−	0.984353i	$-0.443617\pi$
$17$	24.7016	0.352412	0.176206	+	0.984353i	$0.443617\pi$
$18$	−43.5560	−0.570347
$19$	94.6224	1.14252	0.571259	−	0.820770i	$-0.306455\pi$
$19$	94.6224	1.14252	0.571259	+	0.820770i	$0.306455\pi$
$20$	−4.09784	−0.0458152
$21$	112.505	1.16908
$22$	76.6018	0.742344
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−76.6312	−0.651762
$25$	197.935	1.58348
$26$	−41.5801	−0.313636
$27$	−145.004	−1.03355
$28$	7.46352	0.0503740
$29$	−57.5965	−0.368807	−0.184403	−	0.982851i	$-0.559035\pi$
$29$	−57.5965	−0.368807	−0.184403	+	0.982851i	$0.559035\pi$
$30$	−177.187	−1.07833
$31$	88.8691	0.514882	0.257441	−	0.966294i	$-0.417121\pi$
$31$	88.8691	0.514882	0.257441	+	0.966294i	$0.417121\pi$
$32$	−10.3165	−0.0569909
$33$	91.7948	0.484225
$34$	70.8553	0.357400
$35$	−588.173	−2.84055
$36$	−3.46255	−0.0160303
$37$	−305.467	−1.35726	−0.678629	−	0.734482i	$-0.737425\pi$
$37$	−305.467	−1.35726	−0.678629	+	0.734482i	$0.737425\pi$
$38$	271.420	1.15869
$39$	−49.8269	−0.204582
$40$	400.624	1.58360
$41$	−179.205	−0.682613	−0.341306	−	0.939952i	$-0.610869\pi$
$41$	−179.205	−0.682613	−0.341306	+	0.939952i	$0.610869\pi$
$42$	322.717	1.18563
$43$	−96.5826	−0.342528	−0.171264	−	0.985225i	$-0.554785\pi$
$43$	−96.5826	−0.342528	−0.171264	+	0.985225i	$0.554785\pi$
$44$	6.08959	0.0208645
$45$	272.871	0.903938
$46$	−65.9745	−0.211465
$47$	218.484	0.678067	0.339034	−	0.940774i	$-0.389900\pi$
$47$	218.484	0.678067	0.339034	+	0.940774i	$0.389900\pi$
$48$	−226.084	−0.679841
$49$	728.258	2.12320
$50$	567.769	1.60589
$51$	84.9085	0.233129
$52$	−3.30548	−0.00881514
$53$	−519.174	−1.34555	−0.672774	−	0.739848i	$-0.734897\pi$
$53$	−519.174	−1.34555	−0.672774	+	0.739848i	$0.734897\pi$
$54$	−415.937	−1.04818
$55$	−479.898	−1.17654
$56$	−729.669	−1.74118
$57$	325.252	0.755802
$58$	−165.213	−0.374026
$59$	−37.2884	−0.0822803	−0.0411401	−	0.999153i	$-0.513099\pi$
$59$	−37.2884	−0.0822803	−0.0411401	+	0.999153i	$0.513099\pi$
$60$	−14.0858	−0.0303078
$61$	96.3052	0.202141	0.101071	−	0.994879i	$-0.467773\pi$
$61$	96.3052	0.202141	0.101071	+	0.994879i	$0.467773\pi$
$62$	254.917	0.522169
$63$	−496.989	−0.993884
$64$	496.586	0.969894
$65$	260.493	0.497079
$66$	263.309	0.491077
$67$	497.552	0.907248	0.453624	−	0.891193i	$-0.350131\pi$
$67$	497.552	0.907248	0.453624	+	0.891193i	$0.350131\pi$
$68$	5.63276	0.0100452
$69$	−79.0596	−0.137937
$70$	−1687.15	−2.88075
$71$	−19.6235	−0.0328011	−0.0164005	−	0.999866i	$-0.505221\pi$
$71$	−19.6235	−0.0328011	−0.0164005	+	0.999866i	$0.505221\pi$
$72$	338.515	0.554089
$73$	−208.235	−0.333865	−0.166932	−	0.985968i	$-0.553386\pi$
$73$	−208.235	−0.333865	−0.166932	+	0.985968i	$0.553386\pi$
$74$	−876.219	−1.37647
$75$	680.378	1.04751
$76$	21.5770	0.0325664
$77$	874.054	1.29361
$78$	−142.926	−0.207477
$79$	−446.200	−0.635461	−0.317730	−	0.948181i	$-0.602921\pi$
$79$	−446.200	−0.635461	−0.317730	+	0.948181i	$0.602921\pi$
$80$	1181.95	1.65183
$81$	−88.4513	−0.121332
$82$	−514.042	−0.692273
$83$	501.151	0.662752	0.331376	−	0.943499i	$-0.392487\pi$
$83$	501.151	0.662752	0.331376	+	0.943499i	$0.392487\pi$
$84$	25.6549	0.0333236
$85$	−443.897	−0.566440
$86$	−277.043	−0.347375
$87$	−197.981	−0.243974
$88$	−595.347	−0.721184
$89$	1102.82	1.31347	0.656733	−	0.754124i	$-0.271938\pi$
$89$	1102.82	1.31347	0.656733	+	0.754124i	$0.271938\pi$
$90$	782.718	0.916731
$91$	−474.444	−0.546541
$92$	−5.24475	−0.00594351
$93$	305.476	0.340606
$94$	626.711	0.687663
$95$	−1700.40	−1.83640
$96$	−35.4615	−0.0377008
$97$	1814.37	1.89919	0.949595	−	0.313478i	$-0.101494\pi$
$97$	1814.37	1.89919	0.949595	+	0.313478i	$0.101494\pi$
$98$	2088.98	2.15325
$99$	−405.500	−0.411659
$100$	45.1357	0.0451357
$101$	−1386.24	−1.36570	−0.682852	−	0.730557i	$-0.739261\pi$
$101$	−1386.24	−1.36570	−0.682852	+	0.730557i	$0.739261\pi$
$102$	243.556	0.236428
$103$	1372.33	1.31281	0.656407	−	0.754407i	$-0.272075\pi$
$103$	1372.33	1.31281	0.656407	+	0.754407i	$0.272075\pi$
$104$	323.159	0.304696
$105$	−2021.77	−1.87909
$106$	−1489.23	−1.36459
$107$	1657.83	1.49784	0.748919	−	0.662662i	$-0.230573\pi$
$107$	1657.83	1.49784	0.748919	+	0.662662i	$0.230573\pi$
$108$	−33.0656	−0.0294605
$109$	−821.369	−0.721770	−0.360885	−	0.932610i	$-0.617525\pi$
$109$	−821.369	−0.721770	−0.360885	+	0.932610i	$0.617525\pi$
$110$	−1376.57	−1.19319
$111$	−1050.01	−0.897857
$112$	−2152.73	−1.81620
$113$	−267.142	−0.222395	−0.111197	−	0.993798i	$-0.535469\pi$
$113$	−267.142	−0.222395	−0.111197	+	0.993798i	$0.535469\pi$
$114$	932.972	0.766498
$115$	413.319	0.335150
$116$	−13.1339	−0.0105125
$117$	220.109	0.173924
$118$	−106.960	−0.0834447
$119$	808.484	0.622803
$120$	1377.09	1.04759
$121$	−617.847	−0.464198
$122$	276.247	0.205002
$123$	−615.995	−0.451564
$124$	20.2650	0.0146762
$125$	−1310.68	−0.937846
$126$	−1425.59	−1.00795
$127$	−1446.21	−1.01048	−0.505239	−	0.862979i	$-0.668596\pi$
$127$	−1446.21	−1.01048	−0.505239	+	0.862979i	$0.668596\pi$
$128$	1506.97	1.04061
$129$	−331.990	−0.226590
$130$	747.211	0.504114
$131$	1459.33	0.973297	0.486649	−	0.873598i	$-0.338219\pi$
$131$	1459.33	0.973297	0.486649	+	0.873598i	$0.338219\pi$
$132$	20.9322	0.0138024
$133$	3097.00	2.01913
$134$	1427.20	0.920088
$135$	2605.78	1.66126
$136$	−550.685	−0.347212
$137$	−1889.31	−1.17821	−0.589105	−	0.808056i	$-0.700520\pi$
$137$	−1889.31	−1.17821	−0.589105	+	0.808056i	$0.700520\pi$
$138$	−226.779	−0.139889
$139$	−1506.31	−0.919159	−0.459580	−	0.888137i	$-0.652000\pi$
$139$	−1506.31	−0.919159	−0.459580	+	0.888137i	$0.652000\pi$
$140$	−134.122	−0.0809673
$141$	751.011	0.448557
$142$	−56.2890	−0.0332653
