LMFDB - Embedded newform 483.4.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [483,4,Mod(1,483)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(483, 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("483.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 483 = 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	483.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:	$28.4979225328$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.8184789.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 76x^{2} - 8x + 1048 $ x^4 - x^3 - 76x^2 - 8x + 1048
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.69357$ of defining polynomial
Character	$\chi$	$=$	483.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.70156 q^{2} +3.00000 q^{3} +5.70156 q^{4} -5.05515 q^{5} +11.1047 q^{6} -7.00000 q^{7} -8.50781 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.70156 q^{2} +3.00000 q^{3} +5.70156 q^{4} -5.05515 q^{5} +11.1047 q^{6} -7.00000 q^{7} -8.50781 q^{8} +9.00000 q^{9} -18.7119 q^{10} -31.4918 q^{11} +17.1047 q^{12} -59.0383 q^{13} -25.9109 q^{14} -15.1654 q^{15} -77.1047 q^{16} -116.499 q^{17} +33.3141 q^{18} +122.463 q^{19} -28.8222 q^{20} -21.0000 q^{21} -116.569 q^{22} -23.0000 q^{23} -25.5234 q^{24} -99.4455 q^{25} -218.534 q^{26} +27.0000 q^{27} -39.9109 q^{28} +163.645 q^{29} -56.1358 q^{30} +48.4738 q^{31} -217.345 q^{32} -94.4755 q^{33} -431.230 q^{34} +35.3860 q^{35} +51.3141 q^{36} +405.482 q^{37} +453.303 q^{38} -177.115 q^{39} +43.0082 q^{40} +62.7416 q^{41} -77.7328 q^{42} -376.756 q^{43} -179.553 q^{44} -45.4963 q^{45} -85.1359 q^{46} -277.109 q^{47} -231.314 q^{48} +49.0000 q^{49} -368.104 q^{50} -349.498 q^{51} -336.611 q^{52} -478.867 q^{53} +99.9422 q^{54} +159.196 q^{55} +59.5547 q^{56} +367.388 q^{57} +605.743 q^{58} -64.0099 q^{59} -86.4667 q^{60} +456.987 q^{61} +179.429 q^{62} -63.0000 q^{63} -187.680 q^{64} +298.447 q^{65} -349.707 q^{66} +230.437 q^{67} -664.228 q^{68} -69.0000 q^{69} +130.984 q^{70} -372.871 q^{71} -76.5703 q^{72} -1163.37 q^{73} +1500.92 q^{74} -298.336 q^{75} +698.228 q^{76} +220.443 q^{77} -655.602 q^{78} -794.417 q^{79} +389.776 q^{80} +81.0000 q^{81} +232.242 q^{82} +579.797 q^{83} -119.733 q^{84} +588.922 q^{85} -1394.59 q^{86} +490.936 q^{87} +267.926 q^{88} +983.546 q^{89} -168.407 q^{90} +413.268 q^{91} -131.136 q^{92} +145.421 q^{93} -1025.74 q^{94} -619.067 q^{95} -652.036 q^{96} +1384.16 q^{97} +181.377 q^{98} -283.426 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 12 q^{3} + 10 q^{4} - 23 q^{5} + 6 q^{6} - 28 q^{7} + 30 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + 12 * q^3 + 10 * q^4 - 23 * q^5 + 6 * q^6 - 28 * q^7 + 30 * q^8 + 36 * q^9	\( 4 q + 2 q^{2} + 12 q^{3} + 10 q^{4} - 23 q^{5} + 6 q^{6} - 28 q^{7} + 30 q^{8} + 36 q^{9} - 32 q^{10} - 48 q^{11} + 30 q^{12} - 107 q^{13} - 14 q^{14} - 69 q^{15} - 270 q^{16} - 6 q^{17} + 18 q^{18} + 76 q^{19} - 78 q^{20} - 84 q^{21} - 106 q^{22} - 92 q^{23} + 90 q^{24} + 115 q^{25} - 320 q^{26} + 108 q^{27} - 70 q^{28} + 204 q^{29} - 96 q^{30} - 276 q^{31} - 498 q^{32} - 144 q^{33} - 126 q^{34} + 161 q^{35} + 90 q^{36} - 248 q^{37} - 372 q^{38} - 321 q^{39} - 70 q^{40} - 450 q^{41} - 42 q^{42} + 109 q^{43} - 202 q^{44} - 207 q^{45} - 46 q^{46} - 688 q^{47} - 810 q^{48} + 196 q^{49} - 1152 q^{50} - 18 q^{51} - 534 q^{52} - 633 q^{53} + 54 q^{54} + 581 q^{55} - 210 q^{56} + 228 q^{57} - 308 q^{58} + 585 q^{59} - 234 q^{60} + 1311 q^{61} + 846 q^{62} - 252 q^{63} + 722 q^{64} - 1614 q^{65} - 318 q^{66} + 365 q^{67} - 138 q^{68} - 276 q^{69} + 224 q^{70} - 3049 q^{71} + 270 q^{72} - 1164 q^{73} + 2090 q^{74} + 345 q^{75} - 220 q^{76} + 336 q^{77} - 960 q^{78} - 476 q^{79} + 1614 q^{80} + 324 q^{81} + 1784 q^{82} + 1462 q^{83} - 210 q^{84} - 891 q^{85} - 1032 q^{86} + 612 q^{87} + 50 q^{88} - 1767 q^{89} - 288 q^{90} + 749 q^{91} - 230 q^{92} - 828 q^{93} - 1246 q^{94} - 1927 q^{95} - 1494 q^{96} + 1788 q^{97} + 98 q^{98} - 432 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 12 * q^3 + 10 * q^4 - 23 * q^5 + 6 * q^6 - 28 * q^7 + 30 * q^8 + 36 * q^9 - 32 * q^10 - 48 * q^11 + 30 * q^12 - 107 * q^13 - 14 * q^14 - 69 * q^15 - 270 * q^16 - 6 * q^17 + 18 * q^18 + 76 * q^19 - 78 * q^20 - 84 * q^21 - 106 * q^22 - 92 * q^23 + 90 * q^24 + 115 * q^25 - 320 * q^26 + 108 * q^27 - 70 * q^28 + 204 * q^29 - 96 * q^30 - 276 * q^31 - 498 * q^32 - 144 * q^33 - 126 * q^34 + 161 * q^35 + 90 * q^36 - 248 * q^37 - 372 * q^38 - 321 * q^39 - 70 * q^40 - 450 * q^41 - 42 * q^42 + 109 * q^43 - 202 * q^44 - 207 * q^45 - 46 * q^46 - 688 * q^47 - 810 * q^48 + 196 * q^49 - 1152 * q^50 - 18 * q^51 - 534 * q^52 - 633 * q^53 + 54 * q^54 + 581 * q^55 - 210 * q^56 + 228 * q^57 - 308 * q^58 + 585 * q^59 - 234 * q^60 + 1311 * q^61 + 846 * q^62 - 252 * q^63 + 722 * q^64 - 1614 * q^65 - 318 * q^66 + 365 * q^67 - 138 * q^68 - 276 * q^69 + 224 * q^70 - 3049 * q^71 + 270 * q^72 - 1164 * q^73 + 2090 * q^74 + 345 * q^75 - 220 * q^76 + 336 * q^77 - 960 * q^78 - 476 * q^79 + 1614 * q^80 + 324 * q^81 + 1784 * q^82 + 1462 * q^83 - 210 * q^84 - 891 * q^85 - 1032 * q^86 + 612 * q^87 + 50 * q^88 - 1767 * q^89 - 288 * q^90 + 749 * q^91 - 230 * q^92 - 828 * q^93 - 1246 * q^94 - 1927 * q^95 - 1494 * q^96 + 1788 * q^97 + 98 * q^98 - 432 * 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 $

$2$	3.70156	1.30870	0.654350	−	0.756192i	$-0.272942\pi$
$2$	3.70156	1.30870	0.654350	+	0.756192i	$0.272942\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	5.70156	0.712695
$5$	−5.05515	−0.452146	−0.226073	−	0.974110i	$-0.572589\pi$
$5$	−5.05515	−0.452146	−0.226073	+	0.974110i	$0.572589\pi$
$6$	11.1047	0.755578
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.50781	−0.375996
$9$	9.00000	0.333333
$10$	−18.7119	−0.591724
$11$	−31.4918	−0.863195	−0.431597	−	0.902066i	$-0.642050\pi$
$11$	−31.4918	−0.863195	−0.431597	+	0.902066i	$0.642050\pi$
$12$	17.1047	0.411475
$13$	−59.0383	−1.25956	−0.629780	−	0.776773i	$-0.716855\pi$
$13$	−59.0383	−1.25956	−0.629780	+	0.776773i	$0.716855\pi$
$14$	−25.9109	−0.494642
$15$	−15.1654	−0.261047
$16$	−77.1047	−1.20476
$17$	−116.499	−1.66207	−0.831037	−	0.556217i	$-0.812252\pi$
$17$	−116.499	−1.66207	−0.831037	+	0.556217i	$0.812252\pi$
$18$	33.3141	0.436233
$19$	122.463	1.47868	0.739338	−	0.673334i	$-0.235138\pi$
