LMFDB - Embedded newform 930.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,4,Mod(1,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, 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("930.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	930.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:	$54.8717763053$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 128x^{2} + 114x + 3030 $ x^4 - x^3 - 128x^2 + 114x + 3030
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			1.1
Root			$-5.01222$ of defining polynomial
Character	$\chi$	$=$	930.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -28.5770 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -28.5770 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} +27.3129 q^{11} -12.0000 q^{12} -59.9143 q^{13} -57.1541 q^{14} +15.0000 q^{15} +16.0000 q^{16} -116.792 q^{17} +18.0000 q^{18} +64.4914 q^{19} -20.0000 q^{20} +85.7311 q^{21} +54.6257 q^{22} +78.4474 q^{23} -24.0000 q^{24} +25.0000 q^{25} -119.829 q^{26} -27.0000 q^{27} -114.308 q^{28} +42.7313 q^{29} +30.0000 q^{30} +31.0000 q^{31} +32.0000 q^{32} -81.9386 q^{33} -233.584 q^{34} +142.885 q^{35} +36.0000 q^{36} -80.4817 q^{37} +128.983 q^{38} +179.743 q^{39} -40.0000 q^{40} +164.284 q^{41} +171.462 q^{42} +515.943 q^{43} +109.251 q^{44} -45.0000 q^{45} +156.895 q^{46} -397.173 q^{47} -48.0000 q^{48} +473.647 q^{49} +50.0000 q^{50} +350.377 q^{51} -239.657 q^{52} -458.662 q^{53} -54.0000 q^{54} -136.564 q^{55} -228.616 q^{56} -193.474 q^{57} +85.4626 q^{58} +80.8381 q^{59} +60.0000 q^{60} +376.298 q^{61} +62.0000 q^{62} -257.193 q^{63} +64.0000 q^{64} +299.572 q^{65} -163.877 q^{66} +439.949 q^{67} -467.169 q^{68} -235.342 q^{69} +285.770 q^{70} +456.029 q^{71} +72.0000 q^{72} +208.833 q^{73} -160.963 q^{74} -75.0000 q^{75} +257.966 q^{76} -780.521 q^{77} +359.486 q^{78} +902.789 q^{79} -80.0000 q^{80} +81.0000 q^{81} +328.568 q^{82} +604.939 q^{83} +342.925 q^{84} +583.961 q^{85} +1031.89 q^{86} -128.194 q^{87} +218.503 q^{88} +1354.95 q^{89} -90.0000 q^{90} +1712.17 q^{91} +313.790 q^{92} -93.0000 q^{93} -794.347 q^{94} -322.457 q^{95} -96.0000 q^{96} +1396.74 q^{97} +947.295 q^{98} +245.816 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} - 20 q^{5} - 24 q^{6} - 3 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 - 3 * q^7 + 32 * q^8 + 36 * q^9	\( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} - 20 q^{5} - 24 q^{6} - 3 q^{7} + 32 q^{8} + 36 q^{9} - 40 q^{10} + 43 q^{11} - 48 q^{12} - 20 q^{13} - 6 q^{14} + 60 q^{15} + 64 q^{16} - 28 q^{17} + 72 q^{18} - 73 q^{19} - 80 q^{20} + 9 q^{21} + 86 q^{22} - 47 q^{23} - 96 q^{24} + 100 q^{25} - 40 q^{26} - 108 q^{27} - 12 q^{28} + 204 q^{29} + 120 q^{30} + 124 q^{31} + 128 q^{32} - 129 q^{33} - 56 q^{34} + 15 q^{35} + 144 q^{36} - 156 q^{37} - 146 q^{38} + 60 q^{39} - 160 q^{40} + 128 q^{41} + 18 q^{42} + 685 q^{43} + 172 q^{44} - 180 q^{45} - 94 q^{46} + 156 q^{47} - 192 q^{48} + 877 q^{49} + 200 q^{50} + 84 q^{51} - 80 q^{52} + 591 q^{53} - 216 q^{54} - 215 q^{55} - 24 q^{56} + 219 q^{57} + 408 q^{58} + 212 q^{59} + 240 q^{60} + 1288 q^{61} + 248 q^{62} - 27 q^{63} + 256 q^{64} + 100 q^{65} - 258 q^{66} + 1270 q^{67} - 112 q^{68} + 141 q^{69} + 30 q^{70} + 2465 q^{71} + 288 q^{72} - 359 q^{73} - 312 q^{74} - 300 q^{75} - 292 q^{76} + 1871 q^{77} + 120 q^{78} + 2053 q^{79} - 320 q^{80} + 324 q^{81} + 256 q^{82} + 994 q^{83} + 36 q^{84} + 140 q^{85} + 1370 q^{86} - 612 q^{87} + 344 q^{88} + 1055 q^{89} - 360 q^{90} + 386 q^{91} - 188 q^{92} - 372 q^{93} + 312 q^{94} + 365 q^{95} - 384 q^{96} + 2962 q^{97} + 1754 q^{98} + 387 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 - 12 * q^3 + 16 * q^4 - 20 * q^5 - 24 * q^6 - 3 * q^7 + 32 * q^8 + 36 * q^9 - 40 * q^10 + 43 * q^11 - 48 * q^12 - 20 * q^13 - 6 * q^14 + 60 * q^15 + 64 * q^16 - 28 * q^17 + 72 * q^18 - 73 * q^19 - 80 * q^20 + 9 * q^21 + 86 * q^22 - 47 * q^23 - 96 * q^24 + 100 * q^25 - 40 * q^26 - 108 * q^27 - 12 * q^28 + 204 * q^29 + 120 * q^30 + 124 * q^31 + 128 * q^32 - 129 * q^33 - 56 * q^34 + 15 * q^35 + 144 * q^36 - 156 * q^37 - 146 * q^38 + 60 * q^39 - 160 * q^40 + 128 * q^41 + 18 * q^42 + 685 * q^43 + 172 * q^44 - 180 * q^45 - 94 * q^46 + 156 * q^47 - 192 * q^48 + 877 * q^49 + 200 * q^50 + 84 * q^51 - 80 * q^52 + 591 * q^53 - 216 * q^54 - 215 * q^55 - 24 * q^56 + 219 * q^57 + 408 * q^58 + 212 * q^59 + 240 * q^60 + 1288 * q^61 + 248 * q^62 - 27 * q^63 + 256 * q^64 + 100 * q^65 - 258 * q^66 + 1270 * q^67 - 112 * q^68 + 141 * q^69 + 30 * q^70 + 2465 * q^71 + 288 * q^72 - 359 * q^73 - 312 * q^74 - 300 * q^75 - 292 * q^76 + 1871 * q^77 + 120 * q^78 + 2053 * q^79 - 320 * q^80 + 324 * q^81 + 256 * q^82 + 994 * q^83 + 36 * q^84 + 140 * q^85 + 1370 * q^86 - 612 * q^87 + 344 * q^88 + 1055 * q^89 - 360 * q^90 + 386 * q^91 - 188 * q^92 - 372 * q^93 + 312 * q^94 + 365 * q^95 - 384 * q^96 + 2962 * q^97 + 1754 * q^98 + 387 * 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
$7$	−28.5770	−1.54302	−0.771508	−	0.636220i	$-0.780497\pi$
$7$	−28.5770	−1.54302	−0.771508	+	0.636220i	$0.780497\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	27.3129	0.748649	0.374324	−	0.927298i	$-0.377875\pi$
$11$	27.3129	0.748649	0.374324	+	0.927298i	$0.377875\pi$
$12$	−12.0000	−0.288675
$13$	−59.9143	−1.27825	−0.639125	−	0.769103i	$-0.720703\pi$
$13$	−59.9143	−1.27825	−0.639125	+	0.769103i	$0.720703\pi$
$14$	−57.1541	−1.09108
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	−116.792	−1.66625	−0.833126	−	0.553084i	$-0.813451\pi$
$17$	−116.792	−1.66625	−0.833126	+	0.553084i	$0.813451\pi$
$18$	18.0000	0.235702
$19$	64.4914	0.778702	0.389351	−	0.921089i	$-0.372699\pi$
$19$	64.4914	0.778702	0.389351	+	0.921089i	$0.372699\pi$
$20$	−20.0000	−0.223607
$21$	85.7311	0.890860
$22$	54.6257	0.529375
$23$	78.4474	0.711192	0.355596	−	0.934640i	$-0.384278\pi$
$23$	78.4474	0.711192	0.355596	+	0.934640i	$0.384278\pi$
$24$	−24.0000	−0.204124
$25$	25.0000	0.200000
$26$	−119.829	−0.903859
$27$	−27.0000	−0.192450
$28$	−114.308	−0.771508
$29$	42.7313	0.273621	0.136810	−	0.990597i	$-0.456315\pi$
$29$	42.7313	0.273621	0.136810	+	0.990597i	$0.456315\pi$
$30$	30.0000	0.182574
$31$	31.0000	0.179605
