LMFDB - Embedded newform 169.4.a.l.1.5

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			1.5
Root			$0.850942$ 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)

$f(q)$	$=$	$q+0.149058 q^{2} -6.48858 q^{3} -7.97778 q^{4} -10.2526 q^{5} -0.967177 q^{6} -29.6743 q^{7} -2.38162 q^{8} +15.1017 q^{9} +O(q^{10})$	\(q+0.149058 q^{2} -6.48858 q^{3} -7.97778 q^{4} -10.2526 q^{5} -0.967177 q^{6} -29.6743 q^{7} -2.38162 q^{8} +15.1017 q^{9} -1.52823 q^{10} +38.1011 q^{11} +51.7645 q^{12} -4.42320 q^{14} +66.5245 q^{15} +63.4673 q^{16} -71.3014 q^{17} +2.25103 q^{18} +10.0947 q^{19} +81.7926 q^{20} +192.544 q^{21} +5.67928 q^{22} -198.665 q^{23} +15.4533 q^{24} -19.8851 q^{25} +77.2033 q^{27} +236.735 q^{28} +30.8164 q^{29} +9.91603 q^{30} +151.549 q^{31} +28.5133 q^{32} -247.222 q^{33} -10.6281 q^{34} +304.238 q^{35} -120.478 q^{36} +151.381 q^{37} +1.50470 q^{38} +24.4177 q^{40} -207.265 q^{41} +28.7003 q^{42} -303.215 q^{43} -303.962 q^{44} -154.831 q^{45} -29.6127 q^{46} +12.2241 q^{47} -411.812 q^{48} +537.565 q^{49} -2.96405 q^{50} +462.645 q^{51} -250.726 q^{53} +11.5078 q^{54} -390.633 q^{55} +70.6730 q^{56} -65.5005 q^{57} +4.59343 q^{58} +390.502 q^{59} -530.718 q^{60} -156.528 q^{61} +22.5896 q^{62} -448.132 q^{63} -503.488 q^{64} -36.8505 q^{66} -303.706 q^{67} +568.827 q^{68} +1289.05 q^{69} +45.3491 q^{70} +913.468 q^{71} -35.9664 q^{72} -249.855 q^{73} +22.5645 q^{74} +129.026 q^{75} -80.5336 q^{76} -1130.62 q^{77} -147.055 q^{79} -650.701 q^{80} -908.685 q^{81} -30.8945 q^{82} +1020.16 q^{83} -1536.08 q^{84} +731.021 q^{85} -45.1967 q^{86} -199.954 q^{87} -90.7423 q^{88} -946.062 q^{89} -23.0788 q^{90} +1584.91 q^{92} -983.337 q^{93} +1.82211 q^{94} -103.497 q^{95} -185.011 q^{96} +417.680 q^{97} +80.1286 q^{98} +575.390 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.149058	0.0527001	0.0263500	−	0.999653i	$-0.491612\pi$
$2$	0.149058	0.0527001	0.0263500	+	0.999653i	$0.491612\pi$
$3$	−6.48858	−1.24873	−0.624364	−	0.781134i	$-0.714642\pi$
$3$	−6.48858	−1.24873	−0.624364	+	0.781134i	$0.714642\pi$
$4$	−7.97778	−0.997223
$5$	−10.2526	−0.917016	−0.458508	−	0.888690i	$-0.651616\pi$
$5$	−10.2526	−0.917016	−0.458508	+	0.888690i	$0.651616\pi$
$6$	−0.967177	−0.0658080
$7$	−29.6743	−1.60226	−0.801131	−	0.598489i	$-0.795768\pi$
$7$	−29.6743	−1.60226	−0.801131	+	0.598489i	$0.795768\pi$
$8$	−2.38162	−0.105254
$9$	15.1017	0.559321
$10$	−1.52823	−0.0483268
$11$	38.1011	1.04436	0.522178	−	0.852837i	$-0.325120\pi$
$11$	38.1011	1.04436	0.522178	+	0.852837i	$0.325120\pi$
$12$	51.7645	1.24526
$13$	0	0
$13$	0	0
$14$	−4.42320	−0.0844394
$15$	66.5245	1.14510
$16$	63.4673	0.991676
$17$	−71.3014	−1.01724	−0.508621	−	0.860990i	$-0.669845\pi$
$17$	−71.3014	−1.01724	−0.508621	+	0.860990i	$0.669845\pi$
$18$	2.25103	0.0294763
$19$	10.0947	0.121889	0.0609445	−	0.998141i	$-0.480589\pi$
$19$	10.0947	0.121889	0.0609445	+	0.998141i	$0.480589\pi$
$20$	81.7926	0.914469
$21$	192.544	2.00079
$22$	5.67928	0.0550376
$23$	−198.665	−1.80107	−0.900534	−	0.434786i	$-0.856824\pi$
$23$	−198.665	−1.80107	−0.900534	+	0.434786i	$0.856824\pi$
$24$	15.4533	0.131433
$25$	−19.8851	−0.159081
$26$	0	0
$27$	77.2033	0.550288
$28$	236.735	1.59781
$29$	30.8164	0.197326	0.0986630	−	0.995121i	$-0.468543\pi$
$29$	30.8164	0.197326	0.0986630	+	0.995121i	$0.468543\pi$
$30$	9.91603	0.0603470
$31$	151.549	0.878031	0.439016	−	0.898479i	$-0.355327\pi$
$31$	151.549	0.878031	0.439016	+	0.898479i	$0.355327\pi$
$32$	28.5133	0.157515
$33$	−247.222	−1.30412
$34$	−10.6281	−0.0536088
$35$	304.238	1.46930
$36$	−120.478	−0.557768
$37$	151.381	0.672617	0.336308	−	0.941752i	$-0.390822\pi$
$37$	151.381	0.672617	0.336308	+	0.941752i	$0.390822\pi$
$38$	1.50470	0.00642356
$39$	0	0
$40$	24.4177	0.0965194
$41$	−207.265	−0.789495	−0.394748	−	0.918790i	$-0.629168\pi$
$41$	−207.265	−0.789495	−0.394748	+	0.918790i	$0.629168\pi$
$42$	28.7003	0.105442
$43$	−303.215	−1.07534	−0.537672	−	0.843154i	$-0.680696\pi$
$43$	−303.215	−1.07534	−0.537672	+	0.843154i	$0.680696\pi$
$44$	−303.962	−1.04145
$45$	−154.831	−0.512906
$46$	−29.6127	−0.0949164
$47$	12.2241	0.0379377	0.0189689	−	0.999820i	$-0.493962\pi$
$47$	12.2241	0.0379377	0.0189689	+	0.999820i	$0.493962\pi$
$48$	−411.812	−1.23833
$49$	537.565	1.56725
$50$	−2.96405	−0.00838359
$51$	462.645	1.27026
$52$	0	0
$53$	−250.726	−0.649808	−0.324904	−	0.945747i	$-0.605332\pi$
$53$	−250.726	−0.649808	−0.324904	+	0.945747i	$0.605332\pi$
$54$	11.5078	0.0290002
$55$	−390.633	−0.957691
$56$	70.6730	0.168644
$57$	−65.5005	−0.152206
$58$	4.59343	0.0103991
$59$	390.502	0.861679	0.430840	−	0.902428i	$-0.358217\pi$
$59$	390.502	0.861679	0.430840	+	0.902428i	$0.358217\pi$
$60$	−530.718	−1.14192
$61$	−156.528	−0.328546	−0.164273	−	0.986415i	$-0.552528\pi$
$61$	−156.528	−0.328546	−0.164273	+	0.986415i	$0.552528\pi$
$62$	22.5896	0.0462723
$63$	−448.132	−0.896179
$64$	−503.488	−0.983375
$65$	0	0
$66$	−36.8505	−0.0687270
$67$	−303.706	−0.553786	−0.276893	−	0.960901i	$-0.589305\pi$
$67$	−303.706	−0.553786	−0.276893	+	0.960901i	$0.589305\pi$
$68$	568.827	1.01442
$69$	1289.05	2.24904
$70$	45.3491	0.0774323
$71$	913.468	1.52688	0.763442	−	0.645876i	$-0.223508\pi$
$71$	913.468	1.52688	0.763442	+	0.645876i	$0.223508\pi$
$72$	−35.9664	−0.0588706
$73$	−249.855	−0.400594	−0.200297	−	0.979735i	$-0.564191\pi$
$73$	−249.855	−0.400594	−0.200297	+	0.979735i	$0.564191\pi$
$74$	22.5645	0.0354469
$75$	129.026	0.198649
$76$	−80.5336	−0.121551
$77$	−1130.62	−1.67333
$78$	0	0
$79$	−147.055	−0.209430	−0.104715	−	0.994502i	$-0.533393\pi$
