LMFDB - Embedded newform 177.4.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,4,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("177.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.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:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 $ x^8 - 2x^7 - 49x^6 + 89x^5 + 648x^4 - 1023x^3 - 1476x^2 + 1940*x - 384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.17127$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.17127 q^{2} -3.00000 q^{3} -3.28558 q^{4} +9.58086 q^{5} -6.51382 q^{6} +14.1591 q^{7} -24.5041 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.17127 q^{2} -3.00000 q^{3} -3.28558 q^{4} +9.58086 q^{5} -6.51382 q^{6} +14.1591 q^{7} -24.5041 q^{8} +9.00000 q^{9} +20.8027 q^{10} +19.1455 q^{11} +9.85673 q^{12} +15.5364 q^{13} +30.7433 q^{14} -28.7426 q^{15} -26.9204 q^{16} +100.737 q^{17} +19.5414 q^{18} +74.3408 q^{19} -31.4787 q^{20} -42.4773 q^{21} +41.5701 q^{22} +98.9130 q^{23} +73.5122 q^{24} -33.2071 q^{25} +33.7338 q^{26} -27.0000 q^{27} -46.5208 q^{28} +194.021 q^{29} -62.4080 q^{30} +52.9293 q^{31} +137.581 q^{32} -57.4366 q^{33} +218.728 q^{34} +135.656 q^{35} -29.5702 q^{36} -212.802 q^{37} +161.414 q^{38} -46.6093 q^{39} -234.770 q^{40} -395.318 q^{41} -92.2298 q^{42} +305.894 q^{43} -62.9041 q^{44} +86.2277 q^{45} +214.767 q^{46} -630.864 q^{47} +80.7611 q^{48} -142.520 q^{49} -72.1018 q^{50} -302.211 q^{51} -51.0461 q^{52} -109.277 q^{53} -58.6243 q^{54} +183.431 q^{55} -346.956 q^{56} -223.022 q^{57} +421.272 q^{58} -59.0000 q^{59} +94.4360 q^{60} +240.030 q^{61} +114.924 q^{62} +127.432 q^{63} +514.089 q^{64} +148.852 q^{65} -124.710 q^{66} -100.162 q^{67} -330.980 q^{68} -296.739 q^{69} +294.547 q^{70} +263.939 q^{71} -220.537 q^{72} -296.815 q^{73} -462.051 q^{74} +99.6214 q^{75} -244.253 q^{76} +271.083 q^{77} -101.201 q^{78} +626.260 q^{79} -257.920 q^{80} +81.0000 q^{81} -858.343 q^{82} -7.08512 q^{83} +139.562 q^{84} +965.148 q^{85} +664.179 q^{86} -582.062 q^{87} -469.143 q^{88} -132.516 q^{89} +187.224 q^{90} +219.982 q^{91} -324.986 q^{92} -158.788 q^{93} -1369.78 q^{94} +712.249 q^{95} -412.743 q^{96} +1476.73 q^{97} -309.449 q^{98} +172.310 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9} + 29 q^{10} - 27 q^{11} - 114 q^{12} + 89 q^{13} - 37 q^{14} + 36 q^{15} + 362 q^{16} + 79 q^{17} + 18 q^{18} + 288 q^{19} + 457 q^{20} - 159 q^{21} + 596 q^{22} + 202 q^{23} - 9 q^{24} + 264 q^{25} + 270 q^{26} - 216 q^{27} + 702 q^{28} - 114 q^{29} - 87 q^{30} + 538 q^{31} + 316 q^{32} + 81 q^{33} + 498 q^{34} - 196 q^{35} + 342 q^{36} + 395 q^{37} + 397 q^{38} - 267 q^{39} + 918 q^{40} - 39 q^{41} + 111 q^{42} + 527 q^{43} + 64 q^{44} - 108 q^{45} - 539 q^{46} + 860 q^{47} - 1086 q^{48} + 347 q^{49} - 591 q^{50} - 237 q^{51} - 644 q^{52} - 812 q^{53} - 54 q^{54} + 536 q^{55} - 2218 q^{56} - 864 q^{57} - 1154 q^{58} - 472 q^{59} - 1371 q^{60} - 460 q^{61} - 2014 q^{62} + 477 q^{63} - 451 q^{64} - 986 q^{65} - 1788 q^{66} + 1934 q^{67} - 69 q^{68} - 606 q^{69} - 1028 q^{70} - 1687 q^{71} + 27 q^{72} + 1980 q^{73} - 2400 q^{74} - 792 q^{75} - 940 q^{76} - 821 q^{77} - 810 q^{78} + 3319 q^{79} - 2119 q^{80} + 648 q^{81} + 429 q^{82} + 2057 q^{83} - 2106 q^{84} + 566 q^{85} - 6690 q^{86} + 342 q^{87} + 1189 q^{88} + 1668 q^{89} + 261 q^{90} + 2427 q^{91} - 980 q^{92} - 1614 q^{93} + 332 q^{94} + 2146 q^{95} - 948 q^{96} + 1956 q^{97} - 2026 q^{98} - 243 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9 + 29 * q^10 - 27 * q^11 - 114 * q^12 + 89 * q^13 - 37 * q^14 + 36 * q^15 + 362 * q^16 + 79 * q^17 + 18 * q^18 + 288 * q^19 + 457 * q^20 - 159 * q^21 + 596 * q^22 + 202 * q^23 - 9 * q^24 + 264 * q^25 + 270 * q^26 - 216 * q^27 + 702 * q^28 - 114 * q^29 - 87 * q^30 + 538 * q^31 + 316 * q^32 + 81 * q^33 + 498 * q^34 - 196 * q^35 + 342 * q^36 + 395 * q^37 + 397 * q^38 - 267 * q^39 + 918 * q^40 - 39 * q^41 + 111 * q^42 + 527 * q^43 + 64 * q^44 - 108 * q^45 - 539 * q^46 + 860 * q^47 - 1086 * q^48 + 347 * q^49 - 591 * q^50 - 237 * q^51 - 644 * q^52 - 812 * q^53 - 54 * q^54 + 536 * q^55 - 2218 * q^56 - 864 * q^57 - 1154 * q^58 - 472 * q^59 - 1371 * q^60 - 460 * q^61 - 2014 * q^62 + 477 * q^63 - 451 * q^64 - 986 * q^65 - 1788 * q^66 + 1934 * q^67 - 69 * q^68 - 606 * q^69 - 1028 * q^70 - 1687 * q^71 + 27 * q^72 + 1980 * q^73 - 2400 * q^74 - 792 * q^75 - 940 * q^76 - 821 * q^77 - 810 * q^78 + 3319 * q^79 - 2119 * q^80 + 648 * q^81 + 429 * q^82 + 2057 * q^83 - 2106 * q^84 + 566 * q^85 - 6690 * q^86 + 342 * q^87 + 1189 * q^88 + 1668 * q^89 + 261 * q^90 + 2427 * q^91 - 980 * q^92 - 1614 * q^93 + 332 * q^94 + 2146 * q^95 - 948 * q^96 + 1956 * q^97 - 2026 * q^98 - 243 * 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 $

$2$	2.17127	0.767661	0.383830	−	0.923404i	$-0.374605\pi$
$2$	2.17127	0.767661	0.383830	+	0.923404i	$0.374605\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−3.28558	−0.410697
$5$	9.58086	0.856938	0.428469	−	0.903556i	$-0.359053\pi$
$5$	9.58086	0.856938	0.428469	+	0.903556i	$0.359053\pi$
$6$	−6.51382	−0.443209
$7$	14.1591	0.764520	0.382260	−	0.924055i	$-0.375146\pi$
$7$	14.1591	0.764520	0.382260	+	0.924055i	$0.375146\pi$
$8$	−24.5041	−1.08294
$9$	9.00000	0.333333
$10$	20.8027	0.657838
$11$	19.1455	0.524781	0.262391	−	0.964962i	$-0.415489\pi$
$11$	19.1455	0.524781	0.262391	+	0.964962i	$0.415489\pi$
$12$	9.85673	0.237116
$13$	15.5364	0.331464	0.165732	−	0.986171i	$-0.447001\pi$
$13$	15.5364	0.331464	0.165732	+	0.986171i	$0.447001\pi$
$14$	30.7433	0.586892
$15$	−28.7426	−0.494753
$16$	−26.9204	−0.420631
$17$	100.737	1.43720	0.718598	−	0.695426i	$-0.244784\pi$
$17$	100.737	1.43720	0.718598	+	0.695426i	$0.244784\pi$
$18$	19.5414	0.255887
$19$	74.3408	0.897629	0.448815	−	0.893625i	$-0.351846\pi$
$19$	74.3408	0.897629	0.448815	+	0.893625i	$0.351846\pi$
$20$	−31.4787	−0.351942
$21$	−42.4773	−0.441396
$22$	41.5701	0.402854
$23$	98.9130	0.896730	0.448365	−	0.893851i	$-0.352006\pi$
$23$	98.9130	0.896730	0.448365	+	0.893851i	$0.352006\pi$
$24$	73.5122	0.625234
$25$	−33.2071	−0.265657
$26$	33.7338	0.254452
$27$	−27.0000	−0.192450
$28$	−46.5208	−0.313986
$29$	194.021	1.24237	0.621185	−	0.783664i	$-0.286652\pi$
