LMFDB - Embedded newform 867.4.a.p.1.4

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			1.4
Root			$-0.496675$ 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)

$f(q)$	$=$	$q-1.49668 q^{2} -3.00000 q^{3} -5.75996 q^{4} +0.812858 q^{5} +4.49003 q^{6} +3.87059 q^{7} +20.5942 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.49668 q^{2} -3.00000 q^{3} -5.75996 q^{4} +0.812858 q^{5} +4.49003 q^{6} +3.87059 q^{7} +20.5942 q^{8} +9.00000 q^{9} -1.21658 q^{10} +20.1332 q^{11} +17.2799 q^{12} -20.0363 q^{13} -5.79302 q^{14} -2.43857 q^{15} +15.2569 q^{16} -13.4701 q^{18} +123.098 q^{19} -4.68203 q^{20} -11.6118 q^{21} -30.1329 q^{22} -41.9418 q^{23} -61.7826 q^{24} -124.339 q^{25} +29.9879 q^{26} -27.0000 q^{27} -22.2945 q^{28} -24.9417 q^{29} +3.64975 q^{30} +76.4806 q^{31} -187.588 q^{32} -60.3996 q^{33} +3.14624 q^{35} -51.8397 q^{36} -31.5913 q^{37} -184.238 q^{38} +60.1090 q^{39} +16.7401 q^{40} -108.393 q^{41} +17.3791 q^{42} +233.202 q^{43} -115.967 q^{44} +7.31572 q^{45} +62.7733 q^{46} -51.1712 q^{47} -45.7707 q^{48} -328.019 q^{49} +186.095 q^{50} +115.409 q^{52} +1.55884 q^{53} +40.4102 q^{54} +16.3654 q^{55} +79.7117 q^{56} -369.295 q^{57} +37.3295 q^{58} -588.603 q^{59} +14.0461 q^{60} -726.454 q^{61} -114.467 q^{62} +34.8353 q^{63} +158.703 q^{64} -16.2867 q^{65} +90.3986 q^{66} +266.752 q^{67} +125.826 q^{69} -4.70890 q^{70} +879.263 q^{71} +185.348 q^{72} +334.474 q^{73} +47.2819 q^{74} +373.018 q^{75} -709.043 q^{76} +77.9274 q^{77} -89.9636 q^{78} +824.354 q^{79} +12.4017 q^{80} +81.0000 q^{81} +162.229 q^{82} -224.694 q^{83} +66.8834 q^{84} -349.028 q^{86} +74.8250 q^{87} +414.627 q^{88} -561.132 q^{89} -10.9493 q^{90} -77.5525 q^{91} +241.583 q^{92} -229.442 q^{93} +76.5866 q^{94} +100.062 q^{95} +562.765 q^{96} +1777.87 q^{97} +490.937 q^{98} +181.199 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.49668	−0.529155	−0.264577	−	0.964364i	$-0.585232\pi$
$2$	−1.49668	−0.529155	−0.264577	+	0.964364i	$0.585232\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−5.75996	−0.719995
$5$	0.812858	0.0727042	0.0363521	−	0.999339i	$-0.488426\pi$
$5$	0.812858	0.0727042	0.0363521	+	0.999339i	$0.488426\pi$
$6$	4.49003	0.305508
$7$	3.87059	0.208992	0.104496	−	0.994525i	$-0.466677\pi$
$7$	3.87059	0.208992	0.104496	+	0.994525i	$0.466677\pi$
$8$	20.5942	0.910143
$9$	9.00000	0.333333
$10$	−1.21658	−0.0384718
$11$	20.1332	0.551854	0.275927	−	0.961179i	$-0.411015\pi$
$11$	20.1332	0.551854	0.275927	+	0.961179i	$0.411015\pi$
$12$	17.2799	0.415690
$13$	−20.0363	−0.427468	−0.213734	−	0.976892i	$-0.568563\pi$
$13$	−20.0363	−0.427468	−0.213734	+	0.976892i	$0.568563\pi$
$14$	−5.79302	−0.110589
$15$	−2.43857	−0.0419758
$16$	15.2569	0.238389
$17$	0	0
$17$	0	0
$18$	−13.4701	−0.176385
$19$	123.098	1.48635	0.743177	−	0.669095i	$-0.233318\pi$
$19$	123.098	1.48635	0.743177	+	0.669095i	$0.233318\pi$
$20$	−4.68203	−0.0523467
$21$	−11.6118	−0.120662
$22$	−30.1329	−0.292016
$23$	−41.9418	−0.380238	−0.190119	−	0.981761i	$-0.560887\pi$
$23$	−41.9418	−0.380238	−0.190119	+	0.981761i	$0.560887\pi$
$24$	−61.7826	−0.525472
$25$	−124.339	−0.994714
$26$	29.9879	0.226196
$27$	−27.0000	−0.192450
$28$	−22.2945	−0.150474
$29$	−24.9417	−0.159709	−0.0798543	−	0.996807i	$-0.525445\pi$
$29$	−24.9417	−0.159709	−0.0798543	+	0.996807i	$0.525445\pi$
$30$	3.64975	0.0222117
$31$	76.4806	0.443107	0.221554	−	0.975148i	$-0.428887\pi$
$31$	76.4806	0.443107	0.221554	+	0.975148i	$0.428887\pi$
$32$	−187.588	−1.03629
$33$	−60.3996	−0.318613
$34$	0	0
$35$	3.14624	0.0151946
$36$	−51.8397	−0.239998
$37$	−31.5913	−0.140367	−0.0701834	−	0.997534i	$-0.522358\pi$
$37$	−31.5913	−0.140367	−0.0701834	+	0.997534i	$0.522358\pi$
$38$	−184.238	−0.786511
$39$	60.1090	0.246799
$40$	16.7401	0.0661712
$41$	−108.393	−0.412882	−0.206441	−	0.978459i	$-0.566188\pi$
$41$	−108.393	−0.412882	−0.206441	+	0.978459i	$0.566188\pi$
$42$	17.3791	0.0638487
$43$	233.202	0.827047	0.413524	−	0.910493i	$-0.364298\pi$
$43$	233.202	0.827047	0.413524	+	0.910493i	$0.364298\pi$
$44$	−115.967	−0.397332
$45$	7.31572	0.0242347
$46$	62.7733	0.201205
$47$	−51.1712	−0.158810	−0.0794052	−	0.996842i	$-0.525302\pi$
$47$	−51.1712	−0.158810	−0.0794052	+	0.996842i	$0.525302\pi$
$48$	−45.7707	−0.137634
$49$	−328.019	−0.956322
$50$	186.095	0.526357
$51$	0	0
$52$	115.409	0.307775
$53$	1.55884	0.00404007	0.00202003	−	0.999998i	$-0.499357\pi$
$53$	1.55884	0.00404007	0.00202003	+	0.999998i	$0.499357\pi$
$54$	40.4102	0.101836
$55$	16.3654	0.0401221
$56$	79.7117	0.190213
$57$	−369.295	−0.858147
$58$	37.3295	0.0845105
$59$	−588.603	−1.29881	−0.649403	−	0.760445i	$-0.724981\pi$
$59$	−588.603	−1.29881	−0.649403	+	0.760445i	$0.724981\pi$
$60$	14.0461	0.0302224
$61$	−726.454	−1.52480	−0.762401	−	0.647105i	$-0.775980\pi$
$61$	−726.454	−1.52480	−0.762401	+	0.647105i	$0.775980\pi$
$62$	−114.467	−0.234472
$63$	34.8353	0.0696641
$64$	158.703	0.309967
$65$	−16.2867	−0.0310787
$66$	90.3986	0.168595
$67$	266.752	0.486403	0.243201	−	0.969976i	$-0.421802\pi$
$67$	266.752	0.486403	0.243201	+	0.969976i	$0.421802\pi$
$68$	0	0
$69$	125.826	0.219531
$70$	−4.70890	−0.00804030
$71$	879.263	1.46971	0.734854	−	0.678225i	$-0.237251\pi$
$71$	879.263	1.46971	0.734854	+	0.678225i	$0.237251\pi$
$72$	185.348	0.303381
$73$	334.474	0.536263	0.268131	−	0.963382i	$-0.413594\pi$
$73$	334.474	0.536263	0.268131	+	0.963382i	$0.413594\pi$
$74$	47.2819	0.0742757
$75$	373.018	0.574298
$76$	−709.043	−1.07017
$77$	77.9274	0.115333
$78$	−89.9636	−0.130595