$19$	122.463	1.47868	0.739338	+	0.673334i	$0.235138\pi$
$20$	−28.8222	−0.322242
$21$	−21.0000	−0.218218
$22$	−116.569	−1.12966
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−25.5234	−0.217081
$25$	−99.4455	−0.795564
$26$	−218.534	−1.64839
$27$	27.0000	0.192450
$28$	−39.9109	−0.269373
$29$	163.645	1.04787	0.523934	−	0.851759i	$-0.324464\pi$
$29$	163.645	1.04787	0.523934	+	0.851759i	$0.324464\pi$
$30$	−56.1358	−0.341632
$31$	48.4738	0.280844	0.140422	−	0.990092i	$-0.455154\pi$
$31$	48.4738	0.280844	0.140422	+	0.990092i	$0.455154\pi$
$32$	−217.345	−1.20067
$33$	−94.4755	−0.498366
$34$	−431.230	−2.17516
$35$	35.3860	0.170895
$36$	51.3141	0.237565
$37$	405.482	1.80165	0.900823	−	0.434187i	$-0.142964\pi$
$37$	405.482	1.80165	0.900823	+	0.434187i	$0.142964\pi$
$38$	453.303	1.93514
$39$	−177.115	−0.727208
$40$	43.0082	0.170005
$41$	62.7416	0.238990	0.119495	−	0.992835i	$-0.461872\pi$
$41$	62.7416	0.238990	0.119495	+	0.992835i	$0.461872\pi$
$42$	−77.7328	−0.285582
$43$	−376.756	−1.33616	−0.668079	−	0.744090i	$-0.732883\pi$
$43$	−376.756	−1.33616	−0.668079	+	0.744090i	$0.732883\pi$
$44$	−179.553	−0.615195
$45$	−45.4963	−0.150715
$46$	−85.1359	−0.272883
$47$	−277.109	−0.860011	−0.430006	−	0.902826i	$-0.641488\pi$
$47$	−277.109	−0.860011	−0.430006	+	0.902826i	$0.641488\pi$
$48$	−231.314	−0.695569
$49$	49.0000	0.142857
$50$	−368.104	−1.04115
$51$	−349.498	−0.959599
$52$	−336.611	−0.897683
$53$	−478.867	−1.24108	−0.620542	−	0.784174i	$-0.713087\pi$
$53$	−478.867	−1.24108	−0.620542	+	0.784174i	$0.713087\pi$
$54$	99.9422	0.251859
$55$	159.196	0.390290
$56$	59.5547	0.142113
$57$	367.388	0.853714
$58$	605.743	1.37134
$59$	−64.0099	−0.141244	−0.0706218	−	0.997503i	$-0.522498\pi$
$59$	−64.0099	−0.141244	−0.0706218	+	0.997503i	$0.522498\pi$
$60$	−86.4667	−0.186047
$61$	456.987	0.959199	0.479600	−	0.877487i	$-0.340782\pi$
$61$	456.987	0.959199	0.479600	+	0.877487i	$0.340782\pi$
$62$	179.429	0.367540
$63$	−63.0000	−0.125988
$64$	−187.680	−0.366562
$65$	298.447	0.569505
$66$	−349.707	−0.652211
$67$	230.437	0.420184	0.210092	−	0.977682i	$-0.432624\pi$
$67$	230.437	0.420184	0.210092	+	0.977682i	$0.432624\pi$
$68$	−664.228	−1.18455
$69$	−69.0000	−0.120386
$70$	130.984	0.223650
$71$	−372.871	−0.623263	−0.311631	−	0.950203i	$-0.600875\pi$
$71$	−372.871	−0.623263	−0.311631	+	0.950203i	$0.600875\pi$
$72$	−76.5703	−0.125332
$73$	−1163.37	−1.86523	−0.932615	−	0.360873i	$-0.882479\pi$
$73$	−1163.37	−1.86523	−0.932615	+	0.360873i	$0.882479\pi$
$74$	1500.92	2.35781
$75$	−298.336	−0.459319
$76$	698.228	1.05385
$77$	220.443	0.326257
$78$	−655.602	−0.951696
$79$	−794.417	−1.13138	−0.565689	−	0.824618i	$-0.691390\pi$
$79$	−794.417	−1.13138	−0.565689	+	0.824618i	$0.691390\pi$
$80$	389.776	0.544728
$81$	81.0000	0.111111
$82$	232.242	0.312766
$83$	579.797	0.766759	0.383380	−	0.923591i	$-0.374760\pi$
$83$	579.797	0.766759	0.383380	+	0.923591i	$0.374760\pi$
$84$	−119.733	−0.155523
$85$	588.922	0.751500
$86$	−1394.59	−1.74863
$87$	490.936	0.604986
$88$	267.926	0.324557
$89$	983.546	1.17141	0.585706	−	0.810524i	$-0.300817\pi$
$89$	983.546	1.17141	0.585706	+	0.810524i	$0.300817\pi$
$90$	−168.407	−0.197241
$91$	413.268	0.476069
$92$	−131.136	−0.148607
$93$	145.421	0.162145
$94$	−1025.74	−1.12550
$95$	−619.067	−0.668578
$96$	−652.036	−0.693210
$97$	1384.16	1.44887	0.724433	−	0.689346i	$-0.242102\pi$
$97$	1384.16	1.44887	0.724433	+	0.689346i	$0.242102\pi$
$98$	181.377	0.186957
$99$	−283.426	−0.287732
$100$	−566.995	−0.566995
$101$	177.049	0.174426	0.0872129	−	0.996190i	$-0.472204\pi$
$101$	177.049	0.174426	0.0872129	+	0.996190i	$0.472204\pi$
$102$	−1293.69	−1.25583
$103$	−1455.52	−1.39239	−0.696196	−	0.717852i	$-0.745125\pi$
$103$	−1455.52	−1.39239	−0.696196	+	0.717852i	$0.745125\pi$
$104$	502.287	0.473589
$105$	106.158	0.0986664
$106$	−1772.56	−1.62421
$107$	−329.580	−0.297773	−0.148886	−	0.988854i	$-0.547569\pi$
$107$	−329.580	−0.297773	−0.148886	+	0.988854i	$0.547569\pi$
$108$	153.942	0.137158
$109$	1486.26	1.30603	0.653016	−	0.757344i	$-0.273503\pi$
$109$	1486.26	1.30603	0.653016	+	0.757344i	$0.273503\pi$
$110$	589.273	0.510773
$111$	1216.45	1.04018
$112$	539.733	0.455357
$113$	−1465.24	−1.21981	−0.609903	−	0.792476i	$-0.708792\pi$
$113$	−1465.24	−1.21981	−0.609903	+	0.792476i	$0.708792\pi$
$114$	1359.91	1.11726
$115$	116.268	0.0942790
$116$	933.033	0.746810
$117$	−531.345	−0.419853
$118$	−236.936	−0.184845
$119$	815.496	0.628205
$120$	129.025	0.0981524
$121$	−339.265	−0.254895
$122$	1691.57	1.25530
$123$	188.225	0.137981
$124$	276.376	0.200156
$125$	1134.60	0.811857
$126$	−233.198	−0.164881
$127$	−305.856	−0.213704	−0.106852	−	0.994275i	$-0.534077\pi$
$127$	−305.856	−0.213704	−0.106852	+	0.994275i	$0.534077\pi$
$128$	1044.05	0.720955
$129$	−1130.27	−0.771431
$130$	1104.72	0.745312
$131$	−83.0214	−0.0553711	−0.0276856	−	0.999617i	$-0.508814\pi$
$131$	−83.0214	−0.0553711	−0.0276856	+	0.999617i	$0.508814\pi$
$132$	−538.658	−0.355183
$133$	−857.239	−0.558887
$134$	852.976	0.549895
$135$	−136.489	−0.0870156
$136$	991.155	0.624932
$137$	−1055.46	−0.658205	−0.329103	−	0.944294i	$-0.606746\pi$
$137$	−1055.46	−0.658205	−0.329103	+	0.944294i	$0.606746\pi$
$138$	−255.408	−0.157549
$139$	1175.22	0.717128	0.358564	−	0.933505i	$-0.383266\pi$
$139$	1175.22	0.717128	0.358564	+	0.933505i	$0.383266\pi$
$140$	201.756	0.121796
$141$	−831.328	−0.496528
$142$	−1380.21	−0.815664
$143$	1859.22	1.08725
$144$	−693.942	−0.401587
$145$	−827.250	−0.473789
$146$	−4306.27	−2.44103
$147$	147.000	0.0824786
$148$	2311.88	1.28402
$149$	1729.41	0.950864	0.475432	−	0.879752i	$-0.342292\pi$
$149$	1729.41	0.950864	0.475432	+	0.879752i	$0.342292\pi$
$150$	−1104.31	−0.601111
$151$	−378.487	−0.203979	−0.101990	−	0.994785i	$-0.532521\pi$
$151$	−378.487	−0.203979	−0.101990	+	0.994785i	$0.532521\pi$
$152$	−1041.89	−0.555976
$153$	−1048.49	−0.554025
$154$	815.983	0.426972
$155$	−245.042	−0.126982
$156$	−1009.83	−0.518277
$157$	−2853.73	−1.45065	−0.725325	−	0.688406i	$-0.758311\pi$
$157$	−2853.73	−1.45065	−0.725325	+	0.688406i	$0.758311\pi$
$158$	−2940.58	−1.48064
$159$	−1436.60	−0.716540
$160$	1098.71	0.542880
$161$	161.000	0.0788110
$162$	299.827	0.145411
$163$	−617.784	−0.296863	−0.148431	−	0.988923i	$-0.547422\pi$
$163$	−617.784	−0.296863	−0.148431	+	0.988923i	$0.547422\pi$
