LMFDB - Embedded newform 1110.4.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,4,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, 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("1110.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1110.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:	$65.4921201064$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.8827413.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 25x^{2} + 6x + 120 $ x^4 - x^3 - 25x^2 + 6x + 120
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.84825$ of defining polynomial
Character	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} +21.0284 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} +21.0284 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -8.40515 q^{11} +12.0000 q^{12} -59.4799 q^{13} +42.0567 q^{14} -15.0000 q^{15} +16.0000 q^{16} -135.555 q^{17} +18.0000 q^{18} -146.777 q^{19} -20.0000 q^{20} +63.0851 q^{21} -16.8103 q^{22} -109.189 q^{23} +24.0000 q^{24} +25.0000 q^{25} -118.960 q^{26} +27.0000 q^{27} +84.1135 q^{28} +37.6049 q^{29} -30.0000 q^{30} -290.534 q^{31} +32.0000 q^{32} -25.2154 q^{33} -271.111 q^{34} -105.142 q^{35} +36.0000 q^{36} +37.0000 q^{37} -293.554 q^{38} -178.440 q^{39} -40.0000 q^{40} +144.791 q^{41} +126.170 q^{42} +126.314 q^{43} -33.6206 q^{44} -45.0000 q^{45} -218.379 q^{46} +239.848 q^{47} +48.0000 q^{48} +99.1925 q^{49} +50.0000 q^{50} -406.666 q^{51} -237.920 q^{52} -394.725 q^{53} +54.0000 q^{54} +42.0257 q^{55} +168.227 q^{56} -440.331 q^{57} +75.2098 q^{58} +745.436 q^{59} -60.0000 q^{60} +818.180 q^{61} -581.068 q^{62} +189.255 q^{63} +64.0000 q^{64} +297.399 q^{65} -50.4309 q^{66} -291.878 q^{67} -542.221 q^{68} -327.568 q^{69} -210.284 q^{70} +730.707 q^{71} +72.0000 q^{72} +150.951 q^{73} +74.0000 q^{74} +75.0000 q^{75} -587.108 q^{76} -176.747 q^{77} -356.879 q^{78} -638.943 q^{79} -80.0000 q^{80} +81.0000 q^{81} +289.581 q^{82} -985.546 q^{83} +252.340 q^{84} +677.777 q^{85} +252.628 q^{86} +112.815 q^{87} -67.2412 q^{88} -280.594 q^{89} -90.0000 q^{90} -1250.77 q^{91} -436.757 q^{92} -871.603 q^{93} +479.696 q^{94} +733.885 q^{95} +96.0000 q^{96} -1390.36 q^{97} +198.385 q^{98} -75.6463 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 20 q^{5} + 24 q^{6} - 23 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 20 * q^5 + 24 * q^6 - 23 * q^7 + 32 * q^8 + 36 * q^9	\( 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 20 q^{5} + 24 q^{6} - 23 q^{7} + 32 q^{8} + 36 q^{9} - 40 q^{10} - 6 q^{11} + 48 q^{12} - 59 q^{13} - 46 q^{14} - 60 q^{15} + 64 q^{16} - 172 q^{17} + 72 q^{18} - 256 q^{19} - 80 q^{20} - 69 q^{21} - 12 q^{22} - 103 q^{23} + 96 q^{24} + 100 q^{25} - 118 q^{26} + 108 q^{27} - 92 q^{28} - 345 q^{29} - 120 q^{30} - 686 q^{31} + 128 q^{32} - 18 q^{33} - 344 q^{34} + 115 q^{35} + 144 q^{36} + 148 q^{37} - 512 q^{38} - 177 q^{39} - 160 q^{40} - 672 q^{41} - 138 q^{42} - 421 q^{43} - 24 q^{44} - 180 q^{45} - 206 q^{46} - 344 q^{47} + 192 q^{48} + 271 q^{49} + 200 q^{50} - 516 q^{51} - 236 q^{52} + 63 q^{53} + 216 q^{54} + 30 q^{55} - 184 q^{56} - 768 q^{57} - 690 q^{58} + 121 q^{59} - 240 q^{60} - 390 q^{61} - 1372 q^{62} - 207 q^{63} + 256 q^{64} + 295 q^{65} - 36 q^{66} - 845 q^{67} - 688 q^{68} - 309 q^{69} + 230 q^{70} - 53 q^{71} + 288 q^{72} + 516 q^{73} + 296 q^{74} + 300 q^{75} - 1024 q^{76} - 1954 q^{77} - 354 q^{78} - 235 q^{79} - 320 q^{80} + 324 q^{81} - 1344 q^{82} - 1360 q^{83} - 276 q^{84} + 860 q^{85} - 842 q^{86} - 1035 q^{87} - 48 q^{88} - 2529 q^{89} - 360 q^{90} - 2663 q^{91} - 412 q^{92} - 2058 q^{93} - 688 q^{94} + 1280 q^{95} + 384 q^{96} - 3438 q^{97} + 542 q^{98} - 54 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 20 * q^5 + 24 * q^6 - 23 * q^7 + 32 * q^8 + 36 * q^9 - 40 * q^10 - 6 * q^11 + 48 * q^12 - 59 * q^13 - 46 * q^14 - 60 * q^15 + 64 * q^16 - 172 * q^17 + 72 * q^18 - 256 * q^19 - 80 * q^20 - 69 * q^21 - 12 * q^22 - 103 * q^23 + 96 * q^24 + 100 * q^25 - 118 * q^26 + 108 * q^27 - 92 * q^28 - 345 * q^29 - 120 * q^30 - 686 * q^31 + 128 * q^32 - 18 * q^33 - 344 * q^34 + 115 * q^35 + 144 * q^36 + 148 * q^37 - 512 * q^38 - 177 * q^39 - 160 * q^40 - 672 * q^41 - 138 * q^42 - 421 * q^43 - 24 * q^44 - 180 * q^45 - 206 * q^46 - 344 * q^47 + 192 * q^48 + 271 * q^49 + 200 * q^50 - 516 * q^51 - 236 * q^52 + 63 * q^53 + 216 * q^54 + 30 * q^55 - 184 * q^56 - 768 * q^57 - 690 * q^58 + 121 * q^59 - 240 * q^60 - 390 * q^61 - 1372 * q^62 - 207 * q^63 + 256 * q^64 + 295 * q^65 - 36 * q^66 - 845 * q^67 - 688 * q^68 - 309 * q^69 + 230 * q^70 - 53 * q^71 + 288 * q^72 + 516 * q^73 + 296 * q^74 + 300 * q^75 - 1024 * q^76 - 1954 * q^77 - 354 * q^78 - 235 * q^79 - 320 * q^80 + 324 * q^81 - 1344 * q^82 - 1360 * q^83 - 276 * q^84 + 860 * q^85 - 842 * q^86 - 1035 * q^87 - 48 * q^88 - 2529 * q^89 - 360 * q^90 - 2663 * q^91 - 412 * q^92 - 2058 * q^93 - 688 * q^94 + 1280 * q^95 + 384 * q^96 - 3438 * q^97 + 542 * q^98 - 54 * 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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	6.00000	0.408248
$7$	21.0284	1.13543	0.567713	−	0.823227i	$-0.307828\pi$
$7$	21.0284	1.13543	0.567713	+	0.823227i	$0.307828\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	−8.40515	−0.230386	−0.115193	−	0.993343i	$-0.536749\pi$
$11$	−8.40515	−0.230386	−0.115193	+	0.993343i	$0.536749\pi$
$12$	12.0000	0.288675
$13$	−59.4799	−1.26898	−0.634490	−	0.772931i	$-0.718790\pi$
$13$	−59.4799	−1.26898	−0.634490	+	0.772931i	$0.718790\pi$
$14$	42.0567	0.802867
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	−135.555	−1.93394	−0.966970	−	0.254889i	$-0.917961\pi$
$17$	−135.555	−1.93394	−0.966970	+	0.254889i	$0.917961\pi$
$18$	18.0000	0.235702
$19$	−146.777	−1.77226	−0.886131	−	0.463435i	$-0.846617\pi$
$19$	−146.777	−1.77226	−0.886131	+	0.463435i	$0.846617\pi$
$20$	−20.0000	−0.223607
$21$	63.0851	0.655538
$22$	−16.8103	−0.162908
$23$	−109.189	−0.989893	−0.494947	−	0.868923i	$-0.664812\pi$
$23$	−109.189	−0.989893	−0.494947	+	0.868923i	$0.664812\pi$
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	−118.960	−0.897305
$27$	27.0000	0.192450
$28$	84.1135	0.567713
$29$	37.6049	0.240795	0.120397	−	0.992726i	$-0.461583\pi$
