LMFDB - Embedded newform 169.4.a.k.1.9

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 $ x^9 - 4x^8 - 46x^7 + 145x^6 + 680x^5 - 1501x^4 - 3203x^3 + 4784x^2 + 3584x - 4096
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$5.84018$ 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)

$f(q)$	$=$	$q+4.84018 q^{2} -6.19662 q^{3} +15.4273 q^{4} -15.2399 q^{5} -29.9927 q^{6} -4.31620 q^{7} +35.9495 q^{8} +11.3981 q^{9} +O(q^{10})$	\(q+4.84018 q^{2} -6.19662 q^{3} +15.4273 q^{4} -15.2399 q^{5} -29.9927 q^{6} -4.31620 q^{7} +35.9495 q^{8} +11.3981 q^{9} -73.7636 q^{10} -24.5691 q^{11} -95.5971 q^{12} -20.8912 q^{14} +94.4355 q^{15} +50.5834 q^{16} -127.160 q^{17} +55.1687 q^{18} -51.7462 q^{19} -235.110 q^{20} +26.7459 q^{21} -118.919 q^{22} +87.3684 q^{23} -222.765 q^{24} +107.253 q^{25} +96.6792 q^{27} -66.5874 q^{28} +225.726 q^{29} +457.085 q^{30} +108.720 q^{31} -42.7634 q^{32} +152.246 q^{33} -615.479 q^{34} +65.7783 q^{35} +175.842 q^{36} +115.756 q^{37} -250.461 q^{38} -547.865 q^{40} -191.885 q^{41} +129.455 q^{42} -123.301 q^{43} -379.036 q^{44} -173.705 q^{45} +422.878 q^{46} -36.7339 q^{47} -313.446 q^{48} -324.370 q^{49} +519.124 q^{50} +787.965 q^{51} +119.162 q^{53} +467.944 q^{54} +374.430 q^{55} -155.165 q^{56} +320.651 q^{57} +1092.55 q^{58} -804.553 q^{59} +1456.89 q^{60} +678.886 q^{61} +526.226 q^{62} -49.1964 q^{63} -611.650 q^{64} +736.896 q^{66} -87.4806 q^{67} -1961.74 q^{68} -541.388 q^{69} +318.379 q^{70} -981.049 q^{71} +409.755 q^{72} +263.862 q^{73} +560.281 q^{74} -664.606 q^{75} -798.304 q^{76} +106.045 q^{77} +321.051 q^{79} -770.883 q^{80} -906.832 q^{81} -928.757 q^{82} -1042.54 q^{83} +412.617 q^{84} +1937.91 q^{85} -596.798 q^{86} -1398.74 q^{87} -883.248 q^{88} -344.746 q^{89} -840.762 q^{90} +1347.86 q^{92} -673.698 q^{93} -177.799 q^{94} +788.604 q^{95} +264.989 q^{96} -482.521 q^{97} -1570.01 q^{98} -280.041 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) $ 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9 - 147 * q^10 - 181 * q^11 + 39 * q^12 - 147 * q^14 - 218 * q^15 + 269 * q^16 - 55 * q^17 - 79 * q^18 - 161 * q^19 - 370 * q^20 - 188 * q^21 + 340 * q^22 - 204 * q^23 - 798 * q^24 + 307 * q^25 - 668 * q^27 - 344 * q^28 + 280 * q^29 + 521 * q^30 - 706 * q^31 - 680 * q^32 - 500 * q^33 - 216 * q^34 + 20 * q^35 - 909 * q^36 - 298 * q^37 - 739 * q^38 + 13 * q^40 - 1201 * q^41 - 4 * q^42 - 533 * q^43 - 355 * q^44 + 90 * q^45 + 840 * q^46 - 956 * q^47 - 132 * q^48 + 403 * q^49 + 1156 * q^50 + 470 * q^51 - 278 * q^53 + 2555 * q^54 - 250 * q^55 + 250 * q^56 + 810 * q^57 + 2877 * q^58 - 1377 * q^59 + 3157 * q^60 - 136 * q^61 + 2035 * q^62 + 944 * q^63 + 284 * q^64 + 3279 * q^66 + 931 * q^67 - 1536 * q^68 - 2050 * q^69 + 4854 * q^70 - 2046 * q^71 + 4342 * q^72 + 45 * q^73 - 1990 * q^74 + 2393 * q^75 + 3608 * q^76 - 718 * q^77 + 412 * q^79 + 787 * q^80 - 835 * q^81 + 2757 * q^82 - 3709 * q^83 + 1539 * q^84 + 2106 * q^85 - 125 * q^86 - 786 * q^87 - 636 * q^88 - 1663 * q^89 - 1280 * q^90 + 4010 * q^92 + 1186 * q^93 - 2531 * q^94 - 1614 * q^95 + 3084 * q^96 + 1087 * q^97 + 282 * q^98 - 1357 * 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 $

$2$	4.84018	1.71126	0.855630	−	0.517587i	$-0.173170\pi$
$2$	4.84018	1.71126	0.855630	+	0.517587i	$0.173170\pi$
$3$	−6.19662	−1.19254	−0.596270	−	0.802784i	$-0.703351\pi$
$3$	−6.19662	−1.19254	−0.596270	+	0.802784i	$0.703351\pi$
$4$	15.4273	1.92841
$5$	−15.2399	−1.36309	−0.681547	−	0.731774i	$-0.738692\pi$
$5$	−15.2399	−1.36309	−0.681547	+	0.731774i	$0.738692\pi$
$6$	−29.9927	−2.04075
$7$	−4.31620	−0.233053	−0.116527	−	0.993188i	$-0.537176\pi$
$7$	−4.31620	−0.233053	−0.116527	+	0.993188i	$0.537176\pi$
$8$	35.9495	1.58876
$9$	11.3981	0.422151
$10$	−73.7636	−2.33261
$11$	−24.5691	−0.673443	−0.336722	−	0.941604i	$-0.609318\pi$
$11$	−24.5691	−0.673443	−0.336722	+	0.941604i	$0.609318\pi$
$12$	−95.5971	−2.29971
$13$	0	0
$13$	0	0
$14$	−20.8912	−0.398815
$15$	94.4355	1.62554
$16$	50.5834	0.790365
$17$	−127.160	−1.81417	−0.907086	−	0.420944i	$-0.861699\pi$
$17$	−127.160	−1.81417	−0.907086	+	0.420944i	$0.861699\pi$
$18$	55.1687	0.722410
$19$	−51.7462	−0.624810	−0.312405	−	0.949949i	$-0.601135\pi$
$19$	−51.7462	−0.624810	−0.312405	+	0.949949i	$0.601135\pi$
$20$	−235.110	−2.62861
$21$	26.7459	0.277925
$22$	−118.919	−1.15244
$23$	87.3684	0.792068	0.396034	−	0.918236i	$-0.370386\pi$
$23$	87.3684	0.792068	0.396034	+	0.918236i	$0.370386\pi$
$24$	−222.765	−1.89466
$25$	107.253	0.858024
$26$	0	0
$27$	96.6792	0.689108
$28$	−66.5874	−0.449423
$29$	225.726	1.44539	0.722693	−	0.691169i	$-0.242904\pi$
$29$	225.726	1.44539	0.722693	+	0.691169i	$0.242904\pi$
$30$	457.085	2.78173
$31$	108.720	0.629895	0.314948	−	0.949109i	$-0.398013\pi$
$31$	108.720	0.629895	0.314948	+	0.949109i	$0.398013\pi$
$32$	−42.7634	−0.236237
$33$	152.246	0.803108
$34$	−615.479	−3.10452
$35$	65.7783	0.317673
$36$	175.842	0.814081
$37$	115.756	0.514330	0.257165	−	0.966367i	$-0.417212\pi$
$37$	115.756	0.514330	0.257165	+	0.966367i	$0.417212\pi$
$38$	−250.461	−1.06921
$39$	0	0
$40$	−547.865	−2.16563
$41$	−191.885	−0.730912	−0.365456	−	0.930829i	$-0.619087\pi$
$41$	−191.885	−0.730912	−0.365456	+	0.930829i	$0.619087\pi$
$42$	129.455	0.475602
$43$	−123.301	−0.437284	−0.218642	−	0.975805i	$-0.570163\pi$
$43$	−123.301	−0.437284	−0.218642	+	0.975805i	$0.570163\pi$
$44$	−379.036	−1.29868
$45$	−173.705	−0.575431
$46$	422.878	1.35543
$47$	−36.7339	−0.114004	−0.0570020	−	0.998374i	$-0.518154\pi$
$47$	−36.7339	−0.114004	−0.0570020	+	0.998374i	$0.518154\pi$
$48$	−313.446	−0.942542
$49$	−324.370	−0.945686
$50$	519.124	1.46830
$51$	787.965	2.16347
$52$	0	0
$53$	119.162	0.308833	0.154416	−	0.988006i	$-0.450650\pi$
$53$	119.162	0.308833	0.154416	+	0.988006i	$0.450650\pi$
