LMFDB - Embedded newform 1045.4.a.f.1.4

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:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-4.44156 q^{2} -2.04104 q^{3} +11.7274 q^{4} -5.00000 q^{5} +9.06537 q^{6} -16.0963 q^{7} -16.5556 q^{8} -22.8342 q^{9} +O(q^{10})$	\(q-4.44156 q^{2} -2.04104 q^{3} +11.7274 q^{4} -5.00000 q^{5} +9.06537 q^{6} -16.0963 q^{7} -16.5556 q^{8} -22.8342 q^{9} +22.2078 q^{10} +11.0000 q^{11} -23.9361 q^{12} -2.33044 q^{13} +71.4928 q^{14} +10.2052 q^{15} -20.2869 q^{16} -20.5265 q^{17} +101.419 q^{18} -19.0000 q^{19} -58.6371 q^{20} +32.8532 q^{21} -48.8571 q^{22} +134.968 q^{23} +33.7905 q^{24} +25.0000 q^{25} +10.3508 q^{26} +101.713 q^{27} -188.769 q^{28} -229.308 q^{29} -45.3269 q^{30} +151.778 q^{31} +222.550 q^{32} -22.4514 q^{33} +91.1697 q^{34} +80.4817 q^{35} -267.786 q^{36} +87.5858 q^{37} +84.3896 q^{38} +4.75650 q^{39} +82.7778 q^{40} +361.861 q^{41} -145.919 q^{42} -267.170 q^{43} +129.002 q^{44} +114.171 q^{45} -599.468 q^{46} +95.5432 q^{47} +41.4063 q^{48} -83.9077 q^{49} -111.039 q^{50} +41.8954 q^{51} -27.3300 q^{52} +33.9282 q^{53} -451.765 q^{54} -55.0000 q^{55} +266.484 q^{56} +38.7797 q^{57} +1018.48 q^{58} -618.143 q^{59} +119.680 q^{60} -148.528 q^{61} -674.132 q^{62} +367.547 q^{63} -826.173 q^{64} +11.6522 q^{65} +99.7191 q^{66} +672.521 q^{67} -240.723 q^{68} -275.474 q^{69} -357.464 q^{70} +287.835 q^{71} +378.033 q^{72} +696.551 q^{73} -389.017 q^{74} -51.0259 q^{75} -222.821 q^{76} -177.060 q^{77} -21.1263 q^{78} +1189.08 q^{79} +101.435 q^{80} +408.922 q^{81} -1607.23 q^{82} +599.331 q^{83} +385.283 q^{84} +102.633 q^{85} +1186.65 q^{86} +468.025 q^{87} -182.111 q^{88} -679.190 q^{89} -507.096 q^{90} +37.5115 q^{91} +1582.83 q^{92} -309.785 q^{93} -424.360 q^{94} +95.0000 q^{95} -454.232 q^{96} +1396.16 q^{97} +372.681 q^{98} -251.176 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * q^98 + 1870 * 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 $

$2$	−4.44156	−1.57033	−0.785164	−	0.619288i	$-0.787421\pi$
$2$	−4.44156	−1.57033	−0.785164	+	0.619288i	$0.787421\pi$
$3$	−2.04104	−0.392797	−0.196399	−	0.980524i	$-0.562925\pi$
$3$	−2.04104	−0.392797	−0.196399	+	0.980524i	$0.562925\pi$
$4$	11.7274	1.46593
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	9.06537	0.616821
$7$	−16.0963	−0.869121	−0.434560	−	0.900643i	$-0.643096\pi$
$7$	−16.0963	−0.869121	−0.434560	+	0.900643i	$0.643096\pi$
$8$	−16.5556	−0.731659
$9$	−22.8342	−0.845710
$10$	22.2078	0.702272
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−23.9361	−0.575813
$13$	−2.33044	−0.0497190	−0.0248595	−	0.999691i	$-0.507914\pi$
$13$	−2.33044	−0.0497190	−0.0248595	+	0.999691i	$0.507914\pi$
$14$	71.4928	1.36480
$15$	10.2052	0.175664
$16$	−20.2869	−0.316983
$17$	−20.5265	−0.292848	−0.146424	−	0.989222i	$-0.546776\pi$
$17$	−20.5265	−0.292848	−0.146424	+	0.989222i	$0.546776\pi$
$18$	101.419	1.32804
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−58.6371	−0.655583
$21$	32.8532	0.341388
$22$	−48.8571	−0.473472
$23$	134.968	1.22360	0.611799	−	0.791013i	$-0.290446\pi$
$23$	134.968	1.22360	0.611799	+	0.791013i	$0.290446\pi$
$24$	33.7905	0.287394
$25$	25.0000	0.200000
$26$	10.3508	0.0780751
$27$	101.713	0.724990
$28$	−188.769	−1.27407
$29$	−229.308	−1.46832	−0.734162	−	0.678975i	$-0.762425\pi$
$29$	−229.308	−1.46832	−0.734162	+	0.678975i	$0.762425\pi$
$30$	−45.3269	−0.275851
$31$	151.778	0.879361	0.439681	−	0.898154i	$-0.355092\pi$
$31$	151.778	0.879361	0.439681	+	0.898154i	$0.355092\pi$
$32$	222.550	1.22943
$33$	−22.4514	−0.118433
$34$	91.1697	0.459867
$35$	80.4817	0.388683
$36$	−267.786	−1.23975
$37$	87.5858	0.389163	0.194581	−	0.980886i	$-0.437665\pi$
$37$	87.5858	0.389163	0.194581	+	0.980886i	$0.437665\pi$
$38$	84.3896	0.360258
$39$	4.75650	0.0195295
$40$	82.7778	0.327208
$41$	361.861	1.37837	0.689185	−	0.724585i	$-0.257969\pi$
$41$	361.861	1.37837	0.689185	+	0.724585i	$0.257969\pi$
$42$	−145.919	−0.536092
$43$	−267.170	−0.947514	−0.473757	−	0.880656i	$-0.657103\pi$
$43$	−267.170	−0.947514	−0.473757	+	0.880656i	$0.657103\pi$
$44$	129.002	0.441994
$45$	114.171	0.378213
$46$	−599.468	−1.92145
$47$	95.5432	0.296519	0.148260	−	0.988948i	$-0.452633\pi$
$47$	95.5432	0.296519	0.148260	+	0.988948i	$0.452633\pi$
$48$	41.4063	0.124510
$49$	−83.9077	−0.244629
$50$	−111.039	−0.314065
$51$	41.8954	0.115030
$52$	−27.3300	−0.0728844
$53$	33.9282	0.0879321	0.0439661	−	0.999033i	$-0.486001\pi$
$53$	33.9282	0.0879321	0.0439661	+	0.999033i	$0.486001\pi$
$54$	−451.765	−1.13847
$55$	−55.0000	−0.134840
$56$	266.484	0.635900
$57$	38.7797	0.0901139
$58$	1018.48	2.30575
$59$	−618.143	−1.36399	−0.681995	−	0.731357i	$-0.738887\pi$
$59$	−618.143	−1.36399	−0.681995	+	0.731357i	$0.738887\pi$
$60$	119.680	0.257511
$61$	−148.528	−0.311755	−0.155877	−	0.987776i	$-0.549820\pi$
$61$	−148.528	−0.311755	−0.155877	+	0.987776i	$0.549820\pi$
$62$	−674.132	−1.38089
$63$	367.547	0.735024
$64$	−826.173	−1.61362
$65$	11.6522	0.0222350
$66$	99.7191	0.185978
$67$	672.521	1.22629	0.613145	−	0.789970i	$-0.289904\pi$
$67$	672.521	1.22629	0.613145	+	0.789970i	$0.289904\pi$
$68$	−240.723	−0.429294
$69$	−275.474	−0.480626
$70$	−357.464	−0.610359
$71$	287.835	0.481123	0.240561	−	0.970634i	$-0.422668\pi$
$71$	287.835	0.481123	0.240561	+	0.970634i	$0.422668\pi$
$72$	378.033	0.618772
$73$	696.551	1.11678	0.558391	−	0.829578i	$-0.311419\pi$
$73$	696.551	1.11678	0.558391	+	0.829578i	$0.311419\pi$
$74$	−389.017	−0.611113
$75$	−51.0259	−0.0785595
$76$	−222.821	−0.336307
$77$	−177.060	−0.262050