$31$	31.0000	0.179605
$32$	32.0000	0.176777
$33$	−81.9386	−0.432233
$34$	−233.584	−1.17822
$35$	142.885	0.690057
$36$	36.0000	0.166667
$37$	−80.4817	−0.357598	−0.178799	−	0.983886i	$-0.557221\pi$
$37$	−80.4817	−0.357598	−0.178799	+	0.983886i	$0.557221\pi$
$38$	128.983	0.550625
$39$	179.743	0.737998
$40$	−40.0000	−0.158114
$41$	164.284	0.625777	0.312889	−	0.949790i	$-0.398703\pi$
$41$	164.284	0.625777	0.312889	+	0.949790i	$0.398703\pi$
$42$	171.462	0.629933
$43$	515.943	1.82978	0.914890	−	0.403702i	$-0.132277\pi$
$43$	515.943	1.82978	0.914890	+	0.403702i	$0.132277\pi$
$44$	109.251	0.374324
$45$	−45.0000	−0.149071
$46$	156.895	0.502889
$47$	−397.173	−1.23263	−0.616316	−	0.787499i	$-0.711376\pi$
$47$	−397.173	−1.23263	−0.616316	+	0.787499i	$0.711376\pi$
$48$	−48.0000	−0.144338
$49$	473.647	1.38090
$50$	50.0000	0.141421
$51$	350.377	0.962011
$52$	−239.657	−0.639125
$53$	−458.662	−1.18872	−0.594359	−	0.804200i	$-0.702594\pi$
$53$	−458.662	−1.18872	−0.594359	+	0.804200i	$0.702594\pi$
$54$	−54.0000	−0.136083
$55$	−136.564	−0.334806
$56$	−228.616	−0.545538
$57$	−193.474	−0.449584
$58$	85.4626	0.193479
$59$	80.8381	0.178377	0.0891883	−	0.996015i	$-0.471573\pi$
$59$	80.8381	0.178377	0.0891883	+	0.996015i	$0.471573\pi$
$60$	60.0000	0.129099
$61$	376.298	0.789837	0.394919	−	0.918716i	$-0.370773\pi$
$61$	376.298	0.789837	0.394919	+	0.918716i	$0.370773\pi$
$62$	62.0000	0.127000
$63$	−257.193	−0.514338
$64$	64.0000	0.125000
$65$	299.572	0.571651
$66$	−163.877	−0.305635
$67$	439.949	0.802213	0.401107	−	0.916031i	$-0.368626\pi$
$67$	439.949	0.802213	0.401107	+	0.916031i	$0.368626\pi$
$68$	−467.169	−0.833126
$69$	−235.342	−0.410607
$70$	285.770	0.487944
$71$	456.029	0.762264	0.381132	−	0.924521i	$-0.375534\pi$
$71$	456.029	0.762264	0.381132	+	0.924521i	$0.375534\pi$
$72$	72.0000	0.117851
$73$	208.833	0.334822	0.167411	−	0.985887i	$-0.446459\pi$
$73$	208.833	0.334822	0.167411	+	0.985887i	$0.446459\pi$
$74$	−160.963	−0.252860
$75$	−75.0000	−0.115470
$76$	257.966	0.389351
$77$	−780.521	−1.15518
$78$	359.486	0.521843
$79$	902.789	1.28572	0.642859	−	0.765985i	$-0.277748\pi$
$79$	902.789	1.28572	0.642859	+	0.765985i	$0.277748\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	328.568	0.442491
$83$	604.939	0.800008	0.400004	−	0.916513i	$-0.369009\pi$
$83$	604.939	0.800008	0.400004	+	0.916513i	$0.369009\pi$
$84$	342.925	0.445430
$85$	583.961	0.745170
$86$	1031.89	1.29385
$87$	−128.194	−0.157975
$88$	218.503	0.264687
$89$	1354.95	1.61376	0.806880	−	0.590715i	$-0.201154\pi$
$89$	1354.95	1.61376	0.806880	+	0.590715i	$0.201154\pi$
$90$	−90.0000	−0.105409
$91$	1712.17	1.97236
$92$	313.790	0.355596
$93$	−93.0000	−0.103695
$94$	−794.347	−0.871602
$95$	−322.457	−0.348246
$96$	−96.0000	−0.102062
$97$	1396.74	1.46203	0.731017	−	0.682359i	$-0.239046\pi$
$97$	1396.74	1.46203	0.731017	+	0.682359i	$0.239046\pi$
$98$	947.295	0.976441
$99$	245.816	0.249550
$100$	100.000	0.100000
$101$	604.706	0.595748	0.297874	−	0.954605i	$-0.403722\pi$
$101$	604.706	0.595748	0.297874	+	0.954605i	$0.403722\pi$
$102$	700.753	0.680244
$103$	−1519.61	−1.45370	−0.726852	−	0.686794i	$-0.759017\pi$
$103$	−1519.61	−1.45370	−0.726852	+	0.686794i	$0.759017\pi$
$104$	−479.315	−0.451930
$105$	−428.656	−0.398405
$106$	−917.323	−0.840550
$107$	−1345.86	−1.21598	−0.607989	−	0.793945i	$-0.708024\pi$
$107$	−1345.86	−1.21598	−0.607989	+	0.793945i	$0.708024\pi$
$108$	−108.000	−0.0962250
$109$	−1826.86	−1.60534	−0.802668	−	0.596426i	$-0.796587\pi$
$109$	−1826.86	−1.60534	−0.802668	+	0.596426i	$0.796587\pi$
$110$	−273.129	−0.236744
$111$	241.445	0.206459
$112$	−457.233	−0.385754
$113$	1769.44	1.47305	0.736525	−	0.676410i	$-0.236465\pi$
$113$	1769.44	1.47305	0.736525	+	0.676410i	$0.236465\pi$
$114$	−386.948	−0.317904
$115$	−392.237	−0.318055
$116$	170.925	0.136810
$117$	−539.229	−0.426083
$118$	161.676	0.126131
$119$	3337.58	2.57105
$120$	120.000	0.0912871
$121$	−585.008	−0.439525
$122$	752.597	0.558499
$123$	−492.852	−0.361293
$124$	124.000	0.0898027
$125$	−125.000	−0.0894427
$126$	−514.387	−0.363692
$127$	619.182	0.432626	0.216313	−	0.976324i	$-0.430597\pi$
$127$	619.182	0.432626	0.216313	+	0.976324i	$0.430597\pi$
$128$	128.000	0.0883883
$129$	−1547.83	−1.05642
$130$	599.143	0.404218
$131$	−1227.42	−0.818630	−0.409315	−	0.912393i	$-0.634232\pi$
$131$	−1227.42	−0.818630	−0.409315	+	0.912393i	$0.634232\pi$
$132$	−327.754	−0.216116
$133$	−1842.97	−1.20155
$134$	879.897	0.567250
$135$	135.000	0.0860663
$136$	−934.338	−0.589109
$137$	2485.47	1.54999	0.774993	−	0.631970i	$-0.217754\pi$
$137$	2485.47	1.54999	0.774993	+	0.631970i	$0.217754\pi$
$138$	−470.685	−0.290343
$139$	−2424.62	−1.47952	−0.739762	−	0.672868i	$-0.765062\pi$
$139$	−2424.62	−1.47952	−0.739762	+	0.672868i	$0.765062\pi$
$140$	571.541	0.345029
$141$	1191.52	0.711660
$142$	912.059	0.539002
$143$	−1636.43	−0.956960
$144$	144.000	0.0833333
$145$	−213.656	−0.122367
$146$	417.666	0.236755
$147$	−1420.94	−0.797261
$148$	−321.927	−0.178799
$149$	1416.34	0.778733	0.389367	−	0.921083i	$-0.372694\pi$
$149$	1416.34	0.778733	0.389367	+	0.921083i	$0.372694\pi$
$150$	−150.000	−0.0816497
$151$	−1298.43	−0.699768	−0.349884	−	0.936793i	$-0.613779\pi$
$151$	−1298.43	−0.699768	−0.349884	+	0.936793i	$0.613779\pi$
$152$	515.931	0.275313
$153$	−1051.13	−0.555417
$154$	−1561.04	−0.816833
$155$	−155.000	−0.0803219
$156$	718.972	0.368999
$157$	2804.15	1.42545	0.712726	−	0.701443i	$-0.247460\pi$
$157$	2804.15	1.42545	0.712726	+	0.701443i	$0.247460\pi$
$158$	1805.58	0.909140
$159$	1375.99	0.686306
$160$	−160.000	−0.0790569
$161$	−2241.80	−1.09738
$162$	162.000	0.0785674
$163$	1236.69	0.594264	0.297132	−	0.954836i	$-0.403970\pi$
$163$	1236.69	0.594264	0.297132	+	0.954836i	$0.403970\pi$
$164$	657.136	0.312889
$165$	409.693	0.193300
$166$	1209.88	0.565691
$167$	−1443.53	−0.668885	−0.334443	−	0.942416i	$-0.608548\pi$
$167$	−1443.53	−0.668885	−0.334443	+	0.942416i	$0.608548\pi$
$168$	685.849	0.314967
$169$	1392.73	0.633922
$170$	1167.92	0.526915
$171$	580.422	0.259567
$172$	2063.77	0.914890
$173$	−2179.44	−0.957801	−0.478900	−	0.877869i	$-0.658965\pi$
$173$	−2179.44	−0.957801	−0.478900	+	0.877869i	$0.658965\pi$
$174$	−256.388	−0.111705
