LMFDB - Embedded newform 462.4.a.r.1.1

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:	$3$
Coefficient field:	3.3.1028796.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 295x + 175 $ x^3 - x^2 - 295*x + 175
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.592735$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -17.2278 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -17.2278 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +34.4555 q^{10} +11.0000 q^{11} +12.0000 q^{12} +15.5987 q^{13} +14.0000 q^{14} -51.6833 q^{15} +16.0000 q^{16} -131.738 q^{17} -18.0000 q^{18} -7.96965 q^{19} -68.9111 q^{20} -21.0000 q^{21} -22.0000 q^{22} +49.5445 q^{23} -24.0000 q^{24} +171.796 q^{25} -31.1974 q^{26} +27.0000 q^{27} -28.0000 q^{28} +74.7961 q^{29} +103.367 q^{30} -48.9350 q^{31} -32.0000 q^{32} +33.0000 q^{33} +263.475 q^{34} +120.594 q^{35} +36.0000 q^{36} +202.857 q^{37} +15.9393 q^{38} +46.7961 q^{39} +137.822 q^{40} +87.9068 q^{41} +42.0000 q^{42} +444.616 q^{43} +44.0000 q^{44} -155.050 q^{45} -99.0889 q^{46} +386.811 q^{47} +48.0000 q^{48} +49.0000 q^{49} -343.592 q^{50} -395.213 q^{51} +62.3949 q^{52} -562.265 q^{53} -54.0000 q^{54} -189.505 q^{55} +56.0000 q^{56} -23.9090 q^{57} -149.592 q^{58} -322.679 q^{59} -206.733 q^{60} +346.790 q^{61} +97.8700 q^{62} -63.0000 q^{63} +64.0000 q^{64} -268.731 q^{65} -66.0000 q^{66} +984.254 q^{67} -526.950 q^{68} +148.633 q^{69} -241.189 q^{70} +563.185 q^{71} -72.0000 q^{72} -132.516 q^{73} -405.714 q^{74} +515.388 q^{75} -31.8786 q^{76} -77.0000 q^{77} -93.5923 q^{78} +55.2753 q^{79} -275.644 q^{80} +81.0000 q^{81} -175.814 q^{82} +874.909 q^{83} -84.0000 q^{84} +2269.54 q^{85} -889.232 q^{86} +224.388 q^{87} -88.0000 q^{88} +567.540 q^{89} +310.100 q^{90} -109.191 q^{91} +198.178 q^{92} -146.805 q^{93} -773.623 q^{94} +137.299 q^{95} -96.0000 q^{96} +593.228 q^{97} -98.0000 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} - 24 q^{8} + 27 q^{9} - 14 q^{10} + 33 q^{11} + 36 q^{12} - 15 q^{13} + 42 q^{14} + 21 q^{15} + 48 q^{16} - 40 q^{17} - 54 q^{18} + 41 q^{19} + 28 q^{20} - 63 q^{21} - 66 q^{22} + 266 q^{23} - 72 q^{24} + 330 q^{25} + 30 q^{26} + 81 q^{27} - 84 q^{28} + 39 q^{29} - 42 q^{30} + 332 q^{31} - 96 q^{32} + 99 q^{33} + 80 q^{34} - 49 q^{35} + 108 q^{36} + 553 q^{37} - 82 q^{38} - 45 q^{39} - 56 q^{40} - 320 q^{41} + 126 q^{42} + 290 q^{43} + 132 q^{44} + 63 q^{45} - 532 q^{46} + 33 q^{47} + 144 q^{48} + 147 q^{49} - 660 q^{50} - 120 q^{51} - 60 q^{52} - 482 q^{53} - 162 q^{54} + 77 q^{55} + 168 q^{56} + 123 q^{57} - 78 q^{58} - 443 q^{59} + 84 q^{60} + 546 q^{61} - 664 q^{62} - 189 q^{63} + 192 q^{64} + 287 q^{65} - 198 q^{66} + 661 q^{67} - 160 q^{68} + 798 q^{69} + 98 q^{70} + 948 q^{71} - 216 q^{72} + 1539 q^{73} - 1106 q^{74} + 990 q^{75} + 164 q^{76} - 231 q^{77} + 90 q^{78} + 1568 q^{79} + 112 q^{80} + 243 q^{81} + 640 q^{82} + 52 q^{83} - 252 q^{84} + 3028 q^{85} - 580 q^{86} + 117 q^{87} - 264 q^{88} + 1042 q^{89} - 126 q^{90} + 105 q^{91} + 1064 q^{92} + 996 q^{93} - 66 q^{94} - 1237 q^{95} - 288 q^{96} + 3670 q^{97} - 294 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 - 24 * q^8 + 27 * q^9 - 14 * q^10 + 33 * q^11 + 36 * q^12 - 15 * q^13 + 42 * q^14 + 21 * q^15 + 48 * q^16 - 40 * q^17 - 54 * q^18 + 41 * q^19 + 28 * q^20 - 63 * q^21 - 66 * q^22 + 266 * q^23 - 72 * q^24 + 330 * q^25 + 30 * q^26 + 81 * q^27 - 84 * q^28 + 39 * q^29 - 42 * q^30 + 332 * q^31 - 96 * q^32 + 99 * q^33 + 80 * q^34 - 49 * q^35 + 108 * q^36 + 553 * q^37 - 82 * q^38 - 45 * q^39 - 56 * q^40 - 320 * q^41 + 126 * q^42 + 290 * q^43 + 132 * q^44 + 63 * q^45 - 532 * q^46 + 33 * q^47 + 144 * q^48 + 147 * q^49 - 660 * q^50 - 120 * q^51 - 60 * q^52 - 482 * q^53 - 162 * q^54 + 77 * q^55 + 168 * q^56 + 123 * q^57 - 78 * q^58 - 443 * q^59 + 84 * q^60 + 546 * q^61 - 664 * q^62 - 189 * q^63 + 192 * q^64 + 287 * q^65 - 198 * q^66 + 661 * q^67 - 160 * q^68 + 798 * q^69 + 98 * q^70 + 948 * q^71 - 216 * q^72 + 1539 * q^73 - 1106 * q^74 + 990 * q^75 + 164 * q^76 - 231 * q^77 + 90 * q^78 + 1568 * q^79 + 112 * q^80 + 243 * q^81 + 640 * q^82 + 52 * q^83 - 252 * q^84 + 3028 * q^85 - 580 * q^86 + 117 * q^87 - 264 * q^88 + 1042 * q^89 - 126 * q^90 + 105 * q^91 + 1064 * q^92 + 996 * q^93 - 66 * q^94 - 1237 * q^95 - 288 * q^96 + 3670 * q^97 - 294 * q^98 + 297 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−17.2278	−1.54090	−0.770449	−	0.637501i	$-0.779968\pi$
$5$	−17.2278	−1.54090	−0.770449	+	0.637501i	$0.779968\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	34.4555	1.08958
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	12.0000	0.288675
$13$	15.5987	0.332793	0.166396	−	0.986059i	$-0.446787\pi$
$13$	15.5987	0.332793	0.166396	+	0.986059i	$0.446787\pi$
$14$	14.0000	0.267261
$15$	−51.6833	−0.889638
$16$	16.0000	0.250000
$17$	−131.738	−1.87947	−0.939737	−	0.341898i	$-0.888930\pi$
$17$	−131.738	−1.87947	−0.939737	+	0.341898i	$0.888930\pi$
$18$	−18.0000	−0.235702
$19$	−7.96965	−0.0962297	−0.0481148	−	0.998842i	$-0.515321\pi$
$19$	−7.96965	−0.0962297	−0.0481148	+	0.998842i	$0.515321\pi$
$20$	−68.9111	−0.770449
$21$	−21.0000	−0.218218
$22$	−22.0000	−0.213201
$23$	49.5445	0.449162	0.224581	−	0.974455i	$-0.427899\pi$
$23$	49.5445	0.449162	0.224581	+	0.974455i	$0.427899\pi$
$24$	−24.0000	−0.204124
$25$	171.796	1.37437
$26$	−31.1974	−0.235320
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	74.7961	0.478941	0.239471	−	0.970904i	$-0.423026\pi$
$29$	74.7961	0.478941	0.239471	+	0.970904i	$0.423026\pi$
$30$	103.367	0.629069
$31$	−48.9350	−0.283516	−0.141758	−	0.989901i	$-0.545275\pi$
$31$	−48.9350	−0.283516	−0.141758	+	0.989901i	$0.545275\pi$
$32$	−32.0000	−0.176777
$33$	33.0000	0.174078
$34$	263.475	1.32899
$35$	120.594	0.582405
$36$	36.0000	0.166667
$37$	202.857	0.901337	0.450668	−	0.892691i	$-0.351186\pi$
$37$	202.857	0.901337	0.450668	+	0.892691i	$0.351186\pi$
$38$	15.9393	0.0680447
$39$	46.7961	0.192138
$40$	137.822	0.544790
$41$	87.9068	0.334847	0.167424	−	0.985885i	$-0.446455\pi$
$41$	87.9068	0.334847	0.167424	+	0.985885i	$0.446455\pi$