$79$	824.354	1.17401	0.587007	−	0.809582i	$-0.300306\pi$
$79$	824.354	1.17401	0.587007	+	0.809582i	$0.300306\pi$
$80$	12.4017	0.0173319
$81$	81.0000	0.111111
$82$	162.229	0.218478
$83$	−224.694	−0.297149	−0.148574	−	0.988901i	$-0.547468\pi$
$83$	−224.694	−0.297149	−0.148574	+	0.988901i	$0.547468\pi$
$84$	66.8834	0.0868760
$85$	0	0
$86$	−349.028	−0.437636
$87$	74.8250	0.0922078
$88$	414.627	0.502266
$89$	−561.132	−0.668313	−0.334157	−	0.942518i	$-0.608451\pi$
$89$	−561.132	−0.668313	−0.334157	+	0.942518i	$0.608451\pi$
$90$	−10.9493	−0.0128239
$91$	−77.5525	−0.0893375
$92$	241.583	0.273770
$93$	−229.442	−0.255828
$94$	76.5866	0.0840352
$95$	100.062	0.108064
$96$	562.765	0.598301
$97$	1777.87	1.86099	0.930493	−	0.366310i	$-0.119379\pi$
$97$	1777.87	1.86099	0.930493	+	0.366310i	$0.119379\pi$
$98$	490.937	0.506042
$99$	181.199	0.183951
$100$	716.190	0.716190
$101$	1943.41	1.91462	0.957310	−	0.289064i	$-0.0933441\pi$
$101$	1943.41	1.91462	0.957310	+	0.289064i	$0.0933441\pi$
$102$	0	0
$103$	−197.134	−0.188585	−0.0942923	−	0.995545i	$-0.530059\pi$
$103$	−197.134	−0.188585	−0.0942923	+	0.995545i	$0.530059\pi$
$104$	−412.632	−0.389057
$105$	−9.43872	−0.00877262
$106$	−2.33308	−0.00213782
$107$	1515.24	1.36900	0.684502	−	0.729011i	$-0.260019\pi$
$107$	1515.24	1.36900	0.684502	+	0.729011i	$0.260019\pi$
$108$	155.519	0.138563
$109$	−797.916	−0.701160	−0.350580	−	0.936533i	$-0.614016\pi$
$109$	−797.916	−0.701160	−0.350580	+	0.936533i	$0.614016\pi$
$110$	−24.4937	−0.0212308
$111$	94.7738	0.0810408
$112$	59.0532	0.0498215
$113$	1604.17	1.33547	0.667733	−	0.744401i	$-0.267265\pi$
$113$	1604.17	1.33547	0.667733	+	0.744401i	$0.267265\pi$
$114$	552.715	0.454092
$115$	−34.0927	−0.0276449
$116$	143.663	0.114989
$117$	−180.327	−0.142489
$118$	880.947	0.687269
$119$	0	0
$120$	−50.2204	−0.0382040
$121$	−925.654	−0.695458
$122$	1087.27	0.806856
$123$	325.179	0.238377
$124$	−440.525	−0.319035
$125$	−202.677	−0.145024
$126$	−52.1372	−0.0368631
$127$	−2315.35	−1.61775	−0.808876	−	0.587980i	$-0.799923\pi$
$127$	−2315.35	−1.61775	−0.808876	+	0.587980i	$0.799923\pi$
$128$	1263.18	0.872267
$129$	−699.607	−0.477496
$130$	24.3759	0.0164454
$131$	−1775.94	−1.18446	−0.592231	−	0.805768i	$-0.701753\pi$
$131$	−1775.94	−1.18446	−0.592231	+	0.805768i	$0.701753\pi$
$132$	347.900	0.229400
$133$	476.464	0.310637
$134$	−399.242	−0.257382
$135$	−21.9472	−0.0139919
$136$	0	0
$137$	1489.27	0.928736	0.464368	−	0.885642i	$-0.346281\pi$
$137$	1489.27	0.928736	0.464368	+	0.885642i	$0.346281\pi$
$138$	−188.320	−0.116166
$139$	2460.19	1.50123	0.750614	−	0.660741i	$-0.229758\pi$
$139$	2460.19	1.50123	0.750614	+	0.660741i	$0.229758\pi$
$140$	−18.1222	−0.0109401
$141$	153.514	0.0916892
$142$	−1315.97	−0.777703
$143$	−403.396	−0.235900
$144$	137.312	0.0794630
$145$	−20.2740	−0.0116115
$146$	−500.599	−0.283766
$147$	984.056	0.552133
$148$	181.965	0.101063
$149$	1981.35	1.08938	0.544692	−	0.838636i	$-0.316647\pi$
$149$	1981.35	1.08938	0.544692	+	0.838636i	$0.316647\pi$
$150$	−558.286	−0.303893
$151$	1293.87	0.697306	0.348653	−	0.937252i	$-0.386639\pi$
$151$	1293.87	0.697306	0.348653	+	0.937252i	$0.386639\pi$
$152$	2535.11	1.35280
$153$	0	0
$154$	−116.632	−0.0610291
$155$	62.1678	0.0322157
$156$	−346.226	−0.177694
$157$	−3212.58	−1.63307	−0.816535	−	0.577296i	$-0.804108\pi$
$157$	−3212.58	−1.63307	−0.816535	+	0.577296i	$0.804108\pi$
$158$	−1233.79	−0.621235
$159$	−4.67653	−0.00233253
$160$	−152.482	−0.0753425
$161$	−162.340	−0.0794669
$162$	−121.231	−0.0587949
$163$	626.362	0.300984	0.150492	−	0.988611i	$-0.451914\pi$
$163$	626.362	0.300984	0.150492	+	0.988611i	$0.451914\pi$
$164$	624.340	0.297273
$165$	−49.0963	−0.0231645
$166$	336.293	0.157238
$167$	3740.76	1.73335	0.866673	−	0.498876i	$-0.166254\pi$
$167$	3740.76	1.73335	0.866673	+	0.498876i	$0.166254\pi$
$168$	−239.135	−0.109820
$169$	−1795.55	−0.817271
$170$	0	0
$171$	1107.89	0.495451
$172$	−1343.24	−0.595470
$173$	99.7292	0.0438281	0.0219141	−	0.999760i	$-0.493024\pi$
$173$	99.7292	0.0438281	0.0219141	+	0.999760i	$0.493024\pi$
$174$	−111.989	−0.0487922
$175$	−481.267	−0.207888
$176$	307.170	0.131556
$177$	1765.81	0.749866
$178$	839.832	0.353641
$179$	2170.40	0.906275	0.453138	−	0.891441i	$-0.350305\pi$
$179$	2170.40	0.906275	0.453138	+	0.891441i	$0.350305\pi$
$180$	−42.1383	−0.0174489
$181$	2792.37	1.14671	0.573356	−	0.819306i	$-0.305641\pi$
$181$	2792.37	1.14671	0.573356	+	0.819306i	$0.305641\pi$
$182$	116.071	0.0472733
$183$	2179.36	0.880345
$184$	−863.758	−0.346071
$185$	−25.6792	−0.0102053
$186$	343.400	0.135373
$187$	0	0
$188$	294.744	0.114343
$189$	−104.506	−0.0402206
$190$	−149.760	−0.0571826
$191$	−3351.18	−1.26955	−0.634773	−	0.772699i	$-0.718906\pi$
$191$	−3351.18	−1.26955	−0.634773	+	0.772699i	$0.718906\pi$
$192$	−476.110	−0.178960
$193$	2799.99	1.04429	0.522144	−	0.852857i	$-0.325132\pi$
$193$	2799.99	1.04429	0.522144	+	0.852857i	$0.325132\pi$
$194$	−2660.90	−0.984749
$195$	48.8601	0.0179433
$196$	1889.37	0.688548
$197$	2538.15	0.917949	0.458974	−	0.888449i	$-0.348217\pi$
$197$	2538.15	0.917949	0.458974	+	0.888449i	$0.348217\pi$
$198$	−271.196	−0.0973386
$199$	2724.18	0.970413	0.485207	−	0.874400i	$-0.338744\pi$
$199$	2724.18	0.970413	0.485207	+	0.874400i	$0.338744\pi$
$200$	−2560.67	−0.905332
$201$	−800.257	−0.280825
$202$	−2908.65	−1.01313
$203$	−96.5390	−0.0333779
$204$	0	0
$205$	−88.1082	−0.0300183
$206$	295.046	0.0997904
$207$	−377.477	−0.126746
$208$	−305.692	−0.101904
$209$	2478.37	0.820250
