LMFDB - Embedded newform 1045.4.a.b.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, 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("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + \cdots + 17756160 $ x^20 - 8x^19 - 82x^18 + 700x^17 + 2826x^16 - 25467x^15 - 53768x^14 + 498499x^13 + 623613x^12 - 5673747x^11 - 4539454x^10 + 37893109x^9 + 19879768x^8 - 143049638x^7 - 45064360x^6 + 280461480x^5 + 43097920x^4 - 240447168x^3 - 36068096x^2 + 74703872*x + 17756160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$3.74913$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.74913 q^{2} -7.66455 q^{3} -0.442295 q^{4} +5.00000 q^{5} -21.0708 q^{6} +25.2191 q^{7} -23.2090 q^{8} +31.7453 q^{9} +O(q^{10})$	\(q+2.74913 q^{2} -7.66455 q^{3} -0.442295 q^{4} +5.00000 q^{5} -21.0708 q^{6} +25.2191 q^{7} -23.2090 q^{8} +31.7453 q^{9} +13.7456 q^{10} +11.0000 q^{11} +3.38999 q^{12} -28.9340 q^{13} +69.3307 q^{14} -38.3227 q^{15} -60.2660 q^{16} -120.346 q^{17} +87.2719 q^{18} +19.0000 q^{19} -2.21148 q^{20} -193.293 q^{21} +30.2404 q^{22} +89.9077 q^{23} +177.886 q^{24} +25.0000 q^{25} -79.5433 q^{26} -36.3707 q^{27} -11.1543 q^{28} +151.358 q^{29} -105.354 q^{30} +248.274 q^{31} +19.9926 q^{32} -84.3100 q^{33} -330.847 q^{34} +126.096 q^{35} -14.0408 q^{36} -172.303 q^{37} +52.2334 q^{38} +221.766 q^{39} -116.045 q^{40} -200.930 q^{41} -531.388 q^{42} +351.885 q^{43} -4.86525 q^{44} +158.727 q^{45} +247.168 q^{46} -600.661 q^{47} +461.912 q^{48} +293.005 q^{49} +68.7282 q^{50} +922.398 q^{51} +12.7974 q^{52} +353.369 q^{53} -99.9877 q^{54} +55.0000 q^{55} -585.310 q^{56} -145.626 q^{57} +416.103 q^{58} -638.520 q^{59} +16.9500 q^{60} -467.807 q^{61} +682.537 q^{62} +800.590 q^{63} +537.090 q^{64} -144.670 q^{65} -231.779 q^{66} -669.209 q^{67} +53.2285 q^{68} -689.102 q^{69} +346.653 q^{70} +801.627 q^{71} -736.775 q^{72} -761.026 q^{73} -473.682 q^{74} -191.614 q^{75} -8.40361 q^{76} +277.411 q^{77} +609.664 q^{78} -604.467 q^{79} -301.330 q^{80} -578.359 q^{81} -552.382 q^{82} -1379.76 q^{83} +85.4927 q^{84} -601.730 q^{85} +967.376 q^{86} -1160.09 q^{87} -255.298 q^{88} +493.198 q^{89} +436.360 q^{90} -729.691 q^{91} -39.7658 q^{92} -1902.91 q^{93} -1651.29 q^{94} +95.0000 q^{95} -153.234 q^{96} -859.990 q^{97} +805.509 q^{98} +349.198 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) $ 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * 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 $

$2$	2.74913	0.971964	0.485982	−	0.873969i	$-0.338462\pi$
$2$	2.74913	0.971964	0.485982	+	0.873969i	$0.338462\pi$
$3$	−7.66455	−1.47504	−0.737522	−	0.675324i	$-0.764004\pi$
$3$	−7.66455	−1.47504	−0.737522	+	0.675324i	$0.764004\pi$
$4$	−0.442295	−0.0552869
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−21.0708	−1.43369
$7$	25.2191	1.36171	0.680853	−	0.732420i	$-0.261609\pi$
$7$	25.2191	1.36171	0.680853	+	0.732420i	$0.261609\pi$
$8$	−23.2090	−1.02570
$9$	31.7453	1.17575
$10$	13.7456	0.434675
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	3.38999	0.0815506
$13$	−28.9340	−0.617296	−0.308648	−	0.951176i	$-0.599877\pi$
$13$	−28.9340	−0.617296	−0.308648	+	0.951176i	$0.599877\pi$
$14$	69.3307	1.32353
$15$	−38.3227	−0.659659
$16$	−60.2660	−0.941656
$17$	−120.346	−1.71695	−0.858477	−	0.512853i	$-0.828589\pi$
$17$	−120.346	−1.71695	−0.858477	+	0.512853i	$0.828589\pi$
$18$	87.2719	1.14279
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−2.21148	−0.0247250
$21$	−193.293	−2.00858
$22$	30.2404	0.293058
$23$	89.9077	0.815090	0.407545	−	0.913185i	$-0.366385\pi$
$23$	89.9077	0.815090	0.407545	+	0.913185i	$0.366385\pi$
$24$	177.886	1.51295
$25$	25.0000	0.200000
$26$	−79.5433	−0.599989
$27$	−36.3707	−0.259242
$28$	−11.1543	−0.0752845
$29$	151.358	0.969189	0.484595	−	0.874739i	$-0.338967\pi$
$29$	151.358	0.969189	0.484595	+	0.874739i	$0.338967\pi$
$30$	−105.354	−0.641165
$31$	248.274	1.43843	0.719215	−	0.694788i	$-0.244502\pi$
$31$	248.274	1.43843	0.719215	+	0.694788i	$0.244502\pi$
$32$	19.9926	0.110445
$33$	−84.3100	−0.444742
$34$	−330.847	−1.66882
$35$	126.096	0.608973
$36$	−14.0408	−0.0650037
$37$	−172.303	−0.765578	−0.382789	−	0.923836i	$-0.625036\pi$
$37$	−172.303	−0.765578	−0.382789	+	0.923836i	$0.625036\pi$
$38$	52.2334	0.222984
$39$	221.766	0.910539
$40$	−116.045	−0.458707
$41$	−200.930	−0.765366	−0.382683	−	0.923880i	$-0.625000\pi$
$41$	−200.930	−0.765366	−0.382683	+	0.923880i	$0.625000\pi$
$42$	−531.388	−1.95226
$43$	351.885	1.24795	0.623976	−	0.781444i	$-0.285516\pi$
$43$	351.885	1.24795	0.623976	+	0.781444i	$0.285516\pi$
$44$	−4.86525	−0.0166696
$45$	158.727	0.525812
$46$	247.168	0.792237
$47$	−600.661	−1.86416	−0.932079	−	0.362254i	$-0.882007\pi$
$47$	−600.661	−1.86416	−0.932079	+	0.362254i	$0.882007\pi$
$48$	461.912	1.38898
$49$	293.005	0.854243
$50$	68.7282	0.194393
$51$	922.398	2.53258
$52$	12.7974	0.0341284
$53$	353.369	0.915829	0.457914	−	0.888996i	$-0.348597\pi$
$53$	353.369	0.915829	0.457914	+	0.888996i	$0.348597\pi$
$54$	−99.9877	−0.251974
$55$	55.0000	0.134840
$56$	−585.310	−1.39670
$57$	−145.626	−0.338398
$58$	416.103	0.942017
$59$	−638.520	−1.40895	−0.704476	−	0.709727i	$-0.748818\pi$
$59$	−638.520	−1.40895	−0.704476	+	0.709727i	$0.748818\pi$
$60$	16.9500	0.0364705
$61$	−467.807	−0.981911	−0.490955	−	0.871185i	$-0.663352\pi$
$61$	−467.807	−0.981911	−0.490955	+	0.871185i	$0.663352\pi$
$62$	682.537	1.39810
$63$	800.590	1.60103
$64$	537.090	1.04900
$65$	−144.670	−0.276063
$66$	−231.779	−0.432273
$67$	−669.209	−1.22025	−0.610126	−	0.792304i	$-0.708881\pi$
$67$	−669.209	−1.22025	−0.610126	+	0.792304i	$0.708881\pi$