$164$	357.725	0.170327
$165$	477.587	0.225334
$166$	2146.16	1.00346
$167$	3609.26	1.67241	0.836207	−	0.548414i	$-0.184768\pi$
$167$	3609.26	1.67241	0.836207	+	0.548414i	$0.184768\pi$
$168$	178.664	0.0820490
$169$	1288.52	0.586492
$170$	2179.93	0.983488
$171$	1102.16	0.492892
$172$	−2148.10	−0.952273
$173$	−2360.79	−1.03750	−0.518750	−	0.854926i	$-0.673603\pi$
$173$	−2360.79	−1.03750	−0.518750	+	0.854926i	$0.673603\pi$
$174$	1817.23	0.791746
$175$	696.118	0.300695
$176$	2428.17	1.03994
$177$	−192.030	−0.0815470
$178$	3640.66	1.53303
$179$	−260.331	−0.108704	−0.0543522	−	0.998522i	$-0.517309\pi$
$179$	−260.331	−0.108704	−0.0543522	+	0.998522i	$0.517309\pi$
$180$	−259.400	−0.107414
$181$	2786.16	1.14416	0.572082	−	0.820196i	$-0.306136\pi$
$181$	2786.16	1.14416	0.572082	+	0.820196i	$0.306136\pi$
$182$	1529.74	0.623032
$183$	1370.96	0.553794
$184$	195.680	0.0784005
$185$	−2049.77	−0.814607
$186$	538.286	0.212199
$187$	3668.78	1.43469
$188$	−1579.96	−0.612926
$189$	−189.000	−0.0727393
$190$	−2291.51	−0.874968
$191$	−898.259	−0.340292	−0.170146	−	0.985419i	$-0.554424\pi$
$191$	−898.259	−0.340292	−0.170146	+	0.985419i	$0.554424\pi$
$192$	−563.039	−0.211635
$193$	−4200.40	−1.56659	−0.783294	−	0.621651i	$-0.786462\pi$
$193$	−4200.40	−1.56659	−0.783294	+	0.621651i	$0.786462\pi$
$194$	5123.55	1.89613
$195$	895.342	0.328804
$196$	279.377	0.101814
$197$	−5068.61	−1.83311	−0.916557	−	0.399903i	$-0.869044\pi$
$197$	−5068.61	−1.83311	−0.916557	+	0.399903i	$0.869044\pi$
$198$	−1049.12	−0.376554
$199$	−607.052	−0.216245	−0.108122	−	0.994138i	$-0.534484\pi$
$199$	−607.052	−0.216245	−0.108122	+	0.994138i	$0.534484\pi$
$200$	846.063	0.299129
$201$	691.310	0.242593
$202$	655.357	0.228271
$203$	−1145.52	−0.396057
$204$	−1992.69	−0.683901
$205$	−317.168	−0.108058
$206$	−5387.69	−1.82222
$207$	−207.000	−0.0695048
$208$	4552.13	1.51747
$209$	−3856.57	−1.27639
$210$	392.951	0.129125
$211$	−1114.86	−0.363744	−0.181872	−	0.983322i	$-0.558216\pi$
$211$	−1114.86	−0.363744	−0.181872	+	0.983322i	$0.558216\pi$
$212$	−2730.29	−0.884514
$213$	−1118.61	−0.359841
$214$	−1219.96	−0.389695
$215$	1904.56	0.604139
$216$	−229.711	−0.0723604
$217$	−339.317	−0.106149
$218$	5501.47	1.70920
$219$	−3490.10	−1.07689
$220$	907.665	0.278158
$221$	6877.93	2.09348
$222$	4502.75	1.36128
$223$	−710.587	−0.213383	−0.106692	−	0.994292i	$-0.534026\pi$
$223$	−710.587	−0.213383	−0.106692	+	0.994292i	$0.534026\pi$
$224$	1521.42	0.453812
$225$	−895.009	−0.265188
$226$	−5423.68	−1.59636
$227$	2095.70	0.612761	0.306380	−	0.951909i	$-0.400882\pi$
$227$	2095.70	0.612761	0.306380	+	0.951909i	$0.400882\pi$
$228$	2094.69	0.608438
$229$	5827.95	1.68176	0.840878	−	0.541225i	$-0.182039\pi$
$229$	5827.95	1.68176	0.840878	+	0.541225i	$0.182039\pi$
$230$	430.375	0.123383
$231$	661.328	0.188365
$232$	−1392.26	−0.393993
$233$	3397.06	0.955144	0.477572	−	0.878593i	$-0.341517\pi$
$233$	3397.06	0.955144	0.477572	+	0.878593i	$0.341517\pi$
$234$	−1966.81	−0.549462
$235$	1400.83	0.388851
$236$	−364.956	−0.100664
$237$	−2383.25	−0.653202
$238$	3018.61	0.822132
$239$	−6151.78	−1.66496	−0.832480	−	0.554054i	$-0.813080\pi$
$239$	−6151.78	−1.66496	−0.832480	+	0.554054i	$0.813080\pi$
$240$	1169.33	0.314499
$241$	2792.09	0.746284	0.373142	−	0.927774i	$-0.378280\pi$
$241$	2792.09	0.746284	0.373142	+	0.927774i	$0.378280\pi$
$242$	−1255.81	−0.333581
$243$	243.000	0.0641500
$244$	2605.54	0.683617
$245$	−247.702	−0.0645923
$246$	696.726	0.180576
$247$	−7229.99	−1.86248
$248$	−412.406	−0.105596
$249$	1739.39	0.442689
$250$	4199.81	1.06248
$251$	−2264.50	−0.569458	−0.284729	−	0.958608i	$-0.591904\pi$
$251$	−2264.50	−0.569458	−0.284729	+	0.958608i	$0.591904\pi$
$252$	−359.198	−0.0897912
$253$	724.312	0.179989
$254$	−1132.15	−0.279674
$255$	1766.76	0.433879
$256$	5366.07	1.31008
$257$	−5948.62	−1.44383	−0.721916	−	0.691981i	$-0.756738\pi$
$257$	−5948.62	−1.44383	−0.721916	+	0.691981i	$0.756738\pi$
$258$	−4183.76	−1.00957
$259$	−2838.38	−0.680958
$260$	1701.62	0.405884
$261$	1472.81	0.349289
$262$	−307.309	−0.0724642
$263$	−6134.61	−1.43831	−0.719156	−	0.694848i	$-0.755472\pi$
$263$	−6134.61	−1.43831	−0.719156	+	0.694848i	$0.755472\pi$
$264$	803.779	0.187383
$265$	2420.74	0.561151
$266$	−3173.12	−0.731416
$267$	2950.64	0.676315
$268$	1313.85	0.299463
$269$	−3089.28	−0.700210	−0.350105	−	0.936710i	$-0.613854\pi$
$269$	−3089.28	−0.700210	−0.350105	+	0.936710i	$0.613854\pi$
$270$	−505.222	−0.113877
$271$	4264.24	0.955846	0.477923	−	0.878402i	$-0.341390\pi$
$271$	4264.24	0.955846	0.477923	+	0.878402i	$0.341390\pi$
$272$	8982.65	2.00240
$273$	1239.80	0.274859
$274$	−3906.85	−0.861393
$275$	3131.72	0.686727
$276$	−393.408	−0.0857984
$277$	2445.88	0.530538	0.265269	−	0.964175i	$-0.414539\pi$
$277$	2445.88	0.530538	0.265269	+	0.964175i	$0.414539\pi$
$278$	4350.15	0.938505
$279$	436.264	0.0936145
$280$	−301.058	−0.0642558
$281$	−6801.32	−1.44389	−0.721944	−	0.691951i	$-0.756751\pi$
$281$	−6801.32	−1.44389	−0.721944	+	0.691951i	$0.756751\pi$
$282$	−3077.21	−0.649806
$283$	8289.87	1.74128	0.870639	−	0.491922i	$-0.163705\pi$
$283$	8289.87	1.74128	0.870639	+	0.491922i	$0.163705\pi$
$284$	−2125.95	−0.444196
$285$	−1857.20	−0.386004
$286$	6882.03	1.42288
$287$	−439.191	−0.0903298
$288$	−1956.11	−0.400225
$289$	8659.11	1.76249
$290$	−3062.12	−0.620048
$291$	4152.47	0.836503
$292$	−6633.01	−1.32934
$293$	2479.89	0.494459	0.247229	−	0.968957i	$-0.420480\pi$
$293$	2479.89	0.494459	0.247229	+	0.968957i	$0.420480\pi$
$294$	544.130	0.107940
$295$	323.579	0.0638627
$296$	−3449.77	−0.677411
$297$	−850.279	−0.166122
$298$	6401.52	1.24440
$299$	1357.88	0.262636
$300$	−1700.98	−0.327355
$301$	2637.29	0.505020
$302$	−1400.99	−0.266948
$303$	531.146	0.100705
$304$	−9442.44	−1.78145
$305$	−2310.14	−0.433698
$306$	−3881.07	−0.725052
$307$	−522.159	−0.0970722	−0.0485361	−	0.998821i	$-0.515456\pi$
$307$	−522.159	−0.0970722	−0.0485361	+	0.998821i	$0.515456\pi$
$308$	1256.87	0.232522
$309$	−4366.55	−0.803898
$310$	−907.039	−0.166182
$311$	−2735.90	−0.498838	−0.249419	−	0.968396i	$-0.580240\pi$
$311$	−2735.90	−0.498838	−0.249419	+	0.968396i	$0.580240\pi$
$312$	1506.86	0.273427
$313$	6544.54	1.18185	0.590925	−	0.806726i	$-0.298763\pi$
$313$	6544.54	1.18185	0.590925	+	0.806726i	$0.298763\pi$
$314$	−10563.2	−1.89847