$29$	194.021	1.24237	0.621185	+	0.783664i	$0.286652\pi$
$30$	−62.4080	−0.379803
$31$	52.9293	0.306658	0.153329	−	0.988175i	$-0.451001\pi$
$31$	52.9293	0.306658	0.153329	+	0.988175i	$0.451001\pi$
$32$	137.581	0.760035
$33$	−57.4366	−0.302982
$34$	218.728	1.10328
$35$	135.656	0.655146
$36$	−29.5702	−0.136899
$37$	−212.802	−0.945525	−0.472762	−	0.881190i	$-0.656743\pi$
$37$	−212.802	−0.945525	−0.472762	+	0.881190i	$0.656743\pi$
$38$	161.414	0.689075
$39$	−46.6093	−0.191371
$40$	−234.770	−0.928010
$41$	−395.318	−1.50581	−0.752906	−	0.658128i	$-0.771348\pi$
$41$	−395.318	−1.50581	−0.752906	+	0.658128i	$0.771348\pi$
$42$	−92.2298	−0.338842
$43$	305.894	1.08485	0.542423	−	0.840106i	$-0.317507\pi$
$43$	305.894	1.08485	0.542423	+	0.840106i	$0.317507\pi$
$44$	−62.9041	−0.215526
$45$	86.2277	0.285646
$46$	214.767	0.688384
$47$	−630.864	−1.95789	−0.978947	−	0.204116i	$-0.934568\pi$
$47$	−630.864	−1.95789	−0.978947	+	0.204116i	$0.934568\pi$
$48$	80.7611	0.242851
$49$	−142.520	−0.415510
$50$	−72.1018	−0.203935
$51$	−302.211	−0.829766
$52$	−51.0461	−0.136131
$53$	−109.277	−0.283214	−0.141607	−	0.989923i	$-0.545227\pi$
$53$	−109.277	−0.283214	−0.141607	+	0.989923i	$0.545227\pi$
$54$	−58.6243	−0.147736
$55$	183.431	0.449705
$56$	−346.956	−0.827926
$57$	−223.022	−0.518246
$58$	421.272	0.953718
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	94.4360	0.203194
$61$	240.030	0.503815	0.251907	−	0.967751i	$-0.418942\pi$
$61$	240.030	0.503815	0.251907	+	0.967751i	$0.418942\pi$
$62$	114.924	0.235409
$63$	127.432	0.254840
$64$	514.089	1.00408
$65$	148.852	0.284044
$66$	−124.710	−0.232588
$67$	−100.162	−0.182638	−0.0913191	−	0.995822i	$-0.529108\pi$
$67$	−100.162	−0.182638	−0.0913191	+	0.995822i	$0.529108\pi$
$68$	−330.980	−0.590252
$69$	−296.739	−0.517727
$70$	294.547	0.502930
$71$	263.939	0.441180	0.220590	−	0.975367i	$-0.429202\pi$
$71$	263.939	0.441180	0.220590	+	0.975367i	$0.429202\pi$
$72$	−220.537	−0.360979
$73$	−296.815	−0.475884	−0.237942	−	0.971279i	$-0.576473\pi$
$73$	−296.815	−0.475884	−0.237942	+	0.971279i	$0.576473\pi$
$74$	−462.051	−0.725842
$75$	99.6214	0.153377
$76$	−244.253	−0.368654
$77$	271.083	0.401205
$78$	−101.201	−0.146908
$79$	626.260	0.891896	0.445948	−	0.895059i	$-0.352867\pi$
$79$	626.260	0.891896	0.445948	+	0.895059i	$0.352867\pi$
$80$	−257.920	−0.360454
$81$	81.0000	0.111111
$82$	−858.343	−1.15595
$83$	−7.08512	−0.00936980	−0.00468490	−	0.999989i	$-0.501491\pi$
$83$	−7.08512	−0.00936980	−0.00468490	+	0.999989i	$0.501491\pi$
$84$	139.562	0.181280
$85$	965.148	1.23159
$86$	664.179	0.832793
$87$	−582.062	−0.717283
$88$	−469.143	−0.568305
$89$	−132.516	−0.157828	−0.0789138	−	0.996881i	$-0.525145\pi$
$89$	−132.516	−0.157828	−0.0789138	+	0.996881i	$0.525145\pi$
$90$	187.224	0.219279
$91$	219.982	0.253411
$92$	−324.986	−0.368284
$93$	−158.788	−0.177049
$94$	−1369.78	−1.50300
$95$	712.249	0.769213
$96$	−412.743	−0.438806
$97$	1476.73	1.54577	0.772885	−	0.634546i	$-0.218813\pi$
$97$	1476.73	1.54577	0.772885	+	0.634546i	$0.218813\pi$
$98$	−309.449	−0.318970
$99$	172.310	0.174927
$100$	109.105	0.109105
$101$	−1349.75	−1.32975	−0.664875	−	0.746955i	$-0.731515\pi$
$101$	−1349.75	−1.32975	−0.664875	+	0.746955i	$0.731515\pi$
$102$	−656.183	−0.636978
$103$	1258.15	1.20359	0.601794	−	0.798651i	$-0.294453\pi$
$103$	1258.15	1.20359	0.601794	+	0.798651i	$0.294453\pi$
$104$	−380.705	−0.358954
$105$	−406.969	−0.378249
$106$	−237.270	−0.217412
$107$	−1690.18	−1.52706	−0.763532	−	0.645771i	$-0.776536\pi$
$107$	−1690.18	−1.52706	−0.763532	+	0.645771i	$0.776536\pi$
$108$	88.7106	0.0790387
$109$	−1293.50	−1.13665	−0.568324	−	0.822805i	$-0.692408\pi$
$109$	−1293.50	−1.13665	−0.568324	+	0.822805i	$0.692408\pi$
$110$	398.278	0.345221
$111$	638.406	0.545899
$112$	−381.168	−0.321580
$113$	1086.11	0.904185	0.452093	−	0.891971i	$-0.350678\pi$
$113$	1086.11	0.904185	0.452093	+	0.891971i	$0.350678\pi$
$114$	−484.243	−0.397837
$115$	947.672	0.768442
$116$	−637.470	−0.510238
$117$	139.828	0.110488
$118$	−128.105	−0.0999409
$119$	1426.35	1.09876
$120$	704.310	0.535787
$121$	−964.449	−0.724605
$122$	521.171	0.386759
$123$	1185.95	0.869381
$124$	−173.903	−0.125943
$125$	−1515.76	−1.08459
$126$	276.689	0.195631
$127$	−681.878	−0.476432	−0.238216	−	0.971212i	$-0.576563\pi$
$127$	−681.878	−0.476432	−0.238216	+	0.971212i	$0.576563\pi$
$128$	15.5784	0.0107574
$129$	−917.681	−0.626336
$130$	323.199	0.218049
$131$	−1373.82	−0.916267	−0.458134	−	0.888883i	$-0.651482\pi$
$131$	−1373.82	−0.916267	−0.458134	+	0.888883i	$0.651482\pi$
$132$	188.712	0.124434
$133$	1052.60	0.686255
$134$	−217.479	−0.140204
$135$	−258.683	−0.164918
$136$	−2468.47	−1.55639
$137$	−920.396	−0.573976	−0.286988	−	0.957934i	$-0.592654\pi$
$137$	−920.396	−0.573976	−0.286988	+	0.957934i	$0.592654\pi$
$138$	−644.301	−0.397439
$139$	2435.03	1.48587	0.742936	−	0.669362i	$-0.233433\pi$
$139$	2435.03	1.48587	0.742936	+	0.669362i	$0.233433\pi$
$140$	−445.709	−0.269067
$141$	1892.59	1.13039
$142$	573.083	0.338676
$143$	297.453	0.173946
$144$	−242.283	−0.140210
$145$	1858.88	1.06463
$146$	−644.465	−0.365317
$147$	427.559	0.239895
$148$	699.177	0.388324
$149$	1240.27	0.681925	0.340963	−	0.940077i	$-0.389247\pi$
$149$	1240.27	0.681925	0.340963	+	0.940077i	$0.389247\pi$
$150$	216.305	0.117742
$151$	3384.25	1.82389	0.911943	−	0.410318i	$-0.134582\pi$
$151$	3384.25	1.82389	0.911943	+	0.410318i	$0.134582\pi$
$152$	−1821.65	−0.972076
$153$	906.634	0.479065
$154$	588.596	0.307990
$155$	507.108	0.262787
$156$	153.138	0.0785954
$157$	195.675	0.0994685	0.0497342	−	0.998762i	$-0.484163\pi$
$157$	195.675	0.0994685	0.0497342	+	0.998762i	$0.484163\pi$
$158$	1359.78	0.684674
$159$	327.830	0.163514
$160$	1318.14	0.651303
$161$	1400.52	0.685568
$162$	175.873	0.0852956
$163$	135.992	0.0653479	0.0326739	−	0.999466i	$-0.489598\pi$
$163$	135.992	0.0653479	0.0326739	+	0.999466i	$0.489598\pi$
$164$	1298.85	0.618433
$165$	−550.292	−0.259637
$166$	−15.3837	−0.00719283
$167$	−627.557	−0.290789	−0.145395	−	0.989374i	$-0.546445\pi$
$167$	−627.557	−0.290789	−0.145395	+	0.989374i	$0.546445\pi$
$168$	1040.87	0.478004
$169$	−1955.62	−0.890132
$170$	2095.60	0.945442
$171$	669.067	0.299210
$172$	−1005.04	−0.445543
