LMFDB - Embedded newform 867.4.a.p.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,4,Mod(1,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, 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("867.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	867.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:	$51.1546559750$
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} - 4x^{7} - 36x^{6} + 116x^{5} + 431x^{4} - 860x^{3} - 1756x^{2} + 480x + 544 $ x^8 - 4x^7 - 36x^6 + 116x^5 + 431x^4 - 860x^3 - 1756x^2 + 480*x + 544
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.48829$ of defining polynomial
Character	$\chi$	$=$	867.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.48829 q^{2} -3.00000 q^{3} -1.80841 q^{4} +13.0806 q^{5} -7.46487 q^{6} -4.39839 q^{7} -24.4062 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.48829 q^{2} -3.00000 q^{3} -1.80841 q^{4} +13.0806 q^{5} -7.46487 q^{6} -4.39839 q^{7} -24.4062 q^{8} +9.00000 q^{9} +32.5485 q^{10} -42.5432 q^{11} +5.42522 q^{12} +78.3499 q^{13} -10.9445 q^{14} -39.2419 q^{15} -46.2624 q^{16} +22.3946 q^{18} -97.9336 q^{19} -23.6551 q^{20} +13.1952 q^{21} -105.860 q^{22} +182.797 q^{23} +73.2185 q^{24} +46.1033 q^{25} +194.957 q^{26} -27.0000 q^{27} +7.95408 q^{28} -3.88129 q^{29} -97.6454 q^{30} +192.030 q^{31} +80.1350 q^{32} +127.629 q^{33} -57.5338 q^{35} -16.2757 q^{36} +23.6261 q^{37} -243.687 q^{38} -235.050 q^{39} -319.249 q^{40} +203.331 q^{41} +32.8334 q^{42} -65.1959 q^{43} +76.9354 q^{44} +117.726 q^{45} +454.851 q^{46} +172.223 q^{47} +138.787 q^{48} -323.654 q^{49} +114.718 q^{50} -141.689 q^{52} -20.5667 q^{53} -67.1839 q^{54} -556.492 q^{55} +107.348 q^{56} +293.801 q^{57} -9.65779 q^{58} +68.2662 q^{59} +70.9654 q^{60} +530.411 q^{61} +477.827 q^{62} -39.5855 q^{63} +569.499 q^{64} +1024.87 q^{65} +317.579 q^{66} -12.3217 q^{67} -548.390 q^{69} -143.161 q^{70} +267.175 q^{71} -219.656 q^{72} +1167.19 q^{73} +58.7887 q^{74} -138.310 q^{75} +177.104 q^{76} +187.121 q^{77} -584.872 q^{78} +928.509 q^{79} -605.142 q^{80} +81.0000 q^{81} +505.947 q^{82} +236.643 q^{83} -23.8622 q^{84} -162.226 q^{86} +11.6439 q^{87} +1038.32 q^{88} -1216.07 q^{89} +292.936 q^{90} -344.614 q^{91} -330.571 q^{92} -576.090 q^{93} +428.542 q^{94} -1281.03 q^{95} -240.405 q^{96} -158.507 q^{97} -805.346 q^{98} -382.888 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9	\( 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9} - 72 q^{10} + 12 q^{11} - 72 q^{12} - 60 q^{13} + 56 q^{14} - 72 q^{15} - 100 q^{16} - 36 q^{18} - 140 q^{19} + 224 q^{20} - 84 q^{21} + 484 q^{22} - 96 q^{23} + 144 q^{24} + 220 q^{25} + 500 q^{26} - 216 q^{27} + 348 q^{28} + 380 q^{29} + 216 q^{30} + 124 q^{31} - 448 q^{32} - 36 q^{33} - 592 q^{35} + 216 q^{36} + 1260 q^{37} + 84 q^{38} + 180 q^{39} - 836 q^{40} + 284 q^{41} - 168 q^{42} - 76 q^{43} - 1460 q^{44} + 216 q^{45} + 1056 q^{46} - 184 q^{47} + 300 q^{48} - 536 q^{49} - 360 q^{50} - 1580 q^{52} + 1064 q^{53} + 108 q^{54} - 2148 q^{55} + 244 q^{56} + 420 q^{57} - 532 q^{58} + 1224 q^{59} - 672 q^{60} + 324 q^{61} + 3068 q^{62} + 252 q^{63} - 876 q^{64} + 652 q^{65} - 1452 q^{66} - 624 q^{67} + 288 q^{69} - 1932 q^{70} + 2636 q^{71} - 432 q^{72} + 1640 q^{73} - 120 q^{74} - 660 q^{75} - 1172 q^{76} + 504 q^{77} - 1500 q^{78} + 3788 q^{79} + 1740 q^{80} + 648 q^{81} + 1916 q^{82} + 2032 q^{83} - 1044 q^{84} + 1612 q^{86} - 1140 q^{87} + 4576 q^{88} + 1304 q^{89} - 648 q^{90} - 2872 q^{91} - 1744 q^{92} - 372 q^{93} - 208 q^{94} - 748 q^{95} + 1344 q^{96} + 2376 q^{97} + 604 q^{98} + 108 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9 - 72 * q^10 + 12 * q^11 - 72 * q^12 - 60 * q^13 + 56 * q^14 - 72 * q^15 - 100 * q^16 - 36 * q^18 - 140 * q^19 + 224 * q^20 - 84 * q^21 + 484 * q^22 - 96 * q^23 + 144 * q^24 + 220 * q^25 + 500 * q^26 - 216 * q^27 + 348 * q^28 + 380 * q^29 + 216 * q^30 + 124 * q^31 - 448 * q^32 - 36 * q^33 - 592 * q^35 + 216 * q^36 + 1260 * q^37 + 84 * q^38 + 180 * q^39 - 836 * q^40 + 284 * q^41 - 168 * q^42 - 76 * q^43 - 1460 * q^44 + 216 * q^45 + 1056 * q^46 - 184 * q^47 + 300 * q^48 - 536 * q^49 - 360 * q^50 - 1580 * q^52 + 1064 * q^53 + 108 * q^54 - 2148 * q^55 + 244 * q^56 + 420 * q^57 - 532 * q^58 + 1224 * q^59 - 672 * q^60 + 324 * q^61 + 3068 * q^62 + 252 * q^63 - 876 * q^64 + 652 * q^65 - 1452 * q^66 - 624 * q^67 + 288 * q^69 - 1932 * q^70 + 2636 * q^71 - 432 * q^72 + 1640 * q^73 - 120 * q^74 - 660 * q^75 - 1172 * q^76 + 504 * q^77 - 1500 * q^78 + 3788 * q^79 + 1740 * q^80 + 648 * q^81 + 1916 * q^82 + 2032 * q^83 - 1044 * q^84 + 1612 * q^86 - 1140 * q^87 + 4576 * q^88 + 1304 * q^89 - 648 * q^90 - 2872 * q^91 - 1744 * q^92 - 372 * q^93 - 208 * q^94 - 748 * q^95 + 1344 * q^96 + 2376 * q^97 + 604 * q^98 + 108 * 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 $

$2$	2.48829	0.879744	0.439872	−	0.898061i	$-0.355024\pi$
$2$	2.48829	0.879744	0.439872	+	0.898061i	$0.355024\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−1.80841	−0.226051
$5$	13.0806	1.16997	0.584984	−	0.811045i	$-0.301101\pi$
$5$	13.0806	1.16997	0.584984	+	0.811045i	$0.301101\pi$
$6$	−7.46487	−0.507920
$7$	−4.39839	−0.237491	−0.118745	−	0.992925i	$-0.537887\pi$
$7$	−4.39839	−0.237491	−0.118745	+	0.992925i	$0.537887\pi$
$8$	−24.4062	−1.07861
$9$	9.00000	0.333333
$10$	32.5485	1.02927
$11$	−42.5432	−1.16611	−0.583057	−	0.812431i	$-0.698143\pi$
$11$	−42.5432	−1.16611	−0.583057	+	0.812431i	$0.698143\pi$
$12$	5.42522	0.130511
$13$	78.3499	1.67157	0.835783	−	0.549060i	$-0.185014\pi$
$13$	78.3499	1.67157	0.835783	+	0.549060i	$0.185014\pi$
$14$	−10.9445	−0.208931
$15$	−39.2419	−0.675482
$16$	−46.2624	−0.722850
$17$	0	0
$17$	0	0
$18$	22.3946	0.293248
$19$	−97.9336	−1.18250	−0.591250	−	0.806488i	$-0.701365\pi$
$19$	−97.9336	−1.18250	−0.591250	+	0.806488i	$0.701365\pi$
$20$	−23.6551	−0.264472
$21$	13.1952	0.137115
$22$	−105.860	−1.02588
$23$	182.797	1.65721	0.828603	−	0.559836i	$-0.189136\pi$
$23$	182.797	1.65721	0.828603	+	0.559836i	$0.189136\pi$
$24$	73.2185	0.622736
$25$	46.1033	0.368826
$26$	194.957	1.47055
$27$	−27.0000	−0.192450
$28$	7.95408	0.0536850
