LMFDB - Embedded newform 169.4.a.k.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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 $ x^9 - 4x^8 - 46x^7 + 145x^6 + 680x^5 - 1501x^4 - 3203x^3 + 4784x^2 + 3584x - 4096
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.82555$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.82555 q^{2} +4.44352 q^{3} +15.2860 q^{4} +12.7712 q^{5} -21.4425 q^{6} -26.1871 q^{7} -35.1589 q^{8} -7.25513 q^{9} +O(q^{10})$	\(q-4.82555 q^{2} +4.44352 q^{3} +15.2860 q^{4} +12.7712 q^{5} -21.4425 q^{6} -26.1871 q^{7} -35.1589 q^{8} -7.25513 q^{9} -61.6281 q^{10} -42.3430 q^{11} +67.9236 q^{12} +126.367 q^{14} +56.7490 q^{15} +47.3733 q^{16} -27.3331 q^{17} +35.0100 q^{18} +13.1196 q^{19} +195.220 q^{20} -116.363 q^{21} +204.329 q^{22} -28.7976 q^{23} -156.229 q^{24} +38.1033 q^{25} -152.213 q^{27} -400.296 q^{28} -141.628 q^{29} -273.846 q^{30} +56.0144 q^{31} +52.6687 q^{32} -188.152 q^{33} +131.897 q^{34} -334.441 q^{35} -110.902 q^{36} +313.982 q^{37} -63.3095 q^{38} -449.021 q^{40} -352.375 q^{41} +561.516 q^{42} -320.676 q^{43} -647.255 q^{44} -92.6566 q^{45} +138.964 q^{46} -339.339 q^{47} +210.504 q^{48} +342.765 q^{49} -183.869 q^{50} -121.455 q^{51} +349.461 q^{53} +734.514 q^{54} -540.771 q^{55} +920.710 q^{56} +58.2973 q^{57} +683.432 q^{58} -258.742 q^{59} +867.465 q^{60} +650.732 q^{61} -270.301 q^{62} +189.991 q^{63} -633.142 q^{64} +907.938 q^{66} +894.996 q^{67} -417.813 q^{68} -127.963 q^{69} +1613.86 q^{70} -741.469 q^{71} +255.082 q^{72} -820.643 q^{73} -1515.14 q^{74} +169.313 q^{75} +200.546 q^{76} +1108.84 q^{77} -199.372 q^{79} +605.013 q^{80} -480.475 q^{81} +1700.40 q^{82} -541.696 q^{83} -1778.72 q^{84} -349.076 q^{85} +1547.44 q^{86} -629.325 q^{87} +1488.73 q^{88} -380.615 q^{89} +447.119 q^{90} -440.199 q^{92} +248.901 q^{93} +1637.50 q^{94} +167.553 q^{95} +234.034 q^{96} +1430.50 q^{97} -1654.03 q^{98} +307.204 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) $ 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9 - 147 * q^10 - 181 * q^11 + 39 * q^12 - 147 * q^14 - 218 * q^15 + 269 * q^16 - 55 * q^17 - 79 * q^18 - 161 * q^19 - 370 * q^20 - 188 * q^21 + 340 * q^22 - 204 * q^23 - 798 * q^24 + 307 * q^25 - 668 * q^27 - 344 * q^28 + 280 * q^29 + 521 * q^30 - 706 * q^31 - 680 * q^32 - 500 * q^33 - 216 * q^34 + 20 * q^35 - 909 * q^36 - 298 * q^37 - 739 * q^38 + 13 * q^40 - 1201 * q^41 - 4 * q^42 - 533 * q^43 - 355 * q^44 + 90 * q^45 + 840 * q^46 - 956 * q^47 - 132 * q^48 + 403 * q^49 + 1156 * q^50 + 470 * q^51 - 278 * q^53 + 2555 * q^54 - 250 * q^55 + 250 * q^56 + 810 * q^57 + 2877 * q^58 - 1377 * q^59 + 3157 * q^60 - 136 * q^61 + 2035 * q^62 + 944 * q^63 + 284 * q^64 + 3279 * q^66 + 931 * q^67 - 1536 * q^68 - 2050 * q^69 + 4854 * q^70 - 2046 * q^71 + 4342 * q^72 + 45 * q^73 - 1990 * q^74 + 2393 * q^75 + 3608 * q^76 - 718 * q^77 + 412 * q^79 + 787 * q^80 - 835 * q^81 + 2757 * q^82 - 3709 * q^83 + 1539 * q^84 + 2106 * q^85 - 125 * q^86 - 786 * q^87 - 636 * q^88 - 1663 * q^89 - 1280 * q^90 + 4010 * q^92 + 1186 * q^93 - 2531 * q^94 - 1614 * q^95 + 3084 * q^96 + 1087 * q^97 + 282 * q^98 - 1357 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4.82555	−1.70609	−0.853046	−	0.521836i	$-0.825247\pi$
$2$	−4.82555	−1.70609	−0.853046	+	0.521836i	$0.825247\pi$
$3$	4.44352	0.855156	0.427578	−	0.903978i	$-0.359367\pi$
$3$	4.44352	0.855156	0.427578	+	0.903978i	$0.359367\pi$
$4$	15.2860	1.91075
$5$	12.7712	1.14229	0.571145	−	0.820849i	$-0.306499\pi$
$5$	12.7712	1.14229	0.571145	+	0.820849i	$0.306499\pi$
$6$	−21.4425	−1.45897
$7$	−26.1871	−1.41397	−0.706986	−	0.707228i	$-0.749945\pi$
$7$	−26.1871	−1.41397	−0.706986	+	0.707228i	$0.749945\pi$
$8$	−35.1589	−1.55382
$9$	−7.25513	−0.268708
$10$	−61.6281	−1.94885
$11$	−42.3430	−1.16063	−0.580314	−	0.814393i	$-0.697070\pi$
$11$	−42.3430	−1.16063	−0.580314	+	0.814393i	$0.697070\pi$
$12$	67.9236	1.63399
$13$	0	0
$13$	0	0
$14$	126.367	2.41236
$15$	56.7490	0.976836
$16$	47.3733	0.740208
$17$	−27.3331	−0.389955	−0.194978	−	0.980808i	$-0.562463\pi$
$17$	−27.3331	−0.389955	−0.194978	+	0.980808i	$0.562463\pi$
$18$	35.0100	0.458441
$19$	13.1196	0.158413	0.0792065	−	0.996858i	$-0.474761\pi$
$19$	13.1196	0.158413	0.0792065	+	0.996858i	$0.474761\pi$
$20$	195.220	2.18263
$21$	−116.363	−1.20917
$22$	204.329	1.98014
$23$	−28.7976	−0.261074	−0.130537	−	0.991443i	$-0.541670\pi$
$23$	−28.7976	−0.261074	−0.130537	+	0.991443i	$0.541670\pi$
$24$	−156.229	−1.32876
$25$	38.1033	0.304826
$26$	0	0
$27$	−152.213	−1.08494
$28$	−400.296	−2.70174
$29$	−141.628	−0.906882	−0.453441	−	0.891286i	$-0.649804\pi$
$29$	−141.628	−0.906882	−0.453441	+	0.891286i	$0.649804\pi$
$30$	−273.846	−1.66657
$31$	56.0144	0.324532	0.162266	−	0.986747i	$-0.448120\pi$
$31$	56.0144	0.324532	0.162266	+	0.986747i	$0.448120\pi$
$32$	52.6687	0.290956
$33$	−188.152	−0.992517
$34$	131.897	0.665300
$35$	−334.441	−1.61516
$36$	−110.902	−0.513434
$37$	313.982	1.39509	0.697545	−	0.716541i	$-0.254276\pi$
$37$	313.982	1.39509	0.697545	+	0.716541i	$0.254276\pi$
$38$	−63.3095	−0.270267
$39$	0	0
$40$	−449.021	−1.77491
$41$	−352.375	−1.34224	−0.671118	−	0.741351i	$-0.734186\pi$
$41$	−352.375	−1.34224	−0.671118	+	0.741351i	$0.734186\pi$
$42$	561.516	2.06295
$43$	−320.676	−1.13727	−0.568636	−	0.822589i	$-0.692529\pi$
$43$	−320.676	−1.13727	−0.568636	+	0.822589i	$0.692529\pi$
$44$	−647.255	−2.21767
$45$	−92.6566	−0.306943
$46$	138.964	0.445417
$47$	−339.339	−1.05314	−0.526571	−	0.850131i	$-0.676523\pi$
$47$	−339.339	−1.05314	−0.526571	+	0.850131i	$0.676523\pi$
$48$	210.504	0.632993
$49$	342.765	0.999315
