LMFDB - Embedded newform 354.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,4,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("354.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	354.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:	$20.8866761420$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.30645.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 33x - 28 $ x^3 - 33*x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.12929$ of defining polynomial
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -17.5682 q^{5} +6.00000 q^{6} +6.71947 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -17.5682 q^{5} +6.00000 q^{6} +6.71947 q^{7} -8.00000 q^{8} +9.00000 q^{9} +35.1365 q^{10} +31.2147 q^{11} -12.0000 q^{12} +16.7996 q^{13} -13.4389 q^{14} +52.7047 q^{15} +16.0000 q^{16} +13.1512 q^{17} -18.0000 q^{18} +56.5534 q^{19} -70.2729 q^{20} -20.1584 q^{21} -62.4294 q^{22} +5.31487 q^{23} +24.0000 q^{24} +183.643 q^{25} -33.5993 q^{26} -27.0000 q^{27} +26.8779 q^{28} -284.357 q^{29} -105.409 q^{30} +19.1894 q^{31} -32.0000 q^{32} -93.6441 q^{33} -26.3025 q^{34} -118.049 q^{35} +36.0000 q^{36} -95.2199 q^{37} -113.107 q^{38} -50.3989 q^{39} +140.546 q^{40} +31.8768 q^{41} +40.3168 q^{42} +449.715 q^{43} +124.859 q^{44} -158.114 q^{45} -10.6297 q^{46} -400.529 q^{47} -48.0000 q^{48} -297.849 q^{49} -367.285 q^{50} -39.4537 q^{51} +67.1985 q^{52} +85.2380 q^{53} +54.0000 q^{54} -548.387 q^{55} -53.7557 q^{56} -169.660 q^{57} +568.715 q^{58} +59.0000 q^{59} +210.819 q^{60} -567.114 q^{61} -38.3789 q^{62} +60.4752 q^{63} +64.0000 q^{64} -295.140 q^{65} +187.288 q^{66} +553.313 q^{67} +52.6050 q^{68} -15.9446 q^{69} +236.098 q^{70} -876.646 q^{71} -72.0000 q^{72} -981.851 q^{73} +190.440 q^{74} -550.928 q^{75} +226.214 q^{76} +209.746 q^{77} +100.798 q^{78} -768.295 q^{79} -281.092 q^{80} +81.0000 q^{81} -63.7537 q^{82} +15.5314 q^{83} -80.6336 q^{84} -231.044 q^{85} -899.431 q^{86} +853.072 q^{87} -249.718 q^{88} -879.728 q^{89} +316.228 q^{90} +112.885 q^{91} +21.2595 q^{92} -57.5683 q^{93} +801.058 q^{94} -993.544 q^{95} +96.0000 q^{96} -499.395 q^{97} +595.698 q^{98} +280.932 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} - 6 q^{5} + 18 q^{6} + 6 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 - 6 * q^5 + 18 * q^6 + 6 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} - 6 q^{5} + 18 q^{6} + 6 q^{7} - 24 q^{8} + 27 q^{9} + 12 q^{10} - 45 q^{11} - 36 q^{12} + 39 q^{13} - 12 q^{14} + 18 q^{15} + 48 q^{16} + 72 q^{17} - 54 q^{18} - 3 q^{19} - 24 q^{20} - 18 q^{21} + 90 q^{22} - 117 q^{23} + 72 q^{24} + 363 q^{25} - 78 q^{26} - 81 q^{27} + 24 q^{28} - 3 q^{29} - 36 q^{30} + 39 q^{31} - 96 q^{32} + 135 q^{33} - 144 q^{34} - 333 q^{35} + 108 q^{36} - 24 q^{37} + 6 q^{38} - 117 q^{39} + 48 q^{40} - 504 q^{41} + 36 q^{42} + 201 q^{43} - 180 q^{44} - 54 q^{45} + 234 q^{46} - 663 q^{47} - 144 q^{48} - 861 q^{49} - 726 q^{50} - 216 q^{51} + 156 q^{52} - 1098 q^{53} + 162 q^{54} - 1056 q^{55} - 48 q^{56} + 9 q^{57} + 6 q^{58} + 177 q^{59} + 72 q^{60} - 243 q^{61} - 78 q^{62} + 54 q^{63} + 192 q^{64} - 1668 q^{65} - 270 q^{66} - 330 q^{67} + 288 q^{68} + 351 q^{69} + 666 q^{70} - 2271 q^{71} - 216 q^{72} - 381 q^{73} + 48 q^{74} - 1089 q^{75} - 12 q^{76} + 276 q^{77} + 234 q^{78} - 1113 q^{79} - 96 q^{80} + 243 q^{81} + 1008 q^{82} - 2262 q^{83} - 72 q^{84} + 261 q^{85} - 402 q^{86} + 9 q^{87} + 360 q^{88} - 2055 q^{89} + 108 q^{90} + 978 q^{91} - 468 q^{92} - 117 q^{93} + 1326 q^{94} - 2577 q^{95} + 288 q^{96} - 18 q^{97} + 1722 q^{98} - 405 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 - 6 * q^5 + 18 * q^6 + 6 * q^7 - 24 * q^8 + 27 * q^9 + 12 * q^10 - 45 * q^11 - 36 * q^12 + 39 * q^13 - 12 * q^14 + 18 * q^15 + 48 * q^16 + 72 * q^17 - 54 * q^18 - 3 * q^19 - 24 * q^20 - 18 * q^21 + 90 * q^22 - 117 * q^23 + 72 * q^24 + 363 * q^25 - 78 * q^26 - 81 * q^27 + 24 * q^28 - 3 * q^29 - 36 * q^30 + 39 * q^31 - 96 * q^32 + 135 * q^33 - 144 * q^34 - 333 * q^35 + 108 * q^36 - 24 * q^37 + 6 * q^38 - 117 * q^39 + 48 * q^40 - 504 * q^41 + 36 * q^42 + 201 * q^43 - 180 * q^44 - 54 * q^45 + 234 * q^46 - 663 * q^47 - 144 * q^48 - 861 * q^49 - 726 * q^50 - 216 * q^51 + 156 * q^52 - 1098 * q^53 + 162 * q^54 - 1056 * q^55 - 48 * q^56 + 9 * q^57 + 6 * q^58 + 177 * q^59 + 72 * q^60 - 243 * q^61 - 78 * q^62 + 54 * q^63 + 192 * q^64 - 1668 * q^65 - 270 * q^66 - 330 * q^67 + 288 * q^68 + 351 * q^69 + 666 * q^70 - 2271 * q^71 - 216 * q^72 - 381 * q^73 + 48 * q^74 - 1089 * q^75 - 12 * q^76 + 276 * q^77 + 234 * q^78 - 1113 * q^79 - 96 * q^80 + 243 * q^81 + 1008 * q^82 - 2262 * q^83 - 72 * q^84 + 261 * q^85 - 402 * q^86 + 9 * q^87 + 360 * q^88 - 2055 * q^89 + 108 * q^90 + 978 * q^91 - 468 * q^92 - 117 * q^93 + 1326 * q^94 - 2577 * q^95 + 288 * q^96 - 18 * q^97 + 1722 * q^98 - 405 * 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
$5$	−17.5682	−1.57135	−0.785675	−	0.618639i	$-0.787684\pi$
$5$	−17.5682	−1.57135	−0.785675	+	0.618639i	$0.787684\pi$
$6$	6.00000	0.408248
$7$	6.71947	0.362817	0.181409	−	0.983408i	$-0.441934\pi$
$7$	6.71947	0.362817	0.181409	+	0.983408i	$0.441934\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	35.1365	1.11111
$11$	31.2147	0.855598	0.427799	−	0.903874i	$-0.359289\pi$
$11$	31.2147	0.855598	0.427799	+	0.903874i	$0.359289\pi$
$12$	−12.0000	−0.288675
$13$	16.7996	0.358414	0.179207	−	0.983811i	$-0.442647\pi$
$13$	16.7996	0.358414	0.179207	+	0.983811i	$0.442647\pi$
$14$	−13.4389	−0.256550
$15$	52.7047	0.907219
$16$	16.0000	0.250000
$17$	13.1512	0.187626	0.0938131	−	0.995590i	$-0.470094\pi$
$17$	13.1512	0.187626	0.0938131	+	0.995590i	$0.470094\pi$
$18$	−18.0000	−0.235702
$19$	56.5534	0.682855	0.341428	−	0.939908i	$-0.389090\pi$
$19$	56.5534	0.682855	0.341428	+	0.939908i	$0.389090\pi$
$20$	−70.2729	−0.785675
$21$	−20.1584	−0.209473
$22$	−62.4294	−0.604999
$23$	5.31487	0.0481838	0.0240919	−	0.999710i	$-0.492331\pi$
$23$	5.31487	0.0481838	0.0240919	+	0.999710i	$0.492331\pi$
$24$	24.0000	0.204124
$25$	183.643	1.46914
$26$	−33.5993	−0.253437
$27$	−27.0000	−0.192450
$28$	26.8779	0.181409
$29$	−284.357	−1.82082	−0.910411	−	0.413706i	$-0.864234\pi$
$29$	−284.357	−1.82082	−0.910411	+	0.413706i	$0.864234\pi$
