LMFDB - Embedded newform 169.4.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	169.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:	$9.97132279097$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.56155 q^{2} +8.68466 q^{3} +12.8078 q^{4} -2.80776 q^{5} +39.6155 q^{6} -9.56155 q^{7} +21.9309 q^{8} +48.4233 q^{9} +O(q^{10})$	\(q+4.56155 q^{2} +8.68466 q^{3} +12.8078 q^{4} -2.80776 q^{5} +39.6155 q^{6} -9.56155 q^{7} +21.9309 q^{8} +48.4233 q^{9} -12.8078 q^{10} -39.4233 q^{11} +111.231 q^{12} -43.6155 q^{14} -24.3845 q^{15} -2.42329 q^{16} +2.01515 q^{17} +220.885 q^{18} +60.1922 q^{19} -35.9612 q^{20} -83.0388 q^{21} -179.831 q^{22} +4.46876 q^{23} +190.462 q^{24} -117.116 q^{25} +186.054 q^{27} -122.462 q^{28} +140.693 q^{29} -111.231 q^{30} -136.155 q^{31} -186.501 q^{32} -342.378 q^{33} +9.19224 q^{34} +26.8466 q^{35} +620.194 q^{36} +185.708 q^{37} +274.570 q^{38} -61.5767 q^{40} -310.231 q^{41} -378.786 q^{42} +427.471 q^{43} -504.924 q^{44} -135.961 q^{45} +20.3845 q^{46} +258.617 q^{47} -21.0455 q^{48} -251.577 q^{49} -534.233 q^{50} +17.5009 q^{51} +612.656 q^{53} +848.695 q^{54} +110.691 q^{55} -209.693 q^{56} +522.749 q^{57} +641.779 q^{58} +517.885 q^{59} -312.311 q^{60} -161.311 q^{61} -621.080 q^{62} -463.002 q^{63} -831.348 q^{64} -1561.77 q^{66} +49.8987 q^{67} +25.8096 q^{68} +38.8096 q^{69} +122.462 q^{70} -279.963 q^{71} +1061.96 q^{72} -467.732 q^{73} +847.118 q^{74} -1017.12 q^{75} +770.928 q^{76} +376.948 q^{77} +37.5379 q^{79} +6.80403 q^{80} +308.386 q^{81} -1415.14 q^{82} +76.1553 q^{83} -1063.54 q^{84} -5.65808 q^{85} +1949.93 q^{86} +1221.87 q^{87} -864.587 q^{88} -202.806 q^{89} -620.194 q^{90} +57.2348 q^{92} -1182.46 q^{93} +1179.70 q^{94} -169.006 q^{95} -1619.70 q^{96} +1174.37 q^{97} -1147.58 q^{98} -1909.01 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9}+O(q^{10}) $ 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9	\( 2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9} - 5 q^{10} - 17 q^{11} + 140 q^{12} - 46 q^{14} - 90 q^{15} + 57 q^{16} + 70 q^{17} + 215 q^{18} + 141 q^{19} - 175 q^{20} - 63 q^{21} - 170 q^{22} + 145 q^{23} + 216 q^{24} + 75 q^{25} + 335 q^{27} - 80 q^{28} + 34 q^{29} - 140 q^{30} + 140 q^{31} - 105 q^{32} - 425 q^{33} + 39 q^{34} - 70 q^{35} + 725 q^{36} + 190 q^{37} + 310 q^{38} - 185 q^{40} - 538 q^{41} - 370 q^{42} + 455 q^{43} - 680 q^{44} - 375 q^{45} + 82 q^{46} - 60 q^{47} - 240 q^{48} - 565 q^{49} - 450 q^{50} - 233 q^{51} + 545 q^{53} + 914 q^{54} + 510 q^{55} - 172 q^{56} + 225 q^{57} + 595 q^{58} + 809 q^{59} + 200 q^{60} + 502 q^{61} - 500 q^{62} - 390 q^{63} - 1271 q^{64} - 1598 q^{66} + 475 q^{67} - 505 q^{68} - 479 q^{69} + 80 q^{70} - 127 q^{71} + 1155 q^{72} - 585 q^{73} + 849 q^{74} - 1725 q^{75} + 140 q^{76} + 255 q^{77} + 240 q^{79} + 1065 q^{80} + 122 q^{81} - 1515 q^{82} - 260 q^{83} - 1220 q^{84} + 1205 q^{85} + 1962 q^{86} + 1615 q^{87} - 1020 q^{88} - 921 q^{89} - 725 q^{90} - 1040 q^{92} - 2200 q^{93} + 1040 q^{94} + 1270 q^{95} - 1920 q^{96} + 415 q^{97} - 1285 q^{98} - 2210 q^{99}+O(q^{100}) \) 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9 - 5 * q^10 - 17 * q^11 + 140 * q^12 - 46 * q^14 - 90 * q^15 + 57 * q^16 + 70 * q^17 + 215 * q^18 + 141 * q^19 - 175 * q^20 - 63 * q^21 - 170 * q^22 + 145 * q^23 + 216 * q^24 + 75 * q^25 + 335 * q^27 - 80 * q^28 + 34 * q^29 - 140 * q^30 + 140 * q^31 - 105 * q^32 - 425 * q^33 + 39 * q^34 - 70 * q^35 + 725 * q^36 + 190 * q^37 + 310 * q^38 - 185 * q^40 - 538 * q^41 - 370 * q^42 + 455 * q^43 - 680 * q^44 - 375 * q^45 + 82 * q^46 - 60 * q^47 - 240 * q^48 - 565 * q^49 - 450 * q^50 - 233 * q^51 + 545 * q^53 + 914 * q^54 + 510 * q^55 - 172 * q^56 + 225 * q^57 + 595 * q^58 + 809 * q^59 + 200 * q^60 + 502 * q^61 - 500 * q^62 - 390 * q^63 - 1271 * q^64 - 1598 * q^66 + 475 * q^67 - 505 * q^68 - 479 * q^69 + 80 * q^70 - 127 * q^71 + 1155 * q^72 - 585 * q^73 + 849 * q^74 - 1725 * q^75 + 140 * q^76 + 255 * q^77 + 240 * q^79 + 1065 * q^80 + 122 * q^81 - 1515 * q^82 - 260 * q^83 - 1220 * q^84 + 1205 * q^85 + 1962 * q^86 + 1615 * q^87 - 1020 * q^88 - 921 * q^89 - 725 * q^90 - 1040 * q^92 - 2200 * q^93 + 1040 * q^94 + 1270 * q^95 - 1920 * q^96 + 415 * q^97 - 1285 * q^98 - 2210 * 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 $

$2$	4.56155	1.61275	0.806376	−	0.591403i	$-0.201426\pi$
$2$	4.56155	1.61275	0.806376	+	0.591403i	$0.201426\pi$
$3$	8.68466	1.67136	0.835682	−	0.549214i	$-0.185073\pi$
$3$	8.68466	1.67136	0.835682	+	0.549214i	$0.185073\pi$
$4$	12.8078	1.60097
$5$	−2.80776	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	−2.80776	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$6$	39.6155	2.69550
$7$	−9.56155	−0.516275	−0.258138	−	0.966108i	$-0.583109\pi$
$7$	−9.56155	−0.516275	−0.258138	+	0.966108i	$0.583109\pi$
$8$	21.9309	0.969217
$9$	48.4233	1.79346
$10$	−12.8078	−0.405017
$11$	−39.4233	−1.08060	−0.540299	−	0.841473i	$-0.681689\pi$
$11$	−39.4233	−1.08060	−0.540299	+	0.841473i	$0.681689\pi$
$12$	111.231	2.67580
$13$	0	0
$13$	0	0
$14$	−43.6155	−0.832624
$15$	−24.3845	−0.419736
$16$	−2.42329	−0.0378639
$17$	2.01515	0.0287498	0.0143749	−	0.999897i	$-0.495424\pi$
$17$	2.01515	0.0287498	0.0143749	+	0.999897i	$0.495424\pi$
$18$	220.885	2.89240
$19$	60.1922	0.726792	0.363396	−	0.931635i	$-0.381617\pi$
$19$	60.1922	0.726792	0.363396	+	0.931635i	$0.381617\pi$
$20$	−35.9612	−0.402058
$21$	−83.0388	−0.862884
$22$	−179.831	−1.74274
$23$	4.46876	0.0405131	0.0202565	−	0.999795i	$-0.493552\pi$
$23$	4.46876	0.0405131	0.0202565	+	0.999795i	$0.493552\pi$
$24$	190.462	1.61991
$25$	−117.116	−0.936932
$26$	0	0
$27$	186.054	1.32615
$28$	−122.462	−0.826542
$29$	140.693	0.900899	0.450449	−	0.892802i	$-0.351264\pi$
$29$	140.693	0.900899	0.450449	+	0.892802i	$0.351264\pi$
$30$	−111.231	−0.676931
$31$	−136.155	−0.788845	−0.394423	−	0.918929i	$-0.629055\pi$
$31$	−136.155	−0.788845	−0.394423	+	0.918929i	$0.629055\pi$
$32$	−186.501	−1.03028
$33$	−342.378	−1.80607
$34$	9.19224	0.0463663
$35$	26.8466	0.129654
$36$	620.194	2.87127
$37$	185.708	0.825142	0.412571	−	0.910925i	$-0.364631\pi$
