LMFDB - Embedded newform 1045.4.a.d.1.19

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:	$22$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+3.88965 q^{2} -9.46724 q^{3} +7.12938 q^{4} +5.00000 q^{5} -36.8243 q^{6} +13.8479 q^{7} -3.38640 q^{8} +62.6286 q^{9} +O(q^{10})$	\(q+3.88965 q^{2} -9.46724 q^{3} +7.12938 q^{4} +5.00000 q^{5} -36.8243 q^{6} +13.8479 q^{7} -3.38640 q^{8} +62.6286 q^{9} +19.4483 q^{10} -11.0000 q^{11} -67.4956 q^{12} -74.0808 q^{13} +53.8636 q^{14} -47.3362 q^{15} -70.2070 q^{16} +130.416 q^{17} +243.603 q^{18} -19.0000 q^{19} +35.6469 q^{20} -131.102 q^{21} -42.7862 q^{22} -35.5411 q^{23} +32.0599 q^{24} +25.0000 q^{25} -288.149 q^{26} -337.305 q^{27} +98.7272 q^{28} +90.6883 q^{29} -184.121 q^{30} +134.835 q^{31} -245.989 q^{32} +104.140 q^{33} +507.274 q^{34} +69.2396 q^{35} +446.503 q^{36} -172.386 q^{37} -73.9034 q^{38} +701.341 q^{39} -16.9320 q^{40} +305.991 q^{41} -509.940 q^{42} -82.5436 q^{43} -78.4232 q^{44} +313.143 q^{45} -138.242 q^{46} +88.7617 q^{47} +664.666 q^{48} -151.235 q^{49} +97.2413 q^{50} -1234.68 q^{51} -528.150 q^{52} -345.216 q^{53} -1312.00 q^{54} -55.0000 q^{55} -46.8946 q^{56} +179.878 q^{57} +352.746 q^{58} -283.263 q^{59} -337.478 q^{60} -580.681 q^{61} +524.460 q^{62} +867.277 q^{63} -395.157 q^{64} -370.404 q^{65} +405.067 q^{66} -357.778 q^{67} +929.788 q^{68} +336.476 q^{69} +269.318 q^{70} -422.735 q^{71} -212.086 q^{72} -271.412 q^{73} -670.521 q^{74} -236.681 q^{75} -135.458 q^{76} -152.327 q^{77} +2727.97 q^{78} -702.997 q^{79} -351.035 q^{80} +1502.37 q^{81} +1190.20 q^{82} -127.554 q^{83} -934.674 q^{84} +652.082 q^{85} -321.066 q^{86} -858.568 q^{87} +37.2504 q^{88} +776.183 q^{89} +1218.02 q^{90} -1025.87 q^{91} -253.386 q^{92} -1276.51 q^{93} +345.252 q^{94} -95.0000 q^{95} +2328.84 q^{96} -1655.75 q^{97} -588.251 q^{98} -688.915 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10}) $ 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9 - 20 * q^10 - 242 * q^11 - 196 * q^12 - q^13 - 63 * q^14 - 105 * q^15 + 6 * q^16 + 187 * q^17 - 361 * q^18 - 418 * q^19 + 370 * q^20 - 107 * q^21 + 44 * q^22 - 361 * q^23 + 208 * q^24 + 550 * q^25 - 365 * q^26 - 1467 * q^27 - 773 * q^28 - 319 * q^29 - 45 * q^30 - 402 * q^31 - 873 * q^32 + 231 * q^33 - 717 * q^34 - 205 * q^35 + 725 * q^36 - 838 * q^37 + 76 * q^38 - 607 * q^39 - 390 * q^40 - 392 * q^41 - 1350 * q^42 - 610 * q^43 - 814 * q^44 + 1045 * q^45 - 605 * q^46 - 1866 * q^47 - 1637 * q^48 + 379 * q^49 - 100 * q^50 - 2659 * q^51 - 638 * q^52 - 1303 * q^53 + 2338 * q^54 - 1210 * q^55 + 727 * q^56 + 399 * q^57 + 44 * q^58 - 2417 * q^59 - 980 * q^60 + 918 * q^61 - 1634 * q^62 - 374 * q^63 - 1716 * q^64 - 5 * q^65 + 99 * q^66 - 2339 * q^67 + 4940 * q^68 + 127 * q^69 - 315 * q^70 - 2370 * q^71 - 3306 * q^72 + 2207 * q^73 + 2051 * q^74 - 525 * q^75 - 1406 * q^76 + 451 * q^77 + 1380 * q^78 + 586 * q^79 + 30 * q^80 + 1950 * q^81 - 1566 * q^82 - 2870 * q^83 + 3076 * q^84 + 935 * q^85 - 1246 * q^86 - 1811 * q^87 + 858 * q^88 - 1768 * q^89 - 1805 * q^90 - 2195 * q^91 - 6728 * q^92 - 2916 * q^93 + 672 * q^94 - 2090 * q^95 + 6022 * q^96 - 4022 * q^97 + 1162 * q^98 - 2299 * 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 $

$2$	3.88965	1.37520	0.687600	−	0.726090i	$-0.258664\pi$
$2$	3.88965	1.37520	0.687600	+	0.726090i	$0.258664\pi$
$3$	−9.46724	−1.82197	−0.910986	−	0.412438i	$-0.864677\pi$
$3$	−9.46724	−1.82197	−0.910986	+	0.412438i	$0.864677\pi$
$4$	7.12938	0.891173
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−36.8243	−2.50557
$7$	13.8479	0.747718	0.373859	−	0.927486i	$-0.378034\pi$
$7$	13.8479	0.747718	0.373859	+	0.927486i	$0.378034\pi$
$8$	−3.38640	−0.149659
$9$	62.6286	2.31958
$10$	19.4483	0.615008
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−67.4956	−1.62369
$13$	−74.0808	−1.58049	−0.790243	−	0.612793i	$-0.790046\pi$
$13$	−74.0808	−1.58049	−0.790243	+	0.612793i	$0.790046\pi$
$14$	53.8636	1.02826
$15$	−47.3362	−0.814810
$16$	−70.2070	−1.09698
$17$	130.416	1.86062	0.930312	−	0.366768i	$-0.119536\pi$
$17$	130.416	1.86062	0.930312	+	0.366768i	$0.119536\pi$
$18$	243.603	3.18988
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	35.6469	0.398545
$21$	−131.102	−1.36232
$22$	−42.7862	−0.414638
$23$	−35.5411	−0.322210	−0.161105	−	0.986937i	$-0.551506\pi$
$23$	−35.5411	−0.322210	−0.161105	+	0.986937i	$0.551506\pi$
$24$	32.0599	0.272675
$25$	25.0000	0.200000
$26$	−288.149	−2.17348
$27$	−337.305	−2.40423
$28$	98.7272	0.666346
$29$	90.6883	0.580703	0.290352	−	0.956920i	$-0.406228\pi$
$29$	90.6883	0.580703	0.290352	+	0.956920i	$0.406228\pi$
$30$	−184.121	−1.12053
$31$	134.835	0.781194	0.390597	−	0.920562i	$-0.372269\pi$
$31$	134.835	0.781194	0.390597	+	0.920562i	$0.372269\pi$
$32$	−245.989	−1.35891
$33$	104.140	0.549345
$34$	507.274	2.55873
$35$	69.2396	0.334390
$36$	446.503	2.06715
$37$	−172.386	−0.765948	−0.382974	−	0.923759i	$-0.625100\pi$
$37$	−172.386	−0.765948	−0.382974	+	0.923759i	$0.625100\pi$
$38$	−73.9034	−0.315492
$39$	701.341	2.87960
$40$	−16.9320	−0.0669296
$41$	305.991	1.16556	0.582778	−	0.812632i	$-0.301966\pi$
$41$	305.991	1.16556	0.582778	+	0.812632i	$0.301966\pi$
$42$	−509.940	−1.87346
$43$	−82.5436	−0.292739	−0.146370	−	0.989230i	$-0.546759\pi$
$43$	−82.5436	−0.292739	−0.146370	+	0.989230i	$0.546759\pi$
$44$	−78.4232	−0.268699
$45$	313.143	1.03735
$46$	−138.242	−0.443103
$47$	88.7617	0.275473	0.137736	−	0.990469i	$-0.456017\pi$
$47$	88.7617	0.275473	0.137736	+	0.990469i	$0.456017\pi$
$48$	664.666	1.99867
$49$	−151.235	−0.440918
$50$	97.2413	0.275040
$51$	−1234.68	−3.39000
$52$	−528.150	−1.40849
$53$	−345.216	−0.894700	−0.447350	−	0.894359i	$-0.647632\pi$
$53$	−345.216	−0.894700	−0.447350	+	0.894359i	$0.647632\pi$
$54$	−1312.00	−3.30630
$55$	−55.0000	−0.134840
$56$	−46.8946	−0.111903