$30$	−105.409	−0.641501
$31$	19.1894	0.111178	0.0555891	−	0.998454i	$-0.482296\pi$
$31$	19.1894	0.111178	0.0555891	+	0.998454i	$0.482296\pi$
$32$	−32.0000	−0.176777
$33$	−93.6441	−0.493980
$34$	−26.3025	−0.132672
$35$	−118.049	−0.570113
$36$	36.0000	0.166667
$37$	−95.2199	−0.423083	−0.211541	−	0.977369i	$-0.567848\pi$
$37$	−95.2199	−0.423083	−0.211541	+	0.977369i	$0.567848\pi$
$38$	−113.107	−0.482852
$39$	−50.3989	−0.206930
$40$	140.546	0.555556
$41$	31.8768	0.121423	0.0607113	−	0.998155i	$-0.480663\pi$
$41$	31.8768	0.121423	0.0607113	+	0.998155i	$0.480663\pi$
$42$	40.3168	0.148119
$43$	449.715	1.59491	0.797453	−	0.603381i	$-0.206180\pi$
$43$	449.715	1.59491	0.797453	+	0.603381i	$0.206180\pi$
$44$	124.859	0.427799
$45$	−158.114	−0.523783
$46$	−10.6297	−0.0340711
$47$	−400.529	−1.24305	−0.621523	−	0.783396i	$-0.713486\pi$
$47$	−400.529	−1.24305	−0.621523	+	0.783396i	$0.713486\pi$
$48$	−48.0000	−0.144338
$49$	−297.849	−0.868364
$50$	−367.285	−1.03884
$51$	−39.4537	−0.108326
$52$	67.1985	0.179207
$53$	85.2380	0.220912	0.110456	−	0.993881i	$-0.464769\pi$
$53$	85.2380	0.220912	0.110456	+	0.993881i	$0.464769\pi$
$54$	54.0000	0.136083
$55$	−548.387	−1.34444
$56$	−53.7557	−0.128275
$57$	−169.660	−0.394247
$58$	568.715	1.28752
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	210.819	0.453610
$61$	−567.114	−1.19035	−0.595176	−	0.803595i	$-0.702918\pi$
$61$	−567.114	−1.19035	−0.595176	+	0.803595i	$0.702918\pi$
$62$	−38.3789	−0.0786149
$63$	60.4752	0.120939
$64$	64.0000	0.125000
$65$	−295.140	−0.563194
$66$	187.288	0.349297
$67$	553.313	1.00892	0.504462	−	0.863434i	$-0.331691\pi$
$67$	553.313	1.00892	0.504462	+	0.863434i	$0.331691\pi$
$68$	52.6050	0.0938131
$69$	−15.9446	−0.0278189
$70$	236.098	0.403131
$71$	−876.646	−1.46533	−0.732667	−	0.680587i	$-0.761725\pi$
$71$	−876.646	−1.46533	−0.732667	+	0.680587i	$0.761725\pi$
$72$	−72.0000	−0.117851
$73$	−981.851	−1.57421	−0.787103	−	0.616822i	$-0.788420\pi$
$73$	−981.851	−1.57421	−0.787103	+	0.616822i	$0.788420\pi$
$74$	190.440	0.299165
$75$	−550.928	−0.848209
$76$	226.214	0.341428
$77$	209.746	0.310426
$78$	100.798	0.146322
$79$	−768.295	−1.09418	−0.547088	−	0.837075i	$-0.684264\pi$
$79$	−768.295	−1.09418	−0.547088	+	0.837075i	$0.684264\pi$
$80$	−281.092	−0.392837
$81$	81.0000	0.111111
$82$	−63.7537	−0.0858587
$83$	15.5314	0.0205396	0.0102698	−	0.999947i	$-0.496731\pi$
$83$	15.5314	0.0205396	0.0102698	+	0.999947i	$0.496731\pi$
$84$	−80.6336	−0.104736
$85$	−231.044	−0.294826
$86$	−899.431	−1.12777
$87$	853.072	1.05125
$88$	−249.718	−0.302500
$89$	−879.728	−1.04776	−0.523882	−	0.851791i	$-0.675517\pi$
$89$	−879.728	−1.04776	−0.523882	+	0.851791i	$0.675517\pi$
$90$	316.228	0.370371
$91$	112.885	0.130039
$92$	21.2595	0.0240919
$93$	−57.5683	−0.0641888
$94$	801.058	0.878966
$95$	−993.544	−1.07300
$96$	96.0000	0.102062
$97$	−499.395	−0.522741	−0.261370	−	0.965239i	$-0.584174\pi$
$97$	−499.395	−0.522741	−0.261370	+	0.965239i	$0.584174\pi$
$98$	595.698	0.614026
$99$	280.932	0.285199
$100$	734.570	0.734570
$101$	276.213	0.272121	0.136061	−	0.990701i	$-0.456556\pi$
$101$	276.213	0.272121	0.136061	+	0.990701i	$0.456556\pi$
$102$	78.9074	0.0765980
$103$	−679.825	−0.650341	−0.325171	−	0.945655i	$-0.605422\pi$
$103$	−679.825	−0.650341	−0.325171	+	0.945655i	$0.605422\pi$
$104$	−134.397	−0.126718
$105$	354.147	0.329155
$106$	−170.476	−0.156208
$107$	−457.046	−0.412937	−0.206469	−	0.978453i	$-0.566197\pi$
$107$	−457.046	−0.412937	−0.206469	+	0.978453i	$0.566197\pi$
$108$	−108.000	−0.0962250
$109$	1141.37	1.00296	0.501482	−	0.865168i	$-0.332788\pi$
$109$	1141.37	1.00296	0.501482	+	0.865168i	$0.332788\pi$
$110$	1096.77	0.950666
$111$	285.660	0.244267
$112$	107.511	0.0907043
$113$	1564.57	1.30250	0.651251	−	0.758862i	$-0.274245\pi$
$113$	1564.57	1.30250	0.651251	+	0.758862i	$0.274245\pi$
$114$	339.321	0.278774
$115$	−93.3729	−0.0757136
$116$	−1137.43	−0.910411
$117$	151.197	0.119471
$118$	−118.000	−0.0920575
$119$	88.3693	0.0680740
$120$	−421.637	−0.320750
$121$	−356.643	−0.267951
$122$	1134.23	0.841706
$123$	−95.6305	−0.0701034
$124$	76.7578	0.0555891
$125$	−1030.25	−0.737184
$126$	−120.950	−0.0855168
$127$	362.592	0.253345	0.126673	−	0.991945i	$-0.459570\pi$
$127$	362.592	0.253345	0.126673	+	0.991945i	$0.459570\pi$
$128$	−128.000	−0.0883883
$129$	−1349.15	−0.920820
$130$	590.280	0.398238
$131$	−643.570	−0.429229	−0.214615	−	0.976699i	$-0.568850\pi$
$131$	−643.570	−0.429229	−0.214615	+	0.976699i	$0.568850\pi$
$132$	−374.576	−0.246990
$133$	380.009	0.247752
$134$	−1106.63	−0.713418
$135$	474.342	0.302406
$136$	−105.210	−0.0663359
$137$	−26.2437	−0.0163660	−0.00818302	−	0.999967i	$-0.502605\pi$
$137$	−26.2437	−0.0163660	−0.00818302	+	0.999967i	$0.502605\pi$
$138$	31.8892	0.0196710
$139$	1967.92	1.20084	0.600422	−	0.799684i	$-0.294999\pi$
$139$	1967.92	1.20084	0.600422	+	0.799684i	$0.294999\pi$
$140$	−472.197	−0.285056
$141$	1201.59	0.717673
$142$	1753.29	1.03615
$143$	524.395	0.306658
$144$	144.000	0.0833333
$145$	4995.65	2.86115
$146$	1963.70	1.11313
$147$	893.546	0.501350
$148$	−380.880	−0.211541
$149$	319.439	0.175634	0.0878170	−	0.996137i	$-0.472011\pi$
$149$	319.439	0.175634	0.0878170	+	0.996137i	$0.472011\pi$
$150$	1101.86	0.599774
$151$	743.043	0.400450	0.200225	−	0.979750i	$-0.435833\pi$
$151$	743.043	0.400450	0.200225	+	0.979750i	$0.435833\pi$
$152$	−452.428	−0.241426
$153$	118.361	0.0625420
$154$	−419.492	−0.219504
$155$	−337.124	−0.174700
$156$	−201.596	−0.103465
$157$	1034.57	0.525909	0.262954	−	0.964808i	$-0.415303\pi$
$157$	1034.57	0.525909	0.262954	+	0.964808i	$0.415303\pi$
$158$	1536.59	0.773700
$159$	−255.714	−0.127544
$160$	562.183	0.277778
$161$	35.7131	0.0174819
$162$	−162.000	−0.0785674
$163$	−835.351	−0.401410	−0.200705	−	0.979652i	$-0.564323\pi$
$163$	−835.351	−0.401410	−0.200705	+	0.979652i	$0.564323\pi$
$164$	127.507	0.0607113
$165$	1645.16	0.776215
$166$	−31.0627	−0.0145237
$167$	−819.839	−0.379886	−0.189943	−	0.981795i	$-0.560830\pi$
$167$	−819.839	−0.379886	−0.189943	+	0.981795i	$0.560830\pi$
$168$	161.267	0.0740597
$169$	−1914.77	−0.871540
$170$	462.088	0.208474
$171$	508.981	0.227618
$172$	1798.86	0.797453
$173$	1157.69	0.508773	0.254387	−	0.967103i	$-0.418126\pi$
