LMFDB - Embedded newform 462.4.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,4,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("462.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 462 = 2 \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	462.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:	$27.2588824227$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{697}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 174 $ x^2 - x - 174
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-12.7004$ of defining polynomial
Character	$\chi$	$=$	462.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +16.7004 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +16.7004 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -33.4008 q^{10} -11.0000 q^{11} -12.0000 q^{12} +59.5019 q^{13} +14.0000 q^{14} -50.1011 q^{15} +16.0000 q^{16} +71.4008 q^{17} -18.0000 q^{18} +44.1011 q^{19} +66.8015 q^{20} +21.0000 q^{21} +22.0000 q^{22} -53.4008 q^{23} +24.0000 q^{24} +153.903 q^{25} -119.004 q^{26} -27.0000 q^{27} -28.0000 q^{28} -164.903 q^{29} +100.202 q^{30} -23.5955 q^{31} -32.0000 q^{32} +33.0000 q^{33} -142.802 q^{34} -116.903 q^{35} +36.0000 q^{36} -106.101 q^{37} -88.2023 q^{38} -178.506 q^{39} -133.603 q^{40} +357.004 q^{41} -42.0000 q^{42} -287.408 q^{43} -44.0000 q^{44} +150.303 q^{45} +106.802 q^{46} +26.2958 q^{47} -48.0000 q^{48} +49.0000 q^{49} -307.805 q^{50} -214.202 q^{51} +238.008 q^{52} +275.603 q^{53} +54.0000 q^{54} -183.704 q^{55} +56.0000 q^{56} -132.303 q^{57} +329.805 q^{58} +596.708 q^{59} -200.405 q^{60} -33.1909 q^{61} +47.1909 q^{62} -63.0000 q^{63} +64.0000 q^{64} +993.704 q^{65} -66.0000 q^{66} +262.903 q^{67} +285.603 q^{68} +160.202 q^{69} +233.805 q^{70} -224.809 q^{71} -72.0000 q^{72} +1004.90 q^{73} +212.202 q^{74} -461.708 q^{75} +176.405 q^{76} +77.0000 q^{77} +357.011 q^{78} +362.802 q^{79} +267.206 q^{80} +81.0000 q^{81} -714.008 q^{82} +974.802 q^{83} +84.0000 q^{84} +1192.42 q^{85} +574.817 q^{86} +494.708 q^{87} +88.0000 q^{88} +908.427 q^{89} -300.607 q^{90} -416.513 q^{91} -213.603 q^{92} +70.7864 q^{93} -52.5917 q^{94} +736.506 q^{95} +96.0000 q^{96} -1078.03 q^{97} -98.0000 q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 7 q^{5} + 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 + 7 * q^5 + 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9	\( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 7 q^{5} + 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9} - 14 q^{10} - 22 q^{11} - 24 q^{12} - 13 q^{13} + 28 q^{14} - 21 q^{15} + 32 q^{16} + 90 q^{17} - 36 q^{18} + 9 q^{19} + 28 q^{20} + 42 q^{21} + 44 q^{22} - 54 q^{23} + 48 q^{24} + 123 q^{25} + 26 q^{26} - 54 q^{27} - 56 q^{28} - 145 q^{29} + 42 q^{30} - 364 q^{31} - 64 q^{32} + 66 q^{33} - 180 q^{34} - 49 q^{35} + 72 q^{36} - 133 q^{37} - 18 q^{38} + 39 q^{39} - 56 q^{40} + 450 q^{41} - 84 q^{42} + 6 q^{43} - 88 q^{44} + 63 q^{45} + 108 q^{46} + 343 q^{47} - 96 q^{48} + 98 q^{49} - 246 q^{50} - 270 q^{51} - 52 q^{52} + 340 q^{53} + 108 q^{54} - 77 q^{55} + 112 q^{56} - 27 q^{57} + 290 q^{58} + 639 q^{59} - 84 q^{60} - 700 q^{61} + 728 q^{62} - 126 q^{63} + 128 q^{64} + 1697 q^{65} - 132 q^{66} + 341 q^{67} + 360 q^{68} + 162 q^{69} + 98 q^{70} + 184 q^{71} - 144 q^{72} + 1825 q^{73} + 266 q^{74} - 369 q^{75} + 36 q^{76} + 154 q^{77} - 78 q^{78} + 620 q^{79} + 112 q^{80} + 162 q^{81} - 900 q^{82} + 1844 q^{83} + 168 q^{84} + 1012 q^{85} - 12 q^{86} + 435 q^{87} + 176 q^{88} - 84 q^{89} - 126 q^{90} + 91 q^{91} - 216 q^{92} + 1092 q^{93} - 686 q^{94} + 1077 q^{95} + 192 q^{96} - 44 q^{97} - 196 q^{98} - 198 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 + 7 * q^5 + 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9 - 14 * q^10 - 22 * q^11 - 24 * q^12 - 13 * q^13 + 28 * q^14 - 21 * q^15 + 32 * q^16 + 90 * q^17 - 36 * q^18 + 9 * q^19 + 28 * q^20 + 42 * q^21 + 44 * q^22 - 54 * q^23 + 48 * q^24 + 123 * q^25 + 26 * q^26 - 54 * q^27 - 56 * q^28 - 145 * q^29 + 42 * q^30 - 364 * q^31 - 64 * q^32 + 66 * q^33 - 180 * q^34 - 49 * q^35 + 72 * q^36 - 133 * q^37 - 18 * q^38 + 39 * q^39 - 56 * q^40 + 450 * q^41 - 84 * q^42 + 6 * q^43 - 88 * q^44 + 63 * q^45 + 108 * q^46 + 343 * q^47 - 96 * q^48 + 98 * q^49 - 246 * q^50 - 270 * q^51 - 52 * q^52 + 340 * q^53 + 108 * q^54 - 77 * q^55 + 112 * q^56 - 27 * q^57 + 290 * q^58 + 639 * q^59 - 84 * q^60 - 700 * q^61 + 728 * q^62 - 126 * q^63 + 128 * q^64 + 1697 * q^65 - 132 * q^66 + 341 * q^67 + 360 * q^68 + 162 * q^69 + 98 * q^70 + 184 * q^71 - 144 * q^72 + 1825 * q^73 + 266 * q^74 - 369 * q^75 + 36 * q^76 + 154 * q^77 - 78 * q^78 + 620 * q^79 + 112 * q^80 + 162 * q^81 - 900 * q^82 + 1844 * q^83 + 168 * q^84 + 1012 * q^85 - 12 * q^86 + 435 * q^87 + 176 * q^88 - 84 * q^89 - 126 * q^90 + 91 * q^91 - 216 * q^92 + 1092 * q^93 - 686 * q^94 + 1077 * q^95 + 192 * q^96 - 44 * q^97 - 196 * q^98 - 198 * 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
$5$	16.7004	1.49373	0.746864	−	0.664977i	$-0.231559\pi$
$5$	16.7004	1.49373	0.746864	+	0.664977i	$0.231559\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−33.4008	−1.05622
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−12.0000	−0.288675
$13$	59.5019	1.26945	0.634725	−	0.772738i	$-0.281113\pi$
$13$	59.5019	1.26945	0.634725	+	0.772738i	$0.281113\pi$
$14$	14.0000	0.267261
$15$	−50.1011	−0.862404
$16$	16.0000	0.250000
$17$	71.4008	1.01866	0.509330	−	0.860571i	$-0.329893\pi$
$17$	71.4008	1.01866	0.509330	+	0.860571i	$0.329893\pi$
$18$	−18.0000	−0.235702
$19$	44.1011	0.532500	0.266250	−	0.963904i	$-0.414215\pi$
$19$	44.1011	0.532500	0.266250	+	0.963904i	$0.414215\pi$
$20$	66.8015	0.746864
$21$	21.0000	0.218218
$22$	22.0000	0.213201
$23$	−53.4008	−0.484123	−0.242061	−	0.970261i	$-0.577824\pi$
$23$	−53.4008	−0.484123	−0.242061	+	0.970261i	$0.577824\pi$
$24$	24.0000	0.204124
$25$	153.903	1.23122
$26$	−119.004	−0.897637
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	−164.903	−1.05592	−0.527960	−	0.849270i	$-0.677043\pi$
$29$	−164.903	−1.05592	−0.527960	+	0.849270i	$0.677043\pi$
$30$	100.202	0.609812
$31$	−23.5955	−0.136705	−0.0683527	−	0.997661i	$-0.521774\pi$
$31$	−23.5955	−0.136705	−0.0683527	+	0.997661i	$0.521774\pi$