$315$	318.474	0.0569651
$316$	−4529.42	−0.806328
$317$	−4130.52	−0.731840	−0.365920	−	0.930646i	$-0.619246\pi$
$317$	−4130.52	−0.731840	−0.365920	+	0.930646i	$0.619246\pi$
$318$	−5317.67	−0.937735
$319$	−5153.49	−0.904513
$320$	948.748	0.165739
$321$	−988.739	−0.171919
$322$	595.952	0.103140
$323$	−14266.8	−2.45767
$324$	461.827	0.0791884
$325$	5871.09	1.00206
$326$	−2286.77	−0.388504
$327$	4458.77	0.754038
$328$	−533.794	−0.0898592
$329$	1939.76	0.325054
$330$	1767.82	0.294895
$331$	6616.50	1.09872	0.549359	−	0.835587i	$-0.314872\pi$
$331$	6616.50	1.09872	0.549359	+	0.835587i	$0.314872\pi$
$332$	3305.75	0.546466
$333$	3649.34	0.600549
$334$	13359.9	2.18869
$335$	−1164.89	−0.189984
$336$	1619.20	0.262900
$337$	4993.62	0.807180	0.403590	−	0.914940i	$-0.367762\pi$
$337$	4993.62	0.807180	0.403590	+	0.914940i	$0.367762\pi$
$338$	4769.55	0.767542
$339$	−4395.72	−0.704256
$340$	3357.77	0.535591
$341$	−1526.53	−0.242423
$342$	4079.73	0.645048
$343$	−343.000	−0.0539949
$344$	3205.37	0.502390
$345$	348.805	0.0544320
$346$	−8738.61	−1.35778
$347$	−9965.27	−1.54168	−0.770841	−	0.637027i	$-0.780164\pi$
$347$	−9965.27	−1.54168	−0.770841	+	0.637027i	$0.780164\pi$
$348$	2799.10	0.431171
$349$	−5902.15	−0.905258	−0.452629	−	0.891699i	$-0.649514\pi$
$349$	−5902.15	−0.905258	−0.452629	+	0.891699i	$0.649514\pi$
$350$	2576.73	0.393519
$351$	−1594.03	−0.242403
$352$	6844.60	1.03642
$353$	947.217	0.142820	0.0714098	−	0.997447i	$-0.477250\pi$
$353$	947.217	0.142820	0.0714098	+	0.997447i	$0.477250\pi$
$354$	−710.809	−0.106721
$355$	1884.92	0.281806
$356$	5607.75	0.834860
$357$	2446.49	0.362694
$358$	−963.633	−0.142261
$359$	−2733.14	−0.401810	−0.200905	−	0.979611i	$-0.564388\pi$
$359$	−2733.14	−0.401810	−0.200905	+	0.979611i	$0.564388\pi$
$360$	387.074	0.0566683
$361$	8138.10	1.18648
$362$	10313.1	1.49737
$363$	−1017.80	−0.147164
$364$	2356.27	0.339292
$365$	5880.99	0.843356
$366$	5074.70	0.724750
$367$	−10099.4	−1.43647	−0.718237	−	0.695799i	$-0.755051\pi$
$367$	−10099.4	−1.43647	−0.718237	+	0.695799i	$0.755051\pi$
$368$	1773.41	0.251210
$369$	564.675	0.0796634
$370$	−7587.36	−1.06608
$371$	3352.07	0.469085
$372$	829.129	0.115560
$373$	−1037.41	−0.144008	−0.0720039	−	0.997404i	$-0.522939\pi$
$373$	−1037.41	−0.144008	−0.0720039	+	0.997404i	$0.522939\pi$
$374$	13580.2	1.87758
$375$	3403.81	0.468726
$376$	2357.59	0.323361
$377$	−9661.34	−1.31985
$378$	−699.595	−0.0951939
$379$	5633.27	0.763487	0.381743	−	0.924268i	$-0.375324\pi$
$379$	5633.27	0.763487	0.381743	+	0.924268i	$0.375324\pi$
$380$	−3529.65	−0.476492
$381$	−917.569	−0.123382
$382$	−3324.96	−0.445340
$383$	−1183.15	−0.157849	−0.0789247	−	0.996881i	$-0.525149\pi$
$383$	−1183.15	−0.157849	−0.0789247	+	0.996881i	$0.525149\pi$
$384$	3132.16	0.416244
$385$	−1114.37	−0.147516
$386$	−15548.0	−2.05019
$387$	−3390.81	−0.445386
$388$	7891.86	1.03260
$389$	−5825.10	−0.759239	−0.379620	−	0.925143i	$-0.623945\pi$
$389$	−5825.10	−0.759239	−0.379620	+	0.925143i	$0.623945\pi$
$390$	3314.16	0.430306
$391$	2679.49	0.346566
$392$	−416.883	−0.0537137
$393$	−249.064	−0.0319685
$394$	−18761.8	−2.39900
$395$	4015.90	0.511548
$396$	−1615.97	−0.205065
$397$	−10610.7	−1.34139	−0.670697	−	0.741731i	$-0.734005\pi$
$397$	−10610.7	−1.34139	−0.670697	+	0.741731i	$0.734005\pi$
$398$	−2247.04	−0.283000
$399$	−2571.72	−0.322674
$400$	7667.71	0.958464
$401$	9348.78	1.16423	0.582114	−	0.813107i	$-0.302226\pi$
$401$	9348.78	1.16423	0.582114	+	0.813107i	$0.302226\pi$
$402$	2558.93	0.317482
$403$	−2861.81	−0.353739
$404$	1009.45	0.124312
$405$	−409.467	−0.0502385
$406$	−4240.20	−0.518319
$407$	−12769.4	−1.55517
$408$	2973.46	0.360805
$409$	11373.4	1.37501	0.687506	−	0.726179i	$-0.258706\pi$
$409$	11373.4	1.37501	0.687506	+	0.726179i	$0.258706\pi$
$410$	−1174.02	−0.141416
$411$	−3166.38	−0.380015
$412$	−8298.72	−0.992351
$413$	448.069	0.0533851
$414$	−766.223	−0.0909609
$415$	−2930.96	−0.346687
$416$	12831.7	1.51232
$417$	3525.66	0.414034
$418$	−14275.3	−1.67041
$419$	1459.66	0.170188	0.0850942	−	0.996373i	$-0.472881\pi$
$419$	1459.66	0.170188	0.0850942	+	0.996373i	$0.472881\pi$
$420$	605.267	0.0703191
$421$	−16056.1	−1.85873	−0.929365	−	0.369163i	$-0.879645\pi$
$421$	−16056.1	−1.85873	−0.929365	+	0.369163i	$0.879645\pi$
$422$	−4126.71	−0.476031
$423$	−2493.98	−0.286670
$424$	4074.11	0.466642
$425$	11585.3	1.32229
$426$	−4140.62	−0.470924
$427$	−3198.91	−0.362543
$428$	−1879.12	−0.212221
$429$	5577.67	0.627722
$430$	7049.84	0.790636
$431$	9595.59	1.07240	0.536199	−	0.844092i	$-0.319860\pi$
$431$	9595.59	1.07240	0.536199	+	0.844092i	$0.319860\pi$
$432$	−2081.83	−0.231856
$433$	−6863.20	−0.761719	−0.380859	−	0.924633i	$-0.624372\pi$
$433$	−6863.20	−0.761719	−0.380859	+	0.924633i	$0.624372\pi$
$434$	−1256.00	−0.138917
$435$	−2481.75	−0.273542
$436$	8473.98	0.930803
$437$	−2816.64	−0.308325
$438$	−12918.8	−1.40933
$439$	7540.01	0.819738	0.409869	−	0.912144i	$-0.365574\pi$
$439$	7540.01	0.819738	0.409869	+	0.912144i	$0.365574\pi$
$440$	−1354.41	−0.146747
$441$	441.000	0.0476190
$442$	25459.1	2.73974
$443$	−10067.4	−1.07973	−0.539863	−	0.841753i	$-0.681524\pi$
$443$	−10067.4	−1.07973	−0.539863	+	0.841753i	$0.681524\pi$
$444$	6935.65	0.741332
$445$	−4971.97	−0.529649
$446$	−2630.28	−0.279254
$447$	5188.23	0.548982
$448$	1313.76	0.138547
$449$	530.162	0.0557236	0.0278618	−	0.999612i	$-0.491130\pi$
$449$	530.162	0.0557236	0.0278618	+	0.999612i	$0.491130\pi$
$450$	−3312.93	−0.347051
$451$	−1975.85	−0.206295
$452$	−8354.15	−0.869350
$453$	−1135.46	−0.117767
$454$	7757.38	0.801920
$455$	−2089.13	−0.215253
$456$	−3125.67	−0.320993
$457$	−14568.7	−1.49123	−0.745617	−	0.666375i	$-0.767845\pi$
$457$	−14568.7	−1.49123	−0.745617	+	0.666375i	$0.767845\pi$
$458$	21572.5	2.20091
$459$	−3145.48	−0.319866
$460$	662.911	0.0671922
$461$	−15985.4	−1.61500	−0.807498	−	0.589870i	$-0.799179\pi$
$461$	−15985.4	−1.61500	−0.807498	+	0.589870i	$0.799179\pi$
$462$	2447.95	0.246513
$463$	−13491.0	−1.35416	−0.677082	−	0.735907i	$-0.736756\pi$
$463$	−13491.0	−1.35416	−0.677082	+	0.735907i	$0.736756\pi$
$464$	−12617.8	−1.26243
$465$	−735.127	−0.0733133
$466$	12574.4	1.25000
$467$	1504.25	0.149055	0.0745273	−	0.997219i	$-0.476255\pi$
$467$	1504.25	0.149055	0.0745273	+	0.997219i	$0.476255\pi$
$468$	−3029.50	−0.299228
$469$	−1613.06	−0.158815