$79$	−147.055	−0.209430	−0.104715	+	0.994502i	$0.533393\pi$
$80$	−650.701	−0.909383
$81$	−908.685	−1.24648
$82$	−30.8945	−0.0416065
$83$	1020.16	1.34913	0.674564	−	0.738217i	$-0.264332\pi$
$83$	1020.16	1.34913	0.674564	+	0.738217i	$0.264332\pi$
$84$	−1536.08	−1.99523
$85$	731.021	0.932828
$86$	−45.1967	−0.0566707
$87$	−199.954	−0.246406
$88$	−90.7423	−0.109922
$89$	−946.062	−1.12677	−0.563384	−	0.826195i	$-0.690501\pi$
$89$	−946.062	−1.12677	−0.563384	+	0.826195i	$0.690501\pi$
$90$	−23.0788	−0.0270302
$91$	0	0
$92$	1584.91	1.79607
$93$	−983.337	−1.09642
$94$	1.82211	0.00199932
$95$	−103.497	−0.111774
$96$	−185.011	−0.196694
$97$	417.680	0.437206	0.218603	−	0.975814i	$-0.429850\pi$
$97$	417.680	0.437206	0.218603	+	0.975814i	$0.429850\pi$
$98$	80.1286	0.0825939
$99$	575.390	0.584130
$100$	158.639	0.158639
$101$	850.691	0.838088	0.419044	−	0.907966i	$-0.362365\pi$
$101$	850.691	0.838088	0.419044	+	0.907966i	$0.362365\pi$
$102$	68.9611	0.0669428
$103$	996.615	0.953392	0.476696	−	0.879068i	$-0.341834\pi$
$103$	996.615	0.953392	0.476696	+	0.879068i	$0.341834\pi$
$104$	0	0
$105$	−1974.07	−1.83476
$106$	−37.3727	−0.0342449
$107$	693.925	0.626956	0.313478	−	0.949596i	$-0.398506\pi$
$107$	693.925	0.626956	0.313478	+	0.949596i	$0.398506\pi$
$108$	−615.911	−0.548760
$109$	−1130.33	−0.993267	−0.496634	−	0.867960i	$-0.665431\pi$
$109$	−1130.33	−0.993267	−0.496634	+	0.867960i	$0.665431\pi$
$110$	−58.2271	−0.0504704
$111$	−982.245	−0.839915
$112$	−1883.35	−1.58893
$113$	−263.539	−0.219395	−0.109698	−	0.993965i	$-0.534988\pi$
$113$	−263.539	−0.219395	−0.109698	+	0.993965i	$0.534988\pi$
$114$	−9.76340	−0.00802128
$115$	2036.83	1.65161
$116$	−245.846	−0.196778
$117$	0	0
$118$	58.2076	0.0454106
$119$	2115.82	1.62989
$120$	−158.436	−0.120526
$121$	120.692	0.0906774
$122$	−23.3317	−0.0173144
$123$	1344.85	0.985865
$124$	−1209.02	−0.875593
$125$	1485.44	1.06290
$126$	−66.7977	−0.0472287
$127$	−101.762	−0.0711019	−0.0355510	−	0.999368i	$-0.511319\pi$
$127$	−101.762	−0.0711019	−0.0355510	+	0.999368i	$0.511319\pi$
$128$	−303.155	−0.209339
$129$	1967.43	1.34281
$130$	0	0
$131$	−382.888	−0.255367	−0.127684	−	0.991815i	$-0.540754\pi$
$131$	−382.888	−0.255367	−0.127684	+	0.991815i	$0.540754\pi$
$132$	1972.28	1.30049
$133$	−299.554	−0.195298
$134$	−45.2700	−0.0291845
$135$	−791.531	−0.504623
$136$	169.813	0.107069
$137$	1023.45	0.638243	0.319122	−	0.947714i	$-0.396612\pi$
$137$	1023.45	0.638243	0.319122	+	0.947714i	$0.396612\pi$
$138$	192.144	0.118525
$139$	1018.95	0.621768	0.310884	−	0.950448i	$-0.399375\pi$
$139$	1018.95	0.621768	0.310884	+	0.950448i	$0.399375\pi$
$140$	−2427.14	−1.46522
$141$	−79.3173	−0.0473739
$142$	136.160	0.0804669
$143$	0	0
$144$	958.461	0.554665
$145$	−315.946	−0.180951
$146$	−37.2430	−0.0211113
$147$	−3488.03	−1.95706
$148$	−1207.68	−0.670749
$149$	−1977.31	−1.08716	−0.543581	−	0.839357i	$-0.682932\pi$
$149$	−1977.31	−1.08716	−0.543581	+	0.839357i	$0.682932\pi$
$150$	19.2325	0.0104688
$151$	3488.30	1.87996	0.939979	−	0.341234i	$-0.110845\pi$
$151$	3488.30	1.87996	0.939979	+	0.341234i	$0.110845\pi$
$152$	−24.0418	−0.0128293
$153$	−1076.77	−0.568965
$154$	−168.529	−0.0881847
$155$	−1553.76	−0.805169
$156$	0	0
$157$	1783.21	0.906471	0.453236	−	0.891391i	$-0.350270\pi$
$157$	1783.21	0.906471	0.453236	+	0.891391i	$0.350270\pi$
$158$	−21.9197	−0.0110370
$159$	1626.85	0.811433
$160$	−292.334	−0.144444
$161$	5895.25	2.88578
$162$	−135.447	−0.0656896
$163$	−1643.40	−0.789701	−0.394851	−	0.918745i	$-0.629204\pi$
$163$	−1643.40	−0.789701	−0.394851	+	0.918745i	$0.629204\pi$
$164$	1653.51	0.787303
$165$	2534.66	1.19589
$166$	152.064	0.0710991
$167$	2437.79	1.12959	0.564796	−	0.825231i	$-0.308955\pi$
$167$	2437.79	1.12959	0.564796	+	0.825231i	$0.308955\pi$
$168$	−458.567	−0.210591
$169$	0	0
$170$	108.965	0.0491601
$171$	152.447	0.0681751
$172$	2418.98	1.07236
$173$	2986.64	1.31254	0.656271	−	0.754525i	$-0.272133\pi$
$173$	2986.64	1.31254	0.656271	+	0.754525i	$0.272133\pi$
$174$	−29.8049	−0.0129856
$175$	590.078	0.254890
$176$	2418.17	1.03566
$177$	−2533.81	−1.07600
$178$	−141.018	−0.0593808
$179$	−3374.90	−1.40923	−0.704614	−	0.709591i	$-0.748880\pi$
$179$	−3374.90	−1.40923	−0.704614	+	0.709591i	$0.748880\pi$
$180$	1235.21	0.511482
$181$	2266.57	0.930790	0.465395	−	0.885103i	$-0.345912\pi$
$181$	2266.57	0.930790	0.465395	+	0.885103i	$0.345912\pi$
$182$	0	0
$183$	1015.64	0.410264
$184$	473.145	0.189569
$185$	−1552.04	−0.616800
$186$	−146.574	−0.0577815
$187$	−2716.66	−1.06236
$188$	−97.5215	−0.0378324
$189$	−2290.95	−0.881706
$190$	−15.4271	−0.00589051
$191$	670.719	0.254092	0.127046	−	0.991897i	$-0.459450\pi$
$191$	670.719	0.254092	0.127046	+	0.991897i	$0.459450\pi$
$192$	3266.92	1.22797
$193$	−4595.68	−1.71401	−0.857006	−	0.515307i	$-0.827678\pi$
$193$	−4595.68	−1.71401	−0.857006	+	0.515307i	$0.827678\pi$
$194$	62.2586	0.0230408
$195$	0	0
$196$	−4288.58	−1.56289
$197$	523.115	0.189190	0.0945949	−	0.995516i	$-0.469844\pi$
$197$	523.115	0.189190	0.0945949	+	0.995516i	$0.469844\pi$
$198$	85.7666	0.0307837
$199$	−1573.25	−0.560427	−0.280213	−	0.959938i	$-0.590405\pi$
$199$	−1573.25	−0.560427	−0.280213	+	0.959938i	$0.590405\pi$
$200$	47.3589	0.0167439
$201$	1970.62	0.691527
$202$	126.803	0.0441673
$203$	−914.454	−0.316168
$204$	−3690.88	−1.26673
$205$	2124.99	0.723980
$206$	148.554	0.0502439
$207$	−3000.17	−1.00737
$208$	0	0
$209$	384.620	0.127295
$210$	−294.251	−0.0966918
$211$	−4782.10	−1.56025	−0.780127	−	0.625622i	$-0.784845\pi$
