LMFDB - Embedded newform 354.4.a.d.1.2

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			1.2
Root			$-5.26097$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -7.67779 q^{5} +6.00000 q^{6} +7.46938 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -7.67779 q^{5} +6.00000 q^{6} +7.46938 q^{7} -8.00000 q^{8} +9.00000 q^{9} +15.3556 q^{10} -35.6271 q^{11} -12.0000 q^{12} +66.8601 q^{13} -14.9388 q^{14} +23.0334 q^{15} +16.0000 q^{16} +23.7916 q^{17} -18.0000 q^{18} +16.2418 q^{19} -30.7111 q^{20} -22.4081 q^{21} +71.2541 q^{22} -86.5571 q^{23} +24.0000 q^{24} -66.0516 q^{25} -133.720 q^{26} -27.0000 q^{27} +29.8775 q^{28} +251.735 q^{29} -46.0667 q^{30} +131.451 q^{31} -32.0000 q^{32} +106.881 q^{33} -47.5832 q^{34} -57.3483 q^{35} +36.0000 q^{36} +76.3839 q^{37} -32.4835 q^{38} -200.580 q^{39} +61.4223 q^{40} -406.202 q^{41} +44.8163 q^{42} -528.064 q^{43} -142.508 q^{44} -69.1001 q^{45} +173.114 q^{46} +152.061 q^{47} -48.0000 q^{48} -287.208 q^{49} +132.103 q^{50} -71.3748 q^{51} +267.441 q^{52} -696.191 q^{53} +54.0000 q^{54} +273.537 q^{55} -59.7550 q^{56} -48.7253 q^{57} -503.470 q^{58} +59.0000 q^{59} +92.1334 q^{60} +307.686 q^{61} -262.901 q^{62} +67.2244 q^{63} +64.0000 q^{64} -513.338 q^{65} -213.762 q^{66} -812.626 q^{67} +95.1664 q^{68} +259.671 q^{69} +114.697 q^{70} -425.818 q^{71} -72.0000 q^{72} +502.024 q^{73} -152.768 q^{74} +198.155 q^{75} +64.9671 q^{76} -266.112 q^{77} +401.161 q^{78} -658.325 q^{79} -122.845 q^{80} +81.0000 q^{81} +812.405 q^{82} -1271.71 q^{83} -89.6325 q^{84} -182.667 q^{85} +1056.13 q^{86} -755.205 q^{87} +285.016 q^{88} -527.379 q^{89} +138.200 q^{90} +499.404 q^{91} -346.228 q^{92} -394.352 q^{93} -304.122 q^{94} -124.701 q^{95} +96.0000 q^{96} -1010.54 q^{97} +574.417 q^{98} -320.643 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−7.67779	−0.686722	−0.343361	−	0.939204i	$-0.611565\pi$
$5$	−7.67779	−0.686722	−0.343361	+	0.939204i	$0.611565\pi$
$6$	6.00000	0.408248
$7$	7.46938	0.403308	0.201654	−	0.979457i	$-0.435368\pi$
$7$	7.46938	0.403308	0.201654	+	0.979457i	$0.435368\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	15.3556	0.485586
$11$	−35.6271	−0.976542	−0.488271	−	0.872692i	$-0.662372\pi$
$11$	−35.6271	−0.976542	−0.488271	+	0.872692i	$0.662372\pi$
$12$	−12.0000	−0.288675
$13$	66.8601	1.42644	0.713218	−	0.700942i	$-0.247237\pi$
$13$	66.8601	1.42644	0.713218	+	0.700942i	$0.247237\pi$
$14$	−14.9388	−0.285182
$15$	23.0334	0.396479
$16$	16.0000	0.250000
$17$	23.7916	0.339430	0.169715	−	0.985493i	$-0.445715\pi$
$17$	23.7916	0.339430	0.169715	+	0.985493i	$0.445715\pi$
$18$	−18.0000	−0.235702
$19$	16.2418	0.196111	0.0980557	−	0.995181i	$-0.468738\pi$
$19$	16.2418	0.196111	0.0980557	+	0.995181i	$0.468738\pi$
$20$	−30.7111	−0.343361
$21$	−22.4081	−0.232850
$22$	71.2541	0.690519
$23$	−86.5571	−0.784713	−0.392357	−	0.919813i	$-0.628340\pi$
$23$	−86.5571	−0.784713	−0.392357	+	0.919813i	$0.628340\pi$
$24$	24.0000	0.204124
$25$	−66.0516	−0.528413
$26$	−133.720	−1.00864
$27$	−27.0000	−0.192450
$28$	29.8775	0.201654
$29$	251.735	1.61193	0.805965	−	0.591963i	$-0.201647\pi$
$29$	251.735	1.61193	0.805965	+	0.591963i	$0.201647\pi$
$30$	−46.0667	−0.280353
$31$	131.451	0.761588	0.380794	−	0.924660i	$-0.375651\pi$
$31$	131.451	0.761588	0.380794	+	0.924660i	$0.375651\pi$
$32$	−32.0000	−0.176777
$33$	106.881	0.563807
$34$	−47.5832	−0.240013
$35$	−57.3483	−0.276961
$36$	36.0000	0.166667
$37$	76.3839	0.339390	0.169695	−	0.985497i	$-0.445722\pi$
$37$	76.3839	0.339390	0.169695	+	0.985497i	$0.445722\pi$
$38$	−32.4835	−0.138672
$39$	−200.580	−0.823553
$40$	61.4223	0.242793
$41$	−406.202	−1.54727	−0.773636	−	0.633630i	$-0.781564\pi$
$41$	−406.202	−1.54727	−0.773636	+	0.633630i	$0.781564\pi$
$42$	44.8163	0.164650
$43$	−528.064	−1.87277	−0.936384	−	0.350978i	$-0.885849\pi$
$43$	−528.064	−1.87277	−0.936384	+	0.350978i	$0.885849\pi$
$44$	−142.508	−0.488271
$45$	−69.1001	−0.228907
$46$	173.114	0.554876
$47$	152.061	0.471923	0.235962	−	0.971762i	$-0.424176\pi$
$47$	152.061	0.471923	0.235962	+	0.971762i	$0.424176\pi$
$48$	−48.0000	−0.144338
$49$	−287.208	−0.837342
$50$	132.103	0.373644
$51$	−71.3748	−0.195970
$52$	267.441	0.713218
$53$	−696.191	−1.80432	−0.902162	−	0.431398i	$-0.858020\pi$
$53$	−696.191	−1.80432	−0.902162	+	0.431398i	$0.858020\pi$
$54$	54.0000	0.136083
$55$	273.537	0.670613
$56$	−59.7550	−0.142591
$57$	−48.7253	−0.113225
$58$	−503.470	−1.13981
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	92.1334	0.198240
$61$	307.686	0.645823	0.322912	−	0.946429i	$-0.395338\pi$
$61$	307.686	0.645823	0.322912	+	0.946429i	$0.395338\pi$
$62$	−262.901	−0.538524
$63$	67.2244	0.134436
$64$	64.0000	0.125000
$65$	−513.338	−0.979565
$66$	−213.762	−0.398672
$67$	−812.626	−1.48176	−0.740882	−	0.671636i	$-0.765592\pi$
$67$	−812.626	−1.48176	−0.740882	+	0.671636i	$0.765592\pi$
$68$	95.1664	0.169715
$69$	259.671	0.453054
$70$	114.697	0.195841
$71$	−425.818	−0.711765	−0.355883	−	0.934531i	$-0.615820\pi$
$71$	−425.818	−0.711765	−0.355883	+	0.934531i	$0.615820\pi$
$72$	−72.0000	−0.117851
$73$	502.024	0.804896	0.402448	−	0.915443i	$-0.368159\pi$
$73$	502.024	0.804896	0.402448	+	0.915443i	$0.368159\pi$
$74$	−152.768	−0.239985
$75$	198.155	0.305079
$76$	64.9671	0.0980557
$77$	−266.112	−0.393848
$78$	401.161	0.582340
$79$	−658.325	−0.937562	−0.468781	−	0.883314i	$-0.655307\pi$
$79$	−658.325	−0.937562	−0.468781	+	0.883314i	$0.655307\pi$
$80$	−122.845	−0.171681
$81$	81.0000	0.111111
$82$	812.405	1.09409