$68$	53.2285	0.0949250
$69$	−689.102	−1.20229
$70$	346.653	0.591900
$71$	801.627	1.33994	0.669970	−	0.742389i	$-0.266307\pi$
$71$	801.627	1.33994	0.669970	+	0.742389i	$0.266307\pi$
$72$	−736.775	−1.20597
$73$	−761.026	−1.22016	−0.610078	−	0.792341i	$-0.708862\pi$
$73$	−761.026	−1.22016	−0.610078	+	0.792341i	$0.708862\pi$
$74$	−473.682	−0.744114
$75$	−191.614	−0.295009
$76$	−8.40361	−0.0126837
$77$	277.411	0.410570
$78$	609.664	0.885010
$79$	−604.467	−0.860859	−0.430429	−	0.902624i	$-0.641638\pi$
$79$	−604.467	−0.860859	−0.430429	+	0.902624i	$0.641638\pi$
$80$	−301.330	−0.421122
$81$	−578.359	−0.793359
$82$	−552.382	−0.743908
$83$	−1379.76	−1.82467	−0.912337	−	0.409440i	$-0.865724\pi$
$83$	−1379.76	−1.82467	−0.912337	+	0.409440i	$0.865724\pi$
$84$	85.4927	0.111048
$85$	−601.730	−0.767845
$86$	967.376	1.21296
$87$	−1160.09	−1.42960
$88$	−255.298	−0.309260
$89$	493.198	0.587404	0.293702	−	0.955897i	$-0.405113\pi$
$89$	493.198	0.587404	0.293702	+	0.955897i	$0.405113\pi$
$90$	436.360	0.511071
$91$	−729.691	−0.840576
$92$	−39.7658	−0.0450638
$93$	−1902.91	−2.12174
$94$	−1651.29	−1.81189
$95$	95.0000	0.102598
$96$	−153.234	−0.162911
$97$	−859.990	−0.900193	−0.450097	−	0.892980i	$-0.648610\pi$
$97$	−859.990	−0.900193	−0.450097	+	0.892980i	$0.648610\pi$
$98$	805.509	0.830293
$99$	349.198	0.354503
$100$	−11.0574	−0.0110574
$101$	−1681.00	−1.65609	−0.828046	−	0.560660i	$-0.810548\pi$
$101$	−1681.00	−1.65609	−0.828046	+	0.560660i	$0.810548\pi$
$102$	2535.79	2.46158
$103$	1234.95	1.18139	0.590696	−	0.806894i	$-0.298853\pi$
$103$	1234.95	1.18139	0.590696	+	0.806894i	$0.298853\pi$
$104$	671.528	0.633161
$105$	−966.467	−0.898262
$106$	971.456	0.890152
$107$	−264.976	−0.239404	−0.119702	−	0.992810i	$-0.538194\pi$
$107$	−264.976	−0.239404	−0.119702	+	0.992810i	$0.538194\pi$
$108$	16.0866	0.0143327
$109$	−256.864	−0.225717	−0.112858	−	0.993611i	$-0.536001\pi$
$109$	−256.864	−0.225717	−0.112858	+	0.993611i	$0.536001\pi$
$110$	151.202	0.131060
$111$	1320.62	1.12926
$112$	−1519.86	−1.28226
$113$	−274.637	−0.228634	−0.114317	−	0.993444i	$-0.536468\pi$
$113$	−274.637	−0.228634	−0.114317	+	0.993444i	$0.536468\pi$
$114$	−400.346	−0.328911
$115$	449.539	0.364519
$116$	−66.9449	−0.0535835
$117$	−918.519	−0.725788
$118$	−1755.37	−1.36945
$119$	−3035.02	−2.33799
$120$	889.431	0.676613
$121$	121.000	0.0909091
$122$	−1286.06	−0.954382
$123$	1540.04	1.12895
$124$	−109.810	−0.0795263
$125$	125.000	0.0894427
$126$	2200.92	1.55614
$127$	−2720.56	−1.90087	−0.950437	−	0.310918i	$-0.899364\pi$
$127$	−2720.56	−1.90087	−0.950437	+	0.310918i	$0.899364\pi$
$128$	1316.59	0.909150
$129$	−2697.04	−1.84078
$130$	−397.717	−0.268323
$131$	147.929	0.0986612	0.0493306	−	0.998783i	$-0.484291\pi$
$131$	147.929	0.0986612	0.0493306	+	0.998783i	$0.484291\pi$
$132$	37.2899	0.0245884
$133$	479.164	0.312397
$134$	−1839.74	−1.18604
$135$	−181.853	−0.115937
$136$	2793.11	1.76108
$137$	−1612.88	−1.00583	−0.502913	−	0.864337i	$-0.667738\pi$
$137$	−1612.88	−1.00583	−0.502913	+	0.864337i	$0.667738\pi$
$138$	−1894.43	−1.16858
$139$	1695.31	1.03449	0.517245	−	0.855837i	$-0.326958\pi$
$139$	1695.31	1.03449	0.517245	+	0.855837i	$0.326958\pi$
$140$	−55.7715	−0.0336682
$141$	4603.80	2.74971
$142$	2203.78	1.30237
$143$	−318.274	−0.186122
$144$	−1913.16	−1.10715
$145$	756.790	0.433435
$146$	−2092.16	−1.18595
$147$	−2245.75	−1.26005
$148$	76.2086	0.0423264
$149$	−1195.86	−0.657508	−0.328754	−	0.944416i	$-0.606629\pi$
$149$	−1195.86	−0.657508	−0.328754	+	0.944416i	$0.606629\pi$
$150$	−526.771	−0.286738
$151$	−43.3327	−0.0233534	−0.0116767	−	0.999932i	$-0.503717\pi$
$151$	−43.3327	−0.0233534	−0.0116767	+	0.999932i	$0.503717\pi$
$152$	−440.970	−0.235312
$153$	−3820.42	−2.01871
$154$	762.637	0.399059
$155$	1241.37	0.643285
$156$	−98.0861	−0.0503408
$157$	2262.91	1.15032	0.575159	−	0.818042i	$-0.304940\pi$
$157$	2262.91	1.15032	0.575159	+	0.818042i	$0.304940\pi$
$158$	−1661.76	−0.836724
$159$	−2708.41	−1.35089
$160$	99.9631	0.0493923
$161$	2267.40	1.10991
$162$	−1589.98	−0.771116
$163$	2917.74	1.40206	0.701029	−	0.713133i	$-0.252724\pi$
$163$	2917.74	1.40206	0.701029	+	0.713133i	$0.252724\pi$
$164$	88.8704	0.0423147
$165$	−421.550	−0.198895
$166$	−3793.13	−1.77352
$167$	2734.99	1.26731	0.633653	−	0.773618i	$-0.281555\pi$
$167$	2734.99	1.26731	0.633653	+	0.773618i	$0.281555\pi$
$168$	4486.14	2.06020
$169$	−1359.82	−0.618945
$170$	−1654.23	−0.746317
$171$	603.161	0.269736
$172$	−155.637	−0.0689953
$173$	−2309.92	−1.01514	−0.507572	−	0.861610i	$-0.669457\pi$
$173$	−2309.92	−1.01514	−0.507572	+	0.861610i	$0.669457\pi$
$174$	−3189.24	−1.38952
$175$	630.479	0.272341
$176$	−662.926	−0.283920
$177$	4893.97	2.07827
$178$	1355.87	0.570935
$179$	−857.629	−0.358113	−0.179056	−	0.983839i	$-0.557304\pi$
$179$	−857.629	−0.358113	−0.179056	+	0.983839i	$0.557304\pi$
$180$	−70.2040	−0.0290705
$181$	−1779.79	−0.730887	−0.365443	−	0.930834i	$-0.619083\pi$
$181$	−1779.79	−0.730887	−0.365443	+	0.930834i	$0.619083\pi$
$182$	−2006.01	−0.817009
$183$	3585.53	1.44836
$184$	−2086.66	−0.836038
$185$	−861.513	−0.342377
$186$	−5231.34	−2.06226
$187$	−1323.81	−0.517681
$188$	265.670	0.103064
$189$	−917.238	−0.353012
$190$	261.167	0.0997214
$191$	−1084.88	−0.410990	−0.205495	−	0.978658i	$-0.565880\pi$
$191$	−1084.88	−0.410990	−0.205495	+	0.978658i	$0.565880\pi$
$192$	−4116.56	−1.54733
$193$	4577.50	1.70723	0.853615	−	0.520905i	$-0.174405\pi$
$193$	4577.50	1.70723	0.853615	+	0.520905i	$0.174405\pi$
$194$	−2364.22	−0.874955
$195$	1108.83	0.407205
$196$	−129.595	−0.0472284