$57$	179.878	0.417989
$58$	352.746	0.798583
$59$	−283.263	−0.625047	−0.312523	−	0.949910i	$-0.601174\pi$
$59$	−283.263	−0.625047	−0.312523	+	0.949910i	$0.601174\pi$
$60$	−337.478	−0.726137
$61$	−580.681	−1.21883	−0.609415	−	0.792852i	$-0.708596\pi$
$61$	−580.681	−1.21883	−0.609415	+	0.792852i	$0.708596\pi$
$62$	524.460	1.07430
$63$	867.277	1.73439
$64$	−395.157	−0.771791
$65$	−370.404	−0.706815
$66$	405.067	0.755459
$67$	−357.778	−0.652381	−0.326191	−	0.945304i	$-0.605765\pi$
$67$	−357.778	−0.652381	−0.326191	+	0.945304i	$0.605765\pi$
$68$	929.788	1.65814
$69$	336.476	0.587057
$70$	269.318	0.459852
$71$	−422.735	−0.706612	−0.353306	−	0.935508i	$-0.614943\pi$
$71$	−422.735	−0.706612	−0.353306	+	0.935508i	$0.614943\pi$
$72$	−212.086	−0.347146
$73$	−271.412	−0.435156	−0.217578	−	0.976043i	$-0.569816\pi$
$73$	−271.412	−0.435156	−0.217578	+	0.976043i	$0.569816\pi$
$74$	−670.521	−1.05333
$75$	−236.681	−0.364394
$76$	−135.458	−0.204449
$77$	−152.327	−0.225445
$78$	2727.97	3.96002
$79$	−702.997	−1.00118	−0.500591	−	0.865684i	$-0.666884\pi$
$79$	−702.997	−1.00118	−0.500591	+	0.865684i	$0.666884\pi$
$80$	−351.035	−0.490586
$81$	1502.37	2.06087
$82$	1190.20	1.60287
$83$	−127.554	−0.168686	−0.0843429	−	0.996437i	$-0.526879\pi$
$83$	−127.554	−0.168686	−0.0843429	+	0.996437i	$0.526879\pi$
$84$	−934.674	−1.21406
$85$	652.082	0.832097
$86$	−321.066	−0.402575
$87$	−858.568	−1.05802
$88$	37.2504	0.0451239
$89$	776.183	0.924441	0.462221	−	0.886765i	$-0.347053\pi$
$89$	776.183	0.924441	0.462221	+	0.886765i	$0.347053\pi$
$90$	1218.02	1.42656
$91$	−1025.87	−1.18176
$92$	−253.386	−0.287145
$93$	−1276.51	−1.42331
$94$	345.252	0.378830
$95$	−95.0000	−0.102598
$96$	2328.84	2.47590
$97$	−1655.75	−1.73316	−0.866578	−	0.499041i	$-0.833686\pi$
$97$	−1655.75	−1.73316	−0.866578	+	0.499041i	$0.833686\pi$
$98$	−588.251	−0.606350
$99$	−688.915	−0.699379
$100$	178.235	0.178235
$101$	−1716.27	−1.69084	−0.845421	−	0.534101i	$-0.820650\pi$
$101$	−1716.27	−1.69084	−0.845421	+	0.534101i	$0.820650\pi$
$102$	−4802.49	−4.66193
$103$	−1008.39	−0.964654	−0.482327	−	0.875991i	$-0.660208\pi$
$103$	−1008.39	−0.964654	−0.482327	+	0.875991i	$0.660208\pi$
$104$	250.867	0.236534
$105$	−655.508	−0.609248
$106$	−1342.77	−1.23039
$107$	−1154.03	−1.04266	−0.521329	−	0.853355i	$-0.674564\pi$
$107$	−1154.03	−1.04266	−0.521329	+	0.853355i	$0.674564\pi$
$108$	−2404.77	−2.14259
$109$	−922.565	−0.810694	−0.405347	−	0.914163i	$-0.632849\pi$
$109$	−922.565	−0.810694	−0.405347	+	0.914163i	$0.632849\pi$
$110$	−213.931	−0.185432
$111$	1632.02	1.39554
$112$	−972.221	−0.820234
$113$	1937.57	1.61302	0.806511	−	0.591219i	$-0.201353\pi$
$113$	1937.57	1.61302	0.806511	+	0.591219i	$0.201353\pi$
$114$	699.661	0.574818
$115$	−177.705	−0.144097
$116$	646.552	0.517507
$117$	−4639.58	−3.66606
$118$	−1101.80	−0.859564
$119$	1806.00	1.39122
$120$	160.299	0.121944
$121$	121.000	0.0909091
$122$	−2258.65	−1.67613
$123$	−2896.89	−2.12361
$124$	961.288	0.696179
$125$	125.000	0.0894427
$126$	3373.40	2.38513
$127$	−469.273	−0.327884	−0.163942	−	0.986470i	$-0.552421\pi$
$127$	−469.273	−0.327884	−0.163942	+	0.986470i	$0.552421\pi$
$128$	430.892	0.297546
$129$	781.460	0.533362
$130$	−1440.74	−0.972011
$131$	−1023.60	−0.682690	−0.341345	−	0.939938i	$-0.610882\pi$
$131$	−1023.60	−0.682690	−0.341345	+	0.939938i	$0.610882\pi$
$132$	742.451	0.489561
$133$	−263.111	−0.171538
$134$	−1391.63	−0.897154
$135$	−1686.52	−1.07521
$136$	−441.642	−0.278459
$137$	1753.59	1.09357	0.546787	−	0.837272i	$-0.315851\pi$
$137$	1753.59	1.09357	0.546787	+	0.837272i	$0.315851\pi$
$138$	1308.77	0.807321
$139$	−1969.98	−1.20210	−0.601049	−	0.799212i	$-0.705250\pi$
$139$	−1969.98	−1.20210	−0.601049	+	0.799212i	$0.705250\pi$
$140$	493.636	0.297999
$141$	−840.328	−0.501903
$142$	−1644.29	−0.971733
$143$	814.889	0.476535
$144$	−4396.97	−2.54454
$145$	453.442	0.259698
$146$	−1055.70	−0.598426
$147$	1431.78	0.803340
$148$	−1229.01	−0.682592
$149$	2892.33	1.59026	0.795130	−	0.606439i	$-0.207403\pi$
$149$	2892.33	1.59026	0.795130	+	0.606439i	$0.207403\pi$
$150$	−920.606	−0.501115
$151$	2513.29	1.35449	0.677246	−	0.735757i	$-0.263173\pi$
$151$	2513.29	1.35449	0.677246	+	0.735757i	$0.263173\pi$
$152$	64.3416	0.0343342
$153$	8167.80	4.31587
$154$	−592.500	−0.310032
$155$	674.173	0.349361
$156$	5000.13	2.56622
$157$	−2215.14	−1.12603	−0.563016	−	0.826446i	$-0.690359\pi$
$157$	−2215.14	−1.12603	−0.563016	+	0.826446i	$0.690359\pi$
$158$	−2734.41	−1.37682
$159$	3268.25	1.63012
$160$	−1229.95	−0.607724
$161$	−492.170	−0.240922
$162$	5843.70	2.83410
$163$	−2301.91	−1.10613	−0.553067	−	0.833137i	$-0.686543\pi$
$163$	−2301.91	−1.10613	−0.553067	+	0.833137i	$0.686543\pi$
$164$	2181.53	1.03871
$165$	520.698	0.245675
$166$	−496.142	−0.231977
$167$	−3264.41	−1.51262	−0.756311	−	0.654212i	$-0.773000\pi$
$167$	−3264.41	−1.51262	−0.756311	+	0.654212i	$0.773000\pi$
$168$	443.962	0.203884
$169$	3290.97	1.49794
$170$	2536.37	1.14430
$171$	−1189.94	−0.532148
$172$	−588.485	−0.260881
$173$	513.476	0.225658	0.112829	−	0.993614i	$-0.464009\pi$
$173$	513.476	0.225658	0.112829	+	0.993614i	$0.464009\pi$
$174$	−3339.53	−1.45499
$175$	346.198	0.149544
$176$	772.277	0.330753
$177$	2681.72	1.13882
$178$	3019.08	1.27129
$179$	2893.76	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$179$	2893.76	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$180$	2232.52	0.924455
$181$	−459.403	−0.188658	−0.0943292	−	0.995541i	$-0.530071\pi$
$181$	−459.403	−0.188658	−0.0943292	+	0.995541i	$0.530071\pi$
$182$	−3990.26	−1.62515
$183$	5497.45	2.22067
$184$	120.356	0.0482217
$185$	−861.930	−0.342542
$186$	−4965.18	−1.95734
$187$	−1434.58	−0.560999
$188$	632.816	0.245494
$189$	−4670.97	−1.79769
$190$	−369.517	−0.141092
$191$	−1855.10	−0.702778	−0.351389	−	0.936230i	$-0.614291\pi$