$78$	−21.1263	−0.0306677
$79$	1189.08	1.69345	0.846723	−	0.532034i	$-0.178572\pi$
$79$	1189.08	1.69345	0.846723	+	0.532034i	$0.178572\pi$
$80$	101.435	0.141759
$81$	408.922	0.560936
$82$	−1607.23	−2.16449
$83$	599.331	0.792591	0.396296	−	0.918123i	$-0.370296\pi$
$83$	599.331	0.792591	0.396296	+	0.918123i	$0.370296\pi$
$84$	385.283	0.500451
$85$	102.633	0.130966
$86$	1186.65	1.48791
$87$	468.025	0.576754
$88$	−182.111	−0.220604
$89$	−679.190	−0.808921	−0.404461	−	0.914555i	$-0.632541\pi$
$89$	−679.190	−0.808921	−0.404461	+	0.914555i	$0.632541\pi$
$90$	−507.096	−0.593918
$91$	37.5115	0.0432118
$92$	1582.83	1.79371
$93$	−309.785	−0.345411
$94$	−424.360	−0.465632
$95$	95.0000	0.102598
$96$	−454.232	−0.482916
$97$	1396.16	1.46142	0.730712	−	0.682686i	$-0.239188\pi$
$97$	1396.16	1.46142	0.730712	+	0.682686i	$0.239188\pi$
$98$	372.681	0.384148
$99$	−251.176	−0.254991
$100$	293.186	0.293186
$101$	−236.571	−0.233066	−0.116533	−	0.993187i	$-0.537178\pi$
$101$	−236.571	−0.233066	−0.116533	+	0.993187i	$0.537178\pi$
$102$	−186.081	−0.180635
$103$	−582.182	−0.556933	−0.278467	−	0.960446i	$-0.589826\pi$
$103$	−582.182	−0.556933	−0.278467	+	0.960446i	$0.589826\pi$
$104$	38.5817	0.0363774
$105$	−164.266	−0.152674
$106$	−150.694	−0.138082
$107$	1558.45	1.40805	0.704023	−	0.710177i	$-0.251385\pi$
$107$	1558.45	1.40805	0.704023	+	0.710177i	$0.251385\pi$
$108$	1192.84	1.06278
$109$	145.056	0.127466	0.0637331	−	0.997967i	$-0.479699\pi$
$109$	145.056	0.127466	0.0637331	+	0.997967i	$0.479699\pi$
$110$	244.286	0.211743
$111$	−178.766	−0.152862
$112$	326.545	0.275497
$113$	−1687.19	−1.40458	−0.702290	−	0.711890i	$-0.747839\pi$
$113$	−1687.19	−1.40458	−0.702290	+	0.711890i	$0.747839\pi$
$114$	−172.242	−0.141508
$115$	−674.840	−0.547210
$116$	−2689.19	−2.15246
$117$	53.2136	0.0420478
$118$	2745.52	2.14191
$119$	330.402	0.254520
$120$	−168.952	−0.128526
$121$	121.000	0.0909091
$122$	659.695	0.489557
$123$	−738.571	−0.541420
$124$	1779.97	1.28908
$125$	−125.000	−0.0894427
$126$	−1632.48	−1.15423
$127$	2331.61	1.62911	0.814555	−	0.580087i	$-0.196981\pi$
$127$	2331.61	1.62911	0.814555	+	0.580087i	$0.196981\pi$
$128$	1889.09	1.30448
$129$	545.304	0.372181
$130$	−51.7538	−0.0349162
$131$	−862.636	−0.575335	−0.287668	−	0.957730i	$-0.592880\pi$
$131$	−862.636	−0.575335	−0.287668	+	0.957730i	$0.592880\pi$
$132$	−263.297	−0.173614
$133$	305.831	0.199390
$134$	−2987.04	−1.92568
$135$	−508.567	−0.324225
$136$	339.828	0.214265
$137$	2891.24	1.80303	0.901515	−	0.432748i	$-0.142456\pi$
$137$	2891.24	1.80303	0.901515	+	0.432748i	$0.142456\pi$
$138$	1223.53	0.754740
$139$	−2422.62	−1.47830	−0.739151	−	0.673540i	$-0.764773\pi$
$139$	−2422.62	−1.47830	−0.739151	+	0.673540i	$0.764773\pi$
$140$	943.843	0.569781
$141$	−195.007	−0.116472
$142$	−1278.44	−0.755520
$143$	−25.6348	−0.0149908
$144$	463.235	0.268076
$145$	1146.54	0.656654
$146$	−3093.77	−1.75371
$147$	171.259	0.0960896
$148$	1027.16	0.570485
$149$	−1259.23	−0.692352	−0.346176	−	0.938170i	$-0.612520\pi$
$149$	−1259.23	−0.692352	−0.346176	+	0.938170i	$0.612520\pi$
$150$	226.634	0.123364
$151$	3181.06	1.71438	0.857190	−	0.515001i	$-0.172208\pi$
$151$	3181.06	1.71438	0.857190	+	0.515001i	$0.172208\pi$
$152$	314.556	0.167854
$153$	468.706	0.247664
$154$	786.421	0.411504
$155$	−758.892	−0.393262
$156$	55.7815	0.0286288
$157$	−3168.41	−1.61062	−0.805308	−	0.592857i	$-0.798000\pi$
$157$	−3168.41	−1.61062	−0.805308	+	0.592857i	$0.798000\pi$
$158$	−5281.38	−2.65926
$159$	−69.2487	−0.0345395
$160$	−1112.75	−0.549816
$161$	−2172.49	−1.06345
$162$	−1816.25	−0.880853
$163$	2163.57	1.03966	0.519828	−	0.854271i	$-0.325996\pi$
$163$	2163.57	1.03966	0.519828	+	0.854271i	$0.325996\pi$
$164$	4243.70	2.02059
$165$	112.257	0.0529648
$166$	−2661.96	−1.24463
$167$	−3510.58	−1.62669	−0.813343	−	0.581784i	$-0.802355\pi$
$167$	−3510.58	−1.62669	−0.813343	+	0.581784i	$0.802355\pi$
$168$	−543.903	−0.249780
$169$	−2191.57	−0.997528
$170$	−455.849	−0.205659
$171$	433.849	0.194019
$172$	−3133.22	−1.38899
$173$	−3830.63	−1.68345	−0.841726	−	0.539905i	$-0.818460\pi$
$173$	−3830.63	−1.68345	−0.841726	+	0.539905i	$0.818460\pi$
$174$	−2078.76	−0.905692
$175$	−402.409	−0.173824
$176$	−223.156	−0.0955740
$177$	1261.65	0.535772
$178$	3016.66	1.27027
$179$	−130.792	−0.0546139	−0.0273070	−	0.999627i	$-0.508693\pi$
$179$	−130.792	−0.0546139	−0.0273070	+	0.999627i	$0.508693\pi$
$180$	1338.93	0.554433
$181$	−1828.32	−0.750819	−0.375410	−	0.926859i	$-0.622498\pi$
$181$	−1828.32	−0.750819	−0.375410	+	0.926859i	$0.622498\pi$
$182$	−166.609	−0.0678567
$183$	303.150	0.122456
$184$	−2234.47	−0.895257
$185$	−437.929	−0.174039
$186$	1375.93	0.542408
$187$	−225.792	−0.0882970
$188$	1120.48	0.434676
$189$	−1637.21	−0.630104
$190$	−421.948	−0.161112
$191$	−117.373	−0.0444649	−0.0222324	−	0.999753i	$-0.507077\pi$
$191$	−117.373	−0.0444649	−0.0222324	+	0.999753i	$0.507077\pi$
$192$	1686.25	0.633825
$193$	−4920.91	−1.83531	−0.917654	−	0.397380i	$-0.869920\pi$
$193$	−4920.91	−1.83531	−0.917654	+	0.397380i	$0.869920\pi$
$194$	−6201.11	−2.29492
$195$	−23.7825	−0.00873385
$196$	−984.022	−0.358608
$197$	−279.079	−0.100932	−0.0504659	−	0.998726i	$-0.516071\pi$
$197$	−279.079	−0.100932	−0.0504659	+	0.998726i	$0.516071\pi$
$198$	1115.61	0.400420
$199$	985.493	0.351054	0.175527	−	0.984475i	$-0.443837\pi$
$199$	985.493	0.351054	0.175527	+	0.984475i	$0.443837\pi$
$200$	−413.889	−0.146332
$201$	−1372.64	−0.481684
$202$	1050.74	0.365990
$203$	3691.02	1.27615
$204$	491.325	0.168626
$205$	−1809.31	−0.616426