$175$	−714.426	−0.308603
$176$	437.006	0.187162
$177$	−242.514	−0.102986
$178$	2709.90	1.14110
$179$	17.3214	0.00723274	0.00361637	−	0.999993i	$-0.498849\pi$
$179$	17.3214	0.00723274	0.00361637	+	0.999993i	$0.498849\pi$
$180$	−180.000	−0.0745356
$181$	2072.44	0.851069	0.425535	−	0.904942i	$-0.360086\pi$
$181$	2072.44	0.851069	0.425535	+	0.904942i	$0.360086\pi$
$182$	3424.35	1.39467
$183$	−1128.89	−0.456013
$184$	627.579	0.251444
$185$	402.409	0.159923
$186$	−186.000	−0.0733236
$187$	−3189.93	−1.24744
$188$	−1588.69	−0.616316
$189$	771.580	0.296953
$190$	−644.914	−0.246247
$191$	2613.63	0.990135	0.495067	−	0.868855i	$-0.335143\pi$
$191$	2613.63	0.990135	0.495067	+	0.868855i	$0.335143\pi$
$192$	−192.000	−0.0721688
$193$	3214.77	1.19899	0.599494	−	0.800380i	$-0.295369\pi$
$193$	3214.77	1.19899	0.599494	+	0.800380i	$0.295369\pi$
$194$	2793.48	1.03381
$195$	−898.715	−0.330043
$196$	1894.59	0.690448
$197$	2487.36	0.899579	0.449789	−	0.893135i	$-0.351499\pi$
$197$	2487.36	0.899579	0.449789	+	0.893135i	$0.351499\pi$
$198$	491.631	0.176458
$199$	2207.68	0.786423	0.393212	−	0.919448i	$-0.371364\pi$
$199$	2207.68	0.786423	0.393212	+	0.919448i	$0.371364\pi$
$200$	200.000	0.0707107
$201$	−1319.85	−0.463158
$202$	1209.41	0.421257
$203$	−1221.13	−0.422201
$204$	1401.51	0.481005
$205$	−821.421	−0.279856
$206$	−3039.22	−1.02792
$207$	706.027	0.237064
$208$	−958.629	−0.319562
$209$	1761.44	0.582974
$210$	−857.311	−0.281715
$211$	−621.313	−0.202715	−0.101358	−	0.994850i	$-0.532319\pi$
$211$	−621.313	−0.202715	−0.101358	+	0.994850i	$0.532319\pi$
$212$	−1834.65	−0.594359
$213$	−1368.09	−0.440093
$214$	−2691.73	−0.859826
$215$	−2579.72	−0.818303
$216$	−216.000	−0.0680414
$217$	−885.888	−0.277134
$218$	−3653.73	−1.13514
$219$	−626.499	−0.193310
$220$	−546.257	−0.167403
$221$	6997.53	2.12989
$222$	482.890	0.145989
$223$	−670.359	−0.201303	−0.100652	−	0.994922i	$-0.532093\pi$
$223$	−670.359	−0.201303	−0.100652	+	0.994922i	$0.532093\pi$
$224$	−914.465	−0.272769
$225$	225.000	0.0666667
$226$	3538.88	1.04160
$227$	−161.980	−0.0473613	−0.0236807	−	0.999720i	$-0.507538\pi$
$227$	−161.980	−0.0473613	−0.0236807	+	0.999720i	$0.507538\pi$
$228$	−773.897	−0.224792
$229$	−1315.79	−0.379695	−0.189847	−	0.981814i	$-0.560799\pi$
$229$	−1315.79	−0.379695	−0.189847	+	0.981814i	$0.560799\pi$
$230$	−784.474	−0.224899
$231$	2341.56	0.666941
$232$	341.850	0.0967395
$233$	3489.29	0.981076	0.490538	−	0.871420i	$-0.336800\pi$
$233$	3489.29	0.981076	0.490538	+	0.871420i	$0.336800\pi$
$234$	−1078.46	−0.301286
$235$	1985.87	0.551250
$236$	323.352	0.0891883
$237$	−2708.37	−0.742309
$238$	6675.15	1.81801
$239$	−1014.84	−0.274664	−0.137332	−	0.990525i	$-0.543853\pi$
$239$	−1014.84	−0.274664	−0.137332	+	0.990525i	$0.543853\pi$
$240$	240.000	0.0645497
$241$	−6323.14	−1.69008	−0.845040	−	0.534703i	$-0.820424\pi$
$241$	−6323.14	−1.69008	−0.845040	+	0.534703i	$0.820424\pi$
$242$	−1170.02	−0.310791
$243$	−243.000	−0.0641500
$244$	1505.19	0.394919
$245$	−2368.24	−0.617555
$246$	−985.705	−0.255473
$247$	−3863.96	−0.995375
$248$	248.000	0.0635001
$249$	−1814.82	−0.461885
$250$	−250.000	−0.0632456
$251$	−5904.09	−1.48471	−0.742356	−	0.670006i	$-0.766292\pi$
$251$	−5904.09	−1.48471	−0.742356	+	0.670006i	$0.766292\pi$
$252$	−1028.77	−0.257169
$253$	2142.62	0.532433
$254$	1238.36	0.305913
$255$	−1751.88	−0.430224
$256$	256.000	0.0625000
$257$	8180.37	1.98552	0.992758	−	0.120135i	$-0.0383327\pi$
$257$	8180.37	1.98552	0.992758	+	0.120135i	$0.0383327\pi$
$258$	−3095.66	−0.747005
$259$	2299.93	0.551779
$260$	1198.29	0.285825
$261$	384.581	0.0912069
$262$	−2454.85	−0.578859
$263$	−405.868	−0.0951592	−0.0475796	−	0.998867i	$-0.515151\pi$
$263$	−405.868	−0.0951592	−0.0475796	+	0.998867i	$0.515151\pi$
$264$	−655.509	−0.152817
$265$	2293.31	0.531611
$266$	−3685.95	−0.849623
$267$	−4064.86	−0.931705
$268$	1759.79	0.401107
$269$	4477.28	1.01481	0.507406	−	0.861707i	$-0.330604\pi$
$269$	4477.28	1.01481	0.507406	+	0.861707i	$0.330604\pi$
$270$	270.000	0.0608581
$271$	−7052.71	−1.58089	−0.790446	−	0.612532i	$-0.790151\pi$
$271$	−7052.71	−1.58089	−0.790446	+	0.612532i	$0.790151\pi$
$272$	−1868.68	−0.416563
$273$	−5136.52	−1.13874
$274$	4970.94	1.09600
$275$	682.821	0.149730
$276$	−941.369	−0.205304
$277$	3133.55	0.679699	0.339849	−	0.940480i	$-0.389624\pi$
$277$	3133.55	0.679699	0.339849	+	0.940480i	$0.389624\pi$
$278$	−4849.25	−1.04618
$279$	279.000	0.0598684
$280$	1143.08	0.243972
$281$	−3596.38	−0.763495	−0.381748	−	0.924267i	$-0.624678\pi$
$281$	−3596.38	−0.763495	−0.381748	+	0.924267i	$0.624678\pi$
$282$	2383.04	0.503220
$283$	4130.11	0.867525	0.433762	−	0.901027i	$-0.357186\pi$
$283$	4130.11	0.867525	0.433762	+	0.901027i	$0.357186\pi$
$284$	1824.12	0.381132
$285$	967.371	0.201060
$286$	−3272.86	−0.676673
$287$	−4694.75	−0.965584
$288$	288.000	0.0589256
$289$	8727.42	1.77639
$290$	−427.313	−0.0865264
$291$	−4190.22	−0.844106
$292$	835.331	0.167411
$293$	1628.40	0.324683	0.162342	−	0.986735i	$-0.448095\pi$
$293$	1628.40	0.324683	0.162342	+	0.986735i	$0.448095\pi$
$294$	−2841.88	−0.563748
$295$	−404.190	−0.0797725
$296$	−643.854	−0.126430
$297$	−737.447	−0.144078
$298$	2832.68	0.550648
$299$	−4700.13	−0.909081
$300$	−300.000	−0.0577350
$301$	−14744.1	−2.82338
$302$	−2596.87	−0.494811
$303$	−1814.12	−0.343955
$304$	1031.86	0.194675
$305$	−1881.49	−0.353226
$306$	−2102.26	−0.392739
$307$	−5664.16	−1.05300	−0.526500	−	0.850175i	$-0.676496\pi$
$307$	−5664.16	−1.05300	−0.526500	+	0.850175i	$0.676496\pi$
$308$	−3122.08	−0.577588
$309$	4558.83	0.839297
$310$	−310.000	−0.0567962
$311$	−10502.3	−1.91489	−0.957444	−	0.288618i	$-0.906804\pi$
$311$	−10502.3	−1.91489	−0.957444	+	0.288618i	$0.906804\pi$
$312$	1437.94	0.260922
$313$	−7236.99	−1.30690	−0.653449	−	0.756971i	$-0.726679\pi$
$313$	−7236.99	−1.30690	−0.653449	+	0.756971i	$0.726679\pi$
$314$	5608.31	1.00795
$315$	1285.97	0.230019
$316$	3611.16	0.642859
$317$	−4163.61	−0.737702	−0.368851	−	0.929489i	$-0.620249\pi$
$317$	−4163.61	−0.737702	−0.368851	+	0.929489i	$0.620249\pi$
$318$	2751.97	0.485292
$319$	1167.11	0.204846
$320$	−320.000	−0.0559017
$321$	4037.59	0.702045
$322$	−4483.59	−0.775965
$323$	−7532.09	−1.29751
$324$	324.000	0.0555556