$143$	−387.105	−0.226373
$144$	998.717	0.577961
$145$	1035.03	0.592791
$146$	−597.314	−0.338589
$147$	2503.30	1.40455
$148$	−69.6565	−0.0386873
$149$	−2946.38	−1.61998	−0.809989	−	0.586444i	$-0.800527\pi$
$149$	−2946.38	−1.61998	−0.809989	+	0.586444i	$0.800527\pi$
$150$	1951.63	1.06233
$151$	1979.37	1.06675	0.533374	−	0.845880i	$-0.320924\pi$
$151$	1979.37	1.06675	0.533374	+	0.845880i	$0.320924\pi$
$152$	−2109.47	−1.12566
$153$	−375.080	−0.198192
$154$	2507.18	1.31191
$155$	−1597.01	−0.827582
$156$	−11.3622	−0.00583142
$157$	505.403	0.256915	0.128457	−	0.991715i	$-0.458997\pi$
$157$	505.403	0.256915	0.128457	+	0.991715i	$0.458997\pi$
$158$	−1279.90	−0.644454
$159$	−1784.59	−0.890110
$160$	185.391	0.0916027
$161$	−752.792	−0.368499
$162$	−253.719	−0.123049
$163$	−2604.26	−1.25142	−0.625711	−	0.780055i	$-0.715191\pi$
$163$	−2604.26	−1.25142	−0.625711	+	0.780055i	$0.715191\pi$
$164$	−40.8646	−0.0194572
$165$	−1649.59	−0.778305
$166$	1437.53	0.672131
$167$	−845.304	−0.391686	−0.195843	−	0.980635i	$-0.562744\pi$
$167$	−845.304	−0.391686	−0.195843	+	0.980635i	$0.562744\pi$
$168$	−2508.15	−1.15183
$169$	−1986.88	−0.904359
$170$	−1273.30	−0.574456
$171$	−1436.79	−0.642539
$172$	−22.0240	−0.00976343
$173$	886.843	0.389742	0.194871	−	0.980829i	$-0.437571\pi$
$173$	886.843	0.389742	0.194871	+	0.980829i	$0.437571\pi$
$174$	−567.898	−0.247427
$175$	6478.44	2.79842
$176$	−1756.44	−0.752255
$177$	−128.174	−0.0544303
$178$	3163.38	1.33205
$179$	1103.01	0.460575	0.230288	−	0.973123i	$-0.426033\pi$
$179$	1103.01	0.460575	0.230288	+	0.973123i	$0.426033\pi$
$180$	62.2234	0.0257659
$181$	−1200.43	−0.492967	−0.246483	−	0.969147i	$-0.579275\pi$
$181$	−1200.43	−0.492967	−0.246483	+	0.969147i	$0.579275\pi$
$182$	−1360.92	−0.554275
$183$	331.037	0.133721
$184$	512.751	0.205438
$185$	5489.37	2.18155
$186$	876.244	0.345427
$187$	659.653	0.257961
$188$	49.8214	0.0193277
$189$	−4745.98	−1.82656
$190$	−4877.53	−1.86238
$191$	2415.93	0.915237	0.457618	−	0.889149i	$-0.348703\pi$
$191$	2415.93	0.915237	0.457618	+	0.889149i	$0.348703\pi$
$192$	1706.95	0.641607
$193$	232.884	0.0868568	0.0434284	−	0.999057i	$-0.486172\pi$
$193$	232.884	0.0868568	0.0434284	+	0.999057i	$0.486172\pi$
$194$	5204.44	1.92607
$195$	895.410	0.328829
$196$	166.067	0.0605199
$197$	−1418.48	−0.513008	−0.256504	−	0.966543i	$-0.582571\pi$
$197$	−1418.48	−0.513008	−0.256504	+	0.966543i	$0.582571\pi$
$198$	−1163.16	−0.417485
$199$	1068.21	0.380520	0.190260	−	0.981734i	$-0.439067\pi$
$199$	1068.21	0.380520	0.190260	+	0.981734i	$0.439067\pi$
$200$	−4412.68	−1.56012
$201$	1710.27	0.600165
$202$	−3976.37	−1.38503
$203$	−1885.14	−0.651777
$204$	19.3619	0.00664511
$205$	3220.39	1.09718
$206$	3936.47	1.33139
$207$	349.243	0.117266
$208$	953.412	0.317823
$209$	2526.88	0.836307
$210$	−5799.35	−1.90568
$211$	−1537.74	−0.501717	−0.250859	−	0.968024i	$-0.580713\pi$
$211$	−1537.74	−0.501717	−0.250859	+	0.968024i	$0.580713\pi$
$212$	−118.388	−0.0383535
$213$	−67.4532	−0.0216987
$214$	4755.41	1.51903
$215$	1735.63	0.550553
$216$	3232.65	1.01830
$217$	2908.69	0.909930
$218$	−2356.06	−0.731984
$219$	−715.783	−0.220859
$220$	−109.432	−0.0335360
$221$	−358.065	−0.108987
$222$	−3011.89	−0.910563
$223$	−5359.68	−1.60946	−0.804732	−	0.593638i	$-0.797691\pi$
$223$	−5359.68	−1.60946	−0.804732	+	0.593638i	$0.797691\pi$
$224$	−337.658	−0.100718
$225$	−3005.54	−0.890532
$226$	−766.285	−0.225542
$227$	1108.36	0.324072	0.162036	−	0.986785i	$-0.448194\pi$
$227$	1108.36	0.324072	0.162036	+	0.986785i	$0.448194\pi$
$228$	74.1681	0.0215434
$229$	−5046.34	−1.45621	−0.728104	−	0.685467i	$-0.759598\pi$
$229$	−5046.34	−1.45621	−0.728104	+	0.685467i	$0.759598\pi$
$230$	1185.59	0.339893
$231$	3004.45	0.855750
$232$	1284.03	0.363365
$233$	7102.11	1.99689	0.998444	−	0.0557635i	$-0.0177593\pi$
$233$	7102.11	1.99689	0.998444	+	0.0557635i	$0.0177593\pi$
$234$	631.372	0.176385
$235$	−3926.25	−1.08987
$236$	−8.50296	−0.00234532
$237$	−1533.75	−0.420372
$238$	2319.10	0.631617
$239$	1556.22	0.421187	0.210594	−	0.977574i	$-0.432460\pi$
$239$	1556.22	0.421187	0.210594	+	0.977574i	$0.432460\pi$
$240$	4062.82	1.09272
$241$	3028.88	0.809573	0.404787	−	0.914411i	$-0.367346\pi$
$241$	3028.88	0.809573	0.404787	+	0.914411i	$0.367346\pi$
$242$	−1772.27	−0.470767
$243$	3611.06	0.953291
$244$	21.9607	0.00576184
$245$	−13087.1	−3.41267
$246$	−1766.95	−0.457954
$247$	−1371.61	−0.353334
$248$	−1981.21	−0.507285
$249$	1722.64	0.438426
$250$	−3759.63	−0.951118
$251$	1449.39	0.364480	0.182240	−	0.983254i	$-0.441665\pi$
$251$	1449.39	0.364480	0.182240	+	0.983254i	$0.441665\pi$
$252$	−113.330	−0.0283297
$253$	−614.213	−0.152629
$254$	−4148.40	−1.02478
$255$	−1525.84	−0.374713
$256$	349.976	0.0854433
$257$	−3099.62	−0.752332	−0.376166	−	0.926552i	$-0.622758\pi$
$257$	−3099.62	−0.752332	−0.376166	+	0.926552i	$0.622758\pi$
$258$	−952.299	−0.229797
$259$	−9997.97	−2.39862
$260$	59.4008	0.0141688
$261$	874.572	0.207412
$262$	4186.01	0.987071
$263$	−1087.92	−0.255073	−0.127537	−	0.991834i	$-0.540707\pi$
$263$	−1087.92	−0.255073	−0.127537	+	0.991834i	$0.540707\pi$
$264$	−2046.43	−0.477080
$265$	9329.76	2.16273
$266$	8883.60	2.04770
$267$	3790.79	0.868887
$268$	113.458	0.0258603
$269$	4721.01	1.07006	0.535028	−	0.844834i	$-0.320301\pi$
$269$	4721.01	1.07006	0.535028	+	0.844834i	$0.320301\pi$
$270$	7474.55	1.68477
$271$	1401.15	0.314074	0.157037	−	0.987593i	$-0.449806\pi$
$271$	1401.15	0.314074	0.157037	+	0.987593i	$0.449806\pi$
$272$	−1624.68	−0.362171
$273$	−1630.84	−0.361549
$274$	−5419.41	−1.19488
$275$	5285.85	1.15909