$206$	2585.80	0.874567
$207$	−3081.88	−1.03481
$208$	47.2774	0.0157601
$209$	−209.000	−0.0691714
$210$	729.597	0.239747
$211$	−4028.03	−1.31422	−0.657111	−	0.753794i	$-0.728222\pi$
$211$	−4028.03	−1.31422	−0.657111	+	0.753794i	$0.728222\pi$
$212$	397.891	0.128902
$213$	−587.481	−0.188984
$214$	−6921.94	−2.21109
$215$	1335.85	0.423741
$216$	−1683.92	−0.530446
$217$	−2443.08	−0.764271
$218$	−644.273	−0.200164
$219$	−1421.68	−0.438669
$220$	−645.008	−0.197666
$221$	47.8358	0.0145601
$222$	793.998	0.240044
$223$	1174.67	0.352742	0.176371	−	0.984324i	$-0.443564\pi$
$223$	1174.67	0.352742	0.176371	+	0.984324i	$0.443564\pi$
$224$	−3582.24	−1.06852
$225$	−570.854	−0.169142
$226$	7493.76	2.20565
$227$	−4567.75	−1.33556	−0.667781	−	0.744358i	$-0.732756\pi$
$227$	−4567.75	−1.33556	−0.667781	+	0.744358i	$0.732756\pi$
$228$	454.786	0.132100
$229$	4847.79	1.39891	0.699457	−	0.714675i	$-0.253425\pi$
$229$	4847.79	1.39891	0.699457	+	0.714675i	$0.253425\pi$
$230$	2997.34	0.859298
$231$	361.385	0.102932
$232$	3796.32	1.07431
$233$	1014.64	0.285284	0.142642	−	0.989774i	$-0.454440\pi$
$233$	1014.64	0.285284	0.142642	+	0.989774i	$0.454440\pi$
$234$	−236.351	−0.0660289
$235$	−477.716	−0.132607
$236$	−7249.23	−1.99951
$237$	−2426.96	−0.665181
$238$	−1467.50	−0.399680
$239$	−6565.39	−1.77690	−0.888452	−	0.458970i	$-0.848218\pi$
$239$	−6565.39	−1.77690	−0.888452	+	0.458970i	$0.848218\pi$
$240$	−207.032	−0.0556826
$241$	−1965.73	−0.525409	−0.262705	−	0.964876i	$-0.584615\pi$
$241$	−1965.73	−0.525409	−0.262705	+	0.964876i	$0.584615\pi$
$242$	−537.428	−0.142757
$243$	−3580.88	−0.945324
$244$	−1741.85	−0.457010
$245$	419.539	0.109401
$246$	3280.41	0.850207
$247$	44.2783	0.0114063
$248$	−2512.78	−0.643393
$249$	−1223.25	−0.311328
$250$	555.195	0.140454
$251$	1937.29	0.487173	0.243587	−	0.969879i	$-0.421676\pi$
$251$	1937.29	0.487173	0.243587	+	0.969879i	$0.421676\pi$
$252$	4310.38	1.07749
$253$	1484.65	0.368929
$254$	−10356.0	−2.55824
$255$	−209.477	−0.0514429
$256$	−1781.13	−0.434847
$257$	6482.31	1.57337	0.786683	−	0.617357i	$-0.211796\pi$
$257$	6482.31	1.57337	0.786683	+	0.617357i	$0.211796\pi$
$258$	−2422.00	−0.584446
$259$	−1409.81	−0.338229
$260$	136.650	0.0325949
$261$	5236.05	1.24178
$262$	3831.45	0.903464
$263$	−8166.78	−1.91477	−0.957387	−	0.288809i	$-0.906741\pi$
$263$	−8166.78	−1.91477	−0.957387	+	0.288809i	$0.906741\pi$
$264$	371.695	0.0866525
$265$	−169.641	−0.0393244
$266$	−1358.36	−0.313108
$267$	1386.25	0.317742
$268$	7886.94	1.79765
$269$	−4545.78	−1.03034	−0.515169	−	0.857089i	$-0.672271\pi$
$269$	−4545.78	−1.03034	−0.515169	+	0.857089i	$0.672271\pi$
$270$	2258.83	0.509140
$271$	−142.040	−0.0318389	−0.0159195	−	0.999873i	$-0.505068\pi$
$271$	−142.040	−0.0318389	−0.0159195	+	0.999873i	$0.505068\pi$
$272$	416.420	0.0928279
$273$	−76.5623	−0.0169735
$274$	−12841.6	−2.83135
$275$	275.000	0.0603023
$276$	−3230.60	−0.704563
$277$	8413.20	1.82491	0.912455	−	0.409176i	$-0.134184\pi$
$277$	8413.20	1.82491	0.912455	+	0.409176i	$0.134184\pi$
$278$	10760.2	2.32142
$279$	−3465.73	−0.743685
$280$	−1332.42	−0.284383
$281$	3314.38	0.703627	0.351814	−	0.936070i	$-0.385565\pi$
$281$	3314.38	0.703627	0.351814	+	0.936070i	$0.385565\pi$
$282$	866.134	0.182899
$283$	−2798.04	−0.587725	−0.293863	−	0.955848i	$-0.594941\pi$
$283$	−2798.04	−0.587725	−0.293863	+	0.955848i	$0.594941\pi$
$284$	3375.56	0.705292
$285$	−193.898	−0.0403002
$286$	113.858	0.0235405
$287$	−5824.64	−1.19797
$288$	−5081.75	−1.03974
$289$	−4491.66	−0.914240
$290$	−5092.42	−1.03116
$291$	−2849.60	−0.574044
$292$	8168.74	1.63712
$293$	−7617.14	−1.51876	−0.759382	−	0.650645i	$-0.774499\pi$
$293$	−7617.14	−1.51876	−0.759382	+	0.650645i	$0.774499\pi$
$294$	−760.655	−0.150892
$295$	3090.72	0.609995
$296$	−1450.03	−0.284735
$297$	1118.85	0.218593
$298$	5592.96	1.08722
$299$	−314.534	−0.0608361
$300$	−598.402	−0.115163
$301$	4300.47	0.823504
$302$	−14128.9	−2.69214
$303$	482.849	0.0915477
$304$	385.451	0.0727209
$305$	742.639	0.139421
$306$	−2081.79	−0.388914
$307$	6442.47	1.19769	0.598845	−	0.800865i	$-0.295626\pi$
$307$	6442.47	1.19769	0.598845	+	0.800865i	$0.295626\pi$
$308$	−2076.45	−0.384146
$309$	1188.25	0.218762
$310$	3370.66	0.617551
$311$	3836.29	0.699472	0.349736	−	0.936848i	$-0.386271\pi$
$311$	3836.29	0.699472	0.349736	+	0.936848i	$0.386271\pi$
$312$	−78.7466	−0.0142889
$313$	10822.8	1.95444	0.977218	−	0.212237i	$-0.0680749\pi$
$313$	10822.8	1.95444	0.977218	+	0.212237i	$0.0680749\pi$
$314$	14072.7	2.52919
$315$	−1837.73	−0.328713
$316$	13944.9	2.48247
$317$	−7916.77	−1.40268	−0.701341	−	0.712826i	$-0.747415\pi$
$317$	−7916.77	−1.40268	−0.701341	+	0.712826i	$0.747415\pi$
$318$	307.572	0.0542383
$319$	−2522.39	−0.442716
$320$	4130.87	0.721632
$321$	−3180.85	−0.553077
$322$	9649.24	1.66997
$323$	390.004	0.0671839
$324$	4795.60	0.822292
$325$	−58.2609	−0.00994380
$326$	−9609.62	−1.63260
$327$	−296.064	−0.0500684
$328$	−5990.81	−1.00850
$329$	−1537.90	−0.257711
$330$	−498.596	−0.0831721
$331$	−9141.12	−1.51795	−0.758975	−	0.651120i	$-0.774300\pi$
$331$	−9141.12	−1.51795	−0.758975	+	0.651120i	$0.774300\pi$
$332$	7028.60	1.16188
$333$	−1999.95	−0.329119
$334$	15592.4	2.55443
$335$	−3362.60	−0.548414
$336$	−666.490	−0.108214
$337$	−5397.86	−0.872523	−0.436261	−	0.899820i	$-0.643698\pi$
$337$	−5397.86	−0.872523	−0.436261	+	0.899820i	$0.643698\pi$
$338$	9733.98	1.56645
$339$	3443.62	0.551716
$340$	1203.62	0.191986
$341$	1669.56	0.265137
$342$	−1926.97	−0.304674
$343$	6871.65	1.08173
$344$	4423.16	0.693258
$345$	1377.37	0.214943
$346$	17013.9	2.64357