$191$	−1855.10	−0.702778	−0.351389	+	0.936230i	$0.614291\pi$
$192$	3741.05	1.40618
$193$	−532.455	−0.198585	−0.0992926	−	0.995058i	$-0.531658\pi$
$193$	−532.455	−0.198585	−0.0992926	+	0.995058i	$0.531658\pi$
$194$	−6440.30	−2.38344
$195$	3506.70	1.28780
$196$	−1078.21	−0.392934
$197$	4837.37	1.74948	0.874741	−	0.484591i	$-0.161031\pi$
$197$	4837.37	1.74948	0.874741	+	0.484591i	$0.161031\pi$
$198$	−2679.64	−0.961786
$199$	−3361.11	−1.19730	−0.598651	−	0.801010i	$-0.704296\pi$
$199$	−3361.11	−1.19730	−0.598651	+	0.801010i	$0.704296\pi$
$200$	−84.6600	−0.0299318
$201$	3387.17	1.18862
$202$	−6675.68	−2.32524
$203$	1255.84	0.434202
$204$	−8802.53	−3.02108
$205$	1529.96	0.521252
$206$	−3922.27	−1.32659
$207$	−2225.89	−0.747391
$208$	5200.99	1.73377
$209$	209.000	0.0691714
$210$	−2549.70	−0.837837
$211$	1191.64	0.388797	0.194398	−	0.980923i	$-0.437725\pi$
$211$	1191.64	0.388797	0.194398	+	0.980923i	$0.437725\pi$
$212$	−2461.18	−0.797333
$213$	4002.14	1.28743
$214$	−4488.78	−1.43386
$215$	−412.718	−0.130917
$216$	1142.25	0.359816
$217$	1867.18	0.584113
$218$	−3588.45	−1.11487
$219$	2569.52	0.792841
$220$	−392.116	−0.120166
$221$	−9661.35	−2.94069
$222$	6347.99	1.91914
$223$	−3424.41	−1.02832	−0.514160	−	0.857694i	$-0.671896\pi$
$223$	−3424.41	−1.02832	−0.514160	+	0.857694i	$0.671896\pi$
$224$	−3406.44	−1.01608
$225$	1565.72	0.463916
$226$	7536.48	2.21823
$227$	−2348.87	−0.686785	−0.343392	−	0.939192i	$-0.611576\pi$
$227$	−2348.87	−0.686785	−0.343392	+	0.939192i	$0.611576\pi$
$228$	1282.42	0.372500
$229$	5365.20	1.54822	0.774109	−	0.633052i	$-0.218198\pi$
$229$	5365.20	1.54822	0.774109	+	0.633052i	$0.218198\pi$
$230$	−691.212	−0.198162
$231$	1442.12	0.410755
$232$	−307.107	−0.0869076
$233$	−979.693	−0.275459	−0.137729	−	0.990470i	$-0.543980\pi$
$233$	−979.693	−0.275459	−0.137729	+	0.990470i	$0.543980\pi$
$234$	−18046.3	−5.04157
$235$	443.808	0.123195
$236$	−2019.49	−0.557025
$237$	6655.44	1.82412
$238$	7024.70	1.91321
$239$	−5252.08	−1.42146	−0.710730	−	0.703465i	$-0.751635\pi$
$239$	−5252.08	−1.42146	−0.710730	+	0.703465i	$0.751635\pi$
$240$	3323.33	0.893834
$241$	−2294.89	−0.613388	−0.306694	−	0.951808i	$-0.599223\pi$
$241$	−2294.89	−0.613388	−0.306694	+	0.951808i	$0.599223\pi$
$242$	470.648	0.125018
$243$	−5116.08	−1.35060
$244$	−4139.90	−1.08619
$245$	−756.175	−0.197185
$246$	−11267.9	−2.92039
$247$	1407.54	0.362588
$248$	−456.604	−0.116913
$249$	1207.59	0.307341
$250$	486.206	0.123002
$251$	5928.59	1.49087	0.745437	−	0.666576i	$-0.232241\pi$
$251$	5928.59	1.49087	0.745437	+	0.666576i	$0.232241\pi$
$252$	6183.15	1.54564
$253$	390.952	0.0971500
$254$	−1825.31	−0.450906
$255$	−6173.42	−1.51606
$256$	4837.28	1.18098
$257$	−3068.95	−0.744888	−0.372444	−	0.928055i	$-0.621480\pi$
$257$	−3068.95	−0.744888	−0.372444	+	0.928055i	$0.621480\pi$
$258$	3039.61	0.733479
$259$	−2387.19	−0.572713
$260$	−2640.75	−0.629894
$261$	5679.68	1.34699
$262$	−3981.45	−0.938834
$263$	20.9755	0.00491789	0.00245895	−	0.999997i	$-0.499217\pi$
$263$	20.9755	0.00491789	0.00245895	+	0.999997i	$0.499217\pi$
$264$	−352.658	−0.0822145
$265$	−1726.08	−0.400122
$266$	−1023.41	−0.235899
$267$	−7348.31	−1.68430
$268$	−2550.74	−0.581384
$269$	−2137.32	−0.484442	−0.242221	−	0.970221i	$-0.577876\pi$
$269$	−2137.32	−0.484442	−0.242221	+	0.970221i	$0.577876\pi$
$270$	−6559.99	−1.47862
$271$	4134.18	0.926693	0.463346	−	0.886177i	$-0.346649\pi$
$271$	4134.18	0.926693	0.463346	+	0.886177i	$0.346649\pi$
$272$	−9156.14	−2.04108
$273$	9712.12	2.15313
$274$	6820.87	1.50388
$275$	−275.000	−0.0603023
$276$	2398.87	0.523169
$277$	5260.86	1.14114	0.570568	−	0.821250i	$-0.306723\pi$
$277$	5260.86	1.14114	0.570568	+	0.821250i	$0.306723\pi$
$278$	−7662.54	−1.65313
$279$	8444.51	1.81204
$280$	−234.473	−0.0500444
$281$	6061.92	1.28692	0.643458	−	0.765481i	$-0.277499\pi$
$281$	6061.92	1.28692	0.643458	+	0.765481i	$0.277499\pi$
$282$	−3268.58	−0.690217
$283$	−4072.49	−0.855422	−0.427711	−	0.903915i	$-0.640680\pi$
$283$	−4072.49	−0.855422	−0.427711	+	0.903915i	$0.640680\pi$
$284$	−3013.84	−0.629714
$285$	899.388	0.186930
$286$	3169.63	0.655330
$287$	4237.34	0.871507
$288$	−15406.0	−3.15210
$289$	12095.4	2.46192
$290$	1763.73	0.357137
$291$	15675.4	3.15776
$292$	−1935.00	−0.387799
$293$	−8142.00	−1.62342	−0.811708	−	0.584064i	$-0.801462\pi$
$293$	−8142.00	−1.62342	−0.811708	+	0.584064i	$0.801462\pi$
$294$	5569.11	1.10475
$295$	−1416.32	−0.279529
$296$	583.768	0.114631
$297$	3710.35	0.724904
$298$	11250.1	2.18692
$299$	2632.91	0.509248
$300$	−1687.39	−0.324738
$301$	−1143.06	−0.218886
$302$	9775.80	1.86270
$303$	16248.3	3.08066
$304$	1333.93	0.251665
$305$	−2903.41	−0.545077
$306$	31769.9	5.93517
$307$	8399.44	1.56150	0.780752	−	0.624841i	$-0.214836\pi$
$307$	8399.44	1.56150	0.780752	+	0.624841i	$0.214836\pi$
$308$	−1086.00	−0.200911
$309$	9546.64	1.75757
$310$	2622.30	0.480440
$311$	1665.62	0.303694	0.151847	−	0.988404i	$-0.451478\pi$
$311$	1665.62	0.303694	0.151847	+	0.988404i	$0.451478\pi$
$312$	−2375.02	−0.430958
$313$	−10728.6	−1.93743	−0.968716	−	0.248173i	$-0.920170\pi$
$313$	−10728.6	−1.93743	−0.968716	+	0.248173i	$0.920170\pi$
$314$	−8616.11	−1.54852
$315$	4336.38	0.775643
$316$	−5011.94	−0.892226
$317$	−742.271	−0.131515	−0.0657573	−	0.997836i	$-0.520946\pi$
$317$	−742.271	−0.131515	−0.0657573	+	0.997836i	$0.520946\pi$
$318$	12712.3	2.24174
$319$	−997.571	−0.175089
$320$	−1975.79	−0.345155
$321$	10925.5	1.89969
$322$	−1914.37	−0.331316
$323$	−2477.91	−0.426857
$324$	10711.0	1.83659
$325$	−1852.02	−0.316097
$326$	−8953.64	−1.52115
$327$	8734.14	1.47706
$328$	−1036.21	−0.174436
$329$	1229.16	0.205976
$330$	2025.33	0.337851
$331$	−4780.16	−0.793781	−0.396891	−	0.917866i	$-0.629911\pi$
$331$	−4780.16	−0.793781	−0.396891	+	0.917866i	$0.629911\pi$
$332$	−909.385	−0.150328
$333$	−10796.3	−1.77668