$37$	185.708	0.825142	0.412571	+	0.910925i	$0.364631\pi$
$38$	274.570	1.17214
$39$	0	0
$40$	−61.5767	−0.243403
$41$	−310.231	−1.18171	−0.590853	−	0.806779i	$-0.701209\pi$
$41$	−310.231	−1.18171	−0.590853	+	0.806779i	$0.701209\pi$
$42$	−378.786	−1.39162
$43$	427.471	1.51602	0.758008	−	0.652246i	$-0.226173\pi$
$43$	427.471	1.51602	0.758008	+	0.652246i	$0.226173\pi$
$44$	−504.924	−1.73000
$45$	−135.961	−0.450398
$46$	20.3845	0.0653375
$47$	258.617	0.802622	0.401311	−	0.915942i	$-0.368555\pi$
$47$	258.617	0.802622	0.401311	+	0.915942i	$0.368555\pi$
$48$	−21.0455	−0.0632844
$49$	−251.577	−0.733460
$50$	−534.233	−1.51104
$51$	17.5009	0.0480514
$52$	0	0
$53$	612.656	1.58783	0.793913	−	0.608031i	$-0.208040\pi$
$53$	612.656	1.58783	0.793913	+	0.608031i	$0.208040\pi$
$54$	848.695	2.13875
$55$	110.691	0.271375
$56$	−209.693	−0.500383
$57$	522.749	1.21473
$58$	641.779	1.45293
$59$	517.885	1.14276	0.571381	−	0.820685i	$-0.306408\pi$
$59$	517.885	1.14276	0.571381	+	0.820685i	$0.306408\pi$
$60$	−312.311	−0.671985
$61$	−161.311	−0.338585	−0.169293	−	0.985566i	$-0.554148\pi$
$61$	−161.311	−0.338585	−0.169293	+	0.985566i	$0.554148\pi$
$62$	−621.080	−1.27221
$63$	−463.002	−0.925917
$64$	−831.348	−1.62373
$65$	0	0
$66$	−1561.77	−2.91274
$67$	49.8987	0.0909865	0.0454933	−	0.998965i	$-0.485514\pi$
$67$	49.8987	0.0909865	0.0454933	+	0.998965i	$0.485514\pi$
$68$	25.8096	0.0460276
$69$	38.8096	0.0677120
$70$	122.462	0.209100
$71$	−279.963	−0.467965	−0.233982	−	0.972241i	$-0.575176\pi$
$71$	−279.963	−0.467965	−0.233982	+	0.972241i	$0.575176\pi$
$72$	1061.96	1.73825
$73$	−467.732	−0.749916	−0.374958	−	0.927042i	$-0.622343\pi$
$73$	−467.732	−0.749916	−0.374958	+	0.927042i	$0.622343\pi$
$74$	847.118	1.33075
$75$	−1017.12	−1.56595
$76$	770.928	1.16357
$77$	376.948	0.557886
$78$	0	0
$79$	37.5379	0.0534600	0.0267300	−	0.999643i	$-0.491491\pi$
$79$	37.5379	0.0534600	0.0267300	+	0.999643i	$0.491491\pi$
$80$	6.80403	0.00950892
$81$	308.386	0.423027
$82$	−1415.14	−1.90580
$83$	76.1553	0.100712	0.0503562	−	0.998731i	$-0.483964\pi$
$83$	76.1553	0.100712	0.0503562	+	0.998731i	$0.483964\pi$
$84$	−1063.54	−1.38145
$85$	−5.65808	−0.00722006
$86$	1949.93	2.44496
$87$	1221.87	1.50573
$88$	−864.587	−1.04733
$89$	−202.806	−0.241544	−0.120772	−	0.992680i	$-0.538537\pi$
$89$	−202.806	−0.241544	−0.120772	+	0.992680i	$0.538537\pi$
$90$	−620.194	−0.726380
$91$	0	0
$92$	57.2348	0.0648602
$93$	−1182.46	−1.31845
$94$	1179.70	1.29443
$95$	−169.006	−0.182522
$96$	−1619.70	−1.72198
$97$	1174.37	1.22927	0.614634	−	0.788812i	$-0.289304\pi$
$97$	1174.37	1.22927	0.614634	+	0.788812i	$0.289304\pi$
$98$	−1147.58	−1.18289
$99$	−1909.01	−1.93800
$100$	−1500.00	−1.50000
$101$	970.697	0.956316	0.478158	−	0.878274i	$-0.341305\pi$
$101$	970.697	0.956316	0.478158	+	0.878274i	$0.341305\pi$
$102$	79.8314	0.0774950
$103$	−1899.70	−1.81731	−0.908654	−	0.417550i	$-0.862889\pi$
$103$	−1899.70	−1.81731	−0.908654	+	0.417550i	$0.862889\pi$
$104$	0	0
$105$	233.153	0.216699
$106$	2794.66	2.56077
$107$	−1906.49	−1.72250	−0.861251	−	0.508180i	$-0.830318\pi$
$107$	−1906.49	−1.72250	−0.861251	+	0.508180i	$0.830318\pi$
$108$	2382.94	2.12313
$109$	896.004	0.787354	0.393677	−	0.919249i	$-0.371203\pi$
$109$	896.004	0.787354	0.393677	+	0.919249i	$0.371203\pi$
$110$	504.924	0.437660
$111$	1612.81	1.37911
$112$	23.1704	0.0195482
$113$	−334.882	−0.278788	−0.139394	−	0.990237i	$-0.544516\pi$
$113$	−334.882	−0.278788	−0.139394	+	0.990237i	$0.544516\pi$
$114$	2384.55	1.95906
$115$	−12.5472	−0.0101742
$116$	1801.96	1.44231
$117$	0	0
$118$	2362.36	1.84299
$119$	−19.2680	−0.0148428
$120$	−534.773	−0.406815
$121$	223.196	0.167690
$122$	−735.827	−0.546054
$123$	−2694.25	−1.97506
$124$	−1743.84	−1.26292
$125$	679.806	0.486430
$126$	−2112.01	−1.49327
$127$	620.893	0.433822	0.216911	−	0.976191i	$-0.430402\pi$
$127$	620.893	0.433822	0.216911	+	0.976191i	$0.430402\pi$
$128$	−2300.23	−1.58839
$129$	3712.44	2.53381
$130$	0	0
$131$	−1331.70	−0.888180	−0.444090	−	0.895982i	$-0.646473\pi$
$131$	−1331.70	−0.888180	−0.444090	+	0.895982i	$0.646473\pi$
$132$	−4385.09	−2.89147
$133$	−575.531	−0.375225
$134$	227.616	0.146739
$135$	−522.396	−0.333042
$136$	44.1941	0.0278648
$137$	−622.015	−0.387900	−0.193950	−	0.981011i	$-0.562130\pi$
$137$	−622.015	−0.387900	−0.193950	+	0.981011i	$0.562130\pi$
$138$	177.032	0.109203
$139$	330.580	0.201723	0.100861	−	0.994900i	$-0.467840\pi$
$139$	330.580	0.201723	0.100861	+	0.994900i	$0.467840\pi$
$140$	343.845	0.207573
$141$	2246.00	1.34147
$142$	−1277.07	−0.754711
$143$	0	0
$144$	−117.344	−0.0679073
$145$	−395.033	−0.226246
$146$	−2133.58	−1.20943
$147$	−2184.86	−1.22588
$148$	2378.51	1.32103
$149$	1810.54	0.995470	0.497735	−	0.867329i	$-0.334165\pi$
$149$	1810.54	0.995470	0.497735	+	0.867329i	$0.334165\pi$
$150$	−4639.63	−2.52549
$151$	423.239	0.228097	0.114049	−	0.993475i	$-0.463618\pi$
$151$	423.239	0.228097	0.114049	+	0.993475i	$0.463618\pi$
$152$	1320.07	0.704419
$153$	97.5804	0.0515615
$154$	1719.47	0.899732
$155$	382.292	0.198106
$156$	0	0
$157$	1322.17	0.672105	0.336052	−	0.941843i	$-0.390908\pi$
$157$	1322.17	0.672105	0.336052	+	0.941843i	$0.390908\pi$
$158$	171.231	0.0862178
$159$	5320.71	2.65383
$160$	523.651	0.258739
$161$	−42.7283	−0.0209159
$162$	1406.72	0.682237
$163$	3606.39	1.73297	0.866486	−	0.499201i	$-0.166373\pi$
$163$	3606.39	1.73297	0.866486	+	0.499201i	$0.166373\pi$
$164$	−3973.37	−1.89188
$165$	961.316	0.453566
$166$	347.386	0.162424
$167$	3415.43	1.58260	0.791300	−	0.611429i	$-0.209405\pi$
$167$	3415.43	1.58260	0.791300	+	0.611429i	$0.209405\pi$
$168$	−1821.11	−0.836321
$169$	0	0
$170$	−25.8096	−0.0116442
$171$	2914.71	1.30347
$172$	5474.94	2.42710
$173$	2342.23	1.02934	0.514671	−	0.857388i	$-0.327914\pi$
$173$	2342.23	1.02934	0.514671	+	0.857388i	$0.327914\pi$
$174$	5573.63	2.42837
$175$	1119.82	0.483715
$176$	95.5342	0.0409157
$177$	4497.66	1.90997
$178$	−925.110	−0.389550
$179$	−666.891	−0.278468	−0.139234	−	0.990260i	$-0.544464\pi$
$179$	−666.891	−0.278468	−0.139234	+	0.990260i	$0.544464\pi$
$180$	−1741.36	−0.721073