$32$	−32.0000	−0.176777
$33$	33.0000	0.174078
$34$	−142.802	−0.720302
$35$	−116.903	−0.564576
$36$	36.0000	0.166667
$37$	−106.101	−0.471430	−0.235715	−	0.971822i	$-0.575743\pi$
$37$	−106.101	−0.471430	−0.235715	+	0.971822i	$0.575743\pi$
$38$	−88.2023	−0.376534
$39$	−178.506	−0.732918
$40$	−133.603	−0.528112
$41$	357.004	1.35987	0.679934	−	0.733273i	$-0.262008\pi$
$41$	357.004	1.35987	0.679934	+	0.733273i	$0.262008\pi$
$42$	−42.0000	−0.154303
$43$	−287.408	−1.01929	−0.509644	−	0.860386i	$-0.670223\pi$
$43$	−287.408	−1.01929	−0.509644	+	0.860386i	$0.670223\pi$
$44$	−44.0000	−0.150756
$45$	150.303	0.497909
$46$	106.802	0.342327
$47$	26.2958	0.0816094	0.0408047	−	0.999167i	$-0.487008\pi$
$47$	26.2958	0.0816094	0.0408047	+	0.999167i	$0.487008\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−307.805	−0.870605
$51$	−214.202	−0.588124
$52$	238.008	0.634725
$53$	275.603	0.714283	0.357141	−	0.934050i	$-0.383751\pi$
$53$	275.603	0.714283	0.357141	+	0.934050i	$0.383751\pi$
$54$	54.0000	0.136083
$55$	−183.704	−0.450376
$56$	56.0000	0.133631
$57$	−132.303	−0.307439
$58$	329.805	0.746648
$59$	596.708	1.31669	0.658345	−	0.752716i	$-0.271257\pi$
$59$	596.708	1.31669	0.658345	+	0.752716i	$0.271257\pi$
$60$	−200.405	−0.431202
$61$	−33.1909	−0.0696666	−0.0348333	−	0.999393i	$-0.511090\pi$
$61$	−33.1909	−0.0696666	−0.0348333	+	0.999393i	$0.511090\pi$
$62$	47.1909	0.0966653
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	993.704	1.89621
$66$	−66.0000	−0.123091
$67$	262.903	0.479383	0.239692	−	0.970849i	$-0.422954\pi$
$67$	262.903	0.479383	0.239692	+	0.970849i	$0.422954\pi$
$68$	285.603	0.509330
$69$	160.202	0.279509
$70$	233.805	0.399215
$71$	−224.809	−0.375774	−0.187887	−	0.982191i	$-0.560164\pi$
$71$	−224.809	−0.375774	−0.187887	+	0.982191i	$0.560164\pi$
$72$	−72.0000	−0.117851
$73$	1004.90	1.61116	0.805582	−	0.592484i	$-0.201853\pi$
$73$	1004.90	1.61116	0.805582	+	0.592484i	$0.201853\pi$
$74$	212.202	0.333352
$75$	−461.708	−0.710846
$76$	176.405	0.266250
$77$	77.0000	0.113961
$78$	357.011	0.518251
$79$	362.802	0.516688	0.258344	−	0.966053i	$-0.416823\pi$
$79$	362.802	0.516688	0.258344	+	0.966053i	$0.416823\pi$
$80$	267.206	0.373432
$81$	81.0000	0.111111
$82$	−714.008	−0.961573
$83$	974.802	1.28914	0.644568	−	0.764547i	$-0.277037\pi$
$83$	974.802	1.28914	0.644568	+	0.764547i	$0.277037\pi$
$84$	84.0000	0.109109
$85$	1192.42	1.52160
$86$	574.817	0.720745
$87$	494.708	0.609635
$88$	88.0000	0.106600
$89$	908.427	1.08194	0.540972	−	0.841040i	$-0.318056\pi$
$89$	908.427	1.08194	0.540972	+	0.841040i	$0.318056\pi$
$90$	−300.607	−0.352075
$91$	−416.513	−0.479807
$92$	−213.603	−0.242061
$93$	70.7864	0.0789269
$94$	−52.5917	−0.0577066
$95$	736.506	0.795409
$96$	96.0000	0.102062
$97$	−1078.03	−1.12843	−0.564213	−	0.825629i	$-0.690820\pi$
$97$	−1078.03	−1.12843	−0.564213	+	0.825629i	$0.690820\pi$
$98$	−98.0000	−0.101015
$99$	−99.0000	−0.100504
$100$	615.611	0.615611
$101$	1360.60	1.34044	0.670221	−	0.742161i	$-0.266199\pi$
$101$	1360.60	1.34044	0.670221	+	0.742161i	$0.266199\pi$
$102$	428.405	0.415866
$103$	−1581.43	−1.51284	−0.756422	−	0.654083i	$-0.773055\pi$
$103$	−1581.43	−1.51284	−0.756422	+	0.654083i	$0.773055\pi$
$104$	−476.015	−0.448819
$105$	350.708	0.325958
$106$	−551.206	−0.505074
$107$	1653.72	1.49412	0.747061	−	0.664755i	$-0.231464\pi$
$107$	1653.72	1.49412	0.747061	+	0.664755i	$0.231464\pi$
$108$	−108.000	−0.0962250
$109$	−1814.65	−1.59461	−0.797304	−	0.603579i	$-0.793741\pi$
$109$	−1814.65	−1.59461	−0.797304	+	0.603579i	$0.793741\pi$
$110$	367.408	0.318464
$111$	318.303	0.272180
$112$	−112.000	−0.0944911
$113$	613.176	0.510467	0.255233	−	0.966880i	$-0.417848\pi$
$113$	613.176	0.510467	0.255233	+	0.966880i	$0.417848\pi$
$114$	264.607	0.217392
$115$	−891.813	−0.723148
$116$	−659.611	−0.527960
$117$	535.517	0.423150
$118$	−1193.42	−0.931041
$119$	−499.805	−0.385017
$120$	400.809	0.304906
$121$	121.000	0.0909091
$122$	66.3818	0.0492617
$123$	−1071.01	−0.785121
$124$	−94.3818	−0.0683527
$125$	482.685	0.345381
$126$	126.000	0.0890871
$127$	1475.64	1.03104	0.515520	−	0.856878i	$-0.327599\pi$
$127$	1475.64	1.03104	0.515520	+	0.856878i	$0.327599\pi$
$128$	−128.000	−0.0883883
$129$	862.225	0.588486
$130$	−1987.41	−1.34082
$131$	348.839	0.232658	0.116329	−	0.993211i	$-0.462887\pi$
$131$	348.839	0.232658	0.116329	+	0.993211i	$0.462887\pi$
$132$	132.000	0.0870388
$133$	−308.708	−0.201266
$134$	−525.805	−0.338975
$135$	−450.910	−0.287468
$136$	−571.206	−0.360151
$137$	−43.4008	−0.0270655	−0.0135328	−	0.999908i	$-0.504308\pi$
$137$	−43.4008	−0.0270655	−0.0135328	+	0.999908i	$0.504308\pi$
$138$	−320.405	−0.197642
$139$	2779.25	1.69592	0.847961	−	0.530059i	$-0.177830\pi$
$139$	2779.25	1.69592	0.847961	+	0.530059i	$0.177830\pi$
$140$	−467.611	−0.282288
$141$	−78.8875	−0.0471172
$142$	449.618	0.265712
$143$	−654.521	−0.382754
$144$	144.000	0.0833333
$145$	−2753.94	−1.57726
$146$	−2009.81	−1.13926
$147$	−147.000	−0.0824786
$148$	−424.405	−0.235715
$149$	−198.086	−0.108912	−0.0544558	−	0.998516i	$-0.517342\pi$
$149$	−198.086	−0.108912	−0.0544558	+	0.998516i	$0.517342\pi$
$150$	923.416	0.502644
$151$	−2745.62	−1.47970	−0.739852	−	0.672770i	$-0.765104\pi$
$151$	−2745.62	−1.47970	−0.739852	+	0.672770i	$0.765104\pi$
$152$	−352.809	−0.188267
$153$	642.607	0.339553
$154$	−154.000	−0.0805823
$155$	−394.053	−0.204201
$156$	−714.023	−0.366459
$157$	3495.88	1.77708	0.888540	−	0.458798i	$-0.151720\pi$
$157$	3495.88	1.77708	0.888540	+	0.458798i	$0.151720\pi$
$158$	−725.603	−0.365354
$159$	−826.809	−0.412391
$160$	−534.412	−0.264056
$161$	373.805	0.182981
$162$	−162.000	−0.0785674
$163$	−2640.32	−1.26875	−0.634373	−	0.773027i	$-0.718742\pi$
$163$	−2640.32	−1.26875	−0.634373	+	0.773027i	$0.718742\pi$
$164$	1428.02	0.679934
$165$	551.112	0.260025
$166$	−1949.60	−0.911557
$167$	1983.42	0.919053	0.459527	−	0.888164i	$-0.348019\pi$
$167$	1983.42	0.919053	0.459527	+	0.888164i	$0.348019\pi$
$168$	−168.000	−0.0771517
$169$	1343.48	0.611504
$170$	−2384.84	−1.07593
$171$	396.910	0.177500
$172$	−1149.63	−0.509644
$173$	903.446	0.397039	0.198519	−	0.980097i	$-0.436387\pi$
$173$	903.446	0.397039	0.198519	+	0.980097i	$0.436387\pi$
$174$	−989.416	−0.431077