$50$	−183.869	−0.520061
$51$	−121.455	−0.333473
$52$	0	0
$53$	349.461	0.905701	0.452851	−	0.891586i	$-0.350407\pi$
$53$	349.461	0.905701	0.452851	+	0.891586i	$0.350407\pi$
$54$	734.514	1.85101
$55$	−540.771	−1.32577
$56$	920.710	2.19705
$57$	58.2973	0.135468
$58$	683.432	1.54722
$59$	−258.742	−0.570937	−0.285469	−	0.958388i	$-0.592149\pi$
$59$	−258.742	−0.570937	−0.285469	+	0.958388i	$0.592149\pi$
$60$	867.465	1.86649
$61$	650.732	1.36586	0.682932	−	0.730482i	$-0.260705\pi$
$61$	650.732	1.36586	0.682932	+	0.730482i	$0.260705\pi$
$62$	−270.301	−0.553681
$63$	189.991	0.379946
$64$	−633.142	−1.23661
$65$	0	0
$66$	907.938	1.69333
$67$	894.996	1.63196	0.815979	−	0.578082i	$-0.196199\pi$
$67$	894.996	1.63196	0.815979	+	0.578082i	$0.196199\pi$
$68$	−417.813	−0.745106
$69$	−127.963	−0.223259
$70$	1613.86	2.75562
$71$	−741.469	−1.23938	−0.619691	−	0.784846i	$-0.712742\pi$
$71$	−741.469	−1.23938	−0.619691	+	0.784846i	$0.712742\pi$
$72$	255.082	0.417524
$73$	−820.643	−1.31574	−0.657870	−	0.753131i	$-0.728542\pi$
$73$	−820.643	−1.31574	−0.657870	+	0.753131i	$0.728542\pi$
$74$	−1515.14	−2.38015
$75$	169.313	0.260674
$76$	200.546	0.302687
$77$	1108.84	1.64109
$78$	0	0
$79$	−199.372	−0.283938	−0.141969	−	0.989871i	$-0.545343\pi$
$79$	−199.372	−0.283938	−0.141969	+	0.989871i	$0.545343\pi$
$80$	605.013	0.845532
$81$	−480.475	−0.659087
$82$	1700.40	2.28998
$83$	−541.696	−0.716372	−0.358186	−	0.933650i	$-0.616605\pi$
$83$	−541.696	−0.716372	−0.358186	+	0.933650i	$0.616605\pi$
$84$	−1778.72	−2.31041
$85$	−349.076	−0.445442
$86$	1547.44	1.94029
$87$	−629.325	−0.775526
$88$	1488.73	1.80340
$89$	−380.615	−0.453315	−0.226658	−	0.973974i	$-0.572780\pi$
$89$	−380.615	−0.453315	−0.226658	+	0.973974i	$0.572780\pi$
$90$	447.119	0.523672
$91$	0	0
$92$	−440.199	−0.498847
$93$	248.901	0.277525
$94$	1637.50	1.79676
$95$	167.553	0.180954
$96$	234.034	0.248813
$97$	1430.50	1.49737	0.748687	−	0.662924i	$-0.230685\pi$
$97$	1430.50	1.49737	0.748687	+	0.662924i	$0.230685\pi$
$98$	−1654.03	−1.70492
$99$	307.204	0.311870
$100$	582.446	0.582446
$101$	−801.195	−0.789326	−0.394663	−	0.918826i	$-0.629139\pi$
$101$	−801.195	−0.789326	−0.394663	+	0.918826i	$0.629139\pi$
$102$	586.088	0.568935
$103$	−745.805	−0.713459	−0.356730	−	0.934208i	$-0.616108\pi$
$103$	−745.805	−0.713459	−0.356730	+	0.934208i	$0.616108\pi$
$104$	0	0
$105$	−1486.09	−1.38122
$106$	−1686.34	−1.54521
$107$	−134.546	−0.121561	−0.0607806	−	0.998151i	$-0.519359\pi$
$107$	−134.546	−0.121561	−0.0607806	+	0.998151i	$0.519359\pi$
$108$	−2326.73	−2.07305
$109$	−293.245	−0.257686	−0.128843	−	0.991665i	$-0.541126\pi$
$109$	−293.245	−0.257686	−0.128843	+	0.991665i	$0.541126\pi$
$110$	2609.52	2.26189
$111$	1395.19	1.19302
$112$	−1240.57	−1.04663
$113$	1705.38	1.41972	0.709861	−	0.704342i	$-0.248758\pi$
$113$	1705.38	1.41972	0.709861	+	0.704342i	$0.248758\pi$
$114$	−281.317	−0.231121
$115$	−367.779	−0.298223
$116$	−2164.92	−1.73282
$117$	0	0
$118$	1248.57	0.974071
$119$	715.774	0.551386
$120$	−1995.23	−1.51783
$121$	461.932	0.347056
$122$	−3140.14	−2.33029
$123$	−1565.78	−1.14782
$124$	856.235	0.620098
$125$	−1109.77	−0.794090
$126$	−916.811	−0.648222
$127$	1493.18	1.04329	0.521646	−	0.853162i	$-0.325318\pi$
$127$	1493.18	1.04329	0.521646	+	0.853162i	$0.325318\pi$
$128$	2633.91	1.81881
$129$	−1424.93	−0.972545
$130$	0	0
$131$	−459.112	−0.306205	−0.153102	−	0.988210i	$-0.548926\pi$
$131$	−459.112	−0.306205	−0.153102	+	0.988210i	$0.548926\pi$
$132$	−2876.09	−1.89645
$133$	−343.565	−0.223991
$134$	−4318.85	−2.78427
$135$	−1943.95	−1.23932
$136$	961.000	0.605920
$137$	2503.06	1.56096	0.780478	−	0.625183i	$-0.214976\pi$
$137$	2503.06	1.56096	0.780478	+	0.625183i	$0.214976\pi$
$138$	617.491	0.380901
$139$	1803.73	1.10065	0.550326	−	0.834950i	$-0.314503\pi$
$139$	1803.73	1.10065	0.550326	+	0.834950i	$0.314503\pi$
$140$	−5112.25	−3.08617
$141$	−1507.86	−0.900601
$142$	3578.00	2.11450
$143$	0	0
$144$	−343.699	−0.198900
$145$	−1808.75	−1.03592
$146$	3960.06	2.24477
$147$	1523.08	0.854570
$148$	4799.52	2.66566
$149$	−2925.44	−1.60846	−0.804232	−	0.594316i	$-0.797423\pi$
$149$	−2925.44	−1.60846	−0.804232	+	0.594316i	$0.797423\pi$
$150$	−817.028	−0.444733
$151$	1769.28	0.953524	0.476762	−	0.879032i	$-0.341810\pi$
$151$	1769.28	0.953524	0.476762	+	0.879032i	$0.341810\pi$
$152$	−461.271	−0.246145
$153$	198.305	0.104784
$154$	−5350.78	−2.79986
$155$	715.371	0.370709
$156$	0	0
$157$	−1157.24	−0.588265	−0.294132	−	0.955765i	$-0.595031\pi$
$157$	−1157.24	−0.588265	−0.294132	+	0.955765i	$0.595031\pi$
$158$	962.079	0.484423
$159$	1552.84	0.774516
$160$	672.641	0.332356
$161$	754.126	0.369152
$162$	2318.56	1.12446
$163$	−2909.21	−1.39795	−0.698977	−	0.715144i	$-0.746361\pi$
$163$	−2909.21	−1.39795	−0.698977	+	0.715144i	$0.746361\pi$
$164$	−5386.39	−2.56467
$165$	−2402.93	−1.13374
$166$	2613.98	1.22220
$167$	1272.16	0.589476	0.294738	−	0.955578i	$-0.404768\pi$
$167$	1272.16	0.589476	0.294738	+	0.955578i	$0.404768\pi$
$168$	4091.19	1.87882
$169$	0	0
$170$	1684.48	0.759965
$171$	−95.1845	−0.0425669
$172$	−4901.85	−2.17304
$173$	2371.80	1.04234	0.521169	−	0.853453i	$-0.325496\pi$
$173$	2371.80	1.04234	0.521169	+	0.853453i	$0.325496\pi$
$174$	3036.84	1.32312
$175$	−997.815	−0.431015
$176$	−2005.93	−0.859105
$177$	−1149.72	−0.488240
$178$	1836.68	0.773397
$179$	727.183	0.303643	0.151822	−	0.988408i	$-0.451486\pi$
$179$	727.183	0.303643	0.151822	+	0.988408i	$0.451486\pi$
$180$	−1416.35	−0.586490
$181$	3874.20	1.59098	0.795488	−	0.605969i	$-0.207215\pi$
$181$	3874.20	1.59098	0.795488	+	0.605969i	$0.207215\pi$
$182$	0	0
$183$	2891.54	1.16803
$184$	1012.49	0.405662
