LMFDB - Embedded newform 966.4.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,4,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("966.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	966.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:	$56.9958450655$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.12197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x - 4 $ x^3 - x^2 - 15*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.24392$ of defining polynomial
Character	$\chi$	$=$	966.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +13.7318 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +13.7318 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +27.4635 q^{10} +45.9053 q^{11} -12.0000 q^{12} +5.45531 q^{13} +14.0000 q^{14} -41.1953 q^{15} +16.0000 q^{16} +39.2521 q^{17} +18.0000 q^{18} +27.3958 q^{19} +54.9270 q^{20} -21.0000 q^{21} +91.8107 q^{22} -23.0000 q^{23} -24.0000 q^{24} +63.5611 q^{25} +10.9106 q^{26} -27.0000 q^{27} +28.0000 q^{28} +9.12126 q^{29} -82.3905 q^{30} +29.6741 q^{31} +32.0000 q^{32} -137.716 q^{33} +78.5042 q^{34} +96.1223 q^{35} +36.0000 q^{36} +350.619 q^{37} +54.7916 q^{38} -16.3659 q^{39} +109.854 q^{40} -445.765 q^{41} -42.0000 q^{42} -447.238 q^{43} +183.621 q^{44} +123.586 q^{45} -46.0000 q^{46} +585.065 q^{47} -48.0000 q^{48} +49.0000 q^{49} +127.122 q^{50} -117.756 q^{51} +21.8212 q^{52} -595.290 q^{53} -54.0000 q^{54} +630.361 q^{55} +56.0000 q^{56} -82.1874 q^{57} +18.2425 q^{58} -365.737 q^{59} -164.781 q^{60} +629.292 q^{61} +59.3483 q^{62} +63.0000 q^{63} +64.0000 q^{64} +74.9110 q^{65} -275.432 q^{66} -150.031 q^{67} +157.008 q^{68} +69.0000 q^{69} +192.245 q^{70} -125.507 q^{71} +72.0000 q^{72} -702.089 q^{73} +701.238 q^{74} -190.683 q^{75} +109.583 q^{76} +321.337 q^{77} -32.7319 q^{78} +783.124 q^{79} +219.708 q^{80} +81.0000 q^{81} -891.530 q^{82} +744.142 q^{83} -84.0000 q^{84} +539.000 q^{85} -894.476 q^{86} -27.3638 q^{87} +367.243 q^{88} -40.6181 q^{89} +247.172 q^{90} +38.1872 q^{91} -92.0000 q^{92} -89.0224 q^{93} +1170.13 q^{94} +376.193 q^{95} -96.0000 q^{96} +394.706 q^{97} +98.0000 q^{98} +413.148 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 9 q^{5} - 18 q^{6} + 21 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 9 * q^5 - 18 * q^6 + 21 * q^7 + 24 * q^8 + 27 * q^9	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 9 q^{5} - 18 q^{6} + 21 q^{7} + 24 q^{8} + 27 q^{9} + 18 q^{10} + 53 q^{11} - 36 q^{12} + 61 q^{13} + 42 q^{14} - 27 q^{15} + 48 q^{16} - 2 q^{17} + 54 q^{18} + 63 q^{19} + 36 q^{20} - 63 q^{21} + 106 q^{22} - 69 q^{23} - 72 q^{24} - 72 q^{25} + 122 q^{26} - 81 q^{27} + 84 q^{28} + 312 q^{29} - 54 q^{30} + 220 q^{31} + 96 q^{32} - 159 q^{33} - 4 q^{34} + 63 q^{35} + 108 q^{36} - 44 q^{37} + 126 q^{38} - 183 q^{39} + 72 q^{40} + 171 q^{41} - 126 q^{42} - 627 q^{43} + 212 q^{44} + 81 q^{45} - 138 q^{46} + 353 q^{47} - 144 q^{48} + 147 q^{49} - 144 q^{50} + 6 q^{51} + 244 q^{52} - 504 q^{53} - 162 q^{54} + 1181 q^{55} + 168 q^{56} - 189 q^{57} + 624 q^{58} + 538 q^{59} - 108 q^{60} + 256 q^{61} + 440 q^{62} + 189 q^{63} + 192 q^{64} + 244 q^{65} - 318 q^{66} + 1113 q^{67} - 8 q^{68} + 207 q^{69} + 126 q^{70} + 53 q^{71} + 216 q^{72} + 360 q^{73} - 88 q^{74} + 216 q^{75} + 252 q^{76} + 371 q^{77} - 366 q^{78} + 134 q^{79} + 144 q^{80} + 243 q^{81} + 342 q^{82} + 1506 q^{83} - 252 q^{84} + 577 q^{85} - 1254 q^{86} - 936 q^{87} + 424 q^{88} + 2381 q^{89} + 162 q^{90} + 427 q^{91} - 276 q^{92} - 660 q^{93} + 706 q^{94} + 945 q^{95} - 288 q^{96} - 88 q^{97} + 294 q^{98} + 477 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 9 * q^5 - 18 * q^6 + 21 * q^7 + 24 * q^8 + 27 * q^9 + 18 * q^10 + 53 * q^11 - 36 * q^12 + 61 * q^13 + 42 * q^14 - 27 * q^15 + 48 * q^16 - 2 * q^17 + 54 * q^18 + 63 * q^19 + 36 * q^20 - 63 * q^21 + 106 * q^22 - 69 * q^23 - 72 * q^24 - 72 * q^25 + 122 * q^26 - 81 * q^27 + 84 * q^28 + 312 * q^29 - 54 * q^30 + 220 * q^31 + 96 * q^32 - 159 * q^33 - 4 * q^34 + 63 * q^35 + 108 * q^36 - 44 * q^37 + 126 * q^38 - 183 * q^39 + 72 * q^40 + 171 * q^41 - 126 * q^42 - 627 * q^43 + 212 * q^44 + 81 * q^45 - 138 * q^46 + 353 * q^47 - 144 * q^48 + 147 * q^49 - 144 * q^50 + 6 * q^51 + 244 * q^52 - 504 * q^53 - 162 * q^54 + 1181 * q^55 + 168 * q^56 - 189 * q^57 + 624 * q^58 + 538 * q^59 - 108 * q^60 + 256 * q^61 + 440 * q^62 + 189 * q^63 + 192 * q^64 + 244 * q^65 - 318 * q^66 + 1113 * q^67 - 8 * q^68 + 207 * q^69 + 126 * q^70 + 53 * q^71 + 216 * q^72 + 360 * q^73 - 88 * q^74 + 216 * q^75 + 252 * q^76 + 371 * q^77 - 366 * q^78 + 134 * q^79 + 144 * q^80 + 243 * q^81 + 342 * q^82 + 1506 * q^83 - 252 * q^84 + 577 * q^85 - 1254 * q^86 - 936 * q^87 + 424 * q^88 + 2381 * q^89 + 162 * q^90 + 427 * q^91 - 276 * q^92 - 660 * q^93 + 706 * q^94 + 945 * q^95 - 288 * q^96 - 88 * q^97 + 294 * q^98 + 477 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	13.7318	1.22821	0.614103	−	0.789226i	$-0.289518\pi$
$5$	13.7318	1.22821	0.614103	+	0.789226i	$0.289518\pi$
$6$	−6.00000	−0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	27.4635	0.868472
$11$	45.9053	1.25827	0.629135	−	0.777296i	$-0.283409\pi$
$11$	45.9053	1.25827	0.629135	+	0.777296i	$0.283409\pi$
$12$	−12.0000	−0.288675
$13$	5.45531	0.116387	0.0581935	−	0.998305i	$-0.481466\pi$
$13$	5.45531	0.116387	0.0581935	+	0.998305i	$0.481466\pi$
$14$	14.0000	0.267261
$15$	−41.1953	−0.709105
$16$	16.0000	0.250000
$17$	39.2521	0.560002	0.280001	−	0.960000i	$-0.409665\pi$
$17$	39.2521	0.560002	0.280001	+	0.960000i	$0.409665\pi$
$18$	18.0000	0.235702
$19$	27.3958	0.330791	0.165396	−	0.986227i	$-0.447110\pi$
$19$	27.3958	0.330791	0.165396	+	0.986227i	$0.447110\pi$
$20$	54.9270	0.614103
$21$	−21.0000	−0.218218
$22$	91.8107	0.889732
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−24.0000	−0.204124
$25$	63.5611	0.508489
$26$	10.9106	0.0822981
$27$	−27.0000	−0.192450
$28$	28.0000	0.188982
$29$	9.12126	0.0584060	0.0292030	−	0.999574i	$-0.490703\pi$
$29$	9.12126	0.0584060	0.0292030	+	0.999574i	$0.490703\pi$
$30$	−82.3905	−0.501413
$31$	29.6741	0.171924	0.0859618	−	0.996298i	$-0.472604\pi$
$31$	29.6741	0.171924	0.0859618	+	0.996298i	$0.472604\pi$