$181$	−701.037	−0.287888	−0.143944	−	0.989586i	$-0.545978\pi$
$181$	−701.037	−0.287888	−0.143944	+	0.989586i	$0.545978\pi$
$182$	0	0
$183$	−1400.93	−0.565899
$184$	98.0037	0.0392659
$185$	−521.425	−0.207221
$186$	−5393.86	−2.12633
$187$	−79.4440	−0.0310670
$188$	3312.31	1.28497
$189$	−1778.96	−0.684660
$190$	−770.928	−0.294363
$191$	1300.88	0.492819	0.246409	−	0.969166i	$-0.420749\pi$
$191$	1300.88	0.492819	0.246409	+	0.969166i	$0.420749\pi$
$192$	−7219.97	−2.71384
$193$	519.333	0.193691	0.0968457	−	0.995299i	$-0.469125\pi$
$193$	519.333	0.193691	0.0968457	+	0.995299i	$0.469125\pi$
$194$	5356.94	1.98251
$195$	0	0
$196$	−3222.14	−1.17425
$197$	−3121.05	−1.12876	−0.564379	−	0.825516i	$-0.690884\pi$
$197$	−3121.05	−1.12876	−0.564379	+	0.825516i	$0.690884\pi$
$198$	−8708.03	−3.12552
$199$	1237.06	0.440667	0.220333	−	0.975425i	$-0.429285\pi$
$199$	1237.06	0.440667	0.220333	+	0.975425i	$0.429285\pi$
$200$	−2568.47	−0.908090
$201$	433.353	0.152072
$202$	4427.89	1.54230
$203$	−1345.25	−0.465112
$204$	224.148	0.0769289
$205$	871.056	0.296767
$206$	−8665.57	−2.93087
$207$	216.392	0.0726584
$208$	0	0
$209$	−2372.98	−0.785369
$210$	1063.54	0.349483
$211$	−2531.67	−0.826006	−0.413003	−	0.910730i	$-0.635520\pi$
$211$	−2531.67	−0.826006	−0.413003	+	0.910730i	$0.635520\pi$
$212$	7846.76	2.54206
$213$	−2431.38	−0.782139
$214$	−8696.57	−2.77797
$215$	−1200.24	−0.380723
$216$	4080.33	1.28533
$217$	1301.86	0.407261
$218$	4087.17	1.26981
$219$	−4062.09	−1.25338
$220$	1417.71	0.434463
$221$	0	0
$222$	7356.93	2.22417
$223$	1195.53	0.359008	0.179504	−	0.983757i	$-0.442551\pi$
$223$	1195.53	0.359008	0.179504	+	0.983757i	$0.442551\pi$
$224$	1783.24	0.531909
$225$	−5671.16	−1.68035
$226$	−1527.58	−0.449617
$227$	869.192	0.254142	0.127071	−	0.991894i	$-0.459442\pi$
$227$	869.192	0.254142	0.127071	+	0.991894i	$0.459442\pi$
$228$	6695.25	1.94475
$229$	−4684.64	−1.35183	−0.675916	−	0.736978i	$-0.736252\pi$
$229$	−4684.64	−1.35183	−0.675916	+	0.736978i	$0.736252\pi$
$230$	−57.2348	−0.0164085
$231$	3273.66	0.932430
$232$	3085.52	0.873166
$233$	−4868.99	−1.36900	−0.684502	−	0.729011i	$-0.739980\pi$
$233$	−4868.99	−1.36900	−0.684502	+	0.729011i	$0.739980\pi$
$234$	0	0
$235$	−726.137	−0.201566
$236$	6632.95	1.82953
$237$	326.004	0.0893511
$238$	−87.8920	−0.0239378
$239$	−4807.53	−1.30114	−0.650572	−	0.759444i	$-0.725471\pi$
$239$	−4807.53	−1.30114	−0.650572	+	0.759444i	$0.725471\pi$
$240$	59.0907	0.0158929
$241$	−5875.96	−1.57056	−0.785278	−	0.619143i	$-0.787480\pi$
$241$	−5875.96	−1.57056	−0.785278	+	0.619143i	$0.787480\pi$
$242$	1018.12	0.270443
$243$	−2345.23	−0.619121
$244$	−2066.03	−0.542065
$245$	706.368	0.184197
$246$	−12290.0	−3.18528
$247$	0	0
$248$	−2986.00	−0.764562
$249$	661.383	0.168327
$250$	3100.97	0.784490
$251$	5806.27	1.46011	0.730057	−	0.683386i	$-0.239494\pi$
$251$	5806.27	1.46011	0.730057	+	0.683386i	$0.239494\pi$
$252$	−5930.02	−1.48237
$253$	−176.173	−0.0437783
$254$	2832.24	0.699647
$255$	−49.1385	−0.0120673
$256$	−3841.83	−0.937947
$257$	−1195.86	−0.290256	−0.145128	−	0.989413i	$-0.546359\pi$
$257$	−1195.86	−0.290256	−0.145128	+	0.989413i	$0.546359\pi$
$258$	16934.5	4.08641
$259$	−1775.66	−0.426001
$260$	0	0
$261$	6812.83	1.61572
$262$	−6074.64	−1.43241
$263$	234.184	0.0549066	0.0274533	−	0.999623i	$-0.491260\pi$
$263$	234.184	0.0549066	0.0274533	+	0.999623i	$0.491260\pi$
$264$	−7508.64	−1.75047
$265$	−1720.19	−0.398757
$266$	−2625.32	−0.605145
$267$	−1761.30	−0.403707
$268$	639.091	0.145667
$269$	−2668.27	−0.604785	−0.302393	−	0.953183i	$-0.597785\pi$
$269$	−2668.27	−0.604785	−0.302393	+	0.953183i	$0.597785\pi$
$270$	−2382.94	−0.537114
$271$	−5701.28	−1.27796	−0.638982	−	0.769222i	$-0.720644\pi$
$271$	−5701.28	−1.27796	−0.638982	+	0.769222i	$0.720644\pi$
$272$	−4.88331	−0.00108858
$273$	0	0
$274$	−2837.35	−0.625587
$275$	4617.12	1.01245
$276$	497.065	0.108405
$277$	−7152.49	−1.55145	−0.775725	−	0.631072i	$-0.782615\pi$
$277$	−7152.49	−1.55145	−0.775725	+	0.631072i	$0.782615\pi$
$278$	1507.96	0.325329
$279$	−6593.09	−1.41476
$280$	588.769	0.125663
$281$	6132.87	1.30198	0.650990	−	0.759086i	$-0.274354\pi$
$281$	6132.87	1.30198	0.650990	+	0.759086i	$0.274354\pi$
$282$	10245.3	2.16346
$283$	3377.15	0.709367	0.354683	−	0.934986i	$-0.384589\pi$
$283$	3377.15	0.709367	0.354683	+	0.934986i	$0.384589\pi$
$284$	−3585.70	−0.749198
$285$	−1467.76	−0.305061
$286$	0	0
$287$	2966.29	0.610086
$288$	−9030.99	−1.84776
$289$	−4908.94	−0.999173
$290$	−1801.96	−0.364879
$291$	10199.0	2.05455
$292$	−5990.60	−1.20059
$293$	4704.77	0.938073	0.469037	−	0.883179i	$-0.344601\pi$
$293$	4704.77	0.938073	0.469037	+	0.883179i	$0.344601\pi$
$294$	−9966.34	−1.97704
$295$	−1454.10	−0.286986
$296$	4072.75	0.799742
$297$	−7334.86	−1.43304
$298$	8258.86	1.60545
$299$	0	0
$300$	−13027.0	−2.50704
$301$	−4087.28	−0.782681
$302$	1930.62	0.367864
$303$	8430.17	1.59835
$304$	−145.863	−0.0275192
$305$	452.922	0.0850303
$306$	445.118	0.0831560
$307$	−5130.49	−0.953787	−0.476894	−	0.878961i	$-0.658237\pi$
$307$	−5130.49	−0.953787	−0.476894	+	0.878961i	$0.658237\pi$
$308$	4827.86	0.893159
$309$	−16498.2	−3.03738
$310$	1743.84	0.319496
$311$	7948.94	1.44933	0.724667	−	0.689099i	$-0.241994\pi$
$311$	7948.94	1.44933	0.724667	+	0.689099i	$0.241994\pi$
$312$	0	0
$313$	−8521.87	−1.53893	−0.769465	−	0.638689i	$-0.779477\pi$
$313$	−8521.87	−1.53893	−0.769465	+	0.638689i	$0.779477\pi$
$314$	6031.14	1.08394
$315$	1300.00	0.232529
$316$	480.776	0.0855879
$317$	6662.46	1.18044	0.590222	−	0.807241i	$-0.299040\pi$
$317$	6662.46	1.18044	0.590222	+	0.807241i	$0.299040\pi$
$318$	24270.7	4.27998
$319$	−5546.59	−0.973509
$320$	2334.23	0.407773
$321$	−16557.2	−2.87893
$322$	−194.907	−0.0337322
$323$	121.297	0.0208951
$324$	3949.74	0.677253
$325$	0	0
$326$	16450.8	2.79486
$327$	7781.49	1.31595
$328$	−6803.64	−1.14533
$329$	−2472.78	−0.414374
$330$	4385.09	0.731489
$331$	3911.77	0.649578	0.324789	−	0.945786i	$-0.394707\pi$
$331$	3911.77	0.649578	0.324789	+	0.945786i	$0.394707\pi$
$332$	975.379	0.161238
$333$	8992.61	1.47986
$334$	15579.7	2.55234
$335$	−140.104	−0.0228498
$336$	201.227	0.0326722