$197$	−4420.88	−1.59886	−0.799428	−	0.600763i	$-0.794864\pi$
$197$	−4420.88	−1.59886	−0.799428	+	0.600763i	$0.794864\pi$
$198$	959.991	0.344564
$199$	−1701.79	−0.606214	−0.303107	−	0.952957i	$-0.598024\pi$
$199$	−1701.79	−0.606214	−0.303107	+	0.952957i	$0.598024\pi$
$200$	−580.224	−0.205140
$201$	5129.19	1.79993
$202$	−4621.27	−1.60966
$203$	3817.12	1.31975
$204$	−407.972	−0.140018
$205$	−1004.65	−0.342282
$206$	3395.04	1.14827
$207$	2854.15	0.958343
$208$	1743.74	0.581281
$209$	209.000	0.0691714
$210$	−2656.94	−0.873078
$211$	−4957.87	−1.61760	−0.808800	−	0.588084i	$-0.799883\pi$
$211$	−4957.87	−1.61760	−0.808800	+	0.588084i	$0.799883\pi$
$212$	−156.293	−0.0506333
$213$	−6144.11	−1.97647
$214$	−728.453	−0.232692
$215$	1759.42	0.558101
$216$	844.126	0.265905
$217$	6261.25	1.95872
$218$	−706.152	−0.219388
$219$	5832.92	1.79978
$220$	−24.3262	−0.00745488
$221$	3482.09	1.05987
$222$	3630.56	1.09760
$223$	2512.39	0.754449	0.377224	−	0.926122i	$-0.376879\pi$
$223$	2512.39	0.754449	0.377224	+	0.926122i	$0.376879\pi$
$224$	504.197	0.150393
$225$	793.633	0.235150
$226$	−755.012	−0.222224
$227$	−3873.61	−1.13260	−0.566300	−	0.824199i	$-0.691626\pi$
$227$	−3873.61	−1.13260	−0.566300	+	0.824199i	$0.691626\pi$
$228$	64.4099	0.0187090
$229$	5345.27	1.54247	0.771234	−	0.636551i	$-0.219640\pi$
$229$	5345.27	1.54247	0.771234	+	0.636551i	$0.219640\pi$
$230$	1235.84	0.354299
$231$	−2126.23	−0.605608
$232$	−3512.86	−0.994098
$233$	−1683.22	−0.473268	−0.236634	−	0.971599i	$-0.576044\pi$
$233$	−1683.22	−0.473268	−0.236634	+	0.971599i	$0.576044\pi$
$234$	−2525.13	−0.705439
$235$	−3003.31	−0.833677
$236$	282.414	0.0778966
$237$	4632.97	1.26980
$238$	−8343.67	−2.27244
$239$	2362.20	0.639321	0.319660	−	0.947532i	$-0.396431\pi$
$239$	2362.20	0.639321	0.319660	+	0.947532i	$0.396431\pi$
$240$	2309.56	0.621173
$241$	−6045.16	−1.61578	−0.807890	−	0.589333i	$-0.799390\pi$
$241$	−6045.16	−1.61578	−0.807890	+	0.589333i	$0.799390\pi$
$242$	332.644	0.0883603
$243$	5414.87	1.42948
$244$	206.909	0.0542868
$245$	1465.03	0.382029
$246$	4233.76	1.09730
$247$	−549.746	−0.141617
$248$	−5762.18	−1.47540
$249$	10575.2	2.69147
$250$	343.641	0.0869351
$251$	−5519.96	−1.38811	−0.694057	−	0.719920i	$-0.744179\pi$
$251$	−5519.96	−1.38811	−0.694057	+	0.719920i	$0.744179\pi$
$252$	−354.097	−0.0885159
$253$	988.985	0.245759
$254$	−7479.18	−1.84758
$255$	4611.99	1.13260
$256$	−677.250	−0.165344
$257$	−219.080	−0.0531744	−0.0265872	−	0.999646i	$-0.508464\pi$
$257$	−219.080	−0.0531744	−0.0265872	+	0.999646i	$0.508464\pi$
$258$	−7414.50	−1.78917
$259$	−4345.33	−1.04249
$260$	63.9869	0.0152627
$261$	4804.91	1.13953
$262$	406.676	0.0958951
$263$	−4749.98	−1.11367	−0.556837	−	0.830622i	$-0.687985\pi$
$263$	−4749.98	−1.11367	−0.556837	+	0.830622i	$0.687985\pi$
$264$	1956.75	0.456172
$265$	1766.84	0.409571
$266$	1317.28	0.303638
$267$	−3780.14	−0.866446
$268$	295.988	0.0674640
$269$	2010.85	0.455776	0.227888	−	0.973687i	$-0.426818\pi$
$269$	2010.85	0.455776	0.227888	+	0.973687i	$0.426818\pi$
$270$	−499.938	−0.112686
$271$	−646.709	−0.144962	−0.0724811	−	0.997370i	$-0.523092\pi$
$271$	−646.709	−0.144962	−0.0724811	+	0.997370i	$0.523092\pi$
$272$	7252.78	1.61678
$273$	5592.75	1.23989
$274$	−4434.03	−0.977625
$275$	275.000	0.0603023
$276$	304.787	0.0664710
$277$	−774.834	−0.168069	−0.0840347	−	0.996463i	$-0.526781\pi$
$277$	−774.834	−0.168069	−0.0840347	+	0.996463i	$0.526781\pi$
$278$	4660.62	1.00549
$279$	7881.53	1.69124
$280$	−2926.55	−0.624624
$281$	27.0541	0.00574346	0.00287173	−	0.999996i	$-0.499086\pi$
$281$	27.0541	0.00574346	0.00287173	+	0.999996i	$0.499086\pi$
$282$	12656.4	2.67262
$283$	15.1833	0.00318923	0.00159462	−	0.999999i	$-0.499492\pi$
$283$	15.1833	0.00318923	0.00159462	+	0.999999i	$0.499492\pi$
$284$	−354.556	−0.0740811
$285$	−728.132	−0.151336
$286$	−874.976	−0.180904
$287$	−5067.28	−1.04220
$288$	634.672	0.129856
$289$	9570.17	1.94793
$290$	2080.51	0.421283
$291$	6591.43	1.32782
$292$	336.598	0.0674587
$293$	−3195.85	−0.637214	−0.318607	−	0.947887i	$-0.603215\pi$
$293$	−3195.85	−0.637214	−0.318607	+	0.947887i	$0.603215\pi$
$294$	−6173.87	−1.22472
$295$	−3192.60	−0.630103
$296$	3998.96	0.785253
$297$	−400.078	−0.0781645
$298$	−3287.57	−0.639074
$299$	−2601.39	−0.503152
$300$	84.7498	0.0163101
$301$	8874.23	1.69934
$302$	−119.127	−0.0226987
$303$	12884.1	2.44281
$304$	−1145.05	−0.216031
$305$	−2339.04	−0.439124
$306$	−10502.8	−1.96211
$307$	−4245.06	−0.789181	−0.394590	−	0.918857i	$-0.629113\pi$
$307$	−4245.06	−0.789181	−0.394590	+	0.918857i	$0.629113\pi$
$308$	−122.697	−0.0226991
$309$	−9465.34	−1.74260
$310$	3412.68	0.625250
$311$	8862.03	1.61582	0.807909	−	0.589307i	$-0.200599\pi$
$311$	8862.03	1.61582	0.807909	+	0.589307i	$0.200599\pi$
$312$	−5146.96	−0.933940
$313$	−2013.70	−0.363645	−0.181823	−	0.983331i	$-0.558200\pi$
$313$	−2013.70	−0.363645	−0.181823	+	0.983331i	$0.558200\pi$
$314$	6221.03	1.11807
$315$	4002.95	0.716002
$316$	267.353	0.0475942
$317$	3933.06	0.696855	0.348427	−	0.937336i	$-0.386716\pi$
$317$	3933.06	0.696855	0.348427	+	0.937336i	$0.386716\pi$
$318$	−7445.77	−1.31301
$319$	1664.94	0.292222
$320$	2685.45	0.469129
$321$	2030.92	0.353131
$322$	6233.36	1.07879
$323$	−2286.57	−0.393896
$324$	255.805	0.0438623
$325$	−723.350	−0.123459
$326$	8021.25	1.36275
$327$	1968.75	0.332942
$328$	4663.37	0.785036
$329$	−15148.2	−2.53844
$330$	−1158.90	−0.193318
$331$	−1013.06	−0.168226	−0.0841131	−	0.996456i	$-0.526806\pi$
$331$	−1013.06	−0.168226	−0.0841131	+	0.996456i	$0.526806\pi$
$332$	610.260	0.100881
$333$	−5469.80	−0.900130
$334$	7518.84	1.23177
$335$	−3346.05	−0.545714