$210$	14.1267	0.00464207
$211$	417.230	0.136129	0.0680647	−	0.997681i	$-0.478318\pi$
$211$	417.230	0.136129	0.0680647	+	0.997681i	$0.478318\pi$
$212$	−8.97888	−0.00290883
$213$	−2637.79	−0.848537
$214$	−2267.82	−0.724415
$215$	189.560	0.0601298
$216$	−556.043	−0.175157
$217$	296.025	0.0926060
$218$	1194.22	0.371022
$219$	−1003.42	−0.309612
$220$	−94.2643	−0.0288877
$221$	0	0
$222$	−141.846	−0.0428831
$223$	1074.81	0.322757	0.161379	−	0.986893i	$-0.448406\pi$
$223$	1074.81	0.322757	0.161379	+	0.986893i	$0.448406\pi$
$224$	−726.077	−0.216576
$225$	−1119.05	−0.331571
$226$	−2400.92	−0.706667
$227$	−3375.49	−0.986957	−0.493479	−	0.869758i	$-0.664275\pi$
$227$	−3375.49	−0.986957	−0.493479	+	0.869758i	$0.664275\pi$
$228$	2127.13	0.617862
$229$	−2565.82	−0.740410	−0.370205	−	0.928950i	$-0.620713\pi$
$229$	−2565.82	−0.740410	−0.370205	+	0.928950i	$0.620713\pi$
$230$	51.0258	0.0146284
$231$	−233.782	−0.0665877
$232$	−513.653	−0.145358
$233$	−212.233	−0.0596730	−0.0298365	−	0.999555i	$-0.509499\pi$
$233$	−212.233	−0.0596730	−0.0298365	+	0.999555i	$0.509499\pi$
$234$	269.891	0.0753988
$235$	−41.5949	−0.0115462
$236$	3390.33	0.935134
$237$	−2473.06	−0.677817
$238$	0	0
$239$	−4285.84	−1.15995	−0.579975	−	0.814634i	$-0.696938\pi$
$239$	−4285.84	−1.15995	−0.579975	+	0.814634i	$0.696938\pi$
$240$	−37.2051	−0.0100066
$241$	−1104.81	−0.295299	−0.147650	−	0.989040i	$-0.547171\pi$
$241$	−1104.81	−0.295299	−0.147650	+	0.989040i	$0.547171\pi$
$242$	1385.40	0.368005
$243$	−243.000	−0.0641500
$244$	4184.35	1.09785
$245$	−266.632	−0.0695286
$246$	−486.688	−0.126139
$247$	−2466.44	−0.635368
$248$	1575.06	0.403291
$249$	674.081	0.171559
$250$	303.342	0.0767402
$251$	480.822	0.120913	0.0604566	−	0.998171i	$-0.480744\pi$
$251$	480.822	0.120913	0.0604566	+	0.998171i	$0.480744\pi$
$252$	−200.650	−0.0501579
$253$	−844.424	−0.209836
$254$	3465.33	0.856040
$255$	0	0
$256$	−3160.19	−0.771532
$257$	4146.30	1.00638	0.503189	−	0.864176i	$-0.332160\pi$
$257$	4146.30	1.00638	0.503189	+	0.864176i	$0.332160\pi$
$258$	1047.08	0.252669
$259$	−122.277	−0.0293356
$260$	93.8107	0.0223765
$261$	−224.475	−0.0532362
$262$	2658.00	0.626763
$263$	8011.72	1.87842	0.939209	−	0.343346i	$-0.111560\pi$
$263$	8011.72	1.87842	0.939209	+	0.343346i	$0.111560\pi$
$264$	−1243.88	−0.289983
$265$	1.26712	0.000293730 0
$266$	−713.112	−0.164375
$267$	1683.40	0.385851
$268$	−1536.48	−0.350208
$269$	5577.54	1.26420	0.632098	−	0.774888i	$-0.282194\pi$
$269$	5577.54	1.26420	0.632098	+	0.774888i	$0.282194\pi$
$270$	32.8478	0.00740389
$271$	−3089.50	−0.692524	−0.346262	−	0.938138i	$-0.612549\pi$
$271$	−3089.50	−0.692524	−0.346262	+	0.938138i	$0.612549\pi$
$272$	0	0
$273$	232.657	0.0515790
$274$	−2228.95	−0.491445
$275$	−2503.35	−0.548937
$276$	−724.750	−0.158061
$277$	−585.502	−0.127001	−0.0635007	−	0.997982i	$-0.520227\pi$
$277$	−585.502	−0.127001	−0.0635007	+	0.997982i	$0.520227\pi$
$278$	−3682.10	−0.794381
$279$	688.325	0.147702
$280$	64.7943	0.0138293
$281$	6399.58	1.35860	0.679301	−	0.733860i	$-0.262283\pi$
$281$	6399.58	1.35860	0.679301	+	0.733860i	$0.262283\pi$
$282$	−229.760	−0.0485177
$283$	−6034.81	−1.26761	−0.633803	−	0.773495i	$-0.718507\pi$
$283$	−6034.81	−1.26761	−0.633803	+	0.773495i	$0.718507\pi$
$284$	−5064.52	−1.05818
$285$	−300.185	−0.0623909
$286$	603.752	0.124827
$287$	−419.546	−0.0862892
$288$	−1688.29	−0.345429
$289$	0	0
$290$	30.3436	0.00614427
$291$	−5333.62	−1.07444
$292$	−1926.56	−0.386107
$293$	7539.51	1.50329	0.751643	−	0.659570i	$-0.229262\pi$
$293$	7539.51	1.50329	0.751643	+	0.659570i	$0.229262\pi$
$294$	−1472.81	−0.292164
$295$	−478.450	−0.0944286
$296$	−650.597	−0.127754
$297$	−543.597	−0.106204
$298$	−2965.43	−0.576453
$299$	840.360	0.162539
$300$	−2148.57	−0.413492
$301$	902.632	0.172847
$302$	−1936.50	−0.368983
$303$	−5830.23	−1.10541
$304$	1878.10	0.354330
$305$	−590.504	−0.110860
$306$	0	0
$307$	5367.77	0.997899	0.498950	−	0.866631i	$-0.333719\pi$
$307$	5367.77	0.997899	0.498950	+	0.866631i	$0.333719\pi$
$308$	−448.859	−0.0830394
$309$	591.403	0.108879
$310$	−93.0450	−0.0170471
$311$	9705.57	1.76962	0.884811	−	0.465951i	$-0.154288\pi$
$311$	9705.57	1.76962	0.884811	+	0.465951i	$0.154288\pi$
$312$	1237.90	0.224622
$313$	2858.58	0.516219	0.258109	−	0.966116i	$-0.416901\pi$
$313$	2858.58	0.516219	0.258109	+	0.966116i	$0.416901\pi$
$314$	4808.19	0.864146
$315$	28.3162	0.00506487
$316$	−4748.25	−0.845285
$317$	−5747.90	−1.01840	−0.509202	−	0.860647i	$-0.670059\pi$
$317$	−5747.90	−1.01840	−0.509202	+	0.860647i	$0.670059\pi$
$318$	6.99925	0.00123427
$319$	−502.155	−0.0881357
$320$	129.003	0.0225359
$321$	−4545.71	−0.790395
$322$	242.970	0.0420503
$323$	0	0
$324$	−466.557	−0.0799995
$325$	2491.30	0.425208
$326$	−937.460	−0.159267
$327$	2393.75	0.404815
$328$	−2232.27	−0.375782
$329$	−198.063	−0.0331901
$330$	73.4812	0.0122576
$331$	−5525.96	−0.917626	−0.458813	−	0.888533i	$-0.651725\pi$
$331$	−5525.96	−0.917626	−0.458813	+	0.888533i	$0.651725\pi$
$332$	1294.23	0.213946
$333$	−284.321	−0.0467889
$334$	−5598.71	−0.917208
$335$	216.832	0.0353635
$336$	−177.160	−0.0287644
$337$	−4477.38	−0.723734	−0.361867	−	0.932230i	$-0.617861\pi$
$337$	−4477.38	−0.723734	−0.361867	+	0.932230i	$0.617861\pi$
$338$	2687.35	0.432463
$339$	−4812.51	−0.771031
$340$	0	0
$341$	1539.80	0.244530
$342$	−1658.15	−0.262170
$343$	−2597.24	−0.408856
$344$	4802.62	0.752732
$345$	102.278	0.0159608
$346$	−149.262	−0.0231919
$347$	−9834.94	−1.52152	−0.760760	−	0.649034i	$-0.775173\pi$