$42$	42.0000	0.154303
$43$	444.616	1.57682	0.788411	−	0.615149i	$-0.210904\pi$
$43$	444.616	1.57682	0.788411	+	0.615149i	$0.210904\pi$
$44$	44.0000	0.150756
$45$	−155.050	−0.513633
$46$	−99.0889	−0.317606
$47$	386.811	1.20047	0.600237	−	0.799822i	$-0.295073\pi$
$47$	386.811	1.20047	0.600237	+	0.799822i	$0.295073\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	−343.592	−0.971826
$51$	−395.213	−1.08511
$52$	62.3949	0.166396
$53$	−562.265	−1.45723	−0.728613	−	0.684925i	$-0.759835\pi$
$53$	−562.265	−1.45723	−0.728613	+	0.684925i	$0.759835\pi$
$54$	−54.0000	−0.136083
$55$	−189.505	−0.464598
$56$	56.0000	0.133631
$57$	−23.9090	−0.0555582
$58$	−149.592	−0.338663
$59$	−322.679	−0.712021	−0.356010	−	0.934482i	$-0.615863\pi$
$59$	−322.679	−0.712021	−0.356010	+	0.934482i	$0.615863\pi$
$60$	−206.733	−0.444819
$61$	346.790	0.727900	0.363950	−	0.931419i	$-0.381428\pi$
$61$	346.790	0.727900	0.363950	+	0.931419i	$0.381428\pi$
$62$	97.8700	0.200476
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−268.731	−0.512800
$66$	−66.0000	−0.123091
$67$	984.254	1.79471	0.897357	−	0.441306i	$-0.145485\pi$
$67$	984.254	1.79471	0.897357	+	0.441306i	$0.145485\pi$
$68$	−526.950	−0.939737
$69$	148.633	0.259324
$70$	−241.189	−0.411823
$71$	563.185	0.941376	0.470688	−	0.882300i	$-0.344006\pi$
$71$	563.185	0.941376	0.470688	+	0.882300i	$0.344006\pi$
$72$	−72.0000	−0.117851
$73$	−132.516	−0.212464	−0.106232	−	0.994341i	$-0.533879\pi$
$73$	−132.516	−0.212464	−0.106232	+	0.994341i	$0.533879\pi$
$74$	−405.714	−0.637341
$75$	515.388	0.793492
$76$	−31.8786	−0.0481148
$77$	−77.0000	−0.113961
$78$	−93.5923	−0.135862
$79$	55.2753	0.0787210	0.0393605	−	0.999225i	$-0.487468\pi$
$79$	55.2753	0.0787210	0.0393605	+	0.999225i	$0.487468\pi$
$80$	−275.644	−0.385225
$81$	81.0000	0.111111
$82$	−175.814	−0.236773
$83$	874.909	1.15703	0.578517	−	0.815671i	$-0.303632\pi$
$83$	874.909	1.15703	0.578517	+	0.815671i	$0.303632\pi$
$84$	−84.0000	−0.109109
$85$	2269.54	2.89608
$86$	−889.232	−1.11498
$87$	224.388	0.276517
$88$	−88.0000	−0.106600
$89$	567.540	0.675945	0.337973	−	0.941156i	$-0.390259\pi$
$89$	567.540	0.675945	0.337973	+	0.941156i	$0.390259\pi$
$90$	310.100	0.363193
$91$	−109.191	−0.125784
$92$	198.178	0.224581
$93$	−146.805	−0.163688
$94$	−773.623	−0.848863
$95$	137.299	0.148280
$96$	−96.0000	−0.102062
$97$	593.228	0.620960	0.310480	−	0.950580i	$-0.399510\pi$
$97$	593.228	0.620960	0.310480	+	0.950580i	$0.399510\pi$
$98$	−98.0000	−0.101015
$99$	99.0000	0.100504
$100$	687.185	0.687185
$101$	−911.792	−0.898284	−0.449142	−	0.893460i	$-0.648270\pi$
$101$	−911.792	−0.898284	−0.449142	+	0.893460i	$0.648270\pi$
$102$	790.425	0.767292
$103$	−718.775	−0.687602	−0.343801	−	0.939043i	$-0.611715\pi$
$103$	−718.775	−0.687602	−0.343801	+	0.939043i	$0.611715\pi$
$104$	−124.790	−0.117660
$105$	361.783	0.336252
$106$	1124.53	1.03041
$107$	1624.82	1.46801	0.734007	−	0.679142i	$-0.237648\pi$
$107$	1624.82	1.46801	0.734007	+	0.679142i	$0.237648\pi$
$108$	108.000	0.0962250
$109$	−702.213	−0.617062	−0.308531	−	0.951214i	$-0.599837\pi$
$109$	−702.213	−0.617062	−0.308531	+	0.951214i	$0.599837\pi$
$110$	379.011	0.328521
$111$	608.570	0.520387
$112$	−112.000	−0.0944911
$113$	802.338	0.667944	0.333972	−	0.942583i	$-0.391611\pi$
$113$	802.338	0.667944	0.333972	+	0.942583i	$0.391611\pi$
$114$	47.8179	0.0392856
$115$	−853.541	−0.692114
$116$	299.185	0.239471
$117$	140.388	0.110931
$118$	645.358	0.503475
$119$	922.163	0.710374
$120$	413.467	0.314535
$121$	121.000	0.0909091
$122$	−693.579	−0.514703
$123$	263.720	0.193324
$124$	−195.740	−0.141758
$125$	−806.193	−0.576865
$126$	126.000	0.0890871
$127$	2032.57	1.42017	0.710084	−	0.704117i	$-0.248657\pi$
$127$	2032.57	1.42017	0.710084	+	0.704117i	$0.248657\pi$
$128$	−128.000	−0.0883883
$129$	1333.85	0.910378
$130$	537.462	0.362604
$131$	−58.5353	−0.0390401	−0.0195200	−	0.999809i	$-0.506214\pi$
$131$	−58.5353	−0.0390401	−0.0195200	+	0.999809i	$0.506214\pi$
$132$	132.000	0.0870388
$133$	55.7876	0.0363714
$134$	−1968.51	−1.26905
$135$	−465.150	−0.296546
$136$	1053.90	0.664494
$137$	−1338.19	−0.834518	−0.417259	−	0.908788i	$-0.637009\pi$
$137$	−1338.19	−0.834518	−0.417259	+	0.908788i	$0.637009\pi$
$138$	−297.267	−0.183370
$139$	−342.297	−0.208873	−0.104436	−	0.994532i	$-0.533304\pi$
$139$	−342.297	−0.208873	−0.104436	+	0.994532i	$0.533304\pi$
$140$	482.378	0.291203
$141$	1160.43	0.693094
$142$	−1126.37	−0.665654
$143$	171.586	0.100341
$144$	144.000	0.0833333
$145$	−1288.57	−0.738000
$146$	265.033	0.150235
$147$	147.000	0.0824786
$148$	811.427	0.450668
$149$	−177.281	−0.0974728	−0.0487364	−	0.998812i	$-0.515519\pi$
$149$	−177.281	−0.0974728	−0.0487364	+	0.998812i	$0.515519\pi$
$150$	−1030.78	−0.561084
$151$	−2329.57	−1.25548	−0.627741	−	0.778422i	$-0.716020\pi$
$151$	−2329.57	−1.25548	−0.627741	+	0.778422i	$0.716020\pi$
$152$	63.7572	0.0340223
$153$	−1185.64	−0.626491
$154$	154.000	0.0805823
$155$	843.041	0.436869
$156$	187.185	0.0960690
$157$	2408.40	1.22427	0.612137	−	0.790752i	$-0.290310\pi$
$157$	2408.40	1.22427	0.612137	+	0.790752i	$0.290310\pi$
$158$	−110.551	−0.0556642
$159$	−1686.79	−0.841330
$160$	551.289	0.272395
$161$	−346.811	−0.169767
$162$	−162.000	−0.0785674
$163$	3830.26	1.84055	0.920274	−	0.391274i	$-0.127966\pi$
$163$	3830.26	1.84055	0.920274	+	0.391274i	$0.127966\pi$
$164$	351.627	0.167424
$165$	−568.516	−0.268236
$166$	−1749.82	−0.818146
$167$	−782.568	−0.362616	−0.181308	−	0.983426i	$-0.558033\pi$
$167$	−782.568	−0.362616	−0.181308	+	0.983426i	$0.558033\pi$
$168$	168.000	0.0771517
$169$	−1953.68	−0.889249
$170$	−4539.09	−2.04784
$171$	−71.7269	−0.0320766
$172$	1778.46	0.788411
$173$	−1854.82	−0.815143	−0.407571	−	0.913173i	$-0.633624\pi$
$173$	−1854.82	−0.815143	−0.407571	+	0.913173i	$0.633624\pi$
$174$	−448.777	−0.195527
$175$	−1202.57	−0.519463
$176$	176.000	0.0753778
$177$	−968.037	−0.411085
$178$	−1135.08	−0.477965
$179$	1722.23	0.719136	0.359568	−	0.933119i	$-0.382924\pi$
$179$	1722.23	0.719136	0.359568	+	0.933119i	$0.382924\pi$
$180$	−620.200	−0.256816
$181$	−2361.43	−0.969742	−0.484871	−	0.874586i	$-0.661134\pi$