$32$	32.0000	0.176777
$33$	−137.716	−0.726463
$34$	78.5042	0.395981
$35$	96.1223	0.464218
$36$	36.0000	0.166667
$37$	350.619	1.55788	0.778938	−	0.627101i	$-0.215759\pi$
$37$	350.619	1.55788	0.778938	+	0.627101i	$0.215759\pi$
$38$	54.7916	0.233905
$39$	−16.3659	−0.0671961
$40$	109.854	0.434236
$41$	−445.765	−1.69797	−0.848985	−	0.528417i	$-0.822786\pi$
$41$	−445.765	−1.69797	−0.848985	+	0.528417i	$0.822786\pi$
$42$	−42.0000	−0.154303
$43$	−447.238	−1.58612	−0.793060	−	0.609144i	$-0.791513\pi$
$43$	−447.238	−1.58612	−0.793060	+	0.609144i	$0.791513\pi$
$44$	183.621	0.629135
$45$	123.586	0.409402
$46$	−46.0000	−0.147442
$47$	585.065	1.81576	0.907878	−	0.419235i	$-0.137702\pi$
$47$	585.065	1.81576	0.907878	+	0.419235i	$0.137702\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	127.122	0.359556
$51$	−117.756	−0.323317
$52$	21.8212	0.0581935
$53$	−595.290	−1.54282	−0.771409	−	0.636340i	$-0.780448\pi$
$53$	−595.290	−1.54282	−0.771409	+	0.636340i	$0.780448\pi$
$54$	−54.0000	−0.136083
$55$	630.361	1.54542
$56$	56.0000	0.133631
$57$	−82.1874	−0.190982
$58$	18.2425	0.0412993
$59$	−365.737	−0.807031	−0.403516	−	0.914973i	$-0.632212\pi$
$59$	−365.737	−0.807031	−0.403516	+	0.914973i	$0.632212\pi$
$60$	−164.781	−0.354552
$61$	629.292	1.32086	0.660431	−	0.750887i	$-0.270374\pi$
$61$	629.292	1.32086	0.660431	+	0.750887i	$0.270374\pi$
$62$	59.3483	0.121568
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	74.9110	0.142947
$66$	−275.432	−0.513687
$67$	−150.031	−0.273569	−0.136785	−	0.990601i	$-0.543677\pi$
$67$	−150.031	−0.273569	−0.136785	+	0.990601i	$0.543677\pi$
$68$	157.008	0.280001
$69$	69.0000	0.120386
$70$	192.245	0.328252
$71$	−125.507	−0.209788	−0.104894	−	0.994483i	$-0.533450\pi$
$71$	−125.507	−0.209788	−0.104894	+	0.994483i	$0.533450\pi$
$72$	72.0000	0.117851
$73$	−702.089	−1.12566	−0.562831	−	0.826572i	$-0.690288\pi$
$73$	−702.089	−1.12566	−0.562831	+	0.826572i	$0.690288\pi$
$74$	701.238	1.10158
$75$	−190.683	−0.293576
$76$	109.583	0.165396
$77$	321.337	0.475582
$78$	−32.7319	−0.0475148
$79$	783.124	1.11530	0.557648	−	0.830078i	$-0.311704\pi$
$79$	783.124	1.11530	0.557648	+	0.830078i	$0.311704\pi$
$80$	219.708	0.307051
$81$	81.0000	0.111111
$82$	−891.530	−1.20065
$83$	744.142	0.984098	0.492049	−	0.870567i	$-0.336248\pi$
$83$	744.142	0.984098	0.492049	+	0.870567i	$0.336248\pi$
$84$	−84.0000	−0.109109
$85$	539.000	0.687798
$86$	−894.476	−1.12156
$87$	−27.3638	−0.0337207
$88$	367.243	0.444866
$89$	−40.6181	−0.0483766	−0.0241883	−	0.999707i	$-0.507700\pi$
$89$	−40.6181	−0.0483766	−0.0241883	+	0.999707i	$0.507700\pi$
$90$	247.172	0.289491
$91$	38.1872	0.0439902
$92$	−92.0000	−0.104257
$93$	−89.0224	−0.0992602
$94$	1170.13	1.28393
$95$	376.193	0.406279
$96$	−96.0000	−0.102062
$97$	394.706	0.413158	0.206579	−	0.978430i	$-0.433767\pi$
$97$	394.706	0.413158	0.206579	+	0.978430i	$0.433767\pi$
$98$	98.0000	0.101015
$99$	413.148	0.419424
$100$	254.244	0.254244
$101$	749.885	0.738775	0.369388	−	0.929275i	$-0.379567\pi$
$101$	749.885	0.738775	0.369388	+	0.929275i	$0.379567\pi$
$102$	−235.513	−0.228620
$103$	332.501	0.318081	0.159040	−	0.987272i	$-0.449160\pi$
$103$	332.501	0.318081	0.159040	+	0.987272i	$0.449160\pi$
$104$	43.6425	0.0411490
$105$	−288.367	−0.268016
$106$	−1190.58	−1.09094
$107$	1755.87	1.58641	0.793206	−	0.608954i	$-0.208411\pi$
$107$	1755.87	1.58641	0.793206	+	0.608954i	$0.208411\pi$
$108$	−108.000	−0.0962250
$109$	798.305	0.701502	0.350751	−	0.936469i	$-0.385926\pi$
$109$	798.305	0.701502	0.350751	+	0.936469i	$0.385926\pi$
$110$	1260.72	1.09277
$111$	−1051.86	−0.899440
$112$	112.000	0.0944911
$113$	−544.067	−0.452934	−0.226467	−	0.974019i	$-0.572717\pi$
$113$	−544.067	−0.452934	−0.226467	+	0.974019i	$0.572717\pi$
$114$	−164.375	−0.135045
$115$	−315.830	−0.256099
$116$	36.4850	0.0292030
$117$	49.0978	0.0387957
$118$	−731.473	−0.570657
$119$	274.765	0.211661
$120$	−329.562	−0.250706
$121$	776.300	0.583246
$122$	1258.58	0.933990
$123$	1337.29	0.980324
$124$	118.697	0.0859618
$125$	−843.664	−0.603677
$126$	126.000	0.0890871
$127$	−1.40468	−0.000981457 0	−0.000490729	−	1.00000i	$-0.500156\pi$
$127$	−1.40468	−0.000981457 0	−0.000490729		1.00000i	$0.500156\pi$
$128$	128.000	0.0883883
$129$	1341.71	0.915747
$130$	149.822	0.101079
$131$	−2194.78	−1.46380	−0.731902	−	0.681410i	$-0.761367\pi$
$131$	−2194.78	−1.46380	−0.731902	+	0.681410i	$0.761367\pi$
$132$	−550.864	−0.363232
$133$	191.771	0.125027
$134$	−300.061	−0.193443
$135$	−370.757	−0.236368
$136$	314.017	0.197991
$137$	−1234.48	−0.769843	−0.384921	−	0.922949i	$-0.625771\pi$
$137$	−1234.48	−0.769843	−0.384921	+	0.922949i	$0.625771\pi$
$138$	138.000	0.0851257
$139$	−838.469	−0.511640	−0.255820	−	0.966724i	$-0.582346\pi$
$139$	−838.469	−0.511640	−0.255820	+	0.966724i	$0.582346\pi$
$140$	384.489	0.232109
$141$	−1755.19	−1.04833
$142$	−251.014	−0.148342
$143$	250.428	0.146446
$144$	144.000	0.0833333
$145$	125.251	0.0717346
$146$	−1404.18	−0.795963
$147$	−147.000	−0.0824786
$148$	1402.48	0.778938
$149$	609.791	0.335275	0.167638	−	0.985849i	$-0.446386\pi$
$149$	609.791	0.335275	0.167638	+	0.985849i	$0.446386\pi$
$150$	−381.366	−0.207590
$151$	2067.33	1.11415	0.557075	−	0.830462i	$-0.311924\pi$
$151$	2067.33	1.11415	0.557075	+	0.830462i	$0.311924\pi$
$152$	219.167	0.116952
$153$	353.269	0.186667
$154$	642.675	0.336287
$155$	407.478	0.211158
$156$	−65.4637	−0.0335980
$157$	3476.30	1.76713	0.883563	−	0.468312i	$-0.155138\pi$
$157$	3476.30	1.76713	0.883563	+	0.468312i	$0.155138\pi$
$158$	1566.25	0.788633
$159$	1785.87	0.890746
$160$	439.416	0.217118
$161$	−161.000	−0.0788110
$162$	162.000	0.0785674
$163$	−1070.77	−0.514535	−0.257267	−	0.966340i	$-0.582822\pi$
$163$	−1070.77	−0.514535	−0.257267	+	0.966340i	$0.582822\pi$
$164$	−1783.06	−0.848985
$165$	−1891.08	−0.892246
$166$	1488.28	0.695863
$167$	2373.54	1.09982	0.549909	−	0.835224i	$-0.314662\pi$
$167$	2373.54	1.09982	0.549909	+	0.835224i	$0.314662\pi$
$168$	−168.000	−0.0771517
$169$	−2167.24	−0.986454
$170$	1078.00	0.486346
$171$	246.562	0.110264
$172$	−1788.95	−0.793060
$173$	451.372	0.198365	0.0991825	−	0.995069i	$-0.468377\pi$
$173$	451.372	0.198365	0.0991825	+	0.995069i	$0.468377\pi$
$174$	−54.7275	−0.0238442