$173$	−2869.43	−1.26103	−0.630515	−	0.776177i	$-0.717156\pi$
$173$	−2869.43	−1.26103	−0.630515	+	0.776177i	$0.717156\pi$
$174$	−1263.81	−0.550630
$175$	−470.183	−0.203100
$176$	−515.404	−0.220739
$177$	177.000	0.0751646
$178$	−287.728	−0.121158
$179$	2266.38	0.946353	0.473176	−	0.880968i	$-0.343107\pi$
$179$	2266.38	0.946353	0.473176	+	0.880968i	$0.343107\pi$
$180$	−283.308	−0.117314
$181$	449.089	0.184423	0.0922114	−	0.995739i	$-0.470606\pi$
$181$	449.089	0.184423	0.0922114	+	0.995739i	$0.470606\pi$
$182$	477.640	0.194533
$183$	−720.090	−0.290878
$184$	−2423.77	−0.971102
$185$	−2038.82	−0.810256
$186$	−344.772	−0.135913
$187$	1928.66	0.754213
$188$	2072.75	0.804101
$189$	−382.296	−0.147132
$190$	1546.49	0.590494
$191$	495.207	0.187602	0.0938009	−	0.995591i	$-0.470098\pi$
$191$	495.207	0.187602	0.0938009	+	0.995591i	$0.470098\pi$
$192$	−1542.27	−0.579706
$193$	−2118.19	−0.790002	−0.395001	−	0.918681i	$-0.629256\pi$
$193$	−2118.19	−0.790002	−0.395001	+	0.918681i	$0.629256\pi$
$194$	3206.39	1.18663
$195$	−446.557	−0.163993
$196$	468.260	0.170649
$197$	372.268	0.134635	0.0673173	−	0.997732i	$-0.478556\pi$
$197$	372.268	0.134635	0.0673173	+	0.997732i	$0.478556\pi$
$198$	374.131	0.134285
$199$	989.744	0.352568	0.176284	−	0.984339i	$-0.443592\pi$
$199$	989.744	0.352568	0.176284	+	0.984339i	$0.443592\pi$
$200$	813.710	0.287690
$201$	300.487	0.105446
$202$	−2930.66	−1.02080
$203$	2747.16	0.949816
$204$	992.939	0.340782
$205$	−3787.48	−1.29039
$206$	2731.80	0.923948
$207$	890.217	0.298910
$208$	−418.246	−0.139424
$209$	1423.29	0.471059
$210$	−883.641	−0.290367
$211$	−1268.28	−0.413802	−0.206901	−	0.978362i	$-0.566338\pi$
$211$	−1268.28	−0.413802	−0.206901	+	0.978362i	$0.566338\pi$
$212$	359.037	0.116315
$213$	−791.816	−0.254715
$214$	−3669.84	−1.17227
$215$	2930.72	0.929645
$216$	661.610	0.208411
$217$	749.432	0.234446
$218$	−2808.54	−0.872560
$219$	890.444	0.274752
$220$	−602.675	−0.184692
$221$	1565.09	0.476378
$222$	1386.15	0.419065
$223$	1129.78	0.339262	0.169631	−	0.985508i	$-0.445742\pi$
$223$	1129.78	0.339262	0.169631	+	0.985508i	$0.445742\pi$
$224$	1948.02	0.581062
$225$	−298.864	−0.0885524
$226$	2358.25	0.694108
$227$	−161.210	−0.0471359	−0.0235680	−	0.999722i	$-0.507503\pi$
$227$	−161.210	−0.0471359	−0.0235680	+	0.999722i	$0.507503\pi$
$228$	732.758	0.212842
$229$	118.980	0.0343338	0.0171669	−	0.999853i	$-0.494535\pi$
$229$	118.980	0.0343338	0.0171669	+	0.999853i	$0.494535\pi$
$230$	2057.65	0.589903
$231$	−813.250	−0.231636
$232$	−4754.29	−1.34541
$233$	−4469.51	−1.25668	−0.628342	−	0.777937i	$-0.716266\pi$
$233$	−4469.51	−1.25668	−0.628342	+	0.777937i	$0.716266\pi$
$234$	303.604	0.0848172
$235$	−6044.22	−1.67779
$236$	193.849	0.0534682
$237$	−1878.78	−0.514937
$238$	3096.99	0.843479
$239$	−4682.67	−1.26735	−0.633675	−	0.773600i	$-0.718454\pi$
$239$	−4682.67	−1.26735	−0.633675	+	0.773600i	$0.718454\pi$
$240$	773.761	0.208108
$241$	−3836.61	−1.02547	−0.512735	−	0.858547i	$-0.671367\pi$
$241$	−3836.61	−1.02547	−0.512735	+	0.858547i	$0.671367\pi$
$242$	−2094.08	−0.556251
$243$	−243.000	−0.0641500
$244$	−788.638	−0.206915
$245$	−1365.46	−0.356066
$246$	2575.03	0.667389
$247$	1154.99	0.297531
$248$	−1296.98	−0.332091
$249$	21.2554	0.00540966
$250$	−3291.13	−0.832597
$251$	−331.121	−0.0832677	−0.0416338	−	0.999133i	$-0.513256\pi$
$251$	−331.121	−0.0832677	−0.0416338	+	0.999133i	$0.513256\pi$
$252$	−418.687	−0.104662
$253$	1893.74	0.470587
$254$	−1480.54	−0.365738
$255$	−2895.44	−0.711058
$256$	−4078.89	−0.995822
$257$	1423.82	0.345585	0.172792	−	0.984958i	$-0.444721\pi$
$257$	1423.82	0.345585	0.172792	+	0.984958i	$0.444721\pi$
$258$	−1992.54	−0.480813
$259$	−3013.08	−0.722872
$260$	−489.066	−0.116656
$261$	1746.19	0.414123
$262$	−2982.93	−0.703382
$263$	−5205.39	−1.22045	−0.610225	−	0.792228i	$-0.708921\pi$
$263$	−5205.39	−1.22045	−0.610225	+	0.792228i	$0.708921\pi$
$264$	1407.43	0.328111
$265$	−1046.97	−0.242697
$266$	2285.48	0.526811
$267$	397.548	0.0911218
$268$	329.091	0.0750090
$269$	349.224	0.0791544	0.0395772	−	0.999217i	$-0.487399\pi$
$269$	349.224	0.0791544	0.0395772	+	0.999217i	$0.487399\pi$
$270$	−561.672	−0.126601
$271$	−8231.66	−1.84516	−0.922579	−	0.385808i	$-0.873923\pi$
$271$	−8231.66	−1.84516	−0.922579	+	0.385808i	$0.873923\pi$
$272$	−2711.88	−0.604529
$273$	−659.945	−0.146307
$274$	−1998.43	−0.440619
$275$	−635.768	−0.139412
$276$	974.959	0.212629
$277$	−7242.28	−1.57093	−0.785463	−	0.618908i	$-0.787575\pi$
$277$	−7242.28	−1.57093	−0.785463	+	0.618908i	$0.787575\pi$
$278$	5287.11	1.14065
$279$	476.364	0.102219
$280$	−3324.13	−0.709482
$281$	1769.76	0.375713	0.187856	−	0.982197i	$-0.439846\pi$
$281$	1769.76	0.375713	0.187856	+	0.982197i	$0.439846\pi$
$282$	4109.33	0.867756
$283$	5303.35	1.11396	0.556981	−	0.830525i	$-0.311960\pi$
$283$	5303.35	1.11396	0.556981	+	0.830525i	$0.311960\pi$
$284$	−867.191	−0.181191
$285$	−2136.75	−0.444105
$286$	645.851	0.133531
$287$	−5597.35	−1.15122
$288$	1238.23	0.253345
$289$	5234.96	1.06553
$290$	4036.14	0.817278
$291$	−4430.20	−0.892451
$292$	975.207	0.195444
$293$	138.305	0.0275764	0.0137882	−	0.999905i	$-0.495611\pi$
$293$	138.305	0.0275764	0.0137882	+	0.999905i	$0.495611\pi$
$294$	928.348	0.184158
$295$	−565.271	−0.111564
$296$	5214.51	1.02394
$297$	−516.929	−0.100994
$298$	2692.96	0.523487
$299$	1536.75	0.297233
$300$	−327.314	−0.0629916
$301$	4331.18	0.829386
$302$	7348.14	1.40012
$303$	4049.24	0.767731
$304$	−2001.28	−0.377570
$305$	2299.70	0.431738
$306$	1968.55	0.367760
$307$	2464.57	0.458178	0.229089	−	0.973406i	$-0.426425\pi$
$307$	2464.57	0.458178	0.229089	+	0.973406i	$0.426425\pi$
$308$	−890.665	−0.164774
$309$	−3774.46	−0.694892
$310$	1101.07	0.201731
$311$	−9275.20	−1.69115	−0.845576	−	0.533855i	$-0.820743\pi$
$311$	−9275.20	−1.69115	−0.845576	+	0.533855i	$0.820743\pi$
$312$	1142.12	0.207242
$313$	5699.12	1.02918	0.514590	−	0.857437i	$-0.327944\pi$
$313$	5699.12	1.02918	0.514590	+	0.857437i	$0.327944\pi$
$314$	424.863	0.0763580
$315$	1220.91	0.218382
$316$	−2057.63	−0.366299
$317$	−859.960	−0.152366	−0.0761832	−	0.997094i	$-0.524273\pi$
$317$	−859.960	−0.152366	−0.0761832	+	0.997094i	$0.524273\pi$
$318$	711.809	0.125523
$319$	3714.63	0.651972