$211$	−4782.10	−1.56025	−0.780127	+	0.625622i	$0.784845\pi$
$212$	2000.23	0.648003
$213$	−5927.11	−1.90666
$214$	103.435	0.0330406
$215$	3108.72	0.986108
$216$	−183.869	−0.0579199
$217$	−4497.11	−1.40684
$218$	−168.485	−0.0523452
$219$	1621.21	0.500233
$220$	3116.39	0.955031
$221$	0	0
$222$	−146.412	−0.0442636
$223$	−2090.37	−0.627721	−0.313860	−	0.949469i	$-0.601622\pi$
$223$	−2090.37	−0.627721	−0.313860	+	0.949469i	$0.601622\pi$
$224$	−846.112	−0.252381
$225$	−300.299	−0.0889774
$226$	−39.2827	−0.0115622
$227$	−2174.16	−0.635702	−0.317851	−	0.948141i	$-0.602961\pi$
$227$	−2174.16	−0.635702	−0.317851	+	0.948141i	$0.602961\pi$
$228$	522.549	0.151784
$229$	−5997.16	−1.73058	−0.865291	−	0.501270i	$-0.832866\pi$
$229$	−5997.16	−1.73058	−0.865291	+	0.501270i	$0.832866\pi$
$230$	303.606	0.0870399
$231$	7336.14	2.08953
$232$	−73.3929	−0.0207693
$233$	−1550.00	−0.435810	−0.217905	−	0.975970i	$-0.569922\pi$
$233$	−1550.00	−0.435810	−0.217905	+	0.975970i	$0.569922\pi$
$234$	0	0
$235$	−125.329	−0.0347895
$236$	−3115.34	−0.859286
$237$	954.176	0.261521
$238$	315.381	0.0858953
$239$	3895.06	1.05419	0.527093	−	0.849807i	$-0.323282\pi$
$239$	3895.06	1.05419	0.527093	+	0.849807i	$0.323282\pi$
$240$	4222.13	1.13557
$241$	−4088.19	−1.09271	−0.546356	−	0.837553i	$-0.683985\pi$
$241$	−4088.19	−1.09271	−0.546356	+	0.837553i	$0.683985\pi$
$242$	17.9901	0.00477870
$243$	3811.58	1.00623
$244$	1248.74	0.327633
$245$	−5511.42	−1.43719
$246$	200.462	0.0519551
$247$	0	0
$248$	−360.932	−0.0924161
$249$	−6619.42	−1.68469
$250$	221.418	0.0560147
$251$	5761.42	1.44883	0.724417	−	0.689362i	$-0.242109\pi$
$251$	5761.42	1.44883	0.724417	+	0.689362i	$0.242109\pi$
$252$	3575.10	0.893690
$253$	−7569.36	−1.88095
$254$	−15.1685	−0.00374708
$255$	−4743.29	−1.16485
$256$	3982.72	0.972343
$257$	−4899.09	−1.18909	−0.594547	−	0.804061i	$-0.702669\pi$
$257$	−4899.09	−1.18909	−0.594547	+	0.804061i	$0.702669\pi$
$258$	293.262	0.0707663
$259$	−4492.12	−1.07771
$260$	0	0
$261$	465.378	0.110369
$262$	−57.0727	−0.0134579
$263$	7611.57	1.78460	0.892300	−	0.451444i	$-0.149091\pi$
$263$	7611.57	1.78460	0.892300	+	0.451444i	$0.149091\pi$
$264$	588.789	0.137263
$265$	2570.58	0.595884
$266$	−44.6511	−0.0102922
$267$	6138.60	1.40703
$268$	2422.90	0.552248
$269$	−6596.55	−1.49516	−0.747582	−	0.664170i	$-0.768785\pi$
$269$	−6596.55	−1.49516	−0.747582	+	0.664170i	$0.768785\pi$
$270$	−117.984	−0.0265937
$271$	2904.72	0.651104	0.325552	−	0.945524i	$-0.394450\pi$
$271$	2904.72	0.651104	0.325552	+	0.945524i	$0.394450\pi$
$272$	−4525.30	−1.00878
$273$	0	0
$274$	152.554	0.0336355
$275$	−757.645	−0.166137
$276$	−10283.8	−2.24280
$277$	1161.11	0.251857	0.125928	−	0.992039i	$-0.459809\pi$
$277$	1161.11	0.251857	0.125928	+	0.992039i	$0.459809\pi$
$278$	151.882	0.0327672
$279$	2288.64	0.491101
$280$	−724.579	−0.154649
$281$	8132.21	1.72643	0.863215	−	0.504836i	$-0.168447\pi$
$281$	8132.21	1.72643	0.863215	+	0.504836i	$0.168447\pi$
$282$	−11.8229	−0.00249661
$283$	1131.86	0.237747	0.118873	−	0.992909i	$-0.462072\pi$
$283$	1131.86	0.237747	0.118873	+	0.992909i	$0.462072\pi$
$284$	−7287.45	−1.52264
$285$	671.548	0.139576
$286$	0	0
$287$	6150.44	1.26498
$288$	430.598	0.0881015
$289$	170.889	0.0347830
$290$	−47.0944	−0.00953614
$291$	−2710.15	−0.545951
$292$	1993.29	0.399481
$293$	3834.34	0.764520	0.382260	−	0.924055i	$-0.375146\pi$
$293$	3834.34	0.764520	0.382260	+	0.924055i	$0.375146\pi$
$294$	−519.920	−0.103137
$295$	−4003.65	−0.790174
$296$	−360.531	−0.0707954
$297$	2941.53	0.574696
$298$	−294.734	−0.0572935
$299$	0	0
$300$	−1029.34	−0.198097
$301$	8997.69	1.72298
$302$	519.960	0.0990739
$303$	−5519.78	−1.04654
$304$	640.685	0.120874
$305$	1604.81	0.301282
$306$	−160.501	−0.0299845
$307$	−6574.27	−1.22219	−0.611097	−	0.791556i	$-0.709271\pi$
$307$	−6574.27	−1.22219	−0.611097	+	0.791556i	$0.709271\pi$
$308$	9019.87	1.66868
$309$	−6466.62	−1.19053
$310$	−231.601	−0.0424325
$311$	662.775	0.120844	0.0604220	−	0.998173i	$-0.480755\pi$
$311$	662.775	0.120844	0.0604220	+	0.998173i	$0.480755\pi$
$312$	0	0
$313$	−8353.04	−1.50844	−0.754220	−	0.656622i	$-0.771985\pi$
$313$	−8353.04	−1.50844	−0.754220	+	0.656622i	$0.771985\pi$
$314$	265.803	0.0477711
$315$	4594.49	0.821811
$316$	1173.17	0.208848
$317$	−5258.80	−0.931746	−0.465873	−	0.884852i	$-0.654260\pi$
$317$	−5258.80	−0.931746	−0.465873	+	0.884852i	$0.654260\pi$
$318$	242.496	0.0427626
$319$	1174.14	0.206078
$320$	5162.04	0.901771
$321$	−4502.59	−0.782897
$322$	878.737	0.152081
$323$	−719.769	−0.123991
$324$	7249.29	1.24302
$325$	0	0
$326$	−244.963	−0.0416173
$327$	7334.24	1.24032
$328$	493.626	0.0830974
$329$	−362.743	−0.0607862
$330$	377.811	0.0630237
$331$	2806.48	0.466036	0.233018	−	0.972472i	$-0.425140\pi$
$331$	2806.48	0.466036	0.233018	+	0.972472i	$0.425140\pi$
$332$	−8138.65	−1.34538
$333$	2286.10	0.376209
$334$	363.373	0.0595296
$335$	3113.77	0.507830
$336$	12220.2	1.98413
$337$	10987.6	1.77607	0.888033	−	0.459780i	$-0.152072\pi$
$337$	10987.6	1.77607	0.888033	+	0.459780i	$0.152072\pi$
$338$	0	0
$339$	1709.99	0.273965
$340$	−5831.93	−0.930237
$341$	5774.17	0.916976
$342$	22.7235	0.00359283
$343$	−5773.59	−0.908876
$344$	722.142	0.113184
$345$	−13216.1	−2.06241
$346$	445.183	0.0691711
$347$	3976.96	0.615258	0.307629	−	0.951506i	$-0.400464\pi$
$347$	3976.96	0.615258	0.307629	+	0.951506i	$0.400464\pi$
$348$	1595.19	0.245722
$349$	9033.03	1.38546	0.692732	−	0.721195i	$-0.256407\pi$
$349$	9033.03	1.38546	0.692732	+	0.721195i	$0.256407\pi$
$350$	87.9561	0.0134327
$351$	0	0