$173$	1157.69	0.508773	0.254387	+	0.967103i	$0.418126\pi$
$174$	−1706.14	−0.743347
$175$	1233.98	0.533029
$176$	499.435	0.213900
$177$	−177.000	−0.0751646
$178$	1759.46	0.740881
$179$	−1147.12	−0.478992	−0.239496	−	0.970897i	$-0.576982\pi$
$179$	−1147.12	−0.478992	−0.239496	+	0.970897i	$0.576982\pi$
$180$	−632.456	−0.261892
$181$	−2521.41	−1.03544	−0.517722	−	0.855549i	$-0.673220\pi$
$181$	−2521.41	−1.03544	−0.517722	+	0.855549i	$0.673220\pi$
$182$	−225.769	−0.0919512
$183$	1701.34	0.687250
$184$	−42.5190	−0.0170355
$185$	1672.85	0.664811
$186$	115.137	0.0453883
$187$	410.512	0.160533
$188$	−1602.12	−0.621523
$189$	−181.426	−0.0698242
$190$	1987.09	0.758729
$191$	−697.805	−0.264353	−0.132176	−	0.991226i	$-0.542197\pi$
$191$	−697.805	−0.264353	−0.132176	+	0.991226i	$0.542197\pi$
$192$	−192.000	−0.0721688
$193$	3595.53	1.34099	0.670497	−	0.741912i	$-0.266081\pi$
$193$	3595.53	1.34099	0.670497	+	0.741912i	$0.266081\pi$
$194$	998.790	0.369634
$195$	885.419	0.325160
$196$	−1191.40	−0.434182
$197$	−4313.27	−1.55994	−0.779970	−	0.625817i	$-0.784766\pi$
$197$	−4313.27	−1.55994	−0.779970	+	0.625817i	$0.784766\pi$
$198$	−561.864	−0.201666
$199$	−2338.60	−0.833061	−0.416531	−	0.909122i	$-0.636754\pi$
$199$	−2338.60	−0.833061	−0.416531	+	0.909122i	$0.636754\pi$
$200$	−1469.14	−0.519420
$201$	−1659.94	−0.582503
$202$	−552.426	−0.192419
$203$	−1910.73	−0.660625
$204$	−157.815	−0.0541630
$205$	−560.020	−0.190797
$206$	1359.65	0.459861
$207$	47.8339	0.0160613
$208$	268.794	0.0896035
$209$	1765.30	0.584250
$210$	−708.295	−0.232748
$211$	−5677.99	−1.85255	−0.926277	−	0.376844i	$-0.877009\pi$
$211$	−5677.99	−1.85255	−0.926277	+	0.376844i	$0.877009\pi$
$212$	340.952	0.110456
$213$	2629.94	0.846011
$214$	914.092	0.291991
$215$	−7900.70	−2.50616
$216$	216.000	0.0680414
$217$	128.943	0.0403374
$218$	−2282.73	−0.709203
$219$	2945.55	0.908868
$220$	−2193.55	−0.672222
$221$	220.936	0.0672478
$222$	−571.320	−0.172723
$223$	−2773.65	−0.832903	−0.416451	−	0.909158i	$-0.636726\pi$
$223$	−2773.65	−0.832903	−0.416451	+	0.909158i	$0.636726\pi$
$224$	−215.023	−0.0641376
$225$	1652.78	0.489714
$226$	−3129.15	−0.921008
$227$	−4820.74	−1.40953	−0.704767	−	0.709439i	$-0.748948\pi$
$227$	−4820.74	−1.40953	−0.704767	+	0.709439i	$0.748948\pi$
$228$	−678.641	−0.197123
$229$	4417.48	1.27474	0.637369	−	0.770558i	$-0.280023\pi$
$229$	4417.48	1.27474	0.637369	+	0.770558i	$0.280023\pi$
$230$	186.746	0.0535376
$231$	−629.238	−0.179224
$232$	2274.86	0.643758
$233$	1994.45	0.560775	0.280388	−	0.959887i	$-0.409537\pi$
$233$	1994.45	0.560775	0.280388	+	0.959887i	$0.409537\pi$
$234$	−302.393	−0.0844790
$235$	7036.58	1.95326
$236$	236.000	0.0650945
$237$	2304.88	0.631723
$238$	−176.739	−0.0481356
$239$	−2125.39	−0.575229	−0.287615	−	0.957746i	$-0.592862\pi$
$239$	−2125.39	−0.575229	−0.287615	+	0.957746i	$0.592862\pi$
$240$	843.275	0.226805
$241$	−3103.50	−0.829519	−0.414760	−	0.909931i	$-0.636134\pi$
$241$	−3103.50	−0.829519	−0.414760	+	0.909931i	$0.636134\pi$
$242$	713.286	0.189470
$243$	−243.000	−0.0641500
$244$	−2268.46	−0.595176
$245$	5232.67	1.36450
$246$	191.261	0.0495706
$247$	950.077	0.244745
$248$	−153.516	−0.0393074
$249$	−46.5941	−0.0118586
$250$	2060.49	0.521268
$251$	−2434.80	−0.612284	−0.306142	−	0.951986i	$-0.599038\pi$
$251$	−2434.80	−0.612284	−0.306142	+	0.951986i	$0.599038\pi$
$252$	241.901	0.0604695
$253$	165.902	0.0412260
$254$	−725.184	−0.179142
$255$	693.132	0.170218
$256$	256.000	0.0625000
$257$	3680.10	0.893223	0.446611	−	0.894728i	$-0.352631\pi$
$257$	3680.10	0.893223	0.446611	+	0.894728i	$0.352631\pi$
$258$	2698.29	0.651118
$259$	−639.827	−0.153502
$260$	−1180.56	−0.281597
$261$	−2559.22	−0.606940
$262$	1287.14	0.303511
$263$	1259.69	0.295344	0.147672	−	0.989036i	$-0.452822\pi$
$263$	1259.69	0.295344	0.147672	+	0.989036i	$0.452822\pi$
$264$	749.153	0.174648
$265$	−1497.48	−0.347130
$266$	−760.018	−0.175187
$267$	2639.18	0.604926
$268$	2213.25	0.504462
$269$	6177.17	1.40011	0.700054	−	0.714090i	$-0.253159\pi$
$269$	6177.17	1.40011	0.700054	+	0.714090i	$0.253159\pi$
$270$	−948.684	−0.213834
$271$	−1621.10	−0.363375	−0.181688	−	0.983356i	$-0.558156\pi$
$271$	−1621.10	−0.363375	−0.181688	+	0.983356i	$0.558156\pi$
$272$	210.420	0.0469065
$273$	−338.654	−0.0750779
$274$	52.4873	0.0115725
$275$	5732.35	1.25699
$276$	−63.7785	−0.0139095
$277$	−2668.70	−0.578868	−0.289434	−	0.957198i	$-0.593467\pi$
$277$	−2668.70	−0.578868	−0.289434	+	0.957198i	$0.593467\pi$
$278$	−3935.85	−0.849124
$279$	172.705	0.0370594
$280$	944.393	0.201565
$281$	5032.81	1.06844	0.534221	−	0.845345i	$-0.320605\pi$
$281$	5032.81	1.06844	0.534221	+	0.845345i	$0.320605\pi$
$282$	−2403.17	−0.507471
$283$	−8141.08	−1.71002	−0.855012	−	0.518608i	$-0.826450\pi$
$283$	−8141.08	−1.71002	−0.855012	+	0.518608i	$0.826450\pi$
$284$	−3506.58	−0.732667
$285$	2980.63	0.619499
$286$	−1048.79	−0.216840
$287$	214.195	0.0440542
$288$	−288.000	−0.0589256
$289$	−4740.04	−0.964796
$290$	−9991.31	−2.02314
$291$	1498.18	0.301805
$292$	−3927.41	−0.787103
$293$	−2767.40	−0.551785	−0.275893	−	0.961188i	$-0.588973\pi$
$293$	−2767.40	−0.551785	−0.275893	+	0.961188i	$0.588973\pi$
$294$	−1787.09	−0.354508
$295$	−1036.53	−0.204572
$296$	761.759	0.149582
$297$	−842.797	−0.164660
$298$	−638.878	−0.124192
$299$	89.2879	0.0172697
$300$	−2203.71	−0.424104
$301$	3021.85	0.578659
$302$	−1486.09	−0.283161
$303$	−828.639	−0.157109
$304$	904.855	0.170714
$305$	9963.19	1.87046
$306$	−236.722	−0.0442239
$307$	1231.68	0.228976	0.114488	−	0.993425i	$-0.463477\pi$
$307$	1231.68	0.228976	0.114488	+	0.993425i	$0.463477\pi$
$308$	838.984	0.155213
$309$	2039.47	0.375475
$310$	674.249	0.123532
$311$	−6977.91	−1.27229	−0.636143	−	0.771571i	$-0.719471\pi$
$311$	−6977.91	−1.27229	−0.636143	+	0.771571i	$0.719471\pi$
$312$	403.191	0.0731609
$313$	−3365.20	−0.607707	−0.303853	−	0.952719i	$-0.598273\pi$
$313$	−3365.20	−0.607707	−0.303853	+	0.952719i	$0.598273\pi$
$314$	−2069.14	−0.371874
$315$	−1062.44	−0.190038
$316$	−3073.18	−0.547088
$317$	3196.66	0.566379	0.283189	−	0.959064i	$-0.408608\pi$
$317$	3196.66	0.566379	0.283189	+	0.959064i	$0.408608\pi$
$318$	511.428	0.0901870
$319$	−8876.12	−1.55789
$320$	−1124.37	−0.196419