$325$	−1497.86	−0.255650
$326$	2473.38	0.420208
$327$	5480.59	0.926842
$328$	1314.27	0.221246
$329$	11350.0	1.90197
$330$	819.386	0.136684
$331$	−2554.52	−0.424197	−0.212098	−	0.977248i	$-0.568030\pi$
$331$	−2554.52	−0.424197	−0.212098	+	0.977248i	$0.568030\pi$
$332$	2419.75	0.400004
$333$	−724.335	−0.119199
$334$	−2887.06	−0.472973
$335$	−2199.74	−0.358761
$336$	1371.70	0.222715
$337$	−9488.18	−1.53369	−0.766846	−	0.641831i	$-0.778175\pi$
$337$	−9488.18	−1.53369	−0.766846	+	0.641831i	$0.778175\pi$
$338$	2785.45	0.448251
$339$	−5308.31	−0.850466
$340$	2335.84	0.372585
$341$	846.699	0.134461
$342$	1160.84	0.183542
$343$	−3733.51	−0.587728
$344$	4127.54	0.646925
$345$	1176.71	0.183629
$346$	−4358.87	−0.677267
$347$	7244.61	1.12078	0.560390	−	0.828229i	$-0.310651\pi$
$347$	7244.61	1.12078	0.560390	+	0.828229i	$0.310651\pi$
$348$	−512.775	−0.0789875
$349$	−974.285	−0.149433	−0.0747167	−	0.997205i	$-0.523805\pi$
$349$	−974.285	−0.149433	−0.0747167	+	0.997205i	$0.523805\pi$
$350$	−1428.85	−0.218215
$351$	1617.69	0.245999
$352$	874.011	0.132344
$353$	−4490.13	−0.677013	−0.338506	−	0.940964i	$-0.609922\pi$
$353$	−4490.13	−0.677013	−0.338506	+	0.940964i	$0.609922\pi$
$354$	−485.029	−0.0728220
$355$	−2280.15	−0.340895
$356$	5419.81	0.806880
$357$	−10012.7	−1.48440
$358$	34.6428	0.00511432
$359$	3142.50	0.461991	0.230995	−	0.972955i	$-0.425802\pi$
$359$	3142.50	0.461991	0.230995	+	0.972955i	$0.425802\pi$
$360$	−360.000	−0.0527046
$361$	−2699.86	−0.393623
$362$	4144.89	0.601797
$363$	1755.02	0.253760
$364$	6848.70	0.986179
$365$	−1044.16	−0.149737
$366$	−2257.79	−0.322450
$367$	8832.92	1.25633	0.628167	−	0.778078i	$-0.283805\pi$
$367$	8832.92	1.25633	0.628167	+	0.778078i	$0.283805\pi$
$368$	1255.16	0.177798
$369$	1478.56	0.208592
$370$	804.817	0.113082
$371$	13107.2	1.83421
$372$	−372.000	−0.0518476
$373$	−1006.78	−0.139757	−0.0698785	−	0.997556i	$-0.522261\pi$
$373$	−1006.78	−0.139757	−0.0698785	+	0.997556i	$0.522261\pi$
$374$	−6379.86	−0.882071
$375$	375.000	0.0516398
$376$	−3177.39	−0.435801
$377$	−2560.22	−0.349756
$378$	1543.16	0.209978
$379$	7409.48	1.00422	0.502110	−	0.864804i	$-0.332557\pi$
$379$	7409.48	1.00422	0.502110	+	0.864804i	$0.332557\pi$
$380$	−1289.83	−0.174123
$381$	−1857.55	−0.249777
$382$	5227.26	0.700131
$383$	10157.7	1.35518	0.677588	−	0.735442i	$-0.263025\pi$
$383$	10157.7	1.35518	0.677588	+	0.735442i	$0.263025\pi$
$384$	−384.000	−0.0510310
$385$	3902.60	0.516611
$386$	6429.55	0.847812
$387$	4643.49	0.609927
$388$	5586.95	0.731017
$389$	−5674.58	−0.739622	−0.369811	−	0.929107i	$-0.620577\pi$
$389$	−5674.58	−0.739622	−0.369811	+	0.929107i	$0.620577\pi$
$390$	−1797.43	−0.233375
$391$	−9162.05	−1.18502
$392$	3789.18	0.488220
$393$	3682.27	0.472636
$394$	4974.72	0.636098
$395$	−4513.94	−0.574990
$396$	983.263	0.124775
$397$	−12200.7	−1.54241	−0.771203	−	0.636589i	$-0.780345\pi$
$397$	−12200.7	−1.54241	−0.771203	+	0.636589i	$0.780345\pi$
$398$	4415.36	0.556085
$399$	5528.92	0.693715
$400$	400.000	0.0500000
$401$	−12225.9	−1.52252	−0.761260	−	0.648447i	$-0.775419\pi$
$401$	−12225.9	−1.52252	−0.761260	+	0.648447i	$0.775419\pi$
$402$	−2639.69	−0.327502
$403$	−1857.34	−0.229580
$404$	2418.83	0.297874
$405$	−405.000	−0.0496904
$406$	−2442.27	−0.298541
$407$	−2198.19	−0.267715
$408$	2803.01	0.340122
$409$	−7480.97	−0.904426	−0.452213	−	0.891910i	$-0.649365\pi$
$409$	−7480.97	−0.904426	−0.452213	+	0.891910i	$0.649365\pi$
$410$	−1642.84	−0.197888
$411$	−7456.41	−0.894884
$412$	−6078.44	−0.726852
$413$	−2310.11	−0.275238
$414$	1412.05	0.167630
$415$	−3024.69	−0.357774
$416$	−1917.26	−0.225965
$417$	7273.87	0.854204
$418$	3522.89	0.412225
$419$	13729.7	1.60081	0.800403	−	0.599462i	$-0.204619\pi$
$419$	13729.7	1.60081	0.800403	+	0.599462i	$0.204619\pi$
$420$	−1714.62	−0.199202
$421$	−10555.8	−1.22199	−0.610995	−	0.791635i	$-0.709230\pi$
$421$	−10555.8	−1.22199	−0.610995	+	0.791635i	$0.709230\pi$
$422$	−1242.63	−0.143341
$423$	−3574.56	−0.410877
$424$	−3669.29	−0.420275
$425$	−2919.81	−0.333250
$426$	−2736.18	−0.311193
$427$	−10753.5	−1.21873
$428$	−5383.46	−0.607989
$429$	4909.29	0.552501
$430$	−5159.43	−0.578628
$431$	5505.70	0.615314	0.307657	−	0.951497i	$-0.400455\pi$
$431$	5505.70	0.615314	0.307657	+	0.951497i	$0.400455\pi$
$432$	−432.000	−0.0481125
$433$	16294.3	1.80844	0.904220	−	0.427067i	$-0.140453\pi$
$433$	16294.3	1.80844	0.904220	+	0.427067i	$0.140453\pi$
$434$	−1771.78	−0.195963
$435$	640.969	0.0706485
$436$	−7307.45	−0.802668
$437$	5059.18	0.553807
$438$	−1253.00	−0.136691
$439$	2130.98	0.231677	0.115839	−	0.993268i	$-0.463044\pi$
$439$	2130.98	0.231677	0.115839	+	0.993268i	$0.463044\pi$
$440$	−1092.51	−0.118372
$441$	4262.83	0.460299
$442$	13995.1	1.50606
$443$	−10678.3	−1.14524	−0.572619	−	0.819822i	$-0.694073\pi$
$443$	−10678.3	−1.14524	−0.572619	+	0.819822i	$0.694073\pi$
$444$	965.781	0.103230
$445$	−6774.76	−0.721696
$446$	−1340.72	−0.142343
$447$	−4249.03	−0.449602
$448$	−1828.93	−0.192877
$449$	−5661.35	−0.595046	−0.297523	−	0.954715i	$-0.596161\pi$
$449$	−5661.35	−0.595046	−0.297523	+	0.954715i	$0.596161\pi$
$450$	450.000	0.0471405
$451$	4487.07	0.468487
$452$	7077.75	0.736525
$453$	3895.30	0.404011
$454$	−323.961	−0.0334895
$455$	−8560.87	−0.882066
$456$	−1547.79	−0.158952
$457$	6265.30	0.641309	0.320654	−	0.947196i	$-0.396097\pi$
$457$	6265.30	0.641309	0.320654	+	0.947196i	$0.396097\pi$
$458$	−2631.59	−0.268485
$459$	3153.39	0.320670
$460$	−1568.95	−0.159027
$461$	18808.7	1.90024	0.950118	−	0.311889i	$-0.100962\pi$
$461$	18808.7	1.90024	0.950118	+	0.311889i	$0.100962\pi$
$462$	4683.12	0.471599
$463$	9962.74	1.00002	0.500009	−	0.866020i	$-0.333330\pi$
$463$	9962.74	1.00002	0.500009	+	0.866020i	$0.333330\pi$
$464$	683.700	0.0684052
$465$	465.000	0.0463739
$466$	6978.57	0.693725
$467$	−3198.98	−0.316983	−0.158492	−	0.987360i	$-0.550663\pi$
$467$	−3198.98	−0.316983	−0.158492	+	0.987360i	$0.550663\pi$
$468$	−2156.92	−0.213042
$469$	−12572.4	−1.23783
$470$	3971.73	0.389792
$471$	−8412.46	−0.822985
$472$	646.705	0.0630657
$473$	14091.9	1.36986
$474$	−5416.73	−0.524892
$475$	1612.28	0.155740
$476$	13350.3	1.28553
$477$	−4127.96	−0.396239
$478$	−2029.68	−0.194217