$352$	1086.39	0.164502
$353$	5251.24	0.791772	0.395886	−	0.918300i	$-0.370438\pi$
$353$	5251.24	0.791772	0.395886	+	0.918300i	$0.370438\pi$
$354$	−377.685	−0.0567054
$355$	−9365.38	−1.40018
$356$	7547.48	1.12364
$357$	−13728.7	−2.03529
$358$	−503.057	−0.0742664
$359$	−3850.22	−0.566035	−0.283018	−	0.959115i	$-0.591336\pi$
$359$	−3850.22	−0.566035	−0.283018	+	0.959115i	$0.591336\pi$
$360$	368.748	0.0539853
$361$	−6757.10	−0.985143
$362$	337.851	0.0490527
$363$	−783.117	−0.113231
$364$	0	0
$365$	2561.66	0.367351
$366$	151.390	0.0216210
$367$	−7694.40	−1.09440	−0.547199	−	0.837002i	$-0.684306\pi$
$367$	−7694.40	−1.09440	−0.547199	+	0.837002i	$0.684306\pi$
$368$	−12608.7	−1.78608
$369$	−3130.04	−0.441581
$370$	−231.344	−0.0325054
$371$	7440.11	1.04116
$372$	7844.84	1.09338
$373$	8180.64	1.13560	0.567798	−	0.823168i	$-0.307795\pi$
$373$	8180.64	1.13560	0.567798	+	0.823168i	$0.307795\pi$
$374$	−404.941	−0.0559866
$375$	−9638.41	−1.32727
$376$	−29.1133	−0.00399309
$377$	0	0
$378$	−341.486	−0.0464660
$379$	1503.22	0.203733	0.101867	−	0.994798i	$-0.467518\pi$
$379$	1503.22	0.203733	0.101867	+	0.994798i	$0.467518\pi$
$380$	825.675	0.111464
$381$	660.293	0.0887869
$382$	99.9762	0.0133906
$383$	12405.4	1.65506	0.827531	−	0.561420i	$-0.189745\pi$
$383$	12405.4	1.65506	0.827531	+	0.561420i	$0.189745\pi$
$384$	1967.05	0.261408
$385$	11591.8	1.53447
$386$	−685.024	−0.0903285
$387$	−4579.05	−0.601463
$388$	−3332.16	−0.435991
$389$	−3610.27	−0.470560	−0.235280	−	0.971928i	$-0.575601\pi$
$389$	−3610.27	−0.470560	−0.235280	+	0.971928i	$0.575601\pi$
$390$	0	0
$391$	14165.1	1.83212
$392$	−1280.28	−0.164958
$393$	2484.40	0.318884
$394$	77.9746	0.00997031
$395$	1507.69	0.192050
$396$	−4590.33	−0.582507
$397$	−2381.37	−0.301052	−0.150526	−	0.988606i	$-0.548097\pi$
$397$	−2381.37	−0.301052	−0.150526	+	0.988606i	$0.548097\pi$
$398$	−234.506	−0.0295345
$399$	1943.68	0.243874
$400$	−1262.06	−0.157757
$401$	3092.11	0.385069	0.192534	−	0.981290i	$-0.438329\pi$
$401$	3092.11	0.385069	0.192534	+	0.981290i	$0.438329\pi$
$402$	293.738	0.0364435
$403$	0	0
$404$	−6786.63	−0.835761
$405$	9316.34	1.14304
$406$	−136.307	−0.0166621
$407$	5767.76	0.702451
$408$	−1101.84	−0.133700
$409$	−341.312	−0.0412635	−0.0206318	−	0.999787i	$-0.506568\pi$
$409$	−341.312	−0.0412635	−0.0206318	+	0.999787i	$0.506568\pi$
$410$	316.748	0.0381538
$411$	−6640.74	−0.796992
$412$	−7950.78	−0.950745
$413$	−11587.9	−1.38064
$414$	−447.201	−0.0530887
$415$	−10459.3	−1.23717
$416$	0	0
$417$	−6611.51	−0.776419
$418$	57.3309	0.00670848
$419$	−545.670	−0.0636223	−0.0318111	−	0.999494i	$-0.510128\pi$
$419$	−545.670	−0.0636223	−0.0318111	+	0.999494i	$0.510128\pi$
$420$	15748.7	1.82966
$421$	10508.0	1.21645	0.608226	−	0.793764i	$-0.291881\pi$
$421$	10508.0	1.21645	0.608226	+	0.793764i	$0.291881\pi$
$422$	−712.812	−0.0822255
$423$	184.605	0.0212194
$424$	597.134	0.0683947
$425$	1417.84	0.161824
$426$	−883.485	−0.100481
$427$	4644.85	0.526417
$428$	−5535.98	−0.625214
$429$	0	0
$430$	463.381	0.0519680
$431$	−5518.04	−0.616693	−0.308347	−	0.951274i	$-0.599776\pi$
$431$	−5518.04	−0.616693	−0.308347	+	0.951274i	$0.599776\pi$
$432$	4899.88	0.545707
$433$	6072.31	0.673941	0.336971	−	0.941515i	$-0.390598\pi$
$433$	6072.31	0.673941	0.336971	+	0.941515i	$0.390598\pi$
$434$	−670.331	−0.0741404
$435$	2050.04	0.225959
$436$	9017.54	0.990508
$437$	−2005.47	−0.219530
$438$	241.654	0.0263623
$439$	14472.6	1.57343	0.786717	−	0.617314i	$-0.211779\pi$
$439$	14472.6	1.57343	0.786717	+	0.617314i	$0.211779\pi$
$440$	930.340	0.100801
$441$	8118.13	0.876593
$442$	0	0
$443$	−8593.87	−0.921686	−0.460843	−	0.887482i	$-0.652453\pi$
$443$	−8593.87	−0.921686	−0.460843	+	0.887482i	$0.652453\pi$
$444$	7836.14	0.837582
$445$	9699.55	1.03326
$446$	−311.587	−0.0330809
$447$	12829.9	1.35757
$448$	14940.7	1.57562
$449$	7695.65	0.808865	0.404432	−	0.914568i	$-0.367469\pi$
$449$	7695.65	0.808865	0.404432	+	0.914568i	$0.367469\pi$
$450$	−44.7620	−0.00468912
$451$	−7897.01	−0.824513
$452$	2102.46	0.218786
$453$	−22634.1	−2.34755
$454$	−324.077	−0.0335015
$455$	0	0
$456$	155.997	0.0160203
$457$	−13368.8	−1.36841	−0.684207	−	0.729288i	$-0.739851\pi$
$457$	−13368.8	−1.36841	−0.684207	+	0.729288i	$0.739851\pi$
$458$	−893.926	−0.0912018
$459$	−5504.70	−0.559777
$460$	−16249.3	−1.64702
$461$	−15502.1	−1.56617	−0.783086	−	0.621913i	$-0.786356\pi$
$461$	−15502.1	−1.56617	−0.783086	+	0.621913i	$0.786356\pi$
$462$	1093.51	0.110119
$463$	−9877.67	−0.991478	−0.495739	−	0.868472i	$-0.665103\pi$
$463$	−9877.67	−0.991478	−0.495739	+	0.868472i	$0.665103\pi$
$464$	1955.83	0.195683
$465$	10081.7	1.00544
$466$	−231.040	−0.0229672
$467$	3509.82	0.347784	0.173892	−	0.984765i	$-0.444366\pi$
$467$	3509.82	0.347784	0.173892	+	0.984765i	$0.444366\pi$
$468$	0	0
$469$	9012.28	0.887310
$470$	−18.6813	−0.00183341
$471$	−11570.5	−1.13194
$472$	−930.029	−0.0906950
$473$	−11552.8	−1.12304
$474$	142.228	0.0137822
$475$	−200.735	−0.0193903
$476$	−16879.6	−1.62536
$477$	−3786.38	−0.363451
$478$	580.591	0.0555557
$479$	19882.2	1.89653	0.948266	−	0.317476i	$-0.102835\pi$
$479$	19882.2	1.89653	0.948266	+	0.317476i	$0.102835\pi$
$480$	1896.83	0.180371
$481$	0	0
$482$	−609.379	−0.0575860
$483$	−38251.8	−3.60356
$484$	−962.851	−0.0904255
$485$	−4282.28	−0.400925
$486$	568.148	0.0530283
$487$	−407.728	−0.0379383	−0.0189691	−	0.999820i	$-0.506038\pi$
$487$	−407.728	−0.0379383	−0.0189691	+	0.999820i	$0.506038\pi$
$488$	372.789	0.0345807
$489$	10663.4	0.986122
$490$	−821.522	−0.0757400