$321$	1371.14	0.238409
$322$	−71.4262	−0.0123616
$323$	743.748	0.128121
$324$	324.000	0.0555556
$325$	3085.13	0.526560
$326$	1670.70	0.283839
$327$	−3424.10	−0.579062
$328$	−255.015	−0.0429294
$329$	−2691.34	−0.450998
$330$	−3290.32	−0.548867
$331$	−1870.84	−0.310667	−0.155333	−	0.987862i	$-0.549645\pi$
$331$	−1870.84	−0.310667	−0.155333	+	0.987862i	$0.549645\pi$
$332$	62.1254	0.0102698
$333$	−856.979	−0.141028
$334$	1639.68	0.268620
$335$	−9720.73	−1.58537
$336$	−322.534	−0.0523681
$337$	−8864.60	−1.43289	−0.716447	−	0.697642i	$-0.754233\pi$
$337$	−8864.60	−1.43289	−0.716447	+	0.697642i	$0.754233\pi$
$338$	3829.54	0.616272
$339$	−4693.72	−0.752000
$340$	−924.176	−0.147413
$341$	598.992	0.0951239
$342$	−1017.96	−0.160951
$343$	−4306.16	−0.677874
$344$	−3597.72	−0.563885
$345$	280.119	0.0437133
$346$	−2315.39	−0.359757
$347$	11723.0	1.81362	0.906810	−	0.421540i	$-0.138510\pi$
$347$	11723.0	1.81362	0.906810	+	0.421540i	$0.138510\pi$
$348$	3412.29	0.525626
$349$	−10204.4	−1.56512	−0.782561	−	0.622574i	$-0.786087\pi$
$349$	−10204.4	−1.56512	−0.782561	+	0.622574i	$0.786087\pi$
$350$	−2467.96	−0.376909
$351$	−453.590	−0.0689768
$352$	−998.870	−0.151250
$353$	−8361.05	−1.26066	−0.630331	−	0.776326i	$-0.717081\pi$
$353$	−8361.05	−1.26066	−0.630331	+	0.776326i	$0.717081\pi$
$354$	354.000	0.0531494
$355$	15401.1	2.30255
$356$	−3518.91	−0.523882
$357$	−265.108	−0.0393025
$358$	2294.23	0.338698
$359$	−5684.57	−0.835711	−0.417855	−	0.908514i	$-0.637218\pi$
$359$	−5684.57	−0.835711	−0.417855	+	0.908514i	$0.637218\pi$
$360$	1264.91	0.185185
$361$	−3660.71	−0.533709
$362$	5042.83	0.732169
$363$	1069.93	0.154702
$364$	451.538	0.0650193
$365$	17249.4	2.47363
$366$	−3402.68	−0.485959
$367$	−2441.58	−0.347273	−0.173637	−	0.984810i	$-0.555552\pi$
$367$	−2441.58	−0.347273	−0.173637	+	0.984810i	$0.555552\pi$
$368$	85.0380	0.0120460
$369$	286.892	0.0404742
$370$	−3345.69	−0.470092
$371$	572.754	0.0801507
$372$	−230.273	−0.0320944
$373$	−1023.67	−0.142100	−0.0710501	−	0.997473i	$-0.522635\pi$
$373$	−1023.67	−0.142100	−0.0710501	+	0.997473i	$0.522635\pi$
$374$	−821.024	−0.113514
$375$	3090.74	0.425614
$376$	3204.23	0.439483
$377$	−4777.10	−0.652608
$378$	362.851	0.0493732
$379$	−7521.37	−1.01938	−0.509692	−	0.860357i	$-0.670241\pi$
$379$	−7521.37	−1.01938	−0.509692	+	0.860357i	$0.670241\pi$
$380$	−3974.17	−0.536502
$381$	−1087.78	−0.146269
$382$	1395.61	0.186926
$383$	8542.44	1.13968	0.569841	−	0.821755i	$-0.307005\pi$
$383$	8542.44	1.13968	0.569841	+	0.821755i	$0.307005\pi$
$384$	384.000	0.0510310
$385$	−3684.87	−0.487788
$386$	−7191.06	−0.948226
$387$	4047.44	0.531635
$388$	−1997.58	−0.261370
$389$	−5845.87	−0.761948	−0.380974	−	0.924586i	$-0.624411\pi$
$389$	−5845.87	−0.761948	−0.380974	+	0.924586i	$0.624411\pi$
$390$	−1770.84	−0.229923
$391$	69.8972	0.00904054
$392$	2382.79	0.307013
$393$	1930.71	0.247816
$394$	8626.55	1.10304
$395$	13497.6	1.71933
$396$	1123.73	0.142600
$397$	2784.75	0.352047	0.176024	−	0.984386i	$-0.443676\pi$
$397$	2784.75	0.352047	0.176024	+	0.984386i	$0.443676\pi$
$398$	4677.21	0.589063
$399$	−1140.03	−0.143039
$400$	2938.28	0.367285
$401$	7926.29	0.987083	0.493541	−	0.869722i	$-0.335702\pi$
$401$	7926.29	0.987083	0.493541	+	0.869722i	$0.335702\pi$
$402$	3319.88	0.411892
$403$	322.376	0.0398478
$404$	1104.85	0.136061
$405$	−1423.03	−0.174594
$406$	3821.46	0.467133
$407$	−2972.26	−0.361989
$408$	315.630	0.0382990
$409$	6985.79	0.844560	0.422280	−	0.906466i	$-0.361230\pi$
$409$	6985.79	0.844560	0.422280	+	0.906466i	$0.361230\pi$
$410$	1120.04	0.134914
$411$	78.7310	0.00944894
$412$	−2719.30	−0.325171
$413$	396.449	0.0472348
$414$	−95.6677	−0.0113570
$415$	−272.858	−0.0322749
$416$	−537.588	−0.0633592
$417$	−5903.77	−0.693307
$418$	−3530.60	−0.413127
$419$	7534.64	0.878500	0.439250	−	0.898365i	$-0.355244\pi$
$419$	7534.64	0.878500	0.439250	+	0.898365i	$0.355244\pi$
$420$	1416.59	0.164577
$421$	6931.57	0.802432	0.401216	−	0.915983i	$-0.368588\pi$
$421$	6931.57	0.802432	0.401216	+	0.915983i	$0.368588\pi$
$422$	11356.0	1.30995
$423$	−3604.76	−0.414349
$424$	−681.904	−0.0781042
$425$	2415.13	0.275649
$426$	−5259.88	−0.598220
$427$	−3810.70	−0.431880
$428$	−1828.18	−0.206469
$429$	−1573.19	−0.177049
$430$	15801.4	1.77212
$431$	1881.26	0.210249	0.105124	−	0.994459i	$-0.466476\pi$
$431$	1881.26	0.210249	0.105124	+	0.994459i	$0.466476\pi$
$432$	−432.000	−0.0481125
$433$	12272.5	1.36208	0.681039	−	0.732247i	$-0.261528\pi$
$433$	12272.5	1.36208	0.681039	+	0.732247i	$0.261528\pi$
$434$	−257.886	−0.0285228
$435$	−14987.0	−1.65188
$436$	4565.47	0.501482
$437$	300.574	0.0329026
$438$	−5891.11	−0.642667
$439$	−57.1133	−0.00620927	−0.00310463	−	0.999995i	$-0.500988\pi$
$439$	−57.1133	−0.00620927	−0.00310463	+	0.999995i	$0.500988\pi$
$440$	4387.09	0.475333
$441$	−2680.64	−0.289455
$442$	−441.872	−0.0475514
$443$	14868.0	1.59458	0.797290	−	0.603597i	$-0.206266\pi$
$443$	14868.0	1.59458	0.797290	+	0.603597i	$0.206266\pi$
$444$	1142.64	0.122133
$445$	15455.3	1.64640
$446$	5547.30	0.588951
$447$	−958.317	−0.101402
$448$	430.046	0.0453521
$449$	−5866.46	−0.616604	−0.308302	−	0.951288i	$-0.599761\pi$
$449$	−5866.46	−0.616604	−0.308302	+	0.951288i	$0.599761\pi$
$450$	−3305.57	−0.346280
$451$	995.026	0.103889
$452$	6258.29	0.651251
$453$	−2229.13	−0.231200
$454$	9641.49	0.996690
$455$	−1983.18	−0.204336
$456$	1357.28	0.139387
$457$	8047.63	0.823747	0.411873	−	0.911241i	$-0.364875\pi$
$457$	8047.63	0.823747	0.411873	+	0.911241i	$0.364875\pi$
$458$	−8834.96	−0.901376
$459$	−355.083	−0.0361087
$460$	−373.492	−0.0378568
$461$	−4844.14	−0.489401	−0.244701	−	0.969599i	$-0.578690\pi$
$461$	−4844.14	−0.489401	−0.244701	+	0.969599i	$0.578690\pi$
$462$	1258.48	0.126731
$463$	−6595.55	−0.662033	−0.331016	−	0.943625i	$-0.607392\pi$
$463$	−6595.55	−0.662033	−0.331016	+	0.943625i	$0.607392\pi$
$464$	−4549.72	−0.455205
$465$	1011.37	0.100863
$466$	−3988.90	−0.396528
$467$	4487.76	0.444687	0.222344	−	0.974968i	$-0.428629\pi$
$467$	4487.76	0.444687	0.222344	+	0.974968i	$0.428629\pi$
$468$	604.787	0.0597356
$469$	3717.97	0.366055
$470$	−14073.2	−1.38116
$471$	−3103.71	−0.303634
$472$	−472.000	−0.0460287
$473$	14037.7	1.36460
$474$	−4609.77	−0.446696
$475$	10385.6	1.00321