$336$	11649.0	1.89139
$337$	−136.648	−0.0220882	−0.0110441	−	0.999939i	$-0.503516\pi$
$337$	−136.648	−0.0220882	−0.0110441	+	0.999939i	$0.503516\pi$
$338$	−3738.33	−0.601592
$339$	2104.97	0.337245
$340$	266.142	0.0424518
$341$	2731.01	0.433703
$342$	1658.17	0.262174
$343$	−1260.82	−0.198478
$344$	−8166.87	−1.28002
$345$	−3445.51	−0.537681
$346$	−6350.26	−0.986682
$347$	−2125.53	−0.328831	−0.164415	−	0.986391i	$-0.552574\pi$
$347$	−2125.53	−0.328831	−0.164415	+	0.986391i	$0.552574\pi$
$348$	513.103	0.0790379
$349$	707.436	0.108505	0.0542524	−	0.998527i	$-0.482722\pi$
$349$	707.436	0.108505	0.0542524	+	0.998527i	$0.482722\pi$
$350$	1733.27	0.264706
$351$	1052.35	0.160029
$352$	219.919	0.0333003
$353$	4851.78	0.731541	0.365771	−	0.930705i	$-0.380805\pi$
$353$	4851.78	0.731541	0.365771	+	0.930705i	$0.380805\pi$
$354$	13454.1	2.02000
$355$	4008.14	0.599239
$356$	−218.139	−0.0324757
$357$	23262.1	3.44863
$358$	−2357.73	−0.348073
$359$	−2531.13	−0.372112	−0.186056	−	0.982539i	$-0.559571\pi$
$359$	−2531.13	−0.372112	−0.186056	+	0.982539i	$0.559571\pi$
$360$	−3683.88	−0.539326
$361$	361.000	0.0526316
$362$	−4892.86	−0.710395
$363$	−927.410	−0.134095
$364$	322.739	0.0464728
$365$	−3805.13	−0.545671
$366$	9857.08	1.40775
$367$	−13301.8	−1.89196	−0.945981	−	0.324221i	$-0.894898\pi$
$367$	−13301.8	−1.89196	−0.945981	+	0.324221i	$0.894898\pi$
$368$	−5418.38	−0.767534
$369$	−6378.59	−0.899881
$370$	−2368.41	−0.332778
$371$	8911.66	1.24709
$372$	841.647	0.117305
$373$	8008.92	1.11176	0.555880	−	0.831263i	$-0.312381\pi$
$373$	8008.92	1.11176	0.555880	+	0.831263i	$0.312381\pi$
$374$	−3639.31	−0.503167
$375$	−958.069	−0.131932
$376$	13940.7	1.91207
$377$	−4379.40	−0.598277
$378$	−2521.60	−0.343115
$379$	7127.74	0.966036	0.483018	−	0.875610i	$-0.339541\pi$
$379$	7127.74	0.966036	0.483018	+	0.875610i	$0.339541\pi$
$380$	−42.0180	−0.00567232
$381$	20851.9	2.80387
$382$	−2982.47	−0.399467
$383$	−3693.73	−0.492795	−0.246398	−	0.969169i	$-0.579247\pi$
$383$	−3693.73	−0.492795	−0.246398	+	0.969169i	$0.579247\pi$
$384$	−10091.1	−1.34103
$385$	1387.05	0.183612
$386$	12584.1	1.65936
$387$	11170.7	1.46728
$388$	380.369	0.0497689
$389$	11341.9	1.47830	0.739148	−	0.673543i	$-0.235228\pi$
$389$	11341.9	1.47830	0.739148	+	0.673543i	$0.235228\pi$
$390$	3048.32	0.395789
$391$	−10820.0	−1.39947
$392$	−6800.35	−0.876197
$393$	−1133.81	−0.145530
$394$	−12153.6	−1.55403
$395$	−3022.34	−0.384988
$396$	−154.449	−0.0195994
$397$	−187.777	−0.0237387	−0.0118693	−	0.999930i	$-0.503778\pi$
$397$	−187.777	−0.0237387	−0.0118693	+	0.999930i	$0.503778\pi$
$398$	−4678.43	−0.589218
$399$	−3672.57	−0.460799
$400$	−1506.65	−0.188331
$401$	−2111.79	−0.262987	−0.131494	−	0.991317i	$-0.541977\pi$
$401$	−2111.79	−0.262987	−0.131494	+	0.991317i	$0.541977\pi$
$402$	14100.8	1.74946
$403$	−7183.56	−0.887937
$404$	743.496	0.0915602
$405$	−2891.79	−0.354801
$406$	10493.8	1.28275
$407$	−1895.33	−0.230830
$408$	−21407.9	−2.59767
$409$	13834.5	1.67254	0.836272	−	0.548315i	$-0.184730\pi$
$409$	13834.5	1.67254	0.836272	+	0.548315i	$0.184730\pi$
$410$	−2761.91	−0.332686
$411$	12362.0	1.48364
$412$	−546.213	−0.0653155
$413$	−16102.9	−1.91858
$414$	7846.42	0.931475
$415$	−6898.78	−0.816019
$416$	−578.467	−0.0681771
$417$	−12993.8	−1.52592
$418$	574.568	0.0672321
$419$	−10938.1	−1.27533	−0.637665	−	0.770314i	$-0.720100\pi$
$419$	−10938.1	−1.27533	−0.637665	+	0.770314i	$0.720100\pi$
$420$	427.464	0.0496621
$421$	−2700.18	−0.312586	−0.156293	−	0.987711i	$-0.549954\pi$
$421$	−2700.18	−0.312586	−0.156293	+	0.987711i	$0.549954\pi$
$422$	−13629.8	−1.57225
$423$	−19068.2	−2.19179
$424$	−8201.31	−0.939366
$425$	−3008.65	−0.343391
$426$	−16891.0	−1.92105
$427$	−11797.7	−1.33707
$428$	117.198	0.0132359
$429$	2439.43	0.274538
$430$	4836.88	0.542454
$431$	9018.19	1.00787	0.503934	−	0.863742i	$-0.331886\pi$
$431$	9018.19	1.00787	0.503934	+	0.863742i	$0.331886\pi$
$432$	2191.92	0.244117
$433$	10137.9	1.12516	0.562580	−	0.826743i	$-0.309809\pi$
$433$	10137.9	1.12516	0.562580	+	0.826743i	$0.309809\pi$
$434$	17213.0	1.90380
$435$	−5800.46	−0.639335
$436$	113.610	0.0124792
$437$	1708.25	0.186994
$438$	16035.5	1.74932
$439$	6817.75	0.741215	0.370607	−	0.928790i	$-0.379150\pi$
$439$	6817.75	0.741215	0.370607	+	0.928790i	$0.379150\pi$
$440$	−1276.49	−0.138305
$441$	9301.55	1.00438
$442$	9572.72	1.03015
$443$	2265.15	0.242935	0.121468	−	0.992595i	$-0.461240\pi$
$443$	2265.15	0.242935	0.121468	+	0.992595i	$0.461240\pi$
$444$	−584.105	−0.0624333
$445$	2465.99	0.262695
$446$	6906.88	0.733297
$447$	9165.73	0.969852
$448$	13545.0	1.42844
$449$	7617.52	0.800653	0.400326	−	0.916373i	$-0.368897\pi$
$449$	7617.52	0.800653	0.400326	+	0.916373i	$0.368897\pi$
$450$	2181.80	0.228558
$451$	−2210.23	−0.230766
$452$	121.470	0.0126405
$453$	332.126	0.0344473
$454$	−10649.0	−1.10085
$455$	−3648.46	−0.375917
$456$	3379.84	0.347095
$457$	−14145.7	−1.44794	−0.723970	−	0.689831i	$-0.757685\pi$
$457$	−14145.7	−1.44794	−0.723970	+	0.689831i	$0.757685\pi$
$458$	14694.8	1.49922
$459$	4377.07	0.445107
$460$	−198.829	−0.0201531
$461$	1768.76	0.178697	0.0893487	−	0.996000i	$-0.471521\pi$
$461$	1768.76	0.178697	0.0893487	+	0.996000i	$0.471521\pi$
$462$	−5845.27	−0.588629
$463$	−9809.23	−0.984608	−0.492304	−	0.870423i	$-0.663845\pi$
$463$	−9809.23	−0.984608	−0.492304	+	0.870423i	$0.663845\pi$
$464$	−9121.75	−0.912643
$465$	−9514.54	−0.948873
$466$	−4627.39	−0.460000
$467$	−6733.94	−0.667258	−0.333629	−	0.942704i	$-0.608273\pi$
$467$	−6733.94	−0.667258	−0.333629	+	0.942704i	$0.608273\pi$
$468$	406.257	0.0401265
$469$	−16876.9	−1.66163