$185$	4009.92	1.59360
$186$	−1201.09	−0.473484
$187$	1157.36	0.452593
$188$	−5187.13	−2.01229
$189$	3986.03	1.53408
$190$	−808.537	−0.308723
$191$	2003.77	0.759098	0.379549	−	0.925172i	$-0.376079\pi$
$191$	2003.77	0.759098	0.379549	+	0.925172i	$0.376079\pi$
$192$	−2813.38	−1.05749
$193$	1026.25	0.382751	0.191376	−	0.981517i	$-0.438705\pi$
$193$	1026.25	0.382751	0.191376	+	0.981517i	$0.438705\pi$
$194$	−6902.96	−2.55466
$195$	0	0
$196$	5239.50	1.90944
$197$	1308.19	0.473120	0.236560	−	0.971617i	$-0.423980\pi$
$197$	1308.19	0.473120	0.236560	+	0.971617i	$0.423980\pi$
$198$	−1482.43	−0.532079
$199$	−285.120	−0.101566	−0.0507830	−	0.998710i	$-0.516172\pi$
$199$	−285.120	−0.101566	−0.0507830	+	0.998710i	$0.516172\pi$
$200$	−1339.67	−0.473644
$201$	3976.93	1.39558
$202$	3866.21	1.34666
$203$	3708.82	1.28231
$204$	−1856.56	−0.637182
$205$	−4500.24	−1.53322
$206$	3598.92	1.21723
$207$	208.930	0.0701529
$208$	0	0
$209$	−555.525	−0.183859
$210$	7171.23	2.35648
$211$	−2813.91	−0.918092	−0.459046	−	0.888412i	$-0.651809\pi$
$211$	−2813.91	−0.918092	−0.459046	+	0.888412i	$0.651809\pi$
$212$	5341.86	1.73057
$213$	−3294.73	−1.05986
$214$	649.259	0.207395
$215$	−4095.42	−1.29909
$216$	5351.65	1.68580
$217$	−1466.86	−0.458879
$218$	1415.07	0.439635
$219$	−3646.55	−1.12516
$220$	−8266.21	−2.53322
$221$	0	0
$222$	−6732.55	−2.03540
$223$	−1509.86	−0.453399	−0.226699	−	0.973965i	$-0.572794\pi$
$223$	−1509.86	−0.453399	−0.226699	+	0.973965i	$0.572794\pi$
$224$	−1379.24	−0.411403
$225$	−276.444	−0.0819093
$226$	−8229.40	−2.42217
$227$	643.122	0.188042	0.0940209	−	0.995570i	$-0.470028\pi$
$227$	643.122	0.188042	0.0940209	+	0.995570i	$0.470028\pi$
$228$	891.131	0.258845
$229$	−154.009	−0.0444420	−0.0222210	−	0.999753i	$-0.507074\pi$
$229$	−154.009	−0.0444420	−0.0222210	+	0.999753i	$0.507074\pi$
$230$	1774.74	0.508795
$231$	4927.16	1.40339
$232$	4979.47	1.40913
$233$	−261.668	−0.0735726	−0.0367863	−	0.999323i	$-0.511712\pi$
$233$	−261.668	−0.0735726	−0.0367863	+	0.999323i	$0.511712\pi$
$234$	0	0
$235$	−4333.76	−1.20299
$236$	−3955.12	−1.09092
$237$	−885.912	−0.242811
$238$	−3454.01	−0.940714
$239$	2495.09	0.675288	0.337644	−	0.941274i	$-0.390370\pi$
$239$	2495.09	0.675288	0.337644	+	0.941274i	$0.390370\pi$
$240$	2688.39	0.723061
$241$	−3835.70	−1.02522	−0.512612	−	0.858620i	$-0.671322\pi$
$241$	−3835.70	−1.02522	−0.512612	+	0.858620i	$0.671322\pi$
$242$	−2229.08	−0.592110
$243$	1974.76	0.521321
$244$	9947.07	2.60982
$245$	4377.52	1.14151
$246$	7555.77	1.95829
$247$	0	0
$248$	−1969.40	−0.504263
$249$	−2407.04	−0.612610
$250$	5355.28	1.35479
$251$	−3977.59	−1.00025	−0.500126	−	0.865953i	$-0.666713\pi$
$251$	−3977.59	−1.00025	−0.500126	+	0.865953i	$0.666713\pi$
$252$	2904.19	0.725980
$253$	1219.38	0.303010
$254$	−7205.41	−1.77995
$255$	−1551.13	−0.380923
$256$	−7644.95	−1.86644
$257$	−121.486	−0.0294868	−0.0147434	−	0.999891i	$-0.504693\pi$
$257$	−121.486	−0.0294868	−0.0147434	+	0.999891i	$0.504693\pi$
$258$	6876.09	1.65925
$259$	−8222.29	−1.97262
$260$	0	0
$261$	1027.53	0.243687
$262$	2215.47	0.522413
$263$	1175.42	0.275586	0.137793	−	0.990461i	$-0.455999\pi$
$263$	1175.42	0.275586	0.137793	+	0.990461i	$0.455999\pi$
$264$	6615.22	1.54219
$265$	4463.03	1.03457
$266$	1657.89	0.382150
$267$	−1691.27	−0.387655
$268$	13680.9	3.11826
$269$	−8319.66	−1.88572	−0.942860	−	0.333190i	$-0.891875\pi$
$269$	−8319.66	−1.88572	−0.942860	+	0.333190i	$0.891875\pi$
$270$	9380.62	2.11439
$271$	−3928.56	−0.880602	−0.440301	−	0.897850i	$-0.645128\pi$
$271$	−3928.56	−0.880602	−0.440301	+	0.897850i	$0.645128\pi$
$272$	−1294.86	−0.288648
$273$	0	0
$274$	−12078.7	−2.66313
$275$	−1613.41	−0.353790
$276$	−1956.03	−0.426592
$277$	−6022.61	−1.30637	−0.653183	−	0.757200i	$-0.726567\pi$
$277$	−6022.61	−1.30637	−0.653183	+	0.757200i	$0.726567\pi$
$278$	−8704.01	−1.87781
$279$	−406.392	−0.0872044
$280$	11758.6	2.50967
$281$	2183.71	0.463592	0.231796	−	0.972764i	$-0.425540\pi$
$281$	2183.71	0.463592	0.231796	+	0.972764i	$0.425540\pi$
$282$	7276.26	1.53651
$283$	−8133.25	−1.70838	−0.854190	−	0.519961i	$-0.825946\pi$
$283$	−8133.25	−1.70838	−0.854190	+	0.519961i	$0.825946\pi$
$284$	−11334.1	−2.36815
$285$	744.526	0.154744
$286$	0	0
$287$	9227.67	1.89788
$288$	−382.118	−0.0781823
$289$	−4165.90	−0.847935
$290$	8728.24	1.76738
$291$	6356.46	1.28049
$292$	−12544.3	−2.51405
$293$	3182.55	0.634562	0.317281	−	0.948332i	$-0.397230\pi$
$293$	3182.55	0.634562	0.317281	+	0.948332i	$0.397230\pi$
$294$	−7349.72	−1.45797
$295$	−3304.44	−0.652176
$296$	−11039.3	−2.16772
$297$	6445.17	1.25922
$298$	14116.9	2.74419
$299$	0	0
$300$	2588.11	0.498082
$301$	8397.59	1.60807
$302$	−8537.77	−1.62680
$303$	−3560.13	−0.674997
$304$	621.520	0.117259
$305$	8310.62	1.56021
$306$	−956.931	−0.178772
$307$	−1230.21	−0.228703	−0.114352	−	0.993440i	$-0.536479\pi$
$307$	−1230.21	−0.228703	−0.114352	+	0.993440i	$0.536479\pi$
$308$	16949.7	3.13572
$309$	−3314.00	−0.610119
$310$	−3452.06	−0.632464
$311$	−5881.49	−1.07237	−0.536187	−	0.844099i	$-0.680136\pi$
$311$	−5881.49	−1.07237	−0.536187	+	0.844099i	$0.680136\pi$
$312$	0	0
$313$	6800.24	1.22803	0.614013	−	0.789296i	$-0.289554\pi$
$313$	6800.24	1.22803	0.614013	+	0.789296i	$0.289554\pi$
$314$	5584.31	1.00363
$315$	2426.41	0.434008
$316$	−3047.59	−0.542533
$317$	−2212.50	−0.392008	−0.196004	−	0.980603i	$-0.562796\pi$
$317$	−2212.50	−0.392008	−0.196004	+	0.980603i	$0.562796\pi$
$318$	−7493.30	−1.32139
$319$	5996.94	1.05255
$320$	−8085.97	−1.41256
$321$	−597.858	−0.103954
$322$	−3639.07	−0.629807
$323$	−358.600	−0.0617740
$324$	−7344.53	−1.25935
$325$	0	0
$326$	14038.5	2.38504
$327$	−1303.04	−0.220361
