LMFDB - Embedded newform 169.4.a.l.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:	$9$
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			1.2
Root			$4.83438$ 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)

$f(q)$	$=$	$q-3.83438 q^{2} -0.279163 q^{3} +6.70249 q^{4} +11.3710 q^{5} +1.07042 q^{6} +31.0623 q^{7} +4.97517 q^{8} -26.9221 q^{9} +O(q^{10})$	\(q-3.83438 q^{2} -0.279163 q^{3} +6.70249 q^{4} +11.3710 q^{5} +1.07042 q^{6} +31.0623 q^{7} +4.97517 q^{8} -26.9221 q^{9} -43.6008 q^{10} -20.9478 q^{11} -1.87108 q^{12} -119.105 q^{14} -3.17436 q^{15} -72.6966 q^{16} +114.389 q^{17} +103.229 q^{18} -45.1768 q^{19} +76.2140 q^{20} -8.67143 q^{21} +80.3219 q^{22} +73.9590 q^{23} -1.38888 q^{24} +4.29980 q^{25} +15.0530 q^{27} +208.194 q^{28} -27.2113 q^{29} +12.1717 q^{30} +179.587 q^{31} +238.945 q^{32} +5.84785 q^{33} -438.610 q^{34} +353.209 q^{35} -180.445 q^{36} +354.736 q^{37} +173.225 q^{38} +56.5727 q^{40} +81.4187 q^{41} +33.2496 q^{42} -256.018 q^{43} -140.402 q^{44} -306.131 q^{45} -283.587 q^{46} +463.501 q^{47} +20.2942 q^{48} +621.865 q^{49} -16.4871 q^{50} -31.9331 q^{51} +76.6055 q^{53} -57.7191 q^{54} -238.198 q^{55} +154.540 q^{56} +12.6117 q^{57} +104.338 q^{58} +54.4676 q^{59} -21.2761 q^{60} -494.496 q^{61} -688.606 q^{62} -836.261 q^{63} -334.634 q^{64} -22.4229 q^{66} -611.991 q^{67} +766.688 q^{68} -20.6466 q^{69} -1354.34 q^{70} +16.3056 q^{71} -133.942 q^{72} +321.825 q^{73} -1360.19 q^{74} -1.20035 q^{75} -302.797 q^{76} -650.686 q^{77} +385.324 q^{79} -826.633 q^{80} +722.694 q^{81} -312.190 q^{82} +663.760 q^{83} -58.1202 q^{84} +1300.71 q^{85} +981.671 q^{86} +7.59638 q^{87} -104.219 q^{88} +545.723 q^{89} +1173.82 q^{90} +495.709 q^{92} -50.1341 q^{93} -1777.24 q^{94} -513.706 q^{95} -66.7046 q^{96} -689.189 q^{97} -2384.47 q^{98} +563.958 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.83438	−1.35566	−0.677829	−	0.735219i	$-0.737079\pi$
$2$	−3.83438	−1.35566	−0.677829	+	0.735219i	$0.737079\pi$
$3$	−0.279163	−0.0537249	−0.0268625	−	0.999639i	$-0.508552\pi$
$3$	−0.279163	−0.0537249	−0.0268625	+	0.999639i	$0.508552\pi$
$4$	6.70249	0.837811
$5$	11.3710	1.01705	0.508527	−	0.861046i	$-0.330190\pi$
$5$	11.3710	1.01705	0.508527	+	0.861046i	$0.330190\pi$
$6$	1.07042	0.0728327
$7$	31.0623	1.67721	0.838603	−	0.544744i	$-0.183373\pi$
$7$	31.0623	1.67721	0.838603	+	0.544744i	$0.183373\pi$
$8$	4.97517	0.219873
$9$	−26.9221	−0.997114
$10$	−43.6008	−1.37878
$11$	−20.9478	−0.574182	−0.287091	−	0.957903i	$-0.592688\pi$
$11$	−20.9478	−0.574182	−0.287091	+	0.957903i	$0.592688\pi$
$12$	−1.87108	−0.0450113
$13$	0	0
$13$	0	0
$14$	−119.105	−2.27372
$15$	−3.17436	−0.0546411
$16$	−72.6966	−1.13588
$17$	114.389	1.63196	0.815980	−	0.578080i	$-0.196198\pi$
$17$	114.389	1.63196	0.815980	+	0.578080i	$0.196198\pi$
$18$	103.229	1.35175
$19$	−45.1768	−0.545488	−0.272744	−	0.962087i	$-0.587931\pi$
$19$	−45.1768	−0.545488	−0.272744	+	0.962087i	$0.587931\pi$
$20$	76.2140	0.852098
$21$	−8.67143	−0.0901077
$22$	80.3219	0.778395
$23$	73.9590	0.670501	0.335250	−	0.942129i	$-0.391179\pi$
$23$	73.9590	0.670501	0.335250	+	0.942129i	$0.391179\pi$
$24$	−1.38888	−0.0118127
$25$	4.29980	0.0343984
$26$	0	0
$27$	15.0530	0.107295
$28$	208.194	1.40518
$29$	−27.2113	−0.174242	−0.0871208	−	0.996198i	$-0.527767\pi$
$29$	−27.2113	−0.174242	−0.0871208	+	0.996198i	$0.527767\pi$
$30$	12.1717	0.0740747
$31$	179.587	1.04048	0.520239	−	0.854021i	$-0.325843\pi$
$31$	179.587	1.04048	0.520239	+	0.854021i	$0.325843\pi$
$32$	238.945	1.32000
$33$	5.84785	0.0308479
$34$	−438.610	−2.21238
$35$	353.209	1.70581
$36$	−180.445	−0.835392
$37$	354.736	1.57617	0.788085	−	0.615567i	$-0.211073\pi$
$37$	354.736	1.57617	0.788085	+	0.615567i	$0.211073\pi$
$38$	173.225	0.739496
$39$	0	0
$40$	56.5727	0.223623
$41$	81.4187	0.310133	0.155067	−	0.987904i	$-0.450441\pi$
$41$	81.4187	0.310133	0.155067	+	0.987904i	$0.450441\pi$
$42$	33.2496	0.122155
$43$	−256.018	−0.907962	−0.453981	−	0.891011i	$-0.649997\pi$
$43$	−256.018	−0.907962	−0.453981	+	0.891011i	$0.649997\pi$
$44$	−140.402	−0.481056
$45$	−306.131	−1.01412
$46$	−283.587	−0.908970
$47$	463.501	1.43848	0.719240	−	0.694762i	$-0.244490\pi$
$47$	463.501	1.43848	0.719240	+	0.694762i	$0.244490\pi$
$48$	20.2942	0.0610253
$49$	621.865	1.81302
$50$	−16.4871	−0.0466325
$51$	−31.9331	−0.0876769
$52$	0	0
$53$	76.6055	0.198539	0.0992695	−	0.995061i	$-0.468349\pi$
$53$	76.6055	0.198539	0.0992695	+	0.995061i	$0.468349\pi$
$54$	−57.7191	−0.145455
$55$	−238.198	−0.583974
$56$	154.540	0.368773
$57$	12.6117	0.0293063
$58$	104.338	0.236212
$59$	54.4676	0.120188	0.0600939	−	0.998193i	$-0.480860\pi$
$59$	54.4676	0.120188	0.0600939	+	0.998193i	$0.480860\pi$
$60$	−21.2761	−0.0457789
$61$	−494.496	−1.03793	−0.518965	−	0.854795i	$-0.673683\pi$
$61$	−494.496	−1.03793	−0.518965	+	0.854795i	$0.673683\pi$
$62$	−688.606	−1.41053
$63$	−836.261	−1.67236
$64$	−334.634	−0.653582
$65$	0	0
$66$	−22.4229	−0.0418192
$67$	−611.991	−1.11592	−0.557960	−	0.829868i	$-0.688416\pi$
$67$	−611.991	−1.11592	−0.557960	+	0.829868i	$0.688416\pi$
$68$	766.688	1.36727
$69$	−20.6466	−0.0360226
$70$	−1354.34	−2.31249
$71$	16.3056	0.0272551	0.0136276	−	0.999907i	$-0.495662\pi$
$71$	16.3056	0.0272551	0.0136276	+	0.999907i	$0.495662\pi$
$72$	−133.942	−0.219239
$73$	321.825	0.515983	0.257992	−	0.966147i	$-0.416939\pi$
$73$	321.825	0.515983	0.257992	+	0.966147i	$0.416939\pi$
$74$	−1360.19	−2.13675
$75$	−1.20035	−0.00184805
$76$	−302.797	−0.457016
$77$	−650.686	−0.963021
$78$	0	0