$347$	−11464.3	−1.77359	−0.886794	−	0.462165i	$-0.847073\pi$
$347$	−11464.3	−1.77359	−0.886794	+	0.462165i	$0.847073\pi$
$348$	5488.73	0.845479
$349$	−1438.79	−0.220679	−0.110339	−	0.993894i	$-0.535194\pi$
$349$	−1438.79	−0.220679	−0.110339	+	0.993894i	$0.535194\pi$
$350$	1787.32	0.272961
$351$	−237.036	−0.0360458
$352$	2448.05	0.370686
$353$	6497.89	0.979739	0.489870	−	0.871796i	$-0.337044\pi$
$353$	6497.89	0.979739	0.489870	+	0.871796i	$0.337044\pi$
$354$	−5603.70	−0.841337
$355$	−1439.17	−0.215165
$356$	−7965.15	−1.18582
$357$	−674.362	−0.0999749
$358$	580.922	0.0857617
$359$	−10283.9	−1.51188	−0.755941	−	0.654640i	$-0.772820\pi$
$359$	−10283.9	−1.51188	−0.755941	+	0.654640i	$0.772820\pi$
$360$	−1890.16	−0.276723
$361$	361.000	0.0526316
$362$	8120.61	1.17903
$363$	−246.965	−0.0357089
$364$	439.913	0.0633454
$365$	−3482.75	−0.499440
$366$	−1346.46	−0.192297
$367$	6174.57	0.878228	0.439114	−	0.898431i	$-0.355292\pi$
$367$	6174.57	0.878228	0.439114	+	0.898431i	$0.355292\pi$
$368$	−2738.08	−0.387860
$369$	−8262.80	−1.16570
$370$	1945.09	0.273298
$371$	−546.121	−0.0764236
$372$	−3632.98	−0.506348
$373$	−7009.97	−0.973090	−0.486545	−	0.873656i	$-0.661743\pi$
$373$	−7009.97	−0.973090	−0.486545	+	0.873656i	$0.661743\pi$
$374$	1002.87	0.138655
$375$	255.129	0.0351329
$376$	−1581.77	−0.216951
$377$	534.387	0.0730035
$378$	7271.77	0.989470
$379$	−848.127	−0.114948	−0.0574741	−	0.998347i	$-0.518305\pi$
$379$	−848.127	−0.114948	−0.0574741	+	0.998347i	$0.518305\pi$
$380$	1114.11	0.150401
$381$	−4758.90	−0.639910
$382$	521.318	0.0698244
$383$	−2084.78	−0.278140	−0.139070	−	0.990283i	$-0.544411\pi$
$383$	−2084.78	−0.278140	−0.139070	+	0.990283i	$0.544411\pi$
$384$	−3855.71	−0.512398
$385$	885.299	0.117192
$386$	21856.5	2.88203
$387$	6100.62	0.801322
$388$	16373.3	2.14234
$389$	−3027.01	−0.394539	−0.197269	−	0.980349i	$-0.563207\pi$
$389$	−3027.01	−0.394539	−0.197269	+	0.980349i	$0.563207\pi$
$390$	105.631	0.0137150
$391$	−2770.42	−0.358328
$392$	1389.14	0.178985
$393$	1760.67	0.225990
$394$	1239.54	0.158496
$395$	−5945.41	−0.757332
$396$	−2945.65	−0.373799
$397$	6308.80	0.797555	0.398778	−	0.917048i	$-0.369435\pi$
$397$	6308.80	0.797555	0.398778	+	0.917048i	$0.369435\pi$
$398$	−4377.12	−0.551270
$399$	−624.211	−0.0783199
$400$	−507.173	−0.0633966
$401$	−2907.43	−0.362070	−0.181035	−	0.983477i	$-0.557945\pi$
$401$	−2907.43	−0.362070	−0.181035	+	0.983477i	$0.557945\pi$
$402$	6096.65	0.756401
$403$	−353.710	−0.0437210
$404$	−2774.36	−0.341658
$405$	−2044.61	−0.250858
$406$	−16393.9	−2.00397
$407$	963.444	0.117337
$408$	−693.602	−0.0841627
$409$	−10316.6	−1.24724	−0.623621	−	0.781727i	$-0.714339\pi$
$409$	−10316.6	−1.24724	−0.623621	+	0.781727i	$0.714339\pi$
$410$	8036.13	0.967991
$411$	−5901.12	−0.708225
$412$	−6827.50	−0.816424
$413$	9949.85	1.18547
$414$	13688.4	1.62499
$415$	−2996.65	−0.354458
$416$	−518.639	−0.0611258
$417$	4944.65	0.580673
$418$	928.285	0.108622
$419$	383.904	0.0447611	0.0223806	−	0.999750i	$-0.492875\pi$
$419$	383.904	0.0447611	0.0223806	+	0.999750i	$0.492875\pi$
$420$	−1926.42	−0.223808
$421$	4149.92	0.480414	0.240207	−	0.970722i	$-0.422785\pi$
$421$	4149.92	0.480414	0.240207	+	0.970722i	$0.422785\pi$
$422$	17890.7	2.06376
$423$	−2181.65	−0.250769
$424$	−561.701	−0.0643364
$425$	−513.163	−0.0585696
$426$	2609.33	0.296767
$427$	2390.75	0.270952
$428$	18276.6	2.06409
$429$	52.3215	0.00588836
$430$	−5933.26	−0.665412
$431$	7268.73	0.812349	0.406174	−	0.913796i	$-0.366863\pi$
$431$	7268.73	0.812349	0.406174	+	0.913796i	$0.366863\pi$
$432$	−2063.45	−0.229810
$433$	9973.04	1.10687	0.553434	−	0.832893i	$-0.313317\pi$
$433$	9973.04	1.10687	0.553434	+	0.832893i	$0.313317\pi$
$434$	10851.1	1.20016
$435$	−2340.13	−0.257932
$436$	1701.13	0.186856
$437$	−2564.39	−0.280713
$438$	6314.49	0.688854
$439$	6329.47	0.688130	0.344065	−	0.938946i	$-0.388196\pi$
$439$	6329.47	0.688130	0.344065	+	0.938946i	$0.388196\pi$
$440$	910.556	0.0986569
$441$	1915.96	0.206885
$442$	−212.465	−0.0228641
$443$	6231.93	0.668370	0.334185	−	0.942507i	$-0.391539\pi$
$443$	6231.93	0.668370	0.334185	+	0.942507i	$0.391539\pi$
$444$	−2096.46	−0.224085
$445$	3395.95	0.361761
$446$	−5217.35	−0.553921
$447$	2570.14	0.271954
$448$	13298.4	1.40243
$449$	12107.2	1.27255	0.636275	−	0.771462i	$-0.280474\pi$
$449$	12107.2	1.27255	0.636275	+	0.771462i	$0.280474\pi$
$450$	2535.48	0.265608
$451$	3980.47	0.415594
$452$	−19786.4	−2.05901
$453$	−6492.66	−0.673404
$454$	20287.9	2.09727
$455$	−187.558	−0.0193249
$456$	−642.019	−0.0659327
$457$	−5197.12	−0.531972	−0.265986	−	0.963977i	$-0.585697\pi$
$457$	−5197.12	−0.531972	−0.265986	+	0.963977i	$0.585697\pi$
$458$	−21531.7	−2.19675
$459$	−2087.82	−0.212312
$460$	−7914.13	−0.802170
$461$	9997.95	1.01009	0.505045	−	0.863093i	$-0.331476\pi$
$461$	9997.95	1.01009	0.505045	+	0.863093i	$0.331476\pi$
$462$	−1605.11	−0.161638
$463$	11324.6	1.13671	0.568356	−	0.822783i	$-0.307580\pi$
$463$	11324.6	1.13671	0.568356	+	0.822783i	$0.307580\pi$
$464$	4651.95	0.465434
$465$	1548.93	0.154472
$466$	−4506.57	−0.447989
$467$	−1526.29	−0.151239	−0.0756193	−	0.997137i	$-0.524093\pi$
$467$	−1526.29	−0.151239	−0.0756193	+	0.997137i	$0.524093\pi$
$468$	624.058	0.0616391
$469$	−10825.1	−1.06579
$470$	2121.80	0.208237
$471$	6466.84	0.632646
$472$	10233.7	0.997976
$473$	−2938.87	−0.285686
$474$	10779.5	1.04455
$475$	−475.000	−0.0458831
$476$	3874.77	0.373108
$477$	−774.723	−0.0743651
$478$	29160.6	2.79032
$479$	−4558.69	−0.434847	−0.217423	−	0.976077i	$-0.569765\pi$