$470$	−8256.47	−0.810304
$471$	−17344.2	−1.69677
$472$	14819.4	1.44516
$473$	3870.73	0.376271
$474$	12736.6	1.23420
$475$	475.000	0.0458831
$476$	1342.38	0.129260
$477$	11217.8	1.07679
$478$	6493.98	0.621397
$479$	8174.17	0.779723	0.389862	−	0.920873i	$-0.372523\pi$
$479$	8174.17	0.779723	0.389862	+	0.920873i	$0.372523\pi$
$480$	−766.172	−0.0728558
$481$	4985.41	0.472588
$482$	−16618.9	−1.57048
$483$	−17378.6	−1.63717
$484$	−53.5177	−0.00502608
$485$	−4299.95	−0.402579
$486$	14886.2	1.38940
$487$	−2625.85	−0.244330	−0.122165	−	0.992510i	$-0.538984\pi$
$487$	−2625.85	−0.244330	−0.122165	+	0.992510i	$0.538984\pi$
$488$	10857.3	1.00715
$489$	−22363.2	−2.06810
$490$	4027.55	0.371318
$491$	4279.46	0.393339	0.196669	−	0.980470i	$-0.436987\pi$
$491$	4279.46	0.393339	0.196669	+	0.980470i	$0.436987\pi$
$492$	−681.151	−0.0624160
$493$	−18215.3	−1.66405
$494$	−1511.32	−0.137647
$495$	1745.99	0.158538
$496$	−14962.5	−1.35451
$497$	20216.4	1.82460
$498$	29072.6	2.61601
$499$	−18866.1	−1.69251	−0.846256	−	0.532777i	$-0.821148\pi$
$499$	−18866.1	−1.69251	−0.846256	+	0.532777i	$0.821148\pi$
$500$	−55.2869	−0.00494501
$501$	−20962.5	−1.86933
$502$	−15175.1	−1.34920
$503$	−14067.7	−1.24701	−0.623505	−	0.781819i	$-0.714292\pi$
$503$	−14067.7	−1.24701	−0.623505	+	0.781819i	$0.714292\pi$
$504$	−18580.8	−1.64218
$505$	−8404.98	−0.740627
$506$	2718.85	0.238869
$507$	10422.4	0.912971
$508$	1203.29	0.105093
$509$	21760.7	1.89495	0.947473	−	0.319836i	$-0.103628\pi$
$509$	21760.7	1.89495	0.947473	+	0.319836i	$0.103628\pi$
$510$	12679.0	1.10085
$511$	−19192.4	−1.66149
$512$	−12394.6	−1.06986
$513$	−691.043	−0.0594743
$514$	−602.279	−0.0516836
$515$	6174.75	0.528334
$516$	1192.89	0.101771
$517$	−6607.27	−0.562065
$518$	−11945.9	−1.01326
$519$	17704.5	1.49738
$520$	3357.64	0.283158
$521$	1670.93	0.140508	0.0702540	−	0.997529i	$-0.477619\pi$
$521$	1670.93	0.140508	0.0702540	+	0.997529i	$0.477619\pi$
$522$	13209.3	1.10758
$523$	−9394.47	−0.785452	−0.392726	−	0.919655i	$-0.628468\pi$
$523$	−9394.47	−0.785452	−0.392726	+	0.919655i	$0.628468\pi$
$524$	−65.4283	−0.00545467
$525$	−4832.33	−0.401715
$526$	−13058.3	−1.08245
$527$	−29878.8	−2.46972
$528$	5081.03	0.418794
$529$	−4083.60	−0.335629
$530$	4857.28	0.398088
$531$	−20270.0	−1.65658
$532$	−211.932	−0.0172714
$533$	5813.71	0.472457
$534$	−10392.1	−0.842154
$535$	−1324.88	−0.107065
$536$	15531.6	1.25161
$537$	6573.34	0.528232
$538$	5528.09	0.442998
$539$	3223.06	0.257564
$540$	80.4329	0.00640978
$541$	16788.2	1.33416	0.667081	−	0.744985i	$-0.267543\pi$
$541$	16788.2	1.33416	0.667081	+	0.744985i	$0.267543\pi$
$542$	−1777.88	−0.140898
$543$	13641.3	1.07809
$544$	−2406.03	−0.189628
$545$	−1284.32	−0.100944
$546$	15375.2	1.20512
$547$	22081.1	1.72600	0.863000	−	0.505204i	$-0.168583\pi$
$547$	22081.1	1.72600	0.863000	+	0.505204i	$0.168583\pi$
$548$	713.371	0.0556089
$549$	−14850.7	−1.15448
$550$	756.010	0.0586116
$551$	2875.80	0.222347
$552$	15993.3	1.23319
$553$	−15244.1	−1.17224
$554$	−2130.12	−0.163357
$555$	6603.11	0.505021
$556$	−749.826	−0.0571937
$557$	−14468.3	−1.10061	−0.550305	−	0.834964i	$-0.685488\pi$
$557$	−14468.3	−1.10061	−0.550305	+	0.834964i	$0.685488\pi$
$558$	21667.3	1.64382
$559$	−10181.4	−0.770356
$560$	−7599.29	−0.573444
$561$	10146.4	0.763602
$562$	74.3751	0.00558243
$563$	10092.5	0.755500	0.377750	−	0.925908i	$-0.376698\pi$
$563$	10092.5	0.755500	0.377750	+	0.925908i	$0.376698\pi$
$564$	−2036.24	−0.152023
$565$	−1373.18	−0.102248
$566$	41.7408	0.00309982
$567$	−14585.7	−1.08032
$568$	−18604.9	−1.37438
$569$	20841.0	1.53550	0.767751	−	0.640748i	$-0.221376\pi$
$569$	20841.0	1.53550	0.767751	+	0.640748i	$0.221376\pi$
$570$	−2001.73	−0.147093
$571$	−13792.4	−1.01085	−0.505425	−	0.862871i	$-0.668664\pi$
$571$	−13792.4	−1.01085	−0.505425	+	0.862871i	$0.668664\pi$
$572$	140.771	0.0102901
$573$	8315.10	0.606227
$574$	−13930.6	−1.01298
$575$	2247.69	0.163018
$576$	17050.1	1.23337
$577$	8331.13	0.601091	0.300545	−	0.953768i	$-0.402831\pi$
$577$	8331.13	0.601091	0.300545	+	0.953768i	$0.402831\pi$
$578$	26309.6	1.89332
$579$	−35084.4	−2.51824
$580$	−334.725	−0.0239632
$581$	−34796.3	−2.48467
$582$	18120.7	1.29060
$583$	3887.05	0.276133
$584$	17662.6	1.25151
$585$	−4592.60	−0.324582
$586$	−8785.81	−0.619349
$587$	643.789	0.0452675	0.0226337	−	0.999744i	$-0.492795\pi$
$587$	643.789	0.0452675	0.0226337	+	0.999744i	$0.492795\pi$
$588$	993.286	0.0696640
$589$	4717.20	0.329998
$590$	−8776.87	−0.612437
$591$	33884.0	2.35838
$592$	10384.0	0.720911
$593$	−12754.4	−0.883239	−0.441620	−	0.897202i	$-0.645596\pi$
$593$	−12754.4	−0.883239	−0.441620	+	0.897202i	$0.645596\pi$
$594$	−1099.86	−0.0759730
$595$	−15175.1	−1.04558
$596$	528.923	0.0363516
$597$	13043.4	0.894191
$598$	−7151.56	−0.489045
$599$	14324.9	0.977127	0.488563	−	0.872528i	$-0.337521\pi$
$599$	14324.9	0.977127	0.488563	+	0.872528i	$0.337521\pi$
$600$	4447.15	0.302590
$601$	10691.1	0.725621	0.362810	−	0.931863i	$-0.381817\pi$
$601$	10691.1	0.725621	0.362810	+	0.931863i	$0.381817\pi$
$602$	24396.4	1.65170
$603$	−21244.3	−1.43471
$604$	19.1659	0.00129114
$605$	605.000	0.0406558
$606$	35420.0	2.37432
$607$	−159.550	−0.0106687	−0.00533436	−	0.999986i	$-0.501698\pi$
$607$	−159.550	−0.0106687	−0.00533436	+	0.999986i	$0.501698\pi$
$608$	379.860	0.0253377
$609$	−29256.5	−1.94669
$610$	−6430.31	−0.426812
$611$	17379.5	1.15074
$612$	1689.75	0.111608
$613$	6207.71	0.409016	0.204508	−	0.978865i	$-0.434441\pi$
$613$	6207.71	0.409016	0.204508	+	0.978865i	$0.434441\pi$
$614$	−11670.2	−0.767055
$615$	7700.19	0.504881
$616$	−6438.41	−0.421122