$181$	−2361.43	−0.969742	−0.484871	+	0.874586i	$0.661134\pi$
$182$	218.382	0.0889426
$183$	1040.37	0.420253
$184$	−396.356	−0.158803
$185$	−3494.77	−1.38887
$186$	293.610	0.115745
$187$	−1449.11	−0.566683
$188$	1547.25	0.600237
$189$	−189.000	−0.0727393
$190$	−274.599	−0.104850
$191$	−587.414	−0.222533	−0.111266	−	0.993791i	$-0.535491\pi$
$191$	−587.414	−0.222533	−0.111266	+	0.993791i	$0.535491\pi$
$192$	192.000	0.0721688
$193$	−3532.39	−1.31744	−0.658722	−	0.752386i	$-0.728903\pi$
$193$	−3532.39	−1.31744	−0.658722	+	0.752386i	$0.728903\pi$
$194$	−1186.46	−0.439085
$195$	−806.193	−0.296065
$196$	196.000	0.0714286
$197$	−4156.19	−1.50313	−0.751564	−	0.659660i	$-0.770700\pi$
$197$	−4156.19	−1.50313	−0.751564	+	0.659660i	$0.770700\pi$
$198$	−198.000	−0.0710669
$199$	2126.74	0.757589	0.378795	−	0.925481i	$-0.376339\pi$
$199$	2126.74	0.757589	0.378795	+	0.925481i	$0.376339\pi$
$200$	−1374.37	−0.485913
$201$	2952.76	1.03618
$202$	1823.58	0.635183
$203$	−523.573	−0.181023
$204$	−1580.85	−0.542557
$205$	−1514.44	−0.515966
$206$	1437.55	0.486208
$207$	445.900	0.149721
$208$	249.579	0.0831982
$209$	−87.6662	−0.0290143
$210$	−723.566	−0.237766
$211$	181.344	0.0591671	0.0295836	−	0.999562i	$-0.490582\pi$
$211$	181.344	0.0591671	0.0295836	+	0.999562i	$0.490582\pi$
$212$	−2249.06	−0.728613
$213$	1689.55	0.543504
$214$	−3249.64	−1.03804
$215$	−7659.75	−2.42972
$216$	−216.000	−0.0680414
$217$	342.545	0.107159
$218$	1404.43	0.436329
$219$	−397.549	−0.122666
$220$	−758.022	−0.232299
$221$	−2054.94	−0.625475
$222$	−1217.14	−0.367969
$223$	116.623	0.0350209	0.0175104	−	0.999847i	$-0.494426\pi$
$223$	116.623	0.0350209	0.0175104	+	0.999847i	$0.494426\pi$
$224$	224.000	0.0668153
$225$	1546.17	0.458123
$226$	−1604.68	−0.472308
$227$	2156.71	0.630600	0.315300	−	0.948992i	$-0.397895\pi$
$227$	2156.71	0.630600	0.315300	+	0.948992i	$0.397895\pi$
$228$	−95.6359	−0.0277791
$229$	5396.06	1.55713	0.778563	−	0.627567i	$-0.215949\pi$
$229$	5396.06	1.55713	0.778563	+	0.627567i	$0.215949\pi$
$230$	1707.08	0.489398
$231$	−231.000	−0.0657952
$232$	−598.369	−0.169331
$233$	−3492.13	−0.981875	−0.490937	−	0.871195i	$-0.663346\pi$
$233$	−3492.13	−0.981875	−0.490937	+	0.871195i	$0.663346\pi$
$234$	−280.777	−0.0784400
$235$	−6663.90	−1.84981
$236$	−1290.72	−0.356010
$237$	165.826	0.0454496
$238$	−1844.33	−0.502311
$239$	1811.99	0.490409	0.245205	−	0.969471i	$-0.421145\pi$
$239$	1811.99	0.490409	0.245205	+	0.969471i	$0.421145\pi$
$240$	−826.933	−0.222410
$241$	546.395	0.146043	0.0730215	−	0.997330i	$-0.476736\pi$
$241$	546.395	0.146043	0.0730215	+	0.997330i	$0.476736\pi$
$242$	−242.000	−0.0642824
$243$	243.000	0.0641500
$244$	1387.16	0.363950
$245$	−844.161	−0.220128
$246$	−527.441	−0.136701
$247$	−124.316	−0.0320245
$248$	391.480	0.100238
$249$	2624.73	0.668013
$250$	1612.39	0.407905
$251$	−6093.48	−1.53234	−0.766169	−	0.642639i	$-0.777840\pi$
$251$	−6093.48	−1.53234	−0.766169	+	0.642639i	$0.777840\pi$
$252$	−252.000	−0.0629941
$253$	544.989	0.135428
$254$	−4065.14	−1.00421
$255$	6808.63	1.67205
$256$	256.000	0.0625000
$257$	5967.34	1.44838	0.724188	−	0.689603i	$-0.242215\pi$
$257$	5967.34	1.44838	0.724188	+	0.689603i	$0.242215\pi$
$258$	−2667.70	−0.643735
$259$	−1420.00	−0.340673
$260$	−1074.92	−0.256400
$261$	673.165	0.159647
$262$	117.071	0.0276055
$263$	−5535.60	−1.29787	−0.648934	−	0.760844i	$-0.724785\pi$
$263$	−5535.60	−1.29787	−0.648934	+	0.760844i	$0.724785\pi$
$264$	−264.000	−0.0615457
$265$	9686.57	2.24544
$266$	−111.575	−0.0257185
$267$	1702.62	0.390257
$268$	3937.02	0.897357
$269$	1703.64	0.386145	0.193072	−	0.981184i	$-0.438155\pi$
$269$	1703.64	0.386145	0.193072	+	0.981184i	$0.438155\pi$
$270$	930.300	0.209690
$271$	−4053.71	−0.908654	−0.454327	−	0.890835i	$-0.650120\pi$
$271$	−4053.71	−0.908654	−0.454327	+	0.890835i	$0.650120\pi$
$272$	−2107.80	−0.469868
$273$	−327.573	−0.0726213
$274$	2676.37	0.590094
$275$	1889.76	0.414388
$276$	594.533	0.129662
$277$	−4114.37	−0.892450	−0.446225	−	0.894921i	$-0.647232\pi$
$277$	−4114.37	−0.892450	−0.446225	+	0.894921i	$0.647232\pi$
$278$	684.595	0.147695
$279$	−440.415	−0.0945052
$280$	−964.755	−0.205911
$281$	8176.45	1.73582	0.867911	−	0.496720i	$-0.165462\pi$
$281$	8176.45	1.73582	0.867911	+	0.496720i	$0.165462\pi$
$282$	−2320.87	−0.490091
$283$	−2712.16	−0.569686	−0.284843	−	0.958574i	$-0.591941\pi$
$283$	−2712.16	−0.569686	−0.284843	+	0.958574i	$0.591941\pi$
$284$	2252.74	0.470688
$285$	411.898	0.0856096
$286$	−343.172	−0.0709516
$287$	−615.347	−0.126560
$288$	−288.000	−0.0589256
$289$	12441.8	2.53242
$290$	2577.14	0.521845
$291$	1779.68	0.358511
$292$	−530.066	−0.106232
$293$	−4223.37	−0.842089	−0.421044	−	0.907040i	$-0.638336\pi$
$293$	−4223.37	−0.842089	−0.421044	+	0.907040i	$0.638336\pi$
$294$	−294.000	−0.0583212
$295$	5559.04	1.09715
$296$	−1622.85	−0.318671
$297$	297.000	0.0580259
$298$	354.563	0.0689237
$299$	772.830	0.149478
$300$	2061.55	0.396746
$301$	−3112.31	−0.595983
$302$	4659.14	0.887760
$303$	−2735.38	−0.518625
$304$	−127.514	−0.0240574
$305$	−5974.41	−1.12162
$306$	2371.28	0.442996
$307$	7193.46	1.33730	0.668652	−	0.743575i	$-0.266871\pi$
$307$	7193.46	1.33730	0.668652	+	0.743575i	$0.266871\pi$
$308$	−308.000	−0.0569803
$309$	−2156.32	−0.396987
$310$	−1686.08	−0.308913
$311$	3934.68	0.717412	0.358706	−	0.933451i	$-0.383218\pi$
$311$	3934.68	0.717412	0.358706	+	0.933451i	$0.383218\pi$
$312$	−374.369	−0.0679310
$313$	1599.19	0.288791	0.144396	−	0.989520i	$-0.453876\pi$
$313$	1599.19	0.288791	0.144396	+	0.989520i	$0.453876\pi$
$314$	−4816.79	−0.865692
$315$	1085.35	0.194135
$316$	221.101	0.0393605
$317$	1114.12	0.197399	0.0986994	−	0.995117i	$-0.468532\pi$
$317$	1114.12	0.197399	0.0986994	+	0.995117i	$0.468532\pi$
$318$	3373.59	0.594910
$319$	822.758	0.144406
$320$	−1102.58	−0.192612
$321$	4874.47	0.847558
$322$	693.622	0.120044
$323$	1049.90	0.180861
$324$	324.000	0.0555556
$325$	2679.80	0.457380
$326$	−7660.53	−1.30146
$327$	−2106.64	−0.356261
$328$	−703.254	−0.118386
$329$	−2707.68	−0.453736
$330$	1137.03	0.189672
$331$	−7291.03	−1.21073	−0.605365	−	0.795948i	$-0.706973\pi$