$79$	385.324	0.548764	0.274382	−	0.961621i	$-0.411527\pi$
$79$	385.324	0.548764	0.274382	+	0.961621i	$0.411527\pi$
$80$	−826.633	−1.15526
$81$	722.694	0.991349
$82$	−312.190	−0.420435
$83$	663.760	0.877797	0.438899	−	0.898537i	$-0.355369\pi$
$83$	663.760	0.877797	0.438899	+	0.898537i	$0.355369\pi$
$84$	−58.1202	−0.0754932
$85$	1300.71	1.65979
$86$	981.671	1.23089
$87$	7.59638	0.00936112
$88$	−104.219	−0.126247
$89$	545.723	0.649961	0.324980	−	0.945721i	$-0.394642\pi$
$89$	545.723	0.649961	0.324980	+	0.945721i	$0.394642\pi$
$90$	1173.82	1.37480
$91$	0	0
$92$	495.709	0.561753
$93$	−50.1341	−0.0558996
$94$	−1777.24	−1.95009
$95$	−513.706	−0.554791
$96$	−66.7046	−0.0709168
$97$	−689.189	−0.721408	−0.360704	−	0.932680i	$-0.617464\pi$
$97$	−689.189	−0.721408	−0.360704	+	0.932680i	$0.617464\pi$
$98$	−2384.47	−2.45783
$99$	563.958	0.572524
$100$	28.8194	0.0288194
$101$	471.398	0.464414	0.232207	−	0.972666i	$-0.425405\pi$
$101$	471.398	0.464414	0.232207	+	0.972666i	$0.425405\pi$
$102$	122.444	0.118860
$103$	−335.521	−0.320970	−0.160485	−	0.987038i	$-0.551306\pi$
$103$	−335.521	−0.320970	−0.160485	+	0.987038i	$0.551306\pi$
$104$	0	0
$105$	−98.6029	−0.0916444
$106$	−293.735	−0.269151
$107$	1629.26	1.47203	0.736014	−	0.676967i	$-0.236706\pi$
$107$	1629.26	1.47203	0.736014	+	0.676967i	$0.236706\pi$
$108$	100.893	0.0898927
$109$	−1518.95	−1.33476	−0.667381	−	0.744717i	$-0.732585\pi$
$109$	−1518.95	−1.33476	−0.667381	+	0.744717i	$0.732585\pi$
$110$	913.341	0.791669
$111$	−99.0292	−0.0846796
$112$	−2258.12	−1.90511
$113$	1195.41	0.995173	0.497586	−	0.867414i	$-0.334220\pi$
$113$	1195.41	0.995173	0.497586	+	0.867414i	$0.334220\pi$
$114$	−48.3581	−0.0397294
$115$	840.988	0.681935
$116$	−182.383	−0.145982
$117$	0	0
$118$	−208.850	−0.162934
$119$	3553.17	2.73713
$120$	−15.7930	−0.0120141
$121$	−892.190	−0.670315
$122$	1896.09	1.40708
$123$	−22.7291	−0.0166619
$124$	1203.68	0.871723
$125$	−1372.48	−0.982069
$126$	3206.54	2.26716
$127$	526.425	0.367816	0.183908	−	0.982943i	$-0.441125\pi$
$127$	526.425	0.367816	0.183908	+	0.982943i	$0.441125\pi$
$128$	−628.446	−0.433963
$129$	71.4707	0.0487802
$130$	0	0
$131$	−834.024	−0.556252	−0.278126	−	0.960545i	$-0.589713\pi$
$131$	−834.024	−0.556252	−0.278126	+	0.960545i	$0.589713\pi$
$132$	39.1951	0.0258447
$133$	−1403.30	−0.914896
$134$	2346.61	1.51281
$135$	171.168	0.109125
$136$	569.103	0.358825
$137$	−466.762	−0.291082	−0.145541	−	0.989352i	$-0.546492\pi$
$137$	−466.762	−0.291082	−0.145541	+	0.989352i	$0.546492\pi$
$138$	79.1670	0.0488344
$139$	−713.080	−0.435127	−0.217563	−	0.976046i	$-0.569811\pi$
$139$	−713.080	−0.435127	−0.217563	+	0.976046i	$0.569811\pi$
$140$	2367.38	1.42914
$141$	−129.392	−0.0772822
$142$	−62.5218	−0.0369487
$143$	0	0
$144$	1957.14	1.13261
$145$	−309.420	−0.177213
$146$	−1234.00	−0.699497
$147$	−173.602	−0.0974042
$148$	2377.61	1.32053
$149$	662.814	0.364428	0.182214	−	0.983259i	$-0.441674\pi$
$149$	662.814	0.364428	0.182214	+	0.983259i	$0.441674\pi$
$150$	4.60258	0.00250533
$151$	190.862	0.102862	0.0514310	−	0.998677i	$-0.483622\pi$
$151$	190.862	0.102862	0.0514310	+	0.998677i	$0.483622\pi$
$152$	−224.762	−0.119938
$153$	−3079.58	−1.62725
$154$	2494.98	1.30553
$155$	2042.09	1.05822
$156$	0	0
$157$	−2833.96	−1.44060	−0.720302	−	0.693661i	$-0.755997\pi$
$157$	−2833.96	−1.44060	−0.720302	+	0.693661i	$0.755997\pi$
$158$	−1477.48	−0.743937
$159$	−21.3854	−0.0106665
$160$	2717.05	1.34251
$161$	2297.34	1.12457
$162$	−2771.08	−1.34393
$163$	3565.55	1.71334	0.856672	−	0.515861i	$-0.172528\pi$
$163$	3565.55	1.71334	0.856672	+	0.515861i	$0.172528\pi$
$164$	545.708	0.259833
$165$	66.4959	0.0313739
$166$	−2545.11	−1.18999
$167$	−120.630	−0.0558959	−0.0279480	−	0.999609i	$-0.508897\pi$
$167$	−120.630	−0.0558959	−0.0279480	+	0.999609i	$0.508897\pi$
$168$	−43.1418	−0.0198123
$169$	0	0
$170$	−4987.44	−2.25011
$171$	1216.25	0.543914
$172$	−1715.96	−0.760700
$173$	−2080.28	−0.914223	−0.457112	−	0.889409i	$-0.651116\pi$
$173$	−2080.28	−0.914223	−0.457112	+	0.889409i	$0.651116\pi$
$174$	−29.1274	−0.0126905
$175$	133.562	0.0576932
$176$	1522.83	0.652204
$177$	−15.2053	−0.00645708
$178$	−2092.51	−0.881125
$179$	−553.140	−0.230970	−0.115485	−	0.993309i	$-0.536842\pi$
$179$	−553.140	−0.230970	−0.115485	+	0.993309i	$0.536842\pi$
$180$	−2051.84	−0.849639
$181$	−3305.19	−1.35731	−0.678655	−	0.734457i	$-0.737437\pi$
$181$	−3305.19	−1.35731	−0.678655	+	0.734457i	$0.737437\pi$
$182$	0	0
$183$	138.045	0.0557627
$184$	367.958	0.147425
$185$	4033.71	1.60305
$186$	192.233	0.0757807
$187$	−2396.19	−0.937042
$188$	3106.61	1.20517
$189$	467.582	0.179955
$190$	1969.75	0.752107
$191$	−3318.14	−1.25703	−0.628514	−	0.777798i	$-0.716336\pi$
$191$	−3318.14	−1.25703	−0.628514	+	0.777798i	$0.716336\pi$
$192$	93.4174	0.0351137
$193$	−3902.18	−1.45536	−0.727681	−	0.685916i	$-0.759402\pi$
$193$	−3902.18	−1.45536	−0.727681	+	0.685916i	$0.759402\pi$
$194$	2642.61	0.977983
$195$	0	0
$196$	4168.04	1.51897
$197$	−3632.99	−1.31391	−0.656953	−	0.753932i	$-0.728155\pi$
$197$	−3632.99	−1.31391	−0.656953	+	0.753932i	$0.728155\pi$
$198$	−2162.43	−0.776148
$199$	2957.23	1.05343	0.526715	−	0.850042i	$-0.323424\pi$
$199$	2957.23	1.05343	0.526715	+	0.850042i	$0.323424\pi$
$200$	21.3922	0.00756330
$201$	170.845	0.0599527
$202$	−1807.52	−0.629587
$203$	−845.245	−0.292239
$204$	−214.031	−0.0734567
$205$	925.813	0.315422
$206$	1286.52	0.435125
$207$	−1991.13	−0.668565
$208$	0	0
$209$	946.355	0.313209
$210$	378.081	0.124239
$211$	−3194.05	−1.04212	−0.521061	−	0.853520i	$-0.674464\pi$