$334$	−12697.4	−2.08016
$335$	−1788.89	−0.291754
$336$	9204.25	1.49444
$337$	−501.552	−0.0810720	−0.0405360	−	0.999178i	$-0.512907\pi$
$337$	−501.552	−0.0810720	−0.0405360	+	0.999178i	$0.512907\pi$
$338$	12800.7	2.05996
$339$	−18343.5	−2.93888
$340$	4648.94	0.741542
$341$	−1483.18	−0.235539
$342$	−4628.47	−0.731809
$343$	−6844.13	−1.07740
$344$	279.526	0.0438111
$345$	1682.38	0.262540
$346$	1997.24	0.310325
$347$	12185.2	1.88511	0.942556	−	0.334049i	$-0.108415\pi$
$347$	12185.2	1.88511	0.942556	+	0.334049i	$0.108415\pi$
$348$	−6121.06	−0.942883
$349$	6809.57	1.04443	0.522217	−	0.852812i	$-0.325105\pi$
$349$	6809.57	1.04443	0.522217	+	0.852812i	$0.325105\pi$
$350$	1346.59	0.205652
$351$	24987.8	3.79986
$352$	2705.88	0.409727
$353$	−2082.68	−0.314023	−0.157011	−	0.987597i	$-0.550186\pi$
$353$	−2082.68	−0.314023	−0.157011	+	0.987597i	$0.550186\pi$
$354$	10431.0	1.56610
$355$	−2113.68	−0.316007
$356$	5533.71	0.823837
$357$	−17097.8	−2.53477
$358$	11255.7	1.66168
$359$	−6107.05	−0.897822	−0.448911	−	0.893577i	$-0.648188\pi$
$359$	−6107.05	−0.897822	−0.448911	+	0.893577i	$0.648188\pi$
$360$	−1060.43	−0.155248
$361$	361.000	0.0526316
$362$	−1786.92	−0.259443
$363$	−1145.54	−0.165634
$364$	−7313.79	−1.05315
$365$	−1357.06	−0.194608
$366$	21383.1	3.05387
$367$	7265.51	1.03340	0.516698	−	0.856168i	$-0.327161\pi$
$367$	7265.51	1.03340	0.516698	+	0.856168i	$0.327161\pi$
$368$	2495.23	0.353459
$369$	19163.8	2.70360
$370$	−3352.61	−0.471064
$371$	−4780.53	−0.668983
$372$	−9100.74	−1.26842
$373$	1317.74	0.182923	0.0914613	−	0.995809i	$-0.470846\pi$
$373$	1317.74	0.182923	0.0914613	+	0.995809i	$0.470846\pi$
$374$	−5580.02	−0.771486
$375$	−1183.40	−0.162962
$376$	−300.582	−0.0412270
$377$	−6718.26	−0.917794
$378$	−18168.4	−2.47218
$379$	−6093.61	−0.825878	−0.412939	−	0.910759i	$-0.635498\pi$
$379$	−6093.61	−0.825878	−0.412939	+	0.910759i	$0.635498\pi$
$380$	−677.291	−0.0914324
$381$	4442.72	0.597395
$382$	−7215.70	−0.966460
$383$	242.797	0.0323926	0.0161963	−	0.999869i	$-0.494844\pi$
$383$	242.797	0.0323926	0.0161963	+	0.999869i	$0.494844\pi$
$384$	−4079.36	−0.542120
$385$	−761.636	−0.100822
$386$	−2071.06	−0.273094
$387$	−5169.59	−0.679031
$388$	−11804.5	−1.54454
$389$	12637.3	1.64714	0.823569	−	0.567217i	$-0.191980\pi$
$389$	12637.3	1.64714	0.823569	+	0.567217i	$0.191980\pi$
$390$	13639.9	1.77098
$391$	−4635.14	−0.599512
$392$	512.142	0.0659874
$393$	9690.67	1.24384
$394$	18815.7	2.40589
$395$	−3514.99	−0.447742
$396$	−4911.54	−0.623268
$397$	8944.84	1.13080	0.565401	−	0.824816i	$-0.308721\pi$
$397$	8944.84	1.13080	0.565401	+	0.824816i	$0.308721\pi$
$398$	−13073.6	−1.64653
$399$	2490.93	0.312538
$400$	−1755.17	−0.219397
$401$	12059.8	1.50183	0.750917	−	0.660396i	$-0.229612\pi$
$401$	12059.8	1.50183	0.750917	+	0.660396i	$0.229612\pi$
$402$	13174.9	1.63459
$403$	−9988.66	−1.23467
$404$	−12235.9	−1.50683
$405$	7511.86	0.921647
$406$	4884.80	0.597114
$407$	1896.25	0.230942
$408$	4181.13	0.507345
$409$	−9977.01	−1.20619	−0.603095	−	0.797670i	$-0.706066\pi$
$409$	−9977.01	−1.20619	−0.603095	+	0.797670i	$0.706066\pi$
$410$	5950.99	0.716826
$411$	−16601.7	−1.99246
$412$	−7189.18	−0.859673
$413$	−3922.61	−0.467359
$414$	−8657.93	−1.02781
$415$	−637.772	−0.0754386
$416$	18223.1	2.14774
$417$	18650.3	2.19019
$418$	812.937	0.0951245
$419$	2979.80	0.347429	0.173714	−	0.984796i	$-0.444423\pi$
$419$	2979.80	0.347429	0.173714	+	0.984796i	$0.444423\pi$
$420$	−4673.37	−0.542945
$421$	−13839.9	−1.60217	−0.801085	−	0.598551i	$-0.795744\pi$
$421$	−13839.9	−1.60217	−0.801085	+	0.598551i	$0.795744\pi$
$422$	4635.07	0.534673
$423$	5559.02	0.638981
$424$	1169.04	0.133900
$425$	3260.41	0.372125
$426$	15566.9	1.77047
$427$	−8041.23	−0.911340
$428$	−8227.54	−0.929189
$429$	−7714.75	−0.868232
$430$	−1605.33	−0.180037
$431$	−5674.71	−0.634203	−0.317101	−	0.948392i	$-0.602710\pi$
$431$	−5674.71	−0.634203	−0.317101	+	0.948392i	$0.602710\pi$
$432$	23681.1	2.63741
$433$	−11673.7	−1.29561	−0.647807	−	0.761805i	$-0.724314\pi$
$433$	−11673.7	−1.29561	−0.647807	+	0.761805i	$0.724314\pi$
$434$	7262.68	0.803271
$435$	−4292.84	−0.473163
$436$	−6577.32	−0.722469
$437$	675.281	0.0739200
$438$	9994.55	1.09031
$439$	8086.99	0.879205	0.439603	−	0.898192i	$-0.355119\pi$
$439$	8086.99	0.879205	0.439603	+	0.898192i	$0.355119\pi$
$440$	186.252	0.0201800
$441$	−9471.64	−1.02274
$442$	−37579.3	−4.04404
$443$	−3607.28	−0.386878	−0.193439	−	0.981112i	$-0.561964\pi$
$443$	−3607.28	−0.386878	−0.193439	+	0.981112i	$0.561964\pi$
$444$	11635.3	1.24366
$445$	3880.92	0.413423
$446$	−13319.7	−1.41414
$447$	−27382.4	−2.89741
$448$	−5472.10	−0.577082
$449$	−6455.96	−0.678565	−0.339282	−	0.940685i	$-0.610184\pi$
$449$	−6455.96	−0.678565	−0.339282	+	0.940685i	$0.610184\pi$
$450$	6090.09	0.637977
$451$	−3365.90	−0.351428
$452$	13813.7	1.43748
$453$	−23793.9	−2.46785
$454$	−9136.29	−0.944466
$455$	−5129.33	−0.528498
$456$	−609.137	−0.0625558
$457$	−217.271	−0.0222396	−0.0111198	−	0.999938i	$-0.503540\pi$
$457$	−217.271	−0.0222396	−0.0111198	+	0.999938i	$0.503540\pi$
$458$	20868.7	2.12911
$459$	−43990.1	−4.47338
$460$	−1266.93	−0.128415
$461$	5039.66	0.509154	0.254577	−	0.967052i	$-0.418064\pi$
$461$	5039.66	0.509154	0.254577	+	0.967052i	$0.418064\pi$
$462$	5609.33	0.564870
$463$	12440.1	1.24868	0.624342	−	0.781151i	$-0.285367\pi$
$463$	12440.1	1.24868	0.624342	+	0.781151i	$0.285367\pi$
$464$	−6366.95	−0.637022
$465$	−6382.56	−0.636525
$466$	−3810.66	−0.378810
$467$	−4556.26	−0.451474	−0.225737	−	0.974188i	$-0.572479\pi$
$467$	−4556.26	−0.451474	−0.225737	+	0.974188i	$0.572479\pi$
$468$	−33077.3	−3.26710
$469$	−4954.48	−0.487797
$470$	1726.26	0.169418
$471$	20971.2	2.05160
$472$	959.243	0.0935440
$473$	907.980	0.0882642
$474$	25887.4	2.50853
$475$	−475.000	−0.0458831
$476$	12875.6	1.23982
$477$	−21620.4	−2.07533