$276$	−18.0281	−0.00393176
$277$	4122.82	0.894283	0.447142	−	0.894463i	$-0.352442\pi$
$277$	4122.82	0.894283	0.447142	+	0.894463i	$0.352442\pi$
$278$	−4320.77	−0.932167
$279$	−1349.43	−0.289564
$280$	13112.5	2.79864
$281$	803.897	0.170664	0.0853318	−	0.996353i	$-0.472805\pi$
$281$	803.897	0.170664	0.0853318	+	0.996353i	$0.472805\pi$
$282$	2154.24	0.454905
$283$	−3147.53	−0.661134	−0.330567	−	0.943782i	$-0.607240\pi$
$283$	−3147.53	−0.661134	−0.330567	+	0.943782i	$0.607240\pi$
$284$	−4.47479	−0.000934964 0
$285$	−5844.92	−1.21482
$286$	−1110.39	−0.229577
$287$	−5865.40	−1.20635
$288$	156.650	0.0320510
$289$	−4302.83	−0.875806
$290$	2968.94	0.601181
$291$	6236.67	1.25636
$292$	−47.4844	−0.00951649
$293$	−2812.88	−0.560854	−0.280427	−	0.959875i	$-0.590476\pi$
$293$	−2812.88	−0.560854	−0.280427	+	0.959875i	$0.590476\pi$
$294$	7180.59	1.42442
$295$	670.088	0.132251
$296$	6809.95	1.33723
$297$	−3872.31	−0.756547
$298$	−8451.56	−1.64290
$299$	333.400	0.0644850
$300$	155.148	0.0298583
$301$	−3161.15	−0.605335
$302$	5677.73	1.08184
$303$	−4765.02	−0.903444
$304$	−6223.53	−1.17416
$305$	−1730.64	−0.324906
$306$	−1075.90	−0.200997
$307$	−7885.52	−1.46596	−0.732981	−	0.680249i	$-0.761872\pi$
$307$	−7885.52	−1.46596	−0.732981	+	0.680249i	$0.761872\pi$
$308$	199.313	0.0368730
$309$	4717.22	0.868457
$310$	−4580.96	−0.839294
$311$	−1904.46	−0.347240	−0.173620	−	0.984813i	$-0.555547\pi$
$311$	−1904.46	−0.347240	−0.173620	+	0.984813i	$0.555547\pi$
$312$	1110.82	0.201563
$313$	−3854.89	−0.696137	−0.348069	−	0.937469i	$-0.613162\pi$
$313$	−3854.89	−0.696137	−0.348069	+	0.937469i	$0.613162\pi$
$314$	1449.73	0.260550
$315$	8931.09	1.59749
$316$	−101.748	−0.0181132
$317$	1878.70	0.332865	0.166432	−	0.986053i	$-0.446775\pi$
$317$	1878.70	0.332865	0.166432	+	0.986053i	$0.446775\pi$
$318$	−5119.03	−0.902707
$319$	−1538.11	−0.269961
$320$	−8923.85	−1.55893
$321$	5698.58	0.990853
$322$	−2159.35	−0.373714
$323$	2337.32	0.402638
$324$	−20.1698	−0.00345846
$325$	−2869.20	−0.489707
$326$	−7470.21	−1.26913
$327$	−2823.35	−0.477467
$328$	3995.11	0.672541
$329$	7151.00	1.19832
$330$	−4731.77	−0.789320
$331$	6829.96	1.13417	0.567083	−	0.823661i	$-0.308072\pi$
$331$	6829.96	1.13417	0.567083	+	0.823661i	$0.308072\pi$
$332$	114.279	0.0188911
$333$	4638.36	0.763305
$334$	−2424.72	−0.397229
$335$	−8941.21	−1.45824
$336$	−7399.74	−1.20146
$337$	11048.9	1.78597	0.892985	−	0.450087i	$-0.148607\pi$
$337$	11048.9	1.78597	0.892985	+	0.450087i	$0.148607\pi$
$338$	−5699.26	−0.917157
$339$	−918.267	−0.147119
$340$	−101.223	−0.0161458
$341$	2373.24	0.376886
$342$	−4121.37	−0.651632
$343$	12609.5	1.98499
$344$	2153.17	0.337474
$345$	1420.73	0.221709
$346$	2543.87	0.395258
$347$	−7093.14	−1.09735	−0.548674	−	0.836037i	$-0.684867\pi$
$347$	−7093.14	−1.09735	−0.548674	+	0.836037i	$0.684867\pi$
$348$	−45.1460	−0.00695425
$349$	−5363.16	−0.822588	−0.411294	−	0.911503i	$-0.634923\pi$
$349$	−5363.16	−0.822588	−0.411294	+	0.911503i	$0.634923\pi$
$350$	18583.1	2.83803
$351$	2101.92	0.319636
$352$	−275.500	−0.0417165
$353$	11789.3	1.77757	0.888787	−	0.458320i	$-0.151549\pi$
$353$	11789.3	1.77757	0.888787	+	0.458320i	$0.151549\pi$
$354$	−367.662	−0.0552006
$355$	352.642	0.0527219
$356$	251.478	0.0374391
$357$	2779.06	0.411999
$358$	3163.94	0.467093
$359$	−2645.66	−0.388949	−0.194474	−	0.980908i	$-0.562300\pi$
$359$	−2645.66	−0.388949	−0.194474	+	0.980908i	$0.562300\pi$
$360$	−6083.26	−0.890600
$361$	2094.39	0.305349
$362$	−3443.37	−0.499943
$363$	−2123.77	−0.307077
$364$	−108.189	−0.0155786
$365$	3742.07	0.536628
$366$	949.564	0.135613
$367$	−9775.68	−1.39043	−0.695213	−	0.718804i	$-0.744690\pi$
$367$	−9775.68	−1.39043	−0.695213	+	0.718804i	$0.744690\pi$
$368$	1512.76	0.214289
$369$	2721.13	0.383893
$370$	15746.0	2.21242
$371$	−16992.6	−2.37793
$372$	69.6585	0.00970866
$373$	10981.1	1.52435	0.762174	−	0.647373i	$-0.224132\pi$
$373$	10981.1	1.52435	0.762174	+	0.647373i	$0.224132\pi$
$374$	1892.19	0.261611
$375$	−4505.30	−0.620407
$376$	−4870.78	−0.668062
$377$	834.899	0.114057
$378$	−13613.6	−1.85241
$379$	−2313.77	−0.313589	−0.156795	−	0.987631i	$-0.550116\pi$
$379$	−2313.77	−0.313589	−0.156795	+	0.987631i	$0.550116\pi$
$380$	−387.747	−0.0523447
$381$	−4971.18	−0.668455
$382$	6929.97	0.928189
$383$	9219.25	1.22998	0.614989	−	0.788535i	$-0.289160\pi$
$383$	9219.25	1.22998	0.614989	+	0.788535i	$0.289160\pi$
$384$	5180.00	0.688388
$385$	−15707.1	−2.07924
$386$	668.017	0.0880860
$387$	1466.55	0.192633
$388$	413.736	0.0541346
$389$	5876.90	0.765991	0.382996	−	0.923750i	$-0.374892\pi$
$389$	5876.90	0.765991	0.382996	+	0.923750i	$0.374892\pi$
$390$	2568.44	0.333483
$391$	−568.136	−0.0734830
$392$	−16235.5	−2.09187
$393$	5016.25	0.643858
$394$	−4068.85	−0.520268
$395$	8018.39	1.02139
$396$	−92.4671	−0.0117340
$397$	14268.6	1.80383	0.901916	−	0.431911i	$-0.142160\pi$
$397$	14268.6	1.80383	0.901916	+	0.431911i	$0.142160\pi$
$398$	3064.12	0.385905
$399$	10645.5	1.33570
$400$	−13018.7	−1.62733
$401$	−11556.1	−1.43911	−0.719557	−	0.694434i	$-0.755655\pi$
$401$	−11556.1	−1.43911	−0.719557	+	0.694434i	$0.755655\pi$
$402$	4905.84	0.608659
$403$	−1288.21	−0.159232
$404$	−316.108	−0.0389281
$405$	1589.51	0.195020
$406$	−5407.43	−0.661001
$407$	−8157.48	−0.993492
$408$	−1892.91	−0.229689
$409$	−14148.3	−1.71049	−0.855245	−	0.518223i	$-0.826594\pi$
$409$	−14148.3	−1.71049	−0.855245	+	0.518223i	$0.826594\pi$
$410$	9237.54	1.11271
$411$	−6494.27	−0.779413
$412$	312.936	0.0374205
$413$	−1220.45	−0.145410
$414$	1001.79	0.118925
$415$	−9005.88	−1.06526
$416$	149.544	0.0176250
$417$	−5177.73	−0.608045
