LMFDB - Embedded newform 1682.4.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,4,Mod(1,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, 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("1682.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1682 = 2 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1682.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:	$99.2412126297$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.19816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 42x - 54 $ x^3 - x^2 - 42*x - 54
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$7.53003$ of defining polynomial
Character	$\chi$	$=$	1682.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +6.53003 q^{3} +4.00000 q^{4} +20.9205 q^{5} -13.0601 q^{6} +8.55839 q^{7} -8.00000 q^{8} +15.6413 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +6.53003 q^{3} +4.00000 q^{4} +20.9205 q^{5} -13.0601 q^{6} +8.55839 q^{7} -8.00000 q^{8} +15.6413 q^{9} -41.8409 q^{10} -10.8092 q^{11} +26.1201 q^{12} +54.7046 q^{13} -17.1168 q^{14} +136.611 q^{15} +16.0000 q^{16} +106.127 q^{17} -31.2825 q^{18} +113.636 q^{19} +83.6818 q^{20} +55.8865 q^{21} +21.6184 q^{22} -112.855 q^{23} -52.2402 q^{24} +312.666 q^{25} -109.409 q^{26} -74.1729 q^{27} +34.2336 q^{28} -273.222 q^{30} +102.805 q^{31} -32.0000 q^{32} -70.5845 q^{33} -212.254 q^{34} +179.045 q^{35} +62.5650 q^{36} +105.665 q^{37} -227.272 q^{38} +357.223 q^{39} -167.364 q^{40} -216.958 q^{41} -111.773 q^{42} +102.230 q^{43} -43.2369 q^{44} +327.222 q^{45} +225.711 q^{46} -455.212 q^{47} +104.480 q^{48} -269.754 q^{49} -625.331 q^{50} +693.011 q^{51} +218.819 q^{52} -593.714 q^{53} +148.346 q^{54} -226.134 q^{55} -68.4671 q^{56} +742.048 q^{57} -558.141 q^{59} +546.445 q^{60} +473.986 q^{61} -205.610 q^{62} +133.864 q^{63} +64.0000 q^{64} +1144.45 q^{65} +141.169 q^{66} +193.132 q^{67} +424.507 q^{68} -736.949 q^{69} -358.091 q^{70} -2.38155 q^{71} -125.130 q^{72} -119.013 q^{73} -211.330 q^{74} +2041.71 q^{75} +454.545 q^{76} -92.5096 q^{77} -714.446 q^{78} +964.306 q^{79} +334.727 q^{80} -906.665 q^{81} +433.915 q^{82} +1068.19 q^{83} +223.546 q^{84} +2220.22 q^{85} -204.459 q^{86} +86.4738 q^{88} -772.544 q^{89} -654.445 q^{90} +468.184 q^{91} -451.421 q^{92} +671.318 q^{93} +910.423 q^{94} +2377.32 q^{95} -208.961 q^{96} -1344.03 q^{97} +539.508 q^{98} -169.070 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} - 24 q^{8} + 5 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 20 * q^5 + 4 * q^6 + 24 * q^7 - 24 * q^8 + 5 * q^9	\( 3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} - 24 q^{8} + 5 q^{9} - 40 q^{10} - 10 q^{11} - 8 q^{12} - 4 q^{13} - 48 q^{14} + 130 q^{15} + 48 q^{16} + 66 q^{17} - 10 q^{18} + 164 q^{19} + 80 q^{20} + 88 q^{21} + 20 q^{22} - 204 q^{23} + 16 q^{24} + 79 q^{25} + 8 q^{26} + 142 q^{27} + 96 q^{28} - 260 q^{30} + 86 q^{31} - 96 q^{32} - 130 q^{33} - 132 q^{34} + 24 q^{35} + 20 q^{36} + 42 q^{37} - 328 q^{38} + 394 q^{39} - 160 q^{40} - 562 q^{41} - 176 q^{42} - 18 q^{43} - 40 q^{44} + 422 q^{45} + 408 q^{46} - 654 q^{47} - 32 q^{48} + 539 q^{49} - 158 q^{50} + 556 q^{51} - 16 q^{52} + 712 q^{53} - 284 q^{54} - 142 q^{55} - 192 q^{56} + 828 q^{57} + 184 q^{59} + 520 q^{60} - 322 q^{61} - 172 q^{62} - 784 q^{63} + 192 q^{64} + 1494 q^{65} + 260 q^{66} - 228 q^{67} + 264 q^{68} - 684 q^{69} - 48 q^{70} - 52 q^{71} - 40 q^{72} + 494 q^{73} - 84 q^{74} + 3048 q^{75} + 656 q^{76} - 872 q^{77} - 788 q^{78} + 2110 q^{79} + 320 q^{80} - 1513 q^{81} + 1124 q^{82} - 288 q^{83} + 352 q^{84} + 2704 q^{85} + 36 q^{86} + 80 q^{88} - 914 q^{89} - 844 q^{90} - 2984 q^{91} - 816 q^{92} - 62 q^{93} + 1308 q^{94} + 1900 q^{95} + 64 q^{96} - 218 q^{97} - 1078 q^{98} + 304 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 20 * q^5 + 4 * q^6 + 24 * q^7 - 24 * q^8 + 5 * q^9 - 40 * q^10 - 10 * q^11 - 8 * q^12 - 4 * q^13 - 48 * q^14 + 130 * q^15 + 48 * q^16 + 66 * q^17 - 10 * q^18 + 164 * q^19 + 80 * q^20 + 88 * q^21 + 20 * q^22 - 204 * q^23 + 16 * q^24 + 79 * q^25 + 8 * q^26 + 142 * q^27 + 96 * q^28 - 260 * q^30 + 86 * q^31 - 96 * q^32 - 130 * q^33 - 132 * q^34 + 24 * q^35 + 20 * q^36 + 42 * q^37 - 328 * q^38 + 394 * q^39 - 160 * q^40 - 562 * q^41 - 176 * q^42 - 18 * q^43 - 40 * q^44 + 422 * q^45 + 408 * q^46 - 654 * q^47 - 32 * q^48 + 539 * q^49 - 158 * q^50 + 556 * q^51 - 16 * q^52 + 712 * q^53 - 284 * q^54 - 142 * q^55 - 192 * q^56 + 828 * q^57 + 184 * q^59 + 520 * q^60 - 322 * q^61 - 172 * q^62 - 784 * q^63 + 192 * q^64 + 1494 * q^65 + 260 * q^66 - 228 * q^67 + 264 * q^68 - 684 * q^69 - 48 * q^70 - 52 * q^71 - 40 * q^72 + 494 * q^73 - 84 * q^74 + 3048 * q^75 + 656 * q^76 - 872 * q^77 - 788 * q^78 + 2110 * q^79 + 320 * q^80 - 1513 * q^81 + 1124 * q^82 - 288 * q^83 + 352 * q^84 + 2704 * q^85 + 36 * q^86 + 80 * q^88 - 914 * q^89 - 844 * q^90 - 2984 * q^91 - 816 * q^92 - 62 * q^93 + 1308 * q^94 + 1900 * q^95 + 64 * q^96 - 218 * q^97 - 1078 * q^98 + 304 * 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
$3$	6.53003	1.25670	0.628352	−	0.777929i	$-0.283730\pi$
$3$	6.53003	1.25670	0.628352	+	0.777929i	$0.283730\pi$
$4$	4.00000	0.500000
$5$	20.9205	1.87118	0.935591	−	0.353085i	$-0.114867\pi$
$5$	20.9205	1.87118	0.935591	+	0.353085i	$0.114867\pi$
$6$	−13.0601	−0.888624
$7$	8.55839	0.462110	0.231055	−	0.972941i	$-0.425782\pi$
$7$	8.55839	0.462110	0.231055	+	0.972941i	$0.425782\pi$
$8$	−8.00000	−0.353553
$9$	15.6413	0.579306
$10$	−41.8409	−1.32313
$11$	−10.8092	−0.296282	−0.148141	−	0.988966i	$-0.547329\pi$
$11$	−10.8092	−0.296282	−0.148141	+	0.988966i	$0.547329\pi$
$12$	26.1201	0.628352
$13$	54.7046	1.16710	0.583551	−	0.812076i	$-0.301663\pi$
$13$	54.7046	1.16710	0.583551	+	0.812076i	$0.301663\pi$
$14$	−17.1168	−0.326761
$15$	136.611	2.35152
$16$	16.0000	0.250000
$17$	106.127	1.51409	0.757045	−	0.653363i	$-0.226642\pi$
$17$	106.127	1.51409	0.757045	+	0.653363i	$0.226642\pi$
$18$	−31.2825	−0.409631
$19$	113.636	1.37210	0.686051	−	0.727554i	$-0.259343\pi$
$19$	113.636	1.37210	0.686051	+	0.727554i	$0.259343\pi$
$20$	83.6818	0.935591
$21$	55.8865	0.580735
$22$	21.6184	0.209503
$23$	−112.855	−1.02313	−0.511564	−	0.859245i	$-0.670934\pi$