$328$	12389.1	2.08559
$329$	8886.31	1.48911
$330$	11595.5	1.93427
$331$	1744.14	0.289627	0.144814	−	0.989459i	$-0.453742\pi$
$331$	1744.14	0.289627	0.144814	+	0.989459i	$0.453742\pi$
$332$	−8280.35	−1.36881
$333$	−2277.98	−0.374872
$334$	−6138.87	−1.00570
$335$	11430.2	1.86417
$336$	−5512.50	−0.895034
$337$	8805.54	1.42335	0.711674	−	0.702510i	$-0.247937\pi$
$337$	8805.54	1.42335	0.711674	+	0.702510i	$0.247937\pi$
$338$	0	0
$339$	7577.88	1.21408
$340$	−5335.96	−0.851127
$341$	−2371.82	−0.376661
$342$	459.318	0.0726230
$343$	6.15700	0.000969232 0
$344$	11274.6	1.76711
$345$	−1634.24	−0.255027
$346$	−11445.2	−1.77832
$347$	−1943.15	−0.300616	−0.150308	−	0.988639i	$-0.548027\pi$
$347$	−1943.15	−0.300616	−0.150308	+	0.988639i	$0.548027\pi$
$348$	−9619.85	−1.48183
$349$	5316.91	0.815495	0.407748	−	0.913095i	$-0.366314\pi$
$349$	5316.91	0.815495	0.407748	+	0.913095i	$0.366314\pi$
$350$	4815.01	0.735352
$351$	0	0
$352$	−2230.15	−0.337692
$353$	−5013.67	−0.755951	−0.377976	−	0.925816i	$-0.623380\pi$
$353$	−5013.67	−0.755951	−0.377976	+	0.925816i	$0.623380\pi$
$354$	5548.06	0.832983
$355$	−9469.44	−1.41573
$356$	−5818.07	−0.866171
$357$	3180.56	0.471521
$358$	−3509.06	−0.518043
$359$	−8143.55	−1.19722	−0.598608	−	0.801042i	$-0.704279\pi$
$359$	−8143.55	−1.19722	−0.598608	+	0.801042i	$0.704279\pi$
$360$	3257.70	0.476933
$361$	−6686.88	−0.974905
$362$	−18695.1	−2.71435
$363$	2052.60	0.296787
$364$	0	0
$365$	−10480.6	−1.50296
$366$	−13953.3	−1.99276
$367$	6517.85	0.927054	0.463527	−	0.886083i	$-0.346584\pi$
$367$	6517.85	0.927054	0.463527	+	0.886083i	$0.346584\pi$
$368$	−1364.24	−0.193249
$369$	2556.52	0.360670
$370$	−19350.1	−2.71882
$371$	−9151.38	−1.28064
$372$	3804.70	0.530281
$373$	7247.57	1.00607	0.503036	−	0.864266i	$-0.332216\pi$
$373$	7247.57	1.00607	0.503036	+	0.864266i	$0.332216\pi$
$374$	−5584.93	−0.772165
$375$	−4931.31	−0.679071
$376$	11930.8	1.63639
$377$	0	0
$378$	−19234.8	−2.61728
$379$	6549.25	0.887632	0.443816	−	0.896118i	$-0.353624\pi$
$379$	6549.25	0.887632	0.443816	+	0.896118i	$0.353624\pi$
$380$	2561.21	0.345757
$381$	6634.96	0.892177
$382$	−9669.31	−1.29509
$383$	−8030.33	−1.07136	−0.535679	−	0.844421i	$-0.679944\pi$
$383$	−8030.33	−1.07136	−0.535679	+	0.844421i	$0.679944\pi$
$384$	11703.8	1.55536
$385$	14161.2	1.87460
$386$	−4952.22	−0.653009
$387$	2326.55	0.305595
$388$	21866.6	2.86110
$389$	481.389	0.0627440	0.0313720	−	0.999508i	$-0.490012\pi$
$389$	481.389	0.0627440	0.0313720	+	0.999508i	$0.490012\pi$
$390$	0	0
$391$	787.126	0.101807
$392$	−12051.2	−1.55275
$393$	−2040.07	−0.261853
$394$	−6312.74	−0.807186
$395$	−2546.21	−0.324339
$396$	4695.91	0.595905
$397$	−2802.07	−0.354237	−0.177118	−	0.984190i	$-0.556678\pi$
$397$	−2802.07	−0.354237	−0.177118	+	0.984190i	$0.556678\pi$
$398$	1375.86	0.173281
$399$	−1526.64	−0.191548
$400$	1805.08	0.225635
$401$	4376.79	0.545053	0.272527	−	0.962148i	$-0.412141\pi$
$401$	4376.79	0.545053	0.272527	+	0.962148i	$0.412141\pi$
$402$	−19190.9	−2.38098
$403$	0	0
$404$	−12247.1	−1.50820
$405$	−6136.23	−0.752869
$406$	−17897.1	−2.18773
$407$	−13295.0	−1.61918
$408$	4270.22	0.518156
$409$	12969.0	1.56791	0.783954	−	0.620819i	$-0.213200\pi$
$409$	12969.0	1.56791	0.783954	+	0.620819i	$0.213200\pi$
$410$	21716.2	2.61582
$411$	11122.4	1.33486
$412$	−11400.4	−1.36324
$413$	6775.70	0.807289
$414$	−1008.20	−0.119687
$415$	−6918.10	−0.818304
$416$	0	0
$417$	8014.93	0.941229
$418$	2680.71	0.313679
$419$	7679.05	0.895336	0.447668	−	0.894200i	$-0.352255\pi$
$419$	7679.05	0.895336	0.447668	+	0.894200i	$0.352255\pi$
$420$	−22716.4	−2.63916
$421$	−2963.10	−0.343022	−0.171511	−	0.985182i	$-0.554865\pi$
$421$	−2963.10	−0.343022	−0.171511	+	0.985182i	$0.554865\pi$
$422$	13578.7	1.56635
$423$	2461.95	0.282988
$424$	−12286.7	−1.40730
$425$	−1041.48	−0.118869
$426$	15898.9	1.80823
$427$	−17040.8	−1.93129
$428$	−2056.67	−0.232273
$429$	0	0
$430$	19762.7	2.21637
$431$	−14202.6	−1.58728	−0.793639	−	0.608389i	$-0.791816\pi$
$431$	−14202.6	−1.58728	−0.793639	+	0.608389i	$0.791816\pi$
$432$	−7210.85	−0.803083
$433$	10118.2	1.12298	0.561488	−	0.827485i	$-0.310229\pi$
$433$	10118.2	1.12298	0.561488	+	0.827485i	$0.310229\pi$
$434$	7078.39	0.782889
$435$	−8037.23	−0.885875
$436$	−4482.53	−0.492372
$437$	−377.814	−0.0413576
$438$	17596.6	1.91963
$439$	−16184.3	−1.75954	−0.879768	−	0.475404i	$-0.842302\pi$
$439$	−16184.3	−1.75954	−0.879768	+	0.475404i	$0.842302\pi$
$440$	19012.9	2.06001
$441$	−2486.80	−0.268524
$442$	0	0
$443$	−10025.2	−1.07519	−0.537596	−	0.843203i	$-0.680667\pi$
$443$	−10025.2	−1.07519	−0.537596	+	0.843203i	$0.680667\pi$
$444$	21326.8	2.27956
$445$	−4860.90	−0.517818
$446$	7285.93	0.773540
$447$	−12999.2	−1.37549
$448$	16580.2	1.74852
$449$	10428.3	1.09608	0.548041	−	0.836451i	$-0.315374\pi$
$449$	10428.3	1.09608	0.548041	+	0.836451i	$0.315374\pi$
$450$	1334.00	0.139745
$451$	14920.6	1.55784
$452$	26068.4	2.71273
$453$	7861.84	0.815412
$454$	−3103.42	−0.320816
$455$	0	0
$456$	−2049.67	−0.210492
$457$	12263.8	1.25531	0.627656	−	0.778491i	$-0.284015\pi$
$457$	12263.8	1.25531	0.627656	+	0.778491i	$0.284015\pi$
$458$	743.180	0.0758221
$459$	4160.46	0.423080
$460$	−5621.87	−0.569828
$461$	−4047.98	−0.408966	−0.204483	−	0.978870i	$-0.565551\pi$
$461$	−4047.98	−0.408966	−0.204483	+	0.978870i	$0.565551\pi$
$462$	−23776.3	−2.39431
$463$	10473.9	1.05133	0.525663	−	0.850693i	$-0.323817\pi$
$463$	10473.9	1.05133	0.525663	+	0.850693i	$0.323817\pi$
$464$	−6709.37	−0.671281
$465$	3178.76	0.317014
$466$	1262.69	0.125522
$467$	4906.17	0.486146	0.243073	−	0.970008i	$-0.421845\pi$
$467$	4906.17	0.486146	0.243073	+	0.970008i	$0.421845\pi$