$470$	5185.25	0.508889
$471$	−8561.18	−0.837534
$472$	544.584	0.0531070
$473$	11864.7	1.15336
$474$	−8821.75	−0.854845
$475$	−12178.4	−1.17638
$476$	4649.60	0.447719
$477$	−4309.80	−0.413694
$478$	−22771.2	−2.17893
$479$	−10871.0	−1.03697	−0.518487	−	0.855085i	$-0.673505\pi$
$479$	−10871.0	−1.03697	−0.518487	+	0.855085i	$0.673505\pi$
$480$	3296.14	0.313432
$481$	−23939.0	−2.26928
$482$	10335.1	0.976661
$483$	483.000	0.0455016
$484$	−1934.34	−0.181662
$485$	−6997.12	−0.655099
$486$	899.480	0.0839531
$487$	16079.8	1.49619	0.748096	−	0.663591i	$-0.230968\pi$
$487$	16079.8	1.49619	0.748096	+	0.663591i	$0.230968\pi$
$488$	−3887.96	−0.360655
$489$	−1853.35	−0.171394
$490$	−916.885	−0.0845319
$491$	−19353.9	−1.77888	−0.889441	−	0.457050i	$-0.848906\pi$
$491$	−19353.9	−1.77888	−0.889441	+	0.457050i	$0.848906\pi$
$492$	1073.18	0.0983384
$493$	−19064.6	−1.74163
$494$	−26762.3	−2.43743
$495$	1432.76	0.130097
$496$	−3737.56	−0.338349
$497$	2610.10	0.235571
$498$	6438.47	0.579346
$499$	13483.3	1.20961	0.604806	−	0.796373i	$-0.293251\pi$
$499$	13483.3	1.20961	0.604806	+	0.796373i	$0.293251\pi$
$500$	6469.02	0.578607
$501$	10827.8	0.965569
$502$	−8382.18	−0.745249
$503$	−20352.0	−1.80408	−0.902039	−	0.431655i	$-0.857930\pi$
$503$	−20352.0	−1.80408	−0.902039	+	0.431655i	$0.857930\pi$
$504$	535.992	0.0473710
$505$	−895.007	−0.0788659
$506$	2681.09	0.235551
$507$	3865.57	0.338612
$508$	−1743.86	−0.152306
$509$	−9128.10	−0.794884	−0.397442	−	0.917627i	$-0.630102\pi$
$509$	−9128.10	−0.794884	−0.397442	+	0.917627i	$0.630102\pi$
$510$	6539.79	0.567817
$511$	8143.57	0.704991
$512$	11510.4	0.993541
$513$	3306.49	0.284571
$514$	−22019.2	−1.88954
$515$	7357.85	0.629564
$516$	−6444.30	−0.549795
$517$	8726.67	0.742357
$518$	−10506.4	−0.891170
$519$	−7082.37	−0.599001
$520$	−2539.13	−0.214132
$521$	945.225	0.0794838	0.0397419	−	0.999210i	$-0.487346\pi$
$521$	945.225	0.0794838	0.0397419	+	0.999210i	$0.487346\pi$
$522$	5451.69	0.457115
$523$	−14059.5	−1.17548	−0.587741	−	0.809049i	$-0.699983\pi$
$523$	−14059.5	−1.17548	−0.587741	+	0.809049i	$0.699983\pi$
$524$	−473.352	−0.0394627
$525$	2088.36	0.173606
$526$	−22707.6	−1.88232
$527$	−5647.17	−0.466783
$528$	7284.50	0.600411
$529$	529.000	0.0434783
$530$	8960.53	0.734378
$531$	−576.089	−0.0470812
$532$	−4887.60	−0.398316
$533$	−3704.16	−0.301022
$534$	10922.0	0.885093
$535$	1666.07	0.134637
$536$	−1960.51	−0.157987
$537$	−780.994	−0.0627605
$538$	−11435.2	−0.916365
$539$	−1543.10	−0.123314
$540$	−778.200	−0.0620156
$541$	−23148.5	−1.83961	−0.919806	−	0.392374i	$-0.871654\pi$
$541$	−23148.5	−1.83961	−0.919806	+	0.392374i	$0.871654\pi$
$542$	15784.4	1.25092
$543$	8358.48	0.660583
$544$	25320.6	1.99561
$545$	−7513.24	−0.590518
$546$	4589.21	0.359707
$547$	−6489.79	−0.507282	−0.253641	−	0.967298i	$-0.581628\pi$
$547$	−6489.79	−0.507282	−0.253641	+	0.967298i	$0.581628\pi$
$548$	−6017.78	−0.469100
$549$	4112.88	0.319733
$550$	11592.3	0.898719
$551$	20040.4	1.54946
$552$	587.039	0.0452646
$553$	5560.92	0.427621
$554$	9053.59	0.694314
$555$	−6149.32	−0.470314
$556$	6700.59	0.511094
$557$	1242.29	0.0945021	0.0472510	−	0.998883i	$-0.484954\pi$
$557$	1242.29	0.0945021	0.0472510	+	0.998883i	$0.484954\pi$
$558$	1614.86	0.122513
$559$	22243.1	1.68297
$560$	−2728.43	−0.205888
$561$	11006.3	0.828321
$562$	−25175.5	−1.88962
$563$	15337.3	1.14812	0.574058	−	0.818814i	$-0.305368\pi$
$563$	15337.3	1.14812	0.574058	+	0.818814i	$0.305368\pi$
$564$	−4739.87	−0.353873
$565$	7407.00	0.551531
$566$	30685.5	2.27881
$567$	−567.000	−0.0419961
$568$	3172.32	0.234344
$569$	11775.1	0.867556	0.433778	−	0.901020i	$-0.357180\pi$
$569$	11775.1	0.867556	0.433778	+	0.901020i	$0.357180\pi$
$570$	−6874.54	−0.505163
$571$	−12980.3	−0.951331	−0.475666	−	0.879626i	$-0.657793\pi$
$571$	−12980.3	−0.951331	−0.475666	+	0.879626i	$0.657793\pi$
$572$	10600.5	0.774875
$573$	−2694.78	−0.196468
$574$	−1625.69	−0.118215
$575$	2287.25	0.165887
$576$	−1689.12	−0.122187
$577$	−17255.5	−1.24499	−0.622493	−	0.782626i	$-0.713880\pi$
$577$	−17255.5	−1.24499	−0.622493	+	0.782626i	$0.713880\pi$
$578$	32052.2	2.30657
$579$	−12601.2	−0.904470
$580$	−4716.62	−0.337667
$581$	−4058.58	−0.289808
$582$	15370.6	1.09473
$583$	15080.4	1.07130
$584$	9897.71	0.701318
$585$	2686.03	0.189835
$586$	9179.45	0.647098
$587$	10049.3	0.706606	0.353303	−	0.935509i	$-0.385059\pi$
$587$	10049.3	0.706606	0.353303	+	0.935509i	$0.385059\pi$
$588$	838.130	0.0587821
$589$	5936.23	0.415277
$590$	1197.75	0.0835772
$591$	−15205.8	−1.05835
$592$	−31264.6	−2.17055
$593$	16348.1	1.13210	0.566051	−	0.824371i	$-0.308471\pi$
$593$	16348.1	1.13210	0.566051	+	0.824371i	$0.308471\pi$
$594$	−3147.36	−0.217404
$595$	−4122.45	−0.284040
$596$	9860.34	0.677676
$597$	−1821.15	−0.124849
$598$	5026.28	0.343712
$599$	1849.87	0.126183	0.0630914	−	0.998008i	$-0.479904\pi$
$599$	1849.87	0.126183	0.0630914	+	0.998008i	$0.479904\pi$
$600$	2538.19	0.172702
$601$	18335.1	1.24443	0.622217	−	0.782845i	$-0.286232\pi$
$601$	18335.1	1.24443	0.622217	+	0.782845i	$0.286232\pi$
$602$	9762.11	0.660920
$603$	2073.93	0.140061
$604$	−2157.97	−0.145375
$605$	1715.03	0.115250
$606$	1966.07	0.131792
$607$	16265.7	1.08765	0.543826	−	0.839198i	$-0.316975\pi$
$607$	16265.7	1.08765	0.543826	+	0.839198i	$0.316975\pi$
$608$	−26616.7	−1.77541
$609$	−3436.55	−0.228663
$610$	−8551.11	−0.567581
$611$	16360.1	1.08324
$612$	−5978.06	−0.394851
$613$	19625.5	1.29309	0.646545	−	0.762875i	$-0.276213\pi$
$613$	19625.5	1.29309	0.646545	+	0.762875i	$0.276213\pi$
$614$	−1932.80	−0.127038
$615$	−951.504	−0.0623876
$616$	−1875.49	−0.122671
$617$	2886.80	0.188360	0.0941801	−	0.995555i	$-0.469977\pi$
$617$	2886.80	0.188360	0.0941801	+	0.995555i	$0.469977\pi$
$618$	−16163.1	−1.05206
$619$	767.335	0.0498252	0.0249126	−	0.999690i	$-0.492069\pi$
$619$	767.335	0.0498252	0.0249126	+	0.999690i	$0.492069\pi$
$620$	−1397.12	−0.0904997
$621$	−621.000	−0.0401286
$622$	−10127.1	−0.652829
$623$	−6884.82	−0.442752
$624$	13656.4	0.876111
$625$	6695.09	0.428486
$626$	24225.0	1.54669
$627$	−11569.7	−0.736922
$628$	−16270.7	−1.03387
$629$	−47238.4	−2.99447
$630$	1178.85	0.0745502
$631$	29321.3	1.84986	0.924931	−	0.380135i	$-0.124123\pi$
$631$	29321.3	1.84986	0.924931	+	0.380135i	$0.124123\pi$
$632$	6758.75	0.425393
$633$	−3344.57	−0.210008
$634$	−15289.4	−0.957759