$175$	−1077.32	−0.465358
$176$	−176.000	−0.0753778
$177$	−1790.12	−0.760192
$178$	−1816.85	−0.765051
$179$	519.236	0.216813	0.108407	−	0.994107i	$-0.465425\pi$
$179$	519.236	0.216813	0.108407	+	0.994107i	$0.465425\pi$
$180$	601.214	0.248955
$181$	130.450	0.0535706	0.0267853	−	0.999641i	$-0.491473\pi$
$181$	130.450	0.0535706	0.0267853	+	0.999641i	$0.491473\pi$
$182$	833.027	0.339275
$183$	99.5727	0.0402220
$184$	427.206	0.171163
$185$	−1771.93	−0.704188
$186$	−141.573	−0.0558098
$187$	−785.408	−0.307138
$188$	105.183	0.0408047
$189$	189.000	0.0727393
$190$	−1473.01	−0.562439
$191$	−1409.24	−0.533871	−0.266935	−	0.963714i	$-0.586011\pi$
$191$	−1409.24	−0.533871	−0.266935	+	0.963714i	$0.586011\pi$
$192$	−192.000	−0.0721688
$193$	4202.84	1.56750	0.783749	−	0.621078i	$-0.213305\pi$
$193$	4202.84	1.56750	0.783749	+	0.621078i	$0.213305\pi$
$194$	2156.06	0.797918
$195$	−2981.11	−1.09478
$196$	196.000	0.0714286
$197$	−154.809	−0.0559883	−0.0279941	−	0.999608i	$-0.508912\pi$
$197$	−154.809	−0.0559883	−0.0279941	+	0.999608i	$0.508912\pi$
$198$	198.000	0.0710669
$199$	−1793.01	−0.638710	−0.319355	−	0.947635i	$-0.603466\pi$
$199$	−1793.01	−0.638710	−0.319355	+	0.947635i	$0.603466\pi$
$200$	−1231.22	−0.435302
$201$	−788.708	−0.276772
$202$	−2721.20	−0.947836
$203$	1154.32	0.399100
$204$	−856.809	−0.294062
$205$	5962.10	2.03127
$206$	3162.86	1.06974
$207$	−480.607	−0.161374
$208$	952.030	0.317363
$209$	−485.112	−0.160555
$210$	−701.416	−0.230487
$211$	1722.61	0.562036	0.281018	−	0.959702i	$-0.409328\pi$
$211$	1722.61	0.562036	0.281018	+	0.959702i	$0.409328\pi$
$212$	1102.41	0.357141
$213$	674.427	0.216953
$214$	−3307.44	−1.05650
$215$	−4799.83	−1.52254
$216$	216.000	0.0680414
$217$	165.168	0.0516698
$218$	3629.30	1.12756
$219$	−3014.71	−0.930206
$220$	−734.817	−0.225188
$221$	4248.48	1.29314
$222$	−636.607	−0.192461
$223$	3310.67	0.994167	0.497083	−	0.867703i	$-0.334404\pi$
$223$	3310.67	0.994167	0.497083	+	0.867703i	$0.334404\pi$
$224$	224.000	0.0668153
$225$	1385.12	0.410407
$226$	−1226.35	−0.360954
$227$	4301.85	1.25781	0.628907	−	0.777480i	$-0.283503\pi$
$227$	4301.85	1.25781	0.628907	+	0.777480i	$0.283503\pi$
$228$	−529.214	−0.153719
$229$	−6227.21	−1.79697	−0.898484	−	0.439005i	$-0.855331\pi$
$229$	−6227.21	−1.79697	−0.898484	+	0.439005i	$0.855331\pi$
$230$	1783.63	0.511343
$231$	−231.000	−0.0657952
$232$	1319.22	0.373324
$233$	−3714.88	−1.04451	−0.522254	−	0.852790i	$-0.674909\pi$
$233$	−3714.88	−1.04451	−0.522254	+	0.852790i	$0.674909\pi$
$234$	−1071.03	−0.299212
$235$	439.150	0.121902
$236$	2386.83	0.658345
$237$	−1088.40	−0.298310
$238$	999.611	0.272248
$239$	−1154.15	−0.312368	−0.156184	−	0.987728i	$-0.549919\pi$
$239$	−1154.15	−0.312368	−0.156184	+	0.987728i	$0.549919\pi$
$240$	−801.618	−0.215601
$241$	−7071.87	−1.89020	−0.945102	−	0.326775i	$-0.894038\pi$
$241$	−7071.87	−1.89020	−0.945102	+	0.326775i	$0.894038\pi$
$242$	−242.000	−0.0642824
$243$	−243.000	−0.0641500
$244$	−132.764	−0.0348333
$245$	818.319	0.213390
$246$	2142.02	0.555164
$247$	2624.10	0.675982
$248$	188.764	0.0483327
$249$	−2924.40	−0.744284
$250$	−965.370	−0.244222
$251$	−5835.90	−1.46756	−0.733782	−	0.679385i	$-0.762247\pi$
$251$	−5835.90	−1.46756	−0.733782	+	0.679385i	$0.762247\pi$
$252$	−252.000	−0.0629941
$253$	587.408	0.145969
$254$	−2951.28	−0.729055
$255$	−3577.26	−0.878497
$256$	256.000	0.0625000
$257$	−5769.05	−1.40025	−0.700124	−	0.714021i	$-0.746872\pi$
$257$	−5769.05	−1.40025	−0.700124	+	0.714021i	$0.746872\pi$
$258$	−1724.45	−0.416122
$259$	742.708	0.178184
$260$	3974.82	0.948106
$261$	−1484.12	−0.351973
$262$	−697.679	−0.164514
$263$	117.360	0.0275161	0.0137581	−	0.999905i	$-0.495621\pi$
$263$	117.360	0.0275161	0.0137581	+	0.999905i	$0.495621\pi$
$264$	−264.000	−0.0615457
$265$	4602.67	1.06694
$266$	617.416	0.142317
$267$	−2725.28	−0.624661
$268$	1051.61	0.239692
$269$	−4665.78	−1.05754	−0.528769	−	0.848766i	$-0.677346\pi$
$269$	−4665.78	−1.05754	−0.528769	+	0.848766i	$0.677346\pi$
$270$	901.820	0.203271
$271$	−6719.71	−1.50625	−0.753124	−	0.657879i	$-0.771454\pi$
$271$	−6719.71	−1.50625	−0.753124	+	0.657879i	$0.771454\pi$
$272$	1142.41	0.254665
$273$	1249.54	0.277017
$274$	86.8015	0.0191382
$275$	−1692.93	−0.371227
$276$	640.809	0.139754
$277$	2781.08	0.603245	0.301623	−	0.953427i	$-0.402472\pi$
$277$	2781.08	0.603245	0.301623	+	0.953427i	$0.402472\pi$
$278$	−5558.50	−1.19920
$279$	−212.359	−0.0455685
$280$	935.221	0.199608
$281$	−3428.98	−0.727956	−0.363978	−	0.931407i	$-0.618582\pi$
$281$	−3428.98	−0.727956	−0.363978	+	0.931407i	$0.618582\pi$
$282$	157.775	0.0333169
$283$	−3791.14	−0.796324	−0.398162	−	0.917315i	$-0.630352\pi$
$283$	−3791.14	−0.796324	−0.398162	+	0.917315i	$0.630352\pi$
$284$	−899.236	−0.187887
$285$	−2209.52	−0.459230
$286$	1309.04	0.270648
$287$	−2499.03	−0.513982
$288$	−288.000	−0.0589256
$289$	185.068	0.0376691
$290$	5507.87	1.11529
$291$	3234.09	0.651497
$292$	4019.61	0.805582
$293$	9935.04	1.98093	0.990463	−	0.137779i	$-0.0439963\pi$
$293$	9935.04	1.98093	0.990463	+	0.137779i	$0.0439963\pi$
$294$	294.000	0.0583212
$295$	9965.25	1.96678
$296$	848.809	0.166676
$297$	297.000	0.0580259
$298$	396.172	0.0770122
$299$	−3177.45	−0.614570
$300$	−1846.83	−0.355423
$301$	2011.86	0.385254
$302$	5491.24	1.04631
$303$	−4081.80	−0.773905
$304$	705.618	0.133125
$305$	−554.301	−0.104063
$306$	−1285.21	−0.240101
$307$	−3836.06	−0.713145	−0.356573	−	0.934268i	$-0.616055\pi$
$307$	−3836.06	−0.713145	−0.356573	+	0.934268i	$0.616055\pi$
$308$	308.000	0.0569803
$309$	4744.29	0.873441
$310$	788.106	0.144392
$311$	7481.77	1.36416	0.682078	−	0.731280i	$-0.261077\pi$
$311$	7481.77	1.36416	0.682078	+	0.731280i	$0.261077\pi$
$312$	1428.05	0.259125
$313$	−2170.59	−0.391978	−0.195989	−	0.980606i	$-0.562792\pi$
$313$	−2170.59	−0.391978	−0.195989	+	0.980606i	$0.562792\pi$
$314$	−6991.76	−1.25659
$315$	−1052.12	−0.188192
$316$	1451.21	0.258344
$317$	−3930.00	−0.696312	−0.348156	−	0.937437i	$-0.613192\pi$
$317$	−3930.00	−0.696312	−0.348156	+	0.937437i	$0.613192\pi$
$318$	1653.62	0.291605
$319$	1813.93	0.318372
$320$	1068.82	0.186716
$321$	−4961.16	−0.862632
$322$	−747.611	−0.129387
$323$	3148.85	0.542436
$324$	324.000	0.0555556
$325$	9157.50	1.56297