$617$	8323.57	0.543103	0.271551	−	0.962424i	$-0.412463\pi$
$617$	8323.57	0.543103	0.271551	+	0.962424i	$0.412463\pi$
$618$	−26021.4	−1.69375
$619$	−2015.40	−0.130865	−0.0654327	−	0.997857i	$-0.520843\pi$
$619$	−2015.40	−0.130865	−0.0654327	+	0.997857i	$0.520843\pi$
$620$	−549.052	−0.0355652
$621$	−3270.01	−0.211306
$622$	24362.9	1.57052
$623$	12438.0	0.799871
$624$	−13365.0	−0.857415
$625$	625.000	0.0400000
$626$	−5535.92	−0.353450
$627$	−1601.89	−0.102031
$628$	−1000.87	−0.0635975
$629$	20735.9	1.31446
$630$	11004.6	0.695928
$631$	19440.8	1.22651	0.613255	−	0.789885i	$-0.289860\pi$
$631$	19440.8	1.22651	0.613255	+	0.789885i	$0.289860\pi$
$632$	14029.0	0.882983
$633$	37999.8	2.38603
$634$	10812.5	0.677317
$635$	−13602.8	−0.850096
$636$	1197.92	0.0746863
$637$	−8477.82	−0.527321
$638$	4577.13	0.284029
$639$	25447.9	1.57544
$640$	6582.95	0.406584
$641$	15035.4	0.926465	0.463232	−	0.886237i	$-0.346690\pi$
$641$	15035.4	0.926465	0.463232	+	0.886237i	$0.346690\pi$
$642$	5583.26	0.343230
$643$	−6828.19	−0.418783	−0.209392	−	0.977832i	$-0.567148\pi$
$643$	−6828.19	−0.418783	−0.209392	+	0.977832i	$0.567148\pi$
$644$	−1002.86	−0.0613636
$645$	−13485.2	−0.823223
$646$	−6286.09	−0.382853
$647$	25725.1	1.56315	0.781574	−	0.623813i	$-0.214417\pi$
$647$	25725.1	1.56315	0.781574	+	0.623813i	$0.214417\pi$
$648$	13423.1	0.813748
$649$	−7023.72	−0.424815
$650$	−1988.58	−0.119998
$651$	−47989.7	−2.88919
$652$	−1290.50	−0.0775154
$653$	−31456.1	−1.88511	−0.942553	−	0.334058i	$-0.891582\pi$
$653$	−31456.1	−1.88511	−0.942553	+	0.334058i	$0.891582\pi$
$654$	5412.34	0.323607
$655$	739.645	0.0441226
$656$	12109.3	0.720712
$657$	−24159.0	−1.43460
$658$	−41644.2	−2.46727
$659$	25853.7	1.52825	0.764126	−	0.645068i	$-0.223171\pi$
$659$	25853.7	1.52825	0.764126	+	0.645068i	$0.223171\pi$
$660$	186.450	0.0109963
$661$	−23027.3	−1.35501	−0.677503	−	0.735520i	$-0.736938\pi$
$661$	−23027.3	−1.35501	−0.677503	+	0.735520i	$0.736938\pi$
$662$	−2785.03	−0.163510
$663$	−26688.7	−1.56335
$664$	32022.7	1.87157
$665$	2395.82	0.139708
$666$	−15037.2	−0.874893
$667$	13608.3	0.789976
$668$	−1209.67	−0.0700654
$669$	−19256.3	−1.11284
$670$	−9198.71	−0.530414
$671$	−5145.88	−0.296057
$672$	−3864.44	−0.221836
$673$	−5377.25	−0.307991	−0.153995	−	0.988072i	$-0.549214\pi$
$673$	−5377.25	−0.307991	−0.153995	+	0.988072i	$0.549214\pi$
$674$	−375.664	−0.0214689
$675$	−909.267	−0.0518485
$676$	601.443	0.0342196
$677$	17973.3	1.02034	0.510170	−	0.860073i	$-0.329582\pi$
$677$	17973.3	1.02034	0.510170	+	0.860073i	$0.329582\pi$
$678$	5786.82	0.327790
$679$	−21688.2	−1.22580
$680$	13965.5	0.787579
$681$	29689.4	1.67063
$682$	7507.90	0.421543
$683$	−15292.3	−0.856724	−0.428362	−	0.903607i	$-0.640909\pi$
$683$	−15292.3	−0.856724	−0.428362	+	0.903607i	$0.640909\pi$
$684$	−266.775	−0.0149129
$685$	−8064.42	−0.449819
$686$	−3466.16	−0.192913
$687$	−40969.1	−2.27521
$688$	−21206.7	−1.17514
$689$	−10224.4	−0.565338
$690$	−9472.15	−0.522607
$691$	−13700.2	−0.754240	−0.377120	−	0.926164i	$-0.623086\pi$
$691$	−13700.2	−0.754240	−0.377120	+	0.926164i	$0.623086\pi$
$692$	1021.67	0.0561241
$693$	8806.49	0.482728
$694$	−5843.35	−0.319611
$695$	8476.54	0.462638
$696$	26924.5	1.46634
$697$	24181.1	1.31410
$698$	1944.83	0.105463
$699$	12901.1	0.698091
$700$	−278.858	−0.0150569
$701$	−6677.36	−0.359772	−0.179886	−	0.983687i	$-0.557573\pi$
$701$	−6677.36	−0.359772	−0.179886	+	0.983687i	$0.557573\pi$
$702$	2893.05	0.155543
$703$	−3273.75	−0.175636
$704$	5907.99	0.316287
$705$	23019.0	1.22971
$706$	13338.2	0.711031
$707$	−42393.3	−2.25511
$708$	−2164.58	−0.114901
$709$	4862.08	0.257545	0.128772	−	0.991674i	$-0.458896\pi$
$709$	4862.08	0.257545	0.128772	+	0.991674i	$0.458896\pi$
$710$	11018.9	0.582438
$711$	−19189.0	−1.01216
$712$	−11446.6	−0.602500
$713$	22321.7	1.17245
$714$	63950.5	3.35194
$715$	−1591.37	−0.0832362
$716$	379.325	0.0197989
$717$	−18105.2	−0.943026
$718$	−6958.41	−0.361679
$719$	8401.04	0.435753	0.217876	−	0.975976i	$-0.430087\pi$
$719$	8401.04	0.435753	0.217876	+	0.975976i	$0.430087\pi$
$720$	−9565.82	−0.495135
$721$	31144.4	1.60871
$722$	992.435	0.0511560
$723$	46333.4	2.38335
$724$	787.191	0.0404085
$725$	3783.95	0.193838
$726$	−2549.57	−0.130335
$727$	5803.16	0.296048	0.148024	−	0.988984i	$-0.452709\pi$
$727$	5803.16	0.296048	0.148024	+	0.988984i	$0.452709\pi$
$728$	16935.4	0.862179
$729$	−25886.8	−1.31519
$730$	−10460.8	−0.530372
$731$	−42347.9	−2.14267
$732$	−1585.86	−0.0800754
$733$	16203.9	0.816513	0.408256	−	0.912867i	$-0.366137\pi$
$733$	16203.9	0.816513	0.408256	+	0.912867i	$0.366137\pi$
$734$	−36568.5	−1.83892
$735$	−11228.8	−0.563509
$736$	1797.49	0.0900223
$737$	−7361.30	−0.367920
$738$	−17535.5	−0.874651
$739$	30146.6	1.50062	0.750311	−	0.661085i	$-0.229904\pi$
$739$	30146.6	1.50062	0.750311	+	0.661085i	$0.229904\pi$
$740$	381.043	0.0189289
$741$	4213.56	0.208892
$742$	24499.3	1.21213
$743$	−5502.82	−0.271708	−0.135854	−	0.990729i	$-0.543378\pi$
$743$	−5502.82	−0.271708	−0.135854	+	0.990729i	$0.543378\pi$
$744$	44164.5	2.17627
$745$	−5979.30	−0.294046
$746$	22017.6	1.08059
$747$	−43800.8	−2.14537
$748$	585.513	0.0286210
$749$	−6682.47	−0.325997
$750$	−2633.85	−0.128233
$751$	28375.0	1.37872	0.689359	−	0.724420i	$-0.257892\pi$
$751$	28375.0	1.37872	0.689359	+	0.724420i	$0.257892\pi$
$752$	36199.5	1.75540
$753$	42308.0	2.04753
$754$	−12039.5	−0.581503
$755$	−216.664	−0.0104440
$756$	405.690	0.0195169
$757$	27053.7	1.29892	0.649461	−	0.760395i	$-0.274995\pi$
$757$	27053.7	1.29892	0.649461	+	0.760395i	$0.274995\pi$
$758$	19595.1	0.938952
$759$	−7580.12	−0.362505