$54$	467.944	1.17924
$55$	374.430	0.917966
$56$	−155.165	−0.370265
$57$	320.651	0.745110
$58$	1092.55	2.47343
$59$	−804.553	−1.77532	−0.887660	−	0.460500i	$-0.847670\pi$
$59$	−804.553	−1.77532	−0.887660	+	0.460500i	$0.847670\pi$
$60$	1456.89	3.13472
$61$	678.886	1.42496	0.712479	−	0.701693i	$-0.247572\pi$
$61$	678.886	1.42496	0.712479	+	0.701693i	$0.247572\pi$
$62$	526.226	1.07791
$63$	−49.1964	−0.0983835
$64$	−611.650	−1.19463
$65$	0	0
$66$	736.896	1.37433
$67$	−87.4806	−0.159514	−0.0797571	−	0.996814i	$-0.525414\pi$
$67$	−87.4806	−0.159514	−0.0797571	+	0.996814i	$0.525414\pi$
$68$	−1961.74	−3.49848
$69$	−541.388	−0.944572
$70$	318.379	0.543622
$71$	−981.049	−1.63985	−0.819923	−	0.572473i	$-0.805984\pi$
$71$	−981.049	−1.63985	−0.819923	+	0.572473i	$0.805984\pi$
$72$	409.755	0.670695
$73$	263.862	0.423051	0.211526	−	0.977372i	$-0.432157\pi$
$73$	263.862	0.423051	0.211526	+	0.977372i	$0.432157\pi$
$74$	560.281	0.880153
$75$	−664.606	−1.02323
$76$	−798.304	−1.20489
$77$	106.045	0.156948
$78$	0	0
$79$	321.051	0.457228	0.228614	−	0.973517i	$-0.426581\pi$
$79$	321.051	0.457228	0.228614	+	0.973517i	$0.426581\pi$
$80$	−770.883	−1.07734
$81$	−906.832	−1.24394
$82$	−928.757	−1.25078
$83$	−1042.54	−1.37872	−0.689362	−	0.724417i	$-0.742109\pi$
$83$	−1042.54	−1.37872	−0.689362	+	0.724417i	$0.742109\pi$
$84$	412.617	0.535955
$85$	1937.91	2.47289
$86$	−596.798	−0.748307
$87$	−1398.74	−1.72368
$88$	−883.248	−1.06994
$89$	−344.746	−0.410595	−0.205298	−	0.978700i	$-0.565816\pi$
$89$	−344.746	−0.410595	−0.205298	+	0.978700i	$0.565816\pi$
$90$	−840.762	−0.984712
$91$	0	0
$92$	1347.86	1.52743
$93$	−673.698	−0.751175
$94$	−177.799	−0.195091
$95$	788.604	0.851674
$96$	264.989	0.281722
$97$	−482.521	−0.505079	−0.252539	−	0.967587i	$-0.581266\pi$
$97$	−482.521	−0.505079	−0.252539	+	0.967587i	$0.581266\pi$
$98$	−1570.01	−1.61832
$99$	−280.041	−0.284295
$100$	1654.63	1.65463
$101$	839.779	0.827338	0.413669	−	0.910427i	$-0.364247\pi$
$101$	839.779	0.827338	0.413669	+	0.910427i	$0.364247\pi$
$102$	3813.89	3.70227
$103$	159.823	0.152892	0.0764459	−	0.997074i	$-0.475643\pi$
$103$	159.823	0.152892	0.0764459	+	0.997074i	$0.475643\pi$
$104$	0	0
$105$	−407.603	−0.378838
$106$	576.764	0.528494
$107$	1075.62	0.971818	0.485909	−	0.874009i	$-0.338489\pi$
$107$	1075.62	0.971818	0.485909	+	0.874009i	$0.338489\pi$
$108$	1491.50	1.32889
$109$	−1161.67	−1.02080	−0.510401	−	0.859936i	$-0.670503\pi$
$109$	−1161.67	−1.02080	−0.510401	+	0.859936i	$0.670503\pi$
$110$	1812.31	1.57088
$111$	−717.298	−0.613359
$112$	−218.328	−0.184197
$113$	−1463.48	−1.21834	−0.609172	−	0.793038i	$-0.708498\pi$
$113$	−1463.48	−1.21834	−0.609172	+	0.793038i	$0.708498\pi$
$114$	1552.01	1.27508
$115$	−1331.48	−1.07966
$116$	3482.34	2.78730
$117$	0	0
$118$	−3894.18	−3.03803
$119$	548.851	0.422799
$120$	3394.91	2.58259
$121$	−727.357	−0.546474
$122$	3285.93	2.43848
$123$	1189.04	0.871641
$124$	1677.26	1.21470
$125$	270.461	0.193526
$126$	−238.119	−0.168360
$127$	289.923	0.202571	0.101285	−	0.994857i	$-0.467704\pi$
$127$	289.923	0.202571	0.101285	+	0.994857i	$0.467704\pi$
$128$	−2618.38	−1.80808
$129$	764.048	0.521478
$130$	0	0
$131$	−1201.97	−0.801653	−0.400826	−	0.916154i	$-0.631277\pi$
$131$	−1201.97	−0.801653	−0.400826	+	0.916154i	$0.631277\pi$
$132$	2348.74	1.54872
$133$	223.347	0.145614
$134$	−423.422	−0.272971
$135$	−1473.38	−0.939319
$136$	−4571.35	−2.88228
$137$	779.620	0.486186	0.243093	−	0.970003i	$-0.421838\pi$
$137$	779.620	0.486186	0.243093	+	0.970003i	$0.421838\pi$
$138$	−2620.41	−1.61641
$139$	2031.48	1.23963	0.619814	−	0.784749i	$-0.287208\pi$
$139$	2031.48	1.23963	0.619814	+	0.784749i	$0.287208\pi$
$140$	1014.78	0.612605
$141$	227.626	0.135954
$142$	−4748.45	−2.80621
$143$	0	0
$144$	576.553	0.333653
$145$	−3440.03	−1.97020
$146$	1277.14	0.723951
$147$	2010.00	1.12777
$148$	1785.81	0.991841
$149$	2603.60	1.43151	0.715757	−	0.698350i	$-0.246082\pi$
$149$	2603.60	1.43151	0.715757	+	0.698350i	$0.246082\pi$
$150$	−3216.81	−1.75101
$151$	206.776	0.111438	0.0557192	−	0.998446i	$-0.482255\pi$
$151$	206.776	0.111438	0.0557192	+	0.998446i	$0.482255\pi$
$152$	−1860.25	−0.992671
$153$	−1449.38	−0.765854
$154$	513.279	0.268579
$155$	−1656.88	−0.858606
$156$	0	0
$157$	699.208	0.355433	0.177716	−	0.984082i	$-0.443129\pi$
$157$	699.208	0.355433	0.177716	+	0.984082i	$0.443129\pi$
$158$	1553.94	0.782437
$159$	−738.400	−0.368295
$160$	651.708	0.322013
$161$	−377.100	−0.184594
$162$	−4389.23	−2.12870
$163$	−2944.14	−1.41474	−0.707371	−	0.706842i	$-0.750119\pi$
$163$	−2944.14	−1.41474	−0.707371	+	0.706842i	$0.750119\pi$
$164$	−2960.27	−1.40950
$165$	−2320.20	−1.09471
$166$	−5046.10	−2.35936
$167$	−2740.82	−1.27000	−0.635002	−	0.772510i	$-0.719001\pi$
$167$	−2740.82	−1.27000	−0.635002	+	0.772510i	$0.719001\pi$
$168$	961.500	0.441556
$169$	0	0
$170$	9379.81	4.23176
$171$	−589.806	−0.263764
$172$	−1902.20	−0.843264
$173$	338.812	0.148898	0.0744491	−	0.997225i	$-0.476280\pi$
$173$	338.812	0.148898	0.0744491	+	0.997225i	$0.476280\pi$
$174$	−6770.13	−2.94967
$175$	−462.926	−0.199965
$176$	−1242.79	−0.532266
$177$	4985.51	2.11714
$178$	−1668.63	−0.702635
$179$	−1094.40	−0.456979	−0.228489	−	0.973546i	$-0.573379\pi$
$179$	−1094.40	−0.456979	−0.228489	+	0.973546i	$0.573379\pi$
$180$	−2679.80	−1.10967
$181$	−1420.26	−0.583243	−0.291622	−	0.956534i	$-0.594195\pi$
$181$	−1420.26	−0.583243	−0.291622	+	0.956534i	$0.594195\pi$
$182$	0	0
$183$	−4206.80	−1.69932
$184$	3140.85	1.25840
$185$	−1764.11	−0.701080
$186$	−3260.82	−1.28546
$187$	3124.22	1.22174
$188$	−566.705	−0.219847
$189$	−417.287	−0.160599
$190$	3816.98	1.45744
$191$	−896.779	−0.339731	−0.169866	−	0.985467i	$-0.554333\pi$
$191$	−896.779	−0.339731	−0.169866	+	0.985467i	$0.554333\pi$