$29$	−3.88129	−0.0248530	−0.0124265	−	0.999923i	$-0.503956\pi$
$29$	−3.88129	−0.0248530	−0.0124265	+	0.999923i	$0.503956\pi$
$30$	−97.6454	−0.594251
$31$	192.030	1.11257	0.556284	−	0.830992i	$-0.312227\pi$
$31$	192.030	1.11257	0.556284	+	0.830992i	$0.312227\pi$
$32$	80.1350	0.442688
$33$	127.629	0.673256
$34$	0	0
$35$	−57.5338	−0.277857
$36$	−16.2757	−0.0753503
$37$	23.6261	0.104976	0.0524880	−	0.998622i	$-0.483285\pi$
$37$	23.6261	0.104976	0.0524880	+	0.998622i	$0.483285\pi$
$38$	−243.687	−1.04030
$39$	−235.050	−0.965079
$40$	−319.249	−1.26194
$41$	203.331	0.774512	0.387256	−	0.921972i	$-0.373423\pi$
$41$	203.331	0.774512	0.387256	+	0.921972i	$0.373423\pi$
$42$	32.8334	0.120626
$43$	−65.1959	−0.231216	−0.115608	−	0.993295i	$-0.536882\pi$
$43$	−65.1959	−0.231216	−0.115608	+	0.993295i	$0.536882\pi$
$44$	76.9354	0.263601
$45$	117.726	0.389990
$46$	454.851	1.45792
$47$	172.223	0.534497	0.267248	−	0.963628i	$-0.413886\pi$
$47$	172.223	0.534497	0.267248	+	0.963628i	$0.413886\pi$
$48$	138.787	0.417338
$49$	−323.654	−0.943598
$50$	114.718	0.324473
$51$	0	0
$52$	−141.689	−0.377859
$53$	−20.5667	−0.0533030	−0.0266515	−	0.999645i	$-0.508484\pi$
$53$	−20.5667	−0.0533030	−0.0266515	+	0.999645i	$0.508484\pi$
$54$	−67.1839	−0.169307
$55$	−556.492	−1.36432
$56$	107.348	0.256160
$57$	293.801	0.682717
$58$	−9.65779	−0.0218643
$59$	68.2662	0.150636	0.0753179	−	0.997160i	$-0.476003\pi$
$59$	68.2662	0.150636	0.0753179	+	0.997160i	$0.476003\pi$
$60$	70.9654	0.152693
$61$	530.411	1.11331	0.556657	−	0.830742i	$-0.312084\pi$
$61$	530.411	1.11331	0.556657	+	0.830742i	$0.312084\pi$
$62$	477.827	0.978775
$63$	−39.5855	−0.0791636
$64$	569.499	1.11230
$65$	1024.87	1.95568
$66$	317.579	0.592293
$67$	−12.3217	−0.0224677	−0.0112339	−	0.999937i	$-0.503576\pi$
$67$	−12.3217	−0.0224677	−0.0112339	+	0.999937i	$0.503576\pi$
$68$	0	0
$69$	−548.390	−0.956789
$70$	−143.161	−0.244443
$71$	267.175	0.446590	0.223295	−	0.974751i	$-0.428319\pi$
$71$	267.175	0.446590	0.223295	+	0.974751i	$0.428319\pi$
$72$	−219.656	−0.359537
$73$	1167.19	1.87135	0.935676	−	0.352860i	$-0.114791\pi$
$73$	1167.19	1.87135	0.935676	+	0.352860i	$0.114791\pi$
$74$	58.7887	0.0923520
$75$	−138.310	−0.212942
$76$	177.104	0.267305
$77$	187.121	0.276941
$78$	−584.872	−0.849022
$79$	928.509	1.32235	0.661174	−	0.750233i	$-0.270059\pi$
$79$	928.509	1.32235	0.661174	+	0.750233i	$0.270059\pi$
$80$	−605.142	−0.845712
$81$	81.0000	0.111111
$82$	505.947	0.681372
$83$	236.643	0.312951	0.156475	−	0.987682i	$-0.449987\pi$
$83$	236.643	0.312951	0.156475	+	0.987682i	$0.449987\pi$
$84$	−23.8622	−0.0309951
$85$	0	0
$86$	−162.226	−0.203411
$87$	11.6439	0.0143489
$88$	1038.32	1.25778
$89$	−1216.07	−1.44835	−0.724177	−	0.689614i	$-0.757780\pi$
$89$	−1216.07	−1.44835	−0.724177	+	0.689614i	$0.757780\pi$
$90$	292.936	0.343091
$91$	−344.614	−0.396981
$92$	−330.571	−0.374613
$93$	−576.090	−0.642342
$94$	428.542	0.470220
$95$	−1281.03	−1.38349
$96$	−240.405	−0.255586
$97$	−158.507	−0.165917	−0.0829584	−	0.996553i	$-0.526437\pi$
$97$	−158.507	−0.165917	−0.0829584	+	0.996553i	$0.526437\pi$
$98$	−805.346	−0.830125
$99$	−382.888	−0.388704
$100$	−83.3736	−0.0833736
$101$	1828.17	1.80108	0.900541	−	0.434771i	$-0.143171\pi$
$101$	1828.17	1.80108	0.900541	+	0.434771i	$0.143171\pi$
$102$	0	0
$103$	1341.75	1.28356	0.641781	−	0.766888i	$-0.278196\pi$
$103$	1341.75	1.28356	0.641781	+	0.766888i	$0.278196\pi$
$104$	−1912.22	−1.80297
$105$	172.601	0.160421
$106$	−51.1760	−0.0468930
$107$	−1193.17	−1.07802	−0.539010	−	0.842299i	$-0.681202\pi$
$107$	−1193.17	−1.07802	−0.539010	+	0.842299i	$0.681202\pi$
$108$	48.8270	0.0435035
$109$	1780.94	1.56499	0.782493	−	0.622660i	$-0.213948\pi$
$109$	1780.94	1.56499	0.782493	+	0.622660i	$0.213948\pi$
$110$	−1384.71	−1.20025
$111$	−70.8784	−0.0606079
$112$	203.480	0.171670
$113$	627.008	0.521982	0.260991	−	0.965341i	$-0.415951\pi$
$113$	627.008	0.521982	0.260991	+	0.965341i	$0.415951\pi$
$114$	731.062	0.600616
$115$	2391.10	1.93888
$116$	7.01896	0.00561805
$117$	705.149	0.557189
$118$	169.866	0.132521
$119$	0	0
$120$	957.746	0.728582
$121$	478.921	0.359820
$122$	1319.82	0.979431
$123$	−609.993	−0.447165
$124$	−347.268	−0.251497
$125$	−1032.02	−0.738453
$126$	−98.5003	−0.0696437
$127$	−771.160	−0.538814	−0.269407	−	0.963026i	$-0.586828\pi$
$127$	−771.160	−0.538814	−0.269407	+	0.963026i	$0.586828\pi$
$128$	775.998	0.535853
$129$	195.588	0.133493
$130$	2550.17	1.72050
$131$	2303.37	1.53623	0.768117	−	0.640309i	$-0.221194\pi$
$131$	2303.37	1.53623	0.768117	+	0.640309i	$0.221194\pi$
$132$	−230.806	−0.152190
$133$	430.750	0.280833
$134$	−30.6600	−0.0197658
$135$	−353.177	−0.225161
$136$	0	0
$137$	−626.370	−0.390616	−0.195308	−	0.980742i	$-0.562571\pi$
$137$	−626.370	−0.390616	−0.195308	+	0.980742i	$0.562571\pi$
$138$	−1364.55	−0.841729
$139$	491.350	0.299826	0.149913	−	0.988699i	$-0.452101\pi$
$139$	491.350	0.299826	0.149913	+	0.988699i	$0.452101\pi$
$140$	104.045	0.0628098
$141$	−516.670	−0.308592
$142$	664.810	0.392884
$143$	−3333.25	−1.94924
$144$	−416.362	−0.240950
$145$	−50.7698	−0.0290773
$146$	2904.30	1.64631
$147$	970.962	0.544787
$148$	−42.7257	−0.0237299
$149$	1106.05	0.608126	0.304063	−	0.952652i	$-0.401657\pi$
$149$	1106.05	0.608126	0.304063	+	0.952652i	$0.401657\pi$
$150$	−344.155	−0.187334
$151$	−2836.17	−1.52850	−0.764252	−	0.644918i	$-0.776892\pi$
$151$	−2836.17	−1.52850	−0.764252	+	0.644918i	$0.776892\pi$
$152$	2390.18	1.27546
$153$	0	0
$154$	465.613	0.243637
$155$	2511.88	1.30167
$156$	425.066	0.218157
$157$	3191.74	1.62248	0.811238	−	0.584716i	$-0.198794\pi$
$157$	3191.74	1.62248	0.811238	+	0.584716i	$0.198794\pi$
$158$	2310.40	1.16333
$159$	61.7002	0.0307745
$160$	1048.22	0.517931
$161$	−804.012	−0.393571
$162$	201.552	0.0977493
$163$	−3391.98	−1.62994	−0.814970	−	0.579503i	$-0.803247\pi$
$163$	−3391.98	−1.62994	−0.814970	+	0.579503i	$0.803247\pi$
$164$	−367.705	−0.175079
$165$	1669.48	0.787688
$166$	588.836	0.275316
$167$	−509.484	−0.236078	−0.118039	−	0.993009i	$-0.537661\pi$
$167$	−509.484	−0.236078	−0.118039	+	0.993009i	$0.537661\pi$
$168$	−322.044	−0.147894
$169$	3941.71	1.79413