$211$	−3194.05	−1.04212	−0.521061	+	0.853520i	$0.674464\pi$
$212$	513.447	0.166338
$213$	−4.55191	−0.00146428
$214$	−6247.22	−1.99557
$215$	−2911.18	−0.923446
$216$	74.8914	0.0235913
$217$	5578.39	1.74509
$218$	5824.23	1.80948
$219$	−89.8416	−0.0277211
$220$	−1596.52	−0.489259
$221$	0	0
$222$	379.716	0.114797
$223$	75.1092	0.0225546	0.0112773	−	0.999936i	$-0.496410\pi$
$223$	75.1092	0.0225546	0.0112773	+	0.999936i	$0.496410\pi$
$224$	7422.18	2.21391
$225$	−115.760	−0.0342991
$226$	−4583.65	−1.34911
$227$	4322.79	1.26394	0.631968	−	0.774995i	$-0.282248\pi$
$227$	4322.79	1.26394	0.631968	+	0.774995i	$0.282248\pi$
$228$	84.5297	0.0245531
$229$	−677.923	−0.195626	−0.0978132	−	0.995205i	$-0.531185\pi$
$229$	−677.923	−0.195626	−0.0978132	+	0.995205i	$0.531185\pi$
$230$	−3224.67	−0.924472
$231$	181.647	0.0517382
$232$	−135.381	−0.0383111
$233$	92.9061	0.0261222	0.0130611	−	0.999915i	$-0.495842\pi$
$233$	92.9061	0.0261222	0.0130611	+	0.999915i	$0.495842\pi$
$234$	0	0
$235$	5270.47	1.46301
$236$	365.068	0.100695
$237$	−107.568	−0.0294823
$238$	−13624.2	−3.71062
$239$	−2082.56	−0.563638	−0.281819	−	0.959468i	$-0.590938\pi$
$239$	−2082.56	−0.563638	−0.281819	+	0.959468i	$0.590938\pi$
$240$	230.765	0.0620660
$241$	−6541.25	−1.74838	−0.874189	−	0.485586i	$-0.838606\pi$
$241$	−6541.25	−1.74838	−0.874189	+	0.485586i	$0.838606\pi$
$242$	3421.00	0.908719
$243$	−608.181	−0.160555
$244$	−3314.35	−0.869589
$245$	7071.23	1.84394
$246$	87.1520	0.0225878
$247$	0	0
$248$	893.476	0.228773
$249$	−185.297	−0.0471596
$250$	5262.62	1.33135
$251$	−2409.29	−0.605868	−0.302934	−	0.953012i	$-0.597966\pi$
$251$	−2409.29	−0.605868	−0.302934	+	0.953012i	$0.597966\pi$
$252$	−5605.03	−1.40112
$253$	−1549.28	−0.384989
$254$	−2018.51	−0.498633
$255$	−363.111	−0.0891722
$256$	5086.77	1.24189
$257$	−263.587	−0.0639770	−0.0319885	−	0.999488i	$-0.510184\pi$
$257$	−263.587	−0.0639770	−0.0319885	+	0.999488i	$0.510184\pi$
$258$	−274.046	−0.0661293
$259$	11018.9	2.64356
$260$	0	0
$261$	732.584	0.173739
$262$	3197.97	0.754088
$263$	−7324.62	−1.71732	−0.858660	−	0.512545i	$-0.828703\pi$
$263$	−7324.62	−1.71732	−0.858660	+	0.512545i	$0.828703\pi$
$264$	29.0940	0.00678263
$265$	871.081	0.201925
$266$	5380.77	1.24029
$267$	−152.346	−0.0349191
$268$	−4101.86	−0.934930
$269$	−4683.37	−1.06152	−0.530762	−	0.847521i	$-0.678094\pi$
$269$	−4683.37	−1.06152	−0.530762	+	0.847521i	$0.678094\pi$
$270$	−656.324	−0.147936
$271$	−1629.37	−0.365231	−0.182615	−	0.983184i	$-0.558456\pi$
$271$	−1629.37	−0.365231	−0.182615	+	0.983184i	$0.558456\pi$
$272$	−8315.66	−1.85372
$273$	0	0
$274$	1789.75	0.394608
$275$	−90.0714	−0.0197510
$276$	−138.384	−0.0301801
$277$	6200.66	1.34499	0.672494	−	0.740103i	$-0.265223\pi$
$277$	6200.66	1.34499	0.672494	+	0.740103i	$0.265223\pi$
$278$	2734.22	0.589883
$279$	−4834.86	−1.03747
$280$	1757.28	0.375062
$281$	2951.26	0.626538	0.313269	−	0.949664i	$-0.398576\pi$
$281$	2951.26	0.626538	0.313269	+	0.949664i	$0.398576\pi$
$282$	496.139	0.104768
$283$	5249.17	1.10258	0.551291	−	0.834313i	$-0.314135\pi$
$283$	5249.17	1.10258	0.551291	+	0.834313i	$0.314135\pi$
$284$	109.288	0.0228346
$285$	143.408	0.0298061
$286$	0	0
$287$	2529.05	0.520157
$288$	−6432.90	−1.31619
$289$	8171.77	1.66330
$290$	1186.43	0.240241
$291$	192.396	0.0387576
$292$	2157.03	0.432296
$293$	−2912.33	−0.580684	−0.290342	−	0.956923i	$-0.593769\pi$
$293$	−2912.33	−0.580684	−0.290342	+	0.956923i	$0.593769\pi$
$294$	665.655	0.132047
$295$	619.352	0.122237
$296$	1764.87	0.346558
$297$	−315.328	−0.0616067
$298$	−2541.48	−0.494040
$299$	0	0
$300$	−8.04530	−0.00154832
$301$	−7952.50	−1.52284
$302$	−731.840	−0.139446
$303$	−131.597	−0.0249506
$304$	3284.20	0.619611
$305$	−5622.92	−1.05563
$306$	11808.3	2.20600
$307$	5000.10	0.929546	0.464773	−	0.885430i	$-0.346136\pi$
$307$	5000.10	0.929546	0.464773	+	0.885430i	$0.346136\pi$
$308$	−4361.22	−0.806829
$309$	93.6650	0.0172441
$310$	−7830.14	−1.43459
$311$	−7840.94	−1.42964	−0.714822	−	0.699307i	$-0.753492\pi$
$311$	−7840.94	−1.42964	−0.714822	+	0.699307i	$0.753492\pi$
$312$	0	0
$313$	−7518.43	−1.35772	−0.678860	−	0.734267i	$-0.737526\pi$
$313$	−7518.43	−1.35772	−0.678860	+	0.734267i	$0.737526\pi$
$314$	10866.5	1.95297
$315$	−9509.13	−1.70088
$316$	2582.63	0.459761
$317$	485.238	0.0859738	0.0429869	−	0.999076i	$-0.486313\pi$
$317$	485.238	0.0859738	0.0429869	+	0.999076i	$0.486313\pi$
$318$	81.9998	0.0144601
$319$	570.017	0.100046
$320$	−3805.13	−0.664728
$321$	−454.830	−0.0790845
$322$	−8808.86	−1.52453
$323$	−5167.72	−0.890215
$324$	4843.84	0.830563
$325$	0	0
$326$	−13671.7	−2.32271
$327$	424.034	0.0717099
$328$	405.072	0.0681901
$329$	14397.4	2.41263
$330$	−254.971	−0.0425324
$331$	8222.88	1.36547	0.682735	−	0.730666i	$-0.260790\pi$
$331$	8222.88	1.36547	0.682735	+	0.730666i	$0.260790\pi$
$332$	4448.84	0.735428
$333$	−9550.23	−1.57162
$334$	462.541	0.0757758
$335$	−6958.96	−1.13495
$336$	630.384	0.102352
$337$	7744.78	1.25188	0.625942	−	0.779870i	$-0.284715\pi$
$337$	7744.78	1.25188	0.625942	+	0.779870i	$0.284715\pi$
$338$	0	0
$339$	−333.714	−0.0534656
$340$	8718.02	1.39059
$341$	−3761.96	−0.597423
$342$	−4663.58	−0.737362
$343$	8662.19	1.36360
$344$	−1273.73	−0.199637
$345$	−234.773	−0.0366369
$346$	7976.58	1.23937
$347$	−3532.61	−0.546514	−0.273257	−	0.961941i	$-0.588101\pi$
$347$	−3532.61	−0.546514	−0.273257	+	0.961941i	$0.588101\pi$
$348$	50.9146	0.00784285
$349$	3366.19	0.516298	0.258149	−	0.966105i	$-0.416888\pi$
$349$	3366.19	0.516298	0.258149	+	0.966105i	$0.416888\pi$