$192$	3790.16	1.42464
$193$	2589.97	0.965959	0.482980	−	0.875632i	$-0.339555\pi$
$193$	2589.97	0.965959	0.482980	+	0.875632i	$0.339555\pi$
$194$	−2335.49	−0.864321
$195$	0	0
$196$	−5004.16	−1.82367
$197$	1481.87	0.535932	0.267966	−	0.963428i	$-0.413648\pi$
$197$	1481.87	0.535932	0.267966	+	0.963428i	$0.413648\pi$
$198$	−1355.45	−0.486502
$199$	−3599.45	−1.28220	−0.641101	−	0.767456i	$-0.721522\pi$
$199$	−3599.45	−1.28220	−0.641101	+	0.767456i	$0.721522\pi$
$200$	3855.69	1.36319
$201$	542.084	0.190227
$202$	4064.68	1.41579
$203$	−974.278	−0.336852
$204$	12156.2	4.17207
$205$	2924.30	0.996301
$206$	773.572	0.261638
$207$	995.830	0.334372
$208$	0	0
$209$	1271.36	0.420774
$210$	−1972.87	−0.648291
$211$	3051.22	0.995518	0.497759	−	0.867315i	$-0.334156\pi$
$211$	3051.22	0.995518	0.497759	+	0.867315i	$0.334156\pi$
$212$	1838.35	0.595557
$213$	6079.19	1.95558
$214$	5206.21	1.66303
$215$	1879.09	0.596059
$216$	3475.57	1.09483
$217$	−469.259	−0.146799
$218$	−5622.67	−1.74686
$219$	−1635.05	−0.504506
$220$	5776.45	1.77022
$221$	0	0
$222$	−3471.85	−1.04962
$223$	2599.87	0.780718	0.390359	−	0.920663i	$-0.372351\pi$
$223$	2599.87	0.780718	0.390359	+	0.920663i	$0.372351\pi$
$224$	184.576	0.0550558
$225$	1222.48	0.362216
$226$	−7083.52	−2.08491
$227$	993.208	0.290403	0.145202	−	0.989402i	$-0.453617\pi$
$227$	993.208	0.290403	0.145202	+	0.989402i	$0.453617\pi$
$228$	4946.79	1.43688
$229$	−437.772	−0.126327	−0.0631633	−	0.998003i	$-0.520119\pi$
$229$	−437.772	−0.126327	−0.0631633	+	0.998003i	$0.520119\pi$
$230$	−6444.60	−1.84758
$231$	−657.123	−0.187167
$232$	8114.72	2.29637
$233$	2933.08	0.824689	0.412344	−	0.911028i	$-0.364710\pi$
$233$	2933.08	0.824689	0.412344	+	0.911028i	$0.364710\pi$
$234$	0	0
$235$	559.819	0.155398
$236$	−12412.1	−3.42355
$237$	−1989.43	−0.545263
$238$	2656.53	0.723519
$239$	−5813.48	−1.57340	−0.786700	−	0.617335i	$-0.788212\pi$
$239$	−5813.48	−1.57340	−0.786700	+	0.617335i	$0.788212\pi$
$240$	4776.87	1.28477
$241$	−1148.25	−0.306910	−0.153455	−	0.988156i	$-0.549040\pi$
$241$	−1148.25	−0.306910	−0.153455	+	0.988156i	$0.549040\pi$
$242$	−3520.54	−0.935160
$243$	3008.95	0.794339
$244$	10473.4	2.74791
$245$	4943.36	1.28906
$246$	5755.15	1.49161
$247$	0	0
$248$	3908.44	1.00075
$249$	6460.24	1.64418
$250$	1309.08	0.331174
$251$	−3255.80	−0.818741	−0.409371	−	0.912368i	$-0.634252\pi$
$251$	−3255.80	−0.818741	−0.409371	+	0.912368i	$0.634252\pi$
$252$	−758.968	−0.189724
$253$	−2146.57	−0.533413
$254$	1403.28	0.346652
$255$	−12008.5	−2.94902
$256$	−7780.24	−1.89947
$257$	6320.90	1.53419	0.767095	−	0.641534i	$-0.221702\pi$
$257$	6320.90	1.53419	0.767095	+	0.641534i	$0.221702\pi$
$258$	3698.13	0.892386
$259$	−499.628	−0.119866
$260$	0	0
$261$	2572.84	0.610171
$262$	−5817.74	−1.37184
$263$	−6168.99	−1.44637	−0.723187	−	0.690653i	$-0.757323\pi$
$263$	−6168.99	−1.44637	−0.723187	+	0.690653i	$0.757323\pi$
$264$	5473.15	1.27594
$265$	−1816.01	−0.420968
$266$	1081.04	0.249183
$267$	2136.26	0.489651
$268$	−1349.59	−0.307609
$269$	843.598	0.191209	0.0956043	−	0.995419i	$-0.469522\pi$
$269$	843.598	0.191209	0.0956043	+	0.995419i	$0.469522\pi$
$270$	−7131.40	−1.60742
$271$	2063.06	0.462442	0.231221	−	0.972901i	$-0.425728\pi$
$271$	2063.06	0.462442	0.231221	+	0.972901i	$0.425728\pi$
$272$	−6432.20	−1.43386
$273$	0	0
$274$	3773.50	0.831990
$275$	−2635.12	−0.577831
$276$	−8352.16	−1.82153
$277$	6582.49	1.42781	0.713905	−	0.700242i	$-0.246925\pi$
$277$	6582.49	1.42781	0.713905	+	0.700242i	$0.246925\pi$
$278$	9832.74	2.12133
$279$	1239.20	0.265911
$280$	2364.70	0.504706
$281$	935.025	0.198502	0.0992508	−	0.995062i	$-0.468355\pi$
$281$	935.025	0.198502	0.0992508	+	0.995062i	$0.468355\pi$
$282$	1101.75	0.232653
$283$	−5338.77	−1.12140	−0.560701	−	0.828018i	$-0.689468\pi$
$283$	−5338.77	−1.12140	−0.560701	+	0.828018i	$0.689468\pi$
$284$	−15134.9	−3.16230
$285$	−4886.68	−1.01566
$286$	0	0
$287$	828.215	0.170341
$288$	−487.421	−0.0997276
$289$	11256.8	2.29122
$290$	−16650.3	−3.37152
$291$	2990.00	0.602326
$292$	4070.69	0.815818
$293$	1763.76	0.351673	0.175837	−	0.984419i	$-0.443737\pi$
$293$	1763.76	0.351673	0.175837	+	0.984419i	$0.443737\pi$
$294$	9728.75	1.92991
$295$	12261.3	2.41993
$296$	4161.38	0.817146
$297$	−2375.33	−0.464075
$298$	12601.9	2.44969
$299$	0	0
$300$	−10253.1	−1.97321
$301$	532.192	0.101910
$302$	1000.83	0.190700
$303$	−5203.79	−0.986634
$304$	−2617.50	−0.493828
$305$	−10346.1	−1.94235
$306$	−7015.27	−1.31058
$307$	4736.59	0.880558	0.440279	−	0.897861i	$-0.354879\pi$
$307$	4736.59	0.880558	0.440279	+	0.897861i	$0.354879\pi$
$308$	1636.00	0.302661
$309$	−990.363	−0.182329
$310$	−8019.60	−1.46930
$311$	−4746.90	−0.865505	−0.432753	−	0.901513i	$-0.642458\pi$
$311$	−4746.90	−0.865505	−0.432753	+	0.901513i	$0.642458\pi$
$312$	0	0
$313$	10115.9	1.82679	0.913393	−	0.407078i	$-0.133452\pi$
$313$	10115.9	1.82679	0.913393	+	0.407078i	$0.133452\pi$
$314$	3384.29	0.608238
$315$	749.746	0.134106
$316$	4952.95	0.881725
$317$	5906.13	1.04644	0.523219	−	0.852198i	$-0.324731\pi$
$317$	5906.13	1.04644	0.523219	+	0.852198i	$0.324731\pi$
$318$	−3573.99	−0.630249
$319$	−5545.89	−0.973386
$320$	9321.45	1.62839
$321$	−6665.23	−1.15893
$322$	−1825.23	−0.315888
$323$	6580.07	1.13351
$324$	−13990.0	−2.39883
$325$	0	0
$326$	−14250.2	−2.42099
$327$	7198.41	1.21735
$328$	−6898.16	−1.16124
$329$	158.551	0.0265690
$330$	−11230.2	−1.87334
$331$	2802.82	0.465429	0.232715	−	0.972545i	$-0.425239\pi$
$331$	2802.82	0.465429	0.232715	+	0.972545i	$0.425239\pi$
$332$	−16083.6	−2.65875
$333$	1319.40	0.217125
$334$	−13266.0	−2.17331
$335$	1333.19	0.217433
$336$	1352.90	0.219662
$337$	−244.919	−0.0395892	−0.0197946	−	0.999804i	$-0.506301\pi$
$337$	−244.919	−0.0395892	−0.0197946	+	0.999804i	$0.506301\pi$