$320$	4925.41	0.860434
$321$	5070.54	0.881650
$322$	3040.91	0.526283
$323$	7488.88	1.29007
$324$	−266.132	−0.0456330
$325$	−515.920	−0.0880557
$326$	295.275	0.0501650
$327$	3880.49	0.656244
$328$	9686.89	1.63070
$329$	−8932.47	−1.49685
$330$	−1194.83	−0.199313
$331$	10291.1	1.70891	0.854454	−	0.519527i	$-0.173892\pi$
$331$	10291.1	1.70891	0.854454	+	0.519527i	$0.173892\pi$
$332$	23.2787	0.00384815
$333$	−1915.22	−0.315175
$334$	−1362.60	−0.223227
$335$	−959.640	−0.156510
$336$	1143.50	0.185665
$337$	1861.32	0.300869	0.150434	−	0.988620i	$-0.451933\pi$
$337$	1861.32	0.300869	0.150434	+	0.988620i	$0.451933\pi$
$338$	−4246.18	−0.683319
$339$	−3258.34	−0.522032
$340$	−3171.07	−0.505810
$341$	1013.36	0.160928
$342$	1452.73	0.229692
$343$	−6874.52	−1.08218
$344$	−7495.64	−1.17482
$345$	−2843.01	−0.443660
$346$	−6230.30	−0.968044
$347$	−2342.82	−0.362447	−0.181224	−	0.983442i	$-0.558006\pi$
$347$	−2342.82	−0.362447	−0.181224	+	0.983442i	$0.558006\pi$
$348$	1912.41	0.294586
$349$	−9507.00	−1.45816	−0.729081	−	0.684428i	$-0.760052\pi$
$349$	−9507.00	−1.45816	−0.729081	+	0.684428i	$0.760052\pi$
$350$	−1020.90	−0.155912
$351$	−419.483	−0.0637902
$352$	2634.06	0.398852
$353$	7366.56	1.11071	0.555357	−	0.831612i	$-0.312581\pi$
$353$	7366.56	1.11071	0.555357	+	0.831612i	$0.312581\pi$
$354$	384.315	0.0577009
$355$	2528.76	0.378064
$356$	435.391	0.0648193
$357$	−4279.04	−0.634372
$358$	4920.93	0.726478
$359$	7674.15	1.12821	0.564103	−	0.825704i	$-0.309222\pi$
$359$	7674.15	1.12821	0.564103	+	0.825704i	$0.309222\pi$
$360$	−2112.93	−0.309337
$361$	−1332.44	−0.194262
$362$	975.095	0.141574
$363$	2893.35	0.418351
$364$	−722.767	−0.104075
$365$	−2843.74	−0.407803
$366$	−1563.51	−0.223295
$367$	4845.83	0.689238	0.344619	−	0.938743i	$-0.388008\pi$
$367$	4845.83	0.689238	0.344619	+	0.938743i	$0.388008\pi$
$368$	−2662.77	−0.377192
$369$	−3557.86	−0.501937
$370$	−4426.84	−0.622002
$371$	−1547.26	−0.216522
$372$	521.710	0.0727135
$373$	−3677.62	−0.510510	−0.255255	−	0.966874i	$-0.582159\pi$
$373$	−3677.62	−0.510510	−0.255255	+	0.966874i	$0.582159\pi$
$374$	4187.65	0.578980
$375$	4547.28	0.626188
$376$	15458.7	2.12027
$377$	3014.39	0.411800
$378$	−830.068	−0.112947
$379$	2448.51	0.331851	0.165925	−	0.986138i	$-0.446939\pi$
$379$	2448.51	0.331851	0.165925	+	0.986138i	$0.446939\pi$
$380$	−2340.15	−0.315913
$381$	2045.63	0.275068
$382$	1075.23	0.144014
$383$	7777.55	1.03764	0.518818	−	0.854885i	$-0.326372\pi$
$383$	7777.55	1.03764	0.518818	+	0.854885i	$0.326372\pi$
$384$	−46.7352	−0.00621080
$385$	2597.21	0.343808
$386$	−4599.16	−0.606453
$387$	2753.04	0.361615
$388$	−4851.93	−0.634843
$389$	−13158.8	−1.71512	−0.857558	−	0.514387i	$-0.828019\pi$
$389$	−13158.8	−1.71512	−0.857558	+	0.514387i	$0.828019\pi$
$390$	−969.596	−0.125891
$391$	9964.21	1.28878
$392$	3492.31	0.449971
$393$	4121.45	0.529007
$394$	808.296	0.103354
$395$	6000.11	0.764300
$396$	−566.137	−0.0718420
$397$	−7496.45	−0.947698	−0.473849	−	0.880606i	$-0.657136\pi$
$397$	−7496.45	−0.947698	−0.473849	+	0.880606i	$0.657136\pi$
$398$	2149.00	0.270653
$399$	−3157.80	−0.396210
$400$	893.949	0.111744
$401$	−4346.75	−0.541313	−0.270657	−	0.962676i	$-0.587241\pi$
$401$	−4346.75	−0.541313	−0.270657	+	0.962676i	$0.587241\pi$
$402$	652.438	0.0809469
$403$	822.332	0.101646
$404$	4434.69	0.546124
$405$	776.050	0.0952153
$406$	5964.83	0.729136
$407$	−4074.20	−0.496193
$408$	7405.40	0.898584
$409$	14635.7	1.76941	0.884706	−	0.466149i	$-0.154359\pi$
$409$	14635.7	1.76941	0.884706	+	0.466149i	$0.154359\pi$
$410$	−8223.66	−0.990580
$411$	2761.19	0.331385
$412$	−4133.76	−0.494311
$413$	−835.387	−0.0995320
$414$	1932.90	0.229461
$415$	−67.8816	−0.00802934
$416$	2137.52	0.251924
$417$	−7305.08	−0.857869
$418$	3090.36	0.361613
$419$	−6292.05	−0.733620	−0.366810	−	0.930296i	$-0.619550\pi$
$419$	−6292.05	−0.733620	−0.366810	+	0.930296i	$0.619550\pi$
$420$	1337.13	0.155346
$421$	−10784.3	−1.24845	−0.624223	−	0.781246i	$-0.714584\pi$
$421$	−10784.3	−1.24845	−0.624223	+	0.781246i	$0.714584\pi$
$422$	−2753.79	−0.317660
$423$	−5677.78	−0.652631
$424$	2677.73	0.306702
$425$	−3345.19	−0.381802
$426$	−1719.25	−0.195535
$427$	3398.61	0.385176
$428$	5553.21	0.627160
$429$	−892.359	−0.100428
$430$	6363.40	0.713652
$431$	5954.37	0.665457	0.332728	−	0.943023i	$-0.392031\pi$
$431$	5954.37	0.665457	0.332728	+	0.943023i	$0.392031\pi$
$432$	726.850	0.0809504
$433$	4561.77	0.506292	0.253146	−	0.967428i	$-0.418535\pi$
$433$	4561.77	0.506292	0.253146	+	0.967428i	$0.418535\pi$
$434$	1627.22	0.179975
$435$	−5576.65	−0.614667
$436$	4249.89	0.466818
$437$	7353.28	0.804931
$438$	1933.40	0.210916
$439$	−7301.09	−0.793763	−0.396882	−	0.917870i	$-0.629908\pi$
$439$	−7301.09	−0.793763	−0.396882	+	0.917870i	$0.629908\pi$
$440$	−4494.79	−0.487002
$441$	−1282.68	−0.138503
$442$	3398.25	0.365697
$443$	−15813.1	−1.69595	−0.847973	−	0.530040i	$-0.822177\pi$
$443$	−15813.1	−1.69595	−0.847973	+	0.530040i	$0.822177\pi$
$444$	−2097.53	−0.224199
$445$	−1269.62	−0.135248
$446$	2453.05	0.260438
$447$	−3720.81	−0.393710
$448$	7279.04	0.767639
$449$	1604.95	0.168691	0.0843456	−	0.996437i	$-0.473120\pi$
$449$	1604.95	0.168691	0.0843456	+	0.996437i	$0.473120\pi$
$450$	−648.916	−0.0679782
$451$	−7568.57	−0.790221
$452$	−3568.51	−0.371346
$453$	−10152.8	−1.05302
$454$	−350.030	−0.0361844
$455$	2107.61	0.217157
$456$	5464.96	0.561228
$457$	9914.58	1.01485	0.507423	−	0.861697i	$-0.330598\pi$
$457$	9914.58	1.01485	0.507423	+	0.861697i	$0.330598\pi$
$458$	258.339	0.0263567
$459$	−2719.90	−0.276589
$460$	−3113.65	−0.315597
$461$	−5717.04	−0.577590	−0.288795	−	0.957391i	$-0.593255\pi$
$461$	−5717.04	−0.577590	−0.288795	+	0.957391i	$0.593255\pi$
$462$	−1765.79	−0.177818
$463$	15831.2	1.58906	0.794532	−	0.607222i	$-0.207716\pi$
$463$	15831.2	1.58906	0.794532	+	0.607222i	$0.207716\pi$
$464$	−5223.11	−0.522579
$465$	−1521.33	−0.151720
$466$	−9704.53	−0.964707
$467$	375.189	0.0371770	0.0185885	−	0.999827i	$-0.494083\pi$
$467$	375.189	0.0371770	0.0185885	+	0.999827i	$0.494083\pi$
$468$	−459.415	−0.0453771
$469$	−1418.21	−0.139630
$470$	−13123.6	−1.28798
$471$	−587.024	−0.0574281
$472$	1445.74	0.140986
$473$	5856.49	0.569306
$474$	−4079.35	−0.395296