$337$	−627.211	−0.101384	−0.0506919	−	0.998714i	$-0.516143\pi$
$337$	−627.211	−0.101384	−0.0506919	+	0.998714i	$0.516143\pi$
$338$	0	0
$339$	−2908.34	−0.465957
$340$	−72.4674	−0.0115591
$341$	5367.69	0.852424
$342$	13295.6	2.10217
$343$	5685.08	0.894943
$344$	9374.80	1.46935
$345$	−108.968	−0.0170048
$346$	10684.2	1.66007
$347$	−3823.02	−0.591442	−0.295721	−	0.955274i	$-0.595560\pi$
$347$	−3823.02	−0.591442	−0.295721	+	0.955274i	$0.595560\pi$
$348$	15649.4	2.41063
$349$	−3410.67	−0.523120	−0.261560	−	0.965187i	$-0.584237\pi$
$349$	−3410.67	−0.523120	−0.261560	+	0.965187i	$0.584237\pi$
$350$	5108.10	0.780112
$351$	0	0
$352$	7352.48	1.11332
$353$	5587.64	0.842492	0.421246	−	0.906946i	$-0.361593\pi$
$353$	5587.64	0.842492	0.421246	+	0.906946i	$0.361593\pi$
$354$	20516.3	3.08031
$355$	786.070	0.117522
$356$	−2597.49	−0.386704
$357$	−167.336	−0.0248077
$358$	−3042.06	−0.449100
$359$	−2230.14	−0.327861	−0.163931	−	0.986472i	$-0.552417\pi$
$359$	−2230.14	−0.327861	−0.163931	+	0.986472i	$0.552417\pi$
$360$	−2981.75	−0.436533
$361$	−3235.89	−0.471774
$362$	−3197.82	−0.464292
$363$	1938.38	0.280272
$364$	0	0
$365$	1313.28	0.188330
$366$	−6390.40	−0.912655
$367$	−8699.14	−1.23731	−0.618653	−	0.785664i	$-0.712321\pi$
$367$	−8699.14	−1.23731	−0.618653	+	0.785664i	$0.712321\pi$
$368$	−10.8291	−0.00153398
$369$	−15022.4	−2.11934
$370$	−2378.51	−0.334197
$371$	−5857.94	−0.819756
$372$	−15144.7	−2.11080
$373$	10964.2	1.52199	0.760997	−	0.648755i	$-0.224710\pi$
$373$	10964.2	1.52199	0.760997	+	0.648755i	$0.224710\pi$
$374$	−362.388	−0.0501033
$375$	5903.88	0.813000
$376$	5671.70	0.777914
$377$	0	0
$378$	−8114.84	−1.10419
$379$	−13910.1	−1.88526	−0.942631	−	0.333835i	$-0.891657\pi$
$379$	−13910.1	−1.88526	−0.942631	+	0.333835i	$0.891657\pi$
$380$	−2164.58	−0.292213
$381$	5392.24	0.725074
$382$	5934.03	0.794794
$383$	494.753	0.0660071	0.0330035	−	0.999455i	$-0.489493\pi$
$383$	494.753	0.0660071	0.0330035	+	0.999455i	$0.489493\pi$
$384$	−19976.7	−2.65477
$385$	−1058.38	−0.140104
$386$	2368.97	0.312376
$387$	20699.5	2.71891
$388$	15041.0	1.96802
$389$	−4140.47	−0.539666	−0.269833	−	0.962907i	$-0.586968\pi$
$389$	−4140.47	−0.539666	−0.269833	+	0.962907i	$0.586968\pi$
$390$	0	0
$391$	9.00524	0.00116474
$392$	−5517.30	−0.710881
$393$	−11565.4	−1.48447
$394$	−14236.8	−1.82041
$395$	−105.398	−0.0134256
$396$	−24450.1	−3.10269
$397$	1881.79	0.237895	0.118948	−	0.992901i	$-0.462048\pi$
$397$	1881.79	0.237895	0.118948	+	0.992901i	$0.462048\pi$
$398$	5642.90	0.710686
$399$	−4998.29	−0.627137
$400$	283.807	0.0354759
$401$	−421.765	−0.0525236	−0.0262618	−	0.999655i	$-0.508360\pi$
$401$	−421.765	−0.0525236	−0.0262618	+	0.999655i	$0.508360\pi$
$402$	1976.76	0.245254
$403$	0	0
$404$	12432.5	1.53103
$405$	−865.876	−0.106236
$406$	−6136.41	−0.750110
$407$	−7321.23	−0.891646
$408$	383.811	0.0465722
$409$	−2550.22	−0.308313	−0.154157	−	0.988046i	$-0.549266\pi$
$409$	−2550.22	−0.308313	−0.154157	+	0.988046i	$0.549266\pi$
$410$	3973.37	0.478611
$411$	−5401.99	−0.648322
$412$	−24330.9	−2.90946
$413$	−4951.79	−0.589980
$414$	987.083	0.117180
$415$	−213.826	−0.0252923
$416$	0	0
$417$	2870.98	0.337152
$418$	−10824.5	−1.26661
$419$	12384.8	1.44400	0.722002	−	0.691891i	$-0.243222\pi$
$419$	12384.8	1.44400	0.722002	+	0.691891i	$0.243222\pi$
$420$	2986.17	0.346929
$421$	−10463.0	−1.21124	−0.605622	−	0.795752i	$-0.707076\pi$
$421$	−10463.0	−1.21124	−0.605622	+	0.795752i	$0.707076\pi$
$422$	−11548.3	−1.33214
$423$	12523.1	1.43947
$424$	13436.1	1.53895
$425$	−236.008	−0.0269366
$426$	−11090.9	−1.26140
$427$	1542.38	0.174803
$428$	−24417.9	−2.75767
$429$	0	0
$430$	−5474.94	−0.614012
$431$	3962.39	0.442834	0.221417	−	0.975179i	$-0.428932\pi$
$431$	3962.39	0.442834	0.221417	+	0.975179i	$0.428932\pi$
$432$	−450.863	−0.0502133
$433$	−8394.14	−0.931632	−0.465816	−	0.884882i	$-0.654239\pi$
$433$	−8394.14	−0.931632	−0.465816	+	0.884882i	$0.654239\pi$
$434$	5938.48	0.656812
$435$	−3430.73	−0.378140
$436$	11475.8	1.26053
$437$	268.984	0.0294446
$438$	−18529.4	−2.02140
$439$	10174.5	1.10616	0.553079	−	0.833129i	$-0.313453\pi$
$439$	10174.5	1.10616	0.553079	+	0.833129i	$0.313453\pi$
$440$	2427.56	0.263021
$441$	−12182.2	−1.31543
$442$	0	0
$443$	−5880.74	−0.630705	−0.315353	−	0.948975i	$-0.602123\pi$
$443$	−5880.74	−0.630705	−0.315353	+	0.948975i	$0.602123\pi$
$444$	20656.5	2.20792
$445$	569.431	0.0606598
$446$	5453.48	0.578990
$447$	15723.9	1.66379
$448$	7948.97	0.838289
$449$	−10664.9	−1.12095	−0.560475	−	0.828172i	$-0.689381\pi$
$449$	−10664.9	−1.12095	−0.560475	+	0.828172i	$0.689381\pi$
$450$	−25869.3	−2.70998
$451$	12230.3	1.27695
$452$	−4289.10	−0.446332
$453$	3675.68	0.381233
$454$	3964.87	0.409869
$455$	0	0
$456$	11464.3	1.17734
$457$	14828.9	1.51787	0.758933	−	0.651169i	$-0.225721\pi$
$457$	14828.9	1.51787	0.758933	+	0.651169i	$0.225721\pi$
$458$	−21369.2	−2.18017
$459$	374.928	0.0381266
$460$	−160.702	−0.0162886
$461$	9711.70	0.981169	0.490585	−	0.871394i	$-0.336783\pi$
$461$	9711.70	0.981169	0.490585	+	0.871394i	$0.336783\pi$
$462$	14933.0	1.50378
$463$	−11353.5	−1.13962	−0.569809	−	0.821777i	$-0.692983\pi$
$463$	−11353.5	−1.13962	−0.569809	+	0.821777i	$0.692983\pi$
$464$	−340.941	−0.0341116
$465$	3320.07	0.331107
$466$	−22210.1	−2.20787
$467$	6451.31	0.639252	0.319626	−	0.947544i	$-0.396443\pi$
$467$	6451.31	0.639252	0.319626	+	0.947544i	$0.396443\pi$
$468$	0	0
$469$	−477.109	−0.0469741
$470$	−3312.31	−0.325076
$471$	11482.6	1.12333
$472$	11357.7	1.10758
$473$	−16852.3	−1.63820
$474$	1487.08	0.144101
$475$	−7049.50	−0.680954
$476$	−246.780	−0.0237629
$477$	29666.8	2.84770
$478$	−21929.8	−2.09842
$479$	9566.46	0.912531	0.456266	−	0.889844i	$-0.349187\pi$
$479$	9566.46	0.912531	0.456266	+	0.889844i	$0.349187\pi$
$480$	4547.73	0.432447
$481$	0	0
$482$	−26803.5	−2.53292
$483$	−371.080	−0.0349581
$484$	2858.64	0.268467
$485$	−3297.35	−0.308711
$486$	−10697.9	−0.998489
$487$	−4917.11	−0.457527	−0.228764	−	0.973482i	$-0.573468\pi$
$487$	−4917.11	−0.457527	−0.228764	+	0.973482i	$0.573468\pi$
$488$	−3537.68	−0.328162
$489$	31320.3	2.89643
$490$	3222.14	0.297064