$175$	444.928	0.192191
$176$	734.485	0.314568
$177$	1097.21	0.465940
$178$	−81.2363	−0.0342074
$179$	1.91608	0.000800081 0	0.000400041	−	1.00000i	$-0.499873\pi$
$179$	1.91608	0.000800081 0	0.000400041		1.00000i	$0.499873\pi$
$180$	494.343	0.204701
$181$	3796.95	1.55925	0.779627	−	0.626245i	$-0.215409\pi$
$181$	3796.95	1.55925	0.779627	+	0.626245i	$0.215409\pi$
$182$	76.3744	0.0311057
$183$	−1887.88	−0.762600
$184$	−184.000	−0.0737210
$185$	4814.61	1.91339
$186$	−178.045	−0.0701875
$187$	1801.88	0.704634
$188$	2340.26	0.907878
$189$	−189.000	−0.0727393
$190$	752.385	0.287283
$191$	3450.82	1.30729	0.653646	−	0.756800i	$-0.273238\pi$
$191$	3450.82	1.30729	0.653646	+	0.756800i	$0.273238\pi$
$192$	−192.000	−0.0721688
$193$	−3555.01	−1.32588	−0.662941	−	0.748672i	$-0.730692\pi$
$193$	−3555.01	−1.32588	−0.662941	+	0.748672i	$0.730692\pi$
$194$	789.411	0.292146
$195$	−224.733	−0.0825306
$196$	196.000	0.0714286
$197$	−1324.72	−0.479098	−0.239549	−	0.970884i	$-0.577000\pi$
$197$	−1324.72	−0.479098	−0.239549	+	0.970884i	$0.577000\pi$
$198$	826.296	0.296577
$199$	−1475.79	−0.525708	−0.262854	−	0.964836i	$-0.584664\pi$
$199$	−1475.79	−0.525708	−0.262854	+	0.964836i	$0.584664\pi$
$200$	508.489	0.179778
$201$	450.092	0.157945
$202$	1499.77	0.522393
$203$	63.8488	0.0220754
$204$	−471.025	−0.161659
$205$	−6121.13	−2.08546
$206$	665.003	0.224917
$207$	−207.000	−0.0695048
$208$	87.2850	0.0290968
$209$	1257.61	0.416225
$210$	−576.734	−0.189516
$211$	585.617	0.191069	0.0955345	−	0.995426i	$-0.469544\pi$
$211$	585.617	0.191069	0.0955345	+	0.995426i	$0.469544\pi$
$212$	−2381.16	−0.771409
$213$	376.521	0.121121
$214$	3511.73	1.12176
$215$	−6141.36	−1.94808
$216$	−216.000	−0.0680414
$217$	207.719	0.0649810
$218$	1596.61	0.496037
$219$	2106.27	0.649901
$220$	2521.44	0.772708
$221$	214.133	0.0651770
$222$	−2103.71	−0.636000
$223$	3068.81	0.921536	0.460768	−	0.887521i	$-0.347574\pi$
$223$	3068.81	0.921536	0.460768	+	0.887521i	$0.347574\pi$
$224$	224.000	0.0668153
$225$	572.050	0.169496
$226$	−1088.13	−0.320272
$227$	4555.53	1.33199	0.665994	−	0.745957i	$-0.268008\pi$
$227$	4555.53	1.33199	0.665994	+	0.745957i	$0.268008\pi$
$228$	−328.750	−0.0954912
$229$	−3835.82	−1.10689	−0.553446	−	0.832885i	$-0.686688\pi$
$229$	−3835.82	−1.10689	−0.553446	+	0.832885i	$0.686688\pi$
$230$	−631.661	−0.181089
$231$	−964.012	−0.274577
$232$	72.9700	0.0206496
$233$	−544.549	−0.153110	−0.0765549	−	0.997065i	$-0.524392\pi$
$233$	−544.549	−0.153110	−0.0765549	+	0.997065i	$0.524392\pi$
$234$	98.1956	0.0274327
$235$	8033.97	2.23012
$236$	−1462.95	−0.403516
$237$	−2349.37	−0.643916
$238$	549.530	0.149667
$239$	−6593.05	−1.78439	−0.892194	−	0.451652i	$-0.850835\pi$
$239$	−6593.05	−1.78439	−0.892194	+	0.451652i	$0.850835\pi$
$240$	−659.124	−0.177276
$241$	−742.035	−0.198335	−0.0991674	−	0.995071i	$-0.531618\pi$
$241$	−742.035	−0.198335	−0.0991674	+	0.995071i	$0.531618\pi$
$242$	1552.60	0.412417
$243$	−243.000	−0.0641500
$244$	2517.17	0.660431
$245$	672.856	0.175458
$246$	2674.59	0.693194
$247$	149.453	0.0384998
$248$	237.393	0.0607842
$249$	−2232.42	−0.568169
$250$	−1687.33	−0.426864
$251$	3887.66	0.977638	0.488819	−	0.872385i	$-0.337428\pi$
$251$	3887.66	0.977638	0.488819	+	0.872385i	$0.337428\pi$
$252$	252.000	0.0629941
$253$	−1055.82	−0.262368
$254$	−2.80936	−0.000693995 0
$255$	−1617.00	−0.397100
$256$	256.000	0.0625000
$257$	−5919.83	−1.43684	−0.718422	−	0.695607i	$-0.755135\pi$
$257$	−5919.83	−1.43684	−0.718422	+	0.695607i	$0.755135\pi$
$258$	2683.43	0.647531
$259$	2454.33	0.588821
$260$	299.644	0.0714736
$261$	82.0913	0.0194687
$262$	−4389.55	−1.03507
$263$	−5378.10	−1.26094	−0.630471	−	0.776213i	$-0.717138\pi$
$263$	−5378.10	−1.26094	−0.630471	+	0.776213i	$0.717138\pi$
$264$	−1101.73	−0.256843
$265$	−8174.37	−1.89490
$266$	383.541	0.0884076
$267$	121.854	0.0279302
$268$	−600.122	−0.136785
$269$	563.438	0.127708	0.0638539	−	0.997959i	$-0.479661\pi$
$269$	563.438	0.127708	0.0638539	+	0.997959i	$0.479661\pi$
$270$	−741.515	−0.167138
$271$	−8229.05	−1.84457	−0.922286	−	0.386507i	$-0.873681\pi$
$271$	−8229.05	−1.84457	−0.922286	+	0.386507i	$0.873681\pi$
$272$	628.034	0.140001
$273$	−114.562	−0.0253977
$274$	−2468.95	−0.544361
$275$	2917.79	0.639816
$276$	276.000	0.0601929
$277$	−7056.39	−1.53060	−0.765302	−	0.643671i	$-0.777410\pi$
$277$	−7056.39	−1.53060	−0.765302	+	0.643671i	$0.777410\pi$
$278$	−1676.94	−0.361784
$279$	267.067	0.0573079
$280$	768.978	0.164126
$281$	2775.63	0.589252	0.294626	−	0.955613i	$-0.404805\pi$
$281$	2775.63	0.589252	0.294626	+	0.955613i	$0.404805\pi$
$282$	−3510.39	−0.741279
$283$	−2283.10	−0.479562	−0.239781	−	0.970827i	$-0.577076\pi$
$283$	−2283.10	−0.479562	−0.239781	+	0.970827i	$0.577076\pi$
$284$	−502.028	−0.104894
$285$	−1128.58	−0.234566
$286$	500.856	0.103553
$287$	−3120.35	−0.641773
$288$	288.000	0.0589256
$289$	−3372.27	−0.686398
$290$	250.502	0.0507240
$291$	−1184.12	−0.238537
$292$	−2808.35	−0.562831
$293$	−1830.89	−0.365056	−0.182528	−	0.983201i	$-0.558428\pi$
$293$	−1830.89	−0.365056	−0.182528	+	0.983201i	$0.558428\pi$
$294$	−294.000	−0.0583212
$295$	−5022.21	−0.991200
$296$	2804.95	0.550792
$297$	−1239.44	−0.242154
$298$	1219.58	0.237075
$299$	−125.472	−0.0242684
$300$	−762.733	−0.146788
$301$	−3130.67	−0.599497
$302$	4134.65	0.787823
$303$	−2249.65	−0.426532
$304$	438.333	0.0826978
$305$	8641.28	1.62229
$306$	706.538	0.131994
$307$	−52.1972	−0.00970375	−0.00485187	−	0.999988i	$-0.501544\pi$
$307$	−52.1972	−0.00970375	−0.00485187	+	0.999988i	$0.501544\pi$
$308$	1285.35	0.237791
$309$	−997.504	−0.183644
$310$	814.956	0.149311
$311$	5264.42	0.959864	0.479932	−	0.877306i	$-0.340661\pi$
$311$	5264.42	0.959864	0.479932	+	0.877306i	$0.340661\pi$
$312$	−130.927	−0.0237574
$313$	3816.63	0.689230	0.344615	−	0.938744i	$-0.388010\pi$
$313$	3816.63	0.689230	0.344615	+	0.938744i	$0.388010\pi$
$314$	6952.60	1.24955
$315$	865.101	0.154739
$316$	3132.50	0.557648
$317$	−11060.4	−1.95966	−0.979831	−	0.199827i	$-0.935962\pi$
$317$	−11060.4	−1.95966	−0.979831	+	0.199827i	$0.935962\pi$
$318$	3571.74	0.629853
$319$	418.714	0.0734906
$320$	878.832	0.153526
$321$	−5267.60	−0.915915
$322$	−322.000	−0.0557278
$323$	1075.34	0.185244