$418$	7248.25	0.848142
$419$	−2222.16	−0.259092	−0.129546	−	0.991573i	$-0.541352\pi$
$419$	−2222.16	−0.259092	−0.129546	+	0.991573i	$0.541352\pi$
$420$	−461.029	−0.0535617
$421$	2518.51	0.291555	0.145777	−	0.989317i	$-0.453432\pi$
$421$	2518.51	0.291555	0.145777	+	0.989317i	$0.453432\pi$
$422$	−4410.94	−0.508818
$423$	−3317.56	−0.381337
$424$	11574.2	1.32569
$425$	4889.31	0.558039
$426$	−193.486	−0.0220057
$427$	3152.08	0.357236
$428$	378.039	0.0426944
$429$	−1330.62	−0.149751
$430$	4978.57	0.558344
$431$	9935.17	1.11035	0.555174	−	0.831734i	$-0.312652\pi$
$431$	9935.17	1.11035	0.555174	+	0.831734i	$0.312652\pi$
$432$	9537.22	1.06218
$433$	−8427.47	−0.935331	−0.467665	−	0.883906i	$-0.654905\pi$
$433$	−8427.47	−0.935331	−0.467665	+	0.883906i	$0.654905\pi$
$434$	8343.45	0.922807
$435$	3557.79	0.392145
$436$	−187.299	−0.0205734
$437$	−2176.31	−0.238232
$438$	−2053.19	−0.223985
$439$	4886.04	0.531203	0.265601	−	0.964083i	$-0.414430\pi$
$439$	4886.04	0.531203	0.265601	+	0.964083i	$0.414430\pi$
$440$	10698.6	1.15918
$441$	−11058.2	−1.19406
$442$	−1027.09	−0.110529
$443$	9724.04	1.04290	0.521448	−	0.853283i	$-0.325392\pi$
$443$	9724.04	1.04290	0.521448	+	0.853283i	$0.325392\pi$
$444$	−239.435	−0.0255926
$445$	−19818.1	−2.11116
$446$	−15374.0	−1.63224
$447$	−10127.8	−1.07165
$448$	16253.3	1.71405
$449$	−10093.7	−1.06091	−0.530457	−	0.847712i	$-0.677980\pi$
$449$	−10093.7	−1.06091	−0.530457	+	0.847712i	$0.677980\pi$
$450$	−8621.27	−0.903134
$451$	−4785.66	−0.499662
$452$	−60.9171	−0.00633915
$453$	6803.83	0.705678
$454$	3179.28	0.328658
$455$	8525.95	0.878467
$456$	−7251.02	−0.744650
$457$	3561.77	0.364578	0.182289	−	0.983245i	$-0.441649\pi$
$457$	3561.77	0.364578	0.182289	+	0.983245i	$0.441649\pi$
$458$	−14475.2	−1.47682
$459$	−3581.82	−0.364237
$460$	94.2502	0.00955313
$461$	−7492.00	−0.756914	−0.378457	−	0.925619i	$-0.623545\pi$
$461$	−7492.00	−0.756914	−0.378457	+	0.925619i	$0.623545\pi$
$462$	8618.13	0.867861
$463$	15427.0	1.54849	0.774247	−	0.632884i	$-0.218129\pi$
$463$	15427.0	1.54849	0.774247	+	0.632884i	$0.218129\pi$
$464$	3788.25	0.379020
$465$	−5489.53	−0.547464
$466$	20372.1	2.02515
$467$	11868.7	1.17606	0.588028	−	0.808841i	$-0.299904\pi$
$467$	11868.7	1.17606	0.588028	+	0.808841i	$0.299904\pi$
$468$	50.1919	0.00495753
$469$	16284.9	1.60334
$470$	−11262.3	−1.10530
$471$	1737.26	0.169955
$472$	831.290	0.0810662
$473$	−2579.23	−0.250725
$474$	−4399.51	−0.426321
$475$	18729.1	1.80916
$476$	184.361	0.0177524
$477$	7883.38	0.756719
$478$	4463.96	0.427148
$479$	697.153	0.0665005	0.0332503	−	0.999447i	$-0.489414\pi$
$479$	697.153	0.0665005	0.0332503	+	0.999447i	$0.489414\pi$
$480$	637.258	0.0605973
$481$	4427.95	0.419744
$482$	8688.20	0.821030
$483$	−2587.63	−0.243770
$484$	−140.889	−0.0132315
$485$	−32605.0	−3.05261
$486$	10358.2	0.966782
$487$	14414.0	1.34119	0.670597	−	0.741822i	$-0.266038\pi$
$487$	14414.0	1.34119	0.670597	+	0.741822i	$0.266038\pi$
$488$	−2146.98	−0.199159
$489$	−8951.83	−0.827844
$490$	−37539.7	−3.46096
$491$	17538.9	1.61206	0.806029	−	0.591876i	$-0.201612\pi$
$491$	17538.9	1.61206	0.806029	+	0.591876i	$0.201612\pi$
$492$	−140.467	−0.0128714
$493$	−1422.72	−0.129972
$494$	−3934.41	−0.358335
$495$	7287.00	0.661669
$496$	−5845.12	−0.529140
$497$	−642.278	−0.0579680
$498$	4941.32	0.444630
$499$	−18798.6	−1.68645	−0.843226	−	0.537559i	$-0.819347\pi$
$499$	−18798.6	−1.68645	−0.843226	+	0.537559i	$0.819347\pi$
$500$	−298.878	−0.0267324
$501$	−2905.62	−0.259109
$502$	4157.50	0.369638
$503$	4634.11	0.410785	0.205392	−	0.978680i	$-0.434153\pi$
$503$	4634.11	0.410785	0.205392	+	0.978680i	$0.434153\pi$
$504$	11079.6	0.979219
$505$	24911.3	2.19513
$506$	−1761.84	−0.154789
$507$	−6829.63	−0.598254
$508$	−329.784	−0.0288027
$509$	−2193.57	−0.191018	−0.0955092	−	0.995429i	$-0.530448\pi$
$509$	−2193.57	−0.191018	−0.0955092	+	0.995429i	$0.530448\pi$
$510$	−4376.80	−0.380016
$511$	−6815.56	−0.590025
$512$	−11051.8	−0.953958
$513$	−13720.6	−1.18086
$514$	−8891.13	−0.762979
$515$	−24661.4	−2.11012
$516$	−75.7045	−0.00645873
$517$	5834.60	0.496335
$518$	−28678.7	−2.43257
$519$	3048.41	0.257823
$520$	−5807.30	−0.489744
$521$	−4401.81	−0.370147	−0.185074	−	0.982725i	$-0.559252\pi$
$521$	−4401.81	−0.370147	−0.185074	+	0.982725i	$0.559252\pi$
$522$	2508.67	0.210348
$523$	−9974.09	−0.833913	−0.416957	−	0.908926i	$-0.636903\pi$
$523$	−9974.09	−0.833913	−0.416957	+	0.908926i	$0.636903\pi$
$524$	332.774	0.0277429
$525$	22268.8	1.85122
$526$	−3120.66	−0.258683
$527$	2195.20	0.181451
$528$	−6037.55	−0.497634
$529$	529.000	0.0434783
$530$	26762.0	2.19333
$531$	566.204	0.0462734
$532$	706.216	0.0575533
$533$	2597.69	0.211104
$534$	10873.7	0.881183
$535$	−29791.9	−2.40751
$536$	−11092.2	−0.893861
$537$	3791.46	0.304681
$538$	13542.0	1.08520
$539$	19448.1	1.55415
$540$	594.201	0.0473525
$541$	−9161.72	−0.728083	−0.364042	−	0.931383i	$-0.618603\pi$
$541$	−9161.72	−0.728083	−0.364042	+	0.931383i	$0.618603\pi$
$542$	4019.15	0.318519
$543$	−4126.31	−0.326109
$544$	−254.833	−0.0200843
$545$	14760.3	1.16012
$546$	−4677.99	−0.366666
$547$	1113.51	0.0870390	0.0435195	−	0.999053i	$-0.486143\pi$
$547$	1113.51	0.0870390	0.0435195	+	0.999053i	$0.486143\pi$
$548$	−430.825	−0.0335838
$549$	−1462.34	−0.113682
$550$	15162.2	1.17549
$551$	−5449.92	−0.421369
$552$	1762.52	0.135902
$553$	−14604.2	−1.12302
$554$	11826.1	0.906939
$555$	18869.0	1.44314
$556$	−343.486	−0.0261998
$557$	−7660.96	−0.582775	−0.291387	−	0.956605i	$-0.594117\pi$
$557$	−7660.96	−0.582775	−0.291387	+	0.956605i	$0.594117\pi$
$558$	−3870.78	−0.293661