$23$	−112.855	−1.02313	−0.511564	+	0.859245i	$0.670934\pi$
$24$	−52.2402	−0.444312
$25$	312.666	2.50132
$26$	−109.409	−0.825266
$27$	−74.1729	−0.528688
$28$	34.2336	0.231055
$29$	0	0
$29$	0	0
$30$	−273.222	−1.66278
$31$	102.805	0.595622	0.297811	−	0.954625i	$-0.403743\pi$
$31$	102.805	0.595622	0.297811	+	0.954625i	$0.403743\pi$
$32$	−32.0000	−0.176777
$33$	−70.5845	−0.372339
$34$	−212.254	−1.07062
$35$	179.045	0.864692
$36$	62.5650	0.289653
$37$	105.665	0.469493	0.234746	−	0.972057i	$-0.424574\pi$
$37$	105.665	0.469493	0.234746	+	0.972057i	$0.424574\pi$
$38$	−227.272	−0.970222
$39$	357.223	1.46670
$40$	−167.364	−0.661563
$41$	−216.958	−0.826417	−0.413209	−	0.910636i	$-0.635592\pi$
$41$	−216.958	−0.826417	−0.413209	+	0.910636i	$0.635592\pi$
$42$	−111.773	−0.410642
$43$	102.230	0.362555	0.181278	−	0.983432i	$-0.441977\pi$
$43$	102.230	0.362555	0.181278	+	0.983432i	$0.441977\pi$
$44$	−43.2369	−0.148141
$45$	327.222	1.08399
$46$	225.711	0.723461
$47$	−455.212	−1.41275	−0.706377	−	0.707836i	$-0.749672\pi$
$47$	−455.212	−1.41275	−0.706377	+	0.707836i	$0.749672\pi$
$48$	104.480	0.314176
$49$	−269.754	−0.786455
$50$	−625.331	−1.76870
$51$	693.011	1.90276
$52$	218.819	0.583551
$53$	−593.714	−1.53873	−0.769367	−	0.638807i	$-0.779428\pi$
$53$	−593.714	−1.53873	−0.769367	+	0.638807i	$0.779428\pi$
$54$	148.346	0.373839
$55$	−226.134	−0.554398
$56$	−68.4671	−0.163380
$57$	742.048	1.72433
$58$	0	0
$59$	−558.141	−1.23159	−0.615794	−	0.787907i	$-0.711165\pi$
$59$	−558.141	−1.23159	−0.615794	+	0.787907i	$0.711165\pi$
$60$	546.445	1.17576
$61$	473.986	0.994880	0.497440	−	0.867498i	$-0.334273\pi$
$61$	473.986	0.994880	0.497440	+	0.867498i	$0.334273\pi$
$62$	−205.610	−0.421168
$63$	133.864	0.267703
$64$	64.0000	0.125000
$65$	1144.45	2.18386
$66$	141.169	0.263283
$67$	193.132	0.352162	0.176081	−	0.984376i	$-0.443658\pi$
$67$	193.132	0.352162	0.176081	+	0.984376i	$0.443658\pi$
$68$	424.507	0.757045
$69$	−736.949	−1.28577
$70$	−358.091	−0.611429
$71$	−2.38155	−0.00398082	−0.00199041	−	0.999998i	$-0.500634\pi$
$71$	−2.38155	−0.00398082	−0.00199041	+	0.999998i	$0.500634\pi$
$72$	−125.130	−0.204816
$73$	−119.013	−0.190814	−0.0954071	−	0.995438i	$-0.530415\pi$
$73$	−119.013	−0.190814	−0.0954071	+	0.995438i	$0.530415\pi$
$74$	−211.330	−0.331981
$75$	2041.71	3.14343
$76$	454.545	0.686051
$77$	−92.5096	−0.136915
$78$	−714.446	−1.03712
$79$	964.306	1.37333	0.686664	−	0.726975i	$-0.259074\pi$
$79$	964.306	1.37333	0.686664	+	0.726975i	$0.259074\pi$
$80$	334.727	0.467796
$81$	−906.665	−1.24371
$82$	433.915	0.584365
$83$	1068.19	1.41264	0.706319	−	0.707893i	$-0.250354\pi$
$83$	1068.19	1.41264	0.706319	+	0.707893i	$0.250354\pi$
$84$	223.546	0.290368
$85$	2220.22	2.83314
$86$	−204.459	−0.256365
$87$	0	0
$88$	86.4738	0.104752
$89$	−772.544	−0.920106	−0.460053	−	0.887891i	$-0.652170\pi$
$89$	−772.544	−0.920106	−0.460053	+	0.887891i	$0.652170\pi$
$90$	−654.445	−0.766495
$91$	468.184	0.539330
$92$	−451.421	−0.511564
$93$	671.318	0.748521
$94$	910.423	0.998968
$95$	2377.32	2.56745
$96$	−208.961	−0.222156
$97$	−1344.03	−1.40686	−0.703431	−	0.710763i	$-0.748350\pi$
$97$	−1344.03	−1.40686	−0.703431	+	0.710763i	$0.748350\pi$
$98$	539.508	0.556107
$99$	−169.070	−0.171638
$100$	1250.66	1.25066
$101$	−986.733	−0.972115	−0.486057	−	0.873927i	$-0.661565\pi$
$101$	−986.733	−0.972115	−0.486057	+	0.873927i	$0.661565\pi$
$102$	−1386.02	−1.34546
$103$	548.272	0.524493	0.262247	−	0.965001i	$-0.415537\pi$
$103$	548.272	0.524493	0.262247	+	0.965001i	$0.415537\pi$
$104$	−437.637	−0.412633
$105$	1169.17	1.08666
$106$	1187.43	1.08805
$107$	1387.51	1.25361	0.626803	−	0.779178i	$-0.284363\pi$
$107$	1387.51	1.25361	0.626803	+	0.779178i	$0.284363\pi$
$108$	−296.692	−0.264344
$109$	−1293.32	−1.13649	−0.568246	−	0.822859i	$-0.692378\pi$
$109$	−1293.32	−1.13649	−0.568246	+	0.822859i	$0.692378\pi$
$110$	452.268	0.392019
$111$	689.996	0.590014
$112$	136.934	0.115527
$113$	−302.883	−0.252149	−0.126075	−	0.992021i	$-0.540238\pi$
$113$	−302.883	−0.252149	−0.126075	+	0.992021i	$0.540238\pi$
$114$	−1484.10	−1.21928
$115$	−2360.99	−1.91446
$116$	0	0
$117$	855.650	0.676110
$118$	1116.28	0.870865
$119$	908.274	0.699676
$120$	−1092.89	−0.831389
$121$	−1214.16	−0.912217
$122$	−947.972	−0.703487
$123$	−1416.74	−1.03856
$124$	411.219	0.297811
$125$	3926.05	2.80925
$126$	−267.728	−0.189295
$127$	−2021.68	−1.41256	−0.706281	−	0.707931i	$-0.749629\pi$
$127$	−2021.68	−1.41256	−0.706281	+	0.707931i	$0.749629\pi$
$128$	−128.000	−0.0883883
$129$	667.562	0.455625
$130$	−2288.89	−1.54422
$131$	−854.726	−0.570059	−0.285030	−	0.958519i	$-0.592003\pi$
$131$	−854.726	−0.570059	−0.285030	+	0.958519i	$0.592003\pi$
$132$	−282.338	−0.186170
$133$	972.543	0.634062
$134$	−386.264	−0.249016
$135$	−1551.73	−0.989272
$136$	−849.014	−0.535311
$137$	−365.024	−0.227636	−0.113818	−	0.993502i	$-0.536308\pi$
$137$	−365.024	−0.227636	−0.113818	+	0.993502i	$0.536308\pi$
$138$	1473.90	0.909177
$139$	−1010.10	−0.616370	−0.308185	−	0.951326i	$-0.599722\pi$
$139$	−1010.10	−0.616370	−0.308185	+	0.951326i	$0.599722\pi$
$140$	716.182	0.432346
$141$	−2972.55	−1.77541
$142$	4.76310	0.00281487
$143$	−591.315	−0.345792
$144$	250.260	0.144826
$145$	0	0
$146$	238.026	0.134926
$147$	−1761.50	−0.988341
$148$	422.660	0.234746
$149$	819.765	0.450723	0.225362	−	0.974275i	$-0.427644\pi$
$149$	819.765	0.450723	0.225362	+	0.974275i	$0.427644\pi$
$150$	−4083.43	−2.22274
$151$	1000.25	0.539068	0.269534	−	0.962991i	$-0.413130\pi$
$151$	1000.25	0.539068	0.269534	+	0.962991i	$0.413130\pi$
$152$	−909.090	−0.485111
$153$	1659.96	0.877121
$154$	185.019	0.0968134
$155$	2150.72	1.11452
$156$	1428.89	0.733352
$157$	1702.38	0.865382	0.432691	−	0.901542i	$-0.357564\pi$
$157$	1702.38	0.865382	0.432691	+	0.901542i	$0.357564\pi$
$158$	−1928.61	−0.971090
$159$	−3876.97	−1.93373
$160$	−669.455	−0.330781
$161$	−965.860	−0.472798
$162$	1813.33	0.879436
$163$	3451.76	1.65867	0.829334	−	0.558753i	$-0.188720\pi$
$163$	3451.76	1.65867	0.829334	+	0.558753i	$0.188720\pi$
$164$	−867.831	−0.413209
$165$	−1476.66	−0.696714
$166$	−2136.38	−0.998886
$167$	1409.23	0.652990	0.326495	−	0.945199i	$-0.394132\pi$
$167$	1409.23	0.652990	0.326495	+	0.945199i	$0.394132\pi$
$168$	−447.092	−0.205321