$338$	0	0
$339$	9068.65	1.45292
$340$	29896.7	4.76875
$341$	−2671.17	−0.424199
$342$	−2854.77	−0.451369
$343$	2880.51	0.453448
$344$	−4432.60	−0.694738
$345$	8250.68	1.28754
$346$	1639.91	0.254804
$347$	10356.8	1.60226	0.801129	−	0.598492i	$-0.204233\pi$
$347$	10356.8	1.60226	0.801129	+	0.598492i	$0.204233\pi$
$348$	−21578.7	−3.32397
$349$	8433.58	1.29352	0.646761	−	0.762693i	$-0.276123\pi$
$349$	8433.58	1.29352	0.646761	+	0.762693i	$0.276123\pi$
$350$	−2240.64	−0.342193
$351$	0	0
$352$	1050.66	0.159092
$353$	−4733.72	−0.713741	−0.356871	−	0.934154i	$-0.616156\pi$
$353$	−4733.72	−0.713741	−0.356871	+	0.934154i	$0.616156\pi$
$354$	24130.7	3.62298
$355$	14951.0	2.23526
$356$	−5318.50	−0.791797
$357$	−3401.02	−0.504204
$358$	−5297.08	−0.782010
$359$	−7561.34	−1.11162	−0.555811	−	0.831309i	$-0.687592\pi$
$359$	−7561.34	−1.11162	−0.555811	+	0.831309i	$0.687592\pi$
$360$	−6244.60	−0.914220
$361$	−4181.33	−0.609613
$362$	−6874.30	−0.998081
$363$	4507.15	0.651692
$364$	0	0
$365$	−4021.22	−0.576659
$366$	−20361.6	−2.90798
$367$	−11940.0	−1.69826	−0.849130	−	0.528184i	$-0.822873\pi$
$367$	−11940.0	−1.69826	−0.849130	+	0.528184i	$0.822873\pi$
$368$	4419.39	0.626023
$369$	−2187.12	−0.308555
$370$	−8538.60	−1.19973
$371$	−514.327	−0.0719745
$372$	−10393.4	−1.44858
$373$	−5797.12	−0.804728	−0.402364	−	0.915480i	$-0.631811\pi$
$373$	−5797.12	−0.804728	−0.402364	+	0.915480i	$0.631811\pi$
$374$	15121.8	2.09072
$375$	−1675.94	−0.230788
$376$	−1320.56	−0.181125
$377$	0	0
$378$	−2019.74	−0.274826
$379$	−8846.31	−1.19896	−0.599478	−	0.800391i	$-0.704625\pi$
$379$	−8846.31	−1.19896	−0.599478	+	0.800391i	$0.704625\pi$
$380$	12166.0	1.64238
$381$	−1796.54	−0.241574
$382$	−4340.57	−0.581369
$383$	−1047.61	−0.139766	−0.0698830	−	0.997555i	$-0.522263\pi$
$383$	−1047.61	−0.139766	−0.0698830	+	0.997555i	$0.522263\pi$
$384$	16225.1	2.15621
$385$	−1616.12	−0.213935
$386$	12535.9	1.65301
$387$	−1405.39	−0.184600
$388$	−7444.01	−0.974000
$389$	−11858.4	−1.54562	−0.772808	−	0.634640i	$-0.781148\pi$
$389$	−11858.4	−1.54562	−0.772808	+	0.634640i	$0.781148\pi$
$390$	0	0
$391$	−11109.8	−1.43695
$392$	−11660.9	−1.50247
$393$	7448.14	0.956003
$394$	7172.50	0.917120
$395$	−4892.76	−0.623245
$396$	−4320.28	−0.548238
$397$	−10480.3	−1.32492	−0.662460	−	0.749097i	$-0.730488\pi$
$397$	−10480.3	−1.32492	−0.662460	+	0.749097i	$0.730488\pi$
$398$	−17422.0	−2.19418
$399$	−1384.00	−0.173650
$400$	5425.22	0.678152
$401$	−7936.40	−0.988341	−0.494171	−	0.869365i	$-0.664528\pi$
$401$	−7936.40	−0.988341	−0.494171	+	0.869365i	$0.664528\pi$
$402$	2623.78	0.325528
$403$	0	0
$404$	12955.5	1.59545
$405$	13820.0	1.69561
$406$	−4715.68	−0.576441
$407$	−2844.03	−0.346372
$408$	28326.9	3.43723
$409$	5835.68	0.705516	0.352758	−	0.935715i	$-0.385244\pi$
$409$	5835.68	0.705516	0.352758	+	0.935715i	$0.385244\pi$
$410$	14154.1	1.70493
$411$	−4831.01	−0.579796
$412$	2465.64	0.294838
$413$	3472.61	0.413744
$414$	4820.00	0.572198
$415$	15888.2	1.87933
$416$	0	0
$417$	−12588.3	−1.47830
$418$	6153.60	0.720054
$419$	−8544.29	−0.996219	−0.498109	−	0.867114i	$-0.665972\pi$
$419$	−8544.29	−0.996219	−0.498109	+	0.867114i	$0.665972\pi$
$420$	−6288.22	−0.730556
$421$	16524.6	1.91297	0.956484	−	0.291786i	$-0.0942496\pi$
$421$	16524.6	1.91297	0.956484	+	0.291786i	$0.0942496\pi$
$422$	14768.4	1.70359
$423$	−418.696	−0.0481269
$424$	4283.81	0.490661
$425$	−13638.3	−1.55660
$426$	29424.3	3.34651
$427$	−2930.21	−0.332091
$428$	16594.0	1.87407
$429$	0	0
$430$	9095.11	1.02001
$431$	6836.58	0.764052	0.382026	−	0.924152i	$-0.375226\pi$
$431$	6836.58	0.764052	0.382026	+	0.924152i	$0.375226\pi$
$432$	4890.36	0.544647
$433$	−6290.65	−0.698174	−0.349087	−	0.937090i	$-0.613508\pi$
$433$	−6290.65	−0.698174	−0.349087	+	0.937090i	$0.613508\pi$
$434$	−2271.30	−0.251211
$435$	21316.5	2.34954
$436$	−17921.4	−1.96853
$437$	−4520.98	−0.494892
$438$	−7913.95	−0.863340
$439$	8868.59	0.964180	0.482090	−	0.876122i	$-0.339878\pi$
$439$	8868.59	0.964180	0.482090	+	0.876122i	$0.339878\pi$
$440$	13460.6	1.45843
$441$	−3697.20	−0.399222
$442$	0	0
$443$	−4310.67	−0.462316	−0.231158	−	0.972916i	$-0.574251\pi$
$443$	−4310.67	−0.462316	−0.231158	+	0.972916i	$0.574251\pi$
$444$	−11066.0	−1.18281
$445$	5253.87	0.559680
$446$	12583.8	1.33601
$447$	−16133.5	−1.70714
$448$	2640.00	0.278412
$449$	4604.48	0.483962	0.241981	−	0.970281i	$-0.422203\pi$
$449$	4604.48	0.483962	0.241981	+	0.970281i	$0.422203\pi$
$450$	5917.01	0.619845
$451$	4714.45	0.492228
$452$	−22577.6	−2.34947
$453$	−1281.31	−0.132895
$454$	4807.30	0.496956
$455$	0	0
$456$	11527.2	1.18380
$457$	−16379.2	−1.67656	−0.838281	−	0.545238i	$-0.816439\pi$
$457$	−16379.2	−1.67656	−0.838281	+	0.545238i	$0.816439\pi$
$458$	−2118.89	−0.216178
$459$	−12293.8	−1.25016
$460$	−20541.2	−2.08204
$461$	−9631.51	−0.973068	−0.486534	−	0.873662i	$-0.661739\pi$
$461$	−9631.51	−0.973068	−0.486534	+	0.873662i	$0.661739\pi$
$462$	−3180.59	−0.320291
$463$	17855.0	1.79220	0.896102	−	0.443848i	$-0.146387\pi$
$463$	17855.0	1.79220	0.896102	+	0.443848i	$0.146387\pi$
$464$	11418.0	1.14238
$465$	10267.1	1.02392
$466$	14196.6	1.41126
$467$	8220.28	0.814538	0.407269	−	0.913308i	$-0.366481\pi$
$467$	8220.28	0.814538	0.407269	+	0.913308i	$0.366481\pi$
$468$	0	0
$469$	377.584	0.0371753
$470$	2709.62	0.265927
$471$	−4332.73	−0.423867
$472$	−28923.3	−2.82055
$473$	3029.40	0.294486
$474$	−9629.18	−0.933087
$475$	−5549.93	−0.536102
$476$	8467.29	0.815331
$477$	1358.22	0.130374
$478$	−28138.3	−2.69250
$479$	3164.10	0.301819	0.150909	−	0.988548i	$-0.451780\pi$
$479$	3164.10	0.301819	0.150909	+	0.988548i	$0.451780\pi$
$480$	−4038.39	−0.384013
$481$	0	0
$482$	−5557.73	−0.525203
$483$	2336.74	0.220136
$484$	−11221.2	−1.05383
$485$	7353.55	0.688469
$486$	14563.9	1.35932