$324$	324.000	0.0555556
$325$	346.746	0.0591815
$326$	−2141.54	−0.363831
$327$	−2394.91	−0.405012
$328$	−3566.12	−0.600323
$329$	4095.45	0.686291
$330$	−3782.16	−0.630913
$331$	6972.73	1.15787	0.578936	−	0.815373i	$-0.303468\pi$
$331$	6972.73	1.15787	0.578936	+	0.815373i	$0.303468\pi$
$332$	2976.57	0.492049
$333$	3155.57	0.519292
$334$	4747.07	0.777689
$335$	−2060.18	−0.335999
$336$	−336.000	−0.0545545
$337$	−9451.07	−1.52769	−0.763846	−	0.645398i	$-0.776692\pi$
$337$	−9451.07	−1.52769	−0.763846	+	0.645398i	$0.776692\pi$
$338$	−4334.48	−0.697528
$339$	1632.20	0.261501
$340$	2156.00	0.343899
$341$	1362.20	0.216327
$342$	493.125	0.0779682
$343$	343.000	0.0539949
$344$	−3577.90	−0.560778
$345$	947.491	0.147859
$346$	902.743	0.140265
$347$	1404.40	0.217269	0.108634	−	0.994082i	$-0.465352\pi$
$347$	1404.40	0.217269	0.108634	+	0.994082i	$0.465352\pi$
$348$	−109.455	−0.0168604
$349$	−4669.60	−0.716211	−0.358106	−	0.933681i	$-0.616577\pi$
$349$	−4669.60	−0.716211	−0.358106	+	0.933681i	$0.616577\pi$
$350$	889.855	0.135899
$351$	−147.293	−0.0223987
$352$	1468.97	0.222433
$353$	−6282.45	−0.947255	−0.473627	−	0.880725i	$-0.657056\pi$
$353$	−6282.45	−0.947255	−0.473627	+	0.880725i	$0.657056\pi$
$354$	2194.42	0.329469
$355$	−1723.43	−0.257663
$356$	−162.473	−0.0241883
$357$	−824.294	−0.122202
$358$	3.83216	0.000565743 0
$359$	−8274.06	−1.21640	−0.608200	−	0.793783i	$-0.708108\pi$
$359$	−8274.06	−1.21640	−0.608200	+	0.793783i	$0.708108\pi$
$360$	988.686	0.144745
$361$	−6108.47	−0.890577
$362$	7593.89	1.10256
$363$	−2328.90	−0.336737
$364$	152.749	0.0219951
$365$	−9640.91	−1.38254
$366$	−3775.75	−0.539239
$367$	−205.888	−0.0292841	−0.0146421	−	0.999893i	$-0.504661\pi$
$367$	−205.888	−0.0292841	−0.0146421	+	0.999893i	$0.504661\pi$
$368$	−368.000	−0.0521286
$369$	−4011.88	−0.565990
$370$	9629.22	1.35297
$371$	−4167.03	−0.583130
$372$	−356.090	−0.0496301
$373$	152.876	0.0212215	0.0106107	−	0.999944i	$-0.496622\pi$
$373$	152.876	0.0212215	0.0106107	+	0.999944i	$0.496622\pi$
$374$	3603.76	0.498252
$375$	2530.99	0.348533
$376$	4680.52	0.641966
$377$	49.7593	0.00679770
$378$	−378.000	−0.0514344
$379$	−11194.5	−1.51721	−0.758603	−	0.651553i	$-0.774118\pi$
$379$	−11194.5	−1.51721	−0.758603	+	0.651553i	$0.774118\pi$
$380$	1504.77	0.203140
$381$	4.21404	0.000566645 0
$382$	6901.65	0.924395
$383$	3885.85	0.518427	0.259214	−	0.965820i	$-0.416537\pi$
$383$	3885.85	0.518427	0.259214	+	0.965820i	$0.416537\pi$
$384$	−384.000	−0.0510310
$385$	4412.53	0.584112
$386$	−7110.02	−0.937540
$387$	−4025.14	−0.528707
$388$	1578.82	0.206579
$389$	6311.77	0.822671	0.411336	−	0.911484i	$-0.365062\pi$
$389$	6311.77	0.822671	0.411336	+	0.911484i	$0.365062\pi$
$390$	−449.466	−0.0583579
$391$	−902.799	−0.116769
$392$	392.000	0.0505076
$393$	6584.33	0.845128
$394$	−2649.44	−0.338773
$395$	10753.7	1.36981
$396$	1652.59	0.209712
$397$	−8255.41	−1.04365	−0.521823	−	0.853054i	$-0.674748\pi$
$397$	−8255.41	−1.04365	−0.521823	+	0.853054i	$0.674748\pi$
$398$	−2951.58	−0.371732
$399$	−575.312	−0.0721845
$400$	1016.98	0.127122
$401$	−12951.5	−1.61288	−0.806441	−	0.591315i	$-0.798609\pi$
$401$	−12951.5	−1.61288	−0.806441	+	0.591315i	$0.798609\pi$
$402$	900.183	0.111684
$403$	161.882	0.0200097
$404$	2999.54	0.369388
$405$	1112.27	0.136467
$406$	127.698	0.0156097
$407$	16095.3	1.96023
$408$	−942.051	−0.114310
$409$	3097.26	0.374449	0.187225	−	0.982317i	$-0.440051\pi$
$409$	3097.26	0.374449	0.187225	+	0.982317i	$0.440051\pi$
$410$	−12242.3	−1.47464
$411$	3703.43	0.444469
$412$	1330.01	0.159040
$413$	−2560.16	−0.305029
$414$	−414.000	−0.0491473
$415$	10218.4	1.20867
$416$	174.570	0.0205745
$417$	2515.41	0.295396
$418$	2515.23	0.294315
$419$	6368.21	0.742500	0.371250	−	0.928533i	$-0.378929\pi$
$419$	6368.21	0.742500	0.371250	+	0.928533i	$0.378929\pi$
$420$	−1153.47	−0.134008
$421$	284.879	0.0329790	0.0164895	−	0.999864i	$-0.494751\pi$
$421$	284.879	0.0329790	0.0164895	+	0.999864i	$0.494751\pi$
$422$	1171.23	0.135106
$423$	5265.58	0.605252
$424$	−4762.32	−0.545468
$425$	2494.91	0.284755
$426$	753.042	0.0856456
$427$	4405.04	0.499239
$428$	7023.47	0.793206
$429$	−751.284	−0.0845509
$430$	−12282.7	−1.37750
$431$	6676.27	0.746136	0.373068	−	0.927804i	$-0.378306\pi$
$431$	6676.27	0.746136	0.373068	+	0.927804i	$0.378306\pi$
$432$	−432.000	−0.0481125
$433$	−6732.35	−0.747197	−0.373598	−	0.927591i	$-0.621876\pi$
$433$	−6732.35	−0.747197	−0.373598	+	0.927591i	$0.621876\pi$
$434$	415.438	0.0459485
$435$	−375.753	−0.0414160
$436$	3193.22	0.350751
$437$	−630.104	−0.0689747
$438$	4212.53	0.459549
$439$	6792.07	0.738424	0.369212	−	0.929345i	$-0.379628\pi$
$439$	6792.07	0.738424	0.369212	+	0.929345i	$0.379628\pi$
$440$	5042.89	0.546387
$441$	441.000	0.0476190
$442$	428.265	0.0460871
$443$	−11163.4	−1.19726	−0.598631	−	0.801025i	$-0.704289\pi$
$443$	−11163.4	−1.19726	−0.598631	+	0.801025i	$0.704289\pi$
$444$	−4207.43	−0.449720
$445$	−557.758	−0.0594164
$446$	6137.61	0.651624
$447$	−1829.37	−0.193571
$448$	448.000	0.0472456
$449$	9879.67	1.03842	0.519210	−	0.854647i	$-0.326226\pi$
$449$	9879.67	1.03842	0.519210	+	0.854647i	$0.326226\pi$
$450$	1144.10	0.119852
$451$	−20463.0	−2.13651
$452$	−2176.27	−0.226467
$453$	−6201.98	−0.643254
$454$	9111.07	0.941858
$455$	524.377	0.0540290
$456$	−657.500	−0.0675225
$457$	−6171.28	−0.631686	−0.315843	−	0.948812i	$-0.602287\pi$
$457$	−6171.28	−0.631686	−0.315843	+	0.948812i	$0.602287\pi$
$458$	−7671.65	−0.782691
$459$	−1059.81	−0.107772
$460$	−1263.32	−0.128049
$461$	3077.14	0.310883	0.155441	−	0.987845i	$-0.450320\pi$
$461$	3077.14	0.310883	0.155441	+	0.987845i	$0.450320\pi$
$462$	−1928.02	−0.194155
$463$	−7010.22	−0.703656	−0.351828	−	0.936065i	$-0.614440\pi$
$463$	−7010.22	−0.703656	−0.351828	+	0.936065i	$0.614440\pi$
$464$	145.940	0.0146015
$465$	−1222.43	−0.121912
$466$	−1089.10	−0.108265
$467$	3695.28	0.366161	0.183081	−	0.983098i	$-0.441393\pi$
$467$	3695.28	0.366161	0.183081	+	0.983098i	$0.441393\pi$
$468$	196.391	0.0193978
$469$	−1050.21	−0.103400
$470$	16067.9	1.57693
$471$	−10428.9	−1.02025
$472$	−2925.89	−0.285329
$473$	−20530.6	−1.99577
$474$	−4698.75	−0.455318
$475$	1741.31	0.168204
$476$	1099.06	0.105830
$477$	−5357.61	−0.514273
$478$	−13186.1	−1.26175