$169$	795.598	0.362129
$170$	−4440.44	−2.00333
$171$	1777.41	0.794867
$172$	408.918	0.181278
$173$	1358.39	0.596973	0.298487	−	0.954414i	$-0.403518\pi$
$173$	1358.39	0.596973	0.298487	+	0.954414i	$0.403518\pi$
$174$	0	0
$175$	2675.91	1.15589
$176$	−172.948	−0.0740705
$177$	−3644.67	−1.54774
$178$	1545.09	0.650614
$179$	−1702.56	−0.710924	−0.355462	−	0.934691i	$-0.615677\pi$
$179$	−1702.56	−0.710924	−0.355462	+	0.934691i	$0.615677\pi$
$180$	1308.89	0.541994
$181$	−7.33698	−0.00301300	−0.00150650	−	0.999999i	$-0.500480\pi$
$181$	−7.33698	−0.00301300	−0.00150650	+	0.999999i	$0.500480\pi$
$182$	−936.368	−0.381364
$183$	3095.14	1.25027
$184$	902.843	0.361731
$185$	2210.56	0.878507
$186$	−1342.64	−0.529284
$187$	−1147.15	−0.448598
$188$	−1820.85	−0.706377
$189$	−634.801	−0.244312
$190$	−4754.64	−1.81546
$191$	−1324.78	−0.501873	−0.250936	−	0.968004i	$-0.580738\pi$
$191$	−1324.78	−0.501873	−0.250936	+	0.968004i	$0.580738\pi$
$192$	417.922	0.157088
$193$	−1834.47	−0.684187	−0.342094	−	0.939666i	$-0.611136\pi$
$193$	−1834.47	−0.684187	−0.342094	+	0.939666i	$0.611136\pi$
$194$	2688.06	0.994802
$195$	7473.26	2.74447
$196$	−1079.02	−0.393227
$197$	−4949.99	−1.79021	−0.895107	−	0.445851i	$-0.852901\pi$
$197$	−4949.99	−1.79021	−0.895107	+	0.445851i	$0.852901\pi$
$198$	338.140	0.121366
$199$	3554.04	1.26603	0.633014	−	0.774140i	$-0.281818\pi$
$199$	3554.04	1.26603	0.633014	+	0.774140i	$0.281818\pi$
$200$	−2501.32	−0.884352
$201$	1261.16	0.442563
$202$	1973.47	0.687389
$203$	0	0
$204$	2772.04	0.951381
$205$	−4538.85	−1.54638
$206$	−1096.54	−0.370873
$207$	−1765.20	−0.592705
$208$	875.274	0.291776
$209$	−1228.32	−0.406529
$210$	−2338.34	−0.768386
$211$	−2475.23	−0.807590	−0.403795	−	0.914849i	$-0.632309\pi$
$211$	−2475.23	−0.807590	−0.403795	+	0.914849i	$0.632309\pi$
$212$	−2374.86	−0.769367
$213$	−15.5516	−0.00500271
$214$	−2775.03	−0.886434
$215$	2138.69	0.678407
$216$	593.383	0.186919
$217$	879.844	0.275243
$218$	2586.64	0.803622
$219$	−777.159	−0.239797
$220$	−904.536	−0.277199
$221$	5805.63	1.76710
$222$	−1379.99	−0.417203
$223$	2381.72	0.715211	0.357605	−	0.933873i	$-0.383593\pi$
$223$	2381.72	0.715211	0.357605	+	0.933873i	$0.383593\pi$
$224$	−273.869	−0.0816902
$225$	4890.48	1.44903
$226$	605.767	0.178297
$227$	5452.74	1.59432	0.797160	−	0.603768i	$-0.206334\pi$
$227$	5452.74	1.59432	0.797160	+	0.603768i	$0.206334\pi$
$228$	2968.19	0.862163
$229$	−596.232	−0.172053	−0.0860264	−	0.996293i	$-0.527417\pi$
$229$	−596.232	−0.172053	−0.0860264	+	0.996293i	$0.527417\pi$
$230$	4721.97	1.35373
$231$	−604.090	−0.172062
$232$	0	0
$233$	−5623.04	−1.58102	−0.790509	−	0.612450i	$-0.790184\pi$
$233$	−5623.04	−1.58102	−0.790509	+	0.612450i	$0.790184\pi$
$234$	−1711.30	−0.478082
$235$	−9523.24	−2.64352
$236$	−2232.56	−0.615794
$237$	6296.95	1.72587
$238$	−1816.55	−0.494745
$239$	1564.27	0.423366	0.211683	−	0.977338i	$-0.432106\pi$
$239$	1564.27	0.423366	0.211683	+	0.977338i	$0.432106\pi$
$240$	2185.78	0.587881
$241$	−730.326	−0.195205	−0.0976026	−	0.995225i	$-0.531117\pi$
$241$	−730.326	−0.195205	−0.0976026	+	0.995225i	$0.531117\pi$
$242$	2428.32	0.645035
$243$	−3917.88	−1.03429
$244$	1895.94	0.497440
$245$	−5643.38	−1.47160
$246$	2833.48	0.734374
$247$	6216.43	1.60138
$248$	−822.438	−0.210584
$249$	6975.31	1.77527
$250$	−7852.10	−1.98644
$251$	4244.39	1.06735	0.533673	−	0.845691i	$-0.320811\pi$
$251$	4244.39	1.06735	0.533673	+	0.845691i	$0.320811\pi$
$252$	535.456	0.133851
$253$	1219.88	0.303135
$254$	4043.37	0.998833
$255$	14498.1	3.56042
$256$	256.000	0.0625000
$257$	−235.374	−0.0571292	−0.0285646	−	0.999592i	$-0.509094\pi$
$257$	−235.374	−0.0571292	−0.0285646	+	0.999592i	$0.509094\pi$
$258$	−1335.12	−0.322175
$259$	904.323	0.216957
$260$	4577.78	1.09193
$261$	0	0
$262$	1709.45	0.403093
$263$	−3453.63	−0.809734	−0.404867	−	0.914376i	$-0.632682\pi$
$263$	−3453.63	−0.809734	−0.404867	+	0.914376i	$0.632682\pi$
$264$	564.676	0.131642
$265$	−12420.8	−2.87925
$266$	−1945.09	−0.448349
$267$	−5044.73	−1.15630
$268$	772.528	0.176081
$269$	1921.31	0.435481	0.217741	−	0.976007i	$-0.430131\pi$
$269$	1921.31	0.435481	0.217741	+	0.976007i	$0.430131\pi$
$270$	3103.46	0.699521
$271$	2480.09	0.555921	0.277961	−	0.960592i	$-0.410342\pi$
$271$	2480.09	0.555921	0.277961	+	0.960592i	$0.410342\pi$
$272$	1698.03	0.378522
$273$	3057.25	0.677778
$274$	730.048	0.160963
$275$	−3379.67	−0.741098
$276$	−2947.79	−0.642885
$277$	−4766.90	−1.03399	−0.516995	−	0.855989i	$-0.672949\pi$
$277$	−4766.90	−1.03399	−0.516995	+	0.855989i	$0.672949\pi$
$278$	2020.20	0.435840
$279$	1608.00	0.345047
$280$	−1432.36	−0.305715
$281$	194.329	0.0412552	0.0206276	−	0.999787i	$-0.493434\pi$
$281$	194.329	0.0412552	0.0206276	+	0.999787i	$0.493434\pi$
$282$	5945.09	1.25541
$283$	−2817.42	−0.591797	−0.295898	−	0.955219i	$-0.595619\pi$
$283$	−2817.42	−0.591797	−0.295898	+	0.955219i	$0.595619\pi$
$284$	−9.52621	−0.00199041
$285$	15524.0	3.22653
$286$	1182.63	0.244512
$287$	−1856.81	−0.381895
$288$	−500.520	−0.102408
$289$	6349.89	1.29247
$290$	0	0
$291$	−8776.56	−1.76801
$292$	−476.053	−0.0954071
$293$	−4059.12	−0.809339	−0.404670	−	0.914463i	$-0.632614\pi$
$293$	−4059.12	−0.809339	−0.404670	+	0.914463i	$0.632614\pi$
$294$	3523.00	0.698863
$295$	−11676.6	−2.30453
$296$	−845.320	−0.165991
$297$	801.751	0.156641
$298$	−1639.53	−0.318709
$299$	−6173.71	−1.19410
$300$	8166.86	1.57171
$301$	874.921	0.167540
$302$	−2000.50	−0.381178
$303$	−6443.39	−1.22166
$304$	1818.18	0.343025
$305$	9916.01	1.86160
$306$	−3319.91	−0.620218
$307$	131.572	0.0244600	0.0122300	−	0.999925i	$-0.496107\pi$
$307$	131.572	0.0244600	0.0122300	+	0.999925i	$0.496107\pi$
$308$	−370.038	−0.0684574
$309$	3580.23	0.659133
$310$	−4301.45	−0.788083
$311$	−1517.84	−0.276748	−0.138374	−	0.990380i	$-0.544188\pi$
$311$	−1517.84	−0.276748	−0.138374	+	0.990380i	$0.544188\pi$
$312$	−2857.78	−0.518558
$313$	4244.17	0.766436	0.383218	−	0.923658i	$-0.374816\pi$
$313$	4244.17	0.766436	0.383218	+	0.923658i	$0.374816\pi$
$314$	−3404.77	−0.611917
$315$	2800.50	0.500921
$316$	3857.23	0.686664
$317$	−5596.56	−0.991590	−0.495795	−	0.868439i	$-0.665123\pi$
$317$	−5596.56	−0.991590	−0.495795	+	0.868439i	$0.665123\pi$
$318$	7753.94	1.36736