$559$	1400.03	0.105930
$560$	38685.5	2.91921
$561$	2267.47	0.170647
$562$	2305.94	0.173079
$563$	17217.7	1.28888	0.644441	−	0.764654i	$-0.277090\pi$
$563$	17217.7	1.28888	0.644441	+	0.764654i	$0.277090\pi$
$564$	171.255	0.0127857
$565$	4800.65	0.357460
$566$	−9028.54	−0.670491
$567$	−2895.02	−0.214426
$568$	437.476	0.0323171
$569$	−2385.63	−0.175766	−0.0878830	−	0.996131i	$-0.528010\pi$
$569$	−2385.63	−0.175766	−0.0878830	+	0.996131i	$0.528010\pi$
$570$	−16765.9	−1.23201
$571$	16927.9	1.24065	0.620323	−	0.784347i	$-0.287002\pi$
$571$	16927.9	1.24065	0.620323	+	0.784347i	$0.287002\pi$
$572$	−88.2725	−0.00645255
$573$	8304.44	0.605450
$574$	−16824.6	−1.22343
$575$	−4552.52	−0.330179
$576$	−7540.39	−0.545457
$577$	2886.69	0.208275	0.104137	−	0.994563i	$-0.466792\pi$
$577$	2886.69	0.208275	0.104137	+	0.994563i	$0.466792\pi$
$578$	−12342.5	−0.888200
$579$	800.509	0.0574578
$580$	236.021	0.0168970
$581$	16402.7	1.17125
$582$	17889.6	1.27414
$583$	−13864.5	−0.984920
$584$	4642.30	0.328938
$585$	−3955.44	−0.279551
$586$	−8068.62	−0.568791
$587$	503.810	0.0354250	0.0177125	−	0.999843i	$-0.494362\pi$
$587$	503.810	0.0354250	0.0177125	+	0.999843i	$0.494362\pi$
$588$	570.832	0.0400353
$589$	8409.00	0.588263
$590$	1922.12	0.134122
$591$	−4875.85	−0.339366
$592$	20091.3	1.39484
$593$	−7654.92	−0.530101	−0.265050	−	0.964235i	$-0.585389\pi$
$593$	−7654.92	−0.530101	−0.265050	+	0.964235i	$0.585389\pi$
$594$	−11107.6	−0.767253
$595$	−14528.8	−1.00105
$596$	−671.870	−0.0461760
$597$	3671.84	0.251723
$598$	956.343	0.0653976
$599$	17366.5	1.18460	0.592301	−	0.805717i	$-0.298220\pi$
$599$	17366.5	1.18460	0.592301	+	0.805717i	$0.298220\pi$
$600$	−15168.0	−1.03205
$601$	3872.81	0.262854	0.131427	−	0.991326i	$-0.458044\pi$
$601$	3872.81	0.262854	0.131427	+	0.991326i	$0.458044\pi$
$602$	−9067.63	−0.613902
$603$	−7555.06	−0.510225
$604$	451.361	0.0304066
$605$	11103.0	0.746115
$606$	−13668.3	−0.916229
$607$	−12823.3	−0.857464	−0.428732	−	0.903432i	$-0.641039\pi$
$607$	−12823.3	−0.857464	−0.428732	+	0.903432i	$0.641039\pi$
$608$	−976.167	−0.0651132
$609$	−6479.92	−0.431165
$610$	−4964.27	−0.329504
$611$	−3167.07	−0.209699
$612$	−85.5304	−0.00564928
$613$	−17226.4	−1.13502	−0.567509	−	0.823367i	$-0.692093\pi$
$613$	−17226.4	−1.13502	−0.567509	+	0.823367i	$0.692093\pi$
$614$	−22619.3	−1.48671
$615$	11069.7	0.725809
$616$	−19485.8	−1.27452
$617$	−5892.12	−0.384454	−0.192227	−	0.981351i	$-0.561571\pi$
$617$	−5892.12	−0.384454	−0.192227	+	0.981351i	$0.561571\pi$
$618$	13531.1	0.880747
$619$	−13036.6	−0.846503	−0.423251	−	0.906012i	$-0.639111\pi$
$619$	−13036.6	−0.846503	−0.423251	+	0.906012i	$0.639111\pi$
$620$	−364.171	−0.0235894
$621$	3335.09	0.215511
$622$	−5462.84	−0.352154
$623$	36095.3	2.32123
$624$	3277.23	0.210247
$625$	−1188.48	−0.0760630
$626$	−11057.6	−0.705989
$627$	8685.84	0.553236
$628$	115.248	0.00732310
$629$	−7545.52	−0.478314
$630$	25618.4	1.62010
$631$	−10044.6	−0.633707	−0.316853	−	0.948475i	$-0.602626\pi$
$631$	−10044.6	−0.633707	−0.316853	+	0.948475i	$0.602626\pi$
$632$	9947.37	0.626084
$633$	−5285.79	−0.331898
$634$	5388.95	0.337575
$635$	25989.1	1.62416
$636$	−406.945	−0.0253717
$637$	−10556.6	−0.656620
$638$	−4412.00	−0.273782
$639$	297.972	0.0184469
$640$	−27080.8	−1.67260
$641$	4583.33	0.282419	0.141209	−	0.989980i	$-0.454901\pi$
$641$	4583.33	0.282419	0.141209	+	0.989980i	$0.454901\pi$
$642$	16346.1	1.00488
$643$	19260.0	1.18125	0.590623	−	0.806948i	$-0.298882\pi$
$643$	19260.0	1.18125	0.590623	+	0.806948i	$0.298882\pi$
$644$	−171.661	−0.0105037
$645$	5966.00	0.364203
$646$	6704.50	0.408336
$647$	−1771.91	−0.107667	−0.0538337	−	0.998550i	$-0.517144\pi$
$647$	−1771.91	−0.107667	−0.0538337	+	0.998550i	$0.517144\pi$
$648$	1971.89	0.119542
$649$	−995.784	−0.0602279
$650$	−8230.18	−0.496637
$651$	9998.26	0.601940
$652$	−593.857	−0.0356706
$653$	−28000.8	−1.67803	−0.839017	−	0.544106i	$-0.816869\pi$
$653$	−28000.8	−1.67803	−0.839017	+	0.544106i	$0.816869\pi$
$654$	−8098.66	−0.484224
$655$	−26224.7	−1.56440
$656$	11786.7	0.701515
$657$	3161.94	0.187761
$658$	20512.3	1.21528
$659$	−27664.0	−1.63526	−0.817629	−	0.575745i	$-0.804712\pi$
$659$	−27664.0	−1.63526	−0.817629	+	0.575745i	$0.804712\pi$
$660$	−376.160	−0.0221848
$661$	23392.1	1.37647	0.688234	−	0.725489i	$-0.258386\pi$
$661$	23392.1	1.37647	0.688234	+	0.725489i	$0.258386\pi$
$662$	19591.4	1.15022
$663$	−1230.80	−0.0720972
$664$	−11172.4	−0.652973
$665$	−55654.3	−3.24539
$666$	13304.9	0.774107
$667$	1324.72	0.0769016
$668$	−192.757	−0.0111646
$669$	−18423.2	−1.06470
$670$	−25647.5	−1.47888
$671$	2571.82	0.147964
$672$	−1160.66	−0.0666270
$673$	4318.41	0.247344	0.123672	−	0.992323i	$-0.460533\pi$
$673$	4318.41	0.247344	0.123672	+	0.992323i	$0.460533\pi$
$674$	31693.3	1.81124
$675$	−28701.4	−1.63662
$676$	−453.072	−0.0257779
$677$	18272.8	1.03734	0.518671	−	0.854974i	$-0.326427\pi$
$677$	18272.8	1.03734	0.518671	+	0.854974i	$0.326427\pi$
$678$	−2634.01	−0.149201
$679$	59384.5	3.35636
$680$	9896.04	0.558082
$681$	3809.84	0.214381
$682$	6807.53	0.382220
$683$	−7245.38	−0.405910	−0.202955	−	0.979188i	$-0.565055\pi$
$683$	−7245.38	−0.405910	−0.202955	+	0.979188i	$0.565055\pi$
$684$	−327.635	−0.0183150
$685$	33951.7	1.89376
$686$	36169.9	2.01308
$687$	−17346.2	−0.963314
$688$	6352.45	0.352013
$689$	7525.76	0.416123
$690$	4075.31	0.224847
$691$	−12055.2	−0.663679	−0.331839	−	0.943336i	$-0.607669\pi$
$691$	−12055.2	−0.663679	−0.331839	+	0.943336i	$0.607669\pi$
$692$	202.229	0.0111092
$693$	−13272.0	−0.727509
$694$	−20346.3	−1.11288
$695$	27068.9	1.47738