$468$	0	0
$469$	−23437.4	−2.30754
$470$	20912.8	2.05242
$471$	−5142.20	−0.503058
$472$	9097.07	0.887132
$473$	13578.4	1.31995
$474$	4275.02	0.414257
$475$	499.901	0.0482884
$476$	10941.3	1.05356
$477$	−2535.38	−0.243370
$478$	−12040.2	−1.15210
$479$	1249.42	0.119180	0.0595902	−	0.998223i	$-0.481021\pi$
$479$	1249.42	0.119180	0.0595902	+	0.998223i	$0.481021\pi$
$480$	2988.90	0.284216
$481$	0	0
$482$	18509.4	1.74913
$483$	3350.97	0.315682
$484$	7061.08	0.663137
$485$	18269.2	1.71044
$486$	−9529.32	−0.889421
$487$	−19290.2	−1.79492	−0.897458	−	0.441100i	$-0.854588\pi$
$487$	−19290.2	−1.79492	−0.897458	+	0.441100i	$0.854588\pi$
$488$	−22879.0	−2.12230
$489$	−12927.1	−1.19547
$490$	−21123.9	−1.94751
$491$	−8381.17	−0.770339	−0.385170	−	0.922846i	$-0.625857\pi$
$491$	−8381.17	−0.770339	−0.385170	+	0.922846i	$0.625857\pi$
$492$	−23934.5	−2.19319
$493$	3871.12	0.353644
$494$	0	0
$495$	3923.36	0.356246
$496$	2653.59	0.240221
$497$	19416.9	1.75245
$498$	11615.3	1.04517
$499$	−199.025	−0.0178549	−0.00892743	−	0.999960i	$-0.502842\pi$
$499$	−199.025	−0.0178549	−0.00892743	+	0.999960i	$0.502842\pi$
$500$	−16964.0	−1.51731
$501$	5652.86	0.504094
$502$	19194.1	1.70652
$503$	−7567.90	−0.670847	−0.335424	−	0.942067i	$-0.608879\pi$
$503$	−7567.90	−0.670847	−0.335424	+	0.942067i	$0.608879\pi$
$504$	−6679.86	−0.590367
$505$	−10232.2	−0.901639
$506$	−5884.17	−0.516963
$507$	0	0
$508$	22824.7	1.99347
$509$	−5998.22	−0.522331	−0.261166	−	0.965294i	$-0.584107\pi$
$509$	−5998.22	−0.522331	−0.261166	+	0.965294i	$0.584107\pi$
$510$	7485.04	0.649889
$511$	21490.3	1.86042
$512$	15819.8	1.36552
$513$	−1996.98	−0.171869
$514$	586.238	0.0503071
$515$	−9524.81	−0.814977
$516$	−21781.5	−1.85829
$517$	14368.6	1.22231
$518$	39677.1	3.36547
$519$	10539.1	0.891362
$520$	0	0
$521$	−2863.99	−0.240833	−0.120416	−	0.992723i	$-0.538423\pi$
$521$	−2863.99	−0.240833	−0.120416	+	0.992723i	$0.538423\pi$
$522$	−4958.38	−0.415752
$523$	−2529.09	−0.211452	−0.105726	−	0.994395i	$-0.533717\pi$
$523$	−2529.09	−0.211452	−0.105726	+	0.994395i	$0.533717\pi$
$524$	−7017.98	−0.585080
$525$	−4433.81	−0.368585
$526$	−5672.03	−0.470176
$527$	−1531.05	−0.126553
$528$	−8913.38	−0.734669
$529$	−11337.7	−0.931840
$530$	−21536.6	−1.76508
$531$	1877.20	0.153416
$532$	−5251.73	−0.427991
$533$	0	0
$534$	8161.31	0.661375
$535$	−1718.31	−0.138858
$536$	−31467.0	−2.53576
$537$	3231.25	0.259663
$538$	40147.0	3.21721
$539$	−14513.7	−1.15983
$540$	−29715.1	−2.36803
$541$	−22053.6	−1.75260	−0.876302	−	0.481762i	$-0.839997\pi$
$541$	−22053.6	−1.75260	−0.876302	+	0.481762i	$0.839997\pi$
$542$	18957.5	1.50239
$543$	17215.1	1.36053
$544$	−1439.60	−0.113460
$545$	−3745.08	−0.294352
$546$	0	0
$547$	−2821.80	−0.220569	−0.110285	−	0.993900i	$-0.535176\pi$
$547$	−2821.80	−0.220569	−0.110285	+	0.993900i	$0.535176\pi$
$548$	38261.8	2.98259
$549$	−4721.14	−0.367019
$550$	7785.59	0.603597
$551$	−1858.10	−0.143662
$552$	4499.02	0.346904
$553$	5220.97	0.401479
$554$	29062.4	2.22878
$555$	17818.2	1.36277
$556$	27571.8	2.10307
$557$	−1837.17	−0.139754	−0.0698772	−	0.997556i	$-0.522261\pi$
$557$	−1837.17	−0.139754	−0.0698772	+	0.997556i	$0.522261\pi$
$558$	1961.07	0.148779
$559$	0	0
$560$	−15843.6	−1.19556
$561$	5142.77	0.387038
$562$	−10537.6	−0.790930
$563$	−24877.9	−1.86230	−0.931152	−	0.364631i	$-0.881195\pi$
$563$	−24877.9	−1.86230	−0.931152	+	0.364631i	$0.881195\pi$
$564$	−23049.1	−1.72082
$565$	21779.7	1.62173
$566$	39247.4	2.91465
$567$	12582.2	0.931931
$568$	26069.2	1.92577
$569$	−7111.72	−0.523969	−0.261985	−	0.965072i	$-0.584377\pi$
$569$	−7111.72	−0.523969	−0.261985	+	0.965072i	$0.584377\pi$
$570$	−3592.75	−0.264007
$571$	−11919.4	−0.873578	−0.436789	−	0.899564i	$-0.643884\pi$
$571$	−11919.4	−0.873578	−0.436789	+	0.899564i	$0.643884\pi$
$572$	0	0
$573$	8903.80	0.649148
$574$	−44528.6	−3.23796
$575$	−1097.28	−0.0795823
$576$	4593.52	0.332286
$577$	−11600.6	−0.836984	−0.418492	−	0.908220i	$-0.637441\pi$
$577$	−11600.6	−0.836984	−0.418492	+	0.908220i	$0.637441\pi$
$578$	20102.8	1.44665
$579$	4560.16	0.327312
$580$	−27648.6	−1.97939
$581$	14185.5	1.01293
$582$	−30673.4	−2.18463
$583$	−14797.2	−1.05118
$584$	28852.9	2.04442
$585$	0	0
$586$	−15357.6	−1.08262
$587$	10341.7	0.727167	0.363583	−	0.931562i	$-0.381553\pi$
$587$	10341.7	0.727167	0.363583	+	0.931562i	$0.381553\pi$
$588$	23281.8	1.63287
$589$	734.888	0.0514101
$590$	15945.8	1.11267
$591$	5812.97	0.404592
$592$	14874.4	1.03266
$593$	2782.21	0.192668	0.0963338	−	0.995349i	$-0.469288\pi$
$593$	2782.21	0.192668	0.0963338	+	0.995349i	$0.469288\pi$
$594$	−31101.5	−2.14834
$595$	9141.29	0.629842
$596$	−44718.2	−3.07337
$597$	−1266.94	−0.0868547
$598$	0	0
$599$	10560.6	0.720359	0.360179	−	0.932883i	$-0.382715\pi$
$599$	10560.6	0.720359	0.360179	+	0.932883i	$0.382715\pi$
$600$	−5952.84	−0.405040
$601$	3846.85	0.261092	0.130546	−	0.991442i	$-0.458327\pi$
$601$	3846.85	0.261092	0.130546	+	0.991442i	$0.458327\pi$
$602$	−40523.0	−2.74351
$603$	−6493.31	−0.438520
$604$	27045.2	1.82194
$605$	5899.42	0.396439
$606$	17179.6	1.15161
$607$	6495.50	0.434340	0.217170	−	0.976134i	$-0.430317\pi$
$607$	6495.50	0.434340	0.217170	+	0.976134i	$0.430317\pi$
$608$	690.993	0.0460912
$609$	16480.2	1.09657
$610$	−40103.4	−2.66186
$611$	0	0
$612$	3031.28	0.200216
$613$	3532.60	0.232758	0.116379	−	0.993205i	$-0.462871\pi$
$613$	3532.60	0.232758	0.116379	+	0.993205i	$0.462871\pi$
$614$	5936.46	0.390189
$615$	−19996.9	−1.31114
$616$	−38985.6	−2.54996
$617$	−16297.9	−1.06342	−0.531710	−	0.846926i	$-0.678451\pi$
$617$	−16297.9	−1.06342	−0.531710	+	0.846926i	$0.678451\pi$
$618$	15991.9	1.04092