$319$	0	0
$320$	1338.91	0.233898
$321$	9060.50	1.57541
$322$	1931.72	0.334319
$323$	12059.8	2.07748
$324$	−3626.66	−0.621855
$325$	17104.3	2.91930
$326$	−6903.53	−1.17286
$327$	−8445.42	−1.42824
$328$	1735.66	0.292183
$329$	−3895.88	−0.652848
$330$	2953.32	0.492651
$331$	8383.85	1.39220	0.696100	−	0.717945i	$-0.254917\pi$
$331$	8383.85	1.39220	0.696100	+	0.717945i	$0.254917\pi$
$332$	4272.76	0.706319
$333$	1652.73	0.271980
$334$	−2818.45	−0.461733
$335$	4040.41	0.658959
$336$	894.185	0.145184
$337$	−9887.12	−1.59818	−0.799089	−	0.601213i	$-0.794684\pi$
$337$	−9887.12	−1.59818	−0.799089	+	0.601213i	$0.794684\pi$
$338$	−1591.20	−0.256064
$339$	−1977.84	−0.316877
$340$	8880.88	1.41657
$341$	−1111.24	−0.176472
$342$	−3554.83	−0.562056
$343$	−5244.19	−0.825538
$344$	−817.837	−0.128183
$345$	−15417.3	−2.40591
$346$	−2716.78	−0.422124
$347$	10678.8	1.65206	0.826032	−	0.563623i	$-0.190593\pi$
$347$	10678.8	1.65206	0.826032	+	0.563623i	$0.190593\pi$
$348$	0	0
$349$	−1457.88	−0.223605	−0.111803	−	0.993730i	$-0.535662\pi$
$349$	−1457.88	−0.223605	−0.111803	+	0.993730i	$0.535662\pi$
$350$	−5351.83	−0.817335
$351$	−4057.60	−0.617033
$352$	345.895	0.0523758
$353$	6737.20	1.01582	0.507911	−	0.861410i	$-0.330418\pi$
$353$	6737.20	1.01582	0.507911	+	0.861410i	$0.330418\pi$
$354$	7289.35	1.09442
$355$	−49.8232	−0.00744884
$356$	−3090.18	−0.460053
$357$	5931.06	0.879285
$358$	3405.12	0.502699
$359$	3539.85	0.520407	0.260204	−	0.965554i	$-0.416210\pi$
$359$	3539.85	0.520407	0.260204	+	0.965554i	$0.416210\pi$
$360$	−2617.78	−0.383247
$361$	6054.19	0.882663
$362$	14.6740	0.00213051
$363$	−7928.50	−1.14639
$364$	1872.74	0.269665
$365$	−2489.81	−0.357048
$366$	−6190.29	−0.884075
$367$	4917.53	0.699436	0.349718	−	0.936855i	$-0.386277\pi$
$367$	4917.53	0.699436	0.349718	+	0.936855i	$0.386277\pi$
$368$	−1805.69	−0.255782
$369$	−3393.49	−0.478748
$370$	−4421.12	−0.621198
$371$	−5081.24	−0.711064
$372$	2685.27	0.374260
$373$	2032.31	0.282115	0.141057	−	0.990001i	$-0.454950\pi$
$373$	2032.31	0.282115	0.141057	+	0.990001i	$0.454950\pi$
$374$	2294.30	0.317206
$375$	25637.2	3.53040
$376$	3641.69	0.499484
$377$	0	0
$378$	1269.60	0.172755
$379$	7051.47	0.955699	0.477849	−	0.878442i	$-0.341416\pi$
$379$	7051.47	0.955699	0.477849	+	0.878442i	$0.341416\pi$
$380$	9509.29	1.28373
$381$	−13201.7	−1.77517
$382$	2649.56	0.354878
$383$	−9334.72	−1.24538	−0.622691	−	0.782467i	$-0.713961\pi$
$383$	−9334.72	−1.24538	−0.622691	+	0.782467i	$0.713961\pi$
$384$	−835.844	−0.111078
$385$	−1935.34	−0.256193
$386$	3668.94	0.483793
$387$	1599.00	0.210030
$388$	−5376.12	−0.703431
$389$	1901.32	0.247816	0.123908	−	0.992294i	$-0.460457\pi$
$389$	1901.32	0.247816	0.123908	+	0.992294i	$0.460457\pi$
$390$	−14946.5	−1.94063
$391$	−11977.0	−1.54911
$392$	2158.03	0.278054
$393$	−5581.38	−0.716396
$394$	9899.98	1.26587
$395$	20173.7	2.56975
$396$	−676.280	−0.0858190
$397$	1995.81	0.252309	0.126155	−	0.992011i	$-0.459736\pi$
$397$	1995.81	0.252309	0.126155	+	0.992011i	$0.459736\pi$
$398$	−7108.09	−0.895217
$399$	6350.73	0.796828
$400$	5002.65	0.625331
$401$	12920.6	1.60904	0.804520	−	0.593925i	$-0.202422\pi$
$401$	12920.6	1.60904	0.804520	+	0.593925i	$0.202422\pi$
$402$	−2522.32	−0.312939
$403$	5623.90	0.695152
$404$	−3946.93	−0.486057
$405$	−18967.8	−2.32721
$406$	0	0
$407$	−1142.16	−0.139102
$408$	−5544.09	−0.672728
$409$	9713.54	1.17434	0.587168	−	0.809465i	$-0.300243\pi$
$409$	9713.54	1.17434	0.587168	+	0.809465i	$0.300243\pi$
$410$	9077.71	1.09345
$411$	−2383.62	−0.286071
$412$	2193.09	0.262247
$413$	−4776.79	−0.569129
$414$	3530.40	0.419106
$415$	22347.0	2.64330
$416$	−1750.55	−0.206317
$417$	−6595.97	−0.774595
$418$	2456.64	0.287460
$419$	15925.4	1.85682	0.928411	−	0.371555i	$-0.121175\pi$
$419$	15925.4	1.85682	0.928411	+	0.371555i	$0.121175\pi$
$420$	4676.69	0.543331
$421$	−10849.9	−1.25604	−0.628019	−	0.778198i	$-0.716134\pi$
$421$	−10849.9	−1.25604	−0.628019	+	0.778198i	$0.716134\pi$
$422$	4950.45	0.571053
$423$	−7120.09	−0.818417
$424$	4749.71	0.544025
$425$	33182.2	3.78723
$426$	31.1032	0.00353745
$427$	4056.56	0.459744
$428$	5550.05	0.626803
$429$	−3861.30	−0.434558
$430$	−4277.38	−0.479706
$431$	−532.335	−0.0594934	−0.0297467	−	0.999557i	$-0.509470\pi$
$431$	−532.335	−0.0594934	−0.0297467	+	0.999557i	$0.509470\pi$
$432$	−1186.77	−0.132172
$433$	6995.94	0.776451	0.388225	−	0.921564i	$-0.373088\pi$
$433$	6995.94	0.776451	0.388225	+	0.921564i	$0.373088\pi$
$434$	−1759.69	−0.194626
$435$	0	0
$436$	−5173.28	−0.568246
$437$	−12824.5	−1.40384
$438$	1554.32	0.169562
$439$	3272.22	0.355750	0.177875	−	0.984053i	$-0.443078\pi$
$439$	3272.22	0.355750	0.177875	+	0.984053i	$0.443078\pi$
$440$	1809.07	0.196009
$441$	−4219.29	−0.455598
$442$	−11611.3	−1.24953
$443$	3818.14	0.409493	0.204746	−	0.978815i	$-0.434363\pi$
$443$	3818.14	0.409493	0.204746	+	0.978815i	$0.434363\pi$
$444$	2759.98	0.295007
$445$	−16162.0	−1.72169
$446$	−4763.45	−0.505731
$447$	5353.09	0.566426
$448$	547.737	0.0577637
$449$	4323.19	0.454396	0.227198	−	0.973849i	$-0.427043\pi$
$449$	4323.19	0.454396	0.227198	+	0.973849i	$0.427043\pi$
$450$	−9780.97	−1.02462
$451$	2345.14	0.244853
$452$	−1211.53	−0.126075
$453$	6531.66	0.677449
$454$	−10905.5	−1.12736
$455$	9794.62	1.00918
$456$	−5936.38	−0.609641
$457$	−8367.43	−0.856481	−0.428240	−	0.903665i	$-0.640866\pi$
$457$	−8367.43	−0.856481	−0.428240	+	0.903665i	$0.640866\pi$
$458$	1192.46	0.121660
$459$	−7871.73	−0.800481
$460$	−9443.94	−0.957231
$461$	17249.0	1.74266	0.871328	−	0.490701i	$-0.163259\pi$
$461$	17249.0	1.74266	0.871328	+	0.490701i	$0.163259\pi$
$462$	1208.18	0.121666
$463$	−16774.3	−1.68373	−0.841864	−	0.539690i	$-0.818542\pi$
$463$	−16774.3	−1.68373	−0.841864	+	0.539690i	$0.818542\pi$
$464$	0	0
$465$	14044.3	1.40062
$466$	11246.1	1.11795
$467$	−7701.05	−0.763088	−0.381544	−	0.924351i	$-0.624608\pi$
$467$	−7701.05	−0.763088	−0.381544	+	0.924351i	$0.624608\pi$
$468$	3422.60	0.338055
$469$	1652.90	0.162737
$470$	19046.5	1.86925
$471$	11116.6	1.08753
$472$	4465.13	0.435432
$473$	−1105.02	−0.107419
$474$	−12593.9	−1.22037
$475$	35530.1	3.43207
$476$	3633.10	0.349838
$477$	−9286.44	−0.891398
$478$	−3128.55	−0.299365
$479$	−5988.77	−0.571261	−0.285630	−	0.958340i	$-0.592203\pi$