$491$	20130.1	1.85022	0.925109	−	0.379701i	$-0.123973\pi$
$491$	20130.1	1.85022	0.925109	+	0.379701i	$0.123973\pi$
$492$	−10728.9	−0.983127
$493$	−2197.25	−0.200728
$494$	0	0
$495$	−5899.21	−0.535656
$496$	9618.39	0.870722
$497$	−27106.6	−2.44647
$498$	−986.679	−0.0887834
$499$	772.760	0.0693256	0.0346628	−	0.999399i	$-0.488964\pi$
$499$	772.760	0.0693256	0.0346628	+	0.999399i	$0.488964\pi$
$500$	−11850.5	−1.05994
$501$	−15817.8	−1.41055
$502$	858.787	0.0763537
$503$	−7444.25	−0.659886	−0.329943	−	0.944001i	$-0.607030\pi$
$503$	−7444.25	−0.659886	−0.329943	+	0.944001i	$0.607030\pi$
$504$	1067.28	0.0943262
$505$	−8721.76	−0.768541
$506$	−1128.28	−0.0991264
$507$	0	0
$508$	811.837	0.0709044
$509$	3591.47	0.312748	0.156374	−	0.987698i	$-0.450019\pi$
$509$	3591.47	0.312748	0.156374	+	0.987698i	$0.450019\pi$
$510$	−707.027	−0.0613876
$511$	7414.29	0.641857
$512$	3018.90	0.260582
$513$	779.347	0.0670741
$514$	−730.251	−0.0626653
$515$	−10217.9	−0.874276
$516$	−15695.7	−1.33908
$517$	465.753	0.0396205
$518$	−669.587	−0.0567953
$519$	−19379.0	−1.63901
$520$	0	0
$521$	1822.89	0.153286	0.0766432	−	0.997059i	$-0.475580\pi$
$521$	1822.89	0.153286	0.0766432	+	0.997059i	$0.475580\pi$
$522$	69.3685	0.00581643
$523$	12203.2	1.02028	0.510140	−	0.860091i	$-0.329594\pi$
$523$	12203.2	1.02028	0.510140	+	0.860091i	$0.329594\pi$
$524$	3054.60	0.254658
$525$	−3828.77	−0.318288
$526$	1134.57	0.0940485
$527$	−10805.6	−0.893171
$528$	−15690.5	−1.29326
$529$	27300.8	2.24384
$530$	383.166	0.0314032
$531$	5897.24	0.481955
$532$	2389.78	0.194756
$533$	0	0
$534$	915.009	0.0741504
$535$	−7114.50	−0.574928
$536$	723.313	0.0582880
$537$	21898.3	1.75974
$538$	−983.271	−0.0787952
$539$	20481.8	1.63676
$540$	6314.66	0.503222
$541$	−6728.89	−0.534746	−0.267373	−	0.963593i	$-0.586156\pi$
$541$	−6728.89	−0.534746	−0.267373	+	0.963593i	$0.586156\pi$
$542$	432.973	0.0343132
$543$	−14706.8	−1.16230
$544$	−2033.04	−0.160231
$545$	11588.8	0.910842
$546$	0	0
$547$	−14650.8	−1.14519	−0.572597	−	0.819837i	$-0.694064\pi$
$547$	−14650.8	−1.14519	−0.572597	+	0.819837i	$0.694064\pi$
$548$	−8164.87	−0.636471
$549$	−2363.83	−0.183763
$550$	−112.933	−0.00875544
$551$	311.083	0.0240519
$552$	−3070.04	−0.236720
$553$	4363.75	0.335561
$554$	173.073	0.0132729
$555$	10070.5	0.770216
$556$	−8128.92	−0.620042
$557$	−4208.91	−0.320175	−0.160087	−	0.987103i	$-0.551178\pi$
$557$	−4208.91	−0.320175	−0.160087	+	0.987103i	$0.551178\pi$
$558$	341.141	0.0258811
$559$	0	0
$560$	19309.1	1.45707
$561$	17627.3	1.32660
$562$	1212.17	0.0909830
$563$	−9224.28	−0.690510	−0.345255	−	0.938509i	$-0.612208\pi$
$563$	−9224.28	−0.690510	−0.345255	+	0.938509i	$0.612208\pi$
$564$	632.776	0.0472423
$565$	2701.95	0.201189
$566$	168.714	0.0125293
$567$	26964.6	1.99719
$568$	−2175.54	−0.160710
$569$	14132.9	1.04127	0.520634	−	0.853780i	$-0.325696\pi$
$569$	14132.9	1.04127	0.520634	+	0.853780i	$0.325696\pi$
$570$	100.100	0.00735564
$571$	−17983.4	−1.31801	−0.659003	−	0.752140i	$-0.729022\pi$
$571$	−17983.4	−1.31801	−0.659003	+	0.752140i	$0.729022\pi$
$572$	0	0
$573$	−4352.01	−0.317291
$574$	916.774	0.0666645
$575$	3950.49	0.286516
$576$	−7603.51	−0.550022
$577$	24401.8	1.76059	0.880295	−	0.474427i	$-0.157345\pi$
$577$	24401.8	1.76059	0.880295	+	0.474427i	$0.157345\pi$
$578$	25.4724	0.00183307
$579$	29819.4	2.14033
$580$	2520.55	0.180449
$581$	−30272.7	−2.16166
$582$	−403.970	−0.0287717
$583$	−9552.92	−0.678630
$584$	595.061	0.0421640
$585$	0	0
$586$	571.540	0.0402902
$587$	20586.5	1.44753	0.723763	−	0.690049i	$-0.242411\pi$
$587$	20586.5	1.44753	0.723763	+	0.690049i	$0.242411\pi$
$588$	27826.8	1.95163
$589$	1529.85	0.107022
$590$	−596.777	−0.0416422
$591$	−3394.27	−0.236246
$592$	9607.71	0.667018
$593$	19701.5	1.36432	0.682162	−	0.731201i	$-0.261040\pi$
$593$	19701.5	1.36432	0.682162	+	0.731201i	$0.261040\pi$
$594$	438.459	0.0302865
$595$	−21692.6	−1.49464
$596$	15774.5	1.08414
$597$	10208.2	0.699821
$598$	0	0
$599$	−3645.91	−0.248694	−0.124347	−	0.992239i	$-0.539684\pi$
$599$	−3645.91	−0.248694	−0.124347	+	0.992239i	$0.539684\pi$
$600$	−307.292	−0.0209086
$601$	−6285.73	−0.426623	−0.213311	−	0.976984i	$-0.568425\pi$
$601$	−6285.73	−0.426623	−0.213311	+	0.976984i	$0.568425\pi$
$602$	1341.18	0.0908014
$603$	−4586.47	−0.309744
$604$	−27828.9	−1.87474
$605$	−1237.40	−0.0831526
$606$	−822.769	−0.0551530
$607$	−1831.87	−0.122493	−0.0612466	−	0.998123i	$-0.519508\pi$
$607$	−1831.87	−0.122493	−0.0612466	+	0.998123i	$0.519508\pi$
$608$	287.834	0.0191994
$609$	5933.51	0.394808
$610$	239.210	0.0158776
$611$	0	0
$612$	8590.23	0.567385
$613$	−4396.09	−0.289651	−0.144826	−	0.989457i	$-0.546262\pi$
$613$	−4396.09	−0.289651	−0.144826	+	0.989457i	$0.546262\pi$
$614$	−979.950	−0.0644097
$615$	−13788.2	−0.904054
$616$	2692.72	0.176124
$617$	15140.8	0.987917	0.493959	−	0.869485i	$-0.335549\pi$
$617$	15140.8	0.987917	0.493959	+	0.869485i	$0.335549\pi$
$618$	−963.903	−0.0627409
$619$	20913.0	1.35794	0.678971	−	0.734165i	$-0.262426\pi$
$619$	20913.0	1.35794	0.678971	+	0.734165i	$0.262426\pi$
$620$	12395.6	0.802933
$621$	−15337.6	−0.991106
$622$	98.7921	0.00636849
$623$	28073.7	1.80538
$624$	0	0
$625$	−12743.9	−0.815612
$626$	−1245.09	−0.0794949
$627$	−2495.64	−0.158957
$628$	−14226.1	−0.903954
$629$	−10793.6	−0.684215
$630$	684.847	0.0433095
$631$	−7104.58	−0.448223	−0.224112	−	0.974563i	$-0.571948\pi$
$631$	−7104.58	−0.448223	−0.224112	+	0.974563i	$0.571948\pi$
$632$	350.228	0.0220433
$633$	31029.0	1.94833
$634$	−783.867	−0.0491031
$635$	1043.32	0.0652016