$619$	9123.86	0.592438	0.296219	−	0.955120i	$-0.404274\pi$
$619$	9123.86	0.592438	0.296219	+	0.955120i	$0.404274\pi$
$620$	10935.1	0.708332
$621$	4383.38	0.283251
$622$	28381.4	1.82957
$623$	9967.20	0.640975
$624$	0	0
$625$	−18936.0	−1.21191
$626$	−32814.9	−2.09512
$627$	−2468.48	−0.157228
$628$	−17689.5	−1.12402
$629$	−8582.09	−0.544023
$630$	−11708.8	−0.740458
$631$	−23250.8	−1.46688	−0.733438	−	0.679756i	$-0.762086\pi$
$631$	−23250.8	−1.46688	−0.733438	+	0.679756i	$0.762086\pi$
$632$	7009.68	0.441187
$633$	−12503.7	−0.785112
$634$	10676.5	0.668801
$635$	19069.7	1.19174
$636$	23736.6	1.47990
$637$	0	0
$638$	−28938.6	−1.79575
$639$	5379.45	0.333032
$640$	33638.2	2.07760
$641$	−7478.44	−0.460812	−0.230406	−	0.973095i	$-0.574005\pi$
$641$	−7478.44	−0.460812	−0.230406	+	0.973095i	$0.574005\pi$
$642$	2885.00	0.177355
$643$	−8327.98	−0.510768	−0.255384	−	0.966840i	$-0.582202\pi$
$643$	−8327.98	−0.510768	−0.255384	+	0.966840i	$0.582202\pi$
$644$	11527.5	0.705356
$645$	−18198.1	−1.11093
$646$	1730.44	0.105392
$647$	6278.75	0.381519	0.190760	−	0.981637i	$-0.438905\pi$
$647$	6278.75	0.381519	0.190760	+	0.981637i	$0.438905\pi$
$648$	16893.0	1.02410
$649$	10955.9	0.662645
$650$	0	0
$651$	−6518.01	−0.392413
$652$	−44470.1	−2.67114
$653$	−27163.5	−1.62786	−0.813928	−	0.580966i	$-0.802675\pi$
$653$	−27163.5	−1.62786	−0.813928	+	0.580966i	$0.802675\pi$
$654$	6287.88	0.375957
$655$	−5863.41	−0.349774
$656$	−16693.1	−0.993533
$657$	5953.87	0.353550
$658$	−42881.4	−2.54056
$659$	−10352.0	−0.611921	−0.305961	−	0.952044i	$-0.598978\pi$
$659$	−10352.0	−0.611921	−0.305961	+	0.952044i	$0.598978\pi$
$660$	−36731.1	−2.16630
$661$	−6270.46	−0.368975	−0.184487	−	0.982835i	$-0.559063\pi$
$661$	−6270.46	−0.368975	−0.184487	+	0.982835i	$0.559063\pi$
$662$	−8416.44	−0.494130
$663$	0	0
$664$	19045.4	1.11311
$665$	−4387.73	−0.255863
$666$	10992.5	0.639567
$667$	4078.53	0.236764
$668$	19446.2	1.12634
$669$	−6709.11	−0.387727
$670$	−55156.9	−3.18044
$671$	−27554.0	−1.58526
$672$	−6128.68	−0.351814
$673$	−30301.9	−1.73559	−0.867794	−	0.496923i	$-0.834463\pi$
$673$	−30301.9	−1.73559	−0.867794	+	0.496923i	$0.834463\pi$
$674$	−42491.6	−2.42836
$675$	−5799.83	−0.330719
$676$	0	0
$677$	−10223.3	−0.580375	−0.290188	−	0.956970i	$-0.593718\pi$
$677$	−10223.3	−0.580375	−0.290188	+	0.956970i	$0.593718\pi$
$678$	−36567.5	−2.07134
$679$	−37460.7	−2.11724
$680$	12273.1	0.692136
$681$	2857.72	0.160805
$682$	11445.3	0.642617
$683$	−16978.0	−0.951163	−0.475582	−	0.879672i	$-0.657762\pi$
$683$	−16978.0	−0.951163	−0.475582	+	0.879672i	$0.657762\pi$
$684$	−1454.99	−0.0813346
$685$	31967.1	1.78307
$686$	−29.7109	−0.00165360
$687$	−684.343	−0.0380048
$688$	−15191.5	−0.841818
$689$	0	0
$690$	7886.09	0.435099
$691$	−7935.33	−0.436866	−0.218433	−	0.975852i	$-0.570094\pi$
$691$	−7935.33	−0.436866	−0.218433	+	0.975852i	$0.570094\pi$
$692$	36255.3	1.99165
$693$	−8044.78	−0.440976
$694$	9376.79	0.512879
$695$	23035.8	1.25726
$696$	22126.4	1.20503
$697$	9631.48	0.523412
$698$	−25657.1	−1.39131
$699$	−1162.73	−0.0629160
$700$	−15252.6	−0.823562
$701$	18557.3	0.999857	0.499928	−	0.866067i	$-0.333360\pi$
$701$	18557.3	0.999857	0.499928	+	0.866067i	$0.333360\pi$
$702$	0	0
$703$	4119.33	0.221001
$704$	26809.1	1.43524
$705$	−19257.2	−1.02875
$706$	24193.7	1.28972
$707$	20981.0	1.11608
$708$	−17574.7	−0.932904
$709$	−14163.7	−0.750253	−0.375127	−	0.926974i	$-0.622401\pi$
$709$	−14163.7	−0.750253	−0.375127	+	0.926974i	$0.622401\pi$
$710$	45695.3	2.41537
$711$	1446.47	0.0762964
$712$	13382.0	0.704370
$713$	−1613.08	−0.0847270
$714$	−15348.0	−0.804458
$715$	0	0
$716$	11115.7	0.580186
$717$	11087.0	0.577476
$718$	39297.2	2.04256
$719$	24836.7	1.28825	0.644125	−	0.764920i	$-0.277222\pi$
$719$	24836.7	1.28825	0.644125	+	0.764920i	$0.277222\pi$
$720$	−4389.45	−0.227201
$721$	19530.5	1.00881
$722$	32267.9	1.66328
$723$	−17044.0	−0.876726
$724$	59220.9	3.03995
$725$	−5396.48	−0.276441
$726$	−9904.95	−0.506346
$727$	−10641.0	−0.542852	−0.271426	−	0.962459i	$-0.587495\pi$
$727$	−10641.0	−0.542852	−0.271426	+	0.962459i	$0.587495\pi$
$728$	0	0
$729$	21747.7	1.10490
$730$	50574.7	2.56418
$731$	8765.07	0.443486
$732$	44200.0	2.23180
$733$	22429.1	1.13020	0.565102	−	0.825021i	$-0.308837\pi$
$733$	22429.1	1.13020	0.565102	+	0.825021i	$0.308837\pi$
$734$	−31452.2	−1.58164
$735$	19451.6	0.976166
$736$	−1516.73	−0.0759612
$737$	−37896.8	−1.89409
$738$	−12336.6	−0.615336
$739$	33425.8	1.66386	0.831928	−	0.554884i	$-0.187237\pi$
$739$	33425.8	1.66386	0.831928	+	0.554884i	$0.187237\pi$
$740$	61295.6	3.04496
$741$	0	0
$742$	44160.5	2.18488
$743$	13136.3	0.648619	0.324309	−	0.945951i	$-0.394868\pi$
$743$	13136.3	0.648619	0.324309	+	0.945951i	$0.394868\pi$
$744$	−8751.09	−0.431224
$745$	−37361.3	−1.83733
$746$	−34973.5	−1.71645
$747$	3930.07	0.192495
$748$	17691.5	0.864791
$749$	3523.37	0.171884
$750$	23796.3	1.15856
$751$	20688.2	1.00522	0.502612	−	0.864512i	$-0.332373\pi$
$751$	20688.2	1.00522	0.502612	+	0.864512i	$0.332373\pi$
$752$	−16075.6	−0.779544
$753$	−17674.5	−0.855371
$754$	0	0
$755$	22595.8	1.08920
$756$	60930.3	2.93124
$757$	−6145.85	−0.295079	−0.147539	−	0.989056i	$-0.547135\pi$
$757$	−6145.85	−0.295079	−0.147539	+	0.989056i	$0.547135\pi$
$758$	−31603.8	−1.51438
$759$	5418.33	0.259121
$760$	−5890.98	−0.281169
$761$	−16488.5	−0.785424	−0.392712	−	0.919662i	$-0.628463\pi$
$761$	−16488.5	−0.785424	−0.392712	+	0.919662i	$0.628463\pi$
$762$	−32017.4	−1.52214
$763$	7679.23	0.364360
$764$	30629.6	1.45045
$765$	2532.59	0.119694
$766$	38750.8	1.82784
$767$	0	0
$768$	−33970.5	−1.59610
$769$	−7801.71	−0.365848	−0.182924	−	0.983127i	$-0.558556\pi$