$347$	−9834.94	−1.52152	−0.760760	+	0.649034i	$0.775173\pi$
$348$	−430.989	−0.0663892
$349$	−9155.23	−1.40421	−0.702104	−	0.712075i	$-0.747756\pi$
$349$	−9155.23	−1.40421	−0.702104	+	0.712075i	$0.747756\pi$
$350$	720.300	0.110005
$351$	540.981	0.0822662
$352$	−3776.75	−0.571879
$353$	−7025.07	−1.05923	−0.529613	−	0.848239i	$-0.677663\pi$
$353$	−7025.07	−1.05923	−0.529613	+	0.848239i	$0.677663\pi$
$354$	−2642.84	−0.396795
$355$	714.716	0.106854
$356$	3232.10	0.481182
$357$	0	0
$358$	−3248.38	−0.479560
$359$	8370.77	1.23062	0.615310	−	0.788285i	$-0.289031\pi$
$359$	8370.77	1.23062	0.615310	+	0.788285i	$0.289031\pi$
$360$	150.661	0.0220571
$361$	8294.23	1.20925
$362$	−4179.27	−0.606788
$363$	2776.96	0.401523
$364$	446.699	0.0643226
$365$	271.880	0.0389886
$366$	−3261.80	−0.465838
$367$	6592.31	0.937645	0.468823	−	0.883292i	$-0.344678\pi$
$367$	6592.31	0.937645	0.468823	+	0.883292i	$0.344678\pi$
$368$	−639.902	−0.0906446
$369$	−975.538	−0.137627
$370$	38.4334	0.00540016
$371$	6.03365	0.000844344 0
$372$	1321.58	0.184195
$373$	−10437.4	−1.44886	−0.724432	−	0.689347i	$-0.757898\pi$
$373$	−10437.4	−1.44886	−0.724432	+	0.689347i	$0.757898\pi$
$374$	0	0
$375$	608.032	0.0837297
$376$	−1053.83	−0.144540
$377$	499.739	0.0682702
$378$	156.412	0.0212829
$379$	−1745.83	−0.236616	−0.118308	−	0.992977i	$-0.537747\pi$
$379$	−1745.83	−0.236616	−0.118308	+	0.992977i	$0.537747\pi$
$380$	−576.351	−0.0778057
$381$	6946.06	0.934009
$382$	5015.63	0.671786
$383$	1031.82	0.137660	0.0688298	−	0.997628i	$-0.478073\pi$
$383$	1031.82	0.137660	0.0688298	+	0.997628i	$0.478073\pi$
$384$	−3789.53	−0.503604
$385$	63.3439	0.00838521
$386$	−4190.67	−0.552590
$387$	2098.82	0.275682
$388$	−10240.5	−1.33990
$389$	7514.99	0.979499	0.489750	−	0.871863i	$-0.337088\pi$
$389$	7514.99	0.979499	0.489750	+	0.871863i	$0.337088\pi$
$390$	−73.1276	−0.00949477
$391$	0	0
$392$	−6755.28	−0.870390
$393$	5327.81	0.683849
$394$	−3798.79	−0.485737
$395$	670.083	0.0853557
$396$	−1043.70	−0.132444
$397$	3017.78	0.381507	0.190753	−	0.981638i	$-0.438907\pi$
$397$	3017.78	0.381507	0.190753	+	0.981638i	$0.438907\pi$
$398$	−4077.22	−0.513499
$399$	−1429.39	−0.179346
$400$	−1897.03	−0.237129
$401$	14853.9	1.84979	0.924897	−	0.380219i	$-0.124151\pi$
$401$	14853.9	1.84979	0.924897	+	0.380219i	$0.124151\pi$
$402$	1197.72	0.148600
$403$	−1532.39	−0.189414
$404$	−11194.0	−1.37852
$405$	65.8415	0.00807824
$406$	144.487	0.0176621
$407$	−636.033	−0.0774619
$408$	0	0
$409$	5721.53	0.691714	0.345857	−	0.938287i	$-0.387588\pi$
$409$	5721.53	0.691714	0.345857	+	0.938287i	$0.387588\pi$
$410$	131.869	0.0158843
$411$	−4467.81	−0.536206
$412$	1135.49	0.135780
$413$	−2278.24	−0.271441
$414$	564.960	0.0670682
$415$	−182.644	−0.0216039
$416$	3758.58	0.442980
$417$	−7380.57	−0.866734
$418$	−3709.31	−0.434039
$419$	−6130.96	−0.714838	−0.357419	−	0.933944i	$-0.616343\pi$
$419$	−6130.96	−0.714838	−0.357419	+	0.933944i	$0.616343\pi$
$420$	54.3667	0.00631625
$421$	6388.31	0.739542	0.369771	−	0.929123i	$-0.379436\pi$
$421$	6388.31	0.739542	0.369771	+	0.929123i	$0.379436\pi$
$422$	−624.458	−0.0720335
$423$	−460.541	−0.0529368
$424$	32.1031	0.00367704
$425$	0	0
$426$	3947.91	0.449007
$427$	−2811.81	−0.318672
$428$	−8727.71	−0.985677
$429$	1210.19	0.136197
$430$	−283.710	−0.0318180
$431$	854.435	0.0954911	0.0477456	−	0.998860i	$-0.484796\pi$
$431$	854.435	0.0954911	0.0477456	+	0.998860i	$0.484796\pi$
$432$	−411.936	−0.0458780
$433$	10107.4	1.12178	0.560892	−	0.827889i	$-0.310458\pi$
$433$	10107.4	1.12178	0.560892	+	0.827889i	$0.310458\pi$
$434$	−443.054	−0.0490029
$435$	60.8220	0.00670389
$436$	4595.97	0.504832
$437$	−5162.97	−0.565168
$438$	1501.80	0.163832
$439$	−16418.3	−1.78497	−0.892486	−	0.451075i	$-0.851041\pi$
$439$	−16418.3	−1.78497	−0.892486	+	0.451075i	$0.851041\pi$
$440$	337.033	0.0365168
$441$	−2952.17	−0.318774
$442$	0	0
$443$	5106.09	0.547625	0.273812	−	0.961783i	$-0.411715\pi$
$443$	5106.09	0.547625	0.273812	+	0.961783i	$0.411715\pi$
$444$	−545.894	−0.0583490
$445$	−456.120	−0.0485892
$446$	−1608.65	−0.170788
$447$	−5944.04	−0.628956
$448$	614.276	0.0647808
$449$	−11275.1	−1.18509	−0.592543	−	0.805539i	$-0.701876\pi$
$449$	−11275.1	−1.18509	−0.592543	+	0.805539i	$0.701876\pi$
$450$	1674.86	0.175452
$451$	−2182.30	−0.227850
$452$	−9239.96	−0.961529
$453$	−3881.60	−0.402590
$454$	5052.02	0.522253
$455$	−63.0391	−0.00649521
$456$	−7605.34	−0.781037
$457$	−13048.0	−1.33557	−0.667787	−	0.744352i	$-0.732758\pi$
$457$	−13048.0	−1.33557	−0.667787	+	0.744352i	$0.732758\pi$
$458$	3840.19	0.391791
$459$	0	0
$460$	196.373	0.0199042
$461$	14147.2	1.42929	0.714645	−	0.699487i	$-0.246588\pi$
$461$	14147.2	1.42929	0.714645	+	0.699487i	$0.246588\pi$
$462$	349.896	0.0352352
$463$	7778.27	0.780749	0.390375	−	0.920656i	$-0.372345\pi$
$463$	7778.27	0.780749	0.390375	+	0.920656i	$0.372345\pi$
$464$	−380.532	−0.0380728
$465$	−186.504	−0.0185998
$466$	317.643	0.0315763
$467$	3546.16	0.351384	0.175692	−	0.984445i	$-0.443784\pi$
$467$	3546.16	0.351384	0.175692	+	0.984445i	$0.443784\pi$
$468$	1038.68	0.102592
$469$	1032.49	0.101654
$470$	62.2540	0.00610971
$471$	9637.74	0.942853
$472$	−12121.8	−1.18210
$473$	4695.11	0.456409
$474$	3701.37	0.358670
$475$	−15306.0	−1.47850
$476$	0	0
$477$	14.0296	0.00134669
$478$	6414.51	0.613793
$479$	2338.12	0.223030	0.111515	−	0.993763i	$-0.464430\pi$
$479$	2338.12	0.223030	0.111515	+	0.993763i	$0.464430\pi$
$480$	457.447	0.0434990
$481$	632.973	0.0600023
$482$	1653.54	0.156259
$483$	487.019	0.0458802