$636$	−12978.7	−0.809180
$637$	0	0
$638$	175.015	0.0108603
$639$	13794.9	0.854018
$640$	3108.12	0.191967
$641$	−2696.10	−0.166131	−0.0830653	−	0.996544i	$-0.526471\pi$
$641$	−2696.10	−0.166131	−0.0830653	+	0.996544i	$0.526471\pi$
$642$	−671.148	−0.0412587
$643$	11994.7	0.735653	0.367826	−	0.929894i	$-0.380102\pi$
$643$	11994.7	0.735653	0.367826	+	0.929894i	$0.380102\pi$
$644$	−47031.0	−2.87777
$645$	−20171.2	−1.23138
$646$	−107.288	−0.00653432
$647$	16239.8	0.986788	0.493394	−	0.869806i	$-0.335756\pi$
$647$	16239.8	0.986788	0.493394	+	0.869806i	$0.335756\pi$
$648$	2164.14	0.131197
$649$	14878.6	0.899899
$650$	0	0
$651$	29179.8	1.75676
$652$	13110.7	0.787508
$653$	7316.99	0.438493	0.219246	−	0.975670i	$-0.429640\pi$
$653$	7316.99	0.438493	0.219246	+	0.975670i	$0.429640\pi$
$654$	1093.23	0.0653650
$655$	3925.58	0.234176
$656$	−13154.5	−0.782923
$657$	−3773.23	−0.224061
$658$	−54.0699	−0.00320344
$659$	8573.39	0.506786	0.253393	−	0.967363i	$-0.418453\pi$
$659$	8573.39	0.506786	0.253393	+	0.967363i	$0.418453\pi$
$660$	−20220.9	−1.19257
$661$	14262.5	0.839255	0.419628	−	0.907696i	$-0.362161\pi$
$661$	14262.5	0.839255	0.419628	+	0.907696i	$0.362161\pi$
$662$	418.329	0.0245601
$663$	0	0
$664$	−2429.64	−0.142001
$665$	3071.20	0.179092
$666$	340.762	0.0198262
$667$	−6122.14	−0.355397
$668$	−19448.2	−1.12645
$669$	13563.5	0.783852
$670$	464.133	0.0267627
$671$	−5963.87	−0.343119
$672$	5490.07	0.315155
$673$	−2690.08	−0.154079	−0.0770394	−	0.997028i	$-0.524547\pi$
$673$	−2690.08	−0.154079	−0.0770394	+	0.997028i	$0.524547\pi$
$674$	1637.80	0.0935988
$675$	−1535.20	−0.0875405
$676$	0	0
$677$	−25288.0	−1.43559	−0.717795	−	0.696254i	$-0.754849\pi$
$677$	−25288.0	−1.43559	−0.717795	+	0.696254i	$0.754849\pi$
$678$	254.889	0.0144380
$679$	−12394.4	−0.700518
$680$	−1741.02	−0.0981837
$681$	14107.2	0.793818
$682$	860.688	0.0483247
$683$	−28981.5	−1.62364	−0.811821	−	0.583906i	$-0.801524\pi$
$683$	−28981.5	−1.62364	−0.811821	+	0.583906i	$0.801524\pi$
$684$	−1216.19	−0.0679858
$685$	−10493.0	−0.585279
$686$	−860.601	−0.0478978
$687$	38913.0	2.16103
$688$	−19244.2	−1.06639
$689$	0	0
$690$	−1969.97	−0.108689
$691$	31243.3	1.72005	0.860023	−	0.510255i	$-0.170449\pi$
$691$	31243.3	1.72005	0.860023	+	0.510255i	$0.170449\pi$
$692$	−23826.7	−1.30890
$693$	−17074.3	−0.935929
$694$	592.799	0.0324241
$695$	−10446.8	−0.570172
$696$	476.216	0.0259352
$697$	14778.3	0.803108
$698$	1346.45	0.0730141
$699$	10057.3	0.544208
$700$	−4707.51	−0.254182
$701$	25077.1	1.35114	0.675570	−	0.737296i	$-0.263898\pi$
$701$	25077.1	1.35114	0.675570	+	0.737296i	$0.263898\pi$
$702$	0	0
$703$	1528.15	0.0819846
$704$	−19183.4	−1.02699
$705$	813.205	0.0434426
$706$	782.741	0.0417264
$707$	−25243.7	−1.34284
$708$	20214.1	1.07301
$709$	5894.38	0.312226	0.156113	−	0.987739i	$-0.450104\pi$
$709$	5894.38	0.312226	0.156113	+	0.987739i	$0.450104\pi$
$710$	−1395.99	−0.0737894
$711$	−2220.77	−0.117138
$712$	2253.16	0.118597
$713$	−30107.5	−1.58139
$714$	−2046.37	−0.107260
$715$	0	0
$716$	26924.2	1.40531
$717$	−25273.4	−1.31639
$718$	−573.907	−0.0298301
$719$	−5430.64	−0.281681	−0.140841	−	0.990032i	$-0.544980\pi$
$719$	−5430.64	−0.281681	−0.140841	+	0.990032i	$0.544980\pi$
$720$	−9826.68	−0.508637
$721$	−29573.9	−1.52759
$722$	−1007.20	−0.0519171
$723$	26526.6	1.36450
$724$	−18082.2	−0.928205
$725$	−612.788	−0.0313908
$726$	−116.730	−0.00596730
$727$	17033.7	0.868974	0.434487	−	0.900678i	$-0.356930\pi$
$727$	17033.7	0.868974	0.434487	+	0.900678i	$0.356930\pi$
$728$	0	0
$729$	−197.281	−0.0100229
$730$	381.836	0.0193594
$731$	21619.6	1.09389
$732$	−8102.57	−0.409125
$733$	−7130.74	−0.359318	−0.179659	−	0.983729i	$-0.557499\pi$
$733$	−7130.74	−0.359318	−0.179659	+	0.983729i	$0.557499\pi$
$734$	−1146.91	−0.0576749
$735$	35761.3	1.79466
$736$	−5664.60	−0.283695
$737$	−11571.5	−0.578349
$738$	−466.559	−0.0232714
$739$	−27531.1	−1.37043	−0.685215	−	0.728341i	$-0.740292\pi$
$739$	−27531.1	−1.37043	−0.685215	+	0.728341i	$0.740292\pi$
$740$	12381.8	0.615087
$741$	0	0
$742$	1109.01	0.0548694
$743$	11915.0	0.588314	0.294157	−	0.955757i	$-0.404961\pi$
$743$	11915.0	0.588314	0.294157	+	0.955757i	$0.404961\pi$
$744$	2341.94	0.115403
$745$	20272.4	0.996946
$746$	1219.39	0.0598460
$747$	15406.2	0.754595
$748$	21672.9	1.05941
$749$	−20591.7	−1.00455
$750$	−1436.69	−0.0699471
$751$	−4730.41	−0.229847	−0.114923	−	0.993374i	$-0.536662\pi$
$751$	−4730.41	−0.229847	−0.114923	+	0.993374i	$0.536662\pi$
$752$	775.832	0.0376219
$753$	−37383.4	−1.80920
$754$	0	0
$755$	−35763.9	−1.72395
$756$	18276.7	0.879257
$757$	15910.3	0.763895	0.381948	−	0.924184i	$-0.375253\pi$
$757$	15910.3	0.763895	0.381948	+	0.924184i	$0.375253\pi$
$758$	224.067	0.0107368
$759$	49114.4	2.34880
$760$	246.490	0.0117647
$761$	33817.2	1.61087	0.805436	−	0.592683i	$-0.201931\pi$
$761$	33817.2	1.61087	0.805436	+	0.592683i	$0.201931\pi$
$762$	98.4221	0.00467908
$763$	33541.8	1.59147
$764$	−5350.85	−0.253386
$765$	11039.6	0.521750
$766$	1849.13	0.0872219
$767$	0	0
$768$	−25842.2	−1.21419
$769$	−36061.0	−1.69102	−0.845510	−	0.533960i	$-0.820703\pi$
$769$	−36061.0	−1.69102	−0.845510	+	0.533960i	$0.820703\pi$
$770$	1727.85	0.0808668
$771$	31788.2	1.48485
$772$	36663.3	1.70925
$773$	−463.749	−0.0215781	−0.0107891	−	0.999942i	$-0.503434\pi$
$773$	−463.749	−0.0215781	−0.0107891	+	0.999942i	$0.503434\pi$
$774$	−682.545	−0.0316971
$775$	−3013.57	−0.139678
$776$	−994.755	−0.0460176
$777$	29147.5	1.34576
$778$	−538.140	−0.0247985
$779$	−2092.28	−0.0962308
$780$	0	0
$781$	34804.1	1.59461