$479$	−5988.77	−0.571261	−0.285630	+	0.958340i	$0.592203\pi$
$480$	−4371.56	−0.415695
$481$	5780.37	0.547946
$482$	1460.65	0.138031
$483$	−6307.09	−0.594167
$484$	−4856.64	−0.456108
$485$	−28117.7	−2.63250
$486$	7835.76	0.731352
$487$	5790.34	0.538779	0.269390	−	0.963031i	$-0.413178\pi$
$487$	5790.34	0.538779	0.269390	+	0.963031i	$0.413178\pi$
$488$	−3791.89	−0.351743
$489$	22540.1	2.08446
$490$	11286.8	1.04058
$491$	−19228.8	−1.76738	−0.883692	−	0.468069i	$-0.844950\pi$
$491$	−19228.8	−1.76738	−0.883692	+	0.468069i	$0.844950\pi$
$492$	−5666.96	−0.519281
$493$	0	0
$494$	−12432.9	−1.13235
$495$	−3537.02	−0.321166
$496$	1644.88	0.148905
$497$	−20.3823	−0.00183958
$498$	−13950.6	−1.25530
$499$	−7081.65	−0.635307	−0.317653	−	0.948207i	$-0.602895\pi$
$499$	−7081.65	−0.635307	−0.317653	+	0.948207i	$0.602895\pi$
$500$	15704.2	1.40463
$501$	9202.29	0.820615
$502$	−8488.79	−0.754727
$503$	5312.64	0.470932	0.235466	−	0.971883i	$-0.424338\pi$
$503$	5312.64	0.470932	0.235466	+	0.971883i	$0.424338\pi$
$504$	−1070.91	−0.0946473
$505$	−20642.9	−1.81900
$506$	−2439.76	−0.214349
$507$	5195.28	0.455089
$508$	−8086.74	−0.706281
$509$	−13862.7	−1.20717	−0.603587	−	0.797297i	$-0.706262\pi$
$509$	−13862.7	−1.20717	−0.603587	+	0.797297i	$0.706262\pi$
$510$	−28996.2	−2.51759
$511$	−1018.56	−0.0881771
$512$	−512.000	−0.0441942
$513$	−8428.72	−0.725414
$514$	470.747	0.0403965
$515$	11470.1	0.981423
$516$	2670.25	0.227812
$517$	4920.49	0.418574
$518$	−1808.65	−0.153412
$519$	8870.32	0.750219
$520$	−9155.57	−0.772112
$521$	15447.5	1.29897	0.649487	−	0.760373i	$-0.274984\pi$
$521$	15447.5	1.29897	0.649487	+	0.760373i	$0.274984\pi$
$522$	0	0
$523$	4349.54	0.363656	0.181828	−	0.983330i	$-0.441799\pi$
$523$	4349.54	0.363656	0.181828	+	0.983330i	$0.441799\pi$
$524$	−3418.90	−0.285030
$525$	17473.8	1.45261
$526$	6907.26	0.572568
$527$	10910.3	0.901825
$528$	−1129.35	−0.0930848
$529$	569.329	0.0467928
$530$	24841.5	2.03594
$531$	−8730.03	−0.713467
$532$	3890.17	0.317031
$533$	−11868.6	−0.964514
$534$	10089.5	0.817629
$535$	29027.4	2.34573
$536$	−1545.06	−0.124508
$537$	−11117.8	−0.893422
$538$	−3842.62	−0.307932
$539$	2915.83	0.233012
$540$	−6206.92	−0.494636
$541$	−3206.84	−0.254848	−0.127424	−	0.991848i	$-0.540671\pi$
$541$	−3206.84	−0.254848	−0.127424	+	0.991848i	$0.540671\pi$
$542$	−4960.18	−0.393096
$543$	−47.9107	−0.00378645
$544$	−3396.06	−0.267656
$545$	−27056.9	−2.12658
$546$	−6114.51	−0.479261
$547$	−3289.81	−0.257152	−0.128576	−	0.991700i	$-0.541041\pi$
$547$	−3289.81	−0.257152	−0.128576	+	0.991700i	$0.541041\pi$
$548$	−1460.10	−0.113818
$549$	7413.74	0.576340
$550$	6759.34	0.524035
$551$	0	0
$552$	5895.59	0.454589
$553$	8252.91	0.634628
$554$	9533.79	0.731141
$555$	14435.0	1.10402
$556$	−4040.40	−0.308185
$557$	−313.140	−0.0238207	−0.0119104	−	0.999929i	$-0.503791\pi$
$557$	−313.140	−0.0238207	−0.0119104	+	0.999929i	$0.503791\pi$
$558$	−3215.99	−0.243985
$559$	5592.43	0.423139
$560$	2864.73	0.216173
$561$	−7490.91	−0.563755
$562$	−388.659	−0.0291718
$563$	3425.84	0.256451	0.128225	−	0.991745i	$-0.459072\pi$
$563$	3425.84	0.256451	0.128225	+	0.991745i	$0.459072\pi$
$564$	−11890.2	−0.887707
$565$	−6336.46	−0.471818
$566$	5634.85	0.418463
$567$	−7759.60	−0.574731
$568$	19.0524	0.00140743
$569$	−17763.8	−1.30878	−0.654390	−	0.756158i	$-0.727074\pi$
$569$	−17763.8	−1.30878	−0.654390	+	0.756158i	$0.727074\pi$
$570$	−31047.9	−2.28150
$571$	5741.80	0.420818	0.210409	−	0.977613i	$-0.432520\pi$
$571$	5741.80	0.420818	0.210409	+	0.977613i	$0.432520\pi$
$572$	−2365.26	−0.172896
$573$	−8650.85	−0.630706
$574$	3713.62	0.270041
$575$	−35286.0	−2.55918
$576$	1001.04	0.0724132
$577$	−14477.5	−1.04455	−0.522275	−	0.852777i	$-0.674917\pi$
$577$	−14477.5	−1.04455	−0.522275	+	0.852777i	$0.674917\pi$
$578$	−12699.8	−0.913912
$579$	−11979.1	−0.859821
$580$	0	0
$581$	9141.98	0.652794
$582$	17553.1	1.25017
$583$	6417.59	0.455899
$584$	952.105	0.0674630
$585$	17900.6	1.26512
$586$	8118.24	0.572289
$587$	−18082.8	−1.27148	−0.635738	−	0.771905i	$-0.719304\pi$
$587$	−18082.8	−1.27148	−0.635738	+	0.771905i	$0.719304\pi$
$588$	−7046.00	−0.494170
$589$	11682.3	0.817254
$590$	23353.1	1.62955
$591$	−32323.6	−2.24977
$592$	1690.64	0.117373
$593$	1731.57	0.119911	0.0599553	−	0.998201i	$-0.480904\pi$
$593$	1731.57	0.119911	0.0599553	+	0.998201i	$0.480904\pi$
$594$	−1603.50	−0.110762
$595$	19001.5	1.30922
$596$	3279.06	0.225362
$597$	23208.0	1.59102
$598$	12347.4	0.844354
$599$	11480.5	0.783105	0.391552	−	0.920156i	$-0.371938\pi$
$599$	11480.5	0.783105	0.391552	+	0.920156i	$0.371938\pi$
$600$	−16333.7	−1.11137
$601$	−2924.56	−0.198495	−0.0992474	−	0.995063i	$-0.531644\pi$
$601$	−2924.56	−0.198495	−0.0992474	+	0.995063i	$0.531644\pi$
$602$	−1749.84	−0.118469
$603$	3020.83	0.204009
$604$	4001.00	0.269534
$605$	−25400.8	−1.70692
$606$	12886.8	0.863845
$607$	−3586.09	−0.239794	−0.119897	−	0.992786i	$-0.538256\pi$
$607$	−3586.09	−0.239794	−0.119897	+	0.992786i	$0.538256\pi$
$608$	−3636.36	−0.242556
$609$	0	0
$610$	−19832.0	−1.31635
$611$	−24902.2	−1.64883
$612$	6639.83	0.438561
$613$	−11679.8	−0.769561	−0.384781	−	0.923008i	$-0.625723\pi$
$613$	−11679.8	−0.769561	−0.384781	+	0.923008i	$0.625723\pi$
$614$	−263.145	−0.0172958
$615$	−29638.8	−1.94334
$616$	740.077	0.0484067
$617$	−11063.0	−0.721847	−0.360924	−	0.932595i	$-0.617539\pi$
$617$	−11063.0	−0.721847	−0.360924	+	0.932595i	$0.617539\pi$
$618$	−7160.46	−0.466078
$619$	2463.60	0.159969	0.0799843	−	0.996796i	$-0.474513\pi$
$619$	2463.60	0.159969	0.0799843	+	0.996796i	$0.474513\pi$
$620$	8602.89	0.557259
$621$	8370.81	0.540916
$622$	3035.67	0.195690
$623$	−6611.73	−0.425190
$624$	5715.57	0.366676
$625$	43051.5	2.75530
$626$	−8488.34	−0.541952
$627$	−8020.96	−0.510887
$628$	6809.53	0.432691
$629$	11213.9	0.710854
$630$	−5600.99	−0.354205
$631$	1032.43	0.0651352	0.0325676	−	0.999470i	$-0.489632\pi$
$631$	1032.43	0.0651352	0.0325676	+	0.999470i	$0.489632\pi$
$632$	−7714.45	−0.485545
$633$	−16163.3	−1.01490
$634$	11193.1	0.701160
$635$	−42294.6	−2.64316
$636$	−15507.9	−0.966867
$637$	−14756.8	−0.917873
$638$	0	0
$639$	−37.2505	−0.00230611
$640$	−2677.82	−0.165391
$641$	−12841.6	−0.791282	−0.395641	−	0.918405i	$-0.629478\pi$