$491$	2950.82	0.271220	0.135610	−	0.990762i	$-0.456701\pi$
$491$	2950.82	0.271220	0.135610	+	0.990762i	$0.456701\pi$
$492$	−34507.3	−3.16201
$493$	283.519	0.0259007
$494$	0	0
$495$	5360.04	0.486699
$496$	329.944	0.0298688
$497$	2676.88	0.241599
$498$	3016.93	0.271470
$499$	−13430.1	−1.20484	−0.602418	−	0.798180i	$-0.705796\pi$
$499$	−13430.1	−1.20484	−0.602418	+	0.798180i	$0.705796\pi$
$500$	8706.79	0.778759
$501$	29661.9	2.64510
$502$	26485.6	2.35480
$503$	1320.29	0.117035	0.0585175	−	0.998286i	$-0.481363\pi$
$503$	1320.29	0.117035	0.0585175	+	0.998286i	$0.481363\pi$
$504$	−10154.0	−0.897414
$505$	−2725.49	−0.240164
$506$	−803.623	−0.0706036
$507$	0	0
$508$	7952.25	0.694536
$509$	−20916.4	−1.82143	−0.910713	−	0.413041i	$-0.864467\pi$
$509$	−20916.4	−1.82143	−0.910713	+	0.413041i	$0.864467\pi$
$510$	−224.148	−0.0194616
$511$	4472.24	0.387163
$512$	877.105	0.0757089
$513$	11199.0	0.963837
$514$	−5454.98	−0.468110
$515$	5333.90	0.456388
$516$	47548.0	4.05656
$517$	−10195.5	−0.867311
$518$	−8099.77	−0.687033
$519$	20341.4	1.72040
$520$	0	0
$521$	−10104.2	−0.849661	−0.424831	−	0.905273i	$-0.639666\pi$
$521$	−10104.2	−0.849661	−0.424831	+	0.905273i	$0.639666\pi$
$522$	31077.1	2.60576
$523$	7131.22	0.596227	0.298113	−	0.954530i	$-0.403643\pi$
$523$	7131.22	0.596227	0.298113	+	0.954530i	$0.403643\pi$
$524$	−17056.2	−1.42195
$525$	9725.21	0.808463
$526$	1068.24	0.0885507
$527$	−274.374	−0.0226792
$528$	829.682	0.0683849
$529$	−12147.0	−0.998359
$530$	−7846.76	−0.643097
$531$	25077.7	2.04949
$532$	−7371.27	−0.600724
$533$	0	0
$534$	−8034.26	−0.651080
$535$	5352.98	0.432579
$536$	1094.32	0.0881856
$537$	−5791.72	−0.465421
$538$	−12171.5	−0.975369
$539$	9917.98	0.792575
$540$	−6690.72	−0.533190
$541$	−16831.7	−1.33762	−0.668809	−	0.743435i	$-0.733195\pi$
$541$	−16831.7	−1.33762	−0.668809	+	0.743435i	$0.733195\pi$
$542$	−26006.7	−2.06104
$543$	−6088.27	−0.481165
$544$	−375.828	−0.0296204
$545$	−2515.77	−0.197731
$546$	0	0
$547$	−9560.55	−0.747312	−0.373656	−	0.927567i	$-0.621896\pi$
$547$	−9560.55	−0.747312	−0.373656	+	0.927567i	$0.621896\pi$
$548$	−7966.62	−0.621017
$549$	−7811.19	−0.607238
$550$	21061.2	1.63282
$551$	8468.64	0.654766
$552$	851.129	0.0656276
$553$	−358.920	−0.0276001
$554$	−32626.5	−2.50210
$555$	−4528.40	−0.346342
$556$	4234.00	0.322952
$557$	22827.9	1.73653	0.868267	−	0.496097i	$-0.165234\pi$
$557$	22827.9	1.73653	0.868267	+	0.496097i	$0.165234\pi$
$558$	−30074.7	−2.28166
$559$	0	0
$560$	−65.0571	−0.00490922
$561$	−689.944	−0.0519242
$562$	27975.4	2.09977
$563$	21629.7	1.61916	0.809578	−	0.587013i	$-0.199696\pi$
$563$	21629.7	1.61916	0.809578	+	0.587013i	$0.199696\pi$
$564$	28766.3	2.14766
$565$	940.271	0.0700133
$566$	15405.1	1.14403
$567$	−2948.65	−0.218398
$568$	−6139.83	−0.453559
$569$	−10589.9	−0.780229	−0.390114	−	0.920766i	$-0.627564\pi$
$569$	−10589.9	−0.780229	−0.390114	+	0.920766i	$0.627564\pi$
$570$	−6695.25	−0.491988
$571$	−1757.27	−0.128791	−0.0643954	−	0.997924i	$-0.520512\pi$
$571$	−1757.27	−0.128791	−0.0643954	+	0.997924i	$0.520512\pi$
$572$	0	0
$573$	11297.7	0.823679
$574$	13530.9	0.983917
$575$	−523.365	−0.0379580
$576$	−40256.6	−2.91208
$577$	13580.6	0.979840	0.489920	−	0.871767i	$-0.337026\pi$
$577$	13580.6	0.979840	0.489920	+	0.871767i	$0.337026\pi$
$578$	−22392.4	−1.61142
$579$	4510.23	0.323729
$580$	−5059.49	−0.362214
$581$	−728.163	−0.0519953
$582$	46523.2	3.31349
$583$	−24152.9	−1.71580
$584$	−10257.8	−0.726831
$585$	0	0
$586$	21461.0	1.51288
$587$	957.326	0.0673136	0.0336568	−	0.999433i	$-0.489285\pi$
$587$	957.326	0.0673136	0.0336568	+	0.999433i	$0.489285\pi$
$588$	−27983.1	−1.96259
$589$	−8195.49	−0.573327
$590$	−6632.95	−0.462838
$591$	−27105.2	−1.88657
$592$	−450.026	−0.0312431
$593$	−6729.49	−0.466015	−0.233007	−	0.972475i	$-0.574857\pi$
$593$	−6729.49	−0.466015	−0.233007	+	0.972475i	$0.574857\pi$
$594$	−33458.4	−2.31113
$595$	54.1000	0.00372754
$596$	23188.9	1.59372
$597$	10743.4	0.736514
$598$	0	0
$599$	2281.52	0.155626	0.0778132	−	0.996968i	$-0.475206\pi$
$599$	2281.52	0.155626	0.0778132	+	0.996968i	$0.475206\pi$
$600$	−22306.2	−1.51775
$601$	6401.42	0.434475	0.217237	−	0.976119i	$-0.430295\pi$
$601$	6401.42	0.434475	0.217237	+	0.976119i	$0.430295\pi$
$602$	−18644.4	−1.26227
$603$	2416.26	0.163180
$604$	5420.74	0.365177
$605$	−626.682	−0.0421128
$606$	38454.7	2.57775
$607$	2779.24	0.185841	0.0929207	−	0.995674i	$-0.470380\pi$
$607$	2779.24	0.185841	0.0929207	+	0.995674i	$0.470380\pi$
$608$	−11225.9	−0.748800
$609$	−11683.0	−0.777371
$610$	2066.03	0.137133
$611$	0	0
$612$	1249.79	0.0825485
$613$	−22620.8	−1.49045	−0.745226	−	0.666812i	$-0.767658\pi$
$613$	−22620.8	−1.49045	−0.745226	+	0.666812i	$0.767658\pi$
$614$	−23403.0	−1.53822
$615$	7564.82	0.496005
$616$	8266.80	0.540712
$617$	−21974.0	−1.43378	−0.716889	−	0.697187i	$-0.754435\pi$
$617$	−21974.0	−1.43378	−0.716889	+	0.697187i	$0.754435\pi$
$618$	−75257.5	−4.89854
$619$	−7145.19	−0.463957	−0.231979	−	0.972721i	$-0.574520\pi$
$619$	−7145.19	−0.463957	−0.231979	+	0.972721i	$0.574520\pi$
$620$	4896.30	0.317162
$621$	831.430	0.0537265
$622$	36259.5	2.33742
$623$	1939.14	0.124703
$624$	0	0
$625$	12730.8	0.814773
$626$	−38873.0	−2.48191
$627$	−20608.5	−1.31264
$628$	16934.0	1.07602
$629$	374.231	0.0237227
$630$	5930.02	0.375012
$631$	18883.2	1.19133	0.595666	−	0.803232i	$-0.296888\pi$
$631$	18883.2	1.19133	0.595666	+	0.803232i	$0.296888\pi$
$632$	823.239	0.0518144
$633$	−21986.7	−1.38056
$634$	30391.1	1.90376
$635$	−1743.32	−0.108947
$636$	68146.4	4.24871
$637$	0	0
$638$	−25301.1	−1.57003
$639$	−13556.7	−0.839274
$640$	6458.50	0.398898
$641$	−3631.08	−0.223743	−0.111871	−	0.993723i	$-0.535684\pi$
$641$	−3631.08	−0.223743	−0.111871	+	0.993723i	$0.535684\pi$
$642$	−75526.7	−4.64299
$643$	10772.0	0.660660	0.330330	−	0.943866i	$-0.392840\pi$
$643$	10772.0	0.660660	0.330330	+	0.943866i	$0.392840\pi$
$644$	−547.253	−0.0334857
$645$	−10423.6	−0.636327
$646$	553.301	0.0336987
$647$	15148.3	0.920464	0.460232	−	0.887799i	$-0.347766\pi$
$647$	15148.3	0.920464	0.460232	+	0.887799i	$0.347766\pi$
$648$	6763.18	0.410004
$649$	−20416.7	−1.23487