$350$	−512.127	−0.0782123
$351$	0	0
$352$	−5005.37	−0.757919
$353$	−9967.98	−1.50295	−0.751476	−	0.659761i	$-0.770658\pi$
$353$	−9967.98	−1.50295	−0.751476	+	0.659761i	$0.770658\pi$
$354$	58.3031	0.00875360
$355$	185.411	0.0277199
$356$	3657.70	0.544544
$357$	−991.914	−0.147052
$358$	2120.95	0.313117
$359$	2742.72	0.403219	0.201609	−	0.979466i	$-0.435383\pi$
$359$	2742.72	0.403219	0.201609	+	0.979466i	$0.435383\pi$
$360$	−1523.05	−0.222978
$361$	−4818.05	−0.702443
$362$	12673.4	1.84005
$363$	249.066	0.0360126
$364$	0	0
$365$	3659.47	0.524783
$366$	−529.317	−0.0755952
$367$	−1641.86	−0.233527	−0.116763	−	0.993160i	$-0.537252\pi$
$367$	−1641.86	−0.233527	−0.116763	+	0.993160i	$0.537252\pi$
$368$	−5376.57	−0.761611
$369$	−2191.96	−0.309238
$370$	−15466.8	−2.17319
$371$	2379.54	0.332991
$372$	−336.023	−0.0468333
$373$	11207.5	1.55577	0.777887	−	0.628404i	$-0.216291\pi$
$373$	11207.5	1.55577	0.777887	+	0.628404i	$0.216291\pi$
$374$	9187.91	1.27031
$375$	383.146	0.0527616
$376$	2306.00	0.316284
$377$	0	0
$378$	−1792.89	−0.243958
$379$	3844.47	0.521048	0.260524	−	0.965467i	$-0.416105\pi$
$379$	3844.47	0.521048	0.260524	+	0.965467i	$0.416105\pi$
$380$	−3443.11	−0.464810
$381$	−146.958	−0.0197609
$382$	12723.0	1.70410
$383$	12154.3	1.62156	0.810778	−	0.585354i	$-0.199045\pi$
$383$	12154.3	1.62156	0.810778	+	0.585354i	$0.199045\pi$
$384$	175.439	0.0233146
$385$	−7398.96	−0.979444
$386$	14962.4	1.97297
$387$	6892.53	0.905341
$388$	−4619.28	−0.604403
$389$	−5269.41	−0.686812	−0.343406	−	0.939187i	$-0.611581\pi$
$389$	−5269.41	−0.686812	−0.343406	+	0.939187i	$0.611581\pi$
$390$	0	0
$391$	8460.07	1.09423
$392$	3093.88	0.398634
$393$	232.829	0.0298846
$394$	13930.3	1.78121
$395$	4381.53	0.558123
$396$	3779.92	0.479667
$397$	655.483	0.0828659	0.0414329	−	0.999141i	$-0.486808\pi$
$397$	655.483	0.0828659	0.0414329	+	0.999141i	$0.486808\pi$
$398$	−11339.2	−1.42809
$399$	391.748	0.0491527
$400$	−312.581	−0.0390726
$401$	7699.97	0.958898	0.479449	−	0.877570i	$-0.340837\pi$
$401$	7699.97	0.958898	0.479449	+	0.877570i	$0.340837\pi$
$402$	−655.086	−0.0812754
$403$	0	0
$404$	3159.53	0.389091
$405$	8217.75	1.00826
$406$	3240.99	0.396177
$407$	−7430.94	−0.905008
$408$	−158.872	−0.0192778
$409$	−555.201	−0.0671220	−0.0335610	−	0.999437i	$-0.510685\pi$
$409$	−555.201	−0.0671220	−0.0335610	+	0.999437i	$0.510685\pi$
$410$	−3549.92	−0.427605
$411$	130.303	0.0156383
$412$	−2248.83	−0.268912
$413$	1691.89	0.201580
$414$	7634.75	0.906347
$415$	7547.62	0.892767
$416$	0	0
$417$	199.065	0.0233772
$418$	−3628.69	−0.424605
$419$	3316.99	0.386744	0.193372	−	0.981126i	$-0.438058\pi$
$419$	3316.99	0.386744	0.193372	+	0.981126i	$0.438058\pi$
$420$	−660.885	−0.0767807
$421$	−13270.3	−1.53623	−0.768117	−	0.640310i	$-0.778806\pi$
$421$	−13270.3	−1.53623	−0.768117	+	0.640310i	$0.778806\pi$
$422$	12247.2	1.41276
$423$	−12478.4	−1.43433
$424$	381.125	0.0436535
$425$	491.849	0.0561369
$426$	17.4538	0.00198506
$427$	−15360.2	−1.74082
$428$	10920.1	1.23328
$429$	0	0
$430$	11162.6	1.25188
$431$	13692.2	1.53023	0.765117	−	0.643892i	$-0.222681\pi$
$431$	13692.2	1.53023	0.765117	+	0.643892i	$0.222681\pi$
$432$	−1094.30	−0.121874
$433$	−15189.1	−1.68578	−0.842888	−	0.538089i	$-0.819146\pi$
$433$	−15189.1	−1.68578	−0.842888	+	0.538089i	$0.819146\pi$
$434$	−21389.7	−2.36575
$435$	86.3785	0.00952076
$436$	−10180.7	−1.11828
$437$	−3341.23	−0.365750
$438$	344.487	0.0375804
$439$	−2802.04	−0.304634	−0.152317	−	0.988332i	$-0.548673\pi$
$439$	−2802.04	−0.304634	−0.152317	+	0.988332i	$0.548673\pi$
$440$	−1185.07	−0.128400
$441$	−16741.9	−1.80778
$442$	0	0
$443$	−7539.78	−0.808636	−0.404318	−	0.914618i	$-0.632491\pi$
$443$	−7539.78	−0.808636	−0.404318	+	0.914618i	$0.632491\pi$
$444$	−663.742	−0.0709455
$445$	6205.42	0.661045
$446$	−287.997	−0.0305764
$447$	−185.033	−0.0195789
$448$	−10394.5	−1.09619
$449$	5558.82	0.584269	0.292135	−	0.956377i	$-0.405634\pi$
$449$	5558.82	0.584269	0.292135	+	0.956377i	$0.405634\pi$
$450$	443.867	0.0464979
$451$	−1705.54	−0.178073
$452$	8012.20	0.833766
$453$	−53.2817	−0.00552625
$454$	−16575.2	−1.71347
$455$	0	0
$456$	62.7453	0.00644368
$457$	−4327.71	−0.442980	−0.221490	−	0.975163i	$-0.571092\pi$
$457$	−4327.71	−0.442980	−0.221490	+	0.975163i	$0.571092\pi$
$458$	2599.42	0.265203
$459$	1721.90	0.175101
$460$	5636.71	0.571333
$461$	−11292.7	−1.14090	−0.570450	−	0.821333i	$-0.693231\pi$
$461$	−11292.7	−1.14090	−0.570450	+	0.821333i	$0.693231\pi$
$462$	−696.506	−0.0701394
$463$	8582.54	0.861478	0.430739	−	0.902477i	$-0.358253\pi$
$463$	8582.54	0.861478	0.430739	+	0.902477i	$0.358253\pi$
$464$	1978.17	0.197918
$465$	−570.075	−0.0568529
$466$	−356.238	−0.0354128
$467$	17543.0	1.73831	0.869157	−	0.494536i	$-0.164662\pi$
$467$	17543.0	1.73831	0.869157	+	0.494536i	$0.164662\pi$
$468$	0	0
$469$	−19009.8	−1.87163
$470$	−20209.0	−1.98334
$471$	791.137	0.0773963
$472$	270.986	0.0264261
$473$	5363.01	0.521335
$474$	412.458	0.0399680
$475$	−194.252	−0.0187639
$476$	23815.1	2.29320
$477$	−2062.38	−0.197966
$478$	7985.32	0.764101
$479$	6299.15	0.600867	0.300434	−	0.953803i	$-0.402869\pi$
$479$	6299.15	0.600867	0.300434	+	0.953803i	$0.402869\pi$
$480$	−758.498	−0.0721262
$481$	0	0
$482$	25081.7	2.37020
$483$	−641.331	−0.0604173
$484$	−5979.89	−0.561597
$485$	−7836.78	−0.733711
$486$	2332.00	0.217658
$487$	3571.73	0.332342	0.166171	−	0.986097i	$-0.446860\pi$
$487$	3571.73	0.332342	0.166171	+	0.986097i	$0.446860\pi$
$488$	−2460.20	−0.228213
$489$	−995.368	−0.0920493
$490$	−27113.8	−2.49975