$484$	5331.73	0.500726
$485$	1445.16	0.135302
$486$	363.692	0.0339453
$487$	3327.52	0.309619	0.154810	−	0.987944i	$-0.450524\pi$
$487$	3327.52	0.309619	0.154810	+	0.987944i	$0.450524\pi$
$488$	−14960.7	−1.38779
$489$	−1879.09	−0.173773
$490$	399.062	0.0367914
$491$	−5511.23	−0.506554	−0.253277	−	0.967394i	$-0.581508\pi$
$491$	−5511.23	−0.506554	−0.253277	+	0.967394i	$0.581508\pi$
$492$	−1873.02	−0.171631
$493$	0	0
$494$	3691.46	0.336208
$495$	147.289	0.0133740
$496$	1166.86	0.105632
$497$	3403.27	0.307158
$498$	−1008.88	−0.0907811
$499$	−15156.1	−1.35968	−0.679839	−	0.733361i	$-0.737950\pi$
$499$	−15156.1	−1.35968	−0.679839	+	0.733361i	$0.737950\pi$
$500$	1167.41	0.104417
$501$	−11222.3	−1.00075
$502$	−719.634	−0.0639817
$503$	10720.0	0.950264	0.475132	−	0.879915i	$-0.342400\pi$
$503$	10720.0	0.950264	0.475132	+	0.879915i	$0.342400\pi$
$504$	717.406	0.0634043
$505$	1579.72	0.139201
$506$	1263.83	0.111036
$507$	5386.64	0.471852
$508$	13336.4	1.16477
$509$	12724.1	1.10803	0.554013	−	0.832508i	$-0.313096\pi$
$509$	12724.1	1.10803	0.554013	+	0.832508i	$0.313096\pi$
$510$	0	0
$511$	1294.61	0.112075
$512$	−5375.64	−0.464008
$513$	−3323.66	−0.286049
$514$	−6205.67	−0.532530
$515$	−160.242	−0.0137109
$516$	4029.71	0.343795
$517$	−1030.24	−0.0876400
$518$	183.009	0.0155231
$519$	−299.188	−0.0253042
$520$	−335.411	−0.0282861
$521$	−13530.5	−1.13778	−0.568889	−	0.822415i	$-0.692626\pi$
$521$	−13530.5	−1.13778	−0.568889	+	0.822415i	$0.692626\pi$
$522$	335.966	0.0281702
$523$	7104.82	0.594019	0.297010	−	0.954875i	$-0.404011\pi$
$523$	7104.82	0.594019	0.297010	+	0.954875i	$0.404011\pi$
$524$	10229.3	0.852807
$525$	1443.80	0.120024
$526$	−11990.9	−0.993973
$527$	0	0
$528$	−921.511	−0.0759538
$529$	−10407.9	−0.855419
$530$	−1.89646	−0.000155429 0
$531$	−5297.42	−0.432935
$532$	−2744.42	−0.223657
$533$	2171.80	0.176494
$534$	−2519.50	−0.204175
$535$	1231.67	0.0995324
$536$	5493.55	0.442696
$537$	−6511.20	−0.523238
$538$	−8347.77	−0.668955
$539$	−6604.06	−0.527750
$540$	126.415	0.0100741
$541$	16514.7	1.31242	0.656211	−	0.754577i	$-0.272158\pi$
$541$	16514.7	1.31242	0.656211	+	0.754577i	$0.272158\pi$
$542$	4623.98	0.366452
$543$	−8377.10	−0.662055
$544$	0	0
$545$	−648.592	−0.0509773
$546$	−348.213	−0.0272933
$547$	−775.472	−0.0606157	−0.0303079	−	0.999541i	$-0.509649\pi$
$547$	−775.472	−0.0606157	−0.0303079	+	0.999541i	$0.509649\pi$
$548$	−8578.14	−0.668686
$549$	−6538.09	−0.508267
$550$	3746.70	0.290472
$551$	−3070.28	−0.237383
$552$	2591.27	0.199804
$553$	3190.74	0.245360
$554$	876.306	0.0672034
$555$	77.0376	0.00589201
$556$	−14170.6	−1.08088
$557$	14283.2	1.08653	0.543267	−	0.839560i	$-0.317187\pi$
$557$	14283.2	1.08653	0.543267	+	0.839560i	$0.317187\pi$
$558$	−1030.20	−0.0781574
$559$	−4672.52	−0.353536
$560$	48.0019	0.00362223
$561$	0	0
$562$	−9578.10	−0.718910
$563$	−7158.60	−0.535877	−0.267939	−	0.963436i	$-0.586342\pi$
$563$	−7158.60	−0.535877	−0.267939	+	0.963436i	$0.586342\pi$
$564$	−884.233	−0.0660158
$565$	1303.96	0.0970939
$566$	9032.16	0.670759
$567$	313.518	0.0232214
$568$	18107.7	1.33765
$569$	4328.74	0.318929	0.159464	−	0.987204i	$-0.449023\pi$
$569$	4328.74	0.318929	0.159464	+	0.987204i	$0.449023\pi$
$570$	449.279	0.0330144
$571$	12937.7	0.948207	0.474103	−	0.880469i	$-0.342772\pi$
$571$	12937.7	0.948207	0.474103	+	0.880469i	$0.342772\pi$
$572$	2323.54	0.169847
$573$	10053.6	0.732972
$574$	627.923	0.0456603
$575$	5215.02	0.378228
$576$	1428.33	0.103322
$577$	14654.1	1.05729	0.528645	−	0.848843i	$-0.322700\pi$
$577$	14654.1	1.05729	0.528645	+	0.848843i	$0.322700\pi$
$578$	0	0
$579$	−8399.97	−0.602920
$580$	116.778	0.00836022
$581$	−869.698	−0.0621018
$582$	7982.69	0.568545
$583$	31.3845	0.00222953
$584$	6888.22	0.488076
$585$	−146.580	−0.0103596
$586$	−11284.2	−0.795471
$587$	12094.6	0.850419	0.425209	−	0.905095i	$-0.360201\pi$
$587$	12094.6	0.850419	0.425209	+	0.905095i	$0.360201\pi$
$588$	−5668.12	−0.397533
$589$	9414.64	0.658614
$590$	716.084	0.0499673
$591$	−7614.46	−0.529978
$592$	−481.985	−0.0334619
$593$	−8913.39	−0.617249	−0.308625	−	0.951184i	$-0.599869\pi$
$593$	−8913.39	−0.617249	−0.308625	+	0.951184i	$0.599869\pi$
$594$	813.587	0.0561985
$595$	0	0
$596$	−11412.5	−0.784352
$597$	−8172.55	−0.560268
$598$	−1257.75	−0.0860085
$599$	−17664.7	−1.20494	−0.602470	−	0.798142i	$-0.705817\pi$
$599$	−17664.7	−1.20494	−0.602470	+	0.798142i	$0.705817\pi$
$600$	7682.00	0.522694
$601$	3318.06	0.225202	0.112601	−	0.993640i	$-0.464082\pi$
$601$	3318.06	0.225202	0.112601	+	0.993640i	$0.464082\pi$
$602$	−1350.95	−0.0914626
$603$	2400.77	0.162134
$604$	−7452.62	−0.502057
$605$	−752.425	−0.0505627
$606$	8725.96	0.584931
$607$	15291.9	1.02254	0.511268	−	0.859422i	$-0.329176\pi$
$607$	15291.9	1.02254	0.511268	+	0.859422i	$0.329176\pi$
$608$	−23091.8	−1.54029
$609$	289.617	0.0192707
$610$	883.792	0.0586618
$611$	1025.28	0.0678863
$612$	0	0
$613$	13570.8	0.894160	0.447080	−	0.894494i	$-0.352464\pi$
$613$	13570.8	0.894160	0.447080	+	0.894494i	$0.352464\pi$
$614$	−8033.81	−0.528043
$615$	264.324	0.0173310
$616$	1604.85	0.104970
$617$	17609.2	1.14898	0.574489	−	0.818512i	$-0.305201\pi$
$617$	17609.2	1.14898	0.574489	+	0.818512i	$0.305201\pi$
$618$	−885.138	−0.0576140
$619$	21150.6	1.37337	0.686683	−	0.726957i	$-0.259066\pi$
$619$	21150.6	1.37337	0.686683	+	0.726957i	$0.259066\pi$
$620$	−358.084	−0.0231952
$621$	1132.43	0.0731769
$622$	−14526.1	−0.936403
$623$	−2171.91	−0.139672
$624$	917.077	0.0588340
$625$	15377.7	0.984170
$626$	−4278.36	−0.273159