$641$	−12841.6	−0.791282	−0.395641	+	0.918405i	$0.629478\pi$
$642$	−18121.0	−1.11399
$643$	8449.14	0.518198	0.259099	−	0.965851i	$-0.416574\pi$
$643$	8449.14	0.518198	0.259099	+	0.965851i	$0.416574\pi$
$644$	−3863.44	−0.236399
$645$	13965.7	0.852557
$646$	−24119.7	−1.46900
$647$	−27036.8	−1.64285	−0.821426	−	0.570315i	$-0.806821\pi$
$647$	−27036.8	−1.64285	−0.821426	+	0.570315i	$0.806821\pi$
$648$	7253.32	0.439718
$649$	6033.07	0.364898
$650$	−34208.5	−2.06426
$651$	5745.40	0.345899
$652$	13807.1	0.829334
$653$	27105.2	1.62436	0.812181	−	0.583405i	$-0.198280\pi$
$653$	27105.2	1.62436	0.812181	+	0.583405i	$0.198280\pi$
$654$	16890.8	1.00991
$655$	−17881.3	−1.06668
$656$	−3471.32	−0.206604
$657$	−1861.52	−0.110540
$658$	7791.76	0.461633
$659$	22622.4	1.33724	0.668621	−	0.743603i	$-0.266885\pi$
$659$	22622.4	1.33724	0.668621	+	0.743603i	$0.266885\pi$
$660$	−5906.64	−0.348357
$661$	−21000.2	−1.23572	−0.617862	−	0.786287i	$-0.712001\pi$
$661$	−21000.2	−1.23572	−0.617862	+	0.786287i	$0.712001\pi$
$662$	−16767.7	−0.984434
$663$	37910.9	2.22072
$664$	−8545.51	−0.499443
$665$	20346.0	1.18645
$666$	−3305.47	−0.192319
$667$	0	0
$668$	5636.91	0.326495
$669$	15552.7	0.898809
$670$	−8080.82	−0.465954
$671$	−5123.42	−0.294765
$672$	−1788.37	−0.102660
$673$	−13691.2	−0.784186	−0.392093	−	0.919926i	$-0.628249\pi$
$673$	−13691.2	−0.784186	−0.392093	+	0.919926i	$0.628249\pi$
$674$	19774.2	1.13008
$675$	−23191.3	−1.32242
$676$	3182.39	0.181065
$677$	−9694.60	−0.550360	−0.275180	−	0.961393i	$-0.588737\pi$
$677$	−9694.60	−0.550360	−0.275180	+	0.961393i	$0.588737\pi$
$678$	3955.68	0.224066
$679$	−11502.7	−0.650125
$680$	−17761.8	−1.00167
$681$	35606.5	2.00359
$682$	2222.48	0.124785
$683$	6012.25	0.336826	0.168413	−	0.985717i	$-0.446136\pi$
$683$	6012.25	0.336826	0.168413	+	0.985717i	$0.446136\pi$
$684$	7109.65	0.397433
$685$	−7636.47	−0.425948
$686$	10488.4	0.583744
$687$	−3893.41	−0.216220
$688$	1635.67	0.0906388
$689$	−32478.9	−1.79586
$690$	30834.6	1.70124
$691$	−10304.9	−0.567317	−0.283659	−	0.958925i	$-0.591548\pi$
$691$	−10304.9	−0.567317	−0.283659	+	0.958925i	$0.591548\pi$
$692$	5433.55	0.298487
$693$	−1446.97	−0.0793156
$694$	−21357.5	−1.16819
$695$	−21131.7	−1.15334
$696$	0	0
$697$	−23025.0	−1.25127
$698$	2915.75	0.158113
$699$	−36718.6	−1.98687
$700$	10703.7	0.577943
$701$	11785.8	0.635011	0.317506	−	0.948256i	$-0.397155\pi$
$701$	11785.8	0.635011	0.317506	+	0.948256i	$0.397155\pi$
$702$	8115.20	0.436308
$703$	12007.4	0.644192
$704$	−691.790	−0.0370353
$705$	−62187.0	−3.32212
$706$	−13474.4	−0.718294
$707$	−8444.85	−0.449224
$708$	−14578.7	−0.773872
$709$	29226.1	1.54811	0.774054	−	0.633120i	$-0.218226\pi$
$709$	29226.1	1.54811	0.774054	+	0.633120i	$0.218226\pi$
$710$	99.6463	0.00526713
$711$	15083.0	0.795577
$712$	6180.35	0.325307
$713$	−11602.1	−0.609398
$714$	−11862.1	−0.621749
$715$	−12370.6	−0.647040
$716$	−6810.25	−0.355462
$717$	10214.7	0.532045
$718$	−7079.70	−0.367983
$719$	14706.5	0.762809	0.381404	−	0.924408i	$-0.375441\pi$
$719$	14706.5	0.762809	0.381404	+	0.924408i	$0.375441\pi$
$720$	5235.56	0.270997
$721$	4692.32	0.242374
$722$	−12108.4	−0.624137
$723$	−4769.05	−0.245315
$724$	−29.3479	−0.00150650
$725$	0	0
$726$	15857.0	0.810618
$727$	35975.1	1.83527	0.917637	−	0.397419i	$-0.130094\pi$
$727$	35975.1	1.83527	0.917637	+	0.397419i	$0.130094\pi$
$728$	−3745.47	−0.190682
$729$	−1103.91	−0.0560843
$730$	4979.62	0.252471
$731$	10849.3	0.548941
$732$	12380.6	0.625135
$733$	−10872.8	−0.547879	−0.273939	−	0.961747i	$-0.588327\pi$
$733$	−10872.8	−0.547879	−0.273939	+	0.961747i	$0.588327\pi$
$734$	−9835.06	−0.494576
$735$	−36851.4	−1.84937
$736$	3611.37	0.180865
$737$	−2087.61	−0.104339
$738$	6786.98	0.338526
$739$	1078.25	0.0536727	0.0268363	−	0.999640i	$-0.491457\pi$
$739$	1078.25	0.0536727	0.0268363	+	0.999640i	$0.491457\pi$
$740$	8842.25	0.439253
$741$	40593.4	2.01247
$742$	10162.5	0.502798
$743$	23176.4	1.14436	0.572180	−	0.820128i	$-0.306098\pi$
$743$	23176.4	1.14436	0.572180	+	0.820128i	$0.306098\pi$
$744$	−5370.54	−0.264642
$745$	17149.9	0.843385
$746$	−4064.61	−0.199485
$747$	16707.8	0.818350
$748$	−4588.59	−0.224299
$749$	11874.9	0.579304
$750$	−51274.4	−2.49637
$751$	−8738.85	−0.424614	−0.212307	−	0.977203i	$-0.568098\pi$
$751$	−8738.85	−0.424614	−0.212307	+	0.977203i	$0.568098\pi$
$752$	−7283.39	−0.353189
$753$	27716.0	1.34134
$754$	0	0
$755$	20925.7	1.00869
$756$	−2539.20	−0.122156
$757$	−4121.15	−0.197868	−0.0989338	−	0.995094i	$-0.531543\pi$
$757$	−4121.15	−0.197868	−0.0989338	+	0.995094i	$0.531543\pi$
$758$	−14102.9	−0.675781
$759$	7965.84	0.380951
$760$	−19018.6	−0.907732
$761$	8706.54	0.414733	0.207367	−	0.978263i	$-0.433511\pi$
$761$	8706.54	0.414733	0.207367	+	0.978263i	$0.433511\pi$
$762$	26403.3	1.25524
$763$	−11068.7	−0.525184
$764$	−5299.12	−0.250936
$765$	34727.0	1.64125
$766$	18669.4	0.880619
$767$	−30532.9	−1.43739
$768$	1671.69	0.0785440
$769$	32258.8	1.51272	0.756361	−	0.654154i	$-0.226975\pi$
$769$	32258.8	1.51272	0.756361	+	0.654154i	$0.226975\pi$
$770$	3870.69	0.181156
$771$	−1537.00	−0.0717945
$772$	−7337.88	−0.342094
$773$	−24794.1	−1.15366	−0.576832	−	0.816863i	$-0.695711\pi$
$773$	−24794.1	−1.15366	−0.576832	+	0.816863i	$0.695711\pi$
$774$	−3198.00	−0.148514
$775$	32143.5	1.48984
$776$	10752.2	0.497401
$777$	5905.25	0.272651
$778$	−3802.63	−0.175233
$779$	−24654.2	−1.13393
$780$	29893.1	1.37223
$781$	25.7427	0.00117945
$782$	23953.9	1.09539
$783$	0	0
$784$	−4316.06	−0.196614
$785$	35614.6	1.61929
$786$	11162.8	0.506568
$787$	−11114.7	−0.503427	−0.251714	−	0.967802i	$-0.580994\pi$
$787$	−11114.7	−0.503427	−0.251714	+	0.967802i	$0.580994\pi$
$788$	−19800.0	−0.895107
$789$	−22552.3	−1.01760
$790$	−40347.5	−1.81709
$791$	−2592.20	−0.116521
$792$	1352.56	0.0606832
$793$	25929.2	1.16113
$794$	−3991.62	−0.178410
$795$	−81108.0	−3.61837
$796$	14216.2	0.633014
$797$	12329.8	0.547986	0.273993	−	0.961732i	$-0.411656\pi$
$797$	12329.8	0.547986	0.273993	+	0.961732i	$0.411656\pi$
$798$	−12701.5	−0.563442
$799$	−48310.1	−2.13904
$800$	−10005.3	−0.442176
$801$	−12083.6	−0.533023
$802$	−25841.3	−1.13776
$803$	1286.44	0.0565348
$804$	5044.63	0.221282
$805$	−20206.2	−0.884691