$479$	−4558.69	−0.434847	−0.217423	+	0.976077i	$0.569765\pi$
$480$	2271.16	0.215966
$481$	−204.113	−0.0193488
$482$	8730.89	0.825065
$483$	4434.13	0.417722
$484$	1419.02	0.133266
$485$	−6980.78	−0.653569
$486$	15904.7	1.48447
$487$	−1686.04	−0.156883	−0.0784413	−	0.996919i	$-0.524994\pi$
$487$	−1686.04	−0.156883	−0.0784413	+	0.996919i	$0.524994\pi$
$488$	2458.96	0.228098
$489$	−4415.93	−0.408374
$490$	−1863.40	−0.171796
$491$	−11377.5	−1.04574	−0.522870	−	0.852412i	$-0.675139\pi$
$491$	−11377.5	−1.04574	−0.522870	+	0.852412i	$0.675139\pi$
$492$	−8661.54	−0.793683
$493$	4706.89	0.429995
$494$	−196.665	−0.0179116
$495$	1255.88	0.114036
$496$	−3079.12	−0.278743
$497$	−4633.09	−0.418154
$498$	5433.16	0.488887
$499$	18283.0	1.64020	0.820099	−	0.572221i	$-0.193918\pi$
$499$	18283.0	1.64020	0.820099	+	0.572221i	$0.193918\pi$
$500$	−1465.93	−0.131117
$501$	7165.21	0.638958
$502$	−8604.57	−0.765021
$503$	−3646.66	−0.323254	−0.161627	−	0.986852i	$-0.551674\pi$
$503$	−3646.66	−0.323254	−0.161627	+	0.986852i	$0.551674\pi$
$504$	−6084.94	−0.537788
$505$	1182.85	0.104230
$506$	−6594.14	−0.579339
$507$	4473.07	0.391826
$508$	27343.8	2.38816
$509$	−2257.30	−0.196568	−0.0982839	−	0.995158i	$-0.531335\pi$
$509$	−2257.30	−0.196568	−0.0982839	+	0.995158i	$0.531335\pi$
$510$	930.403	0.0807823
$511$	−11211.9	−0.970618
$512$	−7201.75	−0.621631
$513$	−1932.55	−0.166324
$514$	−28791.5	−2.47070
$515$	2910.91	0.249068
$516$	6395.01	0.545591
$517$	1050.97	0.0894039
$518$	6261.76	0.531131
$519$	7818.45	0.661256
$520$	−192.908	−0.0162685
$521$	−18156.6	−1.52679	−0.763394	−	0.645933i	$-0.776469\pi$
$521$	−18156.6	−1.52679	−0.763394	+	0.645933i	$0.776469\pi$
$522$	−23256.2	−1.94999
$523$	−16592.6	−1.38728	−0.693638	−	0.720324i	$-0.743993\pi$
$523$	−16592.6	−1.38728	−0.693638	+	0.720324i	$0.743993\pi$
$524$	−10116.5	−0.843400
$525$	821.330	0.0682777
$526$	36273.2	3.00682
$527$	−3115.48	−0.257519
$528$	455.470	0.0375412
$529$	6049.34	0.497193
$530$	753.471	0.0617522
$531$	14114.8	1.15354
$532$	3586.60	0.292291
$533$	−843.294	−0.0685312
$534$	−6157.11	−0.498959
$535$	−7792.24	−0.629697
$536$	−11134.0	−0.897227
$537$	266.952	0.0214522
$538$	20190.3	1.61797
$539$	−922.985	−0.0737584
$540$	−5964.18	−0.475291
$541$	6540.32	0.519760	0.259880	−	0.965641i	$-0.416317\pi$
$541$	6540.32	0.519760	0.259880	+	0.965641i	$0.416317\pi$
$542$	630.881	0.0499975
$543$	3731.67	0.294920
$544$	−4568.18	−0.360035
$545$	−725.279	−0.0570046
$546$	340.056	0.0266539
$547$	2700.43	0.211082	0.105541	−	0.994415i	$-0.466343\pi$
$547$	2700.43	0.211082	0.105541	+	0.994415i	$0.466343\pi$
$548$	33906.8	2.64311
$549$	3391.51	0.263654
$550$	−1221.43	−0.0946943
$551$	4356.85	0.336856
$552$	4560.63	0.351655
$553$	−19139.9	−1.47181
$554$	−37367.7	−2.86571
$555$	893.829	0.0683620
$556$	−28411.1	−2.16708
$557$	−16081.4	−1.22332	−0.611662	−	0.791119i	$-0.709499\pi$
$557$	−16081.4	−1.22332	−0.611662	+	0.791119i	$0.709499\pi$
$558$	15393.3	1.16783
$559$	622.624	0.0471094
$560$	−1632.73	−0.123206
$561$	460.849	0.0346828
$562$	−14721.0	−1.10493
$563$	17435.4	1.30518	0.652589	−	0.757712i	$-0.273683\pi$
$563$	17435.4	1.30518	0.652589	+	0.757712i	$0.273683\pi$
$564$	−2286.93	−0.170740
$565$	8435.96	0.628148
$566$	12427.7	0.922921
$567$	−6582.15	−0.487521
$568$	−4765.27	−0.352018
$569$	21828.1	1.60823	0.804114	−	0.594476i	$-0.202640\pi$
$569$	21828.1	1.60823	0.804114	+	0.594476i	$0.202640\pi$
$570$	861.210	0.0632845
$571$	12874.9	0.943606	0.471803	−	0.881704i	$-0.343603\pi$
$571$	12874.9	0.943606	0.471803	+	0.881704i	$0.343603\pi$
$572$	−300.630	−0.0219755
$573$	239.562	0.0174657
$574$	25870.5	1.88121
$575$	3374.20	0.244720
$576$	18865.0	1.36465
$577$	−11684.3	−0.843020	−0.421510	−	0.906824i	$-0.638500\pi$
$577$	−11684.3	−0.843020	−0.421510	+	0.906824i	$0.638500\pi$
$578$	19950.0	1.43566
$579$	10043.7	0.720904
$580$	13445.9	0.962608
$581$	−9647.03	−0.688858
$582$	12656.7	0.901437
$583$	373.211	0.0265125
$584$	−11531.8	−0.817104
$585$	−266.068	−0.0188044
$586$	33831.9	2.38496
$587$	−10055.1	−0.707013	−0.353507	−	0.935432i	$-0.615011\pi$
$587$	−10055.1	−0.707013	−0.353507	+	0.935432i	$0.615011\pi$
$588$	2008.42	0.140860
$589$	−2883.79	−0.201739
$590$	−13727.6	−0.957892
$591$	569.610	0.0396457
$592$	−1776.85	−0.123358
$593$	23154.8	1.60346	0.801730	−	0.597686i	$-0.203913\pi$
$593$	23154.8	1.60346	0.801730	+	0.597686i	$0.203913\pi$
$594$	−4969.42	−0.343262
$595$	−1652.01	−0.113825
$596$	−14767.6	−1.01494
$597$	−2011.43	−0.137893
$598$	1397.02	0.0955325
$599$	−14377.2	−0.980694	−0.490347	−	0.871527i	$-0.663130\pi$
$599$	−14377.2	−0.980694	−0.490347	+	0.871527i	$0.663130\pi$
$600$	844.762	0.0574788
$601$	−9628.46	−0.653499	−0.326750	−	0.945111i	$-0.605953\pi$
$601$	−9628.46	−0.653499	−0.326750	+	0.945111i	$0.605953\pi$
$602$	−19100.8	−1.29317
$603$	−15356.5	−1.03709
$604$	37305.7	2.51316
$605$	−605.000	−0.0406558
$606$	−2144.60	−0.143760
$607$	−13382.7	−0.894869	−0.447435	−	0.894317i	$-0.647662\pi$
$607$	−13382.7	−0.894869	−0.447435	+	0.894317i	$0.647662\pi$
$608$	−4228.45	−0.282050
$609$	−7533.49	−0.501269
$610$	−3298.47	−0.218937
$611$	−222.657	−0.0147426
$612$	5496.72	0.363058
$613$	−6404.88	−0.422008	−0.211004	−	0.977485i	$-0.567673\pi$
$613$	−6404.88	−0.422008	−0.211004	+	0.977485i	$0.567673\pi$
$614$	−28614.6	−1.88077
$615$	3692.86	0.242131
$616$	2931.32	0.191731
$617$	2024.21	0.132077	0.0660385	−	0.997817i	$-0.478964\pi$
$617$	2024.21	0.132077	0.0660385	+	0.997817i	$0.478964\pi$
$618$	−5277.70	−0.343528
$619$	14886.1	0.966596	0.483298	−	0.875456i	$-0.339439\pi$