$475$	−2468.65	−0.238462
$476$	−4686.37	−0.451260
$477$	−983.491	−0.0944046
$478$	−10167.3	−0.972894
$479$	−3601.59	−0.343551	−0.171775	−	0.985136i	$-0.554950\pi$
$479$	−3601.59	−0.343551	−0.171775	+	0.985136i	$0.554950\pi$
$480$	−3954.43	−0.376030
$481$	−3306.18	−0.313407
$482$	−8330.33	−0.787212
$483$	−4201.56	−0.395813
$484$	3168.77	0.297593
$485$	14148.4	1.32463
$486$	−527.619	−0.0492455
$487$	−14132.2	−1.31498	−0.657488	−	0.753465i	$-0.728381\pi$
$487$	−14132.2	−1.31498	−0.657488	+	0.753465i	$0.728381\pi$
$488$	−5881.71	−0.545600
$489$	−407.976	−0.0377286
$490$	−2964.79	−0.273338
$491$	−293.022	−0.0269326	−0.0134663	−	0.999909i	$-0.504287\pi$
$491$	−293.022	−0.0269326	−0.0134663	+	0.999909i	$0.504287\pi$
$492$	−3896.54	−0.357052
$493$	19545.1	1.78553
$494$	2507.80	0.228403
$495$	1650.87	0.149902
$496$	−1424.88	−0.128990
$497$	3737.14	0.337291
$498$	46.1512	0.00415278
$499$	15571.2	1.39692	0.698459	−	0.715650i	$-0.253869\pi$
$499$	15571.2	1.39692	0.698459	+	0.715650i	$0.253869\pi$
$500$	4980.15	0.445438
$501$	1882.67	0.167887
$502$	−718.954	−0.0639213
$503$	12697.7	1.12557	0.562784	−	0.826604i	$-0.309730\pi$
$503$	12697.7	1.12557	0.562784	+	0.826604i	$0.309730\pi$
$504$	−3122.60	−0.275975
$505$	−12931.7	−1.13951
$506$	4111.83	0.361251
$507$	5866.86	0.513918
$508$	2240.36	0.195669
$509$	−4294.84	−0.373999	−0.187000	−	0.982360i	$-0.559876\pi$
$509$	−4294.84	−0.373999	−0.187000	+	0.982360i	$0.559876\pi$
$510$	−6286.80	−0.545851
$511$	−4202.63	−0.363822
$512$	−8981.00	−0.775210
$513$	−2007.20	−0.172749
$514$	3091.49	0.265292
$515$	12054.2	1.03140
$516$	3015.11	0.257234
$517$	−12078.2	−1.02747
$518$	−6542.22	−0.554921
$519$	8608.28	0.728057
$520$	−3647.48	−0.307601
$521$	−22351.1	−1.87950	−0.939751	−	0.341860i	$-0.888943\pi$
$521$	−22351.1	−1.87950	−0.939751	+	0.341860i	$0.888943\pi$
$522$	3791.44	0.317906
$523$	12800.7	1.07024	0.535118	−	0.844777i	$-0.320267\pi$
$523$	12800.7	1.07024	0.535118	+	0.844777i	$0.320267\pi$
$524$	4513.78	0.376308
$525$	1410.55	0.117260
$526$	−11302.3	−0.936891
$527$	5331.95	0.440727
$528$	1546.21	0.127444
$529$	−2383.22	−0.195875
$530$	−2273.25	−0.186309
$531$	−531.000	−0.0433963
$532$	−3458.40	−0.281843
$533$	−6141.83	−0.499122
$534$	863.184	0.0699506
$535$	−16193.4	−1.30860
$536$	2454.38	0.197786
$537$	−6799.14	−0.546377
$538$	758.260	0.0607637
$539$	−2728.62	−0.218052
$540$	849.924	0.0677313
$541$	18834.0	1.49674	0.748372	−	0.663280i	$-0.230836\pi$
$541$	18834.0	1.49674	0.748372	+	0.663280i	$0.230836\pi$
$542$	−17873.2	−1.41646
$543$	−1347.27	−0.106477
$544$	13859.5	1.09232
$545$	−12392.8	−0.974037
$546$	−1432.92	−0.112314
$547$	−16024.5	−1.25258	−0.626289	−	0.779591i	$-0.715427\pi$
$547$	−16024.5	−1.25258	−0.626289	+	0.779591i	$0.715427\pi$
$548$	3024.03	0.235730
$549$	2160.27	0.167938
$550$	−1380.43	−0.107021
$551$	14423.7	1.11519
$552$	7271.31	0.560666
$553$	8867.29	0.681872
$554$	−15725.0	−1.20594
$555$	6116.47	0.467802
$556$	−8000.47	−0.610244
$557$	12161.8	0.925157	0.462578	−	0.886578i	$-0.346924\pi$
$557$	12161.8	0.925157	0.462578	+	0.886578i	$0.346924\pi$
$558$	1034.32	0.0784697
$559$	4752.49	0.359587
$560$	−3651.92	−0.275575
$561$	−5785.99	−0.435445
$562$	3842.64	0.288420
$563$	20165.9	1.50958	0.754790	−	0.655967i	$-0.227739\pi$
$563$	20165.9	1.50958	0.754790	+	0.655967i	$0.227739\pi$
$564$	−6218.26	−0.464248
$565$	10405.9	0.774831
$566$	11515.0	0.855144
$567$	1146.89	0.0849466
$568$	−6467.57	−0.477770
$569$	−18351.0	−1.35205	−0.676024	−	0.736880i	$-0.736298\pi$
$569$	−18351.0	−1.35205	−0.676024	+	0.736880i	$0.736298\pi$
$570$	−4639.46	−0.340922
$571$	−19894.1	−1.45804	−0.729021	−	0.684492i	$-0.760024\pi$
$571$	−19894.1	−1.45804	−0.729021	+	0.684492i	$0.760024\pi$
$572$	−977.304	−0.0714391
$573$	−1485.62	−0.108312
$574$	−12153.4	−0.883748
$575$	−3284.62	−0.238223
$576$	4626.80	0.334693
$577$	916.177	0.0661022	0.0330511	−	0.999454i	$-0.489478\pi$
$577$	916.177	0.0661022	0.0330511	+	0.999454i	$0.489478\pi$
$578$	11366.5	0.817968
$579$	6354.56	0.456108
$580$	−6107.51	−0.437242
$581$	−100.319	−0.00716340
$582$	−9619.18	−0.685099
$583$	−2092.16	−0.148625
$584$	7273.16	0.515352
$585$	1339.67	0.0946813
$586$	300.298	0.0211693
$587$	−1121.91	−0.0788862	−0.0394431	−	0.999222i	$-0.512558\pi$
$587$	−1121.91	−0.0788862	−0.0394431	+	0.999222i	$0.512558\pi$
$588$	−1404.78	−0.0985240
$589$	3934.81	0.275265
$590$	−1227.36	−0.0856432
$591$	−1116.81	−0.0777314
$592$	5728.70	0.397717
$593$	16327.9	1.13070	0.565352	−	0.824850i	$-0.308740\pi$
$593$	16327.9	1.13070	0.565352	+	0.824850i	$0.308740\pi$
$594$	−1122.39	−0.0775292
$595$	13665.6	0.941573
$596$	−4075.00	−0.280065
$597$	−2969.23	−0.203555
$598$	3336.71	0.228174
$599$	24940.5	1.70124	0.850620	−	0.525780i	$-0.176227\pi$
$599$	24940.5	1.70124	0.850620	+	0.525780i	$0.176227\pi$
$600$	−2441.13	−0.166098
$601$	17458.9	1.18496	0.592482	−	0.805584i	$-0.298148\pi$
$601$	17458.9	1.18496	0.592482	+	0.805584i	$0.298148\pi$
$602$	9404.17	0.636687
$603$	−901.460	−0.0608794
$604$	−11119.2	−0.749064
$605$	−9240.25	−0.620941
$606$	8791.99	0.589357
$607$	19831.7	1.32610	0.663052	−	0.748573i	$-0.269261\pi$
$607$	19831.7	1.32610	0.663052	+	0.748573i	$0.269261\pi$
$608$	10227.9	0.682230
$609$	−8241.47	−0.548377
$610$	4993.26	0.331428
$611$	−9801.37	−0.648971
$612$	−2978.82	−0.196751
$613$	4099.73	0.270125	0.135062	−	0.990837i	$-0.456877\pi$
$613$	4099.73	0.270125	0.135062	+	0.990837i	$0.456877\pi$
$614$	5351.26	0.351725
$615$	11362.5	0.745006
$616$	−6642.64	−0.434480
$617$	19906.4	1.29887	0.649434	−	0.760418i	$-0.275006\pi$
$617$	19906.4	1.29887	0.649434	+	0.760418i	$0.275006\pi$
$618$	−8195.39	−0.533442
$619$	−11172.3	−0.725450	−0.362725	−	0.931896i	$-0.618154\pi$
$619$	−11172.3	−0.725450	−0.362725	+	0.931896i	$0.618154\pi$
$620$	−1666.14	−0.107926
$621$	−2670.65	−0.172576
$622$	−20139.0	−1.29823
$623$	−1876.31	−0.120662
$624$	1254.74	0.0804964
$625$	−10371.4	−0.663769
$626$	12374.3	0.790060
$627$	−4269.88	−0.271966
$628$	−642.905	−0.0408514
$629$	−21437.0	−1.35890
$630$	2650.92	0.167643
$631$	23430.1	1.47819	0.739096	−	0.673600i	$-0.235253\pi$
$631$	23430.1	1.47819	0.739096	+	0.673600i	$0.235253\pi$
$632$	−15345.9	−0.965867
$633$	3804.85	0.238909