$806$	−11247.8	−0.491547
$807$	12546.2	0.547271
$808$	7893.86	0.343695
$809$	36175.5	1.57214	0.786070	−	0.618138i	$-0.212113\pi$
$809$	36175.5	1.57214	0.786070	+	0.618138i	$0.212113\pi$
$810$	37935.7	1.64559
$811$	−26581.0	−1.15091	−0.575453	−	0.817835i	$-0.695174\pi$
$811$	−26581.0	−1.15091	−0.575453	+	0.817835i	$0.695174\pi$
$812$	0	0
$813$	16195.0	0.698629
$814$	2284.31	0.0983602
$815$	72212.5	3.10367
$816$	11088.2	0.475691
$817$	11617.0	0.497462
$818$	−19427.1	−0.830381
$819$	7322.98	0.312437
$820$	−18155.4	−0.773189
$821$	10811.1	0.459574	0.229787	−	0.973241i	$-0.426197\pi$
$821$	10811.1	0.459574	0.229787	+	0.973241i	$0.426197\pi$
$822$	4767.24	0.202283
$823$	−28625.1	−1.21240	−0.606202	−	0.795311i	$-0.707308\pi$
$823$	−28625.1	−1.21240	−0.606202	+	0.795311i	$0.707308\pi$
$824$	−4386.17	−0.185436
$825$	−22069.3	−0.931341
$826$	9553.58	0.402435
$827$	9970.68	0.419244	0.209622	−	0.977783i	$-0.432777\pi$
$827$	9970.68	0.419244	0.209622	+	0.977783i	$0.432777\pi$
$828$	−7060.80	−0.296352
$829$	−20201.0	−0.846334	−0.423167	−	0.906052i	$-0.639082\pi$
$829$	−20201.0	−0.846334	−0.423167	+	0.906052i	$0.639082\pi$
$830$	−44694.0	−1.86910
$831$	−31128.0	−1.29942
$832$	3501.10	0.145888
$833$	−28628.1	−1.19076
$834$	13191.9	0.547722
$835$	29481.7	1.22186
$836$	−4913.28	−0.203265
$837$	−7625.33	−0.314898
$838$	−31850.9	−1.31297
$839$	−28023.4	−1.15313	−0.576564	−	0.817052i	$-0.695607\pi$
$839$	−28023.4	−1.15313	−0.576564	+	0.817052i	$0.695607\pi$
$840$	−9353.38	−0.384193
$841$	0	0
$842$	21699.8	0.888153
$843$	1268.98	0.0518456
$844$	−9900.90	−0.403795
$845$	16644.3	0.677610
$846$	14240.2	0.578708
$847$	−10391.3	−0.421544
$848$	−9499.42	−0.384683
$849$	−18397.9	−0.743714
$850$	−66364.4	−2.67797
$851$	−11924.9	−0.480352
$852$	−62.2064	−0.00250136
$853$	26333.8	1.05704	0.528518	−	0.848922i	$-0.322748\pi$
$853$	26333.8	1.05704	0.528518	+	0.848922i	$0.322748\pi$
$854$	−8113.12	−0.325088
$855$	37184.3	1.48734
$856$	−11100.1	−0.443217
$857$	36388.6	1.45042	0.725211	−	0.688527i	$-0.241742\pi$
$857$	36388.6	1.45042	0.725211	+	0.688527i	$0.241742\pi$
$858$	7722.60	0.307279
$859$	−14974.0	−0.594770	−0.297385	−	0.954758i	$-0.596115\pi$
$859$	−14974.0	−0.594770	−0.297385	+	0.954758i	$0.596115\pi$
$860$	8554.76	0.339203
$861$	−12125.0	−0.479930
$862$	1064.67	0.0420682
$863$	−40168.2	−1.58440	−0.792202	−	0.610259i	$-0.791065\pi$
$863$	−40168.2	−1.58440	−0.792202	+	0.610259i	$0.791065\pi$
$864$	2373.53	0.0934597
$865$	28418.1	1.11705
$866$	−13991.9	−0.549034
$867$	41465.0	1.62425
$868$	3519.37	0.137621
$869$	−10423.4	−0.406893
$870$	0	0
$871$	10565.2	0.411009
$872$	10346.6	0.401811
$873$	−21022.3	−0.815004
$874$	25648.9	0.992663
$875$	33600.7	1.29818
$876$	−3108.64	−0.119899
$877$	32161.7	1.23834	0.619169	−	0.785258i	$-0.287469\pi$
$877$	32161.7	1.23834	0.619169	+	0.785258i	$0.287469\pi$
$878$	−6544.43	−0.251553
$879$	−26506.2	−1.01710
$880$	−3618.14	−0.138599
$881$	8870.49	0.339222	0.169611	−	0.985511i	$-0.445749\pi$
$881$	8870.49	0.339222	0.169611	+	0.985511i	$0.445749\pi$
$882$	8438.58	0.322156
$883$	−24647.3	−0.939351	−0.469675	−	0.882839i	$-0.655629\pi$
$883$	−24647.3	−0.939351	−0.469675	+	0.882839i	$0.655629\pi$
$884$	23222.5	0.883549
$885$	−76248.3	−2.89611
$886$	−7636.28	−0.289555
$887$	−32371.7	−1.22541	−0.612703	−	0.790313i	$-0.709918\pi$
$887$	−32371.7	−1.22541	−0.612703	+	0.790313i	$0.709918\pi$
$888$	−5519.97	−0.208601
$889$	−17302.4	−0.652759
$890$	32323.9	1.21742
$891$	9800.35	0.368489
$892$	9526.90	0.357605
$893$	−51728.5	−1.93844
$894$	−10706.2	−0.400523
$895$	−35618.4	−1.33027
$896$	−1095.47	−0.0408451
$897$	−40314.5	−1.50063
$898$	−8646.38	−0.321307
$899$	0	0
$900$	19561.9	0.724516
$901$	−63008.9	−2.32978
$902$	−4690.29	−0.173137
$903$	5713.26	0.210549
$904$	2423.07	0.0891483
$905$	−153.493	−0.00563787
$906$	−13063.3	−0.479029
$907$	43728.0	1.60084	0.800422	−	0.599437i	$-0.204609\pi$
$907$	43728.0	1.60084	0.800422	+	0.599437i	$0.204609\pi$
$908$	21810.9	0.797160
$909$	−15433.7	−0.563152
$910$	−19589.2	−0.713601
$911$	−16033.6	−0.583115	−0.291557	−	0.956553i	$-0.594173\pi$
$911$	−16033.6	−0.583115	−0.291557	+	0.956553i	$0.594173\pi$
$912$	11872.8	0.431082
$913$	−11546.3	−0.418540
$914$	16734.9	0.605623
$915$	64751.8	2.33948
$916$	−2384.93	−0.0860264
$917$	−7315.08	−0.263430
$918$	15743.5	0.566026
$919$	29331.1	1.05282	0.526411	−	0.850230i	$-0.323537\pi$
$919$	29331.1	1.05282	0.526411	+	0.850230i	$0.323537\pi$
$920$	18887.9	0.676864
$921$	859.170	0.0307390
$922$	−34497.9	−1.23224
$923$	−130.282	−0.00464603
$924$	−2416.36	−0.0860308
$925$	33037.8	1.17435
$926$	33548.5	1.19058
$927$	8575.66	0.303842
$928$	0	0
$929$	−10024.6	−0.354034	−0.177017	−	0.984208i	$-0.556645\pi$
$929$	−10024.6	−0.354034	−0.177017	+	0.984208i	$0.556645\pi$
$930$	−28088.6	−0.990387
$931$	−30653.8	−1.07910
$932$	−22492.1	−0.790509
$933$	−9911.51	−0.347790
$934$	15402.1	0.539585
$935$	−23998.9	−0.839408
$936$	−6845.20	−0.239041
$937$	21241.3	0.740578	0.370289	−	0.928917i	$-0.379259\pi$
$937$	21241.3	0.740578	0.370289	+	0.928917i	$0.379259\pi$
$938$	−3305.80	−0.115073
$939$	27714.5	0.963184
$940$	−38092.9	−1.32176
$941$	11953.8	0.414115	0.207057	−	0.978329i	$-0.433611\pi$
$941$	11953.8	0.414115	0.207057	+	0.978329i	$0.433611\pi$
$942$	−22233.2	−0.768999
$943$	24484.8	0.845531
$944$	−8930.25	−0.307897
$945$	−13280.3	−0.457152
$946$	2210.05	0.0759564
$947$	31650.6	1.08607	0.543034	−	0.839711i	$-0.317275\pi$
$947$	31650.6	1.08607	0.543034	+	0.839711i	$0.317275\pi$
$948$	25187.8	0.862934
$949$	−6510.57	−0.222700
$950$	−71060.2	−2.42684
$951$	−36545.7	−1.24614
$952$	−7266.20	−0.247373
$953$	−17373.5	−0.590539	−0.295269	−	0.955414i	$-0.595409\pi$
$953$	−17373.5	−0.590539	−0.295269	+	0.955414i	$0.595409\pi$
$954$	18572.9	0.630313
$955$	−27715.0	−0.939095
$956$	6257.09	0.211683
$957$	0	0
$958$	11977.5	0.403942
$959$	−3124.02	−0.105193
$960$	8743.11	0.293940
$961$	−19222.2	−0.645234
$962$	−11560.7	−0.387457
$963$	21702.4	0.726222
$964$	−2921.30	−0.0976026
$965$	−38378.0	−1.28024
$966$	12614.2	0.420140
$967$	11017.6	0.366394	0.183197	−	0.983076i	$-0.441355\pi$
$967$	11017.6	0.366394	0.183197	+	0.983076i	$0.441355\pi$
$968$	9713.29	0.322517
$969$	78751.1	2.61078