$29$	37.6049	0.240795	0.120397	+	0.992726i	$0.461583\pi$
$30$	−30.0000	−0.182574
$31$	−290.534	−1.68327	−0.841637	−	0.540044i	$-0.818408\pi$
$31$	−290.534	−1.68327	−0.841637	+	0.540044i	$0.818408\pi$
$32$	32.0000	0.176777
$33$	−25.2154	−0.133013
$34$	−271.111	−1.36750
$35$	−105.142	−0.507778
$36$	36.0000	0.166667
$37$	37.0000	0.164399
$37$	37.0000	0.164399
$38$	−293.554	−1.25318
$39$	−178.440	−0.732646
$40$	−40.0000	−0.158114
$41$	144.791	0.551524	0.275762	−	0.961226i	$-0.411070\pi$
$41$	144.791	0.551524	0.275762	+	0.961226i	$0.411070\pi$
$42$	126.170	0.463536
$43$	126.314	0.447970	0.223985	−	0.974593i	$-0.428093\pi$
$43$	126.314	0.447970	0.223985	+	0.974593i	$0.428093\pi$
$44$	−33.6206	−0.115193
$45$	−45.0000	−0.149071
$46$	−218.379	−0.699960
$47$	239.848	0.744370	0.372185	−	0.928159i	$-0.378609\pi$
$47$	239.848	0.744370	0.372185	+	0.928159i	$0.378609\pi$
$48$	48.0000	0.144338
$49$	99.1925	0.289191
$50$	50.0000	0.141421
$51$	−406.666	−1.11656
$52$	−237.920	−0.634490
$53$	−394.725	−1.02301	−0.511506	−	0.859280i	$-0.670912\pi$
$53$	−394.725	−1.02301	−0.511506	+	0.859280i	$0.670912\pi$
$54$	54.0000	0.136083
$55$	42.0257	0.103032
$56$	168.227	0.401434
$57$	−440.331	−1.02322
$58$	75.2098	0.170268
$59$	745.436	1.64487	0.822436	−	0.568857i	$-0.192614\pi$
$59$	745.436	1.64487	0.822436	+	0.568857i	$0.192614\pi$
$60$	−60.0000	−0.129099
$61$	818.180	1.71733	0.858666	−	0.512536i	$-0.171294\pi$
$61$	818.180	1.71733	0.858666	+	0.512536i	$0.171294\pi$
$62$	−581.068	−1.19025
$63$	189.255	0.378475
$64$	64.0000	0.125000
$65$	297.399	0.567506
$66$	−50.4309	−0.0940547
$67$	−291.878	−0.532218	−0.266109	−	0.963943i	$-0.585738\pi$
$67$	−291.878	−0.532218	−0.266109	+	0.963943i	$0.585738\pi$
$68$	−542.221	−0.966970
$69$	−327.568	−0.571515
$70$	−210.284	−0.359053
$71$	730.707	1.22139	0.610697	−	0.791865i	$-0.290889\pi$
$71$	730.707	1.22139	0.610697	+	0.791865i	$0.290889\pi$
$72$	72.0000	0.117851
$73$	150.951	0.242021	0.121010	−	0.992651i	$-0.461387\pi$
$73$	150.951	0.242021	0.121010	+	0.992651i	$0.461387\pi$
$74$	74.0000	0.116248
$75$	75.0000	0.115470
$76$	−587.108	−0.886131
$77$	−176.747	−0.261586
$78$	−356.879	−0.518059
$79$	−638.943	−0.909959	−0.454979	−	0.890502i	$-0.650353\pi$
$79$	−638.943	−0.909959	−0.454979	+	0.890502i	$0.650353\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	289.581	0.389987
$83$	−985.546	−1.30335	−0.651673	−	0.758500i	$-0.725932\pi$
$83$	−985.546	−1.30335	−0.651673	+	0.758500i	$0.725932\pi$
$84$	252.340	0.327769
$85$	677.777	0.864885
$86$	252.628	0.316762
$87$	112.815	0.139023
$88$	−67.2412	−0.0814538
$89$	−280.594	−0.334190	−0.167095	−	0.985941i	$-0.553439\pi$
$89$	−280.594	−0.334190	−0.167095	+	0.985941i	$0.553439\pi$
$90$	−90.0000	−0.105409
$91$	−1250.77	−1.44083
$92$	−436.757	−0.494947
$93$	−871.603	−0.971838
$94$	479.696	0.526349
$95$	733.885	0.792579
$96$	96.0000	0.102062
$97$	−1390.36	−1.45536	−0.727679	−	0.685918i	$-0.759401\pi$
$97$	−1390.36	−1.45536	−0.727679	+	0.685918i	$0.759401\pi$
$98$	198.385	0.204489
$99$	−75.6463	−0.0767954
$100$	100.000	0.100000
$101$	−450.911	−0.444231	−0.222115	−	0.975020i	$-0.571296\pi$
$101$	−450.911	−0.444231	−0.222115	+	0.975020i	$0.571296\pi$
$102$	−813.332	−0.789528
$103$	−1025.52	−0.981041	−0.490521	−	0.871430i	$-0.663193\pi$
$103$	−1025.52	−0.981041	−0.490521	+	0.871430i	$0.663193\pi$
$104$	−475.839	−0.448653
$105$	−315.426	−0.293166
$106$	−789.450	−0.723379
$107$	1416.85	1.28011	0.640055	−	0.768329i	$-0.278912\pi$
$107$	1416.85	1.28011	0.640055	+	0.768329i	$0.278912\pi$
$108$	108.000	0.0962250
$109$	−1183.02	−1.03956	−0.519781	−	0.854299i	$-0.673987\pi$
$109$	−1183.02	−1.03956	−0.519781	+	0.854299i	$0.673987\pi$
$110$	84.0515	0.0728545
$111$	111.000	0.0949158
$112$	336.454	0.283856
$113$	963.070	0.801752	0.400876	−	0.916132i	$-0.368706\pi$
$113$	963.070	0.801752	0.400876	+	0.916132i	$0.368706\pi$
$114$	−880.662	−0.723523
$115$	545.946	0.442694
$116$	150.420	0.120397
$117$	−535.319	−0.422994
$118$	1490.87	1.16310
$119$	−2850.51	−2.19585
$120$	−120.000	−0.0912871
$121$	−1260.35	−0.946922
$122$	1636.36	1.21434
$123$	434.372	0.318423
$124$	−1162.14	−0.841637
$125$	−125.000	−0.0894427
$126$	378.511	0.267622
$127$	2573.56	1.79816	0.899079	−	0.437786i	$-0.144237\pi$
$127$	2573.56	1.79816	0.899079	+	0.437786i	$0.144237\pi$
$128$	128.000	0.0883883
$129$	378.942	0.258635
$130$	594.799	0.401287
$131$	2950.62	1.96791	0.983956	−	0.178411i	$-0.0570957\pi$
$131$	2950.62	1.96791	0.983956	+	0.178411i	$0.0570957\pi$
$132$	−100.862	−0.0665067
$133$	−3086.48	−2.01227
$134$	−583.757	−0.376335
$135$	−135.000	−0.0860663
$136$	−1084.44	−0.683751
$137$	−875.424	−0.545931	−0.272965	−	0.962024i	$-0.588004\pi$
$137$	−875.424	−0.545931	−0.272965	+	0.962024i	$0.588004\pi$
$138$	−655.136	−0.404122
$139$	−351.409	−0.214433	−0.107216	−	0.994236i	$-0.534194\pi$
$139$	−351.409	−0.214433	−0.107216	+	0.994236i	$0.534194\pi$
$140$	−420.567	−0.253889
$141$	719.544	0.429762
$142$	1461.41	0.863656
$143$	499.937	0.292356
$144$	144.000	0.0833333
$145$	−188.024	−0.107687
$146$	301.903	0.171135
$147$	297.578	0.166965
$148$	148.000	0.0821995
$149$	−1707.50	−0.938819	−0.469410	−	0.882981i	$-0.655533\pi$
$149$	−1707.50	−0.938819	−0.469410	+	0.882981i	$0.655533\pi$
$150$	150.000	0.0816497
$151$	−2555.45	−1.37721	−0.688607	−	0.725134i	$-0.741778\pi$
$151$	−2555.45	−1.37721	−0.688607	+	0.725134i	$0.741778\pi$
$152$	−1174.22	−0.626589
$153$	−1220.00	−0.644647
$154$	−353.493	−0.184969
$155$	1452.67	0.752783
$156$	−713.759	−0.366323
$157$	−3312.17	−1.68369	−0.841847	−	0.539716i	$-0.818532\pi$
$157$	−3312.17	−1.68369	−0.841847	+	0.539716i	$0.818532\pi$
$158$	−1277.89	−0.643438
$159$	−1184.17	−0.590636
$160$	−160.000	−0.0790569
$161$	−2296.07	−1.12395
$162$	162.000	0.0785674
$163$	483.400	0.232287	0.116144	−	0.993232i	$-0.462947\pi$
$163$	483.400	0.232287	0.116144	+	0.993232i	$0.462947\pi$
$164$	579.163	0.275762
$165$	126.077	0.0594854
$166$	−1971.09	−0.921605
$167$	2604.88	1.20702	0.603508	−	0.797357i	$-0.293769\pi$
$167$	2604.88	1.20702	0.603508	+	0.797357i	$0.293769\pi$
$168$	504.681	0.231768
$169$	1340.86	0.610313
$170$	1355.55	0.611566
$171$	−1320.99	−0.590754
$172$	505.256	0.223985