$769$	−7801.71	−0.365848	−0.182924	+	0.983127i	$0.558556\pi$
$770$	−68335.8	−3.19825
$771$	−539.826	−0.0252158
$772$	15687.2	0.731341
$773$	−27301.3	−1.27032	−0.635161	−	0.772380i	$-0.719066\pi$
$773$	−27301.3	−1.27032	−0.635161	+	0.772380i	$0.719066\pi$
$774$	−11226.9	−0.521372
$775$	2134.33	0.0989258
$776$	−50294.8	−2.32665
$777$	−36535.9	−1.68690
$778$	−2322.97	−0.107047
$779$	−4623.02	−0.212628
$780$	0	0
$781$	31396.0	1.43846
$782$	−3798.32	−0.173693
$783$	21557.6	0.983916
$784$	16237.9	0.739700
$785$	−14779.3	−0.671969
$786$	9844.49	0.446745
$787$	26016.0	1.17836	0.589181	−	0.808001i	$-0.299450\pi$
$787$	26016.0	1.17836	0.589181	+	0.808001i	$0.299450\pi$
$788$	19997.0	0.904013
$789$	5222.98	0.235669
$790$	12286.9	0.553352
$791$	−44658.9	−2.00745
$792$	−10800.9	−0.484590
$793$	0	0
$794$	13521.5	0.604360
$795$	19831.6	0.884722
$796$	−4358.34	−0.194067
$797$	8178.16	0.363470	0.181735	−	0.983348i	$-0.441829\pi$
$797$	8178.16	0.363470	0.181735	+	0.983348i	$0.441829\pi$
$798$	7366.88	0.326798
$799$	9275.18	0.410678
$800$	2006.85	0.0886910
$801$	2761.41	0.121810
$802$	−21120.4	−0.929911
$803$	34748.5	1.52708
$804$	60791.3	2.66660
$805$	9631.08	0.421678
$806$	0	0
$807$	−36968.6	−1.61258
$808$	28169.1	1.22647
$809$	16162.0	0.702380	0.351190	−	0.936304i	$-0.385777\pi$
$809$	16162.0	0.702380	0.351190	+	0.936304i	$0.385777\pi$
$810$	29610.7	1.28446
$811$	24714.6	1.07009	0.535046	−	0.844823i	$-0.320294\pi$
$811$	24714.6	1.07009	0.535046	+	0.844823i	$0.320294\pi$
$812$	56692.9	2.45016
$813$	−17456.6	−0.753052
$814$	64155.5	2.76247
$815$	−37154.0	−1.59687
$816$	−5753.73	−0.246839
$817$	−4207.15	−0.180159
$818$	−62582.5	−2.67499
$819$	0	0
$820$	−68790.6	−2.92960
$821$	−28565.3	−1.21429	−0.607147	−	0.794589i	$-0.707686\pi$
$821$	−28565.3	−1.21429	−0.607147	+	0.794589i	$0.707686\pi$
$822$	−53671.8	−2.27740
$823$	9276.01	0.392881	0.196441	−	0.980516i	$-0.437062\pi$
$823$	9276.01	0.392881	0.196441	+	0.980516i	$0.437062\pi$
$824$	26221.7	1.10859
$825$	−7169.21	−0.302545
$826$	−32696.5	−1.37731
$827$	−36.2767	−0.00152535	−0.000762676	−	1.00000i	$-0.500243\pi$
$827$	−36.2767	−0.00152535	−0.000762676		1.00000i	$0.500243\pi$
$828$	3193.70	0.134044
$829$	16764.4	0.702355	0.351177	−	0.936309i	$-0.385781\pi$
$829$	16764.4	0.702355	0.351177	+	0.936309i	$0.385781\pi$
$830$	33383.7	1.39610
$831$	−26761.6	−1.11715
$832$	0	0
$833$	−9368.82	−0.389688
$834$	−38676.5	−1.60582
$835$	16247.0	0.673353
$836$	−8491.74	−0.351307
$837$	−8526.14	−0.352099
$838$	−37055.7	−1.52753
$839$	−45316.6	−1.86473	−0.932363	−	0.361525i	$-0.882256\pi$
$839$	−45316.6	−1.86473	−0.932363	+	0.361525i	$0.882256\pi$
$840$	52249.4	2.14616
$841$	−4330.61	−0.177564
$842$	14298.6	0.585227
$843$	9703.37	0.396444
$844$	−43013.3	−1.75424
$845$	0	0
$846$	−11880.3	−0.482803
$847$	−12096.7	−0.490728
$848$	16555.1	0.670407
$849$	−36140.3	−1.46093
$850$	5025.72	0.202801
$851$	−9041.93	−0.364222
$852$	−50363.2	−2.02513
$853$	−17557.7	−0.704766	−0.352383	−	0.935856i	$-0.614629\pi$
$853$	−17557.7	−0.704766	−0.352383	+	0.935856i	$0.614629\pi$
$854$	82231.3	3.29496
$855$	−1215.62	−0.0486237
$856$	4730.49	0.188884
$857$	37943.6	1.51240	0.756201	−	0.654339i	$-0.227053\pi$
$857$	37943.6	1.51240	0.756201	+	0.654339i	$0.227053\pi$
$858$	0	0
$859$	−12833.2	−0.509735	−0.254867	−	0.966976i	$-0.582032\pi$
$859$	−12833.2	−0.509735	−0.254867	+	0.966976i	$0.582032\pi$
$860$	−62602.5	−2.48224
$861$	41003.3	1.62299
$862$	68535.6	2.70804
$863$	−24264.0	−0.957076	−0.478538	−	0.878067i	$-0.658833\pi$
$863$	−24264.0	−0.957076	−0.478538	+	0.878067i	$0.658833\pi$
$864$	−8016.87	−0.315671
$865$	30290.7	1.19065
$866$	−48825.8	−1.91590
$867$	−18511.3	−0.725116
$868$	−22422.3	−0.876801
$869$	8442.00	0.329546
$870$	38784.1	1.51138
$871$	0	0
$872$	10310.2	0.400397
$873$	−10378.5	−0.402357
$874$	1823.16	0.0705598
$875$	29061.8	1.12282
$876$	−55741.0	−2.14990
$877$	−4098.68	−0.157814	−0.0789069	−	0.996882i	$-0.525143\pi$
$877$	−4098.68	−0.157814	−0.0789069	+	0.996882i	$0.525143\pi$
$878$	78098.4	3.00193
$879$	14141.7	0.542649
$880$	−25618.1	−0.981347
$881$	46763.9	1.78833	0.894164	−	0.447740i	$-0.147771\pi$
$881$	46763.9	1.78833	0.894164	+	0.447740i	$0.147771\pi$
$882$	12000.2	0.458127
$883$	−27183.9	−1.03603	−0.518013	−	0.855373i	$-0.673328\pi$
$883$	−27183.9	−1.03603	−0.518013	+	0.855373i	$0.673328\pi$
$884$	0	0
$885$	−14683.3	−0.557712
$886$	48377.0	1.83438
$887$	−21182.0	−0.801827	−0.400914	−	0.916116i	$-0.631307\pi$
$887$	−21182.0	−0.801827	−0.400914	+	0.916116i	$0.631307\pi$
$888$	−49053.2	−1.85374
$889$	−39102.0	−1.47518
$890$	23456.5	0.883444
$891$	20344.8	0.764955
$892$	−23079.7	−0.866331
$893$	−4452.00	−0.166831
$894$	62728.5	2.34671
$895$	9286.99	0.346849
$896$	−68974.5	−2.57174
$897$	0	0
$898$	−50322.3	−1.87002
$899$	−7933.19	−0.294312
$900$	−4225.72	−0.156508
$901$	−9551.85	−0.353183
$902$	−72000.2	−2.65781
$903$	37314.9	1.37515
$904$	−59959.2	−2.20599
$905$	49478.1	1.81736
$906$	−37937.8	−1.39117
$907$	−50810.3	−1.86012	−0.930060	−	0.367409i	$-0.880245\pi$
$907$	−50810.3	−1.86012	−0.930060	+	0.367409i	$0.880245\pi$
$908$	9830.75	0.359300
$909$	5812.77	0.212099
$910$	0	0
$911$	−7309.10	−0.265819	−0.132910	−	0.991128i	$-0.542432\pi$
$911$	−7309.10	−0.265819	−0.132910	+	0.991128i	$0.542432\pi$
$912$	2761.74	0.100274
$913$	22937.1	0.831441
$914$	−59179.7	−2.14168
$915$	36928.4	1.33422
$916$	−2354.18	−0.0849174
$917$	12022.8	0.432964
$918$	−20076.5	−0.721812
$919$	3509.95	0.125988	0.0629939	−	0.998014i	$-0.479935\pi$
$919$	3509.95	0.125988	0.0629939	+	0.998014i	$0.479935\pi$
$920$	12930.7	0.463384
$921$	−5466.48	−0.195577