$83$	−1271.71	−1.68179	−0.840893	−	0.541201i	$-0.817970\pi$
$83$	−1271.71	−1.68179	−0.840893	+	0.541201i	$0.817970\pi$
$84$	−89.6325	−0.116425
$85$	−182.667	−0.233094
$86$	1056.13	1.32425
$87$	−755.205	−0.930649
$88$	285.016	0.345260
$89$	−527.379	−0.628113	−0.314057	−	0.949404i	$-0.601688\pi$
$89$	−527.379	−0.628113	−0.314057	+	0.949404i	$0.601688\pi$
$90$	138.200	0.161862
$91$	499.404	0.575294
$92$	−346.228	−0.392357
$93$	−394.352	−0.439703
$94$	−304.122	−0.333700
$95$	−124.701	−0.134674
$96$	96.0000	0.102062
$97$	−1010.54	−1.05778	−0.528891	−	0.848690i	$-0.677392\pi$
$97$	−1010.54	−1.05778	−0.528891	+	0.848690i	$0.677392\pi$
$98$	574.417	0.592090
$99$	−320.643	−0.325514
$100$	−264.206	−0.264206
$101$	−1157.13	−1.13999	−0.569996	−	0.821647i	$-0.693055\pi$
$101$	−1157.13	−1.13999	−0.569996	+	0.821647i	$0.693055\pi$
$102$	142.750	0.138572
$103$	1666.94	1.59464	0.797321	−	0.603556i	$-0.206250\pi$
$103$	1666.94	1.59464	0.797321	+	0.603556i	$0.206250\pi$
$104$	−534.881	−0.504321
$105$	172.045	0.159903
$106$	1392.38	1.27585
$107$	1158.97	1.04712	0.523561	−	0.851988i	$-0.324603\pi$
$107$	1158.97	1.04712	0.523561	+	0.851988i	$0.324603\pi$
$108$	−108.000	−0.0962250
$109$	−691.474	−0.607625	−0.303813	−	0.952732i	$-0.598260\pi$
$109$	−691.474	−0.607625	−0.303813	+	0.952732i	$0.598260\pi$
$110$	−547.074	−0.474195
$111$	−229.152	−0.195947
$112$	119.510	0.100827
$113$	−952.674	−0.793098	−0.396549	−	0.918014i	$-0.629792\pi$
$113$	−952.674	−0.793098	−0.396549	+	0.918014i	$0.629792\pi$
$114$	97.4506	0.0800621
$115$	664.567	0.538880
$116$	1006.94	0.805965
$117$	601.741	0.475479
$118$	−118.000	−0.0920575
$119$	177.708	0.136895
$120$	−184.267	−0.140177
$121$	−61.7130	−0.0463659
$122$	−615.373	−0.456666
$123$	1218.61	0.893318
$124$	525.802	0.380794
$125$	1466.85	1.04959
$126$	−134.449	−0.0950607
$127$	471.104	0.329163	0.164581	−	0.986363i	$-0.447373\pi$
$127$	471.104	0.329163	0.164581	+	0.986363i	$0.447373\pi$
$128$	−128.000	−0.0883883
$129$	1584.19	1.08124
$130$	1026.68	0.692657
$131$	−83.9060	−0.0559611	−0.0279805	−	0.999608i	$-0.508908\pi$
$131$	−83.9060	−0.0559611	−0.0279805	+	0.999608i	$0.508908\pi$
$132$	427.525	0.281903
$133$	121.316	0.0790934
$134$	1625.25	1.04776
$135$	207.300	0.132160
$136$	−190.333	−0.120007
$137$	2498.91	1.55837	0.779184	−	0.626795i	$-0.215634\pi$
$137$	2498.91	1.55837	0.779184	+	0.626795i	$0.215634\pi$
$138$	−519.343	−0.320358
$139$	−1514.17	−0.923959	−0.461979	−	0.886891i	$-0.652861\pi$
$139$	−1514.17	−0.923959	−0.461979	+	0.886891i	$0.652861\pi$
$140$	−229.393	−0.138480
$141$	−456.184	−0.272465
$142$	851.636	0.503294
$143$	−2382.03	−1.39297
$144$	144.000	0.0833333
$145$	−1932.77	−1.10695
$146$	−1004.05	−0.569148
$147$	861.625	0.483440
$148$	305.536	0.169695
$149$	2353.78	1.29416	0.647078	−	0.762424i	$-0.275991\pi$
$149$	2353.78	1.29416	0.647078	+	0.762424i	$0.275991\pi$
$150$	−396.310	−0.215724
$151$	460.620	0.248243	0.124121	−	0.992267i	$-0.460389\pi$
$151$	460.620	0.248243	0.124121	+	0.992267i	$0.460389\pi$
$152$	−129.934	−0.0693359
$153$	214.124	0.113143
$154$	532.224	0.278492
$155$	−1009.25	−0.522999
$156$	−802.322	−0.411777
$157$	2575.14	1.30903	0.654517	−	0.756047i	$-0.272872\pi$
$157$	2575.14	1.30903	0.654517	+	0.756047i	$0.272872\pi$
$158$	1316.65	0.662956
$159$	2088.57	1.04173
$160$	245.689	0.121396
$161$	−646.528	−0.316482
$162$	−162.000	−0.0785674
$163$	2786.33	1.33891	0.669454	−	0.742854i	$-0.266528\pi$
$163$	2786.33	1.33891	0.669454	+	0.742854i	$0.266528\pi$
$164$	−1624.81	−0.773636
$165$	−820.611	−0.387179
$166$	2543.42	1.18920
$167$	−1342.32	−0.621986	−0.310993	−	0.950412i	$-0.600661\pi$
$167$	−1342.32	−0.621986	−0.310993	+	0.950412i	$0.600661\pi$
$168$	179.265	0.0823250
$169$	2273.28	1.03472
$170$	365.333	0.164822
$171$	146.176	0.0653705
$172$	−2112.26	−0.936384
$173$	2328.40	1.02327	0.511633	−	0.859204i	$-0.329041\pi$
$173$	2328.40	1.02327	0.511633	+	0.859204i	$0.329041\pi$
$174$	1510.41	0.658068
$175$	−493.364	−0.213113
$176$	−570.033	−0.244135
$177$	−177.000	−0.0751646
$178$	1054.76	0.444143
$179$	−2588.37	−1.08080	−0.540402	−	0.841407i	$-0.681728\pi$
$179$	−2588.37	−1.08080	−0.540402	+	0.841407i	$0.681728\pi$
$180$	−276.400	−0.114454
$181$	3235.63	1.32874	0.664372	−	0.747402i	$-0.268699\pi$
$181$	3235.63	1.32874	0.664372	+	0.747402i	$0.268699\pi$
$182$	−998.807	−0.406794
$183$	−923.059	−0.372866
$184$	692.457	0.277438
$185$	−586.459	−0.233067
$186$	788.704	0.310917
$187$	−847.624	−0.331468
$188$	608.245	0.235962
$189$	−201.673	−0.0776167
$190$	249.402	0.0952289
$191$	−5055.46	−1.91518	−0.957592	−	0.288127i	$-0.906968\pi$
$191$	−5055.46	−1.91518	−0.957592	+	0.288127i	$0.906968\pi$
$192$	−192.000	−0.0721688
$193$	−3216.34	−1.19957	−0.599785	−	0.800161i	$-0.704747\pi$
$193$	−3216.34	−1.19957	−0.599785	+	0.800161i	$0.704747\pi$
$194$	2021.08	0.747965
$195$	1540.01	0.565552
$196$	−1148.83	−0.418671
$197$	−140.757	−0.0509064	−0.0254532	−	0.999676i	$-0.508103\pi$
$197$	−140.757	−0.0509064	−0.0254532	+	0.999676i	$0.508103\pi$
$198$	641.287	0.230173
$199$	−1107.10	−0.394372	−0.197186	−	0.980366i	$-0.563180\pi$
$199$	−1107.10	−0.394372	−0.197186	+	0.980366i	$0.563180\pi$
$200$	528.413	0.186822
$201$	2437.88	0.855496
$202$	2314.27	0.806096
$203$	1880.30	0.650105
$204$	−285.499	−0.0979850
$205$	3118.74	1.06255
$206$	−3333.87	−1.12758
$207$	−779.014	−0.261571
$208$	1069.76	0.356609
$209$	−578.646	−0.191511
$210$	−344.090	−0.113069
$211$	−3619.70	−1.18100	−0.590499	−	0.807038i	$-0.701069\pi$
$211$	−3619.70	−1.18100	−0.590499	+	0.807038i	$0.701069\pi$