$170$	0	0
$171$	−881.402	−0.394167
$172$	117.901	0.0522665
$173$	2882.85	1.26693	0.633465	−	0.773771i	$-0.281632\pi$
$173$	2882.85	1.26693	0.633465	+	0.773771i	$0.281632\pi$
$174$	28.9734	0.0126234
$175$	−202.780	−0.0875929
$176$	1968.15	0.842925
$177$	−204.799	−0.0869696
$178$	−3025.95	−1.27418
$179$	2170.25	0.906211	0.453106	−	0.891457i	$-0.350316\pi$
$179$	2170.25	0.906211	0.453106	+	0.891457i	$0.350316\pi$
$180$	−212.896	−0.0881575
$181$	325.490	0.133665	0.0668327	−	0.997764i	$-0.478711\pi$
$181$	325.490	0.133665	0.0668327	+	0.997764i	$0.478711\pi$
$182$	−857.499	−0.349242
$183$	−1591.23	−0.642772
$184$	−4461.37	−1.78748
$185$	309.045	0.122819
$186$	−1433.48	−0.565096
$187$	0	0
$188$	−311.450	−0.120823
$189$	118.757	0.0457051
$190$	−3187.59	−1.21712
$191$	−500.443	−0.189585	−0.0947927	−	0.995497i	$-0.530219\pi$
$191$	−500.443	−0.189585	−0.0947927	+	0.995497i	$0.530219\pi$
$192$	−1708.50	−0.642188
$193$	−3461.93	−1.29117	−0.645584	−	0.763689i	$-0.723386\pi$
$193$	−3461.93	−1.29117	−0.645584	+	0.763689i	$0.723386\pi$
$194$	−394.411	−0.145964
$195$	−3074.60	−1.12911
$196$	585.298	0.213301
$197$	−1041.31	−0.376599	−0.188300	−	0.982112i	$-0.560298\pi$
$197$	−1041.31	−0.376599	−0.188300	+	0.982112i	$0.560298\pi$
$198$	−952.738	−0.341960
$199$	−1667.81	−0.594109	−0.297055	−	0.954860i	$-0.596004\pi$
$199$	−1667.81	−0.594109	−0.297055	+	0.954860i	$0.596004\pi$
$200$	−1125.21	−0.397820
$201$	36.9651	0.0129717
$202$	4549.01	1.58449
$203$	17.0715	0.00590237
$204$	0	0
$205$	2659.70	0.906154
$206$	3338.67	1.12921
$207$	1645.17	0.552402
$208$	−3624.66	−1.20829
$209$	4166.41	1.37893
$210$	429.483	0.141129
$211$	−533.520	−0.174071	−0.0870356	−	0.996205i	$-0.527739\pi$
$211$	−533.520	−0.174071	−0.0870356	+	0.996205i	$0.527739\pi$
$212$	37.1930	0.0120492
$213$	−801.526	−0.257839
$214$	−2968.96	−0.948382
$215$	−852.804	−0.270515
$216$	658.967	0.207579
$217$	−844.623	−0.264225
$218$	4431.51	1.37679
$219$	−3501.56	−1.08043
$220$	1006.36	0.308405
$221$	0	0
$222$	−176.366	−0.0533194
$223$	−3866.75	−1.16115	−0.580576	−	0.814206i	$-0.697172\pi$
$223$	−3866.75	−1.16115	−0.580576	+	0.814206i	$0.697172\pi$
$224$	−352.465	−0.105134
$225$	414.930	0.122942
$226$	1560.18	0.459210
$227$	−3880.71	−1.13468	−0.567339	−	0.823484i	$-0.692027\pi$
$227$	−3880.71	−1.13468	−0.567339	+	0.823484i	$0.692027\pi$
$228$	−531.311	−0.154329
$229$	5773.87	1.66615	0.833074	−	0.553162i	$-0.186579\pi$
$229$	5773.87	1.66615	0.833074	+	0.553162i	$0.186579\pi$
$230$	5949.75	1.70572
$231$	−561.364	−0.159892
$232$	94.7275	0.0268068
$233$	−1930.90	−0.542907	−0.271453	−	0.962452i	$-0.587504\pi$
$233$	−1930.90	−0.542907	−0.271453	+	0.962452i	$0.587504\pi$
$234$	1754.62	0.490183
$235$	2252.79	0.625344
$236$	−123.453	−0.0340513
$237$	−2785.53	−0.763458
$238$	0	0
$239$	833.853	0.225680	0.112840	−	0.993613i	$-0.464005\pi$
$239$	833.853	0.225680	0.112840	+	0.993613i	$0.464005\pi$
$240$	1815.43	0.488272
$241$	1736.31	0.464089	0.232045	−	0.972705i	$-0.425458\pi$
$241$	1736.31	0.464089	0.232045	+	0.972705i	$0.425458\pi$
$242$	1191.69	0.316550
$243$	−243.000	−0.0641500
$244$	−959.199	−0.251666
$245$	−4233.61	−1.10398
$246$	−1517.84	−0.393390
$247$	−7673.09	−1.97663
$248$	−4686.72	−1.20003
$249$	−709.928	−0.180682
$250$	−2567.97	−0.649650
$251$	−4673.83	−1.17534	−0.587668	−	0.809102i	$-0.699954\pi$
$251$	−4673.83	−1.17534	−0.587668	+	0.809102i	$0.699954\pi$
$252$	71.5867	0.0178950
$253$	−7776.75	−1.93249
$254$	−1918.87	−0.474018
$255$	0	0
$256$	−2625.08	−0.640889
$257$	1979.98	0.480574	0.240287	−	0.970702i	$-0.422758\pi$
$257$	1979.98	0.480574	0.240287	+	0.970702i	$0.422758\pi$
$258$	486.679	0.117439
$259$	−103.917	−0.0249308
$260$	−1853.38	−0.442083
$261$	−34.9317	−0.00828435
$262$	5731.47	1.35149
$263$	−4596.79	−1.07776	−0.538879	−	0.842383i	$-0.681152\pi$
$263$	−4596.79	−1.07776	−0.538879	+	0.842383i	$0.681152\pi$
$264$	−3114.95	−0.726181
$265$	−269.026	−0.0623628
$266$	1071.83	0.247061
$267$	3648.22	0.836208
$268$	22.2827	0.00507884
$269$	3848.29	0.872246	0.436123	−	0.899887i	$-0.356351\pi$
$269$	3848.29	0.872246	0.436123	+	0.899887i	$0.356351\pi$
$270$	−878.808	−0.198084
$271$	−7510.38	−1.68348	−0.841740	−	0.539883i	$-0.818468\pi$
$271$	−7510.38	−1.68348	−0.841740	+	0.539883i	$0.818468\pi$
$272$	0	0
$273$	1033.84	0.229197
$274$	−1558.59	−0.343642
$275$	−1961.38	−0.430093
$276$	991.713	0.216283
$277$	1146.43	0.248673	0.124337	−	0.992240i	$-0.460320\pi$
$277$	1146.43	0.248673	0.124337	+	0.992240i	$0.460320\pi$
$278$	1222.62	0.263770
$279$	1728.27	0.370856
$280$	1404.18	0.299699
$281$	−3659.49	−0.776893	−0.388446	−	0.921471i	$-0.626988\pi$
$281$	−3659.49	−0.776893	−0.388446	+	0.921471i	$0.626988\pi$
$282$	−1285.62	−0.271482
$283$	−2518.37	−0.528981	−0.264491	−	0.964388i	$-0.585204\pi$
$283$	−2518.37	−0.528981	−0.264491	+	0.964388i	$0.585204\pi$
$284$	−483.162	−0.100952
$285$	3843.10	0.798757
$286$	−8294.10	−1.71483
$287$	−894.330	−0.183939
$288$	721.215	0.147563
$289$	0	0
$290$	−126.330	−0.0255806
$291$	475.521	0.0957921
$292$	−2110.75	−0.423021
$293$	−1440.56	−0.287229	−0.143615	−	0.989634i	$-0.545873\pi$
$293$	−1440.56	−0.287229	−0.143615	+	0.989634i	$0.545873\pi$
$294$	2416.04	0.479273
$295$	892.967	0.176239
$296$	−576.623	−0.113228
$297$	1148.67	0.224419
$298$	2752.16	0.534995
$299$	14322.1	2.77013
$300$	250.121	0.0481357
$301$	286.757	0.0549116
$302$	−7057.21	−1.34469
$303$	−5484.50	−1.03986
$304$	4530.64	0.854771
$305$	6938.12	1.30254
$306$	0	0
$307$	−7714.19	−1.43411	−0.717055	−	0.697016i	$-0.754510\pi$
$307$	−7714.19	−1.43411	−0.717055	+	0.697016i	$0.754510\pi$
$308$	−338.392	−0.0626028
$309$	−4025.26	−0.741065
$310$	6250.28	1.14514
$311$	−927.991	−0.169201	−0.0846005	−	0.996415i	$-0.526961\pi$
$311$	−927.991	−0.169201	−0.0846005	+	0.996415i	$0.526961\pi$
$312$	5736.66	1.04094
$313$	4461.84	0.805745	0.402872	−	0.915256i	$-0.368012\pi$
$313$	4461.84	0.805745	0.402872	+	0.915256i	$0.368012\pi$
$314$	7941.99	1.42736
$315$	−517.804	−0.0926189
$316$	−1679.12	−0.298918
$317$	−4862.79	−0.861583	−0.430791	−	0.902452i	$-0.641765\pi$
$317$	−4862.79	−0.861583	−0.430791	+	0.902452i	$0.641765\pi$