$479$	15333.3	1.46263	0.731313	−	0.682042i	$-0.238908\pi$
$479$	15333.3	1.46263	0.731313	+	0.682042i	$0.238908\pi$
$480$	−1318.25	−0.125353
$481$	1912.74	0.181316
$482$	−1484.07	−0.140244
$483$	483.000	0.0455016
$484$	3105.20	0.291623
$485$	5420.00	0.507442
$486$	−486.000	−0.0453609
$487$	−9095.79	−0.846344	−0.423172	−	0.906049i	$-0.639083\pi$
$487$	−9095.79	−0.846344	−0.423172	+	0.906049i	$0.639083\pi$
$488$	5034.33	0.466995
$489$	3212.31	0.297067
$490$	1345.71	0.124067
$491$	14841.7	1.36415	0.682073	−	0.731284i	$-0.261078\pi$
$491$	14841.7	1.36415	0.682073	+	0.731284i	$0.261078\pi$
$492$	5349.18	0.490162
$493$	358.029	0.0327075
$494$	298.905	0.0272235
$495$	5673.25	0.515138
$496$	474.786	0.0429809
$497$	−878.549	−0.0792924
$498$	−4464.85	−0.401756
$499$	6799.26	0.609974	0.304987	−	0.952357i	$-0.401348\pi$
$499$	6799.26	0.609974	0.304987	+	0.952357i	$0.401348\pi$
$500$	−3374.66	−0.301838
$501$	−7120.61	−0.634981
$502$	7775.32	0.691294
$503$	10513.9	0.931995	0.465997	−	0.884786i	$-0.345696\pi$
$503$	10513.9	0.931995	0.465997	+	0.884786i	$0.345696\pi$
$504$	504.000	0.0445435
$505$	10297.2	0.907368
$506$	−2111.65	−0.185522
$507$	6501.72	0.569530
$508$	−5.61871	−0.000490729 0
$509$	−5087.78	−0.443049	−0.221525	−	0.975155i	$-0.571103\pi$
$509$	−5087.78	−0.443049	−0.221525	+	0.975155i	$0.571103\pi$
$510$	−3234.00	−0.280792
$511$	−4914.62	−0.425460
$512$	512.000	0.0441942
$513$	−739.687	−0.0636608
$514$	−11839.7	−1.01600
$515$	4565.83	0.390669
$516$	5366.86	0.457873
$517$	26857.6	2.28471
$518$	4908.66	0.416360
$519$	−1354.12	−0.114526
$520$	599.288	0.0505395
$521$	−1548.83	−0.130241	−0.0651205	−	0.997877i	$-0.520743\pi$
$521$	−1548.83	−0.130241	−0.0651205	+	0.997877i	$0.520743\pi$
$522$	164.183	0.0137664
$523$	10212.0	0.853806	0.426903	−	0.904297i	$-0.359605\pi$
$523$	10212.0	0.853806	0.426903	+	0.904297i	$0.359605\pi$
$524$	−8779.10	−0.731902
$525$	−1334.78	−0.110961
$526$	−10756.2	−0.891621
$527$	1164.77	0.0962776
$528$	−2203.46	−0.181616
$529$	529.000	0.0434783
$530$	−16348.7	−1.33989
$531$	−3291.63	−0.269010
$532$	767.083	0.0625136
$533$	−2431.79	−0.197622
$534$	243.709	0.0197496
$535$	24111.1	1.94844
$536$	−1200.24	−0.0967214
$537$	−5.74824	−0.000461927 0
$538$	1126.88	0.0903031
$539$	2249.36	0.179753
$540$	−1483.03	−0.118184
$541$	−5217.67	−0.414649	−0.207325	−	0.978272i	$-0.566476\pi$
$541$	−5217.67	−0.414649	−0.207325	+	0.978272i	$0.566476\pi$
$542$	−16458.1	−1.30431
$543$	−11390.8	−0.900235
$544$	1256.07	0.0989953
$545$	10962.1	0.861588
$546$	−229.123	−0.0179589
$547$	−17929.1	−1.40145	−0.700726	−	0.713430i	$-0.747140\pi$
$547$	−17929.1	−1.40145	−0.700726	+	0.713430i	$0.747140\pi$
$548$	−4937.91	−0.384921
$549$	5663.63	0.440287
$550$	5835.59	0.452419
$551$	249.884	0.0193202
$552$	552.000	0.0425628
$553$	5481.87	0.421542
$554$	−14112.8	−1.08230
$555$	−14443.8	−1.10470
$556$	−3353.88	−0.255820
$557$	10493.0	0.798208	0.399104	−	0.916906i	$-0.369321\pi$
$557$	10493.0	0.798208	0.399104	+	0.916906i	$0.369321\pi$
$558$	534.135	0.0405228
$559$	−2439.82	−0.184604
$560$	1537.96	0.116055
$561$	−5405.64	−0.406821
$562$	5551.25	0.416664
$563$	14588.7	1.09208	0.546040	−	0.837759i	$-0.316135\pi$
$563$	14588.7	1.09208	0.546040	+	0.837759i	$0.316135\pi$
$564$	−7020.78	−0.524163
$565$	−7470.99	−0.556296
$566$	−4566.20	−0.339102
$567$	567.000	0.0419961
$568$	−1004.06	−0.0741712
$569$	8804.22	0.648668	0.324334	−	0.945943i	$-0.394860\pi$
$569$	8804.22	0.648668	0.324334	+	0.945943i	$0.394860\pi$
$570$	−2257.16	−0.165863
$571$	−12325.9	−0.903369	−0.451685	−	0.892178i	$-0.649177\pi$
$571$	−12325.9	−0.903369	−0.451685	+	0.892178i	$0.649177\pi$
$572$	1001.71	0.0732232
$573$	−10352.5	−0.754766
$574$	−6240.71	−0.453802
$575$	−1461.90	−0.106027
$576$	576.000	0.0416667
$577$	9189.41	0.663016	0.331508	−	0.943452i	$-0.392443\pi$
$577$	9189.41	0.663016	0.331508	+	0.943452i	$0.392443\pi$
$578$	−6744.54	−0.485356
$579$	10665.0	0.765498
$580$	501.003	0.0358673
$581$	5208.99	0.371954
$582$	−2368.23	−0.168671
$583$	−27327.0	−1.94128
$584$	−5616.71	−0.397981
$585$	674.199	0.0476491
$586$	−3661.77	−0.258134
$587$	−25329.9	−1.78105	−0.890525	−	0.454935i	$-0.849663\pi$
$587$	−25329.9	−1.78105	−0.890525	+	0.454935i	$0.849663\pi$
$588$	−588.000	−0.0412393
$589$	812.947	0.0568708
$590$	−10044.4	−0.700884
$591$	3974.16	0.276607
$592$	5609.90	0.389469
$593$	5725.34	0.396478	0.198239	−	0.980154i	$-0.436478\pi$
$593$	5725.34	0.396478	0.198239	+	0.980154i	$0.436478\pi$
$594$	−2478.89	−0.171229
$595$	3773.00	0.259963
$596$	2439.16	0.167638
$597$	4427.37	0.303518
$598$	−250.944	−0.0171603
$599$	−16336.7	−1.11436	−0.557178	−	0.830393i	$-0.688116\pi$
$599$	−16336.7	−1.11436	−0.557178	+	0.830393i	$0.688116\pi$
$600$	−1525.47	−0.103795
$601$	−19204.7	−1.30346	−0.651728	−	0.758453i	$-0.725956\pi$
$601$	−19204.7	−1.30346	−0.651728	+	0.758453i	$0.725956\pi$
$602$	−6261.33	−0.423908
$603$	−1350.27	−0.0911898
$604$	8269.30	0.557075
$605$	10660.0	0.716345
$606$	−4499.31	−0.301604
$607$	6344.20	0.424223	0.212111	−	0.977246i	$-0.431966\pi$
$607$	6344.20	0.424223	0.212111	+	0.977246i	$0.431966\pi$
$608$	876.666	0.0584762
$609$	−191.546	−0.0127452
$610$	17282.6	1.14713
$611$	3191.71	0.211330
$612$	1413.08	0.0933337
$613$	−25232.0	−1.66250	−0.831248	−	0.555902i	$-0.812373\pi$
$613$	−25232.0	−1.66250	−0.831248	+	0.555902i	$0.812373\pi$
$614$	−104.394	−0.00686158
$615$	18363.4	1.20404
$616$	2570.70	0.168144
$617$	−1164.73	−0.0759969	−0.0379984	−	0.999278i	$-0.512098\pi$
$617$	−1164.73	−0.0759969	−0.0379984	+	0.999278i	$0.512098\pi$
$618$	−1995.01	−0.129856
$619$	−3604.47	−0.234048	−0.117024	−	0.993129i	$-0.537336\pi$
$619$	−3604.47	−0.234048	−0.117024	+	0.993129i	$0.537336\pi$
$620$	1629.91	0.105579
$621$	621.000	0.0401286
$622$	10528.8	0.678726
$623$	−284.327	−0.0182846
$624$	−261.855	−0.0167990
$625$	−19530.1	−1.24993
$626$	7633.27	0.487359
$627$	−3772.84	−0.240308
$628$	13905.2	0.883563
$629$	13762.5	0.872413
$630$	1730.20	0.109417
$631$	2059.95	0.129961	0.0649803	−	0.997887i	$-0.479302\pi$
$631$	2059.95	0.129961	0.0649803	+	0.997887i	$0.479302\pi$
$632$	6264.99	0.394317
$633$	−1756.85	−0.110314
$634$	−22120.8	−1.38569
$635$	−19.2887	−0.00120543