$212$	−2784.76	−0.902162
$213$	1277.45	0.410938
$214$	−2317.94	−0.740427
$215$	4054.36	1.28607
$216$	216.000	0.0680414
$217$	981.854	0.307155
$218$	1382.95	0.429656
$219$	−1506.07	−0.464707
$220$	1094.15	0.335306
$221$	1590.71	0.484175
$222$	458.303	0.138555
$223$	−4615.80	−1.38608	−0.693042	−	0.720897i	$-0.743730\pi$
$223$	−4615.80	−1.38608	−0.693042	+	0.720897i	$0.743730\pi$
$224$	−239.020	−0.0712955
$225$	−594.464	−0.176138
$226$	1905.35	0.560805
$227$	−942.648	−0.275620	−0.137810	−	0.990459i	$-0.544006\pi$
$227$	−942.648	−0.275620	−0.137810	+	0.990459i	$0.544006\pi$
$228$	−194.901	−0.0566125
$229$	−1694.37	−0.488939	−0.244469	−	0.969657i	$-0.578614\pi$
$229$	−1694.37	−0.488939	−0.244469	+	0.969657i	$0.578614\pi$
$230$	−1329.13	−0.381046
$231$	798.336	0.227388
$232$	−2013.88	−0.569904
$233$	−2075.12	−0.583457	−0.291728	−	0.956501i	$-0.594230\pi$
$233$	−2075.12	−0.583457	−0.291728	+	0.956501i	$0.594230\pi$
$234$	−1203.48	−0.336214
$235$	−1167.49	−0.324080
$236$	236.000	0.0650945
$237$	1974.98	0.541302
$238$	−355.417	−0.0967993
$239$	−6362.89	−1.72210	−0.861048	−	0.508523i	$-0.830192\pi$
$239$	−6362.89	−1.72210	−0.861048	+	0.508523i	$0.830192\pi$
$240$	368.534	0.0991198
$241$	−3069.86	−0.820528	−0.410264	−	0.911967i	$-0.634564\pi$
$241$	−3069.86	−0.820528	−0.410264	+	0.911967i	$0.634564\pi$
$242$	123.426	0.0327856
$243$	−243.000	−0.0641500
$244$	1230.75	0.322912
$245$	2205.12	0.575021
$246$	−2437.21	−0.631671
$247$	1085.93	0.279740
$248$	−1051.60	−0.269262
$249$	3815.13	0.970980
$250$	−2933.71	−0.742176
$251$	−937.411	−0.235732	−0.117866	−	0.993029i	$-0.537605\pi$
$251$	−937.411	−0.235732	−0.117866	+	0.993029i	$0.537605\pi$
$252$	268.898	0.0672181
$253$	3083.78	0.766306
$254$	−942.207	−0.232753
$255$	548.000	0.134577
$256$	256.000	0.0625000
$257$	5289.18	1.28377	0.641887	−	0.766800i	$-0.278152\pi$
$257$	5289.18	1.28377	0.641887	+	0.766800i	$0.278152\pi$
$258$	−3168.38	−0.764554
$259$	570.540	0.136879
$260$	−2053.35	−0.489782
$261$	2265.61	0.537310
$262$	167.812	0.0395705
$263$	6110.59	1.43268	0.716340	−	0.697751i	$-0.245816\pi$
$263$	6110.59	1.43268	0.716340	+	0.697751i	$0.245816\pi$
$264$	−855.049	−0.199336
$265$	5345.20	1.23907
$266$	−242.632	−0.0559275
$267$	1582.14	0.362641
$268$	−3250.51	−0.740882
$269$	292.382	0.0662709	0.0331354	−	0.999451i	$-0.489451\pi$
$269$	292.382	0.0662709	0.0331354	+	0.999451i	$0.489451\pi$
$270$	−414.600	−0.0934510
$271$	−5885.04	−1.31915	−0.659577	−	0.751637i	$-0.729264\pi$
$271$	−5885.04	−1.31915	−0.659577	+	0.751637i	$0.729264\pi$
$272$	380.665	0.0848575
$273$	−1498.21	−0.332146
$274$	−4997.82	−1.10193
$275$	2353.22	0.516017
$276$	1038.69	0.226527
$277$	4631.69	1.00466	0.502331	−	0.864675i	$-0.332476\pi$
$277$	4631.69	1.00466	0.502331	+	0.864675i	$0.332476\pi$
$278$	3028.34	0.653337
$279$	1183.06	0.253863
$280$	458.786	0.0979204
$281$	−2702.22	−0.573670	−0.286835	−	0.957980i	$-0.592603\pi$
$281$	−2702.22	−0.573670	−0.286835	+	0.957980i	$0.592603\pi$
$282$	912.367	0.192662
$283$	874.152	0.183615	0.0918073	−	0.995777i	$-0.470736\pi$
$283$	874.152	0.183615	0.0918073	+	0.995777i	$0.470736\pi$
$284$	−1703.27	−0.355883
$285$	374.102	0.0777541
$286$	4764.06	0.984982
$287$	−3034.08	−0.624028
$288$	−288.000	−0.0589256
$289$	−4346.96	−0.884787
$290$	3865.53	0.782731
$291$	3031.62	0.610711
$292$	2008.09	0.402448
$293$	1685.44	0.336055	0.168028	−	0.985782i	$-0.446260\pi$
$293$	1685.44	0.336055	0.168028	+	0.985782i	$0.446260\pi$
$294$	−1723.25	−0.341844
$295$	−452.989	−0.0894036
$296$	−611.071	−0.119993
$297$	961.930	0.187936
$298$	−4707.56	−0.915106
$299$	−5787.22	−1.11934
$300$	792.619	0.152540
$301$	−3944.31	−0.755303
$302$	−921.239	−0.175534
$303$	3471.40	0.658175
$304$	259.868	0.0490279
$305$	−2362.35	−0.443501
$306$	−428.249	−0.0800044
$307$	−770.260	−0.143196	−0.0715978	−	0.997434i	$-0.522810\pi$
$307$	−770.260	−0.143196	−0.0715978	+	0.997434i	$0.522810\pi$
$308$	−1064.45	−0.196924
$309$	−5000.81	−0.920667
$310$	2018.50	0.369816
$311$	−4479.32	−0.816718	−0.408359	−	0.912821i	$-0.633899\pi$
$311$	−4479.32	−0.816718	−0.408359	+	0.912821i	$0.633899\pi$
$312$	1604.64	0.291170
$313$	203.662	0.0367785	0.0183892	−	0.999831i	$-0.494146\pi$
$313$	203.662	0.0367785	0.0183892	+	0.999831i	$0.494146\pi$
$314$	−5150.28	−0.925627
$315$	−516.134	−0.0923203
$316$	−2633.30	−0.468781
$317$	−630.965	−0.111793	−0.0558967	−	0.998437i	$-0.517802\pi$
$317$	−630.965	−0.111793	−0.0558967	+	0.998437i	$0.517802\pi$
$318$	−4177.14	−0.736612
$319$	−8968.57	−1.57412
$320$	−491.378	−0.0858403
$321$	−3476.91	−0.604556
$322$	1293.06	0.223786
$323$	386.417	0.0665661
$324$	324.000	0.0555556
$325$	−4416.22	−0.753747
$326$	−5572.65	−0.946751
$327$	2074.42	0.350813
$328$	3249.62	0.547043
$329$	1135.80	0.190331
$330$	1641.22	0.273777
$331$	−8083.56	−1.34233	−0.671167	−	0.741306i	$-0.734207\pi$
$331$	−8083.56	−1.34233	−0.671167	+	0.741306i	$0.734207\pi$
$332$	−5086.84	−0.840893
$333$	687.455	0.113130
$334$	2684.63	0.439810
$335$	6239.17	1.01756
$336$	−358.530	−0.0582126
$337$	−6089.17	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$337$	−6089.17	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$338$	−4546.56	−0.731657
$339$	2858.02	0.457895
$340$	−730.667	−0.116547
$341$	−4683.20	−0.743723
$342$	−292.352	−0.0462239
$343$	−4707.26	−0.741016
$344$	4224.51	0.662123
$345$	−1993.70	−0.311122
$346$	−4656.80	−0.723558
$347$	7425.27	1.14873	0.574365	−	0.818599i	$-0.305249\pi$
$347$	7425.27	1.14873	0.574365	+	0.818599i	$0.305249\pi$
$348$	−3020.82	−0.465324