$479$	−9341.30	−0.891054	−0.445527	−	0.895268i	$-0.646984\pi$
$479$	−9341.30	−0.891054	−0.445527	+	0.895268i	$0.646984\pi$
$480$	480.000	0.0456435
$481$	4822.01	0.457099
$482$	−12646.3	−1.19507
$483$	6725.39	0.633573
$484$	−2340.03	−0.219763
$485$	−6983.69	−0.653841
$486$	−486.000	−0.0453609
$487$	−3984.57	−0.370756	−0.185378	−	0.982667i	$-0.559351\pi$
$487$	−3984.57	−0.370756	−0.185378	+	0.982667i	$0.559351\pi$
$488$	3010.39	0.279250
$489$	−3710.07	−0.343098
$490$	−4736.47	−0.436678
$491$	10778.1	0.990649	0.495324	−	0.868708i	$-0.335049\pi$
$491$	10778.1	0.990649	0.495324	+	0.868708i	$0.335049\pi$
$492$	−1971.41	−0.180646
$493$	−4990.68	−0.455921
$494$	−7727.92	−0.703837
$495$	−1229.08	−0.111602
$496$	496.000	0.0449013
$497$	−13032.0	−1.17618
$498$	−3629.63	−0.326602
$499$	9845.68	0.883273	0.441637	−	0.897194i	$-0.354398\pi$
$499$	9845.68	0.883273	0.441637	+	0.897194i	$0.354398\pi$
$500$	−500.000	−0.0447214
$501$	4330.60	0.386181
$502$	−11808.2	−1.04985
$503$	4751.68	0.421207	0.210604	−	0.977572i	$-0.432457\pi$
$503$	4751.68	0.421207	0.210604	+	0.977572i	$0.432457\pi$
$504$	−2057.55	−0.181846
$505$	−3023.53	−0.266427
$506$	4285.25	0.376487
$507$	−4178.18	−0.365995
$508$	2476.73	0.216313
$509$	12578.6	1.09535	0.547677	−	0.836690i	$-0.315512\pi$
$509$	12578.6	1.09535	0.547677	+	0.836690i	$0.315512\pi$
$510$	−3503.77	−0.304214
$511$	−5967.83	−0.516636
$512$	512.000	0.0441942
$513$	−1741.27	−0.149861
$514$	16360.7	1.40397
$515$	7598.05	0.650116
$516$	−6191.32	−0.528212
$517$	−10847.9	−0.922808
$518$	4599.86	0.390166
$519$	6538.31	0.552987
$520$	2396.57	0.202109
$521$	12429.3	1.04518	0.522589	−	0.852585i	$-0.324966\pi$
$521$	12429.3	1.04518	0.522589	+	0.852585i	$0.324966\pi$
$522$	769.163	0.0644930
$523$	8427.37	0.704595	0.352298	−	0.935888i	$-0.385400\pi$
$523$	8427.37	0.704595	0.352298	+	0.935888i	$0.385400\pi$
$524$	−4909.70	−0.409315
$525$	2143.28	0.178172
$526$	−811.736	−0.0672877
$527$	−3620.56	−0.299268
$528$	−1311.02	−0.108058
$529$	−6013.00	−0.494206
$530$	4586.62	0.375905
$531$	727.543	0.0594589
$532$	−7371.89	−0.600774
$533$	−9842.97	−0.799900
$534$	−8129.71	−0.658815
$535$	6729.32	0.543802
$536$	3519.59	0.283625
$537$	−51.9642	−0.00417583
$538$	8954.56	0.717581
$539$	12936.7	1.03381
$540$	540.000	0.0430331
$541$	1399.40	0.111210	0.0556051	−	0.998453i	$-0.482291\pi$
$541$	1399.40	0.111210	0.0556051	+	0.998453i	$0.482291\pi$
$542$	−14105.4	−1.11786
$543$	−6217.33	−0.491365
$544$	−3737.35	−0.294554
$545$	9134.31	0.717928
$546$	−10273.0	−0.805212
$547$	699.642	0.0546883	0.0273442	−	0.999626i	$-0.491295\pi$
$547$	699.642	0.0546883	0.0273442	+	0.999626i	$0.491295\pi$
$548$	9941.88	0.774993
$549$	3386.68	0.263279
$550$	1365.64	0.105875
$551$	2755.80	0.213069
$552$	−1882.74	−0.145172
$553$	−25799.0	−1.98388
$554$	6267.09	0.480620
$555$	−1207.23	−0.0923313
$556$	−9698.50	−0.739762
$557$	18199.4	1.38444	0.692222	−	0.721685i	$-0.256632\pi$
$557$	18199.4	1.38444	0.692222	+	0.721685i	$0.256632\pi$
$558$	558.000	0.0423334
$559$	−30912.4	−2.33892
$560$	2286.16	0.172514
$561$	9569.79	0.720208
$562$	−7192.77	−0.539873
$563$	13979.6	1.04649	0.523243	−	0.852184i	$-0.324722\pi$
$563$	13979.6	1.04649	0.523243	+	0.852184i	$0.324722\pi$
$564$	4766.08	0.355830
$565$	−8847.19	−0.658768
$566$	8260.22	0.613432
$567$	−2314.74	−0.171446
$568$	3648.24	0.269501
$569$	−3032.40	−0.223418	−0.111709	−	0.993741i	$-0.535632\pi$
$569$	−3032.40	−0.223418	−0.111709	+	0.993741i	$0.535632\pi$
$570$	1934.74	0.142171
$571$	1243.51	0.0911372	0.0455686	−	0.998961i	$-0.485490\pi$
$571$	1243.51	0.0911372	0.0455686	+	0.998961i	$0.485490\pi$
$572$	−6545.73	−0.478480
$573$	−7840.89	−0.571655
$574$	−9389.51	−0.682771
$575$	1961.19	0.142238
$576$	576.000	0.0416667
$577$	19368.5	1.39744	0.698718	−	0.715397i	$-0.253754\pi$
$577$	19368.5	1.39744	0.698718	+	0.715397i	$0.253754\pi$
$578$	17454.8	1.25610
$579$	−9644.32	−0.692236
$580$	−854.626	−0.0611834
$581$	−17287.4	−1.23442
$582$	−8380.43	−0.596873
$583$	−12527.4	−0.889932
$584$	1670.66	0.118378
$585$	2696.14	0.190550
$586$	3256.80	0.229586
$587$	215.769	0.0151716	0.00758582	−	0.999971i	$-0.497585\pi$
$587$	215.769	0.0151716	0.00758582	+	0.999971i	$0.497585\pi$
$588$	−5683.77	−0.398630
$589$	1999.23	0.139859
$590$	−808.381	−0.0564077
$591$	−7462.08	−0.519372
$592$	−1287.71	−0.0893994
$593$	−3773.36	−0.261304	−0.130652	−	0.991428i	$-0.541707\pi$
$593$	−3773.36	−0.261304	−0.130652	+	0.991428i	$0.541707\pi$
$594$	−1474.89	−0.101878
$595$	−16687.9	−1.14981
$596$	5665.37	0.389367
$597$	−6623.04	−0.454042
$598$	−9400.25	−0.642818
$599$	12925.5	0.881671	0.440836	−	0.897588i	$-0.354682\pi$
$599$	12925.5	0.881671	0.440836	+	0.897588i	$0.354682\pi$
$600$	−600.000	−0.0408248
$601$	23532.6	1.59720	0.798598	−	0.601864i	$-0.205575\pi$
$601$	23532.6	1.59720	0.798598	+	0.601864i	$0.205575\pi$
$602$	−29488.3	−1.99643
$603$	3959.54	0.267404
$604$	−5193.73	−0.349884
$605$	2925.04	0.196562
$606$	−3628.24	−0.243213
$607$	−21152.0	−1.41439	−0.707195	−	0.707018i	$-0.750040\pi$
$607$	−21152.0	−1.41439	−0.707195	+	0.707018i	$0.750040\pi$
$608$	2063.72	0.137656
$609$	3663.40	0.243758
$610$	−3762.98	−0.249768
$611$	23796.4	1.57561
$612$	−4204.52	−0.277709
$613$	4385.74	0.288970	0.144485	−	0.989507i	$-0.453847\pi$
$613$	4385.74	0.288970	0.144485	+	0.989507i	$0.453847\pi$
$614$	−11328.3	−0.744583
$615$	2464.26	0.161575
$616$	−6244.17	−0.408417
$617$	18732.0	1.22224	0.611120	−	0.791538i	$-0.290719\pi$
$617$	18732.0	1.22224	0.611120	+	0.791538i	$0.290719\pi$
$618$	9117.65	0.593472
$619$	12303.9	0.798924	0.399462	−	0.916750i	$-0.369197\pi$
$619$	12303.9	0.798924	0.399462	+	0.916750i	$0.369197\pi$
$620$	−620.000	−0.0401610
$621$	−2118.08	−0.136869
$622$	−21004.6	−1.35403
$623$	−38720.5	−2.49006
$624$	2875.89	0.184499
$625$	625.000	0.0400000
$626$	−14474.0	−0.924116
$627$	−5284.33	−0.336580
$628$	11216.6	0.712726
$629$	9399.64	0.595847
$630$	2571.93	0.162648
$631$	−9754.58	−0.615410	−0.307705	−	0.951482i	$-0.599561\pi$
$631$	−9754.58	−0.615410	−0.307705	+	0.951482i	$0.599561\pi$
$632$	7222.31	0.454570
$633$	1863.94	0.117038
$634$	−8327.21	−0.521634
$635$	−3095.91	−0.193476
$636$	5503.94	0.343153
$637$	−28378.3	−1.76513
$638$	2334.23	0.144848