$634$	−1867.21	−0.116966
$635$	−6532.97	−0.408273
$636$	−1077.11	−0.0671545
$637$	−2214.25	−0.137726
$638$	8065.46	0.500493
$639$	2375.45	0.147060
$640$	149.254	0.00921844
$641$	−16450.0	−1.01363	−0.506816	−	0.862055i	$-0.669177\pi$
$641$	−16450.0	−1.01363	−0.506816	+	0.862055i	$0.669177\pi$
$642$	11009.5	0.676808
$643$	−11207.2	−0.687355	−0.343677	−	0.939088i	$-0.611673\pi$
$643$	−11207.2	−0.687355	−0.343677	+	0.939088i	$0.611673\pi$
$644$	−4601.52	−0.281561
$645$	−8792.17	−0.536731
$646$	16260.4	0.990335
$647$	28416.4	1.72668	0.863341	−	0.504621i	$-0.168368\pi$
$647$	28416.4	1.72668	0.863341	+	0.504621i	$0.168368\pi$
$648$	−1984.83	−0.120326
$649$	−1129.59	−0.0683207
$650$	−1120.20	−0.0675969
$651$	−2248.30	−0.135357
$652$	−446.812	−0.0268382
$653$	−16811.3	−1.00747	−0.503734	−	0.863859i	$-0.668041\pi$
$653$	−16811.3	−1.00747	−0.503734	+	0.863859i	$0.668041\pi$
$654$	8425.61	0.503773
$655$	−13162.4	−0.785184
$656$	10642.1	0.633391
$657$	−2671.33	−0.158628
$658$	−19394.8	−1.14907
$659$	−7979.14	−0.471659	−0.235829	−	0.971794i	$-0.575781\pi$
$659$	−7979.14	−0.471659	−0.235829	+	0.971794i	$0.575781\pi$
$660$	1808.03	0.106632
$661$	19673.4	1.15765	0.578824	−	0.815453i	$-0.303512\pi$
$661$	19673.4	1.15765	0.578824	+	0.815453i	$0.303512\pi$
$662$	22344.7	1.31186
$663$	−4695.28	−0.275037
$664$	173.614	0.0101469
$665$	10084.8	0.588078
$666$	−4158.46	−0.241947
$667$	19191.2	1.11407
$668$	2061.89	0.119426
$669$	−3389.33	−0.195873
$670$	−2083.64	−0.120146
$671$	4595.50	0.264393
$672$	−5844.07	−0.335476
$673$	−12964.9	−0.742588	−0.371294	−	0.928515i	$-0.621086\pi$
$673$	−12964.9	−0.742588	−0.371294	+	0.928515i	$0.621086\pi$
$674$	4041.44	0.230965
$675$	896.593	0.0511257
$676$	6425.34	0.365575
$677$	26269.4	1.49131	0.745655	−	0.666332i	$-0.232137\pi$
$677$	26269.4	1.49131	0.745655	+	0.666332i	$0.232137\pi$
$678$	−7074.74	−0.400743
$679$	20909.2	1.18177
$680$	−23650.0	−1.33373
$681$	483.629	0.0272139
$682$	2200.28	0.123538
$683$	−8043.98	−0.450650	−0.225325	−	0.974284i	$-0.572344\pi$
$683$	−8043.98	−0.450650	−0.225325	+	0.974284i	$0.572344\pi$
$684$	−2198.27	−0.122885
$685$	−8818.18	−0.491862
$686$	−14926.5	−0.830751
$687$	−356.941	−0.0198226
$688$	−8234.77	−0.456319
$689$	−1697.77	−0.0938751
$690$	−6172.96	−0.340580
$691$	3316.01	0.182557	0.0912785	−	0.995825i	$-0.470905\pi$
$691$	3316.01	0.182557	0.0912785	+	0.995825i	$0.470905\pi$
$692$	9427.72	0.517902
$693$	2439.75	0.133735
$694$	−5086.90	−0.278236
$695$	23329.6	1.27330
$696$	14262.9	0.776772
$697$	−39823.2	−2.16415
$698$	−20642.3	−1.11937
$699$	13408.5	0.725547
$700$	1544.82	0.0834127
$701$	−13786.8	−0.742824	−0.371412	−	0.928468i	$-0.621126\pi$
$701$	−13786.8	−0.742824	−0.371412	+	0.928468i	$0.621126\pi$
$702$	−910.813	−0.0489692
$703$	−15819.9	−0.848731
$704$	9842.50	0.526922
$705$	18132.7	0.968674
$706$	15994.8	0.852652
$707$	−19111.2	−1.01662
$708$	−581.547	−0.0308699
$709$	−5117.29	−0.271063	−0.135532	−	0.990773i	$-0.543274\pi$
$709$	−5117.29	−0.271063	−0.135532	+	0.990773i	$0.543274\pi$
$710$	5490.63	0.290225
$711$	5636.34	0.297299
$712$	3247.18	0.170917
$713$	5235.40	0.274989
$714$	−9290.96	−0.486983
$715$	2849.85	0.149061
$716$	−7446.36	−0.388664
$717$	14048.0	0.731704
$718$	16662.7	0.866079
$719$	−5501.98	−0.285382	−0.142691	−	0.989767i	$-0.545575\pi$
$719$	−5501.98	−0.285382	−0.142691	+	0.989767i	$0.545575\pi$
$720$	−2321.28	−0.120151
$721$	17814.3	0.920167
$722$	−2893.09	−0.149127
$723$	11509.8	0.592055
$724$	−1475.52	−0.0757420
$725$	−6442.87	−0.330044
$726$	6282.24	0.321151
$727$	−23570.5	−1.20245	−0.601226	−	0.799079i	$-0.705321\pi$
$727$	−23570.5	−1.20245	−0.601226	+	0.799079i	$0.705321\pi$
$728$	−5390.45	−0.274428
$729$	729.000	0.0370370
$730$	−6174.53	−0.313054
$731$	30814.8	1.55914
$732$	2365.91	0.119463
$733$	30142.7	1.51889	0.759446	−	0.650571i	$-0.225470\pi$
$733$	30142.7	1.51889	0.759446	+	0.650571i	$0.225470\pi$
$734$	10521.6	0.529101
$735$	4096.39	0.205575
$736$	13608.6	0.681546
$737$	−1917.66	−0.0958450
$738$	−7725.08	−0.385317
$739$	−17267.8	−0.859548	−0.429774	−	0.902937i	$-0.641407\pi$
$739$	−17267.8	−0.859548	−0.429774	+	0.902937i	$0.641407\pi$
$740$	6698.72	0.332770
$741$	−3464.97	−0.171780
$742$	−3359.53	−0.166216
$743$	36299.4	1.79232	0.896161	−	0.443728i	$-0.146344\pi$
$743$	36299.4	1.79232	0.896161	+	0.443728i	$0.146344\pi$
$744$	3890.95	0.191733
$745$	11882.8	0.584368
$746$	−7985.12	−0.391898
$747$	−63.7661	−0.00312327
$748$	−6336.78	−0.309753
$749$	−23931.4	−1.16747
$750$	9873.38	0.480700
$751$	−19115.5	−0.928807	−0.464404	−	0.885624i	$-0.653731\pi$
$751$	−19115.5	−0.928807	−0.464404	+	0.885624i	$0.653731\pi$
$752$	16983.1	0.823550
$753$	993.364	0.0480746
$754$	6545.05	0.316123
$755$	32424.1	1.56296
$756$	1256.06	0.0604266
$757$	7724.96	0.370896	0.185448	−	0.982654i	$-0.440626\pi$
$757$	7724.96	0.370896	0.185448	+	0.982654i	$0.440626\pi$
$758$	5316.38	0.254749
$759$	−5681.22	−0.271693
$760$	−17453.0	−0.833009
$761$	15791.5	0.752223	0.376112	−	0.926574i	$-0.377261\pi$
$761$	15791.5	0.752223	0.376112	+	0.926574i	$0.377261\pi$
$762$	4441.63	0.211159
$763$	−18314.8	−0.868990
$764$	−1627.04	−0.0770475
$765$	8686.33	0.410529
$766$	16887.2	0.796552
$767$	−916.649	−0.0431529
$768$	12236.7	0.574938
$769$	−29233.6	−1.37086	−0.685430	−	0.728139i	$-0.740386\pi$
$769$	−29233.6	−1.37086	−0.685430	+	0.728139i	$0.740386\pi$
$770$	5639.25	0.263928
$771$	−4271.45	−0.199523
$772$	6959.46	0.324451
$773$	−37733.7	−1.75574	−0.877869	−	0.478901i	$-0.841035\pi$
$773$	−37733.7	−1.75574	−0.877869	+	0.478901i	$0.841035\pi$
$774$	5977.61	0.277598
$775$	−1757.63	−0.0814658
$776$	−36186.0	−1.67397
$777$	9039.25	0.417350
$778$	−28571.4	−1.31663
$779$	−29388.3	−1.35166
$780$	1467.20	0.0673514
$781$	5053.25	0.231523
$782$	21635.0	0.989343
$783$	−5238.56	−0.239094
$784$	3836.69	0.174776
$785$	1874.73	0.0852383
$786$	8948.80	0.406098
$787$	−13387.8	−0.606384	−0.303192	−	0.952929i	$-0.598052\pi$
$787$	−13387.8	−0.606384	−0.303192	+	0.952929i	$0.598052\pi$
$788$	−1223.12	−0.0552941
$789$	15616.2	0.704627
$790$	13027.9	0.586723
$791$	15378.4	0.691268
$792$	−4222.29	−0.189435
$793$	3729.21	0.166996
$794$	−16276.8	−0.727510
$795$	3140.90	0.140121
$796$	−3251.88	−0.144799