$331$	−7291.03	−1.21073	−0.605365	+	0.795948i	$0.706973\pi$
$332$	3499.64	0.578517
$333$	1825.71	0.300446
$334$	1565.14	0.256409
$335$	−16956.5	−2.76547
$336$	−336.000	−0.0545545
$337$	3738.08	0.604233	0.302116	−	0.953271i	$-0.402307\pi$
$337$	3738.08	0.604233	0.302116	+	0.953271i	$0.402307\pi$
$338$	3907.36	0.628794
$339$	2407.02	0.385638
$340$	9078.18	1.44804
$341$	−538.285	−0.0854832
$342$	143.454	0.0226816
$343$	−343.000	−0.0539949
$344$	−3556.93	−0.557491
$345$	−2560.62	−0.399592
$346$	3709.65	0.576393
$347$	7204.46	1.11457	0.557285	−	0.830322i	$-0.311843\pi$
$347$	7204.46	1.11457	0.557285	+	0.830322i	$0.311843\pi$
$348$	897.554	0.138258
$349$	−9663.02	−1.48209	−0.741045	−	0.671455i	$-0.765670\pi$
$349$	−9663.02	−1.48209	−0.741045	+	0.671455i	$0.765670\pi$
$350$	2405.15	0.367316
$351$	421.165	0.0640460
$352$	−352.000	−0.0533002
$353$	9636.75	1.45301	0.726505	−	0.687161i	$-0.241143\pi$
$353$	9636.75	1.45301	0.726505	+	0.687161i	$0.241143\pi$
$354$	1936.07	0.290681
$355$	−9702.42	−1.45057
$356$	2270.16	0.337973
$357$	2766.49	0.410135
$358$	−3444.45	−0.508506
$359$	−9454.16	−1.38989	−0.694946	−	0.719062i	$-0.744572\pi$
$359$	−9454.16	−1.38989	−0.694946	+	0.719062i	$0.744572\pi$
$360$	1240.40	0.181597
$361$	−6795.48	−0.990740
$362$	4722.85	0.685711
$363$	363.000	0.0524864
$364$	−436.764	−0.0628919
$365$	2282.96	0.327386
$366$	−2080.74	−0.297164
$367$	−9030.03	−1.28437	−0.642185	−	0.766549i	$-0.721972\pi$
$367$	−9030.03	−1.28437	−0.642185	+	0.766549i	$0.721972\pi$
$368$	792.711	0.112291
$369$	791.161	0.111616
$370$	6989.54	0.982078
$371$	3935.85	0.550780
$372$	−587.220	−0.0818439
$373$	−2189.42	−0.303925	−0.151963	−	0.988386i	$-0.548559\pi$
$373$	−2189.42	−0.303925	−0.151963	+	0.988386i	$0.548559\pi$
$374$	2898.23	0.400705
$375$	−2418.58	−0.333053
$376$	−3094.49	−0.424431
$377$	1166.72	0.159388
$378$	378.000	0.0514344
$379$	−8881.76	−1.20376	−0.601881	−	0.798586i	$-0.705582\pi$
$379$	−8881.76	−1.20376	−0.601881	+	0.798586i	$0.705582\pi$
$380$	549.198	0.0741401
$381$	6097.71	0.819934
$382$	1174.83	0.157355
$383$	9135.19	1.21876	0.609382	−	0.792877i	$-0.291418\pi$
$383$	9135.19	1.21876	0.609382	+	0.792877i	$0.291418\pi$
$384$	−384.000	−0.0510310
$385$	1326.54	0.175602
$386$	7064.77	0.931574
$387$	4001.55	0.525607
$388$	2372.91	0.310480
$389$	14341.4	1.86924	0.934622	−	0.355642i	$-0.115738\pi$
$389$	14341.4	1.86924	0.934622	+	0.355642i	$0.115738\pi$
$390$	1612.39	0.209350
$391$	−6526.87	−0.844189
$392$	−392.000	−0.0505076
$393$	−175.606	−0.0225398
$394$	8312.38	1.06287
$395$	−952.271	−0.121301
$396$	396.000	0.0502519
$397$	1676.44	0.211935	0.105967	−	0.994370i	$-0.466206\pi$
$397$	1676.44	0.211935	0.105967	+	0.994370i	$0.466206\pi$
$398$	−4253.47	−0.535697
$399$	167.363	0.0209990
$400$	2748.74	0.343592
$401$	11505.4	1.43280	0.716398	−	0.697692i	$-0.245789\pi$
$401$	11505.4	1.43280	0.716398	+	0.697692i	$0.245789\pi$
$402$	−5905.52	−0.732689
$403$	−763.323	−0.0943519
$404$	−3647.17	−0.449142
$405$	−1395.45	−0.171211
$406$	1047.15	0.128002
$407$	2231.43	0.271763
$408$	3161.70	0.383646
$409$	12430.2	1.50277	0.751387	−	0.659862i	$-0.229385\pi$
$409$	12430.2	1.50277	0.751387	+	0.659862i	$0.229385\pi$
$410$	3028.88	0.364843
$411$	−4014.56	−0.481809
$412$	−2875.10	−0.343801
$413$	2258.75	0.269119
$414$	−891.800	−0.105869
$415$	−15072.7	−1.78287
$416$	−499.159	−0.0588300
$417$	−1026.89	−0.120593
$418$	175.332	0.0205162
$419$	11755.2	1.37060	0.685299	−	0.728261i	$-0.259671\pi$
$419$	11755.2	1.37060	0.685299	+	0.728261i	$0.259671\pi$
$420$	1447.13	0.168126
$421$	−1881.52	−0.217814	−0.108907	−	0.994052i	$-0.534735\pi$
$421$	−1881.52	−0.217814	−0.108907	+	0.994052i	$0.534735\pi$
$422$	−362.689	−0.0418375
$423$	3481.30	0.400158
$424$	4498.12	0.515207
$425$	−22632.0	−2.58309
$426$	−3379.11	−0.384315
$427$	−2427.53	−0.275120
$428$	6499.29	0.734007
$429$	514.758	0.0579318
$430$	15319.5	1.71807
$431$	16815.2	1.87926	0.939630	−	0.342193i	$-0.111170\pi$
$431$	16815.2	1.87926	0.939630	+	0.342193i	$0.111170\pi$
$432$	432.000	0.0481125
$433$	−4487.62	−0.498063	−0.249031	−	0.968495i	$-0.580112\pi$
$433$	−4487.62	−0.498063	−0.249031	+	0.968495i	$0.580112\pi$
$434$	−685.090	−0.0757728
$435$	−3865.71	−0.426084
$436$	−2808.85	−0.308531
$437$	−394.852	−0.0432228
$438$	795.099	0.0867381
$439$	−16966.1	−1.84453	−0.922263	−	0.386563i	$-0.873662\pi$
$439$	−16966.1	−1.84453	−0.922263	+	0.386563i	$0.873662\pi$
$440$	1516.04	0.164260
$441$	441.000	0.0476190
$442$	4109.87	0.442278
$443$	2231.40	0.239316	0.119658	−	0.992815i	$-0.461820\pi$
$443$	2231.40	0.239316	0.119658	+	0.992815i	$0.461820\pi$
$444$	2434.28	0.260193
$445$	−9777.45	−1.04156
$446$	−233.246	−0.0247635
$447$	−531.844	−0.0562759
$448$	−448.000	−0.0472456
$449$	17612.7	1.85122	0.925608	−	0.378483i	$-0.123554\pi$
$449$	17612.7	1.85122	0.925608	+	0.378483i	$0.123554\pi$
$450$	−3092.33	−0.323942
$451$	966.975	0.100960
$452$	3209.35	0.333972
$453$	−6988.71	−0.724853
$454$	−4313.43	−0.445902
$455$	1881.12	0.193820
$456$	191.272	0.0196428
$457$	17630.5	1.80464	0.902320	−	0.431066i	$-0.141863\pi$
$457$	17630.5	1.80464	0.902320	+	0.431066i	$0.141863\pi$
$458$	−10792.1	−1.10105
$459$	−3556.91	−0.361705
$460$	−3414.16	−0.346057
$461$	−1705.98	−0.172354	−0.0861771	−	0.996280i	$-0.527465\pi$
$461$	−1705.98	−0.172354	−0.0861771	+	0.996280i	$0.527465\pi$
$462$	462.000	0.0465242
$463$	7949.27	0.797913	0.398957	−	0.916970i	$-0.369372\pi$
$463$	7949.27	0.797913	0.398957	+	0.916970i	$0.369372\pi$
$464$	1196.74	0.119735
$465$	2529.12	0.252226
$466$	6984.25	0.694290
$467$	1790.48	0.177417	0.0887085	−	0.996058i	$-0.471726\pi$
$467$	1790.48	0.177417	0.0887085	+	0.996058i	$0.471726\pi$
$468$	561.554	0.0554654
$469$	−6889.78	−0.678338
$470$	13327.8	1.30801
$471$	7225.19	0.706835
$472$	2581.43	0.251737
$473$	4890.78	0.475430
$474$	−331.652	−0.0321377
$475$	−1369.16	−0.132255
$476$	3688.65	0.355187
$477$	−5060.38	−0.485742
$478$	−3623.98	−0.346772
$479$	652.495	0.0622407	0.0311203	−	0.999516i	$-0.490092\pi$
$479$	652.495	0.0622407	0.0311203	+	0.999516i	$0.490092\pi$
$480$	1653.87	0.157267
$481$	3164.31	0.299958