$760$	−2204.85	−0.105235
$761$	−15255.9	−0.726712	−0.363356	−	0.931650i	$-0.618369\pi$
$761$	−15255.9	−0.726712	−0.363356	+	0.931650i	$0.618369\pi$
$762$	57324.5	2.72526
$763$	−6477.89	−0.307360
$764$	479.836	0.0227223
$765$	−19102.1	−0.902795
$766$	−10154.5	−0.478979
$767$	18474.9	0.869741
$768$	5190.82	0.243890
$769$	29950.7	1.40449	0.702244	−	0.711937i	$-0.252182\pi$
$769$	29950.7	1.40449	0.702244	+	0.711937i	$0.252182\pi$
$770$	3813.19	0.178465
$771$	1679.15	0.0784346
$772$	−2024.60	−0.0943874
$773$	29183.8	1.35792	0.678959	−	0.734177i	$-0.262432\pi$
$773$	29183.8	1.35792	0.678959	+	0.734177i	$0.262432\pi$
$774$	30709.7	1.42614
$775$	6206.85	0.287686
$776$	19959.5	0.923328
$777$	33305.0	1.53772
$778$	31180.3	1.43685
$779$	−3817.67	−0.175587
$780$	−490.430	−0.0225131
$781$	8817.90	0.404007
$782$	−29745.7	−1.36023
$783$	−5505.00	−0.251255
$784$	−17658.3	−0.804404
$785$	11314.5	0.514438
$786$	−3116.99	−0.141449
$787$	32179.8	1.45754	0.728770	−	0.684758i	$-0.240092\pi$
$787$	32179.8	1.45754	0.728770	+	0.684758i	$0.240092\pi$
$788$	1955.33	0.0883957
$789$	36406.4	1.64272
$790$	−8308.79	−0.374194
$791$	−6926.10	−0.311332
$792$	−8104.53	−0.363614
$793$	13535.5	0.606130
$794$	−516.222	−0.0230731
$795$	−13542.1	−0.604135
$796$	752.692	0.0335157
$797$	−18685.8	−0.830471	−0.415236	−	0.909714i	$-0.636301\pi$
$797$	−18685.8	−0.830471	−0.415236	+	0.909714i	$0.636301\pi$
$798$	−10096.4	−0.447880
$799$	72287.2	3.20067
$800$	499.815	0.0220889
$801$	15656.7	0.690641
$802$	−5805.58	−0.255614
$803$	−8371.29	−0.367891
$804$	−2268.61	−0.0995123
$805$	11337.0	0.496368
$806$	−19748.5	−0.863042
$807$	−15412.3	−0.672290
$808$	39014.1	1.69865
$809$	8168.39	0.354988	0.177494	−	0.984122i	$-0.443201\pi$
$809$	8168.39	0.354988	0.177494	+	0.984122i	$0.443201\pi$
$810$	−7949.91	−0.344853
$811$	−14029.3	−0.607444	−0.303722	−	0.952761i	$-0.598229\pi$
$811$	−14029.3	−0.607444	−0.303722	+	0.952761i	$0.598229\pi$
$812$	−1688.29	−0.0729649
$813$	4956.73	0.213825
$814$	−5210.50	−0.224359
$815$	14588.7	0.627019
$816$	−55589.3	−2.38482
$817$	6685.81	0.286300
$818$	38032.7	1.62565
$819$	−23164.3	−0.988309
$820$	444.352	0.0189237
$821$	−33845.6	−1.43876	−0.719379	−	0.694618i	$-0.755573\pi$
$821$	−33845.6	−1.43876	−0.719379	+	0.694618i	$0.755573\pi$
$822$	33984.8	1.44204
$823$	13602.5	0.576129	0.288064	−	0.957611i	$-0.406988\pi$
$823$	13602.5	0.576129	0.288064	+	0.957611i	$0.406988\pi$
$824$	−28661.9	−1.21175
$825$	−2107.75	−0.0889485
$826$	−44269.0	−1.86479
$827$	−795.407	−0.0334450	−0.0167225	−	0.999860i	$-0.505323\pi$
$827$	−795.407	−0.0334450	−0.0167225	+	0.999860i	$0.505323\pi$
$828$	−1262.38	−0.0529838
$829$	−41119.6	−1.72273	−0.861365	−	0.507986i	$-0.830390\pi$
$829$	−41119.6	−1.72273	−0.861365	+	0.507986i	$0.830390\pi$
$830$	−18965.6	−0.793141
$831$	5938.75	0.247910
$832$	−15540.2	−0.647547
$833$	−35262.0	−1.46670
$834$	−35721.5	−1.48314
$835$	13675.0	0.566756
$836$	−92.4397	−0.00382427
$837$	−9029.89	−0.372902
$838$	−30070.4	−1.23957
$839$	−43419.5	−1.78666	−0.893331	−	0.449399i	$-0.851638\pi$
$839$	−43419.5	−1.78666	−0.893331	+	0.449399i	$0.851638\pi$
$840$	22430.7	0.921348
$841$	−1479.74	−0.0606723
$842$	−7423.14	−0.303822
$843$	−207.357	−0.00847184
$844$	2192.84	0.0894321
$845$	−6799.11	−0.276801
$846$	−52420.9	−2.13034
$847$	3051.52	0.123791
$848$	−21296.1	−0.862396
$849$	−116.373	−0.00470426
$850$	−8271.17	−0.333763
$851$	−15491.3	−0.624014
$852$	2717.51	0.109273
$853$	−42028.1	−1.68701	−0.843503	−	0.537124i	$-0.819511\pi$
$853$	−42028.1	−1.68701	−0.843503	+	0.537124i	$0.819511\pi$
$854$	−32433.4	−1.29959
$855$	3015.80	0.120630
$856$	6149.81	0.245556
$857$	1774.58	0.0707334	0.0353667	−	0.999374i	$-0.488740\pi$
$857$	1774.58	0.0707334	0.0353667	+	0.999374i	$0.488740\pi$
$858$	6706.30	0.266841
$859$	−24580.9	−0.976357	−0.488178	−	0.872744i	$-0.662338\pi$
$859$	−24580.9	−0.976357	−0.488178	+	0.872744i	$0.662338\pi$
$860$	−778.184	−0.0308557
$861$	38838.4	1.53729
$862$	24792.1	0.979610
$863$	−10354.7	−0.408432	−0.204216	−	0.978926i	$-0.565465\pi$
$863$	−10354.7	−0.408432	−0.204216	+	0.978926i	$0.565465\pi$
$864$	−727.145	−0.0286319
$865$	−11549.6	−0.453986
$866$	27870.3	1.09361
$867$	−73351.0	−2.87328
$868$	−2769.32	−0.108291
$869$	−6649.14	−0.259559
$870$	−15946.2	−0.621410
$871$	19362.9	0.753257
$872$	5961.54	0.231518
$873$	−27300.6	−1.05840
$874$	4696.19	0.181752
$875$	3152.39	0.121795
$876$	−2579.87	−0.0995044
$877$	−35754.0	−1.37665	−0.688327	−	0.725400i	$-0.741655\pi$
$877$	−35754.0	−1.37665	−0.688327	+	0.725400i	$0.741655\pi$
$878$	18742.9	0.720434
$879$	24494.8	0.939919
$880$	−3314.63	−0.126973
$881$	−20224.5	−0.773419	−0.386709	−	0.922202i	$-0.626388\pi$
$881$	−20224.5	−0.773419	−0.386709	+	0.922202i	$0.626388\pi$
$882$	25571.1	0.976219
$883$	−36560.5	−1.39338	−0.696692	−	0.717371i	$-0.745345\pi$
$883$	−36560.5	−1.39338	−0.696692	+	0.717371i	$0.745345\pi$
$884$	−1540.11	−0.0585968
$885$	24469.8	0.929429
$886$	6227.18	0.236124
$887$	−30375.8	−1.14985	−0.574927	−	0.818205i	$-0.694969\pi$
$887$	−30375.8	−1.14985	−0.574927	+	0.818205i	$0.694969\pi$
$888$	−30650.2	−1.15828
$889$	−68610.3	−2.58843
$890$	6779.33	0.255330
$891$	−6361.94	−0.239207
$892$	−1111.22	−0.0417111
$893$	−11412.6	−0.427667
$894$	25197.8	0.942661
$895$	−4288.15	−0.160153
$896$	33203.3	1.23799
$897$	19938.5	0.742170
$898$	20941.5	0.778205
$899$	37578.2	1.39411
$900$	−351.020	−0.0130007
$901$	−42526.5	−1.57243
$902$	−6076.21	−0.224297
$903$	−68017.0	−2.50660
$904$	6374.03	0.234510
$905$	−8898.93	−0.326862
$906$	913.056	0.0334815
$907$	−22391.7	−0.819742	−0.409871	−	0.912144i	$-0.634426\pi$