$349$	876.439	0.134426	0.0672131	−	0.997739i	$-0.478589\pi$
$349$	876.439	0.134426	0.0672131	+	0.997739i	$0.478589\pi$
$350$	986.729	0.150694
$351$	−1805.22	−0.274518
$352$	1140.07	0.172630
$353$	8581.27	1.29387	0.646933	−	0.762546i	$-0.276051\pi$
$353$	8581.27	1.29387	0.646933	+	0.762546i	$0.276051\pi$
$354$	354.000	0.0531494
$355$	3269.34	0.488785
$356$	−2109.52	−0.314057
$357$	−533.125	−0.0790363
$358$	5176.75	0.764244
$359$	1713.00	0.251835	0.125918	−	0.992041i	$-0.459812\pi$
$359$	1713.00	0.251835	0.125918	+	0.992041i	$0.459812\pi$
$360$	552.801	0.0809310
$361$	−6595.21	−0.961540
$362$	−6471.27	−0.939564
$363$	185.139	0.0267693
$364$	1997.61	0.287647
$365$	−3854.43	−0.552740
$366$	1846.12	0.263656
$367$	−3968.90	−0.564509	−0.282254	−	0.959340i	$-0.591082\pi$
$367$	−3968.90	−0.564509	−0.282254	+	0.959340i	$0.591082\pi$
$368$	−1384.91	−0.196178
$369$	−3655.82	−0.515757
$370$	1172.92	0.164803
$371$	−5200.11	−0.727699
$372$	−1577.41	−0.219852
$373$	−2066.26	−0.286828	−0.143414	−	0.989663i	$-0.545808\pi$
$373$	−2066.26	−0.286828	−0.143414	+	0.989663i	$0.545808\pi$
$374$	1695.25	0.234383
$375$	−4400.56	−0.605984
$376$	−1216.49	−0.166850
$377$	16831.0	2.29932
$378$	403.346	0.0548833
$379$	9542.74	1.29334	0.646672	−	0.762768i	$-0.276160\pi$
$379$	9542.74	1.29334	0.646672	+	0.762768i	$0.276160\pi$
$380$	−498.803	−0.0673370
$381$	−1413.31	−0.190042
$382$	10110.9	1.35424
$383$	6369.49	0.849780	0.424890	−	0.905245i	$-0.360313\pi$
$383$	6369.49	0.849780	0.424890	+	0.905245i	$0.360313\pi$
$384$	384.000	0.0510310
$385$	2043.15	0.270464
$386$	6432.68	0.848224
$387$	−4752.57	−0.624256
$388$	−4042.16	−0.528891
$389$	14650.9	1.90959	0.954796	−	0.297260i	$-0.0960729\pi$
$389$	14650.9	1.90959	0.954796	+	0.297260i	$0.0960729\pi$
$390$	−3080.03	−0.399906
$391$	−2059.33	−0.266355
$392$	2297.67	0.296045
$393$	251.718	0.0323091
$394$	281.515	0.0359962
$395$	5054.48	0.643844
$396$	−1282.57	−0.162757
$397$	5305.05	0.670662	0.335331	−	0.942100i	$-0.391152\pi$
$397$	5305.05	0.670662	0.335331	+	0.942100i	$0.391152\pi$
$398$	2214.19	0.278863
$399$	−363.948	−0.0456646
$400$	−1056.83	−0.132103
$401$	3833.36	0.477379	0.238690	−	0.971096i	$-0.423282\pi$
$401$	3833.36	0.477379	0.238690	+	0.971096i	$0.423282\pi$
$402$	−4875.76	−0.604927
$403$	8788.81	1.08636
$404$	−4628.54	−0.569996
$405$	−621.901	−0.0763024
$406$	−3760.61	−0.459694
$407$	−2721.33	−0.331429
$408$	570.998	0.0692858
$409$	7655.50	0.925526	0.462763	−	0.886482i	$-0.346858\pi$
$409$	7655.50	0.925526	0.462763	+	0.886482i	$0.346858\pi$
$410$	−6237.47	−0.751333
$411$	−7496.74	−0.899724
$412$	6667.74	0.797321
$413$	440.693	0.0525063
$414$	1558.03	0.184959
$415$	9763.91	1.15492
$416$	−2139.52	−0.252161
$417$	4542.51	0.533448
$418$	1157.29	0.135419
$419$	−1840.16	−0.214553	−0.107276	−	0.994229i	$-0.534213\pi$
$419$	−1840.16	−0.214553	−0.107276	+	0.994229i	$0.534213\pi$
$420$	688.179	0.0799517
$421$	−3267.46	−0.378257	−0.189128	−	0.981952i	$-0.560566\pi$
$421$	−3267.46	−0.378257	−0.189128	+	0.981952i	$0.560566\pi$
$422$	7239.40	0.835092
$423$	1368.55	0.157308
$424$	5569.52	0.637925
$425$	−1571.47	−0.179359
$426$	−2554.91	−0.290577
$427$	2298.23	0.260466
$428$	4635.88	0.523561
$429$	7146.09	0.804234
$430$	−8108.72	−0.909389
$431$	−1553.19	−0.173584	−0.0867918	−	0.996226i	$-0.527661\pi$
$431$	−1553.19	−0.173584	−0.0867918	+	0.996226i	$0.527661\pi$
$432$	−432.000	−0.0481125
$433$	−14035.6	−1.55775	−0.778876	−	0.627178i	$-0.784210\pi$
$433$	−14035.6	−1.55775	−0.778876	+	0.627178i	$0.784210\pi$
$434$	−1963.71	−0.217191
$435$	5798.30	0.639097
$436$	−2765.89	−0.303813
$437$	−1405.84	−0.153891
$438$	3012.14	0.328598
$439$	−11866.5	−1.29011	−0.645053	−	0.764138i	$-0.723165\pi$
$439$	−11866.5	−1.29011	−0.645053	+	0.764138i	$0.723165\pi$
$440$	−2188.30	−0.237097
$441$	−2584.88	−0.279114
$442$	−3181.42	−0.342363
$443$	7408.68	0.794576	0.397288	−	0.917694i	$-0.369951\pi$
$443$	7408.68	0.794576	0.397288	+	0.917694i	$0.369951\pi$
$444$	−916.607	−0.0979735
$445$	4049.11	0.431339
$446$	9231.60	0.980109
$447$	−7061.34	−0.747181
$448$	478.040	0.0504136
$449$	7863.18	0.826473	0.413237	−	0.910624i	$-0.364398\pi$
$449$	7863.18	0.826473	0.413237	+	0.910624i	$0.364398\pi$
$450$	1188.93	0.124548
$451$	14471.8	1.51098
$452$	−3810.70	−0.396549
$453$	−1381.86	−0.143323
$454$	1885.30	0.194893
$455$	−3834.31	−0.395067
$456$	389.802	0.0400311
$457$	−13663.2	−1.39855	−0.699276	−	0.714852i	$-0.746494\pi$
$457$	−13663.2	−1.39855	−0.699276	+	0.714852i	$0.746494\pi$
$458$	3388.74	0.345732
$459$	−642.373	−0.0653233
$460$	2658.27	0.269440
$461$	3632.33	0.366973	0.183486	−	0.983022i	$-0.441262\pi$
$461$	3632.33	0.366973	0.183486	+	0.983022i	$0.441262\pi$
$462$	−1596.67	−0.160788
$463$	12282.5	1.23287	0.616433	−	0.787407i	$-0.288577\pi$
$463$	12282.5	1.23287	0.616433	+	0.787407i	$0.288577\pi$
$464$	4027.76	0.402983
$465$	3027.75	0.301954
$466$	4150.23	0.412566
$467$	−13023.2	−1.29046	−0.645229	−	0.763989i	$-0.723238\pi$
$467$	−13023.2	−1.29046	−0.645229	+	0.763989i	$0.723238\pi$
$468$	2406.96	0.237739
$469$	−6069.81	−0.597608
$470$	2334.99	0.229159
$471$	−7725.42	−0.755772
$472$	−472.000	−0.0460287
$473$	18813.4	1.82884
$474$	−3949.95	−0.382758
$475$	−1072.79	−0.103628
$476$	710.833	0.0684475
$477$	−6265.71	−0.601441
$478$	12725.8	1.21771
$479$	−6618.55	−0.631335	−0.315667	−	0.948870i	$-0.602228\pi$
$479$	−6618.55	−0.631335	−0.315667	+	0.948870i	$0.602228\pi$
$480$	−737.067	−0.0700883
$481$	5107.04	0.484118
$482$	6139.73	0.580201
$483$	1939.58	0.182721
$484$	−246.852	−0.0231829