$482$	−1092.79	−0.103268
$483$	−1040.43	−0.0980153
$484$	484.000	0.0454545
$485$	−10220.0	−0.956837
$486$	−486.000	−0.0453609
$487$	4992.01	0.464496	0.232248	−	0.972657i	$-0.425392\pi$
$487$	4992.01	0.464496	0.232248	+	0.972657i	$0.425392\pi$
$488$	−2774.32	−0.257351
$489$	11490.8	1.06264
$490$	1688.32	0.155654
$491$	−18585.3	−1.70824	−0.854119	−	0.520077i	$-0.825903\pi$
$491$	−18585.3	−1.70824	−0.854119	+	0.520077i	$0.825903\pi$
$492$	1054.88	0.0966620
$493$	−9853.46	−0.900157
$494$	248.633	0.0226448
$495$	−1705.55	−0.154866
$496$	−782.960	−0.0708789
$497$	−3942.29	−0.355807
$498$	−5249.45	−0.472357
$499$	−18206.9	−1.63337	−0.816684	−	0.577085i	$-0.804190\pi$
$499$	−18206.9	−1.63337	−0.816684	+	0.577085i	$0.804190\pi$
$500$	−3224.77	−0.288432
$501$	−2347.70	−0.209357
$502$	12187.0	1.08353
$503$	−16070.6	−1.42456	−0.712280	−	0.701895i	$-0.752337\pi$
$503$	−16070.6	−1.42456	−0.712280	+	0.701895i	$0.752337\pi$
$504$	504.000	0.0445435
$505$	15708.1	1.38417
$506$	−1089.98	−0.0957617
$507$	−5861.04	−0.513408
$508$	8130.28	0.710084
$509$	−14915.1	−1.29883	−0.649413	−	0.760436i	$-0.724985\pi$
$509$	−14915.1	−1.29883	−0.649413	+	0.760436i	$0.724985\pi$
$510$	−13617.3	−1.18232
$511$	927.615	0.0803039
$512$	−512.000	−0.0441942
$513$	−215.181	−0.0185194
$514$	−11934.7	−1.02416
$515$	12382.9	1.05952
$516$	5335.39	0.455189
$517$	4254.93	0.361956
$518$	2840.00	0.240892
$519$	−5564.47	−0.470623
$520$	2149.85	0.181302
$521$	5598.43	0.470771	0.235385	−	0.971902i	$-0.424365\pi$
$521$	5598.43	0.470771	0.235385	+	0.971902i	$0.424365\pi$
$522$	−1346.33	−0.112888
$523$	−5078.21	−0.424579	−0.212289	−	0.977207i	$-0.568092\pi$
$523$	−5078.21	−0.424579	−0.212289	+	0.977207i	$0.568092\pi$
$524$	−234.141	−0.0195200
$525$	−3607.72	−0.299912
$526$	11071.2	0.917732
$527$	6446.58	0.532860
$528$	528.000	0.0435194
$529$	−9712.35	−0.798253
$530$	−19373.1	−1.58777
$531$	−2904.11	−0.237340
$532$	223.150	0.0181857
$533$	1371.23	0.111435
$534$	−3405.24	−0.275954
$535$	−27992.1	−2.26206
$536$	−7874.03	−0.634527
$537$	5166.68	0.415193
$538$	−3407.29	−0.273046
$539$	539.000	0.0430730
$540$	−1860.60	−0.148273
$541$	17562.1	1.39567	0.697833	−	0.716260i	$-0.254148\pi$
$541$	17562.1	1.39567	0.697833	+	0.716260i	$0.254148\pi$
$542$	8107.42	0.642515
$543$	−7084.28	−0.559881
$544$	4215.60	0.332247
$545$	12097.6	0.950830
$546$	655.146	0.0513510
$547$	15101.0	1.18039	0.590193	−	0.807262i	$-0.299052\pi$
$547$	15101.0	1.18039	0.590193	+	0.807262i	$0.299052\pi$
$548$	−5352.75	−0.417259
$549$	3121.11	0.242633
$550$	−3779.52	−0.293016
$551$	−596.099	−0.0460884
$552$	−1189.07	−0.0916849
$553$	−386.927	−0.0297538
$554$	8228.74	0.631057
$555$	−10484.3	−0.801864
$556$	−1369.19	−0.104436
$557$	8211.86	0.624681	0.312341	−	0.949970i	$-0.398887\pi$
$557$	8211.86	0.624681	0.312341	+	0.949970i	$0.398887\pi$
$558$	880.830	0.0668253
$559$	6935.44	0.524755
$560$	1929.51	0.145601
$561$	−4347.34	−0.327174
$562$	−16352.9	−1.22741
$563$	−11427.9	−0.855465	−0.427732	−	0.903905i	$-0.640687\pi$
$563$	−11427.9	−0.855465	−0.427732	+	0.903905i	$0.640687\pi$
$564$	4641.74	0.346547
$565$	−13822.5	−1.02923
$566$	5424.31	0.402829
$567$	−567.000	−0.0419961
$568$	−4505.48	−0.332827
$569$	−6880.75	−0.506953	−0.253476	−	0.967342i	$-0.581574\pi$
$569$	−6880.75	−0.506953	−0.253476	+	0.967342i	$0.581574\pi$
$570$	−823.796	−0.0605351
$571$	8523.78	0.624709	0.312355	−	0.949966i	$-0.398882\pi$
$571$	8523.78	0.624709	0.312355	+	0.949966i	$0.398882\pi$
$572$	686.343	0.0501704
$573$	−1762.24	−0.128479
$574$	1230.69	0.0894917
$575$	8511.55	0.617315
$576$	576.000	0.0416667
$577$	14371.2	1.03688	0.518441	−	0.855114i	$-0.326513\pi$
$577$	14371.2	1.03688	0.518441	+	0.855114i	$0.326513\pi$
$578$	−24883.6	−1.79069
$579$	−10597.2	−0.760627
$580$	−5154.28	−0.369000
$581$	−6124.36	−0.437317
$582$	−3559.37	−0.253506
$583$	−6184.91	−0.439370
$584$	1060.13	0.0751174
$585$	−2418.58	−0.170933
$586$	8446.74	0.595447
$587$	11781.0	0.828372	0.414186	−	0.910192i	$-0.364066\pi$
$587$	11781.0	0.828372	0.414186	+	0.910192i	$0.364066\pi$
$588$	588.000	0.0412393
$589$	389.995	0.0272826
$590$	−11118.1	−0.775804
$591$	−12468.6	−0.867831
$592$	3245.71	0.225334
$593$	28477.0	1.97202	0.986012	−	0.166677i	$-0.0533036\pi$
$593$	28477.0	1.97202	0.986012	+	0.166677i	$0.0533036\pi$
$594$	−594.000	−0.0410305
$595$	−15886.8	−1.09461
$596$	−709.125	−0.0487364
$597$	6380.21	0.437394
$598$	−1545.66	−0.105697
$599$	9800.46	0.668507	0.334254	−	0.942483i	$-0.391516\pi$
$599$	9800.46	0.668507	0.334254	+	0.942483i	$0.391516\pi$
$600$	−4123.11	−0.280542
$601$	−9667.23	−0.656131	−0.328065	−	0.944655i	$-0.606397\pi$
$601$	−9667.23	−0.656131	−0.328065	+	0.944655i	$0.606397\pi$
$602$	6224.63	0.421423
$603$	8858.29	0.598238
$604$	−9318.28	−0.627741
$605$	−2084.56	−0.140082
$606$	5470.75	0.366723
$607$	−1908.82	−0.127639	−0.0638193	−	0.997961i	$-0.520328\pi$
$607$	−1908.82	−0.127639	−0.0638193	+	0.997961i	$0.520328\pi$
$608$	255.029	0.0170112
$609$	−1570.72	−0.104514
$610$	11948.8	0.793105
$611$	6033.76	0.399509
$612$	−4742.55	−0.313246
$613$	9970.32	0.656929	0.328465	−	0.944516i	$-0.393469\pi$
$613$	9970.32	0.656929	0.328465	+	0.944516i	$0.393469\pi$
$614$	−14386.9	−0.945617
$615$	−4543.31	−0.297893
$616$	616.000	0.0402911
$617$	21216.0	1.38432	0.692159	−	0.721745i	$-0.256660\pi$
$617$	21216.0	1.38432	0.692159	+	0.721745i	$0.256660\pi$
$618$	4312.65	0.280712
$619$	1144.12	0.0742911	0.0371456	−	0.999310i	$-0.488173\pi$
$619$	1144.12	0.0742911	0.0371456	+	0.999310i	$0.488173\pi$
$620$	3372.16	0.218434
$621$	1337.70	0.0864413
$622$	−7869.36	−0.507287
$623$	−3972.78	−0.255483
$624$	748.738	0.0480345
$625$	−7585.60	−0.485479
$626$	−3198.38	−0.204206
$627$	−262.999	−0.0167514
$628$	9633.59	0.612137
$629$	−26723.9	−1.69404
$630$	−2170.70	−0.137274
$631$	24108.9	1.52101	0.760506	−	0.649330i	$-0.224951\pi$
$631$	24108.9	1.52101	0.760506	+	0.649330i	$0.224951\pi$
$632$	−442.203	−0.0278321
$633$	544.033	0.0341602
$634$	−2228.25	−0.139582
$635$	−35016.6	−2.18833
$636$	−6747.18	−0.420665
$637$	764.337	0.0475418
$638$	−1645.52	−0.102111
$639$	5068.66	0.313792
$640$	2205.15	0.136197