$636$	7143.48	0.445373
$637$	267.310	0.0166267
$638$	837.429	0.0519657
$639$	−1129.56	−0.0699293
$640$	1757.66	0.108559
$641$	20939.2	1.29025	0.645125	−	0.764077i	$-0.276805\pi$
$641$	20939.2	1.29025	0.645125	+	0.764077i	$0.276805\pi$
$642$	−10535.2	−0.647650
$643$	−23140.6	−1.41925	−0.709624	−	0.704581i	$-0.751135\pi$
$643$	−23140.6	−1.41925	−0.709624	+	0.704581i	$0.751135\pi$
$644$	−644.000	−0.0394055
$645$	18424.1	1.12473
$646$	2150.69	0.130987
$647$	4123.48	0.250558	0.125279	−	0.992122i	$-0.460017\pi$
$647$	4123.48	0.250558	0.125279	+	0.992122i	$0.460017\pi$
$648$	648.000	0.0392837
$649$	−16789.3	−1.01546
$650$	693.491	0.0418476
$651$	−623.157	−0.0375168
$652$	−4283.08	−0.257267
$653$	7934.30	0.475487	0.237744	−	0.971328i	$-0.423592\pi$
$653$	7934.30	0.475487	0.237744	+	0.971328i	$0.423592\pi$
$654$	−4789.83	−0.286387
$655$	−30138.1	−1.79785
$656$	−7132.24	−0.424493
$657$	−6318.80	−0.375220
$658$	8190.91	0.485281
$659$	−15803.7	−0.934181	−0.467091	−	0.884209i	$-0.654698\pi$
$659$	−15803.7	−0.934181	−0.467091	+	0.884209i	$0.654698\pi$
$660$	−7564.33	−0.446123
$661$	−5586.17	−0.328709	−0.164355	−	0.986401i	$-0.552554\pi$
$661$	−5586.17	−0.328709	−0.164355	+	0.986401i	$0.552554\pi$
$662$	13945.5	0.818739
$663$	−642.398	−0.0376299
$664$	5953.13	0.347931
$665$	2633.35	0.153559
$666$	6311.14	0.367195
$667$	−209.789	−0.0121785
$668$	9494.14	0.549909
$669$	−9206.42	−0.532049
$670$	−4120.37	−0.237587
$671$	28887.8	1.66200
$672$	−672.000	−0.0385758
$673$	−20234.7	−1.15897	−0.579487	−	0.814981i	$-0.696747\pi$
$673$	−20234.7	−1.15897	−0.579487	+	0.814981i	$0.696747\pi$
$674$	−18902.1	−1.08024
$675$	−1716.15	−0.0978587
$676$	−8668.96	−0.493227
$677$	−7512.52	−0.426484	−0.213242	−	0.976999i	$-0.568402\pi$
$677$	−7512.52	−0.426484	−0.213242	+	0.976999i	$0.568402\pi$
$678$	3264.40	0.184909
$679$	2762.94	0.156159
$680$	4312.00	0.243173
$681$	−13666.6	−0.769024
$682$	2724.40	0.152966
$683$	−16956.8	−0.949975	−0.474987	−	0.879993i	$-0.657547\pi$
$683$	−16956.8	−0.949975	−0.474987	+	0.879993i	$0.657547\pi$
$684$	986.249	0.0551319
$685$	−16951.5	−0.945525
$686$	686.000	0.0381802
$687$	11507.5	0.639065
$688$	−7155.81	−0.396530
$689$	−3247.49	−0.179564
$690$	1894.98	0.104552
$691$	22505.5	1.23900	0.619500	−	0.784997i	$-0.287336\pi$
$691$	22505.5	1.23900	0.619500	+	0.784997i	$0.287336\pi$
$692$	1805.49	0.0991825
$693$	2892.04	0.158527
$694$	2808.80	0.153632
$695$	−11513.7	−0.628399
$696$	−218.910	−0.0119221
$697$	−17497.2	−0.950867
$698$	−9339.19	−0.506438
$699$	1633.65	0.0883980
$700$	1779.71	0.0960953
$701$	32977.4	1.77680	0.888402	−	0.459067i	$-0.151816\pi$
$701$	32977.4	1.77680	0.888402	+	0.459067i	$0.151816\pi$
$702$	−294.587	−0.0158383
$703$	9605.49	0.515331
$704$	2937.94	0.157284
$705$	−24101.9	−1.28756
$706$	−12564.9	−0.669810
$707$	5249.19	0.279231
$708$	4388.84	0.232970
$709$	3887.13	0.205902	0.102951	−	0.994686i	$-0.467172\pi$
$709$	3887.13	0.205902	0.102951	+	0.994686i	$0.467172\pi$
$710$	−3446.86	−0.182195
$711$	7048.12	0.371765
$712$	−324.945	−0.0171037
$713$	−682.505	−0.0358486
$714$	−1648.59	−0.0864102
$715$	3438.81	0.179866
$716$	7.66432	0.000400041 0
$717$	19779.1	1.03022
$718$	−16548.1	−0.860125
$719$	6960.02	0.361008	0.180504	−	0.983574i	$-0.442227\pi$
$719$	6960.02	0.361008	0.180504	+	0.983574i	$0.442227\pi$
$720$	1977.37	0.102350
$721$	2327.51	0.120223
$722$	−12216.9	−0.629733
$723$	2226.11	0.114509
$724$	15187.8	0.779627
$725$	579.757	0.0296988
$726$	−4657.80	−0.238109
$727$	9562.26	0.487819	0.243910	−	0.969798i	$-0.421570\pi$
$727$	9562.26	0.487819	0.243910	+	0.969798i	$0.421570\pi$
$728$	305.497	0.0155529
$729$	729.000	0.0370370
$730$	−19281.8	−0.977606
$731$	−17555.0	−0.888231
$732$	−7551.50	−0.381300
$733$	35530.2	1.79037	0.895183	−	0.445699i	$-0.147045\pi$
$733$	35530.2	1.79037	0.895183	+	0.445699i	$0.147045\pi$
$734$	−411.776	−0.0207070
$735$	−2018.57	−0.101301
$736$	−736.000	−0.0368605
$737$	−6887.20	−0.344224
$738$	−8023.77	−0.400215
$739$	26209.5	1.30464	0.652322	−	0.757942i	$-0.273795\pi$
$739$	26209.5	1.30464	0.652322	+	0.757942i	$0.273795\pi$
$740$	19258.4	0.956695
$741$	−448.358	−0.0222279
$742$	−8334.06	−0.412335
$743$	31885.1	1.57436	0.787181	−	0.616721i	$-0.211539\pi$
$743$	31885.1	1.57436	0.787181	+	0.616721i	$0.211539\pi$
$744$	−712.179	−0.0350938
$745$	8373.50	0.411787
$746$	305.752	0.0150058
$747$	6697.27	0.328033
$748$	7207.53	0.352317
$749$	12291.1	0.599607
$750$	5061.98	0.246450
$751$	−16826.8	−0.817600	−0.408800	−	0.912624i	$-0.634053\pi$
$751$	−16826.8	−0.817600	−0.408800	+	0.912624i	$0.634053\pi$
$752$	9361.04	0.453939
$753$	−11663.0	−0.564439
$754$	99.5186	0.00480670
$755$	28388.0	1.36840
$756$	−756.000	−0.0363696
$757$	11050.1	0.530545	0.265273	−	0.964173i	$-0.414538\pi$
$757$	11050.1	0.530545	0.265273	+	0.964173i	$0.414538\pi$
$758$	−22388.9	−1.07283
$759$	3167.47	0.151478
$760$	3009.54	0.143641
$761$	−25066.2	−1.19402	−0.597010	−	0.802234i	$-0.703645\pi$
$761$	−25066.2	−1.19402	−0.597010	+	0.802234i	$0.703645\pi$
$762$	8.42807	0.000400678 0
$763$	5588.13	0.265143
$764$	13803.3	0.653646
$765$	4851.00	0.229266
$766$	7771.70	0.366583
$767$	−1995.21	−0.0939280
$768$	−768.000	−0.0360844
$769$	−39352.7	−1.84538	−0.922688	−	0.385547i	$-0.874013\pi$
$769$	−39352.7	−1.84538	−0.922688	+	0.385547i	$0.874013\pi$
$770$	8825.05	0.413030
$771$	17759.5	0.829563
$772$	−14220.0	−0.662941
$773$	30740.7	1.43036	0.715179	−	0.698942i	$-0.246345\pi$
$773$	30740.7	1.43036	0.715179	+	0.698942i	$0.246345\pi$
$774$	−8050.28	−0.373852
$775$	1886.12	0.0874212
$776$	3157.64	0.146073
$777$	−7363.00	−0.339956
$778$	12623.5	0.581717
$779$	−12212.1	−0.561674
$780$	−898.932	−0.0412653
$781$	−5761.44	−0.263970
$782$	−1805.60	−0.0825678
$783$	−246.274	−0.0112402
$784$	784.000	0.0357143
$785$	47735.7	2.17039
$786$	13168.7	0.597596
$787$	−11194.1	−0.507021	−0.253511	−	0.967333i	$-0.581585\pi$
$787$	−11194.1	−0.507021	−0.253511	+	0.967333i	$0.581585\pi$
$788$	−5298.88	−0.239549
$789$	16134.3	0.728005
$790$	21507.3	0.968604
$791$	−3808.47	−0.171193
$792$	3305.18	0.148289
$793$	3432.98	0.153731
$794$	−16510.8	−0.737969
$795$	24523.1	1.09402
$796$	−5903.15	−0.262854
$797$	19455.8	0.864694	0.432347	−	0.901707i	$-0.357686\pi$