$619$	14886.1	0.966596	0.483298	+	0.875456i	$0.339439\pi$
$620$	−8899.85	−0.576494
$621$	13728.0	0.887097
$622$	−17039.1	−1.09840
$623$	10932.5	0.703050
$624$	−96.4948	−0.00619052
$625$	625.000	0.0400000
$626$	−48069.9	−3.06910
$627$	426.576	0.0271704
$628$	−37157.3	−2.36105
$629$	−1797.83	−0.113966
$630$	8162.40	0.516187
$631$	−26067.9	−1.64461	−0.822303	−	0.569050i	$-0.807311\pi$
$631$	−26067.9	−1.64461	−0.822303	+	0.569050i	$0.807311\pi$
$632$	−19685.9	−1.23903
$633$	8221.34	0.516223
$634$	35162.8	2.20267
$635$	−11658.0	−0.728560
$636$	−812.109	−0.0506324
$637$	195.542	0.0121627
$638$	11203.3	0.695209
$639$	−6572.47	−0.406891
$640$	−9445.47	−0.583383
$641$	−22131.1	−1.36369	−0.681846	−	0.731496i	$-0.738823\pi$
$641$	−22131.1	−1.36369	−0.681846	+	0.731496i	$0.738823\pi$
$642$	14127.9	0.868511
$643$	−5655.76	−0.346876	−0.173438	−	0.984845i	$-0.555488\pi$
$643$	−5655.76	−0.346876	−0.173438	+	0.984845i	$0.555488\pi$
$644$	−25477.7	−1.55895
$645$	−2726.52	−0.166444
$646$	−1732.23	−0.105501
$647$	−19352.8	−1.17595	−0.587973	−	0.808881i	$-0.700074\pi$
$647$	−19352.8	−1.17595	−0.587973	+	0.808881i	$0.700074\pi$
$648$	−6769.94	−0.410414
$649$	−6799.58	−0.411258
$650$	258.769	0.0156150
$651$	4986.41	0.300204
$652$	25373.1	1.52406
$653$	−12448.3	−0.746004	−0.373002	−	0.927831i	$-0.621672\pi$
$653$	−12448.3	−0.746004	−0.373002	+	0.927831i	$0.621672\pi$
$654$	1314.98	0.0786238
$655$	4313.18	0.257298
$656$	−7341.05	−0.436920
$657$	−15905.2	−0.944474
$658$	6830.65	0.404691
$659$	−849.200	−0.0501975	−0.0250987	−	0.999685i	$-0.507990\pi$
$659$	−849.200	−0.0501975	−0.0250987	+	0.999685i	$0.507990\pi$
$660$	1316.48	0.0776426
$661$	−3793.75	−0.223237	−0.111619	−	0.993751i	$-0.535603\pi$
$661$	−3793.75	−0.223237	−0.111619	+	0.993751i	$0.535603\pi$
$662$	40600.8	2.38368
$663$	−97.6345	−0.00571917
$664$	−9922.26	−0.579907
$665$	−1529.15	−0.0891699
$666$	8882.89	0.516824
$667$	−30949.2	−1.79664
$668$	−41170.0	−2.38461
$669$	−2397.54	−0.138556
$670$	14935.2	0.861189
$671$	−1633.81	−0.0939976
$672$	7311.48	0.419712
$673$	−12440.3	−0.712540	−0.356270	−	0.934383i	$-0.615952\pi$
$673$	−12440.3	−0.712540	−0.356270	+	0.934383i	$0.615952\pi$
$674$	23974.9	1.37015
$675$	2542.83	0.144998
$676$	−25701.5	−1.46230
$677$	8522.43	0.483816	0.241908	−	0.970299i	$-0.422227\pi$
$677$	8522.43	0.483816	0.241908	+	0.970299i	$0.422227\pi$
$678$	−15295.0	−0.866374
$679$	−22473.0	−1.27015
$680$	−1699.14	−0.0958222
$681$	9322.95	0.524605
$682$	−7415.46	−0.416353
$683$	22802.3	1.27746	0.638731	−	0.769430i	$-0.279460\pi$
$683$	22802.3	1.27746	0.638731	+	0.769430i	$0.279460\pi$
$684$	5087.93	0.284418
$685$	−14456.2	−0.806339
$686$	−30520.8	−1.69867
$687$	−9894.52	−0.549490
$688$	5420.07	0.300346
$689$	−79.0676	−0.00437190
$690$	−6117.67	−0.337530
$691$	−11655.7	−0.641686	−0.320843	−	0.947132i	$-0.603966\pi$
$691$	−11655.7	−0.641686	−0.320843	+	0.947132i	$0.603966\pi$
$692$	−44923.4	−2.46782
$693$	4043.01	0.221618
$694$	50919.3	2.78511
$695$	12113.1	0.661117
$696$	−7748.42	−0.421987
$697$	−7427.75	−0.403653
$698$	6390.49	0.346538
$699$	−2070.91	−0.112059
$700$	−4719.22	−0.254814
$701$	29100.5	1.56792	0.783960	−	0.620811i	$-0.213197\pi$
$701$	29100.5	1.56792	0.783960	+	0.620811i	$0.213197\pi$
$702$	1052.81	0.0566037
$703$	−1664.13	−0.0892801
$704$	−9087.90	−0.486525
$705$	975.035	0.0520879
$706$	−28860.8	−1.53851
$707$	3807.92	0.202562
$708$	14795.9	0.785403
$709$	28240.3	1.49589	0.747944	−	0.663761i	$-0.231041\pi$
$709$	28240.3	1.49589	0.747944	+	0.663761i	$0.231041\pi$
$710$	6392.18	0.337879
$711$	−27151.7	−1.43216
$712$	11244.4	0.591855
$713$	20485.2	1.07599
$714$	2995.22	0.156993
$715$	128.174	0.00670411
$716$	−1533.86	−0.0800601
$717$	13400.2	0.697963
$718$	45676.7	2.37415
$719$	26917.3	1.39617	0.698086	−	0.716014i	$-0.254035\pi$
$719$	26917.3	1.39617	0.698086	+	0.716014i	$0.254035\pi$
$720$	−2316.18	−0.119887
$721$	9371.01	0.484042
$722$	−1603.40	−0.0826488
$723$	4012.12	0.206379
$724$	−21441.5	−1.10065
$725$	−5732.69	−0.293665
$726$	1096.91	0.0560746
$727$	7401.00	0.377562	0.188781	−	0.982019i	$-0.439546\pi$
$727$	7401.00	0.377562	0.188781	+	0.982019i	$0.439546\pi$
$728$	−621.024	−0.0316163
$729$	−3732.19	−0.189615
$730$	15468.8	0.784284
$731$	5484.08	0.277478
$732$	3555.17	0.179512
$733$	14809.9	0.746272	0.373136	−	0.927777i	$-0.378282\pi$
$733$	14809.9	0.746272	0.373136	+	0.927777i	$0.378282\pi$
$734$	−27424.7	−1.37911
$735$	−856.293	−0.0429726
$736$	30037.1	1.50432
$737$	7397.73	0.369741
$738$	36699.7	1.83053
$739$	−23335.1	−1.16156	−0.580781	−	0.814060i	$-0.697253\pi$
$739$	−23335.1	−1.16156	−0.580781	+	0.814060i	$0.697253\pi$
$740$	−5135.78	−0.255128
$741$	−90.3735	−0.00448037
$742$	2425.63	0.120010
$743$	−14279.1	−0.705045	−0.352522	−	0.935803i	$-0.614676\pi$
$743$	−14279.1	−0.705045	−0.352522	+	0.935803i	$0.614676\pi$
$744$	5128.67	0.252723
$745$	6296.17	0.309629
$746$	31135.2	1.52807
$747$	−13685.2	−0.670303
$748$	−2647.96	−0.129437
$749$	−25085.3	−1.22376
$750$	−1133.17	−0.0551701
$751$	1425.69	0.0692734	0.0346367	−	0.999400i	$-0.488973\pi$
$751$	1425.69	0.0692734	0.0346367	+	0.999400i	$0.488973\pi$
$752$	−1938.28	−0.0939916
$753$	−3954.07	−0.191360
$754$	−2373.51	−0.114639
$755$	−15905.3	−0.766694
$756$	−19200.3	−0.923687
$757$	23916.6	1.14830	0.574151	−	0.818749i	$-0.305332\pi$
$757$	23916.6	1.14830	0.574151	+	0.818749i	$0.305332\pi$
$758$	3767.00	0.180506
$759$	−3030.22	−0.144914
$760$	−1572.78	−0.0750667
$761$	−22441.1	−1.06897	−0.534486	−	0.845177i	$-0.679495\pi$