$491$	−13385.2	−1.23028	−0.615139	−	0.788419i	$-0.710900\pi$
$491$	−13385.2	−1.23028	−0.615139	+	0.788419i	$0.710900\pi$
$492$	−152.341	−0.0139595
$493$	−3112.66	−0.284356
$494$	0	0
$495$	6412.77	0.582288
$496$	−13055.4	−1.18186
$497$	506.488	0.0457125
$498$	710.500	0.0639323
$499$	−19227.7	−1.72495	−0.862476	−	0.506098i	$-0.831087\pi$
$499$	−19227.7	−1.72495	−0.862476	+	0.506098i	$0.831087\pi$
$500$	−9199.05	−0.822788
$501$	33.6754	0.00300300
$502$	9238.13	0.821350
$503$	−18651.2	−1.65332	−0.826658	−	0.562705i	$-0.809761\pi$
$503$	−18651.2	−1.65332	−0.826658	+	0.562705i	$0.809761\pi$
$504$	−4160.54	−0.367708
$505$	5360.26	0.472334
$506$	5940.53	0.521914
$507$	0	0
$508$	3528.35	0.308160
$509$	−15886.1	−1.38338	−0.691688	−	0.722196i	$-0.743133\pi$
$509$	−15886.1	−1.38338	−0.691688	+	0.722196i	$0.743133\pi$
$510$	1392.31	0.120887
$511$	9996.62	0.865410
$512$	−14477.1	−1.24961
$513$	−680.049	−0.0585280
$514$	1010.69	0.0867310
$515$	−3815.21	−0.326443
$516$	479.031	0.0408686
$517$	−9709.33	−0.825949
$518$	−42250.7	−3.58377
$519$	580.736	0.0491166
$520$	0	0
$521$	1824.67	0.153436	0.0767179	−	0.997053i	$-0.475556\pi$
$521$	1824.67	0.153436	0.0767179	+	0.997053i	$0.475556\pi$
$522$	−2809.01	−0.235530
$523$	16206.3	1.35498	0.677488	−	0.735534i	$-0.263069\pi$
$523$	16206.3	1.35498	0.677488	+	0.735534i	$0.263069\pi$
$524$	−5590.04	−0.466034
$525$	−37.2855	−0.00309956
$526$	28085.4	2.32810
$527$	20542.7	1.69802
$528$	−425.119	−0.0350396
$529$	−6697.07	−0.550429
$530$	−3340.06	−0.273741
$531$	−1466.38	−0.119841
$532$	−9405.57	−0.766510
$533$	0	0
$534$	584.151	0.0473384
$535$	18526.4	1.49713
$536$	−3044.76	−0.245361
$537$	154.416	0.0124088
$538$	17957.8	1.43906
$539$	−13026.7	−1.04100
$540$	1147.25	0.0914257
$541$	2429.83	0.193099	0.0965496	−	0.995328i	$-0.469219\pi$
$541$	2429.83	0.193099	0.0965496	+	0.995328i	$0.469219\pi$
$542$	6247.65	0.495128
$543$	922.687	0.0729213
$544$	27332.6	2.15418
$545$	−17272.0	−1.35752
$546$	0	0
$547$	−2409.16	−0.188315	−0.0941573	−	0.995557i	$-0.530016\pi$
$547$	−2409.16	−0.188315	−0.0941573	+	0.995557i	$0.530016\pi$
$548$	−3128.47	−0.243871
$549$	13312.9	1.03493
$550$	345.368	0.0267756
$551$	1229.32	0.0950468
$552$	−102.720	−0.00792041
$553$	11969.1	0.920390
$554$	−23775.7	−1.82334
$555$	−1126.06	−0.0861237
$556$	−4779.41	−0.364554
$557$	−8668.70	−0.659434	−0.329717	−	0.944080i	$-0.606953\pi$
$557$	−8668.70	−0.659434	−0.329717	+	0.944080i	$0.606953\pi$
$558$	18538.7	1.40646
$559$	0	0
$560$	−25677.1	−1.93760
$561$	668.928	0.0503425
$562$	−11316.2	−0.849371
$563$	−7818.35	−0.585265	−0.292632	−	0.956225i	$-0.594531\pi$
$563$	−7818.35	−0.585265	−0.292632	+	0.956225i	$0.594531\pi$
$564$	−867.250	−0.0647479
$565$	13593.0	1.01214
$566$	−20127.3	−1.49473
$567$	22448.5	1.66270
$568$	81.1229	0.00599268
$569$	9117.24	0.671730	0.335865	−	0.941910i	$-0.390971\pi$
$569$	9117.24	0.671730	0.335865	+	0.941910i	$0.390971\pi$
$570$	−549.880	−0.0404069
$571$	−11842.4	−0.867932	−0.433966	−	0.900929i	$-0.642886\pi$
$571$	−11842.4	−0.867932	−0.433966	+	0.900929i	$0.642886\pi$
$572$	0	0
$573$	926.302	0.0675337
$574$	−9697.35	−0.705156
$575$	318.009	0.0230642
$576$	9009.04	0.651696
$577$	−10958.1	−0.790623	−0.395312	−	0.918547i	$-0.629363\pi$
$577$	−10958.1	−0.790623	−0.395312	+	0.918547i	$0.629363\pi$
$578$	−31333.7	−2.25486
$579$	1089.34	0.0781892
$580$	−2073.88	−0.148471
$581$	20617.9	1.47225
$582$	−737.720	−0.0525421
$583$	−1604.72	−0.113997
$584$	1601.13	0.113451
$585$	0	0
$586$	11167.0	0.787209
$587$	23057.8	1.62129	0.810644	−	0.585539i	$-0.199117\pi$
$587$	23057.8	1.62129	0.810644	+	0.585539i	$0.199117\pi$
$588$	−1163.56	−0.0816063
$589$	−8113.18	−0.567568
$590$	−2374.83	−0.165712
$591$	1014.19	0.0705895
$592$	−25788.1	−1.79035
$593$	−9904.56	−0.685888	−0.342944	−	0.939356i	$-0.611424\pi$
$593$	−9904.56	−0.685888	−0.342944	+	0.939356i	$0.611424\pi$
$594$	1209.09	0.0835177
$595$	40403.2	2.78381
$596$	4442.50	0.305322
$597$	−825.549	−0.0565954
$598$	0	0
$599$	21334.5	1.45527	0.727633	−	0.685967i	$-0.240620\pi$
$599$	21334.5	1.45527	0.727633	+	0.685967i	$0.240620\pi$
$600$	−5.97192	−0.000406338 0
$601$	−19484.4	−1.32244	−0.661219	−	0.750193i	$-0.729960\pi$
$601$	−19484.4	−1.32244	−0.661219	+	0.750193i	$0.729960\pi$
$602$	30492.9	2.06445
$603$	16476.1	1.11270
$604$	1279.25	0.0861789
$605$	−10145.1	−0.681747
$606$	504.592	0.0338245
$607$	19855.7	1.32770	0.663852	−	0.747864i	$-0.268921\pi$
$607$	19855.7	1.32770	0.663852	+	0.747864i	$0.268921\pi$
$608$	−10794.8	−0.720043
$609$	235.961	0.0157005
$610$	21560.4	1.43108
$611$	0	0
$612$	−20640.8	−1.36333
$613$	5153.57	0.339561	0.169781	−	0.985482i	$-0.445694\pi$
$613$	5153.57	0.339561	0.169781	+	0.985482i	$0.445694\pi$
$614$	−19172.3	−1.26015
$615$	−258.453	−0.0169460
$616$	−3237.27	−0.211743
$617$	−924.218	−0.0603040	−0.0301520	−	0.999545i	$-0.509599\pi$
$617$	−924.218	−0.0603040	−0.0301520	+	0.999545i	$0.509599\pi$
$618$	−359.147	−0.0233771
$619$	−15690.6	−1.01883	−0.509417	−	0.860520i	$-0.670139\pi$
$619$	−15690.6	−1.01883	−0.509417	+	0.860520i	$0.670139\pi$
$620$	13687.1	0.886589
$621$	1113.31	0.0719412
$622$	30065.2	1.93811
$623$	16951.4	1.09012
$624$	0	0
$625$	−16144.0	−1.03322
$626$	28828.5	1.84061
$627$	−264.187	−0.0168272
$628$	−18994.6	−1.20695
$629$	40577.8	2.57225
$630$	36461.6	2.30582
$631$	21103.6	1.33141	0.665705	−	0.746215i	$-0.268131\pi$
$631$	21103.6	1.33141	0.665705	+	0.746215i	$0.268131\pi$
$632$	1917.05	0.120659
$633$	891.661	0.0559879
$634$	−1860.59	−0.116551
$635$	5985.98	0.374089