$478$	−20428.8	−1.95479
$479$	9231.33	0.880564	0.440282	−	0.897860i	$-0.354878\pi$
$479$	9231.33	0.880564	0.440282	+	0.897860i	$0.354878\pi$
$480$	11644.2	1.10726
$481$	12770.5	1.21057
$482$	−8926.30	−0.843531
$483$	4659.49	0.438953
$484$	862.655	0.0810157
$485$	−8278.76	−0.775091
$486$	−19899.8	−1.85735
$487$	−6588.07	−0.613006	−0.306503	−	0.951870i	$-0.599159\pi$
$487$	−6588.07	−0.613006	−0.306503	+	0.951870i	$0.599159\pi$
$488$	1966.42	0.182409
$489$	21792.8	2.01534
$490$	−2941.26	−0.271168
$491$	2470.87	0.227106	0.113553	−	0.993532i	$-0.463777\pi$
$491$	2470.87	0.227106	0.113553	+	0.993532i	$0.463777\pi$
$492$	−20653.0	−1.89250
$493$	11827.2	1.08047
$494$	5474.82	0.498631
$495$	−3444.57	−0.312772
$496$	−9466.33	−0.856957
$497$	−5854.01	−0.528346
$498$	4697.10	0.422655
$499$	12846.1	1.15245	0.576224	−	0.817292i	$-0.304525\pi$
$499$	12846.1	1.15245	0.576224	+	0.817292i	$0.304525\pi$
$500$	891.173	0.0797089
$501$	30905.0	2.75595
$502$	23060.1	2.05025
$503$	−5589.68	−0.495490	−0.247745	−	0.968825i	$-0.579690\pi$
$503$	−5589.68	−0.495490	−0.247745	+	0.968825i	$0.579690\pi$
$504$	−2936.94	−0.259567
$505$	−8581.34	−0.756167
$506$	1520.67	0.133601
$507$	−31156.4	−2.72920
$508$	−3345.63	−0.292201
$509$	−1597.92	−0.139149	−0.0695744	−	0.997577i	$-0.522164\pi$
$509$	−1597.92	−0.139149	−0.0695744	+	0.997577i	$0.522164\pi$
$510$	−24012.4	−2.08488
$511$	−3758.49	−0.325374
$512$	15368.2	1.32653
$513$	6408.79	0.551569
$514$	−11937.2	−1.02437
$515$	−5041.93	−0.431406
$516$	5571.33	0.475318
$517$	−976.378	−0.0830582
$518$	−9285.33	−0.787594
$519$	−4861.20	−0.411143
$520$	1254.34	0.105781
$521$	16410.5	1.37995	0.689976	−	0.723832i	$-0.257621\pi$
$521$	16410.5	1.37995	0.689976	+	0.723832i	$0.257621\pi$
$522$	22092.0	1.85238
$523$	−20166.8	−1.68610	−0.843050	−	0.537835i	$-0.819242\pi$
$523$	−20166.8	−1.68610	−0.843050	+	0.537835i	$0.819242\pi$
$524$	−7297.64	−0.608395
$525$	−3277.54	−0.272464
$526$	81.5874	0.00676308
$527$	17584.6	1.45351
$528$	−7311.33	−0.602623
$529$	−10903.8	−0.896181
$530$	−6713.86	−0.550248
$531$	−17740.4	−1.44985
$532$	−1875.82	−0.152870
$533$	−22668.1	−1.84214
$534$	−28582.4	−2.31625
$535$	−5770.16	−0.466291
$536$	1211.58	0.0976348
$537$	−27395.9	−2.20153
$538$	−8313.44	−0.666204
$539$	1663.58	0.132942
$540$	−12023.9	−0.958194
$541$	−15426.3	−1.22593	−0.612967	−	0.790109i	$-0.710024\pi$
$541$	−15426.3	−1.22593	−0.612967	+	0.790109i	$0.710024\pi$
$542$	16080.5	1.27439
$543$	4349.28	0.343730
$544$	−32081.0	−2.52843
$545$	−4612.82	−0.362554
$546$	37776.7	2.96098
$547$	−17400.2	−1.36011	−0.680054	−	0.733162i	$-0.738044\pi$
$547$	−17400.2	−1.36011	−0.680054	+	0.733162i	$0.738044\pi$
$548$	12502.0	0.974564
$549$	−36367.3	−2.82717
$550$	−1069.65	−0.0829276
$551$	−1723.08	−0.133222
$552$	−1139.44	−0.0878585
$553$	−9735.05	−0.748602
$554$	20462.9	1.56929
$555$	8160.10	0.624102
$556$	−14044.8	−1.07128
$557$	13063.1	0.993720	0.496860	−	0.867831i	$-0.334486\pi$
$557$	13063.1	0.993720	0.496860	+	0.867831i	$0.334486\pi$
$558$	32846.2	2.49192
$559$	6114.90	0.462670
$560$	−4861.10	−0.366820
$561$	13581.5	1.02212
$562$	23578.7	1.76977
$563$	−13347.8	−0.999189	−0.499595	−	0.866259i	$-0.666518\pi$
$563$	−13347.8	−0.999189	−0.499595	+	0.866259i	$0.666518\pi$
$564$	−5991.02	−0.447283
$565$	9687.86	0.721365
$566$	−15840.6	−1.17638
$567$	20804.7	1.54095
$568$	1431.55	0.105751
$569$	−7710.89	−0.568115	−0.284058	−	0.958807i	$-0.591681\pi$
$569$	−7710.89	−0.568115	−0.284058	+	0.958807i	$0.591681\pi$
$570$	3498.30	0.257066
$571$	4374.54	0.320611	0.160305	−	0.987067i	$-0.448752\pi$
$571$	4374.54	0.320611	0.160305	+	0.987067i	$0.448752\pi$
$572$	5809.66	0.424675
$573$	17562.7	1.28044
$574$	16481.8	1.19850
$575$	−888.527	−0.0644420
$576$	−24748.1	−1.79023
$577$	−18427.0	−1.32951	−0.664753	−	0.747063i	$-0.731463\pi$
$577$	−18427.0	−1.32951	−0.664753	+	0.747063i	$0.731463\pi$
$578$	47047.0	3.38564
$579$	5040.88	0.361817
$580$	3232.76	0.231436
$581$	−1766.37	−0.126129
$582$	60971.8	4.34255
$583$	3797.38	0.269762
$584$	919.110	0.0651251
$585$	−23197.9	−1.63951
$586$	−31669.5	−2.23252
$587$	8288.52	0.582801	0.291400	−	0.956601i	$-0.405879\pi$
$587$	8288.52	0.582801	0.291400	+	0.956601i	$0.405879\pi$
$588$	10207.7	0.715915
$589$	−2561.86	−0.179218
$590$	−5508.98	−0.384409
$591$	−45796.5	−3.18751
$592$	12102.7	0.840233
$593$	−6525.36	−0.451879	−0.225939	−	0.974141i	$-0.572545\pi$
$593$	−6525.36	−0.451879	−0.225939	+	0.974141i	$0.572545\pi$
$594$	14432.0	0.996887
$595$	9029.98	0.622173
$596$	20620.5	1.41720
$597$	31820.5	2.18145
$598$	10241.1	0.700318
$599$	11679.6	0.796690	0.398345	−	0.917236i	$-0.369585\pi$
$599$	11679.6	0.796690	0.398345	+	0.917236i	$0.369585\pi$
$600$	801.496	0.0545349
$601$	7404.55	0.502559	0.251279	−	0.967915i	$-0.419149\pi$
$601$	7404.55	0.502559	0.251279	+	0.967915i	$0.419149\pi$
$602$	−4446.10	−0.301012
$603$	−22407.1	−1.51325
$604$	17918.2	1.20709
$605$	605.000	0.0406558
$606$	63200.3	4.23653
$607$	−6625.90	−0.443060	−0.221530	−	0.975154i	$-0.571105\pi$
$607$	−6625.90	−0.443060	−0.221530	+	0.975154i	$0.571105\pi$
$608$	4673.80	0.311756
$609$	−11889.4	−0.791104
$610$	−11293.2	−0.749590
$611$	−6575.54	−0.435381
$612$	58231.4	3.84618
$613$	16572.5	1.09193	0.545967	−	0.837807i	$-0.316162\pi$
$613$	16572.5	1.09193	0.545967	+	0.837807i	$0.316162\pi$
$614$	32670.9	2.14738
$615$	−14484.5	−0.949707
$616$	515.841	0.0337400
$617$	−17601.2	−1.14845	−0.574227	−	0.818696i	$-0.694697\pi$
$617$	−17601.2	−1.14845	−0.574227	+	0.818696i	$0.694697\pi$
$618$	37133.1	2.41701
$619$	−1221.24	−0.0792984	−0.0396492	−	0.999214i	$-0.512624\pi$
$619$	−1221.24	−0.0792984	−0.0396492	+	0.999214i	$0.512624\pi$
$620$	4806.44	0.311341
$621$	11988.2	0.774668
$622$	6478.70	0.417640
$623$	10748.5	0.691221
$624$	−49239.0	−3.15888
$625$	625.000	0.0400000
$626$	−41730.5	−2.66435