$635$	1546.15	0.0966253
$636$	−8190.87	−0.510674
$637$	−2892.88	−0.179937
$638$	−19075.9	−1.18374
$639$	−3355.84	−0.207754
$640$	−5277.85	−0.325977
$641$	−15027.9	−0.926001	−0.463000	−	0.886358i	$-0.653227\pi$
$641$	−15027.9	−0.926001	−0.463000	+	0.886358i	$0.653227\pi$
$642$	−3659.88	−0.224991
$643$	19893.0	1.22007	0.610034	−	0.792375i	$-0.291156\pi$
$643$	19893.0	1.22007	0.610034	+	0.792375i	$0.291156\pi$
$644$	917.952	0.0561683
$645$	5713.67	0.348800
$646$	−52809.5	−3.21635
$647$	6993.69	0.424962	0.212481	−	0.977165i	$-0.431846\pi$
$647$	6993.69	0.424962	0.212481	+	0.977165i	$0.431846\pi$
$648$	−689.133	−0.0417773
$649$	2015.79	0.121921
$650$	21732.2	1.31140
$651$	−1017.95	−0.0612851
$652$	−3522.34	−0.211573
$653$	−7296.95	−0.437292	−0.218646	−	0.975804i	$-0.570164\pi$
$653$	−7296.95	−0.437292	−0.218646	+	0.975804i	$0.570164\pi$
$654$	16504.4	0.986810
$655$	419.686	0.0250358
$656$	−4837.67	−0.287926
$657$	−10470.3	−0.621743
$658$	7180.16	0.425398
$659$	17065.9	1.00879	0.504397	−	0.863472i	$-0.331715\pi$
$659$	17065.9	1.00879	0.504397	+	0.863472i	$0.331715\pi$
$660$	2722.99	0.160595
$661$	−3802.18	−0.223733	−0.111867	−	0.993723i	$-0.535683\pi$
$661$	−3802.18	−0.223733	−0.111867	+	0.993723i	$0.535683\pi$
$662$	24491.4	1.43789
$663$	20633.8	1.20867
$664$	−4932.80	−0.288298
$665$	4333.47	0.252699
$666$	13508.3	0.785938
$667$	−3763.84	−0.218495
$668$	20578.4	1.19192
$669$	−2131.76	−0.123197
$670$	−4311.92	−0.248633
$671$	−14391.3	−0.827976
$672$	4564.25	0.262009
$673$	11806.0	0.676206	0.338103	−	0.941109i	$-0.390215\pi$
$673$	11806.0	0.676206	0.338103	+	0.941109i	$0.390215\pi$
$674$	18484.2	1.05636
$675$	−2685.03	−0.153106
$676$	7346.60	0.417990
$677$	−821.416	−0.0466316	−0.0233158	−	0.999728i	$-0.507422\pi$
$677$	−821.416	−0.0466316	−0.0233158	+	0.999728i	$0.507422\pi$
$678$	−16271.0	−0.921659
$679$	−9689.10	−0.547620
$680$	−5010.43	−0.282561
$681$	6287.11	0.353778
$682$	−5650.54	−0.317259
$683$	32952.5	1.84611	0.923053	−	0.384673i	$-0.125686\pi$
$683$	32952.5	1.84611	0.923053	+	0.384673i	$0.125686\pi$
$684$	6284.06	0.351282
$685$	5335.51	0.297605
$686$	−1269.64	−0.0706631
$687$	17483.9	0.970962
$688$	29049.7	1.60975
$689$	28271.5	1.56322
$690$	1291.12	0.0712351
$691$	14866.1	0.818429	0.409214	−	0.912438i	$-0.365803\pi$
$691$	14866.1	0.818429	0.409214	+	0.912438i	$0.365803\pi$
$692$	−13460.2	−0.739422
$693$	1983.98	0.108752
$694$	−36887.1	−2.01760
$695$	−5940.91	−0.324247
$696$	−4176.79	−0.227472
$697$	−7309.36	−0.397219
$698$	−21847.2	−1.18471
$699$	10191.2	0.551453
$700$	3968.96	0.214304
$701$	−7493.89	−0.403766	−0.201883	−	0.979410i	$-0.564706\pi$
$701$	−7493.89	−0.403766	−0.201883	+	0.979410i	$0.564706\pi$
$702$	−5900.42	−0.317232
$703$	49656.4	2.66405
$704$	5910.37	0.316414
$705$	4202.48	0.224503
$706$	3506.18	0.186908
$707$	−1239.34	−0.0659268
$708$	−1094.87	−0.0581182
$709$	−17896.5	−0.947979	−0.473989	−	0.880531i	$-0.657186\pi$
$709$	−17896.5	−0.947979	−0.473989	+	0.880531i	$0.657186\pi$
$710$	6977.14	0.368799
$711$	−7149.75	−0.377126
$712$	−8367.82	−0.440446
$713$	−1114.90	−0.0585599
$714$	9055.82	0.474658
$715$	−9398.65	−0.491594
$716$	−1484.30	−0.0774731
$717$	−18455.3	−0.961266
$718$	−10116.9	−0.525848
$719$	−8470.81	−0.439371	−0.219686	−	0.975571i	$-0.570503\pi$
$719$	−8470.81	−0.439371	−0.219686	+	0.975571i	$0.570503\pi$
$720$	3507.98	0.181576
$721$	10188.6	0.526275
$722$	30123.7	1.55275
$723$	8376.27	0.430867
$724$	15885.5	0.815440
$725$	−16273.8	−0.833645
$726$	−3767.43	−0.192593
$727$	18742.9	0.956173	0.478086	−	0.878313i	$-0.341331\pi$
$727$	18742.9	0.956173	0.478086	+	0.878313i	$0.341331\pi$
$728$	−3516.01	−0.179000
$729$	729.000	0.0370370
$730$	21768.9	1.10370
$731$	43891.9	2.22079
$732$	7816.62	0.394686
$733$	18945.7	0.954671	0.477336	−	0.878721i	$-0.341603\pi$
$733$	18945.7	0.954671	0.477336	+	0.878721i	$0.341603\pi$
$734$	−37383.7	−1.87991
$735$	−743.107	−0.0372924
$736$	4998.94	0.250358
$737$	−7256.87	−0.362701
$738$	2090.18	0.104255
$739$	21028.3	1.04674	0.523369	−	0.852106i	$-0.324675\pi$
$739$	21028.3	1.04674	0.523369	+	0.852106i	$0.324675\pi$
$740$	−11686.9	−0.580567
$741$	−21690.0	−1.07530
$742$	12407.9	0.613892
$743$	10998.3	0.543053	0.271527	−	0.962431i	$-0.412472\pi$
$743$	10998.3	0.543053	0.271527	+	0.962431i	$0.412472\pi$
$744$	−1237.22	−0.0609659
$745$	−8742.42	−0.429929
$746$	−3840.03	−0.188463
$747$	5218.17	0.255586
$748$	20917.8	1.02250
$749$	2307.06	0.112548
$750$	12599.4	0.613422
$751$	−27694.8	−1.34567	−0.672836	−	0.739792i	$-0.734924\pi$
$751$	−27694.8	−1.34567	−0.672836	+	0.739792i	$0.734924\pi$
$752$	21366.4	1.03611
$753$	−6793.50	−0.328777
$754$	−35762.0	−1.72729
$755$	1913.31	0.0922284
$756$	−1077.60	−0.0518410
$757$	11428.6	0.548719	0.274359	−	0.961627i	$-0.411534\pi$
$757$	11428.6	0.548719	0.274359	+	0.961627i	$0.411534\pi$
$758$	20851.9	0.999175
$759$	2172.94	0.103916
$760$	5266.90	0.251382
$761$	−3125.54	−0.148884	−0.0744419	−	0.997225i	$-0.523718\pi$
$761$	−3125.54	−0.148884	−0.0744419	+	0.997225i	$0.523718\pi$
$762$	−3396.44	−0.161470
$763$	−10403.8	−0.493634
$764$	−5121.48	−0.242524
$765$	5300.29	0.250500
$766$	−4379.52	−0.206578
$767$	3779.03	0.177905
$768$	16098.2	0.756373
$769$	−9092.65	−0.426384	−0.213192	−	0.977010i	$-0.568386\pi$
$769$	−9092.65	−0.426384	−0.213192	+	0.977010i	$0.568386\pi$
$770$	−4124.91	−0.193054
$771$	−17845.9	−0.833597
$772$	−23948.9	−1.11650
$773$	35229.9	1.63924	0.819620	−	0.572908i	$-0.194185\pi$
$773$	35229.9	1.63924	0.819620	+	0.572908i	$0.194185\pi$
$774$	−12551.3	−0.582877
$775$	−4820.50	−0.223429
$776$	−11776.2	−0.544767
$777$	−8515.13	−0.393151
$778$	−21562.0	−0.993616
$779$	7683.50	0.353389
$780$	5104.85	0.234337
$781$	11742.4	0.537997
$782$	9918.28	0.453551
$783$	4418.42	0.201662
$784$	−3778.13	−0.172109
$785$	14426.0	0.655906
$786$	−921.927	−0.0418372
$787$	36470.6	1.65189	0.825945	−	0.563750i	$-0.190642\pi$
$787$	36470.6	1.65189	0.825945	+	0.563750i	$0.190642\pi$
$788$	−28899.0	−1.30645
$789$	−18403.8	−0.830410
$790$	14865.1	0.669463
$791$	10256.7	0.461044
$792$	2411.34	0.108186
$793$	−26979.7	−1.20817
$794$	−39276.0	−1.75548
$795$	7262.23	0.323981
$796$	−3461.14	−0.154117
$797$	26018.0	1.15634	0.578171	−	0.815916i	$-0.303767\pi$
$797$	26018.0	1.15634	0.578171	+	0.815916i	$0.303767\pi$