$487$	7487.99	0.696741	0.348371	−	0.937357i	$-0.386735\pi$
$487$	7487.99	0.696741	0.348371	+	0.937357i	$0.386735\pi$
$488$	24405.6	2.26391
$489$	18243.7	1.68714
$490$	23926.7	2.20592
$491$	−16013.7	−1.47187	−0.735933	−	0.677054i	$-0.763256\pi$
$491$	−16013.7	−1.47187	−0.735933	+	0.677054i	$0.763256\pi$
$492$	18343.6	1.68089
$493$	−28703.4	−2.62218
$494$	0	0
$495$	4267.78	0.387520
$496$	5499.44	0.497847
$497$	4234.41	0.382171
$498$	31268.7	2.81362
$499$	−10343.2	−0.927904	−0.463952	−	0.885860i	$-0.653569\pi$
$499$	−10343.2	−0.927904	−0.463952	+	0.885860i	$0.653569\pi$
$500$	4172.48	0.373198
$501$	16983.8	1.51453
$502$	−15758.6	−1.40108
$503$	−18615.3	−1.65013	−0.825063	−	0.565040i	$-0.808861\pi$
$503$	−18615.3	−1.65013	−0.825063	+	0.565040i	$0.808861\pi$
$504$	−1768.58	−0.156308
$505$	−12798.1	−1.12774
$506$	−10389.8	−0.912809
$507$	0	0
$508$	4472.73	0.390641
$509$	3628.41	0.315965	0.157983	−	0.987442i	$-0.449501\pi$
$509$	3628.41	0.315965	0.157983	+	0.987442i	$0.449501\pi$
$510$	−58123.1	−5.04654
$511$	−1138.88	−0.0985935
$512$	−16710.7	−1.44241
$513$	−5002.78	−0.430562
$514$	30594.3	2.62540
$515$	−2435.68	−0.208406
$516$	11787.2	1.00563
$517$	902.521	0.0767753
$518$	−2418.29	−0.205122
$519$	−2099.49	−0.177567
$520$	0	0
$521$	−3105.46	−0.261137	−0.130569	−	0.991439i	$-0.541680\pi$
$521$	−3105.46	−0.261137	−0.130569	+	0.991439i	$0.541680\pi$
$522$	12453.0	1.04416
$523$	1911.23	0.159794	0.0798969	−	0.996803i	$-0.474541\pi$
$523$	1911.23	0.159794	0.0798969	+	0.996803i	$0.474541\pi$
$524$	−18543.1	−1.54592
$525$	2868.58	0.238467
$526$	−29859.0	−2.47512
$527$	−13824.9	−1.14274
$528$	7701.10	0.634748
$529$	−4533.77	−0.372628
$530$	−8789.80	−0.720386
$531$	−9170.35	−0.749452
$532$	3445.64	0.280804
$533$	0	0
$534$	10339.9	0.837920
$535$	−16392.4	−1.32468
$536$	−3144.88	−0.253430
$537$	6781.57	0.544965
$538$	4083.16	0.327208
$539$	7969.50	0.636866
$540$	−22730.2	−1.81140
$541$	15251.2	1.21202	0.606008	−	0.795459i	$-0.292770\pi$
$541$	15251.2	1.21202	0.606008	+	0.795459i	$0.292770\pi$
$542$	9985.55	0.791358
$543$	8800.80	0.695541
$544$	5437.82	0.428575
$545$	17703.6	1.39145
$546$	0	0
$547$	1838.85	0.143736	0.0718681	−	0.997414i	$-0.477104\pi$
$547$	1838.85	0.143736	0.0718681	+	0.997414i	$0.477104\pi$
$548$	12027.4	0.937567
$549$	7737.99	0.601547
$550$	−12754.4	−0.988819
$551$	−11680.4	−0.903091
$552$	−19462.6	−1.50070
$553$	−1385.72	−0.106558
$554$	31860.4	2.44336
$555$	10931.5	0.836066
$556$	31340.3	2.39051
$557$	−6301.29	−0.479343	−0.239672	−	0.970854i	$-0.577040\pi$
$557$	−6301.29	−0.479343	−0.239672	+	0.970854i	$0.577040\pi$
$558$	5997.96	0.455042
$559$	0	0
$560$	3327.29	0.251078
$561$	−19359.6	−1.45698
$562$	4525.69	0.339688
$563$	11759.8	0.880316	0.440158	−	0.897920i	$-0.354922\pi$
$563$	11759.8	0.880316	0.440158	+	0.897920i	$0.354922\pi$
$564$	3511.66	0.262176
$565$	22303.3	1.66072
$566$	−25840.6	−1.91901
$567$	3914.07	0.289904
$568$	−35268.2	−2.60532
$569$	−17219.4	−1.26867	−0.634337	−	0.773056i	$-0.718727\pi$
$569$	−17219.4	−1.26867	−0.634337	+	0.773056i	$0.718727\pi$
$570$	−23652.4	−1.73805
$571$	825.501	0.0605011	0.0302506	−	0.999542i	$-0.490369\pi$
$571$	825.501	0.0605011	0.0302506	+	0.999542i	$0.490369\pi$
$572$	0	0
$573$	5557.00	0.405143
$574$	4008.71	0.291498
$575$	9370.52	0.679614
$576$	−6971.62	−0.504313
$577$	−1073.19	−0.0774308	−0.0387154	−	0.999250i	$-0.512327\pi$
$577$	−1073.19	−0.0774308	−0.0387154	+	0.999250i	$0.512327\pi$
$578$	54484.8	3.92088
$579$	−16049.1	−1.15194
$580$	−53070.3	−3.79935
$581$	4499.83	0.321316
$582$	14472.1	1.03074
$583$	−2927.71	−0.207981
$584$	9485.71	0.672126
$585$	0	0
$586$	8536.93	0.601804
$587$	14939.4	1.05045	0.525225	−	0.850963i	$-0.323981\pi$
$587$	14939.4	1.05045	0.525225	+	0.850963i	$0.323981\pi$
$588$	31008.9	2.17480
$589$	−5625.86	−0.393565
$590$	59346.7	4.14113
$591$	−9182.57	−0.639121
$592$	5855.34	0.406509
$593$	−6296.13	−0.436006	−0.218003	−	0.975948i	$-0.569954\pi$
$593$	−6296.13	−0.436006	−0.218003	+	0.975948i	$0.569954\pi$
$594$	−11497.0	−0.794154
$595$	−8364.40	−0.576314
$596$	40166.6	2.76055
$597$	22304.4	1.52908
$598$	0	0
$599$	−3518.96	−0.240035	−0.120017	−	0.992772i	$-0.538295\pi$
$599$	−3518.96	−0.240035	−0.120017	+	0.992772i	$0.538295\pi$
$600$	−23892.2	−1.62566
$601$	5380.65	0.365194	0.182597	−	0.983188i	$-0.441550\pi$
$601$	5380.65	0.365194	0.182597	+	0.983188i	$0.441550\pi$
$602$	2575.90	0.174395
$603$	−997.110	−0.0673391
$604$	3190.00	0.214899
$605$	11084.8	0.744895
$606$	−25187.3	−1.68839
$607$	20708.6	1.38474	0.692369	−	0.721543i	$-0.256567\pi$
$607$	20708.6	1.38474	0.692369	+	0.721543i	$0.256567\pi$
$608$	2212.84	0.147603
$609$	6037.23	0.401709
$610$	−50077.1	−3.32387
$611$	0	0
$612$	−22360.1	−1.47688
$613$	−8563.66	−0.564246	−0.282123	−	0.959378i	$-0.591039\pi$
$613$	−8563.66	−0.564246	−0.282123	+	0.959378i	$0.591039\pi$
$614$	22925.9	1.50687
$615$	−18120.8	−1.18813
$616$	3812.28	0.249352
$617$	−30449.6	−1.98680	−0.993400	−	0.114703i	$-0.963409\pi$
$617$	−30449.6	−1.98680	−0.993400	+	0.114703i	$0.963409\pi$
$618$	−4793.53	−0.312013
$619$	−26233.1	−1.70339	−0.851693	−	0.524041i	$-0.824424\pi$
$619$	−26233.1	−1.70339	−0.851693	+	0.524041i	$0.824424\pi$
$620$	−25561.2	−1.65575
$621$	8446.70	0.545820
$622$	−22975.8	−1.48111
$623$	1487.99	0.0956905
$624$	0	0
$625$	−17528.4	−1.12182
$626$	48962.7	3.12611
$627$	−7878.13	−0.501790
$628$	10786.9	0.685421
$629$	−14719.6	−0.933084
$630$	3628.90	0.229490
$631$	20639.5	1.30213	0.651066	−	0.759021i	$-0.274322\pi$
$631$	20639.5	1.30213	0.651066	+	0.759021i	$0.274322\pi$
$632$	11541.6	0.726425
$633$	−18907.2	−1.18719
$634$	28586.7	1.79073
$635$	−4418.38	−0.276123
$636$	−11391.5	−0.710226
$637$	0	0
$638$	−26843.1	−1.66572
$639$	−11182.1	−0.692262
$640$	39903.8	2.46459