$782$	2111.43	0.0965530
$783$	2379.12	0.108586
$784$	34117.8	1.55420
$785$	−18282.5	−0.831249
$786$	370.321	0.0168052
$787$	−13712.8	−0.621101	−0.310551	−	0.950557i	$-0.600513\pi$
$787$	−13712.8	−0.621101	−0.310551	+	0.950557i	$0.600513\pi$
$788$	−4173.29	−0.188664
$789$	−49388.3	−2.22848
$790$	224.733	0.0101211
$791$	7820.34	0.351529
$792$	−1370.36	−0.0614819
$793$	0	0
$794$	−354.963	−0.0158655
$795$	−16679.4	−0.744097
$796$	12551.1	0.558870
$797$	4005.41	0.178016	0.0890081	−	0.996031i	$-0.471630\pi$
$797$	4005.41	0.178016	0.0890081	+	0.996031i	$0.471630\pi$
$798$	289.722	0.0128522
$799$	−871.598	−0.0385919
$800$	−566.991	−0.0250577
$801$	−14287.1	−0.630225
$802$	460.904	0.0202931
$803$	−9519.76	−0.418362
$804$	−15721.2	−0.689607
$805$	−60441.4	−2.64631
$806$	0	0
$807$	42802.3	1.86705
$808$	−2026.02	−0.0882120
$809$	−22419.9	−0.974342	−0.487171	−	0.873307i	$-0.661971\pi$
$809$	−22419.9	−0.974342	−0.487171	+	0.873307i	$0.661971\pi$
$810$	1388.68	0.0602385
$811$	−14185.7	−0.614213	−0.307107	−	0.951675i	$-0.599361\pi$
$811$	−14185.7	−0.614213	−0.307107	+	0.951675i	$0.599361\pi$
$812$	7295.32	0.315290
$813$	−18847.5	−0.813051
$814$	859.733	0.0370192
$815$	16849.1	0.724169
$816$	29362.8	1.25969
$817$	−3060.87	−0.131073
$818$	−50.8754	−0.00217459
$819$	0	0
$820$	−16952.7	−0.721969
$821$	−30014.4	−1.27590	−0.637949	−	0.770079i	$-0.720217\pi$
$821$	−30014.4	−1.27590	−0.637949	+	0.770079i	$0.720217\pi$
$822$	−989.858	−0.0420015
$823$	11791.4	0.499418	0.249709	−	0.968321i	$-0.419665\pi$
$823$	11791.4	0.499418	0.249709	+	0.968321i	$0.419665\pi$
$824$	−2373.56	−0.100348
$825$	4916.04	0.207460
$826$	−1727.27	−0.0727596
$827$	−26657.2	−1.12087	−0.560436	−	0.828198i	$-0.689366\pi$
$827$	−26657.2	−1.12087	−0.560436	+	0.828198i	$0.689366\pi$
$828$	23934.7	1.00458
$829$	−23450.6	−0.982475	−0.491238	−	0.871026i	$-0.663455\pi$
$829$	−23450.6	−0.982475	−0.491238	+	0.871026i	$0.663455\pi$
$830$	−1559.04	−0.0651990
$831$	−7533.96	−0.314501
$832$	0	0
$833$	−38329.1	−1.59427
$834$	−985.500	−0.0409174
$835$	−24993.6	−1.03585
$836$	−3068.42	−0.126942
$837$	11700.1	0.483170
$838$	−81.3367	−0.00335290
$839$	−11212.1	−0.461362	−0.230681	−	0.973029i	$-0.574095\pi$
$839$	−11212.1	−0.461362	−0.230681	+	0.973029i	$0.574095\pi$
$840$	4701.49	0.193115
$841$	−23439.4	−0.961062
$842$	1566.30	0.0641071
$843$	−52766.5	−2.15584
$844$	38150.6	1.55592
$845$	0	0
$846$	27.5169	0.00111826
$847$	−3581.44	−0.145289
$848$	−15912.9	−0.644399
$849$	−7344.18	−0.296881
$850$	211.341	0.00852815
$851$	−30074.1	−1.21143
$852$	47285.2	1.90137
$853$	−10447.0	−0.419342	−0.209671	−	0.977772i	$-0.567239\pi$
$853$	−10447.0	−0.419342	−0.209671	+	0.977772i	$0.567239\pi$
$854$	692.353	0.0277422
$855$	−1562.97	−0.0625177
$856$	−1652.67	−0.0659894
$857$	−37041.9	−1.47646	−0.738230	−	0.674549i	$-0.764338\pi$
$857$	−37041.9	−1.47646	−0.738230	+	0.674549i	$0.764338\pi$
$858$	0	0
$859$	15972.0	0.634409	0.317204	−	0.948357i	$-0.397256\pi$
$859$	15972.0	0.634409	0.317204	+	0.948357i	$0.397256\pi$
$860$	−24800.7	−0.983369
$861$	−39907.6	−1.57961
$862$	−822.510	−0.0324998
$863$	−32985.8	−1.30110	−0.650549	−	0.759464i	$-0.725461\pi$
$863$	−32985.8	−1.30110	−0.650549	+	0.759464i	$0.725461\pi$
$864$	2201.32	0.0866787
$865$	−30620.7	−1.20362
$866$	905.128	0.0355167
$867$	−1108.83	−0.0434345
$868$	35876.9	1.40293
$869$	−5602.94	−0.218719
$870$	305.576	0.0119080
$871$	0	0
$872$	2692.02	0.104545
$873$	6307.66	0.244538
$874$	−298.932	−0.0115693
$875$	−44079.5	−1.70304
$876$	−12933.6	−0.498843
$877$	12279.5	0.472804	0.236402	−	0.971655i	$-0.424032\pi$
$877$	12279.5	0.472804	0.236402	+	0.971655i	$0.424032\pi$
$878$	2157.26	0.0829201
$879$	−24879.4	−0.954677
$880$	−24792.4	−0.949719
$881$	30939.0	1.18316	0.591579	−	0.806247i	$-0.298505\pi$
$881$	30939.0	1.18316	0.591579	+	0.806247i	$0.298505\pi$
$882$	1210.07	0.0461965
$883$	31309.8	1.19327	0.596636	−	0.802512i	$-0.296503\pi$
$883$	31309.8	1.19327	0.596636	+	0.802512i	$0.296503\pi$
$884$	0	0
$885$	25978.0	0.986712
$886$	−1280.99	−0.0485729
$887$	−4033.26	−0.152676	−0.0763380	−	0.997082i	$-0.524323\pi$
$887$	−4033.26	−0.152676	−0.0763380	+	0.997082i	$0.524323\pi$
$888$	2339.34	0.0884042
$889$	3019.73	0.113924
$890$	1445.80	0.0544531
$891$	−34621.9	−1.30177
$892$	16676.5	0.625977
$893$	123.399	0.00462419
$894$	1912.40	0.0715440
$895$	34601.3	1.29229
$896$	8995.93	0.335416
$897$	0	0
$898$	1147.10	0.0426272
$899$	4670.18	0.173258
$900$	2395.72	0.0887303
$901$	17877.1	0.661012
$902$	−1177.11	−0.0434519
$903$	−58382.2	−2.15154
$904$	627.650	0.0230922
$905$	−23238.2	−0.853550
$906$	−3373.80	−0.123716
$907$	8809.38	0.322504	0.161252	−	0.986913i	$-0.448447\pi$
$907$	8809.38	0.322504	0.161252	+	0.986913i	$0.448447\pi$
$908$	17345.0	0.633936
$909$	12846.9	0.468760
$910$	0	0
$911$	−33046.9	−1.20186	−0.600930	−	0.799302i	$-0.705203\pi$
$911$	−33046.9	−1.20186	−0.600930	+	0.799302i	$0.705203\pi$
$912$	−4157.14	−0.150939
$913$	38869.3	1.40897
$914$	−1992.73	−0.0721155
$915$	−10412.9	−0.376219
$916$	47844.0	1.72578
$917$	11362.0	0.409165
$918$	−820.522	−0.0295003
$919$	−4512.85	−0.161986	−0.0809930	−	0.996715i	$-0.525809\pi$
$919$	−4512.85	−0.161986	−0.0809930	+	0.996715i	$0.525809\pi$
$920$	−4850.95	−0.173838
$921$	42657.7	1.52619
$922$	−2310.72	−0.0825374
$923$	0	0
$924$	−58526.1	−2.08373
$925$	−3010.23	−0.107001
$926$	−1472.35	−0.0522510
$927$	15050.6	0.533252
$928$	878.676	0.0310818
$929$	33818.5	1.19435	0.597174	−	0.802111i	$-0.296290\pi$
$929$	33818.5	1.19435	0.597174	+	0.802111i	$0.296290\pi$