$797$	19455.8	0.864694	0.432347	+	0.901707i	$0.357686\pi$
$798$	−1150.62	−0.0510422
$799$	22965.0	1.01683
$800$	2033.95	0.0898889
$801$	−365.563	−0.0161255
$802$	−25902.9	−1.14048
$803$	−32229.6	−1.41639
$804$	1800.37	0.0789727
$805$	−2210.81	−0.0967962
$806$	323.763	0.0141490
$807$	−1690.31	−0.0737322
$808$	5999.08	0.261196
$809$	29099.7	1.26464	0.632319	−	0.774708i	$-0.282103\pi$
$809$	29099.7	1.26464	0.632319	+	0.774708i	$0.282103\pi$
$810$	2224.54	0.0964969
$811$	−6804.12	−0.294605	−0.147303	−	0.989091i	$-0.547059\pi$
$811$	−6804.12	−0.294605	−0.147303	+	0.989091i	$0.547059\pi$
$812$	255.395	0.0110377
$813$	24687.2	1.06496
$814$	32190.6	1.38609
$815$	−14703.5	−0.631954
$816$	−1884.10	−0.0808293
$817$	−12252.5	−0.524674
$818$	6194.52	0.264776
$819$	343.685	0.0146634
$820$	−24484.5	−1.04273
$821$	−35218.9	−1.49713	−0.748567	−	0.663059i	$-0.769258\pi$
$821$	−35218.9	−1.49713	−0.748567	+	0.663059i	$0.769258\pi$
$822$	7406.86	0.314287
$823$	−36151.1	−1.53117	−0.765583	−	0.643337i	$-0.777549\pi$
$823$	−36151.1	−1.53117	−0.765583	+	0.643337i	$0.777549\pi$
$824$	2660.01	0.112459
$825$	−8753.38	−0.369398
$826$	−5120.31	−0.215688
$827$	−5471.38	−0.230059	−0.115029	−	0.993362i	$-0.536696\pi$
$827$	−5471.38	−0.230059	−0.115029	+	0.993362i	$0.536696\pi$
$828$	−828.000	−0.0347524
$829$	19839.0	0.831167	0.415583	−	0.909555i	$-0.363577\pi$
$829$	19839.0	0.831167	0.415583	+	0.909555i	$0.363577\pi$
$830$	20436.7	0.854662
$831$	21169.2	0.883695
$832$	349.140	0.0145484
$833$	1923.35	0.0800003
$834$	5030.81	0.208876
$835$	32592.8	1.35080
$836$	5030.46	0.208112
$837$	−801.202	−0.0330867
$838$	12736.4	0.525027
$839$	24653.2	1.01445	0.507225	−	0.861813i	$-0.330671\pi$
$839$	24653.2	1.01445	0.507225	+	0.861813i	$0.330671\pi$
$840$	−2306.93	−0.0947581
$841$	−24305.8	−0.996589
$842$	569.757	0.0233196
$843$	−8326.88	−0.340205
$844$	2342.47	0.0955345
$845$	−29760.0	−1.21157
$846$	10531.2	0.427978
$847$	5434.10	0.220446
$848$	−9524.64	−0.385704
$849$	6849.30	0.276875
$850$	4989.81	0.201352
$851$	−8064.23	−0.324839
$852$	1506.08	0.0605606
$853$	−27717.8	−1.11259	−0.556296	−	0.830984i	$-0.687778\pi$
$853$	−27717.8	−1.11259	−0.556296	+	0.830984i	$0.687778\pi$
$854$	8810.08	0.353015
$855$	3385.73	0.135426
$856$	14046.9	0.560881
$857$	31543.3	1.25729	0.628645	−	0.777693i	$-0.283610\pi$
$857$	31543.3	1.25729	0.628645	+	0.777693i	$0.283610\pi$
$858$	−1502.57	−0.0597865
$859$	16748.0	0.665234	0.332617	−	0.943062i	$-0.392068\pi$
$859$	16748.0	0.665234	0.332617	+	0.943062i	$0.392068\pi$
$860$	−24565.5	−0.974041
$861$	9361.06	0.370528
$862$	13352.5	0.527598
$863$	39073.8	1.54124	0.770619	−	0.637296i	$-0.219947\pi$
$863$	39073.8	1.54124	0.770619	+	0.637296i	$0.219947\pi$
$864$	−864.000	−0.0340207
$865$	6198.13	0.243633
$866$	−13464.7	−0.528348
$867$	10116.8	0.396292
$868$	830.876	0.0324905
$869$	35949.6	1.40334
$870$	−751.505	−0.0292855
$871$	−818.463	−0.0318399
$872$	6386.44	0.248018
$873$	3552.35	0.137719
$874$	−1260.21	−0.0487725
$875$	−5905.65	−0.228168
$876$	8425.06	0.324950
$877$	11372.9	0.437896	0.218948	−	0.975737i	$-0.429738\pi$
$877$	11372.9	0.437896	0.218948	+	0.975737i	$0.429738\pi$
$878$	13584.1	0.522144
$879$	5492.66	0.210765
$880$	10085.8	0.386354
$881$	13707.3	0.524189	0.262095	−	0.965042i	$-0.415587\pi$
$881$	13707.3	0.524189	0.262095	+	0.965042i	$0.415587\pi$
$882$	882.000	0.0336718
$883$	−7930.53	−0.302246	−0.151123	−	0.988515i	$-0.548289\pi$
$883$	−7930.53	−0.302246	−0.151123	+	0.988515i	$0.548289\pi$
$884$	856.530	0.0325885
$885$	15066.6	0.572270
$886$	−22326.7	−0.846592
$887$	−3379.90	−0.127943	−0.0639717	−	0.997952i	$-0.520377\pi$
$887$	−3379.90	−0.127943	−0.0639717	+	0.997952i	$0.520377\pi$
$888$	−8414.85	−0.318000
$889$	−9.83275	−0.000370956 0
$890$	−1115.52	−0.0420137
$891$	3718.33	0.139808
$892$	12275.2	0.460768
$893$	16028.3	0.600636
$894$	−3658.75	−0.136876
$895$	26.3111	0.000982664 0
$896$	896.000	0.0334077
$897$	376.417	0.0140114
$898$	19759.3	0.734273
$899$	270.665	0.0100414
$900$	2288.20	0.0847481
$901$	−23366.4	−0.863981
$902$	−40926.0	−1.51074
$903$	9392.00	0.346120
$904$	−4352.53	−0.160136
$905$	52138.7	1.91508
$906$	−12404.0	−0.454850
$907$	−27965.5	−1.02379	−0.511896	−	0.859048i	$-0.671057\pi$
$907$	−27965.5	−1.02379	−0.511896	+	0.859048i	$0.671057\pi$
$908$	18222.1	0.665994
$909$	6748.96	0.246258
$910$	1048.75	0.0382042
$911$	−1357.11	−0.0493557	−0.0246779	−	0.999695i	$-0.507856\pi$
$911$	−1357.11	−0.0493557	−0.0246779	+	0.999695i	$0.507856\pi$
$912$	−1315.00	−0.0477456
$913$	34160.1	1.23826
$914$	−12342.6	−0.446669
$915$	−25923.8	−0.936629
$916$	−15343.3	−0.553446
$917$	−15363.4	−0.553266
$918$	−2119.61	−0.0762066
$919$	−31780.9	−1.14076	−0.570379	−	0.821382i	$-0.693204\pi$
$919$	−31780.9	−1.14076	−0.570379	+	0.821382i	$0.693204\pi$
$920$	−2526.64	−0.0905445
$921$	156.592	0.00560246
$922$	6154.29	0.219827
$923$	−684.680	−0.0244166
$924$	−3856.05	−0.137289
$925$	22285.7	0.792162
$926$	−14020.4	−0.497560
$927$	2992.51	0.106027
$928$	291.880	0.0103248
$929$	−37901.2	−1.33853	−0.669266	−	0.743023i	$-0.733391\pi$
$929$	−37901.2	−1.33853	−0.669266	+	0.743023i	$0.733391\pi$
$930$	−2444.87	−0.0862047
$931$	1342.39	0.0472559
$932$	−2178.20	−0.0765549
$933$	−15793.3	−0.554178
$934$	7390.56	0.258915
$935$	24743.0	0.865436
$936$	392.782	0.0137163
$937$	41733.1	1.45503	0.727514	−	0.686093i	$-0.240676\pi$
$937$	41733.1	1.45503	0.727514	+	0.686093i	$0.240676\pi$
$938$	−2100.43	−0.0731145
$939$	−11449.9	−0.397927
$940$	32135.9	1.11506
$941$	6773.55	0.234656	0.117328	−	0.993093i	$-0.462567\pi$
$941$	6773.55	0.234656	0.117328	+	0.993093i	$0.462567\pi$
$942$	−20857.8	−0.721426
$943$	10252.6	0.354051
$944$	−5851.79	−0.201758
$945$	−2595.30	−0.0893388
$946$	−41061.2	−1.41122
$947$	32914.3	1.12943	0.564715	−	0.825286i	$-0.308986\pi$
$947$	32914.3	1.12943	0.564715	+	0.825286i	$0.308986\pi$
$948$	−9397.49	−0.321958
$949$	−3830.11	−0.131012
$950$	3482.62	0.118938
$951$	33181.2	1.13141
$952$	2198.12	0.0748334
$953$	−25158.6	−0.855160	−0.427580	−	0.903978i	$-0.640634\pi$
$953$	−25158.6	−0.855160	−0.427580	+	0.903978i	$0.640634\pi$
$954$	−10715.2	−0.363646
$955$	47385.9	1.60562
$956$	−26372.2	−0.892194
$957$	−1256.14	−0.0424298