$326$	5280.64	0.897139
$327$	5443.96	0.920647
$328$	−2856.03	−0.480786
$329$	−184.071	−0.0308455
$330$	−1102.22	−0.183865
$331$	−1768.92	−0.293741	−0.146871	−	0.989156i	$-0.546920\pi$
$331$	−1768.92	−0.293741	−0.146871	+	0.989156i	$0.546920\pi$
$332$	3899.21	0.644568
$333$	−954.910	−0.157143
$334$	−3966.85	−0.649869
$335$	4390.57	0.716068
$336$	336.000	0.0545545
$337$	6018.75	0.972885	0.486442	−	0.873713i	$-0.338294\pi$
$337$	6018.75	0.972885	0.486442	+	0.873713i	$0.338294\pi$
$338$	−2686.95	−0.432399
$339$	−1839.53	−0.294718
$340$	4769.68	0.760800
$341$	259.550	0.0412182
$342$	−793.820	−0.125511
$343$	−343.000	−0.0539949
$344$	2299.27	0.360373
$345$	2675.44	0.417509
$346$	−1806.89	−0.280749
$347$	−80.4197	−0.0124414	−0.00622069	−	0.999981i	$-0.501980\pi$
$347$	−80.4197	−0.0124414	−0.00622069	+	0.999981i	$0.501980\pi$
$348$	1978.83	0.304818
$349$	−3017.07	−0.462751	−0.231375	−	0.972865i	$-0.574323\pi$
$349$	−3017.07	−0.462751	−0.231375	+	0.972865i	$0.574323\pi$
$350$	2154.64	0.329058
$351$	−1606.55	−0.244306
$352$	352.000	0.0533002
$353$	4972.53	0.749748	0.374874	−	0.927076i	$-0.377686\pi$
$353$	4972.53	0.749748	0.374874	+	0.927076i	$0.377686\pi$
$354$	3580.25	0.537537
$355$	−3754.40	−0.561303
$356$	3633.71	0.540972
$357$	1499.42	0.222290
$358$	−1038.47	−0.153310
$359$	−6368.09	−0.936198	−0.468099	−	0.883676i	$-0.655061\pi$
$359$	−6368.09	−0.936198	−0.468099	+	0.883676i	$0.655061\pi$
$360$	−1202.43	−0.176037
$361$	−4914.09	−0.716444
$362$	−260.900	−0.0378801
$363$	−363.000	−0.0524864
$364$	−1666.05	−0.239904
$365$	16782.3	2.40664
$366$	−199.145	−0.0284413
$367$	1895.80	0.269645	0.134823	−	0.990870i	$-0.456954\pi$
$367$	1895.80	0.269645	0.134823	+	0.990870i	$0.456954\pi$
$368$	−854.412	−0.121031
$369$	3213.03	0.453290
$370$	3543.86	0.497936
$371$	−1929.22	−0.269974
$372$	283.145	0.0394635
$373$	−6255.46	−0.868353	−0.434176	−	0.900828i	$-0.642961\pi$
$373$	−6255.46	−0.868353	−0.434176	+	0.900828i	$0.642961\pi$
$374$	1570.82	0.217179
$375$	−1448.06	−0.199406
$376$	−210.367	−0.0288533
$377$	−9812.02	−1.34044
$378$	−378.000	−0.0514344
$379$	8145.25	1.10394	0.551970	−	0.833864i	$-0.313876\pi$
$379$	8145.25	1.10394	0.551970	+	0.833864i	$0.313876\pi$
$380$	2946.02	0.397705
$381$	−4426.92	−0.595271
$382$	2818.49	0.377504
$383$	10497.1	1.40046	0.700230	−	0.713918i	$-0.253081\pi$
$383$	10497.1	1.40046	0.700230	+	0.713918i	$0.253081\pi$
$384$	384.000	0.0510310
$385$	1285.93	0.170226
$386$	−8405.68	−1.10839
$387$	−2586.67	−0.339762
$388$	−4312.12	−0.564213
$389$	−8068.41	−1.05163	−0.525816	−	0.850599i	$-0.676240\pi$
$389$	−8068.41	−1.05163	−0.525816	+	0.850599i	$0.676240\pi$
$390$	5962.22	0.774126
$391$	−3812.85	−0.493157
$392$	−392.000	−0.0505076
$393$	−1046.52	−0.134325
$394$	309.618	0.0395897
$395$	6058.92	0.771791
$396$	−396.000	−0.0502519
$397$	2999.88	0.379244	0.189622	−	0.981857i	$-0.439274\pi$
$397$	2999.88	0.379244	0.189622	+	0.981857i	$0.439274\pi$
$398$	3586.02	0.451636
$399$	926.124	0.116201
$400$	2462.44	0.307805
$401$	3632.38	0.452350	0.226175	−	0.974087i	$-0.427378\pi$
$401$	3632.38	0.452350	0.226175	+	0.974087i	$0.427378\pi$
$402$	1577.42	0.195707
$403$	−1403.97	−0.173541
$404$	5442.40	0.670221
$405$	1352.73	0.165970
$406$	−2308.64	−0.282206
$407$	1167.11	0.142142
$408$	1713.62	0.207933
$409$	−2523.44	−0.305076	−0.152538	−	0.988298i	$-0.548745\pi$
$409$	−2523.44	−0.305076	−0.152538	+	0.988298i	$0.548745\pi$
$410$	−11924.2	−1.43633
$411$	130.202	0.0156263
$412$	−6325.72	−0.756422
$413$	−4176.96	−0.497662
$414$	961.214	0.114109
$415$	16279.6	1.92562
$416$	−1904.06	−0.224409
$417$	−8337.75	−0.979141
$418$	970.225	0.113529
$419$	3154.35	0.367781	0.183890	−	0.982947i	$-0.441131\pi$
$419$	3154.35	0.367781	0.183890	+	0.982947i	$0.441131\pi$
$420$	1402.83	0.162979
$421$	−7403.35	−0.857049	−0.428524	−	0.903530i	$-0.640966\pi$
$421$	−7403.35	−0.857049	−0.428524	+	0.903530i	$0.640966\pi$
$422$	−3445.23	−0.397420
$423$	236.663	0.0272031
$424$	−2204.82	−0.252537
$425$	10988.8	1.25420
$426$	−1348.85	−0.153409
$427$	232.336	0.0263315
$428$	6614.88	0.747061
$429$	1963.56	0.220983
$430$	9599.66	1.07660
$431$	13004.1	1.45333	0.726666	−	0.686991i	$-0.241069\pi$
$431$	13004.1	1.45333	0.726666	+	0.686991i	$0.241069\pi$
$432$	−432.000	−0.0481125
$433$	10572.3	1.17338	0.586689	−	0.809812i	$-0.300431\pi$
$433$	10572.3	1.17338	0.586689	+	0.809812i	$0.300431\pi$
$434$	−330.336	−0.0365361
$435$	8261.81	0.910629
$436$	−7258.61	−0.797304
$437$	−2355.03	−0.257795
$438$	6029.42	0.657755
$439$	−850.549	−0.0924703	−0.0462352	−	0.998931i	$-0.514722\pi$
$439$	−850.549	−0.0924703	−0.0462352	+	0.998931i	$0.514722\pi$
$440$	1469.63	0.159232
$441$	441.000	0.0476190
$442$	−8496.96	−0.914387
$443$	−7085.18	−0.759881	−0.379940	−	0.925011i	$-0.624056\pi$
$443$	−7085.18	−0.759881	−0.379940	+	0.925011i	$0.624056\pi$
$444$	1273.21	0.136090
$445$	15171.1	1.61613
$446$	−6621.35	−0.702982
$447$	594.258	0.0628802
$448$	−448.000	−0.0472456
$449$	5180.48	0.544503	0.272252	−	0.962226i	$-0.412232\pi$
$449$	5180.48	0.544503	0.272252	+	0.962226i	$0.412232\pi$
$450$	−2770.25	−0.290202
$451$	−3927.04	−0.410016
$452$	2452.70	0.255233
$453$	8236.85	0.854307
$454$	−8603.70	−0.889409
$455$	−6955.93	−0.716701
$456$	1058.43	0.108696
$457$	9046.82	0.926023	0.463012	−	0.886352i	$-0.346769\pi$
$457$	9046.82	0.926023	0.463012	+	0.886352i	$0.346769\pi$
$458$	12454.4	1.27065
$459$	−1927.82	−0.196041
$460$	−3567.25	−0.361574
$461$	−3131.97	−0.316422	−0.158211	−	0.987405i	$-0.550573\pi$
$461$	−3131.97	−0.316422	−0.158211	+	0.987405i	$0.550573\pi$
$462$	462.000	0.0465242
$463$	−3757.25	−0.377136	−0.188568	−	0.982060i	$-0.560385\pi$
$463$	−3757.25	−0.377136	−0.188568	+	0.982060i	$0.560385\pi$
$464$	−2638.44	−0.263980
$465$	1182.16	0.117895
$466$	7429.77	0.738578
$467$	1289.71	0.127796	0.0638981	−	0.997956i	$-0.479647\pi$
$467$	1289.71	0.127796	0.0638981	+	0.997956i	$0.479647\pi$
$468$	2142.07	0.211575
$469$	−1840.32	−0.181190
$470$	−878.301	−0.0861979
$471$	−10487.6	−1.02600
$472$	−4773.66	−0.465521
$473$	3161.49	0.307327
$474$	2176.81	0.210937
$475$	6787.28	0.655625
$476$	−1999.22	−0.192509
$477$	2480.43	0.238094
$478$	2308.31	0.220878
$479$	−4998.41	−0.476792	−0.238396	−	0.971168i	$-0.576622\pi$