$476$	353.477	0.0340370
$477$	767.142	0.0736373
$478$	4250.77	0.406749
$479$	−11061.6	−1.05516	−0.527578	−	0.849507i	$-0.676900\pi$
$479$	−11061.6	−1.05516	−0.527578	+	0.849507i	$0.676900\pi$
$480$	−1686.55	−0.160375
$481$	−1599.66	−0.151639
$482$	6207.00	0.586559
$483$	−107.139	−0.0100932
$484$	−1426.57	−0.133976
$485$	8773.48	0.821409
$486$	486.000	0.0453609
$487$	10974.5	1.02115	0.510576	−	0.859832i	$-0.329432\pi$
$487$	10974.5	1.02115	0.510576	+	0.859832i	$0.329432\pi$
$488$	4536.91	0.420853
$489$	2506.05	0.231754
$490$	−10465.3	−0.964850
$491$	−15665.9	−1.43990	−0.719952	−	0.694024i	$-0.755836\pi$
$491$	−15665.9	−1.43990	−0.719952	+	0.694024i	$0.755836\pi$
$492$	−382.522	−0.0350517
$493$	−3739.65	−0.341634
$494$	−1900.15	−0.173061
$495$	−4935.48	−0.448148
$496$	307.031	0.0277946
$497$	−5890.59	−0.531648
$498$	93.1882	0.00838526
$499$	9805.79	0.879694	0.439847	−	0.898073i	$-0.355033\pi$
$499$	9805.79	0.879694	0.439847	+	0.898073i	$0.355033\pi$
$500$	−4120.99	−0.368592
$501$	2459.52	0.219328
$502$	4869.60	0.432950
$503$	15264.4	1.35310	0.676549	−	0.736398i	$-0.263475\pi$
$503$	15264.4	1.35310	0.676549	+	0.736398i	$0.263475\pi$
$504$	−483.802	−0.0427584
$505$	−4852.58	−0.427598
$506$	−331.804	−0.0291512
$507$	5744.32	0.503184
$508$	1450.37	0.126673
$509$	−2921.84	−0.254437	−0.127218	−	0.991875i	$-0.540605\pi$
$509$	−2921.84	−0.254437	−0.127218	+	0.991875i	$0.540605\pi$
$510$	−1386.26	−0.120362
$511$	−6597.52	−0.571149
$512$	−512.000	−0.0441942
$513$	−1526.94	−0.131416
$514$	−7360.20	−0.631604
$515$	11943.3	1.02191
$516$	−5396.59	−0.460410
$517$	−12502.4	−1.06355
$518$	1279.65	0.108542
$519$	−3473.08	−0.293740
$520$	2361.12	0.199119
$521$	2736.92	0.230147	0.115074	−	0.993357i	$-0.463290\pi$
$521$	2736.92	0.230147	0.115074	+	0.993357i	$0.463290\pi$
$522$	5118.43	0.429172
$523$	4134.84	0.345705	0.172853	−	0.984948i	$-0.444702\pi$
$523$	4134.84	0.345705	0.172853	+	0.984948i	$0.444702\pi$
$524$	−2574.28	−0.214615
$525$	−3701.94	−0.307745
$526$	−2519.37	−0.208840
$527$	252.365	0.0208599
$528$	−1498.31	−0.123495
$529$	−12138.8	−0.997678
$530$	2994.96	0.245458
$531$	531.000	0.0433963
$532$	1520.04	0.123876
$533$	535.519	0.0435195
$534$	−5278.37	−0.427748
$535$	8029.48	0.648869
$536$	−4426.51	−0.356709
$537$	3441.35	0.276546
$538$	−12354.3	−0.990025
$539$	−9297.26	−0.742971
$540$	1897.37	0.151203
$541$	9283.88	0.737792	0.368896	−	0.929471i	$-0.379736\pi$
$541$	9283.88	0.737792	0.368896	+	0.929471i	$0.379736\pi$
$542$	3242.19	0.256945
$543$	7564.24	0.597813
$544$	−420.840	−0.0331679
$545$	−20051.8	−1.57601
$546$	677.308	0.0530881
$547$	−8585.30	−0.671080	−0.335540	−	0.942026i	$-0.608919\pi$
$547$	−8585.30	−0.671080	−0.335540	+	0.942026i	$0.608919\pi$
$548$	−104.975	−0.00818302
$549$	−5104.03	−0.396784
$550$	−11464.7	−0.888829
$551$	−16081.4	−1.24336
$552$	127.557	0.00983548
$553$	−5162.53	−0.396986
$554$	5337.39	0.409322
$555$	−5018.54	−0.383829
$556$	7871.70	0.600422
$557$	3298.59	0.250926	0.125463	−	0.992098i	$-0.459958\pi$
$557$	3298.59	0.250926	0.125463	+	0.992098i	$0.459958\pi$
$558$	−345.410	−0.0262050
$559$	7555.06	0.571636
$560$	−1888.79	−0.142528
$561$	−1231.54	−0.0926836
$562$	−10065.6	−0.755503
$563$	15412.6	1.15376	0.576879	−	0.816830i	$-0.304271\pi$
$563$	15412.6	1.15376	0.576879	+	0.816830i	$0.304271\pi$
$564$	4806.35	0.358836
$565$	−27486.8	−2.04669
$566$	16282.2	1.20917
$567$	544.277	0.0403130
$568$	7013.17	0.518074
$569$	19052.4	1.40373	0.701863	−	0.712312i	$-0.252352\pi$
$569$	19052.4	1.40373	0.701863	+	0.712312i	$0.252352\pi$
$570$	−5961.26	−0.438052
$571$	−1784.65	−0.130797	−0.0653987	−	0.997859i	$-0.520832\pi$
$571$	−1784.65	−0.130797	−0.0653987	+	0.997859i	$0.520832\pi$
$572$	2097.58	0.153329
$573$	2093.42	0.152624
$574$	−428.391	−0.0311510
$575$	976.037	0.0707888
$576$	576.000	0.0416667
$577$	−15717.4	−1.13401	−0.567005	−	0.823714i	$-0.691898\pi$
$577$	−15717.4	−1.13401	−0.567005	+	0.823714i	$0.691898\pi$
$578$	9480.09	0.682214
$579$	−10786.6	−0.774223
$580$	19982.6	1.43057
$581$	104.362	0.00745212
$582$	−2996.37	−0.213408
$583$	2660.68	0.189012
$584$	7854.81	0.556566
$585$	−2656.26	−0.187731
$586$	5534.79	0.390171
$587$	−16015.2	−1.12610	−0.563049	−	0.826424i	$-0.690372\pi$
$587$	−16015.2	−1.12610	−0.563049	+	0.826424i	$0.690372\pi$
$588$	3574.19	0.250675
$589$	1085.23	0.0759186
$590$	2073.05	0.144654
$591$	12939.8	0.900632
$592$	−1523.52	−0.105771
$593$	2180.71	0.151014	0.0755069	−	0.997145i	$-0.475943\pi$
$593$	2180.71	0.151014	0.0755069	+	0.997145i	$0.475943\pi$
$594$	1685.59	0.116432
$595$	−1552.49	−0.106968
$596$	1277.76	0.0878170
$597$	7015.81	0.480968
$598$	−178.576	−0.0122116
$599$	18727.1	1.27741	0.638704	−	0.769453i	$-0.279471\pi$
$599$	18727.1	1.27741	0.638704	+	0.769453i	$0.279471\pi$
$600$	4407.42	0.299887
$601$	−22639.9	−1.53661	−0.768305	−	0.640084i	$-0.778900\pi$
$601$	−22639.9	−1.53661	−0.768305	+	0.640084i	$0.778900\pi$
$602$	−6043.70	−0.409174
$603$	4979.82	0.336308
$604$	2972.17	0.200225
$605$	6265.59	0.421045
$606$	1657.28	0.111093
$607$	−15404.1	−1.03004	−0.515020	−	0.857178i	$-0.672216\pi$
$607$	−15404.1	−1.03004	−0.515020	+	0.857178i	$0.672216\pi$
$608$	−1809.71	−0.120713
$609$	5732.19	0.381412
$610$	−19926.4	−1.32262
$611$	−6728.74	−0.445525
$612$	473.445	0.0312710
$613$	24191.5	1.59394	0.796971	−	0.604017i	$-0.206434\pi$
$613$	24191.5	1.59394	0.796971	+	0.604017i	$0.206434\pi$
$614$	−2463.36	−0.161910
$615$	1680.06	0.110157
$616$	−1677.97	−0.109752
$617$	19350.7	1.26261	0.631306	−	0.775534i	$-0.282519\pi$
$617$	19350.7	1.26261	0.631306	+	0.775534i	$0.282519\pi$
$618$	−4078.95	−0.265501
$619$	5393.83	0.350236	0.175118	−	0.984547i	$-0.443969\pi$
$619$	5393.83	0.350236	0.175118	+	0.984547i	$0.443969\pi$
$620$	−1348.50	−0.0873500
$621$	−143.502	−0.00927298
$622$	13955.8	0.899642
$623$	−5911.30	−0.380146
$624$	−806.382	−0.0517326
$625$	−4855.72	−0.310766
$626$	6730.39	0.429714
$627$	−5295.89	−0.337317
$628$	4138.28	0.262954
$629$	−1252.26	−0.0793814
$630$	2124.88	0.134377
$631$	12544.3	0.791411	0.395705	−	0.918378i	$-0.370500\pi$
$631$	12544.3	0.791411	0.395705	+	0.918378i	$0.370500\pi$
$632$	6146.36	0.386850
$633$	17034.0	1.06957
$634$	−6393.31	−0.400490
$635$	−6370.10	−0.398094