$958$	30666.7	1.03423
$959$	−8641.34	−0.290973
$960$	−2636.50	−0.0886381
$961$	−28910.4	−0.970442
$962$	3825.47	0.128210
$963$	15802.8	0.528804
$964$	−2968.14	−0.0991674
$965$	−48816.5	−1.62846
$966$	966.000	0.0321745
$967$	25734.2	0.855799	0.427899	−	0.903826i	$-0.359254\pi$
$967$	25734.2	0.855799	0.427899	+	0.903826i	$0.359254\pi$
$968$	6210.40	0.206208
$969$	−3226.03	−0.106951
$970$	10840.0	0.358816
$971$	−16969.6	−0.560846	−0.280423	−	0.959876i	$-0.590475\pi$
$971$	−16969.6	−0.560846	−0.280423	+	0.959876i	$0.590475\pi$
$972$	−972.000	−0.0320750
$973$	−5869.28	−0.193382
$974$	−18191.6	−0.598455
$975$	−1040.24	−0.0341684
$976$	10068.7	0.330215
$977$	47396.8	1.55205	0.776027	−	0.630700i	$-0.217232\pi$
$977$	47396.8	1.55205	0.776027	+	0.630700i	$0.217232\pi$
$978$	6424.62	0.210058
$979$	−1864.59	−0.0608708
$980$	2691.42	0.0877290
$981$	7184.74	0.233834
$982$	29683.4	0.964597
$983$	−40149.7	−1.30272	−0.651361	−	0.758768i	$-0.725802\pi$
$983$	−40149.7	−1.30272	−0.651361	+	0.758768i	$0.725802\pi$
$984$	10698.4	0.346597
$985$	−18190.7	−0.588431
$986$	716.057	0.0231277
$987$	−12286.4	−0.396230
$988$	597.811	0.0192499
$989$	10286.5	0.330729
$990$	11346.5	0.364258
$991$	24954.5	0.799906	0.399953	−	0.916536i	$-0.369026\pi$
$991$	24954.5	0.799906	0.399953	+	0.916536i	$0.369026\pi$
$992$	949.572	0.0303921
$993$	−20918.2	−0.668498
$994$	−1757.10	−0.0560682
$995$	−20265.2	−0.645677
$996$	−8929.70	−0.284085
$997$	14637.0	0.464952	0.232476	−	0.972602i	$-0.425317\pi$
$997$	14637.0	0.464952	0.232476	+	0.972602i	$0.425317\pi$
$998$	13598.5	0.431316
$999$	−9466.71	−0.299813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.4.a.f.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.4.a.f.1.3	✓	3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	966.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +13.7318 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +13.7318 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +27.4635 q^{10} +45.9053 q^{11} -12.0000 q^{12} +5.45531 q^{13} +14.0000 q^{14} -41.1953 q^{15} +16.0000 q^{16} +39.2521 q^{17} +18.0000 q^{18} +27.3958 q^{19} +54.9270 q^{20} -21.0000 q^{21} +91.8107 q^{22} -23.0000 q^{23} -24.0000 q^{24} +63.5611 q^{25} +10.9106 q^{26} -27.0000 q^{27} +28.0000 q^{28} +9.12126 q^{29} -82.3905 q^{30} +29.6741 q^{31} +32.0000 q^{32} -137.716 q^{33} +78.5042 q^{34} +96.1223 q^{35} +36.0000 q^{36} +350.619 q^{37} +54.7916 q^{38} -16.3659 q^{39} +109.854 q^{40} -445.765 q^{41} -42.0000 q^{42} -447.238 q^{43} +183.621 q^{44} +123.586 q^{45} -46.0000 q^{46} +585.065 q^{47} -48.0000 q^{48} +49.0000 q^{49} +127.122 q^{50} -117.756 q^{51} +21.8212 q^{52} -595.290 q^{53} -54.0000 q^{54} +630.361 q^{55} +56.0000 q^{56} -82.1874 q^{57} +18.2425 q^{58} -365.737 q^{59} -164.781 q^{60} +629.292 q^{61} +59.3483 q^{62} +63.0000 q^{63} +64.0000 q^{64} +74.9110 q^{65} -275.432 q^{66} -150.031 q^{67} +157.008 q^{68} +69.0000 q^{69} +192.245 q^{70} -125.507 q^{71} +72.0000 q^{72} -702.089 q^{73} +701.238 q^{74} -190.683 q^{75} +109.583 q^{76} +321.337 q^{77} -32.7319 q^{78} +783.124 q^{79} +219.708 q^{80} +81.0000 q^{81} -891.530 q^{82} +744.142 q^{83} -84.0000 q^{84} +539.000 q^{85} -894.476 q^{86} -27.3638 q^{87} +367.243 q^{88} -40.6181 q^{89} +247.172 q^{90} +38.1872 q^{91} -92.0000 q^{92} -89.0224 q^{93} +1170.13 q^{94} +376.193 q^{95} -96.0000 q^{96} +394.706 q^{97} +98.0000 q^{98} +413.148 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 9 q^{5} - 18 q^{6} + 21 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 9 * q^5 - 18 * q^6 + 21 * q^7 + 24 * q^8 + 27 * q^9	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 9 q^{5} - 18 q^{6} + 21 q^{7} + 24 q^{8} + 27 q^{9} + 18 q^{10} + 53 q^{11} - 36 q^{12} + 61 q^{13} + 42 q^{14} - 27 q^{15} + 48 q^{16} - 2 q^{17} + 54 q^{18} + 63 q^{19} + 36 q^{20} - 63 q^{21} + 106 q^{22} - 69 q^{23} - 72 q^{24} - 72 q^{25} + 122 q^{26} - 81 q^{27} + 84 q^{28} + 312 q^{29} - 54 q^{30} + 220 q^{31} + 96 q^{32} - 159 q^{33} - 4 q^{34} + 63 q^{35} + 108 q^{36} - 44 q^{37} + 126 q^{38} - 183 q^{39} + 72 q^{40} + 171 q^{41} - 126 q^{42} - 627 q^{43} + 212 q^{44} + 81 q^{45} - 138 q^{46} + 353 q^{47} - 144 q^{48} + 147 q^{49} - 144 q^{50} + 6 q^{51} + 244 q^{52} - 504 q^{53} - 162 q^{54} + 1181 q^{55} + 168 q^{56} - 189 q^{57} + 624 q^{58} + 538 q^{59} - 108 q^{60} + 256 q^{61} + 440 q^{62} + 189 q^{63} + 192 q^{64} + 244 q^{65} - 318 q^{66} + 1113 q^{67} - 8 q^{68} + 207 q^{69} + 126 q^{70} + 53 q^{71} + 216 q^{72} + 360 q^{73} - 88 q^{74} + 216 q^{75} + 252 q^{76} + 371 q^{77} - 366 q^{78} + 134 q^{79} + 144 q^{80} + 243 q^{81} + 342 q^{82} + 1506 q^{83} - 252 q^{84} + 577 q^{85} - 1254 q^{86} - 936 q^{87} + 424 q^{88} + 2381 q^{89} + 162 q^{90} + 427 q^{91} - 276 q^{92} - 660 q^{93} + 706 q^{94} + 945 q^{95} - 288 q^{96} - 88 q^{97} + 294 q^{98} + 477 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 9 * q^5 - 18 * q^6 + 21 * q^7 + 24 * q^8 + 27 * q^9 + 18 * q^10 + 53 * q^11 - 36 * q^12 + 61 * q^13 + 42 * q^14 - 27 * q^15 + 48 * q^16 - 2 * q^17 + 54 * q^18 + 63 * q^19 + 36 * q^20 - 63 * q^21 + 106 * q^22 - 69 * q^23 - 72 * q^24 - 72 * q^25 + 122 * q^26 - 81 * q^27 + 84 * q^28 + 312 * q^29 - 54 * q^30 + 220 * q^31 + 96 * q^32 - 159 * q^33 - 4 * q^34 + 63 * q^35 + 108 * q^36 - 44 * q^37 + 126 * q^38 - 183 * q^39 + 72 * q^40 + 171 * q^41 - 126 * q^42 - 627 * q^43 + 212 * q^44 + 81 * q^45 - 138 * q^46 + 353 * q^47 - 144 * q^48 + 147 * q^49 - 144 * q^50 + 6 * q^51 + 244 * q^52 - 504 * q^53 - 162 * q^54 + 1181 * q^55 + 168 * q^56 - 189 * q^57 + 624 * q^58 + 538 * q^59 - 108 * q^60 + 256 * q^61 + 440 * q^62 + 189 * q^63 + 192 * q^64 + 244 * q^65 - 318 * q^66 + 1113 * q^67 - 8 * q^68 + 207 * q^69 + 126 * q^70 + 53 * q^71 + 216 * q^72 + 360 * q^73 - 88 * q^74 + 216 * q^75 + 252 * q^76 + 371 * q^77 - 366 * q^78 + 134 * q^79 + 144 * q^80 + 243 * q^81 + 342 * q^82 + 1506 * q^83 - 252 * q^84 + 577 * q^85 - 1254 * q^86 - 936 * q^87 + 424 * q^88 + 2381 * q^89 + 162 * q^90 + 427 * q^91 - 276 * q^92 - 660 * q^93 + 706 * q^94 + 945 * q^95 - 288 * q^96 - 88 * q^97 + 294 * q^98 + 477 * q^99

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