$761$	−22441.1	−1.06897	−0.534486	+	0.845177i	$0.679495\pi$
$762$	21136.9	1.00487
$763$	−2334.87	−0.110784
$764$	−1376.48	−0.0651823
$765$	−2343.53	−0.110759
$766$	9259.68	0.436770
$767$	1440.54	0.0678162
$768$	3635.36	0.170807
$769$	−2966.80	−0.139123	−0.0695615	−	0.997578i	$-0.522160\pi$
$769$	−2966.80	−0.139123	−0.0695615	+	0.997578i	$0.522160\pi$
$770$	−3932.10	−0.184030
$771$	−13230.6	−0.618014
$772$	−57709.5	−2.69043
$773$	18391.6	0.855758	0.427879	−	0.903836i	$-0.359261\pi$
$773$	18391.6	0.855758	0.427879	+	0.903836i	$0.359261\pi$
$774$	−27096.2	−1.25834
$775$	3794.46	0.175872
$776$	−23114.2	−1.06927
$777$	2877.48	0.132856
$778$	13444.6	0.619555
$779$	−6875.36	−0.316220
$780$	−278.908	−0.0128032
$781$	3166.18	0.145064
$782$	12305.0	0.562693
$783$	−23323.6	−1.06452
$784$	1702.23	0.0775433
$785$	15842.1	0.720289
$786$	−7820.12	−0.354879
$787$	4993.78	0.226187	0.113093	−	0.993584i	$-0.463924\pi$
$787$	4993.78	0.226187	0.113093	+	0.993584i	$0.463924\pi$
$788$	−3272.88	−0.147959
$789$	16668.7	0.752118
$790$	26406.9	1.18926
$791$	27157.6	1.22075
$792$	4158.36	0.186567
$793$	346.135	0.0155001
$794$	−28020.9	−1.25242
$795$	346.244	0.0154465
$796$	11557.3	0.514620
$797$	−27110.3	−1.20489	−0.602443	−	0.798162i	$-0.705806\pi$
$797$	−27110.3	−1.20489	−0.602443	+	0.798162i	$0.705806\pi$
$798$	2772.47	0.122988
$799$	−1961.17	−0.0868350
$800$	5563.75	0.245885
$801$	15508.7	0.684113
$802$	12913.5	0.568568
$803$	7662.06	0.336722
$804$	−16097.5	−0.706114
$805$	10862.5	0.475591
$806$	1571.02	0.0686562
$807$	9278.09	0.404714
$808$	3916.56	0.170525
$809$	14651.2	0.636721	0.318361	−	0.947970i	$-0.396868\pi$
$809$	14651.2	0.636721	0.318361	+	0.947970i	$0.396868\pi$
$810$	9081.26	0.393929
$811$	−16985.0	−0.735418	−0.367709	−	0.929941i	$-0.619858\pi$
$811$	−16985.0	−0.735418	−0.367709	+	0.929941i	$0.619858\pi$
$812$	43286.1	1.87074
$813$	289.910	0.0125062
$814$	−4279.19	−0.184257
$815$	−10817.9	−0.464948
$816$	−849.928	−0.0364625
$817$	5076.24	0.217375
$818$	45821.7	1.95858
$819$	−856.544	−0.0365447
$820$	−21218.5	−0.903636
$821$	18345.3	0.779847	0.389923	−	0.920847i	$-0.372501\pi$
$821$	18345.3	0.779847	0.389923	+	0.920847i	$0.372501\pi$
$822$	26210.1	1.11215
$823$	1129.79	0.0478517	0.0239258	−	0.999714i	$-0.492383\pi$
$823$	1129.79	0.0478517	0.0239258	+	0.999714i	$0.492383\pi$
$824$	9638.36	0.407485
$825$	−561.285	−0.0236866
$826$	−44192.8	−1.86158
$827$	−42737.9	−1.79703	−0.898514	−	0.438944i	$-0.855353\pi$
$827$	−42737.9	−1.79703	−0.898514	+	0.438944i	$0.855353\pi$
$828$	−36142.5	−1.51696
$829$	−32743.8	−1.37182	−0.685910	−	0.727686i	$-0.740596\pi$
$829$	−32743.8	−1.37182	−0.685910	+	0.727686i	$0.740596\pi$
$830$	13309.8	0.556614
$831$	−17171.6	−0.716820
$832$	1925.34	0.0802275
$833$	1722.33	0.0716391
$834$	−21962.0	−0.911847
$835$	17552.9	0.727476
$836$	−2451.03	−0.101400
$837$	15437.9	0.637528
$838$	−1705.13	−0.0702896
$839$	−11541.7	−0.474926	−0.237463	−	0.971397i	$-0.576316\pi$
$839$	−11541.7	−0.474926	−0.237463	+	0.971397i	$0.576316\pi$
$840$	2719.52	0.111705
$841$	28193.0	1.15597
$842$	−18432.1	−0.754408
$843$	−6764.76	−0.276383
$844$	−47238.4	−1.92655
$845$	10957.8	0.446108
$846$	9689.92	0.393790
$847$	−1947.66	−0.0790110
$848$	−688.300	−0.0278730
$849$	5710.90	0.230857
$850$	2279.24	0.0919734
$851$	11821.3	0.476179
$852$	−6889.64	−0.277037
$853$	18224.8	0.731543	0.365771	−	0.930705i	$-0.380805\pi$
$853$	18224.8	0.731543	0.365771	+	0.930705i	$0.380805\pi$
$854$	−10618.7	−0.425484
$855$	−2169.25	−0.0867680
$856$	−25801.0	−1.03021
$857$	−7004.17	−0.279181	−0.139590	−	0.990209i	$-0.544579\pi$
$857$	−7004.17	−0.279181	−0.139590	+	0.990209i	$0.544579\pi$
$858$	−232.389	−0.00924666
$859$	49926.8	1.98310	0.991549	−	0.129733i	$-0.0414119\pi$
$859$	49926.8	1.98310	0.991549	+	0.129733i	$0.0414119\pi$
$860$	15666.1	0.621174
$861$	11888.3	0.470560
$862$	−32284.5	−1.27565
$863$	−23919.2	−0.943477	−0.471738	−	0.881739i	$-0.656373\pi$
$863$	−23919.2	−0.943477	−0.471738	+	0.881739i	$0.656373\pi$
$864$	22636.3	0.891322
$865$	19153.1	0.752863
$866$	−44295.8	−1.73814
$867$	9167.64	0.359111
$868$	−28651.0	−1.12037
$869$	13079.9	0.510593
$870$	10393.8	0.405038
$871$	−1567.27	−0.0609699
$872$	−2401.48	−0.0932619
$873$	−31880.1	−1.23594
$874$	11389.9	0.440811
$875$	2012.04	0.0777365
$876$	−16672.7	−0.643057
$877$	−41369.1	−1.59286	−0.796429	−	0.604732i	$-0.793280\pi$
$877$	−41369.1	−1.59286	−0.796429	+	0.604732i	$0.793280\pi$
$878$	−28112.7	−1.08059
$879$	15546.8	0.596567
$880$	1115.78	0.0427420
$881$	−1043.41	−0.0399017	−0.0199508	−	0.999801i	$-0.506351\pi$
$881$	−1043.41	−0.0399017	−0.0199508	+	0.999801i	$0.506351\pi$
$882$	−8509.86	−0.324878
$883$	29290.9	1.11633	0.558163	−	0.829731i	$-0.311506\pi$
$883$	29290.9	1.11633	0.558163	+	0.829731i	$0.311506\pi$
$884$	560.990	0.0213441
$885$	−6308.26	−0.239604
$886$	−27679.5	−1.04956
$887$	33779.7	1.27870	0.639352	−	0.768914i	$-0.279203\pi$
$887$	33779.7	1.27870	0.639352	+	0.768914i	$0.279203\pi$
$888$	2959.57	0.111843
$889$	−37530.4	−1.41589
$890$	−15083.3	−0.568082
$891$	4498.15	0.169129
$892$	13775.8	0.517095
$893$	−1815.32	−0.0680262
$894$	−11415.4	−0.427057
$895$	653.962	0.0244241
$896$	−30407.5	−1.13375
$897$	641.975	0.0238962
$898$	−53774.9	−1.99832
$899$	−34804.0	−1.29119
$900$	−6694.65	−0.247950
$901$	−696.429	−0.0257507
$902$	−17679.5	−0.652619
$903$	−8777.41	−0.323470
$904$	27932.4	1.02767
$905$	9141.62	0.335777
$906$	28837.5	1.05746
$907$	−19281.2	−0.705866	−0.352933	−	0.935649i	$-0.614816\pi$
$907$	−19281.2	−0.705866	−0.352933	+	0.935649i	$0.614816\pi$