$627$	−7435.10	−0.473571
$628$	18504.4	1.17580
$629$	0	0
$630$	−42.3801	−0.00268010
$631$	5652.30	0.356600	0.178300	−	0.983976i	$-0.442940\pi$
$631$	5652.30	0.356600	0.178300	+	0.983976i	$0.442940\pi$
$632$	16976.9	1.06852
$633$	−1251.69	−0.0785944
$634$	8602.74	0.538893
$635$	−1882.05	−0.117617
$636$	26.9366	0.00167941
$637$	6572.29	0.408797
$638$	751.563	0.0466374
$639$	7913.37	0.489903
$640$	1026.78	0.0634175
$641$	21389.9	1.31802	0.659011	−	0.752134i	$-0.270975\pi$
$641$	21389.9	1.31802	0.659011	+	0.752134i	$0.270975\pi$
$642$	6803.45	0.418241
$643$	5815.95	0.356701	0.178350	−	0.983967i	$-0.442924\pi$
$643$	5815.95	0.356701	0.178350	+	0.983967i	$0.442924\pi$
$644$	935.071	0.0572158
$645$	−568.681	−0.0347160
$646$	0	0
$647$	−24007.9	−1.45881	−0.729404	−	0.684084i	$-0.760202\pi$
$647$	−24007.9	−1.45881	−0.729404	+	0.684084i	$0.760202\pi$
$648$	1668.13	0.101127
$649$	−11850.5	−0.716751
$650$	−3728.67	−0.225001
$651$	−888.076	−0.0534661
$652$	−3607.82	−0.216707
$653$	1070.36	0.0641447	0.0320724	−	0.999486i	$-0.489789\pi$
$653$	1070.36	0.0641447	0.0320724	+	0.999486i	$0.489789\pi$
$654$	−3582.66	−0.214210
$655$	−1443.59	−0.0861153
$656$	−1653.74	−0.0984265
$657$	3010.26	0.178754
$658$	296.436	0.0175627
$659$	12742.8	0.753245	0.376623	−	0.926367i	$-0.377085\pi$
$659$	12742.8	0.753245	0.376623	+	0.926367i	$0.377085\pi$
$660$	282.793	0.0166783
$661$	1443.34	0.0849312	0.0424656	−	0.999098i	$-0.486479\pi$
$661$	1443.34	0.0849312	0.0424656	+	0.999098i	$0.486479\pi$
$662$	8270.57	0.485566
$663$	0	0
$664$	−4627.38	−0.270448
$665$	387.297	0.0225846
$666$	425.537	0.0247586
$667$	1046.10	0.0607273
$668$	−21546.7	−1.24800
$669$	−3224.44	−0.186344
$670$	−324.527	−0.0187128
$671$	−14625.9	−0.841467
$672$	2178.23	0.125040
$673$	−4792.95	−0.274524	−0.137262	−	0.990535i	$-0.543830\pi$
$673$	−4792.95	−0.274524	−0.137262	+	0.990535i	$0.543830\pi$
$674$	6701.18	0.382967
$675$	3357.16	0.191433
$676$	10342.3	0.588432
$677$	22167.3	1.25843	0.629216	−	0.777230i	$-0.283376\pi$
$677$	22167.3	1.25843	0.629216	+	0.777230i	$0.283376\pi$
$678$	7202.76	0.407995
$679$	6881.42	0.388932
$680$	0	0
$681$	10126.5	0.569820
$682$	−2304.58	−0.129394
$683$	30315.2	1.69836	0.849179	−	0.528105i	$-0.177097\pi$
$683$	30315.2	1.69836	0.849179	+	0.528105i	$0.177097\pi$
$684$	−6381.38	−0.356723
$685$	1210.56	0.0675230
$686$	3887.22	0.216348
$687$	7697.45	0.427476
$688$	3557.94	0.197159
$689$	−31.2335	−0.00172700
$690$	−153.077	−0.00844573
$691$	26216.1	1.44328	0.721642	−	0.692267i	$-0.243388\pi$
$691$	26216.1	1.44328	0.721642	+	0.692267i	$0.243388\pi$
$692$	−574.437	−0.0315561
$693$	701.347	0.0384444
$694$	14719.7	0.805119
$695$	1999.78	0.109146
$696$	1540.96	0.0839223
$697$	0	0
$698$	13702.4	0.743043
$699$	636.698	0.0344522
$700$	2772.08	0.149678
$701$	−10843.2	−0.584228	−0.292114	−	0.956384i	$-0.594359\pi$
$701$	−10843.2	−0.584228	−0.292114	+	0.956384i	$0.594359\pi$
$702$	−809.673	−0.0435315
$703$	−3888.84	−0.208635
$704$	3195.21	0.171057
$705$	124.785	0.00666619
$706$	10514.3	0.560494
$707$	7522.15	0.400141
$708$	−10171.0	−0.539900
$709$	5358.13	0.283820	0.141910	−	0.989880i	$-0.454676\pi$
$709$	5358.13	0.283820	0.141910	+	0.989880i	$0.454676\pi$
$710$	−1069.70	−0.0565423
$711$	7419.19	0.391338
$712$	−11556.1	−0.608261
$713$	−3207.74	−0.168486
$714$	0	0
$715$	−327.903	−0.0171509
$716$	−12501.4	−0.652514
$717$	12857.5	0.669698
$718$	−12528.3	−0.651188
$719$	−8174.28	−0.423990	−0.211995	−	0.977271i	$-0.567996\pi$
$719$	−8174.28	−0.423990	−0.211995	+	0.977271i	$0.567996\pi$
$720$	111.615	0.00577729
$721$	−763.026	−0.0394127
$722$	−12413.8	−0.639879
$723$	3314.43	0.170491
$724$	−16083.9	−0.825628
$725$	3101.23	0.158864
$726$	−4156.21	−0.212468
$727$	22324.8	1.13890	0.569451	−	0.822026i	$-0.307156\pi$
$727$	22324.8	1.13890	0.569451	+	0.822026i	$0.307156\pi$
$728$	−1597.13	−0.0813099
$729$	729.000	0.0370370
$730$	−406.915	−0.0206310
$731$	0	0
$732$	−12553.0	−0.633844
$733$	−7293.42	−0.367515	−0.183758	−	0.982972i	$-0.558826\pi$
$733$	−7293.42	−0.367515	−0.183758	+	0.982972i	$0.558826\pi$
$734$	−9866.55	−0.496159
$735$	799.897	0.0401424
$736$	7867.79	0.394036
$737$	5370.58	0.268423
$738$	1460.06	0.0728261
$739$	−12956.5	−0.644945	−0.322472	−	0.946579i	$-0.604514\pi$
$739$	−12956.5	−0.644945	−0.322472	+	0.946579i	$0.604514\pi$
$740$	147.911	0.00734774
$741$	7399.32	0.366830
$742$	−9.03041	−0.000446788 0
$743$	14307.3	0.706439	0.353219	−	0.935540i	$-0.385087\pi$
$743$	14307.3	0.706439	0.353219	+	0.935540i	$0.385087\pi$
$744$	−4725.17	−0.232840
$745$	1610.55	0.0792028
$746$	15621.3	0.766673
$747$	−2022.24	−0.0990495
$748$	0	0
$749$	5864.87	0.286112
$750$	−910.026	−0.0443059
$751$	23289.9	1.13164	0.565819	−	0.824530i	$-0.308560\pi$
$751$	23289.9	1.13164	0.565819	+	0.824530i	$0.308560\pi$
$752$	−780.713	−0.0378586
$753$	−1442.46	−0.0698092
$754$	−747.947	−0.0361255
$755$	1051.73	0.0506971
$756$	601.951	0.0289587
$757$	−615.306	−0.0295425	−0.0147712	−	0.999891i	$-0.504702\pi$
$757$	−615.306	−0.0295425	−0.0147712	+	0.999891i	$0.504702\pi$
$758$	2612.95	0.125206
$759$	2533.27	0.121149
$760$	2060.69	0.0983539
$761$	2678.42	0.127586	0.0637928	−	0.997963i	$-0.479680\pi$
$761$	2678.42	0.127586	0.0637928	+	0.997963i	$0.479680\pi$
$762$	−10396.0	−0.494235
$763$	−3088.41	−0.146537
$764$	19302.7	0.914067
$765$	0	0
$766$	−1544.30	−0.0728432
$767$	11793.4	0.555197
$768$	9480.58	0.445444
$769$	−8111.73	−0.380385	−0.190193	−	0.981747i	$-0.560911\pi$
$769$	−8111.73	−0.380385	−0.190193	+	0.981747i	$0.560911\pi$