$485$	7758.71	0.726402
$486$	486.000	0.0453609
$487$	4946.81	0.460290	0.230145	−	0.973156i	$-0.426080\pi$
$487$	4946.81	0.460290	0.230145	+	0.973156i	$0.426080\pi$
$488$	−2461.49	−0.228333
$489$	−8358.98	−0.773019
$490$	−4410.25	−0.406602
$491$	14394.7	1.32306	0.661532	−	0.749917i	$-0.269906\pi$
$491$	14394.7	1.32306	0.661532	+	0.749917i	$0.269906\pi$
$492$	4874.43	0.446659
$493$	5989.17	0.547138
$494$	−2171.85	−0.197806
$495$	2461.83	0.223538
$496$	2103.21	0.190397
$497$	−3180.60	−0.287061
$498$	−7630.26	−0.686586
$499$	6880.93	0.617300	0.308650	−	0.951176i	$-0.400123\pi$
$499$	6880.93	0.617300	0.308650	+	0.951176i	$0.400123\pi$
$500$	5867.41	0.524797
$501$	4026.95	0.359104
$502$	1874.82	0.166688
$503$	−14239.9	−1.26228	−0.631140	−	0.775669i	$-0.717413\pi$
$503$	−14239.9	−1.26228	−0.631140	+	0.775669i	$0.717413\pi$
$504$	−537.795	−0.0475304
$505$	8884.23	0.782858
$506$	−6167.55	−0.541860
$507$	−6819.83	−0.597395
$508$	1884.41	0.164581
$509$	2273.69	0.197995	0.0989977	−	0.995088i	$-0.468436\pi$
$509$	2273.69	0.197995	0.0989977	+	0.995088i	$0.468436\pi$
$510$	−1096.00	−0.0951602
$511$	3749.80	0.324621
$512$	−512.000	−0.0441942
$513$	−438.528	−0.0377417
$514$	−10578.4	−0.907765
$515$	−12798.4	−1.09508
$516$	6336.77	0.540621
$517$	−5417.49	−0.460853
$518$	−1141.08	−0.0967880
$519$	−6985.20	−0.590782
$520$	4106.70	0.346328
$521$	12788.4	1.07537	0.537685	−	0.843146i	$-0.319299\pi$
$521$	12788.4	1.07537	0.537685	+	0.843146i	$0.319299\pi$
$522$	−4531.23	−0.379936
$523$	17226.1	1.44024	0.720119	−	0.693851i	$-0.244087\pi$
$523$	17226.1	1.44024	0.720119	+	0.693851i	$0.244087\pi$
$524$	−335.624	−0.0279805
$525$	1480.09	0.123041
$526$	−12221.2	−1.01306
$527$	3127.42	0.258506
$528$	1710.10	0.140952
$529$	−4674.86	−0.384225
$530$	−10690.4	−0.876154
$531$	531.000	0.0433963
$532$	485.263	0.0395467
$533$	−27158.7	−2.20708
$534$	−3164.28	−0.256426
$535$	−8898.33	−0.719081
$536$	6501.01	0.523882
$537$	7765.12	0.624003
$538$	−584.765	−0.0468606
$539$	10232.4	0.817700
$540$	829.201	0.0660799
$541$	12162.5	0.966559	0.483280	−	0.875466i	$-0.339445\pi$
$541$	12162.5	0.966559	0.483280	+	0.875466i	$0.339445\pi$
$542$	11770.1	0.932783
$543$	−9706.90	−0.767151
$544$	−761.331	−0.0600033
$545$	5308.99	0.417270
$546$	2996.42	0.234863
$547$	−20942.2	−1.63697	−0.818487	−	0.574525i	$-0.805187\pi$
$547$	−20942.2	−1.63697	−0.818487	+	0.574525i	$0.805187\pi$
$548$	9995.65	0.779184
$549$	2769.18	0.215274
$550$	−4706.45	−0.364879
$551$	4088.62	0.316118
$552$	−2077.37	−0.160179
$553$	−4917.28	−0.378127
$554$	−9263.39	−0.710404
$555$	1759.38	0.134561
$556$	−6056.68	−0.461979
$557$	−713.531	−0.0542788	−0.0271394	−	0.999632i	$-0.508640\pi$
$557$	−713.531	−0.0542788	−0.0271394	+	0.999632i	$0.508640\pi$
$558$	−2366.11	−0.179508
$559$	−35306.4	−2.67138
$560$	−917.572	−0.0692402
$561$	2542.87	0.191373
$562$	5404.45	0.405646
$563$	5768.11	0.431788	0.215894	−	0.976417i	$-0.430733\pi$
$563$	5768.11	0.431788	0.215894	+	0.976417i	$0.430733\pi$
$564$	−1824.73	−0.136233
$565$	7314.43	0.544638
$566$	−1748.30	−0.129835
$567$	605.020	0.0448120
$568$	3406.55	0.251647
$569$	−18140.6	−1.33655	−0.668274	−	0.743916i	$-0.732966\pi$
$569$	−18140.6	−1.33655	−0.668274	+	0.743916i	$0.732966\pi$
$570$	−748.205	−0.0549804
$571$	−10377.2	−0.760551	−0.380275	−	0.924873i	$-0.624171\pi$
$571$	−10377.2	−0.760551	−0.380275	+	0.924873i	$0.624171\pi$
$572$	−9528.12	−0.696487
$573$	15166.4	1.10573
$574$	6068.16	0.441254
$575$	5717.24	0.414653
$576$	576.000	0.0416667
$577$	7437.89	0.536643	0.268322	−	0.963329i	$-0.413531\pi$
$577$	7437.89	0.536643	0.268322	+	0.963329i	$0.413531\pi$
$578$	8693.92	0.625639
$579$	9649.02	0.692572
$580$	−7731.07	−0.553474
$581$	−9498.88	−0.678279
$582$	−6063.24	−0.431838
$583$	24803.2	1.76200
$584$	−4016.19	−0.284574
$585$	−4620.04	−0.326522
$586$	−3370.87	−0.237627
$587$	−10561.8	−0.742646	−0.371323	−	0.928504i	$-0.621096\pi$
$587$	−10561.8	−0.742646	−0.371323	+	0.928504i	$0.621096\pi$
$588$	3446.50	0.241720
$589$	2134.99	0.149356
$590$	905.979	0.0632179
$591$	422.272	0.0293908
$592$	1222.14	0.0848475
$593$	18914.9	1.30985	0.654926	−	0.755693i	$-0.272700\pi$
$593$	18914.9	1.30985	0.654926	+	0.755693i	$0.272700\pi$
$594$	−1923.86	−0.132891
$595$	−1364.41	−0.0940088
$596$	9415.12	0.647078
$597$	3321.29	0.227691
$598$	11574.4	0.791495
$599$	−3246.52	−0.221451	−0.110726	−	0.993851i	$-0.535317\pi$
$599$	−3246.52	−0.221451	−0.110726	+	0.993851i	$0.535317\pi$
$600$	−1585.24	−0.107862
$601$	−17520.8	−1.18917	−0.594584	−	0.804033i	$-0.702683\pi$
$601$	−17520.8	−1.18917	−0.594584	+	0.804033i	$0.702683\pi$
$602$	7888.62	0.534080
$603$	−7313.64	−0.493921
$604$	1842.48	0.124121
$605$	473.819	0.0318405
$606$	−6942.81	−0.465400
$607$	5968.63	0.399109	0.199555	−	0.979887i	$-0.436050\pi$
$607$	5968.63	0.399109	0.199555	+	0.979887i	$0.436050\pi$
$608$	−519.737	−0.0346679
$609$	−5640.91	−0.375338
$610$	4724.70	0.313603
$611$	10166.8	0.673168
$612$	856.497	0.0565717
$613$	−4602.69	−0.303264	−0.151632	−	0.988437i	$-0.548453\pi$
$613$	−4602.69	−0.303264	−0.151632	+	0.988437i	$0.548453\pi$
$614$	1540.52	0.101255
$615$	−9356.21	−0.613461
$616$	2128.90	0.139246
$617$	6339.50	0.413645	0.206822	−	0.978379i	$-0.433688\pi$
$617$	6339.50	0.413645	0.206822	+	0.978379i	$0.433688\pi$
$618$	10001.6	0.651010
$619$	−14416.3	−0.936092	−0.468046	−	0.883704i	$-0.655042\pi$
$619$	−14416.3	−0.936092	−0.468046	+	0.883704i	$0.655042\pi$
$620$	−4037.00	−0.261500
$621$	2337.04	0.151018
$622$	8958.65	0.577507
$623$	−3939.20	−0.253323