$173$	−1584.86	−0.696502	−0.348251	−	0.937401i	$-0.613224\pi$
$173$	−1584.86	−0.696502	−0.348251	+	0.937401i	$0.613224\pi$
$174$	225.629	0.0983041
$175$	525.709	0.227085
$176$	−134.482	−0.0575965
$177$	2236.31	0.949668
$178$	−561.189	−0.236308
$179$	−1557.43	−0.650323	−0.325162	−	0.945658i	$-0.605419\pi$
$179$	−1557.43	−0.650323	−0.325162	+	0.945658i	$0.605419\pi$
$180$	−180.000	−0.0745356
$181$	1637.66	0.672521	0.336261	−	0.941769i	$-0.390838\pi$
$181$	1637.66	0.672521	0.336261	+	0.941769i	$0.390838\pi$
$182$	−2501.53	−1.01882
$183$	2454.54	0.991502
$184$	−873.514	−0.349980
$185$	−185.000	−0.0735215
$186$	−1743.21	−0.687194
$187$	1139.36	0.445553
$188$	959.391	0.372185
$189$	567.766	0.218513
$190$	1467.77	0.560438
$191$	1942.23	0.735786	0.367893	−	0.929868i	$-0.380079\pi$
$191$	1942.23	0.735786	0.367893	+	0.929868i	$0.380079\pi$
$192$	192.000	0.0721688
$193$	4319.95	1.61118	0.805588	−	0.592477i	$-0.201850\pi$
$193$	4319.95	1.61118	0.805588	+	0.592477i	$0.201850\pi$
$194$	−2780.72	−1.02909
$195$	892.198	0.327649
$196$	396.770	0.144596
$197$	1392.38	0.503568	0.251784	−	0.967783i	$-0.418983\pi$
$197$	1392.38	0.503568	0.251784	+	0.967783i	$0.418983\pi$
$198$	−151.293	−0.0543025
$199$	4298.15	1.53109	0.765547	−	0.643380i	$-0.222469\pi$
$199$	4298.15	1.53109	0.765547	+	0.643380i	$0.222469\pi$
$200$	200.000	0.0707107
$201$	−875.635	−0.307276
$202$	−901.822	−0.314119
$203$	790.769	0.273405
$204$	−1626.66	−0.558281
$205$	−723.953	−0.246649
$206$	−2051.04	−0.693701
$207$	−982.704	−0.329964
$208$	−951.678	−0.317245
$209$	1233.68	0.408304
$210$	−630.851	−0.207299
$211$	−1039.78	−0.339247	−0.169624	−	0.985509i	$-0.554255\pi$
$211$	−1039.78	−0.339247	−0.169624	+	0.985509i	$0.554255\pi$
$212$	−1578.90	−0.511506
$213$	2192.12	0.705172
$214$	2833.69	0.905174
$215$	−631.570	−0.200338
$216$	216.000	0.0680414
$217$	−6109.46	−1.91123
$218$	−2366.03	−0.735082
$219$	452.854	0.139731
$220$	168.103	0.0515159
$221$	8062.81	2.45413
$222$	222.000	0.0671156
$223$	4482.81	1.34615	0.673075	−	0.739574i	$-0.264973\pi$
$223$	4482.81	1.34615	0.673075	+	0.739574i	$0.264973\pi$
$224$	672.908	0.200717
$225$	225.000	0.0666667
$226$	1926.14	0.566924
$227$	−4786.80	−1.39961	−0.699805	−	0.714334i	$-0.746730\pi$
$227$	−4786.80	−1.39961	−0.699805	+	0.714334i	$0.746730\pi$
$228$	−1761.32	−0.511608
$229$	5344.79	1.54233	0.771165	−	0.636635i	$-0.219674\pi$
$229$	5344.79	1.54233	0.771165	+	0.636635i	$0.219674\pi$
$230$	1091.89	0.313032
$231$	−530.240	−0.151027
$232$	300.839	0.0851338
$233$	966.476	0.271742	0.135871	−	0.990727i	$-0.456617\pi$
$233$	966.476	0.271742	0.135871	+	0.990727i	$0.456617\pi$
$234$	−1070.64	−0.299102
$235$	−1199.24	−0.332893
$236$	2981.74	0.822436
$237$	−1916.83	−0.525365
$238$	−5701.02	−1.55270
$239$	−4418.30	−1.19580	−0.597899	−	0.801571i	$-0.703998\pi$
$239$	−4418.30	−1.19580	−0.597899	+	0.801571i	$0.703998\pi$
$240$	−240.000	−0.0645497
$241$	−3238.30	−0.865550	−0.432775	−	0.901502i	$-0.642466\pi$
$241$	−3238.30	−0.865550	−0.432775	+	0.901502i	$0.642466\pi$
$242$	−2520.71	−0.669575
$243$	243.000	0.0641500
$244$	3272.72	0.858666
$245$	−495.963	−0.129330
$246$	868.744	0.225159
$247$	8730.28	2.24897
$248$	−2324.27	−0.595127
$249$	−2956.64	−0.752487
$250$	−250.000	−0.0632456
$251$	−3690.56	−0.928071	−0.464036	−	0.885817i	$-0.653599\pi$
$251$	−3690.56	−0.928071	−0.464036	+	0.885817i	$0.653599\pi$
$252$	757.021	0.189238
$253$	917.752	0.228058
$254$	5147.11	1.27149
$255$	2033.33	0.499341
$256$	256.000	0.0625000
$257$	7906.53	1.91905	0.959524	−	0.281626i	$-0.0908737\pi$
$257$	7906.53	1.91905	0.959524	+	0.281626i	$0.0908737\pi$
$258$	757.884	0.182883
$259$	778.050	0.186663
$260$	1189.60	0.283753
$261$	338.444	0.0802649
$262$	5901.23	1.39152
$263$	3888.46	0.911684	0.455842	−	0.890061i	$-0.349338\pi$
$263$	3888.46	0.911684	0.455842	+	0.890061i	$0.349338\pi$
$264$	−201.724	−0.0470274
$265$	1973.62	0.457505
$266$	−6172.97	−1.42289
$267$	−841.783	−0.192945
$268$	−1167.51	−0.266109
$269$	−4249.18	−0.963112	−0.481556	−	0.876415i	$-0.659928\pi$
$269$	−4249.18	−0.963112	−0.481556	+	0.876415i	$0.659928\pi$
$270$	−270.000	−0.0608581
$271$	−7696.98	−1.72531	−0.862654	−	0.505795i	$-0.831199\pi$
$271$	−7696.98	−1.72531	−0.862654	+	0.505795i	$0.831199\pi$
$272$	−2168.88	−0.483485
$273$	−3752.30	−0.831866
$274$	−1750.85	−0.386031
$275$	−210.129	−0.0460772
$276$	−1310.27	−0.285758
$277$	−4289.00	−0.930328	−0.465164	−	0.885224i	$-0.654005\pi$
$277$	−4289.00	−0.930328	−0.465164	+	0.885224i	$0.654005\pi$
$278$	−702.819	−0.151627
$279$	−2614.81	−0.561091
$280$	−841.135	−0.179527
$281$	−3060.77	−0.649787	−0.324893	−	0.945751i	$-0.605328\pi$
$281$	−3060.77	−0.649787	−0.324893	+	0.945751i	$0.605328\pi$
$282$	1439.09	0.303888
$283$	−1465.29	−0.307783	−0.153892	−	0.988088i	$-0.549181\pi$
$283$	−1465.29	−0.307783	−0.153892	+	0.988088i	$0.549181\pi$
$284$	2922.83	0.610697
$285$	2201.66	0.457596
$286$	999.874	0.206727
$287$	3044.71	0.626215
$288$	288.000	0.0589256
$289$	13462.2	2.74013
$290$	−376.049	−0.0761460
$291$	−4171.08	−0.840251
$292$	603.806	0.121010
$293$	−4356.94	−0.868720	−0.434360	−	0.900739i	$-0.643025\pi$
$293$	−4356.94	−0.868720	−0.434360	+	0.900739i	$0.643025\pi$
$294$	595.155	0.118062
$295$	−3727.18	−0.735609
$296$	296.000	0.0581238
$297$	−226.939	−0.0443378
$298$	−3415.01	−0.663846
$299$	6494.57	1.25616
$300$	300.000	0.0577350
$301$	2656.18	0.508636
$302$	−5110.90	−0.973838
$303$	−1352.73	−0.256477
$304$	−2348.43	−0.443065
$305$	−4090.90	−0.768014
$306$	−2440.00	−0.455834
$307$	−3937.78	−0.732055	−0.366027	−	0.930604i	$-0.619282\pi$
$307$	−3937.78	−0.732055	−0.366027	+	0.930604i	$0.619282\pi$
$308$	−706.986	−0.130793
$309$	−3076.55	−0.566404
$310$	2905.34	0.532298
$311$	−5501.98	−1.00318	−0.501589	−	0.865106i	$-0.667251\pi$
$311$	−5501.98	−1.00318	−0.501589	+	0.865106i	$0.667251\pi$
$312$	−1427.52	−0.259030
$313$	−8673.59	−1.56633	−0.783163	−	0.621816i	$-0.786395\pi$
$313$	−8673.59	−1.56633	−0.783163	+	0.621816i	$0.786395\pi$
$314$	−6624.34	−1.19055
$315$	−946.277	−0.169259
$316$	−2555.77	−0.454979
$317$	65.6874	0.0116384	0.00581920	−	0.999983i	$-0.498148\pi$
$317$	65.6874	0.0116384	0.00581920	+	0.999983i	$0.498148\pi$
$318$	−2368.35	−0.417643
$319$	−316.075	−0.0554758