$318$	153.528	0.0270737
$319$	165.123	0.0289815
$320$	7449.41	1.30136
$321$	3579.51	0.622395
$322$	−2000.61	−0.346242
$323$	0	0
$324$	−146.481	−0.0251168
$325$	3612.19	0.616518
$326$	−8440.23	−1.43393
$327$	−5342.83	−0.903545
$328$	−4962.53	−0.835397
$329$	−757.505	−0.126938
$330$	4154.14	0.692964
$331$	4773.38	0.792655	0.396327	−	0.918109i	$-0.370285\pi$
$331$	4773.38	0.792655	0.396327	+	0.918109i	$0.370285\pi$
$332$	−427.946	−0.0707428
$333$	212.635	0.0349920
$334$	−1267.74	−0.207688
$335$	−161.176	−0.0262865
$336$	−610.440	−0.0991139
$337$	−3778.93	−0.610834	−0.305417	−	0.952219i	$-0.598796\pi$
$337$	−3778.93	−0.610834	−0.305417	+	0.952219i	$0.598796\pi$
$338$	9808.12	1.57838
$339$	−1881.02	−0.301366
$340$	0	0
$341$	−8169.57	−1.29738
$342$	−2193.19	−0.346766
$343$	2932.21	0.461587
$344$	1591.18	0.249392
$345$	−7173.30	−1.11941
$346$	7173.37	1.11457
$347$	−6840.93	−1.05833	−0.529165	−	0.848519i	$-0.677495\pi$
$347$	−6840.93	−1.05833	−0.529165	+	0.848519i	$0.677495\pi$
$348$	−21.0569	−0.00324358
$349$	−1856.99	−0.284820	−0.142410	−	0.989808i	$-0.545485\pi$
$349$	−1856.99	−0.284820	−0.142410	+	0.989808i	$0.545485\pi$
$350$	−504.577	−0.0770593
$351$	−2115.45	−0.321693
$352$	−3409.20	−0.516224
$353$	9899.93	1.49269	0.746346	−	0.665558i	$-0.231806\pi$
$353$	9899.93	1.49269	0.746346	+	0.665558i	$0.231806\pi$
$354$	−509.599	−0.0765109
$355$	3494.82	0.522496
$356$	2199.16	0.327402
$357$	0	0
$358$	5400.20	0.797234
$359$	−2268.74	−0.333536	−0.166768	−	0.985996i	$-0.553333\pi$
$359$	−2268.74	−0.333536	−0.166768	+	0.985996i	$0.553333\pi$
$360$	−2873.24	−0.420647
$361$	2731.99	0.398307
$362$	809.913	0.117591
$363$	−1436.76	−0.207742
$364$	623.202	0.0897380
$365$	15267.5	2.18942
$366$	−3959.45	−0.565475
$367$	11286.8	1.60536	0.802678	−	0.596413i	$-0.203408\pi$
$367$	11286.8	1.60536	0.802678	+	0.596413i	$0.203408\pi$
$368$	−8456.62	−1.19791
$369$	1829.98	0.258171
$370$	768.994	0.108049
$371$	90.4605	0.0126590
$372$	1041.81	0.145202
$373$	−14045.4	−1.94972	−0.974859	−	0.222821i	$-0.928473\pi$
$373$	−14045.4	−1.94972	−0.974859	+	0.222821i	$0.928473\pi$
$374$	0	0
$375$	3096.06	0.426346
$376$	−4203.31	−0.576514
$377$	−304.099	−0.0415435
$378$	295.501	0.0402088
$379$	−7085.67	−0.960334	−0.480167	−	0.877177i	$-0.659424\pi$
$379$	−7085.67	−0.960334	−0.480167	+	0.877177i	$0.659424\pi$
$380$	2316.63	0.312739
$381$	2313.48	0.311084
$382$	−1245.25	−0.166787
$383$	13226.0	1.76454	0.882269	−	0.470745i	$-0.156015\pi$
$383$	13226.0	1.76454	0.882269	+	0.470745i	$0.156015\pi$
$384$	−2327.99	−0.309375
$385$	2447.67	0.324012
$386$	−8614.30	−1.13590
$387$	−586.763	−0.0770719
$388$	286.645	0.0375056
$389$	−6279.21	−0.818429	−0.409214	−	0.912438i	$-0.634197\pi$
$389$	−6279.21	−0.818429	−0.409214	+	0.912438i	$0.634197\pi$
$390$	−7650.51	−0.993329
$391$	0	0
$392$	7899.16	1.01777
$393$	−6910.12	−0.886946
$394$	−2591.07	−0.331311
$395$	12145.5	1.54711
$396$	692.418	0.0878670
$397$	3398.14	0.429591	0.214795	−	0.976659i	$-0.431092\pi$
$397$	3398.14	0.429591	0.214795	+	0.976659i	$0.431092\pi$
$398$	−4149.99	−0.522664
$399$	−1292.25	−0.162139
$400$	−2132.85	−0.266606
$401$	9794.58	1.21975	0.609873	−	0.792499i	$-0.291221\pi$
$401$	9794.58	1.21975	0.609873	+	0.792499i	$0.291221\pi$
$402$	91.9800	0.0114118
$403$	15045.5	1.85973
$404$	−3306.07	−0.407136
$405$	1059.53	0.129997
$406$	42.4787	0.00519257
$407$	−1005.13	−0.122414
$408$	0	0
$409$	14212.5	1.71825	0.859125	−	0.511766i	$-0.171009\pi$
$409$	14212.5	1.71825	0.859125	+	0.511766i	$0.171009\pi$
$410$	6618.11	0.797184
$411$	1879.11	0.225522
$412$	−2426.44	−0.290150
$413$	−300.262	−0.0357746
$414$	4093.66	0.485972
$415$	3095.44	0.366142
$416$	6278.57	0.739982
$417$	−1474.05	−0.173104
$418$	10367.2	1.21310
$419$	−6723.88	−0.783969	−0.391984	−	0.919972i	$-0.628211\pi$
$419$	−6723.88	−0.783969	−0.391984	+	0.919972i	$0.628211\pi$
$420$	−312.134	−0.0362632
$421$	2696.63	0.312175	0.156088	−	0.987743i	$-0.450112\pi$
$421$	2696.63	0.312175	0.156088	+	0.987743i	$0.450112\pi$
$422$	−1327.55	−0.153138
$423$	1550.01	0.178166
$424$	501.955	0.0574932
$425$	0	0
$426$	−1994.43	−0.226832
$427$	−2332.96	−0.264402
$428$	2157.74	0.243687
$429$	9999.76	1.12539
$430$	−2122.03	−0.237984
$431$	−9276.76	−1.03677	−0.518383	−	0.855149i	$-0.673466\pi$
$431$	−9276.76	−1.03677	−0.518383	+	0.855149i	$0.673466\pi$
$432$	1249.09	0.139113
$433$	−2404.97	−0.266918	−0.133459	−	0.991054i	$-0.542608\pi$
$433$	−2404.97	−0.266918	−0.133459	+	0.991054i	$0.542608\pi$
$434$	−2101.67	−0.232450
$435$	152.310	0.0167878
$436$	−3220.67	−0.353766
$437$	−17901.9	−1.95965
$438$	−8712.89	−0.950498
$439$	12788.8	1.39038	0.695189	−	0.718827i	$-0.255321\pi$
$439$	12788.8	1.39038	0.695189	+	0.718827i	$0.255321\pi$
$440$	13581.8	1.47157
$441$	−2912.89	−0.314533
$442$	0	0
$443$	−5158.91	−0.553289	−0.276645	−	0.960972i	$-0.589223\pi$
$443$	−5158.91	−0.553289	−0.276645	+	0.960972i	$0.589223\pi$
$444$	128.177	0.0137005
$445$	−15907.0	−1.69453
$446$	−9621.60	−1.02152
$447$	−3318.14	−0.351102
$448$	−2504.88	−0.264161
$449$	−5578.55	−0.586343	−0.293172	−	0.956060i	$-0.594711\pi$
$449$	−5578.55	−0.586343	−0.293172	+	0.956060i	$0.594711\pi$
$450$	1032.47	0.108158
$451$	−8650.35	−0.903168
$452$	−1133.89	−0.117994
$453$	8508.51	0.882482
$454$	−9656.35	−0.998226
$455$	−4507.77	−0.464456
$456$	−7170.55	−0.736386
$457$	−8938.73	−0.914959	−0.457479	−	0.889220i	$-0.651248\pi$
$457$	−8938.73	−0.914959	−0.457479	+	0.889220i	$0.651248\pi$
$458$	14367.1	1.46578
$459$	0	0
$460$	−4324.08	−0.438285
$461$	7265.66	0.734047	0.367024	−	0.930212i	$-0.380377\pi$
$461$	7265.66	0.734047	0.367024	+	0.930212i	$0.380377\pi$
$462$	−1396.84	−0.140664
$463$	−9395.66	−0.943096	−0.471548	−	0.881840i	$-0.656305\pi$
$463$	−9395.66	−0.943096	−0.471548	+	0.881840i	$0.656305\pi$
$464$	179.558	0.0179650
$465$	−7535.63	−0.751519
$466$	−4804.63	−0.477619
$467$	14717.4	1.45833	0.729164	−	0.684339i	$-0.239909\pi$
$467$	14717.4	1.45833	0.729164	+	0.684339i	$0.239909\pi$
$468$	−1275.20	−0.125953
$469$	54.1957	0.00533587
$470$	5605.60	0.550143
$471$	−9575.23	−0.936737
$472$	−1666.12	−0.162477
$473$	2773.64	0.269624
$474$	−6931.21	−0.671647