$624$	−3209.29	−0.205888
$625$	−3005.73	−0.192367
$626$	−407.324	−0.0260063
$627$	1735.94	0.110569
$628$	10300.6	0.654517
$629$	1817.29	0.115199
$630$	1032.27	0.0652803
$631$	14400.5	0.908520	0.454260	−	0.890869i	$-0.349904\pi$
$631$	14400.5	0.908520	0.454260	+	0.890869i	$0.349904\pi$
$632$	5266.60	0.331478
$633$	10859.1	0.681849
$634$	1261.93	0.0790499
$635$	−3617.03	−0.226043
$636$	8354.29	0.520863
$637$	−19202.8	−1.19441
$638$	17937.1	1.11307
$639$	−3832.36	−0.237255
$640$	982.757	0.0606982
$641$	−887.186	−0.0546673	−0.0273336	−	0.999626i	$-0.508702\pi$
$641$	−887.186	−0.0546673	−0.0273336	+	0.999626i	$0.508702\pi$
$642$	6953.83	0.427485
$643$	5485.30	0.336422	0.168211	−	0.985751i	$-0.446201\pi$
$643$	5485.30	0.336422	0.168211	+	0.985751i	$0.446201\pi$
$644$	−2586.11	−0.158241
$645$	−12163.1	−0.742513
$646$	−772.835	−0.0470693
$647$	902.122	0.0548162	0.0274081	−	0.999624i	$-0.491275\pi$
$647$	902.122	0.0548162	0.0274081	+	0.999624i	$0.491275\pi$
$648$	−648.000	−0.0392837
$649$	−2102.00	−0.127135
$650$	8832.44	0.532980
$651$	−2945.56	−0.177336
$652$	11145.3	0.669454
$653$	19620.0	1.17579	0.587893	−	0.808939i	$-0.299958\pi$
$653$	19620.0	1.17579	0.587893	+	0.808939i	$0.299958\pi$
$654$	−4148.84	−0.248062
$655$	644.212	0.0384297
$656$	−6499.24	−0.386818
$657$	4518.21	0.268299
$658$	−2271.60	−0.134584
$659$	13786.3	0.814928	0.407464	−	0.913221i	$-0.366413\pi$
$659$	13786.3	0.814928	0.407464	+	0.913221i	$0.366413\pi$
$660$	−3282.44	−0.193589
$661$	27331.2	1.60826	0.804130	−	0.594453i	$-0.202631\pi$
$661$	27331.2	1.60826	0.804130	+	0.594453i	$0.202631\pi$
$662$	16167.1	0.949173
$663$	−4772.13	−0.279539
$664$	10173.7	0.594601
$665$	−931.437	−0.0543152
$666$	−1374.91	−0.0799950
$667$	−21789.4	−1.26490
$668$	−5369.27	−0.310993
$669$	13847.4	0.800256
$670$	−12478.3	−0.719523
$671$	−10962.0	−0.630673
$672$	717.060	0.0411625
$673$	−16258.8	−0.931248	−0.465624	−	0.884983i	$-0.654170\pi$
$673$	−16258.8	−0.931248	−0.465624	+	0.884983i	$0.654170\pi$
$674$	12178.3	0.695983
$675$	1783.39	0.101693
$676$	9093.11	0.517359
$677$	−26264.1	−1.49101	−0.745503	−	0.666502i	$-0.767791\pi$
$677$	−26264.1	−1.49101	−0.745503	+	0.666502i	$0.767791\pi$
$678$	−5716.04	−0.323781
$679$	−7548.11	−0.426612
$680$	1461.33	0.0824112
$681$	2827.94	0.159129
$682$	9366.40	0.525891
$683$	−19180.9	−1.07458	−0.537288	−	0.843399i	$-0.680551\pi$
$683$	−19180.9	−1.07458	−0.537288	+	0.843399i	$0.680551\pi$
$684$	584.704	0.0326852
$685$	−19186.1	−1.07017
$686$	9414.53	0.523977
$687$	5083.10	0.282289
$688$	−8449.02	−0.468192
$689$	−46547.4	−2.57375
$690$	3987.40	0.219997
$691$	−24723.0	−1.36108	−0.680540	−	0.732711i	$-0.738255\pi$
$691$	−24723.0	−1.36108	−0.680540	+	0.732711i	$0.738255\pi$
$692$	9313.60	0.511633
$693$	−2395.01	−0.131283
$694$	−14850.5	−0.812275
$695$	11625.5	0.634503
$696$	6041.64	0.329034
$697$	−9664.20	−0.525190
$698$	−1752.88	−0.0950536
$699$	6225.35	0.336859
$700$	−1973.46	−0.106557
$701$	9796.12	0.527809	0.263905	−	0.964549i	$-0.414990\pi$
$701$	9796.12	0.527809	0.263905	+	0.964549i	$0.414990\pi$
$702$	3610.45	0.194113
$703$	1240.61	0.0665583
$704$	−2280.13	−0.122068
$705$	3502.48	0.187108
$706$	−17162.5	−0.914902
$707$	−8643.08	−0.459768
$708$	−708.000	−0.0375823
$709$	17940.8	0.950328	0.475164	−	0.879897i	$-0.342389\pi$
$709$	17940.8	0.950328	0.475164	+	0.879897i	$0.342389\pi$
$710$	−6538.68	−0.345623
$711$	−5924.93	−0.312521
$712$	4219.03	0.222072
$713$	−11378.0	−0.597628
$714$	1066.25	0.0558871
$715$	18288.7	0.956586
$716$	−10353.5	−0.540402
$717$	19088.7	0.994253
$718$	−3426.01	−0.178075
$719$	8993.62	0.466489	0.233244	−	0.972418i	$-0.425066\pi$
$719$	8993.62	0.466489	0.233244	+	0.972418i	$0.425066\pi$
$720$	−1105.60	−0.0572268
$721$	12451.0	0.643132
$722$	13190.4	0.679912
$723$	9209.59	0.473732
$724$	12942.5	0.664372
$725$	−16627.5	−0.851765
$726$	−370.278	−0.0189288
$727$	−35089.3	−1.79008	−0.895040	−	0.445985i	$-0.852853\pi$
$727$	−35089.3	−1.79008	−0.895040	+	0.445985i	$0.852853\pi$
$728$	−3995.23	−0.203397
$729$	729.000	0.0370370
$730$	7708.86	0.390846
$731$	−12563.5	−0.635673
$732$	−3692.24	−0.186433
$733$	−10956.5	−0.552095	−0.276048	−	0.961144i	$-0.589025\pi$
$733$	−10956.5	−0.552095	−0.276048	+	0.961144i	$0.589025\pi$
$734$	7937.79	0.399168
$735$	−6615.37	−0.331989
$736$	2769.83	0.138719
$737$	28951.5	1.44700
$738$	7311.64	0.364696
$739$	31753.5	1.58061	0.790304	−	0.612714i	$-0.209922\pi$
$739$	31753.5	1.58061	0.790304	+	0.612714i	$0.209922\pi$
$740$	−2345.84	−0.116533
$741$	−3257.78	−0.161508
$742$	10400.2	0.514561
$743$	−3388.40	−0.167306	−0.0836531	−	0.996495i	$-0.526659\pi$
$743$	−3388.40	−0.167306	−0.0836531	+	0.996495i	$0.526659\pi$
$744$	3154.81	0.155458
$745$	−18071.8	−0.888725
$746$	4132.51	0.202818
$747$	−11445.4	−0.560595
$748$	−3390.50	−0.165734
$749$	8656.79	0.422313
$750$	8801.12	0.428495
$751$	24144.0	1.17314	0.586568	−	0.809900i	$-0.300479\pi$
$751$	24144.0	1.17314	0.586568	+	0.809900i	$0.300479\pi$
$752$	2432.98	0.117981
$753$	2812.23	0.136100
$754$	−33662.1	−1.62586
$755$	−3536.54	−0.170474
$756$	−806.693	−0.0388084
$757$	−31519.4	−1.51333	−0.756667	−	0.653801i	$-0.773174\pi$
$757$	−31519.4	−1.51333	−0.756667	+	0.653801i	$0.773174\pi$
$758$	−19085.5	−0.914532
$759$	−9251.33	−0.442427
$760$	997.606	0.0476145
$761$	16804.7	0.800488	0.400244	−	0.916409i	$-0.368925\pi$
$761$	16804.7	0.800488	0.400244	+	0.916409i	$0.368925\pi$
$762$	2826.62	0.134380
$763$	−5164.88	−0.245060
$764$	−20221.8	−0.957592
$765$	−1644.00	−0.0776980
$766$	−12739.0	−0.600885