$641$	19827.7	1.22176	0.610878	−	0.791725i	$-0.290817\pi$
$641$	19827.7	1.22176	0.610878	+	0.791725i	$0.290817\pi$
$642$	−9748.93	−0.599314
$643$	−942.319	−0.0577939	−0.0288969	−	0.999582i	$-0.509199\pi$
$643$	−942.319	−0.0577939	−0.0288969	+	0.999582i	$0.509199\pi$
$644$	−1387.24	−0.0848837
$645$	−22979.2	−1.40280
$646$	−2099.81	−0.127888
$647$	−2620.21	−0.159213	−0.0796066	−	0.996826i	$-0.525366\pi$
$647$	−2620.21	−0.159213	−0.0796066	+	0.996826i	$0.525366\pi$
$648$	−648.000	−0.0392837
$649$	−3549.47	−0.214682
$650$	−5359.60	−0.323416
$651$	1027.64	0.0618682
$652$	15321.1	0.920274
$653$	5802.60	0.347738	0.173869	−	0.984769i	$-0.444373\pi$
$653$	5802.60	0.347738	0.173869	+	0.984769i	$0.444373\pi$
$654$	4213.28	0.251915
$655$	1008.43	0.0601568
$656$	1406.51	0.0837118
$657$	−1192.65	−0.0708214
$658$	5415.36	0.320840
$659$	13941.5	0.824103	0.412052	−	0.911160i	$-0.364812\pi$
$659$	13941.5	0.824103	0.412052	+	0.911160i	$0.364812\pi$
$660$	−2274.07	−0.134118
$661$	14780.5	0.869738	0.434869	−	0.900494i	$-0.356795\pi$
$661$	14780.5	0.869738	0.434869	+	0.900494i	$0.356795\pi$
$662$	14582.1	0.856115
$663$	−6164.81	−0.361118
$664$	−6999.27	−0.409073
$665$	−961.096	−0.0560447
$666$	−3651.42	−0.212447
$667$	3705.73	0.215122
$668$	−3130.27	−0.181308
$669$	349.869	0.0202193
$670$	33913.0	1.95548
$671$	3814.69	0.219470
$672$	672.000	0.0385758
$673$	−2994.43	−0.171511	−0.0857554	−	0.996316i	$-0.527330\pi$
$673$	−2994.43	−0.171511	−0.0857554	+	0.996316i	$0.527330\pi$
$674$	−7476.17	−0.427257
$675$	4638.50	0.264497
$676$	−7814.72	−0.444625
$677$	−18613.6	−1.05669	−0.528344	−	0.849030i	$-0.677187\pi$
$677$	−18613.6	−1.05669	−0.528344	+	0.849030i	$0.677187\pi$
$678$	−4814.03	−0.272687
$679$	−4152.59	−0.234701
$680$	−18156.4	−1.02392
$681$	6470.14	0.364077
$682$	1076.57	0.0604457
$683$	9237.50	0.517515	0.258758	−	0.965942i	$-0.416687\pi$
$683$	9237.50	0.517515	0.258758	+	0.965942i	$0.416687\pi$
$684$	−286.908	−0.0160383
$685$	23054.0	1.28591
$686$	686.000	0.0381802
$687$	16188.2	0.899007
$688$	7113.86	0.394205
$689$	−8770.61	−0.484954
$690$	5121.24	0.282554
$691$	4478.39	0.246550	0.123275	−	0.992373i	$-0.460660\pi$
$691$	4478.39	0.246550	0.123275	+	0.992373i	$0.460660\pi$
$692$	−7419.30	−0.407571
$693$	−693.000	−0.0379869
$694$	−14408.9	−0.788120
$695$	5897.02	0.321851
$696$	−1795.11	−0.0977635
$697$	−11580.6	−0.629336
$698$	19326.0	1.04800
$699$	−10476.4	−0.566886
$700$	−4810.29	−0.259731
$701$	18521.8	0.997946	0.498973	−	0.866618i	$-0.333711\pi$
$701$	18521.8	0.997946	0.498973	+	0.866618i	$0.333711\pi$
$702$	−842.331	−0.0452873
$703$	−1616.70	−0.0867354
$704$	704.000	0.0376889
$705$	−19991.7	−1.06799
$706$	−19273.5	−1.02743
$707$	6382.54	0.339520
$708$	−3872.15	−0.205543
$709$	−32316.2	−1.71179	−0.855895	−	0.517150i	$-0.826993\pi$
$709$	−32316.2	−1.71179	−0.855895	+	0.517150i	$0.826993\pi$
$710$	19404.8	1.02570
$711$	497.478	0.0262403
$712$	−4540.32	−0.238983
$713$	−2424.46	−0.127345
$714$	−5532.98	−0.290009
$715$	−2956.04	−0.154615
$716$	6888.91	0.359568
$717$	5435.97	0.283138
$718$	18908.3	0.982802
$719$	22295.0	1.15641	0.578207	−	0.815890i	$-0.303753\pi$
$719$	22295.0	1.15641	0.578207	+	0.815890i	$0.303753\pi$
$720$	−2480.80	−0.128408
$721$	5031.42	0.259889
$722$	13591.0	0.700559
$723$	1639.18	0.0843180
$724$	−9445.70	−0.484871
$725$	12849.7	0.658242
$726$	−726.000	−0.0371135
$727$	−11823.2	−0.603160	−0.301580	−	0.953441i	$-0.597514\pi$
$727$	−11823.2	−0.603160	−0.301580	+	0.953441i	$0.597514\pi$
$728$	873.528	0.0444713
$729$	729.000	0.0370370
$730$	−4565.93	−0.231497
$731$	−58572.7	−2.96359
$732$	4161.48	0.210126
$733$	7471.43	0.376485	0.188243	−	0.982123i	$-0.439721\pi$
$733$	7471.43	0.376485	0.188243	+	0.982123i	$0.439721\pi$
$734$	18060.1	0.908187
$735$	−2532.48	−0.127091
$736$	−1585.42	−0.0794014
$737$	10826.8	0.541126
$738$	−1582.32	−0.0789242
$739$	334.930	0.0166720	0.00833600	−	0.999965i	$-0.497347\pi$
$739$	334.930	0.0166720	0.00833600	+	0.999965i	$0.497347\pi$
$740$	−13979.1	−0.694434
$741$	−372.949	−0.0184894
$742$	−7871.71	−0.389460
$743$	31080.2	1.53462	0.767310	−	0.641276i	$-0.221595\pi$
$743$	31080.2	1.53462	0.767310	+	0.641276i	$0.221595\pi$
$744$	1174.44	0.0578724
$745$	3054.16	0.150196
$746$	4378.85	0.214907
$747$	7874.18	0.385678
$748$	−5796.45	−0.283341
$749$	−11373.8	−0.554857
$750$	4837.16	0.235504
$751$	−4775.62	−0.232044	−0.116022	−	0.993247i	$-0.537014\pi$
$751$	−4775.62	−0.232044	−0.116022	+	0.993247i	$0.537014\pi$
$752$	6188.98	0.300118
$753$	−18280.4	−0.884696
$754$	−2333.45	−0.112704
$755$	40133.3	1.93457
$756$	−756.000	−0.0363696
$757$	−2628.31	−0.126192	−0.0630962	−	0.998007i	$-0.520097\pi$
$757$	−2628.31	−0.126192	−0.0630962	+	0.998007i	$0.520097\pi$
$758$	17763.5	0.851188
$759$	1634.97	0.0781891
$760$	−1098.40	−0.0524250
$761$	−25838.8	−1.23082	−0.615411	−	0.788207i	$-0.711010\pi$
$761$	−25838.8	−1.23082	−0.615411	+	0.788207i	$0.711010\pi$
$762$	−12195.4	−0.579781
$763$	4915.49	0.233228
$764$	−2349.66	−0.111266
$765$	20425.9	0.965360
$766$	−18270.4	−0.861796
$767$	−5033.38	−0.236955
$768$	768.000	0.0360844
$769$	9148.78	0.429016	0.214508	−	0.976722i	$-0.431185\pi$
$769$	9148.78	0.429016	0.214508	+	0.976722i	$0.431185\pi$
$770$	−2653.08	−0.124169
$771$	17902.0	0.836220
$772$	−14129.5	−0.658722
$773$	−4945.06	−0.230092	−0.115046	−	0.993360i	$-0.536702\pi$
$773$	−4945.06	−0.230092	−0.115046	+	0.993360i	$0.536702\pi$
$774$	−8003.09	−0.371660
$775$	−8406.84	−0.389655
$776$	−4745.82	−0.219543
$777$	−4259.99	−0.196688
$778$	−28682.7	−1.32176
$779$	−700.587	−0.0322222
$780$	−3224.77	−0.148033
$781$	6195.03	0.283836
$782$	13053.7	0.596932
$783$	2019.50	0.0921723
$784$	784.000	0.0357143
$785$	−41491.3	−1.88648
$786$	351.212	0.0159380
$787$	22864.1	1.03560	0.517800	−	0.855502i	$-0.326751\pi$
$787$	22864.1	1.03560	0.517800	+	0.855502i	$0.326751\pi$
$788$	−16624.8	−0.751564
$789$	−16606.8	−0.749325
$790$	1904.54	0.0857729
$791$	−5616.37	−0.252459
$792$	−792.000	−0.0355335
$793$	5409.47	0.242240
$794$	−3352.88	−0.149860
$795$	29059.7	1.29640
$796$	8506.94	0.378795