$636$	−1022.86	−0.0637718
$637$	−5003.75	−0.311234
$638$	17752.2	1.10160
$639$	−7889.81	−0.488445
$640$	2248.73	0.138889
$641$	20583.3	1.26832	0.634158	−	0.773204i	$-0.281347\pi$
$641$	20583.3	1.26832	0.634158	+	0.773204i	$0.281347\pi$
$642$	−2742.27	−0.168581
$643$	13356.7	0.819186	0.409593	−	0.912268i	$-0.365671\pi$
$643$	13356.7	0.819186	0.409593	+	0.912268i	$0.365671\pi$
$644$	142.852	0.00874096
$645$	23702.1	1.44693
$646$	−1487.50	−0.0905956
$647$	−16856.8	−1.02428	−0.512142	−	0.858901i	$-0.671148\pi$
$647$	−16856.8	−1.02428	−0.512142	+	0.858901i	$0.671148\pi$
$648$	−648.000	−0.0392837
$649$	1841.67	0.111389
$650$	−6170.26	−0.372334
$651$	−386.828	−0.0232888
$652$	−3341.40	−0.200705
$653$	−18088.2	−1.08399	−0.541996	−	0.840381i	$-0.682331\pi$
$653$	−18088.2	−1.08399	−0.541996	+	0.840381i	$0.682331\pi$
$654$	6848.20	0.409459
$655$	11306.4	0.674469
$656$	510.030	0.0303557
$657$	−8836.66	−0.524735
$658$	5382.68	0.318904
$659$	−24013.8	−1.41949	−0.709746	−	0.704458i	$-0.751190\pi$
$659$	−24013.8	−1.41949	−0.709746	+	0.704458i	$0.751190\pi$
$660$	6580.64	0.388108
$661$	−11195.2	−0.658765	−0.329383	−	0.944197i	$-0.606841\pi$
$661$	−11195.2	−0.658765	−0.329383	+	0.944197i	$0.606841\pi$
$662$	3741.68	0.219675
$663$	−662.808	−0.0388255
$664$	−124.251	−0.00726185
$665$	−6676.08	−0.389304
$666$	1713.96	0.0997215
$667$	−1511.32	−0.0877341
$668$	−3279.36	−0.189943
$669$	8320.95	0.480877
$670$	19441.5	1.12103
$671$	−17702.3	−1.01846
$672$	645.069	0.0370299
$673$	−19748.3	−1.13112	−0.565558	−	0.824708i	$-0.691339\pi$
$673$	−19748.3	−1.13112	−0.565558	+	0.824708i	$0.691339\pi$
$674$	17729.2	1.01321
$675$	−4958.35	−0.282736
$676$	−7659.09	−0.435770
$677$	−10205.5	−0.579361	−0.289681	−	0.957123i	$-0.593549\pi$
$677$	−10205.5	−0.579361	−0.289681	+	0.957123i	$0.593549\pi$
$678$	9387.44	0.531744
$679$	−3355.67	−0.189659
$680$	1848.35	0.104237
$681$	14462.2	0.813794
$682$	−1197.98	−0.0672628
$683$	17024.6	0.953775	0.476888	−	0.878964i	$-0.341765\pi$
$683$	17024.6	0.953775	0.476888	+	0.878964i	$0.341765\pi$
$684$	2035.92	0.113809
$685$	461.055	0.0257168
$686$	8612.32	0.479330
$687$	−13252.4	−0.735971
$688$	7195.45	0.398727
$689$	1431.97	0.0791779
$690$	−560.237	−0.0309100
$691$	−25966.2	−1.42953	−0.714763	−	0.699367i	$-0.753465\pi$
$691$	−25966.2	−1.42953	−0.714763	+	0.699367i	$0.753465\pi$
$692$	4630.77	0.254387
$693$	1887.71	0.103475
$694$	−23446.1	−1.28242
$695$	−34572.9	−1.88694
$696$	−6824.57	−0.371674
$697$	419.220	0.0227821
$698$	20408.8	1.10671
$699$	−5983.34	−0.323764
$700$	4935.92	0.266515
$701$	32481.0	1.75006	0.875029	−	0.484070i	$-0.160842\pi$
$701$	32481.0	1.75006	0.875029	+	0.484070i	$0.160842\pi$
$702$	907.180	0.0487739
$703$	−5385.01	−0.288904
$704$	1997.74	0.106950
$705$	−21109.8	−1.12772
$706$	16722.1	0.891423
$707$	1856.01	0.0987302
$708$	−708.000	−0.0375823
$709$	21568.1	1.14246	0.571231	−	0.820789i	$-0.306466\pi$
$709$	21568.1	1.14246	0.571231	+	0.820789i	$0.306466\pi$
$710$	−30802.2	−1.62815
$711$	−6914.65	−0.364725
$712$	7037.82	0.370440
$713$	101.989	0.00535699
$714$	530.216	0.0277911
$715$	−9212.70	−0.481868
$716$	−4588.47	−0.239496
$717$	6376.16	0.332109
$718$	11369.1	0.590937
$719$	1089.21	0.0564959	0.0282479	−	0.999601i	$-0.491007\pi$
$719$	1089.21	0.0564959	0.0282479	+	0.999601i	$0.491007\pi$
$720$	−2529.82	−0.130946
$721$	−4568.06	−0.235955
$722$	7321.42	0.377389
$723$	9310.50	0.478923
$724$	−10085.7	−0.517722
$725$	−52220.1	−2.67504
$726$	−2139.86	−0.109391
$727$	−29261.0	−1.49275	−0.746375	−	0.665525i	$-0.768208\pi$
$727$	−29261.0	−1.49275	−0.746375	+	0.665525i	$0.768208\pi$
$728$	−903.077	−0.0459756
$729$	729.000	0.0370370
$730$	−34498.8	−1.74912
$731$	5914.32	0.299246
$732$	6805.37	0.343625
$733$	28862.9	1.45440	0.727199	−	0.686426i	$-0.240822\pi$
$733$	28862.9	1.45440	0.727199	+	0.686426i	$0.240822\pi$
$734$	4883.15	0.245559
$735$	−15698.0	−0.787796
$736$	−170.076	−0.00851777
$737$	17271.5	0.863235
$738$	−573.783	−0.0286196
$739$	−38244.8	−1.90373	−0.951865	−	0.306517i	$-0.900836\pi$
$739$	−38244.8	−1.90373	−0.951865	+	0.306517i	$0.900836\pi$
$740$	6691.38	0.332405
$741$	−2850.23	−0.141303
$742$	−1145.51	−0.0566751
$743$	−1258.93	−0.0621613	−0.0310806	−	0.999517i	$-0.509895\pi$
$743$	−1258.93	−0.0621613	−0.0310806	+	0.999517i	$0.509895\pi$
$744$	460.547	0.0226942
$745$	−5611.98	−0.275982
$746$	2047.33	0.100480
$747$	139.782	0.00684654
$748$	1642.05	0.0802663
$749$	−3071.10	−0.149821
$750$	−6181.48	−0.300954
$751$	20065.0	0.974945	0.487473	−	0.873138i	$-0.337919\pi$
$751$	20065.0	0.974945	0.487473	+	0.873138i	$0.337919\pi$
$752$	−6408.46	−0.310761
$753$	7304.40	0.353502
$754$	9554.20	0.461463
$755$	−13053.9	−0.629247
$756$	−725.702	−0.0349121
$757$	20152.3	0.967568	0.483784	−	0.875188i	$-0.339262\pi$
$757$	20152.3	0.967568	0.483784	+	0.875188i	$0.339262\pi$
$758$	15042.7	0.720813
$759$	−497.706	−0.0238018
$760$	7948.35	0.379364
$761$	−19217.4	−0.915414	−0.457707	−	0.889103i	$-0.651329\pi$
$761$	−19217.4	−0.915414	−0.457707	+	0.889103i	$0.651329\pi$
$762$	2175.55	0.103428
$763$	7669.38	0.363893
$764$	−2791.22	−0.132176
$765$	−2079.40	−0.0982754
$766$	−17084.9	−0.805877
$767$	991.178	0.0466615
$768$	−768.000	−0.0360844
$769$	−9327.94	−0.437418	−0.218709	−	0.975790i	$-0.570184\pi$
$769$	−9327.94	−0.437418	−0.218709	+	0.975790i	$0.570184\pi$
$770$	7369.73	0.344918
$771$	−11040.3	−0.515702
$772$	14382.1	0.670497
$773$	24269.2	1.12924	0.564621	−	0.825350i	$-0.309022\pi$
$773$	24269.2	1.12924	0.564621	+	0.825350i	$0.309022\pi$
$774$	−8094.88	−0.375923
$775$	3524.00	0.163336
$776$	3995.16	0.184817
$777$	1919.48	0.0886242
$778$	11691.7	0.538778
$779$	1802.75	0.0829141
$780$	3541.68	0.162580
$781$	−27364.2	−1.25374
$782$	−139.794	−0.00639263
$783$	7677.65	0.350417
$784$	−4765.58	−0.217091
$785$	−18175.6	−0.826387
$786$	−3861.42	−0.175232
$787$	−13391.7	−0.606561	−0.303281	−	0.952901i	$-0.598082\pi$
$787$	−13391.7	−0.606561	−0.303281	+	0.952901i	$0.598082\pi$
$788$	−17253.1	−0.779970
$789$	−3779.06	−0.170517
$790$	−26995.2	−1.21575
$791$	10513.1	0.472570
$792$	−2247.46	−0.100833
$793$	−9527.31	−0.426639
$794$	−5569.51	−0.248935
$795$	4492.44	0.200416
$796$	−9354.41	−0.416531
$797$	−3841.85	−0.170747	−0.0853735	−	0.996349i	$-0.527208\pi$