$479$	−4998.41	−0.476792	−0.238396	+	0.971168i	$0.576622\pi$
$480$	1603.24	0.152453
$481$	−6313.22	−0.598457
$482$	14143.7	1.33658
$483$	−1121.42	−0.105644
$484$	484.000	0.0454545
$485$	−18003.5	−1.68556
$486$	486.000	0.0453609
$487$	−6613.85	−0.615404	−0.307702	−	0.951483i	$-0.599560\pi$
$487$	−6613.85	−0.615404	−0.307702	+	0.951483i	$0.599560\pi$
$488$	265.527	0.0246309
$489$	7920.96	0.732511
$490$	−1636.64	−0.150889
$491$	−3491.36	−0.320902	−0.160451	−	0.987044i	$-0.551295\pi$
$491$	−3491.36	−0.320902	−0.160451	+	0.987044i	$0.551295\pi$
$492$	−4284.05	−0.392560
$493$	−11774.2	−1.07562
$494$	−5248.20	−0.477991
$495$	−1653.34	−0.150125
$496$	−377.527	−0.0341764
$497$	1573.66	0.142029
$498$	5848.81	0.526288
$499$	−5026.14	−0.450904	−0.225452	−	0.974254i	$-0.572386\pi$
$499$	−5026.14	−0.450904	−0.225452	+	0.974254i	$0.572386\pi$
$500$	1930.74	0.172691
$501$	−5950.27	−0.530616
$502$	11671.8	1.03772
$503$	−18052.7	−1.60026	−0.800129	−	0.599827i	$-0.795236\pi$
$503$	−18052.7	−1.60026	−0.800129	+	0.599827i	$0.795236\pi$
$504$	504.000	0.0445435
$505$	22722.5	2.00226
$506$	−1174.82	−0.103215
$507$	−4030.43	−0.353052
$508$	5902.56	0.515520
$509$	−20110.6	−1.75125	−0.875627	−	0.482988i	$-0.839551\pi$
$509$	−20110.6	−1.75125	−0.875627	+	0.482988i	$0.839551\pi$
$510$	7154.52	0.621191
$511$	−7034.32	−0.608963
$512$	−512.000	−0.0441942
$513$	−1190.73	−0.102480
$514$	11538.1	0.990125
$515$	−26410.5	−2.25978
$516$	3448.90	0.294243
$517$	−289.254	−0.0246062
$518$	−1485.42	−0.125995
$519$	−2710.34	−0.229231
$520$	−7949.63	−0.670412
$521$	−18317.0	−1.54027	−0.770135	−	0.637881i	$-0.779811\pi$
$521$	−18317.0	−1.54027	−0.770135	+	0.637881i	$0.779811\pi$
$522$	2968.25	0.248883
$523$	−8234.95	−0.688507	−0.344253	−	0.938877i	$-0.611868\pi$
$523$	−8234.95	−0.688507	−0.344253	+	0.938877i	$0.611868\pi$
$524$	1395.36	0.116329
$525$	3231.96	0.268674
$526$	−234.720	−0.0194568
$527$	−1684.73	−0.139256
$528$	528.000	0.0435194
$529$	−9315.36	−0.765625
$530$	−9205.35	−0.754443
$531$	5370.37	0.438897
$532$	−1234.83	−0.100633
$533$	21242.4	1.72629
$534$	5450.56	0.441702
$535$	27617.7	2.23181
$536$	−2103.22	−0.169488
$537$	−1557.71	−0.125177
$538$	9331.55	0.747792
$539$	−539.000	−0.0430730
$540$	−1803.64	−0.143734
$541$	13286.4	1.05587	0.527937	−	0.849284i	$-0.322966\pi$
$541$	13286.4	1.05587	0.527937	+	0.849284i	$0.322966\pi$
$542$	13439.4	1.06508
$543$	−391.350	−0.0309290
$544$	−2284.82	−0.180075
$545$	−30305.4	−2.38191
$546$	−2499.08	−0.195880
$547$	−1047.91	−0.0819114	−0.0409557	−	0.999161i	$-0.513040\pi$
$547$	−1047.91	−0.0819114	−0.0409557	+	0.999161i	$0.513040\pi$
$548$	−173.603	−0.0135328
$549$	−298.718	−0.0232222
$550$	3385.86	0.262497
$551$	−7272.39	−0.562277
$552$	−1281.62	−0.0988212
$553$	−2539.61	−0.195290
$554$	−5562.16	−0.426559
$555$	5315.79	0.406563
$556$	11117.0	0.847961
$557$	18004.4	1.36960	0.684802	−	0.728729i	$-0.259889\pi$
$557$	18004.4	1.36960	0.684802	+	0.728729i	$0.259889\pi$
$558$	424.718	0.0322218
$559$	−17101.3	−1.29393
$560$	−1870.44	−0.141144
$561$	2356.22	0.177326
$562$	6857.96	0.514743
$563$	16934.6	1.26769	0.633844	−	0.773461i	$-0.281476\pi$
$563$	16934.6	1.26769	0.633844	+	0.773461i	$0.281476\pi$
$564$	−315.550	−0.0235586
$565$	10240.3	0.762498
$566$	7582.27	0.563086
$567$	−567.000	−0.0419961
$568$	1798.47	0.132856
$569$	13058.5	0.962110	0.481055	−	0.876690i	$-0.340254\pi$
$569$	13058.5	0.962110	0.481055	+	0.876690i	$0.340254\pi$
$570$	4419.03	0.324725
$571$	−18777.9	−1.37624	−0.688118	−	0.725599i	$-0.741563\pi$
$571$	−18777.9	−1.37624	−0.688118	+	0.725599i	$0.741563\pi$
$572$	−2618.08	−0.191377
$573$	4227.73	0.308230
$574$	4998.05	0.363440
$575$	−8218.52	−0.596062
$576$	576.000	0.0416667
$577$	7907.19	0.570504	0.285252	−	0.958453i	$-0.407923\pi$
$577$	7907.19	0.570504	0.285252	+	0.958453i	$0.407923\pi$
$578$	−370.136	−0.0266361
$579$	−12608.5	−0.904995
$580$	−11015.7	−0.788628
$581$	−6823.61	−0.487248
$582$	−6468.18	−0.460678
$583$	−3031.63	−0.215364
$584$	−8039.22	−0.569632
$585$	8943.34	0.632071
$586$	−19870.1	−1.40073
$587$	−18639.1	−1.31059	−0.655296	−	0.755372i	$-0.727456\pi$
$587$	−18639.1	−1.31059	−0.655296	+	0.755372i	$0.727456\pi$
$588$	−588.000	−0.0412393
$589$	−1040.59	−0.0727956
$590$	−19930.5	−1.39072
$591$	464.427	0.0323248
$592$	−1697.62	−0.117858
$593$	−551.427	−0.0381861	−0.0190931	−	0.999818i	$-0.506078\pi$
$593$	−551.427	−0.0381861	−0.0190931	+	0.999818i	$0.506078\pi$
$594$	−594.000	−0.0410305
$595$	−8346.94	−0.575111
$596$	−792.344	−0.0544558
$597$	5379.03	0.368759
$598$	6354.89	0.434567
$599$	−7181.98	−0.489896	−0.244948	−	0.969536i	$-0.578771\pi$
$599$	−7181.98	−0.489896	−0.244948	+	0.969536i	$0.578771\pi$
$600$	3693.66	0.251322
$601$	16258.9	1.10352	0.551760	−	0.834003i	$-0.313956\pi$
$601$	16258.9	1.10352	0.551760	+	0.834003i	$0.313956\pi$
$602$	−4023.72	−0.272416
$603$	2366.12	0.159794
$604$	−10982.5	−0.739852
$605$	2020.75	0.135793
$606$	8163.60	0.547233
$607$	−29398.1	−1.96578	−0.982892	−	0.184182i	$-0.941037\pi$
$607$	−29398.1	−1.96578	−0.982892	+	0.184182i	$0.941037\pi$
$608$	−1411.24	−0.0941335
$609$	−3462.96	−0.230420
$610$	1108.60	0.0735835
$611$	1564.65	0.103599
$612$	2570.43	0.169777
$613$	14600.0	0.961974	0.480987	−	0.876728i	$-0.340278\pi$
$613$	14600.0	0.961974	0.480987	+	0.876728i	$0.340278\pi$
$614$	7672.12	0.504270
$615$	−17886.3	−1.17276
$616$	−616.000	−0.0402911
$617$	−22010.6	−1.43617	−0.718083	−	0.695957i	$-0.754980\pi$
$617$	−22010.6	−1.43617	−0.718083	+	0.695957i	$0.754980\pi$
$618$	−9488.59	−0.617616
$619$	6060.24	0.393508	0.196754	−	0.980453i	$-0.436960\pi$
$619$	6060.24	0.393508	0.196754	+	0.980453i	$0.436960\pi$
$620$	−1576.21	−0.102100
$621$	1441.82	0.0931695
$622$	−14963.5	−0.964603
$623$	−6358.99	−0.408937
$624$	−2856.09	−0.183229
$625$	−11176.8	−0.715316
$626$	4341.18	0.277170
$627$	1455.34	0.0926963
$628$	13983.5	0.888540
$629$	−7575.70	−0.480227
$630$	2104.25	0.133072
$631$	26677.2	1.68304	0.841522	−	0.540223i	$-0.181660\pi$
$631$	26677.2	1.68304	0.841522	+	0.540223i	$0.181660\pi$
$632$	−2902.41	−0.182677
$633$	−5167.84	−0.324492
$634$	7860.00	0.492367
$635$	24643.8	1.54009
$636$	−3307.24	−0.206196
$637$	2915.59	0.181350
$638$	−3627.86	−0.225123
$639$	−2023.28	−0.125258