$641$	−4431.77	−0.273080	−0.136540	−	0.990635i	$-0.543598\pi$
$641$	−4431.77	−0.273080	−0.136540	+	0.990635i	$0.543598\pi$
$642$	−32260.9	−1.98323
$643$	−14176.1	−0.869439	−0.434719	−	0.900566i	$-0.643152\pi$
$643$	−14176.1	−0.869439	−0.434719	+	0.900566i	$0.643152\pi$
$644$	−5817.63	−0.355973
$645$	−11644.0	−0.710824
$646$	31848.7	1.93974
$647$	5650.79	0.343362	0.171681	−	0.985153i	$-0.445080\pi$
$647$	5650.79	0.343362	0.171681	+	0.985153i	$0.445080\pi$
$648$	−32600.1	−1.97632
$649$	19767.2	1.19558
$650$	0	0
$651$	2907.82	0.175064
$652$	−45420.2	−2.72821
$653$	−18436.3	−1.10485	−0.552425	−	0.833562i	$-0.686298\pi$
$653$	−18436.3	−1.10485	−0.552425	+	0.833562i	$0.686298\pi$
$654$	34841.6	2.08320
$655$	18317.8	1.09273
$656$	−9706.18	−0.577687
$657$	3007.52	0.178591
$658$	767.415	0.0454665
$659$	−15331.9	−0.906294	−0.453147	−	0.891436i	$-0.649699\pi$
$659$	−15331.9	−0.906294	−0.453147	+	0.891436i	$0.649699\pi$
$660$	−35794.4	−2.11106
$661$	22295.1	1.31192	0.655961	−	0.754795i	$-0.272264\pi$
$661$	22295.1	1.31192	0.655961	+	0.754795i	$0.272264\pi$
$662$	13566.2	0.796471
$663$	0	0
$664$	−37478.9	−2.19046
$665$	−3403.78	−0.198485
$666$	6386.12	0.371557
$667$	19721.3	1.14484
$668$	−42283.4	−2.44909
$669$	−16110.4	−0.931037
$670$	6452.88	0.372084
$671$	−16679.7	−0.959629
$672$	−1143.75	−0.0656562
$673$	−16171.7	−0.926259	−0.463129	−	0.886291i	$-0.653274\pi$
$673$	−16171.7	−0.926259	−0.463129	+	0.886291i	$0.653274\pi$
$674$	−1185.45	−0.0677475
$675$	10369.1	0.591272
$676$	0	0
$677$	33614.4	1.90828	0.954141	−	0.299358i	$-0.0967725\pi$
$677$	33614.4	1.90828	0.954141	+	0.299358i	$0.0967725\pi$
$678$	43893.9	2.48633
$679$	2082.66	0.117710
$680$	69666.7	3.92882
$681$	−6154.53	−0.346318
$682$	−12928.9	−0.725915
$683$	22477.5	1.25926	0.629632	−	0.776893i	$-0.283206\pi$
$683$	22477.5	1.25926	0.629632	+	0.776893i	$0.283206\pi$
$684$	−9099.12	−0.508646
$685$	−11881.3	−0.662717
$686$	13942.2	0.775968
$687$	2712.70	0.150649
$688$	−6236.97	−0.345614
$689$	0	0
$690$	39934.7	2.20332
$691$	7618.93	0.419447	0.209723	−	0.977761i	$-0.432744\pi$
$691$	7618.93	0.419447	0.209723	+	0.977761i	$0.432744\pi$
$692$	5226.95	0.287137
$693$	1208.71	0.0662558
$694$	50128.9	2.74188
$695$	−30959.5	−1.68973
$696$	−50283.8	−2.73851
$697$	24400.2	1.32600
$698$	40820.0	2.21355
$699$	−18175.2	−0.983474
$700$	−7141.70	−0.385616
$701$	−18164.3	−0.978684	−0.489342	−	0.872092i	$-0.662763\pi$
$701$	−18164.3	−0.978684	−0.489342	+	0.872092i	$0.662763\pi$
$702$	0	0
$703$	−5989.95	−0.321359
$704$	15027.7	0.804514
$705$	−3468.99	−0.185319
$706$	−22912.0	−1.22140
$707$	−3624.66	−0.192814
$708$	76912.9	4.08272
$709$	−16706.5	−0.884945	−0.442472	−	0.896782i	$-0.645899\pi$
$709$	−16706.5	−0.884945	−0.442472	+	0.896782i	$0.645899\pi$
$710$	72365.7	3.82512
$711$	3659.36	0.193019
$712$	−12393.4	−0.652336
$713$	9498.72	0.498920
$714$	−16461.5	−0.862825
$715$	0	0
$716$	−16883.6	−0.881244
$717$	36023.9	1.87634
$718$	−36598.2	−1.90227
$719$	4549.39	0.235972	0.117986	−	0.993015i	$-0.462356\pi$
$719$	4549.39	0.235972	0.117986	+	0.993015i	$0.462356\pi$
$720$	−8786.58	−0.454800
$721$	−689.830	−0.0356319
$722$	−20238.4	−1.04321
$723$	7115.26	0.366002
$724$	−21910.8	−1.12473
$725$	24209.8	1.24018
$726$	21815.4	1.11521
$727$	−6246.70	−0.318676	−0.159338	−	0.987224i	$-0.550936\pi$
$727$	−6246.70	−0.318676	−0.159338	+	0.987224i	$0.550936\pi$
$728$	0	0
$729$	5839.14	0.296659
$730$	−19463.4	−0.986813
$731$	15679.0	0.793309
$732$	−64899.6	−3.27699
$733$	3447.16	0.173702	0.0868511	−	0.996221i	$-0.472320\pi$
$733$	3447.16	0.173702	0.0868511	+	0.996221i	$0.472320\pi$
$734$	−57791.6	−2.90617
$735$	−30632.1	−1.53725
$736$	−3736.17	−0.187116
$737$	2149.32	0.107424
$738$	−10586.0	−0.528018
$739$	−37032.9	−1.84341	−0.921703	−	0.387895i	$-0.873202\pi$
$739$	−37032.9	−1.84341	−0.921703	+	0.387895i	$0.873202\pi$
$740$	−27215.4	−1.35197
$741$	0	0
$742$	−2489.43	−0.123167
$743$	−15298.0	−0.755356	−0.377678	−	0.925937i	$-0.623277\pi$
$743$	−15298.0	−0.755356	−0.377678	+	0.925937i	$0.623277\pi$
$744$	−24219.1	−1.19343
$745$	−39678.5	−1.95129
$746$	−28059.1	−1.37710
$747$	−11883.0	−0.582029
$748$	48198.4	2.35603
$749$	−4642.62	−0.226485
$750$	−8111.86	−0.394938
$751$	−9844.54	−0.478339	−0.239169	−	0.970978i	$-0.576875\pi$
$751$	−9844.54	−0.478339	−0.239169	+	0.970978i	$0.576875\pi$
$752$	−1858.12	−0.0901048
$753$	20174.9	0.976382
$754$	0	0
$755$	−3151.24	−0.151901
$756$	−6437.62	−0.309701
$757$	5921.31	0.284298	0.142149	−	0.989845i	$-0.454599\pi$
$757$	5921.31	0.284298	0.142149	+	0.989845i	$0.454599\pi$
$758$	−42817.7	−2.05173
$759$	13301.4	0.636116
$760$	28349.9	1.35310
$761$	17872.5	0.851348	0.425674	−	0.904877i	$-0.360037\pi$
$761$	17872.5	0.851348	0.425674	+	0.904877i	$0.360037\pi$
$762$	−8695.58	−0.413396
$763$	5013.99	0.237901
$764$	−13834.9	−0.655142
$765$	22088.4	1.04393
$766$	−5070.61	−0.239176
$767$	0	0
$768$	48211.2	2.26520
$769$	11265.2	0.528262	0.264131	−	0.964487i	$-0.414915\pi$
$769$	11265.2	0.528262	0.264131	+	0.964487i	$0.414915\pi$
$770$	−7822.29	−0.366099
$771$	−39168.2	−1.82958
$772$	39956.3	1.86277
$773$	27603.7	1.28439	0.642196	−	0.766540i	$-0.278023\pi$
$773$	27603.7	1.28439	0.642196	+	0.766540i	$0.278023\pi$
$774$	−6802.35	−0.315898
$775$	11660.6	0.540465
$776$	−17346.4	−0.802447
$777$	3096.00	0.142945
$778$	−57396.7	−2.64495
$779$	9929.31	0.456681
$780$	0	0
$781$	24103.5	1.10434
$782$	−53773.4	−2.45899
$783$	21823.0	0.996027
$784$	−16407.7	−0.747437
$785$	−10655.8	−0.484488
$786$	36050.3	1.63597
$787$	24163.1	1.09444	0.547218	−	0.836990i	$-0.315687\pi$
$787$	24163.1	1.09444	0.547218	+	0.836990i	$0.315687\pi$
$788$	22861.2	1.03350
$789$	38226.9	1.72486
$790$	−23681.8	−1.06653
$791$	6316.70	0.283939
$792$	−10067.3	−0.451675