$320$	−320.000	−0.0559017
$321$	4250.54	0.739072
$322$	−4592.15	−0.794753
$323$	19896.4	3.42745
$324$	324.000	0.0555556
$325$	−1487.00	−0.253796
$326$	966.801	0.164252
$327$	−3549.05	−0.600192
$328$	1158.33	0.194993
$329$	5043.61	0.845177
$330$	252.154	0.0420626
$331$	1002.83	0.166527	0.0832633	−	0.996528i	$-0.473466\pi$
$331$	1002.83	0.166527	0.0832633	+	0.996528i	$0.473466\pi$
$332$	−3942.18	−0.651673
$333$	333.000	0.0547997
$334$	5209.76	0.853489
$335$	1459.39	0.238015
$336$	1009.36	0.163885
$337$	861.240	0.139213	0.0696065	−	0.997575i	$-0.477826\pi$
$337$	861.240	0.139213	0.0696065	+	0.997575i	$0.477826\pi$
$338$	2681.71	0.431556
$339$	2889.21	0.462892
$340$	2711.11	0.432442
$341$	2441.98	0.387803
$342$	−2641.99	−0.417726
$343$	−5126.87	−0.807071
$344$	1010.51	0.158381
$345$	1637.84	0.255589
$346$	−3169.72	−0.492501
$347$	9933.99	1.53684	0.768421	−	0.639944i	$-0.221043\pi$
$347$	9933.99	1.53684	0.768421	+	0.639944i	$0.221043\pi$
$348$	451.259	0.0695115
$349$	−6039.29	−0.926291	−0.463145	−	0.886282i	$-0.653279\pi$
$349$	−6039.29	−0.926291	−0.463145	+	0.886282i	$0.653279\pi$
$350$	1051.42	0.160573
$351$	−1605.96	−0.244215
$352$	−268.965	−0.0407269
$353$	−6246.80	−0.941880	−0.470940	−	0.882165i	$-0.656085\pi$
$353$	−6246.80	−0.941880	−0.470940	+	0.882165i	$0.656085\pi$
$354$	4472.61	0.671516
$355$	−3653.53	−0.546224
$356$	−1122.38	−0.167095
$357$	−8551.52	−1.26777
$358$	−3114.86	−0.459848
$359$	6703.75	0.985545	0.492772	−	0.870158i	$-0.335984\pi$
$359$	6703.75	0.985545	0.492772	+	0.870158i	$0.335984\pi$
$360$	−360.000	−0.0527046
$361$	14684.5	2.14091
$362$	3275.32	0.475544
$363$	−3781.06	−0.546706
$364$	−5003.06	−0.720417
$365$	−754.757	−0.108235
$366$	4909.08	0.701098
$367$	−6309.22	−0.897380	−0.448690	−	0.893687i	$-0.648109\pi$
$367$	−6309.22	−0.897380	−0.448690	+	0.893687i	$0.648109\pi$
$368$	−1747.03	−0.247473
$369$	1303.12	0.183841
$370$	−370.000	−0.0519875
$371$	−8300.42	−1.16155
$372$	−3486.41	−0.485919
$373$	107.082	0.0148646	0.00743232	−	0.999972i	$-0.497634\pi$
$373$	107.082	0.0148646	0.00743232	+	0.999972i	$0.497634\pi$
$374$	2278.72	0.315054
$375$	−375.000	−0.0516398
$376$	1918.78	0.263175
$377$	−2236.73	−0.305564
$378$	1135.53	0.154512
$379$	957.796	0.129812	0.0649059	−	0.997891i	$-0.479325\pi$
$379$	957.796	0.129812	0.0649059	+	0.997891i	$0.479325\pi$
$380$	2935.54	0.396290
$381$	7720.67	1.03817
$382$	3884.47	0.520279
$383$	−12814.6	−1.70965	−0.854825	−	0.518917i	$-0.826336\pi$
$383$	−12814.6	−1.70965	−0.854825	+	0.518917i	$0.826336\pi$
$384$	384.000	0.0510310
$385$	883.733	0.116985
$386$	8639.90	1.13927
$387$	1136.83	0.149323
$388$	−5561.44	−0.727679
$389$	−10799.4	−1.40758	−0.703791	−	0.710407i	$-0.748511\pi$
$389$	−10799.4	−1.40758	−0.703791	+	0.710407i	$0.748511\pi$
$390$	1784.40	0.231683
$391$	14801.2	1.91439
$392$	793.540	0.102244
$393$	8851.85	1.13617
$394$	2784.76	0.356076
$395$	3194.72	0.406946
$396$	−302.585	−0.0383977
$397$	11598.7	1.46631	0.733154	−	0.680063i	$-0.238048\pi$
$397$	11598.7	1.46631	0.733154	+	0.680063i	$0.238048\pi$
$398$	8596.29	1.08265
$399$	−9259.45	−1.16179
$400$	400.000	0.0500000
$401$	2134.66	0.265836	0.132918	−	0.991127i	$-0.457565\pi$
$401$	2134.66	0.265836	0.132918	+	0.991127i	$0.457565\pi$
$402$	−1751.27	−0.217277
$403$	17280.9	2.13604
$404$	−1803.64	−0.222115
$405$	−405.000	−0.0496904
$406$	1581.54	0.193326
$407$	−310.990	−0.0378752
$408$	−3253.33	−0.394764
$409$	−11777.4	−1.42385	−0.711923	−	0.702258i	$-0.752175\pi$
$409$	−11777.4	−1.42385	−0.711923	+	0.702258i	$0.752175\pi$
$410$	−1447.91	−0.174407
$411$	−2626.27	−0.315193
$412$	−4102.07	−0.490521
$413$	15675.3	1.86763
$414$	−1965.41	−0.233320
$415$	4927.73	0.582874
$416$	−1903.36	−0.224326
$417$	−1054.23	−0.123803
$418$	2467.37	0.288715
$419$	9178.34	1.07015	0.535073	−	0.844806i	$-0.320284\pi$
$419$	9178.34	1.07015	0.535073	+	0.844806i	$0.320284\pi$
$420$	−1261.70	−0.146583
$421$	−8782.75	−1.01673	−0.508367	−	0.861141i	$-0.669751\pi$
$421$	−8782.75	−1.01673	−0.508367	+	0.861141i	$0.669751\pi$
$422$	−2079.55	−0.239884
$423$	2158.63	0.248123
$424$	−3157.80	−0.361689
$425$	−3388.88	−0.386788
$426$	4384.24	0.498632
$427$	17205.0	1.94990
$428$	5667.39	0.640055
$429$	1499.81	0.168792
$430$	−1263.14	−0.141660
$431$	−3868.44	−0.432334	−0.216167	−	0.976356i	$-0.569356\pi$
$431$	−3868.44	−0.432334	−0.216167	+	0.976356i	$0.569356\pi$
$432$	432.000	0.0481125
$433$	−9338.71	−1.03647	−0.518233	−	0.855240i	$-0.673410\pi$
$433$	−9338.71	−1.03647	−0.518233	+	0.855240i	$0.673410\pi$
$434$	−12218.9	−1.35144
$435$	−564.073	−0.0621730
$436$	−4732.06	−0.519781
$437$	16026.5	1.75435
$438$	905.709	0.0988046
$439$	−8312.44	−0.903716	−0.451858	−	0.892090i	$-0.649239\pi$
$439$	−8312.44	−0.903716	−0.451858	+	0.892090i	$0.649239\pi$
$440$	336.206	0.0364272
$441$	892.733	0.0963970
$442$	16125.6	1.73533
$443$	−2877.99	−0.308662	−0.154331	−	0.988019i	$-0.549322\pi$
$443$	−2877.99	−0.308662	−0.154331	+	0.988019i	$0.549322\pi$
$444$	444.000	0.0474579
$445$	1402.97	0.149455
$446$	8965.63	0.951871
$447$	−5122.51	−0.542028
$448$	1345.82	0.141928
$449$	7671.63	0.806340	0.403170	−	0.915125i	$-0.367908\pi$
$449$	7671.63	0.806340	0.403170	+	0.915125i	$0.367908\pi$
$450$	450.000	0.0471405
$451$	−1216.99	−0.127064
$452$	3852.28	0.400876
$453$	−7666.34	−0.795135
$454$	−9573.61	−0.989673
$455$	6253.83	0.644360
$456$	−3522.65	−0.361761
$457$	−7621.87	−0.780166	−0.390083	−	0.920780i	$-0.627554\pi$
$457$	−7621.87	−0.780166	−0.390083	+	0.920780i	$0.627554\pi$
$458$	10689.6	1.09059
$459$	−3659.99	−0.372187
$460$	2183.79	0.221347
$461$	18327.6	1.85163	0.925817	−	0.377972i	$-0.123378\pi$
$461$	18327.6	1.85163	0.925817	+	0.377972i	$0.123378\pi$
$462$	−1060.48	−0.106792
$463$	−13217.8	−1.32675	−0.663374	−	0.748288i	$-0.730876\pi$
$463$	−13217.8	−1.32675	−0.663374	+	0.748288i	$0.730876\pi$
$464$	601.678	0.0601987
$465$	4358.01	0.434619
$466$	1932.95	0.192151
$467$	−9667.26	−0.957917	−0.478959	−	0.877837i	$-0.658986\pi$
$467$	−9667.26	−0.957917	−0.478959	+	0.877837i	$0.658986\pi$
$468$	−2141.28	−0.211497
$469$	−6137.73	−0.604294
$470$	−2398.48	−0.235391
$471$	−9936.51	−0.972082
$472$	5963.49	0.581550
$473$	−1061.69	−0.103206
$474$	−3833.66	−0.371489