$696$	4413.69	0.240374
$697$	−4426.64	−0.240561
$698$	−15384.0	−0.834229
$699$	24412.6	1.32099
$700$	1477.30	0.0797664
$701$	−15301.9	−0.824455	−0.412228	−	0.911081i	$-0.635249\pi$
$701$	−15301.9	−0.824455	−0.412228	+	0.911081i	$0.635249\pi$
$702$	6029.27	0.324160
$703$	−28904.0	−1.55069
$704$	13261.3	0.709948
$705$	−13496.0	−0.720975
$706$	33817.2	1.80273
$707$	−45371.8	−2.41355
$708$	−29.2279	−0.00155148
$709$	−12173.6	−0.644835	−0.322418	−	0.946597i	$-0.604496\pi$
$709$	−12173.6	−0.644835	−0.322418	+	0.946597i	$0.604496\pi$
$710$	1011.54	0.0534680
$711$	6775.30	0.357375
$712$	−24585.7	−1.29408
$713$	−2043.99	−0.107360
$714$	7971.61	0.417829
$715$	6956.44	0.363854
$716$	251.522	0.0131283
$717$	5349.32	0.278625
$718$	−7588.95	−0.394453
$719$	2639.63	0.136914	0.0684572	−	0.997654i	$-0.478192\pi$
$719$	2639.63	0.136914	0.0684572	+	0.997654i	$0.478192\pi$
$720$	−17947.3	−0.928970
$721$	44916.5	2.32008
$722$	6007.67	0.309671
$723$	10411.4	0.535551
$724$	−273.736	−0.0140515
$725$	−11400.4	−0.584000
$726$	−6091.94	−0.311423
$727$	−32759.5	−1.67123	−0.835613	−	0.549319i	$-0.814887\pi$
$727$	−32759.5	−1.67123	−0.835613	+	0.549319i	$0.814887\pi$
$728$	10577.0	0.538476
$729$	14800.7	0.751956
$730$	10734.0	0.544222
$731$	−2385.74	−0.120711
$732$	75.4871	0.00381159
$733$	29178.0	1.47028	0.735139	−	0.677917i	$-0.237117\pi$
$733$	29178.0	1.47028	0.735139	+	0.677917i	$0.237117\pi$
$734$	−28041.1	−1.41010
$735$	−44985.2	−2.25756
$736$	237.278	0.0118834
$737$	13287.1	0.664092
$738$	7805.45	0.389326
$739$	−13704.9	−0.682194	−0.341097	−	0.940028i	$-0.610798\pi$
$739$	−13704.9	−0.682194	−0.341097	+	0.940028i	$0.610798\pi$
$740$	1251.75	0.0621830
$741$	−4714.74	−0.233739
$742$	−48742.5	−2.41158
$743$	−991.593	−0.0489610	−0.0244805	−	0.999700i	$-0.507793\pi$
$743$	−991.593	−0.0489610	−0.0244805	+	0.999700i	$0.507793\pi$
$744$	−6810.14	−0.335581
$745$	52947.6	2.60383
$746$	31498.9	1.54592
$747$	−7609.71	−0.372724
$748$	150.422	0.00735292
$749$	54260.9	2.64706
$750$	−12923.2	−0.629186
$751$	9440.73	0.458718	0.229359	−	0.973342i	$-0.426337\pi$
$751$	9440.73	0.458718	0.229359	+	0.973342i	$0.426337\pi$
$752$	−14370.2	−0.696844
$753$	4982.09	0.241112
$754$	2394.87	0.115671
$755$	−35570.1	−1.71461
$756$	−1082.24	−0.0520643
$757$	10480.1	0.503177	0.251589	−	0.967834i	$-0.419047\pi$
$757$	10480.1	0.503177	0.251589	+	0.967834i	$0.419047\pi$
$758$	−6636.94	−0.318027
$759$	−2111.28	−0.100968
$760$	37908.0	1.80930
$761$	−31314.9	−1.49168	−0.745838	−	0.666128i	$-0.767951\pi$
$761$	−31314.9	−1.49168	−0.745838	+	0.666128i	$0.767951\pi$
$762$	−14259.6	−0.677915
$763$	−26883.5	−1.27555
$764$	550.909	0.0260880
$765$	6740.34	0.318559
$766$	26445.0	1.24739
$767$	540.519	0.0254459
$768$	1203.00	0.0565227
$769$	−20862.1	−0.978292	−0.489146	−	0.872202i	$-0.662691\pi$
$769$	−20862.1	−0.978292	−0.489146	+	0.872202i	$0.662691\pi$
$770$	−45055.1	−2.10867
$771$	−10654.6	−0.497684
$772$	53.1051	0.00247577
$773$	−20340.2	−0.946426	−0.473213	−	0.880948i	$-0.656906\pi$
$773$	−20340.2	−0.946426	−0.473213	+	0.880948i	$0.656906\pi$
$774$	4206.75	0.195360
$775$	17590.3	0.815308
$776$	−40448.8	−1.87117
$777$	−34366.8	−1.58674
$778$	16857.6	0.776832
$779$	−16956.8	−0.779898
$780$	204.183	0.00937296
$781$	−524.043	−0.0240099
$782$	−1629.67	−0.0745230
$783$	8351.71	0.381182
$784$	−47899.2	−2.18200
$785$	−9082.30	−0.412944
$786$	14388.9	0.652970
$787$	31293.8	1.41741	0.708707	−	0.705503i	$-0.249279\pi$
$787$	31293.8	1.41741	0.708707	+	0.705503i	$0.249279\pi$
$788$	−323.460	−0.0146228
$789$	−3739.60	−0.168737
$790$	23000.4	1.03584
$791$	−8743.58	−0.393029
$792$	9040.03	0.405585
$793$	−1396.01	−0.0625140
$794$	40928.9	1.82936
$795$	32069.9	1.43069
$796$	243.587	0.0108464
$797$	32015.4	1.42289	0.711446	−	0.702741i	$-0.248041\pi$
$797$	32015.4	1.42289	0.711446	+	0.702741i	$0.248041\pi$
$798$	30536.2	1.35460
$799$	5396.89	0.238959
$800$	−2041.99	−0.0902442
$801$	−16745.7	−0.738677
$802$	−33148.2	−1.45948
$803$	−5560.91	−0.244384
$804$	389.997	0.0171071
$805$	13528.0	0.592296
$806$	−3695.19	−0.161486
$807$	16227.9	0.707867
$808$	30904.2	1.34555
$809$	5234.16	0.227470	0.113735	−	0.993511i	$-0.463719\pi$
$809$	5234.16	0.227470	0.113735	+	0.993511i	$0.463719\pi$
$810$	4559.43	0.197780
$811$	−2377.40	−0.102937	−0.0514685	−	0.998675i	$-0.516390\pi$
$811$	−2377.40	−0.102937	−0.0514685	+	0.998675i	$0.516390\pi$
$812$	−429.873	−0.0185783
$813$	4816.29	0.207767
$814$	−23399.4	−1.00755
$815$	46799.7	2.01144
$816$	−5584.62	−0.239584
$817$	−9138.87	−0.391345
$818$	−40583.9	−1.73470
$819$	7204.18	0.307368
$820$	734.353	0.0312740
$821$	5174.70	0.219974	0.109987	−	0.993933i	$-0.464919\pi$
$821$	5174.70	0.219974	0.109987	+	0.993933i	$0.464919\pi$
$822$	−18628.5	−0.790443
$823$	44178.3	1.87115	0.935576	−	0.353126i	$-0.114881\pi$
$823$	44178.3	1.87115	0.935576	+	0.353126i	$0.114881\pi$
$824$	−30594.1	−1.29344
$825$	18169.4	0.766762
$826$	−3500.81	−0.147468
$827$	5766.56	0.242470	0.121235	−	0.992624i	$-0.461314\pi$
$827$	5766.56	0.242470	0.121235	+	0.992624i	$0.461314\pi$
$828$	79.6387	0.00334255
$829$	38465.6	1.61154	0.805770	−	0.592229i	$-0.201752\pi$
$829$	38465.6	1.61154	0.805770	+	0.592229i	$0.201752\pi$
$830$	−25833.0	−1.08033
$831$	14171.7	0.591589
$832$	−7198.33	−0.299949
$833$	17989.1	0.748242
$834$	−14852.1	−0.616650
$835$	15190.5	0.629566
$836$	576.211	0.0238381
$837$	−12886.3	−0.532159
$838$	−6374.17	−0.262759
$839$	−17261.8	−0.710301	−0.355151	−	0.934809i	$-0.615570\pi$
$839$	−17261.8	−0.710301	−0.355151	+	0.934809i	$0.615570\pi$
$840$	45072.4	1.85136
$841$	−21071.6	−0.863981