$640$	−2137.65	−0.132028
$641$	16397.4	1.01039	0.505195	−	0.863005i	$-0.331421\pi$
$641$	16397.4	1.01039	0.505195	+	0.863005i	$0.331421\pi$
$642$	9922.32	0.609973
$643$	−13095.7	−0.803176	−0.401588	−	0.915820i	$-0.631542\pi$
$643$	−13095.7	−0.803176	−0.401588	+	0.915820i	$0.631542\pi$
$644$	1495.22	0.0914906
$645$	14399.5	0.879037
$646$	−6297.71	−0.383560
$647$	−4191.06	−0.254664	−0.127332	−	0.991860i	$-0.540641\pi$
$647$	−4191.06	−0.254664	−0.127332	+	0.991860i	$0.540641\pi$
$648$	−648.000	−0.0392837
$649$	−6563.79	−0.396997
$650$	−18315.0	−1.10519
$651$	−495.505	−0.0298316
$652$	−10561.3	−0.634373
$653$	31908.1	1.91219	0.956095	−	0.293058i	$-0.0946729\pi$
$653$	31908.1	1.91219	0.956095	+	0.293058i	$0.0946729\pi$
$654$	−10887.9	−0.650996
$655$	5825.75	0.347528
$656$	5712.06	0.339967
$657$	9044.12	0.537055
$658$	368.142	0.0218110
$659$	−16343.7	−0.966102	−0.483051	−	0.875592i	$-0.660471\pi$
$659$	−16343.7	−0.966102	−0.483051	+	0.875592i	$0.660471\pi$
$660$	2204.45	0.130012
$661$	−12561.7	−0.739174	−0.369587	−	0.929196i	$-0.620501\pi$
$661$	−12561.7	−0.739174	−0.369587	+	0.929196i	$0.620501\pi$
$662$	3537.83	0.207706
$663$	−12745.4	−0.746594
$664$	−7798.41	−0.455779
$665$	−5155.54	−0.300636
$666$	1909.82	0.111117
$667$	8805.93	0.511195
$668$	7933.69	0.459527
$669$	−9932.02	−0.573982
$670$	−8781.15	−0.506336
$671$	365.100	0.0210053
$672$	−672.000	−0.0385758
$673$	−7968.16	−0.456390	−0.228195	−	0.973616i	$-0.573282\pi$
$673$	−7968.16	−0.456390	−0.228195	+	0.973616i	$0.573282\pi$
$674$	−12037.5	−0.687933
$675$	−4155.37	−0.236949
$676$	5373.90	0.305752
$677$	−4770.62	−0.270827	−0.135413	−	0.990789i	$-0.543236\pi$
$677$	−4770.62	−0.270827	−0.135413	+	0.990789i	$0.543236\pi$
$678$	3679.05	0.208397
$679$	7546.21	0.426505
$680$	−9539.36	−0.537967
$681$	−12905.6	−0.726199
$682$	−519.100	−0.0291457
$683$	−22592.5	−1.26571	−0.632855	−	0.774271i	$-0.718117\pi$
$683$	−22592.5	−1.26571	−0.632855	+	0.774271i	$0.718117\pi$
$684$	1587.64	0.0887500
$685$	−724.809	−0.0404285
$686$	686.000	0.0381802
$687$	18681.6	1.03748
$688$	−4598.53	−0.254822
$689$	16398.9	0.906747
$690$	−5350.88	−0.295224
$691$	6224.12	0.342658	0.171329	−	0.985214i	$-0.445194\pi$
$691$	6224.12	0.342658	0.171329	+	0.985214i	$0.445194\pi$
$692$	3613.78	0.198519
$693$	693.000	0.0379869
$694$	160.839	0.00879738
$695$	46414.6	2.53324
$696$	−3957.66	−0.215539
$697$	25490.3	1.38524
$698$	6034.14	0.327214
$699$	11144.7	0.603047
$700$	−4309.27	−0.232679
$701$	17785.2	0.958258	0.479129	−	0.877745i	$-0.340953\pi$
$701$	17785.2	0.958258	0.479129	+	0.877745i	$0.340953\pi$
$702$	3213.10	0.172750
$703$	−4679.18	−0.251036
$704$	−704.000	−0.0376889
$705$	−1317.45	−0.0703803
$706$	−9945.06	−0.530152
$707$	−9524.19	−0.506640
$708$	−7160.50	−0.380096
$709$	132.807	0.00703481	0.00351740	−	0.999994i	$-0.498880\pi$
$709$	132.807	0.00703481	0.00351740	+	0.999994i	$0.498880\pi$
$710$	7508.79	0.396901
$711$	3265.21	0.172229
$712$	−7267.42	−0.382525
$713$	1260.02	0.0661822
$714$	−2998.83	−0.157183
$715$	−10930.7	−0.571730
$716$	2076.95	0.108407
$717$	3462.46	0.180346
$718$	12736.2	0.661992
$719$	14392.2	0.746507	0.373253	−	0.927729i	$-0.378242\pi$
$719$	14392.2	0.746507	0.373253	+	0.927729i	$0.378242\pi$
$720$	2404.85	0.124477
$721$	11070.0	0.571802
$722$	9828.18	0.506602
$723$	21215.6	1.09131
$724$	521.800	0.0267853
$725$	−25379.0	−1.30007
$726$	726.000	0.0371135
$727$	−2743.83	−0.139977	−0.0699883	−	0.997548i	$-0.522296\pi$
$727$	−2743.83	−0.139977	−0.0699883	+	0.997548i	$0.522296\pi$
$728$	3332.11	0.169637
$729$	729.000	0.0370370
$730$	−33564.5	−1.70175
$731$	−20521.2	−1.03831
$732$	398.291	0.0201110
$733$	−19617.1	−0.988503	−0.494252	−	0.869319i	$-0.664558\pi$
$733$	−19617.1	−0.988503	−0.494252	+	0.869319i	$0.664558\pi$
$734$	−3791.60	−0.190668
$735$	−2454.96	−0.123201
$736$	1708.82	0.0855817
$737$	−2891.93	−0.144539
$738$	−6426.07	−0.320524
$739$	−18197.2	−0.905813	−0.452906	−	0.891558i	$-0.649613\pi$
$739$	−18197.2	−0.905813	−0.452906	+	0.891558i	$0.649613\pi$
$740$	−7087.72	−0.352094
$741$	−7872.30	−0.390278
$742$	3858.44	0.190900
$743$	−30932.7	−1.52733	−0.763667	−	0.645611i	$-0.776603\pi$
$743$	−30932.7	−1.52733	−0.763667	+	0.645611i	$0.776603\pi$
$744$	−566.291	−0.0279049
$745$	−3308.11	−0.162684
$746$	12510.9	0.614018
$747$	8773.21	0.429712
$748$	−3141.63	−0.153569
$749$	−11576.0	−0.564725
$750$	2896.11	0.141001
$751$	14741.1	0.716261	0.358130	−	0.933672i	$-0.383414\pi$
$751$	14741.1	0.716261	0.358130	+	0.933672i	$0.383414\pi$
$752$	420.733	0.0204023
$753$	17507.7	0.847299
$754$	19624.0	0.947832
$755$	−45852.9	−2.21027
$756$	756.000	0.0363696
$757$	−6181.81	−0.296806	−0.148403	−	0.988927i	$-0.547413\pi$
$757$	−6181.81	−0.296806	−0.148403	+	0.988927i	$0.547413\pi$
$758$	−16290.5	−0.780604
$759$	−1762.22	−0.0842750
$760$	−5892.05	−0.281220
$761$	−20491.8	−0.976122	−0.488061	−	0.872810i	$-0.662296\pi$
$761$	−20491.8	−0.976122	−0.488061	+	0.872810i	$0.662296\pi$
$762$	8853.85	0.420920
$763$	12702.6	0.602705
$764$	−5636.98	−0.266935
$765$	10731.8	0.507200
$766$	−20994.2	−0.990274
$767$	35505.3	1.67147
$768$	−768.000	−0.0360844
$769$	−25261.1	−1.18458	−0.592288	−	0.805726i	$-0.701775\pi$
$769$	−25261.1	−1.18458	−0.592288	+	0.805726i	$0.701775\pi$
$770$	−2571.86	−0.120368
$771$	17307.2	0.808434
$772$	16811.4	0.783749
$773$	−25411.8	−1.18241	−0.591203	−	0.806523i	$-0.701347\pi$
$773$	−25411.8	−1.18241	−0.591203	+	0.806523i	$0.701347\pi$
$774$	5173.35	0.240248
$775$	−3631.40	−0.168315
$776$	8624.24	0.398959
$777$	−2228.12	−0.102875
$778$	16136.8	0.743615
$779$	15744.3	0.724130
$780$	−11924.4	−0.547389
$781$	2472.90	0.113300
$782$	7625.71	0.348715
$783$	4452.37	0.203212
$784$	784.000	0.0357143
$785$	58382.5	2.65447
$786$	2093.04	0.0949824
$787$	22862.1	1.03551	0.517755	−	0.855529i	$-0.326768\pi$
$787$	22862.1	1.03551	0.517755	+	0.855529i	$0.326768\pi$
$788$	−619.236	−0.0279941
$789$	−352.081	−0.0158864
$790$	−12117.8	−0.545739
$791$	−4292.23	−0.192938
$792$	792.000	0.0355335
$793$	−1974.92	−0.0884383
$794$	−5999.76	−0.268166
$795$	−13808.0	−0.616000
$796$	−7172.05	−0.319355
$797$	15105.5	0.671346	0.335673	−	0.941979i	$-0.391036\pi$
$797$	15105.5	0.671346	0.335673	+	0.941979i	$0.391036\pi$
$798$	−1852.25	−0.0821665