$767$	3944.75	0.185706
$768$	−768.000	−0.0360844
$769$	−20736.7	−0.972413	−0.486206	−	0.873844i	$-0.661620\pi$
$769$	−20736.7	−0.972413	−0.486206	+	0.873844i	$0.661620\pi$
$770$	−4086.30	−0.191247
$771$	−15867.5	−0.741187
$772$	−12865.4	−0.599785
$773$	38933.1	1.81155	0.905773	−	0.423764i	$-0.139291\pi$
$773$	38933.1	1.81155	0.905773	+	0.423764i	$0.139291\pi$
$774$	9505.15	0.441415
$775$	−8682.52	−0.402433
$776$	8084.32	0.373982
$777$	−1711.62	−0.0790271
$778$	−29301.9	−1.35029
$779$	−6597.44	−0.303438
$780$	6160.05	0.282776
$781$	15170.6	0.695068
$782$	4118.66	0.188342
$783$	−6796.84	−0.310216
$784$	−4595.33	−0.209336
$785$	−19771.4	−0.898943
$786$	−503.436	−0.0228460
$787$	3530.33	0.159902	0.0799509	−	0.996799i	$-0.474524\pi$
$787$	3530.33	0.159902	0.0799509	+	0.996799i	$0.474524\pi$
$788$	−563.030	−0.0254532
$789$	−18331.8	−0.827159
$790$	−10109.0	−0.455267
$791$	−7115.88	−0.319863
$792$	2565.15	0.115087
$793$	20572.0	0.921225
$794$	−10610.1	−0.474230
$795$	−16035.6	−0.715376
$796$	−4428.39	−0.197186
$797$	−558.941	−0.0248415	−0.0124208	−	0.999923i	$-0.503954\pi$
$797$	−558.941	−0.0248415	−0.0124208	+	0.999923i	$0.503954\pi$
$798$	727.895	0.0322897
$799$	3617.78	0.160185
$800$	2113.65	0.0934111
$801$	−4746.41	−0.209371
$802$	−7666.73	−0.337558
$803$	−17885.6	−0.786015
$804$	9751.52	0.427748
$805$	4963.90	0.217335
$806$	−17577.6	−0.768170
$807$	−877.147	−0.0382615
$808$	9257.08	0.403048
$809$	−19060.1	−0.828329	−0.414164	−	0.910202i	$-0.635926\pi$
$809$	−19060.1	−0.828329	−0.414164	+	0.910202i	$0.635926\pi$
$810$	1243.80	0.0539540
$811$	17689.9	0.765940	0.382970	−	0.923761i	$-0.374901\pi$
$811$	17689.9	0.765940	0.382970	+	0.923761i	$0.374901\pi$
$812$	7521.21	0.325053
$813$	17655.1	0.761614
$814$	5442.67	0.234355
$815$	−21392.8	−0.919457
$816$	−1142.00	−0.0489925
$817$	−8576.69	−0.367271
$818$	−15311.0	−0.654446
$819$	4494.63	0.191765
$820$	12474.9	0.531273
$821$	−1782.96	−0.0757925	−0.0378962	−	0.999282i	$-0.512066\pi$
$821$	−1782.96	−0.0757925	−0.0378962	+	0.999282i	$0.512066\pi$
$822$	14993.5	0.636201
$823$	39521.9	1.67393	0.836967	−	0.547253i	$-0.184326\pi$
$823$	39521.9	1.67393	0.836967	+	0.547253i	$0.184326\pi$
$824$	−13335.5	−0.563791
$825$	−7059.67	−0.297923
$826$	−881.386	−0.0371276
$827$	−22849.4	−0.960764	−0.480382	−	0.877059i	$-0.659502\pi$
$827$	−22849.4	−0.960764	−0.480382	+	0.877059i	$0.659502\pi$
$828$	−3116.06	−0.130786
$829$	3383.92	0.141771	0.0708857	−	0.997484i	$-0.477417\pi$
$829$	3383.92	0.141771	0.0708857	+	0.997484i	$0.477417\pi$
$830$	−19527.8	−0.816652
$831$	−13895.1	−0.580042
$832$	4279.05	0.178304
$833$	−6833.15	−0.284219
$834$	−9085.02	−0.377205
$835$	10306.0	0.427131
$836$	−2314.59	−0.0957555
$837$	−3549.17	−0.146568
$838$	3680.32	0.151712
$839$	−231.775	−0.00953726	−0.00476863	−	0.999989i	$-0.501518\pi$
$839$	−231.775	−0.00953726	−0.00476863	+	0.999989i	$0.501518\pi$
$840$	−1376.36	−0.0565344
$841$	38981.4	1.59832
$842$	6534.91	0.267468
$843$	8106.67	0.331208
$844$	−14478.8	−0.590499
$845$	−17453.7	−0.710564
$846$	−2737.10	−0.111233
$847$	−460.957	−0.0186997
$848$	−11139.0	−0.451081
$849$	−2622.46	−0.106010
$850$	3142.95	0.126826
$851$	−6611.57	−0.266324
$852$	5109.82	0.205469
$853$	−5986.12	−0.240282	−0.120141	−	0.992757i	$-0.538335\pi$
$853$	−5986.12	−0.240282	−0.120141	+	0.992757i	$0.538335\pi$
$854$	−4596.45	−0.184177
$855$	−1122.31	−0.0448913
$856$	−9271.77	−0.370213
$857$	−11213.0	−0.446940	−0.223470	−	0.974711i	$-0.571739\pi$
$857$	−11213.0	−0.446940	−0.223470	+	0.974711i	$0.571739\pi$
$858$	−14292.2	−0.568679
$859$	−18991.4	−0.754339	−0.377169	−	0.926144i	$-0.623103\pi$
$859$	−18991.4	−0.754339	−0.377169	+	0.926144i	$0.623103\pi$
$860$	16217.4	0.643035
$861$	9102.24	0.360283
$862$	3106.38	0.122742
$863$	10769.7	0.424804	0.212402	−	0.977182i	$-0.431871\pi$
$863$	10769.7	0.424804	0.212402	+	0.977182i	$0.431871\pi$
$864$	864.000	0.0340207
$865$	−17876.9	−0.702699
$866$	28071.1	1.10150
$867$	13040.9	0.510832
$868$	3927.42	0.153577
$869$	23454.2	0.915568
$870$	−11596.6	−0.451910
$871$	−54332.3	−2.11364
$872$	5531.79	0.214828
$873$	−9094.86	−0.352594
$874$	2811.68	0.108818
$875$	10956.5	0.423310
$876$	−6024.28	−0.232354
$877$	25605.6	0.985905	0.492952	−	0.870056i	$-0.335918\pi$
$877$	25605.6	0.985905	0.492952	+	0.870056i	$0.335918\pi$
$878$	23733.0	0.912242
$879$	−5056.31	−0.194022
$880$	4376.59	0.167653
$881$	17350.8	0.663521	0.331761	−	0.943364i	$-0.392357\pi$
$881$	17350.8	0.663521	0.331761	+	0.943364i	$0.392357\pi$
$882$	5169.75	0.197363
$883$	13990.2	0.533190	0.266595	−	0.963809i	$-0.414101\pi$
$883$	13990.2	0.533190	0.266595	+	0.963809i	$0.414101\pi$
$884$	6362.84	0.242087
$885$	1358.97	0.0516172
$886$	−14817.4	−0.561850
$887$	14225.8	0.538507	0.269253	−	0.963069i	$-0.413223\pi$
$887$	14225.8	0.538507	0.269253	+	0.963069i	$0.413223\pi$
$888$	1833.21	0.0692777
$889$	3518.85	0.132754
$890$	−8098.21	−0.305003
$891$	−2885.79	−0.108505
$892$	−18463.2	−0.693042
$893$	2469.74	0.0925496
$894$	14122.7	0.528337
$895$	19873.0	0.742212
$896$	−956.080	−0.0356478
$897$	17361.7	0.646253
$898$	−15726.4	−0.584405
$899$	33090.7	1.22763
$900$	−2377.86	−0.0880688
$901$	−16563.5	−0.612441
$902$	−28943.6	−1.06842
$903$	11832.9	0.436074
$904$	7621.39	0.280402
$905$	−24842.5	−0.912478
$906$	2763.72	0.101345
$907$	−8779.59	−0.321413	−0.160706	−	0.987002i	$-0.551377\pi$
$907$	−8779.59	−0.321413	−0.160706	+	0.987002i	$0.551377\pi$
$908$	−3770.59	−0.137810
$909$	−10414.2	−0.379997
$910$	7668.63	0.279354