$797$	−3841.85	−0.170747	−0.0853735	+	0.996349i	$0.527208\pi$
$798$	2280.05	0.101144
$799$	−5267.45	−0.233228
$800$	−5876.56	−0.259710
$801$	−7917.55	−0.349254
$802$	−15852.6	−0.697973
$803$	−30648.2	−1.34689
$804$	−6639.76	−0.291252
$805$	−627.416	−0.0274702
$806$	−644.751	−0.0281767
$807$	−18531.5	−0.808352
$808$	−2209.71	−0.0962094
$809$	−31741.6	−1.37945	−0.689725	−	0.724072i	$-0.742268\pi$
$809$	−31741.6	−1.37945	−0.689725	+	0.724072i	$0.742268\pi$
$810$	2846.05	0.123457
$811$	−3754.50	−0.162563	−0.0812814	−	0.996691i	$-0.525901\pi$
$811$	−3754.50	−0.162563	−0.0812814	+	0.996691i	$0.525901\pi$
$812$	−7642.92	−0.330313
$813$	4863.29	0.209795
$814$	5944.52	0.255965
$815$	14675.6	0.630755
$816$	−631.260	−0.0270815
$817$	25433.0	1.08909
$818$	−13971.6	−0.597194
$819$	1015.96	0.0433462
$820$	−2240.08	−0.0953987
$821$	−1536.30	−0.0653073	−0.0326537	−	0.999467i	$-0.510396\pi$
$821$	−1536.30	−0.0653073	−0.0326537	+	0.999467i	$0.510396\pi$
$822$	−157.462	−0.00668141
$823$	9199.20	0.389628	0.194814	−	0.980840i	$-0.437590\pi$
$823$	9199.20	0.389628	0.194814	+	0.980840i	$0.437590\pi$
$824$	5438.60	0.229930
$825$	−17197.0	−0.725726
$826$	−792.897	−0.0334000
$827$	24586.3	1.03380	0.516899	−	0.856047i	$-0.327086\pi$
$827$	24586.3	1.03380	0.516899	+	0.856047i	$0.327086\pi$
$828$	191.335	0.00803064
$829$	−10879.5	−0.455802	−0.227901	−	0.973684i	$-0.573186\pi$
$829$	−10879.5	−0.455802	−0.227901	+	0.973684i	$0.573186\pi$
$830$	545.717	0.0228218
$831$	8006.09	0.334210
$832$	1075.18	0.0448017
$833$	−3917.08	−0.162928
$834$	11807.5	0.490242
$835$	14403.1	0.596935
$836$	7061.19	0.292125
$837$	−518.115	−0.0213963
$838$	−15069.3	−0.621193
$839$	−22245.5	−0.915376	−0.457688	−	0.889113i	$-0.651322\pi$
$839$	−22245.5	−0.915376	−0.457688	+	0.889113i	$0.651322\pi$
$840$	−2833.18	−0.116374
$841$	56470.1	2.31539
$842$	−13863.1	−0.567405
$843$	−15098.4	−0.616866
$844$	−22712.0	−0.926277
$845$	33639.2	1.36949
$846$	7209.52	0.292989
$847$	−2396.45	−0.0972173
$848$	1363.81	0.0552280
$849$	24423.2	0.987283
$850$	−4830.26	−0.194913
$851$	−506.082	−0.0203857
$852$	10519.8	0.423006
$853$	4994.18	0.200466	0.100233	−	0.994964i	$-0.468041\pi$
$853$	4994.18	0.200466	0.100233	+	0.994964i	$0.468041\pi$
$854$	7621.41	0.305385
$855$	−8941.89	−0.357668
$856$	3656.37	0.145995
$857$	9784.98	0.390022	0.195011	−	0.980801i	$-0.437526\pi$
$857$	9784.98	0.390022	0.195011	+	0.980801i	$0.437526\pi$
$858$	3146.37	0.125193
$859$	42743.2	1.69777	0.848883	−	0.528581i	$-0.177276\pi$
$859$	42743.2	1.69777	0.848883	+	0.528581i	$0.177276\pi$
$860$	−31602.8	−1.25308
$861$	−642.586	−0.0254347
$862$	−3762.52	−0.148668
$863$	34060.8	1.34350	0.671752	−	0.740776i	$-0.265542\pi$
$863$	34060.8	1.34350	0.671752	+	0.740776i	$0.265542\pi$
$864$	864.000	0.0340207
$865$	−20338.6	−0.799461
$866$	−24545.1	−0.963135
$867$	14220.1	0.557025
$868$	515.771	0.0201687
$869$	−23982.1	−0.936176
$870$	29973.9	1.16806
$871$	9295.46	0.361613
$872$	−9130.94	−0.354601
$873$	−4494.55	−0.174247
$874$	−601.149	−0.0232656
$875$	−6922.71	−0.267463
$876$	11782.2	0.454434
$877$	46307.8	1.78301	0.891507	−	0.453008i	$-0.149649\pi$
$877$	46307.8	1.78301	0.891507	+	0.453008i	$0.149649\pi$
$878$	114.227	0.00439062
$879$	8302.19	0.318573
$880$	−8774.19	−0.336111
$881$	−19976.7	−0.763940	−0.381970	−	0.924175i	$-0.624754\pi$
$881$	−19976.7	−0.763940	−0.381970	+	0.924175i	$0.624754\pi$
$882$	5361.28	0.204675
$883$	49011.7	1.86792	0.933961	−	0.357376i	$-0.116328\pi$
$883$	49011.7	1.86792	0.933961	+	0.357376i	$0.116328\pi$
$884$	883.744	0.0336239
$885$	3109.58	0.118110
$886$	−29735.9	−1.12754
$887$	−43962.3	−1.66416	−0.832080	−	0.554655i	$-0.812850\pi$
$887$	−43962.3	−1.66416	−0.832080	+	0.554655i	$0.812850\pi$
$888$	−2285.28	−0.0863614
$889$	2436.43	0.0919180
$890$	−30910.5	−1.16418
$891$	2528.39	0.0950665
$892$	−11094.6	−0.416451
$893$	−22651.3	−0.848820
$894$	1916.63	0.0717023
$895$	20152.8	0.752664
$896$	−860.092	−0.0320688
$897$	−267.864	−0.00997069
$898$	11732.9	0.436005
$899$	−5456.66	−0.202436
$900$	6611.13	0.244857
$901$	1120.99	0.0414489
$902$	−1990.05	−0.0734606
$903$	−9065.55	−0.334089
$904$	−12516.6	−0.460504
$905$	44296.8	1.62704
$906$	4458.26	0.163483
$907$	4834.39	0.176983	0.0884913	−	0.996077i	$-0.471795\pi$
$907$	4834.39	0.176983	0.0884913	+	0.996077i	$0.471795\pi$
$908$	−19283.0	−0.704767
$909$	2485.92	0.0907071
$910$	3966.36	0.144488
$911$	9041.52	0.328824	0.164412	−	0.986392i	$-0.447427\pi$
$911$	9041.52	0.328824	0.164412	+	0.986392i	$0.447427\pi$
$912$	−2714.57	−0.0985617
$913$	484.807	0.0175737
$914$	−16095.3	−0.582477
$915$	−29889.6	−1.07991
$916$	17669.9	0.637369
$917$	−4324.45	−0.155732
$918$	710.167	0.0255327
$919$	12357.3	0.443560	0.221780	−	0.975097i	$-0.428813\pi$
$919$	12357.3	0.443560	0.221780	+	0.975097i	$0.428813\pi$
$920$	746.983	0.0267688
$921$	−3695.03	−0.132199
$922$	9688.28	0.346059
$923$	−14727.3	−0.525196
$924$	−2516.95	−0.0896122
$925$	−17486.4	−0.621568
$926$	13191.1	0.468128
$927$	−6118.42	−0.216780
$928$	9099.43	0.321879
$929$	33139.6	1.17037	0.585186	−	0.810899i	$-0.301022\pi$
$929$	33139.6	1.17037	0.585186	+	0.810899i	$0.301022\pi$
$930$	−2022.75	−0.0713209
$931$	−16844.4	−0.592967
$932$	7977.79	0.280388
$933$	20933.7	0.734555
$934$	−8975.53	−0.314441
$935$	−7211.97	−0.252253
$936$	−1209.57	−0.0422395
$937$	−8742.57	−0.304810	−0.152405	−	0.988318i	$-0.548702\pi$
$937$	−8742.57	−0.304810	−0.152405	+	0.988318i	$0.548702\pi$
$938$	−7435.94	−0.258840
$939$	10095.6	0.350860
$940$	28146.3	0.976630
$941$	−21635.8	−0.749530	−0.374765	−	0.927120i	$-0.622277\pi$
$941$	−21635.8	−0.749530	−0.374765	+	0.927120i	$0.622277\pi$
$942$	6207.42	0.214701
$943$	169.421	0.00585060
$944$	944.000	0.0325472
$945$	3187.33	0.109718
$946$	−28075.5	−0.964917
$947$	37305.5	1.28011	0.640056	−	0.768328i	$-0.278911\pi$
$947$	37305.5	1.28011	0.640056	+	0.768328i	$0.278911\pi$
$948$	9219.54	0.315862
$949$	−16494.7	−0.564217
$950$	−20771.2	−0.709377
$951$	−9589.97	−0.326999
$952$	−706.955	−0.0240678
$953$	46246.7	1.57196	0.785979	−	0.618253i	$-0.212159\pi$
$953$	46246.7	1.57196	0.785979	+	0.618253i	$0.212159\pi$
$954$	−1534.28	−0.0520695
$955$	12259.2	0.415391
$956$	−8501.55	−0.287615
$957$	26628.4	0.899449