$922$	19533.8	0.697733
$923$	0	0
$924$	75316.5	2.68153
$925$	11963.7	0.425260
$926$	−50542.4	−1.79366
$927$	5410.91	0.191713
$928$	−7459.34	−0.263863
$929$	6624.68	0.233960	0.116980	−	0.993134i	$-0.462679\pi$
$929$	6624.68	0.233960	0.116980	+	0.993134i	$0.462679\pi$
$930$	−15339.3	−0.540855
$931$	4496.95	0.158304
$932$	−3999.84	−0.140579
$933$	−26134.5	−0.917048
$934$	−23675.0	−0.829410
$935$	14780.9	0.516992
$936$	0	0
$937$	36109.1	1.25895	0.629473	−	0.777022i	$-0.283271\pi$
$937$	36109.1	1.25895	0.629473	+	0.777022i	$0.283271\pi$
$938$	113098.	3.93687
$939$	30217.0	1.05015
$940$	−66245.8	−2.29862
$941$	47894.5	1.65921	0.829604	−	0.558352i	$-0.188566\pi$
$941$	47894.5	1.65921	0.829604	+	0.558352i	$0.188566\pi$
$942$	24814.0	0.858263
$943$	10147.5	0.350423
$944$	−12257.4	−0.422612
$945$	50906.3	1.75236
$946$	−65523.4	−2.25195
$947$	34541.3	1.18526	0.592631	−	0.805474i	$-0.298089\pi$
$947$	34541.3	1.18526	0.592631	+	0.805474i	$0.298089\pi$
$948$	−13542.0	−0.463950
$949$	0	0
$950$	−2412.30	−0.0823845
$951$	−9831.29	−0.335228
$952$	−25165.8	−0.856753
$953$	37217.0	1.26503	0.632517	−	0.774546i	$-0.282022\pi$
$953$	37217.0	1.26503	0.632517	+	0.774546i	$0.282022\pi$
$954$	12234.6	0.415211
$955$	25590.5	0.867110
$956$	38139.9	1.29030
$957$	26647.5	0.900097
$958$	−6029.14	−0.203332
$959$	−65548.0	−2.20715
$960$	−35930.2	−1.20796
$961$	−26653.4	−0.894679
$962$	0	0
$963$	976.148	0.0326645
$964$	−58632.4	−1.95894
$965$	13106.4	0.437213
$966$	−16170.3	−0.538583
$967$	−19683.1	−0.654567	−0.327283	−	0.944926i	$-0.606133\pi$
$967$	−19683.1	−0.654567	−0.327283	+	0.944926i	$0.606133\pi$
$968$	−16241.0	−0.539262
$969$	−1593.44	−0.0528264
$970$	−88159.0	−2.91816
$971$	33750.5	1.11545	0.557726	−	0.830025i	$-0.311674\pi$
$971$	33750.5	1.11545	0.557726	+	0.830025i	$0.311674\pi$
$972$	30186.2	0.996112
$973$	−47234.6	−1.55629
$974$	93086.1	3.06229
$975$	0	0
$976$	30827.3	1.01102
$977$	−23945.2	−0.784110	−0.392055	−	0.919942i	$-0.628236\pi$
$977$	−23945.2	−0.784110	−0.392055	+	0.919942i	$0.628236\pi$
$978$	62380.5	2.03958
$979$	16116.4	0.526130
$980$	66914.6	2.18113
$981$	2127.53	0.0692423
$982$	40443.8	1.31427
$983$	53283.1	1.72886	0.864429	−	0.502756i	$-0.167680\pi$
$983$	53283.1	1.72886	0.864429	+	0.502756i	$0.167680\pi$
$984$	55051.2	1.78350
$985$	16707.1	0.540440
$986$	−18680.3	−0.603348
$987$	39486.5	1.27342
$988$	0	0
$989$	9234.71	0.296913
$990$	−18932.4	−0.607789
$991$	−46669.4	−1.49597	−0.747983	−	0.663717i	$-0.768978\pi$
$991$	−46669.4	−1.49597	−0.747983	+	0.663717i	$0.768978\pi$
$992$	2950.20	0.0944245
$993$	7750.12	0.247676
$994$	−93697.4	−2.98984
$995$	−3641.32	−0.116018
$996$	−36793.9	−1.17054
$997$	−28293.1	−0.898748	−0.449374	−	0.893344i	$-0.648353\pi$
$997$	−28293.1	−0.898748	−0.449374	+	0.893344i	$0.648353\pi$
$998$	960.405	0.0304620
$999$	−47792.3	−1.51359

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.k.1.2	✓	9
3.2	odd	2		1521.4.a.bh.1.8		9
13.2	odd	12		169.4.e.h.147.3		36
13.3	even	3		169.4.c.l.22.8		18
13.4	even	6		169.4.c.k.146.2		18
13.5	odd	4		169.4.b.g.168.16		18
13.6	odd	12		169.4.e.h.23.16		36
13.7	odd	12		169.4.e.h.23.3		36
13.8	odd	4		169.4.b.g.168.3		18
13.9	even	3		169.4.c.l.146.8		18
13.10	even	6		169.4.c.k.22.2		18
13.11	odd	12		169.4.e.h.147.16		36
13.12	even	2		169.4.a.l.1.8	yes	9
39.38	odd	2		1521.4.a.bg.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.2	✓	9	1.1	even	1	trivial
169.4.a.l.1.8	yes	9	13.12	even	2
169.4.b.g.168.3		18	13.8	odd	4
169.4.b.g.168.16		18	13.5	odd	4
169.4.c.k.22.2		18	13.10	even	6
169.4.c.k.146.2		18	13.4	even	6
169.4.c.l.22.8		18	13.3	even	3
169.4.c.l.146.8		18	13.9	even	3
169.4.e.h.23.3		36	13.7	odd	12
169.4.e.h.23.16		36	13.6	odd	12
169.4.e.h.147.3		36	13.2	odd	12
169.4.e.h.147.16		36	13.11	odd	12
1521.4.a.bg.1.2		9	39.38	odd	2
1521.4.a.bh.1.8		9	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-4.82555 q^{2} +4.44352 q^{3} +15.2860 q^{4} +12.7712 q^{5} -21.4425 q^{6} -26.1871 q^{7} -35.1589 q^{8} -7.25513 q^{9} +O(q^{10})\)	\(q-4.82555 q^{2} +4.44352 q^{3} +15.2860 q^{4} +12.7712 q^{5} -21.4425 q^{6} -26.1871 q^{7} -35.1589 q^{8} -7.25513 q^{9} -61.6281 q^{10} -42.3430 q^{11} +67.9236 q^{12} +126.367 q^{14} +56.7490 q^{15} +47.3733 q^{16} -27.3331 q^{17} +35.0100 q^{18} +13.1196 q^{19} +195.220 q^{20} -116.363 q^{21} +204.329 q^{22} -28.7976 q^{23} -156.229 q^{24} +38.1033 q^{25} -152.213 q^{27} -400.296 q^{28} -141.628 q^{29} -273.846 q^{30} +56.0144 q^{31} +52.6687 q^{32} -188.152 q^{33} +131.897 q^{34} -334.441 q^{35} -110.902 q^{36} +313.982 q^{37} -63.3095 q^{38} -449.021 q^{40} -352.375 q^{41} +561.516 q^{42} -320.676 q^{43} -647.255 q^{44} -92.6566 q^{45} +138.964 q^{46} -339.339 q^{47} +210.504 q^{48} +342.765 q^{49} -183.869 q^{50} -121.455 q^{51} +349.461 q^{53} +734.514 q^{54} -540.771 q^{55} +920.710 q^{56} +58.2973 q^{57} +683.432 q^{58} -258.742 q^{59} +867.465 q^{60} +650.732 q^{61} -270.301 q^{62} +189.991 q^{63} -633.142 q^{64} +907.938 q^{66} +894.996 q^{67} -417.813 q^{68} -127.963 q^{69} +1613.86 q^{70} -741.469 q^{71} +255.082 q^{72} -820.643 q^{73} -1515.14 q^{74} +169.313 q^{75} +200.546 q^{76} +1108.84 q^{77} -199.372 q^{79} +605.013 q^{80} -480.475 q^{81} +1700.40 q^{82} -541.696 q^{83} -1778.72 q^{84} -349.076 q^{85} +1547.44 q^{86} -629.325 q^{87} +1488.73 q^{88} -380.615 q^{89} +447.119 q^{90} -440.199 q^{92} +248.901 q^{93} +1637.50 q^{94} +167.553 q^{95} +234.034 q^{96} +1430.50 q^{97} -1654.03 q^{98} +307.204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) \) 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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