$650$	0	0
$651$	11306.2	0.680682
$652$	46189.8	2.77444
$653$	7358.89	0.441004	0.220502	−	0.975387i	$-0.429230\pi$
$653$	7358.89	0.441004	0.220502	+	0.975387i	$0.429230\pi$
$654$	35495.7	2.12231
$655$	3739.11	0.223052
$656$	751.780	0.0447441
$657$	−22649.1	−1.34494
$658$	−11279.7	−0.668282
$659$	28333.3	1.67482	0.837411	−	0.546574i	$-0.184068\pi$
$659$	28333.3	1.67482	0.837411	+	0.546574i	$0.184068\pi$
$660$	12312.3	0.726146
$661$	−1109.68	−0.0652975	−0.0326488	−	0.999467i	$-0.510394\pi$
$661$	−1109.68	−0.0652975	−0.0326488	+	0.999467i	$0.510394\pi$
$662$	17843.8	1.04761
$663$	0	0
$664$	1670.15	0.0976121
$665$	1615.96	0.0942317
$666$	41020.3	2.38664
$667$	628.724	0.0364982
$668$	43744.1	2.53369
$669$	10382.8	0.600032
$670$	−639.091	−0.0368511
$671$	6359.39	0.365874
$672$	15486.8	0.889013
$673$	20979.1	1.20161	0.600806	−	0.799395i	$-0.294846\pi$
$673$	20979.1	1.20161	0.600806	+	0.799395i	$0.294846\pi$
$674$	−2861.06	−0.163507
$675$	−21790.0	−1.24251
$676$	0	0
$677$	30941.9	1.75656	0.878282	−	0.478142i	$-0.158690\pi$
$677$	30941.9	1.75656	0.878282	+	0.478142i	$0.158690\pi$
$678$	−13266.5	−0.751473
$679$	−11228.8	−0.634641
$680$	−124.087	−0.00699780
$681$	7548.64	0.424764
$682$	24485.0	1.37475
$683$	−5426.21	−0.303995	−0.151997	−	0.988381i	$-0.548571\pi$
$683$	−5426.21	−0.303995	−0.151997	+	0.988381i	$0.548571\pi$
$684$	37330.9	2.08682
$685$	1746.47	0.0974150
$686$	25932.8	1.44332
$687$	−40684.5	−2.25940
$688$	−1035.89	−0.0574023
$689$	0	0
$690$	−497.065	−0.0274245
$691$	33792.7	1.86040	0.930199	−	0.367056i	$-0.119634\pi$
$691$	33792.7	1.86040	0.930199	+	0.367056i	$0.119634\pi$
$692$	29998.7	1.64795
$693$	18253.1	1.00054
$694$	−17438.9	−0.953849
$695$	−928.192	−0.0506595
$696$	26796.7	1.45938
$697$	−625.164	−0.0339738
$698$	−15557.9	−0.843662
$699$	−42285.5	−2.28810
$700$	14342.3	0.774413
$701$	6905.96	0.372089	0.186045	−	0.982541i	$-0.440433\pi$
$701$	6905.96	0.372089	0.186045	+	0.982541i	$0.440433\pi$
$702$	0	0
$703$	11178.2	0.599707
$704$	32774.5	1.75459
$705$	−6306.25	−0.336889
$706$	25488.3	1.35873
$707$	−9281.37	−0.493723
$708$	57604.9	3.05781
$709$	2007.13	0.106318	0.0531589	−	0.998586i	$-0.483071\pi$
$709$	2007.13	0.106318	0.0531589	+	0.998586i	$0.483071\pi$
$710$	3585.70	0.189534
$711$	1817.71	0.0958782
$712$	−4447.71	−0.234108
$713$	−608.445	−0.0319585
$714$	−763.312	−0.0400088
$715$	0	0
$716$	−8541.38	−0.445819
$717$	−41751.8	−2.17469
$718$	−10172.9	−0.528759
$719$	−12787.4	−0.663270	−0.331635	−	0.943408i	$-0.607600\pi$
$719$	−12787.4	−0.663270	−0.331635	+	0.943408i	$0.607600\pi$
$720$	329.474	0.0170538
$721$	18164.1	0.938231
$722$	−14760.7	−0.760854
$723$	−51030.7	−2.62497
$724$	−8978.72	−0.460900
$725$	−16477.5	−0.844081
$726$	8842.03	0.452009
$727$	−6090.70	−0.310717	−0.155359	−	0.987858i	$-0.549653\pi$
$727$	−6090.70	−0.310717	−0.155359	+	0.987858i	$0.549653\pi$
$728$	0	0
$729$	−28693.9	−1.45780
$730$	5990.60	0.303729
$731$	861.420	0.0435852
$732$	−17942.7	−0.905988
$733$	38846.5	1.95747	0.978737	−	0.205117i	$-0.0657574\pi$
$733$	38846.5	1.95747	0.978737	+	0.205117i	$0.0657574\pi$
$734$	−39681.6	−1.99547
$735$	6134.57	0.307860
$736$	−833.427	−0.0417399
$737$	−1967.17	−0.0983198
$738$	−68525.5	−3.41797
$739$	−14457.5	−0.719661	−0.359830	−	0.933018i	$-0.617165\pi$
$739$	−14457.5	−0.719661	−0.359830	+	0.933018i	$0.617165\pi$
$740$	−6678.29	−0.331755
$741$	0	0
$742$	−26721.3	−1.32206
$743$	1277.80	0.0630929	0.0315464	−	0.999502i	$-0.489957\pi$
$743$	1277.80	0.0630929	0.0315464	+	0.999502i	$0.489957\pi$
$744$	−25932.4	−1.27786
$745$	−5083.56	−0.249996
$746$	50013.7	2.45460
$747$	3687.69	0.180623
$748$	−1017.50	−0.0497373
$749$	18229.0	0.889285
$750$	26930.9	1.31117
$751$	−13007.9	−0.632042	−0.316021	−	0.948752i	$-0.602347\pi$
$751$	−13007.9	−0.632042	−0.316021	+	0.948752i	$0.602347\pi$
$752$	−626.706	−0.0303904
$753$	50425.5	2.44038
$754$	0	0
$755$	−1188.35	−0.0572829
$756$	−22784.6	−1.09612
$757$	−10723.2	−0.514850	−0.257425	−	0.966298i	$-0.582874\pi$
$757$	−10723.2	−0.514850	−0.257425	+	0.966298i	$0.582874\pi$
$758$	−63451.7	−3.04046
$759$	−1530.00	−0.0731694
$760$	−3706.44	−0.176904
$761$	−13621.8	−0.648870	−0.324435	−	0.945908i	$-0.605174\pi$
$761$	−13621.8	−0.648870	−0.324435	+	0.945908i	$0.605174\pi$
$762$	24597.0	1.16936
$763$	−8567.19	−0.406491
$764$	16661.4	0.788988
$765$	−273.983	−0.0129489
$766$	2256.84	0.106453
$767$	0	0
$768$	−33365.0	−1.56765
$769$	8495.15	0.398365	0.199183	−	0.979962i	$-0.436171\pi$
$769$	8495.15	0.398365	0.199183	+	0.979962i	$0.436171\pi$
$770$	−4827.86	−0.225953
$771$	−10385.6	−0.485123
$772$	6651.50	0.310094
$773$	−34262.5	−1.59423	−0.797113	−	0.603830i	$-0.793641\pi$
$773$	−34262.5	−1.59423	−0.797113	+	0.603830i	$0.793641\pi$
$774$	94422.0	4.38492
$775$	15946.0	0.739094
$776$	25754.9	1.19143
$777$	−15421.0	−0.712002
$778$	−18887.0	−0.870348
$779$	−18673.5	−0.858854
$780$	0	0
$781$	11037.1	0.505681
$782$	41.0779	0.00187844
$783$	26176.5	1.19473
$784$	609.644	0.0277717
$785$	−3712.33	−0.168788
$786$	−52756.2	−2.39408
$787$	12642.6	0.572629	0.286315	−	0.958136i	$-0.407570\pi$
$787$	12642.6	0.572629	0.286315	+	0.958136i	$0.407570\pi$
$788$	−39973.6	−1.80711
$789$	2033.81	0.0917688
$790$	−480.776	−0.0216522
$791$	3202.00	0.143932
$792$	−41866.2	−1.87834
$793$	0	0
$794$	8583.89	0.383666
$795$	−14939.3	−0.666468
$796$	15843.9	0.705494
$797$	19084.4	0.848184	0.424092	−	0.905619i	$-0.360593\pi$
$797$	19084.4	0.848184	0.424092	+	0.905619i	$0.360593\pi$
$798$	−22800.0	−1.01142
$799$	521.154	0.0230752
$800$	21842.3	0.965304
$801$	−9820.53	−0.433198
$802$	−1923.90	−0.0847075
$803$	18439.5	0.810357
$804$	5550.28	0.243462
$805$	119.971	0.00525269
$806$	0	0
$807$	−23173.0	−1.01082
$808$	21288.2	0.926878
$809$	11610.0	0.504558	0.252279	−	0.967655i	$-0.418820\pi$
$809$	11610.0	0.504558	0.252279	+	0.967655i	$0.418820\pi$
$810$	−3949.74	−0.171333
$811$	−9613.36	−0.416240	−0.208120	−	0.978103i	$-0.566734\pi$
$811$	−9613.36	−0.416240	−0.208120	+	0.978103i	$0.566734\pi$
$812$	−17229.6	−0.744630
$813$	−49513.7	−2.13594
$814$	−33396.2	−1.43800
$815$	−10125.9	−0.435208
$816$	−42.4099	−0.00181941