$907$	−22391.7	−0.819742	−0.409871	+	0.912144i	$0.634426\pi$
$908$	1713.28	0.0626180
$909$	−53363.7	−1.94715
$910$	−10030.1	−0.365378
$911$	16151.0	0.587383	0.293691	−	0.955900i	$-0.405116\pi$
$911$	16151.0	0.587383	0.293691	+	0.955900i	$0.405116\pi$
$912$	8776.32	0.318655
$913$	−15177.3	−0.550160
$914$	−38888.4	−1.40735
$915$	17927.7	0.647727
$916$	−2364.19	−0.0852783
$917$	3730.64	0.134348
$918$	12033.1	0.432628
$919$	−800.844	−0.0287458	−0.0143729	−	0.999897i	$-0.504575\pi$
$919$	−800.844	−0.0287458	−0.0143729	+	0.999897i	$0.504575\pi$
$920$	−10433.3	−0.373887
$921$	32536.5	1.16408
$922$	4862.56	0.173687
$923$	−23194.3	−0.827139
$924$	940.420	0.0334822
$925$	−4307.57	−0.153116
$926$	−26966.8	−0.957003
$927$	39203.9	1.38902
$928$	3026.04	0.107042
$929$	29875.6	1.05510	0.527550	−	0.849524i	$-0.323111\pi$
$929$	29875.6	1.05510	0.527550	+	0.849524i	$0.323111\pi$
$930$	−26156.7	−0.922270
$931$	5567.10	0.195977
$932$	744.481	0.0261655
$933$	−67923.5	−2.38340
$934$	−18512.5	−0.648551
$935$	−6619.03	−0.231514
$936$	21317.9	0.744441
$937$	−13857.0	−0.483124	−0.241562	−	0.970385i	$-0.577660\pi$
$937$	−13857.0	−0.483124	−0.241562	+	0.970385i	$0.577660\pi$
$938$	−46396.7	−1.61504
$939$	15434.1	0.536393
$940$	1328.35	0.0460914
$941$	−50388.2	−1.74560	−0.872799	−	0.488079i	$-0.837698\pi$
$941$	−50388.2	−1.74560	−0.872799	+	0.488079i	$0.837698\pi$
$942$	−47681.4	−1.64920
$943$	−18065.2	−0.623842
$944$	38481.1	1.32675
$945$	−4586.19	−0.157872
$946$	10641.1	0.365722
$947$	33194.8	1.13905	0.569527	−	0.821972i	$-0.307126\pi$
$947$	33194.8	1.13905	0.569527	+	0.821972i	$0.307126\pi$
$948$	−2049.14	−0.0702035
$949$	22019.5	0.753198
$950$	1305.84	0.0445967
$951$	−30145.2	−1.02789
$952$	70439.7	2.39807
$953$	−48838.2	−1.66005	−0.830023	−	0.557729i	$-0.811673\pi$
$953$	−48838.2	−1.66005	−0.830023	+	0.557729i	$0.811673\pi$
$954$	30839.2	1.04660
$955$	−5424.39	−0.183800
$956$	−1044.79	−0.0353461
$957$	−12761.0	−0.431039
$958$	22471.8	0.757862
$959$	−40675.6	−1.36964
$960$	−20582.8	−0.691986
$961$	31848.9	1.06908
$962$	13705.5	0.459339
$963$	−8411.74	−0.281479
$964$	2673.74	0.0893314
$965$	22887.5	0.763496
$966$	−47775.9	−1.59127
$967$	−54246.9	−1.80399	−0.901997	−	0.431742i	$-0.857899\pi$
$967$	−54246.9	−1.80399	−0.901997	+	0.431742i	$0.857899\pi$
$968$	−2808.28	−0.0932455
$969$	17525.6	0.581014
$970$	−11821.1	−0.391292
$971$	31307.6	1.03472	0.517358	−	0.855769i	$-0.326915\pi$
$971$	31307.6	1.03472	0.517358	+	0.855769i	$0.326915\pi$
$972$	−2394.97	−0.0790315
$973$	42754.2	1.40867
$974$	−7218.80	−0.237480
$975$	5544.15	0.182108
$976$	28192.9	0.924623
$977$	40964.8	1.34143	0.670716	−	0.741714i	$-0.265987\pi$
$977$	40964.8	1.34143	0.670716	+	0.741714i	$0.265987\pi$
$978$	−61479.3	−2.01011
$979$	5425.18	0.177109
$980$	−647.974	−0.0211212
$981$	−8154.23	−0.265387
$982$	11764.8	0.382311
$983$	−41841.4	−1.35761	−0.678806	−	0.734318i	$-0.737502\pi$
$983$	−41841.4	−1.35761	−0.678806	+	0.734318i	$0.737502\pi$
$984$	−35742.7	−1.15796
$985$	−22104.4	−0.715030
$986$	−50076.3	−1.61740
$987$	116104.	3.74430
$988$	243.150	0.00782959
$989$	31637.2	1.01719
$990$	4799.96	0.154094
$991$	−20277.4	−0.649982	−0.324991	−	0.945717i	$-0.605361\pi$
$991$	−20277.4	−0.649982	−0.324991	+	0.945717i	$0.605361\pi$
$992$	4963.64	0.158867
$993$	7764.66	0.248141
$994$	55577.4	1.77345
$995$	−8508.94	−0.271107
$996$	−4677.36	−0.148803
$997$	−30826.7	−0.979230	−0.489615	−	0.871939i	$-0.662863\pi$
$997$	−30826.7	−0.979230	−0.489615	+	0.871939i	$0.662863\pi$
$998$	−51865.4	−1.64506
$999$	6266.77	0.198470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.b.1.16	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.b.1.16	✓	20	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+2.74913 q^{2} -7.66455 q^{3} -0.442295 q^{4} +5.00000 q^{5} -21.0708 q^{6} +25.2191 q^{7} -23.2090 q^{8} +31.7453 q^{9} +O(q^{10})\)	\(q+2.74913 q^{2} -7.66455 q^{3} -0.442295 q^{4} +5.00000 q^{5} -21.0708 q^{6} +25.2191 q^{7} -23.2090 q^{8} +31.7453 q^{9} +13.7456 q^{10} +11.0000 q^{11} +3.38999 q^{12} -28.9340 q^{13} +69.3307 q^{14} -38.3227 q^{15} -60.2660 q^{16} -120.346 q^{17} +87.2719 q^{18} +19.0000 q^{19} -2.21148 q^{20} -193.293 q^{21} +30.2404 q^{22} +89.9077 q^{23} +177.886 q^{24} +25.0000 q^{25} -79.5433 q^{26} -36.3707 q^{27} -11.1543 q^{28} +151.358 q^{29} -105.354 q^{30} +248.274 q^{31} +19.9926 q^{32} -84.3100 q^{33} -330.847 q^{34} +126.096 q^{35} -14.0408 q^{36} -172.303 q^{37} +52.2334 q^{38} +221.766 q^{39} -116.045 q^{40} -200.930 q^{41} -531.388 q^{42} +351.885 q^{43} -4.86525 q^{44} +158.727 q^{45} +247.168 q^{46} -600.661 q^{47} +461.912 q^{48} +293.005 q^{49} +68.7282 q^{50} +922.398 q^{51} +12.7974 q^{52} +353.369 q^{53} -99.9877 q^{54} +55.0000 q^{55} -585.310 q^{56} -145.626 q^{57} +416.103 q^{58} -638.520 q^{59} +16.9500 q^{60} -467.807 q^{61} +682.537 q^{62} +800.590 q^{63} +537.090 q^{64} -144.670 q^{65} -231.779 q^{66} -669.209 q^{67} +53.2285 q^{68} -689.102 q^{69} +346.653 q^{70} +801.627 q^{71} -736.775 q^{72} -761.026 q^{73} -473.682 q^{74} -191.614 q^{75} -8.40361 q^{76} +277.411 q^{77} +609.664 q^{78} -604.467 q^{79} -301.330 q^{80} -578.359 q^{81} -552.382 q^{82} -1379.76 q^{83} +85.4927 q^{84} -601.730 q^{85} +967.376 q^{86} -1160.09 q^{87} -255.298 q^{88} +493.198 q^{89} +436.360 q^{90} -729.691 q^{91} -39.7658 q^{92} -1902.91 q^{93} -1651.29 q^{94} +95.0000 q^{95} -153.234 q^{96} -859.990 q^{97} +805.509 q^{98} +349.198 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * q^99

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