$475$	−4515.06	−0.436138
$476$	0	0
$477$	−185.101	−0.0177677
$478$	2074.87	0.198540
$479$	4564.64	0.435415	0.217707	−	0.976014i	$-0.430142\pi$
$479$	4564.64	0.435415	0.217707	+	0.976014i	$0.430142\pi$
$480$	−3144.65	−0.299027
$481$	1851.10	0.175474
$482$	4320.44	0.408280
$483$	2412.03	0.227229
$484$	−866.084	−0.0813377
$485$	−2073.37	−0.194117
$486$	−604.655	−0.0564356
$487$	−13642.7	−1.26943	−0.634713	−	0.772748i	$-0.718882\pi$
$487$	−13642.7	−1.26943	−0.634713	+	0.772748i	$0.718882\pi$
$488$	−12945.3	−1.20083
$489$	10175.9	0.941046
$490$	−10534.4	−0.971220
$491$	15728.0	1.44561	0.722805	−	0.691052i	$-0.242852\pi$
$491$	15728.0	1.44561	0.722805	+	0.691052i	$0.242852\pi$
$492$	1103.12	0.101082
$493$	0	0
$494$	−19092.9	−1.73893
$495$	−5008.43	−0.454772
$496$	−8883.77	−0.804220
$497$	−1175.14	−0.106061
$498$	−1766.51	−0.158954
$499$	7997.35	0.717456	0.358728	−	0.933442i	$-0.383211\pi$
$499$	7997.35	0.717456	0.358728	+	0.933442i	$0.383211\pi$
$500$	1866.31	0.166928
$501$	1528.45	0.136300
$502$	−11629.9	−1.03400
$503$	−1139.68	−0.101025	−0.0505127	−	0.998723i	$-0.516086\pi$
$503$	−1139.68	−0.101025	−0.0505127	+	0.998723i	$0.516086\pi$
$504$	966.131	0.0853867
$505$	23913.6	2.10721
$506$	−19350.8	−1.70010
$507$	−11825.1	−1.03584
$508$	1394.57	0.121799
$509$	−11491.0	−1.00065	−0.500323	−	0.865839i	$-0.666786\pi$
$509$	−11491.0	−1.00065	−0.500323	+	0.865839i	$0.666786\pi$
$510$	0	0
$511$	−5133.74	−0.444429
$512$	−12739.9	−1.09967
$513$	2644.21	0.227572
$514$	4926.76	0.422782
$515$	17551.0	1.50173
$516$	−353.702	−0.0301761
$517$	−7326.92	−0.623284
$518$	−258.576	−0.0219327
$519$	−8648.55	−0.731463
$520$	−25013.1	−2.10942
$521$	2732.20	0.229750	0.114875	−	0.993380i	$-0.463353\pi$
$521$	2732.20	0.229750	0.114875	+	0.993380i	$0.463353\pi$
$522$	−86.9201	−0.00728810
$523$	6802.34	0.568730	0.284365	−	0.958716i	$-0.408217\pi$
$523$	6802.34	0.568730	0.284365	+	0.958716i	$0.408217\pi$
$524$	−4165.44	−0.347267
$525$	608.341	0.0505718
$526$	−11438.2	−0.948151
$527$	0	0
$528$	−5904.45	−0.486663
$529$	21247.6	1.74633
$530$	−669.415	−0.0548633
$531$	614.396	0.0502119
$532$	−778.972	−0.0634825
$533$	15931.0	1.29465
$534$	9077.84	0.735649
$535$	−15607.4	−1.26125
$536$	300.726	0.0242339
$537$	−6510.74	−0.523201
$538$	9575.66	0.767353
$539$	13769.3	1.10034
$540$	638.689	0.0508977
$541$	−4630.22	−0.367964	−0.183982	−	0.982930i	$-0.558899\pi$
$541$	−4630.22	−0.367964	−0.183982	+	0.982930i	$0.558899\pi$
$542$	−18688.0	−1.48103
$543$	−976.469	−0.0771718
$544$	0	0
$545$	23295.9	1.83098
$546$	2572.50	0.201635
$547$	9258.38	0.723692	0.361846	−	0.932238i	$-0.382147\pi$
$547$	9258.38	0.723692	0.361846	+	0.932238i	$0.382147\pi$
$548$	1132.73	0.0882991
$549$	4773.70	0.371105
$550$	−4880.49	−0.378372
$551$	380.109	0.0293887
$552$	13384.1	1.03200
$553$	−4083.95	−0.314045
$554$	2852.66	0.218769
$555$	−927.135	−0.0709093
$556$	−888.561	−0.0677759
$557$	4344.73	0.330506	0.165253	−	0.986251i	$-0.447156\pi$
$557$	4344.73	0.330506	0.165253	+	0.986251i	$0.447156\pi$
$558$	4300.44	0.326258
$559$	−5108.09	−0.386492
$560$	2661.65	0.200849
$561$	0	0
$562$	−9105.88	−0.683467
$563$	25687.6	1.92292	0.961461	−	0.274941i	$-0.0886583\pi$
$563$	25687.6	1.92292	0.961461	+	0.274941i	$0.0886583\pi$
$564$	934.349	0.0697574
$565$	8201.67	0.610702
$566$	−6266.44	−0.465368
$567$	−356.270	−0.0263879
$568$	−6520.72	−0.481696
$569$	−14544.0	−1.07156	−0.535778	−	0.844359i	$-0.679981\pi$
$569$	−14544.0	−1.07156	−0.535778	+	0.844359i	$0.679981\pi$
$570$	9562.76	0.702702
$571$	−4063.66	−0.297826	−0.148913	−	0.988850i	$-0.547577\pi$
$571$	−4063.66	−0.297826	−0.148913	+	0.988850i	$0.547577\pi$
$572$	6027.88	0.440626
$573$	1501.33	0.109457
$574$	−2225.35	−0.161820
$575$	8427.53	0.611222
$576$	5125.49	0.370767
$577$	5830.29	0.420656	0.210328	−	0.977631i	$-0.432547\pi$
$577$	5830.29	0.420656	0.210328	+	0.977631i	$0.432547\pi$
$578$	0	0
$579$	10385.8	0.745456
$580$	91.8125	0.00657295
$581$	−1040.85	−0.0743229
$582$	1183.23	0.0842725
$583$	874.974	0.0621573
$584$	−28486.5	−2.01846
$585$	9223.81	0.651893
$586$	−3584.52	−0.252688
$587$	15551.3	1.09347	0.546737	−	0.837304i	$-0.315870\pi$
$587$	15551.3	1.09347	0.546737	+	0.837304i	$0.315870\pi$
$588$	−1755.90	−0.123149
$589$	−18806.2	−1.31561
$590$	2221.96	0.155045
$591$	3123.92	0.217430
$592$	−1093.00	−0.0758819
$593$	−19860.1	−1.37530	−0.687652	−	0.726040i	$-0.741359\pi$
$593$	−19860.1	−1.37530	−0.687652	+	0.726040i	$0.741359\pi$
$594$	2858.21	0.197431
$595$	0	0
$596$	−2000.18	−0.137468
$597$	5003.42	0.343009
$598$	35637.6	2.43700
$599$	57.5323	0.00392439	0.00196219	−	0.999998i	$-0.499375\pi$
$599$	57.5323	0.00392439	0.00196219	+	0.999998i	$0.499375\pi$
$600$	3375.62	0.229682
$601$	2049.33	0.139091	0.0695456	−	0.997579i	$-0.477845\pi$
$601$	2049.33	0.139091	0.0695456	+	0.997579i	$0.477845\pi$
$602$	713.535	0.0483082
$603$	−110.895	−0.00748924
$604$	5128.95	0.345520
$605$	6264.59	0.420978
$606$	−13647.0	−0.914806
$607$	13173.6	0.880888	0.440444	−	0.897780i	$-0.354821\pi$
$607$	13173.6	0.880888	0.440444	+	0.897780i	$0.354821\pi$
$608$	−7847.91	−0.523478
$609$	−51.2144	−0.00340773
$610$	17264.1	1.14590
$611$	13493.7	0.893446
$612$	0	0
$613$	10162.6	0.669597	0.334798	−	0.942290i	$-0.391332\pi$
$613$	10162.6	0.669597	0.334798	+	0.942290i	$0.391332\pi$
$614$	−19195.1	−1.26165
$615$	−7979.11	−0.523168
$616$	−4566.92	−0.298712
$617$	13324.3	0.869395	0.434697	−	0.900577i	$-0.356855\pi$
$617$	13324.3	0.869395	0.434697	+	0.900577i	$0.356855\pi$
$618$	−10016.0	−0.651947
$619$	−3263.56	−0.211912	−0.105956	−	0.994371i	$-0.533790\pi$
$619$	−3263.56	−0.211912	−0.105956	+	0.994371i	$0.533790\pi$
$620$	−4542.50	−0.294244
$621$	−4935.51	−0.318930
$622$	−2309.11	−0.148854
$623$	5348.77	0.343971
$624$	10874.0	0.697607
$625$	−19262.4	−1.23279
$626$	11102.4	0.708849
$627$	−12499.2	−0.796125
$628$	−5771.97	−0.366762
$629$	0	0
$630$	−1288.45	−0.0814809
$631$	−19762.6	−1.24681	−0.623406	−	0.781898i	$-0.714252\pi$
$631$	−19762.6	−1.24681	−0.623406	+	0.781898i	$0.714252\pi$
$632$	−22661.4	−1.42630
$633$	1600.56	0.100500
$634$	−12100.0	−0.757972
$635$	−10087.3	−0.630395
$636$	−111.579	−0.00695660