$627$	−1978.65	−0.126028
$628$	−15792.6	−1.00349
$629$	−22482.0	−1.42514
$630$	16867.0	1.06666
$631$	8524.59	0.537811	0.268905	−	0.963167i	$-0.413338\pi$
$631$	8524.59	0.537811	0.268905	+	0.963167i	$0.413338\pi$
$632$	2380.63	0.149836
$633$	−11281.6	−0.708376
$634$	−2887.18	−0.180859
$635$	−2346.37	−0.146634
$636$	23300.6	1.45272
$637$	11203.6	0.696865
$638$	−3880.20	−0.240782
$639$	−26475.3	−1.63904
$640$	2154.46	0.133067
$641$	−5872.51	−0.361857	−0.180928	−	0.983496i	$-0.557910\pi$
$641$	−5872.51	−0.361857	−0.180928	+	0.983496i	$0.557910\pi$
$642$	42496.4	2.61246
$643$	15358.7	0.941972	0.470986	−	0.882141i	$-0.343898\pi$
$643$	15358.7	0.941972	0.470986	+	0.882141i	$0.343898\pi$
$644$	−3508.87	−0.214703
$645$	3907.30	0.238527
$646$	−9638.21	−0.587013
$647$	−16744.7	−1.01747	−0.508734	−	0.860924i	$-0.669886\pi$
$647$	−16744.7	−1.01747	−0.508734	+	0.860924i	$0.669886\pi$
$648$	−5087.63	−0.308427
$649$	3115.90	0.188459
$650$	−7203.71	−0.434697
$651$	−17677.0	−1.06424
$652$	−16411.2	−0.985756
$653$	5722.00	0.342909	0.171454	−	0.985192i	$-0.445153\pi$
$653$	5722.00	0.342909	0.171454	+	0.985192i	$0.445153\pi$
$654$	33972.8	2.03125
$655$	−5118.00	−0.305308
$656$	−21482.7	−1.27860
$657$	−16998.2	−1.00938
$658$	4781.02	0.283258
$659$	−20383.0	−1.20487	−0.602434	−	0.798169i	$-0.705802\pi$
$659$	−20383.0	−1.20487	−0.602434	+	0.798169i	$0.705802\pi$
$660$	3712.26	0.218938
$661$	3580.00	0.210659	0.105330	−	0.994437i	$-0.466410\pi$
$661$	3580.00	0.210659	0.105330	+	0.994437i	$0.466410\pi$
$662$	−18593.2	−1.09161
$663$	91466.4	5.35786
$664$	431.950	0.0252454
$665$	−1315.55	−0.0767142
$666$	−41993.8	−2.44328
$667$	−3223.16	−0.187108
$668$	−23273.2	−1.34801
$669$	32419.7	1.87357
$670$	−6958.16	−0.401219
$671$	6387.49	0.367491
$672$	32249.6	1.85127
$673$	−3756.33	−0.215150	−0.107575	−	0.994197i	$-0.534309\pi$
$673$	−3756.33	−0.215150	−0.107575	+	0.994197i	$0.534309\pi$
$674$	−1950.86	−0.111490
$675$	−8432.62	−0.480847
$676$	23462.6	1.33492
$677$	−9655.41	−0.548135	−0.274068	−	0.961710i	$-0.588369\pi$
$677$	−9655.41	−0.548135	−0.274068	+	0.961710i	$0.588369\pi$
$678$	−71349.6	−4.04154
$679$	−22928.7	−1.29591
$680$	−2208.21	−0.124531
$681$	22237.3	1.25130
$682$	−5769.06	−0.323913
$683$	−256.376	−0.0143630	−0.00718151	−	0.999974i	$-0.502286\pi$
$683$	−256.376	−0.0143630	−0.00718151	+	0.999974i	$0.502286\pi$
$684$	−8483.56	−0.474236
$685$	8767.97	0.489061
$686$	−26621.3	−1.48164
$687$	−50793.6	−2.82081
$688$	5795.14	0.321130
$689$	25573.9	1.41406
$690$	6543.87	0.361045
$691$	−11734.5	−0.646023	−0.323011	−	0.946395i	$-0.604695\pi$
$691$	−11734.5	−0.646023	−0.323011	+	0.946395i	$0.604695\pi$
$692$	3660.77	0.201100
$693$	−9540.04	−0.522938
$694$	47396.0	2.59240
$695$	−9849.91	−0.537595
$696$	2907.45	0.158343
$697$	39906.3	2.16866
$698$	26486.8	1.43631
$699$	9274.99	0.501877
$700$	2468.18	0.133269
$701$	26300.4	1.41705	0.708526	−	0.705685i	$-0.249361\pi$
$701$	26300.4	1.41705	0.708526	+	0.705685i	$0.249361\pi$
$702$	97193.9	5.22556
$703$	3275.33	0.175721
$704$	4346.73	0.232704
$705$	−4201.64	−0.224458
$706$	−8100.90	−0.431844
$707$	−23766.7	−1.26427
$708$	19119.0	1.01488
$709$	21504.4	1.13909	0.569546	−	0.821959i	$-0.307119\pi$
$709$	21504.4	1.13909	0.569546	+	0.821959i	$0.307119\pi$
$710$	−8221.47	−0.434572
$711$	−44027.8	−2.32232
$712$	−2628.47	−0.138351
$713$	−4792.17	−0.251708
$714$	−66504.5	−3.48581
$715$	4074.45	0.213113
$716$	20630.7	1.07682
$717$	49722.7	2.58986
$718$	−23754.3	−1.23468
$719$	20213.7	1.04846	0.524231	−	0.851576i	$-0.324353\pi$
$719$	20213.7	1.04846	0.524231	+	0.851576i	$0.324353\pi$
$720$	−21984.8	−1.13795
$721$	−13964.1	−0.721289
$722$	1404.16	0.0723789
$723$	21726.2	1.11758
$724$	−3275.26	−0.168127
$725$	2267.21	0.116141
$726$	−4455.73	−0.227779
$727$	−17074.9	−0.871077	−0.435538	−	0.900170i	$-0.643442\pi$
$727$	−17074.9	−0.871077	−0.435538	+	0.900170i	$0.643442\pi$
$728$	3473.99	0.176861
$729$	7871.17	0.399897
$730$	−5278.49	−0.267624
$731$	−10765.0	−0.544678
$732$	39193.4	1.97900
$733$	25559.4	1.28794	0.643969	−	0.765051i	$-0.277286\pi$
$733$	25559.4	1.28794	0.643969	+	0.765051i	$0.277286\pi$
$734$	28260.3	1.42113
$735$	7158.89	0.359265
$736$	8742.73	0.437855
$737$	3935.56	0.196700
$738$	74540.5	3.71799
$739$	−30123.5	−1.49947	−0.749737	−	0.661736i	$-0.769820\pi$
$739$	−30123.5	−1.49947	−0.749737	+	0.661736i	$0.769820\pi$
$740$	−6145.03	−0.305265
$741$	−13325.5	−0.660626
$742$	−18594.6	−0.919985
$743$	32077.2	1.58385	0.791923	−	0.610621i	$-0.209080\pi$
$743$	32077.2	1.58385	0.791923	+	0.610621i	$0.209080\pi$
$744$	4322.78	0.213012
$745$	14461.6	0.711186
$746$	5125.56	0.251555
$747$	−7988.56	−0.391280
$748$	−10227.7	−0.499947
$749$	−15981.0	−0.779615
$750$	−4603.03	−0.224105
$751$	35691.1	1.73420	0.867101	−	0.498132i	$-0.165980\pi$
$751$	35691.1	1.73420	0.867101	+	0.498132i	$0.165980\pi$
$752$	−6231.69	−0.302189
$753$	−56127.4	−2.71633
$754$	−26131.7	−1.26215
$755$	12566.4	0.605747
$756$	−33301.1	−1.60205
$757$	−16579.7	−0.796037	−0.398018	−	0.917377i	$-0.630302\pi$
$757$	−16579.7	−0.796037	−0.398018	+	0.917377i	$0.630302\pi$
$758$	−23702.0	−1.13575
$759$	−3701.24	−0.177004
$760$	321.708	0.0153547
$761$	−24230.6	−1.15422	−0.577108	−	0.816668i	$-0.695819\pi$
$761$	−24230.6	−1.15422	−0.577108	+	0.816668i	$0.695819\pi$
$762$	17280.6	0.821538
$763$	−12775.6	−0.606170
$764$	−13225.7	−0.626297
$765$	40839.0	1.93011
$766$	944.397	0.0445463
$767$	20984.4	0.987878
$768$	−45795.7	−2.15170
$769$	42540.3	1.99486	0.997428	−	0.0716769i	$-0.0228350\pi$
$769$	42540.3	1.99486	0.997428	+	0.0716769i	$0.0228350\pi$
$770$	−2962.50	−0.138651
$771$	29054.5	1.35716
$772$	−3796.07	−0.176974
$773$	32587.7	1.51630	0.758149	−	0.652082i	$-0.226104\pi$
$773$	32587.7	1.51630	0.758149	+	0.652082i	$0.226104\pi$
$774$	−20107.9	−0.933804
$775$	3370.87	0.156239