$636$	−143.335	−0.00893650
$637$	0	0
$638$	−2185.66	−0.135629
$639$	−438.980	−0.0271765
$640$	−7146.06	−0.441364
$641$	−6695.38	−0.412561	−0.206280	−	0.978493i	$-0.566136\pi$
$641$	−6695.38	−0.412561	−0.206280	+	0.978493i	$0.566136\pi$
$642$	1743.99	0.107212
$643$	−9647.69	−0.591707	−0.295854	−	0.955233i	$-0.595604\pi$
$643$	−9647.69	−0.591707	−0.295854	+	0.955233i	$0.595604\pi$
$644$	15397.9	0.942175
$645$	812.694	0.0496121
$646$	19815.0	1.20683
$647$	−23747.5	−1.44299	−0.721493	−	0.692422i	$-0.756544\pi$
$647$	−23747.5	−1.44299	−0.721493	+	0.692422i	$0.756544\pi$
$648$	3595.52	0.217971
$649$	−1140.98	−0.0690096
$650$	0	0
$651$	−1557.28	−0.0937551
$652$	23898.0	1.43546
$653$	19833.5	1.18858	0.594292	−	0.804249i	$-0.297432\pi$
$653$	19833.5	1.18858	0.594292	+	0.804249i	$0.297432\pi$
$654$	−1625.91	−0.0972142
$655$	−9483.70	−0.565738
$656$	−5918.86	−0.352275
$657$	−8664.19	−0.514494
$658$	−55205.1	−3.27070
$659$	10807.4	0.638841	0.319421	−	0.947613i	$-0.396512\pi$
$659$	10807.4	0.638841	0.319421	+	0.947613i	$0.396512\pi$
$660$	445.688	0.0262854
$661$	13431.5	0.790356	0.395178	−	0.918605i	$-0.370683\pi$
$661$	13431.5	0.790356	0.395178	+	0.918605i	$0.370683\pi$
$662$	−31529.7	−1.85111
$663$	0	0
$664$	3302.32	0.193004
$665$	−15956.9	−0.930498
$666$	36619.2	2.13058
$667$	−2012.52	−0.116829
$668$	−808.520	−0.0468302
$669$	−20.9677	−0.00121175
$670$	26683.3	1.53861
$671$	10358.6	0.595961
$672$	−2072.00	−0.118942
$673$	21581.4	1.23611	0.618054	−	0.786136i	$-0.287921\pi$
$673$	21581.4	1.23611	0.618054	+	0.786136i	$0.287921\pi$
$674$	−29696.4	−1.69713
$675$	64.7251	0.00369077
$676$	0	0
$677$	17482.4	0.992474	0.496237	−	0.868187i	$-0.334715\pi$
$677$	17482.4	0.992474	0.496237	+	0.868187i	$0.334715\pi$
$678$	1279.59	0.0724811
$679$	−21407.8	−1.20995
$680$	6471.27	0.364944
$681$	−1206.76	−0.0679048
$682$	14424.8	0.809902
$683$	−2842.23	−0.159231	−0.0796155	−	0.996826i	$-0.525369\pi$
$683$	−2842.23	−0.159231	−0.0796155	+	0.996826i	$0.525369\pi$
$684$	8151.92	0.455697
$685$	−5307.56	−0.296046
$686$	−33214.1	−1.84857
$687$	189.251	0.0105100
$688$	18611.6	1.03134
$689$	0	0
$690$	900.208	0.0496672
$691$	−22875.5	−1.25937	−0.629686	−	0.776850i	$-0.716816\pi$
$691$	−22875.5	−1.25937	−0.629686	+	0.776850i	$0.716816\pi$
$692$	−13943.0	−0.765946
$693$	17517.8	0.960241
$694$	13545.4	0.740886
$695$	−8108.43	−0.442547
$696$	37.7933	0.00205826
$697$	9313.38	0.506125
$698$	−12907.2	−0.699923
$699$	−25.9359	−0.00140342
$700$	895.195	0.0483360
$701$	11980.3	0.645492	0.322746	−	0.946486i	$-0.395394\pi$
$701$	11980.3	0.645492	0.322746	+	0.946486i	$0.395394\pi$
$702$	0	0
$703$	−16025.9	−0.859782
$704$	7009.85	0.375275
$705$	−1471.32	−0.0786002
$706$	38221.0	2.03749
$707$	14642.7	0.778918
$708$	−101.914	−0.00540981
$709$	−4204.77	−0.222727	−0.111363	−	0.993780i	$-0.535522\pi$
$709$	−4204.77	−0.222727	−0.111363	+	0.993780i	$0.535522\pi$
$710$	−710.936	−0.0375788
$711$	−10373.7	−0.547180
$712$	2715.06	0.142909
$713$	13282.1	0.697641
$714$	3803.38	0.199353
$715$	0	0
$716$	−3707.41	−0.193509
$717$	581.373	0.0302814
$718$	−10516.7	−0.546627
$719$	−11537.0	−0.598411	−0.299205	−	0.954189i	$-0.596722\pi$
$719$	−11537.0	−0.598411	−0.299205	+	0.954189i	$0.596722\pi$
$720$	22254.7	1.15192
$721$	−10422.0	−0.538332
$722$	18474.3	0.952272
$723$	1826.07	0.0939315
$724$	−22153.0	−1.13717
$725$	−117.003	−0.00599364
$726$	−955.015	−0.0488208
$727$	−33899.7	−1.72940	−0.864698	−	0.502293i	$-0.832490\pi$
$727$	−33899.7	−1.72940	−0.864698	+	0.502293i	$0.832490\pi$
$728$	0	0
$729$	−19342.9	−0.982723
$730$	−14031.8	−0.711426
$731$	−29285.6	−1.48176
$732$	925.245	0.0467186
$733$	378.221	0.0190585	0.00952927	−	0.999955i	$-0.496967\pi$
$733$	378.221	0.0190585	0.00952927	+	0.999955i	$0.496967\pi$
$734$	6295.51	0.316582
$735$	−1974.03	−0.0990653
$736$	17672.1	0.885059
$737$	12819.9	0.640741
$738$	8404.81	0.419221
$739$	−22663.6	−1.12814	−0.564069	−	0.825728i	$-0.690765\pi$
$739$	−22663.6	−1.12814	−0.564069	+	0.825728i	$0.690765\pi$
$740$	27035.9	1.34305
$741$	0	0
$742$	−9124.06	−0.451422
$743$	−16407.3	−0.810130	−0.405065	−	0.914288i	$-0.632751\pi$
$743$	−16407.3	−0.810130	−0.405065	+	0.914288i	$0.632751\pi$
$744$	−249.425	−0.0122908
$745$	7536.86	0.370643
$746$	−42974.0	−2.10910
$747$	−17869.8	−0.875264
$748$	−16060.4	−0.785064
$749$	50608.7	2.46889
$750$	−1469.13	−0.0715267
$751$	21957.9	1.06692	0.533458	−	0.845827i	$-0.320892\pi$
$751$	21957.9	1.06692	0.533458	+	0.845827i	$0.320892\pi$
$752$	−33694.9	−1.63395
$753$	672.584	0.0325502
$754$	0	0
$755$	2170.30	0.104616
$756$	3133.96	0.150769
$757$	20823.5	0.999792	0.499896	−	0.866085i	$-0.333371\pi$
$757$	20823.5	0.999792	0.499896	+	0.866085i	$0.333371\pi$
$758$	−14741.2	−0.706364
$759$	432.501	0.0206835
$760$	−2555.77	−0.121984
$761$	28456.1	1.35550	0.677749	−	0.735294i	$-0.262956\pi$
$761$	28456.1	1.35550	0.677749	+	0.735294i	$0.262956\pi$
$762$	563.494	0.0267890
$763$	−47182.0	−2.23867
$764$	−22239.8	−1.05315
$765$	−35017.9	−1.65500
$766$	−46604.2	−2.19828
$767$	0	0
$768$	−1420.04	−0.0667203
$769$	13450.0	0.630713	0.315356	−	0.948973i	$-0.397876\pi$
$769$	13450.0	0.630713	0.315356	+	0.948973i	$0.397876\pi$
$770$	28370.4	1.32779
$771$	73.5836	0.00343716
$772$	−26154.3	−1.21932
$773$	−26577.8	−1.23666	−0.618329	−	0.785919i	$-0.712190\pi$
$773$	−26577.8	−1.23666	−0.618329	+	0.785919i	$0.712190\pi$
$774$	−26428.6	−1.22733
$775$	772.190	0.0357908
$776$	−3428.83	−0.158618
$777$	−3076.07	−0.142025
$778$	20204.9	0.931083
$779$	−3678.24	−0.169174
$780$	0	0