$475$	−3669.43	−0.354452
$476$	−11402.0	−1.09792
$477$	−3552.52	−0.341004
$478$	−8836.59	−0.845557
$479$	11055.1	1.05453	0.527263	−	0.849702i	$-0.323218\pi$
$479$	11055.1	1.05453	0.527263	+	0.849702i	$0.323218\pi$
$480$	−480.000	−0.0456435
$481$	−2200.76	−0.208619
$482$	−6476.61	−0.612036
$483$	−6888.22	−0.648913
$484$	−5041.41	−0.473461
$485$	6951.80	0.650855
$486$	486.000	0.0453609
$487$	6633.61	0.617243	0.308622	−	0.951185i	$-0.400132\pi$
$487$	6633.61	0.617243	0.308622	+	0.951185i	$0.400132\pi$
$488$	6545.44	0.607169
$489$	1450.20	0.134111
$490$	−991.925	−0.0914502
$491$	5347.67	0.491522	0.245761	−	0.969331i	$-0.420962\pi$
$491$	5347.67	0.491522	0.245761	+	0.969331i	$0.420962\pi$
$492$	1737.49	0.159211
$493$	−5097.54	−0.465683
$494$	17460.6	1.59026
$495$	378.232	0.0343439
$496$	−4648.55	−0.420818
$497$	15365.6	1.38680
$498$	−5913.27	−0.532089
$499$	15161.4	1.36016	0.680078	−	0.733140i	$-0.261946\pi$
$499$	15161.4	1.36016	0.680078	+	0.733140i	$0.261946\pi$
$500$	−500.000	−0.0447214
$501$	7814.64	0.696871
$502$	−7381.11	−0.656245
$503$	−9708.10	−0.860562	−0.430281	−	0.902695i	$-0.641586\pi$
$503$	−9708.10	−0.860562	−0.430281	+	0.902695i	$0.641586\pi$
$504$	1514.04	0.133811
$505$	2254.55	0.198666
$506$	1835.50	0.161261
$507$	4022.57	0.352364
$508$	10294.2	0.899079
$509$	4454.79	0.387928	0.193964	−	0.981009i	$-0.437866\pi$
$509$	4454.79	0.387928	0.193964	+	0.981009i	$0.437866\pi$
$510$	4066.66	0.353088
$511$	3174.26	0.274797
$512$	512.000	0.0441942
$513$	−3962.98	−0.341072
$514$	15813.1	1.35697
$515$	5127.59	0.438735
$516$	1515.77	0.129318
$517$	−2015.96	−0.171493
$518$	1556.10	0.131991
$519$	−4754.58	−0.402125
$520$	2379.20	0.200644
$521$	−12718.2	−1.06947	−0.534737	−	0.845018i	$-0.679589\pi$
$521$	−12718.2	−1.06947	−0.534737	+	0.845018i	$0.679589\pi$
$522$	676.888	0.0567559
$523$	10096.3	0.844127	0.422064	−	0.906566i	$-0.361306\pi$
$523$	10096.3	0.844127	0.422064	+	0.906566i	$0.361306\pi$
$524$	11802.5	0.983956
$525$	1577.13	0.131108
$526$	7776.93	0.644658
$527$	39383.4	3.25535
$528$	−403.447	−0.0332534
$529$	−244.699	−0.0201117
$530$	3947.25	0.323505
$531$	6708.92	0.548291
$532$	−12345.9	−1.00614
$533$	−8612.13	−0.699874
$534$	−1683.57	−0.136433
$535$	−7084.24	−0.572483
$536$	−2335.03	−0.188168
$537$	−4672.29	−0.375464
$538$	−8498.36	−0.681023
$539$	−833.728	−0.0666256
$540$	−540.000	−0.0430331
$541$	−9323.72	−0.740957	−0.370479	−	0.928841i	$-0.620806\pi$
$541$	−9323.72	−0.740957	−0.370479	+	0.928841i	$0.620806\pi$
$542$	−15394.0	−1.21998
$543$	4912.98	0.388280
$544$	−4337.77	−0.341876
$545$	5915.08	0.464906
$546$	−7504.59	−0.588218
$547$	15655.9	1.22376	0.611880	−	0.790950i	$-0.290413\pi$
$547$	15655.9	1.22376	0.611880	+	0.790950i	$0.290413\pi$
$548$	−3501.69	−0.272965
$549$	7363.62	0.572444
$550$	−420.257	−0.0325815
$551$	−5519.53	−0.426751
$552$	−2620.54	−0.202061
$553$	−13435.9	−1.03319
$554$	−8578.00	−0.657842
$555$	−555.000	−0.0424476
$556$	−1405.64	−0.107216
$557$	17844.0	1.35740	0.678701	−	0.734415i	$-0.262543\pi$
$557$	17844.0	1.35740	0.678701	+	0.734415i	$0.262543\pi$
$558$	−5229.62	−0.396751
$559$	−7513.14	−0.568465
$560$	−1682.27	−0.126944
$561$	3418.09	0.257240
$562$	−6121.54	−0.459469
$563$	−16641.7	−1.24576	−0.622882	−	0.782316i	$-0.714038\pi$
$563$	−16641.7	−1.24576	−0.622882	+	0.782316i	$0.714038\pi$
$564$	2878.17	0.214881
$565$	−4815.35	−0.358554
$566$	−2930.59	−0.217636
$567$	1703.30	0.126158
$568$	5845.65	0.431828
$569$	20517.9	1.51169	0.755847	−	0.654749i	$-0.227226\pi$
$569$	20517.9	1.51169	0.755847	+	0.654749i	$0.227226\pi$
$570$	4403.31	0.323569
$571$	−10141.1	−0.743240	−0.371620	−	0.928385i	$-0.621198\pi$
$571$	−10141.1	−0.743240	−0.371620	+	0.928385i	$0.621198\pi$
$572$	1999.75	0.146178
$573$	5826.70	0.424806
$574$	6089.42	0.442801
$575$	−2729.73	−0.197979
$576$	576.000	0.0416667
$577$	4254.51	0.306963	0.153482	−	0.988152i	$-0.450951\pi$
$577$	4254.51	0.306963	0.153482	+	0.988152i	$0.450951\pi$
$578$	26924.5	1.93756
$579$	12959.8	0.930212
$580$	−752.098	−0.0538434
$581$	−20724.4	−1.47985
$582$	−8342.16	−0.594147
$583$	3317.72	0.235688
$584$	1207.61	0.0855673
$585$	2676.59	0.189169
$586$	−8713.87	−0.614278
$587$	11715.5	0.823768	0.411884	−	0.911236i	$-0.364871\pi$
$587$	11715.5	0.823768	0.411884	+	0.911236i	$0.364871\pi$
$588$	1190.31	0.0834823
$589$	42643.8	2.98320
$590$	−7454.36	−0.520154
$591$	4177.14	0.290735
$592$	592.000	0.0410997
$593$	−19721.5	−1.36571	−0.682854	−	0.730555i	$-0.739261\pi$
$593$	−19721.5	−1.36571	−0.682854	+	0.730555i	$0.739261\pi$
$594$	−453.878	−0.0313516
$595$	14252.5	0.982012
$596$	−6830.01	−0.469410
$597$	12894.4	0.883977
$598$	12989.1	0.888236
$599$	10395.6	0.709103	0.354552	−	0.935036i	$-0.384633\pi$
$599$	10395.6	0.709103	0.354552	+	0.935036i	$0.384633\pi$
$600$	600.000	0.0408248
$601$	−10286.9	−0.698186	−0.349093	−	0.937088i	$-0.613510\pi$
$601$	−10286.9	−0.698186	−0.349093	+	0.937088i	$0.613510\pi$
$602$	5312.35	0.359660
$603$	−2626.91	−0.177406
$604$	−10221.8	−0.688607
$605$	6301.77	0.423476
$606$	−2705.47	−0.181356
$607$	−13123.5	−0.877542	−0.438771	−	0.898599i	$-0.644586\pi$
$607$	−13123.5	−0.877542	−0.438771	+	0.898599i	$0.644586\pi$
$608$	−4696.87	−0.313295
$609$	2372.31	0.157850
$610$	−8181.80	−0.543068
$611$	−14266.1	−0.944592
$612$	−4879.99	−0.322323
$613$	−8765.70	−0.577558	−0.288779	−	0.957396i	$-0.593249\pi$
$613$	−8765.70	−0.577558	−0.288779	+	0.957396i	$0.593249\pi$
$614$	−7875.55	−0.517641
$615$	−2171.86	−0.142403
$616$	−1413.97	−0.0924847
$617$	8074.52	0.526852	0.263426	−	0.964680i	$-0.415147\pi$
$617$	8074.52	0.526852	0.263426	+	0.964680i	$0.415147\pi$
$618$	−6153.11	−0.400508
$619$	5569.14	0.361620	0.180810	−	0.983518i	$-0.442128\pi$
$619$	5569.14	0.361620	0.180810	+	0.983518i	$0.442128\pi$
$620$	5810.68	0.376391
$621$	−2948.11	−0.190505
$622$	−11004.0	−0.709354
$623$	−5900.45	−0.379448
$624$	−2855.03	−0.183162
$625$	625.000	0.0400000
$626$	−17347.2	−1.10756
$627$	3701.05	0.235735
$628$	−13248.7	−0.841847
$629$	−5015.55	−0.317938
$630$	−1892.55	−0.119684
$631$	−12823.1	−0.808998	−0.404499	−	0.914538i	$-0.632554\pi$
$631$	−12823.1	−0.808998	−0.404499	+	0.914538i	$0.632554\pi$
$632$	−5111.55	−0.321719
$633$	−3119.33	−0.195864
$634$	131.375	0.00822959