$637$	−25358.3	−1.57729
$638$	410.873	0.0254963
$639$	2404.58	0.148863
$640$	10150.6	0.626931
$641$	−14441.2	−0.889851	−0.444926	−	0.895567i	$-0.646770\pi$
$641$	−14441.2	−0.889851	−0.444926	+	0.895567i	$0.646770\pi$
$642$	8906.87	0.547548
$643$	5391.15	0.330647	0.165324	−	0.986239i	$-0.447133\pi$
$643$	5391.15	0.330647	0.165324	+	0.986239i	$0.447133\pi$
$644$	1453.98	0.0889671
$645$	2558.41	0.156182
$646$	0	0
$647$	7534.23	0.457807	0.228904	−	0.973449i	$-0.426486\pi$
$647$	7534.23	0.457807	0.228904	+	0.973449i	$0.426486\pi$
$648$	−1976.90	−0.119846
$649$	−2904.26	−0.175658
$650$	8988.18	0.542378
$651$	2533.87	0.152550
$652$	6134.08	0.368449
$653$	24410.8	1.46289	0.731447	−	0.681898i	$-0.238845\pi$
$653$	24410.8	1.46289	0.731447	+	0.681898i	$0.238845\pi$
$654$	−13294.5	−0.794888
$655$	30129.6	1.79735
$656$	−9406.59	−0.559856
$657$	10504.7	0.623784
$658$	−1884.89	−0.111673
$659$	26163.0	1.54654	0.773268	−	0.634079i	$-0.218621\pi$
$659$	26163.0	1.54654	0.773268	+	0.634079i	$0.218621\pi$
$660$	−3019.09	−0.178058
$661$	−22478.6	−1.32272	−0.661360	−	0.750069i	$-0.730020\pi$
$661$	−22478.6	−1.32272	−0.661360	+	0.750069i	$0.730020\pi$
$662$	11877.6	0.697333
$663$	0	0
$664$	−5775.54	−0.337552
$665$	5634.49	0.328566
$666$	529.098	0.0307840
$667$	−709.488	−0.0411866
$668$	921.354	0.0533657
$669$	11600.2	0.670391
$670$	−401.053	−0.0231254
$671$	−22565.4	−1.29825
$672$	1057.40	0.0606993
$673$	−3131.01	−0.179334	−0.0896668	−	0.995972i	$-0.528580\pi$
$673$	−3131.01	−0.179334	−0.0896668	+	0.995972i	$0.528580\pi$
$674$	−9403.07	−0.537378
$675$	−1244.79	−0.0709807
$676$	−7128.21	−0.405565
$677$	−5849.49	−0.332074	−0.166037	−	0.986120i	$-0.553097\pi$
$677$	−5849.49	−0.332074	−0.166037	+	0.986120i	$0.553097\pi$
$678$	−4680.53	−0.265125
$679$	697.175	0.0394037
$680$	0	0
$681$	11642.1	0.655107
$682$	−20328.3	−1.14136
$683$	9950.48	0.557459	0.278729	−	0.960370i	$-0.410087\pi$
$683$	9950.48	0.557459	0.278729	+	0.960370i	$0.410087\pi$
$684$	1593.93	0.0891018
$685$	−8193.32	−0.457008
$686$	7296.18	0.406078
$687$	−17321.6	−0.961951
$688$	3016.12	0.167134
$689$	−1611.40	−0.0890994
$690$	−17849.3	−0.984796
$691$	2308.23	0.127075	0.0635376	−	0.997979i	$-0.479762\pi$
$691$	2308.23	0.127075	0.0635376	+	0.997979i	$0.479762\pi$
$692$	−5213.36	−0.286391
$693$	1684.09	0.0923137
$694$	−17022.2	−0.931059
$695$	6427.18	0.350787
$696$	−284.183	−0.0154769
$697$	0	0
$698$	−4620.72	−0.250569
$699$	5792.69	0.313447
$700$	366.709	0.0198005
$701$	16620.6	0.895511	0.447755	−	0.894156i	$-0.352224\pi$
$701$	16620.6	0.895511	0.447755	+	0.894156i	$0.352224\pi$
$702$	−5263.85	−0.283007
$703$	−2313.79	−0.124134
$704$	−24228.3	−1.29707
$705$	−6758.37	−0.361043
$706$	24633.9	1.31319
$707$	−8040.99	−0.427740
$708$	370.359	0.0196595
$709$	−6641.97	−0.351825	−0.175913	−	0.984406i	$-0.556288\pi$
$709$	−6641.97	−0.351825	−0.175913	+	0.984406i	$0.556288\pi$
$710$	8696.14	0.459662
$711$	8356.59	0.440783
$712$	29679.7	1.56221
$713$	35102.5	1.84376
$714$	0	0
$715$	−43601.1	−2.28054
$716$	−3924.69	−0.204850
$717$	−2501.56	−0.130296
$718$	−5645.29	−0.293427
$719$	−2327.63	−0.120731	−0.0603657	−	0.998176i	$-0.519227\pi$
$719$	−2327.63	−0.120731	−0.0603657	+	0.998176i	$0.519227\pi$
$720$	−5446.28	−0.281904
$721$	−5901.55	−0.304834
$722$	6797.99	0.350408
$723$	−5208.93	−0.267942
$724$	−588.618	−0.0302152
$725$	−178.941	−0.00916646
$726$	−3575.08	−0.182760
$727$	−19275.9	−0.983362	−0.491681	−	0.870775i	$-0.663617\pi$
$727$	−19275.9	−0.983362	−0.491681	+	0.870775i	$0.663617\pi$
$728$	8410.70	0.428188
$729$	729.000	0.0370370
$730$	37990.1	1.92613
$731$	0	0
$732$	2877.60	0.145299
$733$	−1125.62	−0.0567202	−0.0283601	−	0.999598i	$-0.509029\pi$
$733$	−1125.62	−0.0567202	−0.0283601	+	0.999598i	$0.509029\pi$
$734$	28084.8	1.41230
$735$	12700.8	0.637383
$736$	14648.4	0.733625
$737$	524.204	0.0261999
$738$	4553.52	0.227124
$739$	−24622.2	−1.22563	−0.612817	−	0.790225i	$-0.709964\pi$
$739$	−24622.2	−1.22563	−0.612817	+	0.790225i	$0.709964\pi$
$740$	−558.879	−0.0277633
$741$	23019.3	1.14121
$742$	225.092	0.0111366
$743$	18473.4	0.912144	0.456072	−	0.889943i	$-0.349256\pi$
$743$	18473.4	0.912144	0.456072	+	0.889943i	$0.349256\pi$
$744$	14060.2	0.692836
$745$	14467.8	0.711489
$746$	−34949.1	−1.71525
$747$	2129.78	0.104317
$748$	0	0
$749$	5248.03	0.256020
$750$	7703.90	0.375075
$751$	755.202	0.0366947	0.0183473	−	0.999832i	$-0.494160\pi$
$751$	755.202	0.0366947	0.0183473	+	0.999832i	$0.494160\pi$
$752$	−7967.46	−0.386361
$753$	14021.5	0.678581
$754$	−756.687	−0.0365476
$755$	−37098.9	−1.78830
$756$	−214.760	−0.0103317
$757$	8082.28	0.388052	0.194026	−	0.980996i	$-0.437845\pi$
$757$	8082.28	0.388052	0.194026	+	0.980996i	$0.437845\pi$
$758$	−17631.2	−0.844848
$759$	23330.3	1.11572
$760$	31265.2	1.49225
$761$	34666.5	1.65133	0.825664	−	0.564162i	$-0.190801\pi$
$761$	34666.5	1.65133	0.825664	+	0.564162i	$0.190801\pi$
$762$	5756.61	0.273675
$763$	−7833.29	−0.371670
$764$	905.006	0.0428560
$765$	0	0
$766$	32910.2	1.55234
$767$	5348.65	0.251798
$768$	7875.24	0.370017
$769$	−37237.0	−1.74617	−0.873083	−	0.487572i	$-0.837883\pi$
$769$	−37237.0	−1.74617	−0.873083	+	0.487572i	$0.837883\pi$
$770$	6090.52	0.285048
$771$	−5939.93	−0.277460
$772$	6260.58	0.291870
$773$	27700.3	1.28889	0.644444	−	0.764652i	$-0.277089\pi$
$773$	27700.3	1.28889	0.644444	+	0.764652i	$0.277089\pi$
$774$	−1460.04	−0.0678036
$775$	8853.22	0.410345
$776$	3868.55	0.178960
$777$	311.751	0.0143938
$778$	−15624.5	−0.720007
$779$	−19912.9	−0.915860
$780$	5560.13	0.255237
$781$	−11366.5	−0.520774
$782$	0	0
$783$	104.795	0.00478297
$784$	14973.0	0.682080
$785$	41750.1	1.89825
$786$	−17194.4	−0.780285
$787$	15137.2	0.685621	0.342811	−	0.939405i	$-0.388621\pi$
$787$	15137.2	0.685621	0.342811	+	0.939405i	$0.388621\pi$
$788$	1883.11	0.0851306
$789$	13790.4	0.622244
$790$	30221.6	1.36106
$791$	−2757.83	−0.123966
$792$	9344.84	0.419261
$793$	41557.7	1.86098
$794$	8455.56	0.377930
$795$	807.078	0.0360052
$796$	3016.08	0.134299
$797$	30546.6	1.35761	0.678805	−	0.734318i	$-0.262498\pi$
$797$	30546.6	1.35761	0.678805	+	0.734318i	$0.262498\pi$
$798$	−3215.50	−0.142641
$799$	0	0