$639$	4104.27	0.254088
$640$	−640.000	−0.0395285
$641$	25484.6	1.57033	0.785166	−	0.619285i	$-0.212578\pi$
$641$	25484.6	1.57033	0.785166	+	0.619285i	$0.212578\pi$
$642$	8075.19	0.496421
$643$	3251.40	0.199413	0.0997065	−	0.995017i	$-0.468210\pi$
$643$	3251.40	0.199413	0.0997065	+	0.995017i	$0.468210\pi$
$644$	−8967.18	−0.548690
$645$	7739.15	0.472447
$646$	−15064.2	−0.917480
$647$	4757.08	0.289057	0.144529	−	0.989501i	$-0.453833\pi$
$647$	4757.08	0.289057	0.144529	+	0.989501i	$0.453833\pi$
$648$	648.000	0.0392837
$649$	2207.92	0.133541
$650$	−2995.72	−0.180772
$651$	2657.66	0.160003
$652$	4946.76	0.297132
$653$	24467.2	1.46627	0.733136	−	0.680082i	$-0.238056\pi$
$653$	24467.2	1.46627	0.733136	+	0.680082i	$0.238056\pi$
$654$	10961.2	0.655376
$655$	6137.12	0.366102
$656$	2628.55	0.156444
$657$	1879.50	0.111607
$658$	22700.1	1.34490
$659$	−23064.5	−1.36338	−0.681688	−	0.731643i	$-0.738754\pi$
$659$	−23064.5	−1.36338	−0.681688	+	0.731643i	$0.738754\pi$
$660$	1638.77	0.0966501
$661$	−113.832	−0.00669827	−0.00334913	−	0.999994i	$-0.501066\pi$
$661$	−113.832	−0.00669827	−0.00334913	+	0.999994i	$0.501066\pi$
$662$	−5109.04	−0.299952
$663$	−20992.6	−1.22969
$664$	4839.51	0.282845
$665$	9214.86	0.537349
$666$	−1448.67	−0.0842866
$667$	3352.16	0.194597
$668$	−5774.13	−0.334443
$669$	2011.08	0.116222
$670$	−4399.49	−0.253682
$671$	10277.8	0.591311
$672$	2743.40	0.157483
$673$	19323.0	1.10676	0.553378	−	0.832930i	$-0.313339\pi$
$673$	19323.0	1.10676	0.553378	+	0.832930i	$0.313339\pi$
$674$	−18976.4	−1.08448
$675$	−675.000	−0.0384900
$676$	5570.91	0.316961
$677$	19389.2	1.10072	0.550361	−	0.834927i	$-0.314490\pi$
$677$	19389.2	1.10072	0.550361	+	0.834927i	$0.314490\pi$
$678$	−10616.6	−0.601370
$679$	−39914.7	−2.25594
$680$	4671.69	0.263457
$681$	485.941	0.0273441
$682$	1693.40	0.0950785
$683$	16846.3	0.943788	0.471894	−	0.881655i	$-0.343570\pi$
$683$	16846.3	0.943788	0.471894	+	0.881655i	$0.343570\pi$
$684$	2321.69	0.129784
$685$	−12427.3	−0.693174
$686$	−7467.03	−0.415587
$687$	3947.38	0.219217
$688$	8255.09	0.457445
$689$	27480.4	1.51948
$690$	2353.42	0.129845
$691$	−16483.6	−0.907473	−0.453737	−	0.891136i	$-0.649909\pi$
$691$	−16483.6	−0.907473	−0.453737	+	0.891136i	$0.649909\pi$
$692$	−8717.75	−0.478900
$693$	−7024.69	−0.385059
$694$	14489.2	0.792512
$695$	12123.1	0.661664
$696$	−1025.55	−0.0558526
$697$	−19187.1	−1.04270
$698$	−1948.57	−0.105665
$699$	−10467.9	−0.566424
$700$	−2857.70	−0.154302
$701$	−10807.0	−0.582275	−0.291137	−	0.956681i	$-0.594034\pi$
$701$	−10807.0	−0.582275	−0.291137	+	0.956681i	$0.594034\pi$
$702$	3235.37	0.173948
$703$	−5190.38	−0.278462
$704$	1748.02	0.0935811
$705$	−5957.60	−0.318264
$706$	−8980.26	−0.478720
$707$	−17280.7	−0.919248
$708$	−970.057	−0.0514929
$709$	−32908.8	−1.74318	−0.871591	−	0.490234i	$-0.836911\pi$
$709$	−32908.8	−1.74318	−0.871591	+	0.490234i	$0.836911\pi$
$710$	−4560.29	−0.241049
$711$	8125.10	0.428572
$712$	10839.6	0.570550
$713$	2431.87	0.127734
$714$	−20025.5	−1.04963
$715$	8182.16	0.427966
$716$	69.2855	0.00361637
$717$	3044.53	0.158577
$718$	6284.99	0.326677
$719$	4546.02	0.235797	0.117898	−	0.993026i	$-0.462384\pi$
$719$	4546.02	0.235797	0.117898	+	0.993026i	$0.462384\pi$
$720$	−720.000	−0.0372678
$721$	43425.9	2.24309
$722$	−5399.73	−0.278334
$723$	18969.4	0.975768
$724$	8289.77	0.425535
$725$	1068.28	0.0547241
$726$	3510.05	0.179435
$727$	8969.88	0.457599	0.228799	−	0.973474i	$-0.426520\pi$
$727$	8969.88	0.457599	0.228799	+	0.973474i	$0.426520\pi$
$728$	13697.4	0.697334
$729$	729.000	0.0370370
$730$	−2088.33	−0.105880
$731$	−60258.1	−3.04887
$732$	−4515.58	−0.228006
$733$	−36583.0	−1.84341	−0.921707	−	0.387887i	$-0.873205\pi$
$733$	−36583.0	−1.84341	−0.921707	+	0.387887i	$0.873205\pi$
$734$	17665.8	0.888363
$735$	7104.71	0.356546
$736$	2510.32	0.125722
$737$	12016.3	0.600576
$738$	2957.11	0.147497
$739$	34654.0	1.72499	0.862496	−	0.506064i	$-0.168900\pi$
$739$	34654.0	1.72499	0.862496	+	0.506064i	$0.168900\pi$
$740$	1609.63	0.0799613
$741$	11591.9	0.574680
$742$	26214.4	1.29698
$743$	19764.7	0.975901	0.487951	−	0.872871i	$-0.337745\pi$
$743$	19764.7	0.975901	0.487951	+	0.872871i	$0.337745\pi$
$744$	−744.000	−0.0366618
$745$	−7081.71	−0.348260
$746$	−2013.57	−0.0988231
$747$	5444.45	0.266669
$748$	−12759.7	−0.623718
$749$	38460.8	1.87627
$750$	750.000	0.0365148
$751$	−2145.42	−0.104244	−0.0521222	−	0.998641i	$-0.516599\pi$
$751$	−2145.42	−0.104244	−0.0521222	+	0.998641i	$0.516599\pi$
$752$	−6354.77	−0.308158
$753$	17712.3	0.857199
$754$	−5120.43	−0.247314
$755$	6492.17	0.312946
$756$	3086.32	0.148477
$757$	−15745.7	−0.755995	−0.377997	−	0.925807i	$-0.623387\pi$
$757$	−15745.7	−0.755995	−0.377997	+	0.925807i	$0.623387\pi$
$758$	14819.0	0.710091
$759$	−6427.87	−0.307400
$760$	−2579.66	−0.123124
$761$	−9456.30	−0.450448	−0.225224	−	0.974307i	$-0.572311\pi$
$761$	−9456.30	−0.450448	−0.225224	+	0.974307i	$0.572311\pi$
$762$	−3715.09	−0.176619
$763$	52206.3	2.47706
$764$	10454.5	0.495067
$765$	5255.65	0.248390
$766$	20315.3	0.958254
$767$	−4843.36	−0.228010
$768$	−768.000	−0.0360844
$769$	9769.51	0.458124	0.229062	−	0.973412i	$-0.426434\pi$
$769$	9769.51	0.458124	0.229062	+	0.973412i	$0.426434\pi$
$770$	7805.21	0.365299
$771$	−24541.1	−1.14634
$772$	12859.1	0.599494
$773$	−8099.70	−0.376877	−0.188438	−	0.982085i	$-0.560343\pi$
$773$	−8099.70	−0.376877	−0.188438	+	0.982085i	$0.560343\pi$
$774$	9286.98	0.431284
$775$	775.000	0.0359211
$776$	11173.9	0.516907
$777$	−6899.79	−0.318570
$778$	−11349.2	−0.522991
$779$	10594.9	0.487294
$780$	−3594.86	−0.165021
$781$	12455.5	0.570668
$782$	−18324.1	−0.837939
$783$	−1153.74	−0.0526583
$784$	7578.36	0.345224
$785$	−14020.8	−0.637481
$786$	7364.54	0.334204
$787$	−14769.0	−0.668944	−0.334472	−	0.942406i	$-0.608558\pi$
$787$	−14769.0	−0.668944	−0.334472	+	0.942406i	$0.608558\pi$
$788$	9949.43	0.449789
$789$	1217.60	0.0549402
$790$	−9027.89	−0.406580
$791$	−50565.3	−2.27294
$792$	1966.53	0.0882291
$793$	−22545.7	−1.00961
$794$	−24401.4	−1.09065
$795$	−6879.93	−0.306926
$796$	8830.72	0.393212
$797$	26200.3	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$797$	26200.3	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$798$	11057.8	0.490530