$799$	1877.54	0.0831323
$800$	−4924.88	−0.217651
$801$	8175.85	0.360648
$802$	−7264.76	−0.319860
$803$	−11053.9	−0.485784
$804$	−3154.83	−0.138386
$805$	6242.69	0.273324
$806$	2807.95	0.122712
$807$	13997.3	0.610569
$808$	−10884.8	−0.473918
$809$	14429.4	0.627084	0.313542	−	0.949574i	$-0.398484\pi$
$809$	14429.4	0.627084	0.313542	+	0.949574i	$0.398484\pi$
$810$	−2705.46	−0.117358
$811$	33367.0	1.44473	0.722364	−	0.691513i	$-0.243055\pi$
$811$	33367.0	1.44473	0.722364	+	0.691513i	$0.243055\pi$
$812$	4617.27	0.199550
$813$	20159.1	0.869632
$814$	−2334.22	−0.100509
$815$	−44094.3	−1.89516
$816$	−3427.24	−0.147031
$817$	−12675.0	−0.542770
$818$	5046.88	0.215721
$819$	−3748.62	−0.159936
$820$	23848.4	1.01564
$821$	32922.3	1.39951	0.699754	−	0.714384i	$-0.253293\pi$
$821$	32922.3	1.39951	0.699754	+	0.714384i	$0.253293\pi$
$822$	−260.405	−0.0110495
$823$	24054.1	1.01880	0.509400	−	0.860530i	$-0.329867\pi$
$823$	24054.1	1.01880	0.509400	+	0.860530i	$0.329867\pi$
$824$	12651.4	0.534871
$825$	5078.79	0.214328
$826$	8353.91	0.351900
$827$	−42511.6	−1.78751	−0.893756	−	0.448553i	$-0.851939\pi$
$827$	−42511.6	−1.78751	−0.893756	+	0.448553i	$0.851939\pi$
$828$	−1922.43	−0.0806872
$829$	−14105.7	−0.590968	−0.295484	−	0.955348i	$-0.595481\pi$
$829$	−14105.7	−0.590968	−0.295484	+	0.955348i	$0.595481\pi$
$830$	−32559.1	−1.36162
$831$	−8343.24	−0.348284
$832$	3808.12	0.158681
$833$	3498.64	0.145523
$834$	16675.5	0.692357
$835$	33123.9	1.37282
$836$	−1940.45	−0.0802774
$837$	637.077	0.0263090
$838$	−6308.70	−0.260060
$839$	−13589.3	−0.559184	−0.279592	−	0.960119i	$-0.590199\pi$
$839$	−13589.3	−0.559184	−0.279592	+	0.960119i	$0.590199\pi$
$840$	−2805.66	−0.115244
$841$	2803.88	0.114965
$842$	14806.7	0.606025
$843$	10286.9	0.420286
$844$	6890.46	0.281018
$845$	22436.5	0.913421
$846$	−473.325	−0.0192355
$847$	−847.000	−0.0343604
$848$	4409.65	0.178571
$849$	11373.4	0.459758
$850$	−21977.5	−0.886851
$851$	5665.88	0.228230
$852$	2697.71	0.108477
$853$	−19055.7	−0.764895	−0.382448	−	0.923977i	$-0.624919\pi$
$853$	−19055.7	−0.764895	−0.382448	+	0.923977i	$0.624919\pi$
$854$	−464.673	−0.0186192
$855$	6628.55	0.265136
$856$	−13229.8	−0.528252
$857$	30491.5	1.21537	0.607684	−	0.794179i	$-0.292099\pi$
$857$	30491.5	1.21537	0.607684	+	0.794179i	$0.292099\pi$
$858$	−3927.12	−0.156259
$859$	33454.4	1.32881	0.664405	−	0.747373i	$-0.268685\pi$
$859$	33454.4	1.32881	0.664405	+	0.747373i	$0.268685\pi$
$860$	−19199.3	−0.761269
$861$	7497.08	0.296748
$862$	−26008.2	−1.02766
$863$	−35521.3	−1.40111	−0.700555	−	0.713598i	$-0.747064\pi$
$863$	−35521.3	−1.40111	−0.700555	+	0.713598i	$0.747064\pi$
$864$	864.000	0.0340207
$865$	15087.9	0.593068
$866$	−21144.6	−0.829704
$867$	−555.205	−0.0217483
$868$	660.673	0.0258349
$869$	−3990.82	−0.155787
$870$	−16523.6	−0.643912
$871$	15643.2	0.608553
$872$	14517.2	0.563779
$873$	−9702.27	−0.376142
$874$	4710.07	0.182289
$875$	−3378.80	−0.130542
$876$	−12058.8	−0.465103
$877$	−33245.4	−1.28006	−0.640032	−	0.768348i	$-0.721079\pi$
$877$	−33245.4	−1.28006	−0.640032	+	0.768348i	$0.721079\pi$
$878$	1701.10	0.0653864
$879$	−29805.1	−1.14369
$880$	−2939.27	−0.112594
$881$	49749.3	1.90249	0.951247	−	0.308429i	$-0.0998031\pi$
$881$	49749.3	1.90249	0.951247	+	0.308429i	$0.0998031\pi$
$882$	−882.000	−0.0336718
$883$	−11584.0	−0.441488	−0.220744	−	0.975332i	$-0.570849\pi$
$883$	−11584.0	−0.441488	−0.220744	+	0.975332i	$0.570849\pi$
$884$	16993.9	0.646569
$885$	−29895.7	−1.13552
$886$	14170.4	0.537317
$887$	20072.1	0.759813	0.379907	−	0.925025i	$-0.375956\pi$
$887$	20072.1	0.759813	0.379907	+	0.925025i	$0.375956\pi$
$888$	−2546.43	−0.0962303
$889$	−10329.5	−0.389696
$890$	−30342.2	−1.14278
$891$	−891.000	−0.0335013
$892$	13242.7	0.497083
$893$	1159.68	0.0434570
$894$	−1188.52	−0.0444630
$895$	8671.44	0.323860
$896$	896.000	0.0334077
$897$	9532.34	0.354822
$898$	−10361.0	−0.385022
$899$	3890.95	0.144350
$900$	5540.50	0.205204
$901$	19678.3	0.727612
$902$	7854.08	0.289925
$903$	−6035.57	−0.222427
$904$	−4905.41	−0.180477
$905$	2178.56	0.0800198
$906$	−16473.7	−0.604086
$907$	−4502.35	−0.164827	−0.0824134	−	0.996598i	$-0.526263\pi$
$907$	−4502.35	−0.164827	−0.0824134	+	0.996598i	$0.526263\pi$
$908$	17207.4	0.628907
$909$	12245.4	0.446814
$910$	13911.9	0.506784
$911$	11085.1	0.403144	0.201572	−	0.979474i	$-0.435395\pi$
$911$	11085.1	0.403144	0.201572	+	0.979474i	$0.435395\pi$
$912$	−2116.85	−0.0768597
$913$	−10722.8	−0.388689
$914$	−18093.6	−0.654797
$915$	1662.90	0.0600807
$916$	−24908.9	−0.898484
$917$	−2441.88	−0.0879366
$918$	3855.64	0.138622
$919$	−22731.0	−0.815916	−0.407958	−	0.913001i	$-0.633759\pi$
$919$	−22731.0	−0.815916	−0.407958	+	0.913001i	$0.633759\pi$
$920$	7134.50	0.255671
$921$	11508.2	0.411735
$922$	6263.94	0.223744
$923$	−13376.6	−0.477026
$924$	−924.000	−0.0328976
$925$	−16329.2	−0.580435
$926$	7514.50	0.266676
$927$	−14232.9	−0.504282
$928$	5276.88	0.186662
$929$	−46642.3	−1.64724	−0.823619	−	0.567143i	$-0.808048\pi$
$929$	−46642.3	−1.64724	−0.823619	+	0.567143i	$0.808048\pi$
$930$	−2364.32	−0.0833646
$931$	2160.96	0.0760714
$932$	−14859.5	−0.522254
$933$	−22445.3	−0.787595
$934$	−2579.43	−0.0903656
$935$	−13116.6	−0.458780
$936$	−4284.14	−0.149606
$937$	13272.8	0.462758	0.231379	−	0.972864i	$-0.425676\pi$
$937$	13272.8	0.462758	0.231379	+	0.972864i	$0.425676\pi$
$938$	3680.64	0.128121
$939$	6511.78	0.226309
$940$	1756.60	0.0609511
$941$	−32269.1	−1.11790	−0.558949	−	0.829202i	$-0.688795\pi$
$941$	−32269.1	−1.11790	−0.558949	+	0.829202i	$0.688795\pi$
$942$	20975.3	0.725490
$943$	−19064.3	−0.658344
$944$	9547.33	0.329173
$945$	3156.37	0.108653
$946$	−6322.98	−0.217313
$947$	10366.8	0.355730	0.177865	−	0.984055i	$-0.443081\pi$
$947$	10366.8	0.355730	0.177865	+	0.984055i	$0.443081\pi$
$948$	−4353.62	−0.149155
$949$	59793.6	2.04529
$950$	−13574.6	−0.463597
$951$	11790.0	0.402016
$952$	3998.44	0.136124
$953$	−17371.2	−0.590460	−0.295230	−	0.955426i	$-0.595396\pi$
$953$	−17371.2	−0.590460	−0.295230	+	0.955426i	$0.595396\pi$
$954$	−4960.85	−0.168358
$955$	−23534.9	−0.797457
$956$	−4616.62	−0.156184
$957$	−5441.79	−0.183812
$958$	9996.82	0.337143
$959$	303.805	0.0102298
$960$	−3206.47	−0.107800
$961$	−29234.3	−0.981312