$635$	−12867.8	−0.804161
$636$	−4736.70	−0.295318
$637$	−5899.96	−0.366978
$638$	−632.149	−0.0392273
$639$	6576.36	0.407131
$640$	−640.000	−0.0395285
$641$	−3131.10	−0.192935	−0.0964673	−	0.995336i	$-0.530754\pi$
$641$	−3131.10	−0.192935	−0.0964673	+	0.995336i	$0.530754\pi$
$642$	8501.08	0.522603
$643$	−17336.0	−1.06324	−0.531621	−	0.846982i	$-0.678417\pi$
$643$	−17336.0	−1.06324	−0.531621	+	0.846982i	$0.678417\pi$
$644$	−9184.29	−0.561975
$645$	−1894.71	−0.115665
$646$	39792.8	2.42357
$647$	−2065.20	−0.125489	−0.0627445	−	0.998030i	$-0.519985\pi$
$647$	−2065.20	−0.125489	−0.0627445	+	0.998030i	$0.519985\pi$
$648$	648.000	0.0392837
$649$	−6265.50	−0.378956
$650$	−2973.99	−0.179461
$651$	−18328.4	−1.10345
$652$	1933.60	0.116144
$653$	−17769.3	−1.06488	−0.532438	−	0.846469i	$-0.678724\pi$
$653$	−17769.3	−1.06488	−0.532438	+	0.846469i	$0.678724\pi$
$654$	−7098.09	−0.424400
$655$	−14753.1	−0.880077
$656$	2316.65	0.137881
$657$	1358.56	0.0806736
$658$	10087.2	0.597630
$659$	−5812.33	−0.343575	−0.171788	−	0.985134i	$-0.554954\pi$
$659$	−5812.33	−0.343575	−0.171788	+	0.985134i	$0.554954\pi$
$660$	504.309	0.0297427
$661$	4570.78	0.268960	0.134480	−	0.990916i	$-0.457064\pi$
$661$	4570.78	0.268960	0.134480	+	0.990916i	$0.457064\pi$
$662$	2005.65	0.117752
$663$	24188.4	1.41689
$664$	−7884.37	−0.460802
$665$	15432.4	0.899915
$666$	666.000	0.0387492
$667$	−4106.05	−0.238361
$668$	10419.5	0.603508
$669$	13448.4	0.777200
$670$	2918.78	0.168302
$671$	−6876.93	−0.395649
$672$	2018.72	0.115884
$673$	−13512.7	−0.773960	−0.386980	−	0.922088i	$-0.626482\pi$
$673$	−13512.7	−0.773960	−0.386980	+	0.922088i	$0.626482\pi$
$674$	1722.48	0.0984384
$675$	675.000	0.0384900
$676$	5363.43	0.305156
$677$	13124.1	0.745051	0.372526	−	0.928022i	$-0.378492\pi$
$677$	13124.1	0.745051	0.372526	+	0.928022i	$0.378492\pi$
$678$	5778.42	0.327314
$679$	−29237.0	−1.65245
$680$	5422.21	0.305783
$681$	−14360.4	−0.808065
$682$	4883.97	0.274218
$683$	7022.03	0.393398	0.196699	−	0.980464i	$-0.436978\pi$
$683$	7022.03	0.393398	0.196699	+	0.980464i	$0.436978\pi$
$684$	−5283.97	−0.295377
$685$	4377.12	0.244148
$686$	−10253.7	−0.570685
$687$	16034.4	0.890464
$688$	2021.02	0.111992
$689$	23478.2	1.29818
$690$	3275.68	0.180729
$691$	17893.0	0.985068	0.492534	−	0.870293i	$-0.336071\pi$
$691$	17893.0	0.985068	0.492534	+	0.870293i	$0.336071\pi$
$692$	−6339.44	−0.348251
$693$	−1590.72	−0.0871954
$694$	19868.0	1.08671
$695$	1757.05	0.0958972
$696$	902.517	0.0491520
$697$	−19627.1	−1.06662
$698$	−12078.6	−0.654987
$699$	2899.43	0.156890
$700$	2102.84	0.113543
$701$	31210.1	1.68158	0.840792	−	0.541358i	$-0.182090\pi$
$701$	31210.1	1.68158	0.840792	+	0.541358i	$0.182090\pi$
$702$	−3211.91	−0.172686
$703$	−5430.75	−0.291358
$704$	−537.929	−0.0287983
$705$	−3597.72	−0.192196
$706$	−12493.6	−0.666009
$707$	−9481.92	−0.504391
$708$	8945.23	0.474834
$709$	−8731.46	−0.462506	−0.231253	−	0.972894i	$-0.574283\pi$
$709$	−8731.46	−0.462506	−0.231253	+	0.972894i	$0.574283\pi$
$710$	−7307.07	−0.386239
$711$	−5750.49	−0.303320
$712$	−2244.76	−0.118154
$713$	31723.2	1.66626
$714$	−17103.0	−0.896450
$715$	−2499.69	−0.130745
$716$	−6229.72	−0.325162
$717$	−13254.9	−0.690395
$718$	13407.5	0.696885
$719$	34281.3	1.77813	0.889065	−	0.457781i	$-0.151356\pi$
$719$	34281.3	1.77813	0.889065	+	0.457781i	$0.151356\pi$
$720$	−720.000	−0.0372678
$721$	−21565.0	−1.11390
$722$	29369.0	1.51385
$723$	−9714.91	−0.499725
$724$	6550.64	0.336261
$725$	940.122	0.0481590
$726$	−7562.12	−0.386579
$727$	10371.9	0.529124	0.264562	−	0.964369i	$-0.414773\pi$
$727$	10371.9	0.529124	0.264562	+	0.964369i	$0.414773\pi$
$728$	−10006.1	−0.509412
$729$	729.000	0.0370370
$730$	−1509.51	−0.0765337
$731$	−17122.5	−0.866347
$732$	9818.16	0.495751
$733$	38235.0	1.92666	0.963330	−	0.268320i	$-0.0864684\pi$
$733$	38235.0	1.92666	0.963330	+	0.268320i	$0.0864684\pi$
$734$	−12618.4	−0.634544
$735$	−1487.89	−0.0746688
$736$	−3494.06	−0.174990
$737$	2453.28	0.122616
$738$	2606.23	0.129996
$739$	23061.9	1.14797	0.573983	−	0.818868i	$-0.305398\pi$
$739$	23061.9	1.14797	0.573983	+	0.818868i	$0.305398\pi$
$740$	−740.000	−0.0367607
$741$	26190.9	1.29844
$742$	−16600.8	−0.821343
$743$	28279.2	1.39632	0.698159	−	0.715943i	$-0.254003\pi$
$743$	28279.2	1.39632	0.698159	+	0.715943i	$0.254003\pi$
$744$	−6972.82	−0.343597
$745$	8537.51	0.419853
$746$	214.165	0.0105109
$747$	−8869.91	−0.434449
$748$	4557.45	0.222777
$749$	29794.0	1.45347
$750$	−750.000	−0.0365148
$751$	−24276.0	−1.17955	−0.589776	−	0.807567i	$-0.700784\pi$
$751$	−24276.0	−1.17955	−0.589776	+	0.807567i	$0.700784\pi$
$752$	3837.57	0.186093
$753$	−11071.7	−0.535822
$754$	−4473.47	−0.216066
$755$	12777.2	0.615909
$756$	2271.06	0.109256
$757$	−21714.8	−1.04259	−0.521293	−	0.853378i	$-0.674550\pi$
$757$	−21714.8	−1.04259	−0.521293	+	0.853378i	$0.674550\pi$
$758$	1915.59	0.0917908
$759$	2753.26	0.131669
$760$	5871.08	0.280219
$761$	10047.8	0.478622	0.239311	−	0.970943i	$-0.423078\pi$
$761$	10047.8	0.478622	0.239311	+	0.970943i	$0.423078\pi$
$762$	15441.3	0.734095
$763$	−24876.9	−1.18035
$764$	7768.94	0.367893
$765$	6099.99	0.288295
$766$	−25629.2	−1.20890
$767$	−44338.4	−2.08731
$768$	768.000	0.0360844
$769$	−8808.95	−0.413080	−0.206540	−	0.978438i	$-0.566220\pi$
$769$	−8808.95	−0.413080	−0.206540	+	0.978438i	$0.566220\pi$
$770$	1767.47	0.0827208
$771$	23719.6	1.10796
$772$	17279.8	0.805588
$773$	−4340.66	−0.201970	−0.100985	−	0.994888i	$-0.532199\pi$
$773$	−4340.66	−0.201970	−0.100985	+	0.994888i	$0.532199\pi$
$774$	2273.65	0.105587
$775$	−7263.35	−0.336655
$776$	−11122.9	−0.514546
$777$	2334.15	0.107770
$778$	−21598.7	−0.995311
$779$	−21251.9	−0.977446
$780$	3568.79	0.163825
$781$	−6141.70	−0.281392
$782$	29602.4	1.35368
$783$	1015.33	0.0463410
$784$	1587.08	0.0722978
$785$	16560.9	0.752971
$786$	17703.7	0.803397
$787$	11979.0	0.542572	0.271286	−	0.962499i	$-0.412551\pi$
$787$	11979.0	0.542572	0.271286	+	0.962499i	$0.412551\pi$
$788$	5569.52	0.251784
$789$	11665.4	0.526361
$790$	6389.43	0.287754
$791$	20251.8	0.910330
$792$	−605.171	−0.0271513
$793$	−48665.3	−2.17926
$794$	23197.5	1.03684
$795$	5920.87	0.264141
$796$	17192.6	0.765547
$797$	−4062.47	−0.180552	−0.0902761	−	0.995917i	$-0.528775\pi$