$793$	0	0
$794$	−50726.7	−2.26728
$795$	11253.1	0.502021
$796$	−55529.8	−2.47262
$797$	30385.6	1.35046	0.675228	−	0.737609i	$-0.264045\pi$
$797$	30385.6	1.35046	0.675228	+	0.737609i	$0.264045\pi$
$798$	−6698.79	−0.297161
$799$	4671.10	0.206823
$800$	−4586.51	−0.202697
$801$	−3929.43	−0.173333
$802$	−38413.6	−1.69131
$803$	−6482.87	−0.284901
$804$	8362.89	0.366836
$805$	5746.94	0.251619
$806$	0	0
$807$	−5227.46	−0.228024
$808$	30189.6	1.31444
$809$	−41976.9	−1.82426	−0.912131	−	0.409899i	$-0.865564\pi$
$809$	−41976.9	−1.82426	−0.912131	+	0.409899i	$0.865564\pi$
$810$	66891.2	2.90162
$811$	9674.05	0.418868	0.209434	−	0.977823i	$-0.432838\pi$
$811$	9674.05	0.418868	0.209434	+	0.977823i	$0.432838\pi$
$812$	−15030.5	−0.649590
$813$	−12784.0	−0.551480
$814$	−13765.6	−0.592733
$815$	44868.3	1.92843
$816$	39857.9	1.70993
$817$	6380.35	0.273219
$818$	28245.7	1.20732
$819$	0	0
$820$	45114.0	1.92128
$821$	−25426.0	−1.08085	−0.540423	−	0.841394i	$-0.681736\pi$
$821$	−25426.0	−1.08085	−0.540423	+	0.841394i	$0.681736\pi$
$822$	−23382.9	−0.992181
$823$	14388.3	0.609408	0.304704	−	0.952447i	$-0.401442\pi$
$823$	14388.3	0.609408	0.304704	+	0.952447i	$0.401442\pi$
$824$	5745.56	0.242908
$825$	16328.8	0.689086
$826$	16808.1	0.708024
$827$	−3850.25	−0.161894	−0.0809469	−	0.996718i	$-0.525794\pi$
$827$	−3850.25	−0.161894	−0.0809469	+	0.996718i	$0.525794\pi$
$828$	15363.0	0.644807
$829$	−1918.58	−0.0803802	−0.0401901	−	0.999192i	$-0.512796\pi$
$829$	−1918.58	−0.0803802	−0.0401901	+	0.999192i	$0.512796\pi$
$830$	76901.7	3.21602
$831$	−40789.2	−1.70272
$832$	0	0
$833$	41247.1	1.71564
$834$	−60929.7	−2.52977
$835$	41769.6	1.73113
$836$	19613.7	0.811426
$837$	10511.0	0.434066
$838$	−41355.9	−1.70479
$839$	−31412.1	−1.29257	−0.646285	−	0.763096i	$-0.723678\pi$
$839$	−31412.1	−1.29257	−0.646285	+	0.763096i	$0.723678\pi$
$840$	−14653.1	−0.601882
$841$	26563.1	1.08914
$842$	79981.9	3.27359
$843$	−5793.99	−0.236721
$844$	47072.0	1.91977
$845$	0	0
$846$	−2026.56	−0.0823577
$847$	3139.42	0.127357
$848$	6027.61	0.244091
$849$	33082.3	1.33732
$850$	−66012.0	−2.66376
$851$	10113.4	0.407384
$852$	93785.5	3.77117
$853$	−18315.5	−0.735184	−0.367592	−	0.929987i	$-0.619818\pi$
$853$	−18315.5	−0.735184	−0.367592	+	0.929987i	$0.619818\pi$
$854$	−14182.7	−0.568295
$855$	8988.56	0.359535
$856$	38668.1	1.54398
$857$	9579.31	0.381824	0.190912	−	0.981607i	$-0.438856\pi$
$857$	9579.31	0.381824	0.190912	+	0.981607i	$0.438856\pi$
$858$	0	0
$859$	−7136.55	−0.283465	−0.141732	−	0.989905i	$-0.545267\pi$
$859$	−7136.55	−0.283465	−0.141732	+	0.989905i	$0.545267\pi$
$860$	28989.3	1.14945
$861$	−5132.13	−0.203139
$862$	33090.3	1.30749
$863$	−17239.2	−0.679986	−0.339993	−	0.940428i	$-0.610425\pi$
$863$	−17239.2	−0.679986	−0.339993	+	0.940428i	$0.610425\pi$
$864$	−4134.34	−0.162793
$865$	−5163.44	−0.202962
$866$	−30447.8	−1.19476
$867$	−69754.0	−2.73237
$868$	−7239.41	−0.283089
$869$	−7887.94	−0.307917
$870$	103176.	4.02067
$871$	0	0
$872$	−41761.3	−1.62181
$873$	−5499.81	−0.213219
$874$	−21882.3	−0.846889
$875$	−1167.36	−0.0451019
$876$	−25224.5	−0.972895
$877$	33048.7	1.27249	0.636247	−	0.771486i	$-0.280486\pi$
$877$	33048.7	1.27249	0.636247	+	0.771486i	$0.280486\pi$
$878$	42925.6	1.64996
$879$	−10929.4	−0.419384
$880$	18939.9	0.725529
$881$	19527.8	0.746775	0.373387	−	0.927676i	$-0.378196\pi$
$881$	19527.8	0.746775	0.373387	+	0.927676i	$0.378196\pi$
$882$	−17895.1	−0.683173
$883$	−26361.3	−1.00467	−0.502337	−	0.864672i	$-0.667526\pi$
$883$	−26361.3	−1.00467	−0.502337	+	0.864672i	$0.667526\pi$
$884$	0	0
$885$	−75978.4	−2.88586
$886$	−20864.4	−0.791144
$887$	21405.8	0.810301	0.405151	−	0.914250i	$-0.367219\pi$
$887$	21405.8	0.810301	0.405151	+	0.914250i	$0.367219\pi$
$888$	−25786.5	−0.974479
$889$	−1251.37	−0.0472098
$890$	25429.7	0.957758
$891$	22280.1	0.837723
$892$	40109.0	1.50555
$893$	1900.84	0.0712308
$894$	−78089.1	−2.92136
$895$	16678.5	0.622905
$896$	11301.5	0.421379
$897$	0	0
$898$	22286.5	0.828184
$899$	24541.0	0.910442
$900$	18859.5	0.698501
$901$	−15152.7	−0.560276
$902$	22818.8	0.842330
$903$	−3297.79	−0.121532
$904$	−52611.5	−1.93565
$905$	21644.5	0.795015
$906$	−6201.78	−0.227418
$907$	28887.5	1.05754	0.528772	−	0.848764i	$-0.322653\pi$
$907$	28887.5	1.05754	0.528772	+	0.848764i	$0.322653\pi$
$908$	15322.5	0.560018
$909$	9571.86	0.349261
$910$	0	0
$911$	−31360.9	−1.14054	−0.570271	−	0.821456i	$-0.693162\pi$
$911$	−31360.9	−1.14054	−0.570271	+	0.821456i	$0.693162\pi$
$912$	16219.6	0.588909
$913$	25614.4	0.928492
$914$	−79278.4	−2.86903
$915$	64111.0	2.31633
$916$	−6753.64	−0.243610
$917$	5187.94	0.186828
$918$	−59504.0	−2.13935
$919$	−1104.61	−0.0396493	−0.0198246	−	0.999803i	$-0.506311\pi$
$919$	−1104.61	−0.0396493	−0.0198246	+	0.999803i	$0.506311\pi$
$920$	−47866.0	−1.71532
$921$	−29350.8	−1.05010
$922$	−46618.2	−1.66517
$923$	0	0
$924$	−10137.6	−0.360935
$925$	12415.2	0.441308
$926$	86421.2	3.06693
$927$	1821.68	0.0645433
$928$	−9652.81	−0.341454
$929$	38435.8	1.35742	0.678708	−	0.734409i	$-0.262540\pi$
$929$	38435.8	1.35742	0.678708	+	0.734409i	$0.262540\pi$
$930$	49694.4	1.75220
$931$	16784.9	0.590874
$932$	45249.5	1.59034
$933$	29414.7	1.03215
$934$	39787.6	1.39389
$935$	−47612.7	−1.66535
$936$	0	0
$937$	21008.7	0.732469	0.366235	−	0.930522i	$-0.380647\pi$
$937$	21008.7	0.732469	0.366235	+	0.930522i	$0.380647\pi$
$938$	1827.57	0.0636166
$939$	−62684.3	−2.17852
$940$	8636.50	0.299672
$941$	12695.6	0.439814	0.219907	−	0.975521i	$-0.429425\pi$
$941$	12695.6	0.439814	0.219907	+	0.975521i	$0.429425\pi$
$942$	−20971.2	−0.725348
$943$	−16764.7	−0.578932
$944$	−40697.0	−1.40315
$945$	6359.40	0.218911
$946$	14662.8	0.503942
$947$	48719.4	1.67177	0.835886	−	0.548902i	$-0.184954\pi$
$947$	48719.4	1.67177	0.835886	+	0.548902i	$0.184954\pi$