$799$	46386.8	2.05387
$800$	800.000	0.0353553
$801$	12194.6	0.537920
$802$	−24451.7	−1.07658
$803$	5703.82	0.250664
$804$	−5279.38	−0.231579
$805$	11209.0	0.490763
$806$	−3714.69	−0.162338
$807$	−13431.8	−0.585902
$808$	4837.65	0.210629
$809$	5946.11	0.258411	0.129205	−	0.991618i	$-0.458757\pi$
$809$	5946.11	0.258411	0.129205	+	0.991618i	$0.458757\pi$
$810$	−810.000	−0.0351364
$811$	−17256.1	−0.747155	−0.373577	−	0.927599i	$-0.621869\pi$
$811$	−17256.1	−0.747155	−0.373577	+	0.927599i	$0.621869\pi$
$812$	−4884.53	−0.211100
$813$	21158.1	0.912728
$814$	−4396.37	−0.189303
$815$	−6183.45	−0.265763
$816$	5606.03	0.240503
$817$	33273.9	1.42485
$818$	−14961.9	−0.639526
$819$	15409.6	0.657453
$820$	−3285.68	−0.139928
$821$	27253.3	1.15852	0.579262	−	0.815141i	$-0.303341\pi$
$821$	27253.3	1.15852	0.579262	+	0.815141i	$0.303341\pi$
$822$	−14912.8	−0.632779
$823$	−40811.5	−1.72855	−0.864276	−	0.503017i	$-0.832223\pi$
$823$	−40811.5	−1.72855	−0.864276	+	0.503017i	$0.832223\pi$
$824$	−12156.9	−0.513962
$825$	−2048.46	−0.0864465
$826$	−4620.23	−0.194623
$827$	−40986.5	−1.72339	−0.861694	−	0.507429i	$-0.830596\pi$
$827$	−40986.5	−1.72339	−0.861694	+	0.507429i	$0.830596\pi$
$828$	2824.11	0.118532
$829$	32615.3	1.36644	0.683218	−	0.730214i	$-0.260580\pi$
$829$	32615.3	1.36644	0.683218	+	0.730214i	$0.260580\pi$
$830$	−6049.39	−0.252985
$831$	−9400.64	−0.392424
$832$	−3834.52	−0.159781
$833$	−55318.3	−2.30092
$834$	14547.7	0.604013
$835$	7217.66	0.299135
$836$	7045.77	0.291487
$837$	−837.000	−0.0345651
$838$	27459.3	1.13194
$839$	−1271.01	−0.0523004	−0.0261502	−	0.999658i	$-0.508325\pi$
$839$	−1271.01	−0.0523004	−0.0261502	+	0.999658i	$0.508325\pi$
$840$	−3429.25	−0.140857
$841$	−22563.0	−0.925132
$842$	−21111.6	−0.864077
$843$	10789.1	0.440804
$844$	−2485.25	−0.101358
$845$	−6963.64	−0.283499
$846$	−7149.12	−0.290534
$847$	16717.8	0.678194
$848$	−7338.59	−0.297179
$849$	−12390.3	−0.500866
$850$	−5839.61	−0.235644
$851$	−6313.58	−0.254321
$852$	−5472.35	−0.220047
$853$	31243.6	1.25412	0.627059	−	0.778972i	$-0.284259\pi$
$853$	31243.6	1.25412	0.627059	+	0.778972i	$0.284259\pi$
$854$	−21507.0	−0.861773
$855$	−2902.11	−0.116082
$856$	−10766.9	−0.429913
$857$	−23122.3	−0.921635	−0.460817	−	0.887495i	$-0.652444\pi$
$857$	−23122.3	−0.921635	−0.460817	+	0.887495i	$0.652444\pi$
$858$	9818.59	0.390677
$859$	26449.0	1.05056	0.525278	−	0.850931i	$-0.323961\pi$
$859$	26449.0	1.05056	0.525278	+	0.850931i	$0.323961\pi$
$860$	−10318.9	−0.409151
$861$	14084.3	0.557480
$862$	11011.4	0.435093
$863$	23030.3	0.908412	0.454206	−	0.890897i	$-0.349923\pi$
$863$	23030.3	0.908412	0.454206	+	0.890897i	$0.349923\pi$
$864$	−864.000	−0.0340207
$865$	10897.2	0.428342
$866$	32588.6	1.27876
$867$	−26182.3	−1.02560
$868$	−3543.55	−0.138567
$869$	24657.7	0.962551
$870$	1281.94	0.0499561
$871$	−26359.2	−1.02543
$872$	−14614.9	−0.567572
$873$	12570.6	0.487345
$874$	10118.4	0.391601
$875$	3572.13	0.138011
$876$	−2505.99	−0.0966549
$877$	5763.60	0.221919	0.110960	−	0.993825i	$-0.464608\pi$
$877$	5763.60	0.221919	0.110960	+	0.993825i	$0.464608\pi$
$878$	4261.97	0.163820
$879$	−4885.20	−0.187456
$880$	−2185.03	−0.0837015
$881$	5185.33	0.198295	0.0991476	−	0.995073i	$-0.468388\pi$
$881$	5185.33	0.198295	0.0991476	+	0.995073i	$0.468388\pi$
$882$	8525.65	0.325480
$883$	−13625.0	−0.519273	−0.259637	−	0.965706i	$-0.583603\pi$
$883$	−13625.0	−0.519273	−0.259637	+	0.965706i	$0.583603\pi$
$884$	27990.1	1.06494
$885$	1212.57	0.0460567
$886$	−21356.6	−0.809806
$887$	5491.49	0.207876	0.103938	−	0.994584i	$-0.466856\pi$
$887$	5491.49	0.207876	0.103938	+	0.994584i	$0.466856\pi$
$888$	1931.56	0.0729943
$889$	−17694.4	−0.667548
$890$	−13549.5	−0.510316
$891$	2212.34	0.0831832
$892$	−2681.44	−0.100652
$893$	−25614.3	−0.959853
$894$	−8498.05	−0.317917
$895$	−86.6069	−0.00323458
$896$	−3657.86	−0.136385
$897$	14100.4	0.524858
$898$	−11322.7	−0.420761
$899$	1324.67	0.0491437
$900$	900.000	0.0333333
$901$	53568.1	1.98070
$902$	8974.14	0.331271
$903$	44232.4	1.63008
$904$	14155.5	0.520802
$905$	−10362.2	−0.380610
$906$	7790.60	0.285679
$907$	29546.3	1.08166	0.540831	−	0.841131i	$-0.318110\pi$
$907$	29546.3	1.08166	0.540831	+	0.841131i	$0.318110\pi$
$908$	−647.922	−0.0236807
$909$	5442.36	0.198583
$910$	−17121.7	−0.623715
$911$	23211.4	0.844158	0.422079	−	0.906559i	$-0.361301\pi$
$911$	23211.4	0.844158	0.422079	+	0.906559i	$0.361301\pi$
$912$	−3095.59	−0.112396
$913$	16522.6	0.598925
$914$	12530.6	0.453474
$915$	5644.47	0.203935
$916$	−5263.18	−0.189847
$917$	35076.1	1.26316
$918$	6306.78	0.226748
$919$	51429.5	1.84603	0.923016	−	0.384762i	$-0.125717\pi$
$919$	51429.5	1.84603	0.923016	+	0.384762i	$0.125717\pi$
$920$	−3137.90	−0.112449
$921$	16992.5	0.607950
$922$	37617.4	1.34367
$923$	−27322.7	−0.974364
$924$	9366.25	0.333471
$925$	−2012.04	−0.0715195
$926$	19925.5	0.707119
$927$	−13676.5	−0.484568
$928$	1367.40	0.0483698
$929$	39854.8	1.40753	0.703764	−	0.710434i	$-0.251501\pi$
$929$	39854.8	1.40753	0.703764	+	0.710434i	$0.251501\pi$
$930$	930.000	0.0327913
$931$	30546.2	1.07531
$932$	13957.1	0.490538
$933$	31506.9	1.10556
$934$	−6397.96	−0.224141
$935$	15949.6	0.557871
$936$	−4313.83	−0.150643
$937$	−13214.1	−0.460712	−0.230356	−	0.973106i	$-0.573989\pi$
$937$	−13214.1	−0.460712	−0.230356	+	0.973106i	$0.573989\pi$
$938$	−25144.9	−0.875276
$939$	21711.0	0.754538
$940$	7943.47	0.275625
$941$	37946.7	1.31459	0.657293	−	0.753635i	$-0.271701\pi$
$941$	37946.7	1.31459	0.657293	+	0.753635i	$0.271701\pi$
$942$	−16824.9	−0.581938
$943$	12887.7	0.445048
$944$	1293.41	0.0445942
$945$	−3857.90	−0.132802
$946$	28183.8	0.968640
$947$	34345.2	1.17853	0.589266	−	0.807939i	$-0.299417\pi$
$947$	34345.2	1.17853	0.589266	+	0.807939i	$0.299417\pi$
$948$	−10833.5	−0.371155
$949$	−12512.1	−0.427987
$950$	3224.57	0.110125
$951$	12490.8	0.425912
$952$	26700.6	0.909004
$953$	−7199.34	−0.244711	−0.122355	−	0.992486i	$-0.539045\pi$
$953$	−7199.34	−0.244711	−0.122355	+	0.992486i	$0.539045\pi$
$954$	−8255.91	−0.280183
$955$	−13068.2	−0.442802
$956$	−4059.37	−0.137332
$957$	−3501.34	−0.118268
$958$	−18682.6	−0.630070
$959$	−71027.4	−2.39165
$960$	960.000	0.0322749
$961$	961.000	0.0322581