$776$	5607.04	0.259383
$777$	22600.1	1.04347
$778$	49154.7	2.26514
$779$	−5813.83	−0.267397
$780$	25000.6	1.14765
$781$	4650.09	0.213052
$782$	−18029.1	−0.824448
$783$	−30589.6	−1.39615
$784$	10617.7	0.483680
$785$	−11075.7	−0.503577
$786$	37693.3	1.71053
$787$	−22637.6	−1.02534	−0.512672	−	0.858585i	$-0.671344\pi$
$787$	−22637.6	−1.02534	−0.512672	+	0.858585i	$0.671344\pi$
$788$	34487.4	1.55909
$789$	−198.580	−0.00896026
$790$	−13672.1	−0.615735
$791$	26831.4	1.20608
$792$	2332.94	0.104668
$793$	43017.3	1.92634
$794$	34792.3	1.55508
$795$	16341.2	0.729011
$796$	−23962.7	−1.06700
$797$	40760.8	1.81157	0.905785	−	0.423737i	$-0.139282\pi$
$797$	40760.8	1.81157	0.905785	+	0.423737i	$0.139282\pi$
$798$	9688.85	0.429802
$799$	11576.0	0.512551
$800$	−6149.73	−0.271782
$801$	48611.3	2.14431
$802$	46908.3	2.06532
$803$	2985.53	0.131204
$804$	24148.4	1.05927
$805$	−2460.85	−0.107744
$806$	−38852.4	−1.69791
$807$	20234.6	0.882640
$808$	5811.97	0.253050
$809$	−40739.0	−1.77046	−0.885232	−	0.465150i	$-0.846001\pi$
$809$	−40739.0	−1.77046	−0.885232	+	0.465150i	$0.846001\pi$
$810$	29218.5	1.26745
$811$	4357.33	0.188664	0.0943320	−	0.995541i	$-0.469928\pi$
$811$	4357.33	0.188664	0.0943320	+	0.995541i	$0.469928\pi$
$812$	8953.40	0.386949
$813$	−39139.3	−1.68841
$814$	7375.73	0.317591
$815$	−11509.6	−0.494678
$816$	86683.4	3.71878
$817$	1568.33	0.0671590
$818$	−38807.1	−1.65875
$819$	−64248.6	−2.74118
$820$	10907.6	0.464526
$821$	33540.3	1.42578	0.712889	−	0.701277i	$-0.247386\pi$
$821$	33540.3	1.42578	0.712889	+	0.701277i	$0.247386\pi$
$822$	−64574.8	−2.74003
$823$	−26927.3	−1.14049	−0.570247	−	0.821474i	$-0.693152\pi$
$823$	−26927.3	−1.14049	−0.570247	+	0.821474i	$0.693152\pi$
$824$	3414.80	0.144369
$825$	2603.49	0.109869
$826$	−15257.6	−0.642711
$827$	17188.9	0.722752	0.361376	−	0.932420i	$-0.382307\pi$
$827$	17188.9	0.722752	0.361376	+	0.932420i	$0.382307\pi$
$828$	−15869.2	−0.666055
$829$	4192.46	0.175646	0.0878228	−	0.996136i	$-0.472009\pi$
$829$	4192.46	0.175646	0.0878228	+	0.996136i	$0.472009\pi$
$830$	−2480.71	−0.103743
$831$	−49805.9	−2.07912
$832$	29273.6	1.21981
$833$	−19723.5	−0.820383
$834$	72543.1	3.01195
$835$	−16322.1	−0.676465
$836$	1490.04	0.0616437
$837$	−45480.4	−1.87817
$838$	11590.4	0.477784
$839$	37658.2	1.54959	0.774795	−	0.632213i	$-0.217853\pi$
$839$	37658.2	1.54959	0.774795	+	0.632213i	$0.217853\pi$
$840$	2219.81	0.0911795
$841$	−16164.6	−0.662784
$842$	−53832.2	−2.20330
$843$	−57389.6	−2.34473
$844$	8495.68	0.346485
$845$	16454.8	0.669898
$846$	21622.6	0.878726
$847$	1675.60	0.0679743
$848$	24236.6	0.981472
$849$	38555.3	1.55855
$850$	12681.9	0.511746
$851$	6126.79	0.246796
$852$	28532.8	1.14732
$853$	−12132.5	−0.486996	−0.243498	−	0.969901i	$-0.578295\pi$
$853$	−12132.5	−0.486996	−0.243498	+	0.969901i	$0.578295\pi$
$854$	−31277.6	−1.25327
$855$	−5949.72	−0.237984
$856$	3908.01	0.156043
$857$	7515.88	0.299577	0.149788	−	0.988718i	$-0.452141\pi$
$857$	7515.88	0.299577	0.149788	+	0.988718i	$0.452141\pi$
$858$	−30007.7	−1.19399
$859$	−9723.17	−0.386205	−0.193103	−	0.981179i	$-0.561855\pi$
$859$	−9723.17	−0.386205	−0.193103	+	0.981179i	$0.561855\pi$
$860$	−2942.42	−0.116670
$861$	−40115.9	−1.58786
$862$	−22072.7	−0.872155
$863$	27534.5	1.08608	0.543039	−	0.839707i	$-0.317274\pi$
$863$	27534.5	1.08608	0.543039	+	0.839707i	$0.317274\pi$
$864$	82973.4	3.26714
$865$	2567.38	0.100917
$866$	−45406.5	−1.78173
$867$	−114510.	−4.48556
$868$	13311.8	0.520545
$869$	7732.97	0.301868
$870$	−16697.6	−0.650693
$871$	26504.5	1.03108
$872$	3124.17	0.121328
$873$	−103697.	−4.02019
$874$	2626.61	0.101655
$875$	1730.99	0.0668779
$876$	18319.1	0.706559
$877$	13663.6	0.526099	0.263049	−	0.964782i	$-0.415272\pi$
$877$	13663.6	0.526099	0.263049	+	0.964782i	$0.415272\pi$
$878$	31455.6	1.20908
$879$	77082.3	2.95782
$880$	3861.38	0.147917
$881$	−10289.0	−0.393469	−0.196734	−	0.980457i	$-0.563034\pi$
$881$	−10289.0	−0.393469	−0.196734	+	0.980457i	$0.563034\pi$
$882$	−36841.4	−1.40648
$883$	−24660.3	−0.939845	−0.469923	−	0.882708i	$-0.655718\pi$
$883$	−24660.3	−0.939845	−0.469923	+	0.882708i	$0.655718\pi$
$884$	−68879.5	−2.62066
$885$	13408.6	0.509295
$886$	−14031.1	−0.532034
$887$	−18602.8	−0.704193	−0.352097	−	0.935964i	$-0.614531\pi$
$887$	−18602.8	−0.704193	−0.352097	+	0.935964i	$0.614531\pi$
$888$	−5526.67	−0.208855
$889$	−6498.46	−0.245165
$890$	15095.4	0.568538
$891$	−16526.1	−0.621375
$892$	−24413.9	−0.916410
$893$	−1686.47	−0.0631978
$894$	−106508.	−3.98451
$895$	14468.8	0.540378
$896$	5966.96	0.222480
$897$	−24926.4	−0.927836
$898$	−25111.4	−0.933162
$899$	12227.9	0.453642
$900$	11162.6	0.413429
$901$	−45021.9	−1.66470
$902$	−13092.2	−0.483284
$903$	10821.6	0.398804
$904$	−6561.39	−0.241403
$905$	−2297.02	−0.0843706
$906$	−92549.9	−3.39378
$907$	19077.9	0.698424	0.349212	−	0.937044i	$-0.386449\pi$
$907$	19077.9	0.698424	0.349212	+	0.937044i	$0.386449\pi$
$908$	−16746.0	−0.612044
$909$	−107487.	−3.92204
$910$	−19951.3	−0.726790
$911$	−40926.5	−1.48842	−0.744212	−	0.667943i	$-0.767175\pi$
$911$	−40926.5	−1.48842	−0.744212	+	0.667943i	$0.767175\pi$
$912$	−12628.7	−0.458527
$913$	1403.10	0.0508607
$914$	−845.107	−0.0305839
$915$	27487.2	0.993115
$916$	38250.5	1.37973
$917$	−14174.7	−0.510459
$918$	−171106.	−6.15179
$919$	−36374.3	−1.30564	−0.652818	−	0.757515i	$-0.726413\pi$
$919$	−36374.3	−1.30564	−0.652818	+	0.757515i	$0.726413\pi$
$920$	601.782	0.0215654
$921$	−79519.5	−2.84501
$922$	19602.5	0.700189
$923$	31316.6	1.11679
$924$	10281.4	0.366054
$925$	−4309.65	−0.153190
$926$	48387.7	1.71719
$927$	−63153.9	−2.23759
$928$	−22308.4	−0.789125
$929$	22313.4	0.788028	0.394014	−	0.919104i	$-0.371086\pi$
$929$	22313.4	0.788028	0.394014	+	0.919104i	$0.371086\pi$
$930$	−24825.9	−0.875348
$931$	2873.46	0.101154
$932$	−6984.61	−0.245481
$933$	−15768.9	−0.553322