$817$	25730.4	1.10183
$818$	−11632.9	−0.497233
$819$	0	0
$820$	11156.3	0.475115
$821$	26481.5	1.12571	0.562856	−	0.826555i	$-0.309703\pi$
$821$	26481.5	1.12571	0.562856	+	0.826555i	$0.309703\pi$
$822$	−24641.5	−1.04558
$823$	13814.5	0.585107	0.292553	−	0.956249i	$-0.405495\pi$
$823$	13814.5	0.585107	0.292553	+	0.956249i	$0.405495\pi$
$824$	−41662.0	−1.76136
$825$	40098.1	1.69216
$826$	−22587.8	−0.951491
$827$	44401.0	1.86696	0.933479	−	0.358633i	$-0.116757\pi$
$827$	44401.0	1.86696	0.933479	+	0.358633i	$0.116757\pi$
$828$	2771.50	0.116324
$829$	24337.4	1.01963	0.509815	−	0.860284i	$-0.329714\pi$
$829$	24337.4	1.01963	0.509815	+	0.860284i	$0.329714\pi$
$830$	−975.379	−0.0407902
$831$	−62116.9	−2.59303
$832$	0	0
$833$	−506.966	−0.0210868
$834$	13096.1	0.543743
$835$	−9589.73	−0.397445
$836$	−30392.5	−1.25735
$837$	−25332.2	−1.04613
$838$	56494.0	2.32882
$839$	24680.1	1.01556	0.507778	−	0.861488i	$-0.330467\pi$
$839$	24680.1	1.01556	0.507778	+	0.861488i	$0.330467\pi$
$840$	5113.26	0.210029
$841$	−4594.43	−0.188381
$842$	−47727.4	−1.95344
$843$	53261.9	2.17608
$844$	−32425.0	−1.32241
$845$	0	0
$846$	57124.8	2.32150
$847$	−2134.10	−0.0865744
$848$	−1484.65	−0.0601214
$849$	29329.4	1.18561
$850$	−1076.56	−0.0434421
$851$	829.885	0.0334290
$852$	−31140.6	−1.25218
$853$	10151.7	0.407490	0.203745	−	0.979024i	$-0.434689\pi$
$853$	10151.7	0.407490	0.203745	+	0.979024i	$0.434689\pi$
$854$	7035.65	0.281914
$855$	−8183.81	−0.327345
$856$	−41811.1	−1.66948
$857$	2028.92	0.0808713	0.0404357	−	0.999182i	$-0.487125\pi$
$857$	2028.92	0.0808713	0.0404357	+	0.999182i	$0.487125\pi$
$858$	0	0
$859$	6655.76	0.264367	0.132184	−	0.991225i	$-0.457801\pi$
$859$	6655.76	0.264367	0.132184	+	0.991225i	$0.457801\pi$
$860$	−15372.3	−0.609526
$861$	25761.2	1.01967
$862$	18074.6	0.714182
$863$	−45690.8	−1.80224	−0.901121	−	0.433568i	$-0.857254\pi$
$863$	−45690.8	−1.80224	−0.901121	+	0.433568i	$0.857254\pi$
$864$	−34699.2	−1.36631
$865$	−6576.42	−0.258503
$866$	−38290.3	−1.50249
$867$	−42632.5	−1.66998
$868$	16673.9	0.652014
$869$	−1479.87	−0.0577688
$870$	−15649.4	−0.609846
$871$	0	0
$872$	19650.1	0.763117
$873$	56866.8	2.20464
$874$	1226.99	0.0474868
$875$	−6500.00	−0.251132
$876$	−52026.3	−2.00663
$877$	−30447.5	−1.17234	−0.586168	−	0.810189i	$-0.699364\pi$
$877$	−30447.5	−1.17234	−0.586168	+	0.810189i	$0.699364\pi$
$878$	46411.6	1.78396
$879$	40859.3	1.56786
$880$	−268.237	−0.0102753
$881$	32542.0	1.24446	0.622230	−	0.782835i	$-0.286227\pi$
$881$	32542.0	1.24446	0.622230	+	0.782835i	$0.286227\pi$
$882$	−55569.6	−2.12146
$883$	27641.9	1.05348	0.526741	−	0.850026i	$-0.323414\pi$
$883$	27641.9	1.05348	0.526741	+	0.850026i	$0.323414\pi$
$884$	0	0
$885$	−12628.4	−0.479658
$886$	−26825.3	−1.01717
$887$	40099.9	1.51795	0.758976	−	0.651119i	$-0.225700\pi$
$887$	40099.9	1.51795	0.758976	+	0.651119i	$0.225700\pi$
$888$	35370.4	1.33666
$889$	−5936.70	−0.223971
$890$	2597.49	0.0978293
$891$	−12157.6	−0.457121
$892$	15312.1	0.574761
$893$	15566.8	0.583339
$894$	71725.4	2.68328
$895$	1872.47	0.0699328
$896$	21993.8	0.820044
$897$	0	0
$898$	−48648.4	−1.80781
$899$	−19156.1	−0.710670
$900$	−72634.9	−2.69018
$901$	1234.60	0.0456497
$902$	55789.3	2.05940
$903$	−35496.7	−1.30814
$904$	−7344.26	−0.270206
$905$	1968.35	0.0722984
$906$	16766.8	0.614835
$907$	−36824.9	−1.34812	−0.674062	−	0.738674i	$-0.735452\pi$
$907$	−36824.9	−1.34812	−0.674062	+	0.738674i	$0.735452\pi$
$908$	11132.4	0.406874
$909$	47004.3	1.71511
$910$	0	0
$911$	34520.5	1.25545	0.627725	−	0.778435i	$-0.283986\pi$
$911$	34520.5	1.25545	0.627725	+	0.778435i	$0.283986\pi$
$912$	−1266.77	−0.0459946
$913$	−3002.29	−0.108830
$914$	67642.6	2.44794
$915$	3933.47	0.142117
$916$	−59999.8	−2.16424
$917$	12733.2	0.458545
$918$	1710.25	0.0614888
$919$	23522.8	0.844336	0.422168	−	0.906518i	$-0.361269\pi$
$919$	23522.8	0.844336	0.422168	+	0.906518i	$0.361269\pi$
$920$	−275.171	−0.00986101
$921$	−44556.6	−1.59412
$922$	44300.4	1.58238
$923$	0	0
$924$	41928.3	1.49279
$925$	−21749.5	−0.773102
$926$	−51789.7	−1.83792
$927$	−91989.6	−3.25926
$928$	−26239.4	−0.928180
$929$	24563.2	0.867482	0.433741	−	0.901038i	$-0.357193\pi$
$929$	24563.2	0.867482	0.433741	+	0.901038i	$0.357193\pi$
$930$	15144.7	0.533994
$931$	−15143.0	−0.533073
$932$	−62360.9	−2.19174
$933$	69033.8	2.42236
$934$	29428.0	1.03096
$935$	223.060	0.00780197
$936$	0	0
$937$	−12115.6	−0.422411	−0.211206	−	0.977442i	$-0.567739\pi$
$937$	−12115.6	−0.422411	−0.211206	+	0.977442i	$0.567739\pi$
$938$	−2176.36	−0.0757576
$939$	−74009.6	−2.57211
$940$	−9300.19	−0.322701
$941$	−14898.3	−0.516123	−0.258062	−	0.966128i	$-0.583084\pi$
$941$	−14898.3	−0.516123	−0.258062	+	0.966128i	$0.583084\pi$
$942$	52378.4	1.81166
$943$	−1386.35	−0.0478745
$944$	−1254.99	−0.0432695
$945$	4994.91	0.171941
$946$	−76872.7	−2.64201
$947$	−7434.32	−0.255103	−0.127552	−	0.991832i	$-0.540712\pi$
$947$	−7434.32	−0.255103	−0.127552	+	0.991832i	$0.540712\pi$
$948$	4175.38	0.143049
$949$	0	0
$950$	−32156.7	−1.09821
$951$	57861.2	1.97295
$952$	−422.564	−0.0143859
$953$	−23528.6	−0.799754	−0.399877	−	0.916569i	$-0.630947\pi$
$953$	−23528.6	−0.799754	−0.399877	+	0.916569i	$0.630947\pi$
$954$	135327.	4.59263
$955$	−3652.56	−0.123764
$956$	−61573.8	−2.08309
$957$	−48170.2	−1.62709
$958$	43637.9	1.47169
$959$	5947.43	0.200263
$960$	20272.0	0.681536
$961$	−11252.7	−0.377723
$962$	0	0
$963$	−92318.7	−3.08923
$964$	−75257.9	−2.51441
$965$	−1458.17	−0.0486425
$966$	−1692.70	−0.0563787
$967$	−23558.0	−0.783427	−0.391713	−	0.920087i	$-0.628118\pi$
$967$	−23558.0	−0.783427	−0.391713	+	0.920087i	$0.628118\pi$
$968$	4894.88	0.162528
$969$	1053.42	0.0349234
$970$	−15041.0	−0.497875
$971$	−262.338	−0.00867026	−0.00433513	−	0.999991i	$-0.501380\pi$
$971$	−262.338	−0.00867026	−0.00433513	+	0.999991i	$0.501380\pi$
$972$	−30037.1	−0.991194
$973$	−3160.86	−0.104144
$974$	−22429.7	−0.737878
$975$	0	0
$976$	390.903	0.0128202
$977$	−33144.4	−1.08535	−0.542673	−	0.839944i	$-0.682588\pi$
$977$	−33144.4	−1.08535	−0.542673	+	0.839944i	$0.682588\pi$
$978$	142869.	4.67122
$979$	7995.28	0.261011
$980$	9047.00	0.294894