$770$	−94.8053	−0.00443707
$771$	−12438.9	−0.581033
$772$	−16127.8	−0.751883
$773$	−14345.3	−0.667481	−0.333740	−	0.942665i	$-0.608311\pi$
$773$	−14345.3	−0.667481	−0.333740	+	0.942665i	$0.608311\pi$
$774$	−3141.25	−0.145879
$775$	−9509.54	−0.440765
$776$	36613.9	1.69376
$777$	366.831	0.0169369
$778$	−11247.5	−0.518307
$779$	−13343.0	−0.613689
$780$	−281.432	−0.0129191
$781$	17702.4	0.811064
$782$	0	0
$783$	673.425	0.0307359
$784$	−5004.54	−0.227977
$785$	−2611.37	−0.118731
$786$	−7974.01	−0.361862
$787$	−5402.76	−0.244711	−0.122355	−	0.992486i	$-0.539045\pi$
$787$	−5402.76	−0.244711	−0.122355	+	0.992486i	$0.539045\pi$
$788$	−14619.7	−0.660919
$789$	−24035.2	−1.08451
$790$	−1002.90	−0.0451664
$791$	6209.09	0.279102
$792$	3731.64	0.167422
$793$	14555.5	0.651803
$794$	−4516.64	−0.201876
$795$	−3.80135	−0.000169585 0
$796$	−15691.2	−0.698693
$797$	−24962.7	−1.10944	−0.554719	−	0.832038i	$-0.687174\pi$
$797$	−24962.7	−1.10944	−0.554719	+	0.832038i	$0.687174\pi$
$798$	2139.34	0.0949018
$799$	0	0
$800$	23324.6	1.03081
$801$	−5050.19	−0.222771
$802$	−22231.4	−0.978826
$803$	6734.03	0.295939
$804$	4609.45	0.202193
$805$	−131.959	−0.00577758
$806$	2293.49	0.100229
$807$	−16732.6	−0.729884
$808$	40023.0	1.74258
$809$	−32350.4	−1.40591	−0.702953	−	0.711236i	$-0.748136\pi$
$809$	−32350.4	−1.40591	−0.702953	+	0.711236i	$0.748136\pi$
$810$	−98.5433	−0.00427464
$811$	−18155.3	−0.786088	−0.393044	−	0.919520i	$-0.628578\pi$
$811$	−18155.3	−0.786088	−0.393044	+	0.919520i	$0.628578\pi$
$812$	556.061	0.0240319
$813$	9268.51	0.399829
$814$	951.935	0.0409893
$815$	509.143	0.0218828
$816$	0	0
$817$	28706.9	1.22928
$818$	−8563.26	−0.366024
$819$	−697.972	−0.0297792
$820$	507.500	0.0216130
$821$	31795.1	1.35159	0.675797	−	0.737088i	$-0.263800\pi$
$821$	31795.1	1.35159	0.675797	+	0.737088i	$0.263800\pi$
$822$	6686.86	0.283736
$823$	−33625.8	−1.42421	−0.712103	−	0.702075i	$-0.752257\pi$
$823$	−33625.8	−1.42421	−0.712103	+	0.702075i	$0.752257\pi$
$824$	−4059.82	−0.171639
$825$	7510.04	0.316929
$826$	3409.79	0.143634
$827$	4800.81	0.201863	0.100931	−	0.994893i	$-0.467818\pi$
$827$	4800.81	0.201863	0.100931	+	0.994893i	$0.467818\pi$
$828$	2174.25	0.0912566
$829$	−4599.00	−0.192678	−0.0963389	−	0.995349i	$-0.530713\pi$
$829$	−4599.00	−0.192678	−0.0963389	+	0.995349i	$0.530713\pi$
$830$	273.359	0.0114318
$831$	1756.51	0.0733243
$832$	−3179.83	−0.132501
$833$	0	0
$834$	11046.3	0.458636
$835$	3040.71	0.126022
$836$	−14275.3	−0.590576
$837$	−2064.98	−0.0852760
$838$	9176.06	0.378260
$839$	−30117.6	−1.23930	−0.619652	−	0.784877i	$-0.712726\pi$
$839$	−30117.6	−1.23930	−0.619652	+	0.784877i	$0.712726\pi$
$840$	−194.383	−0.00798434
$841$	−23766.9	−0.974493
$842$	−9561.23	−0.391332
$843$	−19198.8	−0.784389
$844$	−2403.23	−0.0980126
$845$	−1459.52	−0.0594191
$846$	689.280	0.0280117
$847$	−3582.83	−0.145345
$848$	23.7831	0.000963108 0
$849$	18104.4	0.731853
$850$	0	0
$851$	1325.00	0.0533728
$852$	15193.6	0.610943
$853$	43091.2	1.72968	0.864839	−	0.502050i	$-0.167421\pi$
$853$	43091.2	1.72968	0.864839	+	0.502050i	$0.167421\pi$
$854$	4208.36	0.168627
$855$	900.554	0.0360214
$856$	31205.1	1.24599
$857$	−11944.5	−0.476100	−0.238050	−	0.971253i	$-0.576508\pi$
$857$	−11944.5	−0.476100	−0.238050	+	0.971253i	$0.576508\pi$
$858$	−1811.26	−0.0720691
$859$	−11237.0	−0.446335	−0.223168	−	0.974780i	$-0.571640\pi$
$859$	−11237.0	−0.446335	−0.223168	+	0.974780i	$0.571640\pi$
$860$	−1091.86	−0.0432932
$861$	1258.64	0.0498191
$862$	−1278.81	−0.0505296
$863$	22755.7	0.897583	0.448791	−	0.893637i	$-0.351855\pi$
$863$	22755.7	0.897583	0.448791	+	0.893637i	$0.351855\pi$
$864$	5064.88	0.199434
$865$	81.0656	0.00318649
$866$	−15127.5	−0.593597
$867$	0	0
$868$	−1705.09	−0.0666759
$869$	16596.9	0.647884
$870$	−91.0308	−0.00354740
$871$	−5344.74	−0.207921
$872$	−16432.4	−0.638156
$873$	16000.9	0.620329
$874$	7727.30	0.299061
$875$	−784.481	−0.0303089
$876$	5779.67	0.222919
$877$	36523.0	1.40626	0.703132	−	0.711060i	$-0.251784\pi$
$877$	36523.0	1.40626	0.703132	+	0.711060i	$0.251784\pi$
$878$	24572.9	0.944526
$879$	−22618.5	−0.867923
$880$	249.686	0.00956466
$881$	−18972.9	−0.725555	−0.362777	−	0.931876i	$-0.618171\pi$
$881$	−18972.9	−0.725555	−0.362777	+	0.931876i	$0.618171\pi$
$882$	4418.43	0.168681
$883$	28154.1	1.07300	0.536501	−	0.843899i	$-0.319746\pi$
$883$	28154.1	1.07300	0.536501	+	0.843899i	$0.319746\pi$
$884$	0	0
$885$	1435.35	0.0545184
$886$	−7642.16	−0.289778
$887$	14939.4	0.565521	0.282760	−	0.959191i	$-0.408750\pi$
$887$	14939.4	0.565521	0.282760	+	0.959191i	$0.408750\pi$
$888$	1951.79	0.0737588
$889$	−8961.79	−0.338098
$890$	682.664	0.0257112
$891$	1630.79	0.0613171
$892$	−6190.89	−0.232384
$893$	−6299.09	−0.236048
$894$	8896.30	0.332815
$895$	1764.23	0.0658900
$896$	4889.25	0.182297
$897$	−2521.08	−0.0938422
$898$	16875.1	0.627094
$899$	−1907.55	−0.0707680
$900$	6445.71	0.238730
$901$	0	0
$902$	3266.19	0.120568
$903$	−2707.89	−0.0997930
$904$	33036.6	1.21546
$905$	2269.80	0.0833708
$906$	5809.49	0.213032
$907$	−51116.4	−1.87133	−0.935663	−	0.352894i	$-0.885198\pi$
$907$	−51116.4	−1.87133	−0.935663	+	0.352894i	$0.885198\pi$
$908$	19442.7	0.710605
$909$	17490.7	0.638206
$910$	94.3491	0.00343697
$911$	−9940.59	−0.361522	−0.180761	−	0.983527i	$-0.557856\pi$
$911$	−9940.59	−0.361522	−0.180761	+	0.983527i	$0.557856\pi$
$912$	−5634.30	−0.204573
$913$	−4523.80	−0.163983
$914$	19528.5	0.706725
$915$	1771.51	0.0640048
$916$	14779.0	0.533092
$917$	−6873.93	−0.247543