$798$	−9519.36	−0.422283
$799$	32283.1	1.42940
$800$	21614.0	0.955213
$801$	8851.91	0.390471
$802$	34605.1	1.52363
$803$	36636.5	1.61006
$804$	3941.55	0.172895
$805$	−813.879	−0.0356341
$806$	−10593.2	−0.462939
$807$	−9267.83	−0.404267
$808$	−1506.30	−0.0655833
$809$	9082.04	0.394694	0.197347	−	0.980334i	$-0.436767\pi$
$809$	9082.04	0.394694	0.197347	+	0.980334i	$0.436767\pi$
$810$	−1515.67	−0.0657471
$811$	−16350.5	−0.707944	−0.353972	−	0.935256i	$-0.615169\pi$
$811$	−16350.5	−0.707944	−0.353972	+	0.935256i	$0.615169\pi$
$812$	−6531.23	−0.282268
$813$	12792.7	0.551858
$814$	−47266.6	−2.03525
$815$	3122.99	0.134225
$816$	26947.9	1.15609
$817$	−46138.6	−1.97575
$818$	42099.4	1.79948
$819$	3719.41	0.158690
$820$	−1808.35	−0.0770127
$821$	13198.1	0.561046	0.280523	−	0.959847i	$-0.409492\pi$
$821$	13198.1	0.561046	0.280523	+	0.959847i	$0.409492\pi$
$822$	−11720.6	−0.497325
$823$	−10005.0	−0.423759	−0.211879	−	0.977296i	$-0.567958\pi$
$823$	−10005.0	−0.423759	−0.211879	+	0.977296i	$0.567958\pi$
$824$	12383.3	0.523533
$825$	9395.16	0.396482
$826$	1658.56	0.0698650
$827$	−26573.7	−1.11736	−0.558680	−	0.829383i	$-0.688692\pi$
$827$	−26573.7	−1.11736	−0.558680	+	0.829383i	$0.688692\pi$
$828$	−1180.22	−0.0495357
$829$	−31676.3	−1.32710	−0.663549	−	0.748133i	$-0.730951\pi$
$829$	−31676.3	−1.32710	−0.663549	+	0.748133i	$0.730951\pi$
$830$	−10849.1	−0.453709
$831$	7337.65	0.306306
$832$	11080.3	0.461707
$833$	−5708.47	−0.237439
$834$	13050.4	0.541846
$835$	−18245.4	−0.756176
$836$	−21988.5	−0.909674
$837$	1308.79	0.0540484
$838$	5403.02	0.222726
$839$	−16540.5	−0.680623	−0.340311	−	0.940313i	$-0.610532\pi$
$839$	−16540.5	−0.680623	−0.340311	+	0.940313i	$0.610532\pi$
$840$	−903.173	−0.0370981
$841$	2390.74	0.0980255
$842$	−59432.6	−2.43252
$843$	−20404.0	−0.833629
$844$	−6356.43	−0.259238
$845$	−6513.68	−0.265180
$846$	−9231.63	−0.375166
$847$	2374.86	0.0963412
$848$	36922.9	1.49521
$849$	24869.6	1.00533
$850$	42883.8	1.73048
$851$	−9326.09	−0.375669
$852$	−6377.84	−0.256457
$853$	4329.29	0.173777	0.0868887	−	0.996218i	$-0.472308\pi$
$853$	4329.29	0.173777	0.0868887	+	0.996218i	$0.472308\pi$
$854$	−11841.0	−0.474460
$855$	−5571.60	−0.222859
$856$	2804.00	0.111961
$857$	−19109.8	−0.761704	−0.380852	−	0.924636i	$-0.624369\pi$
$857$	−19109.8	−0.761704	−0.380852	+	0.924636i	$0.624369\pi$
$858$	20646.1	0.821499
$859$	−41024.1	−1.62948	−0.814740	−	0.579826i	$-0.803121\pi$
$859$	−41024.1	−1.62948	−0.814740	+	0.579826i	$0.803121\pi$
$860$	10859.0	0.430567
$861$	−1317.57	−0.0521519
$862$	35518.7	1.40345
$863$	30515.4	1.20366	0.601829	−	0.798625i	$-0.294439\pi$
$863$	30515.4	1.20366	0.601829	+	0.798625i	$0.294439\pi$
$864$	−5868.32	−0.231070
$865$	11934.1	0.469102
$866$	−25404.5	−0.996861
$867$	25977.3	1.01757
$868$	−1934.63	−0.0756518
$869$	25017.6	0.976600
$870$	−9186.36	−0.357985
$871$	−13604.6	−0.529247
$872$	−12644.8	−0.491063
$873$	12457.4	0.482955
$874$	−10426.0	−0.403505
$875$	−7942.23	−0.306853
$876$	−19899.0	−0.767495
$877$	5825.94	0.224319	0.112160	−	0.993690i	$-0.464223\pi$
$877$	5825.94	0.224319	0.112160	+	0.993690i	$0.464223\pi$
$878$	27909.8	1.07279
$879$	7439.66	0.285476
$880$	−12274.7	−0.470206
$881$	−6587.10	−0.251901	−0.125951	−	0.992037i	$-0.540198\pi$
$881$	−6587.10	−0.251901	−0.125951	+	0.992037i	$0.540198\pi$
$882$	1632.39	0.0623190
$883$	50190.6	1.91285	0.956426	−	0.291975i	$-0.0943125\pi$
$883$	50190.6	1.91285	0.956426	+	0.291975i	$0.0943125\pi$
$884$	39214.9	1.49201
$885$	970.738	0.0368712
$886$	−37265.3	−1.41304
$887$	10385.6	0.393139	0.196569	−	0.980490i	$-0.437020\pi$
$887$	10385.6	0.393139	0.196569	+	0.980490i	$0.437020\pi$
$888$	−10349.3	−0.391103
$889$	2140.99	0.0807724
$890$	−18404.1	−0.693152
$891$	−2550.84	−0.0959105
$892$	−4051.46	−0.152077
$893$	−33935.5	−1.27168
$894$	19204.5	0.718452
$895$	1316.01	0.0491503
$896$	−7308.38	−0.272495
$897$	4073.64	0.151633
$898$	1962.43	0.0729254
$899$	7932.50	0.294287
$900$	−5102.95	−0.188998
$901$	55787.7	2.06277
$902$	−7313.72	−0.269978
$903$	7911.88	0.291574
$904$	12466.0	0.458642
$905$	−14084.4	−0.517329
$906$	−4202.98	−0.154122
$907$	27202.9	0.995875	0.497938	−	0.867213i	$-0.334091\pi$
$907$	27202.9	0.995875	0.497938	+	0.867213i	$0.334091\pi$
$908$	11948.8	0.436712
$909$	1593.44	0.0581419
$910$	−7733.05	−0.281701
$911$	−47046.7	−1.71101	−0.855503	−	0.517798i	$-0.826752\pi$
$911$	−47046.7	−1.71101	−0.855503	+	0.517798i	$0.826752\pi$
$912$	−28327.3	−1.02852
$913$	−18258.9	−0.661862
$914$	−53926.8	−1.95158
$915$	−6930.41	−0.250396
$916$	33228.4	1.19858
$917$	581.150	0.0209283
$918$	−11643.2	−0.418609
$919$	45639.4	1.63820	0.819099	−	0.573652i	$-0.194474\pi$
$919$	45639.4	1.63820	0.819099	+	0.573652i	$0.194474\pi$
$920$	−989.189	−0.0354485
$921$	−1566.48	−0.0560447
$922$	−59170.9	−2.11355
$923$	22013.7	0.785037
$924$	3770.60	0.134247
$925$	−40323.4	−1.43332
$926$	−49937.6	−1.77220
$927$	−13099.7	−0.464131
$928$	−35567.5	−1.25815
$929$	−8934.94	−0.315550	−0.157775	−	0.987475i	$-0.550432\pi$
$929$	−8934.94	−0.315550	−0.157775	+	0.987475i	$0.550432\pi$
$930$	−2721.12	−0.0959451
$931$	6000.67	0.211240
$932$	19368.5	0.680726
$933$	−8207.70	−0.288004
$934$	5568.08	0.195068
$935$	−18546.2	−0.648691
$936$	4520.58	0.157863
$937$	−44654.4	−1.55688	−0.778440	−	0.627719i	$-0.783989\pi$
$937$	−44654.4	−1.55688	−0.778440	+	0.627719i	$0.783989\pi$
$938$	−5970.83	−0.207841
$939$	19633.6	0.682342
$940$	7986.91	0.277132
$941$	6744.79	0.233660	0.116830	−	0.993152i	$-0.462727\pi$
$941$	6744.79	0.233660	0.116830	+	0.993152i	$0.462727\pi$
$942$	−31689.7	−1.09608
$943$	−1443.06	−0.0498329
$944$	4935.46	0.170165
$945$	955.423	0.0328888
$946$	43918.1	1.50941
$947$	−2363.47	−0.0811008	−0.0405504	−	0.999177i	$-0.512911\pi$
$947$	−2363.47	−0.0811008	−0.0405504	+	0.999177i	$0.512911\pi$
$948$	−13588.3	−0.465534
$949$	68683.2	2.34937
$950$	−45078.9	−1.53953
$951$	−12391.6	−0.422528
$952$	−6938.08	−0.236202
$953$	29812.2	1.01334	0.506670	−	0.862140i	$-0.330876\pi$
$953$	29812.2	1.01334	0.506670	+	0.862140i	$0.330876\pi$
$954$	−15953.0	−0.541402
$955$	4540.83	0.153862
$956$	−35074.8	−1.18661
$957$	−15460.5	−0.522221
$958$	−40239.9	−1.35709
$959$	7388.23	0.248778
$960$	2846.24	0.0956897
$961$	−27441.3	−0.921127
$962$	−88611.7	−2.96981
$963$	−2966.22	−0.0992576
$964$	15919.3	0.531873