$962$	12626.4	0.423173
$963$	14883.5	0.498041
$964$	−28287.5	−0.945102
$965$	70189.0	2.34141
$966$	2242.83	0.0747018
$967$	−19022.6	−0.632600	−0.316300	−	0.948659i	$-0.602441\pi$
$967$	−19022.6	−0.632600	−0.316300	+	0.948659i	$0.602441\pi$
$968$	−968.000	−0.0321412
$969$	−9446.56	−0.313176
$970$	36007.0	1.19187
$971$	−16470.6	−0.544352	−0.272176	−	0.962248i	$-0.587743\pi$
$971$	−16470.6	−0.544352	−0.272176	+	0.962248i	$0.587743\pi$
$972$	−972.000	−0.0320750
$973$	−19454.8	−0.640998
$974$	13227.7	0.435157
$975$	−27472.5	−0.902384
$976$	−531.055	−0.0174166
$977$	−42550.4	−1.39335	−0.696677	−	0.717385i	$-0.745339\pi$
$977$	−42550.4	−1.39335	−0.696677	+	0.717385i	$0.745339\pi$
$978$	−15841.9	−0.517964
$979$	−9992.70	−0.326219
$980$	3273.27	0.106695
$981$	−16331.9	−0.531536
$982$	6982.72	0.226912
$983$	56421.9	1.83070	0.915351	−	0.402656i	$-0.131913\pi$
$983$	56421.9	1.83070	0.915351	+	0.402656i	$0.131913\pi$
$984$	8568.09	0.277582
$985$	−2585.37	−0.0836312
$986$	23548.3	0.760580
$987$	552.213	0.0178086
$988$	10496.4	0.337991
$989$	15347.8	0.493460
$990$	3306.67	0.106155
$991$	51713.1	1.65764	0.828820	−	0.559516i	$-0.189013\pi$
$991$	51713.1	1.65764	0.828820	+	0.559516i	$0.189013\pi$
$992$	755.055	0.0241663
$993$	5306.75	0.169592
$994$	−3147.33	−0.100430
$995$	−29944.0	−0.954058
$996$	−11697.6	−0.372142
$997$	−555.880	−0.0176579	−0.00882894	−	0.999961i	$-0.502810\pi$
$997$	−555.880	−0.0176579	−0.00882894	+	0.999961i	$0.502810\pi$
$998$	10052.3	0.318837
$999$	2864.73	0.0907268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.4.a.k.1.2	✓	2
3.2	odd	2		1386.4.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.k.1.2	✓	2	1.1	even	1	trivial
1386.4.a.x.1.1		2	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:	\( N \)	\(=\)	\( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	462.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +16.7004 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +16.7004 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -33.4008 q^{10} -11.0000 q^{11} -12.0000 q^{12} +59.5019 q^{13} +14.0000 q^{14} -50.1011 q^{15} +16.0000 q^{16} +71.4008 q^{17} -18.0000 q^{18} +44.1011 q^{19} +66.8015 q^{20} +21.0000 q^{21} +22.0000 q^{22} -53.4008 q^{23} +24.0000 q^{24} +153.903 q^{25} -119.004 q^{26} -27.0000 q^{27} -28.0000 q^{28} -164.903 q^{29} +100.202 q^{30} -23.5955 q^{31} -32.0000 q^{32} +33.0000 q^{33} -142.802 q^{34} -116.903 q^{35} +36.0000 q^{36} -106.101 q^{37} -88.2023 q^{38} -178.506 q^{39} -133.603 q^{40} +357.004 q^{41} -42.0000 q^{42} -287.408 q^{43} -44.0000 q^{44} +150.303 q^{45} +106.802 q^{46} +26.2958 q^{47} -48.0000 q^{48} +49.0000 q^{49} -307.805 q^{50} -214.202 q^{51} +238.008 q^{52} +275.603 q^{53} +54.0000 q^{54} -183.704 q^{55} +56.0000 q^{56} -132.303 q^{57} +329.805 q^{58} +596.708 q^{59} -200.405 q^{60} -33.1909 q^{61} +47.1909 q^{62} -63.0000 q^{63} +64.0000 q^{64} +993.704 q^{65} -66.0000 q^{66} +262.903 q^{67} +285.603 q^{68} +160.202 q^{69} +233.805 q^{70} -224.809 q^{71} -72.0000 q^{72} +1004.90 q^{73} +212.202 q^{74} -461.708 q^{75} +176.405 q^{76} +77.0000 q^{77} +357.011 q^{78} +362.802 q^{79} +267.206 q^{80} +81.0000 q^{81} -714.008 q^{82} +974.802 q^{83} +84.0000 q^{84} +1192.42 q^{85} +574.817 q^{86} +494.708 q^{87} +88.0000 q^{88} +908.427 q^{89} -300.607 q^{90} -416.513 q^{91} -213.603 q^{92} +70.7864 q^{93} -52.5917 q^{94} +736.506 q^{95} +96.0000 q^{96} -1078.03 q^{97} -98.0000 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 7 q^{5} + 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 + 7 * q^5 + 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9	\( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 7 q^{5} + 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9} - 14 q^{10} - 22 q^{11} - 24 q^{12} - 13 q^{13} + 28 q^{14} - 21 q^{15} + 32 q^{16} + 90 q^{17} - 36 q^{18} + 9 q^{19} + 28 q^{20} + 42 q^{21} + 44 q^{22} - 54 q^{23} + 48 q^{24} + 123 q^{25} + 26 q^{26} - 54 q^{27} - 56 q^{28} - 145 q^{29} + 42 q^{30} - 364 q^{31} - 64 q^{32} + 66 q^{33} - 180 q^{34} - 49 q^{35} + 72 q^{36} - 133 q^{37} - 18 q^{38} + 39 q^{39} - 56 q^{40} + 450 q^{41} - 84 q^{42} + 6 q^{43} - 88 q^{44} + 63 q^{45} + 108 q^{46} + 343 q^{47} - 96 q^{48} + 98 q^{49} - 246 q^{50} - 270 q^{51} - 52 q^{52} + 340 q^{53} + 108 q^{54} - 77 q^{55} + 112 q^{56} - 27 q^{57} + 290 q^{58} + 639 q^{59} - 84 q^{60} - 700 q^{61} + 728 q^{62} - 126 q^{63} + 128 q^{64} + 1697 q^{65} - 132 q^{66} + 341 q^{67} + 360 q^{68} + 162 q^{69} + 98 q^{70} + 184 q^{71} - 144 q^{72} + 1825 q^{73} + 266 q^{74} - 369 q^{75} + 36 q^{76} + 154 q^{77} - 78 q^{78} + 620 q^{79} + 112 q^{80} + 162 q^{81} - 900 q^{82} + 1844 q^{83} + 168 q^{84} + 1012 q^{85} - 12 q^{86} + 435 q^{87} + 176 q^{88} - 84 q^{89} - 126 q^{90} + 91 q^{91} - 216 q^{92} + 1092 q^{93} - 686 q^{94} + 1077 q^{95} + 192 q^{96} - 44 q^{97} - 196 q^{98} - 198 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 + 7 * q^5 + 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9 - 14 * q^10 - 22 * q^11 - 24 * q^12 - 13 * q^13 + 28 * q^14 - 21 * q^15 + 32 * q^16 + 90 * q^17 - 36 * q^18 + 9 * q^19 + 28 * q^20 + 42 * q^21 + 44 * q^22 - 54 * q^23 + 48 * q^24 + 123 * q^25 + 26 * q^26 - 54 * q^27 - 56 * q^28 - 145 * q^29 + 42 * q^30 - 364 * q^31 - 64 * q^32 + 66 * q^33 - 180 * q^34 - 49 * q^35 + 72 * q^36 - 133 * q^37 - 18 * q^38 + 39 * q^39 - 56 * q^40 + 450 * q^41 - 84 * q^42 + 6 * q^43 - 88 * q^44 + 63 * q^45 + 108 * q^46 + 343 * q^47 - 96 * q^48 + 98 * q^49 - 246 * q^50 - 270 * q^51 - 52 * q^52 + 340 * q^53 + 108 * q^54 - 77 * q^55 + 112 * q^56 - 27 * q^57 + 290 * q^58 + 639 * q^59 - 84 * q^60 - 700 * q^61 + 728 * q^62 - 126 * q^63 + 128 * q^64 + 1697 * q^65 - 132 * q^66 + 341 * q^67 + 360 * q^68 + 162 * q^69 + 98 * q^70 + 184 * q^71 - 144 * q^72 + 1825 * q^73 + 266 * q^74 - 369 * q^75 + 36 * q^76 + 154 * q^77 - 78 * q^78 + 620 * q^79 + 112 * q^80 + 162 * q^81 - 900 * q^82 + 1844 * q^83 + 168 * q^84 + 1012 * q^85 - 12 * q^86 + 435 * q^87 + 176 * q^88 - 84 * q^89 - 126 * q^90 + 91 * q^91 - 216 * q^92 + 1092 * q^93 - 686 * q^94 + 1077 * q^95 + 192 * q^96 - 44 * q^97 - 196 * q^98 - 198 * q^99

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