$797$	19847.5	0.882101	0.441050	−	0.897482i	$-0.354606\pi$
$797$	19847.5	0.882101	0.441050	+	0.897482i	$0.354606\pi$
$798$	−334.726	−0.0148486
$799$	−50957.6	−2.25626
$800$	−5497.48	−0.242956
$801$	5107.86	0.225315
$802$	−23010.8	−1.01314
$803$	−1457.68	−0.0640603
$804$	11811.0	0.518089
$805$	5974.78	0.261594
$806$	1526.65	0.0667169
$807$	5110.93	0.222941
$808$	7294.34	0.317591
$809$	−638.036	−0.0277282	−0.0138641	−	0.999904i	$-0.504413\pi$
$809$	−638.036	−0.0277282	−0.0138641	+	0.999904i	$0.504413\pi$
$810$	2790.90	0.121064
$811$	−17924.7	−0.776103	−0.388052	−	0.921638i	$-0.626852\pi$
$811$	−17924.7	−0.776103	−0.388052	+	0.921638i	$0.626852\pi$
$812$	−2094.29	−0.0905114
$813$	−12161.1	−0.524612
$814$	−4462.85	−0.192166
$815$	−65986.9	−2.83610
$816$	−6323.40	−0.271279
$817$	−3543.44	−0.151737
$818$	−24860.4	−1.06262
$819$	−982.719	−0.0419279
$820$	−6057.75	−0.257983
$821$	16807.2	0.714465	0.357233	−	0.934015i	$-0.383720\pi$
$821$	16807.2	0.714465	0.357233	+	0.934015i	$0.383720\pi$
$822$	8029.12	0.340691
$823$	−32922.8	−1.39443	−0.697215	−	0.716862i	$-0.745578\pi$
$823$	−32922.8	−1.39443	−0.697215	+	0.716862i	$0.745578\pi$
$824$	5750.20	0.243104
$825$	5669.27	0.239247
$826$	−4517.51	−0.190296
$827$	−37778.7	−1.58850	−0.794252	−	0.607588i	$-0.792137\pi$
$827$	−37778.7	−1.58850	−0.794252	+	0.607588i	$0.792137\pi$
$828$	1783.60	0.0748604
$829$	−40731.3	−1.70646	−0.853230	−	0.521534i	$-0.825360\pi$
$829$	−40731.3	−1.70646	−0.853230	+	0.521534i	$0.825360\pi$
$830$	30145.5	1.26068
$831$	−12343.1	−0.515256
$832$	998.318	0.0415991
$833$	−6455.14	−0.268496
$834$	2053.78	0.0852719
$835$	13481.9	0.558755
$836$	−350.665	−0.0145072
$837$	−1321.25	−0.0545626
$838$	−23510.5	−0.969160
$839$	24335.1	1.00136	0.500679	−	0.865633i	$-0.333084\pi$
$839$	24335.1	1.00136	0.500679	+	0.865633i	$0.333084\pi$
$840$	−2894.27	−0.118883
$841$	−18794.5	−0.770615
$842$	3763.04	0.154018
$843$	24529.3	1.00218
$844$	725.378	0.0295836
$845$	33657.6	1.37024
$846$	−6962.61	−0.282954
$847$	−847.000	−0.0343604
$848$	−8996.24	−0.364307
$849$	−8136.47	−0.328908
$850$	45264.0	1.82652
$851$	10050.4	0.404846
$852$	6758.21	0.271752
$853$	−5181.85	−0.207999	−0.104000	−	0.994577i	$-0.533164\pi$
$853$	−5181.85	−0.207999	−0.104000	+	0.994577i	$0.533164\pi$
$854$	4855.06	0.194539
$855$	1235.69	0.0494267
$856$	−12998.6	−0.519021
$857$	23157.8	0.923051	0.461525	−	0.887127i	$-0.347302\pi$
$857$	23157.8	0.923051	0.461525	+	0.887127i	$0.347302\pi$
$858$	−1029.52	−0.0409639
$859$	44044.9	1.74947	0.874734	−	0.484603i	$-0.161036\pi$
$859$	44044.9	1.74947	0.874734	+	0.484603i	$0.161036\pi$
$860$	−30639.0	−1.21486
$861$	−1846.04	−0.0730696
$862$	−33630.4	−1.32884
$863$	−9520.28	−0.375520	−0.187760	−	0.982215i	$-0.560123\pi$
$863$	−9520.28	−0.375520	−0.187760	+	0.982215i	$0.560123\pi$
$864$	−864.000	−0.0340207
$865$	31954.5	1.25605
$866$	8975.24	0.352184
$867$	37325.4	1.46209
$868$	1370.18	0.0535794
$869$	608.029	0.0237353
$870$	7731.43	0.301287
$871$	15353.1	0.597267
$872$	5617.70	0.218164
$873$	5339.05	0.206987
$874$	789.704	0.0305631
$875$	5643.35	0.218034
$876$	−1590.20	−0.0613331
$877$	13687.3	0.527011	0.263506	−	0.964658i	$-0.415121\pi$
$877$	13687.3	0.527011	0.263506	+	0.964658i	$0.415121\pi$
$878$	33932.2	1.30428
$879$	−12670.1	−0.486180
$880$	−3032.09	−0.116150
$881$	−20450.3	−0.782051	−0.391026	−	0.920380i	$-0.627880\pi$
$881$	−20450.3	−0.782051	−0.391026	+	0.920380i	$0.627880\pi$
$882$	−882.000	−0.0336718
$883$	9377.27	0.357384	0.178692	−	0.983905i	$-0.442813\pi$
$883$	9377.27	0.357384	0.178692	+	0.983905i	$0.442813\pi$
$884$	−8219.75	−0.312738
$885$	16677.1	0.633441
$886$	−4462.79	−0.169222
$887$	−8610.33	−0.325937	−0.162969	−	0.986631i	$-0.552107\pi$
$887$	−8610.33	−0.325937	−0.162969	+	0.986631i	$0.552107\pi$
$888$	−4868.56	−0.183985
$889$	−14228.0	−0.536773
$890$	19554.9	0.736496
$891$	891.000	0.0335013
$892$	466.492	0.0175104
$893$	−3082.75	−0.115521
$894$	1063.69	0.0397931
$895$	−29670.1	−1.10812
$896$	896.000	0.0334077
$897$	2318.49	0.0863011
$898$	−35225.5	−1.30901
$899$	−3660.15	−0.135787
$900$	6184.66	0.229062
$901$	74071.4	2.73882
$902$	−1933.95	−0.0713897
$903$	−9336.94	−0.344091
$904$	−6418.71	−0.236154
$905$	40682.1	1.49427
$906$	13977.4	0.512548
$907$	−33607.7	−1.23035	−0.615173	−	0.788392i	$-0.710914\pi$
$907$	−33607.7	−1.23035	−0.615173	+	0.788392i	$0.710914\pi$
$908$	8626.86	0.315300
$909$	−8206.13	−0.299428
$910$	−3762.24	−0.137052
$911$	43842.9	1.59449	0.797245	−	0.603656i	$-0.206290\pi$
$911$	43842.9	1.59449	0.797245	+	0.603656i	$0.206290\pi$
$912$	−382.543	−0.0138896
$913$	9624.00	0.348859
$914$	−35261.0	−1.27607
$915$	−17923.2	−0.647567
$916$	21584.2	0.778563
$917$	409.747	0.0147558
$918$	7113.83	0.255764
$919$	28546.4	1.02466	0.512328	−	0.858790i	$-0.328783\pi$
$919$	28546.4	1.02466	0.512328	+	0.858790i	$0.328783\pi$
$920$	6828.32	0.244699
$921$	21580.4	0.772093
$922$	3411.96	0.121873
$923$	8784.95	0.313283
$924$	−924.000	−0.0328976
$925$	34850.0	1.23877
$926$	−15898.5	−0.564210
$927$	−6468.97	−0.229201
$928$	−2393.48	−0.0846656
$929$	−21453.6	−0.757664	−0.378832	−	0.925465i	$-0.623674\pi$
$929$	−21453.6	−0.757664	−0.378832	+	0.925465i	$0.623674\pi$
$930$	−5058.25	−0.178351
$931$	−390.513	−0.0137471
$932$	−13968.5	−0.490937
$933$	11804.0	0.414198
$934$	−3580.97	−0.125453
$935$	24965.0	0.873201
$936$	−1123.11	−0.0392200
$937$	−38499.0	−1.34227	−0.671135	−	0.741336i	$-0.734193\pi$
$937$	−38499.0	−1.34227	−0.671135	+	0.741336i	$0.734193\pi$
$938$	13779.6	0.479657
$939$	4797.58	0.166734
$940$	−26655.6	−0.924904
$941$	−20531.6	−0.711277	−0.355639	−	0.934624i	$-0.615737\pi$
$941$	−20531.6	−0.711277	−0.355639	+	0.934624i	$0.615737\pi$
$942$	−14450.4	−0.499808
$943$	4355.29	0.150401
$944$	−5162.86	−0.178005
$945$	3256.05	0.112084
$946$	−9781.56	−0.336179
$947$	−31889.6	−1.09427	−0.547134	−	0.837045i	$-0.684281\pi$
$947$	−31889.6	−1.09427	−0.547134	+	0.837045i	$0.684281\pi$
$948$	663.304	0.0227248
$949$	−2067.09	−0.0707065
$950$	2738.31	0.0935185
$951$	3342.37	0.113968
$952$	−7377.30	−0.251155
$953$	38924.3	1.32307	0.661533	−	0.749916i	$-0.269906\pi$