$930$	1502.76	0.0529866
$931$	5426.58	0.191030
$932$	12365.5	0.434600
$933$	−4300.47	−0.150901
$934$	523.168	0.0183282
$935$	27852.7	0.974204
$936$	0	0
$937$	19737.5	0.688150	0.344075	−	0.938942i	$-0.388193\pi$
$937$	19737.5	0.688150	0.344075	+	0.938942i	$0.388193\pi$
$938$	1343.35	0.0467613
$939$	54199.3	1.88363
$940$	999.844	0.0346929
$941$	21587.6	0.747860	0.373930	−	0.927457i	$-0.378010\pi$
$941$	21587.6	0.747860	0.373930	+	0.927457i	$0.378010\pi$
$942$	−1724.68	−0.0596531
$943$	41176.3	1.42193
$944$	24784.1	0.854507
$945$	23488.1	0.808539
$946$	−1722.04	−0.0591843
$947$	−5830.13	−0.200057	−0.100028	−	0.994985i	$-0.531893\pi$
$947$	−5830.13	−0.200057	−0.100028	+	0.994985i	$0.531893\pi$
$948$	−7612.20	−0.260794
$949$	0	0
$950$	−29.9213	−0.00102187
$951$	34122.1	1.16350
$952$	−5039.08	−0.171552
$953$	8839.77	0.300470	0.150235	−	0.988650i	$-0.451997\pi$
$953$	8839.77	0.300470	0.150235	+	0.988650i	$0.451997\pi$
$954$	−564.391	−0.0191539
$955$	−6876.58	−0.233006
$956$	−31073.9	−1.05126
$957$	−7618.47	−0.257336
$958$	2963.60	0.0999474
$959$	−30370.2	−1.02263
$960$	−33494.3	−1.12607
$961$	−6823.95	−0.229061
$962$	0	0
$963$	10479.4	0.350669
$964$	32614.7	1.08968
$965$	47117.4	1.57178
$966$	−5701.75	−0.189908
$967$	−10346.1	−0.344063	−0.172032	−	0.985091i	$-0.555033\pi$
$967$	−10346.1	−0.344063	−0.172032	+	0.985091i	$0.555033\pi$
$968$	−287.442	−0.00954414
$969$	4670.28	0.154831
$970$	−638.310	−0.0211288
$971$	−17420.0	−0.575730	−0.287865	−	0.957671i	$-0.592945\pi$
$971$	−17420.0	−0.575730	−0.287865	+	0.957671i	$0.592945\pi$
$972$	−30408.0	−1.00343
$973$	−30236.5	−0.996236
$974$	−60.7753	−0.00199935
$975$	0	0
$976$	−9934.37	−0.325811
$977$	41735.0	1.36665	0.683327	−	0.730113i	$-0.260532\pi$
$977$	41735.0	1.36665	0.683327	+	0.730113i	$0.260532\pi$
$978$	1589.46	0.0519687
$979$	−36046.0	−1.17675
$980$	43968.9	1.43320
$981$	−17069.9	−0.555555
$982$	3000.55	0.0975067
$983$	34123.8	1.10720	0.553602	−	0.832781i	$-0.313253\pi$
$983$	34123.8	1.10720	0.553602	+	0.832781i	$0.313253\pi$
$984$	−3202.93	−0.103766
$985$	−5363.26	−0.173490
$986$	−327.518	−0.0105784
$987$	2353.69	0.0759054
$988$	0	0
$989$	60238.2	1.93677
$990$	−879.327	−0.0282291
$991$	−54296.7	−1.74046	−0.870228	−	0.492649i	$-0.836029\pi$
$991$	−54296.7	−1.74046	−0.870228	+	0.492649i	$0.836029\pi$
$992$	4321.16	0.138303
$993$	−18210.1	−0.581952
$994$	−4040.46	−0.128929
$995$	16129.9	0.513921
$996$	52808.3	1.68001
$997$	46879.3	1.48915	0.744575	−	0.667538i	$-0.232652\pi$
$997$	46879.3	1.48915	0.744575	+	0.667538i	$0.232652\pi$
$998$	115.186	0.00365347
$999$	11687.1	0.370133

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.l.1.5	yes	9
3.2	odd	2		1521.4.a.bg.1.5		9
13.2	odd	12		169.4.e.h.147.10		36
13.3	even	3		169.4.c.k.22.5		18
13.4	even	6		169.4.c.l.146.5		18
13.5	odd	4		169.4.b.g.168.9		18
13.6	odd	12		169.4.e.h.23.9		36
13.7	odd	12		169.4.e.h.23.10		36
13.8	odd	4		169.4.b.g.168.10		18
13.9	even	3		169.4.c.k.146.5		18
13.10	even	6		169.4.c.l.22.5		18
13.11	odd	12		169.4.e.h.147.9		36
13.12	even	2		169.4.a.k.1.5	✓	9
39.38	odd	2		1521.4.a.bh.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.5	✓	9	13.12	even	2
169.4.a.l.1.5	yes	9	1.1	even	1	trivial
169.4.b.g.168.9		18	13.5	odd	4
169.4.b.g.168.10		18	13.8	odd	4
169.4.c.k.22.5		18	13.3	even	3
169.4.c.k.146.5		18	13.9	even	3
169.4.c.l.22.5		18	13.10	even	6
169.4.c.l.146.5		18	13.4	even	6
169.4.e.h.23.9		36	13.6	odd	12
169.4.e.h.23.10		36	13.7	odd	12
169.4.e.h.147.9		36	13.11	odd	12
169.4.e.h.147.10		36	13.2	odd	12
1521.4.a.bg.1.5		9	3.2	odd	2
1521.4.a.bh.1.5		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)

\(f(q)\)	\(=\)	\(q+0.149058 q^{2} -6.48858 q^{3} -7.97778 q^{4} -10.2526 q^{5} -0.967177 q^{6} -29.6743 q^{7} -2.38162 q^{8} +15.1017 q^{9} +O(q^{10})\)	\(q+0.149058 q^{2} -6.48858 q^{3} -7.97778 q^{4} -10.2526 q^{5} -0.967177 q^{6} -29.6743 q^{7} -2.38162 q^{8} +15.1017 q^{9} -1.52823 q^{10} +38.1011 q^{11} +51.7645 q^{12} -4.42320 q^{14} +66.5245 q^{15} +63.4673 q^{16} -71.3014 q^{17} +2.25103 q^{18} +10.0947 q^{19} +81.7926 q^{20} +192.544 q^{21} +5.67928 q^{22} -198.665 q^{23} +15.4533 q^{24} -19.8851 q^{25} +77.2033 q^{27} +236.735 q^{28} +30.8164 q^{29} +9.91603 q^{30} +151.549 q^{31} +28.5133 q^{32} -247.222 q^{33} -10.6281 q^{34} +304.238 q^{35} -120.478 q^{36} +151.381 q^{37} +1.50470 q^{38} +24.4177 q^{40} -207.265 q^{41} +28.7003 q^{42} -303.215 q^{43} -303.962 q^{44} -154.831 q^{45} -29.6127 q^{46} +12.2241 q^{47} -411.812 q^{48} +537.565 q^{49} -2.96405 q^{50} +462.645 q^{51} -250.726 q^{53} +11.5078 q^{54} -390.633 q^{55} +70.6730 q^{56} -65.5005 q^{57} +4.59343 q^{58} +390.502 q^{59} -530.718 q^{60} -156.528 q^{61} +22.5896 q^{62} -448.132 q^{63} -503.488 q^{64} -36.8505 q^{66} -303.706 q^{67} +568.827 q^{68} +1289.05 q^{69} +45.3491 q^{70} +913.468 q^{71} -35.9664 q^{72} -249.855 q^{73} +22.5645 q^{74} +129.026 q^{75} -80.5336 q^{76} -1130.62 q^{77} -147.055 q^{79} -650.701 q^{80} -908.685 q^{81} -30.8945 q^{82} +1020.16 q^{83} -1536.08 q^{84} +731.021 q^{85} -45.1967 q^{86} -199.954 q^{87} -90.7423 q^{88} -946.062 q^{89} -23.0788 q^{90} +1584.91 q^{92} -983.337 q^{93} +1.82211 q^{94} -103.497 q^{95} -185.011 q^{96} +417.680 q^{97} +80.1286 q^{98} +575.390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + q^{3} + 37 q^{4} + 30 q^{5} + 48 q^{6} + 38 q^{7} + 60 q^{8} + 66 q^{9}+O(q^{10}) \) 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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