$781$	−341.566	−0.0156494
$782$	−32439.1	−1.48340
$783$	−409.613	−0.0186952
$784$	−45207.5	−2.05938
$785$	−32225.0	−1.46517
$786$	−892.754	−0.0405133
$787$	−8281.40	−0.375095	−0.187548	−	0.982256i	$-0.560054\pi$
$787$	−8281.40	−0.375095	−0.187548	+	0.982256i	$0.560054\pi$
$788$	−24350.0	−1.10080
$789$	2044.76	0.0922629
$790$	−16800.4	−0.756624
$791$	37132.1	1.66911
$792$	2805.79	0.125883
$793$	0	0
$794$	−2513.37	−0.112338
$795$	−243.174	−0.0108484
$796$	19820.8	0.882575
$797$	20668.8	0.918605	0.459302	−	0.888280i	$-0.348099\pi$
$797$	20668.8	0.918605	0.459302	+	0.888280i	$0.348099\pi$
$798$	−1502.11	−0.0666343
$799$	53019.3	2.34754
$800$	1027.42	0.0454059
$801$	−14692.0	−0.648085
$802$	−29524.6	−1.29994
$803$	−6741.52	−0.296268
$804$	1145.09	0.0502290
$805$	26123.0	1.14375
$806$	0	0
$807$	1307.42	0.0570303
$808$	2345.28	0.102112
$809$	7238.85	0.314591	0.157296	−	0.987552i	$-0.449722\pi$
$809$	7238.85	0.314591	0.157296	+	0.987552i	$0.449722\pi$
$810$	−31510.0	−1.36685
$811$	13101.3	0.567260	0.283630	−	0.958934i	$-0.408461\pi$
$811$	13101.3	0.567260	0.283630	+	0.958934i	$0.408461\pi$
$812$	−5665.24	−0.244841
$813$	454.861	0.0196220
$814$	28493.1	1.22688
$815$	40543.9	1.74256
$816$	2321.42	0.0995908
$817$	11566.1	0.495283
$818$	2128.85	0.0909946
$819$	0	0
$820$	6205.25	0.264264
$821$	12092.9	0.514062	0.257031	−	0.966403i	$-0.417256\pi$
$821$	12092.9	0.514062	0.257031	+	0.966403i	$0.417256\pi$
$822$	−499.631	−0.0212003
$823$	27041.2	1.14532	0.572659	−	0.819794i	$-0.305912\pi$
$823$	27041.2	1.14532	0.572659	+	0.819794i	$0.305912\pi$
$824$	−1669.27	−0.0705727
$825$	25.1446	0.00106112
$826$	−6487.35	−0.273273
$827$	−25572.3	−1.07526	−0.537628	−	0.843182i	$-0.680680\pi$
$827$	−25572.3	−1.07526	−0.537628	+	0.843182i	$0.680680\pi$
$828$	−13345.5	−0.560131
$829$	−17505.8	−0.733415	−0.366707	−	0.930336i	$-0.619515\pi$
$829$	−17505.8	−0.733415	−0.366707	+	0.930336i	$0.619515\pi$
$830$	−28940.5	−1.21029
$831$	−1730.99	−0.0722593
$832$	0	0
$833$	71134.3	2.95877
$834$	−763.293	−0.0316914
$835$	−1371.68	−0.0568491
$836$	6342.93	0.262410
$837$	2703.33	0.111638
$838$	−12718.6	−0.524293
$839$	−28293.4	−1.16424	−0.582119	−	0.813103i	$-0.697776\pi$
$839$	−28293.4	−1.16424	−0.582119	+	0.813103i	$0.697776\pi$
$840$	−490.566	−0.0201502
$841$	−23648.5	−0.969640
$842$	50883.3	2.08261
$843$	−823.881	−0.0336607
$844$	−21408.1	−0.873100
$845$	0	0
$846$	47847.0	1.94446
$847$	−27713.4	−1.12426
$848$	−5568.95	−0.225517
$849$	−1465.37	−0.0592361
$850$	−1885.94	−0.0761025
$851$	26235.9	1.05682
$852$	−30.5091	−0.00122679
$853$	−15866.0	−0.636861	−0.318430	−	0.947946i	$-0.603156\pi$
$853$	−15866.0	−0.636861	−0.318430	+	0.947946i	$0.603156\pi$
$854$	58896.8	2.35996
$855$	13830.0	0.553190
$856$	8105.86	0.323660
$857$	23623.3	0.941607	0.470804	−	0.882238i	$-0.343964\pi$
$857$	23623.3	0.941607	0.470804	+	0.882238i	$0.343964\pi$
$858$	0	0
$859$	33403.1	1.32677	0.663387	−	0.748277i	$-0.269118\pi$
$859$	33403.1	1.32677	0.663387	+	0.748277i	$0.269118\pi$
$860$	−19512.2	−0.773673
$861$	−706.017	−0.0279454
$862$	−52501.2	−2.07447
$863$	−45490.5	−1.79434	−0.897169	−	0.441687i	$-0.854380\pi$
$863$	−45490.5	−1.79434	−0.897169	+	0.441687i	$0.854380\pi$
$864$	3596.85	0.141629
$865$	−23654.9	−0.929814
$866$	58240.8	2.28534
$867$	−2281.25	−0.0893604
$868$	37389.0	1.46206
$869$	−8071.70	−0.315090
$870$	−331.208	−0.0129069
$871$	0	0
$872$	−7557.03	−0.293479
$873$	18554.4	0.719326
$874$	12811.6	0.495833
$875$	−42632.4	−1.64713
$876$	−602.162	−0.0232251
$877$	−19570.1	−0.753518	−0.376759	−	0.926311i	$-0.622961\pi$
$877$	−19570.1	−0.753518	−0.376759	+	0.926311i	$0.622961\pi$
$878$	10744.1	0.412979
$879$	813.015	0.0311972
$880$	17316.1	0.663326
$881$	−3016.95	−0.115373	−0.0576864	−	0.998335i	$-0.518372\pi$
$881$	−3016.95	−0.115373	−0.0576864	+	0.998335i	$0.518372\pi$
$882$	64194.8	2.45074
$883$	17163.8	0.654141	0.327071	−	0.945000i	$-0.393939\pi$
$883$	17163.8	0.654141	0.327071	+	0.945000i	$0.393939\pi$
$884$	0	0
$885$	−172.900	−0.00656720
$886$	28910.4	1.09623
$887$	28464.7	1.07751	0.538754	−	0.842463i	$-0.318895\pi$
$887$	28464.7	1.07751	0.538754	+	0.842463i	$0.318895\pi$
$888$	−492.687	−0.0186188
$889$	16352.0	0.616903
$890$	−23793.9	−0.896151
$891$	−15138.8	−0.569215
$892$	503.418	0.0188965
$893$	−20939.5	−0.784674
$894$	709.487	0.0265423
$895$	−6289.76	−0.234909
$896$	−19520.9	−0.727845
$897$	0	0
$898$	−21314.6	−0.792070
$899$	−4886.80	−0.181295
$900$	−775.877	−0.0287362
$901$	8762.80	0.324008
$902$	6539.70	0.241406
$903$	2220.04	0.0818144
$904$	5947.36	0.218812
$905$	−37583.4	−1.38046
$906$	204.302	0.00749172
$907$	−26610.1	−0.974171	−0.487085	−	0.873354i	$-0.661940\pi$
$907$	−26610.1	−0.974171	−0.487085	+	0.873354i	$0.661940\pi$
$908$	28973.4	1.05894
$909$	−12691.0	−0.463073
$910$	0	0
$911$	−52648.2	−1.91472	−0.957362	−	0.288890i	$-0.906714\pi$
$911$	−52648.2	−1.91472	−0.957362	+	0.288890i	$0.906714\pi$
$912$	−916.827	−0.0332886
$913$	−13904.3	−0.504015
$914$	16594.1	0.600529
$915$	1569.71	0.0567137
$916$	−4543.77	−0.163898
$917$	−25906.7	−0.932949
$918$	−6602.41	−0.237377
$919$	46923.5	1.68429	0.842146	−	0.539250i	$-0.181292\pi$
$919$	46923.5	1.68429	0.842146	+	0.539250i	$0.181292\pi$
$920$	4184.06	0.149939
$921$	−1395.84	−0.0499398
$922$	43300.6	1.54667
$923$	0	0
$924$	1217.49	0.0433468
$925$	1525.30	0.0542178
$926$	−32908.7	−1.16787
$927$	9032.92	0.320043
$928$	−6502.00	−0.229999
$929$	41163.7	1.45376	0.726878	−	0.686767i	$-0.240971\pi$
$929$	41163.7	1.45376	0.726878	+	0.686767i	$0.240971\pi$