$797$	38966.4	1.73182	0.865911	−	0.500199i	$-0.166740\pi$
$797$	38966.4	1.73182	0.865911	+	0.500199i	$0.166740\pi$
$798$	−6856.44	−0.304155
$799$	−63551.4	−2.81388
$800$	−4568.67	−0.201909
$801$	−1192.64	−0.0526092
$802$	−9437.99	−0.415545
$803$	−5682.67	−0.249735
$804$	−987.272	−0.0433064
$805$	13418.2	0.587489
$806$	1785.51	0.0780295
$807$	−1047.67	−0.0456998
$808$	33074.2	1.44003
$809$	−22024.2	−0.957145	−0.478572	−	0.878048i	$-0.658846\pi$
$809$	−22024.2	−0.957145	−0.478572	+	0.878048i	$0.658846\pi$
$810$	1685.01	0.0730931
$811$	4588.64	0.198679	0.0993397	−	0.995054i	$-0.468327\pi$
$811$	4588.64	0.198679	0.0993397	+	0.995054i	$0.468327\pi$
$812$	−9026.00	−0.390087
$813$	24695.0	1.06530
$814$	−8846.20	−0.380908
$815$	1302.92	0.0559991
$816$	8135.64	0.349025
$817$	22740.4	0.973789
$818$	31778.1	1.35831
$819$	1979.84	0.0844702
$820$	12444.1	0.529958
$821$	−45052.8	−1.91517	−0.957584	−	0.288154i	$-0.906959\pi$
$821$	−45052.8	−1.91517	−0.957584	+	0.288154i	$0.906959\pi$
$822$	5995.29	0.254391
$823$	−17701.8	−0.749752	−0.374876	−	0.927075i	$-0.622315\pi$
$823$	−17701.8	−0.749752	−0.374876	+	0.927075i	$0.622315\pi$
$824$	−30829.9	−1.30341
$825$	1907.30	0.0804895
$826$	−1813.85	−0.0764068
$827$	10882.0	0.457562	0.228781	−	0.973478i	$-0.426526\pi$
$827$	10882.0	0.457562	0.228781	+	0.973478i	$0.426526\pi$
$828$	−2924.88	−0.122761
$829$	26577.8	1.11349	0.556746	−	0.830683i	$-0.312050\pi$
$829$	26577.8	1.11349	0.556746	+	0.830683i	$0.312050\pi$
$830$	−147.389	−0.00616381
$831$	21726.9	0.906975
$832$	7987.10	0.332816
$833$	−14357.0	−0.597169
$834$	−15861.3	−0.658552
$835$	−6012.53	−0.249188
$836$	−4676.34	−0.193463
$837$	−1429.09	−0.0590163
$838$	−13661.8	−0.563171
$839$	−13400.2	−0.551402	−0.275701	−	0.961243i	$-0.588910\pi$
$839$	−13400.2	−0.551402	−0.275701	+	0.961243i	$0.588910\pi$
$840$	9972.39	0.409619
$841$	13255.0	0.543483
$842$	−23415.7	−0.958383
$843$	−5309.29	−0.216918
$844$	4167.05	0.169947
$845$	−18736.5	−0.762788
$846$	−12328.0	−0.500999
$847$	−13655.7	−0.553975
$848$	2941.77	0.119128
$849$	−15910.0	−0.643146
$850$	−7263.32	−0.293094
$851$	−21048.9	−0.847880
$852$	2601.57	0.104611
$853$	−3838.21	−0.154065	−0.0770327	−	0.997029i	$-0.524545\pi$
$853$	−3838.21	−0.154065	−0.0770327	+	0.997029i	$0.524545\pi$
$854$	7379.31	0.295685
$855$	6410.24	0.256404
$856$	41416.2	1.65371
$857$	44265.5	1.76439	0.882194	−	0.470886i	$-0.156066\pi$
$857$	44265.5	1.76439	0.882194	+	0.470886i	$0.156066\pi$
$858$	−1937.55	−0.0770944
$859$	−26682.8	−1.05984	−0.529921	−	0.848047i	$-0.677779\pi$
$859$	−26682.8	−1.05984	−0.529921	+	0.848047i	$0.677779\pi$
$860$	−9629.12	−0.381803
$861$	16792.0	0.664659
$862$	12928.6	0.510845
$863$	−3259.02	−0.128550	−0.0642748	−	0.997932i	$-0.520473\pi$
$863$	−3259.02	−0.128550	−0.0642748	+	0.997932i	$0.520473\pi$
$864$	−3714.69	−0.146269
$865$	−27491.6	−1.08063
$866$	9904.84	0.388661
$867$	−15704.9	−0.615186
$868$	−2462.32	−0.0962862
$869$	11990.1	0.468050
$870$	−12108.4	−0.471855
$871$	−1556.16	−0.0605379
$872$	31695.9	1.23092
$873$	13290.6	0.515257
$874$	15966.0	0.617914
$875$	−21461.8	−0.829190
$876$	−2925.62	−0.112840
$877$	−37049.0	−1.42652	−0.713259	−	0.700900i	$-0.752782\pi$
$877$	−37049.0	−1.42652	−0.713259	+	0.700900i	$0.752782\pi$
$878$	−15852.7	−0.609341
$879$	−414.916	−0.0159212
$880$	−4938.02	−0.189160
$881$	7446.48	0.284765	0.142383	−	0.989812i	$-0.454524\pi$
$881$	7446.48	0.284765	0.142383	+	0.989812i	$0.454524\pi$
$882$	−2785.04	−0.106323
$883$	23801.2	0.907106	0.453553	−	0.891229i	$-0.350156\pi$
$883$	23801.2	0.907106	0.453553	+	0.891229i	$0.350156\pi$
$884$	−5142.24	−0.195647
$885$	1695.81	0.0644114
$886$	−34334.6	−1.30191
$887$	20503.7	0.776154	0.388077	−	0.921627i	$-0.373140\pi$
$887$	20503.7	0.776154	0.388077	+	0.921627i	$0.373140\pi$
$888$	−15643.5	−0.591174
$889$	−9654.77	−0.364242
$890$	−2756.68	−0.103825
$891$	1550.79	0.0583090
$892$	−3711.97	−0.139334
$893$	−46899.0	−1.75746
$894$	−8078.89	−0.302235
$895$	21713.9	0.810966
$896$	220.576	0.00822426
$897$	−4610.26	−0.171608
$898$	3484.79	0.129498
$899$	10269.4	0.380982
$900$	981.942	0.0363682
$901$	−11008.2	−0.407034
$902$	−16433.4	−0.606622
$903$	−12993.5	−0.478846
$904$	−26614.2	−0.979176
$905$	4302.66	0.158039
$906$	−22044.4	−0.808362
$907$	−10620.2	−0.388797	−0.194399	−	0.980923i	$-0.562276\pi$
$907$	−10620.2	−0.388797	−0.194399	+	0.980923i	$0.562276\pi$
$908$	529.666	0.0193586
$909$	−12147.7	−0.443250
$910$	4576.20	0.166703
$911$	−6121.02	−0.222611	−0.111305	−	0.993786i	$-0.535503\pi$
$911$	−6121.02	−0.222611	−0.111305	+	0.993786i	$0.535503\pi$
$912$	6003.85	0.217990
$913$	−135.648	−0.00491709
$914$	21527.2	0.779057
$915$	−6899.09	−0.249264
$916$	−390.919	−0.0141008
$917$	−19452.0	−0.700504
$918$	−5905.65	−0.212326
$919$	−50071.6	−1.79729	−0.898645	−	0.438677i	$-0.855447\pi$
$919$	−50071.6	−1.79729	−0.898645	+	0.438677i	$0.855447\pi$
$920$	−23221.8	−0.832174
$921$	−7393.72	−0.264529
$922$	−12413.2	−0.443393
$923$	4100.66	0.146235
$924$	2672.00	0.0951323
$925$	7066.54	0.251185
$926$	34373.8	1.21986
$927$	11323.4	0.401196
$928$	26693.6	0.944245
$929$	39484.7	1.39446	0.697228	−	0.716849i	$-0.254416\pi$
$929$	39484.7	1.39446	0.697228	+	0.716849i	$0.254416\pi$
$930$	−3303.21	−0.116469
$931$	−10595.0	−0.372974
$932$	14684.9	0.516117
$933$	27825.6	0.976387
$934$	814.637	0.0285393
$935$	18478.3	0.646314
$936$	−3426.35	−0.119651
$937$	−50370.5	−1.75617	−0.878085	−	0.478504i	$-0.841179\pi$
$937$	−50370.5	−1.75617	−0.878085	+	0.478504i	$0.841179\pi$
$938$	−3079.31	−0.107189
$939$	−17097.3	−0.594197
$940$	19858.8	0.689065
$941$	−18333.3	−0.635119	−0.317560	−	0.948238i	$-0.602863\pi$
$941$	−18333.3	−0.635119	−0.317560	+	0.948238i	$0.602863\pi$
$942$	−1274.59	−0.0440853
$943$	−39102.1	−1.35031
$944$	1588.30	0.0547615
$945$	−3662.72	−0.126083
$946$	12716.0	0.437034
$947$	−12078.7	−0.414473	−0.207236	−	0.978291i	$-0.566447\pi$
$947$	−12078.7	−0.414473	−0.207236	+	0.978291i	$0.566447\pi$
$948$	6172.88	0.211483
$949$	−4611.44	−0.157738
$950$	−5360.10	−0.183058
$951$	2579.88	0.0879688
$952$	−34951.3	−1.18989
$953$	40916.7	1.39079	0.695395	−	0.718628i	$-0.255230\pi$
$953$	40916.7	1.39079	0.695395	+	0.718628i	$0.255230\pi$