$908$	−53568.0	−1.95784
$909$	5401.90	0.197106
$910$	833.047	0.0303464
$911$	−9931.37	−0.361187	−0.180593	−	0.983558i	$-0.557802\pi$
$911$	−9931.37	−0.361187	−0.180593	+	0.983558i	$0.557802\pi$
$912$	−786.720	−0.0285646
$913$	6592.64	0.238975
$914$	23083.3	0.835370
$915$	−1515.75	−0.0547642
$916$	56852.1	2.05071
$917$	13885.3	0.500036
$918$	9273.18	0.333399
$919$	22360.2	0.802606	0.401303	−	0.915945i	$-0.368557\pi$
$919$	22360.2	0.802606	0.401303	+	0.915945i	$0.368557\pi$
$920$	11172.4	0.400371
$921$	−13149.3	−0.470450
$922$	−44406.5	−1.58617
$923$	−670.781	−0.0239209
$924$	4238.12	0.150892
$925$	2189.65	0.0778326
$926$	−50298.8	−1.78501
$927$	13293.7	0.471004
$928$	−51032.4	−1.80520
$929$	−22123.5	−0.781323	−0.390661	−	0.920534i	$-0.627754\pi$
$929$	−22123.5	−0.781323	−0.390661	+	0.920534i	$0.627754\pi$
$930$	−6879.64	−0.242572
$931$	1594.25	0.0561217
$932$	11899.1	0.418206
$933$	−7830.00	−0.274751
$934$	6779.12	0.237494
$935$	1128.96	0.0394876
$936$	−880.981	−0.0307647
$937$	34597.4	1.20624	0.603121	−	0.797650i	$-0.293924\pi$
$937$	34597.4	1.20624	0.603121	+	0.797650i	$0.293924\pi$
$938$	48080.4	1.67365
$939$	−22089.6	−0.767698
$940$	−5602.38	−0.194393
$941$	19146.3	0.663284	0.331642	−	0.943405i	$-0.392397\pi$
$941$	19146.3	0.663284	0.331642	+	0.943405i	$0.392397\pi$
$942$	−28722.8	−0.993461
$943$	48839.6	1.68657
$944$	12540.2	0.432362
$945$	8186.06	0.281791
$946$	13053.2	0.448621
$947$	−34120.7	−1.17083	−0.585414	−	0.810734i	$-0.699068\pi$
$947$	−34120.7	−1.17083	−0.585414	+	0.810734i	$0.699068\pi$
$948$	−28462.0	−0.975107
$949$	−1623.27	−0.0555253
$950$	2109.74	0.0720516
$951$	16158.4	0.550970
$952$	−5469.99	−0.186222
$953$	−7005.75	−0.238131	−0.119065	−	0.992886i	$-0.537990\pi$
$953$	−7005.75	−0.238131	−0.119065	+	0.992886i	$0.537990\pi$
$954$	3440.98	0.116778
$955$	586.864	0.0198853
$956$	−76995.1	−2.60481
$957$	5148.28	0.173898
$958$	20247.7	0.682852
$959$	−46538.3	−1.56705
$960$	−8431.24	−0.283455
$961$	−6754.32	−0.226723
$962$	906.580	0.0303839
$963$	−35585.9	−1.19080
$964$	−23052.9	−0.770212
$965$	24604.5	0.820775
$966$	−19694.4	−0.655961
$967$	−19648.3	−0.653408	−0.326704	−	0.945127i	$-0.605938\pi$
$967$	−19648.3	−0.653408	−0.326704	+	0.945127i	$0.605938\pi$
$968$	−2003.22	−0.0665145
$969$	−796.012	−0.0263897
$970$	31005.5	1.02632
$971$	6388.43	0.211138	0.105569	−	0.994412i	$-0.466334\pi$
$971$	6388.43	0.211138	0.105569	+	0.994412i	$0.466334\pi$
$972$	−41994.5	−1.38578
$973$	38995.3	1.28482
$974$	7488.65	0.246357
$975$	118.913	0.00390590
$976$	3013.17	0.0988210
$977$	−33593.0	−1.10004	−0.550019	−	0.835152i	$-0.685379\pi$
$977$	−33593.0	−1.10004	−0.550019	+	0.835152i	$0.685379\pi$
$978$	19613.6	0.641281
$979$	−7471.09	−0.243899
$980$	4920.11	0.160375
$981$	−3312.23	−0.107800
$982$	50533.7	1.64215
$983$	−16401.3	−0.532167	−0.266083	−	0.963950i	$-0.585730\pi$
$983$	−16401.3	−0.532167	−0.266083	+	0.963950i	$0.585730\pi$
$984$	12227.5	0.396135
$985$	1395.39	0.0451380
$986$	−20905.9	−0.675234
$987$	3138.90	0.101228
$988$	519.270	0.0167208
$989$	−36059.4	−1.15938
$990$	−5578.06	−0.179073
$991$	−52043.2	−1.66822	−0.834111	−	0.551597i	$-0.814019\pi$
$991$	−52043.2	−1.66822	−0.834111	+	0.551597i	$0.814019\pi$
$992$	33778.3	1.08111
$993$	18657.3	0.596247
$994$	20578.1	0.656639
$995$	−4927.46	−0.156996
$996$	−14345.6	−0.456384
$997$	−16321.1	−0.518448	−0.259224	−	0.965817i	$-0.583467\pi$
$997$	−16321.1	−0.518448	−0.259224	+	0.965817i	$0.583467\pi$
$998$	−81204.9	−2.57565
$999$	8908.64	0.282139

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.f.1.4	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.f.1.4	✓	23	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)

\(f(q)\)	\(=\)	\(q-4.44156 q^{2} -2.04104 q^{3} +11.7274 q^{4} -5.00000 q^{5} +9.06537 q^{6} -16.0963 q^{7} -16.5556 q^{8} -22.8342 q^{9} +O(q^{10})\)	\(q-4.44156 q^{2} -2.04104 q^{3} +11.7274 q^{4} -5.00000 q^{5} +9.06537 q^{6} -16.0963 q^{7} -16.5556 q^{8} -22.8342 q^{9} +22.2078 q^{10} +11.0000 q^{11} -23.9361 q^{12} -2.33044 q^{13} +71.4928 q^{14} +10.2052 q^{15} -20.2869 q^{16} -20.5265 q^{17} +101.419 q^{18} -19.0000 q^{19} -58.6371 q^{20} +32.8532 q^{21} -48.8571 q^{22} +134.968 q^{23} +33.7905 q^{24} +25.0000 q^{25} +10.3508 q^{26} +101.713 q^{27} -188.769 q^{28} -229.308 q^{29} -45.3269 q^{30} +151.778 q^{31} +222.550 q^{32} -22.4514 q^{33} +91.1697 q^{34} +80.4817 q^{35} -267.786 q^{36} +87.5858 q^{37} +84.3896 q^{38} +4.75650 q^{39} +82.7778 q^{40} +361.861 q^{41} -145.919 q^{42} -267.170 q^{43} +129.002 q^{44} +114.171 q^{45} -599.468 q^{46} +95.5432 q^{47} +41.4063 q^{48} -83.9077 q^{49} -111.039 q^{50} +41.8954 q^{51} -27.3300 q^{52} +33.9282 q^{53} -451.765 q^{54} -55.0000 q^{55} +266.484 q^{56} +38.7797 q^{57} +1018.48 q^{58} -618.143 q^{59} +119.680 q^{60} -148.528 q^{61} -674.132 q^{62} +367.547 q^{63} -826.173 q^{64} +11.6522 q^{65} +99.7191 q^{66} +672.521 q^{67} -240.723 q^{68} -275.474 q^{69} -357.464 q^{70} +287.835 q^{71} +378.033 q^{72} +696.551 q^{73} -389.017 q^{74} -51.0259 q^{75} -222.821 q^{76} -177.060 q^{77} -21.1263 q^{78} +1189.08 q^{79} +101.435 q^{80} +408.922 q^{81} -1607.23 q^{82} +599.331 q^{83} +385.283 q^{84} +102.633 q^{85} +1186.65 q^{86} +468.025 q^{87} -182.111 q^{88} -679.190 q^{89} -507.096 q^{90} +37.5115 q^{91} +1582.83 q^{92} -309.785 q^{93} -424.360 q^{94} +95.0000 q^{95} -454.232 q^{96} +1396.16 q^{97} +372.681 q^{98} -251.176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * q^98 + 1870 * q^99

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