$930$	2185.88	0.0770731
$931$	−28093.9	−0.988980
$932$	622.702	0.0218855
$933$	2188.90	0.0768075
$934$	−67266.5	−2.35656
$935$	−27247.1	−0.953022
$936$	0	0
$937$	−23380.3	−0.815156	−0.407578	−	0.913170i	$-0.633627\pi$
$937$	−23380.3	−0.815156	−0.407578	+	0.913170i	$0.633627\pi$
$938$	72891.0	2.53729
$939$	2098.87	0.0729434
$940$	35325.3	1.22573
$941$	−34506.2	−1.19540	−0.597700	−	0.801720i	$-0.703918\pi$
$941$	−34506.2	−1.19540	−0.597700	+	0.801720i	$0.703918\pi$
$942$	−3033.52	−0.104923
$943$	6021.65	0.207945
$944$	−3959.61	−0.136519
$945$	5316.87	0.183024
$946$	−20563.8	−0.706753
$947$	−483.439	−0.0165889	−0.00829443	−	0.999966i	$-0.502640\pi$
$947$	−483.439	−0.0165889	−0.00829443	+	0.999966i	$0.502640\pi$
$948$	−720.975	−0.0247006
$949$	0	0
$950$	744.835	0.0254375
$951$	−135.460	−0.00461893
$952$	17677.6	0.601823
$953$	−11962.7	−0.406619	−0.203310	−	0.979114i	$-0.565170\pi$
$953$	−11962.7	−0.406619	−0.203310	+	0.979114i	$0.565170\pi$
$954$	7907.94	0.268374
$955$	−37730.6	−1.27846
$956$	−13958.3	−0.472222
$957$	−159.127	−0.00537499
$958$	−24153.3	−0.814571
$959$	−14498.7	−0.488204
$960$	1062.25	0.0357125
$961$	2460.54	0.0825934
$962$	0	0
$963$	−43863.2	−1.46778
$964$	−43842.6	−1.46481
$965$	−44371.7	−1.48018
$966$	2459.11	0.0819052
$967$	7695.43	0.255913	0.127957	−	0.991780i	$-0.459158\pi$
$967$	7695.43	0.255913	0.127957	+	0.991780i	$0.459158\pi$
$968$	−4438.79	−0.147385
$969$	1442.64	0.0478267
$970$	30049.2	0.994661
$971$	−44177.9	−1.46008	−0.730039	−	0.683406i	$-0.760498\pi$
$971$	−44177.9	−1.46008	−0.730039	+	0.683406i	$0.760498\pi$
$972$	−4076.33	−0.134515
$973$	−22149.9	−0.729797
$974$	−13695.4	−0.450542
$975$	0	0
$976$	35948.2	1.17897
$977$	3412.61	0.111749	0.0558746	−	0.998438i	$-0.482205\pi$
$977$	3412.61	0.111749	0.0558746	+	0.998438i	$0.482205\pi$
$978$	3816.62	0.124787
$979$	−11431.7	−0.373196
$980$	47394.8	1.54487
$981$	40893.3	1.33091
$982$	51324.0	1.66784
$983$	6313.78	0.204861	0.102431	−	0.994740i	$-0.467338\pi$
$983$	6313.78	0.204861	0.102431	+	0.994740i	$0.467338\pi$
$984$	−113.081	−0.00366351
$985$	−41310.7	−1.33631
$986$	11935.1	0.385489
$987$	−4019.22	−0.129618
$988$	0	0
$989$	−18934.8	−0.608789
$990$	−24589.0	−0.789384
$991$	12177.5	0.390344	0.195172	−	0.980769i	$-0.437474\pi$
$991$	12177.5	0.390344	0.195172	+	0.980769i	$0.437474\pi$
$992$	42911.5	1.37343
$993$	−2295.52	−0.0733598
$994$	−1942.07	−0.0619705
$995$	33626.7	1.07139
$996$	−1241.95	−0.0395108
$997$	17506.2	0.556096	0.278048	−	0.960567i	$-0.410313\pi$
$997$	17506.2	0.556096	0.278048	+	0.960567i	$0.410313\pi$
$998$	73726.4	2.33845
$999$	5339.86	0.169115

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.l.1.2	yes	9
3.2	odd	2		1521.4.a.bg.1.8		9
13.2	odd	12		169.4.e.h.147.4		36
13.3	even	3		169.4.c.k.22.8		18
13.4	even	6		169.4.c.l.146.2		18
13.5	odd	4		169.4.b.g.168.15		18
13.6	odd	12		169.4.e.h.23.15		36
13.7	odd	12		169.4.e.h.23.4		36
13.8	odd	4		169.4.b.g.168.4		18
13.9	even	3		169.4.c.k.146.8		18
13.10	even	6		169.4.c.l.22.2		18
13.11	odd	12		169.4.e.h.147.15		36
13.12	even	2		169.4.a.k.1.8	✓	9
39.38	odd	2		1521.4.a.bh.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.8	✓	9	13.12	even	2
169.4.a.l.1.2	yes	9	1.1	even	1	trivial
169.4.b.g.168.4		18	13.8	odd	4
169.4.b.g.168.15		18	13.5	odd	4
169.4.c.k.22.8		18	13.3	even	3
169.4.c.k.146.8		18	13.9	even	3
169.4.c.l.22.2		18	13.10	even	6
169.4.c.l.146.2		18	13.4	even	6
169.4.e.h.23.4		36	13.7	odd	12
169.4.e.h.23.15		36	13.6	odd	12
169.4.e.h.147.4		36	13.2	odd	12
169.4.e.h.147.15		36	13.11	odd	12
1521.4.a.bg.1.8		9	3.2	odd	2
1521.4.a.bh.1.2		9	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)

\(f(q)\)	\(=\)	\(q-3.83438 q^{2} -0.279163 q^{3} +6.70249 q^{4} +11.3710 q^{5} +1.07042 q^{6} +31.0623 q^{7} +4.97517 q^{8} -26.9221 q^{9} +O(q^{10})\)	\(q-3.83438 q^{2} -0.279163 q^{3} +6.70249 q^{4} +11.3710 q^{5} +1.07042 q^{6} +31.0623 q^{7} +4.97517 q^{8} -26.9221 q^{9} -43.6008 q^{10} -20.9478 q^{11} -1.87108 q^{12} -119.105 q^{14} -3.17436 q^{15} -72.6966 q^{16} +114.389 q^{17} +103.229 q^{18} -45.1768 q^{19} +76.2140 q^{20} -8.67143 q^{21} +80.3219 q^{22} +73.9590 q^{23} -1.38888 q^{24} +4.29980 q^{25} +15.0530 q^{27} +208.194 q^{28} -27.2113 q^{29} +12.1717 q^{30} +179.587 q^{31} +238.945 q^{32} +5.84785 q^{33} -438.610 q^{34} +353.209 q^{35} -180.445 q^{36} +354.736 q^{37} +173.225 q^{38} +56.5727 q^{40} +81.4187 q^{41} +33.2496 q^{42} -256.018 q^{43} -140.402 q^{44} -306.131 q^{45} -283.587 q^{46} +463.501 q^{47} +20.2942 q^{48} +621.865 q^{49} -16.4871 q^{50} -31.9331 q^{51} +76.6055 q^{53} -57.7191 q^{54} -238.198 q^{55} +154.540 q^{56} +12.6117 q^{57} +104.338 q^{58} +54.4676 q^{59} -21.2761 q^{60} -494.496 q^{61} -688.606 q^{62} -836.261 q^{63} -334.634 q^{64} -22.4229 q^{66} -611.991 q^{67} +766.688 q^{68} -20.6466 q^{69} -1354.34 q^{70} +16.3056 q^{71} -133.942 q^{72} +321.825 q^{73} -1360.19 q^{74} -1.20035 q^{75} -302.797 q^{76} -650.686 q^{77} +385.324 q^{79} -826.633 q^{80} +722.694 q^{81} -312.190 q^{82} +663.760 q^{83} -58.1202 q^{84} +1300.71 q^{85} +981.671 q^{86} +7.59638 q^{87} -104.219 q^{88} +545.723 q^{89} +1173.82 q^{90} +495.709 q^{92} -50.1341 q^{93} -1777.24 q^{94} -513.706 q^{95} -66.7046 q^{96} -689.189 q^{97} -2384.47 q^{98} +563.958 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + q^{3} + 37 q^{4} + 30 q^{5} + 48 q^{6} + 38 q^{7} + 60 q^{8} + 66 q^{9}+O(q^{10}) \) 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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