$911$	−47069.8	−1.71185	−0.855924	−	0.517102i	$-0.827011\pi$
$911$	−47069.8	−1.71185	−0.855924	+	0.517102i	$0.827011\pi$
$912$	−779.605	−0.0283062
$913$	45307.3	1.64233
$914$	27326.4	0.988926
$915$	7087.05	0.256055
$916$	−6777.47	−0.244469
$917$	−626.726	−0.0225696
$918$	1284.75	0.0461906
$919$	21786.5	0.782012	0.391006	−	0.920388i	$-0.372127\pi$
$919$	21786.5	0.782012	0.391006	+	0.920388i	$0.372127\pi$
$920$	−5316.54	−0.190523
$921$	2310.78	0.0826740
$922$	−7264.66	−0.259489
$923$	−28470.3	−1.01529
$924$	3193.34	0.113694
$925$	−5045.28	−0.179338
$926$	−24565.0	−0.871768
$927$	15002.4	0.531547
$928$	−8055.52	−0.284952
$929$	−53325.9	−1.88328	−0.941639	−	0.336623i	$-0.890715\pi$
$929$	−53325.9	−1.88328	−0.941639	+	0.336623i	$0.890715\pi$
$930$	−6055.50	−0.213514
$931$	−4664.77	−0.164212
$932$	−8300.46	−0.291728
$933$	13438.0	0.471532
$934$	26046.5	0.912492
$935$	6507.88	0.227626
$936$	−4813.93	−0.168107
$937$	31039.8	1.08221	0.541103	−	0.840956i	$-0.318007\pi$
$937$	31039.8	1.08221	0.541103	+	0.840956i	$0.318007\pi$
$938$	12139.6	0.422572
$939$	−610.986	−0.0212341
$940$	−4669.97	−0.162040
$941$	−38663.6	−1.33942	−0.669712	−	0.742621i	$-0.733582\pi$
$941$	−38663.6	−1.33942	−0.669712	+	0.742621i	$0.733582\pi$
$942$	15450.8	0.534411
$943$	35159.7	1.21417
$944$	944.000	0.0325472
$945$	1548.40	0.0533011
$946$	−37626.7	−1.29318
$947$	−21611.0	−0.741565	−0.370783	−	0.928720i	$-0.620910\pi$
$947$	−21611.0	−0.741565	−0.370783	+	0.928720i	$0.620910\pi$
$948$	7899.90	0.270651
$949$	33565.4	1.14813
$950$	2145.59	0.0732759
$951$	1892.89	0.0645440
$952$	−1421.67	−0.0483997
$953$	45129.3	1.53398	0.766990	−	0.641659i	$-0.221754\pi$
$953$	45129.3	1.53398	0.766990	+	0.641659i	$0.221754\pi$
$954$	12531.4	0.425283
$955$	38814.7	1.31520
$956$	−25451.6	−0.861048
$957$	26905.7	0.908817
$958$	13237.1	0.446421
$959$	18665.3	0.628503
$960$	1474.13	0.0495599
$961$	−12511.7	−0.419984
$962$	−10214.1	−0.342323
$963$	10430.7	0.349040
$964$	−12279.5	−0.410264
$965$	24694.4	0.823772
$966$	−3879.17	−0.129203
$967$	54862.7	1.82447	0.912237	−	0.409663i	$-0.134354\pi$
$967$	54862.7	1.82447	0.912237	+	0.409663i	$0.134354\pi$
$968$	493.704	0.0163928
$969$	−1159.25	−0.0384319
$970$	−15517.4	−0.513644
$971$	−224.278	−0.00741239	−0.00370619	−	0.999993i	$-0.501180\pi$
$971$	−224.278	−0.00741239	−0.00370619	+	0.999993i	$0.501180\pi$
$972$	−972.000	−0.0320750
$973$	−11309.9	−0.372640
$974$	−9893.62	−0.325474
$975$	13248.7	0.435176
$976$	4922.98	0.161456
$977$	−47467.6	−1.55437	−0.777186	−	0.629271i	$-0.783354\pi$
$977$	−47467.6	−1.55437	−0.777186	+	0.629271i	$0.783354\pi$
$978$	16718.0	0.546607
$979$	18789.0	0.613379
$980$	8820.50	0.287511
$981$	−6223.26	−0.202542
$982$	−28789.4	−0.935548
$983$	−38293.9	−1.24251	−0.621255	−	0.783609i	$-0.713377\pi$
$983$	−38293.9	−1.24251	−0.621255	+	0.783609i	$0.713377\pi$
$984$	−9748.86	−0.315836
$985$	1080.71	0.0349585
$986$	−11978.3	−0.386885
$987$	−3407.41	−0.109887
$988$	4343.71	0.139870
$989$	45707.7	1.46959
$990$	−4923.66	−0.158065
$991$	−47899.0	−1.53538	−0.767691	−	0.640820i	$-0.778594\pi$
$991$	−47899.0	−1.53538	−0.767691	+	0.640820i	$0.778594\pi$
$992$	−4206.42	−0.134631
$993$	24250.7	0.774997
$994$	6361.19	0.202983
$995$	8500.05	0.270824
$996$	15260.5	0.485490
$997$	−7466.54	−0.237179	−0.118590	−	0.992943i	$-0.537837\pi$
$997$	−7466.54	−0.237179	−0.118590	+	0.992943i	$0.537837\pi$
$998$	−13761.9	−0.436497
$999$	−2062.37	−0.0653157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.4.a.d.1.2	✓	3
3.2	odd	2		1062.4.a.l.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.4.a.d.1.2	✓	3	1.1	even	1	trivial
1062.4.a.l.1.2		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -7.67779 q^{5} +6.00000 q^{6} +7.46938 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -7.67779 q^{5} +6.00000 q^{6} +7.46938 q^{7} -8.00000 q^{8} +9.00000 q^{9} +15.3556 q^{10} -35.6271 q^{11} -12.0000 q^{12} +66.8601 q^{13} -14.9388 q^{14} +23.0334 q^{15} +16.0000 q^{16} +23.7916 q^{17} -18.0000 q^{18} +16.2418 q^{19} -30.7111 q^{20} -22.4081 q^{21} +71.2541 q^{22} -86.5571 q^{23} +24.0000 q^{24} -66.0516 q^{25} -133.720 q^{26} -27.0000 q^{27} +29.8775 q^{28} +251.735 q^{29} -46.0667 q^{30} +131.451 q^{31} -32.0000 q^{32} +106.881 q^{33} -47.5832 q^{34} -57.3483 q^{35} +36.0000 q^{36} +76.3839 q^{37} -32.4835 q^{38} -200.580 q^{39} +61.4223 q^{40} -406.202 q^{41} +44.8163 q^{42} -528.064 q^{43} -142.508 q^{44} -69.1001 q^{45} +173.114 q^{46} +152.061 q^{47} -48.0000 q^{48} -287.208 q^{49} +132.103 q^{50} -71.3748 q^{51} +267.441 q^{52} -696.191 q^{53} +54.0000 q^{54} +273.537 q^{55} -59.7550 q^{56} -48.7253 q^{57} -503.470 q^{58} +59.0000 q^{59} +92.1334 q^{60} +307.686 q^{61} -262.901 q^{62} +67.2244 q^{63} +64.0000 q^{64} -513.338 q^{65} -213.762 q^{66} -812.626 q^{67} +95.1664 q^{68} +259.671 q^{69} +114.697 q^{70} -425.818 q^{71} -72.0000 q^{72} +502.024 q^{73} -152.768 q^{74} +198.155 q^{75} +64.9671 q^{76} -266.112 q^{77} +401.161 q^{78} -658.325 q^{79} -122.845 q^{80} +81.0000 q^{81} +812.405 q^{82} -1271.71 q^{83} -89.6325 q^{84} -182.667 q^{85} +1056.13 q^{86} -755.205 q^{87} +285.016 q^{88} -527.379 q^{89} +138.200 q^{90} +499.404 q^{91} -346.228 q^{92} -394.352 q^{93} -304.122 q^{94} -124.701 q^{95} +96.0000 q^{96} -1010.54 q^{97} +574.417 q^{98} -320.643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} - 6 q^{5} + 18 q^{6} + 6 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) 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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