$918$	0	0
$919$	−20224.2	−0.725937	−0.362968	−	0.931801i	$-0.618237\pi$
$919$	−20224.2	−0.725937	−0.362968	+	0.931801i	$0.618237\pi$
$920$	−702.113	−0.0251608
$921$	−16103.3	−0.576137
$922$	−21173.8	−0.756315
$923$	−17617.2	−0.628253
$924$	1346.58	0.0479428
$925$	3928.03	0.139625
$926$	−11641.5	−0.413137
$927$	−1774.21	−0.0628615
$928$	4678.76	0.165504
$929$	1189.09	0.0419943	0.0209972	−	0.999780i	$-0.493316\pi$
$929$	1189.09	0.0419943	0.0209972	+	0.999780i	$0.493316\pi$
$930$	279.135	0.00984215
$931$	−40378.6	−1.42143
$932$	1222.45	0.0429643
$933$	−29116.7	−1.02169
$934$	−5307.44	−0.185937
$935$	0	0
$936$	−3713.69	−0.129686
$937$	37390.7	1.30363	0.651815	−	0.758378i	$-0.274008\pi$
$937$	37390.7	1.30363	0.651815	+	0.758378i	$0.274008\pi$
$938$	−1545.30	−0.0537909
$939$	−8575.73	−0.298039
$940$	239.585	0.00831320
$941$	37121.1	1.28599	0.642994	−	0.765871i	$-0.277692\pi$
$941$	37121.1	1.28599	0.642994	+	0.765871i	$0.277692\pi$
$942$	−14424.6	−0.498915
$943$	4546.21	0.156993
$944$	−8980.25	−0.309621
$945$	−84.9485	−0.00292421
$946$	−7027.06	−0.241511
$947$	51139.6	1.75482	0.877409	−	0.479743i	$-0.159270\pi$
$947$	51139.6	1.75482	0.877409	+	0.479743i	$0.159270\pi$
$948$	14244.8	0.488025
$949$	−6701.63	−0.229235
$950$	22908.1	0.782353
$951$	17243.7	0.587976
$952$	0	0
$953$	−1138.42	−0.0386958	−0.0193479	−	0.999813i	$-0.506159\pi$
$953$	−1138.42	−0.0386958	−0.0193479	+	0.999813i	$0.506159\pi$
$954$	−20.9977	−0.000712607 0
$955$	−2724.04	−0.0923013
$956$	24686.3	0.835159
$957$	1506.47	0.0508852
$958$	−3499.41	−0.118018
$959$	5764.36	0.194099
$960$	−387.010	−0.0130111
$961$	−23941.7	−0.803656
$962$	−947.355	−0.0317505
$963$	13637.1	0.456335
$964$	6363.67	0.212614
$965$	2275.99	0.0759241
$966$	−728.910	−0.0242777
$967$	−4270.42	−0.142014	−0.0710070	−	0.997476i	$-0.522621\pi$
$967$	−4270.42	−0.142014	−0.0710070	+	0.997476i	$0.522621\pi$
$968$	−19063.1	−0.632966
$969$	0	0
$970$	−2162.93	−0.0715954
$971$	−8703.18	−0.287640	−0.143820	−	0.989604i	$-0.545939\pi$
$971$	−8703.18	−0.287640	−0.143820	+	0.989604i	$0.545939\pi$
$972$	1399.67	0.0461877
$973$	9522.39	0.313745
$974$	−4980.22	−0.163836
$975$	−7473.91	−0.245494
$976$	−11083.4	−0.363496
$977$	10690.5	0.350072	0.175036	−	0.984562i	$-0.443996\pi$
$977$	10690.5	0.350072	0.175036	+	0.984562i	$0.443996\pi$
$978$	2812.38	0.0919530
$979$	−11297.4	−0.368811
$980$	1535.79	0.0500603
$981$	−7181.24	−0.233720
$982$	8248.51	0.268045
$983$	−43935.4	−1.42556	−0.712778	−	0.701390i	$-0.752563\pi$
$983$	−43935.4	−1.42556	−0.712778	+	0.701390i	$0.752563\pi$
$984$	6696.81	0.216958
$985$	2063.16	0.0667387
$986$	0	0
$987$	594.189	0.0191623
$988$	14206.6	0.457462
$989$	−9780.94	−0.314475
$990$	−220.444	−0.00707693
$991$	16939.0	0.542971	0.271486	−	0.962442i	$-0.412485\pi$
$991$	16939.0	0.542971	0.271486	+	0.962442i	$0.412485\pi$
$992$	−14346.9	−0.459187
$993$	16577.9	0.529792
$994$	−5093.59	−0.162534
$995$	2214.37	0.0705531
$996$	−3882.68	−0.123522
$997$	−11150.0	−0.354187	−0.177093	−	0.984194i	$-0.556669\pi$
$997$	−11150.0	−0.354187	−0.177093	+	0.984194i	$0.556669\pi$
$998$	22683.7	0.719480
$999$	852.964	0.0270136

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.4.a.p.1.4		8
17.8	even	8		51.4.e.a.13.5	yes	16
17.15	even	8		51.4.e.a.4.4	✓	16
17.16	even	2		867.4.a.q.1.4		8
51.8	odd	8		153.4.f.b.64.4		16
51.32	odd	8		153.4.f.b.55.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.e.a.4.4	✓	16	17.15	even	8
51.4.e.a.13.5	yes	16	17.8	even	8
153.4.f.b.55.5		16	51.32	odd	8
153.4.f.b.64.4		16	51.8	odd	8
867.4.a.p.1.4		8	1.1	even	1	trivial
867.4.a.q.1.4		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)

\(f(q)\)	\(=\)	\(q-1.49668 q^{2} -3.00000 q^{3} -5.75996 q^{4} +0.812858 q^{5} +4.49003 q^{6} +3.87059 q^{7} +20.5942 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.49668 q^{2} -3.00000 q^{3} -5.75996 q^{4} +0.812858 q^{5} +4.49003 q^{6} +3.87059 q^{7} +20.5942 q^{8} +9.00000 q^{9} -1.21658 q^{10} +20.1332 q^{11} +17.2799 q^{12} -20.0363 q^{13} -5.79302 q^{14} -2.43857 q^{15} +15.2569 q^{16} -13.4701 q^{18} +123.098 q^{19} -4.68203 q^{20} -11.6118 q^{21} -30.1329 q^{22} -41.9418 q^{23} -61.7826 q^{24} -124.339 q^{25} +29.9879 q^{26} -27.0000 q^{27} -22.2945 q^{28} -24.9417 q^{29} +3.64975 q^{30} +76.4806 q^{31} -187.588 q^{32} -60.3996 q^{33} +3.14624 q^{35} -51.8397 q^{36} -31.5913 q^{37} -184.238 q^{38} +60.1090 q^{39} +16.7401 q^{40} -108.393 q^{41} +17.3791 q^{42} +233.202 q^{43} -115.967 q^{44} +7.31572 q^{45} +62.7733 q^{46} -51.1712 q^{47} -45.7707 q^{48} -328.019 q^{49} +186.095 q^{50} +115.409 q^{52} +1.55884 q^{53} +40.4102 q^{54} +16.3654 q^{55} +79.7117 q^{56} -369.295 q^{57} +37.3295 q^{58} -588.603 q^{59} +14.0461 q^{60} -726.454 q^{61} -114.467 q^{62} +34.8353 q^{63} +158.703 q^{64} -16.2867 q^{65} +90.3986 q^{66} +266.752 q^{67} +125.826 q^{69} -4.70890 q^{70} +879.263 q^{71} +185.348 q^{72} +334.474 q^{73} +47.2819 q^{74} +373.018 q^{75} -709.043 q^{76} +77.9274 q^{77} -89.9636 q^{78} +824.354 q^{79} +12.4017 q^{80} +81.0000 q^{81} +162.229 q^{82} -224.694 q^{83} +66.8834 q^{84} -349.028 q^{86} +74.8250 q^{87} +414.627 q^{88} -561.132 q^{89} -10.9493 q^{90} -77.5525 q^{91} +241.583 q^{92} -229.442 q^{93} +76.5866 q^{94} +100.062 q^{95} +562.765 q^{96} +1777.87 q^{97} +490.937 q^{98} +181.199 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9}+O(q^{10}) \) 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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