$842$	7224.22	0.295681
$843$	2763.29	0.112898
$844$	−350.655	−0.0143010
$845$	35705.0	1.45360
$846$	−9516.28	−0.386733
$847$	−20222.2	−0.820358
$848$	34147.2	1.38281
$849$	−10819.2	−0.437355
$850$	14024.8	0.565936
$851$	7025.75	0.283008
$852$	−15.3815	−0.000618500 0
$853$	29038.7	1.16561	0.582805	−	0.812612i	$-0.301955\pi$
$853$	29038.7	1.16561	0.582805	+	0.812612i	$0.301955\pi$
$854$	9041.59	0.362291
$855$	25819.7	1.03277
$856$	−36958.9	−1.47574
$857$	−9865.16	−0.393217	−0.196609	−	0.980482i	$-0.562993\pi$
$857$	−9865.16	−0.393217	−0.196609	+	0.980482i	$0.562993\pi$
$858$	−3816.84	−0.151870
$859$	−24476.6	−0.972214	−0.486107	−	0.873899i	$-0.661584\pi$
$859$	−24476.6	−0.972214	−0.486107	+	0.873899i	$0.661584\pi$
$860$	395.779	0.0156930
$861$	−20161.6	−0.798030
$862$	28498.6	1.12606
$863$	−37353.6	−1.47339	−0.736693	−	0.676227i	$-0.763614\pi$
$863$	−37353.6	−1.47339	−0.736693	+	0.676227i	$0.763614\pi$
$864$	1495.92	0.0589032
$865$	−15936.9	−0.626442
$866$	−24173.8	−0.948567
$867$	−14790.4	−0.579365
$868$	663.276	0.0259367
$869$	−11915.7	−0.465148
$870$	10205.4	0.397695
$871$	−7212.34	−0.280575
$872$	18311.2	0.711120
$873$	−27550.3	−1.06808
$874$	−6242.66	−0.241603
$875$	−42898.7	−1.65742
$876$	−163.222	−0.00629538
$877$	−25920.7	−0.998040	−0.499020	−	0.866590i	$-0.666307\pi$
$877$	−25920.7	−0.998040	−0.499020	+	0.866590i	$0.666307\pi$
$878$	14015.4	0.538720
$879$	−9668.92	−0.371018
$880$	31564.0	1.20912
$881$	−9337.06	−0.357064	−0.178532	−	0.983934i	$-0.557135\pi$
$881$	−9337.06	−0.357064	−0.178532	+	0.983934i	$0.557135\pi$
$882$	−31720.0	−1.21096
$883$	−70.8656	−0.00270081	−0.00135041	−	0.999999i	$-0.500430\pi$
$883$	−70.8656	−0.00270081	−0.00135041	+	0.999999i	$0.500430\pi$
$884$	−81.6505	−0.00310656
$885$	2303.34	0.0874870
$886$	27893.0	1.05765
$887$	−1058.85	−0.0400819	−0.0200409	−	0.999799i	$-0.506380\pi$
$887$	−1058.85	−0.0400819	−0.0200409	+	0.999799i	$0.506380\pi$
$888$	23408.3	0.884608
$889$	−47334.7	−1.78578
$890$	−56847.2	−2.14104
$891$	−2362.08	−0.0888135
$892$	−1222.18	−0.0458762
$893$	20673.5	0.774705
$894$	−29051.2	−1.08682
$895$	−19821.6	−0.740293
$896$	49323.1	1.83903
$897$	1146.02	0.0426583
$898$	−28953.3	−1.07593
$899$	−5118.55	−0.189892
$900$	−685.362	−0.0253838
$901$	−12824.4	−0.474187
$902$	−13727.4	−0.506734
$903$	−10866.1	−0.400443
$904$	5955.54	0.219113
$905$	21572.1	0.792356
$906$	19516.5	0.715664
$907$	29050.2	1.06350	0.531751	−	0.846900i	$-0.321534\pi$
$907$	29050.2	1.06350	0.531751	+	0.846900i	$0.321534\pi$
$908$	252.742	0.00923736
$909$	21049.3	0.768055
$910$	24456.3	0.890899
$911$	16041.6	0.583404	0.291702	−	0.956509i	$-0.405778\pi$
$911$	16041.6	0.583404	0.291702	+	0.956509i	$0.405778\pi$
$912$	−21392.6	−0.776732
$913$	13383.2	0.485125
$914$	10216.8	0.369738
$915$	−5948.87	−0.214933
$916$	−1150.73	−0.0415078
$917$	47763.9	1.72007
$918$	−10274.3	−0.369392
$919$	−30151.2	−1.08226	−0.541130	−	0.840939i	$-0.682003\pi$
$919$	−30151.2	−1.08226	−0.541130	+	0.840939i	$0.682003\pi$
$920$	−9214.35	−0.330204
$921$	−27105.5	−0.969767
$922$	−21490.5	−0.767626
$923$	284.455	0.0101440
$924$	685.112	0.0243923
$925$	−60462.8	−2.14919
$926$	44251.6	1.57041
$927$	−20838.1	−0.738311
$928$	594.192	0.0210186
$929$	43339.8	1.53061	0.765303	−	0.643670i	$-0.222589\pi$
$929$	43339.8	1.53061	0.765303	+	0.643670i	$0.222589\pi$
$930$	−15746.5	−0.555212
$931$	68909.5	2.42580
$932$	1619.51	0.0569194
$933$	−6546.32	−0.229707
$934$	34044.8	1.19270
$935$	−11854.2	−0.414626
$936$	−4907.00	−0.171357
$937$	27291.9	0.951533	0.475766	−	0.879572i	$-0.342171\pi$
$937$	27291.9	0.951533	0.475766	+	0.879572i	$0.342171\pi$
$938$	46712.5	1.62603
$939$	−13250.7	−0.460511
$940$	−895.311	−0.0310658
$941$	−4358.15	−0.150980	−0.0754898	−	0.997147i	$-0.524052\pi$
$941$	−4358.15	−0.150980	−0.0754898	+	0.997147i	$0.524052\pi$
$942$	4983.25	0.172360
$943$	4121.72	0.142335
$944$	2452.54	0.0845587
$945$	85287.3	2.93587
$946$	−7398.40	−0.254274
$947$	39067.8	1.34059	0.670293	−	0.742097i	$-0.266169\pi$
$947$	39067.8	1.34059	0.670293	+	0.742097i	$0.266169\pi$
$948$	−349.746	−0.0119823
$949$	3018.51	0.103251
$950$	53723.6	1.83476
$951$	6457.78	0.220197
$952$	−18024.0	−0.613614
$953$	−52563.8	−1.78668	−0.893341	−	0.449379i	$-0.851645\pi$
$953$	−52563.8	−1.78668	−0.893341	+	0.449379i	$0.851645\pi$
$954$	22613.1	0.767428
$955$	−43415.2	−1.47108
$956$	354.869	0.0120055
$957$	−5287.06	−0.178585
$958$	1999.75	0.0674416
$959$	−61837.4	−2.08220
$960$	−30674.6	−1.03127
$961$	−21893.3	−0.734896
$962$	12701.4	0.425684
$963$	−25173.3	−0.842365
$964$	690.682	0.0230761
$965$	−4185.02	−0.139607
$966$	−7422.49	−0.247220
$967$	−2440.46	−0.0811580	−0.0405790	−	0.999176i	$-0.512920\pi$
$967$	−2440.46	−0.0811580	−0.0405790	+	0.999176i	$0.512920\pi$
$968$	13774.0	0.457348
$969$	8034.24	0.266354
$970$	−93525.9	−3.09581
$971$	45490.7	1.50347	0.751733	−	0.659468i	$-0.229218\pi$
$971$	45490.7	1.50347	0.751733	+	0.659468i	$0.229218\pi$
$972$	823.439	0.0271727
$973$	−49301.5	−1.62439
$974$	41345.9	1.36017
$975$	−9862.52	−0.323952
$976$	−6334.21	−0.207739
$977$	−33244.4	−1.08862	−0.544311	−	0.838884i	$-0.683209\pi$
$977$	−33244.4	−1.08862	−0.544311	+	0.838884i	$0.683209\pi$
$978$	−25677.9	−0.839559
$979$	29450.6	0.961437
$980$	−2984.28	−0.0972749
$981$	12472.1	0.405914
$982$	50309.6	1.63487
$983$	1171.32	0.0380053	0.0190027	−	0.999819i	$-0.493951\pi$
$983$	1171.32	0.0380053	0.0190027	+	0.999819i	$0.493951\pi$
$984$	13732.7	0.444901
$985$	25490.7	0.824569
$986$	−4081.02	−0.131811
$987$	24580.6	0.792716
$988$	−312.772	−0.0100715
$989$	2221.40	0.0714220