$965$	21233.6	0.708327
$966$	1787.85	0.0595479
$967$	−11175.2	−0.371635	−0.185818	−	0.982584i	$-0.559493\pi$
$967$	−11175.2	−0.371635	−0.185818	+	0.982584i	$0.559493\pi$
$968$	2886.40	0.0958394
$969$	−42800.5	−1.41894
$970$	−25900.3	−0.857328
$971$	−4477.24	−0.147973	−0.0739863	−	0.997259i	$-0.523572\pi$
$971$	−4477.24	−0.147973	−0.0739863	+	0.997259i	$0.523572\pi$
$972$	1385.48	0.0457194
$973$	−8226.54	−0.271049
$974$	59520.4	1.95807
$975$	17613.3	0.578540
$976$	−35235.8	−1.15561
$977$	21591.9	0.707047	0.353524	−	0.935426i	$-0.384983\pi$
$977$	21591.9	0.707047	0.353524	+	0.935426i	$0.384983\pi$
$978$	−6860.30	−0.224303
$979$	−30973.7	−1.01116
$980$	−1412.29	−0.0460346
$981$	13376.3	0.435344
$982$	−71639.8	−2.32802
$983$	−48547.2	−1.57519	−0.787597	−	0.616191i	$-0.788675\pi$
$983$	−48547.2	−1.57519	−0.787597	+	0.616191i	$0.788675\pi$
$984$	−1601.38	−0.0518803
$985$	25622.6	0.828836
$986$	−70568.7	−2.27927
$987$	5819.29	0.187670
$988$	−41222.2	−1.32738
$989$	8665.39	0.278608
$990$	5303.46	0.170258
$991$	7693.43	0.246609	0.123305	−	0.992369i	$-0.460651\pi$
$991$	7693.43	0.246609	0.123305	+	0.992369i	$0.460651\pi$
$992$	−10535.6	−0.337202
$993$	19849.5	0.634345
$994$	9661.44	0.308292
$995$	3068.74	0.0977743
$996$	9917.25	0.315502
$997$	41546.8	1.31976	0.659879	−	0.751372i	$-0.270607\pi$
$997$	41546.8	1.31976	0.659879	+	0.751372i	$0.270607\pi$
$998$	49909.3	1.58302
$999$	10948.0	0.346727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.4.a.b.1.4	✓	4
3.2	odd	2		1449.4.a.d.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.4.a.b.1.4	✓	4	1.1	even	1	trivial
1449.4.a.d.1.1		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	483.a (trivial)

\(f(q)\)	\(=\)	\(q+3.70156 q^{2} +3.00000 q^{3} +5.70156 q^{4} -5.05515 q^{5} +11.1047 q^{6} -7.00000 q^{7} -8.50781 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.70156 q^{2} +3.00000 q^{3} +5.70156 q^{4} -5.05515 q^{5} +11.1047 q^{6} -7.00000 q^{7} -8.50781 q^{8} +9.00000 q^{9} -18.7119 q^{10} -31.4918 q^{11} +17.1047 q^{12} -59.0383 q^{13} -25.9109 q^{14} -15.1654 q^{15} -77.1047 q^{16} -116.499 q^{17} +33.3141 q^{18} +122.463 q^{19} -28.8222 q^{20} -21.0000 q^{21} -116.569 q^{22} -23.0000 q^{23} -25.5234 q^{24} -99.4455 q^{25} -218.534 q^{26} +27.0000 q^{27} -39.9109 q^{28} +163.645 q^{29} -56.1358 q^{30} +48.4738 q^{31} -217.345 q^{32} -94.4755 q^{33} -431.230 q^{34} +35.3860 q^{35} +51.3141 q^{36} +405.482 q^{37} +453.303 q^{38} -177.115 q^{39} +43.0082 q^{40} +62.7416 q^{41} -77.7328 q^{42} -376.756 q^{43} -179.553 q^{44} -45.4963 q^{45} -85.1359 q^{46} -277.109 q^{47} -231.314 q^{48} +49.0000 q^{49} -368.104 q^{50} -349.498 q^{51} -336.611 q^{52} -478.867 q^{53} +99.9422 q^{54} +159.196 q^{55} +59.5547 q^{56} +367.388 q^{57} +605.743 q^{58} -64.0099 q^{59} -86.4667 q^{60} +456.987 q^{61} +179.429 q^{62} -63.0000 q^{63} -187.680 q^{64} +298.447 q^{65} -349.707 q^{66} +230.437 q^{67} -664.228 q^{68} -69.0000 q^{69} +130.984 q^{70} -372.871 q^{71} -76.5703 q^{72} -1163.37 q^{73} +1500.92 q^{74} -298.336 q^{75} +698.228 q^{76} +220.443 q^{77} -655.602 q^{78} -794.417 q^{79} +389.776 q^{80} +81.0000 q^{81} +232.242 q^{82} +579.797 q^{83} -119.733 q^{84} +588.922 q^{85} -1394.59 q^{86} +490.936 q^{87} +267.926 q^{88} +983.546 q^{89} -168.407 q^{90} +413.268 q^{91} -131.136 q^{92} +145.421 q^{93} -1025.74 q^{94} -619.067 q^{95} -652.036 q^{96} +1384.16 q^{97} +181.377 q^{98} -283.426 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 12 q^{3} + 10 q^{4} - 23 q^{5} + 6 q^{6} - 28 q^{7} + 30 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 12 * q^3 + 10 * q^4 - 23 * q^5 + 6 * q^6 - 28 * q^7 + 30 * q^8 + 36 * q^9	\( 4 q + 2 q^{2} + 12 q^{3} + 10 q^{4} - 23 q^{5} + 6 q^{6} - 28 q^{7} + 30 q^{8} + 36 q^{9} - 32 q^{10} - 48 q^{11} + 30 q^{12} - 107 q^{13} - 14 q^{14} - 69 q^{15} - 270 q^{16} - 6 q^{17} + 18 q^{18} + 76 q^{19} - 78 q^{20} - 84 q^{21} - 106 q^{22} - 92 q^{23} + 90 q^{24} + 115 q^{25} - 320 q^{26} + 108 q^{27} - 70 q^{28} + 204 q^{29} - 96 q^{30} - 276 q^{31} - 498 q^{32} - 144 q^{33} - 126 q^{34} + 161 q^{35} + 90 q^{36} - 248 q^{37} - 372 q^{38} - 321 q^{39} - 70 q^{40} - 450 q^{41} - 42 q^{42} + 109 q^{43} - 202 q^{44} - 207 q^{45} - 46 q^{46} - 688 q^{47} - 810 q^{48} + 196 q^{49} - 1152 q^{50} - 18 q^{51} - 534 q^{52} - 633 q^{53} + 54 q^{54} + 581 q^{55} - 210 q^{56} + 228 q^{57} - 308 q^{58} + 585 q^{59} - 234 q^{60} + 1311 q^{61} + 846 q^{62} - 252 q^{63} + 722 q^{64} - 1614 q^{65} - 318 q^{66} + 365 q^{67} - 138 q^{68} - 276 q^{69} + 224 q^{70} - 3049 q^{71} + 270 q^{72} - 1164 q^{73} + 2090 q^{74} + 345 q^{75} - 220 q^{76} + 336 q^{77} - 960 q^{78} - 476 q^{79} + 1614 q^{80} + 324 q^{81} + 1784 q^{82} + 1462 q^{83} - 210 q^{84} - 891 q^{85} - 1032 q^{86} + 612 q^{87} + 50 q^{88} - 1767 q^{89} - 288 q^{90} + 749 q^{91} - 230 q^{92} - 828 q^{93} - 1246 q^{94} - 1927 q^{95} - 1494 q^{96} + 1788 q^{97} + 98 q^{98} - 432 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 12 * q^3 + 10 * q^4 - 23 * q^5 + 6 * q^6 - 28 * q^7 + 30 * q^8 + 36 * q^9 - 32 * q^10 - 48 * q^11 + 30 * q^12 - 107 * q^13 - 14 * q^14 - 69 * q^15 - 270 * q^16 - 6 * q^17 + 18 * q^18 + 76 * q^19 - 78 * q^20 - 84 * q^21 - 106 * q^22 - 92 * q^23 + 90 * q^24 + 115 * q^25 - 320 * q^26 + 108 * q^27 - 70 * q^28 + 204 * q^29 - 96 * q^30 - 276 * q^31 - 498 * q^32 - 144 * q^33 - 126 * q^34 + 161 * q^35 + 90 * q^36 - 248 * q^37 - 372 * q^38 - 321 * q^39 - 70 * q^40 - 450 * q^41 - 42 * q^42 + 109 * q^43 - 202 * q^44 - 207 * q^45 - 46 * q^46 - 688 * q^47 - 810 * q^48 + 196 * q^49 - 1152 * q^50 - 18 * q^51 - 534 * q^52 - 633 * q^53 + 54 * q^54 + 581 * q^55 - 210 * q^56 + 228 * q^57 - 308 * q^58 + 585 * q^59 - 234 * q^60 + 1311 * q^61 + 846 * q^62 - 252 * q^63 + 722 * q^64 - 1614 * q^65 - 318 * q^66 + 365 * q^67 - 138 * q^68 - 276 * q^69 + 224 * q^70 - 3049 * q^71 + 270 * q^72 - 1164 * q^73 + 2090 * q^74 + 345 * q^75 - 220 * q^76 + 336 * q^77 - 960 * q^78 - 476 * q^79 + 1614 * q^80 + 324 * q^81 + 1784 * q^82 + 1462 * q^83 - 210 * q^84 - 891 * q^85 - 1032 * q^86 + 612 * q^87 + 50 * q^88 - 1767 * q^89 - 288 * q^90 + 749 * q^91 - 230 * q^92 - 828 * q^93 - 1246 * q^94 - 1927 * q^95 - 1494 * q^96 + 1788 * q^97 + 98 * q^98 - 432 * q^99

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