$934$	−17722.3	−0.620867
$935$	−7172.90	−0.250887
$936$	15711.5	0.548660
$937$	7076.87	0.246736	0.123368	−	0.992361i	$-0.460631\pi$
$937$	7076.87	0.246736	0.123368	+	0.992361i	$0.460631\pi$
$938$	−19271.2	−0.670818
$939$	101570.	3.52994
$940$	3164.08	0.109788
$941$	−46090.1	−1.59670	−0.798350	−	0.602194i	$-0.794293\pi$
$941$	−46090.1	−1.59670	−0.798350	+	0.602194i	$0.794293\pi$
$942$	81570.8	2.82136
$943$	−10875.3	−0.375554
$944$	19887.1	0.685666
$945$	−23354.9	−0.803951
$946$	3531.72	0.121381
$947$	−2222.18	−0.0762526	−0.0381263	−	0.999273i	$-0.512139\pi$
$947$	−2222.18	−0.0762526	−0.0381263	+	0.999273i	$0.512139\pi$
$948$	47449.2	1.62561
$949$	20106.4	0.687758
$950$	−1847.58	−0.0630985
$951$	7027.26	0.239616
$952$	−6115.83	−0.208209
$953$	−3131.21	−0.106432	−0.0532160	−	0.998583i	$-0.516947\pi$
$953$	−3131.21	−0.106432	−0.0532160	+	0.998583i	$0.516947\pi$
$954$	−84095.9	−2.85399
$955$	−9275.52	−0.314292
$956$	−37444.1	−1.26677
$957$	9444.25	0.319006
$958$	35906.6	1.21095
$959$	24283.7	0.817685
$960$	18705.2	0.628863
$961$	−11610.6	−0.389736
$962$	49672.8	1.66478
$963$	−72275.5	−2.41853
$964$	−16361.1	−0.546635
$965$	−2662.27	−0.0888100
$966$	18123.8	0.603648
$967$	−59575.5	−1.98120	−0.990599	−	0.136798i	$-0.956319\pi$
$967$	−59575.5	−1.98120	−0.990599	+	0.136798i	$0.956319\pi$
$968$	−409.754	−0.0136054
$969$	23459.0	0.777720
$970$	−32201.5	−1.06590
$971$	−19390.3	−0.640848	−0.320424	−	0.947274i	$-0.603825\pi$
$971$	−19390.3	−0.640848	−0.320424	+	0.947274i	$0.603825\pi$
$972$	−36474.5	−1.20362
$973$	−27280.2	−0.898830
$974$	−25625.3	−0.843006
$975$	17533.5	0.575920
$976$	40767.9	1.33704
$977$	−19420.1	−0.635931	−0.317966	−	0.948102i	$-0.603000\pi$
$977$	−19420.1	−0.635931	−0.317966	+	0.948102i	$0.603000\pi$
$978$	84766.2	2.77150
$979$	−8538.02	−0.278729
$980$	−5391.06	−0.175726
$981$	−57779.0	−1.88047
$982$	9610.83	0.312315
$983$	−41521.0	−1.34722	−0.673609	−	0.739088i	$-0.735257\pi$
$983$	−41521.0	−1.34722	−0.673609	+	0.739088i	$0.735257\pi$
$984$	9810.03	0.317817
$985$	24186.8	0.782392
$986$	46003.8	1.48586
$987$	−11636.8	−0.375282
$988$	10034.9	0.323129
$989$	2933.69	0.0943235
$990$	−13398.2	−0.430124
$991$	39011.3	1.25049	0.625245	−	0.780429i	$-0.284999\pi$
$991$	39011.3	1.25049	0.625245	+	0.780429i	$0.284999\pi$
$992$	−33167.9	−1.06157
$993$	45255.0	1.44625
$994$	−22770.0	−0.726582
$995$	−16805.6	−0.535449
$996$	8609.36	0.273894
$997$	43005.3	1.36609	0.683045	−	0.730376i	$-0.260655\pi$
$997$	43005.3	1.36609	0.683045	+	0.730376i	$0.260655\pi$
$998$	49967.0	1.58485
$999$	58146.6	1.84152

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.d.1.19	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.d.1.19	✓	22	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)

\(f(q)\)	\(=\)	\(q+3.88965 q^{2} -9.46724 q^{3} +7.12938 q^{4} +5.00000 q^{5} -36.8243 q^{6} +13.8479 q^{7} -3.38640 q^{8} +62.6286 q^{9} +O(q^{10})\)	\(q+3.88965 q^{2} -9.46724 q^{3} +7.12938 q^{4} +5.00000 q^{5} -36.8243 q^{6} +13.8479 q^{7} -3.38640 q^{8} +62.6286 q^{9} +19.4483 q^{10} -11.0000 q^{11} -67.4956 q^{12} -74.0808 q^{13} +53.8636 q^{14} -47.3362 q^{15} -70.2070 q^{16} +130.416 q^{17} +243.603 q^{18} -19.0000 q^{19} +35.6469 q^{20} -131.102 q^{21} -42.7862 q^{22} -35.5411 q^{23} +32.0599 q^{24} +25.0000 q^{25} -288.149 q^{26} -337.305 q^{27} +98.7272 q^{28} +90.6883 q^{29} -184.121 q^{30} +134.835 q^{31} -245.989 q^{32} +104.140 q^{33} +507.274 q^{34} +69.2396 q^{35} +446.503 q^{36} -172.386 q^{37} -73.9034 q^{38} +701.341 q^{39} -16.9320 q^{40} +305.991 q^{41} -509.940 q^{42} -82.5436 q^{43} -78.4232 q^{44} +313.143 q^{45} -138.242 q^{46} +88.7617 q^{47} +664.666 q^{48} -151.235 q^{49} +97.2413 q^{50} -1234.68 q^{51} -528.150 q^{52} -345.216 q^{53} -1312.00 q^{54} -55.0000 q^{55} -46.8946 q^{56} +179.878 q^{57} +352.746 q^{58} -283.263 q^{59} -337.478 q^{60} -580.681 q^{61} +524.460 q^{62} +867.277 q^{63} -395.157 q^{64} -370.404 q^{65} +405.067 q^{66} -357.778 q^{67} +929.788 q^{68} +336.476 q^{69} +269.318 q^{70} -422.735 q^{71} -212.086 q^{72} -271.412 q^{73} -670.521 q^{74} -236.681 q^{75} -135.458 q^{76} -152.327 q^{77} +2727.97 q^{78} -702.997 q^{79} -351.035 q^{80} +1502.37 q^{81} +1190.20 q^{82} -127.554 q^{83} -934.674 q^{84} +652.082 q^{85} -321.066 q^{86} -858.568 q^{87} +37.2504 q^{88} +776.183 q^{89} +1218.02 q^{90} -1025.87 q^{91} -253.386 q^{92} -1276.51 q^{93} +345.252 q^{94} -95.0000 q^{95} +2328.84 q^{96} -1655.75 q^{97} -588.251 q^{98} -688.915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9 - 20 * q^10 - 242 * q^11 - 196 * q^12 - q^13 - 63 * q^14 - 105 * q^15 + 6 * q^16 + 187 * q^17 - 361 * q^18 - 418 * q^19 + 370 * q^20 - 107 * q^21 + 44 * q^22 - 361 * q^23 + 208 * q^24 + 550 * q^25 - 365 * q^26 - 1467 * q^27 - 773 * q^28 - 319 * q^29 - 45 * q^30 - 402 * q^31 - 873 * q^32 + 231 * q^33 - 717 * q^34 - 205 * q^35 + 725 * q^36 - 838 * q^37 + 76 * q^38 - 607 * q^39 - 390 * q^40 - 392 * q^41 - 1350 * q^42 - 610 * q^43 - 814 * q^44 + 1045 * q^45 - 605 * q^46 - 1866 * q^47 - 1637 * q^48 + 379 * q^49 - 100 * q^50 - 2659 * q^51 - 638 * q^52 - 1303 * q^53 + 2338 * q^54 - 1210 * q^55 + 727 * q^56 + 399 * q^57 + 44 * q^58 - 2417 * q^59 - 980 * q^60 + 918 * q^61 - 1634 * q^62 - 374 * q^63 - 1716 * q^64 - 5 * q^65 + 99 * q^66 - 2339 * q^67 + 4940 * q^68 + 127 * q^69 - 315 * q^70 - 2370 * q^71 - 3306 * q^72 + 2207 * q^73 + 2051 * q^74 - 525 * q^75 - 1406 * q^76 + 451 * q^77 + 1380 * q^78 + 586 * q^79 + 30 * q^80 + 1950 * q^81 - 1566 * q^82 - 2870 * q^83 + 3076 * q^84 + 935 * q^85 - 1246 * q^86 - 1811 * q^87 + 858 * q^88 - 1768 * q^89 - 1805 * q^90 - 2195 * q^91 - 6728 * q^92 - 2916 * q^93 + 672 * q^94 - 2090 * q^95 + 6022 * q^96 - 4022 * q^97 + 1162 * q^98 - 2299 * q^99

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