$797$	−4062.47	−0.180552	−0.0902761	+	0.995917i	$0.528775\pi$
$798$	−18518.9	−0.821506
$799$	−32512.6	−1.43957
$800$	800.000	0.0353553
$801$	−2525.35	−0.111397
$802$	4269.33	0.187974
$803$	−1268.77	−0.0557583
$804$	−3502.54	−0.153638
$805$	11480.4	0.502646
$806$	34561.9	1.51041
$807$	−12747.5	−0.556053
$808$	−3607.29	−0.157059
$809$	6344.63	0.275730	0.137865	−	0.990451i	$-0.455976\pi$
$809$	6344.63	0.275730	0.137865	+	0.990451i	$0.455976\pi$
$810$	−810.000	−0.0351364
$811$	−7277.14	−0.315086	−0.157543	−	0.987512i	$-0.550357\pi$
$811$	−7277.14	−0.315086	−0.157543	+	0.987512i	$0.550357\pi$
$812$	3163.08	0.136702
$813$	−23090.9	−0.996107
$814$	−621.981	−0.0267818
$815$	−2417.00	−0.103882
$816$	−6506.65	−0.279140
$817$	−18540.0	−0.793919
$818$	−23554.7	−1.00681
$819$	−11256.9	−0.480278
$820$	−2895.81	−0.123325
$821$	35718.7	1.51838	0.759191	−	0.650868i	$-0.225595\pi$
$821$	35718.7	1.51838	0.759191	+	0.650868i	$0.225595\pi$
$822$	−5252.54	−0.222875
$823$	−44027.0	−1.86474	−0.932372	−	0.361500i	$-0.882265\pi$
$823$	−44027.0	−1.86474	−0.932372	+	0.361500i	$0.882265\pi$
$824$	−8204.14	−0.346850
$825$	−630.386	−0.0266027
$826$	31350.6	1.32061
$827$	19663.9	0.826821	0.413410	−	0.910545i	$-0.364337\pi$
$827$	19663.9	0.826821	0.413410	+	0.910545i	$0.364337\pi$
$828$	−3930.81	−0.164982
$829$	22295.4	0.934080	0.467040	−	0.884236i	$-0.345320\pi$
$829$	22295.4	0.934080	0.467040	+	0.884236i	$0.345320\pi$
$830$	9855.46	0.412154
$831$	−12867.0	−0.537125
$832$	−3806.71	−0.158623
$833$	−13446.1	−0.559278
$834$	−2108.46	−0.0875418
$835$	−13024.4	−0.539794
$836$	4934.73	0.204152
$837$	−7844.42	−0.323946
$838$	18356.7	0.756708
$839$	19412.4	0.798796	0.399398	−	0.916778i	$-0.369219\pi$
$839$	19412.4	0.798796	0.399398	+	0.916778i	$0.369219\pi$
$840$	−2523.40	−0.103650
$841$	−22974.9	−0.942018
$842$	−17565.5	−0.718940
$843$	−9182.30	−0.375155
$844$	−4159.10	−0.169624
$845$	−6704.28	−0.272940
$846$	4317.26	0.175450
$847$	−26503.2	−1.07516
$848$	−6315.60	−0.255753
$849$	−4395.88	−0.177699
$850$	−6777.77	−0.273501
$851$	−4040.00	−0.162737
$852$	8768.48	0.352586
$853$	24198.7	0.971332	0.485666	−	0.874144i	$-0.338577\pi$
$853$	24198.7	0.971332	0.485666	+	0.874144i	$0.338577\pi$
$854$	34410.0	1.37879
$855$	6604.97	0.264193
$856$	11334.8	0.452587
$857$	−11420.4	−0.455208	−0.227604	−	0.973754i	$-0.573089\pi$
$857$	−11420.4	−0.455208	−0.227604	+	0.973754i	$0.573089\pi$
$858$	2999.62	0.119354
$859$	−7777.79	−0.308934	−0.154467	−	0.987998i	$-0.549366\pi$
$859$	−7777.79	−0.308934	−0.154467	+	0.987998i	$0.549366\pi$
$860$	−2526.28	−0.100169
$861$	9134.14	0.361545
$862$	−7736.87	−0.305706
$863$	−18527.6	−0.730807	−0.365404	−	0.930849i	$-0.619069\pi$
$863$	−18527.6	−0.730807	−0.365404	+	0.930849i	$0.619069\pi$
$864$	864.000	0.0340207
$865$	7924.31	0.311485
$866$	−18677.4	−0.732892
$867$	40386.7	1.58201
$868$	−24437.8	−0.955616
$869$	5370.41	0.209642
$870$	−1128.15	−0.0439629
$871$	17360.9	0.675375
$872$	−9464.12	−0.367541
$873$	−12513.2	−0.485119
$874$	32053.0	1.24051
$875$	−2628.55	−0.101556
$876$	1811.42	0.0698654
$877$	4292.82	0.165289	0.0826443	−	0.996579i	$-0.473663\pi$
$877$	4292.82	0.165289	0.0826443	+	0.996579i	$0.473663\pi$
$878$	−16624.9	−0.639024
$879$	−13070.8	−0.501556
$880$	672.412	0.0257580
$881$	−3618.54	−0.138379	−0.0691895	−	0.997604i	$-0.522041\pi$
$881$	−3618.54	−0.138379	−0.0691895	+	0.997604i	$0.522041\pi$
$882$	1785.47	0.0681630
$883$	−30343.3	−1.15644	−0.578218	−	0.815883i	$-0.696252\pi$
$883$	−30343.3	−1.15644	−0.578218	+	0.815883i	$0.696252\pi$
$884$	32251.3	1.22707
$885$	−11181.5	−0.424704
$886$	−5755.97	−0.218257
$887$	−37679.6	−1.42633	−0.713166	−	0.700996i	$-0.752739\pi$
$887$	−37679.6	−1.42633	−0.713166	+	0.700996i	$0.752739\pi$
$888$	888.000	0.0335578
$889$	54117.7	2.04168
$890$	2805.94	0.105680
$891$	−680.817	−0.0255985
$892$	17931.3	0.673075
$893$	−35204.2	−1.31922
$894$	−10245.0	−0.383271
$895$	7787.15	0.290833
$896$	2691.63	0.100358
$897$	19483.7	0.725242
$898$	15343.3	0.570169
$899$	−10925.5	−0.405324
$900$	900.000	0.0333333
$901$	53507.0	1.97844
$902$	−2433.97	−0.0898475
$903$	7968.53	0.293661
$904$	7704.56	0.283462
$905$	−8188.30	−0.300761
$906$	−15332.7	−0.562246
$907$	−20439.8	−0.748283	−0.374141	−	0.927372i	$-0.622063\pi$
$907$	−20439.8	−0.748283	−0.374141	+	0.927372i	$0.622063\pi$
$908$	−19147.2	−0.699805
$909$	−4058.20	−0.148077
$910$	12507.7	0.455632
$911$	12061.4	0.438653	0.219326	−	0.975652i	$-0.429614\pi$
$911$	12061.4	0.438653	0.219326	+	0.975652i	$0.429614\pi$
$912$	−7045.30	−0.255804
$913$	8283.66	0.300273
$914$	−15243.7	−0.551661
$915$	−12272.7	−0.443413
$916$	21379.2	0.771165
$917$	62046.7	2.23442
$918$	−7319.99	−0.263176
$919$	10546.8	0.378572	0.189286	−	0.981922i	$-0.439383\pi$
$919$	10546.8	0.378572	0.189286	+	0.981922i	$0.439383\pi$
$920$	4367.57	0.156516
$921$	−11813.3	−0.422652
$922$	36655.3	1.30930
$923$	−43462.4	−1.54993
$924$	−2120.96	−0.0755135
$925$	925.000	0.0328798
$926$	−26435.6	−0.938152
$927$	−9229.66	−0.327014
$928$	1203.36	0.0425669
$929$	−21294.6	−0.752047	−0.376024	−	0.926610i	$-0.622709\pi$
$929$	−21294.6	−0.752047	−0.376024	+	0.926610i	$0.622709\pi$
$930$	8716.03	0.307322
$931$	−14559.2	−0.512522
$932$	3865.90	0.135871
$933$	−16505.9	−0.579185
$934$	−19334.5	−0.677350
$935$	−5696.81	−0.199257
$936$	−4282.55	−0.149551
$937$	−36614.8	−1.27658	−0.638289	−	0.769797i	$-0.720358\pi$
$937$	−36614.8	−1.27658	−0.638289	+	0.769797i	$0.720358\pi$
$938$	−12275.5	−0.427301
$939$	−26020.8	−0.904319
$940$	−4796.96	−0.166446
$941$	−8302.14	−0.287611	−0.143806	−	0.989606i	$-0.545934\pi$
$941$	−8302.14	−0.287611	−0.143806	+	0.989606i	$0.545934\pi$
$942$	−19873.0	−0.687365
$943$	−15809.6	−0.545950
$944$	11927.0	0.411218
$945$	−2838.83	−0.0977219
$946$	−2123.37	−0.0729777
$947$	−15699.7	−0.538724	−0.269362	−	0.963039i	$-0.586813\pi$
$947$	−15699.7	−0.538724	−0.269362	+	0.963039i	$0.586813\pi$
$948$	−7667.32	−0.262682
$949$	−8978.57	−0.307120
$950$	−7338.85	−0.250636
$951$	197.062	0.00671943
$952$	−22804.1	−0.776349
$953$	−11503.5	−0.391014	−0.195507	−	0.980702i	$-0.562635\pi$
$953$	−11503.5	−0.391014	−0.195507	+	0.980702i	$0.562635\pi$
$954$	−7105.05	−0.241126
$955$	−9711.17	−0.329054