$800$	3694.49	0.163275
$801$	−10944.7	−0.482785
$802$	24371.8	1.07306
$803$	−49655.8	−2.18221
$804$	−66.8480	−0.00293227
$805$	−10517.0	−0.460466
$806$	37437.7	1.63609
$807$	−11544.9	−0.503591
$808$	−44618.5	−1.94267
$809$	7978.71	0.346745	0.173372	−	0.984856i	$-0.444534\pi$
$809$	7978.71	0.346745	0.173372	+	0.984856i	$0.444534\pi$
$810$	2636.43	0.114364
$811$	−28418.2	−1.23046	−0.615228	−	0.788349i	$-0.710936\pi$
$811$	−28418.2	−1.23046	−0.615228	+	0.788349i	$0.710936\pi$
$812$	−30.8721	−0.00133424
$813$	22531.1	0.971958
$814$	−2501.06	−0.107693
$815$	−44369.3	−1.90698
$816$	0	0
$817$	6384.87	0.273413
$818$	35364.9	1.51162
$819$	−3101.52	−0.132327
$820$	−4809.82	−0.204837
$821$	11905.8	0.506111	0.253055	−	0.967452i	$-0.418565\pi$
$821$	11905.8	0.506111	0.253055	+	0.967452i	$0.418565\pi$
$822$	4675.77	0.198402
$823$	26123.4	1.10644	0.553222	−	0.833034i	$-0.313398\pi$
$823$	26123.4	1.10644	0.553222	+	0.833034i	$0.313398\pi$
$824$	−32747.1	−1.38446
$825$	5884.14	0.248315
$826$	−747.138	−0.0314725
$827$	32453.5	1.36459	0.682296	−	0.731076i	$-0.260982\pi$
$827$	32453.5	1.36459	0.682296	+	0.731076i	$0.260982\pi$
$828$	−2975.14	−0.124871
$829$	15493.0	0.649090	0.324545	−	0.945870i	$-0.394789\pi$
$829$	15493.0	0.649090	0.324545	+	0.945870i	$0.394789\pi$
$830$	7702.35	0.322112
$831$	−3439.30	−0.143572
$832$	44620.2	1.85929
$833$	0	0
$834$	−3667.87	−0.152288
$835$	−6664.38	−0.276204
$836$	−7534.56	−0.311708
$837$	−5184.81	−0.214114
$838$	−16731.0	−0.689692
$839$	6756.78	0.278033	0.139017	−	0.990290i	$-0.455606\pi$
$839$	6756.78	0.278033	0.139017	+	0.990290i	$0.455606\pi$
$840$	−4212.54	−0.173031
$841$	−24373.9	−0.999382
$842$	6710.01	0.274634
$843$	10978.5	0.448539
$844$	964.821	0.0393490
$845$	51560.1	2.09908
$846$	3856.87	0.156740
$847$	−2106.48	−0.0854540
$848$	951.467	0.0385301
$849$	7555.12	0.305407
$850$	0	0
$851$	4318.78	0.173967
$852$	1449.48	0.0582847
$853$	−13547.3	−0.543787	−0.271893	−	0.962327i	$-0.587650\pi$
$853$	−13547.3	−0.543787	−0.271893	+	0.962327i	$0.587650\pi$
$854$	−5805.07	−0.232606
$855$	−11529.3	−0.461163
$856$	29120.7	1.16276
$857$	18125.0	0.722446	0.361223	−	0.932479i	$-0.382359\pi$
$857$	18125.0	0.722446	0.361223	+	0.932479i	$0.382359\pi$
$858$	24882.3	0.990056
$859$	12781.7	0.507689	0.253844	−	0.967245i	$-0.418305\pi$
$859$	12781.7	0.507689	0.253844	+	0.967245i	$0.418305\pi$
$860$	1542.22	0.0611502
$861$	2682.99	0.106197
$862$	−23083.3	−0.912088
$863$	16669.2	0.657504	0.328752	−	0.944416i	$-0.393372\pi$
$863$	16669.2	0.657504	0.328752	+	0.944416i	$0.393372\pi$
$864$	−2163.65	−0.0851953
$865$	37709.5	1.48227
$866$	−5984.27	−0.234820
$867$	0	0
$868$	1527.42	0.0597282
$869$	−39501.7	−1.54201
$870$	378.990	0.0147689
$871$	−965.405	−0.0375562
$872$	−43466.0	−1.68801
$873$	−1426.56	−0.0553056
$874$	−44545.2	−1.72399
$875$	4539.23	0.175376
$876$	6332.24	0.244231
$877$	−49509.1	−1.90628	−0.953139	−	0.302534i	$-0.902167\pi$
$877$	−49509.1	−1.90628	−0.953139	+	0.302534i	$0.902167\pi$
$878$	31822.2	1.22318
$879$	4321.67	0.165832
$880$	25744.7	0.986196
$881$	21527.6	0.823248	0.411624	−	0.911354i	$-0.364962\pi$
$881$	21527.6	0.823248	0.411624	+	0.911354i	$0.364962\pi$
$882$	−7248.11	−0.276708
$883$	−21464.3	−0.818043	−0.409022	−	0.912525i	$-0.634130\pi$
$883$	−21464.3	−0.818043	−0.409022	+	0.912525i	$0.634130\pi$
$884$	0	0
$885$	−2678.90	−0.101752
$886$	−12836.9	−0.486753
$887$	−20864.2	−0.789799	−0.394899	−	0.918724i	$-0.629221\pi$
$887$	−20864.2	−0.789799	−0.394899	+	0.918724i	$0.629221\pi$
$888$	1729.87	0.0653723
$889$	3391.86	0.127963
$890$	−39581.3	−1.49075
$891$	−3446.00	−0.129568
$892$	6992.66	0.262479
$893$	−16866.4	−0.632042
$894$	−8256.49	−0.308880
$895$	28388.2	1.06024
$896$	−3413.14	−0.127260
$897$	−42966.3	−1.59934
$898$	−13881.1	−0.515832
$899$	−745.325	−0.0276507
$900$	−750.362	−0.0277912
$901$	0	0
$902$	−21524.6	−0.794557
$903$	−860.271	−0.0317032
$904$	−15302.9	−0.563015
$905$	4257.61	0.156384
$906$	21171.6	0.776358
$907$	−23379.4	−0.855899	−0.427950	−	0.903803i	$-0.640764\pi$
$907$	−23379.4	−0.855899	−0.427950	+	0.903803i	$0.640764\pi$
$908$	7017.91	0.256495
$909$	16453.5	0.600361
$910$	−11216.6	−0.408602
$911$	6297.94	0.229045	0.114523	−	0.993421i	$-0.463466\pi$
$911$	6297.94	0.229045	0.114523	+	0.993421i	$0.463466\pi$
$912$	−13591.9	−0.493502
$913$	−10067.5	−0.364936
$914$	−22242.2	−0.804929
$915$	−20814.4	−0.752023
$916$	−10441.5	−0.376634
$917$	−10131.1	−0.364842
$918$	0	0
$919$	5078.99	0.182307	0.0911537	−	0.995837i	$-0.470945\pi$
$919$	5078.99	0.182307	0.0911537	+	0.995837i	$0.470945\pi$
$920$	−58357.6	−2.09130
$921$	23142.6	0.827984
$922$	18079.1	0.645773
$923$	20933.2	0.746504
$924$	1015.18	0.0361437
$925$	1089.24	0.0387179
$926$	−23379.1	−0.829683
$927$	12075.8	0.427854
$928$	−311.028	−0.0110021
$929$	−11589.3	−0.409292	−0.204646	−	0.978836i	$-0.565604\pi$
$929$	−11589.3	−0.409292	−0.204646	+	0.978836i	$0.565604\pi$
$930$	−18750.8	−0.661145
$931$	31696.6	1.11581
$932$	3491.85	0.122725
$933$	2783.97	0.0976883
$934$	36621.1	1.28295
$935$	0	0
$936$	−17210.0	−0.600990
$937$	−3371.38	−0.117543	−0.0587717	−	0.998271i	$-0.518718\pi$
$937$	−3371.38	−0.117543	−0.0587717	+	0.998271i	$0.518718\pi$
$938$	134.855	0.00469420
$939$	−13385.5	−0.465197
$940$	−4073.96	−0.141360
$941$	38717.9	1.34130	0.670652	−	0.741772i	$-0.266014\pi$
$941$	38717.9	1.34130	0.670652	+	0.741772i	$0.266014\pi$
$942$	−23826.0	−0.824089
$943$	37168.3	1.28353
$944$	−3158.16	−0.108887
$945$	1553.41	0.0534736
$946$	6901.62	0.237200
$947$	−48266.7	−1.65624	−0.828119	−	0.560553i	$-0.810589\pi$
$947$	−48266.7	−1.65624	−0.828119	+	0.560553i	$0.810589\pi$
$948$	5037.37	0.172580
$949$	91448.8	3.12809
$950$	−11234.8	−0.383689
$951$	14588.4	0.497435
$952$	0	0
$953$	26497.9	0.900682	0.450341	−	0.892857i	$-0.351302\pi$
$953$	26497.9	0.900682	0.450341	+	0.892857i	$0.351302\pi$
$954$	−460.584	−0.0156310
$955$	−6546.12	−0.221809
$956$	−1507.95	−0.0510151
$957$	−495.368	−0.0167325
$958$	11358.2	0.383053
$959$	2755.02	0.0927677
$960$	−22348.2	−0.751340
$961$	7084.54	0.237808
$962$	4606.09	0.154372
$963$	−10738.5	−0.359340
$964$	−3139.95	−0.104908