$990$	20902.4	0.671033
$991$	−3714.32	−0.119061	−0.0595304	−	0.998226i	$-0.518960\pi$
$991$	−3714.32	−0.119061	−0.0595304	+	0.998226i	$0.518960\pi$
$992$	−916.814	−0.0293436
$993$	23477.1	0.750276
$994$	−1842.34	−0.0587883
$995$	−19196.2	−0.611618
$996$	392.818	0.0124969
$997$	19521.0	0.620098	0.310049	−	0.950721i	$-0.399655\pi$
$997$	19521.0	0.620098	0.310049	+	0.950721i	$0.399655\pi$
$998$	−53922.9	−1.71032
$999$	44293.9	1.40280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.4.a.b.1.3	✓	4
3.2	odd	2		207.4.a.e.1.2		4
4.3	odd	2		368.4.a.l.1.2		4
5.2	odd	4		575.4.b.g.24.6		8
5.3	odd	4		575.4.b.g.24.3		8
5.4	even	2		575.4.a.i.1.2		4
7.6	odd	2		1127.4.a.c.1.3		4
8.3	odd	2		1472.4.a.bf.1.3		4
8.5	even	2		1472.4.a.y.1.2		4
23.22	odd	2		529.4.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.a.b.1.3	✓	4	1.1	even	1	trivial
207.4.a.e.1.2		4	3.2	odd	2
368.4.a.l.1.2		4	4.3	odd	2
529.4.a.g.1.3		4	23.22	odd	2
575.4.a.i.1.2		4	5.4	even	2
575.4.b.g.24.3		8	5.3	odd	4
575.4.b.g.24.6		8	5.2	odd	4
1127.4.a.c.1.3		4	7.6	odd	2
1472.4.a.y.1.2		4	8.5	even	2
1472.4.a.bf.1.3		4	8.3	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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	23.a (trivial)

\(f(q)\)	\(=\)	\(q+2.86845 q^{2} +3.43737 q^{3} +0.228032 q^{4} -17.9704 q^{5} +9.85995 q^{6} +32.7301 q^{7} -22.2935 q^{8} -15.1845 q^{9} +O(q^{10})\)	\(q+2.86845 q^{2} +3.43737 q^{3} +0.228032 q^{4} -17.9704 q^{5} +9.85995 q^{6} +32.7301 q^{7} -22.2935 q^{8} -15.1845 q^{9} -51.5473 q^{10} +26.7049 q^{11} +0.783832 q^{12} -14.4956 q^{13} +93.8848 q^{14} -61.7710 q^{15} -65.7723 q^{16} +24.7016 q^{17} -43.5560 q^{18} +94.6224 q^{19} -4.09784 q^{20} +112.505 q^{21} +76.6018 q^{22} -23.0000 q^{23} -76.6312 q^{24} +197.935 q^{25} -41.5801 q^{26} -145.004 q^{27} +7.46352 q^{28} -57.5965 q^{29} -177.187 q^{30} +88.8691 q^{31} -10.3165 q^{32} +91.7948 q^{33} +70.8553 q^{34} -588.173 q^{35} -3.46255 q^{36} -305.467 q^{37} +271.420 q^{38} -49.8269 q^{39} +400.624 q^{40} -179.205 q^{41} +322.717 q^{42} -96.5826 q^{43} +6.08959 q^{44} +272.871 q^{45} -65.9745 q^{46} +218.484 q^{47} -226.084 q^{48} +728.258 q^{49} +567.769 q^{50} +84.9085 q^{51} -3.30548 q^{52} -519.174 q^{53} -415.937 q^{54} -479.898 q^{55} -729.669 q^{56} +325.252 q^{57} -165.213 q^{58} -37.2884 q^{59} -14.0858 q^{60} +96.3052 q^{61} +254.917 q^{62} -496.989 q^{63} +496.586 q^{64} +260.493 q^{65} +263.309 q^{66} +497.552 q^{67} +5.63276 q^{68} -79.0596 q^{69} -1687.15 q^{70} -19.6235 q^{71} +338.515 q^{72} -208.235 q^{73} -876.219 q^{74} +680.378 q^{75} +21.5770 q^{76} +874.054 q^{77} -142.926 q^{78} -446.200 q^{79} +1181.95 q^{80} -88.4513 q^{81} -514.042 q^{82} +501.151 q^{83} +25.6549 q^{84} -443.897 q^{85} -277.043 q^{86} -197.981 q^{87} -595.347 q^{88} +1102.82 q^{89} +782.718 q^{90} -474.444 q^{91} -5.24475 q^{92} +305.476 q^{93} +626.711 q^{94} -1700.40 q^{95} -35.4615 q^{96} +1814.37 q^{97} +2088.98 q^{98} -405.500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9	\( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9} - 70 q^{10} + 8 q^{11} - 67 q^{12} + 111 q^{13} - 144 q^{14} + 10 q^{15} + 64 q^{16} + 98 q^{17} + 49 q^{18} + 96 q^{19} + 140 q^{20} + 180 q^{21} + 220 q^{22} - 92 q^{23} - 188 q^{24} + 184 q^{25} - 229 q^{26} - 155 q^{27} + 282 q^{28} + 21 q^{29} - 406 q^{30} - 193 q^{31} - 432 q^{32} - 418 q^{33} + 666 q^{34} - 752 q^{35} - 629 q^{36} + 170 q^{37} + 748 q^{38} - 291 q^{39} - 26 q^{40} - 125 q^{41} + 640 q^{42} + 2 q^{43} + 830 q^{44} + 168 q^{45} - 46 q^{46} - 677 q^{47} + 551 q^{48} + 1220 q^{49} + 414 q^{50} - 340 q^{51} + 2247 q^{52} - 230 q^{53} + 641 q^{54} - 972 q^{55} - 2174 q^{56} + 1322 q^{57} - 1835 q^{58} - 1140 q^{59} - 804 q^{60} + 754 q^{61} + 443 q^{62} - 1092 q^{63} - 805 q^{64} + 1318 q^{65} - 398 q^{66} + 488 q^{67} + 284 q^{68} - 161 q^{69} - 3820 q^{70} - 401 q^{71} + 1503 q^{72} + 1509 q^{73} + 1366 q^{74} + 1401 q^{75} - 3832 q^{76} + 736 q^{77} - 1907 q^{78} - 838 q^{79} + 2846 q^{80} - 932 q^{81} - 949 q^{82} + 142 q^{83} + 2614 q^{84} + 112 q^{85} + 918 q^{86} + 2223 q^{87} - 404 q^{88} + 2342 q^{89} + 1784 q^{90} + 292 q^{91} - 460 q^{92} - 509 q^{93} + 1567 q^{94} - 956 q^{95} + 799 q^{96} + 1062 q^{97} + 2478 q^{98} - 1498 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9 - 70 * q^10 + 8 * q^11 - 67 * q^12 + 111 * q^13 - 144 * q^14 + 10 * q^15 + 64 * q^16 + 98 * q^17 + 49 * q^18 + 96 * q^19 + 140 * q^20 + 180 * q^21 + 220 * q^22 - 92 * q^23 - 188 * q^24 + 184 * q^25 - 229 * q^26 - 155 * q^27 + 282 * q^28 + 21 * q^29 - 406 * q^30 - 193 * q^31 - 432 * q^32 - 418 * q^33 + 666 * q^34 - 752 * q^35 - 629 * q^36 + 170 * q^37 + 748 * q^38 - 291 * q^39 - 26 * q^40 - 125 * q^41 + 640 * q^42 + 2 * q^43 + 830 * q^44 + 168 * q^45 - 46 * q^46 - 677 * q^47 + 551 * q^48 + 1220 * q^49 + 414 * q^50 - 340 * q^51 + 2247 * q^52 - 230 * q^53 + 641 * q^54 - 972 * q^55 - 2174 * q^56 + 1322 * q^57 - 1835 * q^58 - 1140 * q^59 - 804 * q^60 + 754 * q^61 + 443 * q^62 - 1092 * q^63 - 805 * q^64 + 1318 * q^65 - 398 * q^66 + 488 * q^67 + 284 * q^68 - 161 * q^69 - 3820 * q^70 - 401 * q^71 + 1503 * q^72 + 1509 * q^73 + 1366 * q^74 + 1401 * q^75 - 3832 * q^76 + 736 * q^77 - 1907 * q^78 - 838 * q^79 + 2846 * q^80 - 932 * q^81 - 949 * q^82 + 142 * q^83 + 2614 * q^84 + 112 * q^85 + 918 * q^86 + 2223 * q^87 - 404 * q^88 + 2342 * q^89 + 1784 * q^90 + 292 * q^91 - 460 * q^92 - 509 * q^93 + 1567 * q^94 - 956 * q^95 + 799 * q^96 + 1062 * q^97 + 2478 * q^98 - 1498 * q^99

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