$970$	56235.5	1.86146
$971$	−17365.0	−0.573912	−0.286956	−	0.957944i	$-0.592643\pi$
$971$	−17365.0	−0.573912	−0.286956	+	0.957944i	$0.592643\pi$
$972$	−15671.5	−0.517144
$973$	−8644.82	−0.284831
$974$	−11580.7	−0.380975
$975$	111691.	3.66870
$976$	7583.78	0.248720
$977$	−29193.4	−0.955967	−0.477983	−	0.878369i	$-0.658632\pi$
$977$	−29193.4	−0.955967	−0.477983	+	0.878369i	$0.658632\pi$
$978$	−45080.2	−1.47393
$979$	8350.60	0.272611
$980$	−22573.5	−0.735800
$981$	−20229.2	−0.658377
$982$	38457.7	1.24973
$983$	−50359.8	−1.63401	−0.817003	−	0.576634i	$-0.804366\pi$
$983$	−50359.8	−1.63401	−0.817003	+	0.576634i	$0.804366\pi$
$984$	11333.9	0.367187
$985$	−103556.	−3.34982
$986$	0	0
$987$	−25440.2	−0.820436
$988$	24865.7	0.800692
$989$	−11537.2	−0.370941
$990$	7074.04	0.227099
$991$	−20823.5	−0.667488	−0.333744	−	0.942664i	$-0.608312\pi$
$991$	−20823.5	−0.667488	−0.333744	+	0.942664i	$0.608312\pi$
$992$	−3289.75	−0.105292
$993$	54746.8	1.74958
$994$	40.7645	0.00130078
$995$	74352.2	2.36897
$996$	27901.2	0.887635
$997$	27651.2	0.878356	0.439178	−	0.898400i	$-0.355270\pi$
$997$	27651.2	0.878356	0.439178	+	0.898400i	$0.355270\pi$
$998$	14163.3	0.449230
$999$	−7837.48	−0.248215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.4.a.d.1.3		3
29.28	even	2		58.4.a.d.1.1	✓	3
87.86	odd	2		522.4.a.k.1.1		3
116.115	odd	2		464.4.a.i.1.3		3
145.144	even	2		1450.4.a.h.1.3		3
232.115	odd	2		1856.4.a.s.1.1		3
232.173	even	2		1856.4.a.r.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.4.a.d.1.1	✓	3	29.28	even	2
464.4.a.i.1.3		3	116.115	odd	2
522.4.a.k.1.1		3	87.86	odd	2
1450.4.a.h.1.3		3	145.144	even	2
1682.4.a.d.1.3		3	1.1	even	1	trivial
1856.4.a.r.1.3		3	232.173	even	2
1856.4.a.s.1.1		3	232.115	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:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1682.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +6.53003 q^{3} +4.00000 q^{4} +20.9205 q^{5} -13.0601 q^{6} +8.55839 q^{7} -8.00000 q^{8} +15.6413 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +6.53003 q^{3} +4.00000 q^{4} +20.9205 q^{5} -13.0601 q^{6} +8.55839 q^{7} -8.00000 q^{8} +15.6413 q^{9} -41.8409 q^{10} -10.8092 q^{11} +26.1201 q^{12} +54.7046 q^{13} -17.1168 q^{14} +136.611 q^{15} +16.0000 q^{16} +106.127 q^{17} -31.2825 q^{18} +113.636 q^{19} +83.6818 q^{20} +55.8865 q^{21} +21.6184 q^{22} -112.855 q^{23} -52.2402 q^{24} +312.666 q^{25} -109.409 q^{26} -74.1729 q^{27} +34.2336 q^{28} -273.222 q^{30} +102.805 q^{31} -32.0000 q^{32} -70.5845 q^{33} -212.254 q^{34} +179.045 q^{35} +62.5650 q^{36} +105.665 q^{37} -227.272 q^{38} +357.223 q^{39} -167.364 q^{40} -216.958 q^{41} -111.773 q^{42} +102.230 q^{43} -43.2369 q^{44} +327.222 q^{45} +225.711 q^{46} -455.212 q^{47} +104.480 q^{48} -269.754 q^{49} -625.331 q^{50} +693.011 q^{51} +218.819 q^{52} -593.714 q^{53} +148.346 q^{54} -226.134 q^{55} -68.4671 q^{56} +742.048 q^{57} -558.141 q^{59} +546.445 q^{60} +473.986 q^{61} -205.610 q^{62} +133.864 q^{63} +64.0000 q^{64} +1144.45 q^{65} +141.169 q^{66} +193.132 q^{67} +424.507 q^{68} -736.949 q^{69} -358.091 q^{70} -2.38155 q^{71} -125.130 q^{72} -119.013 q^{73} -211.330 q^{74} +2041.71 q^{75} +454.545 q^{76} -92.5096 q^{77} -714.446 q^{78} +964.306 q^{79} +334.727 q^{80} -906.665 q^{81} +433.915 q^{82} +1068.19 q^{83} +223.546 q^{84} +2220.22 q^{85} -204.459 q^{86} +86.4738 q^{88} -772.544 q^{89} -654.445 q^{90} +468.184 q^{91} -451.421 q^{92} +671.318 q^{93} +910.423 q^{94} +2377.32 q^{95} -208.961 q^{96} -1344.03 q^{97} +539.508 q^{98} -169.070 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} - 24 q^{8} + 5 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 20 * q^5 + 4 * q^6 + 24 * q^7 - 24 * q^8 + 5 * q^9	\( 3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} - 24 q^{8} + 5 q^{9} - 40 q^{10} - 10 q^{11} - 8 q^{12} - 4 q^{13} - 48 q^{14} + 130 q^{15} + 48 q^{16} + 66 q^{17} - 10 q^{18} + 164 q^{19} + 80 q^{20} + 88 q^{21} + 20 q^{22} - 204 q^{23} + 16 q^{24} + 79 q^{25} + 8 q^{26} + 142 q^{27} + 96 q^{28} - 260 q^{30} + 86 q^{31} - 96 q^{32} - 130 q^{33} - 132 q^{34} + 24 q^{35} + 20 q^{36} + 42 q^{37} - 328 q^{38} + 394 q^{39} - 160 q^{40} - 562 q^{41} - 176 q^{42} - 18 q^{43} - 40 q^{44} + 422 q^{45} + 408 q^{46} - 654 q^{47} - 32 q^{48} + 539 q^{49} - 158 q^{50} + 556 q^{51} - 16 q^{52} + 712 q^{53} - 284 q^{54} - 142 q^{55} - 192 q^{56} + 828 q^{57} + 184 q^{59} + 520 q^{60} - 322 q^{61} - 172 q^{62} - 784 q^{63} + 192 q^{64} + 1494 q^{65} + 260 q^{66} - 228 q^{67} + 264 q^{68} - 684 q^{69} - 48 q^{70} - 52 q^{71} - 40 q^{72} + 494 q^{73} - 84 q^{74} + 3048 q^{75} + 656 q^{76} - 872 q^{77} - 788 q^{78} + 2110 q^{79} + 320 q^{80} - 1513 q^{81} + 1124 q^{82} - 288 q^{83} + 352 q^{84} + 2704 q^{85} + 36 q^{86} + 80 q^{88} - 914 q^{89} - 844 q^{90} - 2984 q^{91} - 816 q^{92} - 62 q^{93} + 1308 q^{94} + 1900 q^{95} + 64 q^{96} - 218 q^{97} - 1078 q^{98} + 304 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 20 * q^5 + 4 * q^6 + 24 * q^7 - 24 * q^8 + 5 * q^9 - 40 * q^10 - 10 * q^11 - 8 * q^12 - 4 * q^13 - 48 * q^14 + 130 * q^15 + 48 * q^16 + 66 * q^17 - 10 * q^18 + 164 * q^19 + 80 * q^20 + 88 * q^21 + 20 * q^22 - 204 * q^23 + 16 * q^24 + 79 * q^25 + 8 * q^26 + 142 * q^27 + 96 * q^28 - 260 * q^30 + 86 * q^31 - 96 * q^32 - 130 * q^33 - 132 * q^34 + 24 * q^35 + 20 * q^36 + 42 * q^37 - 328 * q^38 + 394 * q^39 - 160 * q^40 - 562 * q^41 - 176 * q^42 - 18 * q^43 - 40 * q^44 + 422 * q^45 + 408 * q^46 - 654 * q^47 - 32 * q^48 + 539 * q^49 - 158 * q^50 + 556 * q^51 - 16 * q^52 + 712 * q^53 - 284 * q^54 - 142 * q^55 - 192 * q^56 + 828 * q^57 + 184 * q^59 + 520 * q^60 - 322 * q^61 - 172 * q^62 - 784 * q^63 + 192 * q^64 + 1494 * q^65 + 260 * q^66 - 228 * q^67 + 264 * q^68 - 684 * q^69 - 48 * q^70 - 52 * q^71 - 40 * q^72 + 494 * q^73 - 84 * q^74 + 3048 * q^75 + 656 * q^76 - 872 * q^77 - 788 * q^78 + 2110 * q^79 + 320 * q^80 - 1513 * q^81 + 1124 * q^82 - 288 * q^83 + 352 * q^84 + 2704 * q^85 + 36 * q^86 + 80 * q^88 - 914 * q^89 - 844 * q^90 - 2984 * q^91 - 816 * q^92 - 62 * q^93 + 1308 * q^94 + 1900 * q^95 + 64 * q^96 - 218 * q^97 - 1078 * q^98 + 304 * q^99

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