$962$	9644.02	0.323218
$963$	−12112.8	−0.405326
$964$	−25292.6	−0.845040
$965$	−16073.9	−0.536203
$966$	13450.8	0.448004
$967$	6528.47	0.217106	0.108553	−	0.994091i	$-0.465378\pi$
$967$	6528.47	0.217106	0.108553	+	0.994091i	$0.465378\pi$
$968$	−4680.06	−0.155396
$969$	22596.3	0.749119
$970$	−13967.4	−0.462336
$971$	29560.8	0.976984	0.488492	−	0.872568i	$-0.337547\pi$
$971$	29560.8	0.976984	0.488492	+	0.872568i	$0.337547\pi$
$972$	−972.000	−0.0320750
$973$	69288.6	2.28293
$974$	−7969.14	−0.262164
$975$	4493.57	0.147600
$976$	6020.77	0.197459
$977$	−37419.4	−1.22534	−0.612668	−	0.790340i	$-0.709904\pi$
$977$	−37419.4	−1.22534	−0.612668	+	0.790340i	$0.709904\pi$
$978$	−7420.13	−0.242607
$979$	37007.6	1.20814
$980$	−9472.95	−0.308778
$981$	−16441.8	−0.535112
$982$	21556.2	0.700495
$983$	17121.1	0.555523	0.277762	−	0.960650i	$-0.410407\pi$
$983$	17121.1	0.555523	0.277762	+	0.960650i	$0.410407\pi$
$984$	−3942.82	−0.127736
$985$	−12436.8	−0.402304
$986$	−9981.36	−0.322385
$987$	−34050.1	−1.09810
$988$	−15455.8	−0.497688
$989$	40474.4	1.30133
$990$	−2458.16	−0.0789145
$991$	5253.65	0.168403	0.0842017	−	0.996449i	$-0.473166\pi$
$991$	5253.65	0.168403	0.0842017	+	0.996449i	$0.473166\pi$
$992$	992.000	0.0317500
$993$	7663.56	0.244910
$994$	−26063.9	−0.831688
$995$	−11038.4	−0.351699
$996$	−7259.26	−0.230942
$997$	39459.5	1.25346	0.626728	−	0.779238i	$-0.284394\pi$
$997$	39459.5	1.25346	0.626728	+	0.779238i	$0.284394\pi$
$998$	19691.4	0.624569
$999$	2173.01	0.0688197

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.4.a.n.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.4.a.n.1.1	✓	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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	930.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -28.5770 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -28.5770 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} +27.3129 q^{11} -12.0000 q^{12} -59.9143 q^{13} -57.1541 q^{14} +15.0000 q^{15} +16.0000 q^{16} -116.792 q^{17} +18.0000 q^{18} +64.4914 q^{19} -20.0000 q^{20} +85.7311 q^{21} +54.6257 q^{22} +78.4474 q^{23} -24.0000 q^{24} +25.0000 q^{25} -119.829 q^{26} -27.0000 q^{27} -114.308 q^{28} +42.7313 q^{29} +30.0000 q^{30} +31.0000 q^{31} +32.0000 q^{32} -81.9386 q^{33} -233.584 q^{34} +142.885 q^{35} +36.0000 q^{36} -80.4817 q^{37} +128.983 q^{38} +179.743 q^{39} -40.0000 q^{40} +164.284 q^{41} +171.462 q^{42} +515.943 q^{43} +109.251 q^{44} -45.0000 q^{45} +156.895 q^{46} -397.173 q^{47} -48.0000 q^{48} +473.647 q^{49} +50.0000 q^{50} +350.377 q^{51} -239.657 q^{52} -458.662 q^{53} -54.0000 q^{54} -136.564 q^{55} -228.616 q^{56} -193.474 q^{57} +85.4626 q^{58} +80.8381 q^{59} +60.0000 q^{60} +376.298 q^{61} +62.0000 q^{62} -257.193 q^{63} +64.0000 q^{64} +299.572 q^{65} -163.877 q^{66} +439.949 q^{67} -467.169 q^{68} -235.342 q^{69} +285.770 q^{70} +456.029 q^{71} +72.0000 q^{72} +208.833 q^{73} -160.963 q^{74} -75.0000 q^{75} +257.966 q^{76} -780.521 q^{77} +359.486 q^{78} +902.789 q^{79} -80.0000 q^{80} +81.0000 q^{81} +328.568 q^{82} +604.939 q^{83} +342.925 q^{84} +583.961 q^{85} +1031.89 q^{86} -128.194 q^{87} +218.503 q^{88} +1354.95 q^{89} -90.0000 q^{90} +1712.17 q^{91} +313.790 q^{92} -93.0000 q^{93} -794.347 q^{94} -322.457 q^{95} -96.0000 q^{96} +1396.74 q^{97} +947.295 q^{98} +245.816 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} - 20 q^{5} - 24 q^{6} - 3 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q + 8 * q^2 - 12 * q^3 + 16 * q^4 - 20 * q^5 - 24 * q^6 - 3 * q^7 + 32 * q^8 + 36 * q^9	\( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} - 20 q^{5} - 24 q^{6} - 3 q^{7} + 32 q^{8} + 36 q^{9} - 40 q^{10} + 43 q^{11} - 48 q^{12} - 20 q^{13} - 6 q^{14} + 60 q^{15} + 64 q^{16} - 28 q^{17} + 72 q^{18} - 73 q^{19} - 80 q^{20} + 9 q^{21} + 86 q^{22} - 47 q^{23} - 96 q^{24} + 100 q^{25} - 40 q^{26} - 108 q^{27} - 12 q^{28} + 204 q^{29} + 120 q^{30} + 124 q^{31} + 128 q^{32} - 129 q^{33} - 56 q^{34} + 15 q^{35} + 144 q^{36} - 156 q^{37} - 146 q^{38} + 60 q^{39} - 160 q^{40} + 128 q^{41} + 18 q^{42} + 685 q^{43} + 172 q^{44} - 180 q^{45} - 94 q^{46} + 156 q^{47} - 192 q^{48} + 877 q^{49} + 200 q^{50} + 84 q^{51} - 80 q^{52} + 591 q^{53} - 216 q^{54} - 215 q^{55} - 24 q^{56} + 219 q^{57} + 408 q^{58} + 212 q^{59} + 240 q^{60} + 1288 q^{61} + 248 q^{62} - 27 q^{63} + 256 q^{64} + 100 q^{65} - 258 q^{66} + 1270 q^{67} - 112 q^{68} + 141 q^{69} + 30 q^{70} + 2465 q^{71} + 288 q^{72} - 359 q^{73} - 312 q^{74} - 300 q^{75} - 292 q^{76} + 1871 q^{77} + 120 q^{78} + 2053 q^{79} - 320 q^{80} + 324 q^{81} + 256 q^{82} + 994 q^{83} + 36 q^{84} + 140 q^{85} + 1370 q^{86} - 612 q^{87} + 344 q^{88} + 1055 q^{89} - 360 q^{90} + 386 q^{91} - 188 q^{92} - 372 q^{93} + 312 q^{94} + 365 q^{95} - 384 q^{96} + 2962 q^{97} + 1754 q^{98} + 387 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 - 12 * q^3 + 16 * q^4 - 20 * q^5 - 24 * q^6 - 3 * q^7 + 32 * q^8 + 36 * q^9 - 40 * q^10 + 43 * q^11 - 48 * q^12 - 20 * q^13 - 6 * q^14 + 60 * q^15 + 64 * q^16 - 28 * q^17 + 72 * q^18 - 73 * q^19 - 80 * q^20 + 9 * q^21 + 86 * q^22 - 47 * q^23 - 96 * q^24 + 100 * q^25 - 40 * q^26 - 108 * q^27 - 12 * q^28 + 204 * q^29 + 120 * q^30 + 124 * q^31 + 128 * q^32 - 129 * q^33 - 56 * q^34 + 15 * q^35 + 144 * q^36 - 156 * q^37 - 146 * q^38 + 60 * q^39 - 160 * q^40 + 128 * q^41 + 18 * q^42 + 685 * q^43 + 172 * q^44 - 180 * q^45 - 94 * q^46 + 156 * q^47 - 192 * q^48 + 877 * q^49 + 200 * q^50 + 84 * q^51 - 80 * q^52 + 591 * q^53 - 216 * q^54 - 215 * q^55 - 24 * q^56 + 219 * q^57 + 408 * q^58 + 212 * q^59 + 240 * q^60 + 1288 * q^61 + 248 * q^62 - 27 * q^63 + 256 * q^64 + 100 * q^65 - 258 * q^66 + 1270 * q^67 - 112 * q^68 + 141 * q^69 + 30 * q^70 + 2465 * q^71 + 288 * q^72 - 359 * q^73 - 312 * q^74 - 300 * q^75 - 292 * q^76 + 1871 * q^77 + 120 * q^78 + 2053 * q^79 - 320 * q^80 + 324 * q^81 + 256 * q^82 + 994 * q^83 + 36 * q^84 + 140 * q^85 + 1370 * q^86 - 612 * q^87 + 344 * q^88 + 1055 * q^89 - 360 * q^90 + 386 * q^91 - 188 * q^92 - 372 * q^93 + 312 * q^94 + 365 * q^95 - 384 * q^96 + 2962 * q^97 + 1754 * q^98 + 387 * q^99

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