$953$	38924.3	1.32307	0.661533	+	0.749916i	$0.269906\pi$
$954$	10120.8	0.343472
$955$	10119.8	0.342901
$956$	7247.96	0.245205
$957$	2468.27	0.0833730
$958$	−1304.99	−0.0440108
$959$	9367.31	0.315418
$960$	−3307.73	−0.111205
$961$	−27396.4	−0.919619
$962$	−6328.61	−0.212103
$963$	14623.4	0.489338
$964$	2185.58	0.0730215
$965$	60855.1	2.03005
$966$	2080.87	0.0693072
$967$	43233.1	1.43773	0.718863	−	0.695152i	$-0.244663\pi$
$967$	43233.1	1.43773	0.718863	+	0.695152i	$0.244663\pi$
$968$	−968.000	−0.0321412
$969$	3149.71	0.104420
$970$	20440.0	0.676586
$971$	−5241.26	−0.173224	−0.0866118	−	0.996242i	$-0.527604\pi$
$971$	−5241.26	−0.173224	−0.0866118	+	0.996242i	$0.527604\pi$
$972$	972.000	0.0320750
$973$	2396.08	0.0789464
$974$	−9984.02	−0.328448
$975$	8039.40	0.264068
$976$	5548.64	0.181975
$977$	17573.5	0.575461	0.287730	−	0.957711i	$-0.407099\pi$
$977$	17573.5	0.575461	0.287730	+	0.957711i	$0.407099\pi$
$978$	−22981.6	−0.751401
$979$	6242.94	0.203805
$980$	−3376.64	−0.110064
$981$	−6319.91	−0.205687
$982$	37170.7	1.20791
$983$	−10850.5	−0.352061	−0.176030	−	0.984385i	$-0.556326\pi$
$983$	−10850.5	−0.352061	−0.176030	+	0.984385i	$0.556326\pi$
$984$	−2109.76	−0.0683504
$985$	71601.9	2.31617
$986$	19706.9	0.636507
$987$	−8123.04	−0.261965
$988$	−497.265	−0.0160123
$989$	22028.3	0.708249
$990$	3411.10	0.109507
$991$	−35028.4	−1.12282	−0.561410	−	0.827538i	$-0.689741\pi$
$991$	−35028.4	−1.12282	−0.561410	+	0.827538i	$0.689741\pi$
$992$	1565.92	0.0501190
$993$	−21873.1	−0.699015
$994$	7884.58	0.251593
$995$	−36638.9	−1.16737
$996$	10498.9	0.334007
$997$	51553.3	1.63762	0.818811	−	0.574064i	$-0.194634\pi$
$997$	51553.3	1.63762	0.818811	+	0.574064i	$0.194634\pi$
$998$	36413.7	1.15497
$999$	5477.13	0.173462

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.4.a.r.1.1	✓	3
3.2	odd	2		1386.4.a.bh.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.r.1.1	✓	3	1.1	even	1	trivial
1386.4.a.bh.1.3		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	462.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -17.2278 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -17.2278 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +34.4555 q^{10} +11.0000 q^{11} +12.0000 q^{12} +15.5987 q^{13} +14.0000 q^{14} -51.6833 q^{15} +16.0000 q^{16} -131.738 q^{17} -18.0000 q^{18} -7.96965 q^{19} -68.9111 q^{20} -21.0000 q^{21} -22.0000 q^{22} +49.5445 q^{23} -24.0000 q^{24} +171.796 q^{25} -31.1974 q^{26} +27.0000 q^{27} -28.0000 q^{28} +74.7961 q^{29} +103.367 q^{30} -48.9350 q^{31} -32.0000 q^{32} +33.0000 q^{33} +263.475 q^{34} +120.594 q^{35} +36.0000 q^{36} +202.857 q^{37} +15.9393 q^{38} +46.7961 q^{39} +137.822 q^{40} +87.9068 q^{41} +42.0000 q^{42} +444.616 q^{43} +44.0000 q^{44} -155.050 q^{45} -99.0889 q^{46} +386.811 q^{47} +48.0000 q^{48} +49.0000 q^{49} -343.592 q^{50} -395.213 q^{51} +62.3949 q^{52} -562.265 q^{53} -54.0000 q^{54} -189.505 q^{55} +56.0000 q^{56} -23.9090 q^{57} -149.592 q^{58} -322.679 q^{59} -206.733 q^{60} +346.790 q^{61} +97.8700 q^{62} -63.0000 q^{63} +64.0000 q^{64} -268.731 q^{65} -66.0000 q^{66} +984.254 q^{67} -526.950 q^{68} +148.633 q^{69} -241.189 q^{70} +563.185 q^{71} -72.0000 q^{72} -132.516 q^{73} -405.714 q^{74} +515.388 q^{75} -31.8786 q^{76} -77.0000 q^{77} -93.5923 q^{78} +55.2753 q^{79} -275.644 q^{80} +81.0000 q^{81} -175.814 q^{82} +874.909 q^{83} -84.0000 q^{84} +2269.54 q^{85} -889.232 q^{86} +224.388 q^{87} -88.0000 q^{88} +567.540 q^{89} +310.100 q^{90} -109.191 q^{91} +198.178 q^{92} -146.805 q^{93} -773.623 q^{94} +137.299 q^{95} -96.0000 q^{96} +593.228 q^{97} -98.0000 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} - 24 q^{8} + 27 q^{9} - 14 q^{10} + 33 q^{11} + 36 q^{12} - 15 q^{13} + 42 q^{14} + 21 q^{15} + 48 q^{16} - 40 q^{17} - 54 q^{18} + 41 q^{19} + 28 q^{20} - 63 q^{21} - 66 q^{22} + 266 q^{23} - 72 q^{24} + 330 q^{25} + 30 q^{26} + 81 q^{27} - 84 q^{28} + 39 q^{29} - 42 q^{30} + 332 q^{31} - 96 q^{32} + 99 q^{33} + 80 q^{34} - 49 q^{35} + 108 q^{36} + 553 q^{37} - 82 q^{38} - 45 q^{39} - 56 q^{40} - 320 q^{41} + 126 q^{42} + 290 q^{43} + 132 q^{44} + 63 q^{45} - 532 q^{46} + 33 q^{47} + 144 q^{48} + 147 q^{49} - 660 q^{50} - 120 q^{51} - 60 q^{52} - 482 q^{53} - 162 q^{54} + 77 q^{55} + 168 q^{56} + 123 q^{57} - 78 q^{58} - 443 q^{59} + 84 q^{60} + 546 q^{61} - 664 q^{62} - 189 q^{63} + 192 q^{64} + 287 q^{65} - 198 q^{66} + 661 q^{67} - 160 q^{68} + 798 q^{69} + 98 q^{70} + 948 q^{71} - 216 q^{72} + 1539 q^{73} - 1106 q^{74} + 990 q^{75} + 164 q^{76} - 231 q^{77} + 90 q^{78} + 1568 q^{79} + 112 q^{80} + 243 q^{81} + 640 q^{82} + 52 q^{83} - 252 q^{84} + 3028 q^{85} - 580 q^{86} + 117 q^{87} - 264 q^{88} + 1042 q^{89} - 126 q^{90} + 105 q^{91} + 1064 q^{92} + 996 q^{93} - 66 q^{94} - 1237 q^{95} - 288 q^{96} + 3670 q^{97} - 294 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 - 24 * q^8 + 27 * q^9 - 14 * q^10 + 33 * q^11 + 36 * q^12 - 15 * q^13 + 42 * q^14 + 21 * q^15 + 48 * q^16 - 40 * q^17 - 54 * q^18 + 41 * q^19 + 28 * q^20 - 63 * q^21 - 66 * q^22 + 266 * q^23 - 72 * q^24 + 330 * q^25 + 30 * q^26 + 81 * q^27 - 84 * q^28 + 39 * q^29 - 42 * q^30 + 332 * q^31 - 96 * q^32 + 99 * q^33 + 80 * q^34 - 49 * q^35 + 108 * q^36 + 553 * q^37 - 82 * q^38 - 45 * q^39 - 56 * q^40 - 320 * q^41 + 126 * q^42 + 290 * q^43 + 132 * q^44 + 63 * q^45 - 532 * q^46 + 33 * q^47 + 144 * q^48 + 147 * q^49 - 660 * q^50 - 120 * q^51 - 60 * q^52 - 482 * q^53 - 162 * q^54 + 77 * q^55 + 168 * q^56 + 123 * q^57 - 78 * q^58 - 443 * q^59 + 84 * q^60 + 546 * q^61 - 664 * q^62 - 189 * q^63 + 192 * q^64 + 287 * q^65 - 198 * q^66 + 661 * q^67 - 160 * q^68 + 798 * q^69 + 98 * q^70 + 948 * q^71 - 216 * q^72 + 1539 * q^73 - 1106 * q^74 + 990 * q^75 + 164 * q^76 - 231 * q^77 + 90 * q^78 + 1568 * q^79 + 112 * q^80 + 243 * q^81 + 640 * q^82 + 52 * q^83 - 252 * q^84 + 3028 * q^85 - 580 * q^86 + 117 * q^87 - 264 * q^88 + 1042 * q^89 - 126 * q^90 + 105 * q^91 + 1064 * q^92 + 996 * q^93 - 66 * q^94 - 1237 * q^95 - 288 * q^96 + 3670 * q^97 - 294 * q^98 + 297 * q^99

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