$965$	−45284.3	−1.51063
$966$	6001.84	0.199903
$967$	13003.7	0.432442	0.216221	−	0.976344i	$-0.430627\pi$
$967$	13003.7	0.432442	0.216221	+	0.976344i	$0.430627\pi$
$968$	−11688.6	−0.388106
$969$	0	0
$970$	−5159.15	−0.170774
$971$	1427.34	0.0471734	0.0235867	−	0.999722i	$-0.492491\pi$
$971$	1427.34	0.0471734	0.0235867	+	0.999722i	$0.492491\pi$
$972$	439.443	0.0145012
$973$	−2161.15	−0.0712058
$974$	−33947.1	−1.11677
$975$	−10836.6	−0.355947
$976$	−24538.1	−0.804759
$977$	−14938.8	−0.489187	−0.244593	−	0.969626i	$-0.578654\pi$
$977$	−14938.8	−0.489187	−0.244593	+	0.969626i	$0.578654\pi$
$978$	25320.7	0.827880
$979$	51735.6	1.68895
$980$	7656.08	0.249556
$981$	16028.5	0.521662
$982$	39135.8	1.27177
$983$	−45925.7	−1.49014	−0.745068	−	0.666989i	$-0.767583\pi$
$983$	−45925.7	−1.49014	−0.745068	+	0.666989i	$0.767583\pi$
$984$	14887.6	0.482316
$985$	−13621.0	−0.440609
$986$	0	0
$987$	2272.52	0.0732877
$988$	13876.1	0.446818
$989$	−11917.6	−0.383172
$990$	−12462.4	−0.400083
$991$	13418.5	0.430122	0.215061	−	0.976601i	$-0.431005\pi$
$991$	13418.5	0.430122	0.215061	+	0.976601i	$0.431005\pi$
$992$	15388.3	0.492520
$993$	−14320.1	−0.457639
$994$	−2924.09	−0.0933064
$995$	−21816.0	−0.695089
$996$	1283.84	0.0408434
$997$	−45532.5	−1.44637	−0.723185	−	0.690655i	$-0.757322\pi$
$997$	−45532.5	−1.44637	−0.723185	+	0.690655i	$0.757322\pi$
$998$	19899.7	0.631177
$999$	−637.905	−0.0202026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.4.a.p.1.6		8
17.8	even	8		51.4.e.a.13.3	yes	16
17.15	even	8		51.4.e.a.4.6	✓	16
17.16	even	2		867.4.a.q.1.6		8
51.8	odd	8		153.4.f.b.64.6		16
51.32	odd	8		153.4.f.b.55.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.e.a.4.6	✓	16	17.15	even	8
51.4.e.a.13.3	yes	16	17.8	even	8
153.4.f.b.55.3		16	51.32	odd	8
153.4.f.b.64.6		16	51.8	odd	8
867.4.a.p.1.6		8	1.1	even	1	trivial
867.4.a.q.1.6		8	17.16	even	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:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

\(f(q)\)	\(=\)	\(q+2.48829 q^{2} -3.00000 q^{3} -1.80841 q^{4} +13.0806 q^{5} -7.46487 q^{6} -4.39839 q^{7} -24.4062 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.48829 q^{2} -3.00000 q^{3} -1.80841 q^{4} +13.0806 q^{5} -7.46487 q^{6} -4.39839 q^{7} -24.4062 q^{8} +9.00000 q^{9} +32.5485 q^{10} -42.5432 q^{11} +5.42522 q^{12} +78.3499 q^{13} -10.9445 q^{14} -39.2419 q^{15} -46.2624 q^{16} +22.3946 q^{18} -97.9336 q^{19} -23.6551 q^{20} +13.1952 q^{21} -105.860 q^{22} +182.797 q^{23} +73.2185 q^{24} +46.1033 q^{25} +194.957 q^{26} -27.0000 q^{27} +7.95408 q^{28} -3.88129 q^{29} -97.6454 q^{30} +192.030 q^{31} +80.1350 q^{32} +127.629 q^{33} -57.5338 q^{35} -16.2757 q^{36} +23.6261 q^{37} -243.687 q^{38} -235.050 q^{39} -319.249 q^{40} +203.331 q^{41} +32.8334 q^{42} -65.1959 q^{43} +76.9354 q^{44} +117.726 q^{45} +454.851 q^{46} +172.223 q^{47} +138.787 q^{48} -323.654 q^{49} +114.718 q^{50} -141.689 q^{52} -20.5667 q^{53} -67.1839 q^{54} -556.492 q^{55} +107.348 q^{56} +293.801 q^{57} -9.65779 q^{58} +68.2662 q^{59} +70.9654 q^{60} +530.411 q^{61} +477.827 q^{62} -39.5855 q^{63} +569.499 q^{64} +1024.87 q^{65} +317.579 q^{66} -12.3217 q^{67} -548.390 q^{69} -143.161 q^{70} +267.175 q^{71} -219.656 q^{72} +1167.19 q^{73} +58.7887 q^{74} -138.310 q^{75} +177.104 q^{76} +187.121 q^{77} -584.872 q^{78} +928.509 q^{79} -605.142 q^{80} +81.0000 q^{81} +505.947 q^{82} +236.643 q^{83} -23.8622 q^{84} -162.226 q^{86} +11.6439 q^{87} +1038.32 q^{88} -1216.07 q^{89} +292.936 q^{90} -344.614 q^{91} -330.571 q^{92} -576.090 q^{93} +428.542 q^{94} -1281.03 q^{95} -240.405 q^{96} -158.507 q^{97} -805.346 q^{98} -382.888 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9	\( 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9} - 72 q^{10} + 12 q^{11} - 72 q^{12} - 60 q^{13} + 56 q^{14} - 72 q^{15} - 100 q^{16} - 36 q^{18} - 140 q^{19} + 224 q^{20} - 84 q^{21} + 484 q^{22} - 96 q^{23} + 144 q^{24} + 220 q^{25} + 500 q^{26} - 216 q^{27} + 348 q^{28} + 380 q^{29} + 216 q^{30} + 124 q^{31} - 448 q^{32} - 36 q^{33} - 592 q^{35} + 216 q^{36} + 1260 q^{37} + 84 q^{38} + 180 q^{39} - 836 q^{40} + 284 q^{41} - 168 q^{42} - 76 q^{43} - 1460 q^{44} + 216 q^{45} + 1056 q^{46} - 184 q^{47} + 300 q^{48} - 536 q^{49} - 360 q^{50} - 1580 q^{52} + 1064 q^{53} + 108 q^{54} - 2148 q^{55} + 244 q^{56} + 420 q^{57} - 532 q^{58} + 1224 q^{59} - 672 q^{60} + 324 q^{61} + 3068 q^{62} + 252 q^{63} - 876 q^{64} + 652 q^{65} - 1452 q^{66} - 624 q^{67} + 288 q^{69} - 1932 q^{70} + 2636 q^{71} - 432 q^{72} + 1640 q^{73} - 120 q^{74} - 660 q^{75} - 1172 q^{76} + 504 q^{77} - 1500 q^{78} + 3788 q^{79} + 1740 q^{80} + 648 q^{81} + 1916 q^{82} + 2032 q^{83} - 1044 q^{84} + 1612 q^{86} - 1140 q^{87} + 4576 q^{88} + 1304 q^{89} - 648 q^{90} - 2872 q^{91} - 1744 q^{92} - 372 q^{93} - 208 q^{94} - 748 q^{95} + 1344 q^{96} + 2376 q^{97} + 604 q^{98} + 108 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9 - 72 * q^10 + 12 * q^11 - 72 * q^12 - 60 * q^13 + 56 * q^14 - 72 * q^15 - 100 * q^16 - 36 * q^18 - 140 * q^19 + 224 * q^20 - 84 * q^21 + 484 * q^22 - 96 * q^23 + 144 * q^24 + 220 * q^25 + 500 * q^26 - 216 * q^27 + 348 * q^28 + 380 * q^29 + 216 * q^30 + 124 * q^31 - 448 * q^32 - 36 * q^33 - 592 * q^35 + 216 * q^36 + 1260 * q^37 + 84 * q^38 + 180 * q^39 - 836 * q^40 + 284 * q^41 - 168 * q^42 - 76 * q^43 - 1460 * q^44 + 216 * q^45 + 1056 * q^46 - 184 * q^47 + 300 * q^48 - 536 * q^49 - 360 * q^50 - 1580 * q^52 + 1064 * q^53 + 108 * q^54 - 2148 * q^55 + 244 * q^56 + 420 * q^57 - 532 * q^58 + 1224 * q^59 - 672 * q^60 + 324 * q^61 + 3068 * q^62 + 252 * q^63 - 876 * q^64 + 652 * q^65 - 1452 * q^66 - 624 * q^67 + 288 * q^69 - 1932 * q^70 + 2636 * q^71 - 432 * q^72 + 1640 * q^73 - 120 * q^74 - 660 * q^75 - 1172 * q^76 + 504 * q^77 - 1500 * q^78 + 3788 * q^79 + 1740 * q^80 + 648 * q^81 + 1916 * q^82 + 2032 * q^83 - 1044 * q^84 + 1612 * q^86 - 1140 * q^87 + 4576 * q^88 + 1304 * q^89 - 648 * q^90 - 2872 * q^91 - 1744 * q^92 - 372 * q^93 - 208 * q^94 - 748 * q^95 + 1344 * q^96 + 2376 * q^97 + 604 * q^98 + 108 * q^99

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