$954$	−2135.43	−0.0724707
$955$	4744.51	0.160763
$956$	15385.3	0.520497
$957$	−11143.9	−0.376416
$958$	−7820.03	−0.263730
$959$	−13032.0	−0.438816
$960$	−14776.2	−0.496772
$961$	−26989.5	−0.905961
$962$	−7178.62	−0.240590
$963$	−15211.6	−0.509021
$964$	12605.5	0.421157
$965$	−20294.0	−0.676983
$966$	−9122.73	−0.303850
$967$	27358.4	0.909812	0.454906	−	0.890539i	$-0.349673\pi$
$967$	27358.4	0.909812	0.454906	+	0.890539i	$0.349673\pi$
$968$	23632.9	0.784701
$969$	−22466.6	−0.744822
$970$	30720.0	1.01687
$971$	−25783.7	−0.852151	−0.426076	−	0.904688i	$-0.640104\pi$
$971$	−25783.7	−0.852151	−0.426076	+	0.904688i	$0.640104\pi$
$972$	798.395	0.0263462
$973$	34477.8	1.13598
$974$	−30684.9	−1.00945
$975$	1547.76	0.0508390
$976$	−6461.70	−0.211920
$977$	17329.6	0.567475	0.283738	−	0.958902i	$-0.408425\pi$
$977$	17329.6	0.567475	0.283738	+	0.958902i	$0.408425\pi$
$978$	−885.826	−0.0289628
$979$	−2537.09	−0.0828249
$980$	4486.33	0.146235
$981$	−11641.5	−0.378883
$982$	−636.231	−0.0206751
$983$	−23593.4	−0.765527	−0.382764	−	0.923846i	$-0.625028\pi$
$983$	−23593.4	−0.765527	−0.382764	+	0.923846i	$0.625028\pi$
$984$	−29060.7	−0.941484
$985$	3566.65	0.115374
$986$	42437.7	1.37068
$987$	26797.4	0.864206
$988$	−3794.81	−0.122195
$989$	30256.9	0.972813
$990$	3584.50	0.115074
$991$	27698.5	0.887863	0.443931	−	0.896061i	$-0.353583\pi$
$991$	27698.5	0.887863	0.443931	+	0.896061i	$0.353583\pi$
$992$	7282.07	0.233071
$993$	−30873.2	−0.986639
$994$	8114.34	0.258925
$995$	9482.60	0.302129
$996$	−69.8362	−0.00222173
$997$	21554.5	0.684691	0.342346	−	0.939574i	$-0.388779\pi$
$997$	21554.5	0.684691	0.342346	+	0.939574i	$0.388779\pi$
$998$	33809.3	1.07236
$999$	5745.65	0.181966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.c.1.6	✓	8
3.2	odd	2		531.4.a.f.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.6	✓	8	1.1	even	1	trivial
531.4.a.f.1.3		8	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q+2.17127 q^{2} -3.00000 q^{3} -3.28558 q^{4} +9.58086 q^{5} -6.51382 q^{6} +14.1591 q^{7} -24.5041 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.17127 q^{2} -3.00000 q^{3} -3.28558 q^{4} +9.58086 q^{5} -6.51382 q^{6} +14.1591 q^{7} -24.5041 q^{8} +9.00000 q^{9} +20.8027 q^{10} +19.1455 q^{11} +9.85673 q^{12} +15.5364 q^{13} +30.7433 q^{14} -28.7426 q^{15} -26.9204 q^{16} +100.737 q^{17} +19.5414 q^{18} +74.3408 q^{19} -31.4787 q^{20} -42.4773 q^{21} +41.5701 q^{22} +98.9130 q^{23} +73.5122 q^{24} -33.2071 q^{25} +33.7338 q^{26} -27.0000 q^{27} -46.5208 q^{28} +194.021 q^{29} -62.4080 q^{30} +52.9293 q^{31} +137.581 q^{32} -57.4366 q^{33} +218.728 q^{34} +135.656 q^{35} -29.5702 q^{36} -212.802 q^{37} +161.414 q^{38} -46.6093 q^{39} -234.770 q^{40} -395.318 q^{41} -92.2298 q^{42} +305.894 q^{43} -62.9041 q^{44} +86.2277 q^{45} +214.767 q^{46} -630.864 q^{47} +80.7611 q^{48} -142.520 q^{49} -72.1018 q^{50} -302.211 q^{51} -51.0461 q^{52} -109.277 q^{53} -58.6243 q^{54} +183.431 q^{55} -346.956 q^{56} -223.022 q^{57} +421.272 q^{58} -59.0000 q^{59} +94.4360 q^{60} +240.030 q^{61} +114.924 q^{62} +127.432 q^{63} +514.089 q^{64} +148.852 q^{65} -124.710 q^{66} -100.162 q^{67} -330.980 q^{68} -296.739 q^{69} +294.547 q^{70} +263.939 q^{71} -220.537 q^{72} -296.815 q^{73} -462.051 q^{74} +99.6214 q^{75} -244.253 q^{76} +271.083 q^{77} -101.201 q^{78} +626.260 q^{79} -257.920 q^{80} +81.0000 q^{81} -858.343 q^{82} -7.08512 q^{83} +139.562 q^{84} +965.148 q^{85} +664.179 q^{86} -582.062 q^{87} -469.143 q^{88} -132.516 q^{89} +187.224 q^{90} +219.982 q^{91} -324.986 q^{92} -158.788 q^{93} -1369.78 q^{94} +712.249 q^{95} -412.743 q^{96} +1476.73 q^{97} -309.449 q^{98} +172.310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9} + 29 q^{10} - 27 q^{11} - 114 q^{12} + 89 q^{13} - 37 q^{14} + 36 q^{15} + 362 q^{16} + 79 q^{17} + 18 q^{18} + 288 q^{19} + 457 q^{20} - 159 q^{21} + 596 q^{22} + 202 q^{23} - 9 q^{24} + 264 q^{25} + 270 q^{26} - 216 q^{27} + 702 q^{28} - 114 q^{29} - 87 q^{30} + 538 q^{31} + 316 q^{32} + 81 q^{33} + 498 q^{34} - 196 q^{35} + 342 q^{36} + 395 q^{37} + 397 q^{38} - 267 q^{39} + 918 q^{40} - 39 q^{41} + 111 q^{42} + 527 q^{43} + 64 q^{44} - 108 q^{45} - 539 q^{46} + 860 q^{47} - 1086 q^{48} + 347 q^{49} - 591 q^{50} - 237 q^{51} - 644 q^{52} - 812 q^{53} - 54 q^{54} + 536 q^{55} - 2218 q^{56} - 864 q^{57} - 1154 q^{58} - 472 q^{59} - 1371 q^{60} - 460 q^{61} - 2014 q^{62} + 477 q^{63} - 451 q^{64} - 986 q^{65} - 1788 q^{66} + 1934 q^{67} - 69 q^{68} - 606 q^{69} - 1028 q^{70} - 1687 q^{71} + 27 q^{72} + 1980 q^{73} - 2400 q^{74} - 792 q^{75} - 940 q^{76} - 821 q^{77} - 810 q^{78} + 3319 q^{79} - 2119 q^{80} + 648 q^{81} + 429 q^{82} + 2057 q^{83} - 2106 q^{84} + 566 q^{85} - 6690 q^{86} + 342 q^{87} + 1189 q^{88} + 1668 q^{89} + 261 q^{90} + 2427 q^{91} - 980 q^{92} - 1614 q^{93} + 332 q^{94} + 2146 q^{95} - 948 q^{96} + 1956 q^{97} - 2026 q^{98} - 243 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9 + 29 * q^10 - 27 * q^11 - 114 * q^12 + 89 * q^13 - 37 * q^14 + 36 * q^15 + 362 * q^16 + 79 * q^17 + 18 * q^18 + 288 * q^19 + 457 * q^20 - 159 * q^21 + 596 * q^22 + 202 * q^23 - 9 * q^24 + 264 * q^25 + 270 * q^26 - 216 * q^27 + 702 * q^28 - 114 * q^29 - 87 * q^30 + 538 * q^31 + 316 * q^32 + 81 * q^33 + 498 * q^34 - 196 * q^35 + 342 * q^36 + 395 * q^37 + 397 * q^38 - 267 * q^39 + 918 * q^40 - 39 * q^41 + 111 * q^42 + 527 * q^43 + 64 * q^44 - 108 * q^45 - 539 * q^46 + 860 * q^47 - 1086 * q^48 + 347 * q^49 - 591 * q^50 - 237 * q^51 - 644 * q^52 - 812 * q^53 - 54 * q^54 + 536 * q^55 - 2218 * q^56 - 864 * q^57 - 1154 * q^58 - 472 * q^59 - 1371 * q^60 - 460 * q^61 - 2014 * q^62 + 477 * q^63 - 451 * q^64 - 986 * q^65 - 1788 * q^66 + 1934 * q^67 - 69 * q^68 - 606 * q^69 - 1028 * q^70 - 1687 * q^71 + 27 * q^72 + 1980 * q^73 - 2400 * q^74 - 792 * q^75 - 940 * q^76 - 821 * q^77 - 810 * q^78 + 3319 * q^79 - 2119 * q^80 + 648 * q^81 + 429 * q^82 + 2057 * q^83 - 2106 * q^84 + 566 * q^85 - 6690 * q^86 + 342 * q^87 + 1189 * q^88 + 1668 * q^89 + 261 * q^90 + 2427 * q^91 - 980 * q^92 - 1614 * q^93 + 332 * q^94 + 2146 * q^95 - 948 * q^96 + 1956 * q^97 - 2026 * q^98 - 243 * q^99

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