$958$	22123.3	0.746108
$959$	−176.343	−0.00593788
$960$	3373.10	0.113402
$961$	−29422.8	−0.987639
$962$	3199.32	0.107225
$963$	−4113.41	−0.137646
$964$	−12414.0	−0.414760
$965$	−63167.1	−2.10717
$966$	214.279	0.00713696
$967$	−4220.96	−0.140369	−0.0701845	−	0.997534i	$-0.522359\pi$
$967$	−4220.96	−0.140369	−0.0701845	+	0.997534i	$0.522359\pi$
$968$	2853.15	0.0947351
$969$	−2231.24	−0.0739710
$970$	−17547.0	−0.580824
$971$	34547.4	1.14179	0.570895	−	0.821023i	$-0.306596\pi$
$971$	34547.4	1.14179	0.570895	+	0.821023i	$0.306596\pi$
$972$	−972.000	−0.0320750
$973$	13223.4	0.435686
$974$	−21949.0	−0.722064
$975$	−9255.39	−0.304010
$976$	−9073.82	−0.297588
$977$	−41781.5	−1.36818	−0.684088	−	0.729399i	$-0.739800\pi$
$977$	−41781.5	−1.36818	−0.684088	+	0.729399i	$0.739800\pi$
$978$	−5012.11	−0.163875
$979$	−27460.4	−0.896465
$980$	20930.7	0.682252
$981$	10272.3	0.334321
$982$	31331.8	1.01817
$983$	15767.6	0.511604	0.255802	−	0.966729i	$-0.417660\pi$
$983$	15767.6	0.511604	0.255802	+	0.966729i	$0.417660\pi$
$984$	765.044	0.0247853
$985$	75776.6	2.45121
$986$	7479.30	0.241571
$987$	8074.02	0.260384
$988$	3800.31	0.122372
$989$	2390.18	0.0768487
$990$	9870.96	0.316889
$991$	−25035.0	−0.802485	−0.401242	−	0.915972i	$-0.631421\pi$
$991$	−25035.0	−0.802485	−0.401242	+	0.915972i	$0.631421\pi$
$992$	−614.062	−0.0196537
$993$	5612.52	0.179364
$994$	11781.2	0.375932
$995$	41085.1	1.30903
$996$	−186.376	−0.00592928
$997$	−51179.0	−1.62573	−0.812867	−	0.582450i	$-0.802094\pi$
$997$	−51179.0	−1.62573	−0.812867	+	0.582450i	$0.802094\pi$
$998$	−19611.6	−0.622038
$999$	2570.94	0.0814223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.4.a.d.1.1	✓	3
3.2	odd	2		1062.4.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.4.a.d.1.1	✓	3	1.1	even	1	trivial
1062.4.a.l.1.3		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -17.5682 q^{5} +6.00000 q^{6} +6.71947 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -17.5682 q^{5} +6.00000 q^{6} +6.71947 q^{7} -8.00000 q^{8} +9.00000 q^{9} +35.1365 q^{10} +31.2147 q^{11} -12.0000 q^{12} +16.7996 q^{13} -13.4389 q^{14} +52.7047 q^{15} +16.0000 q^{16} +13.1512 q^{17} -18.0000 q^{18} +56.5534 q^{19} -70.2729 q^{20} -20.1584 q^{21} -62.4294 q^{22} +5.31487 q^{23} +24.0000 q^{24} +183.643 q^{25} -33.5993 q^{26} -27.0000 q^{27} +26.8779 q^{28} -284.357 q^{29} -105.409 q^{30} +19.1894 q^{31} -32.0000 q^{32} -93.6441 q^{33} -26.3025 q^{34} -118.049 q^{35} +36.0000 q^{36} -95.2199 q^{37} -113.107 q^{38} -50.3989 q^{39} +140.546 q^{40} +31.8768 q^{41} +40.3168 q^{42} +449.715 q^{43} +124.859 q^{44} -158.114 q^{45} -10.6297 q^{46} -400.529 q^{47} -48.0000 q^{48} -297.849 q^{49} -367.285 q^{50} -39.4537 q^{51} +67.1985 q^{52} +85.2380 q^{53} +54.0000 q^{54} -548.387 q^{55} -53.7557 q^{56} -169.660 q^{57} +568.715 q^{58} +59.0000 q^{59} +210.819 q^{60} -567.114 q^{61} -38.3789 q^{62} +60.4752 q^{63} +64.0000 q^{64} -295.140 q^{65} +187.288 q^{66} +553.313 q^{67} +52.6050 q^{68} -15.9446 q^{69} +236.098 q^{70} -876.646 q^{71} -72.0000 q^{72} -981.851 q^{73} +190.440 q^{74} -550.928 q^{75} +226.214 q^{76} +209.746 q^{77} +100.798 q^{78} -768.295 q^{79} -281.092 q^{80} +81.0000 q^{81} -63.7537 q^{82} +15.5314 q^{83} -80.6336 q^{84} -231.044 q^{85} -899.431 q^{86} +853.072 q^{87} -249.718 q^{88} -879.728 q^{89} +316.228 q^{90} +112.885 q^{91} +21.2595 q^{92} -57.5683 q^{93} +801.058 q^{94} -993.544 q^{95} +96.0000 q^{96} -499.395 q^{97} +595.698 q^{98} +280.932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} - 6 q^{5} + 18 q^{6} + 6 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 - 6 * q^5 + 18 * q^6 + 6 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} - 6 q^{5} + 18 q^{6} + 6 q^{7} - 24 q^{8} + 27 q^{9} + 12 q^{10} - 45 q^{11} - 36 q^{12} + 39 q^{13} - 12 q^{14} + 18 q^{15} + 48 q^{16} + 72 q^{17} - 54 q^{18} - 3 q^{19} - 24 q^{20} - 18 q^{21} + 90 q^{22} - 117 q^{23} + 72 q^{24} + 363 q^{25} - 78 q^{26} - 81 q^{27} + 24 q^{28} - 3 q^{29} - 36 q^{30} + 39 q^{31} - 96 q^{32} + 135 q^{33} - 144 q^{34} - 333 q^{35} + 108 q^{36} - 24 q^{37} + 6 q^{38} - 117 q^{39} + 48 q^{40} - 504 q^{41} + 36 q^{42} + 201 q^{43} - 180 q^{44} - 54 q^{45} + 234 q^{46} - 663 q^{47} - 144 q^{48} - 861 q^{49} - 726 q^{50} - 216 q^{51} + 156 q^{52} - 1098 q^{53} + 162 q^{54} - 1056 q^{55} - 48 q^{56} + 9 q^{57} + 6 q^{58} + 177 q^{59} + 72 q^{60} - 243 q^{61} - 78 q^{62} + 54 q^{63} + 192 q^{64} - 1668 q^{65} - 270 q^{66} - 330 q^{67} + 288 q^{68} + 351 q^{69} + 666 q^{70} - 2271 q^{71} - 216 q^{72} - 381 q^{73} + 48 q^{74} - 1089 q^{75} - 12 q^{76} + 276 q^{77} + 234 q^{78} - 1113 q^{79} - 96 q^{80} + 243 q^{81} + 1008 q^{82} - 2262 q^{83} - 72 q^{84} + 261 q^{85} - 402 q^{86} + 9 q^{87} + 360 q^{88} - 2055 q^{89} + 108 q^{90} + 978 q^{91} - 468 q^{92} - 117 q^{93} + 1326 q^{94} - 2577 q^{95} + 288 q^{96} - 18 q^{97} + 1722 q^{98} - 405 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 - 6 * q^5 + 18 * q^6 + 6 * q^7 - 24 * q^8 + 27 * q^9 + 12 * q^10 - 45 * q^11 - 36 * q^12 + 39 * q^13 - 12 * q^14 + 18 * q^15 + 48 * q^16 + 72 * q^17 - 54 * q^18 - 3 * q^19 - 24 * q^20 - 18 * q^21 + 90 * q^22 - 117 * q^23 + 72 * q^24 + 363 * q^25 - 78 * q^26 - 81 * q^27 + 24 * q^28 - 3 * q^29 - 36 * q^30 + 39 * q^31 - 96 * q^32 + 135 * q^33 - 144 * q^34 - 333 * q^35 + 108 * q^36 - 24 * q^37 + 6 * q^38 - 117 * q^39 + 48 * q^40 - 504 * q^41 + 36 * q^42 + 201 * q^43 - 180 * q^44 - 54 * q^45 + 234 * q^46 - 663 * q^47 - 144 * q^48 - 861 * q^49 - 726 * q^50 - 216 * q^51 + 156 * q^52 - 1098 * q^53 + 162 * q^54 - 1056 * q^55 - 48 * q^56 + 9 * q^57 + 6 * q^58 + 177 * q^59 + 72 * q^60 - 243 * q^61 - 78 * q^62 + 54 * q^63 + 192 * q^64 - 1668 * q^65 - 270 * q^66 - 330 * q^67 + 288 * q^68 + 351 * q^69 + 666 * q^70 - 2271 * q^71 - 216 * q^72 - 381 * q^73 + 48 * q^74 - 1089 * q^75 - 12 * q^76 + 276 * q^77 + 234 * q^78 - 1113 * q^79 - 96 * q^80 + 243 * q^81 + 1008 * q^82 - 2262 * q^83 - 72 * q^84 + 261 * q^85 - 402 * q^86 + 9 * q^87 + 360 * q^88 - 2055 * q^89 + 108 * q^90 + 978 * q^91 - 468 * q^92 - 117 * q^93 + 1326 * q^94 - 2577 * q^95 + 288 * q^96 - 18 * q^97 + 1722 * q^98 - 405 * q^99

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