$956$	−17673.2	−0.597899
$957$	−948.224	−0.0320290
$958$	22110.1	0.745663
$959$	−18408.7	−0.619863
$960$	−960.000	−0.0322749
$961$	54619.1	1.83341
$962$	−4401.51	−0.147516
$963$	12751.6	0.426703
$964$	−12953.2	−0.432775
$965$	−21599.7	−0.720539
$966$	−13776.4	−0.458851
$967$	−42092.1	−1.39978	−0.699892	−	0.714249i	$-0.746768\pi$
$967$	−42092.1	−1.39978	−0.699892	+	0.714249i	$0.746768\pi$
$968$	−10082.8	−0.334788
$969$	59689.2	1.97884
$970$	13903.6	0.460224
$971$	−13220.7	−0.436945	−0.218473	−	0.975843i	$-0.570107\pi$
$971$	−13220.7	−0.436945	−0.218473	+	0.975843i	$0.570107\pi$
$972$	972.000	0.0320750
$973$	−7389.57	−0.243472
$974$	13267.2	0.436457
$975$	−4460.99	−0.146529
$976$	13090.9	0.429333
$977$	58165.7	1.90469	0.952346	−	0.305020i	$-0.0986631\pi$
$977$	58165.7	1.90469	0.952346	+	0.305020i	$0.0986631\pi$
$978$	2900.40	0.0948309
$979$	2358.44	0.0769928
$980$	−1983.85	−0.0646651
$981$	−10647.1	−0.346521
$982$	10695.3	0.347558
$983$	−38060.6	−1.23494	−0.617470	−	0.786595i	$-0.711842\pi$
$983$	−38060.6	−1.23494	−0.617470	+	0.786595i	$0.711842\pi$
$984$	3474.98	0.112579
$985$	−6961.89	−0.225202
$986$	−10195.1	−0.329288
$987$	15130.8	0.487963
$988$	34921.1	1.12448
$989$	−13792.1	−0.443442
$990$	756.463	0.0242848
$991$	−18530.2	−0.593978	−0.296989	−	0.954881i	$-0.595982\pi$
$991$	−18530.2	−0.593978	−0.296989	+	0.954881i	$0.595982\pi$
$992$	−9297.09	−0.297564
$993$	3008.48	0.0961442
$994$	30731.2	0.980617
$995$	−21490.7	−0.684726
$996$	−11826.5	−0.376243
$997$	24540.0	0.779527	0.389763	−	0.920915i	$-0.372557\pi$
$997$	24540.0	0.779527	0.389763	+	0.920915i	$0.372557\pi$
$998$	30322.8	0.961776
$999$	999.000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.4.a.l.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.4.a.l.1.4	✓	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} +21.0284 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} +21.0284 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -8.40515 q^{11} +12.0000 q^{12} -59.4799 q^{13} +42.0567 q^{14} -15.0000 q^{15} +16.0000 q^{16} -135.555 q^{17} +18.0000 q^{18} -146.777 q^{19} -20.0000 q^{20} +63.0851 q^{21} -16.8103 q^{22} -109.189 q^{23} +24.0000 q^{24} +25.0000 q^{25} -118.960 q^{26} +27.0000 q^{27} +84.1135 q^{28} +37.6049 q^{29} -30.0000 q^{30} -290.534 q^{31} +32.0000 q^{32} -25.2154 q^{33} -271.111 q^{34} -105.142 q^{35} +36.0000 q^{36} +37.0000 q^{37} -293.554 q^{38} -178.440 q^{39} -40.0000 q^{40} +144.791 q^{41} +126.170 q^{42} +126.314 q^{43} -33.6206 q^{44} -45.0000 q^{45} -218.379 q^{46} +239.848 q^{47} +48.0000 q^{48} +99.1925 q^{49} +50.0000 q^{50} -406.666 q^{51} -237.920 q^{52} -394.725 q^{53} +54.0000 q^{54} +42.0257 q^{55} +168.227 q^{56} -440.331 q^{57} +75.2098 q^{58} +745.436 q^{59} -60.0000 q^{60} +818.180 q^{61} -581.068 q^{62} +189.255 q^{63} +64.0000 q^{64} +297.399 q^{65} -50.4309 q^{66} -291.878 q^{67} -542.221 q^{68} -327.568 q^{69} -210.284 q^{70} +730.707 q^{71} +72.0000 q^{72} +150.951 q^{73} +74.0000 q^{74} +75.0000 q^{75} -587.108 q^{76} -176.747 q^{77} -356.879 q^{78} -638.943 q^{79} -80.0000 q^{80} +81.0000 q^{81} +289.581 q^{82} -985.546 q^{83} +252.340 q^{84} +677.777 q^{85} +252.628 q^{86} +112.815 q^{87} -67.2412 q^{88} -280.594 q^{89} -90.0000 q^{90} -1250.77 q^{91} -436.757 q^{92} -871.603 q^{93} +479.696 q^{94} +733.885 q^{95} +96.0000 q^{96} -1390.36 q^{97} +198.385 q^{98} -75.6463 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 20 q^{5} + 24 q^{6} - 23 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 20 * q^5 + 24 * q^6 - 23 * q^7 + 32 * q^8 + 36 * q^9	\( 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 20 q^{5} + 24 q^{6} - 23 q^{7} + 32 q^{8} + 36 q^{9} - 40 q^{10} - 6 q^{11} + 48 q^{12} - 59 q^{13} - 46 q^{14} - 60 q^{15} + 64 q^{16} - 172 q^{17} + 72 q^{18} - 256 q^{19} - 80 q^{20} - 69 q^{21} - 12 q^{22} - 103 q^{23} + 96 q^{24} + 100 q^{25} - 118 q^{26} + 108 q^{27} - 92 q^{28} - 345 q^{29} - 120 q^{30} - 686 q^{31} + 128 q^{32} - 18 q^{33} - 344 q^{34} + 115 q^{35} + 144 q^{36} + 148 q^{37} - 512 q^{38} - 177 q^{39} - 160 q^{40} - 672 q^{41} - 138 q^{42} - 421 q^{43} - 24 q^{44} - 180 q^{45} - 206 q^{46} - 344 q^{47} + 192 q^{48} + 271 q^{49} + 200 q^{50} - 516 q^{51} - 236 q^{52} + 63 q^{53} + 216 q^{54} + 30 q^{55} - 184 q^{56} - 768 q^{57} - 690 q^{58} + 121 q^{59} - 240 q^{60} - 390 q^{61} - 1372 q^{62} - 207 q^{63} + 256 q^{64} + 295 q^{65} - 36 q^{66} - 845 q^{67} - 688 q^{68} - 309 q^{69} + 230 q^{70} - 53 q^{71} + 288 q^{72} + 516 q^{73} + 296 q^{74} + 300 q^{75} - 1024 q^{76} - 1954 q^{77} - 354 q^{78} - 235 q^{79} - 320 q^{80} + 324 q^{81} - 1344 q^{82} - 1360 q^{83} - 276 q^{84} + 860 q^{85} - 842 q^{86} - 1035 q^{87} - 48 q^{88} - 2529 q^{89} - 360 q^{90} - 2663 q^{91} - 412 q^{92} - 2058 q^{93} - 688 q^{94} + 1280 q^{95} + 384 q^{96} - 3438 q^{97} + 542 q^{98} - 54 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 20 * q^5 + 24 * q^6 - 23 * q^7 + 32 * q^8 + 36 * q^9 - 40 * q^10 - 6 * q^11 + 48 * q^12 - 59 * q^13 - 46 * q^14 - 60 * q^15 + 64 * q^16 - 172 * q^17 + 72 * q^18 - 256 * q^19 - 80 * q^20 - 69 * q^21 - 12 * q^22 - 103 * q^23 + 96 * q^24 + 100 * q^25 - 118 * q^26 + 108 * q^27 - 92 * q^28 - 345 * q^29 - 120 * q^30 - 686 * q^31 + 128 * q^32 - 18 * q^33 - 344 * q^34 + 115 * q^35 + 144 * q^36 + 148 * q^37 - 512 * q^38 - 177 * q^39 - 160 * q^40 - 672 * q^41 - 138 * q^42 - 421 * q^43 - 24 * q^44 - 180 * q^45 - 206 * q^46 - 344 * q^47 + 192 * q^48 + 271 * q^49 + 200 * q^50 - 516 * q^51 - 236 * q^52 + 63 * q^53 + 216 * q^54 + 30 * q^55 - 184 * q^56 - 768 * q^57 - 690 * q^58 + 121 * q^59 - 240 * q^60 - 390 * q^61 - 1372 * q^62 - 207 * q^63 + 256 * q^64 + 295 * q^65 - 36 * q^66 - 845 * q^67 - 688 * q^68 - 309 * q^69 + 230 * q^70 - 53 * q^71 + 288 * q^72 + 516 * q^73 + 296 * q^74 + 300 * q^75 - 1024 * q^76 - 1954 * q^77 - 354 * q^78 - 235 * q^79 - 320 * q^80 + 324 * q^81 - 1344 * q^82 - 1360 * q^83 - 276 * q^84 + 860 * q^85 - 842 * q^86 - 1035 * q^87 - 48 * q^88 - 2529 * q^89 - 360 * q^90 - 2663 * q^91 - 412 * q^92 - 2058 * q^93 - 688 * q^94 + 1280 * q^95 + 384 * q^96 - 3438 * q^97 + 542 * q^98 - 54 * q^99

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