$948$	−30691.5	−1.05149
$949$	0	0
$950$	−26862.7	−0.917410
$951$	−36598.0	−1.24792
$952$	19730.9	0.671725
$953$	−11347.3	−0.385704	−0.192852	−	0.981228i	$-0.561774\pi$
$953$	−11347.3	−0.385704	−0.192852	+	0.981228i	$0.561774\pi$
$954$	6574.00	0.223104
$955$	13666.8	0.463085
$956$	−89686.3	−3.03417
$957$	34365.7	1.16080
$958$	15314.8	0.516491
$959$	−3365.00	−0.113307
$960$	−57761.4	−1.94192
$961$	−17970.9	−0.603232
$962$	0	0
$963$	12260.0	0.410254
$964$	−17714.4	−0.591849
$965$	−39470.8	−1.31669
$966$	11310.2	0.376709
$967$	−10585.2	−0.352014	−0.176007	−	0.984389i	$-0.556318\pi$
$967$	−10585.2	−0.352014	−0.176007	+	0.984389i	$0.556318\pi$
$968$	−26148.1	−0.868215
$969$	−40774.2	−1.35176
$970$	35592.5	1.17815
$971$	−53730.8	−1.77580	−0.887900	−	0.460036i	$-0.847836\pi$
$971$	−53730.8	−1.77580	−0.887900	+	0.460036i	$0.847836\pi$
$972$	46420.0	1.53181
$973$	−8768.30	−0.288899
$974$	36243.2	1.19231
$975$	0	0
$976$	34340.4	1.12624
$977$	9534.31	0.312210	0.156105	−	0.987740i	$-0.450106\pi$
$977$	9534.31	0.312210	0.156105	+	0.987740i	$0.450106\pi$
$978$	88302.8	2.88713
$979$	8470.11	0.276513
$980$	76262.7	2.48584
$981$	−13240.8	−0.430933
$982$	−77508.9	−2.51875
$983$	8972.94	0.291142	0.145571	−	0.989348i	$-0.453498\pi$
$983$	8972.94	0.291142	0.145571	+	0.989348i	$0.453498\pi$
$984$	42745.3	1.38483
$985$	−22583.4	−0.730526
$986$	−138929.	−4.48723
$987$	−982.480	−0.0316846
$988$	0	0
$989$	−10772.6	−0.346359
$990$	20656.8	0.663148
$991$	−34900.6	−1.11872	−0.559362	−	0.828924i	$-0.688954\pi$
$991$	−34900.6	−1.11872	−0.559362	+	0.828924i	$0.688954\pi$
$992$	−4649.26	−0.148804
$993$	−17368.0	−0.555043
$994$	20495.3	0.653995
$995$	54855.1	1.74776
$996$	99664.2	3.17066
$997$	−48154.9	−1.52967	−0.764836	−	0.644225i	$-0.777180\pi$
$997$	−48154.9	−1.52967	−0.764836	+	0.644225i	$0.777180\pi$
$998$	−50062.8	−1.58788
$999$	11191.2	0.354429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.k.1.9	✓	9
3.2	odd	2		1521.4.a.bh.1.1		9
13.2	odd	12		169.4.e.h.147.17		36
13.3	even	3		169.4.c.l.22.1		18
13.4	even	6		169.4.c.k.146.9		18
13.5	odd	4		169.4.b.g.168.2		18
13.6	odd	12		169.4.e.h.23.2		36
13.7	odd	12		169.4.e.h.23.17		36
13.8	odd	4		169.4.b.g.168.17		18
13.9	even	3		169.4.c.l.146.1		18
13.10	even	6		169.4.c.k.22.9		18
13.11	odd	12		169.4.e.h.147.2		36
13.12	even	2		169.4.a.l.1.1	yes	9
39.38	odd	2		1521.4.a.bg.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.9	✓	9	1.1	even	1	trivial
169.4.a.l.1.1	yes	9	13.12	even	2
169.4.b.g.168.2		18	13.5	odd	4
169.4.b.g.168.17		18	13.8	odd	4
169.4.c.k.22.9		18	13.10	even	6
169.4.c.k.146.9		18	13.4	even	6
169.4.c.l.22.1		18	13.3	even	3
169.4.c.l.146.1		18	13.9	even	3
169.4.e.h.23.2		36	13.6	odd	12
169.4.e.h.23.17		36	13.7	odd	12
169.4.e.h.147.2		36	13.11	odd	12
169.4.e.h.147.17		36	13.2	odd	12
1521.4.a.bg.1.9		9	39.38	odd	2
1521.4.a.bh.1.1		9	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+4.84018 q^{2} -6.19662 q^{3} +15.4273 q^{4} -15.2399 q^{5} -29.9927 q^{6} -4.31620 q^{7} +35.9495 q^{8} +11.3981 q^{9} +O(q^{10})\)	\(q+4.84018 q^{2} -6.19662 q^{3} +15.4273 q^{4} -15.2399 q^{5} -29.9927 q^{6} -4.31620 q^{7} +35.9495 q^{8} +11.3981 q^{9} -73.7636 q^{10} -24.5691 q^{11} -95.5971 q^{12} -20.8912 q^{14} +94.4355 q^{15} +50.5834 q^{16} -127.160 q^{17} +55.1687 q^{18} -51.7462 q^{19} -235.110 q^{20} +26.7459 q^{21} -118.919 q^{22} +87.3684 q^{23} -222.765 q^{24} +107.253 q^{25} +96.6792 q^{27} -66.5874 q^{28} +225.726 q^{29} +457.085 q^{30} +108.720 q^{31} -42.7634 q^{32} +152.246 q^{33} -615.479 q^{34} +65.7783 q^{35} +175.842 q^{36} +115.756 q^{37} -250.461 q^{38} -547.865 q^{40} -191.885 q^{41} +129.455 q^{42} -123.301 q^{43} -379.036 q^{44} -173.705 q^{45} +422.878 q^{46} -36.7339 q^{47} -313.446 q^{48} -324.370 q^{49} +519.124 q^{50} +787.965 q^{51} +119.162 q^{53} +467.944 q^{54} +374.430 q^{55} -155.165 q^{56} +320.651 q^{57} +1092.55 q^{58} -804.553 q^{59} +1456.89 q^{60} +678.886 q^{61} +526.226 q^{62} -49.1964 q^{63} -611.650 q^{64} +736.896 q^{66} -87.4806 q^{67} -1961.74 q^{68} -541.388 q^{69} +318.379 q^{70} -981.049 q^{71} +409.755 q^{72} +263.862 q^{73} +560.281 q^{74} -664.606 q^{75} -798.304 q^{76} +106.045 q^{77} +321.051 q^{79} -770.883 q^{80} -906.832 q^{81} -928.757 q^{82} -1042.54 q^{83} +412.617 q^{84} +1937.91 q^{85} -596.798 q^{86} -1398.74 q^{87} -883.248 q^{88} -344.746 q^{89} -840.762 q^{90} +1347.86 q^{92} -673.698 q^{93} -177.799 q^{94} +788.604 q^{95} +264.989 q^{96} -482.521 q^{97} -1570.01 q^{98} -280.041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9 - 147 * q^10 - 181 * q^11 + 39 * q^12 - 147 * q^14 - 218 * q^15 + 269 * q^16 - 55 * q^17 - 79 * q^18 - 161 * q^19 - 370 * q^20 - 188 * q^21 + 340 * q^22 - 204 * q^23 - 798 * q^24 + 307 * q^25 - 668 * q^27 - 344 * q^28 + 280 * q^29 + 521 * q^30 - 706 * q^31 - 680 * q^32 - 500 * q^33 - 216 * q^34 + 20 * q^35 - 909 * q^36 - 298 * q^37 - 739 * q^38 + 13 * q^40 - 1201 * q^41 - 4 * q^42 - 533 * q^43 - 355 * q^44 + 90 * q^45 + 840 * q^46 - 956 * q^47 - 132 * q^48 + 403 * q^49 + 1156 * q^50 + 470 * q^51 - 278 * q^53 + 2555 * q^54 - 250 * q^55 + 250 * q^56 + 810 * q^57 + 2877 * q^58 - 1377 * q^59 + 3157 * q^60 - 136 * q^61 + 2035 * q^62 + 944 * q^63 + 284 * q^64 + 3279 * q^66 + 931 * q^67 - 1536 * q^68 - 2050 * q^69 + 4854 * q^70 - 2046 * q^71 + 4342 * q^72 + 45 * q^73 - 1990 * q^74 + 2393 * q^75 + 3608 * q^76 - 718 * q^77 + 412 * q^79 + 787 * q^80 - 835 * q^81 + 2757 * q^82 - 3709 * q^83 + 1539 * q^84 + 2106 * q^85 - 125 * q^86 - 786 * q^87 - 636 * q^88 - 1663 * q^89 - 1280 * q^90 + 4010 * q^92 + 1186 * q^93 - 2531 * q^94 - 1614 * q^95 + 3084 * q^96 + 1087 * q^97 + 282 * q^98 - 1357 * q^99

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