$981$	43387.5	1.41208
$982$	13460.3	0.437410
$983$	4866.80	0.157911	0.0789557	−	0.996878i	$-0.474841\pi$
$983$	4866.80	0.157911	0.0789557	+	0.996878i	$0.474841\pi$
$984$	−59087.3	−1.91426
$985$	8763.17	0.283470
$986$	1293.28	0.0417714
$987$	−21475.3	−0.692569
$988$	0	0
$989$	1910.26	0.0614184
$990$	24450.1	0.784924
$991$	12533.9	0.401770	0.200885	−	0.979615i	$-0.435618\pi$
$991$	12533.9	0.401770	0.200885	+	0.979615i	$0.435618\pi$
$992$	25393.1	0.812733
$993$	33972.4	1.08568
$994$	12210.7	0.389639
$995$	−3473.37	−0.110666
$996$	8470.83	0.269487
$997$	−3560.92	−0.113115	−0.0565574	−	0.998399i	$-0.518012\pi$
$997$	−3560.92	−0.113115	−0.0565574	+	0.998399i	$0.518012\pi$
$998$	−61262.1	−1.94310
$999$	34551.8	1.09426

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.j.1.2		2
3.2	odd	2		1521.4.a.l.1.1		2
13.2	odd	12		169.4.e.g.147.4		8
13.3	even	3		169.4.c.f.22.1		4
13.4	even	6		13.4.c.b.3.2	✓	4
13.5	odd	4		169.4.b.e.168.1		4
13.6	odd	12		169.4.e.g.23.1		8
13.7	odd	12		169.4.e.g.23.4		8
13.8	odd	4		169.4.b.e.168.4		4
13.9	even	3		169.4.c.f.146.1		4
13.10	even	6		13.4.c.b.9.2	yes	4
13.11	odd	12		169.4.e.g.147.1		8
13.12	even	2		169.4.a.f.1.1		2
39.17	odd	6		117.4.g.d.55.1		4
39.23	odd	6		117.4.g.d.100.1		4
39.38	odd	2		1521.4.a.t.1.2		2
52.23	odd	6		208.4.i.e.113.2		4
52.43	odd	6		208.4.i.e.81.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.c.b.3.2	✓	4	13.4	even	6
13.4.c.b.9.2	yes	4	13.10	even	6
117.4.g.d.55.1		4	39.17	odd	6
117.4.g.d.100.1		4	39.23	odd	6
169.4.a.f.1.1		2	13.12	even	2
169.4.a.j.1.2		2	1.1	even	1	trivial
169.4.b.e.168.1		4	13.5	odd	4
169.4.b.e.168.4		4	13.8	odd	4
169.4.c.f.22.1		4	13.3	even	3
169.4.c.f.146.1		4	13.9	even	3
169.4.e.g.23.1		8	13.6	odd	12
169.4.e.g.23.4		8	13.7	odd	12
169.4.e.g.147.1		8	13.11	odd	12
169.4.e.g.147.4		8	13.2	odd	12
208.4.i.e.81.2		4	52.43	odd	6
208.4.i.e.113.2		4	52.23	odd	6
1521.4.a.l.1.1		2	3.2	odd	2
1521.4.a.t.1.2		2	39.38	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q+4.56155 q^{2} +8.68466 q^{3} +12.8078 q^{4} -2.80776 q^{5} +39.6155 q^{6} -9.56155 q^{7} +21.9309 q^{8} +48.4233 q^{9} +O(q^{10})\)	\(q+4.56155 q^{2} +8.68466 q^{3} +12.8078 q^{4} -2.80776 q^{5} +39.6155 q^{6} -9.56155 q^{7} +21.9309 q^{8} +48.4233 q^{9} -12.8078 q^{10} -39.4233 q^{11} +111.231 q^{12} -43.6155 q^{14} -24.3845 q^{15} -2.42329 q^{16} +2.01515 q^{17} +220.885 q^{18} +60.1922 q^{19} -35.9612 q^{20} -83.0388 q^{21} -179.831 q^{22} +4.46876 q^{23} +190.462 q^{24} -117.116 q^{25} +186.054 q^{27} -122.462 q^{28} +140.693 q^{29} -111.231 q^{30} -136.155 q^{31} -186.501 q^{32} -342.378 q^{33} +9.19224 q^{34} +26.8466 q^{35} +620.194 q^{36} +185.708 q^{37} +274.570 q^{38} -61.5767 q^{40} -310.231 q^{41} -378.786 q^{42} +427.471 q^{43} -504.924 q^{44} -135.961 q^{45} +20.3845 q^{46} +258.617 q^{47} -21.0455 q^{48} -251.577 q^{49} -534.233 q^{50} +17.5009 q^{51} +612.656 q^{53} +848.695 q^{54} +110.691 q^{55} -209.693 q^{56} +522.749 q^{57} +641.779 q^{58} +517.885 q^{59} -312.311 q^{60} -161.311 q^{61} -621.080 q^{62} -463.002 q^{63} -831.348 q^{64} -1561.77 q^{66} +49.8987 q^{67} +25.8096 q^{68} +38.8096 q^{69} +122.462 q^{70} -279.963 q^{71} +1061.96 q^{72} -467.732 q^{73} +847.118 q^{74} -1017.12 q^{75} +770.928 q^{76} +376.948 q^{77} +37.5379 q^{79} +6.80403 q^{80} +308.386 q^{81} -1415.14 q^{82} +76.1553 q^{83} -1063.54 q^{84} -5.65808 q^{85} +1949.93 q^{86} +1221.87 q^{87} -864.587 q^{88} -202.806 q^{89} -620.194 q^{90} +57.2348 q^{92} -1182.46 q^{93} +1179.70 q^{94} -169.006 q^{95} -1619.70 q^{96} +1174.37 q^{97} -1147.58 q^{98} -1909.01 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9}+O(q^{10}) \) 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9	\( 2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9} - 5 q^{10} - 17 q^{11} + 140 q^{12} - 46 q^{14} - 90 q^{15} + 57 q^{16} + 70 q^{17} + 215 q^{18} + 141 q^{19} - 175 q^{20} - 63 q^{21} - 170 q^{22} + 145 q^{23} + 216 q^{24} + 75 q^{25} + 335 q^{27} - 80 q^{28} + 34 q^{29} - 140 q^{30} + 140 q^{31} - 105 q^{32} - 425 q^{33} + 39 q^{34} - 70 q^{35} + 725 q^{36} + 190 q^{37} + 310 q^{38} - 185 q^{40} - 538 q^{41} - 370 q^{42} + 455 q^{43} - 680 q^{44} - 375 q^{45} + 82 q^{46} - 60 q^{47} - 240 q^{48} - 565 q^{49} - 450 q^{50} - 233 q^{51} + 545 q^{53} + 914 q^{54} + 510 q^{55} - 172 q^{56} + 225 q^{57} + 595 q^{58} + 809 q^{59} + 200 q^{60} + 502 q^{61} - 500 q^{62} - 390 q^{63} - 1271 q^{64} - 1598 q^{66} + 475 q^{67} - 505 q^{68} - 479 q^{69} + 80 q^{70} - 127 q^{71} + 1155 q^{72} - 585 q^{73} + 849 q^{74} - 1725 q^{75} + 140 q^{76} + 255 q^{77} + 240 q^{79} + 1065 q^{80} + 122 q^{81} - 1515 q^{82} - 260 q^{83} - 1220 q^{84} + 1205 q^{85} + 1962 q^{86} + 1615 q^{87} - 1020 q^{88} - 921 q^{89} - 725 q^{90} - 1040 q^{92} - 2200 q^{93} + 1040 q^{94} + 1270 q^{95} - 1920 q^{96} + 415 q^{97} - 1285 q^{98} - 2210 q^{99}+O(q^{100}) \) 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9 - 5 * q^10 - 17 * q^11 + 140 * q^12 - 46 * q^14 - 90 * q^15 + 57 * q^16 + 70 * q^17 + 215 * q^18 + 141 * q^19 - 175 * q^20 - 63 * q^21 - 170 * q^22 + 145 * q^23 + 216 * q^24 + 75 * q^25 + 335 * q^27 - 80 * q^28 + 34 * q^29 - 140 * q^30 + 140 * q^31 - 105 * q^32 - 425 * q^33 + 39 * q^34 - 70 * q^35 + 725 * q^36 + 190 * q^37 + 310 * q^38 - 185 * q^40 - 538 * q^41 - 370 * q^42 + 455 * q^43 - 680 * q^44 - 375 * q^45 + 82 * q^46 - 60 * q^47 - 240 * q^48 - 565 * q^49 - 450 * q^50 - 233 * q^51 + 545 * q^53 + 914 * q^54 + 510 * q^55 - 172 * q^56 + 225 * q^57 + 595 * q^58 + 809 * q^59 + 200 * q^60 + 502 * q^61 - 500 * q^62 - 390 * q^63 - 1271 * q^64 - 1598 * q^66 + 475 * q^67 - 505 * q^68 - 479 * q^69 + 80 * q^70 - 127 * q^71 + 1155 * q^72 - 585 * q^73 + 849 * q^74 - 1725 * q^75 + 140 * q^76 + 255 * q^77 + 240 * q^79 + 1065 * q^80 + 122 * q^81 - 1515 * q^82 - 260 * q^83 - 1220 * q^84 + 1205 * q^85 + 1962 * q^86 + 1615 * q^87 - 1020 * q^88 - 921 * q^89 - 725 * q^90 - 1040 * q^92 - 2200 * q^93 + 1040 * q^94 + 1270 * q^95 - 1920 * q^96 + 415 * q^97 - 1285 * q^98 - 2210 * q^99

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